a simple theoretical model for photoelectrochemical solar cell

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So l i d S t a t e Co mm u n i ca t i o n s , V o l . 51 , N o . 1 0 , p p . 8 2 9 - 8 3 2 , 1 9 8 4 .Pr in ted in Grea t B r i t a in .

0 0 3 8 - 1 0 9 8 / 8 4 $ 3 .0 0 + . 00Pergamon Press L td .

A S I M P L E T H E O R E T I C A L M O D E L F O R P H O T O E L E C T R O C H E M I C A L SO L A R C E L L

S. Chandra , D.P. Singh and S.N. Sahu

D ep a r t men t o f P h y s ic s , Ban a ras H i n d u U n i v e r s it y , V a ran asi 2 2 1 0 0 5 , I n d i a

(Rece ived 5 March 1 9 8 4 ; in rev ised form 15 M ay 1 9 8 4 by R . F ieschi )

A n ex p re s s io n is d e r iv ed fo r t h e q u a n t u m e f f i c i en cy o f a p h o t o e l ec t ro -chem ical so lar ce ll cons ider ing the space charge recom bina t ion , charget r an s f e r v e l o c i ty an d t h e d a rk cu r r en t . T h e t h e o ry i s ap p li ed t o ex p l a in t h eq u an t u m e f f i c i en cy v s b an d b en d i n g cu rv e fo r n -Cd Se ]S2- , S] - junct io ns o la r c e l l f o r d i f f e r en t l i gh t i n t en s i t y an d e l ec t ro l y t e co n ce n t r a t i o n .

1 . I N T R O D U C T I O N

PH O T O E L E C T RO CH E M ICA L s o la r c el ls (PE SCs ) u s in g

s e m i c o n d u c t o r / e l e c t ro l y t e j u n c t i o n ( in s te a d o f a p - n

j u n c t i o n ) a r e o f g ro w in g i n t e re s t f o r s o l a r en e rg y co n -vers ion and s to rage [1 -4 ] . I t es sen t ia l ly cons i s t s o f a

s emi co n d u c t o r p h o t o e l ec t ro d e d i p p ed i n an e l ec t ro l t y e

an d a me t a l l ic co u n t e r e l ec t ro d e d i p p ed i n t h e s ame

e l ec t ro l y t e . Cha rg e t r an s f e r a t t h e s em i co n d u c t o r /

e l ec t ro l y t e i n t e r f ace p ro d u ces a b an d b en d i n g ( V b) o r

space charge l ayer (SCL) . In sp i t e o f a l arge am oun t o f

ex p e r i men t a l w o rk , t h e t h eo re t i c a l s i t u a t io n i s f a r f r o m

b e i n g s a t is f ac t o ry . Mo s t o f t h e mo d e l s o f PE SC en d u p

i n ma t h em a t i ca l co m p l ex i t y o r e l se r eco u r s e t o o v e r-

s i mp l i f ic a t i o n i s ad o p t ed . T h e f i r s t a t t em p t w as mad e b y

Bu t l e r [ 5 ] b a sed o n G a r t n e r ' s [ 6 ] ap p ro ac h d ev e l o p ed

fo r p - n j u n c t i o n s o l a r c ell . A ll t h e p h o t o g en e ra t ed ca r -

r ie r s a r e a s s u med t o co n t r i b u t e t o t h e p h o t o cu r r en t

across the junct i on neg lec t ing al l loss mechan isms .

Bu t l e r ' s t h eo ry f a i led fo r s emi co n d u c t o r / e l ec t ro l y t e

junc t ion so lar ce l l because the space charge recom bina-

t i o n a t t h e i n t e r f ace d u r i n g t h e ch a rg e t ran s f e r c an n o t b e

neg lec ted . Wi lson's mod el [7 ] t akes in to acc oun t the

r eco m b i n a t i o n a t t h e i n t e r f ace b u t t h e s p ace ch a rg e

reco m b i n a t i o n an d t h e d a rk cu r r en t co n t r i b u t i o n h av e

b een o mi t t ed . O n t h e co n t r a ry Re i ch man n an d Ru s s ak

[8 ] d i d n o t co n s i d e r th e s u r f ace r eco m b i n a t i o n w h i l e t h e

s p ace ch a rg e r eco m b i n a t i o n w as co n s i d e r ed .Mo re co mp l e t e mo d e l s t ak i n g i n t o acco u n t o f d a rk

cu r r en t c o n t r i b u t i o n , s p ace ch a rg e an d s u r f ace r eco m-

b i n a t i o n h ave b een g iv en b y H an em an an d M cCan n [9 ]

us ing curv i l inear quas i -Fermi - level as used by

Pan ayo ta tos and Card [10] and E1 Guiba ly et al . [11 ,

12] us ing " f i a t " quas i -Fermi - level concep t . The energy

level d iagram of an e lec t ro ly te near the i l lumina ted in ter -

face unde r fo rw ard b ias V is shown in Fig . 1 (E l Guibaly

[11 , 12] . Al l the po ten t i a l d rop has been cons idered

ac ros s t h e s p ace ch arg e l ay e r . A t t h e s e m i co n d u c t o r -

e l ec t ro l y t e i n t e r face , i n ac t u a l p r ac t i c e , d o u b l e l ay e rs

a l so ex i s t o n t h e e l ec t ro l y t e si d e ( t e rmed a s H e l mh o l t z

an d G o u y l ay e r ) ap a r t f r o m t h e s p ace ch a rg e l ay e r i n t h e

s emi co n d u c t o r s i de . T h e H e l m h o l t z lay e r is d u e t o t h e

fo rm a t i o n o f a d i p o l a r s t r u c tu r e a t t h e i n t e r f ace an d

ad s o rp t i o n o f i o n s f ro m t h e e l ec t ro l y t e o n t h e e l ec t ro d e

s u r face [1 3 ] . Fo r h ig h l y co n c en t r a t ed e l ec t ro l y t e , w e

can n eg l ec t t h e p o t en t i a l d ro p ac ros s t h e H e l mh o l t z

layer [14] bu t in genera l th i s i s no t t rue . Under fo rward

b ias , the quas i -Fermi - level fo r e lec t rons (Ern) and fo r

h o l e s (E ~ ) i n t h e d ep l e t i o n r egi o n (D R ) o f t h e s emi -

co n d u c t o r a r e a s s u med t o b e f l a t. T h e s ep a ra t i o n

b e t w een t h em i s q V u n d e r d a rk an d q U under i l lumi -

nat ion w here U > V. Th is assumpt ion represen t s the fac t

t h a t t h e m i n o r i t y ca r r i e r co n ce n t r a t i o n u n d e r i l lu mi -

nat ion i s l a rger than i t s va lue in the dark .

In t h e p r e s en t p ap e r w e h av e tr i ed t o d ev e l o p a

ma t h em a t i ca l l y si mp l e t h eo ry fo r a PE SC b y i n c lu d i n gs p ace ch arg e r eco mb i n a t i o n an d o p p o s i n g d a rk cu r r en t

based on E1 Guiba ly 's approac h [11 , 12] . In E1 Guiba ly 's

d e r i v a ti o n fo r p h o t o c u r r en t u n d e r b ia s, an ex p re s s i o nbased on Fig . 1 , the to ta l c ur ren t is

J = ( S ' t / S ) ~ - ~ { e x p ( q V / k T ) + B e x p [ q ( v - - U ) / T ] }

-- (S~/S)Io{1 -- [ ex p ( - - aW ) / (1 + aL p ) ]} (1 a )

= J d - - J z ( a b )

whe re S~ = norm al ized sur face charge t rans fer ve lo c i ty ;

S r = n o rma l i zed su r f ace r eco m b i n a t i o n v e l o c i t y ; B = a

s p ace ch a rge r eco m b i n a t i o n p a r am e t e r an d S =

S~ + S r + exp [ - - q ( V b - - lO /k T] + B e x p [ - - q ( V b - - V - -

U / 2 ) / k T ] .

T h e s eco n d t e rm i n eq u a t i o n (1 ) h a s b een i d en t i f ied

a s t h e p h o t o c u r r e n t b e c a u se o f t h e o c c u r r e n c e o f p h o t o n

f lux t e rm , Io . The f i r s t t e rm has been ide n t i f i ed as the

d a rk cu r r en t , Ja . This is e r roneou s s ince the p resen ce o f

U in the f i r s t t e rm a lso im pl ic i t ly ma kes i t l igh t in tens i ty

d ep en d en t . A co r r ec t ap p ro ach w o u l d b e t o a r r i v e a t a

d e t a i led ex p re s s i o n fo r cu r r en t u n d e r i l l u mi n a t i o n an d

8 2 9

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8 3 0 T H E O R E T I C A L M O D E L F O R P H O T O E L E C T R O C H E M I C A L S O L A R C E L L

[ S e m ic o n d u c t o r ] [ ,E t e c t ro t y t e I J M e t a l

Ee- ; :

I r v . . . . . . . . . . . . . . . . . ' ~ - , . - . . t ~

Efn£i

[ f , r e a D z

I = W I - - 0X

F i g . 1 . S c h e m a t i c o f t h e e n e r g y b a n d d i a g r a m o f a n - t y p es e m i c o n d u c t o r / e l e c t r o l y t e i n t e r f a c e ( s e e t e x t ) .

t h e n s u b t r a c t t h e s t a n d a r d d a r k c u r r e n t c o n t r i b u t i o n .

T h i s is w h a t h a s b e e n a t t e m p t e d i n t h is p a p e r . F u r t h e r , a

s imple de r iv a t ion i s g iven he re fo r ca l cu l a t i ng the h o le

dens i ty a t t he edge o f t he space cha rge r eg ion . The f ina le x p r e s s i o n f o r q u a n t u m e f f i c i en c y o f a P E S C i s i n a

s i m p l e c al c u la b l e f o r m a n d e x p l a i n s t h e e x p e r i m e n t a l

r e su l t s o f t he n -CdSe /S 2 - , S~- e l ec t ro ly t e s ys t em.

2 . T H E O R Y

C o n s i d e r a s e m i c o n d u c t o r / e l e c t r o l y t e j u n c t i o n a s

s h o w n i n F ig . 1 . T h e d e n s i t y o f h o l e s ( m i n o r i t y c a r ri e rs )

i n d a r k ( P . o ) a n d u n d e r i l l u m i n a t i o n ( P n ) i n t h e n e u t r a l

r eg ion a re r e l a t ed a s [15] .

d 2P n P n - - P n o + G ( x ) = 0, (2)D p d x 2 T p

w h e r e D p and rp a re t he d i f fus ion coe f f i c i en t and l i f e

t i m e f o r h o l e s.

G ( x ) i s t h e p h o t o g e n e r a t i o n r a t e g iv e n b y [ 6 ]

G ( x ) = O d o ( 1 - r ) e - ~ , ( 3)

wh e re a i s t he abs orp t ion coe f f i c i en t , Io i s t he i n c iden t

p h o t o n f l u x a n d r i s t h e r e f l e c t a n c e .

S o l v in g e q u a t i o n ( 2 ) u n d e r t h e b o u n d a r y c o n d i t i o n s

( i ) a t x = 0% Pn = Pno, (i i) at x = W , Pn = Pn ( W ) =

Pno eqU/kT [16] ; t he ho le dens i ty a t any po in t i ns ide t he

s e m i c o n d u c t o r i s o b t a i n e d a s :

P n ( X ) - - P nO = P n o [e q U / k T - - 1 ] e ( W - x ) / L p

eft0(1 -- r)rp [ e W O _ o . L p _ x / W ) / L p _ e-aX] (4)+

w h e r e L p = ho le d i f fus ion l eng th = (D pr p) 1/2.

p , ( W ) = h o l e d e n s i t y a t t h e e d g e o f d e p l e t i o n l a y e r

o f w i d t h , W .

W = ( 2 e s e o / q N D ) l n ( V b - V) 1 /2 .

F u r t h e r , t h e h o l e f l u x a t a n y p o i n t i n t h e d e p l e t i o n

r e g i o n is g i v e n b y f o l l o w i n g c o n t i n u i t y e q u a t i o n [ 1 5 ]

] d~ { F p ( x ) } = V ( x ) - R ( x ) ,

w h e r e G ( x ) a n d R ( x ) a r e g e n e r a t i o n a n d r e c o m b i n a t i o n

ra t e s . In t eg ra t ing equa t ion (5 ) wi th in t he l imi t x = 0 t o

x = W we ob ta in t he ho le f l ux a t t he su r face , Fp(0) ,

F p(O ) = F p ( W ) - - I o ( 1 - -r ) 1 l ~ _ L p ] f R ( x ) d xo (6)

w h e r e

F p ( W ) = -- D dpn = DpPn° (eqU/kT-- 1)P d x x = W L p

~Lp/o (1 - - r ) e -~w(7)

a L p + 1

I t i s ve ry d i f f i cu l t t o exac t ly eva lua t e t he i n t eg ra l appea r -

ing in equa t ion (6 ) because o f i l l -de f ined gene ra l na tu re

o f R ( x ) . H o w e v e r , u n d e r t h e a s s u m p t i o n o f f la t q ua s i-

F e r m i - le v e l s o f e l e c t r o n s a n d h o l e s s e p a r a t e d b y qU , th ee x p r e s s i o n s g i ve n b y S a h et al. [17] and Grove [16] can

b e w r i t t e n a s

W

J R 1 . ~ A / ~ IA / ~ q U / 2 k T( X ) d x = 2 O Vth ~ V t , , i . . . . ( 8 )

0

whe re a i s t he cap tu re c ross - sec t ion fo r e l ec t rons and

h o l e s a s s u m e d a s e q u a l, vth i s t he ca r r i e r t he rma l

ve loc i ty , N t i s t he t r ap d ens i ty nea r t he i n t r ins i c Fe rmi -

level E i , and rt i i s t he i n t r i ns i c ca r r i e r con cen t ra t i on .

F r o m e q u a t i o n s ( 6 ) , ( 7 ) a n d ( 8 ) w e g e t

( e ° tp ( 0 ) = D p p . o ( e q U / k T 1 ) - - Io ( 1 - - r ) 1

+ B W e q U / z k T , (9)

w h e r e B = ½aVthNtn .

N o w w e o b t a i n a n a l t e r n a t e e p x r e s s i o n f o r F p ( 0 ) i n

t e r m s o f 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 ( S t ) a n d s u r f a c e

t r a n s f e r v e l o c i ty (S t ) a t t he i n t e r face g iven be low:

Fp(O) = - - (Sr + S t )[Pn(O ) -p do (O )] , ( 1 0 )

w h e r e Pdo(O) a n d Pn(O) a re r e spec t ive ly t he su r face con-

c e n t r a t i o n s o f h o l e s in d a r k a t e q u i l ib r i u m a n d i n t h e

pre sence o f l igh t and app l i ed b i a s , V.

In gene ra l Sr ~ S t . H e n c e , S r c a n b e n e g l e c t e d a n d

( S t + S t ) can be s imply w r i t t en a s S . Fr om Fig . 1 i t is

c l ea r t ha t

P d o ( O ) = P .o e ° V b / k T

an d

p , ( O ) = Pn o ea V# kT [ e q ( v - V)/kT]. ( 1 1 )

T h e n f r o m e q u a t i o n s ( 1 0 ) a n d ( 11 )

F p ( 0 ) = - - SPn o eq Vb /kT[e q ( U - v ) / kT - 1)] . (12)

V o l . 5 1 , N o . 1 0

(5 )

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V o l. 5 1, N o . 1 0 T H E O R E T I C A L M O D E L F O R P H O T O E L E C T R O C H E M I C A L S O L A R C E L L 8 31

1 0 1 0

0 8

0 .6

.2

~ 0 4

E

o

( a )

. . ,~ =__ ,~- : : - : b 0 .5

. g " , ' " . ~ 0 .6

zf " b ) ~ 0 4

~ 0 2

I I IO ~ 0 2 0 3 O a 0 5 0 . 6 0. 7 0 . 8

B a n d b e n d i ng ~ V b - V ( v o lt s )

0 1 0 . 2 0 3 O z . 0 .5 0 6 O r7 0 8

B a n d b e n d i n g , V b - V ( v o t t s )

F i g . 2 . T h e o r e t i c a l ( s o l i d l i n e ) a n d e x p e r i m e n t a l ( d o t t e dl in e ) q u a n t u m e f f i c ie n c y a t d i ff e r e n t i n t e n s i ty o fi l lu m i n a t i o n a s a f u n c t i o n o f b a n d b e n d i n g . T h e a d j u s t -a b l e p a r a m e t e r s : S = 1 × 1 0 3 c m s e c - l , B = 2 . 6 × 1 01 4c m - a - s e c - 1. I n t e n s i t y o f i l l u m i n a t i o n i s : ( i ) f o r c u r v e a ,7 . 1 m W c m - 2 ( e q u i v a l e n t t o I o = 3 . 7 m A c m - = ) a n d ( ii )f o r c u r v e b , 0 . 7 1 m W c m - 2 ( e q u i v a l e n t t o I 0 =0 . 3 7 m A c m - 2 ) .

F i g . 3 . T h e o r e t i c a l ( s o l id l in e s ) a n d e x p e r i m e n t a l ( d o t t e dl in e ) q u a n t u m e f f i c i e n c y f o r d i f f e r e n t e l e c t r o l y t ec o n c e n t r a t i o n s a t I o = 0 . 3 7 m A c m -~ ( ~ 0 . 71 m W c m - Z ) .T h e a d j u s t a b l e p a r a m e t e r s a r e : B = 2 . 6 x 1 0 x4 c m - 3sec -1, S = 103 cm sec -1 fo r cu rve a c o r re spo nd in g to t hee l e c t r o l y t e ( 2 . 5 M N a 2 S + 1 M S + 1 M K O H ) a n d S =l 0 s c m s e c - x f o r c u r v e b c o r r e s p o n d i n g t o t h e e l e c t ro -l y t e ( 0 . 2 5 M N a z S + 0 . 1 M S + 1 M K O H ) .

T h u s e q u a t i n g e q u a t i o n s ( 9 ) a n d ( 1 2 ) a n d s o l v in g f o r

e qU / zkT we g e t

e q U / 2 k T = X

= _ L p 1 + - d - L p i] ) ]I

2 ~D p p n ° + S P n o e q ( V b - V ) / k T( L , , )

T h e n i n t r o d u c i n g X i n e q u a t i o n ( 1 2 ) w e g e t

F p ( O ) = - - S P n o e q ( V b - V ) / k T [ ~ - - e q V / k T ] . ( 1 4 )

T h e a b o v e f l u x i s r e a c h i n g a t t h e s u r f a c e i n t h e

p r e s e n c e o f li g h t . T h e h o l e f l u x r e a c h i n g a t t h e s u r f a c e i n

t h e d a r k i n p r e s e n c e o f a p p l i e d b i a s F i s g i v e n b y [ 1 6 ]

F f f (O ) = D p P n o ( eqV/kT _ 1 ) + B W e qv /2kT . (15 )

L pT h e r e f o r e , t h e n e t h o l e f l u x r e a c h in g t h e s u r f ac e o f t h e

s e m i c o n d u c t o r d u e t o p h o t o e x c i t a t i o n i s g i v e n b y

F~(O) = Fp(O)--Fap(O)

= _ { S P n ° e q ( V b - V ) l h T ( X 2 _ e q V / k T )

+ D p P n ° ( e q V / k T - - 1 ) + B W e q V / 2~ T } . ( 1 6 )L p

T h e q u a n t u m e f f i c ie n c y is g i v en b y

f F ~ ( 0 ) j

7 7 - Io

(13)

3 . R E S U L T S A N D D I S C U S S I O N

I n o r d e r t o c h e c k t h e v a l i d it y o f o u r t h e o r y w e h a v e

a p p l i e d o u r r e s u l t s t o c a l c u l a te t h e e f f e c t o f ( i) i ll u m i n a -

t i o n i n t e n s i t y , a n d ( i i ) e l e c t r o l y t e c o n c e n t r a t i o n

c o n t r o l li n g t h e c h a r g e t r an s f e r v e l o c i t y o n t h e q u a n t u m

e f f e ic i e n c y o f n - C d S e p h o t o a n o d e a n d p o l y s u l f id e

e l e c t r o l y t e j u n c t i o n P E S C . T h e v a l u e s o f c o n s t a n t s a r e

[8 ] : Ca r r i e r l if e t ime , rp = 10 -9 sec , i n t r i ns i c ca r r i e r con-

c e n t r a t i o n , n i = 1 07 c m - 3 , h o l e d i f f u s i o n l e n g t h , L p - -

1 × 1 0 - 4 c m , r e f l e c t a n c e , r = 0 . 1 , a b s o r p t i o n c o e f f i c i e n t ,

a = 5 . 2 × 1 0 4 c m - 1, i n i ti a l b a n d b e n d i n g , F ~ ( w i t h o u t

a p p l i e d b i a s ) = 0 . 7 V , d o n o r c o n c e n t r a t i o n , N D =

1 . 0 x 1 0 1 7 c m - 3 , d i f f u s i o n c o e f f i c i e n t , D p = 1 0 c m 2 s e c - I ,

w a v e l e n g t h o f l i g h t, X = 6 5 0 n m .

F i g u r e 2 s h o w s t h e e f f e c t o f i n t e n s i t y o f l i g h t a t

( k = 6 5 0 n m ) o n q u a n t u m e f f i ci e n c y o f P E S C f o r di f-

f e r e n t b a n d b e n d i n g s . F o r t h e o r e t i c a l c a l c u l a t i o n s ,

q u a n t i t i e s S a n d B a r e tr e a t e d a s p a r a m e t e r s w h i l e o t h e r

c o n s t a n t s a r e t a k e n a s a b o v e . I t is c le a r t h a t t h e a g r e e -

m e n t b e t w e e n o u r t h e o r y a n d e x p e r i m e n t [ 8 ] is

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83 2 T H E O R E T I C A L M O D E L F O R P H O T O E L E C T R O C H E M I C A L S O L A R C E L L V o l. 5 1, N o . 10

r ea s o n ab l y g o o d f o r l o w an d h i g h e r v a l u es o f b an d b en d -

i n g. B u t f o r m o d e r a t e v a l ues o f b an d b en d i n g t h e

theore t ica l va lues ar e h igher than exper imenta l va lues .

T h e d i s c r ep an cy i s a t t r i b u t ed t o t h e f o l lo w i n g r ea s o n s :

( i ) T h e s u r f ace r eco m b i n a t i o n p l ay s an i m p o r t an t

r o l e in t h e m o d e r a t e b an d b en d i n g r eg io n b ecau s e o f t h e

c o m p e t i t i o n b e t w e e n S r a nd S t [ 1 8 ]. H o w ev e r w e h av e

neg lec ted Sr in the p resen t case .

( i i ) the charge t r ans fer parameter 'S ' i s t r ea ted in

o u r m o d e l a s in d ep en d en t o f ap p l ied b i a s an d b a n d

b en d i n g w h i ch i s n o t r i g o r o u sl y t r u e i n v i ew o f t h e ab o v e

[ 11 ] . An othe r p o in t which eme rges f rom Fig . 2 i s tha t as

we lower the in ten s i ty , the e f f ic iency a l so decreases .

T h i s i s a co n s eq u en ce o f t h e n o n - l i n ea r d ep en d en c e o f

b ~( O ) o n I 0 a s g i v en i n eq u a t i o n ( 1 6 ) .

F i g u r e 3 s h o w s t h e e f f ec t o f ch a r g e t r an s f e r v e l o c i t y

o n q u a n t u m e f f i c i en cy f o r d i f fe r en t b an d b en d i n g . T h e

v a l u es o f p a r am e t e r S f o r d i f f e r en t co n cen t r a t i o n s a r e

a s s u m ed t o b e r e l a t ed a s :

S = kCredox,

where k i s a cons tan t , and Creaox i s t h e co n cen t r a t i o n o f

t h e e l ec t r o l y t e . Fo r h i g h e r co n c en t r a t i o n o f e l ec t r o l y t e ,

t h e r e i s a r ea s o n ab le ag r eem en t b e t w ee n t h e t h e o r y an d

ex p e r i m en t w h i l e i t is n o t s o a t l o w er co n cen t r a t i o n s .

T h e p r o b ab l e r ea s o n s f o r d i s ag r eem en t a r e :

( i ) O u r a s s u m p t i o n t h a t S i s i n d ep en d e n t o f b an d

bend ing , i s no t r igorous ly t rue as po in ted ou t ear l ie r .

( ii ) t h e p o t en t i a l d r o p ac r o s s t h e H e l m h o l t z l ay e r

can n o t b e n eg l ec t ed f o r t h e l es s co n ce n t r a t ede l ec t r o l y t e , an d

( i ii ) the ser ies r es i s tance ca n a l so no t be neg lec ted

f o r l es s co n d u c t i n g e l ec t r o l y t e a t l o w er co n cen t r a t i o n s .

We have no t cons idered the ser ies r es i s tance ef f ec t in our

m o d e l w h i c h i s k n o w n t o b e q u i te c o m p l i c a t e d f o r p - n

junct ion so lar ce l l s .

R E F E R E N C E S

1. S . Chandra & R.K . Pand ey , Phys. Status . Sol idi .(a ) 7 2 , 4 1 5 ( 1 9 8 2 ) .

2 . S . Chand ra , Photoelect roc hemica l So lar Cell ,G o r d o n an d B r each Sc ien ce Pu b l is h e rs , N ew Y o r k(1984) ( in p ress ) .

3. R . Me mm ing ,Phi l ips Teeh. Rev . 38 , 160

( 1 9 7 8 / 7 9 ) .4 . H . Ger i scher , in Solar Energy C onvers ion , T o p i c s

i n A p p l . p h y s . 3 1 . ( E d i t ed b y B .O . Se r ap h in ) ,Spr inger Ver lag, Ber l in (1979) .

5 . M .A . B u t t l e r ,Z Appl . Phys . 4 8 , 1 9 1 4 ( 1 9 7 7 ) .6 . W . W .G~irtner, Phys . Rev . 116 , 84 (1959) .7. R.H . Wilson, J . App l . Phys . 4 8 , 4 2 9 2 ( 1 9 7 7 ) .8 . J . R e i ch m an & K . R u s s ak , P h o t o e f f ect s a t S em i -

conductor /Elect ro ly te In ter face , A C S S y m p . 1 4 6 ,p . 3 5 9 ( ed i t ed b y A . J . N o z i k ) ( 1 9 8 1 ) .

9 . D . H a n e m a n & J . F . M c C a n n , P h y s . R e v . 25, 1241( 1 9 8 2 ) .

10 . P . Pa nay o ta tos & H.C. Card , Solid State Electr .23 , 41 (1980) .

I 1 . F . E1 Guib aly , K. Colbow & B.L . Fu n t , J . Appl .Phys . 5 2 , 3 4 8 0 ( 1 9 8 1 ) .

12 . F . E1 Guib aly & K. Colbo w, J . App l . Phys . 5 3 ,1 7 3 7 ( 1 9 8 2 ) .

13. M.A. Butle r & D.S. Gin ley, J . E lect rochem. Soc.1 2 5 , 2 2 8 ( 1 9 7 8 ) .

1 4 . V .A . M y am l i n & Y u .V . P l e s k o v , Elect rochemis t ryo f S e m i c o n d u c to r s , A cad em i c P re ss , N ew Y o r k( 1 9 7 6 ) .

15 . H .J . Hovel , S em i co n d u c t o r s a n d S em i m e t a l s , V o l .11, Solar Cells ( E d i t ed b y R .K . Wi l l a rd s o n & A .C.B ee r ), A cad em i c P r e ss , N ew Y o r k ( 1 9 7 5 ) .

16 . A . Grove , P h ys ic s a n d Tech n o l o g y o f S em i co n d u c -

tor Devices, p . 1 8 6 . Wi l ey , N ew Y o r k ( 1 9 6 7 ) .1 7 . C . Sah , R . N o y c e & W. Sch o ck l ey , Proc. IRE 4 5 ,

1228 (1957) .18 . H . Ger i scher , J . Electroanal. Chem. 150 , 553

( 1 9 8 3 )