Solar Cells, 25 (1988) 61 - 72 61

A METHOD FOR THE DETERMINATION OF THE MATERIAL PARAMETERS T, D, L0, S AND a FROM MEASURED A.C. SHORT-CIRCUIT PHOTOCURRENT

SUDHA GUPTA, FEROZ AHMED and SURESH GARG

Department of Physics and Astrophysics, University of Delhi, Delhi 110007 (India)

(Received October 24, 1987 ; accepted May 4, 1988)

Summary

A procedure is described to determine accurately the material param- eters L0, D, T, S and absorption coefficient a which characterize the base region of a solar cell f rom measured a.c. short-circuit photocurrent (both amplitude and phase) at certain low (cot < 1) and high (cot > 50) frequen- cies. The method is general in the sense that it is valid for any arbitrary value of back surface recombination velocity S and thickness of the base region Xj. Earlier methods consisted in obtaining only one or two material parameters using theoretical expressions for the phase shift under different approximations, in particular, taking S ~ ~ and Xj /Lo >> 1. Here, we have only assumed that the back surface of the solar cell is illuminated uniformly with light of short wavelength such that aX~ >> 1. In order to clearly bring out the frequency dependence of amplitude and phase at both low and high frequencies, numerical results have been presented for an n+/p junction solar cell. The results have been given for two limiting values of Xj and S. The method is equally valid for a p+/n junction solar cell.

1. Introduct ion

The characterization of the base region in a homojunct ion solar cell is very important from the point of view of the cell's performance. The parameters that characterize a material are the minori ty carrier lifetime T, diffusion constant D, diffusion length L0 and surface recombination velocity S. The absorption coefficient a is another important parameter since it determines the ability of a cell to absorb light. Recently, the a.c. photo- current (or EBIC) method has been analysed both theoretically and experi- mentally to determine one or more of these parameters [1 - 11]. The meth- od is based on the principle that when an intensity-modulated light (or electron) beam is incident on the free surface of a pn junction, an a.c. short- circuit current Jsc is generated which exhibits a characteristic phase shift with respect to the incident beam. This phase shift (and also the amplitude IJscl ) can be measured experimentally at different modulation frequencies

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and as a function of beam position. Many workers [2, 4, 8] have only carried out measurements of phase shift at certain modulation frequencies for a fixed beam position. Material parameters such as the carrier lifetime and diffusion constant were determined using simple limiting expressions for the phase shift. As can be easily seen (Section 2), these approximate expressions are valid only under particular conditions. For example, in most cases, a large value of surface recombination velocity has been assumed [2, 4, 8] and also the thickness of the absorber region has been taken to be either much smaller [8] or much larger [2] than the diffusion length. Re- cently, some studies, both theoretical and experimental, have been made in which the beam position relative to the junction has been varied and both the phase and amplitude of Jsc have been studied [3, 6]. To deduce material parameters from the measured results, various procedures have been suggested which are based on certain assumptions and some of which are quite involved. For example, Kamm and Bernt [3] have given a proce- dure to determine D, T and S which seems quite involved and also not so accurate. To deduce the above parameters from measured amplitude and phase values, their method requires that theoretical curves for amplitude and phase for various L 0 and S/D values should be available. Further, in the region close to the actual values of L 0 and S/D, a large number of such curves needs to be available to obtain correct values of T and ~. On the other hand, Fuyuki and Matsunami [6] used a semi-empirical relation for phase shift per unit distance to determine D and r, while L0 was obtained from the low frequency results of the decay of amplitude as a function of beam position. Von Roos [5] has shown theoretically that the parameters T, D and S can be determined if the time~lependent short-circuit current generated at the junction is mixed with certain phase shifted original modu- lated beam and the resulting d.c. component is processed via a least squares filter over a full cycle. This leads to two equations (eqns. (A5) and (A6) of ref. 5) and four unknowns. It is shown that in the limit XjlLo>~ 1, two unknowns drop out and one can obtain T, D and S. However, only T and S can be determined, provided D is known, if one assumes XjlLo'~ 1. Note that the method involves numerical integration to obtain the two equations mentioned above which are then used to determine these parameters.

From the brief survey presented above, one can easily see that there is a need for more detailed theoretical investigations of the information contained in the intensity modulated beam experiment and for simple and accurate methods for determining all the important material parameters. Recently, Ahmed and co-workers [9- 11] have suggested a method for determining L0 and D from the measured local (space-dependent) values of the a.c. photocurrent at different frequencies. In this paper, we describe a procedure to obtain simultaneously the material parameters L0, D, v, S and a that characterize the base region of a homojunction solar cell from measured amplitude and phase values of short-circuit current generated at the junction at certain low and high modulation frequencies of the incident beam. The procedure is quite general and is valid for any value of surface

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recombination velocity and thickness of the base region. Note that this procedure can also be used to determine the material parameters in other photovoltaic devices. One only requires that the surface is illuminated uniformly with light of short wavelength such that aXj >~ 1.

2. Mathematical formulat ion

Before an expression for Js¢ is written and the procedure for evaluating the parameters characterizing the base region of a solar cell is described, we will list the conditions under which the present model will be valid. These are as follows.

(1) The back surface of the solar cell is uniformly illuminated by modu- lated monochromatic light (or in other words, the beam diameter is assumed to be larger than the diffusion length) so that a one<limensional (l-D) model can be applied.

(2) The input light intensity is such that the low injection level condi- t ion is satisfied, in order that the recombination rate may be described by the expression given by Shockley and Read [12].

(3) The base region is doped uniformly so that there is no electric field in this region and also so that the various material parameters can be taken to be space independent.

(4) The optical absorption coefficient is large, so that the contribution to the induced photocurrent is entirely due to the carriers generated in the base region.

(5) At the junction edge (X = Xj), the excess minority carriers are swept away to the other side of the junct ion by the electric field in the depletion region. Hence the concentration of the excess minority carriers at X = Xj may be taken to be zero.

Let us consider an n+/p junction solar cell, whose back surface is illuminated uniformly by an intensity modulated light beam of short wave- length {Fig. 1). A 1-D continui ty equation governing the excess concentra- t ion 5n(X, t) of the minori ty carriers (here, electrons) at a distance X from the illuminated surface and at t ime t in the p-region, may be written as

O25n(X, t) g(X, t) 1 OSn(X, t) OX ~ Ko25n(X, t) + - (1)

D D 3t

where g(X, t) is the generation rate of the minority carriers at a distance X and time t. D is the minori ty carrier diffusion coefficient, r is the lifetime and K02 , given by

K02 = (Dr) -1 (2)

is the inverse of the square o f the minori ty carrier diffusion length. The optical generation rate of minori ty carriers can be written as

g(X, t) = aF exp(--aX) exp(icot) (3)

64

X~

X=Xj

X=0

n +

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Fig. 1. A back-surface illuminated n+/p junction solar cell.

where a is the optical absorption coefficient, F is the amplitude of the modulated photon flux at X = 0 and co is the frequency of modulation of the incident photon beam.

With g(X, t) given by eqn. (3), the solution of eqn. (1) for excess minority carrier concentration may be assumed to be of the form

5n(X, t) = 5n(X, co) exp(icot) (4)

On combining eqns. (1), (3) and (4) we obtain

3 2 aP 3X 2 dn(X, c o ) - p25n(X, co)- -~ exp(--aX) (5)

where p2 is the square of the inverse complex relaxation length and may be written as

p2 = K02 + io2 (6)

with

co 0 2 - (7)

D

The boundary conditions to be satisfied by 6n(X, co) at X = 0 and X = X~ are

3~n(X,3x (.o)x=0= /Sl~n(0, ~ o ) \ / ~ ] (8a)

and

dn(X~, a)) -- 0 (8b)

If we define dimensionless parameters

X SLo x= =- , s= ,a =aLo,p =pLoand =oL o (9)

Lo D

65

then the solution of eqn. (5), subject to the boundary conditions (8), is given by

LoF/D [ ( 1 ) { l + ( s / a ' ) ) s i n h p ' ( x j - - x ) ( 1 ) - - +

fin(x, ¢o) 1 -- (p'/a') 2 (s/p') sinh p'x i + cosh p'xi

{cosh p'x + (s/p') sinh p'x} e-a'xJ -- {(s/p') sinh p'x~ + cosh p'xj) e -a'x] × l (s/p') sinh p'xj + cosh p'xj

(10)

The resulting short-circuit current density Js¢ is given by

85n(x, 6o) x= (11) jsc( a) ) = --qD aX xj

where q is the charge of the minority carrier. Combining eqns. (10) and (11), we obtain

[ l + (sla') 1 I l l Jse = qF 1 -- (p'/a') 2 (s/p') sinh p'xj + cosh p'xj 1 -- (-p'/a') 2

l ( ~ ) sinh p'xt + (s/p') c°sh p'Xi l exp(--aYxl)] (12a) )< 1+ (s/p') sinh p'x| + cosh p'xj

Alternatively, the above expression may also be written as

( I ) [I+(S/aD} 1 Js¢ = ~ = qF [ i = ~ (S/pD) sinh pXj + eosh pXj

I (P/a)(sinh pX| + (S/pD) c°sh pXi) l ] 1 1 + exp(--aX l)

1 -- (p/a) 2 (S/pD) sinh pX l + cosh pX i (125)

o r qFLa

X (SL/D) sinh(Xj/L) + cosh(X i/L)

where

1 L -

P

(12c)

(12d)

Expressions (12b) and (12c} are the same as those given by Morahashi et al. [8] and Muller et al. [4] respectively.

If we choose a sufficiently high energy of the incident beam, such that at the highest frequency of modulation used p' (= (1 + ic0r) In) and

66

are bo th much smaller than a', then the cont r ibu t ion of the second

where

A = a ' 1 + -- s + s i n h a x j c o s ~ ' x j a'

+ ( 1 +

and

s) , I ~; Ca' c o s h a x j cos~ 'x j s i n h a ' x | s in~ 'x j ) exp(- -a 'x j )

B = ~' 1 + - - s + cosh a 'x t sin ~'xj + --7 sinh o~'x t cos ~'xj a

(20a)

1/xj t e rm in eqn. (12a) can be neglected and the equa t ion reduces to

1 Jsc = qF{1 + (s/a')} (13) (s/p') sinh p'xj + cosh p'xi If we put

p' = a ' + i~' (14)

then eqn. (13) may be rewri t ten as

Jsc = q F ( 1 + (s/a')}(a' + i~')[(~' + i~')(cosh a'xj cos ~'xj

+ i sinh a 'xj sin ~'xj) + s{sinh a 'xj cos ~'xi

+ i cosh a'xj sin ~'xj}] -1 (15)

One may also [8] rewri te eqn. (13) as

Jsc = qF{1 + (s/a')}(cosh ~'xj cos ~'xj) -]

( tanh~'x~+itan~'xJ) -1 ' ' ( 1 6 ) × l + i t a n h a x j t a n ~ x j + s a ' + i ~ '

As is evident, jsc is complex and hence may be wr i t ten in the fo rm

js~(~) = Aj(co) exp{i0j(¢o)} (17)

where Aj(c~) and 0j(¢o) are the ampl i tude and phase of the short-circuit pho tocu r ren t respect ively. If we take j ~ , as given b y eqns. (12), then we get the fol lowing expressions for A~(eo) and 0g(~)

Aj(co)=qF(A2+B2)'/2(C2+D2) - 'n 1-- ~ +~a,2 ] ~ (18)

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+ 1 + ((x' sinh (x'xj sin }'xj + }' cosh oe'xj cos }'xj) exp(--a 'x j )

C = oe' cos }'x~ cosh a 'xj - - ~ sin }'x| sinh a 'xj + s cos }'x| sinh a 'xj

D = a ' sin }'x~ sinh o~'xj + }' cos }'xi cosh a'xj + s sin }'xj cosh a'xl

In t he approx imat ion a' >> a ' and a' >> }' these reduce to

A j ( ~ ) = ql-' 1 + (o~,2 + + },2 + s2) sinh2o~,xl

and

+ (a,2 + },2) cos2},xj + s 2 sin2},xj + 2s(a' cosh o~'xt sinh a 'xj

+ }' cos }'xj sin }'xl)} - ' /2

tan 0 j(G))

~' cos }'xj sinh a 'xj -- c~' sin }'xj cosh o~'xj 0d'2 + }'2

sin }'xj sinh a 'xj

(20b)

(20c)

(20d)

(21)

}' sin }'xj cosh a 'xj + a ' cos }'x~ sinh a 'xj +

which can also [8] be wr i t t en as

0/'2 + }'2 cos }'xj cosh a'xj

(22a)

8 a,2 + },2 (}' t anh a 'xj - - a ' t an }'xj) -- t anh a'x~ t an ~kj

tan 0 j ( ( . O ) ---- (22b)

1 + S

a , : + },2 (a ' t anh a'xj + }' tan }'xj)

We will now show how expressions (21) and (22) can be used to obta in all five mater ia l parameters .

3. Determination of material parameters

In order to show explici t ly how one can obtain the various material parameters f rom the exper imenta l ly measured values o f Aj(co) and 0j(co) at certain low and high frequencies , we wri te eqns. (21) and (22a) in te rms of actual mater ial parameters a, S, etc.

Aj(~)=qP 1 + (a2+}2),/2 a 2 + } 2 + D-2 sinh2aXj

68

8 2 + (o~: + }2) cos2}Xj + ~ sin2}Xj +

+ } cos }X i sin }Xj) l_l/21 /

and

tan Oj(w)

2S ~ - (a cosh ozX i sinh c~fj

(23)

sinh ozX i cos ~X i - - c~ cosh o~,Y l sin ~X i -- (DIS)(t~ 2 + ~2) sinh oU~ i sin ~X i

cosh ol.X i sin ~X i + a sinh oz.,,Y i cos }Xj + (D/S)(o~ 2 + ~2) cosh oU~ i cos ~X i

(24a)

where

(3, ~, a = - - and ~ - (24b)

Lo Lo

3.1. High f requency region In the high f r e q u e n c y region (cot > 50) we m a y t ake

o~ = ~ = \ 2 D ]

Also

1 sinh vdfj -~ cosh ovY 1 = ~ exp(~Yj)

Under this a p p r o x i m a t i o n , express ions fo r r ewr i t t en as

AjH(60) = qF + w 1/2 2 + +

and

O~a(w) = I~'-~] J + tan-1 r ) ) -~w

1+~

I f S is large

D ( 2 w l l n

t hen eqn. (27a) can be wr i t t en as

O j ~ ( ~ ) - - - X j + - 4

(25a)

(25b)

a m p l i t u d e and phase m a y be

ex l

(27a)

(27b)

69

It can be seen that eqn. (27b) can be used to determine the value of D if the value of phase is known at a certain high frequency [8]. However, if S is not too large, then D can be determined using eqn. (27a), provided one measures the phase at any two high frequencies. If Wl and to2 are two such frequencies, then f rom eqn. (27a), we have

f 1 (tOlt (- 1 ( t O l / i n anl0jH(tO1) + t ~ ] X,~

an 10zlt(to2)+ ~2-D][ tO2/'/2X~Jf -- (28)

Equation (28) is a transcendental equation, containing only one unknown parameter, namely D: hence, it may be solved to obtain the value of D. Equation (27a) can then be used to get the value of S. Once S and D are known, the value of a can be obtained from the expression for the amplitude A~ (eqn. (26)).

3.2. Low frequency region The remaining two parameters, L0 and 7, can be easily obtained from

the measured values of amplitude and phase at certain low frequencies. For tOT < 1 one finds

1 tOt -~ L--0 and ~-~ 2Lo (29)

In this approximation, expressions (23) and (24a) reduce to

AjL(to) = qF 1 + cosh(Xj/Lo) + (SLo/D) sinh(Xj/Lo) (30)

and

tot 1 -- (D/SLo)(Xj/Lo) sinh(Xi/Lo) -- (Xj/Lo) cosh(Xj/Lo) 0jL(to) = - - (31)

2 sinh(Xj/Lo) + (D/SLo) cosh(Xj/Lo)

respectively, where it has been assumed that ~Xj < 1. From eqn. (30) the value of L 0 can be determined since a, D and S are known. Substituting all the known values into eqn. (31), the value of T can be calculated. Knowing Lo and D, v can also be obtained using the expression Lo = (Dr) 1/2

4. Results and discussion

The values for the amplitude and phase of the short-circuit photo- current generated when a high energy modulated photon beam illuminates

7 0

uniformly the back surface of an n+/p junction solar cell have been cal- culated using the general expressions (18) and (19) in the normalised fre- quency range cot = 0.01 - cot = 100. (Note that the minority carrier lifetime r in the base region of a silicon solar cell is sufficiently high (ps) that the frequencies in the high frequency region ( c o t > 50) will correspond to frequencies in the MHz region which can be easily obtained. Muller e t al. [4] have reported their phase-shift measurements in GaAs in the frequency range 1 - 1 0 0 MHz.) Results have been given for two thicknesses of the base region and for two values of surface recombination velocity. In order to make the results material independent, normalised parameters (s = S L o / D, a' = aLo, x~ = X~/Lo and W = coT) have been used.

In Fig. 2 the amplitude Aj of the photocurrent is plotted as a function of normalised frequency (cot = 0.01 - cot = 100) for xj = 0.5 and 2.5 and

1 0 2

io-

\ \ \ \ \ \ \

\\\~ 1 /

/

t

t

I

~5 L I L I0 -z i(j-I I0 0 I01 10 2 Nor molised f r eq uency~ t.a3 E. =

5 ~

Fig. 2. V a r i a t i o n o f c a l c u l a t e d a m p l i t u d e o f a.c. s h o r t - c i r c u i t p h o t o c u r r e n t w i t h n o r m a l - i sed f r e q u e n c y . S o l i d curves c o r r e s p o n d to s = 1 and d a s h e d cu rv es to s = 103 Curves a a n d b are for x] = 0 .5 a n d xj = 2 .5 r e s p e c t i v e l y .

71

s = 1 and 103. From Fig. 2 it is evident that in all cases the amplitude re- mains constant up to a certain normalised frequency Wc (= weT) and then falls of f rapidly as the frequency increases. This is also seen from expressions of the amplitude in the low and high frequency regions, eqns. (30) and (26) respectively. In the low frequency region (eqn. (30)), amplitude is found to be independent of frequency. Further, W¢ is found to depend both on xj and s. It decreases as x~ increases, whereas it is only slightly increased with increase in s. We also find that over the entire frequency range, the amplitude decreases with increase in base region thickness as well as surface recombination velocity. The effect o f s is small in the high frequency region. The decrease of amplitude with increase in s is due to the lower number of carriers entering the base region, while for a thicker base region (large xt) a smaller number of carriers would be able to diffuse to the junction.

Figure 3 shows the phase 0~(c0) obtained fr.o~a eqn. (19), as a function of normalised frequency for the two above-mentioned values of surface recombination velocity and thickness of the base region. Here, we find that up to a certain We, 0~(co) varies linearly with W as can be seen from eqns.

~o2 r l

/ / / / / /

/" / / / /

/ / l / / / / /

o_ -I / I0 / /

/ /

i(~ 2 / / / /

/ /

/ ( 3 /

/ /

I(]'~ / /

l / /

~6"L r i ~0 "2 i0 -~ i0 0 I01 i0 z

Normofised frequency, OJ ~:-

Fig. 3. V a r i a t i o n o f c a l c u l a t e d p h a s e s h i f t o f a.c . s h o r t - c i r c u i t p h o t o c u r r e n t w i t h n o r m a l - i sed f r e q u e n c y . S o l i d curves c o r r e s p o n d t o s = 1 a n d d a s h e d curves t o s = 10 3. Cu rv es a and b are f o r x I = 0 .5 a n d xj = 2 .5 r e s p e c t i v e l y .

72

(27a) or (27b). Here also we find that Wc depends on both s and x i. Once again, it decreases as the thickness of the absorber region is increased where- as, as before, it increases only slightly with increase in s. Further, we note that over the entire frequency range, the phase increases with increase in base thickness, as expected, because the excess carriers generated on the surface require a longer time to reach the junction. Phase decreases as s increases over the entire frequency range: again, its effect is very small in the high frequency region.

5. Conclusion

From the study described above we find that most of the important material parameters (including absorption coefficient} characterizing the base region of a solar cell can be determined accurately from measurements of the amplitude and phase of the short-circuit a.c. current at certain low and high frequencies. The procedure described for obtaining these param- eters is quite general, is valid for other photovoltaic devices and can easily be experimentally verified.

Acknowledgments

We are grateful to a referee for his valuable comments and suggestions. We also wish to thank Prof. A. Mansingh and Prof. L. S. Kothari for many useful discussions. One of us (S.G.) wishes to thank C.S.I.R. for financial assistance.

References

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276. 9 F. Ahmed and S. Garg, Simultaneous Determination of Diffusion Length, Lifetime

and Diffusion Constant of Minority Carriers using a Modulated Beam, ICTP Report, Trieste, Italy, IC/86/129.

10 R. Kaur, S. Garg and F. Ahmed, Sol. Cells, 20 (1987) 279. 11 R. Kaur, S. Garg and F. Ahmed, Proc. Int. Conf. on Current Trends in the Physics

o f Materials, I.I.T. Kanpur, India, 1987, p. 233. 12 W. Shoekley and W. T. Read, Jr., Phys. Rev., 87 (1952) 835.