Limiting Efficiencies for Photo Voltaic Energy Conversion in Multigap Systems

8/8/2019 Limiting Efficiencies for Photo Voltaic Energy Conversion in Multigap Systems

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ELSEVIER Solar Energy Ma terials and So lar Cells 43 (199 6) 203-222~ S d w ~

Lim iting ef f ic iencies for pho tovoltaic energyconv ers ion in m ultigap system s

A n t o n i o M a r t i * , G e r a r d o L . A r a f i j oInstituto de Energ(a Solar, Universidad Politdcnica de Madrid, E.T.S.L de Telecoraunicaci6n, CiudadUniversitaria, 28040 Madrid, Spain

Received 9 September 1995; revised 1 February 1996; accepted 25 M arch 1996

A b s t r a c t

M u l t i g a p s y s t e m s a r e b e t t e r m a t c h e d t o t h e s u n ' s s p e c t r u m t h a n s i n g l e g a p s y s t e m s a n d a r e ,the re fo re , m ore e f f i c ien t a s pho tovo l ta ic conver te r s . T h is paper r ev iews the d i f f e ren t thermody-namic a p p r o a c h e s u s e d i n t h e p a s t f o r c o m p u t i n g t h e l i m i t i n g e f fi c i e n c y fo r t h e c o n v e r s i o n o f s o l a re n e r g y i n t o w o r k . W i t h i n t h i s t h e r m o d y n a m i c c o n t e x t , t h e l i m i t r a n g e s f r o m 8 5 . 4 % t o 9 5 . 0 %d e p e n d i n g o n t h e a s s u m p t i o n s m a d e . D e t a i l e d b a l a n c e t h e o r y p r o v i d e s a m o r e a c c u r a t e m o d e l o fthe pho tov o l ta ic conver te r . I t leads to a l im i t o f 86 . 8% fo r a s ys tem wi th an in f in i t e num ber o fce l l s , a s a l r ead y po in ted ou t by o ther au thor s . In th i s work , how ever , we us e the conc ep ts o f anglea n d energy r es t r i c t ion to em phas ize tha t th i s l im i t i s independen t o f the l igh t concen t r a t ion .S y s t e m s w i t h a finite n u m b e r o f c e i l s a r e a l s o s t u d i e d a n d t h e i r li m i t i n g e f fi c i e n c y is f o u n d t o b eh i g h e r t h a n p r e v i o u s l y r e p o r t e d . D a t a f o r A M 1 . 5 D i r e c t s p e c t r u m , n e v e r c o m p u t e d b e f o r e , a r ei n c l u d e d .Keywords: Photovo ltaics; Mu ltigap; Limiting efficiency; Thermod ynam ics; Solar cells

1 . I n t r o d u c t i o n

A t o p i c o f i n te r e s t i n p h o t o v o l t a i c s is t he e l u c i d a t i o n o f th e m a x i m u m e f f i c i e n c y o ft h e p h o t o v o l t a i c c o n v e r s i o n p r o c e s s e s . A r e v i e w o f t h e l i te r a tu r e s h o w s t h a t th e d i f f e r e n ts t u d i e s c a n b e g r o u p e d i n t w o m a i n c a t e g o r i e s . I n t h e f i r s t , a u t h o r s s t u d i e d t h e l i m i t i n ge f f i c i e n c y o f single g a p s o l a r c e ll s . T h e p r e s e n t a u t h o r s d e v o t e d a s t u d y t o t h is s u b j e c t[1 ]. I n t h e s e c o n d c a t e g o r y , t h e l i m i t i n g e f fi c i e n c y o f m u l t i g a p s y s t e m s w a s u s e d a s a

* Corresponding author. Email: [email protected]/96/$15.00 Copyright 1996 Elsevier Science B.V . All r ights reserved.PH S 0 9 2 7 - 0 2 4 8 ( 9 6 ) 0 0 0 1 5 - 3


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204 A. Mar ff , G .L Araf i jo / Solar Energy Mater ia l s and Solar Ce ll s 43 (1996) 203 -22 2

w a y t o a p p r o a c h t h e a b s o l u t e l i m i t o f p h o t o v o l t a ic e n e r g y c o n v e r s i o n ( P V E C ) a n d t h is i st h e c a t e g o r y t o w h i c h t h is p a p e r b e lo n g s .

A l t h o u g h t h e q u e s t i o n o f t h e li m i ti n g e f f i c i e n c y o f P V E C i s o n e t h a t o u g h t to h a v e aun iqu e an sw er , d i f f e ren t figures app ear in the l i t e r a tu re a s w i l l be de ta i l ed l a te r .S o m e t i m e s , a u t h o r s m a d e d i f f e r e n t a s s u m p t i o n s a n d u s e d d i f f e r e n t a p p r o a c h e s l e a d i n gt h e m t o d i f f e r e n t r e su l ts . M o s t o f t h e t i m e , t h e n e w f i g u r e a r o s e f r o m t h e c o n s i d e r a t i o no f a n e w a s p e c t p r e v i o u s l y o v e r l o o k e d w h i c h p r o g r e s s i v e l y c o m p l e t e d o u r u n d e r st a n d i n go f t h e p h o t o v o l t a i c p r o c e s se s .

T w o c o m p l e m e n t a r y a p p r o a c h e s h a v e b e e n u s e d i n t h e p a s t t o a n s w e r t h e q u e s t io n o fth e l imiting efficiency. The f ir s t, b a s e d o n t h e r m o d y n a m i c s , is i n d e p e n d e n t o f t h e n a t u r eo f t h e c o n v e r t e r a n d i s b r i e f l y r e v i e w e d i n t h is p a p e r ~ n t h e f ir s t s e c t io n . T h e o t h e r,desc r ibed in S ec t ion 2 , i s based on detailed balance a rgum ent s . I t u ses a m ore r e f inedm o de l tha t m akes som e assum pt ions abo u t the na tu re o f the conve r t e r : t he so la r ce l l. InS e c t i o n 3 , w e m a k e s o m e r e m a r k s a b o u t t h e d i f f e r e n c e s , c o n c e r n i n g t h e l i m i t i n ge f f i c i e n c y , b e t w e e n a s y s t e m w i t h a n infinite n u m b e r o f c e l ls a n d a s y s t e m w i t h a f in i ten u m b e r o f c e l ls . T h e l im i t in g e f f i c i e n c y f o r t h e f in i te s y s t e m i s h i g h e r t h a n p r e v i o u s l yr e p o r t e d .

2 . A br ie f thermo dynam ic rev iewThermodynamics i s t h e n a t u r a l f r a m e w o r k f o r t h e s t u d y o f t h e u p p e r b o u n d s o f t h e

e f f i c i e n c y o f th o s e p r o c e s s e s i n v o l v i n g e n e r g y c o n v e r s i o n : h e a t i n to m e c h a n i c a l w o r k ,l igh t i n to hea t , hea t i n to chem ica l ene rgy , e t c . , and in the case w e a re now dea l ing w i th ,light in to electrical work. B y u s i n g thermodynamics , w e c la im the re i s a un ique andd e f i n i t i v e a n s w e r t o t h e p r o b l e m b e i n g a n a l y z e d . H o w e v e r , w h e n r e v i e w i n g t h e l i t e r a -t u r e, s o m e c o n f u s i o n a p p e a r s b e c a u s e d i f f e r e n t f i g u r e s f o r t h e l i m i ti n g e f f i c i e n c y o fP V E C a r e r e p o r t e d , a l l o b t a i n e d b y u s i n g thermodynamic c o n c e p t s .

A n a l y s e s i n v o l v i n g thermodynamics r ep resen t an a t t em pt to ob ta in a r e su l t t ha t i sindepender~ t o f the na tu re o f the con ver t e r . T o un der s t and the d i f f e ren t hyp o these s anddef in i t ions tha t l ead to d i f f e ren t r e su l t s , w e f ind the genera l desc r ip t ion g iven byL a n d s b e r g o f a n e n e r g y c o n v e r t e r in [ 2 ] v e r y u s e f u l. I n d e e d , m o s t o f th e d i s c u s s i o n a n dexp lana t ion tha t fo l low s has a r i sen f rom R ef . [3 ] .

L a n d s b e r g ' s d e s c r i p t io n o f a g e n e r a l c o n v e r t e r is g r a p h i c a l ly s k e t c h e d i n F i g . 1 . T h ec o n v e r t e r , a s su m e d t o b e a t a te m p e r a t u r e Tc , r e c e i v e s p h o t o n s c h a r a c t e r iz e d b y e n e r g yand en t ropy f luxes E ~ and S~ respec t ive ly f rom a n a rb i t r a ry source . In the genera l case ,i t a l s o e m i ts p h o t o n s c h a r a c t e r i z e d b y a n e n e r g y a n d e n t r o p y f lu x e s , E ~ a n d S ~ , to w a r d sa s ink . T he general c o n v e r t e r p r o d u c e s h e a t ( Q ' ) a n d w o r k ( W ' ) and , i f t he p rocess i sno t s t eady-S ta t e , has in t e rna l ene rgy and en t ropy va ry in g a t a r a te E ~ and S ~ ,r e s p e c t i v e l y . H o w e v e r , f r o m n o w o n , w e w i l l a s s u m e a l l p r o c e s s e s a r e s t e a d y - s t a t e s otha t E ~ and S ~ a re ze ro .

I f t hese in t e rchange p rocesses occur i r r eve r s ib ly , w e use S g to r ep resen t the co r re -s p o n d i n g t o t a l ra t e o f e n t r o p y i n c r ea s e . H o w e v e r , w e s h a ll a s s u m e f o r t h e m o m e n t t h a tt h e i n t e r c h a n g e p r o c e s s e s o c c u r r e v e r s i b l y s o t h a t t h e a d d i t i o n a l e n t r o p y d u e t oi r revers ib i l i t ies , S 'g , wi l l be ta ken as ze ro .


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A. Mar t[ , G .L Araf i jo / Solar Energy Mater ia l s and So lar Ce ll s 43 (1996) 20 3-2 22 205

Q ' W '

E ~ EoSs ~ S ' oF i g . I . S k e t c h o f a g e n e r a l e n e r g y c o n v e r t e r a s d e s c r i b e d b y L a n d s b e r g [ 2] .

Th e c o mp u t a t i o n o f t h e l i m i t i n g e f f i c i e n c i e s i s s i mp l i f i e d b y a s s u mi n g t h e s u n i s ablack body s o u r c e o f p h o t o n s a t a t e m p e r a t u r e Ts c l o s e t o 6 0 0 0 K. No t h i n g e s s e n t i a l w i l lb e l o s t i n t h i s p r e l i mi n a r y d i s c u s s i o n i f , f o r s i mp l i c i t y , we a l s o a s s u me t h a t t h e s u nc o m p l e t e l y s u r ro u n d s t h e c o n v e r te r . H e n c e , t h e e n e r g y a n d e n t r o p y f l u x e s th e c o n v e r t e rrece ives can be wr i t t en a s [2 ] :

4= - 5 ( )T h e a s s u m p t i o n t h a t t h e s u n c o m p l e t e l y s u r r o u n d s t h e c o n v e r t e r i s e q u i v a l e n t t o t h e

s o l a r c e l l b e i n g i l l u mi n a t e d a t th e m a x i m u m p o s s i b l e l i g h t c o n c e n t r a t i o n w i t h a n i d e a lr e f l e c t o r p l a c e d a t i ts r e a r s id e . Th i s m a x i m u m c o n c e n t r a t io n i s g i v e n [ 4 ] b yn 2

g m a x s in20s (2)wh e r e 0 s is t h e a n g l e s u b t e n d e d b y t h e s o l a r d is c a t t h e Ea r t h a n d n i s th e r e f r a c t i o ni n d e x o f t h e me d i u m t h a t s u r r o u n d s t h e c o n v e r t e r . As i t w i l l b e s h o w n l a t e r i n th e n e x ts e c ti o n s , t h e m a x i m u m e f f i c i e n c y i s in d e p e n d e n t o f t h e c o n c e n tr a ti o n i f w e a l lo wr e s t r i c t i o n o f t h e angle w i t h w h i c h p h o t o n s a r e emitted f r o m t h e c o n v e r t er . T h e r e f o r e ,t h e a s s u m p t i o n t h a t t h e s u n c o m p l e t e l y s u r r o u n d s t h e c o n v e r t e r b e c o m e s u n n e c e s s a r y i nt h e e n d . Ho we v e r , we s h a l l r e t a i n t h i s a s s u mp t i o n t o s i mp l i f y t h e p r e l i mi n a r y d i s c u s s i o n .O n e o f t h e c a u s e s o f d i v e r g e n c e i n t h e c o m p u t a t i o n s o f t he l i m i t in g e f f ic i e n c ie s f r o mb a s i c t h e r m o d y n a m i c a p p r o a c h e s c o m e s f r o m t h e d i f f e r e n t w o r k i n g a s s u m p t i o n s a b o u tt h e v a l u e o f th e e n e r g y a n d e n t r o p y f l u x e s e n t e ri n g t h e s i n k ( E 0 a n d S o ) . F o rs i mp l i f i c a t i o n , we wi l l a s s u me t h a t t h e p h o t o n e n e r g y a n d e n t r o p y f l u x e s i n t o t h e s i n kc a n b e c h a r a c t e r i z e d a s t h e o n e s t h a t e m e r g e f r o m a b l a c k b o d y a t a n e f f e c t i v et e m p e r a t u r e TO T h e r e f o r e ,

4S o = -~ , , - r8 , ( 3 )s i n c e t h e c o n v e r t e r i s a s s u me d a l s o t o e mi t r a d i a t i o n i n a l l d i r e c t i o n s .


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206 A. Mar tf , G .L . Ara~jo / Solar Energy Ma ter ia l s and Solar Ce ll s 43 (1996) 20 3-2 22

T w o b a l a n c e e q u a t i o n s c o r r e s p o n d i n g t o t h e e n e r g y a n d e n t r o p y c o n s e r v a t i o n c a nn o w b e w r i t t e n :

E ' s = Q ' + W ' + E 'o ,T c S ' = Q ' + T c S ' , ( 4 )

a n d , h e n c e ,Q ' = r c S ' - T c S 'o ,w ' = E l - r c s l + r c s ; - E b . ( 5 )

T w o m a i n d e f i n i t i o n s f o r e f f i c i e n c y n o w b e c o m e a v a i la b le :W '

~71 = E l _ E~) (6 a )W 'n 2 = ( 6 b )

T h e f i r st o n e c o r r e s p o n d s t o t h e r a t io b e t w e e n t h e p o w e r t h e c o n v e r t e r p r o d u c e s a sw o r k a n d t h e p o w e r i t c o n s u m e s . N o t i c e t h a t t h e e n e r g y i t c o n s u m e s i s t he d i f f e r enceb e t w e e n t h e p o w e r r e c e i v e d f r o m t h e s u n a n d t h e p o w e r e m i t t e d . T h e s e c o n d d e f i n i t i o ni s t h e d e f i n i t io n o f e f f i c i e n c y c o m m o n l y u s e d i n p h o t o v o l t a ic s : t h e r a ti o b e t w e e n t h ep o w e r e x t r a c t e d f r o m t h e c e l l a n d t h e p o w e r r e c e i v e d f r o m t h e s u n . T h i s d i f f e r e n c e i nd e f i n i ti o n s h a s b e e n t h e s o u r c e o f s o m e c o n f u s i o n [5 ].

W i th t he f i r s t de f in i t i on , Eq . 6a , t he e f f i c i enc y inc reases w i th TO ( see F ig . 2 ) . Them a x i m u m e f f i c ie n c y is , t h e re f o re , a c h i e v e d w h e n To = Ts (poin t A in Fig . 2) s ince , a th i g h e r te m p e r a t u r e s , W ' b e c o m e s n e g a t iv e w h i c h m e a n s t h a t t h e s u r ro u n d i n g s a re d o i n gw o r k o n t h e s y s t e m i n s t e a d o f t h e v i c e v e r s a . A t t h i s p o i n t , t h e m a x i m u m e f f i c i e n c y

r /

Cl

/ ! IPTo

Fig. 2. Qualitative behaviour of the power W', heat Q' and the efficiencies defined by Eq. (6) as a function ofthe emission temperature To. Curve labelled as ~lt)v corresponds also to definition 2 but, following [3], doesnot assume the photovolta lc process as reversible. See Table 1 for identification of the dot labels.


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A. Mart[, G.L Araftjo / Solar Energy Materials and Solar Cells 43 (1996) 203 -222 207o b t a i n e d i s t h e s a m e a s th e e f f i c i e n c y o f a C a m o t e n g i n e [ 6] o p e r a t i n g b e t w e e n h e a tr e s e r v o i r s a t t h e s u n ' s a n d E a r t h ' s t e m p e r a t u r e :

T c~71,MAX = 1 T s " ( 7 )

N o t i c e , h o w e v e r , t h a t a l t h o u g h t h e e f f i c i e n c y is m a x i m u m w h e n To = T s , t h e nW ' = 0 w h i c h i s o f n o p r a c t i c a l u s e . I n t h is c o n t e x t , d e f i n i t i o n 2 i n ( 6 ) is p r e f e r r e d .S u b s t i t u t i n g ( 5 ) a n d ( 1 ) in t o E q . ( 6 b ) , th e f o l l o w i n g e q u a t i o n i s o b t a i n e d :44 Tc + - 1 ( 8 )

172 = 1 - - "3" - ~ " ~ 3 T O t ~ ' s ] "E q . ( 8 ) r e a c h e s i ts m a x i m u m f o r T o = Tc g i v i n g th e n :

! [ 5 / 4T c + ( 9 )= 3 T s l '

w h i c h c o r r e s p o n d s t o th e L a n d s b e r g l i m i t i n g e f f ic i e n c y [ 2] , p o i n t B i n F i g . 2 . N o t i c e t h a tw i t h d e f i n i t i o n 6b , m a x i m i z i n g t h e e f f i c i e n c y i s e q u i v a l e n t t o m a x i m i z i n g W ' , H e n c e ,w h e n T O --- Tc , W ' a l so i s m a x i m u m . H e n r y [ 7 ] a s s u m e d t h a t t h e m a x i m u m e f f ic i e n c yw a s f o r T o = 0 s i n c e , i n t u i t i v e l y , f o r t h i s v a l u e n o e m i s s i o n l o s s e s o c c u r . H e f o u n d t h e nt h a t :

4 T cn ' 2, M A X = 1 - - - - - - , ( 1 0 )3 T s

w h i c h c o r r e s p o n d s t o p o i n t C i n F i g . 2 .H o w e v e r , w h e n e n t r o p y f l u x c o n s e r v a t i o n i s t a k e n in t o a c c o u n t a s i n L a n d s b e r g ' s

a p p r o a c h d e s c r i b e d a b o v e , i t i s f o u n d t h a t f o r T o = Tc , a l t h o u g h s o m e e m i s s i o n l o s s e se x i s t, t h e s e lo s s e s a r e c o m p e n s a t e d b e c a u s e t h e Q ' l o s s e s a r e l o w e r t h a n i n th e c a s e inw h i c h TO = 0 . T a b l e 1 i n c l u d e s t h e m a i n r e s u l t s t h a t h a v e b e e n d e s c r i b e d a b o v e . N o t i c e ,h o w e v e r , t h a t in th i s d i s c u s s io n , s t i l l a l l t h e p r o c e s s e s h a v e b e e n a s s u m e d r e v e r s i b le . T h ei r r e v e r s i b l e n a t u r e o f p h o t o v o l t a i c e n e r g y c o n v e r s i o n h a s b e e n p o i n t e d o u t b y s e v e r a l

T a b l e 1Sum mary of the ma in re su l t s about the l im i t ing e f f i c i ency of the p hotovol t a i c ene rgy convers ion cons ide r ingthe rmodynamic approaches . Resu lt s re fe r t o the sun assum ed as a b l ack body a t T = 6000K and Ear th a tT = 300K. Dot s re fe r t o F ig . 2Au thor Ref. Def. S 'g TO Max. e f f i c i ency Resu l tCarnot DO T A [6] "qj 0 T 1 - r~ 95.0%H enr y DO T C [7] "q2 0 0 1 - 4_ ~ 93.3%3 T s

4 T ~ . ~ I T ~ 4Land sberg DO T B [2] "q2 0 TC 1 - ~ r , + 7( r , ) 93.3%A. De Vos and Pauwels DO T D [3] "q2 ( I -1) TDV = 2540K (1 -2) 85 .4%

s'~ 2 4 ~ - ' ~ 'o s r ~ + ~ ] ( I - 1 )r _ ~ r 4n n v = (1 - r o )[ 1 - ( ~ ) ] ( 1 -2 )


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208 A. MarE, G .L Araf i jo / Solar Energy Mater ia l s and Solar Ce l ls 43 (1996) 2 03- 222

a u t h o r s [ 3 , 8 - 1 1 ] a n d a d e t a i l e d d i s c u s s io n i s o u t s id e t h e s c o p e o f t h is p a p e r . I n a n y c a s e ,t h e c o m p u t a t i o n o f t h e e n t r o p y g e n e r a t i o n r a t e S 'g r e q u i r e s a d d i t i o n a l h y p o t h e s e s , s o m eo f t h e m b a s e d o n a s s u mp t i o n s f r o m t h e d e t a i l e d b a l a n c e t h e o r y . T h e y wi l l l e a d u s t o t h ecurv e l abe l l ed a s ~qDv in F ig . 2 a s w i l l d i scussed in the nex t sec t ion .

3 . The deta i l ed ba la nce a ppro a ch fo r a m ul t ig a p sy s temP r e v i o u s r e s u l t s w e r e o b t a i n e d a s s u m i n g t h a t t h e e n t r o p y g e n e r a t i o n r a t e , S ' g , wa s

z e r o . S i n c e n o a s s u mp t i o n wa s ma d e a b o u t t h e n a t u r e o f t h e c o n v e r t e r , t h i s h y p o t h e s i sa l l o w e d u s t o c o m p u t e a n u p p e r b o u n d f o r t h e e f f i c i e n c y . W e s h a l l n o w p r e s e n t ad i f f e r e n t a p p r o a c h .

I f s o m e a p p r o p r i a te a s s u m p t i o n s a r e m a d e a b o u t t h e n a tu r e o f t h e c o n v e r t e r, t h e w o r kr a t e W ' c a n b e c o m p u t e d d i r e c t ly . I n o u r c a s e , t h e c o n v e r t e r w i l l b e a s y s t e m o f o n e o rm o r e s o l ar ce l l s a n d t h e f r a m e w o r k o f th e d e t a i l e d b a l a n c e t h e o r y [ 1 2 ,1 3 ] w i l l p r o v i d et h e n e c e s s a r y s e t o f a s s u mp t i o n s . T h i s wa s r e v i e we d i n d e t a i l i n o t h e r p a p e r s [ 1 , 1 4 ] a tt h e t i me t h a t th e l i mi t i n g e f f i c i e n c y o f a s y s t e m c o n s t i t u te d b y o n e c e l l wa s s t u di e d . F o rt h e c o n v e n i e n c e o f th e r e a d e r , we wi ll, b r i e f l y s u m m a r i z e h e r e t h e s e a s s u m p t i o n s a n d t h ec o n c l u s i o n s t h e y l e a d u s t o .

The bas ic s t ruc tu re o f the so la r ce l l cons ide red i s i l lu s t r a ted in F ig . 3b . I t cons i s t s in as e m i c o n d u c t o r s l a b s a n d wi c h e d b e t w e e n s o m e r e g i o n s p + a n d n + a r b i t r a r il y t h in . T h ef o l l o wi n g a s s u mp t i o n s a r e ma d e r e g a r d i n g t h e o p e r a t i o n o f t h e c e l l :

( a ) On l y r a d i a t i v e r e c o mb i n a t i o n p r o c e s s e s t a k e p l a c e . Wh e n d e s c r i b i n g t h e r a d i a t i v eg e n e r a t i o n - r e c o m b i n a t i o n m e c h a n i s m s , p h o t o n a b s o r p t i o n a n d s p o n t a n e o u s a n d s t i m u -

o l l

p+ rl +

( a ) ( b )Fig. 3. (a) Elementary geometry for illuminating the cells in a concentration system. 0 s is the angle of the sunsphere when it is seen from Earth, 0 x is the max imum angle with which photons reach the cell after theconcentrator , 0 E is the angle with which photons are emitted from the cell. (b) Basic solar cell st ructure usedin the formulat ion of the detailed balance approach.


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A. Mar f f, G .L . Ara(t jo / Solar Energy Ma ter ia l s and So lar Ce ll s 43 (1996) 20 3-2 22 2 09

l a t e d p h o t o n e m i s s i o n p r o c e s s e s a r e c o n s i d e r e d . I n a d d i t i o n , e a c h p h o t o n i s a s s u m e d t op r o d u c e o n l y o n e e l e c t r o n - h o l e p a i r wh e n a b s o r b e d a n d v i c e v e r s a : e a c h e l e c t r o n - h o l ep a i r p r o d u c e s o n l y o n e p h o t o n wh e n r e c o m b i n e d . R e c e n t l y , a t t e n t i o n h a s b e e n g i v e n t othe poss ib i l i ty tha t o n e p h o t o n g e n e r a t e s t w o ( o r m o r e ) e l e c t ro n - h o l e p a i rs b y m e a n s o fo p t i c a l l y i n d u c e d i m p a c t i o n i z a t i o n [ 1 5 - 1 7 ] . I n t h i s wo r k , h o we v e r , t h e o n e t o o n eh y p o t h e s i s w i l l s t i l l b e c o n s i d e r e d a s t h e m o s t l i k e l y a b s o r p t i o n p r o c e s s . F u r t h e r m o r e ,f r o m t h e p o i n t o f v i e w o f t h e l i m i t i n g e f f i c i e n c y , i t h a s b e e n r e c o g n i z e d [ 1 8 ] t h a t t h em a x i m u m e f f i c i e n c y o f t h e h o t c a r r i e r s o l a r e n e r g y c o n v e r t e r i s e s s e n t i a l l y e q u i v a l e n t t otha t o f an in f in i t e s t ack o f o n e to o n e so la r ce l l s .

( b ) C a r r i e r m o b i l i t y i s a s s u m e d t o a p p r o a c h i n f i n i t y s o o h m i c l o s s e s a r e n e g l e c t e d .P s e u d o - F e r m i l e v e l s p l i t t i n g b e c o m e s c o n s t a n t t h r o u g h t h e d e v i c e a n d e q u a l t o t h ee x t e r n a l v o l t a g e a p p l i e d .

Wi t h t h e s e h y p o t h e s i s , t h e c u r r e n t v e r s u s v o l t a g e c h a r a c t e r i s t i c o f t h e s o l a r c e l l wa sp r o v e d i n [ 1 ] t o b e g i v e n b y :

I = q( FA8 s - - FEM ), (1 1)where FABs a n d F E ~ a r e th e n u m b e r o f p h o to n s a b s o r b e d a n d e m i t t e d p e r un i t o f t im ei n t h e c e l l , r e s p e c t i v e l y . T h e s e v a l u e s c a n b e c o m p u t e d f r o m t h e f o l l o wi n g r e l a t i o n s h i p s :

F AB S = q f a b s cos 0 d Ed $2 d S2 E 2 1 0 < 0 xh 3 c 2 Ee x p - 1

~r s

2 E 2 1h 3 c 2 E

e x p k T c

b S7 r / 2 > 0 > 0 x

- 1

( 1 2 )

wh e r e t h e e q u a t i o n f o r 0 < 0 x ( s e e F i g. 3 a f o r t he d e f i n i t io n o f th e a n g l e 0 ) c o r r e s p o n d st o p h o t o n s c o m i n g f r o m t h e s u n c h a r a c t e r i z e d a s a b l a c k b o d y a t a t e m p e r a t u r e T s ; a n dt h e e q u a t i o n f o r 0 > 0 x c o r r e s p o n d s t o p h o t o n s c o m i n g f r o m t h e e n v i r o n m e n t wh i c h i sa l s o t a k e n t o b e a t t e m p e r a t u r e T c . ~ i s the so l id ang le tha t 0 sus ta ins . The o the r t e rm inE q . ( 1 1 ) i s g i v e n b y

= f e b ( V ) c o s 0 d E d ~ 2 dSEU2 E : 1

b ( V ) = h 3c 2 E - q V ( 1 3 )e x p - - 1k:rc

T h e t e r m s a a n d e i n ( 1 2 ) a n d ( 1 3 ) a r e t h e a b s o r p t i e i t y a n d t h e e m i s s i z i t y ,r e s p e c t i v e l y , a n d i n t h e m o s t g e n e r a l c a s e t h e y h a v e a n g u l a r d e p e n d e n c e . I n p r i n c i p l e ,t h e r e i s n o a p r i o r i r e a s o n t o a s s u m e e = a u n d e r n o n e q u i l i b r i u m c o n d i t i o n s . Ho we v e r ,i n th e f r a m e w o r k o f [1 ], it wa s p r o v e d t h a t e = a s ti ll h o l d s i n n o n e q u i l i b r i u m c o n d i t io n sp r o v i d i n g h y p o t h e s e s " a - b " a r e s a t is f ie d . I t h a s to b e e m p h a s i z e d t ha t , wh e n t h e


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210 A. M artl, G.L Arafijo / Solar Energy Ma terials and Solar C ells 43 (1996 ) 203-222c 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 o f th e s o l a r c e l l is w r i t t e n a s in E q s . ( 1 1 ) t o ( 1 3 ) , p h o t o nr e c y c l i n g e f f e c t s h a v e a l r e a d y b e e n t a k e n i n t o a c c o u n t .

G i v e n t h e s o la r s p e c tr u m , t h e u s a g e o f o n l y o n e c e ll d o e s n o t p r o v i d e t h e m a x i m u mp o s s i b l e e f f ic i e n c y s i n c e, f o r e x a m p l e , n o p o w e r i s ex t r a c te d f r o m p h o t o n s w i t h e n e r g i e sl o w e r t h a n t h e b a n d g a p e n e r g y . T w o s o l a r c e ll s c a n b e t t e r m a t c h t h e s o l a r s p e c t r u m , a n dw h e n t h e s c h e m e i s t a k e n t o th e l im i t , a n i n fi n i te n u m b e r o f c e l ls p r o v i d e s t h e b e s ta p p r o a c h [ 1 9 - 2 1 ] .

F i g . 4 s k e t c h e s a s y s t e m t h a t , b a s e d o n s p e c t r a l s p l i t t i n g , c a s t s t h e s u n l i g h t b e t w e e n nc e ll s . T h e s p l it t er i s a s s u m e d t o b e f r e e o f l o s se s . E a c h c e ll i s c h a r a c t e r i z e d b y a n e n e r g yg a p E , > E , _ i a n d r e c e i v e s p h o t o n s w i t h e n e r g i e s i n b e t w e e n E n a n d E n + 1. I n t h el i m i t in g c a s e o f i n f i n it e l y m a n y c e l l s, t h e g a p o f e a c h c e l l is d i ff e r e n t f r o m t h e f o r m e r b ya n a m o u n t d E . T h e c e l l s a r e s u p p o s e d t o h a v e a p e r f e c t b a c k r e f l e c to r a n d t h e r e f o r e , n op h o t o n s a r e e m i t t e d f r o m t h e re a r s i d e o f th e c e ll s . F o r a c o m p l e t e g e n e r a l i z a ti o n , t h es y s t e m i s s u p p o s e d a l s o t o b e c a p a b l e o f p r o v i d i n g c o n c e n t r a t e d l ig h t to t h e c e ll s . I f Xi s t h e c o n c e n t r a t i o n , t h e i n v a r i a n c e o f ra d i a n c e [ 4] e s t a b l i s h e s t h a t p h o t o n s h a v e t oi m p i n g e o v e r t h e c e ll s w i t h i n a s e m i a n g l e 0 x s u c h t h a t

s in Z 0 x = X s i n 2 0 s , ( 1 4 )a s s u m i n g t h e c e ll s a r e s u r r o u n d e d b y a i r ( o f r e fr a c t i v e i n d e x o n e ) . T h e s y m b o l 0 s i s t h es e m i a n g l e w i t h w h i c h p h o t o n s f r o m t h e s u n r e a c h t h e E a r t h ( S e e F ig , 3 a ).

T h e c o n c e p t s o f a n g l e a n d e n e r g y r e s t ri c t io n o f t h e e m i t t e d p h o t o n s [ 1 - 1 4 ] w i l l b eu s e d h e r e . T h e y m e a n t h a t , i d e a l l y , i n o r d e r t o a c h i e v e t h e m a x i m u m e f f i c i e n c y , w ea d m i t t h e p o s s i b i l i t y t h a t th e e m i s s i v i t y ( a b s o r p t i v i t y ) o f t h e c e l l c a n b e t a i lo r e d i n t h es e n s e t h a t th e e m i s s i o n o f p h o t o n s w i t h a n a n g l e g r e a t e r th a n 0 x i s r e s t ri c t e d ( e = a = 0f o r 0 > 0 x ) , a s i s t h e e m i s s i o n o f p h o t o n s w i t h e n e r g i e s h i g h e r t h a n E + d E . C o n s i d e r -i n g e a c h i n d i v i d u a l c e l l, t h e re a r e n o p h o t o n s i m p i n g i n g w i t h th e s e a n g l e s a n d e n e r g i e s

20s

" ~ / ' ~ \ S p e c t r a l

~ 1+1C e l ln

Fig. 4. G eneral sketch of a multigap converter based on the spli t t ing of the solar spectrum. W hen the l imitingefficiency is discussed, 0E is m ade equa l to 0 .


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A. Mart[ , G.L. Ara{~jo Solar Energy Materials and Solar C ells 43 (1996) 203-222 211so the re i s no ne ed to hav e an ab so rp t iv i t y ( em is s iv i ty ) equa l t o un i ty i n t hese r anges andi t can b e se t equa l t o ze ro . By do ing th i s , t he pho tog ene ra t ed cu r r en t o f t he ce l l is no ta f f ec t ed bu t , t he em is s ion o f pho ton s , and , t he re fo re , t he lo s ses a r e r educed , and thee f f i c i e ncy inc reased . E l sew here , t he abso rp t iv i t y ( emis s iv i ty ) w i ll be a s sum ed equa l t oun i ty . Reade r s i n t e r e s t ed i n a d i scuss ion abou t when the a s sumpt ion o f un i t abso rp t iv i t yd o e s n o t l e a d t o m a x i m u m e f f i c i e n c y c a n r e f e r to [ l] .

W i th t hese cons ide ra t i ons , and t ak ing the num ber o f ce i l s to i n f in i ty , i f V ( E ) is theope ra t ing vo l t age o f t he ce l l w i th gap E , t he e l emen ta l power ex t r ac t ed f rom th i s ce l lpe r un i t o f a r ea i s

d w ' = s i n2 O x f ( E , V ) d E , ( 1 5 )wi th

2 r r 1 1 V d E , ( 1 6 )- 1 e xp ~ - 1

where t he i n t eg ra t i on in t he so l id ang le has been ca r r i ed ou t a l r eady . The to t a l power i sob ta ined by in t eg ra t i ng (15 ) i n ene rg i e s :

w ' ; s in 2 O x f f ( E , V ) d E . ( 1 7 )Th e e f f i c i enc y , i n t he pho tovo l t a i c s ense , is ob t a ined by ap p ly ing de f in i t i on 6b .

T h e r e f o r e ,s i n 2 O x f y ( e , V ) d e f f ( E , V ) d e

= ( i s )"02 = Xsin Z0s o" Ts or T4 'wh ere Eq , (14 ) has bee n used . Th i s p rov es t ha t t he e f f i c i enc y i s i ndepe nden t o f t heconcen t r a t i on p rov id ing the ang le o f emis s ion o f pho tons i s r e s t r i c t ed t o t ha t o f t heincom ing l i gh t a s a s su m ed abov e (0 E = 0 x i n F ig . 3a ) .

Two d i f f e r en t app roaches have been used in t he pas t t o max imize (18 ) . I n t he f i r s tone , desc r ibed by De Vos and Pauwe l s [3 ] , f o r i l l u s t r a t i ve and s imp l i fy ing pu rposes , t heope ra t ing vo l t age o f each ce l l was l i nked to t he pho ton ene rgy in o rde r t o cha rac t e r i zet h e e m i s s i o n o f p h o t o n s f r o m t h e s y s t e m b y m e a n s o f a n e f f e c t i v e t e m p e r a t u r e T o :

E - q V Ek-----~c = k-~o. (1 9 )

Then , Eq . (18 ) r e su l t ed in Eq . ( I -2 ) in Tab le 1 and To w a s o p t i m i z e d t o m a x i m i z e th ee f f i c ie n c y . T h e r e s u l t w a s a n o p t i m u m e f f i c i e n c y o f 8 5 . 4 % [ 3] w h i c h c o r r e s p o n d s t o a ne q u i v a l e n t t e m p e r a t u r e To = TDV = 254 0K (Po in t D in F ig . 2 ) . Fo r t he i r com puta t ionsthey used max imum concen t r a t i on . In t h i s pape r , however , i t i s emphas i zed tha t i t i s no tneces sa ry t o a s sume tha t t he ce l l ope ra t e s a t t he max imum concen t r a t i on to ob t a in t hem a x i m u m e f f i c i e n c y p r o v i d i n g t h a t t h e a n g l e o f e m i s s i o n o f p h o t o n s i s r e st r ic t e d


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212 A. Martl, G.L. Arafijo / Solar Energy Materials and Solar Cells 43 (1996) 203-222( 0 E = 0 x ) . T h e r e f o r e , t h e f i g u r e 8 5 . 4 % i s i n d e p e n d e n t o f t h e c o n c e n t r a ti o n . I n a n y c a s e ,t h e s i m p l i f i c a t i o n i n E q . ( 1 9 ) a l l o w e d D e V o s a n d P a u w e l s t o c l e a r l y i l l u s t r a t e h o w t h ep h o t o v o l t a i c c o n v e r s i o n s h o u l d b e c o n s i d e r e d a s a n i r r e v e r s i b l e p r o c e s s e s a t t h e o p e r a t -i n g v o l t a g e th a t c o r r e s p o n d s t o th e m a x i m u m e f f i c i e n c y . W i t h s o m e a d d i ti o n a l c a l cu l a -t io n s i n v o l v i n g e n t r o p y f l u x c o n s e r v a t i o n , th e y c o m p u t e d t h e e n t r o p y g e n e r a t i o n S'g d u et o t he i r r eve r s i b i l i t i e s i n t he conve r s i on p r oces s es ( Eq . ( I - 1 ) i n Tab l e 1 , Eq . ( 31 ) i n Re f .[3 ]). T h e s e r e s u lt s h a v e a l s o b e e n i n c l u d e d i n th e s u m m a r y o f T a b l e 1 f o r c o m p a r i s o nw i t h t h e o t h e r t h e r m o d y n a m i c a p p r o a c h e s d i s c u s s e d i n S e c t i o n 1 . I n l a t e r d i s c u s s i o n s[ 9 - 1 1 ], th e n a t u r e o f t h e i r re v e r s i b il i ti e s w a s d e s c r i b e d m o r e a n d t h e y w e r e a s s o c i a t e d t ot h e c o u p l i n g o f t h e c o n v e r t e r t o th e e x t e r n a l w o r l d b y m e a n s o f th e i n t e r c h a n g e o f b o t he n e r g y a n d m a t t e r (endoreversiblep r o c e s s e s ) .

A s l i g h t l y m o r e c o m p l i c a t e d b u t o t h e r w i s e m o r e r e a l i s t i c a p p r o a c h t o m a x i m i z i n g( 18 ) , a l s o des c r i bed i n [ 3 ] , cons i s t s i n m ax i m i z i ng t he ope r a t i ng vo l t age f o r eachi nd i v i dua l ce l l s o t ha t V = V m ( E) w i t h V m ( E) s a t i s f y i ng

0-~ f ( E ,Vm(E))= 0 . ( 20 )C o m p u t i n g ( 1 8 ) w i t h V = V m ( E ), w e a l s o o b t a in e d t h e m a x i m u m e f f i c i e n c y a s 8 6 .8 %

w i t h t h e n o v e l t y t h a t t h i s f i g u r e i s i n d e p e n d e n t o f t h e c o n c e n t r a t i o n p r o v i d i n g t h a t0 E = 0 x , a s m e n t i o n e d a l r e a d y .

F o r t h e d i s c u s s i o n a b o v e , t h e s p l it ti n g s y s t e m s k e t c h e d i n F ig . 4 h a s b e e n u s e d a s t h es t a rt in g p o i n t f o r s i m p l ic i ty . S o m e o b j e c t i o n s t o t h e m o d e l c a n a r is e f r o m t h e f a c t th a t a nadd i t i ona l ex t e r na l e l em en t , t he s p l i tt e r , ha s be en i n t r od uced i n t he s y s t em and t ha t t h iss p l it te r h a s b e e n a s s u m e d t o b e f r e e o f l o ss e s . I t c a n b e p r o v e d ( s e e n e x t s e c t io n ) t h a t as t a c k e d s y s t e m a s t h e o n e s k e t c h e d i n F i g . 5 , w h i c h w i l l b e d i s c u s s e d i n t h e n e x t t w os e c t io n s , l e a d s to t h e s a m e e q u a t i o n s ( 1 5 t o 1 8 ) w h e n t h e n u m b e r o f c e l ls i n c r e a s e s t oi n f i n it y and t he r e f o r e , t o t he s am e va l ue o f t he l i m i t i ng e f f i c i ency w i t hou t t he neces s i t yo f a n e x t e r n a l e l e m e n t . H o w e v e r , t h i s s y s t e m h a s i n t e r es t in g f e a t u r e s re g a r d i n g t h el im i t i n g e f f i c i e n c y w h e n t h e n u m b e r o f c e ll s c o n s i d e r e d i s fi n it e .

4 . L i m i t i n g e f f i ci e n c y o f a sy s t e m w i t h a f i n it e n u m b e r o f ce l lsI n a s t a c k e d s y s t e m , N c e l l s a r e o r d e r e d b y d e c r e a s i n g g a p a s s k e t c h e d i n F i g . 5 a .

Th e f i r s t ( n = 1 ) i s t he ce l l w i t h t he l ow es t ban dga p . I t is , t he r e f o r e , p l ace d a t the bo t t omof t he s t ack . Sun l i gh t t r ave l s f r om l e f t t o r i gh t . A pe r f ec t back r e f l ec t o r i s p l aced a t t her ea r s i de o f t he l a s t ce l l t o p r even t em i s s i on l o s s es i n t ha t d i r ec t i on .

A t ce l l n + 1 , pho t ons w i t h e ne r g i e s l o w e r t han t he ga p o f t he ce l l, E n + l , a re a l l ow edt o p a s s t h r o u g h ; t h e n , t h e c e l l n a b s o r b s t h o s e p h o t o n s f r o m t h e s u n s p e c t r u m w i t he n e r g i e s E , + l > E > E , . W e w i ll a n a l y z e n o w t w o d i f f e r e n t c a s e s d i f f e ri n g in t h e w a yt he r ad i a t i ve l o s s es a r e hand l ed .

I n c a s e " a " , F i g . 5 a, d u e to ra d i a t i v e r e c o m b i n a t i o n , c e l l n e m i t s p h o t o n s p r o d u c i n gs o m e e n e r g y l o s s e s f o r t h e c el l. H o w e v e r , c e l l n + 1 a n d c e l l n - 1 b e n e f i t f r o m t h e s el um i nes c enc e l o s s es in ce l l n s i nce ce l l n - 1 can co l l ec t a l l t he pho t ons t ha t ce l l n


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A. MarE, G.L . Ara{i jo / Solar Energy Mater ia l s and Solar Ce ll s 43 (1996) 20 3-2 22 213( a )

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c e l l N n + l n n -1 1Fig. 5. Sketch of a stacked multigap system. In (a) no reflector is placed at the rear side of t h e c e l l s and so,photons emitted from o n e c e l l can be absorbed by the others. In (b) a reflector is placed at the rear side. In thiscase, although t h e c e l l s d o n o t absorb photons from their neighbours, the radiative losses in each individualc e l l also decreases, In text it is discussed which approach give the maximum efficiency.

e m i t s f r o m t h e r e a r s i d e a n d c e l l n + 1 c a n c o l l e c t t h e p o r t i o n o f t h e p h o t o n s e m i t t e df r o m t h e f r o n t s id e o f c e l l n w i t h e n e r g i e s h i g h e r t h a n E n 1. T h i s e f f e c t i s c a l l e dradiative coupling b e t w e e n t h e c e l l s . T h e n , o n l y t h e p o r t i o n o f th e e m i t t e d p h o t o n s w i t he n e r g i e s E n _ i < E < E , a r e w a s t e d .

I n c a s e " b " , a r e f l e c t o r is p l a c e d a t th e r ea r s i d e o f t h e c e l l s ( F i g . 5 b ) . T h e e m i s s i o nT o ss e s o f c e l l n w i l l b e r e d u c e d . H o w e v e r , c e l l s n - 1 a n d n + 1 w i l l n o t b e n e f i t f r o m


12/20

2 1 4 A . M a r t f , G . L . A r a f i j o / S o l a r E n e r g y M a t e r i a l s a n d S o l a r C e l l s 4 3 ( 1 9 9 6 ) 2 0 3 - 2 2 2

t h e l u mi n e s c e n t p h o t o n s c o mi n g f r o m c e l l n a s i n t h e p r e v i o u s c a s e . T h e f o l l o wi n gq u e s t i o n a r i s e s : h a s t h i s c o n f i g u r a t i o n a h i g h e r l i mi t i n g e f f i c i e n c y t h a n t h e p r e v i o u sc o n f i g u r a t i o n ? T h e a n s we r i s " y e s " a s i t wa s a n t i c i p a t e d i n Re f . [ 2 2 ] f o r a s y s t e m wi t ht wo g a p s . T h e d i s c u s s i o n i s g e n e r a l i z e d h e r e t o a h i g h e r n u mb e r o f g a p s a t t h e t i me t h a tnew resu l t s a re p resen ted .T o d i s c u s s t h e s e c o n f i g u r a t i o n s s i mu l t a n e o u s l y , we d e f i n e t h e a p p r o p r i a t e s t e pfunc t ion to desc r ibe the re f l ec to r a t the r ea r s ide o f ce l l n :

1 (O) E > E . ( 2 1 )rn = E < E .

T h e v a l u e " 1 " d e s c r i b e s t h e s i t u a t i o n f o r c a s e b , i n wh i c h r e f l e c t o r s a r e i n t r o d u c e d ,wh i l e " 0 " c o r r e s p o n d s t o c a s e a. T h e a b s o r p t i v i t y ( e mi s s i v i t y ) o f c e l l n i s g i v e n a l s o b ya s tep func t ion .

[ 1 E > E,~ ( 2 2 )a , = E < E nC o n s i d e r i n g t h e d i ff e r e n t s o u r c e s f o r th e p h o t o n r a d i a n c e t h a t c e l l n r e c e i v e s a n d i ts

l u m i n e s c e n t l o s s e s , t h e m a x i m u m p o w e r d e li v e r e d b y ce l l n i s g i v e n b y :

P . = q r r s i n Z O x V m , . f E ( b s , " + b ar n ( V m , . + l ) + b f . ( V m , . ) + 1 )+ b L . ( V m , . ) - 2 b E ( V m . . ) ) d ,

( 2 3 )w h e r e I ' m ,x i s t h e o p t i mu m o p e r a t i o n v o l t a g e i n c e l l x t o a c h i e v e t h e ma x i mu m

T a b l e 2D e s c r i p t i o n o f t h e d i f f e r e n t s p e c tr a l p h o t o n r a d i a n c e i n v o l v e d i n a s t a c k e d s y s t e m o f c e l l sN o t a t io n V a l u e M e a n i n gb s . n ( a n - a , , + i ) b sb JR , . ( V , , ) r , ,+ , a . b ( V , , , )b ~ . . (V , , ) r , , a n b ( V , )b ~ , . (V ~ + i ) a . ( 1 - r . + l ) a , , + t b ( V . + l )b ~ , . . ( V . _ 1 a , , (1 - r . ) a n _ l b ( V . _ l )

2 E 2b ( V . ) a , , h 3 c 2 e x p ( ( E - e V . ) / k r c ) - 1b f . ( v . ) (1 - a . + i ) a . b ( V . )

C o m i n g f r o m t h e s u n a n d a b s o r b e d i n c e l l nE m i t t e d f r o m c e l l n , r e f l e c t e d in r e f l e c t o r n + 1a n d a b s o r b e d a g a i n i n c e l l nE m i t t e d f r o m c e l l n , r e f l e c t e d i n r e f le c t o rn a n d a b s o r b e d a g a i n i n c e ll nE m i t t e d f r o m c e l l n + 1 a n d a b s o r b e d a g a i n i n c e l l nE m i t t e d f r o m c e l l n - 1 a n d a b s o r b e d a g a i n i n c e l l nE m i t t e d f r o m c e l l n . b i a s e d a t V ,,R a d i a t i v e l o s s e s in c e l l n n o t a b s o r b e d i n a n y c e l l

( N o t e : B e c a u s e c e l l s w i t h n = N + l a n d n = 0 d o e s n o t e x is t, i t i s n e c e s s ar y t o m a k e a N + l = 0 , a 0 = 1 ,V N + i = 0 a n d V0 = 0 t o g e n e r a l i z e t h e e q u a t i o n s a b o v e )


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A. M arff, G .L. Ara{tjo /S ol ar Energy Materials and Solar Ceils 43 (1996) 20 3-2 22 2 1 5e f f i c i e n c y . N o t i c e t h a t in E q . ( 2 3 ) , t h e c o n c e p t o f a n g l e r e s tr i c ti o n h a s b e e n a p p l i e d . T h em e a n i n g o f th e s p e c t ra l p h o t o n r a d i a n c e s b ( V m , . , ) i s s u m m a r i z e d i n T a b l e 2 .

A f t e r s o m e m a n i p u l a t i o n , a n d t a k i n g i n t o a c c o u n t t h a t a , a , + 1 = a . , a , + i r,, + 1 = r,, ~ ra n d a , , r , , + l = r , , + l a n d t h e e q u a t i o n s i n T a b l e 2 , E q . ( 2 3 ) c a n b e r e a r r a n g e d a s :

P ,, = q r r s i n 2 O x V . f E ( ( a , - - a . + , ) b s + [ a , , + l b ( V m . . + l) - a . b ( V m . , , ) ]

- r . + 1 b ( V m . n + , ) - b ( V m , . ) ] + a ~ (1 - r . ) [ b ( V m . , , _ 1 ) - b ( V m . . ) ] ) d E .( 2 4 )

T o a c h i e v e t h e m a x i m u m e f f i c i e n c y , t h e b a n d g a p o f th e c e ll s w i ll s ti ll r e q u i r eo p t i m i z a t i o n . F i g . 6 p l o t s th e o p t i m u m e f f i c i e n c y t h a t r e s u l ts a f t e r t h e o p t i m i z a t i o n o fb a n d g a p s a n d v o l t a g es . F o r c o m p a r i s o n , w e a l s o p lo t th e m a x i m u m e f f ic i e n c ie s w h e nt e r re s t ri a l A M 1 . 5 d i r e c t n o r m a l i r r a d i a n c e [ 2 3 ] i s c o n s i d e r e d ( i n s t e a d o f b l a c k b o d yr a d i a t i o n a t Ts = 6 0 0 0 K ) a s c h a r a c t e r i z i n g t h e i ll u m i n a t i o n f r o m t h e s u n . A l s o , t h ev a l u e s c o r r e s p o n d i n g t o X = 1 s u n a n d v a l u e s w i t h o u t r e s tr i c ti n g t h e a n g l e o f e m i s s i o n( 0 z = 9 0 ) h a s b e e n p l o tt e d . F o r r e f e r e n c e , T a b l e 3 s u m m a r i z e s t h e c o r r e s p o n d i n gv a l u es . T h e f o l l o w i n g r e m a r k s c a n b e p o i n t e d o u t:

( a ) A s i t c a n b e o b s e r v e d , b o t h a p p r o a c h e s ( w i t h a n d w i t h o u t r e f l e c to r s ) g i v e t h es a m e r e s u l t w h e n t h e n u m b e r o f c e l ls i n c r e a s e t o w a r d s i n f in i ty . T h i s r e su l t ca n b ee x p e c t e d f r o m E q . ( 2 4 ) s i n c e , w h e n n u m b e r o f c e l l s te n d s t o i n f i n i ty , V m ,.+ 1 - V m ,~ ~ 0 .E q . ( 2 4 ) c a n b e w r i t t e n a s :

d P ( E ) = q ~ s i n 2 0 x bs(E) 2 E 2 1h 3 c e E - q V m ( E )e x p k Td E , ( 2 5 )

w h i c h i s in d e p e n d e n t o f w h e t h e r r e f l e c t o r s a re u s e d o r n o t. N o t i c e t h a t E q . ( 2 5 ) i si d e n t ic a l t o E q . ( 1 5 ) w i t h P p l a y i n g t h e r o l e o f w ' a n d , th e r e f o r e , t h e v a l u e f o r t h el i m i t in g e f f i c i e n c y f o r a n in f i n i te n u m b e r o f g a p s , 8 6 . 8 % f o r t h e s u n a s s u m e d a s a b l a c kb o d y a t T s = 6 0 0 0 K , i s o b t a i n e d a s b e f o re . T h e f i g u r e c o r r e s p o n d i n g t o A M 1 . 5 d i r ec tn o r m a l i r r a d i a n c e [ 2 3 ] h a s b e e n c o m p u t e d h e r e f o r th e f ir s t t i m e t o o u r k n o w l e d g e . I tg i v e s 8 5 . 0 % . L i n e a r i n t e r p o l a t io n o f th e d a t a i n R e f . [ 2 3] a n d t r a p e z o i d a l i n t e g r a t i o nh a v e b e e n u s e d i n t h e c o m p u t a t i o n s .

( b ) H o w e v e r , w h e n t h e n u m b e r o f c e l ls i s f i n it e , t h e a p p r o a c h w i t h r e f l e c to r s g i v e s as l i g h t l y h i g h e r e f f i c i e n c y t h a n t h e a p p r o a c h w i t h o u t r e f l e c t o r s t h a t i n R e f . [ 2 0 ] w a sa s s u m e d t o le a d t o t h e m a x i m u m e f f ic i e n c y . I f w e a s s u m e t h a t t he o p t i m u m v o l t a g e ss a t i s f i e s V m . n + ~ > Vm,n, t h i s r e s u l t c a n b e p r o v e d a s f o l l o w s :

I f t h e s y s t e m c o n s i s t s o f N c e l l s a n d h a s r e f le c t o r s , t h e p o w e r d e l i v e r e d i s g i v e n b y :N

P w R ( V m ) = ~ _ , P ~ ( V m ) , ( 2 6 )n= l

w h e r e Vm r e p re s e n t s t h e s e t o f o p t i m u m v o l t a g e s th a t m a x i m i z e s t h e p o w e r a n d P,', i s


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2 16 A. Marti, G.L. Ara{~jo Solar Energy Materials and Solar Cells 43 (1996) 203-222( a )

O ~ t imu m e f f ic i e n cy (%)9 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 E = e x8 0 i nf in i te n u m b e r o f g a p sX = I s u n e E = 9 oo ~ /

7 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 0 ~6 04 f f

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~ . ~ X = l s u n~ - ~ e E = 9 0 o

( B l a ck b o d y 6 0 0 0 K ), I I

2 3N u m b e r o f g ap s

( b )Op t i mu m e f f i c i e n cy ( %)9 0

8 0 _-0 E =0 x ~ ' i n f in it e n u m b e r o f g a p s7 0 - X = 1 s un ( ~ E = 9 0 / . . . . . .

/

. . . . . . . - 2 == . . . . . . . . . . . . . /6 0 ~ "

6 o , . . . . i- " " ~ - =1 s u n i4 0 - . . - 0 E = 9 0 o

3 0 (AM1.5 no rma l d i rec t )20 L2 3

N u m b e r o f g a p sF i g . 6 . M a x i m u m e f f i c i e n c y a s a f u n c t i o n o f t h e n u m b e r o f c e l l s: ( a ) f o r t h e s u n a s s u m e d a s a b l a c k b o d y a t6 0 0 0 K , ( b ) f o r A M 1 . 5 d i r e c t n o r m a l ir r a d i a n c e [ 23 ]. I n b o t h ca s e s , d o t s m a r k e d a s " + " c o r r e s p o n d s to t h ec a s e w i t h r e f l e c t o r s ( i l lu s t r at e d i n F i g . 5 a ) a n d d o t s m a r k e d a s " X " t o t h e c a s e w i t h o u t r e f l e c to r s (F i g . 5 b ) .B o t h v a l u e s a r e v e r y c l o s e b u t t h e o n e c o r r e s p o n d i n g t o t h e c a s e w i t h r e f l e c to r s i s s li g h t ly h i g h e r .

g i v e n b y E q . ( 2 4 ) w i t h t h e a p p r o p r i a t e s v a l u e s f o r r ( E q . ( 2 1 )) . C o n v e r s e l y , i f t h es y s t e m h a s n o r e f l e c to r s , t h e m a x i m u m d e l i v e r a b l e p o w e r i s g i v e n b y :

N

PNR(Vm) = ~ P . ( V m ) , ( 2 7 )n= l


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A. Mar t f, G .L . Ara{t jo / Solar Energy Mater ia l s and Solar Ce l ls 43 (1996) 2 03 -22 2 2 1 7T a b l e 3O p t i m u m e f f ic i e n cy a n d b a n d g a p s f o r m u l t ig a p s y s t e m s w i t h d i ff e r e n t n u m b e r o f c e ll s f o r t h e s un a s s u m e d a sa b l a c k b o d y a t 6 0 0 0 K a n d f o r A M 1 . 5 d i r e c t n o r m a l i r r a d i a n c e [ 2 3] (t a b l e b e l o w ) . T h e v a l u e s o f t h e e f f i c i e n c ya r e p l o tt e d i n F ig . 6 . " R e f l e c t o r s " m e a n s w h e t h e r b a c k r e f l e c t o r s h a v e b e e n p l a c e d a t t h e r e a r s i d e o f a ll th ec e l l s in t h e s t a c k o r no t . L a s t c e ll in th e s t a c k a l w a y s h a s a b a c k r e f l e c t o r . T h e c a s e l a b e l e d a s " ' M a x i m u mc o n c e n t r a t i o n " i s e q u i v a l e n t a l s o to t h e c a s e o f o p e r a t i o n a t o n e s u n w h e n t h e a n g l e o f th e e m i t t e d p h o t o n sf r o m t h e c e l l is r e s tr i c t e d to t h a t o f t h e i n c o m i n g p h o t o n s f r o m t h e s u nN . c e ll s D e s c r i p ti o n R e f l e c to r s ? O p t i m u m g a p s ( e V ) E f f ( % )

E l E 2 E 3 E 4( B l a c k b o d y a t 6 0 0 0 K )

1 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 1 . 3 1 - - - 3 t . 01 1 s u n n o a n g u l a r r e s t i c t i o n N o 1 .3 1 - - - 3 1 . 0

M a x i m u m c o n c e n t r a t i o n Y e s 1 .1 1 - - - 4 0 . 8M a x i m u m c o n c e n t r a t i o n N o 1 .1 1 - - - 4 0 . 81 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 9 8 1 . 8 7 - - 4 2 . 9

2 1 s u n n o a n g u l a r r e s t i c t i o n N o 0 . 9 8 1 .8 8 - - 4 2 . 7M a x i m u m c o n c e n t r a t i o n Y e s 0 . 7 7 1 . 70 - - 5 5 . 9M a x i m u m c o n c e n t r a t i o n N o 0 . 7 8 1 .7 1 - - 5 5 . 61 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 8 2 1 . 4 4 2 . 2 6 - 4 9 . 3

3 1 s u n n o a n g u l a r r e s t i c t i o n N o 0 . 8 3 1 . 4 5 2 . 2 6 - 4 9 . 1M a x i m u m c o n c e n t r a t i o n Y e s 0 . 6 2 1 . 26 2 . 1 0 - 6 3 . 8M a x i m u m c o n c e n t r a t i o n N o 0 . 6 3 1 . 27 2 . 11 - 6 3 . 51 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 7 2 1 .2 1 1 . 7 7 2 . 5 5 5 3 . 3

4 l s u n n o a n g u l a r r e s t i c t i o n N o 0 . 7 3 1 . 2 3 1 . 7 8 2 . 5 6 5 3 . 0M a x i m u m c o n c e n t r a t i o n Y e s 0 . 5 2 1 .0 3 1 .6 1 2 . 41 6 8 . 8M a x i m u m c o n c e n t r a t i o n N o 0 . 5 3 1 . 05 1 . 68 2 . 41 6 8 . 4I s u n , n o a n g u l a r r e s t r i c t i o n Y e s . . . . 6 9 . 9

:e 1 s u n n o a n g u l a r r e s t i c t i o n N o . . . . 6 9 . 9M a x i m u m c o n c e n t r a t i o n Y e s . . . . 8 6 . 8M a x i m u m c o n c e n t r a t i o n N o . . . . 8 6 . 8

( A M 1 . 5 d i r e c t n o r m a l i r r a d ia n c e )1 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 1 . 1 3 - - - 3 2 . 5

1 1 s u n n o a n g u l a r r e s t i c t i o n N o 1 . 1 3 - - - 3 2 . 5M a x i m u m c o n c e n t r a t i o n Y e s 0 . 9 4 - - - 4 4 . 6M a x i m u m c o n c e n t r a t i o n N o 0 . 9 4 - - - 4 4 . 61 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 9 4 1 . 6 4 - - 4 4 . 3

2 1 s u n n o a n g u l a r r e s t i c t io n N o 0 . 9 4 1 .6 4 - - 4 4 . 1M a x i m u m c o n c e n t r a t i o n Y e s 0 . 7 1 1 .4 1 - - 5 9 . 7M a x i m u m c o n c e n t r a t i o n N o 0 . 71 1 .4 1 - - 5 9 . 41 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 7 1 1 . 1 6 1 . 8 3 - 5 0 . 13 1 s u n n o a n g u l a r r e s t i c t i o n N o 0 . 7 1 1 . 1 6 1 .8 3 - 4 9 . 7M a x i m u m c o n c e n t r a t i o n Y e s 0 . 6 9 1 . 16 1 . 84 - 6 7 . 0M a x i m u m c o n c e n t r a t i o n N o 0 . 6 9 1 . 16 1 . 83 - 6 6 . 61 s u n , n o a n g u l a r r e s t r i c t i o n Y e s 0 . 7 1 1 . 13 1 . 55 2 . 1 3 5 4 . 0

4 1 s u n n o a n g u l a r r e s t i c t i o n N o 0 . 7 1 1 . 1 3 1 . 5 5 2 . 1 3 5 3 . 6M a x i m u m c o n c e n t r a t i o n Y e s 0 . 5 3 1 .1 3 1 . 55 2 . 1 3 7 1 . 0M a x i m u m c o n c e n t r a t i o n N o 0 . 5 3 1 .1 3 1 .5 5 2 . 1 3 7 0 . 71 s u n , n o a n g u l a r r e s t r i c t i o n Y e s . . . . 6 5 . 4

:c 1 s u n n o a n g u l a r r e s t i c t i o n N o . . . . 6 5 . 4M a x i m u m c o n c e n t r a t i o n Y e s . . . . 8 5 . 0M a x i m u m c o n c e n t r a t i o n N o . . . . 8 5 . 0


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218 A. Martf, G.L. Ara~jo Solar Energy Materials and Solar Cells 43 (1996) 203-222Vm b e i n g t h e o p t i mu m s e t o f v o l t a g e s f o r t h e n e w a r r a n g e me n t a n d P m g i v e n b y E q .(24) and wi th r = 0 in Eq . (21 ) . No t ice tha t , in genera l , Vm ~ Vm and , the re fo re ,

PwR( Vm) ~ PwR( Vm), ( 2 8 )

( a )O p t i m u m g a p s ( e V )3

( b )

I2 3N u m b e r o f g a p s

O p t i m u m g a p s ( e V )2 .52- - (AM 1-15 .nrma l d i rect ) t "

i I" iT , . . . . . . . . . . . . . . . . . . . . . . . .0 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . ~- -

0 I I I i1 2 3 4N u m b e r o f g a p s

Fig. 7. (a) Optimum gaps as a function of the number of cells in the stack for the sun assumed as a black-bodyat Ts = 600OK. Reflectors at the rear side of each cell as described in Fig. 5b have been assumed. Maximumconcentration (or, equivalently, total restriction of the photons emitted from the cell) has been considered. Barsindicate the tolerance in the value of the optimum gap to keep the m aximum efficiency not less than 1% of itsmaxim um value when the other values of the gaps remain constant and equal to their optimum value (squaredots). (b) The same for AM1.5 direct normal irradiance.


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A. M artl, G.L. Arat~jo Solar Energy M aterials and Sola r Cells 43 (1996) 203-22 2 219s i nce Vm a r e no t t he op t i m um vo l t ages f o r t he con f i gu r a t i on w i t h r e f l ec t o r s . A d i r ec tc a l c u l a t i o n s h o w s t h a t

P w R ( V m ) - P N R ( V m ) =e T r s in a O x f E { r l V m , ] (b E . 1 ( V m . 1 ) - b ( O ) )

+ E r n ( b E , n - l ( g m . . - l ) - - b E ,n ( V m , n ) )( V m . n - I - - V n a , . ) ) d E , ( 2 9 )n = 2and , there fore , bec au se Vm, > Vm. _ 1

P w R ( Vm ) >-- PN R( Vm )' ( 3 0 )w hi ch t oge t he r w i t h Eq . ( 28 ) l eads t o

Pw R( Vm ) >__P N R ( V m ) , ( 3 1 )a s w e w a n t e d t o s h o w .

( c ) T h e s e n s i t i v i t y o f t h e m a x i m u m e f f i c i e n c y w i t h r e s p e c t t o t h e v a l u e o f t h eo p t i m u m b a n d g a p s i s lo w e r a s th e b a n d g a p o f t h e c e ll in t h e s t a c k is h ig h e r . I n o t h e rw o r d s , t h e e f f i c ie n c y i s m o r e a n d m o r e s e n s i ti v e t o th e v a l u e s o f th e g a p o f t h e c e ll s a sw e c o n s i d e r t h e c e l l s t o w a r d s t h e b o t t o m o f t h e s t a c k. T h i s w a s p o i n t e d o u t i n R e f . [ 24 ]f o r a s y s t em w i t h t w o ce l l s and i t is i l lu s t r a t ed he r e , i n F i g . 7 , f o r a s y s t em up t o 4 ce l l s .T h e s e n s i t i v i t y t o t h e b o t t o m c e l l g a p i s h i g h e r w h e n A M 1 . 5 d i r e c t n o r m a l i r r a d i a n c es p e c t r u m i s c o n s id e r e d , i n s t e a d o f b l a c k b o d y r a d i a t io n a t 6 0 0 0 K a s i ll u s tr a te d i n F i g.7b .

I r r a d i a n c e ( W m 2 m i c "1)1 ,2 0 01,10 0 2 ,1 eV

io i j0 04 0 020 0l o o L I ,

- i0 i ~ r . I0 .2 0 .6

( d a s h e d : 1 s u n o p t i m u m g a p sf o r a s y s t e m o f f o u r c e l l s )1,13 eV

II,,I

,,

0,71 eV

,

1 1 .4 1 .8 2 .2 2 .6 3 3 .4 3 .8W a v e l e n g t h ( m i c )

4 .2

F ig . 8 . P lo t o f the da ta cor re s pon ding to t e r re s tr i a l d i rec t norm al so la r spec t ra l i r rad iance fo r a i r m ass 1 .5 [23] .T h e v e r t ic a l l in e s s h o w t h e o p t i m u m g a p s f o r a s y s t e m o f f o u r c e l l s at o n e s u n .


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220 A. Mar t l, G .L . Araf i jo / Solar Energy Mater ia l s and So lar Ce ll s 43 (1996) 20 3-2 22( d ) W h e n t h e s u n i s a s s u m e d a s a b l a c k b o d y a t 6 0 0 0 K , t h e v a l u e s o f th e o p t i m u m

g a p s d e p e n d o n t h e c o n c e n t r a t i o n ( i f n o a n g u l a r r e s t r i c t i o n i s p r o v i d e d ) a n d a l s o o nw h e t h e r r e f l e c t o r s h a v e b e e n p l a c e d a t t h e r e a r s i d e o f t h e c e l l s . T h e n u m e r i c a l r e s u l t ss h o w t h a t t h e o p t i m u m g a p s s h i f t t o w a r d s v a l u e s a b o u t 1 0 m e V h i g h e r w h e n n or e f l e c t o r s a r e p l a c e d i n c o m p a r i s o n w i t h t h e c a s e w i t h r e f l e c t o r s . A l s o , w h e n v a l u e s a to n e s u n a r e c o m p a r e d w i t h v a l u e s a t m a x i m u m c o n c e n t r a t i o n ( o r t o t a l a n g u l a r r e s t r i c -t i o n ) t h e o p t i m u m g a p s s h i f t s t o w a r d s v a l u e s i n t h e r a n g e o f 1 0 0 - 2 0 0 m e V h i g h e r .H o w e v e r , w h e n A M 1 . 5 d i r e c t n o r m a l i r r a d i a n c e i s c o n s i d e r e d , t h e r e i s n o s u c h s h i f tw hen t he cas es w i t h r e f l ec t o r s and w i t hou t r e f l ec t o r s a r e com par ed . A l s o , i n t he s t ackeds y s t e m , w h e n m o r e t h a n o n e c e l l is c o n s i d e re d , o n l y th e o p t i m u m g a p o f th e b o t t o m c e l li s s ens i t i ve t o concen t r a t i on . D ue t o t he va l l eys and c r e s t s t ha t appea r i n t he A M 1. 5d i r e c t n o r m a l i r ra d i a n c e a s a r e s u l t o f th e a t m o s p h e r i c a l a b s o r p t i o n , o p t i m u m g a p s s e e m st o be l ocked t o t hem as i l l u s t r a t ed i n F i g . 8 .

5 . C o n c l u s i o n sT h e l i m i t i n g e f f i c i e n c y o f p h o t o v o l t a i c e n e r g y c o n v e r s i o n i s 8 6 . 8 % f o r t h e s u n

a s s u m e d a s b l a c k b o d y a t 6 0 0 0 K , 8 5 . 0 % i f A M 1 . 5 d i r e c t n o r m a l i r ra d i a n c e i s c o n s i d -e r e d . T h e s e r e s u l t s a r e i n d e p e n d e n t o f t h e c o n c e n t r a t i o n p r o v i d i n g t h e e m i s s i o n o fp h o t o n s c a n b e r e s tr i c te d i n a n g l e t o t h a t s t ri c tl y n e c e s s a r y t o r e c e i v e t h e p h o t o n s f r o mt he s un . Th i s l i m i t i s a t t r i bu t ed t o a s y s t em cons i s t i ng o f an i n f i n i t e num ber o f ce l l s . Thef i g u r e s t h a t r e s u l t d i f f e r f r o m t h e o n e s o b t a i n e d f r o m s t a n d a r d t h e r m o d y n a m i c a p -p r o a c h e s ( 9 3 . 3 - 9 5 % ) d u e t o t h e d i f f e r e n t a s s u m p t i o n s a n d e v e n t h e d e f i n i t i o n o fe f f i c i ency i t s e l f t ha t i s u s ed ( s ee Tab l e 1 ) . Fo r s ys t em s cons i s t i ng o f a s t ack o f a f i n i t en u m b e r o f c e ll s , th e u s e o f b a c k r e f le c t o r s i n c r e a s e s t h e l i m i ti n g e f f i c ie n c y s l i g h tl y w h e nc o m p a r e d t o t h e s i t u a ti o n w i t h o u t r e f l e c to r s . T h e s e n s i ti v i ty o f t h e o p t i m u m e f f ic i e n c y i sh i g h e r f o r th e b o t t o m c e l ls o f t h e s t a ck . C a l c u l a t io n s i n v o l v i n g A M 1 . 5 d i r ec t n o r m a li r r a d i a n c e h a v e b e e n i n c l u d e d a n d s h o w s o m e p e c u l i a r i t i e s w h e n c o m p a r e d w i t h r e s u l t sc o r r e s p o n d i n g t o b l a c k b o d y r a d i a t i o n a t 6 0 0 0 K : t h e s e n s i t i v i t y o f t h e e f f i c i e n c y t o t h ev a l u e o f t h e g a p o f th e b o t t o m c e l l in c r e a s e s a n d t h e v a l u e o f th e o p t i m u m g a p f o r c e ll so t he r t han t he bo t t om one does no t s h i f t s i gn i f i can t l y w i t h concen t r a t i on .

6 . L i s t o f sy mb o lsE~E3E~E~s~s~s~Q,W

E n e r g y f l u x t h a t t h e c o n v e r t e r r e c e i v e s f r o m t h e s u n .E n e r g y f l u x e m i t t e d f r o m t h e c o n v e r t e r t o t h e s i n k .I n t e r na l ene r gy va r i a t i on r a t e i n t he conve r t e r .E n e r g y g a p o f c e l l n t h i n a s ta c k e d s y s t e m .E n t r o p y f l u x t h a t t h e c o n v e r t e r r e c e i v e s f r o m t h e s u n .E n t r o p y f l u x e m i t t e d f r o m t h e c o n v e r t e r t o t h e s i n k .I n t e r na l en t r opy va r i a t i on r a t e i n t he conve r t e r .R a t e o f h e a t c o n d u c t i o n o u t o f c o n v e r t e r .W o r k p r o d u c e d p e r u n i t o f t im e ( p o w e r) .


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A, M ar tf , G .L . Arat~jo / Solar Energy M ater ia l s and So lar Ce ll s 43 (1996) 2 03 -22 2 221Tc C o n v e r t e r t e m p e r a t u r e .Ts T e m p e r a t u r e o f t h e s u n.To E f f e c t i v e t e m p e r a t u r e f o r c h a r a c t e r i z i n g t h e p h o t o n s e m i t t e d f r o m t h e c o n v e r t e r .

S t e f a n - B o l t z m a n c o n s t a n t .X L i g h t c o n c e n t r a t i o n .n R e f r a c t i o n i n d e x .0 s S e m i a n g l e w i t h w h i c h p h o t o n s f r o m t h e s u n r e a c h t h e E a r th .0 x M a x i m u m s e m i a n g l e w i t h w h i c h p h o t o n s i m p i n g e o n t h e c e ll .0 E M a x i m u m s e m i a n g l e f o r t h e p h o t o n s e m i t t e d f r o m t h e c e ll .0 A n g l e t h a t d e f i n e s r a y t r a j e c t o r i e s ( F i g . 3 a ) .

S o l i d a n g l e s u s t a i n e d b y 0 ."q~ E f f i c i e n c y ( t h e r m o d y n a m i c d e f i n it i o n ).~q2 E f f i c i e n c y ( p h o t o v o l t a i c d e f i n i t i o n ) .FABs N u m b e r o f p h o t o n s a b s o r b e d p e r u n i t o f t im e in t h e c e ll .FEM N u m b e r o f p h o t o n s e m i t t e d p e r u n i t o f t i m e f r o m t h e c e ll .a A b s o r p t i v i t y .e E m i s s i v i t y .V V o l t a g e .P w R P o w e r d e l i v e r e d b y a s y s t e m w i t h r e f le c t o r s.P NR P o w e r d e l i v e r e d b y a s y s t e m w i t h o u t r e f le c t o r s .q E l e c t r o n e l e c t ri c c h a r g e .h P l a n c k ' s c o n s ta n t .c S p e e d o f t h e l i g h t i n v a c u u m .k B o l t z m a n ' s c o n s t an t .

Acknow l e dg emen t sT h i s w o r k h a s b e e n p a r t i a l l y s u p p o r t e d b y t h e S p a n i s h P R O N T I C u n d e r c o n t r a c t

T I C - 0 7 7 8 - C 0 4 - 0 1 a n d " A c c i 6 n I n t e g r a d a " H B 9 4 0 2 9 . T h e a u th o rs w o u l d l ik e to th a n kD r . P . A . D a v i e s , M r . P . G r i f f in a n d t h e r e v i e w e r s f o r t h e i r c o m m e n t s a n d c a r e f u l re a d i n go f th e m a n u s c r i p t . A n t o n i o w i s h e s t o r e m e m b e r h i s t u to r , fr i e n d a n d c o a u t h o r o f th i sp a p e r , G e r a r d o , w h o d i e d o n 8 N o v e m b e r 1 9 9 5 .

References[1] G.L. Arafijo and A. MartL Ab solute limiting efficiencies for p hotovoltaic energy conve rsion, Sol. E nergyMater. Sol. Cells 33 (1994) 213-240.[2] P.T. Landsberg and G. Ton ge, Thermodynamic ene rgy conversion efficiencies, J . Appl. P hys. 51(7)(1980) RI-R 20.[3] A . D e V os and H. Pauw els, On the thermodinam ic limit of photovoltaic energy conversion, App l. Phys.25 (1981) 119-125.[4] W .R. M cCluney, Introduction to Radiometry and Photometry, Ch. 5 (Artech House, Boston-London,1994).[5] J.E. P arrott, Therm odynam ic limits to ph otovoltaic en erg y conversion, in: A. Luqu e and G .L. Arafijo(Eds.), P hysical Limitations to Photovo ltaic Energy C onversion (Adam Hilger, 1990),


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2 2 2 A. Mar t l, G .L . Araf i jo / Solar Energy Mater ia l s and So lar Cel ls 43 (1996) 2 03- 22 2[ 6 ] H . B . C a l l e n , T e r m o d i n f i m i c a , E d . A C , M a d r i d , 1 9 8 1 .[ 7 ] C . H . H e n r y , L i m i t i n g e f f i c i e n c i e s o f i d e a l s i n g l e a n d m u l t i p l e e n e r g y g a p t e r r e s t r i a l s o l a r c e l l s , J . A p p l .

P h y s . 3 2 ( 1 9 8 0 ) 5 1 0 - 5 1 9 .[ 8 ] J .E . P a r r o t t, T h e r m o d y n a m i c s o f s o l a r c e l l e ff i c ie n c y , S o l . E n e r g y M a t e r . S o l . C e l l s 2 5 ( 1 9 9 2 ) 7 3 - 8 5 .[ 9 ] A . D e V o s , I s a so l a r c e l l a n e n d o r e v e r s i b l e e n g i n e ? , S o l a r C e l ls 3 1 ( 1 9 9 1 ) 1 8 1 - 1 9 6 .

[ 1 0] A . D e V o s , E n d o r e v e r s i b l e T h e r m o d y n a m i c s o f S o l a r E n e r g y C o n v e r s i o n ( O x f o r d U n i v e r s i t y P r es s , N e wY o r k , 1 9 9 2 ) ,

[ 1 1] A . D e V o s , P . T . L a n d s h e r g , P . B a r u c h a n d J .E . P a r r o t t , E n t r o p y f l u x e s , e n d o r e v e r s i b i l i t i e s a n d s o l a re n e r g y c o n v e r s i o n , J . A p p l . P h y s . 7 4 ( 6 ) ( 1 9 9 3 ) 3 6 3 1 - 3 6 3 7 .

[ 1 2 ] W . S c h o c k l e y a n d H . J . Q u e i s s e r , D e t a i l e d b a l a n c e l i m i t o f e f f i c i e n c y o f p - n j u n c t i o n s o l a r c e l ls , J . A p p l .P h y s . 3 2 ( 1 9 6 1 ) 5 1 0 - 5 1 9 .

[ 1 3 ] J . E . P a r r o t t , S e l f c o n s i s t e n t d e t a i l e d b a l a n c e t r e a t m e n t o f t h e s o l a r c e l l , I E E E P r o c . 1 3 3 , P t . J . N o . 5( 1 9 8 6 ) 3 1 4 - 3 1 8 .

[ 1 4 ] G . L . A r a f i j o a n d A . M a r t l , G e n e r a l i z e d d e t a i l e d b a l a n c e t h e o r y t o c a l c u l a t e t h e m a x i m u m e f f i c i e n c y o fs o l a r ce l ls , P r o c . 1 l t h E u r o p e a n P h o t o v o l t a i c S p e c i a l is t s C o n f ., M o n t r e a u x , 1 9 9 2 , p p. 1 4 2 - 1 4 5 .

[ 15 ] P . T . L a n d s b e r g , H . N u s s b a u m e r a n d G . W i l l e k e , B a n d - b a n d i m p a c t i o n i z a ti o n a n d s o l a r c e l l e f f ic i e n c y , J.A p p l . P h y s . 7 4 ( 2 ) ( 1 9 9 3 ) 1 4 5 1 - 1 4 5 2 .

[ 1 6 ] J .H . W e r n e r , R . B r e n d e l a n d J . Q u e i s s e r , N e w u p p e r e f f i c i e n c y l i m i t s f o r s e m i c o n d u c t o r s o l a r c e l l s , 1 s tW o r l d C o n f . o n P h o t o v o l t a i c E n e r g y C o n v e r s i o n , H a w a i i , 1 9 9 4 .

[ 1 7 ] J . H . W e r n e r , S . K o l o d i n s k i a n d H . Q u e i s s e r , N o v e l o p t i m i z a t i o n p r i n c i p l e s a n d e f f i c i e n c y l i m i t s f o rs e m i c o n d u c t o r s o l a r c e l l s , P h y s . R e v . L e t t . ( 1 9 9 4 ) 3 8 5 1 - 3 8 5 4 .

[ 1 8 ] R . T . R o s s a n d J . N o z i k , E f f i c i e n c y o f h o t c a r r i e r s o l a r e n e r g y c o n v e r t e r s , J . A p p l . P h y s . 5 3 ( 5 ) ( 1 9 8 2 )3 8 1 3 - 3 8 1 8 .

[ 1 9 ] A . D e V o s , D e t a i l e d b a l a n c e l i m i t o f th e e f f i c i e n c y o f t a n d e m s o l a r c e l l s , J . P h y s . D : A p p l . P h y s . 1 3( 1 9 8 0 ) 8 3 9 - 8 4 6 .

[ 2 0 ] H . P a u w e l s a n d A . D e V o s , D e t e r m i n a t i o n o f t h e m a x i m u m e f f i c i e n c y s o l a r ce l l s tr u c t u re , S o l i d S t a teE l e c t r o n i c s 2 4 ( 9 ) ( 1 9 8 1 ) 8 3 5 , 8 4 3 .

[ 2 1 ] J . E . P a r r o t t , T h e l i m i t i n g e f f i c i e n c y o f a n e d g e i l l u m i n a t e d m u l t i g a p s o l a r c e ll , J . P h y s . D : A p p l . P h y s . 1 2( 1 9 7 9 ) 4 4 1 - 4 5 0 .[ 2 2 ] G . L . A r a f i jo a n d A . M a r t l , O n t h e d e t a i l e d b a l a n c e l i m i t o f i d e a l m u l t i p l e b a n d g a p s o l a r c e l ls , P r o c . 2 2 n dI E E E P h o t o v o l t a i c S p e c i a l i s t s C o n f . , N e w Y o r k , 1 9 9 1 , p p . 2 9 0 - 2 9 4 .

[ 2 3] E 8 9 1 - 8 7 , A S T M S t a n d a r d , 1 9 1 6 R a c e S t ., P h i l a d e l p h i a , P A 1 9 1 0 3 .[ 2 4] G . L . A r a f ij o a n d A . M a r t l , L i m i t i n g e f f i c i e n c ie s o f i d e a l m u l t ip l e b a n d g a p s o l a r ce l ls u n d e r c o n c e n t r a t e d

s u n l i g h t , P r o c . 9 t h E u r o p e a n P h o t o v o l t a i c S c i e n c e a n d E n g i n e e r i n g C o n f ., F r e i b u r g , 1 9 9 1 , pp . 1 1 1 - 1 1 4 .

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