PARAMETER SENSITIVITY ANALYSIS OF PHOTON RECYCLING IN GALIUM ARSENIDE SOLAR CELLS:

METHODOLOGICAL DEVELOPMENT

GRACE CAREY, ILYA KORSUNSKY, ARJUNEN KUTAYIAH, KATHLEEN MCGOVERN, LAUREN SWADDELL

Outline

• 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• 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

• 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

Solar Energy• Sustainable and renewable resource

which does not emit greenhouse gases• ~1% of U.S. energy • Solar energy hitting the earth is

approximately 274 million giga-watt/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

• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions

N type

P type

Space

Charged

Region

Electric field

N type

P type

N type

P type

Contacts

Photon

Valence Band, Ev

Conducing Band, Ec

Band Gap, Eg = Ec - Ev

Radiative Recombination

GaAs•Semiconductor •Direct Band Gap

• No energy is lost to phonons (lattice vibrations) as a result of radiative recombinations.

• Good for optical devices

Photon Recycling•Re-absorption of photons generated in a semiconductor device as a product of radiative recombinations. •Increases efficiency by 1-2%

Outline

• 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:

– 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

• 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

• Motivation: Environmental Impact• Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression)• Implementation• Results• Design Conclusions• Future Directions

The Model

GPR (x) = 2π ∫d E ∫ dμ α b(E, x, μ)∞

EG

bn(E,x) = 2

h3c2

n2 E 2

(E -qφ(x)

) - 1kT

ˆΦ= exp

2αwμ

- RF RR

ΨOF = RR ∫bn exp(αx'

)μdx’ + exp(

2αw

)μ∫ bn exp(

- α x’

)μdx’

ΨOR = RF ∫bn exp(α x'

)μ dx’ +∫ bn exp(

- α x’

)μdx’

b(E,x, μ) = {+ exp x [ ΨOF + ∫ bn exp dx’ ] if 1 ≤ μ < 0

- exp x [ ΨOR + ∫ bn exp dx’ ] if 0 > μ ≥ -1

αμ (

- α x

)μ

RF

Φ (α x'

)μαμ (

- α x

)μ

RR

Φ (α x'

)μ

1

-1

∞

0

x

w

Photon recycling rate (GPR): function map

α =4 log(10) π κ

λ

RF = κF * FSF + ρF * (1 – FSF)

E = h*c

λ

μ = cosθ

Experimental variablesConstants

Functions

KEY

Φ RF μ α ΨOF ΨOR bn

E b μ α

GPR

α μ RF bn μ α RF bn E

E E

Photon recycling rate (GPR): function dependency chart

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.

Parameters

GPR

Output

k l k F rF SF W q

n hat T

Tria

ls

Input

-1.5

0

1.5

Results of PLS regression:

k l k F rF SF W q

n hat T

Input Matrix

GPR

Output

-1.5

0

1.5

GPR

BPLS

k

l

kF

r

FSF

W

q

nhat

T

* =

Results of PLS Regression:

-1

-0.5

0

0.5

1

k l k F rF SF W q

n hat T

GPR Regression Coefficients

These give quantitative insight into how changing input parameters affects output.

Significant parameters include wavelength of light, temperature, and the front reflectance.

Accuracy of regression

-2 -1 0 1 2-2

-1

0

1

2

True GPR

Pre

dic

ted G P

R

Predicted vs True GPR

Predicted GPR

True GPR

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

• The Catalyst Scholarship Program!• Dr. Haydee Salmun• All of our wonderful advisors• Dr. Eric Sobie and Amrita Sarkar