Grain boundary effects in polycrystalline silicon solar cells I. Solution of the three-dimensional diffusion equation by the Green's function method

Solar Cells, 8 (1983) 201 - 223

201

GRAIN B O U N D A R Y E F F E C T S IN P O L Y C R Y S T A L L I N E SILICON SOLAR CELLS I: SOLUTION OF THE T H R E E - D I M E N S I O N A L D I F F U S I O N EQUATION BY THE G R E E N ' S FUNCTION METHOD

N. C. HALDER and T. R. WILLIAMS Department of Physics, University of South Florida, Tampa, FL 33620 (U.S.A.)

(Received March 22, 1982; accepted May 26, 1982)

Summary For the polycrystalline silicon solar cell problem, the three
1. Introduction Experimentally, single~rystal (or epitaxial) h o m o j u n c t i o n or heterojunction silicon has been successfully used [ 1 - 3] for the photovoltaic conversion of solar energy. However, because of the somewhat difficult and elaborate manufacturing process and the high production cost of singlecrystal silicon, polycrystalline silicon has been suggested [4 - 6] as a possible replacement with some sacrifice o f efficiency. Although recently, with the advent of new chemical vapor deposition doping techniques, amorphous silicon is also being studied [ 7 - 9] with great interest, polycrystalline silicon still retains its unique position in the selection of solar cell materials and has the best potential to become the most important and technologically viable material for solar cell application in the near future. On the theoretical side the mechanism of photovoltaic conversion, especially the efficiency, for single crystals has been well established [10] and in fact it is used t o d a y in the practical development o f single,crystal cells. For polycrystaUine cells, while there have been m a n y attempts [11 14] to understand the limiting conditions of the efficiency, efforts to achieve higher theoretical as well as practical limits are at present continuing. 0379-6787/83/0000-0000/$03.00

© Elsevier Sequoia/Printed in The Netherlands

202 The primary reason is that for single-crystal silicon the original three
2. Green's function method for polycrystalline solar cells

2.1. Models for polycrystalline n - p junction cells As pointed o u t earlier, a polycrystalline semiconductor material is made up of many individual grains. The grain orientation can be random as in Fig. l(a) or can be relatively ordered as in the fibrous orientation of Fig. l(b). The grain orientation is important because at the grain boundaries recombination o f minority carriers takes place. The physical processes which take place at the grain boundaries have been studied [13, 16, 20, 21] and definite conclusions have been reached as to the o p t i m u m grain orientation. In the depletion region it was found that a grain boundary that was vertical with respect to the junction was least detrimental to the photocurrent [ 2 2 ] . This

203

Random

(a)

Fibrous

(b)

Fig. 1. Grain orientations for polycrystalline (thin film) n-p junction solar cells: (a) random orientation and (b) fibrous orientation.

corresponds to the grain boundaries observed in the bulk, which leads to the assumption that the fibrous orientation will provide the best chance for a substantial conversion efficiency. Generally, when a polycrystalline solar cell is referred to it is a thin film device which is being discussed. The advantage o f such a device is that it is possible t o make the overall thickness of the cell smaller than or o f the order of the grain size itself. Then carriers can have just as much o f a chance, or perhaps even a greater chance, of being collected as o f recombining at the grain boundary. It is also assumed [22] that, if the grain size is greater than a diffusion length for the bulk material, then the grain boundaries will have little effect on the photocurrent. The larger the grain size, the less is the effect and the more the grains behave like a single crystal. For the sake of mathematical manipulations, we will derive expressions for the photocurrent and the dark current for a fibrously oriented rectangular-grained polycrystalline n - p junction solar cell on t h e basis of the ideal model, as shown in Fig. 2(a). It is different from an ideal single-crystal model, as depicted in

Fig. 2(b). It will be necessary here to introduce a few new variables which are particularly important for a polycrystalline model. For example, the grain size and the recombination velocity at the grain boundaries will be considered. This recombination velocity has been estimated [16] to vary from 10 2 to 10 6 cm s- l , depending on the interface state density at the grain boundary and the doping concentration of the semiconductor. In addition, a new effective diffusion length and lifetime, representing the difference between those observed in the single-crystal device and those observed in the polycrystalline device, will be incorporated.

2.2. Green's function approach to the solution of diffusion equations For a polycrystalline n - p junction cell the diffusion equation to be solved in the n region of the cell is a partial differential equation o f the form

~2(pn --Pno)

Pn--Pno _ Lp 2

od~(1 - - R ) exp(--~z) Dp

(1)

where the boundary conditions appropriate to this upper n region and consistent with the center grain o f Fig. 2(a) are

204 -b

b z=0

~ 0

_

~

~

or

Diffused Epitoxial

Depletion

zj+W

H~

Bose Region

(a)

(b)

Fig. 2. Ideal orientations for theoretical models: (a)fibrously oriented rectangular grains for polycrystalline samples and (b) an ideal block for a single-crystalor epitaxial sample.

~(Pn --Pno)

az

Sp -~-~- (Pn --Pno)

Pn --Pn0 = 0 ~(Pn --Pn0)

bx

_

+

Spg (Pn --Pn0)

z =0

(2)

Z = zj

(3)

X = +-a

(4)

Y = -+b

(5)

Dp

b(Pn --Pn0) _ +Sp~ (Pn - - P n 0 ) ~Y Dp

Here the assumptions are made that the recombination velocity is the same at all points on the boundaries and that the density change is occurring at the boundaries in an analogous fashion to that at the surfaces of the cell. We will use the Green's function m e t h o d to obtain a solution with the above boundary conditions. A similar approach has been used by Balda [23] with respect to a two-dimensional case for a p - i - n solar cell model. Following the procedure outlined in Appendix A, we find that the hole density may be expressed as (Pn --Pn0)( x', Y', Z') = (G, f)

(6)

where the primed variables represent arbitrary points in the volume, the right-hand volume integral is integrated with respect to the unprimed variables x, y and z, G is the symbol for the Green's function and c~F(1 -- R) exp(--o~) f =

(7)

Dp

which is the right-hand side of eqn. (1). The Green's function is found by solving the following equation (see Appendix A): G V 2 G - - - - = 5 ( x - - x ' ) 5(y - - y ' ) 5 ( z - - z ' ) Lp 2

(8)

with the boundary conditions

OG _ ~z

SpG

Dp

z =0

(9)

205 G=O ~G

SpgG

~X

Dp

(I0)

x=+a

(II)

y = -+ b

(12)

SpgG

~G ~y

Z =Zj

- -+ - -

Dp

With the aid of the s y m m e t r y inherent in the x and y directions, eqn. (8) is separable. With diffusion taking place in a symmetrical fashion away from x = 0, y = 0, and towards the grain boundaries, the cosine function is a prime partial solution candidate. Another property of the cosine function which is essential to the solution of this problem is its orthogonality which allows 6(x -- x') and 8(y -- y') to be written in cosine form. Following this reasoning, we devise an expansion for G of the form

G(x, y, z Ix', y', z') = G(x)G(y)G(zlx', y', z')

(13)

letting oo

G = ~ Amn(Z)Ma cos(rex) N b cos(ny)

(14)

??lpn

where Am,~(z) includes the primed variables x', y' and z', Ma =

ma + sin(ma) cos(ma)

(15)

and Nb =

I

f 1/2

n

nb+ sin(nb) cos(nb)

(16)

n and m, which are n o t integers, are determined. By the substitution o f eqn. (14) into the boundary conditions eqns. (11) and {12), the following relations can be obtained [24] :

ma tan(ma) = nb tan(nb)

=

Spg Dp

a

(17)

Spg__b Dp

(18)

which are satisfied on the intervals

sTr <~ ma <<.(s +-~ sTr <<.nb <<.(s +-~1 )~r

s = 0, 1, 2 ....

The orthonormality of cos(mx) and cos(ny) on the intervals --a - a and --b b is m e t if t h e y are multiplied by Ma and N b respectively (see Appendix B). Then the completeness relation allows the writing o f

206

5(x--x') = ~, Ma cos(mx)M~

cos(mx')

(19)

712

and

6(y --y') = ~ Nb

(20)

cos(ny) Nb cos(ny')

tl

Substituting eqns. (14), (19) and (20) into eqn. (8) and separating, we obtain

Am,(Z)

d2Am,(Z) dz 2

- 5{z --z') Ma cos(mx')

~,p2

Nb

cos(ny')

(21)

where ~,p2 =

Lp 2 1 + Lp2(n 2 + m 2)

(22)

which is h o m o g e n e o u s when z ¢ z'. The general solution to eqn. (21) in the two regions may be written as z

z

'A sinh( Z-f-]+ B cosh( ] A~.(z)

=

\Tp/

Z<

z'

\~/p/

(23)

where the coefficients can be determined by making use of the boundary conditions (9) and (10), rewritten as dAm.(Z) dz

- Sp-Amn(Z) Dp

z =0

(24)

z =z i

(25)

and

Amn(Z) = 0

in addition to two matching conditions at z = z', obtained by integrating eqn. (21) from z' - - e to z' + e as e -~ O: ['+¢ I d2Am"(Z) lim ~--,0d,_~ ~ ~-Y

z

3,1,2

5(z--z')M~

= lim f e"~0

Am.(Z) t dz

,

cos(mx')

Nb

cos(ny')dz

--6

which gives

e~olimtdAmn(Z)dz z'-ez'+e--/'+e Amn(z) ~3P2 dzl= Ma c°s(mx')

cos(ny')

207

Now, the matching conditions appear if we require that the following expression holds: Am,(z) is continuous at z = z'

(26)

which makes the second term drop out, exposing a condition on the slope [25] lim dAm"(z) z'+~ = Ma cos(mx') Nb cos(ny')

e-*0

dz

(27)

z'-e

These two conditions suggest equal to each other and that ative of the first is equal to M~ Combining the solutions function on either side of z':

that at z = z' the two parts of eqn. (23) are the derivative of the second minus the derivcos(mx') Nb cos(ny'). of these equations, we obtain the Green's

= ~ Ma2Nb27p cos(mx') cos(ny') cos(mx) cos(ny) X oo

G,

m, n

(SpTp/Dp +

Kz)

XlKzsinh(~p)--c°sh(Z'tIISp7P\Tp/) ~

G2

Dp sinh(Zt + c o s h ( Z t I \ 9'p]

z
\')'p ] )

Ma2Nb27p cos(rex') cos(ny') cos(rex) cos(ny) × m, . Sp'),p/Dp + Kz

×

t SpTp s i n h ( £ / + c°sh / z ' )I {Kz sinh( z / - - c o s h ( Z t} Dp \Tp/ \'Yp \~'p/ \ ~'p /

z'
where cosh(zj/Tp) Kz-

(29)

sinh(zj/Tp)

3. Derivation o f the photocurrent equations

3.1. Expression for the hole current density in the n region

Now that the Green's function has been obtained, it is possible to solve for the hole density. The substitution of eqns. (7} and (28) into eqn. (6) allows us to write P. -- P.o =

aF(1 -- R) ~ Dp a

M a 2 N 27P b cos(rex') cos(ny') X m,. Np + Kz

b

X f f cos(mx)cos(ny) dx dy (C, + C2) - - a --b

(30)

208 w h e r e Np = Sp"yp/Dp and

C1

= tK~

sin( z ' l

-cosh(

\Tp/

z

]If z' lNp

\TP/)0

sinh( z ] +

t

coshtZllexp - z

\Tp/

\7p / }

(31)

and

c2:l psin ( )

+ cosh( z ' / I fZJlK z s i n h ( Z t \ T p / } z' \~'p]

cosh(Z--If exp(--az)dz \7p/}

(32)

Integrating eqns. (31) and {32), we obtain 7p(Kz sinh(z/Tp) -- cosh(z'/Tp)) C1 :

X

× --exp(--o~z') (NpO~Tp + 1) sinh ~ \Tp]

+ (Np

+o p, coshz --

\~/p ] )

+ Np

+

~Tp

(33)

and ')'p(Np sinh(z'/?p) + cosh(z'/Tp) ) C2 = (a27p: -- 1) sinh(zj/Tp) × I

The polycrystalline hole density in the n region can now be written Pn --PnO -

4aF(1 -- R) ~ Ma2Nb23,p sin(ma) sin(nb) ~ X Dp ,,, n mn(Np + Kz) X cos(rex') cos(ny') (C1 + C2)

(35)

where C1 and C2 are given by eqns. (33) and (34) respectively. We are now interested in the current density

Jp = --qDp Y(Pn --Pno)

(36)

which should be evaluated at the junction edge (z' = zj ). At this position the gradient of pn --P~o in the x' and y' directions is zero. Therefore eqn. (36) can be written as Jp = --qDp

a(Pn --Pn0) ~z'

z' = zj

(37)

The polycrystaUine hole current density per unit bandwidth for the n region at the junction edge becomes

l

209 oo

Jp = 4 q ~ F ( 1 --R) ~ 7PMa2Nb2 sin(ma) sin(nb) cos(mx) cos(ny) × m. n mn(o~27p2 -- 1) X [ Np + o~-),p-- exp(--~z~ )(Np cosh(z~/'),p) + sinh(z i/Tp)) Np sinh(zj/'),p) + cosh(zj/~,p)

] 0~7p exp(--o~zj )J

(38) where we have replaced x' and y' with x and y respectively. It is worth noting the similarities between the polycrystalline photocurrent density of eqn. (38) and the single-crystal counterpart as reported earlier [ 10]. The polycrystalline case includes the influences due to the grain boundaries in the form of x and y contributions and the location of 7p in place of Lp in the single-crystal case.

3.2. Expression for the electron current density in the p region N o w that the hole current density in the upper n region has been found, it is necessary to solve for the electron current density in the lower p region of the cell. The diffusion equation appropriate to this case is V:(np --np0) --

np -- np0 L, 2

-

--aF(1 - - R ) exp(--az) Dn

(39)

It can be seen that this equation is very similar to eqn. (1) which describes hole diffusion. The boundary conditions are also similar and take the form z = zj + W

(40)

(np - - np0 )

z =H

(41)

(rip -- np0 )

x = ~a

(42)

y = ~b

(43)

np --np0 = 0 (}(rip -- npo)

Sn -

~z ~(np -- np0 )

~x

_

_

Dn +_ Sng

D~

~(np -- np0 ) _ + Sng

~y

Dn

(np -- np0)

It would be expected, if the similarities between the conditions on np np0 and those on Pn -- P~o were noted, that the same procedure would work for the electron density. Therefore, using the Green's function method, we can write (np -- np0)(x' , y', z') = (G, f)

(44)

where f is the right-hand side of eqn. (39) and G is the Green's function for the p region which is f o u n d b y solving G V 2 G - - - - = ~ ( x - - x ' ) ~(y - - y ' ) 5 ( z - - z ' ) Ln 2

(45)

with the following boundary conditions on G: G=0

z=zj +W

(46)

210 ~G

Sn -

~z 8G --= ~x

G

z =H

(47)

G

x = ~a

(48)

G

y =~b

(49)

Dn +

Sng Dn

~G --=+

Sng

()y

Dn

-

Following the same reasoning as before, we can rewrite eqn. (45):

G = ~ A~z(z)Ka cos(kx) Lb cos(/y)

(50)

k,l

where Akl(z) includes the primed variables x', y ' and z'. The values of k and l, which are again n o t integers, are determined by inserting eqn. {50) into the boundary conditions (48) and (49) and obtaining the relationships

ka tan(ka) = Sng a

(51)

D,

lb tan(/b) =

Sng

D~

b

(52)

which are satisfied in the intervals

s~ < ka < (s +-~)~r

s~ < lb <- (s + ~)lr

s = 0, 1, 2 , . . .

(53)

If we proceed in the same manner as t h a t outlined in Section 2.1, then after some algebra the electron density can be determined at any point in the p region. We are interested in the current density Jn at the junction edge z' = zj + W. Therefore, the solution of the equation -- rip0) Jn = qDn ~(np5z'

z' = z i + W

(54)

gives the electron current density per unit bandwidth in the polycrystalline material at the junction edge: Jn = 4qa.F(1 - - R ) exp{--c~(z~ + W)} × × ~ 7nKa2Lb 2 sin(ka) sin(/b) cos(kx) cos(/y) × k, l k/(o~27~2 -- 1)

[ ×

OL~'n - -

N"(c°sh(H'/'Yn)--exp(--o~H')}+sinh(H'/Tn)+°~/nexp(--°lH')] N n sinh(H'/Tn) + cosh(H'/Tn)

(55) where Nn = SnTn/Dn and H ' = H -- (z i + W) and x' and y ' are replaced by x

211 and y respectively. Equation (55) is quite different from the single-crystal equation. We shall study its effect later in Section 4. 3.3. Photocurrent in the depletion region As for the photocurrent collected in the depletion region of the polycrystalline cell, we will assume that it remains essentially unchanged from the single-crystal case. It has been shown [26] that the photocurrent per unit bandwidth is equal to the number of photons absorbed. Thus, the photocurrent collected in the depletion region is Jd~ = qF(1 -- R ) exp(--~zj)(1 -- exp(--~W)}

(56)

Hence, for fibrously oriented grains with grain boundaries running vertically through the junction, the total photocurrent density is the sum of eqns. (38), (55) and (56).

4. Derivations of the dark current equations For the polycrystalline dark current we expect to find (i) an x and y contribution due to the grain boundary effects and (ii) all the Lps and Lns of the single crystal replaced by their effective values. We will now start with the appropriate diffusion equation. 4.1. Diffusion equations and boundary conditions The diffusion equations appropriate for the two regions are

•2(pn --Pn0) Pn --Pn0 _0 2

(57)

Lp

for the n side and V 2(np - -

npo)

np - - np0 _ 0

(58)

L. 2

for the p side. The boundary conditions applicable to the n side are a(p. --P.o) _ Sp

Dp (Pn

c}z

--Pn0)

Z= 0

(59)

P , =P~0 exP/qv-~V~J/

z = z,

(60)

~(P. -- P.o ) ax

_

x = ;a

(61)

~(P. - - P . o ) ~y

_ _+ S p g

y

(62)

\RI/

+ Spg (P~ --P~0) Dp

Dp

(Pn--Pn0)

~b

212

Likewise, the boundary conditions suitable for the p side are nr, = nr,o exP \ k T ] O(np --npo) _ DZ

S~

(np--npo)

3x -- npo)

(63)

z =H

(64)

x =~-a

(65)

y __ ~- b

(66)

Dn

Sng O(np--npo ) _+__(np_npo)

3(np

z = zi + W

Dn _ _+ Sng ( n p - - n p o )

3y

Dn

where Vj is (as before) the junction voltage, which at the "built in" potential allows Pno and npo again to be expressed as ni2/Nd and ni2/Na respectively. It will be convenient again to use the Green's function m e t h o d to solve these equations. Making use of the surface integrals in Appendix A, we see that Pn -- Pn0 =

(Pn -- Pno) ~-Z z = zj

and np--np0 =

--(np--npO)~zz z=z~+w

Performing the necessary operation as .outlined in Section 2.2 and substituting proper boundary conditions, we find that M,,2Nb 2 cos(rex') cos(ny')

Pn

Pn0

×

z~,, Np sinh(zj/~p) + cosh(zj/Tp)

× f f cos(mx)cos(ny)dx dy Np sinh

+cosh

X

Then the hole density becomes 4Ma2Nb 2 sin(ma) sin(nb) Pn - - P n 0 = ~ m, n

mn

X

× Pnolexp(~-~l- 1f cos(mx)cos(ny) ×

x t Np sinh(z/~p) +cosh(z/"gp) f Np sinh(z~/3"p) + cosh(zi/~'p)

(70)

213 where the x', y' and z' have been replaced with x, y and z and the values for m and n can be determined. Ms and Nb are the same as before.

4.2. Current densities of the dark current Now the hole current density associated with this density distribution expression is a vector quantity with x and y as well as z components. In our case, we are only interested in that portion of the dark current which is opposing the photocurrent. Therefore, the current density expression is given by (P,, - P no)

Jp = --qDp

~z

z' = z i

(71)

which yields

M~2Nf Jp = --4qDp ~, sin(ma) sin(nb) cos(rex) cos(ny) X m, n mn'yp × Np sinh(zj/Tp) + cosh{zj/-~--p)f Pn°lexp \ k-T ] -- 1

(72)

Once again performing the Green's function analysis and using the appropriate boundary conditions, we obtain a

oo

np

-

npo = -

~_, K ~ 2 L ~ 2

cos(kx')cosqy')ff

k, I

b

cos(kx) cos(/y) dx dy ×

--a--b

([Nncosh(H'/3'n)+sinh(H'/Tn) sinh ×

~-~ sinh(H'/7~) + cosh(H'/3'~)

iz, -- (zl + W) (73)

This simplifies to

4Ko2Lb2 np --n~ = -- ~_~ kl sin(ka) sin(/b) × k,!

e

qY~

X np01 x p ( - - ~ - ) - 1 f cos(kx)cos(/y) X

x([Nnc°sh(H'/3'n)+sinh(H'/3'n) Nn sinh(H'/7~) + cosh(H'/Tn) sinh

--c°shlZ--(zJ+W) f)Tn

lz--(zj 7n

+w ll (74)

214 where x', y' and z' have been replaced by x, y and z and the discrete values for k and l are to be determined as before. The current density opposing the photocurrent is then ~(np -- rip0) Jn = qDn z' = zj + W bz which leads us to Jr,

oo

=

- - 4 q D n ~ K.2Lb 2 sin(ka) sin(lb) ~, z klTn

t

X npo exp

×

f

-- 1 cos(kx) cos(/y) ×

X l Nn cosh(H'h'n) + sinh(H'/Tn) l Nn sinh(H'/Tn) + cosh(H'/~/n)

(75)

The negative signs in eqns. (72) and (75) indicate that the dark current is in the opposite direction to the photocurrent. If we note that Pn0 = ni2/Nd and np0 = ni2./Na, the total dark current can be written as Jd~k=Jp + Jn = J0 l e x p ( ~ Vj) T --1 f

(76)

where 4qDpni 2 o~ Ma2Nb 2 2.)" ~ sin(ma) sin(nb) cos(rex) cos(ny) × J0 - m Nd × t No cosh(z~h'p) + sinh(zj/?p) I + Np sinh(zj/Tp) + cosh(zj/'yp) Ka2Lb2 sin(ka) sin(/b) cos(kx) cos(/y) × + 4qDnni - - Na 2 : 2 koo ,l x t Nn cosh(H'/Tn) + sinh(Ht/Tn) l N n sinh(H'/Tn) + cosh(H'/'yn)

(77)

5. D i s c u s s i o n a n d c o n c l u s i o n s

The current expressions for the polycrystalline cell derived here for an n-p junction can be applied also to a p - n junction by simply interchanging Sp, Dp, Lp, rp, 7p and Nd with Sn, Dn, Ln, Zn, ~'n and Na respectively throughout the analysis. We shall now briefly comment on the polycrystalline theory developed here by comparing it with the single,crystal theory [10]. In Figs. 3 and 4 we

215 50

....

i ....

I ....

I ....

I ....

I''

'1

40

....

''''l''''J

AMO 40

I''''1

100 50

30

AM1

....

I ....

\

H =20/Jm

A

~ 30

. . . . .

300

AM2

20 i

10 10

iL,pl,,,Ll,,.i

.1

....

.2

I ....

I ....

.3 .4 .5 Voltage (Volts)

0

tl,

.6

.7

....

0

I,,,,I,,

.1

,,

.2

I ....

.3 .4 .5 Voltage (Volts)

,11

.6

....

I

.7

Fig. 3. I - V c h a r a c t e r i s t i c s for a single-crystal silicon solar cell u n d e r AM 0, A M 1 a n d AM 2 irradiance c o n d i t i o n s (cell t h i c k n e s s H = 4 5 0 pro; j u n c t i o n d e p t h z i = 0.5 pro). Fig. 4. I - V c h a r a c t e r i s t i c s for single-crystal silicon solar cells u n d e r t h e m o s t f a v o r a b l e c o n d i t i o n , AM 1 ( t h i c k n e s s e s H = 20, 5 0 , 1 0 0 a n d 300 pro; j u n c t i o n d e p t h z i = 0.5 pro).

show the current-voltage (I-V) characteristics of a single-crystal n - p junction silicon solar cell for three solar irradiance conditions. The effect o f the cell thickness for this single-crystal cen is displayed in Figs. 5 and 6. These calculations have been made using the data given in Table 1 and the standard single-crystal theory [ 1 0 ] . While a detailed calculation of the present polycrystalline theory will be described in Part II, we report here only (i) the I - V characteristics and (ii) the efficiency plots for various grain sizes and thicknesses. These are shown in Figs. 7 and 8. From Fig. 5 it is seen that a single-crystal silicon solar cell operating under air mass (AM) 2 conditions is more efficient than a similar cell under AM 1 conditions which, in turn, is more efficient than a cell at AM 0. For the actual p o w e r produced, however, the opposite is true; the most p o w e r o u t p u t is generated b y the cell operating under AM 0 conditions and the least b y an AM 2 cell. The reason for this seeming discrepancy is that, while many more photons are present in the AM 0 spectrum, the fraction available [29, 30] to silicon (i.e. the fraction of the p h o t o n s with an energy above 1.12 eV) is lower than the fraction available at AM 1 and AM 2 because of atmospheric absorption in the low frequency region [3]. The same relationship exists [31] (but is n o t shown here) with respect to the efficiency and power o u t p u t of a polycrystalline cen. The fact that a solar cell is more efficient while at the same time less powerful under AM 2 conditions than under AM 1 or AM 0 conditions is a consequence o f the solar radiation itself and does not result from the inherent features o f the cell. We find from Fig. 7 that the efficiency improves with the increase in grain size and that it approaches the single-crystal value after a b o u t 500 pm.

216 20

19

r

,

,

r

19 17

300

15

100

13

50

-~

18

g

~'~AM1

"~

17

..~ \

.

-\

2

AM0

15 14 13 .2

.4

.6

.8

Junction Depth (microns)

1.0

0

0.2

0.4

0.6

0.8

1.0

Junction Depth (nlicroos)

Fig. 5. The plots of efficiency as a function o f junction depth for single-crystal silicon solar cells under AM 0, AM 1 and AM 2 irradiance conditions (cell thickness H = 450 pm). Fig. 6. The plots of efficiency as a function o f junction depth for single-crystal silicon solar cells under AM 1 irradiance conditions (thicknesses H = 20, 50, 100 and 300 pro; junction depth zj = 450/Jm).

Figure 8 illustrates efficiency as a function of grain boundary recombination velocity for two different grain sizes (10 and 100 pm) at three different cell thicknesses (20, 50 and 100 #m). It is seen that for a grain size of 10 ~zm the cell thickness is significant only for recombination velocities of less than 100 m s-] . In contrast, for a grain size of 1 0 0 / J m the cell thickness has a substantial effect over the entire range of recombination velocities studied here. In conclusion, we may state that this formulation is one step beyond the previously developed theories based on the single-crystal theory [ 10] for uniform doping and constant lifetime and mobility, involving derivations of the photocurrent and the dark current from the minority carrier transport and continuity equation. While the single-crystal analysis was done in one dimension, the polycrystalline case involves three dimensions and necessitates the use o f the Green's function m e t h o d to solve the partial differential equations for the minority carrier. The derivation involves all the singlecell parameters plus some additional parameters such as the hole-electron pair recombination velocity at the grain boundaries and the grain size. The polycrystaUine n - p junction cell model on which the derivation is based is established from the evidence that the fibrously oriented grain structure gives the highest current flow, contrary to the random grain orientation, and has the best chance of producing solar cell efficiency. In addition, thin film material in ribbon or cast ingot form contains a preponderance o f vertically oriented grains [ 3 2 ] , indicating that the fibrous grains are practically feasible.

217

TABLE

1

Typical solar cell parameters for silicon at 300 K Parameter

Value

Upper n region Resistivity of base Sp

1 ~ cm 100 m s -1

Dp Lp

1.295 X 10 -4 m 2 s-I 6.233 × 10 -7 m

7p Na

3.0 1.5

X 10 - 9 × 1022

S m -3

Lower p region Resistivity of base Sn Dn Ln Tn Nd

1 ~ cm oo 2 7 . 0 )< 10 - 4 m 2 s - 1 1 6 4 . 0 × 10 - 6 m 10.0 10 - 6 s 5.0 × 1025 m - 3

Wa ni b

0 . 2 8 × 10 - 6 m 1.6 X 1016 m - 3

x

The data are taken from Hovel [10] ;however, some modified numbers were selected for Tp and Lp for polycrystalline cells. a F r o m ref. 27. b F r o m ref. 28. .

35

~'''[

....

. . . .

-

---- - - ~

30

1 ....

500

I ....

I ....

I ....

.

.

.

I ....

15

. . . . . . . . .

...~

.... %?

Single

Cryml •

2 0 u m ~o 0" " =

25 10

~o £ Grain Size = 10 # m 5

10 5 0 ,1

.2

.3 .4 .5 Voltage (Volts)

.6

101 102 103 104 Grain Boundaw Recombination VelociW (m/sec)

Fig. 7. I-V characteristics for polycrystalline silicon solar cells under A M 1 irradiance conditions (cell thickness H = 50/~m; junction depth zj = 0.3 pro; grain boundary velocity Sg = 100 m s-l; grain size X g = 2a). (See Fig. 4 for comparison with the single-crystal result.) Fig. 8. T h e plots of efficiency as a function of grain boundary recombination velocity for polycrystalline silicon solar cells under A M 1 irradiance conditions (thicknesses H = 20, 50 and I00 p m ; grain sizes X g = I0 and I00 pro).

218

Nomenclature On

Dp f F(~.) G H H' Jdark

J~ Jn Jp

J~ J0 k Ln Lp ni np np0

Na Nd

N~ Np Pn Pn0 R(}~) Sg Sn Sng Sp Spg T

yj W

Zg zj 7n ")'p Tn Tp

diffusion coefficient of electrons in p-type material (cm 2 s-1 ) diffusion coefficient of holes in n-type material (cm 2 s-1 ) forcing term applicable to Green's function method solar irradiance (photons cm -2 s- 1) Green's function total cell thickness cell thickness minus the junction depth plus the depletion width forward dark current density (mA cm -2) current density from the depletion region current density of electrons in p-type material current density of holes in n-type material short-circuit current density saturation dark current density Boltzmann's constant Ln 2 = Dnrn, diffusion length of electrons in p-type material Lp 2 = Dp~'p, diffusion length of holes in n-type material intrinsic carrier density electron carrier density in potype material electron density in thermal equilibrium acceptor density donor density Sn'Yn/D n Sp~p/Dp hole carrier density in n-type material hole density in thermal equilibrium reflectance grain boundary recombination velocity when Sn~ = Spg recombination velocity of electrons at surface of p-type material recombination velocity of electrons at grain boundaries in p-type material recombination velocity of holes at surface of n-type material recombination velocity of holes at grain boundaries in n-type material temperature junction voltage width of depletion region grain size for a square grain junction depth absorption coefficient (cm -1) effective polycrystalline diffusion length for electrons in p-type material effective polycrystalline diffusion length for holes in n-type material wavelength lifetime of electrons in p-type material lifetime of holes in n-type material

References 1 J. J. Loferski, J. Appl. Phys., 27 (1956) 777. 2 M. Wolf, Energy Convers., 11 (1971) 63. 3 V. L. Dalai, H. Kressel and P. H. Robinson, J. Appl. Phys., 46 (1975) 1283.

219 4 L. M. Magid, Proc. Natl. Workshop on Low Cost PolycrystaUine Silicon Solar Cells, Dallas, TX, May 18, 1976, Southern Methodist University,p. 21. D. M. Warchauer, Proc. Natl. Workshop on Low Cost PolycrystaUine Silicon Solar Cells, Dallas, TX, May 18, 1976, Southern Methodist University, p. 372. 5 R. Van Overstraeten, Proc. 15th Photovoltaic Specialists' Conf., Orlando, FL, May 1981, IEEE, New York, 1981, p. 372. 6 T. L. Chu, J. Vac. Sci. Technol., 12 (1975) 912. T. L. Chu, S. S. Chu, K. Y. Duh and H. T. Yoo, Proc. Natl. Workshop on Low Cost Polycrystalline Silicon Solar Cells, Dallas, TX, May 18, 1976, Southern Methodist University, p. 408. 7 D. E. Carlson and, C. R. Wronski, J. Appl. Phys. Lett., 28 (1976) 671. C. R. Wronski, IEEE Trans. Electron Devices, 24 (1977) 351. D. E. Carlson, IEEE Trans. Electron Devices, 24 (1977) 449. 8 H. Okamo, Y. Nitta, T. Adachi and Y. Homakawa, Surf. Sci., 86 (1979) 486. J. J. Loferski, Surf. Sci., 86 (1979) 424. 9 M. H. Brodsky, M. A. Frisch and J. F. Zeigler, Appl. Phys. Lett., 30 (1977) 561. W. E. Spear and P. G. LeComber, Philos. Mag., 3 (1976) 935. 10 H. J. Hovel, Semiconductors and Semimetals, Vol. 11, Solar Cells, Academic Press, New York, 1975, p. 93. 11 P. Rappaport and J. J. Wysocki, in S. Larach (ed.), Photovoltaic Materials and Devices, Van Nostrand, Princeton, NJ, 1965, p. 239. 12 B. Ellis and T. S. Moss, Solid-State Electron., 13 (1970) 1. 13 A. K. Ghosh, C. Fishman and T. Feng, J. Appl. Phys., 51 (1980) 446. 14 H. Pauwels and A. De Vos, Solid-State Electron., 24 (1981) 835. Proc. 15th Photovoltaic Specialists' Conf., Orlando, FL, May 1981, IEEE, New York, 1981, p. 377. 15 R. B. Hilborn, Jr., and T. Lin, Proc. Natl. Workshop on Low Cost Polycrystalline Silicon Solar Cells; Dallas, TX, May 18, 1976, Southern Methodist University, p. 246. 16 H. C. Card and E. Yang, IEEE Trans. Electron Devices, 29 (1977) 397. 17 M . A . Green, Appl. Phys. Lett., 2 7 ( 1 9 7 5 ) 287. 18 M. A y m a n Shibib, F. A. Lindholm and F. Therez, IEEE Trans. Electron Devices, 26 (1979) 959. F. A. Lindholm and J. G. Fossum, Proc. 15th Photovoltaic Specialists' Conf., Orlando, FL, May 1981, IEEE, New York, 1981, p. 432. 19 W. Z. Shen and C. Y. Wu, J. Appl. Phys., 51 (1980) 466. 20 L. L. Kazmerski, Solid-State Electron., 21 (1978) 1545. 21 L . M . Fraas, J. Appl. Phys., 49 (1978) 871. 22 M. A. Green, Solid-State Electron., 21 (1978) 1139. 23 R. J. Balda, A silicon junction solar energy converter, Ph.D. Thesis, University of Arizona, 1975. 24 W. Shockley, Electrons and Holes in Semiconductors, Van Nostrand, New York, 1950, p. 295. 25 M. D. Greenberg, Application of Green's Functions in Science and Engineering, Prentice-Hall, Englewood Cliffs, NJ, 1971, p. 51. 26 M. Airman, Elements of Solid State Energy Conversion, Van Nostrand, New York, 1969, p. 24O. 27 E. Y. Wang and R. N. Legge, J. Electrochem. Soc., 122 (1975) 1562. 28 S. M. Sze, Physics of Semiconductor Devices, Wiley, New York, 1972. 29 C. D. Mathers, J. Appl. Phys., 48 (1977) 3181. 30 Handbook of Geophysics, U.S. Air Force-Macmillan, New York, 1961, Chap. 16.3. 31 R . N . Hall, Solid-State Electron., 24 (1981) 595. 32 H. F. Matare, Solid-State Electron., 22 (1979) 651.

220

Appendix

A

Derivation of the Green’s function solving formula and the associated boundary conditions The diffusion equation for holes in n-type material is a partial differential equation which can be written in simpler form UP,

-&lo)

(-4.1)

=f

where the operator

L is

L=VZ--I

(A.2)

LP We wish the Green’s function LG = 6(x -x’)

G to be expressed

as

S(y - y’) 6(z -2’)

(A.3)

To obtain the solving formula we multiply eqns. (A.l) pn -JQ, respectively. In the form of inner products (G, UP, (@,-

-LO))

= (G, f)

(A.4)

~no), LG) = ((P, -P,o),

6(x -x’)

= tin -Pno)(L If we take the difference @, -P&(X’,

and (A.3) by G and

S(Y -Y’)

A(2 -2’)) (A.5)

Y’, 2’)

of these two

Y’, 2’) -G,

f) = 0,

-~no),

LG) - (G, L(P, -P,o))

The right-hand

side of eqn. (A.6) can be expanded

-pno) fl {@,

v2G -

(‘4.6)

to give

-P,o)) dv

G V2@,

V

and if the second Green’s formula

sssv2v (u

[Al]

is applied

vV2u)dV=j-(uVv--Vu)

*ds

(A.7)

s

V

and eqn. (A.6) becomes

(P, -~no)(x’, Y', 2’) = (G, f) +#(A,

-AO)

VG -G

V@, -~0))

- CUJ(A-8)

s

which is the solving formula. The surface integral in eqn. (A.@ can be written H {(P, -P,,o)

VG -

G VCP,

-P,o))

. 6 dfi’

(A.91

S

where A is the unit normal

at each of the six boundary

surfaces

(3~ = +a,

221 y = + b, z = 0 and z i = 0). With the grain orientation as shown in Fig. 2(a), the surface integral becomes - - (Pn - - P n o ) ~-X + G

bx

(Pn --Pn0) ~XX -- G

+ffl +fft ÷fft

bx

dy dz +

x =-a x =<,

b(pn --Pn0) f --(Pn--Pno) bG by + G by y=-b d x d z +

I y=b d x d z + (Pn --Pno) bG ~ y + G b(pn--p~o) by bG + G --(Pn --Pn0) ~

b(pnbZ--Pn0)fz= o dx dy +

+fSt (Pn --Pno) ~bG + G b(Pn--Pn0) f ~ = ~ i dx dy ~z If the boundary conditions on Pn -- Pn0 are known, this allows the determination of the boundary conditions for G. For the photocurrent, where the conditions on pn -- Pn0 are known, all the surface integrals can be made to vanish if the conditions on G are given. This leaves the hole density as simply

(Pn --P~o)(X', y', z') =
(A.10)

For the dark current, where the boundary conditions on p~ --Pn0 are also known, all the surface integrals vanish except the one at z = z i if the conditions on G are the same as before. Also, in this case the forcing term f equals zero which leaves, for the hole density,

Z~Zj

Completely analogous reasoning can be used for the Green's function and the electron density in the p region; this gives

(np -- npo)(X', y', z') =

(A.12)

for the photocurrent case and (np -- npo)(X', y', z')

-- (rip -- npo)

dx dy

(A.13)

z=zj+W for the dark current. (A similar use of this m e t h o d in two dimensions is given by Balda [A2] ; Greenberg [A3] gives a general discussion o f Green's functions.)

222

References for A p p e n d i x A A1 C. D. Mathers, J. AppL Phys., 48 (1977) 3181. A2 R. J. Balda, A silicon junction solar energy converter, Ph.D. Thesis, University of Arizona, 1975. A3 M. D. Greenberg, Application of Green's Functions in Science and Engineering, Prentice Hall, Englewood Cliffs, NJ, 1971.

Appendix B

Orthogonality, orthonormality and completeness The o r t h o g o n a l i t y c o n d i t i o n for a set of real f u n c t i o n s of the f o r m cos(mix) on the interval (--a, a) is cos(mix cos(m/x) dx = 0

i¢ j

(B.1)

where i, j = 1, 2, 3, . . . , and mi and mj axe d e t e r m i n e d by

ia tan(mi, ya)

~NPga Dp which is satisfied on t h e interval m

i

(B.2)

=

'

sir ~< m s + l a ~< (s + ½)~

s = 0, 1, 2, ...

(B.3)

for given values of Spg, Dp and a. If eqn. (B.1) is integrated, this gives sin{(m/--my)a}

mi - - m j

+

sin{(mi + mi)a}

mi + mj

(8.4)

and with the proper t r i g o n o m e t r i c identities (B.4) can be written

2 cos(mia) cos(mia)(mi tan(mia) -- my tan(mja)} mi 2 -- my 2 If the c o n d i t i o n (B.2) is t h e n substituted into this expression a value o f zero is o b t a i n e d . Thus the o r t h o g o n a l i t y c o n d i t i o n (B.1) is met, since w h e n i ¢ j the e q u a t i o n has been shown to equal zero and w h e n i = j the integral of cos2(mx) on t h e interval --a - a has a value. T h e same can be shown to be t r u e for cos(ny) and cos(kx) a n d cos(/y). The o r t h o n o r m a l i t y c o n d i t i o n can be m e t b y n o t i n g t h a t

Mai cos(mix) Maj cos(myx) dx = 5~

(B.5)

--a

where l m,,~ l 1/2 Mai'j = mi,ja + sin(mi,ja) cos(mi,ja)l

(B.6)

223 and likewise for cos(ny), cos(kx) and cos(/y). The completeness relation allows for the interchanging of the continuous variable x and the discrete variable m in eqn. (B.5) which gives [B1] 0o

8 ( x - - x ' ) = ~ M a cos(rex) Ma cos(rex')

(B.7)

tn

and so forth for cos(ny), cos(kx) and cos(/y). Reference for A p p e n d i x B B1 W. Shockley, Electrons and Holes in Semiconductors, Van Nostrand, New York, 1950.