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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  by  NICHOLAS GARRY TARR B.Sc,  The U n i v e r s i t y o f 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  in THE  FACULTY OF GRADUATE STUDIES  Department o f E l e c t r i c a l  Engineering  We accept t h i s t h e s i s as  conforming  to  THE  the r e q u i r e d s t a n d a r d  UNIVERSITY OF BRITISH COLUMBIA May, 1981  0  N. Garry T a r r , 1981  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree that for  understood for  Library  shall  for reference  and  study.  I  for extensive  that  h i s or  her  copying or  f i n a n c i a l gain  be  shall  Elct-Jri G&,[  publication  not  be  Date  DE-6  (2/79)  of  Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  Eft^in&e-n  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5  the  representatives.  permission.  Department o f  copying of  g r a n t e d by  the  University  the  p u r p o s e s may by  the  I agree that  permission  department or  f u l f i l m e n t of  advanced degree at  Columbia,  scholarly  in partial  written  ii  ABSTRACT  T h i s t h e s i s i s concerned p r i m a r i l y w i t h  an e x p e r i m e n t a l  e t i c a l i n v e s t i g a t i o n of the p r o p e r t i e s o f the tor  (MIS)  MIS  of use  i n the p r o d u c t i o n  metal-insulator-semiconduc-  of p h o t o v o l t a i c  cells.  j u n c t i o n s s t u d i e d are d i v i d e d i n t o two b a s i c c l a s s e s : those i n  which the semiconductor s u r f a c e i s d e p l e t e d e q u i l i b r i u m , and Junctions while  theor-  t u n n e l j u n c t i o n . P a r t i c u l a r emphasis i s p l a c e d on those p r o p -  e r t i e s which might be The  and  or s t r o n g l y i n v e r t e d a t  those i n which the s u r f a c e i s accumulated at e q u i l i b r i u m .  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 ,  those i n the l a t t e r group are termed n e g a t i v e b a r r i e r s . Recent t h e o r e t i c a l s t u d i e s have p r e d i c t e d t h a t i t s h o u l d be  to form p o s i t i v e b a r r i e r MIS  possible  j u n c t i o n s i n which the dark c u r r e n t  moderate forward b i a s i s dominated by  the i n j e c t i o n of m i n o r i t y  flow  at  carriers  i n t o the b u l k semiconductor. T h i s p r e d i c t i o n i s q u i t e remarkable, i n t h a t i t appears to c o n t r a d i c t the abundant e x p e r i m e n t a l c a t i n g t h a t the dark c u r r e n t i n n o n - i d e a l by m a j o r i t y  c a r r i e r thermionic  experiments p r o v i d i n g the ence of m i n o r i t y  emission.  Schottky  evidence  diodes i s dominated  In t h i s t h e s i s two  diodes are r e p o r t e d . The  first  experiments i n v o l v e d the measurement of the c u r r e n t - v o l t a g e istics  independent  f i r s t i n c o n t r o v e r t i b l e evidence f o r the  c a r r i e r MIS  of A l - S i O ^ - p S i diodes at v a r i o u s  indi-  exist-  of these character-  temperatures. From these meas-  urements, an a c t i v a t i o n energy d e s c r i b i n g the temperature dependence of the dark c u r r e n t was  e x t r a c t e d . T h i s a c t i v a t i o n energy was  exactly with  t h a t expected f o r a m i n o r i t y  c u r r e n t , and  to be s i g n i f i c a n t l y l a r g e r than t h a t p o s s i b l e f o r a m a j o r i t y  c a r r i e r thermionic  emission  carrier  found t o agree  injection-diffusion  c u r r e n t . In the second experiment, i t was  shown t h a t an a l l o y e d aluminum back s u r f a c e  f i e l d r e g i o n c o u l d be  used  iii  to enhance the o p e n - c i r c u i t v o l t a g e s  of A l - S i O ^ - p S i s o l a r c e l l s .  This  demonstration t h a t a m o d i f i c a t i o n t o the r e a r s u r f a c e of an MIS  solar  c e l l c o u l d a l t e r the c e l l o p e n - c i r c u i t v o l t a g e p r o v i d e d  irrefu-  t a b l e e v i d e n c e f o r the e x i s t e n c e N e g a t i v e b a r r i e r MIS these j u n c t i o n s are of use d u c t o r s . A simple MIS  of m i n o r i t y  j u n c t i o n s do not  c a r r i e r MIS  diodes.  f u n c t i o n as r e c t i f i e r s .  i n forming l o w - r e s i s t a n c e  contacts  t o semicon-  j u n c t i o n i s developed h e r e . T h i s model p r e d i c t s t h a t w i t h  low e f f e c t i v e s u r f a c e  b a r r i e r MIS  c o n t a c t which p r e s e n t s  r e c o m b i n a t i o n v e l o c i t y to m i n o r i t y  which o f f e r s n e g l i g i b l e impedance to the  t i o n were demonstrated e x p e r i m e n t a l l y  back s u r f a c e  i n induced  by  b a r r i e r MIS  field solar cells.  For both types of s u b s t r a t e , i t was b a r r i e r MIS  an enhancement i n c e l l o p e n - c i r c u i t v o l t a g e a c o n v e n t i o n a l back s u r f a c e  Induced  could  and  provide  comparable t o t h a t d i f f u s i o n or  obtained alloying.  t u n n e l j u n c t i o n o u t l i n e d above,  t h i s t h e s i s i n c l u d e s a comprehensive i n v e s t i g a t i o n of the under which the p r i n c i p l e of dark c u r r e n t and d e s c r i p t i o n of the  photocurrent  conditions superposition  c h a r a c t e r i s t i c s of homojunction  solar cells.  In p a r t i c u l a r , i t i s shown t h a t the s u p e r p o s i t i o n  should  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 r e c o m b i n a t i o n  apply  p h o t o g e n e r a t i o n occur  junc-  found t h a t the minor-  back c o n t a c t  f i e l d formed by  In a d d i t i o n t o the s t u d i e s o f the MIS  an a c c u r a t e  The  incorporating this structure  back s u r f a c e  i t y c a r r i e r r e f l e c t i n g negative  yet  carriers.  f i e l d c e l l s were s u c c e s s f u l l y f a b r i c a t e d on both n-  p-type s i l i c o n .  provides  a very  carriers,  flow of m a j o r i t y  c a r r i e r r e f l e c t i n g p r o p e r t i e s o f the n e g a t i v e  as the back c o n t a c t  a suitable  and b a r r i e r m e t a l work f u n c t i o n , i t s h o u l d  be p o s s i b l e to form a n e g a t i v e  minority  Instead,  a n a l y t i c model of c u r r e n t flow i n the n e g a t i v e b a r r i e r  c h o i c e of i n s u l a t o r t h i c k n e s s  with  further  i n the d e p l e t i o n r e g i o n . T h i s c o n t r a d i c t s  principle and  the  conclusions  drawn r e c e n t l y by  o t h e r i n v e s t i g a t o r s . I t i s a l s o found  the s u p e r p o s i t i o n p r i n c i p l e may cells ities,  that  s e r i o u s l y o v e r e s t i m a t e the e f f i c i e n c y  f a b r i c a t e d on s u b s t r a t e s w i t h v e r y poor l i f e t i m e s and a p o i n t which had not been a p p r e c i a t e d  previously.  low  mobil-  of  TABLE OF CONTENTS  PAGE:  1.  INTRODUCTION  1  2.  PHOTOVOLTAIC DEVICE THEORY  12  2.1  I n t r o d u c t i o n to P h o t o v o l t a i c Devices  12  2.2  The  15  Superposition P r i n c i p l e  2.2.1  2.2.2  P r e v i o u s Research on the Principle  Superposition 18  An A n a l y t i c D e r i v a t i o n of  the  Superposition P r i n c i p l e  20  2.2.3  Quasi-Fermi L e v e l s i n the D e p l e t i o n Region  32  2.2.4  Numerical A n a l y s i s of S i l i c o n and  GaAs  Homojunction c e l l s 2.2.5 2.3 3.  38  A Case of S u p e r p o s i t i o n  Back Surface  Breakdown  F i e l d Regions  POSITIVE BARRIER SCHOTTKY AND 3.1  J u n c t i o n B a r r i e r Heights  3.2  Tunnelling i n  MIS  59 JUNCTIONS: THEORY  64 65  Metal-Insulator-Semiconductor  Structures  70  3.2.1  The  S e m i c l a s s i c a l Model of Conduction  3.2.2  Models f o r the T u n n e l l i n g Process  71  3.2.3  Expressions  f o r the Tunnel Currents  74  3.2.4  An Estimate  f o r the T u n n e l l i n g P r o b a b i l i t y  Factor 3.3  55  The 3.3.1 3.3.2  Schottky  70  79 B a r r i e r Diode  The M a j o r i t y C a r r i e r Thermionic Current M i n o r i t y C a r r i e r Flow  83 Emission 84 89  vi  PAGE:  3.4  3.3.3  The M i n o r i t y C a r r i e r I n j e c t i o n R a t i o  92  3.3.4  Current Flow Through S u r f a c e S t a t e s  93  T r a n s i t i o n t o the MIS Diode  93  3.4.1  The M i n o r i t y C a r r i e r MIS Diode  94  3.4.2  An 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 s and  3.5  4.  Current Flows  98  The MIS S o l a r C e l l  103  3.5.1  L i g h t C o u p l i n g i n t o the Semiconductor  103  3.5.2  O p t i m a l l y E f f i c i e n t MIS C e l l s  106  3.5.3  The C h a r a c t e r i s t i c s o f T h i c k - I n s u l a t o r C e l l s  107  POSITIVE BARRIER MIS JUNCTIONS: EXPERIMENT  113  4.1  Previous Experimental  113  4.2  New E x p e r i m e n t a l MIS Diodes  Research on the MIS J u n c t i o n  Evidence  f o r Minority Carrier 118  4.3  MIS S o l a r C e l l s w i t h Back S u r f a c e F i e l d s  4.4  V a r i a t i o n o f MIS S o l a r C e l l C h a r a c t e r i s t i c s w i t h I n s u l a t o r Thickness  5.  140  MINORITY CARRIER REFLECTING NEGATIVE BARRIER MIS CONTACTS 5.1  C u r r e n t Flow i n the Negative  144  Induced Back S u r f a c e F i e l d S o l a r C e l l s on n - S i l i c o n Substrates  5.3  150  Induced Back Surface F i e l d S o l a r 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 S u b s t r a t e s  157  5.3.2  Minority Carrier Reflecting Pt-SiO^-pSi Contacts  6.  143  B a r r i e r MIS J u n c t i o n :  Theory 5.2  133  SUMMARY  161 169  APPENDIX A  N u m e r i c a l S o l u t i o n o f the B a s i c  Semiconductor  Equations APPENDIX B  C a l c u l a t i o n o f the Shadow Area f o r an E l l i p s o i d a l Constant Energy  APPENDIX C REFERENCES  Surface of A r b i t r a r y  Orientation  F a b r i c a t i o n Procedure f o r MIS J u n c t i o n s  viii  LIST OF TABLES  TABLE:  PAGE:  2.1  C e l l p r o p e r t i e s used i n n u m e r i c a l  2.2  Changes i n q u a s i - f e r m i under v a r i o u s  2.3  analysis  l e v e l s across  operating  40  depletion  region  conditions  51  True performance parameters, and those p r e d i c t e d by the s u p e r p o s i t i o n p r i n c i p l e  2.4  P r o p e r t i e s of N P +  54  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 4.1  Values o f A,  57 and  corresponding  t o the  c h a r a c t e r i s t i c s o f F i g . 4.3  130  4.2  Open-circuit voltages  f o r Al-SiO^-pSi  5.1  Open-circuit voltages  for selected P N  negative  +  cells cells  137 with  b a r r i e r MIS back c o n t a c t s  153  5.2  Open-circuit voltages  f o r s e l e c t e d N PIM and MISIM c e l l s  A.l  Normalization  A.2  Data used t o compute m o b i l i t y  185  A.3  Data used t o compute p h o t o g e n e r a t i o n d i s t r i b u t i o n  187  A.4  Parameters used f o r g r i d c o n s t r u c t i o n  191  A.5  Explanation  192  +  factors  o f v a r i a b l e s used i n FORTRAN programs  166 176  LIST OF FIGURES  FIGURE: 2.1  C u r r e n t - v o l t a g e c h a r a c t e r i s t i c s under one-sun  illumi-  n a t i o n f o r a t y p i c a l commercial  cell,  illustrating  the parameters  silicon  solar  used to d e s c r i b e c e l l  performance. 2.2  Simplified  2.3  Geometry of N P  2.4  Dark c u r r e n t - v o l t a g e 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 2.5  solar c e l l equivalent c i r c u i t . solar  cell.  (b) GaAs c e l l s .  Band diagrams f o r s i l i c o n one-sun i l l u m i n a t i o n , illumination,  2.6  (c) V=V  (a) S h o r t - c i r c u i t ,  (b) Maximum power p o i n t , one-sun i n dark.  mp  P l o t o f the t r u e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c J (V), and the curve J (a) S i l i c o n  2.7  cell,  Band diagram  - J,.(V) , under one-sun D (b) GaAs c e l l .  sc cell,  illumination,  f o r c e l l i n which most p h o t o g e n e r a t i o n  occurs  i n e m i t t e r , w h i l e most r e c o m b i n a t i o n o c c u r s i n d e p l e t i o n region,  (a) At forward b i a s i n the dark,  illumination 2.8  J  2.9  Band diagram  at same forward  3.2  3.3  bias.  - J„(V) and t r u e J ( 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 w i t h low m o b i l i t i e s and v e r y s h o r t l i f e t i m e s . +  T  for N PP +  moderate forward 3.1  (b) Under  +  back s u r f a c e f i e l d  cell  under  bias.  E q u i l i b r i u m band diagrams f o r MIS  o r n o n - i d e a l Schottky  diodes,  (a) n-type  s u b s t r a t e , (b) p-type s u b s t r a t e .  A slice  through the c o n s t a n t energy  silicon  c o n d u c t i o n band.  s u r f a c e s f o r the  Shadow o f the c o n d u c t i o n band c o n s t a n t energy s u r f a c e s for  a silicon  sample o f <100>  orientation.  X  FIGURE: 3-4  PAGE:  Band diagram f o r d e v i c e o f F i g . 3.1(a) under forward  3.5  moderate  bias.  87  Dark c u r r e n t - v o l t a g e  characteristics  f o r MIS  diodes  with  various insulator thicknesses. 3.6  97  (a) S t r u c t u r e of MIS s o l a r c e l l w i t h b a r r i e r layer,  semi-transparent  (b) S t r u c t u r e of i n v e r s i o n  layer  solar  cell. 3.7  104  Band diagram f o r t h i c k - i n s u l a t o r MIS  cell  at t e r m i n a l  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 3.8  Illuminated current-voltage c h a r a c t e r i s t i c s s o l a r c e l l s with various i n s u l a t o r  109 f o r MIS  t h i c k n e s s e s , as  p r e d i c t e d by t h e o r y . 4.1  110  Capacitance-voltage  c h a r a c t e r i s t i c f o r reverse-biased  A l - S i O - p S i dot d i o d e . x 4.2  Dark c u r r e n t - v o l t a g e  124  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 . x 4.3  4.4  J  -V c h a r a c t e r i s t i c s f o r small-area Al-SiO - p S i solar sc oc x c e l l a t v a r i o u s temperatures. (a) Temperature dependence  dependence 4.5  125  of  • 0>)  127  Temperature  of JQ-^-  131,132  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 w i t h v a r i o u s i n s u l a t o r t h i c k n e s s e s , as measured experimentally. 5.1  142  Band diagrams f o r (a) a n e g a t i v e (b) the c o r r e s p o n d i n g  b a r r i e r MIS j u n c t i o n  p o s i t i v e b a r r i e r MIS j u n c t i o n  and  formed  by d e p o s i t i n g the same metal on a s u b s t r a t e of the o p p o s i t e doping type. 5.2  Structure of P N +  c e l l w i t h n e g a t i v e b a r r i e r Mg-SiC^-pSi  back c o n t a c t . 5.3  Capacitance-voltage  152 c h a r a c t e r i s t i c f o r reverse-biased  P t - S i O - p S i dot d i o d e . x r  145  160  FIGURE:  PAGE  5.4  S t r u c t u r e o f (a) N PIM and (b) MISIM s o l a r c e l l s .  164  B.l  Shadow o f an  206  +  ellipsoid.  ACKNOWLEDGEMENT  The  research  program d e s c r i b e d i n t h i s t h e s i s c o u l d n e v e r have been  completed w i t h o u t the h e l p of many o t h e r s . 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 a s s i s t a n c e o f P e t e r A. who  s u p p l i e d the s u b s t r a t e s  thanks are a l s o due  to my  I l e s of A p p l i e d S o l a r Energy  Corporation,  used i n many of the experiments. S p e c i a l  s u p e r v i s o r , P r o f . David L. P u l f r e y , f o r 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 d i s c u s s i o n s with  f e l l o w graduate students  D a n i e l S.  Camporese, Timothy P.  Lester,  Jurgen K. K l e t a and  David J . Smith, and w i t h P r o f . Lawrence Young, h e l p e d  r e f i n e experimental  techniques  Specialized circuitry  and  clarify  the theory  greatly simplified  d i n g . Much of the vacuum system f i x t u r i n g was F l e t c h e r and  the N a t u r a l Sciences acknowledged.  junction.  f o r 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 A l a n Kot  David G.  of the MIS  the task of data  b u i l t by  recor-  technicians  Derek G. Daines. F i n a l l y , the f i n a n c i a l support and  Engineering  Research C o u n c i l i s g r a t e f u l l y  of  xiii  LIST OF  D  (D ) p  n  SYMBOLS  +  E^,  energy of c o n d u c t i o n  Ey  energy of v a l e n c e band edge metal f e r m i energy  Ep^  ^Fp^  Planck's  •h  h/2ir n  (J )  band edge  level  e l e c t r o n (hole) q u a s i - f e r m i energy  h  J  constant  n  ( p) L  (p)  N  level  [Js]  e l e c t r o n (hole) c u r r e n t d e n s i t y Boltzmann's c o n s t a n t  L^  2 —1 [m s ]  e l e c t r o n (hole) d i f f u s i o n c o e f f i c i e n t  [Am  [J °K "*"]  e l e c t r o n (hole) d i f f u s i o n l e n g t h e l e c t r o n (hole) c o n c e n t r a t i o n acceptor concentration donor c o n c e n t r a t i o n  ]  [m  [m  -3  -3  [m]  [m  ]  ]  ] _3  n.  intrinsic carrier  q  e l e c t r o n charge  S  back s u r f a c e recombination  Sp  f r o n t s u r f a c e recombination  T  a b s o l u t e temperature  V  terminal voltage  W  depletion region width  e  permittivity  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 l a y e r  Eg  p e r m i t t i v i t y of semiconductor 2 -1 -1 e l e c t r o n (hole) m o b i l i t y [m V s ]  U T  n  ( p)  n  (T ) p  U  concentration  [m  1  [C] velocity velocity  [°K]  [V] [m]  [Fm  e l e c t r o n (hole) l i f e t i m e  [s]  [ms [ms  ^] ^]  XIV  ty  electrostatic potential  £  electric field  includes  only  [V]  [Vm" ] 1  those symbols not  defined  i n text  1  CHAPTER 1 INTRODUCTION  In the p a s t decade the need to develop new  energy s u p p l i e s  r e p l a c e r a p i d l y d i m i n i s h i n g o i l and n a t u r a l gas  to  r e s e r v e s has become the  w o r l d ' s foremost t e c h n o l o g i c a l 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 a c c e p t a b l e mental viewpoint  than the d i r e c t g e n e r a t i o n  from an  of e l e c t r i c i t y  enviro-  from s u n l i g h t  through the p h o t o v o l t a i c e f f e c t . P h o t o v o l t a i c c e l l s have s u p p l i e d e l e c t r i c a l energy to v i r t u a l l y c o s t o f these d e v i c e s  has  a l l s p a c e c r a f t ever launched, but  so f a r p r e v e n t e d t h e i r use  o f l a r g e amounts o f e l e c t r i c i t y economically  competitive  with  on E a r t h . In o r d e r  f o r the  present  levels  high  generation  to make p h o t o v o l t a i c s  c o a l - f i r e d or f i s s i o n p l a n t s f o r the  e r a t i o n of e l e c t r i c i t y on a massive s c a l e , i t i s e s t i m a t e d of s o l a r c e l l s must be  the  reduced by more than an order  gen-  t h a t the  cost  of magnitude from  [1-3]. I f t h i s c o s t r e d u c t i o n i s to be brought about, a  technology f o r s o l a r c e l l p r o d u c t i o n employed i n i n d u s t r y must be ate both an i n e x p e n s i v e  radically different  developed. T h i s new  s u b s t r a t e and  a simple  from t h a t  technology must  means o f forming  now  incorporthe  r e c t i f y i n g j u n c t i o n 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 f a b r i c a t e d 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 the m i c r o e l e c t r o n i c s i n d u s t r y . The p u r i f i c a t i o n and  the  c r y s t a l l i z a t i o n of these h i g h q u a l i t y s u b s t r a t e s  l a r g e i t has been estimated electricity  amount of energy consumed i n  i s so  t h a t a t y p i c a l c e l l would have t o generate  f o r over a dozen years  the s i l i c o n i t c o n t a i n s  in  [ 4 ] . Before  just  to repay the energy used to r e f i n e  l a r g e - s c a l e p h o t o v o l t a i c power  gen-  2  e r a t i o n can become a r e a l i t y , i t i s e s s e n t i a l t h a t an i n e x p e n s i v e s u b s t r a t e be developed  on which s o l a r c e l l s of g r e a t e r than 10% e f f i c i e n c y and  an energy payback time on the o r d e r of months can be t h e r e has  fabricated.  as y e t been no s i n g l e , dramatic breakthrough  with  Although  i n the s e a r c h f o r  such a s u b s t r a t e , steady p r o g r e s s has been made i n d e v i s i n g low-cost techniques  f o r the r e f i n i n g and  r e q u i r e e x c e s s i v e energy i n p u t s  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  [ 5 ] . At p r e s e n t p r o s p e c t s appear good f o r  the e v e n t u a l development of an i n e x p e n s i v e s i l i c o n s u b s t r a t e capable  of  meeting the e f f i c i e n c y and energy payback-time goals o u t l i n e d above. There i s a f a i r l y h i g h p r o b a b i l i t y t h a t such  a f u t u r e low-cost  "solar  s i l i c o n s u b s t r a t e w i l l have a l a r g e - g r a i n e d p o l y c r y s t a l l i n e  grade"  (so-called  granular) s t r u c t u r e . Even i f an i n e x p e n s i v e s i l i c o n s u b s t r a t e becomes a v a i l a b l e ,  signif-  i c a n t r e d u c t i o n s i n s o l a r c e l l f a b r i c a t i o n c o s t s w i l l s t i l l be r e q u i r e d to make p h o t o v o l t a i c s c o m p e t i t i v e w i t h t e c h n o l o g i e s . A l l p r e s e n t day devices —  c o n v e n t i o n a l power g e n e r a t i o n  commercial s i l i c o n c e l l s  are homojunction  t h a t i s , they c o n t a i n a m e t a l l u r g i c a l j u n c t i o n formed between  p- and n-type r e g i o n s o f the same semiconductor.  The p r o c e s s o f  solid  s t a t e d i f f u s i o n i s c u r r e n t l y used to form t h i s pn j u n c t i o n . The  diffusion  i s normally  ranging  from 850  c a r r i e d out i n q u a r t z - t u b e  furnaces at temperatures  to 1000°C. In o r d e r to a v o i d c o n t a m i n a t i o n  of the s i l i c o n  with  unwanted i m p u r i t i e s , a l l s u b s t r a t e s must be s u b j e c t e d to e l a b o r a t e c l e a n i n g procedures  b e f o r e they are exposed to these h i g h temperatures.  to t h i s requirement t i v e l y slow and  f o r extreme c l e a n l i n e s s , and because i t i s a  rela-  l a b o r - i n t e n s i v e process, i t i s u n l i k e l y that s o l i d state t  d i f f u s i o n c o u l d be a v i a b l e means o f j u n c t i o n f o r m a t i o n throughput,  Due  low-cost  for future high-  c e l l p r o d u c t i o n . Moreover, s o l i d s t a t e d i f f u s i o n  3  may  prove t o be i n c o m p a t i b l e w i t h g r a n u l a r s i l i c o n s u b s t r a t e s due  t o the  p r e f e r e n t i a l d i f f u s i o n o f dopants a l o n g g r a i n b o u n d a r i e s , which l e a d s to s h o r t i n g of the c e l l  [ 6 ] . Even i f t h i s problem  ure t o the h i g h temperatures  i s not encountered,  necessary f o r d i f f u s i o n i n v a r i a b l y  m i n o r i t y c a r r i e r l i f e t i m e s i n a s u b s t r a t e [ 7 ] , and c i e n c y o f the f i n i s h e d One  thus lowers  reduces the  effi-  cell.  p r o m i s i n g a l t e r n a t i v e technique f o r s o l a r c e l l  i n v o l v e s the replacement  fabrication  of the d i f f u s e d pn j u n c t i o n w i t h a  metal-semi-  conductor j u n c t i o n , or Schottky b a r r i e r . Schottky diodes are now produced  expos-  routinely  i n the e l e c t r o n i c s i n d u s t r y by the d e p o s i t i o n of metal onto c l e a n  s i l i c o n s u r f a c e s under h i g h vacuum c o n d i t i o n s . S i n c e the metal d e p o s i t i o n i s c a r r i e d out at o r near room temperature, w i t h h i g h temperature  the c o m p l i c a t i o n s a s s o c i a t e d  p r o c e s s i n g o u t l i n e d above are a v o i d e d  completely.  F u r t h e r , the technology o f vacuum d e p o s i t i o n i s f a r more r e a d i l y  adapted  to automated mass p r o d u c t i o n than i s s o l i d s t a t e d i f f u s i o n . Because of these i n h e r e n t 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 j u n c t i o n i n s i l i c o n s o l a r c e l l s has been g i v e n serious c o n s i d e r a t i o n i n recent years [8], Schottky b a r r i e r diodes formed on s i l i c o n s u b s t r a t e s were s t u d i e d i n t e n s i v e l y i n the 1960's. By the end o f the decade an e s s e n t i a l l y  complete  u n d e r s t a n d i n g o f the mechanisms r e s p o n s i b l e f o r c o n d u c t i o n i n these d e v i c e s had been a c h i e v e d  [ 9 ] . I t was  determined  t h a t under moderate  forward  b i a s the diode c u r r e n t i s dominated by the flow o f m a j o r i t y c a r r i e r s from the semiconductor  i n t o the m e t a l . For j u n c t i o n s formed on  doped s u b s t r a t e s , t h i s m a j o r i t y c a r r i e r flow was d e s c r i b e d by Bethe's  t h e r m i o n i c e m i s s i o n theory  moderately  shown to be a c c u r a t e l y [10]. I t was  l i s h e d t h a t at forward b i a s p a r t of the diode c u r r e n t r e s u l t s  also estabfrom  the  4  flow o f m i n o r i t y minority  c a r r i e r s from the metal i n t o the semiconductor. These  c a r r i e r s can e i t h e r recombine i n the d e p l e t i o n r e g i o n o r d i f f u s e  i n t o the q u a s i - n e u t r a l base. Experiments performed by Yu and Snow r e v e a l e d t h a t under r e v e r s e b i a s o r s m a l l forward b i a s d e p l e t i o n r e g i o n -generation diodes  processes  recombination  o f t e n dominate the c u r r e n t flow i n s i l i c o n  Schottky  [11]. I n c o n t r a s t , a t h e o r e t i c a l a n a l y s i s undertaken by S c h a r f e t t e r  i n d i c a t e d t h a t only a n e g l i g i b l e f r a c t i o n o f the c u r r e n t flow r e s u l t from i n j e c t e d m i n o r i t y  c a r r i e r s d i f f u s i n g i n t o the base  p r e d i c t i o n was l a t e r c o n f i r m e d e x p e r i m e n t a l l y measured the magnitude of t h i s m i n o r i t y  injection-diffusion  Schottky  b a r r i e r emitters [13].  Yu and Snow found t h a t the m i n o r i t y c a r r i e r i n j e c t i o n r a t i o — the r a t i o o f the m i n o r i t y  [12]. T h i s  by Yu and Snow, who d i r e c t l y  carrier  c u r r e n t by c o n s t r u c t i n g t r a n s i s t o r s w i t h  should  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  that i s , t o the  -4 t o t a l diode c u r r e n t —  was l e s s than 10  over the normal range o f d i o d e  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 o f Schottky  i e n t l y summarized by comparing the c u r r e n t flows  b a r r i e r s can be conveni n a Schottky  diode w i t h  those i n a o n e - s i d e d d i f f u s e d j u n c t i o n pn diode formed on an i d e n t i c a l substrate.  ( I n a o n e - s i d e d pn diode the d i f f u s e d s u r f a c e  l a y e r , or  e m i t t e r , i s much more h e a v i l y doped than the s u b s t r a t e ) . At moderate forward b i a s the c u r r e n t i n the pn diode i s dominated by the flow o f min o r i t y c a r r i e r s s u p p l i e d from the e m i t t e r i n t o the d e p l e t i o n r e g i o n and q u a s i - n e u t r a l base. Under the same b i a s c o n d i t i o n s t h e r e i s an i d e n t i c a l minority diode,  c a r r i e r flow i n t o the d e p l e t i o n r e g i o n and base o f the Schottky  except i n t h i s d e v i c e  t r a n s f e r between the m i n o r i t y Schottky  these c a r r i e r s a r e s u p p l i e d through e l e c t r o n c a r r i e r band and the m e t a l . However, i n the  diode t h e r e e x i s t s an a d d i t i o n a l and f a r l a r g e r c u r r e n t component  5  a r i s i n g from the e m i s s i o n of m a j o r i t y c a r r i e r s from the  semiconductor  i n t o the m e t a l . T h i s t h e r m i o n i c e m i s s i o n c u r r e n t has no c o u n t e r p a r t i n the homojunction  d e v i c e , so at a g i v e n forward b i a s the dark c u r r e n t  d e n s i t y i n the Schottky diode i s f a r l a r g e r than i n the pn d i o d e . For reasons which w i l l be made c l e a r i n Chapter  2, from t h i s r e s u l t i t  f o l l o w s t h a t the o p e n - c i r c u i t v o l t a g e 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  substantially  lower than t h a t of a pn j u n c t i o n s o l a r c e l l formed on a s i m i l a r substrate  [14].  In the e a r l y 1970's i t was  discovered empirically  t h a t the open-  c i r c u i t v o l t a g e s 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  c o u l d be  greatly  i n c r e a s e d i f a v e r y t h i n oxide l a y e r were d e l i b e r a t e l y i n t r o d u c e d between the metal and  the s u b s t r a t e t o form a m e t a l - i n s u l a t o r - s e m i c o n d u c t o r  j u n c t i o n . In p a r t i c u l a r , Anderson, Delahoy and Milano solar cells  formed by d e p o s i t i n g chromium on p-type  found t h a t  (MIS)  MIS  s i l i c o n oxidized for  a few minutes a t 600°C c o u l d g i v e e f f i c i e n c i e s and o p e n - c i r c u i t v o l t a g e s 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 j u n c t i o n 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 c o n v e n t i o n a l t e c h n i q u e s a p-type  s i l i c o n s u b s t r a t e would, i n c o n t r a s t , have had  an energy  on  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 e x p l a n a t i o n f o r the remarkably  high  efficiencies  r e p o r t e d by Anderson ejt a l . for. t h e i r C r - S i O ^ - p S i s o l a r c e l l s was p r o v i d e d i n 1974  first  by Green, Shewchun and s e v e r a l co-workers a t McMaster  U n i v e r s i t y . The McMaster group employed n u m e r i c a l methods t o s o l v e f o r the c o n d u c t i o n c h a r a c t e r i s t i c s of the MIS  d i o d e , assuming t h a t c u r r e n t  flows i n these d e v i c e s as a r e s u l t of e l e c t r o n s t u n n e l l i n g  directly  between the m e t a l and  numerical  the semiconductor  bands  [16,17]. The  6  a n a l y s i s was a p p l i e d only t o those MIS diodes with  formed on s i l i c o n  substrates  s i l i c o n d i o x i d e i n s u l a t i n g l a y e r s , b u t a wide v a r i e t y o f m e t a l work  f u n c t i o n s , i n s u l a t o r t h i c k n e s s e s , and s u b s t r a t e doping l e v e l s was c o n s i d e r e d . The most i n t e r e s t i n g r e s u l t s were o b t a i n e d when the m e t a l work f u n c t i o n was s e l e c t e d t o g i v e s t r o n g i n v e r s i o n o f the semiconductor s u r f a c e a t e q u i l i b r i u m . I n such cases  i t was found t h a t over the range from  r e v e r s e b i a s t o s m a l l forward b i a s t h e diode by  c u r r e n t would be dominated  the flow o f c a r r i e r s between the m e t a l and the m i n o r i t y band i n the  semiconductor  [16]. W i t h i n  be e l e c t r i c a l l y e q u i v a l e n t MIS diodes  t h i s b i a s range the MIS j u n c t i o n would thus t o a one-sided  m e t a l l u r g i c a l pn j u n c t i o n .  i n which the main component o f c u r r e n t flow a t moderate  forward  b i a s r e s u l t s from the i n j e c t i o n o f m i n o r i t y c a r r i e r s i n t o the semiconductor were termed m i n o r i t y c a r r i e r MIS, o r minMIS, d i o d e s . The p r o s p e c t  of producing  of a pn diode by the simple  a j u n c t i o n with the e l e c t r i c a l p r o p e r t i e s  d e p o s i t i o n o f m e t a l on s i l i c o n i s o f obvious  importance n o t o n l y i n the development o f p h o t o v o l t a i c power but  generation,  f o r the m i c r o e l e c t r o n i c s i n d u s t r y as a whole. However, c o n c l u s i v e  experimental  support  f o r the e x i s t e n c e o f m i n o r i t y  c l e a r l y r e q u i r e d . The s u g g e s t i o n  c a r r i e r MIS diodes i s  t h a t the i n t r o d u c t i o n o f a v e r y  thin,  t u n n e l l a b l e i n t e r f a c i a l i n s u l a t i n g l a y e r i n a metal-semiconductor j u n c t i o n can somehow e l i m i n a t e t h e t h e r m i o n i c e m i s s i o n m i n o r i t y c a r r i e r flows  to dominate the diode  c u r r e n t and thus  allow  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 t h a t most Schottky  b a r r i e r s c o n t a i n an i n t e r -  f a c i a l l a y e r o f n a t i v e oxide u n i n t e n t i o n a l l y i n t r o d u c e d d u r i n g  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 the s u b s t r a t e i s etched oxide  diodes,  i n h y d r o f l u o r i c a c i d t o remove a l l t r a c e s o f  from the s u r f a c e , and then q u i c k l y t r a n s f e r r e d i n a i r t o the vacuum  7  system used f o r m e t a l d e p o s i t i o n . E l l i p s o m e t r y r e v e a l s t h a t a new oxide  10 A t h i c k i s formed almost immediately upon exposure  l a y e r roughly  of the e t c h e d  s i l i c o n s u r f a c e to the atmosphere  ated i n t o the j u n c t i o n . D e s p i t e  dark c u r r e n t i n s i l i c o n Schottky emission.  [18], and  the presence of t h i s t h i n  l a y e r , t h e r e i s overwhelming e x p e r i m e n t a l  thermionic  native  diodes  evidence  thus i n c o r p o r interfacial  i n d i c a t i n g that  i s dominated by m a j o r i t y  Indeed, the m e t a l - e m i t t e r  the  carrier  t r a n s i s t o r s which Yu  and  Snow employed i n t h e i r m i n o r i t y c a r r i e r i n j e c t i o n r a t i o measurements were f a b r i c a t e d on c h e m i c a l l y etched contained  s u b s t r a t e s , and  so almost  certainly  i n t e r f a c i a l n a t i v e oxide l a y e r s .  Since the m a j o r i t y of Schottky  diodes  l a y e r s , the d i s t i n c t i o n between these  contain i n t e r f a c i a l  devices  and MIS  diodes  oxide  may  appear  u n c l e a r a t t h i s p o i n t . For the moment, those j u n c t i o n s i n which an f a c i a l l a y e r i s d e l i b e r a t e l y i n t r o d u c e d between the m e t a l and conductor w i l l be a m e t a l and Schottky  termed MIS  junctions, while  a semiconductor w i l l be  diodes  c o n t a i n i n g an i n t e r f a c i a l l a y e r w i l l be  j u n c t i o n s i s suggested. The  MOS  the semi-  barriers.  termed n o n - i d e a l .  f o r d i s t i n g u i s h i n g between  Schottky  i n s u l a t o r t h i c k n e s s i n the MIS  c o n s i d e r e d here w i l l always be s m a l l enough t h a t a p p r e c i a b l e c u r r e n t s can  inter-  a l l other j u n c t i o n s between  r e f e r r e d t o as Schottky  In Chapter 3 a more p r e c i s e c r i t e r i o n and MIS  [13]  flow between the m e t a l and  c a p a c i t o r s c o n s t i t u t e a completely  diodes  tunnel  the semiconductor; t h i c k - i n s u l a t o r different  c l a s s of  device.  By mid-1978, at the commencement of the r e s e a r c h program d e s c r i b e d 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 e x p e r i m e n t a l had been p u b l i s h e d  [19]. However, no i n c o n t r o v e r t i b l e evidence  e x i s t e n c e o f m i n o r i t y c a r r i e r MIS g o a l of the p r e s e n t  data on MIS  diodes had  r e s e a r c h program was  junctions f o r the  ever been r e p o r t e d . The  first  t h e r e f o r e t o e s t a b l i s h unequiv-  8  ocally  t h a t minMIS diodes w i t h p r o p e r t i e s s u i t a b l e f o r p h o t o v o l t a i c  energy c o n v e r s i o n  c o u l d i n f a c t be made. The two independent experiments  undertaken t o accomplish  t h i s g o a l a r e d e s c r i b e d i n Chapter 4. In the  f i r s t o f these e x p e r i m e n t s , the temperature dependence o f the c u r r e n t voltage  c h a r a c t e r i s t i c s o f A l - S i O ^ - p S i diodes  was i n v e s t i g a t e d [20]. T h i s study interest indeed  formed on 10 item s u b s t r a t e s  r e v e a l e d t h a t over the b i a s range of  f o r s o l a r c e l l o p e r a t i o n the dark c u r r e n t i n these  diodes i s  dominated by the i n j e c t i o n o f m i n o r i t y c a r r i e r e l e c t r o n s from the  m e t a l i n t o the semiconductor. The second experiment i n v o l v e d the f a b r i c a t i o n o f MIS s o l a r c e l l s fields at  on 10 ficm s u b s t r a t e s i n c o r p o r a t i n g back s u r f a c e  [21]. A back s u r f a c e f i e l d  the r e a r o f a s o l a r c e l l ;  i s simply  a high-low j u n c t i o n formed  the s t r u c t u r e i s f r e q u e n t l y employed t o  i n c r e a s e the o p e n - c i r c u i t v o l t a g e  o f commercial d i f f u s e d - j u n c t i o n d e v i c e s .  For the A l - S i O - p S i c e l l s examined h e r e , x  the use o f a back s u r f a c e  field  was found t o i n c r e a s e the o p e n - c i r c u i t v o l t a g e by as much as 50 mV over the v a l u e r e c o r d e d w i t h  an ohmic back c o n t a c t . The f a c t t h a t a m o d i f i c a -  t i o n t o the r e a r s u r f a c e o f an MIS s o l a r c e l l  c o u l d produce such a d r a -  m a t i c change i n o p e n - c i r c u i t v o l t a g e p r o v i d e s  further irrefutable  evi-  dence f o r the e x i s t e n c e o f minMIS d i o d e s . Chapter 4 c l o s e s w i t h an i n v e s t i g a t i o n o f the r e l a t i o n s h i p between i n s u l a t o r t h i c k n e s s 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 A l - S i O - p S i s o l a r x c e l l s . P a r t i c u l a r a t t e n t i o n i s p a i d t o the r a t h e r b i z a r r e i l 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 of c e l l s w i t h  relatively  l a y e r s , a f e a t u r e which has been l a r g e l y o v e r l o o k e d  by o t h e r i n v e s t i g a -  t o r s . The r e s u l t s o f t h i s experiment are i n r e a s o n a b l y with  thick insulating  good agreement  the p r e d i c t i o n s o f the models p r e s e n t l y used t o d e s c r i b e the MIS  t u n n e l j u n c t i o n . I n a d d i t i o n , the data o b t a i n e d may p r o v i d e some guidance  9  i n the d e s i g n of p r a c t i c a l MIS d e v i c e s . The t h e o r e t i c a l background  f o r the experiments d e s c r i b e d i n Chapter 4  i s p r e s e n t e d i n Chapters 2 and 3. In Chapter 3 a u n i f i e d theory o f c u r r e n t flow i n S c h o t t k y b a r r i e r and MIS  j u n c t i o n s i s developed, drawing  heavily  on b o t h the r e s u l t s Green e t a l . o b t a i n e d through n u m e r i c a l a n a l y s i s 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 The m a t e r i a l p r e s e n t e d i n Chapter 3 d i f f e r s  from these e a r l i e r  [16,  [26-28].  treat-  ments i n t h a t p u r e l y a n a l y t i c methods are used i n the development  o f the  t h e o r y , and t h a t allowance i s made f o r s t r o n g i n v e r s i o n o f the  semicon-  ductor surface.  approach,  (Although Card and Rhoderick chose an a n a l y t i c  t h e i r r e s u l t s are v a l i d  only f o r the case i n which  s u r f a c e i s d e p l e t e d ) . Chapter 3 i n c l u d e s the f i r s t i n v e s t i g a t i o n of the p r o p e r t i e s o f MIS  the  semiconductor  detailed  theoretical  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 l a y e r s . I t i s found t h a t the a n a l y t i c model can e x p l a i n the unusual i l l u m i n a t e d c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s r e p o r t e d f o r t h i c k i n s u l a t o r MIS  c e l l s i n Chapter  S i n c e a l l the experiments of MIS  4. d e s c r i b e d i n t h i s t h e s i s i n v o l v e the use  j u n c t i o n s i n s o l a r c e l l s , Chapter 2 i s devoted t o a  fundamental  d i s c u s s i o n of p h o t o v o l t a i c d e v i c e t h e o r y . P a r t i c u l a r emphasis i s p l a c e d on the p r i n c i p l e o f dark c u r r e n t and p h o t o c u r r e n t s u p e r p o s i t i o n , which s t a t e s t h a t the c u r r e n t f l o w i n g i n an i l l u m i n a t e d c e l l s u b j e c t t o a b i a s V i s g i v e n by  the a l g e b r a i c sum  o f the s h o r t - c i r c u i t p h o t o c u r r e n t and  the  c u r r e n t which would flow at b i a s V i n the dark. T h i s p r i n c i p l e has s e r v e d as the t h e o r e t i c a l f o u n d a t i o n o f p h o t o v o l t a i c s s i n c e the e a r l y 1950's [29]. P r e v i o u s attempts  to j u s t i f y  the s u p e r p o s i t i o n p r i n c i p l e  [30,31] are  reviewed i n Chapter 2, and found t o c o n t a i n s e r i o u s f l a w s . In the course of c o r r e c t i n g these f l a w s , a simple argument i s developed  establishing  10  the v a l i d i t y o f the s u p e r p o s i t i o n p r i n c i p l e f o r t y p i c a l homo j u n c t i o n cells  operated i n unconcentrated s u n l i g h t  [32,33]. The c o n c l u s i o n s drawn  i n t h i s a n a l y t i c argument are then c o n f i r m e d by d i r e c t n u m e r i c a l s o l u t i o n of  the b a s i c semiconductor e q u a t i o n s f o r r e p r e s e n t a t i v e s i l i c o n  gallium arsenide s o l a r c e l l s .  and  Chapter 2 thus makes an important  b u t i o n t o p r e s e n t u n d e r s t a n d i n g of homojunction  contri-  solar c e l l operation.  By d e p o s i t i n g a low work f u n c t i o n metal on an n-type s i l i c o n  sub-  s t r a t e , or a h i g h work f u n c t i o n m e t a l on a p-type s u b s t r a t e , i t i s p o s s i b l e t o form an MIS accumulated  j u n c t i o n i n which the semiconductor  at e q u i l i b r i u m . J u n c t i o n s o f t h i s  here as n e g a t i v e b a r r i e r MIS  surface i s  type w i l l be r e f e r r e d  c o n t a c t s , t o d i s t i n g u i s h them from  t i o n a l p o s i t i v e b a r r i e r Schottky and MIS  conven-  j u n c t i o n s i n which the semi-  conductor s u r f a c e i s d e p l e t e d or i n v e r t e d . I t has t h a t n e g a t i v e b a r r i e r metal-semiconductor  to  l o n g been r e c o g n i z e d  junctions should o f f e r  little  r e s i s t a n c e t o c u r r e n t flow, and thus be o f use i n forming ohmic c o n t a c t s to semiconductors  [34]. In 1976  t h a t n e g a t i v e b a r r i e r MIS  Green,  Godfrey and Davies p o s t u l a t e d  c o n t a c t s c o u l d be produced which would p r e s e n t  e s s e n t i a l l y no impedance t o the flow of m a j o r i t y c a r r i e r s , but which would have an extremely low e f f e c t i v e s u r f a c e r e c o m b i n a t i o n v e l o c i t y f o r minority carriers  [35]. Contacts o f t h i s type would thus have  c h a r a c t e r i s t i c s analogous  t o those of m e t a l l u r g i c a l high-low  electrical junctions,  and c o u l d t h e r e f o r e be used t o form i n d u c e d back s u r f a c e f i e l d r e g i o n s i n s o l a r c e l l s . E x p e r i m e n t a l l y , Green e t a l . found t h a t n e g a t i v e b a r r i e r MIS  j u n c t i o n s c o u l d p r o v i d e low r e s i s t a n c e ohmic c o n t a c t s t o  silicon,  but n e v e r observed the p r e d i c t e d i n d u c e d back s u r f a c e f i e l d a c t i o n In  [36].  Chapter 5, the f i r s t s u c c e s s f u l f a b r i c a t i o n o f m i n o r i t y c a r r i e r  r e f l e c t i n g n e g a t i v e b a r r i e r MIS  c o n t a c t s i s r e p o r t e d . As suggested by  11  Green e_t a l . [35,36], evidence  f o r the low e f f e c t i v e s u r f a c e  v e l o c i t y of these j u n c t i o n s was  recombination  o b t a i n e d by employing them as back con-  t a c t s i n induced back s u r f a c e f i e l d s o l a r c e l l s . The c a r r i e d out i n t h i s a r e a i n v o l v e d the use  first  experiment  of n e g a t i v e b a r r i e r Mg-SiO - n S i X  back c o n t a c t s to form induced back s u r f a c e f i e l d s on P N +  cells  with  d i f f u s e d f r o n t j u n c t i o n s [37]. L a t e r , n e g a t i v e b a r r i e r platinum-MIS cont a c t s were used to c r e a t e induced back s u r f a c e f i e l d s on p-type s u b s t r a t e s . In t h i s l a t t e r experiment both d i f f u s e d N P +  silicon  and A l - S i O ^ - p S i  minMIS f r o n t j u n c t i o n s were employed. For a l l the d e v i c e s t r u c t u r e s c o n s i d e r e d i t was  found  t h a t the n e g a t i v e b a r r i e r MIS  back c o n t a c t c o u l d  p r o v i d e an enhancement i n o p e n - c i r c u i t v o l t a g e comparable t o t h a t o b t a i n e d w i t h a c o n v e n t i o n a l back s u r f a c e f i e l d Chapter  5 i n c l u d e s a simple  n e g a t i v e b a r r i e r MIS  formed by d i f f u s i o n or  alloying.  t h e o r e t i c a l a n a l y s i s o f c u r r e n t flow i n the  c o n t a c t based on a s t r a i g h t f o r w a r d e x t e n s i o n of the  r e s u l t s o b t a i n e d i n Chapter  3.  CHAPTER 2 PHOTOVOLTAIC DEVICE THEORY  T h i s 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 f o r the experiments on p h o t o v o l t a i c d e v i c e s  des-  c r i b e d i n Chapters 4 and 5. However, i t s h o u l d be emphasized t h a t the m a t e r i a l on the s u p e r p o s i t i o n p r i n c i p l e p r e s e n t e d t u t e s a s i g n i f i c a n t advance i n p r e s e n t s o l a r c e l l operation,  i n S e c t i o n 2.2 c o n s t i -  u n d e r s t a n d i n g o f homojunction  and i s thus o f importance i n i t s own r i g h t .  t i o n 2.1 o f f e r s a s h o r t i n t r o d u c t i o n t o the terminology  used i n the study  of p h o t o v o l t a i c s . S e c t i o n 2.3 c l o s e s the chapter w i t h a b r i e f of the p r o p e r t i e s o f back s u r f a c e  field  2.1  Devices  Introduction  to Photovoltaic  operated  t o supply +  doped p-type s i l i c o n  surface layer i s customarily is  a l a r g e a r e a photodiode  power t o a l o a d . The v a s t  of c e l l s produced today are p l a n a r N P d e v i c e s phorus i n t o u n i f o r m l y  examination  structures.  A p h o t o v o l t a i c c e l l or s o l a r c e l l i s simply which i s n o r m a l l y  majority  formed by d i f f u s i n g phos-  s u b s t r a t e s . The h e a v i l y doped  termed the e m i t t e r , w h i l e  r e f e r r e d t o as the base. Although s i l i c o n  the b u l k  substrate  c e l l s have been s t u d i e d  most i n t e n s i v e l y i n the p a s t and are the mainstay o f the p r e s e n t v o l t a i c s i n d u s t r y , i t s h o u l d be n o t e d t h a t a v a r i e t y o f o t h e r ductors  Sec-  photo-  semicon-  can be used f o r p h o t o v o l t a i c energy c o n v e r s i o n [ 3 ] .  The  current-voltage  c h a r a c t e r i s t i c s o f a t y p i c a l commercial  silicon  s o l a r c e l l exposed t o t e r r e s t r i a l s u n l i g h t a r e shown i n F i g . 2.1. F o l l o w ing  convention,  obtained ( p» v  m  J m  the axes have been o r i e n t e d so t h a t power output i s  for operation  p)  1  S  i n the f i r s t quadrant. The maximum power p o i n t  d e f i n e d as the o p e r a t i n g p o i n t a t which the power s u p p l i e d  13  F i g u r e 2.1  Current-voltage for  c h a r a c t e r i s t i c under one-sun  illumination  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 ,  illustrating  the parameters used t o d e s c r i b e c e l l  performance.  14  by the c e l l i s greatest. The performance of a solar c e l l i s normally summarized i n terms of four parameters: the s h o r t - c i r c u i t current J  , sc  the (Jpen-circuit voltage  v  » the f i l l factor FF and the energy conversion  e f f i c i e n c y n . The f i l l factor i s defined as the r a t i o  of the maximum  output power to the product of the s h o r t - c i r c u i t current and open-circuit voltage, and usually l i e s between 0.65 ice,  and 0.8  for a w e l l designed dev-  n i s simply the r a t i o of the maximum output power to the t o t a l  incident l i g h t power. For the purpose of measuring and comparing solar c e l l performance, various standardized representations of natural sunlight have been devised [38]. Cells designed for use i n space are normally "air  mass zero" (AMO)  tested under  i l l u m i n a t i o n , which i s equivalent to the s o l a r  irradiance just outside the atmosphere when the earth i s at i t s mean distance from the sun. The choice of a standard i l l u m i n a t i o n condition t  for  t e s t i n g t e r r e s t r i a l c e l l s i s more complicated,  since sunlight i s  attenuated and f i l t e r e d on i t s passage through the atmosphere. The  per-  formance of t e r r e s t r i a l c e l l s i s often measured under " a i r mass one" i l l u m i n a t i o n , which i s obtained when AMO at AMI  (AMI)  sunlight i s f i l t e r e d by passage  normal incidence through an atmosphere of s p e c i f i e d composition.  The  spectrum i s thus meant to simulate t y p i c a l i l l u m i n a t i o n conditions  when the sun i s d i r e c t l y overhead. When the sun i s not at the zenith, its  l i g h t must traverse a longer path through the atmosphere before  reaching the ground, and i s therefore attenuated more severely. This s i t u a t i o n i s represented by spectra with higher a i r mass numbers. For example, AM2  i l l u m i n a t i o n r e s u l t s when AMO  sunlight i s passed through the  standard atmosphere at an angle of 60° to the v e r t i c a l , so that the path traversed by the l i g h t i n the atmosphere i s twice as long as when the  15  sun  i s overhead. C e l l s under t e s t are always i l l u m i n a t e d w i t h  normal i n c i d e n c e with  to t h e i r f r o n t s u r f a c e s . For c e l l s used i n  light  at  conjunction  f o c u s s i n g m i r r o r s or l e n s e s , the r a t i o of the i n c i d e n t l i g h t  s i t y to t h a t o f u n f o c u s s e d s u n l i g h t i s termed the c o n c e n t r a t i o n  inten-  ratio,  o r "number of suns". One-sun i l l u m i n a t i o n thus r e f e r s to u n c o n c e n t r a t e d sunlight. Assuming u n i t quantum e f f i c i e n c y , the p h o t o g e n e r a t i o n r a t e G a t a d i s t a n c e x below the s u r f a c e of an i l l u m i n a t e d s o l a r c e l l i s g i v e n by  [39]  A G(x)  where M(A)  = / 0  dA  [1 - R(A)]  M(A)  a(A)  e  a  u  ;  i s the i n c i d e n t photon f l u x s p e c t r a l d e n s i t y , R(A)  f r a c t i o n of i n c i d e n t photons r e f l e c t e d by the a b s o r p t i o n  coefficient  (2.1)  x  gives  the c e l l s u r f a c e , and  the  a(A)  f o r the s u b s t r a t e m a t e r i a l used. A  is  i s the gap  wavelength of a photon w i t h In order  to minimize R(A),  energy e q u a l  to the semiconductor bandgap.  an a n t i r e f l e c t i o n c o a t i n g i s n o r m a l l y  i t e d on the f r o n t s u r f a c e of the c e l l . The magnitude of a(A) l a r g e l y by  ^ f o r v i s i b l e l i g h t , and  through A  gap  i c o n , a(A)  2.2  is  (GaAs), a(A)  of 10  10~*  m a t e r i a l such as s i l -  d e c r e a s i n g wavelength, and  i s of  -1 cm  for visible light  The  Superposition P r i n c i p l e  The  u l t i m a t e g o a l of any  to r e l a t e the  of  r i s e s almost d i s c o n t i n u o u s l y as A decreases  r i s e s much more s l o w l y w i t h  inated device  i s of the order  [40]. In c o n t r a s t , i n an i n d i r e c t gap  4 the o r d e r  i s determined  the band s t r u c t u r e of the semiconductor. For d i r e c t bandgap  m a t e r i a l s such as g a l l i u m a r s e n i d e cm  depos-  [40].  t h e o r e t i c a l study of p h o t o v o l t a i c  terminal current-voltage  c h a r a c t e r i s t i c s o f an  t o i t s b a s i c m a t e r i a l p r o p e r t i e s . In the p a s t ,  devices illum-  virtually  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 o f s o l a r c e l l performance have been based v  on the assumption t h a t the c u r r e n t J with  a voltage V maintained across  V where J ( V ) n  Equation  V )  =  J  sc "  V  i-i  (V) f l o w i n g i n an i l l u m i n a t e d c e l l  i t s terminals  by  <'  V )  2 2)  i s the c u r r e n t which would flow  (2.2)  i s given  at b i a s V i n the  i s embodied i n the s i m p l i f i e d e q u i v a l e n t  dark.  circuit  commonly  drawn f o r a s o l a r c e l l , which models the i l l u m i n a t e d c e l l by  an  current source i n p a r a l l e l with  (2.2)  a diode  been r e f e r r e d to v a r i o u s l y as the o r as the  ( F i g . 2.2).  Equation  "superposition p r i n c i p l e "  " s h i f t i n g approximation"  by  the c u r r e n t The  [32,33,41]  characteristic i s  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  axis.  s u p e r p o s i t i o n p r i n c i p l e i s a p o w e r f u l t o o l i n the d e s i g n  analysis of p h o t o v o l t a i c c e l l s , voltage  has  [31], the l a t t e r term a r i s i n g from a  g r a p h i c a l i n t e r p r e t a t i o n i n which the i l l u m i n a t e d J-V obtained  ideal  f o r when i t a p p l i e s the t e r m i n a l  c h a r a c t e r i s t i c s of a d e v i c e  l e v e l once the p h o t o c u r r e n t  to t h a t i l l u m i n a t i o n l e v e l  and  c h a r a c t e r i s t i c are determined. For homo-  j u n c t i o n c e l l s , well-known e x p r e s s i o n s the dark c u r r e n t , w h i l e  current  are s p e c i f i e d at any i l l u m i n a t i o n  corresponding  the s i n g l e dark c u r r e n t - v o l t a g e  and  are a v a i l a b l e f o r computation of  an a n a l y t i c t e c h n i q u e f o r computing the p h o t o -  c u r r e n t has been proposed by H o v e l  [39]. The  l a r g e l y reduces the task of d e s i g n i n g  superposition  principle  an o p t i m a l l y e f f i c i e n t  cell  to  t h a t of c r e a t i n g a diode s t r u c t u r e 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 , y e t which has example of the use  the minimum p o s s i b l e dark c u r r e n t .  of the s u p e r p o s i t i o n p r i n c i p l e has  already  been  An  F i g u r e 2.2  Simplified s o l a r c e l l equivalent  circuit  18  encountered i n Chapter 1. There i t was n o t e d t h a t a t a given the dark c u r r e n t f l o w i n g i n a Schottky  forward b i a s  b a r r i e r diode i s i n v a r i a b l y o r d e r s  o f magnitude g r e a t e r than t h a t which would flow i n a pn d i o d e .  Since i t  i s expected t h a t the s h o r t - c i r c u i t c u r r e n t  i n well-designed  pn j u n c t i o n s o l a r c e l l s w i l l be comparable  [ 8 ] , i t f o l l o w s from (2.2)  t h a t a Schottky  c e l l w i l l have f a r lower maximum power p o i n t and open-  c i r c u i t voltages  2.2.1  Previous  Schottky and  than a pn c e l l , and w i l l  thus be much l e s s e f f i c i e n t .  Research on the S u p e r p o s i t i o n P r i n c i p l e  In view o f the fundamental importance o f the s u p e r p o s i t i o n to the study o f p h o t o v o l t a i c s , i t i s remarkable t h a t a r i g o r o u s  principle investi-  g a t i o n o f the c o n d i t i o n s r e q u i r e d f o r (2.2) t o apply was n o t c a r r i e d out u n t i l very  r e c e n t l y . Although the p i o n e e r s  recognized  t h a t the p a r a s i t i c shunt and s e r i e s r e s i s t a n c e s  with  r e a l c e l l s could lead to deviations  i n the f i e l d  of p h o t o v o l t a i c s associated  from (2.2), they simply  a. p o s t e r i o r i t h a t an " i d e a l " c e l l w i t h no c o n t a c t leakage paths c o u l d always be r e p r e s e n t e d  r e s i s t a n c e and no shunt  by the e q u i v a l e n t  F i g . 2.2 [29]. Cummerow d i d d e r i v e an e q u a t i o n  assumed  c i r c u i t of  o f the form o f (2.2) from  f i r s t p r i n c i p l e s [30], b u t 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 t o the case i n which o n l y n e g l i g i b l e amounts o f r e c o m b i n a t i o n and p h o t o g e n e r a t i o n  occur  i n the d e p l e t i o n r e g i o n . In most p r a c t i c a l c e l l s the j u n c t i o n i s formed so c l o s e t o t h e s u r f a c e t h a t a s i g n i f i c a n t i n e v i t a b l y occurs  i n the d e p l e t i o n  region.  In 1976 Lindholm, Fossum and Burgess prehensive  f r a c t i o n o f the p h o t o g e n e r a t i o n  [31] conducted the f i r s t  com-  i n v e s t i g a t i o n o f the range o f v a l i d i t y o f (2.2), and concluded  t h a t the s u p e r p o s i t i o n p r i n c i p l e would apply c e l l provided  t o any homojunction s o l a r  t h r e e b a s i c c o n d i t i o n s were s a t i s f i e d . The f i r s t o f these  c o n d i t i o n s , t h a t the i n t e r n a l shunt and s e r i e s r e s i s t a n c e s  associated  19  with  the c e l l s h o u l d be n e g l i g i b l e , had been a p p r e c i a t e d s i n c e the  of the f i r s t  r e s e a r c h on p h o t o v o l t a i c s i n the 1950's  c o n d i t i o n was tral is,  [29]. The  t h a t the m i n o r i t y c a r r i e r c o n c e n t r a t i o n s  r e g i o n s of the c e l l s h o u l d not exceed low the m i n o r i t y c a r r i e r c o n c e n t r a t i o n s  time  second  i n the q u a s i - n e u -  injection levels —  i n these  that  r e g i o n s s h o u l d be much  l e s s than the m a j o r i t y c a r r i e r c o n c e n t r a t i o n . T h i s c o n d i t i o n i s u s u a l l y satisfied  f o r d e v i c e s exposed to one-sun i l l u m i n a t i o n , although  l e v e l i n j e c t i o n i s f r e q u e n t l y encountered i n c e l l s used i n systems. The  final  c o n d i t i o n was  last  to be v i o l a t e d i n a d e v i c e  p h o t o v o l t a i c conversion Because GaAs has  efficiency  the d e p l e t i o n r e g i o n  sunlight [1].  i n a t y p i c a l GaAs s o l a r c e l l  occurs  [42]. F u r t h e r , GaAs i s a d i r e c t bandgap m a t e r i a l , energies  g r e a t e r than E . S i n c e  GaAs homojunction c e l l s have very shallow  follows that a s i g n i f i c a n t occur  importance,  the h i g h e s t p o t e n t i a l  f o r o p e r a t i o n i n AMI  so s t r o n g l y absorbs a l l photons w i t h  efficient  recombination.  a r e l a t i v e l y wide bandgap, under normal o p e r a t i n g  c o n d i t i o n s much of the recombination  and  not  f a b r i c a t e d on a GaAs sub-  s t r a t e . Of a l l s e m i c o n d u c t o r s , t h i s m a t e r i a l has  in  and  c o n d i t i o n l i s t e d above i s o f c o n s i d e r a b l e  since i t i s l i k e l y  concentrator  t h a t the d e p l e t i o n r e g i o n s h o u l d  c o n t r i b u t e s u b s t a n t i a l l y to both p h o t o g e n e r a t i o n The  high-  front junctions, i t  f r a c t i o n of the t o t a l p h o t o g e n e r a t i o n  i n the d e p l e t i o n r e g i o n . In f a c t , the most e f f i c i e n t  must  solar cells  f a b r i c a t e d to date have been s o - c a l l e d " h e t e r o f a c e " d e v i c e s i n c o r p o r a t i n g a wide-bandgap p G a ^ ^ A l ^ A s "window" l a y e r grown by e p i t a x y over an  ex-  tremely  the  shallow  pGaAs-nGaAs homojunction  majority of photogeneration region.  [43], and  f o r these  cells  undoubtably occurs w i t h i n the d e p l e t i o n  20  2.2.2  An A n a l y t i c D e r i v a t i o n of the S u p e r p o s i t i o n P r i n c i p l e In t h i s s u b s e c t i o n ,  v a l i d i t y of Following  (2.2)  a simple  a n a l y t i c argument e s t a b l i s h i n g the  f o r t y p i c a l homojunction s o l a r c e l l s i s developed.  Lindholm e_t al_. [31] and  on a s t r a i g h t f o r w a r d e x t e n s i o n flow i n the pn diode  Cummerow [30], the argument i s based  of Shockley's s e m i n a l a n a l y s i s of  [44]. J u s t as i n these e a r l i e r t r e a t m e n t s , i t i s  assumed t h a t the s e r i e s r e s i s t a n c e a s s o c i a t e d w i t h regions  i s n e g l i g i b l e , and  these r e g i o n s  t h a t the m i n o r i t y  do not exceed low  Lindholm e t a l . and the q u a s i - f e r m i  Cummerow simply  of an i l l u m i n a t e d c e l l ,  the  quasi-neutral  c a r r i e r concentrations  assumed w i t h o u t j u s t i f i c a t i o n across  the d e p l e t i o n  t h i s assumption i s examined c r i t i c a l l y  found to be  that region  i n sub-  g r o s s l y i n a c c u r a t e i n many s i t u a t i o n s .  In the course of the a n a l y s i s i t i s shown t h a t the s u p e r p o s i t i o n can p r o v i d e  apply  —  n a t i o n occur for  Lindholm et_ a l . does  t h a t i s , when the b u l k of b o t h p h o t o g e n e r a t i o n and  recombi-  i n the d e p l e t i o n r e g i o n  support  [32]. In s u b s e c t i o n  the a n a l y t i c argument i s p r o v i d e d by  and p o t e n t i a l s w i t h i n a s o l a r c e l l .  a p p l i e d to both s i l i c o n and  2.2.4,  d i r e c t numerical  the d i f f e r e n t i a l e q u a t i o n s governing the c u r r e n t butions  principle  an e x c e l l e n t 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 n e c e s s a r y c o n d i t i o n s p e c i f i e d by not  in  i n j e c t i o n l e v e l s . However, whereas  energy l e v e l s are c o n s t a n t  s e c t i o n 2.2.3, and  current  GaAs d e v i c e s .  flows,  carrier  This numerical  In s u b s e c t i o n  s o l u t i o n of  analysis i s  2.2.5,  a n a l y s i s i s used t o show that the s u p e r p o s i t i o n p r i n c i p l e may overestimate poor m i n o r i t y  the e f f i c i e n c y o f c e l l s c a r r i e r l i f e t i m e s and  c o n d i t i o n s s e t f o r t h i n [31]  distri-  numerical  seriously  f a b r i c a t e d on m a t e r i a l w i t h  very  low m o b i l i t i e s , even i f a l l t h r e e  are s a t i s f i e d  [33]. T h i s i s not  t a n t p o i n t i n view o f the c u r r e n t i n t e r e s t i n c e l l s  an  unimpor-  f a b r i c a t e d on  inexpensive, The  low-quality  o p e r a t i o n of any  equations equations,  and  Poisson's  d e v i c e s w i l l be  equation  the f i v e b a s i c  forming  elements connected by  the s p r e a d i n g  [45]. Normally, o n l y  for solar cells.  considered here.  v e n t i o n a l l y modelled by  The  s o l a r c e l l i s governed by  of semiconductor p h y s i c s : the c o n t i n u i t y e q u a t i o n s ,  o p e r a t i o n i s of i n t e r e s t  diode  substrates.  (Real three-dimensional  one-dimensional c e l l s are  con-  one-dimensional  lumped r e s i s t a n c e s r e p r e s e n t i n g , f o r example,  r e s i s t a n c e of a s h a l l o w  b a s i c equations  current  steady-state  Further, only  networks of i d e a l i z e d  the  diffused surface layer  are l i s t e d below i n t h e i r s t e a d y - s t a t e ,  [46]). one-dimen-  s i o n a l form.  Continuity  Equations:  0 =  (l/q)(dJ  0 = -(l/q)(dJ  Current  /dx)  n TJ  P  /dx)  + G - U  (2.3)  + G - U  (2.4)  Equations:  J  n  = Q M n £ + qD (dn/dx) n  n  (2.5)  (2.6)  Poisson's  Equation:  £ = -di|)/dx;  (2.7)  As Lindholm e_t_ a l . [31] h a v e ' p o i n t e d the b a s i c e q u a t i o n s  out, a c u r s o r y i n s p e c t i o n o f  r e v e a l s t h a t the p r i n c i p l e o f dark c u r r e n t and  c u r r e n t s u p e r p o s i t i o n can not be  g e n e r a l l y c o r r e c t , at l e a s t i n a  photostrict  22  mathematical sense. When combined w i t h at  the s u r f a c e s  of a c e l l , equations  a p p r o p r i a t e boundary  conditions  (2.3)-(2.7) can be thought o f as a  system s p e c i f y i n g the t e r m i n a l c u r r e n t J as a response t o two e x c i t a t i o n s : the a p p l i e d t e r m i n a l v o l t a g e are n o n - l i n e a r ,  V and i l l u m i n a t i o n . Since  the b a s i c e q u a t i o n s  the response t o two e x c i t a t i o n s a p p l i e d  simultaneously  w i l l n o t i n g e n e r a l be e q u a l t o the sum o f the responses t o the same excitations applied separately. Fortunately, provides  the s u p e r p o s i t i o n  principle  an e x c e l l e n t approximate d e s c r i p t i o n o f c e l l b e h a v i o u r i n many  circumstances,  even though i t i s n o t r i g o r o u s l y t r u e .  Although the argument developed i n t h i s s e c t i o n i s a p p l i c a b l e t o any p l a n a r homojunction s o l a r c e l l ,  i t i s most e a s i l y p r e s e n t e d  to a s p e c i f i c d e v i c e s t r u c t u r e . The s t r u c t u r e c o n s i d e r e d ventional N P d i f f u s e d junction c e l l with +  Before  any p r o g r e s s  i n reference  here i s a con-  the geometry o f F i g . 2.3.  can be made i n s o l v i n g the b a s i c  equations,  boundary c o n d i t i o n s must be imposed on n, p and ty a t the f r o n t and back f a c e s o f the c e l l .  I t i s c o n v e n t i o n a l l y assumed t h a t a t the f r o n t s u r -  face the e l e c t r o n c o n c e n t r a t i o n and  the e l e c t r o n q u a s i - f e r m i  metal contact to the h o l e  i s f i x e d a t i t s thermal e q u i l i b r i u m v a l u e  l e v e l coincides with  t o the e m i t t e r . Analogous boundary c o n d i t i o n s a r e a p p l i e d  concentration  and q u a s i - f e r m i  These two c o n d i t i o n s a u t o m a t i c a l l y drop a c r o s s  the f e r m i l e v e l i n the  the c e l l  l e v e l a t the back s u r f a c e .  determine the e l e c t r o s t a t i c p o t e n t i a l  f o r any a p p l i e d b i a s V. F u r t h e r boundary  conditions  are imposed by s p e c i f y i n g s u r f a c e r e c o m b i n a t i o n v e l o c i t i e s f o r m i n o r i t y c a r r i e r s . At the f r o n t s u r f a c e ,  - y x ) - s p (x ) F  q  F  n  F  (2.8)  1  N  |  QUASI-NEUTRAL  1  EMITTER  |  | DEPLETION REGION  •  F i g u r e 2.3  Geometry o f N P s o l a r  P  1  QUASI-NEUTRAL  -  BASE  1  cell.  24  while  at the back s u r f a c e  Frequently  an ohmic c o n t a c t i s p r e s e n t  which case S_ -»• recombination  (2.8)  and  (2.9)  at the back of the c e l l , i n  s t a t e t h a t the m i n o r i t y  carrier  r a t e at each s u r f a c e i s p r o p o r t i o n a l to the excess m i n o r i t y  c a r r i e r c o n c e n t r a t i o n t h e r e . T h i s 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 s u p e r p o s i t i o n p r i n c i p l e i s to h o l d . F o l l o w i n g Shockley's method o f a n a l y s i s [44], the d e v i c e of F i g . has been d i v i d e d i n t o two  q u a s i - n e u t r a l regions separated  by  2.3  a depletion  or space-charge r e g i o n . In the d e p l e t i o n r e g i o n the c o n c e n t r a t i o n  of  i o n i z e d dopants i s f a r g r e a t e r than the c o n c e n t r a t i o n o f f r e e charge c a r r i e r s , w h i l e i n the q u a s i - n e u t r a l r e g i o n s dopants i s b a l a n c e d  almost e x a c t l y by  b o u n d a r i e s between these  the charge of the i o n i z e d  t h a t of the m a j o r i t y c a r r i e r s .  r e g i o n s are assumed t o be  t h e r e 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  The  a b r u p t . At e q u i l i b r i u m ,  i n the q u a s i - n e u t r a l base, but  a  l a r g e f i e l d e x i s t s i n the q u a s i - n e u t r a l e m i t t e r as a r e s u l t o f the nonuniform  doping p r o f i l e  t h e r e . In the l o w - l e v e l i n j e c t i o n regime the  a p p l i c a t i o n o f a forward  b i a s V t o the c e l l i s assumed t o r e s u l t i n the  r e d u c 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 drop a c r o s s by an amount V, w h i l e  l e a v i n g the m a j o r i t y  d i s t r i b u t i o n s i n the q u a s i - n e u t r a l r e g i o n s Perhaps the most e l e g a n t way  c a r r i e r and e l e c t r i c  field  unchanged.  to d e s c r i b e the o p e r a t i o n of a s o l a r  c e l l i s i n terms o f the c o n t i n u i t y p r i n c i p l e , Lindholm ejt a l . [31]. In the steady  the d e p l e t i o n r e g i o n  f o l l o w i n g the approach of  s t a t e , the c o n t i n u i t y p r i n c i p l e  t h a t the c u r r e n t f l o w i n g through the t e r m i n a l s of a s o l a r c e l l must  holds equal  the d i f f e r e n c e between the t o t a l r a t e of p h o t o g e n e r a t i o n and  the  total  r a t e of r e c o m b i n a t i o n i n the d e v i c e . That i s ,  J (V) L  = q /  G(x)  dx  -  q /  cell  where G(x)  dx  (2.10)  cell  i s the volume r a t e of p h o t o g e n e r a t i o n and  r a t e of r e c o m b i n a t i o n ,  U(x)  the volume  i n c l u d i n g r e c o m b i n a t i o n at the f r o n t and  b o u n d a r i e s of the c e l l . generation  U(x)  (U(x)  back  i s d e f i n e d here to account f o r the  of c a r r i e r s , as w e l l as  recombination).  Equation  thermal  (2.10) simply  s p e c i f i e s t h a t i n the steady s t a t e each photogenerated e l e c t r o n - h o l e p a i r must e i t h e r recombine w i t h i n the c e l l or c o n t r i b u t e t o c u r r e n t flow i n the e x t e r n a l c i r c u i t . From (2.10) i t can be seen t h a t the  superposition  p r i n c i p l e w i l l a c c u r a t e l y d e s c r i b e the c h a r a c t e r i s t i c s o f a g i v e n if  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  depends on the i l l u m i n a t i o n l e v e l and not on V but  on V,  and  i n t o a term which a term which depends  i s independent of i l l u m i n a t i o n .  U i s , i n g e n e r a l , a complicated t i o n s n and  f u n c t i o n of the c a r r i e r  of m i n o r i t y  concentra-  p. However, w i t h i n a q u a s i - n e u t r a l r e g i o n under l o w - l e v e l  i n j e c t i o n c o n d i t i o n s the r e c o m b i n a t i o n r a t e i s c o n t r o l l e d by c a r r i e r s , and  c a r r i e r concentration  constant  i s the r e c i p r o c a l of the m i n o r i t y  [47]. By d e f i n i t i o n ,  i n a q u a s i - n e u t r a l r e g i o n the m i n o r i t y on the excess m i n o r i t y  supply excess  the p r o p o r t i o n a l i t y  c a r r i e r l i f e t i m e x. Moreover,  c a r r i e r c u r r e n t depends  c a r r i e r concentration  even i f a b u i l t - i n e l e c t r i c f i e l d [31]. T h i s a l l o w s  the  U t h e r e f o r e becomes p r o p o r t i o n a l to the  minority  to be  cell  i s present  for low-level due  linearly  injection,  to non-uniform doping  the m i n o r i t y c a r r i e r c u r r e n t and  c o n t i n u i t y equations  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 e q u a t i o n  governing  /  the excess m i n o r i t y  carrier  distribution.  To take a s p e c i f i c example, i n the u n i f o r m l y base r e g i o n , the d r i f t can be  ignored  ing  (2.11) w i t h  i n the base i s  (2.3), the  follow-  d e s c r i b i n g the excess e l e c t r o n d i s t r i b u t i o n  obtained:  2  to s o l v e  becomes  (2.11)  the e l e c t r o n c o n t i n u i t y e q u a t i o n  d n'/dx p  In o r d e r  (2.5)  = qD (dn'/dx). n ^ n p  d i f f e r e n t i a l equation  n'(x)  P  component i n the e l e c t r o n c u r r e n t e q u a t i o n  f o r l o w - l e v e l i n j e c t i o n , so t h i s e q u a t i o n  J  Combining  doped q u a s i - n e u t r a l  (2.12),  2  = n'/(D x ) p n n  -  G(x)/D  n  .  (2.12)  i t i s n e c e s s a r y to s p e c i f y boundary  on n^ at the b o r d e r s of the base. From (2.9)  and  conditions  (2.5), at the back  contact  -D  (dn'/dx) = S n ' 0 O . n  Obtaining and  p  a p  JQ  a boundary c o n d i t i o n on n^  the d e p l e t i o n r e g i o n p r e s e n t s  problem. In  Shockley's  [44], the excess m i n o r i t y  at the b o u n d a r i e s between the d e p l e t i o n r e g i o n  the q u a s i - n e u t r a l r e g i o n s energy l e v e l s  at the b o r d e r between the base  a more d i f f i c u l t  a n a l y s i s of c u r r e n t flow i n the pn diode r i e r concentrations  (2.13)  R  are found by  assuming t h a t the  f o r b o t h c a r r i e r s are c o n s t a n t  Although Shockley's a n a l y s i s d e a l t only w i t h dark, b o t h Cummerow [30] and  across devices  Lindholm e_t a l . [31]  carand  quasi-fermi  the d e p l e t i o n operated  applied this  region.  i n the assumption  27  w i t h o u t q u a l i f i c a t i o n t o the case o f an i l l u m i n a t e d s o l a r c e l l , and added t h e f u r t h e r t a c i t assumption t h a t t h e 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 d e p l e t i o n  region  depends o n l y on the b i a s V and n o t  on the i l l u m i n a t i o n l e v e l . Taken t o g e t h e r ,  these two assumptions  t h a t a t the boundary x^ between the d e p l e t i o n  region  imply  and the q u a s i -  n e u t r a l base  n^(x ) = n p  p()  [ e x p ( q V / k T ) - 1]  (2.14)  i r r e s p e c t i v e o f the i l l u m i n a t i o n l e v e l . Although i t i s p o s s i b l e t o determine the excess e l e c t r o n 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 solving  (2.12) w i t h boundary c o n d i t i o n s  (2.13) and (2.14), n ( x ) can be p  found more e a s i l y by s u p e r p o s i n g t h e s o l u t i o n s t o t h e f o l l o w i n g two systems:  System 1:  d n'/dx p 2  2  = n'/(D x ) p n n  w i t h B.C.'s (2.13) and (2.14)  System 2:  d n'/dx p 2  2  = n'/(D x ) - G(x)/D p n n n  C l e a r l y , the s o l u t i o n t o the f i r s t  w i t h B.C.'s (2.13) and n ' ( x ) = 0.  system g i v e s  P  P  the excess e l e c t r o n  dis-  t r i b u t i o n when a b i a s V i s a p p l i e d i n the dark, w h i l e the s o l u t i o n t o the second system i s b i a s - i n d e p e n d e n t . R e c a l l i n g t h a t U i s p r o p o r t i o n a l t o n',  P  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  into  two terms, w i t h one term c o r r e s p o n d i n g t o t h e a p p l i c a t i o n o f a b i a s V in  the dark and the second, term depending o n l y  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 t o show t h a t the i n t e g r a l of U over the q u a s i - n e u t r a l e m i t t e r can be s p l i t strictly  illumination-dependent  into strictly  the e x p r e s s i o n  t o t a l hole  field,  pro-  the d r i f t  f o r the h o l e c u r r e n t can not be i g n o r e d . W r i t i n g  c o n c e n t r a t i o n i n the e m i t t e r as a sum  c e n t r a t i o n and  and  terms. S i n c e the non-uniform doping  f i l e i n t h i s r e g i o n g i v e s r i s e to a l a r g e e l e c t r i c in  bias-dependent  term the  of the e q u i l i b r i u m con-  the excess c o n c e n t r a t i o n , the h o l e c u r r e n t e q u a t i o n  (2.6)  becomes  =  The  [-qD (d ;/dx) + q P p i P ^ ] + p  [-qD (dp /dx) + q u £ p  P]  p  r i g h t hand term i n square b r a c k e t s  n0  p  i n (2.15) i s simply  n 0  ].  (2.15)  the v a l u e  of  the h o l e c u r r e n t at thermal e q u i l i b r i u m . The p r i n c i p l e of d e t a i l e d b a l ance s t a t e s t h a t t h e r e can be no net h o l e at  e q u i l i b r i u m , so t h i s term must e q u a l  c u r r e n t anywhere i n the  z e r o . S u b s t i t u t i n g the l e f t hand  term i n (2.15) i n t o the h o l e c o n t i n u i t y e q u a t i o n ing  d i f f e r e n t i a l equation  governing  cell  (2.4)  g i v e s the f o l l o w -  the excess h o l e c o n c e n t r a t i o n i n the  emitter:  X  [  d  (  p  p £  )  /  d  x  ]  - u £(dp^/dx) + p  (dp^/dx) (dD /dx)  + D ( d p ' / d x ) + G(x) p n 2  2  r  Although the q u a n t i t i e s y , p  D, p  (2.16)  p  T  p  - p'/x n p  and  £  = 0  may  a l l depend on p o s i t i o n  w i t h i n the e m i t t e r , they s h o u l d a l l be independent of p^ i n the l o w - l e v e l i n j e c t i o n regime. (2.16) i s t h e r e f o r e l i n e a r i n the excess h o l e t r a t i o n . The boundary c o n d i t i o n s on the h o l e  concen-  c o n c e n t r a t i o n i n the  quasi-  29  n e u t r a l e m i t t e r are analogous  t o those on the e l e c t r o n c o n c e n t r a t i o n i n  the q u a s i - n e u t r a l base. At the f r o n t s u r f a c e ,  - M  p  £  P  ;  + D (d ;/dx) = S p ; ( x ) p  P  F  (2.17)  F  w h i l e a t the boundary between the e m i t t e r and the d e p l e t i o n r e g i o n  P  n  ( x  n  }  =  P ( x ) [ e x p ( q V / k T ) - 1] . n 0  (2.18)  n  By analogy w i t h the s o l u t i o n f o r the e l e c t r o n d i s t r i b u t i o n i n the base, the excess h o l e d i s t r i b u t i o n i n the e m i t t e r of a f o r w a r d - b i a s e d , inated c e l l  illum-  can be found by s u p e r p o s i n g the s o l u t i o n s t o two s p e c i a l  cases o f (2.16). The f i r s t o f these s o l u t i o n s i s f o r G(x)=0 w i t h boundary conditions  (2.17) and (2.18), w h i l e the second  s o l u t i o n i s f o r non-zero  G(x) w i t h boundary c o n d i t i o n (2.17) and t h e a d d i t i o n a l c o n d i t i o n p ' ( x )=0. n n I t s h o u l d be noted t h a t w h i l e an e x p l i c i t s o l u t i o n f o r n'(x) i n the u n i P formly doped base can be o b t a i n e d t r i v i a l l y , t h e s o l u t i o n o f (2.16) f o r non-zero £  r e p r e s e n t s a f a r more d i f f i c u l t  mathematical  problem [48].  To complete the e v a l u a t i o n o f the i n t e g r a l of U over the c e l l , i t i s now n e c e s s a r y In  t o examine the i n t e g r a l of U over the d e p l e t i o n r e g i o n .  Cummerow's a n a l y s i s  [30], recombination  simply i g n o r e d . Lindholm e_t a l . [31] n o t e d U i s n o t l i n e a r i n the excess  i n the d e p l e t i o n r e g i o n was t h a t i n the d e p l e t i o n r e g i o n  c a r r i e r c o n c e n t r a t i o n s , and conse-  q u e n t l y concluded  t h a t w i t h i n t h i s r e g i o n i t would n o t i n g e n e r a l be  p o s s i b l e to s p l i t  U i n t o a term which depends o n l y on the i l l u m i n a t i o n  l e v e l and a term which depends o n l y on the a p p l i e d b i a s . On these i t was f u r t h e r concluded  grounds  t h a t the s u p e r p o s i t i o n p r i n c i p l e s h o u l d apply  30  o n l y to those d e v i c e s significantly  i n which the d e p l e t i o n r e g i o n does not  to both the p h o t o g e n e r a t i o n and  However, t h i s r e a s o n i n g method  [44]  i s not  consistent with  contribute  recombination of the use  carriers.  of Shockley's  to o b t a i n boundary c o n d i t i o n s on the excess m i n o r i t y  concentrations  carrier  at the edges of the d e p l e t i o n r e g i o n . I f i t i s assumed  t h a t the q u a s i - f e r m i  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  d e p l e t i o n r e g i o n depend o n l y on the b i a s V and not l e v e l , then the c a r r i e r c o n c e n t r a t i o n s  on the  illumination  i n t h i s r e g i o n must depend only  on V. T h i s i n t u r n i m p l i e s t h a t f o r a given b i a s V the i n t e g r a l of U over the d e p l e t i o n r e g i o n i s independent of the i l l u m i n a t i o n Combining the c o n c l u s i o n s the q u a s i - n e u t r a l r e g i o n s  and  drawn above r e g a r d i n g  level.  recombination i n  the d e p l e t i o n r e g i o n , i t f o l l o w s t h a t  i n t e g r a l of U over the e n t i r e c e l l can be d i v i d e d i n t o s t r i c t l y dependent and in  strictly  illumination-dependent  an i l l u m i n a t e d c e l l i s t h e r e f o r e g i v e n  V where J  upc  V )  =  J  upc '  V  i n t r o d u c e d by  of the symbol J J  must be  upc  and J J J  sc upc  sc  will  upc  2 19  In accordance w i t h  the  "upc"  At t h i s p o i n t the i n t r o d u c t i o n  seem redundant, s i n c e i f (2.19) h o l d s sc  flowing  <' >  Lindholm e t a l . [31], the s u b s c r i p t  i d e n t i c a l to J  f o r a l l V,  . However, the d i s t i n c t i o n between J  upc  prove u s e f u l i n the remainder of t h i s d i s c u s s i o n . While  i s the c u r r e n t i s best  may  current  V )  stands f o r "uncompensated p h o t o c u r r e n t " .  bias-  by  i s a bias-independent photocurrent.  terminology  terms. The  the  flowing i n a r e a l c e l l with "  the t e r m i n a l s  thought of as a mathematical q u a n t i t y d e f i n e d  by  shorted,  J  upc  =  q  ^  G  X  In  (  x  )  d  x  ~  q  /  F  n  U  X  (  x  )  d  x  "  1 /  F  B  u  ( > p x  d  • (2.20)  x  x  (2.20) the r e c o m b i n a t i o n r a t e U(x) i s t o be e v a l u a t e d  f o r boundary  c o n d i t i o n s n ' ( x )=0 and p ' ( x )=0 w i t h p h o t o g e n e r a t i o n d i s t r i b u t i o n G(x) p p n n appropriate  t o t h e i l l u m i n a t i o n c o n d i t i o n s under c o n s i d e r a t i o n .  Throughout the d e r i v a t i o n of (2.19) i t was assumed t h a t the bounda r i e s between the q u a s i - n e u t r a l r e g i o n s  and the d e p l e t i o n r e g i o n do n o t  move as the b i a s a p p l i e d t o the c e l l changes. T h i s i s , o f c o u r s e , n o t the case i n a r e a l c e l l , s i n c e the d e p l e t i o n r e g i o n c o n t r a c t s as the b i a s V i s i n c r e a s e d . I t might thus be expected t h a t the p h o t o c u r r e n t  ^ p U  C  d e f i n e d i n (2.20) would have some s l i g h t b i a s dependence, s i n c e x^ and x p a r e f u n c t i o n s o f V. F o r example, i n the N P c e l l used as a model h e r e , +  an i n c r e a s e i n forward b i a s r e s u l t s i n an i n c r e a s e i n the w i d t h o f the q u a s i - n e u t r a l base a t the expense o f the d e p l e t i o n r e g i o n , w h i l e the w i d t h o f the h e a v i l y - d o p e d changed. Thus i n F i g . 2.3 x  q u a s i - n e u t r a l e m i t t e r remains v i r t u a l l y unp  moves t o the l e f t w h i l e  X  r  i s stationary.  Now c o n s i d e r a f i x e d p o i n t A i n the diode chosen so t h a t x  lies  t o the  P r i g h t o f A f o r s m a l l forward b i a s and t o the l e f t o f A f o r l a r g e r forward b i a s . In the former case A l i e s all  i n the d e p l e t i o n r e g i o n , and so v i r t u a l l y  c a r r i e r s photogenerated a t A a r e c o l l e c t e d and c o n t r i b u t e t o • p T  U  C  In t h e l a t t e r case A l i e s i n t h e q u a s i - n e u t r a l base, b u t even s o most o f the c a r r i e r s photogenerated a t A w i l l  d i f f u s e t o the j u n c t i o n and be  c o l l e c t e d so l o n g as A l i e s w i t h i n an e l e c t r o n d i f f u s i o n l e n g t h o f x . p  Thus ^ p U  in  C  s h o u l d be e s s e n t i a l l y independent o f b i a s as l o n g as the change  the w i d t h o f the d e p l e t i o n r e g i o n w i t h b i a s i s s m a l l compared t o a  minority  c a r r i e r d i f f u s i o n l e n g t h i n the q u a s i - n e u t r a l base.  2.2.3  Quasi-Fermi L e v e l s The  i n the D e p l e t i o n  argument used to e s t a b l i s h (2.19) was  t i o n s t h a t the e l e c t r o s t a t i c p o t e n t i a l and in  Region based on the twin assump-  the q u a s i - f e r m i  energy l e v e l s  the d e p l e t i o n r e g i o n of a s o l a r c e l l are dependent only on  b i a s and not  on  the i l l u m i n a t i o n l e v e l . The  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  a c c u r a c y o f the  confirmed f a i r l y  remain at low should  t h a t the m i n o r i t y  easily.  Provided  Provided  regions  i n these  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 b i a s  r e s u l t i n the r e d u c t i o n  b a r r i e r by  c a r r i e r concentrations  regions V  of the i l l u m i n a t i o n l e v e l .  free c a r r i e r concentrations  i n the  r e g i o n are always n e g l i g i b l e compared t o the c o n c e n t r a t i o n dopants i n t h a t r e g i o n , i t f o l l o w s accurately describe  is  of the j u n c t i o n e l e c t r o s t a t i c p o t e n t i a l  an amount V as w e l l , r e g a r d l e s s  a l s o t h a t the  applied  assumption  t h a t the s e r i e s r e s i s t a n c e a s s o c i a t e d w i t h the q u a s i - n e u t r a l n e g l i g i b l e and  the  depletion of i o n i z e d  t h a t the d e p l e t i o n a p p r o x i m a t i o n  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  should  across  the j u n c t i o n f o r a l l i l l u m i n a t i o n c o n d i t i o n s . The  constancy o f the q u a s i - f e r m i  can b e s t be and h o l e  checked by  currents  and  l e v e l s across  the d e p l e t i o n  region  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 the  gradients  o f these energy l e v e l s [49]; s p e c i -  fically,  J  n  = y n n  VE„ Fn  (2.21)  P  = y p P  VE„ Fp  (2.22)  and  J  Qualitatively,  (2.21) r e v e a l s  t h a t the g r a d i e n t  of E  must be  large  wherever J i s l a r g e and n i s s m a l l . Thus t h e r e w i l l be a l a r g e drop i n n a c r o s s a r e g i o n through which a l a r g e e l e c t r o n c u r r e n t flows  and i n  which the e l e c t r o n c o n c e n t r a t i o n i s s m a l l . S i m i l a r remarks apply t o Quantitatively,  (2.21) and (2.22) can be used t o compute the change  i n the q u a s i - f e r m i l e v e l s a c r o s s any p a r t of a d e v i c e i f a c c u r a t e mates f o r the c u r r e n t s and c a r r i e r c o n c e n t r a t i o n s a v a i l a b l e . In the case o f a s o l a r c e l l , estimate  esti-  i n that region are  i t i s possible to obtain a f i r s t  f o r the c u r r e n t s and c a r r i e r c o n c e n t r a t i o n s  i n the d e p l e t i o n  r e g i o n under g i v e n o p e r a t i n g c o n d i t i o n s by assuming t h a t the two q u a s i fermi l e v e l s are constant  across  t h i s r e g i o n and a p p l y i n g the a n a l y s i s  o u t l i n e d above. The drops i n the q u a s i - f e r m i l e v e l s a c r o s s r e g i o n , AE_ rn  and AE_, , can then be e s t i m a t e d rp  the e s t i m a t e s  f o r A E ^ and ^  E F  p  from (2.21) and (2.22). I f  c a l c u l a t e d f o l l o w i n g t h i s procedure are  much s m a l l e r than kT, the assumption o f c o n s t a n t s e l f - c o n s i s t e n t , and p r o b a b l y  provides  clearly  approximation t o the  However, i f the c a l c u l a t i o n s  suggest t h a t l a r g e changes i n E ^ ^ and E ^ a c r o s s  the assumption o f c o n s t a n t  quasi-fermi l e v e l s i s  an a c c u r a t e  a c t u a l s o l u t i o n t o the b a s i c e q u a t i o n s .  would be r e q u i r e d t o support  the d e p l e t i o n  the e s t i m a t e d  the d e p l e t i o n r e g i o n  e l e c t r o n and h o l e  currents,  q u a s i - f e r m i l e v e l s i n the d e p l e t i o n r e g i o n i s  untenable.  Equations  (2.21) and (2.22) w i l l now be used t o i n v e s t i g a t e the  b e h a v i o u r of t h e q u a s i - f e r m i l e v e l s i n the d e p l e t i o n r e g i o n o f an N P solar c e l l . is  The b e h a v i o u r of the q u a s i - f e r m i l e v e l s when a forward  a p p l i e d i n the dark w i l l be examined f i r s t ,  i l l u m i n a t i o n w i l l be  bias  and then the e f f e c t o f  considered.  A. Forward B i a s i n the Dark Dark c u r r e n t i n a f o r w a r d - b i a s e d  N P +  c e l l normally  results  from  recombination i n e i t h e r the d e p l e t i o n r e g i o n o r the q u a s i - n e u t r a l base. The magnitude o f the i n j e c t i o n - d i f f u s i o n c u r r e n t r e s u l t i n g from recombin a t i o n i n the base i s given by  J  a  = q^lT n ' ( x ) / / T . n p p n  x  (2.23) '  To support  an i n j e c t i o n - d i f f u s i o n c u r r e n t , e l e c t r o n s must flow a l l the  way  the d e p l e t i o n r e g i o n and e n t e r the q u a s i - n e u t r a l base. On  across  moving from the e m i t t e r  across  free electron concentration  the d e p l e t i o n r e g i o n towards the base the  d e c r e a s e s by many o r d e r s  o f magnitude. T h e r e -  f o r e , f o r t h i s type o f dark c u r r e n t , the e l e c t r o n c o n c e n t r a t i o n edge o f the base i s o f g r e a t e s t importance i n d e t e r m i n i n g  near the  A E „ . However, rn  (2.23) s t a t e s t h a t J, i s p r o p o r t i o n a l t o n ' ( x ) . Thus even though d P P increases exponentially with l e t i o n region increases  V, the e l e c t r o n c o n c e n t r a t i o n  i n step so t h a t AE  i s roughly  J, d  i n the dep-  independent o f  rtl  b i a s . A s i m i l a r r e s u l t holds rent J  f o r the d e p l e t i o n r e g i o n r e c o m b i n a t i o n  f o r the magnitude o f J  . A crude e s t i m a t e rg  cur-  can be o b t a i n e d by  rg  n o t i n g t h a t the r a t e o f c a r r i e r r e c o m b i n a t i o n i s h i g h e s t  a t t h e middle  of the d e p l e t i o n r e g i o n , near the p o i n t where the e l e c t r o n and h o l e concentrations by  are e q u a l .  I f the maximum r e c o m b i n a t i o n r a t e i s m u l t i p l i e d  t h e w i d t h W o f t h e d e p l e t i o n r e g i o n one a r r i v e s a t t h e approximate  formula [50]  J  Here n n=p.  n  =  p  rg ~~ * n=p n  W  /<  2 /  V?  •  i s the e l e c t r o n (or h o l e ) c o n c e n t r a t i o n  To support  C  2  '  2  a t the p o i n t where  a d e p l e t i o n region recombination current i t i s only  * )  35  n e c e s s a r y f o r e l e c t r o n s t o t r a v e l from the e m i t t e r t o t h e c e n t e r o f the d e p l e t i o n r e g i o n , where most recombination takes p l a c e . S i m i l a r l y , h o l e s need only  flow from the base t o t h e zone n e a r the c e n t e r o f the d e p l e t i o n  r e g i o n where the r e c o m b i n a t i o n r a t e i s g r e a t e s t . Thus t h e h o l e and e l e c t r o n c u r r e n t s a r e l a r g e o n l y i n those p a r t s o f t h e d e p l e t i o n r e g i o n where the c o r r e s p o n d i n g the q u a s i - f e r m i  c a r r i e r concentrations  l e v e l s across  the d e p l e t i o n region are therefore  p r i m a r i l y by the c a r r i e r c o n c e n t r a t i o n s n a t i o n , t h a t i s by n J  a r e l a r g e as w e l l . The drops i n  a t t h e p l a c e o f maximum recombi-  . J u s t as i n t h e case o f an i n j e c t i o n - d i f f u s i o n  n=p  current, although J  fixed  J  increases exponentially with b i a s , n _p increases n  p r o p o r t i o n a t e l y , so t h a t A E ^ and A E p  p  a r e roughly  bias  independent.  B. Under I l l u m i n a t i o n When an N P s o l a r c e l l i s f o r w a r d - b i a s e d +  s u p p l i e d from the e m i t t e r recombine w i t h h o l e s and  i n the dark, e l e c t r o n s i n the d e p l e t i o n  region  i n the base. Thus throughout t h e d e p l e t i o n r e g i o n t h e e l e c t r o n  i s d i r e c t e d towards the base. To support f e r m i l e v e l must be s l i g h t l y  the e l e c t r o n q u a s i -  a t each p o i n t i n t h e d e p l e t i o n  region  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 t h e  case i f E ^ were p r e c i s e l y constant n  flow  lower a t t h e edge of the base than a t the  edge o f t h e e m i t t e r . T h e r e f o r e , the e l e c t r o n c o n c e n t r a t i o n  this  flow  across  t h i s r e g i o n . When t h e c e l l i s  exposed t o l i g h t , a t l e a s t some photogenerated e l e c t r o n s must flow  from  the base i n t o t h e d e p l e t i o n r e g i o n . T h i s photogenerated e l e c t r o n c u r r e n t opposes the c u r r e n t due t o e l e c t r o n s i n j e c t e d from the e m i t t e r , w i t h the r e s u l t t h a t the n e t e l e c t r o n flow from t h e e m i t t e r  t o t h e base i s s m a l l e r  under i l l u m i n a t i o n than i n the dark. I f t h e e l e c t r o n q u a s i - f e r m i  level  in  t h e e m i t t e r i s chosen as a r e f e r e n c e p o i n t , i t f o l l o w s t h a t everywhere  in  the d e p l e t i o n r e g i o n o f an i l l u m i n a t e d , f o r w a r d - b i a s e d  c e l l E„_ must  36  be h i g h e r  than i t was  at the same forward  b i a s i n the dark.  Therefore,  throughout the d e p l e t i o n r e g i o n the e l e c t r o n c o n c e n t r a t i o n must be i n the l i g h t than i t was  greater  at the same forward b i a s i n the dark, assuming  t h a t the e l e c t r o s t a t i c p o t e n t i a l depends o n l y on the a p p l i e d b i a s . A completely forward  analogous' argument can be developed to show t h a t , f o r a  given  b i a s , the h o l e c o n c e n t r a t i o n everywhere i n the d e p l e t i o n r e g i o n  must a l s o be  g r e a t e r under i l l u m i n a t i o n than i n the dark. T h i s  increase  i n the c o n c e n t r a t i o n of f r e e c a r r i e r s i n the d e p l e t i o n r e g i o n must l e a d to an i n c r e a s e d r a t e of recombination minority c a r r i e r concentrations r e g i o n s w i l l be the dark. As than would be  i n t h a t r e g i o n . A l s o , the  excess  at the b o u n d a r i e s of the q u a s i - n e u t r a l  g r e a t e r i n the l i g h t  than at the same forward  a r e s u l t , t h e r e w i l l be more recombination  bias i n  i n these  the case i f the q u a s i - f e r m i l e v e l s were, i n f a c t ,  regions constant  a c r o s s the d e p l e t i o n r e g i o n . Thus f o r a g i v e n forward b i a s l e s s c u r r e n t can be drawn from the c e l l t e r m i n a l s (2.19) must o v e r e s t i m a t e Although  than p r e d i c t e d by  the energy c o n v e r s i o n  (2.19) i s never s t r i c t l y  (2.19),  and  efficiency.  c o r r e c t , i t s t i l l provides  e x c e l l e n t 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 homojunction s o l a r c e l l s .  i n the d e p l e t i o n r e g i o n are  g r e a t e r under i l l u m i n a t i o n than i n the dark and  real  significantly  the t o t a l r a t e of recom-  b i n a t i o n i s comparable t o the t o t a l r a t e of p h o t o g e n e r a t i o n .  Whether or  t h i s c o n d i t i o n i s r e a l i z e d i n a g i v e n c e l l depends on the e x t e n t  which the c a r r i e r c o n c e n t r a t i o n s i l l u m i n a t i o n , and and  an  (2.19) w i l l be i n a c c u r a t e only i f at some b i a s  p o i n t the c a r r i e r c o n c e n t r a t i o n s  not  so  i n the d e p l e t i o n r e g i o n i n c r e a s e under  on the r e l a t i o n s h i p between the c a r r i e r  the recombination  to  concentrations  rate.  I t has been shown above t h a t f o r a c e l l forward  b i a s e d i n the  dark  37  the c a r r i e r flows and c a r r i e r concentrations i n the depletion region both increase exponentially with bias, with the result that AE_ and AE_ are Fn Fp roughly bias independent. In contrast, i n an illuminated s o l a r c e l l large photocurrents  flow across the depletion region even at s h o r t - c i r c u i t or  low forward b i a s . Applying  (2.21) and  (2.22) i n 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 s h i f t e d Fn Fp  sub-  s t a n t i a l l y from t h e i r positions i n the dark. This result applies to e s s e n t i a l l y any device, regardless of the choice of substrate material or doping p r o f i l e . Thus the assumption that E ^  and E  p p  are  constant  across the depletion region i s grossly i n error at low forward bias. However, as the forward bias i s increased the c a r r i e r concentrations i n the depletion region r i s e , and so the drops i n E ^ required to support  the photocurrent  and E ^  across this region  decrease. If the bias i s increased  s t i l l further, eventually an operating point w i l l be reached for which AE^  and A E  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 c h a r a c t e r i s t i c s . For most homojunction c e l l s , the bias point at which the quasi-fermi energy l e v e l s become e f f e c t i v e l y constant across the depletion region i s reached when the t o t a l rate of recombination i s s t i l l many orders of magnitude smaller than the t o t a l rate of photogeneration.  I f t h i s i s the case, then the  fact that the quasi-fermi l e v e l s are not constant across the depletion region for operation at s h o r t - c i r c u i t or low forward bias w i l l have no measureable e f f e c t on the accuracy of (2.19). Further, J indistinguishable from J  , so (2.2) and  w i l l be upc (2.19) w i l l be interchangeable.  sc I f the drops i n E ^  n  and Ep  p  across the depletion region of an i l l u m -  inated s o l a r c e l l are both much smaller than kT for operation near the  38  maximum power p o i n t , then (2.2)  should p r o v i d e  o f the d e v i c e c h a r a c t e r i s t i c s . The n u m e r i c a l next s u b s e c t i o n  a very accurate d e s c r i p t i o n  a n a l y s i s d e s c r i b e d i n the  r e v e a l s t h a t t h i s c o n d i t i o n on AE„ and Fn  AE„ a t the Fp  imum power p o i n t 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 j u n c t i o n d e v i c e s . More g e n e r a l l y , (2.2) to those and  devices  long minority  or GaAs homo-  t o be a p p l i c a b l e  f a b r i c a t e d on s u b s t r a t e s w i t h h i g h c a r r i e r m o b i l i t i e s c a r r i e r l i f e t i m e s . From (2.21) and  t h a t the drops i n E p  and  n  E  support  a given photocurrent  when y^  and y  p  i s most l i k e l y  max-  (2.22) i t i s apparent  a c r o s s the d e p l e t i o n r e g i o n r e q u i r e d to  ? p  at some s p e c i f i e d b i a s p o i n t w i l l be  small  are l a r g e , i f o t h e r d e v i c e p r o p e r t i e s are c o n s t a n t .  a d e v i c e w i t h l o n g c a r r i e r l i f e t i m e s , the c a r r i e r c o n c e n t r a t i o n s d e p l e t i o n r e g i o n can be r a t e of r e c o m b i n a t i o n  r a i s e d to r e l a t i v e l y h i g h l e v e l s b e f o r e  For  i n the the  total  becomes comparable t o the t o t a l r a t e o f photogener-  a t i o n . Thus the e f f e c t  of h i g h m o b i l i t i e s i s to minimize the i n c r e a s e i n  c a r r i e r concentrations  i n the d e p l e t i o n r e g i o n under i l l u m i n a t i o n , w h i l e  the e f f e c t of l o n g l i f e t i m e s i s t o minimize the i n c r e a s e i n t o t a l recomb i n a t i o n brought about by 2.2.4  this rise in carrier  N u m e r i c a l A n a l y s i s of S i l i c o n and In s u b s e c t i o n 2.2.2  i t was  l e v e l s were always c o n s t a n t solar c e l l ,  concentration.  GaAs Homojunction  Cells  shown t h a t i f the q u a s i - f e r m i  energy  a c r o s s the d e p l e t i o n r e g i o n of an i l l u m i n a t e d  then (2.2) would a c c u r a t e l y d e s c r i b e the c e l l  characteristics  i n the l o w - l e v e l i n j e c t i o n regime. T h i s assumption o f c o n s t a n t l e v e l s i n the d e p l e t i o n r e g i o n had been used i n a l l p r e v i o u s justify  (2.2)  ments based on  quasi-fermi  attempts  [30,31]. In s u b s e c t i o n 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 (2.21) and  argu-  (2.22) were used to show t h a t f o r o p e r a t i o n  at s h o r t - c i r c u i t o r under low  forward b i a s the q u a s i - f e r m i l e v e l s i n  to  39  f a c t v a r y s h a r p l y over the d e p l e t i o n r e g i o n of any However, i t was  subsequently  illuminated c e l l .  proposed t h a t i n c e l l s w i t h  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 Fn  reasonable  and E„ across Fp  the  dep-  l e t i o n r e g i o n would be very s m a l l f o r o p e r a t i o n n e a r the maximum power p o i n t , and  t h a t consequently  (2.2) would p r o v i d e 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 o p e r a t i n g p o i n t s . The  purpose of t h i s s u b s e c t i o n i s to use d i r e c t n u m e r i c a l  solutions  of the b a s i c semiconductor equations  to p r o v i d e q u a n t i t a t i v e s u p p o r t  the c o n c l u s i o n s  2.2.2  drawn i n s u b s e c t i o n s  chosen f o r the n u m e r i c a l Seidman and  s o l u t i o n of (2.3)-(2.7) was  t h a t developed  a silicon  c e l l resembling  the o t h e r a GaAs c e l l . two  a  the b e h a v i o u r of the e l e c t r o s t a t i c  the q u a s i - f e r m i l e v e l s w i t h i n a d e v i c e . Two  modelled, one  by  I t s h o u l d be s t r e s s e d t h a t t h i s a l g o r i t h m makes no  a r b i t r a r y assumptions c o n c e r n i n g  f o r these  algorithm  of the FORTRAN programs w r i t t e n to implement i t are  g i v e n i n Appendix A.  and  2.2.3. The  Choo [51]; a d e t a i l e d d e s c r i p t i o n of the a l g o r i t h m and  complete l i s t i n g  e n t i a l and  and  for  cells  The  those  d e v i c e s were  available  commercially,  doping p r o f i l e s and m a t e r i a l p r o p e r t i e s  are summarized i n Table  In commercial s i l i c o n c e l l s ,  2.1.  the e m i t t e r i s u s u a l l y formed by  c a r r y i n g out a phosphorus d i f f u s i o n under c o n s t a n t  surface  concentration  c o n d i t i o n s at a temperature c l o s e to 900°C. As a r e s u l t , the doping p r o f i l e i s of the  pot-  emitter  complementary e r r o r f u n c t i o n form, w i t h a phos20  phorus c o n c e n t r a t i o n o f approximately  10  3 atoms/cm  at the s u r f a c e  [52].  However, r e c e n t s t u d i e s have shown t h a t because of the bandgap n a r r o w i n g a s s o c i a t e d w i t h such h i g h i m p u r i t y c o n c e n t r a t i o n s ,  the e f f e c t i v e  doping  near the s u r f a c e o f the e m i t t e r i s c o n s i d e r a b l y lower than the a c t u a l phosphorus c o n c e n t r a t i o n  [53]. To compensate f o r t h i s e f f e c t , i n the  TABLE 2.1  C e l l p r o p e r t i e s used i n n u m e r i c a l  a) NTP  Silicon  doping  profile  analysis  Cell  Gaussian  base doping donor c o n c e n t r a t i o n a t e m i t t e r  5*10 surface  1*10  m e t a l l u r g i c a l j u n c t i o n depth  19  cm"  3  cm"  3  0.5 ym  device width carrier  1 5  250 ym  lifetimes  see Appendix A  carrier mobilities  see Appendix A  S  F  in 10  S  B  infinite  3  cm s -  1  b) P N GaAs C e l l +  doping  profile  Gaussian  base doping acceptor  concentration at emitter  m e t a l l u r g i c a l j u n c t i o n depth device width carrier  lifetimes  carrier mobilities  1*10 surface  1*10  U  cm"  19  cm"  3  3  0.2 ym 10.0 um T  n  =  10~  9  s; x P  see Appendix A  S  F  0  S  B  infinite  41  numerical model a Gaussian doping p r o f i l e with a surface concentration 19 3 of only 10  atoms/cm  was used i n the emitter. The Gaussian p r o f i l e i s  somewhat f l a t t e r near the surface than the complementary error function distribution. Since GaAs c e l l s are s t i l l i n 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 c e l l s prepared by several d i f f e r e n t techniques have shown promisingly high e f f i c i e n c i e s . 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 i n the depletion region than to accurately model a p a r t i c u l a r experimental device. To this end the photogeneration d i s t r i b u t i o n 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 i n the other regions of the c e l l . Some preliminary tests on the accuracy of the solutions to the b a s i c equations obtained using Seidman and Choo's algorithm are described i n Appendix A. These i n i t i a l tests were concerned primarily with e s t a b l i s h i n g an appropriate grid geometry for the applications considered here. To provide a further test of the c a p a b i l i t i e s of the numerical model, the dark current-voltage c h a r a c t e r i s t i c s for both the s i l i c o n and the GaAs c e l l were computed. The results are plotted i n F i g . 2.4. The c h a r a c t e r i s t i c for the s i 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  F i g u r e 2.4  Dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s . is and  drawn s o l i d ; J  r  g  and J  d  curves computed  (2.27) o r (2.28) are drawn dashed,  (a) S i l i c o n c e l l .  (b) GaAs  True J (V) curve  cell.  from (2.26)  Figure 2.4(b)  44  J  D  ( V )  =  J  0  e x  P(q /AkT) v  (2.25)  where A=1.6. In t h i s regime the diode c u r r e n t i s dominated by recombina t i o n i n t h e d e p l e t i o n r e g i o n . F o r forward b i a s e s  g r e a t e r than approx-  i m a t e l y 400 mV, r e c o m b i n a t i o n i n the q u a s i - n e u t r a l base becomes dominant and  the c u r r e n t a g a i n depends e x p o n e n t i a l l y on V, b u t w i t h an A - f a c t o r  very c l o s e t o u n i t y . Over t h e b i a s range from 250 t o 400 mV, a t r a n s i t i o n between the two regimes takes p l a c e , and exponential  does n o t have a simple s i n g l e -  dependence on V. A crude a n a l y t i c a p p r o x i m a t i o n t o t h e dep-  l e t i o n r e g i o n recombination c u r r e n t J  i s g i v e n by (2.24), which can be  w r i t t e n i n the form [50]  J  ( ) = Un.W/(2/TT)] v  r g  exp(qV/2kT)  (2.26)  Under l o w - l e v e l i n j e c t i o n c o n d i t i o n s an a n a l y t i c e x p r e s s i o n e l e c t r o n i n j e c t i o n - d i f f u s i o n current  J  d  ( V )  =  J  0d [  e x  f l o w i n g i n t o the base i s [61]  P ( q / T ) - 1] v  f o r the  k  (2.27a)  where  '0d  =  qD n f  Sinh(L /L ) + (S L /D ) Cosh(L /L )  n l  n  B  n  B  n  n  Cosh(L /L ) + (S L /D )  A  B  qD n. n  n A  n  B  Coth(L /L ). B  n  n  n  R  n  (2.27b)  Sinh(L /L ) B  n  (2.27c)  45  Here L„ i s the base width. Plots of J (V) and J,(V) obtained by subB rg d 3  s t i t u t i n g the parameters l i s t e d i n Table 2.1(a) into (2.26) and (2.27) have been superposed on F i g . 2.4(a). I t can be seen that there i s excellent agreement between the upper branch of the -^(V) c h a r a c t e r i s t i c computed using Seidman and Choo's algorithm and the analytic expression for J (V). However, the expression for J  s p e c i f i e d i n (2.26) c l e a r l y  overestimates the dark current flowing at small forward b i a s . This i s not unexpected,  for i n 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 t h i s recombination rate applies throughout the region [50]. The J ( V ) c h a r a c t e r i s t i c for the GaAs c e l l reveals the importance D  of depletion region recombination currents i n t h i s 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 i n j e c t i o n d i f f u s i o n 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  J  od  =  q  i  S i n h ( L / L ) + ( S L / D ) Cosh (1^/1, )  p D  Cosh(L /L ) + ( S L / D ) S i n h ( L / L )  D  n P  •g-^* °B  B  p  B  qD n —2—P D  p  Coth (L /L ) . P B  B  B  p  p  p  p  B  (2.28b)  p  (2.28c)  46  T h i s a n a l y t i c form f o r ^^(V) be  has  been o v e r p l o t t e d on F i g . 2 . 4 ( b ) . I t  seen t h a t even a t a forward b i a s of 900  mV,  which g i v e s a dark c u r r e n t  d e n s i t y comparable i n magnitude t o the one-sun p h o t o c u r r e n t , 25%  of the t o t a l c u r r e n t  can  l e s s than  f l o w i n g r e s u l t s from r e c o m b i n a t i o n i n the  quasi-  n e u t r a l base. More p r e c i s e checks on the accuracy  of the n u m e r i c a l  analysis  be made by examining the i n d i v i d u a l e l e c t r o n and h o l e c u r r e n t  components  w i t h i n a d e v i c e . For example, the e l e c t r o n c u r r e n t f l o w i n g a c r o s s plane x to  i n t o the q u a s i - n e u t r a l base of an N P +  p  (2.27), w h i l e  the h o l e  current  c e l l should  c r o s s i n g the p l a n e x  can  the  conform c l o s e l y  in a P N  cell  +  n should  agree w i t h  (2.28).  l o c a t i n g the p l a n e s r e g i o n and  x^ and  x ,  90%  definition, J  to be  at which the  of the donor or a c c e p t o r  (x ) f o r the s i l i c o n c e l l n p  determined and p l o t t e d i n F i g . 2.4.  For both d e v i c e s  c e l l under a v a r i e t y of b i a s and in  F i g . 2.5.  The  corresponding  same q u a l i t a t i v e form, and in  x  p  were  concentrathis  there i s e x c e l l e n t carrier  the v a l u e  computed  model.  the n u m e r i c a l  model f o r the  i l l u m i n a t i o n c o n d i t i o n s are  silicon  presented  band diagrams f o r the GaAs c e l l have  the  so are not shown. Under moderate forward b i a s  the dark, the e l e c t r o n and h o l e q u a s i - f e r m i  be very n e a r l y c o n s t a n t  Using  f o r the m i n o r i t y  the n u m e r i c a l  The band diagrams generated by  and  depletion  J (x ) f o r the GaAs c e l l were p n  f l o w i n g i n t o the base and  t h i s c u r r e n t component u s i n g  r  free-carrier  concentration. and  agreement between the a n a l y t i c e x p r e s s i o n  for  a b r u p t . Here X  are not  the planes  i n j e c t i o n - d i f f u s i o n current  t h e r e i s some ambiguity i n  s i n c e the b o u n d a r i e s between the  p  the q u a s i - n e u t r a l regions  defined a r b i t r a r i l y t i o n equals  In a r e a l d e v i c e  across  l e v e l s are b o t h found t o  the d e p l e t i o n r e g i o n . 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 s m a l l forward b i a s , E__ and Fn  E„ Fp  47  (b)  F i g u r e 2.5  Band diagrams f o r s i l i c o n (a) S h o r t - c i r c u i t , one-sun  cell. illumination.  (b) Maximum power p o i n t , one-sun i l l u m i n a t i (c) V=V  i n dark.  F i g u r e 2.5(b)  Figure 2.5(c)  50  s h i f t so as t o g r e a t l y i n c r e a s e the c a r r i e r As  concentrations  i n this  regio  the forward b i a s i s i n c r e a s e d under i l l u m i n a t i o n , t h e drops i n E„ Fn  and  a c r o s s t h e d e p l e t i o n r e g i o n become s m a l l e r ; by t h e time t h e max-  imum power p o i n t i s reached both q u a s i - f e r m i l e v e l s a r e e f f e c t i v e l y constant with  a c r o s s t h e d e p l e t i o n r e g i o n . A l l these  the c o n c l u s i o n s  r e s u l t s a r e i n agreement  drawn from the q u a l i t a t i v e arguments o f s u b s e c t i o n  2.2.3. A c t u a l v a l u e s Fig.  f o r AE_ and AE„ a r e t a b u l a t e d i n T a b l e 2.2. Fn Fp 2.6 p r e s e n t s the " t r u e " J (V) c h a r a c t e r i s t i c ( t h a t i s , t h e Li  J (V) L  curve  obtained  u s i n g the n u m e r i c a l  model) and a p l o t o f t h e curve  J s c - J„(V) under one-sun i l l u m i n a t i o n f o r both t h e s i l i c o n and t h e GaAs D c e l l . To the s c a l e o f the f i g u r e , the d i f f e r e n c e between t h e two curves i s not v i s i b l e  f o r e i t h e r d e v i c e . A more p r e c i s e measure o f t h e accuracy  o f the s u p e r p o s i t i o n p r i n c i p l e i n d e s c r i b i n g the i l l u m i n a t e d c u r r e n t v o l t a g e c h a r a c t e r i s t i c s f o r these which l i s t s output for  the t r u e v a l u e s  two c e l l s i s p r o v i d e d by Table 2.3,  of open-circuit voltage, f i l l  power and maximum power p o i n t v o l t a g e t o g e t h e r w i t h  f a c t o r , maximum the values  these q u a n t i t i e s p r e d i c t e d by (2.2). As e x p e c t e d , the t r u e maximum  output  power i s s l i g h t l y l e s s than that p r e d i c t e d by t h e s u p e r p o s i t i o n  principle,  although  the d i f f e r e n c e i s only 0.4% f o r the s i l i c o n  l e s s than 0.1% f o r the GaAs c e l l .  For the s i l i c o n  cell,  c e l l and  t h e t r u e open-  c i r c u i t v o l t a g e i s found t o be 1.5 mV l e s s than t h a t c a l c u l a t e d u s i n g (2.2), w h i l e  f o r t h e GaAs c e l l  the d i f f e r e n c e i s l e s s than 0.1 mV. In  s h o r t , F i g . 2.6 and T a b l e  2.3 i n d i c a t e t h a t the s u p e r p o s i t i o n p r i n c i p l e  provides  a 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 o f a GaAs  just  as a c c u r a t e  c e l l i n which t h e b u l k in  o f b o t h recombination  and p h o t o g e n e r a t i o n  occur  t h e d e p l e t i o n r e g i o n as i t does f o r a t y p i c a l commercial s i l i c o n  cell.  TABLE 2.2  Changes i n quasi-fermi levels across depletion region under various operating conditions  a) S i l i c o n C e l l Condition:  AE (eV) F n  A E F p  s h o r t - c i r c u i t , one-sun  0.36  max. power point, one-sun  3*10  open-circuit, one-sun  4*10  V = V , dark mp  2*10  (eV)  0.28 -4 -6 -5  9*10  -5  5*10 2*10  -6  b) GaAs C e l l Condition:  AE (eV) F n  A E F p  (eV)  s h o r t - c i r c u i t , one-sun  0.19  0.47  max. power point, one-sun  1*10  2*10  open-circuit, one-sun  2*10  V = V , dark mp  1*10  -6 -6  3*10 2*10  -3 -4 -4  30  J.  <  20 J  L  (V)  or  J  sc  E  cd CC ZD  O  o (o ZC CL.  10 J  —I—  o  200  600  400 BIAS  VOLTAGE  (mV)  (a)  F i g u r e 2.6  P l o t o f the t r u e c u r r e n t - v o l t a g e c h a r a c t e r i s t i c J ( V ) , L  and of the curve J  g  c  - J ( V ) , under one-sun i l l u m i n a t i o n . D  To the s c a l e of the f i g u r e , the two curves (a) S i l i c o n  cell.  (b) GaAs  cell.  coincide,  30  Figure 2.6(b)  TABLE 2.3  True performance parameters, and those p r e d i c t e d by the s u p e r p o s i t i o n  a) S i l i c o n  Cell  Quantity: J  [A  sc  V  oc  m~ ] 2  value:  P r e d i c t e d by  560.0  [mV]  481±2  482±2  161.4  162.1  0.814  0.815  FF  b) GaAs  Cell  Quantity:  True v a l u e :  P r e d i c t e d by  [A m~ ]  301.2  V oc  [mV]  909.1  909.1  V mp  [mV]  789±2  790±2  224.7  224.8  0.821  0.821  2  sc  P max FF  superposition  355.0 558.5  max  J  True  [mV] J  V mp  principle  [mW]  superposition  55  2.2.5  A Case o f S u p e r p o s i t i o n Breakdown For c e r t a i n d e v i c e c o n f i g u r a t i o n s the drop i n one o r b o t h q u a s i -  fermi l e v e l s across be  l a r g e even i f  at forward  the d e p l e t i o n r e g i o n a t the maximum power p o i n t may and E p  p  are e s s e n t i a l l y constant  across t h i s  region  b i a s i n the dark. T h i s c o n d i t i o n can a r i s e i f , f o r example,  a l a r g e photogenerated h o l e  c u r r e n t flows  through a p a r t o f the d e p l e t i o n  r e g i o n i n which the h o l e c o n c e n t r a t i o n i s s m a l l and through which t i v e l y l i t t l e h o l e c u r r e n t would flow  rela-  f o r o p e r a t i o n i n the dark. Such a  s i t u a t i o n might be encountered i n a d e v i c e i n which the dark c u r r e n t i s dominated by recombination e r a t i o n occurs support  i n the d e p l e t i o n r e g i o n w h i l e most photogen-  i n the e m i t t e r . I t was n o t e d i n s u b s e c t i o n 2.2.3 t h a t t o  a d e p l e t i o n r e g i o n recombination  c u r r e n t i n an N P d e v i c e , the +  e l e c t r o n flow need only be l a r g e i n t h a t p a r t o f the d e p l e t i o n r e g i o n nearest  the e m i t t e r , w h i l e  the h o l e  flow need be l a r g e only near the base  ( F i g . 2 . 7 ( a ) ) . However, i f most p h o t o g e n e r a t i o n  takes p l a c e i n the e m i t t e r  then when the d e v i c e i s i l l u m i n a t e d a l a r g e h o l e p h o t o c u r r e n t all  the way a c r o s s  In o r d e r t o support  must  flow  the d e p l e t i o n r e g i o n from the e m i t t e r t o the base. t h i s photocurrent,  the h o l e q u a s i - f e r m i l e v e l i n the  d e p l e t i o n r e g i o n must be d i s p l a c e d from i t s dark p o s i t i o n so as to g r e a t l y i n c r e a s e the h o l e c o n c e n t r a t i o n near the e m i t t e r the r e s u l t o f e x t e n d i n g  ( F i g . 2 . 7 ( b ) ) . T h i s has  the zone o f maximum recombination  from the c e n t e r  of the d e p l e t i o n r e g i o n towards the e m i t t e r . I f the c a r r i e r and m o b i l i t i e s are s u f f i c i e n t l y s m a l l , the r a t e o f t o t a l may be r a i s e d t o such an e x t e n t  lifetimes  recombination  that the true i l 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 d i f f e r measurably from those p r e d i c t e d by (2.19) o r (2.2). T h i s 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 d i s p l a y s the t r u e J ( V ) L  curve  and the curve J  - J .(V) f o r an N P GaAs c e l l w i t h +  r  the p r o p e r t i e s  ZONE OF MAXIMUM RECOMBINATION (a)  F i g u r e 2.7  Band diagram f o r c e l l i n which most p h o t o g e n e r a t i o n occurs i n e m i t t e r w h i l e most recombination occurs i n depletion region. (b)  (a) At forward b i a s i n the dark,  Under i l l u m i n a t i o n a t same forward b i a s .  TABLE 2.4  Properties are  o f N P GaAs c e l l whose  shown i n F i g . 2.8  doping p r o f i l e  abrupt  base doping  1*10  emitter  1*10  doping  metallurgical  junction  depth  cm 19  cm  20.0 um  carrier  lifetimes  T  carrier  mobilities  F' B S  -3  0.4 um  device width  S  characteristics  n  =  T p  = 2*10 v  M  n  = 0.1  ^  p  = 0.005 m  infinite  2 m  2  -  - 1 1 s  s  1  V  - 1  - 1  ; s"  1  58  F i g u r e 2.8 °  J  sc  - J_(V) D  and  GaAs c e l l w i t h  true J ( V ) L T  characteristics  low m o b i l i t i e s  (One-sun i l l u m i n a t i o n ) .  f o r an N P  and v e r y s h o r t l i f e t i m e s .  l i s t e d i n Table  2.4.  ( F i g . 2.8  d e s c r i b e d i n the p r e v i o u s v e r y s h o r t l i f e t i m e s and  was  generated u s i n g the n u m e r i c a l  subsection  and  i n Appendix A ) . The  cell  a l l r e c o m b i n a t i o n occurs  i n the d e p l e t i o n r e g i o n , w h i l e  c o r r e s p o n d to the u n i f o r m  absorption  spectrum w i t h e n e r g i e s  essentially  G(x) was  f a c t o r and  a p p l i c a t i o n of a s m a l l r e v e r s e b i a s i n c r e a s e s drawn from the c e l l , so J  and  J  upc  (2.2). A l s o ,  the p h o t o c u r r e n t  are not e q u a l .  device,  the  which  I t must be  can  empha-  sc  to the v a r i a t i o n i n the w i d t h o f the d e p l e t i o n r e g i o n  b i a s . Since  t h i s i s an N P  e m i t t e r and  the d e p l e t i o n r e g i o n does not move a p p r e c i a b l y  +  2.8  open-circuit voltage  s i z e d t h a t the breakdown of the s u p e r p o s i t i o n p r i n c i p l e e v i d e n t due  s e t to  g r e a t e r than the GaAs bandgap. F i g .  are b o t h s i g n i f i c a n t l y l e s s than those p r e d i c t e d by  i s not  found  i n the e m i t t e r of a l l photons i n  r e v e a l s t h a t f o r t h i s c e l l the t r u e f i l l  be  has  low m o b i l i t i e s , p r o p e r t i e s which might be  i n p o o r - q u a l i t y p o l y c r y s t a l l i n e s u b s t r a t e s . For t h i s d e v i c e ,  the AMI  analysis  the boundary between the  i n F i g . 2. with  quasi-neutral as the b i a s  changes. 2.3  Back S u r f a c e  Field  In a t y p i c a l N P +  Regions  silicon  i l l u m i n a t i o n i s l i m i t e d by order is  to a t t a i n h i g h e r  cell  the o p e n - c i r c u i t v o l t a g e under one-sun  r e c o m b i n a t i o n i n the q u a s i - n e u t r a l base. In  open-circuit voltages  i n c e l l s of t h i s k i n d , i t  t h e r e f o r e n e c e s s a r y to reduce the magnitude of the e l e c t r o n i n j e c t i o n -  d i f f u s i o n dark c u r r e n t that  can be  f l o w i n g i n t o the base. From (2.27) i t i s apparent  reduced by  i n c r e a s i n g the base doping N^,  m a t e r i a l p r o p e r t i e s remain c o n s t a n t . increases  as the s u b s t r a t e  Experimentally,  resistivity  However, f o r m a t e r i a l p r e p a r e d by  i f the  other  i t i s found t h a t  decreases from 10 ftcm to 0.1  V  Q C  Qcm.  the C z o c h r a l s k i t e c h n i q u e the e l e c t r o n  60  l i f e t i m e and  d i f f u s i o n length begin  to drop s h a r p l y  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 , i n c r e a s e s i n s u b s t r a t e beyond t h i s p o i n t l e a d 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 and  V  doping  current  [56,57]. oc An  a l t e r n a t i v e method f o r s u p p r e s s i n g  base i s to use  a lightly  at the r e a r of the c e l l  e l e c t r o n r e c o m b i n a t i o n i n the  doped s u b s t r a t e , but  form a high-low j u n c t i o n  [58,59]. The band diagram f o r such a s t r u c t u r e  under moderate forward b i a s i s shown i n F i g . 2.9. t h i s k i n d are known as "back s u r f a c e  High-low j u n c t i o n s  f i e l d s " , s i n c e the b u i l t - i n  f i e l d at the j u n c t i o n r e p e l s m i n o r i t y  electric  c a r r i e r s from the r e g i o n of  h i g h s u r f a c e r e c o m b i n a t i o n v e l o c i t y at the ohmic back c o n t a c t . In f o r a back s u r f a c e f i e l d s t r u c t u r e to be  of use  i n reducing  of  very order  the e l e c t r o n  i n j e c t i o n - d i f f u s i o n dark c u r r e n t , the e l e c t r o n d i f f u s i o n l e n g t h i n the base must be  s u b s t a n t i a l l y g r e a t e r than the base w i d t h . T h i s  is satisfied  i n most present-day commercial c e l l s  doped s u b s t r a t e s  of h i g h q u a l i t y  f a b r i c a t e d on  forward b i a s the e l e c t r o n c o n c e n t r a t i o n  i t can be seen t h a t at a  minority  lightly  high-low j u n c t i o n thus g r e a t l y reduces the e l e c t r o n  i n the v i c i n i t y of the back s u r f a c e , thereby s c r e e n i n g  c a r r i e r e l e c t r o n s from t h i s r e g i o n of h i g h  i t y . Provided  given  i s f a r lower i n the h e a v i l y doped  r e g i o n of the high-low j u n c t i o n than at the r e a r edge of the  concentration  lightly  [57].  From the band diagram o f F i g . 2.9,  doped r e g i o n . The  condition  the e l e c t r o n c o n c e n t r a t i o n  remains a t low  throughout the base, the high-low j u n c t i o n can be ive recombination v e l o c i t y S eff  recombination injection  d e s c r i b e d by  an  veloclevels effect-  which i s f a r lower than S„. More p r e B  c i s e l y , the magnitude of the e l e c t r o n i n j e c t i o n - d i f f u s i o n c u r r e n t the base of the back s u r f a c e  the  field  c e l l would not be  entering  a l t e r e d i f the  Figure  2.9  Band diagram f o r N PP  back s u r f a c e f i e l d  under moderate forward b i a s .  cell  62  high-low j u n c t i o n were r e p l a c e d by for  e l e c t r o n s . Hauser and  derived expressions  for S  a surface with  Dunbar [60]  and  recombination v e l o c i t y  Godlewski e_t al. [61]  have  f o r the case i n which the back s u r f a c e  field  ef f region i s uniformly  doped, w h i l e  a n a l y s i s o f the more r e a l i s t i c  Fossum [62]  has  presented  an  case i n which the back s u r f a c e  approximate field  doping  p r o f i l e i s non-uniform. Aside field fill  from an enhancement i n o p e n - c i r c u i t v o l t a g e , the back s u r f a c e  structure provides f a c t o r [62].  The  a modest i n c r e a s e i n s h o r t - c i r c u i t  increase i n s h o r t - c i r c u i t  improved c o l l e c t i o n e f f i c i e n c y the base by  for minority  current  c u r r e n t r e s u l t s from an  c a r r i e r s generated deep i n  long-wavelength photons. An e l e c t r o n photogenerated near  middle of the base has  a roughly  e q u a l p r o b a b i l i t y o f d i f f u s i n g to  f r o n t j u n c t i o n or t o the back c o n t a c t back c o n t a c t , thus make no  an e l e c t r o n r e a c h i n g  surface w i l l  r e g i o n . In a c e l l w i t h  the back s u r f a c e w i l l  c o n t r i b u t i o n to the p h o t o c u r r e n t .  a p r o p e r l y designed back s u r f a c e be  field  r e f l e c t e d , and may  c o n t r i b u t e to the p h o t o c u r r e n t . c o n d u c t i v i t y modulation due  The  recombine  the j u n c t i o n  increase i n f i l l  back s u r f a c e  to the i n c r e a s e d m i n o r i t y  field  cells.  and with  field  cell;  and  f a c t o r r e s u l t s from carrier  concentra-  t h i s e f f e c t i s important  regime.  In i n d u s t r y , a r a t h e r u n u s u a l t e c h n i q u e i s c u r r e n t l y used t o +  the  an e l e c t r o n approaching the back  o n l y i n or n e a r the h i g h - l e v e l i n j e c t i o n  +  the  an ohmic  However, i n a c e l l  e v e n t u a l l y reach  t i o n i n the base of the back s u r f a c e  cate N P P  and  Following  formation  fabri-  o f the f r o n t  j u n c t i o n , an o r g a n i c p a s t e c o n t a i n i n g m e t a l l i c aluminum i s a p p l i e d to the back o f the c e l l and 850°C f o r roughly aluminum-silicon  one  then f i r e d at a temperature of a p p r o x i m a t e l y  minute  a l l o y a few  [134]. T h i s procedure forms a l a y e r of microns»deep which behaves i n some ways l i k e  63  a h e a v i l y doped p-type r e g i o n . Crude as t h i s p r o c e s s open-circuit voltages  of N P P +  +  c e l l s formed i n t h i s way on 10 ficm  s u b s t r a t e s are t y p i c a l l y 30-50 mV h i g h e r  than those  N P devices  region  +  may appear, the  l a c k i n g a back s u r f a c e f i e l d  o f otherwise  identical  [134], Although N P +  cells  formed on h i g h q u a l i t y s u b s t r a t e s o f 1 t o 0.1 ftcm r e s i s t i v i t y may g i v e s l i g h t l y higher open-circuit voltages, N PP +  +  devices i n c o r p o r a t i n g  10 ficm s u b s t r a t e s have among the h i g h e s t energy c o n v e r s i o n of a l l s i l i c o n c e l l s . Recently fabrication  o f P NN  conversion ever  reported  ficiency P NN +  Fossum and Burgess have r e p o r t e d the  back s u r f a c e f i e l d  efficiencies  c e l l s g i v i n g the h i g h e s t  energy  and among the h i g h e s t o p e n - c i r c u i t v o l t a g e s  f o r s i l i c o n homojunction d e v i c e s +  efficiencies  [135]. I n these h i g h e f -  c e l l s the back s u r f a c e f i e l d r e g i o n i s formed by a con-  v e n t i o n a l phosphorus d i f f u s i o n  u s i n g a phosphine s o u r c e . The g e t t e r i n g  a c t i o n of t h i s phosphorus d i f f u s i o n  i s b e l i e v e d t o p l a y a major p a r t i n  e s 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 the base r e g i o n .  and d i f f u s i o n  lengths i n  64  CHAPTER 3 POSITIVE BARRIER SCHOTTKY AND MIS JUNCTIONS: THEORY  In t h i s c h a p t e r a u n i f i e d a n a l y t i c model o f c o n d u c t i o n i n Schottky b a r r i e r and MIS j u n c t i o n s i s developed.  The f o u n d a t i o n f o r t h i s model  i s e s t a b l i s h e d i n S e c t i o n s 3.1 and 3.2. In S e c t i o n 3.3 the model i s app l i e d t o compute the c u r r e n t flow i n a n o n - i d e a l Schottky d i o d e . S e c t i o n 3.4 then examines the e f f e c t o f i n c r e a s i n g the t h i c k n e s s o f the i n t e r f a c i a l l a y e r i n the n o n - i d e a l Schottky diode t o form an MIS j u n c t i o n . Assuming o n l y t h a t the i n t e r f a c i a l i n s u l a t i n g l a y e r i n the MIS j u n c t i o n p r e s e n t s r o u g h l y the same b a r r i e r t o e l e c t r o n s a t t e m p t i n g t o t u n n e l i n t o the metal it  from e i t h e r the v a l e n c e o r c o n d u c t i o n band o f the semiconductor,  i s shown t h a t the presence  o f t h i s l a y e r can v i r t u a l l y e l i m i n a t e the  flow o f e l e c t r o n s between the m a j o r i t y c a r r i e r band and the metal w h i l e l e a v i n g the n e t e l e c t r o n flow between the m i n o r i t y c a r r i e r band and the m e t a l e s s e n t i a l l y u n a l t e r e d . S e c t i o n 3.4 thus r e c o n c i l e s the p r e d i c t i o n made by Green et_ a l . t h a t m i n o r i t y c a r r i e r flows can dominate the dark c u r r e n t i n s u i t a b l y prepared MIS diodes w i t h the t h e r m i o n i c e m i s s i o n theory o f c o n d u c t i o n i n Schottky d i o d e s . T h i s i s a r a t h e r important p o i n t i n view o f s u g g e s t i o n s made r e c e n t l y t h a t c u r r e n t t r a n s p o r t proceeds by fundamentally  d i f f e r e n t mechanisms i n Schottky and MIS diodes  t i o n 3.5 extends  the model o f MIS diode o p e r a t i o n developed  to the case o f the MIS s o l a r c e l l .  [63]. Sec-  i n S e c t i o n 3.4  I t s h o u l d be acknowledged t h a t much  of the m a t e r i a l p r e s e n t e d i n t h i s c h a p t e r has been i n t r o d u c e d p r e v i o u s l y , a l b e i t i n somewhat d i f f e r e n t  form, i n p u b l i c a t i o n s by Green e_t a l . [16,17,  22-25] and by Card and Rhoderick  [26-28].  T h i s c h a p t e r c o n s i d e r s o n l y p o s i t i v e b a r r i e r Schottky and MIS j u n c tions —  t h a t i s , j u n c t i o n s i n which the semiconductor  surface i s depleted  65  o r i n v e r t e d a t e q u i l i b r i u m . J u n c t i o n s of t h i s c l a s s are formed by  depos-  i t i n g low work f u n c t i o n metals on p-type s i l i c o n s u b s t r a t e s , o r h i g h work f u n c t i o n metals on n-type s u b s t r a t e s . The p r o p e r t i e s of n e g a t i v e b a r r i e r MIS  3.1  c o n t a c t s are examined i n Chapter 5.  Junction B a r r i e r Heights The band diagrams  c o n v e n t i o n a l l y drawn t o r e p r e s e n t the a v a i l a b l e  e l e c t r o n energy s t a t e s i n a t y p i c a l MIS shown i n F i g . 3.1  or n o n - i d e a l Schottky diode are  [64]. In c o n s t r u c t i n g F i g . 3.1(a), i t has been assumed  t h a t the j u n c t i o n was  formed by d e p o s i t i n g a h i g h work f u n c t i o n metal on  an n-type s i l i c o n s u b s t r a t e . C o n v e r s e l y , i f the j u n c t i o n had been  formed  by d e p o s i t i n g a low work f u n c t i o n metal on a p-type s u b s t r a t e , the band diagram o f F i g . 3.1(b) would be a p p r o p r i a t e . For an n-type the j u n c t i o n b a r r i e r h e i g h t q<b  substrate,  i s d e f i n e d t o be the d i f f e r e n c e i n energy Bn  between the c o n d u c t i o n band edge at the semicondcutor s u r f a c e and the f e r m i l e v e l i n the m e t a l , as shown i n F i g . 3 . 1 ( a ) . For a p-type s u b s t r a t e , the b a r r i e r h e i g h t qa>  i s the energy d i f f e r e n c e between the v a l e n c e Bp  band edge at the s u r f a c e and the m e t a l f e r m i l e v e l , as shown i n F i g . 3.1(b). Both these energy d i f f e r e n c e s are t o be measured a t e q u i librium. Schottky o r i g i n a l l y proposed t h a t 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 o r MIS  junction results  from  the d i f f e r e n c e between the work f u n c t i o n dS of the m e t a l and the e l e c t r o n M affinity  o f the semiconductor  [65]. T h i s model undoubtably has some  r e l e v a n c e f o r j u n c t i o n s i n which an i n t e r f a c i a l l a y e r i s p r e s e n t , although the use of concepts such as work f u n c t i o n and e l e c t r o n a f f i n i t y  seems  fundamentally s u s p e c t when the m e t a l and semiconductor are s e p a r a t e d by an o x i d e f i l m o n l y a few atomic diameters t h i c k . Bardeen l a t e r p o s t u l a t e d  F i g u r e 3.1  E q u i l i b r i u m band diagram f o r MIS diodes.  (a) n-type s u b s t r a t e .  or n o n - i d e a l  Schottky  (b) p-type s u b s t r a t e .  (b)  Figure 3.1(b)  68  t h a t charge s t o r e d i n e l e c t r o n s t a t e s l o c a l i z e d a t the i n t e r f a c e p l a y a major p a r t i n forming the b a r r i e r  [66]. I t i s today  could  generally  accepted t h a t the e f f e c t s o f both b a r r i e r m e t a l work f u n c t i o n and s u r f a c e s t a t e charge can be o f importance i n d e t e r m i n i n g the j u n c t i o n b a r r i e r h e i g h t [67]. For Schottky diodes f a b r i c a t e d on f r e s h l y - e t c h e d s i l i c o n s u b s t r a t e s the b a r r i e r m e t a l work f u n c t i o n and s u r f a c e s t a t e charge are found to be of r o u g h l y e q u a l importance i n d e t e r m i n i n g the b a r r i e r h e i g h t . The i n t e r f a c e s t a t e d i s t r i b u t i o n f o r s i l i c o n s u r f a c e s p r e p a r e d i n t h i s way i s such that d>_ tends t o be s u b s t a n t i a l l y l a r g e r than Bn ° Y  Bp  .  T  Bn  i s found t o  range from about 0.5 V f o r low work f u n c t i o n metals t o 0.9 V f o r h i g h work f u n c t i o n m e t a l s , w h i l e <> j i s u s u a l l y l e s s than 0.6 V [64]. I t i s Bp thus n o t p o s s i b l e t o o b t a i n s t r o n g i n v e r s i o n a t the s u r f a c e o f a  p-type  s u b s t r a t e , n o r accumulation a t t h e s u r f a c e o f an n-type s u b s t r a t e , i n a c o n v e n t i o n a l Schottky b a r r i e r d i o d e . For those MIS diodes formed o x i d i z e d a t temperatures  on s i l i c o n s u b s t r a t e s which have been  g r e a t e r than about 400°C p r i o r t o m e t a l l i z a t i o n ,  the a v a i l a b l e e x p e r i m e n t a l d a t a i n d i c a t e t h a t the j u n c t i o n b a r r i e r h e i g h t i s determined almost e x c l u s i v e l y by the b a r r i e r m e t a l work f u n c t i o n [68]. With an a p p r o p r i a t e c h o i c e o f b a r r i e r m e t a l i t i s p o s s i b l e t o o b t a i n c o n d i t i o n s r a n g i n g from accumulation t o s t r o n g i n v e r s i o n at the semicond u c t o r s u r f a c e i n an MIS diode p r e p a r e d i n t h i s way, i r r e s p e c t i v e of the s u b s t r a t e doping t y p e . I t appears t h a t the growth o f the t h i n oxide l a y e r which forms the i n t e r f a c i a l i n s u l a t o r somehow p a s s i v a t e s the s i l i c o n s u r f a c e , g r e a t l y r e d u c i n g the d e n s i t y o f i n t e r f a c e Throughout  states.  t h i s c h a p t e r , r e f e r e n c e 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^ a p p e a r i n g a c r o s s the semiconductor and the drop iJ>  T  69  a c r o s s the i n s u l a t o r i n an MIS  or Schottky d i o d e . For a diode formed on  n-type m a t e r i a l , ty^ and ty^ w i l l be d e f i n e d t o be p o s i t i v e when these p o t 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 s u b s t r a t e ty and ty w i l l be d e f i n e d as p o s i t i v e i n F i g . 3.1(b). The e q u i l i b r i u m v a l u e s o f ty and ty w i l l be denoted  *so  311(1  as  *I0'  The a p p l i c a t i o n of a b i a s V t o a Schottky o r MIS  diode  displaces  the f e r m i l e v e l at the base c o n t a c t r e l a t i v e t o the metal f e r m i l e v e l by an amount qV.  (By c o n v e n t i o n , V i s taken t o be p o s i t i v e  f o r forward b i a s ) .  T h i s i n t u r n 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 b u t i o n a c r o s s the j u n c t i o n . Here the change hty a c r o s s the semiconductor  A  The  change Aty  ( V )  =  drop  i s d e f i n e d by  *S  ( V )  ~ ^SO  ( 3  -  1 )  i n the p o t e n t i a l drop a c r o s s the i n s u l a t o r i s d e f i n e d  a n a l o g o u s l y . The i n the  *S  i n the p o t e n t i a l  distri-  r e l a t i o n s h i p between Aty  and V i s f r e q u e n t l y w r i t t e n  form  N| s  = V/n  where n i s termed the "diode f a c t o r " or " i d e a l i t y  (3.2)  factor".  (3.2) i s  u s e f u l because n i s o f t e n found t o be e s s e n t i a l l y c o n s t a n t over the normal o p e r a t i n g b i a s range. For an i d e a l Schottky diode w i t h a v a n i s h 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 z e r o , so t h a t n i s u n i t y . For the more g e n e r a l case i n which the p o t e n t i a l drop a c r o s s the i n s u l a t o r can not be i g n o r e d , Card and Rhoderick have d e r i v e d a simple e x p r e s s i o n  f o r n a p p l i c a b l e when the semiconductor s u r f a c e  i s depleted  [26].  Unfor-  t u n a t e l y , no c l o s e d s o l u t i o n s f o r ty and ty a r e a v a i l a b l e f o r t h e case i n which the semiconductor s u r f a c e t e c h n i q u e f o r d e t e r m i n i n g Aty Section  3.2  and AiJ)  i n t h i s case w i l l be examined i n  3.4 [ 1 7 ] .  Tunnelling  3.2.1  i s strongly inverted. A possible  i n Metal-Insulator-Semiconductor  Structures  The S e m i c l a s s i c a l Model o f Conduction  A d i r e c t quantum-mechanical treatment o f c o n d u c t i o n i n e i t h e r i  Schottky b a r r i e r o r MIS diodes i s m a t h e m a t i c a l l y i n t r a c t a b l e . F o r t h i s reason the a n a l y s i s o f c u r r e n t on an e x t e n s i o n  flow i n these d e v i c e s  i s i n v a r i a b l y based  o f the s e m i c l a s s i c a l model of e l e c t r o n dynamics used, a  f o r example, i n the development o f the Boltzmann t r a n s p o r t The  main f e a t u r e s The  o f t h i s model w i l l now be very b r i e f l y  equation [69].  reviewed.  s e m i c l a s s i c a l model of e l e c t r o n dynamics i s based on a knowledge  of the band s t r u c t u r e E ( k ) i n the independent e l e c t r o n  approximation.  E l e c t r o n s moving under the combined i n f l u e n c e o f the p e r i o d i c p o t e n t i a l V(r)  lattice  and an e x t e r n a l l y a p p l i e d e l e c t r o m a g n e t i c f i e l d a r e then  r e p r e s e n t e d as wavepackets c o n s t r u c t e d  from the e i g e n s t a t e s  o f the  independent e l e c t r o n H a m i l t o n i a n f o r the unperturbed c r y s t a l . Thus the wave f u n c t i o n  ty(r,t)  of an e l e c t r o n i s w r i t t e n i n the form  * ( r , t ) = / d k F(k,t) 3  l£(r)  (3.3a)  where the s t a t e s ty+ axe B l o c h waves. I f the e x t e r n a l l y a p p l i e d are weak and s l o w l y allowing  v a r y i n g i n time, i n t r a b a n d  fields  t r a n s i t i o n s can be  ignored,  the expansion t o be r e s t r i c t e d t o the s t a t e s o f a s i n g l e band.  Further,  i n t h i s case Wannier's theorem s t a t e s t h a t the envelope f u n c t i o n  F obeys an e f f e c t i v e S c h r o d i n g e r e q u a t i o n  [E(-iV) + H  where H  ext  e x t  ]F(r,t)  [69,70]  = i*[9F(r,t)/3t]  (3.3b)  i s the c o n t r i b u t i o n t o the H a m i l t o n i a n from the e x t e r n a l  f i e l d s and  F(k,t) = (1//H)  Q being  / d  r exp(ik-r) F(r,t) ,  the volume o f a p r i m i t i v e  (3.3c)  cell.  In a p p l i c a t i o n s o f s e m i c l a s s i c a l dynamics, r e f e r e n c e  i s frequently  made t o a d i s t r i b u t i o n f u n c t i o n f ( k , r ) which g i v e s the p r o b a b i l i t y t h a t the independent e l e c t r o n s t a t e |k^>is occupied ->-  r  by an e l e c t r o n a t p o s i t i o n ->•  [69]. T h i s p r a c t i c e o f a s s i g n i n g an e l e c t r o n 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 o f the u n c e r t a i n t y  principle.  ->-  The  p r a c t i c e i s acceptable  only  so l o n g as k and r a r e taken t o r e p r e s e n t  the mean c r y s t a l momentum and mean p o s i t i o n o f a l o c a l i z e d e l e c t r o n  wavepacket. 3.2.2  Models f o r the T u n n e l l i n g In the p a s t ,  two c o n c e p t u a l l y  Process d i s t i n c t methods have been developed  to determine the r a t e a t which e l e c t r o n s t u n n e l through a t h i n i n s u l a t o r separating  two c o n d u c t o r s .  (The conductors i n q u e s t i o n  c o u l d be m e t a l s ,  semimetals o r semiconductors; t o s i m p l i f y t e r m i n o l o g y , i t w i l l be assumed here t h a t the j u n c t i o n has been formed between a m e t a l and a semiconduct o r ) . I n the o r i g i n a l method developed by Bardeen by H a r r i s o n  [72], time-dependent p e r t u r b a t i o n  [71] and l a t e r extended  theory  i s applied to e s t i -  72  mate the r a t e at which  t r a n s i t i o n s between e l e c t r o n s t a t e s i n the m e t a l  and those i n the semiconductor o c c u r . T h i s approach was and Rhoderick for  employed by  Card  [26], and l a t e r used by Green e t al.[16,17] as the b a s i s  t h e i r n u m e r i c a l a n a l y s i s of the MIS  t u n n e l d i o d e . Here a more i n t u -  i t i v e l y a p p e a l i n g method developed by Duke [73,74] i s employed. To the l e v e l o f a p p r o x i m a t i o n g e n e r a l l y used, the two methods g i v e m a t h e m a t i c a l l y i d e n t i c a l e x p r e s s i o n s f o r the t u n n e l c u r r e n t s . In  Duke's approach, the t u n n e l l i n g o f e l e c t r o n s a c r o s s a m e t a l -  i n s u l a t o r - s e m i c o n d u c t o r j u n c t i o n i s t r e a t e d i n the same manner as the t u n n e l l i n g of f r e e e l e c t r o n s through a one-dimensional square barrier tor  potential  [75]. An e l e c t r o n a p p r o a c h i n g the i n t e r f a c e from the  semiconduc-  i n F i g . 3.1 i s r e p r e s e n t e d by a t r a v e l l i n g B l o c h wave w i t h  It. A s o l u t i o n to the S c h r o d i n g e r e q u a t i o n i s then sought which of the  wavevector consists  t h i s i n c i d e n t B l o c h wave, a r e f l e c t e d wave r e t u r n i n g t o the r i g h t i n semiconductor, and a t r a n s m i t t e d wave p r o p a g a t i n g to the l e f t i n the  m e t a l . I f the energy E o f the i n c i d e n t e l e c t r o n l i e s w i t h i n the energy gap o f the i n s u l a t o r , then i n t h i s r e g i o n the e l e c t r o n w a v e f u n c t i o n must have an e x p o n e n t i a l dependence on x. The r a t i o of the c u r r e n t  density  c a r r i e d by the t r a n s m i t t e d wave t o t h a t c a r r i e d by the i n c i d e n t wave g i v e s the p r o b a b i l i t y The  t h a t the e l e c t r o n w i l l  t u n n e l a c r o s s the i n t e r f a c e .  form and amplitude of the t r a n s m i t t e d and r e f l e c t e d  waves are determined by the requirement t h a t ty and  be  electron  everywhere  c o n t i n u o u s . In g e n e r a l , these w i l l not be simple B l o c h waves. T h i s  can  be seen by c o n s i d e r i n g the s o l u t i o n to the S c h r o d i n g e r e q u a t i o n at an abrupt i n t e r f a c e between two In  regions with d i f f e r e n t l a t t i c e  potentials.  each r e g i o n , the s o l u t i o n s t o the S c h r o d i n g e r e q u a t i o n w i l l be B l o c h  waves of the form  73  <j£(r") = u * ( r ) e x p ( i ^ - r )  where u£(r) u+(r)  has  (3.4)  the p e r i o d i c i t y o f the l a t t i c e .  w i l l have d i f f e r e n t  forms i n the two  Since the  functions  r e g i o n s , i t w i l l not  be p o s s i b l e to connect a B l o c h wave i n one  region with  wave i n the a d j o i n i n g r e g i o n i n such a way  t h a t ty i s continuous  the boundary. I n s t e a d ,  a B l o c h wave i n c i d e n t on  r i s e t o t r a n s m i t t e d and t r a n s m i t t e d and  r e f l e c t e d waves w i t h  in  general  a s i n g l e Bloch across  the i n t e r f a c e w i l l  give  a spectrum of k v a l u e s .  r e f l e c t e d waves must, however, have the same energy  The as  the i n c i d e n t wave. In o r d e r  t o 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  tor-semiconductor invariably  j u n c t i o n at a mathematically  wave w i t h  manageable l e v e l , i t i s  assumed t h a t the l a t t i c e - p e r i o d i c component "£(?)  w a v e f u n c t i o n can be junction  approximated by  a constant  a  plane  the same wavevector. From t h i s assumption i t f o l l o w s t h a t  r e f l e c t e d and  t r a n s m i t t e d waves must a l l have the same v a l u e the band s t r u c t u r e of the m e t a l ,  c o n d u c t o r , t h e r e may junction with  be  inter-  the same v a l u e s o f E and k^.  v e l o c i t y v >0, x  of  i n s u l a t o r and  k^. semi-  s e v e r a l s t a t e s a v a i l a b l e i n each r e g i o n of  each band s t r u c t u r e f o r every  state with  t h e r e must be  same E and lc , but w i t h v <0. t x t h a t o n l y one  of the  the  c o n s e r v e d d u r i n g t u n n e l l i n g . I n o t h e r words, the i n c i d e n t ,  Depending on  and  Bloch  the  [72,76]. Thus the t r u e Bloch w a v e f u n c t i o n i s r e p l a c e d by  f a c e must be  t  of the  i n each r e g i o n of  component k^_ of the e l e c t r o n wavevector l y i n g i n the plane  lc  metal-insula-  As F i g . 3.2  the  indicates, in  energy E, t r a n s v e r s e wavevector  at l e a s t one  For s i m p l i c i t y ,  other s t a t e with  i t w i l l be  such p a i r of s t a t e s a t the same energy and  v e c t o r e x i s t s i n each r e g i o n of the j u n c t i o n . The  the  assumed a t  first  t r a n s v e r s e wave-  c o r r e c t i o n s to  the  74  e x p r e s s i o n s f o r the t u n n e l c u r r e n t s r e q u i r e d when t h i s c o n d i t i o n does n o t h o l d w i l l be c o n s i d e r e d l a t e r . In  the independent e l e c t r o n a p p r o x i m a t i o n , the p r o b a b i l i t y o f an  e l e c t r o n making a t u n n e l l i n g t r a n s i t i o n from a s t a t e conductor t o a s t a t e  |k^>in  opposite t r a n s i t i o n ,  from ( k ^ ^ t o  |k,^> i n the semi-  the m e t a l must e q u a l the p r o b a b i l i t y o f t h e |kg^> [73]. T h i s 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  E x p r e s s i o n s f o r the Tunnel C u r r e n t s I f an e l e c t r o n i n the semiconductor i s t o t u n n e l i n t o the m e t a l ,  the is  P a u l i e x c l u s i o n p r i n c i p l e r e q u i r e s t h a t the s t a t e which the e l e c t r o n t o e n t e r i n the m e t a l be unoccupied. T a k i n g t h i s r e s t r i c t i o n  into  account, and summing the c o n t r i b u t i o n s from a l l o c c u p i e d s t a t e s i n the semiconductor  c o n d u c t i o n band, i t i s found t h a t the e l e c t r o n  component  current  f l o w i n g from the c o n d u c t i o n band i n t o the m e t a l i s g i v e n  by [73]  J  Q*M  =  q  ^  2  v <0 x  v  ,„ .3  ( x  k  )f  ( C  ^' S X  )  U  ~ M f  (  E  ' V  ]  6  (  E  '^t  )  (  3  '  5  )  (2TT)  where f (k,x ) i s the d i s t r i b u t i o n f u n c t i o n f o r c o n d u c t i o n band e l e c t r o n s e v a l u a t e d a t the semiconductor s u r f a c e , and f  M  i s the d i s t r i b u t i o n f u n c -  t i o n f o r e l e c t r o n s i n the m e t a l . For most purposes i t can be assumed t h a t t h a t f„ has i t s e q u i l i b r i u m form, and i s thus i d e n t i c a l t o the FermiM D i r a c d i s t r i b u t i o n f u n c t i o n . The i n t e g r a l i s t o be c a r r i e d out over a l l states  | k ^ i n the c o n d u c t i o n band w i t h v e l o c i t i e s d i r e c t e d towards the  m e t a l , and f o r which s t a t e s i n the m e t a l w i t h the same v a l u e s o f E and lc  t  exist.  75  Noting that  v X  (k) = 1 3E <h dk  ,  (3.6)  x  the i n t e g r a l over k a p p e a r i n g i n (3.5) -*•  over k  t  and  can be  transformed  to an  integral  -> E. S i n c e s p e c i f y i n g E and k^ does n o t u n i q u e l y determine  some care i s n e c e s s a r y  i n making t h i s  t r a n s f o r m a t i o n . As noted  k,  above, f o r  every s t a t e i n the c o n d u c t i o n band w i t h energy E, t r a n s v e r s e wavevector k  and v e l o c i t y  fc  component  v  < x  0»  t h e r e must be a t l e a s t one  other  state  w i t h the same E and k  , but w i t h v >0. However, only the s t a t e w i t h v <0 t x _ x to be i n c l u d e d i n computing J™,- T h e r e f o r e  is  VM  =  -9— 2A  /  d  d  f  k  " ' CM  t  c  (  E  ' V  x  s  }  [  1  ~  f  (  E  M  ' N -  )  ]  e  ( >K E  }  (3  -  7)  band  where the r e g i o n S E e x i s t i n both  /S  E  N D  o f the k  the m e t a l and  p l a n e i s t h a t i n which s t a t e s of energy the semiconductor  c o n d u c t i o n band. H a r r i s o n  [72] d e s c r i b e s t h i s r e g i o n as the o v e r l a p of the "shadows" o f the energy s u r f a c e s f o r the metal and  constant  the c o n d u c t i o n band, where the shadow  o f a c o n s t a n t energy s u r f a c e i s d e f i n e d t o be i t s p r o j e c t i o n on a p l a n e p a r a l l e l t o the i n t e r f a c e . I t i s u s u a l l y  assumed t h a t S  i s equivalent L*JM  t o the shadows o f the semiconductor making t h i s assumption,  any  c o n s t a n t energy s u r f a c e s a l o n e .  i n f l u e n c e the band s t r u c t u r e of the  might have on the c u r r e n t flow i s i g n o r e d . I f , as suggested  metal  in Fig.  t h e r e i s more than one p a i r of s t a t e s i n the c o n d u c t i o n band w i t h same v a l u e s of E and k o f each b r a n c h The  , then the c o n t r i b u t i o n s to J„  from  3.2,  the  the shadow  o f the c o n s t a n t energy s u r f a c e must be taken i n t o  shadows i n the <100> d i r e c t i o n  By  account.  f o r the v a r i o u s branches o f a c o n s t a n t  F i g u r e 3.2  A slice  through  the c o n s t a n t energy  silicon  c o n d u c t i o n band, showing t h a t f o r each s t a t e  w i t h a g i v e n E, k one  s u r f a c e s f o r the  and v >0, there e x i s t s at l e a s t t x o t h e r s t a t e of the same E and k , but w i t h v <0. t x  77  energy  s u r f a c e c l o s e t o t h e c o n d u c t i o n band minimum f o r s i l i c o n a r e  illustrated  i n F i g . 3.3.  By analogy w i t h  (3.7), t h e e l e c t r o n c u r r e n t component  flowing  from the metal i n t o the c o n d u c t i o n band i s given by  Vc  Adding  -^SL. / _ 2 cond. 2TT h , , band  =  d  E  V >£ ) E  / d \ S_. CM  L  t  " f ( » k , x ) ] 6(E,k ) .  (3.8)  E  1  c  t  s  (3.8) t o (3.7), i t i s found t h a t the n e t c u r r e n t J  t  C  flowing  M  between the c o n d u c t i o n band and the metal i s g i v e n by  J  CM  - C+M J  =  +  q  M+C /  2rr h 2  <'>  J  3  dE  ™ band nd  A  / d k 2  S  t  9  [ f ( E , i t , x ) - f (E,£ )] e(E,^ ) . c  t  s  M  t  t  CM  An a n a l y s i s s i m i l a r t o t h a t used  t o d e r i v e (3.7), (3.8)  and (3.9)  can be a p p l i e d t o o b t a i n e x p r e s s i o n s f o r t h e c u r r e n t flows between t h e v a l e n c e band and the m e t a l . As i s u s u a l l y t h e case when d e a l i n g w i t h a v a l e n c e band, these c u r r e n t flows are most c o n v e n i e n t l y d e s c r i b e d i n terms o f the motion o f h o l e s  [69]. From t h i s v i e w p o i n t , the t r a n s f e r o f "  an e l e c t r o n from the m e t a l t o an unoccupied  s t a t e i n the v a l e n c e band  i s e q u i v a l e n t t o the e m i s s i o n o f a h o l e from the semiconductor  i n t o the  m e t a l , w h i l e t h e t r a n s f e r o f a v a l e n c e band e l e c t r o n t o the m e t a l i s e q u i v a l e n t t o the i n j e c t i o n o f a h o l e i n t o t h e semiconductor. u s e f u l to define a hole d i s t r i b u t i o n  It is  f u n c t i o n f , which i s r e l a t e d t o  the e l e c t r o n d i s t r i b u t i o n f u n c t i o n f by  78  F i g u r e 3.3  Shadow o f the c o n d u c t i o n f o r a s i l i c o n sample of  band c o n s t a n t  energy s u r f a c e s  <100> o r i e n t a t i o n .  79  f  f  = 1 - f .  (3.10)  i s thus the p r o b a b i l i t y t h a t a s t a t e i n k-space i s " o c c u p i e d " by a  h o l e . In terms o f f ' , the c u r r e n t flows between the v a l e n c e band and the metal a r e g i v e n by  VM  -=3_  =  /  d  /  E  \  f (E,k ,x )  [1 - f ^ ( E , k ) ] 6 ( E , k ) ,  f^E,t )  [1 " f ; ( E , k , x ) ] e<E,* )  v  t  s  t  t  (3.11)  7 - VM al  2  d  A  S  band J  M  = A _ 2  /  A  y  ^ a l  :  band  / d k 2  t  t  t  s  (3.12)  t  VM S  and  VM = VM VV J  +  =  3.2.4  -q _ 2, 2TT h  / dE val. , band  (3  / d k S, 2  13)  [ f ^ ( E , k , x ) - f ^ ( E , k ) ] 6(E,k ) . t  s  t  t  , VM m  An E s t i m a t e f o r the T u n n e l l i n g P r o b a b i l i t y The  "  Factor  s i m p l e s t p o s s i b l e e s t i m a t e f o r 0 ( E , k ) i s o b t a i n e d through use  of the WKB  t  approximation  [77,78]. In i t s c o n v e n t i o n a l form, the WKB  method i s a p p l i e d t o determine  approximate s o l u t i o n s t o t h e f u l l  Schrod-  i n g e r e q u a t i o n f o r the case i n which the p o t e n t i a l v a r i e s s l o w l y over d i s t a n c e s comparable t o t h e e l e c t r o n wavelength. W i t h i n the i n s u l a t o r i n the MIS j u n c t i o n , the p o t e n t i a l e x p e r i e n c e d by an e l e c t r o n i s a sum o f the r a p i d l y - v a r y i n g l a t t i c e p o t e n t i a l and a more s l o w l y v a r y i n g term a s s o c i a t e d w i t h the e l e c t r o s t a t i c p o t e n t i a l drop hty across the i n s u l a t o r , y  80  so the WKB  method can not be  expected t h a t the WKB  a p p l i e d d i r e c t l y . However, i t might  approximation c o u l d s t i l l be  t i v e Schrodinger equation  (3.3)  a p p l i e d to the e f f e c -  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 shown t h a t the WKB  expression  be  [72]  f o r the p r o b a b i l i t y of an e l e c t r o n  has  tunnel-  l i n g through the i n s u l a t o r i s  x^ 6 ( E , k ) = exp[-2 /  M  t  X  where k ^ within  dx  |k (E,k ,x)| I x  ]  t  (3.14)  S  i s the imaginary x-component of the complex wavevector  x  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 e x p r e s s i o n  t i o n a l WKB  the  method gives f o r the p r o b a b i l i t y of a f r e e e l e c t r o n  k^ conven-  penetrating  a p o t e n t i a l b a r r i e r [78]. J u s t as i n the f r e e e l e c t r o n 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 s m a l l , so t h a t only the decreasing  s o l u t i o n to the S c h r o d i n g e r e q u a t i o n need be  the i n s u l a t o r . F u r t h e r , w i t h i n vary  slowly  to apply  met,  over d i s t a n c e s  x  comparable t o  must a l s o be s a t i s f i e d  M  the e x p o n e n t i a l  considered  within  the i n s u l a t o r the e x t e r n a l p o t e n t i a l must  c e r t a i n a d d i t i o n a l conditions  near x<, and  exponentially  I ^. In p r i n c i p l e ,  f o r (3.14)  on the b e h a v i o u r of the p o t e n t i a l [78]. I f these c o n d i t i o n s are  a p p e a r i n g i n (3.14) w i l l be m u l t i p l i e d by  not  a prefac-  t o r of order u n i t y . However, s i n c e 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 e x p o n e n t i a l ,  the exact  value  of t h i s p r e f a c t o r i s of l i t t l e  impor-  tance . In o r d e r it  to e v a l u a t e  the t u n n e l l i n g p r o b a b i l i t y f a c t o r from  i s n e c e s s a r y to know the r e l a t i o n s h i p between energy and  wavevector k^ i n the  —  forbidden  (3.14),  complex  t h a t i s , the band s t r u c t u r e —  f o r the evanescent s t a t e s  gap  t h e o r e t i c a l analyses  of the i n s u l a t o r . P r e v i o u s  of  81  [16,17,79] have g e n e r a l l y assumed t h a t E  t u n n e l l i n g i n MIS s t r u c t u r e s depends on k^. through The  a simple r e l a t i o n s h i p  Franz d i s p e r s i o n  1_ k  £ 2 m  *  *  where m ^  and m ^  CI  suggested by Franz [80]  relationship i s  =  I  first  +  " CI  [ E  E  ( X ) ]  ft  (3.15)  2  2 n  Vit vi E  ( x )  "  E ]  are s c a l a r e f f e c t i v e masses a s s o c i a t e d  t i o n and v a l e n c e bands i n the i n s u l a t o r , and E  w i t h the conduc-  (x) and E  (x) are the  e n e r g i e s o f the c o n d u c t i o n and v a l e n c e band edges i n the i n s u l a t o r . I t  2 s h o u l d be noted  that  (3.15) p r e d i c t s  states with energies within  a n e g a t i v e v a l u e o f k^ f o r a l l  the f o r b i d d e n 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 g i v e n by  k  = Ak: = • 2  T  Ix  I  -k  2  .  (3.16)  t  The p r o b a b i l i t y o f an e l e c t r o n  t u n n e l l i n g through a h i g h p o t e n t i a l  b a r r i e r i s n o t c r i t i c a l l y dependent on the shape o f the b a r r i e r . Thus i n evaluating  the p r o b a b i l i t y o f an e l e c t r o n w i t h energy near the middle o f  the i n s u l a t o r bandgap t u n n e l l i n g between the metal and the semiconductor, it  i s r e a s o n a b l e t o i g n o r e the dependence o f  and  on p o s i t i o n . To  t h i s l e v e l o f a p p r o x i m a t i o n , i t i s a l s o r e a s o n a b l e t o i g n o r e any dependence 6 might have on the b i a s In the semiconductor, wavevectors pied.  close  applied  t o the j u n c t i o n .  o n l y those s t a t e s  i n the c o n d u c t i o n band w i t h  t o the bottom o f the " v a l l e y s " w i l l n o r m a l l y be o c c u -  I t i s thus r e a s o n a b l e t o apply an approximate  tunnelling  probability  82  factor  a g i v e n v a l l e y , where 6 ^ i s t o  to a l l transitions involving  be e v a l u a t e d a t the v a l l e y minimum. In g e n e r a l , i f t h e r e a r e s e v e r a l v a l l e y s , the one w i t h the s m a l l e s t v a l u e o f  w i l l make the l a r g e s t  contribution  t o the t u n n e l c u r r e n t . F o r example, f o r s i l i c o n  orientation,  tunnel currents involving  inated  by t r a n s i t i o n s  o f <100>  the c o n d u c t i o n band w i l l be dom-  t o and from s t a t e s  i n the two v a l l e y s  centered  about k =0. In t h i s case a p a r t i c u l a r l y simple e s t i m a t e f o r t  found. Assuming that  the e f f e c t i v e masses m ^  are e q u a l t o the f r e e e l e c t r o n  and m ^  mass m, (3.14),  can be  a p p e a r i n g i n (3.15)  (3.15) and (3.16) combine  to g i v e [79]  3  where B  = exp [-2 ( 2 m / * ) 2  C M  i s the d i f f e r e n c e  £  1 / 2  B^  2  (1 - B / E e  g I  between the energy  edge a t the semiconductor  )  1 / 2  d]  (3.17)  o f the c o n d u c t i o n band  s u r f a c e and the mean energy  o f the i n s u l a t o r  c o n d u c t i o n band edge, d i s the t h i c k n e s s o f the i n s u l a t o r , and the i n s u l a t o r bandgap By analogy applicable  energy.  t o (3.17), an approximate t u n n e l l i n g  to a l l transitions  * exp[-2(2m/f» ) 2  m  where B^ i s the d i f f e r e n c e  involving  states  probability  6 ^  i n the semiconductor  can be d e f i n e d . 9 ^ i s g i v e n by [79]  v a l e n c e band f o r which k ^ O  Q  is  1/2  B*  / 2  (1 -  B ^ l E ^ )  1  '  2  between the mean energy  d]  (3.18)  o f the i n s u l a t o r  v a l e n c e band edge and E (x<,). v  Although  the e s t i m a t e s f o r the t u n n e l l i n g  above may appear v e r y crude, t h e r e i s l i t t l e  probability  factor  given  p o i n t i n attempting to  generate more a c c u r a t e MIS  and n o n - i d e a l  expressions  f o r 6 ( E , k ) . In the v a s t m a j o r i t y  Schottky j u n c t i o n s , 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 o x i d e grown by o f the s i l i c o n s u b s t r a t e . At p r e s e n t , a v a i l a b l e concerning grown i n t h i s way SiO  thus v e r y  energy gap  likely  oxidation  r e l i a b l e information  is  thought to have the  stoichiometry  the band s t r u c t u r e of an amorphous i n s u -  a m o b i l i t y gap  analogous i n some r e s p e c t s  to  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  e l e c t r o n s t a t e s can e x i s t at e n e r g i e s is  little  [81]. Such l a y e r s are  , where l<x^2. In g e n e r a l ,  forbidden  the low-temperature  the s t r u c t u r e or even the c o m p o s i t i o n of l a y e r s  l a t o r i s c h a r a c t e r i z e d by the  of  t  within  the m o b i l i t y gap  t h a t the b u l k of the c u r r e n t  [82]. I t  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 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 s t a t e s w i t h i n the m o b i l i t y gap Assigning  such SiO  o f the amorphous S i O ^ l a y e r .  l a y e r s the band s t r u c t u r e of c r y s t a l l i n e SiO  suggested i n F i g . 3.1, Further,  there  i n s u l a t o r and distances rough on  i s no  i s therefore hardly  The  a realistic  approximation.  reason to b e l i e v e t h a t i n a r e a l diode the  semiconductor-insulator  metal-  i n t e r f a c e s w i l l be smooth over  comparable t o an e l e c t r o n wavelength. I f the i n t e r f a c e s are the atomic s c a l e , the t r a n s v e r s e wavevector k^ w i l l not  conserved i n t u n n e l l i n g 3.3  , as z.  X  be.  [72].  Schottky B a r r i e r Diode  Purely  f o r n o t a t i o n a l convenience, i t w i l l be  assumed t h a t  j u n c t i o n s examined i n t h i s s e c t i o n and  i n the remainder of the  have been formed on n-type s u b s t r a t e s ,  unless  the chapter  otherwise s p e c i f i e d . The  a n a l y s i s developed a p p l i e s e q u a l l y w e l l to p-type m a t e r i a l i f the of e l e c t r o n s and h o l e s  are  interchanged.  roles  3.3.1  The M a j o r i t y C a r r i e r Thermionic  Emission  Current  Under moderate forward b i a s , the c u r r e n t i n a t y p i c a l Schottky formed on n-type s i l i c o n i s dominated by  diode  the e m i s s i o n o f e l e c t r o n s from  the c o n d u c t i o n band i n t o the metal. In most Schottky diodes t h i s m a j o r i t y c a r r i e r flow i s a c c u r a t e l y i n t r o d u c e d by Bethe on the assumption tions  d e s c r i b e d by  [10] i n 1942.  The  the t h e r m i o n i c e m i s s i o n  t h e r m i o n i c e m i s s i o n theory i s based  t h a t the t r a n s m i s s i o n c o e f f i c i e n t  6 a p p e a r i n g i n equa-  (3.7-3.9) i s e q u a l t o u n i t y ; thus the p r o b a b i l i t y  i n c i d e n t on the i n t e r f a c e may i g n o r e d . T h i s assumption  t h a t an  must always o v e r e s t i m a t e the e l e c t r o n  flow,  c o n t a c t p a r t of the wave-  r e p r e s e n t i n g the i n c i d e n t s e m i c l a s s i c a l e l e c t r o n w i l l be  on c o l l i s i o n w i t h the i n t e r f a c e  of p e r f e c t t r a n s m i s s i o n i s not  it,  the  unreasonable.  (3.9), i t i s n e c e s s a r y  f u n c t i o n f o r e l e c t r o n s a t the semiconductor  t o know the  accuracy of t h i s approximation  surface,  the  q u a s i - f e r m i energy  E^  i s c o n s i d e r e d below. Measuring  E r e l a t i v e t o the energy  approx-  semiconductor  a p p e a r i n g i n the F e r m i - D i r a c  t i o n f u n c t i o n with a position-dependent  t r o n energy  distribution  f u n c t i o n f o r c o n d u c t i o n band e l e c t r o n s can be E  conduction  s u r f a c e . As a f i r s t  i m a t i o n , i t i s r e a s o n a b l e t o assume t h a t throughout  by r e p l a c i n g the f e r m i energy  layer  assumption  In o r d e r t o e v a l u a t e the net c u r r e n t flow J _ . between the CM  the d i s t r i b u t i o n  reflected  [83]. However, i f the i n t e r f a c i a l  i s so t h i n t h a t e l e c t r o n s can f r e e l y t u n n e l through  band and the m e t a l from  electron  s u f f e r quantum-mechanical r e f l e c t i o n i s  s i n c e even i n an i n t i m a t e metal-semiconductor packet  theory  obtained distribu[64]. The the e l e c -  of the c o n d u c t i o n band edge a t the  (3.9) then becomes E  max  dE  [f (E) - f (E)] c  M  a (E) m  (3.19)  where  f ( E ) = exp(-[E + E ( x ) - E ( x ) ] / k T ) c  c  s  F n  s  ,  (3.20)  f ( E ) = exp(-[E + E ( x ) - E ^ / k T ) M  and  c  (3.21)  g  the a r e a c r ^ o f the c o n s t a n t energy  s u r f a c e 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  t o the F e r m i - D i r a c  f u n c t i o n has been taken. T h i s i s v a l i d o n l y i f E (x ) l i e s above C S and E  F n  ( x ) by a t l e a s t kT. The upper l i m i t on the energy s  be r a i s e d t o i n f i n i t y w i t h o u t  E ^ FM  i n t e g r a l can  introducing appreciable error.  I f there i s a s i n g l e v a l l e y i n the c o n d u c t i o n band, and i f the e f f e c t i v e mass t e n s o r a s s o c i a t e d w i t h t h i s v a l l e y has one p r i n c i p a l a x i s p e r p e n d i c u l a r t o the i n t e r f a c e , then f o r those s t a t e s 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 m  *  *  x *  k m  2  *  +  k m  2  (3.23)  *  *  Here m^, m^ and m^ are the p r i n c i p a l v a l u e s of the e f f e c t i v e mass [ 8 4 ] . In t h i s case i t can be seen immediately energy  t h a t the shadow o f the c o n s t a n t  s u r f a c e i s an e l l i p s e , and t h a t the shadow a r e a i s given by  I t then f o l l o w s  J™.  CM  that  ~ 4T Trrqq/ / mm  m m^ z  _y v  4Trqm* k  / dE E  T  2  [f (E) - f (E) ] c  exp(-[E (x ) - E  2  c  (3.25)  M  s  F n  (x )]/kT) s  [1 - e x p ( - [ E ( x ) F n  s  E ]/kT)] M  where  m  * y  m  *  (3.26)  z  For a c o n d u c t i o n band v a l l e y i n which the p r i n c i p a l axes of the  effective  mass t e n s o r 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 t o the i n t e r f a c e , i t i s shown i n Appendix B t h a t  (3.25) s t i l l h o l d s , a l t h o u g h the e x p r e s s i o n  * for m  g  i s more c o m p l i c a t e d than  v a l l e y , the c o n t r i b u t i o n s t o J  (3.26) [ 8 5 ] . I f t h e r e i s more than  one  from each v a l l e y must be summed.  F o l l o w i n g Shockley's a n a l y s i s Of c u r r e n t flow i n the pn diode [ 4 4 ] , it  i s r e a s o n a b l e t o base the computation  e m i s s i o n c u r r e n t on the assumption  o f the e l e c t r o n t h e r m i o n i c  that E  t i o n r e g i o n and the q u a s i - n e u t r a l base  F n  i s c o n s t a n t a c r o s s the d e p l e -  [86]. T h i s 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 g i v e s the band diagram  f o r the d e v i c e o f F i g . 3.1(a)  when a forward b i a s V i s a p p l i e d . Once the e l e c t r o n c u r r e n t has been c a l c u l a t e d , the methods o u t l i n e d i n S e c t i o n 2.2 the s e l f - c o n s i s t e n c y o f t h i s assumption If E  Vri  can be used t o e s t a b l i s h  on the constancy o f  i s c o n s t a n t a c r o s s the semiconductor,  then  Ep n  87  Figure  3.4  Band diagram f o r the d e v i c e of F i g . 3.1(a) under moderate forward b i a s .  E  Fn S>  " FM  ( x  E  ^  =  (3.27)  V  and  E  C  ( X  S>  " Rt<*S> E  =  q  *S0 "  q A  ^S  +  =  q  *Bn " &  (  3  '  2  8  )  n where q<j>g i s the energy d i f f e r e n c e between the f e r m i l e v e l and c o n d u c t i o n band edge a t the back c o n t a c t . D e f i n i n g the  the  effective  * Richardson  constant  for electrons A  by e  & A  ^ = 4iTqm  g  e  2 k  (3.29)  (3.25) becomes  * J  Th  ( V )  =  J  CM  =  A  e  T  e x  2 P ( - 9 > / k T ) exp(qV/nkT) [1 - exp(-qV/kT)]. B n  (3.30)  (3.30) i s the c o n v e n t i o n a l e x p r e s s i o n f o r the e l e c t r o n t h e r m i o n i c current,  emission  [87].  In d e r i v i n g  (3.30), i t was  assumed t h a t the e l e c t r o n d i s t r i b u t i o n  -y  i n k-space r e t a i n s i t s e q u i l i b r i u m form throughout even i n the immediate v i c i n i t y  the  semiconductor,  of the i n t e r f a c e . However, s i n c e a v e r y  l a r g e e l e c t r o n c u r r e n t flows a c r o s s the i n t e r f a c e i n t o the metal when a forward b i a s V i s a p p l i e d t o the d i o d e , y e t v i r t u a l l y no e l e c t r o n s e n t e r the c o n d u c t i o n band from the m e t a l , the s u r f a c e must be , that t h i s e f f e c t by a t l e a s t kT/q  i t seems c l e a r t h a t the r e g i o n n e a r  d e p l e t e d of e l e c t r o n s w i t h v >0.  c o u l d be  Bethe  [10] argued  i g n o r e d p r o v i d e d the b a r r i e r p o t e n t i a l drops  between the semiconductor s u r f a c e and  the p l a n e  x=x., A  which by  d e f i n i t i o n i s s e p a r a t e d from the s u r f a c e by  the mean f r e e  path  89  A f o r electron-phonon  c o l l i s i o n s . The  t h e r m i o n i c e m i s s i o n c u r r e n t i s then  e n v i s a g e d as b e i n g composed of e l e c t r o n s s c a t t e r e d towards the from x=x  . Since r e l a t i v e l y  few e l e c t r o n s s c a t t e r e d from x=x  A  interface w i l l have  A  s u f f i c i e n t energy  t o overcome the remaining p o r t i o n of the b a r r i e r  reach the i n t e r f a c e , the e l e c t r o n d i s t r i b u t i o n a t x=x^ significantly  i n f l u e n c e d by the presence  which the e f f e c t i v e mass t e n s o r has  s h o u l d not  and be  of the m e t a l . For the case i n  one p r i n c i p a l a x i s normal t o the  i n t e r f a c e , a t l e a s t , i t can be shown t h a t the e x p r e s s i o n f o r the thermi o n i c e m i s s i o n c u r r e n t (3.30) i s u n a f f e c t e d by  t h i s m o d i f i c a t i o n to the  t h e o r y . As a g e n e r a l comment, the problem of s e l e c t i n g the c o r r e c t f o r f^(k,Xg) controversy  i n the Schottky diode i s s t i l l  approximation  f o r f ^ [86,89,90].  M i n o r i t y C a r r i e r Flow By making approximations  of  the s u b j e c t of c o n s i d e r a b l e  [88]. The q u a s i - e q u i l i b r i u m d i s t r i b u t i o n f u n c t i o n (3.20) i s  the most w i d e l y used 3.3.2  form  analogous  t o those used i n the  (3.25), an e x p r e s s i o n f o r the net c u r r e n t flow  band and  derivation  between the v a l e n c e  the metal f o r the Schottky diode can be o b t a i n e d . T h i s e x p r e s s i o n  is  J  VM  =  "\ ' ^ ( - t V V " T  V S  [1 - e x p ( - [  X  E F M  (3.31)  ) ] / k T )  - E  (x )]/kT)] s  where *  A,  *  = 4iTqm,  I t i s o f t e n convenient  k  2  to w r i t e (3.31) i n the  (3.32)  form  90  J  VM  A  =  T  2  P  ( X  S  N  )  "  C L  E X  P("t FM E  " E (x )]/kT)] F p  s  ,  (3.33)  v on the h o l e c o n c e n t r a t i o n p ( X g )  which s t r e s s e s the dependence of  at  the semiconductor s u r f a c e . For a t y p i c a l Schottky ductor  diode  formed on n-type s i l i c o n the  semicon-  s u r f a c e i s i n v e r t e d at e q u i l i b r i u m . There are t h e r e f o r e many  unoccupied s t a t e s a t the semiconductor s u r f a c e w i t h e n e r g i e s near  the  v a l e n c e band edge. F u r t h e r , the band diagram of F i g . 3.1(a) i n d i c a t e s t h a t t h e r e are a l s o many o c c u p i e d Thus from (3.11) and  (3.12) the c u r r e n t components  between the v a l e n c e band and though at e q u i l i b r i u m these When a s m a l l forward the h o l e q u a s i - f e r m i from the  s t a t e s i n the metal at these  and J j ^ y f l o w i n g  the m e t a l must be extremely l a r g e , even two  c u r r e n t components c a n c e l e x a c t l y .  b i a s i s a p p l i e d t o a Schottky  o r MIS  l e v e l at the semiconductor s u r f a c e must be  f e r m i l e v e l a t the base c o n t a c t by  F i g . 3.4.  (Ad) i s d e f i n e d t o be p o s i t i v e when the m i n o r i t y  at the s u r f a c e ) . T h i s displacement  of E  displaced  carrier  quasi-  concentration  (x ) r e s u l t s i n a net rp  diode,  an amount qAd>, as shown i n  f e r m i l e v e l s h i f t s so as t o i n c r e a s e the m i n o r i t y c a r r i e r  holes  energies.  flow  of  o  from the m e t a l i n t o the semiconductor. The net h o l e c u r r e n t  J  t T W  VM  c r o s s i n g the i n t e r f a c e  can be  accounts f o r recombination  d i v i d e d i n t o a component ^ g »  i n the d e p l e t i o n r e g i o n , and  which accounts f o r recombination equal  r  +  s t r a t e and  homojunction diode  s u b j e c t to a forward  a component J ^ ,  i n the q u a s i - n e u t r a l base.  to the h o l e c u r r e n t which would flow a c r o s s  r e g i o n boundary o f a P N  which  must be  the e m i t t e r / d e p l e t i o n  formed on an i d e n t i c a l s u b -  b i a s A<j>. (Minor d i f f e r e n c e s i n the w i d t h  o f the d e p l e t i o n r e g i o n between the Schottky  diode  and  the P N +  diode  may  91  r e s u l t i n very s m a l l d i f f e r e n c e s between the two h o l e c u r r e n t s ; however, these d i f f e r e n c e s can be i g n o r e d f o r a l l p r a c t i c a l p u r p o s e s ) .  J  VM  =  J  rg  *  ( A  }  +  J  d  ( A  where a n a l y t i c approximations (2.28). From F i g . 3.4,  W  r  - FM E  = -A£  a  T  2  F i g . 3.4  ( 3  for J  "  q  (  V  "  L  ^  and  3 4 )  are g i v e n i n (2.26) and  that  •  ( 3  s  (1 - exp[q(V - Aty)/kT])  .  '  3 5 )  (3.36)  V  suggests  t h a t the h o l e q u a s i - f e r m i l e v e l a t the semiconduc-  t o r s u r f a c e s h o u l d c o i n c i d e w i t h the f e r m i l e v e l i n the metal so t h a t Aty i s e q u a l to V. T h i s " p i n n i n g " of E„ r  p l a i n e d by  '  (3.33) becomes  p(x ) N  }  i t i s apparent  In terms of Aty and V,  J  *  Thus  (x„) to p  b  [12,91],  can be  ex-  rrl  the f o l l o w i n g i n d i r e c t argument. I f E  (x ) d i d not c o i n c i d e p b , but i n s t e a d l a y c l o s e r t o the f e r m i l e v e l a t the back c o n t a c t , r  with E  rrl  the b a l a n c e between J would be  T I  V-*-M  an enormous net  and J., „ would be d e s t r o y e d . As a r e s u l t , M->-V  flow of h o l e s from the metal i n t o the semicon-  d u c t o r . I f d e s i r e d , the magnitude of t h i s h o l e flow c o u l d be from (3.36). The  there  estimated  flow c o u l d be s u s t a i n e d o n l y i f the r a t e at which h o l e s  were i n j e c t e d i n t o the semiconductor  were matched by the r a t e at which  these excess h o l e s recombined w i t h e l e c t r o n s i n the d e p l e t i o n r e g i o n base. However, s i n c e Aty i s always l e s s than o r e q u a l t o V, an upper  and  92  bound on the h o l e  r e c o m b i n a t i o n c u r r e n t i s s e t by the sum o f J  (V) and  J , ( V ) . F o r a s u b s t r a t e w i t h r e a s o n a b l e 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 , a t moderate forward b i a s J (V) and J,(V) w i l l b o t h be exrg d tremely s m a l l compared t o e i t h e r steady-state  o r J y ^ - Thus the only  possible  s o l u t i o n i s t o have a c o n d i t i o n o f q u a s i - e q u i l i b r i u m between  the metal and the h o l e s  a t the semiconductor s u r f a c e , which i s e q u i v a l e n t  to p i n n i n g E ( X g ) t o E ^ . In f a c t , E ^ C X g ) w i l l be d i s p l a c e d j u s t enough F p  from E„., t h a t the d i f f e r e n c e between J . ^ w and J., i s e q u a l t o J + J,. FM V+M M+V rg d T  3.3.3  The M i n o r i t y The m i n o r i t y  is  defined  Carrier Injection  n  Ratio  c a r r i e r i n j e c t i o n r a t i o y o f a Schottky b a r r i e r diode  t o be the r a t i o o f the m i n o r i t y  c a r r i e r current  c r o s s i n g the  boundary between the d e p l e t i o n r e g i o n and the base t o the t o t a l flow.  Thus  Y =  J J Th J  +  >  d  + T rg J  +  J  + T d  l  a  r  g  e  J  V J  T Th  Y can r e a d i l y be computed from the e x p r e s s i o n s  +  J  .  d  (3.37)  + T d  f o r J , J , and J Th d rg m  above. C a l c u l a t i o n s o f p r e c i s e l y t h i s type l e a d S c h a r f e t t e r clude  current  given  [12] t o con-  t h a t y i s extremely s m a l l f o r t y p i c a l s i l i c o n Schottky diodes  o p e r a t e d a t moderate forward b i a s . In p r i n c i p l e , however, i f the j u n c t i o n b a r r i e r h e i g h t were made l a r g e enough, J__ c o u l d be made s m a l l e r than J , . d  in  T h i s c o n d i t i o n would be achieved strate, since  most e a s i l y w i t h a l i g h t l y  doped sub-  decreases as the doping l e v e l i n c r e a s e s . F o r example,  f o r a 10 ftcm p-type s u b s t r a t e w i t h a 10 usee e l e c t r o n l i f e t i m e , a b a r r i e r height  <f>g o f a p p r o x i m a t e l y 0.99 V would be r e q u i r e d t o make p  equal  93  to  J ^ . Although  such a h i g h b a r r i e r h e i g h t i s not i m p o s s i b l e i n t h e o r y ,  it  i s f a r h i g h e r than the v a l u e s o f a)  r e c o r d e d f o r Schottky  diodes  a  fabricated using conventional  3.3.4  techniques.  C u r r e n t Flow Through S u r f a c e Up  States  to t h i s p o i n t , o n l y the d i r e c t  conduction  and v a l e n c e bands and  flow of c a r r i e r s between the  the metal has been c o n s i d e r e d . I f the  s u r f a c e s t a t e d e n s i t y i s l a r g e , and i f these s t a t e s communicate  readily  w i t h one  important  or both bands and w i t h the m e t a l ,  t h e r e may  be  a third  c u r r e n t component r e s u l t i n g from e l e c t r o n s t u n n e l l i n g between s t a t e s and likely of  the metal  [16]. The magnitude o f t h i s c u r r e n t component i s  to be very s e n s i t i v e to the d e n s i t y and d i s t r i b u t i o n i n energy  the s u r f a c e s t a t e s , which i n t u r n depends c r i t i c a l l y on d e v i c e  c a t i o n procedures. w i l l not be  3.4  these  For t h i s reason,  considered  c u r r e n t flow through  fabri-  surface states  here.  T r a n s i t i o n to the MIS  Diode  I f the i n t e r f a c i a l i n s u l a t i n g l a y e r i n the j u n c t i o n of F i g . 3.1(a) i s made p r o g r e s s i v e l y t h i c k e r , a p o i n t w i l l e v e n t u a l l y be reached  at  which t h i s l a y e r can no l o n g e r be c o n s i d e r e d t r a n s p a r e n t t o e l e c t r o n s . For t h e o r e t i c a l purposes, the t r a n s i t i o n d i o d e , i t i s no ficient  i t i s u s e f u l to d e f i n e t h i s p o i n t as marking  from n o n - i d e a l Schottky longer reasonable  diode t o MIS  d i o d e . In an  to assume t h a t the t r a n s m i s s i o n c o e f -  6(E,^ ) i s u n i t y . In t h i s s e c t i o n and i n S e c t i o n 3.5, t  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 c o n d u c t i o n band 9(E,k ) w i l l t  approximated by a c o n s t a n t , 6 . pM  MIS  Similarly, a l l tunnelling  be  transitions  between the v a l e n c e band and the m e t a l w i l l be d e s c r i b e d by a c o n s t a n t tunnelling probability  factor 0^.  Thus the e x p r e s s i o n s  f o r the net  94  c u r r e n t flows between the semiconductor  J  CM  =  VM  =  6  CM  A  e  j  2  e x  P(  _ <  l+ £/  k T  B  bands and the metal become  ) exp(qV/nkT) [1 - exp(-qV/kT)]  (3.38)  - A<},)/kT]) .  (3.39)  and  J  " VM  \  0  l  2  P ( X  S N  E s t i m a t e s f o r 0_, and CM  }  ( 1  "  e x  Pt9(V  V  6  VM  t I W  i n terms o f the t h i c k n e s s and band s t r u c t u r e  of the i n s u l a t o r are g i v e n i n e q u a t i o n s model f o r the t u n n e l l i n g p r o c e s s may  (3.17) and  appear,  both the e x i s t e n c e o f m i n o r i t y c a r r i e r MIS  (3.18). Crude as  this  i t i s capable of e x p l a i n i n g  diodes and, on a q u a l i t a t i v e  l e v e l , the main f e a t u r e s a p p e a r i n g i n the c o n d u c t i o n c h a r a c t e r i s t i c s a l l MIS  of  diodes.  In the p r e v i o u s s e c t i o n i t was  p o i n t e d out t h a t i n a Schottky  the e l e c t r o n d i s t r i b u t i o n f u n c t i o n a t the semiconductor to d e v i a t e s i g n i f i c a n t l y  from i t s e q u i l i b r i u m form.  diode  surface i s l i k e l y  In an MIS  however, t h e r e i s no doubt t h a t f ( k , x ) can be a c c u r a t e l y  diode,  approximated  by the F e r m i - D i r a c f u n c t i o n a t moderate forward b i a s , s i n c e most e l e c t r o n s i n c i d e n t on the semiconductor  3.4.1  The M i n o r i t y C a r r i e r MIS  s u r f a c e are r e f l e c t e d .  Diode  From (3.38) i t f o l l o w s t h a t the e l e c t r o n t h e r m i o n i c e m i s s i o n c u r r e n t f l o w i n g i n the MIS  diode i s reduced by a f a c t o r 6  r e l a t i v e to i t s  v a l u e i n a S c h o t t k y diode w i t h the same b a r r i e r h e i g h t . However, so l o n g as J„V-*M  and J  w  „ b o t h remain M+V  l a r g e compared t o the sum  region recombination current J  rg  o f the d e p l e t i o n  and the h o l e i n j e c t i o n - d i f f u s i o n c u r r e n t  95  J ^ , then the c o n d i t i o n of q u a s i - e q u i l i b r i u m between the metal and h o l e s at the semiconductor still  a p p l y i n the MIS  w i l l be p i n n e d t o  s u r f a c e which h e l d f o r the Schottky diode must  j u n c t i o n . Thus at moderate forward b i a s E„ ( x ) Fp S i n the MIS diode [16]. P r o v i d e d 6 is sufficiently c  s m a l l , i n t h i s b i a s regime  the sum o f J  b  J^.  the  rg  and J,, w i l l be much l a r g e r d  than  6  I f t h i s i s the c a s e , then a m i n o r i t y c a r r i e r MIS  diode has been  formed. I f the forward b i a s a p p l i e d t o the MIS  diode i s g r a d u a l l y i n c r e a s e d ,  a b i a s p o i n t w i l l e v e n t u a l l y be reached at which the sum becomes comparable t o be pinned t o E  Above t h i s b i a s p o i n t , E  , but w i l l i n s t e a d l i e  base c o n t a c t ; thus i n F i g . 3.4 Green e t a l . have termed s i n c e over t h i s range  F p  of J  (Xg)  rg  and J , d  w i l l no l o n g e r  c l o s e r t o the f e r m i l e v e l a t the  Aty becomes s i g n i f i c a n t l y s m a l l e r than  t h i s b i a s range the " t u n n e l l i m i t e d " regime  V. [16],  the net h o l e c u r r e n t e n t e r i n g the semiconductor i s  l i m i t e d by the r a t e at which h o l e s are s u p p l i e d by t u n n e l l i n g a c r o s s the i n s u l a t o r , r a t h e r than by r e c o m b i n a t i o n p r o c e s s e s w i t h i n the semiconduct o r . C o r r e s p o n d i n g l y , the b i a s range i n which E  (x„) i s p i n n e d t o E rp  i s termed  the "semiconductor  l i m i t e d " regime  When o p e r a t e d i n the semiconductor c a r r i e r MIS formed regime,  b  rn  [16].  l i m i t e d regime,  the m i n o r i t y  diode i s e l e c t r i c a l l y e q u i v a l e n t to a one-sided pn  junction  on an i d e n t i c a l s u b s t r a t e . At the onset o f the t u n n e l l i m i t e d the minMIS diode can be modelled approximately as a pn  connected i n s e r i e s w i t h a voltage-dependent  junction  resistance. Further into  the t u n n e l l i m i t e d regime, both m i n o r i t y and m a j o r i t y c a r r i e r flows be i m p o r t a n t . The maximum forward b i a s which can be a p p l i e d t o the diode b e f o r e the t u n n e l l i m i t e d regime hence on the i n s u l a t o r t h i c k n e s s [16].  i s e n t e r e d depends on  may MIS  a n  d  The  dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s f o r a h y p o t h e t i c a l s e t o f  MIS diodes tor  f a b r i c a t e d on i d e n t i c a l s u b s t r a t e s b u t w i t h  thicknesses  a r e sketched  very s m a l l forward  insula-  i n F i g . 3.5 [16]. For a l l the d i o d e s , a t  b i a s 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 d e p l e t i o n r e g i o n . As the forward c h a r a c t e r i s t i c o f the d e v i c e w i t h those o f t h e o t h e r d i o d e s .  r i e r thermionic emission.  b i a s i s i n c r e a s e d , the  the t h i n n e s t i n s u l a t o r r i s e s above  I n t h i s d e v i c e , a s u b s t a n t i a l f r a c t i o n o f the  dark c u r r e n t f l o w i n g a t moderate forward  moderate forward  different  b i a s r e s u l t s from m a j o r i t y  car-  F o r t h e o t h e r d e v i c e s , the c u r r e n t flow a t  b i a s i s dominated by m i n o r i t y  carrier  injection-diffusion  i n the q u a s i - n e u t r a l b a s e . These d e v i c e s a r e thus m i n o r i t y c a r r i e r MIS diodes. still  I f the forward  b i a s a p p l i e d t o the minMIS diodes  i s increased  f u r t h e r , t h e m i n o r i t y c a r r i e r flows e v e n t u a l l y become t u n n e l 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 t o the t u n n e l l i m i t e d regime occurs  first  f o r those  d e v i c e s w i t h the t h i c k e s t 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 b o t h the  r a t i o o f the m i n o r i t y c a r r i e r c u r r e n t component t o the m a j o r i t y  carrier  component i n the semiconductor l i m i t e d regime, and on the maximum b i a s which can be a p p l i e d b e f o r e Devices  forward  the t u n n e l l i m i t e d regime i s e n t e r e d .  o f the h i g h e s t q u a l i t y are formed when the t u n n 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 m i n o r i t y c a r r i e r band i s g r e a t e r than that f o r the majority  c a r r i e r band, and when the semiconductor s u r f a c e i s s t r o n g l y  i n v e r t e d . Both these thermionic emission rier  conditions help current, while  flows which m a i n t a i n  t o suppress the m a j o r i t y  simultaneously  carrier  strengthening  the c a r -  the s t a t e o f q u a s i - e q u i l i b r i u m between the  m e t a l and t h e m i n o r i t y c a r r i e r s a t the semiconductor s u r f a c e . Although  97  Figure  3.5  Dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s with various i n s u l a t o r increases  a->d.  f o r MIS  thicknesses; insulator  diodes thickness  little  can be done t o c o n t r o l the r a t i o o f 6 ^ t o  an a p p r o p r i a t e  c h o i c e o f b a r r i e r m e t a l work f u n c t i o n can a s s i s t i n a c h i e v i n g h i g h  junc-  t i o n b a r r i e r h e i g h t s and hence s t r o n g s u r f a c e i n v e r s i o n .  3.4.2  An 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 s and C u r r e n t Up  t o t h i s p o i n t n o t h i n g has  p o t e n t i a l d i s t r i b u t i o n across  been s a i d  r e g a r d i n g the  Flows electrostatic  the MIS j u n c t i o n . I n the case o f a t h i c k -  i n s u l a t o r MOS c a p a c i t o r , i t i s p o s s i b l e t o i n t e g r a t e P o i s s o n ' s over the semiconductor t o o b t a i n an e x p r e s s i o n £,(Xg)  j u s t i n s i d e the semiconductor s u r f a c e  shown t h a t t h i s t e c h n i q u e  equation  f o r the e l e c t r i c  field  [92]. Green ejt a l . [17] have  can be extended t o the case  o f the MIS t u n n e l  d i o d e . Here the r e s u l t s o b t a i n e d by Green e_t a l . w i l l be d e r i v e d u s i n g a somewhat l e s s r i g o r o u s argument. For a u n i f o r m l y - d o p e d  d%/dx  2  n-type s u b s t r a t e , P o i s s o n ' s  = -(q/e ) g  e q u a t i o n becomes  [p - n + N ] .  I f ty and the q u a s i - f e r m i p o t e n t i a l s  (3.40)  D  <j>  n  and  <j>  p  are measured r e l a t i v e t o  an a p p r o p r i a t e r e f e r e n c e p o i n t , i n the non-degenerate case concentrations  the c a r r i e r  a r e g i v e n by  n = n  ±  exp[q(^  p = n  ±  exp[q(qS  - d^/kT]  (3.41)  and  Provided  the n e t  drop a c r o s s  p  - i|/)/kT] .  c u r r e n t flow through the diode  (3.42)  i s s m a l l , the  potential  the q u a s i - n e u t r a l base can be i g n o r e d . F u r t h e r , i n Chapter 2  99  i t was argued t h a t each q u a s i - f e r m i across rier of  l e v e l s h o u l d be e s s e n t i a l l y  constant  t h a t p a r t o f the d e p l e t i o n r e g i o n i n which the c o r r e s p o n d i n g  c o n c e n t r a t i o n i s l a r g e . However, i n e v a l u a t i n g the r i g h t - h a n d  carside  (3.40) the charge c o n t r i b u t i o n from each f r e e c a r r i e r need be c o n s i d -  ered  only where the c o n c e n t r a t i o n  purpose o f e v a l u a t i n g fermi l e v e l s technique  (3.40),  are c o n s t a n t  used t o s o l v e  o f t h a t c a r r i e r i s l a r g e . Thus f o r the  i t i s reasonable  across  t o assume t h a t the q u a s i -  the d e p l e t i o n r e g i o n . A p p l y i n g  (3.40) f o r the MOS  capacitor  the same  [92], i t i s then  found t h a t  t£(Xg)]  =  2  (2kT/ )  [p(x )  E g  where the p l a n e  - p(x ) + n(x ) - n ( x ) + N^qi^/kT) ]  s  n  g  n  (3.43)  x marks the boundary between the d e p l e t i o n r e g i o n and n  the q u a s i - n e u t r a l b a s e . Under normal o p e r a t i n g c o n d i t i o n s , the terms p(x N  n  ) and n ( x ) can be i g n o r e d . L>  Thus  D >  [£(Xg)]  p(x  Further, n(x ) i s very n e a r l y equal to n  S  2  =  ( 2 k T / e ) [ p ( x ) + N ( q ^ / k T - 1) ] . s  s  (3.44)  D  ) i s r e l a t e d t o the e q u i l i b r i u m h o l e  concentration p  nO  i n the q u a s i -  n e u t r a l base by  P(x ) s  = P  exp(qA(J>/kT) exp(q<|; /kT) .  n Q  s  I f the charge s t o r e d i n s u r f a c e s t a t e s can be i g n o r e d , s t a t i c p o t e n t i a l drop i|> a c r o s s  *i  =  ^  ( x  s  }d  E  the e l e c t r o -  the i n s u l a t o r i s r e l a t e d t o £,(x*) by  s i ' / e  (3.45)  (3.46)  100  where d i s the t h i c k n e s s o f the i n t e r f a c i a l l a y e r . From F i g . (3.4),  *M  =  V  ^0  +  +  +  X  S  *I '  +  Taken t o g e t h e r , e q u a t i o n s and  (3.47) a r e s u f f i c i e n t  ( 3  (3.34),  to uniquely  (3.39), (3.44),  '  (3.45),  4 7  >  (3.46),  determine the v a l u e s o f ty ,ty,  p(Xg) and hty i n the MIS diode a t any o p e r a t i n g p o i n t . U n f o r t u n a t e l y , there i s no c l o s e d s o l u t i o n f o r t h i s system o f coupled n o n - l i n e a r equat i o n s . However,  a c o n s i d e r a b l e s i m p l i f i c a t i o n o f the system i s p o s s i b l e .  (3.44), (3.45), (3.46) and (3.47) can be combined t o e l i m i n a t e p ( x ) and ty  giving  v  *M -  V  *0  +  +  + d  X  S  *S  +  1 2  k  T  E  £  < ' > 3  /  [p  2  s  n Q  48  e x p ( q ^ / k T ) exp(qA<|>/kT) + N^(qi|»g/kT - 1 ) ]  1 / 2  s  2 I  At e q u i l i b r i u m , A<j> and V must both be z e r o . In t h i s case to a s i n g l e transcendental equation  for  (3.48) reduces  which can be s o l v e d  itera-  tive ly . (3.34) and (3.39) can be combined t o g i v e  " VM0 J  P  (  X  S  }  (  1  ~  e  x  P ^ (  V  " 4>) A  / k T  ])  =  J  + ) + J.(A<j>)  (3.49a)  o  where p ^ ( X g ) constant  i s the s u r f a c e h o l e c o n c e n t r a t i o n a t e q u i l i b r i u m and the  J-^HQ ^  S  gi  v  e  n  by  101  VM0  J  =  V \  T  N  (3.49) i s simply a statement conductor  *W  *  <' > 3 49b  V  o f h o l e c u r r e n t c o n t i n u i t y a t the semi-  s u r f a c e . P r o v i d e d the n e t h o l e c u r r e n t e n t e r i n g the semicon-  d u c t o r i s n o t z e r o , (3.49) can be s o l v e d f o r p (x ) i n terms o f A<j>, and then  (3.45) can be used,to s o l v e f o r \p i n terms o f A<j>. S u b s t i t u t i n g the  r e s u l t i n g e x p r e s s i o n f o r if^ i n (3.48) y i e l d s a s i n g l e t r a n s c e n d e n t a l e q u a t i o n f o r Ad), which can be s o l v e d by i t e r a t i o n . Once Ad) i s known, Tpg, p ( X g ) , immediately  and the c u r r e n t flows a c r o s s the i n t e r f a c e can be found a t any b i a s p o i n t V.  F o l l o w i n g the above p r o c e d u r e ,  r e a s o n a b l y a c c u r a t e approximate  a n a l y t i c s o l u t i o n s f o r the c u r r e n t flows and p o t e n t i a l drops  i n the MIS  diode can be o b t a i n e d , even f o r o p e r a t i o n i n the t u n n e l l i m i t e d •This a n a l y t i c treatment by  Card and Rhoderick  i n t o account  o f the MIS j u n c t i o n i s s i m i l a r t o t h a t  [26-28]. However, the method proposed  regime. developed  here  takes  t h e p o s s i b i l i t y o f s t r o n g i n v e r s i o n a t the semiconductor  s u r f a c e , a matter which was o v e r l o o k e d by Card and Rhoderick.  In c o n t r a s t  to the approach taken h e r e , and t o t h a t f o l l o w e d by Card and Rhoderick, Green ejt a l . [16,17,22-25] chose t o r e l y e n t i r e l y on n u m e r i c a l  analysis  to s o l v e f o r the c a r r i e r c o n c e n t r a t i o n s , p o t e n t i a l s and c u r r e n t flows i n the MIS d i o d e . In t h i s n u m e r i c a l approach,  a technique s i m i l a r t o t h a t  o u t l i n e d i n Appendix A i s used t o s o l v e the f i v e b a s i c e q u a t i o n s  governing  the c a r r i e r c o n c e n t r a t i o n s 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 semiconduct o r , s u b j e c t t o the boundary c o n d i t i o n s imposed by (3.38), (3.39), and  (3.47). Even though no g e n e r a l 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 b u t i o n i n an MIS j u n c t i o n i s a v a i l a b l e , some important  distri-  conclusions  102  concerning  the b e h a v i o u r 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  c a r r i e r concentrations  can be  the  drawn f o r the case i n which E„  (x„) i s  rp  to E^..  pinned  In p a r t i c u l a r , i t can be shown by  argument t h a t under the a p p l i c a t i o n of a forward t r a t i o n at the semiconductor s u r f a c e can not value i f E  F p  (Xg)  coincides with E ^  a p p l i e d to the MIS  diode,  and  AIJJ  the f o l l o w i n g i n d i r e c t b i a s V the h o l e  concen-  f a l l below i t s e q u i l i b r i u m  [16]. From (3.47), when a b i a s V i s  Aip  J.  b  must s a t i s f y the r e l a t i o n s h i p D  V = -Ai|i - Aiji  .  (3.50)  The h o l e c o n c e n t r a t i o n at the s u r f a c e c o u l d decrease only i f -Ai|>g were g r e a t e r than V, f o r i f E„  (x„) i s pinned  rp  must be qV.  i t f o l l o w s t h a t E^, (x )  rJXL  rp  b  d i s p l a c e d from the f e r m i l e v e l at the back c o n t a c t by an amount  But w i t h  having  to E ^ ,  b  t h i s c o n d i t i o n on A ^ , g  Aifj^ p o s i t i v e . However, had  p o t e n t i a l drop across  (3.50) c o u l d be s a t i s f i e d only  by  the s u r f a c e become l e s s i n v e r t e d , the  the i n s u l a t o r would be s m a l l e r than at e q u i l i b r i u m .  In t h i s  case Aij;^ would be n e g a t i v e , which i s a c o n t r a d i c t i o n . T h e r e f o r e  if E  g  F p  ( x ) i s pinned  to E ^ ,  (3.50) can be s a t i s f i e d o n l y by h a v i n g  surface concentration of holes In f a c t , the n u m e r i c a l the h o l e its  remain at i t s e q u i l i b r i u m v a l u e  a n a l y s i s c a r r i e d out by  shown t h a t  e q u i l i b r i u m v a l u e u n t i l the t u n n e l l i m i t e d regime i s e n t e r e d .  a t h i c k - i n s u l a t o r MOS  equal  diode  operated  in this  c a p a c i t o r . I t f o l l o w s t h a t AiJ>  It  is essentially  i s e f f e c t i v e l y unity for a minority regime.  c l o s e to  to the b e h a v i o u r expected  to V throughout the semiconductor l i m i t e d regime. Thus the  f a c t o r n d e f i n e d i n (3.2) MIS  or i n c r e a s e .  c o n c e n t r a t i o n at the semiconductor s u r f a c e remains very  s h o u l d be n o t e d t h a t t h i s i s e x a c t l y o p p o s i t e for  Green e_t a l . has  the  diode  carrier  103  3.5  The MIS S o l a r C e l l  3.5.1  L i g h t C o u p l i n g i n t o the Semiconductor  The  f i r s t problem which must be overcome i n o r d e r t o form an e f f i -  c i e n t Schottky b a r r i e r o r MIS s o l a r c e l l i s the c o u p l i n g o f l i g h t the semiconductor.  I n 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 t o t h i s  problem was a c h i e v e d by making the evaporated as t o be s e m i - t r a n s p a r e n t .  l a y e r so t h i n  i n F i g . 3.6(a). However, t h i s  t i o n was f a r from i d e a l . From the v i e w p o i n t a l o n e , the b e s t MIS s o l a r c e l l s  [19].  b a r r i e r metal  The t h i n b a r r i e r l a y e r was then o v e r l a i d  a t h i c k c o n t a c t g r i d , as i l l u s t r a t e d  f u n c t i o n metals  into  are those  solu-  of e l e c t r i c a l properties  formed by d e p o s i t i n g low work  such as aluminum, chromium o r magnesium on p-type  U n f o r t u n a t e l y , these low work f u n c t i o n metals  sorbers of v i s i b l e l i g h t ;  with  f o r example, Hovel  silicon  are very s t r o n g ab-  [93] has c a l c u l a t e d t h a t o  even when o v e r l a i d w i t h an o p t i m i z e d a n t i - r e f l e c t i o n c o a t i n g , a 75 A t h i c k l a y e r o f aluminum can be expected  to t r a n s m i t only about 60% of i n c i d e n t  l i g h t a t v i s i b l e wavelengths i n t o a s i l i c o n s u b s t r a t e . The  t r a n s m i t t a n c e of evaporated  l a y e r s o f low work f u n c t i o n metals  can be i n c r e a s e d t o some e x t e n t by p a r t i a l o x i d a t i o n d u r i n g d e p o s i t i o n [35].  This p a r t i a l oxidation i s e a s i l y  accomplished  by c a r r y i n g out the  e v a p o r a t i o n s l o w l y under a r e l a t i v e l y h i g h oxygen p r e s s u r e . However, t h e i n c o r p o r a t i o n o f oxygen may produce d e l e t e r i o u s changes i n the work f u n c t i o n o f the l a y e r . A l t e r n a t i v e l y , a composite b a r r i e r l a y e r can be formed i n which an u l t r a - t h i n l a y e r o f low work f u n c t i o n metal a reasonably  i s overlaid  t r a n s p a r e n t l a y e r o f h i g h work f u n c t i o n metal.  ponents a r e c o r r e c t l y  with  I f the com-  chosen, the r e s u l t i n g s t a c k may have h i g h  conduc-  t i v i t y , h i g h o p t i c a l t r a n s m i t t a n c e , and a work f u n c t i o n i n the d e s i r e d range. F o l l o w i n g t h i s approach, Anderson e t a l . have f a b r i c a t e d MIS c e l l s  104  MOOOym  <  Al  (Mum)  Al  MOA)  Si0  (^20A)  x  INSULATOR• p Si  (^300ym)  Al  (Mum)  OHMIC CONTACT  (a)  ,GRID  FINGER  -MOOym-  1+/+/+/+/+  ± A  Al  (Mym)  SiO  (MJOOA)  Si0  x  (-V20A)  INSULATORp Si  (MiOOym)  Al  (Mym)  OHMIC CONTACT  (b)  F i g u r e 3.6  (a) S t r u c t u r e  o f MIS s o l a r c e l l w i t h t h i n , semi-  transparent b a r r i e r (b)  Structure  layer.  of inversion  layer  cell.  105  w i t h the s t r u c t ure  10A Cr - 60A Cu - 30A Cr - SiO  — pSi  g i v i n g photo—  2 current  d e n s i t i e s as h i g h as 26 mA/cm  [94]. In t h i s s t r u c t u r e , the  chromium l a y e r c l o s e s t t o the s i l i c o n induces the j u n c t i o n , w h i l e the copper p r o v i d e s r e a s o n a b l e sheet c o n d u c t i v i t y w i t h o u t a s e r i o u s  loss i n  t r a n s m i t t a n c e . The p h o t o c u r r e n t r e c o r d e d by Anderson e_t a l . i s p r o b a b l y close  t o the upper l i m i t which can be a c h i e v e d w i t h s e m i - t r a n s p a r e n t  b a r r i e r metal l a y e r s . Recently Godfrey and Green [95,96] and Thomas et_ a l . [97] have p r o duced i n v e r s i o n l a y e r c e l l s MIS j u n c t i o n s .  of e x c e p t i o n a l l y  I n the i n v e r s i o n l a y e r c e l l  10 t o 20% o f the s u r f a c e ,  and f u n c t i o n s  high e f f i c i e n c y  the MIS j u n c t i o n  as a c o n t a c t  incorporating covers  only  g r i d (see  F i g . 3 . 6 ( b ) ) . A t h i n l a y e r o f d i e l e c t r i c i s d e p o s i t e d over the r e s t of the s u r f a c e  i n such a way t h a t a h i g h c o n c e n t r a t i o n  charge i s p r e s e n t near the i n t e r f a c e the  thermal e v a p o r a t i o n o f S i O ) .  i n d u c e s an i n v e r s i o n  ( t h i s can e a s i l y be accomplished by  The f i x e d charge i n the d i e l e c t r i c  l a y e r a t the s i l i c o n s u r f a c e ,  v e r y s h a l l o w induced j u n c t i o n . I f t h e t h i c k n e s s of the d i e l e c t r i c a r e s e l e c t e d  of fixed p o s i t i v e  i n effect creating a  and i n d e x of r e f r a c t i o n  t o 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 i n c i d e n t on the c e l l can be t r a n s m i t t e d r e s u l t , photocurrent d e n s i t i e s close  i n t o the s u b s t r a t e .  As a  t o the t h e o r e t i c a l maximum f o r s i l -  i c o n can be a c h i e v e d . Although the i n v e r s i o n l a y e r c e l l w i l l n o t be g i v e n further consideration  i n t h i s t h e s i s , the s t r u c t u r e o f f e r s a means o f  u t i l i z i n g MIS j u n c t i o n s  t o form c e l l s w i t h e f f i c i e n c i e s e q u a l t o o r s u r -  p a s s i n g those o f the b e s t homojunction  devices.  For t h e remainder o f t h i s s e c t i o n , i t w i l l be assumed t h a t a means has  been found t o 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-  t o r 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 f a b r i c a t e d on p-type s u b s t r a t e s n-type m a t e r i a l w i l l be and  3.4.  I t w i l l be  [19], c e l l s  c o n s i d e r e d here f o r c o n s i s t e n c y w i t h  formed  on  Sections  3.3  assumed from the o u t s e t t h a t the b a r r i e r m e t a l has  been chosen to give s t r o n g i n v e r s i o n o f the semiconductor s u r f a c e at e q u i l i b r i u m . The  r e l a t i o n s h i p between c e l l performance and  o f the i n t e r f a c i a l l a y e r w i l l then be  3.5.2  Optimally  E f f i c i e n t MIS  the  thickness  considered.  Cells  In o r d e r to a t t a i n the h i g h e s t p o s s i b l e o p e n - c i r c u i t v o l t a g e i n an MIS  solar c e l l ,  the m a j o r i t y ligible  the i n t e r f a c i a l l a y e r must be  c a r r i e r thermionic/emission  t h i c k enough to suppress  dark c u r r e n t component to neg-  l e v e l s . However, i n o r d e r t o o b t a i n h i g h  l a t o r must s i m u l t a n e o u s l y  be t h i n enough t h a t a n e t  magnitude to the one-sun p h o t o c u r r e n t r i e r band and  fill  the m e t a l w i t h o u t  can  f a c t o r s the i n s u -  current equal i n  flow between the m i n o r i t y  s i g n i f i c a n t displacement  c a r r i e r q u a s i - f e r m i l e v e l at the s u r f a c e from E  of the  car-  minority  . This i s equivalent  r e q u i r i n g t h a t the m i n o r i t y c a r r i e r flow i n the c e l l be  to  semiconductor  2 l i m i t e d up to c u r r e n t d e n s i t i e s comparable to 30 mA/cm . I f these c o n d i t i o n s on i n s u l a t o r t h i c k n e s s can be met, of the MIS  then the  c e l l w i l l be e s s e n t i a l l y the same as those  s i d e d homojunction c e l l w i t h a very shallow i d e n t i c a l substrate l i k e l y t o be  two  characteristics of an i d e a l one-  e m i t t e r r e g i o n formed on  [98]. In f a c t , the performance of the MIS  cell is  s l i g h t l y b e t t e r than t h a t 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 i n v e r s i o n l a y e r of the c e l l w i l l not have been degraded by heavy doping e f f e c t s . As a c a r r i e r s photogenerated very near the s i l i c o n s u r f a c e by  an  MIS  result,  short-wavelength  photons w i l l have a h i g h e r p r o b a b i l i t y of c r o s s i n g the d e p l e t i o n r e g i o n and  c o n t r i b u t i n g to the p h o t o c u r r e n t  i n the MIS  cell  [99].  Similarly,  107  the dark c u r r e n t component r e s u l t i n g from recombination  near t h e s u r f a c e  s h o u l d be s m a l l e r i n the MIS c e l l than i n the homojunction  cell.  In Chapter 2 i t was found t h a t the s u p e r p o s i t i o n p r i n c i p l e provide  should  an e x c e l l e n t approximate d e s c r i p t i o n o f a s i l i c o n homojunction  c e l l operated  i n t h e l o w - l e v e l i n j e c t i o n regime. I t f o l l o w s t h a t the  s u p e r p o s i t i o n p r i n c i p l e s h o u l d a l s o be a p p l i c a b l e w i t h i n the semiconductor in  a m i n o r i t y c a r r i e r MIS s o l a r c e l l . T h e r e f o r e  the net h o l e flow  from  the semiconductor s u r f a c e i n t o the d e p l e t i o n r e g i o n and base must be g i v e n by  J  The  - rg J  ( A  *>  requirement o f h o l e  " VM0 J  VM  p  (  V  (  1  "  e x  +  J  d  ( A  *  }  " u c J  P  •  < - > 3  51  c u r r e n t c o n t i n u i t y a t the s u r f a c e then g i v e s  P t q ( V " f>)/kT]) = J (Acfr) + J (Ad)) - J A<  rg  d  ,  (3.52)  W which r e p l a c e s  (3.49) f o r an i l l u m i n a t e d c e l l . Assuming that the m a j o r i t y  c a r r i e r thermionic  emission  current i s n e g l i g i b l e ,  the t e r m i n a l c u r r e n t J ( V ) . I f E „ (x„) i s pinned L  rn  o  must be e q u a l t o  to  E l 7 M  rM  at a l l operating  p o i n t s , then Ad) i s always e q u a l t o the b i a s V a p p l i e d a t t h e t e r m i n a l s . In t h i s case, s u p e r p o s i t i o n w i l l h o l d a t the c e l l 3.5.3  The C h a r a c t e r i s t i c s o f T h i c k - I n s u l a t o r  terminals.  Cells  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 t h i c k enough t o s e r i o u s l y impede the flow o f a c u r r e n t one-sun p h o t o c u r r e n t ,  the c e l l c o n v e r s i o n  comparable i n magnitude t o t h e e f f i c i e n c y i s reduced. The f i r s t  performance parameter t o be degraded i s the f i l l short-circuit  current  f a c t o r , f o l l o w e d by the  [25]. I n p r i n c i p l e , t h e r e s h o u l d be no d e g r a d a t i o n  108  i n o p e n - c i r c u i t v o l t a g e , so l o n g as the m a j o r i t y  c a r r i e r thermionic  emis-  s i o n c u r r e n t 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 t o one-sun i l l u m i n a t i o n a t t e r m i n a l s h o r t - c i r c u i t i s  shown i n F i g . 3.7. I n o r d e r t o s u p p o r t the m e t a l , E  F p  (Xg)  E  F p  (Xg)  the h o l e p h o t o c u r r e n t  flowing i n t o  must be d i s p l a c e d from E ^ . T h i s displacement  has two consequences. F i r s t , i n o r d e r t o s a t i s f y  of  (3.48) and  (3.45), t h e h o l e c o n c e n t r a t i o n a t the semiconductor s u r f a c e must i n c r e a s e d r a m a t i c a l l y o v e r i t s e q u i l i b r i u m v a l u e . Secondly, from (3.51) i t can be seen t h a t the s h o r t - c i r c u i t c u r r e n t f o r the t h i c k - i n s u l a t o r c e l l must be l e s s than t h a t f o r a c e l l i n which E „ (x„) i s always pinned t o E_.„. HowFp S FM ever, unless the i n s u l a t o r i s e x c e s s i v e l y t h i c k , t h i s suppression of J sc s h o u l d be i n s i g n i f i c a n t . As l o n g as A<j> a t s h o r t - c i r c u i t i s roughly or more l e s s than V  oc  100 mV  , J w i l l be v e r y n e a r l y e q u a l t o J sc upc  When a s m a l l forward b i a s V i s a p p l i e d t o the i l l u m i n a t e d t h i c k i n s u l a t o r c e l l , E „ (x„) i s d i s p l a c e d even f u r t h e r from the f e r m i Fp  level  S  at t h e back c o n t a c t than a t s h o r t c i r c u i t . I f A<f> a t s h o r t - c i r c u i t was s m a l l compared t o V  , i t i s p o s s i b l e t o apply  b i a s b e f o r e J (V) drops s i g n i f i c a n t l y below J Li  a f a i r l y large  forward  . However, a b i a s p o i n t  SC  i s e v e n t u a l l y reached a t which A(J> i s comparable t o V  » and any f u r t h e r  i n c r e a s e i n forward b i a s p a s t t h i s p o i n t causes a sharp drop i n c u r r e n t output.  As J  VM  decreases,  E  rp  (x ) moves c l o s e r t o E , w h i l e o rM  the s u r f a c e  h o l e c o n c e n t r a t i o n drops r a p i d l y towards i t s e q u i l i b r i u m v a l u e . At t e r minal open-circuit  i s z e r o , so E p ( X g ) must a l i g n w i t h E ^ p  M  regardless  o f i n s u l a t o r t h i c k n e s s . Voc s h o u l d t h e r e f o r e be independent o f i n s u l a t o r q c  thickness. F i g . 3.8 shows t h e c u r r e n t - v o l t a g e 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 s e t of MIS s o l a r c e l l s w i t h  various i n s u l a t o r t h i c k -  109  F i g u r e 3.7  Band diagram f o r t h i c k - i n s u l a t o r MIS short-circuit  under one-sun  c e l l at t e r m i n a l  illumination.  kOQ .  200..  600  BIAS VOLTAGE (mV)  F i g u r e 3.8  Illuminated current-voltage c e l l s with various i n s u l a t o r theory. For a l l c e l l s  c h a r a c t e r i s t i c s f o r MIS s o l a r thicknessesas  (x )=10 S e =3e_., and J =30mA/cm . I 0 upc ( a )  (  d  )  J  J  VM0  = l m A / c m 2  VM0~ '  •  (  b  )  J  P r i  VM0  o  = 3 m A / c m 2  p r e d i c t e d by  cm" , d=20A, V  *  (  c  )  J  VM0  =550mV, oc  = 1 0 m A / c m 2  -  Ill  n e s s e s . These c h a r a c t e r i s t i c s were generated to  s i m u l t a n e o u s l y s o l v e (3.45),  t i o n 3.4. in  The  (3.48) and  using i t e r a t i v e  (3.52), as o u t l i n e d i n Sec-  c e l l m a t e r i a l p r o p e r t i e s used i n t h i s c a l c u l a t i o n are  the f i g u r e c a p t i o n . Rather  than s e t t i n g  ity  listed  <f> and Xg d i r e c t l y , the e q u i M  s u r f a c e p ^ ( X g ) was  l i b r i u m h o l e c o n c e n t r a t i o n a t the semiconductor ified.  techniques  Both the d e p l e t i o n r e g i o n recombination  c a r r i e r thermionic emission current  current J  spec-  and the major-  were assumed t o be  negligible.  In  t h i s case, the c e l l o p e n - c i r c u i t v o l t a g e depends only on the magnitude  of  the h o l e i n j e c t i o n - d i f f u s i o n dark c u r r e n t J ^ . The e x p r e s s i o n used  compute  to  J(A<J>) was t h a t given i n (2.28). J ^ ^ was s e t to g i v e a s p e c i f i e d d  open-circuit voltage. The most remarkable  f e a t u r e of F i g . 3.8  i s the way  c h a r a c t e r i s t i c s of those c e l l s w i t h r e l a t i v e l y concave-upwards as V approaches V c e l l w i t h v e r y l a r g e shunt c h a r a c t e r i s t i c may  •  i n which the  t h i c k i n s u l a t o r s become  a conventional diffused-junction  conductance or s e r i e s r e s i s t a n c e , the J  become almost  a straight  (V)  l i n e c o n n e c t i n g the s h o r t -  c i r c u i t and o p e n - c i r c u i t p o i n t s , 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 b i a s V a t which the  c u r r e n t f i r s t b e g i n s t o f a l l s h a r p l y i n F i g . 3.8 to  the d i f f e r e n c e between V  and  i s approximately  equal  the v a l u e of Ad> at e q u i l i b r i u m .  oc Numerical  a n a l y s i s c a r r i e d out by Shewchun, Singh and  has shown t h a t the f i l l even drop below 0.25,  f a c t o r of a t h i c k - i n s u l a t o r MIS  Green  [25]  solar c e l l  may  which i s the lower l i m i t on t h i s parameter i n a  c o n v e n t i o n a l homojunction  d e v i c e . The  the c h a r a c t e r i s t i c s o f F i g . 3.8  lowest  fill  i s indeed s l i g h t l y  factor associated with l e s s than  0.25.  I t s h o u l d be noted t h a t f o r the d e v i c e w i t h the s m a l l e s t v a l u e of J  n  c o n s i d e r e d i n F i g . 3.8,  the h o l e c o n c e n t r a t i o n a t the  semiconductor  112  s u r f a c e becomes comparable to N v  f o r o p e r a t i o n near s h o r t - c i r c u i t . c  Since  the a n a l y s i s 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 a t non-degenerate  l e v e l s (see  S e c t i o n 3.4), the c h a r a c t e r i s t i c shown f o r t h i s d e v i c e 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 o b t a i n e d u s i n g the a n a l y s i s of S e c t i o n 3.4  should  be a t l e a s t q u a l i t a t i v e l y c o r r e c t p r o v i d e d the semiconductor s u r f a c e i s not s t r o n g l y degenerate.  113  CHAPTER 4 POSITIVE BARRIER MIS JUNCTIONS: EXPERIMENT  In t h i s chapter  the f i r s t  conclusive experimental  e x i s t e n c e o f m i n o r i t y c a r r i e r MIS diodes gathered  i s presented.  evidence  f o r the  T h i s evidence  was  through two independent experiments which a r e d e s c r i b e d i n Sec-  t i o n s 4.2 and 4.3 r e s p e c t i v e l y . The f i r s t  o f these experiments i n v o l v e d  the measurement o f the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s o f A l - S i O ^ - p S i diodes  a t v a r i o u s temperatures spanning the range from 0 t o 50°C [ 2 0 ] .  Since the temperature dependence of an i n j e c t i o n - d i f f u s i o n c u r r e n t i s s t r o n g e r than t h a t o f any t h e r m i o n i c e m i s s i o n provided  c u r r e n t , t h i s experiment  a d e f i n i t i v e t e s t f o r the c h a r g e - t r a n s p o r t  mechanisms dominating  the dark c u r r e n t . In the second experiment, A l - S i O ^ - p S i s o l a r c e l l s were formed on 10 ficm s u b s t r a t e s w i t h  a l l o y e d aluminum back s u r f a c e  [21]. The presence o f the back s u r f a c e f i e l d  r e g i o n was found t o i n c r e a s e  the o p e n - c i r c u i t v o l t a g e by up t o 50 mV r e l a t i v e t o the v a l u e with  fields  recorded  an ohmic back c o n t a c t , a r e s u l t which can be e x p l a i n e d o n l y i f the  dark c u r r e n t i n these  c e l l s i s dominated by m i n o r i t y c a r r i e r  d i f f u s i o n . S e c t i o n 4.1 p r o v i d e s  a b r i e f review o f p r e v i o u s  injectionexperimental  r e s e a r c h on the p o s i t i v e b a r r i e r MIS j u n c t i o n , p l a c i n g p a r t i c u l a r emphasis  on i n v e s t i g a t i o n s of a fundamental n a t u r e .  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 t h i c k n e s s 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 experiment a l MIS s o l a r c e l l s i s examined i n S e c t i o n 4.4.  4.1  Previous  Experimental  Research on the MIS J u n c t i o n  As o f mid-1978, when the f i r s t  o f the experiments d i s c u s s e d i n  t h i s chapter was undertaken, no unambiguous demonstration o f the e x i s t ence o f m i n o r i t y  c a r r i e r MIS diodes had been r e p o r t e d . However, over the  i  114  y e a r s a c o n s i d e r a b l e body of i n d i r e c t evidence  s u p p o r t i n g the MIS  diode theory i n t r o d u c e d by Green e t a l . [ 1 6 ] had a t u r e . An overview  o f t h i s evidence  minority  i n 1963,  c r y s t a l to generate  In c o n t r a s t , i n j e c t i o n e l e c t r o l u m i n e s c e n c e was b a r r i e r diodes observed  Since 1963,  j u n c t i o n i n the semiconductors  as w e l l as CdS an MIS  formed on CdS.  sufficient  electroluminescence.  never observed  i n Schottky  a number of o t h e r groups have  electroluminescence associated with minority c a r r i e r  from an MIS  at  J a k l e v i c e_t a l . [101a]  j u n c t i o n c o u l d be used t o i n j e c t  c a r r i e r s i n t o a CdS  [101b]. Although  liter-  i s g i v e n below.  In a p r o p h e t i c experiment conducted demonstrated t h a t an MIS  appeared i n the  tunnel  ZnS,  ZnSe, GaP,  the o b s e r v a t i o n of  c o n t a c t c o n c l u s i v e l y r e v e a l s the presence  injection  GaN  and  GaAs,  electroluminescence of i n j e c t e d m i n o r i t y  c a r r i e r s , i t p r o v i d e s v i r t u a l l y no i n f o r m a t i o n c o n c e r n i n g the  relative  magnitudes of the m a j o r i t y and m i n o r i t y c a r r i e r c u r r e n t s f l o w i n g i n the contact. In the e a r l y 1970's, Card and systematic experimental  Rhoderick  [26,102] undertook the  first  study of the e f f e c t of the i n t r o d u c t i o n 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 l a y e r on the p r o p e r t i e s of  metal-semiconductor  j u n c t i o n s . The  the Au-SiO^-nSi j u n c -  s t r u c t u r e chosen f o r t h i s study was  t i o n , i n which the i n t e r f a c i a l S i O ^ l a y e r was  grown by low  temperature  o x i d a t i o n of the s i l i c o n s u b s t r a t e p r i o r t o b a r r i e r metal d e p o s i t i o n . A f t e r c o r r e c t i o n f o r v a r i a t i o n s i n the b a r r i e r h e i g h t  with T  t h i c k n e s s , i t was  found  t h a t the f o r w a r d - b i a s e d  insulator  Bn  dark c u r r e n t  decreased  by s e v e r a l o r d e r s of magnitude as the i n s u l a t i n g l a y e r t h i c k n e s s  was  o  i n c r e a s e d over the range from 8 t o 26 A. S i n c e the dark c u r r e n t i n diodes of  t h i s type i s dominated by t h e r m i o n i c e m i s s i o n , t h i s experiment  demonstrated t h a t the i n t r o d u c t i o n of an i n t e r f a c i a l i n s u l a t i n g  clearly  layer  115  c o u l d suppress the m a j o r i t y  c a r r i e r thermionic  metal-semiconductor j u n c t i o n . However, no  emission  attempt was  current i n a  made to measure  the magnitude of the i n j e c t e d m i n o r i t y c a r r i e r c u r r e n t i n these Card and  Rhoderick next turned  t h e i r a t t e n t i o n to the d i r e c t meas-  urement o f the m i n o r i t y 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 T h i s measurement was  Snow had  c a r r i e r i n j e c t i o n i n Schottky i s equivalent a Schottky  junctions  accomplished u s i n g the m e t a l - e m i t t e r  s t r u c t u r e which Yu and  devices.  [27].  transistor  employed e a r l i e r to i n v e s t i g a t e m i n o r i t y  diodes  [13]. The  metal-emitter  transistor  to a c o n v e n t i o n a l p l a n a r b i p o l a r t r a n s i s t o r , except t h a t  o r MIS  j u n c t i o n r e p l a c e s the d i f f u s e d e m i t t e r . P r o v i d e d  that  the base w i d t h i s s h o r t compared t o the m i n o r i t y c a r r i e r d i f f u s i o n i n the base, the r a t i o of the c o l l e c t o r c u r r e n t to the e m i t t e r  length  current  i n t h i s d e v i c e i s e q u a l t o the m i n o r i t y c a r r i e r i n j e c t i o n r a t i o of e m i t t e r j u n c t i o n . Card and  Rhoderick confirmed  Yu and  the  Snow's f i n d i n g  -4 t h a t y i s t y p i c a l l y l e s s than 10 c h e m i c a l l y etched  s u b s t r a t e s , and  made as l a r g e as 0.2  f o r Au-nSi Schottky  diodes  formed  then went on to show t h a t y c o u l d  i n Au-SiO^-nSi diodes w i t h  relatively  biases  (£ 1 V) had  flow, so these MIS  to be  a p p l i e d to o b t a i n a p p r e c i a b l e  diodes would not have been of use  be  t h i c k (^O  i n t e r f a c i a l l a y e r s . With such t h i c k i n t e r f a c i a l l a y e r s r e l a t i v e l y forward  on  A)  large  current  for photovoltaic  ene r gy conve r s i on. F o l l o w i n g the development of t h e i r MIS  t u n n e l diode  theory,  Green  e t a l . themselves c a r r i e d out a number of fundamental experiments on MIS  s t r u c t u r e . In the f i r s t of these experiments, A l - S i O ^ - p S i  the  diodes  w i t h d i f f e r e n t i n s u l a t o r t h i c k n e s s e s were f a b r i c a t e d on a s e t of subs 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 t e r i s t i c s o f these  [ 6 8 ] . The  dark c u r r e n t - v o l t a g e  charac-  diodes were found t o 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 t h i c k n e s s p r e d i c t e d t h e o r e t i c a l l y a subsequent experiment, A l - S i O - p S i diodes h a v i n g t h i c k n e s s e s were f a b r i c a t e d on s u b s t r a t e s w i t h i t i e s . While a t h e r m i o n i c  emission  (see F i g . 3.5).  In  the same i n s u l a t o r  a wide range of  c u r r e n t s h o u l d be  resistiv-  independent of  the  s u b s t r a t e doping l e v e l , the c u r r e n t f l o w i n g under moderate forward i n these MIS  diodes was  found to decrease as the s u b s t r a t e doping  i n c r e a s e d . T h i s i s e x a c t l y the b e h a v i o u r which would be d e p l e t i o n r e g i o n recombination  injection-  s u b s t i t u t e d f o r aluminum as  b a r r i e r m e t a l i n these d i o d e s , no s i g n i f i c a n t change i n the v o l t a g e c h a r a c t e r i s t i c s at moderate forward  b i a s was  the  current-  observed, f o r a  given s u b s t r a t e doping. T h i s r e s u l t s t r o n g l y suggested t h a t the  diode  dominated by m i n o r i t y c a r r i e r f l o w s , which depend o n l y  on s u b s t r a t e p r o p e r t i e s . In c o n t r a s t , the magnitude of a emission  level  expected f o r a  c u r r e n t or a m i n o r i t y c a r r i e r  d i f f u s i o n c u r r e n t . When magnesium was  dark c u r r e n t was  bias  thermionic  c u r r e n t would have depended s t r o n g l y on the b a r r i e r m e t a l work  function. In o t h e r experiments, the s m a l l - s i g n a l c a p a c i t a n c e and M g - S i O - p S i diodes was x  recorded  C of  as a f u n c t i o n of b i a s  Al-SiO^-pSi  [24,68]. In  2 r e v e r s e 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 f o r a 2  Schottky versus these  d i o d e . By  f i n d i n g the v o l t a g e - a x i s i n t e r c e p t of a p l o t of  V, the diode b a r r i e r h e i g h t was diodes  the s i l i c o n s u r f a c e was  estimated.  I t was  1/C  found t h a t i n  s t r o n g l y i n v e r t e d at e q u i l i b r i u m ,  a c o n d i t i o n which must be s a t i s f i e d to produce a m i n o r i t y c a r r i e r  MIS  diode. Although the e x p e r i m e n t a l g e n e r a l agreement w i t h  r e s u l t s o b t a i n e d by  the p r e d i c t i o n s o f the MIS  Green e t a l . were i n t u n n e l diode  theory,  they d i d not prove t h a t the m a j o r i t y c a r r i e r t h e r m i o n i c e m i s s i o n  current  117  c o u l d be made n e g l i g i b l e at c u r r e n t d e n s i t i e s comparable to the one-sun photocurrent. had  Most of the d e v i c e s used i n the experiments d e s c r i b e d above  such t h i c k i n s u l a t i n g l a y e r s t h a t they e n t e r e d  the t u n n e l - l i m i t e d 2  regime l o n g b e f o r e  the dark c u r r e n t d e n s i t y reached the 30 mA/cm  level.  These d e v i c e s would thus have been q u i t e u n s u i t a b l e f o r use as MIS cells.  Indeed, i n 1976  S t . P i e r r e , Singh,  suggested t h a t s i g n i f i c a n t m a j o r i t y always be p r e s e n t In 1977  i n e f f i c i e n t MIS  [104]. At low  obey (2.25) w i t h A=2,  solar cells.  while  obeyed w i t h A=l.  regions  forward  f o r forward  of e x p o n e n t i a l  b i a s e s between 300  C h a r a c t e r i s t i c s of t h i s type  suggested t h a t at low  forward  i n the d e p l e t i o n r e g i o n , w h i l e  dominant. I t was  i n an MIS  are  mV  frequently 2.2.  dominated by  at l a r g e r forward  bias  diode  d e v i c e s were m i n o r i t y current-voltage  the  carrier  character-  i s t i c s o f t h i s "double e x p o n e n t i a l " form can e q u a l l y w e l l be by  500  c u r r e n t f l o w i n g i n t o the base became  thus i n f e r r e d t h a t these  diodes. Unfortunately,  and  Al-SiO^-pSi c e l l s , Pulfrey  b i a s the dark c u r r e n t was  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  found to  as d i s c u s s e d i n S e c t i o n  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 o f these  for  dependence  b i a s the c u r r e n t was  observed f o r s i l i c o n homojunction d i o d e s ,  MIS  [103]  P u l f r e y r e p o r t e d dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s  of c u r r e n t on b i a s  recombination  Loferski  c a r r i e r dark c u r r e n t components would  A l - S i O ^ - p S i s o l a r c e l l s showing two  (2.25) was  Shewchun and  solar  explained  assuming t h a t the main c u r r e n t component i n the h i g h e r b i a s range i s  a majority  c a r r i e r thermionic  P r i o r t o 1978,  open-circuit voltages In 1976,  current.  perhaps the most c o n v i n c i n g evidence  of m i n o r i t y c a r r i e r MIS  cells.  emission  diodes  f o r the  r e s i d e d i n the h i g h e f f i c i e n c i e s  r e p o r t e d by s e v e r a l groups f o r t h e i r MIS  Green, Godfrey and  Davies  [35] and  existence and  solar  S t . P i e r r e et_ a l . [103]  118  r e p o r t e d o p e n - c i r c u i t v o l t a g e s over 600 c a t e d on s u b s t r a t e s w i t h r e s i s t i v i t i e s et a l . concentrated c e l l s , and had o f 600  mV  on improving  achieved  by l a t e 1977  mV  for Al-SiO^-pSi c e l l s  ranging  from 0.1  the performance of t h e i r  e f f i c i e n c i e s of over 10%  and  t o 1 Hem.  open-circuit voltages  little  doubt t h a t the m a j o r i t y  dark c u r r e n t component i n these MIS very low  levels.  4.2  Experimental  New  typical  p-type s i l i c o n s u b s t r a t e would  be expected t o have an o p e n - c i r c u i t v o l t a g e of approximately t h e r e c o u l d be  Anderson  Cr-SiO^-pSi  [ 9 4 ] , u s i n g 2 ficm s u b s t r a t e s . S i n c e a  d i f f u s e d j u n c t i o n c e l l formed on a 1 Hem  fabri-  600  c a r r i e r thermionic  mV, emission  s o l a r c e l l s had been reduced to  Evidence f o r M i n o r i t y C a r r i e r MIS  In p r i n c i p l e , a m i n o r i t y c a r r i e r MIS  diode  Diodes  can be i d e n t i f i e d by  the  temperature dependence of i t s dark c u r r e n t - v o l t a g e 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 r i e r i n j e c t i o n - d i f f u s i o n c u r r e n t s obey the  J = J  where J  Q  J  emission  1]  (4.1)  c u r r e n t on a p-type  0 T h " VM\ 6  from (2.27),  l 2  exp(-q*  car-  law  i s a temperature-dependent c o n s t a n t .  f o r a thermionic  while  [exp(qV/kT) -  n  c u r r e n t s and m i n o r i t y  From (3.30) and  (3.38),  substrate  /kT) ,  f o r an i n j e c t i o n - d i f f u s i o n c u r r e n t i n a l o n g ,  (4.2)  uniformly  doped p-type base r e g i o n  J  n  Od  = qvlT n n  2  x  (4.3)  119  The i n t r i n s i c c a r r i e r c o n c e n t r a t i o n n_^ i s g i v e n by  n.(T) =  1  3/2  2kT  (. m  *  m  c  *  3/4  v  exp[-E  [105]  (T)/2kT]  (A.4)  irh  where E (T) i s the bandgap energy o f s i l i c o n . E  To a good a p p r o x i m a t i o n ,  (T) decreases l i n e a r l y w i t h temperature above about 250°K;  thus  [106]  6  g  E  where E Q g  (  T  )  =  E  g0  "  a  <->  T  4  and a are temperature-independent  5  c o n s t a n t s . There i s some  disagreement c o n c e r n i n g the e x a c t v a l u e s of E  and a. I n the l a t e 1950's, 6  M a c f a r l a n e e t a l . concluded t h a t E _ = 1.206 0  eV on the b a s i s of o p t i c a l v  g  a b s o r p t i o n measurements  [106]. More r e c e n t measurements of the temper-  a t u r e dependence of the c o l l e c t o r c u r r e n t i n d i f f u s e d j u n c t i o n t r a n s i s t o r s c a r r i e d out by Slotboom et^ al_. [107] i n d i c a t e = (1.20±.01)  eV. The important p o i n t i s t h a t E g  bipolar  that  i s significantly  l a r g e r than the bandgap energy of s i l i c o n a t room temperature, which i s known t o l i e between 1.11 is  and 1.12  eV. S u b s t i t u t i n g  (4.5) i n t o  (4.4), i t  found t h a t  n  2  Compared to n^,  oc T  exp(-E  3  and-  g()  /kT)  . I " ' 2  n  (4.6)  are not s t r o n g l y dependent  Near 300°K the e l e c t r o n m o b i l i t y  y  .  7  obeys  on temperature.  the e m p i r i c a l r e l a t i o n s h i p  [108]  (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 m o b i l i t y and the c o e f f i c i e n t , from  (4.7) i t f o l l o w s t h a t  D  To a f i r s t  diffusion  n  « T  approximation,  1  ,  .  7  (4.8)  the m i n o r i t y c a r r i e r l i f e t i m e i s i n v e r s e l y  p r o p o r t i o n a l t o the mean m i n o r i t y c a r r i e r v e l o c i t y , which i s i n t u r n 1/2 proportional to T  [109]. Thus  x  Combining  n  « T  (4.6), (4.8) and  J  - T 2  Q  D  Comparing  4  0  ,  (4.9)  (4.9), i t i s found t h a t  exp(-E  (4.2) and  .  5  g()  /kT) .  (4.10)  (4.10), i t i s seen  that J _ nrrn  and J „ , are both  Uih of the  J  where p has  Q  - T  P  exp(-E /kT)  (4.11)  g0  the v a l u e 2 f o r a t h e r m i o n i c e m i s s i o n c u r r e n t and  i n j e c t i o n - d i f f u s i o n c u r r e n t . S i n c e the temperature dominated by  should y i e l d  f o r an  unimportant.  t h a t an A r r h e n i u s p l o t of l o g ( j Q / T ) v e r -  a s t r a i g h t l i n e , and  P  t h a t the s l o p e of t h i s  s h o u l d be p r o p o r t i o n a l t o an a c t i v a t i o n energy f u s i o n c u r r e n t E. = E A gO  2.4  dependence of JQ i s  the e x p o n e n t i a l f a c t o r , the exact v a l u e of p i s  From (4.11) i t can be seen sus 1/T  (Jd  form  E  . For an  line  injection-dif-  w h i l e f o r a t h e r m i o n i c e m i s s i o n c u r r e n t E. = A  qty^. B  n T  121  Because qdj^ < E  g  < E Q f o r any Schottky g  barrier, this  r e a d i l y d i s t i n g u i s h between m i n o r i t y and m a j o r i t y  technique can  c a r r i e r diodes. I t  s h o u l d be n o t e d i n p a s s i n g t h a t Yu and Snow have used an a c t i v a t i o n energy  a n a l y s i s o f t h i s type  thermionic  emission  t o determine the r e l a t i v e magnitudes of the  c u r r e n t and the d e p l e t i o n r e g i o n recombination  r e n t i n c o n v e n t i o n a l Schottky  b a r r i e r diodes [ 1 1 ] .  In r e a l d i f f u s e d j u n c t i o n , MIS and Schottky o f t e n found t h a t  ( A . l ) applies, i f at a l l ,  range. At low forward  b i a s the diode  A v a l u e s between 1 and 2. At h i g h  only over a very l i m i t e d  c u r r e n t s , which obey  forward  across  ohmic c o n t a c t s , w i t h  the b u l k s u b s t r a t e i t s e l f  Rhoderick has shown t h a t a combination o f d e p l e t i o n r e g i o n c u r r e n t and s e r i e s r e s i s t a n c e e f f e c t s can produce Schottky current-voltage  s e v e r a l decades o f c u r r e n t , b u t i n which the diode c a n t l y g r e a t e r than u n i t y  [89]. When d e a l i n g w i t h  thin  give r i s e to  c h a r a c t e r i s t i c s which do n o t have an e x p o n e n t i a l  t e r i s t i c s i n which an e x p o n e n t i a l  bias  (2.25) w i t h  b i a s v o l t a g e drops  i n t e r n a l s e r i e s resistances associated with  current-voltage  b a r r i e r diodes, i t i s  c h a r a c t e r i s t i c s a r e u s u a l l y domin-  ated by d e p l e t i o n r e g i o n recombination  b a r r i e r m e t a l l a y e r s , and w i t h  cur-  form.  recombination diode  charac-  r e l a t i o n s h i p holds  over  factor A i s s i g n i f i injection-diffusion  c u r r e n t s , d e v i a t i o n s 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 m i n o r i t y c a r r i e r MIS d i o d e s , the  t u n n e l r e s i s t a n c e e f f e c t s d i s c u s s e d i n S e c t i o n A . 3 can a l s o l e a d t o d e v i a t i o n s from ( A . l )  [16].  P r i o r t o 1978, both Shewchun and Green [23] and Vernon and Anderson [111] had p u b l i s h e d data on the temperature dependence o f the c u r r e n t voltage  c h a r a c t e r i s t i c s o f MIS d i o d e s .  voltage  c h a r a c t e r i s t i c s f o r Cr-SiO  Vernon e t a l . r e p o r t e d c u r r e n t -  - p S i s o l a r c e l l s at various  temper-  a t u r e s spanning the range from 20°C t o 120°C, but no (4.1) was  even approximately  obeyed c o u l d be  t e r i s t i c s . Shewchun e_t a l . r e c o r d e d an A l - S i O ^ - p S i diode r e g i o n i n which  r e g i o n i n which  d i s c e r n e d i n these  cur rent-voltage c h a r a c t e r i s t i c s f o r  over the temperature range 200-350°C b u t ,  (4.1) was  t h i s l a t t e r study had  obeyed c o u l d be  a very  charac-  found. The  again,  no  diode s e l e c t e d f o r  t h i c k i n s u l a t i n g l a y e r , and showed t u n n e l 2  l i m i t e d b e h a v i o u r at c u r r e n t d e n s i t i e s of only 0.1  mA/cm . At  dark c u r r e n t d e n s i t y , d e p l e t i o n r e g i o n recombination  this  low  c u r r e n t s would have  been f a r l a r g e r 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 t h e r m i o n i c  emis-  sion currents. In the f a l l of 1978, s c r i b e d above was  the a c t i v a t i o n energy a n a l y s i s technique  a p p l i e d to o b t a i n the f i r s t  the e x i s t e n c e of m i n o r i t y  c a r r i e r MIS  i r r e f u t a b l e evidence  [20]. A l - S i O ^ - p S i s o l a r c e l l s  f a b r i c a t e d on chem-mechanically p o l i s h e d s u b s t r a t e s of 10 Qcm  resistivity  <100> o r 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  the d e v i c e  of  f a b r i c a t i o n procedure are g i v e n i n Appendix C. In summary,  the s u b s t r a t e s were f i r s t  cleaned  f o l l o w i n g standard  the manufacture of i n t e g r a t e d c i r c u i t s , and  formed at the back of the s l i c e s by  procedures used i n  then exposed t o a dry oxygen  flow at 500°C f o r 20 minutes to grow a t h i n oxide was  for  diodes w i t h e l e c t r i c a l p r o p e r t i e s  s u i t a b l e f o r p h o t o v o l t a i c energy c o n v e r s i o n  and  de-  l a y e r . An ohmic c o n t a c t  the d e p o s i t i o n of a t h i c k aluminum  l a y e r , f o l l o w e d by s i n t e r i n g i n dry n i t r o g e n at 500°C f o r 10 minutes. MIS  j u n c t i o n i t s e l f was  produced by d e p o s i t i n g a s e m i - t r a n s p a r e n t  The  aluminum  o  dot approximately Contact  80 A t h i c k onto the f r o n t s u r f a c e of the  t o t h i s b a r r i e r m e t a l l a y e r was  made w i t h a s i n g l e aluminum g r i d  f i n g e r s e v e r a l thousand angstroms t h i c k . The the c o n t a c t  substrates.  t h i n b a r r i e r metal l a y e r  and  f i n g e r were d e f i n e d u s i n g m e t a l shadow masks t o g i v e a t o t a l  123  j u n c t i o n a r e a of about 0.1  cm  2  . The  ohmic c o n t a c t , b a r r i e r m e t a l  c o n t a c t f i n g e r aluminum d e p o s i t i o n s were a l l c a r r i e d out by o r a t i o n from a tungsten In o r d e r  capacitance  thermal evap-  filament.  to ensure t h a t the s i l i c o n s u r f a c e i n the  A l - S i O ^ - p S i c e l l s was  and  completed  s t r o n g l y i n v e r t e d at e q u i l i b r i u m , the s m a l l - s i g n a l  C of these  d e v i c e s was  measured as a f u n c t i o n of  reverse  2 b i a s . A p l o t of 1/C  versus  V f o r 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 p o i n t s l i e almost e x a c t l y on a s t r a i g h t l i n e . the method o f l e a s t squares,  the s l o p e and v o l t a g e - a x i s i n t e r c e p t V  t h i s l i n e were computed. The  slope i s consistent with  15  Using of  a doping d e n s i t y  -3  of 1.0*10  cm  . Theory p r e d i c t s t h a t V  s h o u l d e q u a l the b a r r i e r  <}>gp f o r a j u n c t i o n i n which the s u r f a c e i s only d e p l e t e d  height  at e q u i l i b r i u m .  However, i f the s u b s t r a t e s u r f a c e i s s t r o n g l y i n v e r t e d a t e q u i l i b r i u m , V  s h o u l d be  [24]. The  only s l i g h t l y g r e a t e r than the s t r o n g i n v e r s i o n p o t e n t i a l  v o l t a g e - a x i s i n t e r c e p t i n F i g . 4.2  i n v e r s i o n p o t e n t i a l f o r a substrate with i s 580  mV.  I t can  s u r f a c e had The  thus be  indeed been  concluded  i s 590  mV,  while  a doping d e n s i t y of 1.0*10  t h a t s t r o n g i n v e r s i o n of the  c e l l i s shown i n F i g . 4.2.  Two  as would be expected f o r a m i n o r i t y  c a r r i e r MIS  (2.25) w i t h A=l,  r e g i o n can be A i s equal  ex-  character-  d i o d e . However, current  the s l o p e of the upper r e g i o n of the MIS  c h a r a c t e r i s t i c corresponds to an A v a l u e between 1.1  tor  silicon  r e g i o n s of approximately  i n the l o w - l e v e l 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  must obey  cm  achieved.  p o n e n t i a l dependence of c u r r e n t on v o l t a g e are v i s i b l e i n t h i s  while  strong 15 -3  dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c f o r a r e p r e s e n t a t i v e  Al-SiO^-pSi  istic,  the  and  1.2.  Since  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 to u n i t y , i t i s not p o s s i b l e t o d e r i v e a r e l i a b l e  diode no fac-  estimate  124  BIAS  F i g u r e 4.1  VOLTAGE  (mV)  Capacitance-voltage c h a r a c t e r i s t i c A l - S i O - p S i dot x  diode,  for reverse-biased  125  Figure  4.2  Dark c u r r e n t - v o l t a g e c h a r a c t e r i s t i c Al-SiO -pSi solar  cell.  f o r small-area  for  the parameter  appearing i n e q u a t i o n  As n o t e d above, even i n an MIS dominated by m i n o r i t y u n i t y may  (4.1)  from t h i s  curve.  diode i n which the dark c u r r e n t i s  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  be encountered i f the b i a s regime i n which the d e p l e t i o n  recombination current  i s s i g n i f i c a n t i s not w e l l separated  region  from the  regime  i n which s e r i e s r e s i s t a n c e or t u n n e l r e s i s t a n c e becomes i m p o r t a n t . the d e v i c e  of F i g . 4.2,  dark c u r r e n t  i t appears t h a t a s u b s t a n t i a l f r a c t i o n of  f l o w i n g at b i a s l e v e l s below about 400  d e p l e t i o n region recombination processes, tunnel resistance e f f e c t s set i n at biases should not  be  For  mV  the  r e s u l t e d from  while series resistance  or  g r e a t e r than about 500  mV.  s t r e s s e d , however, t h a t the s e r i e s and/or t u n n e l  It  resistance  was  so l a r g e as to s e r i o u s l y degrade the performance of t h i s d e v i c e when  o p e r a t e d as a s o l a r c e l l . The  here t y p i c a l l y  gave f i l l  small-area  f a c t o r s ranging  Al-SiO -pSi c e l l s considered x from 0.6  to 0.7  under  1  simulated  one-sun i l l u m i n a t i o n . It  i s w e l l known t h a t o r d i n a r y  eliminated  from the c u r r e n t - v o l t a g e  measuring J 6  sc  as a f u n c t i o n of V  the dark [137].  oc  s e r i e s r e s i s t a n c e e f f e c t s can  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  From the d i s c u s s i o n o f S e c t i o n  the J  -V  c h a r a c t e r i s t i c . For to o b t a i n J teristics Fig. 6  sc  -V  oc  4.3  c e l l recorded  3.5,  i t follows also that  c e l l can be e l i m i n a t e d  c h a r a c t e r i s t i c r a t h e r than the dark t h i s reason i t was  decided  by  current-voltage  t h a t i t would be  worthwhile  c h a r a c t e r i s t i c s as w e l l as dark c u r r e n t - v o l t a g e  f o r the MIS  by  r a t h e r than by measuring J and V i n  the e f f e c t s of t u n n e l r e s i s t a n c e f o r an MIS recording  be  charac-  cells.  shows the J  sc  -V  oc  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  at s i x temperatures spanning the  x  -pSi  range 0-50°C. These c h a r -  a c t e r i s t i c s were taken by mounting the e n t i r e t e s t bed,  contact  probe  and  127  F i g u r e 4.3 6  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 sc oc x s o l a r c e l l at v a r i o u s temperatures. The v a l u e quoted 2 f o r A r e f e r s to the r e g i o n l£J <10 mA/cm . sc  l i g h t source  assembly w i t h i n a t e m p e r a t u r e - c o n t r o l l e d Statham oven. The  c e l l temperature bed.  was measured u s i n g a thermocouple mounted i n t h e t e s t  T h i s temperature  i n s e r t e d through  measurement was c o n f i r m e d w i t h a mercury thermometer  the oven w a l l , and s h o u l d be a c c u r a t e t o about ±0.2°C.  In o r d e r t o minimize  any e f f e c t  t h a t h e a t i n g 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 s w i t c h e d on o n l y  l o n g enough t o r e c o r d a s i n g l e J -V p a i r , and then l e f t o f f f o r one s c oc to two minutes. The i l l u m i n a t i o n l e v e l was a l s o v a r i e d randomly from measurement t o measurement, r a t h e r than b e i n g s t e a d i l y i n c r e a s e d . For s h o r t - c i r c u i t  c u r r e n t d e n s i t i e s r a n g i n g from roughly 1 t o  2 10 mA/cm , the c h a r a c t e r i s t i c s  o f F i g . 4.3 are w e l l d e s c r i b e d by (2.25)  2 w i t h an A v a l u e c l o s e t o u n i t y . ' At c u r r e n t d e n s i t i e s below about 1 mA/cm , 2 A becomes s i g n i f i c a n t l y  l a r g e r than u n i t y , w h i l e f o r J  a c t u a l l y drops below one, the c e l l by the l i g h t  p r o b a b l y as a r e s u l t  G  C  > 10 mA/cm , A  of excessive h e a t i n g of  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 o f l e a s t squares was a p p l i e d t o compute the e q u a t i o n o f the l i n e b e s t f i t t i n g the 2 2 d a t a p o i n t s l y i n g i n the range 1 mA/cm < J < 10 mA/cm . From the s l o p e G  of t h i s l i n e ,  the v a l u e o f A a p p r o p r i a t e t o t h i s  l a t e d , w h i l e an e s t i m a t e f o r the parameter the i n t e r c e p t  w  a  s  c u r r e n t range was c a l c u -  was o b t a i n e d by f i n d i n g  o f the l i n e w i t h the c u r r e n t a x i s . T h i s e s t i m a t e f o r  w i l l be termed -JQ^as JQ2>  C  a  second  e s t i m a t e f o r J ^ , which w i l l be r e f e r r e d t o  o b t a i n e d by computing the average o f t h e q u a n t i t y  J  sc  exp(-qV /kT) ^ oc  over a l l p o i n t s 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 Jg^ and  A was assumed to be exactly equal to one. The values of A, r  o  r  each c h a r a c t e r i s t i c shown i n F i g . 4.3 are l i s t e d i n  Table 4.1. The data presented i n Table 4.1 represent the averaging of some 20-25 i n d i v i d u a l J -V measurements at each temperature. sc oc After correction for the T factor appearing i n (4.11), J Q ^ and p  were graphed as a function of r e c i p r o c a l temperature: These Arrhenius plots are shown i n F i g . 4.4. F i g . 4.4(a) reveals that the plot of 2A log[(300/T)  " ] versus reciprocal temperature  i s f i t t e d almost per-  f e c t l y by a straight l i n e . Using the method of least squares, the slope of this l i n e was computed, and from the slope the corresponding activation energy  was calculated. The value of E was found to be (1.19±.013)eV, A  which agrees almost exactly with the best available estimates for E _. g0 From F i g . 4.4(b) i t can be seen that not a l l points i n the graph of 2A log[Jg^(300/T) ' ] versus r e c i p r o c a l temperature are co-linear. However, the data points corresponding to the four highest temperatures at which J -V c h a r a c t e r i s t i c s were recorded do l i e very nearly on a l i n e . Once sc oc again, the method of least squares was used to compute the slope of this l i n e , and the appropriate activation energy. In this case E^ was found to be (1.14±.02)eV which, although somewhat less than  E g  Q»  1  S  still  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 with an electron l i f e t i m e of 10 usee and a substrate doping of 10  -3 cm ,  which are reasonable estimates for these quantities i n 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*10  10  cm" at 300°K, (4.3) predicts J 3  = 6*10" mA/cm . From 8  Q  d  Fig. 4.4(a), i t can be seen that the experimental value of —8 2 i s about 5*10 mA/cm .  2  at 300°K  TABLE A . l Values o f A, J  Q  1  and J  Q  c o r r e s p o n d i n g t o the c h a r a c t e r i s t i c s  2  of F i g . 4.3  1-  (°C)  A:  log  1 0  [J  (300/T) - ]: 2  0 1  4  +  log  1 0  [J  (300/T) * ] 2  Q 2  -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  4  F i g u r e 4.4(a)  Temperature dependence  of J  132  F i g u r e 4.4(b)  Temperature dependence  of J  133  From the above r e s u l t s i t can be concluded  that f o r c u r r e n t d e n s i t i e s  2 above about 1 mA/cm , the dark c u r r e n t i n the A l - S i O - p S i 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  e l e c t r o n s i n j e c t e d from the A l c o n t a c t d i f f u s i n g i n t o the q u a s i - n e u t r a l base. T h i s demonstrates t h a t i t i s p o s s i b l e to form m i n o r i t y c a r r i e r diodes  on p-type s i l i c o n s u b s t r a t e s of r e s i s t i v i t y  imately equal minMIS d e v i c e s 4.3  MIS  g r e a t e r than or approx-  to 10 ficm. However, t h i s experiment does not can be  guarantee t h a t  formed on more h e a v i l y doped s u b s t r a t e s .  S o l a r C e l l s w i t h Back S u r f a c e  In the p r e v i o u s  MIS  s e c t i o n i t was  Fields  shown that A l - S i O - p S i s o l a r c e l l s x  can be produced i n which the o p e n - c i r c u i t v o l t a g e i s l i m i t e d by  recombi-  n a t i o n i n the q u a s i - n e u t r a l base. I t f o l l o w s t h a t f u r t h e r i n c r e a s e s i n the o p e n - c i r c u i t v o l t a g e of these MIS reducing  c e l l s can be  by  t h i s can be  done e i t h e r  i n c r e a s i n g the base doping o r , i f the e l e c t r o n d i f f u s i o n l e n g t h i n the  base i s g r e a t e r than the base w i d t h , by bination velocity t i o n 4.1,  through the use  r e d u c i n g the back s u r f a c e recom-  of a back s u r f a c e f i e l d  the a p p l i c a t i o n of the former technique  v o l t a g e s e x c e e d i n g 600  mV  i n Al-SiO -pSi  (BSF).  c e l l s was  [35,103] and  Cr-SiO  T h i s experiment p r o v i d e s  of MIS  particularly  the e x i s t e n c e of m i n o r i t y c a r r i e r MIS The back s u r f a c e f i e l d  -pSi  [94]  region  to  X  mentioned. In t h i s s e c t i o n the f i r s t use  enhance the o p e n - c i r c u i t v o l t a g e  of a BSF  s o l a r c e l l s i s described  [21].  c o n v i n c i n g a d d i t i o n a l evidence  for  diodes.  s u b s t r a t e s used i n the experiments  t h i s s e c t i o n were prepared  In Sec-  to o b t a i n o p e n - c i r c u i t  X  in  only  the magnitude of the e l e c t r o n i n j e c t i o n - d i f f u s i o n dark c u r r e n t  f l o w i n g i n t o the base. As o u t l i n e d i n S e c t i o n 2.3, by  achieved  described  at A p p l i e d S o l a r Energy C o r p o r a t i o n  (ASEC)  f o l l o w i n g procedures c u r r e n t l y employed i n the commercial p r o d u c t i o n  of  134  back s u r f a c e f i e l d  space c e l l s  [134]. The  s t a r t i n g m a t e r i a l was  doped C z o c h r a l s k i s i l i c o n of 9 to 11 ftcm r e s i s t i v i t y and t i o n . Measurements c a r r i e d out at ASEC u s i n g the s u r f a c e decay technique  800  ym.  450  ym,  while  The  BSF  r e g i o n i t s e l f was  Substrate  thicknesses  c e l l s w i t h and without  by  to  chemical p o l i s h i n g .  formed u s i n g the aluminum paste a l l o y i n g  nique d e s c r i b e d i n S e c t i o n 2.3.  tech-  In o r d e r to compare the p r o p e r t i e s of  back s u r f a c e f i e l d s , a number of s u b s t r a t e s were  regions.  At the U n i v e r s i t y of B r i t i s h  Columbia, A l - S i O - p S i f r o n t j u n c t i o n s  were a p p l i e d t o the ASEC s u b s t r a t e s f o l l o w i n g the p r o c e s s Appendix C. The  photovoltage  ranged from 350  the s u b s t r a t e s u r f a c e s were prepared  BSF  <100> o r i e n t a -  have shown t h a t the e l e c t r o n d i f f u s i o n l e n g t h i n t h i s  m a t e r i a l i s roughly  l e f t without  boron-  c e l l geometry i s i l l u s t r a t e d  t r a n s p a r e n t b a r r i e r m e t a l l a y e r and  i n F i g . 3.6(a).  The  the t h i c k , comb-like c o n t a c t  o v e r l y i n g t h i s l a y e r were both formed by minum. The b a r r i e r m e t a l l a y e r was  described i n  the thermal e v a p o r a t i o n  semigrid of a l u -  t y p i c a l l y made 80 A t h i c k . To minimize  c e l l s e r i e s r e s i s t a n c e , the c o n t a c t g r i d should be made as t h i c k as p o s s i b l e . Here l i m i t a t i o n s on the amount of aluminum which c o u l d be in  a s i n g l e pump-down c y c l e of the e v a p o r a t i o n  t h i c k n e s s t o roughly  1 ym.  The  deposited  system r e s t r i c t e d the  c o n t a c t g r i d was  shadow mask, g i v i n g a f i n g e r s p a c i n g of roughly  grid  defined using a metal 1.2  mm  and  a grid  cover2  age No  of approximately  25%.  T o t a l device  a n t i r e f l e c t i o n c o a t i n g was The  areas  a p p l i e d to 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  i l l u m i n a t i o n from an ELH equipped w i t h  ranged from 1.5  t o 3 cm  .  cells. c e l l s were measured under  lamp, which c o n s i s t s of a tungsten-halogen bulb  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  l u m i n o s i t y which i s s h i f t e d towards n e a r - i n f r a r e d wavelengths compared  135 /  to n a t u r a l  sunlight,  AMI i l l u m i n a t i o n  these lamps are w i d e l y used as crude s i m u l a t o r s o f  [138]. o  An uncoated aluminum l a y e r 80 A t h i c k d e p o s i t e d on a s i l i c o n subs t r a t e can be expected t o t r a n s m i t o n l y about 30% of i n c i d e n t v i s i b l e wavelengths  l i g h t at  [ 9 3 ] . As a r e s u l t , under s i m u l a t e d 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 c u r r e n t  density  of the A l - S i O - p S i c e l l s was only  2 about 10 mA/cm . F o r purposes o f comparison, a modern commercial d i f f u s e d j u n c t i o n c e l l would be expected t o g i v e a p h o t o c u r r e n t d e n s i t y somewhat 2 g r e a t e r than 30 mA/cm trical  rather  under one-sun AMI i l l u m i n a t i o n . S i n c e the e l e c -  than the o p t i c a l p r o p e r t i e s  primary i n t e r e s t h e r e , t h e l i g h t  was t h e r e f o r e  increased  level  o f the MIS j u n c t i o n were o f  t o which the MIS c e l l s were exposed 2  t o give a photocurrent density  o f 30 mA/cm i n  each d e v i c e t e s t e d . The f a b r i c a t i o n o f A l - S i O - p S i s o l a r c e l l s on ASEC BSF s u b s t r a t e s x was f i r s t c a r r i e d out i n e a r l y 1979. A t t h a t applied  to p a i r s o f s u b s t r a t e s ,  w i t h each p a i r c o n s i s t i n g  w i t h an a l l o y e d aluminum back s u r f a c e a BSF r e g i o n .  time MIS j u n c t i o n s  The t h i n i n t e r f a c i a l  field  were  of one s l i c e  and a second s l i c e  lacking  oxide l a y e r was grown by e x p o s i n g the  s u b s t r a t e s t o a d r y oxygen flow a t 500°C f o r 20 minutes. A t o t a l o f s i x pairs  of d e v i c e s was f a b r i c a t e d . When i l l u m i n a t e d 2  density  of 30 mA/cm , the mean o p e n - c i r c u i t  found t o be n e a r l y face  40 mV h i g h e r than that  to give a photocurrent  v o l t a g e o f the BSF c e l l s was  o f the c e l l s  l a c k i n g back  sur-  f i e l d s . However, the mean V v a l u e f o r the MIS c e l l s w i t h back oc  surface  f i e l d s was s t i l l  group o f f o u r ASEC N P P +  ingot.  +  15 mV l e s s than t h a t cells  recorded f o r a c o n t r o l  f a b r i c a t e d on s u b s t r a t e s cut from the same  A more complete d e s c r i p t i o n  o f these r e s u l t s i s g i v e n i n Ref. [ 2 1 ] .  By the f a l l o f 1980, improvements i n both the p r o c e s s used t o produce  136  MIS j u n c t i o n s a t UBC and i n the q u a l i t y o f BSF r e g i o n s  formed a t ASEC  made the f a b r i c a t i o n o f a second s e t o f A l - S i O - p S i s o l a r c e l l s on back x s u r f a c e f i e l d s u b s t r a t e s w o r t h w h i l e . F o r t h i s second f a b r i c a t i o n sequence the o x i d a t i o n temperature was r a i s e d t o 600°C and t h e o x i d a t i o n time extended t o 30 minutes. As a r e s u l t , s l i g h t l y have grown than were p r e s e n t cuit voltages  i n the d e v i c e s  t h i c k e r oxide  made e a r l i e r .  layers  should  The o p e n - c i r -  o f the two d e v i c e s produced i n the course o f t h e second  f a b r i c a t i o n sequence a r e l i s t e d i n the f i r s t  column o f Table  4.2. As  u s u a l , these o p e n - c i r c u i t v o l t a g e s were measured a t a p h o t o c u r r e n t den2 s i t y o f 30 mA/cm for  and a t a c o n t r o l l e d temperature o f 28°C. V oc  a c o n t r o l group o f e i g h t ASEC d i f f u s e d j u n c t i o n space c e l l s f a b r i c a t e d  on i d e n t i c a l 10 ficm s u b s t r a t e s were a l s o recorded same t e s t  The ably w i t h  back  surface  compare very  favour-  +  and two N P c e l l s w i t h +  open-circuit voltages those o f the N P P +  from 579 t o 598 mV w i t h voltages  a t t h i s time, under t h e  c o n d i t i o n s . T h i s c o n t r o l group i n c l u d e d s i x N P P  field cells  546  values  +  ohmic back  +  contacts.  o f the two MIS c e l l s  c o n t r o l c e l l s , which were found t o range  a mean o f 588 mV. In c o n t r a s t , the o p e n - c i r c u i t  o f the two N P c e l l s w i t h +  ohmic back c o n t a c t s were found t o be  and 547 mV. Despite  the use o f a 600°C o x i d a t i o n c y c l e , t h e f i l l  A l - S i O x - p S i back s u r f a c e  f a c t o r s o f the  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  sequence were found t o be among the h i g h e s t  f o r any d e v i c e s  fabricated  d u r i n g the course o f t h i s r e s e a r c h program. When i l l u m i n a t e d t o y i e l d a 2 photocurrent  d e n s i t y o f 30 mA/cm , b o t h c e l l s gave f i l l  f a c t o r s o f 0.70.  To demonstrate c o n c l u s i v e l y t h a t t h e a l l o y e d aluminum back s u r f a c e f i e l d r e g i o n was r e s p o n s i b l e f o r the h i g h V  q c  values  c e l l s , t h i s r e g i o n was removed and r e p l a c e d w i t h  quoted f o r the MIS  an ohmic back  contact  TABLE 4.2  Open-circuit voltages  (mV)  measured at T = 28°C and J  Cell:  for Al-SiO -pSi x  sc  With a l l o y e d aluminum BSF:  =30  mA/cm  cells,  2  With Pd back  #1  594  539  #2  593  543  contact  138  u s i n g only room temperature p r o c e s s i n g . Extreme care was  taken not  to  damage the f r o n t j u n c t i o n d u r i n g t h i s s t e p . F i r s t the f r o n t s i d e of each c e l l was  covered  w i t h a commercial e t c h - r e s i s t a n t adhesive  c e l l s were then dipped  i n 10% HF  tape.  f o r a few minutes to remove a l l t r a c e s  of the aluminum back c o n t a c t m e t a l l i z a t i o n . Next each c e l l was i n an a g i t a t e d s o l u t i o n of one called 10  peeled  then re-immersed i n d i l u t e HF  approximately  c e l l s were  thoroughly  to ensure removal of a l l t r a c e s was  away. F i n a l l y , a l a y e r of p a l l a d i u m a few hundred angstroms t h i c k  d e p o s i t e d on the back of the s l i c e s  the hot  of the  (so-  from the back s u r f a c e . F o l l o w i n g a second washing, the tape  s t e n f i l a m e n t t o form a new to  immersed  to n i n e p a r t s 70% HNO^  um of m a t e r i a l from the back of the s l i c e . The  of oxide  was  p a r t 49% HF  "white e t c h " ) f o r a p e r i o d of one minute, to remove  washed, and  The  by  f l a s h evaporation  r e a r c o n t a c t . The  f i l a m e n t f o r a few  s u b s t r a t e s were o n l y exposed  seconds d u r i n g t h i s e v a p o r a t i o n ,  c e l l s s h o u l d have been minimal. P a l l a d i u m  i c o n formed i n t h i s way  from a tung-  so  heating  c o n t a c t s t o p-type  sil-  have been found to g i v e ohmic c h a r a c t e r i s t i c s 2  out  to c u r r e n t d e n s i t i e s of more than 100  mA/cm , w i t h  r e s i s t a n c e s of  2 l e s s than 0.1 ficm . (In a s u b s i d u a r y c h a r a c t e r i s t i c s of p a l l a d i u m were measured by  contacts  experiment, the  current-voltage  to f r e s h l y - e t c h e d p-type  a p p l y i n g such c o n t a c t s  t o the f r o n t s u r f a c e  silicon of 2  Qcm  s l i c e s w i t h s i n t e r e d aluminum back c o n t a c t s ) . The  o p e n - c i r c u i t voltages  recorded  f o r the two  MIS  cells  following  removal of the back s u r f a c e f i e l d r e g i o n are l i s t e d i n the second column o f Table  4.2.  For both c e l l s , the V  v a l u e measured a f t e r  reprocessing  oc i s seen to be face f i e l d  at l e a s t 50 mV  r e g i o n was  the r e p r o c e s s e d  l e s s than t h a t r e c o r d e d when the back s u r -  i n t a c t . As expected, the o p e n - c i r c u i t v o l t a g e s  c e l l s are v e r y  c l o s e to those  for  of the N P c o n t r o l c e l l s  139  w i t h ohmic back c o n t a c t s . The f a c e o f these f u l l 50 mV  MIS  f a c t t h a t a m o d i f i c a t i o n to the back s u r -  c e l l s c o u l d reduce t h e i r o p e n - c i r c u i t v o l t a g e s by  a  demonstrates c o n c l u s i v e l y t h a t the dark c u r r e n t i n these  v i c e s i s dominated by m i n o r i t y c a r r i e r  de-  injection-diffusion.  To ensure t h a t the r e d u c t i o n i n o p e n - c i r c u i t v o l t a g e brought about on replacement of the back s u r f a c e f i e l d to  the MIS  f r o n t j u n c t i o n , two  r e g i o n d i d not  r e s u l t from damage  A l - S i C ^ - p S i c o n t r o l c e l l s were c a r r i e d  through the back c o n t a c t r e f a b r i c a t i o n procedure o u t l i n e d above a l o n g w i t h each MIS  back s u r f a c e f i e l d  r i c a t e d on 2 ftcm s u b s t r a t e s w i t h open-circuit voltages  ranging  o p e n - c i r c u i t v o l t a g e of any  c e l l . These c o n t r o l c e l l s had been ohmic back c o n t a c t s , and  from 558  t o 572  mV.  originally  concluded  of t a p i n g , e t c h i n g , tape removal and p a l l a d i u m  provide  front junction. region  of the MIS  illumination.  back s u r f a c e f i e l d  (A source  should  response at l o n g wavelengths.  To check f o r 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 of one  pro-  deposition described  a back s u r f a c e f i e l d  a small increase i n photocurrent  on  t h a t the  above do not s i g n i f i c a n t l y a l t e r the p r o p e r t i e s of an MIS As p o i n t e d out i n S e c t i o n 2.3,  c e l l s was  r i c h i n i n f r a r e d was  recorded  photocurrent  under i n f r a r e d  o b t a i n e d by simply  operating  an u n f i l t e r e d tungsten-halogen lamp at o n e - t h i r d i t s r a t e d v o l t a g e ; l i g h t i n t e n s i t y was  monitored and h e l d c o n s t a n t  reference s i l i c o n photodiode). the i n f r a r e d p h o t o c u r r e n t  With the back s u r f a c e f i e l d  f o r t h i s c e l l was  1.75  mA.  the  to w i t h i n ±1% u s i n g a  found t o be  replacement of the back s u r f a c e f i e l d w i t h a p a l l a d i u m the p h o t o c u r r e n t  gave  case d i d the  c o n t r o l c e l l drop by more than 2 mV  replacement of the back c o n t a c t . I t can thus be cesses  In no  fab-  region  2.22  mA.  intact, Following  ohmic c o n t a c t ,  under the same i l l u m i n a t i o n c o n d i t i o n s dropped t o  140  4.4  V a r i a t i o n o f MIS  In S e c t i o n  3.5,  Solar  i t was  C e l l Characteristics with Insulator  Thickness  predicted  thick  t h a t MIS  s o l a r c e l l s with  i n s u l a t i n g l a y e r s would have i l l u m i n a t e d c u r r e n t - v o l t a g e which were concave-upwards over a c e r t a i n b i a s Fig. by  3.8.  Further,  i t was  Shewchun ejt a l . [25]  noted t h a t  predicts  fill  carried  out  f a c t ors  f o r MIS  cells  current-voltage  where concave downwards, and 0.25.  Although MIS  the  observation  of the  x  use  factors greater  for photovoltaic current-voltage  energy concharacteristic  4.1  and  procedure  i n Appendix C, w i t h the e x c e p t i o n that  c o n s t a n t at 30 minutes. The  experiment were o f 2 ficm r e s i s t i v i t y and mechanically polished of aluminum was contact.  The  front surfaces.  d e p o s i t e d on  contact  was  30 minutes i n a n i t r o g e n r i e r l a y e r and  val-  c e l l s w i t h t h i c k i n s u l a t o r s , a group of  temperatures r a n g i n g from 600°C t o 660°C were used. The held  than  junction.  - p S i d e v i c e s were f a b r i c a t e d i n accordance w i t h the  time was  cells every-  i n an e x p e r i m e n t a l c e l l would h e l p c o n f i r m the  s e m i c l a s s i c a l model of the MIS  outlined i n Section dation  c h a r a c t e r i s t i c s which are  of an i l l u m i n a t e d  In o r d e r t o produce MIS Al-SiO  c o n v e n t i o n a l homojunction  c e l l s w i t h i n s u l a t o r s t h i c k enough to cause p h o t o c u r -  r e s e m b l i n g F i g . 3.8 idity  l e s s than 0.25  thus must have f i l l  r e n t s u p p r e s s i o n are c l e a r l y of l i t t l e version,  range, as i l l u s t r a t e d i n  the n u m e r i c a l a n a l y s i s  w i t h very t h i c k i n s u l a t o r s . In c o n t r a s t , always have i l l u m i n a t e d  characteristics  substrates  selected  oxidation for this  <100> o r i e n t a t i o n , and  Following  oxidation,  oxi-  had  a thick  chemlayer  the back of the wafers to form an ohmic  then s i n t e r e d at a temperature of 500°C f o r atmosphere. F i n a l l y , the  t h i c k contact  semi-transparent bar-  g r i d were d e p o s i t e d , g i v i n g a t o t a l  junction  2 a r e a of 2 cm The  and  a g r i d coverage of about  illuminated  current-voltage  25%.  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 were r e c o r d e d  are shown i n F i g . 4.5.  As u s u a l , these  characteristics  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 cells  d e n s i t y of 30 mA/cm . The  f a b r i c a t e d u s i n g o x i d a t i o n temperatures of 650°C or g r e a t e r  seen t o be  a fill  are  F u r t h e r , the d e v i c e o x i d i z e d at 660°C has  f a c t o r of l e s s than 0.25,  c e l l s with  those  concave upwards o v e r a l l or p a r t of the power-output quadrant,  as p r e d i c t e d i n S e c t i o n 3.5.  Card  c h a r a c t e r i s t i c s of  fill  [79] and by  as p r e d i c t e d by  f a c t o r s l e s s than 0.25  Shewchun e t a l . MIS  have a l s o been f a b r i c a t e d by  S t . P i e r r e e t a l . [103].  142  F i g u r e 4.5  Illuminated current-voltage Al-SiO -pSi solar x  characteristics for  c e l l s with various i n s u l a t o r  n e s s e s , as measured e x p e r i m e n t a l l y . ature i s s p e c i f i e d J  upc  f o r each c h a r a c t e r i s t i c .  =30mA/cm , T=28°C. 2  Oxidation  thicktemper-  143  CHAPTER 5 MINORITY CARRIER REFLECTING NEGATIVE BARRIER MIS  CONTACTS  i  According Schottky  to the d e f i n i t i o n g i v e n i n Chapter 1, i n a n e g a t i v e  or MIS  barrier  j u n c t i o n the semiconductor s u r f a c e i s accumulated at  e q u i l i b r i u m . J u n c t i o n s of t h i s k i n d can be  formed by  d e p o s i t i n g a low work  f u n c t i o n m e t a l on an n-type s u b s t r a t e , or a h i g h work f u n c t i o n metal on a p-type s u b s t r a t e , p r o v i d e d  t h a t the s u r f a c e s t a t e d e n s i t y 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 , s t r o n g accumulation achieved  i n MIS  the MIS contacts  j u n c t i o n s to form low  3.1).  t u n n e l j u n c t i o n technology,  t h a t low  only through the use  resistance negative  to s i l i c o n have been s u c c e s s f u l l y f a b r i c a t e d Green, Godfrey and  s e l e c t i o n of b a r r i e r m e t a l and form n e g a t i v e b a r r i e r MIS  negative  r e s i s t a n c e metal-semiconductor  [34]. However, i t i s only r e c e n t l y , and  In 1976  to  (see S e c t i o n  l o n g been known t h a t i t should be p o s s i b l e to use  b a r r i e r Schottky contacts  be  j u n c t i o n s , where the growth of the t h i n i n t e r f a c i a l l a y e r  p a s s i v a t e s the semiconductor s u r f a c e I t has  can o n l y  Davies r e c o g n i z e d  of  barrier  [112].  that with a c o r r e c t  i n s u l a t o r t h i c k n e s s , i t s h o u l d be p o s s i b l e  c o n t a c t s which would not only o f f e r n e g l i g -  i b l e impedance t o the flow of m a j o r i t y  c a r r i e r s , but which would  reflect  m i n o r i t y 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 j u n c t i o n [35]. A n e g a t i v e b a r r i e r j u n c t i o n of t h i s type r e a r c o n t a c t i n an induced et  back s u r f a c e f i e l d  a l d i d succeed i n f a b r i c a t i n g low  contacts  c o u l d thus be solar c e l l .  used as  a  Although Green  resistance negative b a r r i e r  MIS  t o both n- and p-type s i l i c o n , they were unable to produce con-  tacts with In the  demonstrable m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s first  s e c t i o n of t h i s c h a p t e r ,  o f c u r r e n t flow i n the n e g a t i v e b a r r i e r MIS  [36].  an approximate a n a l y t i c model c o n t a c t i s developed by  a  144  simple e x t e n s i o n of the r e s u l t s o b t a i n e d i n Chapter  3 [37]. (The  only  o t h e r t h e o r e t i c a l study o f m i n o r i t y c a r r i e r r e f l e c t i o n at the n e g a t i v e b a r r i e r MIS  j u n c t i o n so f a r r e p o r t e d  u s i n g n u m e r i c a l a n a l y s i s . The  [35] was  undertaken by Green et_ a l .  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 p u b l i s h e d . ) The  remainder of the c h a p t e r i s devoted  mental demonstration  o f the m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s of the  n e g a t i v e b a r r i e r j u n c t i o n . In S e c t i o n 5.2,  the use  of n e g a t i v e  Mg-SiO^-nSi c o n t a c t s to form induced back s u r f a c e f i e l d c e l l s with d i f f u s e d front junctions i s discussed  to the e x p e r i -  regions f o r  [37], S e c t i o n 5.3  c o n s i d e r s the f o r m a t i o n of m i n o r i t y c a r r i e r r e f l e c t i n g MIS on p-type  barrier PN +  then  contacts  s u b s t r a t e s u s i n g p l a t i n u m as the b a r r i e r metal. Induced back  s u r f a c e f i e l d a c t i o n was  o b t a i n e d by a p p l y i n g P t - S i O - p S i back c o n t a c t s X  to both N P +  d i f f u s e d f r o n t j u n c t i o n c e l l s , and to c e l l s w i t h A l - S i O ^ - p S i  minMIS f r o n t j u n c t i o n s . Although experiments mentioned above was  the y i e l d of good d e v i c e s i n a l l the never h i g h , the o p e n - c i r c u i t v o l t a g e s of  the b e s t induced back s u r f a c e f i e l d c e l l s were i n every case  comparable  to those o b t a i n e d w i t h c o n v e n t i o n a l d i f f u s e d or a l l o y e d back s u r f a c e fields. 5.1  C u r r e n t Flow i n the Negative For n o t a t i o n a l convenience,  b a r r i e r MIS p-type  B a r r i e r MIS  J u n c t i o n : Theory  i t w i l l be assumed t h a t the n e g a t i v e  j u n c t i o n s c o n s i d e r e d i n t h i s s e c t i o n have been formed  s u b s t r a t e s . However, the c o n c l u s i o n s drawn here apply e q u a l l y w e l l  to n-type m a t e r i a l i f the r o l e s of e l e c t r o n s and h o l e s are The band diagram f o r a n e g a t i v e b a r r i e r MIS p-type  on  interchanged.  c o n t a c t formed on a  s u b s t r a t e i s shown i n F i g . 5 . 1 ( a ) . T h i s band diagram r e p r e s e n t s  a n o n - e q u i l i b r i u m s i t u a t i o n i n which excess i n t o the semiconductor  e i t h e r through  e l e c t r o n s have been i n t r o d u c e d  i l l u m i n a t i o n o r by i n j e c t i o n  from  M  I  M  pSi  nSi  I  INVERSION  ACCUMULATION  DEPLETION  T F n  Fn  -E  '//  X  M  X  -E,  X  S  (a) ure 5.1  '// M  X  S  (b) Band diagrams f o r (a) a n e g a t i v e b a r r i e r MIS p o s i t i v e b a r r i e r MIS  j u n c t i o n , and  (b) the  corresponding  j u n c t i o n formed by d e p o s i t i n g the same metal on a s u b s t r a t e  of the o p p o s i t e doping type. Note the s i m i l a r i t y between the semiconductor r e g i o n s o f the two  surface  s t r u c t u r e s . Both diagrams r e p r e s e n t n o n - e q u i l i b r i u m s i t u a t i o n s .  a PN  j u n c t i o n to the r i g h t . F i g . 5.1(a) i s i n many r e s p e c t s s i m i l a r t o  the band diagram f o r the c o r r e s p o n d i n g  p o s i t i v e b a r r i e r j u n c t i o n formed  by d e p o s i t i n g the same m e t a l on a s u b s t r a t e of the o p p o s i t e doping which i s shown i n F i g . 5.1(b). T h i s o b s e r v a t i o n i s the key  type,  to the a n a l y s i s  of c u r r e n t flow i n the n e g a t i v e b a r r i e r j u n c t i o n . I n p a r t i c u l a r , the band s t r u c t u r e of the accumulation  l a y e r - i n s u l a t o r - metal r e g i o n i n  F i g . 5.1(a) i s n e a r l y i d e n t i c a l to t h a t of the i n v e r s i o n l a y e r - i n s u l a t o r - metal r e g i o n i n F i g . 5.1(b). D i f f e r e n c e s between the two band s t r u c t u r e s can r e s u l t o n l y from the s t o r a g e of charge i n the d e p l e t i o n r e g i o n of p o s i t i v e b a r r i e r j u n c t i o n . Provided i n v e r t e d i n one ferences  case and  should be  the semiconductor s u r f a c e i s s t r o n g l y  s t r o n g l y accumulated i n the o t h e r , these  dif-  small.  I f a n e g a t i v e b a r r i e r MIS  j u n c t i o n i s to p r o v i d e e s s e n t i a l l y  impedance t o the flow of m a j o r i t y c a r r i e r s between the m e t a l and  no 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 a t the m a j o r i t y  the  c a r r i e r q u a s i - f e r m i l e v e l at the semiconductor  i s e f f e c t i v e l y pinned  thin  surface  to the f e r m i l e v e l i n the metal under normal o p e r -  a t i n g c o n d i t i o n s . F u r t h e r , s i n c e the h o l e  c o n c e n t r a t i o n i n the accumu-  l a t i o n l a y e r i s extremely l a r g e , and s i n c e t h i s r e g i o n can be no more than a few hundred angstroms t h i c k , the accumulation  must be very n e a r l y c o n s t a n t  across  l a y e r . I t f o l l o w s that over the normal o p e r a t i n g  both the h o l e d i s t r i b u t i o n and  range  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 j u n c t i o n r e g i o n must remain " f r o z e n " i n t h e i r e q u i l i b r i u m forms. The  energy d i f f e r e n c e between the c o n d u c t i o n  d u c t o r s u r f a c e and  band edge at the  semicon-  the f e r m i l e v e l i n the m e t a l must t h e r e f o r e be  fixed  at i t s e q u i l i b r i u m v a l u e , which, as i n the case o f the p o s i t i v e b a r r i e r j u n c t i o n , w i l l be denoted as qi})^.  In drawing F i g . 5 . 1 ( a ) , fermi l e v e l i s constant computing the m i n o r i t y  i t has  across  been assumed t h a t the e l e c t r o n q u a s i  the a c c u m u l a t i o n l a y e r . As u s u a l ,  c a r r i e r flow through the j u n c t i o n , the  of t h i s assumption can be  after  accuracy  checked u s i n g the methods of S e c t i o n 2.2.  the r e s u l t s of Chapter 3, the net e l e c t r o n c u r r e n t f l o w i n g between conduction  J  CM  band and  " CM 9  A  e  T  the metal i s g i v e n  -P[-(E (X )  '  C  S  (1 - e x p [ - (  E p n  of kT,  set  to zero.  equal  J  CM  =  9  CM  A  e  where qAd> i s now junction region Since  a l l o w i n g the  t  (5.1)  s  (x ) s  For the s i t u a t i o n s of i n t e r e s t h e r e , E multiples  the  by  - E (x ))/kT] F n  F n  rightmost  -  E^/kT])  ( x ) lies g  above E  exponential  ? M  by many  term i n (5.1)  to  be  Therefore  expC-q'f'Bn/kT) exp(qA<j)/kT)  2  d e f i n e d t o be (Ad) w i l l be  (5.2)  the s e p a r a t i o n between E_ and Fn  taken as p o s i t i v e when E ^  the e l e c t r o n c o n c e n t r a t i o n n  lies  (x ) a t the edge o f the P  E„ i n the Fp  above Ep. ) • p  accumulation  A  r e g i o n i s r e l a t e d t o the thermal e q u i l i b r i u m e l e c t r o n c o n c e n t r a t i o n in  From  n ^ p  the q u a s i - n e u t r a l base by  n (x ) = n p  (5.2)  becomes  A  p Q  exp(qA<f,/kT) ,  (5.3)  148  6  CM  CM  A  e  T  '  P  n  (5.4)  (5.4)  ex (-q* /kT) B n  pO  can be w r i t t e n as  J  CM  =  q  S  eff  n  p  ( x  (5.5a)  A>  where  S  e f f  =  9  CM e T" A  q n  pO  Thus so f a r as m i n o r i t y c a r r i e r flows MIS  contact  ity  S ^. e  can be  (5.5b)  exp(-qVkT)  d e s c r i b e d by  are concerned, the n e g a t i v e  an e f f e c t i v e s u r f a c e recombination  veloc-  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 m i n o r i t y c a r r i e r s i n  the q u a s i - n e u t r a l base would be  unchanged i f the n e g a t i v e b a r r i e r  were r e p l a c e d by  recombination  a surface with  at  the edge of the accumulation  is  thus completely  velocity  barrier  layer.  The  velocity ^ ^  quantity S  analogous to the e f f e c t i v e s u r f a c e  positioned  d e f i n e d i n (5.5b) recombination  used t o d e s c r i b e the c u r r e n t flow i n s o l a r c e l l s w i t h  back s u r f a c e f i e l d s From (5.5)  (see S e c t i o n  contact  diffused  2.3).  i t can be seen t h a t the e f f e c t i v e s u r f a c e  v e l o c i t y at a n e g a t i v e b a r r i e r c o n t a c t can be  recombination  reduced by s e l e c t i n g  b a r r i e r m e t a l work f u n c t i o n to maximize ty , or by  increasing  the  the oxide  DTI  t h i c k n e s s i n o r d e r t o reduce the t u n n e l l i n g However, i f the oxide i s made too t h i c k majority  c a r r i e r flow may  junction  i s to be  probability  factor  6^.  the impedance of the c o n t a c t  become e x c e s s i v e . I f the n e g a t i v e  used t o form an induced back s u r f a c e f i e l d  to  barrier solar  cell,  149  a majority  c a r r i e r c u r r e n t e q u a l i n magnitude t o the one-sun  must be a b l e t o pass between the m e t a l and the semiconductor s i g n i f i c a n t displacement  of  photocurrent without  ( x ) from E ^ . c  b  rp  rrl  In d e r i v i n g (5.5), no allowance was made f o r the recombination electrons with holes  through the p r o c e s s  s t a t e s . In g e n e r a l , i f such p r o c e s s e s  of  o f c a r r i e r t r a p p i n g by s u r f a c e  a r e p o s s i b l e the e l e c t r o n c u r r e n t  f l o w i n g i n t o the j u n c t i o n r e g i o n w i l l be g r e a t e r than t h a t s p e c i f i e d by (5.5). Thus (5.5b) g i v e s a lower bound on the e f f e c t i v e s u r f a c e  recombi-  n a t i o n v e l o c i t y o f a n e g a t i v e b a r r i e r MIS c o n t a c t . A p a r t i c u l a r l y simple s u r f a c e recombination  r e l a t i o n s h i p e x i s t s between t h e e f f e c t i v e  velocity  d e f i n e d above and the value  i o n i c emission  f o r a n e g a t i v e b a r r i e r MIS c o n t a c t  of J _, associated with UTn rt  c u r r e n t i n the c o r r e s p o n d i n g  the e l e c t r o n therm-  positive barrier junction.  From (5.5b) and (4.2),  S  (5.6)  =  J  0Th  / n P  0  •  <'> 5  emphasizes t h a t the problem of m i n i m i z i n g  recombination to  eff  the e f f e c t i v e  surface  v e l o c i t y i n a n e g a t i v e b a r r i e r MIS c o n t a c t i s e q u i v a l e n t  the problem o f m i n i m i z i n g  c u r r e n t i n the c o r r e s p o n d i n g  the m a j o r i t y  c a r r i e r thermionic  emission  p o s i t i v e b a r r i e r j u n c t i o n . I n the case o f  the p o s i t i v e b a r r i e r d e v i c e , i n c r e a s i n g the i n s u l a t o r t h i c k n e s s the m a j o r i t y rier  car-  the t u n n e l l i m i t e d regime i s  S i m i l a r l y , i n the n e g a t i v e b a r r i e r c o n t a c t an i n c r e a s e i n the  i n s u l a t o r t h i c k n e s s lowers majority  reduces  c a r r i e r c u r r e n t , but a l s o lowers the maximum m i n o r i t y  c u r r e n t which can be i n j e c t e d b e f o r e  entered.  6  while  simultaneously  r e d u c i n g the maximum  c a r r i e r c u r r e n t which can be passed through the c o n t a c t  before  t u n n e l r e s i s t a n c e e f f e c t s become a p p r e c i a b l e .  5.2  Induced Back Surface  F i e l d S o l a r C e l l s on n - S i l i c o n  When the experiments on n e g a t i v e b a r r i e r MIS t h i s chapter were begun i n the f a l l of 1979,  Substrates  junctions described i n  a s u b s t a n t i a l body o f  evi-  dence had been p u b l i s h e d i n d i c a t i n g t h a t m i n o r i t y c a r r i e r MIS  diodes  c o u l d be  silicon  formed by  substrates  d e p o s i t i n g aluminum o r magnesium on p-type  [20,21,19]. In c o n t r a s t , i n only one  c i r c u i t v o l t a g e e x c e e d i n g 450 formed on n-type s i l i c o n these MIS to  [19]. The  decided  c a r r i e r MIS  low  t h a t the  diode  f o r an MIS  an open-  solar c e l l  o p e n - c i r c u i t voltages obtained difficult,  i f not  s u b s t r a t e . Furthermore, i t was  on n-type s i l i c o n . For t h i s  reason  carrier  c o n t a c t s h o u l d be made u s i n g an n-type decided  t h a t the most c o n v i n c i n g  the m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s of the n e g a t i v e  j u n c t i o n c o u l d be  for  impossible,  f i r s t attempt t o f a b r i c a t e a m i n o r i t y  r e f l e c t i n g n e g a t i v e b a r r i e r MIS  for  been r e p o r t e d  c e l l s suggested t h a t i t would be  form a m i n o r i t y  i t was  mV  i n s t a n c e had  evidence barrier  o b t a i n e d by employing t h i s s t r u c t u r e as the back con-  t a c t i n an induced  back s u r f a c e f i e l d  solar c e l l .  To determine i f a n e g a t i v e b a r r i e r MIS  back c o n t a c t c o u l d enhance  s o l a r c e l l o p e n - c i r c u i t v o l t a g e i n the same manner as a d i f f u s e d back s u r f a c e f i e l d , a group of 2cm  X 2cm  P NN +  BSF  +  s u b s t r a t e s of <100> o r i e n t a t i o n were o b t a i n e d C o r p o r a t i o n . These c e l l s had been p r e p a r e d i n d u s t r i a l procedures. for  use  Four of the c e l l s  as c o n t r o l s . The  then p r o t e c t e d w i t h r e g i o n was  etched  cells  f a b r i c a t e d on 10 ticm  from A p p l i e d S o l a r Energy  i n accordance w i t h  from t h i s group were r e t a i n e d  f r o n t s u r f a c e s of the remaining  tape, a f t e r which the d i f f u s e d N  away. Any  current  t r a c e s of n a t i v e oxide  +  seven c e l l s were  back s u r f a c e  l e f t on the back  field  151  s u r f a c e a f t e r t h i s e t c h i n g p r o c e s s were removed by immersing the s l i c e s i n 10% HF f o r one minute. The s l i c e s were then washed t h o r o u g h l y , and the p r o t e c t i v e tape was p e e l e d away. The samples were next o x i d i z e d i n dry oxygen a t 500°C f o r 20 minutes. F o l l o w i n g o x i d a t i o n , a t h i c k l a y e r of magnesium was d e p o s i t e d  on the back o f the s l i c e s by thermal e v a p o r a -  t i o n t o form the n e g a t i v e b a r r i e r MIS j u n c t i o n . Since magnesium o x i d i z e s r a p i d l y on exposure t o the atmosphere, t h i s magnesium l a y e r was immediately o v e r l a i d with num.  a p r o t e c t i v e c o a t i n g o f t h e r m a l l y evaporated  alumi-  The s t r u c t u r e of t h e 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  c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s o f the completed MIS back c o n t a c t  c e l l s were measured under s i m u l a t e d  one-sun AMI i l l u m i n a t i o n a t a c o n t r o l -  l e d temperature o f 28°C. As u s u a l , a f i l t e r e d tungsten-halogen bulb was  used as t h e AMI s i m u l a t o r . The s h o r t - c i r c u i t  found t o be 34 mA/cm  2  c u r r e n t d e n s i t y was  i n a l l c e l l s t e s t e d . O p e n - c i r c u i t v o l t a g e s were  found t o range from 565 t o 583 mV. The V values oc &  devices  a r e l i s t e d i n the f i r s t  column o f Table 5.1. Under i d e n t i c a l  o p e r a t i n g c o n d i t i o n s , the o p e n - c i r c u i t v o l t a g e s cells MIS  f o r the four best  f o r the P N N +  +  ranged from 574 t o 583 mV. I t f o l l o w s t h a t the n e g a t i v e  c o n t a c t i s capable  velocity  of producing  an e f f e c t i v e s u r f a c e  control barrier  recombination  f o r m i n o r i t y c a r r i e r s which i s as low as can be a c h i e v e d  using  a c o n v e n t i o n a l d i f f u s e d back s u r f a c e f i e l d . However, i t i s a l s o apparent t h a t some o f t h e MIS c o n t a c t s had r e l a t i v e l y h i g h e f f e c t i v e recombination  surface  v e l o c i t i e s . T h i s i n c o n s i s t e n t performance of the n e g a t i v e  b a r r i e r c o n t a c t has been t e n t a t i v e l y a t t r i b u t e d t o v a r i a t i o n s i n the roughness and degree o f contamination  o f t h e etched back s u r f a c e s .  I t i s expected t h a t a h i g h e r y i e l d MIS  c o n t a c t s would be o b t a i n e d  of minority c a r r i e r  reflecting  i f more a t t e n t i o n were p a i d t o a c h i e v i n g  152  A  A  Ti/Ag  HOOOA/lum)  p  (^0.5ym)  Si  n Si  (^350um)  SiO  (^20A)  Mg  F i g u r e 5.2  S t r u c t u r e of P N c e l l w i t h n e g a t i v e Mg-SiO - n S i back c o n t a c t .  x  (^5000A)  barrier  TABLE 5.1  O p e n - c i r c u i t v o l t a g e s (mV) f o r s e l e c t e d P N c e l l s w i t h n e g a t i v e b a r r i e r MIS back c o n t a c t s  Cell:  1st fabrication:  a  2nd f a b r i c a t i o n :  b  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  +  +  +  +  O x i d a t i o n / a n n e a l a t 500 °C i n 0 Anneal a t 450 °C i n N  2  f o r 20 mins; Mg b a r r i e r  f o r 5 mins; A l b a r r i e r  154  a h i g h s u r f a c e f i n i s h i n the e t c h i n g p r o c e s s , and subjected to  t o the f u l l  c l e a n i n g procedure s p e c i f i e d i n Appendix C p r i o r  f a b r i c a t i o n o f the back j u n c t i o n . (The  tape used here to p r o t e c t  f r o n t c o n t a c t m e t a l l i z a t i o n c o u l d not w i t h s t a n d cedure, so the s l i c e s were simply It PN +  i s , of course,  i f the s l i c e s were  dipped  this  full  the  cleaning pro-  i n 10% HF p r i o r to o x i d a t i o n ) .  p o s s i b l e to form n e g a t i v e  b a r r i e r back c o n t a c t s  c e l l s which have never undergone a back s u r f a c e f i e l d  on  diffusion.  However, i t i s u n l i k e l y t h a t c e l l s formed i n t h i s way  would g i v e open-  c i r c u i t voltages s i g n i f i c a n t l y  +  g r e a t e r than those  ohmic back c o n t a c t s . In g e n e r a l , the h o l e  of P N  cells  with  d i f f u s i o n l e n g t h s i n as-grown  n-type m a t e r i a l are too s h o r t f o r a back s u r f a c e f i e l d enhancing the o p e n - c i r c u i t v o l t a g e of a s t a n d a r d  of use i n  ym t h i c k s o l a r  cell  f a b r i c a t e d on such m a t e r i a l . The h i g h o p e n - c i r c u i t v o l t a g e s ' r e c o r d e d  for  PNN"'" c e l l s are thought to be made p o s s i b l e by +  the phosphorus d i f f u s i o n used to form the N  +  300  to be  the g e t t e r i n g a c t i o n of  back s u r f a c e f i e l d  [135]. T h i s g e t t e r i n g a c t i o n i s b e l i e v e d to r a i s e the h o l e  region  diffusion  l e n g t h to a p o i n t where the m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s of the back s u r f a c e f i e l d dark c u r r e n t  can a s s i s t i n r e d u c i n g  the h o l e  injection-diffusion  component.  In o r d e r t o ensure t h a t the N  +  BSF  d i f f u s i o n had been  completely  removed from the s u b s t r a t e s used i n the experiments d e s c r i b e d above, the MIS  back c o n t a c t m e t a l l i z a t i o n and  seven d e v i c e s . T h i s was  accomplished by  f r o n t s i d e of the c e l l s , and for  t h i n oxide were s t r i p p e d from a l l a p p l y i n g p r o t e c t i v e tape to  then immersing the s u b s t r a t e s i n 10%  a few minutes. F o l l o w i n g washing and  HF  tape removal, the s l i c e s were  exposed to a dry n i t r o g e n flow at 450°C f o r f i v e minutes. New c o n t a c t s were then formed by  the  the d e p o s i t i o n of t h e r m a l l y  MIS  back  evaporated  155  aluminum. When a p p l i e d t o p-type has been found  substrates, this  fabrication  procedure  t o y i e l d A l - S i O ^ - p S i s o l a r c e l l s w i t h very low o p e n - c i r c u i t  v o l t a g e s , demonstrating  the presence  o f a l a r g e m a j o r i t y c a r r i e r therm-  i o n i c e m i s s i o n dark c u r r e n t component. The process would t h e r e f o r e be expected  t o g i v e h i g h e f f e c t i v e s u r f a c e recombination  v e l o c i t i e s when  used t o form n e g a t i v e b a r r i e r c o n t a c t s on n-type m a t e r i a l .  to  The  o p e n - c i r c u i t v o l t a g e s o f the r e p r o c e s s e d  549 mV  (see the second column o f Table 5.1).  son, experiments conducted that V  oc  ranged from 544  F o r purposes o f compari-  at A p p l i e d S o l a r Energy C o r p o r a t i o n have shown  v a l u e s between 545 and 550 mV a r e o b t a i n e d when the N  are removed from P N N +  +  cells  ohmic c o n t a c t s . T h i s confirms MIS  cells  layers  +  and r e p l a c e d w i t h c o n v e n t i o n a l s i n t e r e d t h a t the h i g h V  oc  's recorded  f o r the best  back c o n t a c t d e v i c e s i n the f i r s t stage of t h e experiment d i d n o t  result  from the presence  surface f i e l d  of r e s i d u a l t r a c e s o f t h e o r i g i n a l N  +  back  diffusion.  To p r o v i d e f i n a l c o n f i r m a t i o n o f the e x p e r i m e n t a l  r e s u l t s and t o  check t h a t the low V 's r e c o r d e d a f t e r the second MIS back c o n t a c t f a b oc r i c a t i o n were n o t caused by s p u r i o u s e f f e c t s such as the d e g r a d a t i o n o f the f r o n t  c o n t a c t or the j u n c t i o n d u r i n g r e p r o c e s s i n g , the MIS back con-  t a c t m e t a l l i z a t i o n and oxide were s t r i p p e d from the two b e s t c e l l s y e t a g a i n . These c e l l s were then r e p r o c e s s e d the f i r s t s t a g e o f the experiment  f o l l o w i n g the procedure  used i n  ( t h a t i s , the s u b s t r a t e s were o x i d i z e d  f o r 20 minutes a t 500°C, and magnesium was used f o r the b a r r i e r m e t a l ) . The V v a l u e s o b t a i n e d 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 a r e l i s t e d oc in  the t h i r d  c l o s e l y with (column 1 ) .  column o f Table 5.1, and i t can be seen t h a t they those r e c o r d e d a f t e r t h e f i r s t  MIS c o n t a c t  agree  fabrication  156  As n o t e d i n S e c t i o n 5.1, of use  i n forming  i n order f o r a negative  b a r r i e r MIS  to  be  it  i s e s s e n t i a l t h a t the r e s i s t a n c e of the j u n c t i o n to m a j o r i t y  junction  a p r a c t i c a l induced back s u r f a c e f i e l d s o l a r c e l l , carrier  flow be no g r e a t e r than t h a t of a c o n v e n t i o n a l s i n t e r e d c o n t a c t . As no  f o r m a l experiments have been c a r r i e d out here t o determine the  yet,  effec-  t i v e r e s i s t a n c e of the Mg-SiO - n S i and A l - S i O - n S i n e g a t i v e b a r r i e r X  t i o n s . However, measurements o f the f i l l surface f i e l d s o l a r c e l l s  f a c t o r s of the induced  d e s c r i b e d above i n d i c a t e t h a t t h i s  c o n t a c t r e s i s t a n c e i s not e x c e s s i v e . The .were found to l i e i n the range 0.6  fill  - 0.65,  w i l l be  resistance associated with  back  effective  f a c t o r s of these  except i n those  p a r t of the f r o n t c o n t a c t m e t a l l i z a t i o n was Further experimentation  junc-  X  devices  cases  damaged d u r i n g  i n which  processing.  r e q u i r e d to determine i f the s e r i e s  the n e g a t i v e b a r r i e r MIS  contact  can be made  l e s s than or e q u a l to t h a t of a d i f f u s e d back s u r f a c e f i e l d w i t h a s i n tered 5.3  contact. Induced Back Surface  F i e l d S o l a r C e l l s on p - S i l i c o n  F o l l o w i n g the s u c c e s s f u l p r o d u c t i o n n e g a t i v e b a r r i e r MIS to  contacts  c a r r i e r MIS  f i r s t necessary diodes  on  be l i t t l e  chance of success  diode  turned  this could  f o r forming  be  minority  r e p o r t e d by o t h e r i n v e s t i g a t o r s f o r  f a b r i c a t e d on n-type s i l i c o n  c a r r i e r MIS  reflecting  n-type s i l i c o n . As n o t e d i n the p r e v i o u s s e c t i o n ,  In Chapter 3 i t was ity  t o p-type m a t e r i a l . Before  to develop a technique  the low o p e n - c i r c u i t v o l t a g e s solar cells  of m i n o r i t y c a r r i e r  to n-type s u b s t r a t e s , a t t e n t i o n was  f a b r i c a t i n g s i m i l a r contacts  done, i t was  Substrates  in this  MIS  [19] suggested t h a t t h e r e would  task.  p o i n t e d out t h a t the chances of forming  a minor-  on n-type m a t e r i a l are g r e a t e s t when u s i n g a b a r r i e r  157  m e t a l w i t h the h i g h e s t p o s s i b l e work f u n c t i o n . From the v a l u e s o f vacuum work f u n c t i o n f o r the elements t a b u l a t e d by Sze [113], i t can be seen t h a t the metals  p l a t i n u m , i r r i d i u m , rhenium and p a l l a d i u m have the h i g h e s t  work f u n c t i o n s . Of these m e t a l s , p a l l a d i u m and p l a t i n u m are the most r e a d i l y a v a i l a b l e , and the e a s i e s t t o o b t a i n i n a form s u i t a b l e f o r t h e r mal  evaporation.  5.3.1  minMIS Diodes on n - S i l i c o n In the f i r s t  Substrates  stage of the experiments d e s c r i b e d i n t h i s s e c t i o n ,  s m a l l - a r e a MIS s o l a r c e l l s were f a b r i c a t e d on n-type s i l i c o n s u b s t r a t e s u s i n g e i t h e r p a l l a d i u m o r p l a t i n u m f o r the b a r r i e r m e t a l . The s u b s t r a t e s 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 f r o n t s u r f a c e s of  these wafers had been prepared by the s t a n d a r d i n d u s t r i a l  of  chem-mechanical p o l i s h i n g . F o l l o w i n g c l e a n i n g i n accordance  procedure  technique w i t h the  s p e c i f i e d i n Appendix C, the wafers were exposed t o a dry  oxygen flow f o r 20 minutes. At f i r s t ,  o x i d a t i o n was c a r r i e d out at 500°C,  but i n l a t e r experiments o x i d a t i o n temperatures  as h i g h as 650°C were  i n v e s t i g a t e d . The s u b s t r a t e s were n o t removed from the furnace  immediately  a f t e r o x i d a t i o n , but were i n s t e a d exposed t o a dry n i t r o g e n flow f o r an a d d i t i o n a l 20 minutes. T h i s l a s t s t e p may h e l p reduce Q „ _ , the f i x e d r C  positive  charge a s s o c i a t e d w i t h  allow the attainment  the S i - S i O ^ i n t e r f a c e  of h i g h e r b a r r i e r h e i g h t s  [114], and thus  [136]. F o l l o w i n g the  o x i d a t i o n / a n n e a l i n g sequence, a t h i c k l a y e r of t h e r m a l l y evaporated minum was d e p o s i t e d on the backs o f the wafers t o form a n e g a t i v e  alu-  barrier  MIS back c o n t a c t . (The m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s o f t h i s A l - S i O ~ n S i c o n t a c t were n o t of i n t e r e s t h e r e ; the n e g a t i v e b a r r i e r x  technology  simply o f f e r e d a convenient  technique  MIS  f o r forming a l o w - r e s i s t -  158  ance back c o n t a c t t o the l i g h t l y - d o p e d n-type s u b s t r a t e s ) . The r e c t i f y i n g MIS f r o n t j u n c t i o n was then formed by the d e p o s i t i o n o f a l a y e r o f t h e r m a l l y evaporated  palladium or platinum.  semi-transparent  The procedures  fol-  lowed i n the e v a p o r a t i o n o f these metals a r e d i s c u s s e d i n Appendix C. 0  The b a r r i e r m e t a l l a y e r was made approximately  100 A t h i c k , and was  d e l i n e a t e d u s i n g a m e t a l shadow mask t o g i v e a t o t a l d e v i c e area o f about 2 0.1 cm . F i n a l l y , a s m a l l c o n t a c t dot composed of the same m e t a l used i n the b a r r i e r l a y e r was d e p o s i t e d near t h e edge o f each In 5 ficm n-type s i l i c o n , reasonable e s t i m a t e s 2 -1 -1 and m o b i l i t y a r e 0.3 ys and 0.05 m V S u b s t i t u t i n g these e s t i m a t e s  forT  a m i n o r i t y c a r r i e r MIS s o l a r c e l l  s  cell.  f o r the h o l e  lifetime  r e s p e c t i v e l y (see Appendix A ) . (2.28), i t i s found  that  formed on such a s u b s t r a t e s h o u l d  give  p  and y^ i n t o  an o p e n - c i r c u i t v o l t a g e o f about 490 mV a t a p h o t o c u r r e n t  density of  2 30 mA/cm The istics  and a temperature of 28°C. completed p a l l a d i u m - b a r r i e r c e l l s were found  s i m i l a r t o those expected  f o r Schottky  diodes  t o have c h a r a c t e r fabricated using  c o n v e n t i o n a l t e c h n i q u e s . When i l l u m i n a t e d t o g i v e a p h o t o c u r r e n t  density  2 of 30 mA/cm , these 250  c e l l s y i e l d e d an o p e n - c i r c u i t v o l t a g e o f approximately  mV a t a temperature o f 28°C. T h i s low o p e n - c i r c u i t v o l t a g e i n d i c a t e s  t h a t the dark c u r r e n t i n these d e v i c e s i s dominated by m a j o r i t y t h e r m i o n i c e m i s s i o n . The reason  carrier  f o r the poor performance o f these  cells  was d i s c o v e r e d l a t e r when the p a l l a d i u m l a y e r was removed by e t c h i n g i n aqua r e g i a (a mixture Although  of 3 parts hydrochloric acid t o 1 part n i t r i c  the aqua r e g i a r a p i d l y d i s s o l v e d a l l t r a c e s o f m e t a l l i c  dium, even a f t e r prolonged  acid).  palla-  exposure t o the a c i d a t e l e v a t e d temperatures  a p i n k i s h r e s i d u e remained on the s i l i c o n s u r f a c e where the c e l l s had been. I t thus appears t h a t the condensing p a l l a d i u m a c t u a l l y  penetrated  159  the t h i n oxide l a y e r a t the s u b s t r a t e s u r f a c e and r e a c t e d w i t h the underlying silicon  to form p a l l a d i u m s i l i c i d e . Rather  than p r o d u c i n g an  MIS  j u n c t i o n , t h i s p r o c e s s r e s u l t s i n the f o r m a t i o n of an i n t i m a t e - c o n t a c t P d ^ S i - S i Schottky b a r r i e r . Although  every e f f o r t was  made t o  minimize  h e a t i n g of the s u b s t r a t e s d u r i n g the e v a p o r a t i o n , p a l l a d i u m s i l i c i d e s are known t o form at temperatures In  as low as 250°C  [115].  c o n t r a s t to the r e s u l t s o b t a i n e d w i t h p a l l a d i u m , the  barrier cells  gave o p e n - c i r c u i t v o l t a g e s r a n g i n g from 490  2 a p h o t o c u r r e n t d e n s i t y of 30 mA/cm and a temperature V  oc  t o 510 mV  at  of 28°C. These  v a l u e s are the second h i g h e s t e v e r r e p o r t e d f o r MIS  on n-type s i l i c o n  platinum-  cells  [19], and agree q u i t e w e l l w i t h the crude  fabricated  estimate  made above f o r the o p e n - c i r c u i t v o l t a g e of a minMIS c e l l formed on a 5 ficm n-type s u b s t r a t e . F u r t h e r , when the p l a t i n u m l a y e r was aqua r e g i a , no i n s o l u b l e r e s i d u e was  visible  removed i n  a t the s i l i c o n s u r f a c e . I t  was  t h e r e f o r e t e n t a t i v e l y concluded  t h a t the p l a t i n u m c e l l s were t r u e  MIS  d e v i c e s , w i t h the s t r u c t u r e P t - S i O - n S i . x To determine i f s t r o n g i n v e r s i o n of the semiconductor  s u r f a c e had  been a c h i e v e d i n the P t - S i O ^ - n S i c e l l s , the s m a l l - s i g n a l c a p a c i t a n c e C of  these d e v i c e s was  measured as a f u n c t i o n of r e v e r s e b i a s . The  resulting  2 p l o t of 1/C  versus V f o r a r e p r e s e n t a t i v e diode i s shown i n F i g . 5.3.  Using the method of l e a s t s q u a r e s , the s l o p e and v o l t a g e - a x i s i n t e r c e p t V  was  of the l i n e b e s t f i t t i n g the data of F i g . 5.3 were computed. The 15 -3 found t o correspond  agrees  to a doping d e n s i t y of 1.0*10  c l o s e l y w i t h the doping l e v e l expected  cm  slope  , which  f o r 5 ficm n-type  silicon.  The  s t r o n g i n v e r s i o n p o t e n t i a l f o r m a t e r i a l o f t h i s doping d e n s i t y i s  590  mV,  while V  s u r f a c e was  was  found t o be 620  indeed s t r o n g l y i n v e r t e d .  mV,  c o n f i r m i n g t h a t the  silicon  160  F i g u r e 5.3  Capacitance-voltage c h a r a c t e r i s t i c P t - S i O - n S i dot  diode.  for reverse-biased  161  Once i t had been e s t a b l i s h e d t h a t s t r o n g i n v e r s i o n of the semicond u c t o r s u r f a c e c o u l d be  achieved  i n a Pt-SiO -nSi x  diode, a p r e l i m i n a r y  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 w i t h i n s u l a t o r t h i c k n e s s was  conducted. T h i s was  temperature at which the o x i d a t i o n / a n n e a l i n g was  done by r a i s i n g  the  treatment d e s c r i b e d above  c a r r i e d out, w h i l e h o l d i n g the o x i d a t i o n time c o n s t a n t .  nated  devices  The  illumi-  c u r r e n t - v o l t a g e 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 t h i s  experiment c l o s e l y resembled those i n S e c t i o n 4.4.  F i r s t , no  discussed  s i g n i f i c a n t v a r i a t i o n of o p e n - c i r c u i t voltage  w i t h o x i d a t i o n temperature was a 600°C o x i d a t i o n / a n n e a l i n g s i m i l a r to those  of the A l - S i O ^ - p S i c e l l s  observed. Secondly, c e l l s  sequence were found t o have  of c e l l s i n c o r p o r a t i n g oxides  f a b r i c a t e d using characteristics  grown at 500°C; the  f a c t o r s of these d e v i c e s were always g r e a t e r than 0.5.  However, o x i d a t i o n  at 650°C produced c e l l s whose i l l u m i n a t e d c u r r e n t - v o l t a g e  characteristics  were concave upwards over p a r t of the power-output quadrant. As i n Chapter 3 and i n S e c t i o n 4.4,  fill  discussed  c h a r a c t e r i s t i c s of t h i s k i n d r e s u l t  from  the t u n n e l r e s i s t a n c e a s s o c i a t e d w i t h e x c e s s i v e l y t h i c k i n s u l a t i n g l a y e r s .  5.3.2  Minority Carrier Reflecting Pt-SiO -pSi x  On  Contacts  the b a s i s of the experiments d e s c r i b e d above, i t was  t h a t t h e r e was  a reasonable  i n f a c t m i n o r i t y c a r r i e r MIS t h i s hypothesis  by,  p r o b a b i l i t y t h a t the P t - S i O ^ - n S i diodes.  Rather than attempting  concluded c e l l s were to  confirm  f o r example, measuring the temperature dependence of  the c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of these  d e v i c e s , i t was  immediately attempt the f a b r i c a t i o n of a m i n o r i t y c a r r i e r P t - S i O ~ p S i n e g a t i v e b a r r i e r c o n t a c t . From e q u a t i o n x  be shown t h a t the f o r m a t i o n  decided  to  reflecting  (5.6), i t can  readily  of a m i n o r i t y c a r r i e r r e f l e c t i n g c o n t a c t  on  162  a moderately doped p-type s u b s t r a t e i s p o s s i b l e i f and c a r r i e r MIS  diode  can be  only i f a m i n o r i t y  formed on moderately doped n-type m a t e r i a l . F o l -  lowing the g e n e r a l p a t t e r n of the experiments on n e g a t i v e b a r r i e r to n-type m a t e r i a l d i s c u s s e d i n S e c t i o n 5.2, b i n a t i o n v e l o c i t y of the P t - S i O ^ - p S i  the e f f e c t i v e s u r f a c e recom-  c o n t a c t was  i n v e s t i g a t e d by i n c o r -  p o r a t i n g t h i s s t r u c t u r e i n an induced back s u r f a c e f i e l d s o l a r P t - S i C ^ - p S i back c o n t a c t s were a p p l i e d to both N P +  c e l l s and  diffused  cell.  front junction  t o c e l l s w i t h A l - S i O - p S i minMIS f r o n t j u n c t i o n s . In accordance x r  w i t h the terminology  i n t r o d u c e d by  J  Green et_ a l , d e v i c e s of the  k i n d w i l l be r e f e r r e d t o as MISIM c e l l s . c l a s s w i l l be The  contacts  latter  S i m i l a r l y , c e l l s i n the  former  termed N PIM d e v i c e s . +  s u b s t r a t e s employed i n the experiments on P t - S i O ^ - p S i  contacts  d e s c r i b e d i n t h i s s e c t i o n were s u p p l i e d by A p p l i e d S o l a r Energy  Corpora-  t i o n , and were cut from the same i n g o t s used i n the p r o d u c t i o n of back s u r f a c e f i e l d space c e l l s . These s u b s t r a t e s were i d e n t i c a l to those i n the experiments d i s c u s s e d i n S e c t i o n 4.3. 10 ftcm r e s i s t i v i t y 350  to 400  ym.  and  <100> o r i e n t a t i o n , w i t h  s u b s t r a t e s were of  thicknesses ranging  from  Both the f r o n t and back s u r f a c e s of the s l i c e s were chem-  i c a l l y p o l i s h e d . A shallow N the s u b s t r a t e s by then s h i p p e d  The  used  +  l a y e r was  formed at the f r o n t of some of  d i f f u s i o n from a phosphorus s o u r c e . A l l s l i c e s were  to the U n i v e r s i t y of B r i t i s h  Columbia f o r MIS  contact  pro-  cessing. P r i o r to MIS  j u n c t i o n f a b r i c a t i o n , a l l s u b s t r a t e s were c l e a n e d  s p e c i f i e d i n Appendix C. A t h i n o x i d e l a y e r was  then grown on the  as  slices  by exposure to dry oxygen at 600°C f o r 30 minutes. F o l l o w i n g 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 j u n c t i o n s to n-type s i l i c o n , furnace  tube was  the  f l u s h e d w i t h a s t r o n g flow o f d r y , h i g h - p u r i t y n i t r o g e n  163  f o r 20 minutes b e f o r e the  s l i c e s were removed. For  a diffused front junction,  a thin  (-80  of aluminum was  evaporated on one  front junction.  T h i s b a r r i e r l a y e r was  c o n t a c t g r i d . Contact to the N s u b s t r a t e s was then s i n t e r e d  made by  +  deposition  A thick) semi-transparent  surface  region  those s l i c e s l a c k i n g  to create  a r e c t i f y i n g minMIS  then o v e r l a i d w i t h a t h i c k aluminum of the  diffused front  of a t i t a n i u m - s i l v e r  surface  state  completed by  density  substrates.  P r o c e s s i n g was  the  roughly 500  A t h i c k over the back of a l l the  b a r r i e r MIS  back c o n t a c t . The  junction  g r i d , which  at 500°C i n hydrogen f o r 10 minutes. T h i s  a l s o have h e l p e d reduce the  layer  treatment  was may  at the back of the  deposition  of a p l a t i n u m  NP +  layer  o  is illustrated in Fig. The  structure  of both the N PIM and  lost  light  photocurrent density  The  f i n i s h e d c e l l s were  is slightly  i n c i d e n t on  to r e f l e c t i o n , the  to s i m u l a t e one-sun i l l u m i n a t i o n , but 2  +  MISIM c e l l s  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.  e x p e r i m e n t a l d e v i c e s was  a t i o n i n ASEC N P  negative  5.4.  S i n c e a s i g n i f i c a n t f r a c t i o n of the  density  +  c u r r e n t - v o l t a g e c h a r a c t e r i s t i c s of the  measured under ELH  set  s l i c e s to form the  of 30 mA/cm  these uncoated  light  r a t h e r was  i n t e n s i t y was  a d j u s t e d to g i v e a  i n each c e l l t e s t e d .  l e s s than t h a t p r e s e n t l y  attained  This  photocurrent  under AMI  c e l l s equipped w i t h o p t i m i z e d a n t i - r e f l e c t i o n  ten MISIM and  seven N PIM c e l l s +  f a b r i c a t e d here y i e l d e d  c i r c u i t v o l t a g e s spanning a wide range, from 540  not  t o 586  mV.  illumincoatings. open-  Possible  causes f o r t h i s l a r g e spread i n V v a l u e s are c o n s i d e r e d below. The ° oc s i g n i f i c a n t r e s u l t i s t h a t the V v a l u e s r e c o r d e d f o r the two b e s t N PIM oc d e v i c e s were 582 and 586 mV, w h i l e the two b e s t MISIM c e l l s gave V 's oc of 577 and 581 mV (see Table 5.2). In comparison, under the same t e s t +  A  Ti/Ag  MOOOA/lum)  n  (M3.5ym)  A Si  p Si  (^350um)  SiO  (^20A)  x  Pt  (^500A)  Al  (Mum)  Al  MOA)  (a)  SiO  Z_  x  (-V20A)  p Si  (^350ym)  SiO  (^20A)  Pt  x  (^500A)  (b)  F i g u r e 5.4  S t r u c t u r e o f (a) N PIM and (b) MISIM s o l a r +  cells.  165  c o n d i t i o n s the group of N^PP^  c o n t r o l c e l l s w i t h a l l o y e d aluminum back  s u r f a c e f i e l d s r e f e r r e d t o i n S e c t i o n 4.3 of  588  mV.  ASEC N P  c a t e d on otherwise not exceeding  cells  +  550  gave a mean o p e n - c i r c u i t v o l t a g e  l a c k i n g back s u r f a c e f i e l d  r e g i o n s but  fabri-  i d e n t i c a l 10 ftcm s u b s t r a t e s g i v e o p e n - c i r c u i t v o l t a g e s mV  under these  c o n d i t i o n s . On  the b a s i s of these  V oc  measurements, i t was MIS  t e n t a t i v e l y concluded  t h a t the n e g a t i v e  back c o n t a c t s t o the b e s t N PIM and MISIM d e v i c e s were f u n c t i o n i n g +  as e f f i c i e n t m i n o r i t y c a r r i e r  reflectors.  To p r o v i d e c o n v i n c i n g evidence  t h a t the b e s t MISIM and N PIM +  were indeed e x h i b i t i n g induced back s u r f a c e f i e l d MIS  barrier  a c t i o n , the  back c o n t a c t s t o these d e v i c e s were removed and  platinum  replaced with  c o n t a c t s u s i n g only room temperature p r o c e s s i n g . Removal of the m e t a l l i z a t i o n was  accomplished  by simply  f r o n t m e t a l l i z a t i o n was  adhesive all  tape, and  ohmic  platinum  immersing the c e l l s i n warm  methanol; the methanol r a p i d l y undermined and The  cells  l i f t e d the p l a t i n u m  layers.  then p r o t e c t e d by a p p l y i n g e t c h - r e s i s t a n t  the s l i c e s were dipped  t r a c e s of s i l i c o n oxide  i n 10% HF  s o l u t i o n to remove  from the back s u r f a c e . A f t e r t h i s  oxide  o  e t c h the tape was was  p e e l e d away, and  approximately  d e p o s i t e d on the back of the s l i c e s by  As d i s c u s s e d i n S e c t i o n 4.3,  1,000  A of p a l l a d i u m  f l a s h thermal  evaporation.  p a l l a d i u m c o n t a c t s t o f r e s h l y - e t c h e d p-type  s i l i c o n f a b r i c a t e d i n t h i s f a s h i o n g i v e ohmic c h a r a c t e r i s t i c s w i t h 2 r e s i s t a n c e of approximately  0.1 ficm . S e c t i o n 4.3  a  also describes c o n t r o l  experiments c a r r i e d out to ensure t h a t the processes  of t a p i n g ,  tape  removal and p a l l a d i u m d e p o s i t i o n do not harm an A l - S i O ^ - p S i f r o n t  junc-  t i o n . These p r o c e s s e s  the  can,  of c o u r s e , have v i r t u a l l y no e f f e c t on  p r o p e r t i e s of a d i f f u s e d j u n c t i o n . The V v a l u e s o b t a i n e d f o r the r e p r o c e s s e d  c e l l s with  palladium  166  TABLE 5.2  Cell:  Open-circuit voltages  (mV) f o r s e l e c t e d N PIM and MISIM  With P t MIS back c o n t a c t :  +  With Pd back  N PIM#1  586  543  N PIM#2  582  542  MISIM#1  581  545  MISIM#2  577  +  +  cells  contact:  167  ohmic back c o n t a c t s are l i s t e d i n the second column of Table 5.2. average, the o p e n - c i r c u i t v o l t a g e dropped 40 mV r e f a b r i c a t i o n , demonstrating  a f t e r back c o n t a c t  u n e q u i v o c a l l y t h a t the o r i g i n a l  n e g a t i v e b a r r i e r c o n t a c t had been s i g n i f i c a n t l y  On  Pt-SiO^-pSi  r e d u c i n g the  recombination  r a t e at the c e l l back s u r f a c e . Although  the o p e n - c i r c u i t v o l t a g e s of the b e s t MISIM and N PIM  cells  +  f a b r i c a t e d to date compare f a v o u r a b l y w i t h  those of the N P P +  control  +  cells,  the y i e l d of good d e v i c e s has not been h i g h . T h i s low y i e l d most  likely  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 p l a t i n u m  o r a t i o n used t o c r e a t e the n e g a t i v e b a r r i e r P t - S i O ^ - p S i c o n t a c t Appendix C ) . I f the d e p o s i t i o n r a t e i s too r a p i d , the heat the condensing p l a t i n u m may s u r f a c e t o such an e x t e n t  r e l e a s e d by  t h a t the p l a t i n u m w i l l d i f f u s e through the u n d e r l y i n g  i f the e v a p o r a t i o n proceeds too s l o w l y , tungsten  i n t r o d u c e d i n t o the d e p o s i t e d p l a t i n u m due f i l a m e n t w i t h the molten p l a t i n u m likely  charge.  the b a r r i e r h e i g h t of the MIS  may  t o a l l o y i n g of the  In S e c t i o n 2.3  i t was  be  tungsten  Tungsten contamination  is  thus  reduce  c o n t a c t . From (5.5b), i t can be seen t h a t  such a r e d u c t i o n i n b a r r i e r h e i g h t w i l l i n c r e a s e the e f f e c t i v e v e l o c i t y of the  the  silicon.  to lower the work f u n c t i o n of the d e p o s i t e d l a y e r , and  recombination  (see  r a i s e the temperature of the s u b s t r a t e back  t h i n S i O ^ l a y e r and make i n t i m a t e c o n t a c t w i t h Conversely,  evap-  surface  contact.  noted  t h a t the a p p l i c a t i o n of a m i n o r i t y  c a r r i e r r e f l e c t i n g back c o n t a c t t o a s o l a r c e l l i n which the m i n o r i t y c a r r i e r d i f f u s i o n l e n g t h i s g r e a t e r than the base w i d t h s h o u l d g i v e a s m a l l i n c r e a s e i n long-wavelength p h o t o c u r r e n t the n e g a t i v e b a r r i e r P t - S i O ^ - p S i  response.  To determine i f  c o n t a c t c o u l d p r o v i d e a s i m i l a r photo-  c u r r e n t enhancement, the p h o t o c u r r e n t  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 d e s c r i b e d i n S e c t i o n 4.3, a l i g h t source r i c h i n i n f r a r e d was o b t a i n e d by o p e r a t i n g an u n f i l t e r e d  tungsten-  halogen b u l b a t o n e - t h i r d i t s r a t e d v o l t a g e . The i l l u m i n a t i o n l e v e l was h e l d constant t o w i t h i n ± 1 % by m o n i t o r i n g the lamp output w i t h a r e f e r e n c e s i l i c o n p h o t o d i o d e . With the o r i g i n a l P t - S i O ^ - p S i back c o n t a c t ,  cell  N PIM#1 gave an i n f r a r e d p h o t o c u r r e n t of 7.2 mA. When the p l a t i n u m MIS +  c o n t a c t was r e p l a c e d w i t h a p a l l a d i u m ohmic c o n t a c t , the p h o t o c u r r e n t at the same i l l u m i n a t i o n l e v e l dropped t o 6.2 mA. If  a n e g a t i v e b a r r i e r MIS back c o n t a c t i s t o be of use i n i n c r e a s i n g  the e f f i c i e n c y of a s o l a r c e l l ,  i t i s e s s e n t i a l t h a t i t not add a p p r e c i -  a b l e s e r i e s r e s i s t a n c e t o the c e l l . Although  the e f f e c t i v e r e s i s t a n c e o f  the P t - S i O ^ - p S i c o n t a c t has n o t y e t been s t u d i e d i n any d e t a i l , the f i l l f a c t o r s o b t a i n e d f o r the N PIM and MISIM c e l l s +  been g e n e r a l l y good, g i v e n allowance  f a b r i c a t e d thus f a r have  f o r t h e non-optimized  g r i d s . F o r example, t h e MISIM d e v i c e w i t h the h i g h e s t V reasonable  fill  f a c t o r of 0.68.  q c  front contact had a q u i t e  169  CHAPTER 6 SUMMARY  In t h i s t h e s i s , t h r e e major o r i g i n a l c o n t r i b u t i o n s a r e made t o present in  understanding  g e n e r a l . The f i r s t  on the d i s c o v e r y  o f the MIS t u n n e l diode  and o f p h o t o v o l t a i c  devices  c o n t r i b u t i o n i s developed i n Chapter 2, and c e n t e r s  that previous  t h e o r e t i c a l arguments put forward t o  e s t a b l i s h the v a l i d i t y o f the p r i n c i p l e of dark c u r r e n t and p h o t o c u r r e n t s u p e r p o s i t i o n f o r homojunction s o l a r c e l l s course  o f c o r r e c t i n g these  f l a w s , the f i r s t  c o n t a i n s e r i o u s f l a w s . In the comprehensive i n v e s t i g a t i o n  of the b e h a v i o u r o f the q u a s i - f e r m i energy l e v e l s i n the d e p l e t i o n r e g i o n of an i l l u m i n a t e d 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 p a s t  i t had i n v a r i a b l y been assumed t h a t the q u a s i - f e r m i l e v e l s would always be  constant  across  the d e p l e t i o n r e g i o n , i t i s found t h a t f o r o p e r a t i o n  at  s h o r t c i r c u i t o r s m a l l forward  b i a s E^  over t h i s r e g i o n . However, f o r c e l l s reasonably quasi-fermi for  and E ^  p  change d r a m a t i c a l l y  f a b r i c a t e d on s u b s t r a t e s  with  h i g h 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 t h a t both l e v e l s become e f f e c t i v e l y  constant  across  the d e p l e t i o n r e g i o n  o p e r a t i o n c l o s e t o the maximum power p o i n t . From t h i s r e s u l t i t f o l l o w s  t h a t the s u p e r p o s i t i o n p r i n c i p l e must a c c u r a t e l y d e s c r i b e the c h a r a c t e r istics  o f such c e l l s a t a l l o p e r a t i n g p o i n t s , p r o v i d e d  concentrations  i n the q u a s i - n e u t r a l r e g i o n s  the m i n o r i t y  carrier  remain a t low i n j e c t i o n  levels.  In p a r t i c u l a r , i t i s shown t h a t the s u p e r p o s i t i o n p r i n c i p l e s h o u l d even i f a s i g n i f i c a n t  f r a c t i o n o f b o t h recombination  and  apply  photogeneration  o c c u r i n the d e p l e t i o n r e g i o n . T h i s c o n t r a d i c t s the c o n c l u s i o n s  drawn by  Lindholm e t a l . i n a r e c e n t p u b l i c a t i o n [31]. I t i s a l s o found t h a t the s u p e r p o s i t i o n p r i n c i p l e may s e r i o u s l y o v e r e s t i m a t e  the e f f i c i e n c y o f  170  cells  f a b r i c a t e d on s u b s t r a t e s w i t h very poor l i f e t i m e s and low m o b i l -  ities,  a p o i n t which had n o t been a p p r e c i a t e d p r e v i o u s l y . Although the  main r e s u l t s p r e s e n t e d methods, these  i n Chapter 2 are o b t a i n e d  r e s u l t s are confirmed  o f the d i f f e r e n t i a l equations  u s i n g simple a n a l y t i c  through d i r e c t n u m e r i c a l s o l u t i o n  governing  the p o t e n t i a l s , c a r r i e r concen-  t r a t i o n s and c u r r e n t flows w i t h i n a s o l a r c e l l .  The n u m e r i c a l  is  cells.  a p p l i e d t o both s i l i c o n The  second main c o n t r i b u t i o n made i n t h i s t h e s i s i s the p r e s e n t a t i o n  of the f i r s t ity  and g a l l i u m a r s e n i d e  conclusive experimental  c a r r i e r MIS t u n n e l d i o d e s .  of m i n o r i t y  evidence  (In a m i n o r i t y  dark c u r r e n t flow a t moderate forward  ity  f o r the e x i s t e n c e o f minorc a r r i e r MIS d i o d e , the  b i a s i s dominated by the i n j e c t i o n  c a r r i e r s i n t o the q u a s i - n e u t r a l base) . Although the p o s s i b i l -  o f forming  such d e v i c e s had been proposed on t h e o r e t i c a l grounds by  Green e t a l . i n 1974 [16], u n t i l the completion in  analysis  o f the experiments  Chapter 4 t h e i r e x i s t e n c e remained a s u b j e c t o f c o n s i d e r a b l e  described  contro-  v e r s y . Two independent experiments were c a r r i e d out here t o e s t a b l i s h t h a t these d e v i c e s current-voltage  c o u l d i n f a c t be made. In the f i r s t experiment, the  c h a r a c t e r i s t i c s of A l - S i O ^ - p S i diodes were r e c o r d e d  f u n c t i o n o f temperature  as a  [20]. From t h i s d a t a , an a c t i v a t i o n energy de-  s c r i b i n g the temperature dependence o f the dark c u r r e n t was e x t r a c t e d . T h i s a c t i v a t i o n energy was found t o agree e x a c t l y w i t h 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 c u r r e n t , and t o be  c a n t l y l a r g e r than t h a t p o s s i b l e f o r a m a j o r i t y emission  t h a t expected  carrier  signifi-  thermionic  c u r r e n t . In the second experiment, A l - S i O ^ - p S i s o l a r c e l l s were  f a b r i c a t e d on s u b s t r a t e s w i t h a l l o y e d aluminum back s u r f a c e f i e l d s [ 2 1 ] . When the back s u r f a c e f i e l d and  r e g i o n s were removed by c h e m i c a l  r e p l a c e d w i t h ohmic c o n t a c t s , the o p e n - c i r c u i t v o l t a g e s  etching o f these  171  c e l l s were found  t o drop by as much as 50 mV.  T h i s demonstration  change i n the p r o p e r t i e s o f the back s u r f a c e of an MIS significantly evidence The  s o l a r c e l l could  a l t e r the o p e n - c i r c u i t v o l t a g e p r o v i d e d f u r t h e r i r r e f u t a b l e  f o r the e x i s t e n c e of m i n o r i t y c a r r i e r MIS  diodes.  t h i r d p r i n c i p a l c o n t r i b u t i o n of t h i s t h e s i s i n v o l v e s a t h e o r -  e t i c a l and e x p e r i m e n t a l study of the p r o p e r t i e s of the n e g a t i v e MIS  that a  j u n c t i o n [37]. In Chapter  5, a simple  barrier  a n a l y t i c model of c u r r e n t flow  i n the n e g a t i v e b a r r i e r j u n c t i o n i s developed.  T h i s model p r e d i c t s t h a t  w i t h a s u i t a b l e c h o i c e of i n s u l a t o r t h i c k n e s s and b a r r i e r m e t a l work f u n c t i o n , i t s h o u l d be p o s s i b l e to form n e g a t i v e b a r r i e r MIS which p r e s e n t a very low e f f e c t i v e s u r f a c e recombination  contacts  velocity  to  m i n o r i t y c a r r i e r s , y e t which o f f e r n e g l i g i b l e impedance t o the flow of m a j o r i t y c a r r i e r s . The m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s o f the n e g a t i v e b a r r i e r MIS  j u n c t i o n were demonstrated e x p e r i m e n t a l l y by  i z i n g t h i s s t r u c t u r e t o form induced back s u r f a c e f i e l d s o l a r U s i n g magnesium as the b a r r i e r m e t a l , n e g a t i v e b a r r i e r MIS  [37]. D i f f u s e d f r o n t j u n c t i o n P N +  cells.  contacts  were employed to form induced back s u r f a c e f i e l d s on n-type substrates  util-  silicon  c e l l s were used i n t h i s  experiment. L a t e r , P t - S i O ^ - p S i n e g a t i v e b a r r i e r c o n t a c t s were a p p l i e d to form induced back s u r f a c e f i e l d s  on p-type  m a t e r i a l . In t h i s  experiment, both d i f f u s e d f r o n t j u n c t i o n N P +  cells  and  second  c e l l s w i t h pos-  i t i v e b a r r i e r minMIS f r o n t j u n c t i o n s were used. Although  the y i e l d  of  Mg-SiC^-nSi and P t - S i O ^ - p S i j u n c t i o n s w i t h s t r o n g m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s was  never h i g h , the b e s t induced back s u r f a c e  field  c e l l s i n each c l a s s mentioned above gave o p e n - c i r c u i t v o l t a g e s comparable t o those o b t a i n e d w i t h c o n v e n t i o n a l back s u r f a c e f i e l d s formed by d i f f u s i o n o r a l l o y i n g . The m i n o r i t y c a r r i e r r e f l e c t i n g p r o p e r t i e s of  the  172  n e g a t i v e b a r r i e r MIS back c o n t a c t s were confirmed by r e p l a c i n g c o n t a c t s w i t h ohmic back c o n t a c t s w i t h o u t damaging the f r o n t When t h i s was done, the c e l l o p e n - c i r c u i t v o l t a g e dropped  these  junction.  significantly.  I t was a l s o shown t h a t a P t - S i O - p S i induced back s u r f a c e f i e l d c o u l d x enhance the i n f r a r e d p h o t o c u r r e n t response of an N P c e l l . Although no +  d i r e c t measurements o f the e f f e c t i v e r e s i s t a n c e of the Mg-SiO - n S i and x P t - S i O ^ - p S i n e g a t i v e b a r r i e r c o n t a c t s were made, the f i l l the induced back s u r f a c e f i e l d  factors of  c e l l s i n c o r p o r a t i n g these c o n t a c t s were  found t o be r e a s o n a b l y h i g h . In a d d i t i o n t o the t h r e e main c o n t r i b u t i o n s o u t l i n e d above, a t h e o r e t i c a l model o f c u r r e n t flow i n the p o s i t i v e b a r r i e r MIS j u n c t i o n i s developed  i n Chapter  3. T h i s model i s based  l a r g e l y on e a r l i e r  theoretical  s t u d i e s c a r r i e d out by Green e t a l . [ 1 6 , 1 7 ] and by Card and Rhoderick  r  [26-28]. However, t h e model p r e s e n t e d here has the advantages o f b e i n g p u r e l y a n a l y t i c , u n l i k e t h a t developed by Green e_t^ a l , f o r s t r o n g i n v e r s i o n o f the semiconductor  and o f a l l o w i n g  s u r f a c e , u n l i k e that  proposed  by Card and Rhoderick. The main drawback o f t h e 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 o f s u r f a c e s t a t e s on c u r r e n t flows o r 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 a c r o s s the j u n c t i o n . In any case, s u r f a c e s t a t e e f f e c t s are b e s t h a n d l e d by r e s o r t i n g t o e n t i r e l y  numerical  methods, as d i d Green e t 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 t h a t the i l l u m i n a t e d c u r r e n t - v o l t a g e 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 c e r t a i n b i a s range. E x p e r i m e n t a l c o n f i r m a t i o n o f t h i s p r e d i c t i o n i s p r o v i d e d i n Chapter  4. The development o f the MIS j u n c t i o n model was m o t i v a t e d i n  p a r t by s u g g e s t i o n s made r e c e n t l y t h a t the MIS t u n n e l diode i s a fundam e n t a l l y d i f f e r e n t d e v i c e than the n o n - i d e a l Schottky d i o d e . In Chapter 3  it  i s shown t h a t the o p e r a t i o n of both these  d e v i c e s can be  explained  u s i n g the same b a s i c model. Although the m e t a l - i n s u l a t o r - s e m i c o n d u c t o r  t u n n e l j u n c t i o n has  the s u b j e c t of w e l l over one hundred s c i e n t i f i c papers i n the p a s t much r e s e a r c h remains to be t o p i c s f o r f u t u r e study presented  done on t h i s s t r u c t u r e . A number of p o s s i b l e  a r i s e as s t r a i g h t f o r w a r d e x t e n s i o n s  diodes  can be  formed on n-type s i l i c o n would be  a more fundamental f r o n t , the a n a l y t i c model of the MIS  veloped  i n Chapter 3 s h o u l d be m o d i f i e d  concentrations  of the  to a l l o w  of  junction  f o r degenerate  n a t i o n v e l o c i t i e s and  carrier  supply  lower e f f e c t i v e s u r f a c e  formed u s i n g c o n v e n t i o n a l  techniques.  The  order  only 50  s t r u c t u r e s i s a s u b j e c t of  to 100  ym  to f u n c t i o n e f f i c i e n t l y ,  back s u r f a c e  fields.  recombifield  development of more  importance i n view of the r e c e n t t r e n d towards the use cells  barrier  lower c o n t a c t r e s i s t a n c e s than back s u r f a c e  e f f e c t i v e back s u r f a c e f i e l d  silicon  de-  at the semiconductor s u r f a c e . Above a l l , a comprehensive  j u n c t i o n s can s i m u l t a n e o u s l y  regions  result  interest.  i n v e s t i g a t i o n should be undertaken to determine whether n e g a t i v e MIS  decade  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 t h a t m i n o r i t y  c a r r i e r MIS On  been  considerable of l i g h t w e i g h t  t h i c k f o r space a p p l i c a t i o n s [134].  In  such c e l l s must i n c o r p o r a t e h i g h q u a l i t y  174  APPENDIX A NUMERICAL SOLUTION OF THE  The  o p e r a t i o n of any  equations  BASIC SEMICONDUCTOR EQUATIONS  semiconductor d e v i c e i s governed by  of semiconductor p h y s i c s : the c u r r e n t e q u a t i o n s ,  equations,  and P o i s s o n ' s  The  first  b a s i c equations was  the c o n t i n u i t y  e q u a t i o n . An exact a n a l y t i c s o l u t i o n of  coupled n o n - l i n e a r d i f f e r e n t i a l equations circumstances.  the b a s i c  these  i s p o s s i b l e o n l y i n the  simplest  a p p l i c a t i o n of n u m e r i c a l methods to s o l v e  r e p o r t e d i n 1964  by  Gummel [116], who  u t i o n s f o r the c a r r i e r c o n c e n t r a t i o n s and  obtained  the sol-  flows i n one-dimensional  diode  and b i p o l a r t r a n s i s t o r s t r u c t u r e s under s t e a d y - s t a t e c o n d i t i o n s . The a l g o r i t h m i n t r o d u c e d by DeMari  [117]  Gummel was  l a t e r m o d i f i e d and extended by  and A r a n d j e l o v i c [118]. U n f o r t u n a t e l y , i n 1972  Choo  [119]  d i s c o v e r e d t h a t Gummel's a l g o r i t h m would not converge i n many s i 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 . T h i s f a i l u r e to converge was on a n a l y t i c grounds at t h i s time by Mock  a l s o demonstrated  [120].  In response to the f a i l u r e of Gummel's a l g o r i t h m , Seidman Choo [51] developed  a completely  of the b a s i c equations (  (2.3)-(2.7) ). The  new  method f o r the n u m e r i c a l  i n t h e i r s t e a d y - s t a t e , one-dimensional  a l g o r i t h m d e v i s e d by  i n t u i t i v e l y p l e a s i n g , simple  Seidman and  to implement, and  seen e x t e n s i v e use years  i n the m o d e l l i n g  form  efficient.  i t s v a r i a n t s have  of semiconductor d e v i c e s i n r e c e n t  [121-123]. In t h i s appendix, the a p p l i c a t i o n of Seidman and  a l g o r i t h m to the m o d e l l i n g o f s i l i c o n and is  solution  Choo i s at once  computationally  I t i s thus h a r d l y s u r p r i s i n g t h a t t h i s a l g o r i t h m and  and  GaAs homojunction s o l a r  c o n s i d e r e d i n d e t a i l . For n o t a t i o n a l convenience,  developed  i n r e f e r e n c e t o NP  Choo's cells  the a l g o r i t h m i s  d e v i c e s , a f t e r which the few  simple  modif-  i c a t i o n s r e q u i r e d t o t r e a t PN c e l l s a r e c o n s i d e r e d . Before any p r o g r e s s can be made i n the s o l u t i o n o f ( 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 f r e e c a r r i e r c o n c e n t r a t i o n s n and p  and the r e c o m b i n a t i o n r a t e U must be s p e c i f i e d , as must boundary c o n d i t i o n s on n, p and ty. Band-to-band r e c o m b i n a t i o n i s l i k e l y t o be import a n t i n GaAs, which i s a d i r e c t bandgap m a t e r i a l [125]. However, f o r simplicity  o n l y r e c o m b i n a t i o n through  t r a p p i n g c e n t e r s w i l l be c o n s i d e r e d  h e r e . Seidman and Choo assumed t h a t the dependence o f U on n and p c o u l d be a c c u r a t e l y d e s c r i b e d by a S h o c k l e y - R e a d - H a l l model [124] w i t h a s i n g l e t r a p p i n g l e v e l a t midgap. I n t h i s  case,  (A.l)  The boundary c o n d i t i o n s imposed on (2.3)-(2.7) w i l l be those d e s c r i b e d i n s u b s e c t i o n 2.2.2 (see equations  (2.8)-(2.9) and a s s o c i a t e d d i s c u s s i o n ) .  NORMALIZATION A l l n u m e r i c a l methods f o r the s o l u t i o n of (2.3)-(2.7) b e g i n w i t h a n o r m a l i z a t i o n o f v a r i a b l e s designed t o e l i m i n a t e as many c o n s t a n t from the e q u a t i o n s as p o s s i b l e , and thus minimize The n o r m a l i z a t i o n procedure  computational  factors  time.  f o l l o w e d here i s t h a t i n t r o d u c e d by DeMari  and i s summarized i n T a b l e A . l . F o r the remainder  of t h i s appendix, a l l  symbols w i l l r e f e r t o n o r m a l i z e d q u a n t i t i e s u n l e s s o t h e r w i s e In terms o f n o r m a l i z e d v a r i a b l e s ,  specified.  (2.3)-(2.7) become  dJ /dx = U - G n  (A.2)  dJ /dx = -U + G P  (A. 3)  [117],  176  TABLE A . l  Normalization  factors  V a r i a b l e t o be n o r m a l i z e d :  D i v i d e by: L  n,p  (= [ k T / q n . ] 2  D  £  n.  1  V^  (= kT/q)  J ,J n' p  qD n./L  D ,D n P  D  0  n' p U,G N ,N ,N ,n ,p A  V  D  S  B  T  1  D  (= 1 m V  Q  L  D  D  O i  D  0  / D  n  0 / L  1  / L  D  D  1  )  1 / 2  )  J  n  J  V~  =  n  (  d  4  ,  /  d  x  )  dn/dx]  +  (A.4)  = -D [p(d^/dx) + dp/dx]  p  (A.5)  p  d ^/dx 2  2  = -(p + N  D  - N  A  - n)  (A.6)  w h i l e boundary c o n d i t i o n s (2.8) and (2.9) become  w  • v y v - VPO B X  ) ]  <->  '  (A.8)  A  7  and  - p F> J  ( x  = V  W  " nO P  ( x  F  ) ]  The SRH expression f o r the recombination rate becomes  U = (pn - l ) / ( x n + T p + x + T ) p n p n  (A.9)  f o r a t r a p p i n g c e n t e r a t midgap. At t h i s stage  Seidman and Choo found  i t convenient  to introduce  two new v a r i a b l e s u and v d e f i n e d a c c o r d i n g to  u  =  -ty n  e  (A.10)  and  v = pe .  (A.11)  y  u and v bear t h e f o l l o w i n g simple  r e l a t i o n s h i p t o the q u a s i - f e r m i  178  p o t e n t i a l s d> n  d> : p  and  r  Y  (f> = - l n ( u ) n  (A. 12)  x  and  <{> = l n ( v ) .  (A. 13)  p  In terms of u and  v, the c u r r e n t e q u a t i o n s can be w r i t t e n i n a p a r t i c u l -  a r l y simple form. For example, n o t i n g  du/dx =  it  can be  that  (dn/dx)e~* - ne~*(diji/dx) =  seen by  comparison w i t h  J  n  (A.4)  [-n(d^/dx) +  dn/dxje"^  that  = D e^du/dx). n  (A.14)  Similarly,  Jp = -D e p  Defining a  B =  the  current  (A.15)  quantity  T n p  + xp  +  and  c o n t i n u i t y e q u a t i o n s can be  n  -d_(D dx and  ^(dv/dx).  p  T  + T  n  = x ue^ + x ve ^ + x  p  n  combined t o  e*(du/dx)] + uv = 1 + G n  B  B  + x p  n  (A. 16)  give  (A.17)  -d_[D e *(dv/dx)] + vu = 1 + G. dx B B  (A.18)  P  LINEARIZATION  OF THE BASIC EQUATIONS  Seidman and Choo proposed  t h a t (A.17) and (A.18) c o u l d be  linearized  as f o l l o w s . On any g i v e n i t e r a t i o n , l e t u, v and ip be the e s t i m a t e s f o r u, v and ty o b t a i n e d on the p r e c e d i n g i t e r a t i o n . Then a l i n e a r approximat i o n t o (A.17) i s  -d [D e^(du/dx)] + uv = 1 + G dx B B  (A.19)  n  w h i l e a l i n e a r approximation  to (A.18) i s  -d_[D e ^ ( d v / d x ) ] + vu = 1 + G. dx I B  (A.20)  p  P o i s s o n ' s e q u a t i o n can a l s o 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 o r d e r i n 6, (A.6) becomes  -d 6/dx 2  2  + 6(ue^ + ve  Each of equat i o n s  = d \|//dx 2  2  - ue* + v e " * + N„ - N.. D A  (A.21)  (A.19)—(A.21) i s o f the form  -(ay') ' + gy = f  (A.22)  where ' denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o x. To s o l v e (A.22) n u m e r i c a l l y , a s e t of g r i d p o i n t s {x^, j=l,M} i s i n t r o d u c e d , and d e r i v a t i v e s w i t h r e s p e c t t o x are approximated  by t a k i n g 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 p o i n t ( t h a t i s , a p o i n t f o r which 2 s j £ M - l ) g r i d  spacings  h. = x. - x. ,  (A.23)  h. = x . -  (A.24)  and  x.  are d e f i n e d . Then midway between p o i n t s x. and x. 3 J-l a p p r o x i m a t i o n t o ay' i s  a y  '  =  (  a  j-l  +  a  j  '  )  (  y  j ~  y  )  hT  2  (Here the n o t a t i o n  j - l  a reasonable  = a ( x ^ ) , y^ = y ( x ^ ) , e t c . i s u s e d ) .  S i m i l a r l y , between p o i n t s x^ and .. ^, a reasonable approximation to ay' x  +  is  ay' «  (a, + a, )  • ( y ^ - y,)  +1  2  h  .  +  3  T h e r e f o r e at p o i n t x.  (ay')  ( a  i V  ' ji -V  +  (  (A.25)  (y  +  a  i  +  " i - ^  •  (  y  i  '  y  i - l  }  (h /2 + hJ/2)  Defining  181  X  j  =  (  a  j  +  a  j+l  ) / h  j  (A. 26a)  =  X j  + a^/h  (A.26b)  and n|=x{/Oi| + h ) J  J  J  (A.27a)  J  = , / ( h + h.) J J J  (A.27b)  +  n  J  x  (A.25) becomes  -(ay') | ,  x >  *  t-rij Y  " n~  j + 1  y._  ±  +  (n+ +  (A.28)  ^~)y.].  Thus at any i n t e r i o r grid point the d i f f e r e n t i a l equation (A.22) takes the f i n i t e difference form  A y. + b- y- i + c. y.,, = f. i 3 3 J-l J J+l J  (A.29a)  a  where  a  b  j  =  n  j  +  n  j  +  3  j '  (A.29b)  j = -nj  (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 s p e c i f i e d . S i m i l a r l y , v and ty are fixed at x^. With boundary conditions of this type, (A.29) s t i l l holds with and f. = y, , and a =1, b„ = 0, c = 0 and f 1 1 M M M w  form (A.7) becomes  w  = 1, b  1  = 0,  = 0  = y . In f i n i t e difference M M w  [x"/2 + S  B  exp(^)]  where the v a l u e o f x  M  u  M  (x /2) = S  -  M  n ^ )  B  i s that a p p r o p r i a t e t o (A.19).  (A.30)  S i m i l a r l y , (A.8)  becomes  [X*/2 + S  where x^ i  s  p  expH^)] v  - v  ±  ( |/2) = S  2  X  a p p r o p r i a t e t o (A.20).  p  P  C l e a r l y both  n Q  (x ) F  (A. 31)  (A.30) and (A.31) a r e  of the form (A.29a). SOLUTION OF THE LINEARIZED BASIC EQUATIONS Equation  (A.29a) can be expressed  i n matrix  A y = ?  form as  (A.32a)  where  A =  a  i  b  2  (A.32b)  l  C  a  2  C  2  b  M-l  ^-1 b  M  M-1  C  a  M  and y and f a r e column v e c t o r s w i t h j t h element y^ and f_. r e s p e c t i v e l y . The m a t r i x A has non-zero elements o n l y on the main d i a g o n a l and on the two a d j a c e n t  d i a g o n a l s , and so i s termed a t r i d i a g o n a l band m a t r i x . In  o r d e r t o s o l v e f o r y, i t i s n e c e s s a r y  t o i n v e r t A, T h i s can be done  183 c o n v e n i e n t l y u s i n g the f o l l o w i n g a l g o r i t h m , which i s d e r i v e d by simple row-echelon r e d u c t i o n o f A  [126]:  1. D e f i n e a r r a y s d and y a c c o r d i n g t o : l  d  =  l  a  Yl = c / d 1  2. D e f i n e  1  d. = a. - b . y. .. 3 3 3 J - l  ( f o r i = 2 t o M)  Y. = c./d.  ( f o r i = 2 t o M)  an a r r a y g a c c o r d i n g t o : *1 j  8  =  V  =  ( f  d  i  3 ~ 3 b  8  3-l  ) / d  J  (  f  °  r J  =  2  t  0  M  )  3. y i s g i v e n by: y  M  y  ±  =  % = g  ±  " Y  ±  y  ±  +  1  ( f o r i = M-1 t o 1).  In o r d e r t o minimize the s t o r a g e space r e q u i r e d by a computer program implementing the above a l g o r i t h m , the members o f the f o l l o w i n g p a i r s o f a r r a y s can share  the same memory l o c a t i o n s : d and a Y and c g and b  INITIALIZATION AND COMPUTATIONAL PROCEDURE To c a r r y through one i t e r a t i o n o f Seidman and Choo's a l g o r i t h m , equations  (A.19) and (A.20) a r e f i r s t s o l v e d i n f i n i t e  d e s c r i b e d above t o generate improved e s t i m a t e s values  d i f f e r e n c e form as  f o r u and v. These new  f o r u and v a r e then s u b s t i t u t e d i n (A.21), which i s t u r n s o l v e d  i n f i n i t e d i f f e r e n c e form t o determine t h e c o r r e c t i o n term 6 t o be added  184  to  ty.  Once the c o r r e c t e d v a l u e of ty has been computed, the next  b e g i n s . In o r d e r t o s t a r t to  the i t e r a t i v e p r o c e s s , i t i s c l e a r l y  iteration necessary  have r e a s o n a b l y good i n i t i a l e s t i m a t e s f o r u,v and ty. Here an e s t i m a t e  f o r ty was  o b t a i n e d a t each b i a s p o i n t V by assuming t h a t the  p o t e n t i a l drop V r e g i o n , and  - V over the c e l l appears  electrostatic  only a c r o s s the d e p l e t i o n  then a p p l y i n g the d e p l e t i o n approximation.  Knowing ty, i n i t i a l  e s t i m a t e s f o r u and v were o b t a i n e d by s p e c i f y i n g the p o s i t i o n o f the q u a s i - f e r m i energy taken  to be  l e v e l s . The m a j o r i t y c a r r i e r q u a s i - f e r m i l e v e l  was  c o n s t a n t over each q u a s i - n e u t r a l r e g i o n , and t o extend at t h i s  c o n s t a n t v a l u e a c r o s s the d e p l e t i o n r e g i o n . The f e r m i l e v e l s i n the d e p l e t i o n r e g i o n was  s e p a r a t i o n of the q u a s i -  thus s e t e q u a l to qV.  the m i n o r i t y c a r r i e r q u a s i - f e r m i l e v e l was  Finally,  s e t t o c o i n c i d e w i t h the maj-  o r i t y c a r r i e r q u a s i - f e r m i l e v e l i n each q u a s i - n e u t r a l r e g i o n . Although t h i s l a s t i n i t i a l i z a t i o n c o n d i t i o n must s e r i o u s l y underestimate  the  m i n o r i t y c a r r i e r c o n c e n t r a t i o n s a t any V > 0, i t has not r e s u l t e d i n c o m p u t a t i o n a l problems. In g e n e r a t i n g the r e s u l t s p r e s e n t e d i n s u b s e c t i o n 2.2.4, the i t e r a t i v e p r o c e s s was  c o n t i n u e d u n t i l the s o l u t i o n f o r the  t o t a l c u r r e n t flow i n the diode v a r i e d by l e s s than 0.1%  between s u c c e s -  s i v e i t e r a t i o n s . T h i s g e n e r a l l y r e q u i r e d fewer than 20 i t e r a t i o n s  starting  from the i n i t i a l i z a t i o n o u t l i n e d above.  CARRIER LIFETIMES AND The  MOBILITIES  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 h e r e  s p e c i f i e d by s i m p l y making a p i e c e w i s e - l i n e a r approximation of Sze  was  t o the p l o t  the l o g a r i t h m o f the m o b i l i t y v e r s u s the l o g a r i t h m o f N^ p r e s e n t e d by [127]; the p o i n t s used i n the p i e c e w i s e 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 t o compute m o b i l i t y  a) Data f o r s i l i c o n V  (m" )  p : ( m W  3  10  1.4*10  2 1  ••A22  10  1.1*10  23  10  7.0*10  24  10  3.4*10  25  10  1.2*10  )  1  u :  ( m W  _E  5.8*10"  1  -1  2  _o  5.0*10  A  )  1  2  o 3.3*10"  -9 Z  2  -9  2.0*10  -9  8*10  o  o  b) Data f o r GaAs N : T  (m  - 3  )  u  n  :  ( m W  )  1  (m V" s" )  p^:  10  2 1  7.0*10  _ 1  3.7*10  10  2 2  6.0*10  _ 1  3.2*10"  10 10 10  23  4.5*10  J  24  3.0*10  25  For  1.5*10  N  l  21  < 10 21  = 10  m  -3  -1  1  2  2.3*10  1  - 2  2  _9 _9  -1  1.3*10 _-3  -1  7*10  , p and p are a s s i g n e d the v a l u e s l i s t e d f o r n p  -3  m  Otherwise, p^ and p  p  are computed by l i n e a r  interpolation or extrapolation.  I  186  When m o d e l l i n g s i l i c o n related  cells,  the e l e c t r o n  and h o l e l i f e t i m e s  were  t o t h e doping l e v e l through the formulas  + T  p " 0p T  / ( 1  W  (A  '  33a)  and  \•v  VV  / ( 1+  <-> A 3 b  where  N  T  =  N  A  +  N  D  (A.33c)  as suggested by Fossum [ 1 2 8 ] . - x . and x „ were s e t e q u a l t o 1.7*10~^ s On Op and  3.5*10  7  15 7.1*10  s respectively,  w h i l e NQ^  and  were both s e t e q u a l t o  -3 cm  . Since r e l a t i v e l y l i t t l e  carrier lifetimes  i n GaAs i s a v a i l a b l e ,  information concerning minority x  and x n  independent  o f the doping l e v e l i n t h i s m a t e r i a l . F o l l o w i n g H o v e l [129]  and M i l n e s and Feucht  set  were assumed to be p  e q u a l t o 10  8  [130], x  n  was s e t e q u a l t o 10  _q  s, w h i l e x was ' p  s.  PHOTOGENERATION U n l e s s o t h e r w i s e s t a t e d , the p h o t o g e n e r a t i o n d i s t r i b u t i o n computed a t each  g r i d p o i n t from the formula  G(x ) = E exp[-a(X i  ) x.] M(A.) AA J  which i s s i m p l y a f i n i t e - d i f f e r e n c e  1  spectrum  (A.34)  1  approximation  t a k i n g R(A) = 0. F i f t e e n wavelength i n t e r v a l s solar  G(x) was  t o the i n t e g r a l  (2.1),  AA^ were used t o span the  from the wavelength c o r r e s p o n d i n g t o the s i l i c o n bandgap  TABLE A.3  Data used t o compute p h o t o g e n e r a t i o n  Wavelength I n t e r v a l :  (ym)  a  : (m  0.28 - 0.32  1.8*10  8  0.32 - 0.36  1.2*10  8  0.36 - 0.40  1.2*10  7  0.40 - 0.44  4.4*10  6  0.44 - 0.48  2.3*10  6  0.48 - 0.52  1.4*10  6  0.52 - 0.56  9.0*10  5  0.56 - 0.60  6.3*10  5  0.60 - 0.64  4.0*10  5  0.64 - 0.68  3.0*10  5  0.68 - 0.72  2.0*10  5  0.72 - 0.76  1.6*10  5  0.76 - 0.80  1.0*10  5  0.80 - 1.00  4.1*10  4  1.0.0 - 1.20  5*10  ctg^ i s the a b s o r p t i o n c o e f f i c i e n t For the GaAs c e l l ,  )  distribution  M: (m  s  7.52*10  18  19  4.48*10 6.87*10 1.18*10 1.69*10 1.79*10 1.76*10 1.75*10  19 20 20 20 20 20  1.90*10 ° 2  1.83*10 1.83*10  20 20  1.71*10 ° 2  1.61*10  2  )  7.00*10 5.91*10  20 20 20 t T  for silicon.  a u n i f o r m p h o t o g e n e r a t i o n d i s t r i b u t i o n was  used.  The t o t a l p h o t o g e n e r a t i o n i n the GaAs c e l l was s e t e q u a l t o 1.88*10 t w i t h e n e r g i e s g r e a t e r 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 a b s o r p t i o n c o e f f i c i e n t  e x t r a c t e d from the g r a p h i c a l d a t a  [40], w h i l e the t o t a l photon f l u x M ( A _ ^ ) A A ^ i n the  p r e s e n t e d by Sze v a l was  was  ot(A^)  computed from the AMI  s p e c t r a l composition s p e c i f i e d i n R e f . [ 1 3 1 ] .  These d a t a are summarized i n T a b l e A.3. r e p r e s e n t a t i o n s of a ( A ) and t h i s procedure was  inter-  Although  f a r more a c c u r a t e  the s o l a r s p e c t r a l i r r a d i a n c e are  available,-  deemed adequate f o r the purpose of t e s t i n g the  of the s u p e r p o s i t i o n p r i n c i p l e . Any  validity  r e a d e r i n t e n d i n g t o use the programs  p r e s e n t e d i n t h i s appendix f o r v e r y a c c u r a t e m o d e l l i n g of s o l a r  cell  performance s h o u l d i n v e s t i g a t e the techniques employed t o compute the photogeneration  d i s t r i b u t i o n by  Fossum [128] and by Dunbar and Hauser  r a t h e r than u s i n g the crude approximation  to G(x)  [123],  given here.  GRID SELECTION The s e l e c t i o n of an a p p r o p r i a t e g r i d geometry i s a c e n t r a l p a r t of any n u m e r i c a l a n a l y s i s problem. The f i n e t h a t d e r i v a t i v e s can be  g r i d s p a c i n g must be  adequately  approximated  sufficiently  by f i n i t e  differences,  y e t w i t h too f i n e a g r i d t r u n c a t i o n o r r o u n d - o f f e r r o r s may  seriously  affect  o f the e m i t t e r  and  the computations.  In a t y p i c a l s o l a r c e l l the widths  d e p l e t i o n r e g i o n are r o u g h l y one hundred times s m a l l e r than the w i d t h  of the base. For such a d e v i c e a non-uniform The b a s i c g r i d s t r u c t u r e used here  grid i s clearly required.  c o n t a i n e d 100 e v e n l y - s p a c e d p o i n t s  i n each o f the e m i t t e r , d e p l e t i o n and base r e g i o n s . T h i s g r i d s t r u c t u r e was  found t o p r o v i d e a c c u r a t e s o l u t i o n s f o r the p o t e n t i a l s and c u r r e n t s  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 illuminated s i l i c o n another  100  c e l l s i t was  found advantageous t o add  g r i d p o i n t s i n the r e g i o n e x t e n d i n g from the d e p l e t i o n  r e g i o n / b a s e boundary to a depth  of 10 um i n t o the base. T h i s m o d i f i c a t i o n  189  ensured t h a t there would be a reasonably of maximum p h o t o g e n e r a t i o n  in a silicon  f i n e g r i d throughout the  region  c e l l . To check t h a t 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 v a r i o u s o p e r a t i n g c o n d i t i o n s the between g r i d p o i n t s was In no  circumstance  than 0.1%  doubling  Choo's a l g o r i t h m can be  Another simple  g r i d p o i n t s , 20  c a r r i e d out i n roughly  check on the v a l i d i t y of the n u m e r i c a l  at each g r i d p o i n t . T h i s sum  t h a t i s , the sum  s h o u l d be  constant  Here the t o t a l c u r r e n t was  indeed  h e a v i l y doped e m i t t e r r e g i o n s . In such cases when computing the m a j o r i t y  of the m a j o r i t y p o t e n t i a l s and  and J  n  form of  found to be  —  p  the  independent  through very  t r u n c a t i o n e r r o r became a  c a r r i e r c u r r e n t from the  c a r r i e r q u a s i - f e r m i p o t e n t i a l . The c a r r i e r concentrations  be  throughout a d e v i c e , s i n c e  of p o s i t i o n , except i n the case of very s m a l l c u r r e n t flows  problem  1 second  model can  of J  Choo's a l g o r i t h m i s based on the s t e a d y - s t a t e  b a s i c equations.  iterations  470.  computing the t o t a l c u r r e n t —  Seidman and  the number o f p o i n t s .  d i d the s o l u t i o n f o r the t o t a l c u r r e n t change by more  time on the Amdahl  made by  thereby  on h a l v i n g the g r i d s p a c i n g . With 400  of Seidman and of CPU  halved,  spacing  gradient  s o l u t i o n s f o r the  themselves were u n a f f e c t e d by  this  error. PROGRAMS L i s t i n g s and b r i e f d e s c r i p t i o n s of the f o u r FORTRAN programs w r i t t e n t o implement Seidman and  Choo's a l g o r i t h m are g i v e n below. Although  programs l i s t e d below were w r i t t e n t o apply and p-type base r e g i o n s , PN  t o d e v i c e s w i t h n-type e m i t t e r s  s t r u c t u r e s can be  changing the e l e c t r o n and h o l e m o b i l i t i e s and between the FORTRAN v a r i a b l e s a p p e a r i n g  the  t r e a t e d by simply  inter-  l i f e t i m e s . The r e l a t i o n s h i p  i n the programs and  used i n the d i s c u s s i o n above i s documented i n Table A.5.  The  the n o t a t i o n programs  190  were compiled w i t h All  the FORTRAN G c o m p i l e r ,  and run under MTS c o n t r o l .  i n p u t data s u p p l i e d t o the programs s h o u l d be i n MKS u n i t s .  SC.PARSET  e s t a b l i s h e s 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  p h o t o g e n e r a t i o n a t each g r i d p o i n t . 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 o f s e c t i o n s o f w i d t h WSECT, each c o n t a i n i n g NSEG segments. The i n p u t v a l u e s NSEG used i n o b t a i n i n g the data p r e s e n t e d  o f WSECT and i n s u b s e c t i o n 2.2.4  are l i s t e d  i n Table A.4. The i n p u t of a v a l u e  of NSEG < 0  terminates  the g r i d c o n s t r u c t i o n . I n the form l i s t e d below  the program assumes t h a t the c e l l has been f a b r i c a t e d on a u n i f o r m l y - d o p e d p-type s i l i c o n s u b s t r a t e w i t h o u t a back surface but  field.  The doping p r o f i l e i s assumed t o be Gaussian,  t h e program accepts  depth XJNCT, s u r f a c e  values  f o r the m e t a l l u r g i c a l j u n c t i o n  doping c o n c e n t r a t i o n N  , substrate DU  doping N^ and s u r f a c e r e c o m b i n a t i o n v e l o c i t i e s i n p u t data.  Values f o r a ( A ) and the photon f l u x a r e read i n  from the f i l e  SPDTFL. A l l q u a n t i t i e s computed by t h i s program  are w r i t t e n to the s e q u e n t i a l f i l e SC.INIT  S^, and Sg as  computes i n i t i a l v a l u e s  GPARFL.  f o r u, v and ty i n accordance w i t h the  procedure s p e c i f i e d above. The program r e q u i r e s the t e r m i n a l voltage  VOLT t o be s p e c i f i e d as i n p u t d a t a . Values f o r u, v  and ty a r e w r i t t e n t o the s e q u e n t i a l f i l e  SC.CALC  uses Seidman and Choo's a l g o r i t h m for  t o o b t a i n improved  estimates  u, v and ty. The number of i t e r a t i o n s t o be performed  NCNTRL, t h e t e r m i n a l v o l t a g e  SC.READOUT  UVPSFL.  and the number-of-suns i l l u m i n a t i o n  RSUNS a r e s u p p l i e d as i n p u t  data.  computes and outputs v a l u e s  f o r n, p, ty, J ^ , J  p  and t o t a l  c u r r e n t J^, a t s e l e c t e d g r i d p o i n t s . The i n p u t d a t a  for this  program s p e c i f i e s the g r i d p o i n t JSTART a t which sampling i s to b e g i n , and  the g r i d p o i n t JSTOP a t which sampling i s t o end,  the s p a c i n g  INCR between p o i n t s  sampled.  TABLE A.4  Parameters used f o r g r i d  a) S i l i c o n  cell  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  construction  192  TABLE A.5  E x p l a n a t i o n of v a r i a b l e s used i n FORTRAN programs  FORTRAN VARIABLE: VTH  QUANTITY REPRESENTED: kT/q  NI LDEBYE  D  EPSI Q TO HJM HJP, H(J) HSUM XJ JMAX  h.J h J h J + hT3 x +  +  j  M (number o f g r i d p o i n t s )  VBI WO  bi WQ  XJNCT NA NDO  ( d e p l e t i o n r e g i o n width  at V = 0)  ( m e t a l l u r g i c a l j u n c t i o n depth) N  A  N. DO  NT LNT  log  NNOF  n  PNOF  P  PPOB  P  1 0  (N ) T  nO F> ( x  nO P  0  ( K  ( X  F  )  B  )  B  }  NPOB  n  PFLX  M(A )AA  GO  GQ (photogeneration  p0  ( x  i  ±  at surface)  193 FORTRAN VARIABLE:  QUANTITY REPRESENTED:  MU MUN  MOT LNDT,LMUNDT,LMUPDT  (data used t o compute m o b i l i t y )  DN(J),DNJ DNJP  D n,j+l  DP(J) ,DPJ  D P,J  DPJP  D P,j+1  TN(J),TNJ TP(J),TPJ P»J N(J)  A"  N  U(J),UJ  N  D  u.  V(J),VJ PSI(J),PSIJ PSIJM PSIJP DELTA F(J),FJ XPSIJ XPSIJP  exp (ty ) e x  P(*j  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)  +  + 1  )  194  FORTRAN VARIABLE:  QUANTITY REPRESENTED:  VOLT VBAR  V, . - V bi  RSUNS  (number-of-suns)  NCNTRL  (number of i t e r a t i o n s )  JN JP JT PHIP PHIN  195  C PROGRAM SC. PARSE.T C WRITTEN BY GARRY TARR FEB/81 C I M P L I C I T 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 ( 8 0 1 ) , N ( 8 0 1 ) , D N ( 8 0 1 ) , D P ( 8 0 1 ) , T N ( 8 0 1 ) , T P ( 8 0 1 ) , #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/ #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 '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 f  f  196  600 C 700 C  DO 600 1=1,100 READ(SPDTFL,FREE,ERR=4,END=700) ALPHA-( I ) , P F L X ( I ) GOTO 4 ISPDT=I-1 CDIFF=DLOG(NDO/NA)/XJNCT/XJNCT XJNCT=XJNCT/LDEBYE  c  800  XJ=0.0D0 DO 1000 J=1,JMAX ND=0.0D0 XX=XJ*XJ*CDIFF I F ( X X .GT. 1.0D2) GOTO 800 ND=ND0*DEXP(-XX) 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 T N ( J ) = TN 0/(1.0D0+NT/N0TN)/T0 TP(J)=TP0/(1.0D0+NT/NOTP)/TO  C  900 C 1000 C C 102  C 4 51 52  G(J)=0.0 DO 900 I=1,ISPDT G0 = P F L X ( I ) * A L P H A ( I ) / N I *T0 XX=XJ*ALPHA(I) I F (XX .GT. 1.0D2) GOTO 9'00 G(J)=G(J)+G0*DEXP(-XX) CONTINUE XJ=XJ+H(J) H(J)=H(J)/LDEBYE CONTINUE W R I T E ( L P , 1 0 2 ) XJ,JMAX 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 STOP STOP 4 STOP 51 STOP 52 END  197  C  SUBROUTINE FINDMU(LNT,MU,LNDT,LMUDT,IMUDT) I M P L I C I T REAL*8 (A-H O-Z) REAL*8 LNT,MU,LNDT,LMUDT,LMU .DIMENSION LNDT(IMUDT),LMUDT(IMUDT) f  C C  100 C 200 300 C  DATA TEN/1.0D1/ LMU=LMUDT(1) I F ( L N T .LE. L N D T ( 1 ) ) GOTO 300 ISTOP=IMUDT-l DO 100 I = l , I S T O P I F ( (LNT .GE. L N D T ( I ) ) .AND. (LNT .LT. L N D T ( I + 1 ) ) CONTINUE 1 = 1 STOP  ) GOTO 200  LMU=LMUDT(I)-(LMUDT(I)-LMUDT(I+1))/(LNDT(I+1)-LNDT(I))* #(LNT-LNDT(I)) MU=TEN**LMU RETURN END  C PROGRAM S C . I N I T C WRITTEN BY GARRY TARR FEB/81 C I M P L I C I T REAL*8 (A-H,0~Z) INTEGER FREE,GPARFL,UVPSFL REAL*8 LDEBYE,NI,NA,NDO,NNOF,NPOB REAL*4 H DIMENSION U ( 8 0 1 ) , V ( 8 0 1 ) , P S I ( 8 0 1 ) , H ( 8 0 1 ) , F R E E ( 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 ) = P N 0 F XJ=XJ+H(J) 100 CONTINUE C 200 JJ=J RJ=0.0D0 DO 300 J = J J , J M A X 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 JJ=J DO 500 J = J J , J M A X 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/ *'/ 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 1  200  C  DPJ=DPJP HJM=HJP CHIJMU=CHIJPU CHIJMV=CHIJPV XPSIJ=XPSIJP  0  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  C  C  100 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 CHIJPV=(DPJ/XPSIJ+DPJP/XPSIJP)/HJP ETAJM=CHIJMV/HSUM ETAJP=CHIJPV/HSUM AV(J)=ETAJM+ETAJP+UJ*FJ BV(J)=-ETAJM CV(J)=-ETAJP F(J)=FJ+G(J) DNJ=DNJP DPJ=DPJP HJM=HJP CHIJMU=CHIJPU CHIJMV=CHIJPV XPSIJ=XPSIJP CONTINUE 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  C  200 C  300 C C 1000 C C C c  2 3 5  HJM=H(1) APSI(l)=ONE CPSI(l)=ZERO DO 200 J=2,JSTOP HJP=H(J) PSIJP=PSI(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 CONTINUE 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 , J M A X PSI(J)=DELTA(J)+PSI(J) CONTINUE CONTINUE  REWIND UVPSFL WRITE(UVPSFL) U,V,PSI,VOLT,RSUNS,NCNTRL STOP STOP 2 STOP 3 STOP 5 END  202  C C  SUBROUTINE  INVRT(Y,A,B,C,F,JMAX)  I M P L I C I T REAL*8 (A-H,0-Z) DIMENSION Y ( J M A X ) , A ( J M A X ) , B ( J M A X ) , C ( J M A X ) , F ( J M A X ) DATA ZERO/0.0D0/ ONE/1.0D0/ r  C C C  100  JSTOP=JMAX-l A(1)=A(1) C(1)=C(1)/A(1) DO 100 J=2,JSTOP A(J)=A(J)-B(J)*C(J-1) C(J)=C(J)/A(J) A(JMAX)=A(JMAX)-B(JMAX)*C(JSTOP)  C 200 C  300 C  B(1)=F(1)/A(1) DO 200 J=2,JMAX B(J)=(F(J)-B(J)*B(J-1))/A(J) Y(JMAX)=B(JMAX) DO 300 J = l , J S T O P I=JMAX-J Y(I)=B(I)-C(I)*Y(I+1) RETURN END  203  C PROGRAM SC.READOUT C WRITTEN BY GARRY TARR FEB/81 C I M P L I C I T 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  C C  C 101  C 100 C 200 C 102 C  c  500 103 C 2 3 5  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 FORMAT('1' ,10X,'ITERATION NUMBER: ' , I 2,5X,'VOLTAGE: ', #F8.5,5X,'ILLUMINATION L E V E L : ',F6.2,IX,'SUNS'// #1X,'J:',7X,'PSI:',12X,'PHIP:',16X,'PHIN:',15X,'P:',12X, #'N:',11X,'JP:',11X,'JN:',11X,'JTOT:'/) READ(KR,FREE,ERR=5,END=500) JSTART,JSTOP,INCR DO 200 J=JSTART,JSTOP,INCR CALL NPCRNT(J) CONTINUE WRITE(LP,102) FORMAT(IX) GOTO 100 WRITE(LP,103) FORMAT(1X/1X/11X,'ALL STOP STOP 2 STOP 3 STOP 5 END  QUANTITIES I N MKS  UNITS'///)  204  SUBROUTINE NPCRNT(J) C  C  I M P L I C I T REAL*8 (A-H,0~Z) REAL*8 J N , J P , J T , N I , L D E B Y E 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 DATA  C C  C  C 100  C 200 300 C C 102 C  Q/l.6021D-19/,TWO/2.0D0/,FOUR/4.0D0/,LP/6/  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 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 JP=-(CHIJP*(V(J+1)-V(J))+CHIJM*(V(j)-V(J-l)))*C/FOUR GOTO 300 XPSIJP=DEXP(PSI(J+l)) JN=(DN(J)*XPSIJ+DN(J+l)*XPSIJP)*(U(J+l)-U(J))/H(1)*C/TWO JP=-(DP(J)/XPSIJ+DP(J+l)/XPSIJP)*(V(J+1)-V(J))/H(l)*C/TWO GOTO 300 XPSIJM=DEXP(PSI(J-l)) JN=(DN(J-l)*XPSIJM+DN(J)*XPSIJ)*(U(J)-U(J-l))/H(J-l)*C/TWO JP=-(DP(J-1)/XPSIJM+DP(J)/XPSIJ)*(V(J)-V(J-l))/H(J-l)*C/TWO JT=JN+JP WRITE(LP,10 2) J , P S I J , P H I P , P H I N , P , N , J P , J N , J T FORMAT(1X,I4,2X,F13.10,2X,2(F19.16,2X),2(D12.5,2X), #2(D12.5,2X),D13.6) 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 e x p r e s s i o n f o r the shadow a r e a a o f an e l l i p soidal  constant energy s u r f a c e 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 t o the i n t e r f a c e i s d e r i v e d . The shadow a r e a can be computed most e a s i l y by working i n a c o o r d i n a t e system i n which the e f f e c t i v e mass t e n s o r i s d i a g o n a l . In t h i s c o o r d i n a t e system,  E(k) = ( f i / 2 ) [k /m* + k /m* + k /m*] . x x y y z z 2  Thus the e q u a t i o n  2  2  2  (B.l)  o f t h e constant energy s u r f a c e a t energy E can be  w r i t t e n i n t h e form  F(k  x  ,k ,k ) = 0 y z  (B.2)  where  F(k  ,k ,k ) = k / a y z x 2  x  2  + k /b y 2  2  + k /c z 2  2  - 1  (B.3a)  and  a  2  * 2 = 2Em / f t , x  b  2  * 2 2 = 2Em /h , c y  * ? = 2Em *i . z  (B.3b)  a, b and c a r e , o f c o u r s e , the h a l f - l e n g t h s o f t h e p r i n c i p a l axes o f the e l l i p s o i d a l c o n s t a n t energy s u r f a c e s d e f i n e d by (B.2). I f the normal t o the i n t e r f a c e i s r e p r e s e n t e d by the v e c t o r n = (n ,n ,n^), 1  2  then F i g . B . l shows t h a t the p o i n t s on the e l l i p s o i d  206  Figure  B.l  Shadow of an  ellipsoid.  207  which p r o j e c t to the boundaries  o f the shadow s a t i s f y  VF«fi = 0  (B.4)  or  n.k I  (B.5)  / a + n_k / b + n,k / c = 0 . x 2 y 3 z 2  i s clearly  2  the e q u a t i o n  (B.5) '  2  of a plane i n k-space p a s s i n g through the  o r i g i n and w i t h normal 1 g i v e n by  i = (£ ,^ ,£ ) = ( 1  2  3  /a ,n /b ,n /c )/L 2  n ; L  2  2  (B.6a)  2  3  where  L = (n /a 2  A  + n /b 2  4  + n /c ) 2  4  .  1 / 2  (B.6b)  I t i s w e l l known t h a t the i n t e r s e c t i o n o f an e l l i p s o i d w i t h any p l a n e p a s s i n g through the c e n t e r of the e l l i p s o i d i s an e l l i p s e . u s i n g the c o n v e n t i o n a l techniques  Further,  o f l i n e a r a l g e b r a i t can be shown t h a t  the a r e a A o f t h i s e l l i p s e i s given by  A = TT/UJ/O^C ) + A / ( a c ) + £ / ( a b ) ] 2  2  2  2  2  2  2  1 / 2  .  (B.7)  where a, b and c are the h a l f - l e n g t h s o f the p r i n c i p a l axes o f the e l l i p s o i d and £ i s the normal t o the p l a n e i n q u e s t i o n . From F i g . B . l i t can be seen t h a t the shadow area a i s r e l a t e d t o A by  a =  Combining  A(i-n).  (B.6),(B.7) and  (B.8)  a = TT[n b c  + n a c  2  2  2  In terms of the energy E and  2  2  (B.8)  i t i s found  2  + n a b ] 2  On  r  s u b s t i t u t i n g t h i s expression is s t i l l  g i v e n by  m  * = e  T h i s agrees w i t h direct  2  1 / 2  .  2** 1/2 + n.m m l ' 3 x y  for a into  (3.25), but  that m  £  the e x p r e s s i o n  for J  C  M  ic  x  , m  ic  i s given  Crowell  ic  and m , y z  .  (B.10) v  (3.19), i t i s found  2 * * 2 * * 2**1/2 [n m m + n m m + n m m ] . l y z 2 x z 3xy  3 i n t e g r a t i o n over d k.  (B.9)  the e f f e c t i v e masses m  , 2^ 2 * * 2 * * a = ir(2E/h ) [n.m m + n„m m l y z z x z v  2  that  /  that  by  (B.ll) \ - /  [85] o b t a i n e d u s i n g a  APPENDIX C FABRICATION PROCEDURE FOR  T h i s appendix p r o v i d e s followed MIS in  a t UBC  JUNCTIONS  complete d e t a i l s o f the procedure c u r r e n t l y  i n the manufacture of both p o s i t i v e and n e g a t i v e  j u n c t i o n s . T h i s p r o c e s s has the p a s t ;  MIS  undergone a number of minor  the v e r s i o n d e s c r i b e d here i s t h a t i n use  I t i s e s s e n t i a l that a l l steps  i n the p r o c e s s be  interruption, since thin s i l i c o n  barrier  modifications  as of l a t e  1980.  c a r r i e d through w i t h o u t  oxide l a y e r s grow at the r a t e of  angstroms p e r hour when exposed to the atmosphere  several  [18].  SUBSTRATE CLEANING P r i o r to j u n c t i o n f a b r i c a t i o n , a l l substrates "RCA  clean", a standard  tronics industry sing  [132]. The  safety precautions must be  slices  f o r high  to  microelec-  temperature p r o c e s -  used i n the h a n d l i n g  of c o n c e n t r a t e d  a c i d s and  bases  observed when f o l l o w i n g t h i s p r o c e d u r e . A l l c o n c e n t r a t i o n s a l l chemicals s h o u l d  be  q u a l i t y . I f p o s s i b l e , the r e s i s t i v i t y  s h o u l d be  18 Mficm. However, r e s i s t i v i t i e s  of ACS  as low  1. a) Immerse s i l i c o n  water  as 1 M£2cm have been used finished junctions.  f o r 10 minutes i n a s o l u t i o n of 1 p a r t  NH^OH, 1 p a r t 30% H ^  and  h e l d at a temperature of  5 parts  use,  under these c o n d i t i o n s . F u r t h e r , temperature of the s o l u t i o n not falls  the s i l i c o n may  deionized  (80±5)°C. The  p r e p a r e d immediately b e f o r e  concentration  are  reagent grade or  of the d e i o n i z e d  here w i t h o u t apparent e f f e c t on the p r o p e r t i e s o f the  high,  the  c l e a n r e c i p e used here i s o u t l i n e d below. Normal  quoted by volume, and higher  c l e a n i n g procedure used w i d e l y i n the  to prepare s i l i c o n RCA  are s u b j e c t e d  too low  (DI) water  s o l u t i o n should  since ^02  be  decomposes r a p i d l y  i t i s important t h a t exceed 85°C. I f  the  the  or the temperature r i s e s  be p i t t e d .  30%  too  210  b)  Rinse s i l i c o n or  2.  f o r at l e a s t 10 minutes i n a DI water cascade  equivalent.  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 p a r t 49%  HF  to 9 p a r t s DI water a t room temperature. b)  Rinse s i l i c o n  f o r 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 p a r t 36%  1 p a r t 30% ^2°2  HC1,  ^ p a r t s DI water h e l d at a temperature of  (80±5)°C. b)  Rinse s i l i c o n  c) Dry  s i l i c o n by b l o w i n g water from s u r f a c e w i t h  oil-free The  f o r at l e a s t 10 minutes i n DI water cascade.  nitrogen.  f i r s t s t e p of the RCA  organic  clean i s intended  to remove t h i n l a y e r s of  contaminants from the s i l i c o n s u r f a c e . The  away any n a t i v e oxide  layer  removes heavy metals  [132].  The  which may  e f f e c t i v e n e s s o f the RCA  for  Silicon  long periods  face. Following i c o n s h o u l d be  the t h i r d  c l e a n can be monitored by n o t i n g  step  the  l i s t e d above are c a r r i e d  are u s u a l l y found t o be hydrophobic — tend  to bead up  t h a t i s , water  r a t h e r than w e t t i n g  c o m p l e t i o n of the f i r s t s t e p of the RCA strongly hydrophilic —  the s u r f a c e . I f t h i s i s not  thoroughly  strips  s l i c e s which have been exposed to contaminated atmospheres  d r o p l e t s p l a c e d on a s l i c e  wet  second step  be p r e s e n t , w h i l e  c o n d i t i o n of the s i l i c o n s u r f a c e as the s t e p s out.  a j e t of  the  sur-  c l e a n , the  sil-  t h a t i s , water should r e a d i l y  the case,  then the s u r f a c e has not been  degreased. A f t e r immersion i n h y d r o f l u o r i c a c i d f o r a  few  seconds, the s i l i c o n s h o u l d become h y d r o p h o b i c , i n d i c a t i n g t h a t a l l t r a c e s o f oxide have been removed. Exposure to the HCliH^O^ s o l u t i o n regrows a t h 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 v e r y t h i n o x i d e l a y e r i s grown a t t h e s i l i c o n  surface  d u r i n g t h e l a s t s t e p o f t h e RCA c l e a n , f u r t h e r o x i d a t i o n a t temperatures g r e a t e r than 400°C i s r e q u i r e d t o produce m i n o r i t y c a r r i e r MIS d i o d e s . No f o r m a l e x p e r i m e n t s have been c a r r i e d o u t h e r e t o determine an optimum o x i d a t i o n time, b u t experience  i n d i c a t e s t h a t o x i d a t i o n a t a temperature  of a t l e a s t 500°C f o r a minimum p e r i o d o f 20 minutes i s r e q u i r e d t o p r o duce m i n o r i t y c a r r i e r A l - S i O - p S i d e v i c e s on 1 t o 10 Qcm s u b s t r a t e s . The x A l - S i O - p S i back s u r f a c e f i e l d c e l l s d e s c r i b e d i n S e c t i o n 4.3 were f a b r i x c a t e d u s i n g a 30 minute o x i d a t i o n c y c l e a t 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 v o l t a g e s and among t h e h i g h e s t f i l l f a c t o r s o f any d e v i c e s produced d u r i n g t h i s r e s e a r c h program. As demonstrated i n S e c t i o n 4.4,  c e l l s i n c o r p o r a t i n g o x i d e s grown a t temperatures above  600°C have e x c e s s i v e t u n n e l r e s i s t a n c e , and g i v e c o r r e s p o n d i n g l y poor fill  f a c t o r s under one-sun i l l u m i n a t i o n . I n a l l t h e e x p e r i m e n t s on MIS diodes r e p o r t e d h e r e , h i g h temperature  t r e a t m e n t s were c a r r i e d o u t i n a q u a r t z tube f u r n a c e w i t h gas f l o w s o f approximately  1 L/sec. M e d i c a l 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 w a t e r vapour was  used f o r s i n t e r i n g and a n n e a l i n g s t e p s . S l i c e s were always i n s e r t e d  i n t o t h e f u r n a c e w i t h t h e s i d e on w h i c h t h e MIS j u n c t i o n was t o be formed f a c i n g t h e gas f l o w . OHMIC CONTACT FORMATION When w o r k i n g 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 c o n t a c t s can be c o n v e n i e n t l y formed by d e p o s i t i n g a t h i c k l a y e r o f aluminum on t h e back o f t h e 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 a t a temperature o f 500°C f o r 10 m i n u t e s . T h i s t e c h n i q u e has been found  to y i e l d c o n t a c t s w i t h h i g h l y 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 a p p l i e d t o s u b s t r a t e s w i t h t h i n o x i d e l a y e r s formed f o l l o w i n g the procedure o u t l i n e d above. As a t e s t o f c o n t a c t s i n t e r e d aluminum c o n t a c t s  resistance,  o f t h i s k i n d were a p p l i e d t o both s i d e s o f a  300 ym t h i c k , 2 ficm s u b s t r a t e . The c u r r e n t - v o l t a g e s t r u c t u r e was then recorded  c h a r a c t e r i s t i c of t h i s  u s i n g a f o u r - p o i n t probe technique t o e l i m -  i n a t e the e f f e c t s o f l e a d r e s i s t a n c e . The c h a r a c t e r i s t i c was found t o  2 obey Ohm's law out t o c u r r e n t d e n s i t i e s o f more than 100 mA/cm , w h i l e  2 the  r e s i s t a n c e measured between the two c o n t a c t s was o n l y 0.1 £2cm . T h i s  i s not s u b s t a n t i a l l y greater slice  than the b u l k r e s i s t a n c e a s s o c i a t e d w i t h a  of t h i s doping and t h i c k n e s s . /  BARRIER METAL DEPOSITION In a l l the MIS j u n c t i o n s  described  l a y e r and any o v e r l y i n g c o n t a c t thermal evaporation.  i n t h i s t h e s i s , the b a r r i e r m e t a l  f i n g e r s were d e p o s i t e d  In the m i c r o e l e c t r o n i c s  by the process o f  industry, metallization of  s i l i c o n wafers i s o f t e n c a r r i e d out by the p r o c e s s e s o f e l e c t r o n - b e a m evaporation  o r RF s p u t t e r i n g . These t e c h n i q u e s were not employed here  s i n c e they l e a d t o bombardment o f the s u b s t r a t e w i t h X-rays and h i g h energy e l e c t r o n s . T h i s bombardment i s l i k e l y  to create  a high  density of  i n 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 o f the f i n i s h e d MIS j u n c t i o n . However, there devices  i s a chance t h a t h i g h - q u a l i t y  MIS  c o u l d be produced by magnetron s p u t t e r i n g .  The  vacuum system used f o r e v a p o r a t i o n  was a standard  CHA SEC-600  u n i t equipped w i t h a V a r i a n VHS-6 d i f f u s i o n pump. In the SEC-600 system, b a c k s t r e a m i n g o f 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 t o some extent  through the use of an o p t i c a l l y dense w a t e r - c o o l e d b a f f l e and a  l i q u i d nitrogen  cold trap. Unfortunately,  the CHA t r a p p i n g system i s  213  d e c i d e l y i n f e r i o r t o t h a t used on o t h e r commercially a v a i l a b l e  diffusion  pumps, i n t h a t i t a l l o w s a d i r e c t l i n e o f s i g h t from the w a t e r - c o o l e d b a f f l e i n t o the work chamber. There was thus a r a t h e r h i g h p r o b a b i l i t y of the s i l i c o n s u b s t r a t e s becoming coated w i t h a t h i 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 t o m e t a l l i z a t i o n . T h i s problem was compounded by the use o f r e l a t i v e l y h i g h vapour p r e s s u r e 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 o f t h i s p o s s i b l e hydrocarbon  c o n t a m i n a t i o n on the p r o p -  e r t i e s of the f i n i s h e d MIS j u n c t i o n s i s n o t known. E v a p o r a t i o n s were —6 —6 u s u a l l y c a r r i e d out a p r e s s u r e s r a n g i n g from 1*10 t o 2*10 Torr. o  For d e p o s i t i o n r a t e s g r e a t e r than about  1 A/sec, these p r e s s u r e s were  low enough t o p r e v e n t s i g n i f i c a n t c o n t a m i n a t i o n o f t h e d e p o s i t e d f i l m s by r e a c t i o n w i t h r e s i d u a l gases i n the work chamber. Where p o s s i b l e , the t h i c k n e s s e s of t h e d e p o s i t e d f i l m s were measured w i t h a q u a r t z c r y s t a l o s c i l l a t o r type t h i c k n e s s monitor. The b a r r i e r metals i n v e s t i g a t e d here i n c l u d e d aluminum, magnesium, p a l l a d i u m and p l a t i n u m . Of these m a t e r i a l s , p l a t i n u m i s by f a r t h e most difficult  t o e v a p o r a t e . P l a t i n u m both melts and reaches a vapour p r e s s u r e  -4 of 10  T o r r a t a temperature  o f a p p r o x i m a t e l y 1750°C [133]. T h i s p r e c l u d e s  the s e l f - e v a p o r a t i o n o f p l a t i n u m w i r e . Moreover, the temperature  required  to evaporate p l a t i n u m i s so h i g h that the s e l e c t i o n o f source m a t e r i a l s i s extremely l i m i t e d . Although p l a t i n u m i s known t o a l l o y w i t h t u n g s t e n , t u n g s t e n f i l a m e n t s were chosen as t h e source h e r e . P r i o r t o use,  these  f i l a m e n t s were c l e a n e d by h e a t i n g t o white heat f o r s e v e r a l minutes under h i g h vacuum. When c a r r y i n g out the a c t u a l e v a p o r a t i o n , the procedure f o l l o w e d was t o g r a d u a l l y heat the f i l a m e n t u n t i l the p l a t i n u m charge melted, and then q u i c k l y open the s h u t t e r c o v e r i n g the source w h i l e a p p l y i n g a b r i e f b u r s t o f power t o t h e f i l a m e n t t o f l a s h - e v a p o r a t e the  214  charge. The s u b s t r a t e s were thus exposed o n l y a few seconds. I t was  t o the w h i t e - h o t source f o r  hoped t h a t t h i s procedure would  minimize  b o t h the a l l o y i n g of the p l a t i n u m charge w i t h the f i l a m e n t and the h e a t i n g of the s i l i c o n s u b s t r a t e s . The source temperature inum e v a p o r a t i o n was  so h i g h t h a t the t h i c k n e s s monitor c o u l d not be  used to measure the d e p o s i t i o n In an attempt  r e q u i r e d f o r the p l a t -  rate.  t o a s c e r t a i n the degree  of tungsten c o n t a m i n a t i o n o f  the d e p o s i t e d p l a t i n u m f i l m s , a b l a n k g l a s s microscope s l i d e was  position-  ed i n the path o f the evaporant stream d u r i n g one e v a p o r a t i o n . The ition  of the f i l m d e p o s i t e d on the s l i d e was  compos-  then a n a l y z e d u s i n g X-ray  f l u o r e s c e n c e s p e c t r o s c o p y . (The i n s t r u m e n t used i n t h i s a n a l y s i s was  the  s c a n n i n g e l e c t r o n microscope system o p e r a t e d by the Department o f M e t a l l u r g y ) . No t u n g s t e n l i n e s c o u l d be d e t e c t e d i n the f l u o r e s c e n t spectrum,  X-ray  i n d i c a t i n g t h a t the d e p o s i t e d f i l m c o n t a i n e d l e s s than  approx-  i m a t e l y 1% t u n g s t e n . Compared t o p l a t i n u m , the m a t e r i a l s aluminum, magnesium and dium can be e v a p o r a t e d w i t h r e l a t i v e ease. Here aluminum was  palla-  evaporated  from tungsten f i l a m e n t s . Aluminum b a r r i e r l a y e r s and f r o n t - c o n t a c t were g e n e r a l l y d e p o s i t e d at a r a t e of 2-10  o  A/sec. Magnesium was  grids  evaporated  from a b a f f l e d tantalum boat designed f o r use w i t h SiO. Although the r a t e of magnesium e v a p o r a t i o n was  difficult  t o c o n t r o l , d e p o s i t i o n r a t e s of  o  10-20  A/sec were aimed f o r . F i n a l l y , when c a r r y i n g out the  experiments  on p o s i t i v e b a r r i e r Pd-nSi j u n c t i o n s d e s c r i b e d i n S e c t i o n 5.3, dium was  the p a l l a -  d e p o s i t e d by the s e l f - e v a p o r a t i o n o f t h i n w i r e s . S i n c e the  -4 vapour p r e s s u r e o f p a l l a d i u m reaches 10  T o r r a t 1200°C, y e t t h i s  does n o t melt u n t i l 1550°C [133], p a l l a d i u m w i r e s can r e a d i l y be  metal  self-  e v a p o r a t e d . T h i s procedure p e r m i t s p a l l a d i u m f i l m s of extremely h i g h  p u r i t y to be d e p o s i t e d . When d e p o s i t i n g p a l l a d i u m on f r e s h l y - e t c h e d s i l i c o n to form an ohmic c o n t a c t , a t u n g s t e n basket was evaporation  source.  used as  the  REFERENCES  1. 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