PARAMETER SENSITIVITY ANALYSIS OF PHOTON
RECYCLING IN GALIUM ARSENIDE SOLAR CELLS:
METHODOLOGICAL DEVELOPMENT
GRACE CAREY, ILYA KORSUNSKY, ARJUNEN KUTAYIAH,
KATHLEEN MCGOVERN, LAUREN SWADDELL
Outline
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Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
U.S. Energy Consumption and
Production predictions
Source: U.S. Energy Information Administration, Annual Energy Outlook 2011, Early Release, December 16, 2011.
U.S. energy consumption in 2009
U.S. Primary Energy Flow
Source: U.S. Department of Energy, Department of Fossil Fuels, 2011
Carbon Dioxide Emissions
Ice Core Data and The Keeling Curve
Vostok Ice-Core Data
Alternatives to fossil fuels?
The suspense is
terrible… I hope
it’ll last
Nuclear Energy
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400 nuclear plants in the world
100 nuclear plants in the US alone
Powers ~15% of US energy needs
Relies on the use of uranium and
other fissible materials to generate
electricity
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Uranium is a finite mineral resource
Cooling methods often employ the
use of local water systems
endangering aquatic life
Nuclear power plants in the US
produce 2000 metric tons of
radioactive waste
Nuclear disasters can emit large
amounts of radiation which can be
lethal and detrimental to the
environment
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Solar Energy
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Sustainable and renewable resource
which does not emit greenhouse gases
~1% of U.S. energy
Solar energy hitting the earth is
approximately 274 million gigawatt/year = 8.2 million quads of
Btu/year
Solar cells currently have an average
efficiency of 15%  369 thousand
quads of Btu/year can be collected if all
land mass of earth had solar panels
Total potential for solar energy is
444,000 TWh
The world’s total energy consumption is
132,000 TWh
The total annual energy consumption in
the US is less than 0.5% the theoretical
amount of sunlight received
Solar Cell Efficiency Tables
Solar Cell Efficiency Tables
Solar Cell Efficiency Tables
Solar Cell Efficiency Tables
Outline
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Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
N type
Contacts
P type
N type
Photon
Electric field
Space
Charged
Region
P type
N type
P type
Radiative Recombination
Conducing
Band, Ec
Band Gap,
Eg = Ec - E v
Valence Band, Ev
Photon Recycling
•Re-absorption of photons generated in a semiconductor
device as a product of radiative recombinations.
•Increases efficiency by 1-2%
GaAs
•Semiconductor
•Direct Band Gap
•No energy is lost to phonons (lattice vibrations) as a result of
radiative recombinations.
•Good for optical devices
Outline
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Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
Modeling: Motivation
• Goal: create the best solar cell we can!
– Efficacy
– Cost
– Environmental Impact
• Need some design guidelines
• Computational model handles complexity
The Model
The Model
• Output: Photon Recycling Rate
• Inputs:
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Temperature
Front Surface Reflection
Width
Angle of Incidence
Refractive index
Light Wavelength
Internal Surface Reflectance
Reflectance of Metal Grid
Front Internal Shadow Factor
How do we use the model?
• Optimize Photon Recycling over the input
parameters
Dealing with Complexity
• 9 input parameters => 9 dimensional
hypercube
• Are all the parameters important?
• Sensitivity analysis gives importance of each
parameter
• Cut down search space
Outline
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Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
The simplest and most powerful
relationship between independent and
dependent variables is linear.
The dependent variable can be
predicted from the independent
variable by fitting the data to as
follows:
The problem is almost always more
complicated.
If the dependent variable is a function of multiple
independent variables, we have:
This describes multilinear dependencies for only
one sample; for n samples y can be written as a
column vector and the values of x form the rows
of matrix X:
In multiple linear regression, the
solution for the b vector take the form:
Can anyone see a potential problem here?
 The formula for b depends on the invertability of
the product matrix of the X row vector and the X
matrix!
Partial least squares (PLS) regression
does not depend on the invertability
of input data.
Assume a linear relationship between
independent parameter matrix X and dependent
output matrix Y:
PLS regression uses a variation of the NIPALS
algorithm to find the best approximation of this
linear relationship in the form of regression matrix,
B.
What does the PLS algorithm look like?
The Model
The previous complexity can be
reduced to the following:
The regression coefficients (Bpls) can
give us the following information:
(1) Significance of independent parameters to output(s) of
interest
(2) Prediction of dependent parameters from independent
parameters (unlike PCA)
(3) Indication of parameters to be tested in future
experiments
(4) Unreasonable results indicate that a mathematical
model needs to be reevaluated in some regard
Outline
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Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
The Model
Photon recycling rate (GPR): function map
KEY
∞
1
EG
-1
GPR (x) = 2π ∫d E ∫ dμ α
b(E,x, μ) =
{
2αw
Φ= exp
μ
ΨOF = RR
ΨOR = RF
∫
∫
α
+ μ exp
α
- μ exp
- RF RR
αx'
bn exp μ
(
α x'
bn exp μ
(
(
-αx
μ
(
-αx
μ
b(E, x, μ)
Functions
)x [
RF
Φ ΨOF +
)x[
RR
Φ ΨOR +
bn(E,x) =
)dx’ + exp(
2
h 3c2
2αw
μ
)dx’ +∫ b exp(
n
Experimental variables
Constants
∞
∫b
0
n
exp
w
∫b
n
exp
x
(
α x'
μ
) dx’ ] if 1 ≤ μ < 0
(
α x'
μ
) dx’ ] if 0 > μ ≥ -1
ˆn2 E 2
E -qφ(x)
( kT ) - 1
)∫ b exp (
n
- α x’
μ
) dx’
α=
- α x’
μ
4 log(10) π κ
λ
μ = cosθ
) dx’
E=
h*c
λ
RF = κF * FSF + ρF * (1 – FSF)
Photon recycling rate (GPR): function dependency chart
GPR
E
α
Φ
RF
μ
RF
b
μ
bn
E
α
μ
α
μ
α
ΨOF
ΨOR
RF
bn
bn
E
E
Photon recycling rate (GPR): code sample
Outline
•
•
•
•
•
•
•
•
Motivation: Environmental Impact
Solar Cells: Behind the Physics
Modeling and Optimization
Sensitivity Analysis (PLS regression)
Implementation
Results
Design Conclusions
Future Directions
We can apply the PLS algorithm to our
input and output data.
Output
Trials
Input
1.5
0
-1.5
GPR
Parameters
Results of PLS regression:
BPLS
Input Matrix
Output
k
1.5
l
kF
r
*F
=
SF
0
W
q
nhat
-1.5
T
GPR
GPR
Results of PLS Regression:
GPR Regression Coefficients
1
0.5
These give quantitative insight
into how changing input
parameters affects output.
0
-0.5
-1
Significant parameters include
wavelength of light, temperature,
and the front reflectance.
Accuracy of regression
Predicted vs True G
PR
2
Predicted GPR
True GPR
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
  
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Predicted GPR
1
0
-1
-2
-2
-1
0
True GPR
1
2
Conclusions:
1. PLS regression is an accurate tool for both determining
parameter sensitivities from our simulated data sets and
predicting the output variable data of interest.
2. As conserving energy is of optimal interest to the environment,
photon recycling is an important physical phenomenon to
energy conservation and solar cell efficiency.
3. From our regression analysis, the parameters which should be
maximized in future cell design are wavelength of light directed
at the solar cell, temperature, and front reflectance.
Future Directions
• Function for cost
• Function for environmental impact
• Convex optimization
Questions
• Any?
• No?
• Thanks!
Acknowledgements
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The Catalyst Scholarship Program!
Dr. Haydee Salmun
All of our wonderful advisors
Dr. Eric Sobie and Amrita Sarkar