Solid-State Electronics Vol. 31, No. 9, pp. 1401-1407, 1988 Printed in Great Britain. All rights reserved

0038-l lOI/ $3.00 + 0.00 Copyright 0 1988 Pergamon Press plc

A GENERAL THEORY OF AN INTENSITY-MODULATED BEAM METHOD FOR DETERMINATION OF DIFFUSION LENGTH, DIFFUSION CONSTANT, LIFETIME, SURFACE

RECOMBINATION VELOCITY AND ABSORPTION COEFFICIENT IN A SEMICONDUCTOR MATERIAL

SUDHA GUPTA, RAVDEEP KAUR, SURESH GARG~ and FEROZ AHMED

Department of Physics and Astrophysics, University of Delhi, Delhi-110 007, India

(Received 7 May 1987; in revised form 20 November 1987)

Abstract-Using a modulated light (or electron) beam, some efforts, both theoretical and experimental, have been made to determine parameters such as minority carrier lifetime, r, diffusion constant, D, and surface recombination velocity, S, in a semiconductor material. However, none of these workers considered a general space and frequency dependent response of the system. Moreover, the parameters were not determined independent of each other. Here we give a general theory of such a modulated-light (electron)-beam method. Space and frequency dependent expressions for the amplitude A(x, o) and phase shift 0(x, o) of modulated minority carrier concentration and current have been obtained. (These are the two basic quantities which are to be measured in a modulated beam experiment.) We have also obtained frequency dependent expressions for amplitude attenuation coefficient, a(o) and phase shift per unit distance, t(o). These can be determined experimentally from the slope of the measured space dependent amplitude and phase curves at different frequencies. It is shown that it is useful to construct quantities like 2a(w)t(w)[ = u2(o)] and a*(o) - r2(o)( = ki) which help in directly determining carrier lifetime, diffusion constant and diffusion length, L,, . The procedure for determining S and a, the optical absorption coefficient, knowing the measured values of A and 0 at a certain distance, has been given. It may be pointed out that the present analysis enables us to determine all 5 parameters mentioned above simultaneously and independent of each other from the measurements in a modulated light beam experiment.

NOTATION

lifetime of minority carriers (s) diffusion constant (m2 s-l) surface recombination velocity (ms-‘) amplitude of the modulated minority carrier concentration (m--‘) phase of the modulated minority carrier concen- tration with respect to the phase of the incident beam (rad) distance from the incident plane (m) diffusion length of minority carriers for UI = 0 (m) distance from the incident plane in units of 4 angular frequency of modulation of the incident beam (rad.s-‘) amplitude attenuation coefficient (m-l) phase shift per unit length (m-‘) inverse of real diffusion length of minority carriers for 0 = 0 (m-l) optical absorption coefficient (m-l) modulated minority carrier concentration (m-‘) rate of generation of excess carriers (m-‘s-l) amplitude of modulated photon flux in the incidence plane (m-* s-l) inverse of complex diffusion length of the minority carriers (m-‘) modulated minority carrier current density (Am-‘) amplitude of the modulated current density (Amv2) phase of the modulated current density with respect to the phase of the incident beam

1. INTRODUCTION

In the past, several method@-461 have been used to determine parameters like diffusion length &,, life- time 2, diffusion constant D, surface recombination velocity, S and optical absorption coefficient, a in materials used for solar cells and other devices. The intensity modulated light (or electron) beam method has recently been used to infer some of these par- ameters by several workers[32-45]. However, none of them studied the general space and frequency de- pendent behaviour of the amplitude ,4(x, w) and phase 0(x, w) of the modulated carrier density or current generated in the semi-conductor. In most cases the phase between the short circuit current and the incident modulated beam was measured as a function of frequency in the low or high frequency region[35-39,421 to obtain values of carrier lifetime and diffusion constant. The diffusion length was then obtained using the relationship L, = fi. Fuyuki and Matsunami[40] have reported the measurements of amplitude and phase of the modulated short- circuit current. They obtained D and z using a semi- empirical expression for l(w) and L, was obtained from the amplitude measurements at low frequencies.

Recently, we reported a method for simultaneous determination of 7, L,, and D by analysing the space

ton study leave from Hindu College, University of Delhi, and frequency dependence of A and 0[45]. In this, we Delhi. had taken surface recombination velocity to be

1401

1402 SUDHA GUPTA et al.

infinite. In actual practice, S is finite and will influence the space dependence of A and 0. In this paper, we have extended our earlier study to the case of finite surface recombination velocity. Space and frequency dependent expressions for A and (3 for both, minority carrier concentration and current density have been obtained. Using these expressions for A and 0, a procedure for extracting various material parameters from measurements of space and frequency dependent amplitude and phase has been described. To make the procedure clear, curves show- ing spatial variations of A and 0 have been simulated for P-type Si at different frequencies using the theoretical expressions. One may regard these curves as obtained experimentally. Slopes of these amplitude and phase curves, at a frequency w, give the values of a(w) and t(w) respectively. Once E(W) and c(o) are known, ki and a’(~) can be calculated. The diffusion length can then be directly obtained from k,( = l/L,), which is independent of frequency. The values of D and 7 can be determined from the linear frequency dependence of cr2(o) and rr’(co)/kt (eqns (3 1) and (32), Section 2) respectively. It is shown that a can be determined by knowing the amplitude and phase values at a certain low frequency and at a fixed distance from the source plane, in addition to the values of a(w) and t(w) at that frequency. The surface recombination velocity, S can be obtained from the ratio of the amplitudes at a fixed point at any two frequencies, w, and w2 and the values of ~((0) and t(o) at these frequencies. Thus, it can be seen that from the measurements of “local” values of amplitude and phase in a modulated beam experi- ment, one can determine all the important material parameters simultaneously and independent of each other.

With regards to the measurements of A(x, o) and Q(.u. o) experimentally, we may mention that Goucher[46] has reported measurement of d.c. photo voltage, VP, at different space points, x, inside a filament of n-type Ge using a potential probe. Then, by plotting log VP vs x, they obtained the diffusion length of minority carriers (holes) from the slope of the curve. The value of diffusion length so obtained agreed with earlier reported values. More recently, Rieder et a/.[471 have studied minority carrier injec- tion and extraction in n-type Ge by measuring local values of dc. voltage using a single as well as a pair of movable potential probes. Similar procedure can be used to measure local values of a.c. photo- voltage due to a.c. photocurrent generated inside a semi-conductor when a modulated beam is used.

2. MATHEMATICAL FORMULATION

Consider a semi-infinite slab of a semiconductor uniformly illuminated by a monochromatic sinu- soidally modulated light beam, as shown in Fig. 1. The time dependent 1 -D diffusion equation describing the modulated density &(X, t) for excess

Fig. I. A uniformly illuminated semi-infinite slab of a semiconductor.

minority carriers is:

a%(x, t)

ax2 - k@l(X, t)

(1) - l aa-), D D dr

where ki= (Dr)-’ and g(X, t) gives the rate of generation of excess carriers. We take it to be of the form:

g(X, t) = UT exp(-aX)exp(iot), (2)

where a is the optical absorption coefficient and r is the amplitude of the modulated photon flux at X = 0.

We take a solution of the form:

&2(X, t) = Sn(X, w)exp(iot). (3)

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

On combining eqns (1) and (3), we obtain:

a%(x, w)

ax2 - p*&(X, 0) = -$ exp(-OX), (4)

where p2 is the square of the inverse complex relax- ation length and may be written as:

p2 = ki + ia2, (5)

with:

u2 = w/D. (6)

We have solved eqn (4) using the method of finite Fourier transformation[45]. For a semi-infinite slab, the result is:

Sn(X, w) = 6n(O, w)exp( -pX)

al-

+ D(a2-pZ) [exp( - PX) - exp( - aX)l. (7)

We take the following boundary conditions:

6n(X,~)+0 as X-+cc @a)

and

ah (x, w )

ax = (S/D)~dn(O, w). @b) x=0

Modulated beam method for measuring lifetime 1403

The solution given by eqn (7), subject to boundary The amplitude and phase of the modulated carrier conditions (8), takes the form: density are respectively given by:

&(X,W& 1

(n2--p2) p +; ( >

-4(x, 0) =&.c’fi i

exp(-2a’x) 0

x {u exp( - pX) - p exp( - ax) + (a’+ S’)*

(a’+S’)Z+51ZexP(-2W

+ S/D[exp(-pX) - exp(-WI}. (9) 2(u’ + S’)

It may be noted that for S + co, this expression reduces to eqn (11) of Ref. [45] with X, = 0. Since 6n(X, w) is a complex quantity, one can also express it as:

&(X,w)=A(X,o)exp[-iB(X,o)], (lo) and

- [

(x, + S,)2 + 5,2 [(a’ + S’) cos <‘x

l/Z -<‘sinl’x]exp(-(u’+a’)x) , (16)

where the amplitude A (X, o) and phase Q(X, W) are f?(x, 0) = tan-‘{[a’2[(a’ + S’)2 + t’2] given by:

xexp(-(a’-a’)x)+(u’+S’)

A(X, w) x (C, sin c’x + C, cos t’x)]

= J{Re[Gn(X, w)]}’ + {Im@(X, w)]}‘, (lla) x [(a’ + S’)(C, cos t’x - C, sin {‘x)

and - (a” - l)[(a’ + S’)2 + g”]

(1 lb) x exp( -(a’ - a’)x)]-‘}. (17)

If we put: An expression for complex carrier current density can readily be obtained using the relation:

p =a +it, (12)

then the real and imaginary parts of 6n(x, w) [eqn (9)] can be written as:

ReVn (x. w )I

= (l-/&D) u’C (a,+S,)2+512’{(a’+S’)

x [C, cos t’x - C, sin <‘x]exp(-a’x)

-(a’* - l)[(a’ + S’)2 + r’2]

x exp ( - a’x)} ,

and

ImW (x, w )I

= (-T&D) u’C (a,+S?Z+5’2.{(a’+S’)

x [C, sin t’x + C, cos l’x]exp(-a’x)

+ a’2[(a’ + S’)2 + t’2]exp(-u’x)},

where

x [u't'(a' sin <‘x + 5’ c0S t’x)

+ a’(a’ + S’)(a’ cos t’x + t’ sin l’x)]

I

r/2 x exp[-(a’+ a’)x] , (19)

(13b) and

c = [(a” - 1)2 + a’4]-1,

C, = (a’ + S’)(a” - 1) + t’a’2,

c* = <‘(a’2 - 1) -(a’ + S’)a’2,

0,(x, w) = tan-’ {[u’a’2[(a’ + S’)* + t’2]

xexp(-(a’-a’)x)+(u’+S’)

(14)

where the various dimensionless quantities have been defined as:

x=X/L,, u’=uL,,

a’ = aLo, 5’ = <Lo,

ts’ = aLo and S’ = SL,/D. (15)

J(x, co) = -qD&Lh(x, w), (18)

where q is the charge of the minority carriers. We obtain following expressions for the amplitude A/(x, o) and phase 6,(x, w) of the current density, J(x, w):

(134 + (a ’ + sy2

af2 + rf2

(a’+S’)2+t’2

2(u’ + S’) ~exp(-2a’x)-(~,+~,)~+~,2

x [a’(C, sin l’x + C, cos {‘x)

- t’(C, cos <‘x - C, sin {‘x)]]

X [(a’ + S’)[a’(C, cos t’x - C2 sin r’x)

+ t’(C, sin <‘x + C, cos <‘x)1

- u’(u’2 - l)[(a’ + S’)2 + <‘*I

x exp( -(a’ - a’)x)]-I}. (20)

1404 SUDHA GUPTA et al.

Table 1. Transport parameters for a p-type semiconductor at 300K for photon energy of 1.9eV

Material

Si Ge GaAs InAs InSb CdTe InP

Band

gap W

1.10 0.72 I .42 0.36 0.18 1.56 1.35

D

& (&s-‘) (2) & (al, a’

10-a 9 x 10-d 3 x 10-d 3.33 x IO’ 4x 10s 1.2 x 102 0.5 x lo-’ 1.01 x IO-1 7.1 x 10-d 1.41 x 103 IO’ 7.1 x 103 4.0 x 10-8 2.21 x 10-2 2.97 x 10-S 3.36 x IO4 6x 10” 1.78 x IO2 5.0 x 10rn’O 8.58 x 10-2 6.55 x 10m6 1.53 x 105 4.21 x lo6 2.76 x 10’

10-l 2.0 x lo-’ 1.41 x 10-4 7.07 x 10’ 3.61 x IO’ 5.09 x IO’ 2.3 x lO-9 2.73 x IO-’ 2.5 x 1O-6 4.0 x 105 4x 106 IO

4 x 10-8 1.2 x 10-2 2.19 x lO-5 4.56 x lo4 4.87 x lo6 1.07 x 102

A(x, w) and 6(x, w) [or A,@, CO) and 6,(x, o)] are the two basic quantities which an experimental& should measure in a modulated beam experiment in order to determine all the parameters of interest.

It is observed (Table 1) that for all materials, a $ L;’ and hence a’ 9 1. Further, even at very high frequencies 5 ‘, CI’ < a’ and a’* < a’*. Under these conditions and at distances not too close to the source plane, eqns (16), (17), (19) and (20) take simple forms:

r a’+S’ A(x, 0) =k,Da’@’ + q2 + 5’2]1’2 exp( -cz’x) (21)

B(x,o)={‘x+tan’x <‘(a + S’) a’* -- a’(a’+ S’) .‘* 1

and

A,@, 0) = Iqlr U’-tS

u’[(cr’ + S’)2 + 5’2]l’2

x Jmexp(-cc’x),

OXx,w)= -5’~ +tan’ [

<‘(a + S’)

u’(cr’ + S’)

r (2

S’(GI’ + S’)

(22)

(23)

(24)

The above expressions for amplitude and phase may be referred to as “asymptotic” expressions.

Frequency dependence of a’(~), t’(w) and CT*(O)

The frequency dependent expressions for a’(w) and t’(w) can be obtained by squaring eqn (12) and comparing it with eqns (5) and (6):

{

(1 + ID%*)“* + 1 ‘!I a’(w) = L,cc(w) =

2 1 (25)

and

5’(w) = &5(o) = (1 + w*7*)‘1* - 1 112

2 (26)

From these equations, it is clearly seen that both CL’(O) and t’(w) increase with frequency, but the difference of their squares is constant i.e.

LX’*(w) - 5’2(w) = 1. (27)

At very low and very high frequencies, IX’(W) and

t’(o) are given by:

and

cc’(w) = 1 (for 02 Q 1)

= Jwzi2 (for 07 % 1)

(2ga)

(28b)

l’(w) = 1/2wr (for 0.~7 6 1) Wa)

= Joti (for 07 + 1). (2W

The above limiting expressions for r’(w) are the same as obtained by Fuyuki and Matsunami[40].

As we show, it is useful to calculate real and imaginary parts of p*. These are given as:

Re(p*) = ki = u*(m) - t’(w) = k,

and

Im(p*) = o* = ~c((w)<(w) =g.

Also,

Im(p*) u* Re(pZ)=G=Wr.

(30)

(31)

(32)

We note that Re(p*) is independent of frequency whereas Im(p*) and [Im(p*)]/[Re(p*)] vary linearly with w in the entire frequency range.

Determination of various parameters

We will now outline the procedure that an experi- mentalist should follow for extracting the various material parameters from his measurements of space and frequency dependent amplitude and phase. TO

do so, we found it necessary to “simulate” curves for amplitude and phase at different frequencies. These curves for a p-type Si are given in Figs 2-5. For the following discussion, one may regard these curves as the experimental curves.

The straight line curves obtained in Figs 2 and 3 (drawn on a semilog graph) show that, in all cases, the amplitude decays exponentially in the entire distance range except very close to the source plane. We also note that for a fixed value of CO, all amplitude curves are parallel to each other. The negative of the slopes of these curves give us the value of amplitude attenuation coefficient, u(w), which depends only on w. Similarly, the phase shift per unit distance, t(w) can be obtained from the slopes of the curves in

Modulated beam method for measuring lifetime

w = lo5 rod s-’

c - S. IOzms-1

10q5- d-S~103mh

e-s.104ms-’

10’4, b I I I I 1 2 3 4

x(in units of Lo)

Fig. 2. Variation of calculated amplitude with distance for o = 2.5 x lO’rad.s-‘.

Figs 4 and 5. t(o) is also independent of S and depends only on o.

We will now show that if one knows a(o) and e(w), one can obtain all the 5 material parameters &,, D, t, S and a simultaneously and independent of each other. Lo, 7 and D can be easily obtained, if we construct the quantities ki[ = a*(o) - c*(o)] and a*[=2a(w)c(o)] from a(w) and t;(w). ki is indepen- dent of frequency and the diffusion length, L,, , can be directly obtained from it (L,, = l/k,). The values of D and 7 can be determined from the linear frequency dependence of a*(o) and a*(o)/ko2 [eqns (31) and (32)] respectively.

The absorption coefficient, a, can be determined independently, without involving other parameters, if one knows the values of A, 6 and a, { at a certain low frequency. At low frequencies, eqn (21) for the amplitude can be written as:

____ exp( - ax). (33)

w=2.5x103 rod s-’

b- S=lO* mr-’

Fig. 3. Variation of calculated amplitude with distance for w = 10srad.s-‘.

W= 2.5 X103 rod r-’

b -S=lO’ ms-’

c--s=102mr-’

d-S.103ms-

e--s*104m*-’

x (in units of Lo)

Fig. 4. Variation of calculated phase shift with distance for o =2.5 x lO”rad.s-‘.

On combining eqns (22) and (33), one obtains:

tan[fI(*,w)-SXJ=C.FA(X,o)exp(aX)-<. a

(34)

This can be rearranged to give the value of a:

[i

2aC a= w A(X, w)exp(aX) II ’ z r

- tan[e(X, w) - &Y]

(35)

We may note that the expressions A (X, w)exp(aX) and 0(X, o) - &Y in eqn (35) give the values of the amplitude and the phase as given by eqns (21) and (22) at X = 0. We may, therefore, rewrite eqn (35) as:

2a5 a= 0.) A@, 0) 11 -tanB(O,o) ’ (36) _-

2a r

where A(0, w) and f3(0, w) refer to the values of amplitude and phase at X = 0. If S is large, then a can be expressed in terms of 8, a and 5 only. From eqn (22), we find that for S 3 100

t 24 t?=&Y+tan-’ --.z , [ 1 (37)

a

This can be used to get the value of a directly at any frequency.

Surface recombination velocity, S can be deter- mined independently if one knows amplitude or phase at two frequencies, besides the values of a(o) and e(o) at these frequencies. As seen from Figs 2-5, the effect of change in S is more prominent in amplitude values than in those of phase. To get an expression for S, we first define the ratio r of the amplitude at two different frequencies o, and w2 at some fixed distance X from the source plane. If we use

1406 SUDHA GUPTA et al.

6r

a-S~loOm*- w.lOSrad C’

7- b-S=lO’ mC’

c - s =I02 nls-’ 6-

d-S=103ms-’

0 1 2 3

x (in units of Lo)

Fig. 5. Variation of calculated phase shift with distance for w = IO5 rad.s-‘.

eqn (21) for the amplitude, we get:

A(X WI)

r=A

where

y12(X) = exp](a, - ~2)Xl. (39)

Here CX, , (, and CC*, t2 are the values of CI and 5 for frequencies w, and o2 respectively.

Squaring and rearranging terms in eqn (33), we obtain:

SW2 - ---l [r*r:2(X) - l] + 2 [r*r:*(X)tl, - LX*]

4647 I

+r2r:2(X)(r:+5:)-(tL:+r:)=0. (40)

This is a quadratic equation for S and will have two roots. The negative root is discarded from physical considerations. If CO, > w2, the positive root is:

- (r2rf2(X)t: - (i)(r2ri2(X) - l)]“*

- (r’rf2(X)cq -cx,)}. (41)

The above expression for S can be used by an experimentalist to extract the value of S. We may note that the above expression of S contains only experimentally measurable quantities, like A, E and 5.

3. CONCLUSION

From the above study we may conclude that from space and frequency dependent measurements of amplitude and phase of modulated carrier concen- tration (or current) generated by an intensity modu- lated light beam, one can obtain all the important parameters (T, &,, D, S and a), which characterize a

semi-conductor material, simultaneously and inde- pendent of each other.

Acknowledgements-We wish to thank Professor N. K. Ray of Computer Centre, Department of Chemistry for allowing us to use the computer. We also wish to thank Professor Abhai Mansingh for many useful discussion.

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REFERENCES

F. S. Goucher, G. L. Pearson, M. Sparks, G. K. Teal and W. Shockley, Phys. Rev. 81, 637 (1951). L. B. Valdes, Proc. Inst. Radio Engrs 40, 1420 (1952). W. van Roosbroeck, J. appl. Phys. 26, 380 (1955). D. B. Wittry and D. F. Kyser, J. appl. Phys. 36 1387 (1965). H. Higuchi and H. Tamura, Jap. J. appl. Phys. 4, 316 (1965). K. L. Ashley and J. R. Biard, IEEE Trans. Electron Dev. 14, 429 (1967). W. H. Hackett Jr, J. appl. Phys. 43, 1649 (1972). H. C. Casey Jr, B. I. Miller and E. Pinkas, J. appl. Phys. 44, 1281 (1973). M. L. Young and M. C. Rowland, Physics. Status Solidi a 10, 603 (1973). L. Jastrzebski, J. Lagowski and H. C. Gatos, Appl. Phys. Lzett. 27, 531 (1975). C. Hu and C. Drowley, Solid-St. Electron. 21, 965 (1978). M. K. Alam and Y. T. Yeow, Proc. ofthe NASE CODE II Con& Dublin, p. 155 (1981). T. Fuyuki and H. Matsunami, Jap. J. appl. Phys. 20, 743 (1981). J. Dresner, D. J. Szostak and B. Goldstein, Appl. Phys. Lett. 38, 998 (1981). R. T. Swimm and K. A. Dumas, J. appl. Phys. 53,7502 (1982). D. I. Ioannou and C. A. Dimitriadis, IEEE Trans. Electron Dev. 29, 445 (1982). D. E. Burk, D. .Kannkr, J.‘E. Muyshoudt and D. S. Shaulis, J. appl. Phys. 54, 169 (1983). S. Hasegawa, T. Watanabe, A. Tanaka and T. Sukegawa, Rev. Sci. Instrum. 54, 1165 (1983). J. C. Manifacier, Y. Moreau and H. K. Henisch, J. appl. Phys. 54, 5158 (1983). C. T. Ho and J. D. Mathias, J. appl. Phys. 54, 5993 (1983). E. S. Nartowitz and A. M. Goodman, J. Electrochem. Sot. 132, 2992 (1985). R. M. Kingston, Proc. IRE 42, 829 (1954). D. T. Stevenson and R. J. Keyes, J. appl. Phys. 26, 190 (1955). R. H. Dean and C. J. Nusese, IEEE Trans. ED-18, 151 (1971). B. G. Streetman, Solid-State Electronic Devices, p. 184. Prentice-Hall, Englewood Cliffs (1972). D. L. Lile and D. M. Davis, Solid-St. Electron. 18, 699 (1975). S. R. Dhariwal. L. S. Kothari and S. C. Jain, Solid-St. Electron. 20, 247 (1977). J. E. Mahan, T. W. Ekstedt, R. I. Frank and R. Kaplow, IEEE Trans. Electron. Dev. ED-26,733 (1979). H. T. Weaver and R. D. Nasbv. Solid-St. Electron. 22.

I

687 (1979). T. Jung and A. Neugroschel, IEEE Trans, Electron. Dev. ED-31, 588 (1984). D. Gaquet, J. P. Nougier and G. Gineste, Physics B, C 129, 524 (1985). R. J. Carbone and P. R. Longaker, Appl. Phys. Lett. 4, 32 (1964). V. P. Sushkov and Yu A. Moma, Sov. Phys. Semicond. 1, 1273 (1968).

34. 35.

36.

37.

38.

39.

40.

Modulated beam method for measuring lifetime 1407

C. J. Hwang, J. appl. Phys. 42, 4408 (1971). 41. J. Pietzch. Solid-St. Electron. 25. 295 (1982). H. Reich1 and H. Bernt. Solid-St. Electron. 18, 453 42.

(1975). C. Munakata, N. Honma and H. Bole, Jap. J. uppl. Phys. 22, 103 (1983).

G. Schwab, H. Bemt and H. Reichl, Solid-St. Electron. 43.

20, 91 (1977). J. Muller, H. Bernt and H. Reich], Solid-St. Electron. 21, 999 (1978). 44. J. D. Kamm and H. Bemt. Solid-St. Elecfron. 21, 957 45.

(1978). J. Muller, H. Reich1 and H. Bernt, Solid-St. Electron. 46. 22. 257 (1979). 47.

T.‘Fuyuki and H. Matsunami, J. appl. Phys. 52, 3428 (1981).

Mokoto Morahashi, Nobuhiko Sawaki, Toshifumi Somatani and Isamu Akasaki, Jap. J. appl. Phys. 22, 276 (1983). Oldwig von. Roos., J. uppl. Phys. 57, 2196 (1985). Ravdeep Kaur, Suresh Garg and Feroz Ahmed, Solar Cells 20, 279 (1987). F. S. Goucher, Phys. Rev. 81, 475 (1951). G. Reider, H. K. Henisch, S. Rahimi and I. C. Manifacier, Phys. Rev. B. 21, 723 (1980).