Solar Cells, 20 (1987) 279 - 287 279

A THEORY FOR THE SIMULTANEOUS DETERMINATION OF THE MINORITY CARRIER LIFETIME, DIFFUSION LENGTH AND DIFFUSION CONSTANT IN A SEMICONDUCTING MEDIUM USING A MODULATED LIGHT BEAM

RAVDEEP KAUR, SURESH GARG* and FEROZ AHMED

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

(Received April 29, 1986;accepted in revised form November 12, 1986)

Summary

We report a new method for the simultaneous determination of the minority carrier lifetime, diffusion length and diffusion constant in a semi- conducting medium. The method consists of a space-dependent study of the amplitude and phase of the minority carrier density (or current) induced by a sinusoidally modulated photon beam. To illustrate our method, we have solved a one-dimensional time~lependent diffusion equation using a finite Fourier transformation. The procedure is exact and can readily be extended to multidimensional systems.

1. Introduction

A knowledge of the diffusion length L and lifetime r of minority charge carriers in semiconductor materials is vital in determining the performance of solar cells. Several methods have been used to determine these parameters [1]. The beam-induced current (BIC) method is widely employed to deter- mine L [2 - 8]. The value of r can then be obtained through the relation L = (Dr) 1/2, provided the diffusion constant D is known. When one uses a modulated beam whose intensity varies sinusoidally with time, the induced current exhibits a phase lag with respect to the phase of the incident excita- tion. Phase shift analysis of such a modulated current has been used to determine the value of r [9 - 13]. By combining the phase shift and conven- tional electron-beam-induced current methods, efforts have been made to simultaneously obtain r and D [14- 16]. Using a one-dimensional (1D) diffusion equation, Fuyuki and Matsunami [16] obtained expressions for phase shift per unit distance, which we will denote by the symbol ~(~), for the limiting cases of very low and very high frequencies. (They found that ~(c~) varies as r 1/2 for cor ~ 1, whereas it is constant, independent of r, for cor >> 1. Here co is the angular frequency.) By combining these approximate

*On study leave from Hindu College, University of Delhi, Delhi-ll0007, India.

0379-6787/87/$3.50 © Elsevier Sequoia/Printed in The Netherlands

280

expressions Fuyuki and Matsunami generated an empirical relation between L~(w) and 097 for the entire frequency range so that at any modulat ion frequency it is possible to calculate 7, knowing L and ~(¢~). To obtain D, they used the relation connecting it with L and 7 mentioned above. We present here a theory for the simultaneous determination of all the three above-mentioned parameters independent of each other, wi thout resort to any empirical scheme. It consists of a space-dependent s tudy of the amplitude A ( x , 09) and the phase O(x, 09) of the minority carrier density induced by a modulated light beam of different frequencies. From this one can easily calculate the amplitude at tenuat ion factor a ( ~ ) and phase lag per unit distance ~(¢o). As shown in Section 3 these can be combined to yield all the parameters of interest.

To illustrate our method, we have solved a t ime-dependent 1D diffusion equation, with an intensity-modulated monochromat ic light source in a semi- infinite slab (0 ~< x < ~ ) of a semiconducting material using a finite Fourier transformation. As will be observed, the procedure is exact and straight- forward and can readily be extended to multidimensional finite systems.

2. Mathematical formulation

Let us consider a semi-infinite slab of a semiconductor uniformly illuminated by a monochromat ic sinusoidaUy modula ted light beam, as shown in Fig. 1. The t ime-dependent 1D diffusion equation describing the modulated density ~n(x, t) for excess charge carriers is

a 2 g(x, t) 1 a ~x 2 5n(x, t) -- ~:Sn(x , t) + - - - 5n(x, t) (1)

D D at

where ~2 = (Dr)-1 and g(x, t) gives the rate of generation of excess carriers and may be taken to be of the form

g(x , t) = aF exp{--a(x -- x0)) exp(iwt)

= Q(x, co) exp(iwt) (2)

\ x ,.

\

Fig. 1. A u n i f o r m l y i l luminated semi- inf in i te slab o f a s e mi c o n d u c to r .

281

Here a is the optical absorption coefficient and F = F(k) is the amplitude of modulated photon flux at x = x 0, a point in the incidence plane. (The origin is identified as point x 0 in order to give a more general formulation.) We take a solution of the form

~n(x, t) = ~n(x, co) exp(io~t) (3)

That is, we assume that the excess minori ty carrier density has the same frequency as the incident photon beam.

On combining eqns. (1) and (3) we obtain a2

ax 2 5n(x, w) -- p25n(x, w) = --Ql(X, (.D) (4)

where Ql(x, co) = Q(x, co)/D and the square of the inverse complex relaxa- t ion length is given by

p: = ~2 + io 2 (5)

with o 2 = ¢o/D. We have solved eqn. (4) using the method of finite Fourier transforma-

t ion originally proposed by Kobayashi et al. [17] to solve a steady state neutron diffusion equation. I t consists of Fourier transforming the given equation over a finite region and demanding that the t ransformed density be finite and non-singular in the given domain. We multiply eqn. (4) by exp(ikx) and integrate over x in the region x0 < x < xl. This gives

B(k, w) + l(k, ~-~(k, w) = (6)

p: + k 2

where

w) = :' x o

O (k, w) = f ' 3%

exp(--ikx) 8n(x, co) dx

exp(--ikx) Ql(x, co) dx

(7a)

(7b)

and

B(k, co) = [exp(--ikx)(Sn'(x, co) + ikSn(x, o~)~]~ =x, (7c) ~ X 0

The prime over 5n(x, w) indicates the gradient in x. From eqn. (7a) we note that ~'n(k, co) is finite and will have no

singularities in the k-plane. Therefore, the numerator in eqn. (6) must be zero for k = -+ ip. Thus one obtains a set of linear algebraic equations

B(+ip) + ~l(+ip) - 0 (8)

On combining eqns. (7a) to (8), one can eliminate 8n'(xl, co) and express 8n'(xo, co) in terms of ~n(xo, co) and 8n(xl, co).

282

To calculate 5n(x, w) at any x (x0 < x < xl) , we replace xl by x in eqns. (7a) - (7c) and repeat the procedure used in arriving at eqn. (8). This leads to a set of linear algebraic equations for 8n(Xo, ¢~), 5n(x, col 5n'(x0, w) and 5n'(x, ¢o). As before, eliminating 5n'(x, w) and substituting for 5n'(x0, w) one obtains, on simplification

sinh P(xl -- x) sinh p(x -- Xo) 8n(x, ~o) = 8n(x0, w) + ~n(xl, w)

sinh p(xl -- Xo) sinh P(Xm -- Xo)

X

+ s i n h p ( x l - - x ) fQl(x', )sinhp(x'--xo)dx' p sinh P(xl -- Xo) Xo

sinh p(x -- x o) x l + f QI(x', w) sinh p(xl - - x ' ) dx' (9)

p sinh P(xl -- Xo)

As will be noted, this method has the advantage that it requires no inverse Fourier transformation, which usua!ly complicates the mathematics, particularly for multidimensional systems.

For a semi-infinite slab geometry, the boundary conditions can be written as

~n(x, ~o) ~ 0 as x >oo (10)

~n(x ,~o)=O for x = x 0

The latter condition implies that no net photogeneration takes place at any point in the incidence plane. The solution given by eqn. (9), subject to the boundary conditions (10), takes the form

aF ~n(x, ~ ) - [exp{--p(x -- Xo)} -- exp(--a(x -- Xo))] (11)

D(a 2 _ p2)

where p is the inverse complex relaxation length. Obviously, 5n(x, ¢o) is a complex quantity.

The corresponding expression for current J(x, co) can readily be obtained using the relation

d J(x, ¢o) = - - q D - ~ 6n(x, ¢o)

where q is the charge of the minority carriers. Proceeding further, we note that one can also write 5n(x, w) as

5n(x, o~) = A(x , ¢~) exp{i0(x, co)} (12)

where

A(x, ¢o) = [{Re 5n(x, ¢o)} ~- + {Ira 8n(x, ~o)}2] "/2 (13)

and

O(x' c ° )= tan - l l Im ~n(x' aO ~n(x, ¢o) (14)

283

are respectively the amplitude and phase of the modulated carrier density. If we now put

p = a(co) + if(co) (15)

and separate t h e real and hnag inary parts o f eqn. (11) , we obtain

a r R e an(x, co) = D((a 5 _/£2)2 + o+) [ e x p ( - - a ( x - - X o ) ) ( ( a 5 - - / £z ) c o s ~(x - - X o )

+ o 5 sin ~(x - - Xo)) - - (a 2 - - ~z ) e x p ( - - a ( x - - Xo))] ( 16 )

and aP

Im 5n(x, co) = D((a 2 _/£2)2 -I- Or 4 ) [exp{--a(x --Xo))(o 2 cos ~(x --Xo)

_ (a 5 _ / £ 2 ) sin ~(x - Xo)) - o 2 e x p ( - - a ( x - - Xo))] (17 )

Hence, the amplitude and phase of the modulated carrier density are respectively given by

aF A(x, 6o) = D((a 2 _/£2)2 + o+)1/2 [exp(--2~(x --Xo)) + exp(--2a(x --Xo)}

and

- -2 exp{--(a + a)(x --Xo)) cos ~(x -- Xo)] 1/2 (18 )

O(x, co)

• 1[ °2c°s~(x--xo)--(a2--/£2)sin~(X--Xo)--°2exp{--(a--a)(X--Xo )) ] ---- ' l ; an - 2 ~ . . . . . . .

[ (a - / £ ) cos +(x - Xo) + 0 2 sin t ( x - Xo) - ( a 2 - /£2) exp{--(a - a ) ( x - Xo)) ]

(19)

These are the two basic quantities an experimentalist should measure in a modulated beam experiment. A direct comparison with corresponding calculated quantities can be used to test various theoretical models and approximations.

For co = 0, o 2 = ~ = 0 and eqn. (19) implies that 8(x, 0) = 0. Similarly, the expression for amplitude takes a compact form

aI' A(x, O) - D(a2 _/£2) [exp(--~(x -- Xo)) -- exp{--a(x -- Xo))] (20)

This is the same as that obtained for the time-independent excess carrier density [1].

Following the procedure outlined above one obtains similar expressions for the amplitude Aj(x, 0.3) and phase Oj(x, co) of the modulated carrier current

284

aqF Aj(x , 69) = {(a2_~2)2+ (o2)2}1/2 [--2 exp(--(a +cO(X--Xo)}

x {a~cos~(x -Xo) + at sin ~ ( x - X o ) )

+ (o~2 + ~2) exp{--2o~(x -- Xo)} + a 2 exp{--2a(x -- Xo)}l 1/2 (21)

and

O j ( x , G.)) ----- tan- l ( [ (o~o 2 -/- (a 2 -- ~2)~) c o s ~ X O )

+ {~o 2 -- (a 2 -- t~2)o~} sin ~(x - - X o ) - - a o 2 exp{--(a--cO(x--Xo)}]

× [((a 2 _ ,~2)~ _ o2~) cos ~ ( X - X o )

+ ( ( a 2 _ ,~2)~ + o 2 ~ ) sin ~(x - X o )

a(a 2 _ ~2) exp{--(a -- cO(x -- xo))] -1} (22) l

/

3. Results and discussion

As ment ioned earlier, A(x , w ) and O(x, ~ ) are the two basic quantities that an experimentalist should measure as a funct ion of distance in a modulated beam experiment. To study their space dependence at various frequencies, and to show that this information is sufficient to determine T, L and D simultaneously, we have carried out some numerical calculations for p-type silicon. The values of different parameters used here are given in Table 1. The value of F(X) has been taken to be equal to 1021 photons m -2 s -1 and the absorption coefficient a is 4 × 105 m -1, which correspond to an incident photon energy E of 1.9 eV (k = 0.66 #m). The results have been obtained for six different frequencies (0, 104, 2.5 × 104, 5 × 104, 7.5 × 10 4 and 10 s rad s-l).

In Figs. 2 and 3 we have plot ted A(x , ¢o) and 0(x, o~) as functions of distance from the incidence plane. From Fig. 2 we note that in all cases A(x , 6o) decays as a single exponential beyond about 0.5L. The amplitude

T A B L E 1

Va lues o f p a r a m e t e r s fo r p - t y p e s i l icon

Parameter Value

T 10 - 4 s L 3 0 0 / ~ m D 9 × 10 - 4 m 2 s -1

285

t : D

.. .I f t .

I

k

Iv O.O t.O 2.0 3.0 DISTANCE (in units of L}

Fig. 2. Variat ion of calculated ampl i tude A(x, 02) with distance: curve a, co = 0 rad s - l ; curve b, 0 2 = 1 0 4 tad s - t ; curve c, 02= 2 . 5 X 1 0 4 rad s - l ; curve d, 0 2 - - 5 X 1 0 4 md s - l ; curve e, 02 = 7.5 X 10 4 rad 8-I; curve f, 02 = lO s rad s -1.

at tenuat ion factor per unit distance ~ ( ~ ) can be calculated from the linear por t ion of the curve, since

d ~ (w) ffi - - ~ - In A(x, co)

Similarly, f rom the plot o f 0(x, co) vs. x (Fig. 3), we note that O(x, co) increases linearly with distance for all frequencies. From the slopes of these curves, one can readily calculate ~(w) since

d (co) = o(x, co)

Once ~(c~) and ~(oJ) are known, one can readily determine L, r and D. To this end, we first note that for co = 0, the inverse of ~ ( ~ ) gives the diffusion length of the minori ty charge carriers. F m ~ e r , on combining eqns. (5) and (15) we find that

R e ( p 2 ) = s: 2 = c~2(~) -- ~j:~(co) ffi (D'r) -1

and

Im(p 2) = 0 2 = 2ax(co)/j(co) = colD

2 8 6

o

ol o

o ~

o

X L~

II

3

7, X tO

II

3

I!

3

o. o o o o o o o. Q _0 aS ~ ~ ..d ~ ,~ ,,~ ~ - -

= ( p o J ) 3 S V H d

o

...1

°i c

o ~

o

"13

,'c~L ,,,,.,q

II

~ j

~ r.-.I

' N X ' ~

.~,~ ~

287

Thus if one plots o 2 and o2/~ 2 as a funct ion of frequency, the slopes of these curves will give the diffusion constant and lifetime of minori ty carriers. This is illustrated in Fig. 4. As expected, the values of these parameters obtained from Fig. 4 turn out to be the same as those given in Table 1.

An extension of this s tudy to a finite, multidimensional system with surface recombinations will be reported in a separate paper.

Acknowledgments

This work is, to a large extent, the outcome of the visit of one of us (F.A.) to the International Centre for Theoretical Physics, Trieste, Italy in 1985 as an associate. F.A. is grateful to Professor Abdus Salam, Director of the ICTP, for his encouragement and interest. We are thankful to Professor L. S. Kothari for some useful discussions. We also extend our thanks to one of the referees for his suggestions.

References

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