Optoelectronic Physics and Engineering of Atomically Thin Photovoltaics - CaltechTHESIS

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Optoelectronic Physics and Engineering of Atomically Thin Photovoltaics
Citation
Wong, Joeson
(2022)
Optoelectronic Physics and Engineering of Atomically Thin Photovoltaics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/pxk0-3d19.
Abstract
Materials that are atomically thin behave substantially different than those of their bulk counterparts. However, when most materials become thinner, their surface-to-volume ratio increases and the number of unpassivated dangling bonds at the surface approaches the number of internal crystalline bonds, which prevents examining the intrinsic properties of most ultrathin materials. The recent discovery of layered materials, whose crystal structures have naturally passivated basal planes, has enabled the possibility to examine materials’ thicknesses that approach a single atomic layer.
In this thesis, we examine and explore the consequence of this new regime of thickness for active layers in photovoltaic applications. Specifically, we focus on the three aspects that define photovoltaic operation and explore their differences in these ultrathin materials: optical absorption of photons, subsequent carrier generation and transport, and finally, free energy extraction of collected carriers. We first discuss the implications of band-edge abruptness on the maximum efficiency of a solar cell. Then, we show that optical absorption in these ultrathin materials is dominated by cavity wave optics, and design structures that enable near-unity absorption in both ultrathin (~10 nm) and atomically-thin (~7 Å) active layers. Using these optical design rules, we design heterostructures with record incident photon to electron conversion efficiency (>50%). Next, we examine new methods of creating electrical junctions by using thickness to vary the amount of band bending in a material. We spatiotemporally image these 'band-bending junctions' for the first time. Finally, we argue that photoluminescence can be used as a direct readout of the open circuit voltage potential, and motivate examination of monolayer materials which have substantially higher radiative efficiency. We therefore examine the strain tuning of photoluminescence properties of both monolayer TMDC and heterobilayer TMDC systems. This work illustrates that van der Waals materials are an ideal system for examining the novel optoelectronic physics of atomically thin photovoltaics.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
van der Waals materials; photovoltaics; optics; photonics; optoelectronics; atomically thin; nanotechnology; 2D materials;
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Awards:
Demetriades-Tsafka-Kokkalis Prize in Nanotechnology or Related Fields, 2022.
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Thesis Committee:
Falson, Joseph (chair)
Nadj-Perge, Stevan
Heinz, Tony F.
Atwater, Harry Albert
Defense Date:
22 November 2021
Non-Caltech Author Email:
joesonwong11 (AT) gmail.com
Funders:
Funding Agency
Grant Number
Department of Energy (DOE)
DE-SC0019140
NSF
1144469
Record Number:
CaltechTHESIS:12132021-004403821
Persistent URL:
DOI:
10.7907/pxk0-3d19
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DOI
Article adapted for ch. 1
DOI
Article adapted for ch. 2
DOI
Article adapted for ch. 2
DOI
Article adapted for ch. 3
DOI
Article adapted for ch. 4
DOI
Article adapted for ch. 5
DOI
Article adapted for ch. 6
ORCID:
Author
ORCID
Wong, Joeson
0000-0002-6304-7602
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
14446
Collection:
CaltechTHESIS
Deposited By:
Joeson Wong
Deposited On:
25 Jan 2022 23:57
Last Modified:
20 Feb 2025 21:14
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Optoelectronic Physics and Engineering of Atomically
Thin Photovoltaics

Thesis by

Joeson Wong

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2022
Defended November 22, 2021

ii

Joeson Wong
ORCID: 0000-0002-6304-7602

iii

ACKNOWLEDGEMENTS
It’s amazing that six years has passed by so quickly. I have gained many new
experiences and perspectives—some I would have never thought would be part
of a PhD program. Certain lessons were definitely more painful than others, but
assuredly, those are lessons I will never forget. None of those lessons, however,
would have been possible without the amazing people I have had a chance to
interact with and learn from throughout my time at Caltech, so it is my pleasure to
acknowledge their contributions to this PhD journey.
I would first and foremost like to acknowledge my PhD advisor and scientific mentor
for the past six years, Harry Atwater. Harry has an infectious enthusiasm and curious
nature for scientific problems which I have absorbed, in part, through osmosis. The
energy he brings towards science is akin to the quote from Michigan football coach
Jim Harbaugh, “Attack each day with enthusiasm unknown to mankind”. I am
reminded of those words whenever we engage in a scientific discussion. I have also
gained a deep respect for Harry’s “superman” ability to juggle and excel at both his
social and academic responsibilities, something that I hope to echo in my life as
well. Harry also has a deep affinity to solving issues in today’s society, which is
reflected by both the diversity of his research group and in his research directions.
There are many other things I could say about Harry, but perhaps the most important
thing I can say is just ‘thank you’. Thank you for taking a chance on this boy from
Michigan and teaching me so much these past six years, I am forever indebted to
having you as my scientific mentor.
Two other special mentors I must acknowledge are Deep Jariwala and Artur Davoyan,
both postdocs in the group that soon became professors of their own. Both of them
contributed substantially to the works discussed in Chapters 3, 4, and 5. Deep has
taught me so much regarding experimental procedures and the scientific process,
and always had a seemingly limitless knowledge of 2D materials literature and other
current events. His intentional but playful attitude towards experimental research is
something I have tried to emulate throughout my PhD. I am also grateful to have
learned early on from Deep that collaborating with others is often the most efficient
way to get things done. Artur also deserves a special acknowledgement. I remember
my first impression of him was that he was extremely harsh and intimidating with
his probing questions! Over the years however, I have developed a deep respect
and friendship with Artur, and I regard him as one of the most intelligent and

iv
outside-the-box thinkers I know of. His visionary perspective on solving impactful
problems is contagious. The long nights we would have doing or chatting about
science together and the advice he has given me throughout my PhD are some of
the most valuable.
I would also like to thank my thesis committee members, Professor Joseph Falson,
Professor Stevan Nadj-Perge, and Professor Tony Heinz as well as my candidacy
committee members, Professor Marco Bernardi, Professor Stevan Nadj-Perge, and
Professor David Hsieh, for their helpful suggestions and feedback throughout my
PhD. I specifically want to thank Professor Falson for his career advice in a life
in academia, I hope I can put it to good use. I would also like to specifically
thank Professor Nadj-Perge and his group for interactions early on that enabled
making higher quality heterostructures of 2D materials. Finally, I would like to
thank Professor Heinz for seemingly knowing everything there is about the optical
properties of layered materials and laying the foundation for studying these materials.
I would also like to thank the National Science Foundation for supporting me for
the first few years of my PhD, which enabled me to carve out a unique independent
research direction. For the last few years, I have been supported by the Photonics
at Thermodynamics Limit EFRC grant, which has also been a deeply scientifically
enriching experience.
I have also had the chance to interact with many other folks throughout the years,
both in and out of the Atwater group. In particular, I would like to thank those who
have joined me as major contributors to the variety of projects as it pertains to this
thesis. Stefan “Brochenko” Omelchenko contributed significantly to the initial ideas
and examining the analysis I did in Chapter 2, and I am grateful to have someone to
bounce my wild ideas off of as I went through that project. I have always thought that
Stefan personified the word “chill”, and I very much appreciated that. Cora Went has
been a partner-in-crime throughout most of my years as a PhD student and doubles as
an amazing travel buddy. She has contributed significantly to many of my projects,
including the measurements of TMDC band tails in Chapter 2, ongoing work with
strong coupling and absorbers as it pertains to Chapter 3, and for continuing the
carrier selective contacts and stroboSCAT work discussed in the Outlook. I hope we
can get more soup dumplings together in the future. Bolin Liao and the late Ahmed
Zewail contributed significantly to the USEM measurements of the band bending
junction work in Chapter 5. Seeing movies of electrons and holes moving around
in a semiconductor is something I used to only dream of. In a similar vein, I would

like to thank the Advanced Light Source beamline scientists: Eli Rotenberg, Aaron
Bostwick, and Chris Jozwiak. They taught me both the instrumental procedures and
the principles of ARPES, which was used for the work in Chapter 5, and I certainly
felt like a kid in a candy store the first time I saw bandstructure in real time. Hannah
Weaver, Dipti Jasrasaria, Professor Eran Rabani, and Professor Naomi Ginsberg have
also been essential to the stroboSCAT work. I am particularly grateful to Hannah,
who has been persistently working on this project with me for many years. Chace
(Chullhee) Cho and his advisor Professor SungWoo Nam also deserve a special
acknowledgement for their contributions to Chapter 6. I am grateful that SungWoo
decided to do a sabbatical in Harry’s group in 2018, where Chace subsequently
joined me as an officemate. I have learned tremendously about mechanical strain
and its effects on materials as I collaborated with you both. The nights with Korean
BBQ (Honey Pig!) and soju definitely fueled a variety of late-night endeavors as
well. Last but certainly not least, I would like to thank Souvik Biswas for his
contributions to our work on monolayer black phosphorus. Souvik has a relentless
effort towards research, and I have yet to see a single person work harder than he has
throughout my PhD. He shares very much the same attitude I have towards science,
and I am deeply appreciative of our scientific discussions and our occasional (Ed
Sheeran) jam sessions while doing some late-night lab work.
I would also be remiss to not acknowledge other co-authors, including Michelle
Sherrott, William Whitney, Giulia Tagliabue, Ognjen Ilic, Zakaria Al Balushi,
Hamidreza Akbari, Takashi Taniguchi, Kenji Watanabe, Albert Davydov, Amir
Taqieddin, Narayana Aluru, Alberto Crepaldi, Andrey Krayev, Kiyoung Jo, George
Rossman, Arky Yang, Tony Low, Aditya Mohite, and Michael Enright. Thank you
all for either being a part of my projects or allowing me to be a part of and learn
from yours.
I have also been given the opportunity to mentor a few undergraduate researchers
throughout my time at Caltech, which have deeply impacted my views on mentorship.
Kevin Tat was a SURF student in 2016 and contributed to the devices fabricated in
Chapter 4, for which he is a co-author on. I’m thankful he was patient with me as I
was teaching him things I had only recently learned. Sara Anjum was a SURF with
me throughout the summer of 2018 and subsequently became a graduate student in
the group the following year. She contributed to our knowledge on 2D materials
transfer and I wish her the best of luck throughout the remainder of her PhD. Tyler
Colenbrander was a SURF student with me throughout the summer of 2020 in the

vi
middle of the COVID-19 pandemic. He made several important contributions to the
analysis on monolayer absorbers in Chapter 3 and allowed me to “experiment” with
different methods of teaching and mentorship. Thank you all for reminding me the
thrilling feeling of being a teacher.
I would also like to give an acknowledgement and word of encouragement to the
newly formed “2D-PV” team: Susana Torres-Londono, Rachel Tham, and Miles
Johnson. Particularly with Susana, who joined Caltech early in part to work with
me. I feel honored, humbled, and utterly responsible for all of you—so don’t be
a stranger, even after I depart. Best of luck with your PhD journey, and always
remember that it’s a marathon. Don’t forget to enjoy the running too.
Harry’s group is a special place, especially being both large and diverse which
contrasts with a small school like Caltech. I first have to thank several of my
forebearers from when I first joined the group, you all set in place the joyful culture
of the A-team. Cris Flowers and Dagny Fleischman subtly coerced me into being
the next ebeam guru in my first year, and I have learned a tremendous amount from
working with them and the ebeam system. Will Whitney and Michelle Sherrott were
my first officemates in 244 Watson and quickly helped me “grow up” as a graduate
student, sharing with me lessons they learned throughout their PhD. Kelly Mauser
was always a joy to spend time with and my go-to conference buddy. Benjamin
“Benji” Vest was the founding member of the Bourbon and Burgers night (aka
B&B&B) and I greatly appreciated his spirit to increase group social activities and
his knowledge of optics. Thank you for showing me around Paris, too. Phil Jahelka
has an amazing knowledge of semiconductors and anything related to chemistry,
despite not being a chemist by training. I appreciated deeply his insights and the fun
gatherings we had at his home. Mike Kelzenberg was a source of infinite wisdom on
electronics and instrumentation, as well as random historical anecdotes of the group,
and I appreciate his ‘wizardry’ greatly. Rebecca Glaudell was always thoughtful in
terms of keeping the lab organized and running smoothly, and I thank her immensely
for her duties as my go-to safety officer whenever I was unsure of safety procedures.
The ‘Taiwanese Squad’, consisting of Sophia (Wen-Hui) Cheng, Pin Chieh Wu,
Yu-Jung Lu (aka Yuri the Beautiful), and Yi-Rung Lin were always fun to be around
late at night, and I enjoyed our food outings together. Giulia Tagliabue was always
very warm and I’m grateful for some of the tips and tricks she showed me in the
lab, as well as those in the climbing gym. I would also like to thank those that made
my experience in Harry’s group particularly warm when I first joined the group,

vii
including my interactions with Laura Kim, Jeremy Brouillet, Sisir Yalamanchilli,
Carissa Eisler, Sunita Darbe, and John Lloyd.
There are also many of my contemporaries that I would like to acknowledge as
well. Alex Welch and Magel Su became the next generation of ebeam gurus, and
I’m especially thankful for Alex for taking a nice picture when the ebeam was
spouting water—no one would have believed us otherwise. I’ve appreciated my
many coffee hours with many members of the group, including (but not limited to):
Hamid (Hamidreza) Akbari, Souvik Biswas, Komron Shayegan, and Parker Wray. I
especially want to thank Parker for his advice on my personal life on many occasions.
Megan Phelan was always a source of encouragement and thoughtfulness, and I’m
thankful for our friendship and occasional wine nights together. Samuel Loke always
had a multitude of board games to play at his place and is one of the most thoughtful
activity hosts I know of. Finally, I want to thank Cora Went and Souvik Biswas
once again, not just for their contributions throughout my thesis, but for taking the
baton in many occasions.
I would also like to acknowledge and thank the administrative staff that has made
Caltech such a wonderful and truly unique place, including: Kam Flower, Jonathan
Gross, Angie Riley, Lyann Lau, Liz Hormigoso, Tiffany Kimoto, Jennifer Blankenship, Carrie Hofmann, and Christy Jenstad. I especially want to thank Jonathan
Gross and Christy Jenstad. They were one of the first people I met when I was
visiting Caltech and their warmth was something that instantly drew me here.
Outside of Harry’s group, I’ve had the fortune to have some amazing friends. I have
a special gratitude to Peishi Cheng, who has been my roommate and close friend
for over four years and always has something good cooking in the kitchen. I’ve
enjoyed many of our foodventures together (including the one and only Tsujita) and
appreciate your thoughtfulness on many of the personal aspects of my life. I would
also like to give a special thank you to Jash Banker, who has also been one of my
closest friends throughout my PhD. I have enjoyed so many of our times together
while exploring Los Angeles and I am grateful that I met someone with so many
mutual interests. I hope I can visit you soon in Germany! A special thank you needs
to be given to Joanne Wai, who has made me grow tremendously as an individual. I
hope you’re surviving the Utah weather! The ‘Yangji’ squad also deserves a special
shoutout, consisting of Jash, Michelle, Kai, Pai, Claire, Nori, Shiori, and Sam.
Somehow we ended up being an amalgamation of folks from the Asian diaspora,
and I am glad to have both connected culturally with my roots and made some

viii
great friends along the way. Our game nights together were so much fun. I would
also like to thank many of the other friends that I have shared special moments
with throughout these last several years: Ali Naqvi, Jessica Chen, Tom Liu, Vanna
Quon, Vivienne Do, Qui Nguyen, Tara Kwan, Irene Hong, Daniel Pan, Lynn Yi,
Erika Ye, Brynn Holbrook, Steven Wood, Aya Mimura, Thom Bohdanowicz, Austin
Liu, Bryce Edwards, Anthony Ardizzi, Megan Schill, Elise Tookmanian, Rebecca
Gallivan, Eowyn Lucas, Amylynn Chen, Wei-Lin Tan, Siobhan McArdle, Katherine
Rinaldi, Oscar and Hannah Zamora, Jane Siu, Louis Tsui, Jordan Goldstein, and
Mohammad Islam. I’m sorry if I missed anyone specifically, but if you ever put a
smile on my face, I am thankful for you, too.
I would also like to express my deepest gratitude for the scientific mentors I had
during undergrad that made this PhD journey even remotely possible: Jeremy
Feldblyum, Adam Matzger, Vidya Ganapati, Eli Yablonovitch, Alan Teran, Jamie
Philips, Aaron Rosenberg, Katja Nowack, John Kirtley, and Kam Moler. Jeremy
Feldblyum deserves a special acknowledgement, as his mentorship in my first ever
research experience was likely what made me continue pursuing research opportunities throughout undergrad and eventually a PhD.
Much of this thesis would also not be possible without the support and encouragement of my family. Dad, thank you for always supporting me in this academic path
and your general cheerfulness in the face of adversity. Your passion for esoteric
topics, while hardly scientific, is likely how I developed an interest in topics as
esoteric as the ones in science. Mom, thank you for encouraging me to seek academic opportunities when I was younger and supporting me with all of your food
and advice the last few years. I’m grateful that being at Caltech has allowed me to
reconnect with you. I would also like to acknowledge my sister and my cousins,
specifically: Mickey, Rose, Roby, Wayman, and Jerry. Our game nights during the
COVID-19 pandemic was nostalgic of our younger years together.

ix

ABSTRACT
Materials that are atomically thin behave substantially different than those of their
bulk counterparts. However, when most materials become thinner, their surface-tovolume ratio increases and the number of unpassivated dangling bonds at the surface
approaches the number of internal crystalline bonds, which prevents examining the
intrinsic properties of most ultrathin materials. The recent discovery of layered
materials, whose crystal structures have naturally passivated basal planes, has enabled the possibility to examine materials’ thicknesses that approach a single atomic
layer. In this thesis, we examine and explore the consequence of this new regime of
thickness for active layers in photovoltaic applications. Specifically, we focus on the
three aspects that define photovoltaic operation and explore their differences in these
ultrathin materials: optical absorption of photons, subsequent carrier generation and
transport, and finally, free energy extraction of collected carriers. We first discuss
the implications of band-edge abruptness on the maximum efficiency of a solar cell.
Then, we show that optical absorption in these ultrathin materials is dominated by
cavity wave optics, and design structures that enable near-unity absorption in both
ultrathin (∼10 nm) and atomically-thin (∼7 Å) active layers. Using these optical
design rules, we design heterostructures with record incident photon to electron
conversion efficiency (>50%). Next, we examine new methods of creating electrical
junctions by using thickness to vary the amount of band bending in a material. We
spatiotemporally image these ‘band-bending junctions’ for the first time. Finally,
we argue that photoluminescence can be used as a direct readout of the open circuit voltage potential, and motivate examination of monolayer materials which have
substantially higher radiative efficiency. We therefore examine the strain tuning of
photoluminescence properties of both monolayer TMDC and heterobilayer TMDC
systems. This work illustrates that van der Waals materials are an ideal system for
examining the novel optoelectronic physics of atomically thin photovoltaics.

PUBLISHED CONTENT AND CONTRIBUTIONS
* indicates equal contribution
[1] Chullhee Cho*, Joeson Wong*, Amir Taqieddin, Souvik Biswas, Narayana R.
Aluru, SungWoo Nam, and Harry A. Atwater. Highly strain-tunable interlayer excitons in mos2 /wse2 heterobilayers.
Nano Letters, 4 2021.
doi:10.1021/acs.nanolett.1c00724.
J.W. generated the ideas, participated in fabrication and optical measurements
of the samples, analyzed the data, and assisted with writing the manuscript.
[2] Deep Jariwala, Artur R. Davoyan, Giulia Tagliabue, Michelle C. Sherrott, Joeson Wong, and Harry A. Atwater. Near-unity absorption in van der waals
semiconductors for ultrathin optoelectronics. Nano Letters, 16(9):5482–5487,
2016. doi:10.1021/acs.nanolett.6b01914.
J.W. contributed to the fabrication and measurement of the samples.
[3] Deep Jariwala, Artur R. Davoyan, Joeson Wong, and Harry A. Atwater. Van der
waals materials for atomically-thin photovoltaics: Promise and outlook. ACS
Photonics, 11 2017. doi:10.1021/acsphotonics.7b01103.
J.W. performed the modified detailed balance calculations and contributed to
the writing and presentation of the manuscript.
[4] Joeson Wong, Tyler Colenbrander, Cora Went, Susana Torres-Londono, Rachel
Tham, and Harry A. Atwater. Perfect absorption in monolayer ws2 . In Preparation, 2022.
J.W. generated the ideas, fabricated the samples, created and automated the
optical set-up, performed all the measurements, data analysis, and calculations,
and wrote the manuscript.
[5] Joeson Wong*, Deep Jariwala*, Giulia Tagliabue, Kevin Tat, Artur R. Davoyan,
Michelle C. Sherrott, and Harry A. Atwater. High photovoltaic quantum efficiency in ultrathin van der waals heterostructures. ACS Nano, 11(7):7230–7240,
2017. doi:10.1021/acsnano.7b03148.
J.W. generated the ideas, fabricated and measured the devices, analyzed the
data, performed the calculations, and wrote the manuscript.
[6] Joeson Wong*, Stefan T. Omelchenko*, and Harry A. Atwater. Impact of
semiconductor band tails and band filling on photovoltaic efficiency limits. ACS
Energy Letters, 6(1):52–57, 12 2020. doi:10.1021/acsenergylett.0c02362.
J.W. generated the ideas, performed all the derivations, calculations, analysis,
and wrote the manuscript.
[7] Joeson Wong, Artur Davoyan, Bolin Liao, Andrey Krayev, Kiyoung Jo, Eli
Rotenberg, Aaron Bostwick, Chris M. Jozwiak, Deep Jariwala, Ahmed H. Ze-

xi
wail, and Harry A. Atwater. Spatiotemporal imaging of thickness-induced bandbending junctions. Nano Letters, 6 2021. doi:10.1021/acs.nanolett.1c01481.
J.W. generated the ideas, fabricated the samples, performed the ARPES measurements, assisted in the KPFM and SUEM measurements, analyzed the data,
performed the calculations, and wrote the manuscript.
[8] Joeson Wong, Stefan T. Omelchenko, and Harry A. Atwater. Fundamental
photovoltaic efficiency limits due to semiconductor band tails. In 2021 IEEE
48th Photovoltaic Specialists Conference (PVSC), pages 1315–1317, 2021.
doi:10.1109/PVSC43889.2021.9518548.
J.W. generated the ideas, performed all the derivations, calculations, analysis,
and wrote the manuscript.

xii

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . x
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Physics and Engineering of Conventional Photovoltaics . . . . . . . 1
1.2 Entering Flatland: van der Waals Materials . . . . . . . . . . . . . . 14
1.3 What’s Different in Atomically-thin Photovoltaics? . . . . . . . . . . 16
1.4 Scope of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 19

I Absorption Defines the Limits

21

Chapter II: Impact of the Semiconductor Band-Edge on Photovoltaic Efficiency Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Photovoltaic Efficiency Limit for Semiconductors with Band Tails . .
2.3 Generalized Voltage Loss for Semiconductors with Nonabrupt Band
Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Outlook on Examining Semiconductor Band Tails . . . . . . . . . .
2.5 Importance of the Direct-Indirect Gap Splitting on the Efficiency
Potential of Ultrathin TMDC Photovoltaics . . . . . . . . . . . . . .
2.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter III: Optical Design of Cavity Coupling to Excitonic TMDCs . . . . .
3.1 Linear Dielectric Function of TMDCs . . . . . . . . . . . . . . . . .
3.2 Multilayer Near Unity Absorption . . . . . . . . . . . . . . . . . . .
3.3 Monolayer Near Unity Absorption . . . . . . . . . . . . . . . . . . .
3.4 Monolayer Near Unity Absorption at Room Temperature . . . . . . .
3.5 Experimental Demonstration of Near Unity Absorption in Monolayer
WS2 at Room Temperature . . . . . . . . . . . . . . . . . . . . . . .
3.6 Efficiency Limits of Excitonic Multijunctions . . . . . . . . . . . . .

70
75

II Traversing through Flatland

80

22
22
24
26
31
33
39
58
58
62
66
68

Chapter IV: High photovoltaic quantum efficiency in ultrathin van der Waals
heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Van der Waals Heterostructure Device Fabrication . . . . . . . . . . 83

xiii
4.3 Spatial Photocurrent Map and IV Measurements . . . . . . . . . . . 86
4.4 Spectral Response Measurements . . . . . . . . . . . . . . . . . . . 86
4.5 Electromagnetic Simulations and Error Estimation . . . . . . . . . . 87
4.6 Prototypical Optoelectronic Device Characterization . . . . . . . . . 88
4.7 Absorption in van der Waals heterostructures . . . . . . . . . . . . . 90
4.8 Carrier collection efficiency in van der Waals semiconductor junctions 93
4.9 High Photovoltaic Quantum Efficiency Outlook . . . . . . . . . . . . 100
4.10 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter V: Spatiotemporal Imaging of Thickness-Induced Band Bending
Junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Correlation between Electronic Properties and Thickness in Ultrathin
Semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.3 Spatiotemporal Imaging of Charge Carrier Separation due to Thickness111
5.4 Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . . . 119
5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

III The Luminescence is the Voltage

123

Chapter VI: Highly strain tunable interlayer excitons in MoS2 /WSe2 Heterobilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2 Local Strain Engineering of Intra- vs. Inter-layer excitons . . . . . . 125
6.3 Tuning Interlayer Coupling through Strain Engineering . . . . . . . . 132
6.4 ab initio strain calculations of TMDC heterobilayers . . . . . . . . . 135
6.5 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . 139
6.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

IV What’s next for Flatland?

142

Chapter VII: Conclusions and Outlook . . . . . . . . . . . . . . . . . . . . . 143
7.1 Outlook from Semiconductor Band Tails Work . . . . . . . . . . . . 144
7.2 Outlook on Unity Absorbance and Cavity Coupling to Excitonic
Absorbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.3 Outlook on Achieving High-Efficiency, TMDC-based Photovoltaic
Devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Appendices

176

Appendix A: Microscopic Physics of Excitonic Systems . . . . . . . . . . . . 177
A.1 Formation and Dissociation of Excitons . . . . . . . . . . . . . . . . 177
Appendix B: Macroscopic Optical Properties of Layered Structures . . . . . . 184
B.1 Boundary conditions on Electromagnetic Fields . . . . . . . . . . . 184
B.2 Optical Waves in Homogenous Media . . . . . . . . . . . . . . . . . 188

xiv
B.3 Transfer Matrix Method for Layered Media . . . . . . . . . . . . . . 192
B.4 Lorentz Oscillator Model . . . . . . . . . . . . . . . . . . . . . . . 198
B.5 Reflection, Transmission, and Absorption of a 2D exciton . . . . . . 200
Appendix C: Thermodynamics Considerations of Photovoltaic Systems . . . 205
C.1 Derivation of Blackbody Radiation . . . . . . . . . . . . . . . . . . 205
C.2 The Chemical Potential of a Photon . . . . . . . . . . . . . . . . . . 210
C.3 The Validity of a Thermalized Population as the Major Contribution
to Photoluminescence Under Steady State Excitation . . . . . . . . . 214
C.4 The Validity of A Single Quasi-Fermi Level to Describe Carriers
Occupying Multiple Energy Levels . . . . . . . . . . . . . . . . . . 216
Appendix D: Computer Code . . . . . . . . . . . . . . . . . . . . . . . . . . 220

xv

LIST OF ILLUSTRATIONS

Number
Page
1.1 Typical Solar Cell IV Curves . . . . . . . . . . . . . . . . . . . . . . 3
1.2 NREL Efficiency Chart . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 𝐽𝑠𝑐 vs. 𝑉𝑜𝑐 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Detailed Balance Efficiency Limit . . . . . . . . . . . . . . . . . . . 14
1.5 van der Waals Heterostructures . . . . . . . . . . . . . . . . . . . . 15
1.6 van der Waals Photovoltaics Schematic . . . . . . . . . . . . . . . . 16
1.7 Dimensionality Effects on Exciton Screening . . . . . . . . . . . . . 17
2.1 Accounting for band filling in modified reciprocity relations . . . . . 23
2.2 Effects of band tailing on photovoltaic limiting efficiencies . . . . . . 25
2.3 Dependence of photovoltaic figures of merit on the Urbach parameter 26
2.4 Effects of thickness on photovoltaic figures of merit . . . . . . . . . . 27
2.5 Analysis of a Gaussian band tail distribution . . . . . . . . . . . . . 28
2.6 Effects of band tail states on photoluminescence . . . . . . . . . . . 29
2.7 Generalized voltage losses parametrized as a two-bandgap absorber . 30
2.8 Voltage loss due to a nonabrupt band-edge . . . . . . . . . . . . . . 32
2.9 Unique Bandstructure and Absorption Edge of Bulk TMDCs . . . . . 34
2.10 Absorption Toy Model for Bulk TMDCs . . . . . . . . . . . . . . . 36
2.11 Photovoltaic Figures of Merit of WS2 for Varying 𝐶 𝐴𝐸 and 𝐸 𝑅𝐸 0 . . 38
2.12 Photovoltaic Figures of Merit of MoSe2 for Varying 𝐶 𝐴𝐸 and 𝐸 𝑅𝐸 0 . 39
2.13 The importance of including band filling effects . . . . . . . . . . . . 44
2.14 Effects of band tails and band filling on ideality factor and currentvoltage relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.15 Analysis of a two-bandgap toy model . . . . . . . . . . . . . . . . . 50
2.16 Effects of a sub-unity collection efficiency below the bandgap . . . . 54
2.17 Different band edges that map onto a two-bandgap model . . . . . . . 57
3.1 First measurements of the bulk MoS2 dielectric function . . . . . . . 59
3.2 Room temperature dielectric function of monolayer TMDCs . . . . . 60
3.3 Bandstructure as a function of thickness . . . . . . . . . . . . . . . . 61
3.4 Designing ultrathin absorbing cavities . . . . . . . . . . . . . . . . . 62
3.5 Near Unity Absorption in ultrathin TMDCs . . . . . . . . . . . . . . 64
3.6 Angle dependence of absorption in ultrathin TMDC/Ag structures . . 66

xvi
3.7 Metallic Optical Cavity for Monolayer Perfect Absorption . . . . . . 70
3.8 Dielectric-Metal Optical Cavity for Monolayer Perfect Absorption . . 71
3.9 Relationship between Absorption and Excitonic Linewidth . . . . . . 72
3.10 Geometric tolerance of monolayer perfect absorption optical cavities 73
3.11 Experimental Demonstration of Near-Unity Absorption in WS2 at
Room Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.12 Schematic of Reflectance Measurement Set-up . . . . . . . . . . . . . . 76
3.13 Limiting Efficiency of Excitonic Multijunctions . . . . . . . . . . . . 77
3.14 Optimal Absorption Spectra of Excitonic Multijunctions . . . . . . . 79
4.1 Achieving High External Quantum Efficiency in van der Waals heterostructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.2 High Photovoltaic Quantum Efficiency Sample Images . . . . . . . . 85
4.3 Optoelectronic Performance Characteristics . . . . . . . . . . . . . . 89
4.4 Power dependent device characteristics . . . . . . . . . . . . . . . . 91
4.5 Absorbance in van der Waals heterostructures . . . . . . . . . . . . . 92
4.6 Charge transport and collection in vertical PN and Schottky junction
geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.7 Few-layer graphene as a transparent top contact . . . . . . . . . . . . 97
4.8 Absorbance and EQE of thick and thin PN heterojunctions . . . . . . 99
4.9 Thickness dependence on charge collection efficiency . . . . . . . . 100
4.10 Estimated 1 Sun AM 1.5G Performance . . . . . . . . . . . . . . . . 103
5.1 Thickness-dependent surface potentials due to vertical band bending . 108
5.2 Experimental observation of thickness-dependent surface doping in MoS2 /Au 109
5.3 Contact potential difference of other MoS2 flakes on Au . . . . . . . 110
5.4 Photoemission spectroscopy of another MoS2 flake on Au . . . . . . 112
5.5 Scanning Ultrafast Electron Microscopy Imaging of a Band Bending Junction 113
5.6 Scanning ultrafast electron microscopy of bulk MoS2 on Au . . . . . 114
5.7 Scanning ultrafast electron microscopy of monolayer MoS2 on Au . . 115
5.8 Schematic depiction of carrier transport at a band bending junction . 116
5.9 Simulated carrier dynamics for various material parameters . . . . . 117
5.10 Hole dominated charge transport . . . . . . . . . . . . . . . . . . . . 118
5.11 Simulation of Carrier Dynamics at a Bend Bending Junction . . . . . 119
6.1 Fabrication of wrinkled MoS2 /WSe2 heterobilayer . . . . . . . . . . 126
6.2 Sample fabrication and characterization of wrinkled heterostructures 127
6.3 Optical characterization of flat heterobilayers of MoS2 /WSe2 on PDMS128

xvii
6.4
6.5
6.6
6.7

6.8
6.9
6.10
6.11
6.12
6.13
6.14

7.1
7.2
A.1
A.2
A.3
A.4
A.5

B.1

B.2

Photoluminescent mapping of the fabricated wrinkled MoS2 /WSe2
heterobilayer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Photoluminescence of strained heterobilayers . . . . . . . . . . . . . 130
Optical characterization of different wrinkled MoS2 /WSe2 heterobilayer samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Photoluminescence (PL) measurement on wrinkled heterobilayers of
MoS2 /WSe2 via vertical scanning (XZ-PL scanning) accounting for
the finite depth of focus . . . . . . . . . . . . . . . . . . . . . . . . 132
Raman spectroscopy and deformation potentials of strained heterobilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Raman mapping of strained heterobilayers . . . . . . . . . . . . . . . 135
Wrinkle geometry-driven strain analysis . . . . . . . . . . . . . . . . 135
Density functional theory calculation of strained MoS2 /WSe2 heterobilayers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Weighted band structures obtained using DFT calculations . . . . . . 137
Band hybridization as a function of strain with different interlayer
distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Effects of strain, interlayer spacing, and different stacking configurations on the electronic band structure of MoS2 /WSe2 heterobilayer
system obtained using DFT calculations . . . . . . . . . . . . . . . . 139
I-V performance of a carrier selective contact device . . . . . . . . . 150
X-ray photoemission spectroscopy of TiO𝑥 and NiO𝑥 . . . . . . . . . 151
Exciton Wavefunction under an Electric Field . . . . . . . . . . . . . 179
Exciton Dissociation Efficiency under an Electric Field . . . . . . . . 180
Exciton Binding Energy under an Electric Field . . . . . . . . . . . . 180
Exciton Binding Energy vs. Screening Length . . . . . . . . . . . . 181
Wavefunctions of the excitons in the 2D Keldysh potential, for 𝑛 =
1, 2, 3, 4 for each row, i.e. 𝑠, 𝑝, 𝑑, 𝑓 -like wavefunctions. Wavefunctions are similar to the hydrogenic model. . . . . . . . . . . . . . . 183
A schematic of a Gaussian pillbox (dotted box) with height ℎ and
area 𝐴 sandwiched between two materials with dielectric constant 𝜖 1
and 𝜖2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
A schematic of a Gaussian loop (dotted rectangle) with height ℎ and
length 𝑙 drawn between two materials with dielectric constant 𝜖 1 and
𝜖2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

xviii
B.3

Schematic of a one-dimensional stack consisting of 𝑁 layers. The
arrows represent the reflected and transmitted electromagnetic waves
in each layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
C.1 Schematic depiction of the relative timescales relevant to electronelectron scattering (equilibration), electron-phonon coupling (thermalization), and electron-hole recombination. Adapted from Y.
Takeda et al., 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

xix

LIST OF TABLES
Number
Page
2.1 Parameter values for ultrathin WS2 modified detailed balance calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2 Tabulated values of Urbach Energies (Experiment) and Δ𝑉𝑜𝑐 loss
(Calculated) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1 Room temperature values for the excitonic sheet conductor model of
various TMDCs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Chapter 1

INTRODUCTION
“You don’t have to be great to get started, but you
have to get started to be great.”
— Les Brown

1.1

Physics and Engineering of Conventional Photovoltaics

Photovoltaic devices are systems that convert incident radiation (typically sunlight,
in which case is referred to as a solar cell) into electrical energy. In many ways,
photovoltaic systems are essentially light-powered batteries where load matching
is necessary to maximize the output power of the solar cell. In this section, we
first introduce the internal physics that is common to the operation of every solar
cell. Next, we introduce and describe the typical current-voltage behavior of solar
cells in the form of the modified diode equation. We then connect the three main
factors that make up the solar cell power conversion efficiency to internal physical
processes and discuss how they are conventionally engineered. Finally, we describe
the efficiency limits of a solar cell by invoking detailed balance arguments, which
motivates and further elucidates the behavior of a conventional solar cell.
Current Voltage Characteristics of a Typical Solar Cell
The current-voltage (𝐼 −𝑉) characteristics of solar cells can be typically represented
with an equivalent circuit model:
𝑉 − 𝐼 𝑅𝑠
𝑞(𝑉 − 𝐼 𝑅𝑠 )
𝐼 (𝑉) = 𝐼0 exp
− 𝐼𝐿
(1.1)
−1 +
𝑛𝑖𝑑 𝑘 𝑏 𝑇
𝑅𝑠ℎ
where 𝑞 is the fundamental unit of charge (= 1.6 × 10−19 C), 𝑘 𝑏 𝑇 is the thermal
energy (= 25.8 meV at room temperature), 𝐼0 is the recombination current prefactor,
𝑛𝑖𝑑 is the diode ideality factor, 𝑅𝑠 is the equivalent internal series resistance of the
solar cell, 𝑅𝑠ℎ is the equivalent internal shunt resistance, 𝐼 𝐿 is the photogenerated
current, and 𝐼, 𝑉 is the output current, voltage of the solar cell, respectively. Because
the current-voltage curves are strongly asymmetric with voltage (as opposed to the
symmetric case of an ideal resistor), the current-voltage relation is rectifying and
therefore reminescent of a diode, where current flows nearly entirely in one direction.

The power 𝑃 generated from the solar cell can be found by the usual expression
𝑃(𝑉) = 𝐼 (𝑉)𝑉

(1.2)

where we note that the expressions for the power is dependent on the voltage across
the solar cell. Note that Equation 1.2 has powers that are negative, since the current
itself is negative, i.e., the power is generated from the solar cell, as opposed to the
current through a resistor where the power is dissipated. The maximum power point
of the solar cell is denoted by 𝑃𝑚 𝑝 𝑝 = 𝐼𝑚 𝑝 𝑝 𝑉𝑚 𝑝 𝑝 , which specifies a specific load
condition to maximize the energy harvested from a solar cell. For a given load 𝑅 𝐿 ,
the output current is given by:
− 𝐼 (𝑉) =

𝑅𝐿

(1.3)

where the current 𝐼 (𝑉) from the solar cell is now dissipated in the resistive load,
so the negative sign in front of the current accounts for this factor. The specific
current and voltage for a given load resistance can be solved graphically, as shown
in Figure 1.1. Thus, it is clear that the load that maximizes the energy harvested is
given simply by
𝑉𝑚 𝑝 𝑝
(1.4)
𝑅𝑐ℎ =
𝐼𝑚 𝑝 𝑝
Therefore, once the current-voltage relations of a solar cell is carefully characterized,
we can maximize its output power by matching the load resistance to its characteristic
resistance. To note, the maximum power point can also be written as
𝑃𝑚 𝑝 𝑝 = 𝐼𝑚 𝑝 𝑝 𝑉𝑚 𝑝 𝑝 = 𝐼 𝑠𝑐𝑉𝑜𝑐 𝐹𝐹

(1.5)

which separates the power generation of the solar cell into three distinct but interrelated terms. Here, 𝐼 𝑠𝑐 refers to the short circuit current, which describes the
photogenerated current of the solar cell under short-circuit conditions (i.e., when
the external load is 𝑅 𝐿 = 0), 𝑉𝑜𝑐 is the open-circuit voltage, which describes the
photovoltage when the solar cell is under open-circuit conditions (i.e., when the
external load is 𝑅 𝐿 = ∞), and 𝐹𝐹 is the fill factor, which describes the square-ness
of the 𝐼 − 𝑉 curve and can be seen as being equal to
𝐹𝐹 =

𝐼𝑚 𝑝 𝑝 𝑉𝑚 𝑝 𝑝
𝐼 𝑠𝑐𝑉𝑜𝑐

(1.6)

Thus, the power conversion efficiency 𝜂 can be calculated as
𝜂=

𝑃𝑚 𝑝 𝑝 𝐼 𝑠𝑐𝑉𝑜𝑐 𝐹𝐹
𝑃𝑠𝑢𝑛
𝑃𝑠𝑢𝑛

(1.7)

where 𝑃𝑠𝑢𝑛 is the incident power from the sun. The above terms are typically
normalized to the area of the solar cell so that the terms are not a priori dependent
on cell size. Therefore, currents are typically referred to as current densities (e.g.
𝐽𝑠𝑐 is the short-circuit current density, in units of mA/cm2 ), resistances are typically
quoted as sheet resistances (𝑟 𝑠 and 𝑟 𝑠ℎ , in units of Ω-cm2 ), and incident power 𝑃𝑠𝑢𝑛
is typically quoted in units of power density (e.g. W/m2 ).

𝑰𝒔𝒄

Photogenerated Current
𝑽𝒎𝒑𝒑 , 𝑰𝒎𝒑𝒑

Under
Illumination

𝑽𝒐𝒄
Dark Current

Figure 1.1: Typical Solar Cell IV Curves. Typical current-voltage curves of a
solar cell under illumination (blue) and in the dark (orange), along with its powervoltage (yellow) curve. Here, we plot the generated current and power as a positive
quantity. The purple curve corresponds to the current-voltage curve of a resistive
load 𝑅 𝐿 . In this case, the load resistance is matched to the characteristic resistance,
𝑅 𝐿 = 𝑅𝑐ℎ = 𝑉𝑚 𝑝 𝑝 /𝐼𝑚 𝑝 𝑝 , which maximizes the energy harvesting of the solar cell,
i.e., it operates at its max power point. The short-circuit current 𝐼 𝑠𝑐 and open-circuit
voltage 𝑉𝑜𝑐 points are also labelled.

Engineering for Maximum Efficiency
Since solar cells are made to generate electrical power, the power conversion efficiency 𝜂 becomes an important figure of merit to describing the technological

potential of a solar cell technology as well as a method to reducing the overall capital cost of commercializing solar cell modules. Since 1976, NREL has kept track of
the maximum efficiencies achieved in different solar cell technologies, reproduced
in Figure 1.2. Here, we briefly discuss the typical engineering considerations in
maximizing the efficiency potential of a solar cell material, which involves simultaneously optimizing the 𝐽𝑠𝑐 , 𝑉𝑜𝑐 , and 𝐹𝐹 of the solar cell.

Figure 1.2: NREL Efficiency Chart. Maximum power conversion efficiencies
achieved in different photovoltaic technology, plotted as a function of the year the
record was achieved.

Jsc
The short circuit current density 𝐽𝑠𝑐 describes the photogenerated current when
the solar cell is shorted as illustrated in Figure 1.3a. Thus, to maximize 𝐽𝑠𝑐 , we
must optimize both the optical absorption (generation of electron-hole pairs) and
subsequent collection of the electron-hole pairs. To see this more explicitly, the
short circuit current density can be written a
∫ ∞
𝐽𝑠𝑐 = 𝑞
𝐸𝑄𝐸 (𝐸)𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸
(1.8)

where 𝐸 is the energy of the incident photon, 𝑆 𝑠𝑢𝑛 (𝐸) is the incident spectral solar
flux (in units of photons/m2 /sec), and 𝐸𝑄𝐸 (𝐸) is the external quantum efficiency,
which defines the number of incident photons that are converted to collected electrons. Therefore, the maximum 𝐸𝑄𝐸 is unity, and given the typical solar fluence

𝐸𝑐
𝐸𝐹𝑛

𝐸𝑐
𝐸𝐹𝑛

𝐸𝑣

𝐸𝐹𝑝

𝐸𝐹𝑝

𝐸𝑣

𝑅𝐿 = 0

𝑅𝐿 = ∞

Figure 1.3: 𝐽𝑠𝑐 vs. 𝑉𝑜𝑐 . Schematic depiction of the solar cell under short-circuit a
and open-circuit b conditions.

𝑆 𝑠𝑢𝑛 (𝐸) (specified as AM 1.5G), the maximum short circuit current density achievable is ∼70 mA/cm2 .
The 𝐸𝑄𝐸 is a product of two terms:
𝐸𝑄𝐸 (𝐸) = 𝑎(𝐸)𝐼𝑄𝐸 (𝐸)

(1.9)

where 𝑎(𝐸) describes the probability of a photon to being absorbed at photon
energy 𝐸, and 𝐼𝑄𝐸 (𝐸) describes the internal quantum efficiency, or the probability
of the generated electron-hole pair to be collected. Therefore, 𝐼𝑄𝐸 is purely a
description of the electronic geometry, and is wavelength-independent a priori.
However, different photon energies may result in different amounts of generation
and recombination in different places, and therefore there is a subtle photon energy
dependence1.
To maximize absorbance, it is necessary to engineer the optical configuration and/or
thickness of the active layer. Absorption in the ray-optics regime (i.e., 𝐿
𝜆) is
typically described by the Beer-Lambert law:
𝑎(𝐸) = (1 − 𝑅(𝐸))(1 − exp(−𝛼(𝐸)𝐿))

(1.10)

1 For example, in a typical solar cell, bluer photons are absorbed closer to the surface due to the
increased absorption coefficient at higher energies, and therefore the generated electron-hole pairs
with that energy are less likely to collected due to their distance from the junction of the solar cell
and finite surface recombination velocity.

where 𝑅(𝐸) describes the reflectance of the semiconductor, primarily dictated by
the air-semiconductor interface, 𝛼(𝐸) is the absorption coefficient, a relatively intrinsic property of a semiconductor, and 𝐿 is the thickness of the semiconductor or
its equivalent absorption path length. Therefore, absorption is maximized by (1)
minimizing reflectance, e.g., with the use of an anti-reflection coating, (2) maximizing absorption coefficient, e.g., with the use of a direct bandgap semiconductor, and
(3) increasing the thickness of the semiconductor.
On the other hand, maximizing the carrier collection efficiency requires both the
breaking the symmetry of electron and hole transport (e.g., with an electric field,
or more generally, through the asymmetries in the electrical conductivity of the
respective charge carrier) and maximizing the diffusion length 𝐿 𝐷 of the respective
carriers. In contrast to absorbance, carrier collection efficiency typically decreases
with an increase in the active layer thickness, because carriers are only collected
within a diffusion length 𝐿 𝐷 of the electrical junction of the solar cell. Therefore,
our main constraint for the thickness of the solar cell active layer is 𝐿 . 𝐿 𝐷 . A
further increase in the absorption path length can therefore be achieved with light
trapping geometries, e.g., with the addition of surface texturing and a back reflector,
𝐿 𝑒 𝑓 𝑓 = 4𝑛𝑟2 𝐿, [213] where 𝑛𝑟 is the refractive index of the semiconductor.
Voc
We next turn our attention to the physics of the open circuit voltage 𝑉𝑜𝑐 , which is
schematically depicted in Figure 1.3b. To understand the physics of the open-circuit
voltage, it is necessary to recall that it is an electrochemical quantity, and therefore
related to the electrochemical potentials of the charge carriers:
𝑞𝑉 = 𝐸 𝑓 ,𝑙𝑒 𝑓 𝑡 − 𝐸 𝑓 ,𝑟𝑖𝑔ℎ𝑡 = 𝐸 𝑓𝑛 ,𝑙𝑒 𝑓 𝑡 − 𝐸 𝑓 𝑝 ,𝑟𝑖𝑔ℎ𝑡

(1.11)

where 𝐸 𝑓 refers to the Fermi level, specifically of the metal contacts (one on the
left and the other on the right), and 𝐸 𝑓𝑛, 𝑝 corresponds to the quasi-Fermi levels that
describe the local carrier concentration 𝑛 = 𝑔𝑛 (𝐸) 𝑓 (𝐸, 𝐸 𝑓 ,𝑛 )𝑑𝐸, where

𝑓 (𝐸, 𝐸 𝑓𝑛 ) =
exp

𝐸−𝐸 𝑓𝑛
𝑘 𝑏𝑇

(1.12)
+1

is the Fermi occupation factor and 𝑔𝑛 (𝐸) is the density of states for the electrons.
A similar expression holds for holes. Here, quasi-Fermi levels are necessary to
describe the carrier population out of equilibrium2, e.g., under photoexcitation
2 To see a more detailed derivation for when a single quasi-Fermi level is an accurate picture, see

section C.4.

(i.e., 𝑛 = 𝑛0 + Δ𝐺, where 𝑛0 is the equilibrium carrier population and Δ𝐺 is the
photogenerated carrier population). If the quasi-Fermi levels are sufficiently far from
the band-edges of the semiconductor (e.g., (𝐸 𝑐 − 𝐸 𝑓𝑛 )/𝑘 𝑏 𝑇 & 3, which is almost
always true for solar cells under 1 sun illumination), it is possible to approximate
the above expression for carriers as
𝐸 𝑐 − 𝐸 𝑓𝑛
(1.13)
𝑛 = 𝑁𝑐 exp −
𝑘 𝑏𝑇
where 𝑁𝑐 is the effective density of the states for the conduction band, 𝐸 𝑐 is the
𝐸 𝑓 −𝐸 𝑣
conduction band energy, and similarly for holes, we have 𝑝 = 𝑁𝑣 exp − 𝑘𝑝𝑏 𝑇 .
Thus,
𝐸𝑔
Δ𝜇
Δ𝜇
𝑛𝑝 = 𝑁𝑐 𝑁𝑣 exp −
exp
= 𝑛𝑖 exp
(1.14)
𝑘 𝑏𝑇
𝑘 𝑏𝑇
𝑘 𝑏𝑇
where Δ𝜇 = 𝐸 𝑓𝑛 − 𝐸 𝑓 𝑝 is the quasi-Fermi level splitting (or the internal voltage)
and 𝑛𝑖 is the intrinsic carrier population (which is a property of the semiconductor).
Since a small but finite gradient of the quasi-Fermi levels are necessary to drive
current flow
𝜎𝑖
(1.15)
𝐽𝑖 = ∇𝐸 𝑓𝑖
where 𝑖 = 𝑛, 𝑝 refers to the electron and hole current densities, then we generally
have 𝑞𝑉 < Δ𝜇 as a constraint. At open circuit, this condition still holds, since only
the total current 𝐽 = 𝐽𝑛 + 𝐽 𝑝 must be zero. Therefore, we generally have 𝑞𝑉𝑜𝑐 < Δ𝜇.
To summarize the above analysis, we have connected the external voltage 𝑉 to the
electrochemical potentials of the electron and hole. The external open circuit voltage
𝑉𝑜𝑐 will be limited by the internal quasi-Fermi level splitting Δ𝜇, which is related
to the steady state populations of the electrons and holes. Therefore, to maximize
the open circuit voltage 𝑉𝑜𝑐 , we must maximize generation and minimize carrier
recombination.
Another way of understanding the open circuit voltage is from the diode model of
the solar cell Equation 1.1, where we shall set 𝑅𝑠 = 0 and 𝑅𝑠ℎ = ∞ for simplicity.3.
Therefore, we have
𝐽𝑠𝑐
𝑛𝑖𝑑 𝑘 𝑏 𝑇
𝑉𝑜𝑐 =
ln
+1
(1.16)
𝐽0
where we have normalized the current densities to their areas 𝐽 = 𝐼/𝐴 and denoted
the photogenerated current 𝐽 𝐿 as 𝐽𝑠𝑐 , which is the same in this scenario. It is clear
3 More generally, since open circuit implies 𝐼 = 0, series resistance 𝑅

𝑠 has no direct effect on
the open circuit voltage. However, 𝑅𝑠ℎ can have a dramatic effect on the open circuit voltage. The
physical causes and implications of a finite 𝑅𝑠ℎ and 𝑅𝑠 will be discussed in the fill factor section
below.

then that the open circuit voltage 𝑉𝑜𝑐 is increased with generation (in the form of
𝐽𝑠𝑐 ) and decreasing with increased recombination current density 𝐽0 . The rate of
change is partially modulated by 𝑛𝑖𝑑 , which depends on the dominant form of carrier
recombination.
There are three main forms of recombination that dominate traditional photovoltaic
materials, which are (1) Shockley-Reed-Hall (trap-assisted) recombination, (2) Radiative recombination, (3) Auger-Meitner recombination. Trap-assisted recombination stems from trap states (typically due to defects in the crystal structure) which
allows electrons to relax prematurely in a non-radiative way from the conduction
band to the valence band. Typically, the non-radiative relaxation (in the form of
phonon emission) stems from a small but continuous set of states near the band-edge,
in the form of a semiconductor band tail. This continuity of states comes from the
breaking of the crystalline symmetry due to the random positioning of defects, and
the eventual trap-assisted recombination can be parametrized as

𝑆𝑅𝐻

𝑛𝑝 − 𝑛𝑖2
𝜏𝑛𝑆𝑅𝐻 ( 𝑝 + 𝑝 1 ) + 𝜏𝑝𝑆𝑅𝐻 (𝑛 + 𝑛1 )

(1.17)

where 𝑅 𝑆𝑅𝐻 is the Shockley-Reed-Hall (SRH) recombination rate (in units of
1/cm3 /s), 𝑛 and 𝑝 are the electron and hole populations, 𝑛1 and 𝑝 1 are the density of
available trap states for the electrons and holes, and 𝜏𝑖𝑆𝑅𝐻 is the Shockley-Reed-Hall
recombination lifetime. For doped semiconductors (e.g., 𝑛 = 𝑁 𝐷
𝑝, 𝑛1 , 𝑝 1 ), we
have
𝑆𝑅𝐻
(1.18)
𝑅n-type
≈ 𝑆𝑅𝐻
𝜏𝑝
so that the minority carrier dominates the SRH recombination. Furthermore, SRH
recombination is typically a single-carrier process.
Radiative recombination is a fundamental process where the recombination of an
electron and hole results in an emitted photon. The rate is simply given as
𝑅 𝑅𝑎𝑑 = 𝑘 𝑟 (𝑛𝑝 − 𝑛𝑖2 )

(1.19)

where 𝑘 𝑟 is radiative recombination rate coefficient. 𝑘 𝑟 is substantially larger in
materials that have a direct bandgap (e.g. GaAs) compared to those with an indirect
bandgap (e.g. Si). Note that radiative recombination is generally a two-carrier
process.
Finally, Auger-Meitner recombination refers to the energy transfer of an electronhole interaction that is transferred directly to another electron or hole. Since the

energetically “hot” electron or hole is then subsequently thermalized to the bandedge, this process is also typically non-radiative. Its parametrization is given by
𝑅 𝐴𝑢𝑔−𝑀𝑒𝑖𝑡 = (𝐶𝑛 𝑛 + 𝐶 𝑝 𝑝)(𝑛𝑝 − 𝑛𝑖2 )

(1.20)

where 𝐶𝑛 and 𝐶 𝑝 are the Auger-Meitner coefficients for the electrons and holes.
Here, it is clear that Auger-Meitner is a three-carrier process.
Since these three recombination modalities are common to many traditional semiconductors, it is common to use an “ABC” recombination model:
𝑅 = 𝑅 𝑆𝑅𝐻 + 𝑅 𝑅𝑎𝑑 + 𝑅 𝐴𝑢𝑔−𝑀𝑒𝑖𝑡 ≈ 𝐴𝑥 + 𝐵𝑥 2 + 𝐶𝑥 3

(1.21)

where 𝑥 is the minority carrier type (either 𝑛 or 𝑝). This model captures the
effects of the different types of recombination in a straightforward manner. If
there are larger asymmetries between the electron and hole or the solar cell is
operated under high injection, it may be necessary to use the full expressions of
each recombination model. Furthermore, since 𝑛 ∼ exp(Δ𝜇/(2𝑘 𝑏 𝑇)), it is clear that
the ideality factor 𝑛𝑖𝑑 approaches 2 for SRH-dominated recombination, approaches
1 for Radiative-dominated recombination, and approaches 1/2 for Auger-Meitnerdominated recombination.
Of additional interest is the recombination due to the surface of a material, which
is usually due to the breaking of crystalline symmetry at the interface and therefore
results in surface states. Some of these states can be passivated with appropriate termination of chemical bonds, but nonetheless, the interface can typically be
characterized by a surface recombination velocity 𝑆𝑥 , where 𝑥 = 𝑛, 𝑝. In general,
this additional surface recombination acts as a boundary condition on the current
density:
Jx · 𝑛ˆ = ±𝑞𝑆𝑥 (𝑥 − 𝑥0 )
(1.22)
where 𝑛ˆ is the normal vector of the interface, and 𝑥 0 corresponds to the carrier
concentration in the dark. Here, the negative sign is associated with electrons 𝑛 and
the positive sign is associated with holes 𝑝. We note that generally the effective
lifetime 𝜏𝑒 𝑓 𝑓 due to both bulk and surface recombination is given as:
𝜏𝑒 𝑓 𝑓

𝜏𝑏 𝜏𝑠

(1.23)

where 𝜏𝑏 is the bulk lifetime and 𝜏𝑠 is the surface recombination lifetime. This can
be approximated as
 2
1 𝐿
𝜏𝑠 =
(1.24)
2𝑆 𝐷 𝜋

10
for when there are two surfaces with approximately equal surface recombination
velocities (i.e., 𝑆 = 𝑆1 ≈ 𝑆2 [179]). Here, 𝐿 is the thickness of the solar cell and 𝐷
is the diffusion coefficient.
FF
We finally discuss the mechanisms that affect the fill factor of a solar cell, which
describes the “squareness” of the 𝐼 − 𝑉 curve. From inspection of Equation 1.1, the
largest effects are due to the finite series resistance 𝑅𝑠 and shunt resistance 𝑅𝑠ℎ in a
solar cell.
In the case of series resistance, this may come from the resistance in the bulk of the
active layer or at the contacts. Since the resistance is given as 𝑅 = 𝜌𝐿/𝐴, where 𝜌 is
the resistivity, 𝐿 is the thickness of the semiconductor, 𝐴 is the cross-sectional area,
it can be modified either geometrically or through carrier doping. The resistivity is
related to the conductivity as
𝜎=

= 𝑒(𝑛𝜇𝑛 + 𝑝𝜇 𝑝 )

(1.25)

and is therefore a measure of both the local carrier concentration and the mobility
of charge carriers. Large bandgap materials typically have a lower conductivity due
to their reduced intrinsic carrier concentration, dopant states that are further from
the band-edge, and a reduction in the overall solar-generated carriers. The effect of
series resistance can be first observed as a change in the slope of the 𝐽 − 𝑉 curves
near 𝑉𝑜𝑐 , and for large series resistances, will also impact 𝐽𝑠𝑐 .
Another practical concern is the shunt resistance, which is a measure of other
effective electrical circuits that are in parallel to the dominant electrical junction.
For example, in an ideal solar cell, there is a single electrical path for all the carriers
to follow. However, due to spatial heterogeneity and defects, it may be possible
for certain carriers to follow a different electrical path, resulting in an effective
shunt resistance. As an example, if the two sides of the solar cell contacts were
connected, the shunt resistance 𝑅𝑠ℎ → 0, and therefore diode behavior would no
longer be observed. In other words, shunt resistance is often a measure of carrier
selectivity, i.e., as the transport of electrons and holes become more symmetric, the
diode behavior transitions to resistive behavior, and therefore the carrier selectivity
is lost. Thus, the effects of shunt resistance are first observed in the slope of the
short circuit current 𝐽𝑠𝑐 and then for large shunt resistances at 𝑉𝑜𝑐 .

11
It is important to note that both descriptions of the series and shunt resistances
are perturbative to the diode model of a solar cell. It is important to note that for
a given solar cell, the electrical characteristics may be drastically different due to
the nature of the electrical interfaces and the specific geometry of the solar cell.
Therefore, it is necessary to observe diode-like rectifying behavior before ascribing
series and shunt resistances, which may be meaningless in certain situations. Thus,
carrier selectivity and the aforementioned symmetry breaking of electron and hole
transport is fundamental to the operation of a solar cell.
Single Junction Detailed Balance Efficiency Limit
To further elucidate the mechanisms intrinsic to solar cell operation as well as to
ascribe practical efficiencies achievable, it is useful to consider the thermodynamic
efficiency limit of a solar cell. Consider a hypothetical scenario where the solar cell
can be considered as a zero-dimensional object, where local gradients internal to
the solar cell are ignored. Then, the solar cell becomes a steady state flux balance
expression where
𝐽𝑔𝑒𝑛 − 𝐽𝑟𝑒𝑐𝑜𝑚𝑏 − 𝐽𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 = 0
(1.26)
where electron-hole pairs are generated at a rate 𝐽𝑔𝑒𝑛 , collected electrically at a rate
𝐽𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 , and recombines internally as 𝐽𝑟𝑒𝑐𝑜𝑚𝑏 . Note that the generation of carriers
is directly related to the absorbance of the solar cell, i.e.,
∫ ∞
𝐽𝑔𝑒𝑛 = 𝑞
𝑎(𝐸)𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸
(1.27)

where 𝑎(𝐸) is the absorbance of the solar cell and 𝑆 𝑠𝑢𝑛 (𝐸) describes the solar flux.
Further, 𝐽𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 = 𝐽 = 𝐽𝑔𝑒𝑛 − 𝐽𝑟𝑒𝑐𝑜𝑚𝑏 is the current density that we observe in
our electrical circuit. Then, what is the form of 𝐽𝑟𝑒𝑐𝑜𝑚𝑏 ? Clearly there are various
recombination mechanisms that are possible, as described in section 1.1. However,
which mechanism is thermodynamically necessary?
Let us consider an alternative situation where the solar cell is instead in the dark,
without solar radiation. In this case, we must have 𝐽𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑒𝑑 = 𝐽 = 0, since we are
in thermodynamic equilibrium. Thus, we have
𝐽𝑔𝑒𝑛,𝑑𝑎𝑟 𝑘 = 𝐽𝑟𝑒𝑐𝑜𝑚𝑏,𝑑𝑎𝑟 𝑘

(1.28)

In this case, without solar flux, the only generation of electron-hole pairs is from the

12
ambient blackbody radiation:
𝑆 𝑏𝑏 (𝐸)𝑑𝐸 =

2𝜋
𝐸2
𝑑𝐸
ℎ3 𝑐2 exp 𝐸 − 1

(1.29)

𝑘 𝑏 𝑇𝑐

where 𝑇𝑐 is the temperature of the solar cell (e.g. 300 K). Therefore, in the dark, we
have
∫ ∞
𝐽𝑔𝑒𝑛,𝑑𝑎𝑟 𝑘 = 𝐽𝑟𝑒𝑐𝑜𝑚𝑏,𝑑𝑎𝑟 𝑘 = 𝑞
𝑎(𝐸)𝑆 𝑏𝑏 (𝐸)𝑑𝐸
(1.30)

In other words, since the solar cell has a finite radiative absorption, it must have some
finite radiative recombination. To understand the differences under illumination, we
note that the derivation for the law of mass action should still hold, i.e., 𝑛𝑝 =
𝑛𝑖2 exp(Δ𝜇/𝑘 𝑏 𝑇). Furthermore, even under illumination, if we examine open-circuit
conditions, we must still have 𝐽 = 0. Thus, since the generation is increased by a
factor of exp(Δ𝜇/𝑘 𝑏 𝑇), so must the recombination rate, i.e.,
∫ ∞
Δ𝜇
Δ𝜇
𝐽𝑟𝑒𝑐𝑜𝑚𝑏 = 𝐽𝑟𝑒𝑐𝑜𝑚𝑏,𝑑𝑎𝑟 𝑘 exp
= 𝑞 exp
𝑎(𝐸)𝑆 𝑏𝑏 (𝐸)𝑑𝐸
(1.31)
𝑘 𝑏𝑇
𝑘 𝑏𝑇 0
Finally, we note that the ambient blackbody flux contributes a negligible flux even
under solar illumination:
∫ ∞
∫ ∞
𝐽𝑔𝑒𝑛 = 𝑞
𝑎(𝐸)𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸 + 𝑞
𝑎(𝐸)𝑆 𝑏𝑏 (𝐸)𝑑𝐸
(1.32)

Thus, the 𝐽 − 𝑉 characteristics of a solar cell that is only limited by radiative
recombination (which is necessary, due to finite absorption) is given by
𝑞𝑉
−1
(1.33)
𝐽 (𝑉) = 𝐽𝑠𝑐 − 𝐽0 exp
𝑘 𝑏𝑇
∫∞
∫∞
where 𝐽𝑠𝑐 = 𝑞 0 𝑎(𝐸)𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸 and 𝐽0 = 𝑞 0 𝑎(𝐸)𝑆 𝑏𝑏 (𝐸)𝑑𝐸. We have used
the fact that in this hypothetical zero-dimensional solar cell, the internal and external
voltages are the same, i.e., 𝑞𝑉 = Δ𝜇. Finally, note that the form of the 𝐽𝑠𝑐 implicitly
assumes the carrier collection efficiency (𝐼𝑄𝐸) is unity. The final assumption in the
detailed balance efficiency limit is an explicit functional form for the absorbance,
which is characterized by a step-function:
𝑎(𝐸) =

0

if 𝐸 < 𝐸 𝑔

1

if 𝐸 ≥ 𝐸 𝑔

(1.34)

Since we now have an explicit form for every parameter in the 𝐽 (𝑉) expression,
we can calculate the maximum power conversion efficiency by optimizing 𝑃(𝑉) =

13
𝑉 𝐽 (𝑉) for every bandgap 𝐸 𝑔 . The solar spectrum typically used as a reference is
referred to as the Air Mass 1.5 Global (abbreviated as AM 1.5G) spectrum, i.e.,
𝑆 𝑠𝑢𝑛 (𝐸) = 𝑆 𝐴𝑀1.5𝐺 (𝐸), as shown in Figure 1.4a. This spectrum is approximately
equal to that of a blackbody spectrum with a blackbody temperature 𝑇𝑠 ≈ 5760
K. The dips in the spectrum in the infrared part of the spectrum is mostly due to
absorption of carbon dioxide and water vapor, whereas in the UV there is some
minor absorption from ozone. The calculation of the maximum power conversion
efficiency is shown in Figure 1.4b, where we find the maximum efficiency to be
approximately 33.7%. The incident power density of the sun can be calculated as
∫ ∞
𝑃𝑠𝑢𝑛 =
𝐸 𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸
(1.35)

Upon inspection, it is of a curious nature to understand where all the energy went.
Surely the limiting efficiency must be larger! However, the surprisingly low efficiency is limited primarily by carrier thermalization and the subsequent recombination of those carriers. In general, the two main loss mechanisms in a conventional
single-junction solar cell is due to the imperfect absorption of photons below the
bandgap of a solar cell:
∫ 𝐸𝑔
𝐸 𝑆 𝑠𝑢𝑛 (𝐸)𝑑𝐸
(1.36)
𝜂𝑎𝑏𝑠,𝑙𝑜𝑠𝑠 = ∫0 ∞
(𝐸)𝑑𝐸
𝑠𝑢𝑛
and for photons above the band-edge, it is due to carrier thermalization to the
band-edge of the semiconductor:
∫∞
∫∞
(𝐸)𝑑𝐸
𝑆(𝐸)𝑑𝐸
𝑠𝑢𝑛
𝐸𝑔
𝐸𝑔
∫∞
𝜂𝑡ℎ𝑒𝑟𝑚,𝑙𝑜𝑠𝑠 =
(1.37)
(𝐸)𝑑𝐸
𝑠𝑢𝑛
The relative contribution of these loss mechanisms is shown in Figure 1.4b. While
near-unity absorbance across the solar spectrum has been demonstrated [217], the
problem of carrier thermalization to the band-edge is a fundamental challenge to
high efficiency. Attempts to mediate the loss due to carrier thermalization has
been successfully demonstrated with multijunction solar cells, where solar cells
with multiple different bandgaps are optically connected in series, as shown in
Figure 1.2. The maximum solar cell efficiency experimentally demonstrated with six
junctions under solar concentration is 47.1% (by NREL), while the experimentally
demonstrated efficiency maximum of a single junction under 1 sun illumination is
29.1% (demonstrated by Alta Devices). More discussion on the physics of carrier
thermalization and whether it is appropriate in most semiconductors is described in

14
section C.3. However, it is clear from experimental demonstrations that these losses
are difficult to avoid, and the exploration of new materials physics is necessary to
achieve higher power conversion efficiencies.

Other Losses

Thermalization
Losses
Imperfect
Absorption
Loss

Generated
Power

Figure 1.4: Detailed Balance Efficiency Limit a The AM 1.5G solar flux
𝑆 𝐴𝑀1.5𝐺 (𝐸)𝑑𝐸, which is typically used as an approximation for the incident solar spectrum. b The detailed balance efficiency limit of a solar cell (highlighted
in blue) under AM 1.5G illumination. Also listed are different efficiency loss
mechanisms.

1.2

Entering Flatland: van der Waals Materials

Van der Waals (vdW) materials are materials whose out-of-plane forces are characterized by van der Waals interactions, making their bonding strength highly
anisotropic between the in-plane vs. the out-of-plane directions. These materials were first thoroughly characterized in the 1960s and 1970s [52], where a number
of their electrical and optical properties were examined for both bulk and ultrathin
layers. Because of their weak out-of-plane interactions, their crystals can be cleaved
with the application of an adhesive surface, such as with Scotch tape. The field of
van der Waals (“2D”) materials rapidly grew in 2004, when Novoselov and Geim
cleaved few-layer graphene and examined its field-effect properties [140]. Since
then, hundreds of other 2D materials have been discovered4 with a variety of entirely new properties found and various technological applications that have been
4 Many of these materials were actually examined already in the 1960s, as noted in the earlier

citation, but most had not been cleaved down to a single monolayer, stacked together (to form vdW
heterostructures), and implemented into device geometries.

15

Figure 1.5: van der Waals Heterostructures. Schematic depiction of heterostructuring different van der Waals materials together, showing that different combinations of materials can be combined to form entirely new materials. This arbitrary
heterostructuring stems from the van der Waals interactions along the basal plane,
adapted from [56].

proposed. Furthermore, these different layered materials can be stacked together in
various configurations, forming a van der Waals heterostructure (Figure 1.5).
Of considerable interest is the family of transition metal dichcalocogenides (TMDCs),
whose crystal structure in the 2H form results in semiconducting materials. The
most air stable forms of these materials have an elemental composition of MX2 ,
where M = Mo, W and X = S, Se. While these materials were initially studied in the
1960s [52], the Heinz [117] and Wang [178] groups discovered that these materials
became direct bandgap in their monolayer form, which drastically increased their
light-matter coupling and the possibility to use them for light-emitting applications.
Following this discovery, these materials were subsequently proposed to be used
as active layers for photovoltaic applications in 2013 [14], where their strong lightmatter coupling and atomically-thin layers could result in device structures with
record high specific powers. However, several fundamental science questions arose,
particularly on the physical mechanism of the photovoltaic behavior and how to engineer these materials. Experimental demonstrations soon followed after the initial
proposal where these materials were used as active layers in photovoltaic devices

16
[11, 28, 54, 101, 155, 166]. However, besides the initial demonstration, very little
work had been done on optimizing and understanding the photovoltaic behavior in
these materials.
1.3

What’s Different in Atomically-thin Photovoltaics?

Figure 1.6: van der Waals Photovoltaics Schematic. Schematic depiction of a
photovoltaic device formed purely from van der Waals materials.
The central theme of this thesis is to consider the use of these materials as atomicallythin active layers in photovoltaic devices. In doing so, we must study and understand
the absorption, transport, and recombination properties of these materials. Due to
the nature of these materials, we expect a variety of different physics and engineering
design rules to emerge, as opposed to those discussed in section 1.1. Specifically, we
can delineate two main attributes that causes substantial differences when compared
to conventional materials used for photovoltaics (e.g. Si or GaAs): (1) the vastly
different thicknesses of the active layers and (2) the van der Waals interactions in
the out-of-plane direction. We briefly describe implications of these two properties
below.
Effects of Thickness
Arguably the largest difference between using van der Waals materials as active
layers compared to that of conventional materials is their thickness. The typical
thickness of GaAs solar cells are usually a few microns thick, whereas Si solar

17

Figure 1.7: Dimensionality Effects on Exciton Screening. Schematic depiction
of excitons in bulk (3D) materials vs. monolayer (2D) materials. Due to the lack of
electrons in the out-of-plane direction, the electric field lines between electrons and
holes have substantially less screening, increasing the electron-hole interactions and
therefore the binding energy of excitons. Figure from [29].

cells are of order ∼100 𝜇m or more. In contrast, the van der Waals materials of
interest studied in this thesis is routinely 𝐿
𝜆, i.e., their thicknesses are deeply
subwavelength, perhaps only a few nm. This suggests that different optical physics
must be considered (specifically, wave optics), and that novel light trapping schemes
must be employed to achieve near-unity absorption in these ultrathin materials.
Further, this regime of thickness requires many different electronic considerations
as well. For example, these materials routinely have thicknesses that are either
comparable or even thinner than their electronic screening length. While this effect
has been utilized to create 2D devices with large gate tunability, this property
also has implications for the transport of carriers and types of electrical junctions
formed. For example, in an atomically thin pn junction, the two sides of the junction
are completely depleted. Thus, carrier separation in those types of junctions are
expected to be extremely efficient. Furthermore, the large surface to volume ratio
in these materials suggests that interfacial recombination, e.g. parametrized as a
surface recombination velocity, will likely be dominant or comparable to the “bulk”
recombination rates.
In addition, there are a variety of differences due to the presence of quantum mechanical effects. Of particular note is the electric screening of photogenerated electrons

18
and holes, which can form bound states referred to as excitons5. The large binding
energies comes from the fact that the Bohr radius of the exciton is comparable to
the thickness of the material, which results in less electrostatic screening [29] (Figure 1.7). Since excitons are electrically neutral, these excitons must be dissociated
to generate free carriers and therefore electrical current. Therefore, unlike traditional solar cells where free carriers are directly generated from photoexcitation, we
generate excitons that must be dissociated first. Indeed, since monolayer TMDCs
have exciton binding energies that are routinely a few hundred meV, these excitonic
quasiparticles dominate the optoelectronic physics at this thickness regime. It is
also important to note that for a given 2D material, there are typically many different
types of excitons, each with their own binding energy. For example, while the A
exciton in bulk TMDCs have binding energies of approximately 50 meV, the indirect
optical transitions in the bulk have minimal excitonic contribution.
Effects of van der Waals interactions
The van der Waals interactions in these layered materials uniquely enable the creation
of atomically-thin flakes. A simple estimate of the relative in-plane to out-of-plane
bond strength can be made from the relative dimensions of exfoliated flakes. For
flakes that are a few nm in thickness, the typical flake dimensions are roughly a few
microns to a few 10s of microns in thickness. Therefore, the relative dimensional
anisotropy is ∼1000, and we can estimate the bonding strength between the in-plane
covalent bonds and the out-of-plane van der Waals interactions to be of the same
order of magnitude, assuming linear scaling. Interestingly, the transport anisotropy
in these materials are roughly of the same order of magnitude, with the mobilities
in TMDCs to be roughly 1000 times lower in the out-of-plane direction compared
to the in-plane direction [121]. There are also large differences between the inplane and out-of-plane dielectric constants, notably the absence of strong excitonic
resonances in the out-of-plane dielectric function [197].
Despite these differences between the vertical and in-plane directions, there are
several distinguishing features of the van der Waals interaction. First, it enables the
formation of high quality van der Waals heterostructures, which do not suffer from
the same lattice matching problem that is typical in traditional semiconductors (such
as III-V semiconductors). This enables a larger variety of different materials to be
heterostructured, and allows for “designer” materials (Figure 1.5). Furthermore, as
5 A primer on the microscopic formation and dissociation of excitons is described in section A.1.

19
was noted in the original works of Novoselov and Geim [140], the presence of these
van der Waals interactions in the out-of-plane direction results in atomically thin
materials that are almost completely absent of dangling bonds. This is due to the
inherent nature of the crystal structure, which is substantially different than nonlayered materials. Thus, we are able to create materials with nearly zero intrinsic
surface states, which enables the study of truly “two-dimensional” materials.
1.4

Scope of the Thesis

As discussed earlier in section 1.1, the three pillars of understanding the properties
and power conversion efficiency of photovoltaic devices stems from a holistic understanding of the absorption of photons and subsequent generation of electron-hole
pairs, transport and collection of those generated electron-hole pairs, and eventual
recombination of those electron-hole pairs. In this thesis, we explore, understand,
and engineer these three pillars in atomically-thin materials in our quest to use them
as photovoltaic-active layers.
In Part I of this thesis, we discuss the interplay between optical absorption and the
thermodynamic efficiency limits of atomically-thin materials. Chapter 2 focuses
directly on how the optical absorption limits the fundamental efficiency achievable
in photovoltaic materials. This analysis is of general validity to all optoelectronic
materials and is derived from optoelectronic reciprocity relations. We then use
these relations to analyze the effects of semiconductor band tails on the fundamental
photovoltaic efficiency limits. Afterward, we focus our attention to atomically-thin
materials, which have unique bandstructures and defect states. This analysis lays
the foundation for choosing appropriate materials when optimizing a photovoltaic
device. Chapter 3 then focuses on engineering the optical properties of ultrathin
transition metal dichalcogenide materials, which have strongly excitonic properties.
These excitonic properties result in immense light-matter coupling, enabling nearly
perfect absorption in both ultrathin (∼10 nm) and atomically-thin (∼7 Å) with the
use of simple optical cavities. Finally, we discuss conceptually the possibilities of an
excitonic multijunction solar cell assuming near-unity absorbance can be achieved.
In Part II of this thesis, we discuss the transport of both free carriers and excitons in atomically-thin materials. Chapter 4 focuses on vertical heterostructures of
atomically-thin materials that utilizes the optical design of Chapter 3 and the electronic optimization of heterojunctions and metal contacts. This initial optimization
results in unprecedented photovoltaic quantum efficiencies of (>50%), which is a

20
record for van der Waals materials. We then turn our attention to new methods of
separating electrons and holes in ultrathin materials, in the form of a “band-bending
junction”, which utilizes the finite band-bending in materials whose thicknesses are
thinner than their electrostatic screening length. Chapter 5 focuses on the conceptual physics and experimental demonstration of these band-bending junctions by
utilizing various surface-sensitive probes and numerical calculations.
In Part III of this thesis, we discuss the intimate relationship between photoluminescence, recombination dynamics, and the internal photovoltage of a semiconductor
and therefore motivate examining the photoluminescence properties of atomicallythin semiconductors. Specifically, we examine the photoluminescence properties
in monolayer and heterobilayer systems and examine how they can be modified,
which modifies their recombination dynamics. Chapter 6 therefore focuses on how
strain can be utilized to engineer the photoluminescence properties of MoS2 /WSe2
heterobilayers, which have quasiparticles referred to as interlayer excitons.
Finally, in Part IV of this thesis, we give an outlook and perspective on the work
shown here, and discuss some of the remaining grand challenges and research
opportunities for the field of van der Waals materials as well as that specific to using
them for photovoltaic applications.

Part I
Absorption Defines the Limits

21

22
Chapter 2

IMPACT OF THE SEMICONDUCTOR BAND-EDGE ON
PHOTOVOLTAIC EFFICIENCY LIMITS
“If I have seen farther it is by standing on the
shoulders of giants.”
— Sir Isaac Newton

2.1

Introduction

Since the seminal work of Shockley and Queisser, assessing the detailed balance
between absorbed and emitted radiative fluxes from a photovoltaic absorber has been
the standard method for evaluating solar cell efficiency limits [17, 163, 175]. The
principle of detailed balance is one dictated by reciprocity and steady state, so that
photons can be absorbed and emitted with equal probability. This basic principle
has also been extended to evaluate the effects of multiple junctions [5, 120], hot
carriers [167, 210], nanostructured geometries [4, 212], multiexciton generation [68,
91], sub-unity radiative efficiency [125], and many other solar cell configurations
and nonidealities to estimate limiting efficiencies via modifications to the detailed
balance model.
Another important modification to the Shockley-Queisser model is to examine the
assumption of an abrupt, step-like onset of the densities of electronic states and
absorption coefficient. Specifically, it has long been recognized from spectroscopic
measurements of semiconductors that band edges are often not abrupt and that the
density of states and absorption functions can be characterized by a band tail. This
was first recognized by Urbach [188], who found the absorption coefficient for a
variety of materials below their bandgaps to be characterized by an exponential tail:
𝐸 − 𝐸𝑔
(2.1)
𝛼(𝐸 < 𝐸 𝑔 ) = 𝛼0 exp
where 𝛼0 is the absorption coefficient value at the energy of the bandgap, 𝐸 𝑔 is the
bandgap of the material, and 𝛾 is referred to as the Urbach parameter, which describes
the rate at which the absorption coefficient goes to zero. The magnitude of the Urbach
parameter can be influenced by impurities and disorder and is typically attributed
to fluctuations in the electrostatic potential within a semiconductor. Urbach tails

23
have been observed in a wide range of absorber materials including amorphous,
organic, perovskite, and II-VI, III-V and group IV semiconductors [35, 39, 83, 84,
96, 133, 180]. While the Urbach exponential tail is the most prominent functional
form observed for band tail states, other forms such as Gaussian band tails have
been reported, and different functional forms have been attributed to the underlying
physics of those systems [87]. In most cases, the band tail can be characterized by
an exponential with an argument raised to some power.

Figure 2.1: Accounting for band filling in modified reciprocity relations. Experimentally measured a-Si:H EQE and EL from [163] (open circles). Solid lines
correspond to the Rau reciprocity relation, whereas the dashed line is a fit that
includes band filling effects with asymmetric effective masses in the parabolic approximation (𝑚 ℎ /𝑚 𝑒 = 1.818, Δ𝜇 = 1.164 V, and 𝐸 𝑔 = 2.439 eV. The dotted line
includes band filling with the same fitted parameters except 𝑚 ℎ /𝑚 𝑒 = 1, i.e., assumes symmetric effective masses. All spectra are normalized by exp(Δ𝜇/𝑘 𝑏 𝑇)
and the various reciprocity relations overlay for 𝐸 > 𝐸 𝑔,𝐴𝑏𝑠 .
Recent detailed balance analyses have also suggested how this important effect,
i.e., a departure from a step-like absorbance spectra can also degrade the limiting
efficiency of solar cells [17, 32, 60, 82, 133, 161]. However, a key element missing
from previous analyses of photovoltaic efficiency is the effect of band filling for
semiconductors with nonabrupt band edges, wherein the electron-hole quasi-Fermi
level splitting can thereby modify the absorption spectrum, and therefore the radiative emission spectrum as well. This voltage-dependent absorption effect was
first recognized by Parrott [145] as being necessary to make the detailed balance
formulation self-consistent. Perhaps the most intuitive description of why this is

24
necessary is found by examining the generalized Planck’s law [209]:
𝑆 𝑃𝐿 (𝐸) = 𝑎(𝐸)𝑆 𝑏𝑏 (𝐸, 𝛿𝜇)
where
𝑆 𝑏𝑏 (𝐸, Δ𝜇) =

2𝜋
𝐸2
ℎ3 𝑐2 exp 𝐸−Δ𝜇 − 1

(2.2)

(2.3)

𝑘 𝑏𝑇

and 𝑆 𝑃𝐿 (𝐸) describes the luminescence flux, 𝑎(𝐸) is the absorbance, 𝐸 is the photon
energy, Δ𝜇 is the quasi-Fermi level splitting, ℎ is Planck’s constant, and 𝑐 is the
speed of light. A clear singularity occurs at 𝐸 = Δ𝜇, which is typically ignored in
detailed balance calculations because for a step-like absorbance function, we have
𝐸 ≥ 𝐸 𝑔 > Δ𝜇. As a result, the -1 in the denominator is neglected and Boltzmann
statistics are assumed. Clearly, the situation must change if we consider energies
𝐸 < 𝐸 𝑔 , as is the case when band tails affect the luminescence spectrum. In this
case, the absorptivity must be modified such that 𝑎(𝐸 = Δ𝜇) = 0, and in general the
absorption coefficient is occupation dependent:
𝛼(𝐸, Δ𝜇) = 𝛼0𝐾 (𝐸)( 𝑓𝑣 − 𝑓𝑐 )

(2.4)

where 𝛼0𝐾 (𝐸) is the absorption coefficient without band-filling and ( 𝑓𝑣 − 𝑓𝑐 ) is the
band-filling factor [16, 87, 209]. This contribution of band-filling has also been
recognized in experiments as being necessary to accurately fit photoluminescence
spectra under high level injection [58, 136]. We suggest that this contribution is
also important for systems with large band tails, and as an example, we have used
this modified reciprocity relation to fit the electroluminescence spectrum of a-Si:H
which the Rau reciprocity relation [160, 163] was previously unable to fit completely
(see Figure 2.1).
2.2

Photovoltaic Efficiency Limit for Semiconductors with Band Tails

By using the generalized Planck’s law (Equation 2.2) and accounting for band
filling (Equation 2.4), we can calculate the detailed balance limit for photovoltaic
efficiency with band tails in the radiative limit (See section 2.6 and section 2.6).
In Figure 2.2, we consider the case of a band tail parameterized as an exponential
Urbach tail and analyze the effects of varying the Urbach parameter. While the
spectral response of this modified absorbance appears to be similar to the step
function response originally used by Shockley and Queisser (black dashed line), the
maximum achievable efficiency drops rapidly from the Shockley-Queisser limit for
Urbach parameters larger than the thermal energy, 𝑘 𝑏 𝑇. These effects are relatively

25

Figure 2.2: Effects of band tailing on photovoltaic limiting efficiencies. a The
spectral absorbance of a photovoltaic cell with a bandgap of 𝐸 𝑔 = 1.5 eV and a
thickness 𝛼0 𝐿 = 1 plotted for various Urbach parameters (𝛾) in units of 𝑘 𝑏 𝑇. The
dashed line represents the step function absorbance typically used in the ShockleyQueisser (S-Q) limit. b The detailed balance efficiencies as a function of the bandgap
energy. Different colored lines correspond to different Urbach parameters, with the
coloring scheme equal to the legend shown in a.

insensitive to the choice of bandgap and thickness (see Figure 2.3 and Figure 2.4), and
Figure 2.3 suggests the efficiency drop is primarily due to a voltage loss mechanism.
To analyze the cause of the voltage loss, we examine the luminescence spectrum
by using (Equation 2.2) and plot these spectra for various Urbach parameters (Figure 2.6a). In addition, we plot the distribution of bandgaps 𝑃(𝐸 𝑔 ) = 𝜕𝐸 𝑎| 𝐸=𝐸 𝑔
proposed by Rau et al. [163] recently, which generalizes the definition of the photovoltaic bandgap for arbitrary absorbance spectra. While the luminescence spectrum
is narrow and overlaps significantly with the absorption edge for 𝛾 < 𝑘 𝑏 𝑇, this is
not the case for 𝛾 > 𝑘 𝑏 𝑇. In this limit, the luminescence spectrum is significantly
broadened and shifts away from the absorption band-edge and suggests the definition
of a second bandgap, defined by the luminescence spectra. This idea is schematically depicted in Figure 2.6b, where the absorption bandgap is defined as before,
i.e., 𝑃(𝐸 𝑔,𝐴𝑏𝑠 ) = max(𝑃 𝐸 𝑔 ), while the second bandgap, 𝐸 𝑔,𝑃𝐿 , is defined by the
luminescence spectra 𝑆 𝑃𝐿 . We note that this analysis is modified significantly with
the inclusion of band-filling effects, which we describe in section 2.6 and section 2.6
of the Appendix (see also Figure 2.13 and Figure 2.14). We also observe the effects
of broadening followed by luminescence spectral shifts in band tails parameterized

26

Figure 2.3: Dependence of photovoltaic figures of merit on the Urbach parameter. a The detailed balance limited value of conversion efficiency, open circuit
voltage, short circuit current, and fill factor for different bandgaps and Urbach parameters assuming a thickness of 𝛼0 𝐿 = 1. b Linecuts of a at specific bandgap
values.
by a Gaussian band tail (Figure 2.5), where the onset of efficiency loss occurs at
approximately 𝛾 = 2𝑘 𝑏 𝑇 instead.
2.3

Generalized Voltage Loss for Semiconductors with Nonabrupt Band Edges

The similarity between the effects of broadening followed by luminescence shifting
for increasing band tail energies suggests a general picture for the voltage loss
mechanism, for any band tail functional form. A general trend is the observation of
a Stokes shift, i.e., the shift between the absorbance and luminescence spectra, that
occurs precisely at the onset of efficiency loss. However, it is unclear whether the
voltage loss is just directly proportional to the observed Stokes shift Δ𝐸 𝑔 .
To develop an understanding of this loss mechanism, we consider a simpler absorbance spectrum as a two bandgap model, represented by the sum of two step
function absorbances:
𝑎(𝐸) = 𝑎 1 𝜃 (𝐸 − 𝐸 𝑔,1 )𝜃 (𝐸 𝑔,2 − 𝐸) + 𝑎 2 𝜃 (𝐸 − 𝐸 𝑔,2 )

(2.5)

Here, 𝑎 1,2 is the sub-gap and above-gap absorbances respectively, while 𝜃 (𝐸 −

27

Figure 2.4: Effects of thickness on photovoltaic figures of merit. a Absorbance
of a photovoltaic cell plotted with different normalized thicknesses (𝛼0 𝐿) for 𝛾 =
0.5𝑘 𝑏 𝑇 and b 𝛾 = 2𝑘 𝑏 𝑇 assuming a bandgap 𝐸 𝑔 = 1.5 eV. c Conversion efficiency,
open circuit voltage, short circuit current, and fill factor calculated for different
normalized thicknesses assuming a bandgap 𝐸 𝑔 = 1.5𝑒𝑉. The different colored
lines correspond to the same legend shown in a.

𝐸 𝑔 ) is the Heaviside step function, typically considered in the SQ analysis. The
photovoltaic bandgap, i.e., that defined by absorption, is given by 𝐸 𝑔,2 , while 𝐸 𝑔,1
defines the luminescence bandgap. The SQ limit is recovered in the limit that
𝐸 𝑔,1 → 𝐸 𝑔,2 or 𝑎 1 → 0. By varying 𝑎 1 and 𝐸 𝑔,1 and fixing 𝐸 𝑔,2 to 1.34 eV and
𝑎 2 = 1, we can analyze the effects of this simple model as we deviate from the
SQ limit (see section 2.6 and Figure 2.15 for more details). Interestingly, we find
qualitatively similar effects of voltage and efficiency loss in this absorbance model
compared to the full effects of the Urbach band tail, albeit parametrized by 𝑎 1 and
𝐸 𝑔,1 instead of the Urbach parameter 𝛾. However, we also find that the quantitative
bandgap-voltage relation can be significantly affected by the actual functional form
used to more accurately model the band tail state distribution, as illustrated in

28

Figure 2.5: Analysis of a Gaussian band tail distribution. a Calculated absorbance
(solid line), photoluminescence (dashed line) and distribution of bandgaps (dotted
line) for an increasing Gaussian tail (𝛾). Here, the Gaussian tail distribution is
calculated by taking 𝜃 = 2 in Eqn. 14. b Fraction of integrated photoluminescence
below the band gap (solid blue line) and Stokes shift Δ𝐸 𝑔 (solid orange line) for
a Gaussian tail distribution. c Calculated detailed balance efficiency for different
bandgaps plotted for increasing Gaussian tail widths. The different colored lines
correspond to the same values of the Gaussian tail displayed in a.

Figure 2.7.
Non-abrupt band-edge absorbances can be mapped onto the two bandgap model and
therefore there is a general relation that explains the voltage loss mechanism for any
absorbance spectrum given by
Δ𝐸 𝑔
𝑎¯ 𝑆𝐺
𝑎¯ 𝑆𝐺
𝑘 𝑏𝑇
ln
exp
+1−
(2.6)
Δ𝑉𝑜𝑐,𝑟𝑎𝑑 =
𝑎¯ 𝐴𝐺
𝑘 𝑏𝑇
𝑎¯ 𝐴𝐺
where 𝑎¯ 𝑆𝐺 is the weighted sub-gap absorbance, 𝑎¯ 𝐴𝐺 is the weighted above-gap
absorbance, and Δ𝐸 𝑔 = 𝐸 𝑔,𝐴𝑏𝑠 − 𝐸 𝑔,𝑃𝐿 describes the observed Stokes shift between
the absorption and luminescence (see definitions in section 2.6 of the Appendix).

29

Figure 2.6: Effects of band tail states on photoluminescence. a The normalized
spectral photoluminescence (dashed line) of a photovoltaic cell operating at the
radiative limit under 1 sun AM 1.5G illumination for increasing Urbach parameter
(𝛾) with an offset included for clarity. The corresponding absorbance (solid line)
and effective distribution of bandgaps (dotted line) is also plotted, where they are
normalized to their peak value. b Schematic depiction of the density of states profile
along with carrier excitation and recombination; 𝛾 describes the effective width of
the band tail. For small band tailing (𝛾 < 𝑘 𝑏 𝑇), the effect of band tailing is to simply
broaden the luminescence peak. For systems with large band tailing (𝛾 > 𝑘 𝑏 𝑇), the
luminescence shifts to energies below the nominal absorption band edge.

Here, Δ𝑉𝑜𝑐,𝑟𝑎𝑑 is a voltage loss due purely to the non-abruptness of the absorption spectrum, for a semiconductor with assumed unity radiative efficiency. More
generally, although non-radiative losses parametrized by a non-unity external radiative efficiency have not been accounted for (see some discussion of radiative
efficiency effects in section 2.6), band edge non-abruptness by itself can contribute
significantly to voltage loss. Indeed, Equation 2.6 results in no net voltage loss as
Δ𝐸 𝑔 → 0, and suggests that a finite Stokes shift should be directly correlated to a
voltage loss. The magnitude of the voltage loss is scaled by the ratio 𝑎¯ 𝑆𝐺 /𝑎¯ 𝐴𝐺 , and
clearly as the ratio approaches 0 or 1, Equation 2.6 recovers the appropriate losses
of 0 and Δ𝐸 𝑔 /𝑞, respectively.
To observe whether this two bandgap model can quantitatively describe the more
complex band edge functional forms seen in experiments, we choose appropriate
definitions for 𝑎¯ 𝑆𝐺 , 𝑎¯ 𝐴𝐺 , and 𝐸 𝑔,𝑃𝐿 and use Rau’s definition for 𝐸 𝑔,𝐴𝑏𝑠 (see more
details in section 2.6, section 2.6, and with Figure 2.17). We consider both power

30

Figure 2.7: Generalized voltage losses parametrized as a two-bandgap absorber:
a Schematic depiction representing a general absorbance and luminescence spectrum as a simpler two-bandgap step function absorbance. Black solid lines are
the absorption spectrum, whereas the red dashed line corresponds to the bandgap
distribution 𝑃(𝐸 𝑔 ). Blue dashed lines correspond to the luminescence spectrum
𝑆 𝑃𝐿 . Typically, 𝑎¯ 𝑆𝐺
1, which is not visible on a linear scale but still contributes
to the luminescence spectra due to carrier thermalization. b Calculated voltage loss
Δ𝑉𝑜𝑐 = 𝑉𝑜𝑐,𝑆𝑄 (𝐸 𝑔,𝐴𝑏𝑠 )−𝑉𝑜𝑐,𝑟𝑎𝑑 versus observed bandgap shift Δ𝐸 𝑔 = 𝐸 𝑔,𝐴𝑏𝑠 −𝐸 𝑔,𝑃𝐿 ,
normalized to the energy scale 𝑘 𝑏 𝑇 ln ( 𝑎¯ 𝐴𝐺 /𝑎¯ 𝑆𝐺 ). Every plotted point corresponds
to a different absorption spectrum, with 𝑉𝑜𝑐,𝑟𝑎𝑑 calculated using the complete absorption spectra and the full reciprocity relations. The dashed line represents the
two bandgap model, i.e., Equation 2.6 , where we have chosen 𝑎¯ 𝑆𝐺 /𝑎¯ 𝐴𝐺 = 0.1.

law band-edges, as a parametrization of indirect band-edges, as well as exponential
band tails. We find reasonable qualitative agreement but quantitative disagreement between the calculations utilizing the full absorbance spectra and that given
by Equation 2.6 (Figure 2.7b), suggesting that the two-bandgap model is a reasonable first-order representation of the voltage loss mechanism, but importantly,
consideration of the actual band tail functional form yields more accurate results.
Furthermore, we find that the dimensionless parameter 𝜉 = Δ𝐸 𝑔 /(𝑘 𝑏 𝑇 ln( 𝑎¯ 𝐴𝐺 /𝑎¯ 𝑆𝐺 )
describes the physical regime of voltage loss. Generally, for 𝜉 < 1, voltage loss is
minimal since the emission spectrum can be considered as simply a broadening of

31
a single photovoltaic bandgap. In this regime, the efficiency penalty is negligible
and generally the detailed balance efficiency limit for 𝐸 𝑔,𝐴𝑏𝑠 can be achieved given
sufficient absorption above the photovoltaic bandgap. However, for 𝜉 > 1, it is appropriate to define a second bandgap given by the emission spectrum (Figure 2.7a),
resulting in a substantial voltage and efficiency penalty due to additional thermalization losses. Thus, the tuning of the band tail parameter 𝛾 merely sweeps through
different values of 𝜉, and we find that 𝜉 > 1 is equivalent to 𝛾 > 𝑘 𝑏 𝑇 in the case
of an Urbach tail (section 2.6). The discrepancy in Equation 2.6 for large 𝜉 can be
attributed to neglecting higher order terms (section 2.6).
The correlation between the magnitude of the bandgap shift (Δ𝐸 𝑔 ) and open circuit
voltage has already been recognized in the organic photovoltaics literature, where
the presence of low energy charge transfer states generally results in cells with a
lower voltage and efficiency [9, 112, 133, 156, 159]. Here, we have developed
a unified picture with an arbitrarily-shaped band tail and by explicitly including
band-filling effects, for both large and small band tails, the voltage loss mechanism
can be qualitatively captured with a simple two bandgap model. In addition, by
extracting the weighted absorbance ratios, a 𝑎¯ 𝐴𝐺 /𝑎¯ 𝑆𝐺 , we can estimate the voltage
losses in the radiative limit using Equation 2.6. We therefore suggest that any
radiative transition below the photovoltaic absorption edge 𝐸 𝑔,𝐴𝑏𝑠 , measurable in
luminescence measurements, should result in an efficiency penalty. This efficiency
penalty can be viewed as either stemming from a voltage penalty, due to carrier
thermalization within the band tails, or equivalently interpreted as being due to
incomplete absorption at the lower energy transition.
2.4

Outlook on Examining Semiconductor Band Tails

To emphasize the implications of these results for various photovoltaic technologies,
we have calculated the predicted voltage losses due to a nonabrupt band-edge for
several different material systems and plotted experimentally-measured values for
these in Figure 2.8 (See Table 2.2 for references and individual datapoints) [39, 82,
133]. Since Urbach parameters are much more commonly reported than both high
sensitivity EQE and EL spectra, we have used the observed Urbach parameters to
calculate the voltage loss directly rather than through Equation 2.6. As expected,
we find that semiconductors with large band tails (𝛾 > 𝑘 𝑏 𝑇) or equivalently, large
Stokes shifts (Δ𝐸 𝑔
𝑘 𝑏 𝑇) have a substantially modified maximum achievable 𝑉𝑜𝑐 ,
which should be assessed when examining their efficiency potential (e.g., CIGS,
a-Si, kesterites, and OPVs). It should be noted that a more accurate calculation can

32

Figure 2.8: Voltage loss due to a nonabrupt band-edge. Expected open circuit
voltage loss as a function of the observed Urbach parameter 𝛾, plotted for different
materials. The top and bottom black solid lines represent the calculated voltage
loss assuming a bandgap of 2.2 and 0.8 eV, respectively. The gray area in between
represents the voltage loss expected for bandgap values in between 0.8 and 2.2 eV,
which correspond to most of the materials considered for photovoltaics. The colored
data points indicate the expected voltage loss for an experimentally-measured Urbach
parameter. The dashed line corresponds to the region of the inset, where the voltage
loss is minimal and approximately the same irrespective of bandgap.

be made by using the directly measured EQE and EL spectra for a given device.
The analysis presented here should be applicable to any system with nonabrupt
band-edges that obeys the optoelectronic reciprocity relations and should be employed to evaluate the radiative limits on the open circuit voltage. We demonstrated
here that the voltage dependence of the absorbance or EQE, specifically via band
filling, must be included to self-consistently apply the generalized Planck’s law for
semiconductors. We also suggest that in order to accurately estimate efficiency
limits, the abruptness of the band-edge should be experimentally characterized by
measuring both the absorption and luminescence spectra of photovoltaic materials,
and in a completed photovoltaic device, photocurrent and electroluminescence spectra should be used to assess the effects of transport on the reciprocity relations. The
magnitude of the voltage loss can then be estimated directly from the spectroscopic
measurements by applying reciprocity relations. Additional experimental details
and nonidealities for a given photovoltaic material or device may modify the maximum efficiency potential even further, such as reduction in the external radiative

33
efficiency or the finite mobility of charge carriers. However, our analysis suggests
the important role that band edge abruptness and band filling can play in defining
the limit on open circuit voltage and efficiency potential of emerging and established
photovoltaic materials.
2.5

Importance of the Direct-Indirect Gap Splitting on the Efficiency Potential
of Ultrathin TMDC Photovoltaics

Based on the analysis from above, it is clear that the abruptness of the band-edge can
have drastic effects on the efficiency potential of a solar cell material. Furthermore,
any weakly absorbing state can effectively have a similar impact on the efficiency
limit of a solar cell. Thus, we now turn our attention to apply the above analysis to
ultrathin (∼10 nm) but electronically bulk TMDCs, for which we are interested in
using as active layers. This material system effectively behaves as a system that has
two band-edges, because of the weakly absorbing indirect edge (typically around 1.3
eV for all the TMDCs) and the strongly absorbing direct-edge (A exciton, which is
between 1.55 to 2.0 eV). The bandstructure that gives rise to this unique absorption
edge is shown schematically in Figure 2.9a and the corresponding experimental
photocurrent spectra (which is proportional to absorbance) is shown in Figure 2.9b.
It is clear that while the majority of the photocurrent (and generally, absorbance),
occurs at and above the 𝐴 exciton, the indirect-edge results in a second, low-energy,
band-edge, which will have deleterious effects on the open-circuit voltage potential
and limiting efficiency.
To analyze this further, it is useful to develop a simple model that contains the
essential features of the absorption profile and analyze the effects it may have on
the limiting efficiency of an ultrathin TMDC. One of the simplest models for the
absorption-edge of an exciton is given by the Elliott expression:
𝛼𝑒𝑥𝑐 (𝐸) = 𝛼0,𝑒𝑥𝑐

𝐸𝑏

 

Õ 4𝜋 
𝜃 (Δ)
1 
 +
𝛿 Δ+ 2 
 1 − exp −2𝜋/√Δ
𝑛3
𝑛 

(2.7)

where Δ = (𝐸 − 𝐸 𝑔,𝑑𝑖𝑟 )/𝐸 𝑏 is dimensionless and represents the photon energy
relative to the exciton binding energy. Here, 𝐸 𝑔,𝑑𝑖𝑟 is the quasiparticle bandgap
of the direct band-edge, 𝐸 𝑏 is the exciton binding energy, 𝛿(𝑥) is the Dirac-delta
function, 𝜃 (𝑥) is the Heaviside step function, and 𝛼0,𝑒𝑥𝑐 represents the effective
oscillator strength. We further assume a Gaussian broadening given as
𝛼𝑒𝑥𝑐,𝑔𝑎𝑢𝑠𝑠 (𝐸) = 𝛼𝑒𝑥𝑐 (𝐸) ∗ 𝑁 (𝜎𝐸 )

(2.8)

34

EQE (log)

EQE (linear)

Exciton

Indirect Edge

Exciton

Energy (eV)

Figure 2.9: Unique Bandstructure and Absorption Edge of Bulk TMDCs. a
Schematic bandstructure for bulk TMDC layers, with the associated excitonic transitions overlaid as dashed ovals. The first direct-edge exciton, i.e., the 𝐴 exciton,
occurrs at the 𝐾 point (blue oval), while the indirect-edge exciton occurs between
the Γ and 𝑄 points (purple oval). b External quantum efficiency (EQE) spectra of a
20 nm Au/16 nm WS2 /Ag sample, measured using Fourier transform photocurrent
spectroscopy. Appropriate excitons and edges are also labelled. Left axis is a linear
scale, and right axis is a log scale.

where 𝑁 (𝜎𝐸 ) represents a normalized Gaussian with width 𝜎𝐸 , and ‘∗’ operator
represents the convolution operation. Another essential feature is the indirect-edge
exciton, which is typically represented by a power-law near the indirect-edge. Both
linear and parabolic expressions are commonly observed. We shall use
𝛼𝑖𝑛𝑑 (𝐸) = 𝛼0,𝑖𝑛𝑑

𝐸 − 𝐸 𝑔,𝑖𝑛𝑑 2
𝑘 𝑏𝑇

(2.9)

where we could consider convoluting the indirect absorption edge with some band
tailing form (e.g., see Equation 2.28), but as we shall soon see, those effects are
substantially smaller than the effects of the absorption into the indirect edge. The
effects of the power law exponent are also weak. The total absorption coefficient of
the bulk TMDC layer is then just given as the sum as the individual components:
𝛼𝑇 𝑀 𝐷𝐶 = 𝛼𝑒𝑥𝑐,𝑔𝑎𝑢𝑠𝑠 (𝐸) + 𝛼𝑖𝑛𝑑 (𝐸)

(2.10)

Finally, we are interested in relating the absorption coefficient to the overall absorbance. In general, for ultrathin materials, wave optics is the appropriate formalism
that relates the two quantities (e.g. see Appendix B). However, the specific optical
structures surrounding the ultrathin TMDC layer critically describes this connection.

35
Instead, for sake of simplicity, we consider a non-dispersive relationship inspired by
the work of [220] that parametrizes the effects of absorption enhancement as
𝛼𝐿
𝑎=
(2.11)
𝛼𝐿 + 1/𝐶 𝐴𝐸
where 𝐶 𝐴𝐸 can be interpreted as the enhancement factor relative to a single pass
absorption. This expression is derived strictly in the limit that the single pass
absorption 𝛼𝐿 is infinitesimal. Another possible expression is motivated by [162],
which is given as
𝑎 = 1 − exp(−𝛼𝐿𝐶 𝐴𝐸 )
(2.12)
where 𝛼 in both these situations is the absorption coefficient, and 𝐿 is the thickness
of the active layer. Both expressions reproduce fairly similar results, the main
difference being the rate at which 𝑎 rises to unity as 𝐶 𝐴𝐸 is large. For small 𝛼𝐿, note
that both expressions are equivalent (i.e., 𝑎 ≈ 𝐶 𝐴𝐸 𝛼𝐿 from a first order expansion).
Given the faster rise in absorbance and the natural form of the exponential in
wave-like expressions, we use Equation 2.12 as our relationship between 𝛼 and 𝑎.
However, similar results are obtained in both situations. In the ray-optics limit, the
maximum absorption enhancement is 𝐶 𝐴𝐸,ray-optics = 4𝑛𝑟2 , where 𝑛𝑟 is the real part
of the refractive index of the active layer. For TMDCs, the ray-optics limit on the
enhancement factor can be nearly 100, but nanophotonic structures can offer values
that are even higher.
Once the absorbance is specified, we can calculate the luminescence spectral flux
by the reciprocity relations (Equation 2.20), which may be compared to photoluminescence spectra (see Figure 2.10a). Here, it is clear that the presence of the indirect
edge results in a substantial Stokes shift between absorption and emission, although
this is partially modulated by the amount of absorption enhancement 𝐶 𝐴𝐸 Figure 2.10b. We further consider that, in general, the radiative efficiency is sub-unity.
Therefore, we can parametrize its effects on the dark current as:
𝑆 𝑃𝐿 (𝐸, Δ𝜇)𝑑𝐸
𝐽𝑟𝑎𝑑
(2.13)
𝐽𝑑𝑎𝑟 𝑘 =
𝐸 𝑅𝐸
𝐸 𝑅𝐸
where the external radiative efficiency 𝐸 𝑅𝐸 is given as:
𝐽𝑟𝑎𝑑
𝐸 𝑅𝐸 =
𝐽𝑟𝑎𝑑 + 𝐽𝑛𝑟𝑎𝑑,0

𝐽𝑟 𝑎𝑑
𝐽𝑟 𝑎𝑑,0
𝐽𝑟 𝑎𝑑
𝐽𝑟 𝑎𝑑,0 + 𝐸 𝑅𝐸 0 − 1

(2.14)

where we have used the fact that 𝐽𝑟𝑎𝑑,0 /𝐸 𝑅𝐸 0 = 𝐽𝑟𝑎𝑑,0 + 𝐽𝑛𝑟𝑎𝑑,0 . In the weakly
absorbing limit (i.e., 𝛼𝐿
1), we have 𝐽𝑟𝑎𝑑 /𝐽𝑟𝑎𝑑,0 = 𝐶 𝐴𝐸 , so that
𝐶 𝐴𝐸 𝐸 𝑅𝐸 0
𝐸 𝑅𝐸 =
(2.15)
𝐶 𝐴𝐸 𝐸 𝑅𝐸 0 + (1 − 𝐸 𝑅𝐸 0 )

36
In other words, we are taking into account the Purcell factor effect on the radiative
efficiency for a finite increase in the absorption enhancement (which, in the limit of
infinitesimal absorption, is equivalent to an enhancement in the radiative efficiency,
i.e., is a Purcell factor).

Figure 2.10: Absorption Toy Model for Bulk TMDCs. a Representative absorbance spectra from the toy model developed above, where the indirect edge is
clearly visualized below the exciton and results in substantial luminescence. The
first direct-edge exciton is also modelled. We ignore the higher order excitons, which
do not substantially change the results shown here. b The calculated absorbance
curves as a function of different absorption enhancement factors 𝐶 𝐴𝐸 . Different
colors of the curve correspond to different amounts of 𝐶 𝐴𝐸 , given by the color bar
on the right (plotted in a log scale). The AM 1.5G solar flux spectra is also plotted
in black (normalized, units of photons/m2 /s/eV).
We can now examine the effects of the specific absorption spectra on the efficiency
limits of an ultrathin layer of WS2 using the modified detailed balance calculations
(section 2.6). The effects of the different initial radiative efficiencies (𝐸 𝑅𝐸 0 ) and
absorption enhancement factors 𝐶 𝐴𝐸 is shown in Figure 2.11. We first note that,
somewhat surprisingly, the current densities for reasonable absorption enhancement
factors are far below their detailed balance value. At the ray-optics trapping limit
of TMDCs, we would only achieve ∼60% of the detailed balance 𝐽𝑠𝑐 . This fact
stems from the small thicknesses (𝐿 = 10 nm) considered in this analysis as well
as the equally small absorption coefficient at the indirect edge 𝛼0,𝑖𝑛𝑑 = 2 × 103 .
The low indirect-edge absorption coefficient was estimated from experimental data
(Figure 2.9), and the 10 nm regime of thickness is a common regime of thickness
considered in device structures. To first order, this dramatic reduction in the 𝐽𝑠𝑐
scales with the power conversion efficiency, so that the practical efficiency limit
for ultrathin WS2 structures is closer to ∼20%. Therefore, the absorption into the

37
indirect edge is a critical limiting factor in the efficiency potential of electronically
bulk TMDCs, a fact that has not been recognized in the TMDC photovoltaics
literature. To achieve >95% of the efficiency potential in electronically bulk WS2 ,
absorption enhancement factors 𝐶 𝐴𝐸 would need to be at least 1500.
It is clear that with an indirect edge at 1.3 eV, the detailed balance efficiency
potential of ∼33% can only be reached when there is complete absorption above
the band-edge. Also of importance is the limiting open-circuit voltage potential
(Figure 2.11b). We see that for different values of 𝐸 𝑅𝐸 0 , the open circuit voltage
drops by 𝑘 𝑏 𝑇/𝑞 ln(𝐸 𝑅𝐸 1 /𝐸 𝑅𝐸 2 ), i.e., 60 mV per decade. It is useful to note two
other regimes in the open-circuit plot. As the absorption enhancement factor 𝐶 𝐴𝐸
increases, the 𝑉𝑜𝑐 increases due to the Purcell factor and the overall increase in 𝐸 𝑅𝐸.
However, at very large 𝐶 𝐴𝐸 , the optical absorption spectra gradually shifts from a
system that has its main absorption edge at 𝐸 𝑔,𝑑𝑖𝑟 to one that has its main absorption
edge at 𝐸 𝑔,𝑖𝑛𝑑 (see Figure 2.10b). This resultant shift in the effective band-edge
results in a reduction in 𝑉𝑜𝑐 . At intermediate values of 𝐶 𝐴𝐸 , there is a competition
between both effects. The resultant Stokes shift Δ𝐸 𝑔 is reduced as 𝐶 𝐴𝐸 increases,
and therefore the overall efficiency potential increases with 𝐶 𝐴𝐸 (Figure 2.11c),
since the system transitions from two band-edges to a single abrupt band-edge (c.f.
section 2.6).
Table 2.1: Parameter values for ultrathin WS
. 2 modified detailed balance calculations
Parameter

Value

𝐸 𝑔,𝑑𝑖𝑟 (eV)
𝐸 𝑏 (meV)
𝜎𝐸 (meV)
𝛼0,𝑒𝑥𝑐 (1/m)
𝛼0,𝑖𝑛𝑑 (1/m)
𝐸 𝑔,𝑖𝑛𝑑 (eV)
𝐿 (nm)

2.05
100
30
6.14×104
2×103
1.3
10

It is important now to summarize some of our main observations from this analysis. It is rather clear that there is a substantial efficiency impact due to the large
energetic difference between the direct and indirect edge of WS2 . The magnitude
of this effect is partially due to the magnitude of the indirect edge absorption 𝑎 0,𝑖𝑛𝑑
(c.f. Equation 2.6), but only scales logarithmically with those values. Thus, one

38

Figure 2.11: Photovoltaic Figures of Merit of WS2 for Varying 𝐶 𝐴𝐸 and 𝐸 𝑅𝐸 0 .
Calculated photovoltaic figures of merit from detailed balance calculations using the
materials parameters in Table 2.1, where we plot the a short-circuit current density
𝐽𝑠𝑐 b open-circuit voltage 𝑉𝑜𝑐 , and the c power conversion efficiency 𝜂. All of the
figures of merit are normalized to the equivalent detailed balance value for a step
function absorbance at 𝐸 𝑔,𝑖𝑛𝑑 .

consequence of this analysis is to compare these results if the direct band-edge was
substantially closer to the indirect edge, e.g. if 𝐸 𝑔,𝑑𝑖𝑟 = 1.55 eV. This is shown
in Figure 2.12, and this situation is akin to that of using MoSe2 as an active layer.
While at first glance these figures look qualitatively quite similar, it soon becomes
readily apparently that the efficiency potential is substantially higher for moderate
𝐶 𝐴𝐸 , with >95% of the maximum efficiency achieved when 𝐶 𝐴𝐸 ≈ 25. These values
of absorption enhancement is readily attainable in a variety of optical structures, and
is far below the ray-optics limit. This result can be traced back to the abruptness of
the band-edge and the two-bandgap model derived earlier. In this case, the Stokes
shift between the direct and indirect edge is minimized, and maximum efficiency is
achieved in a much more tolerable geometry.
In conclusion, we have found that electronically bulk TMDCs suffer from the existence of both a direct and indirect band-edge. We find that the presence of the
additional band-edge to have deleterious effects on the maximum solar photovoltaic
efficiency potential. This penalty is minimized when the direct to indirect band-edge
energy splitting is minimized, and therefore, materials like WSe2 , MoSe2 , or MoTe2
should be the main materials considered for achieving maximum solar photovoltaic
efficiency in ultrathin active layers. These results could be refined by carefully
measuring the experimental parameters of the various absorption coefficients, such
as the parameters depicted in Table 2.1. However, we suspect that those values will
not substantially change the conclusions presented here.

39

Figure 2.12: Photovoltaic Figures of Merit of MoSe2 for Varying 𝐶 𝐴𝐸 and 𝐸 𝑅𝐸 0 .
Calculated photovoltaic figures of merit from detailed balance calculations using
the materials parameters in Table 2.1, except 𝐸 𝑔,𝑑𝑖𝑟 = 1.55 eV. We plot the a shortcircuit current density 𝐽𝑠𝑐 b open-circuit voltage 𝑉𝑜𝑐 , and the c power conversion
efficiency 𝜂. All of the figures of merit are normalized to the equivalent detailed
balance value for a step function absorbance at 𝐸 𝑔,𝑖𝑛𝑑 .

2.6

Appendix

Optoelectronic Reciprocity Relations
The connection between absorption and emission has been known for quite some
time. Kirchhoff in 1860 [90] is often cited as being the first to recognize the relation
between the two processes, noting that the absorption and emission probability of a
photon must be equal, i.e. 𝑎(𝐸) = 𝑒(𝐸), through arguments of thermal equilibrium.
A surface with 𝑒(𝐸) = 1 for all energies is known as a perfect black body. However,
the precise spectral dependence of a perfect black body emitter was not derived until
Planck did so in 1906 [153]. He theorized a cavity with perfectly absorbing walls
filled with a gas of photons with a small hole that would leak out a spectral flux
characteristic of a black body:
𝐼 𝑏𝑏 (𝐸) = 𝐸 × 𝑆 𝑏𝑏 (𝐸) =

𝐸3
2𝜋
ℎ3 𝑐2 exp 𝐸 − 1

(2.16)

𝑘 𝑏𝑇

which relates the temperature of a black body to its spectral characteristics, often
referred to as thermal radiation. Here, 𝑆 𝑏𝑏 (𝐸) is the energy-resolved photon flux per
unit area per unit time of black-body radiation and 𝐼 𝑏 refers to the spectrally resolved
intensity of the radiation. The above expression can be generalized to non-black
bodies by combining it with Kirchhoff’s law:
𝐼 (𝐸) = 𝑎(𝐸)𝐼 𝑏𝑏 (𝐸)

(2.17)

40
To form a general law for thermal radiation with surfaces characterized by an absorptivity. In an analogous manner, van Roosbroeck and Shockley [191] generalized
Planck’s law to semiconductors and related the absorption coefficient (𝛼) to the
internal photon emission rate per unit volume:
𝑅(𝐸) = 4𝑛𝑟2 (𝐸)𝛼(𝐸)𝑆 𝑏𝑏 (𝐸) =

8𝜋𝑛𝑟2 𝐸 2 𝛼(𝐸)
 
ℎ3 𝑐2 exp 𝐸 − 1

(2.18)

𝑘 𝑏𝑇

which holds for systems at thermal equilibrium. It was not until Lasher and Stern
[97] considered the situations of spontaneous emission were the above expressions
further generalized to include non-equilibrium, steady-state conditions in terms of
the quasi-Fermi level splitting Δ𝜇, which is exactly equal to the chemical potential
of the photon in a spontaneous emission process:
𝑅(𝐸, Δ𝜇) =

8𝜋𝑛𝑟2
𝐸 2 𝛼(𝐸)
ℎ3 𝑐2 exp 𝐸−Δ𝜇 − 1

(2.19)

𝑘 𝑏𝑇

Here, we note that that this expression is valid only when quasi-thermal equilibrium
holds, where exactly two different quasi-Fermi levels accurately describe the energy
dependence of the two separate populations of electrons and holes (e.g. after the
electron-electron interactions subsequent to the excitation of carriers, the carriers
will be distributed according to the Fermi-Dirac distribution), resulting in a single
quasi-Fermi level splitting Δ𝜇. We assume this to be true in the case of the carriers in
the band tails described here with the carriers above the respective band edges. For
example, in the case of band tails caused by some ensemble of defects, an impurity
band may be formed. If this impurity band is several 𝑘 𝑏 𝑇 away from the band edges,
the electrons in this band would likely thermalize amongst themselves, forming a
separate quasi-Fermi level. Therefore, these relations would need to be modified to
include this effect. Wurfel [209] then generalized the Lasher-Stern relation to an
external flux of radiative emission from a semiconductor surface:

where

𝑆 𝑃𝐿 (𝐸) = 𝑎(𝐸)𝑆 𝑏𝑏 (𝐸, Δ𝜇)

(2.20)

𝐸2
2𝜋
𝑆 𝑏𝑏 (𝐸, Δ𝜇) = 3 2
ℎ 𝑐 exp 𝐸−Δ𝜇 − 1

(2.21)

𝑘 𝑏𝑇

is the spectral flux of a photon gas with chemical potential Δ𝜇 and temperature 𝑇.
Here, 𝐸 is the energy of the emitted photon, 𝑘 𝑏 is the Boltzmann constant, ℎ is
Planck’s constant, and 𝑐 is the speed of light. Wurfel’s expression, with a relation

41
that connects the absorbance to the absorption coefficient (e.g. through the BeerLambert law of 𝑎(𝐸) = 1 − exp(−𝛼𝐿) or more complex light-trapping geometries)
suggests a complete set of self-consistent expressions that connect external properties (e.g. absorbance, external luminescence) of the semiconductor to its internal
properties (e.g. bandgap, absorption coefficient, quasi-fermi level splitting, internal
luminescence). External properties are therefore geometry dependent and can be
carefully engineered from the internal properties using photonic design. Moreover, external properties are typically the only properties that are experimentally
accessible.
We note that the above expression has an apparent divergence at 𝐸 = Δ𝜇. The
resolution requires including an occupation factor in the absorption coefficient:
𝛼(𝐸) = 𝛼0𝐾 (𝐸)( 𝑓𝑣 − 𝑓𝑣 )

(2.22)

where 𝑓𝑣 and 𝑓𝑐 are the occupation for the holes and electrons, respectively. In the
case of a semiconductor with equal effective mass for the holes and electrons and
described by a parabolic dispersion, the occupation factor has a simple form:
𝐸 − Δ𝜇
(2.23)
𝑓𝑣 − 𝑓𝑐 = tanh
4𝑘 𝑏 𝑇
While real systems may have more complex occupation factors (typically not representable analytically due to a fairly complex band structure), we note that 𝑓𝑣 − 𝑓𝑐
is generally a function with limiting values from -1 to 1 with a value of zero at
𝐸 = Δ𝜇, which is captured by the simple expression above. For simplicity and to
capture the physics of the band filling irrespective of other materials properties, we
use the simple expression above when calculating band filling effects.
It was suggested more recently by Rau [160] that the principle of optical reciprocity
can be further generalized to an optoelectronic reciprocity by including the serial
collection/injection with Donolato’s theorem [46] to describe photovoltaic cells and
LEDs:
𝑆 𝐸 𝐿 (𝐸) = 𝐸𝑄𝐸 (𝐸)𝑆 𝑏𝑏 (𝐸, Δ𝜇)
(2.24)
where 𝐸𝑄𝐸 (𝐸) = 𝑎(𝐸) × 𝐼𝑄𝐸 (𝐸) and describes the process of absorbing a photon
with probability 𝑎(𝐸) with a subsequent collection probability of 𝐼𝑄𝐸 (𝐸). Thus,
the LED quantum efficiency 𝑄 𝐿𝐸 𝐷 (𝐸) = 𝜂𝑖𝑛 𝑗 (𝐸) × 𝑒(𝐸) is a detailed balance pair
with the photovoltaic quantum efficiency, taking the injection and collection efficiencies to be detailed balance pairs. We note that while the above generalized

42
Planck’s law (Wurfel’s expression) holds quite generally by any system that can be
characterized by two distinct quasi-Fermi levels and a thermodynamic temperature,
Rau’s reciprocity relation strictly holds only in systems where carrier transport under illumination is well modelled as a linear perturbation of thermal equilibrium
(qualitatively, the law of superposition in the current-voltage curves needs to hold).
We also note that previous examples of using optoelectronic reciprocity for photovoltaic analysis (e.g. modified detailed balance models) has often approximated the
black-body flux as
© 2𝜋
𝐸2
 
exp
𝑆 𝑏𝑏 (𝐸, Δ𝜇) ≈ 𝑆 𝑏𝑏 (𝐸, 0) exp
=­ 3 2
(2.25)
𝑘 𝑏𝑇
𝑘 𝑏𝑇
ℎ 𝑐 exp 𝐸 − 1 ®
𝑘 𝑏𝑇
While the above expression has no singularities and generally results in numerically
accurate results for most systems of interest (e.g. idealistic systems with 𝑎(𝐸 ≤
𝐸 𝑔 ) = 0 will generally have (𝐸 − 𝜇)/𝑘 𝑏 𝑇
1), De Vos and Pauwels [38] noted
the subtle differences this approximation has in analyzing entropy generation in
the detailed balance limit. We show in this paper that accounting for band filling
effects has qualitative and quantitative differences on the luminescence spectra of
semiconductors with significant band tailing, which we emphasize in Figure 2.1
with a-Si:H as an example. Therefore, we use the full expression above without any
approximations.

Modified Detailed Balance Limit Calculations
With the above expressions of optoelectronic reciprocity in hand, we can assemble
a modified detailed balance model for solar cells that account for carrier generation,
extraction, and recombination:
𝐸𝑄𝐸 (𝐸, Δ𝜇)𝑆 𝑏𝑏 (𝐸, Δ𝜇)𝑑𝐸 𝐽 (Δ𝜇)
(2.26)
𝐸𝑄𝐸 (𝐸, Δ𝜇)𝑆(𝐸)𝑑𝐸 =
𝜂 𝑒𝑥𝑡 (Δ𝜇)
where the left-hand side describes carrier injection (e.g. from sunlight or other
light source) and the right-hand side describes carrier extraction, either through
radiative recombination, non-radiative recombination (parametrized by 𝜂 𝑒𝑥𝑡 (Δ𝜇)),
or usefully as carrier collection (𝐽 (Δ𝜇)/𝑞). In steady state, these populations must be
balanced. In our analysis in the main text, we consider the modified detailed balance
expression in the radiative limit i.e. 𝜂 𝑒𝑥𝑡 = 1, Δ𝜇 = 𝑞𝑉, 𝐸𝑄𝐸 (𝐸, Δ𝜇) = 𝑎(𝐸, Δ𝜇)
(see section 2.6 for a short analysis on non-unity radiative or collection efficiency),
with absorptivity described by a Beer-Lambert expression:
𝑎(𝐸) = 1 − exp(−2𝛼𝐿)

(2.27)

43
with a perfect back reflector and perfect anti-reflection coating to describe the optical
configuration. To parametrize the band edge density of states, we take inspiration
from Katahara and Hillhouse [87] and convolve a sub-gap exponential density of
states with a parabolic density of states above the bandgap, giving
𝐸 − 𝐸𝑔
𝛼0𝐾 (𝐸) = 𝛼0
(2.28)
𝑘 𝑏𝑇
with
∫ ∞
√
0 𝜃
exp −|𝑥 |
(2.29)
𝑥 − 𝑥 0 𝑑𝑥 0®®
𝐺 (𝑥) = real ­ 
2Γ 1 + 1𝜃 −∞
And the simplified expression above (Equation 2.23) to account for band filling.
Here, 𝛾 is the energy width parameter (i.e. the Urbach parameter, for 𝜃 = 1).
𝐸 𝑔 is the bandgap, Γ is the Gamma function, 𝛼0 scales the absorption coefficient
(i.e., 𝛼(𝐸 = 𝐸 𝑔 ) = 𝛼0 𝜋𝛾/16𝑘 𝑏 𝑇, and 𝜃 describes the power of the sub-gap
exponential distribution. Our expression has an extra factor of 𝑘 𝑏 𝑇 compared to
the Katahara model, where 𝑘 𝑏 𝑇 is the thermal energy, so that 𝛼0 has the usual units
of absorption coefficient. Using a simple piecewise continuous function for the
absorption coefficient above and below the gap yields similar results, as long as the
absorption coefficient below the gap is still modeled as an Urbach tail. Thus, for a
given set of materials parameters (e.g. 𝛼0 𝐿, 𝛾, 𝐸 𝑔 ) and a specific voltage 𝑉 = Δ𝜇, we
can calculate the appropriate absorption coefficient and consequently the absorption
and luminescence characteristics. The current-voltage curve of the photovoltaic cell
in the detailed balance limit is then calculated using Equation 2.26. Specific figures
of merit can then be extracted from the current-voltage curves.
Band filling Contribution to Photoluminescence
In general, we are interested in the contribution of including the band filling on
the luminescence spectrum of a semiconductor with significant band tails. Let us
examine the case where we are weakly absorbing, which is generally true in the
spectral region of a band tail. In this limit, we can take 𝑎 ≈ 2𝛼𝐿, where we assume
a planar system with a perfect mirror and a perfect antireflection coating as above.
In this case, the external luminescence flux by reciprocity becomes
𝑆 𝑃𝐿 (𝐸, Δ𝜇) = 𝑎(𝐸, Δ𝜇)𝑆 𝑏𝑏 (𝐸, Δ𝜇) ≈ 2𝛼(𝐸, Δ𝜇)𝐿𝑆 𝑏𝑏 (𝐸, Δ𝜇)

(2.30)

44

Figure 2.13: The importance of including band filling effects. a Calculated absorbance (solid line), photoluminescence (dashed line), and distribution of bandgaps
(dotted line) for different Urbach parameters (𝛾) without including band filling effects. b Calculated efficiency, open circuit voltage, short circuit current, and fill
factor with (orange solid line) and without (blue solid line) including band filling
effects.
For systems with intrinsic doping and equal
effective masses, we have 𝛼(𝐸, Δ𝜇) =
𝐸−Δ𝜇
𝛼(𝐸, 0)( 𝑓𝑣 − 𝑓𝑐 ) = 𝛼(𝐸, 0) tanh 4𝑘 𝑏 𝑇 . To see this, note that generally speaking,
𝑓𝑣 − 𝑓𝑐 =
exp

 𝐸 −𝐸 

𝑓𝑝

𝑘 𝑏𝑇

+1

exp

𝐸 𝑒 −𝐸 𝑓𝑛
𝑘 𝑏𝑇

(2.31)
+1

And for intrinsic doping and equal effective masses, 𝐸 𝑓 𝑝 − 𝐸𝑖 = −Δ𝜇/2 and 𝐸 𝑓𝑛 −
𝐸𝑖 = Δ𝜇/2 by symmetry arguments. Here, 𝐸 𝑓 𝑝/𝑛 is the quasi-Fermi level for the
holes/electrons, 𝐸𝑖 is the Fermi level of the intrinsic system (at mid-gap), and
Δ𝜇 = 𝐸 𝑓𝑛 − 𝐸 𝑓 𝑝 is the quasi-Fermi level splitting. By symmetry of the electron and
hole in this case, we must have 𝐸 𝑒 − 𝐸𝑖 = 𝐸/2 and 𝐸 ℎ − 𝐸𝑖 = −𝐸/2, where 𝐸 is the
energy of the photon. Thus,
𝑓𝑣 − 𝑓𝑐 =

exp − 𝐸−Δ𝜇
2𝑘 𝑏 𝑇 + 1

exp

𝐸−Δ𝜇
2𝑘 𝑏 𝑇

(2.32)
+1

45
For simplicity in analysis, let us set 𝑥 = (𝐸 − Δ𝜇)/𝑘 𝑏 𝑇. Thus,
𝑓𝑣 − 𝑓𝑐 =

𝑒 −𝑥/4 (𝑒 −𝑥/4 + 𝑒 𝑥/4 )

𝑒 𝑥/4 (𝑒 𝑥/4 + 𝑒 −𝑥/4 )

[exp(𝑥/4) + exp(−𝑥/4)] [exp(𝑥/4) − exp(−𝑥/4)]

[exp(𝑥/4) + exp(−𝑥/4)] 2
[exp(𝑥/4) − exp(−𝑥/4)]
[exp(𝑥/4) + exp(−𝑥/4)]
sinh(𝑥/4)
cosh(𝑥/4)

(2.33)

= tanh(𝑥/4)
𝐸 − Δ𝜇
= tanh
4𝑘 𝑏 𝑇
We have argued already above that tanh((𝐸 − Δ𝜇)/4𝑘 𝑏 𝑇) should serve as a good
approximation to 𝑓𝑣 − 𝑓𝑐 for most systems and should capture the main physics of
band filling. It may be modified to yield more accurate results in the case of high
doping or a large mismatch between the electron and hole effective masses under
the parabolic bands approximation. For the purposes of this work, let us proceed
with the simple expression so that the luminescence becomes
 tanh 𝐸−Δ𝜇
4𝑘 𝑏 𝑇
4𝜋𝐿
(2.34)
𝑆 𝑃𝐿 (𝐸, Δ𝜇) = 3 2 𝛼(𝐸, 0)𝐸 2 ­­
𝐸−Δ𝜇
ℎ 𝑐
exp 𝑘 𝑏 𝑇 − 1
where the term on the left is a sole function of 𝐸 and the term on the right includes
both 𝐸 and Δ𝜇. Note that by taking 𝑥 = (𝐸 − Δ𝜇)/𝑘 𝑏 𝑇, we have
tanh(𝑥/4)
sinh(𝑥/4)
exp(𝑥) − 1 cosh(𝑥/4) (exp(𝑥) − 1)
[exp(𝑥/4) − exp(−𝑥/4)]
exp(𝑥) − 1 [exp(𝑥/4) + exp(−𝑥/4)]
[exp(𝑥/2) − 1]
exp(𝑥) − 1 [exp(𝑥/2) + 1]
[exp(𝑥/2) − 1]
[exp(𝑥/2) − 1] [exp(𝑥/2) + 1] [exp(𝑥/2) + 1]
(exp(𝑥/2) + 1) 2

(2.35)

Let us double check that there are no singularities as 𝑥 → 0, since tanh(0)/(exp(0) −
1) = 0/0. To do so, we shall use L’Hôpital’s rule, i.e.,
𝑓 (𝑥)
𝑓 0 (𝑥)
= lim = 0
𝑥→𝑐 𝑔(𝑥)
𝑥→𝑐
𝑔 (𝑥)
lim

(2.36)

46
with 𝑓 (𝑥) = tanh(𝑥/4) and 𝑔(𝑥) = exp(𝑥) − 1, giving 𝑓 0 (𝑥) = sech2 (𝑥/4)/4 and
𝑔0 (𝑥) = exp(𝑥). Thus, lim𝑥→0 tanh(𝑥/4)
exp(𝑥)−1 = 1/4, so that there are no singularities and
the luminescence can be rewritten as
𝑆 𝑃𝐿 (𝐸, Δ𝜇) =

𝛼(𝐸, 0)𝐸 2
4𝜋𝐿
ℎ3 𝑐2 (exp 𝐸−Δ𝜇 + 1) 2

(2.37)

2𝑘 𝑏 𝑇

which is positive definite and is a good approximation for the luminescence with
significant band tailing while explicitly including the band filling effects. Note that
when (𝐸 − Δ𝜇)/𝑘 𝑏 𝑇
1, we have
𝑆 𝑃𝐿 (𝐸, Δ𝜇) ≈

4𝜋𝐿
𝛼(𝐸, 0)𝐸 2 exp(−𝐸/𝑘 𝑏 𝑇) exp(Δ𝜇/𝑘 𝑏 𝑇)
ℎ3 𝑐 2

(2.38)

which recovers the expression without band filling contribution, suitable for low
injection and sharp band edges and has been the standard expression used in most
detailed balance analyses of solar cells. It is clear from Equation 2.37 that the
luminescence spectra and radiative current will scale non-linearly with Δ𝜇. Further

more, for 𝛼(𝐸, 0) ∼ exp (𝐸 − 𝐸 𝑔 )/𝛾 , as in the case of Urbach tails, we can take a
derivative of the luminescence flux and find that the peak position will occur at
𝑚𝑎𝑥
𝐸 𝑃𝐿
= Δ𝜇 − 2𝑘 𝑏 𝑇 ln ­ 2𝑘 𝑇 𝑘 𝑇 − 1®
𝑚𝑎𝑥 + 𝛾
« 𝐸 𝑃𝐿

(2.39)

𝑚𝑎𝑥

For 𝛾 > 𝑘 𝑏 𝑇. A simpler but approximate solution can be found by taking 𝐸 𝑃𝐿
𝑘 𝑏 𝑇, and neglecting that term, so that
𝑘 𝑏𝑇
𝑚𝑎𝑥
𝐸 𝑃𝐿 ≈ Δ𝜇 + 2𝑘 𝑏 𝑇 ln
(2.40)
𝛾 − 𝑘 𝑏𝑇

which shows that the luminescence peak depends directly on Δ𝜇, for 𝛾 > 𝑘 𝑏 𝑇.
Effects of band tails on J-V characteristics
While Equation 2.37 suggests a rather complex dependence of the band filling
characteristics on current, we find that the 𝐽 − 𝑉 characteristics can be well fitted to
a modified diode expression in most cases:
𝑞𝑉
𝐽𝑟𝑎𝑑 (𝑉) ∼ 𝐽0 (𝛾, 𝐸 𝑔 ) exp
(2.41)
𝑛𝑒 𝑓 𝑓 (𝛾, 𝐸 𝑔 )𝑘 𝑏 𝑇
In other words, the effect of band filling and band tails is to modify the recombination
current prefactor 𝐽0 and effective ideality factor 𝑛𝑒 𝑓 𝑓 , which manifest in the voltage
loss as described in the main text and in section 2.6. Of particular interest is 𝑛𝑒 𝑓 𝑓 ,

47

Figure 2.14: Effects of band tails and band filling on ideality factor and currentvoltage relationships. a Fitted 𝑛𝑒 𝑓 𝑓 for varying Urbach parameter (𝛾) and bandgap
𝐸 𝑔 . Fits were performed for the range 3𝑘 𝑏 𝑇 < 𝑞𝑉 < 𝐸 𝑔 − 3𝑘 𝑏 𝑇. Linecuts of a
occur at 𝐸 𝑔 = 0.8 (blue), 1.34 (orange), and 2.0 eV (yellow). b Corresponding
linecuts of a plotted for varying Urbach parameter (𝛾). Note the transition that
occurs at 𝛾 = 𝑘 𝑏 𝑇 to larger effective ideality factors, corresponding to the onset of
band tailing and band filling effects. Dashed lines represent the fit, while solid lines
represent the 95% confidence interval. 𝐽 − 𝑉 characteristics for different bandgaps
of 0.8 eV c, 1.34 eV d, and 2.0 eV e. The different lines in a given plot represent
different Urbach parameters. The legend in c is the same for d and e. All plots have
voltages normalized to 𝑘 𝑏 𝑇/𝑞 and current densities normalized to their radiative
dark current 𝐽0 , which is a function of 𝛾. Thicknesses were assumed to be 𝛼0 𝐿 = 1.
Note that for Urbach parameters typically observed in experiment (i.e., 𝛾 ∼ 3𝑘 𝑏 𝑇),
𝑛𝑒 𝑓 𝑓 is generally less than 3. For larger Urbach parameters, a modified ideality
factor no longer describes the voltage scaling appropriately, since 𝐸 𝑔,𝑃𝐿 → 𝑘 𝑏 𝑇.

which should be measurable in electroluminescence measurements, because nonradiative dark current occurs in parallel to the radiative dark current. Thus, we would
expect the 𝑛𝑒 𝑓 𝑓 estimated here in Figure 2.14 to be accurate even in systems far away
from the radiative limit, as long as we measure the radiative current flux through
voltage-dependent electroluminescence. We note that the calculated 𝑛𝑒 𝑓 𝑓 for a-Si

48
(assuming 𝐸 𝑔 ∼ 1.7 eV and 𝛾 ∼ 50 meV) is around 1.7, which is quite similar to
the value measured by Rau et al [163]. To get an approximate analytic expression
for 𝑛𝑒 𝑓 𝑓 , we use Equation 2.37 and assume that 𝐸 2 varies slowly compared to the
exponentials in the integrand and that we are in the weakly absorbing limit. Thus,
 −2
𝐸−𝐸
𝐸−𝑉
𝐽𝑟𝑎𝑑 (𝑉) ∼ 𝑑𝐸 exp 𝛾 𝑔 (exp 2𝑘
and with some rewriting, we find that
 𝑏𝑇
𝑘 𝑇𝑥+𝑉−𝐸
𝐽𝑟𝑎𝑑 (𝑉) ∼ 𝑑𝑥 exp 𝑏 𝛾 𝑔 (exp(𝑥/2) + 1) −2 ∼ exp(𝑉/𝛾). That is, we expect
(2.42)
𝑘 𝑏𝑇
which seems to hold somewhat well for small 𝛾 just above 𝑘 𝑏 𝑇, as observed in
Figure 2.14. Furthermore, using the diode approximation from above, we can also
calculate the modified fill factor expression as
𝑞𝑉𝑜𝑐
𝑞𝑉𝑜𝑐
ln
𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇
𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇
𝐹𝐹 (𝑛𝑒 𝑓 𝑓 , 𝑉𝑜𝑐 ) ≈
(2.43)
𝑜𝑐
1 + 𝑛𝑒 𝑞𝑉
𝑓 𝑓 𝑏
𝑛𝑒 𝑓 𝑓 ≈

which reduces the fill factor slightly compared to the case without band tails and is
an additional efficiency loss mechanism.
Two bandgap model for band tails
To develop a simple picture for the apparent bandgap shift, voltage loss, and effects
of band tailing, we use a simplistic model of the absorbance parametrized by two
step functions. We will refer to this as the “two bandgap model”, whose absorbance
can be seen in Figure 2.15a and is given by:
𝐴(𝐸) = 𝑎 1 𝜃 (𝐸 − 𝐸 𝑔,1 )𝜃 (𝐸 𝑔,2 − 𝐸) + 𝑎 2 𝜃 (𝐸 − 𝐸 𝑔,2 )

(2.44)

where 𝑎 2 = 1 and 𝐸 𝑔,2 = 1.34 eV. The above model represents a simplistic picture
of a system with band tails as it deviates from the Shockley-Queisser limit. Here,
𝐸 𝑔,2 defines the absorption bandgap, 𝐸 𝑔,1 is the lower bandgap that forms as a result
of band tailing, and 𝑎 1 is the effective sub-gap absorption. We then calculate the
typical photovoltaic figures of merit in Figure 2.15b while varying Δ𝐸 𝑔 = 𝐸 𝑔,2 −𝐸 𝑔,1
and 𝑎 1 . The result is qualitatively similar to what is seen with a band tail (e.g. see
Figure 2.3 for comparison), where the efficiency loss is essentially all in the voltage.
Moreover, there is a specific transition point where the voltage loss is linear with
the bandgap separation, dependent on the value of 𝑎 1 . To see this, recall that
𝑉𝑜𝑐 = 𝑘 𝑏 𝑇/𝑞 ln(𝐽𝑠𝑐 /𝐽0 + 1), where 𝐽0 = 𝐴(𝐸)𝑆 𝐵𝐵 (𝐸)𝑑𝐸 and 𝐴(𝐸) is given in
Equation 2.44. The loss due to a lower bandgap 𝐸 𝑔,1 is then
𝐽𝑠𝑐
𝑘 𝑏𝑇
𝐽𝑠𝑐
𝑘 𝑏𝑇
𝐽1
𝑘 𝑏𝑇
ln
+1 −
ln
+1 ≈−
ln
+1
(2.45)
Δ𝑉𝑜𝑐 =
𝐽2
𝐽0
𝐽2

49

where 𝐽1,2 = 𝐴1,2 (𝐸)𝑆 𝐵𝐵 (𝐸)𝑑𝐸, 𝐴1,2 (𝐸) = 𝑎 1,2 𝜃 (𝐸−𝐸 𝑔1,2 ), and we have assumed
𝐽𝑠𝑐
𝐽0 , 𝐽2 . Thus, from the perspective of the voltage loss in the detailed balance
analysis, 𝐸 𝑔,1 does not appear as a photovoltaic bandgap until 𝐽1 > 𝐽2 . This occurs
when

𝐸 𝑔,1 2
+2
𝑘 𝑏𝑇

𝑎 1 𝑒 −𝐸 𝑔,1 /𝑘 𝑏 𝑇

𝑎 2 𝑒 −𝐸 𝑔,2 /𝑘 𝑏 𝑇



𝐸 𝑔,1
𝑘 𝑏𝑇

+2

𝐸 𝑔,2
𝐸 𝑔,2 2
𝑘 𝑏𝑇
𝑘 𝑏𝑇 + 2

 >1

(2.46)

Assuming 𝐸 𝑔1,2
𝑘 𝑏 𝑇, we can neglect the terms outside of the exponential to first
order because it shows up logarithmically with Δ𝐸 𝑔 . Thus, the transition to a new
bandgap occurs when
 
𝑎2
(2.47)
Δ𝐸 𝑔 > 𝑘 𝑏 𝑇 ln
𝑎1
In other words, from the perspective of the Shockley-Queisser limit and voltage loss,
the Stokes shift is not apparent until Equation 2.47 is satisfied. At this point, the
voltage loss scales linearly with increasing 𝐸 𝑔,1 . To see this clearly, we plot the voltage loss with bandgap shift with energies and voltages normalized to 𝑘 𝑏 𝑇 ln(𝑎 2 /𝑎 1 )
in Figure 2.15c. We see that indeed the transition occurs under the condition of
𝑜𝑐
Equation 2.47, where thereafter 𝜕Δ𝑉
𝜕Δ𝐸 𝑔 ≈ 1. This is true irrespective of the value of
𝑎 1 . Moreover, while Equation 2.47 is derived for two discrete bandgaps, we can
generalize the concept to how sharp a continuous absorption spectrum should be to
avoid a Stokes shifted voltage loss. Let us define 𝑎 2 = Δ𝑎 + 𝑎 1 and take the limit as
Δ𝑎, Δ𝐸 → 0. Thus, the generalized continuous form of Equation 2.47 becomes
𝑘 𝑏 𝑇 𝜕𝑎
<1
𝑎 𝜕𝐸

(2.48)

𝐸−𝐸
In the case of weakly absorbing Urbach band tails, 𝑎 ∼ 𝛼𝐿 ∼ 𝐶 exp 𝛾 𝑔 . Thus,
Equation 2.48 predicts a Stokes shift should occur when 𝛾 > 𝑘 𝑏 𝑇, which is what we
observe in Figure 2.6.
General Expression for Voltage Loss due to Nonabrupt Band Edges
The plots of Figure 2.7 and Figure 2.15 in the main text suggests a general relation
between bandgap shifts and voltage loss, irrespective of the exact functional form
of the band edge. To see this, note that the majority of the luminescence of
the step-function absorbance is concentrated within 𝑘 𝑏 𝑇 of the band edge and its
integral varies exponentially with the bandgap energy. Thus, the effective bandgap
of the luminescence, 𝐸 𝑔,𝑃𝐿 must be chosen to integrate to nearly the majority of the

50

Figure 2.15: Analysis of a two-bandgap toy. a Absorbance and emission of the twobandgap toy model, parametrized by two step-functions. Solid lines correspond to
absorbance, whereas dashed lines correspond to emission. b Plot of the photovoltaic
figures of merit (𝜂, 𝑉𝑜𝑐 , 𝐽𝑠𝑐 ) for varying bandgap difference Δ𝐸 𝑔 = 𝐸 𝑔,2 − 𝐸 𝑔,1 and
values of the lower bandgap absorbance 𝑎 1 . 𝑎 2 is assumed to be 1 while 𝐸 𝑔,2 = 1.34
eV. Colors correspond to the same as the legend in c. c Voltage loss versus bandgap
difference in normalized units of𝑘 𝑏 𝑇 ln(𝑎 2 /𝑎 1 ), showing the transition to the Stokes
shift behavior for large enough band gap separation, dependent on 𝑎 2 /𝑎 1 .
luminescence flux. Thus, we pragmatically define it as
∫∞
𝑆 (𝐸, Δ𝜇)𝑑𝐸
𝐸 𝑔, 𝑃𝐿 𝑃𝐿
≥ 0.90
max(𝐸 𝑔,𝑃𝐿 ) ∈ ∫ ∞
(𝐸,
Δ𝜇)𝑑𝐸
𝑃𝐿

(2.49)

While this definition of 𝐸 𝑔,𝑃𝐿 is not unique, it parametrizes the luminescence typically assumed under step-function absorbance to a greater variety of luminescence
spectra and is somewhat less sensitive to noise. We further define the above-gap

51
absorbance as
∫∞

∫∞
𝑎(𝐸,
Δ𝜇)𝑆
(𝐸,
Δ𝜇)𝑑𝐸
𝑆 (𝐸, Δ𝜇)𝑑𝐸
𝐵𝐵
𝐸 𝑔, 𝐴𝑏𝑠
𝐸 𝑔, 𝐴𝑏𝑠 𝑃𝐿
∫∞
∫∞
𝑎¯ 𝐴𝐺 =
exp(Δ𝜇/𝑘 𝑏 𝑇) 𝐸
𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸 exp(Δ𝜇/𝑘 𝑏 𝑇) 𝐸
𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸
𝑔, 𝐴𝑏𝑠
𝑔, 𝐴𝑏𝑠
(2.50)
and below-gap absorbance as
∫ 𝐸 𝑔, 𝐴𝑏𝑠
∫ 𝐸 𝑔, 𝐴𝑏𝑠
𝐸 𝑔, 𝐴𝑏𝑠
𝑆 𝑃𝐿 (𝐸, Δ𝜇)𝑑𝐸
𝑎(𝐸,
𝑚𝑢)𝑆
(𝐸,
Δ𝜇)𝑑𝐸
𝐵𝐵
𝐸 𝑔, 𝑃𝐿
𝐸 𝑔, 𝑃𝐿
𝐸 𝑔, 𝑃𝐿
𝑎¯ 𝑆𝐺 =
∫𝐸
∫𝐸
exp(Δ𝜇/𝑘 𝑏 𝑇) 𝐸 𝑔, 𝐴𝑏𝑠 𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸
exp(Δ𝜇/𝑘 𝑏 𝑇) 𝐸 𝑔, 𝐴𝑏𝑠 𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸
𝑔, 𝑃𝐿
𝑔, 𝑃𝐿
(2.51)
where both values are apparently dependent on Δ𝜇. Accurate estimation of the
𝑆 𝑃𝐿 (𝐸,Δ𝜇)
quantity exp(Δ𝜇/𝑘
can be achieved by taking 𝐸
Δ𝜇 and fitting the luminescence
𝐵 𝑇)
spectra to the high energy absorption/EQE, or by fitting the full spectrum with
the band filling factor. Alternatively, since Equation 2.6 of the main text only
requires knowledge of the ratio 𝑎¯ 𝐴𝐺 /𝑎¯ 𝑆𝐺 , we can simply use the directly measured
luminescence spectrum:
∫∞
∫ 𝐸 𝑔, 𝐴𝑏𝑠
(𝐸,
Δ𝜇)𝑑𝐸
𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸
𝑃𝐿
𝐸 𝑔, 𝐴𝑏𝑠
𝐸 𝑔, 𝑃𝐿
𝑎¯ 𝐴𝐺
∫∞
=∫𝐸
(2.52)
𝑔, 𝐴𝑏𝑠
𝑎¯ 𝑆𝐺
𝑆 𝐵𝐵 (𝐸, 0)𝑑𝐸
𝑆 𝑃𝐿 (𝐸, Δ𝜇)𝑑𝐸
𝐸 𝑔, 𝑃𝐿

𝐸 𝑔, 𝐴𝑏𝑠

And the definitions of 𝐸 𝑔,𝐴𝑏𝑠 and 𝐸 𝑔,𝑃𝐿 to estimate the weighted absorbance ratio.
These definitions work well because the integrated number of recombination electrons is what matters in the detailed balance analysis, which is achieved by the
appropriate definitions of weighted absorption and bandgaps. Therefore, the voltage loss is given by a form that is quite similar to Equation 2.45:
2
  𝐸 𝑔, 𝑃𝐿 + 2 𝐸 𝑔, 𝑃𝐿 + 2 
𝐸 𝑔,𝐴𝑏𝑠 − 𝐸 𝑔,𝑃𝐿  𝑘 𝑏 𝑇
𝑘 𝑏𝑇
𝑎¯ 𝑆𝐺 ª®
𝑘 𝑏 𝑇 ©­ 𝑎¯ 𝑆𝐺
ln ­
exp
Δ𝑉𝑜𝑐 =

 𝐸 𝑔, 𝐴𝑏𝑠 2
𝑎¯ 𝐴𝐺
𝑘 𝑏𝑇
𝑎¯ 𝐴𝐺 ®
𝐸 𝑔, 𝐴𝑏𝑠
 𝑘 𝑇
𝑘 𝑏𝑇
 𝑏
(2.53)
Noting the logarithmic dependence on the argument and assuming Δ𝐸 𝑔 = 𝐸 𝑔,𝐴𝑏𝑠 −
𝐸 𝑔,𝑃𝐿
𝐸 𝑔,𝐴𝑏𝑠 , as well as 𝐸 𝑔,𝐴𝑏𝑠 , 𝐸 𝑔,𝑃𝐿
𝑘 𝑏 𝑇, we arrive at a simple expression
that only depends on the observed bandgap shifts and the ratio of the above-gap and
sub-gap absorbances:
Δ𝐸 𝑔
𝑎¯ 𝑆𝐺
𝑘 𝑏𝑇
𝑎¯ 𝑆𝐺
𝑎¯ 𝑆𝐺
Δ𝑉𝑜𝑐
, Δ𝐸 𝑔 ≈
ln
exp
+1−
(2.54)
𝑎¯ 𝐴𝐺
𝑎¯ 𝐴𝐺
𝑘 𝑏𝑇
𝑎¯ 𝐴𝐺
Note that this expression recovers the expected values of voltage loss as 𝑎¯ 𝑆𝐺 /𝑎¯ 𝐴𝐺 →
0, 1 and as Δ𝐸 𝑔 → 0. Furthermore, the functional form of the sub-gap ab-

52
sorbance is captured by its effect of varying the value of 𝑎¯ 𝑆𝐺 . From an experimental standpoint, another method to estimate the voltage loss is by using
the modified 𝐽 − 𝑉 characteristics
found in section 2.6. It is clear then that
𝑞𝑉
𝐽𝑟𝑎𝑑 (𝑉) ≈ 𝐽0,𝑟𝑎𝑑 exp 𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇 = 𝑆 𝐸 𝐿 (𝐸, 𝑉)𝑑𝐸. Furthermore, it is possible to
estimate 𝑛𝑒 𝑓 𝑓 directly from the slope of voltage-dependent electroluminescence
𝑆 𝐸 𝐿 (𝐸, 𝑉). Integrating over 𝑆 𝐸 𝐿 (𝐸, 𝑉) and dividing by exp 𝑞𝑉/𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇 then
yields 𝐽0,𝑟𝑎𝑑 . Note that the 𝑉𝑜𝑐 loss due to an imperfect
 band edge can
 be equiv𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇
𝐽𝑠𝑐,𝑆𝑄
𝐽𝑠𝑐
𝑘 𝑏𝑇
alently written in the form of Δ𝑉𝑜𝑐 = 𝑞 ln 𝐽0,𝑟 𝑎𝑑,𝑆𝑄 −
ln 𝐽0,𝑟
using
𝑎𝑑
Equation 2.41, which can be expanded to yield
(𝑛𝑒 𝑓 𝑓 − 1)𝑘 𝑏 𝑇
𝐽𝑠𝑐,𝑆𝑄
𝐽0,𝑟𝑎𝑑
𝑘 𝑏𝑇
𝑘 𝑏𝑇
𝐽𝑠𝑐
Δ𝑉𝑜𝑐 =
ln
ln
ln
(2.55)
𝐽𝑠𝑐
𝐽0,𝑟𝑎𝑑,𝑆𝑄
𝐽0,𝑟𝑎𝑑
where the first term is the voltage loss due to incomplete absorption above the
bandgap. The second term is the voltage loss due to band tailing, while the third
term is a voltage gain due to band filling effects (e.g. see Figure 2.13).
Effects of Sub-Unity Radiative and Quantum Efficiencies
We have thus far only analyzed the situation assuming the reciprocity between absorption and photoluminescence, which holds quite generally but concerns primarily
the internal open circuit voltage of a device i.e. the quasi Fermi level splitting. To
analyze the effects of a system with sub-unity quantum efficiencies, which may
be particularly relevant for localized states below the absorption gap, we assume
Donolato’s theorem still holds and apply Equation 2.24. Therefore, by reciprocity,
the injection efficiency into these localized states would be relatively low, lowering
the electroluminescence recombination rate and increasing the limiting 𝑉𝑜𝑐 (Figure 2.16). This situation would be analogous to considering free carrier absorption
in the absorption band tail, where 𝐼𝑄𝐸 → 0, and therefore the absorption of freecarriers do not lead to photovoltaic current [89]. Thus, photogenerated carriers that
do not contribute to photovoltaic current, whether they are localized states or free
carriers, would not result in a loss to the open circuit voltage in the radiative limit.
In general, the effect of band tails on the radiative limit should be determined via
photocurrent spectroscopies, which captures this effect experimentally directly.
To analyze the voltage loss effects away from the radiative limit, i.e., sub-unity
radiative efficiency, we note that generally Equation
2.41 holds
 and
 the discussion
𝐽0,𝑟 𝑎𝑑
𝑞𝑉
in section 2.6 suggests that 𝐽 (𝑉) = 𝐽𝑠𝑐 − 𝜂𝑒𝑥𝑡 exp 𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇 − 1 which is quite
similar to Equation 2.26. Thus, it is readily apparent that the loss due to non-radiative

53
recombination is modified with an ideality factor 𝑛𝑒 𝑓 𝑓 ≥ 1, so that
Δ𝑉𝑜𝑐,𝑛𝑟 = −

𝑛𝑒 𝑓 𝑓 𝑘 𝑏 𝑇
|ln(𝜂 𝑒𝑥𝑡 )|

(2.56)

It should be noted that 𝜂 𝑒𝑥𝑡 is generally a function of voltage as well and should
be measured/calculated at the operating voltage. This radiative ideality factor has
already been recognized by Rau et al. to be relevant in amorphous Si [163] when
analyzing its non-radiative losses. In many devices, sub-unity radiative efficiencies
and sub-unity quantum efficiencies are both present and are likely competing to
provide the observed voltage. In contrast, concentration benefits the voltage by a
𝑘 𝑇
similar factor Δ𝑉𝑜𝑐,𝑐𝑜𝑛𝑐 = 𝑒 𝑓 𝑓𝑞 𝑏 |ln(𝐶)||, where 𝐶 > 1 is the concentration factor.
Parametrization of the Band edge Functional Form
In Figure 2.7, we considered various band edge functional forms to argue that there
exists a general expression that relates Δ𝑉𝑜𝑐,𝑟𝑎𝑑 to the existence of a Stokes shift,
i.e. Δ𝐸 𝑔 . We considered two main types of band edges: exponential tails and
indirect edge power laws. Exponential tails are the main form of band edges we
have discussed in this article and we have thus far used the analysis described in
section 2.6. For the calculations in Figure 2.17 and Figure 2.7, we vary 𝜃, 𝐸 𝑔 , and
𝛾 to generate various functional forms for the band tail given by Equation 2.28.
Furthermore, we consider only the absorption spectra that yield a luminescence
bandgap above 4𝑘 𝑏 𝑇, since we assume that 𝐸 𝑔
𝑘 𝑏 𝑇 in deriving Equation 2.6. We
further consider a general power law form for a semiconductor band edge that has
a weak oscillator strength (e.g. an indirect transition) with a higher energy direct
transition with larger oscillator strengths:
𝛼(𝐸) = 𝛼0,𝑖𝑛𝑑 (𝐸 − 𝐸 𝑔,𝑖𝑛𝑑 ) 𝑛 𝜃 (𝐸 − 𝐸 𝑔,𝑖𝑛𝑑 )𝜃 (𝐸 𝑔,𝑑𝑖𝑟 − 𝐸) + 𝛼0,𝑑𝑖𝑟 𝜃 (𝐸 − 𝐸 𝑔,𝑑𝑖𝑟 ) (2.57)
where 𝐸 𝑔,𝑖𝑛𝑑 and 𝐸 𝑔,𝑑𝑖𝑟 represent the indirect and direct band edge, respectively,
while 𝛼0,𝑖𝑛𝑑 and 𝛼0,𝑑𝑖𝑟 represent the absorption coefficients of the indirect and direct
gaps, respectively. 𝑛 parametrizes the different energetic scaling relations of the
indirect edge, typically 𝑛 < 3 experimentally.
For both forms of band edges, we calculate 𝑉𝑜𝑐,𝑟𝑎𝑑 from the complete modified
detailed balance analysis, including band filling effects and assuming 𝜂 𝑒𝑥𝑡 = 1
(Equation 2.26), 𝐸 𝑔,𝐴𝑏𝑠 is then derived from the calculated absorption spectrum using Rau’s definition, and therefore 𝑉𝑜𝑐,𝑆𝑄 (𝐸 𝑔,𝐴𝑏𝑠 ) is calculated using a step-function
at 𝐸 𝑔,𝐴𝑏𝑠 . 𝐸 𝑔,𝑃𝐿 , 𝑎¯ 𝑆𝐺 , 𝑎¯ 𝐴𝐺 is then calculated from the definitions in section 2.6

54

Figure 2.16: Effects of a sub-unity collection efficiency below the bandgap.
Calculated power conversion efficiency, open circuit voltage, short circuit current
density, and fill factor assuming that the collection efficiency below the bandgap
(𝐼𝑄𝐸 𝑆𝐺 ) is less than 1 and given by a constant average value. That is, we take the ex

ternal quantum efficiency to be 𝐸𝑄𝐸 (𝐸) = 𝑎(𝐸) 𝐼𝑄𝐸 𝑆𝐺 𝜃 (𝐸 𝑔 − 𝐸) + 𝜃 (𝐸 − 𝐸 𝑔 ) .
The “Urbach” curve is calculated assuming the collection efficiency decays
with a similar Urbach parameter
to that used in the absorption calculation (i.e.
𝐼𝑄𝐸 𝑆𝐺 (𝛾, 𝐸) = exp (𝐸 − 𝐸 𝑔 )/𝛾 ), which may approximate the mobility-edge better than a constant.
by examining the luminescence spectra, 𝑆 𝑃𝐿 . The results of these different band
edges map well onto a simple relation described by Equation 2.54, suggesting a two
bandgap model is an adequate representation of most experimentally observed band
edge forms.
Table 2.2: Tabulated values of Urbach Energies (Experiment)
and Δ𝑉𝑜𝑐 loss (Calculated) .
Material
Name

Bandgap
(eV)

Urbach Energy
(meV)

Calculated Δ𝑉𝑜𝑐
(mV)

References

55
c-Si

1.12

9.6

24.0

[34]

c-Si

1.12

8.6

20.7

[34]

c-Si

1.12

11

28.9

[63]

GaAs

1.42

6.9

14.6

[181]

GaAs

1.42

7.5

16.3

[84]

GaAs

1.42

5.9

11.7

[12]

InP

1.355

9.4

22.5

[182]

InP

1.361

10.6

26.6

[182]

InP

1.34

7.1

16.3

[12]

a-Si:H

1.72

42

273.1

[33]

a-Si:H

1.64

52

382.8

[33]

a-Si:H

1.69

67

559.1

[35]

a-Si:H

1.70

43

283.5

[183]

a-Si:H

1.69

47

329.0

[184]

a-Si:H

1.7

48

341.3

[192]

a-Si:H

1.8

51

385.3

[192]

a-Si:H

1.85

51

389.1

[192]

CdTe

1.45

17

52.8

[158]

CdTe

1.5

7.2

15.5

[119]

CdTe

1.5

21.1

[129]

CdTe

1.5

10.6

26.5

[172]

CIGS

1.53

24

94.2

[71]

CIGS

18

56.0

[71]

CIGS

1.18

23

84.9

[71]

CIGS

1.2

31

143.6

[186]

CIGS

1.67

25

102.5

[124]

CIGS

1.08

21.9

[174]

Kesterite

1.5

69

551.4

[77]

Kesterite

1.1

54

346.9

[77]

Kesterite

1.38

45

286.6

[214]

Kesterite

1.54

65

516.7

[214]

Kesterite

1.68

56.8

441.8

[135]

Perovskite

1.57

15

44.7

[39]

Perovskite

2.23

23

90.2

[168]

56
Perovskite

1.57

14

40.5

[224]

Perovskite

1.57

14.4

42.2

[224]

Perovskite

1.57

15.8

48.3

[224]

Organic

1.66

37

214.9

[62]

Organic

50

386.8

[94]

Organic

1.31

25.6

104.8

[112]

Organic

1.47

27

115.9

[159]

Organic

1.88

36

211.0

[193]

Organic

1.71

27

118.7

[110]

Organic

1.67

24

95.2

[156]

57

Figure 2.17: Different band edges that map onto a two-bandgap model. Stokes
shift Δ𝐸 𝑔 and radiative voltage loss Δ𝑉𝑜𝑐 calculated from the full detailed balance
analysis with the appropriate definitions of 𝐸 𝑔,𝐴𝑏𝑠 , 𝐸 𝑔,𝑃𝐿 , 𝑎 𝐴𝐺 , 𝑎 𝑆𝐺 , as described
in Section S6. We vary the parameters for the exponential band tail model a
and the indirect edge power law model b. For the exponential band tail model
we take 𝛼0 𝐿 = 10, whereas for the indirect edge model we take 𝛼0,𝑑𝑖𝑟 𝐿 = 100,
𝛼0,𝑖𝑛𝑑 𝐿 = 0.1. Both forms map well onto the generalized
expression c. The colorbar
𝑎 𝑆𝐺
for the generalized expression in c is log10 𝑎 𝐴𝐺 , i.e., describes the ratio of the
sub-gap to above-gap absorption. The different ratios plots are overlaid, showing
the similarity irrespective of 𝑎 𝑆𝐺 /𝑎 𝐴𝐺 , assuming it is sufficiently small.

58
Chapter 3

OPTICAL DESIGN OF CAVITY COUPLING TO EXCITONIC
TMDCS
“Nothing in life is to be feared. It is only to be
understood.”
— Marie Curie

3.1

Linear Dielectric Function of TMDCs

The optical properties of the bulk semiconducting transition metal dichalcogenides
of interest (i.e., MX2 , where M = Mo, W and X = S, Se) were first examined in the
1970s [52] as shown in Figure 3.1, where a variety of sharp excitonic resonances
were quickly observed [134]. The direct optical transitions of these materials were
found to be excitonic, particularly when E ⊥ c and the excitonic effect becomes
stronger as their thickness approaches the exciton Bohr radius. More recently, [104]
extracted the dielectric properties for monolayer TMDCs by assuming they are made
up of a superposition of Lorentzian oscillators1 and fitting the reflection spectrum
to the dielectric function:
𝜀(𝐸) = 1 +

𝑓𝑘
𝐸 2 − 𝐸 2 − 𝑖𝐸𝛾 𝑘
𝑘=1 𝑘

(3.1)

where 𝑓 𝑘 and 𝛾 𝑘 are the oscillator strength and the linewidth of the 𝑘th oscillator,
and 𝐸 𝑘 runs over the full spectral range and is the energy of each oscillator. The
linear dielectric function of each monolayer TMDC material extracted from microreflectance measurements is plotted in Figure 3.2, and this method of dielectric
function extraction has become the standard method of extracting dielectric function
in micron-sized samples (as opposed to ellipsometry, which typically requires much
larger samples). The dimensionality effect on the excitonic resonances is evident
between the monolayer and bulk materials due to the strength and linewidth of the
excitonic transitions. It should be noted that the first two optical transitions are
typically referred to as the 𝐴 and 𝐵 excitons, which are similar in their wavefunction
nature but differ in the specific spin state (specifically, the spin-orbit splitting in the
1 The simplest model for the electronic response from an electromagnetic field is a damped

harmonic oscillator, for which the Lorentz oscillator model is the solution, as described in section B.4.

59

Figure 3.1: First measurements of the bulk MoS2 dielectric function. Dielectric
function of MoS2 for E ⊥ c, adapted from [134].
valence band is the main contribution to the 𝐴 − 𝐵 splitting). To illustrate these
different excitonic states more clearly, the bandstructure of monolayer MoS2 and
its associated optical transitions are shown in Figure 3.3a. Also labelled is the
𝐶 exciton, whose peculiar nature comes from the band-nesting phenomenon (i.e.,
parallel lines that result in a large value of the joint density of optical states) along the
Γ − 𝐾 line of the Brillouin zone. Incidentally, since the largest effects in the quantum
confinement occur closer to the Γ and min 2 points in the valence and conduction
band, respectively, the 𝐶 exciton is more sensitive to quantum confinement than the 𝐴
and 𝐵 excitons are. Specifically, the 𝐴 and 𝐵 excitonic wavefunctions are largely inplane, and therefore the main effect of thickness is to modify their dielectric screening
environment, and therefore the Coulomb interactions. To first order, the decrease
in the attractive electron-hole interactions (which describes the magnitude of the
exciton binding energy 𝐸 𝑏 ) are compensated by the decreased repulsive electronelectron interactions (which partially describes the magnitude of the quasiparticle
bandgap). Thus, the actual optical transition energy of the 𝐴 and 𝐵 excitons are
relatively insensitive to thickness, as seen in Figure 3.2. The bandstructure of
2 The Í

min point is also often referred to as the 𝑄 or Λ point in the 2D materials literature.

60

Figure 3.2: Room temperature dielectric function of monolayer TMDCs. Room
temperature dielectric function of a,e MoSe2 , b,f WSe2 , c,g MoS2 , and d,h WS2 ,
adapted from [104]

other TMDCs are quite similar in nature, with some subtleties. For example, for
tungsten-based compounds, the spin-orbit splitting in the valence band is larger
and the conduction band spin-orbit splitting is also of an opposite sign, so that
the lowest-energy excitonic states for tungsten compounds are optically dark [199].
Other essential differences is the energetic difference between the Γ − 𝐾 points in
the valence band and the min −𝐾 splitting in the conduction band, which partially
dictates the amount of intervalley scattering in the optical transition (which can be
tuned with strain, as we shall see in Chapter 6).

61

𝐸𝐸𝑔𝑔

𝐸𝐸𝑏𝑏

𝐸𝐸𝑏𝑏
𝐵𝐵

𝐴𝐴

𝐶𝐶

Figure 3.3: Bandstructure as a function of thickness. Calculated bandstructure
of a monolayer, b bilayer, and c bulk MoS2 using the quasi-particle self-consistent
𝐺𝑊 method, adapted from [25].

62
3.2

Multilayer Near Unity Absorption

Knowing the different dielectric functions of both the monolayer and optically ‘bulk’
layered materials, we are now able to calculate various linear optical responses of
the materials, including its reflectance 𝑅, transmittance 𝑇, and absorbance 𝐴. For
layered stacks that can be effectively modelled as 1D optical media, it is possible to
use the transfer matrix method to completely solve for their optical properties3.

Figure 3.4: Designing ultrathin absorbing cavities. a Calculated total absorption
assuming 𝑛TMDC = 5.0 + 0.01𝑖 and 𝑛metal = 10 + 10𝑖, for varying thickness of
the TMDC layer. b Absorption spectrum for 𝑡 = 25 nm, the absorption in the
TMDC and metal layers is also shown. c Calculated total absorption assuming
𝑛TMDC = 5.0 + 1.0𝑖 and 𝑛metal = 0.05 + 5𝑖, for varying thickness of the TMDC layer.
d Absorption spectrum for 𝑡 = 15 nm, the absorption in the TMDC and metal layers
is also shown.
We are now interested in designing optical structures that can achieve near-unity
3 A detailed derivation and discussion of the transfer matrix method, and a primer on Maxwell’s

equations and wave optics, can be found in Appendix B.

63
absorption, which is an essential feature to operate solar photovoltaics with high
efficiency. Let us suppose we have a sub-wavelength thick structure suspended in air,
so that it forms an air/TMDC/air structure. This situation is a two-port structure, i.e.,
light can be incident/reflected from both sides. However, under typical photovoltaic
operation, we would expect illumination from only one side. By decomposing the
incident wave into even and odd modes, where only the even modes have a non-zero
electric field intensity at the TMDC (odd modes have zero intensity at the center, by
definition), we would expect a maximum absorbance of only 50% [150]. Thus, to
maximize absorption, additional symmetry breaking must be done.
The simplest method to improve the maximum absorbance is to reduce a two-port
structure into a single-port structure by including a back mirror, e.g. we optically
consider a three-layer stack of air/TMDC/metal. For a three layer stack, the explicit
expression for the reflectivity has a simple analytic expression:

𝑟˜ =

𝑟˜12 + 𝑟˜23 𝑒 2𝑖 𝛽

1 + 𝑟˜12𝑟˜23 𝑒 2𝑖 𝛽˜

(3.2)

where

𝑛˜ 𝑎 − 𝑛˜ 𝑏
2𝜋 𝑛˜ 2
, 𝛽˜ =
(3.3)
𝑛˜ 𝑎 + 𝑛˜ 𝑏
is the interfacial reflectance and the phase accumulation due to propagation, respectively. It is important to note that in the absence of transmittance (e.g. with a
metallic substrate), we simply have 𝐴 = 1 − 𝑅 − 𝑇 = 1 − 𝑅 = 1 − |𝑟˜ | 2 . Thus, to
achieve near-unity absorption, we must equivalently have 𝑟˜ → 0. Thus, to achieve
unity absorbance, we require
𝑟˜𝑎𝑏 =

𝑟˜12
exp 2𝑖 𝛽˜ = −
𝑟˜23

(3.4)

There are a few scenarios where this equation can be nearly satisfied. Let us first
consider the scenario with an ideal metal as the back mirror and a lowly absorbing
TMDC layer (e.g., 𝑛˜ 2 = 𝑛˜ TMDC = 5.0 + 0.01𝑖 for the sake of illustration). In this
scenario, we would expect 𝑟˜23 → −1 because an ideal metal can be modelled as
having an index 𝑛˜ metal = 𝜂 + 𝑖𝜅 with 𝜂 → ∞, 𝜅 → ∞. Similarly, since the middle
layer (i.e., the TMDC) is lowly absorbing, we would expect 𝑟˜12 → −1. Therefore,
to satisfy Equation 3.4, one must have 𝑡 ≈ 𝑚𝜆/(4𝜂2 ), where 𝑚 is an odd integer
(see Figure 3.4a). This condition is one that is akin to that of a ‘Salisbury screen’
type geometry. However, it is important to note that while the total absorption
can be close to unity, the absorption within the semiconductor is actually closer

64

Figure 3.5: Near Unity Absorption in ultrathin TMDCs. Calculated and experimental absorption spectra of a,b WSe2 , c,d WS2 , and e,f MoS2 on silver substrates
(i.e., the optical stack is air/TMDC/Ag). Calculations take into account the real,
dispersive nature of the silver and TMDC. The dotted lines in the calculated data
are the active layer absorption, i.e., the absorption within the TMDC layer. Experimental data was extracted from normalized reflectance measurements, 𝐴 = 1 − 𝑅.
Figured adapted from [80].

to 0 (since the loss in the TMDC is low compared to that of the metal and the

65
thickness is small for the first order resonance4, see Figure 3.4b). This situation
is certainly not advantageous for photovoltaic operation, and for real TMDCs, the
optical loss is significantly higher above its band-edge, which is the operational
point of photovoltaic behavior.
A surprisingly different analysis occurs if we consider the significant loss of a TMDC
layer (e.g. 𝑛˜ TMDC = 5.0 + 1.0𝑖). In this scenario, 𝑟˜12 is no longer on the real axis, and
for a finite conductivity of a real metal (e.g., 𝑛metal = 0.05 + 5𝑖, which is a similar
value to that of silver in the visible part of the spectrum), neither is 𝑟˜23 . Thus, it is
possible to rewrite Equation 3.4 as
4𝜋(𝜂2 + 𝑖𝜅 2 )
exp 𝑖𝜋 + 𝑖
𝑡 = 𝑟 0 𝑒 −𝑖𝜙
(3.5)
where we have defined 𝑟 0 𝑒 −𝑖𝜙 = 𝑟˜12 /𝑟˜23 . Defining further 𝑡 = 𝑡0 − Δ𝑡, where
𝑡0 = 𝜆/(4𝜂2 ), we can rewrite the expression above as
4𝜋𝜂2
Δ𝑡 = 𝑟 0 exp(−𝑖𝜙)
(3.6)
exp(−𝛼2 𝑡) exp −𝑖
In other words, we require 𝑟 0 = exp(−𝛼2 𝑡), where 𝛼2 = 4𝜋𝜅 2 /𝜆 is the absorption
coefficient of the TMDC layer, and 𝜙 = 4𝜋𝜂2 Δ𝑡/𝜆. In other words, the substantial
loss in the TMDC layer as well as the finite conductivity of the metal results in
non-trivial phase shifts at the interface (𝑟˜𝑎𝑏 ), which enables designing absorbing
geometries with thicknesses below 𝜆/(4𝜂2 ) where the loss is a critical component
of the design [88]. Because of the significant loss in the TMDC layer and the finite
but small loss in the metal, we are able to achieve nearly complete absorption in
the active layer (see Figure 3.4c,d). Thus, by using these design rules of the nontrivial interfacial phase shifts and taking into account the real materials dispersions of
metals (e.g. noble ones like Ag or Au) and TMDCs (e.g. WSe2 , MoS2 , WS2 ), we are
able to design near-unity absorption within layers of TMDCs that are approximately
10-15 nm thick, as shown in Figure 3.5.
Although the absorption peaks in our structure are dependent on path length, they
are highly insensitive to the angle of incidence due to the large refractive index of
the TMDC layer, as a can be seen for the case of 13 nm WSe2 on Ag (Figure 3.6).
The peak absorption stays over 80% even at a 60◦ incident angle (Figure 3.6b)
suggesting relatively low sensitivity to the angle of incident light. This feature of
4 For an ideal metal where the skin depth approaches zero, it is clear that the absorbance in the

metal must also approach zero. In this case, unity absorbance is not generally achieved for the first
order resonance, and larger thicknesses are required.

66
TMDC/Ag heterostructures is highly advantageous for off-normal light collection
and combined with their near-unity active layer absorbance, these structures may be
of a particular interest for photovoltaic applications and solar energy harvesting.

Figure 3.6: Angle dependence of absorption in ultrathin TMDC/Ag structures.
a Contour plot of calculated absorption spectra at varying angles for 13 nm WSe2
on Ag back reflector. The insensitivity of the absorption as a function of incident
angle is apparent. b Line cut from a at 520 nm showing the angle dependence of
peak absorption. Figured adapted from [80].

3.3

Monolayer Near Unity Absorption

We are now also interested in the possibility of achieving near-unity absorption
in a single monolayer of a TMDC (∼7 Å). It may be tempting to assume that,
with a judicious choice of a substrate, it would be possible to satisfy Equation 3.4
for the specific optical properties of a TMDC monolayer. However, in the limit
of monolayer absorption, the phase accumulation through the monolayer must be

67
small, so that exp 2𝑖 𝛽˜ → 1, i.e., 𝑟˜12 = −𝑟˜23 . This expression can only be satisfied
if 𝑛˜ 1 = 𝑛˜ 3 , which is the same as the TMDC being suspended in air or immersed in a
material with index 𝑛˜ 1 . In this case, it is clear that the maximum absorbance would
be only 50% by symmetry arguments, as described previously.

To yield unity absorption in an atomically thin material, it is still necessary to
break the optical symmetry by, for example, removing an optical port with a back
mirror. However, as shown above, a simple three layer optical stack cannot yield
unity absorption in this regime of thicknesses. Thus, we shall consider the next
simplest structure: a four-layer structure, where we have an ideal 2D exciton on a
dielectric spacer with index 𝑛2 and thickness 𝑑, which is on top of a back mirror.
This mirror will be parametrized by an interface reflection 𝑟 23 and transmission
amplitude 𝑡23 = 0. The analysis continues similar to the above case, except now we
have a more complicated scattering matrix in this air/exciton/dielectric spacer/mirror
system:
𝑆 = 𝐽1,2
𝐿 2 𝐽2,3
(3.7)
where
1 + 𝑍0 𝜎/2
𝑍0 𝜎/2
−𝑍0 𝜎/2 1 − 𝑍0 𝜎/2
exp(−𝑖𝑞𝑑)
𝐿2 =
exp(𝑖𝑞𝑑)
1 1 𝑟 23
𝐽2,3 =
𝑡23 𝑟 23 1

𝐽1,2

(3.8)

and we have implicitly taken 𝑛1 = 𝑛2 = 1 (i.e., the exciton is suspended over air,
which does not come with a loss of generality in our qualitative results, as we shall
soon see), so that 𝑟 12 = 0 and 𝑡12 = 1. We have not specified 𝑟 23 yet, but we
will soon. 𝑞 = 2𝜋𝑛2 /𝜆 = 2𝜋/𝜆 and 𝑑 is the spacing of the exciton from the back
mirror. Furthermore, we have parametrized the optical properties of the monolayer
TMDC as an interfacial sheet conductor, with an infinitesimal thickness. Thus, we
are implicitly assuming there is little to no phase propagation through the TMDC
monolayer. The sheet conductor model5 for an excitonic material is given as
𝜎2𝐷 (𝜔) =

1 𝜔
𝑖𝛾𝑟
𝑍0 𝜔0 𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2

(3.9)

5 the Lorentzian sheet conductor model for a 2D exciton and some analysis of their optics is

discussed in section B.4 and section B.5.

68
where 𝑍0 is the impedance of free space, 𝜔0 is the energy of the exciton, 𝛾𝑟 is the
radiative rate of the exciton (which is proportional to its oscillator strength), and 𝛾𝑛𝑟
is the non-radiative rate of the exciton (which is what typically dictates its linewidth).
Multiplying the matrices together and examining the reflection amplitude gives us
the expression:
𝑟=

− 𝑍02𝜎 [𝑟 23 exp(2𝑖𝑞𝑑) + 1] + 𝑟 23 exp(2𝑖𝑞𝑑)
𝑍0 𝜎
2 [𝑟 23 exp(2𝑖𝑞𝑑) + 1] + 1

(3.10)

Note that if we now consider an ideal mirror on the back, we would expect 𝑟 23 = −1.
This is because an ideal mirror would cause a null in the electric field at the surface
of the mirror, and the reflected wave would be opposite in sign (or equivalently for
an ideal metallic mirror, 𝜂 → ∞ and 𝜅 → ∞). Thus, if the exciton was placed
exactly at 𝑑 = 𝜆/4, our reflectivity expression would be modified to
𝑟=

1 − 𝑍0 𝜎
1 + 𝑍0 𝜎

(3.11)

where the absorbance, 𝐴 = 1 − |𝑟 | 2 (note that 𝑇 = 0 since a perfect metal forces
𝑡23 = 0), is therefore maximized when 𝑍0 𝜎 = 1. In other words, when the surface
impedance of the exciton is matched with that of the free-space impedance, the
absorption is unity. This condition is most easily achieved when 𝜔 = 𝜔0 , i.e., the
impedance is highest at the excitonic resonance, and therefore the condition for unity
absorbance is given as 𝜎(𝜔0 )𝑍0 = 1 = 2𝛾𝑟 /𝛾𝑛𝑟 . Thus, perfect absorption in this case
requires 2𝛾𝑟 = 𝛾𝑛𝑟 . Notice for a fixed 𝛾𝑟 , the mirror has now reduced the required
𝛾𝑛𝑟 to achieve this critical coupling condition by a factor of two6. Moreover, the
absorbance can now reach a value of 100%, and this is from removing the superfluous
port with the back mirror.
3.4

Monolayer Near Unity Absorption at Room Temperature

The above derivation suggests a straightforward method of achieving unity absorbance, with the caveat that we can satisfy the expression 𝛾𝑛𝑟 = 2𝛾𝑟 . Table 3.1
shows the extracted values for the radiative and non-radiative rates of the exciton
from the monolayer dielectric function data shown in [104]. It is clear that at typical
room temperature conditions, the expression above is far from ever being satisfied.

6 c.f. section B.5 where we required 𝛾

𝑟 = 𝛾 𝑛𝑟 to achieve 50% absorbance.

69
Table 3.1: Room temperature values for the excitonic sheet
conductor model of various TMDCs.
ℏ𝛾𝑟 (meV)
ℏ𝛾𝑛𝑟 (meV)
ℏ𝜔0 (eV)

WS2

MoS2

WSe2

MoSe2

4.11
38.6
2.01

3.35
76.2
1.87

1.47
53.2
1.65

2.38
72.7
1.55

Experimentally, there are a few well known methods of tuning 𝛾𝑛𝑟 in TMDCs,
including via temperature, strain, charge concentration, and/or van der Waals encapsulation. In contrast, 𝛾𝑟 is typically set by the optical transition rates of the
excitonic absorption, and is much less sensitive compared to 𝛾𝑛𝑟 . Epstein and others
demonstrated that with hBN encapsulation and cryogenic temperatures (∼100K),
near-unity absorption can be achieved in WS2 [51]. However, for any practical application (such as a photovoltaic device), this must be achieved at room temperature.
To do so, we examine the more general analysis of an exciton in an arbitrary photonic
structure7, whose excitonic absorbance approaches unity when
𝛾𝑛𝑟 = 2

|𝐸 (𝜔0 , 𝑥0 , 𝑦 0 , 𝑧0 )| 2
|𝐸 0 | 2

(3.12)

𝛾𝑟

For the Salisbury screen geometry (described above) where we simply have a single
mirror on one side, the electric field intensity at the surface is exactly that of the
incident wave, i.e., |𝐸 (𝜔0 ,𝑥0 ,𝑦20 ,𝑧0 )| = 1. In general, to achieve the absorption at more
|𝐸 0 |
reasonable non-radiative rates, the electric field at the exciton must be significantly
enhanced.
To do so, we consider a more complex layered structure which has a metallic
back mirror and a partially transmissive/partially reflective top mirror, which is the
geometry of an optical cavity. In this situation, it is possible to build up an electric
field intensity that is far above the incident electric field, due to the formation
of standing waves with significant quality factors. A simple optimized geometry
using a thin piece of silver as the top mirror is shown in Figure 3.7, where roughly
85% absorbance is achieved in the exciton. Note also the electric field intensity
is substantially higher than in the Salisbury screen geometry, and suggests that
𝐹𝑎𝑏𝑠 ≈ 10, which enables near-unity absorption at room temperature conditions.

|𝐸 ( 𝜔0 ,𝑥0 ,𝑦0 ,𝑧0 ) |
7 See section B.5 for derivation, where we also define 𝐹
𝑎𝑏𝑠 = 2
|𝐸 | 2

70

hBN

hBN

Ag
WS2

Ag

Figure 3.7: Metallic Optical Cavity for Monolayer Perfect Absorption. a Calculated absorption spectrum of monolayer WS2 in an metallic optical cavity and its
associated b electric field profile, showing the substantial increase in the electric
field intensity. Roughly 85% absorbance in the exciton is achieved in this optical
geometry

3.5

Experimental Demonstration of Near Unity Absorption in Monolayer
WS2 at Room Temperature

Due to experimental challenges with fabricating ultrathin low-loss metallic mirrors
(such as the one considered in Figure 3.7), we consider a top mirror that is composed
of lossless dielectric elements. The simplest form of a dielectric mirror is that of a
distributed Bragg reflector (DBR), where alternating high and low refractive indices
form a unit cell, and these unit cells are cascaded serially to increase the interference
effects. As the number of unit cells increases, the reflectance can approach unity.
However, as suggested by the scenario with a thin metal, the actual reflectance of
the top mirror is likely small, and could be achieved in a low number of unit cells.
It is also important to note that by employing materials with large index contrasts
for the high and low index materials, it may be possible to use a lower number of
unit cells, which could drastically lower the fabrication of these unity absorbing
structures. Therefore, we examine the use of GaS (𝑛𝑟 ≈ 2.7) and Mica (𝑛𝑟 ≈ 1.55)
as the materials of choice for a top dielectric mirror. These materials also happen
to be layered materials, can be exfoliated and stacked like the TMDCs considered
thus far. Figure 3.8 shows the results of using a top dielectric mirror composed of
GaS and Mica, where >90% absorbance can be achieved in the TMDC layer. Also
shown is a significant increase in the electric field intensity, as needed.

71

Figure 3.8: Dielectric-Metal Optical Cavity for Monolayer Perfect Absorption. a Calculated absorption spectrum of a heterostructure stack consisting of 58
nm GaS/99 nm Mica/Monolayer WS2 /77 nm Mica/Ag. The parameters for WS2
assumed a sheet conductor model with values from Table 3.1. The parasitic absorption in the silver is less than 5%. The refractive index of GaS and Mica was assumed
to be 𝑛GaS = 2.7 and 𝑛Mica = 1.55. b Electric field profile at the exciton frequency,
with the different shaded regions corresponding to different materials (aqua = GaS,
green = Mica, orange = WS2 , and grey = silver).
The results shown in Figure 3.8 is particularly promising since both parasitic loss is
minimized and the absorption efficiency is achieved with room temperature parameters. To more carefully understand the potential temperature dependence and effects
of linewidth on the excitonic absorption, we calculate the peak excitonic absorption
as a function of different non-radiative rates (Figure 3.9). The radiative rate is fixed
to that from Table 3.1. We also perform a similar calculation assuming the Salisbury
Screen geometry, as studied earlier. We note two important distinctions between
the two optical geometries: (1) Unity absorbance can be achieved for much larger
non-radiative rates 𝛾𝑛𝑟 . For the cavity geometry, this is achieved at approximately
𝛾𝑛𝑟 = 23 meV, compared to roughly 8 meV for the Salisbury screen geometry. (2)
The optical cavity geometry is significantly more tolerant to small perturbations of
the non-radiative rate. In other words, for the cavity geometry, we can achieve >90%
excitonic absorbance for any 𝛾𝑛𝑟 between roughly 14 to 44 meV, while the Salisbury
screen geometry achieves >90% absorbance for roughly 4 to 15 meV. We further
analyze the linewidth dependence by fitting the resultant absorption spectrum to a
Lorentzian with total linewidth 𝛾𝑇 as we vary 𝛾𝑛𝑟 . This is useful to compare to
experimental spectra, where we typically measure the total linewidth 𝛾𝑇 rather than
any individual component. Here, it is even more evident that the optical cavity

72
geometry is significantly more tolerant to small perturbations. These linewidths are
readily achievable even at room temperature (Figure 3.9).

Figure 3.9: Relationship between Absorption and Excitonic Linewidth. a Calculated peak excitonic absorbance for WS2 in an optical cavity geometry (Figure 3.8)
and a Salisbury Screen geometry (where we remove the top mirror but keep the
same thickness for the Mica). b Same as in a, except we plot the total linewidth 𝛾𝑇 ,
which is extracted from fitting the resultant absorption spectrum for each 𝛾𝑛𝑟 to a
Lorentzian with total linewidth 𝛾𝑇 . Generally, we have 𝛾𝑇 ≈ 𝐹𝑃,𝐴𝑏𝑠 𝛾𝑟 + 𝛾𝑛𝑟 .
We next examine the effects of tuning the geometric parameters of the optical cavity
on the final excitonic absorption (Figure 3.10). We analyze the effects of varying
one of the parameters at a given time, assuming the others are at an optimal value
(again, excitonic parameters given by Table 3.1). For every parameter, we find that
we generally have a tolerance of ± ∼4 nm to achieving >90% absorbance. While
this tolerance may seem experimentally daunting, it is possible to use a variety of
methods to ensure the appropriate thickness (e.g. atomic force microscopy). We
also develop a method of using reflection contrast spectroscopy itself as a method
to accurately assert the thickness of a given dielectric layer, which is akin to the
methods used in ellipsometry Figure 3.12.
Using the analysis described above and this optical cavity geometry, we are able to experimentally demonstrated near-unity absorbance in monolayer WS2 (Figure 3.11).
Instead of using a typical silver substrate, we cap the silver with a thin layer of
SiO2 , which dramatically reduces the rate at which silver tarnishes (usually to form
silver sulfide). In this case, the SiO2 can also act as a dielectric spacer. Our anal-

73

Figure 3.10: Geometric tolerance of monolayer perfect absorption optical cavities. Calculated peak excitonic absorbance where the other parameters are fixed
but the a Top GaS thickness is varied b Top Mica thickness is varied and c Bottom
Mica thickness is varied.
ysis where we encapsulate the WS2 on both sides with Mica is still essentially the
same, except the finite SiO2 will reduce our thickness for the bottom spacer. We
therefore using conventional van der Waals heterostructure fabrication techniques
(specifically, PDMS exfoliation and subsequent layered stacking), to form a van der
Waals heterostructure (Figure 3.11a). Here, ever layer was subsequently analyzed
in both reflection and PL spectroscopy as the heterostructure was assembled, and
we performed one vacuum anneal once the heterostructure was formed. We note
that the heterostructure formed in this way results in a variety of optical geometries
that have gone through the same amount of processing and uses the same crystalline
flakes. Therefore, the properties between the different heterostructures can be quantitatively compared (Figure 3.11b). We analyze the absorption and emission of the
WS2 in a cavity geometry, i.e., similar to what we have discussed, and that without
the top dielectric mirror. We observe the not only is the emission enhanced in the
cavity geometry relative to the Salisbury screen geometry, the absorption at the
exciation wavelength (𝜆 = 532 nm) is decreased in the cavity geometry. Therefore,
the luminescence yield has improved by more than a factor 6 due to the presence
of this optical cavity. This factor of 6 is already comparing that of the Salisbury
screen geometry, suggesting that the luminescence quantum yield has substantially
increased. Our analytic analysis suggests that, if we were to achieve near-unity
absorption, the quantum yield would be approximately 50%.
We next turn our attention to reflection spectroscopy to examine the absorbance
properties in this sample Figure 3.11c. We perform reflectance mapping mea-

74

7.5 nm SiO2/Ag

74 nm GaS

66 nm GaS
Cavity
Geometry

1L WS2
Salisbury
Screen
60 nm Mica

Figure 3.11: Experimental Demonstration of Near-Unity Absorption in WS2 at
Room Temperature. a Optical micrograph of fabricated heterostructure, with different flakes and their corresponding thicknesses overlaid (thicknesses are measured
with AFM). In this configuration, we use SiO2 as a capping layer to Ag to reduce
the ambient tarnishing. b Measured enhancement of the photoluminescence in the
optical geometry (66 nm GaS/92 nm Mica/1L WS2 /60 nm Mica/7.5 nm SiO2/Ag)
compared to that in the Salisbury screen geometry (1L WS2 /60 nm Mica/7.5 nm
SiO2/Ag). c Experimental microreflectance measurements as a function of position
over the region of interest (see a). At each pixel, a reflection contrast spectrum 𝑅/𝑅0
is measured. Here 𝑅0 is the reflectance on the substrate. It is clear where there is
minimal reflectance, i.e., maximum absorbance. d Spectral response of a monolayer
WS2 within an optical cavity, achieving roughly 90% absorbance experimentally.
surements, where the reflection spectrum 𝑅 is taken at every spot, and therefore
Figure 3.11c represents a linecut of a hyperspectral image. We find that near the
excitonic frequency, there is minimal reflectance where the full heterostructure is
formed. Equivalently, with appropriate normalized of the substrate reflectance and
that of the sample, we find that we indeed achieve near-unity absorbance in WS2
in this dielectric-metal optical cavity Figure 3.11d. It is likely possible to achieve
absorbances that are even slightly higher, since the thicknesses observed here are

75
slightly offset from the optimal values. However, the fact that we do observe nearunity absorbance experimentally demonstrates the tolerance of this optical cavity
design to small perturbations. Moreover, given this simple optical geometry, it
should be possible to directly integrate electrical contacts to these samples to create
a fully functioning photovoltaic device.
3.6

Efficiency Limits of Excitonic Multijunctions

Having designed structures that have the possibility of achieving unity absorption,
We now consider the prospect of using excitonic van der Waals materials as active
layer absorbers in a multijunction solar cell and consider their efficiency potential.
While the absorbance of each individual van der Waals layer is typically far below unity (typically between 5 - 20%), the wide array of different van der Waals
materials along with their lattice-mismatch free hetero-structuring suggests that a
‘metamaterial’ with near-unity absorbance over the solar spectrum can be theoretically engineered. Assuming carrier selective contacts and negligible resistive losses
between the subcells, theoretical efficiencies of these van der Waals multijunctions
can far exceed the single-junction Shockley-Queisser limit.
To consider the maximum power efficiency potential of a van der Waals heterostructure, we consider a stack with 𝑁 layers and consider the 𝐼 − 𝑉 characteristics of the
𝑖th layer. We assume we are in the radiative limit (i.e., the materials have an internal
luminescence yield of 100%) and that the materials are spaced far enough away
from one another that their luminescent coupling and individual absorbances can be
described with ray optics. Further, we assume that anti-reflection coatings have been
applied between the different layers of the van der Waals materials, so that there is no
reflection when the light interacts with the layered material,i.e., 𝐴(𝐸) = 1 − 𝑇 (𝐸).
Further, we approximate the absorbance of the van der Waals materials as being
dominated by a narrow absorbance peak (e.g., an excitonic absorber), which we
parameterize as a Gaussian:
(𝐸 − 𝐸 𝑒𝑥𝑐 ) 2
(3.13)
𝐴(𝐸) = 𝐴𝑒𝑥𝑐 exp −
2𝜎𝑒𝑥𝑐
Let’s consider first the current density of the 𝑖th layer. Here, we enumerate 𝑖 = 1 as
the first cell (top cell) and the 𝑁th cell as the bottom cell. The current for the top
cell can be written as
𝐽𝑠𝑐,𝑖 =
𝐴𝑖 (𝐸) 𝑆 𝑠𝑢𝑛,𝑡𝑜 𝑝 (𝐸) + 𝑆 𝑃𝐿,𝑡𝑜 𝑝 (𝐸, 𝑉𝑘 ) + 𝑆 𝑃𝐿,𝑏𝑜𝑡 (𝐸, 𝑉𝑘 ) 𝑑𝐸
(3.14)

76

Figure 3.12: Schematic of Reflectance Measurement Set-up. a Schematic depiction
of reflectance measurement set-up used throughout the analysis with monolayer absorbers.
Here, a stabilized white light source (ThorLabs SLS201L) is fiber-coupled into an inverted
microscope (Leica). A 90:10 beamsplitter is used to simultaneously image the sample
and the light source. A flip mirror is used that directs the reflected light to either the
camera (used for imaging) or to a spectrometer (Princeton). The observed spectra at the
output of the spectrometer is a convolution of many different factors, including the light
source spectral flux, reflectance of the actual sample, and the wavelength-dependent optical
efficiencies of the entire set-up. To remove all these effects besides the reflectance of the
sample, we measure a reference spectrum 𝑅0 nearby the sample under the same conditions
as 𝑅. Therefore, the reflectance contrast spectrum 𝑅/𝑅0 has normalized away the effects
of the set-up optical efficiency and lamp spectra. Here, we use the substrate (either Ag or
SiO2 /Ag) as a featureless reference spectrum. b Achieved spatial resolution of the optical
set-up described in a, using a ‘knife-edge’ measurement of the reflectance to quantify a spot
size. The spatial resolution roughly 1 𝜇m and the spot diameter is roughly 2 𝜇m. c Example
reflectance contrast spectra 𝑅/𝑅0 and fitting procedure used to extract out the thickness of
the dielectric layer. In this case, we examined hBN exfoliated on SiO2 /Si. We are able
to achieve precise fits down to a nm or so in resolution, although this accuracy is partially
dictated by the index of the material and the optical properties of the substrate.

where 𝑆 𝑠𝑢𝑛,𝑡𝑜 𝑝 (𝐸) is the flux from the sunlight that is transmitted through the 𝑖 − 1
number of layers, 𝑆 𝑃𝐿,𝑡𝑜 𝑝 (𝐸, 𝑉𝑘 ) is the luminescent flux from the 𝑘 = 1 to the
𝑘 = 𝑖 − 1 van der Waals layers, dependent on the electrical voltage 𝑉𝑘≠𝑖 , and

77

Figure 3.13: Limiting Efficiency of Excitonic Multijunctions. Maximum power
conversion efficiency as a function of the number of excitonic absorbers. Series
constrained refers to a two-terminal device where the current density between every
absorber must be matched, whereas unconstrained has 2𝑁 𝑒𝑥𝑐 number of terminals
and each absorber can have an arbitrary current-voltage curve. Also shown is
the efficiency maximum optimization for a step-function absorber response, which
is what is traditionally considered. The single junction limit is 33.7% and the
multijunction limit is 68%.
similarly for 𝑆 𝑃𝐿,𝑏𝑜𝑡 (𝐸, 𝑉𝑘 ) for 𝑘 = 𝑖 + 1 to 𝑘 = 𝑁. Explicitly, we have:
𝑗=𝑖−1

©Ö
𝑇 𝑗 (𝐸) ®
𝑆 𝑠𝑢𝑛,𝑡𝑜 𝑝 (𝐸) = 𝑆 𝑠𝑢𝑛 (𝐸) ­
(3.15)
« 𝑗=1
𝑗=𝑖−1
𝑘=𝑖−1
©Ö
𝑆 𝑃𝐿,𝑡𝑜 𝑝 (𝐸, 𝑉𝑘 ) =
𝑆 𝑃𝐿,𝑘 (𝐸, 𝑉𝑘 ) ­
𝑇 𝑗 (𝐸) ®
(3.16)
𝑘=1
𝑗=𝑘+1
𝑗=𝑘−1
𝑘=𝑁
©Ö
𝑆 𝑃𝐿,𝑏𝑜𝑡 (𝐸, 𝑉𝑘 ) =
𝑆 𝑃𝐿,𝑘 (𝐸, 𝑉𝑘 ) ­
𝑇 𝑗 (𝐸) ®
(3.17)
𝑘=𝑖+1
« 𝑗=𝑖+1
Î 𝑗=𝑖−1
Î 𝑗=𝑖
where we take 𝑗=𝑖 𝑇 𝑗 (𝐸) = 1 and 𝑗=𝑖+1 𝑇 𝑗 (𝐸) = 1 for the products, which
corresponds to unity coupling for the nearest neighbor. The luminescent flux of
each absorber assuming quasi-equilibrium of electrons and holes with the photon
gas is given by the usual reciprocity relation:
𝑆 𝑃𝐿,𝑘 (𝐸, 𝑉𝑘 ) = (exp(𝑞𝑉𝑘 /𝑘𝑇) − 1) 𝐴 𝑘 (𝐸)𝑆 𝐵𝐵 (𝐸)

(3.18)

78
The net current extracted from each cell is then given by
𝐽𝑖 = 𝐽𝑠𝑐,𝑖 (𝑉𝑘≠𝑖 ) − 𝐽𝑟𝑎𝑑 (𝑉𝑖 ) = 𝐽𝑖 (𝑉1 , 𝑉2 , · · · , 𝑉𝑖 , · · · , 𝑉𝑁−1 , 𝑉𝑁 )

(3.19)

where we note that 𝐽𝑖 is dependent on the voltages of the other layers due to
luminescent coupling, and the radiative recombination current is given by the usual
detailed balance expression:
𝐽𝑟𝑎𝑑 (𝑉𝑖 ) = (exp(𝑞𝑉𝑖 /𝑘𝑇) − 1)
𝐴𝑖 (𝐸)𝑆 𝐵𝐵 (𝐸)𝑑𝐸
(3.20)
Therefore, the net power from each cell is given by 𝑃𝑖 = 𝐽𝑖𝑉𝑖 and the total power for
the entire multijunction is
𝑃𝑡𝑜𝑡𝑎𝑙 =
𝑃𝑖
(3.21)

For a given set of 𝑁 absorbers defined by their absorptance 𝐴𝑖 (𝐸), we can calculate
the power of each subcell 𝑃𝑖 using the above expressions. The transmittance is given
as 𝑇 𝑗 (𝐸) = 1 − 𝐴 𝑗 (𝐸) and the voltage of each subcell 𝑉𝑖 is optimized to yield the
maximum power of the multijunction solar cell. We run an optimization to yield
the limiting efficiency as a function of the number of subcells in Figure 3.13. We
find that once we have approximately 6 or more excitonic absorbers, we can readily
surpass the single-junction detailed balance limit. The absorption spectra for a given
number of absorbers is shown in Figure 3.14. Interestingly, we find that even under
a series constraint (i.e., the 𝐽𝑠𝑐 must be matched between the different absorbers),
the limiting efficiency can still be very appreciable. By examining Figure 3.14, we
see that this is due to the linewidth of the exciton acting as another tuning knob that
can modulate the integrated absorption.

79

Figure 3.14: Optimal Absorption Spectra of Excitonic Multijunctions. Optimized absorption spectra of a unconstrained and b series-constrained excitonic
multijunctions.

Part II
Traversing through Flatland

80

81
Chapter 4

HIGH PHOTOVOLTAIC QUANTUM EFFICIENCY IN
ULTRATHIN VAN DER WAALS HETEROSTRUCTURES
“You can dream, create, design, and build the most
wonderful place in the world... but it requires
people to make the dream reality.”
— Walt Disney

4.1

Introduction

Owing to their naturally passivated basal planes and strong light-matter interactions,
transition metal dichalcogenides are of considerable interest as active elements of
optoelectronic devices such as light-emitting devices, photodetectors and photovoltaics. [14, 151] Ultrathin transition metal dichalcogenide (TMD) photovoltaic devices a few atomic layers in thickness have been realized using TMDs such as molybdenum disulfide (MoS2 ) and tungsten diselenide (WSe2 ). [11, 19, 54, 101, 155, 219]
Complete absorption of the solar spectrum is a challenge as the thickness is reduced
to the ultrathin limit, [8, 50, 73] whereas efficient carrier collection is challenging in
thicker bulk TMD crystals. The active layers in conventional photovoltaics typically
range from a few microns in direct gap materials (gallium arsenide) to a hundred
microns thick or more in indirect gap materials (silicon). [27]
Efficient ultrathin and ultralight (<100 g/m2 ) photovoltaics have long been sought
for many applications where weight and flexibility are important design considerations, such as applications in space power systems, internet-of-things devices,
as well as portable and flexible electronics. [59, 107, 202] Conventional photovoltaic materials are mechanically fragile when thinned down to the ultrathin (<
10 nm) regime, and interfacial reactions mean that a large fraction of the crystal
consists of surface-modified regions rather than intrinsically bulk material. Surface
oxides and dangling bonds in ultrathin films often result in increased nonradiative
recombination losses, lowering photovoltaic efficiencies. By contrast, transition
metal dichalcogenides have intrinsically high absorption and their layered crystallographic structures suggest the possibility of achieving intrinsically passive basal
planes in high quality crystals.

82
(b)

(a)

vs.

Metal
Metal

(c)

(d)

Top Contact

TMD

WSe2

WSe2
Ef

Au

vs.

vs.
MoS2

Transparent Top Contact

TMD

Figure 4.1: Achieving High External Quantum Efficiency in van der Waals heterostructures. a A schematic of the van der Waals device stack where nanophotonic
light trapping combined with efficient exciton dissociation and carrier collection
yields EQEs >50%. b A schematic of comparing near-unity absorption in a single
semiconducting layer on metal with a heterostructure of different semiconductors
on metal. c A schematic of comparing a pn heterojunction with a Schottky junction
for exciton dissociation and charge carrier separation in van der Waals materials. d
A schematic of comparing vertical and lateral carrier collection schemes in van der
Waals materials.
Photovoltaics that can approach the Shockley-Queisser limit, [154, 175] have two
prerequisites: first, that at open circuit, every above-bandgap photon that is absorbed
is extracted as an emitted photon at the band-edge of the material, i.e., it has perfect
external radiative efficiency. [125] Amani et al. have recently demonstrated that
superacid-treated monolayers of MoS2 and WS2 exhibit internal radiative efficiency
> 99%, [2] suggesting that the condition of very high external radiative efficiency
might be satisfied in transition metal dichalcogenides. The second prerequisite is
that at short circuit, the photovoltaic device must convert every incident abovebandgap photon into an extracted electron, i.e., it has external quantum efficiency
(EQE) approaching unity.
To understand the path to high EQE, we can deconvolute the external quantum

83
efficiency into the product of two terms: the absorbance and internal quantum efficiency (IQE). High EQE devices exhibit both high absorption and internal quantum
efficiency, i.e., carrier generation and collection efficiency per absorbed photon.
To date, reports of van der Waals based photovoltaic devices have not considered
both of these concepts and separately evaluated them as criteria for high efficiency
photovoltaics.
Coupling electromagnetic simulations with absorption and EQE measurements enables quantitative characterization of few-atomic-layer thickness optoelectronic devices in van der Waals heterostructures. In this paper, we demonstrate external
quantum efficiencies > 50% (Figure 4.1a) , indicating that van der Waals heterostructures have considerable potential for efficient photovoltaics. We show that
high EQE results from both high optical absorption and efficient electronic charge
carrier collection. We analyze the optical response using electromagnetic simulations to explain how near-unity absorption can be achieved in heterostructures
(Figure 4.1b). We find that experimental absorption results for van der Waals heterostructures match well with these electromagnetic simulations. Thus, we can
separate optical absorption and electronic transport to quantitatively compare their
effects on charge collection efficiency for both pn heterojunctions and Schottky
junctions (Figure 4.1c). In addition, we analyze the role of few-layer graphene as a
transparent top contact (Figure 4.1d). Finally, we outline important considerations
for designing high efficiency photovoltaic devices. By simultaneously maximizing
both external radiative efficiency and external quantum efficiency in a single device, van der Waals materials based photovoltaic devices could in principle achieve
efficiencies close to the Shockley-Queisser limit for their bandgaps.
4.2

Van der Waals Heterostructure Device Fabrication

Atomically smooth metal substrates were prepared using the template stripping
technique. [122, 198] We prepared the substrates using polished silicon wafers
(University Wafer) with native oxide and then cleaned the silicon substrates via
sonication in acetone (10 minutes) followed by sonication in isopropyl alcohol (10
minutes). Samples were then blow dried with nitrogen gas before cleaning with
oxygen plasma (5 minutes, 100 W, 300 mTorr under O2 flow).
Metal was then deposited via electron beam evaporation on the polished and cleaned
surface of the silicon wafer. For gold (Plasmaterials, 99.99% purity), base pressures
of ∼3e-7 was achieved before depositing at 0.3 Å/s. This continued until a thickness

84
of ∼20 nm was achieved. Then, the rate was slowly ramped to 1 Å/s and then held
there until a total thickness of 120 nm was reached. For silver (Plasmaterials, 99.99%
purity), following McPeak et al., [123] we deposited at a base pressure of ∼3e-7 at
40 Å/s for a final thickness of 150 nm. After deposition of the metal, an adhesive
handle was formed using a thermal epoxy (Epo-Tek 375, Epoxy Technology). 1
g of part A Epo-Tek 375 and 0.1 g of part B Epo-Tek 375 was mixed in a glass
vial and was let to settle for ∼30 min, Afterward, individual droplets of the mixture
was added directly onto the metallic surface before placing cleaned silicon chips
(∼ 1 cm2 ) on top. The droplet of epoxy was let to settle under the weight of the
silicon chip before placing on a hot plate (∼80 C) for 2 hours. Individual chips were
then cleaved with a razor blade, forming the final substrate consisting of atomically
smooth metal/themal epoxy/silicon. Typical RMS surface roughness of the metal
was < 0.3 nm using this technique (examined via AFM).
The bottom-most layer of the van der Waals heterostructure (e.g. MoS2 ) was
directly exfoliated onto the metallic susbtrates prepared using the above technique.
Exfoliation was performed using bulk crystals purchased from HQ Graphene using
Scotch tape. Subsequent layers were formed using a visco-elastic dry transfer
technique [22] using a home-built set-up at room temperature. Dry transfer was
performed using PF-20-X4 Gel Film from Gel-Pak as the transparent polymer. Van
der Waals materials were directly exfoliated onto the polymer using Nitto tape
and then mechanically transferred onto the MoS2 /metal substrate. Samples were
examined in the optical microscope during each layer of the process and an AFM
scan was performed afterwards to extract out the thicknesses of individual layers.
Thicknesses were then corroborated with optical measurements and calculations.
A top electrode was patterned using standard photolithography techniques. NR-9
1000 PY was used as a negative resist. The resist was spun at 5000 RPM for 55 s
before baking at 150 C for 1 minute. A mask aligner with a pre-patterned mask was
used to define the features and aligned on top of the van der Waals heterostructure.
After exposure for ∼18s under 10 mW of UV light (𝜆 = 365 nm), the resist was
post-baked at 105 C for 1 min and cooled to room temperature. Finally, the resist
was developed using RD-6 developer for 10-15 seconds before rinsing in deionized
water for 35 seconds. The sample was then blown dry with nitrogen and examined
under an optical microscope.
Electron beam deposition was then used to form the top ring electrodes (10 nm
Ti/90 nm Au). Base pressures of ∼3e-7 was achieved before the beginning of the

85
deposition. For titanium, a deposition rate of 0.3 Å/s was used for the entirety of 10
nm. Immediately afterwards, gold was deposited at a rate of 0.3 Å/s for 15 nm. The
rate was slowly ramped to 0.6 Å/s for 10 nm, and then to 0.9 Å/s for another 10 nm.
At 35 nm of total gold thickness, the rate was finally ramped to 1.0 Å/s until the
total gold thickness was 90 nm. The resist was then lifted-off using heated acetone
(40-45 C) for 30 minutes. If needed, the samples were sonicated in 5 s intervals in
acetone to remove the resist. The sample was then rinsed in isopropyl alcohol and
blow dried with nitrogen. Images of the three samples studied in this chapter are
shown in Figure 4.2.

Figure 4.2: High Photovoltaic Quantum Efficiency Sample Images. The heterostructure designs, optical images, composite reflection & photocurrent maps,
and the photocurrent maps for all the samples analyzed in this paper. The outlines
in the optical images correspond to specific materials with the appropriate thickness
and materials labeled (scale bar = 20 𝜇m). The composite reflection and photocurrent map is made by superimposing a reflection mode scan with a photocurrent scan
(𝜆 = 633 nm). The bright white regions in the composite image correspond to high
photocurrent.

86
4.3

Spatial Photocurrent Map and IV Measurements

Samples were contacted on the top electrode and bottom metallic substrate using
piezoelectric controlled probes (MiBots, Imina Technologies) with ∼3 𝜇m2 tip
diameter under a confocal microscope (Axio Imager 2 LSM 710, Zeiss) with a
long working distance objective (50x, NA = 0.55). Samples were first checked for
photoresponse using dark IV and white light illumination. Voltage sweeps were
performed using a Keithley 236 source measure unit and in-house written scripts.
High resolution spatial photocurrent maps were performed using the same confocal
microscope with an automated stage. The microscope was modified to measure
photocurrent maps. Near-diffraction limited laser light (∼6 𝜇m2 spot size) was
coupled in and focused to perform high resolution spatial photocurrent maps (<1
𝜇m lateral resolution), and power-dependent IV measurements were performed at
particular locations of the device using the spatial photocurrent maps. Illumination
power was modified using neutral density filters in the microscope and the incident
power was measured by a photodetector and cross-referenced with the EQE spectrum
of the measured device.
4.4

Spectral Response Measurements

Quantitative absorbance and external quantum efficiency measurements were performed using a home-built optical set-up. A supercontinuum laser (Fianium) was
coupled to a monochromater to provide monochromatic incident light. A series
of apertures and mirrors were used to collimate the beam before being focused on
the sample with a long working distance (NA = 0.55) 50x objective to provide a
small spot size (∼1 𝜇m lateral resolution). Importantly, the small spot sizes allow
us to probe individual regions on a particular sample. In addition, the relatively
angle-insensitive light trapping structure [80] used in this work along with a small
NA objective allowed us assume that the collected signal is close to the normal
incidence response. During all measurements, the light is first passed through a
chopper (∼103 Hz) and a small fraction was split into a photodetector connected to
a lock-in amplifier. The other beam-split light is used for probing the sample which
is eventually sent to a photodetector (for absorbance measurements) or the sample
itself is used as a photodetector (for external quantum efficiency measurements).
Thus, the sample or photodetector is connected to a second lock-in amplifier for
homodyne lock-in detection.
For absorption measurements, the reflected signal was collected by the same 50x objective and passed through a beam splitter before being collected by a photodetector.

87
The same spectral measurement was done with a calibrated silver mirror (Thorlabs)
in order to obtain the absolute reflection spectrum. In the absence of transmission,
the absorption is simply 𝐴𝑏𝑠(𝜆) = 1 − 𝑅(𝜆). Reflection from the objective itself
and other optical losses was subtracted as a background. As mentioned before, a
reference spectrum was collected using a small amount of beam-split light at the
same time as the sample, background, and mirror scans to account for power fluctuations in the laser beam between scans. As a second reference, the metallic back
substrate was measured during all absorption scans to check if the normalization
was accurate.
For external quantum efficiency measurements, the sample itself was used as a
photodetector. The top ring electrode and bottom metallic substrate was probed
using MiBots. Laser light was then focused on a particular spot and the current was
collected by the probes and sent through a lock-in amplifier for homodyne detection,
as in the reflection spectrum case. After measurement of the current signals from
the sample, another spectral scan was performed with the optical system in the same
configuration using a NIST calibrated photodetector (818-ST2-UV/DB, Newport).
Power fluctuations between scans were again accounted for by using a small amount
of beam-split light and sending it to a photodetector. The measured currents were
normalized to this photodetector’s current before being normalized to the calibrated
photoresponse to yield the absolute EQE. Despite the various steps of calibration
used for normalization, we still estimate measurement errors of 𝛿 𝐴𝑏𝑠/𝐴𝑏𝑠 ≈ 0.02
and 𝛿𝐸𝑄𝐸/𝐸𝑄𝐸 ≈ 0.05 stemming from the assumption of normal incidence for
both absorption and external quantum efficiency measurements while using a NA =
0.55 objective, fluctuations in the laser power during the measurement, and sample
contact stability. In addition, we have observed in our laser that there is relatively
little power for 𝜆 < 450 nm. Additionally, there is relatively high absorbance in the
50x objective for 𝜆 > 700 nm. Combined with the fact that the simulated parasitic
absorption accounts for a larger fraction of the total absorption for 𝜆 < 450 nm
and 𝜆 > 700 nm, the significantly noisier spectra in the active layer IQE at these
wavelengths can be attributed to the factors described above.
4.5

Electromagnetic Simulations and Error Estimation

Calculations were performed using the transfer matrix method (section B.3) with optical constants taken from literature for each of the transition metal dichalcogenides
(TMDs). [104] We assumed that for the TMD thicknesses analyzed in this paper,
their optical response can be represented by the bulk optical permittivities. Permit-

88
tivities of Ag and Au were taken from McPeak [123] and Olman [141], respectively.
The optical response of few-layer graphene was assumed to be like graphite, with
its dielectric constant taken from Djurisic. [44] Hexagonal Boron Nitride (hBN)
was assumed to be a lossless, non-dispersive dielectric in the visible with refractive
index of n = 2.2. [55]
Given that there is sample-to-sample variation of the dielectric constant, it is likely
that the literature values of the dielectric constant differ from the samples measured
here. This difference we estimate leads to absorption simulation errors of ∼5%. Assuming this is true, the estimated error for the active layer IQE can be approximated
as
s
2 
2
𝛿 𝐴𝑏𝑠 𝑝 2
𝛿𝐼𝑄𝐸
𝛿 𝐴𝑏𝑠
𝛿𝐸𝑄𝐸
(4.1)
𝐼𝑄𝐸
𝐴𝑏𝑠
𝐴𝑏𝑠 𝑝
𝐸𝑄𝐸
which is about 7%.
4.6

Prototypical Optoelectronic Device Characterization

We analyzed the optoelectronic device characteristics of a high-performance device
consisting of a vertical van der Waals heterostructure device of 0.6 nm thick fewlayer graphene (FLG)/9 nm WSe2 /3 nm MoS2 /Au (see Figure 4.2 for optical and
photocurrent images). Its optoelectronic and device characteristics are shown in
Figure 4.3. First, we find that this device exhibits an EQE > 50% (Figure 4.3a)
with absorbance greater than 90% from approximately 500 nm to 600 nm. Spectral
features such as the exciton resonances of MoS2 and WSe2 are well reproduced in the
external quantum efficiency spectrum. In addition, we observe a maximum singlewavelength power conversion efficiency (PCE) of 3.4% under 740 W/cm2 of 633 nm
laser illumination (Figure 4.3b). Since the high-performance device is electrically in
parallel with other devices, typical macroscopically large spot size (∼cm) AM 1.5G
illumination measurements would yield device characteristics substantially different
from the high-performing one. Thus, we estimated the AM 1.5G performance using
extracted device parameters of a diode fit under laser illumination (see section 4.10
for details). We estimate the AM 1.5G PCE of this device to be ∼0.4%. This value
is presently too low to be useful for photovoltaics, but the high EQE values reported
here indicate promise for high efficiency devices, when device engineering efforts
are able to also achieve correspondingly high open circuit voltages in van der Waals
based photovoltaics.
Further measurements were performed at different laser powers under 633 nm laser
illumination (Figure 4.4), yielding various power-dependent characteristics. Ex-

89

Figure 4.3: Optoelectronic Performance Characteristics. a Spectral characteristics of the experimentally measured absorbance (blue) and external quantum
efficiency (red). The vertical solid line indicates the excitation wavelength (633 nm)
for the measurements in b and Figure 3. The grey region indicates loss in photocurrent from the reflected photons. b I-V (light blue) and power-voltage (orange)
characteristics of the device, excited at 𝜆 = 633 nm with ∼45 𝜇W incident power
with a spot size area of ∼6 𝜇m2 . We observe a maximum single wavelength power
conversion efficiency of 3.4%. The yellow region indicates generated power from
the device.
amination of the short-circuit current 𝐼 𝑠𝑐 yielded nearly linear dependence on laser
power, as expected in ideal photovoltaic devices Figure 4.4a. The dashed blue line
represents the fit to the expression 𝐼 𝑠𝑐 = 𝐴𝑃 𝜏 , where 𝐴 is a constant of proportionality, 𝑃 is the incident power, and 𝜏 represents the degree of nonlinearity in this device
(𝜏 = 1 is the linear case). [149] We find that 𝜏 = 0.98 in our device, indicating nearly
linear behavior under short circuit conditions. In addition, in an ideal photovoltaic
device, the open circuit voltage is expected to grow logarithmically with the input

90
power, since 𝑉𝑜𝑐 = (𝑛𝑘 𝑏 𝑇/𝑞) ln(𝐽 𝐿 /𝐽𝑑𝑎𝑟 𝑘 + 1) ≈ (𝑛𝑘 𝑏 𝑇)/𝑞 ln(𝐽 𝐿 /𝐽𝑑𝑎𝑟 𝑘 ) for large
illumination current densities 𝐽 𝐿 . Here, 𝐽𝑑𝑎𝑟 𝑘 is the dark current density, 𝑛 is the
ideality factor, 𝑘 𝑏 is the Boltzmann constant, 𝑇 is the temperature of the device, and
𝑞 is the fundamental unit of charge, so that (𝑘 𝑏 𝑇)/𝑞 ≈ 0.0258𝑉 at room temperature. In Figure 4.4b we see that the experimental data match well with the diode fit
(dashed black line, see section 4.10 for fitting details), suggesting an ideality factor
of 𝑛 = 1.75 and a dark current density 𝐽𝑑𝑎𝑟 𝑘 = 0.65 mA/(cm2 ) assuming a 30 𝜇m
× 30 𝜇m device area. Also, since the power conversion efficiency (PCE) is given
as 𝑃𝐶𝐸 = 𝐽𝑠𝑐𝑉𝑜𝑐 𝐹𝐹/𝑃𝑖𝑛 , where 𝐽𝑠𝑐 is the short circuit current density, 𝑉𝑜𝑐 is the
open circuit voltage, 𝐹𝐹 is the fill fraction, and 𝑃𝑖𝑛 in the incident power density,
we would expect the power conversion efficiency to scale roughly logarithmically
as well. This is true for laser powers up to ∼740 W/cm2 (Figure 3 c). However,
for larger input power, the PCE decreases with increasing power. Such a drop in
PCE can be attributed to series resistances in the device, either at the contacts or
at the junction. This is corroborated by the match between the experimental data
(dots) and the fitted expression (dashed line), yielding the diode fitting parameters
in the lower right hand corner of the plot in Figure 4.4c. The fit for the 𝑉𝑜𝑐 was
simultaneously done with the PCE, therefore yielding the same set of parameters
and a good match between experiment and extracted device parameters. Finally,
we observed a decrease in the EQE at 633 nm with increasing power (Figure 4.4d).
Using the above fitted parameters, series resistance can only be used to partially explain a decrease in the EQE at higher powers. Thus, the additional decrease in EQE
at higher powers may be due to the onset of carrier density-dependent nonradiative
processes such as Auger or biexcitonic recombination which are not accounted for
in the diode fit used above, where the dark current is fixed for all powers.
4.7

Absorption in van der Waals heterostructures

We first investigate the absorption and optical properties of van der Waals heterostructures. We formed a heterostructure composed of hexagonal boron nitride
(hBN)/ FLG/WSe2 /MoS2 /Au. The composite heterostructure has various regions
(inset of Figure 4.5a), corresponding to different vertical heterostructures. Given
the sensitivity of the performance of van der Waals materials to different environmental conditions and device fabrication procedures,[170] the samples fabricated
here allow us to study optical and electronic features of different heterostructures
in a systematic manner by probing specific heterostructures fabricated on the same
monolithic substrate. This is enabled by the small spot size of our laser, which ad-

91

Figure 4.4: Power dependent device characteristics. Power dependent device
characteristics at 𝜆 = 633 nm excitation for the a short circuit current (light blue), b
open circuit voltage (black), c maximum power conversion efficiency (green), and d
external quantum efficiency (red). The dashed lines in a, b, and c correspond to fits.
The area of the spot size of the laser in all of the above measurements is estimated
to be ∼ 6 𝜇m2 .

ditionally allows us to properly normalize the spectral response without artificially
including geometric factors (see Methods for details).
As an example, consider the optical response at the location of the blue dot in
the inset of Figure 4.5a. The vertical heterostructure there is composed of 1.5 nm
FLG/4 nm WSe2 /5 nm MoS2 /Au. This location can be probed spectrally for its

92

Gold

90

90

70

70

50

30

10

50

30

(d)
FLG

1.5 nm FLG

90

4 nm WSe2

Au

5 nm MoS2
Au

WSe2

30

0.6 nm FLG

FLG

4 nm WSe2

WSe2
Absorbance (%)

Absorbance (%)

70

50

10

(b)
90

Silver

(c)

Absorbance (%)

Absorbance (%)

(a)

70

50

9.5 nm MoS2
Ag

MoS2

Ag

30

MoS2
10
400 450 500 550 600 650 700 750 800
Wavelength (nm)

10
400 450 500 550 600 650 700 750 800
Wavelength (nm)

Figure 4.5: Absorbance in van der Waals heterostructures. a Experimentally
measured absorbance of the 1.5 nm FLG/4 nm WSe2 /5 nm MoS2 /Au stack as a
function of wavelength. The inset is an optical micrograph of the fabricated van
der Waals heterostructure (scale bar = 20 𝜇m) with the blue dot corresponding to
the spot of spectral measurement. b The simulated absorbance of the structure
in a partitioned into the fraction of absorbance going into individual layers of the
heterostructure stack. The inset is a cross-sectional schematic of the simulated and
measured heterostructure. c Experimentally measured absorbance of the 0.6 nm
FLG/4 nm WSe2 /9.5 nm MoS2 /Ag stack as a function of wavelength. The inset
is an optical micrograph of the fabricated van der Waals heterostructure (scale bar
= 20 𝜇m) with the red dot corresponding to the spot of spectral measurement. d
Same as in b except the simulated absorbance is for the sample fabricated on silver
as shown in c.
absorption characteristics (Figure 4.5a), revealing near-unity absorption in van der
Waals heterostructures. The peaks at ∼610, ∼670, and ∼770 nm correspond to the
resonant excitation of the MoS2 B exciton, MoS2 A exciton, and WSe2 A exciton,
respectively.[104] On the other hand, the broad mode at ∼550 nm corresponds to

93
the photonic mode that leads to near-unity absorption.[80] Measurements of the
absorption can be corroborated with electromagnetic simulations, unveiling both
the accuracy between simulation and experimental results as well as the fraction
of photon flux absorbed into individual layers of the heterostructure stack (Figure 4.5b). Despite the near-unity absorption observed in the heterostructure stack,
there is parasitic absorption in both the underlying gold substrate and in the few-layer
graphene that accounts for 20% of the total absorbance. Such parasitic absorption
can be reduced by using a silver back reflector, as shown in Figure Figure 4.5c and
Figure 4.5d. We find that the simulated and measured absorbance is also in good
agreement for the case of a silver back reflector. Thus, the optical response of a
van der Waals heterostructure can be modelled accurately using full wave electromagnetic simulations and our method of measurement yields accurate and reliable
results.
To note, the subwavelength dimension of the total heterostructure thickness is critical for achieving near-unity absorption. Indeed, the entire stack can be treated
as a single effective medium, where small phase shifts are present between layers
and therefore the material discontinuities are effectively imperceptible to the incident light (see Supplementary Information S3 for details). Ultimately, the van der
Waals heterostructure-on-metal behaves as a single absorbing material with effective medium optical properties. Therefore, as previously demonstrated, near-unity
absorption at different wavelengths can be achieved for a semiconducting layer
with the appropriate thickness [80, 88] (∼ 10 − 15 nm total thickness for TMD
heterostructures).
4.8

Carrier collection efficiency in van der Waals semiconductor junctions

As discussed above, another criterion for high EQE is efficient carrier collection.
Given the large exciton binding energies in TMDs (∼ 50 − 100 meV in the bulk),
[92, 169] the large internal electric field at the semiconductor heterojunction may
play a role in exciton dissociation and subsequent carrier collection. Charge carrier
separation in TMDs can be accomplished using either a pn junction or a Schottky
junction, and we find that a pn heterojunction dramatically enhances the EQE when
compared with a Schottky junction. The heterostructure described in Figure 4.5a
and b can be probed as an optoelectronic device with the formation of a top electrode
(see inset of Figure 4.6). Since the back reflector (gold) can simultaneously serve
as a back contact to the entire vertical heterostructure, we can use this scheme to
compare the electronic performance of various vertical heterostructures. Given the

94
work function between WSe2 (p-type) and Au, it is expected that a Schottky junction
[130] will form between the two materials (See Figure 4.1c), whereas WSe2 (ptype) on top of MoS2 (n-type) is expected to form a pn heterojunction. [101] High
spatial resolution scanning photocurrent microscopy allows us to examine the two
heterostructure devices in detail (Figure 4.6a). We observe large photocurrent for
the pn heterojunction geometry (yellow region) compared to the Schottky junction
geometry (light blue region). The decrease of the photocurrent in the left-side of
the yellow region in Figure 4.6a is due to shadowing from the electrical probes. A
line cut of the spatial photocurrent map shown in Figure 4.6b provides a clearer
distinction between the two junctions, demonstrating 6x more photocurrent for the
pn junction relative to the Schottky junction.
The photocurrent density is directly related to the external quantum efficiency and
therefore the product of the absorbance and IQE. In order to quantitatively compare
the electronic differences between the two junctions, we need to normalize out the
different optical absorption in the two devices, i.e. compute the IQE of each device

𝐼𝑄𝐸 𝐸𝑥 𝑝 (𝜆) =

𝐸𝑄𝐸 (𝜆)
𝐴𝑏𝑠(𝜆)

(4.2)

where 𝐸𝑄𝐸 (𝜆) and 𝐴𝑏𝑠(𝜆) are the experimentally measured EQE and absorbance
of their respective devices (Figure 4.6c,i and Figure 4.6d,i). A plot of the experimentally derived IQE (i.e. 𝐼𝑄𝐸 𝐸𝑥 𝑝 ) is shown in purple in Figure 4.6c,ii
and Figure 4.6d,ii. This plot also confirms that a pn junction geometry (with
𝐼𝑄𝐸 𝐸𝑥 𝑝 ∼ 40%) formed of van der Waals materials is more efficient for carrier
collection than a Schottky junction geometry (with 𝐼𝑄𝐸 𝐸𝑥 𝑝 ∼ 10%).
Embedded in the above analysis is yet another convolution of the optical and electronic properties. As per Figure 4.5b, we found that absorption in FLG and Au
accounted for ∼20% of the absorbance of the total heterostructure. Assuming very
few photons absorbed in those layers ultimately are extracted as free carriers (i.e.
𝐼𝑄𝐸 𝐴𝑢 ≈ 𝐼𝑄𝐸 𝐹 𝐿𝐺 ≈ 0), the IQE defined above convolutes the parasitic optical loss
with the electronic loss in the device. [6] Thus another useful metric we shall define
is 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 , the active layer IQE:

𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 (𝜆) =

𝐸𝑄𝐸 (𝜆)
𝐴𝑏𝑠(𝜆) − 𝐴𝑏𝑠 𝑃 (𝜆)

(4.3)

95
where the additional term 𝐴𝑏𝑠 𝑃 (𝜆) corresponds to the parasitic absorption in the
other layers of the device that do not contribute to current (i.e., Au and FLG in this
device). Thus, 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 (𝜆) is a measure of the carrier generation and collection
efficiency only in the active layer (i.e., WSe2 and MoS2 ) of the device and is purely
an electronic efficiency as defined above. We shall use this quantity to accurately
compare electronic geometries. Given the good agreement between simulations and
experiment shown in Figure 4.5, a simple method of estimating the parasitic absorption described above is therefore through electromagnetic simulations. 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒
of the Schottky and pn heterojunction geometries calculated with Equation 4.3 is
shown in Figure 4.6c,ii and Figure 4.6d,ii with dotted green curves.
Effects of a pn heterojunction
Analysis of these plots reveals several important points. First, 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 for the
pn junction geometry is 3x higher than in the Schottky junction geometry when
spectrally averaged. Though yet to be fully clarified, we attribute higher IQE in pn
heterojunctions to the larger electric fields in a pn heterojunction that may lead to a
higher exciton dissociation efficiency and consequently IQE. Second, compared to
the IQE which included the parasitic absorption (purple dots in Figure 4.6c,ii and
Figure 4.6d,ii), the active layer IQE curves (green dots) are spectrally flat within
measurement error and calculations (𝛿𝐼𝑄𝐸/𝐼𝑄𝐸 ≈ 0.07). Thus, the few broad
peaks around the exciton energies of WSe2 (∼770 nm) and MoS2 (∼610 nm and
∼670 nm) in 𝐼𝑄𝐸 𝐸𝑥 𝑝 are not attributed to, e.g., resonant excitonic transport phenomena, but rather as a simple convolution of the optical and electronic effects when
calculating the electronic IQE. In other words, consideration of parasitic absorption
is critical when analyzing the electronic characteristics of thin optoelectronic devices. However, 𝐼𝑄𝐸 𝐸𝑥 𝑝 is still a useful metric, as it effectively sets a lower bound
on the true IQE. Generally, we expect 𝐼𝑄𝐸 𝐸𝑥 𝑝 ≤ 𝐼𝑄𝐸𝑇𝑟𝑢𝑒 ≤ 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 , as electromagnetic simulations tend to slightly overestimate the absorption when compared
with experimental results. Thus in this paper, we shall plot both expressions when
comparing different electronic device geometries. Finally, it is important to mention that an active layer IQE of 70% is achieved in van der Waals heterostructures
without complete optimization of the electronic configuration of the device, such as
the band profiles and the specific choice of contacts. With careful electronic design,
we suggest it may be possible to achieve active layer IQEs > 90%.

96
FLG + PN
FLG +
Schottky

λ = 633 nm

FLG +
Schottky

FLG + PN

1.5 nm FLG

1.5 nm FLG
4 9nm
WSe2
nm WSe2

4 nm WSe2

5 nm MoS2

Au

Au

Figure 4.6: Charge transport and collection in vertical PN and Schottky junction geometries. a Spatial photocurrent map of the fabricated van der Waals
heterostructure device using a 633 nm laser excitation. The inset is an optical image
of the device (scale bar = 20 𝜇m). b The line profile of the dotted red line arrow in a,
illustrating the different photocurrent intensities depending on the device geometry
(Schottky and pn junction). c (i) Experimentally measured spectral characteristics
of the absorbance (blue) and external quantum efficiency (red) in the 1.5 nm FLG/4
nm WSe2 /Au (Schottky geometry) device along with the (ii) experimentally derived
internal quantum efficiency (purple) and the calculated active layer internal quantum
efficiency (green). The inset is a cross-sectional schematic of the measured device,
at the orange dot in a. d Same as in c except with a 1.5 nm FLG/4 nm WSe2 /5
nm MoS2 /Au (pn geometry) device. The inset is a cross-sectional schematic of the
measured device, at the purple dot in a.

Optically transparent contacts for carrier extraction
As another aspect of analysis, we studied the role of vertical carrier collection
compared to lateral carrier collection in van der Waals heterostructures. Graphene
and its few-layer counterpart can form a transparent conducting contact allowing
for vertical carrier collection, in contrast to in-plane collection (see Figure 4.1d).

97

FLG + PN

PN

PN

0.6 nm FLG

0.6 nm FLG

9 nm WSe2

FLG + PN

9 nm WSe2

3 nm MoS2

5 nm MoS2

Au

Au

4 nm WSe2

4 nm WSe2

9.5 nm MoS2

9.5 nm MoS2

Ag

Ag

Figure 4.7: Few-layer graphene as a transparent top contact. a Cross-sectional
schematic of the two structures (with and without few layer graphene) compared
on gold and b silver back reflectors. c Experimentally derived internal quantum
efficiency (dots) and active layer internal quantum efficiency (dashed line) for a pn
junction geometry with (blue) and without (red) few layer graphene on gold. d is
the same as c except on silver, corresponding to the sample shown in b. e I-V curves
of a pn junction geometry with (blue) and without (red) few-layer graphene under
633 nm (∼180 𝜇W) laser illumination on a gold and f silver substrate. The shaded
yellow and orange regions correspond to where there is a net generated power in the
device.
The strong, in-plane covalent bonds of van der Waals materials suggest that in-plane
conduction may be favorable when contrasted with the weak out-of-plane van der
Waals interaction. However, the length scale for carrier transport in-plane (∼ 𝜇m) is
orders of magnitude larger than in the vertical direction (∼nm). Therefore, transport
in a regime in between these two limiting cases is not surprising.
Silver exhibits lower absorption in the visible than gold, suggesting it could be

98
an optimal back reflector for photovoltaic devices, as seen in Figure 4.5. Thus,
we contrast the case of in-plane and out-of-plane conduction concurrently with the
presence of two different back reflectors that simultaneously function as an electronic
back contact (gold vs. silver) to a pn heterojunction, as in Figure 4.7a and b. Optical
and photocurrent images of the devices are shown in Figure 4.2.
Our results in Figure 4.7c and d show the distinctions between the various contacting
schemes. In the case of both silver and gold, a transparent top contact such as fewlayer graphene seems to enhance the carrier collection efficiency. This is particularly
true in the case of silver, where 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒 enhancements of ∼5x is apparent. In
the case of gold, the IQE is enhanced by about ∼1.5× when parasitic absorption is
taken into account. By analyzing the work functions of gold (∼4.83 eV [3]) and
silver (∼4.26 eV [49]), along with the electron affinity of MoS2 (∼4.0 eV [72]), the
Schottky-Mott rule suggests in both cases that a Schottky barrier should form equal
to 𝜙 𝐵 = 𝜙 𝑀 − 𝜒, [10] where 𝜙 𝐵 is the Schottky barrier height, 𝜙 𝑀 is the work function
of the metal, and 𝜒 is the electron affinity of the semiconductor. However, several
reports [78, 79, 109] have indicated that gold appears to form an electrically Ohmic
contact to MoS2 , which we observe here. Conversely, the above data suggests that
silver and MoS2 follow the traditional Schottky-Mott rule, leading to the formation
of a small Schottky barrier of ∼0.26 eV. Given that the energy barrier is about 10𝑘 𝑏 𝑇,
very few electrons can be extracted out of the pn heterojunction when silver is used
as a back contact, leading to very low IQEs. By taking into account just the active
layer (dashed lines), we see that gold is ∼ 2× better as an electronic contact than
silver.
Finally, we examine the role of vertical carrier collection on the I-V characteristics
of the two devices (Figure 4.7e and f). In the case of gold, we see purely an
enhancement of the short circuit current with vertical carrier collection. On the
other hand, vertical carrier collection for silver drastically increases both the short
circuit current density and the open circuit voltage. This phenomenon is consistent
with the previously described nature of gold (Ohmic) and silver (Schottky) contacts.
Namely, on silver in the absence of a transparent top contact, due to both the
Schottky barrier and the large in-plane propagation distance, carriers are collected
with poor efficiency leading to a small 𝐼 𝑠𝑐 . Consequently, a high recombination
rate of the generated carriers which are inefficiently extracted leads to small 𝑉𝑜𝑐
values. On the other hand, even in the absence of a top transparent electrode, gold
enables efficient extraction of electrons from the pn heterojunction as an Ohmic

99
contact. Thus, the short circuit current and open-circuit voltage in gold are higher
compared to the silver back contact, even in the absence of a transparent electrode.
When introducing few-layer graphene as a transparent top contact, the propagation
distance is significantly reduced in the silver device and carriers can be extracted
with much higher efficiency, leading to a large enhancement of both the current and
voltage. Whereas for gold, the few-layer graphene enhances the already high carrier
collection (yielding larger 𝐼 𝑠𝑐 ) but only has a negligibly small enhancement effect
on the open-circuit voltage. Overall, these results demonstrate that vertical carrier
collection plays a crucial role in high photovoltaic device performance in van der
Waals heterostructures.

Figure 4.8: Absorbance and EQE of thick and thin PN heterojunctions. a Experimentally measured absorbance (blue) and EQE (red) of the thin pn heterojunction
(1.5 nm FLG/4 nm WSe2 /5 nm MoS2 /Au). b Same as in a except for a thick pn
heterojunction (11 nm hBN/1.5 nm FLG/4 nm WSe2 /9 nm MoS2 /Au).

Thickness dependence on charge collection efficiency
As a final point of analysis, we briefly examined the effect of thickness on 𝐼𝑄𝐸 𝐴𝑐𝑡𝑖𝑣𝑒
under vertical carrier collection. We compared the optoelectronic characteristics of
a thicker pn heterojunction (11 nm hBN/1.5 nm FLG/4 nm WSe2 /9 nm MoS2 /Au)
with a thinner pn heterojunction (1.5 nm FLG/4 nm WSe2 /5 nm MoS2 /Au). The
experimentally measured absorbance and EQE are plotted in Figure 4.8 for reference.
By normalizing out the differences in absorption between the pn junctions, we see
a somewhat surprising result when we analyze the active layer IQE (dashed lines,
Figure 4.9). In particular, despite the roughly 50% more length in active layer
thicknesses (13 nm vs. 9 nm) and qualitatively different absorbance and EQE

100

90

90

70

70

50

50

30

30

10

10
450

550

650

FLG + Thin PN IQE (%)

FLG + Thick PN IQE (%)

spectra, the thick pn junction exhibits nearly the same active layer IQE compared
to the thin pn junction. In fact, it appears to be slightly more efficient, but this
is within the error bar of the measurement and simulations (𝛿IQE/IQE0 ≈ .07,
see section 4.5 for details of errors). This observation is corroborated with the
experimentally derived IQE (dotted curves, Figure 4.9), which has nearly the same
spectrum between the two thicknesses, but differ in magnitude due to differences
in parasitic absorption. This result suggests that in the ultrathin limit (∼10 nm)
of van der Waals heterostructures with vertical carrier collection, the IQE has
a weak dependence on active layer thickness. This weak dependence may be
due to a combination of increased scattering competing with charge transfer, [26,
228] tunneling, [57, 101, 218] and exciton quenching [15, 20] effects as the vdW
heterostructure becomes thicker. The exact role of each of these effects, as well as
possibly other effects, will require a new theoretical framework and experimental
measurements to analyze their relative contributions to charge collection efficiency.

750

Wavelength (nm)

Figure 4.9: Thickness dependence on charge collection efficiency. The estimated
active layer (dashed lines) and experimentally derived (solid dots) internal quantum
efficiency of the thin pn junction device (1.5 nm FLG/4 nm WSe2 /5 nm MoS2 /Au,
green) and the thick pn junction device (11 nm hBN/1.5 nm FLG/4 nm WSe2 /9 nm
MoS2 /Au, purple).

4.9

High Photovoltaic Quantum Efficiency Outlook

Our results suggest important challenges that must be addressed to enable high photovoltaic efficiency. For example, despite the usefulness of gold as an electrical back

101
contact, we found from electromagnetic simulations that it accounts for nearly 20%
of the parasitic loss in the heterostructures reported here. Schemes using optically
transparent carrier selective contacts could be used to avoid this parasitic optical
loss. Another open question is the role and importance of exciton dissociation and
transport. Indeed, the large exciton binding energies in transition metal dichalcogenides (∼ 50−100 meV in the bulk) [92, 169] suggests that a significant exciton
population is generated immediately after illumination. However, it is not yet clear
whether such an exciton population fundamentally limits the internal quantum efficiency of the device, posing an upper limit on the maximum achievable EQE in van
der Waals materials based photovoltaic devices. Finally, the problem of open-circuit
voltage must also be addressed. For example, the type-II band alignment between
ultrathin MoS2 and WSe2 suggests a renormalized bandgap of ∼ 400− 500 meV,
[85] given by the minimum conduction band energy and maximum valence band
energy of the two materials. In accordance with the Shockley-Queisser limit, this
would severely reduce the maximum power conversion efficiency attainable by a
factor of ∼3. Therefore, to achieve higher open circuit voltages, a monolithic device
structure may be required to avoid low energy interlayer recombination states.
However, our results described here also suggest a different approach in addressing
the optical and electronic considerations for ultrathin van der Waals heterostructures when compared with conventional photovoltaic structures. For example, our
observation that ultrathin van der Waals heterostructures can be optically treated as
a single effective medium is a regime of optics that is uncommon for the visible to
near-infrared wavelengths analyzed in photovoltaic devices. Likewise, our observation of weak thickness dependence of the charge collection efficiency represents
a realm of electronic transport that is also quite unconventional and unexplored
when compared to traditional photovoltaic structures. Thus, the combination of the
above observations may enable entirely different photovoltaic device physics and
architectures moving forward.
To summarize, we have shown that external quantum efficiencies > 50% and active
layer internal quantum efficiencies > 70% are possible in vertical van der Waals
heterostructures. We experimentally demonstrated absorbance > 90% in van der
Waals heterostructures with good agreement to electromagnetic simulations. We
further used the active layer internal quantum efficiency to quantitatively compare
the electronic charge collection efficiencies of different device geometries made with
van der Waals materials. By further reducing parasitic optical losses and performing

102
a careful study on exciton dissociation and charge transport while simultaneously
engineering the band profiles and contacts, van der Waals photovoltaic devices may
be able to achieve external quantum efficiencies > 90%. Our results presented here
show a promising and exciting route to designing and achieving efficient ultrathin
photovoltaics composed of van der Waals heterostructures.
4.10

Appendix

Diode Equation Fitting
A diode model is commonly used to examine the characteristics of photovoltaic
devices. Here, we assume a single diode model with a series and shunt resistance
as a simple model to understand the photovoltaic device characteristics
𝑉 − 𝐼 𝑅𝑠
𝑞(𝑉 − 𝐼 𝑅𝑠
−1 +
𝐼 (𝑉) = 𝐼 𝑑𝑎𝑟 𝑘 exp
− 𝐼𝐿
(4.4)
𝑛𝑘 𝑏 𝑇
𝑅𝑠ℎ
where 𝐼 𝑑𝑎𝑟 𝑘 is the dark current, 𝑞 is the fundamental charge constant (1.602 × 10−19
C), 𝑛 is the ideality factor, 𝑘 𝑏 is the Boltzmann constant (1.38 × 10 ( − 23) J/K), 𝑇
is the thermodynamic temperature (300 K, for this case), 𝑅𝑠 is the series resistance,
𝑅𝑠ℎ is the shunt resistance, and 𝐼 𝐿 is the generated current from the photovoltaic
effect under illumination. Here, 𝑉 is the applied voltage and 𝐼 is the measured
current. At short circuit, 𝑉 = 0 and 𝐼 = 𝐼 𝑠𝑐 . Thus,
𝐼 𝑅𝑠
−𝑞𝐼 𝑅𝑠
− 𝐼 𝑠𝑐
(4.5)
𝐼 𝐿 = 𝐼 𝑑𝑎𝑟 𝑘 exp
−1 −
𝑛𝑘 𝑏 𝑇
𝑅𝑠ℎ
For the case 𝑅𝑠 = 0, we recover the usual expression 𝐼 𝐿 = −𝐼 𝑠𝑐 . We use the
above two expressions along with the measured short circuit current 𝐼 𝑠𝑐 to perform
a four parameter (𝑛, 𝐼 𝑑𝑎𝑟 𝑘 , 𝑅𝑠 , 𝑅𝑠ℎ ) fit to the open circuit voltage 𝑉𝑜𝑐 and the power
conversion efficiency 𝜂 = 𝑃𝑑𝑒𝑣𝑖𝑐𝑒 /𝑃𝑖𝑛𝑝𝑢𝑡 as a function of input power. Here, we have
explicitly measured the input power of the laser illumination. The fitted parameters
are listed in Figure 4.4b and Figure 4.4c in the main manuscript, and are used to
generate the dashed lines in those plots. Note that we use the same fitted parameters
for both data sets. It is also important to note that by fitting the parameters under
illumination at various powers, we expect the fitted parameters to represent primarily
the device characteristics that are probed by laser illumination, and not all the other
devices that are in parallel (which would be the case if we fitted to the dark IV).

103

Figure 4.10: a Estimated 1 Sun AM 1.5G power conversion efficiency of the device
measured in Figure 4.3 and Figure 4.4 as a function of estimated active area. The
blue line corresponds to a 30 × 30 𝜇m2 estimated active area used for the plot in b. b
The estimated J-V curve of the device studied in Figure 4.3 and Figure 4.4 in the dark
(black line) and under 1 Sun AM 1.5G illumination (blue) assuming a 30 × 30 𝜇m2
active area. Estimated device characteristics are in the bottom right-hand corner of
the plot.

Simulated AM1.5G and Effects of Active Area on Efficiency Estimation
To estimate the power conversion efficiency under AM 1.5G illumination for the
particular device, we use the expression:
∫ 800 𝑛𝑚
𝐼 𝑠𝑐 = −𝑞 𝐴

𝐸𝑄𝐸 𝑒𝑥 𝑝 (𝜆)𝑆 𝐴𝑀1.5𝐺 (𝜆)𝑑𝜆

(4.6)

400 𝑛𝑚

where 𝑞 is the fundamental charge constant (1.602 × 10−19 C), 𝐴 is the estimated
active area, 𝐸𝑄𝐸 𝑒𝑥 𝑝 is the experimentally measured EQE for the device, and 𝑆 𝐴𝑀1.5𝐺
is the solar photon flux (in units of photons m−2 s−1 nm−1 ). Using the above fitted
parameters and the calculated 𝐼 𝑠𝑐 , we can simulate the 𝐼 (𝑉) characteristics of the
device. We take 𝐽 (𝑉) = 𝐼 (𝑉)/𝐴 and calculate the power conversion efficiency 𝜂 as
𝜂= ∫∞

𝐽𝑚 𝑉𝑚
ℎ𝑐

(4.7)

𝑆 𝐴𝑀1.5𝐺 (𝜆)𝑑𝜆

where 𝐽𝑚 , 𝑉𝑚 is the current density and voltage at the maximum power point, respectively, and the denominator of the
above expression represents the total incident
∫ ∞  ℎ𝑐 
−2
power of solar irradiation (𝑆 = 0
𝜆 𝑆 𝐴𝑀1.5𝐺 (𝜆)𝑑𝜆 = 1000 W m ). We plot this

104
as a function of estimated active area 𝐴 in Figure 4.10a. Note that with increasing
estimated active area, we observe an increase in the power conversion efficiency.
Here, the active area effectively reduces the dark current density 𝐽𝑑𝑎𝑟 𝑘 = 𝐼 𝑑𝑎𝑟 𝑘 /𝐴
for increasing 𝐴, and therefore leads to a concentration-like effect on the power
conversion efficiency. Thus, there is a logarithmic dependence of 𝜂 on the active
area A and therefore 𝜂 varies weakly with 𝐴. Moreover, the above analysis for 𝐴 also
allows us to estimate the appropriate area for the simulated device performance, as
this is not the area under illumination, but rather the area from which dark current,
series resistance, and shunt resistance contribute to the total measured current (i.e.,
the total sample size). We estimate this area to be in the range of 202 − 402 𝜇m2
from the optical image (Figure 4.2) and plot the 𝐽 − 𝑉 characteristics assuming a
30 × 30 𝜇m2 active area below (Figure 4.10b). Typical photovoltaic figures of merit
are also shown. We achieve 𝐽𝑠𝑐 >8 mA/cm2 under 1 sun illumination. This value
depends only on the experimentally measured EQE and does not depend on any
fitting parameters, as evident in Equation 4.6. However, the expected 𝑉𝑜𝑐 and 𝐹𝐹
are sub-optimal, due to the type-II band alignment and high series resistance of
the device. Thus, despite having fairly large short circuit current densities, device
performance is limited primarily by the open circuit voltage and fill fraction, leading
to an overall predicted 𝜂 𝐴𝑀1.5𝐺 ≈ 0.4%.
The above analysis differs from the typical experimental scenario where we estimate
the input power as 𝑃𝑖𝑛𝑝𝑢𝑡 = 𝑆 𝐴, where 𝑆 =1000 W m−2 and 𝐴 is the illumination
area. Thus, the experimental efficiency is given as 𝜂 = 𝑃𝑚,𝑒𝑥 𝑝 /𝑃𝑖𝑛𝑝𝑢𝑡 , where
𝑃𝑚,𝑒𝑥 𝑝 is the maximum power of the experimentally measured device. In the
experimental case, 𝐴 is optimally the solar illumination area through some wellcalibrated aperture. [176] In this case, the power conversion efficiency is inversely
proportional to the estimated active area and therefore leads to larger 𝐽𝑠𝑐 and 𝜂 for
smaller 𝐴. This is a common source of error in estimating 𝜂 for small devices,
as |𝛿𝜂|/𝜂 = |𝛿 𝐴|/𝐴, with the error in efficiency 𝛿𝜂 depending linearly with the
error in active area estimation 𝛿 𝐴. Particularly for micron and nano-scale devices
such as in van der Waals materials, particular care must be taken to avoid errors in
measuring and calculating the power conversion efficiency, as discussed by Snaith
et al. in [176]. Here, we show a distribution of efficiencies based on our active area
estimation, leading to AM 1.5G power conversion efficiencies between 0.25% to
0.5%. For our above calculation methodology,
we can derive the error dependence
𝑛𝑘 𝑏 𝑇
to be roughly |𝛿𝜂|/𝜂 ≈ |𝛿 𝐴|/𝐴 𝑞𝑉𝑜𝑐 , where the extra factor of 𝑛𝑘 𝑏 𝑇/(𝑞𝑉𝑜𝑐 )

105
comes from the dependence of 𝜂 with an estimated 𝑉𝑜𝑐 , rather than 𝐽𝑠𝑐 . The low
values of absolute efficiency and logarithmic dependence on active area using our
calculation methodology imply a weak dependence of the error on estimated active
area, and thus suggests our calculated performance is a reasonable estimate for an
experimental AM 1.5G measurement.

106
Chapter 5

SPATIOTEMPORAL IMAGING OF THICKNESS-INDUCED
BAND BENDING JUNCTIONS
“Discovery consists in seeing what everyone else
has seen and thinking what no one else has
thought.”
— Albert Szent-Györgyi

5.1

Introduction

Band bending in semiconductors is a fundamental consequence of incomplete
screening of external fields and is critical to the operation of nearly every electronic and optoelectronic device. Its existence was theoretically proposed by the
works of Mott and Schottky [128, 171] who argued that the electrostatic landscape
must have electronic bands that “bend” to compensate the difference in Fermi levels
at an interface to minimize the overall free energy in the system. Mott also discovered that a characteristic length scale for the band bending in semiconductors is
given by
𝐿𝐷 =

𝜖 𝑆 𝜖0 𝑘 𝑏𝑇
𝑞 2 𝜌0

(5.1)

which is now commonly referred to as the Debye screening length, named after Peter
Debye who discovered the same length scale earlier in electrolytes. [41] 𝐿 𝐷 usually
ranges from 10s of nm to a few microns, depending on the doping concentration
𝜌0 and static dielectric constant 𝜖 𝑆 Here, 𝜖0 is the permittivity of free space, 𝑘 𝑏 𝑇 is
the thermal energy, and 𝑞 is the fundamental unit of charge. In most semiconductor
systems, the region where band bending occurs is referred to as the depletion or
accumulation region.
Layered van der Waals materials such as the semiconducting transition metal
dichalcogenides (TMDCs) are a unique system for studying band bending physics
because of their highly passivated surfaces and the ability to form a wide assortment
of heterostructures, which has enabled a variety of applications including transistors,
solar cells, optical modulators, metasurfaces, and lasers. [37, 79, 143, 157, 190, 207]
Furthermore, these materials can be easily cleaved to yield layers over a wide range

107
of thicknesses, ranging from a single monolayer to 100s of nm. The heterogeneity in sample thicknesses produced during mechanical exfoliation has led to novel
‘thickness’ junctions, i.e., junctions formed from the difference in quantum confinement in few-layer thick samples. [72, 185] Apparent thickness junctions have
also been formed with materials whose thicknesses are thicker than the quantum
confinement regime [146, 216], however, the mechanistic explanation for charge
carrier separation for this regime of thicknesses remains unclear.
Further, despite considerable research on layered materials in the atomically thin
limit in recent years, there has been to our knowledge no direct observation of ‘vertical’ band bending (i.e., in the direction perpendicular to heterostructure interfaces).
This is primarily due to the weak out of plane as opposed to in-plane transport in
layered materials and the difficulty to probe buried interfaces. Meanwhile, direct
observation of band bending can be used to estimate depletion widths, interface
barrier heights, and consequently be used to deduce the electrostatic landscape and
performance of the corresponding device.
In this chapter, we show evidence for the first direct measurement of vertical band
bending in a MoS2 -Au interface. We directly observe correlations between the thickness and surface Fermi levels in samples with identical electronic bandstructures
and preparation methods and find that the MoS2 -Au interface results in a strong
electron transfer to the MoS2 layer. The direct observation of a surface potential
difference between materials with differing thicknesses suggests that a new type of
homojunction, arising solely from the differences in thickness and band bending,
can be used to separate charge carriers. We directly observe this charge carrier
separation spatiotemporally by utilizing scanning ultrafast electron microscopy and
corroborate these observations with numerical simulations. The electrostatic landscape of materials that are comparable to or thinner than their electrostatic screening
length can therefore be carefully tailored by control of their thicknesses, interfaces,
and local geometry.
5.2

Correlation between Electronic Properties and Thickness in Ultrathin
Semiconductors

To examine the interplay between interfaces, thicknesses, and band bending, we
first consider theoretically a semiconductor on a metallic substrate surrounded by
vacuum and solve Poisson’s equation
∇2 𝜙 = −

(5.2)

108
Thick

Thin

Figure 5.1: Thickness-dependent surface potentials due to vertical band bending. Calculated 𝐸 𝑐 − 𝐸 𝑓 band diagram for 10- and 100-nm-thick flakes of MoS2
on Au assuming strong electron transfer at the MoS2 /Au interface. Schematic band
diagrams of a material dominated by its interface properties (bottom left) and bulk
properties (bottom right), which depend on the thickness of the material relative
to its electrostatic screening length (𝐿 𝐷 ). Insets correspond to a schematic of a
semiconductor (e.g., MoS2 ) on its substrate (e.g., Au), with the blue representing
excess electron concentration relative to its bulk value. The surface of the material
refers to the semiconductor-vacuum interface.
to calculate the energy band diagrams for differing thicknesses, as depicted in Figure 5.1. As expected, we find the characteristic length scale to be the Debye screening
length 𝐿 𝐷 , which we estimate to be approximately 40 nm for carrier concentrations
corresponding to typical values for MoS2 . Furthermore, our calculations suggest
that layers that are thinner than their Debye screening length will have a surface
potential that is directly related to its thickness. Importantly, this analysis requires

109

Figure 5.2: Experimental observation of thickness-dependent surface doping in
MoS2 /Au. a Topographic image of MoS2 exfoliated onto a gold substrate with corresponding surface potential b mapped over the same area. The blue and red outlines correspond to
a MoS2 thickness of approximately 15 nm and 60 nm, respectively. The upper left region
corresponds to monolayer MoS2 /Au. Scale bar is 1 𝜇m c Linecut of the topography and
surface potential. The dashed lines are guides for the eye. d Proposed energy diagram at the
surface for the thin and thick MoS2 . e Optical micrograph image of MoS2 exfoliated onto a
gold substrate. The thick (red outline) and thin (blue outline) regions are ∼30 nm and ∼5
nm thick, respectively. Intensity map of photoemitted electrons at the sulfur 2p core level
for binding energies of 161.9 eV f and 162.2 eV g. Intensity map of photoemitted electrons
from the valence band of MoS2 resolved to its Γ point for binding energies of 1.0 eV h and
1.3 eV i. The thin (blue) and thick (red) flake outlines are superimposed. All scale bars in e
– i are 50 𝜇m.

use of a material, like MoS2 , with an absence of surface states and other contaminants at the top (basal-plane) semiconductor-vacuum interface that is typical in most
three-dimensional materials due to the formation of dangling bonds. Van der Waals
materials are therefore ideal for probing this thickness-dependent surface potential
because of their naturally passivated surfaces.

110
To examine the effects of varying the semiconductor thickness on surface potential,
we directly exfoliated MoS2 on Au (see Methods for details). These exfoliated
samples produce a variety of thicknesses that can be determined with atomic force
microscopy (AFM) and Kelvin Probe Force Microscopy (KPFM), as shown in
Figure 5.2a and Figure 5.2b. These images clearly show a direct correlation between
two different thicknesses of a MoS2 flake (with thicknesses of about 15 nm and
60 nm), with correspondingly different surface potentials (50 mV and -230 mV),
as shown in Figure 5.2c. Importantly, these thicknesses are outside the realm
where there are strong quantum confinement effects and therefore can be considered
electronically as ‘bulk’ materials. [226] Since these two flakes were fabricated
under the same conditions, we therefore attribute the difference in observed surface
potentials to vertical band bending at the MoS2 -Au interface. We further show
this correlation between thickness and surface potential for a variety of thicknesses
measured on other samples (See Figure 5.3).

Figure 5.3: Contact potential difference of other MoS2 flakes on Au. Measured contact potential difference of MoS2 on Au for a variety of thicknesses.
OmegaScope-R (AIST-NT) setup was used for KPFM with concurrent AFM measurement. Au tip was biased by 3 V and connected to a lock-in amplifier while the
sample was grounded. R𝑎 (arithmetic average) values were obtained from a standard
sized region of each layer of flakes comprising 300 × 300 points and the standard
deviations in R𝑎 values were plotted as error bars. The lateral areas (sizes) of the
flakes ranged from 20-50 𝜇m2 .
The relative surface potential difference between the 15 nm and 60 nm sample is
about 280 mV, suggesting strong electron transfer from the gold substrate to the
ultrathin MoS2 (Figure 5.2d). Strong electron transfer at the MoS2 -Au interface
appears contradictory to the well-known work function of Au (∼5.1 eV), which

111
instead would suggest hole doping of the neighboring MoS2 layer. However, recent
works has shown that the MoS2 -Au interface induces strong electron transfer [194,
195], particularly if the interface remains pristine during the exfoliation process,
which has also enabled large area monolayer exfoliation of TMDCs and other 2D
materials. [43, 76, 108, 116, 194]
To further examine this correlation between thickness and electronic properties, we
perform spatially resolved angle-resolved photoemission spectroscopy (ARPES) at
Beamline 7.0.2 at the Advanced Light Source (see methods) on another flake of MoS2
on Au (Figure 5.2e). We find a direct correlation between the position of the sulfur
2p core levels and the thicknesses of the corresponding MoS2 flakes (Figure 5.2f, g).
The larger sulfur 2p binding energies for the thinner MoS2 is suggestive of electron
transfer at the MoS2 -Au interface. Finally, we examine electron binding energies
and momenta that correspond to the valence band edge, which occurs at the Γ point
in Brillouin zone in electronically bulk samples. We find that while the thicker
sample has a Fermi level of about 1 eV above the valence band edge, the Fermi level
at the surface of the thinner sample is about 1.3 eV above its valence band edge
(Figure 5.2h, i). Assuming an electronic bandgap of approximately 1.3 eV [117],
we find direct evidence of strong electron transfer and vertical band bending at the
MoS2 -Au interface, which corroborates the KPFM results shown earlier. Similar
results are also observed on another MoS2 sample (See Figure 5.4). Given that the
native doping of bulk MoS2 is typically n-type, these results suggest the MoS2 -Au
interface would yield Ohmic n-type contacts by degenerately doping the neighboring
MoS2 region [78].
5.3

Spatiotemporal Imaging of Charge Carrier Separation due to Thickness

To investigate the effects of the different vertical band bending profiles on the charge
carrier dynamics, we perform scanning ultrafast electron microscopy (SUEM) on
another sample of MoS2 /Au with a thin-thick junction of 10 nm and 100 nm,
respectively (Figure 5.5b). Figure 5.5a shows a conceptual depiction of the SUEM
measurement technique performed over this thin-thick junction. Briefly, SUEM
is a pump-probe technique that uses an optical pump (∼514 nm) and an electron
pulse (∼2 ps pulse width) that records the secondary electron emission with the
presence of the optical pump as a function of pump-probe time delay. Contrast
images can be formed at different time delays by subtracting the static SEM image
(Figure 5.5c) from a similar SEM image formed with the optical pump on at a
specific time delay Δ𝑡. The contrast in the secondary electron emission as a result of

112

Figure 5.4: Photoemission spectroscopy of another MoS2 flake on Au. a Optical
micrograph image of MoS2 exfoliated onto a gold substrate. Intensity map of
photoemitted electrons at the sulfur 2p core level for binding energies of 161.9 eV
b and 162.2 eV c. Intensity map of photoemitted electrons from the valence band
of MoS2 resolved to its Γ point for binding energies of 1.0 eV d and 1.3 eV e. All
scale bars are 50 𝜇m. Red and blue overlays correspond to thick and thin portions
of the sample, respectively.

the optical pump beam has been interpreted as direct images of electron (blue) and
hole (red) carrier populations under ultrafast excitation, which has been previously
used to image carrier separation in pn junctions. [131] Similar methods to directly
image ultrafast carrier dynamics has also been utilized in photoemission electron
microscopy. [118, 206]
By examining the contrast images formed at different time delays via SUEM, we find
direct evidence for carrier separation at a thin-thick junction (Figure 5.5d). First, the
lack of carrier dynamics at negative time delays suggests appropriate background
subtraction. At longer time delays, we initially find the appearance of excess holes
(red contrast) on every thickness of MoS2 for Δ𝑡 < 27 ps. The oblong contrast
profile is due to the beam shape of oblique excitation. At Δ𝑡 = 33 ps, there is a
simultaneous occurrence of both excess holes and electrons on the thin (10 nm)
sample, with the excess electrons located near the thin-thick junction. At slightly
longer time delays (40 ps < Δ𝑡 <60 ps), this population of excess electrons appears

113

Figure 5.5: Scanning Ultrafast Electron Microscopy Imaging of a Band Bending Junction. a Conceptual depiction of the band bending junction and its measurement via scanning
ultrafast electron microscopy (SUEM). In a sample with different thicknesses of MoS2 , band
bending induced by a gold substrate enables lateral carrier separation between electrons and
holes. In SUEM, an optical pulse generates electron-hole pairs that subsequently evolve in
space and time. An electron pulse is raster scanned across the surface of the sample for a
given time delay Δ𝑡 after the optical pulse. An image of the detected secondary electron (SE)
intensity is formed. Contrast images are formed that correspond to the difference between
the SE image at Δt relative to the SE image without an optical pulse. Contrast images
are interpreted as images of the net charge density, i.e., increased (decreased) SE intensity
corresponds to an increase in the local surface electron (hole) density. b Optical image
of MoS2 exfoliated onto a gold substrate. c Static scanning electron micrograph over the
flake in b, with highlighted regions of thick (pink border, ∼100 nm), intermediate thickness
(light green border, ∼30 nm) and thin (light blue border, ∼10 nm) MoS2 on one sample. d
Contrast images over the same area as c for different time delays, with corresponding MoS2
flake outline. Blue and red contrasts are interpreted as excess electrons and holes due to
photoexcitation, respectively. All scale bars are 50 𝜇m.

to increase and spread before monotonically decreasing with a similar time constant
(single exponential fit yields 𝜏 ∼ 75 ps) to that of the excess hole population on
the thick MoS2 (Δ𝑡 > 60 ps). Interestingly, the intermediate thickness of MoS2
(light green outline, Figure 5.5d), yields little to no excess electrons, suggesting the
dominant path for carrier transport is between the thin (10 nm) and thick (100 nm)

114
layers of MoS2 . Further measurements on monolayer MoS2 and bulk (
100 nm
thick) MoS2 (See Figure 5.6 and Figure 5.7) yield little carrier dynamics and no
appearance of excess electrons, as observed at this thin-thick junction.

Figure 5.6: Scanning ultrafast electron microscopy of bulk MoS2 on Au. a
Optical micrograph image of a very thick MoS2 exfoliated onto a gold substrate. b
Static SEM image of the same flake (scale bar = 100 𝜇m). Contrast images formed at
different pump-probe time delays of c -333 ps d 0 ps e 67 ps and f 133 ps, suggesting
that vertical carrier separation in very thick MoS2 on Au is unobservable through
SUEM because its thickness is much larger than the band-bending Debye screening
length. In the contrast images, c – f, the color bar is blue (red) for excess electrons
(holes). Contrast images are taken over the same region as the static SEM image in
b.
We interpret the carrier dynamics in these contrast images as direct evidence of
carrier separation at a thin-thick layer interface, which we refer to hereafter as a
“band bending junction”. The dynamics can be qualitatively explained as follows
(See Figure 5.8 for a schematic): (1) optical excitation results in generation of
electron-hole pairs, which rapidly separate vertically due to vertical band bending
and thicknesses in the semiconductor that are small compared to carrier diffusion
lengths. The strong electron transfer at the MoS2 -Au interface results in a band
profile schematically depicted in Figure 5.1, which causes holes (electrons) to move
toward the MoS2 -vacuum (MoS2 -Au) interface for both thicknesses. Thus, the
timescale for the initial contrast is primarily due to the vertical carrier drift velocity
and thickness of the sample. (2) After the carriers are separated, both carriers can
travel on average over one diffusion length within their lifetime. Therefore, if the
holes on the thin side are within a diffusion length of the thin-thick junction, they
can travel laterally and vertically to separate from the remaining electrons on the
thin side. Electrons can also travel within their own diffusion length but will remain

115

Figure 5.7: Scanning ultrafast electron microscopy of monolayer MoS2 on Au.
a Static SEM image of a monolayer MoS2 on Au (scale bar = 100 𝜇m). Contrast
images formed at different pump-probe time delays of b -333 ps c 0 ps d 20 ps e
40 ps and f 60 ps, suggesting that vertical carrier separation in monolayer MoS2 on
Au is unobservable through SUEM because its thickness is much smaller than the
band-bending Debye screening length. In the contrast images, b – f, the color bar
is blue (red) for excess electrons (holes). Contrast images are taken over the same
region as the static SEM image in a.

near the MoS2 -Au interface due to the band profile. The lateral transport that results
in carrier separation and the eventual contrast flip therefore follows the timescales
of carrier diffusion. (3) Once some holes move across the thin-thick interface,
remaining holes follow along due to the gradient in the quasi-Fermi level in the
holes. (4) After the electrons and holes have completely separated, they recombine
primarily monomolecularly through Shockley-Read-Hall recombination kinetics via
trap states within the bulk of MoS2. These trap states are naturally available due to
defects in the crystalline lattice (e.g. sulfur vacancies) [1, 2].
To model the lateral carrier separation dynamics of this band bending junction, we
turn to time-dependent drift-diffusion equations (see section 5.5). To simplify the
numerical modelling substantially, we make the following assumptions: (1) The

116

Figure 5.8: Schematic depiction of carrier transport at a band bending junction.
In a band bending junction, carriers transport vertically before separating laterally.
The population of holes (electrons) are depicted by red (blue) circles. Arrows
represent the direction of the flow of carriers. Time is increasing from a to f, where
a corresponds to when carriers are first generated and f corresponds to when carriers
have fully separated due to this thickness junction.

MoS2 -Au interface results in strong electron transfer to MoS2 . (2) The dynamics
can be qualitatively modelled by semiconductor drift-diffusion equations. Ab-initio
calculations combined with the Boltzmann transport equations may yield more
accurate results while considering the unique bandstructures of these systems. (3) the
carrier dynamics can be effectively modelled in two dimensions, due to the prominent
dynamics at the thin (10 nm) - thick (100 nm) interface. (4) The anisotropic
mobilities in the vertical vs. horizontal directions is effectively compensated by
scaling the horizontal dimension by the ratio of the square root of the mobilities. (5)
While the carrier dynamics in other semiconductor materials at these time scales
has been shown to exhibit super diffusion and therefore a time-dependent mobility
[106, 132], we fix our mobility to a single value which represents effectively a time
and spatially averaged quantity. (6) Since we are interested in primarily the lateral
carrier dynamics, we assume carrier excitation occurs at 27 ps in the simulation. (7)
Secondary electron contrast is primarily due to the net charge density near the surface
of the MoS2 layer, with the secondary electron emission decaying exponentially from

117
the surface of the semiconductor with a length scale 𝜆 𝑆𝐸 .

Figure 5.9: Simulated carrier dynamics for various material parameters. Net
negative surface charge density of the thin (10 nm, solid line) and thick (100 nm,
dashed line) areas of the sample for varying a electron affinity b metal workfunction
c bandgap d carrier lifetime e effective mass f and carrier mobility. In all cases, the
thin and thick portion of the sample exhibits rapid onset of hole surface charges. The
thin portion of the sample then exhibits lateral hole transport to the thick portion of
the sample, which results in electrons (holes) separated to the thin (thick) portion
of the sample. Dashed lines correspond to the region of the inset. For the bandgap
sweep c, all 𝐸 𝑔 < 1.4 eV overlay with the 1.4 eV curve since the semiconductor
is already degenerately doped and there is minimal change to the band profile.
Similar effects occur for 𝜒𝑒 > 4.1 eV and 𝜙 𝑀 < 3.7 eV. For the effective mass
sweep e, varying the effective mass of the electron does little to the carrier dynamics
because the transport is dominated by the minority carrier, i.e., the holes (see also
Figure 5.10).
With these simulation assumptions in mind, we find excellent agreement between
the experimental secondary electron emission contrast and the calculated timedependent net negative surface charge density (Figure 5.11a). Snapshots of the
charge density at specific times are also shown in Figure 5.11b, showing lateral
carrier separation at the thin-thick interface shortly after excitation. The agreement
has been achieved from a variety of simplifications in the theoretical modelling,
which suggests that the extracted material parameters used to achieve this matching

118
(specifically, the electron and hole mobility of MoS2 ) should not be taken to be
necessarily physical. Improved modelling that explicitly includes (1) the anisotropic
mobility, (2) the large aspect ratio between lateral and vertical directions, and (3) a
time-dependent mobility due to hot carrier dynamics would substantially enhance the
accuracy of a material parameter fit. However, we find that the qualitative picture
of the band bending junction is robust (See Figure 5.9) to a variety of material
constants, i.e., lateral carrier separation is achieved independent of the specific
choice of material parameters, suggesting that this junction should be observable
in other van der Waals materials as well. Our calculations also suggest the current
density at the thin-thick interface is dominated by the hole current (See Figure 5.10),
which is expected since it acts as a minority carrier in this electron-doped system.
Finally, we emphasize that the carrier separation demonstrated here is in the absence
of quasiparticle bandgap or exciton binding energy differences, which has previously
led to novel junctions[72, 185, 189]. Here, the carrier separation is enabled by the
unique interplay between thickness and band bending.

Figure 5.10: Hole dominated charge transport. a Simulated charge density
plots for varying electron lifetime. All curves are overlaid on top of one another,
showing that the minority carrier, i.e., the holes, dominate the carrier flow. b
Simulated electron 𝐽𝑛 , hole 𝐽 𝑝 , and total 𝐽 current density at the thin-thick junction,
showing again that the carrier transport dynamics are dominated by the holes.
Current densities are resolved to the 𝑥 direction, showing only the lateral transport
characteristics. Dashed box refers to the inset.

119

27.3 ps

28.3 ps

31 ps

35 ps

100 ps

400 ps

More Holes

signlog10(δρ(t)) (pC/cm3)

More Electrons

Figure 5.11: Simulation of Carrier Dynamics at a Bend Bending Junction. a
Experimentally measured secondary electron contrast intensity (dots, right axis) on
the thin (blue) and thick (red) MoS2 as a function of different pump-probe delay
times, along with the simulated net negative surface charge density as a function of
time (solid lines, left axis). We assume the thin and thick MoS2 is 10 and 100 nm
thick, respectively, for the simulation. b Simulated cross-sectional maps of the net
charge density at different time steps, plotted with a signed log function to examine
the orders of magnitude change in carrier density more easily (scale bar = 50 nm).
Red corresponds to net positive charge (i.e., holes), while blue corresponds to net
negative charge (i.e., electrons).

5.4

Conclusions and Outlook

Our results suggest that the interplay between thickness and band bending for materials thinner than their screening length can result in the formation of a new type of
homojunction, which we refer to as a band bending junction. These band bending
junctions are formed by combining the highly passivated surfaces of van der Waals
materials with vertical band bending in materials whose thicknesses are comparable
to or below the characteristic electrostatic screening length. Thus, while these results were obtained with the MoS2 /Au system, they are likely generalizable to other
van der Waals heterostructures and perhaps also 3D bonded semiconductors if one
can generate chemically passivated yet electronically active surfaces in the ultrathin
thickness (<100 nm) limit. Furthermore, these band bending junctions may find use
as new photodetector geometries or could be useful for sensing applications. From
the fundamental perspective, they may enable the formation of two-dimensional
electron/hole gases or enable a wide variety of surface-sensitive measurement techniques to indirectly examine vertical carrier transport in layered van der Waals
materials. More generally, the observation of charge carrier transport at these band

120
bending junctions should be considered while modelling and interpreting ultrathin
optoelectronic devices that are geometrically inhomogeneous.
5.5

Appendix

Sample Preparation
Large area ultrathin flakes of MoS2 were fabricated utilizing the strength of the
Au-S bonds, which has been known to yield large area samples. [43, 116, 194] To
summarize our procedure, we first created atomically smooth Au substrates using
template stripping techniques which routinely yields <3 Å RMS. [198, 207] Then,
bulk crystals of MoS2 (HQgraphene) were cleaved from the native crystal using
thermal release tape (Semiconductor Equipment Corp., No. 3195MS) and directly
pressed onto the template stripped Au substrate. With the tape and bulk MoS2
directly pressed onto the Au substrate, the entire stack was then heated to ∼120 C on
a hot plate for ∼2 minutes to release the bulk crystal from the tape and to promote
adhesion between the MoS2 and Au substrate. The MoS2 /Au sample was then
sonicated in Acetone for ∼5 seconds to release the bulk crystal from the ultrathin
layer of MoS2 that would remain adhered to the Au substrate. The sample was
finally rinsed with Isopropanol and blow dried with N2 . We found it was necessary
to template strip the Au substrate immediately before pressing with the MoS2 bulk
crystal to yield large area flakes, similar to what has been observed previously. [194]
Kelvin Probe Force Microscopy Measurements
Scanning probe microscopy was performed on OmegaScope-R SPM (AIST-NT,
now-Horiba Scientific). HQ NSC-14-Cr/Au probes (Mikromasch) were used for
characterization. Kelvin probe imaging was performed in frequency modulation
mode which allowed improved spatial resolution of the distribution of the contact
potential difference (which reflects the distribution of the surface potential on the
sample). The value of the surface potential of the probes was not calibrated, so it
was the contrast in the CPD images, not the absolute value of the surface potential,
which bore the physical meaning in the CPD images.
Photoemission Spectroscopy Measurements
Photoemission spectroscopy measurements were performed at Beamline 7.0.2 (MAESTRO) at the Advanced Light Source. Samples were characterized at the microARPES UHV endstation, with synchrotron beam spot sizes of approximately 10
𝜇m. Incident photon energies were 145 eV and 330 eV for the valence band and

121
core level measurements, respectively. Measurements were performed at ∼72 K and
multiple frames were averaged together to achieve sufficient signal to noise ratios.
Scanning Ultrafast Electron Microscopy Measurements
Scanning ultrafast electron microscopy is a newly developed technique that can
directly image the dynamics of photoexcited carriers in both space and time with
subpicosecond temporal resolution and nanometer spatial resolution. Details of the
setup can be found elsewhere [126, 215] and are briefly summarized here (also
illustrated in Figure 3a). Compared to optical pump-probe spectroscopy, SUEM
is a photon-pump-electron-probe technique, with subpicosecond electron pulses
generated by illuminating a photocathode (ZrO-coated tungsten tip) with an ultrafast
ultraviolet (UV) laser beam (wavelength 257 nm, pulse duration 300 fs, repetition
rate 5 MHz, fluence 300 𝜇J/cm2 ). A typical probing electron pulse consists of
tens to hundreds of electrons, estimated by measuring the beam current through a
Faraday cup, and is accelerated to 30 keV before impacting the sample. The probing
electron pulses arrive at the sample after the optical pump pulses (wavelength 515
nm, fluence 80 𝜇J/cm2 ) by a given time controlled by a mechanical delay stage
(-700 ps to 3.6 ns with 1 ps resolution). The probing electron pulses induce the
emission of secondary electrons from the sample, which are subsequently collected
by an Everhart-Thornley detector. To form an image, the probing electron pulses are
scanned across the sample surface and the secondary electrons emitted from each
location are counted. Because the yield of secondary electrons depends on the local
average electron energy, more/less secondary electrons are emitted from regions of
the sample surface where there is a net accumulation of electrons/holes. Typically,
a reference SEM image is taken long before the pump optical pulse arrives and is
then subtracted from images taken at other delay times to remove the background.
In the resulting “contrast images”, blue/red contrasts are observed at places with net
accumulation of electrons/holes due to higher/lower yield of secondary electrons.
In this fashion, the dynamics of electrons and holes after excitation by the optical
pump pulse can be monitored in real space and time.
Electromagnetic and Transport Simulations
The coupled Drift-Diffusion and Poisson equations were solved using the CHARGE
solver in Lumerical DEVICE, which uses finite-element meshing to solve the coupled differential equations iteratively. Due to the anisotropic mobility known for
these materials [103, 121], solving the multi-dimensional coupled differential equa-

122
tions is computationally costly. Instead, we argue that since we are interested in
a qualitative picture of the carrier transport and because the lateral extent of the
junction is dominated by the in-plane diffusion length, a proper rescaling of the
in-plane dimensions by a factor of 𝐿 k /𝐿 ⊥ = 𝜇 k /𝜇⊥ 1000 allows us to treat the
problem with an approximate isotropic, spatially, and temporally averaged mobility
(𝜇 = 2000 cm2 /(V-s)). Since the experimentally observed in-plane junction width
is ∼10 𝜇m, the rescaled in-plane width should be about 300 nm. In our simulations
our total lateral span is 500 nm. Band bending was captured assuming an Ohmic
contact at the Au-MoS2 interface with a metal work-function of 3.8 eV and semiconductor electron affinity of 4.0 eV. Fermi-Dirac statistics were included due to the
high level of modulation doping. Furthermore, we used a bandgap value of 1.3 eV
[117], hole effective mass of 0.785𝑚 0 [221], an electron effective mass of 0.686𝑚 0
(for the electron effective mass, we took the geometric mean of the effective masses
in the transverse and longitudinal directions at the Q point [221]), dielectric constant of 7 [98], and a Shockley-Read-Hall recombination lifetime of 75 ps. Exciton
dynamics were deemed not relevant, since majority of the transport should occur at
the lowest energy conduction band and highest energy valence band, with exciton
binding energies
𝑘 𝑏 𝑇. [25] Native doping of MoS2 was assumed to be n-type
with a doping level of 1016 cm−3 , similar to that quoted from the supplier. Optical
generation values were calculated assuming the system is optically one-dimensional
over a specific thickness (either 10 nm or 100 nm), and therefore 1D transfer matrix calculations were applicable for each region. These generation rates were then
directly imported into Lumerical DEVICE. Calculated volumetric charge densities
were exponentially weighted from the surface with a characteristic length scale of
𝜆 𝑆𝐸 = 4 nm to yield surface charge densities. The SUEM signal is expected to be
proportional to the net negative charge density 𝐼𝑆𝑈𝐸 𝑀 ∝ −𝛿𝜌 = −𝑞(𝛿 𝑝 − 𝛿𝑛), where
positive/negative SUEM signal scales with the net electron/hole population.

Part III
The Luminescence is the Voltage

123

124
Chapter 6

HIGHLY STRAIN TUNABLE INTERLAYER EXCITONS IN
MOS2 /WSE2 HETEROBILAYERS
“Happiness only real when shared.”
— Christopher McCandless

6.1

Introduction

Strain engineering of nanomaterials has received substantial interest because as the
physical length scales of nanomaterials become smaller, size effects enable fewer
defects, such as grain boundaries, as well as diminishing bending rigidity, resulting
in a superior yield strength with a highly elastic response compared to their bulk
counterparts [64, 65, 227]. Two-dimensional (2D) materials are particularly suited
for strain engineering because they combine high in-plane mechanical strength and
extremely small bending rigidity with substantial strain tunability of electronic band
structure [42, 99, 114, 137, 147, 222]. The unique mechanical strength, flexibility,
and tunability of 2D materials have therefore enabled their applications for wearable
and flexible technologies [18, 86, 165], as well as fundamental studies of material
properties under carefully engineered strain conditions. The semiconducting layered
transition metal dichalcogenides (TMDCs) have emerged as particularly interesting
candidates for strain engineering, since they have exhibited exciton funneling [53,
102, 127], strain-mediated phase transitions [177], and the ability to form sitecontrolled quantum emitters via localized strains [115].
Strain engineering has been explored in TMDC monolayers, multilayers, and heterostructures. Multilayer TMDCs are particularly interesting because they are
known to exhibit interlayer exciton transitions, i.e., optical transitions where the
electron-hole pairs are located in different constituent layers bounded by strong
Coulomb interaction [164]. These interlayer excitons have been observed to be
strain tunable in homobilayers of molybdenum disulfide (MoS2 ) [21, 138] and
heterobilayers of molybdenum diselenide and tungsten diselenide (MoSe2 /WSe2 )
[70] with deformation potentials of approximately 47 meV/% and 22 meV/%, respectively. Furthermore, other studies on multilayer structures have suggested that
interlayer coupling should be an additional degree of freedom tunable via strain due

125
to the Poisson effect [75, 144], but these effects on interlayer exciton transitions have
yet to be observed.
In this chapter, we investigate the strain characteristics of momentum-space indirect
Γ − 𝐾 interlayer exciton in wrinkled molybdenum disulfide and tungsten diselenide
(MoS2 /WSe2 ) heterobilayer. We spatially probed the effects of local strain in the
wrinkled heterobilayer via photoluminescence (PL) and Raman measurements, and
find that the Γ − 𝐾 momentum-space indirect transition gives rise to a larger deformation potential (107 meV/%) compared to its intralayer counterpart because of
the strain-sensitive orbital nature at the Γ point of the valence band and 𝐾 point
of the conduction band. Furthermore, we find that the interlayer exciton exhibits
a non-monotonic dependence in PL intensity with strain, which can be explained
by a competition between interlayer coupling and band structure modulation effects
under strain. This hypothesis is supported by ab initio band structure calculations as
well as Raman measurements where we can directly measure the interlayer coupling.
6.2

Local Strain Engineering of Intra- vs. Inter-layer excitons

For local strain engineering, we prepared wrinkled vertical heterostructures of
MoS2 /WSe2 on an elastomeric substrate of polydimethylsiloxane (PDMS) by using
a combination of our heat-assisted PDMS-to-PDMS (PTP) assembly method and
strain-release mechanism (see section 6.6 and Figure 6.2 for more fabrication details). When a pre-stretched PDMS with the assembled heterobilayer of MoS2 /WSe2
via PTP method (Figure 6.1a,c) is released, the heterobilayer deforms into a periodic
wrinkled structure (Figure 6.1b) with a periodicity of about 4 𝜇m (Figure 6.2) as
shown in an optical micrograph of the fabricated crumpled structure (Figure 6.1d).
The periodic wrinkled geometry induces local stretching and compression of the
heterobilayer lattice at the crests and valleys, respectively, resulting in a periodic
local strain profile of alternating tensile and compressive strain within the TMDC
heterostructure (Figure 6.1b).
To characterize the optical properties of MoS2 /WSe2 heterobilayer, we carried out
steady-state PL spectroscopy and Raman spectroscopy under 532 nm excitation
at room temperature over the fabricated MoS2 /WSe2 heterobilayer and adjacent
isolated constituent monolayers before the release of the pre-stretch (flat state) (Figure 6.1e). We observed the intralayer emission at the exciton energies characteristic
of MoS2 (∼1.88 eV or 680 nm) and WSe2 (∼1.65 eV or 750 nm) in adjacent isolated
monolayers. In the heterobilayer, we observed a new PL emission peak redshifted

126

Figure 6.1: Fabrication of wrinkled MoS2 /WSe2 heterobilayer. a Schematic
depiction of a vertical van der Waals heterobilayer on a pre-stretched elastomeric
substrate. b After the release of the mechanically pre-stretched elastomeric substrate,
a wrinkled heterostructure exhibiting heterogeneous strain profile of alternating
tension (at crest) and compression (at valley) is fabricated. c Optical microscopy
image of a flat MoS2 /WSe2 heterobilayer with the estimated twist angle of 50.7◦ .
Scale bar is 10 𝜇m. d Optical microscopy image of a crumpled MoS2 /WSe2
heterobilayer. Scale bar is 10 𝜇m. e Photoluminescence spectra of flat monolayer
MoS2 , WSe2 , and MoS2 /WSe2 .
by about 75 meV (or 30 nm) with respect to the intralayer A exciton of the WSe2
monolayer, which corresponds to a Γ − 𝐾 optical transition energy of MoS2 /WSe2
[95, 187]. A confocal PL map (Figure 6.2) integrated over the two characteristic
intralayer exciton peaks of MoS2 and WSe2 showed significant PL intensity quenching over the heterobilayer area, indicative of rapid charge transfer due to the type-II
nature of this interface. We further observed uniform emission of the interlayer
exciton at the Γ − 𝐾 optical transition energy of MoS2 /WSe2 across a relatively large
area (Figure 6.3) as well as the out-of-plane Raman mode (A21𝑔 ) of WSe2 at 309 cm−1
in our MoS2 /WSe2 heterobilayer (Figure 6.3), suggesting there is strong interlayer
coupling[30] between the MoS2 and WSe2 layers in the heterobilayer region.
To experimentally explore the effect of local strain on the band structure and the
resultant optical properties in terms of exciton emission, we first spatially resolved
the PL energy shift of the intralayer and the interlayer excitons in the wrinkled
monolayers and MoS2 /WSe2 heterobilayer (Figure 6.5 and Figure 6.4). Typical PL

127

Figure 6.2: Sample fabrication and characterization of wrinkled heterostructures. a, Schematic illustrations of the sample fabrication process of wrinkled
heterostructures via PDMS-to-PDMS (PTP) dry transfer method. Our PTP method
involves a one-time direct stamping transfer process after the exfoliation of individual monolayers and minimizes any solvent and/or polymer residues at the van
der Waals interface. b, A height profile of the fabricated wrinkled structure at the
heterobilayer area via confocal laser scanning microscopy. c, A confocal photoluminescence (PL) mapping integrated over both intralayer exciton energies of MoS2
and WSe2 monolayers. The red color indicates the emission of intralayer excitons
and the black color in the heterojunction indicates the PL quenching.

spectra measured at room temperature across a single wrinkle profile are shown in
Figure 6.5a for the intralayer excitons of WSe2 and MoS2 , and in Figure 6.5b for
the interlayer exciton of the MoS2 /WSe2 heterobilayer. In both cases, we observed a
gradual shift of the emission peak energy along with the wrinkle profile and observed
that the energy peak shifts within the structure can be modulated up to approximately
107 meV for the interlayer exciton, compared to approximately 54 meV for WSe2
and 55 meV for MoS2 intralayer excitons under uniaxial deformation. We observed
greater tunability of the peak energy shifts in the interlayer exciton than in the
intralayer excitons. We further observed the peak energies redshifted (blueshifted)

128

Figure 6.3: Optical characterization of flat heterobilayers of MoS2 /WSe2 on
PDMS. a, An optical micrograph of the fabricated flat heterobilayer of MoS2 /WSe2
on PDMS substrate via our PTP (PDMS-to-PDMS) assembly process. A scale
bar, 10 𝜇m. The estimated twist angle is approximately 53◦ using the sharpedges estimation of the heterobilayer. b, A photoluminescence (PL) mapping of the
fabricated heterobilayer. A color bar indicates corresponding emission wavelengths.
The relatively uniform peak positions over the heterobilayer region indicates that
there is no substantial fabrication-induced heterogeneity. c, Normalized PL intensity
spectra of intra- and interlayer excitons. d, Normalized Raman intensity spectra at
adjacent monolayers of MoS2 and WSe2 and at heterobilayer of MoS2 /WSe2 . A
magnified (x10) inset shows the emergence of the out-of-plane vibration mode of
WSe2 (∼309 cm−1 ).
at the crest (valley) of the wrinkled structure relative to the peak energies in the flat
state for both intralayer and interlayer excitons. These observations are indicative of
both tensile (local stretching of atomic lattices at the crests) and compressive (at the
valleys) strain applied via our fabricated wrinkled structure, which suggests band
structure modulation via local strain profile [69]. The observation of luminescence
peak energy shifts along the wrinkle profile allows us to correlate the spatial position
with a locally applied strain.
We also observed strikingly different qualitative trends for variation of PL intensity

129

Figure 6.4: Photoluminescent mapping of the fabricated wrinkled MoS2 /WSe2
heterobilayer. a Photoluminescent (PL) intensity and b wavelength mapping over
the MoS2 intralayer exciton portion of the spectrum in the fabrciated MoS2 /WSe2
heterobilayer. Dotted lines indicate boundaries of the constituent monolayers. c
PL intensity and d wavelength mapping over WSe2 intralayer exciton and interlayer
exciton portion of the spectrum. PL intensities of intralayer excitons were substantially quenched in the heterojunction area (MoS2 /WSe2 ). All scale bars are 5 𝜇m.
The spatial resolution is ∼0.4 𝜇m, which allows us to resolve individual spectra
along the wrinkled structure. e PL wavelength (top) and intensity (bottom) mapping
over the corresponding wrinkle from valley (left) to crest (right) indicated in blue
rectangle in d. Pixel dimensions are 200 nm.

with strain between the intralayer and interlayer excitons for our wrinkled structure.
Specifically, we observed that the PL intensity of the WSe2 (MoS2 ) intralayer exciton
was highest (lowest) at the crests and lowest (highest) at the valleys. Thus, the
PL intensity of WSe2 (MoS2 ) monotonically increases (decreases) with increasing
tensile strain (Figure 6.5a). In contrast, the maximum peak intensity of the interlayer
exciton in the heterobilayer region occurs at an intermediate position between the
crest and valley, with a peak energy of ∼1.6 eV. By comparing this peak energy
with the flat-state peak energy, we find that the maximum peak intensity of the
interlayer exciton occurs when the heterobilayer is under mild compression. Thus,
by examining the relationship between the peak energy with the peak intensity,
we find that the MoS2 /WSe2 interlayer exciton PL intensity is non-monotonic with
local strain. In other words, the interlayer exciton PL intensity appears to decrease
with both increasing tensile and compressive strain, which is in direct contrast to
the monotonic peak intensity vs. peak energy behavior observed for intralayer
monolayer emission.
To substantiate our findings, we show scatter plots between the peak intensity vs.
peak energy (Figure 6.5c), where each point corresponds to an individually fitted

130

Figure 6.5: Photoluminescence of strained heterobilayers. Relative photoluminescence spectra (smoothed) of a WSe2 and MoS2 intralayer A excitons and b
MoS2 /WSe2 interlayer exciton over a representative wrinkle (from the valley to the
crest). The monolayer WSe2 luminescence saturated the detector in these mapping
measurements, creating a top-hat effect in the spectra. Scatter plots over multiple
wrinkles that show how the c normalized intensity and d linewidth of WSe2 , MoS2 ,
and MoS2 /WSe2 vary with exciton energy. Each point refers to a separate spectrum
measured and fitted to a Gaussian function, over the relevant region of the sample.

spectrum with the peak intensities normalized to the brightest spot within that
region of the sample. We observed this non-monotonic interlayer exciton intensity
trend across multiple spots throughout the entire sample. We also found that this
interlayer exciton intensity trend also appears in various other samples we measured
(see Figure 6.6), as well as when we account for the finite depth of focus in our
measurements (see Figure 6.7). Exciton funneling effects are not a dominant effect
since we did not observe highest intensity at the lowest energy of emission in our
heterobilayers.
To gain additional insight on the mechanism of intensity modulation, we show
similar scatter plots in Figure 6.5d between the fitted peak linewidth (full width at

131

Figure 6.6: Optical characterization of different wrinkled MoS2 /WSe2 heterobilayer samples. Optical micrographs (left), and photoluminescence (PL) intensity
of interlayer exciton measurements (right) in a, sample A, and b, sample B wrinkled
MoS2 /WSe2 heterobilayers. Dotted black arrows indicate the direction of PL spectra
taken towards the crest (left) and towards the valley (right) over a wrinkle in the
heterobilayer samples. The estimated twist angles are 33.5◦ (sample A) and 51.4◦
(sample B).

half maximum) and peak energy. Interestingly, while the interlayer exciton shows
no obvious correlation between the peak linewidth and peak energy, the WSe2 and
MoS2 intralayer excitons show a clear behavior of positive and negative correlation,
respectively. Moreover, these trends in linewidth contrast with the trends in intensity
for the individual monolayers, i.e., peak intensity and linewidth are apparently
inversely correlated, which can be observed more clearly in Figure 6.5a of the
individual spectra. By correlating the peak energy shift to an applied strain, the WSe2
(MoS2 ) peak linewidth decreases (increases) with applied tensile strain. Similarly,
the WSe2 (MoS2 ) peak intensity increases (decreases) with applied tensile strain.

132

Figure 6.7: Photoluminescence (PL) measurement on wrinkled heterobilayers
of MoS2 /WSe2 via vertical scanning (XZ-PL scanning) accounting for the finite
depth of focus. PL spectra taken over a wrinkle (valley-to-crest) a, in adjacent
wrinkled monolayer WSe2 . PL spectra b, in adjacent wrinkled monolayer MoS2 , c,
in wrinkled heterobilayer of MoS2 /WSe2 . Dotted black arrows in a and b indicate
the direction of PL spectra taken from the valley to the crest. Dotted black arrows
in c indicate the direction of PL spectra taken towards the crest (left) and towards
the valley (right) over a wrinkle in the fabricated heterobilayer sample.

These effects of strain on the PL intensity and linewidth of the individual monolayers
are well [7, 36, 42, 114, 137] and can be explained by the decreased (increased)
scattering to the neighboring 𝑄 (Γ) valley in the WSe2 (MoS2 ) conduction (valence)
band from the 𝐾 point. In other words, interband transitions in WSe2 (MoS2 )
become effectively more direct (indirect) as tensile strain is applied. In contrast
to the intralayer excitons, the lack of linewidth modulation with local strain in
interlayer exciton emission suggests that there is little modulation of the non-radiative
scattering channels.
6.3

Tuning Interlayer Coupling through Strain Engineering

To better understand the cause of the non-monotonic intensity variation for the
interlayer exciton, we turn to Raman spectroscopy measurements for additional insight as to the mechanisms that may be at play, i.e., investigating the in-plane and
out-of-plane strain resulting from topological deformation of the wrinkled heterobilayer (Figure 6.8a). We show that the Raman characteristics of the MoS2 E2𝑔
(Figure 6.8b) and WSe2 E2𝑔 (Figure 6.8c) modes [47, 105, 201] are well modulated
by our wrinkled sample profile, where the peaks are redshifted (blueshifted) at the
crest (valley), which corresponds well with the topographical features seen in the
optical micrograph as shown in Figure 6.8a. Because the E2𝑔 mode is an in-plane
vibrational mode, it is therefore sensitive to in-plane strain. We did not use the WSe2
E2𝑔 mode for a quantitative measure of in-plane strain because of the degeneracy

133

Figure 6.8: Raman spectroscopy and deformation potentials of strained heterobilayers. a An optical microscopy image of the wrinkled MoS2 /WSe2 heterobilayer.
Vertical dotted white lines indicate crests of wrinkles. Magenta dotted lines indicate
the boundary between adjacent monolayer WSe2 and heterobilayer of MoS2 /WSe2
areas. Spatially resolved Raman mapping of b MoS2 E2𝑔 mode and c WSe2 E2𝑔
mode peak positions. d peak position separation between MoS2 A1𝑔 and E2𝑔 modes
(A1𝑔 - E2𝑔 ). All the scale bars are 5 𝜇m. e Raman peak shifts of MoS2 E2𝑔 and A1𝑔 E2𝑔 modes as a function of position along the a-a’ line indicated in b and d. C and
V are the crests and valleys of the wrinkle, respectively. f experimentally derived
deformational potentials of MoS2 and WSe2 intralayer excitons, and MoS2 /WSe2
ILE obtained from measured Raman and PL spectra. Dashed lines correspond to
linear fits of the deformation potential curve.

between the WSe2 E2𝑔 and A1𝑔 modes. We conclude that there is a uniform and
controlled tension (compression) at the crest (valley) with our wrinkled geometry,
which corroborates our observations in PL spectroscopy measurements.

134
To study the effects of local strain on the interlayer coupling in the heterobilayer, we
examined the mode separation between the MoS2 E2𝑔 (in-plane) and A1𝑔 (out-ofplane) modes. The wavenumber separation between A1𝑔 and E2𝑔 has been used as
a quantitative measure of the interlayer mechanical coupling strength in artificially
stacked bilayers, where larger peak separation is indicative of stronger interlayer
coupling [40, 111]. The mode separation has also been used as an effective proxy
for the number of layers in naturally exfoliated MoS2 , i.e., ∼19 cm−1 for monolayer
and ∼21 cm−1 for bilayer MoS2 [152, 225]. We show the map of the wavenumber
separation between the MoS2 A1𝑔 and E2𝑔 modes, Δ(A1𝑔 - E2𝑔 ) in Figure 6.8d. In our
strained heterobilayer, we observed the A1𝑔 and E2𝑔 mode separation changes from
a value of ∼20 cm−1 at the crest to ∼17 cm−1 at the valley (Figure 6.8e). We also
observed modulation of the integrated intensity ratio (A1𝑔 /E2𝑔 ) at the crest and valley
(Figure 6.9), which is comparable to the ratio observed between a naturally exfoliated
bilayer and a monolayer of MoS2 [100]. Therefore, we conclude that the modulation
of the E2𝑔 mode together with the A1𝑔 mode along the wrinkle profile suggests the
interlayer coupling in the heterobilayer is tuned between stronger interlayer coupling
(larger Δ) and weaker interlayer coupling (smaller Δ) at different locations of the
wrinkled heterobilayer. We therefore postulate that in our wrinkled strain profile
of the heterobilayer, we are observing simultaneous modulation of the interlayer
coupling between MoS2 and WSe2 as well as in-plane strain in the heterobilayer.
Specifically, there appears to be stronger interlayer coupling at the crests, which are
in tension, as compared to at the valleys, which are in compression. This trend is
highlighted by the linecut in Figure 6.8e. Thus, the combination of these two effects,
in-plane strain and out-of-plane interlayer coupling in our wrinkled structure, are
hypothesized to result in the non-monotonic intensity profile of the interlayer exciton
observed in Figure 6.5.
Together with the Grüneisen parameters of the E2𝑔 Raman modes [36, 47, 105,
201] and the PL peak shifts observed earlier, we can also extract an experimental
deformation potential for our system, which yields values of approximately 55
meV/% for MoS2 , 54 meV/% for WSe2 , and 107 meV/% for the Γ − 𝐾 transition
interlayer exciton under uniaxial strain (Figure 6.8f). The estimated strain values
from the Raman modes corroborates well with those estimated from the surface
morphology (see Figure 6.10 and section 6.6). The deformation potentials of
the intralayer excitons also correspond well with reported values [7, 36], and the
interlayer exciton deformation potential is notably almost twice that of the intralayer
excitons.

135

Figure 6.9: Raman mapping of strained heterobilayers. Spatially resolved Raman
mapping of integrated intensity ratio between MoS2 A1𝑔 and E2𝑔 modes (I 𝐴1𝑔 /I𝐸2𝑔 ).
White arrows indicate the location of the crests in the wrinkled heterobilayer. We observed decreased integrated intensity ratio at the crests, indicating stronger interlayer
coupling. Scale bar indicates 5 𝜇m.

Figure 6.10: Wrinkle geometry-driven strain analysis. a Measured (black dots)
and fit (blue line) data of the fabricated wrinkled geometry. The average amplitude
(A) and wavelength (𝜆) of wrinkles obtained by a sinusoidal fit to the measured
wrinkle geometry profile. b Measured (black line) and fit (red line) data of the
X-ray reflection intensity to estimate the skin layer thickness formed on the PDMS
substrate.
6.4 ab initio strain calculations of TMDC heterobilayers
To investigate whether the experimental observations described earlier are consistent with ab initio calculations, we examine the electronic band structure of the
strained heterobilayer by performing density functional theory (DFT) calculations
using the Perdew-Burke-Ernzerhof (PBE) functional [148] (see section 6.6 for more
details). Although PBE functional underestimates the absolute value of the bandgap
for TMDC monolayers, it has shown the ability to predict accurate deformation po-

136

Figure 6.11: Density functional theory calculation of strained MoS2 /WSe2 heterobilayers. a The band structure of MoS2 /WSe2 heterobilayers under different
in-plane strain values with AB stacking. The color scale describes the origin of
the electronic states where the purely blue (red) are completely localized electronic
states that come from the MoS2 (WSe2 ) layer, and intermediate colors have electronic
states which are delocalized across both layers (i.e., hybridized state that come from
both MoS2 and WSe2 layers). b Electronic transition energy for K-K transitions of
monolayer MoS2 and monolayer WSe2 , Γ−𝐾 transition of MoS2 /WSe2 , and the K-K
transition of MoS2 /WSe2 , as a function of strain. The inset schematic depicts the
AB stacking order. c The band structure of MoS2 /WSe2 heterobilayers as a function
of interlayer distance. d Schematic depiction of the non-monotonic behavior of the
interlayer exciton PL intensity due to the competition between in-plane strain and
out-of-plane interlayer coupling.

tential constants that match with the experimental measurements and the high-level
simulations using the coupled GW and Bethe-Salpeter equations (BSE) approach
[45, 53]. We coupled PBE with D3 corrections to capture accurate van der Waals
interactions in the layered materials [66, 67].
The band structure under different strain conditions is shown in Figure 6.11a for
AB stacking, where we project the weighted orbital contributions onto either of the
constitutive layers. As noted previously, the interlayer exciton energetic transition

137
we observe is presumed to be the Γ−𝐾 transition, because the MoS2 and WSe2 layers
are not intentionally aligned to their crystallographic axes and therefore the effective
oscillator strength of the 𝐾 − 𝐾 transition would be substantially weaker [95]. We
further observe that the theoretically calculated deformation potential (Figure 6.11b)
for the Γ − 𝐾 transition provides a much closer match to the experimentally observed
value (Figure 6.8f), compared to the 𝐾 − 𝐾 transition. The increased deformation
potential constant comes from the fact that the nature of the wavefunctions at
the Γ point (valence band) and 𝐾 point (conduction band) are of similar orbital
nature (specifically, the d2𝑧 orbital) [23] and is therefore affected in a similar way
under mechanical strain. We also find that the Γ point is particularly sensitive to
interlayer coupling and band hybridization, as observed previously [31, 203, 223]
(Figure 6.12). Lastly, we remark that this Γ − 𝐾 interlayer exciton is MoS2 -like in
character and has strain characteristics akin to a natural bilayer of MoS2 , where the
Γ− 𝐾 indirect transition of a natural MoS2 bilayer was also observed to have a higher
deformation potential compared to the direct 𝐾 − 𝐾 transition and a decreasing PL
intensity under tensile strain [36].

Figure 6.12: Weighted band structures obtained using DFT calculations. a,
AA stacking of MoS2 /WSe2 heterobilayer, b, Superposition of band structures of
individual monolayers of MoS2 and WSe2 energy levels aligned with respect to the
vacuum level, and c, AB stacking of MoS2 /WSe2 heterobilayer.
The band structure calculations also enable us to understand the effects of strain on
the band hybridization and therefore the PL intensity trends of the interlayer exciton
observed in Figure 6.5. Specifically, we presume that the oscillator strength (𝜎𝑘 )
of this interlayer exciton transition is proportional to the MoS2 character (𝜁) at the

138
Γ point [95], i.e., 𝜎𝑘 ∼ 𝜁 = |h𝑀𝑜𝑆2 |Γi| 2 . Therefore, by assuming the oscillator
strength scales with the MoS2 character at the Γ point, i.e., Δ𝜎𝑘 /𝜎𝑘,0 ∼ Δ𝜁/𝜁0 we can
calculate relative oscillator strength modulations by examining the band structure
wavefunction projections (weighted orbital contributions). The calculations show
an increase of band hybridization (less the MoS2 contribution) with tensile strain
(Figure 6.11a). The relative modulation in 𝜁 would be approximately -0.02/% with
strain (Figure 6.13), where the negative sign signifies a decrease in 𝜁 with tensile
strain. Similarly, we performed calculations of the band structure as a function
of interlayer spacing (Figure 6.11c) and found an increase in band hybridization
with increased interlayer distance. We calculate the relative modulation in 𝜁 to be
approximately -0.13/Åwith interlayer distance (Figure 6.13). Thus, we predict that
the relative oscillator strength and therefore the PL intensity would decrease with
both tensile strain and increased interlayer distance.

Figure 6.13: Band hybridization as a function of strain with different interlayer
distances Band hybridization factors are extracted from weighted projections of orbital contribution for each strain- and interlayer distance-dependent band structures.
Our theoretical calculations corroborate well with the experimental trends of PL intensity observed for the interlayer exciton if we hypothesize that, (1) as tensile strain
is applied, the interlayer coupling is sufficiently strong so that the in-plane tensile
strain reduces the PL intensity due to the decrease in the oscillator strength, and (2)
as compressive strain is applied, the out-of-plane interlayer coupling is weakened,
so that the PL intensity also decreases due to the substantially reduced interlayer
coupling. Therefore, the PL intensity would likely be maximized near the zero-strain
configuration, as we observed experimentally. These effects are depicted schemat-

139
ically in Figure 6.11d and suggest that multilayer and heterobilayers TMDCs may
exhibit a positive Poisson effect where the reduced out-of-plane interlayer coupling
is always accompanied by compressive in-plane strain and vice versa. The calculated effects of larger strain values, other interlayer spacings, and different stacking
configurations on the electronic band structure are also shown in Figure 6.14, and
we find that the description given above is of general applicability for this relatively
twist-angle insensitive MoS2 /WSe2 heterobilayer [95, 187].

Figure 6.14: Effects of strain, interlayer spacing, and different stacking configurations on the electronic band structure of MoS2 /WSe2 heterobilayer system
obtained using DFT calculations. a, Transition energies as a function of uniaxial strain for different configurations of MoS2 /WSe2 heterobilayer. b, Transition
energies for AB-stacked heterobilayer system as a function of interlayer distance
for various strains. c, Transition energies for AA-stacked heterobilayer system as
a function of interlayer distance for various strains. The insets depict the stacking
order of the heterobilayer systems considered.

6.5

Conclusion and Outlook

In summary, our findings suggest that interlayer excitons in transition metal dichalcogenide heterobilayers are particularly attractive for strain engineering. Specifically,
we found that momentum-space indirect Γ − 𝐾 interlayer excitons in MoS2 /WSe2
heterobilayers have a deformation potential of approximately twice that of the constituent intralayer excitons. In addition, we observed that the interlayer coupling
of the heterobilayer can be directly tuned in our locally strained structures. We
showed that the simultaneous modulation of the out-of-plane interlayer coupling
and the in-plane strain in our wrinkled structures can explain the non-monotonic
dependence of the interlayer exciton PL peak intensity with peak energy. More
generally, the possibility of coupling between in-plane and out-of-plane effects in
multilayered structures, akin to a Poisson effect, should be considered when strain
engineering layered 2D materials. The existence of multiple knobs that strain can
tune in the interlayer exciton system highlights the potential and promise for the next

140
generation of interlayer exciton strain-based devices.
6.6

Appendix

Sample preparation via PTP assembly process
Wrinkled vdW heterostructures were fabricated via a combination of PDMS-toPDMS (PTP) assembly process and strain-release mechanism using a stretchable
substrate of PDMS. As the first step of the PTP dry assembly process, a monolayer
of WSe2 (HQ Graphene) is exfoliated directly on a uniaxially pre-stretched PDMS
substrate (∼120%) with a skin layer which was formed by O2 plasma treatment (Figure 6.2). This stiff skin layer on the PDMS is used to guide conformal out-of-plane
deformation (i.e., wrinkling) of the transferred vdW heterobilayer when the substrate is contracted after the release of the pre-stretched PDMS. The stiff skin layer
enables an increased transfer of strain by reducing the Young’s modulus mismatch
between the PDMS and the TMDC heterobilayer [113]. Next, a monolayer of MoS2
(HQ Graphene), is mechanically exfoliated directly on a separate PDMS substrate.
Then the exfoliated monolayer MoS2 /PDMS directly transfers onto the monolayer
WSe2 /PDMS via a gentle, heat-assisted (∼70◦ ) PTP transfer stamping (Figure 6.2),
forming a vertically stacked heterostructure (Figure 6.3). Target exfoliated monolayers can be integrated into a heterostructure without contacting any other materials
that are essential for avoiding lithographic or etching process-induced polymeric or
solvent residues. We note that our PTP assembly process works with either commercially available PDMS slabs (e.g., Gel-Pak) or home-made PDMS (Dow Inc.
Sylgard) substrates. Mechanical contraction after the release of the pre-stretched
PDMS resulted in periodic wrinkled heterobilayer. We note that the assembled
heterobilayer was constrained by a polymethylmethacrylate (PMMA) capping layer,
which acts as a clamp to prevent slippage of the heterobilayer when it conforms to
the underlying PDMS layer.
Electronic structure calculations
We performed ab initio DFT calculations for the different MoS2 and WSe2 monolayers, and MoS2 /WSe2 stackings using the Vienna Ab initio simulation (VASP) package to compute the atomic electronic structures [93]. The Perdew-Burke-Ernzerhof
(PBE) exchange-correlation functional was applied [148]. We used D3 corrections
for van der Waals interactions [66, 67]. We considered ground-state band structure
as an approximate description for the optical transitions in the MoS2 /WSe2 heterobilayer. The ultrasoft pseudopotentials were used with a 450-eV energy cutoff in all

141
simulations. The structure relaxation was performed using Gamma-point-centered
k-point of 4 × 4 × 1. All structures were fully relaxed until the force on each atom
reached less than 0.01 eV/Åand the total energy is converged within 10−6 eV. Each
unit cell consisted of 12 atoms of MoS2 as a monolayer or 12 atoms of WSe2 as
a monolayer, or both for the MoS2 /WSe2 stacking. During the relaxation of the
MoS2 /WSe2 stacking structure, the volume of the unit cell is kept variable to determine the optimized lattice constant and to reduce the mismatch between the MoS2
and WSe2 allowing determination of the unstrained structure. In all simulations, the
vacuum is kept at least 15 Åto avoid spurious interactions in the aperiodic direction.
Once the structures were fully relaxed and the unstrained configurations were determined, we applied strain in one of the periodic directions. After obtaining all the
relaxed configurations of the unstrained and strained structures, the band structures
were computed using a total of 90 K-points in the G-K-M-G path.
Strain tuning coefficient
We also estimated maximum strain accumulated at the crest of wrinkles by using
measured wrinkle geometry as shown in the wrinkle height profile (Figure 6.2b).
In equilibrium wrinkles, the shear traction on the interface vanishes, and thus, inplane membrane energy will balance the bending energy of a stiff layer, which is
MoS2 /WSe2 /skin layer [74]. As a consequence, with the elastic plate theory, the
maximum uniaxial tensile strain (𝜀) can be estimated as 𝜀 ∼ (𝜋 2 𝑡ℎ)/((1 − 𝑣 2 )𝜆2 )
[22, 196], where 𝑡 is the thickness of the stiff layer (skin layer + flake layers), 𝑣 is
the effective Poisson’s ratio, and ℎ and 𝜆 are measured wrinkle geometry of height
(crest-to-valley) and wavelength, respectively. First, we fitted our measured height
profile to a sinusoidal profile to obtain the average amplitude (∼0.44 𝜇m, and thus
the height of ∼0.88 𝜇m), and wavelength (∼4.36 𝜇m) of wrinkles (Figure 6.10a).
The thickness of the formed silica skin layer was determined by X-ray reflection
analysis, which is approximately 15 nm (Figure 6.10b). The effective Poisson’s
ratio is determined based on Poisson’s ratios of MoS2 (0.125), WSe2 (0.19), and
silica skin layer (0.17). The maximum uniaxial tensile strain is then estimated as
about 0.76%. This geometry-driven estimated strain value is slightly lower than our
Raman-driven estimated strain (∼0.8%), and we attributed the discrepancy to the
use of averaged geometry values of wrinkles, where the actual height profile shows
a smaller radius of curvature at the crest compared to the fitted curve as shown in
Figure 6.10a.

Part IV
What’s next for Flatland?

142

143
Chapter 7

CONCLUSIONS AND OUTLOOK
“The first principle is that you must not fool yourself
– and you are the easiest person to fool.”
— Richard P. Feynman

In this thesis, we have demonstrated the plethora of different physics and design considerations for photovoltaic systems with atomically thin active layers. Specifically,
we have demonstrated the following:
1. In Chapter 2, we laid out the theoretical framework that shows the impact
of band-edge abruptness on the efficiency potential of a photovoltaic active
layer. This analysis is applicable to all photovoltaic systems that obey the
optoelectronic reciprocity relations, which includes both atomically-thin and
conventional systems. We further discussed the role of the indirect and direct
band-edge in electronically bulk transition metal dichalcogenides, and showed
that materials that minimized this energetic difference had a substantially
higher efficiency potential (i.e., is much closer to its detailed balance limit).
2. In Chapter 3, we laid out the theoretical framework and experimentally
achieved near-unity absorbance in both ultrathin (∼10 nm) and atomicallythin (∼7 Å) transition metal dichalcogenides at room temperature. In the case
of the ultrathin active layers, this unity absorbance was achieved by using loss
as a critical component of the design space. In the case of the atomically-thin
active layers, we required an impedance matching of the surface conductivity
to free space, which ultimately related the radiative and non-radiative rates of
the excitonic system. Finally, with these demonstrations of unity absorbance,
we considered theoretically the possibility of creating mulitjunction excitonic
absorbers with efficiency potentials that far surpass the single-junction limit,
which takes advantage of the van der Waals coupling in these layered materials. In a van der Waals heterostructure, the solar absorbance can further be
distributed amongst many different layers.
3. In Chapter 4, we used the optical architecture demonstrated in the previous
chapter for ultrathin unity absorbers and combined it with an electronic geom-

144
etry suitable for high exciton dissociation efficiency and subsequent carrier
collection efficiency. Specifically, we demonstrated that a combination of a
MoS2 -WSe2 pn heterojunction with a transparent top graphene contact would
enable both internal quantum efficiencies above 70% and external quantum
efficiencies above 50%. These values remain the highest values for this thickness regime of active layers (subsequent work led by Cora Went demonstrated
internal quantum efficiencies above 90% [204]).
4. In Chapter 5, we theoretically and experimentally demonstrated the concept
of a ‘band bending junction’, a junction that separates charge carriers due
to the extent of vertical band bending. The difference in the vertical band
bending, in this case, was due to a difference in thicknesses in the active
layer. Therefore, materials that are effectively the same, i.e., with the same
quasi-particle bandgap, can separate carriers at the thickness step due to the
local symmetry breaking of the band profile.
5. In Chapter 6, we theoretically and experimentally demonstrated the strain tuning of interlayer excitons in a MoS2 -WSe2 heterobilayer. We demonstrated a
deformation potential of approximately 100 meV/%, which is roughly twice
that of the conventional intralayer excitons of the constitutient materials. Furthermore, we postulated that an additional effect strain may have on van
der Waals heterostructures is there interlayer coupling. Therefore, van der
Waals heterostructures may have significant potential for strain engineering
applications.
Combined, these results demonstrate the exciting possibilities, physics, and different
engineering challenges that atomically-thin materials face as active layers in photovoltaic systems. We briefly surmise possible future directions and give an outlook
on each of the main results demonstrated in this thesis:
7.1

Outlook from Semiconductor Band Tails Work

In the analysis related to the semiconductor band tails, there are a few interesting
research directions to pursue. One aspect is to quantitatively constrain the efficiency
potential of the different TMDC systems by accurately and systematically measuring their low-energy absorption coefficients and radiative efficiencies. Specifically,
we have found that the absorption coefficient at the indirect band-edge to be quite
critical to the specific efficiency penalty, and so the indirect band-edge of materials like WSe2 , MoSe2 , and MoTe2 should be carefully measured. To perform

145
these measurements, it is possible to do them directly through a highly sensitive
absorbance measurement, such as photothermal deflection spectroscopy, fourier
transform photocurrent spectroscopy, or just typical photocurrent spectroscopy with
lock-in detection. However, a potentially critical component is that the specific
indirect band-edge absorption coefficient will likely depend on thickness. This can
be intuited from examining the effects of thickness on the bandstructure. While
this is readily evident in the one-to-seven layer regime, it likely extends for 10s of
nm, since typically quantum-well physics takes over in that regime of thickness.
Therefore, it may be important to devise an experimental method of extracting the
indirect band-edge as a function of thickness. One method to do so is purely from
photoluminescence measurements. As we know from Chapter 2, the luminescence
is a direct read-out on the absorption profile near the band-edge. Therefore, with
some careful quantitative measurements and analysis, it should be possible to use
thickness-dependent photoluminescence data to extract out the thickness-dependent
absorption coefficient of the different TMDCs.
Another research direction to pursue in regards to semiconductor band tails is examining the role of band tails in excitonic spectra and understanding their microscopic
origin. Specifically, while the Lorentzian oscillator model seems to suggest that
nearly all resonances should ‘look’ like a Lorentzian, it is well documented that
there are significant differences both on the low and high energy side of the resonance. On the high energy side of an excitonic resonance, at room temperature,
the spectrum should have a Boltzmann tail, which describes the occupation factor
and thermalization of excitons as they scatter with each and off the crystalline lattice. On the low energy side, the dynamics of the tail are related to the excitonic
dissociation and dephasing process. Specifically, it can be shown that under the
application of an electric field, excitonic absorption resonances also exhibit a band
tail that is due to the tunneling of an electron out of its Coulomb potential with the
hole. This tunneling is exponentially suppressed, which results in an exponential
tail near the band-edge [48]. This physics is similar to that of the band tail in the
Franz-Keldysh effect, but is modified in the presence of excitons. Disorder in the
energetic landscape furthers increases this band-tailing, and gives ample opportunity for the excitons to dissociate and dephase. A highly sensitive spectroscopic
technique near the excitonic absorption edge, as a function of external parameters
such as the electric field and/or temperature, may provide a lucid description of the
internal physics.

146
Finally, we postulate a completely different application of semiconductor band tails,
which is to achieve low-threshold (solar-driven) lasing. In particular, it is well
known that the condition for lasing is some form of population inversion. In the
case of semiconductors, this is sometimes referred to as the Bernard-Duraffourg
condition:
Δ𝜇 = 𝐸 𝑓𝑛 − 𝐸 𝑓 𝑝 > 𝐸 𝑔
(7.1)
In conventional solar systems, this condition is never reached since the solar fluence
is substantially lower than the injection densities required. Even in the radiative
limit, i.e., no non-radiative recombination, we have Δ𝜇 = 𝑞𝑉𝑜𝑐 < 𝐸 𝑔 , and is usually
several hundred millivolts away. However, we also found in Chapter 2 that this
condition is routinely satisfied at 1-sun conditions when we have semiconductor
band tails (in the radiative limit). The simple intuitive explanation for this is that the
system is akin to that of a 4-level system, which is well-studied in laser systems and
substantially relaxes the condition for population inversion by providing (1) a large
source of carriers at the high energy state (2) rapid transfer of carriers to the low
energy states (3) sufficiently long lifetimes at the low energy state. An analogous
system would be a semiconductor with a large Stokes shift between the absorption
and emission edges with near-unity radiative efficiency. This is certainly uncommon
experimentally given the typical nature of band tails and the coupling of that internal
physics to that of radiative efficiency. However, recent work in the quantum dots
literature suggests that they can act as both near-unity radiative emitters with a
sizable Stokes shift. In this case, the unity radiative efficiency is achieved by careful
surface passivation and epitaxy of the nanocrystal core-shell-ligand structure, and
the Stokes shift stems not from defective energy states, but from a type-I band
alignment between the core and the shell.
7.2

Outlook on Unity Absorbance and Cavity Coupling to Excitonic Absorbers

The work in Chapter 3 describes the methods of achieving critical coupling to active
layers of TMDCs, which results in unity absorbance. This condition can also be
intuited as a matching of radiative and non-radiative rates of the system. However,
of substantial interest is the prospect of strong coupling to the excitonic systems.
This occurs when 𝜅 > 𝛾1 , 𝛾2 , where 𝜅 describes the coupling rate constant and 𝛾1,2
describe the internal loss constant of each individual system (in this case, the two
systems are that of the exciton and of the optical cavity). When the exchange in
energy between the systems is faster than their individual decay rates, the individual

147
modes hybridize and form new eigenstates of the system. While strong coupling
typically does not result in unity absorbance (as we are moving away from the critical
coupling regime), it is of considerable interest to understand whether the presence of
exciton-polariton states affect the transport or recombination dynamics. Some of this
work has begun in the organic photovoltaics literature [139, 200], where improved
energy harvesting and modified recombination dynamics have been achieved. In
the case of layered TMDCs, the excitons are more akin to that of a Mott-Wannier
type, and therefore it’d be of interest to explore the differences and consequences of
‘strongly coupled’ photovoltaics.
Another research direction of interest is the consequences of the unity absorbance
in the monolayer excitonic absorber. Specifically, let us consider the situation where
we excite the exciton on resonance with a coherent source, e.g., a laser. In this case,
the excitons that are generated must obey momentum conservation (this is generally
true) and therefore must be generated at the intercept between the light cone and
the excitonic bandstructure in the dispersion relations. If the excitons do not scatter
strongly with phonons or other quasiparticles, they will remain precisely where
they were generated, and if they were to emit (which they would be able to do so
efficiently, by reciprocity), their angular emission profile would be exactly the same
as the angular profile of the excitation source. In other words, there is a conservation
of etendue due to the momentum conservation of the excitons, and this would yield
several hundred millivolts in the open-circuit voltage. In other words, the momentum
conervation would enable natural angle restriction. This physics requires weak
interaction of the exciton with other quasiparticles (such as phonons). Equivalently,
a strong light-matter interaction (i.e., large optical absorption/emission rates) would
also significantly enhance these effects. The effects of the cavity enhances this
physics, and the main experimental difficulty would be to discriminate between the
excitation source (i.e., the laser) and emission (luminescence from the exciton),
since the energies would be quite similar near resonance. One way to do this
is to do polarization-resolved measurements, or examining the side bands of the
absorption/emission profiles and using sufficiently sharp energy filters. Another
possibility is to couple and examine the light sufficiently off-incidence.
Finally, we discuss the prospects and microscopic origin of near-unity absorbance,
reflectance, and quantum yield in monolayer TMDCs. The analysis described in
Chapter 3 show a direct relationship between the absorbance (and equivalently,
reflectance, since 𝐴 = 1 − 𝑅) between the radiative and non-radiative rates of the

148
exciton. It is also understood that the quantum yield of an emitter is generally given
as
𝛾𝑟
QY =
(7.2)
𝛾𝑟 + 𝛾𝑛𝑟
Thus, we consider a few different regimes of non-radiative rates. For very low nonradiatve rates, the sheet conductivity model predicts a large increase in reflectance
due to the admittance approaching zero. In the limit of no non-radiative recombination, 𝑟˜ → −1, which is the same reflectance amplitude of a perfect mirror. This
can also be observed by analyzing the expression for reflectivity of a suspended
excitonic system, where
𝑅(𝜔 = 𝜔 𝑒𝑥𝑐 ) =

𝛾𝑟2
= QY2
(𝛾𝑟 + 𝛾𝑛𝑟 ) 2

(7.3)

In other words, the reflectance from a suspended exciton can be used to infer the
quantum yield of the system. Similarly, we understand from Chapter 3 that nearunity absorbance occurs when 𝛾𝑟 ≈ 𝛾𝑛𝑟 (i.e., QY ≈ 50%). Finally, we should note
that the total linewidth is typically given as 𝛾𝑇 = 𝛾𝑟 + 𝛾𝑛𝑟 . Therefore, one would
expect that as the non-radiative rates decreases, the quantum yield increases and the
reflectance increases. This can be observed even in the dielectric lorentzian oscillator model (Equation 3.1), where the decreased non-radiative rate would reduce
the scattering rate 𝛾 𝑘 . These observations also correspond quite well with recent
experiments where excitons are strain tuned [7] to reduce their non-radiative scattering rates. However, while near-unity reflectance and absorbance has recently been
demonstrated by our group and others [51, 173, 208] (where the total linewidth corresponds quite well to specific reflectance/absorbance measured), the Javey group
has demonstrated near-unity quantum yield at room temperature with no apparent
change in the linear dielectric function [2]. This peculiar observation suggests that
the internal physics of the unity quantum yield observed by the Javey group is likely
quite different than that due to the pure radiative and non-radiative broadening of an
excitonic transition. Recent analysis from the Tisdale group has also suggested that
the unity radiative efficiency is likely not a consequence of the intrinsic properties of
the exciton, but rather a consequence of highly radiative, long lifetime trap states that
effectively protect the exciton from non-radiative recombination by capturing them
into this defect state and subsequently thermalizing the excitons back to the bandedge [61]. WS2 may serve as a highly interesting candidate since unity absorbance
and quantum yield has been demonstrated at room temperature. Therefore, by further varying the material’s properties with electrical gating, temperature, strain, and

149
cavity coupling, it may be possible to discriminate the origin of the near-unity quantum yield. If these two effects are decoupled, it may also pave way for a material that
can exhibit near-unity absorbance and quantum yield simultaneously, which would
be an ideal optoelectronic material for many applications, including photovoltaics.
7.3

Outlook on Achieving High-Efficiency, TMDC-based Photovoltaic Devices

In Chapter 3 and Chapter 4, we demonstrated that ultrathin TMDC active layers
can achieve both near-unity absorption and carrier collection efficiency. Therefore,
the main challenge that remains is addressing the open-circuit voltage. Aside from
circuit-level degradation of photovoltaic behavior (e.g. see section 1.1), what else
could degrade the open-circuit voltage? Furthermore, despite the low relatively low
radiative efficiency of electronically bulk TMDCs, their radiative efficencies still
have been characterized as being approximately 10−4 . In other words, using the
well-known voltage penalty due to a non-radiative component (c.f. Chapter 2), we
would expect a 𝑞Δ𝑉𝑜𝑐 = 𝑘 𝑏 𝑇 ln(𝐸 𝑅𝐸) ≈ −240 meV voltage penalty. For a bandgap
of 𝐸 𝑔 ≈ 1.3 eV, the radiative limit 𝑉𝑜𝑐 is roughly 𝑉𝑜𝑐,𝑟𝑎𝑑 = 1.0 V. Therefore, even
accounting for the non-unity radiative efficiency, we would expect an open-circuit
voltage limit that approaches 700 mV. However, in the cells described in Chapter 4,
we have drastically lower open circuit voltages, and more generally, low open circuit
voltages has been observed throughout the literature [81]. What causes such a
discrepancy?
The main difference between the case where the radiative efficiency is approximately
10−4 and that of the device architecture is the electronic materials that surround the
active layer. More specifically, the insulating substrates like PDMS or SiO2 that is
typically used to characterize the optical properties of TMDCs do not drastically
modify the surface recombination. However, near a metal (like that in a device),
carriers are rapidly quenched and additional surface recombination occurs at these
interface. Therefore, the open circuit voltage potential is substantially reduced
compared to the previous situation, and there is an effective decrease in the external
radiative efficiency even further. To mitigate this issue, we explored ‘carrier selective
contacts’, i.e., large bandgap materials that are predominantly conducting for a
single carrier type (i.e., either electrons or holes). These types of contacts ‘repel’
the presence of the other charge carrier, and therefore, by reducing the overall
overlap between the electron and hole, we can significantly reduce the presence
of additional surface recombination. The situation with a dielectric like SiO2 is
one where we effectively repel carriers of both types, and so that the internal

150

Figure 7.1: I-V performance of a carrier selective contact device. a 𝐼 − 𝑉
curves in the dark and under illumination for a Au/Ti/TiO𝑥 /WSe2 /NiO𝑥 /Au device.
Inset is a cross-sectional schematic of the device under study. b Laser powerdependent properties of the short-circuit current (blue, left) and of the open-circuit
voltage (orange, right) of the device under study. The inset is an image where the
photocurrent image (blue-white scale) and reflection image are superimposed (grey
scale), showing the region where the device was illuminated (red dot, 𝜆 = 633 nm)

luminescence is maintained but because of the lack of conductivity, no photocurrent
could be extracted in a hypothetical device with SiO2 . Therefore, carrier selective
contacts require both passivation and conductivity (which would result in carrier
selectivity [142]). An examination of these carrier selective contacts is shown in
Figure 7.1, where we demonstrate open circuit voltages that approach 700 mV. In
this geometry, we utilize TiO𝑥 and NiO𝑥 as selective contact materials, whose band
diagrams are shown in Figure 7.2. This electrical geometry is optimal for blocking
carriers of the ‘wrong’ type, since the bandgap of the contact materials are so large
that the effective conductivity of the other material type approaches zero [211].
However, the overall conductivity of the contacts could be substantially improved,
as evidenced by the low short-circuit current.
The low short-circuit current observed in Figure 7.1 suggests that carefully controlling the doping of these selective contacts will be important. Exploring other
materials as selective contacts may be of great interest too, borrowing specifically
from the existing literature of perovskites and organic photovoltaics [205]. It may
be of interest to use other 2D materials, such as large bandgap materials like GaS,
to create atomically-pristine van der Waals interfaces. The main challenge in every
case will be to achieve both high passivation and high carrier conductivity. How-

151
𝐸𝑓 – 𝐸𝑣 = 0.29 ±0.08 eV

𝐸𝑓 – 𝐸𝑣 = 3.08 ±0.18 eV

Figure 7.2: X-ray photoemission spectroscopy of TiO𝑥 and NiO𝑥 . Photoemission
spectrum near the valence band-edge of a NiO𝑥 and b TiO𝑥 . Also shown is a linear
fit to the background (blue line) and the valence band (red line). The intercept can be
used to quantify the valence band edge relative to the Fermi level. Binding energies
are referenced to the Au 4f core level. c Drawn band diagram using literature values
of the work-function, electron affinity, and bandgap. 𝐸 𝑓 − 𝐸 𝑣 is taken from the
values in a and b.
ever, it is important to note that this scheme of carrier selective contacts has been
an extremely effective method of achieving near the maximum power conversion
efficiency potential of a given material (limited by the material’s optical properties,
c.f. Chapter 2) in other photovoltaic active layers, and could be applied to both
ultrathin and atomically-thin TMDC materials if successful.

152
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Appendices

176

177
Appendix A

MICROSCOPIC PHYSICS OF EXCITONIC SYSTEMS
A.1

Formation and Dissociation of Excitons

Introduction to the 3D Hydrogenic Model for the Exciton
An exciton is a quasiparticle in condensed matter systems that was first theorized
when sharp resonances in absorption appeared below the quasiparticle bandgap.
These sharp resonances were attributed to electron-hole interactions, i.e. electronhole pairs were correlated with Coulomb interactions. There correlations describes
the motion and dynamics of a new quasiparticle, called an exciton. The simplest
model for an exciton is analogous to a Hydrogen atom (more precisely, positronium),
where the appropriate Hamiltonian to describe this system is given simply by
 2
ℏ 2
1 𝑒2
− ∇ −
𝜓 = 𝐸𝜓
(A.1)
2𝜇
4𝜋𝜀 0 𝜀 𝑠 𝑟
where note the difference between the Hydrogen atom and the exciton in this simple
model is primarily the reduced effective mass
= ∗+ ∗
𝜇 𝑚𝑒 𝑚 ℎ

(A.2)

and the static dielectric screening 𝜀 𝑆 . We are also at the present, not interested
in the center of mass momentum 𝐾, as the solutions are merely a plane-wave
exp(𝑖𝐾 · 𝑟)/ 𝑉 with energies ℏ2 𝐾/2𝜇 and do not contribute to the main physics.
In practice, we must also consider that the static dielectric screening must be spatiotemporally dependent, i.e.
𝜀 𝑠 = 𝜀 𝑠 (𝜔, 𝑘)
(A.3)
to my knowledge, it is not particularly clear what the exact dependence is for this
relation, but a rough physical picture suggests that the frequencies to consider are
2𝜋/𝜏 . 𝜔 . 𝐸 𝑏 /ℏ i.e. the frequencies of the dielectric should be faster than the
exciton lifetime 𝜏, or else the interactions are too slow to be of interest to renormalize
the electron-hole interaction. The highest frequency will be given by the energy of
the exciton itself, i.e. its binding energy 𝐸 𝑏 . In terms of its spatial dependence, we
suspect the maximum extent will be governed by the wavefunction of the exciton,
whose natural length scale will be its Bohr radius 𝑎 0 . In other words, we would
expect the interactions to be interesting only for 𝑟 . 𝑎 0 i.e. 𝑘 & 2𝜋/𝑎 0 . Despite not

178
knowing the exact functional dependence of 𝜀 𝑠 , we shall solve for the eigenenergies
anyway assuming a constant value (or an averaged value). To do so, we note that
we can define the renormalized Rydberg energies and Bohr radius as
Ry ≡

𝑒4 𝜇
𝑒2
2 (4𝜋𝜀 𝑠 𝜀0 ) 2 ℏ2 2 (4𝜋𝜀 𝑠 𝜀0 )𝑎 0

𝑎0 ≡

4𝜋𝜀 𝑠 𝜀0 ℏ2
𝑒2 𝜇

(A.4)

and in these new energy and spatial units, i.e. 𝐸 = 𝜖Ry and 𝑥 = 𝑋𝑎 0 , 𝑦 = 𝑌 𝑎 0 ,
𝑧 = 𝑍𝑎 0 , 𝑟 = 𝑅𝑎 0 , we can divide (A.1) by the Rydberg energy to yield
𝑎0 
−𝑎 20 ∇2 − 2
𝜓 = 𝜖𝜓
(A.5)
which is conveniently compatible with the dimensionless units of the Bohr radius,
i.e.,
𝜓 = 𝜖𝜓
(A.6)
−∇ −
The solutions for the differential equation can now be solved in various ways, e.g.
rewriting the Laplacian into spherical coordinates, using separation of variables
and expanding in terms of spherical harmonics, Legendre polynomials, and solving
the resultant differential equation in terms of 𝑅. This mathematical analysis is not
particularly interesting, and is repeated in many textbooks - what is found is that
discrete bound states indeed exist, with energies given by the relation
𝜖𝑛 = − 2 ,

𝑛 = 1, 2, 3, . . .

(A.7)

i.e., the solutions are exactly the same as the Hydrogen atom, with a new Rydberg
energy. Notice these energies are negative, in other words, if a free electron-hole
pair can be generated at the quasiparticle gap 𝐸 𝑔 , then the excitonic absorption
occurs at ℏ𝜔 = 𝐸 𝑔 − Ry/𝑛2 .
Field Dissociation of a 3D Exciton
We are now interested in the question of whether it is possible to dissociate an
exciton with an electric field 𝐹. We shall assume this field 𝐹 is uniform throughout
the exciton, and is akin to the situation of a field throughout the bulk of a material.
However, this analysis represents a qualitative picture of the effects e.g. of fields at
an interface (e.g. a built-in field in a semiconductor junction). The analysis begins
similar to what is described above, in analogy to field dissociation in a Hydrogen
atom. The field modifies the Hamiltonian with an extra term
 2
1 𝑒2
ℏ 2
− 𝑒𝐹𝑧 𝑧 𝜓 = 𝐸𝜓
(A.8)
− ∇ −
2𝜇
4𝜋𝜀 0 𝜀 𝑠 𝑟

179

Figure A.1: Exciton Wavefunction under an Electric Field. Wavefunction of an
exciton in an elecric field for different electric field values (see Figure A.2) for the
corresponding electric field strength for a given color.

where we rotate the system such that the field occurs in the 𝑧 direction and 𝐹𝑧
describes the magnitude of that field. The subsequent analysis proceeds similarly to
before, where we conveniently can rewrite the Hamiltonian into excitonic units to
get
(A.9)
−∇ − − F 𝑍 𝜓 = 𝜖𝜓
where we have defined the electric field in atomic units to be F = 𝑒𝐹𝑧 𝑎 0 /𝐸 𝑏,0 . Here,
notice that we have used the previous notion that the binding energy of the exciton is
given by the largest energy in the Rydberg series (𝑛 = 1), i.e. 𝐸 𝑏,0 = Ry. This equation, to the author’s knowledge, cannot be solved analytically. A numerical solution
can be found, however. Below, we plot in Figure A.1 the normalized wavefunction in∫ ∫
|𝜓(𝑋, 𝑌 , 𝑍)| 2 𝑑𝑋 𝑑𝑌
tegrated over the two other spatial dimensions |𝜓(𝑧, 𝐹𝑧 )| 2 =
to analyze the effects of field on the wavefunction properties. We consider a finite
difference grid with dimensions given by [−5, 5] × [−5, 5] × [−5, 5] all in excitonic
radii units. Interestingly, we find that as the field increases, the electron partially
escapes the Coulombic attraction with the hole, i.e. the exciton dissociates. To
analyze this dissociation effect, we note that if 𝑅 ≈ 𝑍, the potential for 𝑍 > 0 is
maximized at 𝑍 = 𝑍 𝑚𝑎𝑥 = 2/𝐹. We consider any part of the wavefunction that is
past this maximum potential effectively ionized, giving the dissociation efficiency
∫∞
to be 𝜂 𝑑𝑖𝑠𝑠 = 𝑍 |𝜓(𝑍)| 2 𝑑𝑍. We plot this efficiency for different field strengths in
𝑚𝑎𝑥

180

Figure A.2: Exciton Dissociation Efficiency under an Electric Field. Dissociation
efficiency of an exciton, calculated via a tunneling probability, of an exciton in an
electric field.

Figure A.3: Exciton Binding Energy under an Electric Field. Quadratic Stark
shift effect of the 1s exciton in an electric field.
Figure A.2. Lastly, we analyze the consequences of the electric field on the binding
energy 𝐸 𝑏 . We first note that for a 1s exciton that obeys spherical symmetry, there
is no preferred orientation. Therefore, to first order, we expect no field dependence.
To second order, we expect there to be a field dependence given as Δ𝐸 𝑏 ≈ −𝛽𝐹𝑧2 .
We can solve for this dependence numerically, which is plotted below in Figure A.3.

181

Figure A.4: Exciton Binding Energy vs. Screening Length.Binding energy of the
exciton 𝐸 𝑏 relative to the energy difference of the first and second excited state Δ12
for a given screening length 𝑟 0 in units of the Bohr radius 𝑎 0 . For a 2D Hydrogenic
exciton, the first excited exciton has a binding energy of 4Ry and second excited
exciton has a binding energy of (4/9)Ry. Thus, the difference in transition energies
is Δ12 = (32/9)Ry and 𝐸 𝑏 /Δ12 = 9/8, which is referred to as the "2D Coulomb
Limit" in this plot. For a given screening length 𝑟 0 /𝑎 0 , it is possible to use an
experimentally measured Δ12 to deduce the binding energy of the exciton 𝐸 𝑏 .
2D Keldysh Problem
Recall the Hamiltonian for a 2D Coulomb Problem, given as
 2
ℏ 2
𝑒2
𝜓 = 𝐸𝜓
− ∇ −
2𝜇
4𝜋𝜀 𝑠 𝜖0𝑟

(A.10)

Recall that there are two important definitions in this problem, given by the Rydberg
energy and Bohr radius:
Ry ≡

𝑒4 𝜇
𝑒2
2 (4𝜋𝜀 𝑠 𝜀0 ) 2 ℏ2 2 (4𝜋𝜀 𝑠 𝜀0 )𝑎 0

𝑎0 ≡

4𝜋𝜀 𝑠 𝜀0 ℏ2
𝑒2 𝜇

(A.11)

We can define dimensionless units 𝑥 = 𝑎 0 𝑋, 𝑦 = 𝑎 0𝑌 , 𝑟 = 𝑎 0 𝜌 and 𝐸 = 𝜖Ry and
divide the time-independent Schrodinger equation by the Rydberg energy to get
 4
ℏ (4𝜋𝜀 𝑠 𝜀0 ) 2 2 2(4𝜋𝜀 𝑠 𝜀0 )ℏ2 1
∇ −
𝜓 = 𝜖𝜓
(A.12)
𝑒 4 𝜇2
𝑒2 𝜇
notice that the terms neatly reorganize into the bohr radius, i.e.,
𝑎0 
−𝑎 20 ∇2 − 2
𝜓 = 𝜖𝜓

(A.13)

182
Rewriting the length units in terms of the bohr radius gives
−∇ 𝜌 −
𝜓 = 𝜖𝜓

(A.14)

which is the Coulomb problem in excitonic units. Note that we use 𝜌 to denote the
2D nature of the expression, as opposed to when we used 𝑅 for the 3D Hydrogenic
model. Let’s now see if we can do the same with the Keldysh potential, given as
  
 
𝜋𝑒 2
𝑉𝑒ℎ (𝑟) = −
𝐻0
− 𝑌0
(A.15)
4𝜋𝜖 𝑆 𝜖 0 (𝜀1 + 𝜀2 )𝑟 0
𝑟0
𝑟0
which is an effective electrostatic interaction for two charges within a thin 2D
dielectric continuum [13, 29]. The screening length 𝑟 0 gives a crossover length
scale between a 1/𝑟 Coulomb interaction at large separation and a weaker log(𝑟)
interaction at small separation. To solve the binding energy problem for these 2D
excitons, we use similar analysis to above, where we can again divide by the Rydberg
energy to get
  
 
2𝜋𝑎 0
𝜈𝑒ℎ (𝑟) = −
𝐻0
− 𝑌0
(A.16)
(𝜀1 + 𝜀2 )𝑟 0
𝑟0
𝑟0
rewriting in terms of dimensionless units, where 𝑟 = 𝑎 0 𝜌, we have
 
 
2𝜋𝑎 0
𝑎0 𝜌
𝑎0 𝜌
−∇ 𝜌 −
𝐻0
− 𝑌0
𝜓 = 𝜖𝜓
(𝜀1 + 𝜀2 )𝑟 0
𝑟0
𝑟0

(A.17)

which can be solved numerically by discretizing the differential equation, as before.
The solutions can be used to infer the exciton binding energy for an observed value
of Δ12 , as seen in Figure A.4. Here, Δ12 = |𝐸 1 − 𝐸 2 | is the energetic differences
between the first two excitons in a hydrogen-like series. Also depicted in Figure A.5
are the excitonic wavefunction solutions of the 2D Keldysh potential.

183

Figure A.5: Wavefunctions of the excitons in the 2D Keldysh potential, for 𝑛 =
1, 2, 3, 4 for each row, i.e. 𝑠, 𝑝, 𝑑, 𝑓 -like wavefunctions. Wavefunctions are similar
to the hydrogenic model.

184
Appendix B

MACROSCOPIC OPTICAL PROPERTIES OF LAYERED
STRUCTURES
B.1

Boundary conditions on Electromagnetic Fields

Let us do a brief introduction and primer on Maxwell’s equations before we go into
the scattering matrix method. The method is generically simple and only requires
us to recall Maxwell’s equations and its boundary conditions. Let’s state Maxwell’s
equations in differential form:
∇·E=

𝜖0

∇·B=0
∇×E=−

(B.1)

𝜕B
𝜕𝑡

∇ × B = 𝜇0 J + 𝜇0 𝜖 0

𝜕E
𝜕𝑡

which is what we normally see in introductory electromagnetism courses. Of course,
we can also state how they change in matter by using the constitutive relations
D = 𝜖E

(B.2)

B = 𝜇H
where 𝜖 and 𝜇 are the dielectric permitivity and magnetic permeability respectively.
Then, we can reformulate Maxwell’s equations in a particularly simple form. Moreover, let’s take there to be no free electrical charges or source of currents, which
is usually the case when we’re considering the type of optical experiments we’re
interested in. So, Maxwell’s equations become
∇·D=0
∇·B=0
𝜕B
𝜕𝑡
𝜕D
∇×H=
𝜕𝑡
∇×E=−

(B.3)

which looks particularly simple and free of phenomenological parameters! (Of
course, this is an illusion, as they’re embedded in the D and H fields.) Then, let’s

185

Figure B.1: A schematic of a Gaussian pillbox (dotted box) with height ℎ and area
𝐴 sandwiched between two materials with dielectric constant 𝜖1 and 𝜖2 .
derive the boundary conditions from using Stokes’ and Gauss’s theorem:
(∇ × F) · 𝑑A =
F · 𝑑r (Stokes)

𝜕𝑆

(∇ · F) 𝑑𝑉 =

(B.4)
F · 𝑑A

(Gauss)

𝜕𝐸

for which we shall apply to Maxwell’s equations.
Perpendicular Components
Let’s look at the first two Maxwell’s equations in (B.3), which essentially look the
same except for what we call D and B (incidentally, because of the symmetry of
the two fields, textbooks typically stick to a basis of using either D and B or E
and H. Personally, I’ve found using E and H rather convenient and pretty). Then,
suppose we draw a Gaussian pillbox (which we shall abbreviate as G.P.) as we do
in Figure B.1 which we can use as our volume in Gauss’s theorem. Then
(∇ · D) 𝑑𝑉 =
D · 𝑑A = 0
(B.5)
Vol(G.P.)

Surf(G.P.)

In the limit as the thickness ℎ goes to 0, there must not be any contribution from
the sidewalls of the pillbox, for the flux is proportional to the area of the sidewall,
which is going to 0 as ℎ → 0. Thus, the only contribution can come from the top
and bottom areas, each with area 𝐴. If we call the direction perpendicular to the top

186
and bottom surface 𝑛ˆ top and 𝑛ˆ bottom , then we have
D1 · 𝑛ˆ top = −D2 · 𝑛ˆ bottom

(B.6)

where D1,2 is the displacement field in the layer with dielectric constant 𝜖1,2 , respectively, and we have used the fact that the surfaces have equal areas and divided them
out. However, notice that actually 𝑛ˆ top = −𝑛ˆ bottom because the outward pointing
normals for the top and bottom surfaces are in opposite directions, so let’s just call
one of them 𝑛ˆ (it doesn’t matter which one, as long as we stay consistent). Moreover,
notice that in the limit of looking at one point, there is only one normal vector at that
point and the surface looks flat in that limit. Then, the above boundary condition
holds point-by-point for all points between the two surfaces. Lastly, as we said
before, the first two Maxwell’s equations are exactly the same under interchange of
D and B. So, we have
D1 · 𝑛ˆ = D2 · 𝑛ˆ
B1 · 𝑛ˆ = B2 · 𝑛ˆ

(B.7)

which are our first two boundary conditions that relate the components perpendicular
to the boundary.
Parallel Components
In a very analogous way, we can derive the boundary conditions on the components
of the fields parallel to the interface. If we examine the third and fourth of Maxwell’s
equations in (B.3), we see that they are essentially equivalent up to a redefinition:
E ↔ H and B ↔ −D. So, let us just derive the boundary conditions on, say, the
third equation. Let us draw a rectangular loop as in Figure B.2 with length 𝑙 and
height ℎ. We shall integrate over this loop and use Stokes’ equation:
(∇ × E) · 𝑑A
E · 𝑑r =
Loop

Surf(Loop)

𝜕B
· 𝑑A
𝜕𝑡

(B.8)

Surf(Loop)

=−

𝜕𝜙B
𝜕𝑡

where we have interchanged the time derivative and the surface integral and defined
the magnetic flux as 𝜙B =
B · 𝑑A. Then, notice what happens when we take
ℎ → 0. It necessarily takes 𝜙B → 0 since the magnetic flux is proportional to the

187

Figure B.2: A schematic of a Gaussian loop (dotted rectangle) with height ℎ and
length 𝑙 drawn between two materials with dielectric constant 𝜖1 and 𝜖2 .
area it threads. Moreover, the contribution of the line integral across the interfaces
(i.e., the sides of the rectangular loop with width ℎ) also goes to 0. Then, in this
limit, we have
E1 · 𝑝ˆtop = −E2 · 𝑝ˆbottom
(B.9)
where we have divided out the factor of 𝑙 associated with the length of the loop.
Again, notice that in fact 𝑝ˆtop = − 𝑝ˆbottom , in the limit that we also take 𝑙 → 0,
so that all wiggles of the loop are smoothed out. Thus, these parallel vectors are
actually point by point constraints on the fields, and we shall arbitrarily denote the
top surface 𝑝ˆtop = 𝑝.
ˆ As before, we can equivalently do the same with the 𝐻 field,
so that the boundary conditions on the parallel components are
E1 · 𝑝ˆ = E2 · 𝑝ˆ
H1 · 𝑝ˆ = H2 · 𝑝ˆ

(B.10)

Using E and H as our basis fields, we shall then summarize the boundary conditions
using the constitutive relations in (B.2) and write things in a more succinct form:
(𝜖1 E1 − 𝜖2 E2 ) · 𝑛ˆ = 0
(E1 − E2 ) × 𝑛ˆ = 0
(𝜇1 H1 − 𝜇2 H2 ) · 𝑛ˆ = 0

(B.11)

(H1 − H2 ) × 𝑛ˆ = 0
which are boundary conditions on the E and H fields! As a reminder, 𝑛ˆ is a unit
vector normal to the interface between 𝜖 1 and 𝜖2 .

188
B.2

Optical Waves in Homogenous Media

The motion of the wave must come from some wave equation, which we shall
derive now. Let’s begin with Maxwell’s equations, which governs any classical
electro-magnetic interaction:
∇ · D = 𝜌𝑓
∇·B=0
∇×E=−

(B.12)

𝜕B
𝜕𝑡

∇ × H = J𝑓 +

𝜕D
𝜕𝑡

where we shall use the bold-face of letters to refer to things that act like vectors
(e.g. D, B) and non bold-faced things act like scalars (e.g. 𝜌 𝑓 ). At the moment we
are interested in getting a wave equation for the electric and magnetic field. Our
previous math courses have taught us that the generic wave equation looks like
𝜕2 𝑓
1 𝜕2 𝑓
𝜕𝑥 2 𝑣 2 𝜕𝑡 2

(B.13)

because any function of the form 𝑓 (𝑥 ± 𝑣𝑡) solves the PDE (partial differential
equation). You can check this by inserting this back above, but more specifically,
"waving" functions like sin(𝑥 ± 𝑣𝑡), cos(𝑥 ± 𝑣𝑡), and exp(𝑖(𝑥 ± 𝑣𝑡)) also satisfy the
wave equation, as you can easily see. With this in mind, let’s find the wave equation
for electromagnetic waves. Let’s focus first on the electric field (we shall show that
this is generally much stronger than the magnetic field in a moment):
∇ × (∇ × E) = −∇ ×

𝜕B
𝜕𝑡

(∇ × B)
𝜕𝑡 
𝜕D
= −𝜇0
J𝑓 +
𝜕𝑡
𝜕𝑡
=−

(assuming non-magnetic materials, i.e., B = 𝜇0 H)
(B.14)

where in the last line, we use the constitutive relation between B and H, i.e. B =
𝜇0 H + M, where the magnetization M → 0 for non-magnetic materials. We shall
also assume that some Ohm’s law holds, that is, the current is proportional to the
electric field applied (i.e., J 𝑓 = 𝜎E). Electrical engineers usually write this as
𝑉 = 𝐼 𝑅, but this expression says the same thing. We may also use the constitutive
relation for the D-field, i.e., D = 𝜖0 E+P. Here, P plays the role of polarizability, i.e.,
the amount of polarization that is induced from some external electric field (could

189
be static or an AC-field). We shall not assume any form for P for now. Inserting
these expressions into above, we finally arrive at
∇ × ∇ × E = −𝜇0 𝜎

𝜕E 1 𝜕 2 E
𝜕2P
− 2 2 − 𝜇0 2
𝜕𝑡
𝑐 𝜕𝑡
𝜕𝑡

(B.15)

where we have also used the fact that 𝜇0 𝜖 0 = 1/𝑐2 , with 𝑐 being the speed of light.
This is the equation we’ve been seeking! While being admittedly complex looking,
equation (B.15) simplifies to the wave-equation (B.13) referred to above in 1D, and
if we take 𝜎 → 0, and P → 0. In fact the curls are a more general expression
for the wave equation, if we do some further manipulation of the curls and use the
first Maxwell as well, but we will ignore this for now. What we do know is that
the presence of pure material properties (i.e., 𝜎 and P) modify the wave equation
to be something slightly different. So the waves could move differently in matter,
compared to vacuum. Let’s figure this out. In the wave equation above, we have yet
to specify how P relates to E. We shall focus our attention to a class of materials
that are specified as linear materials. That is, they obey
(B.16)

P = 𝜖0 𝜒E

and as you can see, they are referred to as linear because they are linear with E.
There are non-linear materials that obey 𝑃 ∼ 𝜒 (2) 𝐸 2 , but those are challenging
equations to deal with and linear optics describes most of the world around us. So,
(B.15) is now completely in terms of E, which is still unknown. 𝜒 is sometimes
referred to as the electric susceptibility.
Like every differential equation, they can more or less only be solved if you know the
solution already (or re-write the problem into something you know how to solve).
So, let’s assume we know the solution is actually a plane wave:
E = E0 exp(𝑖(k · r − 𝜔𝑡))

(B.17)

This expression is referred to a plane-wave because for a given time 𝑡 0, k · r defines
a plane with normal vector k. Exponentials also turn out to be a great function,
because from Fourier math we know that this is a good basis function that can be
expanded into any other wave-form (i.e., exponentials form a complete basis set).
Thus, our results can be applied to any wave, and not just this simple plane-wave.
Let’s see what we find if we insert this plane-wave into our wave-equation. We get
− k × k × E0 = −𝜇0 𝜎(−𝑖𝜔)E0 −

(−𝑖𝜔) 2
𝜖𝑟 E0
𝑐2

(B.18)

190
where we have divided out the exponential factor because that can never be 0, and
we have defined 𝜖𝑟 = 1 + 𝜒. Let’s further simplify this expression by using the
well-known "back of the cab" rule (A × B × C = B · (A · C) − C · (A · B)). This gives
𝜔2
(B.19)
𝑘 = 𝑖𝜎𝜇0 𝜔 + 𝜖𝑟 2
where we have used the fact that k · E0 = 0 for a plane-wave and that |k| ≡ 𝑘.
This expression describes the relationship between 𝑘 and 𝜔, for a given set of
material parameters. We can redefine another parameter that encapsulates all the
other parameters as
𝜔2
𝑘 = 𝜀 𝑐𝑜𝑚 𝑝𝑙𝑒𝑥 2
(B.20)
where
𝑖𝜎
≡ 𝜀1 + 𝑖𝜀 2
(B.21)
𝜀 𝑐𝑜𝑚 𝑝𝑙𝑒𝑥 = 𝜖𝑟 +
𝜖0 𝜔
This equation describes the complex dielectric constant (which we will, from this
point forward, refer to as 𝜀 ≡ 𝜀 𝑐𝑜𝑚 𝑝𝑙𝑒𝑥 for brevity) as some component that has some
real response 𝜖𝑟 but that 𝜎 is related to the imaginary part of this dielectric constant
(which we shall soon find out is related to loss). Let us further define a constant that
is useful to us, i.e.,
(B.22)
𝑛≡ 𝜀

so that 𝑘 = 𝜔𝑛/𝑐 and that 𝑛 ≡ 𝜂 + 𝑖𝜅. This constant 𝑛 will be referred to as the
complex refractive index, and either this quantity or the complex dielectric constant
is enough to describe the optical response of almost every material. These two
quantities can be related to each other with the following:
|𝜀| + 𝜀1
𝜀1 = 𝜂2 − 𝜅 2 ,
𝜂=
(B.23)
|𝜀| − 𝜀1
𝜀2 = 2𝜂𝜅,
𝜅=
where |𝜀| = 𝜀12 + 𝜀 22 .
Let’s briefly examine how the real and imaginary parts of 𝑛 relate to the plane wave
expression that we used earlier to arrive at these properties. In other words, let’s
insert the 𝑘 into the plane-wave expression:
 𝜔

E = E0 exp 𝑖
(𝜂 + 𝑖𝜅)𝑥 − 𝜔𝑡

 𝜔𝜅 
𝜔𝜂
(B.24)
E0
exp 𝑖
𝑥− 𝑡
exp − 𝑥
|{z}
{z
} | {z }
initial amplitude

travelling wave

decaying amplitude

191
and notice that there are three components. The first term E0 is just the initial
amplitude of the wave, where-as the second term describes a traveling wave moving
at velocity 𝑐/𝜂. The last term describes a decaying amplitude term that has a 1/𝑒
distance of 𝑐/(𝜔𝜅). Since we are typically interested in intensities (which go as the
field squared), a common definition for the decay rate is
𝛼=

2𝜔𝜅

(B.25)

where the factor of 2 comes from squaring, and 𝛼 refers to the absorption coefficient
of a material, in units of 1/length. We finally have arrived at the two contributions
of the complex refractive index, i.e. 𝑛 modifies the speed of the wave, whereas 𝜅
describes the decay of the wave.
In summary, 𝑛 or equivalently, 𝜀 completely describes the linear optical response of
any material. In general, 𝜅 and 𝜀 2 both describe loss/absorption within a material,
whereas 𝜂 describes the speed and refractive properties of the wave.

192
B.3

Transfer Matrix Method for Layered Media

Figure B.3: Schematic of a one-dimensional stack consisting of 𝑁 layers. The
arrows represent the reflected and transmitted electromagnetic waves in each layer.
We assume a problem definition similar to what’s shown in Figure B.3. That is,
we have an electromagnetic plane wave propagating in the 𝑧 direction towards a
one-dimensional stack of 𝑁 layers. We shall assume the materials are isotropic,
nonmagnetic, and linear. The electric field in the 𝑗th layer is then given by
𝐸 𝑗 (𝑧) = 𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧 + 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧

(B.26)

where 𝑞 𝑗 is the wavevector in the 𝑗th layer and is given by 𝑞 𝑗 = 2𝜋𝑛 𝑗 /𝜆 = 2𝜋(𝜂 𝑗 +
𝑖𝜅 𝑗 )/𝜆. Thus, the optical response of the materials in encapsulated in the wavevector
of the field. 𝐴 𝑗 and 𝐵 𝑗 are the field amplitudes of the forward and backward
propagating waves, respectively. By Fourier’s theorem, we can simply assume a
monochromatic wave is incident on the stack i.e. 𝐸 (𝑧, 𝑡) = 𝐸 (𝑧)𝑒 −𝑖𝜔𝑡 and assume
the same for the H field. Faraday’s law in Maxwell’s equations then gives
𝑞𝑗
𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧 − 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧
(B.27)
𝜔𝜇0
where we have assumed the material stack is composed of non-magnetic materials
(i.e.𝜇𝑟 = 1). Since the electric and magnetic fields are in a plane perpendicular to
the direction of propagation, the relevant boundary conditions are the ones parallel
to an interface. Maxwell’s curl equations are
𝐻 (𝑧) =

∇×E=−

𝜕B
𝜕𝑡

∇ × H = J𝑓 +

𝜕D
𝜕𝑡

(B.28)

193
and in the absence of a sheet current we must then have that the parallel components
of E and H fields are continuous at the boundary. Between the 𝑗th and 𝑗 + 1th layer
(i.e. at 𝑧 𝑗 in Figure B.3), this implies a condition on the field amplitudes:

𝑞𝑗

𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧𝑖 + 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 = 𝐴 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 + 𝐵 𝑗+1 𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗
𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧 𝑗 − 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 = 𝑞 𝑗+1 𝐴 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 − 𝐵 𝑗+1 𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗

which allows us to form matrices
𝑒𝑖𝑞 𝑗 𝑧 𝑗
𝑒 −𝑖𝑞 𝑗 𝑧 𝑗
𝐴𝑗
𝑒𝑖𝑞 𝑗+1 𝑧𝑖
𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗
𝐴 𝑗+1
𝑞 𝑗 𝑒𝑖𝑞 𝑗 𝑧 𝑗 −𝑞 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 𝐵 𝑗
𝑞 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 −𝑞 𝑗+1 𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗 𝐵 𝑗+1

(B.29)

(B.30)

Note that in (B.30) we can rewrite it as
𝑒𝑖𝑞 𝑗 𝑧 𝑗
𝐴𝑗
𝑒𝑖𝑞 𝑗+1 𝑧 𝑗
𝐴 𝑗+1
𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 𝐵 𝑗
𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗 𝐵 𝑗+1
𝑞 𝑗 −𝑞 𝑗
𝑞 𝑗+1 −𝑞 𝑗+1
| {z } |
{z
{z
}|
{z
𝑀𝑗

𝑃 𝑗, 𝑗

𝑀 𝑗+1

𝑃 𝑗+1, 𝑗

(B.31)
so if we now take 𝑀 −1
𝑗 𝑀 𝑗+1 , we have
𝐴𝑗
𝑗+1
−1
= 𝑃−1
𝑗, 𝑗 𝑀 𝑗 𝑀 𝑗+1 𝑃 𝑗+1,𝑖
𝐵𝑗
𝐵 𝑗+1

(B.32)

Now, let us define an interface matrix 𝐽 𝑗, 𝑗+1 = 𝑀 −1
𝑗 𝑀 𝑗+1 which, calculating explicitly, gives us
−𝑞
−1
𝐽 𝑗, 𝑗+1 = 𝑀 −1
𝑗 𝑀 𝑗+1 =
−2𝑞 𝑗 −𝑞 𝑗 1 𝑞 𝑗+1 −𝑞 𝑗+1
1 𝑞 𝑗 + 𝑞 𝑗+1 𝑞 𝑗 − 𝑞 𝑗+1
(B.33)
2𝑞 𝑗 𝑞 𝑗 − 𝑞 𝑗+1 𝑞 𝑗 + 𝑞 𝑗+1
𝑟 𝑗, 𝑗+1
𝑡 𝑗, 𝑗+1 𝑟 𝑗, 𝑗+1
where, in the last step, we have factored out a 𝑞 𝑗 + 𝑞 𝑗+1 from the entire matrix and
have defined
𝑞 𝑗 − 𝑞 𝑗+1
2𝑞 𝑗
𝑡 𝑗, 𝑗+1 ≡
, 𝑟 𝑗, 𝑗+1 ≡
(B.34)
𝑞 𝑗 + 𝑞 𝑗+1
𝑞 𝑗 + 𝑞 𝑗+1
Therefore, in total we have
𝐴𝑗
𝑗+1
= 𝑃−1
𝑗, 𝑗 𝐽 𝑗, 𝑗+1 𝑃 𝑗+1, 𝑗
𝐵𝑗
𝐵 𝑗+1

(B.35)

194
Acting recursively, we have
𝐴 𝑗+1
𝐴 𝑗+2
−1
= 𝑃 𝑗+1, 𝑗+1 𝐽 𝑗+1, 𝑗+2 𝑃 𝑗+2, 𝑗+1
𝐵 𝑗+1
𝐵 𝑗+2

(B.36)

so that
𝐴𝑗
𝑗+2
−1
= 𝑃−1
𝑗, 𝑗 𝐽 𝑗, 𝑗+1 𝑃 𝑗+1, 𝑗 𝑃 𝑗+1, 𝑗+1 𝐽 𝑗+1, 𝑗+2 𝑃 𝑗+2, 𝑗+1
𝐵𝑗
𝐵 𝑗+2

(B.37)

Now notice what 𝑃 𝑗+1, 𝑗 𝑃−1
𝑗+1, 𝑗+1 gives you, i.e.,
𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗+1
𝑒 −𝑖𝑞 𝑗+1 (𝑧 𝑗+1 −𝑧 𝑗 )
𝑒𝑖𝑞 𝑗+1 𝑧 𝑗+1
𝑒𝑖𝑞 𝑗+1 (𝑧 𝑗+1 −𝑧 𝑗 )
(B.38)
Note that 𝑧 𝑗 refer to the actual coordinates. We can define the 𝑗th layer as having
some thickness 𝑡 𝑗+1 ≡ 𝑧 𝑗+1 − 𝑧 𝑗 . Thus,
−𝑖𝑞 𝑗+1 𝑡 𝑗+1
𝐿 𝑗+1 ≡ 𝑃 𝑗+1, 𝑗 𝑃−1
(B.39)
𝑗+1,𝑖+1 =
𝑒𝑖𝑞 𝑗+1 𝑡 𝑗+1
𝑃 𝑗+1, 𝑗 𝑃−1
𝑗+1, 𝑗+1 =

𝑒𝑖𝑞 𝑗+1 𝑧 𝑗
−𝑖𝑞
𝑒 𝑗+1 𝑧 𝑗

In total then we have
𝐴𝑗
𝑗+2
= 𝑃−1
𝑗, 𝑗 𝐽 𝑗, 𝑗+1 𝐿 𝑗+1 𝐽 𝑗+1, 𝑗+2 𝑃 𝑗+2, 𝑗+1
𝐵 𝑗+2
𝐵𝑗
If we do this now 𝑁 times and begin at 0, it’s clear we have
𝐴𝑁
𝐴0
= 𝐽0,1 𝐿 1 𝐽1,2 𝐿 2 𝐽2,3 · · · 𝐿 𝑁−1 𝐽𝑁−1,𝑁 𝑃 𝑁,𝑁−1
𝑃0,0
𝐵𝑁
𝐵0
{z
| {z }
| {z }

(B.40)

(B.41)

𝐸𝑁

𝐸0

where we define 𝐸 +𝑗 as the right moving wave and 𝐸 −𝑗 as the left moving wave, in
the total 𝐸 field, 𝐸 𝑗 = 𝐸 +𝑗 + 𝐸 −𝑗 . Thus,
𝐸 0+
𝐸 𝑁+
=𝑆 −
𝐸 0−
𝐸𝑁

(B.42)

where
𝑁−1

©Ö
𝑆=­
𝐽 𝑗−1, 𝑗 𝐿 𝑗 ® 𝐽𝑁−1,𝑁
« 𝑗=1

(B.43)

195
where successive terms are multiplied to the right of the previous terms. We can
now easily define other terms that are interesting by looking at the elements of 𝑆.
That is,
𝐸 0+ = 𝑆11 𝐸 𝑁+ + 𝑆12 𝐸 𝑁−

(B.44)

𝐸 0− = 𝑆21 𝐸 𝑁+ + 𝑆22 𝐸 𝑁−

and enforce that 𝐸 𝑁− = 0, since there is no left-propagating wave at the end of the
stack. In this scenario, we can calculate the reflection and transmission amplitudes
as
𝐸 0− 𝑆21
𝐸 𝑁+
𝑟= + =
, 𝑡= + =
(B.45)
𝐸0
𝑆11
𝐸0
𝑆11
which can be simply found by examining at the elements of 𝑆.
Accounting for Sheet Conductors at the Interface
We consider an infinitesimally thin sheet conductor at the 𝑗th interface, e.g. in the
case of a 2D material. The conductivity induces a current with an electric field
given by Ohm’s law J = 𝜎E𝛿(𝑧 − 𝑧 𝑗 ). By inserting this current in (B.28), we have
a new boundary condition for the 𝐻-field at the interface:
𝐻 𝑗 (𝑧 𝑗 ) = 𝐻 𝑗+1 (𝑧 𝑗 ) + 𝜎𝐸 (𝑧 𝑗 )

(B.46)

which gives a new set of conditions when including the electric field continuity:

𝑞𝑗

𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧 𝑗 + 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 = 𝐴 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 + 𝐵 𝑗+1 𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗
𝐴 𝑗 𝑒𝑖𝑞 𝑗 𝑧 𝑗 − 𝐵 𝑗 𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 = 𝑞 𝑗+1 𝐴 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 − 𝐵 𝑗+1 𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗
+ 𝜔𝜇0 𝜎 𝐴 𝑗+1 𝑒𝑖𝑞 𝑗+1 𝑧 𝑗 + 𝐵 𝑗+1 𝑒

(B.47)

−𝑖𝑞 𝑗+1 𝑧 𝑗

where note that the additional 𝐸 (𝑧 𝑗 ) term in the 𝐻-field boundary condition can be
evaluated in either material, because of the continuity of the electric field. The 𝑗th
transfer matrix with a sheet conductor at 𝑧 𝑗 is then renormalized to
𝑒𝑖𝑞 𝑗 𝑧 𝑗
𝐴𝑗
𝑞 𝑗 −𝑞 𝑗
𝑒 −𝑖𝑞 𝑗 𝑧 𝑗 𝐵 𝑗
{z
| {z } |
𝑀𝑗

𝑃 𝑗, 𝑗

𝐴 𝑗+1
𝑒𝑖𝑞 𝑗+1 𝑧 𝑗
(B.48)
𝑞 𝑗+1 + 𝜔𝜇0 𝜎 −𝑞 𝑗+1 + 𝜔𝜇0 𝜎
𝑒 −𝑖𝑞 𝑗+1 𝑧 𝑗 𝐵 𝑗+1
{z
{z
}|
𝑀 𝑗+1

𝑃 𝑗+1, 𝑗

196
and the interface matrix 𝐽 𝑗, 𝑗+1 becomes modified to
𝐽 𝜎𝑗, 𝑗+1 =

𝑡 𝑗, 𝑗+1

1 + (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )
𝑟 𝑗, 𝑗+1 + (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )
𝑟 𝑗, 𝑗+1 − (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )
1 − (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )

(B.49)

where

2𝑞 𝑗
𝑞 𝑗 − 𝑞 𝑗+1
, 𝑟 𝑗, 𝑗+1 ≡
(B.50)
𝑞 𝑗 + 𝑞 𝑗+1
𝑞 𝑗 + 𝑞 𝑗+1
as usual, 𝑍0 is the impedance of free space, and 𝑛 𝑗 = 𝜂 𝑗 + 𝑖𝜅 𝑗 is the refractive index
in the 𝑗th layer. The procedure for calculating reflection and transmittance of a
one-dimensional stack is then the same as above, except replacing 𝐽 𝑗, 𝑗+1 with 𝐽 𝜎𝑗, 𝑗+1
whenever there is sheet conductor at the 𝑗th interface.
𝑡 𝑗, 𝑗+1 ≡

Absorption within a Layer
To examine the optical losses and therefore optoelectronic losses in a given system,
one would like to decompose the fraction of absorption going into various layers in
a given stack. To do so, recall that the time-averaged power density absorbed by a
lossy non-magnetic medium is given by
h𝑄i = 𝜔𝜀00 |E(𝑧)| 2

(B.51)

where the factor of 1/2 comes from time averaging the power in a harmonic and
we have explicitly assumed one-dimensionality, so E is only a function of 𝑧. The
incident power flux is given by the usual equation
𝐼 = 𝑐𝜂0 𝜀 0 |E0 | 2

(B.52)

so that the fraction of absorption going into an individual layer is given by
∫ 𝑧𝑗
h𝑄i 𝑑𝑧 𝜂 𝑗 𝛼 𝑗 ∫ 𝑧 𝑗
4𝜋𝜂
𝑧 𝑗−1
2 −𝛼 𝑗 𝑧
2 +𝛼 𝑗 𝑧
∗ 𝑖 𝜆 𝑗𝑧
𝐴𝑗 𝑒
𝑑𝑧
Abs 𝑗 =
+ |𝐵𝑖 | 𝑒
+ 2 Re 𝐴 𝑗 𝐵 𝑗 𝑒
𝜂0 𝑧 𝑗−1
(B.53)
where 𝜂 𝑗 is the real part of the refractive index in the 𝑗th layer and 𝛼 𝑗 ≡ 4𝜋𝜅 𝑗 /𝜆 is
the absorption coefficient of the 𝑗th material. Here, the field amplitudes 𝐴 𝑗 and 𝐵 𝑗
are the normalized field amplitudes, which can be calculated explicitly by taking
𝐴0
(B.54)
𝐵0
and calculating the 𝑛th amplitudes by using
𝐴𝑛
= 𝐽𝑛−1 𝐽𝑛−2 · · · 𝐽0
𝐵𝑛

(B.55)

197
Note that 𝑟 is explicitly found by using the full transfer matrix. To summarize,
one can calculate the reflection amplitude 𝑟 by calculating 𝐽total . After finding this
value, the same set of transfer matrices 𝐽𝑖 can be used to find the field amplitudes in
the 𝑖th material. Explicitly evaluating the expression in (B.53) gives the fraction of
absorption in the 𝑖th layer.
Similarly, for calculating the absorption going into a sheet conductor (e.g. a mono∫
layer of graphene), we use the power flux dissipated by a conductor Re(E∗ · J)𝑑𝑧
and using the form for the current density in an infinitesimally thin material
(J = 21 𝜎E𝛿(𝑧 − 𝑧𝑖 )), we have
Re(E∗ · 𝜎E𝛿(𝑧 − 𝑧𝑖 ))𝑑𝑧
Abs𝜎 =
(B.56)
oi
4𝜋𝜂
Re(𝜎) h
2 +𝛼𝑖 𝑧𝑖
2 −𝛼𝑖 𝑧𝑖
∗ 𝑖 𝜆 𝑖 𝑧𝑖
| 𝐴𝑖 | 𝑒
+ 2 Re 𝐴𝑖 𝐵𝑖 𝑒
+ |𝐵𝑖 | 𝑒
𝑐𝜂0 𝜀0
where note that, again, we can use the electric field of the 𝑖th or 𝑖 + 1th material for
which the sheet conductor is sandwiched between, as the electric field is continuous
across the boundary and therefore the values are equivalent.

198
B.4

Lorentz Oscillator Model

Let’s consider a damped harmonic oscillator as a canonical example for the motion
of a bound electron-hole pair in a semiconductor (i.e., an exciton). The equation of
motion for this system is given from Newton’s equations as
𝑚 𝑥¥ + 2𝑚𝛾 𝑥¤ + 𝑚𝜔20 𝑥 = 𝑒E

(B.57)

where 𝑥¤ is the first time derivative of the position 𝑥, 𝜔0 is the characteristic frequency
of oscillation, and 𝛾 describes the damping of the oscillation in terms of Ohmic
losses. We’ll see the factor of 2 here as simplifying our definitions later on. E is
the electric field driving our electron-hole pair. Often we are dealing with harmonic
signals, i.e.,
E (𝑡) = E (𝜔) exp(−𝑖𝜔𝑡)
(B.58)
and we can take as an ansatz that the motion also follows the same harmonic motion
𝑥(𝑡) = 𝑥(𝜔) exp(−𝑖𝜔𝑡)

(B.59)

so that the above equation of motion becomes
− 𝑚𝜔2 𝑥(𝜔) + 2𝑚𝛾(−𝑖𝜔𝑥(𝜔)) + 𝑚𝜔20 𝑥(𝜔) = 𝑒E (𝜔)

(B.60)

where we have divided out the time dependence, exp(−𝑖𝜔𝑡). Solving for 𝑥(𝜔)/E (𝜔)
yields
−𝑒
𝑥(𝜔)
= 
(B.61)
E (𝜔) 𝑚 𝜔2 − 𝜔2 + 𝑖2𝛾𝜔

and if we recall the linear polarization response is given as
𝑃 = 𝜀0 𝜒E

(B.62)

with the polarization being the average dipole moment per unit volume, i.e. 𝑃 =
𝜇𝑛 = 𝑒𝑥𝑛, where 𝜇 defines the average dipole moment per unit volume, 𝑛 is the
density of electrons, and 𝑥 is the displsacement of the electrons. Clearly, we can
solve for 𝜒 as
𝑛𝑒 𝑥
𝑛𝑒 2
=−
(B.63)
𝜒=
𝜀0 E 𝜀0 E
𝑚𝜀0 𝜔2 − 𝜔0 + 𝑖2𝛾𝜔
Let us define 𝜔2𝑜𝑠𝑐 ≡ 𝑛𝑒 2 /(𝑚𝜀 0 ) as being some measure of the natural strength of
𝜒(𝜔), which has units of frequency. Thus, we have
𝜔2𝑜𝑠𝑐
𝜒(𝜔) = 2
𝜔0 − 𝜔2 − 𝑖2𝛾𝜔

(B.64)

199
which is the Lorentzian oscillator model (occasionally, oscillator strength is written
as the parameter 𝑓𝑜𝑠𝑐 = 𝜔2𝑜𝑠𝑐 , 2𝛾 → 𝛾, and 𝜔0 is the characteristic oscillator
frequency. It is important to note that this model obeys Kramer-Kronig consistency,
i.e., obeys causality, which comes from the physical nature of the model we are
solving.
Note also that we can rewrite the denominator by expanding as
𝜔2 − 𝜔20 + 𝑖2𝛾𝜔 = 𝜔2 − 𝜔20 + 𝑖2𝛾𝜔 + 𝛾 2 − 𝛾 2 = (𝜔 + 𝑖𝛾) 2 − (𝜔20 − 𝛾 2 )

(B.65)

which, if we define a new frequency 𝜔02
0 = 𝜔0 − 𝛾 , then the expression is of the
form 𝑎 2 − 𝑏 2 , where 𝑎 = 𝜔 + 𝑖𝛾 and 𝑏 = 𝜔00 . Recall furthermore we can perform a
partial fraction decomposition,
(B.66)
𝑎 2 − 𝑏 2 2𝑏 𝑎 − 𝑏 𝑎 + 𝑏

which allows us to rewrite our expression for 𝜒 finally as
𝜔2𝑜𝑠𝑐
𝜒(𝜔) = − 0
2𝜔0 𝜔 − 𝜔00 + 𝑖𝛾 𝜔 + 𝜔00 + 𝑖𝛾

(B.67)

which is optical susceptibility of a damped harmonic oscillator in separated partial
fractions. In order to further simplify this expression, we will consider the fairly
realistic scenario where 𝜔0
𝛾 and that we are mostly interested in the contribution
from the resonant contribution, i.e., 𝜔 ∼ 𝜔0 . In this case, the susceptibility simplifies
as
𝜔2
≡ 𝜒𝑅 (𝜔) + 𝑖 𝜒𝐼 (𝜔)
(B.68)
𝜒(𝜔) = − 𝑜𝑠𝑐
2𝜔0 𝜔 − 𝜔0 + 𝑖𝛾
which is the equation of a Lorentzian. Let us look more specifically at the real and
imaginary parts, given as
𝜒𝑅 (𝜔) = −

𝜔2𝑜𝑠𝑐
𝜔 − 𝜔0
2𝜔0 (𝜔 − 𝜔0 ) 2 + 𝛾 2

𝜒𝐼 (𝜔) =

𝜔2𝑜𝑠𝑐
2𝜔0 (𝜔 − 𝜔0 ) 2 + 𝛾 2

(B.69)

We are now interested in looking at the susceptibility of an 2D exciton. We shall
assume that it can be well modeled as a Lorentzian oscillator, giving a similar form
to the one above. Here, however, we shall use the fact that 𝜔2𝑜𝑠𝑐 /2 → 𝑐𝛾𝑟 /(𝑑),
where 𝛾𝑟 describes the radiative rate of the exciton decay, which is a measure of
its oscillator strength, 𝑐 is the speed of light, and 𝑑 is the thickness of the material.
Furthermore, we shall take 𝛾 → 𝛾𝑛𝑟 /2 as the damping coefficient, to finally yield
𝜒(𝜔) = −

𝛾𝑟
𝜔0 𝑑 𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2

(B.70)

200
which is the susceptibility of an excitonic material, parametrized as a Lorentzian1.
Note here that 𝜔0 here refers to the exciton energy. Furthermore, we have derived
an expression where
𝑖𝜎
𝜀 𝑐𝑜𝑚 𝑝𝑙𝑒𝑥 = 𝜖𝑟 +
(B.71)
𝜖0 𝜔
where for a 2D excitonic material, we have 𝜎 = 𝜎2𝐷 /𝑑 and assuming a single
resonance in the entire dielectric spectrum, we have 𝜖𝑟 = 𝜖 ∞ → 1. Thus,
𝜎2𝐷 (𝜔) = −𝑖𝜖0 𝜔𝑑𝜒(𝜔)

(B.72)

where, defining 𝑍0 = 1/(𝑐𝜖0 ) = 𝑐𝜇0 = 𝜇0 /𝜖0 = 4𝛼/𝐺 0 , where 𝛼 = 𝑒 2 𝑍0 /(2ℎ) is
the fine structure constant and 𝐺 0 = 2𝑒 2 /ℎ is the conductance quantum. 𝑍0 is the
impedance of free space. Thus, we finally have
𝜎2𝐷 (𝜔) =

1 𝜔
𝑖𝛾𝑟
𝑍0 𝜔0 𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2

(B.73)

2 𝛾𝑟
𝑍0 𝛾𝑛𝑟

(B.74)

On resonance, note that
R [𝜎2𝐷 (𝜔)] =
B.5

I [𝜎2𝐷 (𝜔)] = 0

Reflection, Transmission, and Absorption of a 2D exciton

Let us consider a 2D excitonic system that is parametrized as a sheet conductor with
an optical conductivity given as
𝜎2𝐷 (𝜔) =

1 𝜔
𝑖𝛾𝑟
𝑍0 𝜔0 𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2

(B.75)

notice that this sheet conductivity expression describes the optical response of the
exciton, and has terms that are directly dependent on 𝛾𝑟 and 𝛾𝑛𝑟 . Thus, in terms of
the optical response of the 2D exciton, we would certainly expect it to depend on
𝛾𝑟 and 𝛾𝑛𝑟 . We shall find that for the absorbance of a 2D exciton, optimizing the
relative ratio of the radiative and non-radiative decay rate can be used to maximize
the absorption in a given geometry.
Suspended 2D Exciton
We are now interested in analyzing what is the reflection, transmission, and absorption of the 2D exciton suspended in vacuum. In an earlier section we derived the
1 A much more thorough derivation is shown in [24, 173] for the form of the above expression.

Here, we simply motivate the form of the expression by showing the Lorentzian oscillator model.

201
scattering and transfer matrices when we have a sheet conductor between interface
𝑗 and 𝑗 + 1. The interface matrix is given as
(𝑍
𝜎)/(𝑛
(𝑍
𝜎)/(𝑛
𝑗+1
𝑗, 𝑗+1
𝑗+1
𝐽 𝜎𝑗, 𝑗+1 =
(B.76)
𝑡 𝑗, 𝑗+1 𝑟 𝑗, 𝑗+1 − (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )
1 − (𝑍0 𝜎)/(𝑛 𝑗 + 𝑛 𝑗+1 )
where 𝑟 𝑗, 𝑗+1 and 𝑡 𝑗, 𝑗+1 are the Fresnel coefficients for the interface. Since we are
considering a sheet conductor suspended in air, we have that 𝑛1 = 𝑛2 = 1 and
𝑆 = 𝐽1,2 , so that 𝑟 1,2 = 0 and 𝑡1,2 = 1. This drastically simplifies our expressions,
and inserting in the 2D sheet conductivity from above, we have for the scattering
matrix:
1 + 𝑍0 𝜎2𝐷 /2
𝑍0 𝜎2𝐷 /2
𝑆2𝐷 𝐸𝑥𝑐𝑖𝑡𝑜𝑛 =
(B.77)
−𝑍0 𝜎2𝐷 /2 1 − 𝑍0 𝜎2𝐷 /2
from which we can readily read off the reflectance and transmittance as
𝑍0 𝜎2𝐷 2
𝑆21 2
= −
𝑅 = |𝑟 | =
𝑆11
2 + 𝑍0 𝜎2𝐷

1 2
𝑇 = |𝑡| =
𝑆11
2 + 𝑍0 𝜎2𝐷

(B.78)

Note that we are interested in how these expressions look as 𝜔 → 𝜔0 , i.e. we
are operating near resonance. The sheet conductivity then approaches 𝜎2𝐷 →
2𝛾𝑟 /(𝑍0 𝛾𝑛𝑟 ), resulting in the reflection and transmission coefficients:
𝑇 (𝜔0 ) =

𝛾𝑛𝑟
(𝛾𝑛𝑟 + 𝛾𝑟 ) 2

(B.79)

𝐴(𝜔0 ) = 1 − 𝑇 (𝜔0 ) − 𝑅(𝜔0 ) =

2𝛾𝑟 𝛾𝑛𝑟
(𝛾𝑛𝑟 + 𝛾𝑟 ) 2

(B.80)

𝑅(𝜔0 ) =

𝛾𝑟2
(𝛾𝑛𝑟 + 𝛾𝑟 ) 2

as well as the absorbance

Examining this expression for absorbance, we examine what occurs for a fixed 𝛾𝑟 .
As 𝛾𝑛𝑟 → 0, the absorbance goes to 0. In the other limit, as 𝛾𝑛𝑟 → ∞, we also have
absorbance going to 0. Therefore, if a value of 𝛾𝑛𝑟 maximizes the absorption, it
must be some value in between. A simple derivative test shows that this maximum
occurs when 𝛾𝑟 = 𝛾𝑛𝑟 , yielding an absorbance of 50%. In some ways this maximum
absorption value is intuitively obvious: a suspended exciton is a symmetric two port
system where only one of the ports is utilized. Thus, illumination from one side of
this two port system will only ever reach 50% absorption at its maximum [150].
The absorbance spectrum of this free-standing system can be calculated explicitly.
Consider the scenario where 𝜔 ≈ 𝜔0
𝛾𝑟 , 𝛾𝑛𝑟 . In this case, the sheet conductivity
is given by
𝑖𝛾𝑟
𝜎2𝐷 (𝜔) =
(B.81)
𝑍0 𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2

202
and the corresponding reflectance given as
𝑟=−

𝛾𝑟2
𝑖𝛾𝑟
=⇒ 𝑅 =
2(𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2) + 𝑖𝛾𝑟
4(𝜔 − 𝜔0 ) 2 + (𝛾𝑟 + 𝛾𝑛𝑟 ) 2

(B.82)

with transmittance given by
4(𝜔 − 𝜔0 ) 2 + 𝛾𝑛𝑟
2(𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2)
𝑡=
=⇒ 𝑇 =
2(𝜔 − 𝜔0 + 𝑖𝛾𝑛𝑟 /2) + 𝑖𝛾𝑟
4(𝜔 − 𝜔0 ) 2 + (𝛾𝑟 + 𝛾𝑛𝑟 ) 2

(B.83)

so that the absorbance is given as
𝐴=

2𝛾𝑟 𝛾𝑛𝑟
4(𝜔 − 𝜔0 ) 2 + (𝛾𝑟 + 𝛾𝑛𝑟 ) 2

(B.84)

2D Exciton with a single mirror
It is clear that from the above analysis, the simplest method of reaching near-unity
absorbance is to remove the additional port for which no light is being coupled
through. This can be done with a mirror. Consider now the situation where we have
an ideal 2D exciton with a single mirror a certain distance 𝑑 away from the exciton.
This mirror will be parametrized by an interface reflection 𝑟 23 and transmission
amplitude 𝑡23 . The analysis continues similar to the above case, except now we have
a more complicated scattering matrix in this air/exciton/air/mirror system:
𝑆 = 𝐽1,2
𝐿 2 𝐽2,3

(B.85)

where
1 + 𝑍0 𝜎/2
𝑍0 𝜎/2
𝐽1,2
−𝑍0 𝜎/2 1 − 𝑍0 𝜎/2
exp(−𝑖𝑞𝑑)
𝐿2 =
exp(𝑖𝑞𝑑)
1 1 𝑟 23
𝐽2,3 =
𝑡23 𝑟 23 1

(B.86)

where we have implicitly taken 𝑛1 = 𝑛2 = 1, so that 𝑟 12 = 0 and 𝑡12 = 1. We have
not specified 𝑟 23 or 𝑡 23 yet, but we will soon. 𝑞 = 2𝜋/𝜆 and 𝑑 is the spacing from
the exciton and the back mirror. Multiplying the matrices together and examining
the reflection amplitude gives us the expression:
𝑟=

− 𝑍02𝜎 [𝑟 23 exp(2𝑖𝑞𝑑) + 1] + 𝑟 23 exp(2𝑖𝑞𝑑)
𝑍0 𝜎
2 [𝑟 23 exp(2𝑖𝑞𝑑) + 1] + 1

(B.87)

203
Note that if we now consider a perfect mirror on the back, we would expect 𝑟 23 = −1.
This is because a perfect mirror would cause a null in the electric field at the surface
of the mirror, and the reflected wave would be opposite in sign. Moreover, if the
exciton was placed exactly at 𝑑 = 𝜆/4, our reflectivity would be modified to:
𝑟=

1 − 𝑍0 𝜎
1 + 𝑍0 𝜎

(B.88)

where the absorbance, 𝐴 = 1 − |𝑟 | 2 (note that 𝑇 = 0 since a perfect metal forces
𝑡23 = 0) is therefore maximized when 𝑍0 𝜎 = 1. This occurs for example, when
𝜔 = 𝜔0 and therefore 𝜎(𝜔0 )𝑍0 = 2𝛾𝑟 /𝛾𝑛𝑟 . Thus, perfect absorption in this case
requires 2𝛾𝑟 = 𝛾𝑛𝑟 . Notice for a fixed 𝛾𝑟 , the mirror has now reduced the required 𝛾𝑛𝑟
to achieve this critical coupling by now a factor of two! Moreover, the absorbance
can now reach a value of 100%, and this is from removing the superfluous port with
the back mirror.
2D Exciton in an arbitrary photonic structure
We consider now a slightly different problem of a 2D excitonic absorber in an arbitrary photonic environment. We know from Maxwell’s equations that the absorbance
within the layer is still given by
Re(𝜎) |𝐸 (𝑥0 , 𝑦 0 , 𝑧0 )| 2
𝐴2𝐷 𝐸𝑥𝑐𝑖𝑡𝑜𝑛 =
𝑐𝜂0 𝜀0
|𝐸 0 | 2

(B.89)

where, considering the incident medium to be vacuum or air, 𝜂0 → 1 and 𝑐𝜀 0 = 1/𝑍0
is the impedance of free space. Therefore,
|𝐸 (𝑥 0 , 𝑦 0 , 𝑧0 )| 2

|𝐸 (𝑥 0 , 𝑦 0 , 𝑧0 )| 2
𝛾𝑟 𝛾𝑛𝑟
2𝜔0 (𝜔 − 𝜔0 ) 2 + (𝛾𝑛𝑟 /2) 2
|𝐸 0 | 2
|𝐸 0 | 2
(B.90)
Considering now the situation where 𝜔 → 𝜔0 , i.e., we operate at the excitonic
resonance, we have:
2𝛾𝑟 |𝐸 (𝜔0 , 𝑥0 , 𝑦 0 , 𝑧0 )| 2
𝐴(𝜔0 ) =
(B.91)
𝛾𝑛𝑟
|𝐸 0 | 2
𝐴2𝐷 𝐸𝑥𝑐𝑖𝑡𝑜𝑛 = 𝑍0 Re(𝜎)

We now make the ansatz that we can engineer the electric field intensities to be
sufficiently high such that the absorption reaches its maximum value, i.e., 𝐴(𝜔0 ) =
𝐴0 (in many cases, 𝐴0 can approach 1). In this case, this optimum is therefore
reached when
|𝐸 (𝜔0 , 𝑥0 , 𝑦 0 , 𝑧0 )| 2
𝛾𝑛𝑟 = 2
𝛾𝑟
(B.92)
|𝐸 0 | 2

204
where the electric field intensity enhancement is proportional to the LDOS and
therefore the Purcell factor. Thus, this physics is akin to one where we increase the
radiative rate of emitters by placing them in a photonic structure. Here, it is clear
that a similar enhancement in LDOS would also “effectively increase” the radiative
rate, although there are slightly different multiplicative factors that differentiate the
two factors. Nonetheless, we can define 𝛾𝑛𝑟 = 𝐹𝑝,𝑎𝑏𝑠 𝛾𝑟 , where 𝐹𝑝,𝑎𝑏𝑠 is a constant
that determines the relative 𝛾𝑛𝑟 and 𝛾𝑟 to achieve perfect absorption. Notice that the
𝛾𝑟 defined so far is that of an exciton suspended in vacuum. Thus, it is possible to
interpret a new radiative rate 𝛾𝑟0 = 𝐹𝑝,𝑎𝑏𝑠 𝛾𝑟 as the radiative rate of the exciton within
a photonic medium that is modified compared to its native counterpart.

205
Appendix C

THERMODYNAMICS CONSIDERATIONS OF PHOTOVOLTAIC
SYSTEMS
C.1

Derivation of Blackbody Radiation

The well-known and universal blackbody radiation expression is given by the expression
2𝜋
𝐸2
𝑆 𝐵𝐵 (𝐸)𝑑𝐸 = 3 2
𝑑𝐸
(C.1)
ℎ 𝑐 exp(𝐸/𝑘𝑇) − 1
where 𝑆 𝐵𝐵 (𝐸)𝑑𝐸 describes the spectral flux of photons emitted by a blackbody in a
given spectral window 𝑑𝐸. Let us derive this expression from first principles. First,
let us consider the occupation factor of a photon, given by
𝑓 𝐵𝐸 (𝐸) =

exp(𝐸/𝑘𝑇) − 1

(C.2)

This expression says that if a photon has some energy 𝐸, the probability of it being
occupied is given by 𝑓 𝐵𝐸 (𝐸). Interestingly, this probability appears to tend towards
infinity at low energies, which represents the fact that when many particles interact
with one another, they prefer to occupy the lowest energy available state. Moreover,
bosons are particles with symmetric wavefunctions, i.e., their wavefunctions can be
arbitrarily exchanged with one another with no penalty. This allows them to occupy
the same states as one another.
To understand where this factor comes from, consider we have some small system
𝐴 nearby some thermal reservoir 𝐴0, where 𝐴
𝐴0. We are now interested in
understanding the probability 𝑃𝑟 of finding the system in a particular microstate 𝑟
of energy 𝐸𝑟 . Assuming the system 𝐴 interacts weakly with the thermal reservoir
𝐴0, it is possible to consider their energies as separate and additive. Therefore, by
conservation of energy, we have:
𝐸𝑟 + 𝐸 0 = 𝐸 0

(C.3)

where 𝐸 0 is some fixed constant. Thus, if 𝐴 has some energy 𝐸𝑟 , it must be the
case that the reservoir 𝐴0 has energy 𝐸 0 − 𝐸𝑟 . Therefore, if 𝐴 is in one definite
microstate 𝑟, then the total number of possible states of the combined system is
Ω0 (𝐸 0 = 𝐸 0 − 𝐸𝑟 ). Here, Ω0 is a function for system 𝐴0 that describes the number of

206
states are available for a given energy 𝐸 0. According to the fundamental statistical
postulate, the probability of occurrence for system 𝐴 in state 𝑟 is proportional to the
total number of possible configurations where 𝐴 can be in state 𝑟, which is given by
𝑃𝑟 = 𝐶 0Ω0 (𝐸 0 − 𝐸𝑟 )

(C.4)

here, 𝐶 0 is just a proportional constant that can be calculated by normalizing 𝑃𝑟 ,
i.e. 𝑟 𝑃𝑟 = 1. Let us now use the assumption that 𝐴 is much smaller than 𝐴0.
Therefore, the energies of a particular microstate of 𝐴, i.e. 𝑟, would have energies
𝐸𝑟
𝐸 0 . It is therefore possible to expand the function ln(Ω0), which varies slowly
as a function of 𝐸 0. That is,
ln(Ω0 (𝐸 0 − 𝐸𝑟 )) ≈ ln(Ω0 (𝐸 0 )) −
Note that

𝜕 ln(Ω0)
𝐸𝑟
𝜕𝐸 0 𝐸 0=𝐸0

𝜕 ln(Ω0)
=𝛽
𝜕𝐸 0 𝐸 0=𝐸0

(C.5)

(C.6)

where 𝛽 is a constant independent of the system energy 𝐸𝑟 . This constant turns out
to be exactly the inverse of the thermal energy. It is straightforward to see this by
considering the definition of entropy 𝑆, given by
𝑆 = 𝑘 ln(Ω)

(C.7)

where Ω is the number of microstates and 𝑘 is the Boltzmann constant. Recall the
thermodynamic definition of temperature is given by

Thus, it follows that

𝜕𝑆
𝑇 𝜕𝐸

(C.8)

𝜕 ln(Ω0)
𝜕𝐸 0 𝐸 0=𝐸0
𝑘𝑇

(C.9)

Inserting this into (C.5) and solving for Ω0, we have
Ω0 (𝐸 0 − 𝐸𝑟 ) = Ω0 (𝐸 0 ) exp(−𝛽𝐸𝑟 )

(C.10)

where Ω0 (𝐸 0 ) is a constant independent of the microstate 𝑟. Thus,
𝑃𝑟 = 𝐶 exp(−𝛽𝐸𝑟 )

(C.11)

where 𝐶 is again a normalization constant given by 𝐶 = 1/( 𝑟 exp(−𝛽𝐸𝑟 )). This
derivation is quite general and applies to all systems that are small compared to a

207
thermal reservoir 𝐴0. Knowing this fundamental probability relation with this exponential dependence, it is possible to calculate many macroscopic thermodynamic
quantities of a given system. For example, the average energy of a system 𝐸¯ would
be given as
𝑟 𝐸 𝑟 exp(−𝛽𝐸 𝑟 )
(C.12)
𝐸= Í
𝑟 exp(−𝛽𝐸 𝑟 )
where it is useful to define a function called the partition function
𝑍=
exp(−𝛽𝐸𝑟 )
(C.13)

because thermodynamic quantities like the average energy can be calculated from
this partition function as
1 𝜕𝑍
𝜕 ln(𝑍)
𝐸¯ = −
=−
𝑍 𝜕𝛽
𝜕𝛽

(C.14)

We are now interested in calculating the occupation factor for a photon. To do so,
let us assume we have a gas of photons that can be treated as identical particles in
equilibrium at the temperature 𝑇. We shall also assume they are weakly interacting,
so that the if each energetic photon state 𝜖𝑟 is occupied by 𝑛𝑟 number of photons,
then the total energy for a given configuration 𝑅 is given as
𝐸𝑅 =
𝑛𝑟 𝜖 𝑟
(C.15)

Therefore, suppose we are interested in the average number of photons 𝑛 𝑠 that occupy
some energetic state 𝜖 𝑠 . This number is given by
𝑛𝑠 𝑃𝑅
𝑛¯ 𝑠 = Í𝑅
(C.16)
𝑅 𝑃𝑅
where 𝑃 𝑅 = exp(−𝛽𝐸 𝑅 ) describes the probability of the entire system of gas photons
to be in some specific state 𝑅. To perform this calculation, we notice that we can
separate the terms out in the exponential, i.e.,
Í
 Í
𝑛𝑠 𝑛 𝑠 exp(−𝛽𝑛 𝑠 𝜖 𝑠 )
𝑛1 ,𝑛2 ,··· exp(−𝛽(𝑛1 𝜖 1 + 𝑛2 𝜖 2 + · · · ))
 Í
𝑛¯ 𝑠 = Í
(C.17)
𝑛𝑠 exp(−𝛽𝑛 𝑠 𝜖 𝑠 )
𝑛1 ,𝑛2 ,··· exp(−𝛽(𝑛1 𝜖 1 + 𝑛2 𝜖 2 + · · · ))
where the second exponential on the right is a sum over all 𝑛𝑟 except 𝑛 𝑠 , which has
been factored out. Note that these sums are identical on the top and bottom, because
𝑛𝑟 can assume any non-negative integer value 𝑛𝑟 = 0, 1, 2, 3 for each 𝑟. Therefore,
we have
𝑛 𝑛 𝑠 exp(−𝛽𝑛 𝑠 𝜖 𝑠 )
𝑛¯ 𝑠 = Í𝑠
(C.18)
𝑛𝑠 exp(−𝛽𝑛 𝑠 𝜖 𝑠 )

208
This calculation can be easily done by noting that
1 𝜕
1 𝜕
𝑛¯ 𝑠 = −
ln
exp(−𝛽𝑛 𝑠 𝜖 𝑠 ) = −
ln
𝛽 𝜕𝜖 𝑠
𝛽 𝜕𝜖 𝑠
1 − exp(−𝛽𝜖 𝑠 )

(C.19)

where the sum was computed using the geometric series, 𝑛 𝑥 𝑛 = 1/(1 − 𝑥), which
is true as long as 𝑥 = exp(−𝛽𝑛 𝑠 𝜖 𝑠 ) < 1, i.e., the energies are positive. Since this is
true, we finally have for the average photon occupation factor:
𝛽 exp(−𝛽𝜖 𝑠 )
exp(−𝛽𝜖 𝑠 )
𝑛¯ 𝑠 = − (1 − exp(−𝛽𝜖 𝑠 )) −
(C.20)
1 − exp(−𝛽𝜖 𝑠 )
(1 − exp(−𝛽𝜖 𝑠 )) 2
therefore, finally, we have for the occupation factor of a given energy state 𝜖 𝑠 :
𝑛¯ 𝑠 =

exp(𝛽𝜖 𝑠 ) − 1

(C.21)

the subscript 𝑠 can be dropped since our analysis applies to any energetic state 𝜖 𝑠 .
Thus, the occupation factor is
𝑓 𝐵𝐸 (𝐸) =

exp(𝐸/𝑘𝑇) − 1

(C.22)

where 𝑓 𝐵𝐸 (𝐸) is a renaming of the average number of photons 𝑛¯ 𝑠 , i.e., the BoseEinstein occupation factor, and describes the average number of bosons (in this case,
photons), that occupy a state with energy 𝐸.
To finally derive the blackbody distribution, we realize that if we have a perfectly
black absorber/emitter, the photons should be able to couple to every available
mode of free space 𝜌𝑣 (𝐸), with an occupation factor given by 𝑓 𝐵𝐸 (𝐸). The photons
have a velocity given by the speed of light 𝑐, and the hypothetical spectrum can
be considered in the limit of having perfect mirrors that surround the blackbody
emitter, so that thermal equilibrium is achieved. If we were able to peak through the
perfect mirrors with an infinitesimally small hole, the blackbody radiation through
that surface would be given as
𝑆 𝐵𝐵 (𝐸)𝑑𝐸 = 𝑐𝜌𝑣 (𝐸) 𝑓 𝐵𝐸 (𝐸)𝑑𝐸

(C.23)

where 𝜌𝑣 (𝐸)𝑑𝐸 is the density of photonic states in vacuum for an energy bandwidth
𝑑𝐸 when surrounded by these perfect mirrors, and the factor of 4 comes from
examining the relative amount of volumetric flux assuming a Lambertian surface,
i.e.,
∫ 𝜋/2
∫ 2𝜋
cos(𝜃)
sin(𝜃)𝑑𝜃
𝑑𝜙 1
(C.24)
∫𝜋
∫ 2𝜋
sin(𝜃)𝑑𝜃
𝑑𝜙

209
This expression is analogous to the current density of electrons, i.e., 𝐽 = 𝑞𝑛𝑣, if
we take the density of photons 𝑛𝛾 = 𝜌𝑣 (𝐸) 𝑓 𝐵𝐸 (𝐸)𝑑𝐸. In the current density
expression, 𝑞 is the fundamental unit of charge, 𝑛 is the density of electrons, and 𝑣 is
the velocity of charge carriers. To calculate this density of photonic states, assume
we are in an isotropic environment (specifically, vacuum) with perfect mirrors that
surround a 3D box of length 𝐿 and therefore a volume of 𝑉 = 𝐿 3 . The size of
the box will turn out to not be important when we normalize and take 𝐿 to be
large. A parallelpiped analysis will not change the results, since the volumes will be
normalized away anyway. The perfect mirrors, as mentioned before, are necessary
for the blackbody emitter to reach thermal equilibrium, otherwise, electromagnetic
radiation would actually cool the emitter. In this case, waves will propagate to the
edges of the box and can be approximated as plane waves, which has wavevectors of
𝑘 𝑖,𝑛 =

𝑛𝜋

(C.25)

where 𝑛 is an integer, as enforced by the boundary conditions of the system, and
𝑖 = 𝑥, 𝑦, 𝑧. Therefore, the spacing in 𝑘-space for this isotropic system would be given
as
Δ𝑘 𝑖 =
(C.26)
Now consider the situation
where we are interested in the number of possible states
Í q 2
with a given 𝑘 = 𝑖 𝑘 𝑖 , where 𝑘 is the magnitude of the wavevector. What is the
density of available states in 𝑘-space for a given 𝑘? What is 𝜌𝑣 (𝑘)𝑑𝑘? To answer
this question, let us first try to calculate the total number of states 𝑘 0 < 𝑘. This is
given as
𝐿3
Volume with radius 𝑘
= 2 × 𝜋𝑘 3 ×
= 2 𝑘 3𝑉
Volumetric spacing for each 𝑘 point
(2𝜋)
3𝜋
(C.27)
This total number of states has a factor of 2 because each 𝑘 point can actually have
two states, given by the two polarizations of light. Thus, we can calculate the density
of states 𝜌𝑣 (𝑘) (states per unit volume per unit 𝑘) can be simply calculated as
𝑁 (𝑘 0 < 𝑘) =

𝜌𝑣 (𝑘) =

1 𝑑𝑁 𝑘 2
= 2
𝑉 𝑑𝑘

(C.28)

Finally, to convert this expression to a density of electronic states, i.e. 𝜌𝑣 (𝐸), we
simply have to use the dispersion relation of the photon in this environment, i.e., as
𝐸 = ℏ𝜔 = ℏ𝑐𝑘

(C.29)

210
Thus,
𝜌𝑣 (𝑘)𝑑𝑘 = 𝜌𝑣 (𝐸)𝑑𝐸 =

1 𝐸2
𝑑𝐸
𝜋 2 (ℏ𝑐) 3

(C.30)

Using finally the fact that ℏ = ℎ/(2𝜋), we have
𝐸2
2𝜋
𝑆 𝐵𝐵 (𝐸)𝑑𝐸 = 𝑐𝜌𝑣 (𝐸) 𝑓 𝐵𝐸 (𝐸)𝑑𝐸 = 3 2
𝑑𝐸
ℎ 𝑐 exp(𝐸/𝑘𝑇) − 1

(C.31)

This expression is the well-known blackbody radiation, which was derived in the
limit of considering only far-field radiation with a uniform density of states (i.e., in
vacuum). The derivation of the photon gas also requires that a single temperature
𝑇 characterizes the entire system of photons, and that this system is in thermal
equilibrium. This characterizes the emission from a surface.
Consider there is a probability of emission given by 𝑒(𝐸) and a probability of
absorption 𝑎(𝐸) that characterizes the emission and absorption of a photon of
energy 𝐸 for a given surface. It is clear that under thermal equilibrium, the net flux
through the surface must be exactly equal, i.e.,
[𝑎(𝐸) − 𝑒(𝐸)] 𝑆 𝐵𝐵 (𝐸)𝑑𝐸 = 0
(C.32)
Since it is possible to construe a system with a wavelength selective mirror, this
balance must actually occur at each wavelength to achieve thermal equilibrium.
Thus, we must have
𝑎(𝐸) = 𝑒(𝐸)
(C.33)
C.2

The Chemical Potential of a Photon

We are now interested in considering the spontaneous emission rate of a semiconductor, which is seemingly a different phenomena compared to thermal radiation.
While thermal radiation is known to give broad and dim spectra and is known to
be due to the thermal shaking of atoms, the spontaneous emission from a semiconductor is relatively bright, narrow in bandwidth, and is due to electronic interband
transitions. What separates the two phenomena and can they be related?
Let us first consider a similar situation to what we considered in a blackbody emitter,
where a homogeneous semiconductor is surrounded by a perfectly reflecting wall.
The total number of occupied electrons in the conduction band is given by the usual
expression
𝑛(𝐸) = 𝑔(𝐸) 𝑓 (𝐸) = 𝑔𝑐 (𝐸) 𝑓𝑒 (𝐸) = 𝑔𝑐 (𝐸)

exp (𝐸 − 𝐸 𝑓 ,𝑛 )/𝑘𝑇 + 1

(C.34)

211
where 𝑔𝑐 (𝐸) is the density of electronic states at the conduction band and 𝑓 (𝐸) is
the Fermi-Dirac occupation factor. A similar expression is true for occupied holes
in the valence band, i.e.,
exp −(𝐸 − 𝐸 𝑓 ,𝑝 )/𝑘𝑇 + 1
(C.35)
Note that 𝐸 𝑓 ,𝑛 and 𝐸 𝑓 ,𝑝 are referred to as the electron and hole quasi-Fermi levels.
In assigning a single chemical potential and temperature 𝑇 to all the electrons/holes,
we take into account the rapid chemical equilibrium achieved between carrier-carrier
interaction within a given conduction/valence band, as well as the thermal equilibrium between the carriers and the lattice which acts as a thermal reservoir. This
analysis is similar to derivation above for the Boltzmann factor 𝑃𝑟 = exp(−𝛽𝐸𝑟 ).
𝑝(𝐸) = 𝑔(𝐸)(1 − 𝑓 (𝐸)) = 𝑔𝑣 (𝐸) 𝑓 ℎ (𝐸) = 𝑔𝑣 (𝐸)

Notice also that generation and recombination results in a pair-wise formation of
electrons and holes, and the flux of absorption and emission processes must be
balanced in the perfect cavity to achieve thermal equilibrium. The absorption rate
per unit volume per photon energy interval 𝑟 𝑎 (𝐸) can be written as
𝑟 𝑎 (ℏ𝜔) = 𝑛𝛾 (ℏ𝜔)×
∫ ∞
𝑀 (𝐸 𝑒 , 𝐸 ℎ , ℏ𝜔)𝑔𝑐 (𝐸 𝑒 )(1− 𝑓𝑒 (𝐸 𝑒 ))𝑔𝑣 (𝐸 ℎ )(1− 𝑓 ℎ (𝐸 ℎ ))𝛿(ℏ𝜔−(𝐸 𝑒 −𝐸 ℎ ))𝑑𝐸 𝑒 𝑑𝐸 ℎ

(C.36)
where 𝑀 (𝐸 𝑒 , 𝐸 ℎ , ℏ𝜔) is the transition matrix element and describes the relative rate
of an electron with energy 𝐸 𝑒 and hole with energy 𝐸 ℎ to couple with a photon
with energy ℏ𝜔. The density of photons is given as 𝑛𝛾 (ℏ𝜔). The delta function
𝛿(ℏ𝜔 − (𝐸 𝑒 − 𝐸 ℎ )) conserves the energy of the transition, which can be integrated
to give the condition ℏ𝜔 = 𝐸 𝑒 − 𝐸 ℎ . The expressions 1 − 𝑓 𝑗 (𝐸 𝑗 ) describes the fact
that absorption of photons requires unoccupied states. Thus, integrating the delta
function gives
∫ ∞
𝑟 𝑎 (ℏ𝜔) = 𝑛𝛾 (ℏ𝜔)
𝑀 (𝐸 ℎ , ℏ𝜔)𝑔𝑐 (𝐸 ℎ +ℏ𝜔)𝑔𝑣 (𝐸 ℎ )(1− 𝑓𝑒 (𝐸 ℎ +ℏ𝜔))(1− 𝑓 ℎ (𝐸 ℎ ))𝑑𝐸 ℎ

(C.37)
We can drop the subscript ℎ, since it is just an integration variable. Therefore,
∫ ∞
𝑟 𝑎 (ℏ𝜔) = 𝑛𝛾 (ℏ𝜔)
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸)(1 − 𝑓𝑒 (𝐸 + ℏ𝜔))(1 − 𝑓 ℎ (𝐸))𝑑𝐸

(C.38)
We can derive a similar expression for a stimulated emission process, where an
electron transitions from an excited state 𝐸 𝑒 to a lower energy state 𝐸 ℎ . This process

212
is stimulated by the density of photons 𝑛𝛾 (ℏ𝜔), and is therefore a time-reversal
process to absorption. It requires an occupied electron and hole in the conduction
and valence band. In other words,
∫ ∞
𝑟 𝑠𝑡 (ℏ𝜔) = 𝑛𝛾 (ℏ𝜔)
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸) 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)𝑑𝐸 (C.39)

Finally, we have the spontaneous emission rate 𝑟 𝑠𝑝 (ℏ𝜔), which does not depend
on the density of photons in the cavity, but by the density of available photonic
states in the semiconductor 𝜌 𝑠 (ℏ𝜔). It has a similar form to the stimulated emission
otherwise:
∫ ∞
𝑟 𝑠𝑝 (ℏ𝜔) = 𝜌 𝑠 (ℏ𝜔)
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸) 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)𝑑𝐸 (C.40)

where we have already derived from above that
𝜌𝑣 (ℏ𝜔)𝑑 (ℏ𝜔) =

(ℏ𝜔) 2
𝑑 (ℏ𝜔)
𝜋 2 (ℏ𝑐) 3

(C.41)

Note in our derivation that we made use that the speed of light was 𝑐. In a
semiconductor with refractive index 𝑛𝑟 , the speed of light would decrease to 𝑐/𝑛𝑟 .
Therefore, the density of photonic states in a semiconductor is given by
𝜌 𝑠 (ℏ𝜔)𝑑 (ℏ𝜔) =

𝑛𝑟3 (ℏ𝜔) 2
𝑑 (ℏ𝜔)
𝜋 2 (ℏ𝑐) 3

(C.42)

We now know that these rates must balance in steady state, so that
𝑟 𝑎 − 𝑟 𝑠𝑡 − 𝑟 𝑠𝑝 = 0

(C.43)

It is therefore possible to solve for the photon density within this perfect cavity,
given by 𝑛𝛾 (ℏ𝜔). Solving for it gives
𝑛𝛾 (ℏ𝜔) = 𝜌 𝑠 (ℏ𝜔)×
∫∞
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸) 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)𝑑𝐸
∫∞
(1− 𝑓𝑒 (𝐸+ℏ𝜔)) (1− 𝑓 ℎ (𝐸))
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸) 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
− 1 𝑑𝐸
𝑓𝑒 (𝐸+ℏ𝜔) 𝑓 ℎ (𝐸)
(C.44)
The term in the brackets can be dramatically simplified if written out, it is given as
 
(1 − 𝑓𝑒 (𝐸 + ℏ𝜔))(1 − 𝑓 ℎ (𝐸))
(1 − 𝑓𝑒 (𝐸 + ℏ𝜔))(1 − 𝑓 ℎ (𝐸)) − 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
−1 =
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
(C.45)

213
which can be simplified further as
(1 − 𝑓𝑒 (𝐸 + ℏ𝜔))(1 − 𝑓 ℎ (𝐸)) − 𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
1 − 𝑓𝑒 (𝐸 + ℏ𝜔) − 𝑓 ℎ (𝐸)
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
1 − 𝑓𝑒 (𝐸 + ℏ𝜔)
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)
𝑓𝑒 (𝐸 + ℏ𝜔)
1 − 𝑓𝑒 (𝐸 + ℏ𝜔)
−1
𝑓𝑒 (𝐸 + ℏ𝜔)
𝑓 ℎ (𝐸)
(C.46)
where we can further simplify the term within the brackets as
exp −(𝐸 − 𝐸 𝑓 ,𝑝 )/𝑘𝑇 + 1
1 − 𝑓𝑒 (𝐸 + ℏ𝜔)
−1 =
−1
𝑓 ℎ (𝐸)
exp −(𝐸 + ℏ𝜔 − 𝐸 𝑓 ,𝑛 )/𝑘𝑇 + 1
exp −(𝐸 − 𝐸 𝑓 ,𝑝 )/𝑘𝑇 − exp −(𝐸 + ℏ𝜔 − 𝐸 𝑓 ,𝑛 )/𝑘𝑇
exp −(𝐸 + ℏ𝜔 − 𝐸 𝑓 ,𝑛 )/𝑘𝑇 + 1
exp (ℏ𝜔 − 𝐸 𝑓 ,𝑛 + 𝐸 𝑓 ,𝑝 )/𝑘𝑇 − 1
1 + exp (𝐸 + ℏ𝜔 − 𝐸 𝑓 ,𝑛 )/𝑘𝑇

= 𝑓𝑒 (𝐸 + ℏ𝜔) exp (ℏ𝜔 − (𝐸 𝑓 ,𝑛 − 𝐸 𝑓 ,𝑝 ))/𝑘𝑇 − 1
(C.47)
therefore combining the terms and referring to equation (C.45), we have
 
(1 − 𝑓𝑒 (𝐸 + ℏ𝜔))(1 − 𝑓 ℎ (𝐸))
− 1 = exp((ℏ𝜔 − Δ𝜇)/𝑘𝑇) − 1
𝑓𝑒 (𝐸 + ℏ𝜔) 𝑓 ℎ (𝐸)

(C.48)

where we define Δ𝜇 = 𝐸 𝑓 ,𝑛 − 𝐸 𝑓 ,𝑝 as the quasi-Fermi level splitting. Interestingly,
this simplified term of (C.45) results in a term that is independent of 𝐸, which
allows equation (C.44) to be dramatically simplified by taking the 𝐸-independent
term outside of the integral:
𝑛𝛾 (ℏ𝜔) =

𝜌 𝑠 (ℏ𝜔)
exp((ℏ𝜔 − Δ𝜇)/𝑘𝑇) − 1

(C.49)

where this term is independent of the properties of the semiconductor, except through
the quasi-Fermi level splitting. Therefore, since the electron-hole generation and
recombination reach an equilibrium with the population of photons, it is appropriate
to define a chemical potential for the photons, 𝜇 𝛾 = Δ𝜇 = 𝐸 𝑓 ,𝑛 − 𝐸 𝑓 ,𝑝 , so that the
density of photons can be written as
𝑛𝛾 (ℏ𝜔, 𝜇𝛾 ) =

𝜌 𝑠 (ℏ𝜔)
exp (ℏ𝜔 − 𝜇 𝛾 )/𝑘𝑇 − 1

(C.50)

214
This expression is of general validity, since the details of the system, such as the
transition matrix elements, drop out in the derivation.
Knowing this density of photons 𝑛𝛾 , it is possible to define an absorption coefficient
𝛼(ℏ𝜔) given by
𝑟 𝑎 − 𝑟 𝑠𝑡 = 𝛼(ℏ𝜔) 𝑗 𝛾 = 𝛼(ℏ𝜔)𝑛𝛾
(C.51)
𝑛𝑟
where 𝑐/𝑛𝑟 is the velocity of light within a semiconductor with index 𝑛𝑟 , which
results in a photon flux of 𝑗 𝛾 . By examining the earlier definitions for 𝑟 𝑎 and 𝑟 𝑠𝑡 , we
arrive at
∫ ∞
𝑛𝑟
𝛼(ℏ𝜔) =
𝑀 (𝐸, ℏ𝜔)𝑔𝑐 (𝐸 + ℏ𝜔)𝑔𝑣 (𝐸) [1 − 𝑓𝑒 (𝐸 + ℏ𝜔) − 𝑓 ℎ (𝐸)] 𝑑𝐸
(C.52)
Furthermore equation (C.51) relates the absorption coefficient to the spontaneous
emission rate because, in equilibrium, we have 𝑟 𝑎 − 𝑟 𝑠𝑡 = 𝑟 𝑠𝑝 , which we used earlier.
Therefore,
𝑛2
(ℏ𝜔) 2 𝛼(ℏ𝜔)
(C.53)
𝑟 𝑠𝑝 (ℏ𝜔) = 2 𝑟3 2
𝜋 ℏ 𝑐 exp (ℏ𝜔 − 𝜇 𝛾 )/𝑘𝑇 − 1
Using again the fact that ℏ = ℎ/(2𝜋), we can rewrite this expression as
 
(ℏ𝜔) 2 𝛼(ℏ𝜔)
2𝜋
𝑟 𝑠𝑝 (ℏ𝜔) = 4𝑛𝑟2 3 2
(C.54)
ℎ 𝑐 exp (ℏ𝜔 − 𝜇 𝛾 )/𝑘𝑇 − 1
which is the blackbody spectrum modified by a factor of 4𝑛𝑟2 𝛼(ℏ𝜔) because we
are looking at the internal volumetric emission rate inside a semiconductor, and
the chemical potential of the photon 𝜇 𝛾 = 𝐸 𝑓 ,𝑛 − 𝐸 𝑓 ,𝑝 is used here to describe the
non-equilibrium (but steady-state) nature of the process.
C.3

The Validity of a Thermalized Population as the Major Contribution to
Photoluminescence Under Steady State Excitation

There is a general picture in semiconductor optoelectronics that quasi-fermi levels
can form given enough carrier thermalization and electron-electron interactions.
Let’s be a bit more rigorous and see when this is true, and how much of luminescence may occur from non-thermalized carriers. Let’s considered a coupled set of
equations governed by different rates:
𝜕𝑛
=𝐺−
𝜕𝑡
𝜏𝑒𝑒 𝜏𝑟𝑎𝑑,𝑛
𝜕𝑛𝑒𝑒
𝑛𝑒𝑒
𝑛𝑒𝑒
(C.55)
𝜕𝑡
𝜏𝑒𝑒 𝜏𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒𝑒
𝜕𝑛𝑒 𝑝 𝑛𝑒𝑒
𝑛𝑒 𝑝
𝜕𝑡
𝜏𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒 𝑝

215

Figure C.1: Schematic depiction of the relative timescales relevant to electronelectron scattering (equilibration), electron-phonon coupling (thermalization), and
electron-hole recombination. Adapted from Y. Takeda et al., 2014
where 𝑛 corresponds to the electron population that follows the same spectral dependence as the excitation, 𝑛𝑒𝑒 is the population of carriers that have undergone
electron-electron interactions, i.e. follows a Fermi-Dirac distribution, and 𝑛𝑒 𝑝 is the
population of electrons that have undergone both electron-electron interactions and
subsequently thermalized with the lattice through electron-phonon interactions. The
coupled equations can be solved algebraicly under steady state conditions, wheere
𝜕𝑡 = 0. Thus, the first expression is simply:
𝑛 = 𝐺𝜏𝑒 𝑓 𝑓 ,1 ,

where

𝜏𝑒 𝑓 𝑓 ,1

𝜏𝑒𝑒 𝜏𝑟𝑎𝑑,𝑛

(C.56)

This allows solutions to the second expression rather simply, yielding
𝑛𝑒𝑒 =

𝐺𝜏𝑒 𝑓 𝑓 ,1
𝜏𝑒 𝑓 𝑓 ,2 ,
𝜏𝑒𝑒

where

𝜏𝑒 𝑓 𝑓 ,2

𝜏𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒𝑒

(C.57)

and finally, the last population can be solved as
𝑛𝑒 𝑝 = 𝜏𝑟𝑎𝑑,𝑒 𝑝

𝐺𝜏𝑒 𝑓 𝑓 ,1 𝜏𝑒 𝑓 𝑓 ,2
𝜏𝑒𝑒 𝜏𝑒 𝑝

(C.58)

Each population can contribute to the average photoluminescence rate with the
population density scaled by their respective radiative rates, i.e.
𝑛𝑒 𝑝
𝑛𝑒𝑒
𝑟 𝑠𝑝 =
(C.59)
𝜏𝑟𝑎𝑑,𝑛 𝜏𝑟𝑎𝑑,𝑒𝑒 𝜏𝑟𝑎𝑑,𝑒 𝑝
To understand the fraction of luminescence from the different electron populations,
all we have to do is examine the relative rates. To see if the majority of the

216
luminescence comes from electrons that have gone through both electron-electron
and electron-phonon coupling, we can analyze their relative luminescence rates.
Comparing for the completely uncoupled population:
𝑟 𝑠𝑝,𝑒 𝑝 𝑛𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑛
𝜏𝑟𝑎𝑑,𝑛 𝜏𝑟𝑎𝑑,𝑒 𝑝 𝜏𝑒 𝑓 𝑓 ,2 𝜏𝑟𝑎𝑑,𝑛 𝜏𝑒 𝑓 𝑓 ,2
𝑟 𝑠𝑝,𝑛
𝑛 𝜏𝑟𝑎𝑑,𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒 𝑝
𝜏𝑒𝑒 𝜏𝑒 𝑝
𝜏𝑒𝑒 𝜏𝑒 𝑝

(C.60)

Note that if 𝜏𝑒 𝑝
𝜏𝑟𝑎𝑑,𝑒𝑒 , 𝜏𝑒 𝑓 𝑓 ,2 ≈ 𝜏𝑒 𝑝 , and thus the ratio becomes 𝜏𝑟𝑎𝑑,𝑛 /𝜏𝑒𝑒 .
Clearly, if we assume that 𝜏𝑒𝑒
𝜏𝑟𝑎𝑑,𝑛 , 𝑟 𝑠𝑝,𝑒 𝑝 /𝑟 𝑠𝑝,𝑛
1. Comparing the populations for those that have and have not gone through electron-phonon coupling, we
arrive at a somewhat similar expression:
𝑟 𝑠𝑝,𝑒 𝑝 𝑛𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒𝑒 𝜏𝑟𝑎𝑑,𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒𝑒

𝑟 𝑠𝑝,𝑒𝑒
𝑛𝑒𝑒 𝜏𝑟𝑎𝑑,𝑒 𝑝
𝜏𝑒 𝑝 𝜏𝑟𝑎𝑑,𝑒 𝑝

(C.61)

where the last equality comes from the fact that we have already assumed that
𝜏𝑒 𝑝
𝜏𝑟𝑎𝑑,𝑒𝑒 . In conclusion, it is clear that the majority of the luminescence
comes from carriers that completely thermalized to their respective band-edges.
For somewhat realistic values of for example, 𝜏𝑟𝑎𝑑 ∼ 𝑛𝑠 (we have assumed all the
radiative lifetimes from the different subpopulations to be the same, for simplicity)
and for 𝜏𝑒𝑒 ∼ 𝑓 𝑠 and 𝜏𝑒 𝑝 ∼ 𝑝𝑠, we have the ratios:
𝑟 𝑠𝑝,𝑒 𝑝
∼ 106 ,
𝑟 𝑠𝑝,𝑛

𝑟 𝑠𝑝,𝑒 𝑝
∼ 103
𝑟 𝑠𝑝,𝑒𝑒

(C.62)

With more realistic numbers, it is clear that the other populations contribute very
little to the observed steady state luminescence. However, in some nanostructured
and exotic systems, we have 𝜏𝑟𝑎𝑑 ∼ 𝜏𝑒 𝑝 ∼ 𝑝𝑠. In this case, non-thermalized ("hot")
carriers can play a large role to the luminescence observed, while still obeying
Fermi-Dirac statistics.
C.4

The Validity of A Single Quasi-Fermi Level to Describe Carriers Occupying Multiple Energy Levels

The analysis above suggests that given sufficient electron-electron coupling and the
relative types of electron populations, the majority of the observed photoluminescence should occur from the thermalized carriers in steady state. However, our
analysis a priori does not suggest that a single quasi Fermi level is sufficient to describe all the electrons in the band. To do so, we develop a three-level system model
which has energies 𝐸 1 , 𝐸 2 , 𝐸 3 and 𝑛1 , 𝑛2 , 𝑛3 number of electrons occupying each
level, respectively. The total number of available states per level will be denoted
as 𝑁1 , 𝑁2 , 𝑁3 . Consider the situation where there are a total of 𝑛 = 𝑛1 + 𝑛2 + 𝑛3 .

217
Assuming these are all the states that are involved, we shall make the reasonable
assumption that turning on the light does not change the total number of electrons
𝑛, i.e.,
𝑑𝑛
=0
(C.63)
𝑑𝑡
We shall also assume that photogeneration only couples to one of the states, e.g. 𝐸 1 .
Thus, the coupled rate equations in this scenario become
𝑑𝑛1
= −𝑘 13 𝑛1 (𝑁3 − 𝑛3 ) + 𝑘 31 (𝑁1 − 𝑛1 )𝑛3 − 𝑘 12 𝑛1 (𝑁2 − 𝑛2 ) + 𝑘 21 (𝑁1 − 𝑛1 )𝑛2 + 𝐾 (𝑁1 − 𝑛1 )𝑛3
𝑑𝑡
𝑑𝑛2
= −𝑘 23 𝑛2 (𝑁3 − 𝑛3 ) + 𝑘 32 (𝑁2 − 𝑛2 )𝑛3 + 𝑘 12 𝑛1 (𝑁2 − 𝑛2 ) − 𝑘 21 (𝑁1 − 𝑛1 )𝑛2
𝑑𝑡
𝑑𝑛3
𝑑𝑛1 𝑑𝑛2
=−
𝑑𝑡
𝑑𝑡
𝑑𝑡
(C.64)
note that the expressions are in a bi-linear form, i.e., they depend on both the
occupation of filled states 𝑛𝑖 transitioning to unoccupied states 𝑁 𝑗 − 𝑛 𝑗 with some
rate constant 𝑘 𝑖 𝑗 that describes the relative coupling. These forms are bi-linear due
to the fact that we assume they obey Fermi-Dirac statistics, thus, in equilibrium, we
have 𝑛𝑖 = 𝑛𝑖0 and that
𝑛𝑖0 = 𝑁𝑖 𝑓𝑖0
(C.65)
where 𝑖 = 1, 2, 3 and
𝑓𝑖0 =

−𝐸 𝐹
1 + exp 𝐸𝑖𝑘𝑇

(C.66)

note that in equilibrium, a single 𝐸 𝐹 is sufficient to describe the equilibrium population distribution. We also note that in equilibrium, all reactions must have zero
net rate i.e. there is no net flow of carriers from one state 𝐸𝑖 to another 𝐸 𝑗 . That is,
we have
𝑘 𝑖 𝑗 𝑛𝑖0 (𝑁 𝑗 − 𝑛0𝑗 ) = 𝑘 𝑗𝑖 𝑛0𝑗 (𝑁𝑖 − 𝑛𝑖0 )
(C.67)
Thus, there is an intimate relationship between 𝑘 𝑖 𝑗 and 𝑘 𝑗𝑖 . To see this relationship,
we use our above definitions for the equilibrium carrier populations, so that
1 − 𝑓 𝑗0
𝑓𝑖0
𝑘 𝑗𝑖 = 𝑘 𝑖 𝑗
(C.68)
1 − 𝑓𝑖0
𝑓 𝑗0
and since

𝑓𝑖0

𝐸𝑖 − 𝐸 𝑓
= exp −
𝑘𝑇
1 − 𝑓𝑖

(C.69)

218
we have that
𝑓𝑖0

1 − 𝑓 𝑗0

𝐸𝑖 − 𝐸 𝑗
(C.70)
= exp −
𝑘𝑇
1 − 𝑓𝑖0
𝑓 𝑗0
For compactness, let us define 𝐴𝑖 𝑗 ≡ exp −(𝐸𝑖 − 𝐸 𝑗 )/𝑘𝑇 so that 𝑘 𝑗𝑖 = 𝑘 𝑖 𝑗 𝐴𝑖 𝑗 . Let
us now consider the possibility that we may be out of equilibrium. We first assume
that the carriers continue to have Fermi-Dirac statistics in this situation, where

𝑛𝑖 = 𝑁𝑖 𝑓𝑖

(C.71)

but now each energy level may have its own Fermi level, which we define as a
quasi-Fermi level. That is,
𝑓𝑖 ≡

 𝐸 −𝐸 
1 + exp 𝑘𝑇 𝑓𝑖

(C.72)

so that the coupled rate equation can be written as
𝑑𝑛1
= − 𝑘 13 𝑁1 𝑁3 [ 𝑓1 (1 − 𝑓3 ) − 𝐴13 𝑓3 (1 − 𝑓1 )]
𝑑𝑡
− 𝑘 12 𝑁1 𝑁2 [ 𝑓1 (1 − 𝑓2 ) − 𝐴12 𝑓2 (1 − 𝑓1 )]
+ 𝐾 𝑁1 𝑁3 𝑓3 (1 − 𝑓1 )
𝑑𝑛2
= − 𝑘 23 𝑁2 𝑁3 [ 𝑓2 (1 − 𝑓3 ) − 𝐴23 𝑓3 (1 − 𝑓2 )]
𝑑𝑡
+ 𝑘 12 𝑁1 𝑁2 [ 𝑓1 (1 − 𝑓2 ) − 𝐴12 𝑓2 (1 − 𝑓1 )]
𝑑𝑛1 𝑑𝑛2
𝑑𝑛3
=−
𝑑𝑡
𝑑𝑡
𝑑𝑡

(C.73)

and we now consider the situation where steady state (i.e. not equilibrium) can
occur, where
𝑑𝑛1 𝑑𝑛2 𝑑𝑛3
=0
(C.74)
𝑑𝑡
𝑑𝑡
𝑑𝑡
Consider now the population of carriers in 𝐸 2 in steady state (𝑑𝑛2 /𝑑𝑡 = 0), where
𝑓1 (1 − 𝑓2 ) − 𝐴12 𝑓2 (1 − 𝑓1 ) 𝑘 23 𝑁3
𝑓2 (1 − 𝑓3 ) − 𝐴23 𝑓3 (1 − 𝑓2 ) 𝑘 12 𝑁1

(C.75)

We now consider some relative values for the scales of the interactions. First, the
denominator of the left hand side is at most 1, since 𝑓𝑖 (1 − 𝑓 𝑗 ) ≤ 1 and 𝐴23 < 1 if
𝐸 2 > 𝐸 3 . Let us now consider the right-hand side where 𝑘 12
𝑘 23 , i.e. the coupling
between states 𝐸 1 and 𝐸 2 is much higher than that between 𝐸 2 and 𝐸 3 . This, for
example, would be the scenario in most semiconductors where 𝑘 12 corresponds to
intraband interactions whereas 𝑘 23 would correspond to interband interactions. In

219
this scenario, 𝑘 12
𝑘 23 and therefore the right-hand side is much less than 1 if
𝑁1 ≈ 𝑁3 (which we shall assume). Thus, the numerator of the left-hand side must
be close to vanishing, and if it were to completely vanish, we would have:
𝑓1
𝑓2
= 𝐴12
(C.76)
1 − 𝑓1
1 − 𝑓2
Notice that if we insert out definitions for the quasi-Fermi levels, as defined above,
we would then have
𝐸 1 − 𝐸 𝑓1
𝐸 2 − 𝐸 𝑓2
𝐸1 − 𝐸2
exp −
= exp −
exp −
=⇒ 𝐸 𝑓1 = 𝐸 𝑓2 (C.77)
𝑘𝑇
𝑘𝑇
𝑘𝑇
This important result says that if the rate of scattering between levels is much greater
than the transition rate between either level to a third level, then the two levels will
have almost the same fractional occupancy, or alternatively, the same quasi-Fermi
level. Now, let’s see how different the quasi-Fermi levels can be if it were not
vanishing:
𝑘 23 𝑁3
[ 𝑓2 (1 − 𝑓3 ) − 𝐴23 𝑓3 (1 − 𝑓2 )]
(C.78)
𝑓1 (1 − 𝑓2 ) − 𝐴12 𝑓2 (1 − 𝑓1 ) =
𝑘 12 𝑁1
and then dividing by (1 − 𝑓1 )(1 − 𝑓2 ), we have
 
 


𝑓1
𝑓2
𝑘 23 𝑁3
𝑓2
1 − 𝑓3
𝑓3
− 𝐴12
− 𝐴23
(C.79)
1 − 𝑓1
1 − 𝑓2
𝑘 12 𝑁1
1 − 𝑓2 1 − 𝑓1
1 − 𝑓1
and further simplifying gives us




𝑓1
𝑘 23 𝑁3
1 1 − 𝑓3
𝐴23 1 − 𝑓2
𝑓3
1 1 − 𝑓2
−1=
𝐴12
𝑓2
1 − 𝑓1
𝑘 12 𝑁1 𝐴12 1 − 𝑓1
𝐴12
𝑓2
1 − 𝑓1
(C.80)
note that the left-hand side simplifies to exp (𝐸 𝑓1 − 𝐸 𝑓2 )/𝑘𝑇 − 1. Thus,


 
𝑘 23 𝑁3
1 1 − 𝑓3
𝐴23 1 − 𝑓2
𝑓3
𝐸 𝑓1 −𝐸 𝑓2 = 𝑘𝑇 ln 1 +
(C.81)
𝑘 12 𝑁1 𝐴12 1 − 𝑓1
𝐴12
𝑓2
1 − 𝑓1
Using the same assumptions as before, we something of the form ln(1 + 𝑥) for small
𝑥, thus, if 𝑥
1, 𝐸 𝑓1 − 𝐸 𝑓2
𝑘𝑇. For other situations where this assumption of
𝑘 12 and 𝑘 23 is not true, we can explicitly calculate the difference between the two
energies. Note also that 𝐴12 also shows up as a common multiplicative factor, as
well as occupation factors like 1− 𝑓1 . Thus, it is important to remember that resonant
excitation into a specific energy or sufficiently large enough energy separations (e.g.
if 𝐸 1 − 𝐸 2 & 𝑘𝑇), we may also have substantially different quasi-Fermi levels.

220
Appendix D

COMPUTER CODE
Listing D.1: Modified Detailed Balance Code for Band Tails
10
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function SubGap_ReciprocityDetailedBalance_v8
% D e t a i l e d b a l a n c e a n a l y s i s o f band − e d g e s h a r p n e s s ,
% p a r a m e t r i z e d by a g e n e r a l band − e d g e s h a r p n e s s p a r a m e t e r ( gam ) i n t h e
absorption
% c o e f f i c i e n t , w i t h an e x p o n e n t i a l f a l l − o f f r a i s e d t o some power ( t h e t a ) ,
% w i t h band f i l l i n g p a r a m e t r i z e d s i m p l y by t a n h ( ( E−qV ) / ( 4 ∗ kbT ) )
% W r i t t e n by J o e s o n Wong
% L a s t u p d a t e d on O c t o b e r 1 7 , 2 0 1 9 .
% W r i t t e n f o r MATLAB 2018A . R e q u i r e s Curve F i t t i n g T o o l b o x and P a n e l
% function
% ( h t t p s : / / www. mathworks . com / m a t l a b c e n t r a l / f i l e e x c h a n g e /20003 − p a n e l )
clear all ; close all ; clc ;
% check i f modified a b s o r p t i o n t a b l e matrix i s a v a i l a b l e
i f f o p e n ( ' G t a b l e . mat ' ) == −1 % f i l e n a m e i s n o t a v a i l a b l e
d i s p ( ' f i l e named " G t a b l e . mat " n o t f o u n d . P r e s s any key t o b e g i n
c a l c u l a t i o n of Gtable ' ) ;
pause ;
disp ( ' Calculating Gtable . . . ' ) ;
% G e n e r a t e G v a l s f o r t a b l e l o o k up , o n l y n e e d t o do t h i s o n c e
t h e t a l a b e l s = [ 1 , 5 / 4 , 3 / 2 , 2 ] ; % t y p i c a l t h e t a s of i n t e r e s t , can change
this for other theta
xlabels = −5000:0.01:5000;
Gvals = z e r o s ( l e n g t h ( t h e t a l a b e l s ) , l e n g t h ( x l a b e l s ) ) ;
tic ;
for i = 1: length ( t h e t a l a b e l s )
for j = 1: length ( xlabels )
G v a l s ( i , j ) = G( x l a b e l s ( j ) , t h e t a l a b e l s ( i ) ) ;
end
end
toc ; % d i s p l a y t o t a l time i t took to c a l c u l a t e
f i g u r e ; % d i s p l a y Gvals , make s u r e i t h a s enough n u m e r i c a l a c c u r a c y ( e s p .
small values )
semilogy ( x l a b e l s , Gvals ( 1 , : ) , x l a b e l s , Gvals ( 2 , : ) , x l a b e l s , Gvals ( 3 , : ) ,
x l a b e l s , Gvals ( 4 , : ) ) ;
s a v e ( ' G t a b l e . mat ' , ' G v a l s ' , ' x l a b e l s ' , ' t h e t a l a b e l s ' ) ;
% make a s t r u c t u r e l a b e l l e d a s i m i l a r t o i f I had j u s t o p e n e d t h e f i l e ,
% f o r r u n n i n g r e s t of code
a . Gvals = Gvals ;
a . xlabels = xlabels ;

221
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a . thetalabels = thetalabels ;
d i s p ( ' G t a b l e c a l c u l a t e d , p r e s s any key t o s t a r t s u b g a p a n a l y s i s ' ) ;
pause ;
e l s e % open d a t a f i l e f o r G, i f i t e x i s t s
a = open ( ' G t a b l e . mat ' ) ;
d i s p ( ' P o s s i b l e G t a b l e found , p r e s s any key t o s t a r t s u b g a p a n a l y s i s ' ) ;
pause ;
end
d i s p ( ' S t a r t i n g subgap a n a l y s i s . . . ' ) ;
%% D e f i n i t i o n s
% define physical constants
e = 1.6021766208 e −19; % [C] , f u n d a m e n t a l c h a r g e
Fsun = 2 . 1 6 e −5∗ p i ; % h e m i s p h e r i c a l a n g u l a r r a n g e , s u n ( 1 / 4 6 0 0 0 f a c t o r f o r
concentration )
Fcell = pi ; % emission geometrical factor , c e l l
h = 6 . 6 2 6 0 7 0 0 4 e − 3 4 / e ; % [ eV− s ] , P l a n c k ' s c o n s t a n t
c = 2 9 9 7 9 2 4 5 8 ; % [m/ s ] , s p e e d o f l i g h t
k = 1 . 3 8 0 6 4 8 5 2 e − 2 3 / e ; % [ eV / K] , Boltzmann ' s c o n s t a n t
Tsun = 5 7 6 0 ; % [ K e l v i n ] , T e m p e r a t u r e o f Sun , f o r f u l l b l a c k −body e x p r e s s i o n
Tcell = 300; % [ Kelvin ] , Temperature of Cell
E 2 l a m _ f a c t o r = 1 . 2 3 9 8 4 1 9 3 ∗ 1 e − 6 ; % c o n v e r s i o n f a c t o r b e t w e e n e n e r g y [ eV ] t o
w a v e l e n g t h s [m] , E [ eV ] = E 2 l a m _ f a c t o r . / lambda [m]
numTol = 1 e − 1 ;
relVal = 1;
% Load S o l a r Spectrum , u s e f o r c a l c u l a t i n g SQ l i m i t
ASTMG173_filename = ' ASTMG173 . x l s x ' ;
s h e e t = ' SMARTS2 ' ;
x l R a n g e = [ 'A ' , ' 3 ' , ' : ' , 'D ' , ' 2004 ' ] ;
o u t d o o r _ d a t a = x l s r e a d ( ASTMG173_filename , s h e e t , xlRange ,
l a m b d a _ o u t d o o r = o u t d o o r _ d a t a ( : , 1 ) ∗1 e −9 ; % [m]
AM0 = o u t d o o r _ d a t a ( : , 2 ) / 1 e − 9 ; % [W/m^ 3 ]
AM15G = o u t d o o r _ d a t a ( : , 3 ) / 1 e − 9 ; % [W/m^ 3 ]
AM15D = o u t d o o r _ d a t a ( : , 4 ) / 1 e − 9 ; % [W/m^ 3 ]

' basic ' ) ;

% Define energy / wavelength ranges of i n t e r e s t
E _ r a n g e = 0 . 0 0 1 : 0 . 0 0 1 : 5 ; % [ eV ] r a n g e o f e n e r g i e s o f i n t e r e s t . Avoid s i n g u l a r
point (E = 0)
l a m b d a _ r a n g e = E 2 l a m _ f a c t o r . / E _ r a n g e ; % [m] , r a n g e o f w a v e l e n g t h s
% d e f i n e new r a n g e s b a s e d on s p e c t r a l d a t a & t a r g e t s p e c t r a l r a n g e
newAM0 = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM0, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;
newAM15G = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM15G, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;
newAM15D = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM15D, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;

%% S i m u l a t i o n Parameters
% plotting data
p l o t M a i n F i g s = t r u e ; % d e c i d e w h e t h e r t o p l o t main f i g u r e s
p l o t n e f f F i g = f a l s e ; % decide whether to p l o t / c a l c u l a t e neff f i g u r e
p l o t S I f i g s = f a l s e ; % d e c i d e whether t o p l o t SI f i g u r e s
plotGaussFig = f als e ;

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% l i g h t source
isBB = f a l s e ; % u s e 5760 K b l a c k b o d y a s i l l u m i n a t i o n l i g h t s o u r c e i f t r u e ,
o t h e r w i s e use s p e c i f i e d i r r a d
f l u x = newAM15G . / E _ r a n g e . ^ 3 / e ∗h∗ c ; % f l u x o f i n t e r e s t , i f n o t b l a c k b o d y ( e . g .
AM 15G) . E x t r a c o n v e r s i o n f a c t o r t o u s e e n e r g y i n t e g r a l
% a u x i l l a r y p a r a m e t e r s f o r s i m u l a t i o n , n o t main f o c u s
ERE = 1 ; % e x t e r n a l r a d i a t i v e e f f i c i e n c y
IQE_AG = 1 ; % i n t e r n a l quantum e f f i c i e n c y , a b o v e gap
c o m p R a t i o = 2 ; % f o r c o m p r e s s i n g G, which makes t h i n g s r u n f a s t e r ( and
d e p e n d e n t on ( E−Eg ) / gam , may n o t a c t u a l l y make t h i n g s l e s s a c c u r a t e )
% materials parameters
E g _ s p a c i n g = 0 . 5 ; % s p a c i n g o f b a n d g a p s t o c a l c u l a t e . 0 . 0 2 f o r " good "
resolution
Eg = 0 . 5 : E g _ s p a c i n g : 2 . 5 ; % [ eV ] , b a n d g a p o f m a t e r i a l
L = 1 e − 3 ; % t h i c k n e s s o f m a t e r i a l [ cm ]
a l p h a 0 = 1 e3 ; % [ 1 / cm ] , a b s o r p t i o n o f d i r e c t gap m a t e r i a l , a l p h a 0 L i s what
m a t t e r s , s e t a l p h a 0 L = 10
i s P B = t r u e ; % i n c l u d e p a u l i − b l o c k i n g / band f i l l i n g e f f e c t s
% Subgap p a r a m e t e r s
g a m _ s p a c i n g = 0 . 0 5 ; % 0 . 0 2 f o r " good " r e s o l u t i o n
gam = 1 0 . ^ [ − 3 : g a m _ s p a c i n g : 0 . 0 ] ; % Bandege s h a r p n e s s p a r a m e t e r [ eV ] ,
t h e t a = 1; % " o r d e r " of t h e e x p o n e n t i a l , 1 i s urbach , 2 i s Gaussian , e t c .
IQE_SG = 1 ; % i n t e r n a l quantum e f f i c i e n c y , s u b gap
isIQE_SG_Urbach = 0 ; % d e c i d e w h e t h e r we n e e d t o c a l c u l a t e a s u b g a p IQE t h a t
d r o p s o f f l i k e Urbach
% c o d e i s f a s t e r i f we f i g u r e o u t t h e r i g h t t h e t a f i r s t , c o m p r e s s Gx ,
% and t h e n u s e a s t a b l e l o o k −up d a t a s e t . O t h e r w i s e , p a s s i n g a r o u n d t o o
% much d a t a / m a t r i c e s .
Gx_comp = a . G v a l s ( a . t h e t a l a b e l s == t h e t a , : ) ;
Gx_comp = Gx_comp ( 1 : c o m p R a t i o : end ) ;
x_comp = a . x l a b e l s ( 1 : c o m p R a t i o : end ) ;
% c a l c u l a t e i n p u t power
i f isBB
P i n = a b s ( t r a p z ( E_range , e ∗ E _ r a n g e . ∗ bb ( E_range , Fsun , Tsun , 0 ) ) ) ; % i n p u t
power , W/m^2
else
P i n = a b s ( t r a p z ( E_range , e ∗ E _ r a n g e . ∗ f l u x ) ) ;
end
%% C a l c u l a t i o n f o r main F i g u r e s
i f plotMainFigs
o u t = SQ_FOM( isBB , f l u x , Gx_comp , x_comp , gam , Eg , a l p h a 0 ∗L , isPB , IQE_SG ,
IQE_AG , isIQE_SG_Urbach ) ; % d o e s a l l t h e FOM c a l c u l a t i o n s
V o c _ v a l s 1 = r e s h a p e ( o u t . Voc , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ; % o u t p u t , d a t a , s h a p e
i t for data plots
J s c _ v a l s 1 = r e s h a p e ( o u t . J s c , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ;
PCE_vals1 = r e s h a p e ( o u t . PCE , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ;

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%% −−−−−−−−− P l o t F i g u r e s −−−−−−−−−−−−−−−−−−−−
%% F i g 2 , main r e s u l t , EQE + SQ L i m i t
Fig2 = f i g u r e ;
s e t ( Fig2 , ' P o s i t i o n ' , [ 6 0 , 8 0 , 8 0 0 , 4 0 0 ] ) ;
gamWhich = [ 0 . 1 , 0 . 5 , 1 , 2 , 3 ] ∗ k∗ T c e l l ; % which gamma f o r EQE , PCE
EgWhich = 1 . 5 ; % which Eg f o r EQE
% p l o t EQE
E Q E c o l o r P l o t s = g e t ( gca , ' c o l o r o r d e r ' ) ;
subplot (1 ,2 ,1) ;
h o l d on ;
l e g _ s t r = [ ' S−Q ' ] ;
p l o t ( E_range , E_range >=EgWhich , ' k−− ' ) ;
f o r i = 1 : l e n g t h ( gamWhich )
p l o t ( E_range , EQE_vals ( E_range , 0 , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) , ' C o l o r ' ,
EQEcolorPlots ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( n u m 2 s t r ( gamWhich ( i ) . / k . / T c e l l ) ) ] ;
end
hold off ;
hleg = legend ( leg_str , ' Location ' , ' SouthEast ' ) ;
t i t l e ( h l e g , ' \ gamma ( kT ) ' , ' F o n t W e i g h t ' , ' Normal ' ) ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; x l i m ( [ min ( Eg ) , max ( Eg ) ] ) ;
y l a b e l ( ' Absorbance ' ) ;
a x i s s q u a r e ; box on ;
% PCE a t d i f f e r e n t gamma
subplot (1 ,2 ,2) ;
h o l d on ;
f o r i = 1 : l e n g t h ( gamWhich )
p l o t ( Eg , i n t e r p 2 ( Eg , gam , PCE_vals1 ∗ 1 0 0 , Eg , gamWhich ( i ) ) , ' C o l o r ' ,
EQEcolorPlots ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
end
% c a l c u l a t e SQ l i m i t
PCE_SQ = z e r o s ( 1 , l e n g t h ( Eg ) ) ;
Voc_SQ = z e r o s ( 1 , l e n g t h ( Eg ) ) ;
Jsc_SQ = z e r o s ( 1 , l e n g t h ( Eg ) ) ;
f o r i = 1 : l e n g t h ( Eg )
EQE_SQ = E_range >=Eg ( i ) ;
Vmax_SQ = f m i n b n d (@(V) V. ∗ JV_noBF (V, E_range , EQE_SQ , isBB , f l u x ) , 0 , Eg ( i
));
PCE_SQ ( i ) = −Vmax_SQ . ∗ JV_noBF ( Vmax_SQ , E_range , EQE_SQ , isBB , f l u x ) . / P i n ;
Jsc_SQ ( i ) = −JV_noBF ( 0 , E_range , EQE_SQ , isBB , f l u x ) ;
Voc_SQ ( i ) = f z e r o (@(V) JV_noBF (V, E_range , EQE_SQ , isBB , f l u x ) , Eg ( i ) ) ;
end
p l o t ( Eg , PCE_SQ ∗ 1 0 0 , ' k−− ' )
hold off ;
x l a b e l ( ' Band gap ( eV ) ' ) ; y l a b e l ( ' E f f i c i e n c y (%) ' ) ;
a x i s s q u a r e ; box on ;

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%% F i g 3 , PL a t Voc & f r a c t i o n o f subgap PL / S t o k e s S h i f t
Fig3 = f i g u r e ;
s e t ( Fig3 , ' P o s i t i o n ' , [ 6 0 , 8 0 , 4 0 0 , 5 0 0 ] ) ;
gamWhich = [ 0 . 2 : 0 . 4 : 2 . 6 ] ∗ k∗ T c e l l ; % which gamma f o r PL + EQE
EgWhich = 1 . 5 ; % which Eg f o r EQE
o f f s e t = 1 : 1 . 3 5 : 1 5 ; % o f f s e t f o r p l o t t i n g PL
P L c o l o r P l o t s = g e t ( gca , ' c o l o r o r d e r ' ) ;
h o l d on ;
h1 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h2 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h3 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k : ' , ' L i n e W i d t h ' , 1 . 5 ) ;
% assume t h a t t h e s y s t e m , u n d e r i l l u m i n a t i o n & no c o n t a c t s , r e a c h e s i t s
% Voc . Thus , PL i s t h e PL a t Voc .
f o r i = 1 : l e n g t h ( gamWhich )
Voc_gam = i n t e r p 2 ( Eg , gam , V o c _ v a l s 1 , EgWhich , gamWhich ( i ) ) ;
P L _ s p e c t = J _ s p e c t ( E_range , Voc_gam , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
P L p l o t = p l o t ( E_range , P L _ s p e c t . / max ( P L _ s p e c t ) + o f f s e t ( i ) , ' −− ' , ' C o l o r ' ,
P L c o l o r P l o t s ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
EQE_spect = EQE_vals ( E_range , Voc_gam , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
p l o t ( E_range , EQE_spect . / max ( EQE_spect ) + o f f s e t ( i ) , ' C o l o r ' , P L p l o t .
Color , ' LineWidth ' , 1 . 5 )
PEg = EQE_spect ( 3 : end ) − EQE_spect ( 1 : end − 2 ) ; %b a n d g a p d i s t r i b u t i o n ,
central difference
p l o t ( E _ r a n g e ( 2 : end − 1 ) , PEg . / max ( PEg ) + o f f s e t ( i ) , ' : ' , ' C o l o r ' , P L p l o t .
Color , ' LineWidth ' , 1 . 5 ) ;
i f gamWhich ( i ) == k∗ T c e l l
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' ,
PLplot . Color ) ;
else
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = ' , n u m 2 s t r ( gamWhich ( i ) / ( k∗ T c e l l ) )
, ' kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' , P L p l o t . C o l o r ) ;
end
end
hold off ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; x l i m ( [ 0 . 0 , 2 . 1 ] ) ; y l i m ( [ 0 . 5 , 1 1 . 5 ] ) ;
yticks ([]) ;
l e g e n d ( [ h1 , h2 , h3 ] , { ' PL ' , ' Abs ' , ' P ( E_g ) ' } , ' L o c a t i o n ' , ' N o r t h ' , '
Orientation ' , ' horizontal ' ) ;
legend ( ' boxoff ' ) ;
box on ;
figure ;
J _ s u b g a p _ R a t i o = z e r o s ( s i z e ( gam ) ) ; % f r a c t i o n o f r e c o m b i n a t i o n t h a t i s below
t h e bandgap
P L _ p e a k s U r b a c h = z e r o s ( s i z e ( gam ) ) ; % f i n d o u t p e a k PL p o s i t i o n f o r e a c h gam ,
for Stokes s h i f t c a l c u l a t i o n
f o r i = 1 : l e n g t h ( gam )
Voc_gam = i n t e r p 2 ( Eg , gam , V o c _ v a l s 1 , EgWhich , gam ( i ) ) ;

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P L _ s p e c t = J _ s p e c t ( E_range , Voc_gam , EgWhich , gam ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;

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[ ~ , p k _ i n d x ] = max ( P L _ s p e c t ) ; % f i n d E _ r a n g e i n d x f o r PL p e a k
PL_peaksUrbach ( i ) = E_range ( pk_indx ) ;
J _ s u b g a p _ R a t i o ( i ) = t r a p z ( E _ r a n g e ( E_range EgWhich ) ) . / t r a p z ( E_range , P L _ s p e c t ) ;
end
yyaxis l e f t ;
s e m i l o g x ( gam∗1 e3 , J _ s u b g a p _ R a t i o , ' L i n e W i d t h ' , 1 . 5 ) ; h o l d on ;
p l o t ( o n e s ( 1 , 1 0 0 ) ∗k∗ T c e l l ∗1 e3 , l i n s p a c e ( 0 , 1 , 1 0 0 ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
t e x t ( k∗ T c e l l ∗1 e3 ∗ 0 . 7 , 0 . 2 3 , ' \ gamma = kT ' , ' C o l o r ' , ' b l a c k ' , ' R o t a t i o n ' , 9 0 , '
FontSize ' , 12) ;
x l a b e l ( ' \ gamma (meV) ' ) ; y l a b e l ( ' F r a c t i o n o f PL below band gap ' ) ;
box on ;
ylim ( [ 0 , 1 ] ) ;
yyaxis r i g h t ;
s e m i l o g x ( gam∗1 e3 , EgWhich − PL_peaksUrbach , ' L i n e W i d t h ' , 1 . 5 ) ;
y l a b e l ( ' S t o k e s S h i f t ( eV ) ' ) ;
y l i m ( [ min ( EgWhich − P L _ p e a k s U r b a c h ) , max ( EgWhich − P L _ p e a k s U r b a c h ) ] ) ;
end
if plotneffFig
%% F i g u r e S4 , IV c u r v e + n _ e f f A n a l y s i s
FigS4 = f i g u r e ;
s e t ( FigS4 , ' P o s i t i o n ' , [ 6 0 , 4 0 , 7 0 0 , 6 0 0 ] ) ;
numV_vals = 1 0 0 ;
EgWhich = [ 0 . 8 , 1 . 3 4 , 2 . 0 ] ; % b a n d g a p o f l i n e c u t s
neff_gam_Eg = z e r o s ( l e n g t h ( Eg ) , l e n g t h ( gam ) ) ;
d e l t a n _ g a m _ E g = z e r o s ( l e n g t h ( Eg ) , l e n g t h ( gam ) , 2 ) ;
c o l o r P l o t s = z e r o s ( l e n g t h ( EgWhich ) , 3 ) ;
% panel parameters
p = panel () ;
p . pack ( ' h ' , 2) ;
p ( 1 ) . pack ( ' v ' , [ 0 . 0 6 , 0 . 4 7 , 0 . 4 7 ] ) ;
p ( 2 ) . pack ( ' v ' , 3) ;
p . de . m a r g i n = 0 ;
p ( 1 ) . de . m a r g i n = 2 ;
p ( 2 ) . de . m a r g i n = 2 ;
p (1 ,1) . margintop = 40;
p (1) . marginright = 20;
p (2) . marginleft = 20;
% plot n_eff contour
h1 = p ( 1 , 2 ) . s e l e c t ( ) ;
f o r i = 1 : l e n g t h ( Eg )
V = l i n s p a c e ( 3 ∗ k∗ T c e l l , Eg ( i ) −3∗k∗ T c e l l , numV_vals ) ;
f o r j = 1 : l e n g t h ( gam )
J V _ d a r k = z e r o s ( s i z e (V) ) ;

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f o r m = 1 : l e n g t h (V)
EQEV_gam_Eg = EQE_vals ( E_range , V(m) , Eg ( i ) , gam ( j ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
J V _ d a r k (m) = J V _ F u l l (V(m) , E_range , EQEV_gam_Eg , 0 , 0 ) ;
end
f i t r e s u l t = f i t (V ' , l o g ( J V _ d a r k ) ' , ' p o l y 1 ' ) ; % l o g ( J ) = l o g ( J 0 ) + qV / (
nkT ) , i . e . y = p ( 1 ) ∗x + p ( 2 )
pvals = c o e f f v a l u e s ( f i t r e s u l t ) ; % get values of p
c i = c o n f i n t ( f i t r e s u l t , 0 . 9 5 ) ; % f i n d r a n g e o f p , 95% c o n f i d e n c e . s i z e
( 2 xn ) , where n i n number o f c o e f f i c i e n t s
neff_gam_Eg ( i , j ) = 1 / ( k∗ T c e l l ∗ p v a l s ( 1 ) ) ; % 1 / ( nkT / q ) = p ( 1 )
d e l t a n _ g a m _ E g ( i , j , [ 1 , 2 ] ) = a b s ( 1 . / ( k∗ T c e l l ∗ c i ( [ 2 , 1 ] , 1 ) ) − neff_gam_Eg (
i ,j));

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end
end
s u r f ( gam , Eg , neff_gam_Eg ) ;
a x i s ( [ min ( gam ) , max ( gam ) , min ( Eg ) , max ( Eg ) ] ) ; c o l o r m a p ( ' h o t ' ) ;
view ( [ 0 9 0 ] ) ;
shading i n t e r p ;
s e t ( gca , ' x s c a l e ' , ' l o g ' ) ;
y l a b e l ( ' Band gap ( eV ) ' ) ;
box on ; h o l d on ;
xticklabels ([]) ;
f o r i = 1 : l e n g t h ( EgWhich )
E g P l o t = p l o t 3 ( l o g s p a c e ( − 3 , 0 , 1 0 0 ) , EgWhich ( i ) ∗ o n e s ( 1 , 1 0 0 ) , 1000∗ o n e s
( 1 , 1 0 0 ) , ' −− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
c o l o r P l o t s ( i , : ) = EgPlot . Color ;
end

% Colorbars for respsective contours
cbh = c o l o r b a r ( h1 , ' L o c a t i o n ' , ' N o r t h O u t s i d e ' ) ; c o l o r m a p ( cbh , h o t ) ;
p ( 1 , 1 ) . s e l e c t ( cbh ) ; cbh . L i m i t s = [ 0 . 9 , 7 . 1 ] ; cbh . T i c k s = 1 : 7 ;
x l a b e l ( cbh , ' n_ { e f f } ' ) ;
% plot the n_eff l in e cu t s
h2 = p ( 1 , 3 ) . s e l e c t ( ) ;
h2 . X S c a l e = ' l o g ' ;
hold a l l
gam_mev = gam∗1 e3 ;
leg_str = [];
haveleg = [ ] ;
f o r i = 1 : l e n g t h ( EgWhich )
n _ v a l s = i n t e r p 2 ( Eg , gam , neff_gam_Eg ' , EgWhich ( i ) , gam ) ;
d e l t a n 1 = i n t e r p 2 ( Eg , gam , r e s h a p e ( d e l t a n _ g a m _ E g ( : , : , 1 ) , l e n g t h ( Eg ) , l e n g t h (
gam ) ) ' , EgWhich ( i ) , gam ) ;
d e l t a n 2 = i n t e r p 2 ( Eg , gam , r e s h a p e ( d e l t a n _ g a m _ E g ( : , : , 2 ) , l e n g t h ( Eg ) , l e n g t h (
gam ) ) ' , EgWhich ( i ) , gam ) ;
p l o t _ l e g = p l o t ( gam_mev , n _ v a l s , ' −− ' , ' L i n e W i d t h ' , 1 . 5 , ' C o l o r ' ,
c o l o r P l o t s ( i , : ) ) ; h o l d on ;
p l o t ( gam_mev , n _ v a l s − d e l t a n 1 , ' − ' , ' L i n e W i d t h ' , 1 . 5 , ' C o l o r ' , c o l o r P l o t s ( i
,:) );
p l o t ( gam_mev , n _ v a l s + d e l t a n 2 , ' − ' , ' L i n e W i d t h ' , 1 . 5 , ' C o l o r ' , c o l o r P l o t s ( i
,:) );
x2 = [ gam_mev , f l i p l r ( gam_mev ) ] ;

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inBetween = [ t r a n s p o s e ( n_vals − d e l t a n 1 ) , f l i p l r ( t r a n s p o s e ( n_vals + d e l t a n 2 ) )
];
f i l l P l o t = f i l l ( x2 , i n B e t w e e n , c o l o r P l o t s ( i , : ) ) ;
set ( f i l l P l o t , ' facealpha ' , 0.25) ;
set ( f i l l P l o t , ' edgealpha ' , 0.25) ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( n u m 2 s t r ( EgWhich ( i ) ) ) ] ;
haveleg = [ haveleg , p l o t _ l e g ] ;
end
x l a b e l ( ' \ gamma (meV) ' ) ; y l a b e l ( ' n_ { e f f } ' ) ;
box on ;
p l o t ( o n e s ( 1 , 1 0 0 ) ∗k∗ T c e l l ∗1 e3 , l i n s p a c e ( 0 , 1 0 , 1 0 0 ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
t e x t ( k∗ T c e l l ∗1 e3 ∗ 0 . 7 , 2 . 5 , ' \ gamma = kT ' , ' C o l o r ' , ' b l a c k ' , ' R o t a t i o n ' , 9 0 , '
FontSize ' , 12) ;
ylim ( [ 0 . 6 , 7 . 4 ] ) ; y t i c k s ( 1 : 1 : 7 ) ;
h l e g = l e g e n d ( haveleg , l e g _ s t r , ' L o c a t i o n ' , ' NorthWest ' ) ;
t i t l e ( h l e g , ' E_g ( eV ) ' , ' F o n t W e i g h t ' , ' n o r m a l ' ) ;
s e t ( gca , ' L i n e W i d t h ' , 1 . 2 5 ) ;
% p l o t IVs
gamWhich = [ 0 . 1 , 0 . 5 , 1 , 2 , 3 ] ∗ k∗ T c e l l ; % which gamma f o r IV
J V c o l o r P l o t s = g e t ( gca , ' c o l o r o r d e r ' ) ;
f o r i = 1 : l e n g t h ( EgWhich )
hi = p (2 , i ) . s e l e c t () ;
V = l o g s p a c e ( − 7 , l o g 1 0 ( EgWhich ( i ) ) , numV_vals ) ; % v o l t a g e p o i n t s
V = [ −max (V) , V ] ;
% G e n e r a t e Dark JV c u r v e f o r a g i v e n gamma
% p l o t SQ l i m i t JV
JV_SQ = z e r o s ( 1 , l e n g t h (V) ) ;
f o r m = 1 : l e n g t h (V)
JV_SQ (m) = JV_noBF (V(m) , E_range , E_range >=EgWhich ( i ) , 0 , 0 ) ;
end
s e m i l o g y (V . / ( k∗ T c e l l ) , a b s ( JV_SQ / JV_SQ ( 1 ) ) , ' k−− ' ) ;
h o l d on ;
% p l o t JV
JV_gamma = z e r o s ( l e n g t h ( gamWhich ) , l e n g t h (V) ) ;
leg_str = [];
f o r j = 1 : l e n g t h ( gamWhich )
f o r m = 1 : l e n g t h (V)
EQEV_gam = EQE_vals ( E_range , V(m) , EgWhich ( i ) , gamWhich ( j ) , a l p h a 0
, isPB , Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
JV_gamma ( j ,m) = J V _ F u l l (V(m) , E_range , EQEV_gam , 0 , 0 ) ;
end
s e m i l o g y (V . / ( k∗ T c e l l ) , a b s ( JV_gamma ( j , : ) . / JV_gamma ( j , 1 ) ) , ' C o l o r ' ,
J V c o l o r P l o t s ( j , : ) , ' LineWidth ' , 1 . 5 ) ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( n u m 2 s t r ( gamWhich ( j ) . / k . / T c e l l ) ) ] ;
end
hold off ;
x l i m ( [ 0 , min ( EgWhich ) / ( k∗ T c e l l ) ] ) ;
xticklabels ([]) ;
y l a b e l ( ' J / J _ 0 ( \ gamma ) ' ) ;
box on ;
s e t ( gca , ' XColor ' , c o l o r P l o t s ( i , : ) ) ; s e t ( gca , ' YColor ' , c o l o r P l o t s ( i , : ) ) ;
s e t ( gca , ' L i n e W i d t h ' , 1 . 2 5 ) ;

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y l i m ( [ 1 e −1 , 1 e7 ] ) ;
end
xticklabels (0:5:30) ;
x l a b e l ( ' qV / { kT} ' ) ;
p (2 ,1) . select () ;
h l e g = l e g e n d ( [ ' S−Q ' , l e g _ s t r ] , ' L o c a t i o n ' , ' S o u t h E a s t ' , ' f o n t s i z e ' , 8 ) ;
t i t l e ( h l e g , ' \ gamma ( kT ) ' , ' F o n t W e i g h t ' , ' n o r m a l ' ) ;
end

d i s p ( ' Now p l o t t i n g S I f i g u r e s . P r e s s any key t o c o n t i n u e ' ) ;
pause ;
%% F i g S1 , d i f f e r e n t gamma and Eg , FOM c o n t o u r s / l i n e c u t s
FigS1 = f i g u r e ;
s e t ( FigS1 , ' P o s i t i o n ' , [ 7 0 , 4 0 , 9 0 0 , 6 0 0 ] ) ;
% panel parameters
p = panel () ;
p . pack ( ' h ' , [ 0 . 4 7 , 0 . 0 6 , 0 . 4 7 ] ) ;
p ( 1 ) . pack ( ' v ' , 4) ;
p ( 2 ) . pack ( ' v ' , 4) ;
p ( 3 ) . pack ( ' v ' , 4) ;
p . de . m a r g i n = 0 ;
p ( 1 ) . de . m a r g i n = 2 ;
p ( 2 ) . de . m a r g i n = 2 ;
p (1) . marginright = 2;
p (2) . marginright = 40;
p (3) . marginleft = 40;
EgWhich = [ 0 . 8 , 1 . 3 4 , 2 . 0 ] ; % which Eg f o r FOM l i n e c u t s
F F _ v a l s 1 = PCE_vals1 . ∗ P i n . / ( V o c _ v a l s 1 . ∗ J s c _ v a l s 1 ) ; % c a l c u l a t e F i l l f a c t o r s
% PCE c o n t o u r
h1 = p ( 1 , 1 ) . s e l e c t ( ) ;
s u r f ( gam , Eg , PCE_vals1 ' ∗ 1 0 0 ) ; c o l o r m a p ( h o t ) ;
a x i s ( [ min ( gam ) , max ( gam ) , min ( Eg ) , max ( Eg ) ] ) ;
view ( [ 0 9 0 ] ) ;
shading i n t e r p ;
s e t ( gca , ' x s c a l e ' , ' l o g ' ) ;
xticklabels ([]) ;
y l a b e l ( ' Band gap ( eV ) ' ) ;
yticks (0.7:0.5:2.7) ;
box on ;
h o l d on ;
f o r i = 1 : l e n g t h ( EgWhich )
p l o t 3 ( l o g s p a c e ( − 3 , 0 , 1 0 0 ) , EgWhich ( i ) ∗ o n e s ( 1 , 1 0 0 ) , 1000∗ o n e s ( 1 , 1 0 0 ) , ' −− ' ,
' LineWidth ' , 1 . 5 ) ;
end

% Voc c o n t o u r
h2 = p ( 1 , 2 ) . s e l e c t ( ) ;
s u r f ( gam , Eg , V o c _ v a l s 1 ' ) ;

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a x i s ( [ min ( gam ) , max ( gam ) , min ( Eg ) , max ( Eg ) ] ) ;
view ( [ 0 9 0 ] ) ;
shading i n t e r p ;
s e t ( gca , ' x s c a l e ' , ' l o g ' ) ;
xticklabels ([]) ;
yticks (0.7:0.5:2.7) ;
y l a b e l ( ' Band gap ( eV ) ' ) ;
box on ; h o l d on ;
f o r i = 1 : l e n g t h ( EgWhich )
p l o t 3 ( l o g s p a c e ( − 3 , 0 , 1 0 0 ) , EgWhich ( i ) ∗ o n e s ( 1 , 1 0 0 ) , 1000∗ o n e s ( 1 , 1 0 0 ) , ' −− ' ,
' LineWidth ' , 1 . 5 ) ;
end

% Jsc contour
h3 = p ( 1 , 3 ) . s e l e c t ( ) ;
s u r f ( gam , Eg , J s c _ v a l s 1 ' ∗ 1 e − 1 ) ;
a x i s ( [ min ( gam ) , max ( gam ) , min ( Eg ) , max ( Eg ) ] ) ;
view ( [ 0 9 0 ] ) ;
shading i n t e r p ;
s e t ( gca , ' x s c a l e ' , ' l o g ' ) ;
xticklabels ([]) ;
yticks (0.7:0.5:2.7) ;
y l a b e l ( ' Band gap ( eV ) ' ) ;
box on ; h o l d on ;
f o r i = 1 : l e n g t h ( EgWhich )
p l o t 3 ( l o g s p a c e ( − 3 , 0 , 1 0 0 ) , EgWhich ( i ) ∗ o n e s ( 1 , 1 0 0 ) , 1000∗ o n e s ( 1 , 1 0 0 ) , ' −− ' ,
' LineWidth ' , 1 . 5 ) ;
end

% F i l l Factor contour
h4 = p ( 1 , 4 ) . s e l e c t ( ) ;
s u r f ( gam , Eg , F F _ v a l s 1 ' ) ;
a x i s ( [ min ( gam ) , max ( gam ) , min ( Eg ) , max ( Eg ) ] ) ;
view ( [ 0 9 0 ] ) ;
shading i n t e r p ;
s e t ( gca , ' x s c a l e ' , ' l o g ' ) ;
yticks (0.7:0.5:2.7) ;
y l a b e l ( ' Band gap ( eV ) ' ) ;
box on ; h o l d on ;
f o r i = 1 : l e n g t h ( EgWhich )
p l o t 3 ( l o g s p a c e ( − 3 , 0 , 1 0 0 ) , EgWhich ( i ) ∗ o n e s ( 1 , 1 0 0 ) , 1000∗ o n e s ( 1 , 1 0 0 ) , ' −− ' ,
' LineWidth ' , 1 . 5 ) ;
end
x t i c k l a b e l s ( { ' 10^0 ' , ' 10^1 ' , ' 10^2 ' , ' 10^3 ' } ) ;
x l a b e l ( ' \ gamma (meV) ' ) ;
% Colorbars for respsective contours
cbh = c o l o r b a r ( h1 , ' L o c a t i o n ' , ' E a s t O u t s i d e ' ) ;
p ( 2 , 1 ) . s e l e c t ( cbh ) ;
c o l o r m a p ( cbh , h o t ) ;
x l a b e l ( cbh , ' E f f i c i e n c y (%) ' ) ;
cbh = c o l o r b a r ( h2 ,

' L o c a t i o n ' , ' E a s t O u t s i d e ' ) ; c o l o r m a p ( cbh , h o t ) ;

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p ( 2 , 2 ) . s e l e c t ( cbh ) ;
x l a b e l ( cbh , ' V_{ oc } (V) ' ) ;
cbh = c o l o r b a r ( h3 , ' L o c a t i o n ' , ' E a s t O u t s i d e ' ) ; c o l o r m a p ( cbh , h o t ) ;
p ( 2 , 3 ) . s e l e c t ( cbh ) ;
x l a b e l ( cbh , ' J _ { s c } (mA cm^{ −2}) ' ) ;
cbh = c o l o r b a r ( h4 , ' L o c a t i o n ' , ' E a s t O u t s i d e ' ) ; c o l o r m a p ( cbh , h o t ) ;
p ( 2 , 4 ) . s e l e c t ( cbh ) ;
x l a b e l ( cbh , ' F i l l F a c t o r ' ) ;
% Plot Linecuts
p (3 ,1) . select () ; % eta
leg_str = [];
f o r i = 1 : l e n g t h ( EgWhich )
s e m i l o g x ( gam , i n t e r p 2 ( Eg , gam , PCE_vals1 ∗ 1 0 0 , EgWhich ( i ) , gam ) , ' L i n e W i d t h
' , 1 . 5 ) ; h o l d on ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( EgWhich ( i ) ) ] ;
end
xticklabels ( [ ] ) ; yticks (10:10:30) ;
y l i m ( [ 0 , 3 5 ] ) ; y l a b e l ( ' E f f i c i e n c y (%) ' ) ;
hleg = legend ( l e g _ s t r , ' Location ' , ' NorthEast ' , ' FontSize ' , 8) ; t i t l e ( hleg , '
Band gap ( eV ) ' , ' F o n t W e i g h t ' , ' Normal ' ) ;
box on ;
p ( 3 , 2 ) . s e l e c t ( ) ; % Voc
f o r i = 1 : l e n g t h ( EgWhich )
s e m i l o g x ( gam , i n t e r p 2 ( Eg , gam , V o c _ v a l s 1 , EgWhich ( i ) , gam ) , ' L i n e W i d t h ' , 1 . 5 )
; h o l d on ;
end
x t i c k l a b e l s ( [ ] ) ; y t i c k s ( 0 . 4 : 0 . 4 : 1 . 6 ) ; y l a b e l ( ' V_{ oc } (V) ' ) ;
box on ;
p (3 ,3) . s e l e c t () ; % Jsc
f o r i = 1 : l e n g t h ( EgWhich )
s e m i l o g x ( gam , i n t e r p 2 ( Eg , gam , J s c _ v a l s 1 ∗1 e −1 , EgWhich ( i ) , gam ) , ' L i n e W i d t h ' ,
1 . 5 ) ; h o l d on ;
end
y t i c k s ( 1 0 : 2 0 : 7 0 ) ; y l a b e l ( ' J _ { s c } (mA cm^{ −2}) ' ) ;
xticklabels ([]) ;
box on ;
p (3 ,4) . select () ; % F i l l Factor
f o r i = 1 : l e n g t h ( EgWhich )
s e m i l o g x ( gam , i n t e r p 2 ( Eg , gam , F F _ v a l s 1 , EgWhich ( i ) , gam ) , ' L i n e W i d t h ' , 1 . 5 ) ;
h o l d on ;
end
yticks ( 0 . 1 : 0 . 2 : 0 . 9 ) ; ylabel ( ' F i l l Factor ' ) ;
box on ;
x t i c k l a b e l s ( { ' 10^0 ' , ' 10^1 ' , ' 10^2 ' , ' 10^3 ' } ) ;
x l a b e l ( ' \ gamma (meV) ' ) ;

if plotSIfigs
d i s p ( ' Now c a l c u l a t i n g e f f e c t s o f a l p h a 0 L . P r e s s any key t o c o n t i n u e ' ) ;

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%% F i g S2 , EQE & FOM l i n e c u t s f o r d i f f e r e n t gamma , alpha0L ( t h i c k n e s s )
FigS2 = f i g u r e ;
s e t ( FigS2 , ' P o s i t i o n ' , [ 6 0 , 2 0 , 7 5 0 , 6 0 0 ] ) ;
% materials parameters
EgWhich = 1 . 5 ; % [ eV ] , b a n d g a p o f m a t e r i a l
L = 1 ; % t h i c k n e s s of m a t e r i a l , j u s t s e t t o 1 so t a t alpha0 i s i n u n i t s of 1 /L
a l p h a 0 = [ 0 . 0 1 , 0 . 1 , 0 . 3 , 1 , 1 0 ] ; % a b s o r p t i o n o f d i r e c t gap m a t e r i a l ,
a l p h a 0 L i s what m a t t e r s
% Subgap p a r a m e t e r s
gam = 1 0 . ^ [ − 3 : g a m _ s p a c i n g : 0 ] ; % Bandedge s h a r p n e s s p a r a m e t e r [ eV ] ,

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% panel parameters
p = panel () ;
p . pack ( ' h ' , [ 1 / 2 , 1 / 2 ] ) ;
p ( 1 ) . pack ( ' v ' , 2) ;
p ( 2 ) . pack ( ' v ' , 4) ;
p . de . m a r g i n = 0 ;
p (1) . marginright = 20;
p (2) . marginleft = 20;
p ( 1 , 1 ) . marginbottom = 2;

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pause ;

% f i r s t gamma t o p l o t
gam1 = 0 . 5 ∗ k∗ T c e l l ;
p (1 ,1) . select () ;
h o l d on ;
leg_str = [];
t e x t ( 0 . 7 , 0 . 1 , [ ' \ gamma = ' , n u m 2 s t r ( gam1 . / k . / T c e l l ) , ' kT ' ] , ' F o n t W e i g h t ' , '
Bold ' , ' C o l o r ' , ' Red ' ) ;
for i = 1: length ( alpha0 )
p l o t ( E_range , EQE_vals ( E_range , 0 , EgWhich , gam1 , a l p h a 0 ( i ) , isPB , Gx_comp
, x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) , ' L i n e W i d t h ' , 1 . 5 ) ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( num2str ( alpha0 ( i ) ) ) ] ;
end
hold off ;
h l e g = l e g e n d ( l e g _ s t r , ' L o c a t i o n ' , ' NorthWest ' , ' f o n t s i z e ' , 10) ;
t i t l e ( hleg , ' \ alpha_0L ' , ' FontWeight ' , ' normal ' ) ;
xlim ( [ 0 . 5 , 2 . 5 ] ) ; x t i c k l a b e l s ( [ ] ) ;
yticks (0.1:0.2:0.9) ;
y l a b e l ( ' Absorbance ' ) ;
box on ;
% s e c o n d gamma t o p l o t
gam2 = 2∗ k∗ T c e l l ;
p (1 ,2) . select () ;
h o l d on ;
t e x t ( 0 . 7 , 0 . 1 , [ ' \ gamma =
' Bold ' , ' C o l o r ' , ' Red ' ) ;
for i = 1: length ( alpha0 )

' , n u m 2 s t r ( gam2 . / k . / T c e l l ) , ' kT ' ] ,

' FontWeight ' ,

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p l o t ( E_range , EQE_vals ( E_range , 0 , EgWhich , gam2 , a l p h a 0 ( i ) , isPB , Gx_comp
, x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) , ' L i n e W i d t h ' , 1 . 5 ) ;
end
hold off ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; x l i m ( [ 0 . 5 , 2 . 5 ] ) ;
yticks (0.1:0.2:0.9) ;
y l a b e l ( ' Absorbance ' ) ;
box on ;
% C a l c u l a t e FOM f o r v a r y i n g a l p h a 0 L and gamma , f o r f i x e d Eg and IQE
o u t = SQ_FOM( isBB , f l u x , Gx_comp , x_comp , gam , EgWhich , a l p h a 0 ∗L , isPB , IQE_SG
, IQE_AG , isIQE_SG_Urbach ) ; % d o e s a l l t h e FOM c a l c u l a t i o n s
V o c _ v a l s 2 = r e s h a p e ( o u t . Voc , l e n g t h ( a l p h a 0 ) , l e n g t h ( gam ) ) ' ; % o u t p u t , d a t a ,
shape i t f o r data p l o t s
J s c _ v a l s 2 = r e s h a p e ( o u t . J s c , l e n g t h ( a l p h a 0 ) , l e n g t h ( gam ) ) ' ;
PCE_vals2 = r e s h a p e ( o u t . PCE , l e n g t h ( a l p h a 0 ) , l e n g t h ( gam ) ) ' ;
F F _ v a l s 2 = PCE_vals2 ∗ P i n . / ( V o c _ v a l s 2 . ∗ J s c _ v a l s 2 ) ;
p (2 ,1) . select () ; % eta
for i = 1: length ( alpha0 )
s e m i l o g x ( gam , i n t e r p 2 ( a l p h a 0 , gam , PCE_vals2 ∗ 1 0 0 , a l p h a 0 ( i ) , gam ) , '
LineWidth ' , 1 . 5 ) ;
h o l d on ;
end
h o l d o f f ; box on ;
yticks (10:10:30) ;
ylim ( [ 0 , 3 5 ] ) ;
xticklabels ([]) ;
y l a b e l ( ' E f f i c i e n c y (%) ' ) ;
% j u s t m e n t i o n which Eg i n S I
t i t l e ( [ ' E_g = ' , n u m 2 s t r ( EgWhich ) ,

' eV ' ] ,

' FontWeight ' ,

' normal ' ) ;

p ( 2 , 2 ) . s e l e c t ( ) ; % Voc
f o r i =1: l e n g t h ( alpha0 )
s e m i l o g x ( gam , i n t e r p 2 ( a l p h a 0 , gam , V o c _ v a l s 2 , a l p h a 0 ( i ) , gam ) , ' L i n e W i d t h '
, 1.5) ;
h o l d on ;
end
hold off ;
y l a b e l ( ' V_{ oc } (V) ' ) ; x t i c k l a b e l s ( [ ] ) ;
yticks (0.4:0.4:1.6) ;
box on ;

p (2 ,3) . s e l e c t () ; % Jsc
for i = 1: length ( alpha0 )
s e m i l o g x ( gam , i n t e r p 2 ( a l p h a 0 , gam , J s c _ v a l s 2 ∗1 e −1 , a l p h a 0 ( i ) , gam ) , '
LineWidth ' , 1 . 5 ) ;
h o l d on ;
end
x t i c k l a b e l s ( [ ] ) ; y l a b e l ( ' J _ { s c } (mA cm^{ −2}) ' ) ; y t i c k s ( 1 0 : 2 0 : 7 0 ) ;
box on ;

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p ( 2 , 4 ) . s e l e c t ( ) ; % FF
for i = 1: length ( alpha0 )
s e m i l o g x ( gam , i n t e r p 2 ( a l p h a 0 , gam , F F _ v a l s 2 , a l p h a 0 ( i ) , gam ) , ' L i n e W i d t h ' ,
1.5) ;
h o l d on ;
end
ylabel ( ' F i l l Factor ' ) ;
yticks (0.1:0.2:0.9) ;
box on ;
x t i c k l a b e l s ( { ' 10^0 ' , ' 10^1 ' , ' 10^2 ' , ' 10^3 ' } ) ;
x l a b e l ( ' \ gamma (meV) ' ) ;

%% F i g S3 , e f f e c t o f i n c l u d i n g P a u l i − b l o c k i n g t o FOM
FigS3 = f i g u r e ; %
s e t ( FigS3 , ' P o s i t i o n ' , [ 6 0 , 4 0 , 7 0 0 , 5 0 0 ] ) ;
% panel parameters
p = panel () ;
p . pack ( ' h ' , [ 1 / 2 , 1 / 2 ] ) ;
p ( 1 ) . pack ( ' v ' , 1) ;
p ( 2 ) . pack ( ' v ' , 4) ;
p . de . m a r g i n = 0 ;
p (1) . marginright = 20;
p (2) . marginleft = 20;
% materials parameters
EgWhich = 1 . 5 ; % [ eV ] , b a n d g a p o f m a t e r i a l
L = 1e −3; % t h i c k n e s s of m a t e r i a l , u n i t s of 1 / a l p h a 0
a l p h a 0 = 1 e3 ; % [ 1 / cm ] , a b s o r p t i o n o f d i r e c t gap m a t e r i a l , a l p h a 0 L i s what
matters
i s P B = f a l s e ; % i n c l u d e p a u l i − b l o c k i n g / band f i l l i n g e f f e c t s
% Subgap p a r a m e t e r s
gam = 1 0 . ^ [ − 3 : g a m _ s p a c i n g : 0 ] ; % Bandege s h a r p n e s s p a r a m e t e r [ eV ] ,
IQE_SG = 1 ; % i n t e r n a l quantum e f f i c i e n c y , s u b gap
% PL p a r a m e t e r s
gamWhich = [ 0 . 2 : 0 . 4 : 2 . 6 ] ∗ k∗ T c e l l ; % which gamma f o r EQE , PCE
offset = 1:1.35:15;
p (1 ,1) . select () ;
h o l d on ;
% assume t h a t t h e s y s t e m , u n d e r i l l u m i n a t i o n & no c o n t a c t s , r e a c h e s i t s
% Voc . Thus , PL i s t h e PL a t Voc .
h1 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h2 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h3 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k : ' , ' L i n e W i d t h ' , 1 . 5 ) ;
f o r i = 1 : l e n g t h ( gamWhich )
P L _ s p e c t = J _ s p e c t ( E_range , 0 , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB , Gx_comp
, x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;

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P L p l o t = p l o t ( E_range , P L _ s p e c t . / max ( P L _ s p e c t ) + o f f s e t ( i ) , ' −− ' , ' C o l o r ' ,
P L c o l o r P l o t s ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
EQE_spect = EQE_vals ( E_range , 0 , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
p l o t ( E_range , EQE_spect . / max ( EQE_spect ) + o f f s e t ( i ) , ' C o l o r ' , P L p l o t .
Color , ' LineWidth ' , 1 . 5 )
PEg = EQE_spect ( 3 : end ) − EQE_spect ( 1 : end − 2 ) ; % smooth o u t EQE f i r s t ?
p l o t ( E _ r a n g e ( 2 : end − 1 ) , PEg . / max ( PEg ) + o f f s e t ( i ) , ' : ' , ' C o l o r ' , P L p l o t .
Color , ' LineWidth ' , 1 . 5 ) ;
i f gamWhich ( i ) == k∗ T c e l l
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' ,
PLplot . Color ) ;
else
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = ' , n u m 2 s t r ( gamWhich ( i ) / ( k∗ T c e l l ) )
, ' kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' , P L p l o t . C o l o r ) ;
end
end
hold off ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; x l i m ( [ 0 . 0 , 2 . 1 ] ) ; y l i m ( [ 0 . 5 , 1 1 . 5 ] ) ;
yticks ([]) ;
l e g e n d ( [ h1 , h2 , h3 ] , { ' PL ' , ' Abs ' , ' P ( E_g ) ' } , ' L o c a t i o n ' , ' N o r t h ' , '
Orientation ' , ' horizontal ' ) ;
legend ( ' boxoff ' ) ;
box on ;
% C a l c u l a t e FOM f o r no band − f i l l i n g
Voc_vals_noBF = z e r o s ( 1 , l e n g t h ( gam ) ) ;
PCE_vals_noBF = Voc_vals_noBF ;
J s c _ v a l s _ n o B F = Voc_vals_noBF ;
FF_vals_noBF = Voc_vals_noBF ;

f o r i = 1 : l e n g t h ( gam )
EQE_noBF = EQE_vals ( E_range , 0 , EgWhich , gam ( i ) , a l p h a 0 , isPB , Gx_comp ,
x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
Voc_vals_noBF ( i ) = Voc_noBF ( E_range , EQE_noBF , isBB , f l u x ) ;
J s c _ v a l s _ n o B F ( i ) = Jsc_noBF ( E_range , EQE_noBF , isBB , f l u x ) ;
VmaxnoBF = f m i n b n d (@(V) V. ∗ JV_noBF (V, E_range , EQE_noBF , isBB , f l u x ) , 0 ,
EgWhich ) ;
PCE_vals_noBF ( i ) = −VmaxnoBF . ∗ JV_noBF ( VmaxnoBF , E_range , EQE_noBF , isBB ,
f l u x ) . / Pin ;
FF_vals_noBF ( i ) = PCE_vals_noBF ( i ) ∗ P i n . / ( Voc_vals_noBF ( i ) . ∗ J s c _ v a l s _ n o B F ( i
));
end

p (2 ,1) . select () ; % eta
s e m i l o g x ( gam , PCE_vals_noBF ∗ 1 0 0 , gam , i n t e r p 2 ( Eg , gam , PCE_vals1 ∗ 1 0 0 , EgWhich ,
gam ) , ' L i n e W i d t h ' , 1 . 5 )
box on ;
yticks (10:10:30) ;
ylim ( [ 0 , 3 5 ] ) ;
xticklabels ([]) ;
y l a b e l ( ' E f f i c i e n c y (%) ' ) ;

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l e g e n d ( { ' No Band − F i l l i n g ' , ' Band − F i l l i n g ' } , ' L o c a t i o n ' , ' S o u t h W e s t ' , ' f o n t s i z e
' , 10) ;

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p ( 2 , 2 ) . s e l e c t ( ) ; % Voc
s e m i l o g x ( gam , Voc_vals_noBF , gam , i n t e r p 2 ( Eg , gam , V o c _ v a l s 1 , EgWhich , gam ) , '
LineWidth ' , 1 . 5 ) ;
y l a b e l ( ' V_{ oc } (V) ' ) ; x t i c k l a b e l s ( [ ] ) ;
yticks (0.4:0.4:1.6) ;
box on ;
y l a b e l ( ' V_{ oc } (V) ' ) ; x t i c k l a b e l s ( [ ] ) ;
yticks (0.4:0.4:1.6) ;

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p (2 ,3) . s e l e c t () ; % Jsc
s e m i l o g x ( gam , J s c _ v a l s _ n o B F ∗1 e −1 , gam , i n t e r p 2 ( Eg , gam , J s c _ v a l s 1 ∗1 e −1 ,
EgWhich , gam ) , ' L i n e W i d t h ' , 1 . 5 )
x t i c k l a b e l s ( [ ] ) ; y l a b e l ( ' J _ { s c } (mA cm^{ −2}) ' ) ; y t i c k s ( 1 0 : 2 0 : 7 0 ) ;
box on ;

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p ( 2 , 4 ) . s e l e c t ( ) ; % FF
s e m i l o g x ( gam , FF_vals_noBF , gam , i n t e r p 2 ( Eg , gam , F F _ v a l s 1 , EgWhich , gam ) , '
LineWidth ' , 1 . 5 ) ;
ylabel ( ' F i l l Factor ' ) ;
yticks (0.1:0.2:0.9) ;
box on ;

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x t i c k l a b e l s ( { ' 10^0 ' , ' 10^1 ' , ' 10^2 ' , ' 10^3 ' } ) ;
x l a b e l ( ' \ gamma (meV) ' ) ;

d i s p ( ' Now c a l c u l a t i n g e f f e c t s o f d i f f e r e n t s u b g a p IQE . P r e s s any key t o
continue ' ) ;
pause ;
%% F i g S8 , IQE Subgap dependence w i t h gamma , FOM
FigS4 = f i g u r e ;
s e t ( FigS4 , ' P o s i t i o n ' , [ 6 0 , 3 0 , 4 5 0 , 6 0 0 ] ) ;
% materials parameters
EgWhich = 1 . 5 ; % [ eV ] , b a n d g a p o f m a t e r i a l
L = 1e −3; % t h i c k n e s s of m a t e r i a l , u n i t s of 1 / a l p h a 0
a l p h a 0 = 1 e3 ; % a b s o r p t i o n o f d i r e c t gap m a t e r i a l , a l p h a 0 L i s what m a t t e r s
i s P B = t r u e ; % i n c l u d e p a u l i − b l o c k i n g / band f i l l i n g e f f e c t s
% panel parameters
p = panel () ;
p . pack ( ' h ' , 1) ;
p ( 1 ) . pack ( ' v ' , { 0 . 1 , 0 . 2 2 5 , 0 . 2 2 5 , 0 . 2 2 5 , 0 . 2 2 5 } ) ;
p . de . m a r g i n = 0 ;
% Subgap p a r a m e t e r s
gam = 1 0 . ^ [ − 3 : g a m _ s p a c i n g : 0 ] ; % Bandege s h a r p n e s s p a r a m e t e r [ eV ] ,
IQE_SG = 1 0 . ^ [ − 6 : 2 : 0 ] ; % i n t e r n a l quantum e f f i c i e n c y , s u b gap
o u t = SQ_FOM( isBB , f l u x , Gx_comp , x_comp , gam , EgWhich , a l p h a 0 ∗L , isPB , IQE_SG
, IQE_AG , isIQE_SG_Urbach ) ; % d o e s a l l t h e FOM c a l c u l a t i o n s

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V o c _ v a l s 3 = r e s h a p e ( o u t . Voc , l e n g t h ( IQE_SG ) , l e n g t h ( gam ) ) ' ; % o u t p u t , d a t a ,
shape i t f o r data p l o t s
J s c _ v a l s 3 = r e s h a p e ( o u t . J s c , l e n g t h ( IQE_SG ) , l e n g t h ( gam ) ) ' ;
PCE_vals3 = r e s h a p e ( o u t . PCE , l e n g t h ( IQE_SG ) , l e n g t h ( gam ) ) ' ;
F F _ v a l s 3 = P i n ∗ PCE_vals3 . / ( J s c _ v a l s 3 . ∗ V o c _ v a l s 3 ) ;
% c a l c u l a t e u r b a c h IQE , n e e d t o s e t IQE_SG t o s i n g l e number t o h a v e t h e
% right size output
o u t = SQ_FOM( isBB , f l u x , Gx_comp , x_comp , gam , EgWhich , a l p h a 0 ∗L , isPB , 1 ,
IQE_AG , 1 ) ; % d o e s a l l t h e FOM c a l c u l a t i o n s
V o c _ v a l s 4 = r e s h a p e ( o u t . Voc , [ 1 , l e n g t h ( gam ) ] ) ;
J s c _ v a l s 4 = r e s h a p e ( o u t . J s c , [ 1 , l e n g t h ( gam ) ] ) ;
PCE_vals4 = r e s h a p e ( o u t . PCE , [ 1 , l e n g t h ( gam ) ] ) ;
F F _ v a l s 4 = P i n . ∗ PCE_vals4 . / ( J s c _ v a l s 4 . ∗ V o c _ v a l s 4 ) ;
p (1 ,2) . select () ; % eta
leg_str = [];
f o r i = 1 : l e n g t h ( IQE_SG )
s e m i l o g x ( gam , i n t e r p 2 ( IQE_SG , gam , PCE_vals3 ∗ 1 0 0 , IQE_SG ( i ) , gam ) , '
LineWidth ' , 1 . 5 ) ;
h o l d on ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( [ ' 10^{ ' , n u m 2 s t r ( l o g 1 0 ( IQE_SG ( i ) ) ) , ' } ' ] ) ] ;
end
s e m i l o g x ( gam , PCE_vals4 ∗ 1 0 0 , ' k−− ' ) ; l e g _ s t r = [ l e g _ s t r , ' Urbach ' ] ;
hold off ;
ylim ( [ 0 3 5 ]) ;
xticklabels ([]) ;
yticks (10:10:30) ;
hleg = legend ( leg_str , ' Location ' , ' NorthOutside ' , ' Orientation ' , ' Horizontal '
, ' FontSize ' , 8) ;
hleg . Position (2) = hleg . Position (2) + 0.075;
t i t l e ( h l e g , ' IQE_ {SG} ' , ' F o n t W e i g h t ' , ' Normal ' ) ;
y l a b e l ( ' E f f i c i e n c y (%) ' ) ;
box on ;
p ( 1 , 3 ) . s e l e c t ( ) ; % Voc
f o r i = 1 : l e n g t h ( IQE_SG )
s e m i l o g x ( gam , i n t e r p 2 ( IQE_SG , gam , V o c _ v a l s 3 , IQE_SG ( i ) , gam ) , ' L i n e W i d t h '
, 1.5) ;
h o l d on ;
end
s e m i l o g x ( gam , V o c _ v a l s 4 , ' k−− ' ) ; y t i c k s ( 0 . 4 : 0 . 4 : 1 . 6 ) ;
hold off ;
xticklabels ([]) ;
y l a b e l ( ' V_{ oc } (V) ' ) ;
box on ;
p (1 ,4) . s e l e c t () ; % Jsc
f o r i = 1 : l e n g t h ( IQE_SG )
s e m i l o g x ( gam , i n t e r p 2 ( IQE_SG , gam , J s c _ v a l s 3 ∗1 e −1 , IQE_SG ( i ) , gam ) , '
LineWidth ' , 1 . 5 ) ;
h o l d on ;
end
s e m i l o g x ( gam , J s c _ v a l s 4 ∗1 e −1 , ' k−− ' ) ;
hold off ;

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xticklabels ( [ ] ) ; yticks (10:20:70) ;
y l a b e l ( ' J _ { s c } (mA cm^{ −2}) ' ) ;
box on ;
p (1 ,5) . select () ; % F i l l Factor
f o r i = 1 : l e n g t h ( IQE_SG )
s e m i l o g x ( gam , i n t e r p 2 ( IQE_SG , gam , F F _ v a l s 3 , IQE_SG ( i ) , gam ) , ' L i n e W i d t h ' ,
1.5) ;
h o l d on ;
end
s e m i l o g x ( gam , F F _ v a l s 4 , ' k−− ' ) ;
yticks (0.1:0.2:0.9) ;
box on ; h o l d o f f ;
x t i c k l a b e l s ( { ' 10^0 ' , ' 10^1 ' , ' 10^2 ' , ' 10^3 ' } ) ;
x l a b e l ( ' \ gamma (meV) ' ) ; y l a b e l ( ' F i l l F a c t o r ' ) ;

end
d i s p ( ' Now c a l c u l a t i n g e f f e c t s o f a G a u s s i a n T a i l . P r e s s any key t o c o n t i n u e ' ) ;
pause ;
i f plotGaussFig
%% F i g S7 , t h e t a = 2 ( Gaussian T a i l ) ;
% gam , Eg , FOM c o n t o u r s
FigS7 = f i g u r e ; %
s e t ( FigS7 , ' P o s i t i o n ' , [ 6 0 , 4 0 , 7 5 0 , 6 0 0 ] ) ;
% panel parameters
p = panel () ;
p . pack ( ' h ' , [ 0 . 4 7 , 0 . 4 7 ] ) ;
p ( 1 ) . pack ( ' v ' , 1) ;
p ( 2 ) . pack ( ' v ' , 2) ;
p . de . m a r g i n = 0 ;
p (1) . marginright = 20;
p (2) . marginleft = 20;
p ( 2 , 1 ) . marginbottom = 20;

%% Gaussian S i m u l a t i o n Parameters
% materials parameters
Eg = 0 . 5 : E g _ s p a c i n g : 2 . 5 ; % [ eV ] , b a n d g a p o f m a t e r i a l
L = 1 e − 3 ; % t h i c k n e s s o f m a t e r i a l [ cm ]
a l p h a 0 = 1 e3 ; % [ 1 / cm ] , a b s o r p t i o n o f d i r e c t gap m a t e r i a l , a l p h a 0 L i s what
matters
i s P B = t r u e ; % i n c l u d e p a u l i − b l o c k i n g / band f i l l i n g e f f e c t s
isIQE_SG_Urbach = f a l s e ;
% Subgap p a r a m e t e r s
gam = 1 0 . ^ [ − 3 : g a m _ s p a c i n g : l o g 1 0 ( 0 . 5 5 ) ] ; % Bandege s h a r p n e s s p a r a m e t e r [ eV ] ,
IQE_SG = 1 ; % i n t e r n a l quantum e f f i c i e n c y , s u b gap
% c o d e i s f a s t e r i f we f i g u r e o u t t h e r i g h t t h e t a f i r s t , c o m p r e s s Gx ,
% and t h e n u s e a s t a b l e l o o k −up d a t a s e t . O t h e r w i s e , p a s s i n g a r o u n d t o o

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% much d a t a .
t h e t a = 2; % " o r d e r " of t h e e x p o n e n t i a l , 1 i s urbach , 2 i s Gaussian , e t c .
Gx_comp = a . G v a l s ( a . t h e t a l a b e l s == t h e t a , : ) ;
Gx_comp = Gx_comp ( 1 : c o m p R a t i o : end ) ;
x_comp = a . x l a b e l s ( 1 : c o m p R a t i o : end ) ;
% c a l c u l a t e SQ l i m i t
PCE_SQ = z e r o s ( 1 , l e n g t h ( Eg ) ) ;
f o r i = 1 : l e n g t h ( Eg )
EQE_SQ = E_range >=Eg ( i ) ;
Vmax_SQ = f m i n b n d (@(V) V. ∗ JV_noBF (V, E_range , EQE_SQ , isBB , f l u x ) , 0 , Eg ( i
));
PCE_SQ ( i ) = −Vmax_SQ . ∗ JV_noBF ( Vmax_SQ , E_range , EQE_SQ , isBB , f l u x ) . / P i n ;
end
o u t = SQ_FOM( isBB , f l u x , Gx_comp , x_comp , gam , Eg , a l p h a 0 ∗L , isPB , IQE_SG ,
IQE_AG , isIQE_SG_Urbach ) ; % d o e s a l l t h e FOM c a l c u l a t i o n s
V o c _ v a l s _ t h e t a 2 = r e s h a p e ( o u t . Voc , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ; % o u t p u t , d a t a ,
shape i t f o r data p l o t s
J s c _ v a l s _ t h e t a 2 = r e s h a p e ( o u t . J s c , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ;
P C E _ v a l s _ t h e t a 2 = r e s h a p e ( o u t . PCE , l e n g t h ( Eg ) , l e n g t h ( gam ) ) ' ;

% S p e c t r a l P l o t s , PL , EQE , P ( Eg )
p (1 ,1) . select () ;
gamWhich = [ 0 . 5 : 1 : 6 . 5 ] ∗ k∗ T c e l l ; % which gamma f o r EQE , PCE
EgWhich = 1 . 5 ; % which Eg f o r EQE
offset = 1:1.35:15;
P L c o l o r P l o t s = g e t ( gca , ' c o l o r o r d e r ' ) ;
h o l d on ;
% assume t h a t t h e s y s t e m , u n d e r i l l u m i n a t i o n & no c o n t a c t s , r e a c h e s i t s
% Voc . Thus , PL i s t h e PL a t Voc .
h1 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h2 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
h3 = p l o t ( E_range , −1∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) , ' k : ' , ' L i n e W i d t h ' , 1 . 5 ) ;
f o r i = 1 : l e n g t h ( gamWhich )
Voc_gam = i n t e r p 2 ( Eg , gam , V o c _ v a l s _ t h e t a 2 , EgWhich , gamWhich ( i ) ) ;
P L _ s p e c t = J _ s p e c t ( E_range , Voc_gam , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
P L p l o t = p l o t ( E_range , P L _ s p e c t . / max ( P L _ s p e c t ) + o f f s e t ( i ) , ' −− ' , ' C o l o r ' ,
P L c o l o r P l o t s ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
EQE_spect = EQE_vals ( E_range , 0 , EgWhich , gamWhich ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
p l o t ( E_range , EQE_spect . / max ( EQE_spect ) + o f f s e t ( i ) , ' − ' , ' C o l o r ' , P L p l o t
. Color , ' LineWidth ' , 1 . 5 )
PEg = EQE_spect ( 3 : end ) − EQE_spect ( 1 : end − 2 ) ;
p l o t ( E _ r a n g e ( 2 : end − 1 ) , PEg . / max ( PEg ) + o f f s e t ( i ) , ' : ' , ' C o l o r ' , P L p l o t .
Color , ' LineWidth ' , 1 . 5 ) ;
i f gamWhich ( i ) == k∗ T c e l l
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' ,
PLcolorPlots ( i , : ) ) ;

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else
t e x t ( 1 . 6 5 , o f f s e t ( i ) + 0 . 5 , [ ' \ gamma = ' , n u m 2 s t r ( gamWhich ( i ) / ( k∗ T c e l l ) )
, ' kT ' ] , ' F o n t S i z e ' , 1 0 , ' C o l o r ' , P L c o l o r P l o t s ( i , : ) ) ;

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end
end
hold off ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; x l i m ( [ 0 . 3 , 2 . 1 ] ) ; y l i m ( [ 0 . 5 , 1 1 . 5 ] ) ;
yticks ([]) ;
l e g e n d ( [ h1 , h2 , h3 ] , { ' PL ' , ' Abs ' , ' P ( E_g ) ' } , ' L o c a t i o n ' , ' N o r t h ' , '
Orientation ' , ' horizontal ' ) ;
legend ( ' boxoff ' ) ;
box on ;

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p (2 ,1) . select () ;
J _ s u b g a p _ R a t i o = z e r o s ( s i z e ( gam ) ) ; % f r a c t i o n o f r e c o m b i n a t i o n t h a t i s below
t h e bandgap
P L _ p e a k s G a u s s i a n = z e r o s ( s i z e ( gam ) ) ;
EgPV = z e r o s ( s i z e ( gam ) ) ;

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f o r i = 1 : l e n g t h ( gam )
Voc_gam = i n t e r p 2 ( Eg , gam , V o c _ v a l s _ t h e t a 2 , EgWhich , gam ( i ) ) ;
P L _ s p e c t = J _ s p e c t ( E_range , Voc_gam , EgWhich , gam ( i ) , a l p h a 0 , isPB ,
Gx_comp , x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;

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EQE_spect = EQE_vals ( E_range , 0 , EgWhich , gam ( i ) , a l p h a 0 , isPB , Gx_comp ,
x_comp , L , IQE_SG , IQE_AG , isIQE_SG_Urbach ) ;
PEg = EQE_spect ( 3 : end ) − EQE_spect ( 1 : end − 2 ) ;
[ ~ , p k _ i d x ] = max ( PEg ) ;
EgPV ( i ) = E _ r a n g e ( p k _ i d x ) ;
J _ s u b g a p _ R a t i o ( i ) = t r a p z ( E _ r a n g e ( E_range i ) ) ) . / t r a p z ( E_range , P L _ s p e c t ) ;
[ ~ , p k _ i n d x ] = max ( P L _ s p e c t ) ; % f i n d E _ r a n g e i n d x f o r PL p e a k
P L _ p e a k s G a u s s i a n ( i ) = f i n d E g P L ( E_range , P L _ s p e c t , r e l V a l , numTol ) ; %
E_range ( pk_indx ) ;

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end
yyaxis l e f t ;
s e m i l o g x ( gam∗1 e3 , J _ s u b g a p _ R a t i o , ' L i n e W i d t h ' , 1 . 5 ) ; h o l d on ;
x l a b e l ( ' \ gamma (meV) ' ) ; y l a b e l ( ' F r a c t i o n o f PL below E_{g , Abs } ' ) ;
box on ;
ylim ( [ 0 , 1 ] ) ;

yyaxis r i g h t ;
s e m i l o g x ( gam∗1 e3 , EgPV − P L _ p e a k s G a u s s i a n , ' L i n e W i d t h ' , 1 . 5 ) ;
y l a b e l ( ' \ D e l t a E_g ( eV ) ' ) ;
y l i m ( [ min ( EgPV − P L _ p e a k s G a u s s i a n ) , max ( EgPV − P L _ p e a k s G a u s s i a n ) ] ) ;
x l i m ( [ min ( gam ) , max ( gam ) ] ∗ 1 e3 ) ;

% P l o t new SQ l i m i t f o r G a u s s i a n T a i l s
p (2 ,2) . select () ;
l e g _ s t r = [ ' S−Q ' ] ;
h o l d on ;
p l o t ( Eg , PCE_SQ ∗ 1 0 0 , ' k−− ' )

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f o r i = 1 : l e n g t h ( gamWhich )
p l o t ( Eg , i n t e r p 2 ( Eg , gam , P C E _ v a l s _ t h e t a 2 ∗ 1 0 0 , Eg , gamWhich ( i ) ) , ' C o l o r ' ,
P L c o l o r P l o t s ( i , : ) , ' LineWidth ' , 1 . 5 ) ;
l e g _ s t r = [ l e g _ s t r , s t r i n g ( n u m 2 s t r ( gamWhich ( i ) . / k . / T c e l l ) ) ] ;
end
hold off ;
hleg = legend ( l e g _ s t r , ' Location ' , ' SouthEast ' ) ;
t i t l e ( h l e g , ' \ gamma ( kT ) ' , ' F o n t W e i g h t ' , ' Normal ' ) ;
x l a b e l ( ' Band gap ( eV ) ' ) ; y l a b e l ( ' E f f i c i e n c y (%) ' ) ;
box on ;
figure ;
VocPLPeak = z e r o s ( s i z e ( P L _ p e a k s G a u s s i a n ) ) ;
f o r i = 1 : l e n g t h ( gam )
VocPLPeak ( i ) = Voc_noBF ( E_range , E_range > P L _ p e a k s G a u s s i a n ( i ) , isBB , f l u x ) ;
end
s e m i l o g x ( gam∗1 e3 , i n t e r p 2 ( Eg , gam , V o c _ v a l s _ t h e t a 2 , EgWhich , gam ) , ...
gam∗1 e3 , P L _ p e a k s G a u s s i a n , ...
gam∗1 e3 , VocPLPeak , ' L i n e W i d t h ' , 1 . 5 ) ;
l e g e n d ( ' Voc_ {BF} ' , ' PL p e a k ' , ' Voc_ {SQ , PL} ' ) ;
x l a b e l ( ' \ gamma (meV) ' ) ; y l a b e l ( ' V_{ oc } o r E_{ p e a k } ' ) ;
end

% Save a l l v a r i a b l e s d u r i n g t h i s r u n
f i l e n a m e = ' S u b g a p D a t a _ v 2 0 . mat ' ;
save ( filename ) ;

%% sub − f u n c t i o n d e f i n i t i o n s
% main f u n c t i o n s f o r c a l c u l a t i n g PV f i g u r e o f m e r i t s
f u n c t i o n o u t = SQ_FOM( isBB , f l u x , Gx , x , gam , Eg , a l p h a 0 L , isPB , IQE_SG ,
IQE_AG , isIQE_SG_Urbach )
% i n i t i a l i z e data for storage
Voc = z e r o s ( l e n g t h ( a l p h a 0 L ) , l e n g t h ( IQE_SG ) , l e n g t h ( IQE_AG ) , l e n g t h ( Eg ) ,
l e n g t h ( gam ) ) ;
J s c = Voc ;
PCE = Voc ;
% run loop over a l l parameters of i n t e r e s t
f o r i1 = 1: l e n g t h ( alpha0L )
f o r i 2 = 1 : l e n g t h ( IQE_SG )
f o r i 3 = 1 : l e n g t h ( IQE_AG )
f o r i 4 = 1 : l e n g t h ( Eg )
V o c _ g u e s s = Eg ( i 4 ) ;
f o r i 5 = 1 : l e n g t h ( gam )
EQEV = @(V) EQE_vals ( E_range , V, Eg ( i 4 ) , gam ( i 5 ) , a l p h a 0 L (
i 1 ) , isPB , Gx , x , 1 , IQE_SG ( i 2 ) , IQE_AG ( i 3 ) ,
isIQE_SG_Urbach ) ;
Voc ( i 1 , i 2 , i 3 , i 4 , i 5 ) = f z e r o (@(V) J V _ F u l l (V, E_range , EQEV(
V) , isBB , f l u x ) , V o c _ g u e s s ) ;

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V o c _ g u e s s = Voc ( i 1 , i 2 , i 3 , i 4 , i 5 ) ; % u s e l a s t Voc p o i n t f o r
next optimization

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J s c ( i 1 , i 2 , i 3 , i 4 , i 5 ) = − J V _ F u l l ( 0 , E_range , EQEV ( 0 ) , isBB ,
flux ) ;
Vmax = f m i n b n d (@(V) V. ∗ J V _ F u l l (V, E_range , EQEV(V) , isBB ,
f l u x ) , 0 , Voc_guess ) ;

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PCE ( i 1 , i 2 , i 3 , i 4 , i 5 ) = −Vmax . ∗ J V _ F u l l ( Vmax , E_range , EQEV(
Vmax ) , isBB , f l u x ) . / P i n ;
end
end
end
end
end
% s a v e d a t a a s o u t p u t s t r u c t , r e s i z e m a t r i x i n main f u n c t i o n
o u t . Voc = Voc ;
out . Jsc = Jsc ;
o u t . PCE = PCE ;
end
f u n c t i o n o u t = G( x , t h e t a ) % c a l c u l a t i o n s " u n i t l e s s " a b s o r p t i o n c o e f f i c i e n t
f = @(xp ) exp ( − a b s ( xp ) . ^ t h e t a ) . ∗ s q r t ( x−xp ) ; % complex numbers t a k e up
precision !

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i f x < 50 % somewhat a r b i t r a r y d i v i s i o n , b u t makes t h e i n t e g r a l s c o n v e r g e
p r o p e r l y and r u n i n t i m e l y manner
o u t = 1 . / ( 2 ∗ gamma ( 1 + 1 / t h e t a ) ) ∗ i n t e g r a l ( f , − i n f , x , ' AbsTol ' , 1 e −12 , '
RelTol ' , 1e −12) ;
else
o u t = r e a l ( 1 . / ( 2 ∗ gamma ( 1 + 1 / t h e t a ) ) ∗ i n t e g r a l ( f , − i n f , i n f ) ) ;
end

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end
f u n c t i o n o u t = a l p h a ( E , V, Eg , gam , a l p h a 0 , isPB , Gx , x l a b e l s )
xq = ( E−Eg ) . / gam ; % c a l c u l a t e d i m e n s i o n l e s s x
GE = i n t e r p 1 ( x l a b e l s , Gx , xq ) ; % t a b l e l o o k up
o u t = a l p h a 0 ∗ s q r t ( gam / ( k∗ T c e l l ) ) ∗GE ;
i f isPB % i n c l u d e p a u l i − b l o c k i n g e f f e c t
o c c F a c t o r = t a n h ( ( E−V) / ( 4 ∗ k∗ T c e l l ) ) ;
out = out .∗ occFactor ;
end
end
% s p e c t r a l current , i . e . J_rad (E)
f u n c t i o n o u t = J _ s p e c t ( E_range , V, Eg , gam , a l p h a 0 , isPB , Gx , x l a b e l s , L ,
IQE_SG , IQE_AG , isIQE_SG_Urbach )
temp = EQE_vals ( E_range , V, Eg , gam , a l p h a 0 , isPB , Gx , x l a b e l s , L , IQE_SG ,
IQE_AG , isIQE_SG_Urbach ) . ∗ bb ( E_range , F c e l l , T c e l l , V) ;
s i n g u l a r I n d e x = f i n d ( E _ r a n g e == V) ;
temp ( s i n g u l a r I n d e x ) = ( temp ( s i n g u l a r I n d e x + 1 ) + temp ( s i n g u l a r I n d e x − 1 ) ) / 2 ; %
l i n e a r i n t e r p o l a t i o n t o g e t r i d o f s i n g u l a r i t y a t E = qV

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o u t = temp ;
end
% EQE
f u n c t i o n o u t = EQE_vals ( E_range , V, Eg , gam , a l p h a 0 , isPB , Gx , x l a b e l s , L ,
IQE_SG , IQE_AG , isIQE_SG_Urbach )
% s i m p l e p a r a m e t r i z a t i o n w i t h an IQE t h a t f a l l s o f f l i k e t h e Urbach
% parameter
i f isIQE_SG_Urbach
IQE_SG = z e r o s ( 1 , l e n g t h ( E _ r a n g e ) ) ; % do t h i s i n two s t e p s s o t h a t we
don ' t g e t w e i r d NaN e r r o r s from 0∗ exp ( l a r g e number )
IQE_SG ( E_range <=Eg ) = IQE_AG . ∗ exp ( ( E _ r a n g e ( E_range <=Eg ) −Eg ) / gam ) ;
end
o u t = d u b p a s s ( a l p h a ( E_range , V, Eg , gam , a l p h a 0 , isPB , Gx , x l a b e l s ) , L ) . ∗ ( (
E_range <=Eg ) . ∗ IQE_SG + ( E_range >Eg ) . ∗ IQE_AG ) ;
end
f u n c t i o n Eg = f i n d E g P L ( E_range , s p e c , r e l V a l , numTol )
cumRat1 = f l i p l r ( c u m t r a p z ( E_range , f l i p l r ( s p e c ) ) ) . / t r a p z ( E_range , s p e c ) ;
Eg = max ( E _ r a n g e ( a b s ( cumRat1 − r e l V a l ) end
% F u l l JV , i n c l u d i n g band − f i l l i n g e f f e c t s
f u n c t i o n o u t = J V _ F u l l (V, E_range , EQE , isBB , f l u x )
% e i t h e r assume b l a c k b o d y i l l u m i n a t i o n , o r some s p e c t r u m g i v e n by
% f l u x . Must be same s i z e a s E _ r a n g e and EQE
i f isBB
J s c = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ ( bb ( E_range , Fsun , Tsun , 0 ) +bb (
E_range , F c e l l , T c e l l , 0 ) ) ) ) ;
else
J s c = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ ( f l u x +bb ( E_range , F c e l l , T c e l l , 0 ) ) ) )
end
% c o r r e c t f o r s i n g u l a r i t y by i n t e r p o l a t i o n
J s p e c = EQE . ∗ bb ( E_range , F c e l l , T c e l l , V) ;
s i n g u l a r I n d e x = f i n d ( E _ r a n g e == V) ;
J s p e c ( s i n g u l a r I n d e x ) = ( J s p e c ( s i n g u l a r I n d e x +1) + J s p e c ( s i n g u l a r I n d e x −1) )
/ 2 ; % l i n e a r i n t e r p o l a t i o n to get r i d of apparent s i n g u l a r i t y a t E =
qV
J r a d = −e ∗ a b s ( t r a p z ( E_range , J s p e c ) ) ;
out = Jsc − Jrad ;
end
% Bose − E i n s t e i n b l a c k −body s p e c t r u m w i t h some c h e m i c a l p o t e n t i a l , no a p p r o x
f u n c t i o n s p e c t r u m = bb ( E_range , F , T , V)
s p e c t r u m = 2∗F . / ( h ^3∗ c ^ 2 ) . ∗ ...
E _ r a n g e . ^ 2 . / ( exp ( ( E_range −V) / ( k∗T ) ) − 1 ) ; % b l a c k b o d y p h o t o n
f l u x [ P h o t o n s / ( s e c ∗m^2∗ w a v e l e n g t h ) ]
end

% Voc o f c e l l a s s u m i n g no band − f i l l i n g , u s i n g " d i o d e " a p p r o x i m a t i o n
f u n c t i o n o u t = Voc_noBF ( E_range , EQE , isBB , f l u x )

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J r a d = e ∗ a b s ( t r a p z ( E_range , EQE . ∗ bb ( E_range , F c e l l , T c e l l , 0 ) ) ) ;
J s c = Jsc_noBF ( E_range , EQE , isBB , f l u x ) ;
o u t = k∗ T c e l l ∗ l o g ( 1 + J s c / J r a d ) ;

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end
f u n c t i o n o u t = Jsc_noBF ( E_range , EQE , isBB , f l u x )
i f isBB
o u t = e ∗ a b s ( t r a p z ( E_range , EQE . ∗ bb ( E_range , Fsun , Tsun , 0 ) ) ) ;
else
o u t = e ∗ a b s ( t r a p z ( E_range , EQE . ∗ f l u x ) ) ;
end
end
f u n c t i o n o u t = JV_noBF (V, E_range , EQE , isBB , f l u x )
J s c = − Jsc_noBF ( E_range , EQE , isBB , f l u x ) ;
J r a d = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ bb ( E_range , F c e l l , T c e l l , 0 ) ) ) ∗ ( exp (V . / ( k
∗ T c e l l ) ) −1) ;
out = Jsc − Jrad ;
end

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% Beer − L a m b e r t a b s o r p t i o n law , d o e s n o t i n c l u d e i n t e r f e r e n c e e f f e c t s o r
% b a c k w a r d p r o p a g a t i n g waves
f u n c t i o n out = dubpass ( alpha , L)
o u t = −expm1 ( −2∗ a l p h a ∗L ) ; %a b s o r b a n c e , u n i t l e s s . expm1 f o r h i g h p r e c i s i o n
of small alpha
end

end

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Listing D.2: Modified Detailed Balance Code for Excitonic Multijunctions
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f u n c t i o n ExcitonicTandem_DetailedBalance_Sweep_v2
% Requires :
% MATLAB 2020 a , G l o b a l O p t i m i z a t i o n Toolbox , P a r a l l e l Computing T o o l b o x
% Description :
% T h i s f u n c t i o n c a l c u l a t e s t h e d e t a i l e d b a l a n c e e f f i c i e n c y o f an o p t i m i z e d
% m u l t i j u n c t i o n e x c i t o n i c s o l a r c e l l from 1 t o numExc number o f a b s o r b e r s
% and p l o t s t h e i r o p t i m i z e d EQE s p e c t r a and JV c u r v e s .
% A s s u m p t i o n s and D e t a i l s :
% 1 . Abs = EQE and i s d e s c r i b e d by a G a u s s i a n w i t h an a b s o r p t i o n maximum ,
% p e a k p o s i t i o n , and l i n e w i d t h
% 2 . D e t a i l e d b a l a n c e r e l a t i o n s h o l d s , ERE = 1 , and no o p t i c a l c o u p l i n g
% e x i s t s between s u b c e l l s
% 3 . I n c i d e n t s p e c t r u m i s t h e s o l a r AM1. 5G
% 4 . S p e c t r a l windowing from t o p a b s o r b e r s m o d e l l e d by s u b s t r a c t i n g t h e
% EQE r e s p o n s e o f t h e t o p a b s o r b e r s from t h e i n c i d e n t s p e c t r u m , which i s
% e f f e c t i v e l y a ray o p t i c s approximation
% 5 . E n f o r c e s c u r r e n t m a t c h i n g a t t h e maximum power p o i n t
% ( i . e . J ( V_max ) _ i = J ( V_max ) _ j ) i n t h e o p t i m i z a t i o n p r o c e s s
% 6 . O p t i m i z a t i o n p e r f o r m e d by a G l o b a l O p t i m i z a t i o n R o u t i n e ( s u r r o g a t e o p t )
% i n MATLAB' s G l o b a l O p t i m i z a t i o n T o o l b o x
% 7 . A l l d a t a i s s a v e d i n a new d i r e c t o r y l a b e l l e d by t h e d a t e and t i m e o f
% when t h i s s c r i p t i s r a n .
% W r i t t e n by J o e s o n Wong
% L a s t U p d a t e d on A p r i l 1 2 , 2020
% c l e a r f i l e s , f i g u r e s , and make d i r e c t o r y
clear all ; close all ; clc ;
time = clock ;
time = f i x ( time ) ;
dir_name = [ ] ;
f o r i _ t i m e = 1 : l e n g t h ( t i m e ) −1 % no n e e d t o add s e c o n d s m a r k e r on t h e r e . . .
dir_name = [ dir_name , num2str ( time ( i _ t i m e ) ) , ' _ ' ] ;
end
d i r _ n a m e ( end ) = [ ] ;
mkdir ( dir_name ) ;
%% D e f i n i t i o n s
% define physical constants
e = 1.6021766208 e −19; % [C] , f u n d a m e n t a l c h a r g e
Fsun = 2 . 1 6 e −5∗ p i ; % h e m i s p h e r i c a l a n g u l a r r a n g e , s u n ( 1 / 4 6 0 0 0 f a c t o r f o r
concentration )
Fcell = pi ; % emission geometrical factor , c e l l
h = 6 . 6 2 6 0 7 0 0 4 e − 3 4 / e ; % [ eV− s ] , P l a n c k ' s c o n s t a n t
c = 2 9 9 7 9 2 4 5 8 ; % [m/ s ] , s p e e d o f l i g h t
k = 1 . 3 8 0 6 4 8 5 2 e − 2 3 / e ; % [ eV / K] , Boltzmann ' s c o n s t a n t
Tsun = 5 7 6 0 ; % [ K e l v i n ] , T e m p e r a t u r e o f Sun , f o r f u l l b l a c k −body e x p r e s s i o n
Tcell = 300; % [ Kelvin ] , Temperature of Cell
E 2 l a m _ f a c t o r = 1 . 2 3 9 8 4 1 9 3 ∗ 1 e − 6 ; % c o n v e r s i o n f a c t o r b e t w e e n e n e r g y [ eV ] t o
w a v e l e n g t h s [m] , E [ eV ] = E 2 l a m _ f a c t o r . / lambda [m]

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% Load S o l a r Spectrum , u s e f o r c a l c u l a t i n g SQ l i m i t
ASTMG173_filename = ' ASTMG173 . x l s x ' ;
s h e e t = ' SMARTS2 ' ;
x l R a n g e = [ 'A ' , ' 3 ' , ' : ' , 'D ' , ' 2004 ' ] ;
o u t d o o r _ d a t a = x l s r e a d ( ASTMG173_filename , s h e e t , xlRange ,
l a m b d a _ o u t d o o r = o u t d o o r _ d a t a ( : , 1 ) ∗1 e −9 ; % [m]
AM0 = o u t d o o r _ d a t a ( : , 2 ) / 1 e − 9 ; % [W/m^ 3 ]
AM15G = o u t d o o r _ d a t a ( : , 3 ) / 1 e − 9 ; % [W/m^ 3 ]
AM15D = o u t d o o r _ d a t a ( : , 4 ) / 1 e − 9 ; % [W/m^ 3 ]

' basic ' ) ;

% Define energy / wavelength ranges of i n t e r e s t
E _ r a n g e = 0 . 0 0 1 : 0 . 0 0 1 : 5 ; % [ eV ] r a n g e o f e n e r g i e s o f i n t e r e s t . Avoid s i n g u l a r
point (E = 0)
l a m b d a _ r a n g e = E 2 l a m _ f a c t o r . / E _ r a n g e ; % [m] , r a n g e o f w a v e l e n g t h s
% d e f i n e new r a n g e s b a s e d on s p e c t r a l d a t a & t a r g e t s p e c t r a l r a n g e
newAM0 = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM0, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;
newAM15G = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM15G, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;
newAM15D = i n t e r p 1 ( l a m b d a _ o u t d o o r , AM15D, l a m b d a _ r a n g e , ' l i n e a r ' , 0 ) ;

%% S i m u l a t i o n Parameters
% Light source information
f l u x = newAM15G . / E _ r a n g e . ^ 3 / e ∗h∗ c ; % f l u x o f i n t e r e s t , i f n o t b l a c k b o d y ( e . g .
AM 15G) . E x t r a c o n v e r s i o n f a c t o r t o u s e e n e r g y i n t e g r a l
isBB = f a l s e ;
% I n i t i a l i z e E x c i t o n O p t i m i z a t i o n Problem
numExc = 5 ;
IQE_exc = 1 ; % i n t e r n a l quantum e f f i c i e n c y ( d i s s o c i a t i o n ∗ c o l l e c t i o n e f i c i e n c y
of e x c i t o n s )
maxPCE_exc_uncon = z e r o s ( numExc , 1 ) ;
maxPCE_exc_con = maxPCE_exc_uncon ;
x _ e x c _ u n c o n = z e r o s ( numExc , 3∗numExc ) ;
x_exc_con = x_exc_uncon ;
% O p t i o n s f o r p l o t t i n g and o p t i m i z a t i o n
plotSpec = true ;
p l o t I V = f a l s e ; %p l o t S p e c n e e d s t o be t r u e t o r u n p l o t I V
doSeriesCon = f a l s e ; % perform o p t i m i z a t i o n with s e r i e s c o n s t r a i n t
C u r T o l = 5 e1 ; % r o u g h l y t h e t o l e r a n c e we a l l o w i n c u r r e n t m a t c h i n g i s 1 / C u r T o l
i n p e r c e n t a g e ( i . e . 1 / 2 0 = 5%)
% C a l c u l a t e I n p u t Power
i f isBB
P i n = a b s ( t r a p z ( E_range , e ∗ E _ r a n g e . ∗ bb ( E_range , Fsun , Tsun , 0 ) ) ) ; % i n p u t
power , W/m^2
else
P i n = a b s ( t r a p z ( E_range , e ∗ E _ r a n g e . ∗ f l u x ) ) ;
end
%% C a l c u l a t e and P l o t Optimized E x c i t o n Parameters
f o r i d x = 1 : numExc

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% i n i t a l i z e s t a r t i n g point for optimzation
Apeak0 = 0 . 5 ∗ o n e s ( 1 , i d x ) ; % p e a k v a l u e o f a b s o r b a n c e
E_exc0 = l i n s p a c e ( 0 . 7 , 3 . 2 , i d x ) ; % e x c i t o n e x c i t a t i o n e n e r g y
Exc_LW0 = 80 e −3∗ o n e s ( 1 , i d x ) ; % L i n e w i d t h o f e x c i t o n
x0 = [ Apeak0 , E_exc0 , Exc_LW0 ] ;
i f i d x == 1
oldPCE_uncon = 0 ;
oldPCE_con = 0 ;
else
oldPCE_uncon = maxPCE_exc_uncon ( i d x − 1 ) ;
oldPCE_con = maxPCE_exc_con ( i d x − 1 ) ;
end
% Perform Optimization
[ maxPCE , x ] = o p t i m E x c i t o n ( f l u x , i d x , 0 , x0 , oldPCE_uncon ) ;
maxPCE_exc_uncon ( i d x ) = −maxPCE ;
x_exc_uncon ( idx , 1:3∗ idx ) = x ;
i f doSeriesCon
% u s e r e s u l t s from u n c o n s t r a i n e d a s i n i t i a l s t a r t i n g p o i n t f o r
% c o n s t r a i n e d . A l s o f o r c e t h e c o n s t r a i n e d PCEs t o i n c r e a s e w i t h
% number o f e x c i t o n i c a b s o r b e r s .
[ maxPCE , x ] = o p t i m E x c i t o n ( f l u x , i d x , 1 , x , oldPCE_con ) ;
maxPCE_exc_con ( i d x ) = −maxPCE ;
x_exc_con ( idx , 1:3∗ idx ) = x ;
end
end
figure ;
SQ1JPCE = 3 3 . 7 ; % s i n g l e j u n c t i o n SQ l i m i t
p l o t ( 1 : numExc , maxPCE_exc_uncon ∗ 1 0 0 , ' −o ' , ' M a r k e r F a c e C o l o r ' , ' b ' , ' M a r k e r S i z e
' , 1 0 ) ; h o l d on ;
i f doSeriesCon
p l o t ( 1 : numExc , maxPCE_exc_con ∗ 1 0 0 , ' −o ' , ' M a r k e r F a c e C o l o r ' , ' r ' , '
MarkerSize ' , 10) ;
p l o t ( 1 : numExc , SQ1JPCE∗ o n e s ( 1 , numExc ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
l e g e n d ( { ' U n c o n s t r a i n e d ' , ' S e r i e s C o n s t r a i n e d ' , ' SQ L i m i t ' } , ' L o c a t i o n ' , '
NorthWest ' ) ;
else
p l o t ( 1 : numExc , SQ1JPCE∗ o n e s ( 1 , numExc ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
l e g e n d ( { ' U n c o n s t r a i n e d ' , ' SQ L i m i t ' } , ' L o c a t i o n ' , ' N o r t h W e s t ' ) ;
end
x l a b e l ( ' Number o f E x c i t o n i c A b s o r b e r s ' ) ; y l a b e l ( ' Maximum PCE (%) ' ) ;
x t i c k s ( 1 : numExc ) ;
s a v e f i g ( [ d i r _ n a m e , ' / maxPCE_vs_numExc . f i g ' ] ) ;
%% P l o t S p e c t r a l Data o f Optimized E x c i t o n i c Absorbers
i f plotSpec
% Use a d i f f e r e n t c o l o r f o r a d i f f e r e n t number o f e x c i t o n i c a b s o r b e r s
colorLines = [
0.4470
0.7410
0.8500
0.3250
0.0980
0.9290
0.6940
0.1250

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0.4940
0.4660
0.3010
0.6350
0.2500
0.7500
0.7500
0.0000
0.0000
1.0000
0.5000
0.8000
0.5000
0.1050
0.7500
0.0000
0.0000

0.1840
0.6740
0.7450
0.0780
0.7500
0.2500
0.7500
0.0000
0.5000
0.0000
0.5000
0.1050
0.8000
0.5000
0.5000
0.7500
0.0000

0.5560
0.1880
0.9330
0.1840
0.7500
0.2500
0.7500
1.0000
0.0000
0.0000
0.7500
0.5000
0.1050
0.8000
0.0000
0.0000
0.7500];

figure ;
h o l d on ;
s e t ( gcf , ' P o s i t i o n ' , [300 , 75 , 300 , 5 0 0 ] ) ;
A b s S p e c t r a _ u n c o n = z e r o s ( numExc , numExc , l e n g t h ( E _ r a n g e ) ) ;
f o r i 1 = 1 : numExc
for i2 = 1: i1
s p e c = e x c G a u s s ( E_range , x _ e x c _ u n c o n ( i 1 , i 2 ) , ...
x _ e x c _ u n c o n ( i 1 , i 1 + i 2 ) , x _ e x c _ u n c o n ( i 1 , 2∗ i 1 + i 2 ) ) ;
AbsSpectra_uncon ( i1 , i2 , : ) = spec ;
p l o t ( E_range , ( i 1 − 1 ) ∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) + s p e c , ...
' − ' , ' Color ' , c o l o r L i n e s ( i1 , : ) , ' LineWidth ' , 1 . 5 ) ;
end
end
hold off ;
x l i m ( [ 0 . 5 , 3 . 5 ] ) ; box on ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; y l a b e l ( ' E x c i t o n i c A b s o r p t i o n S p e c t r a ' ) ;
s a v e f i g ( [ dir_name , ' / AbsSpectra_optim_uncon . f i g ' ] ) ;
i f doSeriesCon
figure ;
h o l d on ;
s e t ( gcf , ' P o s i t i o n ' , [300 , 75 , 300 , 5 0 0 ] ) ;
A b s S p e c t r a _ c o n = z e r o s ( numExc , numExc , l e n g t h ( E _ r a n g e ) ) ;
f o r i 1 = 1 : numExc
for i2 = 1: i1
s p e c = e x c G a u s s ( E_range , x _ e x c _ c o n ( i 1 , i 2 ) , ...
x _ e x c _ c o n ( i 1 , i 1 + i 2 ) , x _ e x c _ c o n ( i 1 , 2∗ i 1 + i 2 ) ) ;
AbsSpectra_con ( i1 , i2 , : ) = spec ;
p l o t ( E_range , ( i 1 − 1 ) ∗ o n e s ( 1 , l e n g t h ( E _ r a n g e ) ) + s p e c , ...
' − ' , ' Color ' , c o l o r L i n e s ( i1 , : ) , ' LineWidth ' , 1 . 5 ) ;
end
end
hold off ;
x l i m ( [ 0 . 5 , 3 . 5 ] ) ; box on ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; y l a b e l ( ' E x c i t o n i c A b s o r p t i o n S p e c t r a ' ) ;

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s a v e f i g ( [ dir_name ,
end

%% Perform A n a l y s i s t o P l o t J−V C h a r a c t e r i s t i c s o f Optimized E x c i t o n i c
Absorbers
i f plotIV
% I n i t i a l i z e Device Output s t o r a g e
Voc_uncon = z e r o s ( numExc , numExc ) ;
J s c _ u n c o n = Voc_uncon ;
PCE_uncon = Voc_uncon ;
% Do u n c o n s t r a i n e d IV
f o r i 1 = 1 : numExc
EQE_vals = [ z e r o s ( 1 , l e n g t h ( E _ r a n g e ) ) ;
IQE_exc ∗ r e s h a p e ( A b s S p e c t r a _ u n c o n ( i 1 , 1 : i 1 , : ) , [ i 1 , l e n g t h (
E_range ) ] ) ] ;
T r a n s W i n d o w _ v a l s = 1−cumsum ( EQE_vals , 1 ) ; % f o r c a l c u l a t i n g t h e f l u x
the lower l a y e r sees
T r a n s W i n d o w _ v a l s ( TransWindow_vals < 0 ) = 0 ;
figure ;
for i2 = 1: i1
i n f l u x = TransWindow_vals ( i2 , : ) . ∗ f l u x ;
J s c _ u n c o n ( i 1 , i 2 ) = − J V _ b o l t z ( 0 , E_range , EQE_vals ( i 2 + 1 , : ) , isBB ,
influx ) ;
Voc_uncon ( i 1 , i 2 ) = f z e r o (@(V) J V _ b o l t z (V, E_range , EQE_vals ( i 2
+ 1 , : ) , isBB , i n f l u x ) , 0 . 8 ∗ x _ e x c _ u n c o n ( i 1 , i 1 + i 2 ) ) ;
Vmax = f m i n b n d (@(V) V. ∗ J V _ b o l t z (V, E_range , EQE_vals ( i 2 + 1 , : ) , isBB
, i n f l u x ) , 0 , Voc_uncon ( i 1 , i 2 ) ) ;
PCE_uncon ( i 1 , i 2 ) = −Vmax . ∗ J V _ b o l t z ( Vmax , E_range , EQE_vals ( i 2 + 1 , : )
, isBB , i n f l u x ) . / P i n ;

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V _ v a l s = l i n s p a c e ( 0 , Voc_uncon ( i 1 , i 2 ) , 1 0 0 ) ;
J _ v a l s = J V _ b o l t z ( V_vals , E_range , EQE_vals ( i 2 + 1 , : ) , isBB , i n f l u x )
p l o t ( V_vals , − J _ v a l s ∗1 e −1 , ' L i n e W i d t h ' , 1 . 5 , ' C o l o r ' , c o l o r L i n e s (
i 1 , : ) ) ; h o l d on ;
t e x t ( Vmax − 0 . 1 , J s c _ u n c o n ( i 1 , i 2 ) ∗1 e − 1 + 0 . 3 , [ ' \ e t a = ' , n u m 2 s t r (
r o u n d ( PCE_uncon ( i 1 , i 2 ) , 3 ) ∗1 e2 ) , '% ' ] , ' C o l o r ' , c o l o r L i n e s ( i 1
,:) );

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' / AbsSpectra_optim_con . f i g ' ] ) ;

end
x l a b e l ( ' V o l t a g e (V) ' ) ; y l a b e l ( ' C u r r e n t D e n s i t y (mA/ cm ^ 2 ) ' ) ;
y l i m ( [ 0 , 1 . 1 ∗ max ( J s c _ u n c o n ( i 1 , : ) ) ∗1 e − 1 ] ) ;
s a v e f i g ( [ d i r _ n a m e , ' / JV_optim_uncon_numExc = ' , n u m 2 s t r ( i 1 ) , ' . f i g ' ] ) ;
end
i f doSeriesCon
% Do c o n s t r a i n e d IV
Voc_con = z e r o s ( numExc , numExc ) ;
J s c _ c o n = Voc_con ;
PCE_con = Voc_con ;
f o r i 1 = 1 : numExc
EQE_vals = [ z e r o s ( 1 , l e n g t h ( E _ r a n g e ) ) ;

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IQE_exc ∗ r e s h a p e ( A b s S p e c t r a _ c o n ( i 1 , 1 : i 1 , : ) , [ i 1 , l e n g t h (
E_range ) ] ) ] ;
T r a n s W i n d o w _ v a l s = 1−cumsum ( EQE_vals , 1 ) ; % f o r c a l c u l a t i n g t h e f l u x
the lower l a y e r sees
T r a n s W i n d o w _ v a l s ( TransWindow_vals < 0 ) = 0 ;

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figure ;
for i2 = 1: i1
i n f l u x = TransWindow_vals ( i2 , : ) . ∗ f l u x ;
J s c _ c o n ( i 1 , i 2 ) = − J V _ b o l t z ( 0 , E_range , EQE_vals ( i 2 + 1 , : ) , isBB ,
influx ) ;
Voc_con ( i 1 , i 2 ) = f z e r o (@(V) J V _ b o l t z (V, E_range , EQE_vals ( i 2 + 1 , : ) ,
isBB , i n f l u x ) , 0 . 8 ∗ x _ e x c _ c o n ( i 1 , i 1 + i 2 ) ) ;
Vmax = f m i n b n d (@(V) V. ∗ J V _ b o l t z (V, E_range , EQE_vals ( i 2 + 1 , : ) , isBB
, i n f l u x ) , 0 , Voc_con ( i 1 , i 2 ) ) ;
PCE_con ( i 1 , i 2 ) = −Vmax . ∗ J V _ b o l t z ( Vmax , E_range , EQE_vals ( i 2 + 1 , : ) ,
isBB , i n f l u x ) . / P i n ;

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V _ v a l s = l i n s p a c e ( 0 , Voc_con ( i 1 , i 2 ) , 1 0 0 ) ;
J _ v a l s = J V _ b o l t z ( V_vals , E_range , EQE_vals ( i 2 + 1 , : ) , isBB , i n f l u x )
p l o t ( V_vals , − J _ v a l s ∗1 e −1 , ' L i n e W i d t h ' , 1 . 5 , ' C o l o r ' , c o l o r L i n e s (
i 1 , : ) ) ; h o l d on ;
t e x t ( Vmax − 0 . 1 , J s c _ c o n ( i 1 , i 2 ) ∗1 e − 1 + 0 . 3 , [ ' \ e t a = ' , n u m 2 s t r ( r o u n d (
PCE_con ( i 1 , i 2 ) , 3 ) ∗1 e2 ) , '% ' ] , ' C o l o r ' , c o l o r L i n e s ( i 1 , : ) ) ;

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end
x l a b e l ( ' V o l t a g e (V) ' ) ; y l a b e l ( ' C u r r e n t D e n s i t y (mA/ cm ^ 2 ) ' ) ;
y l i m ( [ 0 , 1 . 1 ∗ max ( J s c _ c o n ( i 1 , : ) ) ∗1 e − 1 ] ) ;
s a v e f i g ( [ d i r _ n a m e , ' / JV_optim_con_numExc = ' , n u m 2 s t r ( i 1 ) , ' . f i g ' ] ) ;
end
end
end
end
% s a v e a l l v a r i a b l e s i n t h i s sweep
close all ;
f i l e n a m e = [ d i r _ n a m e , ' / E x c i t o n i c T a n d e m _ V a r i a b l e D a t a . mat ' ] ;
save ( filename ) ;
%%%%%%%%%%%%%%%%%%%%%%%%% Sub F u n c t i o n s %%%%%%%%%%%%%%%%%%%%%%%%%%%
f u n c t i o n [ maxPCE , x ] = o p t i m E x c i t o n ( f l u x , numExc , i s S e r i e s , x0 , oldPCE )
% l e n g t h o f x i s numExc∗3
% S e t −up O p t i m i z a t i o n P a r a m e t e r s
maxExc_LW = 150 e − 3 ;
maxEg = 5 ;
maxA = 0 . 5 ;
l b = z e r o s ( 1 , 3∗numExc ) ;
ub = [ maxA∗ o n e s ( 1 , numExc ) , maxEg∗ o n e s ( 1 , numExc ) , maxExc_LW∗ o n e s ( 1 , numExc )
];
o b j c o n s t r = @(x ) c o n o p t ( x , f l u x , numExc , oldPCE , i s S e r i e s ) ; % r e w r i t e
optimization for surrogateopt

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o p t i o n s = o p t i m o p t i o n s ( ' s u r r o g a t e o p t ' , ' I n i t i a l P o i n t s ' , x0 , ...
' U s e P a r a l l e l ' , t r u e , ' M a x F u n c t i o n E v a l u a t i o n s ' , max ( 5 0 0 , 2 0 0 ∗ numExc ) , ...
' PlotFcn ' , ' s u r r o g a t e o p t p l o t ' ) ;
% Perform Optimization
rng d e f a u l t ;
[ x , maxPCE ] = s u r r o g a t e o p t ( o b j c o n s t r , l b , ub , o p t i o n s ) ;
end
% S u b f u n c t i o n t h a t c a l c u l a t e s PCE and Jmax a s t h e o b j e c t i v e f u n c t i o n
% and n o n l i n e a r c o n s t r a i n t f o r s u r r o g a t e o p t . I n c l u d e s s p e c t r a l
% windowing e f f e c t s . E n f o r c e s l a r g e r PCE v a l u e s a s a c o n s t r a i n t a s w e l l
f u n c t i o n f = c o n o p t ( x , f l u x , numExc , oldPCE , i s S e r i e s )
A b s S p e c t r a = z e r o s ( numExc , l e n g t h ( E _ r a n g e ) ) ;
f o r i = 1 : numExc
A b s S p e c t r a ( i , : ) = e x c G a u s s ( E_range , x ( i ) , x ( numExc+ i ) , x ( 2 ∗ numExc+ i ) ) ;
end
EQE = [ z e r o s ( 1 , l e n g t h ( E _ r a n g e ) ) ;
IQE_exc ∗ A b s S p e c t r a ] ;
TransWindow = 1−cumsum ( EQE , 1 ) ; % f o r c a l c u l a t i n g t h e f l u x t h e l o w e r l a y e r
sees
TransWindow ( TransWindow < 0 ) = 0 ;

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J m a x _ v a l s = z e r o s ( numExc , 1 ) ;
newPCE = 0 ;
f o r i = 1 : numExc
i n p u t f l u x = TransWindow ( i , : ) . ∗ f l u x ;
[ Vmax_val , P o u t ] = f m i n b n d (@(V) V. ∗ J V _ b o l t z (V, E_range , EQE( i + 1 , : ) ,
isBB , i n p u t f l u x ) , 0 , x ( numExc+ i ) ) ;
J m a x _ v a l s ( i ) = J V _ b o l t z ( Vmax_val , E_range , EQE( i + 1 , : ) , isBB , i n p u t f l u x
);
newPCE = newPCE − P o u t . / P i n ;
end

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f . F v a l = −newPCE ;
m i n I n e q = r e a l m i n ; % l o g ( m i n I n e q ) i s t h e l o w e s t v a l u e we c a n r e a c h i n t h e
c o n s t r a i n t , s o t h e r e a r e no i n f i n i t i e s f o r l o g ( 0 ) . Use f l o a t i n g p o i n t
precision
if isSeries
% Perform c u r r e n t matching using n o n l i n e a r i n e q u a l i t i e s ( Ineq
% <= 0 ) by r e w r i t i n g e q u a l i t y c o n s t r a i n t w i t h a l o g f u n c t i o n
f . I n e q = [ oldPCE − newPCE ; l o g ( m i n I n e q + a b s ( ( J m a x _ v a l s − J m a x _ v a l s ( 1 ) )
. / Jmax_vals ( 1 ) ) ∗ CurTol ) ] ;
else
f . I n e q = oldPCE − newPCE ;
end

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end
% Gaussian s p e c t r a for excitons
f u n c t i o n o u t = e x c G a u s s ( E_range , Apeak , E_exc , Exc_LW )
o u t = Apeak ∗ exp ( − 0 . 5 ∗ ( ( E _ r a n g e − E_exc ) . / Exc_LW ) . ^ 2 ) ;
end

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% Bose − E i n s t e i n b l a c k −body s p e c t r u m w i t h some e l e c t r o c h e m i c a l p o t e n t i a l
f u n c t i o n s p e c t r u m = bb ( E_range , F , T , V)
s p e c t r u m = 2∗F . / ( h ^3∗ c ^ 2 ) . ∗ ...
E _ r a n g e . ^ 2 . / ( exp ( ( E_range −V) / ( k∗T ) ) − 1 ) ; % b l a c k b o d y p h o t o n
f l u x [ P h o t o n s / ( s e c ∗m^2∗ w a v e l e n g t h ) ]
end
% C a l c u l a t e JV c h a r a c t e r i s t i c s u n d e r B o l t z m a n n a p p r o x i m a t i o n
f u n c t i o n o u t = J V _ b o l t z (V, E_range , EQE , isBB , f l u x )
i f isBB
J p h o t o n = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ bb ( E_range , Fsun , Tsun , 0 ) ) ) ;
else
J p h o t o n = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ f l u x ) ) ;
end
J r a d = −e ∗ a b s ( t r a p z ( E_range , EQE . ∗ bb ( E_range , F c e l l , T c e l l , 0 ) ) ) ∗ ( exp (V . / ( k
∗ T c e l l ) ) −1) ;
out = Jphoton − Jrad ;
end
end

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Listing D.3: Transfer Matrix Code for Monolayer Excitonic Absorbers
1 % 1D, Normal I n c i d e n c e , T r a n s f e r M a t r i x Method
2 % A c c o u n t i n g f o r TMDC a s a s h e e t c o n d u c t i v i t y p a r a m e t r i z e d by a L o r e n t z i a n
3 % Also a c c o u n t f o r a b s o r p t i o n i n i n d i v i d u a l l a y e r s
4 %
5 % Assumed s t r u c t u r e i s GaS / Mica /TMDC/ Mica / Ag . P e r f o r m s t h e f o l l o w i n g s w e e p s :
6 %
7 % − Sweeps t h i c k n e s s o f GaS and Mica
8 % − Sweeps r a d i a t i v e and non − r a d i a t i v e r a t e s o f E x c i t o n
9 % − Examines t h e a b s o r p t i o n s p e c t r a a t e i t h e r t h e optimum t h i c k n e s s o r t h e
10 % s p e c i f i e d t h i c k n e s s ( i f t h i c k n e s s sweep n o t r a n )
11 % − Examines t h e E l e c t r i c F i e l d P r o f i l e a t e i t h e r t h e optimum t h i c k n e s s o r
12 % t h e s p e c i f i e d t h i c k n e s s ( i f t h i c k n e s s sweep n o t r a n ) . R e q u i r e s r u n n i n g
13 % a b s o r p t i o n s p e c t r a c a l c u l a t i o n .
14 % − P e r f o r m s a L o r e n t z i a n t o t a l l i n e w i d t h f i t a s t h e non − r a d i a t i v e r a t e i s
15 % v a r i e d .
16 %
17 % J o e s o n Wong
18 % L a s t U p d a t e d : 1 1 / 1 4 / 2 0 2 1
19
20 %% Load i n Data and I n i t i a l i z e V a r i a b l e s
21
22 % l o a d m a t e r i a l d a t a
23 c l e a r a l l ;
24 c l o s e a l l ;
25 a = l o a d ( ' Au_nk . mat ' ) ;
26 b = l o a d ( ' Ag_nk . mat ' ) ;
27
28 % w a v e l e n g t h s o f i n t e r e s t
29 lambda = 550 e − 9 : 1 e − 9 : 7 0 0 e − 9 ; % W a v e l e n g t h s o f i n t e r e s t ( u n i t s o f m^ −1)
30 A g _ i n t e r p _ n t i l d e = i n t e r p 1 ( b . lambda , b . A g _ i n t e r p _ n t i l d e , lambda ) ;
31
32 % p h y s i c a l c o n s t a n t s
33 c = 2 9 9 7 9 2 4 5 8 ; % [m/ s ] , s p e e d o f l i g h t
34 c e = 1 . 6 0 2 1 7 6 5 6 5 3 5 e − 1 9 ; % c h a r g e o f e l e c t r o n , [ C o r J / eV ]
35 h b a r = 1 . 0 5 4 5 7 1 8 1 7 e − 3 4 ; % r e d u c e d P l a n c k ' s c o n s t a n t , [ J − s ]
36 mu0 = 4∗ p i ∗1 e − 7 ; % [ Henry / m e t e r ] , p e r m e a b i l i t y o f f r e e s p a c e
37 e p s 0 = 8 . 8 5 4 1 8 7 8 2 1 e − 1 2 ; % [ F a r a d s / m e t e r ] , p e r m i t t i v t y o f f r e e s p a c e
38 G0 = 2∗ c e ^ 2 . / ( 2 ∗ p i ∗ h b a r ) ; % 1 / Ohms
39 n_GaS = 2 . 7 ;
40 n_Mica = 1 . 5 5 ;
41 n_SiO2 = 1 . 4 9 ;
42
43 % l o a d and d e f i n e TMDC (WS2) / hBN d a t a
44 g a m m a _ r _ d e f a u l t = 4 . 1 1 e − 3 ;
45 g a m m a _ n r _ d e f a u l t = 3 8 . 6 e − 3 ;
46 o m e g a _ e x c _ d e f a u l t = 2 . 0 1 ; % [ eV ]
47 omega_peak = o m e g a _ e x c _ d e f a u l t ∗ c e . / h b a r ; % f r e q u e n c y , i n u n i t s o f r a d / s
48 l a m b d a _ p e a k = 2∗ p i ∗ c . / omega_peak ; % w a v e l e n g t h
49 k _ p e a k = f i n d ( a b s ( lambda − l a m b d a _ p e a k ) ∗1 e9 wavelength index of peak
50 o m e g a _ e n e r g y = 2∗ p i ∗ h b a r ∗ c . / lambda . / c e ; % f r e q u e n c y , i n u n i t s o f e n e r g y
51 s i g m a _ s = f u n _ e x c _ s i g m a ( g a m m a _ r _ d e f a u l t , g a m m a _ n r _ d e f a u l t , omega_energy ,
o m e g a _ e x c _ d e f a u l t ) ; % c o n d u c t i v i t y o f TMDC, i n [ 1 / Ohms ]

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% define thicknesses
d1 = 58 e − 9 ; % t o p GaS
d2 = 99 e − 9 ; % t o p Mica
d3 = 77 e − 9 ; % b o t t o m Mica
d4 = 0 e − 9 ; % b o t t o m SiO2 , i f any
t h i c k n e s s e s = [ d1 , d2 , d3 , d4 ] ; %t h i c k n e s s o f e a c h l a y e r
z = cumsum ( [ 0 , t h i c k n e s s e s ] ) ; % u n i t s o f [m] , z p o s i t i o n o f l a y e r s . S t a r t a t z = 0 .
% d e f i n e what t o c a l c u l a t e
doThickSweep = f a l s e ;
doGamSweep = f a l s e ;
doSpectraCalc = true ;
doFieldPlot = true ;
doGamNR_FWHM = t r u e ;
% d e f i n i t i o n o f l a y e r s , e a c h l a y e r i s a s e p a r a t e column
n t i l d e = [ o n e s ( l e n g t h ( lambda ) , 1 ) , ... % f i r s t l a y e r , which i s a i r u s u a l
n_GaS∗ o n e s ( l e n g t h ( lambda ) , 1 ) , ...
n_Mica ∗ o n e s ( l e n g t h ( lambda ) , 1 ) , ...
n_Mica ∗ o n e s ( l e n g t h ( lambda ) , 1 ) , ...
n_SiO2 ∗ o n e s ( l e n g t h ( lambda ) , 1 ) , ...
r e s h a p e ( A g _ i n t e r p _ n t i l d e , [ l e n g t h ( lambda ) , 1 ] ) ...
] ; % l a s t l a y e r i s s u b s t r a t e / a i r , s h o u l d h a v e l e n g t h ( t ) +2 number o f l a y e r s
% S t a c k o r d e r i n g i s : [ n t i l d e ( 1 ) , hasTMDC , n t i l d e ( 2 ) , hasTMDC , . . . ]
% n t i l d e ( 1 ) u s u a l l y a i r . hasTMDC ( 1 ) i s a l a y e r o f TMDC b e t w e e n a i r and
% f i r s t l a y e r o f d i e l e c t r i c i f hasTMDC ( 1 ) = 1 . O t h e r w i s e , n o t h i n g a t
% interface .
hasTMDC = [ 0 , 0 , 1 , 0 , 0 ] ; % 1 i f i n t e r f a c e h a s g r a p h e n e , 0 i f no g r a p h e n e . T h e r e a r e
l e n g t h ( z ) number o f i n t e r f a c e s .
ParIndex = [1 ,2 ,4 ,5 ,6];
ActIndex = 3;
% D e f i n e n o r m a l i z e d E− F i e l d i n t e n s i t y f o r p l a n e waves
E l e c t r i c I n t e n s i t y = @(z , An , Bn , qp ) a b s ( An ) ^2∗ exp ( 1 i ∗ z ∗ ( qp − c o n j ( qp ) ) )
a b s ( Bn ) ^2∗ exp ( −1 i ∗ z ∗ ( qp − c o n j ( qp ) ) ) + ...
2∗ r e a l ( An∗ c o n j ( Bn ) ∗ exp ( 1 i ∗ z ∗ ( qp + c o n j ( qp ) ) ) ) ;

+ ...

% Some s i m p l e e r r o r C h e c k i n g
i f l e n g t h ( hasTMDC ) ~= l e n g t h ( z )
e r r o r ( ' Make s u r e t h e number o f TMDC l o c a t i o n s m a t c h e s number o f i n t e r f a c e s ' ) ;
e l s e i f l e n g t h ( t h i c k n e s s e s ) +2 ~= s i z e ( n t i l d e , 2 )
e r r o r ( ' Make s u r e you h a v e s p e c i f i e d a t h i c k n e s s f o r e a c h l a y e r ' ) ;
end
if doFieldPlot
i f ~doSpectraCalc
e r r o r ( ' Make s u r e t o p e r f o r m s p e c t r u m c a l c u l a t i o n b e f o r e f i e l d c a l c u l a t i o n '
);
end
end

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i f doThickSweep
d1 = 0 e − 9 : 1 e − 9 : 1 2 0 e − 9 ;
d2 = 40 e − 9 : 1 e − 9 : 1 4 0 e − 9 ;
d3 = 30 e − 9 : 1 e − 9 : 1 3 0 e − 9 ;
T o t a l A b s = z e r o s ( l e n g t h ( d1 ) , l e n g t h ( d2 ) , l e n g t h ( d3 ) ) ;
AbsTMDC = T o t a l A b s ;
sigma_exc = fun_excsigma ( gamma_r_default , gamma_nr_default , 0 ,
o m e g a _ e x c _ d e f a u l t , o m e g a _ e x c _ d e f a u l t ) ; % c o n d u c t i v i t y o f TMDC, i n [ 1 / Ohms ]
maxAbsTMDC = 0 ; % f o r f i n d i n g max a c t i v e l a y e r a b s o r p t i o n
d1_max = 0 ;
d2_max = 0 ;
d3_max = 0 ;
f o r i d x _ d 1 = 1 : l e n g t h ( d1 )
f o r i d x _ d 2 = 1 : l e n g t h ( d2 )
f o r i d x _ d 3 = 1 : l e n g t h ( d3 )
t h i c k n e s s e s = [ d1 ( i d x _ d 1 ) , d2 ( i d x _ d 2 ) , d3 ( i d x _ d 3 ) , d4 ] ; %
t h i c k n e s s of each l a y e r
z = cumsum ( [ 0 , t h i c k n e s s e s ] ) ; % u n i t s o f [m] , z p o s i t i o n o f
layers . Start at z = 0.
J = [1 , 0 ;
0 , 1 ] ; % i n i t a l i z e J as i d e n t i t y matrix
% f i r s t l a y e r i s a i r , c o o r d i n a t e s are such t h a t the a i r −
dielectric
% stack s t a r t s at z = 0
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k_peak , j ) / l a m b d a _ p e a k ;
qp = 2∗ p i ∗ n t i l d e ( k_peak , j + 1 ) / l a m b d a _ p e a k ;
J = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q− omega_peak ∗mu0∗
s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗
( 1 − ( q+ omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+
omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ] ∗ J ;
end
r = −J ( 2 , 1 ) / J ( 2 , 2 ) ; % r e f l e c t i o n amplitude
T o t a l A b s ( i d x _ d 1 , i d x _ d 2 , i d x _ d 3 ) = 1− a b s ( r ) . ^ 2 ;
% C a l c u l a t e a b s o r p t i o n and E− f i e l d i n i n d i v i d u a l l a y e r s , go
through
% l a y e r s a g a i n u s i n g t h e f a c t t h a t we s t a r t w i t h [ 1 ; r ] f o r t h e
% f i e l d amplitude
% C o e f f i c i e n t s o f E− f i e l d i n f i r s t l a y e r , e . g . a i r
A( 1 : 2 , 1 ) = [ 1 ; r ] ;
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k_peak , j ) / l a m b d a _ p e a k ;
qp = 2∗ p i ∗ n t i l d e ( k_peak , j + 1 ) / l a m b d a _ p e a k ;
J n = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q− omega_peak ∗mu0∗
s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗
( 1 − ( q+ omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+

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omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ] ;
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A ( 1 : 2 , j + 1 ) = J n ∗A ( 1 : 2 , j ) ; % u p d a t e E− f i e l d c o e f f i c i e n t s
% A b s o r p t i o n i n TMDC
i f hasTMDC ( j )
AbsTMDC( i d x _ d 1 , i d x _ d 2 , i d x _ d 3 ) = r e a l ( s i g m a _ e x c ) / ( c ∗
n t i l d e ( k_peak , 1 ) ∗ e p s 0 ) ∗ E l e c t r i c I n t e n s i t y ( z ( j ) ,A( 1 ,
j + 1 ) , A( 2 , j + 1 ) , qp ) ;
end
end
i f AbsTMDC( i d x _ d 1 , i d x _ d 2 , i d x _ d 3 ) > maxAbsTMDC
d1_max = d1 ( i d x _ d 1 ) ;
d2_max = d2 ( i d x _ d 2 ) ;
d3_max = d3 ( i d x _ d 3 ) ;
maxAbsTMDC = AbsTMDC( i d x _ d 1 , i d x _ d 2 , i d x _ d 3 ) ;
end
end
end
end
FigSpecX = f i g u r e ;
s u r f ( d1 ∗1 e9 , d2 ∗1 e9 , t r a n s p o s e ( s q u e e z e (AbsTMDC ( : , : , a b s ( d3 −d3_max ) d i f f ( d3 ) ) ) / 2 ) ) ) ) ;
view ( [ 0 , 9 0 ] ) ;
shading i n t e r p ; axis t i g h t ;
daspect ([1 , 1 , 1]) ;
x l a b e l ( ' Top GaS T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' Top Mica T h i c k n e s s ( nm ) ' ) ;
a x i s s q u a r e ; box on ;
hc = c o l o r b a r ; x l a b e l ( hc , ' Peak TMDC A b s o r p t i o n ' ) ;
colormap ( ' p a r u l a ' ) ; c a x i s ( [ 0 , 1 ] ) ;
FigSpecY = f i g u r e ;
s u r f ( d2 ∗1 e9 , d3 ∗1 e9 , t r a n s p o s e ( s q u e e z e (AbsTMDC( a b s ( d1 −d1_max ) ) ) ) /2 ,: ,:) ) ) ) ;
view ( [ 0 , 9 0 ] ) ;
shading i n t e r p ; axis t i g h t ;
daspect ([1 , 1 , 1]) ;
x l a b e l ( ' Top Mica T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' Bottom Mica T h i c k n e s s ( nm ) ' ) ;
a x i s s q u a r e ; box on ;
hc = c o l o r b a r ; x l a b e l ( hc , ' Peak TMDC A b s o r p t i o n ' ) ;
colormap ( ' p a r u l a ' ) ; c a x i s ( [ 0 , 1 ] ) ;
FigSpecZ = f i g u r e ;
s u r f ( d1 ∗1 e9 , d3 ∗1 e9 , t r a n s p o s e ( s q u e e z e (AbsTMDC ( : , a b s ( d2 −d2_max ) d2 ) ) ) / 2 , : ) ) ) ) ;
view ( [ 0 , 9 0 ] ) ;
shading i n t e r p ; axis t i g h t ;
daspect ([1 , 1 , 1]) ;
x l a b e l ( ' Top GaS T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' Bottom Mica T h i c k n e s s ( nm ) ' ) ;
a x i s s q u a r e ; box on ;
hc = c o l o r b a r ; x l a b e l ( hc , ' Peak TMDC A b s o r p t i o n ' ) ;

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colormap ( ' p a r u l a ' ) ; c a x i s ( [ 0 , 1 ] ) ;

figure ;
p l o t ( d1 ∗1 e9 , s q u e e z e (AbsTMDC ( : , a b s ( d2 −d2_max ) d3_max ) axis tight ;
x l a b e l ( ' Top GaS T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' E x c i t o n i c Absorbance ' ) ;
a x i s s q u a r e ; box on ;
figure ;
p l o t ( d2 ∗1 e9 , s q u e e z e (AbsTMDC( a b s ( d1 −d1_max ) d3_max ) axis tight ;
x l a b e l ( ' Top Mica T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' E x c i t o n i c Absorbance ' ) ;
a x i s s q u a r e ; box on ;
figure ;
p l o t ( d3 ∗1 e9 , s q u e e z e (AbsTMDC( a b s ( d1 −d1_max ) d2_max ) axis tight ;
x l a b e l ( ' Bottom Mica T h i c k n e s s ( nm ) ' ) ;
y l a b e l ( ' E x c i t o n i c Absorbance ' ) ;
a x i s s q u a r e ; box on ;
% i f r a n , s e t d1 , d2 , d3 t o o p t i m a l v a l u e s
d1 = d1_max ;
d2 = d2_max ;
d3 = d3_max ;
t h i c k n e s s e s = [ d1 , d2 , d3 , d4 ] ; %t h i c k n e s s o f e a c h l a y e r
z = cumsum ( [ 0 , t h i c k n e s s e s ] ) ; % u n i t s o f [m] , z p o s i t i o n o f l a y e r s . S t a r t a t z
= 0.

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229 %% Peak A b s o r p t i o n f o r v a r y i n g gamma_nr , gamma_r
230 i f doGamSweep
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gamma_r = [ 1 e −3 , 3 e −3 , 5 e − 3 ] ;
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gamma_nr = 0 . 2 e − 3 : 0 . 2 e − 3 : 5 0 e − 3 ;
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omega_peak = o m e g a _ e x c _ d e f a u l t ∗ c e . / h b a r ; % f r e q u e n c y , i n u n i t s o f r a d / s
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l a m b d a _ p e a k = 2∗ p i ∗ c . / omega_peak ; % w a v e l e n g t h
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k _ p e a k = f i n d ( a b s ( lambda − l a m b d a _ p e a k ) 236
T o t a l A b s = z e r o s ( l e n g t h ( gamma_r ) , l e n g t h ( gamma_nr ) ) ;
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AbsTMDC = T o t a l A b s ;
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F i g S p e c 2 = f i g u r e ; h o l d on ;
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legH = z e r o s ( l e n g t h ( gamma_r ) , 1 ) ;
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f o r i d x _ g a m _ r = 1 : l e n g t h ( gamma_r )
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f o r i d x _ g a m _ n r = 1 : l e n g t h ( gamma_nr )
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s i g m a _ e x c = f u n _ e x c s i g m a ( gamma_r ( i d x _ g a m _ r ) , gamma_nr ( i d x _ g a m _ n r ) , 0 ,
o m e g a _ e x c _ d e f a u l t , o m e g a _ e x c _ d e f a u l t ) ; % c o n d u c t i v i t y o f TMDC, i n
[ 1 / Ohms ]
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J = [1 , 0 ;
0 , 1 ] ; % i n i t a l i z e J as i d e n t i t y matrix
% f i r s t l a y e r i s a i r , c o o r d i n a t e s are such t h a t the a i r − d i e l e c t r i c
% stack s t a r t s at z = 0
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k_peak , j ) / l a m b d a _ p e a k ;
qp = 2∗ p i ∗ n t i l d e ( k_peak , j + 1 ) / l a m b d a _ p e a k ;
J = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+ omega_peak ∗
mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) )
/ qp ) ,
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+ omega_peak ∗mu0∗
s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ] ∗ J ;
end

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% r e f l e c t i o n amplitude & transmission
r = −J ( 2 , 1 ) / J ( 2 , 2 ) ;
T o t a l A b s ( idx_gam_r , i d x _ g a m _ n r ) = 1 − a b s ( r ) . ^ 2 ; % a s s u m i n g T = 0
% C a l c u l a t e a b s o r p t i o n and E− f i e l d i n i n d i v i d u a l l a y e r s , go t h r o u g h
% l a y e r s a g a i n u s i n g t h e f a c t t h a t we s t a r t w i t h [ 1 ; r ] f o r t h e
% f i e l d amplitude
% C o e f f i c i e n t s o f E− f i e l d i n f i r s t l a y e r , e . g . a i r
A( 1 : 2 , 1 ) = [ 1 ; r ] ;
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k_peak , j ) / l a m b d a _ p e a k ;
qp = 2∗ p i ∗ n t i l d e ( k_peak , j + 1 ) / l a m b d a _ p e a k ;
J n = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+ omega_peak ∗
mu0∗ s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q− omega_peak ∗mu0∗ s i g m a _ e x c ∗hasTMDC ( j )
) / qp ) ,
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+ omega_peak ∗mu0∗
s i g m a _ e x c ∗hasTMDC ( j ) ) / qp ) ] ;
A ( 1 : 2 , j + 1 ) = J n ∗A ( 1 : 2 , j ) ; % u p d a t e E− f i e l d c o e f f i c i e n t s
% A b s o r p t i o n i n TMDC
i f hasTMDC ( j )
AbsTMDC( idx_gam_r , i d x _ g a m _ n r ) = r e a l ( s i g m a _ e x c ) / ( c ∗ n t i l d e (
k_peak , 1 ) ∗ e p s 0 ) ∗ E l e c t r i c I n t e n s i t y ( z ( j ) ,A( 1 , j + 1 ) , A( 2 , j + 1 )
, qp ) ;
end
end

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end
a = p l o t ( gamma_nr ∗1 e3 , T o t a l A b s ( idx_gam_r , : ) , ' L i n e W i d t h ' , 1 . 5 ) ;
p l o t ( gamma_nr ∗1 e3 , AbsTMDC( idx_gam_r , : ) , ' −− ' , ' L i n e W i d t h ' , 1 . 5 ,
a . Color ) ;
legH ( i d x _ g a m _ r ) = a ;

' Color ' ,

end
box on ;
x l a b e l ( ' $ \ gamma_{ n r }$ (meV) ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ; y l a b e l ( ' A b s o r b a n c e ' ) ;
h l e g = l e g e n d ( legH , s t r i n g ( gamma_r ∗1 e3 ) , ' l o c a t i o n ' , ' s o u t h e a s t ' ) ;
t i t l e ( h l e g , ' $ \ gamma_{ r }$ (meV) ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
end

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i f doSpectraCalc
% I n i t i a l i z a t i o n of matrices
T o t a l A b s = z e r o s ( l e n g t h ( lambda ) , 1 ) ; % t o t a l a b s o r p t i o n
% a b s o r p t i o n i n i n d i v i d u a l l a y e r s , add i n a d d i t i o n a l l a y e r s f o r g r a p h e n e ,
ignore air
AbsLayer = z e r o s ( l e n g t h ( lambda ) , min ( s i z e ( n t i l d e ) ) +sum ( hasTMDC ) −1) ;
A = z e r o s ( 2 ∗ l e n g t h ( lambda ) , min ( s i z e ( n t i l d e ) ) ) ; % E− f i e l d c o e f f i c i e n t s , odd
c o e f f i c i e n t s a r e Ai , e v e n c o e f f i c i e n t s a r e Bi
% C a l c u l a t e s p e c t r a l l y , m o n o c h r o m a t i c wave a s s u m p t i o n
f o r k = 1 : l e n g t h ( lambda )
J = [1 , 0 ;
0 , 1 ] ; % i n i t a l i z e J as i d e n t i t y matrix
omega = 2∗ p i ∗ c . / lambda ( k ) ; % f r e q u e n c y , u s e f u l f o r n o t r e p e a t i n g
calculations .
% f i r s t l a y e r i s a i r , c o o r d i n a t e s are such t h a t the a i r − d i e l e c t r i c
% stack s t a r t s at z = 0
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k , j ) / lambda ( k ) ;
qp = 2∗ p i ∗ n t i l d e ( k , j + 1 ) / lambda ( k ) ;
J = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q−omega∗mu0∗ s i g m a _ s ( k ) ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+omega∗mu0∗
s i g m a _ s ( k ) ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q−omega∗mu0∗ s i g m a _ s ( k ) ∗hasTMDC ( j ) ) / qp )
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+omega∗mu0∗ s i g m a _ s ( k ) ∗
hasTMDC ( j ) ) / qp ) ] ∗ J ;
end
% r e f l e c t i o n amplitude & transmission
r = −J ( 2 , 1 ) / J ( 2 , 2 ) ;
TotalAbs ( k ) = 1 − abs ( r ) .^2 ; % assuming T = 0
% C a l c u l a t e a b s o r p t i o n and E− f i e l d i n i n d i v i d u a l l a y e r s , go t h r o u g h
% l a y e r s a g a i n u s i n g t h e f a c t t h a t we s t a r t w i t h [ 1 ; r ] f o r t h e
% f i e l d amplitude
% C o e f f i c i e n t s o f E− f i e l d i n f i r s t l a y e r , e . g . a i r
A( 2 ∗ k − 1 : 2 ∗ k , 1 ) = [ 1 ; r ] ;
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k , j ) / lambda ( k ) ;
qp = 2∗ p i ∗ n t i l d e ( k , j + 1 ) / lambda ( k ) ;
J n = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q−omega∗mu0∗ s i g m a _ s ( k ) ∗
hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+omega∗mu0∗
s i g m a _ s ( k ) ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q−omega∗mu0∗ s i g m a _ s ( k ) ∗hasTMDC ( j ) ) / qp
) ,
exp ( −1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+omega∗mu0∗ s i g m a _ s ( k ) ∗
hasTMDC ( j ) ) / qp ) ] ;
A( 2 ∗ k − 1 : 2 ∗ k , j + 1 ) = J n ∗A( 2 ∗ k − 1 : 2 ∗ k , j ) ; % u p d a t e E− f i e l d

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coefficients
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% Absorption in D i e l e c t r i c layer
% s p e c i a l case for l a s t layer , since i n f i n i t e s are not nice for
MATLAB. Could v e c t o r i z e ?
i f j == l e n g t h ( z )
AbsLayer ( k , j +sum ( hasTMDC ( 1 : j ) ) ) = r e a l ( n t i l d e ( k , end ) ) / n t i l d e (
k , 1 ) ∗ a b s (A( 2 ∗ k −1 , end ) ) ^2∗ exp ( −4∗ p i ∗ imag ( n t i l d e ( k , end ) ) ∗ z (
end ) / lambda ( k ) ) ;
else
AbsLayer ( k , j +sum ( hasTMDC ( 1 : j ) ) ) = 4∗ p i ∗ r e a l ( n t i l d e ( k , j + 1 ) ) ∗
imag ( n t i l d e ( k , j + 1 ) ) / ( lambda ( k ) ∗ n t i l d e ( k , 1 ) ) ∗...
i n t e g r a l (@( z ) E l e c t r i c I n t e n s i t y ( z , A( 2 ∗ k −1 , j + 1 ) , A( 2 ∗ k , j + 1 ) ,
qp ) , z ( j ) , z ( j + 1 ) ) ;
end

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% A b s o r p t i o n i n TMDC
i f hasTMDC ( j )
AbsLayer ( k , j +sum ( hasTMDC ( 1 : j ) ) −1) = r e a l ( s i g m a _ s ( k ) ) / ( c ∗ n t i l d e
( k , 1 ) ∗ e p s 0 ) ∗ E l e c t r i c I n t e n s i t y ( z ( j ) ,A( 2 ∗ k −1 , j + 1 ) , A( 2 ∗ k , j
+ 1 ) , qp ) ;
end

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end
end
% Make some s p e c t r a l a b s o r p t i o n p l o t s
F i g S p e c 3 = f i g u r e ; h o l d on ;
D i e l e c t r i c A b s = sum ( AbsLayer ( : , [ 1 , 2 , 4 ] ) , 2 ) ;
TMDCAbs = AbsLayer ( : , 3 ) ;
B o t M e t a l A b s = AbsLayer ( : , end ) ;
h = a r e a ( r e p m a t ( 1 2 3 9 . 8 4 . / ( lambda ' ∗ 1 e9 ) , [ 1 , 3 ] ) , [ BotMetalAbs , TMDCAbs ,
D i e l e c t r i c A b s ] , ' E d g e C o l o r ' , ' none ' ) ; h o l d on ;
alpha 0.7;
p l o t ( 1 2 3 9 . 8 4 . / ( lambda ∗1 e9 ) , BotMetalAbs , ' C o l o r ' , h ( 1 ) . F a c e C o l o r , ' L i n e W i d t h ' ,
1.5) ;
p l o t ( 1 2 3 9 . 8 4 . / ( lambda ∗1 e9 ) , B o t M e t a l A b s +TMDCAbs , ' C o l o r ' , h ( 2 ) . F a c e C o l o r , '
LineWidth ' , 1 . 5 ) ;
p l o t ( 1 2 3 9 . 8 4 . / ( lambda ∗1 e9 ) , B o t M e t a l A b s +TMDCAbs+ D i e l e c t r i c A b s , ' C o l o r ' , h ( 3 ) .
FaceColor , ' LineWidth ' , 1 . 5 ) ;
a x i s s q u a r e ; box on ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ;
y l a b e l ( ' Absorbance ' ) ; a x i s ( [ 1 . 5 , 3 , 0 , 1 ] ) ;
l e g e n d ( [ h ( 3 ) , h ( 2 ) , h ( 1 ) ] , { ' D i e l e c t r i c s ' , 'TMDC ' , ' Bottom M e t a l ' } , ' L o c a t i o n '
, ' NorthWest ' ) ;

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end
if doFieldPlot
numLayers = l e n g t h ( t h i c k n e s s e s ) + 1 ; % number o f l a y e r s i n s t a c k . P l u s 1 from
metal b a c k r e f l e c t o r
n u m l i n s p a c e P t s = 1 0 0 0 ; % number o f d e f a u l t p o i n t s i n l i n s p a c e , which i s
u s u a l l y 100 ( a s o f 1 2 / 1 5 / 1 6 )
E F i e l d D i s t = z e r o s ( l e n g t h ( lambda ) , n u m l i n s p a c e P t s ∗ numLayers ) ; % l i n s p a c e h a s
100 p t s , s o u s u a l l y h a v e l e n g t h 100∗ numLayers
e x t r a L e n g t h = 50 e − 9 ;% T h i c k n e s s o f b a c k m e t a l a n a l y s i s
L a y e r C o o r d i n a t e s = cumsum ( [ 0 , t h i c k n e s s e s , e x t r a L e n g t h ] ) ;

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% Generate A p p r o p r i a t s P o i n t s f o r sampling , numlinspacepts per l a y e r
zPoints = [ ] ;
f o r i = 1 : numLayers
% add e x t r a p o i n t , t o remove l a t e r
zLayer = l i n s p a c e ( L a y e r C o o r d i n a t e s ( i ) , L a y e r C o o r d i n a t e s ( i +1) ,
n u m l i n s p a c e P t s +1) ;
z L a y e r ( end ) = [ ] ; % remove end p t , t o a v o i d r e p e a t v a l u e s
z P o i n t s = [ zPoints , zLayer ] ;
end
f o r l a m _ i d x = 1 : l e n g t h ( lambda )
f o r i = 1 : numLayers
E F i e l d D i s t ( l a m _ i d x , ( n u m l i n s p a c e P t s ∗ ( i − 1 ) + 1 : n u m l i n s p a c e P t s ∗ i ) ) = ...
E l e c t r i c I n t e n s i t y ( z P o i n t s ( n u m l i n s p a c e P t s ∗ ( i − 1 ) + 1 : n u m l i n s p a c e P t s ∗ i ) , ...
A( 2 ∗ l a m _ i d x −1 , i + 1 ) , A( 2 ∗ l a m _ i d x , i + 1 ) , 2∗ p i ∗ n t i l d e ( l a m _ i d x , i + 1 ) /
lambda ( l a m _ i d x ) ) ;
end
end
% Efield contour Plot
figure ;
s u r f ( lambda ∗1 e9 , z P o i n t s ∗1 e9 , E F i e l d D i s t ' ) ;
x l a b e l ( ' W a v e l e n g t h ( nm ) ' ) ;
y l a b e l ( ' D i s t a n c e from s u r f a c e o f h e t e r o s t r u c t u r e ( nm ) ' ) ;
view ( [ 0 , 9 0 ] ) ;
axis t i g h t ; shading i n t e r p ;
hc = c o l o r b a r ; x l a b e l ( hc , ' $ | E | ^ 2 / | E_0 | ^ 2 $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
colormap ( p a r u l a ) ;
% E f i e l d a t t h e e x c i t o n peak p o s i t i o n
f i g u r e ; h o l d on ;
p l o t ( z P o i n t s ∗1 e9 , E F i e l d D i s t ( k_peak , : ) , ' k− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
y l i m s p l o t = ylim ;
x l i m s p l o t = xlim ;
yvals = [ ylimsplot (1) , ylimsplot (1) , ylimsplot (2) , ylimsplot (2) ] ;
MetalColor = [ 0 . 2 , 0 . 2 , 0 . 2 ] ;
GaSColor = [ 0 . 2 , 0 . 7 , 0 . 7 ] ;
TMDCColor = [ 1 , 0 . 3 , 0 . 1 ] ;
MicaColor = [ 0 . 1 , 0 . 9 , 0 . 2 ] ;
SiO2Color = [ 0 . 7 , 0 . 7 , 0 . 7 ] ;
F a c e C o l o r s = [ GaSColor ; M i c a C o l o r ; TMDCColor ; M i c a C o l o r ; S i O 2 C o l o r ; M e t a l C o l o r
];
FaceAlpha = 0 . 5 ;
TMDC_thick = 0 . 7 e − 9 ;
f o r i = 1 : numLayers
i f hasTMDC ( i )
x v a l s = [ L a y e r C o o r d i n a t e s ( i ) −TMDC_thick / 2 , L a y e r C o o r d i n a t e s ( i ) +
TMDC_thick / 2 , L a y e r C o o r d i n a t e s ( i ) +TMDC_thick / 2 , L a y e r C o o r d i n a t e s ( i )
−TMDC_thick / 2 ] ;
p a t c h ( x v a l s ∗1 e9 , y v a l s , F a c e C o l o r s ( i +sum ( hasTMDC ( 1 : i ) ) − 1 , : ) , ' E d g e C o l o r
' , ' none ' ) ;
x v a l s = [ L a y e r C o o r d i n a t e s ( i ) , L a y e r C o o r d i n a t e s ( i +1) , L a y e r C o o r d i n a t e s ( i
+1) , L a y e r C o o r d i n a t e s ( i ) ] ;

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p a t c h ( x v a l s ∗1 e9 , y v a l s , F a c e C o l o r s ( i +sum ( hasTMDC ( 1 : i ) ) , : ) , ' F a c e A l p h a ' ,
FaceAlpha , ' E d g e C o l o r ' , ' none ' ) ;

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else
x v a l s = [ L a y e r C o o r d i n a t e s ( i ) , L a y e r C o o r d i n a t e s ( i +1) , L a y e r C o o r d i n a t e s ( i
+1) , L a y e r C o o r d i n a t e s ( i ) ] ;
p a t c h ( x v a l s ∗1 e9 , y v a l s , F a c e C o l o r s ( i +sum ( hasTMDC ( 1 : i ) ) , : ) , ' F a c e A l p h a ' ,
FaceAlpha , ' E d g e C o l o r ' , ' none ' ) ;

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end
end
x l i m ( [ 0 , L a y e r C o o r d i n a t e s ( end ) ∗1 e9 ] ) ; box on ;
x l a b e l ( ' D i s t a n c e from s u r f a c e o f h e t e r o s t r u c t u r e ( nm ) ' ) ;
y l a b e l ( ' $ | E | ^ 2 / | E_0 | ^ 2 $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
end

i f doGamNR_FWHM
gamma_nr = 5 e − 3 : 1 e − 3 : 5 0 e − 3 ;
f _ l o r e n t z = @(x , x d a t a ) x ( 1 ) . ∗ ( x ( 3 ) . ^ 2 . / ( ( x ( 2 ) − x d a t a ) . ^ 2 + x ( 3 ) . ^ 2 ) ) + x ( 4 ) ;
LB = [ 0 , o m e g a _ e x c _ d e f a u l t . ∗ 0 . 9 5 , 0 , 0 ] ;
UB = [ 1 , o m e g a _ e x c _ d e f a u l t . ∗ 1 . 0 5 , 0 . 1 , 1 ] ;
l s q O p t s = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' S t e p T o l e r a n c e ' , 1 e −18 , '
F u n c t i o n T o l e r a n c e ' , 1 e −18 , ' O p t i m a l i t y T o l e r a n c e ' , 1 e −18 , '
MaxFunctionEvaluations ' , 300) ;
m s _ l s q O p t s = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' S t e p T o l e r a n c e ' , 1 e −6 , '
F u n c t i o n T o l e r a n c e ' , 1 e −6 , ' O p t i m a l i t y T o l e r a n c e ' , 1 e −6 , '
MaxFunctionEvaluations ' , 10) ;
num_MS_startpts = 500;
do_MS = f a l s e ;
% I n i t i a l i z a t i o n of matrices
FWHM = z e r o s ( 1 , l e n g t h ( gamma_nr ) ) ; % t o t a l a b s o r p t i o n
AbsExc = z e r o s ( l e n g t h ( lambda ) , l e n g t h ( gamma_nr ) ) ;
T o t a l A b s = AbsExc ;
fig_gamnr = f i g u r e ;
f o r i d x _ g a m _ n r = 1 : l e n g t h ( gamma_nr )
sigma_gam_nr = f u n _ e x c s i g m a ( g a m m a _ r _ d e f a u l t , gamma_nr ( i d x _ g a m _ n r ) , 0 ,
omega_energy , o m e g a _ e x c _ d e f a u l t ) ; % c o n d u c t i v i t y o f TMDC, i n [ 1 / Ohms ]
% C a l c u l a t e s p e c t r a l l y , m o n o c h r o m a t i c wave a s s u m p t i o n
f o r k = 1 : l e n g t h ( lambda )
J = [1 , 0 ;
0 , 1 ] ; % i n i t a l i z e J as i d e n t i t y matrix
omega = 2∗ p i ∗ c . / lambda ( k ) ; % f r e q u e n c y , u s e f u l f o r n o t r e p e a t i n g
calculations .
% f i r s t l a y e r i s a i r , c o o r d i n a t e s are such t h a t the a i r − d i e l e c t r i c
% stack s t a r t s at z = 0
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k , j ) / lambda ( k ) ;
qp = 2∗ p i ∗ n t i l d e ( k , j + 1 ) / lambda ( k ) ;

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J = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q−omega∗mu0∗ sigma_gam_nr (
k ) ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+
omega∗mu0∗ sigma_gam_nr ( k ) ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q−omega∗mu0∗
sigma_gam_nr ( k ) ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ (
q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+omega∗mu0∗ sigma_gam_nr (
k ) ∗hasTMDC ( j ) ) / qp ) ] ∗ J ;

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end
% r e f l e c t i o n amplitude & transmission
r = −J ( 2 , 1 ) / J ( 2 , 2 ) ;
TotalAbs ( k , idx_gam_nr ) = 1 − abs ( r ) . ^ 2 ; % assuming T = 0
% C o e f f i c i e n t s o f E− f i e l d i n f i r s t l a y e r , e . g . a i r
A_gam_nr = [ 1 ; r ] ;
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k , j ) / lambda ( k ) ;
qp = 2∗ p i ∗ n t i l d e ( k , j + 1 ) / lambda ( k ) ;
J n = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q−omega∗mu0∗ sigma_gam_nr
( k ) ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q+
omega∗mu0∗ sigma_gam_nr ( k ) ∗hasTMDC ( j ) ) / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ ( 1 − ( q−omega∗mu0∗
sigma_gam_nr ( k ) ∗hasTMDC ( j ) ) / qp ) ,
exp ( −1 i ∗ (
q − qp ) ∗ z ( j ) ) ∗ ( 1 + ( q+omega∗mu0∗ sigma_gam_nr (
k ) ∗hasTMDC ( j ) ) / qp ) ] ;

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A_gam_nr = J n ∗ A_gam_nr ; % u p d a t e E− f i e l d c o e f f i c i e n t s
% A b s o r p t i o n i n TMDC
i f hasTMDC ( j )
AbsExc ( k , i d x _ g a m _ n r ) = r e a l ( sigma_gam_nr ( k ) ) / ( c ∗ n t i l d e ( k
, 1 ) ∗ e p s 0 ) ∗ E l e c t r i c I n t e n s i t y ( z ( j ) , A_gam_nr ( 1 ) , A_gam_nr
( 2 ) , qp ) ;
end
end
end
x d a t a = 1 2 3 9 . 8 4 . / ( lambda ∗1 e9 ) ;
y d a t a = r e s h a p e ( TotalAbs ( : , idx_gam_nr ) , s i z e ( x d a t a ) ) ;
i f i d x _ g a m _ n r == 1
x0 = [ max ( y d a t a ) , o m e g a _ e x c _ d e f a u l t , ( g a m m a _ r _ d e f a u l t + gamma_nr (
i d x _ g a m _ n r ) ) / 2 , min ( y d a t a ) ] ;
end
i f do_MS
s t p t s = [ ( UB( 1 ) −LB ( 1 ) ) ∗ r a n d ( n u m _ M S _ s t a r t p t s ,
(UB( 2 ) −LB ( 2 ) ) ∗ r a n d ( n u m _ M S _ s t a r t p t s ,
(UB( 3 ) −LB ( 3 ) ) ∗ r a n d ( n u m _ M S _ s t a r t p t s ,
(UB( 4 ) −LB ( 4 ) ) ∗ r a n d ( n u m _ M S _ s t a r t p t s ,

1)
1)
1)
1)

+ LB ( 1 ) , ...
+ LB ( 2 ) , ...
+ LB ( 3 ) , ...
+ LB ( 4 ) ] ;

s t a r t p t s = CustomStartPointSet ( stp ts ) ;
problem = createOptimProblem ( ' l s q c u r v e f i t ' , ' o b j e c t i v e ' , f _ l o r e n t z ,
x0 ' , x0 , ...

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' xdata ' , xdata , ' ydata ' , ydata ,
ms_lsqOpts ) ;
ms = M u l t i S t a r t ;
x0 = r u n ( ms , problem , s t a r t p t s ) ;

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' ub ' , UB,

' options ' ,

end
x _ f i t = l s q c u r v e f i t ( f _ l o r e n t z , x0 , x d a t a , y d a t a , LB , UB, l s q O p t s ) ;
FWHM( i d x _ g a m _ n r ) = 2∗ x _ f i t ( 3 ) ;
x0 = x _ f i t ;
f i g u r e ( fig_gamnr ) ;
p l o t ( x d a t a , y d a t a , ' k . ' , x d a t a , f _ l o r e n t z ( x _ f i t , x d a t a ) , ' b−− ' ) ;
axis_gamnr = gca ;
x l a b e l ( a x i s _ g a m n r , ' E n e r g y ( eV ) ' ) ;
y l a b e l ( axis_gamnr , ' Absorbance ' ) ;
l e g e n d ( axis_gamnr , { ' S i m u l a t i o n ' , ' F i t ' } , ' L o c a t i o n ' , ' NorthWest ' ) ;
t i t l e ( a x i s _ g a m n r , [ ' \ gamma_{ n r } : ' , n u m 2 s t r ( r o u n d ( gamma_nr ( i d x _ g a m _ n r ) ∗1 e3
) ) , ' meV , ' , ...
' F i t t e d \ gamma_{T } : ' , n u m 2 s t r ( r o u n d (FWHM( i d x _ g a m _ n r ) ∗1 e3 ) ) , ' meV ' ] ) ;
drawnow ;

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end
figure ;
p l o t ( gamma_nr ∗1 e3 , FWHM∗1 e3 , gamma_nr ∗1 e3 , (FWHM − gamma_nr ) ∗1 e3 , ' L i n e W i d t h ' ,
1.5) ;
a x i s s q u a r e ; box on ;
x l a b e l ( ' \ gamma_{ n r } (meV) ' ) ;
h l e g = l e g e n d ( { ' \ gamma_{T} ' , ' \ gamma_{T} − \ gamma_{ n r } ' } , ' L o c a t i o n ' , '
NorthWest ' ) ;
y l a b e l ( ' \ gamma (meV) ' ) ;

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' l b ' , LB ,

% Make some s p e c t r a l a b s o r p t i o n p l o t s
figure ;
omega_peak = o m e g a _ e x c _ d e f a u l t ∗ c e . / h b a r ; % f r e q u e n c y , i n u n i t s o f r a d / s
l a m b d a _ p e a k = 2∗ p i ∗ c . / omega_peak ; % w a v e l e n g t h
k _ p e a k = f i n d ( a b s ( lambda − l a m b d a _ p e a k ) p l o t (FWHM∗1 e3 , T o t a l A b s ( k_peak , : ) , ' k− ' , ' L i n e W i d t h ' , 1 . 5 ) ; h o l d on ;
p l o t (FWHM∗1 e3 , AbsExc ( k_peak , : ) , ' k−− ' , ' L i n e W i d t h ' , 1 . 5 ) ;
a x i s s q u a r e ; box on ;
x l a b e l ( ' \ gamma_T (meV) ' ) ;
y l a b e l ( ' Absorbance ' ) ;

end
f u n c t i o n e x c _ s i g m a = f u n _ e x c _ s i g m a ( gamma_r , gamma_nr , omega , omega_exc )
% a l l omegas and gammas a r e w i t h t h e same u n i t s , e i t h e r i n r a d / s o r e n e r g y .
% which o n l y h a s a d i f f e r e n t o f h b a r , which i s c a n c e l l e d e v e r y w h e r e o f
% relevance
ce = 1.60217656535 e −19; % c h a r g e of e l e c t r o n , [C]
hbar = 1.054571817 e −34; % reduced Planck ' s c o n s t a n t , [ J −s ]
s i g m a _ 0 = c e ^ 2 / ( 4 ∗ h b a r ) ; % c o n d u c t a n c e quantum , [ 1 / Ohm]

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alpha = 0.0072973525693; % f i n e s t r u c t u r e constant , [ u n i t l e s s ]
p0 = gamma_r . / ( 4 ∗ p i ∗ omega_exc ∗ a l p h a ) ; % o s c i l l a t o r s t r e n g t h , [ u n i t l e s s ]
s i g m a _ l o r e n t z = 4∗ s i g m a _ 0 ∗ p0 . ∗ omega ∗1 i . / ( omega − omega_exc + 1 i ∗ gamma_nr / 2 ) ;
exc_sigma = s i g m a _ l o r e n t z ;
end
f u n c t i o n eps = GaS_epsilon (E)
default_fk = 94.0;
default_Ek = 4.86;
default_gammak = 0 . 0 4 9 8 ;
default_eps_bg = 2.187;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;
end
f u n c t i o n e p s = e p s i l o n ( fk , Ek , gammak , E , N, e p s _ b g )
eps = eps_bg ∗ ones ( s i z e (E) ) ;
f o r i d x = 1 :N
e p s = e p s + f k ( i d x ) . / ( Ek ( i d x ) ^2 −E. ^ 2 − 1 i ∗E∗gammak ( i d x ) ) ;
end
end

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Listing D.4: PDE Solver and Fitting for Excitons coupled to Heat
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function fit_and_solve_ExcitonSeebeck_v6
clear ;
close all ;
rng ( ' d e f a u l t ' ) ;
%% Load stroboSCAT Data
s a m p l e _ d a t a = l o a d ( ' s t r o b o S C A T _ e n c a p s _ d a t a . mat ' ) ;
%% G u e s s e s f o r Parameters ( x0 )
muX0 = 0 ; % [ cm ^ 2 / ( V− s ) , e x c i t o n m o b i l i t y ]
S0 = 0 ; % [ uV / K, S e e b e c k c o e f f i c i e n t f o r e x c i t o n s ]
tauT0 = 10; % [ ns ] , h e a t decay time
tauX0 = 1 0 ; % [ n s ] , e x c i t o n l i f e t i m e
RA0 = 0 ; % [ cm ^ 2 / s ] , Auger c o e f f i c i e n t / B i e x c i t o n r e c o m b i n a t i o n r a t e
Pavg0 = 0 . 4 8 ; % [uW] , t i m e − a v e r a g e d pump power , 480 nW f o r 1 . 8 5 e13 g e n e r a t i o n
a l p h a 0 = 1 ; % f i t t i n g p a r a m e t e r t h a t s c a l e s t h e 520 and 705 d a t a . . .
j u s t p l o t x 0 = t r u e ; % i f t r u e , o n l y c a l c u l a t e u s i n g t h e x0 s p e c i f i e d a b o v e
%% M u l t i S t a r t Parameters
doMS = t r u e ; % p e r f o r m m u l t i s t a r t o p t i m i z a t i o n i f t r u e
p l o t M S L i v e = t r u e ; % p l o t t h e " b e s t " v a l u e and " l i v e " v a l u e o f o p t i m i z a t i o n
n u m _ M S _ s t a r t p t s = 1 0 0 0 ; % number o f m u l t i s t a r t p o i n t s
maxfunceval = 50;
OptimTol = 1 e − 6 ;
StepTol = 1e −6;
m a x I t e r = 5 e1 ;
FuncTol = 1e −6;
ms_lsqOpts = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' MaxFunctionEvaluations ' , maxfunceval
, ...
' O p t i m a l i t y T o l e r a n c e ' , OptimTol , ' S t e p T o l e r a n c e ' , S t e p T o l , ...
' M a x I t e r a t i o n s ' , maxIter , ' F u n c t i o n T o l e r a n c e ' , FuncTol ) ;
%% l s q c u r v e f i t Parameters
maxfunceval = 1000;
OptimTol = 1 e − 1 8 ;
StepTol = 1e −15;
m a x I t e r = 1 e4 ;
FuncTol = 1e −12;
l s q O p t s = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' M a x F u n c t i o n E v a l u a t i o n s ' , m a x f u n c e v a l , ...
' O p t i m a l i t y T o l e r a n c e ' , OptimTol , ' S t e p T o l e r a n c e ' , S t e p T o l , ...
' M a x I t e r a t i o n s ' , maxIter , ' F u n c t i o n T o l e r a n c e ' , FuncTol ) ;
%% F i t t i n g Bounds and Parameter D e f i n i t i o n s
r p h y s _ e n d = 3 e − 4 ; % [ cm ]
t p h y s _ e n d = 10 e − 9 ; % [ s ]
xdata = sample_data . position_705 ;
y d a t a = [ s a m p l e _ d a t a . stroboSCAT_520 ( : ) ; s a m p l e _ d a t a . stroboSCAT_705 ( : ) ] ;
f u n = @(x , x d a t a ) f i t f u n c ( x , x d a t a ( : ) , y d a t a ( : ) , s a m p l e _ d a t a . t i m e p t s , r p h y s _ e n d ,

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tphys_end ) ;
LB = [ 1 ,
0,
0.01 ,
0.01 , 0 ,
0 , 0];
UB = [ 1 0 0 ,
1 e3 ,
5,
5,
5 , 50 , 1 0 ] ;
x0 = [ muX0 , S0 , tauT0 , tauX0 , RA0 , Pavg0 , a l p h a 0 ] ;
%% S t a r t t h e o p t i m i z a t i o n and p l o t t i n g
tic ;
if justplotx0
x o p t = x0 ;
else
i f doMS % w r i t e c u s t o m MS
% initialize
i f plotMSLive
fig_ms = f i g u r e ;
end
best_resnorm = inf ;
x b e s t = x0 ;
x_ms = z e r o s ( n u m _ M S _ s t a r t p t s , l e n g t h ( x0 ) ) ;
resnorm_ms = z e r o s ( n u m _ M S _ s t a r t p t s , 1 ) ;
% s t a r t i n g p o i n t s a r e random p o i n t s w i t h i n t h e l e n g t h ( LB ) − d i m e n s i o n a l
s p a c e , bounded by LB , UB
s t p t s = z e r o s ( n u m _ M S _ s t a r t p t s , l e n g t h ( x0 ) ) ;
f o r i d x = 1 : l e n g t h ( x0 )
s t p t s ( : , i d x ) = (UB( i d x ) −LB ( i d x ) ) ∗ r a n d ( n u m _ M S _ s t a r t p t s , 1 ) + LB ( i d x )
end
f o r idx = 1: num_MS_startpts
% t r y running a l o c a l o p t i m i z e r f o r each p o i n t
try
[ x_ms ( i d x , : ) , resnorm_ms ( i d x ) ] = l s q c u r v e f i t ( fun , s t p t s ( i d x , : )
, x d a t a , y d a t a , LB , UB, m s _ l s q O p t s ) ;
catch
resnorm_ms ( i d x ) = i n f ;
end
% i f a new b e s t v a l u e i s found , p l o t and s a v e i t .
i f resnorm_ms ( i d x ) < b e s t _ r e s n o r m
x b e s t = x_ms ( i d x , : ) ;
b e s t _ r e s n o r m = resnorm_ms ( i d x ) ;
plotnewbest = true ;
else
plotnewbest = false ;
end
i f plotMSLive
f i g u r e ( fig_ms ) ;
s e m i l o g y ( i d x , resnorm_ms ( i d x ) , ' o ' , ' M a r k e r E d g e C o l o r ' , ' b ' , '
M a r k e r F a c e C o l o r ' , ' b ' ) ; h o l d on ;
semilogy ( idx , best_resnorm , ' o ' , ' MarkerEdgeColor ' , ' r ' , '
MarkerFaceColor ' , ' r ' ) ;
xlabel ( ' Fitting Iteration ' ) ;

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y l a b e l ( ' Normalized Residual ' ) ;
legend ( ' Live ' , ' Best ' ) ;
box on ;
axis tight ;
drawnow ;

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i f plotnewbest
p l o t F i t t e d D a t a ( xbest , xdata , ydata , sample_data . timepts , 1 ) ;
% p l o t s t h e d a t a , s a v e s i t , and c l o s e s i t
end

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end
% save pa r am e te r s of o p t i m i z a t i o n f o r each run
s a v e ( s t r c a t ( e x t r a c t A f t e r ( m f i l e n a m e , ' S e e b e c k _ ' ) , ' _ M S f i t _ v a l s . mat ' )
, ' idx ' , ' num_MS_startpts ' , ' ms_lsqOpts ' , ' lsqOpts ' , ' rphys_end
' , ' t p h y s _ e n d ' , ...
' LB ' , 'UB ' , ' x d a t a ' , ' y d a t a ' , ' b e s t _ r e s n o r m ' , ' x b e s t ' , '
resnorm_ms ' , ' x_ms ' , ' s t p t s ' ) ;
end
% run f i n a l o p t i m i z a t i o n f o r " f i n e " tuning
x o p t = l s q c u r v e f i t ( fun , x b e s t , x d a t a , y d a t a , LB , UB, l s q O p t s ) ;

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else
% r u n o n l y one o p t i m i z a t i o n
x o p t = l s q c u r v e f i t ( fun , x0 , x d a t a , y d a t a , LB , UB, l s q O p t s ) ;
end
end
disp ( [ ' Total time for optimzation i s :

' , num2str ( toc ) , ' seconds ' ] ) ;

% p l o t t h e f i n a l " o p t i m i z e d " d a t a , o r j u s t " x0 " i f o p t i m i z a t i o n n o t r a n
p l o t F i t t e d D a t a ( x o p t , x d a t a , y d a t a , s a m p l e _ d a t a . t i m e p t s , 0 ) ; % p l o t s t h e d a t a and
saves i t

f u n c t i o n p l o t F i t t e d D a t a ( xopt , xdata , ydata , t i m e p t s , c l o s e f i g s )
%% Draw t h e p l o t s on t o p o f d a t a
y f i t = f i t f u n c ( xopt , xdata , ydata , t i m e p t s , rphys_end , tphys_end ) ;
s t r = { [ ' \ mu_X : ' , n u m 2 s t r ( r o u n d ( x o p t ( 1 ) , 3 ) ) , ' cm ^ 2 / ( V− s ) ' ]
[ ' S : ' , n u m 2 s t r ( r o u n d ( x o p t ( 2 ) , 3 ) ) , ' \ muV / K ' ]
[ ' \ tau_T : ' , num2str ( round ( xopt ( 3 ) , 3 ) ) , ' ns ' ]
[ ' \ tau_X : ' , n u m 2 s t r ( r o u n d ( x o p t ( 4 ) , 3 ) ) , ' n s ' ]
[ ' R_A : ' , n u m 2 s t r ( r o u n d ( x o p t ( 5 ) , 3 ) ) , ' cm ^ 2 / s ' ]
[ ' P_ { avg } : ' , n u m 2 s t r ( r o u n d ( x o p t ( 6 ) , 3 ) ) , ' \muW ' ]
[ ' \ alpha : ' , num2str ( round ( xopt ( 7 ) ,3) ) ] } ;
line_colors = linspecer ( length ( timepts ) ) ;
fig520 = figure ;
f i g 5 2 0 . P o s i t i o n = [200 , 50 , 600 , 5 0 0 ] ;
for idx2 = 1: length ( timepts )
y l i n e _ d a t a = y d a t a ( ( idx2 −1) ∗ l e n g t h ( x d a t a ) +1: idx2 ∗ l e n g t h ( x d a t a ) ) ;
y l i n e _ f i t = y f i t ( ( idx2 −1) ∗ l e n g t h ( x d a t a ) +1: idx2 ∗ l e n g t h ( x d a t a ) ) ;
h = subplot (4 ,3 , idx2 ) ;
p l o t ( xdata , yline_data ,

' o ' , ' C o l o r ' , l i n e _ c o l o r s ( i d x 2 , : ) ) ; h o l d on ;

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p l o t ( xdata , y l i n e _ f i t , ' − ' , ' Color ' , l i n e _ c o l o r s ( idx2 , : ) ) ;
t i t l e ( h , [ num2str ( t i m e p t s ( idx2 ) ) , ' ns ' ] ) ;
pos = g e t ( h , ' P o s i t i o n ' ) ;
s e t ( h , ' P o s i t i o n ' , [ p o s ( 1 ) + 0 . 0 6 , p o s ( 2 ) − 0 . 0 5 , p o s ( 3 : end ) ] ) ;
end
dim = [ 0 , 0 , 0 . 5 , 1 ] ;
a n n o t a t i o n ( ' t e x t b o x ' , dim , ' F o n t S i z e ' , 7 , ' S t r i n g ' , s t r , ' i n t e r p r e t e r ' , '
t e x ' , ' L i n e S t y l e ' , ' none ' ) ;
s g t i t l e ( ' 520 nm P r o b e ' ) ;
try
s a v e a s ( gcf , s t r c a t ( e x t r a c t A f t e r ( mfilename , ' Seebeck_ ' ) , ' _ 5 2 0 _ f i t ' ) , '
png ' ) ;
catch
d i s p ( [ ' F i l e ' , s t r c a t ( e x t r a c t A f t e r ( mfilename , ' Seebeck_ ' ) , ' _ 5 2 0 _ f i t .
png ' ) , ' d i d n o t s a v e ' ] ) ;
end
fig705 = figure ;
f i g 7 0 5 . P o s i t i o n = [200 , 50 , 600 , 5 0 0 ] ;
f o r idx2 = l e n g t h ( t i m e p t s ) +1:2∗ l e n g t h ( t i m e p t s )
y l i n e _ d a t a = y d a t a ( ( idx2 −1) ∗ l e n g t h ( x d a t a ) +1: idx2 ∗ l e n g t h ( x d a t a ) ) ;
y l i n e _ f i t = y f i t ( ( idx2 −1) ∗ l e n g t h ( x d a t a ) +1: idx2 ∗ l e n g t h ( x d a t a ) ) ;
h = subplot (4 ,3 , idx2 − length ( timepts ) ) ;
p l o t ( xdata , y l i n e _ d a t a , ' o ' , ' Color ' , l i n e _ c o l o r s ( idx2 − l e n g t h ( t i m e p t s )
, : ) ) ; h o l d on ;
p l o t ( xdata , y l i n e _ f i t , ' − ' , ' Color ' , l i n e _ c o l o r s ( idx2 − l e n g t h ( t i m e p t s )
,:) ) ;
t i t l e ( h , [ num2str ( t i m e p t s ( idx2 − l e n g t h ( t i m e p t s ) ) ) , ' ns ' ] ) ;
pos = g e t ( h , ' P o s i t i o n ' ) ;
s e t ( h , ' P o s i t i o n ' , [ p o s ( 1 ) + 0 . 0 6 , p o s ( 2 ) − 0 . 0 5 , p o s ( 3 : end ) ] ) ;
end
dim = [ 0 , 0 , 0 . 5 , 1 ] ;
a n n o t a t i o n ( ' t e x t b o x ' , dim , ' F o n t S i z e ' , 7 , ' S t r i n g ' , s t r , ' i n t e r p r e t e r ' , '
t e x ' , ' L i n e S t y l e ' , ' none ' ) ;
s g t i t l e ( ' 705 nm P r o b e ' ) ;
try
s a v e a s ( gcf , s t r c a t ( e x t r a c t A f t e r ( mfilename , ' Seebeck_ ' ) , ' _ 7 0 5 _ f i t ' ) , '
png ' ) ;
catch
d i s p ( [ ' F i l e ' , s t r c a t ( e x t r a c t A f t e r ( mfilename , ' Seebeck_ ' ) , ' _ 7 0 5 _ f i t .
png ' ) , ' d i d n o t s a v e ' ] ) ;
end

%% Examine " Real " P o p u l a t i o n s from c a l c u l a t i o n s
[ Nout , Tout , r o u t , t o u t , t 0 _ i d x , t 0 _ s i m , ~ , ~ ] = s o l v e E x c i t o n S e e b e c k (
rphys_end , tphys_end , xopt ) ;
Nout = Nout ( t 0 _ i d x : end , : ) ;
T o u t = T o u t ( t 0 _ i d x : end , : ) ;
t o u t = t o u t ( t 0 _ i d x : end ) − t 0 _ s i m ∗1 e9 ; % [ n s ]
excpop = f i g u r e ;
for idx2 = 1: length ( timepts )
N _ p h y s _ s l i c e = i n t e r p 2 ( r o u t , t o u t , Nout , x d a t a , t i m e p t s ( i d x 2 ) ) ;

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h = subplot (4 ,3 , idx2 ) ;
p l o t ( xdata , N_phys_slice , ' − ' , ' Color ' , l i n e _ c o l o r s ( idx2 , : ) , '
LineWidth ' , 2) ;
t i t l e ( h , [ num2str ( t i m e p t s ( idx2 ) ) , ' ns ' ] ) ;

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end
s g t i t l e ( ' E x c i t o n P o p u l a t i o n ( 1 / cm ^ { 2 } ) ' ) ;
try
s a v e a s ( excpop , s t r c a t ( e x t r a c t A f t e r ( m f i l e n a m e ,
_ f i t t e d _ e x c ' ) , ' png ' ) ;
catch
d i s p ( [ ' F i l e ' , s t r c a t ( e x t r a c t A f t e r ( mfilename ,
_ f i t t e d _ e x c . png ' ) , ' d i d n o t s a v e ' ] ) ;
end

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' Seebeck_ ' ) , '

temppop = f i g u r e ;
for idx2 = 1: length ( timepts )
T _ p h y s _ s l i c e = i n t e r p 2 ( r o u t , t o u t , Tout , x d a t a , t i m e p t s ( i d x 2 ) ) ;
h = subplot (4 ,3 , idx2 ) ;
p l o t ( xdata , T_p hys _sl ic e , ' − ' , ' Color ' , l i n e _ c o l o r s ( idx2 , : ) , '
LineWidth ' , 2) ;
t i t l e ( h , [ num2str ( t i m e p t s ( idx2 ) ) , ' ns ' ] ) ;
y t i c k f o r m a t ( ' %.1 f ' ) ;
end
s g t i t l e ( ' T e m p e r a t u r e (K) ' ) ;
try
s a v e a s ( temppop , s t r c a t ( e x t r a c t A f t e r ( m f i l e n a m e , ' S e e b e c k _ ' ) , '
_ f i t t e d _ t e m p ' ) , ' png ' ) ;
catch
d i s p ( [ ' F i l e ' , s t r c a t ( e x t r a c t A f t e r ( mfilename , ' Seebeck_ ' ) , '
_ f i t t e d _ t e m p . png ' ) , ' d i d n o t s a v e ' ] ) ;
end

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' Seebeck_ ' ) , '

if closefigs
close ( fig520 ) ;
close ( fig705 ) ;
c l o s e ( excpop ) ;
c l o s e ( temppop ) ;
end
end

f u n c t i o n y = f i t f u n c ( x , xdata , ydata , t p t s , rphys_end , tphys_end )
a l p h a = x ( end − 1 ) ; % u s e d f o r f i t t i n g
[ Nphys , Tphys , r p h y s , t p h y s , z e r o _ i d x , t 0 , N0 , T0 ] = s o l v e E x c i t o n S e e b e c k (
rphys_end , tphys_end , x ) ;
t p h y s _ n e w = t p h y s ( z e r o _ i d x : end ) − t 0 ∗1 e9 ; % [ n s ]
%% 520 nm C a l c u l a t i o n
s p a c e _ c o n v o _ 5 2 0 = 0 . 4 5 ∗ 0 . 5 2 / 1 . 4 ; % ( 0 . 4 5 ) ∗ w a v e l e n g t h . / NA
t i m e _ c o n v o _ 5 2 0 = 0 . 1 0 8 . / 2 . 3 5 5 ; % [ n s ] , FWHM = 108 p s

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% Time C o n v o l u t i o n
Tconvo520 = g a u s s _ t i m e _ c o n v o ( t p h y s , r p h y s , Tphys / T0 −1 , t i m e _ c o n v o _ 5 2 0 ) ;
% R e d e f i n e t = 0 t o be a t t 0 a f t e r t i m e c o n v o l u t i o n
Tconvo520 = Tconvo520 ( z e r o _ i d x : end , : ) ;
% Space c o n v o l u t i o n
Tconvo520 = g a u s s _ s p a c e _ c o n v o ( tphys_new , r p h y s , Tconvo520 , s p a c e _ c o n v o _ 5 2 0
);

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% S a m p l i n g t o be t h e same a s t h e d a t a s e t
[ xmesh , t m e s h ] = m e s h g r i d ( x d a t a , t p t s ) ;
T t _ 5 2 0 = i n t e r p 2 ( r p h y s , tphys_new , Tconvo520 , xmesh , t m e s h ) ;
T t _ 5 2 0 = Tt_520 ' ;
Tt_520 = Tt_520 ( : ) ;

b e t a = − t r a p z ( xdata , y d a t a ( 1 : l e n g t h ( x d a t a ) ) ) . / t r a p z ( xdata , Tt_520 ( 1 : l e n g t h (
x d a t a ) ) ) ; % Assume b e t a t o be t h e f i r s t e q u a l t o t h e p o i n t a t f i r s t
time point / space point
I520 = − b e t a ∗ Tt_520 ;

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%% 705 nm C a l c u l a t i o n
s p a c e _ c o n v o _ 7 0 5 = 0 . 4 5 ∗ 0 . 7 0 5 / 1 . 4 ; % ( 0 . 4 5 ) ∗ w a v e l e n g t h . / NA
t i m e _ c o n v o _ 7 0 5 = 0 . 6 0 . / 2 . 3 5 5 ; % [ n s ] , FWHM = 60 p s
% Time C o n v o l u t i o n
Tconvo705 = g a u s s _ t i m e _ c o n v o ( t p h y s , r p h y s , Tphys / T0 −1 , t i m e _ c o n v o _ 7 0 5 ) ;
Nconvo705 = g a u s s _ t i m e _ c o n v o ( t p h y s , r p h y s , Nphys / N0 , t i m e _ c o n v o _ 7 0 5 ) ; %
n e e d t o make s u r e t p h y s and p r o b e _ p u l s e _ t i m e _ w i d t h h a v e same u n i t s

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% R e d e f i n e t = 0 t o be a t t 0 a f t e r t i m e c o n v o l u t i o n
Nconvo705 = Nconvo705 ( z e r o _ i d x : end , : ) ;
Tconvo705 = Tconvo705 ( z e r o _ i d x : end , : ) ;
% Space c o n v o l u t i o n
Nconvo705 = g a u s s _ s p a c e _ c o n v o ( tphys_new , r p h y s , Nconvo705 , s p a c e _ c o n v o _ 7 0 5
) ; % space convolution
Tconvo705 = g a u s s _ s p a c e _ c o n v o ( tphys_new , r p h y s , Tconvo705 , s p a c e _ c o n v o _ 7 0 5
) ; % space convolution

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% S a m p l i n g t o be t h e same a s t h e d a t a s e t
Nt_705 = i n t e r p 2 ( r p h y s , tphys_new , Nconvo705 , xmesh , t m e s h ) ;
T t _ 7 0 5 = i n t e r p 2 ( r p h y s , tphys_new , Tconvo705 , xmesh , t m e s h ) ;
% t u r n i n t o column v e c t o r w i t h t h e t i m e t r a c e s s t a c k e d v e r t i c a l l y
Nt_705 = Nt_705 ' ;
Nt_705 = Nt_705 ( : ) ;
T t _ 7 0 5 = Tt_705 ' ;
Tt_705 = Tt_705 ( : ) ;
numDatapts = l e n g t h ( x d a t a ) ∗ l e n g t h ( t p t s ) +1;

gamma = t r a p z ( x d a t a , y d a t a ( n u m D a t a p t s : numDatapts −1+ l e n g t h ( x d a t a ) ) ) . /
t r a p z ( x d a t a , Nt_705 ( 1 : l e n g t h ( x d a t a ) ) − a l p h a ∗ b e t a ∗ T t _ 7 0 5 ( 1 : l e n g t h ( x d a t a ) ) ) ;

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I 7 0 5 = gamma ∗ ( Nt_705 − a l p h a ∗ b e t a ∗ T t _ 7 0 5 ) ; % gamma h e r e i s l i k e t h e
a l p h a p a r a m e t e r from Hannah

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gamma = t r a p z ( x d a t a , y d a t a ( n u m D a t a p t s : numDatapts −1+ l e n g t h ( x d a t a ) ) + a l p h a ∗
b e t a ∗ T t _ 7 0 5 ( 1 : l e n g t h ( x d a t a ) ) ) . / t r a p z ( x d a t a , Nt_705 ( 1 : l e n g t h ( x d a t a ) ) ) ;
I 7 0 5 = gamma∗ Nt_705 − a l p h a ∗ b e t a ∗ T t _ 7 0 5 ; % gamma h e r e i s l i k e t h e a l p h a
p a r a m e t e r from Hannah

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y = [ I 5 2 0 ; I 7 0 5 ] ; % 520 d a t a , f o l l o w e d by 705 d a t a , a l l t i m e d a t a p o i n t s
end

f u n c t i o n [ Nphys , Tphys , r p h y s , t p h y s , z e r o _ i d x , t 0 , N0 , T0 ] =
solveExcitonSeebeck ( rphys_end , tphys_end , x )

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% R e w r i t e s o t h a t most numbers a r e o f o r d e r 1 . . .
muX = x ( 1 ) ; % [ cm ^ 2 / ( V− s ) ] , e x c i t o n m o b i l i t y
S = x ( 2 ) ∗1 e − 6 ; % [V / K] , S e e b e c k c o e f f i c e i t n f o r e x c i t o n s
t a u T = x ( 3 ) ∗1 e − 9 ; % [ s ] , h e a t d e c a y t i m e
tauX = x ( 4 ) ∗1 e − 9 ; % [ s ] , e x c i t o n l i f e t i m e
RA = x ( 5 ) ; % [ cm ^ 2 / s ] , Auger − M e i t n e r / B i e x c i t o n R a t e
Pavg = x ( 6 ) ∗1 e − 6 ; % [W] , Time − a v e r a g e d pump power
%% Fundamental C o n s t a n t s
kb = 1 . 3 8 e − 2 3 ; % [ J / K, B o l t z m a n n C o n s t a n t ]
q = 1 . 6 e −19; % [C, Fundamental u n i t of Charge ]
%% Assumed F i x e d Parameters
DT = 0 . 1 5 ; % [ cm ^ 2 / s ] , h e a t d i f f u s i v i t y
h e a t _ c a p = 0 . 5 4 e − 6 ; % [ J / ( cm^2 −K) ] [ mass h e a t c a p ∗ d e n s i t y ∗ t h i c k n e s s ]
E p h o t o n = 1 2 3 9 . 8 / 4 4 0 ∗ q ; % 440 nm Exc , [ J ]
T0 = 3 0 0 ; % [K, e q u i l i b r i u m t e m p e r a t u r e ]
Eg = 1 . 3 8 ; % [ eV ]
Abs = 0 . 3 5 ; % u n i t l e s s a b s o r b a n c e , a t l a s e r w a v e l e n g t h
f r e p = 16 e6 ; % 16 MHz, [ Hz o r 1 / s ]
LT = s q r t (DT∗ t a u T ) ; % [ cm ] , t h e r m a l d i f f u s i o n l e n g t h
l a m b d a _ e x c = 440 e − 7 ; % [ cm ]
NA = 1 . 4 ;
p u m p _ p u l s e _ p h y s _ w i d t h = 0 . 4 5 ∗ l a m b d a _ e x c /NA; % [ cm ] , l a s e r p u l s e w i d t h
s i g p r = p u m p _ p u l s e _ p h y s _ w i d t h / LT ; % [ u n i t l e s s , b u t f a c t o r s o f s q r t (DT∗ t a u T
) , equivalent to spot size ]
s i g p t = 72 e − 1 2 . / 2 . 3 5 5 ; % [ s ] , pump p u l s e w i d t h , i n t i m e , FWHM = 72 p s

%% D e r i v e d Parameters
N0 = 1 . 8 5 e13 ; % Pavg ∗Abs / ( E p h o t o n ∗ f r e p ) / ( 2 ∗ p i ∗ p u m p _ p u l s e _ p h y s _ w i d t h ^ 2 ) ;
% [ 1 / cm ^ 2 , i n t e g r a t e d c a r r i e r d e n s i t y e x c i t e d by l a s e r ]
N0 = Pavg ∗Abs / ( E p h o t o n ∗ f r e p ) / ( 2 ∗ p i ∗ p u m p _ p u l s e _ p h y s _ w i d t h ^ 2 ) ; % [ 1 / cm ^ 2 ,
i n t e g r a t e d c a r r i e r d e n s i t y e x c i t e d by l a s e r ]
x i = muX∗ kb ∗T0 / ( q∗DT) ; % [ u n i t l e s s ] , r a t i o o f e x c i t o n / h e a t d i f f u s i v i t y
c h i = t a u T / tauX ; % [ u n i t l e s s ] , r a t i o o f l i f e t i m e s
p s i = RA∗N0∗ t a u T ; % [ u n i t l e s s ] , f r a c t i o n o f c a r r i e r s l o s s v i a Auger d u r i n g
heat lifetime

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z e t a = q∗Eg∗N0 / ( h e a t _ c a p ∗T0 ) ; % [ u n i t l e s s ] , a v e r a g e h e a t g e n e r a t i o n
r e l a t i v e t o T0
z e t a 2 = ( E p h o t o n − q∗Eg ) ∗N0 / ( h e a t _ c a p ∗T0 ) ; % [ u n i t l e s s ] , d i r e c t g e n e r a t i o n
o f h e a t due t o t h e r m a l i z a t i o n
s = q∗S / kb ; % [ = q ^2∗ S / kb ] , i . e . , i s u n i t l e s s . Reduced S e e b e c k c o e f f ]

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%% S i m u l a t i o n Parameters / Range
t o t _ t _ p t s = 1001;
t o t _ r _ p t s = 51;
r s o l = l i n s p a c e ( 0 , r p h y s _ e n d / LT , t o t _ r _ p t s ) ; % [ u n i t l e s s , i n u n i t s o f
t 0 = 1000 e − 1 2 ; % [ s ] , c e n t e r o f p u l s e i n t i m e
t_end = ( tphys_end + t0 ) / tauT ; % [ u n i t l e s s , in u n i t s of tauT ]
numt1pts = round ( t 0 / tauT / t_end ∗( t o t _ t _ p t s −1) ) +1;
numt2pts = round ( ( t_end − t 0 / tauT ) / t_end ∗( t o t _ t _ p t s −1) ) +1;
t 1 = l i n s p a c e ( 0 , t 0 / tauT , n u m t 1 p t s ) ; % [ u n i t l e s s , i n u n i t s o f t a u T ]
t 2 = l i n s p a c e ( t 0 / tauT , t _ e n d , n u m t 2 p t s ) ;
t s o l = [ t 1 , t 2 ( 2 : end ) ] ;
m = 1 ; % c y l i n d r i c a l symmetry
%% S o l v e PDE i n two p a r t s , f o r t h e p u l s e r i s e and f a l l .
p d e o p t i o n s = o d e s e t ( ' R e l T o l ' , 2 . 2 2 2 e −14 , ' AbsTol ' , 2 . 2 2 2 e − 1 4 ) ;
s o l 1 = p d e p e (m, @(r , t , u , dudx ) s e e b e c k p d e ( r , t , u , dudx , tauT , t 0 , s i g p r ,
s i g p t , x i , c h i , p s i , z e t a , z e t a 2 , s ) , ...
@s e e b e c k i c , @s e e b e c k b c , r s o l , t 1 , p d e o p t i o n s ) ; % pde s o l v e r
s o l 2 = p d e p e (m, @(r , t , u , dudx ) s e e b e c k p d e ( r , t , u , dudx , tauT , t 0 , s i g p r ,
s i g p t , x i , c h i , p s i , z e t a , z e t a 2 , s ) , ...
@( r ) s e e b e c k i c 2 ( r , s o l 1 , r s o l ) , @s e e b e c k b c , r s o l , t 2 , p d e o p t i o n s ) ; %
pde s o l v e r

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LT ]

%% Convert S o l u t i o n t o P h y s i c a l U n i t s
u1 = [ s o l 1 ( : , : , 1 ) ; s o l 2 ( 2 : end , : , 1 ) ] ;
u2 = [ s o l 1 ( : , : , 2 ) ; s o l 2 ( 2 : end , : , 2 ) ] ;
r p h y s = r s o l ∗LT∗1 e4 ; % [ um ]
t p h y s = t s o l ∗ t a u T ∗1 e9 ; % [ n s ]
Nphys = u1 ∗N0 ; % [ 1 / cm ^ 2 ]
Tphys = u2 ∗T0 ; % [K]
zero_idx = numt1pts ;
end

f u n c t i o n [ C , F , S ] = s e e b e c k p d e ( r , t , u , dudx , tauT , t 0 , s i g p r , s i g p t , x i , c h i , p s i
, zeta , zeta2 , s )
tp = t .∗ tauT ; % [ s ] , r e a l time
Gen = ( 1 . / s q r t ( 2 . ∗ p i . ∗ s i g p t ^ 2 ) ) . ∗ exp ( − r . ^ 2 / ( 2 . ∗ s i g p r ^ 2 ) ) . ∗ exp ( − ( t p − t 0 )
. ^ 2 . / ( 2 . ∗ s i g p t ^2) ) ; % [ 1 / s ]
C = [1; 1];
F = [ x i . ∗ u ( 2 ) . ∗ dudx ( 1 ) + s . ∗ x i . ∗ u ( 1 ) . ∗ dudx ( 2 ) ;
dudx ( 2 ) ] ;
S = [ − ( 1 + s ) . ∗ x i . ∗ dudx ( 1 ) . ∗ dudx ( 2 ) + Gen . ∗ t a u T − c h i . ∗ u ( 1 ) − p s i . ∗ u ( 1 ) . ^ 2 ;
z e t a . ∗ c h i . ∗ u ( 1 ) + z e t a . ∗ p s i . ∗ u ( 1 ) . ^ 2 − ( u ( 2 ) −1) + z e t a 2 . ∗ Gen . ∗ t a u T ] ;
% pump p u l s e w i d t h c a n be t h e t 0 . . .

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end
f u n c t i o n u0 = s e e b e c k i c ( r )
u0 = [ 0 ; 1 ] ;
end
f u n c t i o n u0 = s e e b e c k i c 2 ( r , s o l , r s o l )
u0 = [ i n t e r p 1 ( r s o l , s o l ( end , : , 1 ) , r ) ; i n t e r p 1 ( r s o l , s o l ( end , : , 2 ) , r ) ] ;
end
function
pl =
ql =
pr =
qr =
end

f u n c t i o n out = gauss_time_convo ( t p t s , r p t s , n , sigma_convo )
n = reshape (n , length ( t p t s ) , length ( rpts ) ) ;
out = zeros ( s i z e ( n ) ) ;
t 0 _ c t r = mean ( t p t s ) ; % a d d e d s o t h e g a u s s i a n i s n o r m a l i z e d when i n t e g r a t e d
over t
k e r n e l = 1 / s q r t ( 2 ∗ p i ∗ s i g m a _ c o n v o . ^ 2 ) . ∗ exp ( − ( t p t s − t 0 _ c t r ) . ^ 2 . / ( 2 ∗
sigma_convo . ^ 2 ) ) ;
DelT = mean ( d i f f ( t p t s ) ) ;
for i = 1: length ( rpts )
o u t ( : , i ) = conv ( k e r n e l , n ( : , i ) , ' same ' ) . ∗ DelT ;
end
end

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[ p l , q l , pr , q r ] = s e e b e c k b c ( r l , u l , r r , ur , t )
[0; 0]; % ignored since m = 1;
[0; 0]; % ignored since m = 1;
[0; 0];
[1; 1];

f u n c t i o n out = gauss_space_convo ( t p t s , r p t s , n , sigma_convo )
n = reshape (n , length ( t p t s ) , length ( rpts ) ) ;
r 0 _ c t r = mean ( r p t s ) ;
k e r n e l = 1 . / s q r t ( 2 ∗ p i ∗ s i g m a _ c o n v o . ^ 2 ) . ∗ exp ( − ( r p t s − r 0 _ c t r ) . ^ 2 . / ( 2 ∗
sigma_convo . ^ 2 ) ) ;
DelR = mean ( d i f f ( r p t s ) ) ;
nconv = z e r o s ( l e n g t h ( t p t s ) , 2∗ l e n g t h ( r p t s ) −1) ;
for i = 1: length ( t p t s )
nconv ( i , : ) = conv ( k e r n e l , n ( i , : ) , ' f u l l ' ) . ∗ DelR ;
end
[ ~ , max_idx ] = max ( max ( nconv ) ) ;
o u t = nconv ( : , max_idx : max_idx + l e n g t h ( r p t s ) −1) ;
end
end

274
Listing D.5: Sample Code for Controlling Leica Microscope to Perform Reflection
Contrast Measurements and Subsequent Fitting
1 % T h i s p r o g r a m u s e s m i c r o m a n a g e r t o c o n t r o l t h e ASI s t a g e , L e i c a m i c r o s c o p e ,
2 % and C o b o l t l a s e r , and u s e s l i g h t f i e l d t o t a k e a s p e c t r a . I f a r e f e r e n c e
3 % measurement or d a t a f i l e i s given , then t h e p l o t t e d r e f l e c t i v i t y spectrum
4 % i s a p p r o p r i a t e l y n o r m a l i z e d , b u t t h e d a t a t h a t i s s a v e d i s t h e raw s p e c t r u m .
5 % Other p r o p e r t i e s r e l a t e d t o t h e measurement a r e a l s o saved whenever p o s s i b l e .
6 %
7 % W r i t t e n by J o e s o n Wong
8 % L a s t U p d a t e d O c t o b e r 6 , 2021
9 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
10 c l o s e a l l ;
11
12 %% −−−−−−−−−−−−−−Measurement parameters −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
13 e x p o s u r e _ t i m e = 3 5 0 ; % e x p o s u r e t i m e i n ms
14 num_frames = 2 0 ; % number o f m e a s u r e m e n t f r a m e s
15 R e f l _ f i l t e r _ c u b e _ p o s = 0 ; % 0 i s BF ( t o p o f L e i c a B u t t o n s ) , and g o e s down
i n c r e m e n t a l l y u n t i l 5 ( empty ) .
16 l f _ e x p e r i m e n t _ n a m e = ' PL_PIXIS ( VIS ) _300gpmm_BL500_JWong_Reflection . l f e ' ;
17
18 % r e f l 0 ( e i t h e r t a k e a r e f l 0 f i r s t o r u s e a s p e c i f i e d r e f l 0 d a t a f i l e )
19 m e a s _ r e f l 0 = 1 ; % e i t h e r 0 f o r no r e f l 0 measurement , 1 t o r e c o r d r e f l 0 , 2 t o u s e a
specific refl0 file / directory
20 r e f l 0 _ d i r e c t o r y = ' C : \ U s e r s \ Ateam \ D e s k t o p \ U s e r s \ J o e s o n \ 1 0 − 2 1 − 2 0 2 0 \ F o r _ P i c k u p ' ; %
o n l y u s e d when meas_bg = 2
21 r e f l 0 _ f i l e n a m e = '
Ag_refl_25umExc_105umColl_50xLWD_no425nmLP_70msExp_1frame_run3_2020_10_21_20_16
. c s v ' ; % o n l y u s e d when meas_bg = 2
22 r e f l 0 _ w a v e s _ i d x = 3 ;
23 r e f l 0 _ c o u n t s _ i d x = 6 ;
24
25 meas_bg = 2 ; % e i t h e r 0 f o r no bg measurement , 1 t o r e c o r d bg , 2 t o u s e a s p e c i f i c
bg f i l e / d i r e c t o r y
26 b g _ d i r e c t o r y = ' C : \ U s e r s \ Ateam \ D e s k t o p \ U s e r s \ J o e s o n \ 0 1 − 1 7 − 2 0 2 1 \ old_GaS_PDMS ' ; %
o n l y u s e d when meas_bg = 2
27 b g _ f i l e n a m e = ' bg_1 . c s v ' ; % o n l y u s e d when meas_bg = 2
28 b g _ w a v e s _ i d x = 3 ;
29 b g _ c o u n t s _ i d x = 6 ;
30 num_bg_frames = 2 0 ;
31
32 w a v e _ f i t _ m i n = 4 0 0 ; % [ nm ]
33 w a v e _ f i t _ m a x = 8 0 0 ; % [ nm ]
34 numCameraXPixels = 1 3 4 0 ;
35
36 % J o g g i n g P a r a m e t e r s
37 X j o g s t e p = 5 ; % [ um ]
38 Y j o g s t e p = 5 ; % [ um ]
39 Z j o g s t e p = 1 ; % [ um ]
40 S t a g e S p e e d = 100 e − 3 ; % [mm/ s ]
41
42 % F i t t i n g P a r a m e t e r s
43 t _ t h i n F i l m _ m i n = 0 ; % [ nm ]
44 t _ t h i n F i l m _ m a x = 5 ; % [ nm ]
45

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46 t _ o v e r l a y e r _ m i n = 2 7 0 ;
47 t _ o v e r l a y e r _ m a x = 3 0 0 ;
48
49 n u m _ M S _ s t a r t p t s = 5 0 0 ; % number o f p o i n t s f o r m u l t i s t a r t o p t i m i z a t i o n
50 % t _ o v e r l a y e r = 2 8 5 ; % [ nm ]
51
52
53 %−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
54
55 %% I n i t i a l i z e Micromanager and L i g h t f i e l d
56 i f ~ e x i s t ( 'mmc ' , ' v a r ' )
57
d i s p ( ' I n i t i a l i z i n g Equipment C o n n e c t i o n s . . . ' ) ;
58
m i c r o m a n a g e r _ p a t h = ' C : \ Program F i l e s \ Micro −Manager − 2 . 0 gamma ' ;
59
d e f a u l t S t a g e S p e e d = 0 . 0 0 5 ; % [mm/ s ]
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d e f a u l t S t a g e B a c k l a s h = 0 ; % [mm]
61
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addpath ( micromanager_path ) ;
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i m p o r t mmcorej . ∗ ;
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mmc = CMMCore ;
65
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%% Set −up S t a g e
67
mmc . l o a d D e v i c e ( " P2 " , " S e r i a l M a n a g e r " , "COM2" ) ; % s e t u p s e r i a l p o r t
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mmc . l o a d D e v i c e ( " XYStage " , " A S I S t a g e " , " XYStage " ) ; % s e t u p s t a g e
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mmc . s e t P r o p e r t y ( " XYStage " , " P o r t " , " P2 " ) ; % c o n n e c t p o r t t o s t a g e
70
71
%% Set −up L e i c a Microscope
72
mmc . l o a d D e v i c e ( " P12 " , " S e r i a l M a n a g e r " , "COM12 " ) ;
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mmc . l o a d D e v i c e ( " Scope " , " LeicaDMI " , " Scope " ) ;
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mmc . l o a d D e v i c e ( " IL − T u r r e t " , " LeicaDMI " , " IL − T u r r e t " ) ;
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mmc . l o a d D e v i c e ( " O b j e c t i v e T u r r e t " , " LeicaDMI " , " O b j e c t i v e T u r r e t " ) ;
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mmc . l o a d D e v i c e ( " F o c u s D r i v e " , " LeicaDMI " , " F o c u s D r i v e " ) ;
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mmc . l o a d D e v i c e ( " S i d e P o r t " , " LeicaDMI " , " S i d e P o r t " ) ;
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mmc . l o a d D e v i c e ( " IL − S h u t t e r " , " LeicaDMI " , " IL − S h u t t e r " ) ;
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mmc . l o a d D e v i c e ( " TL− S h u t t e r " , " LeicaDMI " , "TL− S h u t t e r " ) ;
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mmc . l o a d D e v i c e ( " T r a n s m i t t e d L i g h t " , " LeicaDMI " , " T r a n s m i t t e d L i g h t " ) ;
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mmc . s e t P r o p e r t y ( " P12 " , " AnswerTimeout " , 5 0 0 ) ;
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mmc . s e t P r o p e r t y ( " P12 " , " BaudRate " , 1 9 2 0 0 ) ;
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mmc . s e t P r o p e r t y ( " P12 " , "DTR" , " D i s a b l e " ) ;
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mmc . s e t P r o p e r t y ( " P12 " , " F a s t USB t o S e r i a l " , " D i s a b l e " ) ;
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mmc . s e t P r o p e r t y ( " P12 " , " S t o p B i t s " , 1 ) ;
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mmc . s e t P r o p e r t y ( " P12 " , " P a r i t y " , " None " ) ;
87
mmc . s e t P r o p e r t y ( " P12 " , " D a t a B i t s " , 8 ) ;
88
mmc . s e t P r o p e r t y ( " P12 " , " DelayBetweenCharsMs " , 0 )
89
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mmc . s e t P r o p e r t y ( " Scope " , " AnswerTimeOut " , 2 5 0 ) ;
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mmc . s e t P r o p e r t y ( " Scope " , " P o r t " , " P12 " ) ;
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mmc . s e t P a r e n t L a b e l ( " IL − T u r r e t " , " Scope " ) ;
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mmc . s e t P a r e n t L a b e l ( " O b j e c t i v e T u r r e t " , " Scope " ) ;
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mmc . s e t P a r e n t L a b e l ( " F o c u s D r i v e " , " Scope " ) ;
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mmc . s e t P a r e n t L a b e l ( " S i d e P o r t " , " Scope " ) ;
96
97
%% Set −up XCite
98
mmc . l o a d D e v i c e ( " P11 " , " S e r i a l M a n a g e r " , "COM11 " ) ;
99
mmc . l o a d D e v i c e ( " X− C i t e " , " XCiteXT600 " , " Turbo / XT600 C o n t r o l l e r " ) ;
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mmc . l o a d D e v i c e ( " LED1 " , " XCiteXT600 " , "LED1 D e v i c e " ) ;

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mmc . s e t P r o p e r t y ( " P11 " , " AnswerTimeout " , 5 0 0 ) ;
mmc . s e t P r o p e r t y ( " P11 " , " BaudRate " , 1 9 2 0 0 ) ;
mmc . s e t P r o p e r t y ( " P11 " , "DTR" , " D i s a b l e " ) ;
mmc . s e t P r o p e r t y ( " P11 " , " F a s t USB t o S e r i a l " , " D i s a b l e " ) ;
mmc . s e t P r o p e r t y ( " P11 " , " S t o p B i t s " , 1 ) ;
mmc . s e t P r o p e r t y ( " P11 " , " P a r i t y " , " None " ) ;
mmc . s e t P r o p e r t y ( " P11 " , " D a t a B i t s " , 8 ) ;
mmc . s e t P r o p e r t y ( " P11 " , " DelayBetweenCharsMs " , 0 )
mmc . s e t P r o p e r t y ( " X− C i t e " , " P o r t " , " P11 " ) ;
mmc . i n i t i a l i z e A l l D e v i c e s ( ) ;
mmc . s e t X Y S t a g e D e v i c e ( " XYStage " ) ; % c h o o s e c o r r e c t xy s t a g e
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , d e f a u l t S t a g e S p e e d ) ; % s e t s p e e d o f
translation stage
mmc . s e t P r o p e r t y ( " XYStage " , " B a c k l a s h −B " , d e f a u l t S t a g e B a c k l a s h ) ; % amount o f
b a c k l a s h w h i l e moving
%% Set −up L a b e l s f o r Microscope
mmc . s e t F o c u s D e v i c e ( " F o c u s D r i v e " ) ;
mmc . s e t F o c u s D i r e c t i o n ( " F o c u s D r i v e " , 0 ) ;
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " IL − T u r r e t " ,

0,
1,
2,
3,
4,
5,

"1 −BF " ) ;
" 2 − 4 0 5 / 4 1 4 nm PL " ) ;
"3 −532 nm PL " ) ;
"4 − Empty " ) ;
"5 − Empty " ) ;
"6 − Empty " ) ;

mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,
mmc . d e f i n e S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ,

0,
1,
2,
3,
4,
5,

"1 −5 x 0 . 1 5 na " ) ;
"2 −10 x 0 . 3 na " ) ;
"3 −20 x 0 . 5 na " ) ;
"4 −50 x 0 . 5 5 na " ) ;
"5 −50 x 0 . 8 na " ) ;
"6 −100 x 0 . 9 na " ) ;

mmc . d e f i n e S t a t e L a b e l ( " S i d e P o r t " , 0 , " Camera " ) ;
mmc . d e f i n e S t a t e L a b e l ( " S i d e P o r t " , 1 , " L e f t ( S p e c t r o m e t e r ) " ) ;
mmc . d e f i n e S t a t e L a b e l ( " S i d e P o r t " , 2 , " R i g h t ( Empty ) " ) ;
% I n i t i a l i z e f o r X− C i t e
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 0 ) ;
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 5 I n t e n s i t y ( 0 . 0 o r 5 . 0 − 1 0 0 . 0 )%" , 5 ) ;
end
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ; % s e t s p e e d o f t r a n s l a t i o n
stage
%i n i t i a l i z e Light F i e l d
i f ~ e x i s t ( ' i n s t a n c e ' , ' var ' )
r u n S e t u p _ L i g h t F i e l d _ E n v i r o n m e n t .m;
i n s t a n c e =lfm ( t r u e ) ;
instance . load_experiment ( lf_experiment_name ) ;
% S t a r t u p d e l a y when l o a d i n g e x p e r i m e n t / s e t t i n g up s p e c t r o m e t e r

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disp ( ' I n i t i a l i z i n g Light Field ' ) ;
run_idx = 1;
end
% where t o s a v e a l l t h e d a t a
[ d a t a _ f i l e n a m e , d a t a _ d i r e c t o r y , ~ ] = u i p u t f i l e ( ' ∗ . mat ' , ' Name o f d a t a f i l e ' , ' t e s t 1 .
mat ' ) ;
i f isempty ( data_filename )
e r r o r ( ' Use a p r o p e r . mat f i l e n a m e ' ) ;
end
d a t a _ d i r e c t o r y = d a t a _ d i r e c t o r y ( 1 : end − 1 ) ;
d a t a _ f i l e n a m e = e x t r a c t B e f o r e ( d a t a _ f i l e n a m e , ' . mat ' ) ;
% u s e s d a t a f i l e n a m e , u n l e s s a l r e a d y e x i s t s . I f so , a p p e n d s a t i m e s t a m p t o
% prevent overwriting
full_data_name = s t r c a t ( data_directory , ' / ' , data_filename ) ;
i f run_idx > 1
c = clock ;
mat_data_name = s t r c a t ( f u l l _ d a t a _ n a m e , ' _run ' , num2str ( r u n _ i d x ) , ' _ ' , num2str (
c ( 1 ) ) , ' _ ' , num2str ( c ( 2 ) ) , ' _ ' , num2str ( c ( 3 ) ) , ' _ ' , num2str ( c ( 4 ) ) , ' _ ' ,
n u m 2 s t r ( c ( 5 ) ) , ' . mat ' ) ;
fig_data_name = s t r c a t ( full_data_name , ' _run ' , num2str ( run_idx ) , ' _ ' , num2str (
c ( 1 ) ) , ' _ ' , num2str ( c ( 2 ) ) , ' _ ' , num2str ( c ( 3 ) ) , ' _ ' , num2str ( c ( 4 ) ) , ' _ ' ,
n u m 2 s t r ( c ( 5 ) ) , ' . png ' ) ;
f i t _ f i g _ n a m e = s t r c a t ( full_data_name , ' _ f i t _ r u n ' , num2str ( run_idx ) , ' _ ' ,
num2str ( c ( 1 ) ) , ' _ ' , num2str ( c ( 2 ) ) , ' _ ' , num2str ( c ( 3 ) ) , ' _ ' , num2str ( c ( 4 ) ) ,
' _ ' , n u m 2 s t r ( c ( 5 ) ) , ' . png ' ) ;
lf_data_name = s t r c a t ( data_filename , ' _run ' , num2str ( run_idx ) , ' _ ' , num2str ( c
( 1 ) ) , ' _ ' , num2str ( c ( 2 ) ) , ' _ ' , num2str ( c ( 3 ) ) , ' _ ' , num2str ( c ( 4 ) ) , ' _ ' ,
num2str ( c ( 5 ) ) ) ;
run_idx = run_idx +1;
else
m a t _ d a t a _ n a m e = s t r c a t ( f u l l _ d a t a _ n a m e , ' . mat ' ) ;
f i g _ d a t a _ n a m e = s t r c a t ( f u l l _ d a t a _ n a m e , ' . png ' ) ;
f i t _ f i g _ n a m e = s t r c a t ( f u l l _ d a t a _ n a m e , ' _ f i t . png ' ) ;
lf_data_name = data_filename ;
run_idx = run_idx +1;
end

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182
183 %% e i t h e r perform background measurement or u s e a p p r o p r i a t e d a t a f i l e
184 s w i t c h meas_bg
185
case 0
186
bg_waves = nan ;
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bg_counts = 0;
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x0_bg_meas = nan ;
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y0_bg_meas = nan ;
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z0_bg_meas = nan ;
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r a w _ b g _ s p e c t r u m = nan ;
192
case 1
193
d i s p ( ' Move t o d e s i r e d b a c k g r o u n d p o s i t i o n . P r e s s " a , d , w, s " t o move l e f t ,
r i g h t , up , down , r e s p e c t i v e l y . ' ) ;
194
mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Use c a m e r a
195
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 1 ) ;
196
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 5 I n t e n s i t y ( 0 . 0 o r 5 . 0 − 1 0 0 . 0 )%" , 5 ) ;
197

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d i s p ( ' P l e a s e t y p e " s t a r t " t o s t a r t measurement , a d j u s t f o c u s w i t h " r " and
"f" ');
str = [];
while ~strcmp ( str , " s t a r t ")
str = input ( ' ' , ' s ' ) ;
i f strcmp ( str , " a ")
% jog l e f t
mmc . s e t R e l a t i v e X Y P o s i t i o n ( − X j o g s t e p , 0 ) ;
e l s e i f strcmp ( str , "d ") % jog r i g h t
mmc . s e t R e l a t i v e X Y P o s i t i o n ( X j o g s t e p , 0 ) ;
e l s e i f s t r c m p ( s t r , "w" ) % j o g up
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , Y j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " s " ) % j o g down
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , − Y j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " r " ) % j o g up i n z
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , Z j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " f " ) % j o g down i n z
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , − Z j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " d o u b l e " ) % d o u b l e s s i z e and s p e e d o f j o g
di s p ( ' Doubling Step Size ' ) ;
X j o g s t e p = 2∗ X j o g s t e p ;
Y j o g s t e p = 2∗ Y j o g s t e p ;
Z j o g s t e p = 2∗ Z j o g s t e p ;
S t a g e S p e e d = 2∗ S t a g e S p e e d ;
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
e l s e i f s t r c m p ( s t r , " h a l f " ) % h a l v e s s i z e and s p e e d o f j o g
disp ( ' Halving Step Size ' ) ;
Xjogstep = Xjogstep / 2 ;
Yjogstep = Yjogstep / 2 ;
Zjogstep = Zjogstep /2;
StageSpeed = StageSpeed / 2 ;
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
e l s e i f strcmp ( str , " s t a r t ")
break ;
else
d i s p ( ' Not a v a l i d key . Use " a , d , w, s " t o move l e f t , r i g h t , up ,
down ' ) ;
end
end
x0_bg_meas = mmc . g e t X P o s i t i o n ( ) ; % [mm] , c e n t e r o f map , where we b e g a n
measurements
y0_bg_meas = mmc . g e t Y P o s i t i o n ( ) ; % [mm] , c e n t e r o f map , where we b e g a n
measurements
z0_bg_meas = mmc . g e t P o s i t i o n ( " F o c u s D r i v e " ) ; % [mm] , c e n t e r o f map , where
we b e g a n m e a s u r e m e n t s
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 0 ) ;
d i s p ( [ ' P e r f o r m i n g b a c k g r o u n d m e a s u r e m e n t a t x : ' , n u m 2 s t r ( x0_bg_meas ) , ' um
, y : ' , n u m 2 s t r ( y0_bg_meas ) , ' um , z : ' , n u m 2 s t r ( z0_bg_meas ) , ' um ' ] ) ;
d i s p ( [ ' E x p o s u r e Time f o r b a c k g r o u n d : ' , n u m 2 s t r ( e x p o s u r e _ t i m e ) , ' ms ' ] ) ;
d i s p ( [ ' Number o f Frames : ' , n u m 2 s t r ( num_frames ) ] ) ;

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mmc . s e t S t a t e ( " IL − T u r r e t " , R e f l _ f i l t e r _ c u b e _ p o s ) ; % c h a n g e t o a p p r o p r i a t e
f i l t e r cube
mmc . s e t S t a t e ( " S i d e P o r t " , 1 ) ; % Use l e f t s i d e p o r t w i t h s p e c t r o m e t e r
mmc . w a i t F o r S y s t e m ( ) ;

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% s e t s t h e f i l e n a m e and d i r e c t o r y i n l i g h t f i e l d
meas_bg_filename = s t r c a t ( ' bg_for_ ' , lf_data_name ) ;
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationDirectory , data_directory ) ;
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationBaseFileName , meas_bg_filename ) ;

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instance . set_exposure ( exposure_time ) ;
i n s t a n c e . s e t _ f r a m e s ( num_frames ) ;
[ s p e c t r u m , bg_waves ] = a c q u i r e ( i n s t a n c e ) ;
raw_bg_spectrum = squeeze ( spectrum ) ;
i f l e n g t h ( s i z e ( s p e c t r u m ) ) >2
b g _ c o u n t s = mean ( s q u e e z e ( s p e c t r u m ) , 2 ) ;
else
bg_counts = squeeze ( spectrum ) ;
end
d i s p ( ' Background measurement complete ! ' ) ;
mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Go b a c k t o c a m e r a
case 2
try
bg_data = csvread ( s t r c a t ( bg_directory , ' / ' , bg_filename ) , 0 , 0) ;
catch
bg_data = csvread ( s t r c a t ( bg_directory , ' / ' , bg_filename ) , 1 , 0) ;
end
bg_waves = b g _ d a t a ( : , b g _ w a v e s _ i d x ) ; % [ nm ]
bg_counts = bg_data ( : , bg_counts_idx ) ; % counts
i f num_bg_frames >1
b g _ c o u n t s = mean ( r e s h a p e ( b g _ c o u n t s , [ numCameraXPixels , num_bg_frames ] )
, 2) ;
end

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x0_bg_meas = nan ;
y0_bg_meas = nan ;
z0_bg_meas = nan ;
r a w _ b g _ s p e c t r u m = nan ;
otherwise
e r r o r ( ' Use 0 f o r no bg s u b t r a c t i o n / measurement , 1 f o r bg meas , 2 f o r bg
datafile ' ) ;
end
switch meas_refl0
case 0
r e f l 0 _ w a v e s = nan ;
refl0_counts = 0;
x 0 _ r e f l 0 _ m e a s = nan ;
y 0 _ r e f l 0 _ m e a s = nan ;

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z 0 _ r e f l 0 _ m e a s = nan ;
r a w _ r e f l 0 _ s p e c t r u m = nan ;
case 1
d i s p ( ' Move t o d e s i r e d r e f e r e n c e p o s i t i o n . P r e s s " a , d , w, s " t o move l e f t ,
r i g h t , up , down , r e s p e c t i v e l y . ' ) ;
mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Use c a m e r a
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 1 ) ;
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 5 I n t e n s i t y ( 0 . 0 o r 5 . 0 − 1 0 0 . 0 )%" , 5 ) ;
d i s p ( ' P l e a s e t y p e " s t a r t " t o s t a r t measurement , a d j u s t f o c u s w i t h " r " and
"f" ');
str = [];
while ~strcmp ( str , " s t a r t ")
str = input ( ' ' , ' s ' ) ;
i f strcmp ( str , " a ")
% jog l e f t
mmc . s e t R e l a t i v e X Y P o s i t i o n ( − X j o g s t e p , 0 ) ;
e l s e i f strcmp ( str , "d ") % jog r i g h t
mmc . s e t R e l a t i v e X Y P o s i t i o n ( X j o g s t e p , 0 ) ;
e l s e i f s t r c m p ( s t r , "w" ) % j o g up
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , Y j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " s " ) % j o g down
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , − Y j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " r " ) % j o g up i n z
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , Z j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " f " ) % j o g down i n z
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , − Z j o g s t e p ) ;
e l s e i f s t r c m p ( s t r , " d o u b l e " ) % d o u b l e s s i z e and s p e e d o f j o g
di s p ( ' Doubling Step Size ' ) ;
X j o g s t e p = 2∗ X j o g s t e p ;
Y j o g s t e p = 2∗ Y j o g s t e p ;
Z j o g s t e p = 2∗ Z j o g s t e p ;
S t a g e S p e e d = 2∗ S t a g e S p e e d ;
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
e l s e i f s t r c m p ( s t r , " h a l f " ) % h a l v e s s i z e and s p e e d o f j o g
disp ( ' Halving Step Size ' ) ;
Xjogstep = Xjogstep / 2 ;
Yjogstep = Yjogstep / 2 ;
Zjogstep = Zjogstep /2;
StageSpeed = StageSpeed / 2 ;
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
e l s e i f strcmp ( str , " s t a r t ")
break ;
else
d i s p ( ' Not a v a l i d key . Use " a , d , w, s " t o move l e f t , r i g h t , up ,
down ' ) ;
end
end
x 0 _ r e f l 0 _ m e a s = mmc . g e t X P o s i t i o n ( ) ; % [mm] , c e n t e r o f map , where we b e g a n
measurements
y 0 _ r e f l 0 _ m e a s = mmc . g e t Y P o s i t i o n ( ) ; % [mm] , c e n t e r o f map , where we b e g a n
measurements

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z 0 _ r e f l 0 _ m e a s = mmc . g e t P o s i t i o n ( " F o c u s D r i v e " ) ; % [mm] , c e n t e r o f map ,
where we b e g a n m e a s u r e m e n t s

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mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 0 ) ;
d i s p ( [ ' Performing r e f e r e n c e measurement a t x : ' , num2str ( x0_refl0_meas ) , '
um , y : ' , n u m 2 s t r ( y 0 _ r e f l 0 _ m e a s ) , ' um , z : ' , n u m 2 s t r ( z 0 _ r e f l 0 _ m e a s ) , '
um ' ] ) ;
d i s p ( [ ' E x p o s u r e Time f o r R e f e r e n c e : ' , n u m 2 s t r ( e x p o s u r e _ t i m e ) , ' ms ' ] ) ;
d i s p ( [ ' Number o f Frames : ' , n u m 2 s t r ( num_frames ) ] ) ;

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mmc . s e t S t a t e ( " IL − T u r r e t " , R e f l _ f i l t e r _ c u b e _ p o s ) ; % c h a n g e t o a p p r o p r i a t e
f i l t e r cube
mmc . s e t S t a t e ( " S i d e P o r t " , 1 ) ; % Use l e f t s i d e p o r t w i t h s p e c t r o m e t e r
mmc . w a i t F o r S y s t e m ( ) ;

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% s e t s t h e f i l e n a m e and d i r e c t o r y i n l i g h t f i e l d
meas_refl0_filename = s t r c a t ( ' r e f l 0 _ f o r _ ' , lf_data_name ) ;
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationDirectory , data_directory ) ;
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationBaseFileName , meas_refl0_filename ) ;

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instance . set_exposure ( exposure_time ) ;
i n s t a n c e . s e t _ f r a m e s ( num_frames ) ; % a l w a y s u s e 1 f r a m e f o r bg
[ spectrum , r e f l 0 _ w a v e s ]= a c q u i r e ( i n s t a n c e ) ;
raw_refl0_spectrum = squeeze ( spectrum ) ;
i f l e n g t h ( s i z e ( s p e c t r u m ) ) >2
r e f l 0 _ c o u n t s = mean ( s q u e e z e ( s p e c t r u m ) , 2 ) ;
else
r e f l 0 _ c o u n t s = squeeze ( spectrum ) ;
end
d i s p ( ' Reference measurement complete ! ' ) ;
mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Go b a c k t o c a m e r a
case 2
try
refl0_data = csvread ( s t r c a t ( refl0_directory , ' / ' , refl0_filename ) , 0 ,
0) ;
catch
refl0_data = csvread ( s t r c a t ( refl0_directory , ' / ' , refl0_filename ) , 1 ,
0) ;
end
r e f l 0 _ w a v e s = r e f l 0 _ d a t a ( : , r e f l 0 _ w a v e s _ i d x ) ; % [ nm ]
refl0_counts = refl0_data (: , refl0_counts_idx ) ;
x 0 _ r e f l 0 _ m e a s = nan ;
y 0 _ r e f l 0 _ m e a s = nan ;
z 0 _ r e f l 0 _ m e a s = nan ;
r a w _ r e f l 0 _ s p e c t r u m = nan ;
otherwise
e r r o r ( ' Use 0 f o r no r e f l 0 n o r m a l i z a t i o n / measurement , 1 f o r r e f l 0 meas , 2
for refl0 datafile ' ) ;

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end

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394 i f meas_bg && m e a s _ r e f l 0
395
bg_counts = reshape ( bg_counts , s i z e ( r e f l 0 _ c o u n t s ) ) ;
396 end
397
398 %% Prepare f o r R e f l e c t i o n measurement
399 d i s p ( ' Move t o d e s i r e d p o s i t i o n f o r r e f l e c t a n c e m e a s u r e m e n t s . P r e s s " a , d , w, s " t o
move l e f t , r i g h t , up , down , r e s p e c t i v e l y . ' ) ;
400 mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 1 ) ;
401 mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 5 I n t e n s i t y ( 0 . 0 o r 5 . 0 − 1 0 0 . 0 )%" , 5 ) ;
402 mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Use c a m e r a
403
404 d i s p ( ' P l e a s e t y p e " s t a r t " t o s t a r t measurement , a d j u s t f o c u s w i t h " r " and " f " ' ) ;
405 s t r = [ ] ;
406 w h i l e ~ s t r c m p ( s t r , " s t a r t " )
407
408
str = input ( ' ' , ' s ' ) ;
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410
i f strcmp ( str , " a ")
% jog l e f t
411
mmc . s e t R e l a t i v e X Y P o s i t i o n ( − X j o g s t e p , 0 ) ;
412
e l s e i f strcmp ( str , "d ") % jog r i g h t
413
mmc . s e t R e l a t i v e X Y P o s i t i o n ( X j o g s t e p , 0 ) ;
414
e l s e i f s t r c m p ( s t r , "w" ) % j o g up
415
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , Y j o g s t e p ) ;
416
e l s e i f s t r c m p ( s t r , " s " ) % j o g down
417
mmc . s e t R e l a t i v e X Y P o s i t i o n ( 0 , − Y j o g s t e p ) ;
418
e l s e i f s t r c m p ( s t r , " r " ) % j o g up i n z
419
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , Z j o g s t e p ) ;
420
e l s e i f s t r c m p ( s t r , " f " ) % j o g down i n z
421
mmc . s e t R e l a t i v e P o s i t i o n ( " F o c u s D r i v e " , − Z j o g s t e p ) ;
422
e l s e i f s t r c m p ( s t r , " d o u b l e " ) % d o u b l e s s i z e and s p e e d o f j o g
423
di sp ( ' Doubling Step Size ' ) ;
424
X j o g s t e p = 2∗ X j o g s t e p ;
425
Y j o g s t e p = 2∗ Y j o g s t e p ;
426
Z j o g s t e p = 2∗ Z j o g s t e p ;
427
S t a g e S p e e d = 2∗ S t a g e S p e e d ;
428
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
429
e l s e i f s t r c m p ( s t r , " h a l f " ) % h a l v e s s i z e and s p e e d o f j o g
430
disp ( ' Halving Step Size ' ) ;
431
Xjogstep = Xjogstep / 2 ;
432
Yjogstep = Yjogstep / 2 ;
433
Zjogstep = Zjogstep /2;
434
StageSpeed = StageSpeed / 2 ;
435
mmc . s e t P r o p e r t y ( " XYStage " , " Speed −S " , S t a g e S p e e d ) ;
436
e l s e i f strcmp ( str , " s t a r t ")
437
break ;
438
else
439
d i s p ( ' Not a v a l i d key . Use " a , d , w, s " t o move l e f t , r i g h t , up , down ' ) ;
440
end
441 end
442
443 mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 0 ) ;
444 x0_meas = mmc . g e t X P o s i t i o n ( ) ; % [mm] , where we b e g a n m e a s u r e m e n t s
445 y0_meas = mmc . g e t Y P o s i t i o n ( ) ; % [mm] , where we b e g a n m e a s u r e m e n t s
446 z0_meas = mmc . g e t P o s i t i o n ( " F o c u s D r i v e " ) ;

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d i s p ( [ ' P e r f o r m i n g r e f l e c t a n c e m e a s u r e m e n t a t x : ' , n u m 2 s t r ( x0_meas ) , ' um , y :
n u m 2 s t r ( y0_meas ) , ' um , z : ' , n u m 2 s t r ( z0_meas ) , ' um ' ] ) ;
d i s p ( [ ' E x p o s u r e Time f o r S p e c t r u m : ' , n u m 2 s t r ( e x p o s u r e _ t i m e ) , ' ms ' ] ) ;
d i s p ( [ ' Number o f Frames : ' , n u m 2 s t r ( num_frames ) ] ) ;

',

mmc . s e t S t a t e ( " IL − T u r r e t " , R e f l _ f i l t e r _ c u b e _ p o s ) ; % c h a n g e t o a p p r o p r i a t e f i l t e r
cube
mmc . s e t S t a t e ( " S i d e P o r t " , 1 ) ; % Use l e f t s i d e p o r t w i t h s p e c t r o m e t e r
o b j e c t i v e _ i n _ u s e = mmc . g e t S t a t e L a b e l ( " O b j e c t i v e T u r r e t " ) ;
% add i n some c o d e t o t u r n o f f t h e X− c i t e l i g h t when we f i g u r e o u t how t o
% control i t again
mmc . w a i t F o r S y s t e m ( ) ;
% S e t s t h e f i l e n a m e and d i r e c t o r y i n l i g h t f i e l d , and t a k e s a s p e c t r u m
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationDirectory , data_directory ) ;
i n s t a n c e . s e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns . E x p e r i m e n t S e t t i n g s .
FileNameGenerationBaseFileName , lf_data_name ) ;
instance . set_exposure ( exposure_time ) ;
i n s t a n c e . s e t _ f r a m e s ( num_frames ) ;
[ s p e c t r u m , waves ] = a c q u i r e ( i n s t a n c e ) ;

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465
466 r a w _ s p e c t r u m = s q u e e z e ( s p e c t r u m ) ;
467
468 i f l e n g t h ( s i z e ( s p e c t r u m ) ) >2
469
r e f l _ c o u n t s = r e s h a p e ( mean ( s q u e e z e ( s p e c t r u m ) , 2 ) , s i z e ( r e f l 0 _ c o u n t s ) ) ;
470 e l s e
471
r e f l _ c o u n t s = reshape ( squeeze ( spectrum ) , s i z e ( r e f l 0 _ c o u n t s ) ) ;
472 end
473
474
475 i f m e a s _ r e f l 0 && meas_bg
476
spectrum_w_norm = ( r e f l _ c o u n t s − b g _ c o u n t s ) . / ( r e f l 0 _ c o u n t s − b g _ c o u n t s ) ;
477 e l s e i f m e a s _ r e f l 0 && ~ meas_bg
478
spectrum_w_norm = ( r e f l _ c o u n t s ) . / ( r e f l 0 _ c o u n t s ) ;
479 e l s e i f ~ m e a s _ r e f l 0 && meas_bg
480
spectrum_w_norm = r e f l _ c o u n t s − b g _ c o u n t s ;
481 e l s e
482
spectrum_w_norm = r e f l _ c o u n t s ;
483 end
484
485 %% Generate R e f l e c t i v i t y C o n t r a s t P l o t
486 % W a v e l e n g t h
487 s p e c F i g = f i g u r e ;
488 s e t ( s p e c F i g , ' P o s i t i o n ' , [ 2 0 0 , 4 0 0 , 5 0 0 , 5 0 0 ] ) ;
489 box on ;
490 x l a b e l ( ' W a v e l e n g t h ( nm ) ' ) ; y l a b e l ( ' R / R_0 ' ) ;
491 x l i m ( [ min ( waves ) , max ( waves ) ] ) ;
492 a x i s t i g h t ; h o l d on ;
493 p s p e c = p l o t ( waves , spectrum_w_norm ) ;
494 s a v e a s ( s p e c F i g , f i g _ d a t a _ n a m e ) ;
495
496 % E n e r g y
497 s p e c F i g 2 = f i g u r e ;

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s e t ( specFig2 , ' P o s i t i o n ' , [600 , 400 , 500 , 500]) ;
box on ;
e n e r g y = 1 2 3 9 . 8 . / waves ;
x l a b e l ( ' E n e r g y ( eV ) ' ) ; y l a b e l ( ' R / R_0 ' ) ;
x l i m ( [ min ( e n e r g y ) , max ( e n e r g y ) ] ) ;
a x i s t i g h t ; h o l d on ;
p s p e c = p l o t ( e n e r g y , spectrum_w_norm ) ;
f i l t e r _ i n _ u s e = mmc . g e t S t a t e L a b e l ( " IL − T u r r e t " ) ;
g r a t i n g _ c e n t e r _ w a v e l e n g t h = i n s t a n c e . g e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns .
SpectrometerSettings . GratingCenterWavelength ) ;
g r a t i n g _ s e l e c t e d = s t r i n g ( i n s t a n c e . g e t ( P r i n c e t o n I n s t r u m e n t s . L i g h t F i e l d . AddIns .
SpectrometerSettings . GratingSelected ) ) ;
d i s p ( ' A c q u i s i t i o n Complete ! ' ) ;
mmc . s e t S t a t e ( " IL − T u r r e t " , 0 ) ; % c h a n g e b a c k t o BF f i l t e r c u b e
mmc . s e t S t a t e ( " S i d e P o r t " , 0 ) ; % Go b a c k t o c a m e r a
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 4 On / O f f S t a t e ( 1 =On 0= O f f ) " , 1 ) ;
mmc . s e t P r o p e r t y ( " LED1 " , "L . 1 5 I n t e n s i t y ( 0 . 0 o r 5 . 0 − 1 0 0 . 0 )%" , 5 ) ;

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515 %% Perform F i t t i n g u s i n g " s i m p l e " e l l i p s o m e t r y
516 d i s p ( ' S t a r t i n g F i t t i n g . . . ' ) ;
517 w a v e _ b o o l = waves > w a v e _ f i t _ m i n & waves < w a v e _ f i t _ m a x ;
518 w a v e _ f i t = waves ( w a v e _ b o o l ) ;
519 e n e r g y _ f i t = 1 2 3 9 . 8 4 . / w a v e _ f i t ;
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521 n t i l d e _ t h i n F i l m = s q r t ( M o S 2 _ e p s i l o n ( e n e r g y _ f i t ) ) ;
522
523 n _ S i = l o a d ( ' n t i l d e _ S i . mat ' ) ;
524 n t i l d e _ s u b = i n t e r p 1 ( n _ S i . l a m b d a _ S i , n _ S i . n t i l d e _ S i , w a v e _ f i t ) ;
525 lambda_SiO2 = 0 . 2 : 0 . 0 0 1 : 3 ;
526 n_SiO2 = s q r t ( 1 + 0 . 6 9 6 1 6 6 3 . / ( 1 − ( 0 . 0 6 8 4 0 4 3 . / lambda_SiO2 ) . ^ 2 )
+ 0 . 4 0 7 9 4 2 6 . / ( 1 − ( 0 . 1 1 6 2 4 1 4 . / lambda_SiO2 ) . ^ 2 ) + 0 . 8 9 7 4 7 9 4 . / ( 1 − ( 9 . 8 9 6 1 6 1 . /
lambda_SiO2 ) . ^ 2 ) ) ;
527 n t i l d e _ o v e r l a y e r = i n t e r p 1 ( lambda_SiO2 ∗1 e3 , n_SiO2 , w a v e _ f i t ) ;
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530 %% M u l t i S t a r t Parameters
531 m a x f u n c e v a l = 1 e2 ;
532 OptimTol = 1 e − 9 ;
533 S t e p T o l = 1 e − 9 ;
534 m a x I t e r = 1 e2 ;
535 F u n c T o l = 1 e − 9 ;
536 m s _ l s q O p t s = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' M a x F u n c t i o n E v a l u a t i o n s ' , m a x f u n c e v a l , ...
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' O p t i m a l i t y T o l e r a n c e ' , OptimTol , ' S t e p T o l e r a n c e ' , S t e p T o l , ...
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' M a x I t e r a t i o n s ' , maxIter , ' F u n c t i o n T o l e r a n c e ' , FuncTol ) ;
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541 %% l s q c u r v e f i t Parameters
542 m a x f u n c e v a l = 1 e3 ;
543 OptimTol = 1 e − 1 5 ;
544 S t e p T o l = 1 e − 1 5 ;
545 m a x I t e r = 1 e3 ;
546 F u n c T o l = 1 e − 1 5 ;
547 l s q O p t s = o p t i m o p t i o n s ( ' l s q c u r v e f i t ' , ' M a x F u n c t i o n E v a l u a t i o n s ' , m a x f u n c e v a l , ...
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' O p t i m a l i t y T o l e r a n c e ' , OptimTol , ' S t e p T o l e r a n c e ' , S t e p T o l , ...

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' MaxIterations ' , maxIter ,

' F u n c t i o n T o l e r a n c e ' , FuncTol ) ;

x0 = [ ( t _ t h i n F i l m _ m i n + t _ t h i n F i l m _ m a x ) / 2 , ( t _ o v e r l a y e r _ m i n + t _ o v e r l a y e r _ m a x ) / 2 ] ;
f u n = @(x , x d a t a ) f i t f u n c ( x , x d a t a , n t i l d e _ t h i n F i l m , n t i l d e _ o v e r l a y e r , n t i l d e _ s u b ) ;
xdata = wave_fit ( : ) ;
y d a t a = spectrum_w_norm ( w a v e _ b o o l ) ;
ydata = ydata ( : ) ;
lb = [ t_thinFilm_min , t_overlayer_min ] ;
ub = [ t _ t h i n F i l m _ m a x , t _ o v e r l a y e r _ m a x ] ;
s t p t s = [ ( t_thinFilm_max − t_thinFilm_min ) ∗ rand ( num_MS_startpts , 1 ) +
t _ t h i n F i l m _ m i n , ...
( t_overlayer_max − t _ o v e r l a y e r _ m i n ) ∗ rand ( num_MS_startpts , 1 ) +
t_overlayer_min ] ;
s t a r t p t s = CustomStartPointSet ( stp ts ) ;
p r o b l e m = c r e a t e O p t i m P r o b l e m ( ' l s q c u r v e f i t ' , ' o b j e c t i v e ' , fun , ' x0 ' , x0 , ...
' x d a t a ' , x d a t a , ' y d a t a ' , y d a t a , ' l b ' , l b , ' ub ' , ub , ' o p t i o n s ' , m s _ l s q O p t s )
ms = M u l t i S t a r t ;
tic ;
x_ms = r u n ( ms , problem , s t a r t p t s ) ;
[ x _ l s q , r e s n o r m ] = l s q c u r v e f i t ( fun , x_ms , x d a t a , y d a t a , t _ t h i n F i l m _ m i n ,
t_thinFilm_max , lsqOpts ) ;
f i t t e d _ t h i c k n e s s = x_lsq (1) ;
oxide_thickness = x_lsq (2) ;
d i s p ( [ ' T o t a l time f o r f i t : ' , num2str ( round ( toc , 1 ) ) , ' seconds ' ] ) ;
% Wavelength
specFigFit = figure ;
s e t ( s p e c F i g F i t , ' P o s i t i o n ' , [200 , 400 , 500 , 500]) ;
box on ;
x l a b e l ( ' W a v e l e n g t h ( nm ) ' ) ; y l a b e l ( ' R / R_0 ' ) ;
x l i m ( [ min ( w a v e _ f i t ) , max ( w a v e _ f i t ) ] ) ;
a x i s t i g h t ; h o l d on ;
p s p e c 1 = p l o t ( x d a t a , y d a t a , ' k . ' ) ; h o l d on ;
p s p e c 2 = p l o t ( x d a t a , f u n ( x _ l s q , x d a t a ) , ' b− ' ) ;
t i t l e ( [ ' F i l m T h i c k n e s s : ' , n u m 2 s t r ( r o u n d ( f i t t e d _ t h i c k n e s s , 1 ) ) , ' nm , Oxide
T h i c k n e s s : ' , n u m 2 s t r ( r o u n d ( o x i d e _ t h i c k n e s s , 1 ) ) , ' nm , F i t t i n g R e s i d u a l : ' ,
num2str ( round ( resnorm , 1 ) ) ] ) ;
l e g e n d ( [ pspec1 , pspec2 ] , { ' Experiment ' , ' F i t ' } , ' O r i e n t a t i o n ' , ' H o r i z o n t a l ' , '
Location ' , ' NorthOutside ' ) ;
saveas ( specFigFit , fit_fig_name ) ;

% s a v e a l l r e l e v a n t d a t a i n a . mat f i l e
s a v e ( mat_data_name , ' l f _ e x p e r i m e n t _ n a m e ' , ' e x p o s u r e _ t i m e ' , ' num_frames ' , ...
' objective_in_use ' , ' filter_in_use ' , ' Refl_filter_cube_pos ' , '
g r a t i n g _ c e n t e r _ w a v e l e n g t h ' , ' g r a t i n g _ s e l e c t e d ' , ...
' x0_meas ' , ' y0_meas ' , ' z0_meas ' ,
' waves ' , ' r a w _ s p e c t r u m ' , '
spectrum_w_norm ' , ...
' b g _ d i r e c t o r y ' , ' b g _ f i l e n a m e ' , ' b g _ w a v e s _ i d x ' , ' b g _ c o u n t s _ i d x ' , ...
' meas_bg ' , ' bg_waves ' , ' b g _ c o u n t s ' , ...
' x0_bg_meas ' , ' y0_bg_meas ' , ' z0_bg_meas ' , ' r a w _ b g _ s p e c t r u m ' , ...

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' refl0_directory ' , ' refl0_filename ' , ' refl0_waves_idx ' , ' refl0_counts_idx '
, ...
' m e a s _ r e f l 0 ' , ' r e f l 0 _ w a v e s ' , ' r e f l 0 _ c o u n t s ' , ...
' x 0 _ r e f l 0 _ m e a s ' , ' y 0 _ r e f l 0 _ m e a s ' , ' z 0 _ r e f l 0 _ m e a s ' , ' r a w _ r e f l 0 _ s p e c t r u m ' , ...
' fitted_thickness ' , ' oxide_thickness ' , ' ntilde_thinFilm ' , ' ntilde_sub ' , '
n t i l d e _ o v e r l a y e r ' , ' t _ o v e r l a y e r ' , ' w a v e _ f i t _ m i n ' , ' w a v e _ f i t _ m a x ' , ...
' numCameraXPixels ' , ' num_bg_frames ' , ' X j o g s t e p ' , ' Y j o g s t e p ' , ' Z j o g s t e p ' , '
StageSpeed ' ) ;

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f u n c t i o n y = f i t f u n c ( x , xdata , n t i l d e _ t h i n F i l m , n t i l d e _ o v e r l a y e r , n t i l d e _ s u b )
R = FourLayer_thinFilmFast ( ntilde_thinFilm , x (1) , ntilde_overlayer , x (2) ,
ntilde_sub , xdata ) ;
R0 = F o u r L a y e r _ t h i n F i l m F a s t ( n t i l d e _ t h i n F i l m , 0 , n t i l d e _ o v e r l a y e r , x ( 2 ) ,
ntilde_sub , xdata ) ;
y = R ( : ) . / R0 ( : ) ;
end
f u n c t i o n R = t h i n F i l m S u b ( n t i l d e _ t h i n F i l m , t _ t h i n F i l m , n t i l d e _ s u b , lambda )
% n t i l d e _ t h i n F i l m has s i z e s i z e ( n t i l d e _ s u b )
% t _ t h i n F i l m i s a s i n g l e s c a l a r [ same u n i t s a s lambda ]
% n t i l d e _ s u b h a s d i m e n s i o n s o f u s u a l l y l e n g t h ( lambda ) x 1 o r 1 x
% l e n g t h ( lambda )
% lambda i s a v e c t o r and h a s u n i t s o f t _ t h i n F i l m
% O u t p u t R i s a v e c t o r w i t h t h e same d i m e n s i o n s a s lambda , i . e . R =
% R ( lambda )
% Reshape m a t r i c e s t o t h e r i g h t s i z e
n _ a i r = ones ( s i z e ( n t i l d e _ s u b ) ) ;
ntilde_thinFilm = reshape ( ntilde_thinFilm , size ( ntilde_sub ) ) ;
lambda = r e s h a p e ( lambda , s i z e ( n t i l d e _ s u b ) ) ;
r12 = ( n_air − n t i l d e _ t h i n F i l m ) . / ( n _ a i r + n t i l d e _ t h i n F i l m ) ;
r23 = ( n t i l d e _ t h i n F i l m − n t i l d e _ s u b ) . / ( n t i l d e _ t h i n F i l m + n t i l d e _ s u b ) ;
b e t a = ( 2 ∗ p i . ∗ n t i l d e _ t h i n F i l m . ∗ t _ t h i n F i l m ) . / lambda ;
r = ( r 1 2 + r 2 3 . ∗ exp ( 2 i ∗ b e t a ) ) . / ( 1 + r 1 2 . ∗ r 2 3 . ∗ exp ( 2 i ∗ b e t a ) ) ; % s i m p l e 3− l a y e r
i n t e r f e r e n c e model
R = abs ( r ) . ^ 2 ;
end
function R = FourLayer_thinFilmFast ( ntilde_thinFilm , t_thinFilm , ntilde_overlayer ,
t _ o v e r l a y e r , n t i l d e _ s u b , lambda )
% Reshape m a t r i c e s t o t h e r i g h t s i z e
n _ a i r = ones ( s i z e ( n t i l d e _ s u b ) ) ;
ntilde_thinFilm = reshape ( ntilde_thinFilm , size ( ntilde_sub ) ) ;
ntilde_overlayer = reshape ( ntilde_overlayer , size ( ntilde_sub ) ) ;
lambda = r e s h a p e ( lambda , s i z e ( n t i l d e _ s u b ) ) ;
r01 = ( n _ a i r − n t i l d e _ t h i n F i l m ) . / ( n _ a i r + n t i l d e _ t h i n F i l m ) ;
r12 = ( n t i l d e _ t h i n F i l m − n t i l d e _ o v e r l a y e r ) . / ( n t i l d e _ t h i n F i l m +

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ntilde_overlayer ) ;
r23 = ( n t i l d e _ o v e r l a y e r − n t i l d e _ s u b ) . / ( n t i l d e _ o v e r l a y e r + n t i l d e _ s u b ) ;
b e t a 1 = ( 2 ∗ p i . ∗ n t i l d e _ t h i n F i l m . ∗ t _ t h i n F i l m ) . / lambda ;
b e t a 2 = ( 2 ∗ p i . ∗ n t i l d e _ o v e r l a y e r ∗ t _ o v e r l a y e r ) . / lambda ;

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S = J 0 1 ∗L1∗ J 1 2 ∗L2∗ J 2 3 ;
r = S(2 ,1) . / S(1 ,1) ;
num = ( r 0 1 + r 1 2 . ∗ exp ( 2 i ∗ b e t a 1 ) ) + exp ( 2 i ∗ b e t a 2 ) . ∗ r 2 3 . ∗ ( r 0 1 . ∗ r 1 2 + exp ( 2 i ∗
beta1 ) ) ;
denom = ( 1 + exp ( 2 i ∗ b e t a 1 ) . ∗ r 0 1 . ∗ r 1 2 ) + exp ( 2 i ∗ b e t a 2 ) . ∗ r 2 3 . ∗ ( r 1 2 + exp ( 2 i ∗
beta1 ) .∗ r01 ) ;
R = a b s ( num . / denom ) . ^ 2 ;

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J 0 1 = [ 1 , r 0 1 ; r01 , 1 ] ;
L1 = [ exp ( −1 i ∗ b e t a 1 ) , 0 ; 0 , exp ( 1 i ∗ b e t a 1 ) ] ;
J 1 2 = [ 1 , r 1 2 ; r12 , 1 ] ;
L2 = [ exp ( −1 i ∗ b e t a 2 ) , 0 ; 0 , exp ( 1 i ∗ b e t a 2 ) ] ;
J 2 3 = [ 1 , r 2 3 ; r23 , 1 ] ;

end
function R = FourLayer_thinFilm ( ntilde_thinFilm , t_thinFilm , ntilde_overlayer ,
t _ o v e r l a y e r , n t i l d e _ s u b , lambda )
% d e f i n i t i o n o f l a y e r s , e a c h l a y e r i s a s e p a r a t e columm v e c t o r . T o t a l
% m a t r i x o f n t i l d e i s m x n , where m i s t h e number o f w a v e l e n g t h s and n i s
% t h e number o f l a y e r s o f m a t e r i a l
n t i l d e = [ o n e s ( l e n g t h ( lambda ) , 1 ) , ... % f i r s t l a y e r , which i s a i r u s u a l
r e s h a p e ( n t i l d e _ t h i n F i l m , [ l e n g t h ( lambda ) , 1 ] ) , ...
r e s h a p e ( n t i l d e _ o v e r l a y e r , [ l e n g t h ( lambda ) , 1 ] ) , ...
r e s h a p e ( n t i l d e _ s u b , [ l e n g t h ( lambda ) , 1 ] ) , ...
] ; % l a s t l a y e r i s s u b s t r a t e / a i r , s h o u l d h a v e l e n g t h ( t ) +2 number o f l a y e r s
z = cumsum ( [ 0 , t _ t h i n F i l m , t _ o v e r l a y e r ] ) ; % u n i t s o f [m] , a c t u a l z p o s i t i o n o f
layers . Start at z = 0.
R = z e r o s ( l e n g t h ( lambda ) , 1 ) ;
% C a l c u l a t e s p e c t r a l l y , m o n o c h r o m a t i c wave a s s u m p t i o n
f o r k = 1 : l e n g t h ( lambda )
J = [1 , 0 ;
0 , 1 ] ; % i n i t a l i z e J as i d e n t i t y matrix
% f i r s t l a y e r i s a i r , c o o r d i n a t e s are such t h a t the a i r − d i e l e c t r i c
% stack s t a r t s at z = 0
for j = 1: length ( z )
q = 2∗ p i ∗ n t i l d e ( k , j ) / lambda ( k ) ;
qp = 2∗ p i ∗ n t i l d e ( k , j + 1 ) / lambda ( k ) ;
J = 1 / 2 ∗ [ exp ( 1 i ∗ ( q − qp ) ∗ z ( j ) ) ∗ ( 1 + q / qp ) ,
exp ( −1 i ∗ ( q + qp ) ∗ z ( j
) ) ∗ (1 − q / qp ) ; ...
exp ( 1 i ∗ ( q + qp ) ∗ z ( j ) ) ∗ (1 − q / qp ) ,
exp ( −1 i ∗ ( q − qp )
∗ z ( j ) ) ∗ ( 1 + q / qp ) ] ∗ J ;

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end
% r e f l e c t i o n amplitude
r = −J ( 2 , 1 ) / J ( 2 , 2 ) ;
% R e f l e c t i o n c o e f f i c i e n t from J
R( k ) = abs ( r ) . ^ 2 ;
end
end
f u n c t i o n eps = GaS_epsilon (E)
default_fk = [0.0081 , 70.2622];
default_Ek = [2.8236 , 4.6093];
default_gammak = [ 0 . 1 , 0 . 0 3 5 2 ] ;
default_eps_bg = 2.8912;
d e f a u l t _ f k = 30; % d e f a u l t
default_Ek = 4.7163;
default_gammak = 3.8859 e −14;
default_eps_bg = 6.0430;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;
end
f u n c t i o n eps = MoS2_epsilon (E)
d e f a u l t _ f k = [ 0 . 9 8 2 4 , 3.3385 , 9.5771 , 30 , 3 0 ] ;
default_Ek = [1.866 , 2.010 , 2.489 , 2.8596 , 3.185];
default_gammak = [0.05811 , 0.1545 , 0.6605 , 0.36189 , 0 . 4 4 8 1 ] ;
default_eps_bg = 7.523;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;
end
f u n c t i o n eps = MoSe2_epsilon (E)
default_fk = [0.7770 ,
1.5210 ,
2.4830 ,
10.1110 ,
26.3230 ,
13.5090 ,
14.2800 ,
18.6230];
default_Ek = [1.5470 ,
1.7490 ,
1.9890 ,
2.3500 ,
2.6250 ,
3.0280 ,
3.3930 ,
3.6610];
default_gammak = [ 0 . 0 5 0 0 ,
0.1120 ,
0.3890 ,
0.5480 ,
0.4730 ,
0.6250 ,
0.2170 ,
0.1400];
default_eps_bg = 2.753;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;
end
f u n c t i o n eps = WSe2_epsilon (E)
d e f a u l t _ f k = [0.6776 , 1.9509 , 8.1767 , 28.0458 , 27.7930];
default_Ek = [1.652 , 2.079 , 2.431 , 2.927 , 3.507];
default_gammak = [ 0 . 0 4 6 2 , 0 . 1 8 5 6 , 0 . 3 2 9 7 , 0 . 4 9 5 0 , 0 . 0 6 0 7 ] ;
default_eps_bg = 5.68552;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;

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end
f u n c t i o n eps = WS2_epsilon (E)
default_fk = [1.963 , 3.505 , 20.212 , 30.000]; % d e f a u l t
default_Ek = [2.012 , 2.403 , 2.862 , 3.219];
default_gammak = [ 0 . 0 3 5 6 , 0.1892 , 0.3568 , 0 . 3 5 8 5 ] ;
default_eps_bg = 6.9233;
eps = e p s i l o n ( d e f a u l t _ f k , default_Ek , default_gammak , E , l e n g t h ( d e f a u l t _ f k ) ,
default_eps_bg ) ;
end
f u n c t i o n e p s = e p s i l o n ( fk , Ek , gammak , E , N, e p s _ b g )
eps = eps_bg ∗ ones ( s i z e (E) ) ;
f o r i d x = 1 :N
e p s = e p s + f k ( i d x ) . / ( Ek ( i d x ) ^2 −E. ^ 2 − 1 i ∗E∗gammak ( i d x ) ) ;
end
end

290
Listing D.6: Excitons dissociating in an Electric Field Code
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numPts = 1 0 0 ; % u s e g = 40 f o r r e a s o n a b l e s p e e d and a c c u r a c y . 200 i s " h i g h
resolution "
p = l i n s p a c e ( − 5 , 5 , numPts ) ;
% 1D mesh
h = p (2) − p (1) ; % spacing
[ Xmesh , Ymesh , Zmesh ] = m e s h g r i d ( p , p , p ) ; % 3D Mesh
X = Xmesh ( : ) ; Y = Ymesh ( : ) ; Z = Zmesh ( : ) ;
% a l l elements as a s i n g l e
column
R = s q r t (X. ^ 2 + Y. ^ 2 + Z . ^ 2 ) ;
% d i s t a n c e from t h e c e n t e r
e = o n e s ( numPts , 1 ) ;
L = s p d i a g s ( [ e −2∗ e e ] , − 1 : 1 , numPts , numPts ) / h ^ 2 ; % 1D f i n i t e
difference Laplacian
I = s p e y e ( numPts ) ;
L3 = k r o n ( k r o n ( L , I ) , I ) + k r o n ( k r o n ( I , L ) , I ) + k r o n ( k r o n ( I , I ) , L ) ; %
extend Laplacian to 3 D
F_vals = l i n s p a c e (0 ,1 ,21) ; % Field values , in e x c i t o n i c u n i t s
EvsF = z e r o s ( s i z e ( F _ v a l s ) ) ; % 1 s e i g e n e n e r g i e s v s . f i e l d
P s i v s F = z e r o s ( g3 , l e n g t h ( F _ v a l s ) ) ; % w a v e f u n c t i o n s v s . f i e l d
TunnelProb = zeros ( s i z e ( F_vals ) ) ; % t u n n e l i n g p r o b a b i l i t y = d i s s o c i a t i o n
efficiency
fig = figure ;
colorLines = l i n s p e c e r ( length ( F_vals ) ) ; % c r e a t e colors for p r e t t y l i n e s
f o r F_idx = 1: l e n g t h ( F_vals )
Vext = − 2 . / R − F _ v a l s ( F _ i d x ) . ∗ Z ;
% p o t e n t i a l energy
H = −L3 + s p d i a g s ( Vext , 0 , numPts ^ 3 , numPts ^ 3 ) ; % H a m i l t o n i a n o f e x c i t o n
[ PSI , E ] = e i g s (H, 1 , ' s m a l l e s t r e a l ' ) ; % S m a l l e s t e i g e n v a l u e o f e x c i t o n
EvsF ( F _ i d x ) = E ; % 1 s e n e r g i e s v s . f i e l d , u n i t s o f Rydberg
P s i v s F ( : , F _ i d x ) = PSI ; % w a v e f u n c t i o n s i n column f o r m a t , v s f i e l d
PSI_3 = r e s h a p e ( PSI , [ numPts , numPts , numPts ] ) ; % s h a p e p r o p e r l y t o
integrate
P S I _ z I n t = s q u e e z e ( t r a p z ( p , t r a p z ( p , a b s ( PSI_3 ) . ^ 2 ) ) ) ; % i n t e g r a t e o v e r
all x,y
PSI_zInt = PSI_zInt . / s q r t ( t r a p z ( p , abs ( PSI_zInt ) . ^ 2 ) ) ; % normalize
p l o t ( p , a b s ( P S I _ z I n t ) . ^ 2 , ' − ' , ' C o l o r ' , c o l o r L i n e s ( F_idx , : ) , ' L i n e W i d t h ' ,
1 . 5 ) ; h o l d on ; % p l o t w a v e f c n s
z0 = s q r t ( 2 / F _ v a l s ( F _ i d x ) ) ; % maximum v a l u e o f p o t e n t i a l f o r z > 0
T u n n e l P r o b ( F _ i d x ) = t r a p z ( p ( p> z0 ) , a b s ( P S I _ z I n t ( p> z0 ) ) . ^ 2 ) ; % t r e a t any
p a r t o f t h e wfcn t h a t i s p a s t t h i s p o i n t a s b e i n g i o n i z e d
end
x l a b e l ( ' $ z / a_0 $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ; y l a b e l ( ' $ | \ p s i ( z ) | ^ 2 $ ' , '
interpreter ' , ' latex ' ) ;
axis square ;
% C r e a t e a n o t h e r f i g u r e f o r how t h e b i n d i n g e n e r g i e s c h a n g e w i t h f i e l d
figure ;
p l o t ( F _ v a l s , EvsF , ' k− ' , ' L i n e W i d t h ' , 0 . 5 ) ; h o l d on ;
f o r F_idx = 1: l e n g t h ( F_vals )
p l o t ( F _ v a l s ( F _ i d x ) , EvsF ( F _ i d x ) , ' o ' , ' M a r k e r F a c e C o l o r ' , c o l o r L i n e s ( F_idx

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, : ) , ' M a r k e r E d g e C o l o r ' , c o l o r L i n e s ( F_idx , : ) ) ; h o l d on ;
end
x l a b e l ( ' $ eF_z a_0 / E_b$ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
y l a b e l ( ' $ ( \ h b a r \ omega −E_g ) / E_{b , 0 } $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
% C r e a t e a n o t h e r f i g u r e f o r how d i s s o c i a t i o n e f f i c i e n c y c h a n g e s w i t h f i e l d
figure ;
s e m i l o g y ( F _ v a l s , T u n n e l P r o b , ' k− ' , ' L i n e W i d t h ' , 0 . 5 ) ; h o l d on ;
f o r F_idx = 1: l e n g t h ( F_vals )
semilogy ( F_vals ( F_idx ) , TunnelProb ( F_idx ) , ' o ' , ' MarkerFaceColor ' ,
c o l o r L i n e s ( F_idx , : ) , ' M a r k e r E d g e C o l o r ' , c o l o r L i n e s ( F_idx , : ) ) ; h o l d on
end
x l a b e l ( ' $ eF_z a_0 / E_b$ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
y l a b e l ( ' $ \ e t a _ { d i s s }$ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;

292
Listing D.7: Solutions to the 2D Keldysh Potential Code
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clear ;
close all ;
numpt_vals = [100 , 100];
rvals = [5 , 15];
eps1 = 1;
eps2 = 1;
a0 = 1 ;
r 0 _ v a l s = [ l o g s p a c e ( −5 , 0 , 21) , l i n s p a c e ( 1 , 1 0 , 2 1 ) ] ;
EN = z e r o s ( 2 , l e n g t h ( r 0 _ v a l s ) ) ; % a
for rho_idx = 1: length ( r0_vals )
for idx = 1:2
numpts = n u m p t _ v a l s ( i d x ) ;
rmax = r v a l s ( i d x ) ;
l i n m e s h = l i n s p a c e ( − rmax , rmax , numpts ) ;
% one d i m e n s i o n s p a c e
lattice
[X, Y] = m e s h g r i d ( l i n m e s h , l i n m e s h ) ; % two d i m e n s i o n s p a c e
d e l t a = linmesh (2) − linmesh (1) ;
% spacing
X = X( : ) ; Y = Y( : ) ;
% a l l e l e m e n t s o f a r r a y a s a s i n g l e column
R = s q r t (X. ^ 2 + Y . ^ 2 ) ;
% d i s t a n c e from t h e c e n t e r
Vext = −2 . / R ;
% p o t e n t i a l energy
Vext = −2∗ p i ∗ a0 . / ( ( e p s 1 + e p s 2 ) ∗ r 0 _ v a l s ( r h o _ i d x ) ) ∗ StruveH0Y0 ( R∗ a0 . / r 0 _ v a l s (
rho_idx ) ) ;
e = o n e s ( numpts , 1 ) ;
L = s p d i a g s ( [ e −2∗ e e ] , − 1 : 1 , numpts , numpts ) / d e l t a ^ 2 ; % 1D f i n i t e
difference Laplacian
I = s p e y e ( numpts ) ;
L2 = k r o n ( L , I ) + k r o n ( I , L ) ; % e x t e n d L a p l a c i a n t o 2 D
H = −L2 + s p d i a g s ( Vext , 0 , numpts ^ 2 , numpts ^ 2 ) ; % H a m i l t o n i a n o f H atom
[ PSI , E ] = e i g s (H, i d x ^ 2 , ' s m a l l e s t r e a l ' ) ;
% Smallest eigenvalue
of H
E = diag (E) ;
EN( i d x , r h o _ i d x ) = mean ( E ( ( i d x − 1 ) ∗ ( i d x − 1 ) + 1 : i d x ∗ i d x ) ) ; % a v e r a g e t h e
energies
end
end
D e l t a 1 2 = a b s (EN ( 1 , : ) − EN ( 2 , : ) ) ;
Eb = a b s (EN ( 1 , : ) ) ;
CoulombVal = 9 / 8 ; % From 2D Coulomb
figure ;
p l o t ( r 0 _ v a l s . / a0 , Eb . / D e l t a 1 2 , ' b− ' , ' L i n e W i d t h ' , 2 ) ; h o l d on ;
p l o t ( l i n s p a c e ( 0 , max ( r 0 _ v a l s ) . / a0 ) , CoulombVal . ∗ o n e s ( 1 , 1 0 0 ) , ' k−− ' ) ;
x l a b e l ( ' $ r _ 0 / a_0 $ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
y l a b e l ( ' $E_b / \ D e l t a _ {12}$ ' , ' i n t e r p r e t e r ' , ' l a t e x ' ) ;
t e x t ( 5 , 1 . 1 7 5 , ' 2D Coulomb L i m i t ' ) ;