Electro-Optic Excitations in van der Waa

Electro-Optic Excitations in van der Waals Materials for Active Nanophotonics - CaltechTHESIS
CaltechTHESIS
A Caltech Library Service
About
Browse
Deposit an Item
Instructions for Students
Electro-Optic Excitations in van der Waals Materials for Active Nanophotonics
Citation
Biswas, Souvik
(2023)
Electro-Optic Excitations in van der Waals Materials for Active Nanophotonics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/tz4z-ed06.
Abstract
van der Waals materials are emerging due to their unique properties such as atomic thickness, diverse quasiparticle optical resonances, and no requirement for lattice matching. While there is a vast variety of materials, semiconductors hold a special place for opto-electronic and linear/non-linear optical studies. Black phosphorus (BP), a 2D quantum-well with direct bandgap and puckered crystal structure, is a compelling platform for this research direction. In this thesis, we investigate fundamental optical excitations in novel low-dimensional quantum materials to achieve strong light-matter interaction and integrate with nanophotonic motifs for low-footprint, reconfigurable optical technology, focusing primarily on black phosphorus and transition metal dichalcogenides.
The thesis begins with the 'thin film limit' of van der Waals materials, between 5 and 20 nm thickness range. Chapters 2 and 3 explore how few-layer black phosphorus hosts interband and intraband optical excitations that can be strongly modified with gate-controlled doping and electric field, displaying epsilon near zero and hyperbolic behavior in the mid and far-infrared. In atomic thickness, strongly bound excitonic quasiparticles dominate the optical response. In Chapter 4, we investigate electrically tunable excitons in tri-layer black phosphorus, demonstrating a reconfigurable birefringent material that, when coupled with a Fabry-Perot cavity, enables the realization of a versatile and broadband polarization modulator. In Chapter 5, we examine the ultimate limit of a monolayer, studying MoTe
via photoluminescence measurements and first-principles GW+BSE calculations, highlighting the Rydberg series associated with the exciton and its gate-tunability to understand strong electron-exciton interactions. In Chapter 6, we show how such excitons in monolayer black phosphorus can be strongly quantum confined at natural edges of exfoliated flakes, leading to highly temporally coherent emission. This emission is gate-tunable and understood via transmission electron microscopy and first-principles GW+BSE calculations of phosphorene nanoribbons to be originating from atomic reconstructions of the edge coupled with strain and screening effects.
Overall, our work highlights the potential of van der Waals materials for various electro-optical excitations and their applications in active nanophotonics.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
van der Waals materials; optics; nanophotonics; electro-optics; nanotechnology; 2D materials; black phosphorus; transition metal dichalcogenide
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Awards:
Materials Research Society Graduate Student Award - Gold (2022)
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Group:
Kavli Nanoscience Institute
Thesis Committee:
Nadj-Perge, Stevan (chair)
Hsieh, David
Faraon, Andrei
da Jornada, Felipe H.
Atwater, Harry Albert
Defense Date:
30 March 2023
Non-Caltech Author Email:
souvikdipon (AT) gmail.com
Funders:
Funding Agency
Grant Number
Department of Energy (DOE)
DE-FG02-07ER46405
Record Number:
CaltechTHESIS:04032023-062047194
Persistent URL:
DOI:
10.7907/tz4z-ed06
Related URLs:
URL
URL Type
Description
DOI
Article adapted for Chapter 2
DOI
Article adapted for Chapter 3
DOI
Article adapted for Chapter 4
DOI
Article adapted for Chapter 5
DOI
Article adapted for Chapter 6
ORCID:
Author
ORCID
Biswas, Souvik
0000-0002-8021-7271
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
15127
Collection:
CaltechTHESIS
Deposited By:
Souvik Biswas
Deposited On:
14 Apr 2023 18:03
Last Modified:
20 Feb 2025 21:11
Thesis Files
PDF (Full thesis)
- Final Version
See Usage Policy.
15MB
Repository Staff Only:
item control page
CaltechTHESIS is powered by
EPrints 3.3
which is developed by the
School of Electronics and Computer Science
at the University of Southampton.
More information and software credits
Electro-Optic Excitations in van der Waals
Materials for Active Nanophotonics

Thesis by

Souvik Biswas

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

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2023
(Defended March 30, 2023)

 2023
Souvik Biswas
ORCID: 0000-0002-8021-7271

ACKNOWLEDGEMENTS

iii

Over the course of the last 2017 days, starting from 2017, I have experienced an amazing
journey and I sincerely appreciate the assistance, direction, and motivation that have brought
me to my current position. Although it's difficult to convey my appreciation in words, I would
like to try my best to express it.
Foremost, I express my gratitude to my academic advisor, Professor Harry A. Atwater,
whose mentorship has been invaluable. I have been fortunate to work with such a successful
and accomplished individual who also possesses remarkable humility and approachability.
Our scientific conversations, particularly in one-on-one settings, were a pleasure and
provided a wealth of new ideas that will keep me occupied for years to come. His unwavering
passion for exploring new concepts, alongside his patience when faced with failed
experiments in the lab, is unmatched. As a result of his guidance, my graduate school journey
has been immensely fulfilling. Above all else, he is a benevolent and empathetic person who
prioritizes the well-being of his students and researchers — a quality that I aspire to emulate
should I have the chance.
I express my deep gratitude to my thesis committee at Caltech for generously providing their
time and invaluable feedback. From the outset of my time at Caltech, Professor Dave Hsieh
has been a constant source of support, inspiring me to explore innovative ideas while
patiently working alongside me to decipher complex ARPES data obtained at the ALS
beamline in Berkeley. Additionally, throughout my time at Caltech, Professors Stevan NadjPerge and Andrei Faraon have regularly offered invaluable insights into my research.
I would like to express my deepest gratitude to a remarkable group of scientists and
professors who have made an indelible impact throughout my PhD journey. First and
foremost, I extend my sincere appreciation to the group of Professor Jeffrey Neaton
(Berkeley) - specifically Aurelie Champagne and Jonah Haber - for engaging in regular
discussions with me on the theoretical aspects of excitons in layered materials. Professor
Felipe Jornada (Stanford) has been an exceptional mentor, providing invaluable support to
me through scientific discussions and guidance on job searches, in addition to his presence

iv
on my thesis committee. Working alongside Supavit Pokawanvit (in Felipe's group at
Stanford) has been a tremendous privilege.
I am thankful for the opportunity to be mentored by an exceptional group of scientists at
Caltech, and I hold them in high regard. My first mentor, William Whitney, introduced me
to black phosphorus, while Joeson Wong, my close associate, trained me on 2D materials,
optics, and automating measurements, among other non-academic pursuits, ranging from Ed
Sheeran to midnight food tours. Professor Zakaria Al Balushi has been a close confidant
since my first year of graduate school, introducing me to the world of TMDCs and phase
transition. Meir Grajower, my closest collaborator for over three years, is an outstanding
teammate, with extensive knowledge of solid-state physics and electronics, coupled with
strong criticism about the "real" impact of scientific work, which struck a perfect balance for
me. I learned a lot about experiments, honesty, and ethics in scientific claims from him. I am
also grateful to Professor Muhammad Alam for our discussions during my early days of PhD.
Additionally, Professors Artur Davoyan and Deep Jariwala have provided me with excellent
feedback and support on my work and career from time to time.
I will also miss the group of professors and colleagues who formed our 'boba' squad,
including Professor Pin Chieh Wu, Professor Wen Hui (Sophia) Cheng, Wei-Hsiang Lin,
and Sisir Yalamanchilli. I must say that Professor Benji Vest is one of the most enjoyable
professors I have encountered, and I am envious of his students for having such an excellent
advisor. His expertise in optics is unmatched and something to aspire to.
I had the opportunity to conduct experiments outside of Caltech and was fortunate to
collaborate with a remarkable group of scientists who provided me with invaluable support.
I would like to express my heartfelt appreciation to Dr. Hans Bechtel for his expertise in
infrared spectroscopy; Dr. Chris Jozwiack, Dr. Aaron Bostwick, and Dr. Eli Rotenberg for
their valuable contributions to ARPES; Dr. Huairuo Zhang, Dr. Wei Chang Yang, Dr. Sergiy
Krylyuk, and Dr. Albert Davydov for their assistance with crystals and electron microscopy;
and Irene Lopez and Professor Rainer Hillenbrand for their expertise in near-field
spectroscopy.

I must acknowledge the contribution of Hamidreza Akbari, who has been an excellent
collaborator and a dear friend. During our coffee breaks, I learned a lot about quantum optics
from him. I also worked closely with Melissa Li and Komron Shayegan, who taught me a
great deal about metasurfaces in visible and thermal radiation engineering. Samuel Seah is
entrusted with the responsibility of carrying forward the legacy of black phosphorus. Given
his intelligence and dedication, I am confident that the future of black phosphorus
metasurfaces is promising. I have thoroughly enjoyed working with Morgan Foley on various
aspects of metasurfaces and photonics. I am also grateful to Claudio Hail, Ramon Gao, and
Lior Michaeli for discussions on optics, metasurfaces, and opto-mechanics.
I would like to express my gratitude to a wonderful group of colleagues whom I had the
pleasure of interacting with in the Atwater group. These include Mike Kelzenberg, Ruzan
Sokhoyan, Marianne Aellen, Aisulu Aitbekova, Areum Kim, Arun Nagpal, Jared Sisler (also
for squash), Prachi Thureja, Kyle Virgil, Parker Wray, Ghazaleh Shirmanesh, Laura Kim,
Yonghwi Kim, Kelly Mauser, Cora Went, Qin (Arky) Yang, Nina Vaidya, Haley Bauser,
and Jeremy Brouillet. I am also grateful to Sumit Goel (my sports buddy), Adrian Tan (my
gossip buddy), Ubamanyu Kanthsamy, Pranav Kulkarni, Jash Banker, and the entire “Indian
gang” for their constant support.
During my PhD journey, I was fortunate to have a supportive group of friends from both my
high school and undergraduate days who helped me during my struggles. From my high
school days, I am grateful for the support of Abhishek Gupta, Soura Mondal, Chirantan
Batabyal, Sohanjit Mallick, Meghadri Sen, Nirjhar Roy, Soumya Basu, Saikat Gupta, Soham
Biswas, Rudrashish Bose, and Srirup Bagchi. From my college days, I owe a debt of gratitude
to (in random order) Samrat Halder, Sayantan Dutta, Sayantan Bhadra, Aranya Goswami,
Subhojit Dutta, Tathagata Srimani, Anusheela Das, Biswajit Paria, Mouktik Raha, Abhisek
Datta, Prof. Avik Dutt, Aniruddhe Pradhan, and Subhrajit Mukherjee (my undergraduate
mentor).

vi
I express my gratitude to the KNI community for their assistance and administrative
support, as well as to the International Student Programs at Caltech. I would like to extend a
special thanks to Laura and Daniel for their invaluable help.
Without a doubt, every individual requires the unflinching support of their family to fully
realize their potential. I am no exception to this indisputable truth. I am deeply grateful to my
beloved parents, Santanu Biswas and Dipa Biswas, for their invaluable guidance and support,
which have been instrumental in shaping my life. It is with certainty that I can declare that
without their selfless contributions, I would not have achieved the level of success that I
currently enjoy. From my childhood until now, I am fully cognizant of the numerous
sacrifices they have made to ensure my education was smooth, and for that, I am forever
grateful. Their constant encouragement and blessings serve as a driving force for me to
continue striving for greater accomplishments. I am grateful to my paternal grandmother
(Kalyani Biswas), late maternal grandparents (Heramba K. Biswas and Nilima Biswas), and
late paternal grandfather (Sunil K. Biswas) for their love and blessings. On my maternal side,
my (late) uncles (Amit Biswas and Sumit Biswas) have left an indelible mark on my life; I
miss them dearly.
Since the year 2015, my best friend, Pallavi, has been my unwavering pillar of strength, and
I am exceedingly proud to have her as my life partner. Through all the ups and downs, she
has remained steadfastly by my side, and for her resolute affection and unwavering
inspiration, I shall remain eternally thankful. I would also like to express my gratitude to my
in-laws, Prabir Banerjee and Suparna Banerjee, for welcoming me into their family.
Throughout the preceding years, I have been blessed with some truly extraordinary instances
that shall undoubtedly be forever etched in my memory as moments of immense significance
and joy.

vii

Dedicated to Ma, Bapi, and Pallavi

ABSTRACT
van der Waals materials are emerging due to their unique properties such as atomic thickness, diverse
quasiparticle optical resonances, and no requirement for lattice matching. While there is a vast
variety of materials, semiconductors hold a special place for opto-electronic and linear/non-linear
optical studies. Black phosphorus (BP), a 2D quantum-well with direct bandgap and puckered crystal
structure, is a compelling platform for this research direction. In this thesis, we investigate
fundamental optical excitations in novel low-dimensional quantum materials to achieve strong lightmatter interaction and integrate with nanophotonic motifs for low-footprint, reconfigurable optical
technology, focusing primarily on black phosphorus and transition metal dichalcogenides.
The thesis begins with the 'thin film limit' of van der Waals materials, between 5 and 20 nm thickness
range. Chapters 2 and 3 explore how few-layer black phosphorus hosts interband and intraband
optical excitations that can be strongly modified with gate-controlled doping and electric field,
displaying epsilon near zero and hyperbolic behavior in the mid and far-infrared. As the material
approaches atomic thickness, strongly bound excitonic quasiparticles dominate the optical response.
In Chapter 4, we investigate electrically tunable excitons in tri-layer black phosphorus,
demonstrating a reconfigurable birefringent material that, when coupled with a Fabry-Perot cavity,
enables the realization of a versatile and broadband polarization modulator.
In Chapter 5, we examine the ultimate limit of a monolayer, studying MoTe2 via photoluminescence
measurements and first-principles GW+BSE calculations, highlighting the Rydberg series associated
with the exciton and its gate-tunability to understand strong electron-exciton interactions. In Chapter
6, we show how such excitons in monolayer black phosphorus can be strongly quantum confined at
natural edges of exfoliated flakes, leading to highly temporally coherent emission. This emission is
gate-tunable and understood via transmission electron microscopy and first-principles GW+BSE
calculations of phosphorene nanoribbons to be originating from atomic reconstructions of the edge
coupled with strain and screening effects.
Overall, our work highlights the potential of van der Waals materials for various electro-optical
excitations and their applications in active nanophotonics.

PUBLISHED CONTENT AND CONTRIBUTIONS
*indicates equal contribution

Michelle C. Sherrott*, William S. Whitney*, Deep M. Jariwala, Souvik Biswas, Cora M. Went,
Joeson Wong, George R. Rossman, Harry A. Atwater. Anisotropic Quantum Well Electro-Optics
in

Few-Layer

Black

Phosphorus.

Nano

Letters

(2019),

19,

269-276.

doi:

10.1021/acs.nanolett.8b03876.
S.B. participated in the device fabrication and measurements with M.C.S. and W.S.W. S.B. helped
with the preparation of the manuscript.

Souvik Biswas, William S. Whitney, Meir Y. Grajower, Kenji Watanabe, Takashi Taniguchi,
Hans A. Bechtel, George R. Rossman, Harry A. Atwater. Tunable intraband optical conductivity
and polarization dependent epsilon-near-zero behaviour in black phosphorus. Science
Advances (2021), 7, 2, eabd4623. doi:10.1126/sciadv.abd4623.

S.B. conceived the experiment with inputs from H.A.A. S.B. fabricated the devices, performed the
measurements, analyzed the data, did analytical calculations, and prepared the manuscript.

Souvik Biswas*, Meir Y. Grajower*, Kenji Watanabe, Takashi Taniguchi, Harry A. Atwater.
Broadband electro-optic polarization conversion with atomically thin black phosphorus.
Science (2021), 374, 6566, 448-453. doi:10.1126/science.abj7053.

S.B. conceived the experiment with inputs from M.Y.G. and H.A.A. S.B. fabricated the devices,
performed the measurements, analyzed the data, did analytical calculations and numerical
simulations, and prepared the manuscript.

Souvik Biswas, Aurelie Champagne, Jonah B. Haber, Supavit Pokawanvit, Joeson Wong,
Sergiy Krylyuk, Hamidreza Akbari, Kenji Watanabe, Takashi Taniguchi, Albert V. Davydov,
Zakaria Y. Al Balushi, Diana Y. Qiu, Felipe H. da Jornada, Jeffrey B. Neaton, Harry A. Atwater.
Rydberg excitons and trions in monolayer MoTe2. In revisions, ACS Nano (2022), arxiv.
doi:10.48550/arXiv.2302.03720.

S.B. conceived the experiment with inputs from H.A.A. S.B. fabricated the devices, performed the
measurements, analyzed the data, did analytical calculations, and prepared the manuscript.

Souvik Biswas*, Joeson Wong*, Supavit Pokawanvit, Huairuo Zhang, WeiChang D. Yang,
Kenji Watanabe, Takashi Taniguchi, Hamidreza Akbari, Albert V. Davydov, Felipe H. da
Jornada, Harry A. Atwater. Signatures of edge-confined excitons in monolayer black
phosphorus. Submitted (2023).

S.B. conceived the experiment with inputs from J.W. and H.A.A. S.B. fabricated the devices,
performed the measurements, analyzed the data, did analytical calculations, and prepared the
manuscript with input from J.W.
This thesis contains excerpts from the aforementioned published/submitted manuscripts.

TABLE OF CONTENTS
ACKNOWLEDGEMENTS ........................................................................................................... iii
ABSTRACT .................................................................................................................................. vii
PUBLISHED CONTENT AND CONTRIBUTIONS ................................................................... ix
LIST OF ILLUSTRATIONS ........................................................................................................ xv
LIST OF TABLES .................................................................................................................... xxvii
CHAPTER 1. INTRODUCTION ................................................................................................... 1
1.1 VAN DER WAALS MATERIALS ....................................................................................... 1
1.2 EXCITONS ........................................................................................................................... 2
1.3 ELECTRICAL TUNING OF EXCITONS ............................................................................ 4
1.4 ELECTRIC-FIELD EFFECTS ON EXCITONS ................................................................... 5
1.5 MECHANICAL TUNING OF EXCITONS .......................................................................... 5
1.6 PLASMONS .......................................................................................................................... 6
1.7 PHONONS ............................................................................................................................ 6
1.8 BLACK PHOSPHORUS ....................................................................................................... 7
1.9 LIGHT-MATTER COUPLING ............................................................................................ 9
1.10 METASURFACES............................................................................................................ 11
1.11 ACTIVE NANOPHOTONICS.......................................................................................... 12
1.12 OPTICAL MODULATORS.............................................................................................. 13
1.13 SCOPE OF THIS THESIS ................................................................................................. 14
CHAPTER 2. INTERBAND EXCITATIONS IN MULTILAYER BLACK PHOSPHORUS ... 16
2.1 ABSTRACT ........................................................................................................................ 16
2.2 INTRODUCTION ............................................................................................................... 16
2.3 EXPERIMENTAL MEASUREMENTS OF THE BLACK PHOSPHORUS OPTICAL
RESPONSE WITH AN APPLIED ELECTRIC FIELD ............................................................ 18
2.4 MEASUREMENTS ON A 3.5 NM FLAKE ....................................................................... 20
2.5 MEASUREMENTS ON A 8.5 NM FLAKE ....................................................................... 23
2.6 VISIBLE FREQUENCY MODULATION ......................................................................... 26
2.7 CONCLUSIONS ................................................................................................................. 28

CHAPTER 3. INTRABAND EXCITATIONS IN MULTILAYER BLACK PHOSPHORUS... 29
3.1 ABSTRACT ........................................................................................................................ 29
3.2 INTRODUCTION ............................................................................................................... 29
3.3 OPTICAL AND ELECTRICAL CHARACTERIZATION OF MULTILAYER BP FIELD
EFFECT HETEROSTRUCTURE............................................................................................. 31
3.4 LOW ENERGY DOPING DEPENDENT INTRABAND RESPONSE IN MULTILAYER
BP .............................................................................................................................................. 37
3.5 MEASUREMENT OF THE MULTILAYER BP COMPLEX PERMITTIVITY AND
TUNABLE EPSILON-NEAR-ZERO AND HYPERBOLICITY ............................................. 39
3.6 DETERMINATION OF CARRIER EFFECTIVE MASSES IN A MULTILAYER BP 2DEG
................................................................................................................................................... 45
3.7 CONCLUSIONS ................................................................................................................. 50
CHAPTER 4. ATOMICALLY THIN ELECTRO-OPTIC POLARIZATION MODULATOR.. 51
4.1 ABSTRACT ........................................................................................................................ 51
4.2 INTRODUCTION ............................................................................................................... 51
4.3 OPTICAL CHARACTERIZATION OF ELECTRICALLY TUNABLE EXCITON IN TLBP
................................................................................................................................................... 53
4.4 THEORETICAL UNDERSTANDING OF TUNABLE ANISOTROPY IN TLBP ........... 55
4.5 CAVITY DESIGN AND BROADBAND NATURE OF BIREFRINGENCE IN TLBP .... 59
4.6 SPATIAL INHOMOGENEITIES IN TLBP-CAVITY SAMPLES .................................... 63
4.7 ELECTRICALLY TUNABLE POLARIZATION DYNAMICS........................................ 67
4.8 CONCLUSIONS ................................................................................................................. 72
CHAPTER 5. RYDBERG EXCITONS AND TRIONS IN MONOLAYER MOTE2 ................. 73
5.1 ABSTRACT ........................................................................................................................ 73
5.2 INTRODUCTION ............................................................................................................... 73
5.3 OPTICAL CHARACTERIZATION ................................................................................... 75
5.4 POWER AND TEMPERATURE DEPENDENT DYNAMICS ......................................... 79
5.5 ELECTRICALLY TUNABLE RYDBERG EXCITON EMISSION .................................. 82
5.6 DOPING DEPENDENT EXCITON EMISSION PROPERTIES ....................................... 83
5.7 DISCUSSION ..................................................................................................................... 86

5.8 CONCLUSIONS ................................................................................................................. 89
CHAPTER 6. SIGNATURES OF EDGE-CONFINED EXCITONS IN MONOLAYER BLACK
PHOSPHORUS............................................................................................................................. 90
6.1 ABSTRACT ........................................................................................................................ 90
6.2 INTRODUCTION ............................................................................................................... 90
6.3 OPTICAL SPECTROSCOPY OF EDGE EXCITONS ....................................................... 92
6.4 POLARIZATION AND TIME-DEPENDENT PHOTOLUMINESCENCE ...................... 96
6.5

ELECTRON

MICROSCOPY

IMAGING

EDGES

AND

THEORETICAL

CALCULATIONS .................................................................................................................... 99
6.6 ELECTRICAL TUNING OF EDGE AND INTERIOR EMISSION................................. 102
6.7 CONCLUSION AND DISCUSSION ............................................................................... 106
CHAPTER 7. OUTLOOK AND FUTURE DIRECTIONS ....................................................... 108
7.1 INTERBAND AND INTRABAND EXCITATIONS IN FEW LAYER BLACK
PHOSPHORUS ....................................................................................................................... 109
7.2 ATOMICALLY THIN ELECTRO-OPTIC POLARIZATION MODULATOR .............. 114
7.3 RYDBERG EXCITONS AND TRIONS IN MOTE2 ......................................................... 116
7.4 MOTE2–SCOPE AND OUTLOOK FOR PHASE TRANSITION .................................... 117
7.5 OUTLOOK FROM QUANTUM-CONFINED EXCITONS IN MONOLAYER BLACK
PHOSPHORUS EDGES WORK ............................................................................................ 120
7.6 CONCLUDING REMARKS ON BP’S POTENTIAL IN COMMERCIAL TECHNOLOGY
................................................................................................................................................. 122

Chapter S1. Supplementary Information for Electrical Control of Linear Dichroism in Black
Phosphorus from the Visible to Mid-Infrared............................................................................. 123
S1.1 Identification of Crystal Axes:................................................................................ 123
S1.2 AFM Characterization of Flake Thickness: ............................................................ 123
S1.3 Tunability for 8.5 nm Flake along Zigzag Axis: .................................................... 124
S1.4 Tunability for 8.5 nm Flake at Lower Energies:..................................................... 125
S1.5 Optical Response of Top Contact Material: ........................................................... 126
S1.6 High reflectance modulation of 6 nm BP flake: ..................................................... 127

Chapter S2. Supplementary Material for Intraband Excitations in Multilayer Black Phosphorus
..................................................................................................................................................... 129
S2.1 Unpolarized measurements ..................................................................................... 129
S2.2 Transfer matrix model ............................................................................................ 130
S2.3 Modulation line shape ............................................................................................ 131
S2.4 AFM data ................................................................................................................ 132
S2.5 Parallel plate capacitor model ................................................................................. 133
S2.6 Thomas Fermi screening model.............................................................................. 134
S2.7 Dirac-plasmonic point ............................................................................................ 135

Chapter S3. Supplementary Information for Atomically Thin Electro-Optic Polarization
Modulator ................................................................................................................................... 136
S3.1 Fabrication process ................................................................................................. 136
S3.2 Example of a typical BP staircase flake ................................................................. 137
S3.3 Optical images of devices studied for this study .................................................... 138
S3.4 Raman spectroscopy to identify BP crystal axes .................................................... 138
S3.5 Charge density calculator ....................................................................................... 139
S3.6 Schematic of the experimental setups used for optical characterization ................ 140
S3.7 Phenomenological tight-binding model for TLBP bandgap ................................... 142
S3.8 Discussion about the excitonic framework in TLBP and doping dependence ....... 143
S3.9 Extracted exciton parameters for TLBP as a function of gate voltage/doping density
......................................................................................................................................... 145
S3.10 Variation of the integrated optical conductivity (loss function) with doping ....... 147
S3.11 Transfer matrix formalism for theoretical design of cavity-based devices .......... 147
S3.12 Jones matrix for TLBP birefringence and calculation of Stokes parameters ....... 148
S3.13 Broadband polarization conversion ...................................................................... 150
S3.14 Spatial variation of refractive index in a non-cavity sample in TLBP ................. 150
S3.15 Effect of different thickness on the cavity resonance and polarization conversion
......................................................................................................................................... 151

S3.16 Effect of the incident polarization state on the polarization conversion .............. 153
S3.17 Numerical modelling of cavity-enabled polarization conversion ......................... 156
S3.18 Additional gating results from other spots on D1 ................................................. 159
S3.19 Full spectral dynamics on the normalized Poincaré sphere .................................. 160
S3.20 Normalized Stokes parameter tuning on electron and hole doping ...................... 161
S3.21 Reflectance changes on electron and hole doping ................................................ 163
S3.22 Azimuthal and ellipticity changes on electron and hole doping ........................... 164
S3.23 Additional gating results from a 5-layer BP device.............................................. 165
S3.24 Choice of three-layer BP (TLBP) ......................................................................... 166
S3.25 Comparison of polarization conversion mechanism with liquid crystals ............. 168
S3.26 Cyclic measurements for electrically tunable devices .......................................... 169
S3.27 Discussion about edge effects in spatial mapping of polarization conversion ..... 170
S3.28 Outlook towards high efficiency polarization modulators based on BP .............. 171

Chapter S4. Supplementary Information for Rydberg Excitons and Trions in Monolayer MoTe2
..................................................................................................................................................... 175
S4.1 Crystal growth, device fabrication, and experimental methods. ............................ 175
S4.2 Choice of optical geometry ..................................................................................... 177
S4.3 Estimation of Purcell enhancement ........................................................................ 178
S4.4 Comparison of monolayer and bilayer PL spectra ................................................. 180
S4.5 Optical image of MoTe2 monolayers...................................................................... 180
S4.6 Charge density calculations .................................................................................... 181
S4.7 Power dependent emission spectrum ...................................................................... 182
S4.8 Gate dependent PL fits............................................................................................ 183
S4.9 Gate dependence of additional spot. ....................................................................... 184
S4.10 Absolute intensity of exciton and trion emission modulation .............................. 185
S4.11 Energy shift of exciton and trion emission ........................................................... 186
S4.12 Comparison of MoTe2 with MoS2, MoSe2, WS2 and WSe2. ................................ 187
S4.13 Computational details ........................................................................................... 188
S4.14 Computation of exciton dispersion ....................................................................... 189
S4.15 Computed exciton absorption spectrum as a function of doping density ............. 190

S4.16 Computation of trion binding energy ................................................................... 190
S4.17 Discussion on importance of MoTe2 optical properties and its Rydberg series ... 192
S4.18 Atomic force microscope image of MoTe2 device ............................................... 193

Chapter S5. Supplementary Information for Signatures of Edge-Confined Excitons in Monolayer
Black Phosphorus ....................................................................................................................... 194
S5.1 Fabrication details................................................................................................... 194
S5.2 Experimental methods ............................................................................................ 195
S5.3 Image of samples studied........................................................................................ 196
S5.4 Visualization of strain and charge inhomogeneity in samples ............................... 200
S5.5 Polarization analysis ............................................................................................... 200
S5.6 Temperature dependent PL spectra ........................................................................ 205
S5.7 Power dependent spectra and statistics ................................................................... 206
S5.8 Spectral diffusion .................................................................................................... 208
S5.9 Gate-dependent PL spectra of additional spots ...................................................... 209
S5.10 Lifetime of PL emission ....................................................................................... 210
S5.11 Electrostatic simulations of capacitance ............................................................... 211
S5.12 Band bending schematic ....................................................................................... 214
S5.13 Screening effects ................................................................................................... 216
S5.14 Discussion on edge exciton emission ................................................................... 217
S5.15 First-Principles Computation Of Optical Spectrum ............................................. 219
S5.16 TEM analysis of monolayer BP ............................................................................ 222

Bibliography ............................................................................................................................... 223

LIST OF ILLUSTRATIONS
Figure 1.1. Conceptual schematic of Lego-like behavior in van der Waals heterostructures.
Different materials can be stacked to form new materials with different properties compared to
their parent. Figure taken from Ref. 1 _____________________________________________ 21
Figure 1.2. Schematic of exciton and absorption. (a) Coulomb field lines for an exciton in a 3D
bulk semiconductor and monolayer 2D semiconductor showing reduced screening from the
environment. (b) Optical absorption showing excitonic state and quasiparticle gap for 2D and 3D
case. Figure taken from Ref. 15 __________________________________________________ 23
Figure 1.3. An overview of possible material resonances in the van der Waals library. Each
quasiparticle induces a susceptibility resonance in the optical response of the material that can
interact with light strongly, creating hybrid light-matter modes–polaritons. From left to right–
electrons/holes in graphene and black phosphorus, lattice vibrations in hBN and topological
insulators, electron-hole quasiparticles in semiconductors, cooper pairs in superconductors and
magnons in magnetic materials. This figure is taken from Ref. 53 _______________________ 26
Figure 1.4. Overview of black phosphorus. (a) Side view of anisotropic puckered crystal structure
of layer BP. (b) Top view of a monolayer BP crystal structure. (c) Angle resolved photoemission
spectroscopy measurement of band structure in bulk BP with the band minima being at the 𝛤/𝑍
point. (d) Layer dependent band gap of BP showing strong interlayer interactions. This figure is
taken from Ref. 54 ____________________________________________________________ 28
Figure 1.5. Different regimes of weak coupling. A ring-resonator system is shown for illustrative
purposes with three regimes identified–critical coupling (𝑡 = 𝑎), over-coupling (𝑡 < 𝑎) and undercoupling (𝑡 > 𝑎). Their optical response (transmission) is also plotted. This figure is taken from
Ref. 59 _____________________________________________________________________ 30
Figure 2.1. Anisotropic electro-optical effects in few-layer BP. Anisotropic electro-optical effects
in few-layer BP. (a) Schematic figure of infrared tunability devices. Few-layer BP is mechanically
exfoliated on 285 nm SiO2/Si and then capped with 45 nm Al2O3 by ALD. A semitransparent top
contact of 5 nm Pd is used to apply field (VG1) while the device floats and 20 nm Ni/200 nm Au
contacts are used to gate (VG2) the contacted device. (b) Crystal structure of BP with armchair and
zigzag axes indicated. (c) Illustration of two field-driven electro-optical effects: the quantum-

11
confined Stark effect, and symmetry-breaking modification of quantum well selection rules. In
the quantum-confined Stark effect, an external field tilts the quantum well energy levels, causing
a red-shifting of the intersubband transition energies. In the observed modification of selection
rules, this field breaks the symmetry of the quantum well and orthogonality of its wavefunctions,
allowing previously forbidden transitions to occur. (d) Illustration of anisotropic Pauli-blocking
(Burstein-Moss effect) in BP. Intersubband transitions are blocked due to the filling of the
conduction band. Along the ZZ axis, all optical transitions are disallowed regardless of carrier
concentration. (e) Raman spectra with excitation laser polarized along AC and ZZ axes. The
strength of the Ag2 peak is used to identify crystal axes. ______________________________ 38
Figure 2.2. Electrically tunable linear dichroism: quantum-confined Stark and Burstein-Moss
effects and forbidden transitions. (a) Optical image of fabricated sample; (b) Zero-bias infrared
extinction of 3.5 nm flake, polarized along armchair (AC) axis. (c) Calculated index of refraction
for 3.5 nm thick BP with a Fermi energy at mid-gap. (d) Tunability of BP oscillator strength with
field applied to floating device, for light polarized along the AC axis. (e) Corresponding tunability
for light polarized along the zigzag (ZZ) axis. (f) Tunability of BP oscillator strength with gating
of contacted device, for light polarized along the AC axis. (g) Corresponding tunability for light
polarized along the ZZ axis. ____________________________________________________ 40
Figure 2.3. Variation of Tunability with BP Thickness. (a) Optical image of fabricated 8.5 nm
sample. (b) Zero-bias extinction of 8.5 nm flake, polarized along AC axis. (c) Calculated index of
refraction for 8.5 nm thick BP. (d) Tuning of BP oscillator strength with field applied to floating
device, for light polarized along the AC axis. (e) Tuning of BP oscillator strength with gating of
contacted device, for light polarized along the AC axis. ______________________________ 43
Figure 2.4. Tunability for 8.5 nm Flake along Zigzag Axis. (a) Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis. (b) Tunability of BP
oscillator strength with gating of contacted device, for light polarized along the ZZ axis. ____ 44
Figure 2.5. Tunability for 8.5 nm Flake at Lower Energies. Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis, measured at lower
photon energies. _____________________________________________________________ 45
Figure 2.6. Tunability in the Visible. (a) Schematic figure of visible tuning device. Few-layer BP
is mechanically exfoliated on 45 nm Al2O3/5 nm Ni on SrTiO3 and then coated with 45 nm Al2O3.
A 5 nm thick semitransparent Ni top contact is used. (b) Optical image of fabricated sample with

12
20 nm thick BP. Dashed white line indicates the boundary of the top Ni contact. (c) Tuning of
extinction with field applied to floating device, for light polarized along the AC axis. (d)
Corresponding tuning for light polarized along the ZZ axis. (e) Calculated index of refraction for
20 nm thick BP for the measured energies. (f) Calculated imaginary index of refraction of several
thicknesses of BP from the infrared to visible. ______________________________________ 46
Figure 3.1. Device schematic and electro-optic characterization. (A) Anisotropic puckered crystal
structure of BP (P atoms are in sp3 hybridization). (B) Device schematic and measurement scheme
for hBN encapsulated BP devices. (C) Optical microscope image of the device discussed in the
main text. (D) Normalized reflection spectrum from the BP device shown in (C). (E) Color-map
of source-drain current variation as a function of both gate voltage and source-drain bias. (F) Gate
voltage modulated source-drain current at one representative source-drain voltage (100mV). (G)
Variation of source-drain current with source-drain voltage showing linear conduction with
systematic increase as gate voltage increases on the positive side, the slight dip is due to the fact
that the MCP is not at 0V). (H), (I) Interband optical modulation along the AC and ZZ axis
respectively showing the anisotropy in the electro-optic effects. (J) Schematic of changes in the
AC axis optical conductivity (real part) upon doping. ________________________________ 51
Figure 3.2. Quantum Well electro-optic effects. Schematic of different electro-optic effects
occurring at energies near and above the band-edge of a multilayer BP thin film. __________ 55
Figure 3.3. Intraband response dominated reflection modulation. (A), (C) Measured (colored lines)
and simulated/fit (black lines) intraband response mediated reflection modulation along the AC
and ZZ axis. The fits have been performed between 750-2000 cm-1 to eliminate any band-edge
effect influence on the optical conductivity so that the Drude model suffices. (B), (D) Fits shown
separately, without offset showing a narrowing and strengthening of the Fano-like response near
the hBN and SiO2 phonons with increasing charge density in BP. (E), (F) Modelled false color plot
of modulation in reflection spectra (zoomed in between 800 and 1600 cm-1) as a function of doping
density for the AC and ZZ direction assuming the following parameters : BP meff=0.14m0 (AC),
0.71m0(ZZ), Si meff=0.26m0 (electrons), 0.386m0 (holes). _____________________________ 56
Figure 3.4. Modelled dielectric function and tunable hyperbolicity. (A), (C) Extracted real and
imaginary part (denoted as 𝜖1 and 𝜖2) of the dielectric function for BP 2DEG along the Armchair
axis for different doping densities. The orange shaded region shows the ENZ behavior. The region
where the real part of the permittivity along the AC axis goes negative while remaining positive

13
for the ZZ direction is the hyperbolic region and extends to frequencies beyond our measurement
window. (B), (D) The same for the Zigzag axis. (E) False color plot of the modelled real part of
the dielectric permittivity along the AC direction assuming BP meff=0.14m0 showing the tunability
of ENZ. (F) Calculated isofrequency contours for in-plane plasmonic dispersion (TM polarized
surface modes) showing the tunability of hyperbolicity. ______________________________ 59
Figure 3.5. Schematic of electrostatic gating in BP device. The formation of an inversion layer is
indicated at the interface of SiO2/Si and bottom hBN/BP (of opposite parity). The BP can be
modelled as two separate parts–1. Actively electronically modulated labeled as “BP 2DEG” which
is ~2.9 nm thick from Thomas-Fermi screening calculations and 2. A non-modulated thick region
labeled as “BP bulk” which extends to the remainder of the physical thickness of the BP flake as
measured by atomic force microscopy (AFM). _____________________________________ 62
Figure 3.6. Refractive index of doped BP. (A) Extracted real part of refractive index of BP 2DEG
as a function of voltage for AC excitation. (B) Extracted imaginary part of refractive index of BP
2DEG as a function of voltage for AC excitation. (C),(D) Same as (A),(B) but for ZZ excitation.
(E),(F) Same as (A),(B) but for unpolarized excitation. _______________________________ 63
Figure 3.7. Optical conductivity of doped BP. (A) Extracted real part of optical conductivity of BP
2DEG as a function of voltage for AC excitation. (B) Extracted imaginary part of optical
conductivity of BP 2DEG as a function of voltage for AC excitation. (C),(D) Same as (A),(B) but
for ZZ excitation. (E),(F) Same as (A),(B) but for unpolarized excitation. Here, 𝜎0 = 𝑒 2 /4ℏ . 64
Figure 3.8. Extracted Drude weight and effective mass for BP. (A) Drude weight evolution
obtained from fitting reflection data for AC and ZZ axis, plotted with expected Drude weight. (B)
Extracted effective mass from the Drude weight fits plotted versus voltage/charge density
assuming a parallel plate capacitor model and 100% gating efficiency. __________________ 65
Figure 3.9. Subband effect in BP dispersion. (A) Calculation of transition energies in 18.68nm BP.
(B) Corresponding effective mass along the Armchair direction. _______________________ 68
Figure 4.1. Schematic of electrically tunable polarization conversion and TLBP birefringence (A)
Schematic of cavity design and polarization conversion. TLBP is incorporated in a dielectric
environment between two mirrors (one partially reflective (top) and one highly reflective
(bottom)). The incoming beam is linearly polarized, and the output beam can be azimuthally
rotated or converted between circular and linear polarization with applied voltage (between the
TLBP and the back electrode/mirror), for a fixed wavelength. (B) Experimentally measured

14
polarized absorption from a TLBP device (non-cavity integrated) for different doping densities,
along the armchair (AC) direction. The zigzag direction remains featureless for all conditions.
(C),(D) Extracted complex refractive index (real and imaginary part, respectively) for TLBP as a
function of doping density for the AC and ZZ direction. ______________________________ 73
Figure 4.2. Coulomb screening and scattering of quasi-1D excitons (A) Due to electrical gating,
free charges increase which reduce the overall attraction between the bound electron-hole pair for
the quasi-1D excitons along the AC direction. This screens the electric field lines between them
and weakens the exciton, leading to a reduction in binding energy and oscillator strength (𝑓0 ). (B)
Due to increased charges, excitons now scatter off them much more readily, leading to reduced
coherence and broadening of spectral transitions, manifested as larger linewidths (𝛤). ______ 75
Figure 4.3. Absorption modulation schematic upon doping. Illustrated modulation in absorption
reflecting changes in the optical density of states upon doping showing a reduction of exciton
oscillator strength and broadening of the transition along with bandgap renormalization and
reduction of quasi-particle (QP) band-edge coupled with a reduction in exciton binding energy
rendering the exciton resonance nearly unchanged spectrally. __________________________ 77
Figure 4.4. Example cavity design for polarization conversion and large anisotropy bandwidth
experimental demonstration. (A) Side view of a typical cavity structure adopted in this work. The
top and bottom mirrors are formed by thin and thick Au films. The cavity is comprised of hBN
encapsulated TLBP and PMMA, which acts as the tunable part of the cavity (in determining the
resonance wavelength). (B),(C),(D) Theoretically calculated complex reflection phasor, reflection
amplitude, and phase spectrum, respectively, for such a typical cavity structure having resonance
~1480 nm, showing difference in both the parameters along AC and ZZ, establishing polarization
conversion. (E) Summary of reflection amplitude spectra from 5 representative devices fabricated
as part of this study showing tunable cavity resonance. The PMMA thickness was tuned
systematically to change the resonance over 90 nm across the telecommunication band (E,S and
C). (F)-(J) Experimentally measured spectral trajectories on the normalized Poincaré sphere
corresponding to the 5 device resonances plotted in (E). All trajectories show strong spectral
polarization conversion (either in the azimuthal orientation or the ellipticity or both). The
difference in the trajectories are intimately related to the critical coupling between the cavity and
the incoming polarization. For all the presented trajectories, the azimuthal orientation was aligned
nearly 45 degrees to the AC and ZZ direction of the TLBP flake. For each normalized Poincaré

15
sphere, the blue arrows mark the beginning of the spectral scan (1410nm for D1-4, 1500 for D5)
and the red arrows mark the end (1520nm for D1-4, 1575 for D5)–also shown as stars in x-axis of
(E). _______________________________________________________________________ 78
Figure 4.5. Amplitude and phase shift dependence on cavity parameters. Effect of the top Au and
PMMA thickness are studied on the cavity performance. (A) Resonance of the cavity (along the
AC direction) showing redshifts with increasing PMMA thickness and blueshifts with increasing
metal thickness. (B) Reflection amplitude of the cavity (along the AC direction) showing the
“critical coupling” trace as a function of top Au and PMMA thickness. (C) Maximum phase shift
difference between the AC and ZZ direction plotted as a function of top Au and PMMA thickness.
Strong phase shift difference traces follow the reflection amplitude trace, highlighting the
importance of critical coupling. _________________________________________________ 81
Figure 4.6. Spatial inhomogeneity in optical anisotropy probed by polarization conversion. (A)
Spatial maps of ellipticity angle (in degrees) of device D4 for 4 different wavelengths near the
resonance (~1490 nm) of the cavity. Black lines indicate the extent of the tri-layer region (Sample
optical image shown in inset of (E)). Scale bar (in white) corresponds to 10μm. (B) Same as (A),
but for azimuthal angle. (C),(D) Ellipticity and azimuthal angle spectral scans for a few points
(marked with appropriately colored stars in (A) and (B)), showing spatial variation of the
resonance in the tri-layer region, as well as flat background response from the bare cavity and
weak polarization conversion from the 6-layer region. (E) Zoom-in spatial colormap of ellipticity
at 1495 nm along with superimposed reflected polarization ellipses at each point, for better
visualization of co-variation of azimuthal and ellipticity angles. White lines correspond to right
handedness, while black lines correspond to left handedness. Scale bar (in black) corresponds to
5um. Inset. Optical image of the device (D4) outlining the 3-layer (3) region. Also shown is the 6layer region (6) and the bare cavity (0). (F) Spatial map of maximum ellipticity for each point
within a spectral window between 1450 nm and 1520 nm. (G) Spatial map of ellipticity resonance
wavelength (filtered for 𝜒>10o to only highlight the 3L region). (H),(I) Histograms of ellipticity
resonance wavelength (filtered for |𝜒|>10o) and maximum ellipticity (in degrees), where 1 pixel
on the map corresponds to 1 sq.μm. ______________________________________________ 83
Figure 4.7. Azimuthal spatial colormaps of sample D2 for different wavelengths. Scale bar
corresponds to 10 μm. 3L region is outlined in black dashed lines for 1456 nm spatial map, while
thicker region is outlined in green. _______________________________________________ 85

16
Figure 4.8. Ellipticity spatial colormaps of sample D2 for different wavelengths. Scale bar
corresponds to 10 μm. 3L region is outlined in black dashed lines for 1466 nm spatial map, while
thicker region is outlined in green. _______________________________________________ 86
Figure 4.9. Electrically tunable polarization dynamics. (A),(B),(C),(D) False colormaps of the
evolution of the intensity (S0) and the three normalized Stokes parameters (s1,s2,s3), determining
the polarization state of the reflected light, as a function of wavelength and positive voltages (for
electron doping). The results are from device D1. Continuous tuning of all the 4 parameters can
be seen around the cavity resonance (~1440nm) for the entire range of doping, illustrating efficient
tuning of the polarization state with voltage. (E) Voltage dependent trajectories on the normalized
Poincaré sphere for 9 different wavelengths showing large dynamic range in tunability of the
reflected polarization state. Each color corresponds to a wavelength (same color code in (F)). The
dark arrows mark the beginning of the voltage scan (0V), and the correspondingly colored arrows
indicate the end of the voltage scan (-40V)–hole doping. (F) Visualization of the measured
reflected polarization ellipse for selected voltages and the same 9 wavelengths as in (E). At 1442
nm a strong change in ellipticity is seen where the state becomes almost circular at -18V and the
ellipticity decreases for higher voltages–acting like a QWP. The change in ellipticity is associated
with a change in the azimuthal orientation of the beam. At 1444 nm however, minimal change in
ellipticity is seen with a strong change in the azimuthal orientation–effectively behaving like a
HWP. The solid (dashed) lines correspond to right (left) handedness. ___________________ 88
Figure 4.10. Normalized Poincaré sphere dynamics and polarization conversion for electron
doping. (A) Voltage dependent trajectories for 9 different wavelengths from 0V to +50V (electron
doping) showing highly versatile polarization generation. Colors correspond to the same 9
wavelengths shown in (B). (B) Two-dimensional map of generated polarization states as a function
of wavelength and voltage for 9 wavelengths and select few voltages. Half-wave plate (HWP) like
operation is seen for 1441 nm, whereas quarter-wave plate (QWP) like operation is seen for 1439.5
nm. _______________________________________________________________________ 90
Figure 5.1. Electro-optic investigation of Rydberg excitons in monolayer MoTe2. (a) Excitonic
energy landscape of Rydberg series in monolayer MoTe2 with the quasiparticle band structure,
exciton state energies 𝛺𝑠 , and exciton binding energies 𝐸𝐵𝑆 obtained using GW-BSE calculations.
(b) Investigated device geometry consisting of hBN encapsulated monolayer MoTe2 on Au
substrate with applied gate voltage. (c) Integrated PL intensity map of investigated sample at 4K.

17
Bright spots indicate monolayer. Inset–optical micrograph of sample. (d) Example PL spectra with
assigned Rydberg states. _______________________________________________________ 94
Figure 5.2. Narrowest emission line obtained for the A1s neutral exciton transition. ________ 96
Figure 5.3. Pump power and temperature dependence of Rydberg excitons. (a) PL intensity
variation with increasing pump power for the 1s exciton. Inset–PL(x100) for the 2s and 3s exciton.
(b) Semi-log scale plot of intensity dependence of PL with pump power and corresponding fits to
a power-law showing excitonic emission (𝐼𝑃𝐿 = 𝐼0 𝑃𝛼 ). (c), (d) Temperature variation of
normalized PL spectrum for the 1s and 2s exciton regions, respectively. (e), (f) Fits to a
temperature model estimating different parameters for the 1s exciton and 2s exciton. _______ 98
Figure 5.4. Temperature dependent emission properties. (a) Temperature dependent linewidth
broadening of the A1s and A2s excitonic state. (b) Intensity ratio of the 2s/1s excitonic state with
temperature (1/T). ___________________________________________________________ 100
Figure 5.5. Gate dependent PL spectrum of Rydberg excitons. (a), (d) PL intensity of different
neutral exciton species and their corresponding trion features as a function of gate voltage near the
1s and 2s/3s resonance, respectively. (b), (e) Derivative of the PL spectra, 𝑑𝑃𝐿/𝑑𝐸, shown in (a),
(d). (c), (f) Line cuts of the PL spectrum at different voltages showing the different exciton and
trion resonances. ____________________________________________________________ 101
Figure 5.6. Gate tunable PL properties of Rydberg excitons. (a), (b) PL intensity (normalized) of
different neutral exciton, trion species, respectively, as a function of charge density. (c) Energy
shifts between the neutral exciton and the trion for 1s and 2s states as a function of charge density.
(d) Evolution of the linewidth and (e) resonance energy of various exciton and trion states as a
function of charge density. ____________________________________________________ 103
Figure 5.7. Doping dependent theoretical results. Doping dependence of (a) the variation in exciton
binding energy, 𝛥𝐸𝑏 , for the ground A1s exciton, and the excited A2s and A3s states, (b) the
variation in exciton energy 𝛥𝛺𝑠 (black curve), the exciton binding energy 𝛥𝐸𝑏 (blue curve), and
the renormalization of the QP band gap 𝛥𝐸𝑔 (red curve) for the ground A1s exciton, (c) the
oscillator strength for the A1s, A2s and A3s states. (inset) Same as (c) plotted in semi-log scale on
the y-axis. _________________________________________________________________ 106
Figure 5.8. Wave-function evolution with doping density. (a)-(f) Evolution of the exciton (A1s)
wavefunction as a function of doping density showing Pauli blocking. (b) 2.3 x1011 cm-2 (c) 1.6
x1012 cm-2 (d) 3.0 x1012 cm-2 (e) 4.5 x1012 cm-2 (f) 5.9x1012 cm-2. _____________________ 107

18
Figure 6.1. Emergence of edge excitons in monolayer BP. (a) Schematic of two different emission
mechanisms originating from interior (orange) excitons and edge (blue) excitons and typical
heterostructure schematic studied in this work. (b) Integrated PL intensity map of a MLBP sample
(#gateD1) with spatial points marked along certain edges showing bright and distinct peaks (scale
bar is 5 𝜇𝑚). (c) Normalized PL spectrum from points marked in (b). (d) Normalized PL spectra
from an edge and the interior of the sample shown in (b) to highlight the spectral feature
differences. (e) Variation of PL spectrum (log scale color) with incident laser power (y-scale log)
for an edge spot containing three distinct peaks. (f) Power law fit to the integrated PL intensity for
the corresponding three peaks in (e) showing linear excitonic behavior (𝛼~1), plotted in log-log
scale. _____________________________________________________________________ 113
Figure 6.2. Lorentzian fit to monolayer BP emission. 2-peaks fit to an interior exciton emission
from the interior of device #gateD1. The peak at ~1.7 eV is the interior exciton while the lower
energy shoulder peak is likely from defects in the native crystal. ______________________ 115
Figure 6.3. Polarization and time-dependent emission dynamics. (a) PL spectrum from one edge
site as a function of polarization (measured by rotating the analyzer on the emission side). The
emission is nearly aligned with each other and the interior background with small azimuthal
mismatch between the peaks. (b) Polar plots, along with dipolar fits, to determine the azimuthal
orientation of different peaks corresponding to emission profiles in (a). (c) Comparison of the
interior exciton orientation versus the edge states (as determined by fitting the peaks to Lorentzian
shaped discrete edge states and an interior contribution). The color of the spots is determined by
the emission energy, as shown in the color bar (in eV). (d) Spectral wandering and blinking seen
in emission as a function of time for one representative edge site. _____________________ 116
Figure 6.4. Temperature dependent emission spectrum from one spot showing diminishing
emission strength as well as linewidth broadening as temperature is increased (measured on sample
#D6). _____________________________________________________________________ 118
Figure 6.5. Structural and theoretical characterization of BP edges. (a) High-resolution TEM image
of free-standing monolayer BP with minimum exposure to air showing an interface (magenta)
between the crystalline (left) and the amorphous (right) region. The crystalline image matches well
with a phase contrast image (inset with orange borderlines) simulated using a multi-slice algorithm
(QSTEM) with a defocus about -5.5 nm away from the Scherzer defocus. (b) Positions of the two
neighboring P atoms (dark contrast) identified and labeled with color-coded dots within the

19
indicated area (red borderlines) in (a), based on the phase contrast variance, allow for the
measurement of projected spacings. (c) Histogram of the projected spacings fitted with two
Gaussian profiles: Peak 1 (turquoise, solid) and Peak 2 (red, dashed), suggesting a bimodal
distribution of the interior (0.295 nm ± 0.008 nm) and edge (0.321 nm ± 0.008 nm) spacings in the
monolayer BP. Inset shows simulated ZZ4-i nanoribbon edge. (d) DFT level computation of band
structure of 25 favorable edge reconstructions in BP. (e), (f) Quasiparticle band structure for the
AC12-i and ZZ4-i edge-terminated structures computed at the GW level of theory and projected
over the edge and (2D) interior states. Optical absorption spectrum computed from first-principles
GW-BSE calculations for the AC12-i and ZZ4-i structures. Dash lines indicate the position of the
edge exciton energy, with an inset showing the electron and hole contribution of the edge exciton.
The oscillator strength of both edge excitons is on the order of 10-2 times smaller than the lowestenergy peak. Iso-contour shows the exciton wavefunction squared with the electron (Fe) and hole
(Fh) coordinates integrated out. _________________________________________________ 119
Figure 6.6. Gate tunable edge emission. (a) Schematic of gate-tunable heterostructure geometry
showing uniform doping region in the interior and fringe-field effects at the edge. (b) PL spectrum
from an edge site (from #gateD1) as a function of gate voltage between -5V and 5V, taken in steps
of 10mV. Inset shows a PL spectrum at 0.16V revealing two additional red-shifted peaks
corresponding to the edge exciton along with the interior exciton emission. (c) Same as (b) but
normalized for each voltage to the maximum emission feature to highlight the dominant spectral
features in each band. (d) PL intensity variation for the three features marked in (c) and (d) in
yellow (interior contribution) and orange and blue (edge contribution), as obtained from
Lorentzian fits to each spectrum at a given voltage. Marked in shaded grey is the “on” voltage
window of the edge excitons. __________________________________________________ 121
Figure 6.7. Comparing interior emission between interior and edge. (a) PL spectra as a function of
gate voltage for a spot in the interior of device #gateD1 showing reduction of excitonic emission
as the voltage is increased. (b) Gate dependent peak intensity of interior exciton emission when
the laser spot is excited on the interior of the sample and the edge. Both show monotonic
dependence with differences only in the absolute intensity of emission. (c) Shifts in peak energy
for the interior exciton as a function of gate voltage to compare the differences between having the
spot on the interior and on the edge. A stronger shift is seen at the edge implying that the in-plane

20
Stark effect also plays a role in determining the interior exciton dynamics at the edge along with
doping induced screening effects. _______________________________________________ 123
Figure 6.8. Linear fits to resonance energies corresponding to the two edge exciton peaks (blue
and orange) and interior exciton peak (yellow), as illustrated in Fig. 6.7. ________________ 124
Figure 7.1. Black phosphorus hyperbolic plasmons device. a) Schematic of gate-tunable black
phosphorus plasmonic resonator device, including a silicon dielectric spacer and gold back
reflector. b) Scanning electron microscope image of nanoribbons fabricated in few-layer black
phosphorus, aligned to its crystal axes. c) Simulated absorption modulation for 100 nm black
phosphorus ribbons in the device geometry illustrated in (a), normalized to the 1x1012 cm-2 doping
case. The observed resonances are a convolution of spectrally narrow Fabry-Pérot modes due to
the dielectric spacer and a single broad surface plasmon mode. b) Electric field intensity profile
surrounding an individual nanoribbon at 30 microns for 7.5x1012 cm-2 doping. ___________ 129
Figure 7.2. Infrared optical results on BP nanoribbons. a) Armchair axis absorption modulation
measured in 100 nm black phosphorus nanoribbons on a Salisbury screen device, as described in
Figure 6.2. Three separate trials indicate nearly identical absorption modulation with doping, while
the (green) baseline without doping shows zero modulation. b) A dark field optical microscope
image of a completed nanoribbon Salisbury screen device. The large red circle is the rim of the
silicon membrane, the yellow finger extending down from the top is a gold contact, and the bright
blue regions are the nano-patterned portion of the black phosphorus flakes. ______________ 130
Figure 7.3. Dual trilayer BP-based metasurfaces. (a) Proposed schematic of twisted BP
metasurface. (b) Optical microscope image of a fabricated heterostructure containing two trilayers
of BP twisted at ~90 degrees. (c) Dark field microscope image of patches of resonators of varying
radii fabricated in a heterostructure containing hBN-BP-hBN-Au. (d)-(f) S0, s1, s3–Stokes
parameter calculated for structure (a) optimized for critical coupling as a function of top and
bottom gate voltage (represented as BP oscillator strength in meV) at a wavelength of 1448.48 nm.
__________________________________________________________________________ 134
Figure 7.4. Schematic of FET-like devices studied for Mo1-xWxTe2 Raman spectroscopy. Two
contacts, source and drain were pre-patterned on the ionic substrates and a back contact was
evaporated using electron-beam evaporation. Exfoliated flakes were then transferred using the drytransfer method. ____________________________________________________________ 138

21
Figure 7.5. ARPES measurements of MoTe2. (a) Pristine band structure of 2H phase of MoTe2
bulk crystal. (b) Band structure of K-dosed crystal showing bandgap renormalization and
appearance of conduction band minimum. VBM–valence band maximum, CBM–conduction band
minimum. _________________________________________________________________ 139
Figure S1.1. AFM Characterization of Flake Thickness. (a) AFM crosscut of ‘3.5 nm’ thick flake,
showing measured thickness of 6.5 nm. (b) AFM crosscut of ‘8.5 nm’ thick flake, showing
measured thickness of 11.5 nm. (c) AFM crosscut of ‘20 nm’ thick flake, showing measured
thickness of 20 nm. Thicknesses have some uncertainty due to Ni/Al2O3 top layers. _______ 143
Figure S2.1. Unpolarized response from BP device. (A) Unpolarized reflection modulation at the
band-edge for different electron and hole densities. (B) Same for below the band-edge region along
with fits (dotted black lines). (C) Assuming a parallel plate capacitor model, extracted effective
mass of free carriers (D) Fits to the experimental data (some curves at very low voltages [close to
MCP] do not fit well due to extremely low signal and have been avoided) _______________ 148
Figure S2.2. SiO2 refractive index. n,k data adopted for SiO2 (dominated by phonons). ____ 150
Figure S2.3. Fano response in the system. (A) Fano like response for BP/SiO2 system. (B) Same
for BP/hBN system. _________________________________________________________ 151
Figure S2.4. AFM data. (A) AFM line scan for top hBN. (B) AFM line scan for BP flake. (C) AFM
line scan for bottom hBN. _____________________________________________________ 152
Figure S2.5. Capacitor model for BP. Charge density induced in BP as calculated from parallel
plate capacitor model. ________________________________________________________ 153
Figure S2.6. Band bending in BP. Thomas Fermi screening calculation in BP as the charge density
varies from 1011/cm2 to 1013/cm2. Inset–zoomed in upto 4 nm. ________________________ 153
Figure S2.7. Isofrequency contours (IFC) around Dirac-plasmonic point. IFCs are calculated for
in-plane propagating plasmon (TM) modes at two frequencies–676.7 cm-1 and 659.9 cm-1 and two
carrier densities–4.22x1012/cm2 (electron) and 6.51x1012/cm2 (hole). For higher doping densities,
the IFCs are less sensitive to small changes in the frequency, however for lower doping densities,
the IFCs are quite sensitive to small changes in the frequency and flip the sign of the hyperbolic
dispersion. They also become almost linear adopting a Dirac-like nature. _______________ 154
Figure S3.1. Schematic of the fabrication process illustrating the pickup process. _________ 156

22
Figure S3.2. Optical image of a BP flake. A typical exfoliated BP staircase flake on PDMS. 1,2
and 3 layers are marked–confirmed with optical contrast. Other thicknesses can also be seen. Scale
bar corresponds to 50 μm. _____________________________________________________ 156
Figure S3.3. Optical images of representative devices investigated for this study. (A)–Non-cavity
device for extracting electrically tunable complex refractive index of TLBP (shown in Fig. 2). (B),
(C)–Passive cavity integrated devices. (D), (E), (F)–Active cavity integrated devices. The white
outlines denote BP, while the blue outlines denote the contacting few layers graphene flake. 157
Figure S3.4. Polarized Raman spectroscopy for BP axis identification. Raman spectrum for TLBP
as a function of incident linear polarization excitation. A1g, B2g and A2g modes are seen clearly.
Strongest response from the A2g mode is seen along the armchair (AC) orientation, whereas along
the zigzag (ZZ) direction it is the weakest. ________________________________________ 158
Figure S3.5. Parallel-plate capacitor model. Estimated charge density versus applied gate voltage
for Device D1, using the parallel plate capacitor model. _____________________________ 159
Figure S3.6. Broadband reflectivity characterization setup. Schematic of the optical setup used to
characterize the complex refractive index of TLBP as a function of doping density. LP–linear
polarizer (wire-grid), PD–photodetector (Ge), Ref. PD–Reference photodetector (Ge), BS–Beam
splitter. Blue arrows denote optics on flip mounts. _________________________________ 160
Figure S3.7. Polarization conversion measurement setup. Schematic of the optical setup used to
characterize the polarization conversion. LP–linear polarizer, HWP–halfwave plate, Ref. PD–
Reference photodetector (InGaAs), BS–Beam splitter. ______________________________ 161
Figure S3.8. Binding energy change with screening length and doping. (A) Calculated binding
energy of the ground state exciton as a function of screening length using the Rytova-Keldysh
potential. (B) Band-bending (screening profile) as a function of doping density in BP. _____ 163
Figure S3.9. Exciton parameter modulation with gate voltage for TLBP. Tuning of the exciton
resonance parameters as a function of applied gate voltage. (A), (B) and (C) show changes in the
resonance wavelength, oscillator strength and the linewidth of the excitonic resonance as a
function of gate voltage, respectively. An inset in (A) shows the relation between the applied gate
voltage and the estimated charge density in the BP 2DEG. ___________________________ 165
Figure S3.10. Integrated real optical conductivity variation with doping. The real part of optical
conductivity is proportional to the loss function (∝ 𝐼𝑚(𝜖)) which dictates the overall optical
response for such thin films. As doping is increased on either side, a drop in the loss function

23
indicates reduced absorption due to screening of the excitons via free charges leading to a reduction
in binding energy and oscillator strength. The integration (over optical measurements bandwidth)
assumes a single excitonic feature and no other oscillators. ___________________________ 166
Figure S3.11. Broadband polarization conversion simulations. (A) Reflectance along the AC and
ZZ direction for a cavity with parameters matching D1. (B), (C), (D) Normalized Stokes
parameters (s1, s2, s3) as a function of cavity length obtained by tuning the PMMA thickness
showing efficient broadband polarization conversion. _______________________________ 169
Figure S3.12. Spatial optical inhomogeneity in TLBP samples. Spatial variation of real (A) and
imaginary (B) part of complex refractive index in a TLBP flake. ______________________ 170
Figure S3.13. Effect of thickness on the cavity resonance and polarization conversion. (A), (B),
(C) Reflection spatial maps at 3 different wavelengths (1460 nm, 1510 nm, and 1560 nm) for
device D4 showing difference in contrast for different thicknesses of BP. (D) Reflection amplitude
spectrum for different thicknesses of BP (2,3 and 6 layers) and bare cavity–illustrating redshift of
cavity resonance with increasing thickness of BP. (E) Ellipticity and (F) Azimuthal angle spectrum
for 3 different thicknesses of BP, showing highest polarization conversion in TLBP (3-layers).
__________________________________________________________________________ 171
Figure S3.14. Polarization conversion dependence on incident polarization. Effect of the input
polarization condition on the evolution of spectral trajectories on the normalized Poincaré sphere
is shown. As the phase delay and the relative amplitudes are tuned between the AC and ZZ
component of the incident light, different trajectories are undertaken. The cavity resonance along
the AC direction is at 1440 nm. The blue (red) arrow denotes the polarization state at 1410 (1520)
nm. ______________________________________________________________________ 173
Figure S3.15. Polarization conversion dependence on incident polarization. Effect of the input
polarization condition on the evolution of spectral trajectories on the normalized Poincaré sphere
is shown. As the phase delay and the relative amplitudes are tuned between the AC and ZZ
component of the incident light, different trajectories are undertaken. The cavity resonance along
the AC direction is at 1495 nm. The blue (red) arrow denotes the polarization state at 1410 (1520)
nm, overlapping. ____________________________________________________________ 174
Figure S3.16. Polarization conversion dependence on exciton parameters in TLBP. Effect of
different exciton parameters on the azimuthal and ellipticity of a typical cavity-based device. (A),

24
(B), (C) Effect on azimuthal angle for different exciton broadening, oscillator strength and
resonance wavelength, respectively. (D), (E), (F) Same as (A)-(C), but for ellipticity. ______ 176
Figure S3.17. Numerical modelling of cavity enabled polarization conversion. (A) Intensity (S0)
variation with oscillator strength of the exciton in TLBP. The cavity parameters correspond to
device D1. (B), (C), (D) s1, s2, s3 showing the same. This agrees with our experimental
observation of the electrically tunable polarization conversion results. __________________ 177
Figure S3.18. Normalized Poincaré sphere dynamics from additional spatial points in device D1.
(A)-(I) Normalized Poincaré sphere trajectories for different wavelengths for different voltages for
different spots in device D1. Each color corresponds to a different wavelength. For each color, the
voltage trajectory direction is marked. Beginning and end voltage values are marked for each
measurement. The dark arrows point the 0V condition for each wavelength and the light (color
coded for each wavelength) arrows denote the highest voltage point. ___________________ 178
Figure S3.19. Spectral and voltage tuning of normalized Poincaré sphere trajectories for Device
D1. (A) Evolution of polarization conversion for positive voltages (electron doping) and (B)
negative voltages (hole doping). The same color represents a spectral scan (from 1410 nm to 1520
nm in steps of 0.5nm), while a color variation shows changes of the spectral trajectory with voltage
from 0 to 30V for (A) and 0 to -30V for (B), in steps of 0.5V. Blue arrows represent polarization
state at 1410 nm for 0V while red arrows represent the same at 1520 nm. _______________ 179
Figure S3.20. Normalized Stokes parameters with electron and hole doping. (A),(B),(C) False
colormaps of the evolution of the three normalized Stokes parameters (s1,s2,s3), determining the
polarization state of the reflected light, as a function of wavelength and positive voltages (for
electron doping). The results are from device D1. Continuous tuning of all the 3 parameters can
be seen around the cavity resonance (~1440nm) for the entire range of doping, illustrating efficient
tuning of the polarization state with voltage. (D),(E),(F) Same as (A),(B),(C) but for negative
voltages (for hole doping), showing similar changes as the electron doped side. The nearly
symmetric nature of the doping dependence shows that at 0V, the device is at charge neutral
conditions. (G),(H),(I) Line cuts taken from the false colormaps for the three normalized Stokes
parameters (s1, s2, s3, respectively) for 5 different voltages (0V, 20V, 40V, -20V, -40V) to visualize
the changes with higher clarity. ________________________________________________ 181
Figure S3.21. Reflectance change (S0) of the cavity upon electron and hole doping. (A), (B) show
the change in the reflectance (Stokes intensity S0) of the device D1 as a function of wavelength for

25
different applied voltages on the hole doping and electron doping side, respectively. The colormaps
are plotted in dB for better clarity. ______________________________________________ 182
Figure S3.22. Azimuthal and ellipticity angle change upon electron and hole doping. (A), (B) show
the changes in the ellipticity angle (χ), in degrees, of the device D1 as a function of wavelength for
different applied voltages on the electron doping and hole doping side, respectively. (C), (D) show
the same as (A) and (B), but for the azimuthal angle (ψ), in degrees. The abrupt jump in the
azimuthal angle is a numerical artifact arising from the indistinguishability between +90o and -90o.
__________________________________________________________________________ 183
Figure S3.23. Gate-dependent reflectivity modulation in 5-layer BP device. (A) Reflection contrast
showing excitonic feature along the AC direction and a featureless spectrum along the ZZ
direction. (B) Relative reflection (w.r.t. 0V) shows strong modulation under different applied
biases, with the strongest tuning near the excitonic resonance. ________________________ 185
Figure S3.24. Thomas-Fermi screening effect in BP. Band-bending in multilayer BP as a function
of thickness for a charge density of 5x1012/cm2. A Thomas-Fermi screening length (𝜆 𝑇𝐹 ) of 2.9
nm is obtained. _____________________________________________________________ 186
Figure S3.25. Dipole interaction with optical field in TLBP. Side view of capacitor geometry of
working BP device. The two plates of the parallel-plate capacitor are the BP and the back Au
electrode/reflector. Free carriers are induced in the BP with applied voltage. Incident optical field
(polarized in the in-plane direction, perpendicular to the vertical capacitor field) is shown, along
with the Poynting vector. A top view of the BP flake along with the dipole orientation is shown,
with the Armchair (AC) and Zigzag (ZZ) axes marked. Incident field (in-plane) can be
decomposed along the AC and ZZ direction, marked as x and y, respectively. The x-component is
strongly influenced by the exciton-enhanced cavity interaction (which is also electrically tuned),
whereas the y-component is only influenced by the cavity and not the exciton and thus is not tuned.
__________________________________________________________________________ 188
Figure S3.26. Cyclic Stokes measurements. (A), (B), (C) Normalized Stokes parameters (s1, s2, s3)
spectra measured as a function of voltage in a cyclic fashion. (D) The applied voltage as a function
of the sequence number. ______________________________________________________ 189
Figure S3.27. Edge effects in spatial mapping. Top view and side view illustration of cavity and
BP, along with the gaussian (diffraction limited) beam at the edge showing sampling from both

26
the regions leading to different polarization conversion at the edge compared to the interior of the
sample (spot size is exaggerated for clarity). ______________________________________ 190
Figure S3.28. High efficiency numerical design for polarization conversion. (A) Double DBR
based Fabry-Perot cavity design incorporating BP. (B) Reflection and (C) phase from the
corresponding cavity structure for pristine and doped armchair (AC) direction. ___________ 192
Figure S3.29. High efficiency numerical design for polarization conversion. (A) Reflection and
(B) phase from the proposed cavity structure in Fig. S3.28 for pristine and doped armchair (AC)
direction for 17-top pairs instead of 20. __________________________________________ 192
Figure S3.30. Amplitude, azimuthal and ellipticity line cuts for hole doping. (A) Amplitude
spectrum for three distinct voltages (0V, -20V, -40V) corresponding to hole doping, showing
variation in the intensity at the resonance. (B), (C) Same as (A) but for azimuthal and ellipticity
angle. A region around 1450 nm can be identified to have nearly constant amplitude modulation,
minimal azimuthal change but large ellipticity modulation. __________________________ 193
Figure S4.1. Schematic of the low-temperature confocal optical PL setup used. BS–Beam Splitter,
QWP/HWP–Quarter/Half Wave Plate. ___________________________________________ 195
Figure S4.2. Schematic and calculation of optical geometry. (a) Schematic of a quarter wavelength
Salisbury screen geometry. (b) Absorption spectrum as a function of bottom hBN thickness
showing the cavity-enhancement. _______________________________________________ 197
Figure S4.3. Details of lumerical simulation. (a) Simulation setup XZ view in Lumerical FDTD.
(b), (c) XZ, YZ monitor for 𝑙𝑜𝑔(𝑅𝑒(𝐸)) profile, respectively. ________________________ 198
Figure S4.4. Comparison of monolayer and bilayer MoTe2 emission. (a) PL spectrum from
monolayer and bilayer regions of the same device. (b) Same as (a) but normalized. _______ 199
Figure S4.5. Optical microscope image of flakes used for the device fabrication. _________ 200
Figure S4.6. Applied gate voltage to sheet charge density conversion assuming parallel plate
capacitor model in the dc limit._________________________________________________ 201
Figure S4.7. Power dependent photoluminescence spectrum over 3 decades of pump intensity. (a)
is in linear and (b) is in log scale. The resonances are labelled in (b). ___________________ 201
Figure S4.8. Fit to gate dependent experimental data around 1s region. (a) Experimental PL data
around the A1s resonance region showing the neutral exciton and the charged trion resonances.
(b) Multi-Lorentzian fit to the PL data shown in (a). ________________________________ 202

27
Figure S4.9. Fit to gate dependent experimental data around 2s region. (a) Experimental PL data
around the A2s/3s resonance region showing the neutral exciton and the charged trion resonances.
(b) Multi-Lorentzian fit to the PL data shown in (a). ________________________________ 202
Figure S4.10. Gate-dependent data from additional spot. (a) Experimental PL data around the A1s
resonance region showing the neutral exciton and the charged trion resonances. (b) Same as (a)
but for 2s/3s resonances. ______________________________________________________ 203
Figure S4.11. Gate dependent emission fit parameters (intensity). (a) Absolute photoluminescence
intensity of Rydberg excitons as a function of charge density in linear scale. (b) Same as (a) in
semi-log scale. (c) Absolute photoluminescence intensity of trions associated with Rydberg
excitons as a function of charge density in linear scale. (d) Same as (c) in semi-log scale.___ 204
Figure S4.12. Gate dependent emission fit parameters (energy). (a) Energy shifts of the A1s
exciton and trion as a function of doping density. (b) Energy shifts of the A2s exciton and trion as
a function of doping density. __________________________________________________ 205
Figure S4.13. Exciton dispersion for a few lowest energy excitonic states in monolayer MoTe2.
__________________________________________________________________________ 208
Figure S4.14. Computation of doping-dependent loss function. (a) Computed imaginary part of
the dielectric function (𝜖2 (ℏ𝜔)) for monolayer MoTe2 with and without electron-hole interactions
and projected oscillator strength of the different Rydberg excitons (A1s, A2s, B1s, A3s from left
to right). (b) Evolution of the imaginary part of the dielectric function as a function of doping
density. ___________________________________________________________________ 209
Figure S4.15. Computed imaginary part of the dielectric function for excitons and trions. __ 210
Figure S4.16. Atomic force microscope image (height sensor) of MoTe2 device. __________ 212
Figure S5.1. Schematic of the low-temperature confocal optical PL setup used. BS–Beam Splitter,
QWP/HWP–Quarter/Half Wave Plate. ___________________________________________ 214
Figure S5.2. Optical microscope images of two gated heterostructures (sample (a), (b) #gateD4
and (c) #gateD1). hBN encapsulated monolayer BP is contacted with few layer graphene to gold
electrodes and a back contact of optically thick gold is used as the counter electrode. Scale bar in
(a) is 20 𝜇m and (c) is 10 𝜇m. __________________________________________________ 215
Figure S5.3. Optical microscope images of two encapsulated heterostructures (sample (a) #D6 and
(b) #D2), containing top and bottom hBN and monolayer BP. Scale bars are 5 μm. ________ 216

28
Figure S5.4. Optical microscope images of three bare monolayer samples on TEM grids (sample
(a) #S1, (b) #S2 and (c) #S3). The TEM hole arrays are 15 μm x 15 μm. ________________ 216
Figure S5.5. Optical microscope images of three fully encapsulated (with monolayer graphene)
monolayer BP samples on TEM grids (sample (a) #H1, (b) #H2 and (c) #H3). The TEM hole arrays
are 15 μm x 15 μm. __________________________________________________________ 217
Figure S5.6. Scanning electron microscope image of a typical TEM holey grid used in
measurements. To prevent charging during imaging ~0.5 nm Ti and ~1.5 nm Au were deposited
right before transfer of 2D flakes. _______________________________________________ 218
Figure S5.7. Spatial maps of energy of photoluminescence of the brightest feature (which
approximately follows the exciton energy) for sample. (a) #gate D1, (b) #D6 and (c) #D2,
respectively. We find ~50 meV, ~100 meV and ~20 meV variation in the PL peak energy for (a),
(b) and (c), respectively. ______________________________________________________ 219
Figure S5.8. Polarization analysis-I. (a) Azimuthal orientation of edge dipole emission pattern
versus interior dipole emission pattern (extracted from fitting the “interior”-like emission
envelope). Color indicates peak energy of emission as noted in color bar (in eV). (b) Difference in
the azimuthal dipole emission angle between the edge and the interior as a function of emission
energy of the edge emission. (c) Same as (b) but plotted (on x-axis) as a function of difference in
energy between the interior and edge (𝛥𝐸 = 𝐸𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 − 𝐸𝑒𝑑𝑔𝑒 ). ______________________ 220
Figure S5.9. Polarization analysis-II. (a) Dipole visibility ((𝐼𝑚𝑎𝑥 − 𝐼𝑚𝑖𝑛 )/(𝐼𝑚𝑎𝑥 + 𝐼𝑚𝑖𝑛 ) of edge
exciton emission versus difference of edge and interior dipole emission angle. (b) Dipole visibility
of edge exciton emission versus interior exciton dipole orientation. (c) Same as (b) but plotted (on
x-axis) as a function of difference in energy between the interior and edge (𝛥𝐸 = 𝐸𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 −
𝐸𝑒𝑑𝑔𝑒 ). ____________________________________________________________________ 220
Figure S5.10. Measured armchair direction at some spatial points (direction of the arrows)
superimposed on log(PL) spatial maps for sample (a)#D6 and (b)#D2, respectively. For both
samples terminations along armchair and zigzag are both seen (for #D6 the longer tear is along
armchair, whereas for #D2 it is along zigzag). _____________________________________ 221
Figure S5.11. Extended polarization dataset 1. (a)-(i) False colormaps of emission spectrum as a
function of collection (analyzer) polarizer angle for different spots from device D6. It can be
clearly seen that for all spots (collected near physical edges of sample) signatures of both interior
and edge emission is observed. _________________________________________________ 222

29
Figure S5.12. Extended polarization dataset 2. (a)-(i) False colormaps of emission spectrum as a
function of collection (analyzer) polarizer angle for different spots from device #gateD1. It can be
clearly seen that for all spots (collected near physical edges of sample) signatures of both interior
and edge emission is observed. _________________________________________________ 223
Figure S5.13. Spatial map of integrated photoluminescence spectrum. (𝐼𝑛𝑡. 𝑃𝐿 = ∫ 𝐼(𝜔)𝑑𝜔 )
Spots are marked as stars in different colors showing temperature dependent emission Scale bar is
5 𝜇𝑚. This map (gate #D1) has been acquired at 5K and the same sample has been cycled from
5K to 300K to acquire temperature dependent spectrum presented next. ________________ 224
Figure S5.14. Temperature dependent (normalized and offset) photoluminescence spectrum from
the selected spots (marked in S10). (a) 5K, (b) 40K, (c) 70K, (d) 100K, (e) 190K and (f) 300K
showing clear broadening of edge features and eventual disappearance of the same. Interior
emission profile takes over for higher temperatures. ________________________________ 225
Figure S5.15. Additional pump power dependent data. (a) False color map and (b) line-cuts of
power dependent photoluminescence spectrum for a specific spatial point in device D6 showing
three distinct emission lines from the edge on top of a broad interior emission envelope. (c) Powerlaw fits of integrated photoluminescence intensity to the three aforementioned peaks showing
linear behavior. _____________________________________________________________ 226
Figure S5.16. Accumulated power law fit exponents across 24 spots showing edge emission
measured in sample #gateD1. Similar behavior was also seen in D6. Exponents lie between ~0.9
and ~1.1 indicating linear behavior (for the ranges of incident power measured). _________ 226
Figure S5.17. False color-maps of photoluminescence spectrum collected from three distinct
spatial spots (a)-(c) showing clear signs of spectral diffusion/temporal fluctuation. ________ 227
Figure S5.18. False color-maps of photoluminescence spectrum collected from interior exciton
emission at an interior spatial spot showing stable emission for two different pumping powers (a)
1.7 μW and (b) 91 μW. _______________________________________________________ 227
Figure S5.19. Gate dependent photoluminescence dynamics for P5 (spatial point #5) for device
#gateD1. (a) False color-map of photoluminescence spectra as a function of gate voltage. (b) Same
as (a) but normalized to 1, for each gate voltage. (c) Spectra from (a) at three distinct voltages (8V,
0V, -8V). (d) Spectra from (b) at three distinct voltages (8V, 0V, -8V). (e) Spectra from (b) for
multiple voltages between 6.1V and -7.9V. _______________________________________ 228

30
Figure S5.20. Gate dependent photoluminescence dynamics for P15 (spatial point #15) for device
#gateD1. (a) False color-map of photoluminescence spectra as a function of gate voltage. (b) Same
as (a) but normalized to 1, for each gate voltage. (c) Spectra from (a) at three distinct voltages (8V,
0V, -8V). (d) Spectra from (b) at three distinct voltages (8V, 0V, -8V). (e) Spectra from (b) for
multiple voltages between 6.1V and -7.9V. _______________________________________ 229
Figure S5.21. Lifetime measurements. (a) PL spectrum of a spatial location (device D6) containing
edge exciton emission signatures over an envelope of interior emission. (b) Time dependent PL
measurements collected without a filter and with two filters placed approximately at the two peaks
from edge states at 1.63 eV and 1.68 eV. Bandwidth of the filters are ~10nm. ____________ 230
Figure S5.22. Simulation setup in Lumerical CHARGE module. hBN is modelled as a continuous
medium (including the top and bottom thickness) and BP is inserted inside the hBN, such that it
matches the experimental structure of device #gateD1. Voltage is applied between the top and
bottom gold. The assumption here is that graphene acts metallic enough that the Au-BP contact
can yield similar results to Au-graphene-BP contact. ________________________________ 232
Figure S5.23. In-plane electric field component distribution cross-section at 5V. A hotspot can be
seen at the BP-hBN interface at x=0 nm. _________________________________________ 232
Figure S5.24. Gate dependent in-plane field simulations. (a) In-plane electric field variation as a
function of spatial position (x) and gate voltage. The BP-hBN interface laterally is at x=0 nm. (b)
Line-cuts of in-plane electric field for different voltages as a function of spatial position. (c) Linecuts of in-plane electric field for two spatial positions near the BP-hBN interface, as a function of
gate voltage showing the sensitivity of the field experienced by the dipole depending on its
location.___________________________________________________________________ 233
Figure S5.25. Atomic structure of monolayer black phosphorus and schematic of how band
bending causes transition from n-type behavior to intrinsic at the edge. _________________ 234

LIST OF TABLES

Table 3.T1 Bounds on effective masses. __________________________________________ 80
Table 5.T1. Experimentally measured zero-temperature exciton energy and exciton-phonon
coupling parameters for the Rydberg states.

110

Table 5.T2. Experimentally measured binding energy and energy shifts for Rydberg trions. _ 117
Table 5.T3. Computed Rydberg exciton binding energy and relative dipole moment as a function
of doping density. ___________________________________________________________ 118
Table S4.T1. Binding energy (meV) of Rydberg excitons of all TMDCs ________________ 219

Chapter 1. INTRODUCTION
1.1 VAN DER WAALS MATERIALS
A new class of materials, called van der Waals materials1,2, has recently excited widespread interest
because of their novel and unique properties. They are essentially layered materials, where each layer
is held together by van der Waals forces of attraction–which are much weaker compared to chemical
or ionic bonds. This allows isolation of individual layers via different methods of ‘exfoliation’–such
as mechanical3 (Scotch-tape being one of the most popular choices of a handle layer) and chemical
methods involving sonication and intercalation4,5. The fact that in-plane bonds are chemical and
hence, in general, stronger than the van der Waals bonds makes exfoliation a reliable technique to
generate large area high-quality samples. Although van der Waals materials have been studied back
in 1960s via optical and electrical characterization6, the field rapidly exploded in 2004 when
Novoselov and Geim were able to isolate graphene in the form of a few layers and eventually a
monolayer7. 2D materials appear in different phases of matter such as metals, semiconductors,
insulators and also exhibit unusual and exotic topological properties in Dirac and Weyl semi-metals,
charge density waves, etc8–10. Furthermore, since these materials can be cleaved and then stacked
without lattice matching requirements (due to van der Waals attraction), one can create so-called
‘heterostructures’ which may have completely different properties from their parent layers11,12.
Taken together, the possibility of exploring phases of matter in 2D materials and their
heterostructures is endless and can be studied for decades.

Figure 1.1. Conceptual schematic of Lego-like behavior in van der Waals heterostructures.
Different materials can be stacked to form new materials with different properties compared to
their parent. Figure taken from Ref. 1
Of considerable interest in the context of opto-electronics is graphene, the family of TMDCs, and
black phosphorus. This thesis will mainly focus on the opto-electronics of black phosphorus and one
candidate TMDC–MoTe2.

1.2 EXCITONS
When a (negatively charged) electron and a (positively charged) hole are attracted to each other
through Coulomb forces of interaction, a stable quasiparticle is formed called the exciton13,14. While
such states exhibit significant similarities to the hydrogen atom, they also show interesting deviations
due to the two-dimensional nature of the host15. Usually, they result from photo-excited charge
oscillations in semiconductors and are redshifted from the band-edge by the exciton binding energy
(𝐸𝐵 ). Excitons display an unusually large oscillator strength and thus host strong light-matter
interactions enabling efficient optical excitation and read-out capabilities.

34
Exciton behaves like an atom (impurity) in the background of a crystal lattice and displays a finestructure akin to a hydrogen atom. These states appear as resonances of finite width in the optical
spectrum and are known as the Rydberg series15. The width of each state is related to the quantum
yield of the transition as well as other competing non-radiative processes like phonon scattering,
electron scattering, etc.
Thus far, excitons have remained practically elusive, since in typical bulk semiconductors, the
binding energy is on the order of ~1-10 meV due to strong self-screening16. This makes it very
difficult to observe such states at room temperature due to thermal fluctuations (kBT ~ 25 meV) and
only allows visualization at very low temperatures (kBT < ~ 1 meV), limiting its use for practical
technology.
From this point of view, some of the transition metal dichalcogenides (TMDCs) and black
phosphorus, offer a way out. They host excitons with binding energy on the order of 100s of meV,
making them attractive for practical and viable room-temperature operation13,14,17. This is enabled
by the strong Coulomb attraction between the electron and hole due to low dimension and reduced
dielectric screening in atomically thin sheets. Naturally, monolayers exhibit the largest binding
energy and strongest resonances, in general.
Group VIB TMDCs such as molybdenum and tungsten based diselenides, ditellurides, and
disulfides: MoS2, MoSe2, WS2, WSe2 and MoTe2 occur in semiconducting phase (2H) and can be
exfoliated down to a monolayer13,14,17–20. Black Phosphorus is another candidate which occurs in the
semiconducting phase21. Both these systems host strong excitons, especially at the monolayer limit.
In TMDCs an indirect to direct bandgap transition takes place at the monolayer thickness due to the
lack of interlayer interactions which play a role in band structure renormalization. Such reduced
interactions manifest as a lower valence band maximum resulting in a direct band gap–high
absorption and intense photoluminescence. Black phosphorus is unique since due to its crystal
structure, it remains a direct bandgap at all thickness with changes in the binding energy (detailed
later)22,23. Such properties are attractive for light emission and absorption applications.

35
Another interesting aspect is the dipole orientation in these layered materials. Due to thickness
constraints, most excitons are in-plane dipoles allowing maximal interaction with normally incident
light radiation. Next, we discuss strategies to engineer excitons in different ways.

Figure 1.2. Schematic of exciton and absorption. (a) Coulomb field lines for an exciton in a 3D
bulk semiconductor and monolayer 2D semiconductor showing reduced screening from the
environment. (b) Optical absorption showing excitonic state and quasiparticle gap for 2D and 3D
case. Figure taken from Ref. 15

1.3 ELECTRICAL TUNING OF EXCITONS
2D materials offer attractive opportunities for electrical tuning of their properties. For example,
MoS2 monolayers and few-layers were demonstrated to have excellent transistor like characteristics
which rapidly exploded the field in that direction24. As these materials are atomically thin,
encapsulating them in a field-effect heterostructure enables electrostatic doping of the active layer.

36
This allows tuning of the charge density and the current in the active layer as well, thus modulating
its electrical and optical properties.
One of the first efforts in that direction include Mak and Xu’s efforts to dope a monolayer and
observe in absorption and PL the emergence of trions–a quasiparticle which consists of an exciton
bound to an electron or hole20,25. Trions also possess a relatively high binding energy (~20-30 meV)
and hence can be observed at room temperatures. Further studies by Chernikov reported more
detailed characterization of the exciton and trion as a function of gate voltage, not only for the ground
state but also for the higher energy Rydberg states for a select few TMDCs26–29.
Another important aspect is that 2D materials, near excitonic resonances, show enhanced refractive
properties upon incident light radiation. This is mainly due to high scattering and strong light-matter
interaction on resonance. Excitons can be modelled approximately as a Lorentzian function with
finite amplitude, resonance energy and linewidth, as mentioned before30,31. Upon electrical gating
the Lorentzian function can be heavily modified which forms the basis for actively tunable refractive
element. Such elements can be cascaded to construct complex metasurfaces, which is what a section
of this thesis highlights.

1.4 ELECTRIC-FIELD EFFECTS ON EXCITONS
Depending upon the electrostatic geometry, 2D materials can be made to purely experience
displacement-field effects while keeping the charge density neutral or constant. Such geometry
usually involves dual gates with symmetric doping condition and has the largest effects on
thicknesses beyond a monolayer (due to in-plane dipole of a monolayer). Stark effect has been
heavily explored in natural bilayers as well as monolayer-monolayer heterostructures32–34. At higher
thicknesses other quantum effects kick in such as Burstein-Moss shifts and band-gap
reduction/normalization35.

Detailed discussion of different effects dominating at different

thicknesses is presented later in the thesis.

1.5 MECHANICAL TUNING OF EXCITONS
Upon mechanical straining the electronic band structure can be modified since it is related to the
lattice structure. Optical properties such as excitons (in monolayer or few-layers) and interband
transitions can be modulated with strain36–39. 2D Materials are placed on flexible substrates which

37
allows dynamic modulation of the strain. Changing temperature can also modify the lattice constant
which can be another mechanism for changing strain. Additionally, the phonon spectrum
(manifested in Raman spectroscopy) can also be tuned via strain. Strain engineering in multilayers
not only modifies lattice constant but also tunes interlayer coupling. Such a strategy has been proved
to be very effective for tuning optical properties in black phosphorus as shown by Yan38. A similar
strategy was also adopted by us to tune interlayer interactions between MoS2 and WSe2 on a flexible
substrate40.
Other forms of tuning include applying magnetic fields or optical pumping which are beyond the
scope of this thesis.

1.6 PLASMONS
Plasmons are quantized charge oscillations in a semiconductor or a metal. Such oscillations are
induced by virtue of Drude-like absorption from free carriers and are termed as intraband absorption–
as the carriers do not change their band index. The absorption process is accompanied by a phonon
since it is not momentum conserving. In terms of the optical response of a plasmonic system, the
real part of the dielectric function becomes negative (thus allowing excitation of confined modes),
and the imaginary part becomes positive (meaning the material becomes lossy). Plasmonic materials
thus suffer a trade-off between being able to confine light to sub-wavelength volumes and intrinsic
loss/efficiency. While mostly explored in graphene, other doped semiconductors, and metals in the
library of 2D materials offer interesting alternatives41–46. For example, black phosphorus is a highly
anisotropic system which has been theorized to be in-plane hyperbolic–meaning plasmonic along
one crystal axes and dielectric along another47–50. Thus far, experiments demonstrating direct
evidence of plasmonic behavior have remained elusive. In this thesis, we establish this fact in chapter
3 and show gate-tunability of the same.

1.7 PHONONS
Phonons are quantized vibrations of the atomic lattice. Usually, they are observed in optical
spectroscopy for polar materials since there is a permanent charge separation which allows the
dipoles to interact strongly with incident light. Hexagonal boron nitride and molybdenum trioxide
are two examples of the most studied phonon systems due to high anisotropy and high-quality

38
phonon resonances51,52. Mathematical modelling of phonons is similar to excitons in that they can
be described by a Lorentzian lineshape.

Figure 1.3. An overview of possible material resonances in the van der Waals library. Each
quasiparticle induces a susceptibility resonance in the optical response of the material that can
interact with light strongly, creating hybrid light-matter modes–polaritons. From left to right–
electrons/holes in graphene and black phosphorus, lattice vibrations in hBN and topological
insulators, electron-hole quasiparticles in semiconductors, cooper pairs in superconductors and
magnons in magnetic materials. This figure is taken from Ref. 53

1.8 BLACK PHOSPHORUS
Black phosphorus (BP) is a relatively new member of the 2D materials family and is the layered
allotrope of phosphorus. It has a distinct crystal structure as compared to the TMDCs and graphene
family. It possesses a highly buckled lattice structure which renders strong anisotropy between
different crystalline directions which manifests as anisotropic optical, electrical, thermal, and
mechanical properties. Despite such exciting properties, BP remains relatively poorly explored
especially in the mono/few layer limit because of highly sensitive nature to oxidation and
degradation requiring inert processing atmosphere and immediate encapsulation after isolation for
best quality devices. This is because in-plane bonds in BP are weaker as compared to graphene or
TMDCs and hence in-plane oxidation is much faster than out-of-plane.
BP is a direct band gap semiconductor and has varying bandgap with number of layers–behaving
like a quantum well. As the material becomes thicker, the quantum confinement changes in the
vertical direction which modulates the energy spacing of the intersubband levels. In the monolayer

39
limit, BP has a band gap of ~2 eV which gradually reduces to a value of ~0.3 eV in the bulk limit.
In stark contrast to TMDCs, no indirect-to-direct transition happens in the bandgap for BP since the
band-minimum occurs at the Γ point of the Brillouin zone–which is strongly affected by interlayer
interactions. Such properties make BP attractive for infrared photonics as it bridges the gap between
graphene (in the mid to far-infrared) and TMDCs (which have band-gaps in the visible frequencies).
The most compelling property of BP (within the scope of this thesis) is the anisotropy and the strong
electrical tunability of its optical properties. It is a unique quantum-well like system with large inplane birefringence, not seen usually in other materials. Thus quantum-well effects like quantumconfined Stark effect, Pauli blocking, Burstein-Moss shifts are expected to be observed in gated
structures for multilayer BP. As one approaches the atomic limit, excitonic effects are expected to
kick in. BP, in fact, possesses one of the highest binding energy for excitons in its monolayer limit
in a free-standing film (~0.8 eV) which gradually decreases for thicker films, until the free-carrier
continuum (quasi-particle band gap) merges with the excitonic resonance, i.e., the exciton Coulomb
field is heavily screened by the BP layers.

Figure 1.4. Overview of black phosphorus. (a) Side view of anisotropic puckered crystal structure
of layer BP. (b) Top view of a monolayer BP crystal structure. (c) Angle resolved photoemission
spectroscopy measurement of band structure in bulk BP with the band minima being at the
Γ (Z)point. (d) Layer dependent band gap of BP showing strong interlayer interactions. This figure
is taken from Ref. 54
This anisotropy inspires compelling new physics, such as in-plane hyperbolic plasmons, and
technology applications, such as polarization modulation and generation.

1.9 LIGHT-MATTER COUPLING
Matter interaction with electromagnetic radiation is one of the most widely studied subjects in solidstate physics and perhaps the most promising for future technology. Strong interaction is usually
achieved when the optical density of states of a system is very high which is reflected in the complex
refractive index or dielectric function of the system. There are two broad ways of achieving the same.

41
1. Material resonance
This involves working around a quasi-particle resonance such as those described previously–
excitons, plasmons and phonons, where each susceptibility resonance can be simply modelled with
a Lorentzian lineshape: 𝐼(𝜔) =

𝐴Γ

Γ 2

(𝜔−𝜔0 )2 +( )

, where 𝐴 is the intensity of the resonance, Γ is the

broadening and 𝜔0 is the resonant energy.
2. Geometric resonance
This involves utilizing a structure which can be thought to induce a mode (electric or magnetic
dipole, quadrupole, or higher order) that interacts strongly with light. This is achieved by using
optical cavities like Fabry-Perot or nanostructures that acts as subwavelength antenna elements55–57.
Light-Matter coupling can be further broadly classified into the following categories:
A. Strong coupling
This is achieved when a material is brought in close proximity with an optical resonator and their
resonances are aligned. Mathematically, if the coupling strength between the material and resonator
is denoted by g, and their respective dissipation rates are denoted as 𝜅 and 𝛾, then the condition for
strong coupling is given 𝑔 >

|𝜅−𝛾|

B. Weak coupling
This regime is achieved when the matter acts as a weak perturbation and the optical response is
governed by the optical resonator (𝑔 ≤

|𝜅−𝛾|

). Despite being weak, such regimes are useful for

building amplitude and phase modulators whereby shifting the resonance to different energies, a
large modulation depth can be achieved. In this regime, three kinds of coupling can be accessed–
critical coupling, under coupling and over coupling–all exhibiting Purcell enhancement of certain
degree. If the total response (𝑟𝑡𝑜𝑡 ) of a system is dominated by non-resonant (𝑟𝑛𝑟 ) and resonant (𝑟𝑟 )
processes, it can be mathematically expressed as–𝑟𝑡𝑜𝑡 = 𝑟𝑟 + 𝑟𝑛𝑟 . Critical coupling is achieved when
𝑟𝑛𝑟 ∼ 𝑟𝑟 and a 𝜋 phase shift exists between the two channels–thereby allowing complete absorption

42
of light. In a complex plane representation of the total scattering channel the real and imaginary part
of 𝑟𝑡𝑜𝑡 lie at the center. However, as the resonant channel begins dominating the non-resonant part
(𝑟𝑛𝑟 > 𝑟𝑟 ), the system goes into under-coupling whereas it goes into over-coupling for the reverse
case.58 An illustrative figure for a ring-resonator system where 𝑡 is the transmission of the system,
𝑎 is the absorption in the ring and 𝑘 is coupling between the waveguide and the ring for the different
regimes and their optical output is shown.

Figure 1.5. Different regimes of weak coupling. A ring-resonator system is shown for illustrative
purposes with three regimes identified–critical coupling (t = a), over-coupling (t < a) and undercoupling (t > a). Their optical response (transmission) is also plotted. This figure is taken from
Ref. 59

1.10 METASURFACES
Subwavelength photonic structures have restructured our understanding of classical Snell’s law,
which is written as:
𝑠𝑖𝑛𝜃1 𝜈1 𝜆1 𝑛2
= =
𝑠𝑖𝑛𝜃2 𝜈2 𝜆2 𝑛1

43
where 𝜃1,2 are the incident and refracted angle, 𝜈1,2 are the speed of light, 𝜆1,2 are the wavelengths
of light, 𝑛1,2 are the refractive indices of each medium, respectively.
At an interface when a phase gradient is imposed, this equation needs to be reframed to account for
the spatial dependence of the accumulated phase. A generalized law of reflection and refraction then
emerges as follows:

𝑛𝑡 sin(𝜃𝑡 ) − 𝑛𝑖 sin(𝜃𝑖 ) =

cos(𝜃𝑡 ) sin(𝜙𝑡 ) =

1 𝑑Φ
𝑛𝑡 𝑘0 𝑑𝑦

sin(𝜃𝑟 ) − sin(𝜃𝑖 ) =

cos(𝜃𝑟 ) sin(𝜙𝑟 ) =

where

𝑑Φ
𝑑𝑥

and

𝑑Φ
𝑑𝑦

1 𝑑Φ
𝑘0 𝑑𝑥

1 𝑑Φ
𝑛𝑖 𝑘0 𝑑𝑥

1 𝑑Φ
𝑛𝑡 𝑘0 𝑑𝑦

are phase gradients along x and y directions, 𝜃𝑡 , 𝜃𝑖 , 𝜃𝑡 are the transmitted, incident,

and reflected angles in the x-direction, and 𝜙𝑡 , 𝜙𝑟 are the transmitted and reflected angles in the ydirection. Incident plane is defined to be x-z. Provided a full 2𝜋 phase space is accessible, light can
be directed or manipulated in arbitrary ways. Via careful engineering of the phase gradient and
polarization at each element, novel applications will benefit like beam steering, spatial light
modulation, focusing or lensing and holography.
A convenient approach to designing phase gradients is by controlling the shape or geometry of each
nanophotonic element, as first demonstrated in previous reports60,61. Since then, the field has
exploded via various ways of engineering such as changing materials of the antenna or their
surrounding.

1.11 ACTIVE NANOPHOTONICS
While modifying geometry provides a way to control phase, amplitude, and polarization of each
metasurface element, it also has certain restrictions which limit functionality–such as lack of real

44
time reconfiguration. Geometric elements are fixed at the time of fabrication and thus, their response
is static. Active nanophotonics offers attractive schemes to tune different properties of light as a
function of time, in a controlled fashion.
Dynamic control at the near-field and far-field level in real time opens up a number of promising
applications such as fast LIDARs and detectors, displays for augmented and virtual reality (AR/VR),
LiFi and thermal radiation engineering. It can also impact computing, information transfer and
processing.
There now exist many approaches to dynamically control optical properties of materials and careful
attention must be paid to the different metrics required for each application–such as speed, energy,
robustness, and cost. Mechanical strain tuning, piezo effects, and thermally-induced phase change
materials62–65 can be useful in generating large tunability at the cost of slow speed. In contrast,
semiconductor charge injection66,67 or electric-field based effects like the Stark68 or Pockels shift69
can provide much faster operation at the cost of lower refractive index modulation. Thus, a judicious
combination of a dynamic tuning mechanism and nanophotonic element can enable the “universal
metasurface” with independent control knobs for different properties of light–amplitude, phase,
polarization, wavevector and wavelength.

1.12 OPTICAL MODULATORS
At the heart of active nanophotonics lies optical modulators66,70–72. The fundamental working
principle involves a dynamically tunable element whose complex refractive indices can be changed
with an external stimulus - such as temperature, electric field, magnetic field, electronic doping,
strain, or optical pumping–which in turn modulates the phase or amplitude of the incident light,
either in reflection or transmission mode (depending upon the application). Usually, such tunable
elements are coupled with optical resonators such as Fabry-Pérot cavities, ring resonators,
waveguides, metasurfaces to amplify the dynamic tunability. Since this thesis is focused on electrooptic effects–graphene, TMDC and black phosphorus are attractive candidates for the same. While
graphene has a very broad-band response thanks to its zero bandgap, black phosphorus and TMDCs
exhibit complementary response to graphene due to their direct/indirect bandgap nature. In graphene,
two major mechanisms are at play upon charge injection–Drude absorption at long wavelengths (mid
to far-infrared) and Pauli blocking (in the near to mid-infrared). For black phosphorus, a much more

45
complex picture emerges upon charge injection, depending upon the thickness (which is discussed
in detail in this thesis).

1.13 SCOPE OF THIS THESIS
This thesis explores the emerging field of van der Waals nanophotonics which combines the study
of atomically thin materials and optics at the subwavelength scale. We leverage the material
resonances of quantum-confined, atomically thin semiconductors as active elements for existing and
novel photonic designs. This enables real time reconfiguration of different properties of light at
subwavelength scales. Furthermore, optical spectroscopy of such quantum materials also reveals key
insights into their fundamental properties.
Chapter 2 introduces electro-optic effects in few-layer black phosphorus under symmetric and
asymmetric gating. By using a dual gate, the charge density as well as the electric field can be
independently controlled in black phosphorus. This allows isolation of the two effects: 1. Doping
(Pauli-blocking) and 2. Field (quantum-confined Stark effect and modification of selection rules of
optical transitions). Both regimes present ways to alter the refractive indices of BP which are
attractive for mid-infrared photonics. In Chapter 3, we focus on the mid to far-infrared properties of
similar structures and investigate the free-carrier optical response. We find highly anisotropic
absorption which can be modulated by injecting charge in BP. Such absorption is Drude-like and
induces a metallic behavior (seen as epsilon-near-zero) along one of the crystal axis (armchair), while
the other axis (zigzag) remains dielectric. This marks the first observation of in-plane electrically
tunable hyperbolicity in BP and opens avenues for plasmon switching, beam steering, etc.
Chapter 4 builds on this work and expands its scope by studying electrically tunable excitons in trilayer black phosphorus (significantly thinner than previous studies) which leads to tuning of the
complex refractive index on the order of 1 (orders of magnitude higher than lithium niobate, barium
titanate, etc.). Such large tuning happens only along the armchair axis due to symmetry arguments
and thus a highly tunable birefringence material emerges. By combining with an optical resonator
(Fabry-Perot cavity), large changes in amplitude and phase can be enabled to create different
polarization states of light across the Poincare sphere across the entire telecommunication band. This
constitutes the first demonstration of an electrically tunable polarization modulator using van der
Waals materials.

46
Since excitons form the basis of strong optical modulation, a careful fundamental investigation of
the same is of utmost importance. In Chapter 5, we approach the ultimate limit of a monolayer, since
they host excitons with the highest binding energy, and focus on the fundamental optical properties.
We study MoTe2, a semiconductor of the TMDC family, using photoluminescence measurements to
investigate the Rydberg series associated with the exciton. Combing it with gate-tunable
heterostructure we find that the emission can be strongly tuned between the neutral excitonic and
charged excitonic (trionic) state. We further perform GW+BSE calculations to support our
understanding of excitons and exciton-electron interactions. Finally, in Chapter 6, we study excitons
in monolayer black phosphorus and strong quantum confinement at natural edges of exfoliated flakes
(which happened to be a serendipitous finding). Such confinement results in highly temporally
coherent emission which is polarized and also tunable in a gated heterostructure. Through
transmission electron microscopy we study how edges of phosphorene reconstruct to minimize
energy and using first-principles GW+BSE calculations of phosphorene nanoribbons we understand
how some configurations, in conjunction with strain, lead to formation of edge (localized) states,
appearing as additional peaks in our experiments.
Overall, our work highlights the opportunities presented by van der Waals materials through various
electro-optical excitations for applications in active nanophotonics.

Chapter 2. INTERBAND EXCITATIONS IN
MULTILAYER BLACK PHOSPHORUS
2.1 ABSTRACT
The incorporation of electrically tunable materials into photonic structures such as waveguides
and metasurfaces enables dynamic, electrical control of light propagation at the nanoscale. Fewlayer black phosphorus is a promising material for these applications due to its in-plane
anisotropic, quantum well band structure, with a direct band gap that can be tuned from 0.3 eV to
2 eV with number of layers and subbands that manifest as additional optical transitions across a
wide range of energies. In this work, we report an experimental investigation of three different,
anisotropic electro-optic mechanisms that allow electrical control of the complex refractive index
in few-layer black phosphorus from the mid-infrared to the visible: Pauli-blocking of intersubband
optical transitions (the Burstein-Moss effect); the quantum-confined Stark effect; and the
modification of quantum well selection rules by a symmetry-breaking, applied electric field. These
effects generate near-unity tuning of the BP oscillator strength for some material thicknesses and
photon energies, along a single in-plane crystal axis, transforming absorption from highly
anisotropic to nearly isotropic. Lastly, the anisotropy of these electro-optical phenomena results
in dynamic control of linear dichroism and birefringence, a promising concept for active control
of the complex polarization state of light, or propagation direction of surface waves.

2.2 INTRODUCTION
Dynamic control of the near and far-field propagation of light is critical for next-generation
optoelectronic devices. Ultra-thin, layered materials are promising building blocks for this
functionality, as they are easily incorporated into atom-scale structures, and their optical properties
can be changed dramatically under applied electric fields73,74. Few-layer black phosphorus (BP) is
particularly compelling due to its high electronic mobility, in-plane anisotropy, and thicknesstunable quantum well band structure, with a direct band gap that varies from 0.3 eV in bulk to 2
eV for monolayers54,75. Recent work using electrostatic gating and potassium ions has further
shown that the electronic band gap of BP may be tuned by an electric field.35,76–78 These unique

48
attributes have already enabled the realization of novel and high-performance optoelectronic
devices, including waveguide-integrated photodetectors79–83.

One of the most unusual features of BP is its large in-plane structural anisotropy, which generates
to a polarization-dependent optical response21,49,84 as well as mechanical85, thermal86, and electrical
transport characteristics87,88 that vary with in-plane crystallographic orientation89. This optical
anisotropy corresponds to a large, broadband birefringence90, wherein the distinct optical index of
refraction along each axis leads to a phase delay between polarization states of light. Moreover,
mirror-symmetry in the x-z plane forbids intersubband optical transitions along the zigzag axis,
and as a result, BP exhibits significant linear dichroism, wherein the material absorption depends
strongly on the polarization state of exciting light84,91.
In this work, we use multiple field-effect device configurations to isolate and characterize three
distinct, anisotropic electro-optic effects that allow significant control of the complex refractive
index in BP. These effects are Pauli-blocking of intersubband optical transitions, also known as a
Burstein-Moss or band-filling effect; the quantum confined Stark effect; and modification of
quantum well selection rules by a symmetry-breaking electric field. The resulting response
approaches near-unity tunability of the BP oscillator strength for some BP thicknesses and photon
energies and tunes along one in-plane crystal axis. As a result, we are able to electrically control
dichroism and birefringence in BP. In some cases, we observe tuning of the black phosphorus
optical response from highly anisotropic to nearly isotropic. We observe this anisotropic tunability
from the visible to mid-infrared (mid-IR) spectral regimes, behavior not seen in traditional electrooptic materials such as graphene92, transparent conducting oxides58,93, silicon67, and quantum
wells94. This opens up the possibility of realizing novel photonic structures in which linear
dichroism in the van der Waals plane can be continuously tuned with low power consumption,
because the switching is electrostatic in nature. By controlling optical losses in the propagation
plane, for example, efficient in-plane beam steering of surface plasmon polaritons or other guided
modes is enabled. Moreover, an electrically tunable polarizer could be realized by modulating the
polarization state of light absorbed in a resonant structure containing BP. Because this tunability
is strongest at infrared wavelengths, it could also enable control of the polarization state of thermal
radiation95–97.

2.3 EXPERIMENTAL MEASUREMENTS OF THE BLACK
PHOSPHORUS OPTICAL RESPONSE WITH AN APPLIED
ELECTRIC FIELD
In order to probe and distinguish the electro-optical tuning mechanisms evident in few-layer BP,
we used a combination of gating schemes wherein the BP either floats in an applied field or is
contacted, as shown in Fig. 2.1a. Samples for infrared measurements were fabricated by
mechanically exfoliating few-layer BP onto 285 nm SiO2/Si in a glove box environment. Contacts
of 20 nm Ni/200 nm Au were fabricated by electron beam lithography, electron beam evaporation,
and liftoff. A top gate dielectric of 45 nm Al2O3 was deposited by atomic layer deposition (ALD)
following the technique in Ref. 98, and a semi-transparent top contact of 5 nm Ni was deposited by
electron beam evaporation and liftoff. Measurements were performed in a Fourier Transform
Infrared Spectrometer coupled to a microscope.
Polarization-dependent optical measurements are taken aligned to the crystal axes, in order to
probe the structural anisotropy shown in Fig. 2.1b. This enables us to isolate the contribution of
charge-carrier density effects–i.e., a Burstein-Moss shift–and external field-effects–i.e.: the
quantum-confined Stark effect and control of forbidden transitions in the infrared–to the tunability
of linear dichroism, qualitatively illustrated in Figures 2.1c and 2.1d36,65,99. In the anisotropic
Burstein-Moss (BM) shift, the optical band gap of the material is changed as a result of band filling
and the consequent Pauli-blocking of intersubband transitions. As the carrier concentration of the
sample is changed, the Fermi level moves into (out of) the conduction or valence band, resulting
in a decrease (increase) of absorptivity due to the disallowing (allowing) of optical transitions100–
102

. Because intersubband optical transitions are only allowed along the armchair axis of BP, this

tunability occurs only for light polarized along this axis. In the quantum-confined Stark Effect, the
presence of a strong electric field results in the leaking of electron and hole wave functions into
the band gap as Airy functions, red-shifting the intersubband transitions energies77. In quantum
well structures, this red-shifting is manifested for multiple subbands, and therefore can be observed
over a wide range of energies above the band gap. To assess the gate-tunable anisotropy of the
optical response of BP, the armchair and zigzag axes, illustrated in Fig. 2.1b, of the samples
considered are identified by a combination of cross-polarized visible microscopy–where the
incident light passes through a linear polarizer, then the sample, and finally through a second,

50
orthogonal linear polarizer and by rotating the sample, the fast and slow optical axes (and hence
crystal axes) are identified100–and either polarization-dependent Raman spectroscopy or infrared
measurements, described below. Representative Raman spectra are presented for the visible
frequency sample on SrTiO3 in Figure 2.1e. The optically active armchair axis exhibits a maximum
intensity of the Ag2 resonant shift at 465 cm-1, whereas this is a minimum for the zigzag axis103,104.

Figure 2.1. Anisotropic electro-optical effects in few-layer BP. Anisotropic electro-optical effects
in few-layer BP. (a) Schematic figure of infrared tunability devices. Few-layer BP is mechanically
exfoliated on 285 nm SiO2/Si and then capped with 45 nm Al2O3 by ALD. A semitransparent top
contact of 5 nm Pd is used to apply field (VG1) while the device floats and 20 nm Ni/200 nm Au
contacts are used to gate (VG2) the contacted device. (b) Crystal structure of BP with armchair and
zigzag axes indicated. (c) Illustration of two field-driven electro-optical effects: the quantumconfined Stark effect, and symmetry-breaking modification of quantum well selection rules. In
the quantum-confined Stark effect, an external field tilts the quantum well energy levels, causing
a red-shifting of the intersubband transition energies. In the observed modification of selection
rules, this field breaks the symmetry of the quantum well and orthogonality of its wavefunctions,
allowing previously forbidden transitions to occur. (d) Illustration of anisotropic Pauli-blocking
(Burstein-Moss effect) in BP. Intersubband transitions are blocked due to the filling of the

51
conduction band. Along the ZZ axis, all optical transitions are disallowed regardless of carrier
concentration. (e) Raman spectra with excitation laser polarized along AC and ZZ axes. The
strength of the Ag2 peak is used to identify crystal axes.

2.4 MEASUREMENTS ON A 3.5 NM FLAKE
To illustrate the mechanisms of tunable dichroism of BP in the mid-infrared, we measure tunability
of transmittance using Fourier-Transform Infrared (FTIR) microscopy as a function of externally
(VG1) or directly applied bias (VG2), presented for a 3.5 nm thick flake, as determined from atomic
force microscopy (AFM), in Figure 2.2. Fig. 2.2b presents the raw extinction of the flake along
the armchair axis at zero bias, obtained by normalizing the armchair axis extinction to that of the
optically inactive zigzag axis. A band edge of approximately 0.53 eV is measured, consistent with
a thickness of 3.5nm. A broad, weak shoulder feature is observed at approximately 0.75 eV. The
corresponding calculated optical constants for the flake are presented in Figure 2.2c for
comparison.
Calculations of the optical constants of BP are based on the formalism developed in Ref.101. Optical
conductivity  is calculated using the Kubo formula within an effective low-energy Hamiltonian
for different thicknesses. The permittivity is calculated as () = ∞ + i/ where  is the
thickness of the BP, and the high-frequency permittivity ∞ is taken from Ref.105. Gate-dependent
optical properties used for Figure 2.5 are from Ref.102.
Figures 2.2d and 2.2e illustrate the influence of an external field on the extinction of BP with
carrier concentration held constant (i.e., the BP is left floating). The extinction data for each
voltage is normalized to the zero bias case and to the peak BP extinction seen in Figure 2.2b, to
obtain a tuning strength percentage that quantifies the observed tunability of the BP oscillator
strength. We note that this normalization scheme underestimates the tuning strength away from
the band edge, where BP extinction is maximal.

Figure 2.2. Electrically tunable linear dichroism: quantum-confined Stark and Burstein-Moss
effects and forbidden transitions. (a) Optical image of fabricated sample; (b) Zero-bias infrared
extinction of 3.5 nm flake, polarized along armchair (AC) axis. (c) Calculated index of refraction
for 3.5 nm thick BP with a Fermi energy at mid-gap. (d) Tunability of BP oscillator strength with
field applied to floating device, for light polarized along the AC axis. (e) Corresponding tunability
for light polarized along the zigzag (ZZ) axis. (f) Tunability of BP oscillator strength with gating
of contacted device, for light polarized along the AC axis. (g) Corresponding tunability for light
polarized along the ZZ axis.

Along the armchair axis, presented in Fig. 2.2d, two tunable features are measured near photon
energies of 0.5 and 0.8 eV. We explain the first feature at 0.5 eV as arising from a shifting of the
BP band edge due to the quantum-confined Stark effect. At negative bias, the band gap effectively
shrinks, and this is manifest as a redistribution of oscillator strength near the band edge to lower
energies. As a result, an increase in absorptance is measured below the zero-bias optical band gap,

53
and a decrease is seen above it. At positive bias, this trend is weakened and reversed. We propose
two explanations for this asymmetry: the first is the influence of electrical hysteresis, and the
second is the presence of a small internal field in the BP at zero bias, which has been observed in
previous works on the infrared optical response of few-layer BP84.
The second, higher energy feature observed in the measured spectrum does not correspond to any
predicted intersubband transition. Rather, we propose it arises due to the modification of quantum
well selection rules that limit the allowed intersubband optical transitions in black phosphorus by
the applied electric field. In a symmetric quantum well, only transitions between states with equal
quantum numbers are allowed, as other states have orthogonal wavefunctions with zero overlap
integrals101. However, a strong applied field breaks the symmetry of the quantum well and the
orthogonality of its wavefunctions, eliminating this selection rule. We note that this feature is
present in the 0 V extinction spectrum, consistent with a zero-bias internal field. As the symmetry
is further broken with an externally-applied electric field, this transition is strengthened. Under
positive bias, the internal and external fields are in competition, resulting in minimal change. This
suppressed tunability can also be attributed to hysteresis, as before.
In Figure 2.2e, no tunability is measured for any applied bias for light polarized along the zigzag
axis. This can be well understood due to the dependence of the Stark effect on the initial oscillator
strength of an optical transition; because no intersubband optical transitions are allowed along this
axis, the field effect is weak. Similar behavior has been observed in excitons in ReS2 based on an
optical Stark effect106. Moreover, while the externally applied field can allow ‘forbidden’
transitions along the armchair axis by breaking the out-of-plane symmetry of the quantum well,
in-plane symmetry properties and thus the selection rule precluding all zig-zag axis intersubband
transitions are unaffected. This selection rule and the corresponding symmetry properties have
been previously described107.
In Figures 2.2f and 2.2g, we present the complementary data set of tunable dichroism
measurements due to a directly applied gate bias with electrical contact made to the BP in a
standard field-effect transistor (FET) geometry. Here, we observe tunability dominated by carrier
concentration effects. At the band gap energy of approximately 0.53 eV, a simple decrease in
absorptance is observed at negative and large positive biases, consistent with an ambipolar BM
shift. Unlike the results of applying field while the BP floats, no tunability of the forbidden

54
transition at 0.75 eV is observed; this is explained in part due to the screening of the electric field
due to the carrier concentration tunability. We additionally may consider the possibility that this
optical transition is disallowed by Pauli-blocking effects, negating the symmetry-breaking effect
of the directly applied field. As in the case for the floating BP measurement, no tunability is
observed along the zigzag axis.

2.5 MEASUREMENTS ON A 8.5 NM FLAKE
The anisotropic electro-optical effects described above change character rapidly as the BP
thickness–and hence band gap and band structure–is varied. Figure 2.3 presents analogous results
on a flake of 8.5 nm thickness, determined by AFM, for which an optical image is presented in
Fig. 2.3a. Due to the increased thickness, the energy separation between subbands is smaller,
resulting in a narrower free-spectral range between absorptance features measured in the zero-bias
spectrum, presented in Fig. 2.3b and for which corresponding calculated optical constants are
presented in Fig. 2.3c. Results for tunability by an external field with the BP left floating are
presented in Fig. 2.3d. As in the thin flake, substantial tuning of the absorptance at each
intersubband transition is observed due to the quantum-confined Stark effect (QCSE) red-shifting
the energy of the subbands. Due to the large Stark coefficient in BP–which increases with thickness
in the few-layer limit–absorption is nearly 100% suppressed, resulting in an approximately
isotropic optical response from the material35,108. Unlike the previous sample, tuning of forbidden
transitions is not apparent; all features correspond to transitions measured in the 0 V normalization
scheme as well as the calculated optical constants for a thickness of 8.5 nm. As before, no tuning
is seen along the zigzag axis, as shown in Figure 2.4. In Fig. 2.3e, the tunability for directly gated,
contacted BP is shown. The observed tuning–a reduction in extinction centered at each of the
calculated intersubband transition energies–is relatively weak and does not persist to high photon
energies. This suggests that the dominant tunability mechanism is the ambipolar BM shift, rather
than the QCSE.

Figure 2.3. Variation of Tunability with BP Thickness. (a) Optical image of fabricated 8.5 nm
sample. (b) Zero-bias extinction of 8.5 nm flake, polarized along AC axis. (c) Calculated index of
refraction for 8.5 nm thick BP. (d) Tuning of BP oscillator strength with field applied to floating
device, for light polarized along the AC axis. (e) Tuning of BP oscillator strength with gating of
contacted device, for light polarized along the AC axis.

Figure 2.4. Tunability for 8.5 nm Flake along Zigzag Axis. (a) Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis. (b) Tunability of BP
oscillator strength with gating of contacted device, for light polarized along the ZZ axis.
Additional measurements at lower energies are presented in Figure 2.5.

Figure 2.5. Tunability for 8.5 nm Flake at Lower Energies. Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis, measured at lower
photon energies.

2.6 VISIBLE FREQUENCY MODULATION
In Figure 2.6, we present results of gate-tunable dichroism at visible frequencies in a 20 nm thick
flake, comparable to those considered for infrared tunability. A new device geometry is used to
enable transmission of visible light, shown schematically in Fig. 2.6a and in an optical image in
Fig. 2.6b. In this configuration, a SrTiO3 substrate is utilized to allow transmission-mode
measurements at visible wavelengths. A symmetric gating scheme is devised based on semitransparent top and back gate electrodes of 5 nm Ni. Samples for visible measurements were
fabricated by depositing a 5 nm thick semi-transparent back contact of Ni, followed by 45 nm
Al2O3 by ALD on a 0.5 mm thick SrTiO3 substrate. Few-layer BP was then mechanically
exfoliated and electrical contacts were fabricated as above. Measurements are performed in a
visible spectrometer. Nickel was selected as the optimum metallic contact through FiniteDifference Time Domain simulations. Only an applied field, floating BP measurement is utilized,
as band-filling effects should be negligible at this energy range. In Fig. 2.6c, we present tunability
results from 1.3 to 2 eV. Due to the QCSE, tunability is observed up to 1.8 eV, corresponding to
red light. Thus, we demonstrate that electro-optic tuning of linear dichroism is possible across an

58
extraordinarily wide range of wavelengths in a single material system, enabling multifunctional
photonic devices with broadband operation.

Figure 2.6. Tunability in the Visible. (a) Schematic figure of visible tuning device. Few-layer BP
is mechanically exfoliated on 45 nm Al2O3/5 nm Ni on SrTiO3 and then coated with 45 nm Al2O3.
A 5 nm thick semitransparent Ni top contact is used. (b) Optical image of fabricated sample with
20 nm thick BP. Dashed white line indicates the boundary of the top Ni contact. (c) Tuning of
extinction with field applied to floating device, for light polarized along the AC axis. (d)
Corresponding tuning for light polarized along the ZZ axis. (e) Calculated index of refraction for
20 nm thick BP for the measured energies. (f) Calculated imaginary index of refraction of several
thicknesses of BP from the infrared to visible.

The decay of BP intersubband oscillator strength at higher photon energies provides a spectral
cutoff for QCSE-based tunability, but for 5 nm BP or thinner this oscillator strength is strong
through the entire visible regime, as illustrated in Fig. 2.6f. We thus suggest that in very thin BP,

59
strong tuning of absorption and dichroism is possible to even higher energies. By selecting a flake
of 2 nm, for example, tunable linear dichroism is possible up to 3 eV from the band gap energy of
0.75 eV. A higher density of features, beginning at lower energies, may be introduced by utilizing
a thicker flake, with slightly decreased tuning strength, as seen for 5 and 10 nm thickness flakes.
We also note that by substituting graphene top and bottom contacts or utilizing nanophotonic
techniques to focus light on the BP, higher absolute tuning strength could be easily realized.
This phenomenon is in stark contrast to the gate-tunability of the optical response of other 2D
materials, where substantial tunability is typically constrained to the narrowband energy of the
primary exciton, as in MoS2 and WS226,73. In another van der Waals materials system, monolayer
graphene, tunability is accessible over a broader wavelength range due to the Pauli-blocking of
optical transitions at 2EF; however, this is limited to the range over which electrostatic gating is
effective, typically between EF ~ 0 to EF ~ 0.5 eV44,74. Moreover, these materials are not dichroic
or birefringent in-plane, and so BP offers a novel phenomenon that can be taken advantage of to
realize previously challenging or impossible photonic devices. The same restriction is true of bulk
tunable materials such as quantum wells, transparent conducting oxides, and transition metal
nitrides.

2.7 CONCLUSIONS
In summary, we have observed and isolated three competing, anisotropic electro-optical effects in
few-layer black phosphorus: the Pauli-blocking of intersubband transitions (Burstein-Moss effect),
the quantum-confined Stark effect, and the modification of quantum well selection rules by a
symmetry breaking electric field. These effects, which produce near-unity changes in the black
phosphorus oscillator strength for some material thicknesses and photon energies, can be tuned to
a broad range of frequencies, from the mid infrared to the visible, by controlling the thickness and
thus band structure of the black phosphorus. Further, these are strongly anisotropic electro-optical
effects, and we thus observe them to allow electrical control of both linear dichroism and
birefringence. We observe that absorption in BP can be tuned from anisotropic to nearly isotropic.
We suggest that these phenomena and this material are a promising platform for controlling the
in-plane propagation of surface or waveguide modes, as well as for polarization-switching, phase
and amplitude control, and reconfigurable far-field metasurfaces. As with van der Waals materials
as a whole, few layer black phosphorus provides a route not only to an improved material platform

60
for optoelectronics, but also to new physics, and the potential new technology paradigms that
follow.

Chapter 3. INTRABAND EXCITATIONS IN
MULTILAYER BLACK PHOSPHORUS

3.1 ABSTRACT
Black phosphorus (BP) offers considerable promise for infrared and visible photonics. Efficient
tuning of the bandgap and higher subbands in BP by modulation of the Fermi level or application
of vertical electric fields has been previously demonstrated, allowing electrical control of its above
bandgap optical properties. Here, we report modulation of the optical conductivity below the
band-gap (5-15 µm) by tuning the charge density in a two-dimensional electron gas (2DEG)
induced in BP, thereby modifying its free carrier dominated intraband response. With a moderate
doping density of 7x1012 cm-2 we were able to observe a polarization dependent epsilon-near-zero
behavior in the dielectric permittivity of BP. The intraband polarization sensitivity is intimately
linked to the difference in effective fermionic masses along the two crystallographic directions, as
confirmed by our measurements. Our results suggest the potential of multilayer BP to allow new
optical functions for emerging photonics applications.

3.2 INTRODUCTION
Hyperbolic photonic materials, in which the dielectric permittivities associated with different
polarization directions have opposite signs, present a unique platform to engineer extremely strong
anisotropic light-matter interactions and tailor novel topological properties of light109,110. They can
enable a wide range of phenomena such as near field enhancement and modification of the local
density of states of emitters111, negative refraction112, hyperlensing113, super-Planckian thermal
emission114, sub diffraction light confinement115, canalization of incident energy47, and more. Such
a wide range of novel functionalities are achieved easily with a relatively new class of materials
known as epsilon-near-zero (ENZ) materials116,117. In addition to passive artificial metamaterial
ENZ-based structures based on periodically arranged metal-dielectric stacks118, hyperbolic
dispersion has also been explored for a wide range of natural materials such as graphite, h-BN,
WTe2 in different spectral ranges 46,51,119. Despite many advances in the ability to engineer ENZ
or hyperbolic metamaterials, the idea of an electrically or optically tunable on-demand hyperbolic

62
material still remains experimentally fairly unexplored and is highly attractive for study of
fundamental phenomena such as achieving active control of optical topological transitions, as well
as applications in optical information processing and switching, and other functions51,119.
Two-dimensional electron gases (2DEG) in atomically thin materials with strong electro-optic
susceptibility offer an ideal platform to achieve highly tunable light-matter interactions42,43,120,121.
These systems have established critical metrological standards in the field of condensed matter
physics (such as the fine structure constant122 and conductance quanta123) and have contributed to
advances in photonics124–126. Black phosphorus (BP), among other two-dimensional materials, has
been heavily explored as an electronic platform for high mobility 2DEG127–132, and while first
principle calculations have been performed for undoped BP133, very little is known experimentally
about its optical properties and their tunability. Bulk BP crystal has a puckered structure, as shown
in Fig. 3.1(A), and possesses an anisotropic direct bandgap that is known to dramatically increase
from 0.3eV to 2eV as the atomically thin limit (monolayer) is reached134,135. In addition, the highly
anisotropic band structure and optical properties of BP are extremely susceptible to perturbations
in the local dielectric environment136, temperature137, electron/hole concentration in the 2DEG in
BP138, electric or magnetic field139,140, strain38,39, etc. While monolayer and few-layer BP can
exhibit strong light-matter interactions by virtue of excitonic resonances in the visible-near
infrared (IR)23,141, multilayer BP holds more potential in the mid-IR because of its lower bandgap
and stronger Drude weight142. Quantum well electro-optic effects and its anisotropy near the bandedge have been studied recently in some detail in multilayer BP143–146. However, absorption below
the optical gap, which should be dominated by free carriers in the 2DEG, is still experimentally
poorly understood and has only been investigated theoretically so far47,133. The free-carrier
response of doped BP films can persist up to mid-infrared frequencies and can be approximated to
first order by a Drude model50,147. Similar behavior has been observed in graphene148, but has not
been experimentally explored in BP. Knowledge about the charge dynamics can provide us with
an understanding of how quasiparticles in BP respond to infrared electromagnetic radiation, and
the exact nature of their respective scattering and damping processes.

A comprehensive

understanding of the polarization-dependent, mid-IR optical properties of BP may facilitate the
development of BP-based photonic devices, which hold promise for novel optoelectronic functions
in emerging technology applications.

63
In this work, we report a comprehensive study of the optical conductivity of a 2DEG induced in
multilayer BP for different hole and electron densities by performing reflection spectroscopy.
Modulation of reflection was observed both above and below the BP band edge. While changes
near or above the optical gap can be understood from an interplay of different electro-optic effects
in BP (such as Pauli blocking, quantum confined Stark effect, etc.) and modelled using the Kubo
formalism, modulation below the band gap is attributed primarily to changes in the intraband
optical conductivity which has a Drude like frequency response (𝜎 =

𝑖𝐷
𝜋(𝜔+𝑖Γ)

). We measured the

Drude weight (D) evolution and thus the change in the optical conductivity as a function of
electron/hole concentration in the 2DEG. As predicted by theory, we observed anisotropy of the
polarization resolved Drude response due to the difference in the effective mass of carriers along
the two principal crystal axes. Changes in the intraband absorption imply a transfer of the spectral
weight from the interband transitions, thereby preserving the oscillator f-sum rule for solid-state

systems, ∫0 Δ𝜎 ′ (𝜔)𝑑𝜔 = 0. Finally, from the extracted complex polarized dielectric function for
BP below the band-edge, we were able to identify an epsilon near zero (ENZ) like regime along
the armchair (AC) direction and hence, a transition from dielectric to metallic like response. No
such transition was seen for the zigzag (ZZ) direction, confirming that doped multilayer BP is
indeed an ideal system to host plasmons with tunable in-plane hyperbolic dispersion in the midIR. Furthermore, our results indicate a possible gate tunable optical topological transition for TM
polarized light from hyperbolic to elliptical, which suggests interesting opportunities for multilayer
BP in mid-IR photonic applications149.

3.3 OPTICAL AND ELECTRICAL CHARACTERIZATION OF
MULTILAYER BP FIELD EFFECT HETEROSTRUCTURE
We employed Fourier-transform infrared micro-spectroscopy to measure reflection spectra of
multilayer BP structures. A typical field effect heterostructure, schematically illustrated in side
view in Fig. 3.1(B), constructed using van der Waals assembly technique is shown in Fig. 3.1(C),
where the BP is 18.7 nm, and the top and bottom h-BN are 36.8 nm and 36.4 nm, respectively.
The carrier density in BP was tuned by applying a gate voltage across the bottom hBN and SiO2
(285nm). Such a geometry allowed independent electrical and optical characterization of the
induced 2DEG in BP. Polarized Raman spectroscopy was used to identify the AC and ZZ axes of

64
BP as indicated in the optical image of the device. All optical and electrical measurements were
performed in ambient at room temperature.

Figure 3.1. Device schematic and electro-optic characterization. (A) Anisotropic puckered crystal
structure of BP (P atoms are in sp3 hybridization). (B) Device schematic and measurement scheme
for hBN encapsulated BP devices. (C) Optical microscope image of the device discussed in the
main text. (D) Normalized reflection spectrum from the BP device shown in (C). (E) Color-map
of source-drain current variation as a function of both gate voltage and source-drain bias. (F) Gate
voltage modulated source-drain current at one representative source-drain voltage (100mV). (G)
Variation of source-drain current with source-drain voltage showing linear conduction with
systematic increase as gate voltage increases on the positive side, the slight dip is due to the fact

65
that the MCP is not at 0V). (H), (I) Interband optical modulation along the AC and ZZ axis,
respectively, showing the anisotropy in the electro-optic effects. (J) Schematic of changes in the
AC axis optical conductivity (real part) upon doping.

The reflection spectrum, shown in Fig. 3.1(D), for the same device was measured at the minimal
conductance point (MCP), confirmed from two-terminal electrical measurements as shown in Fig.
3.1(E)-(G), with light polarized along the AC direction. This spectrum is normalized to that of
optically thick Au (approx. 500 nm) evaporated on the same sample as a reference surface. Three
prominent features dominate the spectrum–a narrow hBN phonon around 1370 cm-1, a broad
dominant SiO2 phonon around 1100 cm-1 (recent studies42 show multiple phonon contributions in
SiO2) and the beginning of band edge absorption around 3000 cm-1 convoluted with an interference
dip coming from the entire stack. Additionally, from our transport measurements, a hole mobility
of 1107 cm2/V.s and an electron mobility of 412 cm2/V.s were obtained at low doping levels,
corresponding to scattering rates on the order of approximately 5-10 meV. As shown in Figs.
3.1(H) and 3.1(I), these reflection spectra can be heavily modified under positive or negative gate
voltages.
Modelling the optical conductivity of the BP electron/hole gas allows us to gain an understanding
of the quasiparticle dynamics under applied voltage. Our BP flakes are between 10-20 nm thick
and described by a sheet conductivity 𝜎 since the effective modulation is confined to only 2-3nm
from the interface of BP/b-hBN. The thickness of this modulated region was estimated from the
results of band bending calculations, using a Thomas-Fermi model143. This sheet conductivity has
contributions from both interband and intraband processes, given as 𝜎 = 𝜎𝑖𝑛𝑡𝑒𝑟𝑏𝑎𝑛𝑑 +
𝜎𝑖𝑛𝑡𝑟𝑎𝑏𝑎𝑛𝑑 = 𝜎1 (𝜔) + 𝑖𝜎2 (𝜔). The interband contribution accounts for absorption above the
band-edge, including all subbands, while the intraband part accounts for free carrier response. One
can explicitly calculate for optical conductivity using the Kubo formalism as follows50:
𝑔 ℏ𝑒 2

∑𝑠𝑠′ 𝑗𝑗 ′ ∫ 𝑑𝑘
𝜎𝑖𝑛𝑡𝑒𝑟𝑏𝑎𝑛𝑑 = −𝑖 (2𝜋)

(3.E1)

𝑓(𝐸𝑠𝑗𝑘 )−𝑓(𝐸 ′ ′ ′ )
𝑠 𝑗 𝑘

𝐸𝑠𝑗𝑘 −𝐸𝑠′ 𝑗′ 𝑘′

⟨𝜙𝑠𝑗𝑘 |𝜈
𝛼 |𝜙𝑠 ′ 𝑗 ′ 𝑘 ′ ⟩⟨𝜙𝑠 ′ 𝑗 ′ 𝑘 ′ |𝜈
𝛽 |𝜙𝑠𝑗𝑘 ⟩
𝐸𝑠𝑗𝑘 −𝐸𝑠′ 𝑗′𝑘′ +ℏ𝜔+𝑖𝜂

66
𝜎𝑖𝑛𝑡𝑟𝑎𝑏𝑎𝑛𝑑,𝑗 =

𝑖𝐷𝑗
𝑖𝜂

𝜋(𝜔+ )

, 𝐷𝑗 = 𝜋𝑒 2 ∑𝑁
𝑖=1

𝑛𝑖
𝑚𝑖,𝑗

(3.E2)
Here, 𝜈̂ 𝛼,𝛽 is the velocity operator defined as ℏ−1 𝜕𝑘𝛼,𝛽 𝐻, 𝑔𝑠 = 2 is used to denote the spin
degeneracy, f(E) is the Fermi-Dirac distribution function; the indices s(s′) refer to conduction
(valence) bands and the indices j(j′) refer to the subbands. H is the low-energy in-plane
Hamiltonian around the Γ point, 𝐸𝑠𝑗𝑘 , 𝜙𝑠𝑗𝑘 are the eigen-energies and eigenfunctions of H; 𝑚𝑖,𝑗

represents the effective mass of carriers in each subband (i) along a specific crystal orientation j,
𝑛𝑖 represents the charge density in each subband and 𝜂 is a phenomenological damping term.
Electrostatic doping of BP primarily brings about two fundamental changes in the optical response:
the emergence of a strong intraband component in the mid to far infrared and a shift of the optical
gap (interband transitions), shown schematically in Fig. 3.1(J). A combination of multiple electrooptical effects at the band-edge has been shown to explain the observed modulation, discussed
next, also summarized in Fig. 3.2.
1. Pauli blocking/Burstein Moss shift
The fermionic nature of electrons and holes in a semiconductor dictates that optical transitions
between occupied states in the valence band and unoccupied states in the conduction band are
blocked if the electron states at the same energy and momentum are already filled, leading to
reduced absorption. This effect is known as Pauli blocking/Burstein Moss shift. Additionally, since
BP has a quantum well electronic band structure, characteristic absorption dips are seen for
different subbands as they are filled with increasing doping.
2. Quantum confined Stark effect
When a quantum well is subjected to an external electric field, the electron states shift to lower
energies and the hole states to higher energies thereby reducing the effective optical bandgap.
Additionally, electrons and holes shift to the opposite sides of the well reducing the overlap integral
which reduces the oscillator strength of each transition.

67
3. Forbidden transitions / ‘mixed’ transition
In an unperturbed quantum well system, certain transitions have allowed dipole transitions and
optical matrix elements which do not vanish. From symmetry arguments those transitions happen
to be between subbands of equal principal quantum number index (j=1 VB to j=1 CB, etc.).
However, upon the application of an external electric field, modification of the overlap integral
between electrons and holes causes the previously vanishing optical transitions to be allowed and
they appear as mixed transitions.
4. Band bending
In multilayer systems for a typical field effect heterostructure geometry, a degenerate charge gas
is induced at the interface of the active material and the gate dielectric; in our case, the BP/b-hBN
interface. However, the charge is not distributed equally in the out of plane direction because the
first layer of charge screens the remaining charges. This gives rise to a thickness-dependent charge
profile approximated by the Thomas-Fermi screening model. For BP it can be seen from
calculations for a charge density of about 5-7x1012/cm2 the effective channel thickness is about 2.9
nm- meaning the induced electron/hole gas is two-dimensional in nature and not threedimensional. In all our fitting routines, it is assumed this is the case, and a sheet conductivity for
the 2DEG is used (with a static dielectric constant as a background for the whole BP).

To summarize, we explain our observation in Figs. 3.1(H) and (I) as follows. As we dope the
system with electrons, we see a suppression of absorption along the AC direction (appearing as a
dip) due to Pauli blocking which increases with applied voltage. Higher lying features such as
subbands show very weak modulation. Additionally, a mild red shift of the band edge is seen at
the highest positive voltages indicative of a Stark shift. However, on the hole side, not only do we
see a strong dip at the onset of band-edge transition and at the subband energies we also see a
stronger red-shift of the band gap due to a more dominant Stark shift. The band-edge shifts to
about 2800 cm-1 which is ~30meV below the pristine gap on the hole side, and to about 2900 cm1

which is ~15meV below the optical gap on the electron side. This asymmetry in the Stark shift

might be from impurities/residual doping in the system causing an additional field which cancels
out in the electron doped case but adds up in the hole doped case. These impurities could also be

68
causing the reduction of prominent subband oscillations on the electron side. Further nanoscale
studies would be needed to elucidate more about the underlying mechanism. The noisy weak
modulation for ZZ around 3000 cm-1 arises from the fact that the interference dip of the entire
stack in our device (which also happens serendipitously to be around the same energy as the bandedge) results in low signal.

Figure 3.2. Quantum Well electro-optic effects. Schematic of different electro-optic effects
occurring at energies near and above the band-edge of a multilayer BP thin film.

All of the observed reflection modulation spectra exhibit strong anisotropy with respect to the BP
crystal axes under AC and ZZ polarized illumination. This strong anisotropy is predicted by theory,
and results from the puckered honeycomb lattice crystal structure of phosphorene150. Our results
in Figs. 3.1(H) and 3.1(I) indicate significant optical modulation in the 2DEG and are in excellent
agreement with results from previous studies23,141–143.

3.4 LOW ENERGY DOPING DEPENDENT INTRABAND
RESPONSE IN MULTILAYER BP

Figure 3.3. Intraband response dominated reflection modulation. (A), (C) Measured (colored lines)
and simulated/fit (black lines) intraband response mediated reflection modulation along the AC

70
and ZZ axis. The fits have been performed between 750-2000 cm-1 to eliminate any band-edge
effect influence on the optical conductivity so that the Drude model suffices. (B), (D) Fits shown
separately, without offset showing a narrowing and strengthening of the Fano-like response near
the hBN and SiO2 phonons with increasing charge density in BP. (E), (F) Modelled false color plot
of modulation in reflection spectra (zoomed in between 800 and 1600 cm-1) as a function of doping
density for the AC and ZZ direction assuming the following parameters : BP meff=0.14m0 (AC),
0.71m0(ZZ), Si meff=0.26m0 (electrons), 0.386m0 (holes).

We now turn our attention to the low photon energy regime, which is dominated by the intraband
conductivity of BP. Fig. 3.3 describes this response, the understanding of which is a central result
of this paper. Figs. 3.3(A) and (B) show reflectance spectra (normalized as before) for light
polarized along the AC and ZZ axes, respectively. Both electron and hole doping can modify the
free carrier response of the 2DEG. As doping increases, a strong spectral feature is observed to
appear around the characteristic hBN (~1360 cm-1) and SiO2 (~1100 cm-1) intrinsic phonon peaks
with both electron and hole doping. We propose that this feature results from an increase in the
free carrier density, which increases the Drude conductivity and thus modifies the optical
properties of BP. This broad intraband modulation interferes with the previously described
hBN/SiO2 phonons, giving rise to an absorption line shape with a Fano-like modulation in the
hBN/SiO2 phonon regime. We hypothesize that this asymmetric Fano-like resonance shape151,152
indicates optical coupling between the narrow phonon resonances and the weak free carrier
absorption continuum. To better understand the nature of the line shape, we performed thin film
transfer matrix calculations to fit the spectra and account for multiple reflections and interferences
in the heterostructure stack. Our model incorporates a Drude-like function for the intraband optical
conductivity of the BP 2DEG, given by equation (3.E2), with which we are able to extract the
Drude weight as a function of doping. Assuming a simple parallel plate capacitor model, we can
estimate the doping density at each gate voltage. For undoped BP we assume a charge density of
1011/cm2 to account for the finite MCP response (coming from any defects or trapped charges).
The contribution to the linewidth of the imaginary component of Drude conductivity from
dephasing associated with finite scattering times was assumed to be on the order of that obtained
from DC transport measurements (approx. 5 meV) which is a valid approximation in the energy

71
ranges considered here. The possible sources of scattering include electron-phonon coupling,
electron-electron repulsion and interaction with defects and impurities. Studies have shown that
crystals of layered materials on substrates with strong phonons can also show losses from electronsurface polar phonon coupling153. There have also been reports of DC transport mobilities which
are not well correlated with optical scattering times, or which even show an anti-correlation154.
Further fundamental spectroscopic studies at far-IR (THz) frequencies will be required to further
elucidate these low energy scattering mechanisms in BP as a function of doping and temperature.
Figs. 3.3(C) and (D) summarize the fitted results without any offset to better understand the impact
of doping on the lineshape of reflectance modulation. Excellent agreement between experimental
data and transfer matrix simulations is visible in Figs. 3.3(A) and (B), which indicates that the
intraband (Drude) model suffices to explain the reflectance modulation observed at photon
energies well below the band-edge. Figs. 3.3(E) and 3.3(F) also show in false colors the changes
in the reflection modulation for AC and ZZ polarization as a function of electron/hole doping
density assuming a constant effective mass for BP. Reflection/transmission in highly subwavelength BP films is mostly dominated by the losses in the material and thus, it is important to
note that we do not incorporate the interband region in our Drude modelling of the sub-bandgap
response because we are working much below the (even the Stark shifted ) band-gap, where the
influence of interband losses is almost negligible to first order. Similar assumptions have been
experimentally validated for studies on graphene148,155. Also, it should be noted that while the
interband anisotropy is primarily governed by the parity of wavefunctions and subsequent
selection rules coming from dipole matrix elements in BP, intraband anisotropy stems from the
difference in fermionic effective mass along the two crystallographic axes.

3.5 MEASUREMENT OF THE MULTILAYER BP COMPLEX
PERMITTIVITY AND TUNABLE EPSILON-NEAR-ZERO AND
HYPERBOLICITY
Figures 3.4(A)-(D) illustrate the experimental real and imaginary parts (denoted as 𝜖1 and 𝜖2 ), of
the

dielectric

function

(obtained

𝜖𝐴𝐶/𝑍𝑍 (𝜔) = 𝜖∞𝐴𝐶/𝑍𝑍 +

𝑖𝜎𝐴𝐶/𝑍𝑍 (𝜔)
𝑡𝜖0 𝜔

, where t =

thickness of the 2DEG (2.9nm), 𝜖∞𝐴𝐶/𝑍𝑍 accounts for oscillators not captured in our “Drude”
spectral window) for BP at different doping densities under polarized excitation conditions along

72
the AC and ZZ directions. At higher energies the dielectric function is dominated by subband
transitions whose oscillator strength diminishes upon doping primarily due to Pauli blocking, along
with the aforementioned electro-optic effects. The lower energy response is mostly dominated by
free carriers.

Figure 3.4. Modelled dielectric function and tunable hyperbolicity. (A), (C) Extracted real and
imaginary part (denoted as 𝜖1 and 𝜖2 ) of the dielectric function for BP 2DEG along the Armchair

73
axis for different doping densities. The orange shaded region shows the ENZ behavior. The region
where the real part of the permittivity along the AC axis goes negative while remaining positive
for the ZZ direction is the hyperbolic region and extends to frequencies beyond our measurement
window. (B), (D) The same for the Zigzag axis. (E) False color plot of the modelled real part of
the dielectric permittivity along the AC direction assuming BP meff=0.14m0 showing the tunability
of ENZ. (F) Calculated isofrequency contours for in-plane plasmonic dispersion (TM polarized
surface modes) showing the tunability of hyperbolicity.
We observe a strong modulation of the dielectric function below the band gap with doping density,
indicating that free carrier response is a significant effect in the mid-IR range. An important finding
of our study is the appearance of an epsilon near zero region (ENZ) in BP for higher gate voltages
/ charge densities along the AC direction, where the real part of the permittivity transitions from
positive to negative. A false color plot showing the variation of the modelled real part of the
dielectric permittivity with doping density along the AC direction is shown in Fig. 3.4(E). The
ENZ region is seen to systematically shift to higher photon energies with increased doping. Here
the effective mass of BP is assumed to be 0.14m0, independent of doping density, for the sake of
simplicity. No such negative permittivity region was identified for the ZZ direction measurements,
implying extreme bianisotropy and the possibility to generate surface plasmon modes and in-plane
hyperbolic photonic dispersion in BP. We further calculate the isofrequency contours for in-plane
plasmons (TM polarized surface modes)117 for two different doping densities (one on the electron
side and one on the hole side) at 750 cm-1 to show that the hyperbolic dispersion is electrically
tunable. Electrically tunable hyperbolic dispersion, illustrated in Fig. 3.4(F) has intriguing
implications, suggesting opportunities for active hyperbolic plasmonics and photonics. We note
that the doping density achieved here is modest (~7x1012 cm-2), and higher doping densities with
larger κ dielectrics may enable the hyperbolic dispersion regime to move to shorter wavelengths.
In the frequency regime accessible in our measurements, we do not observe a negative real
permittivity along the ZZ direction, and we expect it to occur at much lower frequencies (<500
cm-1), which indicates that any surface plasmon modes at frequencies above 500 cm-1 will inherit
a hyperbolic dispersion (as shown in Fig. 3.4(F)) whereas those below will inherit an elliptical
dispersion thereby undergoing a topological transition in photonic dispersion. It should be possible
to electrically tune the transition point, as indicated by our results. Additionally, a Dirac-like
dispersion (also known as a Dirac plasmonic point156) can be engineered in the system at slightly

74
lower frequencies than the spectral window accessed in our measurements. Theoretical studies of
surface plasmons in BP and their corresponding dispersion relations have been discussed
elsewhere47; however, our results provide a concrete step in that direction.
In order to account for screening, we model the BP as two parts–1. An actively electronically
modulated 2DEG (thickness extracted from Thomas-Fermi screening calculations) and 2. A nonmodulated bulk region. It is schematically shown in Fig. 3.5. We also extract refractive index (n,k)
data for the modulated 2DEG (with an assumed thickness of 2.9nm) from the obtained 𝜎 and plot
them in Fig. 3.6 and 3.7, respectively. The static contribution to the anisotropic dielectric function
is modelled by using contribution from higher energy oscillators, taken from bulk BP studies,
previously shown to be a reasonable approximation for other two-dimensional systems. The results
have been extrapolated to much lower photon energies for clarity since the Drude model is
expected to suffice. Here, 𝜎0 =

𝑒2
4ℏ

The following relations are employed:
𝜖𝑗𝑗 (𝜔) = 𝜖∞ +

𝑖𝜎(𝜔)
, 𝑛̃ (𝜔) = √𝜖𝑗𝑗 (𝜔) , 𝑤ℎ𝑒𝑟𝑒 𝑡 = 𝑡ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 2𝐷𝐸𝐺
𝑡𝜖0 𝜔
𝜖∞ (𝐴𝐶/𝑍𝑍) = 12/10

Figure 3.5. Schematic of electrostatic gating in BP device. The formation of an inversion layer is
indicated at the interface of SiO2/Si and bottom hBN/BP (of opposite parity). The BP can be

75
modelled as two separate parts: 1. Actively electronically modulated labeled as “BP 2DEG” which
is ~2.9 nm thick from Thomas-Fermi screening calculations and 2. A non-modulated thick region
labeled as “BP bulk” which extends to the remainder of the physical thickness of the BP flake as
measured by atomic force microscopy (AFM).

Figure 3.6. Refractive index of doped BP. (A) Extracted real part of refractive index of BP 2DEG
as a function of voltage for AC excitation. (B) Extracted imaginary part of refractive index of BP
2DEG as a function of voltage for AC excitation. (C), (D) Same as (A), (B) but for ZZ excitation.
(E), (F) Same as (A), (B) but for unpolarized excitation.

Figure 3.7. Optical conductivity of doped BP. (A) Extracted real part of optical conductivity of BP
2DEG as a function of voltage for AC excitation. (B) Extracted imaginary part of optical
conductivity of BP 2DEG as a function of voltage for AC excitation. (C), (D) Same as (A), (B) but
for ZZ excitation. (E), (F) Same as (A), (B) but for unpolarized excitation. Here, 𝜎0 =

𝑒2
4ℏ

3.6 DETERMINATION OF CARRIER EFFECTIVE MASSES IN A
MULTILAYER BP 2DEG
Finally, we use our experimental results to obtain carrier effective masses that can be compared
with results from theory as shown in Fig. 3.8(A) and (B). We see qualitatively good agreement
with theory (meff ≈0.14m0) for our results along the AC axis, in fact our extracted fermionic
effective mass is slightly heavier than the previously theoretically calculated results. We speculate
this could be an interplay of two effects. Firstly, in our BP thin films the 2DEG is highly confined
and thus the band dispersion is modified, leading to heavier confined fermions131. Additionally,
when the Fermi level moves into either the conduction or valence band with gating, we access not

77
only the minima and maxima of the first subbands in the conduction and valence band,
respectively, but also the higher subbands (because of the broad Fermi-Dirac tail at 300K). In BP,
for higher lying subbands along the AC axis, the effective mass increases gradually, as given by
ℏ2

𝑚𝐴𝐶
= 2𝛾2

𝑖 +𝜂

, where 𝛿 𝑖 is the subband transition energy, 𝛾 denotes the effective coupling between

the conduction/valence bands and 𝜂 is related to the in-plane dispersion of the bands50,147. It is
possible that the carriers participating in the intraband transitions come from a mixture of the
different subbands thus leading to an overall lower perceived effective mass. For the ZZ axis (meff
≈ 0.71m0), we see a slightly larger variation in the extracted effective mass between the electron
and hole side. Electronic confinement leads to heavier fermions along the ZZ direction, which is
in accordance with our optical measurements for hole conductivity. It is possible that nonparabolic band effects give rise to a slightly lower effective mass for electrons, but further detailed
analysis is needed to resolve the observed electron-hole asymmetry. It should be noted that for the
ZZ direction the effective mass is not expected to depend on the subband index. As expected from
theory, the ZZ carriers are found to be much heavier than carriers associated with transport along
the AC direction.

Figure 3.8. Extracted Drude weight and effective mass for BP. (A) Drude weight evolution
obtained from fitting reflection data for AC and ZZ axis, plotted with expected Drude weight. (B)
Extracted effective mass from the Drude weight fits plotted versus voltage/charge density
assuming a parallel plate capacitor model and 100% gating efficiency.

78
From our experiments we are able to extract Drude weights for both polarization (AC and ZZ) and
unpolarized light. In order to explain the trends, we look carefully at how Drude weight should
evolve for a quantum well like system - (𝐷 = 𝜋𝑒 2 ∑𝑖

𝑚𝑖∗

, 𝑖 = 𝑠𝑢𝑏𝑏𝑎𝑛𝑑𝑠). Our measured value for

the effective mass is slightly higher than that expected for few-layer BP. As mentioned in the main
text, the origin can be two-fold. One can come from electron confinement. When the doping
density is low (near charge neutral), the screening length is ~100nm, which means the whole
system is uniformly doped and behaves as bulk. However, as we dope the system the screening
length goes down to ~2-5nm (most of the charge resides in the first 2-3 layers) and the system
becomes highly confined (2D). Previous quantum transport measurements revealed rather
anomalously large average in plane effective masses for BP due to such confinement effects.
Another important factor could be the increasing less dispersive nature of the subbands in BP along
the AC axis. When we dope the system, i.e., increase/decrease its Fermi level, we fill up not only
the first subband, but also higher subbands. From our reflection modulation data in the interband
regime, subband oscillations from the first 4-5 states can be seen, which means the Fermi tail is
broad enough to have significant occupation in those states. If we calculate the transition energies
based on the thickness of our sample, we find that the subbands are closely spaced. When we then
proceed
ℏ2
2𝛾2
+𝜂
𝛿𝑖

calculate

the

effective

mass

for

these

subbands

(given

𝑚𝑥𝑖 =

, 𝑤ℎ𝑒𝑟𝑒 𝛿 𝑖 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑢𝑏𝑏𝑎𝑛𝑑 𝑒𝑛𝑒𝑟𝑔𝑦 𝑔𝑎𝑝) we see an increase for higher energy states. It is

possible that participation of those fermions leads to an overall increase in the measured effective
mass. This can be understood as filling up of the different bands in the system. Since our sample
is ~18nm thick, the subbands are placed closely (lying within a few kBT of each other) and as we
gate the heterostructure we move the Fermi level from one subband to the next (also evident from
our band edge modulation data). For the ZZ direction any increase can come only from
confinement since from theory we do not expect a subband dependent effective mass (given by
𝑚𝑦𝑖 =

ℏ2
2𝜈𝑐

). For a more quantitative prediction of how the effective mass evolves, a first-principles

theoretical framework would be required which accounts for all the electro-optic effects at work,
and this is beyond the scope of the present paper. A table summarizing the possible bounds on
effective mass is presented here.

79
Table 3.T1 Bounds on effective masses.
Type

of Subband lower Subband higher Confinement

Confinement

carrier(polarization) bound

bound

lower bound

higher bound

electron (AC)

0.14m0

0.23m0

0.14m0

0.17m0

electron (ZZ)

0.71m0

1.16m0

hole (AC)

0.12m0

0.23m0

0.12m0

0.15m0

hole (ZZ)

0.72m0

1.2m0

The effective masses for electrons and holes in BP can be bounded both due to the screening effect
leading to highly confined carriers and also due to different carrier masses coming from higher
subbands as the BP is electrostatically gated (i.e., Fermi level is pushed into the conduction or
valence subbands). The bounds are presented in this table.

The transition energies are calculated from the equation141:
𝑛𝜋
𝑁+1

𝛿𝑁𝑛 = 𝐸𝑔0 − 2(𝛾 𝑐 − 𝛾 𝑣 )co s (

where 𝐸𝑔0 is the monolayer optical gap, 𝛾 𝑐 (𝛾 𝑣 ) is the nearest neighbor coupling between adjacent
layers for conduction (valence band), n is the index of transition, N is the number of layers in the
system.
For our case 𝐸𝑔0 = 1.82eV,

𝛾 𝑐 − 𝛾 𝑣 = 0.73 eV. We plot the transition energies and the

corresponding calculated effective mass for the AC direction in Fig. 3.9.

Figure 3.9. Subband effect in BP dispersion. (A) Calculation of transition energies in 18.68nm BP.
(B) Corresponding effective mass along the Armchair direction.
In order to verify all of our results, we performed similar measurements with an unpolarized beam
on the same sample. We see highly consistent and reproducible interband conductivity modulation
effects for light polarized along the AC or ZZ axis. Below the band-edge we see a systematic
strengthening of the Fano-like response for both electron and hole doping in the wavelength range
near the hBN and SiO2 phonons. Fits to those data sets are in excellent agreement with our
theoretical analysis. Additionally, we find that the effective mass obtained is approximately the
average of the in-plane effective masses along the AC and ZZ directions. This indicates that
carriers along both the axes participate in the intraband response and is consistent with BP transport
measurements. Subtle deviations are expected to occur in the estimation of the effective mass
because of the non-symmetric nature of the unpolarized beam in our setup (it is slightly elliptically
polarized) and possibly because of complex cross-scattering mechanisms between the two
crystallographic directions. Overall, both polarized and unpolarized measurements are in good
agreement with theory. We note that it is difficult to eliminate small conductivity changes due to
local fluctuations in the charge density arising from charge puddles (bubbles/grain
boundaries/defects) in heterostructures since our measurements probe a large area (approx. 2500

81
μm2)157. Additionally, studies have shown some systems to have an energy dependent quasiparticle
scattering rate owing to strong electron-electron and electron-phonon interactions158,159. Such
interactions can cause additional broadening in both the interband and the intraband absorption
features and have not been considered in our data analysis. We also note that in thick BP films (>
5 nm) like those measured here, excitonic effects are negligible and hence have not been
considered. We performed measurements on three other BP heterostructures and saw very similar
behavior.

3.7 CONCLUSIONS
We have experimentally explored the below bandgap optical response in gated multilayer BP
heterostructures and identified the dominant contribution to be a Drude-like optical conductivity
due to free carriers in the 2DEG. We find that interband transitions play a negligible role in the
low photon energy response for BP, which greatly simplifies modelling of the optical conductivity
of BP 2DEGs and subsequent photonic devices. We have measured the anisotropy in the intraband
optical conductivity of BP by performing polarized reflection measurements and extracted the
effective masses along the two crystallographic axes as a function of charge density. Our intraband
optical conductivity results are consistent with any changes in the interband regime and DC
transport measurements. Moreover, we demonstrated the existence of a plasmonic regime with
electrically tunable hyperbolic dispersion and an ENZ regime. We also identified the wavelength
regime for the onset of a topological transition for BP surface plasmons between hyperbolic and
elliptical dispersion. Our results provide a foundation for a range of future research directions
investigating BP as a strongly bianisotropic (or hyperbolic) mid-IR material for applications such
as plasmonics, molecular fingerprinting, sensing, and tailoring thermal emission. They also pose
important questions about the different scattering mechanisms in BP amidst a complex phase space
of doping, thickness, substrate and temperature, and the nature of the mid-IR to THz response in
BP, which motivates future work. Finally, our demonstration of BP as a naturally occurring
material with tunable hyperbolic dispersion and bianisotropy suggests applications in novel
photonics such as active polarization-sensitive infrared metasurfaces.

Chapter 4. ATOMICALLY THIN ELECTROOPTIC POLARIZATION MODULATOR
4.1 ABSTRACT
Active polarization control, highly desirable in photonic systems, has been limited mostly to
discrete structures in bulky dielectric media and liquid crystal based variable retarders. Here, we
report versatile electrically reconfigurable polarization conversion across the E, S and C
telecommunication bands (1410-1575 nm) in van der Waals layered materials, using tri-layer black
phosphorus (TLBP) integrated in a Fabry-Pérot cavity. The large electrical tunability of the
exciton-dominated near unity birefringence in TLBP enables spectrally broadband polarization
control. We show that polarization states can be generated over a large fraction of the Poincaré
sphere via spectral tuning, and that electrical tuning enables the state of polarization conversion to
span nearly half the Poincaré sphere. Specifically, we observe both linear to circular and crosspolarization conversion with voltage, demonstrating versatility with a high dynamic range.

4.2 INTRODUCTION
Polarization is a fundamental property of light that plays a crucial role in classical and quantum
optics. In table-top optical experiments, it is usually tailored using polarizers, wave plates, and
phase retarders. Generating arbitrary polarization states on demand is a requirement in a wide
range of photonic processes such as circular dichroism sensing of chiral molecules and proteins,
polarization-sensitive digital holography, imaging, polarization encoding for photonic qubits and
detecting material quasiparticle excitations such as phonons, spins or excitons and their associated
anisotropy160–165. Active control of polarization demands a platform with two key ingredients–a
material whose complex refractive indices can be dynamically tuned30,166 and in-plane symmetry
breaking to generate birefringence, either from an inherent material property167 or via
nanostructured metasurface elements168,169. The former can be achieved using different effects,
such as the Pockels effect, (magneto-optical) Kerr effect, Pauli blocking, quantum confined Stark
effect, free carrier absorption and optical pumping68,69,166,170–172. Despite significant advances in
the exploration of these phenomena, electrically reconfigurable control of polarization remains

83
largely elusive and suffers from limited dynamic range of polarization conversion. Furthermore,
commercial polarization control devices like variable phase retarders which use liquid crystal or
lithium niobate as active media are bulky, difficult to integrate into integrated photonic platforms,
restricting the miniaturization of photonic systems.
Two-dimensional (2D), van der Waals, semimetals and semiconductors have yielded new photonic
phenomena and directions for realization of chip-based photonics173,174. Layered van der Waals
semiconductors are known to exhibit strongly bound excitons and are extremely polarizable in the
presence of external electric fields, enabling unprecedented electric field-induced doping and
refractive index modulation20,175. The atomic-scale thickness of van der Waals materials and lack
of lattice matching requirement make them attractive candidates for future generation
optoelectronics operating in the visible and telecommunications frequency bands for applications
such as coherent control of quantum light, free-space and fiber-based communications and light
detection and ranging.31,176 However, most reports on electro-optic modulation in 2D materials,
thus far have largely been limited to graphene and transition metal dichalcogenides which are
optically isotropic in the layer plane, and hence not suitable for inherent active polarization control.
Black phosphorus (BP), a 2D quantum-well like semiconductor exhibits natural birefringence
owing to its in-plane anisotropic crystal structure23,177,178. Its unique electrically tunable optical
dichroism demonstrated at the few-layer limit combined with its thickness tunable bandgap (~750
nm to ~4 μm), opens up possibilities for polarization-sensitive electro-optic conversion at infrared
wavelengths in the telecommunication band, with a well-chosen thickness of BP179–182.
In this work, we show that TLBP manifests a large birefringence in the telecommunication band,
between the armchair (AC) and zigzag (ZZ) axis (Δ𝑛 = 𝑛𝐴𝐶 − 𝑛𝑍𝑍 ), of ~1.5 because of a strong
excitonic feature at 1398 nm. Upon electrostatic doping, this resonance is highly suppressed giving
rise to near unity complex refractive index tuning. Furthermore, we demonstrate, that by
integrating TLBP into a Fabry–Pérot cavity, the difference in the phase and amplitude of reflected
light along AC and ZZ generates polarization states over a large section of the normalized Poincaré
sphere (such as linear, elliptical and circular) and over a large spectral range. In addition, we
demonstrate active control of the reflected polatization state by electrostatic gating. Specifically,
in electrically reconfigurable cavity-based heterostructures, the polarization state was dynamically
tuned from linear to circular (like a quarter-wave plate (QWP)) and from nearly s to p-polarized

84
(like a half-wave plate (HWP)) by applying voltage at two wavelengths: λ=1442nm and
λ=1444nm, near the cavity resonance, respectively. Our results further establish a large anisotropy
bandwidth (>160nm) and highlight the potential of TLBP for very broadband active polarization
modulation over the telecommunication E, S, and C bands.
A conceptual visualization of polarization conversion is illustrated in Fig. 4.1A. An incoming
linearly polarized light illuminates a Fabry-Pérot cavity incorporating TLBP. The polarization
state of the reflected light can be electrically tuned to alter its ellipticity (circular to linear
polarization) or its azimuthal angle (s- to p-polarization). In this work, the two cavity mirrors are
composed of a highly reflecting thick back gold (Au) mirror and a partially transmitting thin top
Au mirror. The cavity medium is formed by hBN encapsulated TLBP and PMMA, which acts to
adjust the cavity resonance frequency to a critically coupled condition, for maximal polarization
conversion.

4.3 OPTICAL CHARACTERIZATION OF ELECTRICALLY
TUNABLE EXCITON IN TLBP
To characterize the electrically tunable complex refractive index of TLBP, gate dependent
polarized absorption measurements, as shown in Fig. 4.1B, were performed on a sample with the
following configuration: hBN/TLBP(with few-layer graphene contacts)/hBN/Au. The charge
neutral response (0 Volts) is dominated by a strong excitonic feature at 1398 nm, arising from the
optical transition between the lowest (highest) lying conduction (valence) bands, in the quantum
well like bandstructure of TLBP23,182. For both positive and negative voltages (electron and hole
doping, respectively) the peak absorption reduces along with an increase in the linewidth of the
excitonic transition. A stronger change is observed in the hole doping response with applied
voltage, as compared to electron doping.

Figure 4.1. Schematic of electrically tunable polarization conversion and TLBP birefringence (A)
Schematic of cavity design and polarization conversion. TLBP is incorporated in a dielectric
environment between two mirrors (one partially reflective (top) and one highly reflective
(bottom)). The incoming beam is linearly polarized, and the output beam can be azimuthally
rotated or converted between circular and linear polarization with applied voltage (between the
TLBP and the back electrode/mirror), for a fixed wavelength. (B) Experimentally measured
polarized absorption from a TLBP device (non-cavity integrated) for different doping densities,
along the armchair (AC) direction. The zigzag direction remains featureless for all conditions. (C),
(D) Extracted complex refractive index (real and imaginary part, respectively) for TLBP as a
function of doping density for the AC and ZZ direction.

Through a Kramers-Kronig consistent transfer matrix analysis of the gate-dependent absorption,
the complex refractive indices (𝑛̃ = 𝑛 + 𝑖𝑘) of TLBP for different doping densities were
estimated, as illustrated in Fig. 4.1C, D. Quite strikingly, near-unity tuning of the complex
refractive indices (for both 𝑛 and 𝑘) is observed near the excitonic resonance along the armchair
direction. No noticeable feature was seen along the orthogonal direction, zigzag (ZZ), for any
voltages, rendering that polarization passive.
A detailed discussion of the possible origins of the strong electro-optic response is provided next.
At the Γ-point of the band-structure in BP, wave functions are either even or odd with respect to
reflection across reflection plane σh which lies in the x-z plane (where, x is AC, and z is out-of-

86
plane axis). Wave functions along the x (AC) direction are even with respect to σh , whereas those
along the y (ZZ) direction are odd. This implies that in a k.p approximation, the perturbing
Hamiltonian contains vanishing linear terms in the y-direction and close to the Γ-point, the system
becomes quasi-one- dimensional. This induces a rather strong anisotropy, especially pronounced
at the excitonic resonance/optical band-edge due to the higher optical density of states and the
system behaves like an ensemble of one-dimensional exciton chains along the x-direction. In the
absence of excitonic features in TLBP, in the near infrared, the dielectric permittivities for the AC
and ZZ directions are set by the higher energy oscillators (ϵ∞ = 12.5 (𝐴𝐶), 10.2 (𝑍𝑍)), which
because of differences in the crystal symmetry along those two directions, gives rise to a broadband
anisotropy. This anisotropy is exaggerated due to an excitonic feature in the telecom band for
TLBP.

4.4 THEORETICAL UNDERSTANDING OF TUNABLE
ANISOTROPY IN TLBP
Having established the anisotropy in BP, we move on to explain the strong electrical tunability of
the optical properties of BP. When the system (BP) is near charge-neutral/flat-band conditions, the
optical susceptibility is strongly dominated by the neutral exciton resonance–which is manifested
as a strong peak in absorption along the AC direction. As the gate voltage is tuned to
positive/negative values, a two-dimensional electron gas (2DEG) forms at the interface of BP and
the bottom hBN. This 2DEG, which effectively increases the Fermi level of BP, results in the
following changes:
Coulomb screening of the exciton: This is the predominant mechanism of modulation in our
current scheme of electrostatic gating. As the charge density in the 2DEG in TLBP increases, the
free carriers effectively screen the field between the bound electron-hole pairs. This reduces the
interaction strength of the quasi-one-dimensional dipoles in TLBP (along the AC direction),
increases its Bohr radius, and leads to a reduction in the exciton binding energy and thus lowers
the oscillator strength. This mechanism is thus also responsible for tuning the anisotropy in the
system with gate voltage, as it effectively diminishes the excitonic contribution and the difference
of optical response along the AC and ZZ direction are eventually dictated by the higher energy
oscillators (𝝐∞ ).

87
Coulomb scattering of the exciton: This is another important consequence of gating. With more
free carriers available, the probability of an exciton to elastically or inelastically scatter off an
electron or hole increases. This leads to a reduction in coherence lifetime of the exciton which
manifests itself as spectral broadening corresponding to the excitonic transition, seen for both
electron and hole doping in our case.
The combined effects are schematically illustrated in Fig. 4.2.

Figure 4.2. Coulomb screening and scattering of quasi-1D excitons (A) Due to electrical gating,
free charges increase which reduce the overall attraction between the bound electron-hole pair for
the quasi-1D excitons along the AC direction. This screens the electric field lines between them
and weakens the exciton, leading to a reduction in binding energy and oscillator strength (𝑓0 ). (B)
Due to increased charges, excitons now scatter off them much more readily, leading to reduced
coherence and broadening of spectral transitions, manifested as larger linewidths (𝛤).
We also address some other electro-optic effects seen in typical 2D semiconductors that might be
at play:

Trions–As excess free electrons/holes accumulate in the 2DEG, the probability of an exciton
to bind to a free charge to form a trion increases. Compared to the exciton, a trion is lower in
energy which manifests as a redshift of the absorption peak. Generally, trions have lower
oscillator strength compared to excitons and hence can show up as a reduced absorption peak.
This explains the redshift of the excitonic peak upon hole doping (which happens to be more
efficient than electron doping for the investigated non-cavity device).

Band-structure (gap) renormalization–In the presence of excess free carriers, there can be
significant band-structure renormalization which leads to energy shifts in the absorption peak.

88
Typically, band-gap renormalization red shifts the quasiparticle gap. A near exact cancellation
of this effect is expected from the reduction in the binding energy of the exciton leaving the
spectral position of the exciton nearly unchanged for the doping densities achieved here. This
phenomenon is not probed directly through our measurements since the quasi-particle gap is
not tracked.

Pauli blocking–As the Fermi level is increased, the lowest optical transitions below the Fermi
level get blocked because of Pauli’s exclusion principle, resulting in a blue-shift of the
absorption peak. Such features were not seen in our measurements and are expected to occur
at much higher charge densities (>1013/cm2).

Stark shift–The 2DEG induced in BP has its own vertical electric field which can cause a
reduction of the bandgap and a red-shift of the absorption peak. We believe this is a very weak
(and thus negligible) effect in our current scheme of electrostatics where a pure vertical
displacement field (in absence of doping) does not exist, rather the vertical field arises from a
thickness dependent doping profile, due to screening. Since the entire thickness of TLBP is
below the Thomas-Fermi screening length (see S28), it is justified to consider the entire film
to be under uniform doping and have minimal displacement field-dependent energy shifts.

The above-listed 4 effects play a minor role and the most significant effect responsible for large
polarization conversion in the cavity-based devices is a change in the oscillator strength of the
exciton.
The overall effect of the electrostatic doping on the quasi-1D excitonic absorption spectrum can
be schematically represented as follows:

Figure 4.3. Absorption modulation schematic upon doping. Illustrated modulation in absorption
reflecting changes in the optical density of states upon doping showing a reduction of exciton
oscillator strength and broadening of the transition along with bandgap renormalization and
reduction of quasi-particle (QP) band-edge coupled with a reduction in exciton binding energy
rendering the exciton resonance nearly unchanged spectrally.

4.5 CAVITY DESIGN AND BROADBAND NATURE OF
BIREFRINGENCE IN TLBP

Figure 4.4. Example cavity design for polarization conversion and large anisotropy bandwidth
experimental demonstration. (A) Side view of a typical cavity structure adopted in this work. The
top and bottom mirrors are formed by thin and thick Au films. The cavity is comprised of hBN
encapsulated TLBP and PMMA, which acts as the tunable part of the cavity (in determining the
resonance wavelength). (B), (C), (D) Theoretically calculated complex reflection phasor,
reflection amplitude and phase spectrum, respectively, for such a typical cavity structure having
resonance ~1480 nm, showing difference in both the parameters along AC and ZZ, establishing
polarization conversion. (E) Summary of reflection amplitude spectra from 5 representative
devices fabricated as part of this study showing tunable cavity resonance. The PMMA thickness
was tuned systematically to change the resonance over 90 nm across the telecommunication band
(E,S and C). (F)-(J) Experimentally measured spectral trajectories on the normalized Poincaré
sphere corresponding to the 5 device resonances plotted in (E). All trajectories show strong spectral

91
polarization conversion (either in the azimuthal orientation or the ellipticity or both). The
difference in the trajectories are intimately related to the critical coupling between the cavity and
the incoming polarization. For all the presented trajectories, the azimuthal orientation was aligned
nearly 45 degrees to the AC and ZZ direction of the TLBP flake. For each normalized Poincaré
sphere, the blue arrows mark the beginning of the spectral scan (1410nm for D1-4, 1500 for D5)
and the red arrows mark the end (1520nm for D1-4, 1575 for D5)–also shown as stars in x-axis of
(E).
We next designed a heterostructure for polarization conversion by integrating the TLBP in an
optically resonant cavity geometry which enhances the degree of polarization conversion. A
transfer matrix calculation of a typical Fabry-Pérot cavity design, schematically shown in side
view in Fig. 4.4A, yields the complex reflection phasor, amplitude and phase spectra, as illustrated
in Fig. 4.4B, C, D respectively. From the phasor diagram in Fig. 4.4B, a prominent and different
complex reflectivity feature is seen along the two polarizations. A clear resonance from the cavity
is seen at 1479 (1470) nm for the AC (ZZ) direction in the reflection amplitude in Fig. 4.4C, along
with a weaker excitonic absorption feature at shorter wavelengths (1398 nm), seen only along the
AC direction. Interestingly, the reflected phase along the AC and ZZ, in Fig. 4.4D shows strong
differences near the cavity resonance. Taken together, these results indicate the potential for
significant polarization conversion of the reflected light. The cavity parameters used in Fig. 4.4BD are not a unique choice. In fact, the optical anisotropy in TLBP is broadband, enabling operation
over the entire wavelength range of the telecommunication E, S and C-bands with appropriate
changes in the cavity parameters–primarily via adjustment of the thicknesses of the dielectric
medium (hBN or PMMA) and the top Au mirror. Both parameters are important in determining
the resonance wavelength and the reflection extinction ratio.
We present numerical results on the effect of the top Au and PMMA thickness on the cavity
performance. Here, the thickness of the bottom Au, bottom hBN, TLBP and top hBN were fixed
at 100 nm, 120 nm, 1.59 nm, and 52 nm. The refractive indices of Au were adopted from Johnson
and Christy, while n=2.17 was used for hBN with no dispersion. For PMMA, n=1.478 was used.
AC response of TLBP was modelled with a single exciton with the following parameters–
resonance wavelength = 1398.2 nm, oscillator strength = 2.5 meV, broadening/linewidth = 45.1
meV (as extracted from measurements discussed in Fig 2.). 𝛜∞ = 12.5 and 10.2 were used for the

92
AC and ZZ direction, respectively. The ZZ direction permittivity was assumed to be constant, with
no excitonic feature. No thickness dependence was assumed for the complex refractive index for
any of the layers. Transfer matrix calculations were run with sweeps of the PMMA thickness and
the top Au thickness. Fig. 4.5A shows the evolution of the cavity resonance. As the PMMA
thickness is swept, the cavity resonance redshifts due to the overall increase in the optical length
of the cavity. As the top Au thickness is increased, the resonance frequency blueshifts because of
the change in the reflectivity of the top mirror (higher reflectivity for thicker top Au). Fig 4.5B
shows the effect of the top Au and PMMA thickness on the reflection amplitude at the resonance
along the AC direction. A trajectory is seen with low reflectivity highlighting the critical coupling
condition. This is the physical set of parameters which correspond to the maximal energy transfer
to the cavity. Finally, the maximum achieved phase shift difference between the AC and ZZ
directions as a function of top Au and PMMA thickness is discussed in Fig. 4.5C. Strong phase
difference is seen along the “critical coupling” trajectory. In this work, cavities were fabricated
with target critical coupling to the AC direction because that is the electrically tunable polarization
direction, while the ZZ remains passive, for all doping conditions.

Figure 4.5. Amplitude and phase shift dependence on cavity parameters. Effect of the top Au and
PMMA thickness are studied on the cavity performance. (A) Resonance of the cavity (along the
AC direction) showing redshifts with increasing PMMA thickness and blueshifts with increasing
metal thickness. (B) Reflection amplitude of the cavity (along the AC direction) showing the
“critical coupling” trace as a function of top Au and PMMA thickness. (C) Maximum phase shift
difference between the AC and ZZ direction plotted as a function of top Au and PMMA thickness.
Strong phase shift difference traces follow the reflection amplitude trace, highlighting the
importance of critical coupling.

To experimentally demonstrate the broadband nature of the TLBP anisotropy, reflection intensities
(S0) measured from five representative heterostructures are shown in Fig. 4.4E. The PMMA
thickness was sequentially tuned to redshift the cavity resonance, spanning approximately 100 nm
across the E, S, and C–telecommunication band. For each heterostructure device (D1 to D5), a

94
corresponding spectral trajectory on the normalized Poincaré sphere is shown in Fig. 4.4F-J,
respectively. The blue (red) arrows mark the beginning (end) of the measured spectral trajectory,
corresponding to 1410 (1520) nm for D1 to D4 and 1500 (1575) nm for D5. Efficient polarization
conversion can be seen for all the devices, confirming the broadband nature of the anisotropy in
TLBP coupled with the cavity mode. The differences in the trajectories arise from where the cavity
resonance wavelength is with respect to the beginning and ending point of the spectral scans. In
addition, the arc length subtended by the trajectories on the normalized Poincaré sphere is
intimately related to how well the cavity critically couples to the incoming free-space
electromagnetic field and is dominated strongly by the top mirror reflectivity.

4.6 SPATIAL INHOMOGENEITIES IN TLBP-CAVITY SAMPLES
We discuss next the spatial inhomogeneities in typical TLBP-cavity samples. Spatial false
colormaps of ellipticity (Fig. 4.6A) and azimuthal angles (Fig. 4.6B), measured for heterostructure
device D4, are shown for four different wavelengths near the cavity resonance, illustrating that the
polarization conversion has strong spatial variation. The optical image of the sample with
appropriately outlined bare cavity (0), 3 and 6-layer BP regions is shown in the inset of Fig. 4.6E.
To gain further insight, spectral scans are shown in Fig. 4.6C and D for a select few points
(appriopriately labelled) chosen from the aforementioned regions. Strong polarization conversion
is seen on the trilayer region. For example, a linearly polarized input (azimuth ~45o to the AC/ZZ
axes) is converted to nearly circular (42o ellipticity) at the resonance. An associated feature is seen
in the azimuth spectrum as well (with a derivate-like lineshape), peaking at -37o and +42o around
the ellipticity resonance. As expected, very weak (no) polarization conversion effects are seen
spectrally in the 6-layer (bare cavity) region. The weak effect for the 6-layer region presumably
originates from the higher quantum-well subband transitions leading to reduced birefringence.

Figure 4.6. Spatial inhomogeneity in optical anisotropy probed by polarization conversion. (A)
Spatial maps of ellipticity angle (in degrees) of device D4 for 4 different wavelengths near the
resonance (~1490 nm) of the cavity. Black lines indicate the extent of the tri-layer region (Sample
optical image shown in inset of (E)). Scale bar (in white) corresponds to 10μm. (B) Same as (A),
but for azimuthal angle. (C), (D) Ellipticity and azimuthal angle spectral scans for a few points
(marked with appropriately colored stars in (A) and (B)), showing spatial variation of the
resonance in the tri-layer region, as well as flat background response from the bare cavity and
weak polarization conversion from the 6-layer region. (E) Zoom-in spatial colormap of ellipticity
at 1495 nm along with superimposed reflected polarization ellipses at each point, for better
visualization of co-variation of azimuthal and ellipticity angles. White lines correspond to right
handedness, while black lines correspond to left handedness. Scale bar (in black) corresponds to
5um. Inset. Optical image of the device (D4) outlining the 3-layer (3) region. Also shown is the 6layer region (6) and the bare cavity (0). (F) Spatial map of maximum ellipticity for each point
within a spectral window between 1450 nm and 1520 nm. (G) Spatial map of ellipticity resonance
wavelength (filtered for |𝜒|>10o to only highlight the 3L region). (H), (I) Histograms of ellipticity

96
resonance wavelength (filtered for |𝜒|>10o) and maximum ellipticity (in degrees), where 1 pixel
on the map corresponds to 1 sq.μm.

To aid the visualization of co-variation in azimuthal and ellipticity spatially, a false colormap of
ellipticity superimposed with measured reflection polarization ellipses, is shown in Fig. 4.6E. The
white (black) color corresponds to right (left) handedness. Finally, in Fig. 4.6F and G, spatial maps
of the maximum achieved ellipticity over a bandwidth of 70 nm (1450 nm to 1520 nm) and the
corresponding resonance wavelength are plotted, respectively. Quite interestingly, the overall
achieved maximum ellipticity is relatively homogenous although the spectral distribution is quite
broad. The histogram in Fig. 4.6H (filtered for absolute ellipticity values > 10o) confirms a strong
centering of the ellipticity response around the resonace wavelength (~1490 nm). Fig. 4.6I shows
the distribution of maximum ellipticity achieved for all pixels across the entire map presented in
Fig. 4.3F. Two strong peaks are seen near +10o and -35o, corresponding to the peak ellipticity
values observed in most of the trilayer region. Our observation of spatially varying complex
refractive indices are consistent with dielectric disorder seen in typical 2D heterostructures157. We
speculate the origin of such behaviour to be two-fold–fabrication induced trapped hydrocarbons
and strain between constituent layers in the heterostructure and different stacking orders in TLBP
(ABA, AAB and ACA)38,136,183.
Spatial maps of ellipticity and azimuthal angles are presented for device D2 for different
wavelengths in Fig. 4.7 and 4.8. All maps show spatial inhomogeneity attributed to dielectric
disorder that originates from strain or trapped bubbles during the heterostructure assembly of the
device. Similar trends were observed in all the devices.

Figure 4.7. Azimuthal spatial colormaps of sample D2 for different wavelengths. Scale bar
corresponds to 10 μm. 3L region is outlined in black dashed lines for 1456 nm spatial map, while
thicker region is outlined in green.

Figure 4.8. Ellipticity spatial colormaps of sample D2 for different wavelengths. Scale bar
corresponds to 10 μm. 3L region is outlined in black dashed lines for 1466 nm spatial map, while
thicker region is outlined in green.

4.7 ELECTRICALLY TUNABLE POLARIZATION DYNAMICS
We now discuss the electrical tunability of the different polarization parameters, the key result of
this work, which is summarized for device D1 in Fig. 4.9. The Stokes (S) parameters (S0,s1,s2,s3),
which completely characterize the polarization conversion induced by the device, are shown as a
function of wavelength and gate voltage in the form of false colormaps in Fig. 4.9A-D for hole
doping regime. It can be seen that all the three normalized s-parameters tune efficiently around the
resonance (~1440 nm) with increased hole doping owing to strong cavity and TLBP interaction.

100
Figure 4.9. Electrically tunable polarization dynamics. (A),(B),(C),(D) False colormaps of the
evolution of the intensity (S0) and the three normalized Stokes parameters (s1,s2,s3), determining
the polarization state of the reflected light, as a function of wavelength and positive voltages (for
electron doping). The results are from device D1. Continuous tuning of all the 4 parameters can
be seen around the cavity resonance (~1440nm) for the entire range of doping, illustrating efficient
tuning of the polarization state with voltage. (E) Voltage dependent trajectories on the normalized
Poincaré sphere for 9 different wavelengths showing large dynamic range in tunability of the
reflected polarization state. Each color corresponds to a wavelength (same color code in (F)). The
dark arrows mark the beginning of the voltage scan (0V), and the correspondingly colored arrows
indicate the end of the voltage scan (-40V)–hole doping. (F) Visualization of the measured
reflected polarization ellipse for selected voltages and the same 9 wavelengths as in (E). At 1442
nm a strong change in ellipticity is seen where the state becomes almost circular at -18V and the
ellipticity decreases for higher voltages–acting like a QWP. The change in ellipticity is associated
with a change in the azimuthal orientation of the beam. At 1444 nm however, minimal change in
ellipticity is seen with a strong change in the azimuthal orientation–effectively behaving like a
HWP. The solid (dashed) lines correspond to right (left) handedness.

A competition between the excitonic absorption and the cavity resonance governs the overall
optical response of the system, due to the close proximity of the two features.
Hence, maximising the polarization conversion requires carefully adjusting the incident
polarization to balance the losses along the two principle axes of TLBP for a fixed orientation of
the device. To quantify the degree of polarization conversion, traces on the normalized Poincaré
sphere were measured for different input ellipticity and azimuth. The longest arc was found for
nearly linearly polarized input at an angle of ~27o with the AC axes of TLBP, corresponding to a
vertical polarization in the lab frame. Our observations of the electrically-driven changes in the Sparameters are consistent with the measured complex refractive indices of TLBP, including a
stronger hole-doped response compared to electron-doping. For example, in all the three Sparameters a suppression of the overall magnitude of the resonance is seen which is caused by the
reduction in excitonic anisotropy. In addition, the linewidth of the aforementioned features reduce
with increasing voltage due to a suppression of the losses along the AC direction. We note that the

101
polarization rotation at the resonance wavelength is primarily enabled by the higher absorption
along the armchair axis than the zigzag axis (Fig. 4.1D), further enhanced by the cavity.
To visualize the changes in the reflected polarization states better, traces on the normalized
Poincaré sphere for different wavelengths around the resonance are plotted as functions of gate
voltage. Fig. 4.9E shows such traces on the normalized Poincaré sphere for 9 different
wavelengths. The dynamic range corresponds to nearly half the normalized Poincaré sphere in
terms of solid angle subtended by the voltage-driven arcs. Quite intriguingly, two very interesting
traces can be identified. The first one is at 1442 nm, where the ellipticity changes between 0.3o to
43.7o to 16.8o between 0, -18 and -40 V, showing tunable quarter-wave plate operation. In contrast,
at 1444 nm, the azimuthal angle is tuned from 24.6o to 89.4o between 0 and -40V with suppressed
ellipticity changes, demonstrating tunable half-wave plate operation. At other wavelengths it is
possible to demonstrate a wide variety of elliptically polarized states. A two-dimensional map of
different polarization ellipses measured as a function of 9 different wavelengths (same
wavelengths and color code as Fig. 4.9E) is shown in Fig. 4.9F for voltages between 0 and -40V
(hole doping), where the strongest changes are noted. This map better illustrates the quarter-wave
plate and half-wave plate like operation, as well as other intermediate polarization conversion
configurations. A similarly high dynamic range for polarization conversion is also seen for electron
doping (Fig. 4.10). A general trend noted in these voltage dependent polarization conversion
measurements is that upon doping the TLBP, the spectral trajectory on the normalized Poincaré
sphere can be collapsed to a point–a manifestation of electrically tunable anisotropy suppression.
Thus, the larger the trajectory at nearly charge neutral doping conditions, the higher the capability
to access a wide range of polarization states by applying a voltage.

102

Figure 4.10. Normalized Poincaré sphere dynamics and polarization conversion for electron
doping. (A) Voltage dependent trajectories for 9 different wavelengths from 0V to +50V (electron
doping) showing highly versatile polarization generation. Colors correspond to the same 9
wavelengths shown in (B). (B) Two-dimensional map of generated polarization states as a function
of wavelength and voltage for 9 wavelengths and select few voltages. Half-wave plate (HWP) like
operation is seen for 1441 nm, whereas quarter-wave plate (QWP) like operation is seen for 1439.5
nm.

103

4.8 CONCLUSIONS
In conclusion, our results shed light on the versatility of BP as an active medium for electronically
reconfigurable broadband polarization conversion. The Γ-point nature of the direct-band minima
enables BP to have pronounced band-edge optical anisotropy spanning from visible (750 nm) to
the mid-infrared (4 μm)23,182, while free carrier modulation provides access to mid to far-infrared
wavelengths (>5 μm)181. Three-layer black phosphorus is of particular interest for polarization
conversion at telecommunications wavelengths, owing to its near unity birefringence close to the
excitonic resonance. These findings, which are unique to TLBP, combined with a resonant optical
cavity enabled voltage-controlled polarization conversion at the atomically thin limit–represent a
new direction for active control of optical polarization at the nanoscale. The demonstrated high
dynamic range of polarization conversion may open an avenue for realization of densely integrated
arrays of nanoscale BP electro-optic polarization converters, as a fundamental step beyond discrete
dielectric polarization converters in lithium niobate or arrays based on micron-scale liquid crystals
spatial light modulator. Previous reports on high-speed BP electro-photoresponse184 and recent
advances in cm-scale layer-controlled growth of BP thin-films185 suggest that unparalleled
possibilities may emerge for large-area, broadband polarization selective sensing, photodetection,
and active electro-optic modulation.

104

Chapter 5. RYDBERG EXCITONS AND TRIONS
IN MONOLAYER MoTe2
5.1 ABSTRACT
Monolayer transition metal dichalcogenide (TMDC) semiconductors exhibit strong excitonic
optical resonances which serve as a microscopic, non-invasive probe into their fundamental
properties. Like the hydrogen atom, such excitons can exhibit an entire Rydberg series of
resonances. Excitons have been extensively studied in most TMDCs (MoS2, MoSe2, WS2 and
WSe2), but detailed exploration of excitonic phenomena has been lacking in the important TMDC
material molybdenum ditelluride (MoTe2). Here, we report an experimental investigation of
excitonic luminescence properties of monolayer MoTe2 to understand the excitonic Rydberg
series, up to 3s. We report significant modification of emission energies with temperature (4K to
300K), quantifying the exciton-phonon coupling. Furthermore, we observe a strongly gate-tunable
exciton-trion interplay for all the Rydberg states governed mainly by free-carrier screening, Pauli

blocking, and band-gap renormalization in agreement with the results of first-principles GW plus
Bethe-Salpeter equation approach calculations. Our results help bring monolayer MoTe2 closer to
its potential applications in near-infrared optoelectronics and photonic devices.

5.2 INTRODUCTION
Excitons14, excitations which consist of bound electron-hole pairs, in monolayer transition-metal
dichalcogenide (TMDC) semiconductors are a suitable platform to investigate a rich variety of
condensed-matter phenomena–such as Mott insulators186,187, Wigner crystals188, and light-induced
magnetic phases189–via optical spectroscopy due to their high binding energy and large oscillator
strength15,190–193. The pronounced optical resonances due to excitons in TMDCs lie below the
electronic gap and arise from their atomically thin, two-dimensional nature, which features
pronounced quantum confinement and weak dielectric screening14,15,191,193,194. In monolayer
TMDCs, a valley degree of freedom195–197 emerges from the crystal structure with C3 and broken
inversion symmetries, which give rise to lowest-energy excitonic states with two-fold degeneracy
and displaying opposite chiral selection rules. Beyond the lowest-energy optical excitations,
excitons can also exist in hydrogenic internal excited states19,198,199, known as Rydberg excitons,

105
which by virtue of their relatively larger wavefunction191,200 offer a sensitive probe of excitonelectron, exciton-exciton, and other quasiparticle interactions, making them attractive candidates
for optical quantum sensing201–203. Additionally, Rydberg excitons offer a way to realize giant
light-matter interactions (similar to Rydberg atoms) and can be studied in cavity quantum
electrodynamics and nonlinear optical measurements204,205. While typically observed in resonant
reflection measurements28,194, a number of recent studies on extremely high-quality TMDC
samples have reported signatures of Rydberg excitons in photoluminescence (PL)19,27,29,206,207.
Although such states have been extensively characterized in monolayer MoS2, MoSe2, WS2 and
WSe2, 19,27–29,198,206–209 they have not been well explored in monolayer MoTe2 209.
The 2H phase of monolayer MoTe2 is semiconducting, with the smallest bandgap (in the near
infrared) among the Mo-based TMDC materials17,210–216. Several studies have reported a phase
transition to the metallic-1T’ phase, under high carrier doping conditions, which may be useful in
phase change photonics217,218. The optical properties of MoTe2 change dramatically under the
extreme conditions of carrier doping, but even at lower carrier doping (~1011 cm-2), the excitonic
properties are significantly altered and yield information about quasiparticle and exciton
interactions and define exciton-electron dynamics27,206,207,219. Because of the presence of heavy
tellurium atoms, the spin-orbit coupling effects (~230 meV for valence band and ~43 meV for
conduction band) [Champagne et al., submitted] and bright-dark A-exciton splitting (~25 meV)213
in MoTe2 are much more significant compared to the other Mo-based TMDCs. Additionally,
MoTe2 is one of the few van der Waals materials to emit near the silicon band-edge and hence, an
accurate understanding of the photo-physics of the entire Rydberg series under different conditions
of excitation density, temperature and doping can guide future development of near-infrared
optoelectronic and photovoltaic components–such as detectors, modulators, and light emitting
diodes.
In this work, we report results of the experimental characterization of the optical properties of
electrostatically gated monolayer MoTe2, probed via photoluminescence measurements.
Combining high-quality heterostructures and a resonant back-reflector geometry, we identify
different optical transitions corresponding to the excitonic Rydberg series. The evolution of
emission as a function of temperature reveals a semiconductor-like behavior with quantitative
estimation of zero-temperature energies and Rydberg exciton-phonon coupling strengths. By

106
controlling the charge density in the monolayer MoTe2 from charge neutrality up to electron/hole
densities of ~1012 cm-2 we find strong modulation of optical transitions and continuous tuning of
the ground and excited state excitonic manifold–which is computed and illustrated in Fig. 5.1(a).
We also perform first-principles calculations based on many-body perturbation theory (MBPT) to
obtain the excited-state properties of monolayer MoTe2 including many-electron interactions. First
principles GW plus Bethe Salpeter equation (GW-BSE) calculations with a new plasmon pole
model developed [Champagne et al., submitted] to account for the dynamical screening of carriers
show that the strong tunability is attributed to enhanced screening of the excitonic states from the
increased electron density as well as phase space filling which leads to Pauli blocking of optical
transitions. Our ab initio calculations also capture well the trion binding energy close to charge
neutrality. Additionally, a linear linewidth broadening is observed which is attributed to enhanced
exciton-electron scattering with increasing carrier density, in qualitative agreement with explicit
calculations that consider the scattering of excitons to the degenerate Fermi sea.

5.3 OPTICAL CHARACTERIZATION

Figure 5.1. Electro-optic investigation of Rydberg excitons in monolayer MoTe2. (a) Excitonic
energy landscape of Rydberg series in monolayer MoTe2 with the quasiparticle band structure,
exciton state energies 𝛺 𝑆 , and exciton binding energies 𝐸𝑏𝑆 obtained using GW-BSE calculations.
(b) Investigated device geometry consisting of hBN encapsulated monolayer MoTe2 on Au

107
substrate with applied gate voltage. (c) Integrated PL intensity map of investigated sample at 4K.
Bright spots indicate monolayer. Inset–optical micrograph of sample. (d) Example PL spectra with
assigned Rydberg states.

To efficiently probe the optical properties of monolayer MoTe2 samples, we adopt a Salisburyscreen57 geometry, shown schematically in Fig. 5.1(b). The MoTe2 is placed approximately a
quarter wavelength away from a back-reflector of optically thick gold which also acts as the bottom
electrode, to cause a destructive interference of the electromagnetic field at the monolayer, thereby
enhancing light-matter interaction. This configuration results in a mild Purcell enhancement of the
emission220 (Fp ~2, where Fp is the Purcell factor) and allows efficient tuning of the Fermi level in
MoTe2 with the application of a gate voltage across the bottom hBN.
Measurements of the spatial dependence of PL at T = 4K yield bright, uniform emission from the
1150𝑛𝑚

monolayer regions of the device. Fig. 5.1(c) shows integrated PL counts, 𝐼𝑃𝐿 = ∫𝜆=800𝑛𝑚 𝐼(𝜆)𝑑𝜆,
over a bandwidth from 800nm (~1.55eV) to 1150nm (~1.07eV). The bilayer regions exhibit lower
emission and a broader peak, while very faint emission is seen from the multilayer regions. There
have been reports investigating whether bilayer can become direct gap semiconductor at lower
temperatures using emission spectroscopy because of the similar photoluminescence quantum
yield214, 𝑃𝐿𝑄𝑌 =

𝛾𝑟
𝛾𝑟 +𝛾𝑛𝑟

, where 𝛾𝑟 , 𝛾𝑛𝑟 are the radiative and non-radiative rates, respectively. From

our measurements, we find the ratio of the PLQY of the monolayer to bilayer to be three, indicating
that the indirect to direct gap transition might happen when MoTe2 is thinned to a bilayer. A
representative PL spectrum (Fig. 5.1(d)) from one of the brightest monolayer spots shows sharp
emission around 1.172 eV and 1.149 eV with linewidths of 7.13 and 8.77 meV, respectively, which
we attribute to the A1s exciton and trion, as reported previously17,212–216. Some of the cleanest
regions of the sample show extremely narrow linewidths, the narrowest obtained being ~4.48 meV,
indicating high sample quality, as shown in Fig. 5.2.

108

Figure 5.2. Narrowest emission line obtained for the A1s neutral exciton transition.

Ab initio GW-BSE calculations using a modified plasmon pole model to account for dynamical
screening associated with free carriers [Champagne et al., submitted] support our understanding
of the experimental spectra and enable prediction of important optical properties such as the
quasiparticle band gap, optical resonance energies, and exciton binding energies. Within this
formalism, one- and two-particle excitations can be calculated using state-of-the-art GW and GWBSE approach, respectively. Computational details are reported in S4.13 and elsewhere
[Champagne et al., submitted]. In monolayer MoTe2, the lowest interband excitonic transition is
bright and occurs at the degenerate K and K’ points in the Brillouin zone (Fig. 5.1(a)). Due to spinorbit coupling effects, a splitting of both the valence band maximum and conduction band
minimum occurs, resulting in two distinct series of excitons, typically labeled as A and B excitons.
The optical gap computed with the first-principles GW-BSE approach, and which corresponds to
the lowest-energy A1s exciton energy, is found at Eopt = 1.09 eV, in good agreement with the
experimentally measured gap. The computed GW quasiparticle gap is Eg = 1.58 eV, (S4.15).
Thus, we predict an exciton binding energy of Eb = Eg − Eopt = 490 meV for the lowest energy
optically-bright excited state. Because of strong Coulomb interactions in low dimensions, charged
excitons (trions) are expected to form in monolayer MoTe2. We calculate the binding energy of
the negatively charged exciton from first principles by solving the corresponding equation of

109
motion for three-quasiparticle correlated bound states (S4.16) and obtain 20.6 meV (S4.16), in
agreement with former reports17 and measured values of ~23 meV.
Additional luminescence peaks are seen at higher energies at 1.269 eV, 1.29 eV and 1.315 eV, and
have been identified as the A2s trion, A2s exciton and A3s exciton, respectively, in accordance
with previous reports on MoTe2209 and other TMDCs. The assignment of these excitonic and
trionic peaks is in accordance with nomenclature which reflects how the exciton wavefunction
transforms under the crystal symmetry in analogy with the hydrogen atom221. Our ab-initio GWBSE calculations (S4.15) predict higher excited state excitons at 1.31 eV, 1.35 eV, and 1.43 eV,
corresponding to the A2s, B1s, and A3s excitons, respectively. The slight discrepancy in the peak
energies and re-ordering of the B1s and A3s exciton is likely related to the enhanced screening of
MoTe2 by the hBN dielectric, which is not considered in the calculations. The observation of
Rydberg states up to 3s enables investigation of stronger Rydberg interactions, as well as photonmatter coupling, in MoTe2. The linewidths observed for these excited states are exceptionally
narrow, ~10 meV and ~25 meV for the 2s and 3s exciton, respectively. Additionally, the brightness
of the 2s state is ~10% of the 1s state which is comparable to or higher than that for other TMDCs.

110

5.4 POWER AND TEMPERATURE DEPENDENT DYNAMICS

Figure 5.3. Pump power and temperature dependence of Rydberg excitons. (a) PL intensity
variation with increasing pump power for the 1s exciton. Inset–PL(x100) for the 2s and 3s exciton.
(b) Semi-log scale plot of intensity dependence of PL with pump power and corresponding fits to
a power-law showing excitonic emission (𝐼𝑃𝐿 = 𝐼0 𝑃𝛼 ). (c), (d) Temperature variation of
normalized PL spectrum for the 1s and 2s exciton regions, respectively. (e), (f) Fits to a
temperature model estimating different parameters for the 1s exciton and 2s exciton.

To verify that the emission is excitonic in nature, we performed pump-power dependent
photoluminescence measurements. We scanned the incident pump fluence over 2 decades in
intensity and observed an increase in the emission intensity (Fig. 5.3(a)). A mild spectral
broadening is associated with increasing pump density, originating from enhanced exciton-exciton
interactions. We analyze the peaks by fitting to a Lorentzian lineshape profile, 𝐼𝑃𝐿 =
∑𝑖=𝑅𝑦𝑑𝑏𝑒𝑟𝑔 𝑠𝑡𝑎𝑡𝑒𝑠

𝐴𝑖 Γ 𝑖

Γ 2

(𝜔−𝜔𝑖 )2 +( 𝑖 )

-- (5.E1), where 𝐴𝑖 is the oscillator strength, Γ𝑖 is the broadening

(full-width at half maximum) and 𝜔𝑖 is the resonance frequency of each resonance, respectively.

111
We can extract the integrated PL intensity, ∫ 𝐼𝑃𝐿,𝑖 (𝑃) 𝑑𝜔 = 𝐶0,𝑖 𝑃𝛼𝑖 , where 𝑃𝑖 is the incident power
and 𝛼𝑖 is the exponent for each resonance, 𝐼𝑃𝐿,𝑖 is given by equation 5.E1 and 𝐶0,𝑖 is a
dimensionless constant as a function of pump power. Near linear scaling is seen for all the
excitonic states as shown in Fig. 5.3(b), plotted in semi-log scale, with exponents as 𝛼 =
1, 0.93, 0.92 for 1s, 2s and 3s states, respectively. This excludes any defect related emission as no
saturation or non-linearity is observed over 2 orders of magnitude of incident pump power.
Rydberg excitons in MoTe2 also show strong temperature dependence, consistent with previous
observations in other TMDCs. While the lowest energy state has been investigated for MoTe2,
there is lack of knowledge about the excited state dynamics with temperature. Our measurements,
in Fig 5.3(c) and (d) for the 1s and 2s exciton, respectively, show a redshift for excitonic states
with increasing temperature which can be modeled with a semi-empirical semiconductor bandgap
⟨ℏ𝜔⟩

dependence of the form 𝐸𝑒𝑥𝑐 (𝑇) = 𝐸𝑒𝑥𝑐 (0) − 𝑆⟨ℏ𝜔⟩ [coth (

𝑘𝐵 𝑇

) − 1], where 𝐸𝑒𝑥𝑐 (0) is the

resonance energy at zero temperature limit, 𝑆 is a dimensionless constant, 𝑘𝐵 is the Boltzmann
constant and ⟨ℏ𝜔⟩ is the average phonon energy212. From the fits, we extract the parameters
summarized in Table 5.T1, also shown in Fig. 5.3(e), (f). Furthermore, the PLQY drops with
increasing temperature, which is attributed to an increase in accessible non-radiative decay
channels from the phonon contributions (evident from the linewidth broadening with increasing
temperature, shown in Fig. 5.4), while the radiative contribution remains constant. The zero-limit
exciton energy indicates the Rydberg state energy levels, in close agreement with ab initio GWBSE computed energy levels (A1s: 1.09 eV and A2s: 1.31 eV), which correspond to 𝑇 = 0𝐾.
Interestingly, the relative intensity of the 2s exciton state with respect to the 1s state grows with
increasing temperature (see Fig 5.4), possibly stemming from weaker coupling with the phonons.

112

Figure 5.4. Temperature dependent emission properties. (a) Temperature dependent linewidth
broadening of the A1s and A2s excitonic state. (b) Intensity ratio of the 2s/1s excitonic state with
temperature (1/T).

Table 5.T1. Experimentally measured zero-temperature exciton energy and exciton-phonon
coupling parameters for the Rydberg states.
State index

Energy

(𝐸𝑒𝑥𝑐 (0)) S

<ℏ𝜔> (meV)

(eV)
1s

1.176 ± 0.019

1.22 ± 0.434

8.7 ± 7.009

1.292 ± 0.018

1.08 ± 0.304

6.7 ± 5.825

113

5.5 ELECTRICALLY TUNABLE RYDBERG EXCITON EMISSION

Figure 5.5. Gate dependent PL spectrum of Rydberg excitons. (a), (d) PL intensity of different
neutral exciton species and their corresponding trion features as a function of gate voltage near the
1s and 2s/3s resonance, respectively. (b), (e) Derivative of the PL spectra,

𝑑𝑃𝐿
𝑑𝐸

, shown in (a), (d).

(c), (f) Line cuts of the PL spectrum at different voltages showing the different exciton and trion
resonances.

114
Reduced dielectric screening and strong electron-hole Coulomb interactions in two dimensional
semiconductors make their electronic and optical properties highly sensitive to their dielectric
environments22,193,194,199,219,222,223. In particular, the presence of free carriers can significantly affect
the electronic landscape of a TMD monolayer17,27,206–208,219. A key finding of our study,
summarized in Fig. 5.5, is the carrier dependence of the exciton-electron interaction, as quantified
by the gate voltage dependence of the exciton and trion emission properties. We first focus near
the 1s exciton resonance illustrated in Fig. 5.5(a)-(c). A false color map shows the evolution of the
1s exciton and trion peaks as a function of applied gate voltages (Vg = −10V to 10V). At very low
voltages the neutral exciton peak dominates in emission, but with a very small change in the carrier
density the trion peak rapidly emerges as a dominant feature on either side of Vg ~ − 0.65V (which
is identified as the charge neutral condition from the peak in neutral 1s exciton emission intensity).
At higher voltages the emission from the neutral exciton is completely suppressed. While trion
emission grows in intensity for higher voltages, it eventually saturates and shows a slight reduction
at even higher voltages. Such changes are better visualized in the derivative of PL with respect to
𝑑𝑃𝐿

energy (

𝑑𝐸

), shown in Fig. 5.5(b) and (e) for different regions of the gate voltage. The resonances

corresponding to the 2s and 3s excitonic states show a qualitatively similar doping dependence
(Fig. 5.5(d)-(f)). Line-cuts corresponding to near charge neutral condition and finite doping
showing strong exciton and trion emission spectrum are plotted in Fig. 5.5(c) and (f) for the 1s and
2s, 3s states, respectively.

5.6 DOPING DEPENDENT EXCITON EMISSION PROPERTIES
To quantitatively understand the doping induced changes in the emission dynamics, the spectrum
is fitted to a sum of multiple Lorentzian features, as given by (5.E1), corresponding to the different
exciton and trion states.

115

Figure 5.6. Gate tunable PL properties of Rydberg excitons. (a), (b) PL intensity (normalized) of
different neutral exciton, trion species, respectively, as a function of charge density. (c) Energy
shifts between the neutral exciton and the trion for 1s and 2s states as a function of charge density.
(d) Evolution of the linewidth and (e) resonance energy of various exciton and trion states as a
function of charge density.

The evolution of the peak intensity, linewidth and energy are then extracted as a function of carrier
density, with the results presented in Fig. 5.6. Fig. 5.6(a) and (b) quantify the changes in the PL
intensity of the different exciton and trion states, as discussed previously. A crossover-density (Nc )
is defined where the exciton and trion intensities overlap and is identified to be Vg = 0.296V and
Vg = −1.36V on the electron and hole side, respectively, for the 1s state. This corresponds to
charge

densities

Nc− = 2.08x1011 cm−2 and

Nc+ = 1.54x1011 cm−2 ,

respectively.

Additionally, as seen in Fig. 5.6(e), the exciton slightly blue shifts (2s much more than 1s) with
increasing charge density, while the trion redshifts. A qualitatively similar trend is seen for the
features corresponding to the 2s exciton state and a crossover-density of Vg = 1.24V and Vg =
−2.42V, corresponding to Nc− = 4.19x1011 cm−2 and Nc+ = 3.95x1011 cm−2 is identified on
the electron and hole side, respectively. The emission strength from the 3s trion state, which

116
appears red-shifted to the 3s excitonic state, is not high enough to perform further quantitative
analysis. However, from the derivative of PL measurements in Fig. 5.6(e) and (f), it is clear that a
similar qualitative picture also holds true for the 3s state. Further studies with magnetic fields are
required to study quantitative dynamics of the even higher states, so that the visibility is improved.
Our observations are consistent with previous reports of gate-tunable exciton and trion intensities
in other TMDCs27,206,207.
The energy differences (ΔE = Eexciton − Etrion ) between the exciton and trion exhibit an unusual
gate dependence and show striking difference between the 1s and 2s states (Fig 5.6(c)). An overall
slight blue shift of the energy difference ΔE1s ~1 meV, over a doping density of n~1x1012 cm−2
is seen. A linear fit for ΔE1s reveals a rate of change in the exciton energy of 0.103
(0.118

meV
1011 cm−2

) for the electron (hole) doping. A zero-density limit of the energy shift provides

the trion binding energy which is 21.94 meV (22.14 meV) for the negative (positive) trion. The
energy shift is highly exaggerated for the 2s state, where a much larger shift of ΔE2s ~10 meV is
seen over a smaller doping density of n~5x1011 cm−2 . A similar analysis yields
1.57

meV
1011 cm−2

(2.56

meV
1011 cm−2

) for the electron (hole) doped case. The trion binding energies are

estimated to 18.14 meV (13.76 meV) for the negative (positive) side. The lower binding energy
of the 2s trion compared to the 1s state follows from the lower binding energy of the corresponding
neutral state. This unusually larger energy shift is understood as stemming from the larger
wavefunction of the 2s state and thus, higher susceptibility to the electronic landscape which also
is evident from the stronger dependence of the oscillator strength with doping density of the 2s
state as compared to the 1s state. These results are illustrated in Fig. 5.6(c) and summarized in
Table 5.T2. In general, we expect the sensitivity of self-doping to increase dramatically with even
higher lying Rydberg states.
Table 5.T2. Experimentally measured binding energy and energy shifts for Rydberg trions.
State index 𝐸𝑏,𝑡𝑟𝑖𝑜𝑛+ (meV) 𝐸𝑏,𝑡𝑟𝑖𝑜𝑛− (meV)

Δ𝐸
Δ𝑛+

(meV/1011 cm-2)

Δ𝐸
Δ𝑛−

(meV/1011 cm-2)

22.14 ± 0.20

21.94 ± 0.10

0.118 ± 0.03

0.103 ± 0.02

13.67 ± 0.12

18.14 ± 0.09

2.56 ± 0.16

1.57 ± 0.19

117
We also measure the linewidth evolution (Fig. 5.6(d)) and observe that neutral exciton states
exhibit linewidth broadening as a function of doping density. Additionally, while the 2s trion
broadens, the 1s trion remains nearly unchanged with increasing carrier concentration.

5.7 DISCUSSION
To better understand doping induced changes in the optical properties we use first-principles GW
and GW-BSE calculations with a modified plasmon pole model [Champagne et al., submitted] to
compute the exciton spectrum under different carrier densities. For optical excitations close to the
band edge, we find two main effects to support our experimental observation: (i) a dopingindependent ground exciton energy (Fig.5.7(b)) and (ii) a suppression of the exciton oscillator
strength (Fig.5.7(c)). Numerical results are reported in Table 5.T3.
Table 5.T3. Computed Rydberg exciton binding energy and relative dipole moment as a function
of doping density.
Doping

A1s

density (cm-

Δ𝐸𝑏

Rel. Dipole

(meV)

Moment

A2s
Δ𝐸𝑏 (meV)

A3s

Rel. Dipole

Δ𝐸𝑏 (meV)

Moment

Rel. Dipole
Moment

0.25

0.09

2.3x1011

-116

0.58

-78

0.08

-90

0.03

1.6x1012

-246

0.28

-173

0.03

-192

0.02

3.0x1012

-301

0.10

4.5x1012

-353

0.05

5.9x1012

-366

0.02

8.7x1012

-368

0.01

Not detectable

The evolution of the exciton energy with increasing doping density arises from an interplay of
various effects224–226. In the low-doping regime, the doping-independent exciton energy results
from a compensation between the band gap renormalization and exciton binding energy reduction
(Fig.5.7(a) and (b)), expected from the reduced electron-hole Coulomb interaction.

118

Figure 5.7. Doping dependent theoretical results. Doping dependence of (a) the variation in exciton
binding energy, 𝛥𝐸𝑏 , for the ground A1s exciton, and the excited A2s and A3s states, (b) the
variation in exciton energy 𝛥𝛺 𝑠 (black curve), the exciton binding energy 𝛥𝐸𝑏 (blue curve), and
the renormalization of the QP band gap 𝛥𝐸𝑔 (red curve) for the ground A1s exciton, (c) the
oscillator strength for the A1s, A2s and A3s states. (inset) Same as (c) plotted in semi-log scale on
the y-axis.

At higher doping concentration, the ground exciton peak is expected to blueshift slightly, as the
exciton binding energy saturates, while the quasiparticle gap slightly increases due to an increase
of the energy continuum with the free carrier concentration225. In addition, as the doping density
increases, the exciton delocalizes in real space (exciton wave function reported in Fig. 5.8), and
Pauli blocking prevents transitions around the K valley, which is eventually reflected in a decrease
of the oscillator strength of the exciton peak, as shown in Fig.5(c). This argument can be related
to the intrinsically lower oscillator strength (and by reciprocity, lower PLQY) of the higher order
Rydberg states–also due to a more delocalized wave-function in real space. Similarly, the exciton
binding energy and oscillator strength of the A2s and A3s excited states decrease rapidly, and the
corresponding peaks quickly vanish above a doping density of 2x1012 cm−2 .

119

Figure 5.8. Wave-function evolution with doping density. (a)-(f) Evolution of the exciton (A1s)
wavefunction as a function of doping density showing Pauli blocking. (b) 2.3 x1011 cm-2 (c) 1.6
x1012 cm-2 (d) 3.0 x1012 cm-2 (e) 4.5 x1012 cm-2 (f) 5.9x1012 cm-2.

The average lifetime, 𝜏, of an unstable particle is related to the decay rate 𝛾, as 𝜏 = . The PL

linewidth 𝐿, obtained as 𝐿 = = ℏ𝛾, provides information about intrinsic contributions from

radiative exciton lifetime and dephasing from exciton-phonon scattering, as well as extrinsic
inhomogeneous broadening effects (e.g., doping, defects, substrate-induced disorder). In MoTe2
monolayer, the bright exciton state is energetically below the dark states, which, at low
temperature, prevents scattering towards intervalley exciton states that would require the
absorption of a phonon. Therefore, at low temperature, the intrinsic contributions to the linewidth
are dominated by radiative exciton decay227. Using Fermi’s golden rule228, we compute a radiative
exciton lifetime of 0.3 ps, corresponding to a radiative linewidth of 2.2 meV for the ground exciton.
The discrepancy with the experimental zero-doping linewidth of 7.13 meV comes from
inhomogeneous broadening effects, such as the presence of defects or substrate effects.
Furthermore, due to the presence of a back reflector which gives rise to a slight Purcell
enhancement in emission, it is expected that the enhanced radiative rate is higher than the
computed one in vacuum (by approximately ~2). With increasing doping density, charged excited
states, known as trions, emerge and couple to the excitons. Using the microscopic many-body
theory developed in previous studies226, we expect an approximately linear exciton linewidth
broadening with doping density (~8.7 meVcm-2, in our measurements) due to enhanced excitonelectron scattering.

120

5.8 CONCLUSIONS
In summary, we report on the optical luminescence features of a monolayer MoTe2, including
Rydberg excitons up to 3s states, by combining results of experimental photoluminescence
measurements and first principles calculations. We observe a linear dependence of the exciton
peak energy with incident pump fluence and a red shift with increasing temperature, following a
semi-empirical semiconductor relationship. The optical response can further be modulated with
gate voltage, with an efficient exciton to trion conversion. With increasing doping density, we
predict (i) a reduction in the exciton oscillator strength and (ii) a near-constant (mild blueshift)
exciton energy, supporting our experimental measurements. Our understanding of MoTe2 photophysics creates a foundation for understanding and design of future optoelectronic devices in the
near infrared.

121

Chapter 6. SIGNATURES OF EDGE-CONFINED
EXCITONS IN MONOLAYER BLACK
PHOSPHORUS
6.1 ABSTRACT
Quantum confining two-dimensional excitons in van der Waals materials via electrostatic
trapping229,230,

lithographic

patterning231–234,

Moiré

potentials235–238,

and

chemical

implantation239,240 has enabled controlled tailoring of light emission. While such approaches rely
on complex preparation of materials, natural edges provide an omnipresent, yet rich playground
for investigating quantum-confined excitons. Here, we observe that certain edge sites of monolayer
black phosphorus (BP) strongly localize the intrinsic quasi-1D excitons, yielding discrete spectral
lines in photoluminescence, with nearly an order of magnitude linewidth reduction. We find,
through a combination of detailed structural characterization using transmission electron
microscopy of BP edges and rigorous GW+BSE calculations of BP nanoribbons, that certain
atomic reconstructions in conjunction with strain and enhanced sensitivity to the electrostatic
environment can give rise to such distinct emission features. We observe linearly-polarized
emission from edge reconstructions that preserve parent mirror symmetry, which agree well with
our calculations. Furthermore, we demonstrate strong electrical switching of the edge-excitons,
similar to an excitonic transistor. Our results motivate investigation of BP nanoribbons and
quantum dots for tunable quantum light generation, metasurfaces and implementation into solidstate photonic circuits for quantum information processing as well as for the study of exotic phases
that may reside in such edge-based structures.

6.2 INTRODUCTION
The quest to generate tunable light for classical and quantum-optics based applications has
rendered

two-dimensional

(2D)

semiconductors

hosting

strongly

bound

excitonic

quasiparticles13,241 as emerging candidates. A combination of strong oscillator strengths due to
reduced screening200, presence of Rydberg excited states15,242 and polarization-selective
properties197,243 make excitons desirable in many frontiers. For example, a strong confinement of
excitons in real space to discretize their center of mass motion results in a smaller Bohr radius and

122
a longer lifetime–which facilitates dipole-dipole interactions enabling exploration of quantum nonlinear effects244. Owing to the strong susceptibility of excitons to external manipulation, schemes
such as gate-defined electrostatic traps229,230, strain engineered landscapes231–234,237, Moiré
potentials in twisted bilayer systems235–238 and ion-irradiated point defects239,240 have been
explored to achieve exciton control and confinement. One strategy that has been overlooked, thus
far, is the edges of two-dimensional crystals–where the symmetry breaking of the crystalline
structure as well as an effective reduction in dimensionality at the edges directly results in a locally
modified electronic structure. Such modifications can have important implications in building
electronic, optical, thermoelectric, and catalytic devices. Moreover, edges have been known to
host exotic transport properties including the observation of metallic and topological states245–248
as well as display novel optical effects249–251, but little is known about its impact on excitonic
properties.
Not only are edges ubiquitous to every material, but they can also exhibit strong gradients and
inhomogeneities in the dielectric environment and strain landscape–which could, with judicious
schemes, be harnessed to confine excitons. An extremely large deformation potential252–255 and
high sensitivity to environmental screening222 puts monolayer black phosphorus (BP) in a unique
position to have strongly modified excitons at the edges. Furthermore, unlike the well-investigated
transition metal dichalcogenides (i.e., MX2, where M = Mo, W and X = S, Se, Te) that have its
band-edge at the K point14, BP exhibits highly anisotropic optical transitions at the Γ point, which
results in a larger sensitivity to layer thickness and subsequent quantum confinement
effects177,256,257. In fact, several theoretical reports on BP nanoribbons and quantum dots have
predicted the emergence of quantum-confined edge states with distinct opto-electronic
properties258–261. BP edges exhibit vastly different properties depending upon the fracture plane armchair (1,0), zigzag (0,1) or a mixed (such as 1,3) configuration250,262–266. Additionally, edges
undergo atomic reconstruction to minimize their free energy in a non-periodic environment, some
of which perturb lattice symmetries and modify optical selection rules, causing a rotation of the
exciton dipole250,262,265,267. However, despite such predictions for novel electronic states at the
edges of monolayer BP, they have remained elusive likely due to the air-sensitive nature of BP,
which is exasperated in the monolayer limit268.

123
In this work, we report on novel optical properties observed at certain edge sites of high quality
exfoliated monolayer BP heterostructures encapsulated in an inert environment with hexagonal
boron nitride (hBN). We use low-temperature confocal photoluminescence measurements and find
striking differences in the optical spectrum at such edge sites compared to the interior of the
sample. For brevity, we address optical transitions seen at the edges of samples as “edge excitons”
and the transitions from the interior as “interior excitons”, for the remainder of the text.
Transmission electron microscopy on bare and graphene-encapsulated monolayer BP flakes
indicate atomically rough morphology at the edges, along with signatures of edge reconstruction
and strain inhomogeneity. Through rigorous GW+BSE calculations of reconstructed edges in
phosphorene nanoribbons we find the emergence of additional states in the optical absorption,
whose wave functions are strongly confined to the edges. Our results are best explained through
an interplay of emergence of edge reconstruction-driven optical transitions and sensitivity of the
edges to strain and screening effects. Finally, we demonstrate that the edge excitons can be
modulated (switched on and off) with an electrostatic gate through a combination of an in-plane
dc Stark effect and doping-induced screening.

6.3 OPTICAL SPECTROSCOPY OF EDGE EXCITONS
A typical sample in this study is conceptually illustrated in Fig. 6.1(a), where a heterostructure
consisting of a monolayer BP flake, encapsulated in hBN dielectric layers (right schematic)
exhibits distinct emission features at the edges compared to the interior of the crystal (left
schematic). We present spectroscopic data from a total of four devices–two containing electrostatic
gates–showing similar behavior. Confocal, spatially-resolved photoluminescence spectra were
acquired with a continuous-wave 532 nm (2.33 eV) excitation laser (see appendix for details of
the measurement configuration). The false colormap of integrated photoluminescence intensity in
Fig. 6.1(b) shows bright uniform emission from the monolayer regions of sample #gateD1. Quite
strikingly, in contrast to previous reports on monolayer BP excitons22,256,269, we notice significantly
different spectral features at the edges of the flake–plotted in Fig. 6.1(c). Each spectrum has been
normalized to unity, with the color scheme matching that of the spatial location labels (stars) in
Fig. 6.1(b). A series of narrow, sharp emission lines emerge, that appear on top of a relatively
broad background envelope which resembles emission from a typical monolayer exciton in the
interior of the BP flake. For a better comparison, normalized spectra from the interior and one

124
edge-site are plotted together, in Fig. 6.1(d). Clearly, the spectral features corresponding to the
edge and interior display significant differences and merit further investigation. Since the emission
background envelope at the edges qualitatively resembles that of the interior in its lineshape, it
implies that the optical cross section (~500 nm) overlaps with the interior and is larger than the
spatial extent of the edges in these measurements. The spectrally narrow lines emerging at the
edges are quite sensitive to the spatial location in the heterostructure and are, generally, not
energetically even in spacing. The edge emission peaks exhibit typical linewidths between ~3-10
meV, while the interior exciton has a linewidth of ~>30-50 meV–varying slightly across different
heterostructures and spatial locations due to fabrication-induced sample inhomogeneities157. An
approximate order of magnitude reduction in linewidth likely arises from stronger quantum
confinement and decoupling of the excitons from the lattice, resulting in reduced scattering of the
excitonic quasiparticles.

125

Figure 6.1. Emergence of edge excitons in monolayer BP. (a) Schematic of two different emission
mechanisms originating from interior (orange) excitons and edge (blue) excitons and typical
heterostructure schematic studied in this work. (b) Integrated PL intensity map of a MLBP sample
(#gateD1) with spatial points marked along certain edges showing bright and distinct peaks (scale
bar is 5 𝜇𝑚). (c) Normalized PL spectrum from points marked in (b). (d) Normalized PL spectra
from an edge and the interior of the sample shown in (b) to highlight the spectral feature
differences. (e) Variation of PL spectrum (log scale color) with incident laser power (y-scale log)
for an edge spot containing three distinct peaks. (f) Power law fit to the integrated PL intensity for
the corresponding three peaks in (e) showing linear excitonic behavior (𝛼~1), plotted in log-log
scale.

126
Most of the edge emission features showed power scaling analogous to the interior confirming the
observed emission to be excitonic in nature. Emission from one such spot (from sample #D6) has
been shown in Fig. 6.1(e) as a false colormap (in log scale) of photoluminescence spectrum versus
incident power. The three peaks at 1.75, 1.69 and 1.63 eV display near-unity exponent (𝛼 =
1.05, 1.03, and 0.99, respectively) evolution, as shown in Fig. 6.1(f). This establishes that the
discrete features are excitonic in nature and do not show saturation in emission up to ~100𝜇W of
incident power (~1.21 GW/m2)–ruling out point defects or multi-excitonic species. Furthermore,
a similar analysis across twenty-four measured spots on one sample revealed a small variation in
the exponent (𝛼) between 0.9 and 1.1 (appendix), indicating that the intensity linearity should be
a generic feature to all edge excitons.
Interestingly, the power dependence for edge excitons does not show non-linearity or saturation
behavior–meaning they are still in a quasi-1D regime, but an order of magnitude lower linewidth
(higher temporal coherence and longer lifetime) establishes a strong decoupling from the lattice.
We find edge excitons to occur between 1.61 and 1.72 eV, over a bandwidth of 110 meV and
generally red shifted with respect to the interior exciton; occasionally we find blue-shifted features
like in Fig. 6.1(d). Furthermore, the interior emission is best described by a sum of two Lorentzian
oscillators–one corresponding to the excitonic transition and one shoulder peak which is likely
from native defects in the crystal (shown in Fig. 6.2). Further studies on higher quality crystals
may enable even narrower linewidths of both the interior and edge excitons, resulting in complete
spectral separation of the two types of excitons. It is worth noting that not all edges exhibit such
peaks and is likely related to the diversity of edge terminations possible, as discussed later.

127

Figure 6.2. Lorentzian fit to monolayer BP emission. 2-peaks fit to an interior exciton emission
from the interior of device #gateD1. The peak at ~1.7 eV is the interior exciton while the lower
energy shoulder peak is likely from defects in the native crystal.

6.4 POLARIZATION AND TIME-DEPENDENT
PHOTOLUMINESCENCE
BP is known to have strong linear dichroism in its emission properties due to its unique crystal
structure and optical selection rules–motivating investigation of the polarization dynamics of the
edge excitons. Fig. 6.3(a) displays a false colormap of photoluminescence spectra as a function of
polarization angle for a spatial location in sample #D6. The individual peak intensity variation
with emission azimuthal angle, associated with the emission colormap shown in Fig. 6.3(a),
pointed with arrows, were fit to a polar equation of the following form 𝐼 = 𝐼0 cos2 (𝜃 − 𝜃0 ) + 𝑐 ,
where, 𝜃0 is the azimuthal orientation of the emission and 𝐼0 , 𝑐 are constants which denote the
peak, polarized and unpolarized, background component of the total emission, respectively. The
results of the fits, summarized in Fig. 6.3(b), clearly show that all of the emission patterns fit well
to a linear dipole model, establishing that emission observed from these states are linearly
polarized.

128

Figure 6.3. Polarization and time-dependent emission dynamics. (a) PL spectrum from one edge
site as a function of polarization (measured by rotating the analyzer on the emission side). The
emission is nearly aligned with each other and the interior background with small azimuthal
mismatch between the peaks. (b) Polar plots, along with dipolar fits, to determine the azimuthal
orientation of different peaks corresponding to emission profiles in (a). (c) Comparison of the
interior exciton orientation versus the edge states (as determined by fitting the peaks to Lorentzian
shaped discrete edge states and an interior contribution). The color of the spots is determined by
the emission energy, as shown in the color bar (in eV). (d) Spectral wandering and blinking seen
in emission as a function of time for one representative edge site.

Quite interestingly, for Fig. 6.3(a), the polar plots show nearly similar dipole alignment with small
angular mismatch in the azimuthal orientation. To investigate this in more details, we studied
multiple spatial spots across all four devices and the aggregated azimuthal variation between the
local interior exciton (which corresponds to the local armchair direction and is extracted from the

129
background envelope of the emission profile), and the edge excitons is plotted in Fig. 6.3(c), along
with a guide to the eye line (with a slope of 1). The color for each spot denotes the emission energy,
as indicated by the color scale bar. Under the absence of a mechanism responsible for breaking
mirror symmetry and modifying the optical selection rules, the interior and edge excitons are
expected to be nearly aligned in their orientation. A large cluster of states are indeed found to be
lying close to the dashed line–establishing that the edge exciton emission is intimately linked to
the intrinsic anisotropy of BP.
We monitored temporal spectral variation from many such edge sites (appendix), revealing jitter
and blinking behavior, out of which emission from one such site is shown in Fig. 6.3(d), indicating
that the emission is arising from confined states that are more sensitive to local charge fluctuations.
The sensitivity to local charge fluctuations is likely due a combination of charge hopping at the
dangling bonds on the edges, hopping between nearly degenerate edge-reconstruction states, and
a reduced density of states. This is in stark contrast to the interior emission which is stable under
continuous pumping on similar time scales (appendix).
To estimate the binding energy of these states, we performed temperature dependent emission
measurements. Since the absolute PL intensity is dependent not only on the binding energy
(radiative efficiency) of a state but also on geometric factors, we found it unreliable to study the
intensity trends quantitively as a function of temperature–given that the measurement system drifts
at least a spot size over large temperature ranges, making quantitative comparison challenging.
However, most of these edge excitonic states disappear between 40 and 70K or broaden
significantly to merge into the interior exciton envelope as seen in temperature dependent
measurements (shown in Fig. 6.4). We can thus put an upper bound of 3-6 meV as the dissociation
energy of these states.

130

Figure 6.4. Temperature dependent emission spectrum from one spot showing diminishing
emission strength as well as linewidth broadening as temperature is increased (measured on sample
#D6).

6.5 ELECTRON MICROSCOPY IMAGING OF EDGES AND
THEORETICAL CALCULATIONS
To understand the structural origin of the distinct emission peaks, we employ aberration-corrected
transmission electron microscopy (TEM) to analyze the atomic registry of monolayer BP within
the interior region and at the edge. A typical high-resolution TEM (HRTEM) image of the bare
monolayer BP (Fig. 6.5(a)) shows an interface that separates the crystalline and amorphous
regions. The crystalline region matches well with the simulated phase contrast of monolayer BP
(space group 64: Cmca) imaged with an under-focused electron beam about -5.5 nm from the
Scherzer defocus (Fig. 6.5(a) inset within the orange borderlines). Two P atoms, which are located
at different sub-planes, form a pair that display the dark-contrast atomic column along the [010]
zone axis (Fig. 6.5(a)). Positions of the P-atom pairs can be identified and labeled with falsecolored dots by a custom-built algorithm based on the phase contrast variance, allowing for the
measurement of projected spacings. Each of the atomic positions is color-coded based on the mean
value of the projected spacings with the nearest neighboring pairs (Fig. 6.5(b)). The histogram of

131
the projected spacings (Fig. 6.5(c)) can be fitted with two Gaussian profiles, suggesting a bimodal
distribution associated with the interior (0.295 nm ± 0.008 nm) and edge (0.321 nm ± 0.008 nm)
spacings. The interior spacing predominated by the “white” dots in the crystalline region (Fig.
6.5(b)) agrees well with the theoretical value of 0.2975 nm in the monolayer BP structure. Bulk
defects, such as vacancies and atomic dislocations cause local expansion (“red” dots) and
compression (“blue” dots) in spacings. In one case, the condensation of vacancies causes part of
an atomic layer to go missing. In contrast, the edge consists of the “red” dots that distribute along
the zigzag direction which is perpendicular to the P-atom pairs. The increase in the projected
spacing along the zigzag edge with respect to that of the bulk is about 9% in all free-standing
monolayer BP samples we examined using TEM, indicating a tensile strain at the edge. The lattice
expansion appears to arise from the reconstruction of P atoms near the edge surviving the
microscopic fracture and amorphization caused by mechanical exfoliation and transfer. Taken
together, the observations suggest that atomic-scale roughness and reconstruction, and
inhomogeneous strain distributions may be the origin of the distinct optical emission features at
BP edges.
Motivated by these findings, we theoretically investigated a number of BP nanoribbons with
different edge reconstructions, focusing on structures that were previously predicted to display
small formation energy and have a bandgap267. At the density functional theory (DFT) level we
find, out of the twenty-four previously computed edge structures with armchair and zigzag fracture
orientations, five promising structures that accord well with our observed results, highlighted in
Fig. 6.5(d).

132

Figure 6.5. Structural and theoretical characterization of BP edges. (a) High-resolution TEM image
of free-standing monolayer BP with minimum exposure to air showing an interface (magenta)
between the crystalline (left) and the amorphous (right) region. The crystalline image matches well
with a phase contrast image (inset with orange borderlines) simulated using a multi-slice algorithm
(QSTEM) with a defocus about -5.5 nm away from the Scherzer defocus. (b) Positions of the two
neighboring P atoms (dark contrast) identified and labeled with color-coded dots within the
indicated area (red borderlines) in (a), based on the phase contrast variance, allow for the
measurement of projected spacings. (c) Histogram of the projected spacings fitted with two
Gaussian profiles: Peak 1 (turquoise, solid) and Peak 2 (red, dashed), suggesting a bimodal
distribution of the interior (0.295 nm ± 0.008 nm) and edge (0.321 nm ± 0.008 nm) spacings in the
monolayer BP. Inset shows simulated ZZ4-i nanoribbon edge. (d) DFT level computation of band

133
structure of 25 favorable edge reconstructions in BP. (e), (f) Quasiparticle band structure for the
AC12-i and ZZ4-i edge-terminated structures computed at the GW level of theory and projected
over the edge and (2D) interior states. Optical absorption spectrum computed from first-principles
GW-BSE calculations for the ZZ4-i and AC12-i structures. Dash lines indicate the position of the
edge exciton energy, with an inset showing the electron and hole contribution of the edge exciton.
The oscillator strength of both edge excitons is on the order of 10-2 times smaller than the lowestenergy peak. Iso-contour shows the exciton wavefunction squared with the electron (Fe) and hole
(Fh) coordinates integrated out.
Optical absorption, which is directly related to the imaginary part of the dielectric function, for
both armchair and zigzag edges, shows distinct peaks in agreement with experimental observations
originating likely due to quantum confinement effects. Upon projection of the optical absorption
to interior and edge contributions (appendix), we find two likely structures that exhibit edge
excitons maintaining the same dipole orientation as the 2D interior states, one in structure AC12i and another in ZZ4-i as shown in Fig. 6.5(e) and (f), respectively. These edge excitons have
nearly the same polarization direction as the interior exciton as they preserve the inherent mirror
symmetry, but are dimmer than the brightest, lowest energy exciton with the oscillator strength
being lower by two orders of magnitude–further supported by lack of features from such states in
absorption spectroscopy. Their energies are spectrally quite close to the lowest energy exciton
(~0.39 eV and ~0.01 eV above the lowest energy for AC12-i and ZZ4-i, respectively). Further, we
expect a stronger sensitivity of the edge excitons to dielectric screening compared with interior
excitons due to stronger confinement and smaller Bohr radius at the edges194,222. This increased
sensitivity of dielectric screening can also result in a redshift between the edge and interior
excitons, although this is likely a higher-order effect194. Therefore, we hypothesize that the origin
of the narrow excitonic edge emission is a combination of atomic reconstruction at the edges, along
with strain inhomogeneity and increased sensitivity to electrostatic screening effects. It is unclear
whether the linewidth probed over a diffraction limited spot (~500 nm radius) is due to multiple
sites of slightly different reconstruction having nearly degenerate energies or sites with similar
reconstruction experiencing strain gradients. Further, since the electrostatic screening of the hBN
encapsulation likely broadens the intrinsic linewidth of the edge excitons, substantially narrower
linewidths may be achievable with the appropriate dielectric environment.

134

6.6 ELECTRICAL TUNING OF EDGE AND INTERIOR EMISSION

Figure 6.6. Gate tunable edge emission. (a) Schematic of gate-tunable heterostructure geometry
showing uniform doping region in the interior and fringe-field effects at the edge. (b) PL spectrum
from an edge site (from #gateD1) as a function of gate voltage between -5V and 5V, taken in steps
of 10mV. Inset shows a PL spectrum at 0.16V revealing two additional red-shifted peaks
corresponding to the edge exciton along with the interior exciton emission. (c) Same as (b) but
normalized for each voltage to the maximum emission feature to highlight the dominant spectral
features in each band. (d) PL intensity variation for the three features marked in (c) and (d) in
yellow (interior contribution) and orange and blue (edge contribution), as obtained from
Lorentzian fits to each spectrum at a given voltage. Marked in shaded grey is the “on” voltage
window of the edge excitons.

Finally, we demonstrate how the edge excitons can be strongly manipulated using electrostatic
gates; a schematic of the device is shown in Fig. 6.6(a). In short, monolayer BP is encapsulated in

135
hBN and voltage is applied across a back electrode of gold and top electrode of few layer graphene.
Fig. 6.6(b) shows false colormap of the variation of the PL intensity measured on an edge-site, as
the gate voltage is scanned from -5V to 5V. Three clear spectral features emerge which are also
shown in spectrum in the inset of Fig. 6.6(b). Two, spectrally narrow, features of emission (marked
with blue and orange arrows) coming from the edge excitons dominate at very small voltages. At
larger positive or negative voltages, the interior emission feature takes over. To better illustrate the
transfer of spectral weights between the different features, Fig. 6.6(c) shows the normalized PL
spectrum at each voltage as a false colormap. Analyzing the peak intensities extracted from fitting
spectrum at each gate voltage to a sum of Lorentzian features, we find that while the interior
exciton monotonically decreases for increasing electron doping, the two edge exciton states show
a near symmetric behavior with strong turn on at approximately -2.5V and turn off at
approximately +1(2)V, for peak 1 (in blue) and 2 (in orange), respectively, as shown in Fig. 6.6(d).
The monotonically varying gate-dependent background emission originates from the interior
exciton which is further confirmed by measuring a similar dependence at a spatial spot in the
interior of the device (shown in Fig. 6.7).

Figure 6.7. Comparing interior emission between interior and edge. (a) PL spectra as a function of
gate voltage for a spot in the interior of device #gateD1 showing reduction of excitonic emission
as the voltage is increased. (b) Gate dependent peak intensity of interior exciton emission when
the laser spot is excited on the interior of the sample and the edge. Both show monotonic
dependence with differences only in the absolute intensity of emission. (c) Shifts in peak energy
for the interior exciton as a function of gate voltage to compare the differences between having the
spot on the interior and on the edge. A stronger shift is seen at the edge implying that the in-plane

136
Stark effect also plays a role in determining the interior exciton dynamics at the edge along with
doping induced screening effects.

The anomalous gate-dependent emission from the interior and edge excitons reveals significant
differences in the dominant electro-optic effects at play. The monolayer BP (in the interior) is ndoped at zero bias and shows strong quenching of PL as more electrons are added (positive gate
voltages) due to enhanced screening effects270. However, the edge of the sample experiences band
bending effects likely due to redistribution of electronic charges arising from atomic
reconstruction. Our gate-dependent PL measurements indicate an upward band bending, which
aligns the Fermi level of the edge close to intrinsic, leading to bright emission from the edge under
no bias. In addition, strong fringing effects at the edges invalidate a parallel plate capacitor
approximation and lead to emergent in-plane electric fields causing a linear dc Stark effect.
Through finite element simulations of the electric field distribution at the edge of BP for the same
heterostructure, we find an in-plane field strength 𝐹𝑥 ~1 − 50

𝜇𝑚

near the edge at 𝑉𝑔 = 2𝑉. We

also expect doping induced screening effects to play a role in a similar manner for the edge, as it
does for the interior, albeit more strongly due to the localized nature of the edge exciton (see Fig.
6.8). Collectively they lead to 1. reduction of PL intensity and 2. linear energy shift of the
resonance with field strength.

137

Figure 6.8. Linear fits to resonance energies corresponding to the two edge exciton peaks (blue
and orange) and interior exciton peak (yellow), as illustrated in Fig. 6.7.

The linear slopes are 0.76 and 0.95 for the blue and orange curves, respectively. The resonance
energies have been extracted by fitting PL spectrum at each voltage to a sum of Lorentzian curves.
From gate dependent PL measurements (Fig. 6.6), we can, to first order, get an estimate of the inplane projection of the dipole, as follows. The linear stark effect can be represented with the
following equation:

Δ𝐸𝑥 = −

(𝑒.𝑑 𝑐𝑜𝑠𝜃 𝑉𝑥 )
𝑑𝑥

(6.E1)

where Δ𝐸𝑥 is the change in resonance energy, 𝑒. 𝑑 is the dipole moment, 𝜃 is the angle the dipole
subtends with the x-y plane, 𝑉𝑥 is the applied in-plane voltage and 𝑑𝑥 is the distance across which
this voltage is applied. From this relation, we obtain:
Δ𝐸𝑥 = −𝑒. 𝑑 𝑐𝑜𝑠𝜃𝐹𝑥 (6.E2)

138

where 𝐹𝑥 is the in-plane field. We first fit PL spectrum at each voltage to a sum of Lorentzian
features corresponding to the interior and edge excitons (for data presented in Fig. 6.6(c)) and
obtain the evolution of resonance energies for each exciton peak–which we further approximate
by linear fits in a voltage window where emission signal is high from the edge excitons to decouple
doping effects as much as possible. The resonance energies are fit to the following linear equation:

𝐸𝑟𝑒𝑠 (𝑉) = 𝐸0 + 𝛼𝑉

(6.E3)

where 𝐸𝑟𝑒𝑠 (𝑉) is the voltage-dependent resonance energy, 𝐸0 is the resonance energy under no
voltage and 𝛼 is the slope of the fit. We can thus rewrite E2 as follows:
𝛼 (10−3 ) 𝑒𝑉 = −(𝑒. 𝑑 𝑐𝑜𝑠𝜃 𝐹𝑥 ). (6.E4)
𝛼 (10−3 )𝑉 = −(𝑑 𝑐𝑜𝑠𝜃) ∗

[1−50]
𝜇𝑚

𝑉 (6.E5)

𝑑 𝑐𝑜𝑠𝜃 = [1−50] 𝑛𝑚 (6.E6).
The estimate for the in-plane field 𝐹𝑥 ~

[1−50]𝑉
𝜇𝑚

is obtained from electrostatic simulations as detailed
0.76

before. For the edge exciton at ~1.67 eV (𝛼 = 0.76), this yields 𝑑𝑐𝑜𝑠𝜃~ [1−50] = [0.02 −
0.95

0.76] 𝑛𝑚 and ~ [1−50] = [0.02 − 0.95] 𝑛𝑚 for the exciton at ~1.66 eV (𝛼 = 0.95), which is the
range of dipole projection in the x-y plane.

6.7 CONCLUSION AND DISCUSSION
In summary, we observed signatures of localized exciton emission from certain edges of
monolayer BP using low-temperature photoluminescence spectroscopy. These excitons are
distinct from their conventional 2D (quasi-1D in the case of BP) counterparts because of higher
temporal coherence due to reduced scattering and stronger confinement, resulting in the narrowest

139
linewidths reported thus far for any BP-based system. By further tailoring the edges of BP and
maximizing its optical cross section (such as in bottom-up synthesis of BP nanoribbons or quantum
dot nanostructures)271, a variety of novel optical devices are possible that takes advantage of the
narrow polarized optical linewidths and high gate tunability. Resonant optical coupling of edge
excitons would also enable spectral separation and isolation of individual edge states, enabling
quantum optical applications. By creating extremely small nanostructures, it may also be possible
to increase nonlinear interactions between edge excitons, resulting in hybridized excitonic states.
We envisage that the discovery of these edge excitons will enable a wide variety of novel, BPbased, photonic applications.

140

Chapter 7. OUTLOOK AND FUTURE
DIRECTIONS
In this thesis, we have demonstrated that van der Waals materials are attractive for various
photonic and opto-electronic applications31,64,171,173,174,184,272 . They provide ideal testbeds for
optical spectroscopy to investigate fundamental physics 14,45,120,155,235,236,273,274, as well as are
promising candidates for coupling with nanophotonic structures and metasurfaces to achieve
superior control of light 30,124,125,173. Such a unique intersection enables realization of next
generation photonic devices such as optical modulators, light emitting diodes, lasers for optical
computing, information processing, augmented and virtual reality and holography. It also
inspires discovery of new van der Waals materials and heterostructures with exotic properties
that can enable multi-functional control of light 10,275–279.
Next, we briefly propose some possible future research directions for each of the projects
undertaken and described in this thesis, to elucidate what opportunities lie next.

141

7.1 INTERBAND AND INTRABAND EXCITATIONS IN FEW
LAYER BLACK PHOSPHORUS
Few layer BP hosts interband (from conduction band to valence band) and intersubband
transitions in the 1-4 𝜇m range. These transitions can be strongly tuned via multiple electro-optic
effects like Pauli blocking/Burstein-Moss shift, quantum-confined Stark effect, and modification
of symmetry-induced selection rules–enabling a medium whose complex refractive indices can
be altered. When combined with judicious photonic designs, optical modulators, and detectors
with high efficiency/detectivity or responsivity can be constructed in the mid-infrared. For
example, proposals184,280 on a waveguide-integrated BP photodetector with intrinsic responsivity
of ~135 mAW-1 and orders of magnitude reduction in dark current compared to its graphenecounterpart are quite interesting. Similarly, Chang 281 demonstrated mid-infrared LED operation
by combining few layer BP with silicon photonic waveguides. Even higher efficiency light
emission can be achieved by using photonic crystal cavities with adiabatic out-coupling–which
is

common

strategy

used

routinely

photonics

research

(see

In the mid to far-infrared range (beyond 5 𝜇m) BPs hosts intraband or Drude optical transitions–that
primarily arise due to collective charge oscillations of the free electrons or holes. As noted earlier,
such transitions are also strongly polarization dependent (due to the differences in the effective mass
and the scattering rates) and enable dielectric to metallic transition along one crystal axis (armchair)
via charge-injection. Transition to optically metallic is possible for the zigzag direction as well but
under much heavier doping. This opens up the field of plasmonics in BP, more specifically,
hyperbolic plasmonics46,109,111,112,114,115,118,148,149. A structured surface such as a grating is typically used
to excite surface plasmons from free space to account for the momentum mismatch of the two modes.
As previously explored in graphene plasmonics43,44,97, BP can be patterned into nanoribbons which
act as dipole resonators. Varying the ribbon width will modify the surface plasmon resonance
condition, roughly according to the following relation–𝜔𝑝 ~

√𝑊

, where 𝜔𝑝 is the plasmon resonance

frequency and 𝑊 is the width of the nano-ribbon. 100 nm ribbons are relatively straightforward in
terms of fabrication and are expected to yield plasmon resonances at a wavelength of order 20-50
𝜇𝑚. Not many bright sources and detectors exist in this range, and to circumvent that issue, we adopt

142
a Salisbury screen-like device with a back reflector - which enhances the light-matter coupling in the
BP nanoribbons. Through the use of higher order Fabry-Perot modes which correspond to optical
path lengths of 𝐿 =

𝑚𝜆
4𝑛

(with 𝑚 = 5,7,9), we can generate a larger spectral density of resonances. A

schematic of this plasmonic device and FDTD simulation results that predict its resonant optical
response are presented in Figure 7.1.

Figure 7.1. Black phosphorus hyperbolic plasmons device. (a) Schematic of gate-tunable black
phosphorus plasmonic resonator device, including a silicon dielectric spacer and gold back
reflector. (b) Scanning electron microscope image of nanoribbons fabricated in few-layer black
phosphorus, aligned to its crystal axes. (c) Simulated absorption modulation for 100 nm black
phosphorus ribbons in the device geometry illustrated in (a), normalized to the 1x1012 cm-2 doping

143
case. The observed resonances are a convolution of spectrally narrow Fabry-Pérot modes due to
the dielectric spacer and a single broad surface plasmon mode. (d) Electric field intensity profile
surrounding an individual nanoribbon at 30 microns for 7.5x1012 cm-2 doping.

Figure 7.2. Infrared optical results on BP nanoribbons. (a) Armchair axis absorption modulation
measured in 100 nm black phosphorus nanoribbons on a Salisbury screen device, as described in
Figure 6.2. Three separate trials indicate nearly identical absorption modulation with doping, while
the (green) baseline without doping shows zero modulation. (b) A dark field optical microscope
image of a completed nanoribbon Salisbury screen device. The large red circle is the rim of the
silicon membrane, the yellow finger extending down from the top is a gold contact, and the bright
blue regions are the nano-patterned portion of the black phosphorus flakes.

144
Fabrication details:
The fabrication begins with a silicon membrane device, on silicon on insulator chips with a
commercially grown, 50 nm top silicon oxide gate dielectric. Using dry transfer process, a layer
of ~5-10 nm thick hexagonal boron nitride is stamped on top of the silicon oxide. This is mainly
to increase the mobility of black phosphorus devices which is otherwise dominated by interfacial
trap states. Using the same process, a black phosphorus flake, exfoliated onto PDMS, is stamped,
and transferred to the membrane. This is followed by standard fabrication techniques like
electron-beam lithography, reactive ion etching and electron-beam evaporation for metal
deposition to complete the device. Encapsulation with PMMA or h-BN is then used to prevent
degradation of black phosphorus. Using Helium focused ion beam (FIB) irradiation, we have
also made nanoribbons in BP that are about ~10 nm in width but the time required to pattern
multiple of these structures very quickly grows and becomes intractable. The initial experimental
characterization for infrared absorption modulation in the mid to far infrared was performed at
the Advanced Light Source (ALS), Berkeley. We used an FTIR spectrometer (Nicolet is-50)
coupled to an infrared microscope. To boost detectivity we used a helium-cooled, silicon
bolometer detector that can measure up to 50 𝜇m. Some of the initial results are shown in Figure
7.2. While repeatable absorption modulation is seen with electrostatic doping for armchair axis
polarization–the origin may derive from black phosphorus plasmons as well as resonantly
enhanced silicon free carrier absorption, or black phosphorus free carrier absorption. While these
results are encouraging, more measurements are needed to confirm the signatures of plasmons.
Once measured, this may enable in-plane beam steering. By modifying the degree of anisotropy
of the BP through electrostatic gating, the propagation of a surface plasmon polariton could be
redirected, essentially creating a switch to guide plasmon ‘traffic’ in plane. Another way to
overcome fabrication of black phosphorus nanoribbons and still observe plasmons would be to use
the strategy of acoustic plasmons, as proposed in Ref. 48

145
One more interesting problem to study in BP, is the transition arising from bands within the
conduction or valence band, as demonstrated in other systems 282. BP offers a highly anisotropic
environment with regards to these transitions due to the selection rules. Since these transitions
are quite low in energy, BP will emerge as a promising system for anisotropic mid to far-IR
photovoltaics and opto-electronics once these transitions are found experimentally. The only
challenge to observing these transitions lie in the mismatch of optical field oscillations and the
symmetry of these transitions but those can be overcome by using near-field coupling techniques
or photonic modes that allow efficient conversion between free-space and the required
symmetries for electromagnetic oscillations.

146

7.2 ATOMICALLY THIN ELECTRO-OPTIC POLARIZATION MODULATOR
As demonstrated previously, trilayer BP, when integrated with optical resonators or
nanophotonic structures can enable realization of diverse polarization states across the Poincare
sphere via electrical tuning. This is enabled by the large refractive index change around the
exciton resonance due to strong screening of the Coulomb field due to electrons/holes. This work
can be further extended to develop compelling technologies at telecom wavelengths. One
immediate follow up of this work (as also numerically illustrated in the supplemental material
of broadband electro-optic polarization conversion in atomically thin black phosphorus) is to
replace the lossy metallic mirrors in the Fabry-Perot cavity and sandwich BP between high
quality distributed Bragg reflectors (DBRs). Numerically, such structures show working
efficiencies of >90% and phase shift of 2𝜋 which is promising for full polarization control.
Introducing two unit-cells of orthogonally aligned BP layers would enable traversal of the full
Poincare sphere with voltage controls only, with much higher efficiencies. One such work is
underway currently, where two trilayer unit-cells of BP are being combined in a nanophotonic
cavity to traverse the full normalized Poincare sphere at a fixed wavelength by tuning the voltage
only. An initial design of such a structure, along with some preliminary fabricated devices and
results of full-wave simulations of the different Stokes parameter (as a function of BP excitonic
oscillator strength controlled with doping) is illustrated in Figure 7.3.

147
Figure 7.3. Dual trilayer BP-based metasurfaces. (a) Proposed schematic of twisted BP
metasurface. (b) Optical microscope image of a fabricated heterostructure containing two trilayers
of BP twisted at ~90 degrees. (c) Dark field microscope image of patches of resonators of varying
radii fabricated in a heterostructure containing hBN-BP-hBN-Au. (d)-(f) S0, s1, s3–Stokes
parameter calculated for structure (a) optimized for critical coupling as a function of top and
bottom gate voltage (represented as BP oscillator strength in meV) at a wavelength of 1448.48 nm.

Another attractive direction is to pursue generation of vortex beams with certain topological
charge. By creating optical resonators where gate control is independently controllable in each
“pizza pie” sector and incorporating BP as the active element it may be possible to switch
between vortex beams of different topological charge 283. It is now an active area of research in
the Atwater group, following directly from the BP polarization control results.
BP can also be incorporated into photonic structures to achieve polarization-dependent high
absorption at room and low temperatures. Simple Salisbury screen structures can allow trapping
of ~>60% of incoming light by placing the BP layer at a quarter wavelength away from a backreflector.
One challenge with most active materials is the covariation of amplitude and phase (polarization)
due to simultaneous tuning of the real and imaginary part of the refractive index as constrained by
Kramers-Kronig relations. The polarization control results can be improved and made more
technologically compelling by adding a third lossy material which is isotropic and whose complex
refractive properties can be independently tuned. This would allow three independent gate-controlled
parameters which is sufficient to tune the three Stokes parameters without any compromise. A great
choice for such a system is monolayer or few layer graphene. Judicious designs with three gates
would be needed to realize such structures.

148

7.3 RYDBERG EXCITONS AND TRIONS IN MoTe2
In MoTe2, we studied the role of Rydberg excitons in photoluminescence and how charge
injection effects the same. An immediate next step of this work is to perform the measurements
under high magnetic field which increases the visibility of even higher order excitons (n>3). In
other TMDCs such as WSe 2 and WS2, states up to n~12 have been observed through a variety
of measurements such as absorption, emission, and photocurrent spectroscopy 284. As the
quantum number of the Rydberg states increases, the wave function also grows due to an increase
in the Bohr radius which makes it attractive for many applications such as sensing and non-linear
optics.
As seen in our measurements, Rydberg excitons with larger quantum number show higher energy
shifts upon doping–thereby acting as precise self-doping sensors206,207,219. Due to their larger
wavefunction they can also acts as dielectric environment sensors as already demonstrated in
previous reports. This technique can be useful in non-invasive probing of fundamental properties
of many systems such as long-range order, phase transition, etc.
When coupled with optical cavities, Rydberg excitons can enable non-linear properties via
exciton-exciton interactions204. Since the wavefunctions are large, at a certain exciton density,
when the Bohr radius exceeds the inter-exciton distance, strong dipolar repulsive interactions
dominate. Such interactions can shift the exciton energy by more than a linewidth of the
emission. When a cavity is strongly coupled to such excitonic state, this shift manifests as
modified transmission or absorption. This forms the basis of non-linear optical absorption
whereby increasing the pump density the transmission or absorption can be changed in a nonlinear fashion204,205. MoTe2 is a promising candidate as it exhibits strong photoluminescence
from higher order Rydberg excitons in the near-infrared (silicon) wavelength.

149

7.4 MoTe2–SCOPE AND OUTLOOK FOR PHASE TRANSITION
One of the primary reasons why MoTe2 gained attention even after similarly behaved TMDCs
had been well explored was because of the variety of phases it exists in–1T’, Td, and 2H and
possible alloys with W (Mo 1-xWxTe2). A handful of theoretical studies and later, a few
experimental reports, proposed that by doping 2H phase of MoTe 2 with electrons, an instability
can be generated which causes the system to phase transition into the 1T’ phase 217,285. This is
attractive for phase-change photonics since charge injection can be much faster than thermally
or mechanically induced phase transitions. Furthermore, for a monolayer system, at the exciton
resonance a phase transition can induce a complex refractive index change of order ~1 or more–
which is compelling for optical modulation with high depth. However, the methods used
previously involve ionic liquid or laser ablation, making them less attractive for high-speed and
low energy operations. In addition, the characterization methods used to identify the different
phases have not been very rigorous and the origin of new Raman modes could also be attributed
to tellurium formation, rather than the 1T’ phase. Thus, the system merits further investigation
on whether this phase transition occurs through a metastable phase and if so, can it be achieved
on solid-state systems.
We have attempted two experiments in that direction with some promising evidence, described
as follows.

Solid-state ionic substrates (Raman spectroscopy)

Two substrates–lanthanum fluoride (LaF3) and lithium conductive glass ceramic (LiCGC) were
used.
Lanthanum Fluoride (LaF 3) is a solid superionic conductor which has been used extensively in
solid state EDL transistors. LaF 3 conducts fluorine ions, and the application of an external
electric field leads to F- ion migration towards or away from the interface. This results in two
EDLs, based on F- accumulation and depletion at opposite sides of the electrolyte, thus forming
nanoscale capacitors with large electric fields at the interfaces. The highest carrier density given
in literature for LaF 3 EDL gating is 5 x 1013 cm-2, as measured by in gating MoSe 2286. When
using LaF3 as a back gate, the electrochemical window ranges from -2.0 V to 2.0 V.

150
Lithium Conductive Glass Ceramic (Li CGC) is an ion-conductive glass ceramic in which Li +
ions are can migrate within a solid oxide material framework. This substrate has a similar
operating principle to the LaF 3 substrate, where the application of a gate voltage leads to ionic
movement giving regions with excess and depleted Li+ concentrations at the interfaces. Flat
surfaces are necessary to increase the geometric capacitance of the EDL, and Li CGCs is one of
the few commercially available substrates with a polished surface, achieving average surface
roughness of 1.1 nm. The highest reported carrier density for Li GCGs is 1 x 10 14 cm-2 in gating
single-layer graphene287. When using the Li CGCs as a back gate, the electrochemical window
is in the range from -2.0 V to 3.5 V.
Multiple devices were studied in a field-effect transistor architecture, shown in Figure 7.4, with
different alloys of Mo1-xWxTe2 using both substrates. While gating was seen in both cases, phase
transition was not seen clearly except one device on LaF3 substrate–which might be due to
electrostatic and electrochemical effects. Future studies can help clarify further the underlying cause
of such modulation seen in Raman spectroscopy.

151

Figure 7.4. Schematic of FET-like devices studied for Mo1-xWxTe2 Raman spectroscopy. Two
contacts, source and drain were pre-patterned on the ionic substrates and a back contact was
evaporated using electron-beam evaporation. Exfoliated flakes were then transferred using the drytransfer method.
Potassium ion dosing (ARPES)
We used potassium ion dosing on multilayer flakes with the idea that the top-most layer(s) would be
doped by the neighboring K-ions and show signs of structural phase transition. To do so, a technique
called angle resolved photo-emission spectroscopy was used at the Lawrence Berkeley National Lab,
Advanced Light Source–beamline 7. In short, this technique is widely used to measure the band
structure of materials with its sensitivity being highest at the surface. While signatures of doping
were seen in the band structure of multilayer 2H MoTe2 as strong renormalization effects, before we
could reach doping densities required for phase transition, a large coverage of K-monolayer began
forming on the surface. This reduced the visibility of bands from the MoTe2 and enhanced the
appearance of quantum well like states formed at the interface of K and MoTe2.

152

Figure 7.5. ARPES measurements of MoTe2. (a) Pristine band structure of 2H phase of MoTe2
bulk crystal. (b) Band structure of K-dosed crystal showing bandgap renormalization and
appearance of conduction band minimum. VBM–valence band maximum, CBM–conduction band
minimum.

7.5 OUTLOOK FROM QUANTUM-CONFINED EXCITONS IN MONOLAYER
BLACK PHOSPHORUS EDGES WORK
As our experiments point out, edges of monolayer BP can act as sites for exciton trapping, leading
to higher quantum confinement. Such effects are manifested in the photoluminescence spectrum as
narrow, additional peaks on top of the quasi-1D exciton. This concept can be further extended to
develop a multitude of technologies.
First, since the emission is polarized, one can envision developing even narrower linewidth states
(effectively single emitter states) with well-defined polarization. In naturally exfoliated flakes it is
often tricky to control the orientation of the tear/edge of a flake. However, with controlled growth
techniques of phosphorene edges271, polarization and emission energy can be precisely tuned to

153
match the desired metric which is attractive for tunable emission applications in the nearinfrared/visible spectrum (displays).
Second, if single emitter states can be reliably generated, this system can be used to develop qubits.
By incorporating gates, such states can be populated with electrons/holes and turned off–thereby
creating a real-time qubit switch. Single photon sources are well established in other TMDCs and
known to originate from vacancies or strain induced lattice deformations232–234,239,240,288. No such
report exists in BP, but our work is the first in that direction.
Third, coupling these emitters with plasmonic or photonic structures can enable strong enhancement
of the light emission via Purcell effect. In fact, near-field polarization tuning of such dipole antennas
could be used to alter the intrinsic polarization of emission. Polarization conversion structures made
of BP (previously demonstrated in three layers but extendable to monolayer) could be cascaded to
electrically tune the polarization of the emission. Taken together, the polarization and the quantum
yield could be electrically controlled.
Finally, our work explores a single edge but when two such edges are brought in close proximity
coupling between them emerges which is interesting to investigate. This can be achieved by either
fabricating nanoribbons of BP, previously done in transmission electron microscopy, along different
orientations or by controlled growth of BP nanoribbons, like previous reports in graphene. It is
possible such a system can host exotic physics like extended edge states with topological properties
and spin-polarization.

154

7.6 CONCLUDING REMARKS ON BP’S POTENTIAL IN COMMERCIAL
TECHNOLOGY
BP is an extremely promising material for opto-electronic technology as demonstrated in this
thesis and other reports. However, a big challenge which prevents adoption of BP for commercial
application is its air-sensitive nature. BP can easily oxidize in a matter of minutes when thinned
down to a few layers and requires immediate encapsulation. A variety of passivation techniques
have been tested in our lab from atomic layer deposition of aluminum oxide to electron-beam
evaporation of silicon dioxide. The best encapsulant was found to be hexagonal boron nitride
(hBN)–another van der Waals insulator–which can be exfoliated and used to cap BP on both
sides. One important point worth noting is that unless the hBN completely covers the BP,
oxidation can still originate from the exposed parts. In-plane oxidation is much faster than outof-plane oxidation and thus, multilayer flakes are much more stable than monolayers.
Another challenge in adopting BP for commercial technology lies in reliably growing large area,
uniform, and controlled thickness of BP. Unlike TMDCs, where significant progress has been
made in high-quality growth of monolayer films, the only report on reliably growing BP films
has been using pulsed laser deposition. Further research needs to be done on improving the
quality of such films. Eventually, if BP can be grown or transferred on to high-quality hBN films,
large area opto-electronic can be commercialized in the near, mid, and far-infrared wavelengths.
I wish for a ‘bright’ future for black phosphorus.

155

Chapter S1. Supplementary Information for Electrical
Control of Linear Dichroism in Black Phosphorus
from the Visible to Mid-Infrared
S1.1 Identification of Crystal Axes
To identify the principal crystal axes of the BP flakes, cross-polarization microscopy was used.
Incident light passes through a linear polarizer, then the sample, and finally through a second,
orthogonal linear polarizer. This technique has been previously described in this thesis. By
rotating the sample, the fast and slow optical axes (and hence crystal axes) are identified.

S1.2 AFM Characterization of Flake Thickness
To characterize the thickness of the BP flakes, AFM measurements are made of the entire device
stack. Cross-cuts of AFM images of the flakes are shown in Figure S1.1. We note that, as
previously described, AFM measures a thickness 2–3 nm larger than the true value, due to the
presence of thin phosphorus oxide layers at each interface. Moreover, we note that the presence of
the top oxide and nickel coatings prevent perfectly accurate determination of thickness, and
therefore we additionally determine thickness based on the energy levels of the band gap and
intersubband transitions.

156

Figure S1.1. AFM Characterization of Flake Thickness. (a) AFM crosscut of ‘3.5 nm’ thick flake,
showing measured thickness of 6.5 nm. (b) AFM crosscut of ‘8.5 nm’ thick flake, showing
measured thickness of 11.5 nm. (c) AFM crosscut of ‘20 nm’ thick flake, showing measured
thickness of 20 nm. Thicknesses have some uncertainty due to Ni/Al2O3 top layers.

S1.3 Tunability for 8.5 nm Flake along Zigzag Axis
Fourier transform infrared spectroscopy is used to measure electrical tunability of extinction for
light polarized along the zig-zag crystal axis of the 8.5 nm flake, as with the 3.5 nm flake. The
corresponding spectra for tunability of the floating device under an applied field and contacted
device under direct gating are shown in Fig. S1.2a and S1.2b, respectively. No tunability is seen
for this polarization, as with the 3.5 nm flake.

157

Figure S1.2. Tunability for 8.5 nm Flake along Zigzag Axis. (a) Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis. (b) Tunability of BP
oscillator strength with gating of contacted device, for light polarized along the ZZ axis.

S1.4 Tunability for 8.5 nm Flake at Lower Energies
To better understand the behavior of the QCSE at the band edge of the 8.5 nm flake, a second
measurement was made of electrical tunability of extinction for light polarized along the armchair
crystal axis of the 8.5 nm flake using a KBr beam splitter instead of CaF2. With better resolution
at lower photon energies, clear QCSE red shifting of intersubband transitions can be seen at the
lowest transition energies. The tunability strength is plotted in arbitrary units since the extinction
tunability is still normalized to the CaF2 extinction / oscillator strength maximum.

158

Figure S1.3. Tunability for 8.5 nm Flake at Lower Energies. Tunability of BP oscillator strength
with field applied to floating device, for light polarized along the ZZ axis, measured at lower
photon energies.

S1.5 Optical Response of Top Contact Material
In order to verify that no interference effects or spurious absorption features are present in the
fabricated device for visible measurements, we performed full wave Finite Difference Time
Domain (FDTD) simulations using the Lumerical software package. We verify that the
transmittance through 5 nm Ni/90 nm Al2O3/5 nm Ni/0.5 mm SrTiO3 is featureless, and therefore
we can be confident that all tunability is due to the BP. For this reason, we select Ni as the semitransparent top and bottom-contact and 45 nm thick top and bottom gate dielectrics of Al2O3.

159

Figure S1.4. Optical Response of Top Contact Material. FDTD simulation results of transmittance
through Ni/Al2O3/Ni/SrTiO3 superstrate/substrate for visible BP measurements. No features are
observed.

S1.6 High reflectance modulation of 6 nm BP flake
In order to demonstrate that thin BP films can generate technologically compelling absolute
modulation depths, we present armchair-axis FTIR reflectance data for a 6 nm flake. This data,
for which reflectance at 100 V is normalized to reflectance at zero bias, is presented in Figure S5.
The device structure consists of 6 nm BP on 285 nm SiO2 on Si, with a 10 nm Al2O3 cap. Further,
the observed modulation depth can be dramatically enhanced by integrating the BP into a resonant
optical cavity.

160

Figure S1.5. High reflectance modulation of 6 nm BP flake. Armchair-axis FTIR reflectance is
shown for 100 V bias, normalized to the zero-bias reflectance. Data taken at room temperature
under ambient conditions.

161

Chapter S2. Supplementary Material for Intraband
Excitations in Multilayer Black Phosphorus
S2.1 Unpolarized measurements
We also repeated measurements on our device without a polarizer and were able to see consistent
modulation data both in the interband and the intraband regime. From the fits to the Drude weight,
effective masses that are close to the average of the effective masses of the AC and ZZ axis were
obtained, as shown in Fig. S2.1.

Figure S2.1. Unpolarized response from BP device. (A) Unpolarized reflection modulation at the
band-edge for different electron and hole densities. (B) Same for below the band-edge region along

162
with fits (dotted black lines). (C) Assuming a parallel plate capacitor model, extracted effective
mass of free carriers (D) Fits to the experimental data (some curves at very low voltages [close to
MCP] do not fit well due to extremely low signal and have been avoided).

S2.2 Transfer matrix model
We employed a transfer matrix model to account for the multiple reflections in our device. It is
formulated as follows:
For a stack consisting of N layers, we have

( ) = JNtotal ( ) , where JNtotal = ∏ Ji
i=1

k i − ωμ0 σ
k i+1
Ji =
k i − ωμ0 σ
2 i(ki+ki+1 )zi
(1 −
i+1
ei(ki−ki+1 )zi (1 +

k i + ωμ0 σ
k i+1
k i + ωμ0 σ
−i(ki −ki+1 )zi
(1 +
k i+1
e−i(ki+ki+1 )zi (1 −

k = wavevector, z = thickness, i = layer index.
The term ωμ0 σ is invoked only at interfaces containing the 2DEG, else excluded.
Reflection and transmission are given by:
T=

ñN 2
|t| ,
̃1

R = |r|2 ,

where ñ
m = complex refractive index of layer − m.

The complex permittivity of SiO2 is adopted from Kischkat et al. shown in Fig. S2.2, and that of
Si is adopted from Salzberg et al. The permittivity of hBN is modelled using a single Lorentzian
oscillator (we only probe the in-plane phonon since the incident E-field is perpendicular to the caxis), extracted by fitting the reflection data near the hBN phonon.
ϵhBN =

ϵ∞ (1 + ωpl 2 )
ωtl 2 − ω2 − iγω

ϵhBN = 4.95, ωpl = 841.25 cm−1 , ωtl = 1368.4 cm−1 , γ = 9.42 cm−1
hBN)

(top

163
ϵhBN = 4.95, ωpl = 850.12 cm−1 , ωtl = 1364.1 cm−1 , γ = 9.31 cm−1 (bottom hBN)
The 2DEG formed simultaneously of opposite charge in the Si is modelled as : σ =

nSi e2 m−1
eff
ω+iΓ

where nSi is equal to nBP, meff = 0.26m0 (for electrons), 0.386m0 (for holes), Γ=130cm-1.

Figure S2.2. SiO2 refractive index. n,k data adopted for SiO2 (dominated by phonons).

S2.3 Modulation line shape
The observed modulation line shapes below the band edge are reminiscent of Fano-like resonances
in our measurements. A Fano resonance occurs when there is an optical coupling between a sharp
resonance like a narrow linewidth phonon and a broad continuum like the intraband Drude
absorption in BP. We neglect the effect of Si 2DEG to simplify our analysis (it behaves similar to
the BP 2DEG except the anisotropy). We examine the Fano lineshape in the presence of each
phonon system (namely, hBN and SiO2) with BP in Fig. S2.3. For hBN a sharp phonon dominates
the Fano resonance; however, for SiO2 several phonons contribute to the Fano resonance. Overall,
we see a superposition of these resonances in the experimental data since both materials are present
in our heterostructures. For clarity, the charge density in BP was assumed to be 1013/cm2 with the
polarization along the AC direction.

164

Figure S2.3. Fano response in the system. (A) Fano like response for BP/SiO2 system. (B) Same
for BP/hBN system.

S2.4 AFM data
AFM scans to measure the thickness of BP, and top and bottom hBN flakes are shown in Fig. S2.4

Figure S2.4. AFM data. (A) AFM line scan for top hBN. (B) AFM line scan for BP flake. (C) AFM
line scan for bottom hBN.

165

S2.5 Parallel plate capacitor model
The electrostatics of our device is modelled as follows–the voltage applied between BP and the Si
causes the hBN and SiO2 to act as a dielectric between two parallel plates (BP and Si). The
capacitance is calculated as
Ci =

ϵ0 ϵr,i A
, i − denotes the material (hBN and SiO2 ).

hBN and SiO2 are in series, so the capacitance adds up as Ceff =
hBN and SiO2, we obtain Ceff =

C1 C2
C1 +C2

. Assuming ϵ=3.9 for both

10.7nF
cm2

. Plugging this into q = C ∗ (V − VMCP ), we get the charge

density in the induced gas in BP (VMCP = 17V), as plotted in Fig. S2.5. At the highest voltages on
the electron side, we induce up to 4.9x1012/cm2, and on the hole side we induce up to 7.2x1012/cm2.
Even higher charge density may be had by going to higher voltages, however, to be able to reliably
repeat measurements on the same device without any breakdown, such regimes were avoided.

Figure S2.5. Capacitor model for BP. Charge density induced in BP as calculated from parallel
plate capacitor model.

S2.6 Thomas Fermi screening model
The charge induced in BP is not uniformly spread over the entire thickness, rather concentrated in
the first 2-3 layers and then decaying exponentially. A Thomas-Fermi screening model is
employed to understand the charge distribution in BP, shown in Fig. S2.6. A length scale of ~2.9

166
nm for a charge density of about 5-7x1012/cm2 is obtained for the effective thickness of the 2DEG,
which is then used to calculate the dielectric constant and the refractive index.

Figure S2.6. Band bending in BP. Thomas Fermi screening calculation in BP as the charge density
varies from 1011/cm2 to 1013/cm2. Inset–zoomed in up to 4 nm.

S2.7 Dirac-plasmonic point
We note that for the carrier densities obtained in the measurements, there exists a Dirac-plasmonic
point (DPP) where the isofrequency contour dispersion almost becomes linear. However, this
happens at frequencies slightly below the cutoff of our measurements. The DPP is quite sensitive
to the doping and also the frequency as summarized in Fig. S2.7.

167

Figure S2.7. Isofrequency contours (IFC) around Dirac-plasmonic point. IFCs are calculated for
in-plane propagating plasmon (TM) modes at two frequencies:676.7 cm-1 and 659.9 cm-1 and two
carrier densities:4.22x1012/cm2 (electron) and 6.51x1012/cm2 (hole). For higher doping densities,
the IFCs are less sensitive to small changes in the frequency, however for lower doping densities,
the IFCs are quite sensitive to small changes in the frequency and flip the sign of the hyperbolic
dispersion. They also become almost linear adopting a Dirac-like nature.

168

Chapter S3. Supplementary Information for
Atomically Thin Electro-Optic Polarization
Modulator
S3.1 Fabrication process
Mechanically exfoliated tri-layers of BP (exfoliated using Scotch-tape) were identified on PDMS
substrates with the aid of optical contrast. Mild heating of the tapes (~50-70oC) during the
exfoliation process yields large area BP thin flakes. We found ~7% contrast in the grey channel
per monolayer, meaning tri-layers showed around 21% contrast. This was also verified with optical
absorption measurements for 1-5 layers of BP, where the bandgap changes dramatically with
thickness. hBN and few-layer graphene (FLG) flakes were exfoliated (using Scotch-tape) on precleaned SiO2/Si chips (sonicated for 30 minutes in Acetone and Isopropanol (IPA), followed by
oxygen plasma : 70 W, 90 mTorr for 5 minutes). Clean flakes of desired thickness ranges were
identified with a combination of optical microscopy and atomic force measurements (AFM). A
dome-shaped (polycarbonate/polydimethylsiloxane) PC/PDMS stamp was used to pick-up the
individual layers in a top-down approach (hBN-BP-FLG-hBN), at temperatures between 70110oC. Pre-patterned electrodes with back reflectors were prepared using electron-beam
lithography (100 keV, 5nA) and electron-beam evaporation of Ti(3nm)/Au(100nm). Assembled
heterostructures were dropped on the electrodes at 200oC. Subsequently, the PC film was washed
by rinsing the sample overnight in chloroform and finally in IPA. For devices without the top
mirror, this was followed by wire-bonding to chip carriers. For passive cavity samples, PMMA of
desired thickness (adopted from calibration curve by Kayakuam) was spin-coated on the entire
device, followed by baking at 180oC for 3 minutes. This was followed by electron-beam
evaporation of the top metal (Au) of the desired thickness at 1𝐀̇/s at base pressures of ~3x10-8
Torr. For active devices, this was followed by opening windows to the electrical contacts and wirebonding to chip carriers.

169

Figure S3.1. Schematic of the fabrication process illustrating the pickup process.

S3.2 Example of a typical BP staircase flake

Figure S3.2. Optical image of a BP flake. A typical exfoliated BP staircase flake on PDMS. 1, 2
and 3 layers are marked–confirmed with optical contrast. Other thicknesses can also be seen. Scale
bar corresponds to 50 μm.

170

S3.3 Optical images of devices studied for this study

Figure S3.3. Optical images of representative devices investigated for this study. (A)–Non-cavity
device for extracting electrically tunable complex refractive index of TLBP (shown in Fig. 2). (B),
(C)–Passive cavity integrated devices. (D), (E), (F)–Active cavity integrated devices. The white
outlines denote BP, while the blue outlines denote the contacting few layers graphene flake.

S3.4 Raman spectroscopy to identify BP crystal axes
Typical Raman spectra are shown as a function of the incident polarization (linear) of the excitation
laser. The strongest response in the A2g peak is seen for the armchair (AC) orientation, whereas
for the zigzag (ZZ) orientation the same response is the weakest. This combined with linearly
polarized absorption measurements enable robust determination of the BP crystal axes. A 514 nm
laser was used for the excitation.

171

Figure S3.4. Polarized Raman spectroscopy for BP axis identification. Raman spectrum for TLBP
as a function of incident linear polarization excitation. A1g, B2g and A2g modes are seen clearly.
Strongest response from the A2g mode is seen along the armchair (AC) orientation, whereas along
the zigzag (ZZ) direction it is the weakest.

S3.5 Charge density calculator
In the devices investigated in this work, the voltage was applied between the back-electrode\backAu mirror and the TLBP (grounded). The charge accumulated in the TLBP was estimated using a
parallel-plate capacitor model. Since the thickness of the TLBP samples studied (~1.6 nm) are
below the Thomas-Fermi screening length (~3nm) for charge densities accessed in this work
(<1013/cm2), the entire TLBP can be assumed to be equipotential. The bottom hBN is the dielectric
capacitor, which enables the formation of a two-dimensional electron gas at the TLBP. Hence, the
capacitance is calculated as follows :

172

ϵ0 ϵr A

where, C = capacitance, ϵ0 = vacuum permittivity, ϵr = relative dielectric permittivity of hBN (ϵr =
3.9), A = area of the capacitor, d = thickness of the hBN flake.
The charge density is subsequently calculated as:
n=

(V − VCNP )

where, n = induced charge density (cm-2), V = applied voltage, VCNP = voltage at charge neutral
point. The CNP is estimated from reflectivity measurements, where the highest excitonic
absorption is seen. An example gate voltage to charge density (for device D1) conversion is shown.

Figure S3.5. Parallel-plate capacitor model. Estimated charge density versus applied gate voltage
for Device D1, using the parallel plate capacitor model.

S3.6 Schematic of the experimental setups used for optical
characterization
The differential reflectivity measurements were done with the aid of a chopped (mechanical
chopper ~419 Hz) supercontinuum white light source (Fianium Super-K FIU 15) and Ge
photodetector. Input light was polarized with a wire-grid linear polarizer. Lock-in amplifiers were

173
used to improve the overall signal quality, locked to the chopper frequency. Voltage was applied
using Keithley 2400. In all measurements, the BP was grounded while voltage was applied to the
back-electrode. A flat Au surface was used to normalize the reflectivity data. Newport motion
controllers (ESP 301) were used to generate spatial maps.
For the polarization conversion measurements, a tunable laser in the near-infrared (Santec TSL210 covering 1410 to 1520 nm and Newport Velocity 6400 covering 1500 to 1575 nm) was used
as the source. A polarimeter (PAX1000IR2) was used to measure the polarization state of the
reflected light. Motion controllers (MT3-Z8) were used to perform spatial mapping. Keithley 2400
was used to apply voltage. The input polarization state was controlled using a linear polarizer and
a half-wave plate. Labview and python scripts were written to automate data acquisition.

Figure S3.6. Broadband reflectivity characterization setup. Schematic of the optical setup used to
characterize the complex refractive index of TLBP as a function of doping density. LP–linear
polarizer (wire-grid), PD–photodetector (Ge), Ref. PD–Reference photodetector (Ge), BS–Beam
splitter. Blue arrows denote optics on flip mounts.

174

Figure S3.7. Polarization conversion measurement setup. Schematic of the optical setup used to
characterize the polarization conversion. LP–linear polarizer, HWP–halfwave plate, Ref. PD–
Reference photodetector (InGaAs), BS–Beam splitter.

S3.7 Phenomenological tight-binding model for TLBP bandgap
We begin our discussion of the optical properties of TLBP with a simple phenomenological tightbinding model. TLBP is a direct bandgap semiconductor with its band minima at the Γ-point. In
the low energy approximation, coupling among only nearest-neighbours needs to be considered.
For monolayer BP, the Schrödinger equation reads:
H1k ψ1k = E1k ψ1k
H1k is the Hamitonian at the Γ-point for monolayer BP. For N-layers, considering nearest layer
coupling as γk , the Hamiltonian can be constructed as:
𝐻1𝑘
𝛾𝑘
𝐻𝑁𝑘 =

𝛾𝑘
𝐻1𝑘
𝛾𝑘

𝛾𝑘
𝐻1𝑘

⋯ 0
⋯ 0
⋯ 0
⋯ 𝐻1𝑘 )

Solving the N-layer Hamiltonian produces eigenvalues of the following form:

175
nπ

ENk = E1k − 2γk cos (

), where n=1, 2, 3…N.

N+1

nπ

The optical transition energies are given by : EijN = ECB − EVB = Eg0 − 2(γCB − γVB ) cos (

N+1

where Eg0 = ECB1 − EVB1 is the bandgap of monolayer BP. For 3 − layers, using Eg0 =
1.9 eV, γCB − γVB = 0.73 eV, (known from previous studies (23,182)) we achieve E11

0.868 eV = 1429 nm, which is in close agreement with the measured optical bandgap of
1398 nm. Thus, the tight binding model works as a good approximation to estimate the lowest
energy optical transition for TLBP.

S3.8 Discussion about the excitonic framework in TLBP and doping
dependence
While the 1-D tight binding model works as a good approximation to estimate the optical bandgap
(and higher order transitions) of TLBP, it is not sufficient to capture the screening effects which
dictate the optical susceptibility of the system at finite-doping levels, since it does not capture the
electron-hole correlations. A more accurate way to model the susceptibility of TLBP is to consider
excitons in the Wannier-Mott framework, where they obey the following equation:
(−

μ∗x δx 2

μ∗y δy 2

+ Veh (r)) ψi (x, y) = Ei ψi (x, y)

−1

me x

m∗hx

where, μx = ( ∗ +

is the reduced excitonic mass of TLBP in the AC (x) and ZZ (y)

direction, and m∗e and m∗h represent the conduction and valence band effective masses. It is
noteworthy that the optical transitions in the ZZ direction remain disallowed due to symmetry
arguments. Taking into account the polarizability of the 2D-sheet and nonlocal screening from the
enviroment, the e-h interaction potential can be simplified into the Rytova-Keldysh potential as
follows:
Veh (r) = −

2πe2
[H0 ( ) − Y0 ( )]
(ϵa + ϵb )r0
r0
r0

176
where, r = √x 2 + y 2 is the e-h distance, H0 and Y0 are the Struve and Neumann functions,
respectively. ϵa and ϵb are the dielectric function of the environment (in our case, hBN, ϵhBN =
3.9) and r0 =

dϵTLBP
ϵa +ϵb

is the screening length, with d = thickness of TLBP (1.59 nm) and ϵTLBP is

the dielectric function of TLBP. The screening length (all other parameters kept constant)
primarily depends on the dielectric function (ϵTLBP ) or the polarizability of the TLBP. Under static
conditions (no doping), the aforementioned set of equations can be solved numerically to obtain
the binding energies of the entire Rydberg series of excitons along the AC direction. It is
noteworthy, that within our window of optical measurements, only the ground state of the Rydberg
series is experimentally probed. Under finite doping, the polarizability or the dielectric function of
TLBP is heavily modified since the excitonic contribution is suppressed due to a decrease of the
screening length (r0 )–realized from Thomas-Fermi screening calculations. Assuming meff =
0.16m0 , we numerically solve for the binding energy of the ground state exciton as a function of
the screening length (r0 ). The range of screening lengths is extracted from Thomas-Fermi
calculations of band-bending, which for the range of doping densities accessed in these
measurements (n =

1012

7x1012

cm2

2 to

) turns out to be approximately between 2 and 10 nm, decreasing

with increasing charge density. It can be clearly seen that the binding energy drops with the
reduction in screening length, in line with our measurements. Note that the Hamiltonian is only
solved along the AC direction.

177

Figure S3.8. Binding energy change with screening length and doping. (A) Calculated binding
energy of the ground state exciton as a function of screening length using the Rytova-Keldysh
potential. (B) Band-bending (screening profile) as a function of doping density in BP.
Furthermore, the oscillator strength of the excitonic absorption varies inversely with the Bohrradius of the exciton, |𝜙𝑒𝑥 (𝑛)|2 ∝

. In the presence of finite doping with a reduction in the

2 (𝑛)
𝑎𝐵

binding energy, the Bohr-radius increases, leading to a drop in the oscillator strength.

S3.9 Extracted exciton parameters for TLBP as a function of gate
voltage/doping density
Having discussed a theoretical framework for the modulation mechanism, we present here the
results of the different exciton parameters (modelled as a Lorentzian) as a function of doping
density, which enables us to quantify the doping dependence. The optical sheet conductivity of
TLBP is modelled as follows :
̃(ω) =

4iσ0 f0 ω
ω − ω0 +

iΓ

where, σ0 is the universal conductivity, ω0 = exciton frequency/resonance wavelength, f0 =
oscillator strength and Γ = broadening/linewidth of the resonance.
The optical conductivity can be converted to complex refractive index via the following relations:

178

ϵ̃ = ϵ∞ +

iσ
̃(ω)
dBP ϵ0 ω

ñ = n + ik = √ϵ̃
where, ϵ∞ accounts for the contribution of higher (than the exciton) energy resonances, dBP =
thickness of the BP layer and ϵ̃ and ñ are the complex dielectric function and refractive index.
By fitting the gate dependent differential reflectivity measurements using transfer matrix
calculations, we extracted the exciton parameters at each voltage. Fig. S3.9A tracks the changes
in the resonance wavelength corresponding to the excitonic transition. A strong redshift is seen for
negative gate voltages (hole doping), whereas very mild blueshift is seen for positive gate voltages
(electron doping). An inset shows the relationship between the applied gate voltage and estimated
charge density using the capacitor model. On either sides of charge neutral condition (0V), a
reduction of the oscillator strength is seen (more dramatic on the hole side) with a broadening of
the resonances, summarized in Fig. S3.9B and S3.9C, respectively. These observations are in line
with the expected electro-optic effects–the strong Coulomb screening from the excess induced
carriers reduces the binding energy of the exciton and lowers its oscillator strength. Increased
scattering of the excitons with free charges increases the effective linewidth of the transition. The
asymmetry between the electron and hole doping is likely related to the efficiency of contacts to
BP and the presence of defect states that pin the Fermi energy on the electron side, thus limiting
the modulation depth. This also explains why for hole doping a redshift is seen–due to higher
gating efficiency the optical response is dominated by trions which are at lower energies than
excitons, whereas for the electron side the contribution is comparatively less. We note that at room
temperature, optical features are quite broad in BP and since no explicit trion peak was observed,
the absorption was modelled with a single Lorentzian feature corresponding to the exciton.

179

Figure S3.9. Exciton parameter modulation with gate voltage for TLBP. Tuning of the exciton
resonance parameters as a function of applied gate voltage. (A), (B) and (C) show changes in the
resonance wavelength, oscillator strength and the linewidth of the excitonic resonance as a
function of gate voltage, respectively. An inset in (A) shows the relation between the applied gate
voltage and the estimated charge density in the BP 2DEG.

S3.10 Variation of the integrated optical conductivity (loss function)
with doping

Figure S3.10. Integrated real optical conductivity variation with doping. The real part of optical
conductivity is proportional to the loss function (∝ 𝐼𝑚(𝜖)) which dictates the overall optical
response for such thin films. As doping is increased on either side, a drop in the loss function
indicates reduced absorption due to screening of the excitons via free charges leading to a reduction

180
in binding energy and oscillator strength. The integration (over optical measurements bandwidth)
assumes a single excitonic feature and no other oscillators.

S3.11 Transfer matrix formalism for theoretical design of cavity-based
devices
We employed a transfer matrix model to account for the multiple reflections in our cavity-based
devices. It is formulated as follows:
For a stack consisting of N layers, we have

( ) = MNtotal ( ) , where MNtotal = ∏ Mi (ω)
i=1

k i − ωμ0 σ
k i+1
Mi (ω) =
k i − ωμ0 σ
2 i(ki+ki+1 )zi
(1 −
i+1
ei(ki−ki+1 )zi (1 +

k i + ωμ0 σ
k i+1
k i + ωμ0 σ
−i(ki −ki+1 )zi
(1 +
k i+1
e−i(ki+ki+1 )zi (1 −

k = wavevector, z = thickness, i = layer index.
The term ωμ0 σ, where σ is the sheet conductivity, may be invoked only at interfaces containing
2D-thin films (for example, TLBP) where optical conductivity is used, else excluded if refractive
index is used. Both approaches produced consistent results for non-cavity devices. However, for
cavity-based devices a refractive index approach was used due to its accuracy over the sheet
conductivity model due to multiple round trips of light within the cavity (leading to increased
“optical thickness” of thin 2D sheets).
Reflection and transmission are given by:
T(ω) =

ñN 2
|t| ,
̃1

R(ω) = |r|2 ,

where ñ
m = complex refractive index of layer − m.

181

S3.12 Jones matrix for TLBP birefringence and calculation of Stokes
parameters
The polarization state of the reflected light from a device with TLBP can be calculated using the
Jones vector method as follows. First, the reflected light amplitude and phase are calculated using
the transfer matrix method assuming illumination along only the AC or the ZZ direction. Then,
the cavity can be treated as a retarder plate (for both amplitude and phase) and its Jones matrix is
given as:
r eiϕAC
Jcavity = ( AC

rZZ eiϕZZ

For a given rotation (𝜃) between the input optical beam and the TLBP axis, the effective Jones
matrix is:
r eiϕAC
Jout = R(θ) ( AC
) R(−θ)
rZZ eiϕZZ
cos(θ) − sin(θ)
R(θ) = (
sin(θ) cos(θ)
Jout = (

Jout = (

cos(θ)
sin(θ)

r eiϕAC
− sin(θ)
) ∗ ( AC
cos(θ)

rAC eiϕAC cos2 (θ) + rZZ eiϕZZ sin2 (θ)
cos(θ) sin(θ) (rAC eiϕAC − rZZ eiϕZZ )

cos(θ) sin(θ)
)∗(
iϕZZ
− sin(θ) cos(θ)
rZZ e
cos(θ) sin(θ) (rAC eiϕAC − rZZ eiϕZZ )
rAC eiϕAC sin2 (θ) + rZZ eiϕZZ cos2 (θ)

Let Ex = rx eiϕx and Ey = ry eiϕy , then, the Stokes parameters are given as:

S0 = |Ex |2 + |Ey | , S1 = |Ex |2 − |Ey | , S2 = 2Re(Ex Ey∗ ), S3 = −2Im(Ex Ey∗ )
For clarity, the Stokes parameters can be normalized as:
s1 =

S1
S2
S3
, s2 = , s3 =
S0
S0
S0

The azimuthal and ellipticity can then be calculated as:
s2
s3
ψ (azi. ) = arctan ( ) , χ (ell. ) = arctan (
s1
√s12 + s22

182

S3.13 Broadband polarization conversion

Figure S3.11. Broadband polarization conversion simulations. (A) Reflectance along the AC and
ZZ direction for a cavity with parameters matching D1. (B), (C), (D) Normalized Stokes
parameters (s1, s2, s3) as a function of cavity length obtained by tuning the PMMA thickness
showing efficient broadband polarization conversion.

S3.14 Spatial variation of refractive index in a non-cavity sample in
TLBP
Results from a non-cavity device containing TLBP flake encapsulated in hBN on Au are presented
here. Spatial variation of the complex refractive index is seen across the flake. The TLBP studied
here was broader in linewidth.

183

Figure S3.12. Spatial optical inhomogeneity in TLBP samples. Spatial variation of real (A) and
imaginary (B) part of complex refractive index in a TLBP flake.

S3.15 Effect of different thickness on the cavity resonance and
polarization conversion
A staircase sample was fabricated to study the effect of different thicknesses of BP on the cavity
resonance. A systematic red shift is seen as a function of BP thickness, as expected. Adding more
layers increases the effective optical path length inside the cavity which explains the redshift of
resonance. However, very weak polarization conversion was seen from thicknesses other than 3layers. This reiterates the importance of choosing tri-layer flakes for working in the
telecommunications band, since the anisotropy is very high near the excitonic resonance. In Figure
S3.13A, B, C reflection spatial colormaps can be seen at three different wavelengths (1460 nm,
1510 nm, and 1560 nm) showing changes in contrast for different thicknesses. Figure S3.13D
illustrates the systematic redshift of the cavity resonance as a function of thickness. Figure S3.13E,
F illustrates the strongest polarization conversion in the 3L region as quantified by the ellipticity
and azimuthal angle.

184

Figure S3.13. Effect of thickness on the cavity resonance and polarization conversion. (A), (B),
(C) Reflection spatial maps at 3 different wavelengths (1460 nm, 1510 nm, and 1560 nm) for
device D4 showing difference in contrast for different thicknesses of BP. (D) Reflection amplitude
spectrum for different thicknesses of BP (2,3 and 6 layers) and bare cavity–illustrating redshift of
cavity resonance with increasing thickness of BP. (E) Ellipticity and (F) Azimuthal angle spectrum
for 3 different thicknesses of BP, showing highest polarization conversion in TLBP (3-layers).

185

S3.16 Effect of the incident polarization state on the polarization
conversion
We investigate here, numerically, the effect of incident polarization on the spectral trajectory
traversed on the normalized Poincaré sphere. To illustrate conditions close to the experimental
measurements, two sets of simulations are presented–the first one with the cavity resonance along
the AC direction at 1440 nm and the second one at 1495 nm, corresponding to conditions in Device
D1 and D4, respectively. The difference in spectral trajectories arise from the competition between
the excitonic absorption along the AC direction and the cavity resonance and represents a tuning
knob to access different polarization states.

186

Figure S3.14. Polarization conversion dependence on incident polarization. Effect of the input
polarization condition on the evolution of spectral trajectories on the normalized Poincaré sphere
is shown. As the phase delay and the relative amplitudes are tuned between the AC and ZZ
component of the incident light, different trajectories are undertaken. The cavity resonance along
the AC direction is at 1440 nm. The blue (red) arrow denotes the polarization state at 1410 (1520)
nm.

187

Figure S3.15. Polarization conversion dependence on incident polarization. Effect of the input
polarization condition on the evolution of spectral trajectories on the normalized Poincaré sphere
is shown. As the phase delay and the relative amplitudes are tuned between the AC and ZZ
component of the incident light, different trajectories are undertaken. The cavity resonance along
the AC direction is at 1495 nm. The blue (red) arrow denotes the polarization state at 1410 (1520)
nm, overlapping.

188

S3.17 Numerical modelling of cavity-enabled polarization conversion
We discuss here, through numerical modelling, the effect of the different exciton parameters on
the polarization conversion performance of a typical cavity. Fig S3.16 summarizes our findings by
showing the azimuthal and ellipticity dependence on the exciton broadening, oscillator strength
and the exciton resonance wavelength. The most striking impact on the polarization conversion is
seen from the oscillator strength, which dictates how strong the exciton, hence the anisotropy, is.
Impact of the broadening or the resonance frequency is relatively weak on the azimuthal angle.
Similarly, for ellipticity also, a strong dependence is seen on the oscillator strength. Quite
interestingly, while the overall magnitude of the ellipticity resonance is reduced with decreased
oscillator strength, for wavelengths slightly below 1440 nm, a non-monotonic dependence is seen.
This wavelength range can be used to tune ellipticity from a low value to a high vale and back to
low again, making it attractive for tunable quarter-wave plate like operation. This nonmonotonicity stems from the co-variation of amplitude and phase as a function of exciton
parameters, which are both captured in ellipticity. With increasing doping, while the refractive
anisotropy is reduced which causes the overall ellipticity change to decrease, the losses are
quenched too, which increases the Q-factor of the cavity resulting in the lineshape modification
and higher ellipticity for certain wavelength ranges. A small impact of broadening is seen on
ellipticity, whereas a monotonic but sizeable effect is observed with resonance frequency of the
exciton on the same. Cavity parameters used were: top Au = 10.5 nm, PMMA = 140 nm, top hBN
= 53 nm, BP = 1.59 nm, bottom hBN = 119 nm.

189

Figure S3.16. Polarization conversion dependence on exciton parameters in TLBP. Effect of
different exciton parameters on the azimuthal and ellipticity of a typical cavity-based device. (A),
(B), (C) Effect on azimuthal angle for different exciton broadening, oscillator strength and
resonance wavelength, respectively. (D), (E), (F) Same as (A)-(C), but for ellipticity.
Since the polarization dynamics is most strongly tuned with the strength of the oscillator, we can
numerically estimate the cavity spectral variation with doping. We plot the Stokes parameters (S0,
s1, s2, s3) as a function of the oscillator strength. An excellent agreement with experimental
measurements is seen, confirming that screening of the exciton due to free carriers is the major
driving mechanism for polarization conversion.

190

Figure S3.17. Numerical modelling of cavity enabled polarization conversion. (A) Intensity (S0)
variation with oscillator strength of the exciton in TLBP. The cavity parameters correspond to
device D1. (B), (C), (D) s1, s2, s3 showing the same. This agrees with our experimental
observation of the electrically tunable polarization conversion results.

191

S3.18 Additional gating results from other spots on D1
Here, we discuss the gating results from other spatial positions on device D1. The general trend of
the spectral trajectory collapsing in arc length is observed for all points. However, the exact
trajectories traced out on the normalized Poincaré sphere are determined by the local complex
refractive index of the TLBP and its gate tunability.

Figure S3.18. Normalized Poincaré sphere dynamics from additional spatial points in device D1.
(A)-(I) Normalized Poincaré sphere trajectories for different wavelengths for different voltages for
different spots in device D1. Each color corresponds to a different wavelength. For each color, the
voltage trajectory direction is marked. Beginning and end voltage values are marked for each

192
measurement. The dark arrows point the 0V condition for each wavelength and the light (color
coded for each wavelength) arrows denote the highest voltage point.

S3.19 Full spectral dynamics on the normalized Poincaré sphere

Figure S3.19. Spectral and voltage tuning of normalized Poincaré sphere trajectories for Device
D1. (A) Evolution of polarization conversion for positive voltages (electron doping) and (B)
negative voltages (hole doping). The same color represents a spectral scan (from 1410 nm to 1520

193
nm in steps of 0.5nm), while a color variation shows changes of the spectral trajectory with voltage
from 0 to 30V for (A) and 0 to -30V for (B), in steps of 0.5V. Blue arrows represent polarization
state at 1410 nm for 0V while red arrows represent the same at 1520 nm.

S3.20 Normalized Stokes parameter tuning on electron and hole
doping
A competition between the excitonic absorption and the cavity resonance governs the overall
optical response of the system, due to the close proximity of the two features. Hence, maximising
the polarization conversion requires carefully adjusting the incident polarization to balance the
losses along the two principle axes of TLBP for a fixed orientation of the device. To quantify the
degree of polarization conversion, traces on the normalized Poincaré sphere were measured for
different input ellipticity and azimuth. The longest arc was found for nearly linearly polarized input
at an angle of ~27o with the AC axes of TLBP, corresponding to a vertical polarization in the lab
frame.

194

Figure S3.20. Normalized Stokes parameters with electron and hole doping. (A), (B), (C) False
colormaps of the evolution of the three normalized Stokes parameters (s1,s2,s3), determining the
polarization state of the reflected light, as a function of wavelength and positive voltages (for
electron doping). The results are from device D1. Continuous tuning of all the 3 parameters can
be seen around the cavity resonance (~1440nm) for the entire range of doping, illustrating efficient
tuning of the polarization state with voltage. (D), (E), (F) Same as (A), (B), (C) but for negative
voltages (for hole doping), showing similar changes as the electron doped side. The nearly
symmetric nature of the doping dependence shows that at 0V, the device is at charge neutral

195
conditions. (G), (H), (I) Line cuts taken from the false colormaps for the three normalized Stokes
parameters (s1, s2, s3 respectively) for 5 different voltages (0V, 20V, 40V, -20V, -40V) to visualize
the changes with higher clarity.

S3.21 Reflectance changes on electron and hole doping

Figure S3.21. Reflectance change (S0) of the cavity upon electron and hole doping. (A), (B) show
the change in the reflectance (Stokes intensity S0) of the device D1 as a function of wavelength for
different applied voltages on the hole doping and electron doping side, respectively. The colormaps
are plotted in dB for better clarity.

196

S3.22

Azimuthal and ellipticity changes on electron and hole doping

Figure S3.22. Azimuthal and ellipticity angle change upon electron and hole doping. (A), (B) show
the changes in the ellipticity angle (χ), in degrees, of the device D1 as a function of wavelength for
different applied voltages on the electron doping and hole doping side, respectively. (C), (D) show
the same as (A) and (B), but for the azimuthal angle (ψ), in degrees. The abrupt jump in the
azimuthal angle is a numerical artifact arising from the indistinguishability between +90o and -90o.

197

S3.23

Additional gating results from a 5-layer BP device

We present here electrically tunable reflection contrast measurements performed on a 5-layer BP
sample. The device geometry is the same as the one adopted for 3-layer devices (hBN-BP/FLGhBN/Au). Fig. S3.23A shows the reflection contrast along armchair (AC) and zigzag (ZZ)
directions. A strong excitonic feature at 2 μm is seen along the AC direction, while a featureless
spectrum is seen along the ZZ direction. Upon application of gate voltage, strong modulation of
the reflectivity is seen. Fig. S3.23B summarizes the results demonstrating the same. The changes
are visualized better when normalized to the reflectivity at 0 V. The strongest modulation is seen
at 2 μm, which confirms that the origin of this electro-optic response originates from the exciton
dynamics. These measurements were performed using a Fourier transform infrared (FTIR) white
light source coupled with an infrared microscope in ambient conditions. These measurements
demonstrate that efficient polarization conversion can also be achieved at much longer
wavelengths (~2 μm) due to the large excitonic tunability. In fact, the thickness of BP can be tuned,
and the operation wavelength can span the entire visible (750 nm) to mid and far-IR range (~3-20
μm).

198

Figure S3.23. Gate-dependent reflectivity modulation in 5-layer BP device. (A) Reflection contrast
showing excitonic feature along the AC direction and a featureless spectrum along the ZZ
direction. (B) Relative reflection (w.r.t. 0V) shows strong modulation under different applied
biases, with the strongest tuning near the excitonic resonance.

S3.24

Choice of three-layer BP (TLBP)

We motivate here why TLBP was used in the study over other thicknesses of BP. The choice of
three-layers is primarily driven by the compelling technological advantages of constructing
electro-optic modulators in the telecom band such as for optical fiber communication (typical
losses are ~0.2 dB/km at 1.5 μm as compared to 8 dB/km at 640 nm), quantum networks and
photonic integrated circuits (based on Silicon). Previous reports on layer-dependent bandgap in
BP support the fact that BP is well suited for operation from visible to mid-IR. Furthermore, these

199
studies clearly show the evolution of the optical bandgap (excitonic resonance) in BP as a function
of layer thickness. While the optical bandgap is in the visible (~750 nm) for monolayer BP, it
saturates at ~4 μm for >20nm thin films. Three layers hits the sweet spot with an excitonic
resonance ~1400 nm, particularly suited for the window of telecom operation.
Furthermore, the electrical tunability of such optical transitions is expected from similar studies in
monolayer TMDCs and few-layer BP. The key physics driving the polarization conversion
tunability (or in general terms, tuning the exciton polarizability) is the suppression of excitons due
to screening by induced free charges. These effects are most prominent for monolayer TMDCs
owing to their direct bandgap and strong excitons, while for bilayer or multilayer TMDCs, the
bandgap becomes indirect, and the magnitude of electrically driven changes reduce. However, for
BP, because of the band minima being at the Γ-point (arising out of D18
2h point group crystal
symmetry), the direct nature of the bandgap is always maintained and thus the excitonic effects
and its tunability remain quite significant at the few-layer limit, as shown in our work, for both
TLBP and 5-layers BP. The modulation fraction decreases as the thickness of BP is increased,
which stems from the finite Thomas-Fermi screening length that governs the amount of effective
modulation depth in the vertical direction in a semiconductor channel. Such length scales are
typically of order ~2-3 nm for the charge densities accessed here for BP; anything thicker is
effectively not modulated beyond the Thomas-Fermi screening length. The Thomas-Fermi
screening length can be visualized in the following plot:

200

Figure S3.24. Thomas-Fermi screening effect in BP. Band-bending in multilayer BP as a function
of thickness for a charge density of 5x1012/cm2. A Thomas-Fermi screening length (𝝀𝑻𝑭 ) of 2.9
nm is obtained.
To summarize, both the bandgap being conveniently in the telecom band along with the high
electrical tunability of the exciton drives our choice of three-layers BP (TLBP). The same effect
can however be shown in monolayer BP in the visible/NIR (600-900 nm), bilayer BP in the NIR
(900-1200 nm), and so on and so forth. Finally, the degree of polarization conversion will
ultimately depend on the coupling between the cavity and free space and hence the cavity would
need to be designed for the right working wavelength.

S3.25 Comparison of polarization conversion mechanism with liquid
crystals
The distinction between bulk materials and excitonic 2D semiconductors, particularly TLBP in
this case, is quite important. While the electro-optic operation hinges on the physical re-orientation
of the dipoles for bulky materials like liquid crystals, such is not the case for 2D excitons. The
physical mechanism of the operation of our device is the screening effect of the excitons by the
excess induced charges in TLBP which eventually effect the dipole oscillator strength. The vertical
electric field that is generated in the capacitor (hBN in our devices) alters the charge density in the
TLBP–which acts as the other plate of the parallel plate capacitor (the back Au is the first plate).
This causes the quasi-1-D dipoles which are in-plane to be affected by the out of plane electric
field from the capacitor. Thus, when a normally incident light interacts with the heterostructure,

201
the in-plane dipoles influence the in-plane electromagnetic fields. Previous measurements of
tuning the dipole strength in TMDCs have also leveraged this strong vertical-field induced doping
to tune the interaction with an in-plane optical field. However, due to lack of crystal symmetry
breaking in the in-plane direction, such efforts have failed to control polarization.
It is also important to note, that thus far, no evidence points to actual reorientation of the dipoles
(initially along x-direction) with gating, but rather a reduction in the oscillator strength and
enhanced scattering.

Figure S3.25. Dipole interaction with optical field in TLBP. Side view of capacitor geometry of
working BP device. The two plates of the parallel-plate capacitor are the BP and the back Au
electrode/reflector. Free carriers are induced in the BP with applied voltage. Incident optical field
(polarized in the in-plane direction, perpendicular to the vertical capacitor field) is shown, along
with the Poynting vector. A top view of the BP flake along with the dipole orientation is shown,
with the Armchair (AC) and Zigzag (ZZ) axes marked. Incident field (in-plane) can be
decomposed along the AC and ZZ direction, marked as x and y, respectively. The x-component is
strongly influenced by the exciton-enhanced cavity interaction (which is also electrically tuned),
whereas the y-component is only influenced by the cavity and not the exciton and thus is not tuned.

S3.26 Cyclic measurements for electrically tunable devices
Multiple cyclic measurements were done on all gate tunable devices, and the results are highly
consistent. No noticeable hysteresis was observed when the voltages were swept in a cyclic
fashion. The cyclic scheme adopted in this work is as follows: 0V to +20V, back to 0V, to -20V

202
and then back to 0V. The normalized Stokes parameters (s1, s2, s3) and the voltage sequence for
one such device (D1) is shown in Fig. S3.26.

Figure S3.26. Cyclic Stokes measurements. (A), (B), (C) Normalized Stokes parameters (s1, s2, s3)
spectra measured as a function of voltage in a cyclic fashion. (D) The applied voltage as a function
of the sequence number.

S3.27 Discussion about edge effects in spatial mapping of polarization
conversion
“Edge effects” may result from the convolution of an atomically sharp edge encountered at the end
of a BP flake and a gaussian beam, in the spatial maps presented. In the current measurement
configuration, we are limited by the optical diffraction limit, and it is not possible to resolve an

203
atomically sharp edge. Thus, a true edge is not measured, rather it is convolved with the nearest
pixel. So, near the edges, the ellipticity appears reduced because the finite size (~1.5-2 μm)
gaussian beam samples both the BP flake + cavity and the no-BP cavity only region. Furthermore,
the blueshift in the resonance compared to the interior of the BP flake can be understood from the
fact that the beam is sampling less material or reduced cavity length (as it scans half a TLBP and
half a missing BP region, for example) and thus the cavity resonance blueshifts. To prevent a
convolution of the edge effect and true polarization dynamics, all the gate tunable and passive
device measurements shown in the paper have been done in the interior of the samples and not at
the edges. Further higher resolution measurements, such as tip-based near field microscopy, would
be required to address if there are any true edge effects, such as edge lattice reconstruction and
how that influences the polarization conversion. An illustration of the finite size of the beam and
the sampling effect is shown for further clarity in Fig. S3.27.

Figure S3.27. Edge effects in spatial mapping. Top view and side view illustration of cavity and
BP, along with the gaussian (diffraction limited) beam at the edge showing sampling from both
the regions leading to different polarization conversion at the edge compared to the interior of the
sample (spot size is exaggerated for clarity).

S3.28 Outlook towards high efficiency polarization modulators based
on BP
Our current work illustrates the versatility of using BP, especially TLBP (for telecom band), as an
active material for polarization conversion. In the current scheme of operation in a modest quality
factor cavity and sufficiently close to the excitonic feature, the overall reflectivity of the devices
is low. Several ways may be adopted to boost this efficiency to a higher value. Firstly, to increase

204
the overall reflectivity, high-Q cavities (such as photonic crystal cavities, DBR-based Fabry-Pérot
cavities, etc.) can be combined with BP at sufficiently detuned wavelengths (detuned from
excitonic resonance) to reduce the losses in the system. This obviously requires a judicious choice
of thickness for BP depending upon the working wavelength. For example, at 1550 nm, we expect
bilayer (or even monolayer) BP to be a good candidate. Furthermore, to increase the doping
concentration and thus the modulation depth, higher dielectric permittivity materials such as HfO2
can be used. This has been shown to be an effective strategy for low-loss electro-optic modulation
recently for TMDCs. Better growth of higher quality BP crystals showing prominent excitonic
behavior on different substrates is needed however to make this a viable strategy. Here, we show
numerically how such a design would work using experimentally measured complex refractive
indices for TLBP and integrated with an ultra-low loss dielectric mirror based Fabry-Pérot cavity,
operating at a sufficiently detuned wavelength. The structure considered here consists of a top
DBR of 20 pairs of SiO2/SiNx (n=1.45/1.87), TLBP, 10-nm gate dielectric Al2O3 (n=1.73), a buffer
layer of SiNx (n=1.87) to adjust the cavity resonance, a bottom DBR of 40 pairs of
AlAs/Al0.1Ga0.9As (n=3.5/3.05) and a GaAs substrate (n=3.55). The refractive indices for AC BP
and AC-doped BP were taken from measurements done at 0V and 50V (hole doping). As can be
seen for a working wavelength of 2986.6 nm a phase shift of ~2π can be obtained between undoped
and doped conditions, whereas the reflectance stays extremely high (>95%) for the same
wavelength. Similarly, the top and bottom pairs can be appropriately designed to maintain high
reflection all throughout the spectral window, with a slight compromise in the achieved phase shift.
Here, we show the performance of the same design with 17 top pairs (instead of 20), which gives
us a maximal phase shift of ~1.55π with over 90% reflection at the working wavelength, but also
maintaining >65% for the entire spectral range which is already an extremely high reflection
efficiency.

205

Figure S3.28. High efficiency numerical design for polarization conversion. (A) Double DBR
based Fabry-Perot cavity design incorporating BP. (B) Reflection and (C) phase from the
corresponding cavity structure for pristine and doped armchair (AC) direction.

Figure S3.29. High efficiency numerical design for polarization conversion. (A) Reflection and
(B) phase from the proposed cavity structure in Fig. S3.28 for pristine and doped armchair (AC)
direction for 17-top pairs instead of 20.

206
Furthermore, for two-state polarization switching devices (applications requiring switching
between two-states of polarization such as linear to circular polarization), the design can be tuned
to have no amplitude modulation and only polarization modulation. An illustration of this can be
already seen in our current working devices. For example, if we analyze the intensity, ellipticity
and azimuthal for hole doping in Fig. S36, around 1450 nm, minimal change in intensity (<0.3 dB)
and azimuthal angle is seen with significantly large change in ellipticity (~210) between 0V, -20V
and -40V. Such designs may be further optimized by tuning the cavity resonance for wavelength
of interest.

Figure S3.30. Amplitude, azimuthal and ellipticity line cuts for hole doping. (A) Amplitude
spectrum for three distinct voltages (0V, -20V, -40V) corresponding to hole doping, showing
variation in the intensity at the resonance. (B), (C) Same as (A) but for azimuthal and ellipticity
angle. A region around 1450 nm can be identified to have nearly constant amplitude modulation,
minimal azimuthal change but large ellipticity modulation.
Finally, we comment on the possibility of complete independent control of both amplitude and
phase which requires multiple degrees of freedom (at least 2). We believe judicious dual gate
designs, multiple cascaded BP unit-cells and hetero structuring with a broadband lossy material,
like graphene, to balance the losses in the two system to keep the amplitude constant, might pave
the way towards such control.

207

Chapter S4. Supplementary Information for Rydberg
Excitons and Trions in Monolayer Mote2

S4.1 Crystal growth, device fabrication, and experimental methods.
SiO2 (285 nm)/Si chips were cleaned with ultrasonication in acetone and isopropanol for 30 min.
each, followed by oxygen plasma treatment at 70 W, 300 mTorr for 5 min. Monolayer MoTe2, few
layer hBN and graphene flakes were directly exfoliated using Scotch-tape at 100 oC to increase the
yield of monolayer flakes. Monolayer thickness was initially identified using optical contrast (~7
% contrast per layer) and later verified with atomic force microscopy. hBN thickness was
confirmed

with

atomic

force

microscopy.

Flakes

were

assembled

with

polycarbonate/polydimethylsiloxane (PC/PDMS) stamp with pick-up at temperatures between 80

C -110 oC. The entire heterostructure stack was dropped on prefabricated gates at 180 oC. The

polymer was washed off in chloroform overnight, followed by isopropanol for 10 min. Given the
air-sensitive nature of MoTe2 monolayers, exfoliation, identification of MoTe2 flakes and stacking
of the entire heterostructures were done in a nitrogen purged glovebox with oxygen and moisture
levels below 0.5 ppm. Gates were fabricated on SiO2 (285 nm)/Si chips with electron beam
lithography (PMMA 950 A4 spun at 3500 rpm for 1 min. and baked at 180 oC, 10 nA beam current
and dosage of 1350 𝜇C/cm2 at 100 kV), developing in methyl isobutyl ketone: isopropanol (1:3)
for 1 min., followed by isopropanol for 30 s and electron beam evaporating 5 nm Ti/95 nm Au at
0.5

deposition rate. Liftoff was done in warm acetone (60o C) for 10 min., followed by rinse in

isopropanol for 5 min. The gates were precleaned before drop-down of heterostructure by
annealing in high vacuum (2x10-7 Torr) at 300 oC for 6 h. The chip was then wire-bonded with
Aluminum wires on to a custom home-made printed circuit board.

208

Figure S4.1. Schematic of the low-temperature confocal optical PL setup used. BS–Beam Splitter,
QWP/HWP–Quarter/Half Wave Plate.

Low temperature confocal photoluminescence measurements were performed in an attoDRY800
closed-cycle cryostat at base pressures of <2x10-5 mbar. Sample was mounted on a thermally
conducting stage with Apiezon glue and stage was cooled using a closed-cycle circulating liquid
helium loop. Temperature was varied between ~4K and 300K. A 532 nm (Cobolt) continuous
wave laser was used as the excitation source with power ranging between 10 nW and 1 mW,
focused to a diffraction limited spot. Emission was collected in a confocal fashion with a cryogenic
compatible apochromatic objective with an NA of 0.82 (for the visible and NIR range, LT APO
VISIR) and dispersed onto a grating-based spectrometer (with 150 grooves per mm with a Silicon
CCD)–Princeton Instruments HRS 300. Voltage was applied using a Keithley 2400. Data was
acquired with home-written MATLAB codes.
MoTe2 single-crystals were grown by the chlorine-assisted chemical vapor transport (CVT)
method. A vacuum-sealed quartz ampoule with polycrystalline MoTe2 powder and a small amount

209
of TeCl4 transport agent (4 mg per cm3 of ampoules’ volume) was placed in a furnace containing
a temperature gradient so that the MoTe2 charge was kept at 825 °C, and the temperature at the
opposite end of the ampoule was about 710 °C. The ampoule was slowly cooled after 6 days of
growth. The 2H phase of the obtained MoTe2 flakes was confirmed by powder X-ray diffraction
and transmission electron microscopy studies. Flakes obtained from the aforementioned source as
well as commercially available MoTe2 (2D Semiconductors) were investigated with similar
results.
Atomic Force Microscopy was performed using Bruker Dimension Icon in tapping mode. Data
analysis was performed in MATLAB.

S4.2 Choice of optical geometry
The optical geometry adopted here is a Salisbury screen resonant near the A1s exciton transition.
Such an optical cavity is illustrated in Figure S4.2. An optical spacer (dielectric) is placed atop a
back reflector (metal or dielectric mirror) and an absorbing layer is placed on the spacer. The

thickness of the spacer is chosen to be quarter wavelength ( ) such that after one round trip in
4𝑛

the cavity the light has travelled a total distance of

2𝑛

which introduces a 𝜋 phase shift between

the incoming and outgoing electromagnetic wave. This enables near perfect light absorption at the
resonance wavelength enabling strong light-matter interaction. In our structures, since we have a
top hBN the Salisbury screen condition was estimated from a transfer matrix approach to find the
right hBN thicknesses which was used as a guide to select suitable hBN top and bottom flakes.
The modelling assumed the following parameters:
1. hBN dielectric/optical spacer (𝑛 = 2.2, 𝑘 = 0)
2. MoTe2 monolayer absorbing sheet (optical conductivity model) 𝜎(𝜔) =
𝜎0 =

𝑒2

, 𝑝0 =

4ℏ

𝛾𝑟
4𝜋𝜔0 𝛼

4𝑖𝜎0 𝑝0 𝜔

, where

𝑖Γ

(𝜔−𝜔0 )+

𝛾𝑟 (2 𝑚𝑒𝑉) is the oscillator strength, 𝜔0 (1.17 𝑒𝑉)is the resonance

frequency and Γ (7 𝑚𝑒𝑉) is the broadening associated with the optical transition.
3. Refractive index of Au (back reflector) was adopted from Johnson and Christy .
4. A top layer of hBN (5 nm) was added.

210
Calculations were done using a standard 1D transfer matrix model (with stackrt function in
Lumerical as well as home-written MATLAB code). Furthermore, our choice of a back-reflector
of gold suppresses any photoluminescence coming from the silicon substrate, which would be
emitted in a similar wavelength/energy range 𝜆𝑆𝑖 ~1150𝑛𝑚, 𝐸𝑆𝑖 ~1.07 𝑒𝑉 at low temperatures.
In a false color map of absorption spectrum variation with bottom hBN thickness shows high
absorption at the excitonic resonance for thicknesses around 100 nm. Motivated by these
calculations, the bottom hBN for the device was chosen to be 97.3 nm (confirmed by AFM
measurements)–limited by the occurrence of naturally exfoliated thicknesses.

Figure S4.2. Schematic and calculation of optical geometry. (a) Schematic of a quarter wavelength
Salisbury screen geometry. (b) Absorption spectrum as a function of bottom hBN thickness
showing the cavity-enhancement.

S4.3 Estimation of Purcell enhancement
Simulation set-up details:
Initially, the transfer matrix (1D) model was used to optimize the structure for a resonant Salisbury
screen design. The obtained parameters were then fed into Lumerical FDTD (schematic in Figure
S4.3a) as follows:

hBN dielectric/optical spacer (𝑛 = 2.2, 𝑘 = 0), total thickness of 102.3 nm.

211

Refractive index of Au (back reflector) was adopted from Johnson and Christy, assumed
to be optically thick (~150 nm).

A uniform mesh of 2nm was used in all directions around the dipole and the total simulation
span was taken to be 5𝜇𝑚, to eliminate any artifacts.

An x-polarized dipole (in-plane of the simulation) was used to mimic the emission from MoTe2.
Since the physical thickness of a monolayer (𝑡~0.7 𝑛𝑚) is much smaller than the wavelengths of
interest (𝜆~1𝜇𝑚) it is justified to assume the dipole is completely in plane with no out-of-plane
component. Perfectly matched layers (PML) boundary conditions were used on all sides and the
dipole wavelength was chosen to match the A1s exciton emission (𝜆 = 1.172 𝑒𝑉 or 1058 𝑛𝑚).
Field profile (power) monitors were placed around the dipole in XZ and YZ configuration and the
total field (real) intensity recorded as shown in Figure S4.3b and c, respectively. From the dipole
analysis (in built) in Lumerical FDTD, we estimated a Purcell factor–𝐹𝑝 = 2.012.

Figure S4.3. Details of lumerical simulation. (a) Simulation setup XZ view in Lumerical FDTD.
(b), (c) XZ, YZ monitor for log(𝑅𝑒(𝐸)) profile, respectively.

212

S4.4 Comparison of monolayer and bilayer PL spectra

Figure S4.4. Comparison of monolayer and bilayer MoTe2 emission. (a) PL spectrum from
monolayer and bilayer regions of the same device. (b) Same as (a), but normalized.
The trion peak dominates for both the monolayer (1.157 eV) and bilayer (1.149 eV) spectrum
under no applied bias. The exciton peak for the monolayer occurs at ~1.181 eV. The sample is
lightly n-doped as confirmed by gate dependent measurements. The trion peak for the bilayer is 8
meV lower in energy than the monolayer and about 3.5 times less bright. Since the doping density
is the same for the monolayer and the bilayer as they are part of the same flake, the intensities are
linked to the quantum yield differences.

S4.5 Optical image of MoTe2 monolayers
The following two monolayer (ML) region containing flakes were used in the device fabrication
(both show ~7% contrast). A bilayer (BL) region is also seen in the first flake. Scale bar–2.5𝜇𝑚.

213

Figure S4.5. Optical microscope image of flakes used for the device fabrication.

S4.6 Charge density calculations
We assume a parallel-plate capacitor model where the two plates of the capacitor are the back
reflector/electrode (gold) and the monolayer MoTe2. This assumption is justified in the dc limit
since the MoTe2 is semiconducting. This yields the following relation:

𝜖 𝜖

= 0 𝑟,ℎ𝐵𝑁 , where 𝜖𝑟,ℎ𝐵𝑁 = 3.9 and 𝑑ℎ𝐵𝑁 = 97.3 𝑛𝑚.
𝑑ℎ𝐵𝑁

𝜖 𝜖

𝑛 = 𝐶(𝑉 − 𝑉𝐶𝑁𝑃 ) = 0 𝑟,ℎ𝐵𝑁

(𝑉−𝑉𝐶𝑁𝑃 )

𝑑ℎ𝐵𝑁

where

𝑉𝐶𝑁𝑃 = −0.65𝑉 (as

estimated

from

measurements).
The relation between applied gate voltage and sheet charge density in MoTe2 is depicted in Figure
S4.6.

214

Figure S4.6. Applied gate voltage to sheet charge density conversion assuming parallel plate
capacitor model in the dc limit.

S4.7 Power dependent emission spectrum

Figure S4.7. Power dependent photoluminescence spectrum over 3 decades of pump intensity. (a)
is in linear and (b) is in log scale. The resonances are labelled in (b).

215

S4.8 Gate dependent PL fits
Fits to the PL spectrum data are shown in Figure S4.8 and S4.9 showing good match between the
two.

Figure S4.8. Fit to gate dependent experimental data around 1s region. (a) Experimental PL data
around the A1s resonance region showing the neutral exciton and the charged trion resonances.
(b) Multi-Lorentzian fit to the PL data shown in (a).

Figure S4.9. Fit to gate dependent experimental data around 2s region. (a) Experimental PL data
around the A2s/3s resonance region showing the neutral exciton and the charged trion resonances.
(b) Multi-Lorentzian fit to the PL data shown in (a).

216

S4.9 Gate dependence of additional spot
We have measured across 30 spots in the two devices presented and cycled the same spot through
more than 10 sequences of gating between -17V and 17V and also repeated the measurement on a
third device and all of them have yielded similar results. The only observable that varied across
the spots and samples is the photoluminescence quantum yield which is well-known in the van der
Waals community to be inhomogeneous across samples due to imperfections in fabrication
techniques resulting in strain and bubble formation in-between layers in heterostructures. Results
from another spot are summarized in Figure S4.10.

Figure S4.10. Gate-dependent data from additional spot. (a) Experimental PL data around the A1s
resonance region showing the neutral exciton and the charged trion resonances. (b) Same as (a)
but for 2s/3s resonances.

217

S4.10 Absolute intensity of exciton and trion emission modulation

Figure S4.11. Gate dependent emission fit parameters (intensity). (a) Absolute photoluminescence
intensity of Rydberg excitons as a function of charge density in linear scale. (b) Same as (a) in
semi-log scale. (c) Absolute photoluminescence intensity of trions associated with Rydberg
excitons as a function of charge density in linear scale. (d) Same as (c) in semi-log scale.

218

S4.11 Energy shift of exciton and trion emission

Figure S4.12. Gate dependent emission fit parameters (energy). (a) Energy shifts of the A1s
exciton and trion as a function of doping density. (b) Energy shifts of the A2s exciton and trion as
a function of doping density.

219

S4.12 Comparison of MoTe2 with MoS2, MoSe2, WS2, and WSe2
Table S4.T1. Binding energy (meV) of Rydberg excitons of all TMDCs.

Material

A1s

A1s+

440(±80)
289

MoS2

, 26118,

221290,

A1s-

A2s

2718,

18(±1.5)

9118,
48291

A2s+

A2s-

A3s+

A3s-

2)293,

13.1(±0.

13(±0.

16.4(±0.

3)293

5)293

A3s

5318,
20291

291

222

24.3(±0.
MoSe2

231290,
208292

1)293,
23.7(±0.
1)293,
292

22.7(±0.

26.1(±0.
1)293,
23.1(±0.

4)293,
56292

22.6(±1.
0)293,

2)293

292

24.6(±0.

7)293

320(±50)
WS2

289

20.5(±0.

29(±0.

1)293,

17.5(0.

27.1(±0.

293

180290
172294,
167290,
WSe2

170242,
169284,
37013

18.4(±0.
41294,

3)293,

17.8(±0.

20294,

39242,

9.2(±0.

4)293,

13242,

40284

7)293,

18.6295

17284

14.1295

177290,

MoTe2

156296,

24298,

580(±80)

24297,

27297,

288(±18)

13.67(±0

18.14(±0

297

22.14(±0

21.94(±0

, 270

.12)

.09)

.2)

.1)

404(±19)

150

, 490

Not cited numbers are from this work combing experiments and theory, theory only (MoTe2).

220

Table S4.T2. Gate dependence of energy shift of charged Rydberg excitons of all TMDCs.

Material A1s+

A1s-

A2s+

A2s-

A3s+

A3s-

MoS2

−0.75 ± 0.05293, 1.10 ± 0.07293, −7.9 ± 0.6293, 11.1 ± 0.6293,
MoSe2
WS2
WSe2293
MoTe2

−1.91 ± 0.06293, 2.9 ± 0.1293,

−2.9 ± 1.6293, 7.0 ± 1.0293,

0.9292, 0.4299

1.2299

3.9292, 1.8299

4299

−0.18 ± 0.02,

1.2 ± 0.1,

−5.5 ± 0.4,

−1.74 ± 0.03

2.3 ± 0.1

−1.8 ± 0.1

4.7 ± 0.3

-0.40±0.10

0.28±0.05

-8.59±0.54

4.23±0.51

Noted quantity is

𝑑Δ𝐸
𝑑𝐸𝐹

, where 𝐸𝐹 =

𝑛𝜋ℏ2
𝑚∗

−23.8 ± 1.6293 13.7 ± 2.3293

( 𝑛 is the sheet charge density and 𝑚∗ is the effective

mass).
This work (𝑚𝑒∗ = 0.647𝑚0 , 𝑚ℎ∗ = 0.805𝑚0 ). Effective mass obtained from GW+BSE
calculations (MoTe2).

S4.13 Computational details
We first perform density functional theory (DFT) calculations with the Perdew-Burke-Ernzerhof
(PBE) generalized gradient approximation300 with the Quantum Espresso package301,302, which
uses a plane-wave basis set, and norm-conserving pseudopotentials303,304. We use a cut-off energy
of 125 Ry and include a vacuum of along the out of plane direction, to avoid spurious interactions
with repeated unit cells. Both in-plane lattice parameters and atomic coordinates are optimized
within the unit cell. A uniform 24x24x1 k-grid is used in the self consistent density calculation,
whereas the wave function is generated on a 12x12x1 k-grid. Spin orbit coupling (SOC) is
considered. The GW calculations305–307 are performed with the Berkeley GW code308,309 using a
generalized plasmon pole (GPP) model307 for the undoped system. Calculations are done with a
dielectric cut-off energy of 35 Ry, on a 12x12x1 q-grid with the nonuniform neck subsampling

221
(NNS) scheme310, with 6,000 states in the summation over unoccupied states, and using a truncated
Coulomb interaction311. The BSE matrix elements are computed on a uniform 24x24x1 coarse kgrid, then interpolated onto a 288x288x1 fine k-grid, with two valence and four conduction bands.
For the doped systems, we use a newly developed plasmon pole model accounting for both
dynamical screening effects and local fields effects associated with the free carriers, which is used
at both the GW and BSE levels. Further details can be found in Ref. [Champagne et al., submitted].
The doping is introduced as a shift of the Fermi energy above the bottom of the conduction band,
corresponding to doping densities ranging from 0 up to 8.7x1012 𝑐𝑚−2 . The dielectric matrix,
kernel matrix, and BSE Hamiltonian are built for a 48x48x1 k-grid. SOC is not considered in the
calculations for the doped systems. The computed intrinsic radiative linewidth is obtained from a
Fermi’s golden rule following Ref.226,228.

S4.14 Computation of exciton dispersion
We construct BSE Hamiltonian for a series of center-of-mass wavevectors Q in a dense 90 × 90
k-grid. Solving for the lowest energy exciton at each Q yields the exciton dispersion with the
effective mass MX = 0.87 m0, where m0 is the electron mass.

Figure S4.13. Exciton dispersion for a few lowest energy excitonic states in monolayer MoTe2.

222

S4.15 Computed exciton absorption spectrum as a function of doping
density

Figure S4.14. Computation of doping-dependent loss function. (a) Computed imaginary part of
the dielectric function (𝜖2 (ℏ𝜔)) for monolayer MoTe2 with and without electron-hole interactions
and projected oscillator strength of the different Rydberg excitons (A1s, A2s, B1s, A3s from left
to right). (b) Evolution of the imaginary part of the dielectric function as a function of doping
density.

S4.16 Computation of trion binding energy
To obtain charged excitations, we obtain excited-state properties associated with N+3-particle
excitations, which consist of one neutral electron-hole pair plus an additional carrier (which we
restrict here to be an extra electron). We note that such a description for a trion is applicable for a
vanishingly small Fermi surface; for larger carrier doping, one needs to explicitly include the
hybridization with of the neutral electron-hole pair with intraband plasmons in the degenerate
Fermi see.

223
We solve for trion excitations by writing a Dyson’s equation associated with correlated 3-particle
excitations, 𝐿 = 𝐿𝐾𝐿0 where 𝐿0 is non-interacting Green’s function, L is the interacting Green’s
function, and K is the interaction kernel. We obtain an equation of motion equivalent to that derived
in a prior work312, and which we will detail in a subsequent manuscript.
We expand our trion wave function in the electron and hole basis in the same way we constructed
the exciton wave function
𝑛,𝑞
|𝑇(𝑛, 𝑞)⟩ = ∑ 𝐵𝑣𝑐
|0⟩
1 𝑐2 𝑐̂𝑣𝑘1 +𝑘2 −𝑞 𝑐̂𝑐2 𝑘2 𝑐̂𝑐1 𝑘1
𝑣𝑐1 𝑐2
𝑘1 𝑘2

𝑘1 𝑘2

where T is a trion wavefunction with principal quantum number n and wavevector q, B are
expansion coefficients, v and c label valence and conduction bands, respectively, k is a wavevector,
c ̂ is a fermionic destruction operator, and |0⟩ is the many-body ground state. and we arrive at the
Dyson’s-like equation
𝑛,𝑞
𝑛,𝑞
̂ |𝑇(𝑛, 𝑞)⟩ = Ω𝑇𝑛,𝑞 𝐵𝑣𝑐
(𝐸𝑐1𝑘1 + 𝐸𝑐2𝑘2 − 𝐸𝑣𝑘1 +𝑘2 −𝑞 )𝐵𝑣𝑐1𝑐2 − ∑⟨𝑇(𝑛′ , 𝑞)|𝐾
1 𝑐2
𝑘1 𝑘2

𝑛′

𝑘1 𝑘2

where Ω𝑇𝑛,𝑞 is the energy of trion at state n and momentum q and 𝐸𝑚𝑘 is the quasiparticle energy
for an electron in band m and wavevector k. From here, we found the trion binding energy Δ𝐸 =
min
Ωmin
𝑒𝑥𝑐𝑖𝑡𝑜𝑛 − Ω𝑡𝑟𝑖𝑜𝑛 of 20.6 meV for the A1s state. Our optical spectrum associated with the

absorption of excitons and trions is shown in Fig. S4.15.

Figure S4.15. Computed imaginary part of the dielectric function for excitons and trions.

224

S4.17 Discussion on importance of MoTe2 optical properties and its
Rydberg series
Well studied TMDCs like Mo and W based sulfides and selenides have band-gaps and optical
transitions in the visible spectrum (~600-760 nm). Rydberg excitons associated with these
materials are thus at even higher energies. From a technology point of view wavelengths near the
silicon band edge (~1.1 eV or ~1100 nm) and telecom band (~1550nm) are very important for the
development of silicon-based opto-electronics. With the advent of hybrid platforms where new
materials are being integrated into silicon photonics, it is important to find and expand the library
of materials which have strong photo-response in these wavelengths. MoTe2 is one of the few 2D
materials which in its semiconducting 2H polytype exhibits a ground state excitonic optical
transition in the silicon band-edge window. It has thus attracted integration into waveguide-based
photonic and opto-electronic applications; for example–light emitting diode313, waveguideintegrated high-speed314 and strain-engineered64 photodetector. It is evident that MoTe2 has
attracted significant interest, yet these studies are done for bilayer and bulk samples limiting the
achievable performance because the excitonic photo-response should be maximal for a
monolayer214. A careful and detailed understanding of the photo physics of monolayer MoTe2 at
low temperatures under electrostatic doping conditions is lacking which is the primary motivation
of selecting this material.
Rydberg excitons in MoTe2 are in a unique part of the electromagnetic spectrum (~800-1000 nm)
which is exciting for a lot of applications in the near infrared such as quantum optics with rareearth ions315, opto-electronics for health sensing316 and laser technology317, including high-power
applications318. Furthermore, Rydberg excitons provide a way to conveniently study long-range
dipole-dipole interactions because of their large size. As the quantum number increases for the
Rydberg exciton their average radius ⟨𝑟𝑛 ⟩ increases (for a hydrogenic system) as per the formula–

⟨𝑟𝑛 ⟩ = 𝑎𝐵 (3𝑛2 − 𝑙(𝑙 + 1)), where 𝑎𝐵 is the Bohr radius, 𝑛 is the principal quantum number and

𝑙 is the angular momentum. Such an increasing size leads to huge interaction effects and can
provide insights into atomic and molecular physics at the single-particle/quantum level. They can
also provide excellent sensing capabilities since their wavefunction is exceptionally large and are
extra sensitive to the environment186,201,202, as well as to their self. Furthermore, Rydberg excitons
can be engineered to form an ordered array using Rydberg blockade which is very appealing for

225
quantum simulations. They can also play a vital role in non-linear optics when incorporated into
optical cavities to form exciton-polariton modes (due to their large exciton wavefunction and
strong dipole-dipole repulsion as compared to 1s excitons)204,205.

S4.18 Atomic force microscope image of MoTe2 device

Figure S4.16. Atomic force microscope image (height sensor) of MoTe2 device.

226

Chapter S5. Supplementary Information for Signatures
of Edge-Confined Excitons in Monolayer Black
Phosphorus
S5.1 Fabrication details
SiO2 (285 nm)/Si chips were cleaned with ultrasonication in acetone and isopropanol for 30 min.
each, followed by oxygen plasma treatment at 70 W, 300 mTorr for 5 min. Monolayer black
phosphorus (BP) was exfoliated to polydimethylsiloxane (PDMS) stamps. Few layer hexagonal
boron nitride (hBN) and graphene flakes were directly exfoliated using Scotch-tape at 100 oC to
increase the size of flakes. Monolayer thickness was initially identified using optical contrast (~7
% contrast per layer) and later verified with photoluminescence spectroscopy. hBN thickness was
confirmed

with

atomic

force

microscopy.

Flakes

were

assembled

with

polycarbonate/polydimethylsiloxane (PC/PDMS) stamp with pick-up at temperatures between 80

C to 110 oC. The entire heterostructure stack was dropped on prefabricated gates at 180 oC. The

polymer was washed off in chloroform overnight, followed by isopropanol for 10 min. Gates were
fabricated on SiO2 (285 nm)/Si chips with electron beam lithography (PMMA 950 A4 spun at
3500 rpm for 1 min. and baked at 180 oC, 10 nA beam current and dosage of 1350 𝜇C/cm2 at 100
kV), developing in methyl isobutyl ketone: isopropanol (1:3) for 1 min., followed by isopropanol
for 30 s and electron beam evaporating 5 nm Ti/95 nm Au at 0.5

deposition rate. Liftoff was

done in warm acetone (60o C) for 10 min., followed by rinse in isopropanol for 5 min. The gates
were precleaned before drop-down of heterostructure by annealing in high vacuum (2x10-7 Torr)
at 300 oC for 6 h. The chip was then wire-bonded with Aluminum wires on to a custom homemade printed circuit board.
For TEM sample fabrication first the TEM holey grid purchased from NORCADA (NH005D03
with 0.05 mm x 0.05 mm, 200 nm thick membrane, 300 nm diameter holes, 600 nm pitch, 25 x 25
array) was loaded into electron-beam evaporation chamber to deposit ~0.5 nm Ti and ~1.5 nm Au
at 0.1

deposition rate. Right after the deposition, exfoliated flakes of monolayer graphene and

BP on PDMS were transferred using the dry-transfer technique at 50 oC.

227

S5.2 Experimental methods
Low temperature confocal photoluminescence measurements were performed in an attoDRY800
closed-cycle cryostat at base pressures of <2x10-5 mbar. Sample was mounted on a thermally
conducting stage with Apiezon glue and stage was cooled using a closed-cycle circulating liquid
helium loop. Temperature was varied between ~4K and 300K. A 532 nm (Cobolt) continuous
wave laser was used as the excitation source with power ranging between 10 nW and 1 mW,
focused to a diffraction limited spot. Emission was collected in a confocal fashion with a cryogenic
compatible apochromatic objective with an NA of 0.82 (for the visible and NIR range, LT APO
VISIR) and dispersed onto a grating-based spectrometer (with 150 grooves per mm with a Silicon
CCD)–Princeton Instruments HRS 300. Polarization dependent data was acquired using a
combination of linear polarizer and half wave plate on the excitation/emission channel. Voltage
was applied using a Keithley 2400. Data was acquired with home-written MATLAB codes.

Figure S5.1. Schematic of the low-temperature confocal optical PL setup used. BS–Beam Splitter,
QWP/HWP–Quarter/Half Wave Plate.
Atomic Force Microscopy was performed using Bruker Dimension Icon in tapping mode. Data
analysis was performed in MATLAB.

228

S5.3 Image of samples studied
Multiple devices/samples were measured as part of this study. Optical microscope images of
devices studied via photoluminescence are shown below.

Figure S5.2. Optical microscope images of two gated heterostructures (sample (a), (b) #gateD4
and (c) #gateD1). hBN encapsulated monolayer BP is contacted with few layer graphene to gold
electrodes and a back contact of optically thick gold is used as the counter electrode. Scale bar in
(a) is 20 μm and (c) is 10 μm.

229

Figure S5.3. Optical microscope images of two encapsulated heterostructures (sample (a) #D6 and
(b) #D2), containing top and bottom hBN and monolayer BP. Scale bars are 5 μm.
Optical microscope images of devices studied via transmission electron microscopy are shown
below.

Figure S5.4. Optical microscope images of three bare monolayer samples on TEM grids (sample
(a) #S1, (b) #S2 and (c) #S3). The TEM hole arrays are 15 μm x 15 μm.

230

Figure S5.5. Optical microscope images of three fully encapsulated (with monolayer graphene)
monolayer BP samples on TEM grids (sample (a) #H1, (b) #H2 and (c) #H3). The TEM hole arrays
are 15 μm x 15 μm.

Figure S5.6. Scanning electron microscope image of a typical TEM holey grid used in
measurements. To prevent charging during imaging ~0.5 nm Ti and ~1.5 nm Au were deposited
right before transfer of 2D flakes.

231

S5.4 Visualization of strain and charge inhomogeneity in samples
We find spatial variations in the peak emission energy across different samples which indicate that
there is some degree of inhomogeneous strain and charge distribution inducing during stacking of
heterostructures.

Figure S5.7. Spatial maps of energy of photoluminescence of the brightest feature (which
approximately follows the exciton energy) for sample. (a) #gate D1, (b) #D6 and (c) #D2,
respectively. We find ~50 meV, ~100 meV and ~20 meV variation in the PL peak energy for (a),
(b) and (c), respectively.

S5.5 Polarization analysis
To acquire polarization dependent data, the emission polarizer (analyzer) was rotated in steps of 1
or 2o and photoluminescence spectra was collected at each angle. Angle-dependent false color
plots are shown for some spatial spots for different devices. After fitting the PL spectrum at each
emission angle to a sum of Lorentzian peaks, the PL intensity variation as a function of emission
angle has been fit to the following equation–𝐼 = 𝐼0 cos2 (𝜃 − 𝜃0 ) + 𝑐, where 𝜃0 is the azimuthal
orientation of the emission and 𝐼0 , 𝑐 are constants which denote the peak intensity and background
intensity of emission. Following such analysis over multiple spots in different samples, a statistical
set of data has been generated which shows that the emission from edge states is aligned with the
local armchair axes (interior excitons), as expected from theory. Small deviations are likely due to
strain-induced effects whereas a few points that show much larger deviation are likely due to
reconstructions that break mirror-symmetry. Such structures have not been investigated in great
details in this work and merit future studies.

232

Figure S5.8. Polarization analysis-I. (a) Azimuthal orientation of edge dipole emission pattern
versus interior dipole emission pattern (extracted from fitting the “interior”-like emission
envelope). Color indicates peak energy of emission as noted in color bar (in eV). (b) Difference in
the azimuthal dipole emission angle between the edge and the interior as a function of emission
energy of the edge emission. (c) Same as (b), but plotted (on x-axis) as a function of difference in
energy between the interior and edge (Δ𝐸 = 𝐸𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 − 𝐸𝑒𝑑𝑔𝑒 ).

Figure S5.9. Polarization analysis-II. (a) Dipole visibility ( 𝑚𝑎𝑥

−𝐼𝑚𝑖𝑛

𝐼𝑚𝑎𝑥 +𝐼𝑚𝑖𝑛

) of edge exciton emission

versus difference of edge and interior dipole emission angle. (b) Dipole visibility of edge exciton
emission versus interior exciton dipole orientation. (c) Same as (b), but plotted (on x-axis) as a
function of difference in energy between the interior and edge (Δ𝐸 = 𝐸𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 − 𝐸𝑒𝑑𝑔𝑒 ).

233

Figure S5.10. Measured armchair direction at some spatial points (direction of the arrows)
superimposed on log(PL) spatial maps for sample (a)#D6 and (b)#D2, respectively. For both
samples terminations along armchair and zigzag are both seen (for #D6, the longer tear is along
armchair, whereas for #D2, it is along zigzag).
We estimate the macroscopic orientation of BP flakes investigated in this work by superimposing
a false colormap of the emission signal with arrows pointing along the armchair direction (obtained
by performing a full polarization analysis at that spot, as described previously). As can be seen,
fractures along both armchair and zigzag edges are seen.

234

Figure S5.11. Extended polarization dataset 1. (a)-(i) False colormaps of emission spectrum as a
function of collection (analyzer) polarizer angle for different spots from device D6. It can be
clearly seen that for all spots (collected near physical edges of sample) signatures of both interior
and edge emission is observed.

235

Figure S5.12. Extended polarization dataset 2. (a)-(i) False colormaps of emission spectrum as a
function of collection (analyzer) polarizer angle for different spots from device #gateD1. It can be
clearly seen that for all spots (collected near physical edges of sample) signatures of both interior
and edge emission is observed.

236

S5.6 Temperature dependent PL spectra

2 𝑒𝑉

Figure S5.13. Spatial map of integrated photoluminescence spectrum. (𝐼𝑛𝑡. 𝑃𝐿 = ∫1.5 𝑒𝑉 𝐼(𝜔)𝑑𝜔)
Spots are marked as stars in different colors showing temperature dependent emission Scale bar is
5 𝜇𝑚. This map (gate #D1) has been acquired at 5K and the same sample has been cycled from
5K to 300K to acquire temperature dependent spectrum presented next.

237

Figure S5.14. Temperature dependent (normalized and offset) photoluminescence spectrum from
the selected spots (marked in S10). (a) 5K, (b) 40K, (c) 70K, (d) 100K, (e) 190K and (f) 300K
showing clear broadening of edge features and eventual disappearance of the same. Interior
emission profile takes over for higher temperatures.

S5.7 Power dependent spectra and statistics
Power dependence has been investigated across multiple spots and statistics have been collected.
PL spectra as a function of power is shown below. Spectrum at each incident power is fit to the
following set of equations–𝐼𝑃𝐿 (𝜔) = ∑𝑁
𝑖=1

𝐴𝑖 Γ 𝑖

Γ 2

(𝜔−𝜔𝑖 )2 +( 𝑖 )

, where 𝐴𝑖 is the oscillator strength, Γ𝑖 is

the full-width at half maximum and 𝜔𝑖 is the resonance frequency/energy of each resonance), and
thereafter fitting the obtained variation of integrated intensity with power to an exponent law
∫ 𝐼𝑃𝐿,𝑖 (𝑃) 𝑑𝜔 = 𝐶0,𝑖 𝑃𝛼𝑖 , where 𝑃𝑖 is the incident power and 𝛼𝑖 is the exponent for each resonance,
𝐼𝑃𝐿,𝑖 is given by equation 1 and 𝐶0,𝑖 is a proportionality constant. The integration is done across the
full width at half-maxima for each peak.

238

Figure S5.15. Additional pump power dependent data. (a) False color map and (b) line-cuts of
power dependent photoluminescence spectrum for a specific spatial point in device D6 showing
three distinct emission lines from the edge on top of a broad interior emission envelope. (c) Powerlaw fits of integrated photoluminescence intensity to the three aforementioned peaks showing
linear behavior.
A plot of 𝛼 versus 𝜔𝑖 is shown for sample where data was collected for 24 spots showing edge
emission.

Figure S5.16. Accumulated power law fit exponents across 24 spots showing edge emission
measured in sample #gateD1. Similar behavior was also seen in D6. Exponents lie between ~0.9
and ~1.1 indicating linear behavior (for the ranges of incident power measured).

239

S5.8 Spectral diffusion
Emission data was collected as a function of time for multiple spots out of which many showed
signs of temporal fluctuation or spectral diffusion indicating high sensitivity to the environment.
Such signatures allude to the fact that these states are highly localized and quantum-confined.
Additional spots showing spectral diffusion are presented below, which are in stark contrast to
stable emission from the interior excitons.

Figure S5.17. False color-maps of photoluminescence spectrum collected from three distinct
spatial spots (a)-(c) showing clear signs of spectral diffusion/temporal fluctuation.

Figure S5.18. False color-maps of photoluminescence spectrum collected from interior exciton
emission at an interior spatial spot showing stable emission for two different pumping powers (a)
1.7 μW and (b) 91 μW.

240

S5.9 Gate-dependent PL spectra of additional spots
Gate dependent PL spectra from different spots show emergence of edge emission peaks as a
function of gate voltage which get quenched with added electrons/holes. In contrast, the interior
emission shows a monotonic reduction indicating sample is n-doped. The details of the
electrostatic landscape dictate for what voltage conditions the edge peaks line up and is sensitively
linked to the amount of band-bending occurring at the edge of the sample.

Figure S5.19. Gate dependent photoluminescence dynamics for P5 (spatial point #5) for device
#gateD1. (a) False color-map of photoluminescence spectra as a function of gate voltage. (b) Same
as (a) but normalized to 1, for each gate voltage. (c) Spectra from (a) at three distinct voltages (8V,
0V, -8V). (d) Spectra from (b) at three distinct voltages (8V, 0V, -8V). (e) Spectra from (b) for
multiple voltages between 6.1V and -7.9V.

241

Figure S5.20. Gate dependent photoluminescence dynamics for P15 (spatial point #15) for device
#gateD1. (a) False color-map of photoluminescence spectra as a function of gate voltage. (b) Same
as (a) but normalized to 1, for each gate voltage. (c) Spectra from (a) at three distinct voltages (8V,
0V, -8V). (d) Spectra from (b) at three distinct voltages (8V, 0V, -8V). (e) Spectra from (b) for
multiple voltages between 6.1V and -7.9V.

S5.10 Lifetime of PL emission
Time dependent emission spectra were collected both with and without filtering of the edge

emission peak. The data was fit to the following equation: 𝐼𝑃𝐿 (𝑡) = ∑𝑖=1,2,… 𝐼𝑖 𝑒 τI , where 𝐼𝑖 is
the intensity of emission at 𝑡 = 0 and 𝜏𝑖 is the time constant (lifetime) associated with that
emission. Each data set was fit to a sum of three exponentials with the assumption that the first
lifetime is below the resolving capability of the instrument and thus convoluted with the instrument
response function, the second lifetime corresponds to the interior excitons and the third lifetime

242
corresponds to the edge excitons. The instrument limited lifetime is ~100 ps whereas the decay of
interior excitons is ~300-500 ps which corresponds to values observed previously. A third lifetime
which is on the order of 3-5 ns is seen for edge states but has very weak intensity because of
significant coupling to the interior (which has a much shorter lifetime). Upon filtering most of the
interior emission, an increase in the PL intensity for the longest lifetime component can be seen.
To extract true lifetime of such edge states, quality of grown BP crystals must be improved such
that the interior exciton is well blue-shifted and spectrally not overlapping with the edge emission
- to filter out the interior emission completely.

Figure S5.21. Lifetime measurements. (a) PL spectrum of a spatial location (device D6) containing
edge exciton emission signatures over an envelope of interior emission. (b) Time dependent PL
measurements collected without a filter and with two filters placed approximately at the two peaks
from edge states at 1.63 eV and 1.68 eV. Bandwidth of the filters are ~10nm.

S5.11 Electrostatic simulations of capacitance
Finite element simulations were performed using Lumerical CHARGE module to model the
electric-field profile around the BP monolayer. The thickness of each layer is as follows–hBN top
33 nm, BP monolayer 0.7 nm, bottom hBN 42 nm and bottom gold electrode 100 nm. Voltage was
applied between the bottom gold and the BP sheet. A mesh size of nm was used. For BP, the
following parameters were assumed (as extracted from literature)–

243

DC permittivity–12.3

Work function–5.15

Band minimum–Γ point

Effective mass of electron–0.17me

Effective mass of hole–0.15me

Band-gap–2.05 eV

(Hole/electron) mobility–100 cm2/Vs

Voltage was scanned from -5V to 5V in steps of 0.1V and a non-uniform adaptive mesh was used
to account for edge effects and fringing fields.
The field distribution as well as line-cuts are shown below, along with a schematic for the structure
simulated. A hotspot can be seen near the physical edge of the BP monolayer for the in-plane
component of the electric field which is primarily responsible for the in-plane Stark effect.
However, the magnitude is very sensitive to the location of the edge and as seen below, a shift on
the order of a nanometer can reduce the maximal field strength by an order of magnitude. Since
emission emerges from a complex set of edge configurations it is hard to pin-point the exact
magnitude of the field experienced by each state under applied voltage, but the trend can be
explained well from these simulations.

Figure S5.22. Simulation setup in Lumerical CHARGE module. hBN is modelled as a continuous
medium (including the top and bottom thickness) and BP is inserted inside the hBN, such that it

244
matches the experimental structure of device #gateD1. Voltage is applied between the top and
bottom gold. The assumption here is that graphene acts metallic enough that the Au-BP contact
can yield similar results to Au-graphene-BP contact.

Figure S5.23. In-plane electric field component distribution cross-section at 5V. A hotspot can be
seen at the BP-hBN interface at x=0 nm.

Figure S5.24. Gate dependent in-plane field simulations. (a) In-plane electric field variation as a
function of spatial position (x) and gate voltage. The BP-hBN interface laterally is at x=0 nm. (b)
Line-cuts of in-plane electric field for different voltages as a function of spatial position. (c) Linecuts of in-plane electric field for two spatial positions near the BP-hBN interface, as a function of
gate voltage showing the sensitivity of the field experienced by the dipole depending on its
location.

245

S5.12 Band bending schematic
From gate dependent PL measurements, a strong band bending is seen from the interior to the
edge. Here, we schematically represent the band bending to explain the shifts in the Fermi level
between the interior and the edge, also supported by the gate-dependent PL measurements.

Figure S5.25. Atomic structure of monolayer black phosphorus and schematic of how band
bending causes transition from n-type behavior to intrinsic at the edge.

We note that the edge excitons appear brightest at a voltage that is offset from where the interior
excitons show highest luminescence. Based on a standard parallel plate capacitor model we can
approximate the minimum amount of band bending that occurs for each of the edge exciton states.
It is important to note that the band bending can be larger than this since we do not reach charge
neutrality for the interior exciton to prevent the device from breakdown at high voltages. For device
#gateD1, the bottom hBN was ~42 nm in thickness. Employing the relations for a parallel-plate
capacitor we get

246

𝜖 𝜖

= 0 𝑟,ℎ𝐵𝑁 , where 𝜖𝑟,ℎ𝐵𝑁 = 3.9 and 𝑑ℎ𝐵𝑁 = 42 𝑛𝑚.
𝑑ℎ𝐵𝑁

𝜖 𝜖

𝑛 = 𝐶(𝑉 − 𝑉𝐶𝑁𝑃 ) = 0 𝑟,ℎ𝐵𝑁

(𝑉−𝑉𝐶𝑁𝑃 )

𝑑ℎ𝐵𝑁

, where 𝑉𝐶𝑁𝑃 > −5𝑉 (as estimated from PL measurements).

At 𝑉~0𝑉, we thus get a charge density of ~2.5x1012 cm-2. Assuming an effective mass 𝑚𝑒𝑓𝑓 =
0.15𝑚0 , for monolayer BP, we use the parabolic band approximation equation: 𝐸𝐹 =

𝜋ℏ2 𝑛2𝐷
𝑚𝑒𝑓𝑓

, to

get a band bending lower bound of ~40 meV.

S5.13 Screening effects
Excitons in layered 2D sheets are much more prone to screening from the environment as a
consequence of the Coulomb field lines being expelled out of the material itself. The difference in
binding energy is striking as one approaches from a 3D to 2D limit. Strong self-screening in a 3D
material renders the binding energy weaker for excitons in the medium and leads to an increased
Bohr radius due to weaker Coulomb forces. Similarly, the binding energy gets larger in the 2D
limit as the Bohr radius gets smaller. For monolayer black phosphorus, a similar argument holds
despite its excitons being quasi 1D due to optical selection rules. Naturally, as the confinement
gets tighter and the Bohr radius is further reduced at the edges, more of the Coulomb field lines
are expelled outside the medium as compared to the excitons not at the edge. This makes the edge
excitons more sensitive to screening from hBN dielectric environment which leads to a reduction
in the energy–causing a red shift of the resonance. Since this screening is more pronounced more
edges, when excitons get significantly redshifted such that their resonance energy is lower than or
comparable to the interior excitons, they show up as bright peaks in photoluminescence.
In short, mathematically the exciton states can be computed using an effective-mass theory:
[−

1 2
∇ − 𝑊(𝜌)] 𝜓𝑛 (𝜌) = 𝐸𝑛 𝜓𝑛 (𝜌),
2𝜇 𝜌

where 𝜇 is the effective mass of the exciton, 𝜌 is the separation between the electron and hole, 𝜓 is
the wave function describing the exciton, 𝑊 represents the Coulomb interaction and 𝐸 is the
binding energy of the corresponding eigen-state.

247
In the anisotropic 2D limit, the Coulomb interaction is best described by the Rytova-Keldysh
potential,
𝑊(𝜌) =

𝜋𝑒 2
[𝐻0 ( ) − 𝑌0 ( )],
𝜌0 (𝜖𝑎 + 𝜖𝑏 )
𝜌0
𝜌0

where 𝑌0 , 𝐻0 are the Bessel and Struve function of the second kind and 𝜖𝑎 and 𝜖𝑏 are the dielectric
permittivities of the surround top (a) and bottom (b) medium, respectively.
However, the R-K model has unphysical divergence at 𝜌~0, which is a relevant length scale in
our case, as for edge excitons the Bohr radius is much smaller than the interior excitons due to
stronger localization. A better approximation for the potential was shown to be as follows:

𝑒 2 𝐿𝑛2𝐷,𝑎 𝐿𝑛2𝐷,𝑏
𝑒 2 𝐿𝑛2𝐷,𝑎 𝐿𝑛2𝐷,𝑏
𝑒2
𝑊(𝜌) =
+2∑
+ (𝐿2𝐷,𝑎 + 𝐿2𝐷,𝑏 ) ∑
2 + (2𝑛𝑑)2
2 + [(2𝑛 + 1)𝑑]2
𝜖2𝐷 𝜌
√(𝜌
√(𝜌
𝑛=1 2𝐷
𝑛=0 2𝐷
where 𝐿2𝐷,𝑖 = (𝜖2𝐷 − 𝜖𝑖 )/(𝜖2𝐷 + 𝜖𝑖 ) and 𝜖2𝐷 is the dielectric permittivity of the 2D sheet. From
this expression it is clear that other factors remaining constant the Coulomb potential has an inverse
dependence on the electron-hole separation–meaning edge excitons have very strong Coulomb
attraction.
This analysis is derived from Ref.157

S5.14 Discussion on edge exciton emission
We explore the different possibilities that can give rise to discrete lines in the photoluminescence
spectrum from BP edges, elaborating on why the proposed mechanism in this work is able to
explain the complete set of observation.

Point defects–Presence of point defects due to vacancies, interstitials, or foreign atoms are
known to cause additional optical transitions. However, firstly, such defects are expected to
occur in regions other than the edges as well but were found to be absent in all of the samples
investigated as part of this study. Furthermore, such point defects display saturable emission
and hence non-linear dependence on incident pump power, while linear scaling is seen in our

248
measurements–indicating more excitonic like behavior. Additionally, we find most of the
emission states having aligned polarization with the local armchair axis–implying preservation
of the intrinsic BP symmetry. For completeness, we list possible defects that are typically
studied in phosphorene –

Oxygen (O)

ii.

Hydrogen (H)

iii.

Fluorine (F)

iv.

Chlorine (Cl)

Hydroxyl (OH)

vi.

Sulphur (S)

vii.

Selenium (Se)

Of these foreign atoms, it was found that so called Family 1 edges (i.e., H, F, Cl, OH) tend to
form saturated bonds with P atoms which causes the edge states to be energetically far away
from the bandgap. Family 2 edges (pristine, O, S, Se) form weaker unsaturated bonds with the
𝑝𝑧 orbital of the phosphorus atom and push edge states within the band gap of the ribbons.
These calculations were reported for phosphorene nanoribbons along armchair and zigzag
orientation259. Despite our measurements not being consistent solely with point defects, we
cannot completely rule out the possibility of such states–with O being the most likely. For
example, emission states which are not aligned with the local armchair orientation might arise
from such states (or reconstruction that does not preserve the intrinsic BP symmetry like a (1,3)
reconstruction). However, the density of such states due to defects is expected to be low as the
entire processing is done in a completely inert environment (in a nitrogen purged glovebox)
and immediately encapsulated with hBN. All optical measurements were performed under high
vacuum. Future (optical and structural) correlated spectroscopy-based studies must be done to
isolate the effect of edge reconstruction and point defects in BP.

Edge quantum confinement–Since an edge is an abrupt termination of atoms, a strong confining
potential well is created for excitons which can be approximately modelled as a triangular
potential barrier. The Schrödinger’s equation for such a system is as follows –

249
ℏ2 2
− ∇ 𝜓𝑛 (𝑥) + 𝑉(𝑥)𝜓𝑛 (𝑥) = 𝐸𝑛 𝜓𝑛 (𝑥),
2𝜇
where, 𝑉(𝑥) = ∞, if 𝑥 > 0 and 𝑉(𝑥) = 𝛼𝑥, if 𝑥 < 0 and the edge is at 𝑥 = 0. 𝜓𝑛 (𝑥)is the
wave function, 𝜇 is the effective mass and 𝐸𝑛 in the eigen-energy of the 𝑛𝑡ℎ bound state of the
exciton. Upon solving this equation numerically, we find multiple eigen states that correspond
to the quantized energy levels supported in this potential well which could, in principle, also
explain the observation of peaks in our experiment. Here, we note that if this were solely the
case for the observed peaks, this feature would be generic to all edges and not selected edge
sites. Thus, we can assert that this is not the only mechanism leading to the formation of edge
excitons.

Strain induced localization–In TMDCs it is well known that strong strain gradients can cause
significant reduction in the band gap leading to the formation of “defect states” that give rise
to single-photon emission. To verify whether such is the case for BP, we prepared 1. An
intentional strained sample by draping a sheet of monolayer BP on a fabricated nano-pillar
array made of gold and 2. A serendipitous heterostructure containing a large bubble in between
the layers causing strain on the BP. For sample 1, the monolayer was aligned in such a way
that the only interior of BP would be strained. We saw no evidence of localized emission in
the sample in either of the sample (except some of the edge sites)–ruling out strain to be the
only driving factor.

S5.15 First-Principles Computation Of Optical Spectrum
In order to verify the reconstructed edge responsible for distinct emission peaks at various edges
of the different sections in monolayer BP, we investigate a selected number of BP nanoribbons
with different edge reconstructions, focusing on previously predicted structures that display small
formation energy and displaying a gap267. Using these criteria, out of the 24 previously computed
edge structures with an armchair and zigzag fracture orientations, we have selected 5 promising
structures, as discussed earlier. We constructed nanoribbons of two unit cells along the extended
direction (zigzag for AC10-ii, AC12-i, and armchair for ZZ4-i, ZZ8-i, and ZZ10-i) and 8 unit cells

250
along the confined direction (armchair for AC10-ii, AC12-i and zigzag for ZZ4-i, ZZ8-i, and
ZZ10-i) with 13 Å out-of-plane vacuum.
We perform DFT calculations using the Quantum Espresso package with Perdew-Burke-Ernzerhof
(PBE) generalized gradient approximation at an energy cutoff of 30 Ry to obtain the ground-state
wave function basis for our subsequent GW and GW-BSE calculations. We perform GW
calculations with the BerkeleyGW package using the Hyberten-Louie generalized plasmon-pole
model (GPP). To converge the calculation of the dielectric matrix, we use a hybrid stochasticdeterministic compression of the unoccupied states following Ref. YYY, which approximately
captures 170,000 bands, and evaluate the response on a 1x1x8 k-point grid with a dielectric cutoff
of 12 Ry. To capture the long-range dielectric screening (q=0), we utilize the nonuniform neck
subsampling (NNS) method on a 1x1x4 k-point grid with additional 32 k-points. Self-energy
calculations are performed on a 1x1x4 k-point grid and interpolated to a 1x1x32 k-point grid with
truncated Coulomb potential to prevent spurious interactions along both extended and out-of-plane
directions. This yields quasiparticle band structure in which we project the DFT wave function to
edge and bulk state by weighting the probability density with a clamp function that goes from 0 to
1 in an area of 1 unit cell from the edges. Furthermore, we project the wavefunction at each band
index and k-point to the edge and interior state using a clamp function:
⟨𝑒𝑑𝑔𝑒𝑛,𝑘 ⟩ = ⟨𝜓𝑛,𝑘 |𝑓𝑐𝑙𝑎𝑚𝑝 (𝑥)|𝜓𝑛,𝑘 ⟩
⟨𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟𝑛,𝑘 ⟩ = 1 − ⟨𝑒𝑑𝑔𝑒𝑛,𝑘 ⟩
where,
0 , 𝑥 > 𝑥1 + 𝑎 or 𝑥 < 𝑥2 − 𝑎
𝑥−𝑥1
1−
, 𝑥1 + 𝑎 > 𝑥 > 𝑥1

𝑓𝑐𝑙𝑎𝑚𝑝 (𝑥) =

𝑥−𝑥2

, 𝑥2 > 𝑥 > 𝑥2 − 𝑎

1, otherwise.

For 𝑥1 and 𝑥2 are positions of the outermost atom along the confined direction with all other atoms
in the structure lie in-between 𝑥1 and 𝑥2 , and 𝑎 is the length of the primitive unit cell along the
confined direction.

251
The matrix elements of the kernel of the BSE are computed including 20 conduction bands and 20
valence bands on a 1x1x4 k-point grid interpolated onto a 1x1x32 k-point grid.
To further distinguish between edge and interior excitons, we take the two-particle exciton
amplitude

𝜓𝑠 (𝑟𝑒 , 𝑟ℎ ) = ∑𝜈,𝑐,𝑘 𝐴 𝜈𝑐𝑘 𝜓𝑐𝑘 (𝑟𝑒 )𝜓𝜈𝑘 (𝑟ℎ ) for

each

state

and

compute

their

corresponding electron, 𝐹𝑒 (𝑟𝑒 ), and hole, 𝐹ℎ (𝑟ℎ ), projection:

𝐹𝑒 (𝑟𝑒 ) = ∫ 𝑑 3 𝑟ℎ |𝜓(𝑟𝑒 , 𝑟ℎ )|2 = Σ𝜈,𝑐 ′ ,𝑐,𝑘 𝐴𝜈𝑐𝑘
𝐴𝜈𝑐
′ 𝑘 𝜓𝑐 ′ 𝑘 (𝑟𝑒 )𝜓𝑐𝑘 (𝑟𝑒 )

𝐹ℎ (𝑟ℎ ) = ∫ 𝑑3 𝑟𝑒 |𝜓(𝑟𝑒 , 𝑟ℎ )|2 = Σ𝜈,𝑐 ′ ,𝑐,𝑘 𝐴𝜈𝑐𝑘
𝐴𝜈𝑐
′ 𝑘 𝜓𝑐 ′ 𝑘 (𝑟ℎ )𝜓𝑐𝑘 (𝑟ℎ )

S5.16 TEM analysis of monolayer BP
A FEI Titan 80-300 STEM/TEM equipped with a probe spherical-aberration corrector was
employed to conduct annular dark field scanning transmission electron microscopy (ADF-STEM)
imaging analysis. The images were collected with the microscope operating at 300 kV with a probe
convergence semi-angle of 14 mrad and a collection angle of 34-195 mrad.

252

Bibliography
1.

Geim, A. K. & Grigorieva, I. v. Van der Waals heterostructures. Nature vol. 499 419–425.

Liu, Y. Van der Waals heterostructures and devices. Nat. Rev. Mater. 1, 16042 (2016).

Yi, M. & Shen, Z. A review on mechanical exfoliation for the scalable production of
graphene. J Mater Chem A Mater 3, 11700–11715 (2015).

Ang, P. K., Wang, S., Bao, Q., Thong, J. T. L. & Loh, K. P. High-throughput synthesis of
graphene by intercalation-exfoliation of graphite oxide and study of ionic screening in
graphene transistor. ACS Nano 3, 3587–3594 (2009).

Ciesielski, A. & Samorì, P. Graphene via sonication assisted liquid-phase exfoliation.
Chem Soc Rev 43, 381–398 (2013).

Wilson, J. A. & Yoffe, A. D. The transition metal dichalcogenides discussion and
interpretation of the observed optical, electrical and structural properties. Adv Phys 18, 193–
335 (1969).

Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene.
Nature 438, 197–200 (2005).

Armitage, N. P., Mele, E. J. & Vishwanath, A. Weyl and Dirac semimetals in threedimensional solids. Rev Mod Phys 90, 015001 (2018).

Stajic, J. Magnetic Weyl semimetals. Science (1979) 365, 1260.3-1260 (2019).

10.

Yan, B. & Felser, C. Topological Materials: Weyl Semimetals. Annu Rev Condens Matter
Phys 8, 337–354 (2017).

11.

Cao, Y. et al. Unconventional superconductivity in magic-angle graphene superlattices.
Nature 556, 43–50 (2018).

12.

Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene
superlattices. Nature 556, 80–84 (2018).

13.

He, K. et al. Tightly bound excitons in monolayer WSe2. Phys Rev Lett 113, 026803 (2014).

253
14.

Wang, G. et al. Colloquium: Excitons in atomically thin transition metal dichalcogenides.
Rev Mod Phys 90, 021001 (2018).

15.

Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in
monolayer WS2. Phys Rev Lett 113, 076802 (2014).

16.

Mahan, G. D. Excitons in Degenerate Semiconductors. Physical Review 153, 882 (1967).

17.

Yang, J. et al. Robust Excitons and Trions in Monolayer MoTe2. ACS Nano 9, 6603–6609
(2015).

18.

Vaquero, D. et al. Excitons, trions and Rydberg states in monolayer MoS2 revealed by lowtemperature photocurrent spectroscopy. Communications Physics 2020 3:1 3, 1–8 (2020).

19.

Chen, S. Y. et al. Luminescent Emission of Excited Rydberg Excitons from Monolayer
WSe2. Nano Lett 19, 2464–2471 (2019).

20.

Ross, J. S. et al. Electrical control of neutral and charged excitons in a monolayer
semiconductor. Nat Commun 4, 1474 (2013).

21.

Wang, X. et al. Highly anisotropic and robust excitons in monolayer black phosphorus. Nat
Nanotechnol 10, 517–521 (2015).

22.

Li, L. et al. Direct observation of the layer-dependent electronic structure in phosphorene.
Nat Nanotechnol 2016 12:1 12, 21–25 (2016).

23.

Zhang, G. et al. Determination of layer-dependent exciton binding energies in few-layer
black phosphorus. Sci Adv 4, eaap9977 (2018).

24.

Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V. & Kis, A. Single-layer MoS2
transistors. Nat Nanotechnol 2011 6:3 6, 147–150 (2011).

25.

Mak, K. Tightly bound trions in monolayer MoS2. Nat Mater 12, 207–211 (2013).

26.

Chernikov, A. et al. Electrical Tuning of Exciton Binding Energies in Monolayer WS2. Phys
Rev Lett 115, (2015).

27.

Wagner, K. et al. Autoionization and Dressing of Excited Excitons by Free Carriers in
Monolayer WSe2. Phys Rev Lett 125, 267401 (2020).

254
28.

Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in
monolayer WS2. Phys Rev Lett 113, 076802 (2014).

29.

Hill, H. M. et al. Observation of excitonic rydberg states in monolayer MoS2 and WS2 by
photoluminescence excitation spectroscopy. Nano Lett 15, 2992–2997 (2015).

30.

Yu, Y. et al. Giant Gating Tunability of Optical Refractive Index in Transition Metal
Dichalcogenide Monolayers. Nano Lett 17, 3613–3618 (2017).

31.

Datta, I. et al. Low-loss composite photonic platform based on 2D semiconductor
monolayers. Nat Photonics 14, 256–262 (2020).

32.

Karni, O. et al. Infrared Interlayer Exciton Emission in MoS2/WSe2 Heterostructures. Phys
Rev Lett 123, 247402 (2019).

33.

Leisgang, N. et al. Giant Stark splitting of an exciton in bilayer MoS2. Nat Nanotechnol
2020 15:11 15, 901–907 (2020).

34.

Jauregui, L. A. et al. Electrical control of interlayer exciton dynamics in atomically thin
heterostructures. Science (1979) 366, 870–875 (2019).

35.

Liu, Y. et al. Gate-Tunable Giant Stark Effect in Few-Layer Black Phosphorus. Nano Lett
17, 1970–1977 (2017).

36.

Rodin, A. S., Carvalho, A. & Castro Neto, A. H. Strain-induced gap modification in black
phosphorus. Phys Rev Lett 112, (2014).

37.

Aslan, O. Probing the optical properties and strain-tuning of ultrathin Mo1–xWxTe2. Nano
Lett. 18, 2485–2491 (2018).

38.

Huang, S. et al. Strain-tunable van der Waals interactions in few-layer black phosphorus.
Nat Commun 10, 1–7 (2019).

39.

Quereda, J. et al. Strong Modulation of Optical Properties in Black Phosphorus through
Strain-Engineered Rippling. Nano Lett 16, 2931–2937 (2016).

40.

Cho, C. et al. Highly Strain-Tunable Interlayer Excitons in MoS2/WSe2Heterobilayers.
Nano Lett 21, 3956–3964 (2021).

255
41.

AN Grigorenko, M. P. K. N. Graphene plasmonics. Nat Photon. 6, 749–758 (2012).

42.

Ni, G. X. et al. Fundamental limits to graphene plasmonics. Nature 557, 530–533 (2018).

43.

Ju, L. et al. Graphene plasmonics for tunable terahertz metamaterials. Nat Nanotechnol 6,
630–634 (2011).

44.

Brar, V. W., Jang, M. S., Sherrott, M., Lopez, J. J. & Atwater, H. A. Highly confined tunable
mid-infrared plasmonics in graphene nanoresonators. Nano Lett 13, 2541–2547 (2013).

45.

di Pietro, P. et al. Observation of Dirac plasmons in a topological insulator. Nat
Nanotechnol 8, 556–560 (2013).

46.

Wang, C. et al. Van der Waals thin films of WTe2 for natural hyperbolic plasmonic surfaces.
Nat Commun 11, 1–9 (2020).

47.

Correas-Serrano, D., Gomez-Diaz, J. S., Melcon, A. A. & Alù, A. Black phosphorus
plasmonics: anisotropic elliptical propagation and nonlocality-induced canalization.
Journal of Optics 18, 104006 (2016).

48.

Lee, I. H. et al. Anisotropic Acoustic Plasmons in Black Phosphorus. ACS Photonics 5,
2208–2216 (2018).

49.

Liu, Z. & Aydin, K. Localized Surface Plasmons in Nanostructured Monolayer Black
Phosphorus. Nano Lett 16, 3457–3462 (2016).

50.

Low, T. et al. Plasmons and Screening in Monolayer and Multilayer Black Phosphorus.
Phys Rev Lett 113, (2014).

51.

Dai, S. et al. Tunable phonon polaritons in atomically thin van der Waals crystals of boron
nitride. Science 343, 1125–1129 (2014).

52.

Ma, W. et al. In-plane anisotropic and ultra-low-loss polaritons in a natural van der Waals
crystal. Nature 2018 562:7728 562, 557–562 (2018).

53.

Basov, D. N., Fogler, M. M. & García De Abajo, F. J. Polaritons in van der Waals materials.
Science 354, 6309 (2016).

256
54.

Ling, X., Wang, H., Huang, S., Xia, F. & Dresselhaus, M. S. The renaissance of black
phosphorus. Proc Natl Acad Sci U S A 112, 4523–4530 (2015).

55.

Chen, H. T., Taylor, A. J. & Yu, N. A review of metasurfaces: Physics and applications.
Reports on Progress in Physics vol. 79 076401 (2016).

56.

Koshelev, K. et al. Subwavelength dielectric resonators for nonlinear nanophotonics.
Science 367, 288–292 (2020).

57.

Fante, R. L. & Mccormack, M. T. Reflection Properties of the Salisbury Screen. IEEE Trans
Antennas Propag 36, 1443–1454 (1988).

58.

Park, J., Kang, J. H., Kim, S. J., Liu, X. & Brongersma, M. L. Dynamic reflection phase
and polarization control in metasurfaces. Nano Lett 17, 407–413 (2017).

59.

Qian, K. et al. Under-Coupling Whispering Gallery Mode Resonator Applied to Resonant
Micro-Optic Gyroscope. Sensors 2017, Vol. 17, Page 100 17, 100 (2017).

60.

Yu, N. & Capasso, F. Flat optics with designer metasurfaces. Nat Mat 13, 139–150 (2014).

61.

Yu, N. et al. Light propagation with phase discontinuities: Generalized laws of reflection
and refraction. Science (1979) 334, 333–337 (2011).

62.

Wuttig, M. & Yamada, N. Phase-change materials for rewriteable data storage. Nat Mater
6, 824–832 (2007).

63.

Shportko, K. et al. Resonant bonding in crystalline phase-change materials. Nat Mater 7,
653–658 (2008).

64.

Maiti, R. et al. Strain-engineered high-responsivity MoTe2 photodetector for silicon
photonic integrated circuits. Nat Photon 14, 578–584 (2020).

65.

Xu, K. et al. The positive piezoconductive effect in graphene. Nat Commun 6, (2015).

66.

GT Reed, G. M. F. G. D. T. Silicon optical modulators. Nat Photon 4, 518–526 (2010).

67.

Xu, Q., Schmidt, B., Pradhan, S. & Lipson, M. Micrometre-scale silicon electro-optic
modulator. Nature 435, 325–327 (2005).

257
68.

Kuo, Y.-H. et al. Strong quantum-confined Stark effect in germanium quantum-well
structures on silicon. Nature 437, 1334–1336 (2005).

69.

Abel, S. et al. Large Pockels effect in micro- and nanostructured barium titanate integrated
on silicon. Nat Mater 18, 42–47 (2019).

70.

Han, J.-H. Efficient low-loss InGaAsP/Si hybrid MOS optical modulator. Nat Photon 11,
486–490 (2017).

71.

Liu, A. A high-speed silicon optical modulator based on a metal–oxide–semiconductor
capacitor. Nature 427, 615–618 (2004).

72.

Marris-Morini, D. Low loss and high speed silicon optical modulator based on a lateral
carrier depletion structure. Opt Express 16, 334–339 (2008).

73.

Mak, K. F. et al. Tightly bound trions in monolayer MoS2. Nat Mat 2012 12:3 12, 207–211
(2012).

74.

Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature
486, 82–85 (2012).

75.

Li, L. et al. Black phosphorus field-effect transistors. Nat Nanotechnol 9, 372–377 (2014).

76.

Kim, J. et al. Observation of tunable band gap and anisotropic Dirac semimetal state in
black phosphorus. Science 349, 723–726 (2015).

77.

Deng, B. et al. Efficient electrical control of thin-film black phosphorus bandgap. Nat
Commun 8, (2017).

78.

Peng, R. et al. Midinfrared Electro-optic Modulation in Few-Layer Black Phosphorus. Nano
Lett 17, 6315–6320 (2017).

79.

Youngblood, N., Chen, C., Koester, S. J. & Li, M. Waveguide-integrated black phosphorus
photodetector with high responsivity and low dark current. Nat Photon 9, 247–252 (2015).

80.

Wang, T. et al. Visualizing Light Scattering in Silicon Waveguides with Black Phosphorus
Photodetectors. Advanced Materials 28, 7162–7166 (2016).

258
81.

Chen, C., Youngblood, N. & Li, M. Study of black phosphorus anisotropy on silicon
photonic waveguide. 2015 Optoelectronics Global Conference, OGC 2015 (2015).

82.

Chen, C. et al. Three-Dimensional Integration of Black Phosphorus Photodetector with
Silicon Photonics and Nanoplasmonics. Nano Lett 17, 985–991 (2017).

83.

Huang, M. et al. Broadband Black-Phosphorus Photodetectors with High Responsivity.
Advanced Materials 28, 3481–3485 (2016).

84.

Zhang, G. et al. Infrared fingerprints of few-layer black phosphorus. Nat Commun 8, (2017).

85.

Wei, Q. & Peng, X. Superior mechanical flexibility of phosphorene and few-layer black
phosphorus. Appl Phys Lett 104, (2014).

86.

Luo, Z. et al. Anisotropic in-plane thermal conductivity observed in few-layer black
phosphorus. Nat Commun 6, (2015).

87.

Yuan, H. Polarization-sensitive broadband photodetector using a black phosphorus vertical
p–n junction. Nat Nanotechnol 10, 707–713 (2015).

88.

Liao, B. et al. Spatial-Temporal Imaging of Anisotropic Photocarrier Dynamics in Black
Phosphorus. Nano Lett 17, 3675–3680 (2017).

89.

Xia, F., Wang, H. & Jia, Y. Rediscovering black phosphorus as an anisotropic layered
material for optoelectronics and electronics. Nat Commun 5, (2014).

90.

Lan, S., Rodrigues, S., Kang, L. & Cai, W. Visualizing Optical Phase Anisotropy in Black
Phosphorus. ACS Photonics 3, 1176–1181 (2016).

91.

Qiao, J., Kong, X., Hu, Z. X., Yang, F. & Ji, W. High-mobility transport anisotropy and
linear dichroism in few-layer black phosphorus. Nat Commun 5, (2014).

92.

Sherrott, M. C. et al. Experimental Demonstration of >230° Phase Modulation in GateTunable Graphene-Gold Reconfigurable Mid-Infrared Metasurfaces. Nano Lett 17, 3027–
3034 (2017).

93.

Huang, Y. W. et al. Gate-Tunable Conducting Oxide Metasurfaces. Nano Lett 16, 5319–
5325 (2016).

259
94.

Kuo, Y. H. et al. Strong quantum-confined Stark effect in germanium quantum-well
structures on silicon. Nature 437, 1334–1336 (2005).

95.

Inoue, T., Zoysa, M. de, Asano, T. & Noda, S. Realization of dynamic thermal emission
control. Nat Mater 13, 928–931 (2014).

96.

Coppens, Z. J. & Valentine, J. G. Spatial and Temporal Modulation of Thermal Emission.
Advanced Materials 29, (2017).

97.

Brar, V. W. et al. Electronic modulation of infrared radiation in graphene plasmonic
resonators. Nat Commun 6, 7032 (2015).

98.

Wood, J. D. et al. Effective passivation of exfoliated black phosphorus transistors against
ambient degradation. Nano Lett 14, 6964–6970 (2014).

99.

Wang, X. et al. Observation of a giant two-dimensional band-piezoelectric effect on biaxialstrained graphene. NPG Asia Mater 7, e154 (2015).

100. Whitney, W. S. et al. Field Effect Optoelectronic Modulation of Quantum-Confined
Carriers in Black Phosphorus. Nano Lett 17, 78–84 (2020).
101. Low, T. et al. Tunable optical properties of multilayer black phosphorus thin films. Phys
Rev B 90, (2014).
102. Lin, C., Grassi, R., Low, T. & Helmy, A. S. Multilayer Black Phosphorus as a Versatile
Mid-Infrared Electro-optic Material. Nano Lett 16, 1683–1689 (2016).
103. Ling, X. et al. Anisotropic Electron-Photon and Electron-Phonon Interactions in Black
Phosphorus. Nano Lett 16, 2260–2267 (2016).
104. Hsiao, Y., Chang, P. Y., Fan, K. L., Hsu, N. C. & Lee, S. C. Black phosphorus with a unique
rectangular shape and its anisotropic properties. AIP Adv 8, (2018).
105. Wang, X. & Lan, S. Optical properties of black phosphorus. Adv Opt Photonics 8, 618
(2016).
106. Sim, S. et al. Selectively tunable optical Stark effect of anisotropic excitons in atomically
thin ReS2. Nat Commun 7, (2016).

260
107. Yuan, H. et al. Polarization-sensitive broadband photodetector using a black phosphorus
vertical p-n junction. Nat Nanotechnol 10, 707–713 (2015).
108. Dolui, K. & Quek, S. Y. Quantum-confinement and Structural Anisotropy result in
Electrically-Tunable Dirac Cone in Few-layer Black Phosphorous. Sci Rep 5, (2015).
109. Poddubny, A., Iorsh, I., Belov, P. & Kivshar, Y. Hyperbolic metamaterials. Nat Photon 7
958–967 (2013).
110. Krishnamoorthy, H. N. S., Jacob, Z., Narimanov, E., Kretzschmar, I. & Menon, V. M.
Topological Transitions in Metamaterials. Science 336, 205–209 (2012).
111. Lu, D., Kan, J. J., Fullerton, E. E. & Liu, Z. Enhancing spontaneous emission rates of
molecules using nanopatterned multilayer hyperbolic metamaterials. Nat Nanotechnol 9,
48–53 (2014).
112. High, A. A. et al. Visible-frequency hyperbolic metasurface. Nature 522, 192–196 (2015).
113. Liu, Z., Lee, H., Xiong, Y., Sun, C. & Zhang, X. Far-field optical hyperlens magnifying
sub-diffraction-limited objects. Science 315, 1686 (2007).
114. Biehs, S. A., Tschikin, M., Messina, R. & Ben-Abdallah, P. Super-Planckian near-field
thermal emission with phonon-polaritonic hyperbolic metamaterials. Appl Phys Lett 102,
131106 (2013).
115. Caldwell, J. D. et al. Sub-diffractional volume-confined polaritons in the natural hyperbolic
material hexagonal boron nitride. Nat Commun 5, 5221 (2014).
116. Alù, A., Silveirinha, M. G., Salandrino, A. & Engheta, N. Epsilon-near-zero metamaterials
and electromagnetic sources: Tailoring the radiation phase pattern. Phys Rev B 75, 155410
(2007).
117. Silveirinha, M. & Engheta, N. Tunneling of Electromagnetic Energy through
Subwavelength Channels and Bends using Epsilon-Near-Zero Materials. Phys Rev Lett 97,
157403 (2006).
118. Xu, T. & Lezec, H. J. Visible-frequency asymmetric transmission devices incorporating a
hyperbolic metamaterial. Nat Commun 5, 1–7 (2014).

261
119. Sun, J., Zhou, J., Li, B. & Kang, F. Indefinite permittivity and negative refraction in natural
material: Graphite. Appl Phys Lett 98, 101901 (2011).
120. Fei, Z. et al. Gate-tuning of graphene plasmons revealed by infrared nano-imaging. Nature
486, 82–85 (2012).
121. Chen, J. et al. Optical nano-imaging of gate-tunable graphene plasmons. Nature 487, 77–
81 (2012).
122. Klitzing, K. v., Dorda, G. & Pepper, M. New Method for High-Accuracy Determination of
the Fine-Structure Constant Based on Quantized Hall Resistance. Phys Rev Lett 45, 494–
497 (1980).
123. Jeckelmann, B. & Jeanneret, B. The quantum Hall effect as an electrical resistance standard.
Reports on Progress in Physics 64, 1603 (2001).
124. Liu, M. et al. A graphene-based broadband optical modulator. Nature 474, 64–67 (2011).
125. Liu, M., Yin, X. & Zhang, X. Double-layer graphene optical modulator. Nano Lett 12,
1482–1485 (2012).
126. Sensale-Rodriguez, B. et al. Broadband graphene terahertz modulators enabled by intraband
transitions. Nat Commun 3, 780 (2012).
127. Long, G. et al. Achieving Ultrahigh Carrier Mobility in Two-Dimensional Hole Gas of
Black Phosphorus. Nano Lett 16, 7768–7773 (2016).
128. Gillgren, N. et al. Gate tunable quantum oscillations in air-stable and high mobility fewlayer phosphorene heterostructures. 2d Mater 2, 011001 (2015).
129. Chen, X. et al. High-quality sandwiched black phosphorus heterostructure and its quantum
oscillations. Nat Commun 6, 7315 (2015).
130. Li, L. et al. Black phosphorus field-effect transistors. Nat Nanotechnol 9, 372–377 (2014).
131. Li, L. et al. Quantum oscillations in a two-dimensional electron gas in black phosphorus
thin films. Nat Nanotechnol 10, 608-613 (2015).

262
132. Li, L. et al. Quantum Hall effect in black phosphorus two-dimensional electron system. Nat
Nanotechnol 11, 593-597 (2016).
133. Valagiannopoulos, C. A., Mattheakis, M., Shirodkar, S. N. & Kaxiras, E. Manipulating
polarized light with a planar slab of black phosphorus. J Phys Commun 1, 45003 (2017).
134. Castellanos-Gomez, A. Isolation and characterization of few-layer black phosphorus. 2D
Mater. 1, 025001 (2014).
135. Tran, V., Soklaski, R., Liang, Y. & Yang, L. Layer-controlled band gap and anisotropic
excitons in few-layer black phosphorus. Phys Rev B 89, 235319 (2014).
136. Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Environmental Screening Effects in 2D
Materials: Renormalization of the Bandgap, Electronic Structure, and Optical Spectra of
Few-Layer Black Phosphorus. Nano Lett 17, 4706–4712 (2017).
137. Zhang, G. et al. The optical conductivity of few-layer black phosphorus by infrared
spectroscopy. Nat Commun 11, 1847 (2020).
138. Xu, R. et al. Extraordinarily Bound Quasi-One-Dimensional Trions in Two-Dimensional
Phosphorene Atomic Semiconductors. ACS Nano 10, 2046–2053 (2016).
139. Chaves, A., Low, T., Avouris, P., Çaklr, D. & Peeters, F. M. Anisotropic exciton Stark shift
in black phosphorus. Phys Rev B 91, 155311 (2015).
140. Zhou, X., Lou, W. K., Zhai, F. & Chang, K. Anomalous magneto-optical response of black
phosphorus thin films. Phys Rev B 92, 165405 (2015).
141. Li, L. et al. Direct observation of the layer-dependent electronic structure in phosphorene.
Nat Nanotechnol 12, 21–25 (2017).
142. Lin, C., Grassi, R., Low, T. & Helmy, A. S. Multilayer Black Phosphorus as a Versatile
Mid-Infrared Electro-optic Material. Nano Lett 16, 1683–1689 (2016).
143. Whitney, W. S. et al. Field Effect Optoelectronic Modulation of Quantum-Confined
Carriers in Black Phosphorus. Nano Lett 17, 78–84 (2020).
144. Sherrott, M. C. et al. Anisotropic Quantum Well Electro-Optics in Few-Layer Black
Phosphorus. Nano Lett 19, 269–276 (2019).

263
145. Chen, C. et al. Widely tunable mid-infrared light emission in thin-film black phosphorus.
Sci Adv 6, eaay6134 (2020).
146. Peng, R. et al. Midinfrared Electro-optic Modulation in Few-Layer Black Phosphorus. Nano
Lett 17, 6315–6320 (2017).
147. Low, T. et al. Tunable optical properties of multilayer black phosphorus thin films. Phys
Rev B 90, 075434 (2014).
148. Chang, Y. C. et al. Realization of mid-infrared graphene hyperbolic metamaterials. Nat
Commun 7, 10568 (2016).
149. Gomez-Diaz, J. S., Tymchenko, M. & Alù, A. Hyperbolic Plasmons and Topological
Transitions over Uniaxial Metasurfaces. Phys Rev Lett 114, 233901 (2015).
150. Asahina, H. & Morita, A. Band structure and optical properties of black phosphorus.
Journal of Physics C: Solid State Physics (1984).
151. Fan, P., Yu, Z., Fan, S. & Brongersma, M. L. Optical Fano resonance of an individual
semiconductor nanostructure. Nat Mater 13, 471–475 (2014).
152. Luk’Yanchuk, B. et al. The Fano resonance in plasmonic nanostructures and metamaterials.
Nat Mater 9, 707–715 (2010).
153. Scharf, B., Perebeinos, V., Fabian, J. & Avouris, P. Effects of optical and surface polar
phonons on the optical conductivity of doped graphene. Phys Rev B 87, 035414 (2013).
154. Principi, A. et al. Plasmon losses due to electron-phonon scattering: The case of graphene
encapsulated in hexagonal boron nitride. Phys Rev B 90, 165408 (2014).
155. Fei, Z. et al. Infrared nanoscopy of dirac plasmons at the graphene-SiO2 interface. Nano
Lett 11, 4701–4705 (2011).
156. Mattheakis, M., Valagiannopoulos, C. A. & Kaxiras, E. Epsilon-near-zero behavior from
plasmonic Dirac point: Theory and realization using two-dimensional materials. Phys Rev
B 94, 201404 (2016).
157. Raja, A. et al. Dielectric disorder in two-dimensional materials. Nat Nanotechnol 14, 832–
837 (2019).

264
158. González, J., Guinea, F. & Vozmediano, M. A. H. Unconventional Quasiparticle Lifetime
in Graphite. Phys Rev Lett 77, 3589–3592 (1996).
159. Liu, Y. & Ruden, P. P. Temperature-dependent anisotropic charge-carrier mobility limited
by ionized impurity scattering in thin-layer black phosphorus. Phys Rev B 95, 165446
(2017).
160. Fleischer, S., Zhou, Y., Field, R. W. & Nelson, K. A. Molecular orientation and alignment
by intense single-cycle THz pulses. Phys Rev Lett 107, 163603 (2011).
161. Crawford, G. P., Eakin, J. N., Radcliffe, M. D., Callan-Jones, A. & Pelcovits, R. A. Liquidcrystal diffraction gratings using polarization holography alignment techniques. Journal of
Applied Physics 98, 123102 (2005).
162. Demos, S. G. & Alfano, R. R. Optical polarization imaging. Appl Opt 36, 150 (1997).
163. Schubert, M., Tiwald, T. E. & Herzinger, C. M. Infrared dielectric anisotropy and phonon
modes of sapphire. Phys Rev B 61, 8187–8201 (2000).
164. Ling, X. et al. Anisotropic Electron-Photon and Electron-Phonon Interactions in Black
Phosphorus. Nano Lett 16, 2260–2267 (2016).
165. Crespi, A. et al. Integrated photonic quantum gates for polarization qubits. Nat Commun 2,
566 (2011).
166. Feigenbaum, E., Diest, K. & Atwater, H. A. Unity-order index change in transparent
conducting oxides at visible frequencies. Nano Lett 10, 2111–2116 (2010).
167. Niu, S. et al. Giant optical anisotropy in a quasi-one-dimensional crystal. Nat Photon 12,
392–396 (2018).
168. Nicholls, L. H. et al. Ultrafast synthesis and switching of light polarization in nonlinear
anisotropic metamaterials. Nat Photon 11, 628–633 (2017).
169. Dorrah, A. H., Rubin, N. A., Zaidi, A., Tamagnone, M. & Capasso, F. Metasurface optics
for on-demand polarization transformations along the optical path. Nat Photon 15, 287–296
(2021).

265
170. Subkhangulov, R. R. et al. Terahertz modulation of the Faraday rotation by laser pulses via
the optical Kerr effect. Nat Photon 10, 111–114 (2016).
171. Gao, Y. et al. High-speed electro-optic modulator integrated with graphene-boron nitride
heterostructure and photonic crystal nanocavity. Nano Lett 15, 2001–2005 (2015).
172. Yang, Y. et al. Femtosecond optical polarization switching using a cadmium oxide-based
perfect absorber. Nat Photon 11, 390–395 (2017).
173. Sun, Z., Martinez, A. & Wang, F. Optical modulators with 2D layered materials. Nat Photon
10, 227–238 (2016).
174. Xia, F., Wang, H., Xiao, D., Dubey, M. & Ramasubramaniam, A. Two-dimensional
material nanophotonics. Nat Photon 8, 899–907 (2014).
175. Chernikov, A. et al. Electrical Tuning of Exciton Binding Energies in Monolayer WS2. Phys
Rev Lett 115, (2015).
176. van de Groep, J. et al. Exciton resonance tuning of an atomically thin lens. Nat Photon 14,
426–430 (2020).
177. Ling, X., Wang, H., Huang, S., Xia, F. & Dresselhaus, M. S. The renaissance of black
phosphorus. Proceedings of the National Academy of Sciences of the United States of
America 112, 4523–4530 (2015).
178. Low, T. et al. Tunable optical properties of multilayer black phosphorus thin films. Phys
Rev B 90, (2014).
179. Whitney, W. S. et al. Field Effect Optoelectronic Modulation of Quantum-Confined
Carriers in Black Phosphorus. Nano Lett 17, 78–84 (2020).
180. Sherrott, M. C. et al. Anisotropic Quantum Well Electro-Optics in Few-Layer Black
Phosphorus. Nano Lett 19, 269–276 (2019).
181. Biswas, S. et al. Tunable intraband optical conductivity and polarization-dependent epsilonnear-zero behavior in black phosphorus. Sci Adv 7, eabd4623 (2021).
182. Zhang, G. et al. Infrared fingerprints of few-layer black phosphorus. Nat Commun 8, (2017).

266
183. Çaklr, D., Sevik, C. & Peeters, F. M. Significant effect of stacking on the electronic and
optical properties of few-layer black phosphorus. Phys Rev B 92, 165406 (2015).
184. Youngblood, N., Chen, C., Koester, S. J. & Li, M. Waveguide-integrated black phosphorus
photodetector with high responsivity and low dark current. Nat Photon 9, 247–252 (2015).
185. Wu, Z. et al. Large-scale growth of few-layer two-dimensional black phosphorus. Nat
Mater 20, 1203-1209 (2021).
186. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature
2020 587:7833 587, 214–218 (2020).
187. Zhang, Z. et al. Correlated interlayer exciton insulator in heterostructures of monolayer
WSe2 and moiré WS2/WSe2. Nat Phys 2022 18:10 18, 1214–1220 (2022).
188. Smoleński, T. et al. Signatures of Wigner crystal of electrons in a monolayer semiconductor.
Nature 2021 595:7865 595, 53–57 (2021).
189. Wang, X. et al. Light-induced ferromagnetism in moiré superlattices. Nature 2022
604:7906 604, 468–473 (2022).
190. Zhu, B., Chen, X. & Cui, X. Exciton Binding Energy of Monolayer WS2. Scientific Reports
5, 1–5 (2015).
191. Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Optical spectrum of MoS2: Many-body effects
and diversity of exciton states. Phys Rev Lett 111, 216805 (2013).
192. Klots, A. R. et al. Probing excitonic states in suspended two-dimensional semiconductors
by photocurrent spectroscopy. Sci Rep 4, 1–7 (2014).
193. Ugeda, M. M. et al. Giant bandgap renormalization and excitonic effects in a monolayer
transition metal dichalcogenide semiconductor. Nat Mater 13, 1091–1095 (2014).
194. Raja, A. et al. Coulomb engineering of the bandgap and excitons in two-dimensional
materials. Nat Commun 2017 8:1 8, 1–7 (2017).
195. Xu, X., Yao, W., Xiao, D. & Heinz, T. F. Spin and pseudospins in layered transition metal
dichalcogenides. Nat Phys 2014 10:5 10, 343–350 (2014).

267
196. Xiao, D., Liu, G. bin, Feng, W., Xu, X. & Yao, W. Coupled spin and valley physics in
monolayers of MoS2 and other group-VI dichalcogenides. Phys Rev Lett 108, 196802
(2012).
197. Cao, T. et al. Valley-selective circular dichroism of monolayer molybdenum disulphide.
Nat Commun 2012 3:1 3, 1–5 (2012).
198. Diware, M. S. et al. Dielectric function, critical points, and Rydberg exciton series of WSe2
monolayer. Journal of Physics: Condensed Matter 30, 235701 (2018).
199. Chernikov, A. et al. Exciton binding energy and nonhydrogenic Rydberg series in
monolayer WS2. Phys Rev Lett 113, 076802 (2014).
200. Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Screening and many-body effects in twodimensional crystals: Monolayer MoS2. Phys Rev B 93, 235435 (2016).
201. Popert, A. et al. Optical Sensing of Fractional Quantum Hall Effect in Graphene. Nano Lett
22, 7363–7369 (2022).
202. Xu, Y. et al. Creation of moiré bands in a monolayer semiconductor by spatially periodic
dielectric screening. Nat Mater 2021 20:5 20, 645–649 (2021).
203. Xu, Y. et al. Correlated insulating states at fractional fillings of moiré superlattices. Nature
2020 587:7833 587, 214–218 (2020).
204. Walther, V., Johne, R. & Pohl, T. Giant optical nonlinearities from Rydberg excitons in
semiconductor microcavities. Nat Commun 9, 1–6 (2018).
205. Gu, J. et al. Enhanced nonlinear interaction of polaritons via excitonic Rydberg states in
monolayer WSe2. Nat Commun 12, 1–7 (2021).
206. Goldstein, T. et al. Ground and excited state exciton polarons in monolayer MoSe2. J. Chem.
Phys. 153, 071101 (2020).
207. Liu, E. et al. Exciton-polaron Rydberg states in monolayer MoSe2 and WSe2. Nat Commun
12, 1–8 (2021).
208. Arora, A. et al. Excited-State Trions in Monolayer WS2. Phys Rev Lett 123, 167401 (2019).

268
209. Han, B. et al. Exciton States in Monolayer MoSe2 and MoTe2 Probed by Upconversion
Spectroscopy. Phys Rev X 8, 031073 (2018).
210. Chen, B. et al. Environmental changes in MoTe2 excitonic dynamics by defects-activated
molecular interaction. ACS Nano 9, 5326–5332 (2015).
211. Froehlicher, G., Lorchat, E. & Berciaud, S. Direct versus indirect band gap emission and
exciton-exciton annihilation in atomically thin molybdenum ditelluride (MoTe2). Phys Rev
B 94, 085429 (2016).
212. Helmrich, S. et al. Exciton–phonon coupling in mono- and bilayer MoTe2. 2d Mater 5,
045007 (2018).
213. Ruppert, C., Aslan, O. B. & Heinz, T. F. Optical properties and band gap of single- and fewlayer MoTe2 crystals. Nano Lett 14, 6231–6236 (2014).
214. Lezama, I. G. et al. Indirect-to-Direct Band Gap Crossover in Few-Layer MoTe2. Nano Lett
15, 2336–2342 (2015).
215. Arora, A. et al. Valley Zeeman splitting and valley polarization of neutral and charged
excitons in monolayer MoTe2 at high magnetic fields. Nano Lett 16, 3624–3629 (2016).
216. Koirala, S., Mouri, S., Miyauchi, Y. & Matsuda, K. Homogeneous linewidth broadening
and exciton dephasing mechanism in MoTe2. Phys Rev B 93, 075411 (2016).
217. Wang, Y. et al. Structural phase transition in monolayer MoTe2 driven by electrostatic
doping. Nature 550, 487–491 (2017).
218. Duerloo, K. A. N., Li, Y. & Reed, E. J. Structural phase transitions in two-dimensional Moand W-dichalcogenide monolayers. Nat Commun 5, 1–9 (2014).
219. Sidler, M. et al. Fermi polaron-polaritons in charge-tunable atomically thin semiconductors.
Nat Phys 13, 255 (2017).
220. Lu, Y. J. et al. Dynamically controlled Purcell enhancement of visible spontaneous emission
in a gated plasmonic heterostructure. Nat Commun 8, 1–8 (2017).
221. Berghäuser, G. & Malic, E. Analytical approach to excitonic properties of MoS2. Phys Rev
B 89, 125309 (2014).

269
222. Qiu, D. Y., da Jornada, F. H. & Louie, S. G. Environmental Screening Effects in 2D
Materials: Renormalization of the Bandgap, Electronic Structure, and Optical Spectra of
Few-Layer Black Phosphorus. Nano Lett 17, 4706–4712 (2017).
223. Bradley, A. J. et al. Probing the Role of Interlayer Coupling and Coulomb Interactions on
Electronic Structure in Few-Layer MoSe2 Nanostructures. Nano Lett 15, 2594–2599 (2015).
224. Liang, Y. & Yang, L. Carrier plasmon induced nonlinear band gap renormalization in twodimensional semiconductors. Phys Rev Lett 114, 063001 (2015).
225. Gao, S., Liang, Y., Spataru, C. D. & Yang, L. Dynamical Excitonic Effects in Doped TwoDimensional Semiconductors. Nano Lett 16, 5568–5573 (2016).
226. Katsch, F. & Knorr, A. Excitonic theory of doping-dependent optical response in atomically
thin semiconductors. Phys Rev B 105, 045301 (2022).
227. Brem, S. et al. Intrinsic lifetime of higher excitonic states in tungsten diselenide monolayers.
Nanoscale 11, 12381–12387 (2019).
228. Palummo, M., Bernardi, M. & Grossman, J. C. Exciton radiative lifetimes in twodimensional transition metal dichalcogenides. Nano Lett 15, 2794–2800 (2015).
229. Thureja, D. et al. Electrically tunable quantum confinement of neutral excitons. Nature 606,
298–304 (2022).
230. Shanks, D. N. et al. Nanoscale Trapping of Interlayer Excitons in a 2D Semiconductor
Heterostructure. Nano Lett 21, 5641–5647 (2021).
231. Palacios-Berraquero, C. et al. Large-scale quantum-emitter arrays in atomically thin
semiconductors. Nat Commun 2017 8:1 8, 1–6 (2017).
232. Luo, Y. et al. Deterministic coupling of site-controlled quantum emitters in monolayer
WSe2 to plasmonic nanocavities. Nat Nanotechnol 13, 1137–1142 (2018).
233. Yu, L. et al. Site-Controlled Quantum Emitters in Monolayer MoSe2. Nano Lett 21, 2376–
2381 (2021).
234. Branny, A., Kumar, S., Proux, R. & Gerardot, B. D. Deterministic strain-induced arrays of
quantum emitters in a two-dimensional semiconductor. Nat Commun 8, 1–7 (2017).

270
235. Seyler, K. L. et al. Signatures of moiré-trapped valley excitons in MoSe2/WSe2
heterobilayers. Nature 567, 66–70 (2019).
236. Tran, K. et al. Evidence for moiré excitons in van der Waals heterostructures. Nature 567,
71–75 (2019).
237. Bai, Y. et al. Excitons in strain-induced one-dimensional moiré potentials at transition metal
dichalcogenide heterojunctions. Nat Mater 19, 1068–1073 (2020).
238. Baek, H. et al. Highly energy-tunable quantum light from moiré-trapped excitons. Sci Adv
6, 8526–8537 (2020).
239. Fournier, C. et al. Position-controlled quantum emitters with reproducible emission
wavelength in hexagonal boron nitride. Nat Commun 12, 1–6 (2021).
240. Klein, J. et al. Site-selectively generated photon emitters in monolayer MoS2 via local
helium ion irradiation. Nature Communications 10, 1–8 (2019).
241. Mak, K. F., Lee, C., Hone, J., Shan, J. & Heinz, T. F. Atomically thin MoS2: A new directgap semiconductor. Phys Rev Lett 105, 136805 (2010).
242. Chen, S. Y. et al. Luminescent Emission of Excited Rydberg Excitons from Monolayer
WSe2. Nano Lett 19, 2464–2471 (2019).
243. Lundt, N. et al. Optical valley Hall effect for highly valley-coherent exciton-polaritons in
an atomically thin semiconductor. Nat Nanotechnol 14, 770–775 (2019).
244. Chang, D. E., Vuletić, V. & Lukin, M. D. Quantum nonlinear optics — photon by photon.
Nat Photon 2014 8:9 8, 685–694 (2014).
245. Rizzo, D. J. et al. Inducing metallicity in graphene nanoribbons via zero-mode superlattices.
Science 369, 6511 (2020).
246. Kimouche, A. et al. Ultra-narrow metallic armchair graphene nanoribbons. Nat Commun 6,
10177 (2015).
247. Rizzo, D. J. et al. Topological band engineering of graphene nanoribbons. Nature 560,
204–208 (2018).

271
248. Son, Y. W., Cohen, M. L. & Louie, S. G. Half-metallic graphene nanoribbons. Nature 444,
347–349 (2006).
249. Joshi, P. P., Li, R., Spellberg, J. L., Liang, L. & King, S. B. Nanoimaging of the EdgeDependent Optical Polarization Anisotropy of Black Phosphorus. Nano Lett 22, 3180–3186
(2022).
250. Ribeiro, H. B. et al. Edge phonons in black phosphorus. Nat Commun 7, 12191 (2016).
251. Yin, X. et al. Edge nonlinear optics on a MoS2 atomic monolayer. Science 344, 488–490
(2014).
252. Rodin, A. S., Carvalho, A. & Castro Neto, A. H. Strain-induced gap modification in black
phosphorus. Phys Rev Lett 112, 176801 (2014).
253. Quereda, J. et al. Strong Modulation of Optical Properties in Black Phosphorus through
Strain-Engineered Rippling. Nano Lett 16, 2931–2937 (2016).
254. ÇakIr, D., Sahin, H. & Peeters, F. M. Tuning of the electronic and optical properties of
single-layer black phosphorus by strain. Phys Rev B 90, 205421 (2014).
255. Huang, S. et al. Strain-tunable van der Waals interactions in few-layer black phosphorus.
Nat Commun 10, 1–7 (2019).
256. Surrente, A. et al. Excitons in atomically thin black phosphorus. Phys Rev B 93, (2016).
257. Low, T. et al. Tunable optical properties of multilayer black phosphorus thin films. Phys
Rev B 90, (2014).
258. Guo, H., Lu, N., Dai, J., Wu, X. & Zeng, X. C. Phosphorene nanoribbons, phosphorus
nanotubes, and van der Waals multilayers. Journal of Physical Chemistry C 118, 14051–
14059 (2014).
259. Peng, X., Copple, A. & Wei, Q. Edge effects on the electronic properties of phosphorene
nanoribbons. J. Appl. Phys. 116, 144301 (2014).
260. Tran, V. & Yang, L. Scaling laws for the band gap and optical response of phosphorene
nanoribbons. Phys Rev B 89, 245407 (2014).

272
261. de Sousa, J. S. et al. Substrate effects on the exciton fine structure of black phosphorus
quantum dots. Phys Rev B 96, 035122 (2017).
262. Liang, L. et al. Electronic bandgap and edge reconstruction in phosphorene materials. Nano
Lett. 14, 6400–6406 (2014).
263. Sorkin, V. & Zhang, Y. W. The structure and elastic properties of phosphorene edges.
Nanotechnology 26, 235707 (2015).
264. Yao, F. et al. In situ TEM study of edge reconstruction and evolution in monolayer black
phosphorus. Nanoscale 13, 4133–4139 (2021).
265. Lee, Y. et al. Atomic-scale imaging of few-layer black phosphorus and its reconstructed
edge. J Phys D Appl Phys 50, (2017).
266. Masih Das, P. et al. Controlled Sculpture of Black Phosphorus Nanoribbons. ACS Nano 10,
5687–5695 (2016).
267. Liu, Y., Li, D. & Cui, T. Edge reconstructions of black phosphorene: a global search.
Nanoscale 13, 4085–4091 (2021).
268. Favron, A. et al. Photooxidation and quantum confinement effects in exfoliated black
phosphorus. Nat Mater 14, 826–832 (2015).
269. Wang, X. et al. Highly anisotropic and robust excitons in monolayer black phosphorus. Nat
Nanotechnol 10, 517–521 (2015).
270. Yoon, S. et al. Electrical control of anisotropic and tightly bound excitons in bilayer
phosphorene. Phys Rev B 103, L041407 (2021).
271. Watts, M. C. et al. Production of phosphorene nanoribbons. Nature 568, 216–220 (2019).
272. Bie, Y.-Q. et al. A MoTe2-based light-emitting diode and photodetector for silicon photonic
integrated circuits. Nat Nanotechnol 12, 1124–1129 (2017).
273. Lagoin, C. et al. Extended Bose–Hubbard model with dipolar excitons. Nature 609, 485–
489 (2022).

273
274. Andersen, T. I. et al. Moiré Excitons Correlated with Superlattice Structure in Twisted
WSe2/WSe2 Homobilayers. Nat Mat 20, 480-487 (2021).
275. Yang, X., Burch, K. & Ran, Y. Divergent bulk photovoltaic effect in Weyl semimetals.
arXiv, 1712.09363 (2017).
276. Kotov, O. v. & Lozovik, Y. E. Giant tunable nonreciprocity of light in Weyl semimetals.
Phys Rev B 98, 195446 (2018).
277. Okamura, Y. et al. Giant magneto-optical responses in magnetic Weyl semimetal Co3Sn2S2.
Nat Commun 11, 1–8 (2020).
278. Zhao, B., Guo, C., Garcia, C. A. C., Narang, P. & Fan, S. Axion-Field-Enabled
Nonreciprocal Thermal Radiation in Weyl Semimetals. Nano Lett 20, 1923–1927 (2020).
279. Osterhoudt, G. B. et al. Colossal mid-infrared bulk photovoltaic effect in a type-I Weyl
semimetal. Nat Mater 18, 471–475 (2019).
280. Huang, L. Waveguide-integrated black phosphorus photodetector for mid-infrared
applications. ACS Nano 13, 913–921 (2019).
281. Chang, T. Y. et al. Black Phosphorus Mid-Infrared Light-Emitting Diodes Integrated with
Silicon Photonic Waveguides. Nano Lett 20, 6824–6830 (2020).
282. Schmidt, P. et al. Nano-imaging of intersubband transitions in van der Waals quantum
wells. Nat Nanotechnol 13, 1035–1041 (2018).
283. Genevet, P. et al. Ultra-thin plasmonic optical vortex plate based on phase discontinuities.
Appl Phys Lett 100, (2012).
284. Wang, T. et al. Giant Valley-Polarized Rydberg Excitons in Monolayer WSe2 Revealed by
Magneto-photocurrent Spectroscopy. Nano Lett 20, 7635–7641 (2020).
285. Cho, S. et al. Phase patterning for ohmic homojunction contact in MoTe2. Science (1979)
349, 625–628 (2015).
286. Noumbé, U. N. et al. Ionic Glass–Gated 2D Material–Based Phototransistor: MoSe2 over
LaF3 as Case Study. Adv Funct Mater 29, 1907462 (2019).

274
287. Zhao, J. et al. Lithium-ion-based solid electrolyte tuning of the carrier density in graphene.
Sci Rep 6, 1–7 (2016).
288. Hayee, F. et al. Revealing multiple classes of stable quantum emitters in hexagonal boron
nitride with correlated optical and electron microscopy. Nat Mater 19, 534–539 (2020).
289. Hill, H. M. et al. Observation of excitonic rydberg states in monolayer MoS2 and WS2 by
photoluminescence excitation spectroscopy. Nano Lett 15, 2992–2997 (2015).
290. Goryca, M. et al. Revealing exciton masses and dielectric properties of monolayer
semiconductors with high magnetic fields. Nat Commun 10, 1–12 (2019).
291. Robert, C. et al. Optical spectroscopy of excited exciton states in MoS2 monolayers in van
der Waals heterostructures. Phys Rev Mater 2, 011001 (2018).
292. Goldstein, T. et al. Ground and excited state exciton polarons in monolayer MoSe2. J Chem
Phys 153, 071101 (2020).
293. Liu, E. et al. Exciton-polaron Rydberg states in monolayer MoSe2 and WSe2. Nat Commun
12, 1–8 (2021).
294. Liu, E. et al. Magnetophotoluminescence of exciton Rydberg states in monolayer WSe2.
Phys Rev B 99, 205420 (2019).
295. Wagner, K. et al. Autoionization and Dressing of Excited Excitons by Free Carriers in
Monolayer WSe2. Phys Rev Lett 125, 267401 (2020).
296. Han, B. et al. Exciton States in Monolayer MoSe2 and MoTe2 Probed by Upconversion
Spectroscopy. Phys Rev X 8, 031073 (2018).
297. Yang, J. et al. Robust Excitons and Trions in Monolayer MoTe2. ACS Nano 9, 6603–6609
(2015).
298. Arora, A. et al. Valley Zeeman splitting and valley polarization of neutral and charged
excitons in monolayer MoTe2 at high magnetic fields. Nano Lett 16, 3624–3629 (2016).
299. Xiao, K. et al. Many-Body Effect on Optical Properties of Monolayer Molybdenum
Diselenide. Journal of Physical Chemistry Letters 12, 2555–2561 (2021).

275
300. Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made
Simple. Phys Rev Lett 77, 3865 (1996).
301. Giannozzi, P. et al. Advanced capabilities for materials modelling with Quantum
ESPRESSO. Journal of Physics: Condensed Matter 29, 465901 (2017).
302. Giannozzi, P. et al. QUANTUM ESPRESSO: a modular and open-source software project
for quantum simulations of materials. Journal of Physics: Condensed Matter 21, 395502
(2009).
303. Hamann, D. R. Optimized norm-conserving Vanderbilt pseudopotentials. Phys Rev B 88,
085117 (2013).
304. Schlipf, M. & Gygi, F. Optimization algorithm for the generation of ONCV
pseudopotentials. Comput Phys Commun 196, 36–44 (2015).
305. Hedin, L. New Method for Calculating the One-Particle Green’s Function with Application
to the Electron-Gas Problem. Physical Review 139, A796 (1965).
306. Strinati, G., Mattausch, H. J. & Hanke, W. Dynamical aspects of correlation corrections in
a covalent crystal. Phys Rev B 25, 2867 (1982).
307. Hybertsen, M. S. & Louie, S. G. Electron correlation in semiconductors and insulators:
Band gaps and quasiparticle energies. Phys Rev B 34, 5390 (1986).
308. Deslippe, J., Samsonidze, G., Jain, M., Cohen, M. L. & Louie, S. G. Coulomb-hole
summations and energies for GW calculations with limited number of empty orbitals: A
modified static remainder approach. Phys Rev B Condens Matter Mater Phys 87, 165124
(2013).
309. Deslippe, J. et al. BerkeleyGW: A massively parallel computer package for the calculation
of the quasiparticle and optical properties of materials and nanostructures. Comput Phys
Commun 183, 1269–1289 (2012).
310. da Jornada, F. H., Qiu, D. Y. & Louie, S. G. Nonuniform sampling schemes of the Brillouin
zone for many-electron perturbation-theory calculations in reduced dimensionality. Phys
Rev B 95, 035109 (2017).

276
311. Ismail-Beigi, S. Truncation of periodic image interactions for confined systems. Phys Rev
B 73, 233103 (2006).
312. Deilmann, T., Drüppel, M. & Rohlfing, M. Three-particle correlation from a Many-Body
Perspective: Trions in a Carbon Nanotube. Phys Rev Lett 116, 196804 (2016).
313. Bie, Y. A MoTe2-based light-emitting diode and photodetector for silicon photonic
integrated circuits. Nat Nanotechnol 12, 1124–1129 (2017).
314. Flöry, N. et al. Waveguide-integrated van der Waals heterostructure photodetector at
telecom wavelengths with high speed and high responsivity. Nat Nanotechnol 15, 118–124
(2020).
315. Kindem, J. M. et al. Control and single-shot readout of an ion embedded in a nanophotonic
cavity. Nature 580, 201–204 (2020).
316. Biswas, S., Shao, Y., Hachisu, T., Nguyen-Dang, T. & Visell, Y. Integrated Soft
Optoelectronics for Wearable Health Monitoring. Adv Mater Technol 5, 2000347 (2020).
317. Dalir, H. & Koyama, F. 29 GHz directly modulated 980 nm vertical-cavity surface emitting
lasers with bow-tie shape transverse coupled cavity. Appl Phys Lett 103, 091109 (2013).
318. Alford, W. J., Allerman, A. A. & Raymond, T. D. High power and good beam quality at
980 nm from a vertical external-cavity surface-emitting laser. JOSA B, 19, 4, 663–666
(2002).