Design, Modeling, and Optimization of Indium Arsenide
Diodes for Microscale Thermophotovoltaics
BARKER
by
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Michael Masakichi Masaki
APR 2 4 2001
B.S., Electrical Engineering
University of Hawai'i at Manda (1998)
LIBRARIES
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 2000
@ MIT, MM. All Rights Reserved
The author hereby grants to MIT permission to reproduce and
to distribute copies of this thesis document in whole or in
.
Signature of Author.... . . . ......
...
. . . ........ . .........................
Department ofElectrical Engineing and Computer Science
29 September 2000
Certified by........
....
..
.... ...
... . . . . . . . . . . . . . . . . . . . . . . ..
Clifton G. Fonstad,Jr.
Professor of Electrical Engineering
Thesis Supervisor
Accepted by.......
..............
Arthur C. Smith
Chairperson, Department Committee on Graduate Students
Design, Modeling, and Optimization of Indium Arsenide Diodes for
Microscale Thermophotovoltaics
by
Michael Masakichi Masaki
Submitted to the Department of Electrical Engineering and Computer Science
on 29 September 2000, in partial fulfillment of the
requirements for the degree of
Master of Science
Abstract
The band gap of Indium Arsenide is 0.354 eV (3.48 pm) at room temperature, ideal for
photovoltaic applications in the near infrared wavelength range. In order to facilitate the design
of InAs photovoltaics, SimWindows@, a one dimensional Poisson equation solver, was used as
a design and modeling tool for InAs photodiodes and to provide insight into their design. The
validity of the modeling was confirmed by comparing its predictions with experimental data in
the published literature.
Two InAs photovoltaics were also fabricated by Molecular Beam Epitaxy, followed by characterization. The I-V characteristics of the two devices differed greatly from theoretical predictions. Since the discrepancy could not be explained by modifying the models created in
SimWindows@, it was concluded that the results must be due to a more fundamental aspect outside the scope of SimWindows@. Several tests were done to identify this aspect. It was
found that the diode characteristics of the device resulted primarily from a metal-semiconductor
junction on the back side of the wafer, not from the InAs p-n structure grown on the wafer
as originally expected. Finally, a low temperature measurement revealed that the remaining
non-linear I-V characteristic was due to the tunneling nature of the p-n junction.
Thesis Supervisor: Clifton G. Fonstad,Jr.
Title: Professor of Electrical Engineering
2
Acknowledgments
At this time I would like to thank everyone who helped and supported me in this endeavor. Without
their help and support, I would not be writing these words at this moment.
Firstly, I would like to thank God for too many reasons to list here.
I also thank my parents, Melvin Y. and Clara Y. Masaki, for their undying support and motivation,
and their belief that an education is the most important gift of all. I would like to extend my warmest
thanks to my brother Gavin for keeping me in line.
I owe a great debt of gratitude to Professor Clifton G. Fonstad. His unending patience, support,
and guidance were invaluable to me, and I would not be able to conduct the research without the use of
Professor Fonstad's laboratories, experience, and research funding. I would also like to thank Professor
Fonstad for giving me the opportunity to be the teacher's assistant for 6.012 (a rewarding and invaluable
experience), and for reading over this document.
Henry Choy's friendship, honesty, advice, and knowledge was essential to finishing this work. The
many nights spent on debating points and testing if I really understood things drove me to push harder
that I have done before. Also, thank you Henry for teaching me how to use the many pieces of equipment
in the laboratory, growing the samples 9722 and 9725 that were used in this thesis, and tolerating
(barely) my Star Trek fanaticism. Many thanks go to Professor Sheila Prasad for tips (especially on
how to write my thesis) and many cups of coffee. I also thank Karen Young-Waithe for processing the
wafers and providing me with the necessary information in analyzing the photodiodes. I am grateful
for the assistance of Gale Petrich, Joyce Wu, and Hans Callebaut for their assistance in the cryogenic
measurements.
In addition, thank you Dr. P. Aitor Postigo Resa and Dr. M. H. Madhusudhana Reddy for your
knowledge of the art of MBE growth, and Wojciech Giziewicz for your advice and humor. I would also
like to thank my roommate Harry Lee for his advice, comments, and informing me of the existence of
SimWindows@, and also for being a great roommate.
Thank you Junji Himeno and the MIT Kendo club for your words of encouragement and keeping me
in shape, both physically and mentally, and Steve Wang for filling in for me at all the practices I could
not attend.
Finally, I would like to thank Professor Kazutoshi Najita of the University of Hawai'i for encouraging
me to attend graduate school instead of working in industry, Mrs. Suzuki and Mr. Mersereau for believing
that I had a future, and Brandon Rai Mitsuda, Esq., for his friendship and support over the last 13
years.
3
Contents
1
2
Introduction
1.1
Why InAs Photovoltaics?
1.2
SimWindows@
1.3
Overview of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Formulating a Model for Optimizing a Device Design
2.1
2.2
2.3
3
11
SimWindows
17
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1
Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.2
Advantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1.3
InAs Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.4
Shortcomings of SimWindows Simulations . . . . . . . . . . . . . . . . . . 27
Theoretical Results from InAs Model . . . . . . . . . . . . . . . . . . . . . . . . .
17
29
2.2.1
Testing the SimWindows Model . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
Design Parameters from Model: . . . . . . . . . . . . . . . . . . . . . . . . 34
Summary of Chapter 2:
29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Application of SimWindows to MIT Diodes
40
3.1
InAs Photovoltaic Structure: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2
Comparison with the SimWindows Simulations . . . . . . . . . . . . . . . . . . . 40
3.3
3.2.1
9725: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2
9722: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Explaining the Discrepancies
3.3.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Poor Metallization on top of the device: . . . . . . . . . . . . . . . . . . . 52
4
4
3.3.2
Surface Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.3
Bad Metal-Semiconductor Contact on the Bottom: . . . . . . . . . . . . . 55
3.3.4
What Happened to the P-N Junction? . . . . . . . . . . . . . . . . . . . . 57
Conclusion
62
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.1
Summary of Accomplishments.
4.2
Future Avenues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A Optical Properties of Indium Arsenide
64
B Ideal Model of a Homojunction Photovoltaic Device
70
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
B .1
Basic Principles:
B.2
Circuit Model and Current-Voltage characteristics:
. . . . . . . . . . . . . . . . . 71
B .2.1
Circuit M odel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
B.2.2
The open circuit voltage V, and short circuit voltage I .. . . . . . . . . . 72
B.2.3
Maximum Power Generation
B.3 Monochromatic Response
B .4 Spectral Response:
. . . . . . . . . . . . . . . . . . . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5
List of Figures
1-1
5000 C Black Body Spectrum, the energy spectrum of interest for InAs.
1-2 InAs Band Structure.
EO is the bandgap.
. . . . .
14
Eo = 0.354 eV, Ao = 0.46 eV,
Al = 0.28 eV, and El = 2.50 eV . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1-3
Index of Refraction (Solid) and Extinction Coefficent (Grey) of InAs[5] . . . . . . 15
1-4
Absorption Coefficient of InAs [5] . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-1
The model index of refraction data (gray) and the calculated index of refraction
(solid) with respect to enegy.
2-2
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
The model absorption coefficient data (gray) and the calculated index of refraction (solid) with respect to enegy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2-3
Thermal Conductivity versus Temperature. The solid line represents the measured data[11], and the thin line represents the model results. . . . . . . . . . . . 23
2-4
Electron Mobility versus Doping Concentration. Model (black line) and mea-
sured (grey line)[13] data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2-5
Hole Mobility versus doping. Model (solid line) and measured (diamond) [14] data. 25
2-6
The dominant Auger recombination processes in InAs. A: eeh recombination. B:
ehh, recombination[18].
2-7
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
IV characteristic from SimWindows@. Note that the IV characteristic changes
from exponential (ideal) to linear (R).
2-8
Lack of high level injection in SimWindows@.
the ideal diode equation (grey).
relationship (n=1) to resistive.
2-9
. . . . . . . . . . . . . . . . . . . . . . . . 28
SimWindow data (black) and
Note that the plot changes from exponential
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Structures used to verify model. Device A: pin diode. Device B: p-n diode.
6
.
.
. 29
2-10 I-V characteristics of measured and simulated Results. Figure A is the I-V characteristic of the PIN structure and Figure B is I-V characteristic of the P-N
structure. Note that the measured current does not saturate in Figure B. This
is caused by tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2-11 I-V characteristics of measured and simulated Results.
The simulation data
uses a mobility 1.7 times smaller than the previous figure. Figure A is the
I-V
characteristic of the PIN structure and Figure B is I-V characteristic of the P-N
structure. Note that the measured current does not saturate in Figure B. This
is caused by tunneling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2-12 Photocurrent versus the location of the light impulse, calculated (black) and
simulated by SimWindows@ (grey).
. . . . . . . . . . . . . . . . . . . . . . . . . 33
2-13 Minority Carrier Lifetime versus Doping Concentration for P-type InAs. The
data for Auger recombination is approximate. . . . . . . . . . . . . . . . . . . . . 36
2-14 Minority Carrier Lifetime versus Doping Concentration for N-type InAs. . . . . . 36
2-15 Electron Diffusion Length versus Doping. Note the diffusion length is within the
same order of magnitude of the substrate thickness for low p-type doping
. . . . 37
2-16 Hole Diffusion Length versus Doping . . . . . . . . . . . . . . . . . . . . . . . . . 38
3-1
Device Structures used in this analysis . . . . . . . . . . . . . . . . . . . . . . . . 41
3-2
A diagram of the device structure and surface of device 9725. . . . . . . . . . . . 41
3-3
Terminal Characteristics of a device fabricated on growth 9725 plotted on a linear
scale..
. . ................
....
....
.
...
...
....
. .. .
...
42
3-4
Log plot of device 9725 and fit.
3-5
Log Scale Plot of the 9725 Data, with the Results of the SimWindows@ Simulations. 43
3-6
Doping variations of the 0.56 pm 2x10181
3-7
Doping variations of the 1.0 pam 5x10 17 - 1
3-8
Doping varitation of the 1.0 pm 5x 10171
. . . . . . . . . . . . . . . . . . . . . . . . . . . 43
p-type buffer layer.
p-type layer. . . . . . . . . . . . . . . 45
n-type layer. Doping this layer more
n-type made no noticeable changes in the I-V curve.
3-9
. . . . . . . . . 45
. . . . . . . . . . . . . . . . 46
Changes in the IV characteristics with changes in the SHR lifetime.
3-10 The simulated effects of increasing series resistance in device 9725.
. . . . . . . 47
. . . . . . . . 48
3-11 Side profile of the 9722 devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
7
3-12 The metalization pattern of device 9722. The top metal (shown in black) served
as etch mask to define the mesas. Areas A,B,C,F, and H are 4x10- 4cm 2 in area;
D and E are 4.9 x10-cm 2 in area; and G is 2.24x10- 2 cm
2. . . . . . . . . . .
. .
49
3-13 A comparison of the simulated I-V characteristics between the 9722 and 9725
d evices.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3-14 The measured IV characteristics of the 9722 and 9725 devices. The data from
device 9722 was multiplied by 9.78. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3-15 Measurements taken from device 9722 from pad E to the back of the substrate
(gray) and from pad D to the back of the substrate (dotted black)
. . . . . . . . 51
3-16 Measurements taken from 9722. These measurements were taken from A, B, C,
F,or H to the back of the substrate . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3-17 I-V characteristics of several devices after thermal annealing. These plots demonstrate the I-V plots that were symmetrical.
. . . . . . . . . . . . . . . . . . . . . 52
3-18 9722 Measurements. These figures display the more diode-like I-V characteristics. 53
3-19 Surface inversion in InAs. Note that the Fermi Energy at the surface penetrates
the conduction band. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3-20 Effects of Passivation on the InAs photodiodes. Note that the diode-like characteristics dissapear after every treatment. . . . . . . . . . . . . . . . . . . . . . . . 54
3-21 Through-substrate measurement of 9722 (grey) compared with 9725 I-V characteristics (black) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3-22 Compensated 9722 Data (grey) with 9725 Data (Black).
. . . . . . . . . . . . . . 56
3-23 Through Substrate measurement of 9722, before back metal (Grey) and after
(Black). Note that all the diode-like characteristics dissapear. . . . . . . . . . . . 56
3-24 Room Temperature Measurements of device 9722. Note that the I-V characteristics are sym m etrical. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3-25 The differential resistances of the devices. Note that resistance is not symmetrical
about the origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3-26 Linear Plot of the I-V Characteristics of Device 9722 at room temperature (grey)
and 132 K (black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
8
3-27 Log Plot of the I-V Characteristics of Device 9722 at room temperature (grey)
and 132 K (black). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3-28 Tunneling Current in a P-N junction. Figure A demonstrates tunneling directly
from the condunction band to the valence band. Figure B indicates tunneling
by interband states. ........
..................................
60
3-29 Effect of Tunneling on the Size of Power Rectangle. Figure A represents an ideal
diode, Figure B represents a diode with tunneling.
A-i Transitions of interest in InAs.
A-2
Real Permittivity,
61,
. . . . . . . . . . . . . . . . . 61
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
versus photon energy. eia is equation A.5, and deals with
the EO and EO + AO transitions. Eib is equation A.11, and deals with the El
and E 1
El
A-3
+ A, transitions. ei1 is equation A.15, and deals with the E2 transition.
+ Eib
b
=
- 61c .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Imaginary Permittivity, E2, versus photon energy.
with the EO and EO + AO transitions.
6
6
68
2a is equation A.6, and deals
2b is equation A. 12, and deals with the E 1
and E 1 + A 1 transitions. 62c is equation A.16, and deals with the E 2 transitions.
e 2diand E2d2 is equation A.18, and deals with the EL indirect transitions.
6
2a
+
6
2b
- 6 2c
+
6
2d1
+
6
2d2.
2 =
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
B-1 An example side schematic of a p-n junction photovoltaic. . . . . . . . . . . . . . 71
B-2 a) A photovoltaic under no illumination. No carriers are generated. b) A photovoltaic irradiated by light. Excess holes and electrons are generated.
. . . . . . . 72
B-3 a) The cicuit diagram of a photovoltaic in operation. b) The idealized equivalent
circuit of the photovoltiac operation. . . . . . . . . . . . . . . . . . . . . . . . . . 73
B-4 a) I-V curves of a photovoltaic with and without illumination. b) The power
rectangle of a photovoltaic.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
B-5
A photovolatic with impulse illumination of area G at x.. . . . . . . . . . . . . . 76
B-6
The cross section of a photovoltaic. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
9
List of Tables
1.1
Bandgap energy of several semiconductors at room temperature, and the black
body temperature necessary for the black body peak to be absorbed by the
m aterial. [4]
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1
SimWindows@ InAs Material Parameters
. . . . . . . . . . . . . . . . . . . . . . 19
2.2
Values used for the photodiode
2.3
Electron and hole mobility of several semiconductors at 300 K[4]. 'NA' indicates
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
that the material does not have mobility data at room temperature.
A.1
InAs optical porperties calculation parameters.
10
. . . . . . . 35
. . . . . . . . . . . . . . . . . . . 66
Chapter 1
Introduction
The main source of power for most satellites and deep space probes are solar cells. For
example, both the Stardust[1] and Deep Space 1 [2] probes use solar cells as their primary
source of energy. However solar cells harvest increasingly smaller amounts of energy as the
probe travels farther away from the sun. Eventually the solar cells become useless and reduced
to excess weight as the probe leaves the confines of the solar system. A solution to this problem
is to bring the luminous source with the satellite. An example of a luminous source is the black
body spectrum from a hot object, which can be created readily from a sustained nuclear fission
reaction. A photovoltaic is able to use the black body spectra produced from this reaction to
produce electricity. The goal of this project is to analyze and diagnose a photovoltaic created for
such an application. This effort is part of a larger project directed at using InAs photovoltaics in
a new enhanced performance application called microscale themophotovoltaic (MTPV) energy
conversion[3].
1.1
Why InAs Photovoltaics?
The power density of a black body is given by Equation 1.1:
PdA =-
27rhc 2
5
h
e kBTX
)
dA
[ Watt
1M
where c is the speed of light, and kB is the Boltzman constant. Also, from Wien displacement
law, the black body peak is given by:
11
Material
Bandgap Energy [eV]
Blackbody Temperature [K]
Si
1.12
2483
Ge
0.664
1472
C (Diamond)
BN
BP
BAs
AiN
AlP
AlAs
AlSb
GaN
GaP
GaAs
GaSb
InN
InP
InAs
InSb
5.50
6.4
1.98
1.46
6.2
2.41
2.153
1.615
3.44
2.722
1.424
0.75
1.89
1.344
0.354
0.169
12194
14189
4390
3237
13745
5343
4773
3580
7929
6035
3157
1663
4190
2980
785
375
Table 1.1: Bandgap energy of several semiconductors at room temperature, and the black body
temperature necessary for the black body peak to be absorbed by the material.[4]
T- 2 2 1 7
E
[-+1
(1.2)
eV
where E is the black body photon energy in electron volts.
The bandgap of the material must be smaller or equal to than the black body peak in order
for the material to absorb a significant amount of the incident black body radiation. The band
gaps all group IV and III-V semiconductors are shown in Table 1.1.
For TPV and MTPV
applications one typically uses a blackbody temperature less than 1000 K so among the group
IV and III-V semiconductors, the two best candidates are InAs and InSb.
An important figure of merit for a photovoltaic is the open circuit voltage, Vc, which
determines the maximum efficiency of the photovoltaic and must be maximized (this is discussed
in detail in Appendices B.2.2 and B.2.3). Since the maximum value for V, is the band gap of
the material, InAs would be a better choice than InSb since InAs has a direct band gap of 0.354
eV (A
-
3.48 fum) at room temperature, more than twice the bandgap of InSb (0.169 eV at room
temperature). Also the band gap of InSb is too small to build diodes that operate properly at
12
room temperature and the MBE in the laboratory where the research was conducted does not
have Antimony cells. Consequently, InAs was chosen for the initial work. In later work, closely
related ternary alloys might also be of interest, but their use complicates the design too much
at this stage.
The goal is to use a 500'C (773 K) black body (the spectrum is shown in Figure 1-1) as the
radiation source, which is close to 784 K.
The band structure of InAs is shown in Figure 1-2. Other properties of InAs that will be
important are the refractive index, shown in Figure 1-3, and the absorption coefficient, shown
in Figure 1-4.
[5].
The Figures 1-3 and 1-4 are derived in Appendix A by a method used
by Sadao Adachi[5][6] which uses a algebraic method of ascertaining the optical properties of
several semiconductors from the band structure.
1.2
SimWindows@
In order to develop an efficient photovoltaic design, a model must be developed for the
device that takes in consideration the device dimensions, specifications, and the material properties.
One such program is SimWindows@.
SimWindows@ is a one dimensional Poisson
equation solver program that is able to simulate semiconductor devices. SimWindows@ has the
capability to simulate the electrical, optical, and thermal properties of a device. For example,
it can simulate optical generation of carriers in a semiconductor, or calculate the total amount
of heat radiated from the device.
The models developed by SimWindows@ can ideally be adjusted to explain any discrepancies
that may occur between the theoretical and measured result, and can also be used to reveal the
limiting mechanisms of the device such as Auger recombination. SimWindows@ is discussed
further in Section 2.
1.3
Overview of the Thesis
The purpose of this thesis was to create a simulation file using SimWindows@ to diagnose
and design InAs photovoltaics. The model developed for SimWindows@ will be discussed, as
well as the advantages of using SimWindows@ and its shortcomings. The model was tested and
13
500 C Black Body Spectrum
8.OOE-04
8.00E-05
7.00E-04
7.00E-05
5.00E-04
5.00E-05
t! 4.00E-04
4.OOE-05
3.00E-04
3.00E-05
2.00E-04
2.00E-05
P
"E
1.00E-05
0
5
0.00E+00
10
15
Waveiungth
20
25
(m"cmn)
Figure 1-1: 5000 C Black Body Spectrum, the energy spectrum of interest for InAs.
EgL
El
A1
B0
'Ill
L
F
X
Figure 1-2: InAs Band Structure. EO is the bandgap. Eo = 0.354 eV, AO = 0.46 eV, A1 = 0.28
eV, and El = 2.50 eV
14
l I I i i i
I
!i
i 4 .
.
Sn=
4.
3.5-
al
-2.5
2-
1.5.
1.
0.5
0
1
2
4
3
5
6
Phown Energy (Electrmn Volts)
Figure 1-3: Index of Refraction (Solid) and Extinction Coefficent (Grey) of InAs[5]
1.00E+07
1.OO4(6
1.00E+05
1.OtE+04
v
0
I
I
I
2
3
Photon
Energy
Eectron
4
Voas)
Figure 1-4: Absorption Coefficient of InAs [5]
15
5
6
compared to InAs diodes in the literature. SimWindows@ was able to fit most of the data in the
literature within the limitations of the program. Also, some design issues of InAs photodiodes
will be discussed.
The model created by SimWindows@ was then applied to two InAs diodes grown by MBE in
an effort to analyze them. The measurement revealed a diode characteristic, but the saturation
current predicted by SimWindows@ was several orders of magnitude larger than what was
actually measured. In addition, the current though the device did not scale properly with area.
However, the model developed in SimWindows@ was unable to explain the discrepancy between
the theoretical results and the measured results. Therefore, the discrepancy was due to a more
fundamental aspect of the diode that was not included in the SimWindows@ model.
Several tests were done to isolate the problem. The top metal was annealed to provide
a better top contact, which made no change in the I-V characteristics.
to passivate the surface, but this yielded a more ohmic behavior.
Next, HF was used
Next, CP-4 was used to
passivate the surface, and this also brought about no change. Metal was applied to the back
of the substrate, which removed all diode-like behavior from the device. It was concluded, as
will be discussed in Section 3.3.3, that the diode-like behavior was primarily due to the poor
metal-semiconductor junction on the back of the device (substrate to probe station chuck), not
from the p-n junction. In addition, a low temperature measurement revealed that the remaining
non-linear I-V characteristic was due to the tunneling nature of the p-n junction.
16
Chapter 2
Formulating a Model for Optimizing
a Device Design
2.1
SimWindows
2.1.1
Description
SimWindows@ is a one dimensional Poisson equation solving program developed by David
Wells Winston in 1996 as his Ph.D. thesis from the University of Colorado[7]. SimWindows@
was created to simulate VCSELs and as a tool for VCSEL design. However, SimWindows@
can be also used to simulate two terminal devices, such as resistors and diodes, and it can be
used with devices which either emit or absorb light, making it ideal for photovoltaic simulation.
Since it was designed to simulate VCSELs, the program can also simulate heterostructures.
SimWindows@ is able to simulate a user defined device under numerous conditions, such
as different biasing conditions, AMO illumination conditions, and non-uniform temperature. In
addition, the program uses material property files that are defined by the user that allow the
development of more realistic material models. For example, the lattice and carrier temperature and doping concentration can be taken into account when specifying the electron or hole
mobility. Also, SimWindows@ can take into account many other phenomenon, such as finite
surface recombination velocity. Thus SimWindows@ is ideal for designing and simulating a
device, provided that the materials under construction have bee adequately characterized.
17
2.1.2
Advantages
There are several advantages of using SimWindows@ rather than other Poisson Equation
solvers. Firstly, SimWindows@ is free', and Winston's thesis is readily accessible[7]. The inner
workings of the program and the assumptions the program is based on can be found in this
thesis. The thesis indirectly also provides the user knowledge of the limitations of his program
since most aspects of the program are discussed in the thesis. By knowing the models that were
used, one can judge when the results are valid or incorrect.
SimWindows@ is fairly easy to use. It is designed to run under Windows
NT
, and most of the simulation aspects of program are menu driven.
95, 98, and
The most difficult
aspect of SimWindows@ is creating a good material data file. The material data file developed
for InAs is detailed in Table 2.1.
Creating a device then consists simply of specifying the
thickness and doping of each layer. In addition, the material parameters can be changed when
defining the device itself, allowing to user to change parameters in different layers, such as the
mobility.
The program is able to simulate and diagnose most of the aspects needed in a device analysis
through its ability to simulate the electrical, optical, and thermal environment and models of
the device. The ability to control the use of Maxwell-Boltzman or Fermi-Dirac Statistics and
to enable and disable different generation and recombination processes are the most useful of
the electrical properties that can be controlled.
Since the bandgap of InAs is small (0.354
eV at room temperature) compared to Silicon or Gallium Arsenide, it is essential that FermiDirac Statistics is used for InAs. However, for low doped samples of InAs or samples at lower
temperatures, Maxwell-Boltzman statistics can be used. This is useful since the computation
time for Maxwell-Boltzman simulations is much shorter than for simulations that use FermiDirac statistics.
The ability to enable or disable generation and recombination processes is an extremely
useful feature of SimWindows@.
The main recombination mechanisms in the program are
Shockley-Hall-Read, radiative, and Auger recombination, and the main generation mechanism
(besides thermal generation) is optical generation. The user is able to find out the dominant
'SimWindows@ can be downloaded for free at [http://www-ocs.colorado.edu/SimWindows/siml50.exe
18
Material=InAs
Alloy=Default
BAND-GAP Model=Band-gap terms=0.35,0,0,-2.76e-4,83
ELECTRON-AFFINITY Model=Band-gap terms=4.9,0,0,1.38e-4,83
STATICPERMITIVITY Value=15.15
REFRACTIVEINDEX segments=4
start-e=0.00 end-e=0.20 value=3.5
start-e=0.20 end-e=0.35 value=0.867*e+3.33
start-e=0.35 end-e=0.63 value=-0.473*e+3.8
starte=0.63 end.e=10.00 value=3.5
ABSORPTION Segments=4
start-e=0.00 end-e=0.35 value=0
start-e=0.35 end-e=1.15 value=10 ^(0.306*e+3.69)
start-e=1.25 end.e=1.80 value=10 ^(1.6*e-2.2)
start-e=1.50 end.e=10.00 value=1.3e5
THERMALCONDUCTIVITY Value=1/(2/T ^2+.0001*T 2)
DERIVTHERMALCONDUCT Value=-2*T*(.0001*T^4-2)/(.0001*T^4+2)
ELECTRON-MOBILITY model=mobility terms= 10000,0,0,300,-1.66,3e8,0,0,-1.66,2.5e16,0
HOLEMOBILITY model=mobility terms=100,0,0,300,-2.3,3.5e6,0,0,-3.3,5e16,0
ELECTRONDOSMASS Value=0.027
HOLE-DOSMASS Value=0.43
ELECTRON.CONDMASS Value=0.027
HOLE-COND-MASS Value=0.42
ELECTRONSHRLIFETIME Value=1.e-7
HOLESHR-LIFETIME Value=1.e-7
ELECTRON-AUGERCOEFFICIENT Value=1.le-26
HOLEAUGER.COEFFICIENT Value=2.54e-26
RADRECOMB-CONST Value=1.le-10
ELECTRON.ENERGYLIFETIME Value=0.8e-12
HOLEENERGYLIFETIME Value=0.8e-12
ELECTRONCOLLISION-FACTOR Value=0.5
HOLE-COLLISIONFACTOR Value=0.5
Table 2.1: SimWindows@ InAs Material Parameters
19
recombination in the device by enabling or disabling any of the above processes. In addition,
the program also plots the generation and recombination rate of each process throughout the
length of the device. Finite surface recombination velocity and Schottky barriers at the surfaces
can also be simulated, as well as intraband tunneling.
The most useful of the optical aspects (besides simulating optical generation of carriers, and
light generation of the device) is that the user is able to define an incident radiation spectra, and
to be able to incorporate the wavelength dependence of the index of refraction and absorption
coefficient. Modeling either of these by hand or creating a program to do so would be tedious.
In addition, the direction of light, area of illumination, and reflection between boundaries can
also be simulated.
Finally, the program is able to simulate thermal effects in the device.
The band gap,
mobility, electron affinity, refractive index, absorption coefficient, and thermal conductivity can
be evaluated at any temperature as long as the model created by the user incorporates the
thermal effects. In addition, the device can be isothermal, have different temperature reservoirs
at each terminal of the device, or the user is able specify the electron and hole temperature in
the lattice.
2.1.3
InAs Model
The properties used for the InAs model is tabulated in Table .2.1, and are explained further
in this section.
Band Gap, Electron Affinity, and Static Permittivity:
The band gap model[8] that was used is:
Eg = 0. 4 15 -
2.76 x 10--4T 2
x
T+83
[eV]
(2.1)
The electron affinity model[9] is:
X = 4.9 -
10 4 T 2
T+83
[eV]
(2.2)
The temperature dependence is taken from the GaAs model provided with the program.
20
The static permittivity[10] is E = 15.15.
Refractive Index and Absorption Coefficient:
The refractive index and absorption coefficient were taken from two articles from Sadao
Adachi[5][6]. The relations are detailed in Appendix A. Since 500' C black body radiation is
insignificant 1 eV, the model needs to be accurate only to 1 eV. Also, SimWindows@ tended to
crash if the model was too complicated, and, for example, contained too many segments. The
refractive index was specified as:
3.50,
n(E)
E
<
0.20 eV
0.867E + 3.33,
0.20 < E < 0.35 eV
-0.473E + 3.80,
0.35 < E < 0.63 eV
(2.3)
E > 0.63 eV
3.50,
The absorption coefficient was modeled as:
0,
a(E)
E < 0.35 eV
=
1 0 0.306E+3.69,
0.35 < E < 1.15 eV
=
io(E)
1 0 1.6E-2.2
1.15 < E < 1.80 eV
1.3 x 10 5,
E > 1.80 eV
(2.4)
(2.4)
.cm
The index of refraction model is plotted and compared to the data from Figure 1-3 in Figure
2-1, and the absorption coefficient model is plotted and compared to the data from Figure 1-4
in Figure 2-2.
Thermal Conductivity and the Derivative of the Thermal Conductivity:
Thermal conductivity was not essential for the photovoltaic simulations, so a rough model
was made that followed the correct trends in temperature. According to Shalyt [11], the thermal
conductivity increases from 2 to 6 Kelvin as T 2 .2 , and decreases from about 30 K as T-
2
(see
Figure 2-3). T 2 provided a better fit while using the expression:
2
=7 T
1
TWatt
[K
10
21
(
(2.5)
.rn
~.
-
~
- -
-
~-,--~.-
5
4.5
3
r 2.5
2
1.5
0.5
0
0
1I
2
3
4
5
6
Photon Energy (Electron Volts)
Figure 2-1: The model index of refraction data (gray) and the calculated index of refraction
(solid) with respect to enegy.
.
E 07
I
_
I
I
.
.
.
.
.
.
.
.
.
1.OTE+06
E1.00m+05
f
1 .0+04
0
I
2
3
Phown
Energy
4
5
6
(lecben Valtm)
Figure 2-2: The model absorption coefficient data (gray) and the calculated index of refraction
(solid) with respect to enegy.
22
--
The derivative of the thermal conductivity is given by:
&-
-2T
dT
- 2)
1 0 00
T4
Watt
(2.6)
K 2 - cmJ
+2
A plot of the model versus measured results[11] is shown in Figure 2-3.
Thermal Conductivity Versus Temperature
100
____--Thermll
Conductivity Model
-Measured
I/ I
10
E
/
4
1
10
100
Temperature (Kelvin)
Figure 2-3: Thermal Conductivity versus Temperature. The solid line represents the measured
data[11], and the thin line represents the model results.
Electron and Hole Mobility:
The mobility model in the program is of the form:
Peh (T, NA,
ND)
=
A
(-)
B
+ 1±
DTE
NA+ND
G
(2.7)
where, A, B, C, D, E, F, and G are constants. This function does not provide InAs with a good
fit, but it is better to use the built in models whenever possible, since SimWindows@ is able
to evaluate the built in models faster compared to user defined models. Also, the mobility is
not a well defined parameter since it depends greatly on the growth conditions. Thus, only the
23
general shape of the curve is of prime importance.
The electron mobility was defined to be:
Pe (T, NA, ND) = 10000
(
300
-
3 x 108T-1.66
N+ V
+
cm
2
V.S
2.5 x 101U
(2.8)
The model versus measured data is shown in Figure 2-4. The poor fit is due to forcing the
model to saturate at 3.3 x 104
C,2
a value given for a pure piece of material[12].
Voltsr
bC
Electron Mobility Versus Doping Concentration
zI
.7
40
'E
1.OOE+04
r
FU
1.00E+03 1-4-..1.00E +15
1.OOE+16
I I I 11111
1.OOE+17
1.00E+18
1.OOE+20
1.00E+19
Doping Cancentration (1/cma
Figure 2-4: Electron Mobility versus Doping Concentration. Model (black line) and measured
(grey line)[13] data.
The hole mobility was defined to be:
ph (TNA, ND) = 100 (3T0y23
300
3.5 x 10'T-2.3
+1 +
NA +N
5x101
>
[
cm21
.2 1
(2-9)
The model versus measured data is shown in Figure 2-5. There is a lack of mobility data
for holes for lightly doped InAs. The small band gap makes lightly doped p-type InAs intrinsic
at high temperatures[15].
24
Hole Mobility Versus Doping Concentration
1000
1UU
U.
I
1.00E+15
1.OOE+17
1.OOE+16
1.OOE+18
1.00E+19
1.00E+20
3
Doping Concentration (1/cm ]
Figure 2-5: Hole Mobility versus doping. Model (solid line) and measured (diamond) [14] data.
Effective Mass:
The electron density of states mass is given by Equation 2.10:
1
d
me=
Nim1 "m?"
2
2
(2.10)
where N = 1 at the F minimum, and the mt, m, are the transverse and longitudinal masses
of the minima (mt = ml of the r minimum). For InAs, md = 0.027. The electron conduction
effective mass is given by Equation 2.11:
1
=
2
Tnc
3
mt
+
mi
(2.11)
For InAs, mC = 0.027[16].
The hole density of states mass is given by Equation 2.12:
)2
mn =
25
(2.12)
(2.12)an
where mih, mhh are the light and heavy hole effective mass. For InAs, mlh
=
0.43, mhh = 0.026,
and md = 0.43.
The hole conduction effective mass is given by Equation 2.13:
5
5
d
mIAmhh
mh
3
3
m1h
+
(2.13)
hh
For InAs, md = 0.42[17].
Lifetime:
The Shockley-Hall-Read (SHR) lifetime is another parameter that is not well defined since
the density of deep level states depends on the growth conditions, and impurity content. The
SHR lifetime was taken to be 1 x 10- 7 seconds based on the work done by N. P. Esina and N.
V. Zotova[18].
The electron Auger coefficient has been reported to be A
=
1.1 ± 0.1 x 1026
[20]. The
electron Auger recombination process is dominated by the eeh (electron, electron, hole) process,
as shown in Figure 2-6A. The hole electron Auger coefficient is A = (3.38 - 1.7)
x
1026
19]
C
C
'6 C
['V
E8 V
r 7V
r7 V
A)
F
B)
F
Figure 2-6: The dominant Auger recombination processes in InAs. A: eeh recombination. B:
ehh, recombination[18].
26
and is expected to be roughly twice the electron Auger coefficient[18] [19]. Thus, the average
was chosen, A = 2.54 x
10
-26cm S
.
The hole Auger recombination process is dominated by the
ehh, (electron, hole, hole, spin-orbit) process, shown in Figure 2-6B. The ehhs process does not
require an activation energy (this is due to the fact that AO (split orbit band energy, AO = 0.46
eV) and EO (Band gap, Eo = 0.36 eV) are close in energy (see Figure 1-2), and since the
lifetime depends exponentially on this energy, the lifetime is very short [18]). At high doping
levels (ND > 2 x 1017
,the
recombination shifts from SHR to Auger[20].
The radiative
recombination lifetime was reported to be Rd = 1.1 x 1010m[21], and the electron and hole
energy lifetimes were reported to be 0.8 x 10-12s[22].
Scattering Coefficients:
Due to the polar nature of III-V materials, InAs has optical mode phonons.
In III-V
materials, the relative movement of the two different atoms in the basis causes a polarization
in the of the crystal, and a strong interaction may result. The scattering coefficient for polar
scattering is 0. However InAs, GaAs, and InP experience impurity scattering as well, and the
scattering coefficient of ionized impurity scattering is
[23].
In light of this, the scattering
parameter was set to 1, the same value that the program used for the GaAs model.
2.1.4
Shortcomings of SimWindows Simulations
Unfortunately, SimWindows@ is not a perfect simulator. There are several shortcomings
of SimWindows@ simulations. Firstly, SimWindows@ is unable to simulate high level injection
in a device. This can be seen in Figures 2-7 and 2-8. Note that the characteristics change from
exponential to resistive without entering a region where 1 < n < 2. This severely limits the
accuracy of the I-V characteristics in the forward bias regime.
Also, SimWindows@ is unable to simulate interband tunneling.
This prevents accurate
simulation of devices such as Esaki and Zener diodes. Unfortunately, Zener tunneling currents
dominate the I-V characteristics of InAs diode at low temperatures. However, at 300 K, diffusion
currents dominate. [25].
In addition, SimWindows@ is a semiconductor simulator: it is not
designed to simulate metals. This prevents SimWindows@ from simulating Schottky barrier
diodes.
27
Current Density Versus Applied Potential
4.50E+06
4.00E+06
3.50E+06
3.00E+06
-Sin Wu~dsA
deal
E 2.50E+06
2.00E+06
0
1.50E+06
0
1.OOE+06
5.00E+05
0.00E+00
0.00
0.10
0.20
0.30
0.40
Applied
0.50
0.60
Potential (Volts]
0.70
0.90
0.80
1.00
Figure 2-7: IV characteristic from SimWindows@. Note that the IV characteristic changes from
exponential (ideal) to linear (R).
Current Density Versus Applied Potential
1.00E+06
1.00E+04
1.00E+02
1.006+00 .
-1.0
'
0
I
1111
-0.75
-0.50
-0.25
0.00
0.25
Applied Potantial (Voltal
0.50
0.75
I
1.00
Figure 2-8: Lack of high level injection in SimWindows@. SimWindow data (black) and the
ideal diode equation (grey). Note that the plot changes from exponential relationship (n=1) to
resistive.
28
Finally, SimWindows@ is unable to simulate space charge generation and recombination and
avalanche breakdown. In Figure 2-8, the reverse bias characteristic has a slightly higher current
magnitude than what would be expected. However, this additional current is only due to base
width modulation. The inability of SimWindows@ to simulate avalanche breakdown severely
limits the accuracy of simulations of InAs in the reverse bias regime. Like other small band gap
materials, the critical field strength for avalanche breakdown in InAs low (8 x 10 4 -1.2
x 105V
roughly a third of that of Silicon).[9].
In the present application, the parameters of interest are mobility, refractive index, absorption coefficient, and recombination. For the most part, SimWindows@ can be expected to
perform well in these areas.
2.2
2.2.1
Theoretical Results from InAs Model
Testing the SimWindows Model
Dark Analysis:
1x10 19 cm 4 , p+ InAs, 0.1
prm
1x10 18 cm-4 , p InAs, 0.1 gm
IxO1
Undoped InAs, 0.7 gm
18
4
cm ,
cm-3, p+ InAs,
0.1 jim
1x10 18 cmr3 , p InAs, 0.1 jim
8
3
lxlOt cm- , n InAs, 0.2 jim
ND>1x10
9
1x11
8
3
cm- , n InAs, 0.2 pm
1
n InAs,
ND>lxlO
Substrate
8
4
cm ,
n InAs,
Substrate
A)
B)
Figure 2-9: Structures used to verify model. Device A: pin diode. Device B: p-n diode.
The Indium Arsenide model was constructed with the model discussed in Section 2.1.3.
To ensure that the model was valid, it was tested by simulating devices in the literature, and
comparing the real and simulated I-V characteristics. The devices used were reported by C. H.
Kwan, R.-M. Lin, and S.-F. Tang[25][26].
The two diode structures are shown in Figure 2-9.
29
The device area is 3.14x 10- 4cm
2
for both devices. The substrate thickness was assumed to be
300 um for both cases.
The data from the simulation and measurements are shown in Figure 2-10. The difference in
reverse saturation current can be explained by the mobility. The simulated saturation current
is 1.7 times larger than the measured current. The mobility in the model can be 1.7 times larger
than the acutal mobility in the device. The compensated data (the simulated data was divided
by 1.7), is shown in Figure 2-11. The excess reverse current of the measured p-n junction is
due to a shunt leakage current[26], most likely due to tunneling, which SimWindows@ cannot
model. The nonideality factor of the measured data is 1.3, and this is most likely due to high
level injection, another feature which SimWindows@ cannot simulate. A reasonably good fit
can be achieved by SimWindows@ within the capabilities of the program.
Illuminated Analysis:
The SimWindows@ program was tested at irradiated conditions by using localized generation in a photodiode, as discussed in Appendix B.3. Impulse illumination was chosen since the
program can convolve the results of the impulse generation function along the length of device
to find the effects of a generalized generation function. This also tested the optical parameters
of the model. For impulse illumination conditions, the user only specifies the energy of each
photon and the incident power. The generation rate is given by:
P
G = ce(E)~
~
(2.14)
where a(E) is the absorption coefficient of InAs for a photon with energy E, P is the incident
power density, in units of W
eV, P = 100 ",
cm-.s.
, and E is the photon energy in Joules. For this analysis, E = 0.45
and the associated a(E) is 6.73 x 10 3
1.
G was found to be 6.34 x 1024
M is defined as:
M = G -6x
(2.15)
where 6x is the width of the pulse in centimeters. In this case, 8x
calculated to be 1.86806 x 1018
=
2 x 10-
7
cm, and M was
'
The parameters used in the analysis are shown in Table 2.2. The results of the analysis is
30
.~~.I
Current Density Versus Applied Potential
1.0XE-02
-- SimWindows Si mlutation. PIN
diode, Uncompe nsated Data
E
---
Measurement, PIN diode
0)
N
-
1.lOE-03
q
CL
E
4
AjIjA,
0
1.00E-04
a)
AAAA
4
.- 1.
1
1.t_%JaUC
-0.50
-0.40
-0.30
-0.20
-0.10
0.00
0.10
0 .20
Applied Potential (Volts]
A)
Current Density Versus Applied Potential
1.OOE-02
...
.
..
.
..
....
.
-0-- SimWindows Simulation, PN Diode
Uncompensated Data.
-0- Measurement PN Diode
E
.....
.....
1.1111..
...
.......
....
1.00E-03
EL
E
K..
...
II
1.XE-04
t
0
1.OOE-0
-0.50
-0.40
-0.30
-0.20
.0.10
0.00
0.10
020
Applied Potential (Volts)
B)
Figure 2-10: I-V characteristics of measured and simulated Results. Figure A is the I-V
characteristic of the PIN structure and Figure B is I-V characteristic of the P-N structure. Note
that the measured current does not saturate in Figure B. This is caused by tunneling.
31
Current Density Versus Applied Potential
1.00E-02
-0E
SimWindows Simlutation, PIN
diode. Compensated Data
j
-A- Measurement, PIN diode
S1 .00rE-03
-C
aL
E
-TVI :i-'1 4
0 1.00)E-04
C
I
'-C
I~
ii
0
1.00OE-05
-0. 5
-0.4
-0.3
"
-0.2
-0.1
0.0
0.1
0.2
Applied Potential (Volts)
A)
Current Density Versus Applied Potential
1.00E-02
.
-0- SimWindows Simulation, PN Diode,
Compensated Data.
-0- Measurement, PN Diode
E
U
1.01E-03
CL
-
-0.5
-0.4
-
......
-0.3
-0.2
-0.1
On.
0.1
02
Applied Potential (Volts]
B)
Figure 2-11: I-V characteristics of measured and simulated Results. The simulation data uses
a mobility 1.7 times smaller than the previous figure. Figure A is the I-V characteristic of the
PIN structure and Figure B is I-V characteristic of the P-N structure. Note that the measured
current does not saturate in Figure B. This is caused by tunneling.
32
Variable
NA
Value
I x 1011 X 1017
ND
4,
1[Pm
I tm
M
1.868 x 1018
In
12
Table 2.2: Values used for the photodiode
shown in Figure 2-12. There is good collaboration between the calculated
and simulated data,
Photocurrent Versus Location
0.35
II
I I
0.3
IiAr
-Calculated
CD
f,
a, 0.25
-C
D-
E
-
Simulated
1
1 1 F-I
I r
tt
0.2
a)
I
I
I
0_
.4
f
r
4QI.15
~1:!bJ-!
221
r
0
0
40
a
0.05
0
0.00E+00
2.50E-01
5.OOE-01
7.50E-01
1.00E+00
1.25E+00
1.50E+00
1.75E+00
2.011+00
Position(pm]
Figure 2-12: Photocurrent versus the location of the light impulse, calculated (black) and
simulated by SimWindows@ (grey).
which means that SimWindows@ should give accurate results for the generated photocurrent
in the device. The simulated data is slightly larger for regions closer to the junction due to
base width modulation: the calculated data assumed that the depletion region did not change
in size for different biasing conditions.
33
2.2.2
Design Parameters from Model:
Repercussions on Photovoltaic Design:
The ideal diode equation is:
Ye
p
J(V) = J
ekBT
1
nkBT
[Amp]
Ph
e
e
weNA
where ni is the intrinsic concentration (ni = 1015
(2.16)
Am
_)cm2
W*ND
for InAs at room temperature)
,
and w*
and w* are the effective region lengths for the p-type and n-type regions. In order to have
the maximum conversion efficiency for a photovoltaic, the reverse saturation current, J', must
be minimized, as discussed in Appendix B.2.3. As a result, the hole and electron components
must be minimized jointly. The first step would to maximize the dopings NA and ND across
the junction to minimize J. The effects of mobility and electron and hole effect lengths will be
discussed currently. The discussion will be limited to homojunctions.
Mobility:
The ratio between electron and hole mobility in InAs is much larger than the ratios of
the other III-V or group I-V compounds, other than InSb and possibly a-Sn (grey Tin). Table
2.3 lists the electron and hole mobility of all group III-V and group I-V (except for a-Sb), and
ratios, if defined.
Since the electron mobility is two orders of magnitude larger than the hole mobility and
the doping level and other parameters are comparable, the hole contribution can usually be
ignored. Equation 2.16 can be rewritten as:
J (V) = nTkBTBe
-
w*NA
[
cm2
(2.17)
As a general rule then, the p-type regions in a diode structure will dominate the diode characteristics, and the I-V characteristics depend little on the doping and thickness of the n-type
regions.
Minority Carrier Lifetime and its Dependence on Doping:
Before continuing to the electron and hole effective lengths, the minority carrier lifetime
34
e
Material
Si
Ge
C (Diamond)
BN
BP
BAs
AIN
AlP
AlAs
AlSb
GaN
GaP
GaAs
GaSb
InN
InP
InAs
InSb
1450
3900
2000
NA
40
NA
NA
80
300
200
NA
160
9200
3750
NA
5900
33000
70000
l
Ratio of
505
1800
2100
NA
500
NA
14
450
200
400
440
135
402
680
50
150
450
850
2.871
2.167
0.952
0.08
0.178
1.5
0.5
1.185
22.886
5.514
39.333
73.333
82.352
Table 2.3: Electron and hole mobility of several semiconductors at
that the material does not have mobility data at room temperature.
300 K[4]. 'NA' indicates
needs to be examined. Plots of the simulated minority lifetimes are shown in Figures 2-13 and 214. In the model, the dominant recombination mechanism is assumed to be SHR recombination
for NAD
< 2 x 10161,
and for NA,D > 10171,
the recombination is limited by Auger
recombination. Radiative recombination does not affect the recombination time. Note that the
recombination times are roughly the same in both n and p-type materials. Since both sides
of the junction are doped heavily to minimize J, the dominant recombination process will be
Auger recombination.
Electron and Hole Effective Diffusion Lengths:
In photovoltaic design, the electron and hole effective diffusion lengths, w* and w*, need
to be maximized to minimize J. The upper limit of the effective diffusion length is the minority
carrier diffusion lengths, which is defined by:
Le,h -
: e
V qkTTh
35
[cm]
(2.18)
Electron Recombination Time Versus Doping Level
1
flE-fl6
1.&0
-
1.0E-08--
0
E
1.OE 09
-g
0
-
1.OE-1 0
-All
-oG -SHR
-, -- Auger
I.DE- II
-0--Radiative
1.OE- 12
1.OE-13 I1.OE+16
1.OE+17
Doping
3
1.OE+ 18
I.OE+19
Level (1/cm )
Figure 2-13: Minority Carrier Lifetime versus Doping Concentration for P-type InAs. The data
for Auger recombination is approximate.
Hole Recombination Time Versus Doping Level
1.OE-06
1.DE-07
1.OE-08
E
1-
-
.2 1.OE-O9
All
-U-SHR
E
S1.OE-1 1
*Auger
-
1.DE-11
-Radiative
-
1.OE-12 1.oE+ 16
1.DE+ 17
3
Doping Level (1/cm)
1.OE+1S
1.0E+19
Figure 2-14: Minority Carrier Lifetime versus Doping Concentration for N-type InAs.
36
Doping versus Minority Carrier Diffusion Length, Electrons
1000
100
0)
E
1
0
0.1
0.01
1. 0E+16
_
-_ -----
_
__
1.00E+17
_
_
_
__
1.OE+18
__
_
1.00E+19
1.00E+20
Doping Concentration (cm-)
Figure 2-15: Electron Diffusion Length versus Doping. Note the diffusion length is within the
same order of magnitude of the substrate thickness for low p-type doping
where A,,h is the carrier mobility and Te,h is the carrier lifetime.
The carrier lifetimes are
within the same order of magnitude for electrons and holes, but the difference between electron
and hole mobilities is quite large. Table 2.3 lists the electron and hole mobilities of several
semiconductors. The electron mobility of Indium Arsenide is about two orders of magnitude
larger than the hole mobility. Thus the minority diffusion length is about an order of magnitude
larger for electrons than holes at a given doping, as seen in Figures 2-15 and 2-16.
Also note that the electron diffusion length is extremely long. For low dopings (NA
1016
=
) the electrons can travel over 100 pm before recombining.
Ramifications on Photovoltaic Design:
To minimize the saturation current, the effective length, w*, should be the electron
diffusion length and the p-type region should be longer than the electron diffusion length.
Since the p-type region must be much larger, the photovoltaic structure should be a n-on-p, psubstrate structure, opposed to p-on-n, n-substrate structure. Growing upon a p-type substrate
would 'give' the electrons the necessary length to diffuse, allowing the electron diffusion length
37
Doping versus Minority Carrier Diffusion Length, Holes
100.00
10.00
-
_-_-
_-_-
1.00o
0
-J
C
o
.
0.10
__
0.01
1.OOE+16
1.OoE+ 17
1.00E+18
1.OOE+ 19
1.00E+20
Doping Concentration (cm-)
Figure 2-16: Hole Diffusion Length versus Doping.
to be used for w*.
The n-region is much more difficult to design: the n-type region needs to be thin enough
such that a large fraction of the generated carriers are created near to the junction, but yet
thick enough to minimize the saturation current. As seen in Figure 2-12 (and discussed in
Appendix B.3), the photocurrent peaks when the carriers are generated close (or inside) the
junction. Therefore, the n-type region cannot be thicker than the optical absorption coefficient.
The best structure to use would be a n+vp+ pin structure. The v serves as a large region
where the generation is maximized (see Appendix B.3. This is caused by the relative lack or
carriers to recombine with in the v region). In addition a v region prevents leakage tunneling
current, as seen in Figures 2-11 and 2-10[25]. The reverse current for the p-n junction does not
saturate in Figure 2-10B due to a shunt leakage current which was postulated to be a tunneling
current by the authors. The v region decreases the probability of a carrier from tunneling from
the n to p regions as seen in Figure 2-10A by increasing the width of the barrier formed by
the band gap. The v region cannot be longer than the minority carrier diffusion length (in this
case, the hole diffusion length at that doping), since carriers would recombine in a longer v
38
region.
2.3
Summary of Chapter 2:
The simulation software SimWindows@ was proposed in this chapter for use of simulating
devices. Advantages and shortcomings of SimWindows@ were also discussed and a material
model was developed for simulating InAs devices. The model was then tested under dark and
illuminated conditions, and was compared to results found in literature or calculated results.
Finally, InAs photovoltaic device design issues were discussed, and how the doping, mobility,
carrier lifetime, and minority carrier diffusion length influence device design.
The SimWindows@ model developed in this chapter was applied to InAs devices fabricated
in our laboratory. The results are discussed in the next chapter.
39
Chapter 3
Application of SimWindows to MIT
Diodes
3.1
InAs Photovoltaic Structure:
The two devices analyzed for this thesis were grown in a three chambered RIBER 2300
system used for the molecular beam expitaxial (MBE) growth of III-V devices. The devices
were grown by Henry Choy, and cross-sections of the structures are shown in Figure 3-1.
Both of these devices were not grown for photovoltaic operation: these were test structures.
Structure 9725 was designed to be a symmetrical pin diode, and the unintentionally doped
region generates a larger depletion region in the device. The concentration of carriers in the
depletion is very low, which would decrease the carrier recombination rate. Therefore, most
of carriers that are optically generated in this region will contribute to the photo current.
Structure 9722 was designed to be a simple p-n diode structure.
3.2
Comparison with the SimWindows Simulations
3.2.1
9725:
Analysis was first done on a device made from Structure 9725. A diagram representing the
different areas of the device is shown in Figure 3-2. The contact ring and grid are made of gold.
The Cathode region consists of a 0.05 [im thick n+ cap layer doped at 1 x 10191,
40
followed by
1x10'
9
cm- 3,
n+ InAs, 0.05
pm
5x10 17 cm- 3, n InAs, 1.0 um
Unintentionally Doped InAs, 0.013
2x10' 7 Cm- 3, n InAs, 0.3 pm
pm
5x10' 7 cm- 3 , p InAs, 1.0 tm
1x10 17 cm- 3 , p InAs, 1.0 gm
2x10' 8 cm- 3, p InAs,
1x10 18 cm- 3, p InAs,
0.56 pm
NA> 2 -4 xlO 8 cm- 3, p InAs,
0.3
pm
NA> 2- 4xl 0 1 8 cm- 3, p InAs,
Substrate
Substrate
A) 9725
B) 9722
Figure 3-1: Device Structures used in this analysis.
Contact Ring
Grid
Mesa Top
Anode
Substrate
Figure 3-2: A diagram of the device structure and surface of device 9725.
41
a 1 pum thick n-type region doped to 5 x 1O17
, followed by a 0.013 tam unintentionally doped
layer. The anode is constructed from a 1 [im p-type region doped at 5 x 1017
, followed by
a 0.56 pim p-type buffer layer, which was doped at 2 x 10186. The area of the mesa is 0.0961
cm 2 , the area of the contact ring is 0.0216 cm 2 , and the area of the grid and contact ring is
0.0729 cm 2
The results of room temperature current-voltage measurements are shown
in Figures 3-3,
3-4, and 3-5. The different areas referred to in Figure 3-5 can be seen on Figure 3-2.
-V Characteristics of Device 9725
- - 9725, Measurement1
0.014
0.012
--
9725 Measurement 2
---
9 7 25 Fit
o~1For Iit:
E
l s=1.5x1O. Amps
n=2.42
R28
Ohms
0.2
0.002
-0.004
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Applied Potential (Volts)
Figure 3-3: Terminal Characteristics of a device fabricated on growth 9725 plotted on a linear
scale.
The experimental I-V curve can be fit in the forward bias regime by an exponential diode
curve with an ideality factor, n, of 2.42, and a reverse saturation current, I, of 1.5 x
10-5Amps,
in series with a resistor, R,, of 2.420. This fit is shown in Figure 3-3. The currents predicted
by SimWindows@ were several orders of magnitude greater than what was obtained experimentally; this is shown in Figure 3-5. This discrepancy cannot be explained by the current
traveling though the different areas of the device as seen in Figure 3-5. The smallest metallization pattern is the contact ring, and the simulated current using this area is much larger than
the measured data.
42
Current Versus Applied Potential
1.0E<01
- -
9725. Measurement
1
-9725,
Measurement 2
-9725
Fit
1.0E-02
1.E-05
Ep H
1.T-'E
-1
06
-0.8
-0.4
-0.2
0
02
06
14
C8
1
Applied Potential (Volts]
Figure 3-4: Log plot of device 9725 and Ifit.
Current Versus Applied Potential
1.00E+03
9725 Measurement 1
1.00E+02
9725. Measurement 2
1.00E+01
1.00E+00
-
-
Simulated Device, Area
.0961 cm2
-
-
Simulated Device. Area
.0216 cm2
-
Device. Area
.0729 cm2
-
-Simulated
A,
-
1.00E-01
o 1.00E-02
m
1.00E-03
1.00E-04
1.00E-05
II I
1.00E-06
1.00E-07 L-2.00
ILI.I..fi.It
-1.50
-1.00
I I IfI I I I I I I 1 1
-0.50
0.00
Applied Potential
0.50
1.00
1.50
2.00
(Valtel
Figure 3-5: Log Scale Plot of the 9725 Data, with the Results of the SimWindows@ Simulations.
43
The reverse saturation current is given by Equation 3.1:
Js = n kBT
W(
-_
we* ) NA
w*
(7h) ND
Amp
c
(3.1)
(
From the diode equation, the most plausible explanations that could explain the small saturation
current were:
* The presence of doping mistakes or poor mobility in some regions
" Differences in minority carrier lifetime
* The presence of a highly resistive region in series with the diode
Each of these factors can be tested readily by SimWindows@.
Doping Differences and Mobility:
The devices where grown in an Arsenic stable environment:
the growth condition was
such that the limiting growth parameter was the Indium flux. The excess Arsenic pressure in
the growth chamber will tend to make the devices more n-type (For example, unintentionally
doped structures would tend to be n-type).
As a result, lightly doped p-type regions could
be compensated, and possibly become intrinsic or n-type, and n-type regions would be more
n-type. Several simulations were done where the doping of each layer was varied (except the
substrate).
The doping concentration is usually quite accurate so the doping variation was
limited to four orders of magnitude toward n-type for the p-type regions.
Figure 3-6 shows the results of changing the doping of the 0.56 pm buffer layer, Figure
3-7 shows the results of doping the 1.0 pm p-type layer, and Figure 3-8 shows the results of
doping the 1.0 pm n-type layer. The PIN structure that was simulated was similar to the 9725
structure, but the 1.0 pm p-type layer was replaced by a 1.0 pm undoped layer.
The changes in the I-V characteristics were small when the epilayer doping was changed.
As noted in Section 2.2.2, the heavily doped p-regions, will dominate the I-V characteristics.
In the present devices, dominant the region is the substrate. Therefore, doping shifts will not
contribute to decreasing the saturation current.
44
Current Density Versus Applied Potential
100000
m
10000
1000
-
Na-5e16
-
o
Nd-5e15
U
PIN
-
100
Original
|
Tht ii
Na-5e17
-
I La.
1:
1
Na-e15
A
Nd-5e16
1 1
I II
11111
E
I I
10
1
I~
E)
1
II
0.1
0.011
~I I
II
.I.
,L
-1.00E+00
-8.00E-01
-6.00E-01
H+Tf-f
0.00E+00
-2.00E-01
-4.00E-01
Applied
II.1tL L[.L IJJ.LLLLLLIAJ|
I |LL
2.00E-01
4.00E-01
6.00E-01
8.00E-01
1.00E+00
Potetial (Valtl
Figure 3-6: Doping variations of the 0.56 pm 2 x 1018 -I
p-type buffer layer.
Current Density Versus Applied Potendal
100000
10000
Original
-
II1 II II
011r1I1
Na-5e16
-
-
Na-5e15
-
0
Nd-5e16
0
Nd-5e15
1000
100
E
PIN
-F~F- i-T7--i T - h
F
I I
10
I
0.1
0
E01
L-
-1000.+00
.
I I
I. I I II.-. 00E-1
I
-8.000-01
-6.000-01
4.
tIE I
-4.000-01
II I I
II I I
I I
-2.00E-01
0.00E+00
Applied Paietdal
T
200E-01
4.00E-01
6.00E-01
Figure 3-7: Doping variations of the 1.0 pum 5x10 1 7
45
8.00E-01
1.00E+00
(Velte)
p-type layer.
Current Density Versus Applied Potential
100000
10000
-
Na-5e17
Original
- - - Na-5e16
Na-5e15
-100
0
0.1
-1.00E+00
-8.00E-01
-6.00E-01
-4.00E-01
-2.00E-01
0.00E+00
Applied POreial
2.00E-01
4.00E-01
6.00-01
6.00E-01
1.00E+00
Vnal
Figure 3-8: Doping varitation of the 1.0 ptm 5 x1017k n-type layer. Doping this layer more
n-type made no noticeable changes in the I-V curve.
In the same vein, the minority carrier mobility (electrons) cannot solely compensate for the
smaller current. In the literature, the electron mobility is between 10,000-30,0002 at room
temperature. To lower the saturation current, the electron mobility would have to be decreased
by two to three orders of magnitude.
Lifetime:
As stated in the last section and in Section 2.2.2, the most heavily doped p-region will
dominate the device operation. In device 9725, the most heavily doped p-type region is the
substrate. Since the substrate is roughly 300 pim thick, the effective length of the region, W:*,
will be the electron diffusion length, which is about 2 1 um for a region doped to 1018k (see
Figure 2-15). The carrier lifetime for the p-type region doped at 108
is limited by Auger
scattering (see Figure 2-13) and the minority carrier lifetime is between 10-10
-
10-11 seconds.
As seen in Figure 3-9, the IV characteristics will not change until the SHR lifetime falls below
the Auger recombination lifetime for that doping (in this case, 10~10 - 10-
seconds). In doing
the simulation, it was assumed that the SHR lifetime was constant within the device, but ideal
46
(10- 7 seconds) in the substrate.
In other words, the density of interband states is constant
within the epilayers.
This simulation revealed that the SHR lifetime will not affect the IV device characteristics
if the SHR lifetime was less than the modeled value (10-7 seconds). In addition, the measured
saturation current would be greater than the predicted value if the SHR lifetime was less than
the Auger recombination lifetime.
Current Density Versus Applied Potential
"-'El-SHR
10000
Lifetime- 1e-7 (Normal)
---
SHR Litetime-1 e-9
*-'--
1000
SHR Lifetime-11-1
SHRLI-Natime-le-11
SHR Lifetime-1e-12
100
E
10
0.1
0.0 1-.
-5.00E-01
- -4.00E-01
- - -
--
-3.OOE-01
-2.00E-01
-1.00E-01
0.00E+00
Applied
1.OOE-01
2.00E-01
-3.00E-01
4.00E-01
5.OOE-01
Potetial (Valt)
Figure 3-9: Changes in the IV characteristics with changes in the SHR lifetime.
High Series Resistance:
The final possibility is that there was a large series resistance in series with the device,
possibly a Schottky barrier diode between the top contact metal and the n-type region. The
effect must be in series with the device: any device shunting the diode would at the least
contribute to the saturation current. However, a large series resistance will only affect resistive
region of the diode in the forward bias regime, leaving the value of the reverse saturation current
intact as seen in Figure 3-10. In addition, a Schottky contact cannot be formed with n-type
InAs[9]. Therefore, it is unlikely that there is a large fixed or variable series resistance with the
diode.
47
Current Density Versus Applied Potential
Increasing Series Resistance
12+E04
1.0%E+03
1.O+02
E
1.00r=+0
1.0E-01
1.0%E-02
1.C012
-0.5
-0.4
-0.3
-0.2
-0.1
Q0
0.1
0.2
0.3
0.4
0.5
Applied Potential (Volts)
Figure 3-10: The simulated effects of increasing series resistance in device 9725.
SimWindows@ could not explain the large discrepancy between the simulated and measured
I-V characteristics. Therefore, the cause must lie outside the scope of SimWindows@. Before
the origin of this discrepancy is discussed, the results of device 9722 will be examined.
3.2.2
9722:
The structure of the 9722 devices and the metallization pattern are shown in Figures 3-11
and 3-12. The metallization shown in Figure 3-12 served as an etch mask to define the mesas
shown in Figure 3-11, Since the substrate doping is the same between samples 9722 and 9725
(see Figure 3-1), the current density should be approximately the same, as seen in Figure 3-13.
The measured current (taken from pad H from Figure 3-12) was a factor of 9.72 smaller
than the current measured from device 9725. However, the smallest area in device 9725 is the
contact ring, which is 54 times larger than metal pad. Other measurements on the same die
were inconclusive: several seemed resistive (see Figure 3-15) and devices had reverse currents
that were larger than their forward currents (see Figure 3-16)
48
Side:
Top of Mesa
Cr, .03 Micron1.Mirn
micronBottom
InAs Na=1X10
17
of
1/cm 3 1B
Mesa
m
micron
InAs
Substrate N,=2-4x 1"
1 /CM3
300 micron
Figure 3-11: Side profile of the 9722 devices.
Metalization Pattern:
E
D
MB
M
MA
Figure 3-12: The metalization pattern of device 9722. The top metal (shown in black) served
as etch mask to define the mesas. Areas A,B,C,F, and H are 4x10-4cm 2 in area; D and E are
4.9x10- 5 cm 2 in area; and G is 2.24x10- 2cm 2 .
49
~EE
-
--
=~
-
Current Density Versus Applied Potential
1DOE+O4
IO
ILLI IL
1,0E
I I I I I1
II II
I I I I
I11F
1111
IT]
i.O&+02
0
1.OIE+01
II
E
IOOE00
1.
0
1.Om-01
1 nOCIE-
D-02
1.:)I
I1I1I1 II II
1.
4
WIIIIM
-5.0Cm-01
III
-3.OO-01
-4.OGE-01
-2. O-01
-1.01-01
OD..fl
Iii
10101
I
liii
2JDOE-01
II
3.00E01
liii
II
4.00-01
SOOE-01
Applied Potential (Volts)
Figure 3-13: A comparison of the simulated I-V characteristics between the 9722 and 9725
devices.
Current Versus Applied Potential
1.13E-01
.
I
O -
1.01E-02
q lb-
C.
L-i&WW-W -
I
r
I
dF
I
E
<1.00)E-03
I
I I
f
4
--0-9725
-X- 9722
1.0£E-04
1.OE 05
-2.0
-1.5
-1.0
0.0
-0.5
0.5
1.0
1.5
2.0
Applied Potential (Volts]
Figure 3-14: The measured IV characteristics of the 9722 and 9725 devices. The data from
device 9722 was multiplied by 9.78.
50
Current Versus Applied Potential
i i i -bo
-E
-f-
6.00E-06
D
4-
-
- - -
4.00E-06
-J.
I
I
1 1 0,
1 I-'-
4
-
2.00E-06
ewE
fill
0.00E+00
-2.00E-06
-L-L-
--4
4--
~i i i
-4.00E-06
-6.OOE-06
I I I I I
I ii I
±-
'illli
i
I I
F++H-
H-
I I
I
1 4
1 i
I
-iH
H
-1
-8.OOE-06
-02
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Applied Potential [Volts
Figure 3-15: Measurements taken from device 9722 from pad E to the back of the substrate
(gray) and from pad D to the back of the substrate (dotted black)
Current Versus Applied Potertial
3.00E-06
1.00E-06
1111.T2fi
i
I
-1.00E-06
-- C
-3.00E-06
-A- 8
-A
-F
+
H
-500E-06
-7.00E-06
-9.00E-06
-1.10E-05
-2.00E-01
-1.50E-01
-1.00E-01
-5.00E-02
0.00E+00
5.00E-02
1.00E-01
1.50E-01
2.00E-01
Applied Potentil (Voks]
Figure 3-16: Measurements taken from 9722. These measurements were taken
from A, B, C,
F,or H to the back of the substrate
51
3.3
Explaining the Discrepancies
Both measurement from 9722 and 9725 produced measurements that do not
match with the
results from SimWindows@. Nor does adjusting the doping, SHR lifetime,
and or mobility
in this model explain this discrepancy. The same effect seems to be in both
devices The IV characteristics are similar in Figure 3-14 in the reverse and a portion of
the forward bias
region. Thus the effect should also be a factor that both devices share in common.
The three
possibilities explored were that there was poor metallization on the top of the
device, that the
surfaces of the device were inverted, or that there was a poor backside contact.
3.3.1
Poor Metallization on top of the device:
Current Versus Applied Potential
0.0003
+
A
0.0002
-+-C
-
B
11 1
14M
-x--H
0.0001
-- 4
-C
E
0
17
1
i"!M C
rp,W_-' TT1 Z7@
-oi
-
.fidwokiiiiii4o
0.0001
I
I-FTT-F-I
Hill
-0.0002
I
T
-H+44
-0.0003
.0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
I
Applied Potential (Volts)
Figure 3-17: I-V characteristics of several devices after thermal annealing. These plots demonstrate the I-V plots that were symmetrical.
If the metal contact on top of the device was poor, then the effective contact area
could be smaller. To test this hypothesis, the device was subjected to a rapid thermal anneal
at 300
C. The I-V characteristics became more symmetrical after the treatment, as seen in
Figure 3-17, and others maintained their diode characteristics, as demonstrated by Figure 3-18.
The current in Figure 3-18 is much larger than the devices in Figure 3-16. Otherwise, the I-V
52
Current Versus Applied Potential
0.0004
tltfhH
0.0002
I
I I
II
I-
-1-
~Th~HS~
I I I
I
FF
lit"
IH4
0
E
S
I
i
111111111444-
-0.0002
-0.0004
-0.0006
-0.0008
I I
-ii
-0.001
-1
-0.8
i
- 110
-0.6
1
-0.4
I I I
1 - +4
-0.2
0
0.2
Applied Potential (Volts)
0.4
0.6
0.8
1
Figure 3-18: 9722 Measurements. These figures display the more diode-like I-V characteristics.
characteristics did not change. Thus, poor top metallization cannot be the answer.
3.3.2
Surface Inversion
Most III-V materials have their Fermi level pinned at an energy lying within the bandgap
at the surface. However, the Fermi energy of InAs will be pinned in the conduction band at the
surface, as shown in figure 3-19[28]. One problem that might occur would be that the current
Surface
EC
-------------------------------
EF
Figure 3-19: Surface inversion in InAs. Note that the Fermi Energy at the surface penetrates
the conduction band.
53
Current Versus Applied Potential
0.000
-
-0.0002
HFT
-Beor
-0.0002
-0.0002
-0.0008
1
-1
1-0d
-1st CPEtch
HF Et1h
-0.8
-0-6
-. 4
-0.2
0
0.2
0.4
0.6
0.8
1
Applied Potential (Volts)
Figure 3-20: Effects of Passivation on the InAs photodiodes. Note that the diode-like characteristics dissapear after every treatment.
flow was limited to the surface channels created by the surface inversion. The inversion layer
would be very thin, so the effective area would be much smaller than the mesa area. The surface
must be passivated to remove the inverted surface.
Two different techniques were employed to passivate the surface: a HF etch and a CP-4 etch.
In III-V materials, an HF etch is used to remove the oxide from the surface and to provide a
short term passivation[29]. The device was etched for 30 seconds, 1 minute, and two minutes.
The treatment made the photodiodes symmetric, as seen in Figure 3-20.
The CP-4 etch was used by Jack Dixon to prepare a InAs surfaces to have recombination
velocities less than 10 3
[30]. The CP-4 etch consisted of 15 cc of Acetic Acid, 25 cc of
HNO 3 , 15 cc of 48% HF, and 0.3 cc of Bromine. The device was etched for 20 seconds and 5
seconds. The CP-4 decrease the current though the device, but no new features in the curve
(see Figure 3-20). The etching seemed to make the device more symmetric. This also cannot
be the solution.
54
Current Versus Applied Potential
0.016
0.014
-
""- 972
0.012
-"9725
0.01
CL
0.006 - - - - - 0.004
0
-0.002
0.00-0.004
Z
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Applied Potential (Volts]
Figure 3-21: Through-substrate measurement of 9722 (grey) compared with 9725 I-V characteristics (black).
3.3.3
Bad Metal-Semiconductor Contact on the Bottom:
Another die of 9722 was probed from the top of the substrate (bottom of the mesa) to the
bottom of the substrate (see Figure 3-11).
This should have been a resistive contact, but
revealed a diode like characteristic as shown in Figure 3-21. By multiplying the current scale
in Figure 3-21 by 1.25, a good fit can be achieved, as shown in Figure 3-22
The only current path that 9725 and the through-substrate measurement of 9722 share is
the substrate. Therefore, it was concluded that the diode characteristic must be caused by
the substrate. To confirm this hypothesis, the bottom was coated with Ge-Au.
All diode
characteristics disappeared, as demonstrated in Figure 3-23.
The diode affect arose from the poor backside contact. The surface band structure was
inverted (see Figure 3-19). Higher dopings of p-type InAs tend to form non-linear (but symmetric) resistive contacts with metals, while lower dopings produce Schottky contacts. However
in this case, the surface was inverted. The HF etchant passivated the backside, creating a nonlinear resistive behavior. The same effect was generated when depositing metal to the back
side. However, this does not explain why the p-n junction failed to give a rectifying behavior.
55
Current Versus Applied Potential
0.01600
69725 -9722
1::
0.01200
0.01000
0
0.00800
S00600
0.00400
0.-D200
0.00000
-0.00200
-0.00400
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Applied Potential (Volts]
Figure 3-22: Compensated 9722 Data (grey) with 9725 Data (Black).
Current Versus Applied Potential
0.016
0.014
7-
0.012
|-
-
9722
-
9722 aftei
Metal
0.01
0.008
4
0.006
0.004
U
0.002
0
-0.002,
-0.004
L.
-1
-0.8
-0.6
-0.4
-0.2
0
Applied
0.2
0.4
0.6
0.8
1
Potential (Volts]
Figure 3-23: Through Substrate measurement of 9722, before back metal (Grey) and after
(Black). Note that all the diode-like characteristics dissapear.
56
3.3.4
What Happened to the P-N Junction?
The results at room temperature revealed a nonlinear but apparently symmetric resistive
response. This can be seen in Figure 3-24. However the differential resistances of the same
devices were not symmetric about the origin, as demonstrated in Figure 3-25, which suggested
that the devices could contain diode characteristics. At lower temperatures, the symmetry
was
Current Versus Applied Potential
0.002-
0.0015
0.001
o0.0005
4
0
C)
-0.0005
-0.001
-0.0015
-0.002
-0.0025
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Applied Patendal (Volts)
Figure 3-24: Room Temperature Measurements of device 9722. Note that the I-V characteristics
are symmetrical.
broken as seen in Figure 3-26 and 3-27. Device 9722 was cooled to 132 K, and a negative resistance feature appears at roughly 0.04 Volts. This feature is indicative of interband tunneling.
Interband tunneling explains why the reverse current does not saturate since the tunneling current will prevail in the reverse bias condition. Tunneling also explains the symmetry about the
origin: the reverse current of a tunneling junction increases exponentially
with applied voltage,
the same voltage dependence as a forward bias p-n junction.
This effect should not occur since the junction was not doped degenerately (Na = 1 X
1017k, Nd
=
2 x 10171). Normally, tunneling that occurs around the origin (V = 0 Volts)
is due to an electron traveling from the conduction band of the n-type region to the valence
band of the p-type region, tunneling through the forbidden band gap, as shown in Figure 3-
57
Differential Resistance Versus Applied Potential
1200
i i i-fl! i i i i i i i-H-
II-1 1I1 1I
I III III I
1000
-Z
I I LAIt
0
800
III
+4
I
400
I
IA
LL IL -FT
LL FL
-J
-L- -FT
:11:FTT'F
i
600
|II
-1
HHHA+H~H+H
II
1-krFTT200
I I I I-TT-1
±H±ffl
i i I-
X-'ff
N*
1 11
II
1111 17S.J
T-iftLi I f 71 P-T
A I I
0
-0.5
FT-T
-0.4
-0.3
-02
-0.1
0
0.1
0.2
0.3
0.4
0.5
Applied Potential (Volts]
Figure 3-25: The differential resistances of the devices. Note that resistance is not symmetrical
about the origin
28A. However, if there was interband defects, as shown in Figure 3-28B, the carriers are able to
tunnel between states in the forbidden band gap. This is the most probable explanation since
there was an abundance of etch pits that formed on the sample.
At room temperature, the forward current is dominated by injection. However, the injected
current varies with temperature as:
J,(T) cx T
where -y is a constant[31].
2e kT
(3.2)
As seen in Equation 3.2, the current decreases with decreasing
temperature. At sufficiently low temperatures, tunneling will become dominant in the forward
bias regime.
There was a p-n junction, but it operated as a tunneling diode. The reverse current was
dominated by the tunneling current and did not saturate.
This effect may be detrimental
toward photovoltaic operation: if the diode operates at low temperatures, the tunneling effect
will decrease the amount of power generation, as seen in Figure 3-29 by changing the size of
the power rectangle (the power rectangle is discussed in detail in Appendix B.2.2).
58
Current Versus Applied Potential
0.1
aoe
---
0.06
K
0.04
300
12 K
0.02
E
0
-0.02
a4
-006
0.1
-
-0.1s
-
-0.8
-0.6
-0.4
-
.z
-. 2
0
1- .
0.2
I - -
0.6
0.4
0.8
Applied Potential (Volts]
Figure 3-26: Linear Plot of the I-V Characteristics of Device 9722 at room temperature (grey)
and 132 K (black).
Current Versus Applied Potential
0.1
.
0.01
E
0.001
-300
K
-132
K
0.001
0ooo1
-08
-0
-0.4
42
0
Applied Potential
C.2
.4
C.6
0.8
(Volts)
Figure 3-27: Log Plot of the I-V Characteristics of Device 9722 at room temperature (grey)
and 132 K (black).
59
EF
EF
EC
EV
A)
B)
Figure 3-28: Tunneling Current in a P-N junction. Figure A demonstrates tunneling directly
from the condunction band to the valence band. Figure B indicates tunneling by interband
states.
This problem can be solved by decreasing the amount of defects formed in the growth, and
by incorporating an v region between the p and n regions. By decreasing the amount of defects
in the sample there are less destinations for carrier tunneling, thus decreasing the tunneling
current. The amount of defects can be improved by better growth techniques. The v region
will increase the size of the barrier, and thus decreasing the probability of tunneling across the
junction.
60
I
___________
v
I
V
7
B)
Figure 3-29: Effect of Tunneling on the Size of Power Rectangle. Figure A represents an ideal
diode, Figure B represents a diode with tunneling.
61
Chapter 4
Conclusion
4.1
Summary of Accomplishments
InAs has considerable promise as a material for microscale thermophotovoltaic applications.
The main emphasis of this thesis was to develop a method to design photovoltaics and the
tool that was chosen to facilitate in the this task was SimWindows@. A material model was
developed for SimWindows@ to diagnose and design InAs photovoltaics. The model was tested
and compared to InAs diodes in the literature and SimWindows@ was able to fit most of
the data in the literature within the limitations of the program. Some design issues of InAs
photodiodes were also discussed.
The model created by SimWindows@ was applied to two InAs diodes grown by MBE in
an effort to analyze them. The measurement revealed diode characteristics for both devices,
but the saturation current predicted by SimWindows@ was several orders of magnitude larger
than what was actually measured. However, the model developed in SimWindows@ was unable
to explain the discrepancy between the theoretical results and the measured results.
The
discrepancy was due to a more fundamental aspect of the diode that was not included in the
SimWindows@ model.
Several tests were done to isolate the problem. The top metal was annealed to provide a
better top contact, but this yielded a more ohmic behavior. Next, the surfaces were pacified,
which made several changes in the I-V characteristics, as well as an ohm behavior.
Finally,
metal was applied to the back of the substrate, which removed all diode-like behavior from the
62
device. It was concluded that the diode-like behavior was due to the poor metal-semiconductor
junction on the back of the device.
Finally, a low temperature measurement revealed the
presence of a p-n junction on the device, but the p-n junction formed a tunneling junction,
instead of a rectifying junction. The tunneling was most likely due to tunneling due to states
located in the forbidden band region. This problem can be solved by better growth techniques
and the incorporation of a v region between the p and n regions.
4.2
Future Avenues
The first initial steps have been overcome in developing InAs photovoltaics. The next step
would be to perfect the growth process on InAs, and to incorporate a v region. These improvements should improve the I-V characteristics at room temperature.
Next, heterostructures
should be incorporated into the design. Heterostructures will provide carrier confinement, but
may also cause interface states due to the lattice mismatch. Another avenue would be to use
graded semiconductors, which are able to absorb more of the incident radiation, and experience
less loss due to carrier randomization.
Of course, models must be provided with these new materials. Presently, a InGaAs and
InAsP model is being worked on to provide a "window layer" for the photovoltaic, and to
minimize the lattice mismatch for InAs.
63
Appendix A
Optical Properties of Indium
Arsenide
The contents of this appendix are taken from Sadao Adachi's work on the subject[5] [6].
The complex refractive index n* (w)is given by:
n*(w) = n (w) + ik (w) =
= 1ei (w) + 62 (w)
v
(A.1)
Where n is the real part of refractive index, k is the extinction coefficient, e is the complex
permittivity,
6i
is the real part of permittivity, and
E2
is the imaginary part of the permittivity.
From Equation A.1, n and k can be written as:
n(w)
k(w) = 4
47r
ei w)
2
(A.2)
32(w)---ci(w)
2
(A.3)
The absorption coefficient, a (w), is defined as:
(w)=47r
) k
E (w)2 + E2 (w) - 6E (w)
47r
A
2
64
(A.4)
There are several major interband transitions that affect the optical properties of InAs,
shown in Figure A-1,and the associated parameters used in this appendix are shown in Table
L4 5c
X7c
176c
L
L6("
6c
X6c
E
El
8V
L
L45v
AI
EO
X7v
A
r 7V
X6v
L6VA
//
L
X
F
Figure A-1: Transitions of interest in InAs.
A.1.
EO and EO + AO Transitions:
The EO transition corresponds to electron-hole pair generation between the symmetry
points P~v and 1{, from either the heavy or light hole band, and the EO + AO transitions are
from rf to rc. ei and
Cl
E2
are given by:
(w)
AEO
f(xo) +
f[H
E2 (W)
=
H (XO - 1)
+
1
E A
H Eo
f
(xSo)] + 1
E
(
ho-EO + 2H (x'O - 1) V'hw - (Eo + Ao)]
65
(A.5)
(A.6)
Parameters
Units
Value
Eo
eV
0.36
AO
eV
0.40
E1
A1
Ex
A
B1
Bi1
eV
eV
eV
eV
eV2
None
eV2
2.50
0.28
4.45
1.07
0.61
6.59
13.76
B2
None
0
B 21
E2
eV2
0
'
None
0.21
C
None
1.78
7
None
0.108
D
None
20.8
E100
None
2.8
Table A.1: InAs optical porperties calculation parameters.
with:
f(x) = x-2 [2 - V/1+x - H (1 - x) V1+ x
X0 (w) -
Xs0 (W) =
and:
H(x)
{
(A.8)
Eo
(A.9)
EO + AO
1, for
=
(A.7)
x>0
(A.10)
0, for x < 0
El and E, + A 1 Transitions:
The E
transition corresponds to electron-hole pair generation between the symmetry
and LC, and the E + A 1 transitions are from LJ to LC. el and
points L
e1 (w) = -BIX
62 (W)
=
[X-2H
2
£2
are given by:
In (1 - x2) - BiX2 in (1- X2s) + 1
(1 - Xj) H (Bi-- B V/E--hw) (B
66
1
- B
1E
-\I)h
(A.11)
...
(A. 12)
+x
2
H ( - Xis) H B - B21 VJE + A - hw
B2
+BIx(2 H (x1 - 1) + B2xIs Hx1-1]
where:
hw
(A.13)
Xi = El
= iw
(A.14)
X's =FEl
Note for Equation A.11, that w should be replaced by w + il, where F is the damping
energy.
E 2 Transitions:
The E 2 transitions do not correspond to single, clearly defined transitions from critical
points in the band structure. Thus, the E2 will have the same structure as a damped harmonic
oscillator:
C (I - X2)
X 2 ) 2_2+
- X 2(1 +)(X27)
-F1(w) =
-2 (w)
(A.15)
I
C7YX 2
=
-
(A.16)
X(1 + (X27)2
X)
where:
X2=
(A.17)
X2E2
EL (Indirect) Transitions:
The EL transition are indirect band transitions from rV to L.
2
(w)=
(h
D2
W)
- (E hq)]2 H (1
E2 is given by:
X,) H (1 - Xc;)
(A.18)
where:
_g
EL
± hW
Ehwq
hw
Xch
=
67
Ech
(A.19)
(A.20)
is the energy of the photon that is participating in the indirect transition. In this case,
4.084 x 1013 [f]. Ech is the high energy cutoff, and it is assumed that Ech = El.
hwq
Wq =
E, and E2 Equations.
E1 is the sum of Equations A.5,A.11, and A.15.
A.18.
62
is the sum of Equations A.6,A.12,A.16 and
is shown in Figure A-2, and 62 is shown in Figure A-3, along with their components.
61
20
15
lbo
10
.E
5
C-
a)
Ela
0
-5
-10
0
1
2
3
Photon Energy (eV)
4
5
6
Figure A-2: Real Permittivity, el, versus photon energy. eia is equation A.5, and deals with
the EO and EO + AO transitions. 6 1b is equation A.11, and deals with the E 1 and El + A,
transitions. 6 1c is equation A.15, and deals with the E 2 transition. el = 6 1a + 6 1b + 6 1c.
The refractive index, extinction coefficient, and absorption coefficient are plotted on Figures
1-3 and 1-4.
68
100
I I I I I I I
H-
62
10
E
E2dl
Ekc
E
62d2
E2a-
0.1
0
1
2
3
4
5
6
Photon Energy (eV)
Figure A-3: Imaginary Permittivity, -2, versus photon energy. 6 2a is equation A.6, and deals
with the EO and EO + AO transitions. E2b is equation A. 12, and deals with the E 1 and E + A
1
1
transitions. 6 2c is equation A.16, and deals with the E 2 transitions. e2dland 6 2d2 is equation
A.18, and deals with the EL indirect transitions. E2 = -2a + 6 2b + 6 2c + 6 2d1 + E2d2.
69
Appendix B
Ideal Model of a Homojunction
Photovoltaic Device
The information was taken from Sze[32] and Bhattacharya[33].
B.1
Basic Principles:
Photovoltaics are semiconductor devices that generate power from optically generated carriers.
The photovoltaic consists of a p-n junction diode, and a front and rear contact metal as seen
in Figure B-1.
The operation of a photovoltaic is shown in Figure B-2.
In Figure B-2a, the photovoltaic is not exposed to light.
Thermally generated minority
carriers will diffuse into the space-charge region and create the reverse saturation current, I,.
At zero bias, I, is counterbalanced by an equal drift outflux of majority carriers, resulting in
no net current. In this situation, no power is generated by the device and the device operates
as a diode. However, in Figure B-2b), the device is illuminated with photons of energy hv and
electron-hole pairs are created if the incident photon energy is sufficient to create a electron-hole
pair (hv > Eg). The minority carriers that are generated near the space-charge region (within
one diffusion length) will diffuse into the space-charge region, and are swept into the opposite
side of the junction. This action will generate a current through the device, and a voltage will
form across the ends. The device still has diode characteristics, but it now also may generate
70
hv
Front Contact
n-type
I
Back Contact
Figure B-1: An example side schematic of a p-n junction photovoltaic.
power.
B.2
Circuit Model and Current-Voltage characteristics:
B.2.1
Circuit Model
The circuit model is shown in Figure B-3.
The photovoltaic is generating the net current I to the load resistor RL. In addition, a
voltage V forms across the resistor, placing the photovoltaic in forward active operation. The
photocurrent Iph is the current created from the diffusion of optically generated excess minority
carriers across the junction. The current is leaving the anode opposed to the cathode, opposite
of the diode normal operation.. The current-voltage relation of the circuit in Figure B-3b) is
given by.
I
= Iph - Is ek
--
(B.1)
The reverse saturation current I, is given by
Is
qAni
?
Z ThND
71
+ Tn
)
TnNA
(B.2)
E,:
Vbi
Efn -----------
Ef --------------------------------------------
E
E,
Ev
b) Irradiation
a) No illumination
Figure B-2: a) A photovoltaic under no illumination. No carriers are generated. b) A photovoltaic irradiated by light. Excess holes and electrons are generated.
Where A is the device cross-sectional area,
hole lifetime in the cathode,
Te
ni is the intrinsic carrier concentration,
Th
is the
is the electron lifetime in the anode, NA is the doping of the
anode (p-type region), ND is the doping of the cathode (n-type region), W is the effective hole
diffusion length, and W is the effective electron diffusion length. The effective hole diffusion
length, Wp, is the hole diffusion length, Lh
than the cathode length, 12. If
L,
> li, or We
B.2.2
= 11
if Le
12
= VDrhTh,
if the hole diffusion length is much longer
is much larger than Lh, then W, = 12. Likewise, We = Le if
< 11 where 11 is the anode length.
The open circuit voltage V,, and short circuit voltage
'Sc.
The I-V curves are given in Figure B-4a):
The I-V curve without illumination is displaced in the negative I direction by the photocurrent, Iph. As Figure B-4a) indicates, part of the curve lies in the forth quadrant, indicating
power generation. Two important quantities are the open circuit voltage, V0c, and the short
circuit current, Ic. Vc is the voltage formed across the terminals of the photovoltaic if the
photovoltaic is illuminated, but connected to a load with infinite impedance. In order to have
no net current flow, the photocurrent must be balanced by the forward current of the diode.
72
hv
V
RL
p
q.V
sek-T
5
"ph
1)
V
RL
M
a) Photovoltaic in operation
b) Idealized Equivalent Circuit
Figure B-3: a) The cicuit diagram of a photovoltaic in operation. b) The idealized equivalent
circuit of the photovoltiac operation.
V, is the voltage formed to create the forward current. Vc is given by:
Voc = kBT Iph
q
The photocurrent
ph
(IS
)
,p>>
>
(B.3)
'ph
q
is
is usually much larger than the reverse saturation current
1
s so the
approximation in B.3 is valid. The short circuit current is the current produced by the diode if
the terminals are shorted together. In this case, the voltage across the diode terminals is zero,
thus:
(B.4)
ISc = Iph
B.2.3
Maximum Power Generation
The power delivered to the load is given by P = VI, and using Equation B.1:
P=IV=V
f'ph
Is
ekB
T
-1)
(B.5)
The power is the shaded rectangle under the I-V curve, as shown in Figure B-4b). The
maximum power, Pm, that can be delivered by the device is given by Pm = ImVm, where
voltage Vm and current Im are necessary to provide the power Pm. Pm is found by:
73
VM
TI'
Voc
V
Without
Illumination
With
Illumination
Isc
N/
a) The operation of a photovoltaic
with and without illumination
b) The power rectangle of a
photovoltaic
Figure B-4: a) I-V curves of a photovoltaic with and without illumination.
rectangle of a photovoltaic.
b) The power
f
dP
(B.6)
dV=0
V=Vm
dP
dV
d
I
I--Is
ekBT
1)±+
(
IphV]
V=Vm
'phs
=
gBT
fekBT
1)
kB 7 eI
ekB T
0
V=Vm
Iph -
Is
(ekBT
-
1)
qIsVm
-BT
J kBT
Is
ekBT =
1+
74
m
kBT
v
ekBT
=0
(B.7)
(B.8)
kBT In
q
1I+
kBT n
'pI,
qVr
i+[1+kBTj
phi
kBT
s
q
q
In 11 +qVm,
kBT_
Voc
Vm=Voc - kBT In 1qVm
q
IlkBT
(B.9)
Im is given by:
Is
Im = 'ph -
-
e kBT
1)
(B.10)
By using Equation B.7 and Equation B.8:
qIsV
kT
IM
qVm
kBT
[
1+1
qVm
qv
e kB T
kBT
I.k+ T_
1
Is+I'ph1
1 +
V
kBT
ph
Iph5Is Iph
1 kBT
(qv
)
The last approximation comes about from Equations B.3 and B.9. Vm is bounded by Vc,
but since
'ph
> Is, Vo c >
kBT. In order for Equation B.9 to hold, Vm
Im
>
kBT
kBT)
'Iph (
-
(B.11)
q Vm
The maximum power is given by:
Pm
=
VmIm ~
=
Iph M
(Vm
kBT)
q
Iph
= 'ph
VOC
- kBT In (1
I
q
+q1/;
kBT)
kBT]
q
(B.12)
(B.13)
q
where:
m = q [Voc -
kBT n
q
75
(1+
qVm
kBT)
kBT
q]
(B.14)
(m is the energy derived per proton to the load at the maximum power point. The conversion
efficiency, q, is given by:
P
Pm
VmIm
' incident
Pincident
_
(B. 15)
kBT
Pincident
To maximize the power delivered by the photovoltaic, Vm and Im must be maximized. By
Equation B.9, Vm is limited by Vc. However, Vc is limited by <pbi, the built in potential of
the diode. To make
SObi
as large as possible, the doping on each side of the junction, NA and
ND, must be large. From Equation B.10, Im, is maximized if Iph = Ic is made as large as
possible, and 1, is minimized. An important Figure of merit is the fill factor,
ff. The fill
factor
is defined as:
ff =
B.3
(B.16)
Monochromatic Response
This discussion will be limited to a short base photovoltaic with localized monochromatic illumination. A short base photovoltaic is shown in figure B-5.
ni'
n-type
n
PX
G (cm- 3s)
Figure B-5: A photovolatic with impulse illumination of area G at x,.
76
The short base approximation is that the minority carrier diffusion length, L, for electrons
in the p-type region and Lh for holes in the n-type region, is much longer than the length of
their respective regions. The minority carrier diffusion length is given by:
Le
DeTmin =
=
Dh-min
Lh
where Pe and
/1 h
=
q
kBT
Pe min
(B.17)
PhTmin
(B.18)
are the respectively the electron and hole mobilities, and Tmin is the minority
carrier lifetime.
The currents are found by the continuity equations and the definition of carrier flux.
an
4
at
-VFe+G-R
(B.19)
at
-V Fh+G - R
(B.20)
Fe
-De V n - nye
(B.21)
Fh
-Dh V pppAh
(B.22)
ap_
where n and p are respectively the electron and hole concentrations, Fe and Fh are the electron
and hole fluxes, G is the generation rate, R is the recombination rate, and
is the electric
field. By assuming constant doping on each side of the junction, low level injection, quasi-static
excitations, quasi-neutrality, and that the minority carrier flux is dominated by the diffusion
component (opposed to drift), Equations B.19-B.22 can be rewritten as:
0
=
0
=
Fe
dFe
(
x - + G -m
(B.23)
~ + G - -P(B.24)
dx
dii
-De--dx
77
Tmin
(B.25)
Fh
for a p-type region, where ft and
(B.26)
-Dh-±Lp +Plh
dx
P are the electron and hole minority carrier concentrations.
For a n-type regions, the equations will be:
0
d Fe
=
dx
=
0
Fe
=-De-
Fh
=
ii
+ G
(B.27)
ft
Tmin
~ + G -P
dx
T-min
dil
dx
dii
(B.28)
(B.29)
_nte
(B.30)
-Dh+
ax
Since the excitation is an impulse, G is given by:
(B.31)
G = M6(x - x,)
where M is the area under the excitation in real space, 6(x - x,) is a Kronecker-Delta function
centered about x0 .
Let the excitation be in the p-region (x, c
[In + xp,
in +
ip].
This argument can be readily
applied to the excitation being in the n-region). The minority carrier continuity equation is
given by:
- + M6(x - xO) 0- 0=dFe
dx
(B.32)
h
T min
Integrating both sides from x- to x4:
04
0
dFe
JX dx
xdx
[Fe (x) - Fe (XO)] +M
S/
-
dx +
M6(x - x)dx -
78
JX
x0
dx
(B.33)
Tmin
(B.34)
M = Fe (XO) - Fe
(xO-)
(B.35)
where we used the fact that the recombination is only dependent upon the concentration at a
point, and as the limits of integration approach this point, the integral approaches zero.
When x C [In + xp, xO-], there is no generation, so by combining Equations B.23 and B.25
and using the short base approximation:
0
d2dx2
_
X=
= A (x - (In +
'h (x)
where A is a constant. For x E
(B.36)
xp))
[4+, In + Ip]:
d2fi
dx2
0
J+
x)
0
=
-B (x - (In +
(B.37)
Ip))
Since the carriers must be continuous at xo:
X)
-()
=
-
A
(xO - (In +
))
But the flux on each side of x, is given by:
Fe (X--)
Fe (x4)
=
-De
-De
-
I
dx X=,-
I
dxj
+
79
DB (xO
(xO
= DeB
-
(1, + Ip))
(in + xp))
(B.38)
(B.39)
And substituting for Equation B.35:
M
=
B
=
-DeB
1+
(B.40)
De
(x
Un1+1P)
_1
(xo-(ln+xp))
Since there is a built-in field between the p and n regions, only the election flux will enter
the depletion region (holes will be repulsed away from the space charge region, so the current
is given by:
Je
=Je(x
qDe
le + xp) = -qFe (x-)
-M
De
qDedx
(xO-
__ 1) (xO
(XO-1+iPi
(Xo-(ln+xp)
(n
-
+ lp))
(In +
Xp))
-qM (xo - (In + 1p))
((XO - (1n + lp)) - (xo - (
Je
=
qM (x, - (In + 1p))
XP - P
+ xp)))
(B.41)
Note that the excitation is independent of the region type (note that the diffusion coefficient
disappears).
Similarly for x0 C [0, in - xn]
Mx"
-Xn
JA =
(B.42)
For xO E [in - Xn, In + Xp] (generation in the space-charge region), the electric field is high,
so the generated carriers are swept out of the space charge region before they can recombine.
Therefore, all carriers generated in the space-charge region will contribute to the photocurrent.
Thus the current can be written as:
Jtotal = qM
80
(B.43)
B.4
Spectral Response:
di
<
Surface
Back Contact
06
X
Figure B-6: The cross section of a photovoltaic.
Figure B-6 indicates the variables that will be used in this discussion. The photons entering
the photovoltaic will flux per unit wavelength
#1 .
The flux per unit wavelength at a point x
inside the diode is given by:
F(x) = #1(A)e-Aox
(B.44)
where a(A) is the absorption coefficient.
The excess carrier generation rate is given by:
dF(x)
G(A)
a(A)#1(A)e-'(A)x
dx
(B.45)
However, the flux in the semiconductor (#1 (A)) is not the incident flux at the surface,
since reflection at the surface is not accounted for. The flux
01
# 1 (A)
(A) = #0 (A) (1 - R (A))
0
(A),
is given by:
(B.46)
where R (A) is the fraction of electrons that are reflected at the surface. Thus Equation B.45
can be rewritten as:
G (A) = a(A)#o (A) (1 - R (A)) e(A)x
81
(B.47)
In this situation, low level injection, quasi-static operation, uniform doping on either end
of the diode, and quasi-neutrality is assumed. Thus the continuity equations in one dimension
can be written as:
Gn A
where
n
dFe
Te
dx
-0,
Gp (A
-_
0
dF
dx
-
Th
=0
(B. 48)
P and ft are the hole and electron excess carrier concentration respectively.
The minority carrier fluxes per unit wavelength are given by:
-Je-
Dif f usion +F Drift
Fe
-q = F
Fh
JhA
q
-D dft
+no
_1t
do
Diffusion + Fr
p
--
= -Dh
hdx
(B.49)
Pot
'ho
Assuming that the electric field is zero outside of the junction (the voltage drop across the
device will fall solely across the space charge region), then:
Frift
FDrift
-
0
In the n-type region, the excess hole concentration is given by the differential equation:
d2 P
+ o(A)O5 (A) (1 - R (A)) e-(x
dx
Dh-
-
=
0
Th
(B.50)
with the boundary conditions:
(B.51)
A(li) = 0
Dh
dx=
(0) Sp
I
(B.52)
The first boundary condition arises from the depletion region removing excess charge at the
space charge region boundary. The second boundary condition is due to the finite recombination
velocity, SP, at x = 0. The solution of the differential equation is given by:
82
z# (1- R) Th
(B.53)
L(SPLh
a2L
- 1
Dh_
+ aLP) sinh
e--
L
Dh
sin h
sinh
(SPLh
+ cosh
+ cosh
(Ih)
e-"JB.54)
h)
The current density per unit wavelength at the junction is given by:
A (Xj)
=
I
-qDx
X=1i
qa (1 - R) Lh
[SPLh+
L
1
e2L2h -
- e-all
+L
Dh
SPLh
SPLh cosh
(7)
+ sinh
- aLhe-'1i (B.55)
sinhQL) +cosh (h)
In the p-type region, the excess electron concentration is specified by the differential equation:
+ a(A)#5 (A) (1 - R (A)) e-a(A)x
De d2
dx
h =0
Th
(B.56)
with the boundary conditions:
ft(li +w) = 0
De
ii
dx
If
= ft (11 + 12) Sn
(B.57)
(B.58)
X=11+12
The solution of the differential equation is:
ao (1 - R)Tre
2L2 -
1
[cosh
x- (
_+W)
e-a(x-(1+w))Le
83
(B.59)
cosh (-12 )
SnLe
SnLe
-
e-al2
+ sinh
({2 )
-
cLee-l2
+ W
x -
.in
sinh IL,) + cosh (2)
Le
The current density per unit wavelength at the junction is given by:
Je (xj)
qDe
I
dx
J
x=l1+w
qaO (1 - R) Le e-C(l2+w)
X
_
I
1
-2L2
[
SnLe
cosh
(
-- a'21
--2) ,/ -e nLe j
SnLe sinh
~1Le
- sinhL)
L,)
IL,)+
-
aLee-0l2
cosh (9)
(B.60)
However, some generation is taking place in the space charge region. As stated in Section B.3,
carriers generated in the space charge region are swept out before they can recombine. Thus
the photocurrent per unit wavelength in the space charge region is given by:
1i+U
JSCI?
qF = q
(A)
Ii+W
21
=
e-a(A)xdx
Gdx = q0O (A) (1 - R (A)) a (A)
11
q 00 (A) (1 - R (A)) e-a(A)li
(I
-
e-a(A)w)
(B.61)
The total photocurrent density per unit wavelength is given by:
J(A) = Jh (A) + Je (A) + JSCR (A)
(B.62)
The spectral response is defined as:
SR (A)
=
1
[Jh (A) + Je (A) + JSCR (A)]
q 00 (A) (I - R (A))
The photocurrent density is given by:
84
(B.63)
AM
Jph
= q J, o (A) (1 - R(A)) SR(A) d
(B.64)
0
where Am is the longest wavelength that can pass though the photovoltaic without exciting a
photon.
Amn =
ch
Eg
(B.65)
where c is the speed of light, h is Planck's constant, and Eg is the bandgap of the material.
85
Bibliography
[1] "STARDUST Flight System Description,"
[http://stardust.jpl.nasa.gov/spacecraft/scsystems.html]
[2] "Advanced Technologies, Technology, DS1"
[http://nmp.jpl.nasa.gov/dsl/tech/scarlet.html
Nov, 1999
(3] Pan, J. L., Choy, H., Fonstad, C. G., Jr., "Very Large Radiative Transfer Over Small
Distances from a Black Body through Evanescent Modes." IEEE Transactions on Electron
Devices, 47, (1), pp.241, (2000)
[4] Madelung, Otfried, eds. "Semiconductors - Group IV Elements and III-V Compunds,"
Berlin: Springer. 1991
[5] Adachi, Sadao. "Optical Dispersion relations for GaP, GaAs, GaSb, InP, InAs, InSb,
AlGal-.As, and Ini,-GaAsyP1.y." Journal of Applied Physics, 66, 6037 (1989). Note:
All el functions should have 1 added to them.[6]
[6] Adachi, Sadao, "Model Dielectric Constants of GaP, GaAs, GaSb, InP, InAs and InSb,"
Physical Review B, 35, (14), 7454 (1987)
[7] Winston, David W., "Physical Simulation of Optoelectronic Devices," Diss. University of
Colorado, 1996. This document can be downloaded at:
[http://www-ocs.colorado.edu/SimWindows/download.html]
[8] Fang, Z. M., et al., "Photoluminescence of InSb, InAs, and InAsSb Grown by Organometal-
lic Vapor Phase Epitaxy" , Journal of Applied. Physics, 67, (11), 7034 (1990)
86
[9] Milnes, A. G., Polyakov, A. Y. "Indium Arsenide: a Semiconductor for High Speed and
Electro-optical Devices." Material Devices and Engineering, B18, 247 (1993)
[10] Madelung, Otfried, eds. "Semiconductors - Basic Data,"
2
"d ed., Berlin: Springer. 1996.
pg 138
[11] Shalyt, S., "Charge and Heat Transfer in n-type Indium Arsenide at Low Temperatures,"
Soviet Physics- Solid State, 4, (7), 1403. (1963)
[12] Madelung, 302
[13] Harman, T.C., Goering, H.L., Beer, A.C., "Electrical Properties of n-type Indium Ar-
senide", Physical Review, 104, (6), 1562 (1956)
[14] Folberth, Von 0. G., Madelung, 0., Weiss, H., "Die elektrischen Eigenschaften von Indiumarsenid II," Zeitschrift fur Naturforschung, 9a, 954. (1954)
[15] Adachi, S., "Physical Properties of III-V Semiconductor Compounds: InP, InAs, GaAs,
GaP, InGaAs, and InGaAsP," New York: John Wiley & Sons. 1992, pg 234.
[16] Adachi, pg 86-87
[17] Adachi, pg 247
[18] Esina, N.P., Zotova, N.V., "Mechanisms of Recombination of Excess Carriers in Indium
Arsenide and its Solid Solutions," Soviet Physics - Semiconductors, 14, (2), 185 (1980).
(Fizika i Tekhnika Poluprovodnikov, 14, 316 (1980))
[19] Andrushko, A.I., Salikhov, Kh.M., Slobodchikov, S.V., "Mechanisms of Recombination in
Indium Arsenide Crystals," Soviet Physics - Semiconductors, 20, (3), 255 (1986). (Fizika
i Tekhnika Poluprovodnikov, 20, 403 (1986))
[20] Vodopyannov, K.L., Graener, H., Phillips. C.C., Tate, T.J., "Picosecond Dynamics and
Studies of Auger Recombination Processes in Indium Arsenide at Room Temperature."
Physical Review B, 46, (20), 46. (1992)
[21) Varshni, Y.P., "Band-to-Band Radiative Recombination in Groups IV, VI, and III-V Semiconductors(I)," Physica Status Solidi, 19, 459 (1967).
87
[22] Gonzalez, B., et al., "An Energy Relaxation Time Model for Device Simulation," Solid
State Electronics, 43, 1791. (1999)
[23] Hilsum, C., Rose-Innes, A.C.,. "Semiconducting III-V Compounds," Oxford: Pregamon
Press. 1961, pg 115-119
[24] Kizhaev, S. S., Mikhailova, M. P., Molchanov, C. C., Stoyanov, N. D., "Growth of InAs
Photodiode Structures from Metalorgranic Compounds," Technical Physics Letters, 24,
247 (1998)
[25] Kuan, C.H., Lin, R., Tang, S., "Analysis of the Dark Current in the bulk of InAs Photode-
tectors," Journal of Applied Physics, 80, (9), 5454. (1996)
[26] Lin, R., Tang, S., "Room Temperature Unpassivated InAs p-i-n Photodetectors grown by
Molecular Beam Epitaxy," IEEE Transactions on Electron Devices, 44, 211 (1996)
[27] Sze, S.M., "Physics of Semiconductor Devices,"
2 nd
ed., New York: John Wiley and Sons.
1981. pg 807
[28] Millea, M. F., McColl, M., Silver, A. H., "Electrical Characteristics of Metal/InAs Contacts", Journal of Electronic Materials, 5, (3), 321, (1976)
[29] Ingrey, S., "III-V Surface Processing," Journal of Vacuum Science Technology A., 10, (4),
829, (1992)
[30] Dixon, J., "Photoelectromagnetic Effect in Indium Arsenide," Physical Review, 107, (2),
374, (1957)
[31] Sze, S.M., "Physics of Semiconductor Devices,"
2 nd
ed., New York: John Wiley and Sons.
2 nd
ed., New York: John Wiley and Sons.
1981, pg 88
[32] Sze, S.M., "Physics of Semiconductor Devices,"
1981
[33] Bhattacharya, "Semiconductor Optoelectronic Devices," New Delhi: Prentice-Hall of India,
Private Ltd. 1994, pgs. 413-430
88