Grain Boundary effects in polycrystalline silicon solar cells

Solar Cells, 28 (1990) 77 - 94

77

GRAIN BOUNDARY EFFECTS IN POLYCRYSTALLINE SILICON SOLAR CELLS S. BANERJEE* and H. SAHA

Department of Electronics and Telecommunication Engineering, Jadavpur University, Calcutta 700 032 (India)

(Received March 23, 1989; accepted in revised form May 30, 1989)

Summary The polycrystalline silicon solar cell is divided into four regions: (a) skin (emitter) grain region, (b) skin grain boundary region, (c) grain base region and (d) grain boundary base region. Since the grain base region, having a much larger area than other regions, is the dominant photoabsorbing region in silicon solar cells, the photocurrent and dark current contributions from this region have been computed using two~limensional analysis with the help of Green's function technique. The contributions from the other regions, being much smaller, have been computed using the usual one-dimensional analysis. The dependence of the short~ircuit current and open~ircuit voltage of these cells on the different parameters such as grain size, grain boundary potential barrier, grain boundary recombination velocity etc., have also been investigated taking into account the variation of the diffusion length in the grain boundary region. The control of important parameters for the optimization of polycrystalline solar cells is discussed. 1. Introduction The conversion efficiency of polycrystaUine solar cells is less than that of the single~rystal counterpart owing to the presence of the grain boundaries. Recently, a number of investigations have been carried out to understand the influence of the grain boundaries having finite geometrical dimensions [1 - 12]. Because of the presence of many small grains and grain boundary effects, and the fact that the randomly oriented grain boundaries act as electron-hole traps, the continuity equation of the minority carriers for grains and grain boundary regions in an n - p junction polysflicon solar cell with all possible grain boundary effects should be solved to obtain the photogenerated and dark current contributions. This has been attempted in the present paper by dividing the polysilicon solar cell into four different regions: (a) grain-bulk emitter or skin region, (b) grain boundary skin *Present address: Bangabasi College, Calcutta, West Bengal, India. 0379-6787/90/$3.50

© Elsevier Sequoia/Printed in The Netherlands

78

region, (c) grain-base region, and (d) grain boundary base region. Contributions to the photocurrent as well as the reverse saturation dark current have been c o m p u t e d from each of these regions separately. Our theoretical approach involves a solution of the two-dimensional diffusion equation using Green's function technique. The two-dimensional analysis could have been carried o u t in principle in all four regions separately, but, except for the region (c) which has a comparatively much larger photoabsorbing area, the two-dimensional analysis is unnecessary. This is discussed in detail in Section 2. The idea of non-uniform variation of the diffusion length in the vicinity of the grain boundary region has been incorporated in this analytical model. Further, carrier recombination through grain boundary surface states creating grain boundary potential barriers at the interface has also been included here. The grain-base region photocurrent is obtained in terms of grain boundary surface recombination velocity at the interface, diffusion length, cell thickness and grain size. Grain-base region dark currents are obtained in terms of grain boundary potential barriers. An interdependence of these controlling parameters leading to an optimized interface state density for maximum efficiency in polysilicon solar cells has been studied in this paper.

2. PolycrystaUine solar cell model A polycrystalline solar c.~q contains a large number of grains which may be either randomly oriented or relatively ordered [13 - 19]. We however assume, for the sake of simplicity, that the grains are all columnar [14, 15] and of identical dimensions and uniform distribution. Figure 1 shows the cross-section of such a polycrystalline solar cell having p - n junctions distributed over the grains and the grain boundaries [8]. Grain boundaries are of

d VFr: -- ~

;' ,, Area"1 ,:1

d'/

[

d Y

Incident radietion

[X=O

/~

I

Arecl 2 c

o [ Lhj- . . . . ;JJ u ,~ ~-dg-2L~ :

i Skinlregion

g

t L(Y) / _~_ . . . .

(a)

~-_

~J

~

Y(1-~)*~'L b

iI (b)

Fig. I. (a) Cross-section of the polycrystalline cell: A1, grain boundary skin region; A2, grain boundary base region; B1, grain (bulk) skin region; B2, grain (bulk) base region. (b) Grain cross-section and diffusion length variation along the Y direction.

79 finite dimensions and may occupy a significant fraction of the total active area of the polycrystalline cell. Thus four different regions are of special interest where photodiffusion equations are applicable: (a) the grain boundary skin (emitter) region (A1), (b) bulk (grain) skin region (B1), (c) bulk (grain) base region (B2) and (d) grain boundary base region (A2). Light is incident on the n-type skin layer in the case of the silicon cells along the x direction, and photogenerated electrons diffuse along the x and y directions towards the p - n junction, generating the photocurrent. For each of these regions, the concentration and flux of the photocarriers are controlled by the boundary conditions related to this region. The resultant photocurrent will be the sum of these four components contributed to by these four regions. The same arguments are also valid for the dark current. Ideally one should solve the two-dimensional photodiffusion equations in each of these four regions and compute the current contributions from each of them. However, this would become too laborious and perhaps not really necessary. From Fig. 1, since the area of the grain-base region (B2) is much greater than that of the other three regions (A1, A2 and B1) most of the photogeneration takes place in the B2 region. In addition, in silicon cells which mostly have shallow junctions, the grain-base region is the dominant producer of photocarriers. The total contribution of the other regions is comparatively very small [3, 9, 14], less than 5%. So, for a silicon solar cell, a two-dimensional analysis has be6n carried out in the grain-base region while a one
~2An2 ~2An2 + ~X2 ~y2

--'-

An2 a2No exp(--ol2x) L2n D2

(I)

A number of mathematical techniques exist by which eqn. (1) can be solved, but we have used the Green's function technique which appears to be very elegant [20] (see Appendix A). The equation obtained from (1) can be solved with the help of four boundary conditions relevant to this region. However, the procedure can be considerably simplified under short-

8o circuit condition if one assumes that the grain boundaries are electrically symmetrical, i.e.

An2[y=d, = An:[~=-d, = 0

(2)

Further applying the two well known boundary conditions pertaining to the grain-base region for evaluation of the arbitrary constant (i)

An21~= w2 = 0

(ii)

An2[y=d, - D2 d(An2) I Sg

assuming an ohmic contact at the back surface

dy

for W'I ~
(3)

I y = d'

where Sg is the effective recombination velocity at the interface of the grain boundary (y = d') and carrying out the tedious but straightforward analysis P (Appendix A), one obtains two arbitrary constants An and B~. The current density contribution JAn 2[grain, base

from this region is c o m p u t e d and shown in Appendix A. The total current is obtained by integrating the elemental x and y components of currents over the (y-z) and (x-z) planes in the bulk base region. Thus the photocurrent contribution per grain from this region can be obtained as d' ]2 Ihn,[photo, = f Jz~n2lx=w',(dg -- 2Lbn) dy + Ja.2l~=d'(dg -- 2Lbn) d x ~ain, -d' W'~ base

(4)

which leads to the following expression (Appendix A) IAn~[photo,----

f qD2(dg-2Lbn)(2Md'(f2\

2flW~) + 2Mfld'(W2- W~)

base

-- 4 sinh(x/P--nd') ~ [{H seC(anW2) --B~ tan(a.W2)) n=l X OLn Sil2(O/nW~) + O~nS~ cos(o/nW~) ] + 2x/-pn sinh(x/~nd' )

×

II l

21 'w2+w; l 1 ++++"I

Hsec(anW2)--B~ t a n ( a n W 2 ) -

X Sill Oln

an

(W2

2

I

W'I)

2B" sin a . a,

x sinla" (w2-2 wl) 1]) d~

cos (~n

2

2

(5)

81

2.1.2. Dark current The dark current contribution from this region can be obtained as shown in Appendix A. The relevant boundary conditions for evaluation of A'd and Bnd are An:Ix=W2 = 0 and

A n 2 [ y = d, =

nl00ex p t --q--~Cgb/lexp(~q;V~l--1 l /~'1' / ! \Z~IJ

(6)

where nloo = (Sp/Sn)I/2ni

(Appendix A)

The dark current contribution per grain from this region may therefore be obtained as (Appendix A) Iz~,la~k = qD2(dg - 2Lb~)

sinh( x/'Pnd')B'a t~n(ang"2)

base

X

II

(7)

2.2. Grain boundary skin region (A1) one-dimensional case In the case of silicon and similar solar cells where the photocurrent contribution from the skin region is rather small, owing to a smaller photoabsorbing area [11, 12], one-dimensional analysis is sufficient. The photocarriers are assumed to diffuse towards the x-direction only. Figure 1 shows the details of this region. The length 2d is the electrical width of the grain boundary involving the space charge width across it and W1 is the protrusion of the skin region along the grain boundary owing to the enhanced diffusion through grain boundary pipes. Taking into account the fact that the diffusion length varies almost linearly from a value PLbp at y = 0 in the centre of the grain boundary region to Lbp at the edge of the effective width [3], the diffusion length in the grain boundary region can be written as Ly = y ( 1 - p ) + PLbp, when p is a predetermined transparency factor [14]. The mathematical details using standard photodiffusion equations and the boundary conditions are shown in Appendix B. Taking into account the other two regions: the bulk (grain) skin region (B1) and the grain boundary base region (A2) where one
82

3. Discussion The necessary equations for computing the density of photocurrent and dark current contributions of each of the four regions of a polycrystalline solar cell are derived in Section 2. The total photocurrent and dark current density of a solar cell is the sum of these four individual components. The revelant parameters of polycrystalline solar cell are shown in Table 1. In the case of a silicon solar cell, the major contribution to the photocurrent comes from the base grain (bulk) region [3, 9, 14] since the junction is shallow (0.2 - 0.3 pm) and the absorption coefficient of silicon is relatively small. This is demonstrated in Fig. 2, showing the variation of photocurrent contributions in the grain boundary skin region, the grain boundary base region and the grain (bulk) skin region of a typical polycrystalline silicon solar cell with the variation in grain size dg in the range 1 0 - 100 pm. One can see that the total sum of all these three contributions is rather small, less than 5 mA cm -2. It is interesting to note that the photocurrent density from the grain boundary base region decreases with increasing grain size, as opposed to the other three components of the photocurrent. This follows from the fact that with increasing grain size b u t fixed small grain (bulk) diffusion length (Lbn = 10 pm) more and more photocarriers fail to reach the grain boundary junction undergoing recombinations [21]. With increasing grain (bulk) diffusion length (Lbn = 164 pro), however, the photocurrent density increases with grain size as shown in Fig. 2. However, most of the photocurrent contribution comes from the base grain (bulk) region, as can be seen from Fig. 3 which shows the variation P in photocurrent with the variation in the effective grain size dg (d'g = d g 2Lbn), the grain boundary interface recombination velocity Sg being the parameter. An interesting point to note is that the photocurrent contributions from the skin region are almost independent of the grain size while that from the base region is strongly dependent on the grain size in the case of shallow junction silicon solar cells. The strongest parameter affecting the TABLE 1 T y p i c a l s i l i c o n s o l a r cell p a r a m e t e r s at 3 0 0 K [ 1 0 ]

Upper skin (emitter) region

Lower base region

Sp = 104 c m s -1 Dp = 1 . 2 9 5 crn 2 s - 1 Lp = 0.6 p m N d = 5.0 × 1 0 1 9 c r n - 3

Sn Dn Ln Na

W~ W1 W2 2d Ni

= = = = =

0.3 p m 2 pm 30,100,200, 300 and 500 prn 0.5 p m 1.6 × 10 l ° c m - 3

=4×107cms -1 =27.0cm 2s -1 = 164 p m = 1.5 × 1016 c m - 3

83 4

3 I"~

2(0)

1'I •9

2

'7 3

"5

E v "07

E

~ "04

o

a. 0 1

I

10

4'0

I

'

I

I

70

I

I

I

100

Fig. 2. Short-circuit photocurrent of a silicon cell as a function of grain size dg for different regions (cell thickness, W2 = 30 #m): curve 1, A1 region; curve 2, B1 region (Lbn = 10/~m); curve 3, A2 region; curve 2(a), B1 region (Lbn = 164 #m).

35

0 103

32

104

29 --

105

E 26 E 23 lu

~o6

g o

10

.

.

.

.

40.

7,0

,

,

1O0

Fig. 3. Short-circuit photocurrent of a silicon cell in the B2 region as a function of effective grain size dg for different grain boundary recombination velocities Sg (cell thickness, W2 = 30 #m, diffusion length Lbn -- 164/~m).

84

36

33

E 30

E c

27

z: CL

2~

'

,:o

'

~o

'

8o

~Go

Fig. 4. Short-circuit photocurrent of a silicon cell in the B2 region as a function of effective grain size d~ for different cell thicknesses W2 (diffusion length Lbn = 164/~m).

photocurrent of a silicon solar cell is however the effective recombination velocity Sg at the interface [22]. By increasing Sg from 102 cm s-1 to 106 cm s-1, the short~ircuit photocurrent decreases from 28.5 mA c m - : to 20 mA cm -2 at dg = 30/~m which conforms fairly well with the values reported by Halder and Williams [10]. The influence of the thickness W: of the base region is also investigated and depicted in Fig. 4. Increasing base thickness increases the short~ircuit photocurrents, as expected. A very important parameter of a solar cell is the effective diffusion length Lbn of the base region. The influence of variation of L b , on the short~ircuit photocurrent of a silicon solar cell is shown in Fig. 5 [11, 22 24]. One notes that by increasing Lbn from 10/~m to 164/~m, the shortcircuit current increases from 9 mA cm -2 to 25 mA cm -2 for dg = 30/~m. The contributions to the dark current from the four different regions are very interesting to investigate and are depicted in Figs. 6 and 7. One can see that the contribution to both photocurrent and dark current from the grain (bulk) skin region is negligible. The dark current contribution of the grain boundary skin region and the grain boundary base region are almost comparable, lying in the range of 10 -s mA cm -2. Further, b o t h of these are almost independent of the grain size. The dark current contribution of the grain (bulk) base region depends strongly on the grain boundary potential barrier @gb- Figure 7 shows its variation with ~bgb. One notes that for values o f ~gb > 0.4 eV, the dark current contribution is much less than 10 -8 m A c m -2. For 0.3 ~ @gb<~ 0.4 eV, the dark current contribution is comparable with that from the two grain boundary regions. But for @gb 0.3 eV, its dark current contribution increases almost exponentially and dominates the total dark current of the silicon solar cell [25, 26]. Figures 8 and 9 show the variation of Voc with the grain boundary potential barrier @gb using the inteffacial recombination velocity Sg as the

85

30

164

25 ~

~_____~-~-

100

E

6O

"-

c

/

/.0

10 Lbn:lO

L 3 u

2

t

o.

10

i

i

A

40

i

70

i

L

100

Fig. 5. Short-circuit photocurrent of a silicon cell in the B2 region as a function of effective grain size d~ for different diffusion lengths Lbn (grain boundary r e c o m b i n a t i o n velocity Sg = 10 4 cm s - z , cell thickness W2 = 30 pro).

2 10 ~

•

12

10"~

E c

f

2

B lo-lC e

4 L

Y

2 40 I i i i 710 100 10 Fig. 6. Saturation dark current of a silicon cell as a function of grain size d~ (cell thickness W2 = 30 # m ) : regions A1, B1 and A2 correspond to curves 1, 3 and 2 respectively.

parameter, and the grain size d'~ using Cgb as the parameter respectively. The diode exponential factor A has been assumed to be unity in the case of silicon solar cells while computing the open-circuit voltage. Further, the loss factor due to the ratio of the junction area to the ce!l geometricarea (Aj/AI) has been ignored [13, 27, 28]. One notes that Voc is relatively independent of the grain size, as expected, while it increases very rapdily with Cgb up to ¢gb = 0.35 eV. Further increase of Cgb beyond 0.35 eV does

86

,°-C I \ 10

e,4

2 -,

E -i E

g ~

2

10-8! 8' z,

~

2

2 ~o-9 0

.~

,~

.;

,~

'~

Fig. 7. Saturation dark current of a silicon cell as a function of grain boundary potential barrier ~gb for the B2 region (cell thickness W2 = 30 #m, grain size d~ = 20/Ira).

I 2

550

550 J I

&oo

"4

500

> E

2

J

~so

4sc

v

•gb:"i

f,

c~400

~oc

"2

~

"4

10

,

I

]

40

J

I

I

70

V

I

O0

1

Fig. 8. Open-circuit voltage of a silicon cell as a function of grain boundary potential barrier ~bgb for different recombination velocities Sg: curves 1, 2, 3, 4 and 5 correspond with values of Sg of 102, 103, 104, 10 s and 104 cm s-1 respectively. Fig. 9. Open-circuit voltage of a silicon cell as a function of grain size for different values.

~gb

87 n o t increase Voc significantly. The variation of the different c o m p o n e n t s of the dark current as discussed can directly explain these variations of Voc. A very important characteristic of a polycrystalline silicon or similar cell m a y be understood from the above discussions. Since both Sg and Cgb are functions of the inteffacial state density at the grain boundary [2, 9], the increase in Cgb is directly associated with an increase in Sg leading to a decrease in the short~circuit photocurrent. However, an increase in Cgb leads to a decrease in the dark current which in turn causes an increase in the open-circuit voltage. Thus it appears that for polycrystalline silicon and similar solar cells, there might exist an o p t i m u m Nis for which the conversion efficiency would become maximum. Further investigation will be reported subsequently.

4. Conclusion The straightforward b u t rather tedious analysis of the photocurrent and dark current contributions from the four different regions of a polycrystalline solar cell has revealed a number of interesting features. In the case of shallow junction silicon and similar solar cells where the contributions from the base region dominate, the short-circuit current and open-circuit voltage are mutually inter-related, both depending on the interface recombination velocity Sg and grain boundary potential barrier Cgb of the grain (bulk) base region. Since both Sg and Cgb depend on the interfacial state density Nis in the grain boundary region in an opposite fashion, there might exist an o p t i m u m value of Nis for which the efficiency is maximum. It is also concluded that for satisfactory performance of polycrystalline silicon solar cells, Cgb /> 0.4 V and Sg ~< 104 cm s-1. The short-circuit current depends very strongly on the grain (bulk) base diffusion length Lbn of the photocarriers. Lbn should be a b o u t 100 pm for short-circuit current densities exceeding 20 mA cm -2.

Acknowledgment Thanks are due to M. K. Mukherjee of Jadavpur University and S. Dutta, P. Dasgupta and P. Rudra of the University of Kalyani for many helpful discussions.

References

1 H.C. Card and E. S. Yang, IEEE Trans. Electron. Devices, 24 (1977) 397. 2 L. L. Kazmerski, Solid State Electron., 21 (1978) 145. 3 T. Daud, K. M. Koliwad and F. G. Allen, Appl. Phys. Lett., 33 (1978) 1009. 4 J. G. Fossum and F. A. Lindholm, IEEE Trans. Electron. Devices, 27 (1980) 692. 5 H. Powels and A. De Vas, Solid State Electron., 24 (1981) 835.

88 6 A. K. Ghosh, C. Fishmand and T. Feng, J. Appl. Phys., 51 (1980) 446. 7 L . M . Frass, J. Appl. Phys., 49 (1978) 871. 8 M. K. Mukherjee, A. R. Saha and S. N. Das, IEEE Trans. Electron. Devices, 25 (1978) 285. 9 M. Bohm, R. Kern and H. G. Wagemann, Proc. 4th Commission o f the European Communities Conf. on Photovoltaic Solar Energy, May 10 - 14, Stresa, 1982, Reidel, Dordrecht, 1982, p. 192. 10 N. C. Halder and T. R. Williams, Sol. Cells, 8 (1983) 201; 8 (1983) 225. 11 M. Bohm, N. C. Scheor and H. G. Wagemann, Sol. Cells, 13 (1984) 29. 12 S. N. Singh, N. K. Arora and N. P. Singh, Sol. Cells, 13 (1984 - 1985) 271. 13 H. Saha and S. Banerjee, Sol. Cells, 7 (1982 - 1983) 233 - 246. 14 K. M. Koliwad and T. Daud, Proc. 14th IEEE Photovoltaic Specialists' Conf., January 7 - 10, 1980, IEEE, New York, 1980, p. 1204. 15 S. Banerjee, Ph.D. Thesis, University of Kalyani, India, 1986. 16 K. H. Norian and J. W. Edington, Proc. 14th IEEE Photovoltaic Specialists' Conf., January 7 - 10, 1980, IEEE, New York, 1980, p. 700. 17 P. K. Roy and S. J. Luszcz, Proc. 15th IEEE Photovoltaic Specialists' Conf., June, 1981, IEEE, New York, 1981, p. 1025. 18 R. B. Hall, R. W. Birkmire, J. E. Phillips and J. D. Meakin, Proc. 15th IEEE Photovoltaic Specialists' Conf., June, 1981, IEEE, New York, 1981, p. 777. 19 A. Rothwarf, Proc. 12th IEEE Photovoltaic Specialists' Conf., November, 1976, IEEE, New York, 1976, p. 488. 20 P. M. Morse and H. Feshback, Methods o f Mathematical Physics, Part I, McGraw-Hill and Kogakusha, International Students' Edition, 1953, p. 805. 21 D . P . Joshi and R. S. Srivastava, Sol. Cells, 12 (1984) 337. 22 J. Qualid, C. M. Singhal, J. Dugas, J. P. Crest and H. Amzil, J. Appl. Phys., 53 (4) (1984) 1195. 23 H. Amzil, M. Zehaf, J. P. Crest, E. Psaila and S. Martinuzzi, Sol. Cells, 8 (1983) 269. 24 B. L. Sopori and R. A. Pryor, Sol. Cells, 12 (1981) 205. 25 W. Huber and A. Lopez-Otero, Thin Solid Films, 58 (1979) 21. 26 C. H. Wu and E. S. Yang, Appl. Phys. Lett., 40 (1982) 49. 27 S. Banerjee and H. Saha, J. Phys. D, 16 (1983) 185. 28 W. G. Haines and R. H. Bube, IEEE Trans. Electron. Devices, 27 (1980) 2133.

Appendix A The two
~2An 2 +

8y2

An 2 L2b.

An2

o~2No -

D2

exp(--ol2x )

(A1)

T h e G r e e n ' s f u n c t i o n is f o u n d b y s o l v i n g t h e e q u a t i o n 1 ~72g - - gg-~ g = 5 ( r - - r ' ) = 6 ( x - - x ' ) d ( y - - y ' )

1 = ~-2~)2 f

exp{iK(r -- r')} dr'

(A2) If G(K) is the Fourier transform of g(r, r'), it follows that (2~) 2

89 from which one obtains

1

g(r,

(27r)~---~f

r') = --

d2K exp{iK(r -- r')} K 2 + 1/L~n

(A3)

The excess carrier density An2(x, y) is thus given by 1

_[exp{iK(r -- r')} [%go~

w ~ dx' ~' " ' 7 -W',

-d'

-

(A4)

× e x p ( - - % x ' ) + An0(x, y)[ where An0(x, y) is the solution of the homogeneous part of eqn. (A4). Rearranging eqn. (A4), one obtains

An2(x, y) =

~ 2 j exp(--~2x') d x ' f dy' e x p { i K l ( x - - x ' ) } dK1 D2(27r) -w', -a' -~

X[

(K~ + 1/L~n) 1/2

] An°(x' y)

(A5)

The solution of the homogeneous part of eqn. (A4) can be obtained in a straightforward manner and is given by An0(x, y) = ~ (An cos anx + Bn sin % x ) n=l

× {C, exp(x/~ny ) + Dn exp(--x/-p,y)}

(A6)

where 1 Pn

= O~2n +

~ -

L~n

Equation (A5) can be solved with the help of four boundary conditions (mentioned earlier). Under the condition of electrically symmetrical grain boundaries, i.e. An2l~=d, = An2[y=-d = 0 two arbitrary constants Cn and D~ become equal so that the u n k n o w n P arbitrary constants are reduced to two only, i.e. A n and B'. These two arbitrary constants can be relatively easily evaluated by applying the boundary conditions At/21 x = W 2

=0

and

D2 d(An2) I

An2[y = d' - Sg

dy

y =d~

for W'I < x < W2

9O

and carrying out the tedious straightforward analysis (A7)

A~ = Bn tan(anW2) + H sec(~nW2) where H=

2K sin(v~nd') + 2Mfld'X/~n(2d'cos(x/~nd' ) -- (2/x/pn) sin(d'x/~n))

d'V~n + sinh(d'x/~n) cosh(a'x/-~)

K = M ( f o + f2W2 -- flW~) M = --a2Nod'/8D2r~L~n

and Bn = 2M

(fo + f, d'2) -- f,d'

sin}

~

sin

2

+ H sint~n(W~+ W2)fsinl°~n(W2 f - W ' l ) ' ' IS g

cosh(d'.Zn)

__V~n sinh(d,@n)f] (IVan sinh(d,X/~n)__ D2S--~cosh(d,v~n)f × [an(W2 -- W'I)-- cos{a~(W2 + W'1)}sin{an(W2-- W'I)}

,)-i

--tan(anW~) sin{an(W'1 + W2)}sin(a~(W2- Wi)}] +~ C~2

,

2

+ W

{1 -- exp(--a2W2)}

)

exp(--ol2W2

0~2

fl = {exp(a2W'l) -- exp(--~2W2))/4a2 f2 =

I[1 W" exp(--a2W2)I c% ] ~ {1 -- exp(--o~2W2)} --

The current density contribution from this region is

(A8)

91

JA.~l~ai~, = Ja~lx =w'~ ÷ JA.~ly =d' = qD2 base

dan21 dx

+ qD2 x ~W'1

dAn2 I Y

yffid'

= qD2[(f2 -- 2flW~)M -- 2.=1 ~ ((H sec(c~nW2)

--B~ tan(o~nW:)c~n sin(o~nW'l) + B'O~n cos(o~nW'l)) cosh(v/-pny)] + qD2[2Mfld' + 2.~=1 ~ {(/-/sec(otnW2) --B" t.an(otn W2) cos(OtnX)

+ B" sin(o~.x)}v'~ sinh(d'v'~)]

(A9)

Pho tocurren t The total current is obtained by integrating the elemental x and y components of currents over the (y-z) and (x-z) planes in the bulk (grain) base region. Thus the photocurrent contribution per grain is obtained as d'

W2

IAn2[ph°t°' = f-d J~nJx=w,,(dg -grain. - ' base

2Lbn)

dy + f JAn2lyffid,(dg -w'~

2Lbn) dx (AIO)

From eqn. (A10), the photocurrent component is obtained to be /A.Jphoto. = grain, base

f~. qD:(dg-2Lbn)(2Md'(f2--2flW'l) + 2 M f l d ' ( W 2 -

W'I)

__ 4 sinh(d'v/'p-nn) ~ [{H sec(~nW2) -- B~ tan(~nW2)) %¢/~n

nffil

X an sin(c~.Wi) + B ~

cos(~.Wi)] + 2V~n sinh(d'x/~n)

Dark current In the absence of any illumination, and under small bias conditions, the excess electron density in this region is given by

92

A n 2 ( x , y) = ~

{A'd COS(anX) + Bnd sin(anx)} cosh(x/~ny )

(A12)

rt=l

One of the relevant boundary conditions for evaluating A'~d and B'~a, as has been mentioned in Section 2.1.2, is An2[x = w~ = 0 The other boundary condition involving the grain boundary potential barrier ~bgb can be obtained following the arguments advanced by Fossum and Lindholm [4]. Under quasi,equilibrium conditions in the grain boundary space charge region Anely=d' = ( S , / S n ) l / 2 n i exp,.

~

)t exp(--qOgb/KT)

(A13)

where Sp and Sn are the hole and electron recombination velocities at the grain boundary interface, ni is the intrinsic carrier concentration of the semiconductor, EFN and EFe are the quasi-Fermi levels for electron and holes under quasi-equiiibrium conditions. If the majority carrier quasi-Fermi levels are nearly fiat across the quasi-neutral region to the contacts under small bias, then in the junction space-charge region one can assume E F N EFp ~ q V, where V is tke bias voltage developed across the space-charge region. The excess carrier density at the grain boundary interface is then An21~=d, = (Sp/Sn)l/2ni e x p { e x p ( q V / 2 K T ) - - 1} exp(--qC)gb/KT)

(A14)

Thus the other relevant boundary condition is An21y = d' = n 1oo e x p { exp( q V / 2 K T ) -- 1} exp(--qC)gb/KT )

where hi0 0 = (Sp/Sn)l/2ni Thus one obtains A~d = --B~d tan(ten W2), where B'd = [ 2n 10o exp(--qdpgb/KT){exp(q V / 2 K T ) - - 1} X (cos(anW2) -- cos(anW1)}] [tan(anW2){cos(2ol~W'l) -- cos(2a~W2) + 2(W'1 -- W2)a~ --sin(2a~W'l) sin(anW2)] -1

(A15)

The dark current contribution per grain from this region is obtained as /Z~n2ld~k' grain, base

/B~d~

=

qD2(dg - - 2Lb~ ) 2x/Pn sinh(x/~nd') tan(anW2)~ "--::--"~ \~n/

93 The total p h o t o c u r r e n t density and dark current density are obtained by multiplying eqns. ( A l l ) and (A16) by t h e n u m b e r of grains per square centimetre in this region, which is equal to 1/d~.

Appendix B The photodiffusion equation is given by d2Ap a (ix 2

alNo

Ap 1 L2y

Da

exp(--alx)

(B1)

when Apl is the excess carrier density, D~ and ~1 are the diffusion constant of holes and absorption coefficient of this region, and No is the incident flux density of photons. The relevant boundary conditions in this region are S___~1 Apa[x=0- d(Apl) x and APlJx=w, =Plo{exp(qV/KT)--I} (B2) D, dx =o where $I is the surface recombination velocity of the photocarriers at the top surface (x = 0) and P10 is the minority carrier density which remains the same throughout the skin region. The current density in the grain boundary skin region is given by d(Ap,)

J~P'lx=w~=qnl~lx=W-

_ qD, exp(--W,/Ly) L---~

X [plo{exp(qV/KT) --

1} -- Q] 1exp(W1/Ly)

+ qDaQ t t~Ly exp(--WJLy)--Ul

-- --1 m exp(--W1/LY)}

exp(--alW,) 1

(B3)

where

Q =oqNoL2y/Dx(1- oqLy)22

t = D,/SILy

m = (1 + t)/(1 -- t)

t' = (1 + D,oq/S,)/(1

-- t)

The total short-circuit current contribution per grain from this region having varying diffusion length Ly = y(1 -- p) + PLbp is given by

IAp,l~am boundary

=

4/bp J~p,lx -- w,(dg -- 2y) dy

(B4)

0

The multiplying factor 4 in eqn. (B4) arises from the four surfaces of the planes of the grain boundary surrounding each square columnar grain. The light-generated current and dark current per grain are thus

94

[Ap~Iphoto.skin, grain boundary

2qNoLbp --

OQ

(SiLbp _DI) {_2W~ 1 \S1Lbp +D-~I exp \ Lbp]

t

+ [ S--1+ c~ID!-1 exp (-- ~-~p) -- exp(--ohW,) l \D1 + S1Lbp]

× [(dg(1 -- p) + 2pLbp + 8/ohm}oh(1-- P) [ 1 - o~IL2bp~ 4aIL~p ] ×ln(l---~p) + (1-----~3J Iap,]d~k .ki~. : qDffalo[ (~--_-~)2 grain boundary

(l--p)

1--exp --

\W~]

Lbp ]

\ Lbp]J

(B5)