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Metal-insulator-semiconductor tunnel junctions and their application to photovoltaic energy conversion Tarr, Nicholas Garry 1981

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METAL-INSULATOR-SEMICONDUCTOR TUNNEL JUNCTIONS AND THEIR APPLICATION TO PHOTOVOLTAIC ENERGY CONVERSION B.S c , The Uni v e r s i t y of B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of E l e c t r i c a l Engineering We accept this thesis as conforming by NICHOLAS GARRY TARR to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1981 0 N. Garry Tarr, 1981 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 of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood 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 not be a llowed without my w r i t t e n p e r m i s s i o n . Department o f Elct-Jri G&,[ Eft^ in&e-n The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date DE-6 (2/79) i i ABSTRACT This thesis i s concerned p r i m a r i l y with an experimental and theor-e t i c a l i n v e s t i g a t i o n of the properties of the metal-insulator-semiconduc-tor (MIS) tunnel junction. P a r t i c u l a r emphasis i s placed on those prop-e r t i e s which might be of use i n the production of photovoltaic c e l l s . The MIS junctions studied are divided i n t o two basic classes: those i n which the semiconductor surface i s depleted or strongly inverted at equilibrium, and those i n which the surface i s accumulated at equilibrium. Junctions f a l l i n g i n the former category are termed p o s i t i v e b a r r i e r s , while those i n the l a t t e r group are termed negative b a r r i e r s . Recent t h e o r e t i c a l studies have predicted that i t should be possible to form p o s i t i v e b a r r i e r MIS junctions i n which the dark current flow at moderate forward bias i s dominated by the i n j e c t i o n of minority c a r r i e r s into the bulk semiconductor. This p r e d i c t i o n i s quite remarkable, i n that i t appears to contradict the abundant experimental evidence i n d i -cating that the dark current i n non-ideal Schottky diodes i s dominated by majority c a r r i e r thermionic emission. In t h i s thesis two independent experiments providing the f i r s t i n c o n t r o v e r t i b l e evidence for the e x i s t -ence of minority c a r r i e r MIS diodes are reported. The f i r s t of these experiments involved the measurement of the current-voltage character-i s t i c s of Al-SiO^-pSi diodes at various temperatures. From these meas-urements, an a c t i v a t i o n energy describing the temperature dependence of the dark current was extracted. This a c t i v a t i o n energy was found to agree exactly with that expected for a minority c a r r i e r i n j e c t i o n - d i f f u s i o n current, and to be s i g n i f i c a n t l y l a r g e r than that possible for a majority c a r r i e r thermionic emission current. In the second experiment, i t was shown that an alloyed aluminum back surface f i e l d region could be used i i i to enhance the open-circuit voltages of Al-SiO^-pSi s o l a r c e l l s . This demonstration that a modification to the rear surface of an MIS s o l a r c e l l could a l t e r the c e l l open-circuit voltage provided further i r r e f u -table evidence for the existence of minority c a r r i e r MIS diodes. Negative b a r r i e r MIS junctions do not function as r e c t i f i e r s . Instead, these junctions are of use i n forming low-resistance contacts to semicon-ductors. A simple a n a l y t i c model of current flow i n the negative b a r r i e r MIS junction i s developed here. This model predicts that with a s u i t a b l e choice of i n s u l a t o r thickness and b a r r i e r metal work function, i t should be possible to form a negative b a r r i e r MIS contact which presents a very low e f f e c t i v e surface recombination v e l o c i t y to minority c a r r i e r s , yet which o f f e r s n e g l i g i b l e impedance to the flow of majority c a r r i e r s . The minority c a r r i e r r e f l e c t i n g properties of the negative b a r r i e r MIS junc-t i o n were demonstrated experimentally by incorporating t h i s structure as the back contact i n induced back surface f i e l d s o l a r c e l l s . Induced back surface f i e l d c e l l s were s u c c e s s f u l l y fabricated on both n- and p-type s i l i c o n . For both types of substrate, i t was found that the minor-i t y c a r r i e r r e f l e c t i n g negative b a r r i e r MIS back contact could provide an enhancement i n c e l l open-circuit voltage comparable to that obtained with a conventional back surface f i e l d formed by d i f f u s i o n or a l l o y i n g . In a d d i t i o n to the studies of the MIS tunnel junction outlined above, t h i s thesis includes a comprehensive i n v e s t i g a t i o n of the conditions under which the p r i n c i p l e of dark current and photocurrent superposition provides an accurate d e s c r i p t i o n of the c h a r a c t e r i s t i c s of homojunction s o l a r c e l l s . In p a r t i c u l a r , i t i s shown that the superposition p r i n c i p l e should apply even i f a s i g n i f i c a n t f r a c t i o n of both recombination and photogeneration occur i n the depletion region. This contradicts the conclusions drawn recently by other i n v e s t i g a t o r s . I t i s also found that the superposition p r i n c i p l e may s e r i o u s l y overestimate the e f f i c i e n c y of c e l l s fabricated on substrates with very poor l i f e t i m e s and low mobil-i t i e s , a point which had not been appreciated previously. TABLE OF CONTENTS PAGE: 1. INTRODUCTION 1 2. PHOTOVOLTAIC DEVICE THEORY 12 2.1 Introduction to Photovoltaic Devices 12 2.2 The Superposition P r i n c i p l e 15 2.2.1 Previous Research on the Superposition P r i n c i p l e 18 2.2.2 An A n a l y t i c Derivation of the Superposition P r i n c i p l e 20 2.2.3 Quasi-Fermi Levels i n the Depletion Region 32 2.2.4 Numerical Analysis of S i l i c o n and GaAs Homojunction c e l l s 38 2.2.5 A Case of Superposition Breakdown 55 2.3 Back Surface F i e l d Regions 59 3. POSITIVE BARRIER SCHOTTKY AND MIS JUNCTIONS: THEORY 64 3.1 Junction B a r r i e r Heights 65 3.2 Tunnelling i n Metal-Insulator-Semiconductor Structures 70 3.2.1 The Semiclassical Model of Conduction 70 3.2.2 Models f o r the Tunnelling Process 71 3.2.3 Expressions for the Tunnel Currents 74 3.2.4 An Estimate f o r the Tunnelling P r o b a b i l i t y Factor 79 3.3 The Schottky B a r r i e r Diode 83 3.3.1 The Majority C a r r i e r Thermionic Emission Current 84 3.3.2 Minority C a r r i e r Flow 89 v i PAGE: 3.3.3 The Minority C a r r i e r I n j e c t i o n Ratio 92 3.3.4 Current Flow Through Surface States 93 3.4 T r a n s i t i o n to the MIS Diode 93 3.4.1 The Minority C a r r i e r MIS Diode 94 3.4.2 An Anal y t i c Solution f o r the Potentials and Current Flows 98 3.5 The MIS Solar C e l l 103 3.5.1 Light Coupling i n t o the Semiconductor 103 3.5.2 Optimally E f f i c i e n t MIS Ce l l s 106 3.5.3 The Ch a r a c t e r i s t i c s of Thick-Insulator C e l l s 107 4. POSITIVE BARRIER MIS JUNCTIONS: EXPERIMENT 113 4.1 Previous Experimental Research on the MIS Junction 113 4.2 New Experimental Evidence f o r Minority C a r r i e r MIS Diodes 118 4.3 MIS Solar C e l l s with Back Surface Fields 133 4.4 V a r i a t i o n of MIS Solar C e l l C h a r a c t e r i s t i c s with Insulator Thickness 140 5. MINORITY CARRIER REFLECTING NEGATIVE BARRIER MIS CONTACTS 143 5.1 Current Flow i n the Negative B a r r i e r MIS Junction: Theory 144 5.2 Induced Back Surface F i e l d Solar C e l l s on n - S i l i c o n Substrates 150 5.3 Induced Back Surface F i e l d Solar C e l l s on p - S i l i c o n Substrates 156 5.3.1 minMIS Diodes on n - S i l i c o n Substrates 157 5.3.2 Minority C a r r i e r R e f l e c t i n g Pt-SiO^-pSi Contacts 161 6. SUMMARY 169 APPENDIX A Numerical Solution of the Basic Semiconductor Equations APPENDIX B Calcu l a t i o n of the Shadow Area f o r an E l l i p s o i d a l Constant Energy Surface of Ar b i t r a r y Orientation APPENDIX C Fabrication Procedure f o r MIS Junctions REFERENCES v i i i LIST OF TABLES TABLE: PAGE: 2.1 C e l l properties used i n numerical analysis 40 2.2 Changes i n quasi-fermi l e v e l s across depletion region under various operating conditions 51 2.3 True performance parameters, and those predicted by the superposition p r i n c i p l e 54 2.4 Properties of N +P GaAs c e l l whose c h a r a c t e r i s t i c s are shown i n Fig. 2.8 57 4.1 Values of A, and corresponding to the c h a r a c t e r i s t i c s of F i g . 4.3 130 4.2 Open-circuit voltages f o r Al-SiO^-pSi c e l l s 137 5.1 Open-circuit voltages for selected P +N c e l l s with negative b a r r i e r MIS back contacts 153 5.2 Open-circuit voltages f o r selected N +PIM and MISIM c e l l s 166 A . l Normalization factors 176 A.2 Data used to compute mobility 185 A.3 Data used to compute photogeneration d i s t r i b u t i o n 187 A.4 Parameters used for grid construction 191 A.5 Explanation of variables used i n FORTRAN programs 192 LIST OF FIGURES FIGURE: 2.1 Current-voltage c h a r a c t e r i s t i c s under one-sun i l l u m i -nation f o r a t y p i c a l commercial s i l i c o n s o l a r c e l l , i l l u s t r a t i n g the parameters used to describe c e l l performance. 2.2 S i m p l i f i e d s o l a r c e l l equivalent c i r c u i t . 2.3 Geometry of N +P s o l a r c e l l . 2.4 Dark current-voltage c h a r a c t e r i s t i c s f o r (a) s i l i c o n and (b) GaAs c e l l s . 2.5 Band diagrams for s i l i c o n c e l l , (a) S h o r t - c i r c u i t , one-sun i l l u m i n a t i o n , (b) Maximum power point, one-sun i l l u m i n a t i o n , (c) V=V i n dark. mp 2.6 Plot of the true current-voltage c h a r a c t e r i s t i c J (V), and the curve J - J,.(V) , under one-sun i l l u m i n a t i o n , sc D (a) S i l i c o n c e l l , (b) GaAs c e l l . 2.7 Band diagram for c e l l i n which most photogeneration occurs i n emitter, while most recombination occurs i n depletion region, (a) At forward bias i n the dark, (b) Under i l l u m i n a t i o n at same forward b i a s . 2.8 J - J„(V) and true J T ( V ) c h a r a c t e r i s t i c s f o r an N +P sc D L GaAs c e l l with low m o b i l i t i e s and very short l i f e t i m e s . 2.9 Band diagram for N +PP + back surface f i e l d c e l l under moderate forward b i a s . 3.1 Equilibrium band diagrams for MIS or non-ideal Schottky diodes, (a) n-type substrate, (b) p-type substrate. 3.2 A s l i c e through the constant energy surfaces for the s i l i c o n conduction band. 3.3 Shadow of the conduction band constant energy surfaces for a s i l i c o n sample of <100> o r i e n t a t i o n . X FIGURE: PAGE: 3-4 Band diagram f o r device of F i g . 3.1(a) under moderate forward b i a s . 87 3.5 Dark current-voltage c h a r a c t e r i s t i c s f o r MIS diodes with various i n s u l a t o r thicknesses. 97 3.6 (a) Structure of MIS s o l a r c e l l with semi-transparent b a r r i e r layer, (b) Structure of inversion layer s o l a r c e l l . 104 3.7 Band diagram for t h i c k - i n s u l a t o r MIS c e l l at terminal s h o r t - c i r c u i t under one-sun i l l u m i n a t i o n 109 3.8 Illuminated current-voltage c h a r a c t e r i s t i c s f o r MIS s o l a r c e l l s with various i n s u l a t o r thicknesses, as predicted by theory. 110 4.1 Capacitance-voltage c h a r a c t e r i s t i c f o r reverse-biased Al-SiO -pSi dot diode. 124 x 4.2 Dark current-voltage c h a r a c t e r i s t i c f o r small-area Al-SiO -pSi s o l a r c e l l . 125 x 4.3 J -V c h a r a c t e r i s t i c s f o r small-area Al-SiO -pSi s o l a r sc oc x c e l l at various temperatures. 127 4.4 (a) Temperature dependence of • 0>) Temperature dependence of JQ-^- 131,132 4.5 Illuminated current-voltage c h a r a c t e r i s t i c s for Al-SiO -pSi X s o l a r c e l l s with various i n s u l a t o r thicknesses, as measured experimentally. 142 5.1 Band diagrams for (a) a negative b a r r i e r MIS junction and (b) the corresponding p o s i t i v e b a r r i e r MIS junction formed by depositing the same metal on a substrate of the opposite doping type. 145 5.2 Structure of P +N c e l l with negative b a r r i e r Mg-SiC^-pSi back contact. 152 5.3 Capacitance-voltage c h a r a c t e r i s t i c f o r reverse-biased Pt-SiO -pSi dot diode. 160 x r FIGURE: 5.4 Structure of (a) N +PIM and (b) B.l Shadow of an e l l i p s o i d . PAGE MISIM sol a r c e l l s . 164 206 ACKNOWLEDGEMENT The research program described i n this thesis could never have been completed without the help of many others. I am p a r t i c u l a r l y g r a t e f u l f o r the assistance of Peter A. I l e s of Applied Solar Energy Corporation, who supplied the substrates used i n many of the experiments. Special thanks are also due to my supervisor, Prof. David L. Pulfrey, for h i s e n t h u s i a s t i c encouragement throughout the p r o j e c t . Numerous discussions with fellow graduate students Daniel S. Camporese, Timothy P. Lester, Jurgen K. K l e t a and David J. Smith, and with Prof. Lawrence Young, helped r e f i n e experimental techniques and c l a r i f y the theory of the MIS junction. Sp e c i a l i z e d c i r c u i t r y for the measurement of s o l a r c e l l c h a r a c t e r i s t i c s designed and b u i l t by Alan Kot greatly s i m p l i f i e d the task of data recor-ding. Much of the vacuum system f i x t u r i n g was b u i l t by technicians David G. Fletcher and Derek G. Daines. F i n a l l y , the f i n a n c i a l support of the Natural Sciences and Engineering Research Council i s g r a t e f u l l y acknowledged. x i i i LIST OF SYMBOLS+ 2 —1 D (D ) electron (hole) d i f f u s i o n c o e f f i c i e n t [m s ] n p E^ , energy of conduction band edge Ey energy of valence band edge metal fermi energy l e v e l Ep^ ^Fp^ electron (hole) quasi-fermi energy l e v e l h Planck's constant [Js] •h h/2ir J n (J ) electron (hole) current density [Am ] Boltzmann's constant [J °K "*"] L^ ( Lp) electron (hole) d i f f u s i o n length [m] n (p) electron (hole) concentration [m ] - 3 N acceptor concentration [m ] - 3 donor concentration [m ] _ 3 n. i n t r i n s i c c a r r i e r concentration [m 1 q electron charge [C] S back surface recombination v e l o c i t y [ms ^] Sp front surface recombination v e l o c i t y [ms ^] T absolute temperature [°K] V terminal voltage [V] W depletion region width [m] e p e r m i t t i v i t y [Fm c^ j. p e r m i t t i v i t y of i n s u l a t i n g layer Eg p e r m i t t i v i t y of semiconductor 2 -1 -1 U n ( Up) electron (hole) mobility [m V s ] T (T ) electron (hole) l i f e t i m e [s] n p X I V ty e l e c t r o s t a t i c p o t e n t i a l [V] £ e l e c t r i c f i e l d [Vm"1] includes only those symbols not defined i n text 1 CHAPTER 1 INTRODUCTION In the past decade the need to develop new energy supplies to replace r a p i d l y diminishing o i l and n a t u r a l gas reserves has become the world's foremost technological problem. Although many a l t e r n a t i v e energy sources are a v a i l a b l e , perhaps none i s more acceptable from an enviro-mental viewpoint than the d i r e c t generation of e l e c t r i c i t y from sunlight through the photovoltaic e f f e c t . Photovoltaic c e l l s have supplied e l e c t r i c a l energy to v i r t u a l l y a l l spacecraft ever launched, but the high cost of these devices has so f a r prevented t h e i r use f o r the generation of large amounts of e l e c t r i c i t y on Earth. In order to make photovoltaics economically competitive with c o a l - f i r e d or f i s s i o n plants f o r the gen-eration of e l e c t r i c i t y on a massive scale, i t i s estimated that the cost of s o l a r c e l l s must be reduced by more than an order of magnitude from present l e v e l s [1-3]. I f t h i s cost reduction i s to be brought about, a technology for s o l a r c e l l production r a d i c a l l y d i f f e r e n t from that now employed i n industry must be developed. This new technology must incorpor-ate both an inexpensive substrate and a simple means of forming the r e c t i f y i n g junction at which the photogenerated c a r r i e r s are c o l l e c t e d . V i r t u a l l y a l l s o l a r c e l l s produced commercially today are fabricated on s i n g l e c r y s t a l semiconductor-grade s i l i c o n s l i c e s s u i t a b l e f o r use i n the microelectronics industry. The amount of energy consumed i n the p u r i f i c a t i o n and c r y s t a l l i z a t i o n of these high q u a l i t y substrates i s so large i t has been estimated that a t y p i c a l c e l l would have to generate e l e c t r i c i t y f o r over a dozen years j u s t to repay the energy used to r e f i n e the s i l i c o n i t contains [4]. Before large-scale photovoltaic power gen-2 eration can become a r e a l i t y , i t i s e s s e n t i a l that an inexpensive substrate be developed on which s o l a r c e l l s of greater than 10% e f f i c i e n c y and with an energy payback time on the order of months can be fa b r i c a t e d . Although there has as yet been no s i n g l e , dramatic breakthrough i n the search for such a substrate, steady progress has been made i n devising low-cost techniques for the r e f i n i n g and c r y s t a l l i z a t i o n of s i l i c o n which do not require excessive energy inputs [5]. At present prospects appear good for the eventual development of an inexpensive s i l i c o n substrate capable of meeting the e f f i c i e n c y and energy payback-time goals outlined above. There i s a f a i r l y high p r o b a b i l i t y that such a future low-cost "solar grade" s i l i c o n substrate w i l l have a large-grained p o l y c r y s t a l l i n e (so-called granular) structure. Even i f an inexpensive s i l i c o n substrate becomes a v a i l a b l e , s i g n i f -i c a n t reductions i n s o l a r c e l l f a b r i c a t i o n costs w i l l s t i l l be required to make photovoltaics competitive with conventional power generation technologies. A l l present day commercial s i l i c o n c e l l s are homojunction devices — that i s , they contain a m e t a l l u r g i c a l junction formed between p- and n-type regions of the same semiconductor. The process of s o l i d state d i f f u s i o n i s currently used to form t h i s pn junction. The d i f f u s i o n i s normally c a r r i e d out i n quartz-tube furnaces at temperatures ranging from 850 to 1000°C. In order to avoid contamination of the s i l i c o n with unwanted im p u r i t i e s , a l l substrates must be subjected to elaborate clean-ing procedures before they are exposed to these high temperatures. Due to t h i s requirement f o r extreme c l e a n l i n e s s , and because i t i s a r e l a -t i v e l y slow and labor-intensive process, i t i s u n l i k e l y that s o l i d tstate d i f f u s i o n could be a v i a b l e means of junction formation for future high-throughput, low-cost c e l l production. Moreover, s o l i d state d i f f u s i o n 3 may prove to be incompatible with granular s i l i c o n substrates due to the p r e f e r e n t i a l d i f f u s i o n of dopants along grain boundaries, which leads to shorting of the c e l l [6]. Even i f t h i s problem i s not encountered, expos-ure to the high temperatures necessary f o r d i f f u s i o n i n v a r i a b l y reduces minority c a r r i e r l i f e t i m e s i n a substrate [7], and thus lowers the e f f i -ciency of the f i n i s h e d c e l l . One promising a l t e r n a t i v e technique f o r s o l a r c e l l f a b r i c a t i o n involves the replacement of the d i f f u s e d pn junction with a metal-semi-conductor junct i o n , or Schottky b a r r i e r . Schottky diodes are now routinely produced i n the e l e c t r o n i c s industry by the deposition of metal onto clean s i l i c o n surfaces under high vacuum conditions. Since the metal deposition i s c a r r i e d out at or near room temperature, the complications associated with high temperature processing outlined above are avoided completely. Further, the technology of vacuum deposition i s f a r more r e a d i l y adapted to automated mass production than i s s o l i d state d i f f u s i o n . Because of these inherent advantages, the p o s s i b i l i t y of u t i l i z i n g Schottky b a r r i e r s to form the r e c t i f y i n g junction i n s i l i c o n s o l a r c e l l s has been given serious consideration i n recent years [8], Schottky b a r r i e r diodes formed on s i l i c o n substrates were studied i n t e n s i v e l y i n the 1960's. By the end of the decade an e s s e n t i a l l y complete understanding of the mechanisms responsible for conduction i n these devices had been achieved [9]. I t was determined that under moderate forward bias the diode current i s dominated by the flow of majority c a r r i e r s from the semiconductor i n t o the metal. For junctions formed on moderately doped substrates, t h i s majority c a r r i e r flow was shown to be accurately described by Bethe's thermionic emission theory [10]. I t was also estab-l i s h e d that at forward bias part of the diode current r e s u l t s from the 4 flow of minority c a r r i e r s from the metal i n t o the semiconductor. These minority c a r r i e r s can e i t h e r recombine i n the depletion region or d i f f u s e i n t o the quasi-neutral base. Experiments performed by Yu and Snow revealed that under reverse bias or small forward bias depletion region recombination -generation processes often dominate the current flow i n s i l i c o n Schottky diodes [11]. In contrast, a t h e o r e t i c a l analysis undertaken by Scharfetter indicated that only a n e g l i g i b l e f r a c t i o n of the current flow should r e s u l t from i n j e c t e d minority c a r r i e r s d i f f u s i n g i n t o the base [12]. This p r e d i c t i o n was l a t e r confirmed experimentally by Yu and Snow, who d i r e c t l y measured the magnitude of th i s minority c a r r i e r i n j e c t i o n - d i f f u s i o n current by constructing t r a n s i s t o r s with Schottky b a r r i e r emitters [13]. Yu and Snow found that the minority c a r r i e r i n j e c t i o n r a t i o — that i s , the r a t i o of the minority c a r r i e r i n j e c t i o n - d i f f u s i o n current to the -4 t o t a l diode current — was les s than 10 over the normal range of diode operation. The e l e c t r i c a l c h a r a c t e r i s t i c s of Schottky b a r r i e r s can be conven-i e n t l y summarized by comparing the current flows i n a Schottky diode with those i n a one-sided d i f f u s e d junction pn diode formed on an i d e n t i c a l substrate. (In a one-sided pn diode the d i f f u s e d surface layer, or emitter, i s much more heavily doped than the substrate). At moderate forward bias the current i n the pn diode i s dominated by the flow of mi-no r i t y c a r r i e r s supplied from the emitter i n t o the depletion region and quasi-neutral base. Under the same bias conditions there i s an i d e n t i c a l minority c a r r i e r flow i n t o the depletion region and base of the Schottky diode, except i n t h i s device these c a r r i e r s are supplied through e l e c t r o n transfer between the minority c a r r i e r band and the metal. However, i n the Schottky diode there e x i s t s an a d d i t i o n a l and f a r larger current component 5 a r i s i n g from the emission of majority c a r r i e r s from the semiconductor in t o the metal. This thermionic emission current has no counterpart i n the homojunction device, so at a given forward bias the dark current density i n the Schottky diode i s f a r larger than i n the pn diode. For reasons which w i l l be made clear i n Chapter 2, from this r e s u l t i t follows that the open-circuit voltage and hence the energy conversion e f f i c i e n c y of a Schottky b a r r i e r s o l a r c e l l must always be s u b s t a n t i a l l y lower than that of a pn junction s o l a r c e l l formed on a s i m i l a r sub-str a t e [14]. In the early 1970's i t was discovered em p i r i c a l l y that the open-c i r c u i t voltages of 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 could be greatly increased i f a very thin oxide layer were d e l i b e r a t e l y introduced between the metal and the substrate to form a metal-insulator-semiconductor (MIS) junction. In p a r t i c u l a r , Anderson, Delahoy and Milano found that MIS s o l a r c e l l s formed by depositing chromium on p-type s i l i c o n oxidized f o r a few minutes at 600°C could give e f f i c i e n c i e s and open-circuit voltages only s l i g h t l y lower than those of t y p i c a l d i f f u s e d junction c e l l s [15]. A Schottky b a r r i e r s o l a r c e l l f a b r i c a t e d by conventional techniques on a p-type s i l i c o n substrate would, i n contrast, have had an energy conver-sion e f f i c i e n c y close to zero. A p l a u s i b l e explanation for the remarkably high e f f i c i e n c i e s reported by Anderson ejt al. for. t h e i r Cr-SiO^-pSi s o l a r c e l l s was f i r s t provided i n 1974 by Green, Shewchun and s e v e r a l co-workers at McMaster Un i v e r s i t y . The McMaster group employed numerical methods to solve for the conduction c h a r a c t e r i s t i c s of the MIS diode, assuming that current flows i n these devices as a r e s u l t of electrons tunnelling d i r e c t l y between the metal and the semiconductor bands [16,17]. The numerical 6 analysis was applied only to those MIS diodes formed on s i l i c o n substrates with s i l i c o n dioxide i n s u l a t i n g l a y e r s , but a wide v a r i e t y of metal work functions, i n s u l a t o r thicknesses, and substrate doping l e v e l s was consid-ered. The most i n t e r e s t i n g r e s u l t s were obtained when the metal work function was selected to give strong inversion of the semiconductor sur-face at equilibrium. In such cases i t was found that over the range from reverse bias to small forward bias the diode current would be dominated by the flow of c a r r i e r s between the metal and the minority band i n the semiconductor [16]. Within t h i s bias range the MIS junction would thus be e l e c t r i c a l l y equivalent to a one-sided m e t a l l u r g i c a l pn junction. MIS diodes i n which the main component of current flow at moderate forward bias r e s u l t s from the i n j e c t i o n of minority c a r r i e r s i n t o the semiconductor were termed minority c a r r i e r MIS, or minMIS, diodes. The prospect of producing a junction with the e l e c t r i c a l properties of a pn diode by the simple deposition of metal on s i l i c o n i s of obvious importance not only i n the development of photovoltaic power generation, but f o r the microelectronics industry as a whole. However, conclusive experimental support f o r the existence of minority c a r r i e r MIS diodes i s c l e a r l y required. The suggestion that the introduction of a very t h i n , tunnellable i n t e r f a c i a l i n s u l a t i n g layer i n a metal-semiconductor junc-t i o n can somehow eliminate the thermionic emission current and thus allow minority c a r r i e r flows to dominate the diode c h a r a c t e r i s t i c appears para-d o x i c a l when i t i s r e c a l l e d that most Schottky b a r r i e r s contain an i n t e r -f a c i a l layer of native oxide u n i n t e n t i o n a l l y introduced during processing. In a t y p i c a l f a b r i c a t i o n procedure f o r commercial s i l i c o n Schottky diodes, the substrate i s etched i n h y d r o f l u o r i c a c i d to remove a l l traces of oxide from the surface, and then quickly transferred i n a i r to the vacuum 7 system used for metal deposition. Ellipsometry reveals that a new native oxide layer roughly 10 A thick i s formed almost immediately upon exposure of the etched s i l i c o n surface to the atmosphere [18], and thus incorpor-ated i n t o the junction. Despite the presence of t h i s thin i n t e r f a c i a l layer, there i s overwhelming experimental evidence i n d i c a t i n g that the dark current i n s i l i c o n Schottky diodes i s dominated by majority c a r r i e r thermionic emission. Indeed, the metal-emitter t r a n s i s t o r s which Yu and Snow employed i n t h e i r minority c a r r i e r i n j e c t i o n r a t i o measurements [13] were fabricated on chemically etched substrates, and so almost c e r t a i n l y contained i n t e r f a c i a l native oxide l a y e r s . Since the majority of Schottky diodes contain i n t e r f a c i a l oxide l a y e r s , the d i s t i n c t i o n between these devices and MIS diodes may appear unclear at t h i s point. For the moment, those junctions i n which an i n t e r -f a c i a l layer i s d e l i b e r a t e l y introduced between the metal and the semi-conductor w i l l be termed MIS junctions, while a l l other junctions between a metal and a semiconductor w i l l be r e f e r r e d to as Schottky b a r r i e r s . Schottky diodes containing an i n t e r f a c i a l layer w i l l be termed non-ideal. In Chapter 3 a more precise c r i t e r i o n for d i s t i n g u i s h i n g between Schottky and MIS junctions i s suggested. The i n s u l a t o r thickness i n the MIS diodes considered here w i l l always be small enough that appreciable tunnel currents can flow between the metal and the semiconductor; t h i c k - i n s u l a t o r MOS capacitors constitute a completely d i f f e r e n t class of device. By mid-1978, at the commencement of the research program described i n t h i s t h e s i s , a s u b s t a n t i a l body of experimental data on MIS junctions had been published [19]. However, no i n c o n t r o v e r t i b l e evidence for the existence of minority c a r r i e r MIS diodes had ever been reported. The f i r s t goal of the present research program was therefore to e s t a b l i s h unequiv-8 o c a l l y that minMIS diodes with properties s u i t a b l e f o r photovoltaic energy conversion could i n fact be made. The two independent experiments undertaken to accomplish t h i s goal are described i n Chapter 4. In the f i r s t of these experiments, the temperature dependence of the current-voltage c h a r a c t e r i s t i c s of Al-SiO^-pSi diodes formed on 10 item substrates was investigated [20]. This study revealed that over the bias range of i n t e r e s t f o r s o l a r c e l l operation the dark current i n these diodes i s indeed dominated by the i n j e c t i o n of minority c a r r i e r electrons from the metal i n t o the semiconductor. The second experiment involved the f a b r i -cation of MIS s o l a r c e l l s on 10 ficm substrates incorporating back surface f i e l d s [21]. A back surface f i e l d i s simply a high-low junction formed at the rear of a s o l a r c e l l ; the structure i s frequently employed to increase the open-circuit voltage of commercial diffused-junction devices. For the Al-SiO -pSi c e l l s examined here, the use of a back surface f i e l d x was found to increase the open-circuit voltage by as much as 50 mV over the value recorded with an ohmic back contact. The fact that a modifica-t i o n to the rear surface of an MIS s o l a r c e l l could produce such a dra-matic change i n open-circuit voltage provides further i r r e f u t a b l e e v i -dence f o r the existence of minMIS diodes. Chapter 4 closes with an i n v e s t i g a t i o n of the r e l a t i o n s h i p between i n s u l a t o r thickness and e l e c t r i c a l c h a r a c t e r i s t i c s i n Al-SiO -pSi s o l a r x c e l l s . P a r t i c u l a r attention i s paid to the rather b i z a r r e illuminated current-voltage c h a r a c t e r i s t i c s of c e l l s with r e l a t i v e l y thick i n s u l a t i n g l a y e r s , a feature which has been l a r g e l y overlooked by other i n v e s t i g a -tors . The r e s u l t s of th i s experiment are i n reasonably good agreement with the predictions of the models presently used to describe the MIS tunnel j u n c t i o n . In addition, the data obtained may provide some guidance 9 i n the design of p r a c t i c a l MIS devices. The t h e o r e t i c a l background f o r the experiments described i n Chapter 4 i s presented i n Chapters 2 and 3. In Chapter 3 a u n i f i e d theory of current flow i n Schottky b a r r i e r and MIS junctions i s developed, drawing heavily on both the r e s u l t s Green et a l . obtained through numerical analysis [16, 17, 22-25] and on e a r l i e r t h e o r e t i c a l work by Card and Rhoderick [26-28]. The material presented i n Chapter 3 d i f f e r s from these e a r l i e r t r e a t -ments i n that purely a n a l y t i c methods are used i n the development of the theory, and that allowance i s made f o r strong inversion of the semicon-ductor surface. (Although Card and Rhoderick chose an a n a l y t i c approach, t h e i r r e s u l t s are v a l i d only for the case i n which the semiconductor surface i s depleted). Chapter 3 includes the f i r s t d e t a i l e d 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 properties of MIS s o l a r c e l l s with r e l a t i v e l y thick i n s u l a t i n g layers. I t i s found that the a n a l y t i c model can explain the unusual illuminated current-voltage c h a r a c t e r i s t i c s reported for t h i c k -i n s u l a t o r MIS c e l l s i n Chapter 4. Since a l l the experiments described i n t h i s thesis involve the use of MIS junctions i n s o l a r c e l l s , Chapter 2 i s devoted to a fundamental discussion of photovoltaic device theory. P a r t i c u l a r emphasis i s placed on the p r i n c i p l e of dark current and photocurrent superposition, which states that the current flowing i n an illuminated c e l l subject to a bias V i s given by the algebraic sum of the s h o r t - c i r c u i t photocurrent and the current which would flow at bias V i n the dark. This p r i n c i p l e has served as the t h e o r e t i c a l foundation of photovoltaics since the early 1950's [29]. Previous attempts to j u s t i f y the superposition p r i n c i p l e [30,31] are reviewed i n Chapter 2, and found to contain serious flaws. In the course of c o r r e c t i n g these flaws, a simple argument i s developed e s t a b l i s h i n g 10 the v a l i d i t y of the superposition p r i n c i p l e f o r t y p i c a l homo junction c e l l s operated i n unconcentrated sunlight [32,33]. The conclusions drawn i n t h i s a n a l y t i c argument are then confirmed by d i r e c t numerical s o l u t i o n of the b a s i c semiconductor equations for representative s i l i c o n and gallium arsenide s o l a r c e l l s . Chapter 2 thus makes an important c o n t r i -bution to present understanding of homojunction s o l a r c e l l operation. By depositing a low work function metal on an n-type s i l i c o n sub-s t r a t e , or a high work function metal on a p-type substrate, i t i s possible to form an MIS junction i n which the semiconductor surface i s accumulated at equilibrium. Junctions of t h i s type w i l l be referred to here as negative b a r r i e r MIS contacts, to d i s t i n g u i s h them from conven-t i o n a l p o s i t i v e b a r r i e r Schottky and MIS junctions i n which the semi-conductor surface i s depleted or inverted. I t has long been recognized that negative b a r r i e r metal-semiconductor junctions should o f f e r l i t t l e resistance to current flow, and thus be of use i n forming ohmic contacts to semiconductors [34]. In 1976 Green, Godfrey and Davies postulated that negative b a r r i e r MIS contacts could be produced which would present e s s e n t i a l l y no impedance to the flow of majority c a r r i e r s , but which would have an extremely low e f f e c t i v e surface recombination v e l o c i t y for minority c a r r i e r s [35]. Contacts of t h i s type would thus have e l e c t r i c a l c h a r a c t e r i s t i c s analogous to those of m e t a l l u r g i c a l high-low junctions, and could therefore be used to form induced back surface f i e l d regions i n s o l a r c e l l s . Experimentally, Green et a l . found that negative b a r r i e r MIS junctions could provide low resistance ohmic contacts to s i l i c o n , but never observed the predicted induced back surface f i e l d action [36]. In Chapter 5, the f i r s t successful f a b r i c a t i o n of minority c a r r i e r r e f l e c t i n g negative b a r r i e r MIS contacts i s reported. As suggested by 11 Green e_t a l . [35,36], evidence for the low e f f e c t i v e surface recombination v e l o c i t y of these junctions was obtained by employing them as back con-tacts i n induced back surface f i e l d s o l a r c e l l s . The f i r s t experiment c a r r i e d out i n t h i s area involved the use of negative b a r r i e r Mg-SiO -nSi X back contacts to form induced back surface f i e l d s on P +N c e l l s with d i f f u s e d front junctions [37]. Later, negative b a r r i e r platinum-MIS con-tacts were used to create induced back surface f i e l d s on p-type s i l i c o n substrates. In t h i s l a t t e r experiment both d i f f u s e d N +P and Al-SiO^-pSi minMIS front junctions were employed. For a l l the device structures considered i t was found that the negative b a r r i e r MIS back contact could provide an enhancement i n open-circuit voltage comparable to that obtained with a conventional back surface f i e l d formed by d i f f u s i o n or a l l o y i n g . Chapter 5 includes a simple t h e o r e t i c a l analysis of current flow i n the negative b a r r i e r MIS contact based on a straightforward extension of the re s u l t s obtained i n Chapter 3. CHAPTER 2 PHOTOVOLTAIC DEVICE THEORY This chapter i s intended p r i m a r i l y to provide the fundamental t h e o r e t i c a l background for the experiments on photovoltaic devices des-cribed i n Chapters 4 and 5. However, i t should be emphasized that the material on the superposition p r i n c i p l e presented i n Section 2.2 c o n s t i -tutes a s i g n i f i c a n t advance i n present understanding of homojunction s o l a r c e l l operation, and i s thus of importance i n i t s own r i g h t . Sec-tio n 2.1 off e r s a short introduction to the terminology used i n the study of photovoltaics. Section 2.3 closes the chapter with a b r i e f examination of the properties of back surface f i e l d s tructures. 2.1 Introduction to Photovoltaic Devices A photovoltaic c e l l or sol a r c e l l i s simply a large area photodiode which i s normally operated to supply power to a load. The vast majority of c e l l s produced today are planar N +P devices formed by d i f f u s i n g phos-phorus into uniformly doped p-type s i l i c o n substrates. The heavily doped surface layer i s customarily termed the emitter, while the bulk substrate i s r e ferred to as the base. Although s i l i c o n c e l l s have been studied most i n t e n s i v e l y i n the past and are the mainstay of the present photo-v o l t a i c s industry, i t should be noted that a v a r i e t y of other semicon-ductors can be used for photovoltaic energy conversion [3]. The current-voltage c h a r a c t e r i s t i c s of a t y p i c a l commercial s i l i c o n s o l a r c e l l exposed to t e r r e s t r i a l sunlight are shown i n F i g . 2.1. Follow-ing convention, the axes have been oriented so that power output i s obtained for operation i n the f i r s t quadrant. The maximum power point ( v mp» J m p ) 1 S defined as the operating point at which the power supplied 13 Figure 2.1 Current-voltage c h a r a c t e r i s t i c under one-sun i l l u m i n a t i o n for a t y p i c a l commercial s i l i c o n s o l a r c e l l , i l l u s t r a t i n g the parameters used to describe c e l l performance. 14 by the c e l l is greatest. The performance of a solar c e l l i s normally summarized in terms of four parameters: the short-circuit current J , sc the (Jpen-circuit voltage v » the f i l l factor FF and the energy conversion efficiency n . The f i l l factor is defined as the ratio of the maximum output power to the product of the short-circuit current and open-circuit voltage, and usually l i e s between 0.65 and 0.8 for a well designed dev-ice, n is simply the ratio of the maximum output power to the total incident light power. For the purpose of measuring and comparing solar c e l l performance, various standardized representations of natural sunlight have been dev-ised [38]. Cells designed for use in space are normally tested under "air mass zero" (AMO) illumination, which is equivalent to the solar irradiance just outside the atmosphere when the earth is at i t s mean distance from the sun. The choice of a standard illumination condition t for testing terrestrial cells is more complicated, since sunlight is attenuated and fi l t e r e d on i t s passage through the atmosphere. The per-formance of te r r e s t r i a l cells i s often measured under "air mass one" (AMI) illumination, which i s obtained when AMO sunlight is fi l t e r e d by passage at normal incidence through an atmosphere of specified composition. The AMI spectrum is thus meant to simulate typical illumination conditions when the sun is directly overhead. When the sun is not at the zenith, it s light must traverse a longer path through the atmosphere before reaching the ground, and is therefore attenuated more severely. This situation i s represented by spectra with higher air mass numbers. For example, AM2 illumination results when AMO sunlight is passed through the standard atmosphere at an angle of 60° to the vertical, so that the path traversed by the light in the atmosphere is twice as long as when the 15 sun i s overhead. C e l l s under test are always illuminated with l i g h t at normal incidence to t h e i r front surfaces. For c e l l s used i n conjunction with focussing mirrors or lenses, the r a t i o of the incident l i g h t i n t e n -s i t y to that of unfocussed sunlight i s termed the concentration r a t i o , or "number of suns". One-sun i l l u m i n a t i o n thus refers to unconcentrated sunlight. Assuming u n i t quantum e f f i c i e n c y , the photogeneration rate G at a distance x below the surface of an illuminated s o l a r c e l l i s given by [39] A G(x) = / dA [1 - R(A)] M(A) a(A) e a u ; x (2.1) 0 where M(A) i s the incident photon f l u x s p e c t r a l density, R(A) gives the f r a c t i o n of incident photons r e f l e c t e d by the c e l l surface, and a(A) i s the absorption c o e f f i c i e n t f or the substrate material used. A i s the gap wavelength of a photon with energy equal to the semiconductor bandgap. In order to minimize R(A), an a n t i r e f l e c t i o n coating i s normally depos-i t e d on the front surface of the c e l l . The magnitude of a(A) i s determined la r g e l y by the band structure of the semiconductor. For d i r e c t bandgap materials such as gallium arsenide (GaAs), a(A) i s of the order of 10~* cm ^ for v i s i b l e l i g h t , and r i s e s almost discontinuously as A decreases through A [40]. In contrast, i n an i n d i r e c t gap material such as s i l -gap icon, a(A) r i s e s much more slowly with decreasing wavelength, and i s of 4 -1 the order of 10 cm for v i s i b l e l i g h t [40]. 2.2 The Superposition P r i n c i p l e The ultimate goal of any t h e o r e t i c a l study of photovoltaic devices i s to r e l a t e the terminal current-voltage c h a r a c t e r i s t i c s of an i l l u m -inated device to i t s basic material properties. In the past, v i r t u a l l y a l l t h e o r e t i c a l i n v e s t i g a t i o n s of s o l a r c e l l performance have been based v on the assumption that the current J (V) flowing i n an illuminated c e l l i-i with a voltage V maintained across i t s terminals i s given by V V ) = J s c " V V ) <2'2) where J n(V) i s the current which would flow at bias V i n the dark. Equation (2.2) i s embodied i n the s i m p l i f i e d equivalent c i r c u i t commonly drawn for a s o l a r c e l l , which models the illuminated c e l l by an i d e a l current source i n p a r a l l e l with a diode (Fig. 2.2). Equation (2.2) has been referred to variously as the "superposition p r i n c i p l e " [32,33,41] or as the " s h i f t i n g approximation" [31], the l a t t e r term a r i s i n g from a graphical i n t e r p r e t a t i o n i n which the illuminated J-V c h a r a c t e r i s t i c i s obtained by s h i f t i n g the dark c h a r a c t e r i s t i c through an amount J g c along the current axis. The superposition p r i n c i p l e i s a powerful t o o l i n the design and analysis of photovoltaic c e l l s , for when i t applies the terminal current voltage c h a r a c t e r i s t i c s of a device are s p e c i f i e d at any i l l u m i n a t i o n l e v e l once the photocurrent corresponding to that i l l u m i n a t i o n l e v e l and the s i n g l e dark current-voltage c h a r a c t e r i s t i c are determined. For homo-junction c e l l s , well-known expressions are a v a i l a b l e for computation of the dark current, while an a n a l y t i c technique for computing the photo-current has been proposed by Hovel [39]. The superposition p r i n c i p l e l a r g e l y reduces the task of designing an optimally e f f i c i e n t c e l l to that of creating a diode structure which i s e f f e c t i v e i n c o l l e c t i n g phot generated c a r r i e r s , yet which has the minimum possible dark current. An example of the use of the superposition p r i n c i p l e has already been Figure 2.2 S i m p l i f i e d s o l a r c e l l equivalent c i r c u i t 18 encountered i n Chapter 1. There i t was noted that at a given forward bias the dark current flowing i n a Schottky b a r r i e r diode i s i n v a r i a b l y orders of magnitude greater than that which would flow i n a pn diode. Since i t i s expected that the s h o r t - c i r c u i t current i n well-designed Schottky and pn junction s o l a r c e l l s w i l l be comparable [8], i t follows from (2.2) that a Schottky c e l l w i l l have f a r lower maximum power point and open-c i r c u i t voltages than a pn c e l l , and w i l l thus be much les s e f f i c i e n t . 2.2.1 Previous Research on the Superposition P r i n c i p l e In view of the fundamental importance of the superposition p r i n c i p l e to the study of photovoltaics, i t i s remarkable that a rigorous i n v e s t i -gation of the conditions required for (2.2) to apply was not c a r r i e d out u n t i l very recently. Although the pioneers i n the f i e l d of photovoltaics recognized that the p a r a s i t i c shunt and se r i e s resistances associated with r e a l c e l l s could lead to deviations from (2.2), they simply assumed a. p o s t e r i o r i that an " i d e a l " c e l l with no contact resistance and no shunt leakage paths could always be represented by the equivalent c i r c u i t of F i g . 2.2 [29]. Cummerow did derive an equation of the form of (2.2) from f i r s t p r i n c i p l e s [30], but t h i s d e r i v a t i o n was r e s t r i c t e d to the case i n which only n e g l i g i b l e amounts of recombination and photogeneration occur i n the depletion region. In most p r a c t i c a l c e l l s the junction i s formed so close to the surface that a s i g n i f i c a n t f r a c t i o n of the photogeneration i n e v i t a b l y occurs i n the depletion region. In 1976 Lindholm, Fossum and Burgess [31] conducted the f i r s t com-prehensive i n v e s t i g a t i o n of the range of v a l i d i t y of (2.2), and concluded that the superposition p r i n c i p l e would apply to any homojunction s o l a r c e l l provided three b a s i c conditions were s a t i s f i e d . The f i r s t of these conditions, that the i n t e r n a l shunt and se r i e s resistances associated 19 with the c e l l should be n e g l i g i b l e , had been appreciated since the time of the f i r s t research on photovoltaics i n the 1950's [29]. The second condition was that the minority c a r r i e r concentrations i n the quasi-neu-t r a l regions of the c e l l should not exceed low i n j e c t i o n l e v e l s — that i s , the minority c a r r i e r concentrations i n these regions should be much less than the majority c a r r i e r concentration. This condition i s usually s a t i s f i e d f o r devices exposed to one-sun i l l u m i n a t i o n , although high-l e v e l i n j e c t i o n i s frequently encountered i n c e l l s used i n concentrator systems. The f i n a l condition was that the depletion region should not contribute s u b s t a n t i a l l y to both photogeneration and recombination. The l a s t condition l i s t e d above i s of considerable importance, since i t i s l i k e l y to be v i o l a t e d i n a device fabricated on a GaAs sub-s t r a t e . Of a l l semiconductors, t h i s material has the highest p o t e n t i a l photovoltaic conversion e f f i c i e n c y f or operation i n AMI sunlight [1]. Because GaAs has a r e l a t i v e l y wide bandgap, under normal operating conditions much of the recombination i n a t y p i c a l GaAs s o l a r c e l l occurs i n the depletion region [42]. Further, GaAs i s a d i r e c t bandgap material, and so strongly absorbs a l l photons with energies greater than E . Since e f f i c i e n t GaAs homojunction c e l l s have very shallow front junctions, i t follows that a s i g n i f i c a n t f r a c t i o n of the t o t a l photogeneration must occur i n the depletion region. In f a c t , the most e f f i c i e n t s o l a r c e l l s f a b r i c a t e d to date have been s o - c a l l e d "heteroface" devices incorporating a wide-bandgap pGa^^Al^As "window" layer grown by epitaxy over an ex-tremely shallow pGaAs-nGaAs homojunction [43], and for these c e l l s the majority of photogeneration undoubtably occurs within the depletion region. 20 2.2.2 An A n a l y t i c Derivation of the Superposition P r i n c i p l e In t h i s subsection, a simple a n a l y t i c argument e s t a b l i s h i n g the v a l i d i t y of (2.2) for t y p i c a l homojunction s o l a r c e l l s i s developed. Following Lindholm e_t al_. [31] and Cummerow [30], the argument i s based on a straightforward extension of Shockley's seminal analysis of current flow i n the pn diode [44]. Just as i n these e a r l i e r treatments, i t i s assumed that the s e r i e s resistance associated with the quasi-neutral regions i s n e g l i g i b l e , and that the minority c a r r i e r concentrations i n these regions do not exceed low i n j e c t i o n l e v e l s . However, whereas Lindholm et a l . and Cummerow simply assumed without j u s t i f i c a t i o n that the quasi-fermi energy l e v e l s are constant across the depletion region of an illuminated c e l l , t h i s assumption i s examined c r i t i c a l l y i n sub-section 2.2.3, and found to be grossly inaccurate i n many s i t u a t i o n s . In the course of the analysis i t i s shown that the superposition p r i n c i p l e can provide an excellent approximate d e s c r i p t i o n of c e l l c h a r a c t e r i s t i c s even when the t h i r d necessary condition s p e c i f i e d by Lindholm et_ a l . does not apply — that i s , when the bulk of both photogeneration and recombi-nation occur i n the depletion region [32]. In subsection 2.2.4, support for the a n a l y t i c argument i s provided by d i r e c t numerical s o l u t i o n of the d i f f e r e n t i a l equations governing the current flows, c a r r i e r d i s t r i -butions and p o t e n t i a l s within a s o l a r c e l l . This numerical analysis i s applied to both s i l i c o n and GaAs devices. In subsection 2.2.5, numerical analysis i s used to show that the superposition p r i n c i p l e may s e r i o u s l y overestimate the e f f i c i e n c y of c e l l s f a b r i c a t e d on material with very poor minority c a r r i e r l i f e t i m e s and low m o b i l i t i e s , even i f a l l three conditions set f o r t h i n [31] are s a t i s f i e d [33]. This i s not an unimpor-tant point i n view of the current i n t e r e s t i n c e l l s fabricated on inexpensive, low-quality substrates. The operation of any s o l a r c e l l i s governed by the f i v e b a s ic equations of semiconductor physics: the continuity equations, the current equations, and Poisson's equation [45]. Normally, only steady-state operation i s of i n t e r e s t f o r s o l a r c e l l s . Further, only one-dimensional devices w i l l be considered here. (Real three-dimensional c e l l s are con-ve n t i o n a l l y modelled by forming networks of i d e a l i z e d one-dimensional diode elements connected by lumped resistances representing, for example, the spreading resistance of a shallow d i f f u s e d surface layer [46]). The basic equations are l i s t e d below i n t h e i r steady-state, one-dimen-s i o n a l form. Continuity Equations: 0 = ( l / q ) ( d J /dx) + G - U TJ n (2.3) 0 = - ( l / q ) ( d J /dx) + G - U P (2.4) Current Equations: J n = Q M nn£ + qD n(dn/dx) (2.5) (2.6) Poisson's Equation: £ = -di|)/dx; (2.7) As Lindholm e_t_ a l . [31] have'pointed out, a cursory inspection of the basic equations reveals that the p r i n c i p l e of dark current and photo-current superposition can not be generally correct, at least i n a s t r i c t 22 mathematical sense. When combined with appropriate boundary conditions at the surfaces of a c e l l , equations (2.3)-(2.7) can be thought of as a system s p e c i f y i n g the terminal current J as a response to two e x c i t a t i o n s : the applied terminal voltage V and i l l u m i n a t i o n . Since the basic equations are non-linear, the response to two ex c i t a t i o n s applied simultaneously w i l l not i n general be equal to the sum of the responses to the same excit a t i o n s applied separately. Fortunately, the superposition p r i n c i p l e provides an excel l e n t approximate de s c r i p t i o n of c e l l behaviour i n many circumstances, even though i t i s not rigorously true. Although the argument developed i n th i s section i s applicable to any planar homojunction s o l a r c e l l , i t i s most e a s i l y presented i n reference to a s p e c i f i c device structure. The structure considered here i s a con-ventional N +P d i f f u s e d junction c e l l with the geometry of F i g . 2.3. Before any progress can be made i n so l v i n g the basic equations, boundary conditions must be imposed on n, p and ty at the front and back faces of the c e l l . I t i s conventionally assumed that at the front sur-face the electron concentration i s fi x e d at i t s thermal e q u i l i b r i u m value and the electron quasi-fermi l e v e l coincides with the fermi l e v e l i n the metal contact to the emitter. Analogous boundary conditions are applied to the hole concentration and quasi-fermi l e v e l at the back surface. These two conditions automatically determine the e l e c t r o s t a t i c p o t e n t i a l drop across the c e l l f o r any applied bias V. Further boundary conditions are imposed by s p e c i f y i n g surface recombination v e l o c i t i e s f or minority c a r r i e r s . At the front surface, - y x F ) - q s F p n ( x F ) (2.8) 1 N | | P QUASI-NEUTRAL 1 DEPLETION 1 QUASI-NEUTRAL EMITTER | • REGION -1 BASE Figure 2.3 Geometry of N P s o l a r c e l l . 24 while at the back surface Frequently an ohmic contact i s present at the back of the c e l l , i n which case S_ -»• (2.8) and (2.9) state that the minority c a r r i e r recombination rate at each surface i s proportional to the excess minority c a r r i e r concentration there. This p r o p o r t i o n a l i t y i s e s s e n t i a l i f the superposition p r i n c i p l e i s to hold. Following Shockley's method of analysis [44], the device of F i g . 2.3 has been divided i n t o two quasi-neutral regions separated by a depletion or space-charge region. In the depletion region the concentration of ionized dopants i s f a r greater than the concentration of free charge c a r r i e r s , while i n the quasi-neutral regions the charge of the i o n i z e d dopants i s balanced almost exactly by that of the majority c a r r i e r s . The boundaries between these regions are assumed to be abrupt. At equilibrium, there i s e s s e n t i a l l y no e l e c t r i c f i e l d i n the quasi-neutral base, but a large f i e l d e x i s t s i n the quasi-neutral emitter as a r e s u l t of the non-uniform doping p r o f i l e there. In the low-level i n j e c t i o n regime the a p p l i c a t i o n of a forward bias V to the c e l l i s assumed to r e s u l t i n the reduction of the e l e c t r o s t a t i c p o t e n t i a l drop across the depletion region by an amount V, while leaving the majority c a r r i e r and 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 s i n the quasi-neutral regions unchanged. Perhaps the most elegant way to describe the operation of a s o l a r c e l l i s i n terms of the continuity p r i n c i p l e , following the approach of Lindholm ejt a l . [31]. In the steady s t a t e , the continuity p r i n c i p l e holds that the current flowing through the terminals of a s o l a r c e l l must equal the difference between the t o t a l rate of photogeneration and the t o t a l rate of recombination i n the device. That i s , J L ( V ) = q / G(x) dx - q / U(x) dx (2.10) c e l l c e l l where G(x) i s the volume rate of photogeneration and U(x) the volume rate of recombination, i n c l u d i n g recombination at the front and back boundaries of the c e l l . (U(x) i s defined here to account for the thermal generation of c a r r i e r s , as w e l l as recombination). Equation (2.10) simply s p e c i f i e s that i n the steady state each photogenerated electron-hole p a i r must e i t h e r recombine within the c e l l or contribute to current flow i n the external c i r c u i t . From (2.10) i t can be seen that the superposition p r i n c i p l e w i l l accurately describe the c h a r a c t e r i s t i c s of a given c e l l i f and only i f the i n t e g r a l i n v o l v i n g U can be s p l i t i n t o a term which depends on the i l l u m i n a t i o n l e v e l and not on V, and a term which depends on V but i s independent of i l l u m i n a t i o n . U i s , i n general, a complicated function of the c a r r i e r concentra-tions n and p. However, within a quasi-neutral region under low-level i n j e c t i o n conditions the recombination rate i s c o n t r o l l e d by the supply of minority c a r r i e r s , and U therefore becomes proportional to the excess minority c a r r i e r concentration [47]. By d e f i n i t i o n , the p r o p o r t i o n a l i t y constant i s the r e c i p r o c a l of the minority c a r r i e r l i f e t i m e x. Moreover, i n a quasi-neutral region the minority c a r r i e r current depends l i n e a r l y on the excess minority c a r r i e r concentration for low-level i n j e c t i o n , even i f a b u i l t - i n e l e c t r i c f i e l d i s present due to non-uniform doping [31]. This allows the minority c a r r i e r current and continuity equations to be combined, y i e l d i n g a s i n g l e l i n e a r d i f f e r e n t i a l equation governing / the excess minority c a r r i e r d i s t r i b u t i o n . To take a s p e c i f i c example, i n the uniformly doped quasi-neutral base region, the d r i f t component i n the electron current equation (2.5) can be ignored for low-level i n j e c t i o n , so t h i s equation becomes J = qD (dn'/dx). (2.11) n ^ n p Combining (2.11) with the electron continuity equation (2.3), the follow-ing d i f f e r e n t i a l equation describing the excess electron d i s t r i b u t i o n n'(x) i n the base i s obtained: P d 2n'/dx 2 = n'/(D x ) - G(x)/D . (2.12) p p n n n In order to solve (2.12), i t i s necessary to specify boundary conditions on n^ at the borders of the base. From (2.9) and (2.5), at the back contact -D (dn'/dx) = S Rn'0O. (2.13) n p a p JQ Obtaining a boundary condition on n^ at the border between the base and the depletion region presents a more d i f f i c u l t problem. In Shockley's analysis of current flow i n the pn diode [44], the excess minority car-r i e r concentrations at the boundaries between the depletion region and the quasi-neutral regions are found by assuming that the quasi-fermi energy l e v e l s for both c a r r i e r s are constant across the depletion region. Although Shockley's analysis dealt only with devices operated i n the dark, both Cummerow [30] and Lindholm e_t a l . [31] applied t h i s assumption 27 without q u a l i f i c a t i o n to the case of an illuminated s o l a r c e l l , and added the further t a c i t assumption that the e l e c t r o s t a t i c p o t e n t i a l b a r r i e r across the depletion region depends only on the bias V and not on the i l l u m i n a t i o n l e v e l . Taken together, these two assumptions imply that at the boundary x^ between the depletion region and the quasi-n e u t r a l base n^(x p) = n p ( )[exp(qV/kT) - 1] (2.14) i r r e s p e c t i v e of the i l l u m i n a t i o n l e v e l . Although i t i s possible to determine the excess electron d i s t r i b u t i o n i n the base by d i r e c t l y s olving (2.12) with boundary conditions (2.13) and (2.14), n p(x) can be found more e a s i l y by superposing the solutions to the following two systems: System 1: d 2n'/dx 2 = n'/(D x ) with B.C.'s (2.13) and (2.14) p p n n System 2: d 2n'/dx 2 = n'/(D x ) - G(x)/D with B.C.'s (2.13) and n'(x ) = 0. p p n n n P P C l e a r l y , the s o l u t i o n to the f i r s t system gives the excess electron d i s -t r i b u t i o n when a bias V i s applied i n the dark, while the s o l u t i o n to the second system i s bias-independent. R e c a l l i n g that U i s proportional to n', i t follows that the i n t e g r a l of U over the base can be s p l i t i n t o P two terms, with one term corresponding to the a p p l i c a t i o n of a bias V i n the dark and the second, term depending only on the i l l u m i n a t i o n l e v e l . A s i m i l a r argument can be used to show that the i n t e g r a l of U over the quasi-neutral emitter can be s p l i t i n t o s t r i c t l y bias-dependent and s t r i c t l y illumination-dependent terms. Since the non-uniform doping pro-f i l e i n t h i s region gives r i s e to a large e l e c t r i c f i e l d , the d r i f t term i n the expression for the hole current can not be ignored. Writing the t o t a l hole concentration i n the emitter as a sum of the equilibrium con-centration and the excess concentration, the hole current equation (2.6) becomes = [-qDp(dP];/dx) + q P p i P ^ ] + [-qD p(dp n 0/dx) + q u p £ p n 0 ] . (2.15) The r i g h t hand term i n square brackets i n (2.15) i s simply the value of the hole current at thermal equilibrium. The p r i n c i p l e of d e t a i l e d b a l -ance states that there can be no net hole current anywhere i n the c e l l at equilibrium, so t h i s term must equal zero. Substituting the l e f t hand term i n (2.15) i n t o the hole continuity equation (2.4) gives the follow-ing d i f f e r e n t i a l equation governing the excess hole concentration i n the emitter: X [ d ( p p £ ) / d x ] - u p£(dp^/dx) + (dp^/dx) (dD p/dx) (2.16) + D (d 2p'/dx 2) + G(x) - p ' / x = 0 p r n n p Although the quantities y p , D p, T p and £ may a l l depend on p o s i t i o n within the emitter, they should a l l be independent of p^ i n the low-level i n j e c t i o n regime. (2.16) i s therefore l i n e a r i n the excess hole concen-t r a t i o n . The boundary conditions on the hole concentration i n the quasi-29 n e u t r a l emitter are analogous to those on the electron concentration i n the quasi-neutral base. At the front surface, - M p £ P ; + Dp(dP;/dx) = S Fp;(x F) (2.17) while at the boundary between the emitter and the depletion region P n ( x n } = P n 0(x n)[exp(qV/kT) - 1] . (2.18) By analogy with the so l u t i o n f o r the electron d i s t r i b u t i o n i n the base, the excess hole d i s t r i b u t i o n i n the emitter of a forward-biased, i l l u m -inated c e l l can be found by superposing the solutions to two s p e c i a l cases of (2.16). The f i r s t of these solutions i s for G(x)=0 with boundary conditions (2.17) and (2.18), while the second s o l u t i o n i s for non-zero G(x) with boundary condition (2.17) and the a d d i t i o n a l condition p'(x )=0. n n It should be noted that while an e x p l i c i t s o l u t i o n for n'(x) i n the u n i -P formly doped base can be obtained t r i v i a l l y , the s o l u t i o n of (2.16) f o r non-zero £ represents a far more d i f f i c u l t mathematical problem [48]. To complete the evaluation of the i n t e g r a l of U over the c e l l , i t i s now necessary to examine the i n t e g r a l of U over the depletion region. In Cummerow's analysis [30], recombination i n the depletion region was simply ignored. Lindholm e_t a l . [31] noted that i n the depletion region U i s not l i n e a r i n the excess c a r r i e r concentrations, and conse-quently concluded that within t h i s region i t would not i n general be possible to s p l i t U i n t o a term which depends only on the i l l u m i n a t i o n l e v e l and a term which depends only on the applied b i a s . On these grounds i t was further concluded that the superposition p r i n c i p l e should apply 30 only to those devices i n which the depletion region does not contribute s i g n i f i c a n t l y to both the photogeneration and recombination of c a r r i e r s . However, t h i s reasoning i s not consistent with the use of Shockley's method [44] to obtain boundary conditions on the excess minority c a r r i e r concentrations at the edges of the depletion region. I f i t i s assumed that the quasi-fermi energy l e v e l s and the e l e c t r o s t a t i c p o t e n t i a l i n the depletion region depend only on the bias V and not on the i l l u m i n a t i o n l e v e l , then the c a r r i e r concentrations i n t h i s region must depend only on V. This i n turn implies that for a given bias V the i n t e g r a l of U over the depletion region i s independent of the i l l u m i n a t i o n l e v e l . Combining the conclusions drawn above regarding recombination i n the quasi-neutral regions and the depletion region, i t follows that the i n t e g r a l of U over the e n t i r e c e l l can be divided i n t o s t r i c t l y b i a s -dependent and s t r i c t l y illumination-dependent terms. The current flowing i n an illuminated c e l l i s therefore given by V V ) = Jupc ' V V ) <2'19> where J i s a bias-independent photocurrent. In accordance with the upc terminology introduced by Lindholm et a l . [31], the subscript "upc" stands for "uncompensated photocurrent". At t h i s point the introduction of the symbol J may seem redundant, since i f (2.19) holds for a l l V, upc J must be i d e n t i c a l to J . However, the d i s t i n c t i o n between J upc sc upc and J w i l l prove useful i n the remainder of t h i s discussion. While sc J i s the current flowing i n a r e a l c e l l with the terminals shorted, sc " J i s best thought of as a mathematical quantity defined by upc Jupc = q ^ G ( x ) d x ~ q / n U ( x ) d x " 1 / B u ( x > d x • (2.20) X F X F x p In (2.20) the recombination rate U(x) i s to be evaluated for boundary conditions n'(x )=0 and p'(x )=0 with photogeneration d i s t r i b u t i o n G(x) p p n n appropriate to the i l l u m i n a t i o n conditions under consideration. Throughout the derivation of (2.19) i t was assumed that the bound-aries between the quasi-neutral regions and the depletion region do not move as the bias applied to the c e l l changes. This i s , of course, not the case i n a r e a l c e l l , since the depletion region contracts as the bias V i s increased. I t might thus be expected that the photocurrent ^ U p C defined i n (2.20) would have some s l i g h t bias dependence, since x^ and x are functions of V. For example, i n the N +P c e l l used as a model here, p an increase i n forward bias r e s u l t s i n an increase i n the width of the quasi-neutral base at the expense of the depletion region, while the width of the heavily-doped quasi-neutral emitter remains v i r t u a l l y un-changed. Thus i n F i g . 2.3 x p moves to the l e f t while X r i s stationary. Now consider a fixed point A i n the diode chosen so that x l i e s to the P r i g h t of A for small forward bias and to the l e f t of A for larger forward bia s . In the former case A l i e s i n the depletion region, and so v i r t u a l l y a l l c a r r i e r s photogenerated at A are c o l l e c t e d and contribute to • T Up C-In the l a t t e r case A l i e s i n the quasi-neutral base, but even so most of the c a r r i e r s photogenerated at A w i l l d i f f u s e to the junction and be co l l e c t e d so long as A l i e s within an electron d i f f u s i o n length of x p. Thus ^ U p C should be e s s e n t i a l l y independent of bias as long as the change i n the width of the depletion region with bias i s small compared to a minority c a r r i e r d i f f u s i o n length i n the quasi-neutral base. 2.2.3 Quasi-Fermi Levels i n the Depletion Region The argument used to e s t a b l i s h (2.19) was based on the twin assump-tions that the e l e c t r o s t a t i c p o t e n t i a l and the quasi-fermi energy l e v e l s i n the depletion region of a s o l a r c e l l are dependent only on the applied bias and not on the i l l u m i n a t i o n l e v e l . The accuracy of the assumption on the e l e c t r o s t a t i c p o t e n t i a l can be confirmed f a i r l y e a s i l y . Provided that the s e r i e s resistance associated with the quasi-neutral regions i s n e g l i g i b l e and that the minority c a r r i e r concentrations i n these regions remain at low i n j e c t i o n l e v e l s , the a p p l i c a t i o n of a forward bias V should r e s u l t i n the reduction of the junction e l e c t r o s t a t i c p o t e n t i a l b a r r i e r by an amount V as w e l l , regardless of the i l l u m i n a t i o n l e v e l . Provided also that the free c a r r i e r concentrations i n the depletion region are always n e g l i g i b l e compared to the concentration of ionized dopants i n that region, i t follows that the depletion approximation should accurately describe the v a r i a t i o n of the e l e c t r o s t a t i c p o t e n t i a l across the junction for a l l i l l u m i n a t i o n conditions. The constancy of the quasi-fermi l e v e l s across the depletion region can best be checked by e x p l o i t i n g the r e l a t i o n s h i p between the e l e c t r o n and hole currents and the gradients of these energy l e v e l s [49]; s p e c i -f i c a l l y , J = y n VE„ n n Fn (2.21) and J = y p VE„ P P Fp (2.22) Q u a l i t a t i v e l y , (2.21) reveals that the gradient of E must be large wherever J i s large and n i s small. Thus there w i l l be a large drop i n n across a region through which a large electron current flows and i n which the electron concentration i s small. Similar remarks apply to Quan t i t a t i v e l y , (2.21) and (2.22) can be used to compute the change i n the quasi-fermi l e v e l s across any part of a device i f accurate e s t i -mates for the currents and c a r r i e r concentrations i n that region are a v a i l a b l e . In the case of a s o l a r c e l l , i t i s possible to obtain a f i r s t estimate f o r the currents and c a r r i e r concentrations i n the depletion region under given operating conditions by assuming that the two quasi-fermi l e v e l s are constant across t h i s region and applying the analysis outlined above. The drops i n the quasi-fermi l e v e l s across the depletion region, AE_ and AE_, , can then be estimated from (2.21) and (2.22). I f rn rp the estimates f o r A E ^ and ^ E F p calculated following t h i s procedure are much smaller than kT, the assumption of constant quasi-fermi l e v e l s i s s e l f - c o n s i s t e n t , and probably provides an accurate approximation to the actual s o l u t i o n to the basic equations. However, i f the c a l c u l a t i o n s suggest that large changes i n E^^ and E ^ across the depletion region would be required to support the estimated electron and hole currents, the assumption of constant quasi-fermi l e v e l s i n the depletion region i s c l e a r l y untenable. Equations (2.21) and (2.22) w i l l now be used to inve s t i g a t e the behaviour of the quasi-fermi l e v e l s i n the depletion region of an N P so l a r c e l l . The behaviour of the quasi-fermi l e v e l s when a forward bias i s applied i n the dark w i l l be examined f i r s t , and then the e f f e c t of il l u m i n a t i o n w i l l be considered. A. Forward Bias i n the Dark Dark current i n a forward-biased N +P c e l l normally r e s u l t s from recombination i n e i t h e r the depletion region or the quasi-neutral base. The magnitude of the i n j e c t i o n - d i f f u s i o n current r e s u l t i n g from recombi-nation i n the base i s given by J = q^lT n'(x ) / / T . (2.23) a n p p n x ' To support an i n j e c t i o n - d i f f u s i o n current, electrons must flow a l l the way across the depletion region and enter the quasi-neutral base. On moving from the emitter across the depletion region towards the base the free electron concentration decreases by many orders of magnitude. There-fore, for t h i s type of dark current, the electron concentration near the edge of the base i s of greatest importance i n determining AE„ . However, rn (2.23) states that J, i s proportional to n'(x ). Thus even though J , d P P d increases exponentially with V, the electron concentration i n the dep-l e t i o n region increases i n step so that AE i s roughly independent of r t l b i a s . A s i m i l a r r e s u l t holds f o r the depletion region recombination cur-rent J . A crude estimate for the magnitude of J can be obtained by rg rg noting that the rate of c a r r i e r recombination i s highest at the middle of the depletion region, near the point where the electron and hole concentrations are equal. I f the maximum recombination rate i s m u l t i p l i e d by the width W of the depletion region one a r r i v e s at the approximate formula [50] J rg ~~ * nn=p W / < 2 / V ? • C 2 ' 2 * ) Here n n = p i s the electron (or hole) concentration at the point where n=p. To support a depletion region recombination current i t i s only 35 necessary for electrons to t r a v e l from the emitter to the center of the depletion region, where most recombination takes place. S i m i l a r l y , holes need only flow from the base to the zone near the center of the depletion region where the recombination rate i s greatest. Thus the hole and e l e c -tron currents are large only i n those parts of the depletion region where the corresponding c a r r i e r concentrations are large as w e l l . The drops i n the quasi-fermi l e v e l s across the depletion region are therefore f i x e d p r i m a r i l y by the c a r r i e r concentrations at the place of maximum recombi-nation, that i s by n . Just as i n the case of an i n j e c t i o n - d i f f u s i o n J n=p J current, although J increases exponentially with b i a s , n n_p increases proportionately, so that A E ^ and AEp p are roughly bias independent. B. Under Illumination When an N +P sol a r c e l l i s forward-biased i n the dark, electrons supplied from the emitter recombine with holes i n the depletion region and i n the base. Thus throughout the depletion region the electron flow i s directed towards the base. To support t h i s flow the electron quasi-fermi l e v e l must be s l i g h t l y lower at the edge of the base than at the edge of the emitter. Therefore, at each point i n the depletion region the electron concentration i s a c t u a l l y s l i g h t l y l e s s than would be the case i f E ^ n were p r e c i s e l y constant across t h i s region. When the c e l l i s exposed to l i g h t , at l e a s t some photogenerated electrons must flow from the base i n t o the depletion region. This photogenerated electron current opposes the current due to electrons i n j e c t e d from the emitter, with the r e s u l t that the net electron flow from the emitter to the base i s smaller under i l l u m i n a t i o n than i n the dark. I f the electron quasi-fermi l e v e l i n the emitter i s chosen as a reference point, i t follows that everywhere in the depletion region of an illum i n a t e d , forward-biased c e l l E„_ must 36 be higher than i t was at the same forward bias i n the dark. Therefore, throughout the depletion region the electron concentration must be greater i n the l i g h t than i t was at the same forward bias i n the dark, assuming that the e l e c t r o s t a t i c p o t e n t i a l depends only on the applied b i a s . A completely analogous' argument can be developed to show that, for a given forward b i a s , the hole concentration everywhere i n the depletion region must also be greater under i l l u m i n a t i o n than i n the dark. This increase i n the concentration of free c a r r i e r s i n the depletion region must lead to an increased rate of recombination i n that region. Also, the excess minority c a r r i e r concentrations at the boundaries of the quasi-neutral regions w i l l be greater i n the l i g h t than at the same forward bias i n the dark. As a r e s u l t , there w i l l be more recombination i n these regions than would be the case i f the quasi-fermi l e v e l s were, i n f a c t , constant across the depletion region. Thus for a given forward b i a s l e s s current can be drawn from the c e l l terminals than predicted by (2.19), and so (2.19) must overestimate the energy conversion e f f i c i e n c y . Although (2.19) i s never s t r i c t l y c o r r e c t , i t s t i l l provides an excellent approximate d e s c r i p t i o n of the c h a r a c t e r i s t i c s of most r e a l homojunction s o l a r c e l l s . (2.19) w i l l be inaccurate only i f at some bias point the c a r r i e r concentrations i n the depletion region are s i g n i f i c a n t l y greater under i l l u m i n a t i o n than i n the dark and the t o t a l rate of recom-bination i s comparable to the t o t a l rate of photogeneration. Whether or not t h i s condition i s r e a l i z e d i n a given c e l l depends on the extent to which the c a r r i e r concentrations i n the depletion region increase under i l l u m i n a t i o n , and on the r e l a t i o n s h i p between the c a r r i e r concentrations and the recombination rate. It has been shown above that f o r a c e l l forward biased i n the dark 37 the carrier flows and carrier concentrations in the depletion region both increase exponentially with bias, with the result that AE_ and AE_ are Fn Fp roughly bias independent. In contrast, in an illuminated solar c e l l large photocurrents flow across the depletion region even at short-circuit or low forward bias. Applying (2.21) and (2.22) in the quantitative manner suggested above, i t can readily be shown that at low forward bias these photocurrents can not be supported unless E„ and E^ are shifted sub-Fn Fp stantially from their positions in the dark. This result applies to essentially any device, regardless of the choice of substrate material or doping profile. Thus the assumption that E ^ and E p p are constant across the depletion region is grossly in error at low forward bias. How-ever, as the forward bias is increased the carrier concentrations in the depletion region rise, and so the drops in E ^ and E ^ across this region required to support the photocurrent decrease. If the bias is increased s t i l l further, eventually an operating point w i l l be reached for which AE^ and AE F p are both small fractions of kT, and from this point on (2.19) w i l l accurately describe the c e l l characteristics. For most homo-junction c e l l s , the bias point at which the quasi-fermi energy levels become effectively constant across the depletion region i s reached when the total rate of recombination i s s t i l l many orders of magnitude smaller than the total rate of photogeneration. If this i s the case, then the fact that the quasi-fermi levels are not constant across the depletion region for operation at short-circuit or low forward bias w i l l have no measureable effect on the accuracy of (2.19). Further, J w i l l be upc indistinguishable from J , so (2.2) and (2.19) w i l l be interchangeable. s c If the drops in E^ n and Ep p across the depletion region of an illum-inated solar c e l l are both much smaller than kT for operation near the 38 maximum power point, then (2.2) should provide a very accurate d e s c r i p t i o n of the device c h a r a c t e r i s t i c s . The numerical analysis described i n the next subsection reveals that t h i s condition on AE„ and AE„ at the max-Fn Fp imum power point i s e a s i l y s a t i s f i e d f o r t y p i c a l s i l i c o n or GaAs homo-junction devices. More generally, (2.2) i s most l i k e l y to be a p p l i c a b l e to those devices fabricated on substrates with high c a r r i e r m o b i l i t i e s and long minority c a r r i e r l i f e t i m e s . From (2.21) and (2.22) i t i s apparent that the drops i n Ep n and E ? p across the depletion region required to support a given photocurrent at some s p e c i f i e d bias point w i l l be small when y^ and y p are large, i f other device properties are constant. For a device with long c a r r i e r l i f e t i m e s , the c a r r i e r concentrations i n the depletion region can be raised to r e l a t i v e l y high l e v e l s before the t o t a l rate of recombination becomes comparable to the t o t a l rate of photogener-ation. Thus the e f f e c t of high m o b i l i t i e s i s to minimize the increase i n c a r r i e r concentrations i n the depletion region under i l l u m i n a t i o n , while the e f f e c t of long l i f e t i m e s i s to minimize the increase i n t o t a l recom-bination brought about by t h i s r i s e i n c a r r i e r concentration. 2.2.4 Numerical Analysis of S i l i c o n and GaAs Homojunction C e l l s In subsection 2.2.2 i t was shown that i f the quasi-fermi energy l e v e l s were always constant across the depletion region of an i l l u m i n a t e d s o l a r c e l l , then (2.2) would accurately describe the c e l l c h a r a c t e r i s t i c s i n the low-level i n j e c t i o n regime. This assumption of constant quasi-fermi l e v e l s i n the depletion region had been used i n a l l previous attempts to j u s t i f y (2.2) [30,31]. In subsection 2.2.3, e s s e n t i a l l y q u a l i t a t i v e argu-ments based on (2.21) and (2.22) were used to show that for operation at s h o r t - c i r c u i t or under low forward bias the quasi-fermi l e v e l s i n 39 f a c t vary sharply over the depletion region of any illuminated c e l l . However, i t was subsequently proposed that i n c e l l s with reasonable l i f e t i m e s and c a r r i e r m o b i l i t i e s the drops i n and E„ across the dep-Fn Fp l e t i o n region would be very small f o r operation near the maximum power point, and that consequently (2.2) would provide an e x c e l l e n t d e s c r i p t i o n of the c e l l c h a r a c t e r i s t i c s at a l l operating points. The purpose of t h i s subsection i s to use d i r e c t numerical solutions of the basic semiconductor equations to provide q u a n t i t a t i v e support for the conclusions drawn i n subsections 2.2.2 and 2.2.3. The algorithm chosen for the numerical s o l u t i o n of (2.3)-(2.7) was that developed by Seidman and Choo [51]; a detailed d e s c r i p t i o n of the algorithm and a complete l i s t i n g of the FORTRAN programs written to implement i t are given i n Appendix A. I t should be stressed that t h i s algorithm makes no a r b i t r a r y assumptions concerning the behaviour of the e l e c t r o s t a t i c pot-e n t i a l and the quasi-fermi l e v e l s within a device. Two devices were modelled, one a s i l i c o n c e l l resembling those a v a i l a b l e commercially, and the other a GaAs c e l l . The doping p r o f i l e s and material properties for these two c e l l s are summarized i n Table 2.1. In commercial s i l i c o n c e l l s , the emitter i s usually formed by carrying out a phosphorus d i f f u s i o n under constant surface concentration conditions at a temperature close to 900°C. As a r e s u l t , the emitter doping p r o f i l e i s of the complementary err o r function form, with a phos-20 3 phorus concentration of approximately 10 atoms/cm at the surface [52]. However, recent studies have shown that because of the bandgap narrowing associated with such high impurity concentrations, the e f f e c t i v e doping near the surface of the emitter i s considerably lower than the actual phosphorus concentration [53]. To compensate for t h i s e f f e c t , i n the TABLE 2.1 C e l l properties used i n numerical analysis a) NTP S i l i c o n C e l l doping p r o f i l e base doping donor concentration at emitter surface m e t a l l u r g i c a l junction depth device width c a r r i e r l i f e t i m e s c a r r i e r m o b i l i t i e s S F SB Gaussian 5*10 1 5 cm"3 1*10 1 9 cm"3 0.5 ym 250 ym see Appendix A see Appendix A i n 3 - 1 10 cm s i n f i n i t e b) P +N GaAs C e l l doping p r o f i l e base doping acceptor concentration at emitter surface m e t a l l u r g i c a l junction depth device width c a r r i e r l i f e t i m e s c a r r i e r m o b i l i t i e s S F SB Gaussian 1 * 1 0 U cm"3 1*10 1 9 cm"3 0.2 ym 10.0 um T = 10~ 9 s; x n P see Appendix A 0 i n f i n i t e 41 numerical model a Gaussian doping profile with a surface concentration 19 3 of only 10 atoms/cm was used i n the emitter. The Gaussian profile i s somewhat fl a t t e r near the surface than the complementary error function distribution. Since GaAs cells are s t i l l in a developmental stage, i t i s d i f f i c u l t to select a representative design for such a device. In recent years, both P N [43,54] and N P [55] GaAs cells prepared by several different techniques have shown promisingly high efficiencies. The structure and material properties of the P+N GaAs c e l l considered here were chosen more to ensure that the bulk of both photogeneration and recombination would occur in the depletion region than to accurately model a particular experimental device. To this end the photogeneration distribution G(x) was set to correspond to the uniform absorption i n the depletion region of a l l photons i n the solar spectrum with energies greater than the GaAs bandgap; G(x) was made equal to zero in the other regions of the c e l l . Some preliminary tests on the accuracy of the solutions to the basic equations obtained using Seidman and Choo's algorithm are described in Appendix A. These i n i t i a l tests were concerned primarily with establishing an appropriate grid geometry for the applications considered here. To provide a further test of the capabilities of the numerical model, the dark current-voltage characteristics for both the si l i c o n and the GaAs c e l l were computed. The results are plotted in Fig. 2.4. The characteristic for the si l i c o n c e l l strongly resembles those recorded for experimental s i l i c o n diodes, i n that two regimes over which the current depends exponentially on bias can be discerned. When the forward bias i s less than approximately 250 mV, the current obeys the relation Figure 2.4 Dark current-voltage c h a r a c t e r i s t i c s . True J (V) curve i s drawn s o l i d ; J r g and J d curves computed from (2.26) and (2.27) or (2.28) are drawn dashed, (a) S i l i c o n c e l l . (b) GaAs c e l l . Figure 2.4(b) 44 J D ( V ) = J0 e xP(q v/AkT) (2.25) where A=1.6. In t h i s regime the diode current i s dominated by recombin-ation i n the depletion region. For forward biases greater than approx-imately 400 mV, recombination i n the quasi-neutral base becomes dominant and the current again depends exponentially on V, but with an A-factor very close to unity. Over the bias range from 250 to 400 mV, a t r a n s i t i o n between the two regimes takes place, and does not have a simple s i n g l e -exponential dependence on V. A crude a n a l y t i c approximation to the dep-l e t i o n region recombination current J i s given by (2.24), which can be written i n the form [50] Jr g ( v ) = Un.W/(2/TT)] exp(qV/2kT) (2.26) Under low-level i n j e c t i o n conditions an a n a l y t i c expression f o r the electron i n j e c t i o n - d i f f u s i o n current flowing i n t o the base i s [61] J d ( V ) = J0d [ e x P ( q v / k T ) - 1] (2.27a) where '0d = qD nf n l n A Si n h ( L B / L n ) + (S BL n/D n) Cosh(L R/L n) Cosh(L B/L n) + (S BL n/D n) S i n h ( L B / L n ) (2.27b) qDnn. C o t h ( L B / L n ) . (2.27c) n A 45 Here L„ is the base width. Plots of J (V) and J,(V) obtained by sub-B rg d 3 stituting the parameters l i s t e d in Table 2.1(a) into (2.26) and (2.27) have been superposed on Fig. 2.4(a). It can be seen that there i s excel-lent agreement between the upper branch of the -^(V) characteristic computed using Seidman and Choo's algorithm and the analytic expression for J (V). However, the expression for J specified in (2.26) clearly overestimates the dark current flowing at small forward bias. This i s not unexpected, for in the derivation of (2.26) the maximum rate of recombination i n the depletion region i s f i r s t determined, and i t i s then assumed that this recombination rate applies throughout the region [50]. The J D(V) characteristic for the GaAs c e l l reveals the importance of depletion region recombination currents in this material. For forward biases less than approximately 800 mV, (2.25) i s obeyed with A=1.8. By analogy with (2.27), the analytic expression for the hole injection-diffusion current flowing into the uniformly doped n-type base of the GaAs c e l l i s J d ( V ) = J 0 d texpCqWkT) - 1] (2.28a) where Jod = q D P n i p D Sinh(L B/L p) + (S BL p/D p) Cosh (1^/1, ) Cosh(L B/L p) + (S BL p/D p) Sinh(L B/L p) (2.28b) •g-^* qD n Coth (L /L ) . °B — 2 — - B P (2.28c) P D 46 This a n a l y t i c form f o r ^^(V) has been overplotted on F i g . 2.4(b). I t can be seen that even at a forward bias of 900 mV, which gives a dark current density comparable i n magnitude to the one-sun photocurrent, l e s s than 25% of the t o t a l current flowing results from recombination i n the quasi-n e u t r a l base. More precise checks on the accuracy of the numerical analysis can be made by examining the i n d i v i d u a l electron and hole current components within a device. For example, the electron current flowing across the plane x p i n t o the quasi-neutral base of an N +P c e l l should conform c l o s e l y to (2.27), while the hole current crossing the plane x i n a P +N c e l l n should agree with (2.28). In a r e a l device there i s some ambiguity i n l o c a t i n g the planes x^ and x p, since the boundaries between the depletion region and the quasi-neutral regions are not abrupt. Here X r and x p were defined a r b i t r a r i l y to be the planes at which the f r e e - c a r r i e r concentra-t i o n equals 90% of the donor or acceptor concentration. Using t h i s d e f i n i t i o n , J (x ) for the s i l i c o n c e l l and J (x ) f o r the GaAs c e l l were n p p n determined and p l o t t e d i n F i g . 2.4. For both devices there i s excellent agreement between the a n a l y t i c expression f o r the minority c a r r i e r i n j e c t i o n - d i f f u s i o n current flowing into the base and the value computed fo r t h i s current component using the numerical model. The band diagrams generated by the numerical model for the s i l i c o n c e l l under a v a r i e t y of bias and i l l u m i n a t i o n conditions are presented i n F i g . 2.5. The corresponding band diagrams for the GaAs c e l l have the same q u a l i t a t i v e form, and so are not shown. Under moderate forward bias i n the dark, the e l e c t r o n and hole quasi-fermi l e v e l s are both found to be very nearly constant across the depletion region. However, under one-sun i l l u m i n a t i o n at s h o r t - c i r c u i t or small forward b i a s , E__ and E„ Fn Fp 47 (b) Figure 2.5 Band diagrams for s i l i c o n c e l l . (a) S h o r t - c i r c u i t , one-sun i l l u m i n a t i o n . (b) Maximum power point, one-sun i l l u m i n a t i (c) V=V i n dark. Figure 2.5(b) Figure 2.5(c) 50 s h i f t so as to greatly increase the c a r r i e r concentrations i n t h i s regio As the forward bias i s increased under i l l u m i n a t i o n , the drops i n E„ Fn and across the depletion region become smaller; by the time the max-imum power point i s reached both quasi-fermi l e v e l s are e f f e c t i v e l y constant across the depletion region. A l l these r e s u l t s are i n agreement with the conclusions drawn from the q u a l i t a t i v e arguments of subsection 2.2.3. Actual values for AE_ and AE„ are tabulated i n Table 2.2. Fn Fp Fig. 2.6 presents the "true" J (V) c h a r a c t e r i s t i c (that i s , the Li J L ( V ) curve obtained using the numerical model) and a p l o t of the curve J - J„(V) under one-sun i l l u m i n a t i o n for both the s i l i c o n and the GaAs sc D c e l l . To the scale of the fi g u r e , the difference between the two curves i s not v i s i b l e for e i t h e r device. A more precise measure of the accuracy of the superposition p r i n c i p l e i n describing the il l u m i n a t e d current-voltage c h a r a c t e r i s t i c s f o r these two c e l l s i s provided by Table 2.3, which l i s t s the true values of open-circuit voltage, f i l l f a c t o r , maximum output power and maximum power point voltage together with the values for these quantities predicted by (2.2). As expected, the true maximum output power i s s l i g h t l y less than that predicted by the superposition p r i n c i p l e , although the difference i s only 0.4% f o r the s i l i c o n c e l l and less than 0.1% for the GaAs c e l l . For the s i l i c o n c e l l , the true open-c i r c u i t voltage i s found to be 1.5 mV less than that calculated using (2.2), while f o r the GaAs c e l l the difference i s less than 0.1 mV. In short, F i g . 2.6 and Table 2.3 indicate that the superposition p r i n c i p l e provides j u s t as accurate a description of the c h a r a c t e r i s t i c s of a GaAs c e l l i n which the bulk of both recombination and photogeneration occur i n the depletion region as i t does for a t y p i c a l commercial s i l i c o n c e l l . TABLE 2.2 Changes in quasi-fermi levels across depletion region under various operating conditions a) Silicon Cell Condition: short-circuit, one-sun max. power point, one-sun open-circuit, one-sun V = V , dark mp AE F n(eV) 0.36 3*10 -4 4*10 -6 2*10 -5 A EF p ( e V ) 0.28 9*10 -5 5*10 2*10 -6 b) GaAs Cell Condition: short-circuit, one-sun max. power point, one-sun open-circuit, one-sun V = V , dark mp AE F n(eV) 0.19 1*10 2*10 -6 1*10 -6 A EF p ( e V ) 0.47 2*10 -3 3*10 -4 2*10 -4 30 < E 20 J J . (V) o r J L s c cd CC ZD O o (-o ZC CL. 10 J o 200 — I — 400 600 BIAS VOLTAGE (mV) (a) Figure 2.6 Plot of the true current-voltage c h a r a c t e r i s t i c J L ( V ) , and of the curve J g c - J D ( V ) , under one-sun i l l u m i n a t i o n . To the scale of the f i g u r e , the two curves coincide, (a) S i l i c o n c e l l . (b) GaAs c e l l . 30 Figure 2.6(b) TABLE 2.3 True performance parameters, and those predicted by the superposition p r i n c i p l e a) S i l i c o n C e l l Quantity: J [A m~2] sc V [mV] oc J V [mV] mp max FF True value: 355.0 558.5 481±2 161.4 0.814 Predicted by superposition 560.0 482±2 162.1 0.815 b) GaAs C e l l Quantity: True value: Predicted by superposition J [A m~2] 301.2 sc V [mV] 909.1 909.1 oc V [mV] 789±2 790±2 mp P [mW] 224.7 224.8 max FF 0.821 0.821 55 2.2.5 A Case of Superposition Breakdown For c e r t a i n device configurations the drop i n one or both quasi-fermi l e v e l s across the depletion region at the maximum power point may be large even i f and Ep p are e s s e n t i a l l y constant across t h i s region at forward bias i n the dark. This condition can a r i s e i f , for example, a large photogenerated hole current flows through a part of the depletion region i n which the hole concentration i s small and through which r e l a -t i v e l y l i t t l e hole current would flow for operation i n the dark. Such a s i t u a t i o n might be encountered i n a device i n which the dark current i s dominated by recombination i n the depletion region while most photogen-eration occurs i n the emitter. I t was noted i n subsection 2.2.3 that to support a depletion region recombination current i n an N +P device, the electron flow need only be large i n that part of the depletion region nearest the emitter, while the hole flow need be large only near the base (F i g . 2.7(a)). However, i f most photogeneration takes place i n the emitter then when the device i s illuminated a large hole photocurrent must flow a l l the way across the depletion region from the emitter to the base. In order to support t h i s photocurrent, the hole quasi-fermi l e v e l i n the depletion region must be displaced from i t s dark p o s i t i o n so as to greatly increase the hole concentration near the emitter (Fig. 2.7(b)). This has the r e s u l t of extending the zone of maximum recombination from the center of the depletion region towards the emitter. I f the c a r r i e r l i f e t i m e s and m o b i l i t i e s are s u f f i c i e n t l y small, the rate of t o t a l recombination may be raised to such an extent that the true illuminated current-voltage c h a r a c t e r i s t i c s d i f f e r measurably from those predicted by (2.19) or (2.2). This e f f e c t i s i l l u s t r a t e d i n F i g . 2.8, which displays the true J L(V) curve and the curve J - Jr.(V) for an N +P GaAs c e l l with the properties ZONE OF MAXIMUM RECOMBINATION (a) Figure 2.7 Band diagram f o r c e l l i n which most photogeneration occurs i n emitter while most recombination occurs i n depletion region. (a) At forward bias i n the dark, (b) Under i l l u m i n a t i o n at same forward b i a s . TABLE 2.4 Properties of N P GaAs c e l l whose c h a r a c t e r i s t i c s are shown i n F i g . 2.8 doping p r o f i l e base doping emitter doping m e t a l l u r g i c a l junction depth device width c a r r i e r l i f e t i m e s c a r r i e r m o b i l i t i e s SF' SB abrupt 1*10 cm 19 -3 1*10 cm 0.4 um 20.0 um T n = T p = 2*10 - 1 1 s Mn = 0.1 m 2 v - 1 s - 1 ; ^ p = 0.005 m2 V - 1 s " 1 i n f i n i t e 58 Figure 2.8 J - J_(V) and true J T ( V ) c h a r a c t e r i s t i c s for an N P ° sc D L GaAs c e l l with low m o b i l i t i e s and very short l i f e t i m e s . (One-sun i l l u m i n a t i o n ) . l i s t e d i n Table 2.4. (Fig. 2.8 was generated using the numerical analysis described i n the previous subsection and i n Appendix A). The c e l l has very short l i f e t i m e s and low m o b i l i t i e s , properties which might be found i n poor-quality p o l y c r y s t a l l i n e substrates. For t h i s device, e s s e n t i a l l y a l l recombination occurs i n the depletion region, while G(x) was set to correspond to the uniform absorption i n the emitter of a l l photons i n the AMI spectrum with energies greater than the GaAs bandgap. F i g . 2.8 reveals that for t h i s c e l l the true f i l l f a c tor and open-circuit voltage are both s i g n i f i c a n t l y l e s s than those predicted by (2.2). Also, the a p p l i c a t i o n of a small reverse bias increases the photocurrent which can be drawn from the c e l l , so J and J are not equal. I t must be empha-upc sc si z e d that the breakdown of the superposition p r i n c i p l e evident i n F i g . 2. i s not due to the v a r i a t i o n i n the width of the depletion region with b i a s . Since t h i s i s an N +P device, the boundary between the quasi-neutral emitter and the depletion region does not move appreciably as the bias changes. 2.3 Back Surface F i e l d Regions In a t y p i c a l N +P s i l i c o n c e l l the open-circuit voltage under one-sun i l l u m i n a t i o n i s l i m i t e d by recombination i n the quasi-neutral base. In order to a t t a i n higher open-circuit voltages i n c e l l s of t h i s kind, i t i s therefore necessary to reduce the magnitude of the electron i n j e c t i o n -d i f f u s i o n dark current flowing i n t o the base. From (2.27) i t i s apparent that can be reduced by increasing the base doping N^, i f the other material properties remain constant. Experimentally, i t i s found that V Q C increases as the substrate r e s i s t i v i t y decreases from 10 ftcm to 0.1 Qcm. However, for material prepared by the Czochralski technique the electron 60 l i f e t i m e and d i f f u s i o n length begin to drop sharply for r e s i s t i v i t i e s lower than about 0.1 ficm [57]. As a r e s u l t , increases i n substrate doping beyond t h i s point lead to a slow f a l l - o f f i n both s h o r t - c i r c u i t current and V [56,57]. oc An a l t e r n a t i v e method for suppressing electron recombination i n the base i s to use a l i g h t l y doped substrate, but form a high-low junction at the rear of the c e l l [58,59]. The band diagram for such a structure under moderate forward bias i s shown i n F i g . 2.9. High-low junctions of t h i s kind are known as "back surface f i e l d s " , since the b u i l t - i n e l e c t r i c f i e l d at the junction repels minority c a r r i e r s from the region of very high surface recombination v e l o c i t y at the ohmic back contact. In order for a back surface f i e l d structure to be of use i n reducing the electron i n j e c t i o n - d i f f u s i o n dark current, the electron d i f f u s i o n length i n the base must be s u b s t a n t i a l l y greater than the base width. This condition i s s a t i s f i e d i n most present-day commercial c e l l s fabricated on l i g h t l y doped substrates of high q u a l i t y [57]. From the band diagram of F i g . 2.9, i t can be seen that at a given forward bias the electron concentration i s f a r lower i n the heavily doped region of the high-low junction than at the rear edge of the l i g h t l y doped region. The high-low junction thus greatly reduces the electron concentration i n the v i c i n i t y of the back surface, thereby screening the minority c a r r i e r electrons from this region of high recombination veloc-i t y . Provided the electron concentration remains at low i n j e c t i o n l e v e l s throughout the base, the high-low junction can be described by an e f f e c t -ive recombination v e l o c i t y S which i s f a r lower than S„. More pre-e f f B c i s e l y , the magnitude of the electron i n j e c t i o n - d i f f u s i o n current entering the base of the back surface f i e l d c e l l would not be a l t e r e d i f the Figure 2.9 Band diagram for N PP back surface f i e l d c e l l under moderate forward b i a s . 62 high-low junction were replaced by a surface with recombination v e l o c i t y for e l e ctrons. Hauser and Dunbar [60] and Godlewski e_t al. [61] have derived expressions for S for the case i n which the back surface f i e l d ef f region i s uniformly doped, while Fossum [62] has presented an approximate analysis of the more r e a l i s t i c case i n which the back surface f i e l d doping p r o f i l e i s non-uniform. Aside from an enhancement i n open-circuit voltage, the back surface f i e l d structure provides a modest increase i n s h o r t - c i r c u i t current and f i l l f a c t o r [62]. The increase i n s h o r t - c i r c u i t current r e s u l t s from an improved c o l l e c t i o n e f f i c i e n c y for minority c a r r i e r s generated deep i n the base by long-wavelength photons. An electron photogenerated near the middle of the base has a roughly equal p r o b a b i l i t y of d i f f u s i n g to the front junction or to the back contact region. In a c e l l with an ohmic back contact, an electron reaching the back surface w i l l recombine and thus make no contribution to the photocurrent. However, i n a c e l l with a properly designed back surface f i e l d an electron approaching the back surface w i l l be r e f l e c t e d , and may eventually reach the junction and contribute to the photocurrent. The increase i n f i l l f a c t o r r e s u l t s from conductivity modulation due to the increased minority c a r r i e r concentra-t i o n i n the base of the back surface f i e l d c e l l ; t h i s e f f e c t i s important only i n or near the h i g h - l e v e l i n j e c t i o n regime. In industry, a rather unusual technique i s currently used to f a b r i -cate N +PP + back surface f i e l d c e l l s . Following formation of the front j u n c t i o n , an organic paste containing m e t a l l i c aluminum i s applied to the back of the c e l l and then f i r e d at a temperature of approximately 850°C f o r roughly one minute [134]. This procedure forms a layer of aluminum-silicon a l l o y a few microns»deep which behaves i n some ways l i k e 63 a heavily doped p-type region. Crude as t h i s process may appear, the open-circuit voltages of N +PP + c e l l s formed i n th i s way on 10 ficm substrates are t y p i c a l l y 30-50 mV higher than those of otherwise i d e n t i c a l N +P devices lacking a back surface f i e l d region [134], Although N +P c e l l s formed on high q u a l i t y substrates of 1 to 0.1 ftcm r e s i s t i v i t y may give s l i g h t l y higher open-circuit voltages, N +PP + devices incorporating 10 ficm substrates have among the highest energy conversion e f f i c i e n c i e s of a l l s i l i c o n c e l l s . Recently Fossum and Burgess have reported the fa b r i c a t i o n of P NN back surface f i e l d c e l l s giving the highest energy conversion e f f i c i e n c i e s and among the highest open-circuit voltages ever reported f o r s i l i c o n homojunction devices [135]. In these high e f -f i c i e n c y P +NN + c e l l s the back surface f i e l d region i s formed by a con-ventional phosphorus d i f f u s i o n using a phosphine source. The gettering action of t h i s phosphorus d i f f u s i o n i s believed to play a major part i n es t a b l i s h i n g long minority c a r r i e r l i f e t i m e s and d i f f u s i o n lengths i n the base region. 64 CHAPTER 3 POSITIVE BARRIER SCHOTTKY AND MIS JUNCTIONS: THEORY In t h i s chapter a u n i f i e d a n a l y t i c model of conduction i n Schottky b a r r i e r and MIS junctions i s developed. The foundation f o r t h i s model i s established i n Sections 3.1 and 3.2. In Section 3.3 the model i s ap-p l i e d to compute the current flow i n a non-ideal Schottky diode. Section 3.4 then examines the e f f e c t of increasing the thickness of the i n t e r -f a c i a l layer i n the non-ideal Schottky diode to form an MIS junction. Assuming only that the i n t e r f a c i a l i n s u l a t i n g layer i n the MIS junction presents roughly the same b a r r i e r to electrons attempting to tunnel i n t o the metal from e i t h e r the valence or conduction band of the semiconductor, i t i s shown that the presence of t h i s layer can v i r t u a l l y eliminate the flow of electrons between the majority c a r r i e r band and the metal while leaving the net electron flow between the minority c a r r i e r band and the metal e s s e n t i a l l y unaltered. Section 3.4 thus reconciles the p r e d i c t i o n made by Green et_ al . that minority c a r r i e r flows can dominate the dark current i n s u i t a b l y prepared MIS diodes with the thermionic emission theory of conduction i n Schottky diodes. This i s a rather important point i n view of suggestions made recently that current transport proceeds by fundamentally d i f f e r e n t mechanisms i n Schottky and MIS diodes [63]. Sec-t i o n 3.5 extends the model of MIS diode operation developed i n Section 3.4 to the case of the MIS s o l a r c e l l . I t should be acknowledged that much of the material presented i n t h i s chapter has been introduced previously, a l b e i t i n somewhat d i f f e r e n t form, i n publications by Green e_t a l . [16,17, 22-25] and by Card and Rhoderick [26-28]. This chapter considers only p o s i t i v e b a r r i e r Schottky and MIS junc-tions — that i s , junctions i n which the semiconductor surface i s depleted 65 or inverted at equilibrium. Junctions of t h i s class are formed by depos-i t i n g low work function metals on p-type s i l i c o n substrates, or high work function metals on n-type substrates. The properties of negative b a r r i e r MIS contacts are examined i n Chapter 5. 3.1 Junction B a r r i e r Heights The band diagrams conventionally drawn to represent the available electron energy states i n a t y p i c a l MIS or non-ideal Schottky diode are shown i n F i g . 3.1 [64]. In constructing F i g . 3.1(a), i t has been assumed that the junction was formed by depositing a high work function metal on an n-type s i l i c o n substrate. Conversely, i f the junction had been formed by depositing a low work function metal on a p-type substrate, the band diagram of F i g . 3.1(b) would be appropriate. For an n-type substrate, the junction b a r r i e r height q<b i s defined to be the difference i n energy Bn between the conduction band edge at the semicondcutor surface and the fermi l e v e l i n the metal, as shown i n F i g . 3.1(a). For a p-type substrate, the b a r r i e r height qa> i s the energy difference between the valence Bp band edge at the surface and the metal fermi l e v e l , as shown i n F i g . 3.1(b). Both these energy differences are to be measured at equi-l i b r i u m . Schottky o r i g i n a l l y proposed that the e l e c t r o s t a t i c p o t e n t i a l b a r r i e r induced i n the semiconductor at a Schottky or MIS junction r e s u l t s from the difference between the work function dS of the metal and the electron M a f f i n i t y of the semiconductor [65]. This model undoubtably has some relevance for junctions i n which an i n t e r f a c i a l layer i s present, although the use of concepts such as work function and electron a f f i n i t y seems fundamentally suspect when the metal and semiconductor are separated by an oxide f i l m only a few atomic diameters thick. Bardeen l a t e r postulated Figure 3.1 Equilibrium band diagram for MIS or non-ideal Schottky diodes. (a) n-type substrate. (b) p-type substrate. (b) Figure 3.1(b) 68 that charge stored i n electron states l o c a l i z e d at the i n t e r f a c e could play a major part i n forming the b a r r i e r [66]. It i s today generally accepted that the e f f e c t s of both b a r r i e r metal work function and surface state charge can be of importance i n determining the junction b a r r i e r height [67]. For Schottky diodes fabricated on freshly-etched s i l i c o n substrates the b a r r i e r metal work function and surface state charge are found to be of roughly equal importance i n determining the b a r r i e r height. The i n t e r -face state d i s t r i b u t i o n f o r s i l i c o n surfaces prepared i n th i s way i s such that d>_ tends to be s u b s t a n t i a l l y larger than . i s found to YBn ° Bp TBn range from about 0.5 V for low work function metals to 0.9 V for high work function metals, while <j> i s usually less than 0.6 V [64]. I t i s Bp thus not possible to obtain strong inversion at the surface of a p-type substrate, nor accumulation at the surface of an n-type substrate, i n a conventional Schottky b a r r i e r diode. For those MIS diodes formed on s i l i c o n substrates which have been oxidized at temperatures greater than about 400°C p r i o r to m e t a l l i z a t i o n , the a v a i l a b l e experimental data i n d i c a t e that the junction b a r r i e r height i s determined almost e x c l u s i v e l y by the b a r r i e r metal work function [68]. With an appropriate choice of b a r r i e r metal i t i s possible to obtain conditions ranging from accumulation to strong inversion at the semicon-ductor surface i n an MIS diode prepared i n th i s way, i r r e s p e c t i v e of the substrate doping type. It appears that the growth of the thin oxide layer which forms the i n t e r f a c i a l i n s u l a t o r somehow passivates the s i l i c o n surface, greatly reducing the density of i n t e r f a c e states. Throughout t h i s chapter, reference w i l l be made to the e l e c t r o s t a t i c p o t e n t i a l drop IJJ^ appearing across the semiconductor and the drop iJ>T 69 across the i n s u l a t o r i n an MIS or Schottky diode. For a diode formed on n-type material, ty^ and ty^ w i l l be defined to be p o s i t i v e when these pot-e n t i a l drops are i n the d i r e c t i o n shown i n F i g . 3.1(a). S i m i l a r l y , f o r a diode formed on a p-type substrate ty and ty w i l l be defined as p o s i t i v e i n F i g . 3.1(b). The equilibrium values of ty and ty w i l l be denoted as *so 3 1 1 ( 1 *I0' The a p p l i c a t i o n of a bias V to a Schottky or MIS diode displaces the fermi l e v e l at the base contact r e l a t i v e to the metal fermi l e v e l by an amount qV. (By convention, V i s taken to be p o s i t i v e f or forward b i a s ) . This i n turn r e s u l t s i n a change i n the e l e c t r o s t a t i c p o t e n t i a l d i s t r i -bution across the junction. Here the change hty i n the p o t e n t i a l drop across the semiconductor i s defined by A * S ( V ) = * S ( V ) ~ ^SO ( 3 - 1 ) The change Aty i n the p o t e n t i a l drop across the i n s u l a t o r i s defined analogously. The r e l a t i o n s h i p between Aty and V i s frequently written i n the form Ns| = V/n (3.2) where n i s termed the "diode f a c t o r " or " i d e a l i t y f a c t o r " . (3.2) i s useful because n i s often found to be e s s e n t i a l l y constant over the normal operating bias range. For an i d e a l Schottky diode with a vanish-i n g l y t h i n i n t e r f a c i a l l a y e r , ty must go to zero, so that n i s unity. For the more general case i n which the p o t e n t i a l drop across the i n s u l a t o r can not be ignored, Card and Rhoderick have derived a simple expression f o r n applicable when the semiconductor surface i s depleted [26]. Unfor-tunately, no closed solutions f o r ty and ty are av a i l a b l e f o r the case i n which the semiconductor surface i s strongly inverted. A possible technique f o r determining Aty and AiJ) i n t h i s case w i l l be examined i n Section 3.4 [17]. 3.2 Tunnelling i n Metal-Insulator-Semiconductor Structures 3.2.1 The Semiclassical Model of Conduction A d i r e c t quantum-mechanical treatment of conduction i n e i t h e r i Schottky b a r r i e r or MIS diodes i s mathematically i n t r a c t a b l e . For t h i s reason the analysis of current flow i n these devices i s i n v a r i a b l y based on an extension of the s e m i c l a s s i c a l model of electron dynamics used, a for example, i n the development of the Boltzmann transport equation [69]. The main features of t h i s model w i l l now be very b r i e f l y reviewed. The s e m i c l a s s i c a l model of electron dynamics i s based on a knowledge of the band structure E(k) i n the independent electron approximation. Electrons moving under the combined influence of the p e r i o d i c l a t t i c e p o t e n t i a l V(r) and an externally applied electromagnetic f i e l d are then represented as wavepackets constructed from the eigenstates of the independent electron Hamiltonian f o r the unperturbed c r y s t a l . Thus the wave function ty(r,t) of an electron i s written i n the form * ( r , t ) = / d 3k F(k,t) l£(r) (3.3a) where the states ty+ axe Bloch waves. I f the ex t e r n a l l y applied f i e l d s are weak and slowly varying i n time, intraband t r a n s i t i o n s can be ignored, allowing the expansion to be r e s t r i c t e d to the states of a sing l e band. Further, i n t h i s case Wannier's theorem states that the envelope function F obeys an e f f e c t i v e Schrodinger equation [69,70] [E(-iV) + H e x t ] F ( r , t ) = i * [ 9 F ( r , t ) / 3 t ] (3.3b) where H i s the contribution to the Hamiltonian from the external ext f i e l d s and F(k,t) = (1//H) / d r exp(ik-r) F(r,t) , (3.3c) Q being the volume of a p r i m i t i v e c e l l . In a p plications of s e m i c l a s s i c a l dynamics, reference i s frequently made to a d i s t r i b u t i o n function f ( k , r ) which gives the p r o b a b i l i t y that the independent electron state |k^>is occupied by an electron at p o s i t i o n ->- ->• r [69]. This p r a c t i c e of assigning an electron both a wavevector k and a p o s i t i o n r stands i n apparent v i o l a t i o n of the uncertainty p r i n c i p l e . ->-The p r a c t i c e i s acceptable so long as k and r are taken to represent only the mean c r y s t a l momentum and mean p o s i t i o n of a l o c a l i z e d e lectron wavepacket. 3.2.2 Models f o r the Tunnelling Process In the past, two conceptually d i s t i n c t methods have been developed to determine the rate at which electrons tunnel through a t h i n i n s u l a t o r separating two conductors. (The conductors i n question could be metals, semimetals or semiconductors; to si m p l i f y terminology, i t w i l l be assumed here that the junction has been formed between a metal and a semiconduc-tor) . In the o r i g i n a l method developed by Bardeen [71] and l a t e r extended by Harrison [72], time-dependent perturbation theory i s applied to e s t i -72 mate the rate at which t r a n s i t i o n s between electron states i n the metal and those i n the semiconductor occur. This approach was employed by Card and Rhoderick [26], and l a t e r used by Green et al.[16,17] as the basis for t h e i r numerical analysis of the MIS tunnel diode. Here a more i n t u -i t i v e l y appealing method developed by Duke [73,74] i s employed. To the l e v e l of approximation generally used, the two methods give mathematically i d e n t i c a l expressions for the tunnel currents. In Duke's approach, the t u n n e l l i n g of electrons across a metal-insulator-semiconductor junction i s treated i n the same manner as the tu n n e l l i n g of free electrons through a one-dimensional square p o t e n t i a l b a r r i e r [75]. An electron approaching the i n t e r f a c e from the semiconduc-tor i n F i g . 3.1 i s represented by a t r a v e l l i n g Bloch wave with wavevector It. A s o l u t i o n to the Schrodinger equation i s then sought which consists of t h i s incident Bloch wave, a r e f l e c t e d wave returning to the r i g h t i n the semiconductor, and a transmitted wave propagating to the l e f t i n the metal. I f the energy E of the incident electron l i e s within the energy gap of the i n s u l a t o r , then i n t h i s region the electron wavefunction must have an exponential dependence on x. The r a t i o of the current density c a r r i e d by the transmitted wave to that c a r r i e d by the incident wave gives the p r o b a b i l i t y that the electron w i l l tunnel across the i n t e r f a c e . The form and amplitude of the transmitted and r e f l e c t e d electron waves are determined by the requirement that ty and be everywhere continuous. In general, these w i l l not be simple Bloch waves. This can be seen by considering the s o l u t i o n to the Schrodinger equation at an abrupt i n t e r f a c e between two regions with d i f f e r e n t l a t t i c e p o t e n t i a l s . In each region, the solutions to the Schrodinger equation w i l l be Bloch waves of the form 73 <j£(r") = u*(r) exp(i^-r) (3.4) where u£(r) has the p e r i o d i c i t y of the l a t t i c e . Since the functions u+(r) w i l l have d i f f e r e n t forms i n the two regions, i t w i l l not i n general be possible to connect a Bloch wave i n one region with a s i n g l e Bloch wave i n the adjoining region i n such a way that ty i s continuous across the boundary. Instead, a Bloch wave incident on the i n t e r f a c e w i l l give r i s e to transmitted and r e f l e c t e d waves with a spectrum of k values. The transmitted and r e f l e c t e d waves must, however, have the same energy as the incident wave. In order to keep the d e s c r i p t i o n of t u n n e l l i n g i n the metal-insula-tor-semiconductor junction at a mathematically manageable l e v e l , i t i s i n v a r i a b l y assumed that the l a t t i c e - p e r i o d i c component "£(?) of the Bloch wavefunction can be approximated by a constant i n each region of the junction [72,76]. Thus the true Bloch wavefunction i s replaced by a plane wave with the same wavevector. From this assumption i t follows that the component k^_ of the electron wavevector l y i n g i n the plane of the i n t e r -face must be conserved during t u n n e l l i n g . In other words, the i n c i d e n t , r e f l e c t e d and transmitted waves must a l l have the same value of k^. Depending on the band structure of the metal, i n s u l a t o r and semi-conductor, there may be several states a v a i l a b l e i n each region of the junction with the same values of E and k^. As F i g . 3.2 i n d i c a t e s , i n each band structure f o r every state with energy E, transverse wavevector lc and v e l o c i t y v >0, there must be at l e a s t one other state with the t x same E and lc , but with v <0. For s i m p l i c i t y , i t w i l l be assumed at f i r s t t x that only one such p a i r of states at the same energy and transverse wave-vector e x i s t s i n each region of the j u n c t i o n . The corrections to the 74 expressions for the tunnel currents required when th i s condition does not hold w i l l be considered l a t e r . In the independent electron approximation, the p r o b a b i l i t y of an electron making a tun n e l l i n g t r a n s i t i o n from a state |k,^ > i n the semi-conductor to a state |k^>in the metal must equal the p r o b a b i l i t y of the opposite t r a n s i t i o n , from ( k ^ ^ t o |kg^> [73]. This p r o b a b i l i t y w i l l be denoted as 6(E,k t). 3.2.3 Expressions for the Tunnel Currents I f an electron i n the semiconductor i s to tunnel i n t o the metal, the P a u l i exclusion p r i n c i p l e requires that the state which the electron i s to enter i n the metal be unoccupied. Taking this r e s t r i c t i o n i n t o account, and summing the contributions from a l l occupied states i n the semiconductor conduction band, i t i s found that the electron current component flowing from the conduction band i n t o the metal i s given by [73] JQ*M = q ^ 2 v x ( k ) f C ( ^ ' X S ) U ~ f M ( E ' V ] 6 ( E ' ^ t ) ( 3 ' 5 ) v <0 ,„ .3 x (2TT) where f (k,x ) i s the d i s t r i b u t i o n function f o r conduction band electrons evaluated at the semiconductor surface, and f M i s the d i s t r i b u t i o n func-t i o n f o r electrons i n the metal. For most purposes i t can be assumed that that f„ has i t s equilibrium form, and i s thus i d e n t i c a l to the Fermi-M Dirac d i s t r i b u t i o n function. The i n t e g r a l i s to be c a r r i e d out over a l l states | k ^ i n the conduction band with v e l o c i t i e s directed towards the metal, and f o r which states i n the metal with the same values of E and lc t e x i s t . 75 Noting that v (k) = 1 3E , (3.6) X <h dk x the i n t e g r a l over k appearing i n (3.5) can be transformed to an i n t e g r a l -*• -> over k t and E. Since s p e c i f y i n g E and k^ does not uniquely determine k, some care i s necessary i n making t h i s transformation. As noted above, for every state i n the conduction band with energy E, transverse wavevector kfc and v e l o c i t y component v x < 0 » there must be at l e a s t one other state with the same E and k , but with v >0. However, only the state with v <0 t x _ x i s to be included i n computing J™,- Therefore VM = - 9 — / d E / d k t f c ( E ' V x s } [ 1 ~ f M ( E ' N - ) ] e(E>K} (3-7) 2 A " N D' SCM band where the region S of the k plane i s that i n which states of energy E e x i s t i n both the metal and the semiconductor conduction band. Harrison [72] describes t h i s region as the overlap of the "shadows" of the constant energy surfaces f o r the metal and the conduction band, where the shadow of a constant energy surface i s defined to be i t s pro j e c t i o n on a plane p a r a l l e l to the i n t e r f a c e . I t i s usually assumed that S i s equivalent L*JM to the shadows of the semiconductor constant energy surfaces alone. By making t h i s assumption, any influence the band structure of the metal might have on the current flow i s ignored. I f , as suggested i n F i g . 3.2, there i s more than one p a i r of states i n the conduction band with the same values of E and k , then the contributions to J„ from the shadow of each branch of the constant energy surface must be taken i n t o account. The shadows i n the <100> d i r e c t i o n f o r the various branches of a constant Figure 3.2 A s l i c e through the constant energy surfaces for the s i l i c o n conduction band, showing that for each state with a given E, k and v >0, there e x i s t s at l e a s t t x one other state of the same E and k , but with v <0. t x 77 energy surface close to the conduction band minimum for s i l i c o n are i l l u s t r a t e d i n F i g . 3.3. By analogy with (3.7), the electron current component flowing from the metal i n t o the conduction band i s given by V c = -^SL. / d E / d \ V E >£ t ) L 1 " f c ( E » k t , x s ) ] 6(E,k t) . (3.8) _ 2 cond. S_. 2TT h , , CM band Adding (3.8) to (3.7), i t i s found that the net current J C M flowing between the conduction band and the metal i s given by JCM - JC+M + JM+C <3'9> = q / dE / d 2 k t [ f c ( E , i t t , x s ) - f M(E,£ t)] e(E,^ t) . 2 r r 2h ™ndA- SCM band An analysis s i m i l a r to that used to derive (3.7), (3.8) and (3.9) can be applied to obtain expressions for the current flows between the valence band and the metal. As i s usually the case when dealing with a valence band, these current flows are most conveniently described i n terms of the motion of holes [69]. From t h i s viewpoint, the transfe r of " an el e c t r o n from the metal to an unoccupied state i n the valence band i s equivalent to the emission of a hole from the semiconductor i n t o the metal, while the transfe r of a valence band electron to the metal i s equivalent to the i n j e c t i o n of a hole i n t o the semiconductor. I t i s usef u l to define a hole d i s t r i b u t i o n function f , which i s related to the electron d i s t r i b u t i o n function f by 78 Figure 3.3 Shadow of the conduction band constant energy surfaces f o r a s i l i c o n sample of <100> o r i e n t a t i o n . 79 f = 1 - f . (3.10) f i s thus the p r o b a b i l i t y that a state i n k-space i s "occupied" by a hole. In terms of f ' , the current flows between the valence band and the metal are given by VM = -=3_ / d E / d \ f v ( E , k t , x s ) [1 - f ^ ( E , k t ) ] 6(E,k t) , (3.11) 2 A 7al- SVM band J M = A _ / ^ / d 2k t f^E,tt) [1 " f ; ( E , k t , x s ) ] e<E,*t) (3.12) 2 A y a l : SVM band and JVM = VM + VV (3"13) = -q / dE / d 2k [ f ^ ( E , k t , x s ) - f ^ ( E , k t ) ] 6(E,k t) . _ 2, v a l . S, m 2TT h , , VM band 3.2.4 An Estimate f o r the Tunnelling P r o b a b i l i t y Factor The simplest possible estimate f o r 0(E,k t) i s obtained through use of the WKB approximation [77,78]. In i t s conventional form, the WKB method i s applied to determine approximate solutions to the f u l l Schrod-inger equation for the case i n which the p o t e n t i a l varies slowly over distances comparable to the ele c t r o n wavelength. Within the i n s u l a t o r i n the MIS ju n c t i o n , the p o t e n t i a l experienced by an electron i s a sum of the rapidly-varying l a t t i c e p o t e n t i a l and a more slowly varying term associated with the e l e c t r o s t a t i c p o t e n t i a l drop htyy across the i n s u l a t o r , 80 so the WKB method can not be applied d i r e c t l y . However, i t might be expected that the WKB approximation could s t i l l be applied to the e f f e c -t i v e Schrodinger equation (3.3) governing the envelope of the electron wavefunction. Proceeding i n e s s e n t i a l l y t h i s fashion, Harrison [72] has shown that the WKB expression for the p r o b a b i l i t y of an electron tunnel-l i n g through the i n s u l a t o r i s x^ 6(E,k t) = exp[-2 / M dx | k I x ( E , k t , x ) | ] (3.14) XS where k^ x i s the imaginary x-component of the complex wavevector k^ within the i n s u l a t o r . (3.14) i s i d e n t i c a l to the expression the conven-t i o n a l WKB method gives f o r the p r o b a b i l i t y of a free electron penetrating a p o t e n t i a l b a r r i e r [78]. Just as i n the free electron t u n n e l l i n g problem, (3.14) i s v a l i d only i f 6 i s small, so that only the exponentially decreasing s o l u t i o n to the Schrodinger equation need be considered within the i n s u l a t o r . Further, within the i n s u l a t o r the external p o t e n t i a l must vary slowly over distances comparable to I ^. In p r i n c i p l e , for (3.14) to apply c e r t a i n a d d i t i o n a l conditions on the behaviour of the p o t e n t i a l near x<, and x M must also be s a t i s f i e d [78]. I f these conditions are not met, the exponential appearing i n (3.14) w i l l be m u l t i p l i e d by a prefac-tor of order unity. However, since the t u n n e l l i n g p r o b a b i l i t y i s dominated by the exponential, the exact value of t h i s prefactor i s of l i t t l e impor-tance . In order to evaluate the t u n n e l l i n g p r o b a b i l i t y factor from (3.14), i t i s necessary to know the r e l a t i o n s h i p between energy and complex wavevector k^ — that i s , the band structure — for the evanescent states i n the forbidden gap of the i n s u l a t o r . Previous t h e o r e t i c a l analyses of 81 t u n n e l l i n g i n MIS structures [16,17,79] have generally assumed that E depends on k^ . through a simple r e l a t i o n s h i p f i r s t suggested by Franz [80] The Franz dispersion r e l a t i o n s h i p i s 1_ = £ + ft2 (3.15) k I 2 m C I [ E " E C I ( X ) ] 2 n V i t E v i ( x ) " E ] * * where m^ and m^ are s c a l a r e f f e c t i v e masses associated with the conduc-tio n and valence bands i n the i n s u l a t o r , and E (x) and E (x) are the energies of the conduction and valence band edges i n the i n s u l a t o r . I t 2 should be noted that (3.15) predicts a negative value of k^ for a l l states with energies within the forbidden gap. (3.15) reduces to a para-b o l i c r e l a t i o n s h i p between E and k^ near e i t h e r band edge i n the i n s u l a -2 t o r . ,0nce k^ i s known, k ^ must be given by = A 2 -2 k T • k: - k . (3.16) Ix I t The p r o b a b i l i t y of an electron tunnelling through a high p o t e n t i a l b a r r i e r i s not c r i t i c a l l y dependent on the shape of the b a r r i e r . Thus i n evaluating the p r o b a b i l i t y of an electron with energy near the middle of the i n s u l a t o r bandgap tunnelling between the metal and the semiconductor, i t i s reasonable to ignore the dependence of and on p o s i t i o n . To thi s l e v e l of approximation, i t i s also reasonable to ignore any depend-ence 6 might have on the bias applied to the junct i o n . In the semiconductor, only those states i n the conduction band with wavevectors close to the bottom of the " v a l l e y s " w i l l normally be occu-pied. I t i s thus reasonable to apply an approximate t u n n e l l i n g p r o b a b i l i t y 82 factor to a l l t r a n s i t i o n s i n v o l v i n g a given v a l l e y , where 6 ^ i s to be evaluated at the v a l l e y minimum. In general, i f there are several v a l l e y s , the one with the smallest value of w i l l make the largest contribution to the tunnel current. For example, for s i l i c o n of <100> or i e n t a t i o n , tunnel currents i n v o l v i n g the conduction band w i l l be dom-inated by t r a n s i t i o n s to and from states i n the two v a l l e y s centered about k t=0. In t h i s case a p a r t i c u l a r l y simple estimate f o r can be found. Assuming that the e f f e c t i v e masses m^ and m^ appearing i n (3.15) are equal to the free electron mass m, (3.14), (3.15) and (3.16) combine to give [79] 3 C M = exp [-2 ( 2 m / * 2 ) 1 / 2 B ^ 2 (1 - B e / E g I ) 1 / 2 d] (3.17) where B £ i s the difference between the energy of the conduction band edge at the semiconductor surface and the mean energy of the i n s u l a t o r conduction band edge, d i s the thickness of the i n s u l a t o r , and i s the i n s u l a t o r bandgap energy. By analogy to (3.17), an approximate tu n n e l l i n g p r o b a b i l i t y 6 ^ applicable to a l l t r a n s i t i o n s i n v o l v i n g states i n the semiconductor valence band for which k^O can be defined. 9 ^ i s given by [79] Q m * exp[-2(2m/f» 2) 1 / 2 B* / 2 (1 - B ^ l E ^ ) 1 ' 2 d] (3.18) where B^ i s the difference between the mean energy of the i n s u l a t o r valence band edge and Ev(x<,). Although the estimates f o r the tu n n e l l i n g p r o b a b i l i t y factor given above may appear very crude, there i s l i t t l e point i n attempting to generate more accurate expressions f o r 6(E,k t). In the vast majority of MIS and non-ideal Schottky junctions, the i n t e r f a c i a l i n s u l a t o r i s an amorphous f i l m of s i l i c o n oxide grown by the low-temperature oxidation of the s i l i c o n substrate. At present, l i t t l e r e l i a b l e information i s a v a i l a b l e concerning the structure or even the composition of layers grown i n t h i s way [81]. Such layers are thought to have the stoichiometry SiO , where l<x^2. In general, the band structure of an amorphous insu-l a t o r i s characterized by a mobility gap analogous i n some respects to the forbidden energy gap i n c r y s t a l l i n e i n s u l a t o r s . However, l o c a l i z e d electron states can e x i s t at energies within the mobility gap [82]. It i s thus very l i k e l y that the bulk of the current flowing across the i n t e r f a c e i n an M-SiO^-Si diode r e s u l t s from tun n e l l i n g t r a n s i t i o n s i n v o l v i n g states w i t h i n the mobility gap of the amorphous SiO^ layer. Assigning such SiO layers the band structure of c r y s t a l l i n e SiO , as X z. suggested i n F i g . 3.1, i s therefore hardly a r e a l i s t i c approximation. Further, there i s no reason to believe that i n a r e a l diode the metal-i n s u l a t o r and semiconductor-insulator i n t e r f a c e s w i l l be smooth over distances comparable to an electron wavelength. I f the i n t e r f a c e s are rough on the atomic s c a l e , the transverse wavevector k^ w i l l not be. conserved i n t u n n e l l i n g [72]. 3.3 The Schottky B a r r i e r Diode Purely for n o t a t i o n a l convenience, i t w i l l be assumed that the junctions examined i n t h i s section and i n the remainder of the chapter have been formed on n-type substrates, unless otherwise s p e c i f i e d . The analysis developed applies equally w e l l to p-type material i f the roles of electrons and holes are interchanged. 3.3.1 The Majority C a r r i e r Thermionic Emission Current Under moderate forward b i a s , the current i n a t y p i c a l Schottky diode formed on n-type s i l i c o n i s dominated by the emission of electrons from the conduction band in t o the metal. In most Schottky diodes t h i s majority c a r r i e r flow i s accurately described by the thermionic emission theory introduced by Bethe [10] i n 1942. The thermionic emission theory i s based on the assumption that the transmission c o e f f i c i e n t 6 appearing i n equa-tions (3.7-3.9) i s equal to unity; thus the p r o b a b i l i t y that an electron incident on the i n t e r f a c e may s u f f e r quantum-mechanical r e f l e c t i o n i s ignored. This assumption must always overestimate the electron flow, since even i n an intimate metal-semiconductor contact part of the wave-packet representing the incident s e m i c l a s s i c a l electron w i l l be r e f l e c t e d on c o l l i s i o n with the i n t e r f a c e [83]. However, i f the i n t e r f a c i a l layer i s so t h i n that electrons can f r e e l y tunnel through i t , the assumption of perfect transmission i s not unreasonable. In order to evaluate the net current flow J _ . between the conduction CM band and the metal from (3.9), i t i s necessary to know the d i s t r i b u t i o n function f o r electrons at the semiconductor surface. As a f i r s t approx-imation, i t i s reasonable to assume that throughout the semiconductor the d i s t r i b u t i o n function f o r conduction band electrons can be obtained by replacing the fermi energy E appearing i n the Fermi-Dirac d i s t r i b u -t i o n function with a position-dependent quasi-fermi energy E ^ [64]. The accuracy of t h i s approximation i s considered below. Measuring the e l e c -tron energy E r e l a t i v e to the energy of the conduction band edge at the surface, (3.9) then becomes E max dE [ f c ( E ) - f M ( E ) ] am(E) (3.19) where f c ( E ) = exp(-[E + E c ( x s ) - E F n ( x s ) ] / k T ) , (3.20) f M ( E ) = exp(-[E + E c ( x g ) - E ^ / k T ) (3.21) and the area c r ^ of the constant energy surface i s given by (3.22) In w r i t i n g (3.20) and (3.21), the Boltzmann l i m i t to the Fermi-Dirac function has been taken. This i s v a l i d only i f E (x ) l i e s above E ^ C S FM and E F n ( x s ) by at l e a s t kT. The upper l i m i t on the energy i n t e g r a l can be raised to i n f i n i t y without introducing appreciable e r r o r . If there i s a sing l e v a l l e y i n the conduction band, and i f the e f f e c t i v e mass tensor associated with t h i s v a l l e y has one p r i n c i p a l axis perpendicular to the i n t e r f a c e , then f o r those states which have a s i g -n i f i c a n t p r o b a b i l i t y of being occupied E(k) = 2 x * 2 2 k + k * * m m m (3.23) * * * Here m^ , m^  and m^  are the p r i n c i p a l values of the e f f e c t i v e mass [84]. In this case i t can be seen immediately that the shadow of the constant energy surface i s an e l l i p s e , and that the shadow area i s given by It then follows that J™. ~ 4 T r q / m m CM _y z r q/ v m^  / dE E [ f c ( E ) - f M ( E ) ] (3.25) 4Trqm* k 2 T 2 exp(-[E c(x s) - E F n ( x s ) ] / k T ) [1 - e x p ( - [ E F n ( x s ) - E M ] / k T ) ] where m * * m y z (3.26) For a conduction band v a l l e y i n which the p r i n c i p a l axes of the e f f e c t i v e mass tensor have an a r b i t r a r y o r i e n t a t i o n r e l a t i v e to the i n t e r f a c e , i t i s shown i n Appendix B that (3.25) s t i l l holds, although the expression * for mg i s more complicated than (3.26) [85]. I f there i s more than one v a l l e y , the contributions to J from each v a l l e y must be summed. Following Shockley's analysis Of current flow i n the pn diode [44], i t i s reasonable to base the computation of the electron thermionic emission current on the assumption that E F n i s constant across the deple-t i o n region and the quasi-neutral base [86]. This s i t u a t i o n i s i l l u s t r a t e d i n F i g . 3.4, which gives the band diagram f o r the device of F i g . 3.1(a) when a forward bias V i s applied. Once the electron current has been calculated, the methods outlined i n Section 2.2 can be used to e s t a b l i s h the se l f - c o n s i s t e n c y of t h i s assumption on the constancy of Ep n-If EVri i s constant across the semiconductor, then 87 Figure 3.4 Band diagram f o r the device of F i g . 3.1(a) under moderate forward b i a s . EFn ( xS> " EFM = ^ V (3.27) and EC ( XS> " ERt<*S> = q*S0 " q A^S + = q*Bn " & ( 3 ' 2 8 ) n where q<j>g i s the energy difference between the fermi l e v e l and the conduction band edge at the back contact. Defining the e f f e c t i v e * Richardson constant for electrons A by e & ^ 2 A g = 4iTqme k (3.29) (3.25) becomes * 2 J T h ( V ) = JCM = A e T e xP(-9> B n/kT) exp(qV/nkT) [1 - exp(-qV/kT)]. (3.30) (3.30) i s the conventional expression f o r the electron thermionic emission current, [87]. In d e r i v i n g (3.30), i t was assumed that the electron d i s t r i b u t i o n -y i n k-space retains i t s e q u i l i b r i u m form throughout the semiconductor, even i n the immediate v i c i n i t y of the i n t e r f a c e . However, since a very large electron current flows across the i n t e r f a c e i n t o the metal when a forward bias V i s applied to the diode, yet v i r t u a l l y no electrons enter the conduction band from the metal, i t seems cl e a r that the region near the surface must be depleted of electrons with v >0. Bethe [10] argued , that t h i s e f f e c t could be ignored provided the b a r r i e r p o t e n t i a l drops by at l e a s t kT/q between the semiconductor surface and the plane x=x., A which by d e f i n i t i o n i s separated from the surface by the mean free path 89 A for electron-phonon c o l l i s i o n s . The thermionic emission current i s then envisaged as being composed of electrons scattered towards the i n t e r f a c e from x=x . Since r e l a t i v e l y few electrons scattered from x=x w i l l have A A s u f f i c i e n t energy to overcome the remaining portion of the b a r r i e r and reach the i n t e r f a c e , the electron d i s t r i b u t i o n at x=x^ should not be s i g n i f i c a n t l y influenced by the presence of the metal. For the case i n which the e f f e c t i v e mass tensor has one p r i n c i p a l axis normal to the i n t e r f a c e , at l e a s t , i t can be shown that the expression for the therm-i o n i c emission current (3.30) i s unaffected by t h i s modification to the theory. As a general comment, the problem of s e l e c t i n g the correct form for f^(k,Xg) i n the Schottky diode i s s t i l l the subject of considerable controversy [88]. The quasi-equilibrium d i s t r i b u t i o n function (3.20) i s the most widely used approximation f o r f ^ [86,89,90]. 3.3.2 Minority C a r r i e r Flow By making approximations analogous to those used i n the derivation of (3.25), an expression for the net current flow between the valence band and the metal f o r the Schottky diode can be obtained. This expression i s JVM = "\ T' ^ ( - t V V " V X S ) ] / k T ) (3.31) [1 - e x p ( - [ E F M - E (x s)]/kT)] where * * 2 A, = 4iTqm, k (3.32) It i s often convenient to write (3.31) i n the form 90 JVM = A T 2 P ( X S ) C L " E X P ( " t E F M " E F p ( x s ) ] / k T ) ] , (3.33) Nv which stresses the dependence of on the hole concentration p (Xg) at the semiconductor surface. For a t y p i c a l Schottky diode formed on n-type s i l i c o n the semicon-ductor surface i s inverted at equilibrium. There are therefore many unoccupied states at the semiconductor surface with energies near the valence band edge. Further, the band diagram of F i g . 3.1(a) indicates that there are also many occupied states i n the metal at these energies. Thus from (3.11) and (3.12) the current components and J j ^ y flowing between the valence band and the metal must be extremely large, even though at equilibrium these two current components cancel exactly. When a small forward bias i s applied to a Schottky or MIS diode, the hole quasi-fermi l e v e l at the semiconductor surface must be displaced from the fermi l e v e l at the base contact by an amount qAd>, as shown i n F i g . 3.4. (Ad) i s defined to be p o s i t i v e when the minority c a r r i e r quasi-fermi l e v e l s h i f t s so as to increase the minority c a r r i e r concentration at the surface). This displacement of E (x ) r e s u l t s i n a net flow of rp o holes from the metal i n t o the semiconductor. The net hole current J t T W V M crossing the i n t e r f a c e can be divided i n t o a component ^ rg» which accounts for recombination i n the depletion region, and a component J ^ , which accounts for recombination i n the quasi-neutral base. must be equal to the hole current which would flow across the emitter/depletion region boundary of a P +N homojunction diode formed on an i d e n t i c a l sub-s t r a t e and subject to a forward bias A<j>. (Minor differences i n the width of the depletion region between the Schottky diode and the P +N diode may 91 r e s u l t i n very small differences between the two hole currents; however, these differences can be ignored for a l l p r a c t i c a l purposes). Thus JVM = J r g ( A * } + J d ( A * } ( 3 ' 3 4 ) where a n a l y t i c approximations for J and are given i n (2.26) and (2.28). From F i g . 3.4, i t i s apparent that W - EFM " q ( V " L ^ • ( 3 ' 3 5 ) In terms of Aty and V, (3.33) becomes J r a = -A£ T 2 p ( x s ) (1 - exp[q(V - Aty)/kT]) . (3.36) N V F i g . 3.4 suggests that the hole quasi-fermi l e v e l at the semiconduc-tor surface should coincide with the fermi l e v e l i n the metal [12,91], so that Aty i s equal to V. This "pinning" of E„ (x„) to can be ex-r p b r r l plained by the following i n d i r e c t argument. If E (x ) did not coincide r p b with E , but instead lay close r to the fermi l e v e l at the back contact, r r l the balance between J T I and J., „ would be destroyed. As a r e s u l t , there V-*-M M->-V would be an enormous net flow of holes from the metal i n t o the semicon-ductor. I f desired, the magnitude of t h i s hole flow could be estimated from (3.36). The flow could be sustained only i f the rate at which holes were i n j e c t e d i n t o the semiconductor were matched by the rate at which these excess holes recombined with electrons i n the depletion region and base. However, since Aty i s always less than or equal to V, an upper 92 bound on the hole recombination current i s set by the sum of J (V) and J,(V). For a substrate with reasonable doping, m o b i l i t i e s , and c a r r i e r d l i f e t i m e s , at moderate forward bias J (V) and J,(V) w i l l both be ex-rg d tremely small compared to e i t h e r or J y ^ - Thus the only possible steady-state s o l u t i o n i s to have a condition of quasi-equilibrium between the metal and the holes at the semiconductor surface, which i s equivalent to pinning E F p(Xg) to E ^ . In f a c t , E^CXg) w i l l be displaced j u s t enough from E„., that the difference between J . T ^ w and J., i s equal to J + J , . FM V+M M+V n rg d 3.3.3 The Minority Carrier Injection Ratio The minority c a r r i e r i n j e c t i o n r a t i o y of a Schottky b a r r i e r diode i s defined to be the r a t i o of the minority c a r r i e r current crossing the boundary between the depletion region and the base to the t o t a l current flow. Thus Y = J d > J d . (3.37) J + T + T l a r g e V T + T JTh + J r g + J d JTh + J d Y can r e a d i l y be computed from the expressions f o r J m , J , and J given Th d rg above. Calculations of p r e c i s e l y t h i s type lead Scharfetter [12] to con-clude that y i s extremely small f o r t y p i c a l s i l i c o n Schottky diodes operated at moderate forward b i a s . In p r i n c i p l e , however, i f the junction b a r r i e r height were made large enough, J__ could be made smaller than J , . i n d This condition would be achieved most e a s i l y with a l i g h t l y doped sub-s t r a t e , since decreases as the doping l e v e l increases. For example, for a 10 ftcm p-type substrate with a 10 usee electron l i f e t i m e , a b a r r i e r height <f>gp of approximately 0.99 V would be required to make equal 93 to J ^ . Although such a high b a r r i e r height i s not impossible i n theory, i t i s f a r higher than the values of a) recorded f o r Schottky diodes a fabricated using conventional techniques. 3.3.4 Current Flow Through Surface States Up to t h i s point, only the d i r e c t flow of c a r r i e r s between the conduction and valence bands and the metal has been considered. I f the surface state density i s large, and i f these states communicate r e a d i l y with one or both bands and with the metal, there may be a t h i r d important current component r e s u l t i n g from electrons tun n e l l i n g between these states and the metal [16]. The magnitude of t h i s current component i s l i k e l y to be very s e n s i t i v e to the density and d i s t r i b u t i o n i n energy of the surface s t a t e s , which i n turn depends c r i t i c a l l y on device f a b r i -cation procedures. For t h i s reason, current flow through surface states w i l l not be considered here. 3.4 T r a n s i t i o n to the MIS Diode If the i n t e r f a c i a l i n s u l a t i n g layer i n the junction of F i g . 3.1(a) i s made progressively thick e r , a point w i l l eventually be reached at which t h i s l a y e r can no longer be considered transparent to electrons. For t h e o r e t i c a l purposes, i t i s u s e f u l to define t h i s point as marking the t r a n s i t i o n from non-ideal Schottky diode to MIS diode. In an MIS diode, i t i s no longer reasonable to assume that the transmission coef-f i c i e n t 6(E,^ t) i s unity. In t h i s section and i n Section 3.5, for a l l t u n n e l l i n g t r a n s i t i o n s i n v o l v i n g the conduction band 9(E,k t) w i l l be approximated by a constant, 6 p M . S i m i l a r l y , a l l t u n n e l l i n g t r a n s i t i o n s between the valence band and the metal w i l l be described by a constant t u n n e l l i n g p r o b a b i l i t y factor 0 ^ . Thus the expressions for the net 94 current flows between the semiconductor bands and the metal become JCM = 6CM A e j 2 e x P ( _ < l + B £ / k T ) exp(qV/nkT) [1 - exp(-qV/kT)] (3.38) and JVM = "0VM \ l 2 P ( X S } ( 1 " e x P t 9 ( V - A<},)/kT]) . (3.39) N V Estimates for 0_, and 6 t I W i n terms of the thickness and band structure CM VM of the i n s u l a t o r are given i n equations (3.17) and (3.18). Crude as t h i s model for the t u n n e l l i n g process may appear, i t i s capable of explaining both the existence of minority c a r r i e r MIS diodes and, on a q u a l i t a t i v e l e v e l , the main features appearing i n the conduction c h a r a c t e r i s t i c s of a l l MIS diodes. In the previous section i t was pointed out that i n a Schottky diode the e l e c t r o n d i s t r i b u t i o n function at the semiconductor surface i s l i k e l y to deviate s i g n i f i c a n t l y from i t s equilibrium form. In an MIS diode, however, there i s no doubt that f ( k,x ) can be accurately approximated by the Fermi-Dirac function at moderate forward b i a s , since most e l e c -trons incident on the semiconductor surface are r e f l e c t e d . 3.4.1 The Minority C a r r i e r MIS Diode From (3.38) i t follows that the electron thermionic emission current flowing i n the MIS diode i s reduced by a factor 6 r e l a t i v e to i t s value i n a Schottky diode with the same b a r r i e r height. However, so long as J„- and J w „ both remain large compared to the sum of the depletion V-*M M+V region recombination current J and the hole i n j e c t i o n - d i f f u s i o n current rg 95 J ^ , then the condition of quasi-equilibrium between the metal and the holes at the semiconductor surface which held f or the Schottky diode must s t i l l apply i n the MIS junction. Thus at moderate forward bias E„ (x c) Fp S w i l l be pinned to i n the MIS diode [16]. Provided 6 i s s u f f i c i e n t l y small, i n t h i s bias regime the sum of J and J,, w i l l be much larger than b rg d 6 J ^ . I f t h i s i s the case, then a minority c a r r i e r MIS diode has been formed. If the forward bias applied to the MIS diode i s gradually increased, a bias point w i l l eventually be reached at which the sum of J and J , rg d becomes comparable to Above t h i s bias point, E F p ( X g ) w i l l no longer be pinned to E , but w i l l instead l i e closer to the fermi l e v e l at the base contact; thus i n F i g . 3.4 Aty becomes s i g n i f i c a n t l y smaller than V. Green et a l . have termed t h i s bias range the "tunnel l i m i t e d " regime [16], since over t h i s range the net hole current entering the semiconductor i s l i m i t e d by the rate at which holes are supplied by t u n n e l l i n g across the i n s u l a t o r , rather than by recombination processes within the semiconduc-tor . Correspondingly, the bias range i n which E (x„) i s pinned to E rp b r n i s termed the "semiconductor l i m i t e d " regime [16]. When operated i n the semiconductor l i m i t e d regime, the minority c a r r i e r MIS diode i s e l e c t r i c a l l y equivalent to a one-sided pn junction formed on an i d e n t i c a l substrate. At the onset of the tunnel l i m i t e d regime, the minMIS diode can be modelled approximately as a pn junction connected i n s e r i e s with a voltage-dependent resistance. Further into the tunnel l i m i t e d regime, both minority and majority c a r r i e r flows may be important. The maximum forward bias which can be applied to the MIS diode before the tunnel l i m i t e d regime i s entered depends on a n d hence on the i n s u l a t o r thickness [16]. The dark current-voltage c h a r a c t e r i s t i c s f o r a hypothetical set of MIS diodes fabricated on i d e n t i c a l substrates but with d i f f e r e n t i n s u l a -tor thicknesses are sketched i n F i g . 3.5 [16]. For a l l the diodes, at very small forward bias the c h a r a c t e r i s t i c s are dominated by recombina-t i o n i n the depletion region. As the forward bias i s increased, the c h a r a c t e r i s t i c of the device with the thinnest i n s u l a t o r r i s e s above those of the other diodes. In t h i s device, a s u b s t a n t i a l f r a c t i o n of the dark current flowing at moderate forward bias r e s u l t s from majority car-r i e r thermionic emission. For the other devices, the current flow at moderate forward bias i s dominated by minority c a r r i e r i n j e c t i o n - d i f f u s i o n i n the quasi-neutral base. These devices are thus minority c a r r i e r MIS diodes. If the forward bias applied to the minMIS diodes i s increased s t i l l f urther, the minority c a r r i e r flows eventually become tunnel l i m -i t e d . The t r a n s i t i o n from the semiconductor l i m i t e d regime to the tunnel l i m i t e d regime occurs f i r s t f o r those devices with the thickest i n t e r -f a c i a l layers. The " q u a l i t y " of a minority c a r r i e r MIS diode depends on both the r a t i o of the minority c a r r i e r current component to the majority c a r r i e r component i n the semiconductor l i m i t e d regime, and on the maximum forward bias which can be applied before the tunnel l i m i t e d regime i s entered. Devices of the highest q u a l i t y are formed when the tunn e l l i n g p r o b a b i l i t y c o e f f i c i e n t f o r the minority c a r r i e r band i s greater than that for the majority c a r r i e r band, and when the semiconductor surface i s strongly inverted. Both these conditions help to suppress the majority c a r r i e r thermionic emission current, while simultaneously strengthening the car-r i e r flows which maintain the state of quasi-equilibrium between the metal and the minority c a r r i e r s at the semiconductor surface. Although 97 Figure 3.5 Dark current-voltage c h a r a c t e r i s t i c s f or MIS diodes with various i n s u l a t o r thicknesses; i n s u l a t o r thickness increases a->d. l i t t l e can be done to control the r a t i o of 6 ^ to an appropriate choice of b a r r i e r metal work function can a s s i s t i n achieving high junc-ti o n b a r r i e r heights and hence strong surface inversion. 3.4.2 An An a l y t i c Solution for the Potentials and Current Flows Up to th i s point nothing has been sa i d regarding the e l e c t r o s t a t i c p o t e n t i a l d i s t r i b u t i o n across the MIS junct i o n . In the case of a t h i c k -i n s u l a t o r MOS capacitor, i t i s possible to integrate Poisson's equation over the semiconductor to obtain an expression f o r the e l e c t r i c f i e l d £,(Xg) j u s t i n s i d e the semiconductor surface [92]. Green ejt a l . [17] have shown that t h i s technique can be extended to the case of the MIS tunnel diode. Here the r e s u l t s obtained by Green e_t a l . w i l l be derived using a somewhat less rigorous argument. For a uniformly-doped n-type substrate, Poisson's equation becomes d % / d x 2 = -(q/e g) [p - n + N D] . (3.40) If ty and the quasi-fermi p o t e n t i a l s <j>n and <j>p are measured r e l a t i v e to an appropriate reference point, i n the non-degenerate case the c a r r i e r concentrations are given by n = n ± exp[q(^ - d ^ / k T ] (3.41) and p = n ± exp[q(qSp - i|/)/kT] . (3.42) Provided the net current flow through the diode i s small, the p o t e n t i a l drop across the quasi-neutral base can be ignored. Further, i n Chapter 2 99 i t was argued that each quasi-fermi l e v e l should be e s s e n t i a l l y constant across that part of the depletion region i n which the corresponding car-r i e r concentration i s large. However, i n evaluating the right-hand side of (3.40) the charge contribution from each free c a r r i e r need be consid-ered only where the concentration of that c a r r i e r i s large. Thus f or the purpose of evaluating (3.40), i t i s reasonable to assume that the quasi-fermi l e v e l s are constant across the depletion region. Applying the same technique used to solve (3.40) for the MOS capacitor [92], i t i s then found that t £ ( X g ) ] 2 = (2kT/ E g) [p(x s) - p(x n) + n(x g) - n ( x n ) + N^qi^/kT) ] (3.43) where the plane x marks the boundary between the depletion region and n the quasi-neutral base. Under normal operating conditions, the terms p(x ) and n(x ) can be ignored. Further, n(x ) i s very nearly equal to n L> n N D > Thus [ £ ( X g ) ] 2 = (2kT/e s) [p(x s) + N D ( q ^ / k T - 1) ] . (3.44) p(x ) i s r e l a t e d to the equilibrium hole concentration p i n the quasi-S nO ne u t r a l base by P(x s) = P n Q exp(qA(J>/kT) exp(q<|;s/kT) . (3.45) If the charge stored i n surface states can be ignored, the electro-s t a t i c p o t e n t i a l drop i|> across the i n s u l a t o r i s rel a t e d to £,(x*) by *i = ^ ( x s } d E s / e i ' (3.46) 100 where d i s the thickness of the i n t e r f a c i a l layer. From F i g . (3.4), *M = V + ^0 + + XS + *I ' ( 3' 4 7> Taken together, equations (3.34), (3.39), (3.44), (3.45), (3.46), and (3.47) are s u f f i c i e n t to uniquely determine the values of ty , ty , p(Xg) and hty i n the MIS diode at any operating point. Unfortunately, there i s no closed s o l u t i o n f o r t h i s system of coupled non-linear equa-tions . However, a considerable s i m p l i f i c a t i o n of the system i s poss i b l e . (3.44), (3.45), (3.46) and (3.47) can be combined to eliminate p(x ) and tyv giving *M - V + *0 + XS + *S <3'48> + d 2 k T £ s 2 E I 1 / 2 [ p n Q exp(q^ s/kT) exp(qA<|>/kT) + N^(qi|»g/kT - 1 ) ] 1 / 2 At equilibrium, A<j> and V must both be zero. In t h i s case (3.48) reduces to a si n g l e transcendental equation f o r which can be solved i t e r a -t i v e l y . (3.34) and (3.39) can be combined to give "JVM0 P ( X S } ( 1 ~ e x P ^ ( V " A4>) / k T]) = J + ) + J.(A<j>) (3.49a) o where p^ ( X g ) i s the surface hole concentration at equilibrium and the constant J-^HQ ^S g i v e n by 101 JVM0 = V \ T* W * <3'49b> NV (3.49) i s simply a statement of hole current continuity at the semi-conductor surface. Provided the net hole current entering the semicon-ductor i s not zero, (3.49) can be solved for p (x ) i n terms of A<j>, and then (3.45) can be used,to solve f o r \p i n terms of A<j>. Substituting the r e s u l t i n g expression f o r if^  i n (3.48) y i e l d s a si n g l e transcendental equation f o r Ad), which can be solved by i t e r a t i o n . Once Ad) i s known, Tpg, p ( X g ) , and the current flows across the in t e r f a c e can be found immediately at any bias point V. Following the above procedure, reasonably accurate approximate a n a l y t i c solutions f o r the current flows and p o t e n t i a l drops i n the MIS diode can be obtained, even f o r operation i n the tunnel l i m i t e d regime. •This a n a l y t i c treatment of the MIS junction i s s i m i l a r to that developed by Card and Rhoderick [26-28]. However, the method proposed here takes i n t o account the p o s s i b i l i t y of strong i n v e r s i o n at the semiconductor surface, a matter which was overlooked by Card and Rhoderick. In contrast to the approach taken here, and to that followed by Card and Rhoderick, Green ejt a l . [16,17,22-25] chose to r e l y e n t i r e l y on numerical analysis to solve for the c a r r i e r concentrations, p o t e n t i a l s and current flows i n the MIS diode. In t h i s numerical approach, a technique s i m i l a r to that outlined i n Appendix A i s used to solve the f i v e basic equations governing the c a r r i e r concentrations and e l e c t r o s t a t i c p o t e n t i a l i n the semiconduc-tor, subject to the boundary conditions imposed by (3.38), (3.39), and (3.47). Even though no general a n a l y t i c s o l u t i o n f o r the p o t e n t i a l d i s t r i -bution i n an MIS junction i s a v a i l a b l e , some important conclusions 102 concerning the behaviour of the e l e c t r o s t a t i c p o t e n t i a l and the c a r r i e r concentrations can be drawn f o r the case i n which E„ (x„) i s r p b pinned to E^.. In p a r t i c u l a r , i t can be shown by the following i n d i r e c t argument that under the a p p l i c a t i o n of a forward bias V the hole concen-t r a t i o n at the semiconductor surface can not f a l l below i t s equilibrium value i f E F p ( X g ) coincides with E ^ [16]. From (3.47), when a bias V i s applied to the MIS diode, AIJJ and Aip must s a t i s f y the r e l a t i o n s h i p J. D V = -Ai|i - Aiji . (3.50) The hole concentration at the surface could decrease only i f -Ai|>g were greater than V, f o r i f E„ (x„) i s pinned to E ^ , i t follows that E^ , (x ) r p b rJXL r p b must be displaced from the fermi l e v e l at the back contact by an amount qV. But with t h i s condition on A^ g, (3.50) could be s a t i s f i e d only by having Aifj^ p o s i t i v e . However, had the surface become less inverted, the p o t e n t i a l drop across the i n s u l a t o r would be smaller than at equilibrium. In t h i s case Aij;^ would be negative, which i s a c o n t r a d i c t i o n . Therefore i f E F p ( x g ) i s pinned to E ^ , (3.50) can be s a t i s f i e d only by having the surface concentration of holes remain at i t s equilibrium value or increase. In f a c t , the numerical analysis c a r r i e d out by Green e_t a l . has shown that the hole concentration at the semiconductor surface remains very close to i t s e q u i l i b r i u m value u n t i l the tunnel l i m i t e d regime i s entered. It should be noted that t h i s i s exactly opposite to the behaviour expected fo r a t h i c k - i n s u l a t o r MOS capacitor. I t follows that AiJ> i s e s s e n t i a l l y equal to V throughout the semiconductor l i m i t e d regime. Thus the diode f a c t o r n defined i n (3.2) i s e f f e c t i v e l y unity for a minority c a r r i e r MIS diode operated i n t h i s regime. 103 3.5 The MIS Solar C e l l 3.5.1 Light Coupling into the Semiconductor The f i r s t problem which must be overcome i n order to form an e f f i -cient Schottky b a r r i e r or MIS solar c e l l i s the coupling of l i g h t into the semiconductor. In the e a r l i e s t MIS c e l l s , a p a r t i a l s o l u t i o n to t h i s problem was achieved by making the evaporated b a r r i e r metal layer so thi n as to be semi-transparent. The t h i n b a r r i e r layer was then o v e r l a i d with a thick contact g r i d , as i l l u s t r a t e d i n F i g . 3.6(a). However, t h i s s o l u -t i o n was far from i d e a l . From the viewpoint of e l e c t r i c a l properties alone, the best MIS solar c e l l s are those formed by depositing low work function metals such as aluminum, chromium or magnesium on p-type s i l i c o n [19]. Unfortunately, these low work function metals are very strong ab-sorbers of v i s i b l e l i g h t ; for example, Hovel [93] has calculated that o even when o v e r l a i d with an optimized a n t i - r e f l e c t i o n coating, a 75 A thick layer of aluminum can be expected to transmit only about 60% of incident l i g h t at v i s i b l e wavelengths into a s i l i c o n substrate. The transmittance of evaporated layers of low work function metals can be increased to some extent by p a r t i a l oxidation during deposition [35]. This p a r t i a l oxidation i s e a s i l y accomplished by carrying out the evaporation slowly under a r e l a t i v e l y high oxygen pressure. However, the incorporation of oxygen may produce deleterious changes i n the work func-t i o n of the layer. A l t e r n a t i v e l y , a composite b a r r i e r layer can be formed i n which an u l t r a - t h i n layer of low work function metal i s ov e r l a i d with a reasonably transparent layer of high work function metal. I f the com-ponents are c o r r e c t l y chosen, the r e s u l t i n g stack may have high conduc-t i v i t y , high o p t i c a l transmittance, and a work function i n the desired range. Following t h i s approach, Anderson et a l . have fabricated MIS c e l l s MOOOym INSULATOR• 104 Al (Mum) Al M O A ) < S i 0 x (^20A) p Si (^300ym) OHMIC CONTACT Al (Mum) (a) ,GRID FINGER ± A -MOOym-1+/+/+/+/+ INSULATOR-A l (Mym) SiO (MJOOA) S i 0 x (-V20A) p S i (MiOOym) OHMIC CONTACT A l (Mym) (b) Figure 3.6 (a) Structure of MIS s o l a r c e l l with t h i n , semi-transparent b a r r i e r l a y e r . (b) Structure of inversion layer c e l l . 105 with the st r u c t ure 10A Cr - 60A Cu - 30A Cr - SiO — pSi giving photo— 2 current densities as high as 26 mA/cm [94]. In t h i s structure, the chromium layer closest to the s i l i c o n induces the junction, while the copper provides reasonable sheet conductivity without a serious loss i n transmittance. The photocurrent recorded by Anderson e_t a l . i s probably close to the upper l i m i t which can be achieved with semi-transparent b a r r i e r metal lay e r s . Recently Godfrey and Green [95,96] and Thomas et_ a l . [97] have pro-duced inversion layer c e l l s of exceptionally high e f f i c i e n c y incorporating MIS junctions. In the inversion layer c e l l the MIS junction covers only 10 to 20% of the surface, and functions as a contact g r i d (see Fi g . 3.6(b)). A t h i n layer of d i e l e c t r i c i s deposited over the rest of the surface i n such a way that a high concentration of fixed p o s i t i v e charge i s present near the i n t e r f a c e (this can e a s i l y be accomplished by the thermal evaporation of SiO). The fi x e d charge i n the d i e l e c t r i c induces an inversion layer at the s i l i c o n surface, i n e f f e c t c r e a t i n g a very shallow induced junction. I f the thickness and index of r e f r a c t i o n of the d i e l e c t r i c are selected to minimize r e f l e c t i o n , v i r t u a l l y a l l l i g h t incident on the c e l l can be transmitted i n t o the substrate. As a r e s u l t , photocurrent densities close to the t h e o r e t i c a l maximum for s i l -icon can be achieved. Although the inve r s i o n layer c e l l w i l l not be given further consideration i n t h i s t h e s i s , the structure o f f e r s a means of u t i l i z i n g MIS junctions to form c e l l s with e f f i c i e n c i e s equal to or sur-passing those of the best homojunction devices. For the remainder of t h i s s e c t i o n , i t w i l l be assumed that a means has been found to e f f i c i e n t l y couple i n c i d e n t l i g h t i n t o the semiconduc-tor i n MIS s o l a r c e l l s . Although the most e f f i c i e n t MIS c e l l s reported 106 to date have been fabricated on p-type substrates [19], c e l l s formed on n-type material w i l l be considered here f o r consistency with Sections 3.3 and 3.4. I t w i l l be assumed from the outset that the b a r r i e r metal has been chosen to give strong inversion of the semiconductor surface at equilibrium. The r e l a t i o n s h i p between c e l l performance and the thickness of the i n t e r f a c i a l layer w i l l then be considered. 3.5.2 Optimally E f f i c i e n t MIS C e l l s In order to a t t a i n the highest possible open-circuit voltage i n an MIS s o l a r c e l l , the i n t e r f a c i a l layer must be thick enough to suppress the majority c a r r i e r thermionic/emission dark current component to neg-l i g i b l e l e v e l s . However, i n order to obtain high f i l l factors the i n s u -l a t o r must simultaneously be t h i n enough that a net current equal i n magnitude to the one-sun photocurrent can flow between the minority car-r i e r band and the metal without s i g n i f i c a n t displacement of the minority c a r r i e r quasi-fermi l e v e l at the surface from E . This i s equivalent to re q u i r i n g that the minority c a r r i e r flow i n the c e l l be semiconductor 2 l i m i t e d up to current densities comparable to 30 mA/cm . If these two conditions on i n s u l a t o r thickness can be met, then the c h a r a c t e r i s t i c s of the MIS c e l l w i l l be e s s e n t i a l l y the same as those of an i d e a l one-sided homojunction c e l l with a very shallow emitter region formed on an i d e n t i c a l substrate [98]. In f a c t , the performance of the MIS c e l l i s l i k e l y to be s l i g h t l y better than that of the homojunction c e l l , since the c a r r i e r l i f e t i m e s and m o b i l i t i e s i n the inversion layer of the MIS c e l l w i l l not have been degraded by heavy doping e f f e c t s . As a r e s u l t , c a r r i e r s photogenerated very near the s i l i c o n surface by short-wavelength photons w i l l have a higher p r o b a b i l i t y of crossing the depletion region and contributing to the photocurrent i n the MIS c e l l [99]. S i m i l a r l y , 107 the dark current component r e s u l t i n g from recombination near the surface should be smaller i n the MIS c e l l than i n the homojunction c e l l . In Chapter 2 i t was found that the superposition p r i n c i p l e should provide an excellent approximate de s c r i p t i o n of a s i l i c o n homojunction c e l l operated i n the low-level i n j e c t i o n regime. I t follows that the superposition p r i n c i p l e should also be applicable within the semiconductor i n a minority c a r r i e r MIS s o l a r c e l l . Therefore the net hole flow from the semiconductor surface i n t o the depletion region and base must be given by JVM - J r g ( A * > + J d ( A * } " J u P c • < 3- 5 1> The requirement of hole current continuity at the surface then gives "JVM0 p ( V ( 1 " e x P t q ( V " A<f>)/kT]) = J r g(Acfr) + J d(Ad)) - J , (3.52) W which replaces (3.49) for an illuminated c e l l . Assuming that the majority c a r r i e r thermionic emission current i s n e g l i g i b l e , must be equal to the terminal current J (V). I f E„ (x„) i s pinned to E l 7 M at a l l operating L r n o rM points, then Ad) i s always equal to the bias V applied at the terminals. In t h i s case, superposition w i l l hold at the c e l l terminals. 3.5.3 The Ch a r a c t e r i s t i c s of Thick-Insulator C e l l s When the i n s u l a t o r i n a minMIS s o l a r c e l l becomes thick enough to ser i o u s l y impede the flow of a current comparable i n magnitude to the one-sun photocurrent, the c e l l conversion e f f i c i e n c y i s reduced. The f i r s t performance parameter to be degraded i s the f i l l f a c t o r , followed by the s h o r t - c i r c u i t current [25]. In p r i n c i p l e , there should be no degradation 108 i n open-circuit voltage, so long as the majority c a r r i e r thermionic emis-sion current remains n e g l i g i b l e . The band diagram f o r a t h i c k - i n s u l a t o r MIS c e l l exposed to one-sun i l l u m i n a t i o n at terminal s h o r t - c i r c u i t i s shown i n F i g . 3.7. In order to support the hole photocurrent flowing i n t o the metal, E F p ( X g ) must be displaced from E ^ . This displacement of E F p ( X g ) has two consequences. F i r s t , i n order to s a t i s f y (3.48) and (3.45), the hole concentration at the semiconductor surface must increase dramatically over i t s e q u i l i b r i u m value. Secondly, from (3.51) i t can be seen that the s h o r t - c i r c u i t current f o r the t h i c k - i n s u l a t o r c e l l must be less than that for a c e l l i n which E„ (x„) i s always pinned to E_.„. How-Fp S FM ever, unless the i n s u l a t o r i s excessively t h i c k , t h i s suppression of J sc should be i n s i g n i f i c a n t . As long as A<j> at s h o r t - c i r c u i t i s roughly 100 mV or more les s than V , J w i l l be very nearly equal to J oc sc upc When a small forward bias V i s applied to the illuminated t h i c k -i n s u l a t o r c e l l , E„ (x„) i s displaced even further from the fermi l e v e l Fp S at the back contact than at short c i r c u i t . I f A<f> at s h o r t - c i r c u i t was small compared to V , i t i s possible to apply a f a i r l y large forward bias before J (V) drops s i g n i f i c a n t l y below J . However, a bias point Li S C i s eventually reached at which A(J> i s comparable to V » and any further increase i n forward bias past t h i s point causes a sharp drop i n current output. As J decreases, E (x ) moves closer to E , while the surface VM rp o rM hole concentration drops rapidly towards i t s equilibrium value. At t e r -minal open-circuit i s zero, so E p p ( X g ) must a l i g n with E ^ M regardless of i n s u l a t o r thickness. V q c should therefore be independent of i n s u l a t o r oc thickness. F i g . 3.8 shows the current-voltage c h a r a c t e r i s t i c s under one-sun i l l u m i n a t i o n f o r a set of MIS s o l a r c e l l s with various i n s u l a t o r t h i c k -109 Figure 3.7 Band diagram for t h i c k - i n s u l a t o r MIS c e l l at terminal s h o r t - c i r c u i t under one-sun i l l u m i n a t i o n . 200.. kOQ . 600 BIAS VOLTAGE (mV) Figure 3.8 Illuminated current-voltage c h a r a c t e r i s t i c s f o r MIS sol a r c e l l s with various i n s u l a t o r t h i c k n e s s e s a s predicted by theory. For a l l c e l l s P r i(x o)=10 cm" , d=20A, V =550mV, S oc e =3e_., and J =30mA/cm . I 0 upc ( a ) J V M 0 = l m A / c m 2 • ( b ) J V M 0 = 3 m A / c m 2 * ( c ) J V M 0 = 1 0 m A / c m 2 -( d ) JVM0~ ' I l l nesses. These c h a r a c t e r i s t i c s were generated using i t e r a t i v e techniques to simultaneously solve (3.45), (3.48) and (3.52), as outlined i n Sec-tion 3.4. The c e l l material properties used i n t h i s c a l c u l a t i o n are l i s t e d i n the figure caption. Rather than s e t t i n g <f>M and Xg d i r e c t l y , the equi-l i b r i u m hole concentration at the semiconductor surface p^ ( X g ) was spec-i f i e d . Both the depletion region recombination current J and the major-i t y c a r r i e r thermionic emission current were assumed to be n e g l i g i b l e . In t h i s case, the c e l l open-circuit voltage depends only on the magnitude of the hole i n j e c t i o n - d i f f u s i o n dark current J^. The expression used to compute Jd(A<J>) was that given i n (2.28). J ^ ^ was set to give a s p e c i f i e d open-circuit voltage. The most remarkable feature of F i g . 3.8 i s the way i n which the c h a r a c t e r i s t i c s of those c e l l s with r e l a t i v e l y thick i n s u l a t o r s become concave-upwards as V approaches V • a conventional diffused-junction c e l l with very large shunt conductance or seri e s resistance, the J (V) c h a r a c t e r i s t i c may become almost a s t r a i g h t l i n e connecting the short-c i r c u i t and open-circuit points, but the c h a r a c t e r i s t i c w i l l always be concave downwards [100]. As might be expected, the bias V at which the current f i r s t begins to f a l l sharply i n F i g . 3.8 i s approximately equal to the difference between V and the value of Ad> at equilibrium. oc Numerical analysis c a r r i e d out by Shewchun, Singh and Green [25] has shown that the f i l l f a c tor of a t h i c k - i n s u l a t o r MIS s o l a r c e l l may even drop below 0.25, which i s the lower l i m i t on this parameter i n a conventional homojunction device. The lowest f i l l f a c tor associated with the c h a r a c t e r i s t i c s of F i g . 3.8 i s indeed s l i g h t l y less than 0.25. I t should be noted that for the device with the smallest value of J n considered i n F i g . 3.8, the hole concentration at the semiconductor 112 surface becomes comparable to N for operation near s h o r t - c i r c u i t . Since v c the analysis used to generate F i g . 3.8 i s v a l i d only when the c a r r i e r concentrations i n the semiconductor remain at non-degenerate l e v e l s (see Section 3.4), the c h a r a c t e r i s t i c shown f o r th i s device may be i n e r r o r . However, c h a r a c t e r i s t i c s obtained using the analysis of Section 3.4 should be at least q u a l i t a t i v e l y correct provided the semiconductor surface i s not strongly degenerate. 113 CHAPTER 4 POSITIVE BARRIER MIS JUNCTIONS: EXPERIMENT In t h i s chapter the f i r s t conclusive experimental evidence f o r the existence of minority c a r r i e r MIS diodes i s presented. This evidence was gathered through two independent experiments which are described i n Sec-tions 4.2 and 4.3 res p e c t i v e l y . The f i r s t of these experiments involved the measurement of the current-voltage c h a r a c t e r i s t i c s of Al-SiO^-pSi diodes at various temperatures spanning the range from 0 to 50°C [20]. Since the temperature dependence of an i n j e c t i o n - d i f f u s i o n current i s stronger than that of any thermionic emission current, t h i s experiment provided a d e f i n i t i v e t e s t f o r the charge-transport mechanisms dominating the dark current. In the second experiment, Al-SiO^-pSi s o l a r c e l l s were formed on 10 ficm substrates with alloyed aluminum back surface f i e l d s [21]. The presence of the back surface f i e l d region was found to increase the open-circuit voltage by up to 50 mV r e l a t i v e to the value recorded with an ohmic back contact, a r e s u l t which can be explained only i f the dark current i n these c e l l s i s dominated by minority c a r r i e r i n j e c t i o n -d i f f u s i o n . Section 4.1 provides a b r i e f review of previous experimental research on the p o s i t i v e b a r r i e r MIS junction, p l a c i n g p a r t i c u l a r empha-s i s on inv e s t i g a t i o n s of a fundamental nature. F i n a l l y , the r e l a t i o n s h i p between i n s u l a t o r thickness and e l e c t r i c a l c h a r a c t e r i s t i c s i n experimen-t a l MIS s o l a r c e l l s i s examined i n Section 4.4. 4.1 Previous Experimental Research on the MIS Junction As of mid-1978, when the f i r s t of the experiments discussed i n th i s chapter was undertaken, no unambiguous demonstration of the e x i s t -ence of minority c a r r i e r MIS diodes had been reported. However, over the i 114 years a considerable body of i n d i r e c t evidence supporting the MIS tunnel diode theory introduced by Green et al.[16] had appeared i n the l i t e r -ature. An overview of t h i s evidence i s given below. In a prophetic experiment conducted i n 1963, J a k l e v i c e_t a l . [101a] demonstrated that an MIS junction could be used to i n j e c t s u f f i c i e n t minority c a r r i e r s i n t o a CdS c r y s t a l to generate electroluminescence. In contrast, i n j e c t i o n electroluminescence was never observed i n Schottky b a r r i e r diodes formed on CdS. Since 1963, a number of other groups have observed electroluminescence associated with minority c a r r i e r i n j e c t i o n from an MIS junction i n the semiconductors ZnS, ZnSe, GaP, GaN and GaAs, as w e l l as CdS [101b]. Although the observation of electroluminescence at an MIS contact conclusively reveals the presence of i n j e c t e d minority c a r r i e r s , i t provides v i r t u a l l y no information concerning the r e l a t i v e magnitudes of the majority and minority c a r r i e r currents flowing i n the contact. In the early 1970's, Card and Rhoderick [26,102] undertook the f i r s t systematic experimental study of the e f f e c t of the introduction of a t h i n i n t e r f a c i a l i n s u l a t i n g layer on the properties of metal-semiconductor junctions. The structure chosen for t h i s study was the Au-SiO^-nSi junc-t i o n , i n which the i n t e r f a c i a l SiO^ layer was grown by low temperature oxidation of the s i l i c o n substrate p r i o r to b a r r i e r metal deposition. After correction f or v a r i a t i o n s i n the b a r r i e r height with i n s u l a t o r TBn thickness, i t was found that the forward-biased dark current decreased by several orders of magnitude as the i n s u l a t i n g layer thickness was o increased over the range from 8 to 26 A. Since the dark current i n diodes of t h i s type i s dominated by thermionic emission, t h i s experiment c l e a r l y demonstrated that the introduction of an i n t e r f a c i a l i n s u l a t i n g layer 115 could suppress the majority c a r r i e r thermionic emission current i n a metal-semiconductor j u n c t i o n . However, no attempt was made to measure the magnitude of the i n j e c t e d minority c a r r i e r current i n these devices. Card and Rhoderick next turned t h e i r attention to the d i r e c t meas-urement of the minority c a r r i e r i n j e c t i o n r a t i o y i n MIS junctions [27]. This measurement was accomplished using the metal-emitter t r a n s i s t o r structure which Yu and Snow had employed e a r l i e r to investigate minority c a r r i e r i n j e c t i o n i n Schottky diodes [13]. The metal-emitter t r a n s i s t o r i s equivalent to a conventional planar b i p o l a r t r a n s i s t o r , except that a Schottky or MIS junction replaces the d i f f u s e d emitter. Provided that the base width i s short compared to the minority c a r r i e r d i f f u s i o n length i n the base, the r a t i o of the c o l l e c t o r current to the emitter current i n t h i s device i s equal to the minority c a r r i e r i n j e c t i o n r a t i o of the emitter j u n c t i o n . Card and Rhoderick confirmed Yu and Snow's f i n d i n g -4 that y i s t y p i c a l l y l e s s than 10 for Au-nSi Schottky diodes formed on chemically etched substrates, and then went on to show that y could be made as large as 0.2 i n Au-SiO^-nSi diodes with r e l a t i v e l y thick (^O A) i n t e r f a c i a l l a y e r s . With such thick i n t e r f a c i a l layers r e l a t i v e l y large forward biases (£ 1 V) had to be applied to obtain appreciable current flow, so these MIS diodes would not have been of use for photovoltaic ene r gy conve rs i on. Following the development of t h e i r MIS tunnel diode theory, Green et a l . themselves c a r r i e d out a number of fundamental experiments on the MIS structure. In the f i r s t of these experiments, Al-SiO^-pSi diodes with d i f f e r e n t i n s u l a t o r thicknesses were fabricated on a set of sub-s t r a t e s of i d e n t i c a l r e s i s t i v i t y [68]. The dark current-voltage charac-t e r i s t i c s of these diodes were found to e x h i b i t the q u a l i t a t i v e depend-ence on i n s u l a t o r thickness predicted t h e o r e t i c a l l y (see F i g . 3.5). In a subsequent experiment, Al-SiO -pSi diodes having the same i n s u l a t o r thicknesses were fabricated on substrates with a wide range of r e s i s t i v -i t i e s . While a thermionic emission current should be independent of the substrate doping l e v e l , the current flowing under moderate forward bias i n these MIS diodes was found to decrease as the substrate doping l e v e l increased. This i s exactly the behaviour which would be expected for a depletion region recombination current or a minority c a r r i e r i n j e c t i o n -d i f f u s i o n current. When magnesium was substituted for aluminum as the b a r r i e r metal i n these diodes, no s i g n i f i c a n t change i n the current-voltage c h a r a c t e r i s t i c s at moderate forward bias was observed, for a given substrate doping. This r e s u l t strongly suggested that the diode dark current was dominated by minority c a r r i e r flows, which depend only on substrate properties. In contrast, the magnitude of a thermionic emission current would have depended strongly on the b a r r i e r metal work function. In other experiments, the s m a l l - s i g n a l capacitance C of Al-SiO^-pSi and Mg-SiO x-pSi diodes was recorded as a function of bias [24,68]. In 2 reverse b i a s , 1/C was found to depend l i n e a r l y on V, j u s t as for a 2 Schottky diode. By f i n d i n g the voltage-axis intercept of a p l o t of 1/C versus V, the diode b a r r i e r height was estimated. I t was found that i n these diodes the s i l i c o n surface was strongly inverted at equilibrium, a condition which must be s a t i s f i e d to produce a minority c a r r i e r MIS diode. Although the experimental r e s u l t s obtained by Green et a l . were i n general agreement with the predictions of the MIS tunnel diode theory, they d i d not prove that the majority c a r r i e r thermionic emission current 117 could be made n e g l i g i b l e at current densities comparable to the one-sun photocurrent. Most of the devices used i n the experiments described above had such thick i n s u l a t i n g layers that they entered the tunnel-limited 2 regime long before the dark current density reached the 30 mA/cm l e v e l . These devices would thus have been quite unsuitable for use as MIS s o l a r c e l l s . Indeed, i n 1976 St. P i e r r e , Singh, Shewchun and L o f e r s k i [103] suggested that s i g n i f i c a n t majority c a r r i e r dark current components would always be present i n e f f i c i e n t MIS s o l a r c e l l s . In 1977 Pulfrey reported dark current-voltage c h a r a c t e r i s t i c s f or Al-SiO^-pSi s o l a r c e l l s showing two regions of exponential dependence of current on bias [104]. At low forward bias the current was found to obey (2.25) with A=2, while for forward biases between 300 and 500 mV (2.25) was obeyed with A=l. C h a r a c t e r i s t i c s of t h i s type are frequently observed f o r s i l i c o n homojunction diodes, as discussed i n Section 2.2. In i n t e r p r e t i n g the c h a r a c t e r i s t i c s of these Al-SiO^-pSi c e l l s , Pulfrey suggested that at low forward bias the dark current was dominated by recombination i n the depletion region, while at larger forward bias the minority c a r r i e r i n j e c t i o n - d i f f u s i o n current flowing i n t o the base became dominant. It was thus i n f e r r e d that these devices were minority c a r r i e r MIS diodes. Unfortunately, i n an MIS diode current-voltage character-i s t i c s of t h i s "double exponential" form can equally w e l l be explained by assuming that the main current component i n the higher bias range i s a majority c a r r i e r thermionic emission current. P r i o r to 1978, perhaps the most convincing evidence for the existence of minority c a r r i e r MIS diodes resided i n the high e f f i c i e n c i e s and open-circuit voltages reported by s e v e r a l groups for t h e i r MIS s o l a r c e l l s . In 1976, Green, Godfrey and Davies [35] and St. P i e r r e et_ a l . [103] 118 reported open-circuit voltages over 600 mV for Al-SiO^-pSi c e l l s f a b r i -cated on substrates with r e s i s t i v i t i e s ranging from 0.1 to 1 Hem. Anderson et a l . concentrated on improving the performance of t h e i r Cr-SiO^-pSi c e l l s , and had achieved e f f i c i e n c i e s of over 10% and open-circuit voltages of 600 mV by l a t e 1977 [94], using 2 ficm substrates. Since a t y p i c a l d i f f u s e d junction c e l l formed on a 1 Hem p-type s i l i c o n substrate would be expected to have an open-circuit voltage of approximately 600 mV, there could be l i t t l e doubt that the majority c a r r i e r thermionic emission dark current component i n these MIS s o l a r c e l l s had been reduced to very low l e v e l s . 4.2 New Experimental Evidence for Minority C a r r i e r MIS Diodes In p r i n c i p l e , a minority c a r r i e r MIS diode can be i d e n t i f i e d by the temperature dependence of i t s dark current-voltage c h a r a c t e r i s t i c . I d e a l -l y , both majority c a r r i e r thermionic emission currents and minority car-r i e r i n j e c t i o n - d i f f u s i o n currents obey the law where J Q i s a temperature-dependent constant. From (3.30) and (3.38), for a thermionic emission current on a p-type substrate while from (2.27), for an i n j e c t i o n - d i f f u s i o n current i n a long, uniformly doped p-type base region J = J n [exp(qV/kT) - 1] (4.1) J0Th " 6VM\ l 2 exp(-q* /kT) , (4.2) J n = qvlT n 2 Od n x (4.3) 119 The i n t r i n s i c c a r r i e r concentration n_^  i s given by [105] n.(T) = 1 2kT irh 3/2 * * 3/4 (. mc mv exp[-E (T)/2kT] (A.4) where E (T) i s the bandgap energy of s i l i c o n . To a good approximation, E (T) decreases l i n e a r l y with temperature above about 250°K; thus [106] 6 E g ( T ) = Eg0 " a T <4-5> where E gQ and a are temperature-independent constants. There i s some disagreement concerning the exact values of E and a. In the l a t e 1950's, 6 Macfarlane et a l . concluded that E _ = 1.206 eV on the basis of o p t i c a l g0 v absorption measurements [106]. More recent measurements of the temper-ature dependence of the c o l l e c t o r current i n d i f f u s e d junction b i p o l a r t r a n s i s t o r s c a r r i e d out by Slotboom et^ al_. [107] indi c a t e that = (1.20±.01) eV. The important point i s that E g i s s i g n i f i c a n t l y l a r g e r than the bandgap energy of s i l i c o n at room temperature, which i s known to l i e between 1.11 and 1.12 eV. Substituting (4.5) into (4.4), i t i s found that n 2 oc T3 exp(-E g ( )/kT) . (4.6) Compared to n^, and- are not strongly dependent on temperature. Near 300°K the electron mobility obeys the empirical r e l a t i o n s h i p [108] y . I " 2 ' 7 n (4.7) 120 Invoking the E i n s t e i n r e l a t i o n s h i p between the mobility and the d i f f u s i o n c o e f f i c i e n t , from (4.7) i t follows that D n « T 1 , 7 . (4.8) To a f i r s t approximation, the minority c a r r i e r l i f e t i m e i s inversely proportional to the mean minority c a r r i e r v e l o c i t y , which i s i n turn 1/2 proportional to T [109]. Thus x « T 0 , 5 . (4.9) n Combining (4.6), (4.8) and (4.9), i t i s found that J Q D - T 2 - 4 exp(-E g ( )/kT) . (4.10) Comparing (4.2) and (4.10), i t i s seen that Jnrrn_ and J „ , are both Uih (Jd of the form J Q - T P exp(-E g 0/kT) (4.11) where p has the value 2 for a thermionic emission current and 2.4 for an i n j e c t i o n - d i f f u s i o n current. Since the temperature dependence of JQ i s dominated by the exponential f a c t o r , the exact value of p i s unimportant. From (4.11) i t can be seen that an Arrhenius p l o t of log(jQ/T P) ver-sus 1/T should y i e l d a s t r a i g h t l i n e , and that the slope of t h i s l i n e should be proportional to an a c t i v a t i o n energy E . For an i n j e c t i o n - d i f -fusion current E. = E while for a thermionic emission current E. = qty^. A gO A n T B 121 Because qdj^ < E g < E gQ for any Schottky b a r r i e r , t h i s technique can re a d i l y d i s t i n g u i s h between minority and majority c a r r i e r diodes. I t should be noted i n passing that Yu and Snow have used an a c t i v a t i o n en-ergy analysis of t h i s type to determine the r e l a t i v e magnitudes of the thermionic emission current and the depletion region recombination cur-rent i n conventional Schottky b a r r i e r diodes [11]. In r e a l d i f f u s e d junction, MIS and Schottky b a r r i e r diodes, i t i s often found that ( A . l ) applies, i f at a l l , only over a very l i m i t e d bias range. At low forward bias the diode c h a r a c t e r i s t i c s are usually domin-ated by depletion region recombination currents, which obey (2.25) with A values between 1 and 2. At high forward bias voltage drops across i n t e r n a l s e r i e s resistances associated with ohmic contacts, with thin b a r r i e r metal laye r s , and with the bulk substrate i t s e l f give r i s e to current-voltage c h a r a c t e r i s t i c s which do not have an exponential form. Rhoderick has shown that a combination of depletion region recombination current and se r i e s resistance e f f e c t s can produce Schottky diode charac-t e r i s t i c s i n which an exponential current-voltage r e l a t i o n s h i p holds over several decades of current, but i n which the diode factor A i s s i g n i f i -cantly greater than unity [89]. When dealing with i n j e c t i o n - d i f f u s i o n currents, deviations from ( A . l ) w i l l be encountered when the h i g h - l e v e l i n j e c t i o n regime i s reached [110]. In minority c a r r i e r MIS diodes, the tunnel resistance e f f e c t s discussed i n Section A . 3 can also lead to deviations from ( A . l ) [16]. P r i o r to 1978, both Shewchun and Green [23] and Vernon and Anderson [111] had published data on the temperature dependence of the current-voltage c h a r a c t e r i s t i c s of MIS diodes. Vernon et al.reported current-voltage c h a r a c t e r i s t i c s f o r Cr-SiO -pSi s o l a r c e l l s at various temper-atures spanning the range from 20°C to 120°C, but no region i n which (4.1) was even approximately obeyed could be discerned i n these charac-t e r i s t i c s . Shewchun e_t a l . recorded cur rent-voltage c h a r a c t e r i s t i c s f or an Al-SiO^-pSi diode over the temperature range 200-350°C but, again, no region i n which (4.1) was obeyed could be found. The diode selected for t h i s l a t t e r study had a very thick i n s u l a t i n g layer, and showed tunnel 2 l i m i t e d behaviour at current densities of only 0.1 mA/cm . At this low dark current density, depletion region recombination currents would have been f a r larger than e i t h e r the i n j e c t i o n - d i f f u s i o n or thermionic emis-sion currents. In the f a l l of 1978, the a c t i v a t i o n energy analysis technique de-scribed above was applied to obtain the f i r s t i r r e f u t a b l e evidence for the existence of minority c a r r i e r MIS diodes with e l e c t r i c a l properties s u i t a b l e f o r photovoltaic energy conversion [20]. Al-SiO^-pSi s o l a r c e l l s f a b r i c a t e d on chem-mechanically polished substrates of 10 Qcm r e s i s t i v i t y and <100> or i e n t a t i o n were used i n t h i s experiment. Complete d e t a i l s of the device f a b r i c a t i o n procedure are given i n Appendix C. In summary, the substrates were f i r s t cleaned following standard procedures used i n the manufacture of integrated c i r c u i t s , and then exposed to a dry oxygen flow at 500°C for 20 minutes to grow a t h i n oxide layer. An ohmic contact was formed at the back of the s l i c e s by the deposition of a thick aluminum laye r , followed by s i n t e r i n g i n dry nitrogen at 500°C for 10 minutes. The MIS junction i t s e l f was produced by depositing a semi-transparent aluminum o dot approximately 80 A thick onto the front surface of the substrates. Contact to t h i s b a r r i e r metal layer was made with a s i n g l e aluminum g r i d finger several thousand angstroms t h i c k . The t h i n b a r r i e r metal layer and the contact f i n g e r were defined using metal shadow masks to give a t o t a l 123 2 junction area of about 0.1 cm . The ohmic contact, b a r r i e r metal and contact f i n g e r aluminum depositions were a l l c a r r i e d out by thermal evap-oration from a tungsten filament. In order to ensure that the s i l i c o n surface i n the completed Al-SiO^-pSi c e l l s was strongly inverted at equilibrium, the small-signal capacitance C of these devices was measured as a function of reverse 2 b i a s . A p l o t of 1/C versus V for a t y p i c a l c e l l i s shown i n F i g . 4.1; as expected, the data points l i e almost exactly on a s t r a i g h t l i n e . Using the method of least squares, the slope and voltage-axis intercept V of t h i s l i n e were computed. The slope i s consistent with a doping density 15 -3 of 1.0*10 cm . Theory predicts that V should equal the b a r r i e r height <}>gp for a junction i n which the surface i s only depleted at equilibrium. However, i f the substrate surface i s strongly inverted at equilibrium, V should be only s l i g h t l y greater than the strong inversion p o t e n t i a l [24]. The voltage-axis intercept i n F i g . 4.2 i s 590 mV, while the strong 15 -3 inversion p o t e n t i a l f o r a substrate with a doping density of 1.0*10 cm i s 580 mV. It can thus be concluded that strong inversion of the s i l i c o n surface had indeed been achieved. The dark current-voltage c h a r a c t e r i s t i c for a representative Al-SiO^-pSi c e l l i s shown i n F i g . 4.2. Two regions of approximately ex-ponential dependence of current on voltage are v i s i b l e i n t h i s character-i s t i c , as would be expected for a minority c a r r i e r MIS diode. However, while i n the low-level i n j e c t i o n regime an i n j e c t i o n - d i f f u s i o n current must obey (2.25) with A=l, the slope of the upper region of the MIS diode c h a r a c t e r i s t i c corresponds to an A value between 1.1 and 1.2. Since no region can be i d e n t i f i e d i n the c h a r a c t e r i s t i c over which the diode f a c -tor A i s equal to unity, i t i s not possible to derive a r e l i a b l e estimate 124 BIAS VOLTAGE (mV) Figure 4.1 Capacitance-voltage c h a r a c t e r i s t i c f o r reverse-biased Al-SiO -pSi dot diode, x 125 Figure 4.2 Dark current-voltage c h a r a c t e r i s t i c f o r small-area Al-SiO -pSi s o l a r c e l l . for the parameter appearing i n equation (4.1) from t h i s curve. As noted above, even i n an MIS diode i n which the dark current i s dominated by minority c a r r i e r i n j e c t i o n - d i f f u s i o n , A values greater than unity may be encountered i f the bias regime i n which the depletion region recombination current i s s i g n i f i c a n t i s not w e l l separated from the regime i n which seri e s resistance or tunnel resistance becomes important. For the device of F i g . 4.2, i t appears that a s u b s t a n t i a l f r a c t i o n of the dark current flowing at bias l e v e l s below about 400 mV resulted from depletion region recombination processes, while s e r i e s resistance or tunnel resistance e f f e c t s set i n at biases greater than about 500 mV. I t should be stressed, however, that the s e r i e s and/or tunnel resistance was not so large as to s e r i o u s l y degrade the performance of t h i s device when operated as a s o l a r c e l l . The small-area Al-SiO -pSi c e l l s considered 1 x here t y p i c a l l y gave f i l l factors ranging from 0.6 to 0.7 under simulated one-sun i l l u m i n a t i o n . It i s w e l l known that ordinary s e r i e s resistance e f f e c t s can be eliminated from the current-voltage c h a r a c t e r i s t i c s of a s o l a r c e l l by measuring J as a function of V rather than by measuring J and V i n 6 sc oc the dark [137]. From the discussion of Section 3.5, i t follows also that the e f f e c t s of tunnel resistance for an MIS c e l l can be eliminated by recording the J -V c h a r a c t e r i s t i c rather than the dark current-voltage c h a r a c t e r i s t i c . For t h i s reason i t was decided that i t would be worthwhile to obtain J -V c h a r a c t e r i s t i c s as w e l l as dark current-voltage charac-sc oc t e r i s t i c s f o r the MIS c e l l s . F i g . 4.3 shows the J -V c h a r a c t e r i s t i c s for a t y p i c a l Al-SiO -pSi 6 sc oc x c e l l recorded at s i x temperatures spanning the range 0-50°C. These char-a c t e r i s t i c s were taken by mounting the e n t i r e test bed, contact probe and 127 Figure 4.3 J -V c h a r a c t e r i s t i c s f o r a small-area Al-SiO -pSi 6 sc oc x s o l a r c e l l at various temperatures. The value quoted 2 for A refers to the region l£J <10 mA/cm . sc l i g h t source assembly within a temperature-controlled Statham oven. The c e l l temperature was measured using a thermocouple mounted i n the tes t bed. This temperature measurement was confirmed with a mercury thermometer inserted through the oven w a l l , and should be accurate to about ±0.2°C. In order to minimize any e f f e c t that heating by the l i g h t source might have had on the c h a r a c t e r i s t i c s , the l i g h t source was switched on only long enough to record a si n g l e J -V p a i r , and then l e f t o ff f o r one sc oc to two minutes. The i l l u m i n a t i o n l e v e l was also varied randomly from measurement to measurement, rather than being s t e a d i l y increased. For s h o r t - c i r c u i t current densities ranging from roughly 1 to 2 10 mA/cm , the c h a r a c t e r i s t i c s of F i g . 4.3 are w e l l described by (2.25) 2 with an A value close to unity.' At current densities below about 1 mA/cm , 2 A becomes s i g n i f i c a n t l y larger than unity, while for J G C > 10 mA/cm , A act u a l l y drops below one, probably as a re s u l t of excessive heating of the c e l l by the l i g h t source. For each c h a r a c t e r i s t i c shown i n F i g . 4.3, the method of lea s t squares was applied to compute the equation of the l i n e best f i t t i n g the 2 2 data points l y i n g i n the range 1 mA/cm < J G C < 10 mA/cm . From the slope of t h i s l i n e , the value of A appropriate to t h i s current range was cal c u -l a t e d , while an estimate f o r the parameter was obtained by f i n d i n g the intercept of the l i n e with the current axis. This estimate for w i l l be termed -JQ^ - a second estimate for J ^ , which w i l l be referred to as JQ2> w a s obtained by computing the average of the quantity J exp(-qV /kT) sc ^ oc over a l l points i n the range s p e c i f i e d above. In e f f e c t , i n the computa-129 tion of A was assumed to be exactly equal to one. The values of A, Jg^ and r o r each characteristic shown in Fig. 4.3 are lis t e d in Table 4.1. The data presented in Table 4.1 represent the averaging of some 20-25 individual J -V measurements at each temperature. sc oc After correction for the T p factor appearing in (4.11), J Q ^ and were graphed as a function of reciprocal temperature: These Arrhenius plots are shown in Fig. 4.4. Fig. 4.4(a) reveals that the plot of 2 A l o g [ ( 3 0 0 / T ) " ] versus reciprocal temperature is fit t e d almost per-fectly by a straight line. Using the method of least squares, the slope of this line was computed, and from the slope the corresponding activation energy was calculated. The value of E A was found to be (1.19±.013)eV, which agrees almost exactly with the best available estimates for E _. g0 From Fig. 4.4(b) i t can be seen that not a l l points in the graph of 2 A log[Jg^(300/T) ' ] versus reciprocal temperature are co-linear. However, the data points corresponding to the four highest temperatures at which J -V characteristics were recorded do l i e very nearly on a line. Once sc oc again, the method of least squares was used to compute the slope of this line, and the appropriate activation energy. In this case E^ was found to be (1.14±.02)eV which, although somewhat less than E gQ» 1 S s t i l l larger than the s i l i c o n bandgap energy at room temperature. It should be noted that the experimental J Q data agree very well 15 -3 with an electron lifetime of 10 usee and a substrate doping of 10 cm , which are reasonable estimates for these quantities in 10 ftcm material. 15 -3 2 -1 -1 Taking T =10 usee, N. = 10 cm , y = 0.13 m V s and ° n A n n ± = 1.45*1010 cm"3 at 300°K, (4.3) predicts J Q d = 6*10"8 mA/cm2. From Fig. 4.4(a), i t can be seen that the experimental value of at 300°K —8 2 is about 5*10 mA/cm . TABLE A . l Values of A, J Q 1 and J Q 2 corresponding to the c h a r a c t e r i s t i c s of F i g . 4.3 1- (°C) A: l o g 1 0 [ J 0 1 ( 3 0 0 / T ) 2 - 4 ] : + l o g 1 0 [ J Q 2 ( 3 0 0 / T ) 2 * 4 ] -0.2 1.0661.005 -7.7081.047 -8.3221.019 9.8 1.048+.006 -7.1451.058 -7.5621.018 20.0 1.0371.006 -6.5111.048 -6.8131.013 27.9 1.0331.007 -6.0331.056 -6.2811.014 39.0 1.039+.004 -5.3041.028 -5.5681.014 49.6 1.0341.004 -4.7261.025 -4.9381.013 Here T i s measured i n °K, J n i n Am Figure 4.4(a) Temperature dependence of J 132 Figure 4.4(b) Temperature dependence of J 133 From the above r e s u l t s i t can be concluded that for current d e n s i t i e s 2 above about 1 mA/cm , the dark current i n the Al-SiO -pSi diode whose x c h a r a c t e r i s t i c s are shown i n F i g . 4.3 i s c a r r i e d almost e x c l u s i v e l y by electrons i n j e c t e d from the A l contact d i f f u s i n g into the quasi-neutral base. This demonstrates that i t i s possible to form minority c a r r i e r MIS diodes on p-type s i l i c o n substrates of r e s i s t i v i t y greater than or approx-imately equal to 10 ficm. However, th i s experiment does not guarantee that minMIS devices can be formed on more heavily doped substrates. 4.3 MIS Solar Cel l s with Back Surface F i e l d s In the previous section i t was shown that Al-SiO -pSi s o l a r c e l l s x can be produced i n which the open-circuit voltage i s l i m i t e d by recombi-nation i n the quasi-neutral base. I t follows that further increases i n the open-circuit voltage of these MIS c e l l s can be achieved only by reducing the magnitude of the electron i n j e c t i o n - d i f f u s i o n dark current flowing into the base. As outlined i n Section 2.3, t h i s can be done e i t h e r by increasing the base doping or, i f the electron d i f f u s i o n length i n the base i s greater than the base width, by reducing the back surface recom-bination v e l o c i t y through the use of a back surface f i e l d (BSF). In Sec-t i o n 4.1, the a p p l i c a t i o n of the former technique to obtain open-circuit voltages exceeding 600 mV i n Al-SiO -pSi [35,103] and Cr-SiO -pSi [94] X X c e l l s was mentioned. In t h i s section the f i r s t use of a BSF region to enhance the open-circuit voltage of MIS s o l a r c e l l s i s described [21]. This experiment provides p a r t i c u l a r l y convincing a d d i t i o n a l evidence for the existence of minority c a r r i e r MIS diodes. The back surface f i e l d substrates used i n the experiments described i n t h i s section were prepared at Applied Solar Energy Corporation (ASEC) following procedures currently employed i n the commercial production of 134 back surface f i e l d space c e l l s [134]. The s t a r t i n g material was boron-doped Czochralski s i l i c o n of 9 to 11 ftcm r e s i s t i v i t y and <100> o r i e n t a -t i o n . Measurements ca r r i e d out at ASEC using the surface photovoltage decay technique have shown that the electron d i f f u s i o n length i n t h i s material i s roughly 800 ym. Substrate thicknesses ranged from 350 to 450 ym, while the substrate surfaces were prepared by chemical p o l i s h i n g . The BSF region i t s e l f was formed using the aluminum paste a l l o y i n g tech-nique described i n Section 2.3. In order to compare the properties of c e l l s with and without back surface f i e l d s , a number of substrates were l e f t without BSF regions. At the University of B r i t i s h Columbia, Al-SiO -pSi front junctions were applied to the ASEC substrates following the process described i n Appendix C. The c e l l geometry i s i l l u s t r a t e d i n F i g . 3.6(a). The semi-transparent b a r r i e r metal layer and the thick, comb-like contact g r i d o verlying t h i s layer were both formed by the thermal evaporation of a l u -minum. The b a r r i e r metal layer was t y p i c a l l y made 80 A thick. To minimize c e l l s e r i e s resistance, the contact g r i d should be made as thick as pos-s i b l e . Here l i m i t a t i o n s on the amount of aluminum which could be deposited i n a s i n g l e pump-down cycle of the evaporation system r e s t r i c t e d the g r i d thickness to roughly 1 ym. The contact g r i d was defined using a metal shadow mask, giving a finger spacing of roughly 1.2 mm and a g r i d cover-2 age of approximately 25%. T o t a l device areas ranged from 1.5 to 3 cm . No a n t i r e f l e c t i o n coating was applied to the c e l l s . The c h a r a c t e r i s t i c s of the f i n i s h e d MIS c e l l s were measured under i l l u m i n a t i o n from an ELH lamp, which consists of a tungsten-halogen bulb equipped with a d i c h r o i d r e f l e c t o r . Although ELH lamps have a s p e c t r a l luminosity which i s s h i f t e d towards near-infrared wavelengths compared 135 / to natural sunlight, these lamps are widely used as crude simulators of AMI i l l u m i n a t i o n [138]. o An uncoated aluminum layer 80 A thick deposited on a s i l i c o n sub-str a t e can be expected to transmit only about 30% of incident l i g h t at v i s i b l e wavelengths [93]. As a r e s u l t , under simulated one-sun i l l u m i n a -t i o n the s h o r t - c i r c u i t current density of the Al-SiO -pSi c e l l s was only 2 about 10 mA/cm . For purposes of comparison, a modern commercial d i f f u s e d -junction c e l l would be expected to give a photocurrent density somewhat 2 greater than 30 mA/cm under one-sun AMI i l l u m i n a t i o n . Since the e l e c -t r i c a l rather than the o p t i c a l properties of the MIS junction were of primary i n t e r e s t here, the l i g h t l e v e l to which the MIS c e l l s were exposed 2 was therefore increased to give a photocurrent density of 30 mA/cm i n each device tested. The f a b r i c a t i o n of Al-SiO -pSi s o l a r c e l l s on ASEC BSF substrates x was f i r s t c a r r i e d out i n early 1979. At that time MIS junctions were applied to pair s of substrates, with each pair c o n s i s t i n g of one s l i c e with an alloyed aluminum back surface f i e l d and a second s l i c e lacking a BSF region. The t h i n i n t e r f a c i a l oxide layer was grown by exposing the substrates to a dry oxygen flow at 500°C for 20 minutes. A t o t a l of s i x pair s of devices was fabricated. When illuminated to give a photocurrent 2 density of 30 mA/cm , the mean open-circuit voltage of the BSF c e l l s was found to be nearly 40 mV higher than that of the c e l l s lacking back sur-face f i e l d s . However, the mean V value f o r the MIS c e l l s with back oc surface f i e l d s was s t i l l 15 mV less than that recorded for a control group of four ASEC N +PP + c e l l s f a b r i c a t e d on substrates cut from the same ingot. A more complete d e s c r i p t i o n of these results i s given i n Ref. [21]. By the f a l l of 1980, improvements i n both the process used to produce 136 MIS junctions at UBC and i n the qu a l i t y of BSF regions formed at ASEC made the f a b r i c a t i o n of a second set of Al-SiO -pSi s o l a r c e l l s on back x surface f i e l d substrates worthwhile. For t h i s second f a b r i c a t i o n sequence the oxidation temperature was rai s e d to 600°C and the oxidation time extended to 30 minutes. As a r e s u l t , s l i g h t l y thicker oxide layers should have grown than were present i n the devices made e a r l i e r . The open-cir-c u i t voltages of the two devices produced i n the course of the second f a b r i c a t i o n sequence are l i s t e d i n the f i r s t column of Table 4.2. As usual, these open-circuit voltages were measured at a photocurrent den-2 s i t y of 30 mA/cm and at a cont r o l l e d temperature of 28°C. V values oc for a con t r o l group of eight ASEC di f f u s e d junction space c e l l s fabricated on i d e n t i c a l 10 ficm substrates were also recorded at t h i s time, under the same test conditions. This c o n t r o l group included s i x N +PP + back surface f i e l d c e l l s and two N +P c e l l s with ohmic back contacts. The open-circuit voltages of the two MIS c e l l s compare very favour-ably with those of the N +PP + control c e l l s , which were found to range from 579 to 598 mV with a mean of 588 mV. In contrast, the open-circuit voltages of the two N +P c e l l s with ohmic back contacts were found to be 546 and 547 mV. Despite the use of a 600°C oxidation c y c l e , the f i l l factors of the Al-SiO -pSi back surface f i e l d c e l l s produced i n the second f a b r i c a t i o n x sequence were found to be among the highest for any devices fabricated during the course of t h i s research program. When illuminated to y i e l d a 2 photocurrent density of 30 mA/cm , both c e l l s gave f i l l f actors of 0.70. To demonstrate conclusively that the alloyed aluminum back surface f i e l d region was responsible f o r the high V q c values quoted f o r the MIS c e l l s , t h i s region was removed and replaced with an ohmic back contact TABLE 4.2 Open-circuit voltages (mV) for A l - S i O x - p S i c e l l s , 2 measured at T = 28°C and J =30 mA/cm sc C e l l : With alloyed aluminum BSF: With Pd back contact #1 594 539 #2 593 543 138 using only room temperature processing. Extreme care was taken not to damage the front junction during t h i s step. F i r s t the front side of each c e l l was covered with a commercial e t c h - r e s i s t a n t adhesive tape. The c e l l s were then dipped i n 10% HF for a few minutes to remove a l l traces of the aluminum back contact m e t a l l i z a t i o n . Next each c e l l was immersed i n an agitated s o l u t i o n of one part 49% HF to nine parts 70% HNO^ (so-c a l l e d "white etch") f o r a period of one minute, to remove approximately 10 um of material from the back of the s l i c e . The c e l l s were thoroughly washed, and then re-immersed i n d i l u t e HF to ensure removal of a l l traces of oxide from the back surface. Following a second washing, the tape was peeled away. F i n a l l y , a layer of palladium a few hundred angstroms thick was deposited on the back of the s l i c e s by f l a s h evaporation from a tung-sten filament to form a new rear contact. The substrates were only exposed to the hot filament f o r a few seconds during t h i s evaporation, so heating of the c e l l s should have been minimal. Palladium contacts to p-type s i l -i con formed i n t h i s way have been found to give ohmic c h a r a c t e r i s t i c s 2 out to current densities of more than 100 mA/cm , with resistances of 2 less than 0.1 ficm . (In a subsiduary experiment, the current-voltage c h a r a c t e r i s t i c s of palladium contacts to freshly-etched p-type s i l i c o n were measured by applying such contacts to the front surface of 2 Qcm s l i c e s with s i n t e r e d aluminum back contacts). The open-circuit voltages recorded f o r the two MIS c e l l s following removal of the back surface f i e l d region are l i s t e d i n the second column of Table 4.2. For both c e l l s , the V value measured a f t e r reprocessing oc i s seen to be at least 50 mV less than that recorded when the back sur-face f i e l d region was i n t a c t . As expected, the open-circuit voltages for the reprocessed c e l l s are very close to those of the N P control c e l l s 139 with ohmic back contacts. The fact that a modification to the back sur-face of these MIS c e l l s could reduce t h e i r open-circuit voltages by a f u l l 50 mV demonstrates conclusively that the dark current i n these de-v i c e s i s dominated by minority c a r r i e r i n j e c t i o n - d i f f u s i o n . To ensure that the reduction i n open-circuit voltage brought about on replacement of the back surface f i e l d region did not r e s u l t from damage to the MIS front j u n c t i o n , two Al-SiC^-pSi c o n t r o l c e l l s were c a r r i e d through the back contact r e f a b r i c a t i o n procedure outlined above along with each MIS back surface f i e l d c e l l . These co n t r o l c e l l s had been fab-r i c a t e d on 2 ftcm substrates with ohmic back contacts, and o r i g i n a l l y gave open-circuit voltages ranging from 558 to 572 mV. In no case did the open-circuit voltage of any c o n t r o l c e l l drop by more than 2 mV on replacement of the back contact. It can thus be concluded that the pro-cesses of taping, etching, tape removal and palladium deposition described above do not s i g n i f i c a n t l y a l t e r the properties of an MIS front junction. As pointed out i n Section 2.3, a back surface f i e l d region should provide a small increase i n photocurrent response at long wavelengths. To check for the presence of t h i s e f f e c t , the s h o r t - c i r c u i t photocurrent of one of the MIS back surface f i e l d c e l l s was recorded under i n f r a r e d i l l u m i n a t i o n . (A source r i c h i n i n f r a r e d was obtained by simply operating an u n f i l t e r e d tungsten-halogen lamp at one-third i t s rated voltage; the l i g h t i n t e n s i t y was monitored and held constant to within ±1% using a reference s i l i c o n photodiode). With the back surface f i e l d region i n t a c t , the i n f r a r e d photocurrent f o r t h i s c e l l was found to be 2.22 mA. Following replacement of the back surface f i e l d with a palladium ohmic contact, the photocurrent under the same i l l u m i n a t i o n conditions dropped to 1.75 mA. 140 4.4 V a r i a t i o n of MIS Solar C e l l C h a r a c t e r i s t i c s with Insulator Thickness In Section 3.5, i t was predicted that MIS s o l a r c e l l s with thick i n s u l a t i n g layers would have illuminated current-voltage c h a r a c t e r i s t i c s which were concave-upwards over a c e r t a i n bias range, as i l l u s t r a t e d i n Fi g . 3.8. Further, i t was noted that the numerical analysis c a r r i e d out by Shewchun ejt a l . [25] predicts f i l l f a c t ors less than 0.25 f o r MIS c e l l s with very thick i n s u l a t o r s . In contrast, conventional homojunction c e l l s always have illuminated current-voltage c h a r a c t e r i s t i c s which are every-where concave downwards, and thus must have f i l l factors greater than 0.25. Although MIS c e l l s with i n s u l a t o r s thick enough to cause photocur-rent suppression are c l e a r l y of l i t t l e use for photovoltaic energy con-version, the observation of an illuminated current-voltage c h a r a c t e r i s t i c resembling F i g . 3.8 i n an experimental c e l l would help confirm the v a l -i d i t y of the s e m i c l a s s i c a l model of the MIS j u n c t i o n . In order to produce MIS c e l l s with thick i n s u l a t o r s , a group of Al-SiO -pSi devices were fabricated i n accordance with the procedure x outlined i n Section 4.1 and i n Appendix C, with the exception that o x i -dation temperatures ranging from 600°C to 660°C were used. The oxidation time was held constant at 30 minutes. The substrates selected for t h i s experiment were of 2 ficm r e s i s t i v i t y and <100> or i e n t a t i o n , and had chem-mechanically polished front surfaces. Following oxidation, a thick layer of aluminum was deposited on the back of the wafers to form an ohmic contact. The contact was then sin t e r e d at a temperature of 500°C for 30 minutes i n a nitrogen atmosphere. F i n a l l y , the semi-transparent bar-r i e r layer and thick contact g r i d were deposited, g i v i n g a t o t a l junction 2 area of 2 cm and a g r i d coverage of about 25%. The illuminated current-voltage c h a r a c t e r i s t i c s of the completed Al-SiO^-pSi c e l l s are shown i n F i g . 4.5. As usual, these c h a r a c t e r i s t i c s were recorded at a temperature of 28°C under i l l u m i n a t i o n s u f f i c i e n t to 2 give a photocurrent density of 30 mA/cm . The c h a r a c t e r i s t i c s of those c e l l s f a b r i c a t e d using oxidation temperatures of 650°C or greater are seen to be concave upwards over a l l or part of the power-output quadrant, as predicted i n Section 3.5. Further, the device oxidized at 660°C has a f i l l f a c tor of less than 0.25, as predicted by Shewchun et a l . MIS c e l l s with f i l l f actors less than 0.25 have also been fabricated by Card [79] and by St. Pierre et a l . [103]. 142 Figure 4.5 Illuminated current-voltage c h a r a c t e r i s t i c s for Al-SiO -pSi s o l a r c e l l s with various i n s u l a t o r thick-x nesses, as measured experimentally. Oxidation temper-ature i s s p e c i f i e d f o r each c h a r a c t e r i s t i c . J =30mA/cm2, T=28°C. upc 143 CHAPTER 5 MINORITY CARRIER REFLECTING NEGATIVE BARRIER MIS CONTACTS i According to the d e f i n i t i o n given i n Chapter 1, i n a negative b a r r i e r Schottky or MIS junction the semiconductor surface i s accumulated at equilibrium. Junctions of t h i s kind can be formed by depositing a low work function metal on an n-type substrate, or a high work function metal on a p-type substrate, provided that the surface state density has been reduced to n e g l i g i b l e l e v e l s . In p r a c t i c e , strong accumulation can only be achieved i n MIS junctions, where the growth of the t h i n i n t e r f a c i a l layer passivates the semiconductor surface (see Section 3.1). It has long been known that i t should be possible to use negative b a r r i e r Schottky junctions to form low resistance metal-semiconductor contacts [34]. However, i t i s only recently, and only through the use of the MIS tunnel junction technology, that low resistance negative b a r r i e r contacts to s i l i c o n have been s u c c e s s f u l l y fabricated [112]. In 1976 Green, Godfrey and Davies recognized that with a correct s e l e c t i o n of b a r r i e r metal and i n s u l a t o r thickness, i t should be possible to form negative b a r r i e r MIS contacts which would not only o f f e r n e g l i g -i b l e impedance to the flow of majority c a r r i e r s , but which would r e f l e c t minority c a r r i e r s i n the same manner as a m e t a l l u r g i c a l high-low junction [35]. A negative b a r r i e r junction of t h i s type could thus be used as a rear contact i n an induced back surface f i e l d s o l a r c e l l . Although Green et a l did succeed i n f a b r i c a t i n g low resistance negative b a r r i e r MIS contacts to both n- and p-type s i l i c o n , they were unable to produce con-tacts with demonstrable minority c a r r i e r r e f l e c t i n g properties [36]. In the f i r s t section of t h i s chapter, an approximate a n a l y t i c model of current flow i n the negative b a r r i e r MIS contact i s developed by a 144 simple extension of the r e s u l t s obtained i n Chapter 3 [37]. (The only other t h e o r e t i c a l study of minority c a r r i e r r e f l e c t i o n at the negative b a r r i e r MIS junction so f a r reported [35] was undertaken by Green et_ a l . using numerical a n a l y s i s . The r e s u l t s of t h i s i n v e s t i g a t i o n have never been published.) The remainder of the chapter i s devoted to the e x p e r i -mental demonstration of the minority c a r r i e r r e f l e c t i n g properties of the negative b a r r i e r j u n c t i o n . In Section 5.2, the use of negative b a r r i e r Mg-SiO^-nSi contacts to form induced back surface f i e l d regions for P +N c e l l s with d i f f u s e d front junctions i s discussed [37], Section 5.3 then considers the formation of minority c a r r i e r r e f l e c t i n g MIS contacts on p-type substrates using platinum as the b a r r i e r metal. Induced back surface f i e l d a c tion was obtained by applying Pt-SiO -pSi back contacts X to both N +P d i f f u s e d front junction c e l l s , and to c e l l s with Al-SiO^-pSi minMIS front junctions. Although the y i e l d of good devices i n a l l the experiments mentioned above was never high, the open-circuit voltages of the best induced back surface f i e l d c e l l s were i n every case comparable to those obtained with conventional d i f f u s e d or alloyed back surface f i e l d s . 5.1 Current Flow i n the Negative B a r r i e r MIS Junction: Theory For notational convenience, i t w i l l be assumed that the negative b a r r i e r MIS junctions considered i n t h i s section have been formed on p-type substrates. However, the conclusions drawn here apply equally w e l l to n-type material i f the roles of electrons and holes are interchanged. The band diagram f o r a negative b a r r i e r MIS contact formed on a p-type substrate i s shown i n F i g . 5.1(a). This band diagram represents a non-equilibrium s i t u a t i o n i n which excess electrons have been introduced i n t o the semiconductor e i t h e r through i l l u m i n a t i o n or by i n j e c t i o n from M I T ' / / pSi ACCUMULATION X M XS F n - E M I '// DEPLETION nSi INVERSION X M XS F n - E , (a) (b) ure 5.1 Band diagrams for (a) a negative b a r r i e r MIS j u n c t i o n , and (b) the corresponding p o s i t i v e b a r r i e r MIS junction formed by depositing the same metal on a substrate of the opposite doping type. Note the s i m i l a r i t y between the semiconductor surface regions of the two structures. Both diagrams represent non-equilibrium s i t u a t i o n s . a P N junction to the r i g h t . F i g . 5.1(a) i s i n many respects s i m i l a r to the band diagram for the corresponding p o s i t i v e b a r r i e r junction formed by depositing the same metal on a substrate of the opposite doping type, which i s shown i n F i g . 5.1(b). This observation i s the key to the analysis of current flow i n the negative b a r r i e r junction. In p a r t i c u l a r , the band structure of the accumulation layer - i n s u l a t o r - metal region i n F i g . 5.1(a) i s nearly i d e n t i c a l to that of the inversion layer - i n s u l a t o r - metal region i n F i g . 5.1(b). Differences between the two band structures can r e s u l t only from the storage of charge i n the depletion region of the p o s i t i v e b a r r i e r junction. Provided the semiconductor surface i s strongly inverted i n one case and strongly accumulated i n the other, these d i f -ferences should be small. I f a negative b a r r i e r MIS junction i s to provide e s s e n t i a l l y no impedance to the flow of majority c a r r i e r s between the metal and the semiconductor, then the i n t e r f a c i a l i n s u l a t o r must be s u f f i c i e n t l y t h i n that the majority c a r r i e r quasi-fermi l e v e l at the semiconductor surface i s e f f e c t i v e l y pinned to the fermi l e v e l i n the metal under normal oper-ating conditions. Further, since the hole concentration i n the accumu-l a t i o n layer i s extremely large, and since t h i s region can be no more than a few hundred angstroms thick, must be very nearly constant across the accumulation l a y e r . It follows that over the normal operating range both the hole d i s t r i b u t i o n and the e l e c t r o s t a t i c p o t e n t i a l d i s t r i b u t i o n i n the junction region must remain "frozen" i n t h e i r e q uilibrium forms. The energy difference between the conduction band edge at the semicon-ductor surface and the fermi l e v e l i n the metal must therefore be f i x e d at i t s e q u i l i b r i u m value, which, as i n the case of the p o s i t i v e b a r r i e r junction, w i l l be denoted as qi})^. In drawing F i g . 5.1(a), i t has been assumed that the electron quasi fermi l e v e l i s constant across the accumulation layer. As usual, a f t e r computing the minority c a r r i e r flow through the junction, the accuracy of t h i s assumption can be checked using the methods of Section 2.2. From the r e s u l t s of Chapter 3, the net electron current flowing between the conduction band and the metal i s given by JCM " 9CM A e T ' - P [ - ( E C ( X S ) - E F n ( x s ) ) / k T ] (5.1) (1 - e x p [ - ( E p n ( x s ) - E ^ / k T ] ) For the s i t u a t i o n s of i n t e r e s t here, E F n ( x g ) l i e s above E ? M by many multiples of kT, allowing the rightmost exponential term i n (5.1) to be set equal to zero. Therefore JCM = 9CM A e t 2 expC-q'f'Bn/kT) exp(qA<j)/kT) (5.2) where qAd> i s now defined to be the separation between E_ and E„ i n the Fn Fp junction region (Ad) w i l l be taken as p o s i t i v e when E ^ l i e s above Ep.p) • Since the e l e c t r o n concentration n (x ) at the edge of the accumulation P A region i s r e l a t e d to the thermal e q u i l i b r i u m electron concentration n p ^ i n the quasi-neutral base by n p ( x A ) = n p Q exp(qA<f,/kT) , (5.3) (5.2) becomes 148 CM 6CM Ae T ' ex P(-q* B n/kT) n pO (5.4) (5.4) can be written as J C M = q S e f f np ( xA> (5.5a) where S e f f = 9 C M A e T " exp(-qVkT) q n pO (5.5b) Thus so f a r as minority c a r r i e r flows are concerned, the negative b a r r i e r MIS contact can be described by an e f f e c t i v e surface recombination veloc-i t y S e ^ . More s p e c i f i c a l l y , the d i s t r i b u t i o n of minority c a r r i e r s i n the quasi-neutral base would be unchanged i f the negative b a r r i e r contact were replaced by a surface with recombination v e l o c i t y ^ ^ positioned at the edge of the accumulation l a y e r . The quantity S defined i n (5.5b) i s thus completely analogous to the e f f e c t i v e surface recombination v e l o c i t y used to describe the current flow i n s o l a r c e l l s with d i f f u s e d back surface f i e l d s (see Section 2.3). From (5.5) i t can be seen that the e f f e c t i v e surface recombination v e l o c i t y at a negative b a r r i e r contact can be reduced by s e l e c t i n g the b a r r i e r metal work function to maximize ty , or by increasing the oxide DTI thickness i n order to reduce the t u n n e l l i n g p r o b a b i l i t y factor 6^. However, i f the oxide i s made too thick the impedance of the contact to majority c a r r i e r flow may become excessive. If the negative b a r r i e r junction i s to be used to form an induced back surface f i e l d s o l a r c e l l , 149 a majority c a r r i e r current equal i n magnitude to the one-sun photocurrent must be able to pass between the metal and the semiconductor without s i g n i f i c a n t displacement of (x c) from E ^ . r p b r r l In deriving (5.5), no allowance was made for the recombination of electrons with holes through the process of c a r r i e r trapping by surface st a t e s . In general, i f such processes are possible the electron current flowing i n t o the junction region w i l l be greater than that s p e c i f i e d by (5.5). Thus (5.5b) gives a lower bound on the e f f e c t i v e surface recombi-nation v e l o c i t y of a negative b a r r i e r MIS contact. A p a r t i c u l a r l y simple r e l a t i o n s h i p e x i s t s between the e f f e c t i v e surface recombination v e l o c i t y f or a negative b a r r i e r MIS contact defined above and the value of J r t_, associated with the electron therm-UTn i o n i c emission current i n the corresponding p o s i t i v e b a r r i e r junction. From (5.5b) and (4.2), S e f f = J0Th / n P0 • <5'6> (5.6) emphasizes that the problem of minimizing the e f f e c t i v e surface recombination v e l o c i t y i n a negative b a r r i e r MIS contact i s equivalent to the problem of minimizing the majority c a r r i e r thermionic emission current i n the corresponding p o s i t i v e b a r r i e r junction. In the case of the p o s i t i v e b a r r i e r device, increasing the i n s u l a t o r thickness reduces the majority c a r r i e r current, but also lowers the maximum minority car-r i e r current which can be i n j e c t e d before the tunnel l i m i t e d regime i s entered. S i m i l a r l y , i n the negative b a r r i e r contact an increase i n the in s u l a t o r thickness lowers while simultaneously reducing the maximum majority c a r r i e r current which can be passed through the contact before tunnel resistance e f f e c t s become appreciable. 5.2 Induced Back Surface F i e l d Solar C e l l s on n - S i l i c o n Substrates When the experiments on negative b a r r i e r MIS junctions described i n t h i s chapter were begun i n the f a l l of 1979, a s u b s t a n t i a l body of e v i -dence had been published i n d i c a t i n g that minority c a r r i e r MIS diodes could be formed by depositing aluminum or magnesium on p-type s i l i c o n substrates [20,21,19]. In contrast, i n only one instance had an open-c i r c u i t voltage exceeding 450 mV been reported for an MIS s o l a r c e l l formed on n-type s i l i c o n [19]. The low open-circuit voltages obtained for these MIS c e l l s suggested that i t would be d i f f i c u l t , i f not impossible, to form a minority c a r r i e r MIS diode on n-type s i l i c o n . For t h i s reason i t was decided that the f i r s t attempt to fabricate a minority c a r r i e r r e f l e c t i n g negative b a r r i e r MIS contact should be made using an n-type substrate. Furthermore, i t was decided that the most convincing evidence for the minority c a r r i e r r e f l e c t i n g properties of the negative b a r r i e r junction could be obtained by employing t h i s structure as the back con-tact i n an induced back surface f i e l d s o l a r c e l l . To determine i f a negative b a r r i e r MIS back contact could enhance s o l a r c e l l open-circuit voltage i n the same manner as a d i f f u s e d back surface f i e l d , a group of 2cm X 2cm P +NN + BSF c e l l s f a b r i c a t e d on 10 ticm substrates of <100> or i e n t a t i o n were obtained from Applied Solar Energy Corporation. These c e l l s had been prepared i n accordance with current i n d u s t r i a l procedures. Four of the c e l l s from t h i s group were retained for use as controls. The front surfaces of the remaining seven c e l l s were then protected with tape, a f t e r which the d i f f u s e d N + back surface f i e l d region was etched away. Any traces of native oxide l e f t on the back 151 surface a f t e r t h i s etching process were removed by immersing the s l i c e s i n 10% HF for one minute. The s l i c e s were then washed thoroughly, and the protective tape was peeled away. The samples were next oxidized i n dry oxygen at 500°C for 20 minutes. Following oxidation, a thick layer of magnesium was deposited on the back of the s l i c e s by thermal evapora-tion to form the negative b a r r i e r MIS junct i o n . Since magnesium oxidizes r a p i d l y on exposure to the atmosphere, t h i s magnesium layer was immedi-ately o v e r l a i d with a protective coating of thermally evaporated alumi-num. The structure of the completed c e l l s i s i l l u s t r a t e d i n F i g . 5.2. The current-voltage c h a r a c t e r i s t i c s of the completed MIS back contact c e l l s were measured under simulated one-sun AMI i l l u m i n a t i o n at a c o n t r o l -led temperature of 28°C. As usual, a f i l t e r e d tungsten-halogen bulb was used as the AMI simulator. The s h o r t - c i r c u i t current density was 2 found to be 34 mA/cm i n a l l c e l l s tested. Open-circuit voltages were found to range from 565 to 583 mV. The V values for the four best & oc devices are l i s t e d i n the f i r s t column of Table 5.1. Under i d e n t i c a l operating conditions, the open-circuit voltages for the P +NN + c o n t r o l c e l l s ranged from 574 to 583 mV. It follows that the negative b a r r i e r MIS contact i s capable of producing an e f f e c t i v e surface recombination v e l o c i t y for minority c a r r i e r s which i s as low as can be achieved using a conventional d i f f u s e d back surface f i e l d . However, i t i s also apparent that some of the MIS contacts had r e l a t i v e l y high e f f e c t i v e surface recombination v e l o c i t i e s . This inconsistent performance of the negative b a r r i e r contact has been t e n t a t i v e l y a t t r i b u t e d to va r i a t i o n s i n the roughness and degree of contamination of the etched back surfaces. It i s expected that a higher y i e l d of minority c a r r i e r r e f l e c t i n g MIS contacts would be obtained i f more attention were paid to achieving 152 A A T i / A g H O O O A / l u m ) p Si (^0.5ym) n Si (^350um) S i O x (^20A) Mg (^5000A) Figure 5.2 Structure of P N c e l l with negative b a r r i e r Mg-SiO -nSi back contact. TABLE 5.1 Open-circuit voltages (mV) for selected P N c e l l s with negative b a r r i e r MIS back contacts a b C e l l : 1st f a b r i c a t i o n : 2nd f a b r i c a t i o n : 3rd f a b r i c a t i o n P+NIM#1 582 547 585 P+NIM#2 583 549 577 P+NIM#3 581 545 P+NIM#4 570 546 Oxidation/anneal at 500 °C i n 0 2 for 20 mins; Mg b a r r i e r Anneal at 450 °C i n N for 5 mins; A l b a r r i e r 154 a high surface f i n i s h i n the etching process, and i f the s l i c e s were subjected to the f u l l cleaning procedure s p e c i f i e d i n Appendix C p r i o r to f a b r i c a t i o n of the back junc t i o n . (The tape used here to protect the front contact m e t a l l i z a t i o n could not withstand t h i s f u l l cleaning pro-cedure, so the s l i c e s were simply dipped i n 10% HF p r i o r to oxidation). It i s , of course, possible to form negative b a r r i e r back contacts on P +N c e l l s which have never undergone a back surface f i e l d d i f f u s i o n . However, i t i s u n l i k e l y that c e l l s formed i n t h i s way would give open-c i r c u i t voltages s i g n i f i c a n t l y greater than those of P +N c e l l s with ohmic back contacts. In general, the hole d i f f u s i o n lengths i n as-grown n-type material are too short f o r a back surface f i e l d to be of use i n enhancing the open-circuit voltage of a standard 300 ym thick s o l a r c e l l f a b r i c a t e d on such material. The high open-circuit voltages'recorded for P+NN"'" c e l l s are thought to be made possible by the gettering action of the phosphorus d i f f u s i o n used to form the N + back surface f i e l d region [135]. This gettering action i s believed to raise the hole d i f f u s i o n length to a point where the minority c a r r i e r r e f l e c t i n g properties of the back surface f i e l d can a s s i s t i n reducing the hole i n j e c t i o n - d i f f u s i o n dark current component. In order to ensure that the N + BSF d i f f u s i o n had been completely removed from the substrates used i n the experiments described above, the MIS back contact m e t a l l i z a t i o n and thin oxide were stripped from a l l seven devices. This was accomplished by applying protective tape to the front side of the c e l l s , and then immersing the substrates i n 10% HF for a few minutes. Following washing and tape removal, the s l i c e s were exposed to a dry nitrogen flow at 450°C for f i v e minutes. New MIS back contacts were then formed by the deposition of thermally evaporated 155 aluminum. When applied to p-type substrates, t h i s f a b r i c a t i o n procedure has been found to y i e l d Al-SiO^-pSi s o l a r c e l l s with very low open-circuit voltages, demonstrating the presence of a large majority c a r r i e r therm-i o n i c emission dark current component. The process would therefore be expected to give high e f f e c t i v e surface recombination v e l o c i t i e s when used to form negative b a r r i e r contacts on n-type material. The open-circuit voltages of the reprocessed c e l l s ranged from 544 to 549 mV (see the second column of Table 5.1). For purposes of compari-son, experiments conducted at Applied Solar Energy Corporation have shown that V values between 545 and 550 mV are obtained when the N + layers oc are removed from P +NN + c e l l s and replaced with conventional sintered ohmic contacts. This confirms that the high V 's recorded for the best oc MIS back contact devices i n the f i r s t stage of the experiment did not r e s u l t from the presence of r e s i d u a l traces of the o r i g i n a l N + back surface f i e l d d i f f u s i o n . To provide f i n a l confirmation of the experimental r e s u l t s and to check that the low V 's recorded a f t e r the second MIS back contact fab-oc r i c a t i o n were not caused by spurious e f f e c t s such as the degradation of the front contact or the junction during reprocessing, the MIS back con-tact m e t a l l i z a t i o n and oxide were stripped from the two best c e l l s yet again. These c e l l s were then reprocessed following the procedure used i n the f i r s t stage of the experiment (that i s , the substrates were oxidized f o r 20 minutes at 500°C, and magnesium was used f o r the b a r r i e r metal). The V values obtained a f t e r t h i s t h i r d f a b r i c a t i o n sequence are l i s t e d oc i n the t h i r d column of Table 5.1, and i t can be seen that they agree c l o s e l y with those recorded a f t e r the f i r s t MIS contact f a b r i c a t i o n (column 1). 156 As noted i n Section 5.1, i n order for a negative b a r r i e r MIS junction to be of use i n forming a p r a c t i c a l induced back surface f i e l d s o l a r c e l l , i t i s e s s e n t i a l that the resistance of the junction to majority c a r r i e r flow be no greater than that of a conventional sintered contact. As yet, no formal experiments have been c a r r i e d out here to determine the e f f e c -t i v e resistance of the Mg-SiO -nSi and Al-SiO -nSi negative b a r r i e r junc-X X tions. However, measurements of the f i l l factors of the induced back surface f i e l d s o l a r c e l l s described above indicate that t h i s e f f e c t i v e contact resistance i s not excessive. The f i l l factors of these devices .were found to l i e i n the range 0.6 - 0.65, except i n those cases i n which part of the front contact m e t a l l i z a t i o n was damaged during processing. Further experimentation w i l l be required to determine i f the s e r i e s resistance associated with the negative b a r r i e r MIS contact can be made less than or equal to that of a d i f f u s e d back surface f i e l d with a s i n -tered contact. 5.3 Induced Back Surface F i e l d Solar C e l l s on p - S i l i c o n Substrates Following the successful production of minority c a r r i e r r e f l e c t i n g negative b a r r i e r MIS contacts to n-type substrates, attention was turned to f a b r i c a t i n g s i m i l a r contacts to p-type material. Before t h i s could be done, i t was f i r s t necessary to develop a technique for forming minority c a r r i e r MIS diodes on n-type s i l i c o n . As noted i n the previous s e c t i o n , the low open-circuit voltages reported by other investigators for MIS s o l a r c e l l s fabricated on n-type s i l i c o n [19] suggested that there would be l i t t l e chance of success i n t h i s task. In Chapter 3 i t was pointed out that the chances of forming a minor-i t y c a r r i e r MIS diode on n-type material are greatest when using a b a r r i e r 157 metal with the highest possible work function. From the values of vacuum work function f o r the elements tabulated by Sze [113], i t can be seen that the metals platinum, i r r i d i u m , rhenium and palladium have the highest work functions. Of these metals, palladium and platinum are the most re a d i l y a v a i l a b l e , and the ea s i e s t to obtain i n a form suitable f o r ther-mal evaporation. 5.3.1 minMIS Diodes on n - S i l i c o n Substrates In the f i r s t stage of the experiments described i n t h i s s e c t i o n , small-area MIS s o l a r c e l l s were fabricated on n-type s i l i c o n substrates using e i t h e r palladium or platinum f o r the b a r r i e r metal. The substrates used were of <100> o r i e n t a t i o n and 5 ficm r e s i s t i v i t y . The front surfaces of these wafers had been prepared by the standard i n d u s t r i a l technique of chem-mechanical p o l i s h i n g . Following cleaning i n accordance with the procedure s p e c i f i e d i n Appendix C, the wafers were exposed to a dry oxygen flow for 20 minutes. At f i r s t , oxidation was c a r r i e d out at 500°C, but i n l a t e r experiments oxidation temperatures as high as 650°C were investigated. The substrates were not removed from the furnace immediately a f t e r oxidation, but were instead exposed to a dry nitrogen flow f o r an a d d i t i o n a l 20 minutes. This l a s t step may help reduce Q„_, the fi x e d r C p o s i t i v e charge associated with the Si-SiO^ i n t e r f a c e [114], and thus allow the attainment of higher b a r r i e r heights [136]. Following the oxidation/annealing sequence, a thick layer of thermally evaporated a l u -minum was deposited on the backs of the wafers to form a negative b a r r i e r MIS back contact. (The minority c a r r i e r r e f l e c t i n g properties of t h i s A l - S i O x ~ n S i contact were not of i n t e r e s t here; the negative b a r r i e r MIS technology simply offered a convenient technique for forming a l o w - r e s i s t -158 ance back contact to the lightly-doped n-type substrates). The r e c t i f y i n g MIS front junction was then formed by the deposition of a semi-transparent layer of thermally evaporated palladium or platinum. The procedures f o l -lowed i n the evaporation of these metals are discussed i n Appendix C. 0 The b a r r i e r metal layer was made approximately 100 A thick, and was delineated using a metal shadow mask to give a t o t a l device area of about 2 0.1 cm . F i n a l l y , a small contact dot composed of the same metal used i n the b a r r i e r layer was deposited near the edge of each c e l l . In 5 ficm n-type s i l i c o n , reasonable estimates f o r the hole l i f e t i m e 2 -1 -1 and mobility are 0.3 ys and 0.05 m V s respectively (see Appendix A). Substituting these estimates f o r T p and y^ i n t o (2.28), i t i s found that a minority c a r r i e r MIS sol a r c e l l formed on such a substrate should give an open-circuit voltage of about 490 mV at a photocurrent density of 2 30 mA/cm and a temperature of 28°C. The completed palladium-barrier c e l l s were found to have character-i s t i c s s i m i l a r to those expected f o r Schottky diodes fabricated using conventional techniques. When illuminated to give a photocurrent density 2 of 30 mA/cm , these c e l l s y i e l d e d an open-circuit voltage of approximately 250 mV at a temperature of 28°C. This low open-circuit voltage indicates that the dark current i n these devices i s dominated by majority c a r r i e r thermionic emission. The reason f o r the poor performance of these c e l l s was discovered l a t e r when the palladium layer was removed by etching i n aqua regia (a mixture of 3 parts hydrochloric acid to 1 part n i t r i c a c i d ) . Although the aqua regia r a p i d l y dissolved a l l traces of m e t a l l i c p a l l a -dium, even a f t e r prolonged exposure to the acid at elevated temperatures a pinkish residue remained on the s i l i c o n surface where the c e l l s had been. It thus appears that the condensing palladium a c t u a l l y penetrated 159 the t h i n oxide layer at the substrate surface and reacted with the under-l y i n g s i l i c o n to form palladium s i l i c i d e . Rather than producing an MIS junct i o n , t h i s process r e s u l t s i n the formation of an intimate-contact Pd^Si-Si Schottky b a r r i e r . Although every e f f o r t was made to minimize heating of the substrates during the evaporation, palladium s i l i c i d e s are known to form at temperatures as low as 250°C [115]. In contrast to the r e s u l t s obtained with palladium, the platinum-b a r r i e r c e l l s gave open-circuit voltages ranging from 490 to 510 mV at 2 a photocurrent density of 30 mA/cm and a temperature of 28°C. These V values are the second highest ever reported for MIS c e l l s fabricated oc on n-type s i l i c o n [19], and agree quite w e l l with the crude estimate made above for the open-circuit voltage of a minMIS c e l l formed on a 5 ficm n-type substrate. Further, when the platinum layer was removed i n aqua regia, no inso l u b l e residue was v i s i b l e at the s i l i c o n surface. It was therefore t e n t a t i v e l y concluded that the platinum c e l l s were true MIS devices, with the structure Pt-SiO -nSi. x To determine i f strong inversion of the semiconductor surface had been achieved i n the Pt-SiO^-nSi c e l l s , the small-signal capacitance C of these devices was measured as a function of reverse b i a s . The r e s u l t i n g 2 pl o t of 1/C versus V for a representative diode i s shown i n F i g . 5.3. Using the method of l e a s t squares, the slope and voltage-axis intercept V of the l i n e best f i t t i n g the data of F i g . 5.3 were computed. The slope 15 -3 was found to correspond to a doping density of 1.0*10 cm , which agrees c l o s e l y with the doping l e v e l expected for 5 ficm n-type s i l i c o n . The strong inversion p o t e n t i a l for material of t h i s doping density i s 590 mV, while V was found to be 620 mV, confirming that the s i l i c o n surface was indeed strongly inverted. 160 Figure 5.3 Capacitance-voltage c h a r a c t e r i s t i c for reverse-biased Pt-SiO -nSi dot diode. 161 Once i t had been established that strong inversion of the semicon-ductor surface could be achieved i n a Pt-SiO x-nSi diode, a preliminary i n v e s t i g a t i o n of the v a r i a t i o n of the c h a r a c t e r i s t i c s of these devices with i n s u l a t o r thickness was conducted. This was done by r a i s i n g the temperature at which the oxidation/annealing treatment described above was c a r r i e d out, while holding the oxidation time constant. The i l l u m i -nated current-voltage c h a r a c t e r i s t i c s of the c e l l s produced i n this experiment c l o s e l y resembled those of the Al-SiO^-pSi c e l l s discussed i n Section 4.4. F i r s t , no s i g n i f i c a n t v a r i a t i o n of open-circuit voltage with oxidation temperature was observed. Secondly, c e l l s fabricated using a 600°C oxidation/annealing sequence were found to have c h a r a c t e r i s t i c s s i m i l a r to those of c e l l s incorporating oxides grown at 500°C; the f i l l f actors of these devices were always greater than 0.5. However, oxidation at 650°C produced c e l l s whose illuminated current-voltage c h a r a c t e r i s t i c s were concave upwards over part of the power-output quadrant. As discussed i n Chapter 3 and i n Section 4.4, c h a r a c t e r i s t i c s of t h i s kind r e s u l t from the tunnel resistance associated with excessively thick i n s u l a t i n g l a y e r s . 5.3.2 Minority C a r r i e r R e f l e c t i n g Pt-SiO x-pSi Contacts On the basis of the experiments described above, i t was concluded that there was a reasonable p r o b a b i l i t y that the Pt-SiO^-nSi c e l l s were i n fact minority c a r r i e r MIS diodes. Rather than attempting to confirm t h i s hypothesis by, for example, measuring the temperature dependence of the current-voltage c h a r a c t e r i s t i c s of these devices, i t was decided to immediately attempt the f a b r i c a t i o n of a minority c a r r i e r r e f l e c t i n g Pt-SiO x~pSi negative b a r r i e r contact. From equation (5.6), i t can r e a d i l y be shown that the formation of a minority c a r r i e r r e f l e c t i n g contact on 162 a moderately doped p-type substrate i s possible i f and only i f a minority c a r r i e r MIS diode can be formed on moderately doped n-type material. F o l -lowing the general pattern of the experiments on negative b a r r i e r contacts to n-type material discussed i n Section 5.2, the e f f e c t i v e surface recom-bin a t i o n v e l o c i t y of the Pt-SiO^-pSi contact was investigated by i n c o r -porating t h i s structure i n an induced back surface f i e l d s o l a r c e l l . Pt-SiC^-pSi back contacts were applied to both N +P diffused front junction c e l l s and to c e l l s with Al-SiO -pSi minMIS front junctions. In accordance x r J with the terminology introduced by Green et_ a l , devices of the l a t t e r kind w i l l be referred to as MISIM c e l l s . S i m i l a r l y , c e l l s i n the former class w i l l be termed N +PIM devices. The substrates employed i n the experiments on Pt-SiO^-pSi contacts described i n t h i s section were supplied by Applied Solar Energy Corpora-t i o n , and were cut from the same ingots used i n the production of back surface f i e l d space c e l l s . These substrates were i d e n t i c a l to those used i n the experiments discussed i n Section 4.3. The substrates were of 10 ftcm r e s i s t i v i t y and <100> o r i e n t a t i o n , with thicknesses ranging from 350 to 400 ym. Both the front and back surfaces of the s l i c e s were chem-i c a l l y polished. A shallow N + layer was formed at the front of some of the substrates by d i f f u s i o n from a phosphorus source. A l l s l i c e s were then shipped to the University of B r i t i s h Columbia for MIS contact pro-cessing. P r i o r to MIS junction f a b r i c a t i o n , a l l substrates were cleaned as s p e c i f i e d i n Appendix C. A t h i n oxide layer was then grown on the s l i c e s by exposure to dry oxygen at 600°C for 30 minutes. Following the procedure used when f a b r i c a t i n g p o s i t i v e b a r r i e r junctions to n-type s i l i c o n , the furnace tube was flushed with a strong flow of dry, high-purity nitrogen 163 f o r 20 minutes before the s l i c e s were removed. For those s l i c e s lacking a d i f f u s e d front j u n c t i o n , a t h i n (-80 A thick) semi-transparent layer of aluminum was evaporated on one surface to create a r e c t i f y i n g minMIS front j u n c t i o n . This b a r r i e r layer was then o v e r l a i d with a thick aluminum contact g r i d . Contact to the N + region of the d i f f u s e d front junction substrates was made by deposition of a t i t a n i u m - s i l v e r g r i d , which was then sin t e r e d at 500°C i n hydrogen for 10 minutes. This treatment may also have helped reduce the surface state density at the back of the N +P substrates. Processing was completed by the deposition of a platinum layer o roughly 500 A thick over the back of a l l the s l i c e s to form the negative b a r r i e r MIS back contact. The structure of both the N +PIM and MISIM c e l l s i s i l l u s t r a t e d i n F i g . 5.4. The current-voltage c h a r a c t e r i s t i c s of the f i n i s h e d c e l l s were measured under ELH lamp i l l u m i n a t i o n at a c o n t r o l l e d temperature of 28°C. Since a s i g n i f i c a n t f r a c t i o n of the l i g h t incident on these uncoated experimental devices was l o s t to r e f l e c t i o n , the l i g h t i n t e n s i t y was not set to simulate one-sun i l l u m i n a t i o n , but rather was adjusted to give a 2 photocurrent density of 30 mA/cm i n each c e l l tested. This photocurrent density i s s l i g h t l y less than that presently attained under AMI i l l u m i n -ation i n ASEC N +P c e l l s equipped with optimized a n t i - r e f l e c t i o n coatings. The ten MISIM and seven N +PIM c e l l s fabricated here yi e l d e d open-c i r c u i t voltages spanning a wide range, from 540 to 586 mV. Possible causes f o r t h i s large spread i n V values are considered below. The ° oc s i g n i f i c a n t r e s u l t i s that the V values recorded for the two best N +PIM oc devices were 582 and 586 mV, while the two best MISIM c e l l s gave V 's oc of 577 and 581 mV (see Table 5.2). In comparison, under the same test A A T i / A g MOOOA/lum) n S i (M3.5ym) (a) (b) p S i (^350um) S i O x (^20A) P t (^500A) A l (Mum) A l M O A ) S i O (-V20A) x p S i (^350ym) Z _ S i O (^20A) x P t (^500A) Figure 5.4 Structure of (a) N +PIM and (b) MISIM s o l a r c e l l s . 165 conditions the group of N^PP^ c o n t r o l c e l l s with alloyed aluminum back surface f i e l d s r e f e r r e d to i n Section 4.3 gave a mean open-circuit voltage of 588 mV. ASEC N +P c e l l s lacking back surface f i e l d regions but f a b r i -cated on otherwise i d e n t i c a l 10 ftcm substrates give open-circuit voltages not exceeding 550 mV under these conditions. On the basis of these V oc measurements, i t was t e n t a t i v e l y concluded that the negative b a r r i e r MIS back contacts to the best N +PIM and MISIM devices were functioning as e f f i c i e n t minority c a r r i e r r e f l e c t o r s . To provide convincing evidence that the best MISIM and N +PIM c e l l s were indeed e x h i b i t i n g induced back surface f i e l d a c t i o n , the platinum MIS back contacts to these devices were removed and replaced with ohmic contacts using only room temperature processing. Removal of the platinum m e t a l l i z a t i o n was accomplished by simply immersing the c e l l s i n warm methanol; the methanol rapidly undermined and l i f t e d the platinum layers. The front m e t a l l i z a t i o n was then protected by applying e t c h - r e s i s t a n t adhesive tape, and the s l i c e s were dipped i n 10% HF s o l u t i o n to remove a l l traces of s i l i c o n oxide from the back surface. After t h i s oxide o etch the tape was peeled away, and approximately 1,000 A of palladium was deposited on the back of the s l i c e s by f l a s h thermal evaporation. As discussed i n Section 4.3, palladium contacts to freshly-etched p-type s i l i c o n f abricated i n t h i s fashion give ohmic c h a r a c t e r i s t i c s with a 2 resistance of approximately 0.1 ficm . Section 4.3 also describes c o n t r o l experiments c a r r i e d out to ensure that the processes of taping, tape removal and palladium deposition do not harm an Al-SiO^-pSi front junc-t i o n . These processes can, of course, have v i r t u a l l y no e f f e c t on the properties of a d i f f u s e d junction. The V values obtained for the reprocessed c e l l s with palladium 166 TABLE 5.2 Open-circuit voltages (mV) f o r selected N +PIM and MISIM c e l l s C e l l : N+PIM#1 N+PIM#2 MISIM#1 MISIM#2 With Pt MIS back contact: With Pd back contact: 586 582 581 577 543 542 545 167 ohmic back contacts are l i s t e d i n the second column of Table 5.2. On average, the open-circuit voltage dropped 40 mV a f t e r back contact r e f a b r i c a t i o n , demonstrating unequivocally that the o r i g i n a l Pt-SiO^-pSi negative b a r r i e r contact had been s i g n i f i c a n t l y reducing the recombination rate at the c e l l back surface. Although the open-circuit voltages of the best MISIM and N +PIM c e l l s f a b r i c a t e d to date compare favourably with those of the N +PP + c o n t r o l c e l l s , the y i e l d of good devices has not been high. This low y i e l d most l i k e l y r e s u l t s from d i f f i c u l t i e s i n c o n t r o l l i n g the f l a s h platinum evap-oration used to create the negative b a r r i e r Pt-SiO^-pSi contact (see Appendix C). I f the deposition rate i s too rapid, the heat released by the condensing platinum may r a i s e the temperature of the substrate back surface to such an extent that the platinum w i l l d i f f u s e through the t h i n SiO^ layer and make intimate contact with the underlying s i l i c o n . Conversely, i f the evaporation proceeds too slowly, tungsten may be introduced into the deposited platinum due to a l l o y i n g of the tungsten filament with the molten platinum charge. Tungsten contamination i s l i k e l y to lower the work function of the deposited layer, and thus reduce the b a r r i e r height of the MIS contact. From (5.5b), i t can be seen that such a reduction i n b a r r i e r height w i l l increase the e f f e c t i v e surface recombination v e l o c i t y of the contact. In Section 2.3 i t was noted that the a p p l i c a t i o n of a minority c a r r i e r r e f l e c t i n g back contact to a s o l a r c e l l i n which the minority c a r r i e r d i f f u s i o n length i s greater than the base width should give a small increase i n long-wavelength photocurrent response. To determine i f the negative b a r r i e r Pt-SiO^-pSi contact could provide a s i m i l a r photo-current enhancement, the photocurrent generated by c e l l N+PIM#1 was 168 measured under i n f r a r e d i l l u m i n a t i o n . As described i n Section 4.3, a l i g h t source r i c h i n i n f r a r e d was obtained by operating an u n f i l t e r e d tungsten-halogen bulb at one-third i t s rated voltage. The i l l u m i n a t i o n l e v e l was held constant to within ±1% by monitoring the lamp output with a reference s i l i c o n photodiode. With the o r i g i n a l Pt-SiO^-pSi back contact, c e l l N+PIM#1 gave an i n f r a r e d photocurrent of 7.2 mA. When the platinum MIS contact was replaced with a palladium ohmic contact, the photocurrent at the same i l l u m i n a t i o n l e v e l dropped to 6.2 mA. If a negative b a r r i e r MIS back contact i s to be of use i n increasing the e f f i c i e n c y of a sol a r c e l l , i t i s e s s e n t i a l that i t not add appreci-able ser i e s resistance to the c e l l . Although the e f f e c t i v e resistance of the Pt-SiO^-pSi contact has not yet been studied i n any d e t a i l , the f i l l f actors obtained f o r the N +PIM and MISIM c e l l s fabricated thus f ar have been generally good, given allowance for the non-optimized front contact g r i d s . For example, the MISIM device with the highest V q c had a quite reasonable f i l l f a c tor of 0.68. 169 CHAPTER 6 SUMMARY In t h i s t h e s i s , three major o r i g i n a l contributions are made to present understanding of the MIS tunnel diode and of photovoltaic devices i n general. The f i r s t contribution i s developed i n Chapter 2, and centers on the discovery that previous t h e o r e t i c a l arguments put forward to es t a b l i s h the v a l i d i t y of the p r i n c i p l e of dark current and photocurrent superposition for homojunction s o l a r c e l l s contain serious flaws. In the course of cor r e c t i n g these flaws, the f i r s t comprehensive i n v e s t i g a t i o n of the behaviour of the quasi-fermi energy l e v e l s i n the depletion region of an illuminated s o l a r c e l l i s c a r r i e d out [32,33]. While i n the past i t had i n v a r i a b l y been assumed that the quasi-fermi l e v e l s would always be constant across the depletion region, i t i s found that for operation at short c i r c u i t or small forward bias E^ and E ^ p change dramatically over t h i s region. However, for c e l l s fabricated on substrates with reasonably high c a r r i e r l i f e t i m e s and m o b i l i t i e s , i t i s found that both quasi-fermi l e v e l s become e f f e c t i v e l y constant across the depletion region for operation close to the maximum power point. From t h i s r e s u l t i t follows that the superposition p r i n c i p l e must accurately describe the character-i s t i c s of such c e l l s at a l l operating points, provided the minority c a r r i e r concentrations i n the quasi-neutral regions remain at low i n j e c t i o n l e v e l s . In p a r t i c u l a r , i t i s shown that the superposition p r i n c i p l e should apply even i f a s i g n i f i c a n t f r a c t i o n of both recombination and photogeneration occur i n the depletion region. This contradicts the conclusions drawn by Lindholm et a l . i n a recent p u b l i c a t i o n [31]. I t i s also found that the superposition p r i n c i p l e may se r i o u s l y overestimate the e f f i c i e n c y of 170 c e l l s f a b r i c a t e d on substrates with very poor l i f e t i m e s and low mobil-i t i e s , a point which had not been appreciated previously. Although the main r e s u l t s presented i n Chapter 2 are obtained using simple a n a l y t i c methods, these r e s u l t s are confirmed through d i r e c t numerical s o l u t i o n of the d i f f e r e n t i a l equations governing the p o t e n t i a l s , c a r r i e r concen-tratio n s and current flows within a s o l a r c e l l . The numerical analysis i s applied to both s i l i c o n and gallium arsenide c e l l s . The second main contribution made i n this thesis i s the presentation of the f i r s t conclusive experimental evidence f o r the existence of minor-i t y c a r r i e r MIS tunnel diodes. (In a minority c a r r i e r MIS diode, the dark current flow at moderate forward bias i s dominated by the i n j e c t i o n of minority c a r r i e r s i n t o the quasi-neutral base) . Although the p o s s i b i l -i t y of forming such devices had been proposed on t h e o r e t i c a l grounds by Green et a l . i n 1974 [16], u n t i l the completion of the experiments described i n Chapter 4 t h e i r existence remained a subject of considerable contro-versy. Two independent experiments were c a r r i e d out here to e s t a b l i s h that these devices could i n f a c t be made. In the f i r s t experiment, the current-voltage c h a r a c t e r i s t i c s of Al-SiO^-pSi diodes were recorded as a function of temperature [20]. From t h i s data, an a c t i v a t i o n energy de-s c r i b i n g the temperature dependence of the dark current was extracted. This a c t i v a t i o n energy was found to agree exactly with that expected for a minority c a r r i e r i n j e c t i o n - d i f f u s i o n current, and to be s i g n i f i -cantly l a r g e r than that possible f o r a majority c a r r i e r thermionic emission current. In the second experiment, Al-SiO^-pSi s o l a r c e l l s were fabricated on substrates with alloyed aluminum back surface f i e l d s [21]. When the back surface f i e l d regions were removed by chemical etching and replaced with ohmic contacts, the open-circuit voltages of these 171 c e l l s were found to drop by as much as 50 mV. This demonstration that a change i n the properties of the back surface of an MIS s o l a r c e l l could s i g n i f i c a n t l y a l t e r the open-circuit voltage provided further i r r e f u t a b l e evidence f o r the existence of minority c a r r i e r MIS diodes. The t h i r d p r i n c i p a l contribution of t h i s thesis involves a theor-e t i c a l and experimental study of the properties of the negative b a r r i e r MIS junction [37]. In Chapter 5, a simple a n a l y t i c model of current flow i n the negative b a r r i e r junction i s developed. This model predicts that with a s u i t a b l e choice of i n s u l a t o r thickness and b a r r i e r metal work function, i t should be possible to form negative b a r r i e r MIS contacts which present a very low e f f e c t i v e surface recombination v e l o c i t y to minority c a r r i e r s , yet which o f f e r n e g l i g i b l e impedance to the flow of majority c a r r i e r s . The minority c a r r i e r r e f l e c t i n g properties of the negative b a r r i e r MIS junction were demonstrated experimentally by u t i l -i z i n g t h i s structure to form induced back surface f i e l d s o l a r c e l l s . Using magnesium as the b a r r i e r metal, negative b a r r i e r MIS contacts were employed to form induced back surface f i e l d s on n-type s i l i c o n substrates [37]. Diffused front junction P +N c e l l s were used i n t h i s experiment. Later, Pt-SiO^-pSi negative b a r r i e r contacts were applied to form induced back surface f i e l d s on p-type material. In t h i s second experiment, both d i f f u s e d front junction N +P c e l l s and c e l l s with pos-i t i v e b a r r i e r minMIS front junctions were used. Although the y i e l d of Mg-SiC^-nSi and Pt-SiO^-pSi junctions with strong minority c a r r i e r re-f l e c t i n g properties was never high, the best induced back surface f i e l d c e l l s i n each class mentioned above gave open-circuit voltages comparable to those obtained with conventional back surface f i e l d s formed by d i f -fusion or a l l o y i n g . The minority c a r r i e r r e f l e c t i n g properties of the 172 negative b a r r i e r MIS back contacts were confirmed by replacing these contacts with ohmic back contacts without damaging the front junction. When t h i s was done, the c e l l open-circuit voltage dropped s i g n i f i c a n t l y . It was also shown that a Pt-SiO -pSi induced back surface f i e l d could x enhance the i n f r a r e d photocurrent response of an N +P c e l l . Although no di r e c t measurements of the e f f e c t i v e resistance of the Mg-SiO -nSi and x Pt-SiO^-pSi negative b a r r i e r contacts were made, the f i l l factors of the induced back surface f i e l d c e l l s incorporating these contacts were found to be reasonably high. In addition to the three main contributions outlined above, a theor-e t i c a l model of current flow i n the p o s i t i v e b a r r i e r MIS junction i s developed i n Chapter 3. This model i s based l a r g e l y on e a r l i e r t h e o r e t i c a l studies c a r r i e d out by Green et al.[16,17] and by Card and Rhoderick r [26-28]. However, the model presented here has the advantages of being purely a n a l y t i c , unlike that developed by Green e_t^  a l , and of allowing for strong inversion of the semiconductor surface, unlike that proposed by Card and Rhoderick. The main drawback of the model i s i t s i n a b i l i t y to account f o r the e f f e c t s of surface states on current flows or on the e l e c t r o s t a t i c p o t e n t i a l d i s t r i b u t i o n across the junction. In any case, surface state e f f e c t s are best handled by r e s o r t i n g to e n t i r e l y numerical methods, as did Green et a l . Perhaps the most i n t e r e s t i n g p r e d i c t i o n made by the model i s that the illuminated current-voltage c h a r a c t e r i s t i c s of t h i c k - i n s u l a t o r MIS s o l a r c e l l s w i l l be concave-upwards over a ce r t a i n bias range. Experimental confirmation of th i s p r e d i c t i o n i s provided i n Chapter 4. The development of the MIS junction model was motivated i n part by suggestions made recently that the MIS tunnel diode i s a funda-mentally d i f f e r e n t device than the non-ideal Schottky diode. In Chapter 3 i t i s shown that the operation of both these devices can be explained using the same basic model. Although the metal-insulator-semiconductor tunnel junction has been the subject of w e l l over one hundred s c i e n t i f i c papers i n the past decade much research remains to be done on t h i s structure. A number of possible topics f o r future study a r i s e as straightforward extensions of the r e s u l t presented i n t h i s t h e s i s . F i r s t , a d i r e c t demonstration that minority c a r r i e r MIS diodes can be formed on n-type s i l i c o n would be of i n t e r e s t . On a more fundamental front, the a n a l y t i c model of the MIS junction de-veloped i n Chapter 3 should be modified to allow for degenerate c a r r i e r concentrations at the semiconductor surface. Above a l l , a comprehensive i n v e s t i g a t i o n should be undertaken to determine whether negative b a r r i e r MIS junctions can simultaneously supply lower e f f e c t i v e surface recombi-nation v e l o c i t i e s and lower contact resistances than back surface f i e l d regions formed using conventional techniques. The development of more e f f e c t i v e back surface f i e l d structures i s a subject of considerable importance i n view of the recent trend towards the use of lightweight s i l i c o n c e l l s only 50 to 100 ym thick for space applications [134]. In order to function e f f i c i e n t l y , such c e l l s must incorporate high q u a l i t y back surface f i e l d s . 174 APPENDIX A NUMERICAL SOLUTION OF THE BASIC SEMICONDUCTOR EQUATIONS The operation of any semiconductor device i s governed by the basic equations of semiconductor physics: the current equations, the continuity equations, and Poisson's equation. An exact a n a l y t i c s o l u t i o n of these coupled non-linear d i f f e r e n t i a l equations i s possible only i n the simplest circumstances. The f i r s t a p p l i c a t i o n of numerical methods to solve the basic equations was reported i n 1964 by Gummel [116], who obtained s o l -utions for the c a r r i e r concentrations and flows i n one-dimensional diode and b i p o l a r t r a n s i s t o r structures under steady-state conditions. The algorithm introduced by Gummel was l a t e r modified and extended by DeMari [117] and Arandjelovic [118]. Unfortunately, i n 1972 Choo [119] discovered that Gummel's algorithm would not converge i n many si t u a t i o n s of p r a c t i c a l i n t e r e s t . This f a i l u r e to converge was also demonstrated on a n a l y t i c grounds at t h i s time by Mock [120]. In response to the f a i l u r e of Gummel's algorithm, Seidman and Choo [51] developed a completely new method for the numerical s o l u t i o n of the basic equations i n t h e i r steady-state, one-dimensional form ( (2.3)-(2.7) ). The algorithm devised by Seidman and Choo i s at once i n t u i t i v e l y pleasing, simple to implement, and computationally e f f i c i e n t . It i s thus hardly s u r p r i s i n g that t h i s algorithm and i t s variants have seen extensive use i n the modelling of semiconductor devices i n recent years [121-123]. In t h i s appendix, the a p p l i c a t i o n of Seidman and Choo's algorithm to the modelling of s i l i c o n and GaAs homojunction s o l a r c e l l s i s considered i n d e t a i l . For n o t a t i o n a l convenience, the algorithm i s developed i n reference to NP devices, a f t e r which the few simple modif-i c a t i o n s required to treat PN c e l l s are considered. Before any progress can be made i n the s o l u t i o n of (2.3)-(2.7), an e x p l i c i t r e l a t i o n s h i p between the free c a r r i e r concentrations n and p and the recombination rate U must be s p e c i f i e d , as must boundary condi-tant i n GaAs, which i s a d i r e c t bandgap material [125]. However, for s i m p l i c i t y only recombination through trapping centers w i l l be considered here. Seidman and Choo assumed that the dependence of U on n and p could be accurately described by a Shockley-Read-Hall model [124] with a singl e trapping l e v e l at midgap. In th i s case, The boundary conditions imposed on (2.3)-(2.7) w i l l be those described i n subsection 2.2.2 (see equations (2.8)-(2.9) and associated discussion). NORMALIZATION A l l numerical methods f o r the s o l u t i o n of (2.3)-(2.7) begin with a normalization of variables designed to eliminate as many constant factors from the equations as possible, and thus minimize computational time. The normalization procedure followed here i s that introduced by DeMari [117], and i s summarized i n Table A . l . For the remainder of t h i s appendix, a l l symbols w i l l r e f e r to normalized q u a n t i t i e s unless otherwise s p e c i f i e d . In terms of normalized v a r i a b l e s , (2.3)-(2.7) become tions on n, p and ty. Band-to-band recombination i s l i k e l y to be impor-(A.l) dJ /dx = U - G (A.2) n dJ /dx = -U + G P (A. 3) TABLE A . l Normalization factors 176 Variable to be normalized: Divide by: N A,N D,N T,n 1,p 1 L D (= [ £ k T / q 2 n . ] 1 / 2 ) n,p n. 1 V ^ (= kT/q) J ,J n' p D ,D n P n' p U,G qD 0n./L D D Q (= 1 m V 1 ) L D / D 0 D O n i / L D V S B D 0 / L D Jn = V ~ n ( d 4 , / d x ) + dn/dx] (A.4) J p = -Dp[p(d^/dx) + dp/dx] (A.5) d 2^/dx 2 = -(p + N D - N A - n) (A.6) while boundary conditions (2.8) and (2.9) become w • v y v - V X B ) ] <A-7> PO and - Jp ( xF> = V W " P n O ( x F ) ] ' (A.8) The SRH expression for the recombination rate becomes U = (pn - l ) / ( x n + T p + x + T ) (A.9) p n p n for a trapping center at midgap. At t h i s stage Seidman and Choo found i t convenient to introduce two new variables u and v defined according to -ty u = n e (A.10) and v = pe y. (A.11) u and v bear the following simple r e l a t i o n s h i p to the quasi-fermi 178 p o t e n t i a l s d> and d> : r n Yp (f>n = -ln(u) x (A. 12) and <{>p = l n ( v ) . (A. 13) In terms of u and v, the current equations can be written i n a p a r t i c u l -a r l y simple form. For example, noting that du/dx = (dn/dx)e~* - ne~*(diji/dx) = [-n(d^/dx) + dn/dxje"^ i t can be seen by comparison with (A.4) that J = D e^du/dx). (A.14) n n S i m i l a r l y , Jp = -D pe ^(dv/dx). (A.15) Defining a quantity B = T n + x p + T + T = x ue^ + x ve ^ + x + x (A. 16) p n p n p n p n the current and continuity equations can be combined to give -d_(D e*(du/dx)] + uv = 1 + G (A.17) dx n B B and -d_[D e *(dv/dx)] + vu = 1 + G. (A.18) dx P B B LINEARIZATION OF THE BASIC EQUATIONS Seidman and Choo proposed that (A.17) and (A.18) could be l i n e a r i z e d as follows. On any given i t e r a t i o n , l e t u, v and ip be the estimates for u, v and ty obtained on the preceding i t e r a t i o n . Then a l i n e a r approxima-tio n to (A.17) i s -d [D e^(du/dx)] + uv = 1 + G (A.19) dx n B B while a l i n e a r approximation to (A.18) i s -d_[D e ^(dv/dx)] + vu = 1 + G. (A.20) dx p I B Poisson's equation can also be l i n e a r i z e d by l e t t i n g ty = ty~ + 6. To f i r s t order i n 6, (A.6) becomes -d 26/dx 2 + 6(ue^ + ve = d2\|//dx2 - ue* + ve"* + N„ - N.. (A.21) D A Each of equat ions (A.19)—(A.21) i s of the form -(ay') ' + gy = f (A.22) where ' denotes d i f f e r e n t i a t i o n with respect to x. To solve (A.22) numerically, a set of g r i d points {x^, j=l,M} i s introduced, and d e r i v a t -ives with respect to x are approximated by taking f i n i t e d i f f e r e n c e s . At each i n t e r i o r point (that i s , a point f o r which 2sj£M-l ) g r i d spacings h. = x. - x. , (A.23) and h. = x . - x. (A.24) are defined. Then midway between points x. and x. a reasonable 3 J - l approximation to ay' i s a y ' = ( a j - l + a j ) ' ( y j ~ y j - l ) 2 hT (Here the notation = a(x^), y^ = y ( x ^ ) , etc. i s used). S i m i l a r l y , between points x^ and x.. +^, a reasonable approximation to ay' i s ay' « (a, + a, + 1) • ( y ^ - y,) . 2 h + 3 Therefore at point x. (ay') ( a i + V ' ( y j + i - V (A.25) ( a i + " i - ^ • ( y i ' y i - l } (h /2 + hJ/2) Defining 181 and X j = ( a j + a j + l ) / h j (A. 26a) X j = + a^/h (A.26b) n|=x{/Oi| + h ) (A.27a) n = x , / ( h + + h.) (A.27b) J J J J J J J J (A.25) becomes - ( a y ' ) , | x > * t-r i j Y j + 1 " n~ y._± + (n+ + ^~)y.]. (A.28) Thus at any interior grid point the differential equation (A.22) takes the fi n i t e difference form aA y. + b- y- i + c. y.,, = f. (A.29a) i 3 3 J - l J J+l J where a j = n j + n j + 3 j ' (A.29b) b j = -n j (A.29c) and + c j = -nj • (A.29d) The only problem now remaining i s the imposition of boundary conditions at x^ and x^ .^ At x^, the values of u and i> are fixed once the bias V i s specified. Similarly, v and ty are fixed at x^. With boundary conditions of this type, (A.29) s t i l l holds with = 1, b 1 = 0, = 0 and f. = y, , and a w =1, b„ = 0, c w = 0 and f = y w. In fi n i t e difference 1 1 M M M M M form (A.7) becomes [x"/2 + S B e x p ( ^ ) ] u M - (xM/2) = S B n ^ ) (A.30) where the value of xM i s that appropriate to (A.19). S i m i l a r l y , (A.8) becomes [X*/2 + S p expH^)] v± - v 2 (X|/2) = S p P n Q ( x F ) (A. 31) where x^  i s appropriate to (A.20). Clearly both (A.30) and (A.31) are of the form (A.29a). SOLUTION OF THE LINEARIZED BASIC EQUATIONS Equation (A.29a) can be expressed i n matrix form as where A y = ? A = a i C l b2 a2 C2 (A.32a) (A.32b) b M - l ^ - 1 CM-1 bM aM and y and f are column vectors with j t h element y^ and f_. re s p e c t i v e l y . The matrix A has non-zero elements only on the main diagonal and on the two adjacent diagonals, and so i s termed a t r i d i a g o n a l band matrix. In order to solve for y, i t i s necessary to inv e r t A, This can be done 183 conveniently using the following algorithm, which i s derived by simple row-echelon reduction of A [126]: 1. Define arrays d and y according to: d l = a l Y l = c 1 / d 1 d. = a. - b. y. .. (for i = 2 to M) 3 3 3 J - l Y. = c./d. (for i = 2 to M) 2. Define an array g according to: *1 = V d i 8 j = ( f3 ~ b3 8 3 - l ) / d J ( f ° r J = 2 t 0 M ) 3. y i s given by: yM = % y ± = g ± " Y ± y ± + 1 (for i = M-1 to 1). In order to minimize the storage space required by a computer program implementing the above algorithm, the members of the following p a i r s of arrays can share the same memory lo c a t i o n s : d and a Y and c g and b INITIALIZATION AND COMPUTATIONAL PROCEDURE To carry through one i t e r a t i o n of Seidman and Choo's algorithm, equations (A.19) and (A.20) are f i r s t solved i n f i n i t e difference form as described above to generate improved estimates f o r u and v. These new values for u and v are then substituted i n (A.21), which i s turn solved i n f i n i t e d i f f e r e n c e form to determine the correction term 6 to be added 184 to ty. Once the corrected value of ty has been computed, the next i t e r a t i o n begins. In order to s t a r t the i t e r a t i v e process, i t i s c l e a r l y necessary to have reasonably good i n i t i a l estimates for u,v and ty. Here an estimate for ty was obtained at each bias point V by assuming that the e l e c t r o s t a t i c p o t e n t i a l drop V - V over the c e l l appears only across the depletion region, and then applying the depletion approximation. Knowing ty, i n i t i a l estimates f o r u and v were obtained by s p e c i f y i n g the p o s i t i o n of the quasi-fermi energy l e v e l s . The majority c a r r i e r quasi-fermi l e v e l was taken to be constant over each quasi-neutral region, and to extend at t h i s constant value across the depletion region. The separation of the quasi-fermi l e v e l s i n the depletion region was thus set equal to qV. F i n a l l y , the minority c a r r i e r quasi-fermi l e v e l was set to coincide with the maj-o r i t y c a r r i e r quasi-fermi l e v e l i n each quasi-neutral region. Although t h i s l a s t i n i t i a l i z a t i o n condition must s e r i o u s l y underestimate the minority c a r r i e r concentrations at any V > 0, i t has not resulted i n computational problems. In generating the results presented i n subsection 2.2.4, the i t e r a t i v e process was continued u n t i l the s o l u t i o n f or the t o t a l current flow i n the diode varied by le s s than 0.1% between succes-sive i t e r a t i o n s . This generally required fewer than 20 i t e r a t i o n s s t a r t i n g from the i n i t i a l i z a t i o n outlined above. CARRIER LIFETIMES AND MOBILITIES The r e l a t i o n s h i p between the c a r r i e r m o b i l i t i e s and the t o t a l sub-s t r a t e doping l e v e l N T i n the s i l i c o n and GaAs c e l l s modelled here was s p e c i f i e d by simply making a piecewise-linear approximation to the p l o t of the logarithm of the mobility versus the logarithm of N^ presented by Sze [127]; the points used i n the piecewise l i n e a r f i t are l i s t e d i n Table A.2. 185 TABLE A.2 Data used to compute mobility a) Data f or s i l i c o n V ( m " 3 ) p : ( m W 1 ) u : ( m W 1 ) _E 1 0 2 1 1 .4*10 1 5 . 8 * 1 0 " 2 ••A22 - 1 _o 10 1 .1*10 A 5 .0*10 2 23 -9 o 10 7 .0*10 Z 3 . 3 * 1 0 " 2 24 -9 o 10 3 .4*10 2 .0*10 25 -9 o 10 1 .2*10 8*10 b) Data f or GaAs N T: ( m - 3 ) u n : ( m W 1 ) p^ : ( m 2 V " 1 s " 1 ) 1 0 2 1 7 . 0 * 1 0 _ 1 3 . 7 * 1 0 - 2 1 0 2 2 6 . 0 * 1 0 _ 1 3 . 2 * 1 0 " 2 23 - 1 _9 10 J 4 . 5*10 2 .3*10 24 - 1 _9 10 3 .0*10 1 .3*10 25 - 1 _ -3 10 1 .5*10 7*10 21 - 3 For N < 10 m , p and p are assigned the values l i s t e d f o r l n p 21 - 3 = 10 m Otherwise, p^ and p p are computed by l i n e a r i n t e r p o l a t i o n or extrapolation. I 186 When modelling s i l i c o n c e l l s , the electron and hole l i f e t i m e s were r e l a t e d to the doping l e v e l through the formulas Tp " T 0 p / ( 1 + W ( A ' 3 3 a ) and \ • v/(1 + VV <A-3b> where N T = N A + N D (A.33c) as suggested by Fossum [128]. - x . and x „ were set equal to 1.7*10~^ s On Op and 3.5*10 7 s re s p e c t i v e l y , while NQ^ and were both set equal to 15 -3 7.1*10 cm . Since r e l a t i v e l y l i t t l e information concerning minority c a r r i e r l i f e t i m e s i n GaAs i s a v a i l a b l e , x and x were assumed to be n p independent of the doping l e v e l i n this material. Following Hovel [129] _q and Milnes and Feucht [130], x was set equal to 10 s, while x was n ' p set equal to 10 8 s. PHOTOGENERATION Unless otherwise stated, the photogeneration d i s t r i b u t i o n G(x) was computed at each g r i d point from the formula G(x ) = E exp[-a(X ) x.] M(A.) AA (A.34) i J 1 1 which i s simply a f i n i t e - d i f f e r e n c e approximation to the i n t e g r a l (2.1), taking R(A) = 0. F i f t e e n wavelength i n t e r v a l s AA^ were used to span the s o l a r spectrum from the wavelength corresponding to the s i l i c o n bandgap TABLE A.3 Data used to compute photogeneration d i s t r i b u t i o n Wavelength I n t e r v a l : (ym) a : (m ) M: (m s ) 0.28 - 0.32 1.8*108 7.52*10 1 8 0.32 - 0.36 1.2*10 8 19 4.48*10 0.36 - 0.40 1.2*107 19 6.87*10 0.40 - 0.44 4.4*106 20 1.18*10 0.44 - 0.48 2.3*106 20 1.69*10 0.48 - 0.52 1.4*106 20 1.79*10 0.52 - 0.56 9.0*105 20 1.76*10 0.56 - 0.60 6.3*105 20 1.75*10 0.60 - 0.64 4.0*105 1.90*10 2° 0.64 - 0.68 3.0*105 20 1.83*10 0.68 - 0.72 2.0*10 5 20 1.83*10 0.72 - 0.76 1.6*105 1.71*10 2° 0.76 - 0.80 1.0*105 20 1.61*10 0.80 - 1.00 4.1*10 4 20 7.00*10 1.0.0 - 1.20 5*10 2 20 t 5.91*10 T ctg^ i s the absorption c o e f f i c i e n t f o r s i l i c o n . For the GaAs c e l l , a uniform photogeneration d i s t r i b u t i o n was used. The t o t a l photogeneration i n the GaAs c e l l was set equal to 1.88*10 t with energies greater than the s i l i c o n bandgap 188 to the u l t r a v i o l e t ; X i s the c e n t r a l wavelength of the i t h i n t e r v a l . The absorption c o e f f i c i e n t o t ( A ^ ) was extracted from the graphical data presented by Sze [40], while the t o t a l photon f l u x M (A_^ )AA^ i n the i n t e r -v a l was computed from the AMI s p e c t r a l composition s p e c i f i e d i n Ref.[131]. These data are summarized i n Table A.3. Although f a r more accurate representations of a ( A ) and the s o l a r s p e c t r a l irradiance are a v a i l a b l e , -t h i s procedure was deemed adequate f o r the purpose of t e s t i n g the v a l i d i t y of the superposition p r i n c i p l e . Any reader intending to use the programs presented i n t h i s appendix for very accurate modelling of s o l a r c e l l performance should investigate the techniques employed to compute the photogeneration d i s t r i b u t i o n by Fossum [128] and by Dunbar and Hauser [123], rather than using the crude approximation to G(x) given here. GRID SELECTION The s e l e c t i o n of an appropriate g r i d geometry i s a c e n t r a l part of any numerical analysis problem. The g r i d spacing must be s u f f i c i e n t l y f i n e that derivatives can be adequately approximated by f i n i t e d ifferences, yet with too f i n e a g r i d truncation or round-off errors may s e r i o u s l y a f f e c t the computations. In a t y p i c a l s o l a r c e l l the widths of the emitter and depletion region are roughly one hundred times smaller than the width of the base. For such a device a non-uniform g r i d i s c l e a r l y required. The basic g r i d structure used here contained 100 evenly-spaced points i n each of the emitter, depletion and base regions. This g r i d structure was found to provide accurate solutions f o r the potentials and currents i n s i l i c o n c e l l s operated i n the dark and i n GaAs c e l l s . However, when modelling ill u m i n a t e d s i l i c o n c e l l s i t was found advantageous to add another 100 g r i d points i n the region extending from the depletion region/base boundary to a depth of 10 um i n t o the base. This modification 189 ensured that there would be a reasonably f i n e g r i d throughout the region of maximum photogeneration i n a s i l i c o n c e l l . To check that a s u f f i c i e n t l y f i n e g r i d had been chosen, under various operating conditions the spacing between g r i d points was halved, thereby doubling the number of points. In no circumstance d i d the s o l u t i o n f o r the t o t a l current change by more than 0.1% on halving the g r i d spacing. With 400 g r i d points, 20 i t e r a t i o n s of Seidman and Choo's algorithm can be c a r r i e d out i n roughly 1 second of CPU time on the Amdahl 470. Another simple check on the v a l i d i t y of the numerical model can be made by computing the t o t a l current — that i s , the sum of J and J — n p at each g r i d point. This sum should be constant throughout a device, since Seidman and Choo's algorithm i s based on the steady-state form of the basic equations. Here the t o t a l current was indeed found to be independent of p o s i t i o n , except i n the case of very small current flows through very heavily doped emitter regions. In such cases truncation error became a problem when computing the majority c a r r i e r current from the gradient of the majority c a r r i e r quasi-fermi p o t e n t i a l . The solutions f o r the p o t e n t i a l s and c a r r i e r concentrations themselves were unaffected by t h i s e r r o r . PROGRAMS Li s t i n g s and b r i e f descriptions of the four FORTRAN programs written to implement Seidman and Choo's algorithm are given below. Although the programs l i s t e d below were written to apply to devices with n-type emitters and p-type base regions, PN structures can be treated by simply i n t e r -changing the electron and hole m o b i l i t i e s and l i f e t i m e s . The r e l a t i o n s h i p between the FORTRAN variables appearing i n the programs and the notation used i n the discussion above i s documented i n Table A.5. The programs 190 were compiled with the FORTRAN G compiler, and run under MTS con t r o l . A l l input data supplied to the programs should be i n MKS u n i t s . SC.PARSET establishes the g r i d over which computations proceed, and assigns values f o r the doping, l i f e t i m e s , m o b i l i t i e s and photogeneration at each g r i d point. The g r i d i s b u i l t up of an a r b i t r a r y number of sections of width WSECT, each containing NSEG segments. The input values of WSECT and NSEG used i n obtaining the data presented i n subsection 2.2.4 are l i s t e d i n Table A.4. The input of a value of NSEG < 0 terminates the g r i d construction. In the form l i s t e d below the program assumes that the c e l l has been fabricated on a uniformly-doped p-type s i l i c o n substrate without a back surface f i e l d . The doping p r o f i l e i s assumed to be Gaussian, but the program accepts values for the m e t a l l u r g i c a l junction depth XJNCT, surface doping concentration N , substrate DU doping N^ and surface recombination v e l o c i t i e s S^ , and Sg as input data. Values for a (A ) and the photon f l u x are read i n from the f i l e SPDTFL. A l l quantities computed by this program are written to the sequential f i l e GPARFL. SC.INIT computes i n i t i a l values for u, v and ty i n accordance with the procedure s p e c i f i e d above. The program requires the terminal voltage VOLT to be s p e c i f i e d as input data. Values for u, v and ty are written to the sequential f i l e UVPSFL. SC.CALC uses Seidman and Choo's algorithm to obtain improved estimates for u, v and ty. The number of i t e r a t i o n s to be performed NCNTRL, the terminal voltage and the number-of-suns i l l u m i n a t i o n RSUNS are supplied as input data. SC.READOUT computes and outputs values for n, p, ty, J ^ , J p and t o t a l current J^, at selected g r i d points. The input data f o r t h i s program s p e c i f i e s the g r i d point JSTART at which sampling i s to begin, the gr i d point JSTOP at which sampling i s to end, and the spacing INCR between points sampled. TABLE A.4 Parameters used f o r g r i d construction a) S i l i c o n c e l l NSEG: WSECT: 100 0.5D-6 100 0.5D-6 50 2.5D-6 50 6.5D-6 100 240.0D-6 b) GaAs c e l l NSEG: WSECT: 100 0.2D-6 100 0.2D-6 100 9.6D-6 192 TABLE A.5 Explanation of variables used i n FORTRAN programs FORTRAN VARIABLE: VTH NI LDEBYE EPSI Q TO QUANTITY REPRESENTED: kT/q D HJM HJP, H(J) HSUM XJ JMAX VBI WO XJNCT NA NDO NT LNT NNOF PNOF PPOB NPOB PFLX GO h. J h + J h + + hT J 3 x j M (number of g r i d points) b i WQ (depletion region width at V = 0) (met a l l u r g i c a l junction depth) N A N. DO l o g 1 0 ( N T ) nnO ( xF> P n O ( K F ) PP 0 ( X B ) n p 0 ( x B } M(A i)AA ± GQ (photogeneration at surface) 193 FORTRAN VARIABLE: MU MUN MOT LNDT,LMUNDT,LMUPDT DN(J),DNJ DNJP DP(J) ,DPJ DPJP TN(J),TNJ TP(J),TPJ N(J) U(J),UJ V(J),VJ PSI(J),PSIJ PSIJM PSIJP DELTA F(J),FJ XPSIJ XPSIJP CHIJMU.CHIJMV CHIJPU.CHIJPV ETAJM ETAJP AU(J),AV(J),APSI(J),A(J) BU(J),BV(J),BPSI(J),B(J) CU(J),CV(J),CPSI(J),C(J) QUANTITY REPRESENTED: (data used to compute mobility) D n,j+l D P,J D P,j+1 P»J N A " N D u. exp (ty ) e x P ( * j + 1 ) + + 194 FORTRAN VARIABLE: VOLT VBAR RSUNS NCNTRL JN JP JT PHIP PHIN QUANTITY REPRESENTED: V, . - V b i (number-of-suns) (number of i t e r a t i o n s ) 195 C PROGRAM SC. PARSE.T C WRITTEN BY GARRY TARR FEB/81 C IMPLICIT REAL*8 (A-H,0-Z) INTEGER FREE,SPDTFL,GPARFL REAL*4 H,DN,DP,TN,TP,G,N REAL*8 LDEBYE,NI,NA,NDO,NNOF,NPOB, #K,KS,NT,NOTN,NOTP,LNDT,LMUNDT,LMUPDT,MUN,MUP,LNT,ND DIMENSION H(801),N(801),DN(801),DP(801),TN(801),TP(801), #G(801),FREE(1),PFLX(100),ALPHA(100), #LNDT(5),LMUNDT( 5) ,LMUPDT(5) C DATA FREE/'*'/,KR/5/,LP/6/,IMUDT/5/,SPDTFL/4/,GPARFL/3/, #K/1.38054D-23/,Q/1.6021D-19/,EPSI0/8.854D-12/,T/300.0D0/, #NI/l.4 5D16/,KS/11.7D0/, #NA/5.0D21/,ND0/1.0D25/,XJNCT/0.5D-6/,SF/1.0D3/,SB/1.0D30/, #N0TN/7.lD21/,N0TP/7.1D21/,TN0/1.7D~5/,TP0/3.5D-7/f #LNDT/2.1D1, 2.2D1,2.3D1,2.4D1,2.5D1/, #LMUPDT/-1.236D0,-1.30D0,-1.48D0,-1.70D0,-2.10D0/, #LMUNDT/-0.845D0,-0.96D0,-1.16D0,-1.46D0,-1.92D0/ C C J = l c 100 READ(KR,FREE,ERR=51,END=51) NSEG,WSECT IF(NSEG .LE. 0) GOTO 300 C HH=WSECT/NSEG DO 200 JJ=1,NSEG H(J)=HH J=J + 1 200 CONTINUE GOTO 100 300 JMAX=J IF(JMAX .LE. 810) GOTO 400 WRITE(LP,101) 101 FORMAT(IX f'TOO MANY GRID POINTS') STOP 400 H(JMAX)=0.0 C READ(KR,FREE,ERR= 52,END= 50 0) NA,NDO,XJNCT,SF,SB C 500 EPSI=EPSI0*KS VTH=K*T/Q LDEBYE=DSQRT(EPSI*VTH/Q/NI) T0=LDEBYE*LDEBYE NN0F=(ND0-NA)/NI PN0F=NI/(ND0~NA) PP0B=NA/NI NP0B=NI/NA SF=SF*LDEBYE SB=SB*LDEBYE VBI=VTH*DLOG(NN0F*PP0B) W0=DSQRT(2.0D0*EPSI*VBI/Q/NA)/LDEBYE C REWIND SPDTFL 196 DO 600 1=1,100 600 READ(SPDTFL,FREE,ERR=4,END=700) ALPHA-( I ) ,PFLX(I ) GOTO 4 C 700 ISPDT=I-1 CDIFF=DLOG(NDO/NA)/XJNCT/XJNCT XJNCT=XJNCT/LDEBYE C c XJ=0.0D0 DO 1000 J=1,JMAX ND=0.0D0 XX=XJ*XJ*CDIFF IF(XX .GT. 1.0D2) GOTO 800 ND=ND0*DEXP(-XX) 800 NT=ND+NA N(J)=(ND-NA)/NI LNT=DL0G10(NT) CALL FINDMU(LNT,MUN,LNDT,LMUNDT,IMUDT) DN(J)=VTH*MUN CALL FINDMU(LNT,MUP,LNDT,LMUPDT,IMUDT) DP(J)=VTH*MUP TN(J)= TN 0/(1.0D0+NT/N0TN)/T0 TP(J)=TP0/(1.0D0+NT/NOTP)/TO C G(J)=0.0 DO 900 I=1,ISPDT G0 = PF L X ( I ) * A L P H A ( I ) / N I *T0 XX=XJ*ALPHA(I) IF (XX .GT. 1.0D2) GOTO 9'00 G(J)=G(J)+G0*DEXP(-XX) 900 CONTINUE C XJ=XJ+H(J) H(J)=H(J)/LDEBYE 1000 CONTINUE C C WRITE(LP,102) XJ,JMAX 102 FORMAT(1X/1X,'DEVICE WIDTH = ',DI7.6,5X,'NUMBER OF #GRID POINTS = ',14) REWIND GPARFL WRITE(GPARFL) H,JMAX,VTH,NI,LDEBYE,XJNCT,NA,ND0, #NN0F,PN0F,PP0B,NP0B,VBI,W0,SF,SB,N,DN,DP,TN,TP,G C STOP 4 STOP 4 51 STOP 51 52 STOP 52 END 197 SUBROUTINE FINDMU(LNT,MU,LNDT,LMUDT,IMUDT) C IMPLICIT REAL*8 (A-HfO-Z) REAL*8 LNT,MU,LNDT,LMUDT,LMU .DIMENSION LNDT(IMUDT),LMUDT(IMUDT) C DATA TEN/1.0D1/ C LMU=LMUDT(1) IF(LNT .LE. LNDT(1)) GOTO 300 ISTOP=IMUDT-l DO 100 I=l,ISTOP I F ( (LNT .GE. LNDT(I)) .AND. (LNT .LT. LNDT(I+1)) ) GOTO 200 100 CONTINUE 1 = 1 STOP C 200 LMU=LMUDT(I)-(LMUDT(I)-LMUDT(I+1))/(LNDT(I+1)-LNDT(I))* #(LNT-LNDT(I)) 300 MU=TEN**LMU C RETURN END C PROGRAM SC.INIT C WRITTEN BY GARRY TARR FEB/81 C IMPLICIT REAL*8 (A-H,0~Z) INTEGER FREE,GPARFL,UVPSFL REAL*8 LDEBYE,NI,NA,NDO,NNOF,NPOB REAL*4 H DIMENSION U(801),V(801),PSI(801),H(801),FREE(1) C DATA NCNTRL/0/,RSUNS/0.0D0/,FREE/'*'/,KR/5/,UVPSFL/2/, #GPARFL/3/ C C READ(KR,FREE,ERR= 5,END= 5) VOLT C REWIND GPARFL READ(GPARFL,ERR=3,END= 3) H,JMAX,VTH,NI,LDEBYE,XJNCT,NA, #ND0,NNOF,PN0F,PP0B,NPOB,VBI,W0 C VBAR=VBI-VOLT DVBAR=DEXP(VBAR/VTH) W=WO*DSQRT(VBAR/VBI) C XJ=0.0D0 DO 100 J=1,JMAX I F ( X J .GE. XJNCT) GOTO 200 PSI(J)=0.0D0 U(J)=NN0F V( J)=PN0F XJ=XJ+H(J) 100 CONTINUE C 200 J J = J RJ=0.0D0 DO 300 J=JJ,JMAX RJ=RJ+H(J)/W I F ( R J .GT. 1.0D0) GOTO 400 PSI(J)=2.0D0*VBAR*(0.5D0*RJ*RJ-RJ)/VTH U(J)=NN0F V(J)=PP0B/DVBAR 300 CONTINUE C 400 J J = J DO 500 J=JJ,JMAX PSI(J)=-VBAR/VTH U(J)=NP0B*DVBAR V(J)=PP0B/DVBAR 500 CONTINUE C REWIND UVPSFL WRITE(UVPSFL) U,V,PSI,VOLT,RSUNS,NCNTRL C STOP 3 STOP 3 5 STOP 5 END 199 C PROGRAM SC.CALC C WRITTEN BY GARRY TARR FEB/81 C IMPLICIT REAL*8 (A-H,0~Z) INTEGER FREE,GPARFL,UVPSFL REAL*8 LDEBYE,NI,NA,NDO,NNOF,NPOB REAL*4 H,N,DN,DP,TN,TP,G DIMENSION U(801),AU(801),BU(801),CU(801),F(801), #V(801) ,AV(801)',BV(801) ,CV(801) , #PSI (801),APSI(801),BPSI(801),CPSI(801),DELTA(801), #H(801),N(801),DN(801),DP(801),TN(801),TP(801),G(801) #FREE(1) EQUIVALENCE (BU,APSI),(CU,BPSI),(BV,CPSI),(CV,DELTA) C DATA ZERO/0.0D0/,ONE/1.0D0/,TWO/2.0D0/, #KR/5/,UVPSFL/2/,GPARFL/3/,FREE/1 *'/ C C READ(KR,FREE,END=5,ERR=5) NCNTRL,VOLT,RSUNS REWIND UVPSFL READ(UVPSFL,ERR= 2,END= 2) U,V,PSI REWIND GPARFL READ(GPARFL,ERR=3,END=3) H,JMAX,VTH,NI,LDEBYE,XJNCT,NA, #ND0,NNOF,PNOF,PPOB,NPOB,VBI,WO,SF,SB,N,DN,DP,TN,TP,G C DO 50 J=1,JMAX G(J)=G(J)*RSUNS 50 CONTINUE DVBAR=DEXP((VBI-VOLT)/VTH) U(JMAX)=NPOB*DVBAR V(JMAX)=PP0B/DVBAR PSI(JMAX)=-(VBI-VOLT)/VTH JSTOP=JMAX-l C C C DO 1000 ICNTRL=1,NCNTRL C C DNJ=DN(1) DPJ=DP(1) XPSIJ=DEXP(PSI(1)) HJP=H(1) DNJP=DN(2) DPJP=DP(2) XPSIJP=DEXP(PSI(2)) CHIJPU=(DNJ*XPSIJ+DNJP*XPSIJP)/HJP CHIJPV=(DPJ/XPSIJ+DPJP/XPSIJP)/HJP C AU(l)=ONE CU(l)=ZERO C AV(1)=CHIJPV/TWO+SF/XPSIJ CV(1)=-CHIJPV/TWO C DNJ=DNJP 200 DPJ=DPJP HJM=HJP CHIJMU=CHIJPU CHIJMV=CHIJPV 0 XPSIJ=XPSIJP C DO 100 J=2,JSTOP XPSIJP=DEXP(PSI(J+l)) DNJP=DN(J+1) DPJP=DP(J+1) HJP=H(J) HSUM=HJM+HJP UJ=U(J) VJ=V(J) TNJ=TN(J) TPJ=TP(J) FJ=ONE/(TPJ*UJ*XPSIJ+TNJ*VJ/XPSIJ+TPJ+TNJ) C CHIJPU=(DNJ*XPSIJ+DNJP*XPSIJP)/HJP ETAJM=CHIJMU/HSUM ETAJP=CHIJPU/HSUM AU(J)=ETAJM+ETAJP+VJ*FJ BU(J)=-ETAJM CU(J)=-ETAJP C CHIJPV=(DPJ/XPSIJ+DPJP/XPSIJP)/HJP ETAJM=CHIJMV/HSUM ETAJP=CHIJPV/HSUM AV(J)=ETAJM+ETAJP+UJ*FJ BV(J)=-ETAJM CV(J)=-ETAJP C F(J)=FJ+G(J) DNJ=DNJP DPJ=DPJP HJM=HJP CHIJMU=CHIJPU CHIJMV=CHIJPV XPSIJ=XPSIJP 100 CONTINUE C AU(JMAX)=CHIJMU/TWO+SB*XPSIJ BU(JMAX)=-CHIJMU/TWO AV(JMAX)=ONE BV(JMAX)=ZERO F(1)=U(1) F(JMAX)=SB*NP0B CALL INVRT(U,AU,BU,CU,F,JMAX) C F(1)=SF*PN0F F(JMAX)=V(JMAX) CALL INVRT(V,AV,BV,CV,F,JMAX) C C PSIJM=PSI(1) PSIJ=PSI(2) 201 HJM=H(1) APSI(l)=ONE CPSI(l)=ZERO C DO 200 J=2,JSTOP HJP=H(J) P S I J P = P S I ( J + l ) XPSIJ=DEXP(PSIJ) UJ=U(J) VJ=V(J) HSUM=HJM+HJP ETAJM=TWO/HJM/HSUM ETAJP=TWO/HJP/HSUM F(J)=ETAJP*PSIJP+ETAJM*PSIJM-(ETAJP+ETAJM)*PSIJ #+N(j)-UJ*XPSIJ+VJ/XPSIJ APSI(J)=ETAJP+ETAJM+UJ*XPSIJ+VJ/XPSIJ BPSI(J)=-ETAJP CPSI(J)=-ETAJM PSIJM=PSIJ PSIJ=PSIJP HJM=HJP 200 CONTINUE C APSI(JMAX)=ONE BPSI(JMAX)=ZERO F(l)=ZERO F(JMAX)=ZERO CALL INVRT(DELTA,APSI,BPS I ,CPSI,F,JMAX) DO 300 J=l,JMAX PS I ( J ) = D E L T A ( J ) + P S I ( J ) 300 CONTINUE C C 1000 CONTINUE C C C REWIND UVPSFL WRITE(UVPSFL) U,V,PSI,VOLT,RSUNS,NCNTRL c STOP 2 STOP 2 3 STOP 3 5 STOP 5 END 202 SUBROUTINE INVRT(Y,A,B,C,F,JMAX) C IMPLICIT REAL*8 (A-H,0-Z) DIMENSION Y(JMAX),A(JMAX),B(JMAX),C(JMAX),F(JMAX) C DATA ZERO/0.0D0/ rONE/1.0D0/ C C JSTOP=JMAX-l C A(1)=A(1) C(1)=C(1)/A(1) DO 100 J=2,JSTOP A ( J ) = A ( J ) - B ( J ) * C ( J - 1 ) 100 C ( J ) = C ( J ) / A ( J ) A(JMAX)=A(JMAX)-B(JMAX)*C(JSTOP) C B(1)=F(1)/A(1) DO 200 J=2,JMAX 200 B ( J ) = ( F ( J ) - B ( J ) * B ( J - 1 ) ) / A ( J ) C Y(JMAX)=B(JMAX) DO 300 J=l,JSTOP I=JMAX-J 300 Y ( I ) = B ( I ) - C ( I ) * Y ( I + 1 ) C RETURN END 203 C C C C PROGRAM SC.READOUT C WRITTEN BY GARRY TARR FEB/81 C IMPLICIT REAL*8 (A-H,0~Z) INTEGER FREE,UVPSFL,GPARFL REAL*8 LDEBYE,NI,NA,NDO,NNOF,NPOB REAL*4 H,N,DN,DP,TN,TP,G DIMENSION U ( 8 0 1 ) , V ( 8 0 1 ) , P S I ( 8 0 1 ) , . #H(801),N(801),DN(801),DP(801),TN(801),TP(801),G(801), #FREE(1) COMMON.VTH,NI,LDEBYE,U,V,PSI,DN,DP,H,JMAX r DATA FREE/'*'/,KR/5/,LP/6/,GPARFL/3/,UVPSFL/2/ REWIND GPARFL READ(GPARFL,ERR=3,END= 3) H,JMAX,VTH,NI,LDEBYE,XJNCT,NA, #ND0,NNOF,PN0F,PP0B,NPOB,VBI,WO,SF,SB,N,DN,DP,TN,TP,G REWIND UVPSFL READ(UVPSFL,ERR=2,END=2) U,V,PSI,VOLT,RSUNS,NCNTRL WRITE(LP,101) NCNTRL,VOLT,RSUNS 101 FORMAT('1' ,10X,'ITERATION NUMBER: ' , I 2,5X,'VOLTAGE: ', #F8.5,5X,'ILLUMINATION LEVEL: ',F6.2,IX,'SUNS'// #1X,'J:',7X,'PSI:',12X,'PHIP:',16X,'PHIN:',15X,'P:',12X, #'N:',11X,'JP:',11X,'JN:',11X,'JTOT:'/) C 100 READ(KR,FREE,ERR=5,END=500) JSTART,JSTOP,INCR C DO 200 J=JSTART,JSTOP,INCR CALL NPCRNT(J) 200 CONTINUE C WRITE(LP,102) 102 FORMAT(IX) C GOTO 100 c 500 WRITE(LP,103) 103 FORMAT(1X/1X/11X,'ALL QUANTITIES IN MKS UNITS'///) C STOP 2 STOP 2 3 STOP 3 5 STOP 5 END 204 SUBROUTINE NPCRNT(J) C IMPLICIT REAL*8 (A-H,0~Z) REAL*8 JN,JP,JT,NI,LDEBYE REAL*4 N,P,H,DN,DP DIMENSION U(801),V(801),PSI(801),DN(801),DP(801),H(801) COMMON VTH,NI,LDEBYE,U,V,PSI,DN,DP,H,JMAX C DATA Q/l.6021D-19/,TWO/2.0D0/,FOUR/4.0D0/,LP/6/ C C C=Q*NI/LDEBYE XPSIJ=DEXP(PSI(J)) PHIN=-VTH*DL0G(U(J)) PHIP=VTH*DLOG(V(J)) N=U(J)*XPSIJ*NI P=V(J)/XPSIJ*NI PSIJ=PSI(J)*VTH C I F ( J .EQ. 1) GOTO 100 I F ( J .EQ. JMAX) GOTO 200 HJM=H(J-l) HJP=H(J) XPSIJM=DEXP(PSI(J-l) ) XPSIJP=DEXP(PSI(J + l ) ) CHIJM=(DN(J-l)*XPSIJM+DN(J)*XPSIJ)/HJM CHIJP=(DN(J)*XPSIJ+DN(J+l)*XPSIJP)/HJP JN=(CHIJP*(U(J+1)-U(J))+CHIJM*(U(J)-U(J-l)))*C/FOUR CHIJM=(DP(J-1)/XPSIJM+DP(J)/XPSIJ)/HJM CHIJP=(DP(J)/XPSIJ+DP(J+l)/XPSIJP)/HJP J P = - ( C H I J P * ( V ( J + 1 ) - V ( J ) ) + C H I J M * ( V ( j ) - V ( J - l ) ) ) * C / F O U R GOTO 300 C 100 XPSIJP=DEXP(PSI(J+l)) JN=(DN(J)*XPSIJ+DN(J+l)*XPSIJP)*(U(J+l)-U(J))/H(1)*C/TWO J P = - ( D P ( J ) / X P S I J + D P ( J + l ) / X P S I J P ) * ( V ( J + 1 ) - V ( J ) ) / H ( l ) * C / T W O GOTO 300 C 200 XPSIJM=DEXP(PSI(J-l)) J N = ( D N ( J - l ) * X P S I J M + D N ( J ) * X P S I J ) * ( U ( J ) - U ( J - l ) ) / H ( J - l ) * C / T W O J P = - ( D P ( J - 1 ) / X P S I J M + D P ( J ) / X P S I J ) * ( V ( J ) - V ( J - l ) ) / H ( J - l ) * C / T W O 300 JT=JN+JP C C WRITE(LP,10 2) J,PSIJ,PHIP,PHIN,P,N,JP,JN,JT 102 FORMAT(1X,I4,2X,F13.10,2X,2(F19.16,2X),2(D12.5,2X), #2(D12.5,2X),D13.6) C RETURN END 205 APPENDIX B CALCULATION OF THE SHADOW AREA FOR AN ELLIPSOIDAL CONSTANT ENERGY SURFACE OF ARBITRARY ORIENTATION In t h i s appendix an expression for the shadow area a of an e l l i p -s o i d a l constant energy surface whose p r i n c i p a l axes have an a r b i t r a r y o r i e n t a t i o n r e l a t i v e to the in t e r f a c e i s derived. The shadow area can be computed most e a s i l y by working i n a coordinate system i n which the e f f e c t i v e mass tensor i s diagonal. In t h i s coordinate system, E(k) = (fi 2/2) [k2/m* + k2/m* + k2/m*] . (B.l) x x y y z z Thus the equation of the constant energy surface at energy E can be written i n the form F(k ,k ,k ) = 0 (B.2) x y z where F(k ,k ,k ) = k 2 / a 2 + k 2 / b 2 + k 2 / c 2 - 1 (B.3a) x y z x y z and 2 * 2 2 * 2 2 * ? a = 2Em / f t , b = 2Em /h , c = 2Em *i . (B.3b) x y z a, b and c are, of course, the half-lengths of the p r i n c i p a l axes of the e l l i p s o i d a l constant energy surfaces defined by (B.2). I f the normal to the i n t e r f a c e i s represented by the vector n = (n 1,n 2,n^), then F i g . B . l shows that the points on the e l l i p s o i d 206 Figure B . l Shadow of an e l l i p s o i d . 207 which project to the boundaries of the shadow s a t i s f y VF«fi = 0 (B.4) or n.k / a 2 + n_k /b 2 + n,k / c 2 = 0 . (B.5) I x 2 y 3 z ' (B.5) i s c l e a r l y the equation of a plane i n k-space passing through the o r i g i n and with normal 1 given by i = (£ 1,^ 2,£ 3) = ( n ; L / a 2 , n 2 / b 2 , n 3 / c 2 ) / L (B.6a) where L = ( n 2 / a A + n 2 / b 4 + n 2 / c 4 ) 1 / 2 . (B.6b) It i s w e l l known that the i n t e r s e c t i o n of an e l l i p s o i d with any plane passing through the center of the e l l i p s o i d i s an e l l i p s e . Further, using the conventional techniques of l i n e a r algebra i t can be shown that the area A of t h i s e l l i p s e i s given by A = TT/UJ/O^C2) + A 2 / ( a 2 c 2 ) + £ 2 / ( a 2 b 2 ) ] 1 / 2 . (B.7) where a, b and c are the half-lengths of the p r i n c i p a l axes of the e l l i p s o i d and £ i s the normal to the plane i n question. From F i g . B . l i t can be seen that the shadow area a i s related to A by a = A(i-n). (B.8) Combining (B.6),(B.7) and (B.8) i t i s found that a = T T[n 2b 2c 2 + n 2 a 2 c 2 + n 2 a 2 b 2 ] 1 / 2 . (B.9) ic ic ic In terms of the energy E and the e f f e c t i v e masses m , m and m , x y z , v 2 ^ r 2 * * 2 * * 2 * * 1/2 a = ir(2E/h ) [n.m m + n„m m + n.m m l ' . (B.10) l y z z x z 3 x y v / On s u b s t i t u t i n g t h i s expression f o r a i n t o (3.19), i t i s found that i s s t i l l given by (3.25), but that m£ i s given by * 2 * * 2 * * 2 * * 1 / 2 m = [n m m + n m m + n m m ] . ( B . l l ) e l y z 2 x z 3 x y \ - / This agrees with the expression for J C M Crowell [85] obtained using a 3 d i r e c t i n t e g r a t i o n over d k. APPENDIX C FABRICATION PROCEDURE FOR MIS JUNCTIONS This appendix provides complete d e t a i l s of the procedure currently followed at UBC i n the manufacture of both p o s i t i v e and negative b a r r i e r MIS junctions. This process has undergone a number of minor modifications i n the past; the version described here i s that i n use as of l a t e 1980. It i s e s s e n t i a l that a l l steps i n the process be c a r r i e d through without i n t e r r u p t i o n , since thin s i l i c o n oxide layers grow at the rate of several angstroms per hour when exposed to the atmosphere [18]. SUBSTRATE CLEANING P r i o r to junction f a b r i c a t i o n , a l l substrates are subjected to the "RCA clean", a standard cleaning procedure used widely i n the microelec-tronics industry to prepare s i l i c o n s l i c e s f or high temperature proces-sing [132]. The RCA clean recipe used here i s outlined below. Normal safety precautions used i n the handling of concentrated acids and bases must be observed when following t h i s procedure. A l l concentrations are quoted by volume, and a l l chemicals should be of ACS reagent grade or higher q u a l i t y . If p o s s i b l e , the r e s i s t i v i t y of the deionized water should be 18 Mficm. However, r e s i s t i v i t i e s as low as 1 M£2cm have been used here without apparent e f f e c t on the properties of the f i n i s h e d junctions. 1. a) Immerse s i l i c o n for 10 minutes i n a s o l u t i o n of 1 part 30% NH^OH, 1 part 30% H ^ and 5 parts deionized (DI) water held at a temperature of (80±5)°C. The s o l u t i o n should be prepared immediately before use, since ^ 0 2 decomposes r a p i d l y under these conditions. Further, i t i s important that the temperature of the s o l u t i o n not exceed 85°C. I f the concentration f a l l s too low or the temperature r i s e s too high, the s i l i c o n may be p i t t e d . 210 b) Rinse s i l i c o n for at l e a s t 10 minutes i n a DI water cascade or equivalent. 2. a) Immerse s i l i c o n f o r 1 minute i n a s o l u t i o n of 1 part 49% HF to 9 parts DI water at room temperature. b) Rinse s i l i c o n for at l e a s t 10 minutes i n DI water cascade. 3. a) Immerse s i l i c o n f o r 10 minutes i n a s o l u t i o n of 1 part 36% HC1, 1 part 30% ^2°2 ^ parts DI water held at a temperature of (80±5)°C. b) Rinse s i l i c o n for at l e a s t 10 minutes i n DI water cascade. c) Dry s i l i c o n by blowing water from surface with a j e t of o i l - f r e e nitrogen. The f i r s t step of the RCA clean i s intended to remove thin layers of organic contaminants from the s i l i c o n surface. The second step s t r i p s away any native oxide layer which may be present, while the t h i r d step removes heavy metals [132]. The effectiveness of the RCA clean can be monitored by noting the condition of the s i l i c o n surface as the steps l i s t e d above are c a r r i e d out. S i l i c o n s l i c e s which have been exposed to contaminated atmospheres for long periods are usually found to be hydrophobic — that i s , water droplets placed on a s l i c e tend to bead up rather than wetting the sur-face. Following completion of the f i r s t step of the RCA clean, the s i l -icon should be strongly h y d r o p h i l i c — that i s , water should r e a d i l y wet the surface. I f t h i s i s not the case, then the surface has not been thoroughly degreased. After immersion i n h y d r o f l u o r i c a c i d for a few seconds, the s i l i c o n should become hydrophobic, i n d i c a t i n g that a l l traces of oxide have been removed. Exposure to the HCliH^O^ s o l u t i o n regrows a th i n oxide l a y e r , making the s i l i c o n h y d r o p h i l i c once again. 211 OXIDATION Although a very t h i n oxide l a y e r i s grown at the s i l i c o n s u rface during the l a s t step of the RCA c l e a n , f u r t h e r o x i d a t i o n at temperatures greater than 400°C i s r e q u i r e d to produce m i n o r i t y c a r r i e r MIS diodes. No formal experiments have been c a r r i e d out here to determine an optimum o x i d a t i o n time, but experience i n d i c a t e s that o x i d a t i o n at a temperature of at l e a s t 500°C f o r a minimum p e r i o d of 20 minutes i s required to pro-duce m i n o r i t y c a r r i e r Al-SiO -pSi devices on 1 to 10 Qcm s u b s t r a t e s . The x Al-SiO - p S i back surface f i e l d c e l l s described i n Section 4.3 were f a b r i -x cated u s i n g a 30 minute o x i d a t i o n c y c l e at 600°C. These c e l l s gave both the h i g h e s t o p e n - c i r c u i t voltages and among the highest f i l l f a c t o r s of any devices produced during t h i s research program. As demonstrated i n Section 4.4, c e l l s i n c o r p o r a t i n g oxides grown at temperatures above 600°C have excessive tunnel r e s i s t a n c e , and give correspondingly poor f i l l f a c t o r s under one-sun i l l u m i n a t i o n . In a l l the experiments on MIS diodes reported here, high temperature treatments were c a r r i e d out i n a quartz tube furnace w i t h gas flows of approximately 1 L/sec. Medical grade oxygen was used f o r o x i d a t i o n , w h i l e p r e - p u r i f i e d n i t r o g e n c o n t a i n i n g l e s s than 25 ppm oxygen and water vapour was used f o r s i n t e r i n g and annealing steps. S l i c e s were always i n s e r t e d i n t o the furnace w i t h the si d e on which the MIS j u n c t i o n was to be formed f a c i n g the gas flow. OHMIC CONTACT FORMATION When working w i t h p-type s i l i c o n s u b s t r a t e s , ohmic back contacts can be conveniently formed by d e p o s i t i n g a t h i c k l a y e r of aluminum on the back of the s l i c e and then s i n t e r i n g t h i s l a y e r i n a n i t r o g e n atmosphere at a temperature of 500°C f o r 10 minutes. This technique has been found to y i e l d contacts with highly l i n e a r c h a r a c t e r i s t i c s and very low r e s i s t -ance, even when applied to substrates with t h i n oxide layers formed following the procedure outlined above. As a test of contact resistance, s i n t e r e d aluminum contacts of t h i s kind were applied to both sides of a 300 ym thick, 2 ficm substrate. The current-voltage c h a r a c t e r i s t i c of t h i s structure was then recorded using a four-point probe technique to elim-inate the e f f e c t s of lead resistance. The c h a r a c t e r i s t i c was found to 2 obey Ohm's law out to current densities of more than 100 mA/cm , while 2 the resistance measured between the two contacts was only 0.1 £2cm . This i s not s u b s t a n t i a l l y greater than the bulk resistance associated with a s l i c e of t h i s doping and thickness. / BARRIER METAL DEPOSITION In a l l the MIS junctions described i n t h i s t h e s i s , the b a r r i e r metal layer and any overlying contact fingers were deposited by the process of thermal evaporation. In the microelectronics industry, m e t a l l i z a t i o n of s i l i c o n wafers i s often c a r r i e d out by the processes of electron-beam evaporation or RF sputtering. These techniques were not employed here since they lead to bombardment of the substrate with X-rays and high energy electrons. This bombardment i s l i k e l y to create a high density of in t e r f a c e s t a t e s , and thus s e r i o u s l y degrade the performance of the fi n i s h e d MIS junc t i o n . However, there i s a chance that high-quality MIS devices could be produced by magnetron sputtering. The vacuum system used for evaporation was a standard CHA SEC-600 unit equipped with a Varian VHS-6 d i f f u s i o n pump. In the SEC-600 system, backstreaming of f l u i d from the d i f f u s i o n pump i s r e s t r i c t e d to some extent through the use of an o p t i c a l l y dense water-cooled b a f f l e and a l i q u i d nitrogen cold trap. Unfortunately, the CHA trapping system i s 213 decidely i n f e r i o r to that used on other commercially available d i f f u s i o n pumps, i n that i t allows a d i r e c t l i n e of sight from the water-cooled b a f f l e into the work chamber. There was thus a rather high p r o b a b i l i t y of the s i l i c o n substrates becoming coated with a thi n f i l m of d i f f u s i o n pump f l u i d p r i o r to m e t a l l i z a t i o n . This problem was compounded by the use of r e l a t i v e l y high vapour pressure DC 704 f l u i d i n the d i f f u s i o n pump. The e f f e c t of t h i s possible hydrocarbon contamination on the prop-e r t i e s of the f i n i s h e d MIS junctions i s not known. Evaporations were —6 —6 usually c a r r i e d out a pressures ranging from 1*10 to 2*10 Torr. o For deposition rates greater than about 1 A/sec, these pressures were low enough to prevent s i g n i f i c a n t contamination of the deposited films by reaction with r e s i d u a l gases i n the work chamber. Where possib l e , the thicknesses of the deposited films were measured with a quartz c r y s t a l o s c i l l a t o r type thickness monitor. The b a r r i e r metals investigated here included aluminum, magnesium, palladium and platinum. Of these materials, platinum i s by f a r the most d i f f i c u l t to evaporate. Platinum both melts and reaches a vapour pressure -4 of 10 Torr at a temperature of approximately 1750°C [133]. This precludes the self-evaporation of platinum wire. Moreover, the temperature required to evaporate platinum i s so high that the s e l e c t i o n of source materials i s extremely l i m i t e d . Although platinum i s known to a l l o y with tungsten, tungsten filaments were chosen as the source here. P r i o r to use, these filaments were cleaned by heating to white heat for several minutes under high vacuum. When carrying out the actual evaporation, the procedure followed was to gradually heat the filament u n t i l the platinum charge melted, and then quickly open the shutter covering the source while applying a b r i e f burst of power to the filament to flash-evaporate the 214 charge. The substrates were thus exposed to the white-hot source for only a few seconds. I t was hoped that t h i s procedure would minimize both the a l l o y i n g of the platinum charge with the filament and the heating of the s i l i c o n substrates. The source temperature required for the p l a t -inum evaporation was so high that the thickness monitor could not be used to measure the deposition rate. In an attempt to ascertain the degree of tungsten contamination of the deposited platinum f i l m s , a blank glass microscope s l i d e was p o s i t i o n -ed i n the path of the evaporant stream during one evaporation. The compos-i t i o n of the f i l m deposited on the s l i d e was then analyzed using X-ray fluorescence spectroscopy. (The instrument used i n t h i s analysis was the scanning e l e c t r o n microscope system operated by the Department of Metal-lurgy) . No tungsten l i n e s could be detected i n the fluorescent X-ray spectrum, i n d i c a t i n g that the deposited f i l m contained less than approx-imately 1% tungsten. Compared to platinum, the materials aluminum, magnesium and p a l l a -dium can be evaporated with r e l a t i v e ease. Here aluminum was evaporated from tungsten filaments. Aluminum b a r r i e r layers and front-contact grids o were generally deposited at a rate of 2-10 A/sec. Magnesium was evaporated from a b a f f l e d tantalum boat designed f o r use with SiO. Although the rate of magnesium evaporation was d i f f i c u l t to c o n t r o l , deposition rates of o 10-20 A/sec were aimed f o r . F i n a l l y , when carrying out the experiments on p o s i t i v e b a r r i e r Pd-nSi junctions described i n Section 5.3, the p a l l a -dium was deposited by the self-evaporation of t h i n wires. Since the -4 vapour pressure of palladium reaches 10 Torr at 1200°C, yet t h i s metal does not melt u n t i l 1550°C [133], palladium wires can r e a d i l y be s e l f -evaporated. This procedure permits palladium films of extremely high p u r i t y to be deposited. When depositing palladium on freshly-etched s i l i c o n to form an ohmic contact, a tungsten basket was used as the evaporation source. REFERENCES 1. Solar Photovoltaic Energy Conversion, H. Ehrenreich, chairman (American P h y s i c a l Society, New York, 1979). 2. D.L. Pulfrey, Photovoltaic Power Generation (Van Nostrand Reinhold, New York, 1978). 3. J . Ja v e t s k i , E l e c t r o n i c s , p. 105 (July 19, 1979). 4. L.P. Hunt, Proceedings of the 12th IEEE Photovoltaic S p e c i a l i s t s  Conference (IEEE, New York, 1976) p. 347. 5. 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