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The analysis of Schottky-barrier solar cells McOuat, Ronald F. 1976

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THE ANALYSIS OF SCHOTTKY-BARRIER SOLAR CELLS by Ronald F. McOuat B.A.Sc. , U n i v e r s i t y of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE ,-We accept t h i s t h e s i s as conforming to the re q u i r e d standard i n the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA September 1976 © Ronald F. McOuat, 1976 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n 1 f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e ABSTRACT Several models were developed for the analysis of metal-semiconductor solar cells. The models presented are: (i) a limit model to obtain an idea of what the maximum conversion efficiency of metal-semiconductor solar cells i s followed by; ( i i ) a model suitable for the prediction of the performance of metal/single-crystal s i l i c o n solar c e l l s ; and ( i i i ) a general model for calculating the efficiency of solar cells fabricated from materials other than Si such as GaAs. Extensive use of numerical methods were required to arrive at solutions to the equations presented in the latter two models. The operation of the models is demonstrated using n-and p-type Si and GaAs with Au being taken as the barrier metal. Calculations are presented showing the effect on solar energy conversion efficiency of surface recombination velocity, barrier height, minority-carrier lifetime, barrier metal thickness, collecting grid configuration, and c e l l thickness. A comparison of practical and computed data for the Au/n-GaAs system yields good agreement. Based on the results of the calculations, i t i s shown that metal-semiconductor solar cells provide solar energy conversion of medium efficiency and improvements in efficiency depend on the development of high barrier-height systems. i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF ILLUSTRATIONS i v LIST OF TABLES v i ACKNOWLEDGEMENT v i i I INTRODUCTION • 1 I I LIMIT CALCULATIONS 6 i . Model 6 i i . R e s u l ts and D i s c u s s i o n 13 I I I A MODEL FOR SILICON SCHOTTKY-BARRIER SOLAR CELLS 20 i . I n t r o d u c t i o n 20 i i . The Model 21 i i i . Method of S o l u t i o n 25 i v . R e s ults and Di s c u s s i o n 30 IV GENERAL MODEL 43 i . I n t r o d u c t i o n 43 i i . The Model 43 i i i . Method of S o l u t i o n 46 i v . R e s ults and Di s c u s s i o n 53 V CONCLUSIONS 72 REFERENCES . . . . . . . . . . . 76 APPENDIX I 80 APPENDIX I I 102 i i i LIST OF ILLUSTRATIONS i v Figure Page II.I-9 Dependence of AMO sunlight conversion e f f i c i e n c y on 39 metal f i l m thickness and c o l l e c t i o n grid configuration. Refractive index of gold (bulk data) taken from Ref. 33; grid dimensions taken from Ref. 35. I I I - 10 Details of grid structures and resistance model taken 40 from Ref. 35. IV- 1 I l l u s t r a t i o n of the dependence of the boundary condi- 51 t i o n value at the end-point of integration as a func-t i o n of the i n i t i a l condition dn/dx at x = x^ . IV-2 Dependence of conversion e f f i c i e n c y on surface recom-bination v e l o c i t y . Barrier metal Au, 100 A thick. 56 O IV-3 Spectral current density vs. wavelength for 100 A of Au 58 on S i and on GaAs, with a doping density of 10 donor atoms per cm - 3 i n each case. IV-4 Variation of conversion e f f i c i e n c y with mr.tal/semiconduc- 59 tor b a r r i e r height. Semiconductor n-Si, metal taken as having the o p t i c a l properties of 100 A of gold. IV-5 Variation of open-circuit voltage, s h o r t - c i r c u i t current, 61 and f i l l factor with b a r r i e r height. Same conditions as for F i g . IV-4. IV-6 I l l u s t r a t i o n of how f i l l factor varies with <f>^  through 62 I Q as a supplement to F i g . IV-5. IV-7 Dependence of conversion e f f i c i e n c y on b a r r i e r metal 64 (gold) thickness and c o l l e c t i n g grid configuration. SiO a n t i r e f l e c t i o n coating thickness of 640 A for a Au thickness of 40 A and SiO thickness of 600 A for a Au thickness of 200 A. IV-8 The e f f e c t of semiconductor thickness or conversion 66 e f f i c i e n c y . B a r r i e r metal 100 A of Au, semiconductor doping density = l O 1 ^ atoms cm - 3. IV-9 Dependence of e f f i c i e n c y on minority-carrier l i f e t i m e . 67 B a r r i e r metal 100 A of Au, surface recombination velo-c i t y = 1 x 10^ cm sec--*-. v LIST OF TABLES Table Page I I - l Jp values to be expected for five semiconductor under 13 AMO and AMI illumination. IV-1 Comparison of results from the present numerical model 54 and a simpler model of Chapter III. Calculations refer to the Au/n-Si system with a metal thickness = 100 A and a barrier height — 0.8 eV. For the numerical model the surface recombination velocity was taken as 1.0 x 10^ cm s e c - x . A l l current densities displayed are in m A cm - 2 and doping densities i n cm - 3. IV-2 Comparison of practical (Ref. 40) and computed data for 69 uncoated, Au/n-GaAs cells under AMO conditions. IV-3 Comparison of computed data for a GaAs c e l l under AMO 70 conditions using the models of this chapter and Chapter III with = 1 x 10 1 4 cm - 3 and S = 1 x 10 7 cm sec -- 1. A-II-1 Optical properties of SiO. 102 A5II-2 Optical properties of Au. 103 A-II-3 Optical properties of Si (E i s exponent). 103 A-II-4 Optical properties of GaAs. 104 A-II-5 Diffusion coefficients and minority-carrier lifetimes 106 for Si. A-II-6 Diffusion coefficients and minority-carrier lifetimes 107 for GaAs. A-II-7 Incident solar flux density for AMO and AMI conditions 108 (X in ym and N(X) in W m~2 um _ ±). A-II-8 Resistivity of Au films as a function of thickness. 107 v i ACKNOWLEDGEMENT I would l i k e to express my g r a t i t u d e to a l l of the i n d i v i d u a l s who have a s s i s t e d me i n completing t h i s work. S p e c i a l thanks go to my s u p e r v i s o r , Dr. D.L. P u l f r e y , f o r h i s c o n t i n u a l support and encouragement through a l l phases of my work on t h i s t h e s i s . I would a l s o l i k e f d thank Dr. L. Young f o r h i s many h e l p f u l suggestions and Dr. G.F. Schrack f o r s e v e r a l h e l p f u l d i s c u s s i o n s concern-i n g the numerical a n a l y s i s performed i n developing t h i s t h e s i s . I would l i k e to thank Miss S a n n i f e r Louie f o r her e x c e l l e n t t y p i n g of t h i s t h e s i s and f o r her patience w i t h my numerous c o r r e c t i o n s . G r a t e f u l acknowledgement i s made to the N a t i o n a l Research C o u n c i l of Canada f o r f i n a n c i a l support through Postgraduate Scholarships no. 1560 and operating grant no. A7248. v i i I . INTRODUCTION Present trends i n e l e c t r i c a l energy generation i n d i c a t e increases i n both costs and concern over the n o n - r e n e w a b i l i t y of supply sources. Gas and o i l , which are e x t e n s i v e l y used to supply peak power demands are p a r t i c u l a r l y expensive, are becoming scarce when compared to long term demands, and i n the case of o i l are subject to p o l i t i c a l manipulation. Ba s e l i n e power generation using hydro, c o a l or n u c l e a r power have develop-ment problems most of which are based on environmental grounds. A l l the above f a c t o r s have l e d to the present i n t e n s i v e research on developing a l t e r n a t i v e methods of supplying energy to the world. One prominent a l t e r n a t i v e i s s o l a r energy which i s a renewable source of energy and h o p e f u l l y w i l l have minimal environmental impact. The s o l a r energy e f f o r t i s at present s p l i t i n t o two major areas c o n s i s t i n g of p h o t o v o l t a i c and thermal a p p l i c a t i o n s . A p r a c t i c a l s o l a r energy system, e.g. f o r a house, would need to use both methods w i t h thermal conversion s u p p l y i n g the c l i m a t i c c o n t r o l and hot water needs and photo-v o l t a i c conversion supplying the e l e c t r i c a l needs. In order f o r photo-v o l t a i c conversion to become widespread, a s o l a r c e l l of considerably lower cost than those a v a i l a b l e at the present time i s necessary, i . e . p r e s e n t l y a v a i l a b l e s o l a r panels u s i n g S i p-n c e l l s cost approximately $17-20 per peak watt [1] and the goals that current research e f f o r t s are aiming to achieve have been r e c e n t l y set by the N a t i o n a l Science Foundation of the U.S.A.: namely, to produce p h o t o v o l t a i c power f o r the cost of $5 per peak watt by 1979 and 50c per peak watt by 1985 [ 2 ] , The s o l a r c e l l i s a p h o t o v o l t a i c device which converts i l l u m i n a -t i o n from the sun i n t o e l e c t r i c a l energy. Incident r a d i a t i o n from the sun f a l l s on the s o l a r c e l l and some of i t i s absorbed so that e l e c t r o n -h o l e p a i r s are created w i t h i n the s o l a r c e l l m a t e r i a l . In order t o c o l -l e c t these current c a r r i e r s t h a t are generated, a f i e l d must be produced w i t h i n the c e l l to s p a t i a l l y separate the photogenerated c a r r i e r s b e fore recombination i s allowed to occur. The requirement of generation and s e p a r a t i o n of c a r r i e r s to produce a current can be s a t i s f i e d by a semiconductor m a t e r i a l which has been processed to form a diode. The modern s o l a r c e l l i s thus a semiconductor diode w i t h the j u n c t i o n as c l o s e as p o s s i b l e to the i l l u m i n a t e d surface i n order to maximize the generation of p h o t o c a r r i e r s w i t h i n the r e g i o n of b u i l t - i n f i e l d and hence i n c r e a s e the c o l l e c t i o n of photogenerated c a r r i e r s . The i l l u m i n a t e d s u r f a c e must be as transparent as p o s s i b l e to o b t a i n maximum tr a n s m i s s i o n of the i n c i d e n t photons i n t o the semiconductor m a t e r i a l i n order that photogeneration be l a r g e and a h i g h e f f i c i e n c y r e s u l t . Because of the d i l u t e nature of s u n l i g h t ( b a r e l y more than lkW m-^ can be obtained under the most favourable c o n d i t i o n s on earth) the s o l a r c e l l i s a l a r g e area device when compared to e l e c t r o n i c components u s u a l l y considered i n s o l i d s t a t e c i r c u i t r y . S o l a r c e l l s can be c l a s s i f i e d according to the nature of the j u n c t i o n b u i l t i n t o the semiconductor and the three c a t e g o r i e s are p-n j u n c t i o n s , h e t e r o j u n c t i o n s and metal-semiconductor j u n c t i o n s . The present c o m m e r c i a l l y - a v a i l a b l e s o l a r c e l l s are e i t h e r p-n j u n c t i o n devices u s i n g s i n g l e - c r y s t a l S i or h e t e r o j u n c t i o n devices u s i n g C^S-CdS. The S i and CdS s o l a r c e l l s have been produced i n reasonable q u a n t i t i e s at h i g h c o s t f o r space programs and f o r s m a l l i s o l a t e d t e r r e s t r i a l a p p l i c a t i o n s . . I n t e n s i v e research i s underway [3] i n both these areas and a l i k e l y 3. approach to a commercially v i a b l e c e l l f o r t e r r e s t r i a l a p p l i c a t i o n s would i n v o l v e the use of the semiconductor i n a t h i n f i l m form, e.g., i n the case of s i l i c o n an edge-defined f i l m - f e d ribbon [4] or a deposited p o l y -c r y s t a l l i n e f i l m [ 5 ] . Shallow d i f f u s e d j u n c t i o n formation i n these l a t t e r systems has many d i f f i c u l t i e s and i t i s probable that t h e i r widespread u t i l i z a t i o n w i l l r e q u i r e a l t e r n a t e methods of p r o v i d i n g the e l e c t r o s t a t i c inhomogeneity that i s necessary f o r p h o t o v o l t a i c a c t i o n . P o s s i b l e methods are i o n i m p l a n t a t i o n , semiconductor h e t e r o j u n c t i o n s , induced j u n c t i o n s , and Schottky b a r r i e r s . The S c h o t t k y - b a r r i e r s o l a r c e l l seems to be a t t r a c t i v e because of the s i m p l i c i t y of f a b r i c a t i o n , no d e t e r i o r a t i o n i n the semiconductor m i n o r i t y - c a r r i e r p r o p e r t i e s (as would occur w i t h the hi g h temperature processing used i n p-n j u n c t i o n s ) and l i t t l e e f f e c t from the surface recombination of photogenerated c a r r i e r s because the b u i l t - i n f i e l d i s present r i g h t up to the surface of the semiconductor. An a l t e r n a t i v e to the l a r g e - a r e a low-cost methods of producing s o l a r c e l l s mentioned above i s to use a concentrator w i t h a high e f f i c -i e n c y s o l a r c e l l [6] and S c h o t t k y - b a r r i e r s o l a r c e l l s could be u s e f u l i n t h i s a p p l i c a t i o n i f conversion e f f i c i e n c i e s of around 20% can be achieved. There are disadvantages i n v o l v e d w i t h both l a r g e - a r e a and concentrator arrays and i n terms of costs i t i s not yet c l e a r which of the systems w i l l be s u p e r i o r . However, f o r e i t h e r scheme, before the c o n s t r u c t i o n of S c h o t t k y - b a r r i e r s o l a r c e l l s can be considered, i t i s d e s i r a b l e to compute the l i k e l y performance of these s o l a r c e l l s . I t was the purpose of t h i s work therefore to perform a t h e o r e t i c a l i n v e s t i g a t i o n of the Schottky-b a r r i e r j u n c t i o n to determine i t s s u i t a b i l i t y as a means of j u n c t i o n formation i n s o l a r c e l l s . At the outset of t h i s work, there was no reported evidence on the 4. l i k e l y performance of S c h o t t k y - b a r r i e r diodes as s o l a r c e l l s . Previous work w i t h S c h o t t k y - b a r r i e r diodes i n v o l v i n g p h o t o e f f e c t s was l i m i t e d to stu d i e s on the reverse-biased diode a c t i n g as a photodetector (not photo-generator) and examples of experimental and t h e o r e t i c a l s t u d i e s are given i n Refs. [7] and [ 8 ] , The f i r s t o b j e c t i v e of the current work was thus to derive a simple l i m i t model f o r the S c h o t t k y - b a r r i e r s o l a r c e l l and use i t to p r e d i c t the maximum p o s s i b l e conversion e f f i c i e n c i e s that could be expected from S c h o t t k y - b a r r i e r s o l a r c e l l s . On the b a s i s of the r e s u l t s obtained from t h i s model,the d e s i r a b i l i t y of developing a more r e a l i s t i c model could be assessed. The l i m i t model and r e s u l t s are discussed i n Chapter I I . The r e s u l t s i n d i c a t e d that maximum conversion e f f i c i e n c i e s very close t o those p r e d i c t e d f o r p-n j u n c t i o n s were p o s s i b l e and i n the l i g h t of the favourable r e s u l t s obtained, i t was decided to derive more d e t a i l e d r e s u l t s i n order to evaluate more r i g o r o u s l y the a c t u a l e f f i c -i e n c i e s and to consider i n d e t a i l the p o s s i b l e l o s s mechanisms not con-s i d e r e d i n the l i m i t model. This work r e s u l t e d i n two models being derived. One model proved to be accurate f o r the a n a l y s i s of S i Schottky-b a r r i e r s o l a r c e l l s and i s desc r i b e d i n Chapter I I I . The other model i s more general and allows c o n s i d e r a t i o n of other semiconductors,such as GaAs, and i s presented i n Chapter IV. The two m a t e r i a l s S i and GaAs were chosen f o r more d e t a i l e d s t u d i e s because they are at present the most a c t i v e l y s t u d i e d semiconductors f o r S c h o t t k y - b a r r i e r s o l a r c e l l use. The models were a l l w r i t t e n i n the form of FORTRAN programs and run on the IBM 370/168 computer maintained by the UBC Computing Centre. The models of Chapters I I I and IV r e s i d e together i n one FORTRAN program and the e f f i c i e n c y of the p a r t i c u l a r s o l a r c e l l s t u d i e d can be c a l c u l a t e d by e i t h e r model depending on the user's choice. D e t a i l s of the program and a l i s t i n g are given i n Appendix I . The experimental values f o r the phy s i c a l p r o p e r t i e s of the s o l a r c e l l s analyzed are given i n Appendix I I . 6. II. LIMIT CALCULATIONS The limit model discussed in this chapter can best be described as a f e a s i b i l i t y study to determine i f the metal-semiconductor system i s suitable for use as a solar c e l l structure. By making several simplify-ing assumptions, a model was derived which could be easily solved to give an indication of the maximum solar energy conversion efficiencies to be expected from Schottky-barrier solar cells. From these results i t was determined that more detailed calculations should be pursued, and these are presented in Chapters III and IV. i . The Model The I-V characteristic of an illuminated Schottky-barrier solar c e l l can be expressed as I = IQ[exp(eV/kT) - 1] - I , (II-l) where I q is the diode saturation current, I the photogenerated current, V the diode voltage, T the absolute temperature, e the electronic charge and k i s Boltzmann's constant. The equivalent circuit i s assumed to con-s i s t of an ideal diode in parallel with a current source which represents the photogenerated current as shown i n Fig. I I - l . Considering the case of an n-type semiconductor, the appropriate band diagram for the metal-semiconductor system is shown in Fig. II-2, from which the reverse-saturation dark-current I can be found using o ° thermionic emission theory [9]. There are several different theories describing Ifehe current transport mechanism in a Schottky-barrier diode. By analysing experimental data [10,11], i t has been shown that for high mobility semiconductors (silicon, gallium arsenide, etc.) the thermionic emission theory provides a good approximation to the actual diode 7. O . + Fig. I I - l Simple equivalent ci r c u i t of a Schottky-barrier solar c e l l . METAL / A / 1 / . DEPLETION BULK OHMIC l CONTACT 0 x! W. L Fig. II-2 Band diagram of a metal/n-type semiconductor Schottky-barrier. solar c e l l . E f is the Fermi level and E c and Ey are the < conduction and valence band edges respectively. 8. c h a r a c t e r i s t i c s observed and d i f f u s i o n theory a very bad one. The process of t u n n e l l i n g i s al s o a p o s s i b i l i t y but gives the wrong d i r e c t i o n f o r r e c t i f i c a t i o n and has been shown t o be an important current t r a n s p o r t mechanism only f o r semiconductors w i t h a r e l a t i v e l y high doping d e n s i t y [12]. As w i l l be shown l a t e r (Chapters I I I and I V ) , a high doping d e n s i t y i s not d e s i r a b l e f o r a s o l a r c e l l and so t h i s conduction mechanism can be ignored. The d i f f u s i o n theory [9] i m p l i e s a conduction current l i m i t e d by the processes of d i f f u s i o n and d r i f t i n the space-charge region where there i s band bending i n the semiconductor, and the thermionic emission theory. [9] i m p l i e s current l i m i t i n g by the emission of e l e c t r o n s over the b a r r i e r i n t o the metal i n a manner s i m i l a r to the thermionic emission of e l e c t r o n s from a metal i n t o a vacuum. With the high m o b i l i t y semicon-ductors necessary f o r a good s t a r t i n g m a t e r i a l f o r s o l a r c e l l f a b r i c a t i o n and the r e s u l t s of_experimental measurements, the argument i s much i n favour of the thermionic emission theory. The thermionic emission theory was also used i n the models presented i n Chapters I I I and IV. The appro-p r i a t e s a t u r a t i o n current i s given by I o = a A** T2exp(-e<t>bn/kT) , (II-2) where a i s the diode area, A** the e f f e c t i v e Richardson constant (which depends on the current t r a n s p o r t mechanism considered), and ^ n the metal-semiconductor b a r r i e r h e i g h t . From an a n a l y s i s by Sze [9], the conversion e f f i c i e n c y can be w r i t t e n as n = V 2 I ,(e/kT) exp(eV /kT)(a P. ) " \ ( I I - 3 ) mp o mp i n where V i s the operating voltage at the maximum power p o i n t (see F i g . II-3) and P. i s the i n c i d e n t r a d i a t i o n power d e n s i t y . The value of V i n v J m p i s obtained by i t e r a t i v e s o l u t i o n f o r e q u a l i t y i n the equation 9. Ip = 3 0 m A ( j ) b = 0 . 8 0 e V a = 2 c m 2 T = 3 0 0 ° K Fig. II-3 I-V eharacteristic of a solar c e l l . 10. [1 + eV /kT] exp(eV /kT) = 1 + 1 / 1 . (II-4) mp mp p o Equation ( I I - 4 ) f o l l o w s from the b a s i c Eq. ( I I - l ) and i s a r r i v e d at by c o n s i d e r i n g the voltage at which d(VI)/dV = 0, i . e . , when the r e c t a n g u l a r area shown i n F i g . I I - 3 i s at a maximum. P^n» the i n c i d e n t s u n l i g h t power d e n s i t y , has a s p e c t r a l com-p o s i t i o n that depends upon atmospheric c o n d i t i o n s [13] (see F i g . II-4) and an i n t e n s i t y that depends upon l o c a l time and angle of c o l l e c t i o n . For the present c a l c u l a t i o n s only the t o t a l power density need be c o n s i --2 -2 dered and thus values of 0.135 W cm and 0.106 W cm [14] are appro-p r i a t e f o r outside the earth's atmosphere (AMO s u n l i g h t ) and the most favourable conditions on e a r t h (AMI s u n l i g h t ) i®e., at sea l e v e l w i t h the sun p e r p e n d i c u l a r to the p o i n t of observation on a c l e a r day. The AM or a i r mass number represents the e f f e c t i v e number of atmospheres the sun's r a d i a t i o n must pass through t o reach the s o l a r c e l l , where one atmosphere i s the column of a i r from sea l e v e l to outer space at normal i n c i d e n c e . The a i r mass number can be approximated as equal to sec 0 Z up to 0 Z of about 30° where 0 Z i s the z e n i t h angle measured between the angle of the sun's i n c i d e n t r a d i a t i o n and the normal t o the earth's s u r f a c e . Other con d i t i o n s such as the thickness of the ozone ilayer.and-the dust and water vapour content of the atmosphere have to be considered i n f i n d i n g the i n c i d e n t power d e n s i t y i n an a c t u a l a p p l i c a t i o n . For c a l c u l a t i o n purposes, only AMO and AMI conditions were considered. A f t e r s o l v i n g f o r V m p , s u b s t i t u t i o n of Eq. (II-2) i n t o (II-3) allows the s o l a r c e l l conversion e f f i c i e n c y to be c a l c u l a t e d . One of the important system v a r i a b l e s i s the metal-semiconductor b a r r i e r height and f o r the purpose of e v a l u a t i n g the p o s s i b l e performance of p a r t i c u l a r metal-Qi i ; l_ I i l . 1 1— —i 1 : :—I 1— . 4 . 6 • . 8 1 . 0 1 2 . 1.4 1 6 . 7,8 - 2 . 0 . 2 2 . 2 . 4 WAVELENGTH (jjm) • ' r F i g . I I - 4 S p e c t r a l f l u x density of AMO and AMI i l l u m i n a t i o n . 12. semiconductor systems <j)^ n was taken as being related to <j>m, the metal work function,by the expression •bn = C2*m + C3 • ( I I " 5 ) with the values of C2 and c^ taken from Ref. [9]» p.376, for four semi-conductors for which data i s available. These materials, S i , GaAs, GaP, and CdS, with the addition of Ge, represent semiconductors with a w e l l -developed technology and band gap appropriate for consideration as photo-v o l t a i c converters (see Fig. t l - 4 ) . In the absence of values for C2 and c^ for Ge the r e l a t i o n <j>^ n = <J>m - 4.0 was used, i.e., taking an electron a f f i n i t y of 4.0 eV for Ge and neglecting the effects of surface states and Schottky-barrier lowering. The assumptions i n the above model can be regarded as allowing calculation of a maximum possible solar energy conversion e f f i c i e n c y . In this l i m i t model I was found by assuming? that there were zero P r e f l e c t i o n losses and unity quantum e f f i c i e n c y . This means each photon incident on the solar c e l l was considered as being absorbed within the semiconductor to produce one electron-hole .pair. Furthermore, i t was assumed that each current c a r r i e r so generated was collected and there-fore contributed to I . Loferski's [15] data for the photon density available for conversion i n a semiconductor of given was used to determine the values of I under conditions of AMO and AMI illumination P and the values are given i n Table I I - l . In the calculation of e f f i c i e n c y the resistance losses were taken to be zero. With t y p i c a l solar c e l l resistances of a few tenths of an ohm, this may not seem to be much of an assumption but, i n practice, the degradation of f i l l factor [16] (defined as V I / V I , see Fi g . II-3) due to even a r e l a t i v e l y small r e s i s -mp mp oc sc tance (y IQ) can be very s i g n i f i c a n t , leading to a reduction i n e f f i c i e n c y \ 13. Semiconductor V eV J AMO mA cm J AMI mA cm Ge 0.80 77.0 60.0 S i 1.12 58.0 41.8 GaAs 1.43 42.0 28.3 GaP 2.24 16.4 9.3 CdS 2.42 13.0 7.2 Table I I - l J values t o be expected f o r f i v e semi-conductors under AMO and AMI i l l u m i n a t i o n of the s o l a r c e l l . To s o l v e the equations presented i n the above l i m i t model, a sm a l l computer program was w r i t t e n and used to''determine t h e 2 maximum conversion e f f i c i e n c i e s t h a t could be expected from Schottky-b a r r i e r s o l a r c e l l s f a b r i c a t e d from Ge, S i , GaAs, GaP and CdS. i i . R e sults and Discussion 1.- -The conversion e f f i c i e n c y was s t u d i e d f o r the f i v e semiconductors as a f u n c t i o n of b a r r i e r height and b a r r i e r - m e t a l work f u n c t i o n using Eq. (II-5) to f i n d the appropriate b a r r i e r height as discussed e a r l i e r . Also the dependence of e f f i c i e n c y on the Richardson constant, temperature, and i l l u m i n a t i o n c o n d i t i o n s was .studied f o r a p a r t i c u l a r meital-semiconductor system f o r which the temperature dependence of the b a r r i e r height was well-documented.[17]. For a given semiconductor, i t can be seen i n F i g . I I - 5 that the maximum conversion e f f i c i e n c y i n c r e a s e s w i t h b a r r i e r height and, t a k i n g 1A 15. e<j>, = E as the l i m i t i n g case, the maximum possible conversion e f f i -bn max g r ciency i s about 25% and could occur i n a semiconductor with E = 1". 4 -g 1.6 eV. This figure can be compared with about 26% which i s the calcu-lated [18] maximum conversion e f f i c i e n c y i n a homojunction solar c e l l at 20°C and refers to either GaAs or AlSb. For Schottky-barrier solar c e l l s , the conversion e f f i c i e n c y i s low i n small-band-gap materials because of the l i m i t a t i o n on <L • This i s because a low band gap implies <f>, cannot bn bn be very large ( e ^ n < E ) and this i s exponentially related to I q i n Eq. (II-2) . E f f i c i e n c y has a strong dependence on I through the.* l a t t e r effect on.:VrR. (see Eqs. (II-3) and (11-4)^ -what i s .'implied is. that 'a-'low' . Eg leads to a lower maximum t))^ which implies a larger value for I q which results i n a lower e f f i c i e n c y . The maximum conversion e f f i c i e n c y also f a l l s o f f at high values of band gap because fewer of the incident photons i n the solar spectrum have s u f f i c i e n t energy (hv > Eg) to generate the photocurrent necessary for solar c e l l operation. S i l i c o n i s of particu-l a r interest because of i t s advanced processing technology and i t appears that the maximum conversion e f f i c i e n c i e s for both homojunction and Schottky-ba r r i e r c e l l s (n-type Si) are about the same, i . e . , 22%. Some s l i g h t advantage might accrue from using p-type S i as i n this case [19] A** ^ 30 A cm - 2 °K~2, leading to n- =24.4%. * 6 max For the purpose of evaluating the possible performance of p a r t i -cular metal-seiniconductor systems, Fig. II-6 shows for fiv e semiconductors the conversion e f f i c i e n c y dependence on barr i e r metal work function fy^. The value of ^ n was calculated for the f i v e semiconductors from <j>m (as described e a r l i e r ) and the r e s u l t i n g ^ was used to f i n d the e f f i c i e n c y by the same method as before. Figure .11-6 serves only as a guide to the possible attainable conversion e f f i c i e n c i e s because for a p a r t i c u l a r F i g . I I - 6 Conversion e f f i c i e n c y versus b a r r i e r - m e t a l work f u n c t i o n f o r AMO s u n l i g h t T = 300 °K, A** = 120 A cm"2 °K-2. 17. Schottky barrier depends strongly on the semiconductor surface condition and the method of metal deposition, e.g., for the case of Cr on S i , Eq. (II-5) gives <|>kn = 0.69 eV, i.e., ^ = 0.43 eV, whereas Anderson and Delahoy [20] report <()^  in the range 0.35-1.0 eV depending on the deposi-tion method. It i s noteworthy that a d> of 1.0 eV would give a theore-bp t i c a l converson efficiency close to n J althoughc^in.practice such high max-barriersewould_probably indicate..a, non-intimate barrier contact. As the value of the effective Richardson constant A** depends not only on the mechanism of current transport being considered but also on a variety of experimental conditions, Fig. II-7 has been drawn to indicate that the conversion efficiency is relatively insensitive to this factor. Thus, the practice used in Figs. II-5 and II-6 of taking A** equal to the free electron value w i l l not introduce any serious discre-pancies. The conversion efficiency i s , however, strongly temperature dependent as is also shown in Fig. II-7, which considers the case.of a Au/GaAs diode and uses Padovani and Sumner's [17] data for the tempera-ture dependence of the J-V characteristic. In the above calculations the ideality factor y, which often appears in the I-V expression to describe experimental Schottky diode data, viz, I = IQ[exp(eV/YkT) - 1], has been taken as unity. In practice, Y i s somewhat greater than unity and providing the other parameters are not affected by the conditions that cause y to deviate from unity, the increase in y w i l l improve the efficiency of a Schottky-barrier solar c e l l . It can be concluded that Schottky-barrier solar cells are theoretically capable of a maximum solar-energy conversion efficiency very similar to that of conventional homojunction solar cells. This was 18. •Jig. I I - 7 E f f e c t of s u n l i g h t c o n d i t i o n s and the e f f e c t i v e Richardson constant on the temperature dependence of the conversion e f f i c i e n c y . S olar c e l l i s Au on GaAs (see Ref. i l T ^ f o r data used)'. 19. the r e s u l t t h a t j u s t i f i e d i n v e s t i g a t i o n of the more complicated and re a -l i s t i c models formulated i n Chapters I I I - a n d IV. I t was a l s o d e s i r a b l e to d e r ive more d e t a i l e d models i n order to a r r i v e at a b e t t e r understand-i n g of how the important e f f i c i e n c y degradation mechanisms behave and o r i g i n a t e , i n order to a r r i v e at an optimum design f o r an a c t u a l Schottky-b a r r i e r s o l a r c e l l . 20. I I I . A MODEL FOR SILICON SCHOTTKY-BARRIER SOLAR CELLS i . I n t r o d u c t i o n Recent work [21] using s i n g l e c r y s t a l p-type s i l i c o n Schottky-b a r r i e r s o l a r c e l l s has y i e l d e d conversion e f f i c i e n c i e s of 9%, yet the l i m i t c a l c u a t i o n s discussed i n Chapter I I and Ref. [22] suggest t h a t maximum a t t a i n a b l e conversion e f f i c i e n c i e s are around 25%. In order to p r e d i c t conversion e f f i c i e n c i e s r e l e v a n t to p r a c t i c a l m e t a l - s i n g l e c r y s -t a l semiconductor systems a model i s needed that accounts f o r the various l o s s mechanisms not considered i n the l i m i t c a l c u l a t i o n s , e.g., o p t i c a l r e f l e c t i o n and abs o r p t i o n , m i n o r i t y - c a r r i e r recombination w i t h i n and at the surface of the semiconductor, and s e r i e s r e s i s t a n c e . In a d d i t i o n , i t i s necessary to consider the d e t a i l s of current t r a n s p o r t w i t h i n the device and allow f o r the presence of an i n v e r s i o n l a y e r which w i l l l i k e l y e x i s t [8] under the con d i t i o n s of h i g h metal/semiconductor b a r r i e r height r e q u i r e d [22,23] f o r e f f i c i e n t s o l a r energy conversion. Such a model has been developed [24] and i s described i n Chapter IV. For the case of s i n g l e - c r y s t a l - s i l i c o n s o l a r c e l l s , i t appears t h a t recombination at the semiconductor surface and i n the i n v e r s i o n and d e p l e t i o n regions a f f e c t s the t o t a l photogenerated current only s l i g h t l y , e.g., (see Chapter IV) 15 -3 f o r a doping density of 10 donors cm , i n v e r s i o n and d e p l e t i o n region recombination lowers the photocurrent by only 0.22% and v a r y i n g the s u r -4 7 - 1 face recombination v e l o c i t y from 10 to 10 cm sec reduces the photo-current by only 0.9%. Thus, f o r the case of s i l i c o n , which i s a p a r t i -c u l a r l y important case i n view of the advanced s t a t e of the technology and understanding of t h i s m a t e r i a l , i t appears that recombination at the surface and i n the f i e l d regions of the semiconductor can be neglected. 21. This i s the assumption made i n Chapter I I I and Ref. [25]. However, recombination i n the bulk ( f i e l d - f r e e ) region of the semiconductor i s considered, as are the o p t i c a l p r o p e r t i e s of the s o l a r c e l l , the current generation w i t h i n the b a r r i e r metal f i l m , the doping d e n s i t y of the semi-conductor and the s e r i e s r e s i s t a n c e of the c e l l . The r e s u l t s presented i n t h i s chapter are f o r the g o l d / n - s i l i c o n system. i i . The Model Figure I I I - l shows the energy band diagram f o r a metal/n-semi-conductor system along w i t h the four components of the photogenerated current. J m i s the current d e n s i t y due to absorption i n the metal f i l m of r a d i a t i o n between 0.3 and 2.28 ym, i . e . , the e f f e c t i v e l i m i t s of AMO s u n l i g h t , see Arvesen e t a l . [26]. This component i s c a l c u l a t e d using Fowler's method [27] and forms a very s m a l l p a r t ( t y p i c a l l y l e s s than 1%) of the t o t a l photogenerated c u r r e n t . Fowler's r e s u l t s d e rive the r a t i o Ng/N which represents the p r o b a b i l i t y that an e l e c t r o n with.energy. E = hv above i t s thermal energy w i l l be emitted over a b a r r i e r <j> where Ng i s the t o t a l number of e l e c t r o n s per u n i t volume a v a i l a b l e w i t h s u f f i c i e n t energy to escape and N i s the t o t a l number of e l e c t r o n s per u n i t volume i n the metal. Since the b a r r i e r r m e t a l f i l m i s so t h i n 100 A ) , i t was assumed tha t a l l the a v a i l a b l e e l e c t r o n s w i l l "surmountvwthe b a r r i e r i n t o the semiconductor before the e x c i t e d e l e c t r o n i n t e r a c t s w i t h other e l e c t r o n s i n the b a r r i e r metal [28,29]. Although t h i s i s a s i m p l i f y i n g assumption, the b a r r i e r metal current J i s so s m a l l that the exact s o l u t i o n r e q u i r -m n i n g a l o t of computer a n a l y s i s (such as Monte Carlo methods [30]) was not deemed necessary. Therefore, the current J m i s given by 2.28 ym N (A) J m = •/ e N(X) (1 - R(X) - T(X)) ~ — dX , • ( I I I - l ) 0.3 ym Fig. III-2 Equivalent circuit of a solar c e l l including series resistance. 23. where R(X) i s the r e f l e c t a n c e of the s o l a r c e l l , T(X) i s the t r a n s m i t -tance i n t o the semiconductor s u b s t r a t e , and N(X) i s the i n c i d e n t s p e c t r a l photon f l u x d e n s i t y : a l l of these parameters are dependent on the wave-length X. The components and a r i s e from photon absorption w i t h i n the i n v e r s i o n and d e p l e t i o n l a y e r s r e s p e c t i v e l y . Assuming that each absorbed photon generates one c o l l e c t e d current c a r r i e r , and can be w r i t t e n as ( 2.28 ym J ± = / -e N(X) T(X) {1 - exp(-a(X)x.5>c.dX, (III-2) 0. 3 urn and 2.28 ym J d = / -e N(X) T(X) {expC-aCX^) - exp(-a(X)W) } dX, (III-3) 0.3 ym where x^ i s the i n v e r s i o n l a y e r t h i c k n e s s , W i s the depth o f the d e p l e t -i o n l a y e r and a i s the semiconductor absorption c o e f f i c i e n t . This gives a quantum e f f i c i e n c y of u n i t y f o r the i n v e r s i o n and d e p l e t i o n l a y e r s w i t h the argument being that these l a y e r s are h i g h f i e l d regions so mino-r i t y c a r r i e r recombination and surface recombination need not be c o n s i -dered f o r the case of s i l i c o n (This w i l l be j u s t i f i e d i n Chapter I V ) . The b u l k component of the photocurrent d e n s i t y i s a r r i v e d at through s o l u t i o n of the appropriate c o n t i n u i t y equation, i . e . , 2 D ^ -4 - £ - + N(X) T(X) a(X) exp(-a(X)x) = 0, ( I I I - 4 ) p , I x dx p where D i s the hole d i f f u s i o n c o e f f i c i e n t , S T the hole l i f e t i m e and p p p the excess ho l e con c e n t r a t i o n . Using the boundary co n d i t i o n s that p = 0 at x = W and at x = L allows a n a l y t i c a l s o l u t i o n of the above d i f f e r e n -t i a l equation to y i e l d 24. 2.28 ym -e a(X) N(X) T(X) L J, = / - ^ 5 — 5 • E' U«(X)L - l)exp(-a(A)W) + 0.3 ym (cTL - 1) P P 2[exp(-a(X)L) - exp (-a(X) W) (exp (^ -=-^ ) ) ] 2 } dX, ( H I - 5 ) (exp (— ) - e x p ( — ) P P where L i s the h o l e - d i f f u s i o n l e n g t h . P The t o t a l photogenerated current d e n s i t y i s given by J = J + J . + J , + J .' ( I I I - 6 ) p m x d b In e v a l u a t i n g the v a r i o u s current components, data f o r N(X) was' taken f o r the case of AMO s u n l i g h t from Arvesen e t . a l . [26], and the transmittance was computed using Rouard's treatment [31] along w i t h Salzberg's method [32] f o r the case of transmission i n t o an absorbing s u b s t r a t e . The c a l c u l a t i o n s were performed f o r one hundred wavelengths between 0.3 and 2.28 ym and i n t e g r a t i o n was done using Simpson's Rule. To f i n d the d e p l e t i o n l a y e r depth W, Poisson's equation must be s o l v e d , x.e. ,2 H f = " ? [ ND + N V e a [P(-<+ R + e V)/ k T>3» ( I I I - 7 ) dx where e i s the semiconductor p e r m i t t i v i t y , the semiconductor doping d e n s i t y , the e f f e c t i v e d e n s i t y of s t a t e s i n the semiconductor valence band and e, k, T have t h e i r usual meanings. The above equation was s o l v e d using a fourth-order Runge-Kutta method w i t h v a r i a b l e s t e p - s i z e e r r o r c o n t r o l and the boundary co n d i t i o n s V ='~^\irife at x = 0, and at x = W (which i s unknown but see S e c t i o n I I I - i i i ) the double boundary c o n d i t i o n V ='—<f>n/e and dV/dx = 0. From the r e s u l t i n g voltage p r o f i l e , i n t e r p o l a t i o n was used to o b t a i n the p o i n t at which V = - <j> /2e, i . e . , 25. the i n v e r s i o n l a y e r t h i c k n e s s . A f t e r computation of the s o l a r energy conversion e f f i c i e n c y can be c a l c u l a t e d . In t h i s model the c e l l s e r i e s r e s i s t a n c e R was s taken i n t o account and the J-V r e l a t i o n s h i p of the s o l a r c e l l can be derived from F i g . I I I - 2 t o give J = A** T 2exp(- e(j>bn/kT) [exp(e(V- - aJR s)/ykT) - 1] - J , ( I I I - 8 ) where A** i s the e f f e c t i v e Richardson constant, a i s the s o l a r c e l l a rea, f i s the i d e a l i t y f a c t o r taken here to be u n i t y and R g i s the s e r i e s r e s i s t a n c e of the s o l a r c e l l . i i i . Method of S o l u t i o n The equations used i n the model to describe the operation of the S c h o t t k y S b a r r i e r s o l a r c e l l do not lend themselves to a n a l y t i c a l s o l u t i o n s . The p r o p e r t i e s of the s o l a r c e l l that are wavelength-dependent are formed from data p o i n t s that were exp e r i m e n t a l l y determined by other researchers (values given i n Appendix I I ) . Poisson's equation used to f i n d E and V vs distance i n the i n v e r s i o n and d e p l e t i o n regions i s a non-l i n e a r second order d i f f e r e n t i a l equation and as such the s o l u t i o n cannot be found by a n a l y t i c a l methods without some s i m p l i f y i n g assumptions. These f a c t o r s n e c e s s i t a t e d the extensive use of numerical methods i n order to solve the modeling equations and a r r i v e at a value f o r the s o l a r c e l l e f f i c i e n c y . For the wavelength-dependent funct i o n s i n the model, c a l c u l a t i o n s were performed at 100 wavelengths from 0.3 ym to 2.28 ym which r e s u l t e d i n a spacing of 0.02 ym between adjacent v a l u e s . This spacing was chosen as a reasonable compromise between too f i n e a data p o i n t spacing which would r e q u i r e an excessive number of c a l c u l a t i o n s and too coarse a data •26. p o i n t spacing which would not adequately describe the d e t a i l of the v a r i -ous f u n c t i o n s w i t h respect to wavelength. A problem t h a t then presented i t s e l f was to f i n d s u f f i c i e n t data at a l l of these wavelengths f o r the o p t i c a l constants of the various f i l m s such as the a n t i r e f l e c t i o n c o a t i n g s , the b a r r i e r metal f i l m and the semiconductor s u b s t r a t e . Since t h i s much data i s not a v a i l a b l e from experimental r e s u l t s , i t was necessary t o use i n t e r p o l a t i o n on the known data p o i n t s to f i n d a l l 100 data p o i n t s necess-ary f o r the quantum y i e l d c a l c u l a t i o n s . A f t e r t r y i n g v a r i o u s curve f i t -t i n g techniques i t was decided t h a t the use of a cubic s p l i n e i n tension was the most d e s i r a b l e . The cubic s p l i n e i s very s u i t a b l e f o r f i t t i n g a smooth curve through a se t of data p o i n t s and then using the s e t of cubic f u n c t i o n s d e s c r i b i n g the data p o i n t s , the i n t e r p o l a t i o n to any poin t between the known data p o i n t s can be achieved. However, the use of a piecewise cubic s p l i n e as the i n t e r p o l a t i o n curve r e s u l t e d i n some e x t r a -neous i n f l e c t i o n p o i n t s . Hence, the s p l i n e i n ten s i o n r o u t i n e c a l l e d SMOOTH i n the UBC Computing Centre f i l e *NUMLIB was used, i t a u t o m a t i c a l l y detects the extraneous i n f l e c t i o n p o i n t s and the i n t r o d u c t i o n of " t e n s i o n " i n p a r t s of the curve y i e l d e x p o n e n t i a l f u n c t i o n s which can e l i m i n a t e the extraneous i n f l e c t i o n p o i n t s . As a r e s u l t a curve was found u s i n g SMOOTH which passed through the known data p o i n t s and enabled i n t e r p o l a t i o n to the 100 data p o i n t s f o r c a l c u l a t i o n purposes. The assumption i s that the p h y s i c a l p r o p e r t i e s being i n t e r p o l a t e d f o l l o w the smooth curves d e r i v e d by the i n t e r p o l a t i o n r o u t i n e . By using enough experimental data from the l i t e r a t u r e , the e r r o r s caused by i n t e r p o l a t i o n were minimized. The data that were processed by the subroutine SMOOTH were the r e a l and imaginary p a r t s (n and k r e s p e c t i v e l y ) of the complex r e f r a c t i v e index n* (= n-ik) f o r up to e i g h t a n t i r e f l e c t i o n c o a t i n g s , and the complex 27. r e f r a c t i v e index f o r the b a r r i e r metal f i l m . The SMOOTH subroutine was also used on the r e a l p a r t of the r e f r a c t i v e index n and the absorption c o e f f i c i e n t a f o r the semiconductor s u b s t r a t e . The values of k and a need not both be known s i n c e they can be c a l c u l a t e d from each other u s i n g the r e l a t i o n a(X)X = 4irk(X) f ( I I I - 9 ) where X i s the wavelength at which a and k are evaluated. I f a s o l a r s p e c t r a l f l u x d e n s i t y other than AMO or AMI (both of which are contained i n arrays i n the program) i s d e s i r e d , then the known values must be entered i n t o the inp u t data cards. This data i s then processed by SMOOTH to a r r i v e at a known value of s p e c t r a l f l u x d e n s i t y at each of the 100 wavelengths used f o r the c a l c u l a t i o n of quantum e f f i c i e n c y . Since the doping d e n s i t y i s allowed to vary i n the program the m i n o r i t y c a r r i e r p r o p e r t i e s (I>n» Dp> ^ n and x ) are placed i n the input data f o r doping 14 -3 19 -3 d e n s i t i e s between 1 x 10 cm and 1 x 10 cm The data i s then pro-cessed by SMOOTH and i n t e r p o l a t e d to values of log^g between 14.0 and 19.0 at a spacing of 0.1. From t h i s the values of D . D , x and x can r ° n' p' n p be found f o r any doping d e n s i t y by subsequent l i n e a r i n t e r p o l a t i o n be-tween the known values w i t h a l o g ^ ^ N ^ spacing of 0.1 found by SMOOTH. Using the o p t i c a l data the transmittance and r e f l e c t a n c e were then c a l c u -l a t e d f o r the same 100 wavelengths. Using the integrands of Eqs. "(III-2) ,• (III-3) and ( I I I - 5 ) , the s p e c t r a l current d e n s i t i e s i n the i n v e r s i o n , d e p l e t i o n and bulk regions were evaluated at each of the 100 wavelengths used. The t o t a l current density i s then found by i n t e g r a t i o n of the s p e c t r a l current d e n s i t y using Simpson's Rule to evaluate the i n t e g r a l . In order to f i n d the i n v e r s i o n and d e p l e t i o n region edges, i t was necess-ary to f i n d V and E vs x as described i n Se c t i o n i i . The s o l u t i o n to the 28. second-order n o n l i n e a r d i f f e r e n t i a l equation (Eq. ( I I I - 7 ) ) was found using fourth-order Runge-Kutta i n t e g r a t i o n . The UBC Computing Centre subroutine c a l l e d RKC (Runge-Kutta w i t h e r r o r c o n t r o l ) was used as the method of numerical i n t e g r a t i o n . Equation ( I I I - 7 ) has the boundary con-d i t i o n s V = — at x = 0 and at x = W, V = - — and %L = E = 0. This e e dx may seem l i k e one too many boundary c o n d i t i o n s to s o l v e a second order d i f f e r e n t i a l equation but the f a c t that W i s unknown i s the reason f o r the e x t r a c o n d i t i o n . The second order d i f f e r e n t i a l equation (Eq. ( I I I - 7 ) ) i s s p l i t up i n t o a system of two f i r s t order equations. A value of W was estimated as being the usual [9] d e p l e t i o n width found i n textbooks W = ( ^ 2 - ) * , ( 1 1 1 - 1 0 ) e N D where e i s the semiconductor p e r m i t t i v i t y . Since Runge-Kutta numerical dV i n t e g r a t i o n i s an i n i t i a l value problem the values of V and — are given at the estimated W. The numerical i n t e g r a t i o n then proceeds to x = 0 and a value f o r V at x = 0 i s found. This i s compared to the value c a l l e d f o r i n the boundary conditions namely V = - ^ n ^ 6 " ^ t* i e d i f f e r e n c e -4 between the c a l c u l a t e d V at x = 0 and - 4, /e i s l e s s than 1.0 x 10 bn ^bn^ e t * i e n t* i e v & l u e s °f V and E vs x are accepted as accurate enough. I f the d i f f e r e n c e i s l a r g e r , then a new value f o r W i s found (W^ +j) from the o l d value f o r W((W^) by using the i t e r a t i v e equation V, (0) + /e „ e i D bn and a new i n t e g r a t i o n from x = Wfc+1 to x = 0 i s s t a r t e d . Convergence to V(0) = - cp^n/e was q u i t e r a p i d w i t h only two or three i t e r a t i o n s r e q u i r e d to achieve the accepted accuracy. The boundary c o n d i t i o n s at x = W are s a t i s f i e d s i n c e they are used as the i n i t i a l c o n d i t i o n s f o r the numerical 29. i n t e g r a t i o n . From the voltage p r o f i l e the value of x^ was found as the value where V = - <£ /2e, that i s where the Fermi l e v e l i s i n the middle of the semiconductor band gap. The values of V and E were found f o r 100 po i n t s between x = 0 and x = W and by l i n e a r i n t e r p o l a t i o n x^ was found. The d i v i s i o n x^ i s somewhat a r t i f i c i a l i n t h i s model but i t i s the p o i n t where the semiconductor i s i n t r i n s i c and i s very important i n the general model of Chapter IV. With the s e r i e s r e s i s t a n c e term i n the J-V r e l a t i o n s h i p of Eq. ( I I I - 8 ) , a s l i g h t l y d i f f e r e n t approach became necessary i n order to a r r i v e at the maximum power operating p o i n t . The method i n Chapter I I f i n d s the maximum power operating p o i n t by s a t i s f y i n g the equation dP/dV = 0 where the output power i s P = VI. To evaluate dP/dV, I (where I = aJ) must be expressed i n terms of V but upon examination of Eq. ( I I I -8), t h i s cannot be done. However, V can be expressed i n terms of I and so the c o n d i t i o n dP/dl = 0 i s used i n s t e a d . This leads to the i t e r a t i v e s o l u t i o n of the equation " i + i + i > + % - - ^ i + I ° + I > < I I I" 1 2 ) mp o p mp o p f o r the value of I which s a t i s f i e s the e q u a l i t y . I t i s al s o true f o r mp ~t J the case of R ^ 0 that I i s not equal to I as i s the case i n the s p n sc simple model presented i n Chapter I I . I which i s the short c i r c u i t current i s found by s e t t i n g V = 0 i n Eq. ( I I I - 8 ) and by using an i t e r a -t i v e s o l u t i o n of the equation I + I + I - e l -R ( — j ~ E ) = exp( ^ S ) (111-13) o y f o r e q u a l i t y . This completes a b r i e f d e s c r i p t i o n of the problems that needed to be solved i n order to a r r i v e at a s o l u t i o n <td the equations presented 30. i n the model. The results of the model are presented for a p a r t i c u l a r metal/n-silicon Schottky-barrier solar c e l l system. i v . Results:and Discussion To demonstrate the workings of the model, the gold/n-type s i l i -con system was used. .-This arrangement gives a high b a r r i e r height of 0.8 eV [9] and some data are available on o p t i c a l properties [33] and metal r e s i s t i v i t y [34] (see Appendix I I ) . Figure II I - 3 shows the o p t i c a l performance of the system with the absorptance A i n the metal (100 A thick) being calculated from R + T + A = 1, T being the transmittance at the metal/semiconductor interface and R the reflectance of the solar cellw The i n t e r n a l quantum y i e l d of the device i s shown i n Fig. ITI-4 14 -3 -for the case of a semiconductor doping density of 10 donors cm Apparently, a l l the short wavelength (< .36 ym) photons y i e l d a collected current c a r r i e r and are absorbed i n the inversion region. Penetration to the depletion region s t a r t s around 0.37 ym and photogeneration i n the bulk s t a r t s at about 0.4 ym. The t o t a l i n t e r n a l quantum y i e l d only s t a r t s to drop below unity at about 0.6 ym, i . e . , when most of the sunlight i s absorbed i n the bulk region and electron-hole recombination begins to be s i g n i f i c a n t . This l a t t e r factor coupled with increasing transparency to the incident radiation i s responsible for the further decrease i n quantum y i e l d towards longer wavelengths. Figure 111-4 i s e a s i l y converted to a . spectral current density vs wavelength p l o t , see Fig. I I I - 5 . The fea-tures of F i g . III-4 are reproduced for the bulk and depletion components but the inversion layer component f a l l s towards the shorter wavelengths because of the low number of photons present i n this portion of the sun's spectrum. A l l three components i n Fig. ITI-5 show more structure than i n Fig. III-4 because the spectral current density r e f l e c t s the d e t a i l i n the sun's spectrum, for which data (AM0 sunlight) was taken from Ref. [26]. lOOA OF Au ON Si Uj o 0 I —« — ' ! ' — J " • ' ' . 3 -5 - 7.0 • /.5 • • 2.0 2 . 3 \ WAVELENGTH (um). . ^ F i g . I I I - 3 O p t i c a l c h a r a c t e r i s t i c s of the g o l d / s i l i c o n system. R e f r a c t i v e index of gold (bulk) from Ref. 33; r e f r a c t i v e index of s i l i c o n from Refs. 51 and 9. 32. 100A Au ON n-Si N =1Q14 cm~3 WAVELENGTH fjum) Fig. HI-4. I n t e r n a l quantum y i e l d vs. wavelength f o r a g o l d / n - s i l i c o n s o l a r c e l l . 30 h WAVELENGTH '(urn) Fig. III-5 Spectral current density vs. wavelength for a gold/n-silicon solar c e l l under conditions of AMO sunlight (Ref. 26). 34. To i l l u s t r a t e the dependence of the spectral current density on semicon-ductor doping density, Fig. III-6 has been drawn for the inversion layer response over the doping density range of interest i n solar c e l l f a b rica-t i o n (the lower l i m i t being determined by high series resistance and the upper l i m i t by tunneling [12]). The current i s largest for the lowest doping density case because the inversion layer increases i n width as decreases. Also, because the longer wavelengths are absorbed deeper into the material, the spectral response curves s h i f t to lower wavelengths as increases. The various components of the t o t a l photogenerated current are shown i n Fig. III-7. Because of the decrease i n width with doping density of the inversion and depletion layers, the current from these two regions f a l l s as increases. The bulk component i n i t i a l l y increases with doping density as the bulk/depletion layer interface moves closer to the metal/ semiconductor interface, i . e . ^ the bulk region begins to encompass the region of high o p t i c a l absorption and hence photocurrent generation. 15 -3 However, further increase i n (above about 2 x 10 donors cm ) leads to a deterioration i n minority c a r r i e r transport properties ( l i f e t i m e , d iffusion length, mobility) and thus the bulk photocurrent begins to f a l l . The net result i s , then, a decrease i n t o t a l photocurrent with increasing doping density. I t i s , at f i r s t sight perhaps, surprising that the bulk photocurrent forms such a large component of the t o t a l photocurrent. However, th i s i s a consequence of the r e l a t i v e l y low absorption c o e f f i c i e n t for s i l i c o n and the fact that e f f i c i e n t c a r r i e r removal by the e l e c t r i c f i e l d i n the inversion and depletion regions causes a large c a r r i e r concentration gradient allowing bulk photogenerated holes to diffuse towards the metal/semiconductor interface, see Fig. I I I - l . TOO A Au ON n-Sf ND cm~3 xi jJ.IT) TO14 0.77 TO15 0.23 { W16 0,07 1017 0.02 .8 . 1.0 WAVELENGTH . (JJ m) 1.2 7. F i g . I I I - 6 Dependence of the i n v e r s i o n l a y e r component of the s p e c t r a l current d e n s i t y oh semiconductor doping density. 36. 100A Au ON n-S'i 10 10 10 , 10 •DOPING; DENSITY (cm'3) F i g . I I I - 7 Dependence of the va r i o u s components of the photocurrent de n s i t y on semiconductor doping d e n s i t y . 37. •Figures III-4 to III-7 were computed for the case of a gold O f i l m thickness of 100 A: as i s to be expected, increasing the metal f i l m thickness leads to a reduction i n transmittance and hence photogenerated current, see Fig. III-8. The solar energy conversion e f f i c i e n c y , however, passes through a maximum on varying metal f i l m thickness when the c e l l series resistance i s taken into account, see Fig. I I I - 9 . Handy's [35] method, as corrected by Sahai and Milnes [36] was used to estimate R for s the cases of 5-, 10-, and 20-fingered grids. Grid and busbar dimensions, contact resistances etc. are detailed i n Fig. I l l - 1 0 and were taken from Ref. [35]. For the cases of non-zero R g the curves i n i t i a l l y increase with f i l m thickness owing to the rapid decrease i n f i l m sheet resistance O and hence R . However, above a certain thickness (50-70 A) the f i l m s sheet resistance ceases to contribute strongly to R and the reduction i n s transmittance as the f i l m becomes thicker causes the photocurrent to f a l l . (Fig. III-8) and the conversion e f f i c i e n c y t-6 drop. Although increasing the number of grids w i l l decrease R for a given metal thickness, the conversion e f f i c i e n c y w i l l not necessarily increase as a larger number of grids means a lower device active surface area. This interplay, then, of sheet resistance, transmittance and active surface area leads, i n the present example, to the 20-fingered grid giving best performance for f i l m thicknesses up to 45 A, the 10-fingered grid being superior i n the range 45-65 A, and the 5-fingered grid giving highest e f f i c i e n c i e s thereafter. This model allows more accurate prediction of the performance of metal/single-crystal s i l i c o n diodes as photovoltaic solar energy con-verters. The model considers the o p t i c a l properties of the device; the photogenerated current i n the metal and i n the inversion, depletion and bulk regions of the semiconductor; minority-carrier recombination i n the o 0\ I 1 — _ l I L _ 0 50 100 150 200 25.0 METAL FILM THICKNESS (A) F i g . I I I - 8 Dependence of t o t a l photocurrent d e n s i t y on gold f i l m t h i c k n e s s . R e f r a c t i v e index of gold (bulk data) taken from Ref. 33. 39. O N o UJ O U j o CO o 4.0 3.5 3.0 25 2.0 15 10 0.5 A/o.'OF ACTIVE A Rs AT GRIDS (relative) T\ max ,0 . 7 — '• 5 .855 .84 / m .810 ' .77 /20 .720 .67 NQ =10 cm : 01 0 50 WO 150- 200 ME TAL FIL M THICKNESS (A ) 250 300 F i g . I I I - 9 Dependence of AMO s u n l i g h t conversion e f f i c i e n c y on metal f i l m t h i c k n e s s and c o l l e c t i o n g r i d c o n f i g u r a t i o n . R e f r a c t i v e index of gold (bulk data) taken from Ref. 33; g r i d dimensions taken from Ref. 35. 4p. 2 L - s ^ AV t = 0.02cm SOLAR CELL CONTACT' GRID CONFIGURATION No. OF GRIDS ACTIVE AREA (cm2) ; S (cm) r3 (cm) . e (degs.) • 5 1.71 0.4 0.26724 36.81 10 1.62 0.2 0.13762 36.00 20 1.44 0.1 0.06977 35.63 Rj -R2 -R3 -R 6 -R' = R 8 0.002 _n_ 0.4 M~ ? B U L K L / A -N-0.08SL. OJX (A=2cm2) FOR AN EXPLANATION OF THE SYMBOLS USED SEE REFERENCE 35. F i g . 111-10 D e t a i l s of g r i d s t r u c t u r e s and r e s i s t a n c e model taken from Ref. .35. 41. f i e l d - f r e e regions of the semiconductor; series resistance of the solar c e l l . Calculations are presented for the gold/n-type s i l i c o n system. I t i s shown that: the largest component of the photogenerated current stems from photon absorption i n the bulk region of the semiconductor; the photo-current decreases with semiconductor doping density and metal f i l m thick-ness; there i s an optimum metal f i l m thickness as regards conversion e f f i c i e n c y ; increasing the number of c o l l e c t i n g grids can reduce the actual conversion e f f i c i e n c y . From the above calculations performed on a t y p i c a l structure for a s i l i c o n Schottky-barrier solar c e l l i t i s clear that the g o l d - s i l i c o n combination does not show much promise for a high e f f i c i e n c y solar c e l l . The maximum AMO e f f i c i e n c y found for the gold/n-Si solar c e l l was only 3% when series resistance was taken into account. However, with an i n -crease of b a r r i e r height above 0.8 eV and provision of an a n t i r e f l e c t i o n coating, i t i s possible to increase the e f f i c i e n c y considerably. The highest b a r r i e r height recorded i n S i i s 0.9 eV for the hafnium/p-type Si system [37]. However, there i s reason to believe [38] that this value i s greatly i n error and thus i t appears that the P t - S i / S i system yields the highest value of (j ^ ( i . e . , 0.85 eV) . I f we assume that this system has the same o p t i c a l properties as the Au/Si combination considered i n this chapter, then the b a r r i e r height of 0.85 eV would result i n an ef f i c i e n c y of approximately 4.5%. The addition of an a n t i r e f l e c t i o n coating would double this value to 9% which would result i n a respectable solar c e l l as regards e f f i c i e n c y . However, there would s t i l l be a l i m i -tation on the open c i r c u i t voltage which would be about 400 mV compared to high-quality p-n junction solar c e l l s that achieve 650 mV open c i r -c u i t voltage. Thus, although the use of s i l i c o n i s a t t r a c t i v e i n Schottky-b a r r i e r s o l a r c e l l work because of the well-developed technology and a v a i l a b i l i t y of t h i s m a t e r i a l , i t appears to be only of l i m i t e d s u i t a -b i l i t y f o r t e r r e s t r i a l use on account of the expensiveness of the metals such as Au, P t , and perhaps Hf which seem to be necessary to get the moderate e f f i c i e n c i e s of 9-10%. As found before i n Chapter I I , what i s c l e a r l y needed f o r a h i g h - e f f i c i e n c y Schottkyr-barrier s o l a r c e l l i s a l a r g e b a r r i e r h e i g h t. The p o s s i b i l i t y of a t t a i n i n g l a r g e r b a r r i e r heights than reported above e x i s t s w i t h semiconductors of h i g h e r band gap than s i l i c o n . GaAs i s a p a r t i c u l a r example worthy of c o n s i d e r a t i o n , see F i g . I I - 5 , however, the a n a l y s i s of a S c h o t t k y - b a r r i e r s o l a r c e l l f a b r i -cated from GaAs r e q u i r e s a more general model than used i n t h i s chapter, i . e . , one that does not assume zero surface recombination v e l o c i t y and no m i n o r i t y - c a r r i e r recombination i n the i n v e r s i o n and d e p l e t i o n regions. As a r e s u l t , the model presented i n Chapter IV was developed i n order to make more accurate p r e d i c t i o n s of the e f f i c i e n c i e s to be expected from S c h o t t k y - b a r r i e r s o l a r c e l l s f a b r i c a t e d from semiconductors other than S i , p a r t i c u l a r l y GaAs. 43. IV. GENERAL MODEL i . I n t r o d u c t i o n The e a r l y l i m i t c a l c u l a t i o n s of Chapter I I neglected l i k e l y l o s s mechanisms such as photon r e f l e c t i o n , photon absorption outside the semiconductor d e p l e t i o n r e g i o n , c a r r i e r recombination i n the semiconduc-t o r and at the metal-semiconductor i n t e r f a c e , and c e l l s e r i e s r e s i s t a n c e . A more d e t a i l e d model t a k i n g i n t o account the above f a c t o r s , w i t h the exception that only c a r r i e r recombination i n the f i e l d - f r e e r e gion of the semiconductor was considered, was presented i n Chapter I I I and has been found adequate f o r the a n a l y s i s of s i l i c o n S c h o t t k y - b a r r i e r s o l a r c e l l s . However, a general a n a l y s i s , and p a r t i c u l a r l y one that i s a p p l i c a -b l e t o m a t e r i a l s w i t h high absorption c o e f f i c i e n t s and surface recombina-t i o n v e l o c i t i e s , i s r e q u i r e d . I t i s the purpose of t h i s chapter to present such a model. The model described here considers the o p t i c a l p r o p e r t i e s of the s o l a r c e l l , s e r i e s r e s i s t a n c e and a c t i v e surface area e f f e c t s , and c a r r i e r recombination i n a l l regions of the c e l l . In a d d i t i o n c a l c u l a -t i o n s are made f o r d i f f e r e n t semiconductor thicknesses and m i n o r i t y -c a r r i e r l i f e t i m e s so that some i n f o r m a t i o n may be gained as to the p o s s i -b l e performance of t h i n - f i l m s o l a r c e l l systems which are of p o t e n t i a l importance i n t e r r e s t r i a l p h o t o v o l t a i c s . The c a l c u l a t i o n s performed to i l l u s t r a t e the operation of the model are based on both s i l i c o n and g a l l i u m arsenide data, thus a l l o w i n g comparison of these two m a t e r i a l s which, at present, represent the most a c t i v e l y s t u d i e d semiconductors f o r S c h o t t k y - b a r r i e r s o l a r c e l l use. i i . The Model Figure I I I - l shows the energy band diagram f o r the metal/ 44. n-semiconductor s o l a r c e l l which i s used i n t h i s s e c t i o n to i l l u s t r a t e the proposed model. Besides the usual d e p l e t i o n and b u l k regions i n the semiconductor, an i n v e r s i o n region i s shown as t h i s w i l l l i k e l y be present i n the h i g h - b a r r i e r - h e i g h t systems necessary f o r e f f i c i e n t s o l a r energy conversion [22]. The t o t a l photocurrent d e n s i t y i s thus given by J = J + J + J , + J , (IV-1) p m i d b where the various current components are depicted i n F i g . I I I - l and a r i s e from photon absorption i n the metal, i n v e r s i o n r e g i o n , d e p l e t i o n r e g i o n , and b u l k region r e s p e c t i v e l y . The treatment presented i n Chapter I I I i s used to c a l c u l a t e J m and to determine the p o t e n t i a l and f i e l d p r o f i l e s i n the i n v e r s i o n and d e p l e t i o n regions. From the l a t t e r , the i n v e r s i o n - l a y e r t h i c k n e s s x_^  and d e p l e t i o n - l a y e r depth W f o l l o w . The components of the photocurrent generated w i t h i n the semi-conductor are computed by numerical i n t e g r a t i o n of the c o n t i n u i t y equa-t i o n s over the whole of the semiconductor. In the i n v e r s i o n l a y e r , the appropriate d i f f e r e n t i a l equation i s [39] d 2n _ -E(x) dn .1 dE(x) , JL_. " G(x) , T„ ,v dx t t n - n where n i s the excess m i n o r i t y c a r r i e r c o n c e n t r a t i o n , E i s the e l e c t r o -s t a t i c f i e l d , V i s the thermal voltage kT/e, D i s the e l e c t r o n d i f f u s -t n i o n constant, and L n i s the e l e c t r o n d i f f u s i o n l e n g t h . The c a r r i e r gen-e r a t i o n term i s evaluated using Simpson's Rule computation of the i n t e g r a l 2.28 ym G(x) = / a(A) T( A) N(A) exp[-a(A)x] dA , (IV-3) 0.3 ym where a(A) i s the semiconductor absorption c o e f f i c i e n t , N(A) i s the i n c i -dent photon f l u x , and T(A) i s the transmittance i n t o the semiconductor, the l a t t e r being computed as described e a r l i e r i n Chapter I I I . The i n t e -g r a t i o n l i m i t s i n Eq. (IV-3) r e f e r t o the extremes o f the s o l a r r a d i a t i o n , taken here to be the AMO conditions as measured by Arvesen et a l . [26]. Equation (IV - 2 ) , a f t e r s u b s t i t u t i o n of Eq. (IV-3), i s solved using a do u b l e - p r e c i s i o n fourth-order Runge-Kutta i n t e g r a t i o n r o u t i n e w i t h v a r i a b l e - s t e p - s i z e e r r o r c o n t r o l . The boundary c o n d i t i o n s are dn ,S E(0)- Jm _ _ „ , T T T d x ~ (D V ) n " eD - = 0 a t x = 0 » n = 0 at x = x ± , (IV-4) n t n where S i s the surface recombination v e l o c i t y . The m i n o r i t y - c a r r i e r cur-rent density c r o s s i n g the i n v e r s i o n l a y e r edge i s then assumed c o l l e c t e d and i s c a l c u l a t e d from J . = eD . (IV-5) 1 n dx 1 x = x. 1 In the d e p l e t i o n l a y e r the c o n t i n u i t y equation f o r holes i s given by d 2p _ E(x) dp .1 dE(x) 1 G(x) f 7 2 v d x + (v + T 2* — > ( I V _ 6 ) dx t t L p P where D i s the hole d i f f u s i o n constant, L the hole d i f f u s i o n l e n g t h , p p and p the excess m i n o r i t y - c a r r i e r concentration. In the f i e l d - f r e e b u l k region the c o n t i n u i t y equation reduces to i!p_ = _P_ _ GOEI (IV-7) , 2 T 2 D K ' dx L p P Equations (IV-6) and (IV-7) are solved together by the same i n t e g r a t i o n r o u t i n e as described above f o r the i n v e r s i o n l a y e r . The boundary c o n d i t i o n p = 0 at x = x, and x = L i s used, and the d e p l e t i o n and bulk regions are l i n k e d by assuming that both p(x) and dp/dx are continuous at x = W. Thus the hole m i n o r i t y c a r r i e r current d e n s i t y c r o s s i n g the i n v e r s i o n l a y e r edge from the b u l k and d e p l e t i o n l a y e r s i s given by J , + J. = -eD &.\ • (IV-8) d b p dx' x = x. 1 The t o t a l photocurrent d e n s i t y can thus be c a l c u l a t e d from Eqs. ( I V - 1 ) , ( I V - 5 ) , and (IV-8). Once J i s evaluated, the s o l a r c e l l conversion P e f f i c i e n c y can be computed as d e t a i l e d i n Chapter I I I . In t h i s c a l c u l a -t i o n the s o l a r c e l l s e r i e s r e s i s t a n c e i s taken i n t o account u s i n g the method from Handy [35] as modified by Sahai and Milnes [36]. The J-V r e l a t i o n s h i p i s taken as before to be J = A** T 2 exp(- <k/V.)' (exp[(V - aJR )/yV.] - 1} - J , (IV-9) D t S t p where the various parameters above were described f o l l o w i n g Eq. ( I I I - 8 ) . The reverse s a t u r a t i o n current d e n s i t y i s taken as j = A** T 2 exp(- <k/VJ , (IV-10) O D t i n the J-V r e l a t i o n s h i p given by Eq. (IV-9). For the metal/p-type semiconductor system, the above model was used but w i t h an n-type i n v e r s i o n l a y e r described by the hol e m i n o r i t y -c a r r i e r c o n t i n u i t y equation and a p-type d e p l e t i o n and bulk region des-c r i b e d by the e l e c t r o n m i n o r i t y - c a r r i e r c o n t i n u i t y equation. The appro-p r i a t e p o t e n t i a l and f i e l d p r o f i l e s were computed as before but using the r e l e v a n t metal/p-type Schottky diode boundary c o n d i t i o n s . i i i . Method of S o l u t i o n The numerical s o l u t i o n s f o r the model presented i n Chapter I I I were a l s o used f o r t h i s model to such an extent that both models were j o i n e d together i n t o one program (see Appendix I ) . The methods used f o r i n t e r p o l a t i n g the o p t i c a l p r o p e r t i e s of the s o l a r c e l l to wavelength 47. values used f o r c a l c u l a t i o n and f o r i n t e r p o l a t i n g the semiconductor m i n o r i t y - c a r r i e r p r o p e r t i e s to s p e c i f i c doping d e n s i t y values remain i d e n t i c a l to those described i n Chapter I I I . The methods f o r c a l c u l a t i n g the s e r i e s r e s i s t a n c e and f i n d i n g the s o l a r c e l l maximum power op e r a t i n g p o i n t are a l s o i d e n t i c a l to those of the previous model. Also the current component from absorption of l i g h t i n the b a r r i e r metal i s obtained from the same equations as before. The d i f f e r e n c e between t h i s model and the simp l e r model of Chapter I I I l i e s s o l e l y i n the method of e v a l u a t i n g the photogenerated current d e n s i t y . To evaluate the photogenerated current i n the semiconductor, the voltage and f i e l d p r o f i l e s are c a l c u l a t e d using Runge-Kutta i n t e g r a -t i o n as described i n Chapter I I I . The values of x^ and W are found as before and are important f o r the i n t e g r a t i o n of Eqs. (IV-2) and (IV-6) which are the c o n t i n u i t y equations. From t h i s the values o f V and E at 101 p o i n t s between x = 0 and x = W are evaluated. A f i n e enough spacing was obtained by using 101 p o i n t s between x = 0 and x = W to a c c u r a t e l y describe V< and E as a f u n c t i o n of x. I f a value of E at an x value not equal to one of the 101 known p o i n t s i s needed, (such as during the nu-m e r i c a l i n t e g r a t i o n of Eq. (IV-2) or ( I V - 6 ) ) , then l i n e a r i n t e r p o l a t i o n i s performed to obtain the d e s i r e d value from the two known values on e i t h e r side of x. Equation (IV-3) i s i n t e g r a t e d using Simpson's Rule i n t e g r a t i o n to f i n d the m i n o r i t y - c a r r i e r generation as a f u n c t i o n of x i n the semiconductor. The value of G(x) i s found f o r 101 values of x between x = 0 and x = x., f o r 101 values of x between x = x. and x = W l l and f o r 10,001 values of x between x = W and x = L (the p o s i t i o n of the back ohmic contact of the s o l a r c e l l ) . To f i n d G(x) at any p o i n t x, l i n e a r i n t e r p o l a t i o n between the known values of G(x n) on e i t h e r s i d e of 48. the d e s i r e d value of G(x) i s used. The reason f o r using 10,001 p o i n t s f o r the bulk region compared to 101 p o i n t s f o r each of the i n v e r s i o n and d e p l e t i o n regions i s t h a t the depth of the d e p l e t i o n region i s t y p i c a l l y only a few ym whereas the s o l a r c e l l depth can e a s i l y be 250 ym. The l a r g e number of data p o i n t s are necessary i n order to achieve a hig h enough de n s i t y of known values of G(x) to describe the continuous func-t i o n w e l l enough by l i n e a r i n t e r p o l a t i o n . A f t e r V, E and G are evaluated as f u n c t i o n s of x, then numerical i n t e g r a t i o n of the c o n t i n u i t y equations can be used to f i n d J . P To evaluate J , the m i n o r i t y - c a r r i e r c o n t i n u i t y equations are P solved i n t h e i r r e s p e c t i v e regions of the s o l a r c e l l . Since x = x^ i s the p o i n t where the Fermi l e v e l i s i n the middle of the band gap (semi-conductor i s e f f e c t i v e l y i n t r i n s i c ) , then x^ i s the d i v i s i o n p o i n t between n-type and p-type regions. Taking a s o l a r c e l l f a b r i c a t e d from n-type semiconductor m a t e r i a l as an example then the i n v e r s i o n region ( i n v e r t e d w i t h respect to E^) w i l l be considered as a p-type region and the deple-t i o n and bulk regions as n-type. When'electron-hole p a i r s are generated by the i n c i d e n t r a d i a t i o n i n the i n v e r s i o n l a y e r , the current generated depends on how many m i n o r i t y c a r r i e r s ( e l e c t r o n s ) cross x = x ^ i n t o the d e p l e t i o n and bulk regions where they became m a j o r i t y c a r r i e r s and are considered c o l l e c t e d and the r e f o r e c o n t r i b u t e to J . The same i s tr u e P f o r the m i n o r i t y c a r r i e r s (holes) generated i n the d e p l e t i o n and bulk regions which are p u l l e d by the E f i e l d (holes from the bulk move by d i f f u s i o n along a hole gradient caused by the E f i e l d ) i n t o the i n v e r s i o n region and c o n t r i b u t e to . This r e q u i r e s the s o l u t i o n of the e l e c t r o n c o n t i n u i t y equation between x = 0 and x = x^ and the s o l u t i o n o f the hole c o n t i n u i t y equation between x = and x = L. The c o n t i n u i t y equation f o r the i n v e r s i o n l a y e r (Eq. IV-2) i s sol v e d u s i n g d o u b l e - p r e c i s i o n fourth-order Runge-Kutta numerical i n t e g r a -t i o n w i t h v a r i a b l e - s t e p - s i z e e r r o r c o n t r o l . The subprogram subroutine used i s c a l l e d DRKC and i s maintained by the UBC Computing Centre. The problem i n using Runge-Kutta or f o r that matter any numerical i n t e g r a t i o n i s that the methods are appropriate to i n i t i a l value problems. Thus, f o r a second-order d i f f e r e n t i a l equation such as Eq. (IV-2), the value of the f u n c t i o n and i t s f i r s t d e r i v a t i v e must both be known at the s t a r t i n g p o i n t before numerical i n t e g r a t i o n can be invoked to f i n d the s o l u t i o n to the d i f f e r e n t i a l equation. However, Eq. (IV-2) must be solved to the set of boundary c o n d i t i o n s given by Eq. (IV-4) and t h i s r e s u l t s i n a boundary value problem not d i r e c t l y a p p l i c a b l e to numerical i n t e g r a t i o n . A shooting method was developed i n order to solve t h i s type of problem w i t h regards to t h i s p a r t i c u l a r a p p l i c a t i o n . The b a s i s of t h i s shooting method was to s t a r t at one boundary and create an i n i t i a l value problem by t a k i n g one of the known boundary c o n d i t i o n s as an i n i t i a l c o n d i t i o n and guessing at the other i n i t i a l c o n d i t i o n (second-order system) and use t h i s as the s t a r t i n g p o i n t of i n t e g r a t i o n . The d i f f e r e n t i a l equation i s then n u m e r i c a l l y i n t e g r a t e d to the p o s i t i o n of the other boundary con-d i t i o n which i s taken as the end p o i n t of i n t e g r a t i o n . The values f o r the f u n c t i o n and or the f i r s t d e r i v a t i v e are then compared to the boun-dary c o n d i t i o n of the d e s i r e d s o l u t i o n and a new guess i s made f o r the unknown i n i t i a l c o n d i t i o n at the s t a r t i n g p o i n t . By re p e a t i n g t h i s process enough times, the c o r r e c t s o l u t i o n to the d i f f e r e n t i a l equation which s a t i s f i e s the boundary c o n d i t i o n s i s obtained. In the f o l l o w i n g the method used to obt a i n the new value of the i n i t i a l c o n d i t i o n f o r the next i n t e g r a t i o n i s described together w i t h the r e q u i r e d stopping r u l e s f o r when the c o r r e c t s o l u t i o n i s obtained to the d e s i r e d accuracy. For the s o l u t i o n of Eq. (IV-2) to the boundary c o n d i t i o n s of Eq. (IV-4) the i n t e g r a t i o n was s t a r t e d at x = x^. The value of dn/dx i s unknown but from experience w i t h s o l v i n g t h i s problem was found to l i e between 3 5 1 x 10 and 1 x 10 i n numerical v a l u e . In terms of conc e n t r a t i o n -3 -1 gradients the u n i t s are (mm/10) (ym) , i . e . , the concentration n i s expressed i n e l e c t r o n s per tenth of a cubic m i l l i m e t e r volume and the gradient i s the concentration change per ym. The unusual u n i t s of measure f o r the concentration and concentration gradient were necessary to c o r r e c t l y s c a l e the problem f o r the numerical i n t e g r a t i o n . The value 3 of 1 x 10 f o r dn/dx at x = x^ together w i t h n = 0 i s used to s t a r t the f i r s t i n t e g r a t i o n . The numerical i n t e g r a t i o n i s done from x = x^ to x = 0 and the values of n and dn/dx at x = 0 obtained from the i n t e g r a -t i o n are s u b s t i t u t e d i n t o the boundary c o n d i t i o n at x = 0 using Eq. (IV-4). The e v a l u a t i o n of t h i s equation u s i n g n and dn/dx at x = 0 w i l l only r e s u l t i n zero, as c a l l e d f o r by the boundary c o n d i t i o n , i f the c o r r e c t value o f dn/dx at x = x^ i s guessed the f i r s t time. The value o f the boundary c o n d i t i o n obtained i s saved i n the program and f o r the purpose of d i s c u s s i o n we w i l l c a l l i t BC^. The next step i s to set dn/dx at x = x^ to 1 x 10"* and together w i t h n = 0 at x = x^ i n t e g r a t e again to x = 0. The new values found f o r n and dn/dx are then s u b s t i t u t e d i n t o the boundary c o n d i t i o n of Eq. (IV-4) to a r r i v e at BC^, From experience obtained by t r y i n g v a r i ous dn/dx values at x = x^ the boundary c o n d i t i o n at x = 0 (BC) was found to be a monotonic f u n c t i o n of dn/dx at x^ of the form i l l u s t r a t e d by F i g . IV-1. From the two values of dn/dx at x^ and the r e s u l t a n t BC^ and BC2 a new value f o r dn/dx at x^ i s found using l i n e a r i n t e r p o l a t i o n to BC = 0 from the known s t a r t i n g v alues. From F i g . IV-1 I l l u s t r a t i o n of the dependence of the boundary c o n d i t i o n value at the end-point of i n t e g r a t i o n as a f u n c t i o n of the i n i t i a l c o n d i t i o n dn/dx at x = X J . t h i s an i t e r a t i v e equation f o r the next value of dn/dx at x = x, can be found based on the previous two values f o r the boundary value at x = 0 ( c a l l e d BC) and the as s o c i a t e d values of dn/dx at x = x^. This i s i l l u s -t r a t e d i n F i g . IV-1 and the equation f o r the next dn/dx at x = x^ i s BC. ( f k x . ) - f k x . ) ) . He dx 1 ' dx x . ' dn * \ an, N k. k -1 1 1 X dx" ( xi> = dx" ( xi> B t - BCL , ( I V _ : L 1 ) k+1 k k k-1 This process continues u n t i l one of three stopping r u l e s or c o n d i t i o n s i s s a t i s f i e d . The f i r s t c o n d i t i o n t e s t e d i s whether the value of BC i s l e s s than u n i t y . Since the terms of Eq. IV-4 f o r the boundary c o n d i t i o n a a t x = 0 are of the order 1 x 10^^ a d i f f e r e n c e between them of l e s s than 1.0 i s considered s u f f i c i e n t l y , accurate to r e s u l t i n the c o r r e c t s o l u t i o n to the d i f f e r e n t i a l equation. I f t h i s t e s t i s not met, the d i f f e r e n c e between the l a s t two values of dn/dx at x = x, i s compared and i f i t i s l e s s than 2 x 10 ^ times the l a s t value of dn/dx at x = x. then dn/dx I i s not changing s i g n i f i c a n t l y and the l a s t r e s u l t i s taken as c o r r e c t . As a" s a f e t y measure the number of i n t e g r a t i o n s i s not allowed past 30 as a f i n a l stopping r u l e i n order to prevent the program from t r y i n g i n d e -f i n i t e l y to f i n d a s o l u t i o n to no a v a i l . Normally about four i t e r a t i o n s are a l l t h a t i s r e q u i r e d to a r r i v e at the s o l u t i o n . Using Eq. (IV-5) the current J . i s found and t h i s i n c l u d e s 3T s i n c e the l a t t e r i s i n t r o -l m duced i n the boundary c o n d i t i o n as an i n j e c t e d current at x = 0 (Eq. ( I V - 4 ) ) . S i m i l a r l y , the boundary value problem i s solved f o r Eqs. (IV-6) and (IV-7) by the same method, i n t e g r a t i n g from x = x_^  to x = L. I n t e -g r a t i o n i s a c t u a l l y s t a r t e d at x = x^ and ends at x = W usi n g Eq. ( I V - 6 ) , and the values of p and dp/dx at x = W are put i n as the i n i t i a l c o n d i -t i o n s f o r Eq. (IV-7). I n t e g r a t i o n then continues to x = L t o f i n d the value of p at x = L and hence allow comparison to the d e s i r e d boundary 53. value of zero. This r e s u l t s i n the e v a l u a t i o n of J , + J, from Eq. (IV-8) d b once the c o r r e c t value of dp/dx at x = x. i s found t h a t gives a s o l u t i o n to the two l i n k e d d i f f e r e n t i a l equations that i s c o n s i s t e n t w i t h the boundary conditions p = 0 at x = x^ and x = L. Once and + are known they are combined to y i e l d the t o t a l photogenerated current . The accuracy of the numerical i n t e g r a -t i o n was checked by n o t i n g t h a t Eq. (IV-7) can be solved a n a l y t i c a l l y (as was done i n Chapter I I I f o r S i ) by usin g the boundary c o n d i t i o n s p = 0 at- x = W and x = L. Equation (IV-7) was so l v e d n u m e r i c a l l y u s i n g the above boundary co n d i t i o n s and the r e s u l t a n t bulk current d e n s i t y was evaluated by J, = -eD ^ | . (IV-12) b p dx 1 I T x = W The values obtained above by numerical i n t e g r a t i o n were compared to the values of the bulk current c a l c u l a t e d by the a n a l y t i c a l method described 14 i n Chapter I I I . This t e s t was done f o r four doping d e n s i t i e s (N^ = 10 , 10"^, 10"^ and 1 0 ^ cm ^) and good agreement to four s i g n i f i c a n t d i g i t s was obtained f o r the values of J, i n each case. The numerical i n t e g r a -b t i o n of the c o n t i n u i t y equations by DRKC was then accepted w i t h c o n f i -dence as g i v i n g an accurate s o l u t i o n to the d i f f e r e n t i a l equations of t h i s model. i v . R e s ults and Di s c u s s i o n A. Model Check I t would be expected that S c h o t t k y - b a r r i e r s o l a r c e l l s made from semiconductors w i t h low absorption c o e f f i c i e n t s , and f o r which s u r -face recombination v e l o c i t i e s are low, would be l i t t l e a f f e c t e d by c a r r i e r recombination i n the surface and near-surface regions. S i l i c o n i s an 54. Simple Model J m J. l J d J b v i o 1 4 0.041 6.823 5.335 12.-538 i o 1 5 0.041 3.2-95 3.826 15.298 i o 1 6 0.041 1.440 2.312 15.104 i o 1 7 0.041 0.525 1.306 12.630 (a) Simple Model Numerical Model ND J-+J. m i J d + J b J +J. m x J d + J b '•ID 1 4 6.864 17.873 6.863 17.829 io 1 5 3.337 19.124 3.336 19.081 io 1 6 1.482 17.916 1.482 17.377 io 1 7 0.567 13.936 0.567 13.908 (b) Simple Model Numerical Model D i f f e r e n c e ND J P J P % io 1 4 24.737 24.692 0.18 io 1 5 22.461 22.418 0.19 io 1 6 18.897 18.858 0.21 io 1 7 14.502 14.475 0.19 Cc) Table IV-1 Comparison of r e s u l t s from the present numerical model and a sim p l e r model of Chapter I I I . C a l c u l a t i o n s r e f e r to the Au/ n - S i system w i t h metal thickness = 100 A and b a r r i e r height = 0.8 eV. For the numerical model the surface recombination v e l o c i t y was taken as 1 x 10^ cm sec--'-. A l l currents d i s -p layed are i n mA-cm-^ and doping d e n s i t i e s i n cm - 3. important example of the case in point and earlier calculations from Chapter III have modelled silicon Schottky-barrier solar cells by neglec-ting recombination in the surface and f i e l d regions. In this case the numerical calculations described in this chapter would be expected to show only slight differences from 'the results of the earlier model, and thus a comparison of the two sets of results should provide a check on the efficacy of the model. Such data are shown in Table IV-1 for the case of a c e l l comprising 100 A of gold on n-type Si. The barrier height 4 was taken as 0.8 eV and the surface recombination velocity as 1 x 10 cm sec a l l other parameters are described in Appendix II. Table IV-1 (a) serves to indicate the magnitude of the various current components and these are then grouped together in Table IV-1 (b) so as to allow comparison with results from the numerical method of this chapter. The results are in very close agreement, so much so that the total photogenerated current as calculated by the simpler method is apparently overestimated by only about 0.2%,ssee Table IV-1 (c). B. Effect of Surface Recombination Velocity The variation of conversion efficiency with surface recombina-tion velocity i s shown in Fig. (IV-2) for a range of doping densities in both p and n-type gallium arsenide and s i l i c o n . In both cases the bar-ri e r metal was taken to be gold (100 A thick), giving barrier heights of 0.8 and 0.95 eV for silicon and gallium arsenide, respectively. For silicon there is only a very slight decrease of the conver-sion efficiency with surface recombination velocity. This i s a direct consequence of the low absorption coefficient of si l i c o n which ensures that the main contribution to the photocurrent originates from photons absorbed in the bulk region [25]. The principal reason for the p-Si cells 6-4 -2 -4 2Y N=1014 cm3 1015 cm3 nn16 -3 10 cm 10 cm 70° 10\ S(cm sec'1) 10 a -nSi nGaAs pGaAs p Si nSi n GaAs pGaAs p Si n Si n GaAs\ pGaAs p Si n Si w: F i g . IV-2 Dependence of conversion e f f i c i e n c y on sur f a c e / •. recombination, v e l o c i t y . B a r r i e r metal Au, 100 A t h i c k . being more e f f i c i e n t than the n-Si c e l l s l i e s i n the difference between the appropriate Richardson constant values (see Appendix I I ) . For GaAs the surface recombination velo c i t y affects the conver-sion e f f i c i e n c y more strongly, p a r t i c u l a r l y at the lower doping densities where the inversion region i s s i g n i f i c a n t . Because of the high absorp-tion c o e f f i c i e n t i n GaAs, contributes s i g n i f i c a n t l y to J ^ , as can be seen i n Fig. IV-3, which was computed using the simpler model [25] of Chapter I I I solely to stress the r e l a t i v e importance of J\ i n s i l i c o n and gallium arsenide. The surface recombination velo c i t y enters into the calculations through a boundary condition used i n evaluating Eq. (IV-2), hence the effect on J . Again there i s a difference i n e f f i c i e n c y be-tween n- and p-type c e l l s on account of the appropriate values of the Richardson constant (see Appendix I I ) . Where p-type material has been used i n the above and subsequent calculations, the b a r r i e r metal has been taken as gold but with a ba r r i e r height appropriate to the gold/n-type system. C. Effect of Barrier Height The importance of the metal/semiconductor b a r r i e r height i n Schottky-barrier solar c e l l s was demonstrated by the l i m i t calculations [22]; i n th i s section the numerical model i s used to evaluate t h i s effect i n more d e t a i l . For these calculations the semiconductor was taken as n - s i l i c o n and the metal as having the same o p t i c a l properties as 100 A of gold. The b a r r i e r hei'ght was then introduced as a parameter and no attempt was made to relate a p a r t i c u l a r barrier-height value to a speci-f i c metal, which would then have different o p t i c a l properties from those used here. Figure IV-4 shows that the conversion e f f i c i e n c y increases Xj, Mm Wy jjm • 0665 •279 -0478 •320 n Si n GaAs 30 20 10 DER INV \\r 9 . 1.1 1.3 .3 WAVELENGTH (/urn) .5 F i g . IV-3 S p e c t r a l current density vs. wavelength f o r 100 A and on GaAs, w i t h a doping de n s i t y of 1016 ,donor cm-3 i n each case. 59 I I I J _ ' • i 6 ,7 .8 .9 1.0 K {eV) F i g . IV-4 V a r i a t i o n of conversion e f f i c i e n c y with metal/semiconducto b a r r i e r height. Semiconductor n - S i , metal taken as having the o p t i c a l properties of 100 A of gold. 60. s l i g h t l y faster than l i n e a r l y with b a r r i e r height. This can be explained by reference to Fig. IV-5 arid by use of the following expression for the conversion e f f i c i e n c y : n = (FF) V I /P , (IV-13) o c s c i n where V i s the open-circuit voltage, I i s the s h o r t - c i r c u i t current oc r ° sc (= I i f R g = 0), P ± n i s the input power, and (FF) i s the f i l l factor (where FF = (V I )/(V I ) (IV-14) mp mp oc sc and V and 1 ^ are the maximum-power operating point voltage and current, respectively). The open-circuit voltage increases l i n e a r l y with b a r r i e r height (J) [from V = <|> /e +zN.t In (J.//. A* */,T?):];j. provided I - i s mucU-lar-ger than the D OC D p -F j p ° saturation current with the s l i g h t dependence on doping density a result of the J term. J (the photogenerated current density) i s affected by P P the doping density, primarily because of the reduction i n the width of the f i e l d regions, and hence J. and J, with increasing doping density i d [25]. There are two effects causing J, to vary with doping density. b The bulk region moves closer to the surface with larger doping densities causing to increase, but at the same time the di f f u s i o n length de-creases which reduces due to a reduction i n the co l l e c t i o n of minority carriers i n the bulk. The short c i r c u i t current i s es s e n t i a l l y indepen-14 -3 dent of ba r r i e r height, e.g., from Fig. IV-5, for =10 cm , increases by only 0.4% on increasing ^ from 0.6 to 1.0 eV. This very s l i g h t change i s due to ^ ' s effect on the semiconductor potential pro-f i l e , and hence the positions of x^ and W. The photocurrent component due to photon absorption i n the metal i s also affected by (J>^ , but i s so small the eff e c t on i s hardly dicernible. 61 .7 .8 .9 1.0 L f e V ) Fig. IV-5 Variation of open-circuit voltage, short-circuit current, and f i l l factor with barrier height. Same conditions as for Fig. IV-4. 62. 0 • 7 VfV) .2 -3 -i 1— •5 —r-(f)b =.60eV Voc = .0729 V lsc =-30mA Vmp=.0475 V Imp =-20.5 mA FF = .446 0 .4 .5 0b=.8OeV Voc=.273 V lsc=-30mA %=-^v lmp =-2 6.8 m A FF=.704 -10h .5 -30 A0h =.2eV A FF=.258 0 .1 V f V ) . .2 .3 .5 (j)b=1.0eV •• i i i j r — V0C=.473 V ! s c =-30mA Vmp=.400V lmp=-28.2mA FF =-795 -10 A0b =.2eV AFF = .091 -30 Fig. IV-6 Illustration of how f i l l factor varies with ^through I Q as a supplement to Fig. IV-5 Now, from the above d i s c u s s i o n , i t f o l l o w s t h a t , on i n c r e a s i n g <|>k, the J-V c h a r a c t e r i s t i c w i l l be d i s p l a c e d along the voltage a x i s such that AV. 'V' A<j>,/e. Thus, because the J-V c h a r a c t e r i s t i c i s not r e c t a n -oc b g u l a r there w i l l i n i t i a l l y be a r a p i d increase i n f i l l f a c t o r w i t h <}>, b f o l l o w e d by a slower r i s e as tf>, i n c r e a s e s . The e f f e c t of t h i s can be b seen i n F i g . IV-5 and the mechanics of how t h i s comes about can be seen i n F i g . IV-6. The above-mentioned dependencies of f i l l f a c t o r , V , and J oc p on <j>b l e a d to an increase of conversion e f f i c i e n c y w i t h b a r r i e r h e i g h t , see F i g . IV-4, and thus reemphasize the p r a c t i c a l need f o r h i g h - b a r r i e r -height systems. D. E f f e c t of Metal Thickness and G r i d C o n f i g u r a t i o n The c a l c u l a t i o n s presented i n t h i s s e c t i o n r e f e r , t o the Au/n-GaAs system w i t h a. SiGh.antiref l e c t i o n .coating., of -.optimum thickness and a c o l l e c t i n g g r i d network based on Handy's [35] s t r u c t u r e and dimensions ( d e t a i l e d i n F i g . 111-10) . The SiO t h i c k n e s s r e q u i r e d i s i n the range from 640 A of SiO on 40 1 of Au to 600 A of SiO on 200 A of Au. The optimum SiO t h i c k n e s s was found by v a r y i n g the SiO thickness i n steps (smallest step s i z e of 10 A ) , c a l c u l a t i n g J ^ each time, u n t i l the SiO thickness r e s u l t i n g i n the maximum J ^ was found. Figure IV-7 shows r e s u l t s f o r 5-, 10-, and 2 0 - f i n g e r g r i d systems along w i t h data computed assuming no g r i d coverage and no s e r i e s r e s i s t a n c e . The r e s u l t s are s i m i l a r to those already discussed [25] f o r S i i n Chapter I I I and demon-s t r a t e the i n t e r p l a y between s e r i e s r e s i s t a n c e and a c t i v e surface area that e x i s t s at a given metal t h i c k n e s s when the number of g r i d s i s v a r i e d , and between s e r i e s r e s i s t a n c e and o p t i c a l t ransmission that e x i s t s f o r a given g r i d c o n f i g u r a t i o n when the metal th i c k n e s s i s v a r i e d . Thus, above : . C64.) j SiO/Au/n-GaAs, NQ =W15cm 3 I ; I I ) 50 100 150 METAL FILM THICKNESS (A) F i g . IV-7 Dependence of conversion e f f i c i e n c y on b a r r i e r metal (gold) thickness and co l l e c t i n g grid configuration. SiO a n t i -r e f l e c t i o n coating thickness of 640 A for a Au thickness of 40 A and SiO thickness of 600 A for a Au.thickness of 200 A . 65. a c e r t a i n metal t h i c k n e s s , which depends on the g r i d c o n f i g u r a t i o n , the e f f i c i e n c y curves no longer increase due to reduced s e r i e s r e s i s t a n c e but s t a r t to decrease due to reduced photon transmission i n t o the semi-r conductor. E. E f f e c t of Semiconductor Thickness One route towards the goal of reducing the $/watt r a t i o f o r p h o t o v o l t a i c systems i s to reduce the thickness of the semiconductor m a t e r i a l . Figure IV-8 shows r e s u l t s f o r n - S i and n-GaAs, computed assum-i n g no d e t e r i o r a t i o n i n m i n o r i t y - c a r r i e r p r o p e r t i e s w i t h r e d u c t i o n i n semiconductor t h i c k n e s s . As i s to be expected from the r e l a t i v e magnitudes of the absorption c o e f f i c i e n t s , the GaAs c e l l i s l e s s a f f e c t e d by c e l l -l e n gth r e d u c t i o n than i s the S i c e l l . For example, at a c e l l thickness of 10 ym the e f f i c i e n c y i s reduced below the bulk value by 40% f o r the S i c e l l , and by only 3.5% f o r the GaAs c e l l . F. E f f e c t of M i n o r i t y - C a r r i e r L i f e t i m e In the previous s e c t i o n the assumption was made t h a t the mino-r i t y - c a r r i e r p r o p e r t i e s d i d not d e t e r i o r a t e as the c e l l t h i c k n e s s was reduced. This c o n d i t i o n w i l l not be obtained i n p r a c t i c e , p a r t i c u l a r l y as l i k e l y commercial t h i n - f i l m systems w i l l u t i l i z e p o l y c r y s t a l l i n e or reduced-grade semiconductor m a t e r i a l . To i l l u s t r a t e some p o s s i b l e e f f e c t s , the m i n o r i t y — c a r r i e r l i f e t i m e was taken as a parameter and v a r i e d from the b u l k value * (see Appendix I I ) to x/100 f o r GaAs and S i c e l l s : the r e s u l t s are shown i n F i g . IV-8 f o r a range of semiconductor doping d e n s i t i e s . I t i s to be expected that the m i n o r i t y - c a r r i e r l i f e t i m e w i l l have i t s greatest e f f e c t on the bulk component of the photocurrent. Thus (see F i g . IV-3), s i l i c o n c e l l s are more a f f e c t e d by low m i n o r i t y -c a r r i e r l i f e t i m e s than are g a l l i u m arsenide c e l l s ; t h i s i s evinced by GaAs, n-type d = 0.95 eV 100 . 200 300 . SEMICONDUCTOR THICKNESS (jjm) F i g . IV-8 The e f f e c t of semiconductor thickness or conversion e f f i c i e n c y . B a r r i e r metal 100 of Au, semiconductor doping density = 10^5 atoms cm - 3. 67 . 6V 4 2 o UJ o o o ^ A Uj o 2 L 0 % = IO14 cm~3 n GaAs p GaAs pSi n Si ND = 101S cm 3 n GaAs p GaAs ND = 1016 cm'3 nGaAs J7 -3 ND ^10 cm .01 . 1 1 .01 .1 1 MINORITY CARRIER LIFETIME (>Jm/^'BULK ) Fig. IV-9 Dependence of efficiency on minority-carrier lifetime. Barrier metal 100 A of Au, surface recombination velo-city = 1 x 10' cm sec -1 68. F i g . IV-9. The GaAs c e l l s o n l y s t a r t to show a d e t e r i o r a t i o n i n p e r f o r -mance at the h i g h e r doping d e n s i t i e s ( t h i n n e r i n v e r s i o n and d e p l e t i o n regions) when there i s an attendant r e d u c t i o n i n m i n o r i t y - c a r r i e r d i f f u s -i o n l e n g t h (through the m i n o r i t y - c a r r i e r l i f e t i m e and m o b i l i t y ) . G. Comparison of Computed and P r a c t i c a l Data The highest conversion e f f i c i e n c y reported [21] f o r S i Schottky-b a r r i e r s o l a r c e l l s i s 9.5%. However, these c e l l s cannot be meaningfully compared with the data presented here because of the f o l l o w i n g : ( i ) The c e l l s were t e s t e d under sunlight:..conditionsWthatywereknot completely, s p e c i -fie'di. ( i i ) A l a y e r e d m e t a l l i z a t i o n scheme was used f o r which the o p t i c a l p r o p e r t i e s are not known, ( i i i ) The h i g h reported value f o r y (2.5) suggests the presence of an i n t e r f a c i a l l a y e r . A model [23] i n c o r p o r a -t i n g the l a t t e r feature would thus seem more appropriate f o r comparison purposes i n t h i s case. For the case of GaAs some r e s u l t s are a v a i l a b l e [40] and Table IV-2 shows a comparison of measured (under simulated AMO c o n d i t i o n s ) and computed data. I f the s e r i e s r e s i s t a n c e i s assumed to'be zero w i t h no g r i d s t r u c t u r e then the c a l c u l a t i o n s overestimate the e f f i c i e n c y of the s o l a r c e l l . However, i f the three types of g r i d s t r u c t u r e mentioned above (5, 10, and 20 f i n g e r s on a 2-cm by 1-cm c e l l ) are introduced i n t o the c a l c u l a t i o n s , a nonzero s e r i e s r e s i s t a n c e i s taken i n t o account which r e s u l t s i n much b e t t e r agreement between the experimental and c a l c u l a t e d r e s u l t s . The g r i d s t r u c t u r e used does not appjy d i r e c t l y to that of the experimental r e s u l t s but the s e r i e s r e s i s t a n c e c a l c u l a t e d i s a p p r o x i -mately what one would expect e x p e r i m e n t a l l y . H. Model Comparison on GaAs S o l a r C e l l Table IV-3 compares the e f f i c i e n c i e s of a s o l a r c e l l f a b r i c a t e d 69. Values used in comparison calculations Property Units Measured values eV 1.42 g ND -3 cm 15 3.6 x 10 D -2 „ -2 A** Acm °K 8.6 1.07 eV 0.898 Au thickness A 60 W ym 0.55 -2 -7 J mA cm 6.3 x 10 o L ym 4.5 P yn 2 -1 -1 cm v sec 3800 Property Units Additional values needed -1 7 S cm sec 1.0 x 10 L ym 10 n 2 -1 -1 340 PP cm V sec Results of calculations Solution by Numerical Method Property Units Measured R s = 0 R 4 0 Grid Structure assumed s 5 finger 10 finger 20 finger R s ohms (3.0 .566 .277 .130 V oc volts .442 .480 .476 .474 .471 J P mA/cm2 19.0 21.7 18.6 | 17.6 15.6 F.F. .723 .788 .748 .771 .777 n % 4.5 6.09 4.90 4.78 4.24 Table IV-2 Comparison of practical (Ref. 40) and computed data for uncoated, Au/n-GaAs cells under AMO conditions. E f f i c i e n c y f o r T = T of Appendix I I i n % S i Model General Model % D i f f e r e n c e R s = 0 5.78 5.44 6.15 R s / 0,5 f i n g e r g r i d 4.71 4.45 5.95 10 f i n g e r g r i d 4.53 4.27 6.06 20 f i n g e r g r i d 4.02 3.79 6.10 E f f i c i e n c y f o r T = T normal/100 i n % S i Model General Model % D i f f e r e n c e R s = 0 5.69 5.35 6.36 R s ^ 0 , 5 f i n g e r g r i d 4.64 4.37 6.16 10 f i n g e r g r i d '4.47 4.20 6.27 20 f i n g e r g r i d 3.96 3.73 6.31 Table IV-3 Comparison of computed data f o r a GaAs c e l l under AM0 co n d i t i o n s using the models of t h i s chapter and Chapter I I I w i t h N, = 1 x 1 0 1 4 cm - 3 and S = 1 x 1 0 7 cm s e c - 1 . 71. from GaAs as c a l c u l a t e d by the models of Chapters I I I and IV. The r e s u l t s are shown f o r both zero and nonzero s e r i e s r e s i s t a n c e and a l s o f o r the normal l i f e t i m e and the same l i f e t i m e degraded by a f a c t o r of 100. The percentage d i f f e r e n c e between the r e s u l t s of the two models i s on the order of 6% f o r t h i s p a r t i c u l a r system and i s r e f l e c t e d mainly i n the surface recombination v e l o c i t y l o s s e s . The increases i n recombination l o s s e s show up as a d i f f e r e n c e i n e f f i c i e n c y but the d i f f e r e n c e s between the general model of t h i s chapter and the S i model of Chapter I I I f o r both the x and x/100 r e s u l t s are n e a r l y i d e n t i c a l , i . e . , f o r R g = 0 the percentage d i f f e r e n c e between the two models i s 6.36% f o r the x/100 case and 6.15% f o r the x case. The s l i g h t d i f f e r e n c e i n the two values of the percentage d i f f e rences i s the r e s u l t of the increased recombina-t i o n i n the i n v e r s i o n and d e p l e t i o n regions which i s accounted f o r only i n the general model of t h i s chapter. I t appears that the model of Chapter I I I can be a p p l i e d to GaAs w i t h l e s s e r r o r than was o r i g i n a l l y expected. This i s probably because of the e x c e l l e n t m i n o r i t y - c a r r i e r p r o p e r t i e s assumed here f o r GaAs. However, the r e s u l t s of t h i s t e s t i n d i c a t e that the complete, general model w i l l provide more accurate r e s u l t s of c a l c u l a t e d e f f i c i e n c y f o r S c h o t t k y - b a r r i e r s o l a r c e l l s f a b r i -cated from semiconductors where h i g h surface recombination v e l o c i t i e s and poor l i f e t i m e p r o p e r t i e s are important. V. CONCLUSIONS A t h e o r e t i c a l i n v e s t i g a t i o n of the performance of the Schottky-b a r r i e r diode when used as a s o l a r c e l l has been c a r r i e d out. Three models to describe the ope r a t i o n of S c h o t t k y - b a r r i e r s o l a r c e l l s have been developed and cate g o r i z e d as: ( i ) a very s i m p l i f i e d l i m i t i n g case to c a l c u l a t e maximum conversion e f f i c i e n c i e s , ( i i ) a more r e a l i s t i c case t a k i n g i n t o account some l o s s mechanisms that are appropriate' to s i l i c o n S c h o t t k y - b a r r i e r s o l a r c e l l s i n p a r t i c u l a r and ( i i i ) a more complete model that i s a p p l i c a b l e to the metal/semiconductor s o l a r c e l l system i n general. From the r e s u l t s of c a l c u l a t i o n s performed using the three models developed f o r the S c h o t t k y - b a r r i e r s o l a r c e l l s e v e r a l conclusions can be drawn. The r e s u l t s from the l i m i t model imply that Schottky-b a r r i e r s o l a r c e l l s are capable of s i m i l a r h i g h - l e v e l conversion e f f i c -i e n c i e s to those p r e d i c t e d f o r conventional homojunction s o l a r c e l l s . There are s e v e r a l p o t e n t i a l advantages to be gained by usin g a Schottky-b a r r i e r diode as a s o l a r c e l l when compared to conventional p-n homo-j u n c t i o n s , namely: ease of f a b r i c a t i o n ; improved s h o r t wavelength r e s -ponse; reduced c a r r i e r l i f e t i m e degradation during f a b r i c a t i o n ; and, importantly as regards the future of t e r r e s t r i a l p h o t o v o l t a i c s , a p p l i c a -b i l i t y to p o t e n t i a l l y inexpensive, l a r g e - a r e a t h i n - f i l m semiconductor systems. Considering these advantages and the r e s u l t s of the l i m i t model the S c h o t t k y - b a r r i e r s o l a r c e l l was deemed s u i t a b l e f o r f u r t h e r i n v e s t i -gation . From the model of Chapter I I I a p p l i e d to S i , i t was shown t h a t : the l a r g e s t component of the photogenerated current o r i g i n a t e s from photon absorption i n the bulk region of the semiconductor; the photocurrent 73. decreases w i t h increases i n both semiconductor doping d e n s i t y and metal f i l m t h i c k n e s s ; there i s an optimum metal f i l m thickness as regards con-v e r s i o n e f f i c i e n c y ; i n c r e a s i n g the number of c o l l e c t i n g g r i d s can reduce the a c t u a l conversion e f f i c i e n c y . The most important conclusion from the r e s u l t s of Chapter I I I i s that a high b a r r i e r height i s needed to a r r i v e at a good conversion e f f i c i e n c y . From the general model of Chapter IV a p p l i e d mainly to GaAs and a l s o to S i under AMO i l l u m i n a t i o n c o n d i t i o n s i t was found that i n terms o f e f f i c i e n c y GaAs appears to be s u p e r i o r to S i and n-type GaAs i s b e t t e r than p-type GaAs. The advantages d i s p l a y e d f o r GaAs are the r e s u l t of the higher b a r r i e r height obtained i n GaAs S c h o t t k y - b a r r i e r diodes and the sharper o p t i c a l absorption edge of GaAs. A comparison of experimental r e s u l t s f o r the Au/n-GaAs system w i t h computed data shows good agreement. As the experimental data used [40] r e f e r s to n e a r - i d e a l s o l a r c e l l s , i t i s p o s s i b l e to conclude from the c a l c u l a t i o n s of Chapter IV that the highest AMO conversion e f f i c i e n c y l i k e l y to be a t t a i n e d by p r a c t i c a l metal/semiconductor s o l a r c e l l s w i l l be around 10%. To exceed t h i s value w i l l r e q u i r e the development of h i g h e r - b a r r i e r - h e i g h t systems. One p o s s i b l e approach to o b t a i n the r e q u i r e d improvement i n b a r r i e r height i s to introduce an i n t e r f a c i a l l a y e r between the metal and semiconductor. T h e o r e t i c a l work [23] has shown that an i n t e r f a c i a l l a y e r i n the form of a t h i n oxide (10-50 A) which p o s s i b l y contains a p o s i t i v e charge de n s i t y w i l l enhance the b a r r i e r height f o r the metal/ p-type S i system. The Al/p-type S i MIS system has been s t u d i e d by Charlson and L i e n [41] and they have obtained b a r r i e r heights of 0.85 eV i . e . , approximately twice the normal Al/p-type S i b a r r i e r height. The Cu-Cr/p-type S i system has been s t u d i e d by Anderson and Milano [21] to y i e l d conversion e f f i c i e n c i e s of 9.5%. The enhancement of the b a r r i e r height of S c h o t t k y - b a r r i e r s o l a r c e l l s by means of an i n t e r f a c i a l l a y e r i s at present very a t t r a c t i v e as a method of f a b r i c a t i n g h i g h e f f i c i e n c y s o l a r c e l l s . The presence of a t h i n i n s u l a t i n g l a y e r can a l s o improve S c h o t t k y - b a r r i e r s o l a r c e l l performance by c o n t r o l l i n g the mechanism of dark current flow across the j u n c t i o n , as shown by recent t h e o r e t i c a l work on S i [42,43] and GaAs [44]. P r a c t i c a l MIS GaAs s o l a r c e l l s u s i n g a n t i r e f l e c t i o n coatings have been reported w i t h e f f i c i e n c i e s of 15% [45] and e f f i c i e n c i e s i n excess of 10% are i n d i c a t e d [46] by r e s u l t s from MIS Si„solar.cells working i n t h i s mode. However, these experimental e f f o r t s , w h i l e showing promising r e s u l t s , have problems of r e p r o d u c i b i l i t y which would make f a b r i c a t i o n of l a r g e - a r e a s o l a r c e l l s d i f f i c u l t . The s t a b i l i t y of the i n t e r f a c i a l l a y e r [47] w i t h aging of the s o l a r c e l l may also be questionable due to the extreme thinness of the l a y e r ( r e q u i r e d f o r e f f i c i e n t t u n n e l l i n g of the photogenerated current to the S c h o t t k y - b a r r i e r c o n t a c t ) . The models presented i n Chapters I I I and IV could be modified to accept t h i s type of s o l a r c e l l . The r e q u i r e d changes are i n the dark J-V r e l a t i o n s h i p used to c a l c u l a t e the e f f i c i e n c y from the value of the photogenerated current. The m o d i f i c a t i o n s would be r e l a t e d to the f o l l o w i n g areas: the exponen-t i a l r e l a t i o n s h i p between J and V; the c a l c u l a t i o n of J q i n the J-V r e l a t i o n ; and the e v a l u a t i o n of the b a r r i e r h e i g ht c ^ . Another approach t h a t i n c r e a s e s the e f f i c i e n c y of Schottky-b a r r i e r s o l a r c e l l s yet does not depend on an i n t e r f a c i a l l a y e r has been r e c e n t l y proposed [48], The c e l l design uses a g r a t i n g r a t h e r than a continuous metal f i l m f o r formation of the S c h o t t k y - b a r r i e r contact. 75. This reduces I the dark reverse current by having the contact area of the S c h o t t k y - b a r r i e r s o l a r c e l l a s m a l l f r a c t i o n of the a c t i v e photo-generating area. This increases the open c i r c u i t voltage through a reduced I and r e s u l t s i n a b e t t e r conversion e f f i c i e n c y . Current c o l l e c -t i o n by the S c h o t t k y - b a r r i e r contact i s s t i l l obtained i f the g r a t i n g or dot spacing i s sm a l l compared to the semiconductor m i n o r i t y - c a r r i e r d i f f u s i o n l e n g t h . This method a l s o i n c r e a s e s the photogenerated current d e n s i t y by a l l o w i n g more e f f i c i e n t c oupling of the i n c i d e n t l i g h t i n t o the semiconductor because the b a r r i e r metal does not cover the e n t i r e diode area. However, the above technique r e q u i r e s a more c o s t l y f a b r i c a -t i o n method and a p p l i c a t i o n to t h i n - f i l m low-cost semiconductor m a t e r i a l s w i l l r e q u i r e a very f i n e g r a t i n g spacing. The c a l c u l a t i o n s performed i n t h i s work on the S c h o t t k y - b a r r i e r s o l a r c e l l i n d i c a t e that t h i s may become a v i a b l e means of energy produc-t i o n . This depends on the development of h i g h - b a r r i e r - h e i g h t systems or systems w i t h a low reverse s a t u r a t i o n current I to overcome the i n h e r e n t l y low o p e n - c i r c u i t voltage of S c h o t t k y - b a r r i e r s o l a r c e l l s . With the reduc-t i o n i n the manufacturing costs expected from S c h o t t k y - b a r r i e r s o l a r c e l l s , e s p e c i a l l y when they are a p p l i e d to t h i n - f i l m semiconductor m a t e r i a l s , they may then achieve widespread t e r r e s t r i a l u t i l i z a t i o n . 76. REFERENCES [1] L.M. Magid, "Status of the U.S. P h o t o v o l t a i c Conversion Program", Proc. I n t . S o l . Energy Soc. meeting, Winnipeg, 1976, p. 22. [2] G. Kaplan, "For S o l a r Power: Sunny Days Ahead", IEEE Spectrum, 12, 12, 47, (1975). [3] Proc. 11th IEEE P h o t o v o l t a i c S p e c i a l i s t s Conf. (IEEE, New York, 1975). [4] H.E. Bates, D.N. Jewett, and V.E. 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Sumner, "Experimental Study of Gold-Gallium Arsenide Schottky Barriers", J. Appl. Phys., 36, 3744, (1965). [18] J.J. Wysocki and P. Rappaport, "Effect of Temperature on Photovoltaic Solar Cenergy Conversion", J. Appl. Phys., 31, 571, (1960). [19] J.M. Andrews and M.P. Lepselter, "Reverse Current-Voltage Character-i s t i c s of Metal-Silicide Schottky Diodes", IEEE Solid State Devices Conf. , Washington, D.C, 1968 (IEEE, New York, 1968). [20] W.A. Anderson and A.E. Delahoy, "Schottky Barrier Diodes for Solar Energy Conversion", Proc. IEEE, 60, 1457, (1972). [21] W.A. Anderson and R.A. Milano, "I-V Characteristics for Silicon Schottky Solar Cells", Proc. IEEE, 6_3, 206, (1975). [22] D.L. Pulfrey and R.F. McOuat, "Schottky-Barrier Solar Cell Calcula-tions", Appl. Phys. Lett., 24_, 167, (1974). [23] D.L. Pulfrey, "Barrier Height Enhancement in p-Silicon MIS Solar Cells", IEEE Trans. Elec. Dev., ED-23, 587, (1976). [24] R.F. McOuat and D.L. Pulfrey, "A Model for Schottky-Barrier Solar Cell Analysis", J. Appl. Phys., 47, 2113, (1976). [25] R.F. McOuat and D.L. Pulfrey, "Analysis of Silicon Schottky-Barrier Solar Cells", Proc. 11th IEEE Photovoltaic Specialists Conf., 371, (1975). [26]. J.C. Arvesen, R.N. G r i f f i n , and B.D. Pearson, "Determination of Extraterrestrial Solar Spectral Irradiance from a Research Aircraft", Appl. Optics, 8, 2215, (1969). [27] R.H. Fowler, "The Analysis of Photoelectric Sensitivity Curves for Clean Metals at Various Temperatures", Phy. Rev., 38, 45, (1931). [28] S.W. Duckett, "Random-Walk Models of Photoemission", Phy. Rev., 166, 302, (1968). [29] V.L. Dahal, "Simple Model for Internal Photoemission", J. Appl. Phys., 42, 2274, (1971). [30] R. Stuart, F. Wooten, and W.E. Spicer, "Monte Carlo Calculations Pertaining to the Transport of Hot Electrons in Metals", Phy. Rev., 135, A495, (1964). [31] O.S. 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Crowell, "Determination of Hafnium-p-Type Silicon Schottky-Barrier Height", J. Appl. Phys., 45, 2792, (1974). [39] S.C. Tsaur, A.G. Milnes, R. Sahai, and D.L. Feucht, "Theoretical and Experimental Results for GaAs Solar Cells", Gallium Arsenide and  Related Compounds, (Institute of Physics, London, 1972). [40] Y.C.M. Yeh and R.J. Stirn, "Improved Schottky-Barrier Solar Cells", Proc. 11th IEEE Photovoltaic Specialists Conf., 391, (1975). [41] E.J. Charlson and J.C. Lien, "An Al p-Silicon MOS Photovoltaic Ce l l " , J. Appl. Phys., 46, 3982, (1975). [42] M.A. Green, F.D. King, and J. Shewchun, "Minority Carrier MIS Tunnel Diodes and Their Application to Electron and Photo-Voltaic Energy Conversion - I. Theory", Solid St. Electron., 17, 551, (1974). [43] R. Singh and J. Shewchun, "Photovoltaic Effect in MIS Diodes or Schottky Diodes with an Interfacial Layer", Appl. Phys. Lett., 28, 512, (1976). [44] S.J. Fonash, "Metal-Insulator-Semiconductor Solar Cells: Theory and Experimental Results", to be published in Thin-Solid Films. [45] R.J. Stirn and Y.C.M. Yeh, "A 15% Efficient Antireflection-Coated Metal-Oxide-Semiconductor Solar C e l l " , Appl. Phys. Lett., 27_, 95, (1975). [46] M.A. Green, private communication. [47] H.C. Card, "Aluminum-Silicon Schottky Barriers and Ohmic Contacts in Integrated Circuits", IEEE Trans. Elec. Dev., ED-23, 538, (1976). [48] M.A. Green, "Enhancement of Schottky Solar Cell Efficiency above i t s Semiemperical Limit", Appl. Phys. Lett., 2_7, 287, (1975). [49] Gmelin, Handb. Anorg. Chem. SilicumB15, 266, (1959). [50] B.O. Seraphin and H.E. Bennett, "Optical Constants", Semiconductors and Semimetals, _3, 518, (1967). 79. [51] F. Meyer, E.E. de K l u i z e n a a r , and G.A. Boutsma, "EHipsometry and the Clean Surfaces of S i l i c o n and Germanium", Surf. S c i . , 2_7, 88, (1971). [52] H.F. Wolf, Semiconductors, (Wiley, New York, 1971). [53] L.W. Aukerman, M.F. M i l l e a , and M. M c C o l l , " D i f f u s i o n Lengths of E l e c t r o n s and Holes i n GaAs", J . Appl. Phys., _38, 685, (1967). [54] R. Schulze, "Recommendations f o r the Integrated I r r a d i a n c e and the S p e c t r a l D i s t r i b u t i o n of Simulated Solar R a d i a t i o n f o r T e s t i n g Purposes", P u b l i c a t i o n CIE No. 20 (TC 2.2), (1972). APPENDIX I A b r i e f flow chart of the program developed to implement the models of Chapters I ' l l and IV i s given i n t h i s appendix together w i t h a ; l i s t of the user options a v a i l a b l e and a program l i s t i n g . i . Flow Chart I n i t i a l i z e program i n MAIN Read i n p u t data and do cubic s p l i n e i n t e r p o l a t i o n where necessary and p r i n t out the r e s u l t s i f des i r e d . MAIN uses SMOOTH from the UBC Computing Centre l i b r a r y and c a l l s a l l other subroutines to per-form the c a l c u l a t i o n s . Do transmittance and r e f l e c t a n c e c a l c u l a t i o n s f o r the s o l a r c e l l i n subroutine TRREF c a l l e d from MAIN and p r i n t out r e s u l t s i f d e s i r e d . Execute subroutine QUANT c a l l e d from MAIN to f i n d Jp by a n a l y t i c a l or numerical methods depending on which o p t i o n i s s e l e c t e d . QUANT uses subrou-t i n e s DTEVAL, EGEN, VFUNCN, VFUNCP, ENFUNC, PFUNC, BFUNC, ND, RKC, and DRKC where ND i s user s u p p l i e d and RKC and DRKC are i n the UBC Computing Centre l i b r a r y . C a l c u l a t e the s o l a r c e l l e f f i c i e n c y from Jp by execution of the subroutine EFF c a l l e d from MAIN. EFF uses subroutines IMAXP and ISCJ. 81. User w r i t t e n subroutine VARY i s executed. The subroutines TRREF, QUANT and EFF are c a l l e d from VARY as necessary. The parameters are v a r i e d and a l l data t r a n s f e r s between TRREF, QUANT, EFF and MAIN are achieved by using l a b e l e d COMMON statements. i i . Execution Options In the i n p u t data cards a f t e r the t i t l e card i s a c o n t r o l card which i s s t o r e d as array I i n the program. I t c o n t r o l s the program options that are described below. The entry i s i n the s p e c i f i e d column number and i s e i t h e r zero (blank) or a nonzero s i n g l e d i g i t i n t e g e r (such as 1). Column Option 1 I f equal to 1 the transmittance and r e f l e c t a n c e data are d i r e c t input and are cubic s p l i n e i n t e r p o l a t e d . TRREF i s not invoked. I f equal to 0 then TRREF i s executed and the o p t i c a l p r o p e r t i e s are expected. 2 I f equal to 0 the media surrounding the s o l a r c e l l i s a i r and the r e f r a c t i v e index i s taken as 1.0. I f equal to 1 then the user must supply the r e f r a c t i v e index of the medium i n which the s o l a r c e l l r e s i d e s . 3 I f set by TRREF to equal 1 then the sum of the r e f l e c t a n c e and transmittance i s greater than u n i t y i n d i c a t i n g an e r r o r i n the inpu t o p t i c a l data. 4 I f equal to 1 the i n c i d e n t l i g h t i s p o l a r i z e d and i f equal to 0 unpolarized. Ignored i f the angle of incidence i s 0°. 82. Column Option 5 I f equal to 1 assumes S l i g h t p o l a r i z a t i o n and i f equal to 0 assumes p l i g h t p o l a r i z a t i o n . Ignored i f un p o l a r i z e d . 6 I f equal to 0 the in p u t AMO or AMI data i s used f o r the e f f i c -i e n c y c a l c u l a t i o n s . I f equal to 1 i n p u t s o l a r data i s expected and w i l l be cubic s p l i n e i n t e r p o l a t e d . 7 I f equal to 0 then AMO r a d i a t i o n i s used and i f equal to 1 then AMI data i s used. Ignored i f I(6) ( equals 1. 8 I f equal to 1 then computation i s terminated a f t e r TRREF, otherwise execution continues. U s e f u l f o r studying the o p t i c a l p r o p e r t i e s o f s o l a r c e l l s . 9 I f equal to 1 then computation ceases a f t e r QUANT. U s e f u l f o r studying Jp w i t h changes i n the s o l a r c e l l p r o p e r t i e s when the EFF c a l c u l a t i o n s are not d e s i r e d . 10 I f equal to 1 semiconductor i s p-type and i f equal to 0 semi-conductor i s n-type. 11 I f equal to 0 then R g the s e r i e s r e s i s t a n c e i s . assumed equal to 0. I f equal to 1 then R s i s c a l c u l a t e d f o r the ar r a y ( s ) s e l e c t e d by 1(12), 1(13) and 1(14). 12 I f equal to 1 c a l c u l a t e the e f f i c i e n c y f o r a 5 - f i n g e r array. 13 I f equal t o 1 c a l c u l a t e the e f f i c i e n c y f o r a 1 0 - f i n g e r array. 14 I f equal to 1 c a l c u l a t e the e f f i c i e n c y f o r a 2 0 - f i n g e r array. 15 I f equal to 1 use the general model f o r the c a l c u l a t i o n of Jp and i f equal to 0 use the s i l i c o n model. 16 I f equal to 1 c a l c u l a t e and p r i n t s t h e quantum e f f i c i e n c y as a f u n c t i o n of wavelength. Works only i f 1(15) i s equal to 0. 17-20 For expansion of program. 21 I f equal to 1 the input o p t i c a l data i s p r i n t e d . 22 I f equal to 1 the i n t e r p o l a t e d o p t i c a l data i s p r i n t e d . 23 I f equal to 1 the r e f l e c t a n c e and transmittance as a f u n c t i o n of wavelength i s p r i n t e d . 24 I f equal to 1 the s p e c t r a l current d e n s i t i e s as c a l c u l a t e d by the S i model as a f u n c t i o n of wavelength are p r i n t e d . 25 I f equal to 1 the input s o l a r data i s p r i n t e d . 83. Column Option 26 I f equal to 1 the semiconductor absorption c o e f f i c i e n t as a fu n c t i o n of wavelength i s p r i n t e d . 27 I f equal t o 1 the generation of e l e c t r o n - h o l e p a i r s as a func-t i o n of depth i n the s o l a r c e l l i s p r i n t e d . 28 I f equal to 1 the r e s u l t s o f each i t e r a t i o n of the numerical i n t e g r a t i o n of the d i f f e r e n t i a l equations i s p r i n t e d . 29 I f equal to 1 the values of V and E as a f u n c t i o n of x i n the semiconductor band p r o f i l e are p r i n t e d . 30 I f equal to 1, v a r i a t i o n of parameters i s expected and the program continues execution i n the user program VARY u n t i l completion. 31-80 Unused but a v a i l a b l e f o r expansion. i i i . Program L i s t i n g A l i s t i n g of the FORTRAN program f o l l o w s . 84. COMPLEX NF{100,10) INTEGER a (5,11) ,1 (80) ,TITLE(20) REAL ALP (10) , ALPHA (1 00) ,B (10) ,D (10) ,DN (50) /DP (50) , FDN (50) ,FDP (50) , 1FTAUN (50) ,FTAUP (50) ,ISPGW,LAMBDA (100) , P (70 0) , R (100) , R E F (10 0) ,RK (10 20) ,RN (100) ,RND (50) ,S (100) ,S1 (100) ,S2 (100) ,SI (2) ,SIGMA (10) ,SOL (100) 3,TAUN(50) ,TAUP{50) ,TR{100) ,IL(100) ,ZND(50) CGfflMON/CURR/RJL, R1 , R2,S3,R7,R8,RGB,ROM COMMON/DIFTAU/FDN, F.D.P, FTAUN, FTAUP, ZND COMMON/OPTIC/ALPHA,RE.F ,T.R CQMMON/PRQP/EPSR,DNV,DNC, VT ,EGAP COM HON/PROPM/FREEN,EFERM COMMON/VARIA/ASTAR^^EN^I^SPOW^KjLAMBDA^F^PHIBa^PHIM^OL^SOSVEL, 1T 5 WRITE (6,11) READ (5,12,END=150)TITLE WRITE (6 ,13) TITLE WRITE (6,14) WRITE (6,44) READ (5, 16) (I (J) , J = 1 , 80) WRITE (6,46) DO 10 J=1,8Q IF (I (J) . EQ. 0) GO TO 10 WRITE (6,47) J 10 CONTINUE READ (5,17)PHIM,K IF (I (2) .NE.O) GO TO 25 DO 20 J=1,100 20 NF (J, 1) = (1. ,0.) M1 = 2 GO TO 30 25 M1=1 30 .IF (I (21) .EQ.O) GO TO 35 WRITE (6,18) 35 IF (I (1) .NE. 0) M1=K-1 DO 85 M=M1,K READ (5,21) (A(J,M) ,J=1,5) ,D(M) ,NO READ (5, 19) ( W L(J) , R N ( J ) ,RK(J) ,J=1,HG) IF (I (21) .EQ. 0) GO TO 40 WRITE (6,21) (A (J,M) ,J=1,5) ,D(M) WRITE (6,19) (WL (J) ,RN (J) ,RK (J) , J=1 ,NO) 40 IF (WL (1) .LE. LAMBDA (1) ) GO T O 50 DO 45 J=1,NO WL (NO+2-J)=WL(NO+1-J) RN (NO+2-J)=HN(NO + 1-J) 45 SK (NO + 2-J)=RK(NO+1-J) HL(1)=LAMBDA(1) SLR= (WL (1) - W L (2) ) / (WL (3) - W L (2) ) SN (1) = (SN (3) -RN (2) ) *WLR+RN{2) RK (1) = (RK (3) -RK (2) ) *WLR+RK (2) • NO=NO+1 WRITE (6,22) M, (A (J,M) , J=1 ,5) 50 IF (WL (NO) .GE. LAMBDA (100) ) GO TO 55 NO=NO+1 WL (NO) =LAMBDA (100) WLR=(WL(NO)-WL(NO-1))/(WL(NO-1)-WL(NO-2)) 85. BN(NO) = (RN(NO-1)-RN(NO-2))*SLB+EN (NO-1) RK (NO) = (RK (NO-1) -RK (NO-2) ) *WLB+RK (NO-1) WRITE (6,23) H , (A (J,M) , J=1,5) 55 SI (1) =0. SI (2)=0. DO 60 J=1,NO 60 P(J)=0. CALL SMOOTH(WL,RN,P,NO,SI,6,S90) CALL SHTH (LAMBDA,R,S1,S2,100,£90) M0=6*NO+1 M2=7*N0-1 M3=0 DO 65 J=H0,M2 65 M3=P (J) +H3 IF (M3.NE. 0) WRITE (6,24) M, (A (J,fl) ,J=1,5) DO 7 0 J=1,NO 70 P(J)=0. CALL SMOOTH (WL, RK ,P, NO , SI, 6 ,&90) CALL SMTH (LAMBDA,S,S1,S2,100,&90) £53 = 0. DO 75 J=£l1,M2 75 M3=P(J)+M3 IF{M3.NE.0) WRITE{6,26) M, (A{J,M) ,J=1,5) IF(M.EQ.K) GO TO 95 IF (I (1) > NE. 0) GO TO 82 DO 80 J= 1,100 80 NF (J,M) = {1. ,0. ) *R (J) - (0. , 1.) *S (J) GO TO 85 82 CONTINUE DO 83 J=1,100 REF (J) = R (J) 83 TR (J) =S (J) 85 CONTINUE 90 WRITE (6, 27) M, (A (J,M) ,J=1,5) GO TO 150 95 CONTINUE DO 100 J=1,100 ALPHA (J) = S (J) 100 NF (J,M)= (1. ,0.) *R (J) - (0. ,1.) *S (J) *LAMBDA (J) * 1. E-4/4. /3, 141 592 6 IF(I(22).EQ.O) GO TO 108 WRITE (6,28) DO 105 J = 1 , 1 0 0 105 WRITE (6, 29) LAMBDA (J) , (NF (J,M) ,M = 1,K) 108 I F ( I (26) ,EQ.O) GO TO 110 WRITE (6,31) WRITE(6,32) (LAMBDA (J) , ALPHA (J) ,J=T,100) 110 IF (1(6) .EQ.O) GO TO 135 READ (5, 21) (A (J,K+1) ,J=1,5) ,D1,N0 READ (5, 33) (ML (J) ,RN(J) ,J=1,NO) IF (WI (1) . LE. LAMBDA (1) ) GO TO 115 WRITE (6,34) GO TO 150 115 IF (WL (NO) .GE. LAMBDA (100) ) GO TO 120 WRITE (6,34) GO TO 150 120 IF=6 DO 125 J=1,NO 125 P (J) =0. CALL SMOOTH(WL,RN,P,NO,SI, IF , & 130) CALL SMTH (LAMBDA,SOL,S1,S2,100,&130) ISPCH=0. ISPCW=SCL (1) +5. *SOL (100) DO 128 J=2,98,2 128 ISPOW=ISPGW+4.*SOL(J) +2.*SOL(J + 1) ISP0H=ISPOB/1500. IF (I (25) .EQ.O) GO TO 135 WRITE (6, 3 6) (A(J,K + 1) ,J=1,5) WRITE (6,32) (LAMBDA (J) ,SCL(J) ,J=1, 100) GO TO 135 130 WRITE (6,37) GO TO 150 135 IF (I (8) .NE.O) GO TO 140 READ(5,33)EGAP,DNV,DNC,PHIBN,EP SB,ASTAS,T,SURV EL READ (5,33)FREEN,EFERM READ(5,38)NO READ (5, 39) (RND(J) ,DN (J) ,DP(J) ,TAUN(J) ,TAUP (J) ,J=1, DO 136 J=1,NO 136 END (J) = ALOG 10 (BND (J) ) J1=1 DO 137 J=1,NO 137 P(J)=0. CALL SMOOTH(RND , DN , P , NO, SI, 6 , & 142) CALL SMTH (ZND,FDN,S1,S2,50,&142) J1=J1+1 DO 138 J=1,NO 138 P(J)=0. CALL SMOOTH(BND,DP,P,NO,SI,6,6142) CALL SMTH (ZND,FDP,S1,S2,50,S142) J1=J1+1 DO 139 J=1,NO 13 9 P(J)=0. CALL SMOOTH(RND,TAUN,P,NO,SI,6,& 142) CALL SMTH (ZND,FTAUN,S1,S2,50,&142) J1=J1+1 DO 141 J=1,NO 141 P(J)=0. CALL SMOOTH(BND,TAOP,P,NO,SI,6,S142) CALL SMTH (ZND,FTAUP,S1,S2,50,S142) GO TO 143 142 WHITE (6,41) J1 GO'TO 150 143 IF (1(9) . NE. 0) GO TO 140 BEAD(5,42)EN IF (1(11) .EQ.O) GO TO 140 BEAD (5,19)Bl,a2,B3,B7,B8,BOM WHITE (6,43)B1,R2,83,B7,B8,R0H 140 CALL TRREF IF (I (8) .NE.O) GO TO 145 CALL QUANT IF (I (9) . NE. 0) GO TO 145 CALL EFF 145 IF (I (30) .EQ.O) GO TO 5 CALL VABY GO TO 5 150 STOP 11 FORMAT (•1«,33 (»*--*')) 12 FORMAT(20A4) 13 FORMAT('0*,20A4) 14 FORMAT(«0',33('*--*') ) 16 FORMAT (8011) 87. 17 FORMAT(F10. 3 ,12) 18 FORMAT('INPUT OPTICAL DATA, PRINTING FORMAT OF ML , N , K EXCEPT SUBST 1 RATE WHICH IS WL,N,ALPHA') 19 FORMAT (6G10.3) 21 FORMAT (5A4,F10.3,12) 22 FORMAI(«*CAUTION* SHORT WAVELENGTH OPTICAL DATA INSUFFICIENT, LINE 1AR EXTRAPOLATION USED ON FILM »,I2,2X,5A4) 23 FORMAT {* *CAUT.ION* LONG WAVELENGTH OPTICAL DATA INSUFFICIENT, LINE 1AR EXTRAPOLATION USED ON FILM »,I2,2X;5A4) 24 FORMAT(* EXTRANEOUS INFLECTION POINT IN N FOR FILM ',I2,2X,5A4) 26 FORMAT{* EXTRANEOUS INFLECTION POINT IN K FOR FILM •,I2,2X,5A4) 27 FORMAT(*INTERPOLATION ROUTINE FAILED ON FILM ',I2,2X,5A4) 28 FORKAT{'INTERPOLATED OPTICAL DATA *,/,5X,'LAMBDA',2X,5(4X,*N',9X,'K 1',5X),/) 29 FORMAT (11F10.3,/,10X, 1 OF 10. 3) 31 FORMAT{'ABSORPTION COEFF. FOR SUBSTRATE',/,6{2X,'LAMBDA•,5X,'ALPHA 1',2X) ) 32 FORMAT (1X,12G10.3) 33 FORMAT (8G10. 3) 34 FORMAT('INPUT SOLAR DATA INSUFFICIENT, MUST EXTEND FROM 0.3 UM TO 12. 28 UM») 36 FORMAT (* SPECIAL SOLAR BAIA',2X,5A4,/,6 (2 X, ' LAMBDA' ,3X, * SOL POW»,2X 1) ) 37 FORMAT('INTERPOLATION ROUTINE FAILED ON SOLAR DATA GIVEN') 38 FORMAT (12) 39 FORMAT(5G10.3) 41 FORMAT (•INTERPOLATION FAILED ON DATA FOR J1 = »,I2,', J1=1,2,3,4 FOR 1 DN,DP,TAUN,TAUP RESPECTIVELY') 42 FORMAT (F10.3) 43 FORMAT(1X,'H1=CONTACT STRIP RESISTANCE^',G10.3,'OHMS R2=CONTACT R 1ESISTANCE BETWEEN METAL AND TOP ELECTRODE= *,G10.3,'OHMS•,/,1R3=EES 2ISTANCE OF GRID STRIP =*,G10.3,* OHMS R7=CONTACT RESISTANCE OF BUL 3K REGION TO BOTTOM ELECTRODE=',G10.3,•OHMS•,/,•S8=RESISTANCE OF BO 4TT0M ELECTRODE=',G10.3,» OHMS BABIMER METAL RESISTIVITY=',G10.3,'0 5HM-CM') 44 FORMAT('0','INPUT DATA BEING ACCEPTED') 46 FORMAT (1X,'OPTIONS WITH A 1 CODE ARE AS FOLLOWS;') 47 FORMAT(1X,12) END. BLOCK DATA COMPLEX NF (100,10) INTEGER 1 ( 8 0 ) REAL D(10) ,FDN{50) ,FDP (50) ,FTAUN(50) ,FTAUP (50) , IS POW, L AMBD A (1 00) ,S 10L (100) ,ZND(50) COM MO N/DIFTAU/FDN,FDP,FTAUN,FTAUP,ZND COMMO N/VARIA/ASTAR,D,EN,I,ISPOW,K,LAMBDA,NF,PHIBN,PHIM,SOL, SURVEL, 1T DATA LAMBDA/. 3000002,.32,. 34,. 36,.38,. 4,. 4 2,. 44,. 4 6,. 48,.5,.52,.54 1 ,.56,.5 8,.6,.62,.64,.66,.68,.7,.72,.74,. 76,.78,.8,.82,. 84,.86 ,.88, 2.9,.92,.94,. 96,.98, 1., 1.02,1.04,1.06,1.08, 1. 1, 1.1 2,1. 1 4, 1. 16,1.18, 31.2,1.2 2,1.24,1.26,1.28,1.3,1.32,1.34,1.36,1.38,1.4,1.42,1.44,1.46 4,1. 48,1 .5,1. 5 2, 1. 54, 1. 56,1.58,1.6, 1.62,1.6 4,1. 66, 1.68, 1.7, 1.72,1.7 54,1.76,1.78,1.8,1.82,1.84,1.86,1.88,1.9,1.92,1.94,1.96,1.98,2.,2.0 6 2,2.04,2.06,2.0 8,2.1,2.12,2.14,2.16,2.18,2.2,2.22,2.24,2.26,2.2799 798/ DATA ZND/14.1,14.2,14.3,14.4,14.5,14.6,14.7,14.8,14.9,15.,15.1,15. 12,15.3,15.4,15.5,15.6,15.7,15.8,15.9,16.,16.1,16.2,16.3,16.4,16.5, 116.6,16.7,16.8,16.9,17.,17.1,17.2,17.3,17.4,17.5, 17.6,17.7,17.8,17 1.9,18.,18.1,18.2,18.3,18.4,18.5,18.6,18.7,18.8,18.9,18.99998/ END 88. SOBBOOTINE TfiBEF INTEGER I (80) REAL ALPHA(IOO) ,D(10) , ISPOH, LAMBDA (1 00) ,REF(100) ,S01(100) ,TR(100) COMPLEX A,B,CPHI(10) ,DEL(10) , NF (100, 10) , R (10) , RO, T (10), TAD" . COMPLEX CSQRT,CEXP,CONJG COMMON/OPTIC/ALPHA,REF,TB COMMQN/VARIA/ASTAB,D,EN,I,ISPOH,K,LAMBDA,NF,PHIBN,PHIM,SOL,SUEVEL, 1T1 IF (I (1) .NE.O) 3ETURN PHI=PHIM*3.1415926/180. KR1=K-1 KR2=K-2 DO 50 J= 1,100 SIN2=NF ( J , 1) *NF (J , 1) *SIN (PHI) **2 BETA=2.*3.1415926/LAM8EA(J) CPHI (1)=CSQRT ( (1 . ,0.) * (1.-SIN (PHI) **2) ) CPHI (K) =CSQBT (1.-SIN2/(NP (J,K) *NF (J,K) ) ) DO 10 M=2,KB1 CPHI (M) =CSQBT (1.-SIN2/ (NF(J,M) *NF (J,M) ) ) 10 DEL (M)^BETA*CPHI (M) *NF (J,M) *D (M) *1. E-4 IF (PHIM.EQ.O.) GO TO 30 IF (I (4) .EQ.O) GO TO 20 IF (I (5) . NE.O) GO TO 30 C P LIGHT CALCULATION 20 CONTINUE DO 24 M=2,K A=NF (J,M-1) *CPHI (M) B=NF (J,M) *CPHI (M-1) B (M) - (A-B)/(A + B) 24 T(M)=2.*NF(J,M-1) *CPHI (M-1)/(A + B) RO=B(K) TAU=T (K) DO 25 M=1,KR2 A=BO*CEXP ( (0. ,-2.) *DEL (K-M) ) B=1.+B (K-M) *A BO= (fi (K-M) +A) /B 25 TAU=T (K-M) *TAO*CEXP ( (0. ,-1.) *DEL (K-M) ) /B BEF (J)=BO*CGNJG (BO) TR (J) =TAU*CONJG (TAU) *BEAL (NF(J,K) /CPHI (K) ) *CPHI (1) /NF ( J , 1) IF (I (4) .NE.O) GO TO 40 C S LIGHT CALCULATION 30 A=NF (J, 1) *CPHI (1) DO 34 M=2,K B=NF (J, M) *CPHI (M) B (M) = (A-B) / (A+B) T (fl) -2. * A/ (A + B) 34 A=B BO=B (K) TAU=T (K) DO 35 fl=1,KR2 A=BO*CEXP ( (0.,-2.)*DEL(K-M) ) B=1.+R (K-M) *A RO= (R (K-M) +A) /B 3 5 TAU=T (K-M) *TAU*CEXP { (0. ,-1.)*DEL (K-M) ) /B IF (PHIM.EQ.0..OR.I(5) .NE.O) GO TO 36 REF (J) = (REF (J) +RO*CONJG (10) ) /2. TR (J) = (TR <J) +TAU*CONJG (TAU) *BEAL (NF (J,K) *CPHI (K) } / (NF ( J , 1) *CPHI (1) 1>)/2. GO TO 40 36 REF (J) =RO*CONJG (RO) 89. TR (J) =TAU*CONJG (TAU) *HEAL (NF (J,K) *CPHI (K) ) / (NF (J, 1 )*CPHI (1) ) 40 IF (REF (J) +TS (J) . GT. 1 . ) I (3) =1 50 CONTINUE IF (I (3) .EQ.O) GO TO 60 WRITE (6, 13) STOP 60 WRITE (6,9) IF (I (23) .EQ.O) RETURN WRITE (6,11) WRITE (6,12) (LAMBDA (J) , BEF(J) ,TE(J) ,J=1,100) RETURN 9 FORMAT{'0 *,'CALCULATION OF OPTICAL CHARACTERISTICS OF THE FILMS ON 1 THE SUBSTRATE') 11 FORMAT ('REFLECTANCE AND TRANSMITTANCE OF FILMS OS SUBSTRATE VS. WA 1VELENGTH•,/, 4(• LAMBDA REEL. TRANS. •)) 12 FORMAT (1X,12G10.3) 13 FORMAT(1 ERROR IN REF. OR TRANS. CALCULATION, (REF + IS .GT. 1.)') END SUBROUTINE EPF COMPLEX NF{100,10) INTEGER I (80) REAL A (3) ,D(10) ,ISPOW,LAMBDA {100) ,RS(3) ,SOL(100) COMMON/CUBR/RJL,R1,R2,83,R7,R8,ECB,R0M COMMQN/VASIA/ASTAR, D, EN ,1, ISPOW , K,.LAMBDA , NF,PHIBN, PHI 14 , SOL , SUE VEL, 1T DATA A/.8 55,.81,.72/ WRITE (6,18) B=ASTAB*T*T VT=8.6170 87E-5*T EJ0=B*EXP (-PHIBN/VT) DO 5 J=1,3 5 RS(J)=0. M= 1 IF (I (11) .EQ. 0) GO TO 30 M=3 B6=EOB*D (K) /2. *1. E-4 IF (1(12) .EQ.O) GO TO 10 S=. 4 »=. 9 RA3=.26724 THETA=ATAN (S/2./EA3) B4=EOM*BA3/S/D(K-1) *1.E8*COS (THETA) E=W/ (W-B A3) F=.5*ROM*S/D(K-1)* 1.E8/BA3*SIN{THETA) R5=F* ( (. 5+ALOG (E) ) -«*»/ (2. *W~BA3) /RA3*ALOG (E) ) BC= (H2 + B4) *(1 .+R1/(E3 + (E2 + B5) * . 5 ) +R1/ (R1 + R2 + R4) ) / (2. + (B1+B2+B4) / (E 13+(B2 + B5) *.5) ) RP=2.*RC*(BC + R1)/4./ (2.*RC+R1) BS (1) =BC/ (1 . + BC/RP) +R1+S6+B7+B8 10 I F ( I (13) .EQ.O) GO TO 20 S=.2 W=. 9 BA3=. 13762 THETA=ATAN(S/2./RA3) B4 = BOM*BA3/S/D (K- 1) * 1 . E8*COS (THETA) E=W/ (W-BA3) F=.5*BOM*S/D(K-1)*1.E8/BA3*SIN(THETA) B5=F* ( (.5 + ALOG (E) ) -W*W/ (2. *W-RA3) /RA3*ALOG (E) ) RC= (R2+B4) *(1 . + B1/ (B3+ (E2 + B5) *. 5) +81/ (R1+B2+B4) ) / (2.+ (B1+E2+E4) / (fi 13+(B2+B5) *.5) ) 90. RP=2. * B C * (BC+B1) /9./ (2. *RC+B1) RS (2) = RC/(1.+RC/RP) +R1+R6 + R7+R8 20 IF (I (14) . EQ. 0) GO TO 30 S=. 1 W=. 9 BA3=.069772 THETA=ATAN (S/2./RA3) R4=ROM*RA3/S/D (K-1) *1 . E8*COS (THETA) E=W/ (W-RA3) F=.5*ROM*S/D(K-1)*1.E8/BA3*SIN (THETA) R5=F* ((, 5 + ALOG (E) ) -W*W/ (2. *W-RA3) /RA3*ALOG (E) ) RC= (R2+R4) *(1.+B1/ (R3+ (R2 + R5) *.5) +B1/(R1+R2+R4) ) / (2.+ (R1 +R2+B4) / (R 1 3+(B2+R5) *. 5) ) BP=2.*RC*(RC+R1)/19./(2.*BC+B1) RS (3) =RC/ (1. + RC/RP) +R1+R6+R7 + R8 30 WRITE (6,11) WRITE (6,12) PHIBN,EN,ASTAB,!,RJO,ISPOW DO' 50 J= 1, M IF (M.NE. 1) GO TO 35 RJ=RJI WRITE (6,19) SCJ=-RJL GO TO 4 0 35 IF (RS (J) .EQ.O) GO TO 50 RJ=BJ1*A (J) CALL ISCJ (VT,RJO,RJ,EN,RS (J) ,SCJ) 40 CALL IMAXP (VT,RJO,RJ,EN,RS (J) ,RJMP) VMP=EN*VT*ALQG((RJMP+RJ+RJO)/RJO)+RJMP*RS(J)*2. PMP=VMP*RJMP VOC=EN*VT*ALOG ( (RJO + RJ)/RJO) ETA=-PMP/ISPOW*100. FF=PMP/SCJ/VOC IF (RS (J) . EQ. 0) GO TO 45 GO TO (41,42,43),J 41 WRITE (6,13) GO TO 45 42 WRITE (6,14) GO TO 45 43 WRITE (6,16) 45 WRITE (6 , 17) RJ , BS (J) ,VOC,SCJ,VMP,BJMP,PMP,FF,ETA 50 CONTINUE RETURN 11 FORMAT (9X , ' PHIBN (EV) • ,10.x,1 QUALITY FACTOR N',4X,'ASTAR',15X,'TEMP 1. (DEG.K) • ,7X,«REV. CURB. (A/CM2)',2X,'INC. POWER (W/CM2)') 12 FORMAT(1X,6G20.5) 13 FORMAT(1X,'5 FINGER GRID STRUCTURE FRACTIONAL ACTIVE AR3A=. 855') 14 FORMAT(1X,'10 FINGER GRID STRUCTURE FRACTIONAL ACTIVE AREA=.81') 16 FORMAT{1X,* 20 FINGER GRID STRUCTURE FRACTIONAL ACTIVE AREA=.72') 17 FORMAT(1X,'GENERATED CURB.=•,G14.5,»A/CM2 SERIES RES.=',G14.5,'OH 1MS',/,'OPEN CIS. VOLTAGE=*,G14.5,'VOLTS SHORT CIR. CURB.=',G14.5, 2'A/CM2',/,'MAXIMUM POWEB POINT VOLTAGE, CURRENT, AND POWER',/,1X,G 314.5,'VOLTS',G14.5,'A/CM2',G14.5,'W/CM2*,/,'FILL FACTORS•,G14.5,•E 4FFICIENCY=•,G14.5,'PER CENT') 18 FOBMAT ('0','EFFIENCY CALCULATION BASED ON QUANTUM YIELD OBTAINED') 19 FORMAT (•SERIES RESISTANCE ASSUMED=0') END SUBROUTINE IMAXP(VT,RJ0,RJL,EN,RS,RJMP) A=4.*RS/EN/VT RJMP=-RJL STEP=.01 10 BJMP=BJHP+STEP B=1./ (RJMP+8JL+RJ0) IF( (A+B) *RJHP.GT.ALOG(B*RJ0) ) GO TO 20 GO TO 10 20 RJMP=RJMP-STEP STEP=STEP/10. IF (STEP.GT. 1. E-7*BJL) GOTO 10 BETURN END SUBROUTINE ISCJ (VT,RJO,RJI,EN,RS,SCJ) A=-2.*RS/EN/VT SCJ=-BJL STEP=.01 10 SCJ=SCJ+STEP X=.A*SCJ IF (X.GT.170.) X=170. IF( (SCJ + SJL+RJO)/RJO.GT.EXP (X)) GO TO 20 GO TO 10 20 SCJ=SCJ-STEP STEP=STEP/10. IF (STEP.GT.1,E-7*8JL) GO TO 10 BETUBN END SUBROUTINE QUANT COMPLEX NF(100,10) INTEGER I (80) REAL ALPHA (100) ,BULKJ (100) ,D (10) ,DEPJ (100) , SFIELD (101) ,EX1 (100) , EX 12 (100) ,EX3 (100) ,.F (2) , FDN (50) ,FDP (50) , FTAUN (50) ,FTAUP (5 0) ,G (100) ,GB 2 (10 001) ,GN (101) ,GP (101) ,G1 (2) ,ISPOW,LAMBDA (100) ,SEF(100) ,S (2) ,SOL( 3100) ,S0L1 (100) ,SOL2(100) , 1 1 ( 2 ) ,TB(100) ,V(2) ,VOLT(101) ,XB(10001) ,XL 4(101),XN(101),XP (101) , .ZND (50) , Z l NV (100) , ZZD (1 00) ,ZZB(100),ZZIN(100 5) ,ZZT (100) BE AL*8 CB (2) ,CN (2) ,CP (2) ,DCPW1,DCPS2,DES1,DES2,DS (2) ,E1D,FD{2) ,G1D 1 (2) ,HD,HD2,RD3,HMIND,HMIN2,HMIN3,SD,TID(2) ,XD,XI,X2,ZD,ZSED EXTERNAL V FU NCN,VFU NCP,EN FU NC,PFU NC,BFUNC COMMON/BULK/BD,BTAU CGMM0N/CUER/RJL,S1,E2,B3,R7,R8,aCB,ROM COHMON/DIFTAU/FDN,FDP,FTAUN,FTAUP,ZND COM MO N/G ENEB/EFIELD,GB,GN,GP,X B,XL,X N,XP CO M MO N/OPTIC/ ALPHA,BE.F,TB COMMON/PROP/EPSB,DNV,DNC,VT,EGAP CO M.fl 0 N/ P B0PM/FB E EN , E F E R M COM MON/VAEIA/ASTAB,D, EN,I,ISPOW,K,LAMBDA, NF, PHIBN ,PHIM,SOL,SUBVEL, 1T DATA SOL1/.545,.786,1.089,1.16,1.108,1.6 92,1.725,1.735,1.9 46,2.115 1 ,1. 927, 1.9 94, 1.852, 1.712,1.886,1.816, 1.794,1.685, 1.68, 1.578,1 .495, 21. 367,1 .316,1. 269,1. 223,1. 154,1.079,1.027,.96 8,.935,.889,. 835,. 821 3,.792,.7 72,.741,.716,.679,.65,.621,.608,.578,.556,.538,.512,.492,. 447 5,.466,.432,. 424,. 427,. 407 , . 3.9, . 372,. 359 , . 349,. 34,. 3 25, . 31 6 , . 30 3 5,.297,.288,.278,.273,.259,.252,.245,.238,.237,.225,.221,.209,.195, 6.188,.174,.172,.162,.156,.146,.143,.136,.137,.132,.128,.127,.118,. 7115,.109,.102,.099,.095,.089,.092,.084,.077,.084,.079,.061,.072,.0 87/ DATA SOL2/.125,.44,.675,.79,.9,1.12,1.4,1.66,1.83,1.83,1.77,1.7,1. 163,1. 55, 1 .5,1 . 51 ,1. 52,1. 46,1. 38, 1. 33,1.3, 1.23, 1.1 5,1. 0 8,1. 02,. 974, 2.928,.885,.846,.811,.7 8,.752,.726,.701,.677,.654,.631,.608,.585,.5 363,.54,. 517,.495,.472,.45,.428,.406,. 384,. 363,.343,.32 5,. 3 09,. 296, 4.284,.2 73,.26 3,.254,.245,.236,.228,.22,.212,.204,.196,.189,.181,.1 573,. 166,. 159,. 152,. 145,. 139,. 133,. 127 ,. 122,. 1 17,. 1 13,. 109,. 106,. 10 €3,. 1,.0 977,.0 956,.0937,.092,.0902,.0886,. 0869,.0853,. 0 837,.082,.08 92. 704,.0788,.0772,.0756,.074,.0724,.0708,.0 692,.0676/ DATA SPOW1,SPOW2/.135,.106/ WRITE (6,11) IF (I (6) . NE. 0) GO TO 20 IF {I (7) . NE. 0) GO TO 10 ISPCW=SPO»1 DO 5 J = 1 , 1 0 0 5 SOL (J)=SOL1 (J) GO TO 20 10 ISPCW=SPOW2 DO 15 J= 1,100 15 SOL (J) =SOL2 (J) C EVALUATION OF G AT X=0 20 AK=1.38054E-23 AKT=AK*T C=2.997925E8 H=6.6256E-34 EPS0=8.854185E-12 E=1.6021E-19 EMASS=9.109558E-31 PI=3.1415926 A=2.*SQRT (2.) *PI*EMASS**1.5/H/H/H/FEEEN B=H*C VT=AKT/E COSPM=COS (PHIM*3. 14159 26/180.) DO 22 J=1 , 100 22 SOL (J)=SOL (J) *LAMBDA (J) /B*1. E-7*COSPf! IF (I (10) .NE.O) GO TO 80 DO 45 J=1,99 U= (E* 1.E6/LAMBDA(J) -E*PHIBN)/AKT IF (U.NE.O.) GO TO 25 G (J)=A*AKT*AKT*PI*PI/12./SQRT (E*EFERM) *SOL (J) /D (K-1) * (1. -REF (J) -TR 1 (J) ) *1. E2 GO TO 45 25 X=ABS (U) SUM=0. COUNT=1. SIGN=1. XI= 1. EXPX=EXP (-X) 30 XI=XI*EXPX IF (XI.LT.1.E-20) GO TO 35 SUM=SUM+SIGN*XI/COUNT/CCUNT SIGN=-SIGN COUNT=COUNT+1. GO TO 30 35 IF(U.GT.O.) GO TO 4 0 G (J)=A*AKT*AKT/SQRI (E*EFERM-AKT*U) * (1 .-REF (J) - TR (J) ) *SOL (J)/D (K-1) 1*SUH*1.E2 GO TO 45 40 G (J) =A*AKT#AKT/SQRT (E*EFERM-AKT*U) * (1. -REF (J) -Tfi(J) ) *SOL (J)/D (K-1) 1*(PI*PI/6.+U*U/2.-SUH)*1.E2 45 CONTINUE G0=0. DO 50 J=2,96,2 50 G0=G0+4. *G (J) +2. *G (J+1) G0= (G0+4. *G (9 8) +G (9 9) ) * (2. 26-. 3) /3./9 8. BARJ=-E*D (K-1)*1.E-8*G0 C SOLVING FOR V,E, AND DE/DX VS X FOR N TYPE WRITE (6,16) x=o. CALL ND (X,ENDX,DNDX,DNDXO) X=DNDX0 CALL ND (X,ENDX1,DNDX,DNDXO) DDBAR— (ENDX1+ENDX) /2. PHIN=-VT*ALOG(ENDX1/DNC) W=SQRT(2.*EPS0*EPSR*(PHIBN-PHIN)/E/DDBAR) *1.E3 I F (W.LT.DNDXO) W=DNDXO Z=0. E1 = 1. E-4 60 WI = W H= ( Z - I I ) / 6 4 . HMIN=H/100. V(1)=-PHIN V ( 2 ) = 0 . CALL RKC(2,WI,Z,V,F,H,HMIN,E1,VFUNCN,G1,S,TI) DELTA=V (1)+PHIBN I F ( I (28) .NE.O) WRITE (6, 14) V (1) ,V (2) ,DELTA, W I F (ABS (DELTA/PHIBN) .LT. 1.£-4) GO TO 65 I F (ABS (V (1) ) .LT.EGAP+. 01) GO TO 62 W=W*.99 GO TO ,60 62 W=W+DELTA*SQRT(2.*EPSO*EPSR/E/DDBAR/PHIBN) *1.E2 GO TO 60 65 ZH=-W/100. Z=W+ZH WI=W V (1)=-PHIN V(2)=0. DO 70 J = 1,100 X L ( 1 0 1 - J ) = Z I F ( J . E Q . I O O ) XL ( 1 ) =0 . CALL RKC(2,WI,Z,V ,F, H , HMIN,E1 , VFUNCN,G 1, S , TI) V O L T ( 1 0 1 - J ) = V (1) E F I E L D (101-J) =-V (2) * 1 , E4 70 Z=Z + ZH VOLT (10 1) =-PHIN E F I E L D (101)=0. XL (101)-0 EGAP2=-EGAP/2. DO 75 J = 1,101 I F (VOLT (J) .GT.EGAP2) GO TO 105 75 CONTINUE C SOLVING FOR V,E, AND BE/DX VS X FOR P TYPE 80 WRITE ( 6 , 17) X=0. CALL ND (X,ENDX,DNDX,DNDXO) X=DNDX0 CALL ND (X,ENDX1 ,DNDX , DNDXO) DDBAR=(ENDX+ENDX1)/2. PHTP=~VT*ALOG(ENDX1/DNV) W=SQRT (2.*EPS0*EPSR*(PHIBN-PHIP) /E/DDBAR) *1.E3 I F (W.LT.DNDXO) W=DNDXO Z=0. £1 = 1. E-4 fl=(Z-HI)/6 4. HMIN=H/100. 85 WI=W V(1)=PHIP V (2)=0. CALL BKC(2,WI,Z,V,F,H,HMIN,E1,VFUNCP,G1,S,TI) DELTA=V (1)-PHIBN IF (I (28) .NE.O) WHITE (6,14) V (1) ,V (2) ,DELTA,W IF ( ABS (DELTA) /PHIBN. LT. 1.E-4) GO TO 90 IF (ABS (V (1) ) . LT. EGAP+. 01) GO TO 87 W=W*.99 GO TO 85 87 W=W-DELIA*SQRT(2.*EPSO*EPSR/E/DDBAB/PHIBN) *1. E2 GO TO 85 90 ZH=-W/100. Z=W + ZH WI=W V(1)=PHIP V (2)^0. DO 95 J=1,100 XL(101-J)=Z IF(J.EQ.IOO) XL(1)=0. CALL HKC(2,WI,Z,V,F,H,HMIN,E1,VFUNCP, G1, S, TI) VOLT (101-J) =V (1) EFIELD (101-J) =-V (2) *1. E4 95 Z=Z+ZH VOLT (101)=PHIP EFIELD (101) =0. XL(101)=W EGAP2=EGAP/2* DO 100 J=1,101 IF (VOLT (J) .LT.EGAP2) GO TO 105 100 CONTINUE 105 XI= (XL (J) -XL (J-1) ) * (EGAP2-VOLT (J-1) )/(VOLT (J) - VOLT (J-1) ) + XL(J-1) DO 55 J=1,100 55 G (J) —TR (J) *SOL (J) *ALPHA (J) IF (I (15) +1 (27) . EQ.O) GO TO 145 IF(XI.LE.O.) GO TO 120 DO 115 J= 1,101 XN (J) =XI*FLOAT (J-1) *1. E-6 GN(J)=0. DO 110 M=2,96,2 110 GN (J) =GN (J) + 4.*G (M) *EXP (-ALPHA (.8) *XN (J) ) + 2.*G (M+1) *EXP (-ALPHA (M+1) 1*XN (J) ) GN (J) = (GN (J) +G (1) *EXP (-ALPHA (1) *XN (J) ) +4. *G (98) *EXP(-ALPHA (98) *XN ( 1 J) ) +G (9 9) *EXP (-ALPHA (9 9) *XN (J) ) ) * (2. 26-. 3) /98./3, 115 XN(J)-XM(J) *1.E4 120 IF (XI.LE.Q.) XI=0. W1= (W-XI)/100. DO 130 J=1,101 XP (J) = (W1*FL0AT (J-1) +XI) *1. E-4 GP(J)=0. DO 125 M=2,96,2 125 GP (J) =GP (J) + 4. *G (M) *EXP (-ALPHA (M) *XP (J) ) + 2.*G (M+1) *£XP (-ALPHA (M+1) 1*XP (J) ) GP (J) = (GP (J) +G(1) *EXP (-ALPHA (1) *XP (J) ) +4. *G (98) *BXP (-ALPHA (98) *XP ( 1 J) ) +G (99) *EXP (-ALPHA (99) *XP (J) ) ) * (2.26-.3) /98./3. 130 XP (J) =XP {J) * 1. E4 BW= (D (K) -W) /10000. DO 140 J=1,10001 XB (J) = (BW*FLOAT (J-1) +W) *1.E-4 GB(J)=0. DO 135 M=2,96,2 135 GB (J) =GB (J) +4. *G (M) *EXP (-ALPHA (M) *XB (J) ) +2.*G (11 + 1) *EXP (-ALPHA (M+1) 1*XB(J) ) 95. GB (J) = (GB (J) +G (1) *EXP (-ALPHA (1) *XB (J) ) +4. *G (98) *£XP (-ALPHA (9 8) *XB ( 1 J) ) +G (99) *EXP (-ALPHA (99) *XB (J) ) ) * (2. 26-. 3) /98./3. 140 XB (J) =XB(J) *1. E4 145 IF (I (15) .WE. 0) GO TO 150 WRITE (6,21) GO TO 155 150 WRITE (6,23) 155 IF (I (27) .EQ.O) GO TO 160 WRITE (6,12) GO WRITE (6, 18) WRITE (6,14) (XN (J) ,GN (J) ,J=1,101) WRITE (6,18) WRITE (6,14) (XP (J) ,GP (J) , J=1, 101) WRITE (6, 18) WRITE (6, 14) (XB (J) ,GB (J) , J= 1 , 10001 ,50) 160 IF (I (29) .EQ.O) GO TO 165 WRITE (6,13) WRITE (6, 14) (XL(J) ,VOLT(J) ,EFIELD (J) ,J= 1,101) C SOLVING FOR SHORT CTR. J USING MILNES METHOD 165 CALL ND (W,ENW,DNDX,DNDXO) IF (1(10) .EQ.O) GO TO 170 CALL DTEVAL(FDN,ZND,ENW,BD, DERDN) CALL DTEVAL(FTAUN ,ZND , EN W , BTAU , DTAUN) CALL DTEVAL(FDP,ZND,ENW,DP,DESDP) fiOB=VI/ENW/E/DP GO TO 175 170 CALL DTEVAL(FDP,ZND,ENW,BD,DERDP) CALL DTEVAL(FTAUP,ZND,EN1,BTAU,DTAUP) CALL DTEVAL(FDN,ZND,ENW,DN,DERDN) ROB=VI/INW/E/DN 175 EL=SQRT (BD*BTAU) IF (1(15) .NE.O) GO TO 195 DI N=D (K) * 1. E-4 XIN=XI*1.E-4 WN=W*1.E-4 AE=EXP( (DIN-WN)/EL) BE=1./AE DO 180 J-1,10 0 EX 1 (J) = EXP {-ALPHA (J) *XIN) E.X2 (J)=EXP {-ALPHA (J) *WN) EX3 (J) =EXP (-ALPHA (J) *DIN) ZINV (J) =-E*TR (J) *SOL (J) * (1.-EX1 (J) ) DEPJ (J)=-E*TR (J) *SOL ( J ) * (EX1 (J) -EX2 (J) ) 180 BULKJ (J) =-S*ALPHA (J) *TH (J) *SOL (J) *EL/ (ALPHA (J) * ALPHA (J) *EL*EL-1.) * 1 { (ALPHA (J) *EL-1.) *EX2 (J) +2. * (EX3 (J)-EX2 (J) *BE) / (AE-BE) ) ZIN=0. DEP=0. BULK=0. DO 185 J=2,96,2 ZIN=4.*ZINV (J) +2.*ZINV (J+1) +ZIN DEP=4.*DEPJ (J) +2. *DEPJ (J+1) +DEP 185 BULK=4.*BULKJ(J)+2,*BU1KJ(J+1)+BULK ZIN= (ZIN + ZINV (1) +4. *ZINV (98) +ZINV (99) ) * (2. 26-. 3) /98./3. DEP= (DEP+DEPJ (1) +4. *DEPJ (98) +DEPJ (99) ) *{2. 26-.3) /98./3. BULK= (BULK+3ULKJ (1) +4.*BULKJ (98) +BULKJ (99) ) * (2. 26-.. 3) /9S./3. IF (I (16) .EQ.O) GO TO 187 DO 186 J=1,100 ZZIN (J) =-ZINV (J) /E/SOL (J) ZZD (J) •=—DEPJ (J) /E/SOL (J) ZZB (J) =-BULKJ (J) /E/SOL (J) 186 ZZT (J) =Z2IN (J) +Z2D (J) + ZZB (J) 96. 'WRITS (6,34) WRITE (6,3 2) WRITE (6,33) (LAMBDA (J) ,ZZIN (J) ,ZZD(J) ,ZZB (J) ,ZZT (J) ,J=1 ,100) DO 188 J=1,100 ZZIH (J) =ZZIN (J) /TR (J) ZZD (J)=ZZD (J) /TR (J) ZZB (J)-ZZB (J) /TS (J) 188 ZZT (J) = ZZT (J) /TS (J) WRITE (6,36) WRITE (6,3 2) WRITE (6,33) (LAMBDA (J) ,ZZIN (J) ,ZZD (J) ,ZZB (J) ,ZZT (J ) ,J=1 ,100) 187 IF(I(24).EQ.O) GO TO 190 WRITE (6,24) WRITE (6,14) (LAMBDA (J) ,ZINV (J) ,DEPJ (J) ,BULKJ (J) ,J=1,10Q) 190 RJL= BARJ + ZIN +DEP +BULK IF (I (10) .EQ.O) GO TO 192 BARJ=0. ZIH=-ZIH DEP=-DEP BULK--BULK :RJL=Z:IN+DEP+BULK 192 WRITE (6,26)BARJ,ZIN,DEF,BULK,RJL RJL=ABS (RJL) GO TO 275 C NUMERICAL METHOD OF SOLVING FOR QUANTUM YIELD 195 ElD=1.D-7 HD=DBLE(-XI/6 4.) HMIND=HD*1.D-4 HD2=DBLE ( (W-XI) /64.) HMIN2=HD2*1.D-4 HD3=DBLE ( (D (K) - W)/64 . ) HMIN3=HD3*1.D-4 IF (I (10) .NE.0) GO TO 235 C BVP FOR N TYPE .CDN=0. IF (XI.LE.O.) GO TO 215 CALL DTEVAL(FDN,ZND,ENDX,BNO,DNDO) SD=BBLE(SURVEL/DNO) ZSED= (SD-DBLE (EFIELD (1 )-/YT) ) *1.D-4 X1 = 0.D0 X2=-1,D3 ITEB=1 200 XD=BBLE(XI) ZD=0.D0 CN (1) =X1 CN (2) =X2 CALL DRKC(2,XD,ZD,CN,ID,HD,HMIND,ElD,ENFUNC,G1D,DS,TID) IF (ITER. NE. 1) GO TO 205 DES1=CN (2)-ZSED*CN(1)-BELE(BARJ/E/DNO)*1.D-1Q IF (1(28) .NE.O) WRITE (6,31) ITER,X1 ,X2,CN (1) ,CN(2) ,DES1 DCPW1=X2 X2=100.*X2 ITER=ITER+1 GO TO 200 205 DES2=CN (2)-ZSED*CN(1)-EBLE(BARJ/E/DNO)*1.D-10 DCPW2=X2 IF (1(28) .NE.O) WRITE (6 , 5 1) ITER ,X 1, X2,CN (1) ,CN(2) ,DES2 IF (DABS (DES2/CN(2) ) .LT.1.D-10) GO TO 210 IF (DABS ( (DCPW2-DCPW1)/DCPW2).LT.2.D-15) GO TO 210 IF (ITER.GT.30) GO TO 210 X2= (DCPW2-DCPW1)/(DES2-DES1)*(-DES1) +DCPW1 DES1=DES2 DCPW1=DCPW2 ITEB=ITEB+1 GO TO 200 210 WRITE (6, 19) ITER CDN=SNGL (X2)*1.E10 215 X1=0.D0 X2=-X2 IT E 8=1 220 XD=DBLE(XI) ZD= DBLE (W) CP (1)—X1 CP(2)=X2 CALL DRKC(2,XD,ZD,CP,FD,HD2,HMIN2,E1D,PFUNC,G1D,DS,T1D) XD=ZD ZD=DB1E (D (K) ) CB(1)=CP{1) CB(2)=CP(2) CALL DRKC(2,XD,ZD,CB,FD,BD3,HMIN3,E1D,BFUNC,G1D,DS,TID) IF (ITER.NE.1) GO TO 2 25 DES1=CB(1) IF (I (28) .NE.O) WRITE (6,31) ITER , X1, X2 ,C B (1) ,CB (2) ,DSS1 DCPW1=X2 X2=100.*X2 ITER=ITER+1 GO TO 220 225 DES2=CB (1) DCPW2=X2 IF (I (28) .NE.O) WRITE (6 , 31) ITER, X 1, X2, CB (1) ,CB(2) ,BES2 IF (DABS (DES2) . LT. 1. DO) GO TO 230 IF (DABS ( (DCPW2-DCPW1)/DCPW2) .LT.2.D-15) GO TO 230 IF (ITER.GT.30) GO TO 230 X2= (DCPW2-DCPW 1) / (DES2-DES1) * (-DES1) +DCPW1 DES1=DES2 DCPW1=DCPW2 ITER=ITEB+1 GO TO 220 230 WRITE (6,19) ITEH CDP=SNGL (X2) *1.E10 CALL EGEN (XI, FIELD,.DEB E,GEN) CALL ND (XI,ENDX,DNDX,DNDXO) CALL DTEYAL (FDN,ZND,ENDX,DN,DEBDN) CALL DTEYAL(FDP,ZND,ENDX,DP,DERDP) SJN=E*DN*CDN SJP=-E*DP*CDP BJL=SJN+SJP WBITE (6,27)BARJ,SJN,SJE,BJL BJL=ABS (BJL) GO TO 275 C BVP FOR P TYPE 235 CDP=0. IF (XI.LE.O.) GO TO 255 CALL DTEYAL(FDP,ZND,ENDX,ENO,DNDO) SD=DBLS (SUBVEL/DNO) ZSED= (SD+DBLE (E.FIELD (1) /VT) ) *1. D-U X1=0. DO X2=-1.D3 ITE8=1 240 XD=BBLE(XI) ZD= 0.DO CP (1) =X1 CP (2) = X2 CALL DRKC{2,XD,ZD,CP,f D,HD,HMIN,E1D,PF0NC,GTD,DS, TID) IF(ITER.NE.1) GO TO 245 DES1=CP (2) -ZSED*CP (1) IF (I (28) .NE.O)WRITE(6,31)ITER,X1,X2,CP{1) ,CP(2),DES1 DCPH1=X2 X2=100.*X2 ITEB=ITEB+1 GO TO 240 245 DES2-CP (2)-ZSED*CP (1) DCPW2=X2 IF (1(28) .NE.O) WRITE (6,31) ITER,X1,X2,CP (1) ,CP(2) ,DES2 IF (DABS (DES2/CP(2)) .LT.1.D-10) GO TO 250 IF (DABS { (DCPW2-DCPW1)/DCPW2) . LI.2.D-15) GO TO 250 IF (ITER.GT.30) GO TO 250 X2= (DCPW2-DCPW1)/(DES2-DES1)*{-DES1) +DCPW1 DES1=DES2 DCPW1=DCPW2 ITER=ITER+1 GO TO 240 250 WRITE (6,19) ITER CDP = SNGL (X2)*1.E10 255 X1=0.DO X2=-X2 ITER=1 260 XD=DBLE(XI) ZD=DBLE (W) CN (1) =X1 CN (2) =X2 CALL DRKC (2,XD,ZD,CN,FD,HD2,HMIN2,E1D,ENFUNC,G1D, DS,TID) XD=ZD ZD=DBLE (D (K) ) CB(1)=CH(1) CB (2)=CN (2) CALL DRKC (2, XD, ZD,CB,FD, HD3, HMIN3, E1D,BFLTNC,G1D,DS,TID) IF (ITER. NE. 1) GO TO 26 5 DES1 = CB(1) IF (I (28) .NE.O) WRITE{6,31) ITER , X1, X2 ,CB (1) ,CB(2) ,DES1 DCPW1=X2 X2=100.*X2 ITEB=ITEB+1 GO TO 260 265 DES2=CB (1) DCPW2=X2 IF (1(2 8) .NE.O) WRITE (6,31) ITER , X1, X2, CB (1) ,CB (2) ,DES2 IF (DABS (DES2) .LT.1.DO) GO TO 270 IF (DABS( (DCPW2-DCPW1)/DCPW2) .LT.2.D-15) GO TO 270 IF (ITER.GT.30) GO TO 270 X2= (DCPW2-DCPW1) / (DES2-DES 1) * {-DES 1) +DCPW1 DES1=DES2 DCPW1=DCPW2 ITER=ITER+1 GO TO 260 270 WRITE (6,19)ITER CDN=SNGL (X2) *1. E10 CALL EGEN (XI, FIELD, DERE, GEN) CALL ND (XI,ENDX,DNDX,DNDXO) 99. CALL DTEVAL(FDN,ZND,ENDX,DN,DERDN) CALL DTEVAL(FDP,ZND,ENDX,DP,DERDP) SJN=E*DN*CDN SJP--E*DP*CDP RJL=SJN+SJP WRITE (6,28)SJP,SJN,EJL RJL=ABS(RJL) 275 WRITE (6,29) SURVEL,XI,W,B (K) , EL RETURN 11 FORMAT('0','QUA NTUM YIELD CALCULATION') 12 FORMAT('INTERNAL PHOTOEMISSION=',G12.4,' EIEC/CM3/SEC') 1 3 FOR MAT (4 (2 X , ' X (UM)»,5X,'V (VOLTS) ', 2X ,'E (V/CM) ' , 1X) ) 14 FORMAT(1X,12G11. 3) 16 FORMAT(* 0*,•SOLUTION FOR METAL N TYPE SCHOTTKY BARRIER') 17 FORMAT('0','SOLUTION FOR METAL P TYPE SCHOTTKY BARRIER') 18 FORMAT('GENERATION AS A FUNCTION OF X',/,6 (3X,* X* ,6X,* GEN OF PAIRS 1») ,/,6{3X,' (UM) «,4X,' (P/CM3/SEC) «)) 19 FORMAT (1.X, 12, ' ITERATIONS NEEDED TO SOLVE EVP') 21 FORMATC0«,'CALCULATION OF QUANTUM YIELD BY MILNES METHOD') 23 FQRMAT{»0«,'CALCULATION OF QUANTUM YIELD BY NUMERICAL METHOD') 24 FORMAT('INVERSION LAYER, DEPLETION LAYER AND BULK SHORT CIR. CURR 1 VS LAMBDA ' , / , 3 (3X , 'LAMBDA ' 5X, ' J INV*,6X,'J D E P' , 6 X , ' J BULK',2X),/ 2,3 (4X,' (UM) » ,6X, ' (A/CM2) • ,4X, • (A/CM2) ' ,4X, ' (A/CM2) • ,1X) ) 26 FORMAT(1X,'BARRIER METAL INJECTION CURR=•,G14.5,•A/CM2 INVERSION 1 LAYER CURR= *,G14.5, •A/CM2',/,'DEPLETION LAYER CURR=• , G1 4. 5 , •A/CM2 2 BULK REGION CURR=',G14.5,'A/CM2 *,/,/,'TOTAL PHOTOGENERATED CURR=' 3,G14.5,'A/CM2») 27 FORMAT('0','BARRIER J=',G14.5,•A/CM2 JN=•,G14.5, •A/CM2 JP=»,G14. 15,'A/CM2',/,'TOTAL PHOTOGENERATED CURR=', G14. 5 ,'A/CM2') 28 FORMAT('0','JP=',G14.5,'A/CM2 JN=*,G14.5,'A/CM2*,/,'TOTAL PHOTOGE 1NERATED CURR=',G14.5,1 A/CM2') 29 FORMAT(*0','SUR, VEL.=',G14.5,»CM/SEC XI=',G14.5,'UM W=',G14.5,'U 1M CELL THICKNESS=',G14.5,'UM EL=',G14.5,'CM') 31 FORMAT(1X,13,5D25.16) 32 FORMAT(1X,'QUANTUM EFFIENCY VS WAVELENGTH',/,2(3X,'LAMBDA1 ,6X,'INV 1',8X,•DEP',8X,* BULK',6X,'TOTAL',3X)) 33 FORMAT(1X,10G11.3) 34 FORMAT(1X,'EXTERNAL YIELD') 36 FORMAT{1X,'INTERNAL YIELD*) END SUBROUTINE DTEVAL (A,ZN D,DND,DT,DERDT) REAL A (50) , ND, ZND (50) ND=ALOG10 (DND) IF (ND.LE.14.1) GO TO 10 IF (ND.GE.19.) GO TO 20 J=IFTX(10.*ND-140.) 30 DT= (A (J + 1) -A (J) ) / (ZND (J+1) -ZND (J) ) * (ND-ZND (J) ) +A( J) DEBDT= (A (J+1) -A (J) ) / (EXP (2 . 30 3*ZND (J+1) ) -EXP (2.3Q3*ZND (J) ) ) RETURN 10 DT=A(1) DERDT= (A (2) -A (1) ) / (EXP (2. 303*ZND (2) ) -EXP (2.30 3*ZN D (1) ) ) RETURN 20 DT=A(50) DERDT= (A (50) -A (49) ) / (EXP (2. 303*ZND (50) ) -EXP (2. 303 *ZND (49) ) ) RETURN END SUBROUTINE EGEN (X,FIELD,DERE,GEN) REAL EFIELD (101) ,GB (10001) ,GN (101) ,GP(101) ,XB(100 01) ,XL (101) ,XN(10 11) ,XP (101) COM MO N/GENER/EFIELD,GB,GN,GP,XB,XL,XN,XP 100. C EVALUATION OF E AND DE/DX AT X IF (X.GE. XL (101) ) GO TO 10 J='IFIX (100. *X/XL {1 0 1) ) + 1 IF (J.GT.100)J=100 DEB E= (EFIELD (J+1)-EFIELD (J) ) / (XL (J+1) -XL (J) ) FIE1D=DEBE* (X-XL (J) ) +EFIELD (J) DEBE=DEBE*1.E4 GO TO 20 10 FIELD=0. DERE=0. C EVALUATION OF G AT X 20 IF (X.GT.XN (101) ) GO TO 30 J=IFIX{100.*X/XN(101))+1 IF (J.GT. 100) J = 100 GEN= (GN (J+1) -GN (J) ) / (XN (J+1) -XN (J) ) * (X-XN (J) ) +GN (J) BETUBN 30 IF(X.GT.XP(101) ) GO TO 40 J=IFIX (100. * (X-XP (1) ) / (XP (101) -XP (1) ) ) +1 IF (J. GT. 1C0) J = 100 GEN= (GP (J+1) -GP (J) ) / (XP (J+1) -XP (J) ) * (X-XP (J) ) +GP(J) BETUBN 4 0 J=IFIX( 10000. * (X-XB (1) ) / (XB (10001) -XB (1) ) ) +1 IF (J.GT. 10000)J=10000 GEN= (GB (J + 1) -GB (J) ) / (XB (J+1) -XB (J) ) * (X-XB (J) ) +GB(J) BETUBN END SUBROUTINE VFUNCN (X, V , F) EEAL V (2) ,F (2) COMMON/PBOP/EPSB#DNV,DNC,VT/EGAP E=1.6021E-19 EPS0=8.854185E-12 CALL ND (X, SN, DNDX, DNDXO) IF (ABS (V (1) ) . GT. EGAP) V (1) =V{1) / ABS (V (1) ) *EGAP F(1)=V(2) F (2)=-E/EPS0/EPSB*1. E-6* (EN+DNV*EXP (- (EGAP+V (1) )/VT) ) RETURN END SUBBOOTINE VFUNCP (X, V, F) REAL V (2) ,F(2) COM HON/PROP/EPSB,DNV,DNC,VT,EGA P E=1.6021E-19 EPS0=8.854185E-12 CALL ND (X,EN,DNDX,DNDXO) IF (ABS (V (1) ) . GT. EGAP) V (1) =V (1) /ABS (V (1) ) *EGAP F(1) = V(2) F(2)=E/EPSO/EPSB*1.E-6*(EN + DNC*EXP(-(EGAP-V(1))/¥ T)) RETURN END SUBBOUTINE ENFUNC (X,CN,F) BEAL FDN (50) , FDP (50) , FTAUN (50) , FTAUP (50) , ZND (50) BEAL*8 CN (2) ,F (2) ,X COMMON/DIFTAU/FDN,FDP,FTAUN,FTAUP,ZND COMMON/PBOP/EPSE,DNV,DNC,VT,EGAP XS=SNGL(X) CALL ND (XS,EN,DNDX,DNDXO) CALL DTEVAL(FDN,ZND,EN,DN,DEEDN) CALL DTEVAL(FTAUN,ZND,EN,TADN,DTAUN) CALL EG EN (XS,EFIELD,DEBE,G EN) F(1)=CN(2) F(2)=DBLE(-DEBE/VT-EFIELD/VT*DEBDN*DNDX/DN+1./DN/TAUN) *1.D-8*CN (1) 101. T-DBLE(DNDX*DE3DN/DN+EFIELD/VT)*1.D-4*CN(2)-DBLE(GEN/ON)*1. D-14 RETURN END SUBROUTINE PFUNC (X, CP, F) REAL FDN (50) , FDP(50) , FTAUN (50) ,FTAUP{50) ,ZND (50) REAL*8 CP (2) ,F (2) ,X CQMMON/DIFTAU/FDN,FDP,FTAON,FTAUP,ZND COHHON/PROP/EPSR,DNV,BNC,7T,EGAP XS=SNGL (X) CAL I ND (XS,EN,DNDX,DNDXO) CALL DTEVAL (FDP , ZND, EN , DP, DERDP) CALL DTEVAL(FTAUP,ZND,EN,TAUP,DTAUP) CALL EGEN(XS,EFIELD,DERE,GEN) F(1)=CP (2) F(2) =DBLE(1./DP/TAUP + DERE/VT + EFIELD/VT/DP*DERDP*DN DX) *1.D-8*CP(1)-1DBLE(DERDP*DNDX/DP-EFT FLD/VT)*1.B-4*CP(2)-DBLE(GEN/DP)*1.D-14 RETURN END SUBROUTINE BFUNC (X,CB,F) REAL*8 CB(2) ,F(2) ,X COMMON/BULK/BD,BTAU XS=SNGL (X) CALL EGEN (XS,EFIELD,DERE,GEN) F(1)=CB{2) F (2)=DBLE (1. /BD/BTAU) *1.D-8*CB (1) -DBLE (GEN/BD) *1. D-14 RETURN END 102. APPENDIX I I This appendix gives the values of the various e l e c t r i c a l and m a t e r i a l p r o p e r t i e s used i n the computations of Chapters I I I and IV. A l s o i n c l u d e d are references from which the values were obtained. The o p t i c a l p r o p e r t i e s c o n s i s t of ( n - i k ) , the complex index of r e f r a c t i o n , and a, the absorption c o e f f i c i e n t . The absorption c o e f f i c -i e n t i s r e l a t e d to the imaginary p a r t of the index of r e f r a c t i o n through Eq. ( I I I - 9 ) . The index of r e f r a c t i o n f o r SiO was obtained from the Gmelin s e r i e s [49], f o r GaAs from Seraphin and Bennett [50], and f o r Au from Hass and Hadley [33]. The r e a l p a r t of the index of r e f r a c t i o n f o r S i was obtained from the r e s u l t s of Meyer et a l . [51] w h i l e the absorption c o e f f i c i e n t was taken from Sze [9]. The values of a f o r GaAs and k f o r S i were obtained from the known data by the r e l a t i o n s h i p between a and k. The values f o r the o p t i c a l p r o p e r t i e s mentioned above are given i n the f o l l o w i n g four t a b l e s . A (ym) n k 0.24 1.85 0.70 0.30 2.20 0.40 0.35 2.30 0.30 0.40 2.20 0.20 0.50 2.00 0.05 0.60 1.95 0 1.00 1.90 0 5.00 1.80 0 Table A-II-1 O p t i c a l p r o p e r t i e s of SiO X (vim) n k 0.20 1.24 0.92 0.45 1.40 1.88 0.50 0.84 1.84 0.55 0.331 2.324 0.60 0.200 2.897 0.65 0.142 3.374 0.70 0.131 3.842 0.75 0.140 4.266 0.80 0.149 4.654 0.85 0.157 4.993 0.90 0.166 5.335 0.95 0.174 5.691 1.00 0.179 6.044 1.30 0.69 7.26 1.75 1.10 9.59 1.95 1.30 10.7 2.26 1.62 11.8 2.80 2.20 15.3 Table A-II-i I Optical properties of Au A (ym) n a (cm 0.341 5.10 2. E5 0.382 6.30 1. E5 0.472 4.50 1.3 E4 0.551 4.05 6.8 E3 0.589 3.95 4.9 E3 0.70 3.75 2.1 E3 0.798 3.65 9.6 E2 0.983 3.60 1.3 E2 1.127 3.55 0 1.40 3.5 0 1.80 3.5 o Table A-II-3 Optical properties of S i (E i s exponent) X (ym) n a 0.282 3.486 1.126 E6 0.295 3.526 8.835 E5 0.310 3.513 7.528 E5 0.318 3.451 7.220 E5 0.326 3.389 7.112 E5 0.344 3.391 6.963 E5 0.365 3.514 6.944 E5 0.400 4.149 6.714 E5 0.409 4.320 6.142 E5 0.419 4.459 5.788 E5 0.425 4.755 5.718 E5 0.443 4.929 3.390 E5 0.495 4.427 1.046 E5 0.540 4.127 7.284 E4 0.590 3.888 4.898 E4 0.653 3.746 2.972 E4 0.775 3.625 1.130 E4 0.860 3.610 9.059 E3 0.880 3.607 6.496 E2 0.895 3.603 1.994 E l 0.900 3.595 8.922 E0 0.910 3.581 0 1.00 3.500 0 1.50 . 3.381 0 2.00 3.314 0 3.00 3.324 0 Table A-II-4 O p t i c a l p r o p e r t i e s of GaAs The m a t e r i a l p r o p e r t i e s f o r the semiconductors t h a t were used i n c l u d e the m i n o r i t y - c a r r i e r m o b i l i t y , the l i f e t i m e , and the surface recombination v e l o c i t y . The m i n o r i t y - c a r r i e r m o b i l i t i e s (u^ and y^) f o r both S i and GaAs appear i n Sze [9] and the d i f f u s i o n c o e f f i c i e n t s (D^ and 0^) were obtained using E i n s t i e n ' s r e l a t i o n s h i p . The data from Wolf [52] were used f o r the m i n o r i t y - c a r r i e r l i f e t i m e s T and T f o r S i . n p Tsaur e t a l . [39] and Aukerman et a l . [53] were the sources f o r the GaAs data. Using the r e l a t i o n s h i p s L = (D , n n n ' L = CD T y , p P P the d i f f u s i o n lengths L and L were c a l c u l a t e d as needed. The values n p f o r the d i f f u s i o n c o e f f i c i e n t s and the m i n o r i t y - c a r r i e r l i f e t i m e s are given i n Table A-II-5 f o r S i and Table A-II-6 f o r GaAs. The dependence of the m o b i l i t y on the e l e c t r i c f i e l d E was considered f o r c a l c u l a t i o n s i n the d e p l e t i o n l a y e r but was discontinued. This i s because the e l e c -t r i c f i e l d d i s t r i b u t i o n i s c a l c u l a t e d assuming s h o r t - c i r c u i t c o n d i t i o n s and i f the s o l a r c e l l i s at any other operating p o i n t such as the maximum-power p o i n t the e l e c t r i c f i e l d d i s t r i b u t i o n w i l l change and hence so w i l l the m o b i l i t y dependence. The surface recombination v e l o c i t y was allowed t o vary over a wide range (see F i g . IV-2) but the standard values f o r S i and GaAs of 1 x 10^ and 1 x 10^ cm sec \ r e s p e c t i v e l y , obtained from Tsaur et a l . [39] were used f o r most c a l c u l a t i o n s . The values used f o r the e f f e c t i v e Richardson's constant were 110 A cm"2 °K - 2 f o r n - S i [ 9 ] , 32 A cm"2 °K _ 2 f o r p - S i [ 9 ] , 8.6 A cm"2 °K _ 2 f o r n-GaAs [40], and 79 A cm"2 °K~ 2 f o r p-GaAs [ 9 ] . A l l c a l c u l a t i o n s were done at 300 °K. The i n c i d e n t r a d i a t i o n f l u x d e n s i t y values f o r AMO c o n d i t i o n s were obtained from Arvesen e t a l . [26] and f o r AMI c o n d i t i o n s from Ref. [54]. The values are presented i n Table A-II-7. The values f o r the gold f i l m r e s i s t i v i t y as a f u n c t i o n of thickness were c a l c u l a t e d from the sheet r e s i s t a n c e s given i n Ref. [34] and are given i n Table A-II-8. (cm - 3) D n (cm^ s e c _ i ) D P (cm2 s e c - i ) T n (sec) (sec) 1. E14 41.4 15.8 2.8 E-5 2.8 E-5 2. E14 41.0 15.7 2.0 E-5 2.0 E-5 5. E14 40.2 15.4 1.2 E-5 1.2 E-5 1. E15 38.8 15.0 8.4 E-6 8.0 E-6 2. E15 37.5 14.2 6.3 E-6 5.5 E-6 5. E15 33.6 13.2 4.3 E-6 3.6 E-6 1. E16 28.4 12.7 3.2 E-6 2.4 E-6 2. E16 25.9 11.4 2.7 E-6 1.9 E-6 5. E16 21.2 10.3 2.0 E-6 1.1 E-6 1. Elf7 18.4 9.05 1.9 E-6 9.1 E-7 2. E17 15.8 7.76 1.8 E-6 7.7 E-7 5. E17 11.4 6.46 1.5 E-6 5.5 E-7 1. E18 8.53 5.17 1.4 E-6 4.6 E-7 2. E18 6.72 4.39 1.3 E-6 4.0 E-7 5. E18 4.65 3.10 1.2 E-6 3.6 E-7 1. E19 3.36 2.33 1.2 E-6 3.5 E-7 Table A-II-5 D i f f u s i o n c o e f f i c i e n t s and m i n o r i t y - c a r r i e r l i f e t i m e s f o r S i ND (cm - 3) D n (crn^ sec--*-) DP (cm^ sec--*-) (sec) TP (sec) 1. E14 186 9.83 4.83 E-6 2.34 E-5 2.EE14 184 9.83 2.45 E-6 1.17 E-5 5. E14 181 9.57 9.94 E-7 4.81 E-6 1. E15 176 9.31 5.12 E-7 2;47 E-6 2. E15 171 8.79 2.64 E-7 1.31 E-6 5. E15 160 8.27 1.12 E-7 5.56 E-7 1. E16 155 8.02 5.80 E-8 2.87 E-7 2. E16 150 7.76 3.00 E-8 1.48 E-7 5. E16 129 7.24 1.39 E-8 6.35 E-8 1. E17 i 121 6.46 7.41 E-9 3.56 E-8 2. E17 106 5.69 4.25 E-9 2.02 E-8 5. E17 87.9 5.17 2.05 E-9 8.89 E-9 1. E18 77.6 4.65 1.16 E-9 4.94 E-9 2. E18 64.6 3.62 6.96 E-10 3.18 E-9 5. E18 51.7 2.84 3.48 E-10 1.62 E-9 1. E19 44.0 2.07 2.05 E-10 1.11 E-9 Table A-II-6 Diffusion coefficients and minority-carrier life t i m e s for GaAs Au f i l m thickness (A) Au r e s i s t i v i t y (fi-cm) 40 2.1 TE-3 45 2.7 E-4 50 9.75 E-5 60 4.08 E-5 70 2.31 E-5 80 1.6 E-5 100 1.15 E-5 150 8.25 E-6 200 6.80 E-6 Table A=II=8 R e s i s t i v i t y of Au films as a function of thickness 108. X N(X) X N(X) X N(A) AMO AMI AMO AMI AMO AMI 0.30 .545 .125 0.98 .772 .677 1.64 .238 .166 0.32 .786 .440 1.00 .741 .654 1.66 .237 .159 0.34 1.089 .675 1.02 .716 .631 1.68 .225 .152 0.36 1.16 .790 1.04 .679 .608 1.70 .221 .145 0.38 LUO 8 .900 1.06 .650 .585 1.72 .209 .139 0.40 1;692 1.12 1.08 .621 .563 1.74 .195 .133 0.42 1.725 1.40 1.10 .608 .540 1.76 .188 .127 0.44 1.735 1.66 1.12 .578 .517 1.78 .174 .122 0.46 1.946 1.83 1.14 .556 .495 1.80 .172 .117 0.48 2.115 1.83 1.16 .538 .472 1.82 .162 .113 0.50 1.927 1.77 1.18 .512 .450 1.84; .156 .109 0.52 1.994 1.70 1.20 .492 .428 1.86 .146 .106 0.54 1.852 1.63 1.22 .475 .406 1.88 .143 .103 0.56 1. 712 1.55 1.24 .466 .384 1.90 .136 .100 0.58 1.886 1.50 1.26 .432 .363 1.92; .137 .0977 0.60 1.816 1.51 1.28 .424 .343 1.94 .132 .0956 0.62 1.794 1.52 1.30 .427 .325 1.96 .128 ';0937 0.64 1.685 1.46 1.32 .407 .309 1.98 .127 .0920 0.66 1.680 1.38 1.34 .390 .296 2.00 .118 .0902 0.68 1.578 1.33 1.36 .372 .284 2.02 .115 .0886 0. 70 1.495 1.30 1.38 .359 .273 2.04 .109 .0869 0.72 1.367 1.23 1.40 .349 .263 2.06 .102 .0853 0.74 1.316 1.15 1.42 .340 .254 2.08 .099 .0837 0.76 1.269 1.08 1.44, .325 .245 2.10 .095 .0820 0.78 1.223 1.02 1.46 .316 .236 2.12 .089 .0804 0.80 1.154 .974 1.48 .303 .228 2.14 .092 .0788 0.82 1.079 .928 1.50 .297 .220 2.16 .084 .0772 0.84 1.027 .885 1.52 .288 .212 2.18, .077 .0756 0.86 .968 .846 1.54 .278 .204 2.20 .084 .0740 0.88 .935 .811 1.56 .273 .196 2.22 , CO 79 .0724 0.-90 .889 .780 1.58 .259 .189 2.24 .061 .0708 0.92 .835 .752 1.60 .252 .181 2.26 .072 .0692 0.94 .821 .726 1.62 .245 .173 2.28 Q087 .0676 0.96 .792 .701 Table A-II-7 I n c i d e n t s o l a r f l u x d e n s i t y AMO and AMI con d i t i o n s (A i n ym and N(X) i n W m - 2 ym - 1) 

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