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Wire Array Photovoltaics
Citation
Turner-Evans, Daniel B.
(2013)
Wire Array Photovoltaics.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/8E75-WH21.
Abstract
Over the past five years, the cost of solar panels has dropped drastically and, in concert, the number of installed modules has risen exponentially. However, solar electricity is still more than twice as expensive as electricity from a natural gas plant. Fortunately, wire array solar cells have emerged as a promising technology for further lowering the cost of solar.
Si wire array solar cells are formed with a unique, low cost growth method and use 100 times less material than conventional Si cells. The wires can be embedded in a transparent, flexible polymer to create a free-standing array that can be rolled up for easy installation in a variety of form factors. Furthermore, by incorporating multijunctions into the wire morphology, higher efficiencies can be achieved while taking advantage of the unique defect relaxation pathways afforded by the 3D wire geometry.
The work in this thesis shepherded Si wires from undoped arrays to flexible, functional large area devices and laid the groundwork for multijunction wire array cells. Fabrication techniques were developed to turn intrinsic Si wires into full p-n junctions and the wires were passivated with a-Si:H and a-SiNx:H. Single wire devices yielded open circuit voltages of 600 mV and efficiencies of 9%. The arrays were then embedded in a polymer and contacted with a transparent, flexible, Ni nanoparticle and Ag nanowire top contact. The contact connected >99% of the wires in parallel and yielded flexible, substrate free solar cells featuring hundreds of thousands of wires.
Building on the success of the Si wire arrays, GaP was epitaxially grown on the material to create heterostructures for photoelectrochemistry. These cells were limited by low absorption in the GaP due to its indirect bandgap, and poor current collection due to a
diffusion length of only 80 nm. However, GaAsP on SiGe offers a superior combination of materials, and wire architectures based on these semiconductors were investigated for multijunction arrays. These devices offer potential efficiencies of 34%, as demonstrated through an analytical model and optoelectronic simulations. SiGe and Ge wires were fabricated via chemical-vapor deposition and reactive ion etching. GaAs was then grown on these substrates at the National Renewable Energy Lab and yielded ns lifetime components, as required for achieving high efficiency devices.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Solar, photovoltaic, nanowire, microwire, heterostructure, chemical vapor deposition, vapor-liquid-solid, optoelectronic modeling
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Awards:
Graduate Deans’ Award for Outstanding Community Service, 2013.
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Atwater, Harry Albert
Thesis Committee:
Atwater, Harry Albert (chair)
Schwab, Keith C.
Lewis, Nathan Saul
Painter, Oskar J.
Defense Date:
23 May 2013
Non-Caltech Author Email:
daniel.turner.evans (AT) gmail.com
Funders:
Funding Agency
Grant Number
National Science Foundation fellowship
2009063998
Department of Energy
DE-EE0005311
DARPA
UCSD funding source# 10296067, Army prime agreement number W911NF-09-2-0011
British Petroleum
UNSPECIFIED
National Science Foundation
ASU funding source# 12-729, and NSF prime award# EEC-1041895
Record Number:
CaltechTHESIS:05292013-225612828
Persistent URL:
DOI:
10.7907/8E75-WH21
Related URLs:
URL
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UNSPECIFIED
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
7769
Collection:
CaltechTHESIS
Deposited By:
Daniel Turner-Evans
Deposited On:
06 Jun 2013 22:07
Last Modified:
08 Nov 2023 00:12
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Wire Array Photovoltaics
Thesis by
Dan Turner-Evans

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

California Institute of Technology
Pasadena, California
2013

(Defended May 23, 2013)

c 2013
Dan Turner-Evans

To all of my teachers,
both in school and in life.

I am but the product of those who have invested time in me.

iii

Acknowledgements
The last five years have been a time of tremendous personal and professional growth. I have
been blessed to have been surrounded by so many learned people, both in the ways of the
world and in the ways of the lab. They have taught me how to approach an experiment
with an open mind rather than an anticipated answer, how to work through challenges with
patience and reason, and how to be critical but respectful. They are also wonderful and
inspiring people who have given me their friendship and for all these things I am eternally
grateful.
First and foremost, I have to thank Prof. Harry Atwater. When I first saw him speak
at MIT in 2006, I was drawn to him for his infectious enthusiasm and creativity. To this
day, I am impressed by his boundless knowledge and his bottomless well of ideas. He has
also been an invaluable presence in GSC soccer and basketball, and I have been glad to
have had him on my side. As I have grown, I have found father figures in all areas of my
life, and Harry is among them.
I would also like to thank the other members of my committee: Prof. Keith Schwab,
Prof. Nate Lewis, and Prof. Oskar Painter. I was fortunate enough to TA for Keith, an
experience which launched a great friendship. Keith and I have explored the mountains
of California together and somehow have yet to run out of things to talk about. He has
incredible insight into both the physical workings of the world and the behavior of mankind.
I look forward to our outdoor adventures in the years to come. I have also benefited
tremendously from Nate and Oskar’s Labs. The Lewis group has been a second home base
during my time in grad school, the source of countless engaging and inspiring scientists, and
a perennial softball rival. Oskar’s lab has been the source of an excellent roommate and of
constant inspiration.
I have also drawn inspiration from my greatest mentor and role model during graduate
school, Dr. Mike Kelzenberg. From my days as a SURF student to the present, I have
looked to Mike as a guide and have constantly been in awe of his patience, his unending
curiosity, and his ability to visualize and create even the most fantastic machines. I firmly
believe that no challenge is beyond Mike’s abilities. On top of it all, he is also a lovely
person: hilarious, caring, and thoughtful. I will forever aspire to “be like Mike.”
I would also be remiss if I didn’t acknowledge Dr. Morgan Putnam, Prof. Shannon
Boettcher, Prof. Nick Standwitz, Prof. Shane Ardo, and Prof. Adele Tamboli for their
sage counsel. Morgan contributed to much of the work in the first half of this thesis and
contributed even more in perspective and support over the years. Successively over my
graduate career, first Shannon, then Adele, then Nick, and finally Shane served as excellent
sources of scientific insight and as role models. I owe them all many more words of praise
then the above paltry sentence, but suffice it to say that they have served as consummate
examples of the professional scientist, and their leadership was, and still is, inspiring.
Moving to the younger generation, none of the work in the latter half of this thesis
would have been possible without Chris Chen and Hal Emmer. They have been excellent
colleagues and great friends, and I look forward to following their successes in the years to
come. I am also indebted to Max Bryk, Julie Jester, and Ben Lieber, SURF students who
were more than willing to while away their summers working with me and who taught me
much about mentoring.
In general, the Atwater Group is composed of an amazing group of people, and I have
benefited from the knowledge and the camaraderie of every single student, postdoc, and
iv

staff member who has passed through during my tenure. Tiffany Kimoto and Jennifer
Blankenship have been a pleasure to get to know and the source of good advice over the
years. I owe them many times over for all of the logistical drudgeries that they have helped
me through. I also have to thank Lyra Hass and April Neidholdt for their help in the same
vein early on. In terms of friendship and support, I was lucky enough to be placed in an
office with Ana Brown, Carissa Eisler, and Emily Warmann, the best group of coconspirators
around. I will miss our rapport. Dr. Anna Beck, Dr. Matt Bierman, Dr. Ryan Briggs, Dr.
Mike Deceglie, Dr. Vivian Ferry, Prof. Mike Filler, Dr. Ron Grimm, Dr. Brendan Kayes,
Emily Kosten, Dr. Liz Santori, Dr. Matt Sheldon, and Dr. Emily Warren all provided
excellent advice over the years. Also, if you ever need to make music or play volleyball or
soccer, you can find no better folk than Jeff Bosco, Dr. Victor Brar, Dennis Callahan, Dr.
Matt Escarra, Jim Fakonas, Chris Flowers, Amanda Shing, and Sam Wilson (along with
many of the people mentioned previously).
Furthermore, the facilities at Caltech are unparalleled and their greatness is due in
large part to the people who oversee them. Reginalda Montaya has been a sunny and
welcoming presence every morning in Watson. Christy Jenstad and Michelle Aldecua have
kept Applied Physics running flawlessly. Melissa Melendes, Bophan Chhim, Nils Asplund,
Dr. Guy Derose, and Mary Sikora have been a pleasure to work with in the KNI as has
Alireza Ghaffari with the Applied Physics Cleanroom. Rick Gerhart, Mike Roy, and Steve
Olson of the Chem/ChemE shops have also provided invaluable support as have Caltech
Shipping and Receiving, particularly Rick Germond.
Finally, I have to thank all of my friends and family who have kept me grounded and
happy during my time at Caltech. On campus, Jason Rabinovitch and Nick Parziale have
always helped me get out of lab and into life. Off campus, Mark Cotter has given my some
much need humanistic coloring of the world, and I will always thank him for teaching me
how to hug. I am also indebted to his family, who have taken me in and welcomed me as
one of their own. I will forever look back fondly on our Easters and Thanksgivings. And,
last but not least, I have to thank my moms and my sister for their encouragement and
support during my California adventure.
Financial support for my studies and for the work in this thesis was provided by the
National Science Foundation, the Department of Energy, DARPA, and BP. I particularly
owe my gratitude to the NSF for fellowship support.
Daniel B. Turner-Evans
May 2013
Pasadena, CA

Abstract
Over the past five years, the cost of solar panels has dropped drastically and, in concert,
the number of installed modules has risen exponentially. However, solar electricity is still
more than twice as expensive as electricity from a natural gas plant. Fortunately, wire array
solar cells have emerged as a promising technology for further lowering the cost of solar.
Si wire array solar cells are formed with a unique, low cost growth method and use 100
times less material than conventional Si cells. The wires can be embedded in a transparent,
flexible polymer to create a free-standing array that can be rolled up for easy installation
in a variety of form factors. Furthermore, by incorporating multijunctions into the wire
morphology, higher efficiencies can be achieved while taking advantage of the unique defect
relaxation pathways afforded by the 3D wire geometry.
The work in this thesis shepherded Si wires from undoped arrays to flexible, functional
large area devices and laid the groundwork for multijunction wire array cells. Fabrication
techniques were developed to turn intrinsic Si wires into full p-n junctions and the wires
were passivated with a-Si:H and a-SiNx:H. Single wire devices yielded open circuit voltages
of ∼600 mV and efficiencies of 9%. The arrays were then embedded in a polymer and
contacted with a transparent, flexible, Ni nanoparticle and Ag nanowire top contact. The
contact connected >99% of the wires in parallel and yielded flexible, substrate free solar
cells featuring hundreds of thousands of wires.
Building on the success of the Si wire arrays, GaP was epitaxially grown on the material to create heterostructures for photoelectrochemistry. These cells were limited by low
absorption in the GaP due to its indirect bandgap, and poor current collection due to a
diffusion length of only ∼80 nm. However, GaAsx P1−x on Si1−x Gex offers a superior combination of materials, and wire architectures based on these semiconductors were investigated
for multijunction arrays. These devices offer potential efficiencies of 34%, as demonstrated
through an analytical model and optoelectronic simulations. Si1−x Gex and Ge wires were
fabricated via chemical-vapor deposition and reactive ion etching. GaAs was then grown on
these substrates at the National Renewable Energy Lab and yielded ns lifetime components,
as required for achieving high efficiency devices.

Contents

List of Figures

List of Tables

List of Publications

xvi

1 Introduction
1.1

The State of Solar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1

Energy Use and The Availability of Renewables . . . . . . . . . . . .

1.1.2

The Cost of Solar . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.3

Developing Solar Technologies . . . . . . . . . . . . . . . . . . . . . .

1.2

The Solid State Physics of Photovoltaics . . . . . . . . . . . . . . . . . . . .

1.3

Theoretical Maximum Efficiencies . . . . . . . . . . . . . . . . . . . . . . . .

1.4

Wire Array Cells - Previous Work . . . . . . . . . . . . . . . . . . . . . . .

1.5

Heterostructure Multijunction Cells - Previous Work . . . . . . . . . . . . .

2 Si Wire Array Growth and p-n Junction Formation

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2

Vapor-Liquid-Solid Wire Growth . . . . . . . . . . . . . . . . . . . . . . . .

2.3

Improving Array Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1

The Importance of Catalyst Size and Patterning . . . . . . . . . . .

2.3.2

The Effects of Oxygen and Water Vapor . . . . . . . . . . . . . . . .

2.4

In Situ Doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5

Emitter Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.6

Surface Passivation with a-Si:H and a-SiNx :H . . . . . . . . . . . . . . . . .

2.7

Device Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vii

2.8

2.7.1

Diffusion Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7.2

Current-Voltage Curves . . . . . . . . . . . . . . . . . . . . . . . . .

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Flexible Arrays

3.1

Polymer Infill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Ni Nanoparticle Direct Contact . . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Ag Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

On Substrate Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4.1

Comparison to Indium Tin Oxide Contacts . . . . . . . . . . . . . .

3.4.2

Thermal Imaging of Shunts . . . . . . . . . . . . . . . . . . . . . . .

3.5

Peeled Off Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.6

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 GaP on Si

4.1

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

Device Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

Optoelectronic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3.1

GaP/Si Full Field Optical Modeling . . . . . . . . . . . . . . . . . .

4.3.2

GaAs/Si Optical Modeling . . . . . . . . . . . . . . . . . . . . . . .

4.3.3

GaP/Si Device Physics Modeling . . . . . . . . . . . . . . . . . . . .

4.3.4

AlP Window Layers . . . . . . . . . . . . . . . . . . . . . . . . . . .

Structure Growth and Characterization . . . . . . . . . . . . . . . . . . . .

4.4.1

GaP Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2

Modeling the Optical Effects of Surface Roughness . . . . . . . . . .

4.4.3

Single Wire Measurements . . . . . . . . . . . . . . . . . . . . . . . .

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

4.5

5 GaAsx P1−x on Si1−x Gex : Modeling

5.1

Motivation

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Material Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Device Architectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

Analytical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

5.5

Optical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.6

Electronic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.7

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Si1−x Gex Wire Growth

6.1

Previous Work and Overview . . . . . . . . . . . . . . . . . . . . . . . . . .

6.2

Discussion of Ge Chemistry and Catalysts . . . . . . . . . . . . . . . . . . .

6.3

Au Catalyzed Ge Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.4

Influence of HCl, BCl3 , or SiCl4 on Growth . . . . . . . . . . . . . . . . . .

6.5

Ni, In, and Cu Catalyzed Ge Growth . . . . . . . . . . . . . . . . . . . . . .

6.6

Cu Catalyzed Si1−x Gex Wire Growth

. . . . . . . . . . . . . . . . . . . . .

6.7

Reactive Ion Etched Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.8

Masking Wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.9

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 GaAs Growth on Ge

100

7.1

Previous Work and Overview . . . . . . . . . . . . . . . . . . . . . . . . . .

100

7.2

Structure Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

7.3

Material Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

7.3.1

Patterned, Planar Substrates . . . . . . . . . . . . . . . . . . . . . .

102

7.3.2

X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

7.3.3

Photoluminescence . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

7.4

Parameter Optimization for Device Design . . . . . . . . . . . . . . . . . . .

106

7.5

Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

8 Conclusion
8.1

8.2

8.3

111

Si Wire Array Solar Cells . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

8.1.1

State of the Art

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

8.1.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

Multijunction Wire Array Cells . . . . . . . . . . . . . . . . . . . . . . . . .

113

8.2.1

State of the Art

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

8.2.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

The Wide World of Wire Arrays . . . . . . . . . . . . . . . . . . . . . . . .

115

A Si Wire Array Processing Steps

116

A.1 Fabricating Arrays with p-n Junctions . . . . . . . . . . . . . . . . . . . . .

116

A.2 Creating Flexible, Contacted Large Area Arrays . . . . . . . . . . . . . . .

119

B Code for Making Rough GaP for FDTD Simulations

121

C Code for the Analytical Tandem Model

123

C.1 Code to Plot the Mie Optical Generation Profile . . . . . . . . . . . . . . .

123

C.2 Code to Generate Plots at Each Wavelength . . . . . . . . . . . . . . . . . .

125

C.3 Code to Calculate the Internal Fields of a Particle Given Mie Theory . . . .

126

C.4 Code to Plot the Beer-Lambert Optical Generation Profile . . . . . . . . . .

128

C.5 Code to Generate Subcell, Tandem J-V Curves . . . . . . . . . . . . . . . .

129

C.6 Code for the Hemispherical Subcell . . . . . . . . . . . . . . . . . . . . . . .

133

C.7 Code for the Wire Subcell . . . . . . . . . . . . . . . . . . . . . . . . . . . .

137

D Transmission Matrix Method Code

140

E Sentaurus Code

142

E.1 Code from the Tandem Simulations . . . . . . . . . . . . . . . . . . . . . . .

142

E.2 Comparison between Lumerical and Sentaurus FDTD . . . . . . . . . . . .

196

Bibliography

196

List of Figures

1.1

The cost of crystalline Si modules over time . . . . . . . . . . . . . . . . . .

1.2

A Si wire array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Overview of semiconductor bands . . . . . . . . . . . . . . . . . . . . . . . .

1.4

Ideal Si cell IV curve and AM 1.5G spectrum . . . . . . . . . . . . . . . . .

1.5

Predicted Si wire efficiency as a function of wire radius and diffusion length

2.1

Overview of the wire growth process and a Cu/Si phase diagram . . . . . .

2.2

High fidelity wire arrays in (a) square and (b) hexagonal patterns . . . . . .

2.3

Comparison of catalyst size and growth fidelity as a function of the Cu thickness (tCu ) and hole area . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4

45◦ view of a Si wire array showing typical height variation . . . . . . . . .

2.5

Morphologies of the wire tops for different cooling conditions . . . . . . . .

2.6

Localized and large area wire array defects

. . . . . . . . . . . . . . . . . .

2.7

Varying wire cross sections for different growth conditions . . . . . . . . . .

2.8

The growth reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.9

BCl3 doping density measurement . . . . . . . . . . . . . . . . . . . . . . .

2.10 “Booting” the Si wires . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.11 Experimental doping profiles and theoretical junction dependent dark current 24
2.12 Microwave reflectivity measured lifetimes of a-Si:H and a-SiNx :H passivated
wafers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.13 Cross-sectioned SiNx coated wire revealing the coating thickness variation
along the wire length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.14 EQE maps of a-Si:H and a-SiNx :H coated wires . . . . . . . . . . . . . . . .

2.15 Light I-V curves of single wire solar cells . . . . . . . . . . . . . . . . . . . .

3.1

Overview of a transparent, flexible contact for Si wire arrays . . . . . . . . .

3.2

Overview of an infilled wire array . . . . . . . . . . . . . . . . . . . . . . . .

3.3

Measuring the Si/Ni resistance with a nanoprobe . . . . . . . . . . . . . . .

3.4

Examining the Ni/Si interface . . . . . . . . . . . . . . . . . . . . . . . . . .

3.5

Optical properties of flexible, transparent top contact components . . . . .

3.6

Electronic properties of an on substrate Ni np/Ag nw embedded Si wire array
solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

Electronic properties of an on-substrate ITO/Ag nw embedded Si wire array
solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

Thermal images of a large area Si wire array

. . . . . . . . . . . . . . . . .

3.9

Performance of a peeled off wire array . . . . . . . . . . . . . . . . . . . . .

4.1

GaP/Si device overview and band diagram

. . . . . . . . . . . . . . . . . .

4.2

Isoefficiency contour plot as a function of core and shell bandgaps . . . . . .

4.3

Simulations of GaP on a Si grating . . . . . . . . . . . . . . . . . . . . . . .

4.4

Normal incidence GaP/Si and GaAs/Si wire array absorption . . . . . . . .

4.5

Comparison of Beer-Lambert and full field radial optical absorption for GaP/Si
wire cross sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.6

Comparison of Beer-Lambert (B-L) and full field optical absorption (TE and
TM) as a function of wavelength and GaP thickness for a radial cross section
of a GaP on Si wire

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.7

Properties of GaNPAs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

Simulated GaP/Si wire array light I-V characteristics . . . . . . . . . . . . .

4.9

Influence of an AlP window layer on GaP on Si cell performance . . . . . .

4.10 SEMs of GaP on a Si wire array . . . . . . . . . . . . . . . . . . . . . . . .

4.11 XRD plots of Si wires, planar GaP on Si, and GaP on Si wires . . . . . . .

4.12 Optical absorption of (a) a peeled off Si wire array and (b) a peeled off
GaP/Si wire array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.13 Simulation of Rough GaP coated Si wire arrays . . . . . . . . . . . . . . . .

4.14 Finding the GaP diffusion length . . . . . . . . . . . . . . . . . . . . . . . .

4.15 Process diagram for single wire measurements of GaP coated Si wires . . .

xii

5.1

Lattice matched material combinations overlaid on an isoefficiency contour
plot for series connected, two junction cells . . . . . . . . . . . . . . . . . .

5.2

Overview of the multijunction wire array geometries and electronic structure

5.3

Polar coordinate plots comparing Mie Theory and Beer-Lambert absorption
for the upper GaAs0.9 P0.1 cell . . . . . . . . . . . . . . . . . . . . . . . . . .

5.4

Efficiency, short circuit current density, and open circuit voltage of hemispherical GaAs0.9 P0.1 solar cell structures as a function of device radius and
diffusion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.5

Light IV curve of the shell as the outer (r = d1 + d2 ) and inner (r = 0)
surface recombination velocities are varied . . . . . . . . . . . . . . . . . . .

5.6

Light IV curves for the subcells and tandem device . . . . . . . . . . . . . .

5.7

The effect of series resistance (Rs ) on the performance of the second tandem
cell in Table 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.8

2D vs. 3D optical simulation comparison . . . . . . . . . . . . . . . . . . . .

5.9

Comparison of the absorption profiles for FDTD optical simulations run at
25 nm wavelength steps vs. 50 nm wavelength steps . . . . . . . . . . . . .

5.10 Optical properties of multijunction wire arrays . . . . . . . . . . . . . . . .

5.11 Plot of the relative absorption in the core and shell for all three structures
for varying pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.12 Cartoon depicting the loss and absorption mechanisms in a wire array tandem
cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.13 Efficiency, short circuit current density, and open circuit voltage of the tandem wire array solar cell structures as a function of the GaAs0.9 P0.1 lifetime

5.14 Shockley-Reed-Hall recombination for τn = 500 or 5 ps in the GaAs0.9 P0.1
cell for an array with a 7 µm pitch . . . . . . . . . . . . . . . . . . . . . . .

5.15 The influence of a “defective layer on device performance . . . . . . . . . .

6.1

Phase diagrams for Ge and Au or Cu . . . . . . . . . . . . . . . . . . . . . .

6.2

Ge on Si growth morphology as a function of growth temperature and GeCl4 /H2

6.3

flow rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

XRD and EDAX of Ge wires grown on Si . . . . . . . . . . . . . . . . . . .

xiii

6.4

The effects of small amounts of BCl3 , HCl, or SiCl4 on the Ge growth morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.5

Growth attempts with Au, Cu, Ni, and In catalysts . . . . . . . . . . . . . .

6.6

The influence of temperature and flow rate on Ge wires grown from Cu . .

6.7

Suggested mechanism describing the evolution of the Ge wire morphology .

6.8

XRD plot of Cu catalyzed Ge wires grown on Si . . . . . . . . . . . . . . .

6.9

Si1−x Gex wire growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6.10 Reactive ion etched Ge pillars on (111) and (100) substrates . . . . . . . . .

6.11 A Ge wire masked with PECVD SiOx . . . . . . . . . . . . . . . . . . . . .

7.1

GaAs Growth on (311) Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . .

102

7.2

SEMs of structured Ge before and after GaAs growth . . . . . . . . . . . .

103

7.3

GaAs growth on planar, oxide masked Ge . . . . . . . . . . . . . . . . . . .

104

7.4

XRD of GaAs on structured Ge . . . . . . . . . . . . . . . . . . . . . . . . .

105

7.5

PL of GaAs grown on structured Ge . . . . . . . . . . . . . . . . . . . . . .

105

7.6

TRPL of GaAs grown on structured Ge . . . . . . . . . . . . . . . . . . . .

106

7.7

Overview of the simulated structure for optimizing the doping levels for a
GaAs device grown on Ge . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.8

Device values for a GaAs hemispherical cell as a function of emitter and base
doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.9

107

108

Device values for a GaAs hemispherical cell as a function of window and
defect doping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

D.1 TMM calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

E.1 Layout of Sentaurus simulations for finding the optical generation in tandem
wire arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

E.2 Comparison of FDTD simulations from Sentaurus and Lumerical . . . . . .

197

xiv

List of Tables

1.1

U.S. average levelized costs (2011 $/MWhr) for plants entering service in 2018.

1.2

Current and anticipated total installed module costs from DOE’s $1/W chal-

lenge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3

Limiting theoretical solar conversion efficiencies . . . . . . . . . . . . . . . .

2.1

Device properties of single wire solar cells with different surface coatings . .

3.1

AM 1.5G electrical characteristics of an on-substrate Ni np/Ag nw contacted
Si wire array solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

AM 1.5G electrical characteristics of an on-substrate ITO/Ag nw contacted
Si wire array solar cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1

Light IV characteristics for subcells and tandem cell as the geometry of the
shell and wire are varied . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2

Calculated ideal (IQE=1) JSC of layers under course (50 nm wavelength step)
and fine (25 nm step) simulations . . . . . . . . . . . . . . . . . . . . . . . .

5.3

Light IV characteristics for a representative hemispherical tandem . . . . .

5.4

Device performance as a function of surface recombination velocity . . . . .

7.1

Overview of GaAs on Ge growth run . . . . . . . . . . . . . . . . . . . . . .

101

List of Publications

The work in this thesis has primarily been drawn from the following publications:
• M. D. Kelzenberg,∗ D. B. Turner-Evans,∗ M. C. Putnam, S. W. Boettcher, R. M.
Briggs, J. Y. Baek, N. S. Lewis, and H. A. Atwater. High-performance Si microwire
photovoltaics. Energy & Environmental Science, 4, 866-871, 2011.
• D. B. Turner-Evans, H. Emmer, C. T. Chen, and H. A. Atwater. Flexible, transparent
contacts for inorganic nanostructures and thin films. Advanced Materials, Accepted,
2013.
• A. C. Tamboli,∗ D. B. Turner-Evans,∗ M. Malhotra, M. D. Kelzenberg, and H. A. Atwater. GaP/Si wire array solar cells. 35th IEEE Photovoltaic Specialists Conference,
2010.
• D. B. Turner-Evans, M. D. Kelzenberg, C. T. Chen, E. C. Warmann, A. C. Tamboli,
and H. A. Atwater. Optoelectronic design of multijunction wire-array solar cells. 36th
IEEE Photovoltaic Specialists Conference, 2011.
• D. B. Turner-Evans, C. T. Chen, H. Emmer, and H. A. Atwater. Optoelectronic
analysis of multijunction wire array solar cells. Journal of Applied Physics, Accepted,
2013
Small pieces from the following publications have also been included where appropriate:
• N. C. Strandwitz, D. B. Turner-Evans, A. C. Tamboli, C. T. Chen, H. A. Atwater, and N. S. Lewis. Photoelectrochemical behavior of planar and microwire-array Si/GaP electrodes. Advanced Energy Materials, 2,
1109-1116, 2012.
• A. C. Tamboli, C. T. Chen, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, N. S. Lewis, and H. A.
Atwater. Wafer scale growth of silicon microwire arrays for photovoltaics and solar fuel generation. IEEE
Journal of Photovoltaics, 2, 294-297, 2012.
• S. W. Boettcher, E. L. Warren, M. C. Putnam, E. L. Santori, D. B. Turner-Evans, M. D. Kelzenberg, M. G.
Walter, J. R. McKone, B. S. Brunschwig, H. A. Atwater, and N. S. Lewis. Photoelectrochemical hydrogen
evolution using Si microwire arrays. Journal of the American Chemical Society 133(5), 1216-1219, 2011.
• A. C. Tamboli, M. Malhotra, G. M. Kimball, D. B. Turner-Evans, and H. A. Atwater. Conformal GaP layers
on Si wire arrays for solar energy applications. Applied Physics Letters 97(22), 221914-221913, 2010.

xvi

• M. C. Putnam, S. W. Boettcher, M. D. Kelzenberg, D. B. Turner-Evans, J. M Spurgeon, E. L. Warren, R. M.
Briggs, N. S. Lewis, and H. A. Atwater. Si microwire-array solar cells. Energy & Environmental Science 3(8),
1037-1041, 2010.
• M. D. Kelzenberg, S. W. Boettcher, J. A. Petykiewicz, D. B. Turner-Evans, M. C. Putnam, E. L. Warren, J.
M Spurgeon, R. M. Briggs, N. S. Lewis, and H. A. Atwater. Enhanced absorption and carrier collection in Si
wire arrays for photovoltaic applications. Nature Materials 9(3), 239-244, 2010.
• S. W. Boettcher, J. M. Spurgeon, M. C. Putnam, E. L. Warren, D. B. Turner-Evans, M. D. Kelzenberg, J.
R. Maiolo, H. A. Atwater, and N. S. Lewis. Energy-conversion properties of vapor-liquid-solid grown silicon
wire-array photocathodes. Science 327(5962), 185-187, 2010.
• M. C. Putnam, D. B. Turner-Evans, M. D. Kelzenberg, S. W. Boettcher, N. S. Lewis, and H. A. Atwater.
10 µm minority-carrier diffusion lengths in Si wires synthesized by Cu-catalyzed vapor-liquid-solid growth.
Applied Physics Letters 95(16), 163116-163113, 2009.
• M. D. Kelzenberg, M. C. Putnam, D. B. Turner-Evans, N. S. Lewis, and H. A. Atwater. Predicted efficiency
of Si wire array solar cells. 34th IEEE Photovoltaic Specialists Conference, 2009.
• M. D. Kelzenberg, D. B. Turner-Evans, B. M. Kayes, M. A. Filler, M. C. Putnam, N. S. Lewis, and H. A.
Atwater. Photovoltaic measurements in single-nanowire silicon solar cells. Nano Letters 8(2), 710-714, 2008.
• M. D. Kelzenberg, D. B. Turner-Evans, B. M. Kayes, M. A. Filler, M. C. Putnam, N. S. Lewis, and H. A.
Atwater Single-nanowire Si solar cells. 33rd IEEE Photovoltaic Specialists Conference, 2008.

xvii

Chapter 1
Introduction

1.1
1.1.1

The State of Solar
Energy Use and The Availability of Renewables

In 2009, the world generated 20 petawatt-hours of electricity.(1) While this ever-growing
consumption of energy has allowed for vast improvements in quality of life, it has come at
a cost. Humanity’s accelerated combustion of fossil fuels has led to ever-increasing carbon
dioxide emissions. The US alone emitted 5,444.6 Tg of CO2 in 2011.(2) These CO2 emissions
are affecting the world’s climate such that a 2◦ C global increase in temperature now seems
inevitable,(3) and an increasing number of major weather events are linked to the effects of
climate change. However, despite dire warnings from climate scientists, world governments
seem unable to agree on how to tackle the problem; developing countries feel entitled to
use as much electricity as more prosperous countries, and wealthy nations are unwilling to
lower their energy use at the risk of hurting their GDP.
One obvious solution is to switch to a form of electricity generation that emits much less
CO2 than conventional fossil fuel power plants. Hydro and nuclear power have been used
for decades, but hydro cannot be expanded much beyond current levels and nuclear seems
forever stigmatized by safety concerns. Wind energy has seen tremendous growth and is
cost competitive with coal, but has an overall limited glocal capacity of ∼2 TW.(4)
Solar energy, on the other hand, has a huge potential. If all of the sun’s energy that
falls on the earth were captured for one hour, it would provide 14 TW of continuous power
for a year, close to the 15+ TW of power that the world uses. Practically, more than 600
TW of power is available from the sun.(4) Thus, solar is an attractive energy solution for

lowering CO2 emissions.
1.1.2

The Cost of Solar

While the cost of solar generated electricity is currently more expensive than power from
coal or natural gas over much of the U.S. (see Table 1.1), the cost of photovoltaic modules
has dropped drastically over the last decade (see Figure 1.1) and is projected to continue
falling.(5) This drop has been due in large part to the rapid growth of module manufacturing,
particularly in China (financed by very low interest loans from the state), leading to benefits
from economies of scale.
Table 1.1: U.S. average levelized costs (2011 $/megawatthour) for plants entering service
in 2018.(6)
Plant Type

Capacity

Levelized

Fixed

Variable

Transmission

Total System

Factor

Capital

O&M

Investment

Levelized Cost

(%)

Cost

65.7

4.1

29.2

1.2

100.1

84.4

6.8

30.7

1.1

123

15.8

1.7

48.4

1.2

67.1

17.4

1.2

65.6

Advanced Nuclear

83.4

11.6

12.3

1.1

108.4

Geothermal

76.2

1.4

89.6

Biomass

53.2

14.3

42.3

1.2

111

Wind

70.3

13.1

3.2

86.6

Wind

193.4

22.4

5.7

221.5

Solar Photovoltaic

130.4

9.9

144.3

Solar Thermal

214.2

41.4

5.9

261.5

Hydro

78.1

4.1

6.1

90.3

Coal

(+Fuel)

(Conventional)
Coal
(Advanced)
Natural Gas
(Combined Cycle)
Natural Gas
(Adv. Combined Cycle)

(Offshore)

In the Southwest US (which gets more sun than the rest of the nation), in parts of Southern Europe, and in many developing nations with limited electrical grids, solar electricity
is already competitive with peak electricity costs.(5) These areas have seen rapid growth
in the number of solar installations in the last decade, and the overall installed generation
capacity should only continue to rise as the module price falls. As of February 2013, 100
GW of solar was installed on the global grid,(8) and a recent McKinsey report predicted a
50 fold increase in solar installations by 2050.(5)

80 70 -

Cost ($/W)

60 50 40 30 20 High poly Si prices

2013

2009

2005

2001

1997

1993

1989

1985

1977

1981

10 -

Figure 1.1: The cost of crystalline Si modules over time. The plateau in the mid 2000s is
due to a shortage of polycrystalline Si.(7)
1.1.3

Developing Solar Technologies

Further economies of scale will certainly continue to lower the cost of PV modules. However,
in order to gain significant market penetration, new technologies must be developed to make
the cost of solar electricity competitive with that of natural gas at ∼ $1/W . Table 1.2
outlines the component cost of modules as of 2010 and the anticipated relative costs in 2017
for achieving a $1/W installed module.
The high cost of 2010 modules, $1.70, is due in large part to the conventional Si cell
fabrication process. In this process, large mono or polycrystalline ingots are manufactured in
the energy intensive, time consuming Czochralski process. This process, in which large single
crystalline Si boules are slowly pulled from a melt, requires 214 MWh/ton of energy.(10)
After growth, the boules must then be cut into the desired shape and size. The cutting
process both limits the potential thickness of the cells to ∼ 150µm and causes significant
waste due to the thickness, or “kerf,” of the saw. The wafer must then undergo a number of
expensive processing steps, including high temperature emitter formation and contact firing
and plasma-enhanced chemical-vapor deposition of passivation and anti-reflective coatings,
in order to become a fully functional solar cell.

Table 1.2:

Current and anticipated total installed module costs from DOE’s $1/W

challenge.(9)
Component

2010 Cost ($/W)

2017 Goal ($/W)

PV Module - Total

1.70

0.50

PV Module - Semiconductor

0.54

PV Module - Cell Fab

0.45

PV Module - Module Packaging

0.70

Inverter

0.22

0.10

BOS/Installation

1.48

0.40

In contrast, “thin film” solar cells seek to lower module costs by both using far less
material for the cell and by using in situ processing techniques. CdTe, CuInx Ga1−x Se2 ,
and a-Si can all be formed in ∼100s of nm films on metal, glass, or plastic substrates,
and have all exceeded 10% efficiencies.(11) They are deposited directly through chemical
vapor deposition and the junctions are deposited in situ by modulating the material doping
during growth. By utilizing these techniques, First Solar has successfully commercialized
CdTe films and has captured a significant share of the market.
However, Si is still the dominant commercial technology (> 80% of commercial cells
are made with crystalline Si),(9) Si modules have been made with > 20% efficiencies (Sunpower), and Si costs have room to drop significantly if the material used to make cells can
be thinned and fabricated in an inexpensive process.
Along these lines, a few companies, notably 1366 and Solexeil, have developed innovative
kerf-free processes for making thin Si cells. 1366 pulls 200 µm wafers directly from the melt.
Solexeil creates 35 µm thick wafer by growing epitaxial Si on porous Si and then lifting the
wafers off by etching away the porous layer. Both companies have demonstrated that their
products can be incorporated directly into conventional cell fabrication lines and produce
cells of comparable efficiencies to those from standard wafers.
Si wire array solar cells offer another (potentially) low cost approach (Figure 1.2). They
are formed with a unique vapor-liquid-solid growth method and use the planar equivalent
of ∼ 3µm of material. While the wires are grown on a Si wafer, they can be embedded in
a flexible polymer and peeled off, allowing the substrate to be reused for growth.(12; 13)
Additionally, both p and n-type doping can be done during wire growth, lowering the cell

fabrication costs. Due to the extremely small amount of material needed and the ability
to draw upon decades of knowledge about Si, Si wire arrays offer an exciting avenue for
getting closer to the $1/W goal.

30 μm
Figure 1.2: A Si wire array.
However, reducing the module cost alone is not enough to reach a $1/W total cost. The
balance of systems (BOS) and installation costs of solar modules also need to be lowered.
Here as well, Si wire arrays offer an advantage. They can be embedded in a transparent,
flexible polymer to create a freestanding array that can be rolled up for easy installation in
a variety of form factors, unlike the current rigid modules.(12) The combination of reduced
fabrication costs and reduced installation costs are necessary, and Si wire array solar cells
offer solutions on both fronts.
Furthermore, the wire array concept is not limited to Si. By incorporating multijunctions
into the wire array morphology (e.g. by growing a GaAsx P1−x cell on Si1−x Gex wire cells),
higher efficiencies can be achieved. High efficiency (> 20%) solar modules benefit from a
relatively low BOS/installation cost, assuming that additional components are not needed
for tracking or concentration. Fewer modules need to be installed in order to generate a
given amount of power as compared to conventional systems.

While high efficiency modules are currently much more expensive than single junction
systems, the unique geometry of wire arrays may help to mitigate the cost. Conventional
multijunction cells are grown on expensive substrates such as Ge or GaAs, and they require
a significant amount of material to be deposited through metal-organic-chemical-vapordeposition (MOCVD), a costly growth step. Thick layers must be deposited in order to
mitigate defects that arise from growing dissimilar materials on top of one another. In
contrast, wire arrays use much less material than planar wafers, circumventing the substrate
cost. Furthermore, the outer cell layers would only be deposited on the wire tips and, due
to the wire geometry, defects due to lattice mismatch could relax radially, allowing high
quality films to grow axially and requiring less overall MOCVD deposition.
Thus, single junction Si and multijunction wire array cells have the potential to reach
the $1/W goal.

1.2

The Solid State Physics of Photovoltaics

In order to understand the development of wire array solar cells, knowledge of solid state
physics is essential. I offer a brief overview here and encourage the reader to look at
Ashcroft and Mermin (14); Ibach and Luth (15); Kittel (16); Sze and Ng. (17) for further
understanding.
Just as atoms have discrete electronic energy levels, corresponding to the periodic orbit
of electrons around the nucleus, crystals have allowed and forbidden energies, corresponding
to the movement of electrons through the periodic arrangement of atoms that make up the
crystal lattice. However, rather than having discrete levels, crystals have a density of
electronic states over the range of energies. The electrons populate these states up to a
certain level, the Fermi energy. If excited, by light for example, the electrons may move to
occupy higher energy states.
In semiconductors, the density of states is zero for a range of energies above the Fermi
energy (see Figure 1.3); electrons are forbidden from having an energy at these values. The
distance between the Fermi energy and the next allowable state is called the semiconductor’s
bandgap. The continuum of energies below the Fermi energy is called the valence band while
the range of allowed energies above the bandgap is called the conduction band. Light with
energy greater than the bandgap energy can excite an electron into the conduction band
from the valence band. The excited electron leaves behind an empty state. This empty

state behaves as a particle all of its own, with opposite charge of and motion to an electron.
Thus, it is treated as a quasiparticle and dubbed a “hole.”

Conduction Band

Bandgap
Fermi Energy

Valence Band

Energy
Density of States
Figure 1.3: Overview of semiconductor bands.
Solar photovoltaics use the excited electron and hole to generate power. Light of all
energies above the bandgap is absorbed in the material, creating a large population of energetic carriers (electrons and holes). The carriers diffuse throughout the material according
to the transport equations:
~ + qDn ∇n
J~n = qµn nE
~ − qDp ∇p
J~p = qµp pE
where n refers to the electron density, p refers to the hole density, J~ refers to the current, q
~ is the electric field, and D is the
is the fundamental electronic charge, µ is the mobility, E
diffusion coefficient.
~ in the material, one can be built in to give directionality
While there is no innate E
to the photocarrier movement. By replacing the base atoms of the semiconductor crystal

with elements that have more (less) electronic charge, the material’s chemical energy can
be increased (decreased). Material with an excess of negative charge is called n-type and
material with an excess of positive charge is called p-type. Placing p-type and n-type
material together in series creates a built-in electric field. The flow of photoexcited carriers
across the field creates a current and the chemical potential difference between the n-type
and p-type material creates a voltage, allowing power to be extracted. The electric field
can be found from Poisson’s equation.
An ideal current-voltage curve for a Si solar cell under AM1.5G conditions is shown
in Figure 1.4. AM1.5G is the spectrum from the sun after it has passed through 1.5
atmospheres (i.e. at an angle as for latitudes away from the equator and/or times of day
other than noon) and includes both direct and diffuse components. The figure also includes

(a)
JSC

VOC

(b)

1.6

Spectral Irradiance
(W m-2 nm-1)

a power curve, which can be found by multiplying the current times the voltage.

1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0

1000

2000

3000

4000

Wavelength (nm)
Figure 1.4: (a) Current-voltage and power curves for an ideal Si cell under AM1.5G illumination. (b) AM 1.5G spectrum.
A number of points on the current-voltage curve are of particular interest for understanding the cell performance. The current at zero bias is referred to as the short-circuit
current (JSC ) and depends on the number of photons that are absorbed and the number
of generated carriers that are then collected. The voltage at zero current is called the
open-circuit voltage (VOC ). The open-circuit voltage will ideally be close to the bandgap
energy, though it will be roughly 400 mV lower due to thermodynamic losses.∗ It represents

Just as semiconductors can absorb light and create charge carriers, the opposite process can happen,

with electrons and holes recombining to form a photon. This process is inevitable and leads to the voltage
loss.

the maximum electrochemical potential of the excited carriers. The ratio of the maximum
power to the product of JSC and VOC is called the fill factor (F F ). Each of these metrics is
essential to achieving a high efficiency cell and can reveal a great deal about problems and
limitations of the device performance as follows:
1. JSC can be lower than ideal if not all of the incident light is absorbed or if the
generated carriers run into defects and recombine, creating heat, before they can be
collected by the external circuit.
2. VOC is a good indicator of cell quality. Defects within the material lead to recombination which in turn lowers the electrochemical potential and thus the VOC .
3. F F often gives some insight into resistive losses within the cell. Resistance in the cell
is considered either in series, leading to a reduced slope around VOC , or as a shunt,
leading to a noticeable slope around JOC
In general, a great deal of materials science and solid state physics goes in to making a
cell with a high JSC , VOC , and F F . Specifics along these lines will be mentioned throughout
the thesis.

1.3

Theoretical Maximum Efficiencies

Before attempting to fabricate a device, a researcher must understand the theoretical performance limits in order to define realistic goals. To understand the limiting efficiencies of
solar energy converters, we will start with the Carnot cycle and describe further losses from
that point onward, generally following the treatment of Henry (18).
A Carnot engine converts heat to work through isothermal and isentropic expansion and
compression. For the sun, at a temperature of roughly 5800K and assuming the surrounding
temperature is 300K, the Carnot efficiency is:
1 − Tsurrounding /Tsun = 95%.
However, as mentioned, the Carnot cycle is isentropic. In reality, some entropy will be
gained in the photon absorption and radiative emission processes. This leads to additional
losses and the so called Landsberg limit of 93%.(19)

Furthermore, the above processes assume that the hot and cold reservoirs freely exchange
energy. However, while the converter will readily emit photons as a black body, they will
likely be scattered by the atmosphere or absorbed by the surroundings, preventing them
from reaching the sun. This loss of energy leads to an additional drop in performance,
resulting in an ∼ 86% possible conversion efficiency. This is the upper limit for a solar
energy converter.
Additional efficiency losses can come from the entropy difference between the incident
photons, which come in only around the solid angle subtended by the sun, and the emitted
photons, which can take all angles. This loss leads to ∼ 7% further drop in efficiency. By
using concentration, and thus increasing the angles of the incident photons, or by limiting
the range of emission angles from the cell, this loss can be avoided.
Next, if using a variety of semiconductors for energy conversion, the finite number of
bandgaps will lead to thermal losses. Absorbed photons with energies above the bandgap
of a cell will generate hot carriers (carriers occupying energy levels above the conduction
band edge) which will then relax back down to the edge. The excess energy will be lost as
heat. Table 1.3 lists the maximum efficiencies for three, two, and one bandgap cells.

Table 1.3: Limiting theoretical solar conversion efficiencies
Number of Cells

Spectrum

Efficiency

Black Body/AM 1.5D

47% (20)

Black Body/AM 1.5D

41% (20)

AM 1.5G

31% (21)

Finally, the efficiency of an ideal silicon cell is 29%, with additional losses due to Auger
recombination and a bandgap that is slightly off of the ideal value for the solar spectrum.(22)
In Auger recombination, an excited electron recombines and gives its excess energy to
another exciting electron (turning it into a hot electron). The doubly excited electron then
relaxes back to the band edge.
Thus, Si wire array cells will have a limiting efficiency of 29 % and multijunction wire
cells will be limited to 41 % for two materials and 47 % for three materials (assuming that
the wires do not benefit from concentration effects, which could raise the limits).

1.4

Wire Array Cells - Previous Work

Now that the limiting efficiencies have been defined, it is worthwhile to explore the state
of the art for wire array cells. The work of many excellent scientists has opened up the
possibility of even considering making photovoltaic wire arrays. Herein lies a brief survey
of a few of the important players in the history of the technology, but many more great
researchers are left unmentioned.
The growth of undoped Si and Ge wires alone has a rich history. Si “whiskers” were first
reported by Wagner and Ellis (23) in 1964. Wagner and Ellis, then at Bell Labs, grew Si
wires from a Au catalyst using a high temperature cholorosilane growth mechanism and proposed the vapor-liquid-solid (VLS) growth process that is still accepted today. Givargizov
also made important contributions to the area over the next decade, and demonstrated the
first growth of Ge wires.(24) In the modern era, Prof. Charles Lieber of Harvard popularized the growth of nanowires, beginning in the late 1990s, and Dr. Francis Ross at IBM has
done some excellent work in advancing understanding of the physics of wire growth.(25; 26)
In 2005, Kayes suggested the idea of using Si wires in photovoltaics.(27) He posited
that wire arrays would allow the optical absorption and carrier collection directions to be
decoupled, allowing for high efficiencies despite potential low lifetimes in the wires. He
further predicted efficiencies of over 10% for diffusion lengths greater than 10µm, as shown
in Figure 1.5. At the time of writing, Kayes’s paper has well over 600 citations and has
led to an explosion of research into the field of wire array photovoltaics.
The Atwater and Lewis groups have since made impressive gains in creating high efficiency Si wire array cells,(28–36) along with many other groups in the area. To briefly
mention a few, Prof. Joan Redwing’s group at Penn State has helped to advance the understanding of high temperature chlorosilane wire growth.(37; 38) Dr. Loucas Tsakalakos
at GE fabricated some of the earliest Si wire array cells.(39) Dr. Eric Garnett, formerly
at Berkeley and Stanford, now at AMOLF, has done some nice work in furthering the experimental understanding of light trapping in wire array cells.(40) Finally, Leiber’s group
demonstrated high performance single wire cells.(41)
III-V wires have also been grown and demonstrated promising performance in both
single wire and large area devices.(42; 43) To pick a few recent examples, they have been
used to make lasers on Si,(44) thresholdless coaxial lasers,(45) and GaN LEDs on amorphous

Figure 1.5: Predicted Si wire efficiency as a function of wire radius (cell thickness) and
diffusion length. Reprinted with permission from (27). Copyright 2005, American Institute
of Physics.
substrates.(46) For photovoltaics, Dr. Bernd Witzigmann at ETH outlined many important
considerations for both optical emission and absorption in wire arrays. (47) Prof. Lieber,
Prof. Ali Javey at Berkeley, Profs. Fan, McIntyre, Cui, and Harris at Stanford, Prof. Anna
Fontcuberta i Morral at EPFL, Prof. Dan Dapkus at USC, and Prof. Diana Huffaker at
UCLA have all demonstrated high performance III-V wire array cells.(48–56) Finally, a
group at Lund recently demonstrated 13.8% efficient InP cells by leveraging many of the
important design principles outlined in literature.(57; 58)

1.5

Heterostructure Multijunction Cells - Previous Work

Finally, the field of heterostructure solar cells has a long and full history, though the extension to wire morphologies is recent. A number of people at NREL, particularly Dr.
John Geisz and Dr. Dan Freidman have contributed to the growth of high efficiency triple
junction cells on Ge and once held the record for the highest efficiency tandem for a dual
metamorphic structure grown on GaAs.(59) Dr. Richard King at Spectrolab has traded
record efficiencies with the NREL group and has further contributed to growth on Ge.(60)
Solar Junction holds the current record cell with a 44% efficient triple junction stack.(11)
12

Moving to wires, a number of axial and radial III-V heterostructures have been fabricated, but the use of wires to seed low defect density, lattice-mismatched materials is far
more recent; in 2012, a group at ETH demonstrated that high quality Ge could be seeded
on RIE etched Si micropillars.(61) However, the use of masked structures to grow low defect density III-V material does have more of a precedent, with epitaxial lateral overgrown
employed in order to grow high quality GaN LEDs.(62) The lessons learned from these
experiments have been brought to bear on multijunction wire array growth and design.

Chapter 2
Si Wire Array Growth and p-n Junction Formation

2.1

Introduction

While Si wires have been grown since 1964,(23) the ability to grow highly uniform, oriented
arrays of wires is relatively recent, stemming from the work of Kayes et al. (63) in 2007.
Furthermore, while wires were traditionally grown from Au catalyst particles, Kayes et al.
(63) found that they could still achieve high fidelity wire growth by replacing the Au catalyst
with Cu despite the potential for creating Cu silicides. Au is known to have a deep level
trap in Si,(17) causing high levels of nonradiative recombination and limiting the effective
diffusion length to ∼ 2µm in the wires.(30) Cu is much more electronically benign, though
it has a high solubility level in Si.(64; 65) Additionally, the Au, or impurities in the Au, led
to unintentional n-type doping in the wires. The use of Cu, on the other hand, resulted in
both longer lifetimes and low intrinsic doping densities, as reported in Putnam et al. (33).
However, Cu is not as catalytically favorable as Au for initiating oriented wire growth.
Thus, the transition to Cu catalyst from Au spurred the need to more carefully control
the wire growth chemistry in order to consistently achieve high fidelity arrays and opened
up the possibility to add controlled in situ doping to the crystal growth. To that end,
the use of a larger volume of catalyst and the elimination of water vapor and O2 due to
air leaks allowed for consistent growth of high fidelity arrays. Furthermore, by adding dilute BCl3 to the reactor, the wires could reliably be doped p-type. These arrays achieved
3% energy-conversion efficiencies when used as photocathodes in contact with an aqueous
methyl viologen2+ electrolyte (28) and demonstrated near unity internal quantum efficiencies across the solar spectrum.(31) Additionally, the technology was demonstrated to be
scalable; high fidelity arrays were successfully grown across an entire 6 inch wafer.(36)
14

The ability to repeatedly grow controllably doped, high fidelity arrays then allowed
for the rapid development of solid state, Si wire array solar cells. In situ doped arrays
were cleaned and oxidized, and a three dimensional patterning technique was developed to
selectively expose the upper region of the wire for doping. A thin, highly doped, n-type
emitter was diffused into the exposed region to create a p-n junction. Arrays fabricated
in this manner yielded > 5% energy-conversion efficiencies under 1 sun illumination when
used as photocathodes for H2 evolution from H2 O.(29)
To further improve the performance of wire array solar cells, a-Si:H and a-SiNx :H surface passivation layers were deposited on the arrays. The wires have very large surface
area to volume ratios and thus effective surface passivation is essential to making high efficiency devices. Passivated arrays with areas of ∼0.1 mm2 demonstrated efficiencies of up
to 7.9%,(34) and single wire devices revealed VOC s approaching 600 mV and the potential
for achieving up to 17% efficient devices.(32)
This chapter explores the above solid state advances in more detail, demonstrating the
advances that transformed undoped wire arrays into passivated, p-n junction solar cells.

2.2

Vapor-Liquid-Solid Wire Growth

Silicon microwire arrays were grown through the vapor-liquid-solid process using chemical
vapor deposition, as detailed in Figure 2.1 (a). A degenerately doped, p-type, < 111 >
Si wafer with a 500 nm oxide layer was used as the growth substrate. Photoresist was
drop-cast on the oxide coated wafer and photolithographically patterned to create a series
of holes. For the bulk of the work in this thesis, the holes were 3 µm in diameter and
were arrayed in either square or hexagonal patterns with a 7 µm center-to-center spacing.
After patterning, the oxide was selectively etched at the holes with buffered hydrofluoric
acid (BHF). Finally, the substrates were coated with ∼400 nm of Cu to form an array of
catalyst particles, and the resist and excess metal were lifted off.
The wafers were then loaded into a custom built growth reactor (Figure 2.8) under
a N2 ambient. The temperature was raised to 1000◦ C under H2 at atmospheric pressure,
and the substrate was annealed for 20 min. During the anneal, the Cu catalyst particle
forms a liquid alloy with the Si substrate, as outlined in Figure 2.1 (b). SiCl4 was then
introduced to the process flow. During growth, Si diffuses into the catalyst particle, and
the particle becomes supersaturated (the dotted line in Figure 2.1 (b)), and thus begins
15

1500
1400
1300
1200
1100
1000
900
800
700
600
500
400

(b)

Lithography,
Oxide Etch,
Metal Deposition

Temperature (°C)

(a)

Anneal at
1000°C in H2
Introduce
SiCl4, BCl3

0 10 20 30 40 50 60 70 80 90 100

Figure 2.1:

at.%

(a) Overview of the wire growth process. (b) Phase diagram for Cu and

Si, overlaid with the growth process flow. The catalyst particle is supersaturated with
Si (dotted line), forcing crystalline Si to be deposited at the substrate. Reprinted with
permission of ASM International. All rights reserved. www.asminternational.org (66)
to deposit crystalline Si at the substrate interface, forming a single crystal Si wire that is
registered to the substrate and allowing the liquid alloy to return to a more energetically
favorable mixture. Under these conditions, growth in the < 111 > direction is favored, and
thus the wires grow vertically on the substrate, at roughly 5 µm per minute. When the
desired height/time was reached, the H2 and SiCl4 flow were cut off and replaced by N2 .
The growth chamber was purged for the next 20 min, and the furnace was allowed to cool
to 650◦ C. The chamber was then evacuated and refilled with N2 for unloading. High quality
growths of square and hexagonal arrays are shown in Figure 2.2.

(a)

(b)

30 μm

Figure 2.2: High fidelity wire arrays in (a) square and (b) hexagonal patterns.

2.3

Improving Array Fidelity

2.3.1

The Importance of Catalyst Size and Patterning

tCu:

350 nm

500 nm

Pattern
5 μm

Growth
50 μm

Figure 2.3: Comparison of catalyst size and growth fidelity as a function of the Cu thickness
(tCu ) and hole area.
While growing wire arrays from Au was relatively straightforward, Cu proved to be
more susceptible to the perils of processing variability. Small changes in growth conditions
or in the reactor setup led to widely varying array appearance and properties. The growth
substrate also affected wire growth. As shown in Figure 2.3, the volume of catalyst metal
had a significant effect on array fidelity, with more Cu translating to a higher probability of
seeding an oriented wire. The larger contact area between the catalyst and the wire likely
stabilizes the interface by inhibiting modifications to the surface energy by contaminants
and then promoted the growth of wires oriented to the substrate.
Larger catalyst particles also led to faster wire growth, as demonstrated in Figure 2.4.
The height distribution of Figure 2.4 is typical and is a result of catalyst size variation from
the substrate preparation. Kendrick and Redwing (67) observed this phenomenon as well
as variations in growth height with wire packing density. Additionally, wires could be made
to grow faster/taller by moving the sample farther forward in the tube (closer to the gas
source).
As a final note, the cooling rate of the reactor was found to drastically effect wire
electronic quality. Experiments by Dr. Shannon Boettcher suggested that the reactor
17

20 μm
Figure 2.4: 45◦ view of a Si wire array showing typical height variation.
should be cooled to at least 750◦ C in order to obtain high VOC s from the wires. This
process takes around 20 min and likely results in Cu (and other metals that are incorporated
within the wire during growth) diffusing to the surface rather than being quenched in the
bulk. Additionally, the morphology of the top of the wire varied depending on whether
the substrate was immediately cooled (Figure 2.5 (a)) or whether the wire was annealed at
temperature (Figure 2.5 (b)) after the growth.

(a)

(b)

5 μm

Figure 2.5: Morphologies of the wire tops for different cooling conditions. (a) An abrupt
top. (b) A top with rough epitaxial growth.

2.3.2

The Effects of Oxygen and Water Vapor

Oxygen and water vapor, unintentionally introduced through leaks, had the biggest impact
on array fidelity. Their presence caused areas of non-ideal growth either over the entire substrate or in localized regions (Figure 2.6). A leak in the reactor led to the growth shown in
Figure 2.6 (a), with nonoriented wire growth distributed amongst oriented wires. Nonoriented growth occurs more rapidly than oriented growth. Localized growth, as in Figure 2.6
(b), often occurred even if leaks were eliminated, but seemed to be mitigated by cleaning
the boat or by purging the tube with N2 for 20 min before initiating growth, suggesting that
water vapor trapped on the sample or in the boat initiated selective nonoriented growth.

(a)

(b)

50 μm

Figure 2.6: Localized and large area wire array defects. (a) A low fidelity array. (b)
A“spotting” defect.
Water vapor or oxygen likely change the surface energetics, favoring defect formation.
Hydrogen, on the other hand, passivates the surface. Hexagonal or dodecagonal crosssections result from passivated surfaces, while unpassivated surfaces lead to round wire
profiles (Figure 2.7).(68; 69)
In order to combat leaks, the growth reactor was rebuilt according to the schematics in
Figure 2.8. Outside of regular operation, the valve to the turbo pump was opened and the
system was allowed to pump down overnight. If the chamber pressure did not fall below
≈1 x 10−5 , the system had an unacceptable leak rate and the fidelity would likely be poor.
During normal operation, the system was pumped on with the roughing pump and booster
pump or vented through the KOH scrubber to purge the effluent process gas. Leaks most

(a)

(c)

(b)

1 μm

5 μm

Figure 2.7: Varying wire cross sections for different growth conditions. (a) Round wires.
(b) Dodecagonal wires. (c) Dodecagonal wires with hexagonal bases.
often originated from the Ultra-Torr seals at either end of growth tube and could be fixed
by replacing the o-rings and/or retightening the connections. Occasionally, the sources (the
bubbler of SiCl4 , the BCl3 , or the H2 ) were contaminated and had to be replaced. Filters
were added to the BCl3 and H2 lines in order to remove O2 and H2 O and seemed to help.
All in all, every effort was made to keep the reactor leak tight.

(a)
H2

(b)

Boat

MFC
Sample
GeCl4

KOH
Scrubber

Roughing
Pump

SiCl4

Turbo

BCl3

Mixer

Valve

Tube Furnace
Figure 2.8: The growth reactor. (a) Schematic of the growth reactor. (b) Picture of the
growth reactor. The tube furnace is on the left.

2.4

In Situ Doping

While Cu catalyzed wires proved more difficult to grow with high fidelity than Au catalyzed
wires, the transition to Cu led to intrinsic material and the possibility of controllably doping
the arrays. Trimethylboron and diborane have been used to grow p-type wires from silane
or disilane,(70–75), but BCl3 is favored for SiCl4 wire growth (76) and hence was selected
for its demonstrated compatibility with the array growth process. Thus, a tank of 0.02%
BCl3 in H2 was added to the reactor.
Wire array growths were then performed at a variety of BCl3 flow rates. The BCl3
improved the growth fidelity, likely through the additional Cl that it adds to the growth
process. This added Cl will change the partial pressures of the Si-H-Cl species that result
from the SiCl4 decomposition.(37) Some of these species etch Si and will thus “clean” the
wire surfaces, inhibiting defect formation. However, if the BCl3 flow was too great, the
wires grew as cones rather than as cylinders. The Cu catalyst was etched by the Cl, leading
to a decreasing catalyst size, and thus wire diameter, as the wire grew.

(b)

20 μm

Doping Density (cm−3)

(a)

10
10
BCl3 Flow Rate (sccm)

Figure 2.9: BCl3 doping density measurement. (a) Four point contacted Si wire. (b) BCl3
flow rate vs. wire doping density.
After growth, ∼1 mm2 of wires was removed from the growth substrate for single wire
contacting. The substrate was wet with a solvent (IPA or H2 O), a razor blade was scraped
over the surface of the substrate to remove small areas of wires, and the solvent/wire solution

stuck to the razor was deposited in a vial along with further solvent. The solution was then
placed on a SiNx coated Si substrate and allowed to dry. LOR 10A lift off resist and S1813
were spun on the substrate and four point contacts were photolithographically defined over
6-10 wires. 1 µm of Al and Ag was evaporated on the samples and the resist lifted off to
leave the 4 point contacts shown in Figure 2.9 (a).
Next, the four point resistance of the wires, R, was then measured and the resistivity
calculated from ρ = R·A
l , where l is the wire length and A is the wire area. The wire
dimensions were found in the SEM. The wires were assumed to be cylinders for the purpose
of calculating the area. Finally, the wire doping was estimated from standard Si doping
vs. resistivity curves.(77) Figure 2.9 (b) displays the doping as a function of BCl3 flow.
Gating of the wires by applying a potential to the Si wafer confirmed that the wires were
p-type (applying a positive bias to the substrate led to a decrease in wire conductivity).
Additionally, some of the B from the degenerately doped growth substrate likely diffuses
up into the wires to create a highly doped back surface field at the wire base.

2.5

Emitter Formation

Once repeatable, high fidelity, p-type arrays became a reality, fully solid state cells could
be envisioned. While a one step, n-type diffusion would lead to fully radial p-n junctions,
the base would be difficult to contact if the arrays were removed from the growth substrate,
and shunting to the highly doped substrate or to the back surface field would be likely.
Thus, the three-dimensional patterning process of Figure 2.10 was developed to control
the extent of the junction. The processing proceeded as follows:
1. The wire arrays were cleaned by immersing them in the following chemicals: buffered
hydrofluoric acid (BHF, Transene, Inc) for 30 sec, RCA-2 (metals clean) (1:1:6 HCl:H2 O2 :H2 O)
at 75◦ C for 10 min, BHF for 10 sec, and 60 wt% KOH at 40◦ C for 30 sec.
2. A ∼200 nm barrier oxide was grown on the wire arrays by placing them in a 1100◦ C
tube furnace with O2 flowing at 3 lpm for 1 hr and 40 min.
3. The arrays were infilled with a dilute solution of polydimethylsiloxane (PDMS) (1 g
of PDMS base, 0.1 g of PDMS curing agent, 5 g of toluene) by twice spin coating at
3000 rpm for 30 sec. They were then baked at 60◦ C for at least 30 min to crosslink
the polymer. Roughly 5 µm of the base was protected with the polymer.
22

4. The oxide from the exposed tips was removed by immersing the embedded arrays in
BHF for 5 min.
5. The PDMS was removed by placing the array in a 1:1 solution of dimethylformamide:tetran-butylammonium fluoride in tetrahydofuran for 45 min.
6. The arrays were cleaned with a piranha solution (1:3 H2 O2 :H2 SO4 ) for 10 min, dipped
in BHF for 10 sec, cleaned with RCA-2, and finally dipped in BHF again for 10 sec.
7. The arrays were doped by placing them in a solid source, phosphorous doping furnace
at 850◦ C for 10 min.
The oxide layer, referred to as the “boot,” served as a barrier, protecting the underlying
Si from the P doping. The oxide thickness was selected by doubling the values suggested
in Hamilton and Howard (78).

Grow 200 nm
Dry Oxide

Infill w/PDMS

Remove
PDMS

Etch Oxide
(BHF)

60μm

Dope n-type

10 μm

Figure 2.10: “Booting” the Si wires. (top) Schematic of the processing steps required to
create a radial pn junction wire array. (bottom) SEM demonstrating the uniformity of the
oxide “boot” across the array. (right) SEM image giving a close up of the “booted” wires.
The false coloring corresponds to the n and p-type regions indicated in the cartoon.
The emitter profile proved to have a significant impact on device performance, and thus
the doping density and layer thickness were calibrated on planar Si chips. CeP5 O14 solid
source wafers (PH 900 from Saint Gobain) were used to form the n-type layer. The chips
were placed into a tube furnace loaded with the solid source P wafers at 750, 800, and
850◦ C for 2, 5, 10, and 15 min. They were then sent to Solecon Labs where lap angle
measurements were performed to assess the junction profile. Three such profiles are seen
23

in Figure 2.11 (a). They do not exactly follow the Gaussian model found in introductory
textbooks, but they are in line with more rigurous models.(79)
In light of the given profiles and considering the dark current model of King et al. (80), 10
min and 850◦ C were chosen as the doping time and temperature, respectively. The chosen
profile was selected to reach the optimal tradeoff between doping induced degradation and
quasi Fermi level splitting. The higher the doping, the higher the splitting between quasi
Fermi levels and thus the higher potential open circuit voltage. However, the lifetime and
mobility of Si tend to fall off drastically as the doping is increased beyond ∼ 1019 cm−3 .
Thus, the dark current is minimized at ∼ 1018 cm−3 and for as shallow an emitter as
possible, as shown in Figure 2.11 (b). In order to avoid shunting, slightly higher doping
values and a 100 nm emitter were targeted. Due to depletion of the doping wafers over
time, the emitter formation process was calibrated regularly.

Carrier Concentration (cm-3)

(a) 1020

(b)

1019

850oC,
10 min

850oC,
5 min

1018
1017

800oC,
10 min

1016
1015
1014

100

125

150

Depth (nm)

Figure 2.11: (a) Planar doping profiles for a variety of temperatures and times as a function
of depth. (b) Dark current as a function of emitter thickness and surface concentration.
c 1990 IEEE. (80)
Doping through the use of spin-on dopants and a rapid thermal annealer was also
attempted,(81) but was abandoned due to the difficulty of uniformly coating the arrays
with the dopant glass.

2.6

Surface Passivation with a-Si:H and a-SiNx :H

Due to the high surface area to volume ratio of the wires, the development of an effective
surface passivation technique was deemed essential to achieving high performance devices.
Along these lines, a-Si:H and a-SiNx :H have seen extensive use over the years in the Si
processing community as surface passivation layers. a-Si:H passivation can lower the silicon
SRV to 3 cm/s,(82) and a-SiNx :H has achieved SRVs as low as 1 cm/s. However, the exact
value is heavily dependent on the substrate doping and on the injection conditions.(83; 84) aSiNx :H can also act as an effective antireflective coating for Si.(85) These coatings passivate
the Si surface in three ways:
1. They neutralize dangling bonds by attaching a H or a Si atom to each site.
2. They have a larger bandgap than the Si and a type I offset and thus reflect minority
carriers.
3. They retain ionic H, and thus a positive charge, creating a surface field that proves
effective for passivating the surface of n-type material.
The surface passivation abilities of the two materials was first tested on planar Si pieces
by depositing material on both sides of a high lifetime (∼1 ms), 400 µm thick, double side
polished, float zone wafer. The a-Si:H was deposited with plasma enhanced chemical vapor
deposition (PECVD) at 240◦ C and 500 mTorr, using 5% SiH4 in Ar at a total flow rate of
100 sccm, and a 13.56 MHz plasma at 3 W forward power. 10 nm of material was deposited
on each side. The a-SiNx:H was deposited at 350 C and 1 Torr in a parallel-plate reactor
(Plasmalab System100, Oxford Instruments), using a SiH4 /NH3 gas chemistry whose ratio
was chosen to produce films that had a refractive index of ∼ 2.0 (400 sccm 5% SiH4 in N2 ,
30 sccm NH3 ). In situ stress control was performed by alternating between a 3.56-MHz and
50-kHz plasma frequency, both with 20 W of forward power (65% RF duty cycle). ∼80 nm
of material was deposited on each side.
Microwave reflectivity measurements, seen in Figure 2.12 (a) were used to assess the
surface passivation quality of the two materials. In this technique, a pulsed laser diode is
used to excite carriers within a Si wafer. The carriers then diffuse throughout the wafer,
recombining at the surface or at any internal defect sites. Throughout the process, microwaves are reflected off of the underside of the wafer. The microwave reflectivity of the
25

wafer is directly proportional to the excess carrier concentration, and thus the reflected
microwave power will be directly proportional to the number of excited carriers. Thus, by
monitoring the transient of the reflected microwave power, as seen in Figure 2.12 (b), the
carrier lifetime may be found. If the sole source of recombination is the wafer surface, then
the SRV will equal t/2
τ where t is the wafer thickness and τ is the measured lifetime. The
a-Si:H passivated wafer was found to have a lifetime of 0.975 ms and thus a bounded SRV
of 20 cm/sec. The a-SiNx :H wafer had a lifetime of 1.07 ms and thus a bounded SRV of 19

(a)

(b)

Pulsed
Laser

Microwave
Oscillator

Detector

Microwave Reflectivity (A.U.)

cm/sec.

Circulator

0.012

a−Si:H
a−SiNx:H

0.01
0.008
0.006
0.004
0.002
−5

10
15
−3
Time (sec)
x 10

Figure 2.12: (left) Schematic of the microwave reflectivity setup. (right) Microwave reflectivity as a function of time for a-Si:H and a-SiNx :H passivated planar, float zone Si
wafers.

2.7

Device Properties

2.7.1

Diffusion Length

Given the effectiveness of the surface passivation layers on planar Si, they were next applied
to fully cleaned, p-n junction wire arrays. A 30 min deposition time of a-Si:H was chosen
to produce a ∼10 nm thick layer of nominally intrinsic material on the wire sidewalls. For
the a-SiNx :H, deposition was performed for 25 min, producing a coating that varied from
∼30 nm thick at the wire base to ∼130 nm thick at the wire tip, as observed by milling

out wire cross-sections with a focused ion beam and imaging the cuts with SEM (Figure
2.13). Two point contacts were then made to single wire devices as described in the In
Situ Doping section. After metallization, the a-Si:H-coated single-wire devices required a
30 min anneal at 275◦ C in forming gas (5% H2 in N2 ) to produce ohmic contacts through
the a-Si:H layer. For the a-SiNx :H, prior to removing wires from the growth substrate, the
arrays were partially infilled with wax (Quickstick 135, South Bay Tech.) and then etched
for 10 s in 49% BHF to remove the a-SiNx :H from the uppermost ∼10 µm of each wire,
enabling the formation of single-wire contacts.

20 μm

1 μm

Figure 2.13: Cross-sectioned SiNx coated wire revealing the coating thickness variation
along the wire length.
After verifying that the single wires behaved as photodiodes by taking a light biased
current-voltage sweep with a Keithley 246 Source Measure Unit, scanning photocurrent
microscopy (SPCM) was performed to extract the effective wire diffusion lengths, Lef f . In
SPCM, a 650 nm laser is swept over the wire while any generated photocurrent is collected
through the wire contacts and passed through a transimpedance amplifier. The output
voltage is then compared to the voltage output from a calibrated photodiode with known
external quantum efficiency (EQE).∗ This produces a spatially resolved map of minoritycarrier collection within the single-wire solar cell.

EQE is the number of cariers collected as photocurrent per single incident photon.

For unpassivated wires, relatively uniform carrier collection was observed throughout the
radial portion of the wires, but no carrier collection was observed from the axial portion. In
fact, the abrupt spatial transition between the two collection regimes could not be resolved
by the ∼0.5 µm diameter beam spot of the illumination source, indicating that Lef f was
<0.5 µm for the as-fabricated Si wires. In comparison, the SPCM profile of a typical a-Si:Hcoated single-wire solar cell indicated axial-region carrier collection with a characteristic
decay length of ∼10 µm, as shown in Figure 2.14 (a), indicating a surface recombination
velocity <450 cm s−1 , as calculated by following the treatment in the supplementary info
of Allen et al. (86).
The SPCM profile of a typical a-SiNx :H-coated single-wire solar cell (Figure 2.14 (b))
exhibited high carrier collection efficiency throughout the entire axial portion of the wire,
with no apparent decay length. Furthermore, the EQE of the a-SiNx :H-coated devices
was markedly higher than that of the noncoated devices, due to the anti-reflective nature
of the nitride coating. In fact, the EQE was usually greatest within the axial portion of
these wires, because the tapering thickness of the a-SiNx :H in this region yielded a nearly
optimal antireflective coating at the excitation wavelength. While the long Lef f of the aSiNx wire was originally interpreted as stemming from the combination of very low surface
recombination velocities and diffusion lengths significantly greater than 30 µm, further
experiments have suggested that the nitride is instead inverting the p-type region, leading
to an effective radial junction along the length of the wire. If given a negative charge with a
charge gun, the collection in the nitride coated p-type region drops to zero. This inversion
could be exploited in devices; wire arrays could be made with minimal diffused junctions,
instead relying on an inverting layer to separate carriers throughout most of the length.

(a)

(b)

Figure 2.14: EQE maps of (a) a-Si:H and (b) a-SiNx :H coated wires. The emitter lies to
the left of the white arrow.
Overall, the effective diffusion length in the a-Si:H sample is still presumed to be an
accurate reflection of the carrier decay, and may be due to the internal diffusion length, to

Table 2.1: Device properties of single wire solar cells with different surface coatings. The
champion cells are listed in bold.
Wire

Lef f

VOC

JSC

coating

(µm)

(cm s−1 )

(%)

(mV)

(mA cm−2 )

(%)

Original

< 0.5

> 4x105

4.6

451

1.5-4.6

390-496

6.9-16

58-81

7.4

564

3.6-7.4

561-595

7.8-17

77-82

9.0

535

4.8-9.0

462-543

17-26

56-78

(N = 12)
a-Si:H

5-10

450-600

(N = 20)
a-SiNx :H

(N = 13)

the quality of the surface passivation, or to the a-Si:H layer getting thinner down the length
of the wire. As Lef f was also measured as ∼10 µm in Putnam et al. (33) and lifetimes of
tens of nanoseconds have been measured in bulk suspensions of Si wires, the Si diffusion
length is assumed to be 10 µm.
2.7.2

Current-Voltage Curves

The solar power generating capacity of single wire cells was also measured, allowing us to
report the highest open circuit voltages (VOC ), fill factors (FF), and apparent photovoltaic
efficiencies (η) to date for VLS-grown Si wire solar cells, as summarized in Table 2.1.
To improve the absorption of incident sunlight, all devices were fabricated on reflective
substrates consisting of Si wafers that had been coated with 100 nm of evaporated Ag
(to provide high reflectivity) and ∼300 nm of PECVD SiNx (to prevent the contacts from
shorting). Full field, finite difference optical simulations suggest that using reflective Ag
substrates enabled 17 − 22% greater JSC s than would be possible using the SiNx -coated Si
substrates of prior studies.(32) Figure 2.15 plots the current density vs. voltage behavior
of the most efficient device of each surface coating type.
Following the convention of prior single-wire solar cell studies,(30; 41; 87) current density was determined by normalizing the device current by the total non-shaded physical
area of each wire (including both the axial and radial regions and the surface coating thickness). Note, however, that the wave nature of light and the photonic dimensions of microand nanowires enable them to interact with (and potentially absorb) more sunlight than
29

Figure 2.15: (left) AM 1.5G I-V curves for as fabricated, a-Si:H, and a-SiNx :H coated single
wire solar cells. (right) Schematic of the measurement setup.
predicted by their physical area, from a classical ray-optics perspective. This ill-defined
absorption area prevents a true definition of photovoltaic efficiency for single-wire devices,
and results in an apparent EQE exceeding 100% at certain wavelengths for some of the
devices. Nonetheless, for microwires of the diameter range studied herein (1.2 − 1.8µm),
numerical simulations suggest that minimal systematic error (< 4% relative overstatement
of JSC ) is introduced by normalizing the photovoltaic performance of the champion devices
to their physical area.
Comparing the PECVD coatings, the reduced reflectivity of the a-SiNx :H-coated devices
consistently yielded the highest short-circuit current densities (up to 26 mA cm2 ), and
resulted in the device with the greatest apparent photovoltaic efficiency (η = 9.0%) due
to the full collection across the wire length from the inversion layer. The a-Si:H had the
highest VOC as it effectively passivated the surface while the a-SiNx :H simply inverted it.
Finally, the absorption properties of wire arrays from Kelzenberg et al. (31) were used
to calculate the photocurrent that would be expected to pass through a single wire in an
entire, upright array. The illumination on a single, a-Si:H passivated, two point contacted
wire was then increased until the photocurrent reached this expected value. The measured
current-voltage curve suggested that an upright efficiency of 17% would have been achieved
30

for an array made of these wires.

2.8

Summary and Outlook

In summary, the realization of repeatable, high fidelity growth from Cu catalyst opened the
door for the development of high quality p-n junctions within the wires. The addition of
surface passivation layers allowed us to fabricate the highest efficiency Si wire solar cells to
date, with 9% single wire efficiencies and projected 17% array efficiencies. Due in part to
these promising results, Caelux Inc. sought to leverage the technology from 2010 onward
in order to make > 10% efficient, low cost solar modules.
Additionally, the electrical measurements, in particular the diffusion length data, suggest many ripe areas for further investigation. The lifetimes of the VLS grown Si are
currently low, likely stemming from impurities incorporated during the growth process or
stacking faults and dislocations. TEMs of wire cross sections have suggested that the substrate/wire interface is particularly prone to defective regions and additional defects may
reside throughout the wire. Varying the growth temperature to slow the wire growth rate
and annealing the wires for extended periods of time at higher temperatures may help to
mitigate these defects. Gettering the Si, through slow cools or the growth of oxides, doped or
otherwise, may also help, leeching impurities out of the bulk so that they can be removed at
the surface.(88) Mass spectroscopy of wires to determine the primary contaminants would
also be invaluable. Finally, placing Schottky contacts at various points along a cleaned,
passivated wire could allow position dependent diffusion lengths to be measured to see if
different regions of the wire are more or less recombination active.(89)
The emitter profile could also be further optimized. If the diffusion lengths can be increased to ∼100 µm, then axial junctions should increase the VOC by limiting dark current
due to junction area. In this case, the junction should be located at the bottom of the
cell, in order to limit absorption in the highly doped emitter. Also, the junction doping
density and depth have not been fully optimized, leaving further room for device improvement. Shallower, less highly doped junctions with well passivated surfaces should enhance
performance. Finally, while the arrays are believed to have an innate back surface field
due to diffusion of dopants from the wafer, this aspect of the devices has not been well
characterized.
As for the surface passivation layers, the a-Si:H and a-SiNx :H could be combined to
31

create a layer that effectively passivates the surface as well as providing an anti-reflective
coating. Atomic layer deposition of passivation layers should also be explored. Finally, the
inverting properties of the nitride could be used to minimize the p-n junction, streamlining
the fabrication process.
Overall, the potential of Si wire solar cells was demonstrated, but room exists for further
improvements.

Chapter 3
Flexible Arrays

After demonstrating the potential for 17% array efficiencies in single wire devices,(32) the
remaining hurdle to making an efficient, large area device was to fabricate a flexible, transparent top contact. While Spurgeon et al. (35) fabricated substrate-free, flexible, PDMS
embedded wire array photoelectrodes with metal back contacts, these electrodes relied on
the liquid to make a top contact to the devices. In contrast, thin films and nanostructures
often use conductive oxide contacts. However, these materials are brittle and fracture if
bent to a radius beyond 1 cm.(90) Thus, a new contacting scheme was needed for polymer
embedded wire arrays.
Silver nanowire (Ag nw) contacts have attracted the attention of the material science
community for their high transmission and low resistivity (∼ 80% across the visible spectrum
with a sheet resistance of ∼ 20 Ω/), flexibility, and ease of processing.(91–93) While a
number of organic solar cells and organic light emitting diodes have incorporated Ag nw
contacts,(94; 94–97) their application to inorganic devices has been far more limited;(92; 98;
99) depositing Ag nws directly on inorganic semiconductors does not lead to low barriers
contacts.
However, by depositing metal nanoparticles on the semiconductor surface to create localized contacts and then incorporating Ag nws to bridge the devices, a flexible, transparent
top contacting scheme for polymer embedded Si wires was formed. In particular, combining
electroless nickel deposition with dropcast Ag nws, as in Figure 3.1, led to the creation
of flexible, robust, transparent contacts and the demonstration of a monolithic array of
100,000s of single wire solar cells connected in parallel with an overall series resistance of
14.0 Ω cm2 , a fill factor of 55.5%, and a Si wire contact yield of > 99%.

(a)

(c)

1 μm

+ Ni dep.

(b)

5 μm

10 μm

(1,-2,1)
(1,-1,0)
(2,-1,-2)

Figure 3.1: Overview of the two stage top contacting scheme. (a) Ni nanoparticles are
deposited on the Si wires through electroless deposition. (b) Ag nanowires are drop cast
on the array to create a continuous top contact. (c) By varying the deposition time and
temperature, the Ni nps can be limited to coat the wire edges alone. (left) SEM of the
array before and after Ni np deposition. (right) Schematic of the wire crystal morphology
and Ni np sites.

3.1

Polymer Infill

The first step in creating a flexible wire array solar cell was to develop a reliable polymer
infilling process. The polymer preserves the wire array fidelity and offers mechanical robustness and flexibility after the wires are removed from the substrate. To this end, the method
of Plass et al. (12) was modified to embed the wire array in silicone polymer (Dow Corning
93-500 Space Grade Encapsulant), leaving ∼5 µm of the wire tips exposed for contacting.
While Plass et al. (12) used poly(dimethlysiloxane) (PDMS) to infill the wires, PDMS was
found to repel the electroless Ni solution and hence the use of 93-500, which tolerated the
Ni solution and was otherwise functionally equivalent to the PDMS.
When embedding the wires, a 10:1:15 w/w/w ratio of Dow Corning 93-500 Space Grade
Encapsulant base, 93-500 Space Grade Encapsulant curing agent, and toluene was mixed
to create a dilute film that was drop cast on the wires at 3000 rpm for 30 sec multiple times
until the wires were completely covered with polymer. The infilled polymer surface was
then covered with toluene which was rapidly spun off at 3000 rpm to leave ∼5 µm of the
tips exposed. The film was cured at 60◦ C for 10 hours.
While the average height of individual wire arrays varied from 40-80 µm, depending
on growth time, and the individual wires showed ∼5 µm height variation due to slight
34

differences in catalyst size, the polymer infill was still reliably uniform over the bulk of the
sample. Figure 3.2 shows a large area overview of an infilled wire array. However, as is
typical for a drop cast solution, the corners and edges of the array experienced a polymer
buildup and hence led to small inactive areas.

150 μm
Figure 3.2: Overview of an infilled wire array.

3.2

Ni Nanoparticle Direct Contact

After embedding the arrays, the exposed wire tips were coated with ohmic nickel nanoparticle (Ni np) contacts. These nps were formed on the Si wires by immersing the Si in an
aqueous solution of nickel chloride, sodium hypophosphite, and sodium succinate (Nickelex,
Transene, Inc). The hypophosphite is oxidized at the semiconductor surface, allowing the
Ni2+ ions to scavenge the resulting electrons and to nucleate on the semiconductor. The
temperature of the solution and the deposition time were varied in order to optimize the
Ni np coverage. At 80◦ C, the oxidation occurred selectively at the wire edges, as seen in
Figure 3.1. The edge sites likely catalyze the oxidation reaction. By limiting the deposition

time to 30 sec under these conditions, Ni nps could be made to line the edges of the wires
without obstructing the wire tops or sidewalls, an important advantage for coupling light
into the Si wire solar cells.

(b)

(a)

5 μm

100 μm
(d) 1

(c)

Current (µA)

0.8
−1

0.5
−0.5

0.6
0.4
0.2

40
60
Time (s)

−1

Voltage (V)

Figure 3.3: Measuring the Si/Ni resistance with a nanoprobe. (a) An array of embedded
wires fully coated with Ni nanoparticles. (b) The nanoprobe making contact to a small
group of nanowires. (c) Current vs. time for an applied bias of 1V as the nanoprobe is
moved into and out of the wire array. The current plateaus correspond to a finite number
of wires in contact with the probe. (d) Current vs. voltage curves as measured by the
nanoprobe. The top two curves have five wires contacted while the middle curve has four
in contact and the bottom curve has three wires in contact.

In order to measure the series resistance of the Ni/Si contact, polymer infilled Si wires
with diffused p-n diodes were fully coated with Ni nps by immersing them in the electroless
Ni solution for 75 sec, as seen in Figure 3.3 (a). Ga/In was scribed into the wafer for a back
contact and groups of wires were contacted with a nanoprobe (miBot Micromanipulators,
Imina Technologies). The probe was moved to approach the wires by eye under an optical
microscope (Figure 3.3 (b)). When the probe was deemed to be sufficiently close to the
36

array, a bias of 1 V was applied, and the current was measured as a function of time as
the nanoprobe was slowly moved towards the sample in ∼100 nm steps. A jump in current
could be seen each time an additional wire was contacted, as shown in Figure 3.3 (c), where
the probe was repeatedly moved into and out of the array. After confirming that the device
current and hence the contact were stable, the probe voltage was swept from -1 to 1 V and
the current was recorded, leading to the traces seen in Figure 3.3 (d). The number of wires
that were contacted was estimated by counting the number of discrete steps in the current
vs. time curves needed to achieve the current seen at 1 V in the current-voltage plot. The
series resistance of the devices was then calculated by performing a linear fit to the 0.6
to 1 V range. Given this resistance, the number of wires estimated to be in contact, and
the spacing between wires (7 µm), the Ni/Si contribution to the wire sheet resistance was
estimated to be:
Rsheet = Rmeasured ∗ #of wires ∗ (7µm)2 = 1.38 ± 0.03Ω cm2 .
After measuring the sheet resistance, single wires were removed from the growth substrate and dispersed on a heavily doped (ρ = 0.02 Ωcm) Si substrate. The end of a wire
was then removed with a focused ion beam (FIB) (FEI, Nova 600) at 30 kV and 30 pA as
seen in Figure 3.4 in order to investigate the Ni/Si interface. The Ni appears to make
continuous contact to the Si with no apparent interface layer, though TEM will be needed
to confirm.

3.3

Ag Nanowires

Immediately after plating Ni nps onto the wires, Ag nws (Cambrios ClearOhmT M ) were
dropcast onto the embedded array to form an interconnected top contact. Xu and Zhu (93)
demonstrated conductive and stretchable Ag nw networks by embedding the Ag nws in the
top layer of a film of PDMS, and our approach is thus analogous to their technique. As
seen in Figure 3.1, the Ag nws wrap around the Si wires, forming good mechanical contact
to the Ni nps. Optical properties of the silicone polymer on glass and Ag nws on polymer
on glass can be seen in Figure 3.5. Transmission and reflection values were obtained
with integrating sphere measurements, and the remainder of the light was assumed to be
absorbed. The Ag nws absorb mildly in the blue (< 10% at 400 nm) near their plasmon
37

(b)

(a)

1 μm

30 μm
(d)

(c)

50 nm

500 nm

Figure 3.4: Examining the Ni/Si interface. (a) SEM of a Ni coated wire on a Si substrate.
The area surrounded by the dotted box is magnified in the next panel. (b) Close up of the
wire tip. The dotted line signifies the extent of the FIB cut. (c) The wire after being cut.
The area within the dotted box is magnified in the next panel. (d) Close up of the Ni/Si
interface.

resonance,(100) but otherwise transmit broadly across the spectrum. Reflection losses come
from the polymer/air/glass index contrasts. The Ag nw dispersion on polymer on glass had
a sheet resistance of 10 Ω/, as measured with a four point probe.

(a) 1

(b) 1

Transmission

0.8

0.8 Polymer + Glass

Polymer + Si μwires

polymer

0.6
0.4
0.2

400

0.6

Polymer + Glass
+ Ag nws
Ag nws

glass

Absorption

Si μwires

0.4 Polymer + Si μwires
+ Ag nws

0.2

Reflection
Absorption
600

800

Wavelength (nm)

400

1000

Reflection
600

800

Wavelength (nm)

1000

Figure 3.5: Optical properties of the contact components. (a) Transmission, reflection, and
absorption for silicone polymer coated glass slides and Ag nw coated polymer covered glass
slides. (b) Transmission and reflection for polymer embedded Si wire arrays and Ni np and
Ag nw contacted polymer embedded arrays.
The optical properties of polymer embedded Si wire arrays with and without the Ni np
and Ag nw contact layer are also contained in Figure 3.5. These arrays were removed from
the Si wafer growth substrate by mechanical force with a razor blade, leaving free-standing
polymer/wire films. The films were placed on a glass coverslip to keep them flat during
integrating sphere measurement. While the measured absorption of the Si wire/Ni np/Ag
nw structures are a convolution of absorption from all three of the materials, the absorption
profile looks identical to the profile of the Si wires alone, suggesting that the Ag and Ni
do not have a large impact on the overall device behavior. The absorption falls by ∼ 10%
across the spectrum due to reflection losses introduced by the Ni nps and Ag nws. The Ag
nw films are denser on the Si arrays than on the planar polymer coated substrate due to
their three-dimensional morphology, leading to higher reflection losses.

3.4

On Substrate Performance

In parallel to optical measurements, the current-voltage behavior of wires still on the Si
substrate was measured. The wires at the substrate edge were mechanically removed to
minimize shunting to the substrate through Ag nws at the edges, and Ga/In was scratched
into the wafer to form a back contact. The electrical characteristics of an exemplary sample
are shown in Figure 3.6. For the purpose of calculating the current density and the
series resistance, light beam induced current (LBIC) maps of the device were fed into
image analysis software (ImageJ) to set the active area perimeter and to calculate the area.
Internal inactive areas (e.g. the small dark spots seen in Figure 3.6 ) were included in the
total area. The lower right corner of the device is not contacted as the dropcast polymer
completely covers the Si wires in that region. The characteristics of the device are outlined
in Table 3.1.
Table 3.1: AM 1.5G electrical characteristics of an on-substrate, Ni np/Ag nw contacted Si
wire array solar cell.
Area

JSC

VOC

Fill Factor

Efficiency

Rshunt

Rseries

(cm2 )

(mA/cm2 )

(V)

(%)

(Ωcm2 )

0.103

10.0

0.505

55.5

2.80

445

14.0

The series resistance (Rseries ) was calculated from the slope at open circuit voltage and
and the shunt resistance (Rshunt ) was calculated from the slope at short circuit. Though the
overall conversion efficiency is low, at 2.8%, the VOC and JSC can be improved by adding
surface passivation and Al2 O3 scattering particles to the wires. In contrast, the fill factor
(55.5%) and series resistance (14.0 Ω cm2 ) are direct results of the Ni np/Ag nw contacting
scheme. The series resistance is likely dominated by the Ni np/Si wire contact. While this
resistance was found to be 1.38 Ω cm2 for wires fully coated with Ni, the measured device
has ∼ 10 times fewer Ni nanoparticles, leading to the higher resistance. In contrast, the Ag
nws contribute at most ∼ 1 Ω cm2 to the overall series resistance (10 Ω/ over ∼ 0.1 cm2 )
and the resistance due to transport through the emitter (doped ∼ 1 x 1019 cm−3 ) and base
are negligible. The Ag nw/Ni np contact may also be a bottleneck. Optimization of the Ni
nanoparticle coverage or annealing the devices may lower the sheet resistance and increase

the fill factor.

(a)

(b) IPCE @ 488 nm
60%
48%
36%
24%
12%
0%

1 mm

Contacte

(c)

Wires

Figure 3.6: Electronic properties of an on substrate Ni np/Ag nw embedded Si wire array
solar cell. (a) AM 1.5G current-voltage curve. (b) Incident photon to current efficiency
(IPCE) profile map at 488 nm showing the extent of carrier collection. The IPCE map is
overlaid on a reflected confocal image. The cartoon demonstrates the contacted area of the
device. The Si wires in the lower right corner of the device are completely covered with
polymer and hence are not contacted. (c) IPCE across the AM1.5G spectrum.

The incident photon to conversion efficiency (IPCE) curve of Figure 3.6 has a similar
shape to the absorption profiles of Figure 3.5, suggesting that most of the aforementioned
absorption can, in fact, be attributed to the Si wires. The difference between the IPCE
and absorption curves is likely due to the finite diffusion length within the wires which will
lead to recombination losses rather than current collection. Additionally, the substrate (not
present in the integrating sphere measurements), will absorb some of the light, though it is
degenerately doped and thus will not appreciably contribute to the IPCE as any generated
carriers will rapidly recombine. Also of note, despite the evident scratch on the upper left
hand corner of the sample in Figure 3.6, the shunt resistance is large and the rest of the
device collects photocurrent uniformly. In contacting 100,000s of Si nanowires all in parallel,
the loss of individual devices does not substantially decrease the overall device performance,
in contrast to a defective area in a monolithic cell.
3.4.1

Comparison to Indium Tin Oxide Contacts

To compare the Ni np/Ag nw contacting scheme to a more conventional contact for thin
film solar cells, 20 nm of Indium Tin Oxide (ITO) (ρ ≈7 x 10−4 Ω cm) was sputtered onto
41

an infilled wire array. The ITO was sputtered onto the wires at room temperature with RF
magnetron sputtering under a 1.29% O2 , 98.71% Ar plasma at 3 mTorr and 200 W of power.
The optical properties of the ITO may be found in the supplementary information of (34).
The ITO readily fractured in the polymer infilled area between Si wires and hence Ag nws
were spun onto the ITO coated array in order to make contacts over large areas, with the
ITO serving to make localized contact to the Si. The cell had comparable performance to
the device in Figure 3.6, as shown in Table 3.2 and Figure 3.7 with an efficiency of 2.51%,
a fill factor of 54.6%, and a series resistance of 9.27 Ω cm−2 . However, in some regions the
Ag nws fractured along with the ITO, leading to inactive areas, and thus the ITO based
contact proved to be less robust than the Ni np scheme and hence was not pursued further.

Table 3.2: AM 1.5G electrical characteristics of an on-substrate, ITO/Ag nw contacted Si
wire array solar cell.
JSC

VOC

Fill Factor

Efficiency

Rshunt

Rseries

(cm2 )

(mA/cm2 )

(V)

(%)

(Ωcm2 )

0.2658

10.5

0.437

54.6

2.51

448

(a)

Current Density (mA/cm 2)

Area

9.27

(b)

12
10

60%

48%

36%

24%

12%
0.1

0.2
0.3
Voltage (V)

0.4

0.5

2 mm

Figure 3.7: Electronic properties of an on-substrate ITO/Ag nw embedded Si wire array solar cell. (a) AM 1.5G current-voltage power curve. (b) Incident photon to current efficiency
(IPCE) profile map at 488 nm showing the extent of carrier collection.

3.4.2

Thermal Imaging of Shunts

Shunts in solar cells and heat transfer in wires can be visualized through thermal imaging.(101;
102) Under reverse bias, current preferentially flows through the shunt pathways, causing
those areas to heat up. To image shunt pathways in a large area wire array cell, the cell
was biased and imaged with a FLIR ThermaCAM S6000, resulting in the images shown
in Figure 3.8. Clear temperature changes and local hot spots were visible, but, unfortunately, the resolution of the camera was not sufficient to make out individual wires. The
wires measured in Figure 3.8 were from an early device by Dr. Brendan Kayes. Ag nw/Ni
np contacted cells were never imaged due to the limitations encountered when attempting
to measure Dr. Kayes’ cells.

-2 V

500 μm

Figure 3.8: Thermal images of a large area Si wire array. (left) Cell at 0 V forward bias.
The average temperature is 23.3-25.3◦ C. (right) Cell at -2 V bias. The average temperature
is 45.9-49.9◦ C.

3.5

Peeled Off Cells

Finally, polymer embedded, Ni np/Ag nw contacted Si wire arrays were peeled off of the
growth substrate with a razor blade and Au was evaporated on the back to make a free
standing, flexible device. A Ag busbar was also added to aid in making contact with an
electrical probe. Figure 3.9 shows LBIC maps and current-voltage curves of the device on
and off of the growth substrate. The photoresponse of individual wires can be made out
in the magnified region. The ratio of individual wires contacted (as evidenced by a ∼ 3µm
diameter bright spot in the LBIC map) to the total number of wires in the device (1 per
7 x 7 µm area) yielded a > 99% contact yield. The wire array electrical performance was
43

measured both on substrate and after peel-off, and the performance per contacted area is
virtually identical before and after. However, the electrical properties are inferior to the
device shown in Figure 3.6 due to the lower quality of the Si wire arrays used for this sample.
The peeled off curve in Figure 3.9 was measured by flattening the wire array; it naturally
rolled into a cylinder with a ∼1 mm diameter due to strain built up during the polymer
curing process. This insensitivity to rolling and unrolling alludes to the robustness and
flexiblity of the contact. Figure 3.9 shows the cell wrapped around a pencil with radius of
3.61 mm. Attempts to clamp the end of the peeled off film in order to controllably alter the
radius of curvature resulted in the film tearing; the ∼ 100µm thick films yielded under the
shear stress. Further encapsulation of the arrays will be needed to measure the performance
as a function of radius of curvature and number of cycles of rolling and unrolling.

(a) On Substrate

Peeled Off
Evaporate
back contact

(b)

Remove
wires w/razor

60%
48%
36%
24%
12%
2 mm

100 μm

Figure 3.9: Performance of a peeled off wire array. (a) (above) Diagram showing the peel
off and back contacting of wire arrays embedded in polymer. (below) IPCE map at 488 nm
for a cell on and off the substrate showing the high contact yield both before and after the
peel off step. (b) AM 1.5G current-voltage curve for the cell on substrate and peeled off.
(c) Peeled off cell wrapped around a pencil demonstrating the device flexibility.

3.6

Summary and Outlook

In conclusion, a two step contacting process of Ni nps and Ag nws led to flexible, transparent
contacts for use with Si wire arrays and other inorganic materials. Flexible Si wire array
solar cells had conversion efficiencies of up to 2.80%, fill factors of up to 55.5%, and series
resistances of 14.0 Ω cm2 . Optimization of the Ni np coverage and attempts to anneal
the contacts or to “plasmonically weld” (100) them may improve the cell performance.
In this “welding” process, the contact is illuminated with high power white light which
couples to plasmonic modes at the Ag nw junctions and, hopefully, at the Ni np/Ag nw
interfaces, creating high fields and localized heating. The heating will then fuse the metals
together to create a lower resistance contact. Full field optical simulations could help
with understanding of the Ni np absorption and reflection properties. Experimentally,
illuminated the contacts with ∼ 1 W/cm2 did not alter the performance and illumination
with ∼ 1 W/cm2 of 496 nm laser light created a large barrier resistance, likely due to
oxidation of the Ni nanoparticles. Thus, an oxygen free atmosphere may be needed to
realize plasmonic welding for these contacts.
Though n-Si was contacted through this scheme, Ni is also an ohmic contact for pGe, p-Si, n-InGaAs, p-InGaAs, n-InP, n-InSb, and p-SiC, making this technique broadly
applicable to a variety of other inorganic semiconductors.(17) A number of other metals can
also be readily deposited via electroless deposition, further expanding the range of accessible
materials. Deposition along crystal edges, as seen with the Ni nps, should be possible with
other micro- or nano-structured semiconductors. Electroless metal deposition on materials
may also prove useful for exploring their absorption profiles or may act to selectively deposit
catalysts for electrochemical reactions. Overall, this contacting paradigm should be broadly
applicable to a wide array of inorganic device geometries and need not be limited to solar
applications (i.e. LEDs, transistors, etc. could also take advantage of this scheme).

Chapter 4
GaP on Si

4.1

Motivation

The demonstration of Si diffusion lengths on the order of 10 µm and the development of
large area Si wire array cells led to the possibility of high effiency multijunction wire array
structures. Nominally, heterostructure wire arrays may be fabricated regardless of material
quality. However, multijunctions will only outperform their single junction counterparts
if each cell is able to perform at a high level. For example, combining a 20% efficient
GaAs cell (VOC = 0.9V, JSC = 25mA/cm2 , and F F = 89%) with an 11% efficient Ge
cell (VOC = 0.3V, JSC = 50mA/cm2 , and F F = 75%) leads to a 23% efficient tandem
(VOC = 1.2V, JSC = 25mA/cm2 , and F F = 77%), an improvement over the stand alone
performance of either cell, but still well below the single junction efficiency of GaAs.
Multijunctions not only allow for greater than 30% efficiencies, as will be discussed in
the next chapter, but also lead to devices with high voltages, an important requirement
for solar photoelectrochemistry (PEC).(103) Solar PEC offers a solution to the challenge
of storing solar power. Even if solar module costs drop down to the level of grid parity,
solar generated electricity will still not be able to provide more than 20% of the total power
for the U.S. due to the variability of sunlight.(104) Long term, low cost, grid scale power
storage systems are currently lacking. However, generating fuels such as H2 or syngas
(CO + H2 ) instead of electricity gives flexibility to the power generation process; the fuels
can be converted to energy when needed. Additionally, by converting CO2 and water to
syngas, excess CO2 is pulled out of the air, helping to mitigate problems caused by this
greenhouse gas.(105)
Water splitting (H2 O → H2 +O2 ) requires at least 1.7 V, including overpotential require46

ments, to drive 10 mA/cm2 and carbon dioxide reduction (CO2 + H2 O → CO + H2 + O2 )
requires well over 2 V.(89) Thus, Si alone cannot provide this photovoltage. Higher bandgap
materials may be used instead, but due to the limited photon flux in the ultraviolet regime of
the solar spectrum, overall device efficiencies and product yields will be low. Tandem structures offer higher voltages and appreciable currents, allowing for greater overall efficiencies
despite the voltage constraints.
While wire heterostuctures are not new, previous work has primarily been limited to
Si/Ge systems or to III-Vs, and their architectures have been simple radial or coaxial
superlattices.(42; 106; 107) Dr. Ross at IBM fabricated Si on GaP or Si/Ge/GaAs axial structures, but only in thin layers and not with the explicit intent of forming active
devices.(108) For photoelectrochmical devices, Hwang et al. (109) and Shi et al. (110) used
T iO2 on Si wires to generate H2 . However, the bandgap of T iO2 is too large to generate
appreciable photocurrent from the solar spectrum.
When considering the materials requirements for photoelectrochemistry and for growing
heterostructures with a limited number of defects, GaP stands out as an ideal mate for Si
wire arrays. GaP is lattice matched to Si, has a large enough bandgap, 2.32 eV, to give
the desired voltages in combination with Si, and has been used to reduce both CO2 and
H2 O.(111–113) Additionally, GaP’s material properties have been explored extensively due
to its use in light emitting diodes,(114) it can be doped both n-type and p-type,(115; 116)
and epitaxial growth on Si has been achieved.(117–127) Thus, we set out to develop GaP/Si
wire tandem photoelectrodes.

4.2

Device Overview

While a number of different GaP on Si wire array morphologies can be imagined, initial
exploration focused on understanding the simplest and most straightforward to fabricate:
a Si wire array conformally coated with GaP, as seen in Figure 4.1. The GaP shell is
assumed to have a built-in p-n junction while the Si wire may be highly doped to serve as a
back contact to the GaP or may also include a p-n junction, thus creating a multijunction
cell. As seen in the band-diagram in Figure 4.1, the conduction band of Si is theoretically
aligned with the conduction band of GaP (the electron affinity difference between the two
materials is 0.1 − 0.25 eV),(89) allowing for ready transfer of electrons between the two.
In reality, band offsets at interfaces deviate significantly from those that would be expected
47

Energy (eV)

EC
Si
GaP
EV

Position (m)
Figure 4.1: GaP/Si device overview and band diagram.
from a simple affinity argument due to charge transfer and dipole formation at the interface.
Barriers due to band misalignment at the heterojunction may be mitigated through high
doping densities, leading to enhanced tunneling through band spikes.
For stand alone GaP devices, an n+ Si/n GaP/p+ GaP architecture was considered.
For tandem devices, a tunnel junction may be formed for either p Si/n+ Si/p+ GaP/p
GaP/n+ GaP cells or for n Si/p+ Si/n+ GaP/n GaP/p+ GaP devices. The voltage drop
across the tunnel junction will likely be lower for the latter case due to the aforementioned
alignment.

4.3
4.3.1

Optoelectronic Modeling
GaP/Si Full Field Optical Modeling

Integrating the AM 1.5D solar spectrum above the bandgap of GaP leads to a potential
photocurrent of 8.0 mA/cm2 . Si, on the other hand, has 39.4 mA/cm2 of above bandgap
photons available. Thus, even if the outer layer of GaP absorbs all of the available above
bandgap light, it will still limit the current of the device. Extending this thinking along
the continuum of potential bandgaps, Figure 4.2 demonstrates regions in which either the
top or bottom cell limit the overall device performance. In the upper left corner, the low
bandgap core limits the current. In the lower right corner, the high bandgap shell limits
the efficiency.
In order to design an outer GaP layer that absorbs as much light as possible and
thus maximizes the device performance, the effect of GaP thickness and morphology on
absorption were studied through full field, electromagnetic simulations. Finite-difference48

Tandem Efficiency (%)

Core Band GaP (eV)

1.1

0.4
0.35

0.3

0.9

0.25

0.8

0.2
0.7
1.4

1.6

1.8

Shell Band GaP (eV)

2.2

Figure 4.2: Isoefficiency contour plot as a function of core and shell bandgaps.
time-domain (FDTD) simulations with a two dimensional GaP on Si grating gave a rough
estimate of the absorption that could be expected for a wire array. The Si core was 1 µm
thick and 20 µm tall with a 7 µm center to center spacing. The GaP thickness was varied
from 0.5 to 2 µm in 0.5 µm increments, as seen in Figure 4.3 (a). Boundary conditions
for the top, sides, and bottom were fully absorptive (PML), periodic, and fully reflecting,
respectively. Optical constants were taken from Aspnes and Studna (128). A plane wave
source at a varying wavelength and incident angle was used for excitation. The power absorbed in the GaP at each wavelength was calculated and normalized to the incident power.
Both transverse electric (TE) and transverse magnetic (TM) polarizations were considered,
and the two results were averaged.
As seen in Figure 4.3 (b), the outer GaP layer absorbed up to 80% of the above-band
gap incident power, with losses primarily due to absorption by the Si core and by reflection,
especially at normal incidence where much of the light misses the GaP entirely, simply
traveling in between the GaP structures and reflecting back out of the grating. The location
of the direct transition in GaP is evidenced by the rapid increase in absorption at shorter
wavelengths.
49

A full three-dimensional, periodic wire array was also simulated. This structure consisted of a 1 µm diameter, 10 µm tall Si wire with a 0.5 µm thick GaP shell, a 7 µm pitch,
and the same boundary conditions used above. Computational limits restricted the device
geometry and only allowed for normal incidence to be considered. These conditions give
a lower bound for actual absorption, as typical wires can be up to 100 µm long, can have
GaP coatings of more than a micron, and have maximized absorption at oblique angles of
incidence. As a point of comparison, the exponentially decaying Beer-Lambert absorption
expected for a wire array was also calculated. The results are shown in Figure 4.4 (a).
At 400 nm, where the GaP absorption coefficent is large, the full field absorption cross
section is larger than would be expected from mere geometric, Beer-Lambert considerations.
The size of the structure is on the order of the wavelength of the incident light, and the high
index of the GaP and Si direct light into the wire. This phenomenon of a larger absorption
cross section than would be expected from simple geometric considerations is well know, for
example, for small Mie scatterers.(129)

(b)

t = 0.5, 1, 1.5, 2 μm

Ratio of GaP Absorption

(a)

GaP Absorption to Total Power Ratio (TM)

g r ee

Wavelength (nm)

Incid

ent

e (de
Angl

Figure 4.3: Simulations of GaP on a Si grating. (a) Simulation overview. (b) The GaP on
Si grating absorption as a function of wavelength, angle, and GaP thickness.
At 500 nm, however, the GaP absorbs less than would be expected from Beer-Lambert
theory. This loss of power in the GaP corresponds to an increase in the Si absorption; the
GaP refracts and focuses the incident beam, channeling light into the higher index Si core.
This Si absorption enhancement extends over much of the spectrum. Thus a relatively thick
GaP layer is required to maximize shell absorption before light is lost to the Si core.
In order to further explore the channeling of light into the higher index Si core, two
dimensional full field simulations were conducted on the radial cross section of the wire.

(a)

(b)

Figure 4.4: Normal incidence (a) GaP/Si and (b) GaAs/Si wire array absorption. Relative
power refers to the amount of power absorbed in the listed material as normalized to the
total incident power.
Radial absorption becomes relevant if Al2 O3 scattering particles or other optical elements
are incorporated into the array to boost scattering parallel to the substrate. A plane wave
source, periodic side boundary conditions, a perfectly reflecting bottom, and a perfectly
absorbing top were again used. The Si wire was left at 1 µm and the GaP thickness was
varied as before.
Beer-Lambert and full field generation profiles at 500 nm are shown in Figure 4.5. The
full field profiles clearly show focusing into the Si wire core. A comparison of the power
absorbed as a function of wavelength further elucidates the striking difference between the
Beer-Lambert and full field values (Figure 4.6). The Beer-Lambert model shows the power
absorbed in the Si core decreasing consistently as the GaP shell thickness decreases. The
full field values, on the other hand, remain high, as the GaP continues to direct light into
the Si core.
Overall, the optical simulations suggest that making a GaP/Si tandem wire array will
be difficult. Due to the higher index of the Si and the indirect bandgap of GaP, a thick
GaP coating will have to be used in order to get reasonably high photocurrents and prevent the light from being absorbed by the Si. However, experimentally measured diffusion
lengths of GaP are low, on the order of 100 nm,(130) suggesting that thick layers alone
will not be enough. Instead the GaP will have to be highly structured in order to extract
photogenerated carriers before they recombine.

Beer-Lambert

Full Field
Absorbed Power (W cm-3)

Beam Generation (cm-3 s-1)

λ = 500 nm, TM mode

Figure 4.5: Comparison of Beer-Lambert and full field radial optical absorption for GaP/Si
wire cross sections.
4.3.2

GaAs/Si Optical Modeling

In order to circumvent the severe current mismatch of the GaP/Si material system and
the aforementioned conflict between optically thick material requirements and collection
lengths, GaP may be alloyed with N or As to create a direct gap material and to lower
the bandgap, potentially even to the point of reaching a current matched system.(131–134)
GaAsP is a typical ternary alloy. When alloyed with nitrogen, the localized nitrogen states
coalesce in to a band, which in turn anticrosses with the conduction band and leads to
a smaller, direct bandgap. Figure 4.7 (a) demonstrates the relationship between alloy
bandgap and lattice constant. By altering the N, P, and As ratios, GaNPAs can remain
lattice matched to Si over a broad range of bandgaps. In the lab, GaNPAs has been
the source of some study, but so far the material quality has been poor, with diffusion
lengths at most on the order of 1 µm due to hydrogen and carbon incorporation during
growth.(135; 136) However, recently, GaAsP single wire cells have been fabricated with
>10% efficiencies.(137)
As GaNPAs compounds have a smaller, direct band gap and thus would enable a high
efficiency, multijunction, Si wire array-based heterostructure, GaAs was also tested as a
potential shell material in the wire geometry. GaP and GaAs provide useful bounds for the

500 nm GaP

1 μm GaP

1.5 μm GaP

2 μm GaP

Figure 4.6: Comparison of Beer-Lambert (B-L) and full field optical absorption (TE and
TM) as a function of wavelength and GaP thickness for a radial cross section of a GaP on
Si wire.
behavior of GaNPAs, as their bandgaps lie on either side of those of GaNPAs, and GaNPAs
can be either direct or indirect gap depending on the composition.
GaAs was tested as a shell material in an identical simulation setup to the previously
discussed GaP on Si simulations, with only the optical constants changed. As seen in Figure
4.4 (b), the shell absorption increases drastically, and once again, the absorption cross
section is much greater than the geometric, analytical value. The direct gap of GaAs is able
to much more effectively absorb the incident photons, allowing the GaAs to fully collect all
of the above bandgap light that falls on it rather than losing it to the Si core. Beyond the
bandgap of GaAs, at 900 nm and greater, the Si benefits slightly from the GaAs channeling
light into the wire core, but the Si is poorly absorbing at these wavelengths, leading to low
overall collection in the wire.

1.6
1.2

GaP

AlAs
Ga0.5 In 0.5 P

AlSb

GaAs
GaInNAs

2.0

(b)
AlP

Ga NPAs

2.4

GaNPAs

Bandgap (eV)

(a)

InP

0.8

GaSb

0.4 GaNxP1-x
GaNxAs1-x InN P
0.0

5.4

5.6

5.8

InAs
InN x As 1-x

1-x

6.0

6.2

Lattice Constant (Å)

Figure 4.7: Properties of GaNPAs. (a) The bandgap of a variety of materials, including
GaNPAs, as a function of lattice constant. c IOP Publishing. Reproduced by permission
of IOP Publishing. All rights reserved. (b) Optical data for GaNP courtesy of Dr. John
Geisz (NREL)
Finally, the simulations at individual wavelengths were weighted by the solar spectrum
and summed to give an ideal AM 1.5G photocurrent for each layer. The GaP/Si wire array
generates 0.63 mA/cm2 in the GaP shell and 2.60 mA/cm2 in the Si core while the GaAs/Si
wire array generates 4.20 mA/cm2 in the shell and 0.22 mA/cm2 in the core. Thus, the
two structures fall on either side of current matching conditions and suggest that with
an intermediate material and appropriate thickness choices, current matching should be
possible. Additional light trapping techniques should also be incorporated to increase the
overall absorption.
As the simulated structures were only 10 µm tall and 1.5 µm in diameter, they absorbed
a very small amount of light. Real structures, with thicker layers and significantly longer
wires, will likely absorb far more of the incident photons. Unfortunately, the aforementioned
simulations were limited by computational power. Nevertheless, they still offered worthwhile
understanding of channeling due to index contrast and an estimation for necessary layer
thicknesses for full absorption, revealing the importance of direct gap outer layers. The
simulations also demonstrated the enhanced absorption cross section of the wires.

4.3.3

GaP/Si Device Physics Modeling

When discussing the potential photocurrents from the full field simulations, the internal
quantum efficiency was always assumed to be 1. However, real devices will have finite
diffusion lengths due to Shockley-Reed-Hall (SRH), Auger, radiative, defect, and surface
recombination. Thus, a quasi three-dimensional, GaP/Si wire array device physics model
with recombination losses was created. The simulated structure was identical to that used
in the three-dimensional optical simulations.
Electronically, the simulated structure consisted of a GaP radial pn junction on a degenerately doped Si support. The GaP cell had an n-type base with a doping of 1 x 1017 cm−3
and a p-type emitter with a doping of 5 x 1018 cm−3 . The GaP thickness was varied as in
the two-dimensional model. Contacts were placed on the outer GaP emitter and on either
the Si or on the GaP base. The location of the base contact was found to have minimal
influence on the properties of the cells as the conduction band offset between GaP and Si
is small; transport through the Si and across the GaP/Si heterointerface did not inhibit
the device performance. Finally, the GaP diffusion length was set by modifying the SRH
lifetime in the material and the SRV was assumed to be zero. While a surface recombination velocity of zero is highly unrealistic, the intent of the model was to explore limiting

JSC (mA/cm2)

efficiencies as a function of the material diffusion length.
8.5
7.5
6.5
10 5

VOC (V)

Efficiency (%)

13
12
11
00.5 1 1.5 2
GaP Thickness (μm)
Diffusion
Length
(μm)
10
1.75
1.7
1.65
10
1.5
1.6
Diffusion Length (μm) 0 0.5GaP Thickness (μm)
1.55
1.5
10 5
00.5 1 1.5 2
Diffusion Length (μm)
GaP Thickness (μm)

Figure 4.8: Simulated GaP/Si wire array efficiencies, short circuit current densities (JSC ),
open circuit voltages (VOC ) as a function of GaP diffusion length and conformal shell
thickness.

The optical absorption profile was originally assumed to follow Beer-Lambert theory,
decaying exponentially into the material in accordance with the material optical absorption
length. Additionally, all of the light was assumed to fall on the tops of the wires alone,
leading to much higher collection than seen in the optical simulations. More realistic profiles
were eventually incorporated, as will shortly be discussed, but the Beer-Lambert assumption
helped to put an upper bound on potential efficiencies.
As demonstrated in Figure 4.8, optimal efficiencies were found for GaP thicknesses on
the order of the material diffusion length, a result also seen in the modeling of Si wires.(27)
While the open circuit voltage falls off directly with diffusion length, having little dependence on thickness, the short circuit current reaches a maximum value when the diffusion
length is greater than the thickness, as would be expected for a radial junction geometry
where generated carriers have to travel the shell thickness to be collected. The GaP thickness was capped at 2 µm in order to stay within experimentally realizable values. Overall,
efficiencies of up to 13% could be achieved, though these would require intense light trapping
and higher material quality than has yet been realized experimentally (10 µm in contrast
to the ∼ 100 nm that has been measured).
As seen in the aforementioned GaP/Si optical simulations, however, the Beer-Lambert
model does not provide a realistic description of the optical absorption behavior of the
arrays. Thus, optical generation profiles at a variety of wavelengths were obtained from
the full field simulations, weighted appropriately by the solar spectrum, summed, and the
whole inserted into the device physics simulation. As the whole wire array was illuminated,
much of the light failed to strike the wires and reflected out of the array without being
absorbed. This led to a 0.80% efficiency for a 10 µm diffusion length and 0.5 µm GaP
thickness. Scaling the Beer-Lambert model to also account for the reflection losses led to
an efficiency of 0.97%. Again, light trapping is sorely needed.
4.3.4

AlP Window Layers

Unlike Si, with its broad array of surface passivation coatings, III-Vs are far more limited
in surface passivation techniques. Often, a thin, larger bandgap, heavily doped material is
grown on the III-V surface to act as a minority carrier mirror. Ideal window layers have
a conduction (valence) band at the same energy as the corresponding band in the n-type
(p-type) region, allowing majority carriers to exit to the circuit without a loss of potential
56

while reflecting minority carriers.
For GaP, AlP serves as a potential window layer.(138) It is type II with respect to GaP
and will thus have a conduction band spike at the interface, but the difference is small,
as seen in Figure 4.9. To test the efficiencies of an AlP window layer, a GaNP cell was
simulated, with the use of optical data from John Geisz at NREL (Figure 4.7 (b)), and
the cell capped with a 10 nm thick AlP layer, with optical data taken from Monemar
(139). All other parameters were identical to those of the simulations in Figure 4.8. The
surface recombination velocity was then varied and the device efficiency recorded for cells
both with and without the window layer. The results, seen in Figure 4.9, demonstrate the
effectiveness of the window layer. The efficiency is almost entirely decoupled from the state
of the surface.

(a)

(b)

Figure 4.9: Influence of an AlP window layer on GaP on Si cell performance. (a) Band
diagram showing holes being reflected from the AlP. (b) Efficiency as a function of the SRV
for a GaNP cell with and without an AlP window layer.

4.4

Structure Growth and Characterization

Simulating a structure offers valuable physical insight, but nothing compares to actually
making and measuring real devices. Thus, GaP coated wire arrays were grown and their
performance was analyzed.

4.4.1

GaP Growth

Wire arrays were grown as described in Chapter 2 and ranged in height from 10-50 µm with
1-2 µm diameters. Metalorganic chemical vapor deposition (MOCVD) using trimethyl gallium and phosphine precursors was then used to grow GaP on the exposed wire sidewalls.(140;
141) By varying the V/III ratio during growth, the layers were made either p-type or n-type,
with doping likely coming from P vacancies or interstitials. In each growth, several pieces
of planar silicon were included with the wire arrays to compare growth on planar and wire
array substrates.

Figure 4.10: SEMs of GaP on a Si wire array.
SEM images of a GaP-coated Si wire array are shown in Figure 4.10. A cleaved
wire shows the Si core and GaP coating in cross section. The coating was conformal and
rough, both on the wire array samples and on the planar Si substrates, indicating that the
roughness of the layer is caused by the polar on nonpolar epitaxial growth rather than by
the nature of the substrate.(142) X-ray diffraction measurements (XRD), shown in Figure
4.11 demonstrate that the layers are epitaxial, < 111 > oriented GaP films for both the
wire array samples and planar substrates, although there is a small < 220 > peak arising
in all the wire array samples. This peak could be a product of misoriented wires that broke
off during handling of the samples.
To understand light absorption in GaP/Si arrays, the optical absorption of wires embedded in a transparent polymer and peeled off the substrate was studied with an integrating
sphere. Two wire arrays were studied: a Si wire array with 1.5 µm diameter, 30 µm long
wires in a square array, and a GaP-coated array grown on a Si wire array substrate with the
same properties as the bare Si wire array. Figure 4.12 shows optical absorption in both Si
and GaP/Si wire arrays. The optical absorption is significantly enhanced by the addition of

log Intensity (A.U.)

Si wires only

planar GaP on Si

GaP on Si wires
222

111
220
20

311

50 60
2Θ (o)

Figure 4.11: XRD plots of Si wires, planar GaP on Si, and GaP on Si wires.

the GaP coating, leading to nearly 100% absorption in the GaP/Si wire arrays without any
explicit light trapping structure. The absorption enhancement is likely caused by both the
higher fill factor of the GaP coated wires and scattering caused by the rough GaP surface,
evidenced by the enhanced absorption even below the band gap of GaP. These experiments
validate the optical modeling, further suggesting that the wires exhibit a large absorption
cross section and the GaP layer focuses light into the Si core, providing additional pathways
for absorption enhancement over simply geometric considerations.

(a)

(b)

Figure 4.12: Optical absorption of (a) a peeled off Si wire array and (b) a peeled off GaP/Si
wire array.

4.4.2

Modeling the Optical Effects of Surface Roughness

The morphology of the GaP on Si wires proved to be much rougher than the smooth,
conformal layers considered in the initial optoelectronic simulations. Thus, a model was
constructed to match the experimental SEMs. Randomly oriented GaP squares (width =
500 nm) were placed inside an envelope function, to follow the overall shape and roughness of
the actual wires (see the appendix for the code). The top and bottom boundary conditions
were set to produce perfectly matched absorbing layers, while the sides were periodic. A
500 nm plane-wave source was positioned 1 µm above the top of the wire. An SEM, the
model schematic, and absorbed power profiles can be seen in Figure 4.13.
At 500 nm,the GaP layer in the textured structure absorbed 33% of the incident power
while the Si layer absorbed 45% of the incident power. In contrast, for the conformal

(a)

(b)

(c)

Figure 4.13: Simulation of Rough GaP coated Si wire arrays. (a) SEM of a GaP coated Si
wire. (b) Schematic of the simulation setup. (c) Comparison of the absorbed power in a
rough GaP coated Si wire and a smooth GaP on Si wire.
structure, the GaP only absorbed 21% of the power while the Si absorbed 29% of the
power. In the conformal structure, light was channeled into modes in the higher index Si
core whereas the textured structure led to an increased path length in the GaP shell and
therefore enhanced shell absorption. Thus, in order to achieve current matching without
using a thick shell material, texturing will be beneficial.
The optical model was also used to extract the diffusion length from photoelectrochemical measurements. Spectral response data of GaP coated Si wire arrays was collected under
Ar(g) in a sealed glass cell that had a quartz window. The solution contained a low concentration of redox species (0.0050 mM ferrocenium, 0.20 mM ferrocene) which allowed the
generated current to be collected. Light intensities were calibrated with a Si photodiode
(Thor Labs) that was placed at the same location in the cell as the photoelectrodes. The
external quantum efficiency was found at normal incidence for 488 nm illumination and
compared to the model described below. The 488 nm wavelength was chosen as it is in the
indirect region of the GaP absorption spectrum, resulting in long light penetration depths
and thus resulting in significant losses in photocurrent in planar GaP photoelectrodes that
have short minority-carrier diffusion lengths.
For the optical model, a 45 µm tall Si wire on top of a Si substrate was used as such
a geometry corresponded more closely to the experimentally measured structure. A plane
wave of 500 nm light was incident on the top of an individual wire with periodic side
61

boundary conditions to account for coupling effects between wires in the square array. The
FDTD simulations indicated that the composite GaP/Si microwire photoanodes should
absorb a large fraction (88%) of the 500 nm light, with the percentages of 500 nm light
absorbed in the GaP and Si calculated to be 20% and 68%, respectively. The majority
of the light absorption was calculated to occur near the top of the wire, due to strong
scattering arising from the microscale roughness of the MOCVD-grown GaP. Results are
shown in Figure 4.14 (a).

Length (μm)

45
40
35
30
25
20
15
10

(b)

40
38
36
24
23

Ln(Absorption)

-2 0 2
Length (μm)

Ln(Absorption)

45
40
35
30
25
20
15
10

38
37
36
35
34
33
32
31
30
29
28
27

Length (μm)

GaP

(a)

30
28

-2 0 2
Length (μm)

Figure 4.14: Finding the GaP diffusion length. (a) Optical absorption profiles for the GaP
and Si layers. (b) Integrated absorption as a function of depth into the GaP layer from all
surfaces.
Photoelectrochemical measurements suggested that the photoelectrodes had an incident
photon to current efficiency (IPCE) of 1.5% at 488 nm. By integrating the simulated
absorption within a set distance of the surface, theoretical IPCE values as a function of
the effective carrier collection length, Ld , were estimated (Figure 4.14 (b)). This relation
suggested that Ln of the material was ∼80 nm. Such short Ld values are typical of GaP, and
motivate future investigations of smaller absorber layer thicknesses as well as approaches to
produce heteroepitaxial GaP with larger Ld values and/or increased absorption coefficients
(e.g. by bandstructure modification through alloying with N or As).

4.4.3

Single Wire Measurements

Single wire measurements are a useful technique for exploring the electrical properties of
the material. While Si wires can be contacted with a simple one step lithography process,
contacting heterostructre wires requires a two step lithographic process, as seen in Figure
4.15. Different etches and metals are required to contact the core and shell regions. After
exposing one side of the wire, the GaP was etched with a 2:5:2 mixture of HCl:H2 O:H2 O2 to
expose the Si core, which was then contacted with In. The wire top was then exposed in a
separate lithography process, and the GaP surface was cleaned with HCl, and then contacted
with Au/Zn. While a few of these single wire devices were fabricated, they suffered from
shunting and from poor adhesion of the Au to the substrate. However, the process is sound
overall and should be applicable to future experiments. One step lithography was used for
the samples of Strandwitz et al. (89), but the Si base of those wires was already protected
with an oxide mask and the Al contact layer was not expected to be Ohmic to the GaP.

Figure 4.15: Process diagram for single wire measurements of GaP coated Si wires.

4.5

Summary and Outlook

The investigation of GaP on Si proved to be an illuminating foray into wire array based
heterostructures. Full field modeling revealed that either thick layers or a direct gap alloy
is necessary for full absorption in the outer layer. A rough surface also helped to increase
absorption in the outer layer. Furthermore, the wire array absorption benefits from the
outer cladding, which acts to guide light into the Si. Device physics modeling suggested
that efficiencies of greater than 10% could be attained with diffusion lengths on the order
of 10 µm and full absorption, but comparison between photoelectrochemical measurements
of epitaxial GaP grown on Si wire arrays and the model suggested that the actual diffusion
lengths of the structure were significantly smaller, on the order of 100 nm.
If the diffusion length of the GaP remains small despite further heteroepitaxial refinements, increasingly roughened structures may allow for greater collection despite a limited
diffusion length. Additionally, cladding layers could be developed explicitly to channel light
into the underlying Si wire. They would need to have a large enough bandgap to not cause
significant parasitic absorption and contacting the underlying Si would be a challenge, but
they could help the wires to further surpass the Yablonovitch light trapping limit.
In the end, the GaP on Si work revelaed many important lessons for wire array heterostructures, lessons that were invaluable for the tandem wire work of the remaining chapters.

Chapter 5
GaAsx P1−x on Si1−x Gex : Modeling

5.1

Motivation

Multijunction arrays offer the advantages of wire array solar cells along with both higher
efficiencies and higher voltages. While the previous chapter explored GaP/Si wire heterostuctures for solar photoelectrochemistry, different constituent materials must be selected in
order to obtain high efficiencies for solar electricity generation. The limiting efficiency for a
GaP/Si combination is less than that of a single junction Si cell alone due to severe current
limiting by the GaP subcell. Additionally, GaP is an indirect gap material and thus thick
layers are needed to absorb incident sunlight. Nevertheless, the GaP/Si wire heterostructure
revealed that Group IV and III-V materials could be joined in one functional device and
that the growth morphology could be controlled through the use of SiOx masking layers.
The use of lower bandgap III-Vs and the careful tailoring of the geometries could lead to
a high efficiency device. With this goal in mind, this chapter focuses on the design of such
devices with an analytical model and simulations, and Chapters 6 and 7 will explore the experimental efforts directed towards realizing the proposed designs. GaAsx P1−x on Si1−x Gex
was identified as an ideal materials system due to the potential for lattice matching with
bandgap combinations that allow for high detailed balance efficiencies. Three architectures
were developed and explored: a simple coaxial structure, and structures with hemispherical
or spherical GaAsx P1−x layers seeded off of the Si1−x Gex wire tip. An analytical model
revealed that these cells have the potential for >34% efficiencies. Optical modeling demonstrated that current matching can be realized. Finally, device physics modeling stressed
the importance of achieving high lifetime III-V layers, but also showed that defects at the
heterointerface can have minimal impact if doped so as to repel minority carriers.
65

5.2

Material Choice

To achieve a high efficiency device, the bandgaps of the constituent materials must be chosen
so that the photocurrent in each layer is matched. Additionally, lattice-matched material
systems are desirable as mismatch strain relieving dislocations can act as recombination
centers, though recent work suggests that the wire geometry may facilitate high quality
growth of lattice mismatched materials.(61) Dislocations due to lattice mismatch can be
forced to propagate radially outward from the wire interface, allowing high quality material
to be grown axially.
As a first step towards identifying a desirable material system, detailed balance efficiencies for a series connected, two junction cell were calculated by following the treatment of
Henry (18) with the exception that the quadrilateral rule was used to solve the thermal
radiation and radiative recombination integrals. Band gap and lattice constant data for
SiGe, GaAsP, and GaInP was taken from (143–145), respectively and overlaid on the calculated isoefficiency contour plot to create Figure 5.1. The plot reveals that Si0.1 Ge0.9 ,
GaAs0.9 P0.1 , and Ga0.56 In0.44 P are an almost ideal material system for the core, shell, and
window materials of a tandem wire array device. The combination has a detailed balance
efficiency of over 40%. However, despite the specific composition choice, the chosen material combination is somewhat flexible; the limiting efficiency is well over 35% across a broad
range of alloys: gains in open circuit voltage make up for small losses in short circuit current across this range. Practically, Si1−x Gex wires have been successfully synthesized,(146)
and both Si and Ge wires can be grown using high temperature chlorosilane chemistry,(24)
which has been demonstrated to produce high fidelity, ordered arrays.(63)

5.3

Device Architectures

If the GaAs0.9 P0.1 first fully absorbs all of the available above-bandgap photons from the solar spectrum and the Si0.1 Ge0.9 then absorbs all of the remaining photons above its bandgap,
each will contribute ∼28 mA/cm2 of current. Thus, in order to achieve current matching,
the incident light must first pass through the GaAs0.9 P0.1 and the optical path length must
be long enough in the GaAs0.9 P0.1 so that all of the above bandgap photons are absorbed.
Combining this constraint with realistic growth geometries led to the three device designs
depicted in Figure 5.2. In the conformal structure, the III-V layers are deposited directly
66

matche
Lattice
/SiGe
GaInP

ed
order
e mat
Lattic aInP/SiGe

matche
ic
/SiGe
GaAsP

GaAs/Ge

Figure 5.1: Lattice matched material combinations overlaid on an isoefficiency contour plot
for series connected, two junction cells.
on a p-n Si0.1 Ge0.9 radial junction wire array. In the hemisphere structure, the Si0.1 Ge0.9
wire array is infilled with a dielectric material and growth of the III-V layers is then templated from the wire tips, in a method analogous to epitaxial lateral overgrowth.(62) The
ellipsoidal structures allow for a close packed array of GaAs0.9 P0.1 absorbers at the top of
the cell, directing all incident light through the GaAs0.9 P0.1 material before it reaches the
Si0.1 Ge0.9 . Finally, in the sphere structure, the wire sidewalls are protected by a dielectric
and III-V growth proceeds in all directions from the seed region near the top of the wire.

5.4

Analytical Model

While not as accurate as finite element numerical simulations, a simplified analytical model
allows for physical insight into and rapid exploration of the device parameters. An analytical
model for the conformal wire array device geometry may be solved by extension of the model
of Kayes et al. (27) and is not covered here. Instead, the hemisphere design was chosen as a
case study. The spherical design can also be solved through a similar analytical approach,
though the absorption profile should be modified. For the hemisphere, the optical absorption
and current-voltage curves were calculated for a variety of geometries and recombination

(a)

Conformal
(b)

Energy (eV)

1 Si Ge
0.1
0.9

EC
EV

-1

-2

Hemisphere

Sphere

Window Layer
-1Tunnel Junction
-2
700 750 800 850

Ga0.56In0.44P
GaAs0.9P
500
1000
Position (nm)

(c)

d1
d2

x2
x4

Figure 5.2: Overview of the multijunction wire array geometries and electronic structure.
(a) The three multijunction wire array structures under consideration. (b) A representative
band diagram showing the Si0.1 Ge0.9 and GaAs0.9 P0.1 p-n junctions, and the Ga0.56 In0.44 P
tunnel junction and window layers. The 20 nm of Si0.1 Ge0.9 closest to the Ga0.56 In0.44 P is
highly doped to serve as part of a tunnel junction and hence experiences bandgap narrowing.
(c) Cross section of the GaAs0.9 P0.1 hemispherical shell showing the parameters used in the
analytical model. The depletion region boundaries are marked with dotted lines.

parameters. Optical constants for the Si0.1 Ge0.9 were taken from Palik (147). Constants
for the GaAs0.9 P0.1 were generated by shifting GaAs n and k vs. energy values by a
constant energy such that k fell to zero at the desired bandgap. The optical constants
found in the literature were not used as the absorption coefficient did not go to zero at the
expected bandgap and as multiple samples with the same composition had different optical
constants, suggesting that the material was plagued with defects that altered the optical
properties.(148) Electronic material parameters were taken from a database made available
by the Ioffe Institute.(149)
First, the III-V top subcell performance was analyzed. The optical absorption profile
of the III-V layer was calculated by considering the excitation of a sphere of material by
a plane wave, using Mie theory.(129) A polar coordinate grid that covered the hemisphere
was discretized into 25 points in both r and θ and the fields calculated at those points by
summing the first 20 normal modes. The code can be found in the appendix. The absorption
was then calculated by considering the real part of the divergence of the Poynting vector,
~ · P~ = − 1 ω|E|
~ 2 00 , where Pabs is the absorbed power, P~ is the Poynting vector,
Pabs = − 21 ∇
~ is the internal electric field, and 00 is the imaginary part of the
ω is the frequency, E
dielectric constant. The outer profile converges after summing the first 20 modes, but the
inner profile (close to r = 0) continues to oscillate as new modes are added; the modes tend
to be highly structured around the origin. Thus, the seven lowest r values around the origin
were discarded.
For a 2 µm radius sphere, the Mie theory results are plotted in Figure 5.3 alongside
a simple Beer-Lambert (B-L) profile. The absorption at each wavelength was evaluated,
weighted by the solar spectrum, and summed to yield an optical generation profile. For the
Mie model, the plane wave is incident from the x direction (θ = 90◦ on the plot), and the
X-Z optical generation cross section is shown. The Y-Z cross section is similar. For the
B-L absorption profile, the generation is assumed to decay exponentially into the material
with a decay length of α = 2ωk, where ω is the frequency and k is the imaginary part of
the index of refraction. The two absorption profiles are similar, as might be expected for
direct gap GaAs0.9 P0.1 . Photons are absorbed before they can fully occupy the modes of
the structure. Thus, a Beer Lambert photogeneration profile was assumed in the device
physics equations in order to make them analytically tractable.
In addition to the Beer Lambert absorption assumption, the device physics model of the
69

Mie Theory

21
20.5

Beer-Lambert

20
19.5

Optical Generation (cm-3s-1)

21.5

Figure 5.3: Polar coordinate plots comparing Mie Theory and Beer-Lambert absorption for
the upper GaAs0.9 P0.1 cell.
III-V hemisphere relied on three additional assumptions:
1. Transport occurs only in the radial direction.
2. The p-n junction is abrupt.
3. All of the carriers that are generated in the depletion region are collected.
With these assumptions and following the conventions outlined in Figure 5.2, the limits of
the depletion region may be found by maintaining charge conservation;

NA (d32 − x34 ) = ND ((d2 + x2 )3 − d32 ),
and by solving Poisson’s equation in spherical coordinates (assuming angle invariance);
1 d(r2 dV
dr )
= ,
dr
which leads to;

Vbi + V =

qd22 NA
(NA x34 + ND x32 ) + (−NA x24 + ND x22 ),
6
3d2
2

NA ND
where Vbi = kT
q ln ( n2 ) is the built in voltage, V is the applied voltage, NA is the

p-type dopant density, ND is the n-type dopant density, ni is the intrinsic carrier concentration, T is the temperature, k is Bolzmann’s constant, q is the fundamental electrical charge,
and is the dielectric constant.
Since assumption #3 states that all of the charge carriers in the depletion region are
collected,

Jgdep =

qΓ0 (ω)

d22
(eα(ω)(x2 −d1 ) − eα(ω)(x4 −d1 −d2 ) ,
(d1 + d2 )2

and from the model of (150) for recombination:

Jrdep = −qUmax

r2 (V )3 − r1 (V )3
3(d1 + d2 )2

where Γ0 (ω) is the incident photon flux, α = 2ωk is the Beer-Lambert attenuation
qV
ni
coefficient, k is the imaginary part of the index of refraction, Umax = √τn,0
τp,0 sinh 2kT is

the maximum recombination rate of a mid-level trap, τ is the lifetime in the n or p-type

region, r1 (V ) = r(V ) − LC , r2 (V ) = r(V ) + LC , r(V ) = x4 +

ln nA

N N
ln A 2 D

(x2 + (d2 − x4 )) is

πkT (x2 +(d2 −x4 ))
is the
the point where the Fermi level crosses midgap, and LC = π2 kT
qE =
2q(Vbi +V )

recombination collection length.
In the two quasi-neutral regions, we must solve the diffusion equation in spherical coordinates (again, assuming angle invariance):
dn,p

∇2 n, p −

n, p
1 d(r2 dr )
n, p
αΓ0 α(r−(d1 +d2 ))
− 2 =−
Ln,p
dr
Ln,p
Dn,p

where n, p is the minority carrier in the given region, Ln,p is the diffusion length of the
carriers, and Dn,p is the diffusion coefficient.
The current may then be found from the relation J = qDn,p dn,p
dr , where the current is
evaluated at the depletion region boundaries. The boundary conditions are:
1. n(0) = finite,
2. Sn · n(0) = −Dn dn
dr |0 ,
3. n(x4 ) = n0 (eqV /kT − 1),
4. p(d2 + x2 ) = p0 (eqV /kT − 1),
71

5. Sp · p(d1 + d2 ) = −Dp dp
dr |d1 +d2 ,
where Sn and Sp are the electron and hole surface recombination velocities, respectively.
In the n-type emitter, the differential equation can be solved to give
L2p
2αL2p
er/Lp
e−r/Lp
α(r−(d1 +d2 )) αΓ0
p=A
),
+B
+e
)(1 +
Dp 1 − α2 L2p
r(1 − α2 L2p )
where A and B are constants to be determined from the boundary conditions.
The solution takes the same form in the p-type core (with n substituted for p), but an
r/Ln
additional term, ∆ sinh
r/Ln , where ∆ is a constant, must be included to account for the

additional boundary condition. L’Hospitals rule may be used to solve for the limits at r =
0 and combined with the boundary conditions to find the constants.
The device parameters were selected as follows: diffusion lengths of less than 100 nm
have been seen for GaP/Si wire heterostructures,(89) but lengths of a few microns are
regularly achieved for high quality GaAs and hence a diffusion length range of 10 nm 100 µm was explored.(151) The n-type emitter was set to be 100 nm thick with a dopant
density of 9 x 1017 cm−3 . The p-type base was doped at 5 x 1016 cm−3 . Finally, the hole
and electron recombination velocities were set to 100 cm/s.
Device characteristics for III-V hemispherical top subcells with varying radii and diffusion lengths are shown in Figure 5.4. The VOC is relatively invariant of device diameter,
but falls off uniformly with device diffusion length. The JSC , on the other hand, is relatively
insensitive to diffusion length until the diffusion length approaches the physical dimensions
of the device. Overall, for the parameter space surveyed, a maximum efficiency of 29.7% was
achieved for a 10 µm thick device with a 100 µm diffusion length. Higher efficiencies would
be possible with larger radii, but fall outside of the scope of the architectures considered
herein.
Furthermore, the top subcells are relatively insensitive to surface recombination velocity.
As the radius goes up, the surface to volume ratio goes down, and a sphere is already the
shape with minimum surface-to-volume ratio. The model suggests that surface recombination velocities of ∼10,000 cm/s could be tolerated with minimal effect on device performance
(Figure 5.5).
Finally, the hemispherical top subcell was coupled with the wire model of (27) to simulate
current voltage curves for tandem solar cell structures. First, current-voltage curves were
72

Figure 5.4: Efficiency, short circuit current density, and open circuit voltage of hemispherical
GaAs0.9 P0.1 solar cell structures as a function of device radius and diffusion length.
generated for the hemisphere and for a Si0.1 Ge0.9 wire. Kirchhoff’s circuit laws were then
enforced for the two devices in series. Figure 5.6 shows the evolution of device performance
as the dimensions are altered, and Table 5.1 displays the VOC , JSC , efficiency, and fill factor
for the devices.
The combined performance is primarily limited by the short circuit current density of
the Si0.1 Ge0.9 wire cell, with current matching only achieved by making the GaAs0.9 P0.1
layer small enough to allow some of the blue light to reach the Si0.1 Ge0.9 wire and to be
absorbed therein or by making the Si0.1 Ge0.9 wire much longer. Si0.1 Ge0.9 is a very poor
absorber in the infrared and hence the current is 2-4 mA/cm2 lower than possible. As
the wire is made longer to absorb more of the incident light, the recombination losses rise
and the wire efficiency begins to drop due to voltage losses, limiting the gains beyond
a certain point. However, while the model of Kayes et al. (27) assumes Beer-Lambert
absorption, experimental studies have demonstrated the enhanced absorption enjoyed by
wire arrays,(31) and hence higher efficiencies may be possible in real devices.
Finally, the above model assumed a perfect tunnel junction between the two subcells.
However, real devices will experience a finite, though hopefully small, voltage drop across

Current Density (mA/cm 2)

30
25
20
15
10

S = 1E2 cm/s
S = 1E4 cm/s
S = 1E6 cm/s
S = 1E8 cm/s
0.5

Voltage(V)

1.5

Figure 5.5: Light IV curve of the shell as the outer (r = d1 + d2 ) and inner (r = 0) surface
recombination velocities are varied. The velocities are both set to have the values indicated.
the tunnel junction. Thus, a series resistance was added to the model to account for these
losses. For the first cell considered in Table 5.1, as long as the series resistance for the
tunnel junction is kept at or below 0.7 Ω cm2 , the tandem efficiency will remain above 32%.
For the second cell, the efficiency will remain above 33% for a resistance of up to 1 Ω cm2 .
However, for larger values of series resistance, the cell efficiency rapidly began to fall, as
seen in Figure 5.7.

5.5

Optical Modeling

In order to create a more realistic model of all three structures, full optoelectronic finite
element simulations were conducted. For all three geometries, a 1.5 µm diameter, 40 µm
long Si0.1 Ge0.9 wire served as the base and a square array of varying pitch was assumed.
In the conformal structure, the GaAs0.9 P0.1 shell and the Ga0.56 In0.44 P layers were set to
be 500 nm and 20 nm thick, respectively. In the hemispherical and spherical structures,
the III-V semiconductor shell was ellipsoidal, depicting a case in which growth nucleates
simultaneously at the sides and the top of the wire and continues outward. The shell
thickness was set such that a 500 nm gap was left between adjacent wires.
First, full field optical simulations were run using the finite difference time domain
74

(a)

(b)

(c)

Figure 5.6: Light IV curves of the Si0.1 Ge0.9 wire (blue dashed), the GaAs0.9 P0.1 hemisphere
(red dot-dashed), and the tandem combination (black solid) as the structure geometry is
varied. In (a), the hemisphere has a 3 µm radius and the wire is 100 µm long. In (b), the
wire length is increased to 400 µm. In (c), the radius is increased to 5 µm and the length
is 400 µm.

Table 5.1: Light IV characteristics for subcells and tandem cell as the geometry of the shell
and wire are varied.
VOC (V)

JSC (mA/cm2 )

Eff (%)

FF (%)

3 µm GaAsx P1−x shell

1.24

23.7

26.7

90.7

100 µm Si1−x Gex wire

0.33

24.0

5.94

73.9

Tandem

1.57

23.6

32.4

87.1

5 µm GaAsx P1−x shell

1.23

25.5

28.2

90.1

400 µm Si1−x Gex wire

0.30

26.0

5.62

71.7

Tandem

1.52

25.4

33.6

86.7

method with plane wave incident illumination. Optical constants for Si0.1 Ge0.9 and GaAs0.9 P0.1
were found as mentioned in the previous section, and Ga0.56 In0.44 P values were generated
by shifting InP n and k data to the desired bandgap. Optical absorption simulations were
performed at 50 nm steps for both a two dimensional model and a full three dimensional
model with periodic side boundary conditions, a perfectly absorbing top boundary, and a
perfectly reflecting base. Both TE and TM plane wave sources were used and the results
for both polarizations were averaged. Previous experiments demonstrated that coherent
interface effects were suppressed in wires due to mild diameter tapering and small film
variations and hence partial spectral averaging was employed to smooth these frequency

Current Density (mA/cm2)

30
25

20
15
10

10
15
Rs
Eff FF
(Ω cm2) (%) (%)
33.6 86.7
30.5 79.1
10 27.5 71.5
15 24.6 64.1

0.5

1.5
Voltage(V)

Figure 5.7: The effect of series resistance (Rs ) on the performance of the second tandem
cell in Table 5.1.
specific resonances. Spectral averaging will also more closely represent the surface roughness evidenced in experimental structures, again by smoothing the resonances. Finally, 200
spherical, 50 to 250 nm diameter, Al2 O3 scattering particles were randomly distributed in
the empty space to the sides of the wire to scatter light into the structure, and a 100 nm
MgF, 60 nm TiOx dual layer antireflective coating was placed over the outer Ga0.56 In0.44 P
window. The antireflection coating was optimized to allow maximum transmission of the
relevant wavelengths of the solar spectrum by using simple transfer matrix method (TMM)
calculations on a planar cell. The TMM code may be found in the appendix.
The absorption profiles were calculated as described in Ferry et al. (152) and the cross
sections were compared, as seen in Figure 5.8. The absorption features that were visible
in the 3D structure were mirrored in the 2D model, as demonstrated in Figure 5.8 (b).
Overall, the GaAs0.9 P0.1 absorption for the 2D and 3D models were nearly identical while
the Si0.1 Ge0.9 absorption was appreciably lower in the 850-1100 nm range for the 3D model
due to the smaller physical cross section. As experiments have shown that wires can absorb
the majority of light within this range with the incorporation of scattering particles and
anti-reflective coatings, the higher absorption of the 2D model was assumed to be possible.
Thus, though the proposed tandem device structures are three dimensional, two dimensional simulations were employed as they maintain the features seen in three dimensional
simulations with greatly reduced computational complexity, allowing for rapid cell design
iteration.
Device response to the solar spectrum was characterized by 23 optical simulations,
76

λ = 850 nm
(b) 3D
2D

(a)

(TM)(TE)

Optical Generation (A.U.)
Figure 5.8: 2D vs. 3D optical simulation comparison (a) Comparison of 2D (TE and TM)
and 3D absorption rates in the core and shell as a function of wavelength.(b) Absorption
profiles at 850 nm for the 2D and 3D cases.
stepped in wavelength throughout the AM1.5G solar spectrum. Due to the length of time
required to run the optical simulations and, more significantly, to interpolate the finitedifference-time-domain grid onto the finite element grid, optical simulations were run in 50
nm steps for the optoelectronic simulations. However, to make sure that no significant features were missing in the optical response, optical simulations were also run in 25 nm steps
as a check. A comparison of the two absorption profiles for the different wavelength spacings
is shown in Figure 5.9. The absorption was calculated independently in the Si0.1 Ge0.9 ,
in the GaAs0.9 P0.1 , and in the Ga0.56 In0.44 P by considering the real part of the divergence
of the Poynting vector. The optical generation was then integrated over the volume and
multiplied by the fundamental electric charge, yielding an assumed unity internal quantum
efficiency JSC . The differences in “ideal” short circuit densities that would be expected for
the two profiles are listed in Table 5.2. The largest difference between the two values is
1% for the window layers, which do not contribute to the overall current of the device. The
difference in shell and wire absorption between the two profiles is less than 1%. Thus, 50
nm steps were used for the remaining simulations.
After validating the use of a 2D model and 50 nm wavelength steps, all three heterostructure geometries were explored with array pitches from 3-7 µm in 1 µm steps. For all of
77

Conformal

0.6
0.4
0.2

400

600
800
1000
Wavelength (nm)

1200

1400

0.8
0.6
0.4
0.2

400

600
800
1000
Wavelength (nm)

1200

Sphere

% of Incident Photons Absorbed

0.8

Hemisphere

% of Incident Photons Absorbed

1400

0.8
0.6
0.4
0.2

400

600
800
1000
Wavelength (nm)

1200

1400

Figure 5.9: Comparison of the absorption profiles for FDTD optical simulations run at 25
nm wavelength steps vs. 50 nm wavelength steps.
the simulations, the window layer was highly absorbing in the ultraviolet wavelengths and
accounted for a loss of around ∼2 mA/cm2 of all possible photocurrent. The absorption in
the conformal structure was relatively independent of wire pitch. Light incident between the
wires scatters off of the Al2 O3 particles and back into the array, allowing for high absorption
regardless of packing fraction. The absorption profile, plotted in Figure 5.10 for the full
spectrum, reveals that light is directed into guided modes in the wire core, in contrast to
the simple Beer-Lambert absorption assumed for the analytical model. As seen in Figure
5.10, this facilitates high absorption in the infrared part of the solar spectrum despite the
fact that 40 µm of Si0.1 Ge0.9 is not optically thick for these wavelengths. In the blue part of
the spectrum, 500 nm of GaAs0.9 P0.1 absorbed ∼70% of the incident photons, sufficient for
current matching. Simple Beer Lambert absorption gives a slightly higher absorption rate
of 74.5%, suggesting that the optical path length is not enhanced for the III-V layer in this
structure. As the Si0.9 Ge0.9 wire core has a higher index than the GaAs0.9 P0.1 cladding,
incoupled light is guided into the Si0.9 Ge0.9 core rather than residing in modes in the shell.
In contrast to the conformal structure, the hemisphere and sphere designs showed clear
pitch dependence. For large wire array pitches, the III-V layers were optically thick and
hence absorbed all of the blue light. Additionally, the hemispherical and spherical geometry
focused red light onto the wire core where it was absorbed. However the large index contrast
between the base of the III-V sphere or hemisphere and the underlying air or lower index
dielectric caused red light that reached that interface to be strongly reflected.
Lowering the array pitch at fixed wire diameter decreased the relative fraction of light
that impinged on the large index contrast region at the III-V/dielectric rear interface, allowing for greater core absorption. Meanwhile, the lower pitch also led to thinner III-V layers
78

(a)

Y (μm)

(b) 40
30
20
10

3 0

1 2 3 0
X (μm)

Optical
Generation
(cm-3s-1)
1022
1021
1020
1019
1018
1017

Figure 5.10: Optical properties of multijunction wire arrays (a) Percent of incident photons
absorbed as a function of wavelength for the window, shell, and core layers and for all three
structures. (b) The overall, AM1.5G, generation profile for the three structures at a 7 µm
pitch. The plots are distorted laterally in order to encompass the full profile. The insets
show an undistorted view of the top of the structures.

Table 5.2: Calculated ideal (IQE=1) JSC of layers under course (50 nm wavelength step)
and fine (25 nm step) simulations.
Structure

Conformal

Hemisphere

Sphere

Region

Course

Fine

Difference

JSC (mA/cm2 )

(%)

Window

2.67

Shell

20.49

20.59

0.5

Wire

23.32

23.46

0.6

Window

2.02

2.00

Shell

21.53

21.52

0.04

Wire

21.07

20.92

0.7

Window

2.04

2.02

Shell

21.27

21.30

0.1

Wire

22.92

22.94

0.09

which allowed the Si0.1 Ge0.9 wire to absorb some of the blue light, further aiding in current
matching. A compromise was reached for the 4 µm pitch, allowing for current matching
conditions. The spectral profile at this pitch is shown in Figure 5.10. Idealized JSC s for
all pitches are plotted in Figure 5.11, and all optical absorption and loss mechanisms are
summarized in Figure 5.12. Beer Lambert absorption for the 4 µm pitch structure suggests roughly equivalent absorption rates, again suggesting that the optical path length is
not much altered.
For all structures, the current matched, idealized short circuit current densities were ∼21
mA/cm2 in the wire and in the shell. To reach current densities closer to the theoretical
maximum values, a series of graded index layers could be added on the back to couple
red light out to the wire, longer wires could be used to allow for greater absorption in
the red, and window layers with higher bandgaps could be employed to mitigate parasitic
absorption.

5.6

Electronic Modeling

The photogeneration profile was utilized as data input in a drift-diffusion based device
physics simulator (TCAD Sentaurus by Synopsys) to explore photovoltaic device perfor-

Figure 5.11: Plot of the relative absorption in the core and shell for all three structures for
varying pitch.
mance for various geometries. Electronically, all three devices consisted of a base Si0.1 Ge0.9
wire with a radial p-n junction, a highly doped Si0.1 Ge0.9 /Ga0.56 In0.44 P tunnel junction and
surface field, a GaAs0.9 P0.1 p-n junction shell, and a Ga0.56 In0.44 P front surface field. A
representative band diagram is shown in Figure 5.2.
The electrical parameters of the materials were taken from the simulation database,(153)
except for the tunnel junction tunneling masses (0.05 and 0.14 for electrons and holes,
respectively) and effective Richardson constants (0.21 and 0.4 for electrons and holes, respectively) which were taken from an AlGaAs/GaAs interface model.(154) All interface
recombination velocities were set to 100 cm/s for the initial simulations. The Si0.1 Ge0.9
Shockley-Read Hall lifetime was set to 1 µsec (Ln ∼ 100µm), comparable to experimentally
measured values,(32) and Shockley-Read-Hall lifetimes of 1.25 ps, 5 ps, 20 ps, 500 ps, and
50 ns (Ln ∼ 157 nm, 315 nm, 629 nm, 3.15 µm, and 31.5 µm) were considered for the
GaAs0.9 P0.1 . Auger and radiative recombination were also included. A small contact was
located at the bottom center of the Si0.1 Ge0.9 wire. The GaAs0.9 P0.1 contact was either
located at the outer base for the conformal structure or entirely covered the outside of the
window layer for the hemisphere and sphere structures. Finally, cylindrical symmetry was
specified, enabling the two-dimensional model to serve as a quasi-three-dimensional simu81

4 5

Figure 5.12: Cartoon depicting the loss and absorption mechanisms in a wire array tandem
cell. 1: Reflection off of the III-V/infill interface. 2: Light that misses the cell entirely. 3:
Reflection off of the wire surface. 4: Light absorbed directly in the III-V. 5: Light scattered
into the wire. 6: Light guided through the III-V into the wire and absorbed.
lation. In order to facilitate geometrical correspondence from the two-dimensional optical
simulations to the three-dimensional device physics model, after summing and weighting the

single wavelength simulations, the overall generation profile was weighted by Rr where R
is equal to half of the pitch and r is the profile’s x coordinate. Thus when integrated cylindrically, the profile yields the same idealized JSC as when integrated in two dimensions.
The modified profile was then interpolated onto a finite element grid for device physics
simulations.
A simulated voltage variation was imposed at the outer and inner contacts to explore the
light current-voltage (I-V) device performance, and the results are plotted in Figure 5.13.
When the diffusion length becomes comparable to the shell dimensions, a fraction of the
generated carriers can no longer reach the junction before recombining, and the overall short
circuit current density drops, becoming limited by the shell photocurrent. Figure 5.14
plots the SRH recombination for one of the scenarios considered. At higher lifetimes, the
carriers diffuse throughout the shell, leading to more homogeneous recombination whereas
the carriers are more localized for the short diffusion length devices. Since the conformal
structure features a uniform, radial junction, the carrier path length is shorter than the path
length in the hemisphere or sphere structures and hence the short circuit current density in
the conformal structure is reduced less significantly as the minority carrier lifetime decreases.
Looking at the voltage drop as a function of minority carrier lifetime, the decrease in
open circuit voltage for the hemisphere structure is slightly less than for the sphere which
82

Figure 5.13: Efficiency, short circuit current density, and open circuit voltage of the tandem
wire array solar cell structures as a function of the GaAs0.9 P0.1 lifetime.
in turn is significantly less than the open circuit voltage drop for the conformal structure.
The voltage degradation can be directly related to Shockley-Read-Hall recombination and
the overall volume of material. For example, for the 7 µm pitch, the conformal structure
contains 133.3 µm3 of GaAs0.9 P0.1 while the hemisphere contains only 57.5 µm3 per wire.
The larger defective volume leads to a higher dark current in the conformal structure and
JSC
hence a lower VOC as VOC = kT
q ln ( J0 + 1). Thus, in comparing the hemisphere and

conformal geometries, the former retains a higher voltage as the lifetime decreases due to
the lower overall material volume, while the latter shows a smaller incremental current loss
due to the shorter distance to the junction.
In order to investigate the effects of the individual subcells, a contact was placed at
the interface between the two cells, and each cell was simulated independently. The results
from a representative hemispherical tandem structure with a 50 ns lifetime can be found in
Table 5.3. The GaAs0.9 P0.1 and the Si0.1 Ge0.9 bandgaps are 1.54 and 0.79 eV, respectively.
Subtracting 0.4 eV from each as an approximation of the potential drop due to unavoidable
thermodynamic effects, ideal subcell VOC s of ∼1.14 and ∼0.39 V would be expected. The
actual subcell VOC s are below this value. In the case of the GaAs0.9 P0.1 , the addition
of radiative and auger recombination and window layer resistance account for the ∼13
mV difference. For the Si0.1 Ge0.9 wire, SRH recombination causes the voltage loss. The
dimensions of the wire (40 µm in length) are on the order of the diffusion length (100
µm) and hence appreciable recombination would be expected and is in fact seen in the
simulation. Finally, the simulation compares favorably to the analytical model values of
Table 5.1, with a smaller JSC in the Si0.1 Ge0.9 due to its shorter length of 40 µm (limited to

Figure 5.14: Shockley-Reed-Hall recombination for τn = 500 or 5 ps in the GaAs0.9 P0.1 cell
for an array with a 7 µm pitch.

this value by computation memory), reduced VOC s due to additional loss mechanisms, and
slightly reduced fill factors. Additional simulations will likely bring convergence between
the models as the structure and doping profiles are further optimized.

Table 5.3: Light IV characteristics for a representative hemispherical tandem.

4 µm pitch

VOC (V)

JSC (mA/cm2 )

Eff (%)

FF (%)

1.01

23.2

20.8

88.4

0.26

20.0

3.58

67.7

1.32

20.4

22.1

82.1

GaAs0.9 P0.1 shell
40 µm tall
Si0.1 Ge0.9 wire
Tandem

The effects of increasing the surface recombination velocity on device performance were
also explored. The Ga0.56 In0.44 P surface, the Si0.1 Ge0.9 /Ga0.56 In0.44 P interface, and the
GaAs0.9 P0.1 /dielectric interfaces of the sphere and hemisphere structures were given recombination velocities of first 104 and then 106 cm/s. Table 5.4 lists the device results. The
conformal structure’s behavior is relatively unaffected by the SRV change as the GaAs0.9 P0.1
is completely coated with the Ga0.56 In0.44 P window. In fact, the efficiency goes up slightly
at high SRVs due to an increase in fill factor. The contact is set as a small area at the
bottom of the wire, but, when the SRV is high, the contact effectively extends to the entire
outer surface. Minority carriers are reflected by the window layer, but majority carriers recombine at the surface. The hemisphere and sphere designs, on the other hand, experience
a drop in performance with increasing SRV due to increasing recombination at the III-V
dielectric interface.
Finally, in light of the work by Falub et al. (61), we also considered a modified hemispherical device in which a defective GaAs0.9 P0.1 rectangular region was added below the
hemisphere, as shown in Figure 5.15. The hemisphere was set to have a lifetime of 1 ns.
The defective region was set to have a high p-type doping concentration of 1 x 1019 in order
to serve as a back surface field. The SRH lifetime in this region could be varied from 1 ns
to 1 ps with only a 4 mV drop in VOC and negligible change in JSC , as shown in Figure
5.15. The hemisphere is thick enough to absorb all of the above bandgap light and the high
doping of the defective layer repels minority carriers away from the defective region and

Table 5.4: Device performance as a function of surface recombination velocity.

Conformal

Hemisphere

Sphere

SRV (cm/s)

VOC (V)

Eff (%)

104

1.23

21.3

106

1.20

21.4

104

1.27

20.9

106

1.11

17.0

104

1.24

21.7

106

1.06

17.6

towards the junction. Thus, if defects due to lattice mismatch or polar on nonpolar growth
can be grown out within a few microns of the wire base, the cell can achieve high efficiencies
despite their presence.

5.7

Summary and Outlook

An analytical model suggests that GaAsx P1−x on Si1−x Gex tandem wire array multijunction solar cells can achieve efficiencies approaching 34% with diffusion lengths on the order
of 10 µm, optically thick materials, and a low loss tunnel junction. Full field optical modeling revealed that current matching can be realized for a variety of structures with careful
geometric design. The optical modeling also elucidated many of the reflection and absorption loss mechanisms e.g. guiding into the wire and reflection of red light from the lower
III-V air/oxide interface. Electronic modeling emphasized the importance of high lifetime
material in the active layer, but suggested that defects at the heterointerface can have minimal impact if doped so as to repel minority carriers. Passivating the masking oxide/III-V
interface will also be important for attaining high efficiencies.
For actual device fabrication, growing an effective tunnel junction will be important
as will fully enveloping the cell with a window layer. The tunneling masses and effective
Richardson constants for carriers across the tunnel junction are unknown for the materials
that were considered here and hence measuring these values in real devices would be useful
information.
Future simulations should consider refining the optical model to take into account the
actual geometry of the grown devices and should include defect levels, defect distribution,

(a)

Highly
Doped
Defective
Layer

(b)

Optical
Generation
(cm-3s-1)
1.2x1022
3.5x1020
1.0x1019
2.9x1017
8.4x1015
2.4x1014

Figure 5.15: The influence of a“defective” layer on device performance. (a) Optical generation profile and overview for a GaAs0.9 P0.1 on Si0.1 Ge0.9 cell with a“defective” layer. (b)
Light IV curves for 1 ns and 1 ps SRH lifetimes in the ”defective” layer.
and measured electrical properties. The properties of cells made with other materials (e.g.
GaIn1−x Px ) could also be explored as could the addition of a third material to make a triple
junction. Additionally, structures other than wires should be considered both in simulations
and in actual growth. The proper initial structure may allow for rapid defect mitigation
despite the use of greatly mismatched materials.
Finally, while we have specifically explored the implications of III-V on wire growth for
creating tandem wire array cells, the mechanism may also be useful for creating flexible,
standalone III-V cells, LEDs, or other devices on a reusable Si wafer. Wire arrays may
be able to accommodate thicker strained layers than planar films before relaxing through
defects, allowing band structure alteration and e.g. higher mobilities.(155)

Chapter 6
Si1−x Gex Wire Growth

6.1

Previous Work and Overview

In order to realize the simulated tandem wire cells in practice, Si1−x Gex wire arrays had to
be grown. While Si1−x Gex nanowires have been synthesized using silane and germane,(146)
Si1−x Gex wires have not yet been fabricated in ordered arrays or using high temperature
Cl chemistry.
On one end of the alloy continuum, the Atwater Group has grown ordered arrays of Si
wires for years. Chris Chen and Hal Emmer began at these Si growth conditions and slowly
introduced Ge, allowing them to grow wires with up to 10% Ge composition. However,
higher Ge concentrations could not be achieved, likely due to a prohibitively high strain
energy between the Si substrate and Ge rich alloys, which would inhibit wire nucleation.
On the other end of the continuum, Givargizov (24) demonstrated Ge “whisker” growth
from GeCl4 at 650-800◦ C and Song et al. (156) suggested that Si impurities play a key
role in modifying the bulk and surface energetics of the vapor-liquid-solid process for Ge
wire growth. With these prior efforts as guides, Ge wires were grown with Au, Cu, and
Ni catalysts on Si substrates, though array fidelity was less than 50% for all metals. The
addition of small amounts of HCl, BCl3 , and SiCl4 clearly modified the growth morphology,
but did not improve fidelity. Strain-induced dislocations and nucleation barriers at the
wire/substrate interface likely inhibited wire growth normal to the substrate. Further,
growth on Ge substrates did not lead to any growth at all, likely due to the formation of a
stable Au/Ge alloy.
Thus, to achieve high fidelity Ge wire arrays for the exploration of structured III-V on
Ge growth, pillars with a variety of pitches were prepared with reactive ion etching (RIE)
88

from (111) and (100) Ge substrates using Ti hard masks. Etching was observed to occur
preferentially in the < 111 > direction.
Finally, Si1−x Gex wire arrays and CVD and RIE Ge wires were covered with a PECVD
SiOx and selectively patterned to expose the tips. These structures were then sent to NREL
for GaAs growth as detailed in the next chapter.

6.2

Discussion of Ge Chemistry and Catalysts

Theoretically, the growth of Ge wires should follow the physical processes outlined in Chapter 2. Figure 6.1 shows the phase diagrams for Au/Ge and Cu/Ge, respectively, suggesting
that vapor-liquid-solid growth should be possible within these materials; a liquid alloy of
Ge and the catalyst will form and, by supersaturating this alloy, crystalline Ge should be
deposited. However, the actual growth conditions for Ge wires may be radically different
from those optimized for Si wire growth. Ge becomes a liquid at 938◦ C, forcing the use of a
lower growth temperature than was used for growing Si wires. Furthermore, the decomposition of GeCl4 will not produce the same ratios and species of Cl-containing molecules that
will result from SiCl4 decomposition,(37) and the kinetics and energetics of the compounds
may be very different. Thus, varying conditions will prevail within and around the catalyst
alloy and at the growth interface.

(b) 1100
1000

Temperature (°C)

(a) 1200
1000
800
600
400
200

800
700
600
500
400
300
200

10 20 30 40 50 60 70 80 90 100

at.%

900

10 20 30 40 50 60 70 80 90 100

at.%

Figure 6.1: Phase diagrams for Ge and Au or Cu. Reprinted with permission of ASM
International. All rights reserved. www.asminternational.org (157), (158)
Initial attempts to grow Si1−x Gex wires were highly inconsistent. These growths were
performed at high temperatures (∼ 1000◦ C) using Cu catalyst on Si substrates and taking

the vapors from heated GeCl4 . The wafers were often heavily etched and a liquid condensed
on the walls of the reactor. Further investigation suggested that the GeCl4 was decomposing
into GeCl and HCl, the former condensing on the walls of the reactor and the latter etching
the growth substrates.(159) Givargizov (160) used far less GeCl4 for growth of Ge films,
and hence our reactor was modified to bubble H2 through the GeCl4 in order to achieve
lower concentrations, as seen in Figure 2.8. Bubbling led to more repeatable growths and
allowed for forays into growing Si1−x Gex wires.

6.3

Au Catalyzed Ge Growth

As the Au/Ge phase diagram has no germanicides, Au was chosen as the initial catalyst
for attempting Ge wire growth. The temperature and flow rates were swept over a range
from 750-850◦ C and GeCl4 /H2 = 50/900-100/650 sccm, respectively. Two distinct GeCl4
reactions dominate in this range:
1. GeCl4 + Ge ↔ 2GeCl2 ,
2. GeCl4 + H2 ↔ GeCl2 + 2HCl.
The first reaction etches Ge while the second leads to growth. The competition of these
two reactions led to the wide range of morphologies seen in Figure 6.2.
At low temperatures, polycrystalline growth dominated. As the temperature was raised,
wire-like structures emerged, with the growth appearing to be directed by the catalyst.
Finally, for the highest range of temperatures studied, single crystals nucleated at each
catalyst site, but the catalyst did not appear to be active in material deposition. Dailey
et al. (161) saw similar structures for lower temperature, low digermane partial pressure
growth.
By fixing the temperature at 800◦ C and varying the GeCl4 and H2 flow rates, wires were
grown at GeCl4 /H2 = 75/900 sccm. The array fidelity was ∼ 40% with defects primarily
due to lack of wire nucleation or to wire kinking. Many wires kinked almost immediately
after growth and all appeared to diverge from normal at the same height. This height
likely is a critical point for defect relaxation stemming from the Si substrate/Ge wire lattice
mismatch. At lower GeCl4 flows, wire growth rapidly disappeared. At 100/900 flows, wires
were achieved at the sample edge and polygonal crystals were seen in the middle of the
sample.
90

750oC

775oC

800oC

825oC

20 μm

GeCl4/H2: 100/900

75/900

50/900

100/650

Figure 6.2: Ge on Si growth morphology as a function of growth temperature and GeCl4 /H2
flow rates.
The material properties of the as-grown wires were explored using X-ray diffraction
(XRD) and energy dispersive X-ray spectroscopy (EDX) (Figure 6.3). 2Θ XRD scans of
the wires seen in Figure 6.2 revealed a single crystal Ge peak at 27.3◦ and a single crystal
Si peak at 28.44◦ from the substrate. The broad peak in between the Ge and Si peaks also
suggests the presence of strained Ge or Si1−x Gex . Overall, however, the Ge peak sharpness
and intensity relative to the Si peak suggests that the bulk of the wire is crystalline, relaxed
Ge. The EDX scans reinforce that the wires are pure Ge, with a Si peak appearing only at
the substrate.

6.4

Influence of HCl, BCl3 , or SiCl4 on Growth

Since BCl3 and SiCl4 have been found to alter Si and Ge wire growth, respectively, the
gases were independently introduced to the growth process to evaluate their effect on
morphology.(156) Figure 6.4 displays the results, along with the effect of adding HCl.
Each led to uniquely different structures, but none improved the overall fidelity; they altered the chemistry but not in such a way as to prefer < 111 > oriented wire growth.

(a)

(b)

Ge
Si Au

5 μm

10
27

27.5

28 28.5
2Θ (°)

Counts (A.U.)

Intensity (A.U.)

Figure 6.3: (a) XRD plot of Ge wires grown on a Si substrate. (b) EDX of a Au catalyzed
Ge wire for a variety of positions along the wire length.

6.5

Ni, In, and Cu Catalyzed Ge Growth

Due to the deleterious effects of Au on Si device performance, as well as its scarcity and
cost, the above growths were deemed a sufficient demonstration of the feasibility of Ge wire
growth, and Ni, In, and Au catalysts were considered with the hope that growth conditions
could be found with these metals that would lead to high fidelity arrays. All three have a
complicated phase diagram with Ge, as exemplified by Figure 6.1 (b).
The best results from all of the metals can be seen in Figure 6.5. Au, Cu, and Ni all
led to wire growth, while the high vapor pressure of In caused it to evaporate during the
anneal. While the Ni results initially seemed promising, they were not reproducible, and
hence Ni was abandoned. Growths with Cu, on the other hand, consistently led to wire
formation, though the exact morphology varied with temperature and flow rate as seen in
Figure 6.6.
Though Cu catalyzed wire growth for a variety of conditions, the fidelity was universally
low (< 50%) and normally oriented nanowires often grew alongside their larger counterparts.
Furthermore, wire growth could only be initiated if the sample was first annealed at 1000◦ C.
The wire morphology and the prevalence of nano- rather than micro- wire growth can

+ 10 sccm HCl

Std. Conditions

+ 5 sccm BCl3

10 μm
+ 4 sccm SiCl4

Figure 6.4: The effects of small amounts of BCl3 , HCl, or SiCl4 on the Ge growth morphology.
possibly be explained as follows (see Figure 6.7):
1. As the sample is annealed at 1000◦ C, the Cu saturates with Si.
2. The sample is cooled to 800◦ C, which causes some Si to leave the catalyst alloy and
to be deposited on the substrate.
3. GeCl4 is introduced, and Ge penetrates the catalyst particle.
4. Growth only proceeds if the Si/catalyst area is small enough that the strain energy due
to lattice mismatch is less than the energy gain from changing the alloy composition
by expelling crystalline Ge. This leads to initial growth at small diameters (when the
material can relax slightly radially at the interface edges) and explains the abundance
of normally oriented nanowires.

Material:

20 μm

30 μm

10 μm

Figure 6.5: Growth attempts with Au, Cu, Ni, and In catalysts.

GeCl4 flow (sccm):
10

800oC

20 μm

815oC

Figure 6.6: The influence of temperature and flow rate on Ge wires grown from Cu.
5. The sample is cooled to 650◦ C, which causes the remaining Ge to crystallize out of
the Cu.
Additionally, as demonstrated by XRD (Figure 6.8), the crystal quality of the Cu
catalyzed Ge wires was low, inferior to that of the Au grown wires. The Au wires could
be seeded directly at 800◦ C and seemed to grow uniformly after an initial narrowing, likely
due to massive defect relaxation at the base. The morphology of the Cu wires, on the other
hand, varied along the length, suggesting the continued presence of defects.
To test whether nanowire growth might be preferred, nanoscale Cu catalyst particles
were deposited on Si substrates.(162) 250 nm holes were patterned by electron beam lithography and filled with 100 nm of Cu. After growth, the substrates were examined, and no
wires or metal particles were found at the holes. Cu has a high solubility in Si,(163) and
94

Cool from
1000oC to 800oC,
Si crystallizes out

Ge enters catalyst,
wire nucleates at
critical diameter

Cool to 650oC,
Ge crystallizes out

10 μm
Figure 6.7: Suggested mechanism describing the evolution of the Ge wire morphology.

10
Intensity (A.U.)

10
27

27.5

28 28.5
2Θ (°)

Figure 6.8: XRD plot of Cu catalyzed Ge wires grown on Si.
thus all of the catalyst was likely absorbed by the substrate.
Additionally, growths were attempted directly on a < 111 > Ge substrate to eliminate
the effects of strain entirely. Both Au and Cu were used to try to seed growth on the
Ge substrate, but no Ge deposition was observed. Instead, the catalyst particle swelled
and became spherical. Though no AuGe phases exist on the steady state phase diagram,
metastable AuGe phases have been observed during Ge wire growth, and these phases may
be preferred under our conditions.(164) As demonstrated in Song et al. (156), additional
components may be necessary to change the energetics sufficiently to cause growth to occur.

6.6

Cu Catalyzed Si1−x Gex Wire Growth

As a final note on growth, when GeCl4 was first installed, Si1−x Gex growths were attempted,
and, in a few cases, Si1−x Gex wires were grown. Figure 6.9 demonstrates one success with
a 20:1 ratio of SiCl4 to GeCl4 and otherwise standard Si wire growth conditions. The fidelity
95

(a) 107

(b)

Intensity (A.U.)

100 μm

10
27.5

2Θ (°)

28.5

Figure 6.9: Si1−x Gex wire growth. (A) XRD plot of Cu catalyzed SiGe wires grown on Si.
(b) SEM of Si1−x Gex wires.
and crystal quality are both low, but this was the first Si1−x Gex success, demonstrating
the feasibility of the project. Chris Chen and Hal Emmer went on to refine the growth
parameters and to achieve high fidelity, high quality SiGe wires with up to 10% Ge content.

6.7

Reactive Ion Etched Ge

Given the difficulties growing Si1−x Gex wires, Ge pillars were fabricated using reactive ion
etching (RIE) of < 111 > and < 100 > Ge in order to create substrates for exploring the
growth of III-Vs on wires.(165) The Ge was patterned as follows:
1. 10 nm of Ti was sputtered onto the wafers to serve as a hard mask.
2. AZ 5214 (a negative resist) was spun onto the samples at 4000 rpm, baked at 95◦ for
1 min, exposed through the standard wire lithography mask, baked at 115◦ C for 2
min, flood exposed for 60 s, and developed in CD 026 for 45 sec.
3. The exposed Ti was removed by immersing the samples for 6 sec in an HF based Ti
etch from Transene.
4. The samples underwent RIE at 150W in Cl2 for 10 min, forming 3 µm pillars in a
square lattice with 7 µm spacing.

5. The remaining Ti was removed with the etchant and the Ge was lightly etched by
immersion in 6% H2 O2 in H2 O for 5 min in order to clean the surface.
Representative patterned Ge samples can be seen in Figure 6.10. The < 111 > direction etched preferentially under the chosen conditions, leading to the rough substrate seen
for the (100) sample. Overall, the RIE process was straightforward and led to the desired
Ge structures.

(100)

(111)

5 μm
Figure 6.10: Reactive ion etched Ge pillars on (111) and (100) substrates.

6.8

Masking Wires

Finally, after growing Ge and Si1−x Gex wires and etching Ge pillars, they needed to be
masked in order to achieve selective III-V growth at the tips. However, while SiOx is stable,
GeOx is water soluble and would be rapidly removed under traditional MOCVD growth
conditions. Thus, the booting process developed for creating Si pn junctions could not be
applied here. Instead, SiOx was deposited on the wires with PECVD. The wires were then
infilled and etched. The overall process proceeded as follows:
1. The etched Ge wires were coated with ∼300 nm planar equivalent PECVD SiOx
deposited at 350◦ C, 20 W of high frequency power, 1000 mT, 42 sccm of SiH4 /N2 ,
and 838 sccm of N2 O.
2. The wires were infilled with S1813 by spinning it in three times at 3000 rpm (for the
second and third infills, the resist was added while the wafer was spinning). The resist

was then baked at 115◦ C for 5 min. The samples were ashed at 300 mT of O2 and
300W for 15 min to expose the pillar tips.
3. The SiOx was removed at the tips by immersion in BHF for 90 sec.
4. The resist was removed with acetone.
A booted Ge can be seen in Figure 6.11. Additionally, the SiOx protective mask is
visible over much of the pillars in Figure 6.10, with the exposed Ge tip appearing darker
than the oxide.

Exposed Ge

PECVD SiOx

2 μm
Figure 6.11: A Ge wire masked with PECVD SiOx .

6.9

Summary and Outlook

Ge wire growth was attempted under a variety of different flow rates and temperatures for
Au, Cu, Ni, and In catalyst, and on both Si and Ge. Wires were grown on Si with Au,
Cu, and Ni under select conditions, but the fidelities failed to exceed 40%, likely due to the
effects of strain resulting from lattice mismatch. Growth on Ge resulted in spherical alloy
particles rather than crystalline Ge growth. BCl3 , HCl, and SiCl4 were added to attempt

to alter the growth energetics to favor oriented wire growth to no avail. Thus, pillars were
etched out of planar Ge to allow for the exploration of III-V on wire growth.
While the attempts to grow high fidelity arrays of Ge wires in this thesis were unsuccessful, high fidelities should be possible with further efforts. The use of smaller Cu catalyst
particles or the addition of impurities may yet prove effective. Exploring different conditions for wire nucleation (and switching conditions for growth) may also be fruitful. TEM
studies of the Si/Ge interface and of Ge wires would be invaluable for elucidating the defect
distribution and the growth mechanisms. Additionally, physical models for wire nucleation
are sorely lacking and should be developed in order to understand the effects of strain,
catalyst size, etc. on growth.
Finally, some of the non-wire single crystalline Ge structures of Figure 6.2 may prove
useful on their own. If arrays of highly crystalline Ge polygons can be grown and masked to
restrict III-V growth to a few points or surfaces, the desired strain relief and defect guidance
may still be realizable and allow concepts similar to the tandem design of Chapter 5 to be
realized. A broad range of structures could also be made with RIE and other conventional
fabrication techniques, again potentially leading to interesting substrates for III-V growth.

Chapter 7
GaAs Growth on Ge

7.1

Previous Work and Overview

Once structured Ge substrates had been fabricated, III-V growth on these architectures
could be explored, bringing the realization of tandem wire array solar cells one step closer to
reality. While the wire geometry may mitigate the formation and propagation of defects that
originate from lattice mismatched and polar-on-nonpolar growth, leading to long lifetime
III-V material, these traits have yet to be demonstrated in practice. Seeded CVD growth
of axial Ge/GaAs wire heterostuctures has been performed,(108) but, to our knowledge,
conventional epitaxial growth of III-Vs on highly structured Si1−x Gex has not been explored.
Thus, the work in this chapter set out to experimentally investigate the feasibility of III-V
on Si1−x Gex wire growth.
While GaAs0.9 P0.1 was used in the optoelectronic modeling of Chapter 5, GaAs was
selected as a representative III-V for these cursory experiments. The detailed balance
limiting efficiency of the GaAs/Ge material combination is not the maximum for two cells,
but it still exceeds 35% (Figure 5.1). Additionally, the lattice mismatch between GaAs and
Ge growth is low (< 0.1%) as is the coefficient of thermal expansion mismatch, and the
growth of GaAs on Ge has been extensively explored.(166)
As Caltech does not have the necessary facilities or expertise to grow GaAs, the substrates were sent out for growth. CVD grown < 111 > Ge and Si0.9 Ge0.1 wires, < 100 >
and < 111 > RIE fabricated Ge pillars, and oxide masked, planar (100) and (111) Ge were
sent to Dr. Bill McMahon at the National Renewable Energy Lab (NREL) for growth
of undoped, Al0.2 Ga0.8 As-cladded GaAs.(167) While these films could not be probed electronically due to the thick, intrinsic window layers, time resolved photoluminesence studies
100

(TRPL) and X-ray diffraction (XRD) gave insight into the material quality.
Finally, photovoltaic GaAs devices were designed and optimized through optoelectronic
simulations, and substrates were sent to Sumika, a commercial foundry, to grow the designed
layers.

7.2

Structure Overview

CVD grown < 111 > Ge and Si0.9 Ge0.1 wires, < 100 > and < 111 > RIE fabricated Ge
pillars, and oxide masked, planar (100) and (111) Ge were prepared and sent to NREL for
GaAs growth. A piece of planar (311) Ge was also included in each growth run to serve as
a control.
NREL deposited the following layers on the substrates:

Table 7.1: Overview of GaAs on Ge growth run.
Layer

Material

Thickness (µm)

Superlattice 1

GaAs

0.05

Superlattice 2

Al0.2 Ga0.8 As

0.05

Al0.2 Ga0.8 As

0.05

GaAs

1.5

Al0.2 Ga0.8 As

0.2

GaAs

0.01

Repeat previous pair 10 times

The superlattice serves to mitigate defects stemming from the Ge/GaAs interface due
to lattice mismatch and polar-on-nonpolar growth, and the Al0.2 Ga0.8 As serves as a window
layer, confining minority carriers to the bulk. The presence of the window layers allows the
material quality to be assessed through TRPL.

7.3

Material Characterization

After receiving the samples back from NREL, they were characterized with photoluminesence (PL), TRPL, XRD, and SEM. In order to not degrade the material with the electron
beam, SEM was performed last.

101

Figure 7.1 shows full characterization of the (311) planar control sample. The GaAs
film appeared smooth under optical Nomarksi imaging. XRD of the < 311 > direction
revealed the single crystalline, epitaxial nature of the Ge (peak at 53.69◦ ), the GaAs (peak
at 53.74◦ ), and the Al0.2 Ga0.8 As (peak at 53.73◦ ). The lower intensity side peaks stem
from the superlattice. Finally, TRPL of the film (635-690 nm excitation, 30 mW, 20 MHz
source) yielded a lifetime of ∼7 ns. Thus, the GaAs film grown on the (311) Ge was very
high quality.

(a)

(b)

40 μm

Intensity (A.U.)

53.6 53.7 53.8 53.9
2Θ (Degrees)

10
Time (s)

Figure 7.1: GaAs Growth on (311) Ge. (a) Nomarski optical image of GaAs grown on (311)
planar Ge. (b) XRD rocking curve of the structure in (a). (c) TRPL of the film.
Figure 7.2 shows the SEMs of the structured Ge before and after GaAs growth. In
general, the GaAs appears rough, though crystalline, with some smoother crystallites arising
from the Ge wires. For the RIE wires, the {1 -1 0} planes appear favored from the top
down view of the (111) growth while the {1 0 0} planes appear to be favored for the (100)
substrate.
7.3.1

Patterned, Planar Substrates

As an additional control to the (311) Ge, SiOx masked (111) and (100) substrates were
included in the NREL growth runs. These samples restricted the growth to 3 µm holes in a
7 µm pitch square array and either (100) and (111) surfaces, similar to the wire tops.(168)
The samples’ characteristics are featured in Figure 7.3. The films were rough, and the
XRD peaks were slightly broader than for the other samples, as seen in the next section,
suggesting the presence of defects. The PL showed a broad GaAs peak as well as a red
defect peak and a blue peak, likely from the Al0.2 Ga0.8 As.

102

Substrate:

Si0.9Ge0.1 Wires

Ge Wires

Ge (111) - RIE

Ge (100) - RIE

Before
GaAs
Growth:

20 μm

5 μm

7.5 μm

12 μm

After
GaAs
Growth:

Figure 7.2: SEMs of structured Ge before and after GaAs growth.
7.3.2

X-Ray Diffraction

XRD was performed on all of the structured samples and is shown in Figure 7.4. In (a),
the peak from the Si substrate and from the Si0.9 Ge0.1 wires are visible along with the
GaAs peak. In the other three plots, the Ge, GaAs, and Al0.2 Ga0.8 As overlap one another.
The Ge peak is smaller than the GaAs peak in (b) due to the small number and small size
of the CVD grown Ge wires. In (c) and (d), the Ge peak dominates. Overall, the peaks
demonstrated that the III-V layers were epitaxial and of reasonably high quality.
7.3.3

Photoluminescence

All of the films luminesced at room temperate, suggesting that the defect density was
reasonably low in at least a small portion of the material. The PL spectra are displayed
in Figure 7.5. (a) and (b) feature spectra of the GaAs grown on the CVD wires. These
measurements were done at NREL using a 632 nm excitation source, with both visible and
NIR detection. Visible spectra were taken with two different gratings and hence the two
different PL spectra. The GaAs grown on the Si0.9 Ge0.1 wires featured defect luminescence
103

(a)

(b)

35 μm
(c)

10 μm
(d)

Intensity (A.U.)

65.9

66
66.1
2Θ (Degrees)

66.2

Figure 7.3: GaAs growth on planar, oxide masked Ge. (a) SEM of growth on (111) Ge.
The oxide was not fully etched in the area in the upper right. (b) SEM of growth on (100)
Ge. (c) XRD of films on (100) Ge. (d) PL of films on (111) Ge.
around 1000 nm, likely due to Si incorporated within the material.(169) PL can also be
seen in the NIR. While it is tempting to attribute this peak to the CVD grown wires, the
fact that the peaks in (a) and (b) look identical despite the fact that (a) features Si0.9 Ge0.1
with a nominal bandgap of 1.08 eV while (b) features pure Ge with a bandgap of 0.66 eV
suggests that further study is needed. The PL of the RIE etched structures along with PL
from the (311) Ge and oxide masked, patterned Ge are shown in (c). These measurements
were performed with identical excitation as used for the (311) Ge TRPL and with a 1200
lines/mm grating centered at 870 nm and a visible light CCD. The GaAs film on (311) Ge
has a sharp PL peak, indicating that this planar material is high quality with a minimal

104

(a) 10

(b)

(d) 10

Intensity (A.U.)

27.5
28
28.5
2Θ (Degrees)

10
27.1

27.2
27.3
27.4
2Θ (Degrees)

27.5

27.2 27.25 27.3 27.35 27.4
2Θ (Degrees)

10
65.8 65.9

66 66.1 66.2
2Θ (Degrees)

Figure 7.4: XRD of GaAs on structured Ge. (a) GaAs on CVD Si0.9 Ge0.1 wires. (b) GaAs
on CVD Ge wires. (c) GaAs on (111) RIE Ge. (d) GaAs on (100) Ge.
number of defects. The RIE pillars have similarly sharp PL peaks, while the patterned
(100) Ge has a broad peak in the sub-bandgap region (between 900-1000 nm) suggesting
the presence of defects within the film. Similar peak broadening into the red can be seen
for the Si0.9 Ge0.1 wires.

(b)

(a)

0.8

Ge (111) − RIE
Ge (100) − RIE
Ge (100) − Patterned
Ge (311) − Planar

0.6
0.4
0.2
800

850
900
950
Wavelength (nm)

1000

Figure 7.5: PL of GaAs grown on structured Ge. (a) PL of GaAs on CVD Si0.9 Ge0.1 wires.
(b) PL of GaAs on CVD Ge wires. (c) PL of GaAs grown on RIE, oxide masked (patterned),
and (311) Ge.

TRPL of the GaAs films allowed for the extraction of radiative lifetimes. Figure 7.6
shows the time response for the samples. While the curves were highly multiexponential,
they all appeared to have ns lifetime components, suggesting that at least some small
segments of the films were defect free. (a) features curves obtained at NREL under 780 nm
excitation and at 870 nm collection. In (b), the data was obtained as described previously
for the (311) sample. The instrument response function was obtained by placing a substrate
coated with Lambertian reflecting white paint at the sample position. The signal for t < 1.5
sec likely comes from laser scatter within the microscope. More rigorous TRPL experiments
for these films are ongoing.

105

(b)

Si0.9Ge0.1 Wires
Ge Wires

Intensity (A.U.)

(a)
Intensity (A.U.)

−1

−2

Time (ns)

Ge (111) − RIE
Ge (100) − RIE
Ge (100) − Patterned
Instrument Response Function

Time (ns)

Figure 7.6: TRPL of GaAs grown on structured Ge. (a) TRPL of GaAs on CVD Si0.9 Ge0.1
and Ge wires. (b) TRPL of GaAs on RIE, oxide masked/patterned, and (311) Ge.

7.4

Parameter Optimization for Device Design

Given the promising lifetimes measured for the NREL GaAs growths on Ge, we set out
to make full GaAs devices with Sumika, a commercial foundry. Optoelectronic simulations
were run in order to optimize the layer thicknesses and doping densities. Figure 7.7 shows
a general schematic of the simulation setup. The initial growth was assumed to be highly
defective due to lattice mismatch and polar-on-nonpolar growth and hence the presence
of the“defect” layer. The base layer thicknesses was chosen via optical simulations, which
suggested that a 2 µm thick base could generate >26 mA/cm2 , ∼80% of the maximum
value for GaAs. The window and emitter layer thicknesses were chosen to be as thin as
possible without risking shunts due to the three-dimensional nature of the growth. The
contact layer was set to be as highly doped as possible and, as it will be etched away over
most of the sample, was not considered in the model.
For optimizing the doping density,the simulation setup was generally identical to that
of Chapter 5. However, doping dependent lifetimes and mobilities were also included. The
mobility followed the form:

µ = µ0 +

µe
A a
(1 + N
Ne )

with the specific parameters taken from the Sentuarus parameter file. The lifetimes

106

Contact Layer (300 nm)
Window Layer (30 nm)
Emitter Layer (50 nm)
Base Layer (2 μm)
Defect Layer (1 μm)

Figure 7.7: Overview of the simulated structure for optimizing the doping levels for a GaAs
device grown on Ge.
were roughly fit to Hwang (170) to yield:

τ = τ0 (

1018 2
) ,
NA

with τ0 = 1 ns.
Figure 7.8 displays the device performance as a function of the base and emitter doping
for a fixed window and defect doping of 2 x 1019 . The voltage initially increases with
emitter and base doping as the Fermi levels split further and further with increasing carrier
concentration. However, eventually, the lifetime degradation begins to effect the voltage
and increased doping no longer helps. The current, on the other hand, is highest for the
lowest doping levels where both the mobility and lifetime are high. The fill factor depends
on conductivity which will depend on both the carrier concentration and the mobility. For
the range considered, the increase in carrier concentration exceeds the loss in mobility and
hence the fill factor increases consistently with doping. Overall, the highest efficiency occurs
at a base doping of ∼ 1.25 x 1017 and an emitter of doping of ∼ 5 x 1018 where the increase
in voltage is balanced by the loss in current and fill factor.
Having fixed the emitter and base doping, the doping of the window and base were then
varied, as seen in Figure 7.9. Since the defect layer is assumed to have a low lifetime (1

107

VOC (V)
22

20
18

16
17

Base Doping (cm−3)

Efficiency(%)

Emitter Doping (cm−3)

0.98

0.96
0.94
18

0.92

0.9
17

26
24
22

20
10

86
Base Doping (cm−3)

Base Doping (cm−3)

JSC (mA/cm2)

Emitter Doping (cm−3)
Fill Factor(%)

85
84
83

82
81
17

10
Emitter Doping (cm−3)

Emitter Doping (cm−3)

Figure 7.8: Device values for a GaAs hemispherical cell as a function of emitter and base
doping (with the window and defect doping set to 2 x 1019 ).
ps) and does not absorb much of the incident light and the window layer is also inactive,
higher doping leads to higher performance as the generated carriers are further restricted
from traveling into defective regions. The fill factor does see some slight drop as the doping
is increased due to the loss of mobility in the defective layer, but this small loss is more than
made up for by reduced recombination. Hence the window and defect doping were both
set at 5 x 1018 for the growth (the highest level that could be grown without significant
material degradation).
Planar (100) and (111) Ge substrates, oxide masked (111) and (100) Ge substrates, RIE,
PECVD oxide masked (111) and (100) Ge substrates with 4.5, 5.5, and 7 µm spacings, and
a masked array of Si wires were sent to Sumika for growth of the designed layers. The Ge
substrates were all p-type doped at 0.01-0.04 Ωcm, which should allow contact of the GaAs
films through the Ge substrates.

108

VOC (V)

21
20.5

Defect Doping (cm−3)

Efficiency(%)

Window Doping (cm−3)

0.94

0.938
0.936
0.934
18

0.932
0.93
18

27.5

27
26.5
18

26
18

10
Window Doping (cm−3)
Fill Factor(%)

82.4
Defect Doping (cm−3)

Defect Doping (cm−3)

JSC (mA/cm2)

0.942

82.3
82.2
82.1

82
18

10
Window Doping (cm−3)

Window Doping (cm−3)

Figure 7.9: Device values for a GaAs hemispherical cell as a function of window and defect
doping (with the base doping set to 1.25 x 1017 and the emitter doping set to 5 x 1018 ).

7.5

Summary and Outlook

All in all, the initial growths and characterization of GaAs on structured Ge showed promise,
with room temperature PL and ns lifetime components. Further characterization will hopefully lead to more insight into the growth e.g. by revealing if certain crystalline planes or
geometries are more favorable. Along those lines, TEM characterization is ongoing and
should reveal the nature of the defects. If ns lifetimes can be achieved in the films, then
high efficiencies should be possible, as outlined in Chapter 5.
Furthermore, investigating the electrical properties of the devices from Sumika should
provide a further demonstration of the device feasibility. In order to contact the devices,
Au/Ge will be sputtered onto the samples, and they will then be infilled slightly with a
polymer (e.g. S1813), allowing the excess metal and contact layer to be removed from the
tops of the devices, leaving a ring contact around the outside in the areas that is covered
by the polymer. Back contact will be made to the substrates with Ga/In. Individual wires
will then be tested with the miBot nanoprobes in the dark and under illumination. The
current-voltage curves will be fit with a modified version of the simulation of Chapter 5.
109

For further work on III-V on Ge wire devices, close iteration between MOCVD growth
and TEM characterization would be extremely beneficial. Growth conditions can be tweaked
to alter the nucleation and growth, hopefully leading to high quality axial material.

110

Chapter 8
Conclusion

8.1
8.1.1

Si Wire Array Solar Cells
State of the Art

Over the last six years, Si wire array solar cells have gone from concept to concrete reality.
High fidelity arrays of intrinsic, p-type, or n-type wires can be grown from Cu catalyst.(63)
They can then be embedded in a transparent, flexible polymer and peeled off of the growth
substrate, allowing the Si wafer to be reused for further growths. (12; 13; 35) Furthermore,
despite using 100 times less material than conventional Si solar cells, wire arrays can absorb
85% of the sunlight over the course of the day.(31) Building on these advances, the work in
Chapter 2 demonstrated that high quality p-n junctions could be fabricated within these
arrays and that the wire surfaces could be successfully passivated with a-Si:H and a-SiNx :H.
The addition of p-n junctions and surface passivation to the arrays led to the measurement
of diffusion lengths of ∼ 10µm, SRVs of < 500 cm/s, VOC s approaching 600 mV, and single
wire efficiencies of 9%. Projecting these results to large area arrays suggests that efficiencies
of 17% should be possible. Furthermore, the work of Chapter 3 demonstrated a transparent,
flexible top contact for crafting large area devices. Thus, all of the pieces are in place for
making > 10% efficient, large area Si solar cells. With this in mind, in 2010, Dr. Mike
Kelzenberg and Dr. Morgan Putnam founded Caelux Corp. with the intent of building on
the aforementioned developments to commercialize the Si wire array technology.
More recently, and with a different material, a group at Lund fabricated 13.8% efficient
InP wire arrays solar cells. They combined lessons from prior work on the optical properties
of InP nanowires and on InP wire growth and took advantage of the innate low surface

111

recombination velocity of the material to achieve this impressive result.(58) However, their
cells are still on the growth substrate and thus further work is required to realize a >10%,
substrate free wire array cell.
Thus, single junction wire array solar cells have advanced from growth novelties to high
performance devices and commercialization. The next few years should reveal if they can
truly be competitive with conventional solar technologies in the all-important $/W metric.
8.1.2

Future Work

Though semiconductor wires have come a long way from the days of Wagner and Ellis,
opportunities abound for further understanding and technological development. First and
foremost, the limitations on diffusion length should be understood. Are defects due to stacking faults or point defects limiting the lifetime?(171) Are impurities getting incorporated
within the material during growth or processing? TEM and ICP-MS coupled with rigorous
lifetime analysis may help to probe the material and reveal if e.g. gettering or annealing
are able to improve the material quality.(88)
Furthermore, the junction within the wire could be further optimized. Junction depth,
doping, and profile have yet to be widely explored, though Dr. Emily Warren found that
thick emitters were limiting the photocurrent within her Si wire array photoelectrodes.(172)
If the diffusion lengths can be further improved, axial junctions may allow for better device
performance by limiting the junction area. The inverting properties of the nitride could
also be exploited to make inverted junction Si wire array cells. For that matter, optimized
surface passivation/antireflection coatings should be developed for both the n-type and
p-type regions e.g. by combining the a-Si:H and a-SiNx :H or trying ALD layers.
Additionally, while the Ag nw/Ni np contact was able to connect >99% of the wires
and proved relatively robust, its resistance was still high and it required that the wire tips
be fully exposed and coated with Ni. Plasmonic welding or mild anneals may improve the
series resistance. (100) The electroless deposition of other metals may also prove superior to
Ni. Finally, other contacting schemes, such as a double back contact should be tested and
may prove to have lower resistances and not require that as much of the wire be exposed
for contacting, thereby preserving passivation over as much of the surface as possible.
While the Atwater group has focused on Si, much of the research community has begun
to shift to III-V wire arrays. Direct gap wires can absorb all above-bandgap light in a small
112

volume and can take advantage of well developed window layers for surface passivation.
For III-V nanowires, axial junctions are needed as radial junctions will fully deplete the
wire, but in situ MOCVD growth of abrupt junctions is possible.(173) As a final note,
wires can exceed the Shockley-Quiesser limit due to their large absorption cross section,(31)
and understanding the limit of this phenomenon i.e. how much light can realistically be
concentrated in a direct or indirect gap wire, may allow for the fabrication of devices that
recapture some of the 7% entropic losses innate to unconcentrated devices, allowing wire
arrays to perform more efficiently than conventional planar cells without the use of external
optical elements.

8.2
8.2.1

Multijunction Wire Array Cells
State of the Art

Building on the success of single material wires, heterostructure and multijunction wire
arrays are seeing increasing attention in the literature. Heterostructures offer all of the
benefits of single material arrays with the possibility for higher voltages and/or higher efficiencies. GaP on Si arrays demonstrated the potential of wire array heterostructures, with
the successful epitaxial growth of GaP on Si wires yielding VOC s of 750 mV, though performance was limited by the 80 nm diffusion length of the GaP.(89; 141) Optical simulations
suggested that the GaP guided light into the higher index Si wire and that the roughness
of the GaP surface enhanced the optical path length in the GaP as compared to smooth
films.
The desire for higher efficiency, solid state devices led to the consideration of alternate
material systems to GaP/Si. GaAsx P1−x and Si1−x Gex are lattice matched and the combination can yield efficiencies approaching 40%. Modeling of conformal, hemispherical, and
spherical devices made from these materials reiterated that light is channeled into the high
index core at longer wavelengths, but also suggested that only a few µms of direct gap material are needed to absorb all of the blue light within the outer layer. The simulations also
revealed the need for long diffusion lengths, effective surface passivation, and high doping
of defects in order to achieve high efficiencies.
In the lab, significant progress was made towards experimentally realizing the modeled
multijunction wire arrays. CVD Ge and Si0.9 Ge0.1 wires were grown, and GaAs with ns

113

lifetime components was synthesized on these structures as well as on RIE planar Ge. Much
more work remains to develop full tandem devices, but the process is well underway, and
GaAs solar cells currently being grown on structured Ge at Sumika should further elucidate
the potential and limitations of the technology.
8.2.2

Future Work

Work on wire array heterostructures has just begun, and many areas are ripe for further
investigation. The growth of Ge and Si1−x Gex wire arrays could potentially be improved
with the use of additional gases to alter the reaction energetics or with an explicit nucleation
step. For that matter, wire nucleation is not well described or understood in the literature,
and the development of a rigorous model would help to shed light on the possibilities and
limitations of wire growth. Current models simply address steady state growth conditions
and not the initial wire nucleation. (174; 175)
As for the growth of outer layers, while epitaxial films have been grown, the films are
rough and the lifetimes for both GaP and GaAs can be substantially improved. Iterative
experiments with MOCVD growth and TEM and/or TRPL should help to optimize the
growth conditions. Other substrate geometries and exposed crystal facets may also improve
the growth. Additionally, the opportunity to grow strained layers from a variety of different
facets and, potentially, at greater critical thicknesses than planar layers, may allow for
bandgap engineering to enable novel new devices with increased mobilities or desirable
bandgaps for light absorption or emission.
When making active devices, the optimization of tunnel junctions and window layers
for these unique geometries will be essential. The ability to alter the preferred growth
directions through diffusion limitations may also prove useful for placing active device layers.
If the material preferentially grows radially off of the exposed Ge, placing the exposed Ge
structures close to one another can instead force the III-V to grow axially. Thus, after
initial deposition, subsequent layers can either grow conformally off of the previous layer,
for large spacings, or can be forced to grow axially, by pushing into the diffusion limited
regime.
Optically, it may prove interesting to explore the absorption properties of small III-V
ellipsoids as they morph from a perfect sphere to an elongated ellipsoid. At one end, the
sphere has perfect resonances characterized by whispering gallery modes. On the other
114

end, all of the modes of the elongated ellipsoid are ergotically populated.(176) In between,
rich dynamics exist with some areas of phase space fully sampled, while others have weak
localized modes. These “in between” structures may prove useful for absorbing strongly
across the solar spectrum.

8.3

The Wide World of Wire Arrays

While this thesis focused on the use of wire arrays for photovoltaic applications, they also
have potential to be used in a variety of other devices. Semiconductor wires have been
used as LEDs or sensors.(177) Additionally, the wire dimensions are on the order of many
cell types, and have proved useful for interfacing with neurons e.g. as artificial retinas
or as neural probes for recording action potentials.(178; 179) If individual wires can be
electronically addressed, as in the work of Ikedo et al. (180), then large area arrays of
transistors or memristors could potentially be realized. (181) Wires have also seen use in Li
ion batteries, where their geometry allows them to rapidly expand when incorporating Li and
then return to pure Si without fracturing.(182) Finally, they have seen use as antimicrobial
surfaces, where the wires puncture the cell walls.(183)
Wire array heterostructures also have a number of potentially interesting optical applications. The upper cell of the spherical or hemispherical tandem designs could be replaced
by a very high bandgap material, allowing it to serve solely as a lens, directing light to the
underlying wire. Exploring the potential for enhanced second harmonic generation (SHG)
with wire arrays could also prove interesting. Surfaces and strain lead to the symmetry
breaking necessary to observe SHG, (184; 185) and heterostructure wires have both qualities in abundance. By also designing the wire arrays to act as photonic crystals, SHG could
be further amplified.
All in all, I expect to see the exploration of semiconductor wires continue to grow in the
coming years, as some of the above topics are explored and as new, unforeseen avenues of
exploration arise.

115

Appendix A
Si Wire Array Processing Steps

A.1

Fabricating Arrays with p-n Junctions

The following steps describe how to fabricate p-n junction Si wire arrays:
1. Array Patterning
(a) Obtain a degenerately doped (<0.001 Ω cm) p-type Si(111) wafer with at least
300 nm of thermal oxide.
(b) Place the wafer on the spinner and turn on the vacuum. Blow off any dust
particles from the wafer surface with a N2 gun. Spin at 3000 rpm and spray
with IPA to clean. Allow to spin dry.
(c) Bake the wafer on a >100◦ C hotplate for at least 5 min to drive off water.
(d) Place the wafer in the box for MCC primer, add a few drops of primer to the
bottle cap, close the box, and let sit for at least 5 min. The primer promotes
resist adhesion.
(e) Return the wafer to the spinner. Coat the wafer with S1813 photoresist (Microchem), and then spin first at 500 rpm for 5 sec to spread the resist and then
at 3000 rpm for 1 min to attain the desired thickness.
(f) Cure on a hotplate at 115◦ C for 2 mins.
(g) Photolithographically pattern the resist using the Karl Suss MA 6 mask aligner.
Consult the log and peers to determine the optimal exposure time for a given
setup. Different patterns may require different exposure lengths.
116

(h) Place the sample in MF-319 developer (Microchem) for 60 sec. Check with the
optical microscope that the pattern has come out correctly, and, if not, adjust
the exposure time accordingly for subsequent patterns.
(i) Cure on a hotplate at 115◦ C for at least 10 min immediately prior to the next
step.
(j) After development of the pattern, the oxide within the patterned holes can be
removed by immersion of the samples in buffered HF (BHF), (Transene, Inc.).
The etch time depends on the ambient temperature and the oxide thickness.
For 500 nm of oxide, ∼ 6.5 min will suffice.
(k) Evaporate or sputter 6N Cu to a thickness comparable to or slightly greater
than the buffer oxide.
(l) Lift off the resist by submerging the sample in acetone. Rinse sample in acetone,
isopropanol, methanol, and finally DI water before drying with nitrogen.
2. Wire Growth
(a) Cleave the patterned samples to the desired size and transfer to the desired
SiCl4 CVD reactor. Consult the log and your peers to find appropriate growth
conditions. As of 3/31/13, the following conditions were used in the small reactor
(“Big Blue”):
i. Load the wafer, positioned in slot 3 of the boat, 77 cm into the reactor at
750◦ C while purging with N2 .
ii. Ramp to 1000◦ C over 20 min under 500 sccm of H2 .
iii. Anneal the wafer for 20 min at 1000◦ C under H2 .
iv. Growth for 25 min under 1000 (995) sccm H2 , 12 (13) sccm SiCl4 , and
1 (1.05) sccm BCl3 . Values in parentheses refer to the actual values as
opposed to the setpoints.
v. Cool to 750◦ C under N2 before pump purging and unloading.
3. Cleaning
117

(a) Dip the sample in BHF for 10-30 sec to remove the native oxide.
(b) Immerse the sample in an RCA2 clean (6:1:1 DI:HCl:H2 O2 at 75◦ C) for 10-20
min to remove the Cu.
(c) Dip the sample in BHF for 10-30 sec.
(d) Immerse the sample in 40◦ C KOH for 30 sec to remove the near surface Si.
4. Barrier Oxide Growth
(a) Immediately after the previous step, place the sample in a tube furnace flowing
N2 at 5 lpm.
(b) Allow the furnace to heat up to 1100◦ C over 90 min and to stabilize for 10 min
at 1100◦ C.
(c) Flow O2 at 5 lpm for 100 min while holding the furnace at 1100◦ C.
(d) Switch back to N2 while holding the furnace at 1100◦ C for 10 min before shutting
the heaters off.
(e) Allow the sample to cool to ∼ 650 − 750◦ C.
5. PDMS Junction Definition
(a) Prepare a mixture of 0.1 g of PDMS curing agent, 1 g of PDMS, and 5 g of
toluene.
(b) Cover your sample with the solution and immediately spin at 3000 rpm for 30
sec. Repeat this step twice.
(c) Immediately bake at 60◦ C for at least 15 min.
(d) Prepare a PDMS etching solution from a 1:1 mixture of 1.0M tetrabutylammonium fluoride in tetrahydrofuran and dimethylformamide.
(e) Coat your sample with the etching solution, let it sit for 2 sec, and rinse.
(f) Place your sample in BHF for 5 min to remove the barrier oxide. MAKE SURE
THAT NO BUBBLES ARE TRAPPED IN THE WIRES
118

(g) Place your sample in the PDMS etch for at least 30 min.
(h) Piranha clean your sample (10 min in 1:3 H2 SO4 : H2 O2 ). Rinse.
(i) Dip your sample in BHF for 5 sec. Rinse.
(j) RCA-2 clean your sample as described previously. Rinse and blow dry.
6. Diffusion
(a) Set the diffusion furnace to 850◦ C and make sure that the N2 is flowing at 10
slm. When the middle zone reaches 800◦ C, pull the solid source P wafers to the
inlet and allow them to cool for 10 min.
(b) Dip your sample in BHF for 15 sec, rinse, and blow dry. Immediately place it
in an open slot between the doping wafers and set the boat back in the tube
entrance.
(c) Slowly, over 1 min, push your samples into the doping furnace until the line
marked on the side of the pushrod is flush with the end of the tube.
(d) Cap the tube and wait for your desired amount of time minus 2 min.
(e) Slowly, over 1 min, remove the boat from the furnace and allow it to cool at the
end of the tube.
(f) Return the furnace to 400◦ C and the N2 flow rate to 5 lpm. Place the doping
wafers back in the furnace

A.2

Creating Flexible, Contacted Large Area Arrays

The following steps describe how to create flexible, peeled off Si wire array solar cells with
a Ag nanowire, Ni nanoparticle top contact:
1. Prepare a solution of 0.1 g Dow Corning 93-500 Space Grade Encapsulant curing
agent, 1 g of 93-500 polymer, and 1.5 g of toluene.
2. Drop cast the solution on the wires at 3000 rpm for 30 sec multiple times until the
wires are completely covered with polymer. Cover the surface with toluene and spin
it off at 3000 rpm to leave ∼5 µm of the tips exposed.
119

3. Cure the film at 60◦ C for 10 hours.
4. Dip the wires in a 1:1 solution of 1.0M tetrabutylammonium fluoride in water and
dimethylformamide to make the polymer hydrophilic and then immediately rinse.
5. Place the wetted sample in a solution of Nickelex (Transene, Inc.) at 80◦ C until rapid
bubble formation on the wire tips is observed (usually around 30 sec). Rinse and blow
dry.
6. Spin on Clear OhmT M (Cambrios, Inc.) at 3000 rpm for 30 sec.
7. Bake in an oven at 50◦ for 90 sec. Bake on a hot plate at 150◦ for 90 sec.
8. (Optional) Evaporate a Ag bus bar on the surface through a shadow mask.
9. Remove the film from the growth substrate with a razor blade. Immediately evaporate
200 nm of Au on the back.

120

Appendix B
Code for Making Rough GaP for FDTD Simulations

Listing B.1: Code to create a rough textured shell coating (modified from original code by
Dr. Mike Kelzenberg)
maxz = 2 4 ;
minz = 0 ;
rad = 1 ;

unitcellhalfwidth = 3.5;

mint = 0 . 5 ;
maxt = 1 ;
extnt = 0 . 5 ;

numXtals = 2 0 0 0 0 ;

num = 0 ;
isGood = 0 ;

f i d = fopen ( ’ Textured \ Rec \ S c r i p t . l s f ’ , ’ w’ ) ;

while

(num < numXtals )

xGuess = rand ∗ u n i t c e l l h a l f w i d t h ;
yGuess = rand ∗ u n i t c e l l h a l f w i d t h ;
z G u e s s = minz + ( maxz − minz + r a d +maxt ) ∗ rand ;
if

( rand < 0 . 5 )
xGuess = −xGuess ;

end
if

( rand < 0 . 5 )
yGuess = −yGuess ;

end
xRot = rand ∗ 3 6 0 ;
yRot = rand ∗ 3 6 0 ;
zRot = rand ∗ 3 6 0 ;

( zGuessif

( ( maxt−mint ) ∗ z G u e s s /24+ mint+r a d > ( xGuess ˆ2 +yGuess ˆ 2 ) ˆ 0 . 5 )
isGood = 1 ;

end

121

else
if

( ( xGuess ˆ2 +yGuess ˆ 2 ) + ( z G u e s s − maxz ) ˆ2 < ( r a d + maxt ) ˆ 2 )
isGood = 1 ;

end
end

( isGood )
num = num+1;
f p r i n t f ( f i d , ’ a d d r e c t ; \ r \ n s e t ( ” name ” , ” Rect%d ” ) ; \ r \n ’ , num) ;
f p r i n t f ( f i d , ’ s e t ( ” m a t e r i a l ” , ”GaP − Aspnes and Sedna ” ) ; \ r \ n s e t ( ” s e t mesh o r d e r from
material

d a t a b a s e ” , 0 ) ; \ r \n ’ ) ;

f p r i n t f ( f i d , [ ’ s e t ( ” mesh o r d e r ” , 2 ) ; \ r \ n s e t ( ” x ” , ’ num2str ( 1 e −6∗xGuess )
num2str ( 1 e −6∗yGuess )

’) ;\ r \ nset (” y ” , ’

’ ) ; \ r \n ’ ] ) ;

f p r i n t f ( f i d , [ ’ s e t ( ” z ” , ’ num2str ( 1 e −6∗z G u e s s )

’ ) ; \ r \ n s e t ( ” x span ” , ’ num2str ( 1 e −6∗ e x t n t )

’)

; \ r \n ’ ] ) ;
f p r i n t f ( f i d , [ ’ s e t ( ” y span ” , ’ num2str ( 1 e −6∗ e x t n t )

’ ) ; \ r \ n s e t ( ” z span ” , ’ num2str ( 1 e −6∗ e x t n t )

’ ) ; \ r \n ’ ] ) ;
f p r i n t f ( fid , [ ’ s e t (” f i r s t
f p r i n t f ( fid , [ ’ s e t (” second
f p r i n t f ( fid , [ ’ s e t (” t h i r d

a x i s ” , ”x” ) ; \ r \ n s e t (” r o t a t i o n
a x i s ” , ”y” ) ; \ r \ n s e t (” r o t a t i o n
a x i s ” , ”z” ) ;\ r \ nset (” r o t a t i o n

isGood = 0 ;
end

end

fclose ( fid ) ;

122

1 ” , ’ num2str ( xRot )
2 ” , ’ num2str ( yRot )
3 ” , ’ num2str ( zRot )

’ ) ; \ r \n ’ ] ) ;
’ ) ; \ r \n ’ ] ) ;
’ ) ; \ r \n ’ ] ) ;

Appendix C
Code for the Analytical Tandem Model

C.1

Code to Plot the Mie Optical Generation Profile

Listing C.1: Code to calculated and plot Mie absorption for a sphere at a variety of wavelengths
and under white light conditions
% Constants that

define

rad = 2 ; % P a r t i c l e

t h e problem :

radius

N re = 1 ; % n o f

t h e medium

N im = 0 ; % k o f

t h e medium

steps = 10; % Discretization
% Build a t a b l e

the

for

values

plotting

for

plotting

[ th , ph , r ] = m e s h g r i d ( 0 : p i / s t e p s : p i , 0 : p i / s t e p s : 2 ∗ p i , 0 : r a d / s t e p s : r a d ) ;
nmax = 3 0 ; % L i m i t

% Load t h e

weighting

t h e sum

factors

for

the

solar

spectrum

l o a d am15g 50nm . t x t ;

% Load t h e n and k d a t a

for

the

particle :

l o a d GaAsP . t x t ;
% Initialize

an a r r a y t o h o l d t h e n and k d a t a t h a t matches t h e

spectral

% binning :
nk GaAsP = z e r o s ( l e n g t h ( am15g 50nm ( : , 1 ) ) , 3 ) ;

% D e f i n e and

initialize

symbolic

syms E r pw E t h e t a p w E p h i p w

variables
E r I wl

for

use

E theta I wl

the

calculation

E phi I wl ;

E r pw = 0 ;
E theta pw = 0 ;
E phi pw = 0 ;
E r I wl = 0;
E theta I wl = 0;
E phi I wl = 0;

% An a r r a y t o h o l d t h e

radial ,

a x i a l , and a z i m u t h a l

field

values

for

% particle :
Abs = z e r o s ( l e n g t h ( t h ( : , 1 , 1 ) ) , l e n g t h ( ph ( 1 , : , 1 ) ) , l e n g t h ( r ( 1 , 1 , : ) ) ) ;

% P o p u l a t e t h e n and k a r r a y :
for

l =1: l e n g t h ( am15g 50nm ( : , 1 ) )
nk GaAsP ( l , 1 ) = am15g 50nm ( l , 1 ) / 1 0 0 0 ;
f o r p =1: l e n g t h ( GaAsP ( : , 1 ) )

123

the

am15g 50nm ( l , 1 ) / 1 0 0 0 < GaAsP ( p , 1 ) % L i n e a r

interpolation :

nk GaAsP ( l , 2 ) = ( GaAsP ( p , 2 )−GaAsP ( p −1 ,2) ) / ( GaAsP ( p , 1 )−GaAsP ( p −1 ,1) ) ∗nk GaAsP ( l , 1 )+
GaAsP ( p −1 ,2) −(GaAsP ( p , 2 )−GaAsP ( p −1 ,2) ) / ( GaAsP ( p , 1 )−GaAsP ( p −1 ,1) ) ∗GaAsP ( p −1 ,1) ;
nk GaAsP ( l , 3 ) = ( GaAsP ( p , 3 )−GaAsP ( p −1 ,3) ) / ( GaAsP ( p , 1 )−GaAsP ( p −1 ,1) ) ∗nk GaAsP ( l , 1 )+
GaAsP ( p −1 ,3) −(GaAsP ( p , 3 )−GaAsP ( p −1 ,3) ) / ( GaAsP ( p , 1 )−GaAsP ( p −1 ,1) ) ∗GaAsP ( p −1 ,1) ;
break ;
end
end
end

% Calculate

the

fields

at each wavelength :

% f o r q =1: l e n g t h ( am15g 50nm ( : , 1 ) )
f o r q =1:1
% Define wavelength

specific

constants :

wl = am15g 50nm ( q , 1 ) / 1 0 0 0 ; % The w a v e l e n g t h

interest

d i s p ( s t r c a t ( ’ wl = ’ , num2str ( wl ) , ’ um ’ ) ) ; % Write t h e w a v e l e n g t h t o t h e
N r e I = nk GaAsP ( q , 2 ) ; % n f o r

the

particle

N i m I = nk GaAsP ( q , 3 ) ; % k f o r

the

particle

k pw = 2∗ p i ∗ ( N r e+i ∗N im ) / wl ; % Wave v e c t o r

k I = 2∗ p i ∗ ( N r e I+i ∗ N i m I ) / wl ; % Wave v e c t o r
rh pw = k pw ∗ r ; % r h o
r h I = k I ∗ r ; % rho

for

the

t h e medium
of

the

particle

t h e medium
particle

hbar = 1 . 0 5 4 5 7 1 4 8 E−34; % Planck ’ s

% Get t h e e q u a t i o n

console

field

constant

at the given wavelength :

[ E r pw , E t h e t a p w , E phi pw , E r I w l , E t h e t a I w l , E p h i I w l ] = Mie ( am15g 50nm ( q , 2 ) , rad , N re ,
N im , N r e I , N i m I , wl , nmax ) ;

% Initialize
% particle

arrays

to hold the wavelength

specific

p l a n e wave and

fields :

E pw wl = z e r o s ( 3 , l e n g t h ( t h ( : , 1 , 1 ) ) , l e n g t h ( ph ( 1 , : , 1 ) ) , l e n g t h ( r ( 1 , 1 , : ) ) ) ;
E I w l = z e r o s ( 3 , l e n g t h ( t h ( : , 1 , 1 ) ) , l e n g t h ( ph ( 1 , : , 1 ) ) , l e n g t h ( r ( 1 , 1 , : ) ) ) ;

% Convert the s y m b o l i c

variables

allow

calculation

specific

% values :
E r p w e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E r pw ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
E t h e t a p w e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E t h e t a p w ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
E p h i p w e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E p h i p w ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
E r I e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E r I w l ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
E t h e t a I e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E t h e t a I w l ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
E p h i I e v a l = i n l i n e ( v e c t o r i z e ( c h a r ( E p h i I w l ) ) , ’ t h e t a ’ , ’ phi ’ , ’ rho ’ ) ;
f o r a =1: l e n g t h ( ph ( : , 1 , 1 ) )
f o r b =1: l e n g t h ( t h ( 1 , : , 1 ) )
for

c =1: l e n g t h ( r ( 1 , 1 , : ) )
E pw wl ( 1 , a , b , c ) = E r p w e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , rh pw ( a , b , c ) ) ;
E pw wl ( 2 , a , b , c ) = E t h e t a p w e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , rh pw ( a , b , c ) ) ;
E pw wl ( 3 , a , b , c ) = E p h i p w e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , rh pw ( a , b , c ) ) ;
E I w l ( 1 , a , b , c ) = E r I e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , r h I ( a , b , c ) ) ;
E I w l ( 2 , a , b , c ) = E t h e t a I e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , r h I ( a , b , c ) ) ;
E I w l ( 3 , a , b , c ) = E p h i I e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , r h I ( a , b , c ) ) ;

end
end
disp ( strcat ( ’

’ , num2str ( a ) , ’ / ’ , num2str ( l e n g t h ( ph ( : , 1 , 1 ) ) ) ) ) ;

end
XCPlots ( a , wl , ( l e n g t h ( t h ( 1 , : , 1 ) ) +1) / 2 , E pw wl , E I w l ) ;
Abs = −N r e I ∗ N i m I ∗ s q u e e z e ( ( a b s ( E I w l ( 1 , : , : , : ) ) . ˆ 2 + a b s ( E I w l ( 2 , : , : , : ) ) . ˆ 2 + a b s ( E I w l
( 3 , : , : , : ) ) . ˆ 2 ) ) / hbar + Abs ;

end

124

L o g A b s p l o t x z = l o g 1 0 (− s q u e e z e ( Abs ( 1 , : , : ) ) ) ;
L o g A b s p l o t x y = l o g 1 0 (− s q u e e z e ( Abs ( : , 1 , : ) ) ) ;
L o g A b s p l o t y z = l o g 1 0 (− s q u e e z e ( Abs ( ( l e n g t h ( t h ( 1 , : , 1 ) ) +1) / 2 , : , : ) ) ) ;

h = figure ;
subplot (2 ,1 ,1) ;
p o l a r 3 d ( f l i p u d ( L o g A b s p l o t x z ’ ) , 0 , p i , 0 , rad , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ XZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
subplot (2 ,1 ,2) ;
p o l a r 3 d ( f l i p u d ( L o g A b s p l o t y z ’ ) , 0 , p i , 0 , rad , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ YZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
fname = ’AM15G . j p e g ’ ;
end

C.2

Code to Generate Plots at Each Wavelength

Listing C.2: Code to plot absorption cross sections for the Mie theory treatment of a sphere
function

[ ] = XCPlots ( a , wl , l y z , E pw wl , E I w l )

% P l o t xy , xz , and yz

cross

sections

E r p w p l o t x z = s q u e e z e ( E pw wl ( 1 , 1 , : , : ) ) ;
E t h p w p l o t x z = s q u e e z e ( E pw wl ( 2 , 1 , : , : ) ) ;
E p h p w p l o t x z = s q u e e z e ( E pw wl ( 3 , 1 , : , : ) ) ;
E p w p l o t x z = abs ( E r p w p l o t x z ) . ˆ 2 + abs ( E t h p w p l o t x z ) . ˆ 2 + abs ( E p h p w p l o t x z ) . ˆ 2 ;
E r I p l o t x z = squeeze ( E I wl ( 1 , 1 , : , : ) ) ;
E t h I p l o t x z = squeeze ( E I wl ( 2 , 1 , : , : ) ) ;
E ph I plot xz = squeeze ( E I wl ( 3 , 1 , : , : ) ) ;
E I p l o t x z = abs ( E r I p l o t x z ) . ˆ 2 + abs ( E t h I p l o t x z ) . ˆ 2 + abs ( E p h I p l o t x z ) . ˆ 2 ;
h = figure ;
subplot (2 ,1 ,1) ;
polar3d ( f l i p u d ( E pw plot xz ’ ) , 0 , pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ P l a n e Wave : XZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
subplot (2 ,1 ,2) ;
polar3d ( f l i p u d ( E I p l o t x z ’ ) , 0 , pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ Internal

F i e l d : XZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;

fname = s t r c a t ( ’ XZ ’ , num2str ( wl ∗ 1 0 0 0 ) , ’nm . j p e g ’ ) ;
p r i n t ( h , ’ − d j p e g ’ , fname ) ;
close ;

E r p w p l o t x y = s q u e e z e ( E pw wl ( 1 , : , 1 , : ) ) ;
E t h p w p l o t x y = s q u e e z e ( E pw wl ( 2 , : , 1 , : ) ) ;
E p h p w p l o t x y = s q u e e z e ( E pw wl ( 3 , : , 1 , : ) ) ;
E pw plot xy = abs ( E r p w p l o t x y ) . ˆ 2 + abs ( E t h p w p l o t x y ) . ˆ 2 + abs ( E ph pw plot xy ) . ˆ 2 ;
E r I p l o t x y = squeeze ( E I wl ( 1 , : , 1 , : ) ) ;
E th I plot xy = squeeze ( E I wl ( 2 , : , 1 , : ) ) ;
E ph I plot xy = squeeze ( E I wl ( 3 , : , 1 , : ) ) ;
E I p l o t x y = abs ( E r I p l o t x y ) . ˆ 2 + abs ( E t h I p l o t x y ) . ˆ 2 + abs ( E p h I p l o t x y ) . ˆ 2 ;
h = figure ;
subplot (2 ,1 ,1) ;
polar3d ( f l i p u d ( E pw plot xy ’ ) ,0 ,2∗ pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ P l a n e Wave : XY’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
subplot (2 ,1 ,2) ;
polar3d ( f l i p u d ( E I p l o t x y ’ ) ,0 ,2∗ pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ Internal

F i e l d : XY’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;

fname = s t r c a t ( ’ XY ’ , num2str ( wl ∗ 1 0 0 0 ) , ’nm . j p e g ’ ) ;
p r i n t ( h , ’ − d j p e g ’ , fname ) ;

125

close ;

E r p w p l o t y z = s q u e e z e ( E pw wl ( 1 , l y z , : , : ) ) ;
E t h p w p l o t y z = s q u e e z e ( E pw wl ( 2 , l y z , : , : ) ) ;
E p h p w p l o t y z = s q u e e z e ( E pw wl ( 3 , l y z , : , : ) ) ;
E p w p l o t y z = abs ( E r p w p l o t y z ) . ˆ 2 + abs ( E t h p w p l o t y z ) . ˆ 2 + abs ( E p h p w p l o t y z ) . ˆ 2 ;
E r I p l o t y z = squeeze ( E I wl (1 , l yz , : , : ) ) ;
E t h I p l o t y z = squeeze ( E I wl (2 , l yz , : , : ) ) ;
E ph I plot yz = squeeze ( E I wl (3 , l yz , : , : ) ) ;
E I p l o t y z = abs ( E r I p l o t y z ) . ˆ 2 + abs ( E t h I p l o t y z ) . ˆ 2 + abs ( E p h I p l o t y z ) . ˆ 2 ;
h = figure ;
subplot (2 ,1 ,1) ;
polar3d ( f l i p u d ( E pw plot yz ’ ) , 0 , pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ P l a n e Wave : YZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
subplot (2 ,1 ,2) ;
polar3d ( f l i p u d ( E I p l o t y z ’ ) , 0 , pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ Internal

F i e l d : YZ ’ , ’ f o n t s i z e ’ , 1 4 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;

fname = s t r c a t ( ’ YZ ’ , num2str ( wl ∗ 1 0 0 0 ) , ’nm . j p e g ’ ) ;
p r i n t ( h , ’ − d j p e g ’ , fname ) ;
close ;

C.3

Code to Calculate the Internal Fields of a Particle Given Mie Theory

Listing C.3: Code to calculated the internal fields and absorption of a sphere using Mie theory
function

[ E r pw , E t h e t a p w , E phi pw , E r I , E t h e t a I , E p h i I ] = Mie ( E0 , a , N re , N im , N r e I , N i m I ,

wl , nmax )

INTERNAL FIELDS OF A PARTICLE :

MIE THEORY − VECTOR SPHERICAL HARMONICS

Dan Turner−Evans ,

J u l 2012

d t @ c a l t e c h . edu

California

Institute

o f Technology

% This
% the

a MATLAB t o o l

absorption

for

calculating

of a particle

the

internal

fields

and h e n c e

i l l i m u n i a t e d by a p l a n e wave

% E I = Sum {n+1}ˆ/ i n f

i ˆn∗ E 0 ∗ ( 2 n+1) / ( n ∗ ( n+1) ) ∗

( c n ∗M o1n ˆ ( 1 ) − i ∗ d n ∗ N e1n ˆ ( 1 ) )

% E0 :

Field

% a:

Particle

Intensity
radius
of

(W/mˆ 2 )
(um)

% N re :

Real part

% N im :

Imaginary part

the index

% N re I :

Real part

% N im I :

Imaginary part

% wl : The w a v e l e n g t h
% nmax : Upper l i m i t

the index

of
of

refraction

the index

refraction

the index

interest

t h e medium

the
of

particle
the

particle

(um)

t h e sum
s t e p s between a n g l e s ,

% S e t up t h e

constants :

mu I = 1 ; % P e r m e a b i l i t y
mu = 1 ; % P e r m e a b i l i t y

t h e medium

refraction

% s t e p s : The number o f

necessary

refraction

the

free

radii

the

plot

particle
space

k pw = 2∗ p i ∗ ( N r e+i ∗N im ) / wl ; % Wave v e c t o r
k I = 2∗ p i ∗ ( N r e I+i ∗ N i m I ) / wl ; % Wave v e c t o r
x v a l = k pw ∗ a ; % S i z e

parameter

m v a l = ( N r e I+i ∗ N i m I ) / ( N r e+i ∗N im ) ; % R e l a t i v e
% Define

symbolic

variables

allow

for

refractive

differential

126

index

calculation :

syms r h o t h e t a
% Initialize

p h i E r pw E t h e t a p w E p h i p w

the

radial ,

% t h e p l a n e wave and t h e

E r I

z e n i t h , and a z i m u t h a l
internal

E theta I

field

E phi I x ;

components

for

field :

E r pw = 0 ;
E theta pw = 0 ;
E phi pw = 0 ;
E r I = 0;
E theta I = 0;
E phi I = 0;

% Calculate

the

v e c t o r components o f

the

electric

field .

f o r n = 1 : nmax
% S e t up t h e L e g e n d r e components :
% P n ˆm = ( −1) ˆm/ ( 2 ˆ l ∗ l ! ) ∗(1−x ˆ 2 ) ˆ (m/ 2 ) ∗d ˆ ( l+m) / dx ˆ ( l+m) ( xˆ2 −1) ˆ l
% where we s u b s t i t u t e

cos ( theta )

for x

syms x1 ;
p n m i a l = ( x1 ˆ2 −1) ˆn ;
pnm = i n l i n e ( c h a r ( p n m i a l ) , ’ x1 ’ ) ;
dpnmail = d i f f ( f o r m u l a (pnm) , x1 , n+1) ;
d p n m a i l c o s = s u b s ( dpnmail , x1 , c o s ( t h e t a ) ) ;
% m=1 A s s o c i a t e d L e g e n d r e P o l y n o m i a l :
Pn = −1/(2ˆn∗ f a c t o r i a l ( n ) ) ∗(1− c o s ( t h e t a ) ˆ 2 ) ˆ 0 . 5 ∗ d p n m a i l c o s ;
dPn = d i f f ( Pn , t h e t a ) ; % d e r i v a t i v e

% S e t up t h e
% for

the

Spherical

Bessel

o f Pn w . r . t .

theta

f u n c t i o n and c a l c u l a t e

specific

values

coefficients

j n = b e s s e l j ( n +1/2 , r h o ) ∗ s q r t ( p i / ( 2 ∗ r h o ) ) ; % The s p h e r i c a l

Bessel

Function

j n r = i n l i n e ( char ( jn ∗ rho ) ) ;
d j n = d i f f ( f o r m u l a ( j n r ) , r h o ) ; % Take t h e

derivative

j n I v a l = i n l i n e ( v e c t o r i z e ( c h a r ( j n ) ) , ’ rho ’ ) ;
jnI = jnI val ( x val ) ;
mjnI = j n I v a l ( m v a l ∗ x v a l ) ;
d j n I x v a l = i n l i n e ( v e c t o r i z e ( c h a r ( d j n ) ) , ’ rho ’ ) ;
djnI = djnIx val ( x val ) ;
dmjni = d j n I x v a l ( m v a l ∗ x v a l ) ;

% S e t up t h e

S p h e r i c a l Hankel f u n c t i o n

h n I v a l = b e s s e l j ( n +1/2 , x ) ∗ s q r t ( p i / ( 2 ∗ x ) ) + i ∗ b e s s e l y ( n +1/2 , x ) ∗ s q r t ( p i / ( 2 ∗ x ) ) ; ;
hnIx val = i n l i n e ( v e c t o r i z e ( char ( h n I v a l ) ) , ’ x ’ ) ;
hnI = h n I x v a l ( x v a l ) ;
hnIx = i n l i n e ( c h a r ( x∗ h n I v a l ) ) ;
dhnIx = d i f f ( f o r m u l a ( hnIx ) , x ) ;
d h n I x v a l = i n l i n e ( v e c t o r i z e ( c h a r ( dhnIx ) ) , ’ x ’ ) ;
dhnI = d h n I x v a l ( x v a l ) ;

% Calculate

the

scattering

coefficients

cn = ( mu I ∗ j n I ∗ dhnI − mu I ∗ h n I ∗ d j n I ) / ( mu I ∗ mjnI ∗ dhnI−mu∗ h n I ∗ dmjni ) ;
dn = ( mu I ∗ m v a l ∗ j n I ∗ dhnI − mu I ∗ m v a l ∗ h n I ∗ d j n I ) / ( mu I ∗ m v a l ˆ2∗ mjnI ∗ dhnI−mu∗ h n I ∗ dmjni ) ;

% Find M and N
M t h e t a = 1/ s i n ( t h e t a ) ∗ c o s ( p h i ) ∗Pn∗ j n ; % Find t h e

t h e t a component o f M

M phi = −s i n ( p h i ) ∗dPn∗ j n ; % Find t h e p h i component o f M
N r = j n / r h o ∗ c o s ( p h i ) ∗n ∗ ( n+1)∗Pn ; % Find t h e
N t h e t a = c o s ( p h i ) ∗dPn/ r h o ∗ d j n ; % Find t h e

r a d i a l component o f N

t h e t a component o f N

N p h i = −s i n ( p h i ) ∗Pn/ s i n ( t h e t a ) / r h o ∗ d j n ; % Find t h e p h i component o f N

% C a l c u l a t e E f o r a p l a n e wave
E r pw = E0∗(− i ˆ ( n+1) ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ N r + E r pw ) ;
E t h e t a p w = E0 ∗ ( i ˆn ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ ( M t h e t a − i ∗ N t h e t a ) + E t h e t a p w ) ;
E p h i p w = E0 ∗ ( i ˆn ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ ( M phi − i ∗ N p h i ) + E p h i p w ) ;

127

% Calculate

the

internal E fields

for

the

particle

E r I = E0∗(− i ˆ ( n+1) ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ N r ∗dn + E r I ) ;
E t h e t a I = E0 ∗ ( i ˆn ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ ( M t h e t a ∗ cn − i ∗ N t h e t a ∗dn ) + E t h e t a I ) ;
E p h i I = E0 ∗ ( i ˆn ∗ ( 2 ∗ n+1) / ( n ∗ ( n+1) ) ∗ ( M phi ∗ cn − i ∗ N p h i ∗dn ) + E p h i I ) ;

end

C.4

Code to Plot the Beer-Lambert Optical Generation Profile

Listing C.4: Code to calculate and plot the absorption of a sphere using the Beer-Lambert
approximation
% Constants that

define

a = 2; % Particle

t h e problem :

radius

N re = 1 ; % n o f

t h e medium

N im = 0 ; % k o f

t h e medium

steps = 100; % Discretization

for

plotting

% Build a t a b l e

for

plotting

the

values

[ th , r ] = m e s h g r i d ( 0 : p i / s t e p s : p i , 0 : a / s t e p s : a ) ;

% Load t h e

weighting

factors

for

the

solar

spectrum

l o a d am15g 50nm . t x t ;

% Load t h e n and k d a t a

for

the

particle :

l o a d GaAsP . t x t ;
% Initialize

an a r r a y t o h o l d t h e n and k d a t a t h a t matches t h e

spectral

% binning :
nk GaAsP = z e r o s ( l e n g t h ( am15g 50nm ( : , 1 ) ) , 3 ) ;

% An a r r a y t o h o l d t h e

carrier

generation

C sum = z e r o s ( 1 , l e n g t h ( t h ( 1 , : ) ) , l e n g t h ( r ( : , 1 ) ) ) ;

% P o p u l a t e t h e n and k a r r a y :
for

l =1: l e n g t h ( am15g 50nm ( : , 1 ) )
nk GaAsP ( l , 1 ) = am15g 50nm ( l , 1 ) / 1 0 0 0 ;
f o r p =1: l e n g t h ( GaAsP ( : , 1 ) )
if

am15g 50nm ( l , 1 ) / 1 0 0 0 < GaAsP ( p , 1 ) % L i n e a r

interpolation :

% Calculate

the

carrier

generation

at each wavelength

f o r q =1: l e n g t h ( am15g 50nm ( : , 1 ) )
% Define wavelength

specific

constants :

wl = am15g 50nm ( q , 1 ) / 1 0 0 0 ; % The w a v e l e n g t h

interest

d i s p ( s t r c a t ( ’ wl = ’ , num2str ( wl ) , ’ um ’ ) ) ; % Write t h e w a v e l e n g t h t o t h e
N i m I = nk GaAsP ( q , 3 ) ; % k f o r

the

particle

a l p h = 4∗ p i ∗ N i m I / wl ;
hbar = 1 . 0 5 4 5 7 1 4 8 E−34; % Planck ’ s
c = 3E8 ; % The s p e e d

constant

light

f l u x = am15g 50nm ( q , 2 ) / 1 0 00 ∗ wl / ( c ∗ hbar ) ; % The i n c i d e n t

128

flux

photons

console

% Initialize

arrays

to hold the wavelength

specific

carrier

generation

C wl = z e r o s ( 1 , l e n g t h ( t h ( 1 , : ) ) , l e n g t h ( r ( : , 1 ) ) ) ;

f o r b =1: l e n g t h ( t h ( 1 , : ) )
f o r aa =1: l e n g t h ( r ( : , 1 ) )
C wl ( 1 , b , aa ) = f l u x ∗ a l p h ∗ exp ( a l p h ∗ ( r ( aa , 1 )−a ) ) ;
end
end
C sum = C wl + C sum ;
end

L o g C p l o t = l o g 1 0 ( C sum ) ;

h = figure ;
polar3d ( f l i p u d ( s q u e e z e ( Log C plot ) ’ ) , 0 , pi , 0 , a , 1 , ’ s u r f ’ ) ;
t i t l e ( ’ Ln ( C a r r i e r

Generation ) ’ , ’ f o n t s i z e ’ , 1 4 , ’ fontweight ’ , ’ b ’ ) ;

% fname = ’AM15G . j p e g ’ ;
% p r i n t ( h , ’ − d j p e g ’ , fname ) ;

% [ X, Y, Z ] = s p h 2 c a r t ( th , ph , r ) ;
% E ( 4 , a , b , c ) = E r e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , r h ( a , b , c ) ) ∗ s i n ( t h ( a , b , c ) ) ∗ c o s ( ph ( a , b , c ) ) +
E t h e t a e v a l ( t h ( a , b , c ) , ph ( a , b , c ) , r h ( a , b , c ) ) ∗ c o s ( t h ( a , b , c ) ) ∗ c o s ( ph ( a , b , c ) ) − E p h i e v a l ( t h ( a , b ,
c ) , ph ( a , b , c ) , r h ( a , b , c ) ) ∗ s i n ( ph ( a , b , c ) ) ; % Compute t h e x component o f

the

vector

% % x∗ = s i n ( t h e t a ) c o s ( p h i ) r ∗ + c o s ( t h e t a ) c o s ( p h i ) t h e t a ∗ − s i n ( p h i ) p h i ∗
% E ( 5 , a , b , c ) = E r e v a l ( a , b , c ) ∗ s i n ( t h ( a , b , c ) ) ∗ s i n ( ph ( a , b , c ) ) + E t h e t a e v a l ( a , b , c ) ∗ c o s ( t h ( a , b , c ) ) ∗
s i n ( ph ( a , b , c ) ) + E p h i e v a l ( a , b , c ) ∗ c o s ( ph ( a , b , c ) ) ; % Compute t h e y component o f

the

vector

% % y∗ = s i n ( t h e t a ) s i n ( p h i ) r ∗ + c o s ( t h e t a ) s i n ( p h i ) t h e t a ∗ + c o s ( t h e t a ) p h i ∗
% E ( 6 , a , b , c ) = E r e v a l ( a , b , c ) ∗ c o s ( t h ( a , b , c ) ) − E t h e t a e v a l ( a , b , c ) ∗ s i n ( t h ( a , b , c ) ) ; % Compute t h e
z component o f

the

vector

% % z∗ = cos ( theta ) r∗ − sin ( theta ) theta ∗

C.5

Code to Generate Subcell, Tandem J-V Curves

Listing C.5: Code to calculate the current-voltage characteristics of the subcells and series
configuration for a tandem, hemispherical wire array solar cell
% Specify

universal

constants

numpts = 2 0 0 1 ; % The number o f

% Specify

constants

for

the

points

III V

d1 = 0 . 1 E−4; % The t h i c k n e s s

for

outer

the

t h e IV c u r v e s

cell

III V

emitter

III V

base

% d 1 v a l s = [ 0 . 0 5 E− 4 , 0 . 1E− 4 , 0 . 5E− 4 ] ;
d2 = 2 . 9 E−4; % The t h i c k n e s s

the

% d 2 v a l s = [ 1 E−4 ,3E−4 ,5E− 4 ] ;
S p I I I V = 1 0 0 ; % The I I I V

p t y p e SRV

S n I I I V = 1 0 0 ; % The I I I V

n t y p e SRV

L n I I I V = 1E−2; % The e l e c t r o n

diffusion

L p I I I V = 1E−2; % The h o l e

diffusion

% Specify

t h e SiGe w i r e

the

constants

for

r 1 = 0 . 1 E−4; % The t h i c k n e s s

r 2 = 0 . 6 5 E−4; % The t h i c k n e s s
S p w i r e = 1 0 0 ; % The I I I V

L = 1E−2; % The l e n g t h

in
the

the

III V

p type base

n type

emitter

cell
emitter

the wire base

n t y p e SRV

L n w i r e = 1E−1; % The e l e c t r o n
L p w i r e = 1E−1; % The h o l e

the wire

length

diffusion

length

in
the

the wire

129

the

III V

p type base

n type

emitter

% Specify

the

series

resistance

due t o t h e t u n n e l

junction

Rs = 0 ; % i n Ohm−cmˆ2

% Script

to get

i n d i v i d u a l IV c u r v e s

% C a l c u l a t e and p l o t

the J V curves

for

the

III V

cell

J s = z e r o s ( 4 , numpts ) ;
for

cba = 1 : l e n g t h ( d 1 v a l s )
f o r cb = 1 : l e n g t h ( d 2 v a l s )
d1 = d 1 v a l s ( cba ) ;
d2 = d 2 v a l s ( cb ) ;
figure ;
hold ;
t i t l e ( s t r c a t ( ’ d1 = ’ , num2str ( d1 ∗1E7 ) , ’nm, d2 = ’ , num2str ( d2 ∗1E7 ) , ’nm ’ ) , ’ FontWeight ’ , ’ b ’ , ’
FontSize ’ , 2 4 ) ;
for

abc = 1 : 4
%Ln = L n I I I V ∗ 1 0 ˆ ( abc −1) ;
%Lp = L p I I I V ∗ 1 0 ˆ ( abc −1) ;
Ln = L n I I I V ;
Lp = L p I I I V ;
Sp = S p I I I V ∗ 1 0 0 ˆ ( abc −1) ;
Sn = S n I I I V ∗ 1 0 0 ˆ ( abc −1) ;
w l d a t a 1 = c s v r e a d ( ’ wl data GaAsP . c s v ’ ) ;
J I I I V = z e r o s ( numpts , 1 ) ;
V I I I V = z e r o s ( numpts , 1 ) ;
for

i = 1 : numpts
V I I I V ( i , 1 ) = 1 . 5 ∗ ( 2 ∗ i / ( numpts+1) − 1 ) ;
Vval = V I I I V ( i , 1 ) ;
try
J I I I V ( i , 1 ) = S p h e r e I V ( Sp , Sn , Ln , Lp , Vval , w l d a t a 1 , d1 , d2 ) ;
if

J III V ( i ,1) > 0.03
if

J I I I V ( i , 1 ) < −0.01

J I I I V ( i , 1 ) <= 0 && V I I I V ( i , 1 ) > 1
break

else
J III V ( i ,1) = 0;
end
end
catch
continue
end
end
s c a t t e r ( V III V , J I I I V ) ;
J s ( abc , : ) = J I I I V ;
end
%l e g e n d ( s t r c a t ( ’ L = ’ , num2str ( L n I I I V ∗1E4 ) , ’um ’ ) , s t r c a t ( ’ L = ’ , num2str ( L n I I I V ∗1E5 ) , ’um
’ ) , s t r c a t ( ’ L = ’ , num2str ( L n I I I V ∗1E6 ) , ’um ’ ) , s t r c a t ( ’ L = ’ , num2str ( L n I I I V ∗1E7 ) , ’um ’ )
, ’ FontWeight ’ , ’ b ’ , ’ F o n t S i z e ’ , 2 4 ) ;
l e g e n d ( s t r c a t ( ’ S = ’ , num2str ( S n I I I V ) , ’ cm/ s ’ ) , s t r c a t ( ’ S = ’ , num2str ( S n I I I V ∗ 1 0 0 ) , ’ cm/ s ’ )
, s t r c a t ( ’ S = ’ , num2str ( S n I I I V ∗1E4 ) , ’ cm/ s ’ ) , s t r c a t ( ’ S = ’ , num2str ( S n I I I V ∗1E6 ) , ’ cm/ s
’ ) , ’ FontWeight ’ , ’ b ’ , ’ F o n t S i z e ’ , 2 4 ) ;
x l a b e l ( ’ V o l t a g e (V) ’ , ’ FontWeight ’ , ’ b ’ , ’ F o n t S i z e ’ , 2 4 ) ;
y l a b e l ( ’ C u r r e n t D e n s i t y (A/cmˆ 2 ) ’ , ’ FontWeight ’ , ’ b ’ , ’ F o n t S i z e ’ , 2 4 ) ;
end
end
end
if

% C a l c u l a t e and p l o t

the J V curves

for

the wire

cell

figure ;
hold ;
f o r abcd = 1 : 4

130

Ln = L n w i r e ∗ 1 0 ˆ ( abcd −1) ;
Lp = L p w i r e ∗ 1 0 ˆ ( abcd −1) ;

wl data2 = c s v r e a d ( ’ w l d a t a S i G e . csv ’ ) ;
for

abcde = 1 : l e n g t h ( w l d a t a 1 ( : , 4 ) )
N i = w l d a t a ( abcde , 3 ) ;
a l p = 4∗ p i ∗ N i / w l d a t a ( abcde , 1 ) ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

w l d a t a 2 ( abcde , 4 )= w l d a t a 2 ( abcde , 4 ) − w l d a t a 1 ( abcde , 4 ) ∗(1− exp(− a l p ∗ ( d1+d2 ) ) ) ;
end
J w i r e = z e r o s ( numpts , 1 ) ;
V w i r e = z e r o s ( numpts , 1 ) ;
for

i = 1 : numpts
V w i r e ( i , 1 ) = 0 . 0 1 ∗ ( i −(numpts +1) / 2 ) ;
Vval = V w i r e ( i , 1 ) ;
try
J w i r e ( i , 1 ) = Wire IV ( S p w i r e , Ln , Lp , Vval , w l d a t a 2 , r1 , r2 , L ) ;
if

J w i r e ( i , 1 ) > 10

J w i r e ( i , 1 ) < −0.01

J wire ( i ,1) = 0;
end
catch
continue
end
end
s c a t t e r ( V wire , J w i r e ) ;
end
end

% C a l c u l a t e d and p l o t

i n d i v i d u a l and tandem p e r f o r m a n c e

w l d a t a 1 = c s v r e a d ( ’ wl data GaAsP . c s v ’ ) ;
wl data2 = c s v r e a d ( ’ w l d a t a S i G e . csv ’ ) ;
for

abcde = 1 : l e n g t h ( w l d a t a 1 ( : , 4 ) )
N i = w l d a t a 1 ( abcde , 3 ) ;
a l p = 4∗ p i ∗ N i / w l d a t a 1 ( abcde , 1 ) ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

w l d a t a 2 ( abcde , 4 )= w l d a t a 2 ( abcde , 4 ) − w l d a t a 1 ( abcde , 4 ) ∗(1− exp(− a l p ∗ ( d1+d2 ) ) ) ;
end
J I I I V = z e r o s ( numpts , 1 ) ;
V I I I V = z e r o s ( numpts , 1 ) ;
J w i r e = z e r o s ( numpts , 1 ) ;
V w i r e = z e r o s ( numpts , 1 ) ;
for

i = 1 : numpts
% G e n e r a t e IV c u r v e s

for

I I I −V c e l l

V I I I V ( i , 1 ) = 1 . 5 ∗ ( 2 ∗ i / ( numpts+1) − 1 ) ;
Vval1 = V I I I V ( i , 1 ) ;
try
J I I I V ( i , 1 ) = S p h e r e I V ( S p I I I V , S n I I I V , L n I I I V , L p I I I V , Vval1 , w l d a t a 1 , d1 , d2 ) ;
if

J III V ( i ,1) > 0.03

J I I I V ( i , 1 ) < −0.01

J III V ( i ,1) = 0;
end
catch
continue
end

% G e n e r a t e IV c u r v e s

f o r SiGE c e l l

V w i r e ( i , 1 ) = 1 . 2 ∗ ( 2 ∗ i / ( numpts+1) − 1 ) ;
Vval2 = V w i r e ( i , 1 ) ;
try
J w i r e ( i , 1 ) = Wire IV ( S p w i r e , L n w i r e , L p w i r e , Vval2 , w l d a t a 2 , r1 , r2 , L ) ;
if

J w i r e ( i , 1 ) > 10

J w i r e ( i , 1 ) < −0.01

J wire ( i ,1) = 0;
end
catch

131

continue
end
end

% G e n e r a t e t h e tandem IV c u r v e
J = z e r o s ( numpts , 1 ) ;
V = z e r o s ( numpts , 1 ) ;
if

J w i r e ( ( numpts+1) / 2 , 1 ) > J I I I V ( ( numpts +1) / 2 , 1 )
f o r bc = 1 : numpts
step = 1;
if

J I I I V ( bc , 1 ) > J w i r e ( ( numpts+1) / 2 , 1 )

J I I I V ( bc , 1 ) == 0

continue
end
while

J w i r e ( s t e p , 1 ) == 0

step = step + 1;
end
while

J w i r e ( s t e p , 1 ) > J I I I V ( bc , 1 )

s t e p = s t e p +1;
if

s t e p == numpts
break

end
end
J ( bc , 1 ) = J I I I V ( bc , 1 ) ;
if

a b s ( J w i r e ( s t e p , 1 ) − J I I I V ( bc , 1 ) ) > a b s ( J w i r e ( s t e p −1 ,1) − J I I I V ( bc , 1 ) )
V( bc , 1 ) = V I I I V ( bc , 1 ) + V w i r e ( s t e p −1 ,1)−Rs ∗ ( J w i r e ( s t e p , 1 )+J w i r e ( s t e p −1 ,1) ) / 2 ;

else
V( bc , 1 ) = V I I I V ( bc , 1 ) + V w i r e ( s t e p , 1 )−Rs ∗ ( J w i r e ( s t e p , 1 )+J w i r e ( s t e p −1 ,1) ) / 2 ;
end
end
else
f o r bc = 1 : numpts
step = 1;
if

J w i r e ( bc , 1 ) > J I I I V ( ( numpts +1) / 2 , 1 )

J w i r e ( bc , 1 ) == 0

continue
end
while

J I I I V ( s t e p , 1 ) == 0

step = step + 1;
end
while

J I I I V ( s t e p , 1 ) > J w i r e ( bc , 1 )

s t e p = s t e p +1;
if

s t e p == numpts
break

end
end
J ( bc , 1 ) = J w i r e ( bc , 1 ) ;
if

a b s ( J I I I V ( s t e p , 1 ) − J w i r e ( bc , 1 ) ) > a b s ( J I I I V ( s t e p −1 ,1) − J w i r e ( bc , 1 ) )
V( bc , 1 ) = V w i r e ( bc , 1 ) + V I I I V ( s t e p −1 ,1)−Rs ∗ ( J w i r e ( s t e p , 1 )+J w i r e ( s t e p −1 ,1) ) / 2 ;

else
V( bc , 1 ) = V w i r e ( bc , 1 ) + V I I I V ( s t e p , 1 )−Rs ∗ ( J w i r e ( s t e p , 1 )+J w i r e ( s t e p −1 ,1) ) / 2 ;
end
end
end

% figure ;
% hold ;
% s c a t t e r ( V III V , J I I I V ) ;
% s c a t t e r ( V wire , J w i r e ) ;
% s c a t t e r (V, J ) ;
% p l o t (V, J ∗ 1 0 0 0 ) ;
s e t ( gca , ’ f o n t s i z e ’ , 3 0 , ’ f o n t w e i g h t ’ , ’ b ’ ) ;
% l e g e n d ( ’ GaAsP ’ , ’ SiGe ’ , ’ Tandem ’ , ’ L o c a t i o n ’ , ’ South ’ )

132

x l a b e l ( ’ V o l t a g e (V) ’ , ’ F o n t S i z e ’ , 3 0 ) ;
y l a b e l ( ’ C u r r e n t D e n s i t y (mA/cmˆ 2 ) ’ , ’ F o n t S i z e ’ , 3 0 ) ;
s e t ( gca , ’ XLim ’ , [ 0

2 ] , ’ YLim ’ , [ 0

30]) ;

g r i d on ;

% E x t r a c t t h e E f f , JSC , and VOC
e f f o v e r a l l = max (V. ∗ J ) ∗1000
e f f w i r e = max ( V w i r e . ∗ J w i r e ) ∗ 1 0 0 0 ;
e f f I I I V = max ( V I I I V . ∗ J I I I V ) ∗ 1 0 0 0 ;

J S C w i r e = J w i r e ( ( numpts+1) / 2 ) ∗ 1 0 0 0 ;
J S C I I I V = J I I I V ( ( numpts +1) / 2 ) ∗ 1 0 0 0 ;
jsc = 1;
w h i l e V( j s c , 1 ) < 0

J ( j s c , 1 ) == 0

jsc = jsc + 1;
end
if

a b s (V( j s c , 1 ) ) < V( j s c +1 ,1)
J SC overall = J ( jsc , 1 ) ∗1000;

else
J SC overall = J ( jsc , 1 ) ∗1000;
end

voc = 1 ;
while

J w i r e ( voc , 1 ) > 0

V w i r e ( voc , 1 ) <=0

voc = voc +1;
end
VOC wire = V w i r e ( voc , 1 ) ;
voc = 1 ;
while

J I I I V ( voc , 1 ) > 0

V I I I V ( voc , 1 ) < 0

voc = voc +1;
end
VOC III V = V I I I V ( voc , 1 ) ;
voc = 1 ;
w h i l e J ( voc , 1 ) >= 0

| | V( voc , 1 ) <= 0

voc = voc +1;
end
V O C o v e r a l l = V( voc −1 ,1) ;

C.6

Code for the Hemispherical Subcell

Listing C.6: Code to calculate the current at a given voltage for a hemispherical solar cell
function

[ J ] = S p h e r e I V ( Sp , Sn , Ln , Lp , V, w l d a t a , d1 , d2 )

% d1 = 0 . 1 E−4; % The t h i c k n e s s
% d2 = 2E−4; % The t h i c k n e s s

% Sp = 1 0 0 ; % The I I I V

p t y p e SRV

% Sn = 1 0 0 ; % The I I I V

n t y p e SRV

% Ln = 1E−5; % The e l e c t r o n
% Lp = 1E−5; % The h o l e

the

diffusion

III V

length

emitter

base

the

III V

n type

A n a l y t i c a l Model f o r a S p h e r i c a l

Solar

Cell

% w l d a t a = c s v r e a d ( ’ wl data GaAsP . c s v ’ ) ;
% V = 0;

p type base

the

Dan Turner−Evans ,

J u l 2012

d t @ c a l t e c h . edu

133

emitter

California

Institute

o f Technology

calculating

t h e IV c u r v e

% This

a MATLAB t o o l

for

o f a hemi−s p h e r i c a l

solar

% cell

% Specify

the

appropriate

constants

k = 8 . 6 1 7 3 3 2 4 E−5; % The Bolzmann c o n s t a n t

( eV/K)

T = 3 0 0 ; % The t e m p e r a t u r e (K)
q = 1 . 6 0 2 1 9 E−19; % The f u n d a m e n t a l
VT = k∗T ; % The t h e r m a l

voltage

electrical

c h a r g e (C)

( eV )

NA = 5 E16 ; % The p−type ,

c o r e d o p i n g (cmˆ−3)

ND = 9 E17 ; % The n−type ,

shell

d o p i n g (cmˆ−3)

NC = 5 . 7 5 8 E17 ; % E f f e c t i v e

density

states

t h e c o n d u c t i o n band (cmˆ−3)

NV = 9 . 5 4 3 E18 ; % E f f e c t i v e

density

states

the

v a l e n c e band (cmˆ−3)

EG = 1 . 5 3 8 9 ; % The m a t e r i a l bandgap ( eV )
n i = s q r t (NC∗NV) ∗ exp(−EG/ ( 2 ∗VT) ) ; % The i n t r i n s i c

carrier

P h i 0 = VT∗ l o g (NA∗ND/ n i ˆ 2 ) ; % The b u i l t

( eV )

%V = 0 ; % The a p p l i e d

voltage

thickness

% d2 = 1 . 9 E−4; % The c o r e

radius

(cm)

% eps GaP = 1 1 . 1 ; % The d i e l e c t r i c

constant

% eps GaAs = 1 2 . 9 ; % The d i e l e c t r i c

o f GaP

constant

o f GaAs

e p s 0 = 8 . 8 5 4 1 8 7 8 1 7 6 2 0 E−14; % Vacuum p e r m i t t i v t y
e p s = ( 1 1 . 1 + 1 . 8 ∗ 0 . 9 ) ∗ e p s 0 ; % The o v e r a l l

solve

for

the

(cmˆ−3)

(V)

% d1 = 0 . 1 E−4; % E m i t t e r

% First ,

density

limits

the

i n F/cm

permittivity

depletion

region

syms x2 x4 ;
x2 val = 0;
x4 val = 0;
S = s o l v e ( P h i 0 + V == q / e p s ∗NA∗ ( d2 ˆ2/6+ x4 ˆ 3 / ( 3 ∗ d2 )−x4 ˆ 2 / 2 )+q / e p s ∗ND∗ x2 ˆ2∗(1/2+ x2 / ( 3 ∗ d2 ) ) , NA∗ ( d2
ˆ3−x4 ˆ 3 ) == ND∗ ( ( d2+x2 ) ˆ3−d2 ˆ 3 ) ) ;
f o r a = 1 : l e n g t h ( S . x2 )
if

i s r e a l ( S . x2 ( a ) ) && S . x2 ( a )>0 && S . x2 ( a )0 && S . x4 ( a )x 2 v a l = d o u b l e ( S . x2 ( a ) ) ;
x 4 v a l = d o u b l e ( S . x4 ( a ) ) ;

end
end

% Plot the

voltage

variation

across

the

depletion

region

%r 1 = x 4 v a l : x 4 v a l / 5 0 0 0 : d2 ;
%V1 = q∗NA/ ( 6 ∗ e p s ) ∗ r 1 .ˆ2+ q∗NA∗ x 4 v a l ˆ 3 / ( 3 ∗ e p s ) . / r1−q∗NA/ ( 2 ∗ e p s ) ∗ x 4 v a l ˆ 2 ;
%r 2 = d2 : x 2 v a l / 1 0 0 : d2+x 2 v a l ;
%V2 = −q∗ND/ ( 6 ∗ e p s ) ∗ r 2 .ˆ2 − q∗ND∗ ( d2+x 2 v a l ) ˆ 3 / ( 3 ∗ e p s ) . / r 2+q∗ND/ ( 2 ∗ e p s ) ∗ ( d2+x 2 v a l ) ˆ2+ P h i 0+V ;
% figure ;
% p l o t ( r1 , V1 ) ;
% hold ;
% p l o t ( r2 , V2 ) ;

% Specify

the

constants

for

%Ln = 1E−5; % The e l e c t r o n
%Lp = 1E−4; % The h o l e
%N r e = 3 ; % The r e a l

t h e q u a s i −n e u t r a l

region

diffusion

the

diffusion
part

length

index

%N im = 0 . 5 ; % The i m a g i n a r y p a r t
%wl = 6E−5; % The w a v e l e n g t h ,

length

in
the

material

(cm)

refraction

the index

refraction

i n cm .

% a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

%i n c p o w = 1 0 0 ; % AM1. 5G power (mW/cmˆ 2 )
h = 6 . 6 2 6 E−34; % Planck ’ s
c = 3 E10 ; % Speed o f

constant

light

( Jsec )

(cm/ s )

% f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t
mup = 3 6 7 . 5 ; % The h o l e

mobility

photon

(cmˆ2/V/ s e c )

134

flux

( p h o t o n s /cmˆ2/ s e c )

Dp = mup∗k∗T ; % The d i f f u s i o n
mun = 7 6 6 1 ; % The e l e c t r o n

Dn = mun∗k∗T ; % The d i f f u s i o n

% Calculate

the

current

coefficient

mobility

for

holes

(cmˆ2/ s e c )

(cmˆ2/V/ s e c )

coefficient

for

electrons

(cmˆ2/ s e c )

r e g i o n s 2 and 3

if V > 0
Umax = n i ∗ s q r t (Dn∗Dp) / ( Ln∗Lp ) ∗ s i n h (V/ ( 2 ∗ k∗T) ) ;
else
Umax = 0 ;
end
kappa = p i ∗k∗T/ ( P h i 0+V) ;
rV = x 4 v a l + l o g (NA/ n i ) / l o g (NA∗ND/ n i ˆ 2 ) ∗ ( x 2 v a l +(d2−x 4 v a l ) ) ;
r 1 = rV − ( x 2 v a l +(d2−x 4 v a l ) ) /2∗ kappa ;
r 2 = rV + ( x 2 v a l +(d2−x 4 v a l ) ) /2∗ kappa ;
J23 l = 0;
for

i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i , 1 ) ;
N im = w l d a t a ( i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

% J 2 3 l = 3∗ q∗ f l u x / ( a l p h ˆ 3 ∗ ( d1+d2 ) ˆ 3 ) ∗ ( ( a l p h ∗ ( d2+x 2 v a l ) ∗ ( a l p h ∗ ( d2+x 2 v a l ) −2)+2)∗ exp ( a l p h ∗ (
x 2 v a l −d1 ) ) − ( a l p h ∗ x 4 v a l ∗ ( a l p h ∗ x 4 v a l −2)+2)∗ exp ( a l p h ∗ ( x 4 v a l −d1−d2 ) ) ) +J 2 3 l ;
J 2 3 l = q∗ f l u x ∗ ( exp ( a l p h ∗ ( x 2 v a l −d1 ) )−exp ( a l p h ∗ ( x 4 v a l −d1−d2 ) ) ) + J 2 3 l ;
end
J 2 3 r = q∗Umax/ ( 3 ∗ ( d1+d2 ) ˆ 2 ) ∗ ( r 2 ˆ3− r 1 ˆ 3 ) ;

% Calculate
% First ,

the

solve

% symbolic

current
for

the

region 1

constants

g i v e n t h e boundary c o n d i t i o n s

using

logic .

syms A B C ;
exp sum1 = 0 ;
exp sum2 = 0 ;
exp sum3 = 0 ;
exp sum4 = 0 ;
exp sum5 = 0 ;
flux tot = 0;
for

i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i , 1 ) ;
N im = w l d a t a ( i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

flux tot = flux + flux tot ;
exp sum1 = exp(− a l p h ∗ ( d1+d2 ) ) ∗2∗ a l p h ˆ2∗ f l u x ∗Lnˆ4/(1 − a l p h ˆ2∗Ln ˆ 2 ) ˆ2/Dn + exp sum1 ;
exp sum2 = exp ( a l p h ∗ ( x 4 v a l −(d1+d2 ) ) ) ∗ a l p h ∗ f l u x /Dn∗ ( Lnˆ2/(1 − a l p h ˆ2∗Ln ˆ 2 ) ) ∗(1+2∗ a l p h ∗Ln ˆ2/
x 4 v a l /(1− a l p h ˆ2∗Ln ˆ 2 ) ) + exp sum2 ;
exp sum3 = exp(− a l p h ∗ ( d1+d2 ) ) ∗ a l p h ∗ f l u x /Dn∗Lnˆ2/(1 − a l p h ˆ2∗Ln ˆ 2 ) ∗(1+2∗ a l p h ˆ2∗Lnˆ2/(1 − a l p h ˆ2∗Ln
ˆ 2 ) ) + exp sum3 ;
exp sum4 = exp(− a l p h ∗ ( d1+d2 ) ) ∗ a l p h ˆ2∗ f l u x /Dn∗Lnˆ2/(1 − a l p h ˆ2∗Ln ˆ 2 ) ∗(1+ a l p h ˆ2∗Lnˆ2/(1 − a l p h ˆ2∗Ln
ˆ 2 ) ) + exp sum4 ;
exp sum5 = exp ( a l p h ∗ ( x 4 v a l −(d1+d2 ) ) ) ∗ a l p h ∗ f l u x /Dn∗ ( Lnˆ2/(1 − a l p h ˆ2∗Ln ˆ 2 ) ) ∗ ( a l p h +2∗ a l p h ˆ2∗Ln ˆ2/
x 4 v a l /(1− a l p h ˆ2∗Ln ˆ 2 ) −2∗ a l p h ∗Ln ˆ2/ x 4 v a l ˆ2/(1 − a l p h ˆ2∗Ln ˆ 2 ) )+exp sum5 ;
end
S2 = s o l v e (A+B+exp sum1 == 0 , A∗ exp ( x 4 v a l /Ln ) / x 4 v a l + B∗ exp(− x 4 v a l /Ln ) / x 4 v a l+ C∗ s i n h ( x 4 v a l /Ln
) / x 4 v a l + exp sum2 == n i ˆ2/NA∗ ( exp (V/VT) −1) , Sn ∗ (A/Ln + B/Ln + C/Ln + exp sum3 ) == −Dn∗ (A/ ( 2 ∗
Ln ˆ 2 )+B/ ( 2 ∗ Ln ˆ 2 ) + exp sum4 ) ) ;
A v a l = d o u b l e ( S2 . A) ;
B v a l = d o u b l e ( S2 . B) ;

135

C v a l = d o u b l e ( S2 . C) ;

J1 = −q∗Dn∗ ( A v a l ∗ exp ( x 4 v a l /Ln ) / x 4 v a l ∗ ( 1 / Ln−1/ x 4 v a l ) − B v a l ∗ exp(− x 4 v a l /Ln ) / x 4 v a l ∗ ( 1 / Ln+1/
x 4 v a l ) + C v a l ∗ ( c o s h ( x 4 v a l /Ln ) / ( x 4 v a l ∗Ln ) − s i n h ( x 4 v a l /Ln ) / x 4 v a l ˆ 2 ) + exp sum5 ) ;

% Calculate

the

current

%Sp = 1 0 0 ; % S u r f a c e
% First ,

solve

for

region 4

recombination

the

constants

velocity

(cm/ s )

g i v e n t h e boundary c o n d i t i o n .

syms A4 B4 ;
JH4S = 0 ;
JD4S = 0 ;
CBH4S = 0 ;
CDH4S = 0 ;
for

i i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i i , 1 ) ;
N im = w l d a t a ( i i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

JH4S = a l p h ∗ ( a l p h ∗ f l u x /Dp) ∗ ( Lpˆ2/(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ∗(1+2∗ a l p h ∗Lp ˆ 2 / ( ( d1+d2 ) ∗(1− a l p h ˆ2∗Lp ˆ 2 ) ) −2∗
Lp ˆ 2 / ( ( d1+d2 ) ˆ2∗(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ) + JH4S ;
JD4S = a l p h ∗ exp ( a l p h ∗ ( x 2 v a l −d1 ) ) ∗ ( a l p h ∗ f l u x /Dp) ∗ ( Lpˆ2/(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ∗(1+2∗ a l p h ∗Lp ˆ 2 / ( (
x 2 v a l+d2 ) ∗(1− a l p h ˆ2∗Lp ˆ 2 ) ) −2∗Lp ˆ 2 / ( ( x 2 v a l+d2 ) ˆ2∗(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ) + JD4S ;
CBH4S = ( a l p h ∗ f l u x /Dp) ∗ ( Lpˆ2/(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ∗(1+2∗ a l p h ∗Lp ˆ 2 / ( ( d1+d2 ) ∗(1− a l p h ˆ2∗Lp ˆ 2 ) ) ) +
CBH4S ;
CDH4S = exp ( a l p h ∗ ( x 2 v a l −d1 ) ) ∗ ( a l p h ∗ f l u x /Dp) ∗ ( Lpˆ2/(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ∗(1+2∗ a l p h ∗Lp ˆ 2 / ( ( x 2 v a l+d2
) ∗(1− a l p h ˆ2∗Lp ˆ 2 ) ) ) + CDH4S ;
end
JA4S = A4∗ exp ( ( d1+d2 ) /Lp ) / ( d1+d2 ) ∗ ( 1 / Lp−1/( d1+d2 ) ) ;
JB4S = −B4∗ exp ( −( d1+d2 ) /Lp ) / ( d1+d2 ) ∗ ( 1 / Lp+1/( d1+d2 ) ) ;
CB = A4∗ exp ( ( d1+d2 ) /Lp ) / ( d1+d2 )+B4∗ exp ( −( d1+d2 ) /Lp ) / ( d1+d2 )+CBH4S ;
CD = A4∗ exp ( ( x 2 v a l+d2 ) /Lp ) / ( x 2 v a l+d2 )+B4∗ exp ( −( x 2 v a l+d2 ) /Lp ) / ( x 2 v a l+d2 )+CDH4S ;
S3 = s o l v e ( Sp∗CB == −Dp∗ ( JA4S+JB4S+JH4S ) , CD == n i ˆ2/ND∗ ( exp (V/VT) −1) ) ;

%Now ,

calculate

the

carrier

d i s t r i b u t i o n and t h e

current .

A 4 v a l = d o u b l e ( S3 . A4 ) ;
B 4 v a l = d o u b l e ( S3 . B4 ) ;

% r v a l s = x 2 v a l+d2 : ( d1−x 2 v a l ) / 1 0 0 : d1+d2 ;
% C vals = zeros (1 , length ( r v a l s ) ) ;
% CH4S = 0 ;
% for

steppin = 1: length ( r v a l s )

for

i i =1: l e n g t h ( w l d a t a ( : , 1 ) )

wl = w l d a t a ( i i , 1 ) ;

N im = w l d a t a ( i i , 3 ) ;

a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

inc pow = wl data ( i i , 4 ) ;

coefficent

(cmˆ−1)

f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

CH4S = exp ( a l p h ∗ ( r v a l s ( s t e p p i n )−d1−d2 ) ) ∗ ( a l p h ∗ f l u x /Dp) ∗ ( Lpˆ2/(1 − a l p h ˆ2∗Lp ˆ 2 ) ) ∗(1+2∗ a l p h

photon

flux

( p h o t o n s /cmˆ2/ s e c )

∗Lp ˆ 2 / ( r v a l s ( s t e p p i n ) ∗(1− a l p h ˆ2∗Lp ˆ 2 ) ) ) + CH4S ;

end
C v a l s ( 1 , s t e p p i n ) = A 4 v a l ∗ exp ( r v a l s ( s t e p p i n ) /Lp ) / r v a l s ( s t e p p i n )+B 4 v a l ∗ exp(− r v a l s (

s t e p p i n ) /Lp ) / r v a l s ( s t e p p i n )+CH4S ;
% end
% figure ;
% plot ( r vals , C vals ) ;

JA4 = A 4 v a l ∗ exp ( ( d2+x 2 v a l ) /Lp ) / ( d2+x 2 v a l ) ∗ ( 1 / Lp−1/( d2+x 2 v a l ) ) ;
JB4 = −B 4 v a l ∗ exp ( −( d2+x 2 v a l ) /Lp ) / ( d2+x 2 v a l ) ∗ ( 1 / Lp+1/( d2+x 2 v a l ) ) ;
J4 = q∗Dp∗ ( JA4+JB4+JD4S ) ;

136

%The t o t a l

current

J = J1 ∗ ( x 4 v a l ) ˆ 2 / ( d2+d1 ) ˆ2− J 2 3 r+J 2 3 l ∗ d2 ˆ 2 / ( d1+d2 ) ˆ2+J4 ∗ ( d2+x 2 v a l ) ˆ 2 / ( d2+d1 ) ˆ 2 ;

C.7

Code for the Wire Subcell

Listing C.7: Code to calculate the current at a given voltage for a wire solar cell
function

[ J ] = Wire IV ( Sp , Ln , Lp , V, w l d a t a , r1 , r2 , L )

A n a l y t i c a l Model f o r a Wire S o l a r

From work by Brendan Kayes

Dan Turner−Evans , Aug 2012

Cell

d t @ c a l t e c h . edu

California

Institute

o f Technology

calculating

t h e IV c u r v e

% This

% solar

a MATLAB t o o l

for

of a radial

junction

cell

% Specify

the

appropriate

constants

k = 8 . 6 1 7 3 3 2 4 E−5; % The Bolzmann c o n s t a n t

( eV/K)

T = 3 0 0 ; % The t e m p e r a t u r e (K)
q = 1 . 6 0 2 1 9 E−19; % The f u n d a m e n t a l
VT = k∗T ; % The t h e r m a l

voltage

electrical

NA = 1 E16 ; % The p−type ,

c o r e d o p i n g (cmˆ−3)

ND = 1 E18 ; % The n−type ,

shell

NC = 1 E19 ; % E f f e c t i v e

d o p i n g (cmˆ−3)

density

NV = 6 . 3 E18 ; % E f f e c t i v e

c h a r g e (C)

( eV )

density

states
of

states

t h e c o n d u c t i o n band (cmˆ−3)
in

the

v a l e n c e band (cmˆ−3)

EG = 0 . 7 5 7 ; % The m a t e r i a l bandgap ( eV )
n i = s q r t (NC∗NV) ∗ exp(−EG/ ( 2 ∗VT) ) ; % The i n t r i n s i c

carrier

P h i 0 = VT∗ l o g (NA∗ND/ n i ˆ 2 ) ; % The b u i l t

( eV )

% V = 0 ; % The a p p l i e d

voltage

% r 1 = 0 . 1 E−4; % E m i t t e r

voltage

density

(cmˆ−3)

(V)

thickness

% r 2 = 0 . 6 5 E−4; % The c o r e

radius

(cm)
(cm)

% e p s 0 = 8 . 8 5 4 1 8 7 8 1 7 6 2 0 E−14; % Vacuum p e r m i t t i v t y
% e p s = 1 5 . 7 5 ∗ e p s 0 ; % The o v e r a l l

i n F/cm

permittivity

% e p s i l o n p = eps ;
% e p s i l o n n = eps ;

% First ,

solve

for

the

limits

the

depletion

region

x0 = [ r 1 /100 r2−r 1 ] ;
x = f s o l v e (@( x ) d e p l i m i t s ( x , r1 , r2 , V) , x0 ) ;
x2 val = x(1) ;
x4 val = x(2) ;

% Specify

the

constants

for

%Ln = 1E−5; % The e l e c t r o n
%Lp = 1E−4; % The h o l e
%N r e = 3 ; % The r e a l

t h e q u a s i −n e u t r a l

region

diffusion

the

diffusion
part

length

index

%N im = 0 . 5 ; % The i m a g i n a r y p a r t
%wl = 6E−5; % The w a v e l e n g t h ,

length

in
the

material

(cm) r c i

(cm)

refraction

the index

refraction

i n cm .

% a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

%i n c p o w = 1 0 0 ; % AM1. 5G power (mW/cmˆ 2 )
h = 6 . 6 2 6 E−34; % Planck ’ s
c = 3 E10 ; % Speed o f

constant

light

( Jsec )

(cm/ s )

% f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t
mup = 1 7 5 7 ; % The h o l e

mobility

photon

(cmˆ2/V/ s e c )

137

flux

( p h o t o n s /cmˆ2/ s e c )

Dp = mup∗k∗T ; % The d i f f u s i o n
mun = 3 6 5 2 ; % The e l e c t r o n

Dn = mun∗k∗T ; % The d i f f u s i o n

% Calculate

the

current

coefficient

mobility

for

holes

(cmˆ2/ s e c )

(cmˆ2/V/ s e c )

coefficient

for

electrons

(cmˆ2/ s e c )

r e g i o n s 2 and 3

Umax = n i ∗ s q r t (Dn+Dp) / ( Ln∗Lp ) ∗ s i n h ( q∗V/ ( 2 ∗ k∗T) ) ;
kappa = p i ∗k∗T/ ( P h i 0+V) ;
rV = x 4 v a l + l o g (NA/ n i ) / l o g (NA∗ND/ n i ˆ 2 ) ∗ ( x 2 v a l +( r2−x 4 v a l ) ) ;
r 1 r = rV − ( x 2 v a l +( r2−x 4 v a l ) ) /2∗ kappa ;
r 2 r = rV + ( x 2 v a l +( r2−x 4 v a l ) ) /2∗ kappa ;
J23 l = 0;
for

i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i , 1 ) ;
N im = w l d a t a ( i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

J 2 3 l = q∗ f l u x ∗(1− exp(− a l p h ∗L ) ) ∗ ( ( r 2+x 2 v a l ) ˆ2− x 4 v a l ˆ 2 ) / ( r 1+r 2 ) ˆ2 + J 2 3 l ;
end
J 2 3 r = −q∗L∗Umax/ ( r 1+r 2 ) ˆ 2 ∗ ( r 2 r ˆ2− r 1 r ˆ 2 ) ;

% Calculate
% First ,

the

solve

% symbolic

current
for

the

region 1

constants

g i v e n t h e boundary c o n d i t i o n s

using

logic .

J1 l = 0;
B5 = x 4 v a l /Ln ;
B1 = ( r 1+r 2 ) /Lp ;
for

i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i , 1 ) ;
N im = w l d a t a ( i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

J 1 l = 2∗ q∗ f l u x ∗(1− exp(− a l p h ∗L ) ) ∗Ln ˆ2/ Lpˆ2∗B5/B1ˆ2∗ b e s s e l i ( 1 , B5 ) / b e s s e l i ( 0 , B5 )+J 1 l ;
end
J 1 r = −2∗q ∗ ( n i ˆ2/NA) ∗L∗Dn/Lpˆ2∗B5/B1ˆ2∗ b e s s e l i ( 1 , B5 ) / b e s s e l i ( 0 , B5 ) ∗ ( exp (V/VT) −1) ;

% Calculate

the

current

% Sp = 1 0 0 ; % S u r f a c e

region 4

recombination

velocity

(cm/ s )

J4 l = 0;
B2 = ( r 2 + x 2 v a l ) /Lp ;
B4 = Lp∗Sp/Dp ;
f 1 = b e s s e l i ( 1 , B1 )+B4∗ b e s s e l i ( 0 , B1 ) ;
f 2 = b e s s e l k ( 1 , B1 )−B4∗ b e s s e l k ( 0 , B1 ) ;
for

i =1: l e n g t h ( w l d a t a ( : , 1 ) )
wl = w l d a t a ( i , 1 ) ;
N im = w l d a t a ( i , 3 ) ;
a l p h = 4∗ p i ∗N im / wl ; % The a b s o r p t i o n

coefficent

(cmˆ−1)

inc pow = wl data ( i , 4 ) ;
f l u x = i n c p o w / 1 0 00 ∗ wl / ( c ∗h ) ; % The i n c i d e n t

photon

flux

( p h o t o n s /cmˆ2/ s e c )

J 4 l = 2∗ q∗ f l u x ∗(1− exp(− a l p h ∗L ) ) ∗B2/B1 ˆ 2 ∗ ( ( b e s s e l k ( 1 , B2 ) ∗ ( f 1 −B4∗ b e s s e l i ( 0 , B2 ) )− b e s s e l i ( 1 , B2 ) ∗ (
f 2+B4∗ b e s s e l k ( 0 , B2 ) ) ) / ( f 1 ∗ b e s s e l k ( 0 , B2 )+f 2 ∗ b e s s e l i ( 0 , B2 ) ) )+J 4 l ;
end
J 4 r = −2∗q ∗ ( n i ˆ2/ND) ∗L∗Dp/Lpˆ2∗B2/B1 ˆ 2 ∗ ( ( f 1 ∗ b e s s e l k ( 1 , B2 )−f 2 ∗ b e s s e l i ( 1 , B2 ) ) / ( f 1 ∗ b e s s e l k ( 0 , B2 )+f 2 ∗
b e s s e l i ( 0 , B2 ) ) ) ∗ ( exp (V/VT) −1) ;

%The t o t a l

current

% J = zeros (7 ,1) ;
% J (1 ,1) = J1 r + J1 l + J23 r + J23 l + J4 r + J4 l ;

138

% J (2 ,1) = J1 r ;
% J (3 ,1) = J1 l ;
% J (4 ,1) = J23 r ;
% J (5 ,1) = J23 l ;
% J (6 ,1) = J4 r ;
% J (7 ,1) = J4 l ;
J = J1 r + J1 l + J23 r + J23 l + J4 r + J4 l ;

139

Appendix D
Transmission Matrix Method Code

In order to design an effective anti-reflection coating, I relied on the transmission matrix
method. In this technique, the transmission and propagation matrices in each layer of a
material stack are calculated and used to find the overall reflection and transmission as
follows:

Tlayer =

1  1 + n10

1 − nn01

1 − nn10
1 + nn10

,

where n0 and n1 are the real parts of the indexes of refraction of the two materials.
Player = 

e−in1 (1+ik1 ) c t1

ein1 (1+ik1 ) c t1

,

where k1 is the imaginary part of the index of the refraction of the material, t1 is the
thickness of the material, ω is the frequency of the light, and c is the speed of light.
M = T1 · P1 · T2 · P2 · ...
Rtotal =

M (2, 1) · M (2, 1)∗
M (1, 1) · M (1, 1)∗

Ttotal =

M (1, 1) · M (1, 1)∗

The left plot of Figure D.1 shows a comparison between the TMM calculations and
FDTD simulations for a 500 nm slab of GaAsP cladded with 20 nm of GaInP and coated
on top with 100 nm of TiOx and 60 nm of MgF. The right plot of Figure D.1 demonstrates

140

the effect of adding the TiOx /MgF coating to the back of the cell.

(a) 1
0.8

Planar Film, No Back ARC
Planar Film, Back ARC

0.8

Front ARC (2 layers)
Window
Absorber

0.6

Reflection

(b) 1

TMM Calculation
FDTD Simulation

0.4

0.2

Window
Back ARC (2 layers)

0.6

400

600

800

1000

Wavelength (nm)

1200

400

600

800

Wavelength (nm)

1000

1200

Figure D.1: TMM calculations. (a) Comparison between TMM calculations and FDTD
simulations. (b) Change in reflection as a back surface antireflection coating is added.

141

Appendix E
Sentaurus Code

E.1

Code from the Tandem Simulations

The code for the simulations run in Chapter 5 is included below. Much of it is adapted
from the work featured in the thesis of Dr. Mike Kelzenberg.(186) The following code all
appeared in the following layout in Sentaurus Workbench.

Figure E.1: Layout of Sentaurus simulations for finding the optical generation in tandem
wire arrays.

Listing E.1: Code to generate a tandem wire array FEM grid
##########################################################

142

# Dan Turner−Evans

# 08/02/11

# A core

shell

wire

solar

cell

##########################################################

##########################################################
# Set the

geometric

device

parameters

##########################################################
#d e f i n e

core height

@core height@

#d e f i n e

core radius

(∗ @core diameter@

#d e f i n e

p i t c h @pitch@

0.5)

#d e f i n e

tj thickness

#d e f i n e

shell thickness

@tj thickness@

#d e f i n e

w i n d o w t h i c k n e s s @window thickness@

@shell thickness@

#d e f i n e

core emitter t

#d e f i n e

shell emitter t

0.1
0.1

#d e f i n e

oxide thickness

@oxide thickness@

#d e f i n e

e x p o s e d @exposed@

##########################################################
# Set the doping

device

parameters

##########################################################
#d e f i n e

window doping 1 e18

#d e f i n e

window tj doping 5 e19

#d e f i n e

s h e l l e m i t t e r d o p i n g 9 e17

#d e f i n e

s h e l l b a s e d o p i n g 5 e16

#d e f i n e

c o r e t j d o p i n g 5 e19

#d e f i n e

c o r e e m i t t e r d o p i n g 9 e17

#d e f i n e

c o r e b a s e d o p i n g 5 e16

##########################################################
# Set the

grid

spacing

##########################################################
#d e f i n e DopMaxGrid 0 . 0 4
#d e f i n e DopMinGrid 0 . 0 0 5
#d e f i n e MatMaxGrid 0 . 0 4
#d e f i n e MatMinGrid 0 . 0 0 2
#d e f i n e MatRatio 2
#d e f i n e MBMaxGrid 0 . 0 4
#d e f i n e MBMinGrid 0 . 0 0 5
#d e f i n e MBRatio 1 . 5
#d e f i n e RefMaxGrid 0 . 0 4
#d e f i n e RefMinGrid 0 . 0 1
#d e f i n e

RefRatio

#d e f i n e

window grid 0.002

1.5

#d e f i n e

shell emitter grid

#d e f i n e

shell base grid

0.04

#d e f i n e

window tj grid

0.002

#d e f i n e

core tj grid

0.002

#d e f i n e

core emitter grid

#d e f i n e

core base grid

#d e f i n e

Air grid

#d e f i n e

Oxide grid

0.02

0.04

0.5
0.5

##########################################################
# Create the

structure

##########################################################

# Air
( s d e g e o : c r e a t e −r e c t a n g l e
0.5)

( p o s i t i o n 0 0 0) ( p o s i t i o n

window thickness

shell thickness

(∗

pitch

0 . 5 ) (+ 2 c o r e h e i g h t

) 0 ) ” Ambient ” ” A i r ” )

143

(∗

tj thickness

#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

# Window
( s d e g e o : c r e a t e −r e c t a n g l e

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

shell thickness

w i n d o w t h i c k n e s s ) (+ c o r e h e i g h t

shell thickness

) 0 ) ”GaInP” ” w i n d o w o u t e r ” )

(∗

tj thickness

tj thickness
0.5)

0.5)

window thickness

# Shell
( s d e g e o : c r e a t e −r e c t a n g l e
window thickness
0.5)

shell thickness

( s d e g e o : c r e a t e −r e c t a n g l e
window thickness
0.5)

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

0.5)

(∗

s h e l l t h i c k n e s s ) (+ c o r e h e i g h t

(∗

tj thickness

0.5)

(∗

(∗ window thickness

0.5)

) 0 ) ”GaAsP” ” s h e l l e m i t t e r ” )

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

0 . 5 ) (− s h e l l t h i c k n e s s

(∗ window thickness

(∗

tj thickness

0 . 5 ) (− s h e l l t h i c k n e s s

0.5)

(∗

s h e l l e m i t t e r t ) ) (+ c o r e h e i g h t

(∗

tj thickness

s h e l l e m i t t e r t ) ) 0 ) ”GaAsP” ” s h e l l b a s e ” )

# TJ
( s d e g e o : c r e a t e −r e c t a n g l e
window thickness

tj thickness

0.5)

(∗

(∗ window thickness

0.5)

) 0) ”

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

0 . 5 ) ) (+ c o r e h e i g h t

(∗

tj thickness

0.5)

(∗

GaInP” ” w i n d o w i n n e r ” )
( s d e g e o : c r e a t e −r e c t a n g l e
core height

(∗

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

tj thickness

( s d e g e o : c r e a t e −r e c t a n g l e

0.5)

(∗

tj thickness

0 . 5 ) ) (+

) 0 ) ”GaInP” ” w i n d o w t j ” )

( p o s i t i o n 0 0 0) ( p o s i t i o n

core radius

core height

0) ” SiliconGermanium ”

” t j c o r e ”)

# Core
( s d e g e o : c r e a t e −r e c t a n g l e
core height

(∗

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (− c o r e r a d i u s

tj thickness

( s d e g e o : c r e a t e −r e c t a n g l e

(∗

tj thickness

0 . 5 ) ) (−

0 . 5 ) ) 0) ” SiliconGermanium ” ” c o r e e m i t t e r ”)

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (− c o r e r a d i u s (+ ( ∗

c o r e e m i t t e r t ) ) (− c o r e h e i g h t (+ ( ∗

tj thickness

0.5)

tj thickness

0.5)

c o r e e m i t t e r t ) ) 0) ” SiliconGermanium ”

” c o r e b a s e ”)
#e n d i f

#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

# Window
( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

window thickness
shell thickness

( position 0 core height

window thickness
shell thickness

0 ) ( p o s i t i o n (+ c o r e r a d i u s

shell thickness )

) (+ c o r e r a d i u s

core height

(∗

0 ) ( / (+ ( ∗

tj thickness

0.5)

(∗

tj thickness

0.5)

window thickness

) ) ”GaInP” ” w i n d o w o u t e r ” )

# Shell
( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t

( position 0 core height

0 ) ( p o s i t i o n (+ c o r e r a d i u s

tj thickness

0.5)

(∗ window thickness

0.5)

shell thickness )

tj thickness

0.5)

(∗ window thickness

0.5)

shell thickness

0.5)

(∗ window thickness

0.5)

( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness
( / (+ ( ∗

0.5)

(∗

0.5)

tj thickness

) (+ c o r e r a d i u s

(∗

0 ) ( p o s i t i o n (+ c o r e r a d i u s

0 . 5 ) (− s h e l l t h i c k n e s s

(∗ window thickness
0.5)

(∗

0 ) ( / (+ ( ∗
tj thickness

) ) ”GaAsP” ” s h e l l e m i t t e r ” )

( position 0 core height

(∗ window thickness

tj thickness

core radius

shell thickness

core height

0 . 5 ) (− s h e l l t h i c k n e s s

(∗ window thickness

(∗

shell emitter t ))

core height

s h e l l e m i t t e r t ) ) (+

0 . 5 ) (− s h e l l t h i c k n e s s

shell emitter t )

) ) ”GaAsP” ” s h e l l b a s e ” )

# TJ ,

etc . . .

( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

window thickness

( position 0 core height

(∗ window thickness

0.5) )

0.5)

(∗

) (+ c o r e r a d i u s

0 ) ( p o s i t i o n (+ c o r e r a d i u s

core height

tj thickness

0 ) ( / (+ ( ∗
0.5)

(∗

tj thickness

(∗ window thickness

0.5)
0.5)

(∗
) ) ”GaInP

” ” window inner ”)
( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

( position 0 core height

core height

0) (/ (∗

0 ) ( p o s i t i o n (+ c o r e r a d i u s

tj thickness

) ) ”GaInP” ” w i n d o w t j ” )

144

0 . 5 ) (+ c o r e r a d i u s

(∗

tj thickness

0.5)

( s d e g e o : d e f i n e −c o n t a c t −s e t ” T o p c o n t a c t ” 4
( s d e g e o : d e f i n e −2d−c o n t a c t
shell thickness

( f i n d −edge−i d

window thickness )

core height

( d e f i n e removeMe ( s d e g e o : c r e a t e −r e c t a n g l e
core height

(∗

pitch

0.5) )

( c o l o r : r g b 0 1 0 ) ”##” )
(∗

( p o s i t i o n (+ c o r e r a d i u s

tj thickness

0.5)

0) ) ” Top contact ”)

( position 0 core height

0) ( p o s i t i o n

(∗

pitch

0 . 5 ) (−

0 ) ”GaAs” ” temp ” ) )

( e n t i t y : d e l e t e removeMe )

( d e f i n e removeMeToo ( s d e g e o : c r e a t e −r e c t a n g l e
position

(∗

−0.5) (+ c o r e h e i g h t

pitch

(∗

( p o s i t i o n 0 (− c o r e h e i g h t

(∗

pitch

0.5) )

pitch

0 . 5 ) ) 0) (

0 ) ”GaAs” ” temp ” ) )

( e n t i t y : d e l e t e removeMeToo )

( s d e g e o : c r e a t e −r e c t a n g l e

( p o s i t i o n 0 0 0) ( p o s i t i o n

(∗

( s d e g e o : c r e a t e −r e c t a n g l e

( p o s i t i o n 0 0 0) ( p o s i t i o n

core radius

( p o s i t i o n 0 0 0) ( p o s i t i o n

c o r e r a d i u s (− c o r e h e i g h t

pitch

0.5)

core height

0 ) ” Oxide ” ” Oxide ” )

0) ” SiliconGermanium ”

” t j c o r e ”)

# Core
( s d e g e o : c r e a t e −r e c t a n g l e

(∗

tj thickness

0 . 5 ) ) 0) ” SiliconGermanium ” ” c o r e e m i t t e r ”)
( s d e g e o : c r e a t e −r e c t a n g l e
(+ ( ∗

tj thickness

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (− c o r e r a d i u s
0.5)

c o r e e m i t t e r t ) (− c o r e h e i g h t

c o r e e m i t t e r t ) ) 0) ” SiliconGermanium ” ” c o r e b a s e ”)

#e n d i f

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

# Window
( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

window thickness
shell thickness

( position 0 core height

window thickness
shell thickness

0 ) ( p o s i t i o n (+ c o r e r a d i u s

shell thickness )

) (+ c o r e r a d i u s

core height

(∗

0 ) ( / (+ ( ∗

tj thickness

0.5)

(∗

tj thickness

0.5)

window thickness

) ) ”GaInP” ” w i n d o w o u t e r ” )

# Shell
( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t

( position 0 core height

0 ) ( p o s i t i o n (+ c o r e r a d i u s

tj thickness

0.5)

(∗ window thickness

0.5)

shell thickness )

tj thickness

0.5)

(∗ window thickness

0.5)

shell thickness

0.5)

(∗ window thickness

0.5)

( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness
( / (+ ( ∗

0.5)

(∗

0.5)

tj thickness

) (+ c o r e r a d i u s

(∗

0 ) ( p o s i t i o n (+ c o r e r a d i u s

0 . 5 ) (− s h e l l t h i c k n e s s

(∗ window thickness
0.5)

(∗

0 ) ( / (+ ( ∗
tj thickness

) ) ”GaAsP” ” s h e l l e m i t t e r ” )

( position 0 core height

(∗ window thickness

tj thickness

core radius

shell thickness

core height

0 . 5 ) (− s h e l l t h i c k n e s s

(∗ window thickness

(∗

shell emitter t ))

core height

s h e l l e m i t t e r t ) ) (+

0 . 5 ) (− s h e l l t h i c k n e s s

shell emitter t )

) ) ”GaAsP” ” s h e l l b a s e ” )

# TJ ,

etc . . .

( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

window thickness

( position 0 core height

(∗ window thickness

0.5) )

0.5)

(∗

) (+ c o r e r a d i u s

0 ) ( p o s i t i o n (+ c o r e r a d i u s
0 ) ( / (+ ( ∗

core height

tj thickness

0.5)

(∗

tj thickness

(∗ window thickness

0.5)
0.5)

(∗
) ) ”GaInP

” ” window inner ”)
( s d e g e o : c r e a t e −r e c t a n g l e
tj thickness

0.5)

( p o s i t i o n 0 (− c o r e h e i g h t

(∗ window thickness

( s d e g e o : c r e a t e − e l l i p t i c a l −s h e e t
tj thickness

0.5)

e x p o s e d ) 0 ) ( p o s i t i o n (+ c o r e r a d i u s

core height

( position 0 core height

core height

0) (/ (∗

0 ) ( p o s i t i o n (+ c o r e r a d i u s

tj thickness

(∗

0 ) ”GaInP” ” w i n d o w i n n e r 2 ” )

0 . 5 ) (+ c o r e r a d i u s

(∗

tj thickness

0.5)

) ) ”GaInP” ” w i n d o w t j ” )
( s d e g e o : c r e a t e −r e c t a n g l e
tj thickness

0.5)

( p o s i t i o n 0 (− c o r e h e i g h t
core height

( d e f i n e removeMeToo ( s d e g e o : c r e a t e −r e c t a n g l e
position

(∗

pitch

e x p o s e d ) 0 ) ( p o s i t i o n (+ c o r e r a d i u s

(∗

0 ) ”GaInP” ” w i n d o w t j 2 ” )

−0.5) (+ c o r e h e i g h t

( p o s i t i o n 0 (− c o r e h e i g h t

(∗

pitch

0.5) )

(∗

pitch

0 . 5 ) ) 0) (

0 ) ”GaAs” ” temp ” ) )

( e n t i t y : d e l e t e removeMeToo )

( s d e g e o : c r e a t e −r e c t a n g l e

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

145

o x i d e t h i c k n e s s ) (− c o r e h e i g h t

e x p o s e d ) 0 ) ” Oxide ” ” Oxide ” )
( s d e g e o : c r e a t e −r e c t a n g l e

( p o s i t i o n 0 0 0) ( p o s i t i o n

core radius

core height

0) ” SiliconGermanium ”

” t j c o r e ”)

# Core
( s d e g e o : c r e a t e −r e c t a n g l e
core height

(∗

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (− c o r e r a d i u s

tj thickness

( s d e g e o : c r e a t e −r e c t a n g l e

(∗

tj thickness

0 . 5 ) ) (−

0 . 5 ) ) 0) ” SiliconGermanium ” ” c o r e e m i t t e r ”)

( p o s i t i o n 0 0 0) ( p o s i t i o n

c o r e r a d i u s (− c o r e h e i g h t

exposed ) 0) ”

SiliconGermanium ” ” c o r e e m i t t e r 2 ”)
( s d e g e o : c r e a t e −r e c t a n g l e
tj thickness

( p o s i t i o n 0 0 0 ) ( p o s i t i o n (− c o r e r a d i u s (+ c o r e e m i t t e r t

0 . 5 ) ) ) (− c o r e h e i g h t (+ ( ∗

tj thickness

0.5)

(∗

c o r e e m i t t e r t ) ) 0) ”

SiliconGermanium ” ” c o r e b a s e ”)

#e n d i f

##########################################################
# Define the doping

##########################################################
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Window Outer ” ”

NDopantActiveConcentration ” window doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r W i n d o w O u t e r ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r W i n d o w O u t e r ” ” window outer ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Window Inner ” ”

PDopantActiveConcentration ” window doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r W i n d o w I n n e r ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r W i n d o w I n n e r ” ” window inner ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Shell Emitter ” ”

NDopantActiveConcentration ” s h e l l e m i t t e r d o p i n g )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r S h e l l E m i t t e r ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r S h e l l E m i t t e r ” ” s h e l l e m i t t e r ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Shell Base ” ”

PDopantActiveConcentration ” s h e l l b a s e d o p i n g )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r S h e l l B a s e ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r S h e l l B a s e ” ” s h e l l b a s e ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Window TJ ” ”

PDopantActiveConcentration ” window tj doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r W i n d o w T J ” ”
ConstantProfileDefinition for Window TJ ” ” window tj ”)
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Window Inner2 ” ”

PDopantActiveConcentration ” window doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r W i n d o w I n n e r 2 ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r W i n d o w I n n e r 2 ” ” window inner2 ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Window TJ2 ” ”

PDopantActiveConcentration ” window tj doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r W i n d o w T J 2 ” ”
ConstantProfileDefinition for Window TJ2 ” ” window tj2 ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Core Emitter2 ” ”

PhosphorusActiveConcentration ” core emitter doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r C o r e E m i t t e r 2 ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e E m i t t e r 2 ” ” c o r e e m i t t e r 2 ”)
#e n d i f
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e T J ” ” NDopantActiveConcentration

” core tj doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r C o r e T J ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e T J ” ” t j c o r e ”)
( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” ConstantProfileDefinition for Core Emitter ” ”

PhosphorusActiveConcentration ” core emitter doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r C o r e E m i t t e r ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e E m i t t e r ” ” c o r e e m i t t e r ”)

146

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e B a s e ” ” BoronActiveConcentration

” core base doping )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” C o n s t a n t P r o f i l e P l a c e m e n t f o r C o r e B a s e ” ”
C o n s t a n t P r o f i l e D e f i n i t i o n f o r C o r e B a s e ” ” c o r e b a s e ”)

##########################################################
# Define the

contacts

##########################################################
( s d e g e o : d e f i n e −c o n t a c t −s e t ” T o p c o n t a c t ” 4

( c o l o r : r g b 0 1 0 ) ”##” )

( s d e g e o : d e f i n e −c o n t a c t −s e t ” B o t t o m c o n t a c t ” 4

( c o l o r : r g b 0 0 1 ) ”##” )

( s d e g e o : i n s e r t −v e r t e x

0 . 5 ) 0 0) )

( position

(∗

core radius

( s d e g e o : s e t −c u r r e n t −c o n t a c t −s e t ” T o p c o n t a c t ” )
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e g e o : i n s e r t −v e r t e x
window thickness )

( p o s i t i o n (+ c o r e r a d i u s
0.5

( s d e g e o : d e f i n e −2d−c o n t a c t
shell thickness

(∗

tj thickness

0.5)

shell thickness

0) )
( f i n d −edge−i d

window thickness )

( p o s i t i o n (+ c o r e r a d i u s

0.25

(∗

tj thickness

0.5)

0) ) ” Top contact ”)

#e n d i f

#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

# ( s d e g e o : d e f i n e −2d−c o n t a c t
shell thickness

( f i n d −edge−i d

window thickness )

( p o s i t i o n (+ c o r e r a d i u s

core height

(∗

tj thickness

0.5)

0) ) ” Top contact ”)

#e n d i f

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

( s d e g e o : d e f i n e −2d−c o n t a c t
shell thickness

( f i n d −edge−i d

window thickness )

( p o s i t i o n (+ c o r e r a d i u s

core height

(∗

tj thickness

0.5)

0) ) ” Top contact ”)

#e n d i f

( s d e g e o : s e t −c u r r e n t −c o n t a c t −s e t ” B o t t o m c o n t a c t ” )
( s d e g e o : d e f i n e −2d−c o n t a c t

# Intermediate

contacts

( f i n d −edge−i d

for

( position

(∗

core radius

0 . 2 5 ) 0 0) ) ” Bottom contact ”)

s u b c e l l IV

( s d e g e o : d e f i n e −c o n t a c t −s e t ” M i d c o n t a c t ” 4

( c o l o r : r g b 1 0 0 ) ”##” )

( s d e g e o : s e t −c u r r e n t −c o n t a c t −s e t ” M i d c o n t a c t ” )
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e g e o : d e f i n e −2d−c o n t a c t

( f i n d −edge−i d

( position

core radius

(∗

( f i n d −edge−i d

( position

(∗

core radius

(∗

core radius

core height

0.5)

0) ) ” Mid contact

0.5)

core height

0) ) ” Mid contact

0.5)

core height

0) ) ” Mid contact

”)
( s d e g e o : d e f i n e −2d−c o n t a c t
”)
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

( s d e g e o : d e f i n e −2d−c o n t a c t

( f i n d −edge−i d

( position

”)
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

( s d e g e o : d e f i n e −2d−c o n t a c t

( f i n d −edge−i d

( position

c o r e r a d i u s (− c o r e h e i g h t

( f i n d −edge−i d

( position

(∗

(∗ exposed

0 . 5 ) ) 0) )

” Mid contact ”)
( s d e g e o : d e f i n e −2d−c o n t a c t

core radius

0.5)

core height

0) ) ” Mid contact

”)
#e n d i f

##########################################################
# Refine

the

g r i d between d o p i n g

levels

##########################################################
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win . A l l ” ” R e c t a n g l e ” ( p o s i t i o n 0 0 0 ) ( p o s i t i o n
2 core height

(∗

pitch

s h e l l t h i c k n e s s ) 0) )

( s d e d r : d e f i n e −r e f i n e m e n t −s i z e ” Ref . A l l ” DopMaxGrid DopMaxGrid 0 DopMinGrid DopMinGrid 0 )

147

0 . 5 ) (+

( s d e d r : d e f i n e −r e f i n e m e n t −f u n c t i o n ” Ref . A l l ” ” D o p i n g C o n c e n t r a t i o n ” ” MaxTransDiff ” 1 )
( s d e d r : d e f i n e −r e f i n e m e n t −p l a c e m e n t ” Ref . A l l ” ” Ref . A l l ” ”Win . A l l ” )
#e n d i f

##########################################################
# Further

refine

the

g r i d between

materials

##########################################################
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win . H e t e r o ” ” R e c t a n g l e ” ( p o s i t i o n 0 0 0 ) ( p o s i t i o n
(+ 2 c o r e h e i g h t

(∗

pitch

0.5)

s h e l l t h i c k n e s s ) 0) )

( s d e d r : d e f i n e −r e f i n e m e n t −s i z e ” Ref . H e t e r o 1 ” MatMaxGrid MatMaxGrid 0 MatMinGrid MatMinGrid 0 )
( s d e d r : d e f i n e −r e f i n e m e n t −f u n c t i o n ” Ref . H e t e r o 1 ” ” MaxLenInt ” ”Germanium” ”GaAs” MatMinGrid MatRatio
” DoubleSide ”)
( s d e d r : d e f i n e −r e f i n e m e n t −p l a c e m e n t ” Ref . H e t e r o 1 ” ” Ref . H e t e r o 1 ” ”Win . H e t e r o ” )
#e n d i f

##########################################################
# Refine

the

grid

according

region

##########################################################
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win1” ” R e c t a n g l e ” ( p o s i t i o n 0 0 0 ) ( p o s i t i o n (+ c o r e r a d i u s
tj thickness

0.5)

window thickness

shell thickness
shell thickness

w i n d o w t h i c k n e s s ) (+ c o r e h e i g h t

(∗

tj thickness

(∗

0.5)

) 0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Top1” MBMaxGrid MBMaxGrid MBMinGrid MBMinGrid −MBRatio −MBRatio )
( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Top1” ” S i z e . Top1” ”Win1 ” )
#i f

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win7” ” R e c t a n g l e ” ( p o s i t i o n (− c o r e r a d i u s
0 0 ) ( p o s i t i o n (+ c o r e r a d i u s

(∗

tj thickness

(∗

(∗ window thickness

0.5)

tj thickness

0.5) )

core height

0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Top7” MatMinGrid MatMaxGrid MatMinGrid MatMaxGrid 1 1 )
( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Top7” ” S i z e . Top7” ”Win7 ” )

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win8” ” R e c t a n g l e ” ( p o s i t i o n 0 (− c o r e h e i g h t
) 0 ) ( p o s i t i o n (+ c o r e r a d i u s
(∗

tj thickness

0.5)

(∗

tj thickness

(∗ window thickness

0.5)

(∗

tj thickness

0.5)

0 . 5 ) ) (+ c o r e h e i g h t

0 . 5 ) ) 0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Top8” MatMaxGrid MatMinGrid MatMaxGrid MatMinGrid 1 1 )
( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Top8” ” S i z e . Top8” ”Win8 ” )
#e n d i f
( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win2” ” R e c t a n g l e ” ( p o s i t i o n
(+ c o r e r a d i u s
tj thickness

(∗

0.5)

tj thickness

0.5)

window thickness

shell thickness

core radius

core height 0 ) ( position

w i n d o w t h i c k n e s s ) (+ c o r e h e i g h t

(∗

) 0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Top2” RefMaxGrid RefMaxGrid RefMinGrid RefMinGrid −R e f R a t i o −
RefRatio )
( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Top2” ” S i z e . Top2” ”Win2 ” )

( s d e d r : d e f i n e −r e f i n e m e n t −window ”Win3” ” R e c t a n g l e ” ( p o s i t i o n 0 (− c o r e h e i g h t
p o s i t i o n (+ c o r e r a d i u s

t j t h i c k n e s s ) (+ c o r e h e i g h t

core radius ) 0 ) (

t j t h i c k n e s s ) 0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Top3” RefMaxGrid RefMaxGrid RefMinGrid RefMinGrid −R e f R a t i o −
RefRatio )
( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Top3” ” S i z e . Top3” ”Win3 ” )

#e l s e
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” w i n d o w o u t e r ” ” m a x e d g e l e n g t h ” w i n d o w g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” w i n d o w i n n e r ” ” m a x e d g e l e n g t h ” w i n d o w g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” s h e l l e m i t t e r ” ” m a x e d g e l e n g t h ” s h e l l e m i t t e r g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” s h e l l b a s e ” ” m a x e d g e l e n g t h ” s h e l l b a s e g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” w i n d o w t j ” ” m a x e d g e l e n g t h ” w i n d o w t j g r i d )
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” w i n d o w i n n e r 2 ” ” m a x e d g e l e n g t h ” w i n d o w g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” w i n d o w t j 2 ” ” m a x e d g e l e n g t h ” w i n d o w t j g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” c o r e e m i t t e r 2 ” ” m a x e d g e l e n g t h ” c o r e e m i t t e r g r i d )
#e n d i f

148

( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” t j c o r e ” ” m a x e d g e l e n g t h ”

core tj grid )

( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” c o r e e m i t t e r ” ” m a x e d g e l e n g t h ” c o r e e m i t t e r g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” c o r e b a s e ” ” m a x e d g e l e n g t h ” c o r e b a s e g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” A i r ” ” m a x e d g e l e n g t h ” A i r g r i d )
( s d e n o f f s e t : c r e a t e −n o f f s e t −b l o c k ” r e g i o n ” ” Oxide ” ” m a x e d g e l e n g t h ” O x i d e g r i d )
#e n d i f

##########################################################
# G e n e r a t e t h e mesh

##########################################################

( s d e : s a v e −model ”n@node@ ” )
( s d e d r : append−cmd− f i l e
#i f

[ string

” xy . cmd ” )

compare @Structure@ ” c o n f o r m a l ” ] == 0

( s d e : b u i l d −mesh ” snmesh ” ”−a −c boxmethod ” ”n@node@ ” )
#e l s e
( s d e : b u i l d −mesh ” n o f f s e t ” ”” ”n@node@ ” )
#e n d i f

Listing E.2: Code to create DF-ISE files from the generated .tdr file
# C r e a t e a DF−ISE

#i f

[ string

file

from t h e

. tdr

file

compare @Structure@ ” c o n f o r m a l ” ] == 0

tdx −dd −M 0 −S 0 n@node | sde@ msh . t d r n@node | sde@ msh

#e l s e

tdx −dd −M 0 −S 0 n@node | s d e @ m s h p o f . t d r n@node | sde@ msh

#e n d i f

Listing E.3: Code to generate optical data for SiGe and GaAsP
% Program t o

identify

t h e GaInP and GaAsP c o m p o s i t i o n s

% matched t o SiGe a t a g i v e n

c o m p o s i t i o n and t o t h e n

% f o r GaAsP by s h i f t i n g

the

optical

data

o f GaAs and t o

% f o r GaInP by s h i f t i n g

the

optical

data

o f InP

that

create

are

lattice

optical

create

data

optical

data

% 7/25/11
% Dan Turner−Evans

% Navigate to the

appropriate

directory

cd nk ;

% Import the

t a b u l a t e d data

% to a given

composition

relating

t h e GaInP and GaAsP m a t e r i a l

o f SiGe

lma = i m p o r t d a t a ( ’ l a t t i c e m a t c h e d a l l o y s . t x t ’ ) ;
S i G e x = lma . d a t a ( : , 1 ) ;
SiGe Eg = lma . d a t a ( : , 3 ) ;
GaInP x = lma . d a t a ( : , 4 ) ;
GaInP Eg = lma . d a t a ( : , 6 ) ;
GaAsP x = lma . d a t a ( : , 7 ) ;
GaAsP Eg = lma . d a t a ( : , 9 ) ;

% I m p o r t t h e GaAs o p t i c a l

data

GaAs Eg = 1 . 4 3 ;
GaAs data = i m p o r t d a t a ( ’ GaAs . t x t ’ ) ;

149

parameters

% I m p o r t t h e InP o p t i c a l

data

InP Eg = 1 . 3 5 ;
I n P d a t a = i m p o r t d a t a ( ’ InP . t x t ’ ) ;

% Specify

t h e SiGe c o m p o s i t i o n

ChosenSiGex = @SiGex@ ;

% Identify

t h e GaInP and GaAsP c o m p o s i t i o n

that

are

lattice

matched t o t h e

% c h o s e n SiGe c o m p o s i t i o n
found = 0 ;
for

i =1: l e n g t h ( S i G e x ) ,
f i n d = ChosenSiGex − S i G e x ( i , 1 ) ;
if

find < 0 ,
if

a b s ( ChosenSiGex − S i G e x ( i , 1 ) ) < a b s ( ChosenSiGex − S i G e x ( i −1 ,1) ) ,
found = i ;
break

else
f o u n d = i −1;
break
end
end
end

% C r e a t e GaInP o p t i c a l

d a t a from InP

GaInP = I n P d a t a ;
for

i =1: l e n g t h ( GaInP ( : , 1 ) ) ,
GaInP ( i , 1 ) = 1 . 2 4 0 / ( 1 . 2 4 0 . / GaInP ( i , 1 ) + GaInP Eg ( found , 1 ) − InP Eg ) ;

end

% C r e a t e GaAsP o p t i c a l

d a t a from GaAs

GaAsP = GaAs data ;
for

i =1: l e n g t h ( GaAsP ( : , 1 ) ) ,
GaAsP ( i , 1 ) = 1 . 2 4 0 / ( 1 . 2 4 0 . / GaAsP ( i , 1 ) + GaAsP Eg ( found , 1 ) − GaAs Eg ) ;

end

% Write t h e

generated data to a csv

. txt

file

x l s w r i t e ( ’ GaAsP . t x t ’ , GaAsP )
x l s w r i t e ( ’ GaInP . t x t ’ , GaInP )

Listing E.4: Code to add scattering particles to optical simulations
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

maxz = @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ + @ARBot@ + @ARTop@
e x c l u d e Z o n e = @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ +@ w i n d o w t h i c k n e s s @ + @ARBot@
+ @ARTop@ ;
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

maxz = @ c o r e h e i g h t @ ;
excludeZone = @core diameter@ / 2 ;
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

maxz = @ c o r e h e i g h t @ −@exposed@ ;
e x c l u d e Z o n e = @ c o r e d i a m e t e r @ /2 + @ o x i d e t h i c k n e s s @ ;
#e n d i f
minz = 0 ;
noz = 4 ;

u n i t c e l l h a l f w i d t h = @pitch@ / 2 ;

minRad = 0 . 0 5 ;

150

maxRad = 0 . 2 5 ;

numSpheres = 50∗ @ c o r e h e i g h t @ / 1 0 ;

num = 0 ;
f a i l c = 0;

x e s = z e r o s ( 1 , numSpheres ) ;
z e s = z e r o s ( 1 , numSpheres ) ;
r a d s = z e r o s ( 1 , numSpheres ) ;

f i d = fopen ( ’ Scatterer Script n@node@ . l s f ’ , ’w’ ) ;

while

(num < numSpheres )

f a i l c = f a i l c + 1;
if

( f a i l c > 10000)
break ;

end

xGuess = rand ∗ u n i t c e l l h a l f w i d t h ;
z G u e s s = minz + ( maxz − minz ) ∗ rand ∗(1− rand ) ;
r a d G u e s s = minRad + ( maxRad − minRad ) ∗ rand ;

( xGuess − r a d G u e s s < e x c l u d e Z o n e )
continue ;

end

( maxz − xGuess ∗ noz / u n i t c e l l h a l f w i d t h < z G u e s s )
continue ;

end

isGood = 1 ;
f o r n =1:num
if

( ( xGuess − x e s ( n ) ) ˆ 2 + ( z G u e s s − z e s ( n ) ) ˆ 2 ) ˆ 0 . 5 < ( r a d s ( n ) + r a d G u e s s )
isGood = 0 ;
break ;

end
end

( isGood )
num = num+1;
x e s (num) = xGuess ;
z e s (num) = z G u e s s ;
r a d s (num) = r a d G u e s s ;
e x t n t = (2∗ radGuess ) ˆ ( 1 / 3 ) ;

f p r i n t f ( f i d , ’ a d d c i r c l e ; \ r \ n s e t ( ” name ” , ” S p h e r e%d ” ) ; \ r \n ’ , num ) ;
f p r i n t f ( f i d , ’ s e t ( ” m a t e r i a l ” , ” Al2O3 − P a l i k ” ) ; \ r \ n s e t ( ” s e t mesh o r d e r from m a t e r i a l

database

” , 0 ) ; \ r \n ’ ) ;
f p r i n t f ( f i d , [ ’ s e t ( ” mesh o r d e r ” , 3 ) ; \ r \ n s e t ( ” x ” , ’ num2str ( 1 e −6∗xGuess )
num2str ( 1 e −6∗z G u e s s )

’ ) ; \ r \n ’ ] ) ;

f p r i n t f ( f i d , [ ’ s e t ( ” r a d i u s ” , ’ num2str ( 1 e −6∗ r a d G u e s s )

end

fclose ( fid ) ;

151

’ ) ; \ r \n\ r \n ’ ] ) ;

’) ;\ r \ nset (” y ” , ’

Listing E.5: Code to modify Lumerical simulation for the specific structure
###############################################################################
# Dan Turner−Evans

# 07/25/11

# 3D S i m u l a t i o n

c o r e −s h e l l

wire

structure

###############################################################################
clear ;

# Load t h e

initial

simulation

file

upon which

all

future

simulations

are based

load (” lumcad template . f s p ”) ;

switchtolayout ;
redrawoff ;

# Modify t h e w i r e

structure

for

the given

parameters

s e l e c t (” core ”) ;
s e t ( ” x span ” , @ c o r e d i a m e t e r @ ∗1 e −6) ;
s e t ( ” y max ” , @ c o r e h e i g h t @ ∗1 e −6) ;

# Create a conformal ,

#i f

[ string

layered

cylindrical

wire

structure

compare @Structure@ ” c o n f o r m a l ” ] == 0

s e l e c t ( ” t j / window ” ) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) ∗1 e −6) ;

s e l e c t (” s h e l l ”) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ∗ 2 ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;

s e l e c t ( ” window ” ) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗ 2 ) ∗1 e
−6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;

s e l e c t ( ” ARBottom ” ) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗2 +
@ARBot@) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@)
∗1 e −6) ;

s e l e c t ( ”ARTop” ) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗2 +
@ARBot@ + @ARTop@) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@
+ @ARTop@) ∗1 e −6) ;

#e n d i f

#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

s e l e c t ( ” t j / window ” ) ;
delete ;
s e l e c t (” s h e l l ”) ;
delete ;
s e l e c t ( ” window ” ) ;
delete ;
s e l e c t ( ” ARBottom ” ) ;

152

delete ;
s e l e c t ( ”ARTop” ) ;
delete ;

addcustom ( ” t j / window ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ / 2 ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) ∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) / ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2

+ @ w i n d o w t h i c k n e s s @ / 2 ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) ˆ2−x
ˆ2) ”) ;
s e t ( ” make nonsymmetric ” , 1 ) ;
s e t (” equation

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ” GaInP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 3 ) ;

addcustom ( ” s h e l l ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ∗ 2 ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ /2 − @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) / ( @ c o r e d i a m e t e r @ /2

+ @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 +
@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ˆ2−x ˆ 2 ) ” ) ;
s e t ( ” make nonsymmetric ” , 1 ) ;
s e t (” equation

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ” GaAsP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 4 ) ;

addcustom ( ” window ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗ 2 ) ∗1 e −6)
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ − @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) / ( @ c o r e d i a m e t e r @ /2 +

@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 +
@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ˆ2−x ˆ 2 ) ” ) ;
s e t ( ” make nonsymmetric ” , 1 ) ;
s e t (” equation

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ” GaInP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 5 ) ;

addcustom ( ” ARBottom ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗2 +
@ARBot@∗ 2 ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ − @ s h e l l t h i c k n e s s @ − @ARBot@)
∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@)
∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@) / (

@ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@) ∗ s q r t
( ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@) ˆ2−x
ˆ2) ”) ;
s e t ( ” make nonsymmetric ” , 1 ) ;
s e t (” equation

2” ,”0”) ;

153

s e t ( ” m a t e r i a l ” , ” TiOx − S e n t a u r u s ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 6 ) ;

addcustom ( ”ARTop” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗2 +
@ARBot@∗2 + @ARTop@∗ 2 ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ − @ s h e l l t h i c k n e s s @ − @ARBot@ −
@ARTop@) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ + @ARTop@)

/ ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ +
@ARBot@ + @ARTop@) ˆ2−x ˆ 2 ) ” ) ;
s e t ( ” make nonsymmetric ” , 1 ) ;
s e t (” equation

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ”MgF − S e n t a u r u s ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 7 ) ;

addrect (” i n f i l l ”) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , @pitch@ ∗1 e −6) ;
s e t ( ” y min ” , 0 ) ;
s e t ( ” y max ” , @ c o r e h e i g h t @ ∗1 e −6) ;
s e t ( ” m a t e r i a l ” , ” SiO2 ( G l a s s ) − P a l i k ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 8 ) ;

#e n d i f

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

s e l e c t ( ” t j / window ” ) ;
delete ;
s e l e c t (” s h e l l ”) ;
delete ;
s e l e c t ( ” window ” ) ;
delete ;
s e l e c t ( ” ARBottom ” ) ;
delete ;
s e l e c t ( ”ARTop” ) ;
delete ;

addrect (” boot ”) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ o x i d e t h i c k n e s s @ ∗ 2 ) ∗1 e −6) ;
s e t ( ” y min ” , 0 ) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ − @exposed@ ) ∗1 e −6) ;
s e t ( ” m a t e r i a l ” , ” SiO2 ( G l a s s ) − P a l i k ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 3 ) ;

154

s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) / ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2

+ @ w i n d o w t h i c k n e s s @ / 2 ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ / 2 ) ˆ2−x
ˆ2) ”) ;
s e t ( ” m a t e r i a l ” , ” GaInP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 4 ) ;

a d d r e c t ( ” t j / window s i d e ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @exposed@ ) ∗1 e −6) ;
s e t ( ” y max ” , @ c o r e h e i g h t @ ∗1 e −6) ;
s e t ( ” m a t e r i a l ” , ” GaInP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 4 ) ;

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) / ( @ c o r e d i a m e t e r @ /2

+ @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 +
@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ /2 + @ s h e l l t h i c k n e s s @ ) ˆ2−x ˆ 2 ) ” ) ;
s e t ( ” m a t e r i a l ” , ” GaAsP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 5 ) ;

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) / ( @ c o r e d i a m e t e r @ /2 +

@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ∗ s q r t ( ( @ c o r e d i a m e t e r @ /2 +
@ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ ) ˆ2−x ˆ 2 ) ” ) ;
s e t ( ” m a t e r i a l ” , ” GaInP ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 6 ) ;

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@) / (

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ” TiOx − S e n t a u r u s ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 7 ) ;

addcustom ( ”ARTop” ) ;
s e t (” x ” ,0) ;

155

s e t ( ” x span ” , ( @ c o r e d i a m e t e r @ + @ t j t h i c k n e s s @ + @ w i n d o w t h i c k n e s s @ ∗2 + @ s h e l l t h i c k n e s s @ ∗2 +
@ARBot@∗2 + @ARTop@∗ 2 ) ∗1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ − @ t j t h i c k n e s s @ /2 − @ w i n d o w t h i c k n e s s @ − @ s h e l l t h i c k n e s s @ − @ARBot@ −
@ARTop@) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e −6) ;
s e t (” equation

1 ” , ” ( @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ + @ARTop@)

2” ,”0”) ;

s e t ( ” m a t e r i a l ” , ”MgF − S e n t a u r u s ” ) ;
s e t ( ” s e t mesh o r d e r from m a t e r i a l

database ” ,0) ;

s e t ( ” mesh o r d e r ” , 8 ) ;

#e n d i f

# Edit

simulation

region

s e l e c t ( ”FDTD” ) ;
s e t (” simulation

t i m e ” , 1 e −6) ;

s e t (” x ” ,0) ;
s e t ( ” x span ” , @pitch@ ∗1 e −6) ;
s e t ( ” y min ” , 0 ) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e−6+2e −6) ;

# Edit

sources

s e l e c t ( ”pw” ) ;
s e t (” wavelength

s t a r t ” ,@wl@∗1 e −9) ;

s e t ( ” w a v e l e n g t h s t o p ” ,@wl@∗1 e −9) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , @pitch@ ∗1 e −6+0.5 e −6) ;
s e t ( ” y ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e−6+1e −6) ;
cwnorm ;

# Edit monitors

s e l e c t ( ” power ” ) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , @pitch@ ∗1 e −6) ;
s e t ( ” y min ” , 0 ) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e−6+2e −6) ;
s e t (” p a r t i a l

spectral

average ” ,1) ;

s e t ( ” d e l t a ” , 5 0 e +12) ;

s e l e c t (” n a l l ”) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , @pitch@ ∗1 e −6) ;
s e t ( ” y min ” , 0 ) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ + @ t j t h i c k n e s s @ /2 + @ w i n d o w t h i c k n e s s @ + @ s h e l l t h i c k n e s s @ + @ARBot@ +
@ARTop@) ∗1 e−6+2e −6) ;

s e l e c t ( ” n window ” ) ;

156

s e t (” x ” ,0) ;
s e t ( ” x span ” , 0 . 0 1 ∗ 1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ +@ t j t h i c k n e s s @ / 1 0 ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ +3∗ @ t j t h i c k n e s s @ / 1 0 ) ∗1 e −6) ;

s e l e c t (” n s h e l l ”) ;
s e t (” x ” ,0) ;
s e t ( ” x span ” , 0 . 0 1 ∗ 1 e −6) ;
s e t ( ” y min ” , ( @ c o r e h e i g h t @ +@ s h e l l t h i c k n e s s @ ) ∗1 e −6) ;
s e t ( ” y max ” , ( @ c o r e h e i g h t @ +@ s h e l l t h i c k n e s s @ + 0 . 0 1 ) ∗1 e −6) ;

# Add s c a t t e r e r s
S c a t t e r e r S c r i p t n @ n o d e | matlab1@ ;

f o r ( x =1:2)
i f ( x==1)
F i l e D s c r = ” n@node@ ” + ” Tandem Cell WL ” + num2str (@wl@) + ” TE @Structure@ ” ;
s e l e c t ( ”pw” ) ;
s e t ( ” p o l a r i z a t i o n ” , ”TE” ) ;
i f ( x==2)
F i l e D s c r = ” n@node@ ” + ” Tandem Cell WL ” + num2str (@wl@) + ” TM @Structure@ ” ;
s e l e c t ( ”pw” ) ;
s e t ( ” p o l a r i z a t i o n ” , ”TM” ) ;
save ( FileDscr + ” . f s p ”) ;

exit (2) ;

Listing E.6: Code to run Lumerical simulations
nohup / o p t / l u m e r i c a l / f d t d / mpich / c h p 4 / b i n / mpirun −n 16 / o p t / l u m e r i c a l / f d t d / b i n / f d t d −e n g i n e −mpichp4
n@node | lumcad@ Tandem Cell WL@wl@ TE @Structure@ . f s p
nohup / o p t / l u m e r i c a l / f d t d / mpich / c h p 4 / b i n / mpirun −n 16 / o p t / l u m e r i c a l / f d t d / b i n / f d t d −e n g i n e −mpichp4
n@node | lumcad@ Tandem Cell WL@wl@ TM @Structure@ . f s p

Listing E.7: Code to extract data from Lumerical simulations
##############################################################################
# Dan Turner−Evans

# 7/27/11

# G e n e r a t e s an o u t p u t

file

showing the

relative

absorption

the

materials

##############################################################################

##############################################################################
# C r e a t e an o u t p u t

file

##############################################################################

clear ;

p r e f i x f i l e n a m e = ” Tandem Cell ” ;

f o r ( x =1:2)
i f ( x==1)

157

F i l e D s c r = ” n@node | lumcad@ ” + p r e f i x f i l e n a m e + ” WL” + num2str (@wl@) + ” TE @Structure@ ” ;
JPGDscr = ” n@node | lumcad@ ” + p r e f i x f i l e n a m e + ” WL” + num2str (@wl@) + ” TE @Structure@ ” ;
MatlabOut = ” n@node | lumcad@ OptGen ” + num2str (@wl@) + ”nm” + ” TE @Structure@ ” + ” . mat ” ;

load (

FileDscr + ”. fsp ” ) ;

###############################################################################
# Get d a t a from t h e m o n i t o r s

###############################################################################

f = g e t d a t a ( ” power ” , ” f ” ) ; # F r e q u e n c y v e c t o r
x = getdata (” n a l l ” ,” x ”) ; # P o s i t i o n

vectors

associated

with E f i e l d s

y = getdata (” n a l l ” ,” y ”) ; # P o s i t i o n

vectors

associated

with E f i e l d s

E = g e t e l e c t r i c ( ” power ” ) ;

# C r e a t e an empty m a t r i x

the

appropriate

size

BlankMatrix = matrix ( l e n g t h ( x ) , l e n g t h ( y ) , 1 ) ;

E2 = B l a n k M a t r i x ;
E2 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = E ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ;

i n d e x o v e r a l l 0 = g e t d a t a ( ” n a l l ” , ” i n d e x x ” ) ; # The o v e r a l l
i n d e x c o r e = g e t d a t a ( ” n c o r e ” , ” i n d e x x ” ) ; # The c o r e

matrix

index

values

index

i n d e x w i n d o w = g e t d a t a ( ” n window ” , ” i n d e x x ” ) ; # The window i n d e x
i n d e x s h e l l = g e t d a t a ( ” n s h e l l ” , ” i n d e x x ” ) ; # The s h e l l

index

###############################################################################
# Calculate

u n i t volume = −0.5∗w∗ | E| ˆ 2 ∗ imag ( e p s )

Absorption per

###############################################################################

# Create

matrices

that

will

define

the

regions

interest

Abs core = BlankMatrix ;
Abs window = B l a n k M a t r i x ;
A b s s h e l l = BlankMatrix ;

i n d e x o v e r a l l = BlankMatrix ;
i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = i n d e x o v e r a l l 0 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ; # Make s u r e
that the dimensions

are

right

A b s c o r e ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x c o r e
(1 ,1 ,1) ) ; # Specify

the

core

region

Abs window ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x w i n d o w
(1 ,1 ,1) ) ; # Specify

t h e window r e g i o n

A b s s h e l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x s h e l l
(1 ,1 ,1) ) ; # Specify

the

# C r e a t e a m a t r i x w i t h w,

shell

region

the frequency ,

at every

point

w0 = B l a n k M a t r i x ;
w0 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = B l a n k M a t r i x ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) +2∗ p i ∗ f ( 1 ) ;

# Calculate

the

absorption

A b s o r b e d c o r e = r e a l ( A b s c o r e ∗ 0 . 5 ∗ w0 ∗ ( 2 ∗ r e a l ( i n d e x o v e r a l l ) ∗ imag ( i n d e x o v e r a l l ) ∗ e p s 0 ) ∗E2 ) ;
Absorbed window = r e a l ( Abs window ∗ 0 . 5 ∗ w0 ∗ ( 2 ∗ r e a l ( i n d e x o v e r a l l ) ∗ imag ( i n d e x o v e r a l l ) ∗ e p s 0 ) ∗E2 ) ;
A b s o r b e d s h e l l = r e a l ( A b s s h e l l ∗ 0 . 5 ∗ w0 ∗ ( 2 ∗ r e a l ( i n d e x o v e r a l l ) ∗ imag ( i n d e x o v e r a l l ) ∗ e p s 0 ) ∗E2 ) ;

# N o r m a l i z e t h e a b s o r b e d power

relative

t o t h e i n p u t power

A b s o r b e d c o r e N o r m = A b s o r b e d c o r e / ( @pitch@ ∗1 e −6∗0.5∗ e p s 0 ∗ c ) ;
A b s o r b e d s h e l l N o r m = A b s o r b e d s h e l l / ( @pitch@ ∗1 e −6∗0.5∗ e p s 0 ∗ c ) ;
Absorbed window Norm = Absorbed window / ( @pitch@ ∗1 e −6∗0.5∗ e p s 0 ∗ c ) ;

# Calculate

the average

ratio

light

absorbed

158

t h e Ge and i n

t h e GaAs .

AvePow core TE @Structure@ = i n t e g r a t e ( A b s o r b e d c o r e N o r m , 1 : 2 , x , y ) ;
AvePow shell TE @Structure@ = i n t e g r a t e ( Absorbed shell Norm , 1 : 2 , x , y ) ;
AvePow window TE @Structure@ = i n t e g r a t e ( Absorbed window Norm , 1 : 2 , x , y ) ;

image ( x , y , A b s o r b e d c o r e ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ) ;
e x p o r t f i g u r e ( JPGDscr + ” c o r e . j p g ” ) ;
image ( x , y , A b s o r b e d s h e l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ) ;
e x p o r t f i g u r e ( JPGDscr + ” s h e l l . j p g ” ) ;
image ( x , y , Absorbed window ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ) ;
e x p o r t f i g u r e ( JPGDscr + ” window . j p g ” ) ;
image ( x , y , E2 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ) ;
e x p o r t f i g u r e ( JPGDscr + ” E Mag ” + ” . j p g ” ) ;
image ( x , y , i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ) ;
e x p o r t f i g u r e ( ” @Structure@ . j p g ” ) ;

m a t l a b s a v e ( MatlabOut , x , y , f , A b s o r b e d c o r e N o r m , A b s o r b e d s h e l l N o r m , Absorbed window Norm ) ;
i f ( x==2)
F i l e D s c r = ” n@node | lumcad@ ” + p r e f i x f i l e n a m e + ” WL” + num2str (@wl@) + ” TM @Structure@ ” ;
JPGDscr = ” n@node | lumcad@ ” + p r e f i x f i l e n a m e + ” WL” + num2str (@wl@) + ” TM @Structure@ ” ;
MatlabOut = ” n@node | lumcad@ OptGen ” + num2str (@wl@) + ”nm” + ” TM @Structure@ ” + ” . mat ” ;

load (

FileDscr + ”. fsp ” ) ;

###############################################################################
# Get d a t a from t h e m o n i t o r s

###############################################################################

f = g e t d a t a ( ” power ” , ” f ” ) ; # F r e q u e n c y v e c t o r
x = getdata (” n a l l ” ,” x ”) ; # P o s i t i o n

vectors

associated

with E f i e l d s

y = getdata (” n a l l ” ,” y ”) ; # P o s i t i o n

vectors

associated

with E f i e l d s

E = g e t e l e c t r i c ( ” power ” ) ;

# C r e a t e an empty m a t r i x

the

appropriate

size

BlankMatrix = matrix ( l e n g t h ( x ) , l e n g t h ( y ) , 1 ) ;

E2 = B l a n k M a t r i x ;
E2 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = E ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) ;

i n d e x o v e r a l l 0 = g e t d a t a ( ” n a l l ” , ” i n d e x z ” ) ; # The o v e r a l l
i n d e x c o r e = g e t d a t a ( ” n c o r e ” , ” i n d e x z ” ) ; # The c o r e

matrix

index

values

index

i n d e x w i n d o w = g e t d a t a ( ” n window ” , ” i n d e x z ” ) ; # The window i n d e x
i n d e x s h e l l = g e t d a t a ( ” n s h e l l ” , ” i n d e x z ” ) ; # The s h e l l

index

###############################################################################
# Calculate

Absorption per

u n i t volume = −0.5∗w∗ | E| ˆ 2 ∗ imag ( e p s )

###############################################################################

# Create

matrices

that

will

define

the

regions

interest

Abs core = BlankMatrix ;
Abs window = B l a n k M a t r i x ;
A b s s h e l l = BlankMatrix ;

are

right

A b s c o r e ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x c o r e
(1 ,1 ,1) ) ; # Specify

the

core

region

Abs window ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x w i n d o w
(1 ,1 ,1) ) ; # Specify

t h e window r e g i o n

159

A b s s h e l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = ( i n d e x o v e r a l l ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) == i n d e x s h e l l
(1 ,1 ,1) ) ; # Specify

the

# C r e a t e a m a t r i x w i t h w,

shell

region

the frequency ,

at every

point

w0 = B l a n k M a t r i x ;
w0 ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) = B l a n k M a t r i x ( 1 : l e n g t h ( x ) , 1 : l e n g t h ( y ) , 1 ) +2∗ p i ∗ f ( 1 ) ;

# Calculate

the

absorption

# N o r m a l i z e t h e a b s o r b e d power

relative

t o t h e i n p u t power

# Calculate

the average

ratio

light

absorbed

t h e Ge and i n

t h e GaAs .

AvePow core TM @Structure@ = i n t e g r a t e ( A b s o r b e d c o r e N o r m , 1 : 2 , x , y ) ;
AvePow shell TM @St ruc tur e@ = i n t e g r a t e ( A b s o r b e d s h e l l N o r m , 1 : 2 , x , y ) ;
AvePow window TM @Structure@ = i n t e g r a t e ( Absorbed window Norm , 1 : 2 , x , y ) ;

m a t l a b s a v e ( MatlabOut , x , y , f , A b s o r b e d c o r e N o r m , A b s o r b e d s h e l l N o r m , Absorbed window Norm ) ;
A v e P o w c o r e T o t a l = ( AvePow core TE @Structure@ + AvePow core TM @Structure@ ) / 2 ;
A v e P o w s h e l l T o t a l = ( A v e P o w s h e l l T E @ S t r u c t u r e @ + Ave Po w shell TM @St ruc tur e@ ) / 2 ;
AvePow window Total = ( AvePow window TE @Structure@ + AvePow window TM @Structure@ ) / 2 ;
w r i t e ( ” g v a r s . d a t ” , ”1 @node@ P e r A b s c o r e ” + num2str ( A v e P o w c o r e T o t a l ) ) ;
w r i t e ( ” g v a r s . d a t ” , ”1 @node@ Per Abs window ” + num2str ( AvePow window Total ) ) ;
w r i t e ( ” g v a r s . d a t ” , ”1 @node@ P e r A b s s h e l l ” + num2str ( A v e P o w s h e l l T o t a l ) ) ;

exit (2) ;

Listing E.8: Code to convert extracted Lumerical data to the appropraite form for interpolation
FDTDFile1 = s t r c a t ( ’ n ’ ,

num2str ( @node | lumcad@ ) ,

’ OptGen ’ , num2str (@wl@) ,

’ nm TE @Structure@ . mat ’ ) ;

FDTDFile2 = s t r c a t ( ’ n ’ ,

num2str ( @node | lumcad@ ) ,

’ OptGen ’ , num2str (@wl@) ,

’ nm TM @Structure@ . mat ’ ) ;

O u t p u t F i l e = s t r c a t ( ’ n ’ , num2str ( @node@ ) , ’ OptGen ’ , num2str (@wl@) , ’nm . mat ’ ) ;

l o a d ( FDTDFile1 ) ;
Pabs x = x ;
Pabs y = y ;
Ngen pavg = 0 . 5 ∗ s q u e e z e ( A b s o r b e d c o r e N o r m + A b s o r b e d s h e l l N o r m + Absorbed window Norm ) ;

l o a d ( FDTDFile2 ) ;

160

Ngen pavg = Ngen pavg + 0 . 5 ∗ s q u e e z e ( A b s o r b e d c o r e N o r m + A b s o r b e d s h e l l N o r m +
Absorbed window Norm ) ;

save ( OutputFile ,

’ Pabs x ’ ,

’ Pabs y ’ ,

’ Ngen pavg ’ ) ;

Listing E.9: Code to interpolate the Lumerical data onto the FEM grid
#MATLAB o p t i c a l

generation

#M i c h a e l K e l z e n b e r g ,

grid

conversion

for

L u m e r i c a l −> S e n t a u r u s

project

2009

#s e t d e p @previous@
#s e t d e p @node | sde@

%DAT f i l e %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%T h i s s h o u l d be a v a l i d DF−ISE . d a t

file

(i .e.

g e n e r a t e d by mesh o r

%n o f f s e t 3 d .

The meshing program must be s c r i p t e d

%p o s i t i o n

each v e r t e x

%”P M I U s e r F i e l d 1 ” ,

the

grid

store

t h e x− and y−

a s ” P M I U s e r F i e l d 0 ” and

respectively .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

d a t F i l e = ’ n@node | sde@ msh . dat ’ ;
g r d F i l e = ’ n@node | sde@ msh . grd ’ ;

%FDTD MAT f i l e %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%T h i s s h o u l d be t h e Matlab MAT f i l e

g e n e r a t e d by t h e L u m e r i c a l CAD s c r i p t

%i n c l u d i n g :
% Pabs x , P a b s y

X and Y s p e c i f i c a t i o n

% freq

Freq .

simulation

grid

% Pabs ∗

M a t r i x o f power a b s o r p t i o n

% Ngen ∗

Matrix o f

% IntgPwr ∗

T o t a l power a b s o r p t i o n

% Current ∗

Total photocurrent

% Absfrac ∗

Fraction

∗ these

variables ,

optical

(m)

( Hz )
(W/m3)

generation

rate

( p e r cm3 p e r s )

(W/m)

(A p e r um d e v i c e

absorbed

light ,

i .e.

f o l l o w e d by ” p a v g ” , c o r e s p o n d t o

length )

A b s o r p t i o n Quantum E f f i c i e n c y
partial

spectral

averaging

Note t h a t

presently ,

o n l y Pabs x ,

Pabs y , and Ngen pavg a r e u s e d by t h i s

script .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

FDTDFile = s t r c a t ( ’ n ’ , num2str ( @node | matlab2@ ) , ’ OptGen ’ , num2str (@wl@) , ’nm . mat ’ ) ;

%R e g i o n s t o

p r o c e s s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%These a r e t h e

regions

%T h i s must be a

cell

%around e a c h

to perform the

array

optical

r e g i o n names ,

g e n e r a t i o n mesh c o n v e r s i o n .

including

d o u b l e −q u o t e s

(”)

r e g i o n name ! ! !

Example s y n t a x :

regionsToProcess = { ’” Bas e r e gio n ” ’ , ” E m i t t e r r e g i o n ” ’

};

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” c o r e e m i t t e r ” ’

’” window inner ” ’
’” c o r e b a s e ” ’

’” s h e l l e m i t t e r ” ’

’” window inner2 ” ’

’” s h e l l b a s e ” ’

’” window tj2 ” ’

’” window tj

’” c o r e e m i t t e r 2

” ’};
#e l s e
regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” c o r e e m i t t e r ” ’

’” window inner ” ’

’” s h e l l e m i t t e r ” ’

’” s h e l l b a s e ” ’

’” c o r e b a s e ” ’ } ;

#e n d i f

%O p t i c a l

generation

% The o p t i c a l

profile

generation

f u n c t i o n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

profile

mapping f u n c t i o n

located

near

line

270.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

161

’” window tj

o u t p u t F i l e = ’ n@node | sde@ @wl@nm optgen . dat ’ ;
o u t p u t G r i d = ’ n@node | sde@ @wl@nm optgen . grd ’ ;
e x p o r t F i l e = s t r c a t ( ’ n ’ , num2str ( @node | sde@ ) , ’ og ’ , num2str (@wl@) , ’ . mat ’ ) ;

%Number o f

data v a l u e s

to output per

line

i n o u t p u t DAT f i l e

numperline = 10;

try

d i s p(’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−’);
d i s p ( [ ’ OptGenConverter V e r s i o n

2 ’]) ;

d i s p ( [ ’ ( c ) 2009 M i c h a e l K e l z e n b e r g ’ ] ) ;
disp ( [ ’ California

Institute

o f Technology ’ ] ) ;

d i s p(’−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−’);

disp ( ’

’) ;

d i s p ( [ ’ Opening DAT f i l e

datFile

]) ;

grd = fopen ( d a t F i l e ) ;
if

( grd < 1)
e r r o r ( [ ’ Error opening

file

datFile

for

reading . ’ ] ) ;

%e x i t
end

( ˜ isequal (

’DF−ISE t e x t ’ ) )

f g e t l ( grd ) ,

d i s p ( ’ Error with

grid

file

format :

d i s p ( ’ P l e a s e d o u b l e −c h e c k i n p u t

might n o t be a DF ISE t e x t

file .

The

first

line

file . ’) ;

should read : ’ ) ;

DF−ISE t e x t ’ ) ;

disp ( ’
error ( ’ File

parse

error ’ ) ;

% exit
end
fln = 1;

verts = [ ] ;
regions = {};

n l = f g e t l ( grd ) ;

f l n=f l n +1;

w h i l e ( isempty ( regexp ( nl ,
n l = f g e t l ( grd ) ;

’ n b v e r t i c e s ∗= ∗ [ 0 − 9 ] + ’ ) ) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end
tmp=r e g e x p ( n l ,

’ [ 0 − 9 ] + ’ , ’ match ’ ) ;

numverts = str2num ( tmp { 1 } ) ;
disp ( [ ’

File

reports

n l = f g e t l ( grd ) ;

’ num2str ( numverts )

vertices ’ ] ) ;

f l n=f l n +1;

w h i l e ( isempty ( regexp ( nl ,
n l = f g e t l ( grd ) ;

’ n b e d g e s ∗= ∗ [ 0 − 9 ] + ’ ) ) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end
tmp=r e g e x p ( n l ,

’ [ 0 − 9 ] + ’ , ’ match ’ ) ;

numedges = str2num ( tmp { 1 } ) ;
disp ( [ ’

File

reports

n l = f g e t l ( grd ) ;

’ num2str ( numedges )

’ edges ’ ] ) ;

f l n=f l n +1;

w h i l e ( isempty ( regexp ( nl ,
n l = f g e t l ( grd ) ;

’ n b e l e m e n t s ∗= ∗ [ 0 − 9 ] + ’ ) ) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end
tmp=r e g e x p ( n l ,

’ [ 0 − 9 ] + ’ , ’ match ’ ) ;

numelems = str2num ( tmp { 1 } ) ;
disp ( [ ’

File

reports

’ num2str ( numelems )

’ elements ’ ] ) ;

162

n l = f g e t l ( grd ) ;

f l n=f l n +1;

w h i l e ( isempty ( regexp ( nl ,
n l = f g e t l ( grd ) ;

’ n b r e g i o n s ∗= ∗ [ 0 − 9 ] + ’ ) ) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end
’ [ 0 − 9 ] + ’ , ’ match ’ ) ;

tmp = r e g e x p ( n l ,

n u m r e g i o n s = str2num ( tmp { 1 } ) ;
disp ( [ ’

File

reports

’ num2str ( n u m r e g i o n s )

%Advance t o d a t a

section

n l = f g e t l ( grd ) ;

f l n=f l n +1;

w h i l e ( isempty ( regexp ( nl ,
n l = f g e t l ( grd ) ;

regions ’ ] ) ;

file ...

’ Data . ∗ \ { ’ ,

’ once ’ )

) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end

f e o f ( grd ) )
d i s p ( ’ Unexpec ted end−o f − f i l e , no d a t a p r o c e s s e d . ’ ) ;
’ num2st−−r ( f l n ) ] ) ;

disp ( [ ’ Line :
error ( ’ File

parse

error . ’) ;

end

regionArray = [ ] ;
disp ( ’

’) ;

d i s p ( ’ Reading d a t a
%Main r e a d i n g

points . . . ’ ) ;

loop .

Look f o r

PMIUserField 0 or 1 data

sets . . .

w h i l e ˜ f e o f ( grd )

n l = f g e t l ( grd ) ;

f l n=f l n +1;

while

r e g e x p i ( nl ,

( isempty (

n l = f g e t l ( grd ) ;

’ \ s ∗ f u n c t i o n \ s ∗=\ s ∗ P M I U s e r F i e l d [ 0 1 ] ’ ,

’ once ’ )

) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end
if

( f e o f ( grd ) )
break

end

tmp = r e g e x p ( n l ,

’ [ 0 1 ] ’ , ’ match ’ ) ;

axisNumber = str2num ( tmp { 1 } ) ;

n l = f g e t l ( grd ) ;

f l n=f l n +1;

while

r e g e x p i ( nl ,

( isempty (

n l = f g e t l ( grd ) ;

’ \ s ∗ v a l i d i t y \ s ∗=\ s ∗ \ [ \ s ∗ ” . ∗ ” \ s ∗ \ ] ’ ,

’ once ’ )

) && ˜ f e o f ( g r d ) )

f l n=f l n +1;

end

( f e o f ( grd ) )
error ([ ’ File

Parse Error near

line

’ num2str ( f l n ) ] ) ;

break
end
tmp = r e g e x p ( n l ,

’ ” . ∗ ” ’ , ’ match ’ ) ;

regionName = tmp { 1 } ;

n l = f g e t l ( grd ) ;

f l n=f l n +1;

while

r e g e x p i ( nl ,

( isempty (

n l = f g e t l ( grd ) ;

’ \ s ∗ V a l u e s \ s ∗ \ ( \ s ∗[0 −9]+\ s ∗ \ ) ’ ,

f l n=f l n +1;

end

( f e o f ( grd ) )
disp ( [ ’ File

Parse Error near

line

’ num2str ( f l n ) ] ) ;

break
end
tmp = r e g e x p ( n l ,

’ [ 0 − 9 ] + ’ , ’ match ’ ) ;

163

’ once ’ )

) && ˜ f e o f ( g r d ) )

numElems = str2num ( tmp { 1 } ) ;

dataPoints = [ ] ;
while

(1)

n l = f g e t l ( grd ) ;

f l n = f l n +1;

i f ( isempty ( regexp ( nl , ’[0 −9]+ ’)

) )

break
else
t h i s l i n e = r e g e x p ( n l , ’ [ \ . \ − \ e \E\+0 −9][\ s \.\ −\ e \E\ + 0 − 9 ] ∗ ’ , ’ match ’ ) ;
t h i s l i n e = t h i s l i n e {1};
d a t a P o i n t s = [ d a t a P o i n t s str2num ( t h i s l i n e ) ] ;
end
( ˜ i s e m p t y ( r e g e x p ( n l , ’ } ’ , ’ once ’ )

))

break
end
end

disp ( [ ’

Region

’ regionName

num2str ( numElems )

%d a t a
if

p o i n t s more o r

’ read

’ elements

less

’ num2str ( l e n g t h ( d a t a P o i n t s ) )

for

axis

than s t a t e d

’ num2str ( axisNumber )

’/ ’

...

]) ;

header

( numElems ˜= l e n g t h ( d a t a P o i n t s ) )
d i s p ( [ ’ E r r o r : number o f

data

points

disp ( [ ’ Parse

error

line

’ num2str ( f l n ) ] ) ;

error ([ ’ File

structure

near

error

d o e s n o t match

region

file

header ’ ] ) ;

’ regionName ] ) ;

end

existingRegion = 0;
f o r n =1: l e n g t h ( r e g i o n A r r a y )
c a n R e g i o n = r e g i o n A r r a y {n } ;
if

( i s e q u a l ( regionName , c a n R e g i o n . name ) )
existingRegion = n ;

end
end

( existingRegion )
if

( axisNumber == 0 )
r e g io nA r r a y { e x i s t i n g R e g i o n } . xdata = dataPoint s ;

else
r e g io nA r r a y { e x i s t i n g R e g i o n } . ydata = dataPoint s ;
end

˜ isequal (

l e n g t h ( r e gi on A r r a y { e x i s t i n g R e g i o n } . xdata ) ,

length ( regionArray { existingRegion

} . ydata ) )
d i s p ( ’ E r r o r : number o f x d a t a
error ([ ’ File

structure

error

points
in

d o e s n o t match number o f y d a t a p o i n t s ’ ) ;

region

end

else
newRegion . name = regionName ;
if

( axisNumber == 0 )
newRegion . x d a t a = d a t a P o i n t s ;
newRegion . y d a t a = [ ] ;

else
newRegion . y d a t a = d a t a P o i n t s ;
newRegion . x d a t a = [ ] ;
end
newRegion . g d a t a = z e r o s ( s i z e ( d a t a P o i n t s ) ) ;
r e g i o n A r r a y { end+1} = newRegion ;
end

164

’ regionName

]) ;

end

f o r n =1: l e n g t h ( r e g i o n A r r a y )
if

l e n g t h ( r e g i o n A r r a y {n } . x d a t a ) ,

˜ isequal (

d i s p ( ’ E r r o r : number o f x d a t a
error ([ ’ File

structure

error

l e n g t h ( r e g i o n A r r a y {n } . y d a t a ) )

points
in

d o e s n o t match number o f y d a t a p o i n t s ’ ) ;

region

r e g i o n A r r a y {n } . name ] ) ;

end
end

disp ( ’

’) ;

d i s p ( ’ Completed r e a d i n g DAT f i l e ’ ) ;
disp ( [ ’
disp ( ’

Read

’ num2str ( l e n g t h ( r e g i o n A r r a y ) )

region ( s ) ’]) ;

’) ;

f c l o s e ( grd ) ;

regionsToProcess = unique ( regionsToProcess ) ;
f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
reqName = r e g i o n s T o P r o c e s s {n } ;
hasRegion = 0 ;
f o r m=1: l e n g t h ( r e g i o n A r r a y )
if

r e g i o n A r r a y {m} . name )

i s e q u a l ( reqName ,
h a s R e g i o n =1;
break ;

end
end
if

˜ hasRegion
disp ( [ ’ Error :
disp ( [ ’

Vertex

position

not c o n t a i n e d

e r r o r ( [ ’ Unable t o

information

within

process

this

region :

for

requested

region

’ reqName ] ) ;

grid . ’ ] ) ;
’ reqName ] ) ;

end
end

disp ( ’

’) ;

%O p t i c a l
%T h i s

generation

function

% coordinate

profile

f u n c t i o n %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

determines the

( xi ,

OpticalGeneration

value at each

spatial

yi ) .

%The f u n c t i o n must be

called

newoptgen , and t a k e a s a r g u m e n t s x i and y i

%Load any e x t e r n a l

data

sets

this

area .

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
d i s p ( [ ’ L o a d i n g MAT f i l e

’ FDTDFile ] ) ;

l o a d ( FDTDFile ) ;
optGenMatrix = Ngen pavg ’ ;
newoptgen = @( x i ,

yi )

i n t e r p 2 ( Pabs x ,

Pabs y ,

optGenMatrix ,

x i ∗1 e −6 , y i ∗1 e−6 ) ;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

d i s p ( [ ’ Opening o u t p u t

file

outputFile

]) ;

ogo = f o p e n ( o u t p u t F i l e , ’ w ’ ) ;
if

( ogo < 1 )
e r r o r ( [ ’ Error opening

file

outputFile

for

writing . ’ ] ) ;

%e x i t
end

f p r i n t f ( ogo ,

’DF−ISE t e x t \n\n ’ ) ;

f p r i n t f ( ogo ,

’ I n f o {\n

f p r i n t f ( ogo ,

n b v e r t i c e s = %d\n

nb edges

= %d\n

nb faces

= 0\n ’ ,

f p r i n t f ( ogo ,

n b e l e m e n t s = %d\n

nb regions

= %d\n

datasets

= [

version

= 1.0\n

type

165

= d a t a s e t \n

dimension

= 2\n ’ ) ;

numverts , numedges ) ;

’ , numelems ,

numregions ) ;

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ ” O p t i c a l G e n e r a t i o n ”

’) ;

end
’]\n

f p r i n t f ( ogo ,

functions

= [

’) ;

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ O p t i c a l G e n e r a t i o n

’) ;

end
’ ] \ n}\n\ nData {\n\n ’ ) ;

f p r i n t f ( ogo ,

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
reqName = r e g i o n s T o P r o c e s s {n } ;
hasRegion = 0 ;
f o r m=1: l e n g t h ( r e g i o n A r r a y )
if

i s e q u a l ( reqName ,

r e g i o n A r r a y {m} . name )

h a s R e g i o n=m;
break ;
end
end
if

( hasRegion )
reg = regionArray { hasRegion } ;

disp (

[ ’ Proessing

Optical

Generation

for

region

f p r i n t f ( ogo , ’ D a t a s e t ( ” O p t i c a l G e n e r a t i o n ” ) {\n

r e g . name

’... ’]

);

= O p t i c a l G e n e r a t i o n \n

function

type

= s c a l a r \n ’ ) ;
f p r i n t f ( ogo ,

[’

d i m e n s i o n = 1\ n

V a l u e s (%d ) {\n ’ ,

= v e r t e x \n

location

validity

);
f p r i n t f ( ogo ,

l e n g t h ( r e g . xdata ) ) ;

gdata = z e r o s ( s i z e ( r e g . xdata ) ) ;
nl = 1;
f o r nv =1: l e n g t h ( r e g . x d a t a )

o g i = newoptgen ( r e g . x d a t a ( nv ) ,
f p r i n t f ( ogo ,

’ %22e ’ ,

r e g . y d a t a ( nv ) ) ;

ogi ) ;

g d a t a ( nv ) = o g i ;
nl = nl + 1;
if

( nl > 10)
f p r i n t f ( ogo ,

’\n ’ ) ;

nl = 1;
end
end
if

( nl > 1)
f p r i n t f ( ogo , ’ \ n ’ ) ;

end
f p r i n t f ( ogo ,

disp (

[’

}\n}\n\n ’ ) ;

’ num2str ( l e n g t h ( r e g . x d a t a ) )

processed ’ ]

regionArray { hasRegion } . gdata = gdata ;
end
end

f p r i n t f ( ogo , ’ \ n\n } ’ ) ;
f c l o s e ( ogo ) ;
disp ( [ ’ Finished

disp ( ’

writing

output

file

outputFile

]) ;

’) ;

d i s p ( [ ’ Copying from g r i d

file :

grdFile ]) ;

c o p y f i l e ( grdFile , outputGrid ) ;
d i s p ( [ ’ To g r i d

file :

’ outputGrid ] ) ;

166

);

= [

r e g . name

]\n ’

disp ( ’

’) ;

disp ( [ ’ Exporting

generation

save (

exportFile

disp ( ’

profile :

’ regionArray ’ ,

exportFile

’ numverts ’ ,

]) ;

’ numedges ’ ,

’ numelems ’ ,

’ numregions ’ ) ;

’) ;

disp ( ’ Processing

complete ! ’ ) ;

exit (0) ;

c a t c h ME
d i s p (ME) ;
exit (1) ;
end

#end

Listing E.10: Code to weigh and sum the single frequency simulations to get an AM 1.5G
profile
% MATLAB f i l e

for

w e i g h i n g and summing t h e

% c r e a t e an AM1. 5 g a b s o r p t i o n

single

frequency

simulations

profile .

wllow = 3 0 0 ;
wlspacing = 50;
wlnum = 2 2 ;
weight = [ 0 . 0 7 0 3 , 0 . 7 3 6 1 , 1 . 5 3 5 4 , 2 . 5 8 3 4 , 3 . 1 0 4 5 , 3 . 3 7 5 5 , 3 . 5 4 7 3 , 3 . 6 8 3 5 , 3 . 5 5 1 6 , 3 . 2 8 9 5 ,
3.4004 ,3.3025 ,2.8433 ,1.6354 ,2.8614 ,2.7587 ,1.9358 ,1.2055 ,2.1215 ,2.1937 ,

1.9071 ,0.4237 ,0.0241]∗(

@pitch@ ∗ 0 . 5 ) ˆ2∗1E− 2 0 / 1. 6 0 9E−19;

e x p o r t F i l e = s t r c a t ( ’ n ’ , num2str ( @node | sde@ ) , ’ ogAM1 . 5 . mat ’ ) ;

AM15regionArray = [ ] ;

for

j =0:wlnum
wl = w l l o w + j ∗ w l s p a c i n g ;
G e n F i l e = s t r c a t ( ’ n ’ , num2str ( @node | sde@ ) , ’ og ’ , num2str ( wl ) , ’ . mat ’ ) ;
load ( GenFile ) ;
dim = s i z e ( r e g i o n A r r a y ) ;
for

i =1: dim ( 2 )
AM15regionArray { i } . name = r e g i o n A r r a y { i } . name ;
AM15regionArray { i } . x d a t a = r e g i o n A r r a y { i } . x d a t a ;
AM15regionArray { i } . y d a t a = r e g i o n A r r a y { i } . y d a t a ;
if

j == 0
for

l =1: l e n g t h ( r e g i o n A r r a y { i } . g d a t a ( 1 , : ) )
if

r e g i o n A r r a y { i } . x d a t a ( 1 , l ) == 0
AM15regionArray { i } . g d a t a ( 1 , l ) = 0 ;

else
AM15regionArray { i } . g d a t a ( 1 , l ) = r e g i o n A r r a y { i } . g d a t a ( 1 , l ) ∗
w e i g h t ( j +1) ∗1000 ∗1 E6/ r e g i o n A r r a y { i } . x d a t a ( 1 , l ) ;
end
end
else
for

l =1: l e n g t h ( r e g i o n A r r a y { i } . g d a t a ( 1 , : ) )
if

r e g i o n A r r a y { i } . x d a t a ( 1 , l ) == 0
AM15regionArray { i } . g d a t a ( 1 , l ) = 0 ;

else
AM15regionArray { i } . g d a t a ( 1 , l ) = r e g i o n A r r a y { i } . g d a t a ( 1 , l ) ∗
w e i g h t ( j +1) ∗1000 ∗1 E6/ r e g i o n A r r a y { i } . x d a t a ( 1 , l ) +

167

AM15regionArray { i } . g d a t a ( 1 , l ) ;
end
end
end
end
disp ( [ ’ I n i t i a l i z i n g ’ ] ) ;
d i s p ( [ ’ Copying from d a t

file :

’ GenFile ] ) ;

end

g r d F i l e = ’ n@node | sde@ @wl@nm optgen . grd ’ ;

o u t p u t F i l e = ’ n@node | sde@ AM15g optgen . dat ’ ;
o u t p u t G r i d = ’ n@node | sde@ AM15g optgen . grd ’ ;

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” c o r e e m i t t e r ” ’

’” window inner ” ’
’” c o r e b a s e ” ’

’” s h e l l e m i t t e r ” ’

’” window inner2 ” ’

’” s h e l l b a s e ” ’

’” window tj2 ” ’

’” window tj

’” c o r e e m i t t e r 2

” ’};
#e l s e
regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” window inner ” ’

’” s h e l l e m i t t e r ” ’

’” s h e l l b a s e ” ’

’” window tj

’” c o r e b a s e ” ’ } ;

’” c o r e e m i t t e r ” ’

#e n d i f

d i s p ( [ ’ Opening o u t p u t

file

outputFile

]) ;

ogo = f o p e n ( o u t p u t F i l e , ’ w ’ ) ;
if

( ogo < 1 )
e r r o r ( [ ’ Error opening

file

outputFile

for

writing . ’ ] ) ;

%e x i t
end

f p r i n t f ( ogo ,

’DF−ISE t e x t \n\n ’ ) ;

f p r i n t f ( ogo ,

’ I n f o {\n

f p r i n t f ( ogo ,

n b v e r t i c e s = %d\n

nb edges

= %d\n

nb faces

= 0\n ’ ,

f p r i n t f ( ogo ,

n b e l e m e n t s = %d\n

nb regions

= %d\n

datasets

= [

= 1.0\n

version

= d a t a s e t \n

type

dimension

= 2\n ’ ) ;

numverts , numedges ) ;

’ , numelems ,

numregions ) ;

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ ” O p t i c a l G e n e r a t i o n ”

’) ;

end
’]\n

f p r i n t f ( ogo ,

functions

= [

’) ;

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ O p t i c a l G e n e r a t i o n

’) ;

end
’ ] \ n}\n\ nData {\n\n ’ ) ;

f p r i n t f ( ogo ,

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
reqName = r e g i o n s T o P r o c e s s {n } ;
hasRegion = 0 ;
f o r m=1: l e n g t h ( AM15regionArray )
if

i s e q u a l ( reqName ,

AM15regionArray {m} . name )

h a s R e g i o n=m;
break ;
end
end
if

( hasRegion )
r e g = AM15regionArray { h a s R e g i o n } ;

disp (

[ ’ Processing

Optical

Generation

for

region

f p r i n t f ( ogo , ’ D a t a s e t ( ” O p t i c a l G e n e r a t i o n ” ) {\n
= s c a l a r \n ’ ) ;

168

r e g . name

function

’... ’]

);

= O p t i c a l G e n e r a t i o n \n

type

f p r i n t f ( ogo ,

[’

d i m e n s i o n = 1\ n

V a l u e s (%d ) {\n ’ ,

= v e r t e x \n

location

validity

= [

r e g . name

);
f p r i n t f ( ogo ,

l e n g t h ( r e g . xdata ) ) ;

nl = 1;
f o r nv =1: l e n g t h ( r e g . x d a t a )

o g i = AM15regionArray { h a s R e g i o n } . g d a t a ( nv ) ;
f p r i n t f ( ogo ,

’ %22e ’ ,

ogi ) ;

nl = nl + 1;
if

( nl > 10)
’\n ’ ) ;

f p r i n t f ( ogo ,
nl = 1;
end
end
if

( nl > 1)
f p r i n t f ( ogo , ’ \ n ’ ) ;

end
f p r i n t f ( ogo ,

disp (

[’

}\n}\n\n ’ ) ;

’ num2str ( l e n g t h ( r e g . x d a t a ) )

processed ’ ]

);

end
end

f p r i n t f ( ogo , ’ \ n\n } ’ ) ;
f c l o s e ( ogo ) ;
disp ( [ ’ Finished

disp ( ’

writing

output

file

outputFile

]) ;

’) ;

d i s p ( [ ’ Copying from g r i d

file :

grdFile ]) ;

c o p y f i l e ( grdFile , outputGrid ) ;
d i s p ( [ ’ To g r i d

disp ( ’

file :

’ outputGrid ] ) ;

’) ;

disp ( [ ’ Exporting

generation

save (

disp ( ’

exportFile

profile :

’ AM15regionArray ’ ,

exportFile

]) ;

’ numverts ’ ,

’ numedges ’ ,

’ numelems ’ ,

’ numregions ’ ) ;

’) ;

disp ( ’ Processing

complete ! ’ ) ;

exit (0) ;

c a t c h ME
d i s p (ME) ;
exit (1) ;
end

Listing E.11: Code to place weighted data on FEM grid
e x p o r t F i l e = s t r c a t ( ’ n ’ , num2str ( @node | sde@ ) , ’ @wl@nm EQE . mat ’ ) ;

AM15regionArray = [ ] ;

G e n F i l e = s t r c a t ( ’ n ’ , num2str ( @node | sde@ ) , ’ og ’ , num2str (@wl@) , ’ . mat ’ ) ;
load ( GenFile ) ;
dim = s i z e ( r e g i o n A r r a y ) ;
for

i =1: dim ( 2 )
AM15regionArray { i } . name = r e g i o n A r r a y { i } . name ;
AM15regionArray { i } . x d a t a = r e g i o n A r r a y { i } . x d a t a ;

169

]\n ’

AM15regionArray { i } . y d a t a = r e g i o n A r r a y { i } . y d a t a ;
l =1: l e n g t h ( r e g i o n A r r a y { i } . g d a t a ( 1 , : ) )

for

r e g i o n A r r a y { i } . x d a t a ( 1 , l ) == 0
AM15regionArray { i } . g d a t a ( 1 , l ) = 0 ;

else
AM15regionArray { i } . g d a t a ( 1 , l ) = r e g i o n A r r a y { i } . g d a t a ( 1 , l ) ∗ ( @pitch@ / 2 ) ˆ2∗1E
− 1 1 ∗ 6 8 . 9 9 / ( 1 . 6 0 9 E−19∗ r e g i o n A r r a y { i } . x d a t a ( 1 , l ) ) ;
end
end
end
disp ( [ ’ I n i t i a l i z i n g ’ ] ) ;
d i s p ( [ ’ Copying from d a t

file :

’ GenFile ] ) ;

g r d F i l e = ’ n@node | sde@ @wl@nm optgen . grd ’ ;

o u t p u t F i l e = ’ n@node | sde@ @wl@nm EQE . dat ’ ;
o u t p u t G r i d = ’ n@node | sde@ @wl@nm EQE . grd ’ ;

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” c o r e e m i t t e r ” ’

’” window inner ” ’
’” c o r e b a s e ” ’

’” s h e l l e m i t t e r ” ’

’” window inner2 ” ’

’” s h e l l b a s e ” ’

’” window tj2 ” ’

’” window tj

’” c o r e e m i t t e r 2

” ’};
#e l s e
regionsToProcess = { ’” window outer ” ’
”’

’” t j c o r e ” ’

’” window inner ” ’

’” s h e l l e m i t t e r ” ’

’” s h e l l b a s e ” ’

’” window tj

’” c o r e b a s e ” ’ } ;

’” c o r e e m i t t e r ” ’

#e n d i f

d i s p ( [ ’ Opening o u t p u t

file

outputFile

]) ;

ogo = f o p e n ( o u t p u t F i l e , ’ w ’ ) ;
if

( ogo < 1 )
e r r o r ( [ ’ Error opening

file

outputFile

for

writing . ’ ] ) ;

%e x i t
end

f p r i n t f ( ogo ,

’DF−ISE t e x t \n\n ’ ) ;

f p r i n t f ( ogo ,

’ I n f o {\n

f p r i n t f ( ogo ,

n b v e r t i c e s = %d\n

nb edges

= %d\n

nb faces

= 0\n ’ ,

f p r i n t f ( ogo ,

n b e l e m e n t s = %d\n

nb regions

= %d\n

datasets

= [

= 1.0\n

version

= d a t a s e t \n

type

dimension

’ , numelems ,

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ ” O p t i c a l G e n e r a t i o n ”

’) ;

end
’]\n

f p r i n t f ( ogo ,

functions

= [

’) ;

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
f p r i n t f ( ogo , ’ O p t i c a l G e n e r a t i o n

’) ;

end
’ ] \ n}\n\ nData {\n\n ’ ) ;

f p r i n t f ( ogo ,

f o r n =1: l e n g t h ( r e g i o n s T o P r o c e s s )
reqName = r e g i o n s T o P r o c e s s {n } ;
hasRegion = 0 ;
f o r m=1: l e n g t h ( AM15regionArray )
if

i s e q u a l ( reqName ,

AM15regionArray {m} . name )

h a s R e g i o n=m;
break ;
end
end
if

( hasRegion )
r e g = AM15regionArray { h a s R e g i o n } ;

disp (

[ ’ Processing

Optical

Generation

for

region

170

r e g . name

= 2\n ’ ) ;

numverts , numedges ) ;

’... ’]

);

numregions ) ;

f p r i n t f ( ogo , ’ D a t a s e t ( ” O p t i c a l G e n e r a t i o n ” ) {\n

= O p t i c a l G e n e r a t i o n \n

function

type

= s c a l a r \n ’ ) ;
f p r i n t f ( ogo ,

[’

d i m e n s i o n = 1\ n

V a l u e s (%d ) {\n ’ ,

= v e r t e x \n

location

validity

= [

r e g . name

]\n ’

);
f p r i n t f ( ogo ,

l e n g t h ( r e g . xdata ) ) ;

nl = 1;
f o r nv =1: l e n g t h ( r e g . x d a t a )

o g i = AM15regionArray { h a s R e g i o n } . g d a t a ( nv ) ;
f p r i n t f ( ogo ,

’ %22e ’ ,

ogi ) ;

nl = nl + 1;
if

( nl > 10)
’\n ’ ) ;

f p r i n t f ( ogo ,
nl = 1;
end
end
if

( nl > 1)
f p r i n t f ( ogo , ’ \ n ’ ) ;

end
f p r i n t f ( ogo ,

disp (

[’

}\n}\n\n ’ ) ;

’ num2str ( l e n g t h ( r e g . x d a t a ) )

processed ’ ]

);

end
end

f p r i n t f ( ogo , ’ \ n\n } ’ ) ;
f c l o s e ( ogo ) ;
disp ( [ ’ Finished

disp ( ’

writing

output

file

outputFile

]) ;

’) ;

d i s p ( [ ’ Copying from g r i d

file :

grdFile ]) ;

c o p y f i l e ( grdFile , outputGrid ) ;
d i s p ( [ ’ To g r i d

disp ( ’

file :

’ outputGrid ] ) ;

’) ;

disp ( [ ’ Exporting

generation

save (

exportFile

disp ( ’

profile :

’ AM15regionArray ’ ,

exportFile

]) ;

’ numverts ’ ,

’ numedges ’ ,

’ numelems ’ ,

’ numregions ’ ) ;

’) ;

disp ( ’ Processing

complete ! ’ ) ;

exit (0) ;

c a t c h ME
d i s p (ME) ;
exit (1) ;
end

The next two sets of code appeared in another Workbench layout.
Listing E.12: Code to run the device physics simulations for tandem wire array cells
File{
∗−I n p u t
#i f

@ c o r e h e i g h t @ == 40
#i f

@pitch@ == 7
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

171

Grid

= ” . . / 0 1 − Opt/ n1 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n1 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n1 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n3 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n3 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n3 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n4 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n4 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n4 AM15g optgen . d a t ”
#e n d i f
#e n d i f
#i f

@pitch@ == 4
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n6272 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n6272 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n6272 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n6719 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n6719 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n6719 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n7166 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n7166 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n7166 AM15g optgen . d a t ”
#e n d i f
#e n d i f
#i f

@pitch@ == 5
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n9846 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n9846 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n9846 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n10293 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n10293 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n10293 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n10740 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n10740 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n10740 AM15g optgen . d a t ”
#e n d i f
#e n d i f
#i f

@pitch@ == 6
#i f

[ string

compare @Structure@ ” c o n f o r m a l ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n13420 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n13420 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n13420 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n13867 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n13867 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n13867 AM15g optgen . d a t ”
#e n d i f
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

172

Grid

= ” . . / 0 1 − Opt/ n14314 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n14314 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n14314 AM15g optgen . d a t ”
#e n d i f
#e n d i f
#e n d i f
#i f

@ c o r e h e i g h t @ == 80
#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

Grid

= ” . . / 0 1 − Opt/ n607 msh . g r d ”

Doping

= ” . . / 0 1 − Opt/ n607 msh . d a t ”

O p t i c a l G e n e r a t i o n I n p u t = ” . . / 0 1 − Opt/ n607 AM15g optgen . d a t ”
#e n d i f
#e n d i f

P a r a m e t e r s =”@parameter@ ”
∗−Output
Plot

= ” @tdrdat@ ”

C u r r e n t = ” @plot@ ”
Output

= ”@log@”

N o n L o c a l P l o t = ” n@node@ nl ”

Electrode {
{ Name=”T o p c o n t a c t ”

V o l t a g e =0 h R e c V e l o c i t y = 100}

{ Name=”B o t t o m c o n t a c t ”

V o l t a g e =0 e R e c V e l o c i t y = 100}

#i f

Physics {
M o l e F r a c t i o n ( RegionName = [ ” w i n d o w o u t e r ” ” w i n d o w i n n e r ” ” w i n d o w t j ” ]
x F r a c t i o n =0.56
M o l e F r a c t i o n ( RegionName = [ ” s h e l l e m i t t e r ” ” s h e l l b a s e ” ]
x F r a c t i o n= 0 . 9
M o l e F r a c t i o n ( RegionName = [ ” t j c o r e ” ” c o r e e m i t t e r ” ” c o r e b a s e ” ]
x F r a c t i o n= 0 . 9
#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

M o l e F r a c t i o n ( RegionName = [ ” w i n d o w i n n e r 2 ” ” w i n d o w t j 2 ” ]
x F r a c t i o n=
#e n d i f
#e n d i f

Physics {
A r e a F a c t o r = @< 1 E11 / ( 3 . 1 4 1 5 9 2 ∗ ( @pitch@ ∗ 0 . 5 ) ∗ ( @pitch@ ∗ 0 . 5 ) ) >@ ∗ t o g e t
Fermi
Recombination (
SRH
Mobility (

DopingDep
HighFieldSat

ThermionicEmission

e B a r r i e r T u n n e l i n g ”TD NLM” (

173

current

i n mA/cmˆ2

Band2Band
TwoBand

h B a r r i e r T u n n e l i n g ”TD NLM” (
Band2Band
TwoBand

Optics (
OpticalGeneration (
ReadFromFile ( S c a l i n g =0
TimeDependence (
WaveTime = ( 0 . 9 ,

10)

Scaling = 1.0

P h y s i c s ( m a t e r i a l I n t e r f a c e =”S i l i c o n G e r m a n i u m /GaInP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Ambient /GaInP ” ) {
Recombination ( surfaceSRH )

#i f

[ string

compare @Structure@ ” h e m i s p h e r e ” ] == 0

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide / S i l i c o n G e r m a n i u m ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide /GaInP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide /GaAsP ” ) {
Recombination ( surfaceSRH )
#e n d i f

#i f

[ string

compare @Structure@ ” s p h e r e ” ] == 0

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide / S i l i c o n G e r m a n i u m ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide /GaInP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide /GaAsP ” ) {
Recombination ( surfaceSRH )
#e n d i f

P h y s i c s ( m a t e r i a l = ”GaInP ” ) {
Recombination (
Radiative

174

Auger

P h y s i c s ( m a t e r i a l = ”GaAsP ” ) {
Recombination (
Radiative
Auger

Plot {
x M o l e F r a c t i o n Doping D o n o r C o n c e n t r a t i o n A c c e p t o r C o n c e n t r a t i o n
eEffectiveStateDensity

hEffectiveStateDensity

EffectiveIntrinsicDensity

IntrinsicDensity

e D e n s i t y hDensity SpaceCharge
eQuasiFermiPotential

h Q u a s i F e r m i P o t e n t i a l BandGap ConductionBandEnergy ValenceBandEnergy

ElectronAffinity
ElectricField
eLifetime

ElectricField / vector

eCurrent / Vector hCurrent / Vector
eMobility

ElectrostaticPotential

h L i f e t i m e SRH Auger T o t a l R e c o m b i n a t i o n S u r f a c e R e c o m b i n a t i o n R a d i a t i v e R e c o m b i n a t i o n

hMobility

eVelocity

current / vector

hVelocity

SRH Auger T o t a l R e c o m b i n a t i o n S u r f a c e R e c o m b i n a t i o n R a d i a t i v e R e c o m b i n a t i o n
BarrierTunneling
eBarrierTunneling

hBarrierTunneling

NonLocal
OpticalGeneration

NonLocalPlot

((0 ,

ConductionBand
hDensity

0) ) {
ValenceBand

eDensity

hQuasiFermi

eQuasiFermi

NonLocal

Math{
RhsMin = 1E−12
Extrapolate
Derivatives
RelErrControl
I t e r a t i o n s =20
ExtendedPrecision
D i g i t s =7
Notdamped=100
E r r R e f ( e l e c t r o n ) = 1E0
E r r R e f ( h o l e ) = 1E0
ExitOnFailure
N u m b e r o f T h r e a d s = maximum
S t a c k S i z e = 20000000

∗ 20MB;

needed

f o r NewRayTracer

Method=S u p e r
NonLocal ”TD NLM” (
M a t e r i a l I n t e r f a c e = ” S i l i c o n G e r m a n i u m /GaInP”
Length =15e−7

# [ cm ]

distance

to anchor point

P e r m e a t i o n = 15 e−7
DirectCurrent
Cylindrical (0.0)

175

T r a n s i e n t=BE
T r a n s i e n t D i g i t s =7
T r a n s i e n t E r r R e f ( e l e c t r o n ) = 1E0
T r a n s i e n t E r r R e f ( h o l e ) = 1E0

CNormPrint

Solve {
N e w C u r r e n t P r e f i x = ” tmp ”

Coupled
Plot (

{ poisson }

F i l e P r e f i x = ”n@node@ Banddgm ” )

Coupled

{ poisson

electron }

Coupled

{ poisson

hole }

Coupled

{ poisson

electron

Transient

hole }

I n i t i a l S t e p =1e −10 MaxStep =0.2 MinStep = 1 e −40 I n c r e m e n t=2
I n i t i a l T i m e =0 F i n a l T i m e=1
) { Coupled ( I t e r a t i o n s =20) { P o i s s o n

E l e c t r o n Hole

} }

NewCurrentPrefix = ” Light IV ”

Quasistationary

I n i t i a l S t e p =1e−4 MaxStep =1e−3 MinStep = 1 e −30 I n c r e m e n t =1.7 DoZero
#i f

@GaAsP SRHLifeTime@ > 1 e−8
Goal { v o l t a g e = 1 . 5 Name=”B o t t o m c o n t a c t ” }
) { Coupled { P o i s s o n
Plot (

E l e c t r o n Hole

F i l e P r e f i x = ” n@node@ Banddgm Jsc ” Time = ( 0 )

#e n d i f
#i f

@GaAsP SRHLifeTime@ < 1 e−8 && @GaAsP SRHLifeTime@ > 1 e −10
Goal { v o l t a g e = 1 . 3 Name=”B o t t o m c o n t a c t ” }
) { Coupled { P o i s s o n
Plot (

E l e c t r o n Hole

F i l e P r e f i x = ” n@node@ Banddgm Jsc ” Time = ( 0 )

#e n d i f
#i f

@GaAsP SRHLifeTime@ < 1 e −10
Goal { v o l t a g e = 1 . 2 Name=”B o t t o m c o n t a c t ” }
) { Coupled { P o i s s o n
Plot (

E l e c t r o n Hole

F i l e P r e f i x = ” n@node@ Banddgm Jsc ” Time = ( 0 )

#e n d i f

N e w C u r r e n t P r e f i x = ” tmp 2 ”

Quasistationary

I n i t i a l S t e p =1e−2 MaxStep =0.1 MinStep = 1 e −30 I n c r e m e n t =1.5 DoZero
Goal { c u r r e n t = 0 Name=”B o t t o m c o n t a c t ” }
) { Coupled { P o i s s o n

E l e c t r o n Hole

Plot (

F i l e P r e f i x = ” n@node@ Banddgm Voc ” )

System ( ” rm −f tmp ∗ ” ) ∗ remove t h e
System ( ” rm −f tmp2 ∗ ” ) ∗ remove t h e

p l o t we dont need anymore .
p l o t we dont need anymore .

Only modifications to the parameter file are shown:
176

Listing E.13: Parameter file for the device physics simulations
M a t e r i a l = ”GaInP” {

...

# Taken from

Silicon

S c h a r f e t t e r ∗ r e l a t i o n and t r a p

level

f o r SRH r e c o m b i n a t i o n :

{ ∗ t a u = taumin + ( taumax − taumin ) / ( 1 + ( N/ N r e f ) ˆgamma)
∗ t a u (T) = t a u ∗ ( (T/ 3 0 0 ) ˆ Talpha )
∗ t a u (T) = t a u ∗ exp (

( TempDep )

T c o e f f ∗ ( (T/ 3 0 0 ) −1) ) ( ExpTempDep )

taumin

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [s]

taumax

= 1 . 0 0 0 0 e −05 ,

3 . 0 0 0 0 e −06

# [s]
# [ cmˆ( −3) ]

Nref

= 1 . 0 0 0 0 e +16 ,

1 . 0 0 0 0 e +16

gamma

= 1

# [1]

Talpha

= −1.5000 e +00 ,

−1.5000 e +00

Tcoeff

= 2.55

2.55

Etr ap

= 0 . 0 0 0 0 e +00

# [1]

# [ eV ]

Auger ∗

coefficients :

{ ∗ R Auger = ( C n n + C p p ) ( n p − n i e f f ˆ 2 )
∗ w i t h C n , p = (A + B (T/T0 ) + C (T/T0 ) ˆ 2 ) ( 1 + H exp (−{n , p }/N0 ) )

= 1 . 0 0 0 0 e −30 ,

1 . 0 0 0 0 e −30

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [ cmˆ6/ s ]

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [ cmˆ6/ s ]

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [1]

= 1 . 0 0 0 0 e +18 ,

1 . 0 0 0 0 e +18

# [ cmˆ( −3) ]

# Taken from GaAs

BarrierTunneling
{ ∗ Non L o c a l

Barrier

Tunneling

∗ G( r ) = g ∗A∗T/kB∗F( r ) ∗ Pt ( r ) ∗ l n [ ( 1 + exp ( ( E( r )−Es ) /kB/T) ) /(1+ exp ( ( E( r )−Em) /kB/T) ) ]
∗ where :

Pt ( r )

g = As/A, As i s

A is

i s WKB a p p r o x i m a t i o n

for

the Richardson constant

F( r )

the

E( r )

carrier

electric

for

free

for

Es i s

carrier

quasi

fermi

Em i s

carrier

fermi

energy

electrons

energy

semiconductor

i n metal

0.14

# [1]

M a t e r i a l I n t e r f a c e = ” Oxide /GaAsP” {
SurfaceRecombination {
S0 = @SRV@, @SRV@ ∗ [ cm/ s ]
Sref = 0 ∗

[1]

M a t e r i a l I n t e r f a c e = ” Oxide /GaInP” {
SurfaceRecombination {
S0 = @SRV@, @SRV@ ∗ [ cm/ s ]
Sref = 0 ∗

probability
carriers

field

= 0.05

tunneling

energy

the

the Richardson constant

[1]

177

semiconductor

M a t e r i a l I n t e r f a c e = ” Oxide / S i l i c o n G e r m a n i u m ” {
SurfaceRecombination {
S0 = 1 0 0 , 100 ∗ [ cm/ s ]
Sref = 0 ∗

[1]

M a t e r i a l I n t e r f a c e = ” S i l i c o n G e r m a n i u m /GaInP” {
ThermionicEmission {
A = 2, 2

# [1]

B = 4, 4

# [1]

C = 1, 1

# [1]

SurfaceRecombination {
S0 = @SRV@, @SRV@ ∗ [ cm/ s ]
Sref = 0 ∗

[1]

BarrierTunneling
{ ∗ Non L o c a l

Barrier

Tunneling

∗ G( r ) = g ∗A∗T/kB∗F( r ) ∗ Pt ( r ) ∗ l n [ ( 1 + exp ( ( E( r )−Es ) /kB/T) ) /(1+ exp ( ( E( r )−Em) /kB/T) ) ]
∗ where :

Pt ( r )

g = As/A, As i s

A is

i s WKB a p p r o x i m a t i o n

for

the Richardson constant

F( r )

the

E( r )

carrier

electric

for

free

for

probability
carriers

semiconductor

electrons

field

Es i s

carrier

quasi

fermi

Em i s

carrier

fermi

energy

= 0.21

tunneling

energy

the

the Richardson constant

energy

semiconductor

i n metal

0.4

# [1]

M a t e r i a l = ” SiliconGermanium ” {

...

# Taken from

Silicon

S c h a r f e t t e r ∗ r e l a t i o n and t r a p

level

f o r SRH r e c o m b i n a t i o n :

{ ∗ t a u = taumin + ( taumax − taumin ) / ( 1 + ( N/ N r e f ) ˆgamma)
∗ t a u (T) = t a u ∗ ( (T/ 3 0 0 ) ˆ Talpha )
∗ t a u (T) = t a u ∗ exp (

( TempDep )

T c o e f f ∗ ( (T/ 3 0 0 ) −1) ) ( ExpTempDep )

taumin

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [s]

taumax

= 1 . 0 0 0 0 e −05 ,

3 . 0 0 0 0 e −06

# [s]

taumax

= @SiGe SRHLifeTime@

Nref

= 1 . 0 0 0 0 e +16 ,

1 . 0 0 0 0 e +16

gamma

= 1

# [1]

Talpha

= −1.5000 e +00 ,

−1.5000 e +00

Tcoeff

= 2.55

2.55

Etr ap

= 0 . 0 0 0 0 e +00

@SiGe SRHLifeTime@

# [s]

# [ cmˆ( −3) ]

# [1]

# [ eV ]

# Taken from GaAs

BarrierTunneling
{ ∗ Non L o c a l

Barrier

Tunneling

∗ G( r ) = g ∗A∗T/kB∗F( r ) ∗ Pt ( r ) ∗ l n [ ( 1 + exp ( ( E( r )−Es ) /kB/T) ) /(1+ exp ( ( E( r )−Em) /kB/T) ) ]
∗ where :

Pt ( r )

i s WKB a p p r o x i m a t i o n

for

the

tunneling

178

probability

g = As/A, As i s

A is

the Richardson constant

F( r )

the

E( r )

carrier

electric

for

carriers

semiconductor

electrons

field

Es i s

carrier

quasi

fermi

Em i s

carrier

fermi

energy

= 0.05

free

energy

for

energy

semiconductor

i n metal

0.14

# [1]

M a t e r i a l = ”GaAsP” {

...

# Taken from

Silicon

S c h a r f e t t e r ∗ r e l a t i o n and t r a p

level

f o r SRH r e c o m b i n a t i o n :

{ ∗ t a u = taumin + ( taumax − taumin ) / ( 1 + ( N/ N r e f ) ˆgamma)
∗ t a u (T) = t a u ∗ ( (T/ 3 0 0 ) ˆ Talpha )
∗ t a u (T) = t a u ∗ exp (

( TempDep )

T c o e f f ∗ ( (T/ 3 0 0 ) −1) ) ( ExpTempDep )

taumin

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [s]

taumax

= 1 . 0 0 0 0 e −05 ,

3 . 0 0 0 0 e −06

# [s]

taumax

= @GaAsP SRHLifeTime@

Nref

= 1 . 0 0 0 0 e +16 ,

1 . 0 0 0 0 e +16

gamma

= 1

# [1]

Talpha

= −1.5000 e +00 ,

−1.5000 e +00

Tcoeff

= 2.55

2.55

Etr ap

= 0 . 0 0 0 0 e +00

, @GaAsP SRHLifeTime@

# [s]

# [ cmˆ( −3) ]

# [1]

# [ eV ]

Auger ∗

coefficients :

{ ∗ R Auger = ( C n n + C p p ) ( n p − n i e f f ˆ 2 )
∗ w i t h C n , p = (A + B (T/T0 ) + C (T/T0 ) ˆ 2 ) ( 1 + H exp (−{n , p }/N0 ) )

= 1 . 0 0 0 0 e −30 ,

1 . 0 0 0 0 e −30

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [ cmˆ6/ s ]

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [ cmˆ6/ s ]

= 0 . 0 0 0 0 e +00 ,

0 . 0 0 0 0 e +00

# [1]

= 1 . 0 0 0 0 e +18 ,

1 . 0 0 0 0 e +18

# [ cmˆ( −3) ]

# Taken from GaAs

BarrierTunneling
{ ∗ Non L o c a l

Barrier

Tunneling

∗ G( r ) = g ∗A∗T/kB∗F( r ) ∗ Pt ( r ) ∗ l n [ ( 1 + exp ( ( E( r )−Es ) /kB/T) ) /(1+ exp ( ( E( r )−Em) /kB/T) ) ]
∗ where :

Pt ( r )

g = As/A, As i s

A is

i s WKB a p p r o x i m a t i o n

the Richardson constant

F( r )

the

E( r )

carrier

electric

tunneling

for

free

for

probability
carriers

electrons

field

Es i s

carrier

quasi

fermi

Em i s

carrier

fermi

energy

= 0.05

the

energy

for

the Richardson constant

energy

semiconductor

i n metal

0.14

# [1]

179

semiconductor

Listing E.14: Code to extract select parameters for the simulations
# Plot

l i g h t J−V and P−V c u r v e s and e x t r a c t

Photovoltaic

parameters

# o r P l o t d a r k J−V c h a r a c t e r i s t i c s

# #s e t d e p @node|−2@

set N

@node@

set

@node : index@

# proj load
proj load

@plot@ PLT JV ( $N )
L i g h t I V n @ p r e v i o u s @ d e s . p l t PLT JV ( $N )

#− Automatic

alternating

color

assignment

tied

t o node i n d e x

#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−#
s e t COLORS

[ list

orange green

blue red

v i o l e t brown o r a n g e magenta ]

s e t NCOLORS [ l l e n g t h $COLORS ]
set

color

# Plot

[ lindex

$COLORS [ e x p r

l i g h t J−V c h a r a c t e r i s t i c s

c v c r e a t e D S J ( $N )

$ i%$NCOLORS ] ]

and e x t r a c t PV p a r a m e t e r s

”PLT JV ( $N ) B o t t o m c o n t a c t O u t e r V o l t a g e ” ”PLT JV ( $N ) B o t t o m c o n t a c t

TotalCurrent ”

c v i n v J ( $N ) y

c v c r e a t e V( $N )

”PLT JV ( $N ) B o t t o m c o n t a c t O u t e r V o l t a g e ” ”PLT JV ( $N ) B o t t o m c o n t a c t O u t e r V o l t a g e ”

c v c r e a t e W i t h F o r m u l a P( $N )

”∗” A A

c v d i s p l a y P( $N ) y2

c v s e t C u r v e A t t r J ( $N )

” l i g h t −JV” $ c o l o r

solid

c v s e t C u r v e A t t r P( $N )

” l i g h t −PV” $ c o l o r

dashed

g r s e t A x i s A t t r X { V o l t a g e (V) }

circle

16 0 {} b l a c k 1 14 0 5 0

g r s e t A x i s A t t r Y { C u r r e n t D e n s i t y (mA/cmˆ 2 ) } 16 0 30
g r s e t A x i s A t t r Y2 { Power (mW/cmˆ 2 ) } 16 0 26

# Extract

Photovoltaic

# Extract

short

set

J s c ( $N )

ft scalar

3 defcolor 1 defcolor

2 none 3 d e f c o l o r 1 d e f c o l o r

b l a c k 1 14 0 5 0

parameters

circuit

current

density ,

Jsc

[mA/cm ˆ 2 ]

[ c v c o m p u t e ” v e c v a l y ( ,0) ” A A A A ]

Jsc

# E x t r a c t open

[ f o r m a t %.2 f

circuit

$ J s c ( $N ) ]

v o l t a g e , Voc [ V ]

s e t Jmin [ c v c o m p u t e ” vecmin()” A A A A ]
if

{ $Jmin <= 0} {

elseif

s e t Voc ( $N )

[ expr

[ c v c o m p u t e ” v e c z e r o ()” A A A A ] ]

{ $Jmin <= 1 e −6} {
s e t Voc ( $N )

[ expr

[ c v c o m p u t e ” v e c v a l x (, $Jmin ) ” A A A A ] ]

f t s c a l a r Voc [ f o r m a t %.4 f $Voc ( $N ) ]

# Extract

fill

factor

(FF) , maximum power o u t p o u t (Pm [mW/cm2 ] )

s e t Ps 100 ;# I n c i d e n t

light

power d e n s i t y

f o r AM1. 5 g r a d i a t i o n

{ $Voc ( $N ) > 0} {
s e t Pm( $N )

[ c v c o m p u t e ” vecmax()” A A A A ]

## f i l l f a c t o r
s e t FF( $N )

## e f f i c i e n c y
set

E f f ( $N )

in %

[ e x p r $Pm( $N ) / ( $Voc ( $N ) ∗ $ J s c ( $N ) ) ∗ 1 0 0 ]
i n % (mW/cmˆ 2 / ( 1 0 0mW/cmˆ 2 ) ∗100%)

[ e x p r $Pm( $N ) / $Ps ∗ 1 0 0 ]

180

and

efficiency

i n mW/cmˆ2

( eff )

f t s c a l a r Pm [ f o r m a t %.4 f $Pm( $N ) ]
f t s c a l a r FF
ft scalar

[ f o r m a t %.4 f $FF ( $N ) ]

Eff

[ f o r m a t %.4 f

$ E f f ( $N ) ]

The final set of code stood in its own workbench and was used for the defect simulations:
Listing E.15: Code to generate the structure
##########################################################
# Dan Turner−Evans

# 05/24/12

# A 2D GaAsP on SiGe m i c r o w i r e t o

test

t h e E&M p a c k a g e

##########################################################

( sde : c l e a r )

;;−−−−−−−−−−−−−−−−−−−−
( display ” init

parameter ”) ( newline )

; ; geometry
( define

height

( define

diameter 2)

40)

; um

( define

pitch

; um

4 ) ; um

( define

spacing

( define

bottom emitter t

4 ) ; um

( define

masking oxide t

( define

e x p o s e d h 5 ) ; um

0 . 1 ) ; um
0 . 2 ) ; um

( d e f i n e gap 0 . 1 ) ; um
( define

defect h

( define

s q h 2 ) ; um

2 ) ; um

( define

top emitter t

( define

window t 0 . 0 2 ) ; um

0 . 1 ) ; um

( d e f i n e MgF t 0 . 1 ) ; um
( define

TiOx t 0 . 0 6 ) ; um

( d e f i n e fname ”n@node@ msh ” )
( d e f i n e elGridFname ” n@node@ el msh ” ) ; mixed e l e m e n t
( define

grid

file

name f o r

sdevice

simulation

top window d 1 e19 )

( define

t o p e m i t t e r d 1 e19 )

( define

t o p b a s e d 1 e17 )

( define

d e f e c t d 1 e19 )

( define

b o t t o m e m i t t e r d 6 e19 )

( define

bottom base d 1 e17 )

( d e f i n e WireLT @SiGe SRHLifeTime@ )
( d e f i n e TopCellLT @GaAsP SRHLifeTime@ )
( d e f i n e DefectLT @Defect SRHLifeTime@ )

; ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
( display ” create

structure . . .

\n ” )

;;−−−−−−−−−−−−−−−−−−−−
( s d e g e o : s e t −d e f a u l t −b o o l e a n ”ABA” )
( display ”

back

r e f l e c t o r ”) ( newline )

( s d e g e o : c r e a t e −r e c t a n g l e

( position

(∗

pitch

−0.5) 0 0 ) ( p o s i t i o n

(∗

pitch

0 . 5 ) −1 0 ) ” S i l v e r ” ”

(∗

pitch

−0.5) 0 0 ) ( p o s i t i o n

(∗

pitch

0 . 5 ) (+ h e i g h t

Back−R e f l e c t o r ” )

;;−−−−−−−−−−−−−−−−−−−
( display ” f i l l

with gas ”) ( newline )

( s d e g e o : c r e a t e −r e c t a n g l e
defect h

s q h (− ( ∗

( position
pitch

0 . 5 ) gap ) window t ) 0 ) ” Gas” ” ambient ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

t o p ARC” ) ( n e w l i n e )

181

spacing

( sdegeo : c r e a t e −c i r c l e

( p o s i t i o n 0 (+ h e i g h t

defect h

s q h window t TiOx t MgF t ) 0 ) (− ( ∗

pitch

defect h

s q h window t TiOx t ) 0 ) (− ( ∗

0 . 5 ) gap

defect h

s q h window t ) 0 ) (− ( ∗

defect h

s q h ) 0 ) (− ( ∗

pitch

0 . 5 ) gap ) ”GaAsP” ”

defect h

s q h ) 0 ) (− ( ∗

pitch

0 . 5 ) (+ gap

0 . 5 ) gap ) ”MgF” ”Top ARC ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

bottom ARC” ) ( n e w l i n e )

( sdegeo : c r e a t e −c i r c l e

( p o s i t i o n 0 (+ h e i g h t

pitch

) ”TiOx” ”Bottom ARC ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

t o p window ” ) ( n e w l i n e )

( sdegeo : c r e a t e −c i r c l e

( p o s i t i o n 0 (+ h e i g h t

pitch

0 . 5 ) gap ) ”

GaInP” ” T o p W i n d o w C i r c l e ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

top

c e l l ”) ( newline )

( sdegeo : c r e a t e −c i r c l e

( p o s i t i o n 0 (+ h e i g h t

Top Cell Emitter ”)
( sdegeo : c r e a t e −c i r c l e

( p o s i t i o n 0 (+ h e i g h t

t o p e m i t t e r t ) ) ”GaAsP” ” T o p C e l l C i r c ” )
( s d e g e o : c r e a t e −r e c t a n g l e
pitch

( p o s i t i o n (+ ( ∗

0 . 5 ) gap ) (+ h e i g h t

defect h

pitch

−0.5) gap ) (+ h e i g h t

d e f e c t h ) 0 ) ( p o s i t i o n (− ( ∗

s q h ) 0 ) ”GaAsP” ” T o p C e l l R e c t ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

defective

region ”) ( newline )

( s d e g e o : c r e a t e −r e c t a n g l e
pitch

( p o s i t i o n (+ ( ∗

0 . 5 ) gap ) (+ h e i g h t

pitch

−0.5) gap ) (− h e i g h t

e x p o s e d h ) 0 ) ( p o s i t i o n (− ( ∗

d e f e c t h ) 0 ) ”GaAsP” ” D e f e c t R e g i o n ” )

;−−−−−−−−−−−−−−−−−−−−−
( display ”

oxide ”) ( newline )

( s d e g e o : c r e a t e −r e c t a n g l e
diameter

0.5)

( p o s i t i o n (− ( ∗ d i a m e t e r

m a s k i n g o x i d e t ) (− h e i g h t

−0.5) m a s k i n g o x i d e t ) 0 0 ) ( p o s i t i o n (+ ( ∗

e x p o s e d h ) 0 ) ” Oxide ” ” Boot ” )

;;−−−−−−−−−−−−−−−−−−−−
( display ”

wire ”) ( newline )

# ( s d e g e o : c r e a t e −r e c t a n g l e

( position

(∗ diameter

−0.5) 0 0 ) ( p o s i t i o n

(∗ diameter

0.5)

height

0) ”

SiliconGermanium ” ” Wire Emitter ”)
( s d e g e o : c r e a t e −r e c t a n g l e

( position

(∗ diameter

−0.5) 0 0 ) ( p o s i t i o n

(∗ diameter

0.5)

height

0) ”

SiliconGermanium ” ” Wire Base ”)

;;−−−−−−−−−−−−−−−−−−−−
( d i s p l a y ” s a v e boundary ” ) ( n e w l i n e )
( s d e i o : s a v e−t d r −bnd ” a l l ” ( s t r i n g −append fname ” . bnd ” ) )

; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
( d i s p l a y ” p l a c e dopants ,
;;

set

lifetime

...

\n ” )

c o n s t a n t window d o p i n g

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

”WindowDop” ” N D o p a n t A c t i v e C o n c e n t r a t i o n ” t o p w i n d o w d )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ”WindowDop” ”WindowDop” ” T o p W i n d o w C i r c l e ” )

;;

c o n s t a n t top

cell

emitter

doping

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” T o p C e l l Em i t t e r D o p ” ” N D o p a n t A c t i v e C o n c e n t r a t i o n ” t o p e m i t t e r d )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” T o p C e l l E m i t t e r D o p ” ” T o p C e l l E m i t t e r D o p ” ” T o p C e l l E m i t t e r ” )

;;

c o n s t a n t top

cell

base doping

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” TopCellBaseDop ” ” P D o p a n t A c t i v e C o n c e n t r a t i o n ” t o p b a s e d )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” T o p C e l l B a s e C i r c D o p ” ” TopCellBaseDop ” ” T o p C e l l C i r c ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellBaseRectDop ” ” TopCellBaseDop ” ” T o p C e l l R e c t ” )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” TopCellLTe ” ” e L i f e t i m e ” TopCellLT )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” TopCellLTh ” ” h L i f e t i m e ” TopCellLT )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTe1 ” ” TopCellLTe ” ” T o p W i n d o w C i r c l e ” 0 ” R e p l a c e ” )

182

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTh1 ” ” TopCellLTh ” ” T o p W i n d o w C i r c l e ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTe2 ” ” TopCellLTe ” ” T o p C e l l E m i t t e r ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTh2 ” ” TopCellLTh ” ” T o p C e l l E m i t t e r ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTe3 ” ” TopCellLTe ” ” T o p C e l l C i r c ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTh3 ” ” TopCellLTh ” ” T o p C e l l C i r c ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTe4 ” ” TopCellLTe ” ” T o p C e l l R e c t ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” TopCellLTh4 ” ” TopCellLTh ” ” T o p C e l l R e c t ” 0 ” R e p l a c e ” )

;;

c o n s t a n t bottom

cell

base doping

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” WireDop ” ” P D o p a n t A c t i v e C o n c e n t r a t i o n ” b o t t o m b a s e d )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” WireDop ” ” WireDop ” ” W i r e B a s e ” )

;;

G a u s s i a n bottom

cell

emitter

doping

( s d e d r : d e f i n e −r e f i n e m e n t −window ” BottomCelEmitterDopTop ” ” L i n e ”
( position

(∗ diameter

−0.5) h e i g h t

( position

(∗ diameter

0.5)

height

0)
0) )

( s d e d r : d e f i n e −r e f i n e m e n t −window ” B o t t o m C e l E m i t t e r D o p L e f t ” ” L i n e ”
( position

(∗ diameter

−0.5) (− h e i g h t (+ e x p o s e d h

( position

(∗ diameter

−0.5) h e i g h t

0 . 1 ) ) 0)

0) )

( s d e d r : d e f i n e −r e f i n e m e n t −window ” BottomCelEmitterDopRight ” ” L i n e ”
( position

(∗ diameter

0 . 5 ) (− h e i g h t (+ e x p o s e d h

( position

(∗ diameter

0.5)

( s d e d r : d e f i n e −g a u s s i a n − p r o f i l e
” PeakPos ” 0

height

0 . 1 ) ) 0)

0) )

” BottomCelEmitterDop ” ” N D o p a n t A c t i v e C o n c e n t r a t i o n ”

” PeakVal ” b o t t o m e m i t t e r d ” Length ” 0 . 0 5 ” E r f ”

” Factor ” 0 . )

( s d e d r : d e f i n e −a n a l y t i c a l −p r o f i l e −p l a c e m e n t ” BottomCelEmitterDopTop ” ” BottomCelEmitterDop ” ”
BottomCelEmitterDopTop ” ” N e g a t i v e ” ” NoReplace ” ” E v a l ” )
( s d e d r : d e f i n e −a n a l y t i c a l −p r o f i l e −p l a c e m e n t ” BottomCelEmitterDopRight ” ” BottomCelEmitterDop ” ”
BottomCelEmitterDopRight ” ” P o s i t i v e ” ” NoReplace ” ” E v a l ” )
( s d e d r : d e f i n e −a n a l y t i c a l −p r o f i l e −p l a c e m e n t ” B o t t o m C e l E m i t t e r D o p L e f t ” ” BottomCelEmitterDop ” ”
B o t t o m C e l E m i t t e r D o p L e f t ” ” N e g a t i v e ” ” NoReplace ” ” E v a l ” )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” WireLTe ” ” e L i f e t i m e ” WireLT )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

”WireLTh” ” h L i f e t i m e ” WireLT )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” WireLTe ” ” WireLTe ” ” W i r e B a s e ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ”WireLTh” ”WireLTh” ” W i r e B a s e ” 0 ” R e p l a c e ” )

;;

constant

defect

doping

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” DefectDop ” ” P D o p a n t A c t i v e C o n c e n t r a t i o n ” d e f e c t d )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” DefectDop ” ” DefectDop ” ” D e f e c t R e g i o n ” 0 ” R e p l a c e ” )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” De f e c t L T e ” ” e L i f e t i m e ” DefectLT )

( s d e d r : d e f i n e −c o n s t a n t − p r o f i l e

” DefectLTh ” ” h L i f e t i m e ” DefectLT )

( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” D e f e c t L T e ” ” D e f e c t L T e ” ” D e f e c t R e g i o n ” 0 ” R e p l a c e ” )
( s d e d r : d e f i n e −c o n s t a n t −p r o f i l e −r e g i o n ” DefectLTh ” ” DefectLTh ” ” D e f e c t R e g i o n ” 0 ” R e p l a c e ” )

; ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
( display ” define

contacts . . .

\n ” )

( s d e g e o : d e f i n e −c o n t a c t −s e t ” BottomContact ” 4
( s d e g e o : d e f i n e −c o n t a c t −s e t ” TopContact ” 4

( c o l o r : r g b 1 0 0 ) ”##” )

( c o l o r : r g b 0 0 1 ) ”##” )

( s d e g e o : d e f i n e −2d−c o n t a c t

( f i n d −edge−i d

( p o s i t i o n 0 0 0 ) ) ” BottomContact ” )

( s d e g e o : d e f i n e −2d−c o n t a c t

( f i n d −edge−i d

( p o s i t i o n 0 (+ h e i g h t

defect h

0 . 5 ) gap ) ) 0 ) ) ” TopContact ” )
( sde : r e f r e s h )

; ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
( d i s p l a y ” add r e f i n e m e n t s
;;

global

...

\n ” )

refinement

( s d e d r : d e f i n e −r e f i n e m e n t −s i z e ” r e f . g l o b a l ” 0 . 5

0.5 0 0.5

( s d e d r : d e f i n e −r e f i n e m e n t −window ” win . g l o b a l ” ” R e c t a n g l e ”
( position

(∗

pitch

−0.5) 0 0 )

183

0.5 0 )

s q h window t (− ( ∗

pitch

(∗

( position

pitch

0 . 5 ) (+ h e i g h t

spacing

s q h (− ( ∗

defect h

pitch

0 . 5 ) gap ) window t )

0) )
( s d e d r : d e f i n e −r e f i n e m e n t −p l a c e m e n t ” r e f . g l o b a l ” ” r e f . g l o b a l ” ” win . g l o b a l ” )

;;

on d o p i n g

# ( s d e d r : d e f i n e −r e f i n e m e n t −s i z e ” d o p i n g ” 0 . 1

0.05 0 0.1

0.05 0 )

# ( s d e d r : d e f i n e −r e f i n e m e n t −f u n c t i o n ” d o p i n g ” ” D o p i n g C o n c e n t r a t i o n ” ” MaxTransDiff ” 1 )
# ( s d e d r : d e f i n e −r e f i n e m e n t −p l a c e m e n t ” d o p i n g ” ” d o p i n g ” ” win . g l o b a l ” )

;;

bottom c o n t a c t

( s d e d r : d e f i n e −r e f i n e m e n t −window ”BC” ” R e c t a n g l e ” ( p o s i t i o n
diameter

0.5)

0.2

(∗ diameter

−0.5) 0 0 ) ( p o s i t i o n

(∗

0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . BC” 0 . 1

0.1

0.01 1 1.5)

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . BC” ” S i z e . BC” ”BC” )

;;

emitter

( s d e d r : d e f i n e −r e f i n e m e n t −window ” Emit Top ” ” R e c t a n g l e ” ( p o s i t i o n
position

(∗ diameter

0 . 5 ) (− h e i g h t

0.2)

(∗ diameter

−0.5) h e i g h t 0 ) (

0) )

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . Emit Top ” 0 . 1

0.1

0 . 0 1 1 −1.5)

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . Emit Top ” ” S i z e . Emit Top ” ” Emit Top ” )

( s d e d r : d e f i n e −r e f i n e m e n t −window ” E m i t S i d e R ” ” R e c t a n g l e ” ( p o s i t i o n (− ( ∗ d i a m e t e r
height

exposed h ) 0 ) ( p o s i t i o n

(∗ diameter

0.5)

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . E m i t S i d e R ” 0 . 1

height

0.1

0.01

0.5)

0 . 2 ) (−

0) )
0.01

−1.5 −1.5)

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . E m i t S i d e R ” ” S i z e . E m i t S i d e R ” ” E m i t S i d e R ” )

( s d e d r : d e f i n e −r e f i n e m e n t −window ” E m i t S i d e L ” ” R e c t a n g l e ” ( p o s i t i o n (+ ( ∗ d i a m e t e r
height

exposed h ) 0 ) ( p o s i t i o n

(∗ diameter

−0.5) h e i g h t

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . E m i t S i d e L ” 0 . 1

0.1

0.01

−0.5)

0 . 2 ) (−

0) )

0.01

1.5

1.5)

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . E m i t S i d e L ” ” S i z e . E m i t S i d e L ” ” E m i t S i d e L ” )

;;

top

cell

( s d e d r : d e f i n e −r e f i n e m e n t −window ” T o p C e l l ” ” R e c t a n g l e ” ( p o s i t i o n
sq h ) 0 ) ( p o s i t i o n

(∗

pitch

0 . 5 ) (+ h e i g h t

( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . T o p C e l l T ” 0 . 1

(∗

pitch

defect h

sq h (∗

0.1

0 . 0 1 1 −1.05)

0.1

pitch

−0.5) (+ h e i g h t

0.5)

defect h

) 0) )

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . T o p C e l l T ” ” S i z e . T o p C e l l T ” ” T o p C e l l ” )
( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . T o p C e l l L ” 0 . 1

0.1

0.01

0.1

1.05

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . T o p C e l l L ” ” S i z e . T o p C e l l L ” ” T o p C e l l ” )
( s d e d r : d e f i n e −m u l t i b o x−s i z e ” S i z e . T o p C e l l R ” 0 . 1

0.1

0.01

0.1

−1.05 1 )

( s d e d r : d e f i n e −m u l t i b o x−p l a c e m e n t ” P l a c e m e n t . T o p C e l l R ” ” S i z e . T o p C e l l R ” ” T o p C e l l ” )

; ; −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
( d i s p l a y ” b u i l d mesh . . .

\n ” )

( s d e : b u i l d −mesh ” snmesh ” ”−y 1 e 5 ” elGridFname ) ; ; − y s e t s

t h e max a s p e c t

( d i s p l a y ” done . ” ) ( n e w l i n e )

Listing E.16: Code to create an FDTD grid
##########################################################
# Dan Turner−Evans

# 05/04/12

# A Si

microwire to

test

t h e E&M p a c k a g e

##########################################################

tensor {

MESH {

MinCellSize

direction Y = 1

EMW {
parameter

f i l e n a m e = ” @parameter@ ”

184

ratio

the elements

C o m p l e x R e f r a c t i v e I n d e x WavelengthDep

Real

Imag

w a v e l e n g t h = @wl@
NPWX = 5
NPWY = 30
Grading

off

Listing E.17: Code to setup and run the simulation for a TM excitation source
#d e f i n e

h e i g h t 40

#d e f i n e

diameter 2

#d e f i n e

pitch 4

#d e f i n e

spacing 4

#d e f i n e

defect h 2

#d e f i n e

sq h 2

#d e f i n e gap 0 . 1
#d e f i n e

window t 0 . 0 2

Globals {
GridFile

= ”@tdr@”

ParameterFile

= ” @parameter@ ”

InspectFile
LogFile

= ” @plot@ ”
= ”@log@”

TotalTimeSteps = 10000000
NumberOfThreads = maximum

ComplexRefractiveIndex {
WavelengthDep = { Real , Imag }

PECMedia {
R e g i o n = {” B a c k R e f l e c t o r ”}

Boundary {
Type

= Periodic

Sides

= {X}

Boundary {
Type

= CPML

Sides

= {Y}

PlaneWaveExcitation {
BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 5 ∗ s p a c i n g >@,
BoxCorner2

window t + 0 . 5 ∗ s p a c i n g >@,
Theta

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +
0)

= 180

Psi

= 0

Wavelength

= @<1000.∗ wl>@

Intensity

= 0.1

Nrise

= 4

Plot {

185

Name

= ” n@node@ Eabs ”

Quantity = { A b s E l e c t r i c F i e l d ,

AbsMagneticField }

FinalPlot = yes

Extractor {
Name

= ” n@node@ a ”

Quantity

= { AbsorbedPhotonDensity }

Name

= ” total ”

Quantity

= PhotonFluxDensity

BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

Sensor {

window t + 0 . 2 5 ∗ s p a c i n g >@,
BoxCorner2

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 2 5 ∗ s p a c i n g >@,

Mode

= {Integrate}

Name

= ”reflected”

Quantity

= PhotonFluxDensity

BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

Sensor {

window t + 0 . 7 5 ∗ s p a c i n g >@,
BoxCorner2

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 7 5 ∗ s p a c i n g >@,

= {Integrate}

Mode

Sensor {
Name

= ” absorbed total ”

Quantity

= absorbedPhotonDensity

BoxCorner1 = (@@,

0 , 0)

BoxCorner2 = (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap + window t >@,
0)
Mode

= {Integrate}

Sensor {
Name

= ” absorbed Top Cell ”

Quantity

= absorbedPhotonDensity

Region

= {” T o p C e l l E m i t t e r ” , ” T o p C e l l C i r c ” , ” T o p C e l l R e c t ”}

Mode

= {Integrate}

Sensor {
Name

= ” absorbed Defect Region ”

Quantity

= absorbedPhotonDensity

Region

= {” D e f e c t R e g i o n ”}

Mode

= {Integrate}

Sensor {
Name

= ” absorbed Bottom Cell ”

Quantity

= absorbedPhotonDensity

Region

= {” W i r e B a s e ”}

Mode

= {Integrate}

Detector {

186

Tolerance

= 1 e−4

Listing E.18: Code to extract the reflection, transmission, and absorption of the structure for
the TM excitation
#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
# $Id :

// t c a d / s u p p o r t / main / e x a m p l e s / g e t t i n g −s t a r t e d /emw/ s i m p l e 3 d / i n s p e c t i n s . cmd#3 $

# Author :

Gergoe L e t a y

#s e t d e p @node |emw@
load library

extend

# e x t r a c t photon
set

”[ f i l e

flux

tail

[ file

rootname @plot@ ] ] ”

p r o j l o a d ” $gc . p l t ”
s e t Ntot

[ d s g e t V a l u e $ g c ” t o t a l I n t e g r BoxYmin/ P h o t o n F l u x D e n s i t y ” ]

s e t Nr

[ d s g e t V a l u e $ g c ” r e f l e c t e d I n t e g r BoxYmax/ P h o t o n F l u x D e n s i t y ” ]

s e t Na

[ d s g e t V a l u e $ g c ” a b s o r b e d t o t a l I n t e g r Box/ A b s o r b e d P h o t o n D e n s i t y ” ]

set

Na Top Cell Emitter

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

Top Cell Emitter /

AbsorbedPhotonDensity ” ]
set

Na Top Cell Circ

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

T o p C e l l C i r c / AbsorbedPhotonDensity

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

Top Cell Rect / AbsorbedPhotonDensity

” ]
set

Na Top Cell Rect
” ]

set

Na Top Cell

set

Na Defect Region

[ e x p r $ N a T o p C e l l E m i t t e r+$ N a T o p C e l l C i r c+$ N a T o p C e l l R e c t ]
[ d s g e t V a l u e $gc ” a b s o r b e d D e f e c t R e g i o n I n t e g r

Defect Region /

AbsorbedPhotonDensity ” ]
set

Na Bottom Cell

[ d s g e t V a l u e $gc ” a b s o r b e d B o t t o m C e l l I n t e g r Wire Base / AbsorbedPhotonDensity ” ]

s e t Ni [ e x p r $Ntot+$Nr ]

# calculate

relative

s e t R [ expr

1 . ∗ $Nr / $Ni ]

s e t A [ expr

1 . ∗ $Na/ $Ni ]

set

A Top Cell

set

A Defect Region

set

A Bottom Cell

values

1 . ∗ $ N a T o p C e l l / $Ni ]

[ expr

1 . ∗ $ N a D e f e c t R e g i o n / $Ni ]

1 . ∗ $ N a B o t t o m C e l l / $Ni ]

s e t RA [ e x p r $R+$A ]

# write

back t o swb

f t s c a l a r R [ f o r m a t %.3 g $R ]
f t s c a l a r A [ f o r m a t %.3 g $A ]
ft scalar

A Top Cell

ft scalar

A Defect Region

[ f o r m a t %.3 g $ A T o p C e l l ]

ft scalar

A Bottom Cell

[ f o r m a t %.3 g $ A D e f e c t R e g i o n ]

[ f o r m a t %.3 g $ A B o t t o m C e l l ]

f t s c a l a r RA [ f o r m a t %.3 g $RA ]

Listing E.19: Code to setup and run the simulation for a TE excitation source
#d e f i n e

h e i g h t 40

#d e f i n e

diameter 2

#d e f i n e

pitch 4

#d e f i n e

spacing 4

#d e f i n e

defect h 2

#d e f i n e

sq h 2

#d e f i n e gap 0 . 1
#d e f i n e

window t 0 . 0 2

Globals {
GridFile

= ”@tdr@”

ParameterFile

= ” @parameter@ ”

187

InspectFile
LogFile

= ” @plot@ ”
= ”@log@”

TotalTimeSteps = 10000000
NumberOfThreads = maximum

ComplexRefractiveIndex {
WavelengthDep = { Real , Imag }

PECMedia {
R e g i o n = {” B a c k R e f l e c t o r ”}

Boundary {
Type

= Periodic

Sides

= {X}

Boundary {
Type

= CPML

Sides

= {Y}

PlaneWaveExcitation {
BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 5 ∗ s p a c i n g >@,
BoxCorner2

window t + 0 . 5 ∗ s p a c i n g >@,
Theta

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +
0)

= 180

Psi

= 90

Wavelength

= @<1000.∗ wl>@

Intensity

= 0.1

Nrise

= 4

Plot {
Name

= ” n@node@ Eabs ”

Quantity = { A b s E l e c t r i c F i e l d ,

AbsMagneticField }

FinalPlot = yes

Extractor {
Name

= ” n@node@ a ”

Quantity

= { AbsorbedPhotonDensity }

Name

= ” total ”

Quantity

= PhotonFluxDensity

BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

Sensor {

window t + 0 . 2 5 ∗ s p a c i n g >@,
BoxCorner2

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 2 5 ∗ s p a c i n g >@,
Mode

= {Integrate}

Name

= ”reflected”

Sensor {

188

Quantity

= PhotonFluxDensity

BoxCorner1

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

window t + 0 . 7 5 ∗ s p a c i n g >@,

= (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap +

BoxCorner2

window t + 0 . 7 5 ∗ s p a c i n g >@,

= {Integrate}

Mode

Sensor {
Name

= ” absorbed total ”

Quantity

= absorbedPhotonDensity

BoxCorner1 = (@@,

0 , 0)

BoxCorner2 = (@@, @< h e i g h t + d e f e c t h + s q h + p i t c h ∗ 0 . 5 − gap + window t >@,
0)
= {Integrate}

Mode

Sensor {
Name

= ” absorbed Top Cell ”

Quantity

= absorbedPhotonDensity

Region

= {” T o p C e l l E m i t t e r ” , ” T o p C e l l C i r c ” , ” T o p C e l l R e c t ”}

Mode

= {Integrate}

Sensor {
Name

= ” absorbed Defect Region ”

Quantity

= absorbedPhotonDensity

Region

= {” D e f e c t R e g i o n ”}

Mode

= {Integrate}

Sensor {
Name

= ” absorbed Bottom Cell ”

Quantity

= absorbedPhotonDensity

Region

= {” W i r e B a s e ”}

Mode

= {Integrate}

Detector {
Tolerance

= 1 e−4

Listing E.20: Code to extract the reflection, transmission, and absorption of the structure for
the TE excitation
#−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
# $Id :

// t c a d / s u p p o r t / main / e x a m p l e s / g e t t i n g −s t a r t e d /emw/ s i m p l e 3 d / i n s p e c t i n s . cmd#3 $

# Author :

Gergoe L e t a y

#s e t d e p @node | emw1@
load library

extend

# e x t r a c t photon
set

”[ f i l e

flux

tail

[ file

rootname @plot@ ] ] ”

p r o j l o a d ” $gc . p l t ”
s e t Ntot

[ d s g e t V a l u e $ g c ” t o t a l I n t e g r BoxYmin/ P h o t o n F l u x D e n s i t y ” ]

s e t Nr

[ d s g e t V a l u e $ g c ” r e f l e c t e d I n t e g r BoxYmax/ P h o t o n F l u x D e n s i t y ” ]

s e t Na

[ d s g e t V a l u e $ g c ” a b s o r b e d t o t a l I n t e g r Box/ A b s o r b e d P h o t o n D e n s i t y ” ]

set

Na Top Cell Emitter

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

Top Cell Emitter /

AbsorbedPhotonDensity ” ]
set

Na Top Cell Circ

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

” ]

189

T o p C e l l C i r c / AbsorbedPhotonDensity

set

Na Top Cell Rect

[ d s g e t V a l u e $gc ” a b s o r b e d T o p C e l l I n t e g r

Top Cell Rect / AbsorbedPhotonDensity

” ]
set

Na Top Cell

set

Na Defect Region

[ e x p r $ N a T o p C e l l E m i t t e r+$ N a T o p C e l l C i r c+$ N a T o p C e l l R e c t ]
[ d s g e t V a l u e $gc ” a b s o r b e d D e f e c t R e g i o n I n t e g r

Defect Region /

AbsorbedPhotonDensity ” ]
set

Na Bottom Cell

[ d s g e t V a l u e $gc ” a b s o r b e d B o t t o m C e l l I n t e g r Wire Base / AbsorbedPhotonDensity ” ]

s e t Ni [ e x p r $Ntot+$Nr ]

# calculate

relative

s e t R [ expr

1 . ∗ $Nr / $Ni ]

s e t A [ expr

1 . ∗ $Na/ $Ni ]

set

A Top Cell

set

A Defect Region

set

A Bottom Cell

values

1 . ∗ $ N a T o p C e l l / $Ni ]

[ expr

1 . ∗ $ N a D e f e c t R e g i o n / $Ni ]

1 . ∗ $ N a B o t t o m C e l l / $Ni ]

s e t RA [ e x p r $R+$A ]

# write

back t o swb

# f t s c a l a r R [ f o r m a t %.3 g $R ]
# f t s c a l a r A [ f o r m a t %.3 g $A ]
ft scalar

A Top Cell TE

ft scalar

A Defect Region TE

[ f o r m a t %.3 g $ A T o p C e l l ]

ft scalar

A Bottom Cell TE

[ f o r m a t %.3 g $ A D e f e c t R e g i o n ]

[ f o r m a t %.3 g $ A B o t t o m C e l l ]

# f t s c a l a r RA [ f o r m a t %.3 g $RA ]

Listing E.21: Code to map the FDTD grid onto the FEM mesh
File{
∗−I n p u t
= ” n@node | s d e @ e l m s h . t d r ”

Grid

O p t i c a l S o l v e r I n p u t = ”∗ a e m l . t d r ”
I l l u m i n a t i o n S p e c t r u m= ”am15g 50nm TE TM mod . t x t ”
P a r a m e t e r s =”@parameter@ ”
∗−Output
Plot

= ” @tdrdat@ ”

C u r r e n t = ” @plot@ ”
Output

= ”@log@”

Physics {
Optics (
OpticalGeneration (
ComputeFromSpectrum ( )
OpticalSolver (
FromFile (
I d e n t i f y i n g P a r a m e t e r = ( ” Wavelength ” ” P s i ” )

Plot {
OpticalGeneration

Math{
RhsMin = 1E−12
Extrapolate

190

Derivatives
RelErrControl
I t e r a t i o n s =20
ExtendedPrecision
D i g i t s =7
Notdamped=100
E r r R e f ( e l e c t r o n ) = 1E0
E r r R e f ( h o l e ) = 1E0
ExitOnFailure
N u m b e r o f T h r e a d s = maximum
∗ 20MB;

S t a c k S i z e = 20000000

needed

f o r NewRayTracer

Method=S u p e r

CNormPrint

Solve {
Optics

Listing E.22: Code to run the device physics
File{
∗−I n p u t
Grid

= ” n@node | s d e @ e l m s h . t d r ”

L i f e T i m e = ” n@node | s d e @ e l m s h . t d r ”
P a r a m e t e r s =”@parameter@ ”
#i f

@GaAsP SRHLifeTime@ == 1E−07 && @Defect SRHLifeTime@ == 1E−12
OpticalGenerationInput = ” n117 des . tdr ”

#e l s e
O p t i c a l G e n e r a t i o n I n p u t = ” n@node | s d e v i c e @ d e s . t d r ”
#e n d i f
∗−Output
Plot

= ” @tdrdat@ ”

C u r r e n t = ” @plot@ ”
Output

= ”@log@”

N o n L o c a l P l o t = ” n@node@ nl ”

Electrode {
{ Name=”TopContact ”

V o l t a g e =0 h R e c V e l o c i t y = 100}

{ Name=”BottomContact ”

V o l t a g e =0 e R e c V e l o c i t y = 100}

Physics {
A r e a F a c t o r = @< 1 E11 /4 >@ ∗ t o g e t

current

i n mA/cmˆ2

Fermi
Recombination (
SRH
Mobility (
DopingDep
HighFieldSat

ThermionicEmission

e B a r r i e r T u n n e l i n g ”TD NLM” (
Band2Band

191

TwoBand

h B a r r i e r T u n n e l i n g ”TD NLM” (
Band2Band
TwoBand

Optics (
OpticalGeneration (
ReadFromFile (

Datasetname=A b s o r b e d P h o t o n D e n s i t y
S c a l i n g =0
TimeDependence (
WaveTime = ( 0 . 9 ,

10)

Scaling = 1.0

# Physics

( m a t e r i a l I n t e r f a c e =”S i l i c o n G e r m a n i u m /GaInP ” ) {

Recombination ( surfaceSRH )

# }

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Ambient /GaInP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Ambient /GaAsP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide / S i l i c o n G e r m a n i u m ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l I n t e r f a c e =”Oxide /GaAsP ” ) {
Recombination ( surfaceSRH )

P h y s i c s ( m a t e r i a l = ”GaInP ” ) {
Recombination (
Radiative
Auger

P h y s i c s ( m a t e r i a l = ”GaAsP ” ) {
Recombination (
Radiative
Auger

Plot {
x M o l e F r a c t i o n Doping D o n o r C o n c e n t r a t i o n A c c e p t o r C o n c e n t r a t i o n
eEffectiveStateDensity

hEffectiveStateDensity

EffectiveIntrinsicDensity

e D e n s i t y hDensity SpaceCharge

192

IntrinsicDensity

eQuasiFermiPotential

h Q u a s i F e r m i P o t e n t i a l BandGap ConductionBandEnergy ValenceBandEnergy

ElectronAffinity
ElectricField
eLifetime

ElectricField / vector

eCurrent / Vector hCurrent / Vector
eMobility

ElectrostaticPotential

h L i f e t i m e SRH Auger T o t a l R e c o m b i n a t i o n S u r f a c e R e c o m b i n a t i o n R a d i a t i v e R e c o m b i n a t i o n

hMobility

eVelocity

current / vector

hVelocity

SRH Auger T o t a l R e c o m b i n a t i o n S u r f a c e R e c o m b i n a t i o n R a d i a t i v e R e c o m b i n a t i o n
BarrierTunneling
eBarrierTunneling

hBarrierTunneling

NonLocal
OpticalGeneration

NonLocalPlot

((0 ,

ConductionBand
hDensity

0) ) {
ValenceBand

eDensity

hQuasiFermi

eQuasiFermi

NonLocal

S t a c k S i z e = 20000000

needed

f o r NewRayTracer

Method=S u p e r
NonLocal ”TD NLM” (
M a t e r i a l I n t e r f a c e = ” S i l i c o n G e r m a n i u m /GaAsP”
Length =15e−7

# [ cm ]

distance

to anchor point

P e r m e a t i o n = 15 e−7
DirectCurrent

Cylindrical (0.0)

T r a n s i e n t=BE
T r a n s i e n t D i g i t s =7
T r a n s i e n t E r r R e f ( e l e c t r o n ) = 1E0
T r a n s i e n t E r r R e f ( h o l e ) = 1E0

CNormPrint

Solve {
N e w C u r r e n t P r e f i x = ” tmp ”

Coupled
Plot (

{ poisson }

F i l e P r e f i x = ”n@node@ Banddgm ” )

Coupled

{ poisson

electron }

193

Coupled

{ poisson

hole }

Coupled

{ poisson

electron

Transient

hole }

I n i t i a l S t e p =1e −20 MaxStep =0.2 MinStep = 1 e −40 I n c r e m e n t=2
I n i t i a l T i m e =0 F i n a l T i m e=1
) { Coupled ( I t e r a t i o n s =20) { P o i s s o n

E l e c t r o n Hole

} }

NewCurrentPrefix = ” Light IV ”

Quasistationary

I n i t i a l S t e p =1e−4 MaxStep =1e−3 MinStep = 1 e −30 I n c r e m e n t =1.7 DoZero
Goal { v o l t a g e = 1 . 5 Name=”BottomContact ” }
) { Coupled { P o i s s o n
Plot (

E l e c t r o n Hole

F i l e P r e f i x = ” n@node@ Banddgm Jsc ” Time = ( 0 )

N e w C u r r e n t P r e f i x = ” tmp 2 ”

Quasistationary

I n i t i a l S t e p =1e−2 MaxStep =0.1 MinStep = 1 e −30 I n c r e m e n t =1.5 DoZero
Goal { c u r r e n t = 0 Name=”BottomContact ” }
) { Coupled { P o i s s o n

E l e c t r o n Hole

Plot (

F i l e P r e f i x = ” n@node@ Banddgm Voc ” )

System ( ” rm −f tmp ∗ ” ) ∗ remove t h e
System ( ” rm −f tmp2 ∗ ” ) ∗ remove t h e

p l o t we dont need anymore .
p l o t we dont need anymore .

Listing E.23: Code to extract the current-voltage characteristics
# Plot

l i g h t J−V and P−V c u r v e s and e x t r a c t

Photovoltaic

parameters

# o r P l o t d a r k J−V c h a r a c t e r i s t i c s

# #s e t d e p @node | s d e v i c e 1 @

set N

@node@

set

@node : index@

# proj load
proj load

@plot@ PLT JV ( $N )
L i g h t I V n @ p r e v i o u s @ d e s . p l t PLT JV ( $N )

#− Automatic

alternating

color

assignment

tied

t o node i n d e x

[ list

orange green

blue red

v i o l e t brown o r a n g e magenta ]

s e t NCOLORS [ l l e n g t h $COLORS ]
set

color

# Plot

[ lindex

$COLORS [ e x p r

l i g h t J−V c h a r a c t e r i s t i c s

c v c r e a t e D S J ( $N )

$ i%$NCOLORS ] ]

and e x t r a c t PV p a r a m e t e r s

”PLT JV ( $N ) BottomContact O u t e r V o l t a g e ” ”PLT JV ( $N ) BottomContact T o t a l C u r r e n t ”

c v i n v J ( $N ) y

c v c r e a t e V( $N )

”PLT JV ( $N ) BottomContact O u t e r V o l t a g e ” ”PLT JV ( $N ) BottomContact O u t e r V o l t a g e ”

c v c r e a t e W i t h F o r m u l a P( $N )

”∗” A A

c v d i s p l a y P( $N ) y2

c v s e t C u r v e A t t r J ( $N )

” l i g h t −JV” $ c o l o r

solid

circle

194

3 defcolor 1 defcolor

” l i g h t −PV” $ c o l o r

c v s e t C u r v e A t t r P( $N )

g r s e t A x i s A t t r X { V o l t a g e (V) }

dashed

2 none 3 d e f c o l o r 1 d e f c o l o r

16 0 {} b l a c k 1 14 0 5 0

g r s e t A x i s A t t r Y { C u r r e n t D e n s i t y (mA/cmˆ 2 ) } 16 0 30
g r s e t A x i s A t t r Y2 { Power (mW/cmˆ 2 ) } 16 0 26

# Extract

Photovoltaic

# Extract

short

set

J s c ( $N )

ft scalar

b l a c k 1 14 0 5 0

parameters

circuit

current

density ,

Jsc

[mA/cm ˆ 2 ]

[ c v c o m p u t e ” v e c v a l y ( ,0) ” A A A A ]

Jsc

[ f o r m a t %.2 f

# E x t r a c t open

circuit

$ J s c ( $N ) ]

v o l t a g e , Voc [ V ]

s e t Jmin [ c v c o m p u t e ” vecmin()” A A A A ]
if

{ $Jmin <= 0} {

elseif

s e t Voc ( $N )

[ expr

[ c v c o m p u t e ” v e c z e r o ()” A A A A ] ]

{ $Jmin <= 1 e −6} {
s e t Voc ( $N )

[ expr

[ c v c o m p u t e ” v e c v a l x (, $Jmin ) ” A A A A ] ]

f t s c a l a r Voc [ f o r m a t %.4 f $Voc ( $N ) ]

# Extract

fill

factor

(FF) , maximum power o u t p o u t (Pm [mW/cm2 ] )

s e t Ps 100 ;# I n c i d e n t

light

power d e n s i t y

f o r AM1. 5 g r a d i a t i o n

and

efficiency

( eff )

i n mW/cmˆ2

{ $Voc ( $N ) > 0} {
s e t Pm( $N )

[ c v c o m p u t e ” vecmax()” A A A A ]

## f i l l f a c t o r
s e t FF( $N )

## e f f i c i e n c y
set

E f f ( $N )

in %

[ e x p r $Pm( $N ) / ( $Voc ( $N ) ∗ $ J s c ( $N ) ) ∗ 1 0 0 ]
i n % (mW/cmˆ 2 / ( 1 0 0mW/cmˆ 2 ) ∗100%)

[ e x p r $Pm( $N ) / $Ps ∗ 1 0 0 ]

f t s c a l a r Pm [ f o r m a t %.4 f $Pm( $N ) ]
f t s c a l a r FF
ft scalar

[ f o r m a t %.4 f $FF ( $N ) ]

Eff

[ f o r m a t %.4 f

$ E f f ( $N ) ]

Listing E.24: Code to plot optical generation cross sections for TM excitation
#!MC 1120

|MFBD| = ’ / home/ d t /STDB/ D e f e c t s /01−2D’

$ ! VarSet

## Load d a t a i n

Tecplot

using

t h e SWB−L o a d e r

$ !READDATASET ” n@node | emw@ a eml . t d r ” DATASETREADER = ”SWB−L o a d e r ”

$ !TWODAXIS YDETAIL{ISREVERSED = NO}

$ !EXPORTSETUP IMAGEWIDTH = 439

$ !EXPORTSETUP EXPORTFNAME = ’ / home/ d t /STDB/ D e f e c t s /01−2D/ n@node | emw@ @@ nm TM . t i f ’

$ !EXPORT
EXPORTREGION = CURRENTFRAME

$ ! RemoveVar |MFBD|

$ ! QUIT

Listing E.25: Code to plot optical generation cross sections for TE excitation
#!MC 1120

195

$ ! VarSet

|MFBD| = ’ / home/ d t /STDB/ D e f e c t s /01−2D’

## Load d a t a i n

Tecplot

using

t h e SWB−L o a d e r

$ !READDATASET ” n@node | emw1@ a eml . t d r ” DATASETREADER = ”SWB−L o a d e r ”

$ !TWODAXIS YDETAIL{ISREVERSED = NO}

$ !EXPORTSETUP IMAGEWIDTH = 439

$ !EXPORTSETUP EXPORTFNAME = ’ / home/ d t /STDB/ D e f e c t s /01−2D/ n@node | emw1@ @@ nm TE . t i f

$ !EXPORT
EXPORTREGION = CURRENTFRAME

$ ! RemoveVar |MFBD|

$ ! QUIT

E.2

Comparison between Lumerical and Sentaurus FDTD

The Atwater Group has typically used Lumerical for optical absorption simulations and
Sentaurus for device physics modeling. Dr. Mike Kelzenberg successfully coupled the
two to enable full optoelectronic modeling by using Matlab to interpolate the Lumerical
FDTD grid onto the Sentaurus FEM mesh.(186) However, the interpolation process is slow
and awkward. Sentaurus has a built in FDTD simulation, and, though its accuracy and
limitations are not well understood, it has an efficient interpolation process. Thus, I explored
the use of Sentaurus for full optoelectronic simulations.
Figure E.2 displays a comparison between the Lumerical and Sentaurus simulations for
a two dimensional “wire” profile. While there are some discrepancies between the two,
especially at longer wavelengths for TE polarization, in general, they agree quite well.

196

Sentaurus

Lumerical

% of Incident Photons Absorbed

λ = 1 μm, TM

% of Incident Photons Absorbed

Sentaurus

Lumerical

λ = 700 nm, TM

0.8
0.6
0.4

Sentaurus (III−V)
Lumerical (III−V)
Sentaurus (Wire)
Lumerical (Wire)

0.2
400

TM
600 800 1000 1200
Wavelength (nm)

0.8
0.6
0.4

Sentaurus (III−V)
Lumerical (III−V)
Sentaurus (Wire)
Lumerical (Wire)

0.2
400

TE
600 800 1000 1200
Wavelength (nm)

Figure E.2: Comparison of FDTD simulations from Sentaurus and Lumerical. (left) Absorption profiles from the two software programs at 700 nm and 1 µm. (right) % of Incident
Photons Absorbed vs. Wavelength for both TM and TE polarizations.

197

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