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An expert system approach to the optimal design of single-junction and multijunction tandem solar cells

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An expert system approach to the optimal design of single-junction and multijunction tandem solar cells
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Yeh, Chune-Sin, 1955-
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English
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vii, 194 leaves : ill. ; 28 cm.

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Cells ( jstor )
Diffusion length ( jstor )
Doping ( jstor )
Electric current ( jstor )
Electrons ( jstor )
Expert systems ( jstor )
Integers ( jstor )
Minority carriers ( jstor )
Photovoltaic cells ( jstor )
Protons ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph.D
Expert systems (Computer science) ( lcsh )
Heuristic programming ( lcsh )
Semiconductors -- Junctions ( lcsh )
Solar cells ( lcsh )
Structural optimization ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Chune-Sin Yeh.

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AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS












By
CHUNE-SINYEH


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA
1988


-OFs~rr or FLRMIDt oj














ACKNOWLEDGEMENTS


The author wants to express his deep gratitude to the chairman of his supervisory committee, Dr. Sheng S. Li, for his guidance and encouragement through the entire research process. The author also appreciates the other members of his supervisory committee, Drs. Arnost Neugroschel, Dorothea Burk, Gijs Bosman and Yuan-Chieh Chow for their participation on the committee.
Special thanks are given to Dr. R. Y. Loo for the irradiated solar cell samples and measurements and to Mrs. Li, his host family Mr. and Mrs. David Wilmot, and Mrs. Anne White for their kindness to his family. The author is also grateful to his colleagues at the Device Characterization and DLTS Lab for their helpful discussions. The financial support of the Universal Energy Systems Incorporation is greatly appreciated.
Finally, the author thanks his parents and family, especially his late father, for their love, expectation and patience throughout his graduate study.














TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .................................................... ii

A B STR A CT ................................................................. vi

CHAPTERS

ONE INTRODUCTION .................................................... 1

1.1 Introduction ...................................................... 1
1.2 Fabrication Technology of Solar Cells .............................. 2
1.3 A Synopsis of this Research ....................................... 3

TWO MODELING OF PROTON AND ELECTRON IRRADIATED
SO LAR CELLS ...................................................... 6

2.1 Introduction ...................................................... 6
2.2 Displacem ent Defects .............................................. 7
2.2.1 Defect Formation by Proton Bombardment ................. 8
2.2.2 Defect Formation by Electron Bombardment ............... 10
2.3 Degradation Calculation of Short-Circuit Current ................. 13
2.4 Degradation Calculation of Open-Circuit Voltage ................. 15
2.5 Degradation Calculation of Conversion efficiency .................. 16
2.6 Results and Discussion ........................................... 17
2.7 Sum m ary ........................................................ 20

THREE A NEW METHOD FOR OPTIMAL DESIGN OF GAAS
SINGLE-JUNCTION SOLAR CELLS ............................... 36

3.1 Introduction ..................................................... 36
3.2 Device Modeling for GaAs P/N Junction Solar Cells .............. 37
3.3 Effects of Extrinsic Parameters on Device Modeling ............... 40
3.3.1 Effect of Antireflection Coating ............................ 41
3.3.2 Effect of Grid Design ...................................... 41
3.3.3 Effect of Series Resistance ................................. 42









3.3.4 Effect of High Sun Insolation .............................. 43
3.3.5 Effect of Irradiation ....................................... 43
3.4 Constrained Optimization Technique ............................. 43
3.5 Optimal Design of GaAs Single-Junction Solar Cells .............. 45
3.6 Sum m ary ........................................................ 47

FOUR AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN
OF MULTIJUNCTION SOLAR CELLS ............................. 56

4.1 Introduction ..................................................... 56
4.2 Device Modeling of Multijunction Tandem Solar Cells ............ 57
4.3 Concept of the Expert System Approach ......................... 59
4.3.1 Problem Formulation ...................................... 60
4.3.2 Optimization and Heuristic Rules ......................... 63
4.4 Results and Discussion ........................................... 64
4.5 Sum m ary ........................................................ 67

FIVE THEORETICAL CALCULATIONS OF ELECTRON AND HOLE
M O BILITIES ....................................................... 78

5.1 Introduction ..................................................... 78
5.2 Scattering Processes of Al.Gai__As .............................. 79
5.2.1 Polar Optical Scattering ................................... 79
5.2.2. Piezoelectric Scattering ................................... 81
5.2.3. Deformation Potential Scattering ......................... 81
5.2.4. Ionized Impurity Scattering .............................. 82
5.2.5. Space Charge Scattering .................................. 82
5.2.6. Alloy Scattering .......................................... 83
5.2.7. Intervalley Scattering ..................................... 83
5.3. Analysis and Discussion ................................. 84
5.4. Sum m ary ....................................................... 87

SIX SUMMARY, CONCLUSION and RECOMMENDATIONS ........... 97

6.1. Summary and Conclusion ....................................... 97
6.2. Recomm endations ............................................... 99

APPENDIX A A COMPUTER PROGRAM FOR CALCULATING THE
TOTAL NUMBER OF DISPLACEMENT DEFECTS ...... 101

APPENDIX B A COMPUTER PROGRAM FOR CALCULATING THE
DEGRADATION OF SHORT-CIRCUIT CURRENT ....... 111








APPENDIX C


APPENDIX D


AN EXPERT SYSTEM PROGRAM FOR OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS ................................ 143

INPUT PARAMETERS FOR THE OPTIMAL DESIGN


OF (ALGA) AS/GAAS/INo.53GAo.47,AS
THREE-JUNCTION SOLAR CELLS ...................... 182

REFERENCES ............................................................. 188

BIOGRAPHICAL SKETCH ................................................ 194













Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF
SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS By

Chune-Sin Yeh
August 1988

Chairman: Sheng-San Li
Major Department: Electrical Engineering
The use of an expert system approach to the optimal design of single-junction and multijunction solar cells is a potential new design tool in photovoltaics. This dissertation presents the results of a comprehensive study of this new design method. To facilitate the realistic optimal design of the two-terminal monolithic single-junction and multijunction tandem solar cells, a rule-based system was established by adopting the experimental data and/or semiempirical formulae used today for those design parameters. A numerical simulation based on the displacement damage theory was carried out to study the degradation of AlGaAs/GaAs solar cells after proton or electron irradiation. The damage constant of the minority carrier diffusion length, an important design parameter of a solar cell for space application, was calculated. An efficient Box complex optimization technique with minor modifications is analyzed and applied to accelerate the convergence rate of the algorithm. Design rules were implemented in order to reduce the search space of the optimal design and to make a compromise in the tradeoff between the conflicting criteria for selection. The computation time for the optimal design is very much reduced by adopting these








rules. Realistic optimal design of single-junction and multijunction solar cells were obtained and verified from the expert system and then compared with the state of the art technology. Finally, theoretical calculations of the electron and hole mobilities of Al.Gal-,As were performed. The results show that electron mobilities of Al.Gal_ As are quite sensitive to the Al composition, temperature, doping density and defect density.. This knowledge is indispensable to the fabrication of a high efficiency AlGaAs top cell of a GaAs based multijunction solar cells.














CHAPTER 1
INTRODUCTION


1.1 Introduction

For the past three decades, researchers around the world have been involved in studying the degradation of solar cells induced by energetic electrons or protons for space applications. As the number of satellite launches has increased dramatically, the need for a space power system using solar cells has increased significantly. Moreover, by knowing the degradation of performance in the electron or proton irradiated solar cells, one is able to design a solar cell that gives an optimum end of life conversion efficiency for the specific space mission. This should lead to innovations in design and fabrication such as shallow junctions, window layers and multijunction solar cells.
The objective of this dissertation is to use an expert system approach to study the optimum design of single-junction and multijunction tandem solar cells for both space and terrestrial applications. It should be noted that because the theoretical conversion efficiency increases very slowly beyond three-junction tandem cells [1] and the technology of fabricating cells beyond three junctions is difficult in practice [2], the multijunction study in this work was focused only on the two-terminal monolithic structures of single-junction, two-junction and three-junction solar cells.
Although recently multijunction solar cells have become increasingly important for space applications, the multijunction cell structure in itself is not a new concept. In 1955, Jackson [3] proposed that the efficiency of solar cells could be increased significantly by stacking one or more cells composed of different semiconductor materials. Since then, many other researchers [4-26] have proposed varying approaches to the optimal design of the multijunction cells, theoretically and/or experimentally.








However, little progress in multijunction solar cell design has been made due to (1) lack of reliable data on the electrical and optical parameters of the solar cell materials,
(2) lack of adequate device modeling, (3) inefficient optimization algorithm. In an effort to provide an optimal and realistic design of multijunction solar cells for space and terrestrial applications, the research reported here makes use of an expert system approach.
In this chapter an overview of the major solar cell fabrication technologies now in use is presented. Different processes may have varying effects on the quality and efficiency of solar cells. Finally, a synopsis of our research is given.

1.2 Fabrication Technology of Solar Cells

Although the potential performance advantages of the GaAs single-junction and III-V compounds based multijunction solar cells have been discussed [2-22], their practical realization remains a problem, due to the absence of a good fabrication technology. Among those fabrication processes developed for III-V compound, liquid phase epitaxy (LPE), molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) have proven successful [19, 27-31] and hence the use of the multijunction structure in solar cells has become important and promising.

However, since there is no single fabrication technique which is superior to the other two in terms of crystal purity, doping level, throughput and controllability of cell thickness and uniformity, it is still too early to say which of these three technologies will dominate the future development of single-junction and/or multijunction solar cells. Therefore, in this brief review, it seems most useful to describe the advantages and weaknesses of each fabrication process, but not to give details about the processes themselves.
LPE is the reference technology for the growth of epitaxial layer in III-V compounds. It offers high purity crystal and excellent transport properties, minimum contamination, uniform thickness control and reductions in point defect and disloca-








tion. It remains an easy and cheap method for growing high efficiency AlGaAs-GaAs solar cells. However, the throughput is small.
MOCVD is a standard system used for making high efficiency III-V solar cells. It is a flexible technology that allows the growth of a quite complicated multilayer structures with different doping levels and different compositions. It can grow very thin layers with precise doping and thickness. And MOCVD has large throughput. However, materials made by MOCVD may have carbon contamination and high defect density.
The advantages of MBE are similar to those of MOCVD. While, MBE has only a medium throughput, it can produce high purity crystal without contamination. In multijunction solar cells, the most important feature of MOCVD and MBE is the ease of incorporating superlattice or tunnel junction into the cascade solar cell structures. This can reduce the dislocation density in the top cell and hence reduce the recombination loss and increase output voltage.

1.3 A Synopsis of this Research

Based on the survey given in this chapter, it would appeare that the realistic optimal design of single-junction and multijunction solar cells could be realized through the establishment of the expert system. Moreover, it should be apparent that the lack of full knowledge of the material parameters and their relations - mobility, diffusion length, etc. - as well as the complexity of the optimization technique, would limit the development of a general analytical model which accounts for all variations of the design parameters. It is the goal of this research to generate an expert system for solving this optimal design problem.
The first step was to set up a rule-based knowledge data base for the design parameters listed above by adopting the experimental data and/or semiempirical formulae in use today. This reduced the complexity of the numerical simulation required to solve the problem. The next step was the development of a model of the









proton or electron irradiated solar cells. The model developed here is based on the displacement damage model. A computer program for calculating the degradations of short-circuit current, open-circuit voltage and conversion efficiency of a solar cell was coded. This program included the calculations of the damage constant of the minority carrier diffusion length. This parameter is important for the optimal designs of irradiated solar cells.

In addition, an efficient optimization algorithm was implemented for the expert system. Some heuristic rules were applied to reduce the search space of the design problem and to make some compromise in the tradeoff between the conflicting criteria of selection. Realistic optimal designs were obtained from the simulations and their performance compared with the state of the art technology.
In Chapter 2, a numerical model of proton and electron irradiated solar cells is presented. In this model it is assumed that the radiation induced displacement defects form an effective recombination center which reduces the minority carrier diffusion length and hence degrades the short-circuit current I,, open-circuit voltage V," and conversion efficiency 77, of the solar cell. It should be noted that although the numerical model applies to a special case which is limited to the normally incident protons or electrons, this model can be extended to simulate the real space environment where the incoming protons or electrons are omnidirectional.

In Chapter 3, a new computer model for truly optimizing the structure of GaAs single-junction solar cell for both space and terrestrial application is proposed. The model, however, can apply easily to other solar cell systems. It not only takes into account the effects of the intrinsic structural parameters such as junction depth Xj, cell thickness Tj, doping densities NA and ND, surface recombination velocities Sp and Sn, but also incorporates the extrinsic structure parameters. And an efficient Box optimization algorithm [32] with minor modifications is implemented.

In Chapter 4, an expert system approach to the optimal design of multijunction








solar cells for both terrestrial and space applications is described for the first time. An expert system is a knowledge-intensive computer program. The knowledge of an expert system consists of facts and rules. The facts constitute a body of information that is widely shared, publicly available and generally agreed upon by experts in the field [33-341. The rules are those if-then rules that characterize expert level decision making in the field. In general, a good and robust expert system must include as many facts and rules as possible. However, because of the availability of the earlier research reports and the tradeoff between the number of rules and the computation time, this expert system is currently limited to the AlGaAs, GaAs, Ino.53Gao.47As, Si and Ge materials.
In Chapter 5, the theoretical calculations of the electron and hole mobilities in Al.Gal__As axe reported. It was found that the mobilities of Al.Gal_,,As are quite sensitive to the Al composition, temperature, doping density and defect density. This knowledge is indispensable to the fabrication of a high quality and high efficiency AlGaAs top cell of the GaAs based multijunction solar cells. The final chapter, Chapter 6, summarizes the materials and offers suggestions for future research.
In addition to the six chapters in this dissertation, there are four appendices. In Appendix A, a computer program for calculating the total number of displacement defects induced by the energetic proton and electron bombardments is presented. A computer program for manipulating the degradation of short-circuit current, opencircuit voltage, conversion efficiency and damage constant of the minority carrier diffusion length is in Appendix B. An expert system program for optimal design of single-junction and multijunction solar cells is in Appendix C. Finally, the input parameters for (AlGa)As/GaAs/(InGa)As three-junction solar cells are shown in Appendix D.













CHAPTER 2
MODELLING OF PROTON AND ELECTRON IRRADIATED SOLAR CELLS


2.1 Introduction

The advantages offered by single junction GaAs solar cells for space applications are their high power conversion efficiency, their radiation hardness and their relative insensitivity to temperature compared to silicon solar cells. Although GaAs solar cells have not been extensively flown in space, radiation experiments performed on earth have demonstrated their superior hardness as compared to silicon cells. In this chapter we model the radiation degradation characteristics of GaAs single-junction solar cells and GaAs based multijunction cells under proton or electron radiation environments.
An improved numerical model is offered for computing the displacement defect density, the damage constants of the minority carrier diffusion lengths and the degradations of the short-circuit current I,,, open-circuit voltage Vo, and the conversion efficiency 77 in proton or electron irradiated single-junction and multijunction solar cells is presented. The model which we use in this study is based on the displacement damage theory in semiconductors. It is assumed that the radiation induced displacement defects form effective recombination centers which reduce the minority carrier diffusion length and hence degrade the I,, Vo, and 77 of the solar cell. It should be noted that although our numerical model applies to a special case, limited to the normally incident protons or electrons, this model can be extended to simulate the real space environment where the incoming protons or electrons are omnidirectional.
In earlier work, Wilson et al. [35] and Yaung [36] have also used the displacement damage model to study the GaAs solar cells. We have extended their work








and have obtained a better correlation between theory and experimental findings by including both electron and hole capture cross sections in our calculations. The electron and hole capture cross sections were determined for the GaAs and AlGaAs cells after proton or electron irradiation using deep level transient spectroscopy (DLTS) measurements. In addition, the model includes calculations of the normalized opencircuit voltage, conversion efficiency and damage constant of the minority carrier diffusion length. The damage constant of the minority carrier diffusion length for the irradiated solar cell is one of the important parameters in optimal design of a solar cell for space application. In addition to the single-junction GaAs solar cell, a simple model was developed for calculating the displacement damage in the proton and electron irradiated Al0 .Gao.67As/GaAs two-junction solar cell and in the Alo.3sGa.6sAs/GaAs/In.s3Ga.47As or Ge three-junction solar cell.
The calculations indicate that the degradation rate in each cell varies greatly and depends critically not only on the energy, fluence and the direction of the incident proton or electron, but also on the thickness of each cell in the multijunction cells. Excellent agreement was obtained between our calculated values and the measured I,, Vo and 9, in proton or electron irradiated AlGaAs and GaAs solar cells for proton energies from 100 KeV to 10 MeV and fluences from 1010 to 1012 cm-2, and for 1 MeV electron and fluences varying from 1014 to 1016 cm-2 under normal incidence conditions.

2.2 Displacement Defects

A solid may be affected in two ways by the energetic particle bombardment [37]. First, the lattice atoms may be removed from their regular lattice sites and produce displacement damage. Second, the irradiating particles may cause change in the chemical properties of the solid via ion implantation or transmutation. In our model, it is assumed that the dominant defect produced by the incident electrons or protons is due to lattice displacement. Under this assumption, an atom will invariably








be displaced from its lattice site during collisions if its kinetic energy exceeds the threshold energy, Td for the atomic displacement, and will not be displaced if its kinetic energy is less than Td [38].
2.2.1 Defect Formation by Proton Bombardment
When energetic protons collide with atoms, the energy transferred to the struck atoms is the most important consideration in evaluating irradiation damage [38]. The number of defects formed by an energetic proton coming to rest in the solar cell are related to the energy of the proton, the transferred energy and the threshold energy, Td of the solar cell. Given the conservation of energy and momentum, it follows that the maximum energy which can be transferred to the struck atom in a primary head-on collision with energy, E, is


4M1M2E
TM = (M1 + M2)2 (2.1)



where E is the initial energy of the proton and M1 and M2 are the masses of proton and struck atom respectively. This transferred energy may range from zero in a glancing collision to a maximum TM in a head-on collision. As for a proton, the energy transferred in a collision can be calculated by ignoring the screening effect. Therefore, the scattering in a proton collision obeys the Rutherford differential cross section dap which is given by


dop=CdT (2.2) dT
dup = C- (.2
2 M1Z2Z2 Er
C = 41ra0 2-' Z Z2 (2.3) M2 E



where T is the transferred energy. M1 and M2 are the same as those in equation 2.1, Z1 and Z2 are the atomic number of the proton and struck atom respectively, Er is








the Rydberg constant and ao is the Bohr radius. Since the defects occur when the energy transferred is greater than Td, the displacement cross section Orp is given by

fTM 1 1
Ip = d dap = C(Td TM) (2.4)


The average energy transferred, T, in a Rutherford collision which displaces atoms can be calculated as follows:


_ fw Tdap
fTm dap
_ TdTM ln(TM) (2.5) TM - Td n(


If the transferred energy is sufficiently large (T >> Td), additional displacement can be produced by the recoiling nucleus before it comes to rest at an interstitial site. The average recoil displacements , v, produced by one initiating proton collision event is given as a function of TM, on the assumption that half of the recoil energy produces further displacement and the other half is dissipated in other processes described in [35, 381


1.0 for Td < TM < 2Td (2.6) Vp(E) =-(26
{1 ln(2-d ) for 2Td <= TM 1+2(TM-Wd) T


Since the mass of the proton is heavier than that of an electron, the velocity of an energetic proton is slower than an electron with the same energy. Thus, a proton has the potential of multiple scattering before coming to rest. The displacement defects D(E0) induced by an energetic particle under normal incident condition are


DP(Eo) = NapVpdP (2.7)










where P is the path length traveled by a proton in coming to rest, and all the other variables have the same definitions as in Eqns. 2.4 and 2.6. As for the omnidirectional incident irradiation, the total displacement defects Do. are given by [39]


DPo. = 27r jF dE j d(cos 0)[DP(Eo) - DP(Eo(x/ cos 0))] (2.8)



where 0 is the incident angle of the particle toward the solar cell.
2.2.2 Defect Formation by Electron Bombardment
Because of the small mass of an electron, the electron must travel at a relatively high velocity in order to produce displacement. The maximum energy which can be transferred in a collision by an electron with mass m and kinetic energy E is


TM = 2(E + 2mC2)E (2.9) M2C2


where C is the velocity of light, and m and M2 are masses of the electron and struck atom respectively. Consequently, the nonrelativistic Rutherford scattering is inadequate for the electron. Relativistic Coulomb scattering has been treated by MckinleyFeshbach as follows [40]:


= 47r(aZ2Er)2(1 - 2) 1 2T +T T )/137]TM _T (2.10)
d'e - m2C434 TM + 7rZ2F( M - T 2



where # is the electron velocity ratio to the velocity of light C. All other variables have the same definition as those in the previous equations. Integration of equation

2.10 yields the displacement cross section for an incident electron:










4ir(aoZ2E,)2(1 - /32) TM _TaM
m2C44 [T-1 - 2 ln(TITM) +27rafl( yT -1)- 7rai ln(TM/Td)] (2.11)


where a is equal to Z2/137. The average energy transferred during a collision is


T fTm Td e
T l = M dae
TM ln(TM/Td) - #2(TM - Td) + 27ra6(TM - VTm --Td) - 7ra3(TM - Td)
TM/Td - 1 - /2 ln(T/TM) + 2ra( L- 1)- ra ln(TM/Td)
(2.12)




Thus, if the transferred energy is sufficiently large (T >> Td), the mechanism of recoil displacement produced by an electron is similar to that of a proton. The average recoil displacements, V., produced by one initiating electron collision event are given [35, 38]


0.0 for T(E) < Td
Ve(E) = 1 for Td < T < 2Td (2.13)

1 + T(E) for 2Td < T(E)
2Td

The displacement defects D(Eo) induced by an energetic electron under normal incident condition are


DE(Eo) = NaeVdR (2.14)


where R is the penetration depth of an electron in coming to rest. Since the electron








mass is small, we can neglect the effects of multiple scattering. As for omnidirectional incident irradiation, the total displacement defects Dam are given by [39]


DEom = 27r jo dE f1 d(cos 0)[DE(Eo) - DE(Eo(x/ cos 0))] (2.15)



Now, according to Eqns. 2.7, 2.8, 2.14 and 2.15, for numerical calculations of the total number of displacement defects, the threshold energy, path length, penetration depth (range) and the reduced energy of protons or electrons after travelling an x distance must be given. Although the threshold energies for GaAs and Ge are given as 9.5 eV and 27.5 eV respectively, those of InGaAs and AlGaAs are still unknown. In this respect, a linear extrapolation has been made to calculate the two unknown values as follows:



Ta(Al.Ga..As) = 0.5 x [Td(Al) x x + Td(Ga) x (1 - x) + T(As)] (2.16)



and
Td(In.Gal-.As) = Td(InAs) x x + Td(GaAs) x (1 - x) (2.17)



In addition to the threshold energy, the path length or range an electron or proton coming to rest is also unknown. Thus, we adopt the path length and the range from the data given recently by Janni [41] for the proton case and by Page et al. [42] for the electron case. Since these figures are only given for elements, and the GaAs, (AlGa)As and (InGa)As are compound materials, approximations were made in calculating path lengths and ranges based on the following assumptions [411:


1 W. (2.18) PP








1 =Ewi (2.19) R = Ri



where P, and R, are the path length and range of the compound materials respectively, R, and Pi are data for each element and Wi is the weighting function of each element. The least square method was employed to fit those data in order to obtain the expressions for P and R for the solar cells studied here.
As for the reduced energy, E, of protons or electrons after penetrating a distance x, the fitting process is similar to those for path length and range. These empirical formulae were obtained through the computer programs described in Appendices A and B.

2.3 Degradation Calculation of Short-Circuit Current

To derive an expression for the short-circuit current in an irradiated cell, the following simplified assumptions were made [35, 43, 44]: (1) radiation-induced defects do not greatly alter internal cell electric field, (2) radiation-induced defects alter the cell operation mainly through change in minority carrier lifetimes in the bulk, and

(3) radiation-induced displacements within the solar cell form recombination centers for minority carriers of electron-hole pairs produced by photon absorption.
When sunlight impinges on a solar cell, the short-circuit current generated from a solar cell is given by

t
,..(A) = (x)p(x, A)dx (2.20)



where 77(x) is the current collection efficiency, t is the cell thickness, and


p(x) = Ka exp(-ax)


(2.21)










is the photon generation rate at depth x, K is the integrated solar photon flux in the absorption band and a is the absorption coefficient.
After proton or electron irradiations, defects are created within the materials and the solar cell's short-circuit current is decreased. To rewrite the expression for the short-circuit current after proton or electron irradiation, I' , we need to add a loss term [1 - F(x)] to the integral of Eqn. 2.20, to account for the carrier recombinations caused by the newly generated defects. The expression for the recombination loss coefficients, F(x), is [35]


F(x) = 1 - E2[VNOarO[D(Ex) - D(E-)I] (2.22)



and


V = 1A) 7(x)(1 - F(x))p(x,A)dx (2.23)



where E2(z) is the exponential integral of order 2, ar is the electron or hole capture cross section, � is the proton or electron fluence, E. is the reduced proton or electron energy after penetrating a distance, x and xj is the junction depth.
The normalized I., degradation can thus be calculated by using the expression


Ic = A2 I(A)dA
IJ0- ' X IL ,(A)dA (2.24)


A1 and A2 denote the shortwave and long-wave limits of the total useful solar spectra for the solar cell.





15

2.4 Degradation Calculation of Open-Circuit Voltage

Once the damage constants of the minority carrier lifetimes or diffusion lengths are known, calculations of Vo, degradation are straightforward. According to the Shockley, Read and Hall theory [45], the minority carrier lifetime is inversely proportional to the defect density, Nt and is given by [45]

1
='' (2.25) np-NtVtha.,p (.5



where Vth is the thermal velocity and a,,p is the capture cross-section. Thus, the minority carrier lifetime of the solar cell after proton or electron irradiation, rp,n', can be calculated from Eqn. 2.25 by substituting the values of the displacement damage density Nt and the capture cross-sections from the DLTS measurements. It should be noted that Nt can be calculated directly from Eqns. 2.7, 2.8, 2.14 and 2.15 for proton and electron irradiations respectively. Knowing 'p,', the damage constants of the minority carrier lifetimes KTp and K , are given by [46, 47] as


1 1
+ K (2.26)
1 1
+KT, ' (2.27)



Now using the relation that L' = D r, it is easy to obtain the damage constants of the minority carrier diffusion lengths KLn and KLp. The open-circuit voltage before irradiation is expressed by


_ nKBT ln(- + 1) (2.28) q 10


where n is the diode ideality factor and 10 is the saturation current of the diode. KB is








the Boltzman constant and q is the electron charge. In general, the current conduction mechanism through the diode is dominated by diffusion or by recombination currents. When the diffusion current dominates, the diode's ideality factor, n, is equal to 1, and 1o is given by


10 = (qn A) LD + LDn (2.29)



where ni is the intrinsic carrier density, A is the area of a solar cell, Dp and D, are the hole and electron diffusion coefficients, respectively, Lpo is the hole diffusion length and Lno is the electron diffusion length and ND and NA are the donor and acceptor dopant densities. When the recombination current at the junction dominates, the ideality factor, n, is equal to 2 and the saturation current, 10, is


0- qiWA (2.30)



where W is the depletion width, and rp and r are the hole and electron lifetimes in the n-region and p-region, respectively. In general, the n value will vary between 1 and 2 because both mechanisms contribute to the transport mechanism.

As for the open-circuit voltage of the irradiated cell, V'o, it is given by V' nKBT lnI"
o = q 1In( +1) (2.31) q 0
where I'0 is the saturation current after irradiation.


2.5 Degradation Calculation Of Conversion Efficiency

The maximum power conversion efficiency of a solar cell is the product of the short-circuit current, open-circuit voltage and the fill factor. Experimentally it has








been observed that the fill factor of the cell remains unchanged after irradiation [48]. Thus, the normalized conversion efficiency after proton or electron irradiation is


-7C _ Vc (2.32) 7lco IscoVoco

2.6 Results and Discussion

Figure 2.1 shows the baseline design of the GaAs solar cell. The GaAs and Al0.33Ga0.67As solar cells were fabricated at Hughes Research Laboratories (HRL) by using infinite solution LPE technique. The base region of the solar cell is n-type (1 X 101' cm-3). A wide bandgap, p+, (2 x 1018 cm-3), Be doped AlGaAs window layer was grown on the top of the base layer to passivate the GaAs or Alo.33Gao.67As surface and to reduce its surface recombination velocity. During the growth of the window Be was diffused into the base region to form an electrical junction. The thickness of the window layer and the junction depth were measured to be 0.35 and 0.5 pm, respectively. The GaAs and Alo.33Ga0.67As solar cells' conversion efficiency, before the irradiation, were tested at AMO, 1 sun condition to be 16.0% to 17% and 7.5% to 8.0 % respectively. The proton irradiation experiments for various proton energies and fluences were performed at HRL (low energy protons), California Institute of Technology (medium energy protons) and Univeristy of California, Davis (high energy protons). The details of the experiments and results were given in earlier publications [49-53].
Table 2.1 lists all the cell parameters used in calculating the degradation of the short-circuit current and these parameters are identical to the actual GaAs and Al0.33Ga.67As solar cells used in the proton and electron irradiation experiments. Figures 2.2 and 2.3 give the calculated proton range and the number of displacement damage defects per incident proton as a function of proton energy. For example, a 300 KeV proton will penetrate 2 pm deep into GaAs solar cell (Fig. 2.2) and produces about 100 defects per cm3 (Fig. 2.3). Figure 2.4 shows the remaining short-circuit








current after proton irradiation as a function of the proton energy. The solid lines are the calculated short-circuit currents and the circles are the experimental values. Examining both Fig. 2.2 and Fig. 2.4, it is clear that only proton energies greater than 50 KeV are seen to create damage to the cell. The 100 KeV protons, which are stopped at about 0.8 pm below the surface, create damage close to the junction. The 200 KeV protons, which are stopped at about 1.35 am below the surface cause damage throughout the p-region and the junction. The 290 KeV protons, which are stopped at about 2.0 pm below the surface produce damage in the bulk of the n-GaAs layer. Protons with energies greater than 1 MeV will pass through the entire cell, hence, will create less damage to the cell than the low energy protons. Thus, the calculated short-circuit current agrees closely with the experimental values.
The damage constants of the minority carrier diffusion lengths, KLn and KLp, were also deduced from these calculations and are given in Table 2.1. The opencircuit voltage after proton irradiations can then be calculated as a function of proton energies and fluences. Figure 2.5 gives the normalized open-circuit voltage, VoC/Vo, as a function of the proton energy. Again, as with the results obtained for the shortcircuit current, the low energy protons near 200 KeV degrade the open-circuit voltage more severely than other energies. This is because the 200 KeV protons are stopped in the vicinity of the pn junction. From Fig. 2.5, it can be seen that the theory correlates well with the experimental data. It shows that the modeling technique is valid for the GaAs case.
Table 2.3 summarizes the calculated and measured cell characteristics after proton irradiations. It is again gratifying to see that the theory and experimental data are in accord.
Figure 2.6 gives the total number of displacement defects as a function of incident electron energies for GaAs and Alo.33Gao.67As solar cells. It shows that the total number of defects induced by the electrons is much less than that induced by the








protons (Fig. 2.3) with the same energy. This is because the energy left to the cell from the incident electron is much less than that from the proton, owing to the small effective mass of the electron. Therefore, more electron fluences are needed to cause degradation of solar cells. Table 2.4 indicates the short-circuit degradation of 1MeV electron irradiated GaAs and Al0.3Ga0.67As p-n junction cells for three different fluences. It shows that the radiation hardness of Alo.33Ga o.67As is greater than that of GaAs. Table 2.5 summarizes the calculated and measured cell degradation after 1-MeV electron irradiation for both normal incidence and omnidirectional incidence cases. The calculated results again show the strong agreements with the experimental data for electrons fluences of 1014 and 1015 cm-2.
Figure 2.7 gives the flow chart for simulating the degradation of multijunction solar cells. In general, a middle cell or bottom cell may be affected by the incident energetic protons or electrons only when the reduced energy, after penetrating the top cell or middle cell respectively, is greater than zero. Therefore, in addition to the fluences and initial energy of the incident particle, the thickness of each cell is critical in calculating degradation of the multijunction solar cells. Since there are no experimental data on proton or electron irradiated multijunction solar cells available for comparisons, for simplicity the nearly optimal design of the Alo.33Gao.67As/GaAs two-junction solar cell was chosen for discussion in this section. The input parameters of these two-junction solar cells are same as those in Table 2.1. The short-circuit current degradation of proton irradiated Alo.33Gao.67As/GaAs two-junction cells are given in Table 2.6. Table 2.7 lists the I. degradation of 1-MeV electron irradiated Alo.33Gao.67As/ GaAs two-junction solar cells.
Figure 2.8 shows the short-circuit current degradation of the proton irradiated Alo.azGao.65As/GaAs/Ino.53Ga0.4As (or Ge) three-junction solar cell. The detailed discussion of these multijunction systems has been described in previous publications [54].








2.7 Summary

In this chapter a numerical model for computing the displacement damage for single-junction and multijunction solar cells has been developed and applied to the proton and electron irradiated single-junction GaAs and Al.MGaO.67As solar cells, two-junction Alo.33Gao.67As/GaAs and Al0.3 Gao.65As/ GaAs/Ino.s3Gao.47As (or Ge) three-junction solar cells under different fluences and energies. Excellent agreement was obtained between our calculated values and the measured I , Vo and 77, in proton or electron irradiated (AlGa)As and GaAs solar cells for proton energies from 100 KeV to 10 MeV and fluences from 1010 to 1012 cm-2, and for 1 MeV electrons and fluences varying from 1014 to 1016 cm-2 under normal incidence conditions. Moreover, it is shown in this chapter that in order to obtain an optimal multijunction solar cell with specified end of life efficiency, various physical parameters for each cell must be determined. It was pointed out that major difficulties encountered in carrying out the theoretical calculations using the model developed here include some unknown input parameters and the lack of experimental data to facilitate comparisons with the simulations. These uncertainties can be removed once the actual cell structures for the multijunction cells are fabricated and characterized.








Table 2.1 Input parameters for the simulations of proton or electron irradiated A10.33Ga0.67As
and GaAs solar cells.


Electron or hole lifetime r,,p (s) 2 x 10-8 4 x 10-9 Electron diffusion length, Ln (pUm) 6.0 3.41 Hole diffusion length, Lp 3.0 0.5 P-dopant density, NA (cm-3) 2 x 10i8 2 x 1018 N-dopant density, ND 1 X 1017 1 X 1017 Intrinsic density, ni 1.8 x 106 1.66 x 102 Electron capture cross section,a, (cm2) 1.8 x 10-11 5.0 x 10-16 (proton irradiated cell)
Hole capture cross section, ap 1.5 X 10-13 5.0 X 10-15 Electron capture cross section,an (cm2) 1.2 x 10-14 2.0 x 10-15 (electron irradiated cell)
Hole capture cross section, ap 1.4 X 10-13 2.0 x 10-14 Al0.85GaO.l5As window thickness, d (pm) 0.34 0.30 Junction depth, Xj (pm) 0.5 0.55 Cell thickness, t (pm) 10 4 Cell area, A (cm2) 4 4 Threshold energy, Td (eV) 9.5 10.5








Table 2.2 Calculated damage constants of the minority-carrier diffusion length in GaAs
p-n junction solar cell.


Energy (MeV) KLn KL,

0.1 0.105 0

0.3 0.00081 0.34
1.0 0.00025 0.00835 2.0 0.00006 0.00273 5.0 0.000029 0.00169 10 0.000006 0.00109








Table 2.3 Calculated and experimental data of the degradation of Ic, Vo and r7 in proton
irradiated (AlGa)As-GaAs solar cell.


Energy Fluence I ,/I. VOC/VOCO 77c /?co (MeV) cm-2 (%) (%) (%) Cal. Exp. Cal. Exp. Cal. Exp. 1010 0.97 0.97 0.97 0.925 0.94 0.89 0.1 1011 0.80 0.81 0.72 0.81 0.61 0.63 1012 0.49 0.50 0.63 0.66 0.30 0.28 1010 0.92 0.87 0.93 0.94 0.85 0.81 0.3 1011 0.74 0.71 0.89 0.86 0.67 0.62 1012 0.44 0.46 0.85 0.78 0.37 0.31
1010 0.98 - 0.96 - 0.94

1.0 1011 0.95 - 0.92 - 0.88

1012 0.80 - 0.89 - 0.71

1010 0.99 0.98 0.98 0.979 0.97 0.95
2.0 1011 0.96 0.938 0.96 0.94 0.93 0.90
1012 0.83 0.81 0.93 0.87 0.78 0.71 1010 0.99 1.00 0.99 1.00 0.98 1.00

5.0 1011 0.97 0.93 0.96 0.97 0.93 0.90

1012 0.86 0.84 0.93 0.90 0.80 0.76 1010 0.99 1.00 0.995 0.99 0.98 0.99 10.0 1011 0.97 0.96 0.975 0.97 0.945 0.95 1012 0.89 0.89 0.945 0.93 0.843 0.84








Table 2.4 I. degradation of one-MeV electron irradiated Alo.33Ga0.67As and GaAs p-n
junction solar cells.


Fluence 1016 cm-2 1015 cm-2 1014 cm-2

Cal. Exp. Cal. Exp. Cal. Exp. normal incidence
Alo.33Ga0.67As 0.695 - 0.925 0.926 0.986 0.986 GaAs 0.640 - 0.886 0.82 0.976 0.99 omnidirection
Alo.33Ga0.67As - 0.729 - 0.942 GaAs 0.245 - 0.660








Table 2.5 Calculated and experimental data of degradation of Ic, VOC, q, and KLp and
KL. in one-MeV electron irradiated Al0.mGa0.67As solar cell.


Fluence 1015 cm-2 1014 cm-2 Cal. EXP. Cal. Exp. normal Incidence
I./I1 0.925 0.926 0.986 0.986 Vo/Vo. 0.934 0.97 0.991 0.989 l/c/co 0.863 0.848 0.977 0.941 KL. 9.9 x 10-8 - 2.34 x 10-8 KL, 4.3 x 10-6 - 7.04 x 10-7 Omnidirectional
I,�/I,�o 0.729 0.942 VoC/VoCo 0.909 0.922 ?/7/c. 0.662 0.868 K" 2.89 x 10-7 - 2.12 X 10-7 KLP 4.45 x 10-5 - 4.08 x 10- -








Table 2.6 'Sc degradation of proton irradiated Alo.33Gao.s7As/GaAs two-junction solar
cells.


E(MeV) 1010 cm-2 1011 cm-2 1012 cm-2

Top Bottom Top Bottom Top Bottom

0.1 0.993 1.00 0.969 1.00 0.829 1.00 0.3 0.992 1.00 0.963 1.00 0.832 1.00 1.0 0.996 0.992 0.994 0.965 0.980 0.843 2.0 0.996 0.993 0.995 0.974 0.984 0.871 5.0 0.996 0.994 0.995 0.981 0.987 0.900 10.0 0.996 0.995 0.996 0.988 0.992 0.941






27

Table 2.7 'Sc degradation of one-MeV electron irradiated Al.33Gao.67As/GaAs two-junction
solar cells.


Fluence Top cell Bottom cell 1016 cm-2 0.780 0.64 1015 cm-2 0.950 0.886 1014 cm-2 0.990 0.976















AR COATING


(AlGa) As


GaAs


p-n JUNCTION n CONTACT


NUMBER OF FINGERS = 24 p CONTACT: Au-Zn-Ag n CONTACT: Au-Ge-NI-Ag AR COATING :Ta20x p AIxGal-xAs: x _ 0.85 CELL SIZE = 2 x 2 cm2


Fig. 2.1 The cross sectional view of an (AlGa)As-GaAs p-n junction solar cell.













































I I II I


I I I I 1 11I


I I I I I I I I


0.02


0.1 0.2 1
ENERGY (MeV)


Fig. 2.2 The range of an Alo.s5Gao.15As-GaAs solar cell vs. incident proton energies.


101-


81-


6

z


4


2_-


0 I

















400 350 3110 250


200


1l50


100 50

I I I I I I i I I I I I I I
i0-1 tOO 10i E (MeV)





Fig. 2.3 The total number of displacement defects vs. incident proton energies for GaAs,
Ge, (InGa)As and (AlGa)As single-junction solar cells. Y : Ino.53Ga0.47As, X:
GaAs, + : Alo.Gao.67As and * : Ge.






31














1.0 -......




0.8 1




0.6
0
_w

0.4 -12


d - 0.34 gm 0.2 Xj - 0.5 Pm
On = 1.8 x 1014cm2
-rp - 1.5 x 10-3cm2

0 .0 1 1 11 1 - l I I I I I I 1
0.02 0.1 0.2 1 2 10 ENERGY (MeV)


Fig. 2.4 The normalized short-circuit currents in the Alo.85Gao.15As-GaAs p-n junction
solar cells. Solid curves are from our calculations; solid dots are the experimental
data.






























I I I 111111 I I I 111111 I I I I 1111


Voc
Voco0


Voco = Open Circuit
Voltage


Before Irradiation
I I I I tt t1


0.1 1
PROTON ENERGY, MeV


Fig. 2.5 The normalized open-circuit voltage in the Alo.85Ga0.15As-GaAs p-n junction
solar cell.


1.00 0.95 0.90 0.85 0.80


0.75
0.0


1


I I t I I II!


I I I I I I I I . . . .. . .


I I i I 1 i l i1 1 1 1 1 1 t il li


I i I 1 II I11

















15







10














0
2 3 4 E (MeV)





Fig. 2.6 The total number of displacement defects vs. incident electron energies for
GaAs, Ge, (InGa)As and (AlGa)As single-junction solar cells. Y: Ino.53Ga0.47As,
X : GaAs, + : Alo.33Ga 67As and * : Ge.






















































Fig. 2.7 The flowchart for simulating degradation of the irradiated multijunction solar
cells.












1.0




0.9 0.8 0.7


0.5




0.4





0.3 11

1014 1015
F


Fig. 2.8 1., degradation of the electron i
(or Ge) multijunction solar-cells.


1017


luence, e/cm2 1 16


rradiated Alo 35Gao.65As/GaAs/Ino.53Ga.47As













CHAPTER 3
A NEW METHOD FOR OPTIMAL DESIGN OF GAAS SINGLE-JUNCTION SOLAR CELLS

3.1 Introduction

Since simple analytical models can not provide enough information and accuracy to optimize the design of solar cells [55], computer modeling has become an important and essential tool for solar cell design and analysis. The incorporation of an efficient optimum algorithm for computer simulation would enable, using the knowledge of cell parameters from the state of the art and its applications, to optimize the cell parameters in order to attain a maximum conversion efficiency.
Recently, Chen and Wu [56] and Fossum [57] have proposed two different computer aided designs for silicon solar cells while Hovel [20], Hamaker [21] and Kinechtli et al. [19] have established simplified solutions for GaAs solar cells. However, the results obtained by these previous models were less than truly optimum due both to ignorance of the interaction between the parameters and to the lack of an efficient optimum algorithm in computer simulation. In addition, they did not include the extrinsic parameters such as series resistance, contact structures, irradiation, air mass ratio, temperature, etc., in the optimization cycle and hence the results were not realistic.
In this chapter a new computer model is proposed for truly optimizing the structure of a GaAs single-junction solar cell for space application. However, the model can apply easily to other solar cell systems as well. Our model not only takes into account the effects of the intrinsic structural parameters such as junction depth Xj, cell thickness Tj, doping densities NA and ND, surface recombination velocity Sp and Sn, but also incorporates the extrinsic structural parameters. It should be noted that








the electron or proton irradiation on a solar cell for space application will degrade the cell's performance. Therefore, the damage constants of cell parameters, such as minority carrier diffusion length and minority carrier lifetime, should be estimated in advance in order to attain an accurate optimal structure of the cell. The detailed discussion of the calculations of damage constants of the irradiated solar cell has been presented in Chapter 2. In this chapter it is assumed that the values of damage constants for the cell parameters are available and hence can be employed directly in the optimum simulation.
To test the optimum algorithm both GaAs and Si solar cells have been simulated. It turns out that the conversion efficiency of a GaAs cell is better than that of Si. The results also show that there can be several optimum sets of parameters that yield nearly the same values for efficiency while maintaining acceptable characteristics.
Section 3.2 of this chapter outlines the device modeling for a p/n junction solar cell. The effects of extrinsic parameters on solar cell parameters are presented in section 3.3. In section 3.4 we propose a true and efficient optimization technique for the optimal design of a single-junction solar cell. Comparisons between the calculated results and the published data have been done also in this section. Summary is made in section 3.5.

3.2 Device Modeling for GaAs P/N Junction Solar Cells

For simplicity the discussion will be restricted to a p/n junction solar cell with uniform doping. The cross section diagram of the GaAs p/n junction solar cell to be analyzed is the same as that shown in Fig. 2.1. There are four major solar cell parameters involved namely short-circuit current Ir, open-circuit voltage Voc, conversion efficiency 77, and fill factor F.F. in designing a typical p/n junction solar cell. In general, the higher the values of the Irc, Vo, and F.F. a solar cell can produce, the higher q, the solar cell can obtain. However, a solar cell with a low energy bandgap tends to produce higher 1,, and lower Vo, than the cell with a higher energy bandgap.








This tradeoff complicates the design of a solar cell.
Theoretically, when the photon of the light impinges on the p/n junction solar cell having a bandgap energy, Eg, the short-circuit current of the solar cell can be represented as the sum of the currents obtained from the emitter layer IE, base layer IB and depletion region Iw as shown in Fig. 3.1. It is given as [20]


ISV = IE(A) + IB(A) + Iw(A)dA (3.1)
0.35


where A1 (y m) = 1.24 / Eg.
The current component from emitter layer is represented as



IE(A) = qAp(A) (1 - R(A)) aLn (-aLn exp(-aXj) + a2L2 - 1
aLn + SfZ ( ex -X' coh Li exp(-aXj) sinh -L.Sn 7 sinh L + cosh L' (3.2)


where A is area of the cell, q is the electron charge, R(A) is reflection coefficient, a is the absorption coefficient of the cell and is a function of wavelength, Ln is the electron diffusion length, rn is the electron lifetime and p(A) is the solar photon flux density which is [581


p(A) = 4.06 x 1018A3H\ (3.3)



where H\ is the solar radiance. The solar radiance is crucial in evaluating the conversion efficiency of a solar cell for different air mass ratio. Existing software, LOWTRAN6 [59], is used here.
The current component from the base layer is










IB(A) = qAp(A) (1 - R(A)) aLp exp(-a(X1 + Wa))
-a2L ,- 1 ep-(+W)

L P- (cosh - - exp(-ad')) + sinh - + aL exp(-ad') S asinh A + cosh )34



where Lp is the hole diffusion length, rp is the hole lifetime, Wj is the depletion width and d' = Tj - Xj- Wj.
The current component from the depletion region is shown as


Iw = qAp(A)(1 - R(A)) exp(-aXj) (1 - exp(-aWj)) (3.5)


The open-circuit voltage V,, for the solar cell is given by


- nKBT ln(Lc + 1) (3.6) q I0


where KB is the Boltzmann constant, T is the absolute temperature and 10 is the saturation current density, n is the diode ideality factor. For simplicity, we may assume that 1 is either a simple diffusion-dominated current if n is close to 1 or a recombination-dominated current if n is close to 2. The former one is given as


Io = qAni2 Dp Sp cosh(d'/Lp) + -D sinh(d'/Lp) LND Lp cosh(d'/Lp) + Sp sinh(d'/Lp) J qAni2 D, (Sn cosh(Xj/L,) + - sinh(Xj/L,) LENA L ," cosh(Xj/L) + S sinh(X(/L).


where ni, ND and NA are the intrinsic densities, donor and acceptor density of the cell








respectively. Dp is the hole diffusion coefficient whereas D. is the electron diffusion coefficient.
As for the recombination-dominated case, it is


I- qAniWj (3.8)



The fill factor F.F. for a solar cell is given by
F.F_= V,, 1 e - 1)T (3.9) V"C (exp qV.)-1



where Vm is given by the relationship as follows: exp( (1 +KBT1 = m) + 1 (3.10)



Therefore, the conversion efficiency of a single-junction solar cell is given by q JcVcF.F. (3.11)



where Pi, is the incident solar power density. P. for different air mass ratio is calculated from LOWTRAN6 [59].

3.3 Effects of Extrinsic Parameters on Device Modeling

In practical solar cell design, series resistance, contact materials, grid line configuration, antireflection coating and high sun insolation can affect the performance of the solar cell. Therefore, these effects must be taken into account in the optimization cycle in order to ascertain an realistically optimal design. In this section, the relationships between these extrinsic parameters and four major solar cell parameters are described.








3.3.1 Effect of Antireflection Coating
For a single layer of non-absorbing medium, the reflectivity at normal incidence of a given wavelength is given as [60]


R=_ (n1 - .non2 2(3.12) n1 k-2 + non2/



where no, ni and n2 are the refractive indices of air, antireflection coating material and solar cell respectively. The thickness of the antireflection coating is

A0
dAR = 4- (3.13) 4n,
A double layer coating can offer less reflectivity loss over a wider region than a single layer coating can. A single layer has a zero reflectivity loss at only one wavelength and has an acceptable reflectivity loss over only a limited range. The reflection coefficient R of double layers is [601


R n12 n3 - n2 2no2
R= +n2no) (3.14)


where n2 and n3 are the refractive indices of the second coating layer and solar cell respectively and no and nj are the same as defined in Eqn. 3.12. The thicknesses of these two coating layers can be estimated by the following relationship:


n1dl = n2d2 = -� (3.15)
4


3.3.2 Effect of Grid Design
As the area of solar cell increases, it becomes more important to include the interconnect configuration in the grid process [61]. Different grid geometry can affect the sheet resistance of the series resistance. The parallel grid line design in Fig. 3.2








is the standard grid design for a solar cell. The series resistance of this grid design is approximated as [62]

p 2 bpf 2 btR
RP, --b + -h +h a (3.16) 12D wh w


where D is the thickness of window layer, b is finger spacing, w is finger width, h is finger height, pf is emitter resistivity, ps is metal finger resistivity and R, is the contact resistance.
It should be noted that as the width of grid element increases, the series resistance decreases and hence the electrical power loss decreases. However, this is not the only consideration. When less cell area is exposed to the photons, the short-circuit current decreases. The shadow loss is calculated as the total area of the fingers divided by the whole area of the solar cell. Therefore, there is an optimal value where the sum of these two losses is a minimum.
3.3.3 Effect of Series Resistance
When series and shunt resistance problems become important, the ratios Vm/Vo� and Im/I., are reduced and the ratios calculated from Eqns. 3.8 to 3.10 should be modified. For simplicity, if shunt resistance is much higher than the series resistance, which is always true for GaAs, the open-circuit voltage V',, after additional reduction is given as


V = V.- IR (3.17)



and the reduced fill factor F.F'. is given as [63]


F.F.' = F.F. x (1 - vIc) (3.18)








3.3.4 Effect of High Sun Insolation
For high sun insolation, the short-circuit current is estimated simply by multiplying the number of sun insolation by the short-circuit current obtained from one sun insolation. The open-circuit voltage of high sun insolation, V.,x, is given as [64]


nKBT
= V" + ln(X) (3.19)
q


where X is the number of sun insolation. From this equation, it appears that the conversion efficiency of high sun insolation increases with any increase of open-circuit voltage. Again, this is not always true because of the reduction of the fill factor in high sun insolation. Additionally, at high sun insolation, the effect of series resistance becomes more critical because of the high ISc.

3.3.5 Effect of Irradiation
After proton or electron irradiation, the minority carrier lifetime and diffusion lengths will be degraded. The diffusion length is related to the lifetime, L2 = Di-, and it is given by the usual relations [46, 47]:


1 1
- = - + gro (3.20) 7r To
1 1
=L 2 -� + KLO (3.21)



where K, and KL are the minority carrier lifetime and diffusion length damage constants, respectively, and � is the fluence of proton or electron.

3.4 Constrained Optimization Technique

Under the constrained optimization approach, the optimal design of a solar cell can be mathematically expressed in closed form as











Y = Max(f(Xi) I Gk < Xk


where X1,..., XN are the explicit independent variables to be optimized, which might be cell thickness, doping level, etc. The implicit variables XN+I,... XM are dependent functions of the explicit variables. These implicit variables might be the minority carrier diffusion length, the minority carrier lifetime or others. f(Xi) is the objective function which might be the solar cell efficiency expected to be maximized, or perhaps the short-circuit current expected to be optimized. The upper and lower constraints Hk and Gk may be either constants or functions of the independent variables.
The Box complex algorithm [32, 65] is one of the most efficient constrained optimization techniques currently available. This algorithm is a sequential search technique which has proven effective in solving problems with nonlinear objective function subject to nonlinear inequality constraints. Derivatives of the objective function are not required for this algorithm. This procedure will tend to find the global optimum since the initial set of guesses is randomly scattered throughout the feasible region. The algorithm, with some minor modifications for increasing the convergence rate, proceeds as follows [65, 66]: Step 1. Initial feasible starting K guesses, where K is at least equal to (N + 1), are generated. Each guess consists of N points which are generated from random numbers and constraints for each of the independent variables:



Xi= Gi + rid(Hi - Gi),

i= 1,2,...,N, J = 1,2,...,K- 1 (3.23)


where rij are random numbers between 0 and 1.








Step 2. The guess points must satisfy both the explicit and implicit constraints. If a constraint is violated, the point is moved to the upper bound or lower bound of the constraint which is violated. This procedure is repeated as necessary until all the constraints are satisfied.
Step 3. The objective function is evaluated at each guess. The guess having the lowest function values is replaced by a guess which is


Xij(new) = 1.3 (Xi,a- Xi1(old)) + Xi,a, i = 1,2,..., N (3.24)


where Xi,a is the average of the remaining guesses and is defined by


Xi'.K( Xi - Xi(old) , i = 1, 2,..., N (3.25)


Step 4. If a guess repeatedly gives the lowest function value on consecutive trials, it is moved to the average of the best and worst guesses. This minor modification would increase the convergence rate. Moreover, it will save computation time because additional constraint-violation checking is not necessary for this new guess. Step 5. Convergence is assumed when the objective function value at each guess is within the tolerance which the user assigned. A flowchart illustrating the above procedure is given in Fig. 3.3.

3.5 Optimal Design of GaAs Single-Junction Solar Cells

The main task in this section is to optimize the efficiency of a single-junction solar cell. According to a previous study [52] and the work done by Knechtli et al. [19], GaAs single-junction solar cells with bandgap energy 1.43 eV offer some advantages for example high 17, radiation hardness, and relative insensitivity to temperature as compared to silicon solar cells. The optimal design of a Si solar cell is also developed here for comparison purpose.








For the GaAs cell, the design problem for maximizing the efficiency 77 is formulated as follows:


77c = Max (f(ND, NA,Xj, Tj, Eg, Sn, Sp))


(3.26)


such that


ln(2 x 1015cm-3) ln(2 x 1015cm-3)

0.05Im

0.1gm 1.43eV

10cm/s 104cm/s


ln(ND) _ ln(101cm-3) ln(NA) _ ln(101'cm-3) Xj _ 0.5!sm Tj _<10pm Eg < 1.43eV Sn < 106Cm/s Sp < 106Cm/s


in which the parameters to be optimized are intrinsic structural parameters (ND, NA, Xj) and Tj and material parameters (S,, Sp and E.) whose values are constants. The material parameters are determined by the materials and by the top and back contact surface passivations. For the GaAs solar cell, a thin (AlGa)As window layer is grown on the top for reducing the surface recombination velocity [19, 20]. The choices of the upper and lower bounds of the constraints are based on the considerations of contact resistance [67], the state of the art technology and radiation hardness.
Table 3.1 and 3.2 list the comparisons between the results calculated here and the published data for the optimal designs of Si and GaAs solar cells. Our designs show much better performance than those obtained from previous models [19-21, 56, 57, 681. Moreover, they are close to the cell efficiencies made by the state of the art


(3.27) (3.28) (3.29) (3.30) (3.31) (3.32) (3.33)








[69], namely 20% for Si solar cell and 26% for GaAs solar cell respectively. The major discrepancies between our simulations and the previous models are the selections of the constraints. For instance, for an Si solar cell, ND should be greater than 1019 cm-' to obtain a good contact whereas NA is two orders less than ND [67]. In this respect, the values of ND in a p/n junction Si solar cell can be much less than that in n/p Si solar cell and therefore results in better performance.
The effect of series resistance on GaAs solar cell efficiency is described in Table 3.3. Table 3.4 shows the optimal designs of the 300 KeV proton irradiated GaAs solar cell. It is clear that the junction thickness decreases when the proton fluence increases, and the effect of fill factor due to the irradiation is negligible. These results conform to the assumptions made for calculating the degradation of proton or electron irradiated solar cells in Chapter 2. Figure 3.4 shows the temperature and air mass ratio dependence of the efficiency of a GaAs solar cell. It shows that the efficiency of the GaAs solar cell increases as the temperature decreases. This is reasonable because the open-circuit voltage decreases when the temperature increases.


3.6 Summary

A new method of incorporating an efficient optimization algorithm into device modeling techniques for the optimal design of GaAs and Si solar cells has been examined. A modified Box complex optimization technique and a device modeling that considers the effects of extrinsic parameters has been implemented to obtain the optimal design of a GaAs single-junction solar cell. All the parameters to be optimized are adjusted in a systematic way, resulting in a truly optimal design. Comparisons between the calculated results and the published data have been made to verify the optimal designs. The optimal efficiency of the single-junction GaAs solar cell obtained by the simulation is 27.8% at room temperature for AMO which is close to the cell efficiency, namely 26%, provided by the state of the art.








Table 3.1 Comparisons between calculated results and published data for the optimal
designs of Si single-junction solar cell at room temperature.


AMO AM1 AM1[56] ND (cm-3) 5.OX 1016 5.Ox 1016 3.0x 1019 NA (cm-3) 5.Ox 107' 1.16x 1017 6.0x 1016 Xj (pm) 0.5 0.623 0.1 Tj (pm) 300 200 66.4 S. (cm/s) 10000 10000 118
S. (cm/s) 10000 10000 100 V. (V) 0.89 0.88 0.63 J, A/cm2 0.0372 0.0328 0.034 F.F. 0.85 0.85 71C 21.1% 23.8% 18.06% E9 1.12 1.12 1.12 Collection efficiency 0.958 0.944 Sun Insolation 1 1 1 Shadow loss 0.018 0.018 Reflection loss 0.041 0.041 R. (Q) 0.473 0.473








Table 3.2 Comparisons between calculated results and published data for the optimal
designs of GaAs single-junction solar cell at room temperature.


AM0 AMO[68] AMO[25] ND (cm-3) 3.16x 1017 2x 1017 7x 1017 NA (cm-3) 2.60x 1018 5x 10'8 1.5x 1016 X3 (pm) 0.5 0.3 0.5 Tj (pm) 10 2.5 3.0 S. (cm/s) 10000 - S, (cm/s) 10000 - V,, (V) 1.23 1.05 1.04 J., A/cm2 0.0345 0.0341 0.0317 F.F. 0.888 0.85 0.83 7c 27.8% 22.5% 20.3% E9 1.43 1.43 1.43 Collection efficiency 0.999 - Sun Insolation 1 1 1 Shadow loss 0.018 - Reflection loss 0.041 R. (fQ) 0.473 -








Table 3.3 Effect of series resistance on GaAs single-junction solar cell efficiency.


Grid Number R, (11cm2) %

0 0 29.4 24 0.473 27.7 18 0.969 26.9 16 1.22 26.5 12 2 25.4 9 3 23.9








Table 3.4 Effect of irradiation on GaAs single-junction solar cell for 300 KeV proton at
fluences of 1010, 1011 and 1012 cm-2.


None 1010 cm-2 1011 cm-2 1012 cm-2 ND (cm-3) 1017 1017 3x 1017 1017 NA (cm-3) 1018 1018 1018 1018 Xj (pm) 0.5 0.5 0.4 0.175 Tj (pm) 10 10 10 10 V,, (V) 1.209 1.15 1.15 1.08 J.: A/cm2 0.0347 0.0315 0.0283 0.0241 F.F. 0.887 0.883 0.88 0.88 77c 27.5% 23.7% 21.3% 17.0%











DW

















(GaAI)As GaAs E
So+

!E











GaAs GaAs
ii

n n+













*: E
tV


Fig. 3.1 The energy band diagram for a (AlGa)As-GaAs p-n junction solar cell.






53 2a 2a
IW


Fig. 3.2 Parallel gridline pattern for solar cells.



















































Fig. 3.3 Flowchart for the modified Box optimization algorithm.

















34 32




30 )28
C,


26




24
250 290 330 370 410 450

Temperature (K) Fig. 3.4 Temperature and air mass ratio dependence of the efficiency of GaAs singlejunction solar cell.














CHAPTER 4
AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF MULTIJUNCTION SOLAR CELLS

4.1 Introduction

Single-junction solar cells today are those most frequently used. They are made from a single material with a single bandgap to absorb the sun light. The incident photon with energy less than the bandgap of the material can not be converted to electricity. And much of the photon with energy greater than the bandgap is wasted as heat. Therefore, the efficiency of a such single-junction cell is limited. In multijunction cells, solar cells with different bandgaps axe put on top of each other in decreasing bandgap order. In this way, all the incident photons with energies equal to or greater than the bandgap of the top cell are absorbed by the top cell. The incident photons with energies less than the bandgap of the top are transmitted to the next cell. The phenomenon recurs at each cell.
In 1955, Jackson [3] proposed that the efficiency of solar cells could be increased significantly by constructing a system of stacked p/n homojunction photovoltaic cells which are composed of different semiconductor materials. The next year, Loferski [4] was the first to use multiple cell concepts to optimize the efficiency of a photovoltaic system, selecting sets of cells which made use of the energy available from the entire solar spectrum. Recently, many researchers [3-18] have proposed varying approaches to the optimal design of the multijunction cells both theoretically and experimentally. However, their results were not realistic for several reasons. First, data for the electrical and optical parameters of the materials they used were not always available and accurate. Secondly, their designs were based on either a small set of design parameters or an inadequate device modeling. Thirdly, they did not








consider the interrelation of the parameters of the solar cells and hence their results were not optimal.
In this chapter, we propose for the first time an expert system approach to the optimal design of multijunction solar cells for both terrestrial and space applications. An expert system is a knowledge-intensive computer program. The knowledge of an expert system consists of facts and rules. The facts constitute a body of information that is widely shared, publicly available and generally agreed upon by experts in the field [33, 34, 70]. The rules are those if-then rules that characterize expert level decision making in the field. In general, an expert system with a large number of facts and rules is better than one with just a few facts and rules. To assure the usefulness of our expert system, we have implemented as many facts and rules as possible with the help of published literatures and experts in the photovoltaic area. However, because of the availability of other research reports and the tradeoff between the number of rules and the computation time, our expert system is based on the AlGaAs, GaAs, Ino.5Ga0.47As, Si and Ge materials. And less than a hundred rules are actually implemented.
Section 4.2 deals with the device modeling of multijunction tandem solar cells used in the expert system. In section 4.3, the concept of an expert system approach to the optimal design of multijunction solar cells is presented. An optimization technique with heuristic rules is also shown in this section. Results and discussion are in section
4.4, with a summary in section 4.5.

4.2 Device Modeling of Multijunction Tandem Solar Cells

Since the theoretical conversion efficiency increases very slowly beyond threejunction tandem cells [1] and the technology of fabricating cells beyond three junction is difficult in practice [2], the multijunction study focused on those structures that are two-junction and three-junction only. In general, the calculations of I , Vow, F.F. and q, for the multijunction cells are similar to those for single-junction solar cell








presented in Chapter 3. Here, we will only illustrate the three-junction calculations.
Let the top, middle and bottom cells have the bandgaps of Egl, E.2 and Eg3 with corresponding short-circuit currents Iscl, Isc2 and Isc3 respectively. Then, the short-circuit currents can be written as


Isc1 = IEI(A) + IB1(A) + Iw1(A)dA (4.1)
0.3

Isc2 = j IE2(A) + IB2(A) + Iw2(A)dA (4.2) I.3 = IE3(A) + IB3(A) + Iw3(A)dA (4.3)


where A (I m) = 1.24 / Egl, A2 (i m) = 1.24 / Eg2, and A3 (p m) = 0.24 / Eg3 respectively. The current components from emitter, base and space charge of each cell and other notations in Eqn. 4.1 to Eqn. 4.3 are similar to those given for the single-junction solar cell.
The representations of the open-circuit voltage, fill factor and conversion efficiency of each cell are also similar to Eqns. 3.6, 3.9 and 3.11 respectively. For the two-terminal case as shown in Fig. 4.1, the I,, values are all the same for the top, middle and bottom cells, namely the smallest value among the three cells. The total conversion efficiency of the three-junction cells is then



77tot = ?cl + i c2 + 7c3
I.V.1F.F.1 + I5Voc2F.F.2 + I.Vo3F.F.2 (4.4) Pin


Those effects such as series resistance, grid structures, etc., which have been discussed in Chapter 3, are still applicable to the design of multijunction solar cells. They are not repeated here. However, the effect of the tunnel diode should be considered. The role of a tunnel diode in the design of the multijunction solar cell is to connect the two








different p/n junctions. The doping density of the tunnel diode must be quite high in order to have a low impedance to current flow in both directions and the voltage drop across it should be as small as possible [5]. Additionally, the bandgap of the tunnel diode should be as large or larger than the top cell bandgap so that as many photons as possible can be transmitted from the top cell and absorbed at bottom. According to Kane's theory, the tunnel current is given as [71, 72, 73]


2~ 2_ 2mEg 7 -EW
V- 4h2Wt exp 2Vi (4.5)



where m* is the effective mass of the tunnel diode; Wt is the depletion width of the tunnel diode; Eg is the bandgap of the tunnel material; V is the voltage across the tunnel diode and Jt is the tunnel current density. Therefore, the open-circuit voltage of the solar cell should be adjusted by the amount of the voltage across the tunnel diode.
An alternative approach for the intercell ohmic contact of the multijunction solar cells is the metal intercell contact shown in Fig. 4.2. In the metal interconnect contact technique, the top (AlGa)As cell and middle cell will be interconnected by the metal interconnect technique. Grooves will be etched through the top cell to reach the middle cell. Metals will be deposited within the grooves to shorten the base region of the top cell to the emitter layer of the middle cell. The connection between the middle and the bottom cell is made by fabricating a tunnel junction as shown in Fig. 4.3. The metal intercell contact structure has the problems of a complex fabrication process and double shadowing. The total series resistance of the multijunction cells is then the sum of the series resistances of top cell, bottom cell and emitter layer of the middle layer.

4.3 Concept of the Expert System Approach The shortcomings of the conventional computer programs for the optimum de-








sign of high conversion efficiency single-junction and/or multijunction solar cells are twofold. First, they are not flexible: due to the intermixture of data and codes, the programs are not always applicable without code corrections when the goal of the design changes. Second, results obtained from those programs are sometimes unrealistic: data or design criteria often are not properly taken into account.

In this section, we propose a new approach to formulating the design problem and then develop an expert system to aiding the optimal design of high efficiency singlejunction and/or multijunction solar cells for both space and terrestrial applications. Our proposed expert system will apply to the radiation free environment as well as to the proton or electron irradiation environment. A systematic approach to this design problem consists of four tasks (1) constructing a solar cell knowledge data base, (2) searching for all possible designs from the knowledge base, (3) selecting the optimum design by applying heuristic rules, and (4) verifying the optimum design. This expert system will produce practical designs corresponding to user specifications and thus become an important tool for the solar cell designer.
4.3.1 Problem Formulation
An expert system approach to solving for the optimal design of multijunction solar cells will usually be successful if the theoretical rules and/or experienced rules have been formulated quantitatively. The design procedures of the proposed expert system can be formulated into four tasks. The first task is the construction of a solar cell knowledge base. Here the first step is to define the problem domain precisely. This will include an analysis of the desired results from the expert system. Our expert system is currently limited to (AlGa)As, GaAs, Si, Ge and Ino.53Ga0.47As materials. The goal of the proposed solar cell expert system is to optimize the conversion efficiency of a 2-terminal monolithic single-junction or multijunction solar cell based on these five material systems. Then the types of knowledge required must be identified as well as the possible interactive consultation and the approximate number of rules,








which will reduce the search space of the design problem. In this respect, the facts in the solar cell knowledge base will be the fundamental properties of the materials such as the bandgap, minority carrier diffusion length, minority carrier lifetime, contact resistance, etc. The data structure of these facts can be represented as lists, trees, nets, rules or other formations. Since most of the fundamental properties, namely facts, are a function of doping density, temperature etc., the if-then rules representation for the facts is used for the expert system.
The rules consist of antecedent-consequent pairs. The characteristic of a rulesbased system is the separation of data examination from data modification. The examination of data generally occurs in the antecedent of a rule, while data modification is normally managed by the consequent of a rule [70]. For example, if a GaAs p/n junction solar cell is chosen, then the junction thickness of the cell should be less than 0.5 pm in order to obtain high efficiency and high radiation hardness. By the application of this rule, the search space of the problem is reduced and computation time is saved. However, there may be a number of conflict rules in the knowledge base and a selection must be made. Therefore, a conflict resolution process is set up to determine which satisfied rule to be used in the conflict set.
There are two ways of constructing the rules-based system of the expert system. The first method is to let the expert system automatically adopt the solar cell properties from existing data files or tables. These data may be the results obtained from the expert system itself. The other way is through a set of queries under a certain condition C as


C =(C1, C2..., Cn) (4.6) where Cj is a value under condition j, and n is the total number of queries. The queries might be of the following type:


1. Input the cell name please.








2. Input the bandgap of this cell please.

3. Input the contact material please.

The second task is to search the knowledge data base. Let Gi be the mapping function from a description of a solar cell to the value of the material with respect to a property i under a certain condition C. A solar cell is described by a set of tuples, e.g. bandgap energy, short-circuit current, open-circuit voltage, conversion efficiency and so on, as


M= (MI, M2, ...,mk) (4.7)



where 1 is the name of a solar cell and k is the total number of tuples. The retrieved value Vki of a solar cell M, with regard to property i is represented by Gi. And


Vki = Gi(Mk) (4.8)



The set of solar cells which satisfy the requirements on property i is


Si = (siCi _ Gi(s),s E M) (4.9) where M is the universal set of Mj ( j = 1, 2,... ) created in the knowledge data base. The solar cell which satisfies the requirements are obtained by the intersection of the sets as


D = S fl S2 n S3... n Sr (4.10) However, the set D of the selected solar cells can be empty in some cases, and then the procedures for constructing the knowledge data base must be invoked.








The third task is the selection of the optimal designs. The selection of optimal design Dopt is performed by calculating the scalar product of a weighting factor W = (w1, w2,..., Wr ) predetermined by the solar cell designer and a matrix V. as


E(ei,e2, ...,en)= WV. (4.11)



where n is the possible designs and ei is a value of this evaluation for the selected solar cell i. The weighting factors can be the cost of solar cells, complexity of fabrication processes, conversion efficiency, weight of solar cells, preference of materials and radiation hardness. In this expert system, it is assumed that the weighting factors of the radiation hardness and conversion efficiency axe dominant.
The last task is to verify the optimal design. The verification of the optimal design Dopt is made by comparing the calculated results obtained from this expert system with the experimental and/or theoretical data collected from the literature of the photovoltaic field.
4.3.2 Optimization and Heuristic Rules
For the multijunction solar cells, the optimization problem, which is to maximize the total efficiency of each subcell, can be formulated as

n
77tot = Eq/i (4.12) i=1
where n is the number of the subcells of the multijunction solar cell. And


77i = Max (f(NDi, NAi, Xji, Tji, Egi, Si, Spi)) (4.13) It should be noted that the format of Eqn. 4.13 is similar to that of Eqn. 3. 26. Consequently, according to Eqn. 4.12 and 4.13 the number of constraints for the optimal designs of two-junction and three-junction solar cells will be two times








and three times more than those of a single-junction solar cell. For example, a threejunction solar cell needs at least 15 constraints if Sn and SP are fixed. Since the computation time will be increased exponentially instead of linearly as the number of the constraints increase, a couple of heuristic rules have been adopted in this system in order to prune the search space and hence to save the computation time.

A heuristic is a technique that improves the efficiency of the search process and leads to an adequate answer, if not the best one of a difficult problem [74]. A heuristic rule serves as an aid to problem solving by experimental, especially trial-and-error, methods [75]. Therefore, it is possible to construct a special purpose heuristic rule that exploits domain-specific knowledge to solve a particular problem. In this respect, heuristic rules are applied to the selections of the upper bound and/or lower bound of the constraints, and to the combinations of material in the system for multijunction solar cells, in this way to obtaining a quick feasible optimal solution. Then, the solution will be compared with the existing data to justify the validity of heuristic rules.


4.4 Results and Discussion

The design of the multijunction tandem cells is considerably more complex than that of single-junction cells and hence additional parameters must be considered. They are as follows [76, 77]:


1. Bandgap energies must be optimized for multijunction solar cells.

2. Lattice matching is desired.

3. Direct optical transitions are desirable.

4. Metallurgical system must be compatible.


5. A compatible substrate must be available.








6. Must be invariant with changes in the environment.

According to our computer calculations, the optimum bandgap combinations of two-junction and three-junction tandem cells are 1.75/1.10 eV , 2.02/1.43 eV, 2.00/1.43/1.04 eV and 2.00/1.40/1.00 eV under AMO at room temperature. However, considering the lattice matching problem, it is clear that the materials selected for the top and middle cells, (AlGa)As and GaAs are quite favorably lattice matched for the two-junction tandem cells. Although the (AlGa)As/Si and (AlGa)As/(InGa)As two-junction solar cells meet the optimum bandgap energy combinations, they are still not the qualified designs due to the lattice constants and thermal expansion coefficients mismatch. In addition, the radiation hardness of GaAs is greater than that of Si or (InGa)As.
As for the three-junction structure, (InGa)As, Ge and Si can be the possible candidate cells for the bottom cell. However, these materials exhibit lattice mismatch with respect to GaAs and generate and propagate dislocations which may adversely affect solar cell performance.
In a multijunction solar cell it is advantageous to have direct bandgap materials, to reduce the thickness of the required material. This reduction results not only in lower material costs but also in lower epitaxial growth costs for growing the required layers. Obviously, it is also beneficial to minimize thickness to gain a weight advantage for the final structure. Even more significant, however, is the reduction in growth time. In addition, the thinner layers inherent in the direct bandgaps materials tend to lead to lower minority carrier recombination losses and improved radiation hardness. Therefore, the top cell for our optimal design of the GaAs based multijunction solar cell is limited to the direct bandgap material.
A comprehensive understanding of the electronic transport properties of the materials and of the total assembly in the two-junction or three-junction tandem cells is essential to a complete analysis of the relative merits of the final choices. The most








tractable parameter which reflects the quality of the device is probably the minority carrier diffusion length. A long diffusion length would require that the defects and recombination centers be minimized, that the junction quality be kept as high as possible and that the interfaces between layers be kept free of strain and imperfections. Since (AlGa)As, GaAs and (InGa)As are polar materials which are different from the nonpolar material such as Ge and Si, a characterization of the dominant scattering mechanisms that affect the mobility and hence affect the diffusion length should be done to facilitate the study of the optimal designs. Detailed discussion of different scattering mechanisms and calculations of the electron and hole mobilities of the (AlGa)As is given in Chapter 5.
Table 4.1 and Table 4.2 list the results of our calculated two-junction and threejunction solar cells respectively. For those two-junction solar cells except GaAs/Ge in Table 4.1, an Al.Gal-As heavy-doped tunnel junction with Al composition greater than 0.45 has been used for our simulations. However, a heavy-doped GaAs tunnel junction is used for the case of GaAs/Ge. For the three-junction solar cells in Table 4.2, an (AlGa)As tunnel junction is used for connecting the top cell and middle cell whereas a GaAs tunnel junction is used for connecting the middle cell and bottom cell. Here, we assume that it is feasible to obtain an (AlGa)As tunnel junction which is as good as the GaAs tunnel junction. According to the state of the art technology [78, 79], the n-type and p-type doping densities of GaAs can be as high as 2x 1019 cm-3 and 5x 1019 cm-3 respectively. The effect of doping densities of the tunnel junction on the efficiency of a multijunction solar cell is shown in Table 4.3. It is found that the voltage drop across the tunnel junction decreases as the doping densities increase and hence the performance of the solar cell increases. An alternative approach is to use the MIC structure to shorten the top and bottom cell for a two-junction solar cell. However, according to our simulation results, the efficiency of the (AlGa)As/GaAs two-junction solar cell obtained from this method, for example 24%, is much lower








than that in Table 4.1. This inferiority is due to the increases of the shadowing loss and series resistance. It may be desirable for the optimal designs of multijunction solar cells only if the high quality tunnel junction is not feasible. The optimal designs of two-junction and three-junction solar cells are listed in Table 4.4 and Table 4.5 respectively. The temperature and air mass ratio dependence of the efficiency of the Alo.44Gao.56As/ GaAs and AloA4Gao.6As/GaAs/ Ino.53Ga0.47As are shown in Fig. 4.4 and Fig. 4.5.


4.5 Summary

An expert system approach to the optimal design of two-junction and threejunction solar cells has been presented in this chapter for the first time. A rulebased system with a couple of heuristic rules is implemented in the expert system to prune the search space of the design problem and hence to save the computation time. The optimal designs of Alo.44Gao.56As/ GaAs two-junction solar cell and Alo.44Gao.56As/GaAs/ Ino.53Ga0.47As three-junction solar cell with room temperature efficiencies for AMO of 30.01% and 35.3% respectively were obtained in the simulation.








Table 4.1 Simulation results of two-junction solar cells at room temperature for AMO.

(AlGa)As/GaAs GaAs/Ge (AlGa)As/(InGa)As (AlGa)As/Si V 1 (V) 1.52 1.22 1.24 1.44 F.F., 0.91 0.895 0.896 0.91 rci(%) 16.86 22.3 23.5 17.5 Eg (eV) 1.97 1.40 1.52 1.85' V.2 (V) 1.21 0.531 0.55 0.88 F.F.2 0.895 0.80 0.80 0.87 ?7c2(%) 13.1 8.68 9.37 10.3 E.2 (eV) 1.40 0.66 0.744 1.12
J.(A/cm2) 0.0164 0.0275 0.0284 0.0180
Total 77,(%) 30.01 31.0 32.9 27.8








Table 4.2 Simulation results of three-junction solar cells at room temperature for AMO.


(A1Ga)As/GaAs/(InGa)As (AlGa)As/GaAs/Ge Vol (V) 1.50 1.46 F.F.1 0.91 0.91 '77c (%) 16.7 15.85 Eg (eV) 1.98 1.96 V.2 (V) 1.21 1.21 F.F.2 0.896 0.896 77c2(%) 13.2 12.9 Eg2 (eV) 1.40 1.40 V.3 (V) 0.54 0.521 F.F.3 0.81 0.806 77.3(%) 5.34 4.99 Eg3 (eV) 0.744 0.66 J,,(A/cm2) 0.0164 0.0160 Total 77c(%) 35.3 33.78







Table 4.3 The effect of doping densities of the tunnel junction on the efficiencies of twojunction and three-junction solar cells.

Doping density AlO.44Gao.56As/ GaAs Alo.44Gao.56As/ GaAs/Ino.53Gao,47As
ND = 2x 1019
NA = 5X 101' 30.0% 35.3 %
ND = 1019
NA = 5X 1019 27.9% 30.06 %
ND = 1019
NA = 3x 1019 24.6% 24.54 %








Table 4.4 Optimal design of Alo.4Gao.s6As/GaAs two-junction solar cell at room temperature for AMO.


Alo.44Ga0.56As GaAs
NA (cm-3) 8.54 x 1015 1.05x 1018 ND (cm-3) 1.17 x 1015 1.65x 1017 Xj (pm) 0.05 0.477 Tj (Mm) 4.5 7.28 V (V) 1.52 1.21 F.F. 0.912 0.895 7 (%) 16.86 13.16
J.c(A/cm2) 0.0164 0.0164 E. (eV) 1.975 1.406








Table 4.5 Optimal design of Alo.44Gao.56As/GaAs/ Ino.mGao.47As three-junction solar cell
at room temperature for AMO.


Alo.44Ga0.56As GaAs Ino.53Gao.47As NA (cm-3) 3.33 x 1015 10'8 2x 1018 ND (cm-3) 7.94 x 1014 1.8x 1017 5x 1017 Xj (jim) 0.05 0.5 0.36 TJ (Asm) 7.1 10 2.0 Vo, (V) 1.50 1.21 0.541 F.F. 0.912 0.895 0.81 7 (%) 16.76 13.24 5.34 J (A/cm2) 0.0164 0.0164 0.0164 Eg (eV) 1.98 1.406 0.744

























P+ � WINDOW P GaA1As




n GaA1As n+ TUNNEL P+ DIODE P GaAs




n GaAs





n+ SUBSTRATE


Fig. 4.1 The schematic diagram of an AlGaAs/GaAs two-junction solar cell.
























MIDDLE CELL EMITTER CONTACT


TOP EMITTER CONTACT


MIC



AIGaAs


GaAs


1


P N P N 7- P++

P N


Fig. 4.2 The schematic diagram of an AlGaAs/GaAs/InGaAs three-junction solar cell.


TOP CELL


MIDDLE CELL

TUNNEL
DIODE

BOTTOM CELL


---------7 7 u






















































Fig. 4.3 Flowchart for the optimization of multijunction solar cells.






















33 i31 .29 27


I I I I 290 330 370 410


Temperature (K)




Fig. 4.4 Temperature and air mass ratio dependence of the efficiency of the A1GaAs/GaAs
two-junction solar cell.



























i39




%v436
1+ n



33



30
250 290 330 370 410 4W Temperature (K)




Fig. 4.5 Temperature and air mass ratio dependence of the efficiency of the AIGaAs/GaAs
/InGaAs three-junction solar cell.













CHAPTER 5
THEORETICAL CALCULATIONS OF ELECTRON AND HOLE MOBILITIES IN ALXGAI.XAS

5.1 Introduction

Although there exists some experimental and theoretical data for electron and hole mobilities in Al.GajiAs [80-1001, the absence of a well developed experimental characterization of these mobilities as a function of Al composition, dopant density and temperature makes the development of an accurate model difficult. In this chapter, we examine various scattering mechanisms of the Al.Gaj_.As and develop a good approximation for numerical simulations of these mobilities. The reasons why we probe the AlGajiAs alloy system particularly are twofold. First, Al.Ga__.As material is the leading candidate for the top cell of the multijunction solar cells because of its bandgap and lattice matching to the GaAs solar cell. Second, unlike the GaAs, there are few reports on high efficiency Al.GajiAs solar cells. We believe that this is due to the low mobilities and lifetimes of AlGal-.As. Therefore, the characterization of the dominant scattering processes that affect the electron or hole mobility and hence affect the diffusion length of Al.Gaj_.As are needed to facilitate the optimal design of a GaAs based multijunction solar cell.
Instead of doing a linear extrapolation of the experimental mobility data of GaAs and AlAs, which would not be accurate at all, we investigate in detail all the possible scattering mechanisms of Al.Gal-,.As, thus attaining an accurate model for numerical simulation. The modeling of electron mobility is discussed first; it is more complicate than that of hole mobility due to the different bands involved.
For a full analysis of the Hall electron mobilities in the AlGal-.As ternary compound, we first consider all the electrons involved in the conduction process such as









those in the F, X and L minima. It is known that the mobilities in different scattering mechanisms - for example polar optical scattering ppo, piezoelectric scattering utpe, deformation potential scattering Pdp, ionized impurity scattering pii, space charge scattering p,., alloy scattering pa and intervalley scattering pi, - have all been observed in the Al.Gal-,As materials at room temperature [80-83]. According to our calculations, pij and p, are dominant in limiting the electron mobility at temperatures as low as 100 K in the n-type Al.GajiAs with doping concentrations greater than 1018 cm-3. When the temperature increases, pp. and pi, become more dominant than other scatterings. As the temperature increases above room temperature, pp and pi, play the dominant role in the hall electron mobility of AlGal.As. It should be noted that although the model can be applied to all the different Al compositions from 0 to 1, we only show the results for Al composition between 0.10 and 0.45. This is because that bandgap region of the Al.Ga..As is probably the potential top cell for the GaAs based multijunction solar cells.
As for the hole mobility calculations, the intervalley scattering is not considered since different bands are not involved. Our calculations show that /Ldp, Iii and Ya are dominant in limiting the hole mobility in the p-type Al.Ga_.As with doping concentrations greater than 1018 cm-3 at 100 K. As the temperature increases, po becomes more and more dominant in limiting the hole mobility of Al.Gal-.As.

5.2 Scattering Processes Of Al,,Gal-..As

5.2.1 Polar Optical Scattering
In polar semiconductors such as III-V compounds, the interaction of carriers with the optical mode of lattice vibrations is known as polar optical scattering [84]. Because of the strong dipole moment set up by the optical modes in the polar crystals, the coupling between an electron or hole and the optical modes is likely to be much stronger than in non-optical crystals. The temperature dependence of the electron or hole mobility due to the polar optical scattering follows the analysis of Fortini et al.








[851 and is given by


25.54[exp(z) - 1]G(z)Tl/2(m*/mo)-3/2 (5.1)



where z = To/T with T., being the longitudinal optical phonon temperature, and the function G(z) was obtained by [851. T. is denoted by hwLo/KB, where wLo is the longitudinal phonon energy and m* is the effective mass of an electron or hole. The el and Eh are the low and high frequency dielectric constants, respectively. They are given by [86-87]



ci = 10.06x + 12.91(1 - x) (5.2) Ch = 8.16x + 10.91(1 - x) (5.3)


It should be noted that one of the crucial parameters needed for the calculations of electron mobility in the various minima for different values of x is the effective mass. The masses used in the polar optical scattering and others scatterings are the conductivity effective masses. They are given as [88]



mc, = 0.067 + 0.083x (5.4) mx = 0.32 - 0.06x (5.5) mcL = 0.11+0.3x (5.6)



Since the hole mobility does not involve different bands transitions, only one hole mobility effective mass is needed for the calculations. It is defined as [72]


1 VrM- + Mh, (5.7) m a -mlhr/mlh + mhhvfh-h










where mh and mhh are the light hole and heavy hole effective masses. They are given as [881



mjh = 0.087 + 0.063x (5.8) mhh = 0.62 + 0.14x (5.9)



5.2.2 Piezoelectric Scattering
If a III-V compound semiconductor consists of dissimilar atoms such as Al.Ga_. As where the bonds are partly ionic and the unit cell does not contain a center of symmetry, carriers may be scattered by longitudinal acoustic waves due to piezoelectric scattering. Since the lattice constant of Al.Gaj_.As material is almost independent of Al composition, its elastic constants should be nearly the same as for GaAs. With this assumption, p., is given as [89]


Ape = 43 T (m*/mo)3/2 (5.10)


According to our calculation / is negligible. It should be noted that all the parameters with same names in these different scattering processes are defined as the same unless specified otherwise.

5.2.3 Deformation Potential Scattering
The scattering of an electron or hole by the longitudinal acoustical phonon is an important scattering for many semiconductors near room temperature. The scattering is elastic if the electron or hole energy is much greater than the phonon energy and the change in electron or hole energy during scattering is small compared to the average energy of an electron or hole. The Idp due to the acoustical mode scattering has been derived by Bardeen and Shockley as [90]









J dp = 3.17 x 10-5 putT-3/2 E2(m*/Mo)S/2 (5.11)


where the mass density p = 5.37 (1 - x) + 3.60 x [86, 91] and ut is the longitudinal sound velocity. The deformation potential E1 is equal to (6.7 - 1.2x) [88].

5.2.4 Ionized Impurity Scattering
The scattering of an electron or hole by an ionized impurity center in a semiconductor is an example of elastic scattering. This is due to the fact that the mass of an impurity atom is much greater than that of an electron or hole. sii can be calculated from the expression given by [92]


64 V / 2f52(2KBT)3/2
~ (5.12) /i=Niqv- - In (24m*e�c.K2PT2 )
q2h2Ni


where N, is the ionized impurity density, and h, KB and co axe the Planck constant, Boltzmann constant and permittivity in vacuum, respectively.

5.2.5 Space Charge Scattering
The space charge scattering is caused by the crystal inhomogeneities. Such inhomogeneities may act like a mobility killer center and probably result from the grown-in defects. Weisburg [93] and Conwell and Vassel [94] have derived the expression for the pc due to the space charge scattering which is


3.2 x 109
Na Tm*/rO (5.13)


where N. and a are the density and cross section of the space charge scattering centers. Saxena and Mudares [81] showed that the best fitted value of Nsa was 9.5 x 104 cm-1, whereas Stringfellow [83] showed that N~a was a function of the alloy composition x, which is given by










No = 5 x 103 + 6.3 x 105x (5.14)



However, according to our calculations the optimal fitted value is 4.5 x 10' cm-1. This is in close agreement with the capture cross sections and defect densities obtained from our DLTS measurements.
5.2.6 Alloy Scattering
In the III-V ternary compound semiconductors of the type A.B-.C, the constituent elements A and B are randomly distributed among the C atoms. This random distribution constitutes a random potential component to the periodic potential, which causes an additional scattering process known as the alloy scattering. The alloy scattering mobility p. is given by [89, 95, 96]


p. = 52.3T-'/2{(m/mo)'/2x(1 - x)(E)2}-l (5.15)



where Ea = 0.3 + 0.011x [89] is the alloy scattering potential.
5.2.7 Intervalley Scattering
Since there is no band to band transition involved for hole, the intervalley scattering is for electron only. The scattering rate from a k state in the i valley to a state in the j valley has been derived by Fawcett el al. [97], the mobility limited by nonequivalent intervalley scattering can be expressed as [89, 98]


-v 4s2'3tAdp [T/T, + 2/3 + (2/3)(AE/KBT)]1/2+ =Zi(WjW)3/2 ( exp(T/T) - 1 [T/T - 2/3 - (2/3)(AE/KBT)]1/2)_(
1 - exp(-T,/T)


where AE is the subband gap among the minima involved in the process and Zj is








the number of nonequivalent intervalley. For equivalent intervalley scattering, the Zj is changed to (Zj - 1) and AE = 0. Since the masses of group V atoms are larger than those of group III atoms in AlxGal-.As, we have selected the longitudinal optical phonon involved in the process of scattering among the L and X minima, and hence T, = T. can be assumed [991. It should be noted that because there is only one F band minimum, there is no equivalent intervalley scattering in the F band.

5.3 Analysis and Discussion

To analyze the mobility data, the following assumptions are made : (a) the electrons or holes are scattered in a parabolic band, (b) the various scattering mechanisms are independent of each other, and (c) Matthiessen's rule for calculating the electron or hole mobility in AlxGal__As is valid. The F, X, L conduction band structure as a function of alloy composition plays an important role in determining the electron mobility. For the electron mobility in the composition range 0 < x < 0.32, the transport properties are primarily determined by the electrons in the F conduction minimum, and the effects of the L and X minima may be negligible. Similarly, for x is greater than 0.6, the X minima plays the major role. In the intermediate band crossover composition range, i.e., for 0.32 < x < 0.6, the effects of F, L and X minima must be taken into account. Considering the three-valley conduction, nH and /tH can be expressed as [100]


(1 + (nxptx)/(nrjsr) + (nLAL)/(nr/ir))2
= nr1 + (nx/nr)(,Ux/r)2 + (nL/nr)(PL/Yr)2 (5.17)



and


PH = Pr1 + (nx/nr)(Mx/r r)2 + (nL/nr)(YL/ r)2 (5.18)
1 + (nxpx)/(nrPr) + (nLYL)/(nrpr










Assuming that the Boltzmann statistics are valid for the electron concentrations in the crystals studied, the following approximation is also valid.


nr = Nwexp(EF/KBT)


(5.19)


d
rnL _-( L3/2 exp(-AErL/KBT) nr mr


(5.20)


d
nr (Mr)3/2(AE/KT)


(5.21)


where A Erx and A ErL are the I-X and F-L intervalley separation and md is the density of state effective mass. The empirical formulae for the conduction and density of state effective masses are as follows [88]:


d
mr = 0.067 + 0.083x

d
mL = 0.85-0.14x
md = 0.56 +0.1x


(5.22) (5.23) (5.24)


The energy band gaps of the three different conduction band minima are given respectively by


and


and










r { 1.424 + 1.247x for x < 0.45 g 1.424 + 1.247x + 1.147(x - 0.45)2 for 0.45
EX = 1.900-+ 0.125x+ 0.143x2 (5.25) EL = 1.708 + 0.642x (5.26)
5



Figures 5.1 and 5.2 show the results of electron and hole mobilities respectively for the Al composition between 0.10 and 0.45 and three different NsQ values. From Table 5.1, it is clear that our results are in better agreement with the experimental data than those of others. The major discrepancy among these data is the calculation of the space charge scattering. The product of the density of the scattering center
(N.) and the capture cross section (a) used in our calculations is 4.5 x 10' cm-1 which is less than those given in references [81] and [83].
Figures 5.3 and 5.4 show the room temperature electron and hole mobilities of different scattering processes as a function of Al compositions at ND = 1.5x 1016 and NA = 101l cm-3 respectively. According to Fig. 5.3, polar optical phonon scattering, intervalley scattering and space charge scattering are the three dominant scattering processes limiting the electron mobility at room temperature. However, polar optical phonon scattering, alloy scattering and deformation potential scattering are the three dominant scattering processes for hole mobility as shown in Fig. 5.4.
The Al composition and temperature dependence of the electron and hole mobilities of Al.GalixAs at ND = 10" and NA = 1018 cm-3 are shown in Fig. 5.5 and Fig. 5.6 respectively. According to our calculations, ionized impurity scattering and space charge scattering are the two dominant processes at low temperatures 100K and 200K, for the electron mobility case. As the temperature increases, polar optical phonon scattering and intervalley scattering become more dominant. For the hole mobility case, deformation scattering and ionized impurity scattering and alloy








scattering are the three dominant processes at low temperature. When the temperature increases, the polar optical phonon scattering takes over the role of the ionized impurity scattering in limiting the hole mobility of Al.Gal-.As.
Figures 5.7 and 5.8 show the temperature and doping density dependence of the electron and hole mobilities of Al0.*Ga0.62As. It is indicated that the doping level has little effect on either electron or hole mobilities of Al0.sGao.62As if doping density is less than 1018 cm-3. For doping density above 1018 cm-3, however, there is a huge decrease. This is because at low temperature and high doping level, the ionized impurity scattering is the most dominant process among all scattering processes.

5.4 Summary

This chapter presents the results of a comprehensive study of the scattering processes of Al.Gal-.As alloy system. Calculations of the electron and hole mobilities of for Al.Gal-.As as a function of doping density, temperature and Al composition have been carried out. It has been found that polar optical phonon scattering, intervalley scattering and space charge scattering are the three dominant processes for electron mobility of Al Gal-.As at room temperature. As for the low temperature and high doping density such as 10i8 cm -3 or higher, ionized impurity becomes dominant. For the hole mobility case, it was found that the influences of the alloy scattering, polar optical phonon scattering and deformation scattering are significant. Our theoretical calculations show good agreements with the experimental data.








Table 5.1 Electron mobilities of the Alo.3sGa0.62As and Alo.19Gao.81As.

x(%) ND(cm-3) p (cm2/V-s) N.Q(cm-1) experiment

0.38 1.5x 1016 1432 45000 1200
0.38 1.5x 1016 855 244000[78] 0.38 1.5x 1016 1231 95000 [80]

0.19 1.5x 1016 2689 45000 2700
0.19 1.5x 1016 2221 95000
0.19 1.5X 1016 1989 128500




























"4






1000
0




0.10 0.17 0.24 0.31 0.38 0.45 X composition Fig. 5.1 Electron mobilities vs. Al composition and NsQ values. + this study, solid
box: NsQ obtained from [81] and *: NsQ obtained from [83].



























Ui
00









InI
0.10 0.17 0.24 0.31 0.38 0.45 X composition
Fig. 5.2 Hole mobilities vs. Al composition and NsQ values. + this study, solid box:
NsQ obtained from [81] and * : NsQ obtained from [83].
























10




5
10

0
: : :Use 10




3

0.10 0.17 0.24 0.31 0.38 0.45 X composition Fig. 5.3 Room temperature electron mobilities of different scattering processes vs. Al
composition at ND = 1.5 X 1016 CM-3.





92








5






4 Upe

10


Ust
4 q4

0 Udp
10



I IT 10
0.10 0.17 0.24 0.31 0.38 0.45 X composition Fig. 5.4 Room temperature hole mobilities of different scattering processes vs. Al composition at NA = 1018 cm-3.















5







10 10




10 3





0.10 0.16 0.22 0.28 0.34 0.40 X composition Fig. 5.5 Al composition and temperature dependence of electron mobility of AlxGal, As
at ND = 1017 cm-3.




Full Text

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AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS By CHUNE-SIN^H A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1988 fioRm mmit.

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ACKNOWLEDGEMENTS The author wants to express his deep gratitude to the chairman of his supervisory committee, Dr. Sheng S. Li, for his guidance and encouragement through the entire research process. The author also appreciates the other members of his supervisory committee, Drs. Arnost Neugroschel, Dorothea Burk, Gijs Bosman and Yuan-Chieh Chow for their participation on the committee. Special thanks are given to Dr. R. Y. Loo for the irradiated solar cell samples and measurements and to Mrs. Li, his host family Mr. and Mrs. David Wilmot, and Mrs. Anne White for their kindness to his family. The author is also grateful to his colleagues at the Device Characterization and DLTS Lab for their helpful discussions. The financial support of the Universal Energy Systems Incorporation is greatly appreciated. Finally, the author thanks his parents and family, especially his late father, for their love, expectation and patience throughout his graduate study. 11

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ii ABSTRACT vi CHAPTERS ONE INTRODUCTION 1 1.1 Introduction 1 1.2 Fabrication Technology of Solar Cells 2 1.3 A Synopsis of this Research 3 TWO MODELING OF PROTON AND ELECTRON IRRADIATED SOLAR CELLS 6 2.1 Introduction 6 2.2 Displacement Defects 7 2.2.1 Defect Formation by Proton Bombardment 8 2.2.2 Defect Formation by Electron Bombardment 10 2.3 Degradation Calculation of Short-Circuit Current 13 2.4 Degradation Calculation of Open-Circuit Voltage 15 2.5 Degradation Calculation of Conversion efficiency 16 2.6 Results and Discussion 17 2.7 Summary 20 THREE A NEW METHOD FOR OPTIMAL DESIGN OF GAAS SINGLE-JUNCTION SOLAR CELLS 36 3.1 Introduction 36 3.2 Device Modeling for GaAs P/N Junction Solar Cells 37 3.3 Effects of Extrinsic Parameters on Device Modeling 40 3.3.1 Effect of Antirefiection Coating 41 3.3.2 Effect of Grid Design 41 3.3.3 Effect of Series Resistance 42 iii

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3.3.4 Effect of High Sun Insolation 43 3.3.5 Effect of Irradiation 43 3.4 Constrained Optimization Technique 43 3.5 Optimal Design of GaAs SingleJunction Solar Cells 45 3.6 Summary 47 FOUR AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF MULTIJUNCTION SOLAR CELLS 56 4.1 Introduction 56 4.2 Device Modeling of Multijunction Tandem Solar Cells 57 4.3 Concept of the Expert System Approach 59 4.3.1 Problem Formulation 60 4.3.2 Optimization and Heuristic Rules 63 4.4 Results and Discussion 64 4.5 Summary 67 FIVE THEORETICAL CALCULATIONS OF ELECTRON AND HOLE MOBH.ITIES 78 5.1 Introduction 78 5.2 Scattering Processes of AlxGai.^As 79 5.2.1 Polar Optical Scattering 79 5.2.2. Piezoelectric Scattering 81 5.2.3. Deformation Potential Scattering 81 5.2.4. Ionized Impurity Scattering 82 5.2.5. Space Charge Scattering 82 5.2.6. Alloy Scattering 83 5.2.7. Intervalley Scattering 83 5.3. Analysis and Discussion , 84 5.4. Summary 87 SIX SUMMARY, CONCLUSION and RECOMMENDATIONS 97 6.1. Summary and Conclusion 97 6.2. Recommendations 99 APPENDIX A A COMPUTER PROGRAM FOR CALCULATING THE TOTAL NUMBER OF DISPLACEMENT DEFECTS 101 APPENDIX B A COMPUTER PROGRAM FOR CALCULATING THE DEGRADATION OF SHORT-CIRCUIT CURRENT Ill IV

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APPENDIX C AN EXPERT SYSTEM PROGRAM FOR OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS 143 APPENDIX D INPUT PARAMETERS FOR THE OPTIMAL DESIGN OF (ALGA)AS/GAAS/INo.53GAo.47AS THREE-JUNCTION SOLAR CELLS 182 REFERENCES 188 BIOGRAPHICAL SKETCH 194

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS By Chune-Sin Yeh August 1988 Chairman: Sheng-San Li Major Department: Electrical Engineering The use of an expert system approach to the optimal design of single-j unction and multijunction solax cells is a potential new design tool in photovoltaics. This dissertation presents the results of a comprehensive study of this new design method. To facilitate the realistic optimal design of the two-terminal monolithic single-j unction and multijunction tandem solar cells, a rule-based system was established by adopting the experimental data and/or semiempirical formulae used today for those design parameters. A numerical simulation based on the displacement damage theory was carried out to study the degradation of AlGaAs/GaAs solar cells after proton or electron irradiation. The damage constant of the minority carrier diffusion length, an important design parameter of a solar cell for space application, was calculated. An efficient Box complex optimization technique with minor modifications is analyzed and applied to accelerate the convergence rate of the algorithm. Design rules were implemented in order to reduce the search space of the optimal design and to make a compromise in the tradeoff between the conflicting criteria for selection. The computation time for the optimal design is very much reduced by adopting these VI

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rules. Realistic optimal design of singlejunction and multijunction solar cells were obtained and verified from the expert system and then compared with the state of the art technology. Finally, theoretical calculations of the electron and hole mobilities of AlxGai_xAs were performed. The results show that electron mobilities of AlxGai_xAs are quite sensitive to the A1 composition, temperature, doping density and defect density. . This knowledge is indispensable to the fabrication of a high efficiency AlGaAs top cell of a GaAs based multijunction solar cells. Vll

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CHAPTER 1 INTRODUCTION 1.1 IatiQdu.ction For the pa.st three decades, researchers around the world have been involved in studying the degradation of solar cells induced by energetic electrons or protons for space applications. As the number of satellite launches heis increased dramatically, the need for a space power system using solar cells has increased significantly. Moreover, by knowing the degradation of performance in the electron or proton irradiated solar cells, one is able to design a solar cell that gives an optimum end of life conversion efficiency for the specific space mission. This should lead to innovations in design and fabrication such as shallow junctions, window layers and multijunction solar cells. The objective of this dissertation is to use an expert system approach to study the optimum design of single-j unction and multijunction tandem solar cells for both space and terrestrial applications. It should be noted that because the theoretical conversion efficiency increases very slowly beyond three-junction tandem cells [1] and the technology of fabricating cells beyond three junctions is difficult in practice [2], the multijunction study in this work waa focused only on the two-terminal monolithic structures of single-junction, two-junction and three-junction solar cells. Although recently multijunction solar cells have become increasingly important for space applications, the multijunction cell structure in itself is not a new concept. In 1955, Jackson [3] proposed that the efficiency of solar cells could be increased significantly by stacking one or more cells composed of different semiconductor materials. Since then, many other researchers [4-26] have proposed varying approaches to the optimal design of the multijunction cells, theoretically and/or experimentally. 1

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2 However, little progress in mult ij unction solar cell design has been made due to (1) lack of reliable data on the electrical and optical parameters of the solar cell materials, (2) lack of adequate device modeling, (3) inefficient optimization algorithm. In an effort to provide an optimal and realistic design of multijunction solar cells for space and terrestrial applications, the research reported here makes use of an expert system approach. In this chapter an overview of the major solar cell fabrication technologies now in use is presented. Different processes may have varying effects on the quality and efficiency of solar cells. Finally, a synopsis of our research is given. 1.2 Fabrication Technology of Solar Cells Although the potential performance advantages of the GaAs single-junction and III-V compounds beised multijunction solar cells have been discussed [2-22], their practical realization remains a problem, due to the absence of a good fabrication technology. Among those fabrication processes developed for III-V compound, liquid phase epitaxy (LPE), molecular beam epitaxy (MBE) and metal organic chemical vapor deposition (MOCVD) have proven successful [19, 27-31] and hence the use of the multijunction structure in solar cells has become important and promising. However, since there is no single fabrication technique which is superior to the other two in terms of crystal purity, doping level, throughput and controllability of ceU thickness and uniformity, it is still too early to say which of these three technologies will dominate the future development of single-junction and/or multijunction solar cells. Therefore, in this brief review, it seems most useful to describe the advantages and weaknesses of each fabrication process, but not to give details about the processes themselves. LPE is the reference technology for the growth of epitaxial layer in III-V compounds. It offers high purity crystal and excellent transport properties, minimum contamination, uniform thickness control and reductions in point defect and disloca-

PAGE 10

3 tion. It remains an ea^y and cheap method for growing high efficiency AlGaAs-GaAs solar cells. However, the throughput is small. MOCVD is a standard system used for making high efficiency III-V solar cells. It is a flexible technology that allows the growth of a quite complicated multilayer structures with different doping levels and different compositions. It can grow very thin layers with precise doping and thickness. And MOCVD has large throughput. However, materials made by MOCVD may have carbon contamination and high defect density. The advantages of MBE are similar to those of MOCVD. While, MBE has only a medium throughput, it can produce high purity crystal without contamination. In multijunction solar cells, the most important feature of MOCVD and MBE is the ease of incorporating superlattice or tunnel junction into the cascade solar cell structures. This can reduce the dislocation density in the top cell and hence reduce the recombination loss and increase output voltage. 1.3 A Synopsis of this Research Based on the survey given in this chapter, it would appeare that the realistic optimal design of single-j unction and multijunction solar cells could be realized through the establishment of the expert system. Moreover, it should be apparent that the lack of full knowledge of the material parameters and their relations — mobility, diffusion length, etc. — as well as the complexity of the optimization technique, would limit the development of a general analytical model which accounts for all variations of the design parameters. It is the goal of this research to generate an expert system for solving this optimal design problem. The first step was to set up a rule-based knowledge data base for the design parameters listed above by adopting the experimental data and/or semiempirical formulae in use today. This reduced the complexity of the numerical simulation required to solve the problem. The next step was the development of a model of the

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4 proton or electron irradiated solar cells. The model developed here is based on the displacement damage model. A computer program for calculating the degradations of short-circuit current, open-circuit voltage and conversion efficiency of a solar cell was coded. This program included the calculations of the damage constant of the minority carrier diffusion length. This parameter is important for the optimal designs of irradiated solar cells. In addition, an efficient optimization algorithm was implemented for the expert system. Some heuristic rules were applied to reduce the search space of the design problem and to make some compromise in the tradeoff between the conflicting criteria of selection. Realistic optimal designs were obtained from the simulations and their performance compared with the state of the art technology. In Chapter 2, a numerical model of proton and electron irradiated solar cells is presented. In this model it is cissumed that the radiation induced displacement defects form an effective recombination center which reduces the minority carrier diffusion length and hence degrades the short-circuit current Igc, open-circuit voltage Vqc and conversion efficiency rjc of the solar cell. It should be noted that although the numerical model applies to a special ca^e which is limited to the normally incident protons or electrons, this model can be extended to simulate the real space environment where the incoming protons or electrons are omnidirectional. In Chapter 3, a new computer model for truly optimizing the structure of GaAs single-junction solar cell for both space and terrestrial application is proposed. The model, however, can apply easily to other solar cell systems. It not only takes into account the effects of the intrinsic structural parameters such as junction depth Xj, cell thickness Tj, doping densities Na and No, surface recombination velocities Sp and Sn, but also incorporates the extrinsic structure parameters. And an efl[icient Box optimization algorithm [32] with minor modifications is implemented. In Chapter 4, an expert system approach to the optimal design of multijunction

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5 solar cells for both terrestrial and space applications is described for the first time. An expert system is a knowledge-intensive computer program. The knowledge of an expert system consists of facts and rules. The facts constitute a body of information that is widely shared, publicly available and generally agreed upon by experts in the field [33-34]. The rules are those if-then rules that characterize expert level decision making in the field. In general, a good and robust expert system must include as many facts and rules as possible. However, because of the availability of the earlier research reports and the tradeoff between the number of rules and the computation time, this expert system is currently limited to the AlGaAs, GaAs, Ino. 53 Gao. 47 As, Si and Ge materials. In Chapter 5, the theoretical calculations of the electron and hole mobilities in AlxGai_xAs are reported. It was found that the mobilities of AlxGai_xAs are quite sensitive to the A1 composition, temperature, doping density and defect density. This knowledge is indispensable to the fabrication of a high quality and high efficiency AlGaAs top cell of the GaAs based multijunction solar cells. The final chapter. Chapter 6 , summarizes the materials and offers suggestions for future research. In addition to the six chapters in this dissertation, there are four appendices. In Appendix A, a computer program for calculating the total number of displacement defects induced by the energetic proton and electron bombardments is presented. A computer program for manipulating the degradation of short-circuit current, opencircuit voltage, conversion efficiency and damage constant of the minority carrier diffusion length is in Appendix B. An expert system program for optimal design of single-j unction and multijunction solar cells is in Appendix C. Finally, the input parameters for (AlGa)As/GaAs/(InGa)As threejunction solar cells are shown in Appendix D.

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CHAPTER 2 MODELLING OF PROTON AND ELECTRON IRRADIATED SOLAR CELLS 2.1 Introduction The advantages offered by single junction GaAs solar cells for space applications are their high power conversion efficiency, their radiation hardness and their relative insensitivity to temperature compared to silicon solar cells. Although GaAs solar cells have not been extensively flown in space, radiation experiments performed on earth have demonstrated their superior hardness as compared to silicon cells. In this chapter we model the radiation degradation characteristics of GaAs single-junction solar cells and GaAs based multijunction cells under proton or electron radiation environments. An improved numerical model is offered for computing the displacement defect density, the damage constants of the minority carrier diffusion lengths and the degradations of the short-circuit current Igc, open-circuit voltage Voc and the conversion efficiency rjc in proton or electron irradiated singlejunction and multijunction solar cells is presented. The model which we use in this study is based on the displacement damage theory in semiconductors. It is assumed that the radiation induced displacement defects form effective recombination centers which reduce the minority carrier diffusion length and hence degrade the Ec, Voc and rjc of the solar cell. It should be noted that although our numerical model applies to a special case, limited to the normally incident protons or electrons, this model can be extended to simulate the real space environment where the incoming protons or electrons are omnidirectional. In earlier work, Wilson et al. [35] and Yaung [36] have also used the displacement damage model to study the GaAs solar cells. We have extended their work 6

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7 and have obtained a better correlation between theory and experimental findings by including both electron and hole capture cross sections in our calculations. The electron and hole capture cross sections were determined for the GaAs and AlGaAs cells after proton or electron irradiation using deep level transient spectroscopy (DLTS) measurements. In addition, the model includes calculations of the normalized opencircuit voltage, conversion efficiency and damage constant of the minority carrier diffusion length. The damage constant of the minority carrier diffusion length for the irradiated solar cell is one of the important parameters in optimal design of a solar cell for space application. In addition to the single-junction GaAs solar cell, a simple model was developed for calculating the displacement damage in the proton and electron irradiated Alo.33Gao.67As/GaAs two-junction solar cell and in the Alo.35Gao.65As/GaAs/Ino.53Gao.47As or Ge three-junction solar cell. The calculations indicate that the degradation rate in each cell varies greatly and depends critically not only on the energy, fluence and the direction of the incident proton or electron, but also on the thickness of each cell in the multijunction cells. Excellent agreement was obtained between our calculated values and the measured Isc, Voc and Tjc in proton or electron irradiated AlGaAs and GaAs solar cells for proton energies from 100 KeV to 10 MeV and fluences from 10^° to 10^^ cm“^, and for 1 MeV electron and fluences varying from 10^^ to 10^® cm~^ under normal incidence conditions. 2.2 Displacement Defects A solid may be affected in two ways by the energetic particle bombardment [37]. First, the lattice atoms may be removed from their regular lattice sites and produce displacement damage. Second, the irradiating particles may cause change in the chemical properties of the solid via ion implantation or transmutation. In our model, it is assumed that the dominant defect produced by the incident electrons or protons is due to lattice displacement. Under this assumption, an atom will invariably

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8 be displaced from its lattice site during collisions if its kinetic energy exceeds the threshold energy, Tj for the atomic displacement, and will not be displaced if its kinetic energy is less than Tj [38]. 2.2.1 Defect Formation by Proton Bombardment When energetic protons collide with atoms, the energy transferred to the struck atoms is the most important consideration in evaluating irradiation damage [38]. The number of defects formed by an energetic proton coming to rest in the solar cell are related to the energy of the proton, the transferred energy and the threshold energy, Td of the solar cell. Given the conservation of energy and momentum, it follows that the maximum energy which can be transferred to the struck atom in a primary head-on collision with energy, E, is ^ 4M1M2E ^ “ (Ml + M2)2 where E is the initial energy of the proton and Mi and M 2 are the masses of proton and struck atom respectively. This transferred energy may range from zero in a glancing collision to a maximum Tm in a head-on collision. As for a proton, the energy transferred in a collision can be calculated by ignoring the screening effect. Therefore, the scattering in a proton collision obeys the Rutherford differential cross section d
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9 the Rydberg constant and a<) is the Bohr radius. Since the defects occur when the energy transferred is greater than Ta, the displacement cross section cTp is given by CTp = d.7p = C(i J-d (2.4) The average energy transferred, T, in a Rutherford collision which displaces atoms can be calculated as follows: _ ffr TdT, dap = ^ln(— ) (2.5) If the transferred energy is sufficiently large (T >> Ta), additional displacement can be produced by the recoiling nucleus before it comes to rest at an interstitial site. The average recoil displacements , v, produced by one initiating proton collision event is given cLS a function of Tm, on the assumption that half of the recoil energy produces further displacement and the other half is dissipated in other processes described in [35, 38] Vp(E) = 1.0 ^ 2(TM-Td) T^) for Td < Tm < 2Ta for 2Td <= Tm (2.6) Since the mass of the proton is heavier than that of an electron, the velocity of an energetic proton is slower than an electron with the same energy. Thus, a proton has the potential of multiple scattering before coming to rest. The displacement defects D(Eo) induced by an energetic particle under normal incident condition are DP(Eo) = r NapVpdP Jo (2.7)

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10 where P is the path length traveled by a proton in coming to rest, and all the other variables have the same definitions as in Eqns. 2.4 and 2.6. As for the omnidirectional incident irradiation, the total displacement defects Dom are given by [39] DPe = 27t dE d(cos 0)[DP(Eo) DP(Eo(x/ cos 0))] (2.8) Jo Jo where 0 is the incident angle of the particle toward the solar cell. 2.2.2 Defect Formation by Electron Bombardment Because of the small meiss of an electron, the electron must travel at a relatively high velocity in order to produce displacement. The maximum energy which can be transferred in a collision by an electron with mass m and kinetic energy E is 2(E + 2mC^)E — m;c5 — where C is the velocity of light, and m and M 2 are masses of the electron and struck atom respectively. Consequently, the nonrelativistic Rutherford scattering is inadequate for the electron. Relativistic Coulomb scattering has been treated by MckinleyFeshbach as follows [40]: d(7e = 47T(aoZ2E,)^(l-/3^) [1-^V + ^Z2/?(J— Tm Tm Tm ^ )/137]Tm^ (2.10) where ^ is the electron velocity ratio to the velocity of light C. All other variables have the same definition as those in the previous equations. Integration of equation 2.10 yields the displacement cross section for an incident electron:

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11 47r(aoZ2Er)2(l -/5^)fTM , ^32 1 / t. / rr ^ , o o( ^ i / t ^ M = ^2(34^4 ln(T/TM) + 27 ra^(W — -l)7 ra/ 31 n(TM/Td)J ( 2 . 11 ) where a is equal to Z 2 /I 37 . The average energy transferred during a collision is Tm ln(TM/Td) — ^^(Tm — Td) + 27ra^(TM — vA'm ~ Ta) — 7ra/3(TM — Ta) Tw/Td 1 ^ ln(T/TM) + 2ia/?(yS _ i) _ »a/?ln(TM/Td) ( 2 . 12 ) Thus, if the transferred energy is sufficiently large (T >> Ta), the mechanism of recoil displacement produced by an electron is similar to that of a proton. The average recoil displacements, Ve, produced by one initiating electron collision event are given [35, 38] Ve(E) = 0.0 1 for f (E) < Td for Td < T < 2Td 1 + ^ for2Td
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12 mass is small, we can neglect the effects of multiple scattering. As for omnidirectional incident irradiation, the total displacement defects Dom are given by [39] DEom = 27 t dE /' d(cos ^)[DE(Eo) DE(Eo(x/ cos 0))] (2.15) JQ Jo Now, according to Eqns. 2.7, 2.8, 2.14 and 2.15, for numerical calculations of the total number of displacement defects, the threshold energy, path length, penetration depth (range) and the reduced energy of protons or electrons after travelling an x distance must be given. Although the threshold energies for GaAs and Ge are given as 9.5 eV and 27.5 eV respectively, those of InGaAs and AlGaAs are still unknown. In this respect, a linear extrapolation has been made to calculate the two unknown values as follows: Td(Al,Gai_xAs) = 0.5 x [Td(Al) x x + Td(Ga) x (1 x) + Td(As)] (2.16) and Td(InxGai_xAs) = Td(InAs) x x + Td(GaAs) x (1 x) (2.17) In addition to the threshold energy, the path length or range an electron or proton coming to rest is also unknown. Thus, we adopt the path length and the range from the data given recently by Janni [41] for the proton case and by Page et al. [42] for the electron case. Since these figures are only given for elements, and the GaAs, (AlGa)As and (InGa)As are compound materials, approximations were made in calculating path lengths and ranges based on the following assumptions [41]: (2.18)

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13 Rc ^ Ri where Pc and Rc are the path length and range of the compound materials respectively, Rj and Pj are data for each element and W; is the weighting function of each element. The least square method was employed to fit those data in order to obtain the expressions for P and R for the solar cells studied here. As for the reduced energy, of protons or electrons after penetrating a distance X, the fitting process is similar to those for path length and range. These empirical formulae were obtained through the computer programs described in Appendices A and B. (2.19) 2.3 Degradation Calculation of Short-Circuit Current To derive an expression for the short-circuit current in an irradiated cell, the following simplified assumptions were made [35, 43, 44]: (1) radiation-induced defects do not greatly alter internal cell electric field, (2) radiation-induced defects alter the cell operation mainly through change in minority carrier lifetimes in the bulk, and (3) radiation-induced displacements within the solar cell form recombination centers for minority carriers of electron-hole pairs produced by photon absorption. When sunlight impinges on a solar cell, the short-circuit current generated from a solar cell is given by Isco(A) = / Tj{x)p{x, A)dx (2.20) Jo where tj(x) is the current collection efficiency, t is the cell thickness, and p(x) = Kaexp(— ax) (2.21)

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14 is the photon generation rate at depth x, K is the integrated solar photon flux in the absorption band and a is the absorption coefficient. After proton or electron irradiations, defects are created within the materials and the solar cellÂ’s short-circuit current is decreased. To rewrite the expression for the short-circuit current after proton or electron irradiation, IÂ’sc, we need to add a loss term [1 F(x)] to the integral of Eqn. 2.20, to account for the carrier recombinations caused by the newly generated defects. The expression for the recombination loss coefficients, F(x), is [35] F(x) = 1 E2[V6<7,\D{E,) D(E,.)|] (2.22) and = / ^(x)(l F(x))/>(x,A)dx Jo (2.23) where E 2 (z) is the exponential integral of order 2, <7r is the electron or hole capture cross section, (f> is the proton or electron fluence. Ex is the reduced proton or electron energy after penetrating a distance, x and Xj is the junction depth. The normalized I,c degradation can thus be calculated by using the expression ^ Isc(A)dA Isco *'^1 IsCo(A)dA (2.24) Ax and A 2 denote the shortwave and long-wave limits of the total useful solar spectra for the solar cell.

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15 2.4 Degradation Calculation of Open-Circuit Voltage Once the damage constants of the minority carrier lifetimes or diffusion lengths are known, calculations of Voc degradation are straightforward. According to the Shockley, Read and Hall theory [45], the minority carrier lifetime is inversely proportional to the defect density, Nt and is given by [45] 1 ~ N.V,|,<7„,p where Vth is the thermal velocity and <7n,p is the capture cross-section. Thus, the minority carrier lifetime of the solar cell after proton or electron irradiation, rp,n^ can be calculated from Eqn. 2.25 by substituting the values of the displacement damage density Nt and the capture cross-sections from the DLTS measurements. It should be noted that Nt can be calculated directly from Eqns. 2.7, 2.8, 2.14 and 2.15 for proton and electron irradiations respectively. Knowing Tp the damage constants of the minority carrier lifetimes K^p and are given by [46, 47] as i r + (2.26) 'p i i + (2.27) 'n 'n Now using the relation that = D t, it is easy to obtain the damage constants of the minority carrier diffusion lengths Kli, and KLp. The open-circuit voltage before irradiation is expressed by V OC iiKbT q io (2.28) where n is the diode ideality factor and Iq is the saturation current of the diode. Kb is

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16 the Boltzman constant and q is the electron charge. In general, the current conduction mechanism through the diode is dominated by diffusion or by recombination currents. When the diffusion current dominates, the diode’s ideality factor, n, is equal to 1, and lo is given by lo = (qn?A) l^LpoNo UoNa/ (2.29) where nj is the intrinsic carrier density, A is the area of a solar cell. Dp and Dn are the hole and electron diffusion coefficients, respectively, Lpo is the hole diffusion length and Lno is the electron diffusion length and No and Na are the donor and acceptor dopant densities. When the recombination current at the junction dominates, the ideality factor, n, is equal to 2 and the saturation current, Iq, is Io = qnjWA (2.30) where W is the depletion width, and Tp and t„ are the hole and electron lifetimes in the n-region and p-region, respectively. In general, the n value will vary between 1 and 2 because both mechanisms contribute to the transport mechanism. As for the open-circuit voltage of the irradiated cell, V’oc, it is given by v;. = ln(^ + 1) (2.31) 4 ^0 where I’o is the saturation current after irradiation. 2.5 Degradation Calculation Of Conversion Efficiency The maximum power conversion efficiency of a solar cell is the product of the short-circuit current, open-circuit voltage and the fill factor. Experimentally it has

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17 been observed that the fill factor of the cell remains unchanged after irradiation [48]. Thus, the normalized conversion efficiency after proton or electron irradiation is n ' r V' Vco Isco^oco 2.6 Results and Discussion (2.32) Figure 2.1 shows the baseline design of the GaAs solar cell. The GaAs and Alo. 33 Gao. 67 As solar cells were fabricated at Hughes Research Laboratories (HRL) by using infinite solution LPE technique. The base region of the solar cell is n-type (1 X lO^’^ cm~^). A wide bandgap, p+, (2 x 10^® cm"®), Be doped AlGaAs window layer was grown on the top of the base layer to passivate the GaAs or Alo. 33 Gao. 67 As surface and to reduce its surface recombination velocity. During the growth of the window , Be Wcis diffused into the base region to form an electrical junction. The thickness of the window layer and the junction depth were measured to be 0.35 and 0.5 fim, respectively. The GaAs and Alo.33Gao.e7As solar cells’ conversion efficiency, before the irradiation, were tested at AMO, 1 sun condition to be 16.0% to 17% and 7.5% to 8.0 % respectively. The proton irradiation experiments for various proton energies and fluences were performed at HRL (low energy protons), California Institute of Technology (medium energy protons) and Univeristy of California, Davis (high energy protons). The details of the experiments and results were given in earlier publications [49-53]. Table 2.1 lists all the cell parameters used in calculating the degradation of the short-circuit current and these parameters are identical to the actual GaAs and Alo ssGao.erAs solar cells used in the proton and electron irradiation experiments. Figures 2.2 and 2.3 give the calculated proton range and the number of displacement damage defects per incident proton as a function of proton energy. For example, a 300 KeV proton will penetrate 2 fim deep into GaAs solar cell (Fig. 2.2) and produces about 100 defects per cm® (Fig. 2.3). Figure 2.4 shows the remaining short-circuit

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18 current after proton irradiation as a function of the proton energy. The solid lines are the calculated short-circuit currents and the circles are the experimental values. Examining both Fig. 2.2 and Fig. 2.4, it is clear that only proton energies greater than 50 KeV are seen to create damage to the cell. The 100 KeV protons, which are stopped at about 0.8 fim below the surface, create damage close to the junction. The 200 KeV protons, which are stopped at about 1.35 fim below the surface cause damage throughout the p-region and the junction. The 290 KeV protons, which are stopped at about 2.0 fj,m below the surface produce damage in the bulk of the n-GaAs layer. Protons with energies greater than 1 MeV will pass through the entire cell, hence, will create less damage to the cell than the low energy protons. Thus, the calculated short-circuit current agrees closely with the experimental values. The damage constants of the minority carrier diffusion lengths, Klh and I
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19 protons (Fig. 2.3) with the same energy. This is because the energy left to the cell from the incident electron is much less than that from the proton, owing to the small effective mass of the electron. Therefore, more electron fluences are needed to cause degradation of solar cells. Table 2.4 indicates the short-circuit degradation of 1MeV electron irradiated GaAs and Alo.33Gao.67As p-n junction cells for three different fluences. It shows that the radiation hardness of Alo.33Ga o.erAs is greater than that of GaAs. Table 2.5 summarizes the calculated and measured cell degradation after 1-MeV electron irradiation for both normal incidence and omnidirectional incidence cases. The calculated results again show the strong agreements with the experimental data for electrons fluences of 10^“* and 10^® cm~^. Figure 2.7 gives the flow chart for simulating the degradation of multijunction solar cells. In general, a middle cell or bottom cell may be affected by the incident energetic protons or electrons only when the reduced energy, after penetrating the top cell or middle cell respectively, is greater than zero. Therefore, in addition to the fluences and initial energy of the incident particle, the thickness of each cell is critical in calculating degradation of the multijunction solar cells. Since there are no experimental data on proton or electron irradiated multijunction solar cells available for comparisons, for simplicity the nearly optimal design of the Alo.33Gao.67As/ GaAs two-junction solar cell was chosen for discussion in this section. The input parameters of these two-junction solar cells are same as those in Table 2.1. The short-circuit current degradation of proton irradiated Alo.33Gao.67As/GaAs two-junction cells are given in Table 2.6. Table 2.7 lists the Ijc degradation of 1-MeV electron irradiated Alo .ssGao.erAs/GaAs two-junction solar cells. Figure 2.8 shows the short-circuit current degradation of the proton irradiated Alo ,35Gao.65As/GaAs/Ino.53Gao.47As (or Ge) three-junction solar cell. The detailed discussion of these multijunction systems has been described in previous publications [54].

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20 2.7 Summary In this chapter a numerical model for computing the displacement damage for single-junction and multijunction solar cells has been developed and applied to the proton and electron irradiated single-j unction GaAs and AlojaGao.erAs solar cells, two-junction Alo. 33 Ga<)_g 7 As/GaAs and Alo. 35 Gag_ 65 As/GaAs/Ino. 53 Gao. 47 As (or Ge) three-j unction solar cells under different fluences and energies. Excellent agreement was obtained between our calculated values and the measured Igc, V<>c and tJc in proton or electron irradiated (AlGa)As and GaAs solar cells for proton energies from 100 KeV to 10 MeV and fluences from 10^° to 10^^ cm“^, and for 1 MeV electrons and fluences varying from 10^“* to 10^® cm“^ under normal incidence conditions. Moreover, it is shown in this chapter that in order to obtain an optimal multijunction solar cell with specified end of life efficiency, various physical parameters for each cell must be determined. It was pointed out that major difficulties encountered in carrying out the theoretical calculations using the model developed here include some unknown input parameters and the lack of experimental data to facilitate comparisons with the simulations. These uncertainties can be removed once the actual cell structures for the multijunction cells are fabricated and characterized.

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21 Table 2.1 Input parameters for the simulations of proton or electron irradiated Alo^aGao.erAs and GaAs solar cells. Electron or hole lifetime (s) 2 X 10-® 4 X 10-^ Electron diffusion length, L„ (/^ni) 6.0 3.41 Hole diffusion length, Lp 3.0 0.5 P-dopant density, Na (cm“^) 2 X 10^® 2 X 10^* N-dopant density. No X 0 1 X 10^^ Intrinsic density, n. 1.8 X 10® 1.66 X 10^ Electron capture cross section,
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22 Table 2.2 Calculated damage constants of the minority-carrier diffusion length in GaAs p-n junction solar cell. Energy (MeV) Ku Klp 0.1 0.105 0 0.3 0.00081 0.34 1.0 0.00025 0.00835 2.0 0.00006 0.00273 5.0 0.000029 0.00169 10 0.000006 0.00109

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23 Table 2.3 Calculated and experimental data of the degradation of Igc, Vqc and rjc in proton irradiated (AlGa)As-GaAs solar cell. Energy Fluence Isc/lsco ^ oc/^ oco VcIVco (MeV) cm~^ (%) (%) (%) Cal. Exp. Cal. Exp. Cal. Exp. 10'° 0.97 0.97 0.97 0.925 0.94 0.89 0.1 10^1 0.80 0.81 0.72 0.81 0.61 0.63 10^2 0.49 0.50 0.63 0.66 0.30 0.28 10'° 0.92 0.87 0.93 0.94 0.85 0.81 0.3 10" 0.74 0.71 0.89 0.86 0.67 0.62 10" 0.44 0.46 0.85 0.78 0.37 0.31 10'° 0.98 0.96 0.94 1.0 10" 0.95 0.92 0.88 10'2 0.80 0.89 0.71 10'° 0.99 0.98 0.98 0.979 0.97 0.95 2.0 10" 0.96 0.938 0.96 0.94 0.93 0.90 10'2 0.83 0.81 0.93 0.87 0.78 0.71 10'° 0.99 1.00 0.99 1.00 0.98 1.00 5.0 10" 0.97 0.93 0.96 0.97 0.93 0.90 10'2 0.86 0.84 0.93 0.90 0.80 0.76 10'° 0.99 1.00 0.995 0.99 0.98 0.99 10.0 10" 0.97 0.96 0.975 0.97 0.945 0.95 10'2 0.89 0.89 0.945 0.93 0.843 0.84

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24 Table 2.4 Igc degradation of one-MeV electron irradiated Alo. 33 Gao. 67 As and GaAs p-n junction solar cells. Fluence 10 ^® cm -2 10 ^® cm ^ 10 ^^ cm ^ Cal. Exp. Cal. Exp. Cal. Exp. normal incidence Alo. 33 Gao. 67 As GaAs 0.695 0.640 0.925 0.926 0.886 0.82 0.986 0.986 0.976 0.99 omnidirection Alo. 33 Gao. 67 As GaAs 0.729 0.245 0.942 0.660

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25 Table 2.5 Calculated and experimental data of degradation of Isc, Voc, and Klp and Kl„ in one-MeV electron irradiated Alo^sGao.erAs solar cell. Fluence 10^® cm -2 10’'^ cm' -2 Cal. EXP. Cal. Exp. normal Incidence Isc /f SCO 0.925 0.926 0.986 0.986 Vqc/V OCO 0.934 0.97 0.991 0.989 Vc/Vco 0.863 0.848 0.977 0.941 Ku 9.9 X 10-® 2.34 X 10-* Klp 4.3 X 10-® 7.04 X lO-’’ Omnidirectional Isc/Isco 0.729 0.942 Voc/Voco 0.909 0.922 VcIVco 0.662 0.868 Ku 2.89 X 10-’’ 2.12 X 10-'^ Ku 4.45 X 10-® 4.08 X 10-5 -

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26 Table 2.6 Igc degradation of proton irradiated Alo. 33 Gao. 67 As/GaAs two-junction solar cells. E(MeV) 10 ^° cm ^ 10 ^^ cm ^ 10 ^^ cm ^ Top Bottom Top Bottom Top Bottom 0.1 0.993 1.00 0.969 1.00 0.829 1.00 0.3 0.992 1.00 0.963 1.00 0.832 1.00 1.0 0.996 0.992 0.994 0.965 0.980 0.843 2.0 0.996 0.993 0.995 0.974 0.984 0.871 5.0 0.996 0.994 0.995 0.981 0.987 0.900 10.0 0.996 0.995 0.996 0.988 0.992 0.941

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27 Table 2.7 Isc degradation of one-MeV electron irradiated Alo.33Gao.67As/Ga As two-junction solar cells. Fluence Top cell Bottom cell 10^® cm-2 0.780 0.64 10^® cm~^ 0.950 0.886 cm-2 0.990 0.976

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28 AR COATING (AIGa) As GaAs NUMBER OF FINGERS = 24 p CONTACT: Au-Zn-Ag n CONTACT: Au-Ge-NI-Ag AR COATING : TajO* p AlxGa,_^As: x > 0.85 CELL SIZE = 2x2 cm^ Fig. 2.1 The cross sectional view of an (AlGa)As-GaAs p-n junction solar cell.

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29 Fig. 2.2 The range of an Alo.ssGao.isAs-GaAs solar cell vs. incident proton energies.

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D(E) 30 10 "^ 10 *Â’ 10 ^ E (MeV) Fig. 2.3 The total number of displacement defects vs. incident proton energies for GaAs, Ge, (InGa)As and (AlGa)As single-j unction solar cells. Y : Ino.53Gao.47As, X : GaAs, + : Alo.33Gao.67As and * : Ge.

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SCO 31 ENERGY (MeV) Fig. 2.4 The normalized short-circuit currents in the Alo.ssGao.isAs-GaAs p-n junction solar cells. Solid curves are from our calculations; solid dots are the experimental data.

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32 Fig. 2.5 The normalized open-circuit voltage in the Alo.85Gao.15 AsGa As p-n junction solar cell.

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D(E) 33 Fig. 2.6 The total number of displacement defects vs. incident electron energies for GaAs, Ge, (InGa)As and (AlGa)As single-junction solar cells. Y : Ino.53Gao.47 As, X : GaAs, -|; Alo.33Gaog7As and * : Ge.

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34 Fig. 2.7 The flowchart for simulating degradation of the irradiated multijunction solar cells.

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SCO 35 Fig. 2.8 Isc degradation of the electron irradiated Alo;35Gao.65As/GaAs/Ino.53Gao.47As (or Ge) multijunction solar cells.

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CHAPTER 3 A NEW METHOD FOR OPTIMAL DESIGN OF GAAS SINGLE-JUNCTION SOLAR CELLS 3.1 Introduction Since simple analytical models can not provide enough information and accuracy to optimize the design of solar cells [55], computer modeling has become an important and essential tool for solar cell design and analysis. The incorporation of an efficient optimum algorithm for computer simulation would enable, using the knowledge of cell parameters from the state of the art and its applications, to optimize the cell parameters in order to attain a maximum conversion efficiency. Recently, Chen and Wu [56] and Fossum [57] have proposed two different computer aided designs for silicon solar cells while Hovel [20], Hamaker [21] and KinechtU et al. [19] have established simplified solutions for GaAs solar cells. However, the results obtained by these previous models were less than truly optimum due both to ignorance of the interaction between the parameters and to the lack of an efficient optimum algorithm in computer simulation. In addition, they did not include the extrinsic parameters such as series resistance, contact structures, irradiation, air mass ratio, temperature, etc., in the optimization cycle and hence the results were not realistic. In this chapter a new computer model is proposed for truly optimizing the structure of a GaAs single-junction solar cell for space application. However, the model can apply easily to other solar cell systems as well. Our model not only takes into account the effects of the intrinsic structural parameters such as junction depth Xj, cell thickness Tj, doping densities N^ and Nq, surface recombination velocity Sp and Sn, but also incorporates the extrinsic structural parameters. It should be noted that 36

PAGE 44

37 the electron or proton irradiation on a solar cell for space application will degrade the cellÂ’s performance. Therefore, the damage constants of cell parameters, such as minority carrier diffusion length and minority carrier lifetime, should be estimated in advance in order to attain an accurate optimal structure of the cell. The detailed discussion of the calculations of damage constants of the irradiated solar cell has been presented in Chapter 2. In this chapter it is cissumed that the values of damage constants for the cell parameters are available and hence can be employed directly in the optimum simulation. To test the optimum algorithm both GaAs and Si solar cells have been simulated. It turns out that the conversion efficiency of a GaAs cell is better than that of Si. The results also show that there can be several optimum sets of parameters that yield nearly the same values for efficiency while maintaining acceptable characteristics. Section 3.2 of this chapter outlines the device modeling for a p/n junction solar cell. The effects of extrinsic parameters on solar cell parameters are presented in section 3.3. In section 3.4 we propose a true and efficient optimization technique for the optimal design of a single-junction solar ceU. Comparisons between the calculated results and the published data have been done also in this section. Summary is made in section 3.5. 3.2 Device Modeling for GaAs P/N Junction Solar Cells For simplicity the discussion will be restricted to a p/n junction solar cell with uniform doping. The cross section diagram of the GaAs p/n junction solar cell to be analyzed is the same eis that shown in Fig. 2.1. There are four major solar cell parameters involved namely short-circuit current Igc, open-circuit voltage Voc, conversion efficiency rjc and fill factor F.F. in designing a typical p/n junction solar cell. In general, the higher the values of the Isc, Voc and F.F. a solar cell can produce, the higher r/c the solar cell can obtain. However, a solar cell with a low energy bandgap tends to produce higher Igc and lower Voc than the cell with a higher energy bandgap.

PAGE 45

38 This tradeoff complicates the design of a solar cell. Theoretically, when the photon of the light impinges on the p/n junction solar cell having a bandgap energy, Eg, the short-circuit current of the solar cell can be represented as the sum of the currents obtained from the emitter layer Ie, base layer Ib and depletion region Iw as shown in Fig. 3.1. It is given a.s [20] I,c = r Ie(A) Ib(A) Iw(A)dA (3.1) J0.3S where Ai (// m) = 1.24 / Eg. The current component from emitter layer is represented as Ie(A) = qAp(A)(l-R(A))aL„ 1 (-aLnexp(-aXj) -f aLn + Sn^ (l — exp(— oXj) cosh — exp(— aXj) sinh ^ Sn^ sinh ^ -Icosh ^ ) (3.2) where A is area of the cell, q is the electron charge, R(A) is reflection coefficient, a is the absorption coefficient of the cell and is a function of wavelength, L„ is the electron diffusion length, is the electron lifetime and p{\) is the solar photon flux density which is [58] p{\) = 4.06 X IO^^A^Ha (3.3) where Ha is the solar radiance. The solar radiance is crucial in evaluating the conversion efficiency of a solar cell for different air mass ratio. Existing software, LOWTRAN6 [59], is used here. The current component from the base layer is

PAGE 46

Ib(A) = qAp(A)(l -R(A))aL a^Ll 1 Spf^ (cosh ^ (aLp ^ ^exp(-a(Xj + Wj)) — exp(— ad')) + sinh ^ + aLp exp(— ad') Sp^ sinh ^ + cosh ^ )(3.4) where Lp is the hole diffusion length, Tp is the hole lifetime, Wj is the depletion width and d’ = Tj Xj Wj. The current component from the depletion region is shown ais Iw = qA/)(A)(l — R(A)) exp(— aXj) (1 — exp(— aWj)) (3.5) The open-circuit voltage Voc for the solar cell is given by Voe = ^^ln(^ + l) (3.6) q to where Kg is the Boltzmann constant, T is the absolute temperature and Iq is the saturation current density, n is the diode ideality factor. For simplicity, we may assume that Iq is either a simple diffusion-dominated current if n is close to 1 or a recombination-dominated current if n is close to 2. The former one is given as T A 2 Dp /SpCosh(d'/Lp) + g^sinh(d'/Lp)\ lo = qAni \ n ; + LpNo y cosh(d'/Lp) -f Sp sinh(d'/Lp) j A^ 2 Dn / SnCOsh(Xj/Ln) -t^sinh(Xj/Ln) \ L„Na cosh(Xj/L„) -ISnsinh(Xj/Ln)y where n;. No and Na are the intrinsic densities, donor and acceptor density of the ceU

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40 respectively. Dp is the hole diffusion coefficient whereas D„ is the electron diffusion coefficient. As for the recombination-dominated case, it is . qAmWj fo — TJL— The fill factor F.F. for a solar cell is given by F F = yoc\ (expgf)-lj where V„i is given by the relationship as follows: I q Vm \ I q Vm exp(i:r^)(i + ftf) KbT' KbT' ^ + 1 to (3.8) (3.9) (3.10) Therefore, the conversion efficiency of a single-j unction solar cell is given by JscVocF.F. Vc = (3.11) where Pin is the incident solar power density. Pin for different air mass ratio is calculated from LOWTRAN6 [59]. 3.3 Effects of Extrinsic Parameters on Device Modeling In practical solar cell design, series resistance, contact materials, grid line configuration, antireflection coating and high sun insolation can aflfect the performance of the solar cell. Therefore, these effects must be taken into account in the optimization cycle in order to ascertain an realistically optimal design. In this section, the relationships between these extrinsic parameters and four major solar cell parameters are described.

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41 3.3.1 Effect of Antireflection Coating For a single layer of non-absorbing medium, the reflectivity at normal incidence of a given wavelength is given as [60] / ni^ npn2 y \ni2 + non2/ (3.12) where no, ni and n 2 are the refractive indices of air, antireflection coating material and solar cell respectively. The thickness of the antireflection coating is (3.13) A double layer coating can offer less reflectivity loss over a wider region than a single layer coating can. A single layer has a zero reflectivity loss at only one wavelength and has an acceptable reflectivity loss over only a limited range. The reflection coefficient R of double layers is [60] / ni^na n2^np y \ni2u3 4n2'^noJ (3.14) where U 2 and na are the refractive indices of the second coating layer and solar cell respectively and no and ni are the same as defined in Eqn. 3.12. The thicknesses of these two coating layers can be estimated by the following relationship: Uidj — n2d2 — Ao (3.15) 3.3.2 Effect of Grid Design As the area of solar cell increases, it becomes more important to include the interconnect configuration in the grid process [61]. Different grid geometry can affect the sheet resistance of the series resistance. The parallel grid line design in Fig. 3.2

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42 is the standard grid design for a solar cell. The series resistance of this grid design is approximated as [62] R, 12D w 3h w (3.16) where D is the thickness of window layer, b is finger spacing, w is finger width, h is finger height, pf is emitter resistivity, pa is metal finger resistivity and Rc is the contact resistance. It should be noted that as the width of grid element increases, the series resistance decreases and hence the electrical power loss decreases. However, this is not the only consideration. When less cell area is exposed to the photons, the short-circuit current decreases. The shadow loss is calculated as the total area of the fingers divided by the whole area of the solar cell. Therefore, there is an optimal value where the sum of these two losses is a minimum. 3.3.3 Effect of Series Resistance When series and shunt resistance problems become important, the ratios Ym/Yoc and Im/Isc are reduced and the ratios calculated from Eqns. 3.8 to 3.10 should be modified. For simplicity, if shunt resistance is much higher than the series resistance, which is always true for GaAs, the open-circuit voltage VÂ’oc after additional reduction is given as V' = V I R and the reduced fill factor F.FÂ’. is given as [63] F.F.'= F.F. X (1 (3.17) (3.18)

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43 3.3.4 Effect of High Sun Insolation For high sun insolation, the short-circuit current is estimated simply by multiplying the number of sun insolation by the short-circuit current obtained from one sun insolation. The open-circuit voltage of high sun insolation, Vqcx, is given as [64] Vocx = Voc + ^^ln(X) (3.19) where X is the number of sun insolation. From this equation, it appears that the conversion efficiency of high sun insolation increases with any increase of opencircuit voltage. Again, this is not always true because of the reduction of the fill factor in high sun insolation. Additionally, at high sun insolation, the effect of series resistance becomes more critical because of the high Ijc3.3.5 Effect of Irradiation After proton or electron irradiation, the minority carrier lifetime and diffusion lengths will be degraded. The diffusion length is related to the lifetime, = Dr, and it is given by the usual relations [46, 47]: = -^-Kr (3.20) T 'To ^ = jT + KL?i (3.21) where K^and Kl are the minority carrier lifetime and diffusion length damage constants, respectively, and is the fluence of proton or electron. 3.4 Constrained Optimization Technique Under the constrained optimization approach, the optimal design of a solar cell can be mathematically expressed in closed form as

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44 Y = Max (f(Xi) 1 Gk < Xk < Hk, i = 1, 2, N, k = 1, 2, M) (3.22) where Xi,..., Xn are the explicit independent variables to be optimized, which might be cell thickness, doping level, etc. The imphcit variables Xn+i,... Xm are dependent functions of the explicit variables. These implicit variables might be the minority carrier diffusion length, the minority carrier lifetime or others. f(X;) is the objective function which might be the solar cell efficiency expected to be maximized, or perhaps the short-circuit current expected to be optimized. The upper and lower constraints Hk and Gk may be either constants or functions of the independent variables. The Box complex algorithm [32, 65] is one of the most efficient constrained optimization techniques currently available. This algorithm is a sequential search technique which has proven effective in solving problems with nonlinear objective function subject to nonlinear inequality constraints. Derivatives of the objective function are not required for this algorithm. This procedure will tend to find the global optimum since the initial set of guesses is randomly scattered throughout the feasible region. The algorithm, with some minor modifications for increasing the convergence rate, proceeds as follows [65, 66]: Step 1. Initial feasible starting K guesses, where K is at least equal to (N -11), are generated. Each guess consists of N points which are generated from random numbers and constraints for each of the independent variables: Xij= Gi-Frij(Hi-Gi), i = l,2,...,N, J = 1,2,...,K-1 (3.23) where rij are random numbers between 0 and 1.

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45 Step 2. The guess points must satisfy both the explicit and imphcit constraints. If a constraint is violated, the point is moved to the upper bound or lower bound of the constraint which is violated. This procedure is repeated as necessary until all the constraints are satisfied. Step 3. The objective function is evaluated at each guess. The guess having the lowest function values is replaced by a guess which is Xij(new) = 1.3 (Xi,» Xjj(old)) + Xi,^, i = 1, 2, ..., N (3.24) where Xi,a is the average of the remaining guesses and is defined by Xi,. = fexy Xij(old)l , i = 1,2,...,N (3.25) Step 4. If a guess repeatedly gives the lowest function value on consecutive trials, it is moved to the average of the best and worst guesses. This minor modification would increase the convergence rate. Moreover, it will save computation time because additional constraint-violation checking is not necessary for this new guess. Step 5. Convergence is assumed when the objective function value at each guess is within the tolerance which the user assigned. A flowchart illustrating the above procedure is given in Fig. 3.3. 3.5 Optimal Design of GaAs Single-Junction Solar Cells The main task in this section is to optimize the efficiency of a single-j unction solar cell. According to a previous study [52] and the work done by Knechtli et al. [19], GaAs single-j unction solar cells with bandgap energy 1.43 eV offer some advantages for example high rjci radiation hardness, and relative insensitivity to temperature as compared to silicon solar cells. The optimal design of a Si solar cell is also developed here for comparison purpose.

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46 For the GaAs cell, the design problem for maximizing the efficiency rjc is formulated as follows: r/c = Max (f(No, Na, Xj, Tj, Eg, S„, Sp)) (3.26) such that ln(2 X lO^^cm"^) < In(ND) < ln(10^®cm“^) (3.27) ln(2 X lO^^crn”^) < In(NA) < ln(10^^cm“^) (3.28) 0.05/zm < Xj < 0.5/im (3.29) 0.1/zm < Tj < 10//m (3.30) 1.43eV < Eg < 1.43eV (3.31) 10^ cm/s < Sn < 10® cm/s (3.32) 10^ cm/s < Sp < 10® cm/s (3.33) in which the parameters to be optimized are intrinsic structural parameters (No, Na, Xj) and Tj and material parameters (Sn, Sp and Eg) whose values are constants. The material parameters are determined by the materials and by the top and back contact surface passivations. For the GaAs solar cell, a thin (AlGa)As window layer is grown on the top for reducing the surface recombination velocity [19, 20]. The choices of the upper and lower bounds of the constraints are based on the considerations of contact resistance [67], the state of the art technology and radiation hardness. Table 3.1 and 3.2 list the comparisons between the results calculated here and the published data for the optimal designs of Si and GaAs solar cells. Our designs show much better performance than those obtained from previous models [19-21, 56, 57, 68]. Moreover, they are close to the cell efficiencies made by the state of the art

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47 [69], namely 20% for Si solar cell and 26% for GaAs solar cell respectively. The major discrepancies between our simulations and the previous models are the selections of the constraints. For instance, for an Si solar cell. No should be greater than 10^^ cm“^ to obtain a good contact whereas Na is two orders less than No [67]. In this respect, the values of Np in a p/n junction Si solar cell can be much less than that in n/p Si solar cell and therefore results in better performance. The effect of series resistance on GaAs solar cell efficiency is described in Table 3.3. Table 3.4 shows the optimal designs of the 300 KeV proton irradiated GaAs solar cell. It is clear that the junction thickness decreases when the proton fluence increases, and the effect of fill factor due to the irradiation is negligible. These results conform to the assumptions made for calculating the degradation of proton or electron irradiated solar cells in Chapter 2. Figure 3.4 shows the temperature and air mass ratio dependence of the efficiency of a GaAs solar cell. It shows that the efficiency of the GaAs solar cell increases as the temperature decreases. This is reasonable because the open-circuit voltage decreases when the temperature increases. 3.6 Summary A new method of incorporating an efficient optimization algorithm into device modeling techniques for the optimal design of GaAs and Si solar cells has been examined. A modified Box complex optimization technique and a device modeling that considers the effects of extrinsic parameters has been implemented to obtain the optimal design of a GaAs single-j unction solar cell. All the parameters to be optimized are adjusted in a systematic way, resulting in a truly optimal design. Comparisons between the calculated results and the published data have been made to verify the optimal designs. The optimal efficiency of the single-j unction GaAs solar cell obtained by the simulation is 27.8% at room temperature for AMO which is close to the cell efficiency, namely 26%, provided by the state of the art.

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48 Table 3.1 Comparisons between calculated results and published data for the optimal designs of Si single-junction solar cell at room temperature. AMO AMI AMI [56] Nd (cm“^) 5.0x 10^® 5.0x 10^® 3.0x 10^® Na (cm~^) 5.0x 10^^ 1.16X 10^^ 6.0x 10^® Xj (;im ) 0.5 0.623 0.1 Tj (//m ) 300 200 66.4 Sn (cm/s) 10000 10000 118 Sn (cm/s) 10000 10000 100 Voo (V) 0.89 0.88 0.63 Jsc A/cm^ 0.0372 0.0328 0.034 F.F. 0.85 0.85 nc 21.1% 23.8% 18.06% Eg 1.12 1.12 1.12 Collection efficiency 0.958 0.944 Sun Insolation 1 1 1 Shadow loss 0.018 0.018 Reflection loss 0.041 0.041 R, (fi) 0.473 0.473 -

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49 Table 3.2 Comparisons between calculated results and published data for the optimal designs of GaAs single-junction solar cell at room temperature. AMO AM0[68] AM0[25] Nd (cm"^) 3.16X 10^^ 2x 10^^ 7x 10^^ Na (cm-^) 2.60X 5x 10^® 1.5x 10^® Xj {fim ) 0.5 0.3 0.5 Tj (^m ) 10 2.5 3.0 Sn (cm/s) 10000 Sn (cm/s) 10000 Voc (V) 1.23 1.05 1.04 Jsc A/cm^ 0.0345 0.0341 0.0317 F.F. 0.888 0.85 0.83 Vc 27.8% 22.5% 20.3% Eg 1.43 1.43 1.43 Collection efficiency 0.999 Sun Insolation 1 1 1 Shadow loss 0.018 Reflection loss 0.041 R,(«) 0.473 -

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Table 3.3 Effect of series resistance on GaAs single-junction solar cell efficiency. Grid Number Rs (flcm^) r}c % 0 0 29.4 24 0.473 27.7 18 0.969 26.9 16 1.22 26.5 12 2 25.4 9 3 23.9

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51 Table 3.4 Effect of irradiation on GaAs single-j unction solar cell for 300 KeV proton at fluences of 10^°, 10^^ and 10^^ cm“^. None 10^° cm ^ 10^^ cm ^ 10^2 cm-2 Nd (cm-3) 10'^ 10^7 CO X o 10'" Na (cm"^) 10^« 10^® 10^« 10^® Xj (/zm ) 0.5 0.5 0.4 0.175 Tj (/xm ) 10 10 10 10 Voc(V) 1.209 1.15 1.15 1.08 Jsc A/cm^ 0.0347 0.0315 0.0283 0.0241 F.F. 0.887 0.883 0.88 0.88 Vc 27.5% 23.7% 21.3% 17.0%

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52 X w Fig. 3.1 The energy band diagram for a (AlGa)As-GaAs p-n junction solar cell.

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53 "TIV 2a \ 1 / < 2a ) Fig. 3.2 Parallel gridline pattern for solar cells.

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54 Fig. 3.3 Flowchart for the modified Box optimization algorithm.

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E ff iciency (%) 55 Temperature (K) Fig. 3.4 Temperature and air mass ratio dependence of the efficiency of GaAs singlejunction solar cell.

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CHAPTER 4 AN EXPERT SYSTEM APPROACH TO THE OPTIMAL DESIGN OF MULTIJUNCTION SOLAR CELLS 4.1 Introduction Single-junction solar cells today are those most frequently used. They are made from a single material with a single bandgap to absorb the sun light. The incident photon with energy less than the bandgap of the material can not be converted to electricity. And much of the photon with energy greater than the bandgap is wasted as heat. Therefore, the efficiency of a such single-junction cell is limited. In multijunction cells, solar cells with different bandgaps are put on top of each other in decreasing bandgap order. In this way, all the incident photons with energies equal to or greater than the bandgap of the top cell are absorbed by the top cell. The incident photons with energies less, than the bandgap of the top are transmitted to the next cell. The phenomenon recurs at each cell. In 1955, Jackson [3] proposed that the efficiency of solar cells could be increased significantly by constructing a system of stacked p/n homojunction photovoltaic cells which are composed of different semiconductor materials. The next year, Loferski [4] was the first to use multiple cell concepts to optimize the efficiency of a photovoltaic system, selecting sets of cells which made use of the energy available from the entire solar spectrum. Recently, many researchers [3-18] have proposed varying approaches to the optimal design of the multijunction cells both theoretically and experimentally. However, their results were not realistic for several reasons. First, data for the electrical and optical parameters of the materials they used were not always available and accurate. Secondly, their designs were based on either a small set of design parameters or an inadequate device modeling. Thirdly, they did not 56

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57 consider the interrelation of the parameters of the solar cells and hence their results were not optimal. In this chapter, we propose for the first time an expert system approach to the optimal design of multijunction solar cells for both terrestrial and space applications. An expert system is a knowledge-intensive computer program. The knowledge of an expert system consists of facts and rules. The facts constitute a body of information that is widely shared, publicly available and generally agreed upon by experts in the field [33, 34, 70]. The rules are those if-then rules that characterize expert level decision making in the field. In general, an expert system with a large number of facts and rules is better than one with just a few facts and rules. To assure the usefulness of our expert system, we have implemented as many facts and rules as possible with the help of published literatures and experts in the photovoltaic area. However, because of the availability of other research reports and the tradeoff between the number of rules and the computation time, our expert system is based on the AlGaAs, GaAs, Ino. 53 Gao. 47 As, Si and Ge materials. And less than a hundred rules are actually implemented. Section 4.2 deals with the device modeling of multijunction tandem solar cells used in the expert system. In section 4.3, the concept of an expert system approach to the optimal design of multijunction solar cells is presented. An optimization technique with heuristic rules is also shown in this section. Results and discussion are in section 4.4, with a summary in section 4.5. 4.2 Device Modeling of Multijunction Tandem Solar Cells Since the theoretical conversion efficiency increases very slowly beyond threejunction tandem cells [ 1 ] and the technology of fabricating cells beyond three junction is difficult in practice [ 2 ], the multijunction study focused on those structures that are two-junction and three-junction only. In general, the calculations of Igc, Voc, F.F. and rjc for the multijunction cells are similar to those for single-j unction solar cell

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58 presented in Chapter 3. Here, we will only illustrate the three-junction calculations. Let the top, middle and bottom cells have the bandgaps of Egi, Eg 2 and Egs with corresponding short-circuit currents Lci, Lcz and Lcs respectively. Then, the short-circuit currents can be written as Iscl = J Iei(A) + Ibi(A) -M wi(A)dA 0.3 (4.1) Isc 2 = J f Ie2(A) -|Ib2(A) + Iw2(A)dA (4.2) Isc3 = J /'^'lE3(A) + lB3(A)-Mw3(A)dA >2 (4.3) where Ai (// m) = 1.24 / Fgx, A 2 (/i m) = 1.24 / Fg 2 , and A 3 (/z m) = 0.24 / Ej3 respectively. The current components from emitter, base and space charge of each cell and other notations in Eqn. 4.1 to Eqn. 4.3 are similar to those given for the single-j unction solar cell. The representations of the open-circuit voltage, fill factor and conversion efficiency of each cell are also similar to Eqns. 3.6, 3.9 and 3.11 respectively. For the twoterminal case as shown in Fig. 4.1, the Lc values are all the same for the top, middle and bottom cells, namely the smallest value among the three cells. The total conversion efficiency of the three-j unction cells is then Vtot — Vcl + Vc2 + Vc3 _ IscVqciF.F.i -f IscVoc2F.F.2 ~4~ IscVqc3E.F.2 Pin Those effects such as series resistance, grid structures, etc., which have been discussed in Chapter 3, are still applicable to the design of multijunction solar cells. They are not repeated here. However, the effect of the tunnel diode should be considered. The role of a tunnel diode in the design of the multijunction solar cell is to connect the two

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59 different p/n junctions. The doping density of the tunnel diode must be quite high in order to have a low impedance to current flow in both directions and the voltage drop across it should be as small as possible [5]. Additionally, the bandgap of the tunnel diode should be as large or larger than the top cell bandgap so that as many photons as possible can be transmitted from the top cell and absorbed at bottom. According to KaneÂ’s theory, the tunnel current is given as [71, 72, 73] Jt -7r^m*EgWt V " 4h^Wt7T3 2V2h ^ (4.5) where m* is the effective mass of the tunnel diode; Wt is the depletion width of the tunnel diode; Eg is the bandgap of the tunnel material; V is the voltage across the tunnel diode and Jt is the tunnel current density. Therefore, the open-circuit voltage of the solar cell should be adjusted by the amount of the voltage across the tunnel diode. An alternative approach for the intercell ohmic contact of the multijunction solar cells is the metal intercell contact shown in Fig. 4.2. In the metal interconnect contact technique, the top (AlGa)As cell and middle cell will be interconnected by the metal interconnect technique. Grooves will be etched through the top cell to reach the middle cell. Metals will be deposited within the grooves to shorten the base region of the top cell to the emitter layer of the middle cell. The connection between the middle and the bottom cell is made by fabricating a tunnel junction as shown in Fig. 4.3. The metal intercell contact structure has the problems of a complex fabrication process and double shadowing. The total series resistance of the multijunction cells is then the sum of the series resistances of top cell, bottom cell and emitter layer of the middle layer. 4.3 Concept of the Expert System Approach The shortcomings of the conventional computer programs for the optimum de-

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60 sign of high conversion efficiency single-j unction and/or mult ij unction solar cells are twofold. First, they are not flexible: due to the intermixture of data and codes, the programs are not always applicable without code corrections when the goal of the design changes. Second, results obtained from those programs are sometimes unrealistic: data or design criteria often are not properly taken into account. In this section, we propose a new approach to formulating the design problem and then develop an expert system to aiding the optimal design of high efficiency singlejunction and/or multijunction solar cells for both space and terrestrial applications. Our proposed expert system will apply to the radiation free environment a^ well a.s to the proton or electron irradiation environment. A systematic approach to this design problem consists of four tasks ( 1 ) constructing a solar cell knowledge data base, ( 2 ) searching for all possible designs from the knowledge base, (3) selecting the optimum design by applying heuristic rules, and (4) verifying the optimum design. This expert system will produce practical designs corresponding to user specifications and thus become an important tool for the solar cell designer. 4.3.1 Problem Formulation An expert system approach to solving for the optimal design of multijunction solar cells will usually be successful if the theoretical rules and/or experienced rules have been formulated quantitatively. The design procedures of the proposed expert system can be formulated into four tasks. The first ta^k is the construction of a solar cell knowledge base. Here the first step is to define the problem domain precisely. This will include an analysis of the desired results from the expert system. Our expert system is currently limited to (AlGa)As, GaAs, Si, Ge and Ino. 53 Gao. 47 As materials. The goal of the proposed solar cell expert system is to optimize the conversion efficiency of a 2 -terminal monolithic single-j unction or multijunction solar cell based on these five material systems. Then the types of knowledge required must be identified as well as the possible interactive consultation and the approximate number of rules.

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61 which will reduce the search space of the design problem. In this respect, the facts in the solar cell knowledge base will be the fundamental properties of the materials such as the bandgap, minority carrier diffusion length, minority carrier lifetime, contact resistance, etc. The data structure of these facts can be represented as lists, trees, nets, rules or other formations. Since most of the fundamental properties, namely facts, are a function of doping density, temperature etc., the if-then rules representation for the facts is used for the expert system. The rules consist of antecedent-consequent pairs. The characteristic of a rulesbased system is the separation of data examination from data modification. The examination of data generally occurs in the antecedent of a rule, while data modification is normally managed by the consequent of a rule [70]. For example, if a GaAs p/n junction solar cell is chosen, then the junction thickness of the cell should be less than 0.5 ^m in order to obtain high efficiency and high radiation hardness. By the application of this rule, the search space of the problem is reduced and computation time is saved. However, there may be a number of conflict rules in the knowledge base and a selection must be made. Therefore, a conflict resolution process is set up to determine which satisfied rule to be used in the conflict set. There are two ways of constructing the rules-based system of the expert system. The first method is to let the expert system automatically adopt the solar cell properties from existing data files or tables. These data may be the results obtained from the expert system itself. The other way is through a set of queries under a certain condition C as C — (Cl, C 2 , ..., C„) (4.6) where Cj is a value under condition j, and n is the total number of queries. The queries might be of the following type: 1. Input the cell name please.

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62 2. Input the bandgap of this cell please. 3. Input the contact material please. The second task is to search the knowledge data base. Let G-, be the mapping function from a description of a solar cell to the value of the material with respect to a property i under a certain condition C. A solar cell is described by a set of tuples, e.g. bandgap energy, short-circuit current, open-circuit voltage, conversion efficiency and so on, as Ml = (mi,m2,...,mk) (4.7) where 1 is the name of a solar cell and k is the total number of tuples. The retrieved value Vki of a solar cell Mi with regard to property i is represented by Gj. And Vki = Gi(Mk) (4.8) The set of solar cells which satisfy the requirements on property i is Si = (s|Ci
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63 The third task is the selection of the optimal designs. The selection of optimal design Dopt is performed by calculating the scalar product of a weighting factor W = (wi, W 2 ,.”, Wr ) predetermined by the solar cell designer and a matrix Vs as E(ei,e 2 , ...,e„) = WVs (4.11) where n is the possible designs and ei is a value of this evaluation for the selected solar cell i. The weighting factors can be the cost of solax cells, complexity of fabrication processes, conversion efficiency, weight of solar cells, preference of materials and radiation hardness. In this expert system, it is assumed that the weighting factors of the radiation hardness and conversion efficiency are dominant. The last task is to verify the optimal design. The verification of the optimal design Dopt is made by comparing the calculated results obtained from this expert system with the experimental and/or theoretical data collected from the literature of the photovoltaic field. 4.3.2 Optimization and Heuristic Rules For the multijunction solar cells, the optimization problem, which is to maximize the total efficiency of each subcell, can be formulated as n ^tot ~ ^ ^ ^ci (4.12) i=l where n is the number of the subcells of the multijunction solar cell. And rjci = Max(f(NDi,NAi,Xji,Tji,Egi,S„i,Spi)) (4.13) It should be noted that the format of Eqn. 4.13 is similar to that of Eqn. 3. 26. Consequently, according to Eqn. 4.12 and 4.13 the number of constraints for the optimal designs of two-junction and three-j unction solar cells will be two times

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64 and three times more than those of a singlejunction solar cell. For example, a threejunction solar cell needs at leeist 15 constraints if Sn and Sp are fixed. Since the computation time will be increased exponentially instead of linearly as the number of the constraints increase, a couple of heuristic rules have been adopted in this system in order to prune the search space and hence to save the computation time. A heuristic is a technique that improves the efficiency of the search process and leads to an adequate answer, if not the best one of a difficult problem [74]. A heuristic rule serves as an aid to problem solving by experimental, especially trial-and-error, methods [75]. Therefore, it is possible to construct a special purpose heuristic rule that exploits domain-specific knowledge to solve a particular problem. In this respect, heuristic rules are applied to the selections of the upper bound and/or lower bound of the constraints, and to the combinations of material in the system for multijunction solar cells, in this way to obtaining a quick feasible optimal solution. Then, the solution will be compared with the existing data to justify the validity of heuristic rules. 4.4 Results and Discussion The design of the multijunction tandem cells is considerably more complex than that of single-j unction cells and hence additional parameters must be considered. They are as follows [76, 77]: 1. Bandgap energies must be optimized for multijunction solar cells. 2. Lattice matching is desired. 3. Direct optical transitions are desirable. 4. Metallurgical system must be compatible. 5. A compatible substrate must be available.

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65 6. Must be invariant with changes in the environment. According to our computer calculations, the optimum bandgap combinations of two-junction and three-junction tandem cells are 1.75/1.10 eV , 2.02/1.43 eV, 2.00/1.43/1.04 eV and 2.00/1.40/1.00 eV under AMO at room temperature. However, considering the lattice matching problem, it is clear that the materials selected for the top and middle cells, (AlGa)As and GaAs are quite favorably lattice matched for the two-junction tandem cells. Although the (AlGa) As/Si and (AlGa)As/(InGa)As two-junction solar cells meet the optimum bandgap energy combinations, they are still not the qualified designs due to the lattice constants and thermal expansion coefficients mismatch. In addition, the radiation hardness of GaAs is greater than that of Si or (InGa)As. As for the three-junction structure, (InGa)As, Ge and Si can be the possible candidate cells for the bottom ceU. However, these materials exhibit lattice mismatch with respect to GaAs and generate and propagate dislocations which may adversely affect solar cell performance. In a multijunction solar cell it is advantageous to have direct bandgap materials, to reduce the thickness of the required material. This reduction results not only in lower material costs but also in lower epitaxial growth costs for growing the required layers. Obviously, it is also beneficial to minimize thickness to gain a weight advantage for the final structure. Even more significant, however, is the reduction in growth time. In addition, the thinner layers inherent in the direct bandgaps materials tend to lead to lower minority carrier recombination losses and improved radiation hardness. Therefore, the top cell for our optimal design of the GaAs based multij unction solar cell is limited to the direct bandgap material. A comprehensive understanding of the electronic transport properties of the materials and of the total assembly in the two-junction or three-j unction tandem cells is essential to a complete analysis of the relative merits of the final choices. The most

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66 tractable parameter which reflects the quality of the device is probably the minority carrier diffusion length. A long diffusion length would require that the defects and recombination centers be minimized, that the junction quality be kept as high as possible and that the interfaces between layers be kept free of strain and imperfections. Since (AlGa)As, GaAs and (InGa)As are polar materials which are different from the nonpolar material such as Ge and Si, a characterization of the dominant scattering mechanisms that affect the mobility and hence affect the diffusion length should be done to facilitate the study of the optimal designs. Detailed discussion of different scattering mechanisms and calculations of the electron and hole mobilities of the (AlGa)As is given in Chapter 5. Table 4.1 and Table 4.2 list the results of our calculated two-junction and threejunction solar cells respectively. For those two-junction solar cells except GaAs/Ge in Table 4.1, an AlxGai_xAs heavy-doped tunnel junction with A1 composition greater than 0.45 has been used for our simulations. However, a heavy-doped GaAs tunnel junction is used for the case of GaAs/Ge. For the threejunction solar cells in Table 4.2, an (AlGa)As tunnel junction is used for connecting the top cell and middle ceU whereas a GaAs tunnel junction is used for connecting the middle cell and bottom cell. Here, we assume that it is feasible to obtain an (AlGa)As tunnel junction which is as good as the GaAs tunnel junction. According to the state of the art technology [78, 79], the n-type and p-type doping densities of GaAs can be as high as 2x 10^® cm“^ and 5x 10^^ cm~^ respectively. The effect of doping densities of the tunnel junction on the efficiency of a multijunction solar cell is shown in Table 4.3. It is found that the voltage drop across the tunnel junction decreases as the doping densities increase and hence the performance of the solar ceU increases. An alternative approach is to use the MIC structure to shorten the top and bottom cell for a two-junction solar cell. However, according to our simulation results, the efficiency of the (AlGa)As/GaAs two-junction solar cell obtained from this method, for example 24%, is much lower

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67 than that in Table 4.1. This inferiority is due to the increases of the shadowing loss and series resistance. It may be desirable for the optimal designs of multijunction solar cells only if the high quality tunnel junction is not feasible. The optimal designs of two-junction and three-j unction solar cells are listed in Table 4.4 and Table 4.5 respectively. The temperature and air mass ratio dependence of the efficiency of the Alo . 44 Gao. 56 As/GaAs and Alo. 44 Gao. 56 As/GaAs/ Ino. 53 Gao. 47 As are shown in Fig. 4.4 and Fig. 4.5. 4.5 Summary An expert system approach to the optimal design of two-junction and threejunction solar cells has been presented in this chapter for the first time. A rulebased system with a couple of heuristic rules is implemented in the expert system to prune the search space of the design problem and hence to save the computation time. The optimal designs of Alo.44Gao.5eAs/ GaAs two-junction solar cell and Alo .44Gao.56As/GaAs/ Ino.53Gao.47As three-junction solar cell with room temperature efficiencies for AMO of 30.01% and 35.3% respectively were obtained in the simulation.

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68 Table 4.1 Simulation results of two-junction solar cells at room temperature for AMO. (AlGa)As/GaAs GaAs/Ge (AlGa)As/(InGa)As (AlGa)As/Si Vocl (V) 1.52 1.22 1.24 1.44 F.F.i 0.91 0.895 0.896 0.91 rici{%) 16.86 22.3 23.5 17.5 Egi (eV) 1.97 1.40 1.52 1.85Voc2 (V) 1.21 0.531 0.55 0.88 F.F.2 0.895 0.80 0.80 0.87 ric2i%) 13.1 8.68 9.37 10.3 Eg 2 (eV) 1.40 0.66 0.744 1.12 Jsc(A/cm^) 0.0164 0.0275 0.0284 0.0180 Total rjc{%) 30.01 31.0 32.9 27.8

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69 Table 4.2 Simulation results of three-junction solar cells at room temperature for AMO. (AlGa)As/GaAs/(InGa)As (AlGa)As/GaAs/Ge Vocl (V) 1.50 1.46 F.F.i 0.91 0.91 Vcl{%) 16.7 15.85 Egi (eV) 1.98 1.96 Voc2 (V) 1.21 1.21 F.F.2 0.896 0.896 r/c 2 (%) 13.2 12.9 Eg 2 (eV) 1.40 1.40 Voc3 (V) 0.54 0.521 F.F.3 0.81 0.806 Vc3{%) 5.34 4.99 Eg 3 (eV) 0.744 0.66 Jsc(A/cm^) 0.0164 0.0160 Total rjc{%) 35.3 33.78

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70 Table 4.3 The effect of doping densities of the tunnel junction on the efficiencies of twojunction and three-j unction solar cells. Doping density Alo.44Gao_56As/GaAs AIq .44 Gao .56 As / GaAs / Ino. 53 Gao. 47 As Nd = 2 x 10 ^® Na = 5x 10^^ 30.0% 35.3 % Nd = 10 ^® Na = 5x 10^^ 27.9% 30.06 % Nd = 10 '^ Na = 3x 10^^ 24.6% 24.54 %

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71 Table 4.4 Optimal design of Alo. 44 Gao. 56 As/GaAs two-junction solar cell at room temperature for AMO. Alo. 44 Gao. 56 As GaAs Na (cm 8.54 X 10^® 1.05X 10^« Nd (cm"^) 1.17 X 10^® 1.65X 10^^ Xj (/xm) 0.05 0.477 Tj (^m) 4.5 7.28 Voc (V) 1.52 1.21 F.F. 0.912 0.895 T/c (%) 16.86 13.16 Jsc(A/cm^) 0.0164 0.0164 Eg (eV) 1.975 1.406

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72 Table 4.5 Optimal design of Alo. 44 Gao.s 6 As/GaAs/ Ino. 53 Gao. 47 As three-junction solar cell at room temperature for AMO. Alo.44Gao.S6As GaAs I1io.53Gao.47 As Na (cm 3.33 X IQi® 10 '« 2 x 10 ^® Nd (cm"^) 7.94 X 10^“^ 1 . 8 x 10 ^^ 5x 10^^ Xj (urn) 0.05 0.5 0.36 Tj (//m) 7.1 10 2.0 Voc (V) 1.50 1.21 0.541 F.F. 0.912 0.895 0.81 Vc (%) 16.76 13.24 5.34 Jsc(A/cm^) 0.0164 0.0164 0.0164 E, (eV) 1.98 1.406 0.744

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73 \1/ \|/ n n p+ p GaAlAs n GaAlAs n+ P GaAs n GaAs 1 SUBSTRATE ^ WINDOW TUNNEL DIODE Fig. 4.1 The schematic diagram of an AlGaAs/GaAs two-junction solar cell.

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74 MIDDLE CELL TOP EMITTER TOP CELL MIDDLE CELL TUNNEL DIODE BOTTOM CELL P N P N N++ P-HP N Fig. 4.2 The schematic diagram of an AlGaAs/GaAs/InGaAs three-junction solar cell.

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75 Fig. 4.3 Flowchart for the optimization of multijunction solar cells.

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76 Temperature (K) Fig. 4.4 Temperature and air mass ratio dependence of the efficiency of the AlGaAs/GaAs two-j unction solar cell.

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Efflciericy 77 45 42 39 36 33 30 Temperature (K) Fig. 4.5 Temperature and air mass ratio dependence of the efficiency of the AlGaAs / GaAs /InGaAs three-junction solar cell.

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CHAPTER 5 THEORETICAL CALCULATIONS OF ELECTRON AND HOLE MOBILITIES IN ALxGAi.^AS 5.1 Introduction Although there exists some experimental and theoretical data for electron and hole mobilities in AlxGai_xAs [80-100], the absence of a well developed experimental characterization of these mobilities as a function of A1 composition, dopant density and temperature makes the development of an accurate model difficult. In this chapter, we examine various scattering mechanisms of the AlxGai_xAs and develop a good approximation for numerical simulations of these mobilities. The reasons why we probe the AlxGai_xAs alloy system particularly are twofold. First, AlxGai_xAs material is the leading candidate for the top cell of the multij unction solar cells because of its bandgap and lattice matching to the GaAs solar cell. Second, unlike the GaAs, there are few reports on high efficiency AlxGai_xAs solar cells. We believe that this is due to the low mobilities and lifetimes of AlxGai_xAs. Therefore, the characterization of the dominant scattering processes that affect the electron or hole mobility and hence affect the diffusion length of AlxGai_xAs are needed to facilitate the optimal design of a GaAs based multijunction solar cell. Instead of doing a linear extrapolation of the experimental mobility data of GaAs and AlAs, which would not be accurate at all, we investigate in detail all the possible scattering mechanisms of AlxGa^.xAs, thus attaining an accurate model for numerical simulation. The modeling of electron mobility is discussed first; it is more complicate than that of hole mobility due to the different bands involved. For a full analysis of the Hall electron mobilities in the AlxGai_xAs ternary compound, we first consider all the electrons involved in the conduction process such as 78

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79 those in the F, X and L minima. It is known that the mobilities in different scattering mechanisms for example polar optical scattering /^po, piezoelectric scattering deformation potential scattering ionized impurity scattering fin, space charge scattering /igc, alloy scattering and intervalley scattering /Xjv have all been observed in the AlxGai.^As materials at room temperature [80-83]. According to our calculations, and fi^c are dominant in limiting the electron mobility at temperatures as low as 100 K in the n-type AlxGai_xAs with doping concentrations greater than 10^* cm When the temperature increases, fipo and become more dominant than other scatterings. As the temperature increases above room temperature, fipo and Hiy play the dominant role in the hall electron mobility of AlxGai_xAs. It should be noted that although the model can be applied to all the different A1 compositions from 0 to 1, we only show the results for A1 composition between 0.10 and 0.45. This is because that bandgap region of the AlxGaj.xAs is probably the potential top cell for the GaAs based multijunction solar cells. As for the hole mobility calculations, the intervalley scattering is not considered since different bands are not involved. Our calculations show that /idp, and are dominant in limiting the hole mobility in the p-type AlxGai_xAs with doping concentrations greater than 10^* cm~^ at 100 K. As the temperature increases, fip^ becomes more and more dominant in limiting the hole mobility of AlxGai_xAs. 5.2 Scattering Processes Of AlxGai_xAs 5.2.1 Polar Optical Scattering In polar semiconductors such as III-V compounds, the interaction of carriers with the optical mode of lattice vibrations is known as polar optical scattering [84]. Because of the strong dipole moment set up by the optical modes in the polar crystals, the coupling between an electron or hole and the optical modes is likely to be much stronger than in non-optical crystals. The temperature dependence of the electron or hole mobility due to the polar optical scattering follows the analysis of Fortini et al.

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80 [85] and is given by Mpo — ^4[exp(z) l]G(z)Ti/2(mVmo)-^/2 (5.1) where z — To/T with Tq, being the longitudinal optical phonon temperature, and the function G(z) was obtained by [85]. To is denoted by ^wlo/Kb, where u?i,q is the longitudinal phonon energy and m* is the effective mass of an electron or hole. The ei and are the low and high frequency dielectric constants, respectively. They are given by [86-87] 10.06x-M2.91(l -x) (5.2) 8.16X + 10.91(1 -x) (5.3) It should be noted that one of the crucial parameters needed for the calculations of electron mobility in the various minima for different values of x is the effective mass. The masses used in the polar optical scattering and others scatterings are the conductivity effective masses. They are given as [88] mp = 0.067 + 0.083X (5.4) mx = 0.32 0.06X (5.5) = 0.11-h0.3x (5.6) Since the hole mobility does not involve different bands transitions, only one hole mobility effective mass is needed for the calculations. It is defined as [72] J_ _ y/mih -Iy/mhh mu,ymU^ -Imhh>/hh (5.7)

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81 where mui and mhh are the light hole and heavy hole effective masses. They are given as [88] mih = 0.087 -|0.063x (5.8) mhh = 0.62 -10.14x (5.9) 5.2.2 Piezoelectric Scattering If a III-V compound semiconductor consists of dissimilar atoms such as AlxGai_xAs where the bonds are partly ionic and the unit cell does not contain a center of symmetry, carriers may be scattered by longitudinal acoustic waves due to piezoelectric scattering. Since the lattice constant of AlxGai_xAs material is almost independent of A1 composition, its elastic constants should be nearly the same as for GaAs. With this assumption, fip^ is given as [89] ei T (m*/mo)3/2 According to our calculation /Xpg is negligible. It should be noted that all the parameters with same names in these different scattering processes are defined as the same unless specified otherwise. 5.2.3 Deformation Potential Scattering The scattering of an electron or hole by the longitudinal acoustical phonon is an important scattering for many semiconductors near room temperature. The scattering is elastic if the electron or hole energy is much greater than the phonon energy and the change in electron or hole energy during scattering is small compared to the average energy of an electron or hole. The n^p due to the acoustical mode scattering has been derived by Bardeen and Shockley as [90]

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82 /^dp = 3.17 X 10"® (5.11) where the mass density p = 5.37 (1 x) + 3.60 x [86, 91] and is the longitudinal sound velocity. The deformation potential Ei is equal to (6.7 1.2x) [88]. 5.2.4 Ionized Impurity Scattering The scattering of an electron or hole by an ionized impurity center in a semiconductor is an example of elastic scattering. This is due to the fact that the mass of an impurity atom is much greater than that of an electron or hole. can be calculated from the expression given by [92] , ^ 64V?to^,.^(2KBT)3/^ N,q3v^ln(2l!=^g^) where Nj is the ionized impurity density, and h, Kb and Cq are the Planck constant, Boltzmann constant and permittivity in vacuum, respectively. 5.2.5 Space Charge Scattering The space charge scattering is caused by the crystal inhomogeneities. Such inhomogeneities may act like a mobility killer center and probably result from the grown-in defects. Weisburg [93] and Conwell and Vcissel [94] have derived the expression for the /fsc due to the space charge scattering which is 3.2 X 10^ ~ N AFT) — NsCr^Tm*/mo where Ng and
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83 = 5 X 10^ + 6.3 X 10®x (5.14) However, according to our calculations the optimal fitted value is 4.5 x 10“* cm“^. This is in close agreement with the capture cross sections and defect densities obtained from our DLTS measurements. 5.2.6 Alloy Scattering In the III-V ternary compound semiconductors of the type AxBi_xC, the constituent elements A and B are randomly distributed among the C atoms. This random distribution constitutes a random potential component to the periodic potential, which causes an additional scattering process known as the alloy scattering. The alloy scattering mobility fig, is given by [89, 95, 96] fig = 52.3T“^/^{(m*/mo)®^^x(l x)(Ea)^}"^ (5.15) where Eg = 0.3 -t0.01 lx [89] is the alloy scattering potential. 5.2.7 Intervalley Scattering Since there is no band to band transition involved for hole, the intervalley scattering is for electron only. The scattering rate from a k state in the i valley to a state in the j valley has been derived by Fawcett el al. [97], the mobility limited by nonequivalent intervalley scattering can be expressed as [89, 98] 4y2/3/Xdp [T/Tc 2/3 -F (2/3)( AE/KbTc)]^/^ Zj(Tc/T)3/2^ exp(Tc/T)-l ^ [T/T, 2/3 (2/3)(AE/KbT,)]i/ 2 l-exp(-T./T) ) where AE is the subband gap among the minima involved in the process and Zj is

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84 the number of nonequivalent intervalley. For equivalent intervalley scattering, the Zj is changed to (Zj 1) and AE = 0. Since the masses of group V atoms are larger than those of group III atoms in AlxGai_xAs, we have selected the longitudinal optical phonon involved in the process of scattering among the L and X minima, and hence Tc = To can be assumed [99]. It should be noted that because there is only one F band minimum, there is no equivalent intervalley scattering in the F band. 5.3 Analysis and Discussion To analyze the mobility data, the following assumptions are made : (a) the electrons or holes are scattered in a parabolic band, (b) the various scattering mechanisms are independent of each other, and (c) Matthiessen’s rule for calculating the electron or hole mobility in AlxGai_xAs is valid. The F, X, L conduction band structure as a function of alloy composition plays an important role in determining the electron mobility. For the electron mobility in the composition range 0 < x < 0.32, the transport properties are primarily determined by the electrons in the F conduction minimum, and the effects of the L and X minima may be negligible. Similarly, for x is greater than 0.6, the X minima plays the major role. In the intermediate band crossover composition range, i.e., for 0.32 < x < 0.6, the effects of F, L and X minima must be taken into account. Considering the threevalley conduction, nn and /ih can be expressed a.s [100] _ (1 -I(nx/^x)/(Pr^r) + (nL/iL)/(nr/^r)T + (nx/nr)(Mx/Mr)^ + (nL/nr)(/^L/^r)^ and ^ 1 + {nx/nr){fix/firy + {nL/nr){fiL/ “ ^1-1(nx//x)/(nr/^r) + {nLt^h)/{nrfir (5.18)

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85 Assuming that the Boltzmann statistics are valid for the electron concentrations in the crystals studied, the following approximation is also valid. nr = Njexp(EF/KeT) (5.19) and and = exp(-AErL/KBT) (5.20) nr mp — = exp(AErx/KsT) (5.21) nr mr where A Erx and A Err are the F-X and F-L intervalley separation and m*^ is the density of state effective ma^s. The empirical formulae for the conduction and density of state effective masses are as follows [88]: mp = 0.067 0.083x (5.22) mx II p bo Ol 1 o (5.23) = 0.56 -f O.lx (5.24) The energy band gaps of the three different conduction band minima are given respectively by

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86 El 1.424 + 1.247X for x< 0.45 1.424 + 1.247a: + 1.147(a: 0.45)^ for 0.45 < x 1.900 + 0.125 x + 0.143x2 1.708 + 0.642X (5.25) (5.26) Figures 5.1 and 5.2 show the results of electron and hole mobilities respectively for the A1 composition between 0.10 and 0.45 and three different NsQ values. From Table 5.1, it is clear that our results are in better agreement with the experimental data than those of others. The major discrepancy among these data is the calculation of the space charge scattering. The product of the density of the scattering center (Na) and the capture cross section (cr) used in our calculations is 4.5 x 10“* cm“^ which is less than those given in references [81] and [83]. Figures 5.3 and 5.4 show the room temperature electron and hole mobilities of different scattering processes a function of A1 compositions at No = 1.5x 10^® and Na = 10^® cm“^ respectively. According to Fig. 5.3, polar optical phonon scattering, intervalley scattering and space charge scattering are the three dominant scattering processes limiting the electron mobility at room temperature. However, polar optical phonon scattering, alloy scattering and deformation potential scattering are the three dominant scattering processes for hole mobility as shown in Fig. 5.4. The A1 composition and temperature dependence of the electron and hole mobilities of AlxGai_xAs at No = 10^^ and Na = 10^® cm~^ are shown in Fig. 5.5 and Fig. 5.6 respectively. According to our calculations, ionized impurity scattering and space charge scattering are the two dominant processes at low temperatures lOOK and 200K, for the electron mobility case. As the temperature increases, polar optical phonon scattering and intervalley scattering become more dominant. For the hole mobility case, deformation scattering and ionized impurity scattering and alloy

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87 scattering are the three dominant processes at low temperature. When the temperature increases, the polar optical phonon scattering takes over the role of the ionized impurity scattering in limiting the hole mobility of AlxGai_xAs. Figures 5.7 and 5.8 show the temperature and doping density dependence of the electron and hole mobilities of Alo. 38 Gao. 62 As. It is indicated that the doping level has little effect on either electron or hole mobilities of Alo 38 Gao. 62 As if doping density is less than 10^* cm“^. For doping density above 10^® cm“®, however, there is a huge decrease. This is because at low temperature and high doping level, the ionized impurity scattering is the most dominant process among all scattering processes. 5.4 Summary This chapter presents the results of a comprehensive study of the scattering processes of AlxGai_xAs alloy system. Calculations of the electron and hole mobilities of for AlxGai_xAs as a function of doping density, temperature and A1 composition have been carried out. It has been found that polar optical phonon scattering, intervalley scattering and space charge scattering are the three dominant processes for electron mobility of AlxGai_xAs at room temperature. As for the low temperature and high doping density such as 10^® cm or higher, ionized impurity becomes dominant. For the hole mobility case, it was found that the influences of the alloy scattering, polar optical phonon scattering and deformation scattering are significant. Our theoretical calculations show good agreements with the experimental data.

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Table 5.1 Electron mobilities of the Alo. 38 Gao. 62 As and Alo.igGao.siAs. x(%) ND(cm~^) H (cm^/V-s) NsQ(cm"^) experiment 0.38 1.5x 10^® 1432 45000 1200 0.38 1.5x 10^® 855 244000[78] 0.38 1.5x 10^® 1231 95000 [80] 0.19 1.5x 10^® 2689 45000 2700 0.19 1.5x 10^® 2221 95000 0.19 1.5x 10^® 1989 128500 -

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Mobility 89 Fig. 5.1 Electron mobilities vs. A1 composition and NsQ values. + : this study, solid box : NsQ obtained from [81] and * : NsQ obtained from [83].

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Mofaillty 90 X composition Fig. 5.2 Hole mobilities vs. A1 composition and NsQ values. + : this study, solid box : NsQ obtained from [81] and * : NsQ obtained from [83].

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Mobility 91 7 Fig. 5.3 Room temperature electron mobilities of different scattering processes vs. A1 composition at Nd = 1.5 x 10^® cm~^.

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Mobility 92 5 Fig. 5.4 Room temperature hole mobilities of different scattering processes vs. A1 composition at Na = 10^* cm“^.

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Motaility 93 5 Fig. 5.5 A1 composition and temperature dependence of electron mobility of AlxGax_xAs at Nd = 10^^ cm~^.

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Mobility 94 4 X composition Fig. 5.6 A1 composition and temperature dependence of hole mobility of Al^Gai.^As at Na = cm-^

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Mobility 95 11)14 igl5 iq16 iq 17 igl8 igl9 N-doping density Fig. 5.7 Temperature and doping density dependence of electron mobility of Alo. 38 Gao .62 As.

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Mobility 96 10l4 10^^ 10^® 10^^ 10^® 10^® P-doping density Fig. 5.8 Temperature and doping density dependence of hole mobility of Alo,38Gao,62As.

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CHAPTER 6 SUMMARY, CONCLUSION AND RECOMMENDATIONS 6.1 Summary and Conclusion In this dissertation, an expert system has been developed to facilitate the optimal design of single-j unction and multijunction solar cells for both space and terrestrial applications. A rule-base expert system was set up by adopting the experimental data and/or semiempirical formulae used today. In addition, a modified Box complex algorithm with some heuristic rules was implemented to reduce the computation time of the optimization process. This work was necessitated by the lack of an efficient optimization algorithm and by the failure of an inadequate device model to realistically predict the optimal performance of the solar cells. A simple model for calculating the displacement damage of the proton or electron irradiated solar cells under different fiuences, energies and environmental conditions has been presented in Chapter 2. A computer program for computing the degradation of short-circuit current, open-circuit voltage and conversion efficiency of a solar cell has been coded. This program included the calculations of the damage constant of the minority carrier diffusion length, a parameter important for the optimal design of irradiated solar cells. Close agreement was obtained between the calculated and measured Lc, Voc and 7 }^ in the proton and electron irradiated Alo.33Gao.67 As and GaAs single-j unction and twojunction cells for proton energies of 100 KeV up to 10 MeV and fiuences varying from 10^° to 10^^ cm~^, and for 1 MeV electrons with fiuences of 10^“*, 10^® and 10^® cm~^ for both the normal and omnidirectional incident conditions. In Chapter 3, a new computer model for truly optimizing the structure of GaAs single-j unction solar cell for both space and terrestrial applications is proposed. The 97

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98 model, however, can apply easily to other solar cell system as well. This model not only takes into account the effects of the intrinsic structure parameters such as junction depth Xj, cell thickness Tj, doping densities Na and No, surface recombination velocity Sp and Sn, but also incorporates the extrinsic structure parameters. An efficient Box optimization algorithm [15] with minor modifications is implemented in the model. The optimal efficiency of the single-junction GaAs solar cell obtained by the simulation is 27.8% at room temperature for AMO which is close to the cell efficiency namely 26% made by the state of the art [69]. In Chapter 4, an expert system approach to the optimal design of multijunction solar cells for both terrestrial and space applications is described for the first time. An expert systems is a knowledge-intensive computer program. The knowledge of an expert system consists of facts and rules. The facts constitute a body of information that is widely shared, publicly available and generally agreed upon by experts in the field [33, 34, 68 ]. The rules are those if-then rules that characterize expert level decision making in the field. We have developed a rule-base system and a couple of heuristic rules which can prune the search space of the design problem and hence to save the computation time. The optimal designs of Alo. 44 Gao. 56 As/GaAs two-junction solar cell and Alo. 44 Gao. 56 As/GaAs/Ino. 53 Gao. 47 As three-j unction solar cell with room temperature efficiencies for AMO of 30.01% and 35.3% respectively were obtained in the simulation. In Chapter 5, a comprehensive study of the scattering processes of (AlGa)As has been presented. Theoretical calculations of the electron and hole mobilities in AlxGai_xAs were made. It was found that the mobilities of AlxGai_xAs are quite sensitive to the A1 composition, temperature, doping density and defect density. This knowledge is indispensable to the fabrication of a high quality (AlGa)As material and a high efficiency top cell of the GaAs based multijunction solar cells.

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99 6.2 Recommendations Although an expert system has been developed in this dissertation to accomplish the goals of optimizing the performance of single-j unction and multijunction solar cells, there are some uncertainties due to the lack of experimental data. Actual cells should be fabricated and characterized in order to facilitate comparisons with the simulations results. Further extensions of this research include the following: 1. Since the threshold energy Ta plays a major role in the displacement damage model, it is important that an accurate value of Tj be obtained for each material used in a multijunction solar cell. Except for GaAs and Ge, values of Ta for materials used in the present model are still not well known. Also new data are needed to obtain more accurate calculations of the displacement damages in multijunction solar cells. 2. The path length and penetration depth are based on JanniÂ’s data [41]. Further experimental data for (AlGa)As and (InGa)As are needed to improve the computer simulations. 3. Accurate recombination cross section data for (AlGa)As, (InGa)As and Ge are needed for calculations of short-circuit degradation. This can be achieved by using DLTS technique to determine the recombination cross section in each cell. 4. The correlations between the minority carrier lifetime, diffusion length and mobility and doping density, temperature and x composition for InxGax_xAs and AlxGai_xAs should be further improved in order to optimize the top and bottom cells. 5. Additional device structures such as heterojunctions or three-terminal structures etc. should be considered for the expert system in the future.

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100 6. Defect characterization of (InGa)As and (AlGa-)As should be done in order to extract the parameters which affect the quality of these materials. In short, the more facts and rules an expert system has, the more powerful it is. The expert system presented in this dissertation may be considered simply as a kernel. As the information listed just above becomes available , the more accurate results can be derived from the expert system developed by the research described in this dissertation.

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APPENDIX A A COMPUTER PROGRAM FOR CALCULATING THE TOTAL NUMBER OF DISPLACEMENT DEFECTS { The purpose of this prograon is to calculate the total number of displacement defects induced by energetic electron or proton. The input parameters are atomic number, atomic weight aind threshold energy of the cell material. The range of proton energy is from 400 eV to 10 MeV aind is from 200 Kev to 5 MeV for electron. } Program Displace(input, output, out(lfn=103) ,inpa(lfn=104) ); CONST Mp = 1.67264E-27; (* proton mass. Kg *) Me = 9.1095E-31; (* electron mass, Kg *) Er = 13.6; (* Rydberg constant, eV *) Z1 = 1; (* projectile atomic number *) aO = 5.2917E-5 ; (* Bohr's radius, urn *) pai = 3.141592654; Ne = 4.42E10; (* electron concentration, l/(um*um) Vc TYPE 2.99792E8; (* velocity of light, m *) Xout = array[1..50] of real; Yout = array[1..50] of real; pro jectile= (proton, electron) ; InPara = array [1..4] of real; VAR out, inpa : text; Tm, Alpha , Beta , Beta2 , Vd , TmDivTd, PaiAB, Ein : real ; Particle : projectile; CrossS , (* files to store the results *) (* majcimun transfer energy, eV *) (* Z2 / 137 *) ( velocity / (Vc of light) *) (* Beta * Beta ) (* average number of displacement *) (* Tm / Td *) (* pai * alpha * beta *) (* particle energy, eV ) ( proton or electron *) (* crosssection area *) 101

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102 Dtotal , (* total displacements due to particle dajnage * Tbar , (* average recoil energy *) Ed (* displacement with no multiple scattering *) ReduceC , Dave : Xout; (* displacement with multiple scattering *) Ec : real; (* classical energy of M0*Vc**2, eV *) ii : integer; JoeV : real; (* energy conversion, from jouls to eV *) Delta : real; cell. FigNum : integer; keep : xout; Td (* Threshold energy *) Z2 (* Projectile atomic number ) M2 : real ; (* Projectile atomic weight *) Sequence : integer; Range : integer; procedure initial; var i, j : integer; ch, ans : char; begin alpha := Z2 / 137; Ec:= Me sqr(Vc) / ( 1.60218E-19 ); JoeV:= 1 / ( 1.60218E-19) ; Delta:= 4*pai* sqr(aO*Er*Z2) ; end; Function BetaRatioC Evar : real ): real; var square : real ; begin square := sqr( Evar/ Ec + 1); BetaRatio:=sqrt(l 1/square); end; PROCEDURE PreCalC index: projectile); begin case index of proton : Tm:=( 4*M2) * Ein / Sqr(l+M2); electron: begin Tm : =2*Ein/ (Mp*m2*sqr (Vc) * Joev) * (Ein+2*Me*sqr (vc) *Joev) ;

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103 beta:=Betaratio(Ein) ; beta2:=sqr(beta) ; TmDivTd:=Tm / Td; PaiAB : =Pai*alpha*beta ; end; end; (* case *) end; (* Calculation of the displacement cross sections *) function CrossSection(index: projectile) :real; var Ta, Tb : real; begin case index of proton : begin PreCal (index) ; if Td>=Tm then CrossS [ii] :=1E-16 else CrossS[ii]:= delta/(M2*Ein) * ( 1/Td 1/Tm) ; CrossSection:=CrossS [ii] ; end; electron : begin PreCal (index) ; crossS[ii]:= (deltai/sqr(Ec) ) (l-beta2)/ sqr(beta2) * (TmDivTd -1 -beta2*ln(TmDivTd)+ 2*PaiAB* (sqrt(TmDivTd)-l)PaiAB*ln (TmDivTd) ); CrossSection:=CrossS [ii] ; end; end; (* of case *) end; procedure RecoilAve(index:projectile) ; var Numer, Denom : real; begin if index=electron then begin Numer:= Tm*ln (TmDivTd)beta2*(Tm-Td)+ 2*PaiAB*(Tm-sqrt(Tm*Td)) PaiAB* (Tm-Td) ; Denom: = TmDivTd1Beta2*ln(TmDivTd)+ 2*PaiAB*(sqrt (TmDivTd) -1) PaiAB*ln (TmDivTd) ; Tbar[ii] := Numer / Denom; end else Tbar [ii] :=(Td*Tm)/(Tm-Td) * ln(tm/td) ; end;

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(* First derivative of the path length. *) function Dp (Evar: real) ; real; var XX : real ; begin xx:=Evar / 1E6; if cell=l then begin if XX <= 0.150 Then Dp : =1 . 938958*EXP(-0 .452449*LN(XX) ) if XX <= 1 . 25 then Dp : =12 . 74343+EXP (0 . 145 135*LN (XX) ) Dp : =15 . 95597*EXP (0 . 5556635*LN(XX) ) ; end else if cell = -2 then begin if XX <= 0.150 then Dp : = 1 . 940185*exp (-0 . 4521705*ln(xx) ) if XX <= 1.25 then Dp : = 12 . 819498*exp (0 . 1469979*ln(xx) ) Dp : = 16 . 035977*exp(0 . 5565806*ln(xx) ) ; end else if cell=2 then begin IF XX <= 0.150 THEN DP:=2.106838*EXP(-0.454091*LN(XX)) IF XX <= 1.25 THEN DP : =13 . 455826+EXP (0 . 135261*LN(XX) ) DP : =16 . 933599*EXP (0 . 550638*LN(XX) ) ; end else if cell=3 then begin if XX <= 0.150 then Dp : =1 . 947311 1*EXP (-0 . 453661*LN(XX) ) if XX <= 1.25 then Dp : =12 . 496518*EXP(0 . 137668*LN(XX) ) Dp : =15 . 817996*EXP (0 . 545829*LN (XX) ) ; end else begin if XX <= 0.150 then Dp : =2 . 046476 1*EXP ( -0 . 452753+LN (XX) ) if XX <= 1.25 then else else else else ELSE ELSE else else else

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105 Dp : =13 . 265437*EXP (0 . 141380*LN(XX) ) else Dp : =1 . 6697369*EXP (0 . 562229*LN(XX) ) ; end; end; (* first derivative of penetration depth *) function DrE(Evar: real):real; var XX : real; begin XX : = Evar / 1 . 0E6 ; if cell = -2 then begin if XX <= 0.20 then Dre: = 104.1110 + 7388.5995 xx 15681.3270 *sqr(xx) else if XX <= 1.0 then Dre : = 734.929 + 1661.5505 * XX 1069.1373 sqr(xx) else Dre : = 1463.058 81.84514 * XX 2.55829 * sqr(xx) ; end else if cell = = -1 then begin if XX <= 0.20 then Dre : = 103.2968 + 7322.016 * XX 15542.8868 * sqr(xx) else if XX <=1.0 then Dre:= 728.088 + 1647.176 XX 1060.1532 * sqr(xx) else Dre : = 1449.811 81.27775 * XX + 2.540295 * sqr(xx) ; end; if cell = = 0 then begin if XX <= 0.20 then Dre : = 101.9048 + 7298.573 * XX 15467.2737 * sqr(xx) else if XX <=1.0 then Dre : = 727.348 + 1638.159 * XX 1051.4330 * sqr(xx) else Dre: = 1446 . 344 79.51220 * XX + 2.484936 * sqr(xx); end; if cell = = 1 then begin if XX <= 0.20 then Dre : = 103.5243 + 7340.605 * XX 15581.5448 * sqr(xx) else if XX <=1.0 then Dre : = 707.2341 + 1755.050 * XX 1166.50767* sqr(xx) else Dre : = 1453.507 81.43596 * XX + 2.5452710* sqr(xx) ; end; if cell = 2 then

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106 begin if XX o CN O II V then Dre: = 100.0425 + 7057.603 XX 1499.25345 * sqr(xx) else if XX <=1.0 then Dre : = 700.994 + 1589.943 * XX 1024.50218* sqr(xx) else Dre : = 1397.202 78.99068 XX + 2.4680365* sqr(xx); end; if cell = = 3 then begin if XX <= 0.20 then Dre : = 99.67980 + 6.757367 * XX 14485.9866 * sqr(xx) else if XX <=1.0 then Dre : = 678.5045 + 1476.745 XX 963.79863 * sqr(xx) else Dre : = 1321.854 83.65355 XX + 2.756492 * sqr(xx); end; if cell = = 4 then begin if XX <= 0.20 then Dre: = 109.6292 + 7729.241 * XX 16443.7531 * sqr(xx) else if XX <=1.0 then Dre: = 769.090 + 1729.35898+xx 1118.0255 * sqr(xx) else Dre: = 1522.400 86.90735 XX + 2.7739247* sqr(xx); end; end; function VdE( index : pro j ect ile) : real; begin case index of proton: begin Precail (index) ; if Tm > ( 2*Td ) then VDE:=1/2*(TM/(TM-TD))* ( 1+ LN(TM/(2*TD))) ELSE if Tm > Td then VdE:=1.0 else VdE:=0.0; end; electron rbegin particle : =electron ; Precail ( index) ; ii:=l; RecoilAve (particle) ; if TbarLl] > ( 2 Td ) then VdE:= TbarCl] / (2*Td) else if Tbar[l] >= Td then VdE:=1.0 else VdE:=0.0;

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107 end; end; (* case ) end; (* Calculate total number of displacement defect *) function TotalD ( index : projectile) :real; var xO, xl, x2, xi, h : real; n, jj : integer; xOO : real; begin n:=250; case index of proton : begin h:=Ein/ ( 2 * n ) ; if Ein< 250 then x0:=0.0 else xO:=crosssect ion (index) *VdE (index) *Dp(Ein) ; xl:=0.0; x2:=0.0; for jj:=l to ( 2*n -1 ) do begin Ein:=0.0 + jj*h; if Ein< 200 then begin end else if odd(jj) then xl :=xl+crosssect ion (index) *VdE( index) *Dp(Ein) else x2 : =x2+crosssection(index) *VdE(index) *Dp(Ein) ; end; TotalD: = lE-6 * Ne*h*(x0+2*x2+4*xl)/3; end; electron: begin h:=Ein / ( 2 * n ) ; if Ein< 260E3 then begin x0:=0.0 ; end else xO :=crosssection(index)*Vde(index)*Dre(Ein) ; if x0<=0.0 then x0:=0.0; xl:=0.0; x2:=0.0; for jj:=l to (2*n 1) do begin Ein:=0.0 + jj*h; if Ein< 260E3 then begin end else if odd(jj) then xl : =xl+crosssection ( index) *vde( index) *Dre(Ein) else x2 :=x2+crosssection(index)*Vde(index)*Dre(ein) ; end;

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108 ToTalD : = 1 . 0E-6*Ne*H* (x0+2*x2+4*xl) /3 ; end; end; (* case *) end; Procedure SetUpModel; var i, j : integer; Rtemp: xout; begin writelnCÂ’ > Part one. Â’); writelnC' > Td ? '); readln(Td); writelnC' > Z2 ? Â’); readln(Z2) ; writelnC' > M2 ? '); readlnCM2) ; writelnC' > cell ?' ) ;readlnCcell) ; initial; writelnC' 1 = Calculate proton displacement cross section'); writelnC' 2 = Calculate electron displacement cross section'); writelnC' 3 = average transferred energy for proton '); writelnC' 4 = average transferred energy for electron'); writelnC' 5 = total number of displacement defect for proton '); writelnC' 6 = total number of displacement defect for electron'); read CFigNum) ; case figNum of 1 : begin for j :=1 to 50 do begin ii:=j; Ein:=expCCj/10 +2)*lnClO) ); keep[j] :=j*0. 1; IF Ein<=180 then Ein:=180; Rtemp[j]:=C lnCCrossSectionCproton)*lE16)) / LNClO); end; end; 2 : begin for j :=1 to 50 do begin Ein:=200E3*j ; keepCj] :=j*0.1; if Ein<=250 then Ein:=250; if Cein<=360) and Ccell=4) then Ein:=360 ;

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109 Rtemp [j] : =CrossSect ion (elect ron)*lE16 ; end; end; 3 : begin PARTICLE : =proton ; for J:=l to 50 do begin ii:=j ; Ein:=100E3 * j ; PreCal (proton) ; RecoilAve(proton) ; keepCj] :=Ein /1E6; Rtemp [j] :=Tbar[j] ; end; end; 4 : begin particle : =electron ; for j :=1 to 50 do begin ii:=j ; Ein:=100E3 * j ; PreCal (electron) ; RecoilAve(electron) ; keepCj] :=Ein/lE6; Rtemp [j] :=Tbar[j] ; end; end; 5 : begin writeln(' range. 1— > 20 Kev. 2— >1 Mev. 3— >10 Mev ? ') readln(dnum) ; for j :=1 to 50 do begin ii:=j; if dnum=l then Ein:=400*j else if dnum=2 then Ein: =20000* j else Ein:=200E3*j ; Dtotal [j] :=TotalD (proton) ; keep[j]:=j / 10; Rtemp [j] :=Dtotal[j] ; end; end; 6 : begin

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no for j :=1 to 50 do begin Ein:=100E3*j ; Dtotal [j] :=ToTeilD(electron) ; keepCj] :=j/10; RtempCj] :=DtotaLl[j] ; end; end; end; (* of case ) for i:=l to 50 do writeln(out ,keep[i] ,rtemp[i]) ; end; begin (* of main *) writelnC' — > Start to calculation the displacement defect!'); SetUpModel; end .

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APPENDIX B A COMPUTER PROGRAM FOR CALCULATING THE DEGRADATION OF SHORT-CIRCUIT CURRENT { This program is to calculate the short-circuit degradation of single-junction amd multijunction solar cells. In addition, the calculation for daimage constants of minority carrier diffusion length and lifetime is included in this program. GaiAs, Si, Ge, InGaAs and AlGaAs are those materials used in this program. Proton aind electron are the energetical particles that produce the displacement defects. This program is user friendly. User may just follow the questions given by the prograim to enter the necessary information for the program to execute. } Program SolarIrr(input , output, out (lfn=103) , inpa(lfn=104) , Inpb(lfn=105) , Inp2(lfn=106) , inp3(lfn=107) , inp4(lfn=108) ) ; CONST Mp = 1.67264E-27; (* proton mass. Kg *) Me = 9.1095E-31; (* electron mass, Kg *) Er = 13.6; ( Rydberg constant, eV *) Z1 = 1; (* projectile atomic number *) aO = 5.2917E-5 ; (* Bohr's radius, urn *) pai = 3.141592654; Ne = 4.42E10; (* electron concentration, l/(um*um) Vc = 2.99792E8; (* velocity of light, m *) InlO = 2.30258509; TYPE Xout = array [0. .50] of real; Yout = array [0. .50] of real; projectile=(proton, electron) ; InPaira = array [1..4] of real; VAR out, inpa, (* files to store the results *) Inp2, Inp3, Inp4, inpb : text; Ein : real ; (* particle energy, eV *) 111

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(* proton or electron *) Particle : projectile; ReduceC , Kv3, W13, kv2, W12, WLl, Kvl : Xout; Delta : real; ii : integer; Config, Cell : integer; Plen : real; Rlen : real; Device : integer; Sequence : integer; Range : integer; Wd, Tj, Xj, Lp, Lnn, Lpirr, Lnirr, Nintr , NA, ND, Dpp, Dn, TauP, TauN, TauNIrr, TauPIrr, MrestN, MrestP, Mneff, Mpeff, Vtherl, Vther2, VthP, Vthn, Kip, Kin, Ktp, Ktn, CapCn , CapCp, low, high : InPara; tcell , lower, flux, absorpt , Dxij, SigMaR : real; SigmaR, SigmaE : InPara; Name : array [1..3] of integer; Omiter , Aiter, DIST : INTEGER; UpPene : real;

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113 Einn : real; Window : array [1..3] of boolean; Average, BasicC, OMNI, Runn, normal : boolean; absp : array [1 .. 200] of real; IscRatio , VscRatio , Pmratio : real; procedure initial; var i, j : integer; ch, ans : char; begin WRITELN(' Omni directional irradiation ? "Y" or "y"Â’); readln(ch) ; if (Ch='yO or (Ch='YÂ’) then begin 0mni:=true ; writelnCÂ’ what is the omni iteration #, <= 20 '); readln(omiter) ; writeln(out ,omiter) ; end else Omni :=false; writelnC' cell configuration? 1, 2 or 3 ?Â’); readlnCConf ig) ; for i:= 1 to Config do begin writelnC' window layer ? "Y" or "y" '); readlnCch); if Cch ='y') or Cch='Y') then window [i]:= true else window [i]:= false; writelnC i:l, Â’ window layer thickness Wd ? '); readln CWd[i]); writelnCout , ' Wd [',i:l,'] = ', Wd[i]); writelnCi: 1 , ' cell name: -2 , -1 ,0 , 1 ,2 ,3,4,5 '); writelnC' -2 : AlGaAs : A1 =40*/. , -1 : 33*/., 0 : 85Â’/., 1: 35Â’/. '); writelnC' 2 : GaAs, 3: InGaAs , 4 ; Ge, 5: Si '); readln CName [i]) ; writelnC i:l,' Junction depth Xij ? ');

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114 readln (Xj[i]); writeln(out , ' Xj = >, Xj[i]); writelnC i:l,' Cell thickness Tj ? '); readln (Tj [i] ) ; writeln(out , ' Tj Tj[i]); writelnC i: 1, ' low emd high cutoff wavelength readlnC low[i] , high[i]); writelnC 1: 1, Â’ Electron and Hole capture cross sections? readln ( SigmaE[i] ,SigmaH[i] ) ; writelnCout, ' .. SigmaE = ',SigmaE[i], ' .. SigmaH = ' ,SigmaH[i] ) ; end; for i:= 0 to 23 do readlnC inpb, WLl [i] ,Kvl[i] ) ; (* for GaAs ) for i:= 0 to 35 do ( for Si *) readln(inp2,WL2[i] ,Kv2[i]) ; for i:= 0 to 11 do ( for Ge ) readlnC inp3,WL3[i] ,Kv3[i] ) ; for i:= 1 to 182 do read ( inp4 , Atype [i] , Btype [i] ) ; } end; C* GaAs absorption coefficients vs. wavelength *) Function GaiAsAbsorpC Atemp : real ): real; var i : integer; Btemp : read; found : boolean; begin if Atemp <= 0.300 then Atemp:=0.300; if Atemp >= 0.873 then Atemp :=0 .873; found :=false; i:= 1; repeat if Atemp <= WLl[i] then begin Btemp: = Kvl [i-1] (Kvl [i-1] -kvl [i] ) /(WLl [i] -WLl [i-1] ) CAtemp-WLl[i-l]) ; found: = true; end else i:= i + 1 ; until ( i >= 24 ) or ( found ); GaAsAbsorp:= 4.0 * 3.141596 * Btemp / Atemp ; end;

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115 (* Si absorption coefficients vs. wavelength *) Function SiAbsorp( Atemp : real ); real; var i ; integer; Btemp : real; found : boolean; begin if Atemp <= 0.294 then Atemp :=0 . 294; if Atemp >= 1.125 then Atemp :=1 . 125; f ound:=false; i:= 1; repeat if Atemp <= WL2[i] then begin Btemp : = Kv2 [i-1] (Kv2 [i-1] -kv2 [i] ) /(WL2 [i] -WL2 [i-1] ) *(Atemp-WL2[i-l]) ; found: = true; end else i:= i + 1 ; until ( i >= 36 ) or ( found ) ; SiAbsorp:= 4.0 * 3.141596 * Btemp / Atemp ; end; (* Ge absorption coefficients vs. wavelength ) Function GeAbsorpC Atemp : real ): real; var i : integer; Btemp : real; found : boolean; begin if Atemp <= 0.500 then Atemp :=0 . 500 ; found :=false; i:= 1; repeat if Atemp <= WL3[i] then begin Btemp : = Kv3 [i-1] (Kv3 [i-1] -kv3 [i] ) / (WL3 [i] -WL3 [i-1] ) *(Atemp-WL3[i-l]) ; found: = true; end else i:= i + 1 ;

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116 until ( i >= 12 ) or ( found ) ; GeAbsorp:= 4.0 * 3.141596 * Btemp / Atemp ; end; (* (InGa)As absorption coefficients vs. wavelength *) (* In — > 53 •/. ; Ga — > 47 ’/. *) Function InGaAsAbsorp (Atemp : real):real; var Btemp : real; begin if Atemp <= 0.74 then Btemp := 10.0 else if Atemp <= 1.40 then Btemp:= exp((-l . 51515*Atemp+6. 12121 )*lnl0) / 1E4 if Atemp <= 1.50 then Btemp := exp ( (-1 . 5491*Atemp+6 . 16874) *lnl0) / 1E4 if Atemp <= 1.54 then Btemp := exp((-3 . 65495*Atemp+9.32752)*lnl0) / 1E4 if Atemp <= 1.73 then Btemp:= exp((-14.2051*Atemp+25.5749)*lnl0 ) / 1E4 Btemp := lE-3; InGciAsAbsorp:= Btemp; end; (* Path length vs. proton energy *) Function Plength(Evar : real ): real; var XX : real; begin XX := Evar •/ 1E6; if cell=0 then begin if XX <= 0.150 then plength:= 3.300891*EXP(0.550212*LN(XX)) ELSE if XX <= 1.250 then plength:= 10 .796238*EXP(1 . 163227*LN(XX) ) ELSE plength:= 9 . 963561*EXP(l . 565366*LN(XX) ) ; end else IF CELL=-2 THEN begin if XX <= 0.150 then plength:= 3 . 541585*exp(0 . 5478295*ln(xx) ) else else else else else

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117 if XX <= 1.250 then Plength:= 11 . 176566*exp(l . 1469979*ln(xx) ) else Plength:= 10 . 302054*exp(l .5565806*ln(xx) ) ; end else if cell = 1 then begin if XX <= 0.150 then Plength:= 3 . 541146*EXP(0 . 547551*LN(XX) ) ELSE if XX <= 1.250 then Plength:= 11 . 12832*EXP(1 . 145135*LN(XX) ) ELSE Plength:= 10 . 25669*EXP(1 . 555663*LN(XX) ) ; end else if cell = 2 then begin if XX <= 0.150 Then Plength : =3 . 859321+EXP (0 . 545909*LN(XX) ) ELSE if XX <= 1.25 THEN plength : =1 1 . 852628*EXP ( 1 . 135261*LN (XX) ) ELSE plength : =10 . 920408*EXP ( 1 . 550638*LN (XX) ) ; end else if cell = 3 then begin if XX <= 0.150 then Plength := 3 . 564291*EXP(0 . 546339*LN(XX) ) ELSE if XX <= 1.25 then Plength := 10 . 98432*EXP(1 . 137668*LN(XX) ) ELSE plength:= 10 . 23269*EXP(1 . 545829*LN(XX) ) ; end else begin if XX <= 0.150 then Plength := 3.739584*EXP(0.547247*LN(XX)) ELSE if XX <= 1.25 then Plength:= 11 . 62228*EXP(1 . 141380*LN(XX) ) ELSE Plength:= 10 . 68817*EXP(1 . 562229*LN(XX) ) ; end; end; (* Penetration length vs. proton energy *) Function Rlength(Evar : read) :real; Var XX : real; begin xx:=Evar / 1E6 ;

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118 if cell=0 then begin if XX <= 0.175 then rlength:= 5 . 010253*EXP(0 .865712*LN(XX) ) ELSE if XX <= 1.5 then rlength:= 10 . 310898*EXP(1 . 257302*LN(XX) ) ELSE rlength:= 9 . 561796*EXP(1 .579760*LN(XX) ) ; end else if cell=-2 then begin if XX <= 0 . 175 then rlength:= 5.378740*exp(0.873210*ln(xx)) else if XX <= 1.50 then rlength:= 10.632386*exp(1.2475845*ln(xx)) else rlength : = 9 . 8560927*exp ( 1 . 5722024*ln (xx) ) ; end else if cell =1 then begin if XX <= 0.175 then Rlength := 5 .378855*EXP(0 .874058*LN(XX) ) ELSE if XX <= 1.5 then Rlength := 10 . 58202*EXP(1 .24645*LN(XX) ) ELSE Rlength:= 9.809475*EXP(1 .57141*LN(XX)) ; end else if cell = 2 then begin if XX <= 0 . 175 then Rlength : =5 .86137*EXP (0 . 8786719*LN(XX) ) ELSE if XX <= 1.5 then Rlength : =1 1 . 236521*EXP( 1 . 243952*LN (XX) ) ELSE Rlength : =10 . 427194*EXP(1 . 567030*LN(XX) ) ; end else if cell = 3 then begin if XX <= 0.175 then Rlength := 5 . 244473*EXP(0 .877962*LN(XX) ) ELSE if Xx <= 1.5 then Rlength := 10 .34555*EXP(1 .249145*LN(XX) ) ELSE Rlength := 9 .707031*EXP(1 .563826*LN(XX) ) ; end else begin if XX <= 0 . 175 then Rlength:= 5 . 640120*EXP(0 .871511*LN(XX) ) ELSE if XX <= 1.5 then

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Rlength:= 11 . 05239*EXP(1 .242816*LN(XX) ) ELSE Rlength:= 10 . 22688*EXP(1 .577814*LN(XX) ) ; end; end; (* reduced energy vs. path length and penetration depth *) Function eeleft(xx : real; kk : integer) : real; var dx : real; eleft: real; begin if kk=l then dx:=Rlen-xx else dx:=Plen-xx; if KK=1 then begin if Xx >= Rlen then eleft: =0.0 else if cell = 0 then begin if Ein <= 0 . 1E6 then eleft:= 0 . 159627*EXP(1. 164662*LN(DX) ) ELSE if Ein <= 0.45E6 then eleft := 0 . 143353*EXP(0.892092*LN(DX) ) ELSE IF EIN <= 2.0E6 THEN ELEFT := 0 . 179735*EXP(0.722203*LN(DX) ) ELSE eleft := 0.256094*EXP(0.619708*LN(DX)) ; end else if cell = -2 then begin if Ein <= 0 . 1E6 then eleft := 0 . 1488971*exp(l . 1532964*ln(dx) ) else if Ein <= 0.45E6 then eleft := 0 . 1371156*exp(0 . 9002795*ln(dx) ) else if Ein <= 2.0E6 then eleft := 0 . 17390829*exp(0 .726625*ln(dx) ) else eleft := 0 . 2422937*exp(0 . 6300121*ln(dx) ) ; end else if cell = 1 then begin if Ein <= 0.1E6 then Eleft := 0.149104*EXP(1.152006*LN(DX)) ELSE if Ein <= 0.45E6 then Eleft := 0.1374183*EXP(0.901323*LN(DX)) ELSE IF EIN <= 2.0E6 THEN ELEFT:= 0 . 174311*EXP(0.727081*LN(DX) ) ELSE Eleft:= 0.250527*EXP(0.622724*LN(DX)) ;

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120 end else if cell = 2 then begin if Ein <= 0.1E6 then Elef t : = 0 . 136 198+EXP (1.1451 18*LN (DX) ) if Ein <= 0.45E6 then Eleft := 0 . 1290153*EXP(0 . 907960*LN(DX) ) IF EIN <= 2.0E6 THEN ELEFT : = 0 . 168008+EXP (0 . 724845+LN (DX) ) Eleft := 0.240366*EXP(0.6243284*LN(DX)) ; end else if cell = 3 then begin if Ein <= 0 . 1E6 then Elef t : = 0 . 155403*EXP ( 1 . 148223+LN (DX) ) if Ein <= 0.45E6 then Eleft := 0 . 140691*EXP(0. 901968*LN(DX) ) IF EIN <= 2.0E6 THEN ELEFT : = 0 . 178959*EXP (0 . 722529+LN (DX) ) Eleft := 0.248245*EXP(0.627503*LN(DX)) ; end else begin if Ein <= 0.1E6 then Elef t : = 0 . 140368+EXP ( 1 . 155406+LN (DX) ) if Ein <= 0.45E6 then Eleft := 0 . 131029*EXP(0. 906509*LN(DX) ) IF EIN <= 2.0E6 THEN ELEFT := 0. 168896*EXP(0.726630*LN(DX)) Eleft ;= 0.245948*EXP(0.619878*LN(DX)) ; end; end else begin if Xx >= Plen then Eleft: =0.0 else if cell =0 then begin if Ein <= 0 . 07E6 then Eleft := 0 . 1252061*EXP (1 . 875158*LN(DX) ) if Ein <= 0 . 275E6 then Eleft := 0 . 1023260*EXP(1 . 205425*LN(DX) ) IF EIN <= 1.25E6 THEN ELEFT := 0 . 142437*EXP (0 . 812657*LN(DX) ) Eleft:= 0.230867*EXP(0.638220*LN(DX)) ; end else if cell = -2 then ELSE ELSE ELSE ELSE ELSE ELSE ELSE ELSE ELSE ELSE ELSE ELSE

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121 begin if Ein <= 0.07E6 then Eleft:= 0.1080430*exp(1.8799742*ln(dx)) else if Ein <= 0.275E6 then Eleft:= 0.093611105*exp(1.2303236*ln(dx)) else if Ein <= 1.25E6 then Eleft:= 0.1352034*exp(0.82234935*ln(dx)) else Elef t : = 0 . 21969256*exp (0 . 64718419*ln(dx) ) ; end else if cell =1 then begin if Ein <= 0.07E6 then Elef t : = 0 . 107870+EXP ( 1 . 880521*LN(DX) ) ELSE if Ein <= 0.275E6 then Elef t : = 0 . 093579*EXP ( 1 . 233397*LN(DX) ) ELSE IF EIN <= 1.25E6 THEN ELEFT: = 0 . 135318*EXP(0 .823439*LN(DX) ) ELSE Eleft:= 0.224582*EXP(0.642162*LN(DX)) ; end else if cell = 2 then begin if Ein <= 0 . 07E6 then Elef t : = 0 , 0907337+EXP ( 1 . 8837 10*LN (DX) ) ELSE if Ein <= 0.275E6 then Elef t : = 0 . 0837655+EXP ( 1 . 25 1878+LN (DX) ) ELSE IF EIN <= 1.25E6 THEN ELEFT: = 0 . 126622*EXP(0 .829107*LN(DX) ) ELSE Eleft:= 0.214667*EXP(0.644224*LN(DX)) ; end else if cell = 3 then begin if Ein <= 0.07E6 then Eleft := 0 . 105743*EXP(1 .883100*LN(DX) ) ELSE if Ein <= 0.275E6 then Eleft : = 0 . 0925301+EXP ( 1 . 250373*LN(DX) ) ELSE IF EIN <= 1.25E6 THEN ELEFT : = 0 . 135927*EXP (0 . 8255772+LN (DX) ) ELSE Eleft := 0.222678*EXP(0.646363*LN(DX)) ; end else begin if Ein <= 0.07E6 then Eleft := 0 . 097060*EXP(1 .880410*LN(DX) ) ELSE if Ein <= 0.275E6 then Eleft : = 0 . 087422 1+EXP ( 1 . 2431922*LN(DX) ) ELSE

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122 IF EIN <= 1.25E6 THEN ELEFT:= 0. 1300987*EXP(0.8247081*LN(DX)) ELSE Eleft:= 0.220143*EXP(0.6394407*LN(DX)) ; end; end; if eleft<=0.0 then eeleft:=0.00 else eeleft : =eleft*lE6 ; end; (* total number of defects vs. proton energy *) function Dcx (Evar: real) : real; var XX : real ; begin XX := Evar / 1E6; if cell=l then begin if xx<= 0.00032 then Dcx:= 0.0 else if XX <= 0.004 then Dcx: =-3.7668 + 1.5337E4*xx -3.284372E6 *sqr(xx) + 3.16018E8 * sqr(xx) *xx else if XX <= 0.2 then Dcx:=31.119 + 822.902+xx 6.4357E3 * sqr(xx) + 1.7539E4 * sqr(xx) * xx else Dcx:=77.465 + 19.003*xx 0 .8047*sqr(xx) + 0 . 033244*sqr(xx)*xx; end else if cell = -2 then begin if XX <= 0.00032 then Dcx: =0.0 else if XX <= 0.004 then Dcx:= -3.435537 + 2.8478580E4 * xx 7.3789605E6 * sqr(xx) + 7.26184506E8 *sqr(xx)*xx else if XX <= 0.0136 then Dcx:= 158.5301737*exp(0.24944024*ln(xx)) else if XX <= 0.2 then Dcx:= 96.805575 * exp(0 . 13378489*ln(xx) ) else Dcx:= 7.77309E1 + 6.1216539 * xx + 1.0583263 * sqr(xx) 0 . 06695623*sqr(xx) *xx; end else if cell=2 then begin if XX <=0.00032 then

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123 end; Dcx:=0.0 else if XX <= 0.004 then Dcx:=-4. 72641 + 4.0476E4+XX 1 . 05160E7*sqr(xx) + 1.03643E9 *sqr(xx)*xx else if XX <= 0.0136 then Dcx:= 222.64114*exp(0.2473555*ln(xx)) else if XX <= 0.2 then Dcx:= 136.18292*exp(0.1320091*ln(xx)) else Dcx:=108.49 + 8.7396*xx + 1 . 2429*sqr(xx) 0.078368*sqr(xx)*xx; end else if cell=3 then begin if xx<= 0.00032 then Dcx:=0.0 else if XX <= 0.004 then Dcx:=-5. 63716 + 2.2414E4*xx -4.741112E6*sqr(xx) if + 4.53575E8*sqr(xx)*xx XX <= 0.2 then Dcx:=45.234 else + 1.313E3 *xx 1.0286E4* sqr(xx) + 2 .7847E4*sqr(xx)*xx Dcx:=117.71 + 30.862*xx else else .4795*sqr(xx) end begin if xx<= 0.00096 then Dcx:=0.0 else if XX <= 0.004 then Dcx:=-1.3800 + 1.8865E3*xx + 7.39931E4*sqr(xx) 3 . 16944E7*sqr(xx)*xx if XX <= 0.2 then Dcx:=7.4836 + 3.020E2*xx + 5.5966E3*sqr(xx)*xx + 0 . 062118*sqr(xx) *xx; else 2 . 2730E3*sqr(xx) else end; Dcx:=25.334 + 10.060*xx ~ 0 . 6903*sqr(xx) + 0 . 03290*sqr(xx)*xx; (* penetration length vs. electron energy ) Function Erlen (EOO; real) : real; var EO : real ; begin E0:= E00/1.0E6; if cell = -2 then (* 40*/. *) begin if EO <= 0.2 then

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124 Erlen:=-9. 13188E-1 + 104.1110+E0 + 3694. 2997*sqr(E0) 5227 . 1090*sqr(E0)*E0 else if EO <= 1.0 then Erlen:=-50.5646 + 734.9293*E0 + 830 .77527*sqr(E0) 356 .37912*sqr(E0)*E0 else Erlen: =-262. 7650 + 1463.058*E0 40 . 92257*sqr(E0) + 0.852763*sqr(E0)*E0; end else if cell = -1 then ( 33 */, *) begin if EO <= 0.2 then Erlen :=-9.05805E-l + 103.2968+E0 + 3661 . 0080*sqr(E0) 5180 . 9561*sqr(E0)*E0 else if EO <= 1.0 then Erlen: =-50. 0565 + 728.0881*E0 + 823.5880*sqr(E0) 353.3844*sqr(E0)*E0 else Erlen: =-260. 36929 + 1449.811+EO 40 . 6388*sqr (EO) + 0.8467635*sqr(E0)*E0; end else if cell = 0 then begin if EO <= 0.2 then Erlen:=-8.95767E-1 + 101.9048+E0 + 3649.2866*sqr(E0) 5 . 15575E3*E0*sqr(E0) else if EO <= 1.0 then Erlen :=-5.02875El + 7 . 273489E2*E0 + 8 . 190796E2*sqr(E0) 3.504776E2*E0*sqr(E0) else Erlen :=-2.60305E2 + 1 .446344E3+E0 3 . 975610El*sqr(E0) + 8.283120E-l*E0*sqr(E0) ; end else if cell=l then begin if E0<= 0.20 then Erlen := -0.9078 + 103.5243*E0 + 3670.3027*sqr(E0) 5193.8482*sqr(E0)*E0 else if E0<=1.0 then Erlen:= -47.4253 + 707.2341*E0 + 877. 5254*sqr (EO) 388.8358*sqr(E0)*E0 else Erlen:= -261.023 + 1453.507*E0 40 .717*sqr(E0) + 0.84842*sqr(E0) ; end else if cell=2 then begin if E0<= 0.20 then

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125 Erlen:= -0.8763 + 100.042*E0 + 4997.51*sqr(E0)*E0 if E0<=1.0 then 3528 ,800*sqr(E0) else Erlen:= -48.067 + 700.994*E0 + 794 . 971*sqr(E0) 341 . 500*sqr(E0) *E0 else Erlen:= -250.77 + 1397.20+E0 39 .4954*sqr(E0) + 0.82267*sqr(E0)*E0; end else if cell=3 then begin if E0<= 0.20 then Erlen:= -0.8649 + 99.6798*E0 + 3378. 6836*sqr(E0) 4828.662 *sqr(E0)*E0 else if E0<=1.0 then Erlen:= -46.212 + 678.5045+E0+ 738.37 *sqr(E0) 321 .266*sqr(E0)*E0 else Erlen:= -230.19 + 1321.85*E0 41.826 sqr(EO) + 0 . 9188*e0*sqr (EO) ; end else begin if E0<= 0.20 then Erlen:= -0.96021 + 109.62 *E0 + 3864. 6200*sqr(E0) 5481.251 *sqr(E0)*E0 else if E0<=1.0 then Erlen:= -52.739 + 769.09 *E0+ 864.67 *sqr(E0) 372.675*sqr(E0)*E0 else Erlen:= -269.858 + 1522. 4*E0 43.453*sqr(E0) + 0 . 9246*E0*sqr(E0) ; end; end; ( reduced energy vs. penetration length for electron ) Function EnLeft( xx : real ): real; var eleft, dx : real; begin dx:= Rlen xx; dx:= dx / 1.0E4; if ( XX >= rlen ) then Eleft: =0.0 else begin if cell = -2 then begin if Ein <= 0.06 then Eleft:=6.71898E-3 + 55.73670*dx 25122. 9085*sqr(dx)

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+ 6 . 1416611E6*sqr(dx) *dx if Ein <= 0.2 then Eleft:=2.43462E-2 + 22.39569*dx + 3.5767907E4*sqr(dx)*dx if Ein <= 0.75 then Eleft :=7.39327E-2 + 10.6759*dx + 248 . 90519*sqr(dx)*dx Eleft :=1.79908E-1 + 6.89179*dx + 0.3290142*sqr(dx)*dx; end else if cell = -1 then begin if Ein <= 0.06 then Eleft :=6.7175E-3 + 56.2156*dx + 6.300951E6*sqr(dx)*dx if Ein <=0.2 then Eleft :=2.4340E-2 + 22.5947*dx + 3.672471E4*sqr(dx)*dx if Ein <= 0.75 then Eleft:=7.3879E-2 + 10.7762*dx + 256 . 1124*sqr(dx)*dx Eleft :=1.7984E-1 + 6.95545*dx + + 0.34241*sqr(dx)*dx; end else if cell = 0 then begin if Ein <= 0.06 then Eleft : =6 .7295E-3 + 56.6382*dx + 6 .447600E6*sqr(dx)*dx if Ein <=0.2 then Eleft :=2.4392E-2 + 22.7070*dx + 3.732092E4*sqr(dx)*dx if Ein <= 0.75 then Eleft :=7.4360E-2 + 10.7862*dx + 255 . 6569*sqr(dx)*dx Eleft :=1 .8051E-1 + 6.96689*dx + + 0 .30856*sqr(dx)*dx; end else if cell = 1 then begin if Ein <= 0.06 then Elef t : =6 . 7179E-3 + 56.0810*dx + 6 . 255908E6*sqr(dx) *dx if Ein <=0.2 then else 1121 .2664*sqr(dx) else 49 . 637098*sqr(dx) else + 1 .37183*sqr(dx) 25553 . 5627*sqr (dx) else 1141. 1113*sqr(dx) else 50 . 5997*sqr (dx) else 1 .39815*sqr(dx) 25964. 7837*sqr(dx) else 1153. 9870*sqr (dx) else 50 .4696*sqr(dx) else 1 .39212*sqr(dx) 25432 . 1632*sqr(dx) else

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Eleft:=2.4341E-2 + 22.5388*dx + 3.645419E4*sqr(dx)*dx if Ein <= 0.75 then Eleft:=7.3895E-2 + 10.7480*dx + 254. 0705*sqr(dx)*dx Eleft:=l .7986E-1 + 6.93756*dx + + 0.33859*sqr(dx)*dx; end else if cell = 2 then begin if Ein <= 0.06 then Eleft :=6.7121E-3 + 58.2063*dx + 6.992584E6*sqr(dx)*dx if Ein <= 0.2 then Eleft;=2.4316E-2 + 23.4220*dx + 4.088511E4*sqr(dx)*dx if Ein <= 0.75 then Eleft:=7.3654E-2 + 11.1937*dx + 287 .5420*sqr(dx) *dx Eleft :=1.7958E-1 + 7.22016*dx + + 0 .40181*sqr(dx)*dx; end else if cell = 3 then begin if Ein <= 0.06 then Eleft :=6.6644E-3 + 59.7865*dx • + 7.562258E6*sqr(dx)*dx if Ein <=0.2 then Eleft:=2.4878E-2 + 23.8571*dx • + 4.212895E4*sqr(dx)*dx 1135 . 5200*sqr(dx) else 50 .3274*sqr(dx) else 1 .39073*sqr(dx) 27383. 0354*sqr(dx) else 1225 .4735*sqr(dx) else 54.7031*sqr(dx) else 1 . 51002*sqr(dx) 28777 .7318*sqr(dx) else 1240 . 8724*sqr (dx) else if Ein <= 0.75 then Eleft :=7.2921E-2 + 11.7078*dx 59 . 0603*sqr(dx) + 328 . 3620*sqr (dx)*dx else Eleft :=1.7471E-1 + 7.63919*dx + 1 .86041*sqr(dx) + 0.69492*sqr(dx)*dx; end else if cell = 4 then begin if Ein <= 0.06 then Eleft : =6.71 15E-3 + 53.1383*dx 22821. 4950*sqr(dx) + 5.320835E6*sqr(dx)*dx else if Ein <=0.2 then Eleft:=2.4307E-2 + 21.3896*dx 1021 .8507*sqr(dx) + 3 . 114575E4*sqr (dx) *dx else

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128 if Ein <= 0.75 then Eleft:=7.3580E-2 + 10.2288*dx 45 . 5539*sqr(dx) + 219 .4354*sqr(dx)*dx else Eleft:=1.7838E-l + 6.61613*dx + 1 .30966*sqr(dx) + 0.29368*sqr(dx)*dx; end; end; if eleft<=0.0 then Enleft:=0.0 else Enleft :=eleft*l .0E6; end; (* total, number of defects vs. electron energy *) function Dcy(Evar : real) : real; var XX : real ; begin XX := Evar / 1E6; if cell = -1 then (* 33*/, *) begin if XX <0.30 then Dcy:=0.0 else if XX <1.1 then Dcy:= 0.039169 0.43764 *xx + 1.15530 *sqr(xx ) 0.361914 *sqr(xx )*xx else Dcy:=-0. 18590 +0.319210 *xx +0.27616 *sqr(xx )-0. 019077* sqr(xx )*xx ; end else if cell = -2 then ( 40*/. *) begin if XX <0.30 then Dcy:=0.0 else if XX <1.1 then Dcy:= 0.038218 0.424089 *xx + 1.1170 *sqr(xx ) 0.351427 *sqr(xx )*xx else Dcy: =-0.17246 +0.294892 *xx +0.27487 *sqr(xx ) -0.021847* sqr(xx )*xx ; end else if cell = 1 then begin if XX < 0.30 then Dcy:= 0.0 else if XX < 1.1 then Dcy:= 0.03857 0.43231 * xx + 1.14292 * sqr(xx) 0.35809 * sqr(xx) *xx else

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129 Dcy;= -0.78798 + 0.7259 * xx + 0.44538 * sqr(xx) 0.037336* sqr(xx) *xx; end else if cell = 2 then begin if XX < 0.30 then Dcy:= 0.0 else if XX < 1.1 then Dcy:= 0.0466 0.52535 * xx + 1.3945 * sqr(xx) 0.43876 * XX * sqr(xx) else Dcy:= -0.97400 + 0.9147 xx + 0.52926 *sqr(xx) 0.04475 * XX * sqr(xx); end else if cell=3 then begin if XX < 0.32 then Dcy:=0.0 else if XX <1.1 then Dcy:= 0.07027 0.7030 *xx + 1.7493 *sqr(xx ) 0.55049 *sqr(xx )*xx else Dcy:= -1.1404 +1.0277 *xx +0.66216 *sqr(xx ) -0.057361 * sqr(xx )*xx ; end else begin if XX < 0.60 then Dcy:=0.0 else IF XX < 2.2 THEN Dcy:= 0.07911 0.2661 *xx + 0.2598 *sqr(xx ) 0.03838 *xx *sqr(xx ) else Dcy:= 0.1524 0.4971 *xx + 0.32364 *sqr(xx ) -0.0251 *sqr(xx )*xx ; end; end; Function RoX( x : real): real; begin RoX:= absorpt * exp ( (x) * absorpt) ; (* omit (x-lower) ) end; Function omnidx(i : integer; x: real): real; var j : integer; XX, xy, xz, H, xO ,

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130 xl,x2: real; begin H:= (1.0 0 . 01)/(2*oraiter) ; if (particle = proton) then x0:= ABS(Dcx(eeleft(Xj [i] ,2)) Dcx(eeleft(x,2))) + ABS(Dcx(eeleft(xj [i]/0.01,2)) -Dcx(eeleft (x, 2)) ) ; if (particle = electron) then x0:= ABS(Dcy(enleft(xj [i])) Dcy (enleft(x) ) ) + ABS(Dcy(enleft(xj [i]/0.01)) Dcy(enleft(x) ) ) ; xl:= 0.0; x2:= 0.0; for j:= 1 to (2*omiter 1) do begin XX := 0.01 + h * j ; if particle = proton then begin xy:= Dcx(eeleft(xj [i]/xx,2)) ; xz:= Dcx(eeleft(x,2)) ; end else begin xy:= Dcy(enleft(xj [i]/xx)) ; xz : = Dcy (enlef t (x) ) ; end; if odd(j) then xl:= xl + abs(xy xz) else x2:= x2 + abs(xy xz) ; end; 0mnidx:= 2*pai*H*(xO + 2.0*x2 + 4.0*X1) / 3.0; end; function caldcx(x:real; i : integer) : real ; vajT temp, tempi : real; H, xO, XX, xl, x2, xi : real; M, j : integer; begin if not Omni then begin if particle = proton then begin temp:= abs(dcx(eeleft (x, 1) )-Dcx(eeleft (Xj [i] , 1) ) ); templ:=abs(dcx(eeleft(x,2))-dcx(eeleft(Xj [i] ,2)) ) ; Caldcx:= ( temp + tempi ) * 0.5 ; end else

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Caldcx:= abs(dcy(enleft(x))-Dcy(enleft(Xj [i]))) ; end else Caldcx:= Omnidx(i ,x) ; end; Function FandRo(x,u : real ; i : integer) : real; var temp, tempi : real; begin temp:=caldcx(x, i) ; if x>= Xj [i] then sigmaR:=sigmaH[i] else sigmaR:=sigmaE[i] ; FandRo:=exp( -sqrt(6)*sigmaR*temp*FLUX / u) Rox(x) ; end; Function Photoabsorb (upper : real ) : real; var H, X, xO, xl,x2, xi : real; M, I, J : integer; begin H:= ( upper ) / ( 2.0* DIST ); x0:= RoX (upper) + RoX(O.O); xl:=0.0; x2:=0.0; for i:= 1 to (2+DIST-l ) do begin x:= 0.0 + I*H; if odd(i) then xl:=xl + Rox(x) else x2:=x2 + RoX(x) ; end; XI := H * ( xO + 2.0* X2 + 4.0* xl ) / 3.0; Photoabsorb : =Xi ; end; Function FractionLoss (upper : real ; ii: integer) : real; var Hx, Hu, Jl, J2, J3, Kl, K2, K3, L, Jout, Z, X, u : real; I, J, n, m : integer; begin Hx:= (upper ) / ( 2.0*DIST ); J1:=0.0; j2:=0.0; j3;=0.0;

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for i:= 0 to ( 2*DIST ) do begin x:= 0.0 + I*Hx; Hu:=(l0.0) / ( 2.0 * DIST) ; (* instead of lE-6 *) Kl:= 0.0 + FandRo(x,l, ii ); (* FandRo(x,0)=0 . 0 *) k2:=0; K3:=0; for j:=l to ( 2*DIST-1 ) do begin u:=0.0 + j*Hu; Z := FeindRo(x,u, ii) ; if odd(j) then k3:=k3 + z else k2 : =k2 + z ; end; L := ( kl + 2.0*k2 + 4.0*K3 ) * Hu / 3.0; if ( i=0 ) or ( i= (2*DIST)) then + L else if odd(i) then j3:=j3 + L else j2:=j2 + L; end; Jout:=( J1 + 2.0*j2 + 4.0*j3) Hx / 3.0; Fractionloss :=Jout ; end; procedure printout (ij : integer); var i, j : integer; begin writelnCout ,Ij , ' ReduceC[ij] ) ; end; procedure coverglass( i : integer ) ; var temp, tempi : real; ii : integer; begin ii:= cell; cell:=0; if particle = proton then begin Plen:=Plength(Ein) ; Rlen: =Rlength(Ein) ;

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Ein:=Eeleft(Wd[i] ,1); end; cell:= ii; end; Function AlGaAs33( temp : real):real; var i, j : integer; atemp, btemp : real ; begin Atemp := 1.24 / temp + 0.0667 ; if atemp <= 2.0 then Btemp:=exp((27.5373*atemp-51.2296)*lnl0) / 1E4 if atemp <= 2.2658 then Btemp:=exp((l .7156*atemp+0 .4138)*lnl0) / 1E4 if atemp <= 2.3324 then Btemp :=exp((8. 1696*atemp-14.2096)*lnl0) / 1E4 if atemp <= 2.3999 then Btemp:=exp((2.3258*atemp-0 .5796)*lnl0) / 1E4 if atemp <= 3.1316 then btemp:= exp((0.4108*atemp+4.0145)*lnl0) / 1E4 Btemp := 20.0; AlGaAs33 := btemp; end; Function AlGaAs40( temp : real): real; var i, j : integer; atemp , btemp : read ; begin Atemp := 1.24 / temp ; if atemp <= 2.0 then Btemp:=exp((27.5373*atemp-51.2296)*lnl0) / 1E4 if atemp <= 2.2658 then Btemp:=exp((1.7156*atemp+0.4138)*lnl0) / 1E4 if atemp <= 2.3324 then Btemp:=exp((8.1696*atemp-14.2096)*lnl0) / 1E4 if atemp <= 2.3999 then Btemp:=exp((2.3258*atemp-0.5796)*lnl0) / 1E4 if atemp <= 3.1316 then btemp:= exp((0.4108*atemp+4.0145)*lnl0) / 1E4 Btemp := 20.0; else else else else else else else else else else

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134 A1G2lAs 40 := btemp; end; Function GetAbsorpC II : integer ; temp : real) : real; begin if ii =-2 then GetAbsorp:= AlGaLAs40(temp) else if ii = 1 then GetAbsorp:= AlGeLAs33(temp) else if ii = 2 then GetAbsorp:= GaAsAbsorp(temp) else if ii = 3 then Getabsorp:= InGaiAsAbsorpCtemp) else if ii = 4 then GetAbsorp:= GeAbsorp(temp) else GetAbsorp:= SiAbsorp(temp) ; end; function degradation( ii, jj : integer): real; var i, j : integer; xO, xOO, xl, xlO, x2, x20, xi, xiO : real; H : REAL; begin h:= ( highCjj] low[jj] ) / ( 2 * Aiter ); absorpt:=GetAbsorp(ii,low[jj]) ; x0:= fractionloss(UpPene,j j) ; x00:= photoabsorb(Uppene) ; absorpt :=GetAbsorp(ii ,high[jj] ) ; xO:=xO + fractionlossCUpPene, jj ); x00:=x00+ photoabsorb(Uppene) xl:=0.0; x2:=0.0; xl0:=0.0; x20:=0.0; for i:= 1 to ( 2* Aiter -1 ) do begin absorpt : =Get Absorp ( ii , low [j j ] +i*h) ; if odd(i) then begin xlO:=xlO + photoabsorb(Uppene) ; xl := xl + fractionloss(Uppene, j j ); end else begin x2:= x2 + fractionloss(UPPENE,jj ) ; x20:=x20 + photoabsorb(UpPene) ;

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135 end; end; xi:= H * ( xO + 2.0*x2 + 4.0*xl ) / 3.0 ; xiO:=H * ( xOO + 2.0*x20 + 4.0*xl0 ) / 3.0; degradation:=xi / xiO; end; Procedure irradiated; var i, j : integer; beg, endd : integer; ch : char; STOP : BOOLEAN; TEMPE , REFIRST, RESECOND, REDIFF : REAL; begin WRITELN(OUT, ' ENERGY LEVEL ? ',EIN ); for i;= 1 to Config do begin Temper = Ein; cell:= Name[i] ; Stop := false; dist:= 4; Refirst:= 0.0; Resecond := 0.0; repeat coverglass(i) ; Dxij := Xj [i] ; if particle = proton then Rlen:= Rlength(ein) else Rlen:= Erlen(ein); if rlen <= Tj [i] THEN UpPene := Rlen ELSE UPPENE:= Tj [i] ; Resecond := degradat ion (cell , 1) ; if Abs(Resecond Refirst) <= 0.000001 then Stop := true else begin Dist:= Dist + 1 ; Ein := TempE; Refirst := Resecond ; end;

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136 if dist >= 100 then Stop := true; until stop ; ReduceC[i] := Refirst; writeln(out , ' Iteration = \dist); printout (I) ; if Tj [i] > Rlen then writelnCout, ' Only cell #',cell,' .. degrades') else begin if particle = proton then Ein:= eeleft (Tj [i] , 1) else Ein:= enleft (Tj [i] ) ; end; end; end; (* Irradiation calculation for space environment, proton *) Procedure SpaceProton; var i, j : integer; t : real; begin for i:=l to 9 do begin for j :=1 to 3 do begin if i=l then begin ein:=lE5; t:=1.2E14; end else if i=2 then begin ein:=2E5; t:=6.2E13; end else if i=3 then begin ein:=4E5; t:=2.0E13; end else if i=4 then begin ein:=lE6; t:=2.2E12; end else if i=5 then begin ein:=2E6; t:=3.7E13; end else if i=6 then begin ein: =3E6 ; t:=l.lE13; end else if i=7 then begin ein:=4E6 ; t:=7.5E12; end else if i=8 then begin ein: =6E6 ; t:=6.7Ell; end else if i=9 then begin ein:=lE7; t:=9.2E10; end; if J = 1 then flux:= =t* 3 else if j=2 then flux:= =t*7 else flux:=t*10; writeln(out, 'ein= ',ein,' flux= '.flux); irradiated; end; end; end;

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137 procedure SpaceElectron; var i, j : integer; t : real ; begin for i:=l to 7 do begin for j :=1 to 3 do begin if i=l then begin if i=2 then begin if i=3 then begin if i=4 then begin if i=5 then begin if i=6 then begin begin ein:=5E6; t; if j=l then flux:= if 2=2 then flux:= flux: =t* 10; writeln(out , ' ein= irradiated; end; end; end; ein:=lE5; t:=7.4E14; end else ein:=5E5; t:=6.4E13; end else ein:=lE6; t:=1.8E13; end else ein:=2E6; t:=3.4E12; end else ein:=3E6 ; t:=6.1Ell; end else ein:=4E6; t:=8.5E10; end else =8.5E9 ; end; t*3 else t*7 else ',ein, ' flux= '.flux); (* Calculations for daunage constants of minority carrier diffusion *) (* length and minority carrier lifetime. *) Procedure damage; const k = 1.38E-23; var Ans : char; Index : integer; Rplen : real; procedure getmass( ii : integer); var i : integer; percent , MpTemp, MpH, MpL : real ;

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138 begin writelnC' what is the percentage of X (Al) ? '); readln (percent) ; MnEff[ii]:= 0.067 + 0.083 * percent; MpL:= 0.082 + 0.063 * percent; MpH:= 0.62 +0.14 * percent; MpTemp:= exp(l . 5*ln(MpL) ) + exp(l . 5*ln(MpH) ) ; MpEff[ii]:= exp(2/3.0 * ln( MpTemp)); end; procedure getVth(ii : integer); var i : integer; TT, Vthl, Vth2 : real ; begin writelnC' what is the Temperature ? '); readln(TT); writeln(out, ' TT = ' ,TT) ; VthN[ii] := sqrt( 3* K * TT / (MnEff [ii] *Me) ) * 100 ; VthP[ii]:= sqrt( 3* K * TT / (MpEff [ii] *Me) ) 100 ; end; procedure Dinitial; var i, j : integer; ch, ans : char; begin for i:= 1 to Config do begin writelnC i:l,', Ln and Lp ? urn '); readln (Lnn[i] ,Lp [i] ) ; writelnCout, ' Lnn[' ,i: 1, '] = ' ,Lnn[i] , ' Lp[' ,i: 1, '] = ' ,Lp[i] ) ; writelnC i:l, ' , Tn and Tp ? sec '); readln (TauN [i] ,TauP [i] ) ; writelnCout, ' Tn[' , 1 : 1 , ' ,TauN[i] , ' Tp [' , i : 1 , '] = \TauP [i] ) ; GetMass(conf ig) ; writeln(out , 'Mn = Â’,Mneff[i],' Mp = ',Mpeff[i]); GetVth (Config) ; writelnCout, ' VthN (cm) = ',VthN[i], ' VthP(cm) = ',VthP[i]); Dpp[i]:= Lp[i] * Lp[i] lE-8 / TauP[i]; Dn[i] := Lnn[i] * Lnn[i] * lE-8 / TauN[i]; writeln(out, ' Dn = \Dn[i] , ' Dpp = DppCi]); end; end;

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Function geteleft(ii : integer; xx : real): real; begin cell := ii; if particle = electron then geteleft:= enleft(xx) else GetEleft:=(Eeleft(xx,l) + EELEFT(XX,2))/2.0; end; Function GetDefect( ii : integer; xx : real) : real; begin cell := ii; if particle = electron then GetDefect:= Dcy(xx) else GetDefect:= Dcx(xx) ; end; Function GetLen( ii : integer; xx : real):re 2 il; begin cell := ii; if particle = electron then BEGIN rLEN := Erlen(xx) ; gETLEN := RLEN; END ELSE BEGIN RLEN:= RLENGTH(XX); PLEN:= PLENGTH(XX); GetLen:= (rlen + PLEN)/2.0; END; end; Function GetConst(ii, jj : integer): real; var Kl, Kt : real ; begin if ii = 1 then begin Kt := ( 1 / TauNIrrCjj] 1 / TauN[jj] ) / flux Kl := Kt / Dn[jj] ; end else begin Kt := ( 1 / TauPIrrEjj] 1 / TauP[jj] ) / flux

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K1 := Kt / DppEjj] ; end; GetConst:= Kl; end; procedure GetDamage(ii : integer); var i : integer; Etemp : real; Rl, R2, Nt, Ntl, Nt2 : reed; begin if window [ii] then begin RPlen := getLen(0 ,Ein) ; Ein:= getEleft(0,Wd[ii] ) ; end; i := NameCii] ; RpLen:= getLen(i,Ein) ; Ntl:= GetDef ect(i,Ein) ; writelnCout, ' defect ## Â’,Ntl, ' Ein = ',Ein); if RPLen <= Xj [ii] then begin Nt:= Ntl / RpLen 1E4; Etemp := 0.0; end else begin Etemp: = getEleft (i ,Xj [ii] ) ; Nt2:= getDef ect(i, Etemp) ; writeln(out, ' ### 2 ',nt2, ' Ein = '.Etemp); Nt := ( NTl NT2 ) / Xj [ii] 1E4; end; TauNIrr[ii] := 1 / ( SigmaE[ii] Nt * Flux * VthN[ii]); Kln[ii] :=Getconst(l,ii) ; if Etemp >= 0 then begin Ein := Etemp; RPlen:= getLen(i ,Ein) ; Ntl:= GetDef ect(i, Ein) ; writelnCout , ' $$$$ '.Ntl,' Ein = '.Ein); if RPLEn <= Tj [ii] then begin

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141 Nt:= Ntl / RPlen * 1E4; Etemp:= 0.0; end else begin Etemp:= getEleft(i,Tj [ii] ) ; Nt2:= getdefectd ,Etemp) ; Nt := ( Ntl Nt2 ) / Tj[ii] 1E4 ; writelnCout, '$$$0 ',Nt2,' Ein =' ,etemp) ; end; TauPIrrCii] := 1 / ( sigmaH[ii] * Nt * flux * VthP[ii]); Klp[ii]:= Getconst (2,ii) end else Klp[ii] := 0.0; Ein:= Etemp ; end; begin (* deimage *) writelnC' what kind of particle ? Proton or Electron '); readln (particle) ; writelnC' Ein (eV) = '); readln(Ein); writeln(out , ' Ein = ',ein); writelnC' what is the fluence of the particle ? '); readln(flux) ; writeln(out , ' Flux = ',flux); Dinitial; repeat for index :=1 to config do begin RPlen:= getlen(index,Ein) ; getDeunage (index) ; writeln(' Kin = dKln[index], ' Kip = Kip [index] ) ; writeln(out, ' Kin = ' ,Kln[index] , ' Kip = ' , Kip [index] ) ; end; writeln(' more Ein to run ? "y" or "n" Â’); readln (Ans) ; if (Ans = 'yO or (Ans = 'Y') then Runn :=true else Runn :=false; if runn then begin writeln(' Ein (eV) = ? '); readln (Ein) ; writeln(out) ; writeln(out , 'Ein = Ein); end; Until ( not Runn ) end; (* dajnage *)

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142 begin (* of main *) writelnC' — > solair cell starts ! Running sequence ?'); writelnC' 1 = calculation for specified Ein and Flux for Proton.'); writelnC' 2 = Specified Ein and Flux for electron'); writelnC' 3 = Space envirinment for Proton'); writelnC' 4 = Space environment for electron'); writelnC' 5 = Calculations of damage constaint '); readlnC sequence) ; writelnC' Absorption Coeff. iteration ,10< Aiter < 25 '); readlnCaiter) ; initial ; case sequence of 1 : begin particle :=proton; writelnC' specified the Ein and Flux please '); writelnC' Ein <= 1E7 eV; Flux <= 1E12 '); readlnCEin,flux) ; if ein <= 10E6 then irradiated; end; 2 : begin particle : =electron ; writelnC 'specified the Ein and Flux please '); writelnC' Ein <= 5E6; Flux <= 1E16 '); readlnCEin,flux) ; if ein<=5.0E6 then irradiated; end; 3 : begin particle :=proton; SpaceProton; end; 4 : begin particle :=electron; SpaceElectron; end; 5 : begin Damage ; end; end; C* case *) end.

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APPENDIX C AN EXPERT SYSTEM PROGRAM FOR OPTIMAL DESIGN OF SINGLE-JUNCTION AND MULTIJUNCTION TANDEM SOLAR CELLS { This program is the kernel of the expert system. A modified Box complex search optimization algorithm has been implemented in this program. The rule-based has been implemented as pattern-directed module. They are in the form of if-then format. This program can integrate the program in Appendix B into a large system. However, because it takes a little while to calculate the degradation of the irradiated solar cell, this program will take the results from Appendix B namely the damage constant as its input for optimizing the cell design of irradiation case. This prograim is user friendly, the purpose of each procedure is described by the name of the proceudre itself. The program is run in Harris 800 computer system. Program optimaCinput , output, Inpl (lfn=103) , Inp2(lfn=104) , Inp3(lfn=105) , Inp4(lfn=106) ,0ut (lfn=107) , inp5(lfn=108) ) ; Const Qe = 1.602E-19; Pemit = 8.854E-14; InlO = 2.30258509; TYPE IndType = array [1..20] of real; IntType = array [1..20] of integer; ArrayType = array [1..20, 1..20] of real; InpType = array [1..200] of real; DataType = array [1..3] of real; KvType = array [0..40] of real; CellType = (ALGAAS,GE,SI,GAAS,INGAAS) ; RGType = (RECOMBINATION, DIFFUSION) ; Var Inpl, Inp2, Inp3 , Inp4 , out, Inp5 : text; AdaVec, PreGuess , 143

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144 Guess, AveX, Ubound, Lbound: Indtype; InitGuess : ArrayType; IndVar , Wf actor, CellConf , CellNum, Step, lAR, Iter, Insolation, Particle, TunnelNum , AirM : integer; ColEff , IdealF, KLn, KLp, Na, Ndd, Joo , Voc, Vm, FF, Ada, Aj, Eg» Sn, Sp , Mobe, Mobp, Lnn, Lpp, Dnn, Dpp, Tn, Tp, Xj. Tj, Mn, Mp, Wj , Ni : DataType; Airmass , Pam, Rs, Rsl, LossRatio , ReFL , Shadow , Fluxx, Fworst, Fbest, Tratio , AdaTot , High, Low, ( Collecting Efficiency *) (* Ideality Factor *) (* damage constant *) (* doping density *) (* dark current density *) (* open circuit voltage *) (* maximum open circuit voltage *) (* fill factor *) (* conversion efficiency *) (* cell area *) ( bandgap energy *) (* surface velocity *) (* mobilities *) (* minority diffusion length *) (* diffusion constant *) ( minority carrier lifetime *) (* junction depth, cell thickness*) (* effective mass *) (* depletion width *) (* Solar Irradiamce *) (* Series Resistance *) (* losses *) (* Particel fluences *) (* T / 300 *)

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145 Jsc , error, TuRatio, Xlal, Alratio, Teff , Tdegree : real; CellName : array [1..3] RGmodel : array [1..3] Atype : inptype; Btype WLl, Kvl, WL2, Kv2, : array [1..6] WL3, Kv3 : KvType; Tna, Tnd System, Mic, Btunnel , PN, : array [1..2] of Celltype; of RGType; of inptype; of real; (* doping density of tunnel Jn *) GFactor ; boolean; Fxy, Fxx, Fyy : InpType; TunnelCell , TunnelTd, nindex, TdropV, ARtd : array [1..2] of real; Procedure Initial; var i, j : integer; begin Mic:= false; Gfactor:= true; PN := true; Alratio;= 0.00; for i := 0 to 23 do readln(inpl,WLl[i] ,Kvl[i]) ; (* for GaAs *) for i:= 0 to 35 do readln(inp2,WL2[i] ,Kv2[i]) ; (* for Si *) for i:= 0 to 11 do readln(inp3,WL3[i] ,Kv3[i] ) ; ( for Ge *) for i:= 1 to 196 do begin for j := 1 to 6 do read(inp4,Btype[j ,i]) ; AtypeCi] := 1E4 / (1000 + (I-l) * 200);

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146 end; for i:= 1 to 168 do readln(INP5, xlal, Fxy[i]); Fxx[l]:= 0.099; Fxx[2] := 0.198; Fxx[3] := 0.315; Fxx[4] := 0.419; Fxx[5]:= 0.491; fxx[6] := 0.590; Fxx[7] := 0.700; Fxx[8] := 0.804; for i:= 1 to 21 do FyyCi]:= 1.5 + (i-l)*0.1; end; (* of initial ) Procedure Readinput ; var RG, I : integer; ch : char ; begin writelnCÂ’ Welcome to the solar cell expert system. '); writeln(Â’ You could have at most 3 P/N Junctions in your design.Â’); writelnCÂ’ How many p/n junctions you want for this run ?Â’); readln(CellConf) ; writelnCÂ’At what Temperature?Â’); readlnCt degree) ; Tratio:= Tdegree / 300; writelnCÂ’ At what kind of Airmass?Â’); writelnC' 0 : For space application Â’); writelnCÂ’ 1, 1.5, 2.0, 2.5, 3.0 for terrestrial applicationÂ’); readlnCAirmass) ; if CHoundCAirmass*10) = 0 ) then begin AirM:= 1; Pam:= 135.3; end else if CRoundCAirmass*10) = 10) then begin AirM:= 2; Pam:= 103.87 end else if CRoundCAirmass*10) = 15) then begin AirM:= 3; Pam:= 97.39; end else if CRoundCAirmass*10) = 20) then begin AirM:= 4;

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147 Pam:=91.94 ; end else if (Round(Airmass*10) = 25) then begin AirM:= 5; Paju:= 87.32; end else if (Round(Airmass*10) = 30) then begin AirM:= 6; Pam:= 83.2 ; end; writelnC' At what kind of sun insolation?'); readln(insolation) ; writelnC' Cell materials selection. '); writelnC' If answer (y) or (Y) then user assign cell materials'); writelnC' Otherwise, system default optimum materials'); readln(ch) ; if (ch = 'y') or (ch = 'Y') then system: = false else system:= true; end; Procedure CellSelection; var RG, i : integer; begin for i:= 1 to cellConf do begin writelnC' Cell Name [' ,i:2, ']?') ; writelnC '1: AlGaAs; 2: GaAs; 3: InGaAs; 4: Ge; 5: Si'); readlnCcellnum) ; if CellNum = 1 then CellNameEi] := ALGAAS else if CellNum = 2 then CellNajneCi] := GAAS else if CellNum = 3 then CellNameEi] := INGAAS else if CellNum = 4 then CellNcimeEi] := GE else CellName Ei] := SI ; writelnC 'Front surface recombination velocity SpE',i:2,'] ?'); readlnCSpEi] ) ;

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148 writelnC'Back surface recombination velocity Sn [',i:2,'] ?Â’); readln(Sn[i] ) ; writelnC' R-G Model [',i:2,'] ?0; writelnC' 1 : Diffusion model; 2 : Recombination model'); readln(RG) ; if RG <> 1 then RG := 2; if RG = 1 then begin RGmodel[i]:= DIFFUSION; IdealF[i] := 1.0; end else begin RGmodel[i]:= RECOMBINATION; writelnC' Input Ideality Factor; > 1.0 '); readln(IdealF[i]) ; end; end; end; Procedure InputBound; var i, j : integer; begin if not system then begin writelnC' Input # of independent variables '); writelnC' 1 junction > IndVar =5 '); writelnC' 2 junction > IndVar = 10'); writelnCÂ’ 3 junction > IndVar = 15'); readlnClndVar) ; writelnC' Input upper bound and lower bound of the Indep. variables '); for I:= 1 to Indvar do begin if Cl = 1) or Cl = 6) or Ci = 11) then writelnC 'Input upper and lower bound of Nd. ? = LogCNd)') else if Ci = 2) or Cl = 7) or Ci = 12) then writelnC 'Input upper and lower bound of Na. ? = LogCNa)') else if Ci = 3) or Ci = 8) or Ci = 13) then writelnC 'Input upper and lower bound of Junction depth, urn') else if Ci = 4) or Ci = 9) or Ci = 14) then

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149 writ eln( 'Input upper and lower bound of cell thickness ,um' ) else begin writeln( 'Input upper bound aind lower bound of Eg, eV'); writelnC 'Usually Ubound = LBound, Unless AlGaAs'); end; readln (Ubound [i] ,Lbound[i] ) ; writeln(Ubound[i] ,Lbound[i]); end; end; (* system *) end; (* of inputbound *) Procedure TunnelSelection; var i : integer; begin Btunnel:= false; if CellConf >= 2 then begin writelnC' Do you need tunnel junctions in your design?'); writelnC' 0 : Mic ; 1 : 1 tunnel junction '); writelnC' 2 : 2 tunnel junctions.'); writelnC' 3 : MIC + 1 tunnel Junction '); readln (TunnelNum) ; if TunnelNum >=3 then TunnelNum := 3 else if TunnelNum <= 0 then TunnelNum := 0; if tunnelNum = 3 then begin Mic:= true; TunnelNum := 1; end else Mic:= false; if tunnelNum <> 0 then Btunnel:= true else Mic;= true; for i:= 1 to TunnelNum do begin writelnC' Input Tunnel junction. '); writelnC' 1 : AlGaAs, x= ?*/, , 2 : GaAs '); readln (TunnelCell [i] ) ; if TunnelCell [i] = 1 then begin

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150 writelnCÂ’ what is the A1 ration, A1 >= 0.45 readln(Turatio) ; end else TunnelCellCi] := 2; writelnC' what are Nd and Na of tunnel jn? >= lE19/cm2Â’); readln(Tnd[i] , tna[i]); end; end; If Mic then begin Shadow := shadow * 2.0; Rs:= Rs 2 + Rsl; end; end; Procedure contact; Var N, Nline, ConNum, TypeCon, ShapeNum : integer; Wd, 1th, W, A, B, d, h, RhoS, RhoF, Rcl : real; Again : Boolean; begin writelnC' what are the contact structures ?'); writelnC' 1 : regulair front and back contacts,'); writelnC' 2 : metal interconnect contact.'); readlnCConNum) ; writelnC' P-type or N-type contact for front ?'); writelnC' 1 : P-type, i.e. AuZn; 2: N-type, Au/Ge/Ni '); readlnCtypeCon) ; if typecon = 1 then begin writelnC' Au/Zn > 1200 aind 600 Angstrons respectively '); end else begin writelnC' Au/Ge/Ni > 1200, 400, 600 Angstrons respectively'); end; writelnC' What is the grid structure ?'); writelnC' 1 : triangle shape; 2 : rectangle shape .'); readlnCshapenum) ; writelnC' what is the area of the solr cell? Width x Length, cm2 ');

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151 readln(Wd,lth) ; if shapenum = 1 then begin repeat writelnCÂ’ What is the width of the triangle? cm please. 0; writelnC' Width can not > ',Wd,' (cell width) cm'); readln(W) ; writelnC' Input # of grid line '); readln(Nline) ; N := TruncCWd / (2 *. W)); if Nline > N then begin again := true; writelnC' too many grid lines'); writelnC' can not > ' ,N, ' lines for grid width = ',w); end else again := false; until (not again) ; writelnC' Input finger length please. Must be < ',1th, 'cm'); readln(a) ; if a > 1th then a:= 1th; RhoF:= 2E-6; Rcl:= lE-4; Rhos:= 4E-2; Rs:= Rhos + Rcl/sqr(l Nline*w) * (l/(12*sqr(Nline)) w / (4*Nline) + sqr(w) / 3 Nline * sqr(w) * w / 6) + Rhof / (4 * W * Nline) ; shadow := Nline * W * a / 2 / (Wd * 1th); end else begin repeat writelnC' Input the # of fingers '); readlnC Nline ); writelnC' what is the finger spacing? cm '); writelnC' It is usually <= ' ,Wd/Nline, ' cm'); readln(b) ; writelnC' what is the Finger width? cm '); readln(w) ; N;= truncCWd / ( b + w )); if Nline > N then begin again := true; writelnC' too many grid line');

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152 writelnCÂ’ cam not > ' ,N, Â’ lines'); end else again := false; until (not again) ; writelnC' what is the finger height? cm '); readln(h) ; writelnC' what is the finger length? cm; < Cell Length'); readlnC a) ; writelnC' what is the thickness of window or emitter layer'); readlnC d) ; Rhos:= 4E-2; Rhof:= 2E-6; Rcl:= lE-4; Rs:= Rhos * sqr(b)/(12*d) + b / W Rhof / (3*h) * sqr(a) + b / w * Rcl; Rsl:= Rhos sqr(b)/(l2*0.001) + b / W * Rhof / (3*h) * sqr(a) + b / w * Rcl; shadow := Nline * W * a / (Lth Wd) ; end; end; Procedure ARcoating; Var ArMat , I : integer; begin writelnC' what kinds of AR structure '); writelnC' 0 : No AR coating; 1 : single layer; 2 : Double layer '); readln(IAR) ; if lAR >= 2 then lAR := 2; if lAR <= 0 then IAR:= 0; if lAR <> 0 then begin for i:= 1 to lAR do begin writelnC' what kinds of Coating Materials?'); writelnC' 1 : Ta205 (2.15); 2 : Ti02 (2.15) '); writelnC' 3 : Si3N4 (1.93); 4 : MgF2 (1.38) '); writelnC' 5 : Si02 (1.80) 6 : ZnS (2.30) '); readln (ArMat) ; if ArMat = 1 then nindexCi] := 2.15 else

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153 if ArMat = 2 then nindex[i] := 2.15 else if Armat = 3 then nindexCi] := 1.93 else if Armat = 4 then nindex [i] := 1.38 else if Armat = 5 then nindexCi] := 1.80 else nindexCi] := 2.30; end; if lAR = 1 then RefL:= sqr((sqr(nindex[l] )-3.0522)/(sqr(nindex[l] )+3.052)) RefL:= sqr((sqr(nindex[l] )*3.0522 sqr(nindex[2] ) )/ (sqr(nindex[l] )*3.0522 + sqr(nindex [2] ) ) ) ; end else RefL:= 0.0; end; Procedure ARthickness; var i : integer; begin if lAR <> 0 then begin for i:= 1 to lAR do begin if I = 1 then ARtd[i] := (1.24/(Eg[l] + High))/ (4 * Nindex[i]) else ARtd[i] := ( 1.24/ (Eg [1] + High))/ (4 * Nindex[i]); writeln(out, 'AR coating thickness, [ ',1:2,'] ',ARtd[i],' end; end; end; Procedure Highinsolation; Var i : integer; Jscx, Vocx : real; begin for i:= 1 to CellConf do begin Jscx:= Jsc. * Insolation; Vocx:= Voc[i] + 0.0259 * Tratio * In(Insolation) ; Ada[i] := ada[i] * Vocx / Voc[ij; else um') ;

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Voc[i] := Vocx; end; adatot:= 0.0; for i:= 1 to CellConf do adatot:= ada[i] + adatot; end; Procedure Tunnel; forward; Procedure RealAda; var VocT, Du, Vr, DV : real; Vratio , Fratio : array [1..3] of real; i : integer; begin if Btunnel. then tunnel; LossRatio:= (1 RefL) * (1 Shadow); Jsc:= Jsc * LossRatio; DV := Jsc * Rs / CellConf; VocT:= 0.0; for i:= 1 to CellConf do begin Vratio[i]:= (Voc[i] DV) / Voc[i] ; Fratio [i]:= 1 Dv / Voc[i]; Voc[i]:= Voc[i] * Vratio[i]; VocT:= Voct + Voc[i]; FF[i]:= FF[i] * Fratio [i]; ada[i] := ada[i]*vratio[i]*Fratio[i]*LossRat end; if Btunnel then begin Du:= 0.0; for i:= 1 to TunnelNum do Du:= Du + TdropV[i] ; Vr:= 1 Du / VocT; if Vr <= 0.0 then Vr:= 0.0; end else Vr:= 1.0; adatot : = 0.0;

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155 for i:= 1 to CellConf do adatot:= adatot + ada[i] ; AdaTot:= adatot Vr; end; Function Tun Wd (I : integer; ratio, Teg : real):real; var Tmn, Tmp, Tni, Tvbi, Eps, Tnn, Txl, Tlh, Thh : real; begin Eps:= 12.91 * (1 ratio) + 10.06 ratio; Tmn:= 0.067 + 0.083 * ratio; Tlh:= 0.087 + 0.063 * ratio; thh:= 0.62 + 0.14 * ratio; tmp:= exp(2/3*ln(Tlh*sqrt (tlh) + thh*sqrt (thh) ) ) ; Teff:= Tmn * Tmp / (Tmn + Tmp); Txl:= Tmn * Tmp ; Tni:= sqrt(Txl*sqrt(txl)*2.33E31)*tdegree*sqrt(tdegree) * exp(-Teg/(0.0259*2*tratio)) ; Tnn:= (Tna[i] /lE9)*(Tnd[i]/lE9)/(Tna[i] + Tnd[i]) * 1E18; Tvbi:= 0.0259*tratio *ln((Tna[i]/Tni)*(Tnd[i]/Tni)) ; TunWd:= sqrt (2*Eps*Pemit/Qe * Tvbi / Tnn) / lE-8 ; end; Procedure Tunnel; var Tw. TEgl, TEg, Tmass : real; I : integer; begin for i:= 1 to TunnelNum do begin if TunnelCell[i] = 1 then begin if Turatio > 0.45 then Tegl:= 1.424 + 1 . 247*Turatio + 1 . 147*sqr(Turatio-0 .45) else Tegl:= 1.424 + 1 . 247*Turatio ; TEg:= Tegl 2.9725E-4 * (Tdegree 300); end else begin TEg:= 1.519 5 .405E“4*sqr(Tdegree)/(204+Tdegree) ; turatio := 0.0; end;

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156 Tw:= TunWd(i, Turatio, Teg); TdropV[i]:= Jsc / (lElO * sqrt(Teg * Teff) / Tw * exp( 0.40 sqrt(TEg * Teff) * Tw) ); end; end; Procedure Irradiation; var Ch : char; I : integer; begin writeln('Do you consider the irradiation case?'); writelnC' (Y)es, (y)es, otherwise NO '); readln(ch) ; if (ch = 'Y') or (ch = 'y') then begin writeln( Â’ what kind of particle?'); writeln(' 0 : no irradiation; 1 : proton; 2 : electron'); readln (particle) ; if particle <> 1 then particle := 2; writeln(' what is the fluence?'); writeln(' Proton : lElO to 1E12; Electron: 1E14 to 1E16 '); readln (fluxx) ; writeln(' what is the damage constants for Ln and Lp'); writeln(' Proton : Ln: order of lE-8; Lp: lE-9'); writeln(' Electron : Lp: lE-7 ; Lp: lE-8'); for i:= 1 to CellConf do begin writeln(' KLn [',1:2,'] and KLp [',1:2,']'); readln(Kln[i] ,Klp[i]) ; end; end else particle := 0; end; Function Bandgap(Name : CellType) :real; begin if Name = GAAS then Bandgap:= 1.519 5 .405E-4*sqr(Tdegree)/(Tdegree + 204) else if Name = SI then Bandgap:= 1.170 4.73E-4*sqr(Tdegree)/(Tdegree + 636) else if Name = GE then Bandgap:= 0.7437 4.77E-4*sqr(Tdegree)/(Tdegree + 235) else

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157 if Name = INGAAS then Bandgap:= 0.812 3 . 26E-4*Tdegree + 3.31E-7 *sqr(Tdegree) else if Name = ALGAAS then Bandgap:= 1.424 + 1 . 247*alratio (3.95 1 . 15*alratio) +1E-4* (Tdegree 300.0); end; Procedure GetTopBound; begin EgCl] := Bandgap(Cellname[l] ) ; PN:= true; Sn[l] := 1E4; Sp[l]:= 1E4; If CellnameCl] = ALGAAS then begin Ubound[l] := 16.0; Lbound[l]:= 14.5; Ubound[2] := 17.0; Lbound[2] := 15.5; end else begin UboundCl] := 17.5; Lbound[l] := 17.0; Ubound[2] := 18.5; Lbound[2] := 18; end; Ubound [3] : = 0.5; Lbound[3] := 0.05; Ubound [4] := 10.0; Lbound [4] : = 2.0; Ubound [5] := Eg[l]; Lbound [5] := Eg[l]; IdealF[l]:= 1.0; RGmodel[l]:= DIFFUSION; end; Procedure GetMedBound; begin Eg[2] := Bandgap(Cellname[2] ) ; PN:= true; Sn[2] := 1E4; Sp[2]:= 1E4; Ubound [6] := 17.5; Lbound [6] := 17.0;

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Ubound [7] : = 18.5; Lbound[7]:= 18.0; Ubound [8] : = 0.5; Lbound[8] := 0.05; Ubound [9] := 10.0; Lbound [9] : = 2.0; Ubo\ind[10] := Eg[2] ; Lbound [10] := Eg [2] ; IdealF[2]:= 1.0; RGmodel[2]:= DIFFUSION; end; Procedure PruneSearch; begin if CellConf = 1 then begin CellNameEl] := GAAS; GetTopBound; IndVar:= 5; end else if CellConf = 2 then begin CellNameEl] := ALGAAS; GetTopBound; Ubound [5] := 1.98; Lbound [5] := 1.85; CellName [2] = GAAS; GetMedBound; IndVeir:= 10; end else begin CellNameEl] := ALGAAS; GetTopBound; Ubound [5] := 1.98; Lbound [5] := 1.85; CellName [2] := GAAS; GetMedBound; CellName [3] := INGAAS; Eg [3]:= Bandgap(CellnameE3] ) ; PN:= true; Sn[3]:= 1E4; SpE3];= 1E4; UboundEll]:= 17.7 ; Lbound El 1]:= 17.7;

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Ubound[12] := 17.3; Lbound[12] := 17.3; Ubound[l3] := 0.5; Lbound[13]:= 0.05; Ubound[14]:= 10.0; Lbound[14] ;= 2.0; Ubound[15]:= Eg [3]; Lbound[15]:= Eg[3] ; IndVax:= 15; IdealF[3]:= 1.0; RGmodel[3]:= DIFFUSION; end; end; (* PruneSearch *) Function GaAsAbsorp( Atemp : real ): real; var i : integer; Btemp : real; found : booleain; begin if Atemp <= 0.300 then Atemp: =0.300; if Atemp >= 0.873 then Atemp :=0 .873; found: “false; i:= 1; repeat if Atemp <= WLl[i] then begin Btemp : = Kvl [i-1] (Kvl [i-1] -kvl [i] ) /(WLl [i] -WLl [i-l] ) *(Atemp-WLl[i-l]) ; found := true; end else i:= i + 1 ; until ( i >= 24 ) or ( found ); GaAsAbsorp:= 4.0 * 3.141596 * Btemp / Atemp ; end; (* Si absorption coefficients vs. wavelength *) Function SiAbsorpC Atemp : real ): real; var i : integer; Btemp : real; found : booleaoi; begin

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160 if Atemp <= 0.294 then Atemp:=0.294; if Atemp >= 1.125 then Atemp :=1 . 125 ; found :=false; i:= 1; repeat if Atemp <= WL2[i] then begin Btemp: = Kv2 [i-1] (Kv2 [i-l] -kv2[i] ) /(WL2 [i] -WL2 [i-1] ) *(Atemp-WL2[i-l]) ; found:® true; end else i:= i + 1 ; until ( i >= 36 ) or ( found ) ; SiAbsorp:® 4.0 * 3.141596 * Btemp / Atemp ; end; (* Ge absorption coefficients vs. wavelength *) Function GeAbsorpC Atemp : real ): real; var i : integer; Btemp : read; found : boolean; begin if Atemp <® 0.500 then Atemp :®0 . 500 ; found :=false; i:® 1; repeat if Atemp <® WL3[i] then begin Btemp : ® Kv3 [i-1] (Kv3 [i-1] -kv3 [i] ) / (WL3 [i] -WL3 [i1] ) (Atemp-WL3[i-l]) ; found:® true; end else i:® i + 1 ; until ( i >® 12 ) or ( found ) ; GeAbsorp:® 4.0 * 3.141596 * Btemp / Atemp ; end; Function AlGaAsAbspC x, yw : real):real; var il, i2 : integer; flagl, flag2 : boolean; al, a2, a3, y: real;

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161 begin if X <= 0,099 then x:= 0.099 else if X >= 0.804 then x:= 0.804; y:= 1.24 / yw ; if y <= 1.500 then y:= 1.500 else if y >= 3.500 then y:= 3.500; Flagl:= false; il:= 1; while not flagl do begin if (x >= Fxx[il]) and ( X <= Fxx[il+1]) then Flagl := true else Il:= il + 1; end; Flag2:= false; i2:= 1; while not Flag2 do begin if ( y >= Fyy[i2]) eind ( y <= Fyy[i2+1]) then Flag2:= true else I2:= 12 + 1; end; al:= Fxy[(il-1) 21 + i2] + (y Fyy[I2]) / (Fyy[I2+l] Fyy[I2]) * (Fxy[(il-l)*21 + i2 + 1] Fxy[(il-l)*21 + i2]); a2:= FxyCil 21 + i2 ] + (y Fyy[i2]) / (Fyy[l2+l] Fyy[i2]) (Fxy[il*21 + i2 + 1] Fxy[il*21 + i2] ) ; a3:= al (al a2) (x Fxx[il]) / (Fxx[il+1] Fxx[il]); AlGaAsAbsp:= a3; end; Function GaAsAlU(Tx, xx, Ndl. Nal:real; ii : integer): real; begin if ii = 1 then GaAsAlU:= expClnlO * ( -1.5545 + 0.0106*xx + (0.735 + 0.0013*xx)* ln(Nal)/lnl0 (0.0253 + 0.0052*xx)*sqr(ln(Nal)/lnl0))* 300 / Tx) else GaAsAlU:= exp(lnl0 ( -9.723 + 0.0095*xx + (1.576 + 0.0012*xx)* In(Ndl) /InlO (0.0507 + 0.0034*xx)*sqr(ln(Ndl)/lnl0))*exp(0.75*ln(300/Tx))) ; end; Function GaAsAlL(Tx, xx, Ndl, Nalireal; ii : integer) : real ; var Uhl, Uel, UhO, UeO : real; begin

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162 if ii = 1 then begin Uel:= GaAsAlU(Tx, xx, Ndl, Nal, ii) ; UeO:= GaAsAlU(Tx, 0.0, Ndl, Nal, ii) ; GaAsAlL:= sqrt (Uel/UeO) *exp(-9 .72*xx) *( 210.06 + 27 . 254*ln(nal)/ InlO 0.850*sqr(ln(Nal)/lnl0))*exp(0.87*ln(Tx/300)) ; end else begin Uhl:= GaAsAlU(Tx, xx, Ndl, Nal, ii) ; UhO:= GaAsAlUdx, 0.0, Ndl, Nal, ii) ; GaAsAlL:= sqrt(Uhl/UhO)*exp(-9.72*xx)*( 116.92 + 14.466*ln(Ndl)/ InlO 0.438*sqr(ln(Ndl)/lnl0))* Tx / 300 ; end; end; (* (InGa)As absorption coefficients vs. wavelength *) (* In — > 53 •/. ; Ga — > 47 ’/. *) Function InGaAsAbsorpCAtemp : real): real; var Btemp : real; begin if Atemp <= 0.74 then Btemp := 10.0 else if Atemp <= 1.40 then Btemp:= exp((-l . 51515*Atemp+6. 12121 )*lnl0) / 1E4 else if Atemp <= 1.50 then Btemp:= exp((-l . 5491*Atemp+6 . 16874)*lnl0) / 1E4 else if Atemp <= 1.54 then Btemp:= exp((-3.65495*Atemp+9.32752)*lnl0) / 1E4 else if Atemp <= 1.73 then Btemp:= exp((-14.2051*Atemp+25.5749)*lnl0 ) / 1E4 else Btemp := lE-3; InGaAsAbsorp:= Btemp; end; (* Absorption Coefficients Calculations of Each Cell*) Function Absp(Name : CellType; Wlength: real ): real; var Atemp : real; begin Atemp := Wlength; if Name = GAAS then

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163 Absp:=GaAsAbsorp(Atemp) else if Name = ALGAAS then Absp:=AlGaAsAbsp(alratio, Atemp) else if Naime = INGAAS then Absp := INGAASAbsorp (Atemp) else if Name = GE then Absp :=GeAbsorp (Atemp) else if Name = SI then Absp:=SiAbsorp(Atemp) ; end; (* to interpolate the irradian.ce power density of AM? *) function AMX( x : real; ii : integer) : real ; var i , j : integer; XX : real ; found : boolean; begin Found ;=false; i: = l; while ( not Fo\md) and ( i <=195 ) do begin if ( X = AtypeCi] ) then begin Found :=t rue; AMX := Btype[ii,i] * 1E3; end else if ( X < AtypeCi] ) and ( x > Atype[i+1] ) then begin XX:=(x Atype[i+1]) * (Btype [ii, i+1] Btype[ii,i]) / (Atype[i+1] AtypeCi]) + Btype Cii , i+1] ; AMX:=XX * 1E3; Found :=t rue; end else i :=i+l; end; end; function cosh( x : real ) : real; begin if (x) >= 87.0 then (* to avoid out of range of exp(x) *) x:= 87.0 ELSE if X <= 87.0 then

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164 x:= -87; cosh:=( exp(x) + exp(-x) ) / 2.0; end; function sinh( x : real ) : real; begin if X >= 87.0 then (* same as that of function cosh ) x:= 87.0 else if X <= -87.0 then x:= -87.0; sinh:=( exp(x) exp(-x) ) / 2.0; end; (* Calculate the Incident Photon Number vs. sunlight wavelength *) (* XI > wavelenth ; X2 — > irradiance power density. *) Function NphCal(Xl,X2:real) :real; begin NphCal:= 4.06E18 *X1 *sqr(Xl) * X2 * lE-3 ; end; Function QF( ii, Tx : integer; Wlength: real ): real; var i, j : integer; Cl, C2, Dl, D2, Al, A2, A3, Flux : real; begin i:= ii; Flux:= NphCal (Wlength, AMX(Wlength,AirM)) ; if not GFactor then QF:= Qe * Flux else begin if i then (* default AlGaAs,GaAs,InGaLAs,Si or Ge system *) QF:= Qe * Flux else if Tx = 2 then begin Al:= AbspCCellNcime [i-1] , Wlength ); Dl:= Tj[i-1]; Cl:= Al * Dl ; if Cl >= 87.0 then ( to avoid out of range *) Cl:= 87.0; QF:= Qe * Flux * exp(-Cl);

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165 end else if Tx = 3 then begin Al:= Absp(cellName[i-2] ,Wlength) ; Dl:= Tj[i-2]; A2:= Absp(CellNajne[i-l] jWlength) ; D2:= Tj[i-1]; Cl:= A1 D1 ; C2:= A2 D2 ; if Cl >= 87.0 then Cl:= 87.0; if C2 >= 87.0 then C2:= 87.0; QF:= Qe Flux exp(-Cl ) exp(-C2); end; end; end; (* Calculate short-circuit current density of each cell ) Function Jpnd(ii, jj, Tx : integer; Naone : CellType; Wlength : real );real; var XO, XI, XS, X2, X3, X4, X5, X6, Jx, X7, XQF, Absorb, Cl, C2 : real; i, j : integer; begin i:= ii ; j := Jj ; XQF:= QF(i,Tx, Wlength) ; if GFactor then begin Absorb:= Absp(Name, Wlength); if j = 1 then begin if PN then begin X0:= Lnn[i] ; XI := Tn[i] ; XS:= Sn[i]; end else begin X0:= LppCi] ; Xl:=Tp[i]; XS:= SpCi];

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166 end; X2:= Absorb * XO ; X3:= XS * 1E4 * XI / XO ; X4:= coshCXj [i]/X0) ; X5:= sinh(Xj [i]/X0) ; X7:= EXP(-Absorb*Xj [i] ) ; Jx := XQF*X2/(sqr(X2)-l)*( (X2+X3*(l-X7*X4)-X7*X5 )/ ( X3*X5 + X4) X2*X7); end else if j = 2 then begin if PN then begin X0:= Lpp[i] ; Xl:= Tp[i] ; XS:= Sp[i] ; end else begin X0:= Lnn[i] ; XI := Tn[i]; XS:= Sn[i]; end; X2:= Absorb XO ; X3:= XS * 1E4 XI / XO ; X4:= cosh((Tj[i] Xj [i] Wj[i])/X0); X5:= sinh((Tj[i] Xj [i] Wj[i])/X0); Cl:= absorb * (Tj [i] Xj [i] Wj [i] ) ; C2:= absorb * (Xj [i] + Wj [i] ) ; if Cl >= 87.0 then Cl:= 87.0; if C2 >= 87.0 then C2:= 87.0; X6:= exp(-Cl) ; X7:= XQF*X2/(sqr(X2)-l)*EXP(-C2); (* check overflow *) if ((In(abs(x3))/lnl0 + In(abs(x4))/lnl0) >= 37.0 ) ((In(abs(x3))/lnl0 + In(abs(x5))/lnl0) >= 37.0 ) begin x4:= x4 / lElO; x5:= x5 / lElO; x6:= x6 / lElO; end ; Jx :=X7 * (X2 (X3 * (X4 X6) + X5 + X2 * X6) / (x3 * X5 + X4 ) ); or then

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167 end else begin Cl:= Absorb * Xj [i] ; C2:= Absorb * Wj [i] ; if Cl >= 87.0 then Cl:= 87.0 ; if C2 >= 87.0 then C2:= 87.0; Jx := XQF*exp(-Cl)*( 1-exp (-C2)) ; end; end else Jx : = XQF; Jpnd:= Jx; end; Function TransM(ii : integer; Name: CellType; Upper, Lower : real): real; var csub. Cl, cnew, cold, X, XO, XI, X2, X3, H : real; i : integer; cfound : boolean; citer : integer; begin citer := 5; cold:= 0.0; cfound := false; repeat H:= (Upper Lower ) / ( 2.0 * citer ); cnew:= 0.0 ; X1:=0.0; X2:=0.0; cl:= absp (Name, 1 .24/lower) Tj [ii] ; if cl >= 87 then cl:= 87; xO := exp(Cl) ; cl:= absp(Name,l .24/upper) * Tj [ii] ; if cl >= 87 then cl:= 87; x0:= xO + exp(Cl) ; for i:=l to ( 2 * citer 1) do begin X := 1.24 / (Lower + i * H) ; cl :=absp(Name, x) * Tj [ii] ;

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168 if cl >= 87 then cl:= 87; x3:= exp(Cl) ; if odd(i) then XI := XI + X3 else X2:= X2 + X3 ; end; csub:= H* (X0+2.0*X2+4*X1 ) /3.0 ; cnew:= csub; If (absCcnew cold) < 0.0001) or (citer >= 100) then begin CFound:= true ; if citer >= 100 then writelnC' > Exceed 100 iteration in CollEff'); end else begin citer := citer + 1; cold:= cnew; end; Until ( cf ound) ; TransM := cnew; end; ( use Simpson composite method to integrate the Jsc *) Function JscTot(ii, Tx: integer; Name: CellType; Upper, Lower : real): real var Jnew, Jold, X, XO, XI, X2, X3, H : real; Jsub : array [1..3] of real; i, j, k : integer; Jfound : boolean; liter : integer; begin liter := 5; lold:= 0.0; lfound:= false; repeat H:= (Upper Lower ) / ( 2.0 liter ); if not GFactor then K:= 1 else K:= 3; lnew:= 0.0 ;

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169 for j := 1 to k do begin X1:=0.0; X2:=0.0; x0:= Jpnd(ii,j ,Tx, Name, 1.24/Lower ) + Jpnd(ii ,j ,Tx,Name, 1 . 24/Upper) ; for i:=l to ( 2 Jiter 1) do begin X := 1.24 / (Lower + i * H) ; X3:=Jpnd(ii, j ,Tx,Name,X) ; if odd(i) then XI := XI + X3 else X2:= X2 + X3 ; end; JsubCj] :=H* (X0 + 2.0*X2 + 4*X1) / 3.0 ; Jnew:= Jnew + Jsub[j]; end; If (abs(Jnew Jold) < 0.0001) or (Jiter >= 100) then begin JFound:= true ; if Jiter >= 100 then writeln(' > Exceed 100 iteration in Jsc '); end else begin Jiter:= Jiter + 1; Jold:= Jnew; end; Until ( Jfound) ; JscTot:= Jnew; end; (* use Binary search method to obtain the Maximum voltage, Vm *) Procedure GetVm(Xsc,Xoo : real; k : integer); var i : integer; Diff, XVt, X, XI, x2: real; Found : boolean; begin Found :=false; xl:= 0.1; x2 : = Eg [k] ; REPEAT x:= (xl + x2) / 2.0 ; XVt;= X / (IdealFM* 0.0259 * Tratio) ;

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170 Diff:= Ln((l + Xsc/Xoo)/(l + Xvt) ) Xvt ; if (ABS(Diff / xvt) <= 0.001) then Found :=t rue else if Diff >=0.0 then xl;= X else x2:= X ; until (Found) or ( xl > x2) ; if Found=true then Vm[k] :=x else Vm [k] : =0 . 0 ; end; Procedure GetAdaC x : real; k: integer) ; begin if RGmodelCk] = DIFFUSION then Joo[k]:= qe * sqr(Ni[k]) * (Dpp[k] / (LppCk] Ndd[k]) + DnnCk] / (Lnn[k] * Na[k])) else Joo[k]:= Qe Wj [k] lE-4 * Ni [k] / sqrt(Tn[k] * Tp[k]); Voc[k] := IdealFCk] * 0.0259 Tratio * ln( Jsc/Joo[k]+ 1 ); GetVm(Jsc, Joo [k] ,k) ; FF[k] :=Vm[k]/Voc[k]*(l.0-(exp(Vm[k]/ (IdealFCk] *0.0259 * Tratio))-!) / (exp (VocCk]/ (IdealFCk] *0.0259 * Tratio)) 1) ) ; AdaCk] := Jsc * VocCk] * FFCk] / (Pam * lE-3) ; end; Procedure Single Junction; begin Low:= EgCl] ; Jsc:= JscToT(l,l,CellNameCl] .high, low) ; ColeffCl]:= 1 TransMd.CellNameCl] .high, low) ; Jsc:= Jsc * ColeffCl]; GetAda(EgCl] ,1) ; AdaTotÂ’:= Ada Cl] ; end; (* of Single Junction *) procedure Doublejunction; var k, II, i, j : integer; found: boolean; X : real; Jtop, Jbotl, Jbot2, Jbot : real;

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171 begin (* of Double *) Jtop:= JscToT(l,l,CellName[l] ,high,Eg[l]) ; Coleff[l];= 1 TransM(l,CellName[l] ,high,Eg[l]); Jtop;= Jtop * ColeffEl]; Jbotl:= JscToT(2,l.CellName[2] ,Eg[l] ,Eg[2]); Coleff[2]:= 1 TransM(l,CellName[2] ,Eg[l] ,Eg[2]) ; Jbotl:= Jbotl * Coleff[2]; Jbot2:= JscToT(2,2,CellName[2] ,High,Eg[l] ) ; Jbot:= Jbotl + Jbot2; if Jtop >= Jbot then Jsc:= Jbot else Jsc:= Jtop; Getada(Eg[l] ,1) ; GetAda(Eg[2] ,2) ; AdaToT:= Ada[l] + Ada[2] ; end; procedure triple junction; var i. j, k, m, n, 0 : integer; xl, x2 : real; Jtop, Jmid, Jbot 1 , Jbot2 , Jbot3, Jbot : real; found : booleain; Procedure AtripleAda; begin if (Jtop <= Jmid ) and (Jtop <= Jbot ) then Jsc:= Jtop else if (Jtop >= Jmid ) and (Jtop <= Jbot ) then Jsc:= Jmid else if (Jtop <= Jmid ) and (Jtop >= Jbot ) then Jsc:= Jbot else if (Jtop >= Jmid ) and (Jtop >= Jbot ) then begin if Jmid >= Jbot then Jsc:= Jbot else Jsc:= Jmid;

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end; GetAda(Eg[l] , 1) ; GetAda(Eg[2] ,2) ; GetAda(Eg[3] ,3) ; AdaToT:= Ada[l] + Ada[2] + Ada[3] ; end; (* of AtripleAda *) begin (* triplej unction *) x2;= Eg[l] ; xl:= Eg [2] ; Low:= Eg [3] ; Jtop:= JscToT(l,l, CellName [1] ,High,x2) ; Coleff[l]:= 1 TremsM(l ,CellNcime[l] ,high,x2) ; Jtop:= Jtop * ColeffCl]; Jbotl : = JscToT(2 , l,CellNajne [2] ,x2 ,xl) ; Coleff[2]:= 1 TransM(2 .CellName [2] ,x2 ,xl) ; Jbotl := Jbotl * Coleff[2]; Jbot2:= JscToT(2,2,CellNajne[2] ,High,x2) ; Jmid:= Jbotl + Jbot2; Jbotl := JscToT(3, l.CellNajne [3] ,xl .low) ; Coleff[3]:= 1 TransM(3.CellNajne[3] .xl.low) ; Jbotl := Jbotl * Coleff[3]; (* Jbot2:= JscToT(2. 2. CellName [2] .x2.xl); Jbot3:= JscToTCl .3.CellNaLme [l] .high.xl) ; Jbot:= Jbotl + Jbot2 + Jbot3; *) Jbot:= jbotl; ATripleAda; end; procedure GetNi(Name:CellType; i:integer); var Ih. bh. xl. x2 : real; begin if Name = SI then begin Mn[i]:= 1.1 ; Mp[i]:= 0,56; Eg[i]:= Bandgap(NaLme) ; end else if Name = GAAS then begin Mn[i]:= 0.067;

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173 Mp[i]:= 0.641; Eg[i]:= Bajadgap(name) ; Eg[i]:= Eg[i] 1.6E-8 * EXP(l/3 * LN(NA[I]) ); end else if Name = ALGAAS then begin Mn[i]:= 0.067 + 0.083 * alratio; lh:= 0.087 + 0.063 * alratio; hh:= 0.62 + 0.14 * alratio; Mp[i]:= exp(2.0/3.0*ln(lh*sqrt(lh) + hh*sqrt (hh) ) ) ; Eg[i]:= bandgap(name) ; end else If Name = GE then begin Mn[i]:= 0.55 ; Mp[i]:= 0.37 ; Eg[i]:= bandgap(name) ; end else begin Mn[i]:= 0.041; Mp[i] := 0.5; Eg[i]:= bandgap(name) ; end; xl := Mn[i] * Mp[i] ; Ni[i]:= sqrt(xl * sqrt(xl) * 2.33E31) Tdegree * sqrt (Tdegree) * exp(-Eg[i] / (0.0259 * 2 * Tratio)); end;(* of getNi *) Procedure Getlmplicit(ii : integer) ; var Eps, xratio, Vbi : real; i : integer; begin if CellName[ii] = SI then begin Eps:= 12.0; Mobe[ii]:= 88.0 * exp(-0 . 57*ln(Tratio) ) + 7.4E8 * exp(-2.33*ln(Tdegree)) / ( 1+0 . 88*exp (-0 . 146*ln (Tratio) ) *(Ndd [ii] / (1 . 26E17+exp(2 . 4*ln (Tratio) ) ) ) ) ; Mobp[ii]:= 54.3 * exp(-0 . 57*ln(Tratio) ) + 1.36E8 * exp (-2 . 23*ln (Tdegree)) /(l+0.88*exp(-0.146*ln(Tratio))*(Na[ii]/(2.35E17*exp(2.4*ln(Tratio))))); Dnn[ii] := 0.0259 * Tratio * MobeCii] ; DppCii] := 0.0259 * Tratio * Mobp[ii] ;

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174 Tn[ii]:= 12E-6 / (1 + Na[ii] / 5E16) ; Tp[ii]:= 12E-6 / (1 + Ndd[ii] / 5E16); Lnn[ii] := sqrt(Tn[ii]*Dnn[ii] ) 1E4 ; (* um *) Lpp[ii]:= sqrt(Tp[ii]*Dpp[ii]) * 1E4 ; end else if (CellNameCii] = GAAS) or (CellName[ii] = ALGAAS) then begin if CellNameCii] = GAAS then begin Xratio:= 0.0; Eps:= 12.9; end else begin if Eg[ii] <= 1.99 then Alratio:= (Eg[ii] 1.424) / 1.247; Xratio:= Alratio; Eps:= 12.9 *(l Alratio) + 10.06 * alratio; end; Mobe[ii]:= GaAsAlU(Tdegree,xratio ,Ndd[ii] ,Na[ii] , 1) ; Mobp[ii]:= GaAsAlU(Tdegree,xratio ,Ndd[ii] ,Na[ii] ,2) ; Dnn[ii] := 0.0259 * Tratio * Mobe[ii] ; Dpp[ii]:= 0.0259 * Tratio * Mobp[ii] ; LnnEii] := GaAsAlL(Tdegree,xratio,Ndd[ii] ,Na[ii] ,1) ; LppCii] := GaAsAlL(Tdegree,xratio,Ndd[ii] ,Na[ii] ,2) ; Tn[ii];= sqr(Lnn[ii]) / Dnn[ii] * lE-8 ; Tp[ii]:= sqr(Lpp[ii]) / Dpp[ii] * lE-8 ; end else if (CellNameCii] = GE ) then begin MobeCii] := 695.0; MobpCii] := 540.0; DnnCii] := 18.0; DppCii] ;= 14.0; LnnCii] := 232.0; LppCii] := 100.0; TnCii] := 3E-5; TpCii] := 7.2E-6; NddCii] := 2E19; NaCii] := 2E17; Eps:= 16.0; end else begin Eps:= 12.0; MobeCii] := 6000.0;

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175 MobpCii] := 180,0; dim[ii] := 155.4; DppCii] := 4.66; Lnn[ii] := 6.0; LppEii] := 2.0; Tn[ii] := 2.3E-9; TpCii] := 8.59E-9; NddCii] := 5E17; Na[ii] := 2E17; end; GetNi (CellName [ii] , ii) ; Vbi:= 0.0259 * Tratio ln(Na[ii] / sqr(Ni[ii]) * Ndd[ii] ); Wj[ii]:= sqrt(2 Eps * Pemit Vbi / Qe *(1/Na[ii] + 1/Ndd[ii])) *1E4 end; Function Ranu(var lo, hi : real) : real; Alien; Procedure InitialGuess ; var i, j : integer; 11, hh: real; begin 11 := 0 . 0 ; hh:= 1.0; for i:= 1 to (IndVar+1) do for j:= 1 to IndVar do InitGuessEi, j] := LboundEj] + (UboundEj] IboundEj] )*Ranu(ll,hh) ; end; Procedure Radiation(II : integer) ; begin LnnEii] sqrt(l/(l/sqr(LimEii]*lE-4) + KlnEii] * fluxx)) * 1E4; LppEii] := sqrt(l/(l/sqr(LppEii]*lE-4) + KlpEii] * fluxx)) 1E4; TnEii] := 1 /(l/TnEii] + DnnEii] * KlnEii] * fluxx); TpEii] 1 /(1/TpEii] + DppEii] * KlpEii] * fluxx); end; Function GetF(ii : integer) : real; var i, j : integer; begin i:= ii; repeat NddEi] :=exp (Guess E(i-l) *5+1] * InlO) ;

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176 Na [i] :=exp(guess[(i-l)*5+2] InlO) ; Xj[i]:= Guess C(i-1)*5 + 3] ; Tj[i]:= Guess [(i-l)*5 + 4]; if ( i = 1 ) then Eg[l] := Guess[(i-1)*5 + 5] else if i = 2 then Eg[2] := Guess[(i-1)*5 + 5] else if (i = 3) then Eg[3] := Guess[(i-1)*5 + 5]; Getlmplicit(i) ; if particle <> 0 then Radiation(i) ; i:= i 1; until ( i <= 0 ) ; if ii = 1 then Single Junction else if ii = 2 then Doublej unction else Triplejunction; RealAda; if Insolation > 1 then Highinsolation; GetF:= AdaTot; end;(* of GetF ) Procedure FBestWorst (Var best, worst : integer) ; var i, j : integer; Temp : real; found: booleam; begin temp:= AdaVec[l]; found := false; for i:= 2 to (IndVar+1) do begin if (AdaVecCi] > temp ) then begin temp:= AdaVec[i] ; best:= i; found := true; end; end; if not found then best:= 1 ;

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177 temp:= AdaVec[l] ; found := false; for i:= 2 to (IndVar+1) do begin if (AdaVecCi] < temp) then begin Temp:= AdaVec[i] ; worst := i; found := true; end; end; if not found then worst := 1; Fworst:= temp; end; Procedure Deworst (Var best, worst : integer) ; var i, j : integer; al,ll,hh. Sum : real; begin 11:=0.0; hh:=1.0; For i:= 1 to IndVar do PreGuess [i] : = InitGuess [worst , i] ; for i:= 1 to IndVax do begin Sum:= 0.0; for J:= 1 to (IndVar+1) do sum:= sum + InitGuess [j , i] ; AveX[i] :=(Sum PreGuess [i]) / IndVar; InitGuess [worst, i] := 1.3 *(AveX[i] PreGuess [i]) + AveX[i]; Guess [i] := InitGuess [worst ,i] ; end; end; Procedure notBetter(var be, wo : integer); var i, j : integer; begin for I:= 1 to IndVcLT do begin Guess [i] := (Guess [i] + InitGuess [be, i]) * 0.5; InitGuess [wo, i] := Guess [i];

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178 end; end; Procedure CheckBound; var i : integer; begin for i:= 1 to IndVar do begin if Guess [i] >= UboundCi] then Guess [i] := UboundCi] else if Guess [i] <= LboundCi] then Guess [i]:= LboundCi]; end; end; Procedure Printout; var i, j : integer; begin (* main of Printout For i:= 1 to CellConf begin writeln(out, 'Cell # writelnCout, ' Na writelnCout, ’ Nd writelnCout, ' Xj writelnCout, ' Tj WritelnCout, ' Sp writelnCout, ' Sn writelnCout, ' Voc writelnCout, ' F.F writelnCout, ' Ada WRITELNCOUT, ' Ln writelnCout, ’ Lp writelnCout, ' Tn writelnCout, ' Tp writelnCout, ’ Mobe: ’ writelnCout, ' Mobp: ' writelnCout, ' Ni writelnCout, ' Wj writelnCout, ' Mn writelnCout, ' Mp writelnCout, ' Eg O do M:2,' ,NaCi]) ; .NddCi]); .Xj Ci]) ; ,TjCi]); ,SpCi]) ; ,SnCi] ) ; ,VocCi]) ; .FFCi]); ,AdaCi] ) ,LnnCi]) .LppCi] ) ,TnCi]) ; ,TpCi]) ; ,mobeCi] ) ; ,mobpCi] ) ; ,NiCi]) ,Wj Ci]) ,MnCi]) ,MpCi]) ,EgCi] ) — •> > );

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179 writelnCout , ' Col: ' ,ColEff [i] ) ; end; writelnCout, ' Jsc : Â’,jsc); WritelnCout,Â’ ToTal Ada : Â’,AdaTot); writelnCout,Â’ Air Mass :Â’,Airmass); WritelnCout,Â’ Sun Insolation : Â’, insolation) ; writelnCout,Â’ Temperature : Â’ ,Tdegree) ; if Btunnel then for i:= 1 to TunnelNum do writelnCOUT, Â’ Tunnel drop : Â’ ,TdropV[i]) ; end; C* Printout*) Procedure BoxCii : integer) ; var index , count , Be, Wo, i, j : integer; Spt, foiind: boolean; begin InitialGuess; For i:= 1 to ClndVar+1) do begin for j:= 1 to IndVar do Guess[j]:= InitGuess [i, j] ; AdaVec[i] := GetFCii) ; end; FbestWorstCBe,Wo) ; index := wo; Deworst CBe, Wo) ; Checkbound; AdaVec[Wo]:= GetFCii); Found := FALSE; repeat if CAdaVec[Wo] <= Fworst) then begin NotBetterCbe,wo) ; Checkbound ; AdaVec[Wo] := GetFCii); end else begin FBestWorstCBe ,Wo) ; if Cabs C CAdaVec [Wo] -AdaVec [Be] ) / AdaVec [Be] ) <= error) Found := true else then

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180 begin if wo = index then Notbetter(be,wo) else begin DeWorst(Be,Wo) ; Index := wo; end; Checkbound; AdaVec[Wo]:= GetF(ii); end; end; until (Found) ; end; begin Initial; Readinput ; if system then PruneSearch else begin CellSelection; Input Bound; end; ARcoating; Contact ; TunnelSelection; Irradiation; writelnC' The highest Cutoff Eg. < 4.0 eV '); readln(High) ; writelnC' Relative Error for convergence detections. <= lE-4 '); readln (error) ; Step:= 1; Fbest:= 0.15; writeln(Â’ Maiximum Iteration? Â’); readln (Iter) ; repeat Box(CellConf ) ; printout ; Stepr=Step + 1; if (AdaToT >= Fbest) and (AdaToT <= 0.35) then Fbest:= AdaToT; Until (Step >= iter) or (AdaTot > 0.35 ); ARthickness; writeln(out , ' Shadow : shadow);

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181 writelnCout, ' Reflection : ',RefL); writeln(out , ' LossRatio : ' ,LossRatio) ; writelnCout,' Series Resistance : ',Rs); end.

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APPENDIX D INPUT PARAMETERS FOR THE OPTIMAL DESIGN OF (alga)as/gaas/iNo.53GAo.4tAS three-junction solar cells Table D.l Physical parameters for the Ino.53Gao.47As bottom cell. Electron lifetime r„ (s) 2.3 X 10-® Hole lifetime Tp 8.58 xlO-^ Electron dilfusion length Ln (/im) 6.0 Hole diffusion length Lp 2.0 P-dopant density Na (cm~^) 2 X 10^^ N-dopant density Nq 5 X 10^^ Electron mobility /in(cm^/V-s) 6000 Hole mobility /ip 180 Electron effective mass mn(mo) 0.041 Hole effective mciss mp 0.5 Bandgap Energy Eg (eV) at 300 K 0.75 182

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A.t»aorptlorx Coofflclerxt 183 6 (AlGa)As Bandgap (eY) Fig. D.l Absorption coefficients for various A1 composition of (AlGa)As top cell. + : 14 %, X : 22 %, solid square : 30 %, Y : 38 % and * : 46%.

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A.baox'ption Coefficient 184 6 Bandgap Energy (eY) Fig. D .2 Absorption coefficients for GaAs and Ino.53Gao.47 As solar cell materials. X : Irio.53Gao.47 As and -|: GaAs.

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185 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 A1 composition Fig. D.3 The A1 composition dependence of minority carrier mobility for (AlGa)As top cell at room temperature and doping density of 10*Â’Â’ cm~^. + : electron mobility and X : hole mobility.

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186 A 0 4 Doping Density Fig. D.4 The doping density dependence of minority carrier mobility for Alo.44Gao_56As and GaAs. + : Alo.44Gao.56As electron mobility, X : Alo.44Gao.5e As hole mobility, Y : GaAs electron mobility and * : GaAs hole mobility.

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187 1 Fig. D .5 The doping density dependence of minority carrier diffusion length for Alo.44Gao.56 As and GaAs. + : Alo.44Gao.5eAs electron diffusion length, X : Alo.44Gao.5e As hole diffusion length, Y : GaAs electron diffusion length and * : GaAs hole diffusion length.

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BIOGRAPHICAL SKETCH Chune-Sin Yeh was born in Taiwan in 1955. He received the B.S. degree in electrical engineering from National Cheng-Kung University, Taiwan, in 1977. He went to the College of Wilham and Mary, Williamsburg, Virginia, in September 1981 and received an M.S. degree in applied science (computer science division) in May 1983. He also received the M.E. degree in electrical engineering from the University of Florida, in May 1985. Since then he has been working toward the Ph.D. degree in the Department of Electrical Engineering at the University of Florida. From 1977 to 1979, he served in the Republic of China Air Force. He worked for the Department of Computer Science, Taipei City Junior College of Business, Taiwan, from 1979 to 1981. His research interests are computer modeling of multijunction solar cells and computer aided design for VLSI. 194

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Slielig S. Li, airman Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. A most Neugroschel Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Dorothy Er^urk Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Gijs Bosman Associate Professor of Electrical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. vAf . r//i.c7Ar n-C|^ieh Chow YuanAssociate Professor of Computer and Information Sciences

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This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. August 1988 Dean, College of Engineering Dean, Graduate School