Optical constants of germanium sulphide single crystal by interference method

Physica B 159 (1989) 171-180 North-Holland, Amsterdam

OPTICAL CONSTANTS OF GERMANIUM SULPHIDE SINGLE CRYSTAL BY INTERFERENCE METHOD A.M. ELKORASHY Faculty of Engineering,

Fayoum

University, Fayoum,

Egypt

The refractive index, n, of a GeS single crystal was measured by the interference method in the transparency region. An effective thickness smaller than the true one was introduced to account for the scattering by the layer structure of the material in this region. The effective thickness hypothesis gave a satisfactory explanation of the pronounced rise in reflectance for E < E,. The extinction coefficient, k, was calculated from the measured absorption coefficient. Measurements were performed at room temperature using plane polarized light with the plane of polarization parallel to the aand b-crystallographic axes which lie in the plane of cleavage. The real and imaginary parts of the complex dielectric constant (E,, q) as well as the reflectance and its phase change (R, @) were calculated from the values of n and k. It was shown that the GeS single crystal exhibits birefringence. Assuming that GeS binding is partly ionic and partly covalent, the optical constants were satisfactorily fitted to the model of single effective oscillator.

1. Introduction The binary IV-VI compounds formed with Ge, Sn, and Pb as cations and S, Se, and Te as anions form a very interesting class of semiconductors which can be divided into three groups according to their crystal structure [l]. The wellknown lead salts (PbS, PbSe, and PbTe) crystallize in a cubic (NaCl) structure, and are easy to produce both in bulk and epitaxial form, and have been extremely thoroughly studied [l, 21. GeTe and SnTe have a rhombohedral structure at low temperatures, converting to cubic at T = 273 K for SnTe and T = 673 K for GeTe. These compounds have also been extensively studied [l], though not as fully as the lead salts. The remaining four compounds, GeS, GeSe, SnS, and SnSe, comprise the least studied of the three groups. Among the last four compounds GeS receives increasing interest. Germanium sulphide, GeS, is a layer-type semiconductor which has an orthorhombic D:E symmetry and the atomic arrangement shows strongly distorted octahedral coordination. This causes considerable anisotropy of the optical

properties. The lattice parameters of GeS were first determined by Zachariasen [3] as a = 0.430 nm ,

b = 0.365 nm

and

c = 1.044nm. Here the b- and c-axes are inverted with respect to those of Zachariasen [3] to agree with the current literature. According to the recent structure examination of GeS [4], the atomic coordination is 3 + 3 type with nearest neighbour Ge-S distances 0.2441 nm (3 x ), 0.3270nm (2 x ) and 0.3278 nm (1 x ). Much work has been devoted, during the past years, to the study of crystalline GeS, especially to its optical properties [5-91. In previous works [lo-121 optical absorption in a GeS single crystal was studied near the fundamental edge. The aim of the present work is to complete those previous studies by reporting an investigation of the main optical constants of a GeS single crystal in the transparency region by the interference method.

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i Optical constants of a GeS single crystal

2. Experimental

Germanium sulphide single crystals were grown by vapour transport in a temperature gradient, a method introduced by Schaefer [13] and modified by Hrubf [14]. Thin samples for interference measurements were obtained by cleavage. The surfaces of the cleaved samples were mirror like. To account for the anisotropy of the material, measurements were carried out with plane polarized light with the plane of polarization parallel to the a- and b-crystallographic axes which lie in the plane of cleavage. Prior to measurements, samples were orientated both optically and by the back reflection Laue technique. Both X-ray and optical methods showed excellent agreement. Measurements of the refractive index were performed at room temperature in the transparency region with a photon energy between 0.5 and 1.6 eV.

3. Method and results Quite pronounced interference was observed, at room temperature, in transmittance as well as in reflectance responses of enough thin samples, for both the a- and b-axes. We were able to calculate the refractive index of GeS from the values of wavelength corresponding to successive interference maxima or minima in the following way. The condition

For photon energies less than the energy gap (E < E,), the spectral variation of the refractive index can be expressed by the Cauchy-Sellmaier expansion

or n(E) = n, + a;E* +

+ ajE6

+

a;E* ,

(3b)

where n,, a,, a*, . . . and a;, a;, . . . are constants. We chose the values of nd which gave the best least-squares fitting to a straight line at the low energy side (E < 1 eV) where the fourth and higher orders in eq. (3b) can be neglected. Fig. 1 shows the variation of the product nd with the square of photon energy E* as obtained from the interference maxima and minima in the reflectance response. When the true thickness of the sample was substituted for d, the obtained values of II were unreasonably low. Samples showed a pronounced rise in reflectance towards lower photon energy. The one surface reflectance, R, (reflectance of the surface of an infinitely thick slab), was calculated in the following way. In the absence of interference, the sample reflectance R and transmittance T at normal incidence are given by R

=

1

R

s

T

2nd = mh

a;E4

+

Cl- R,)*ed-24 1 - Rt exp(-2cyd)

(4)

I ’

Cl- 4)’ exp(- 4

=

(1)

1 - Rz exp(-2ad)

(5)

’

applies for interference maxima in transmittance and interference minima in reflectance, and the condition 2nd=(m+

$)A

(2)

for interference minima in transmittance and interference magima in reflectance. We assumed several series of successive interference of order m. Using the above equations we were able to calculate values of nd corresponding to each value of m.

u 16 L 15 0

1 SQUARE

OF PHOTON

3

2 ENERGY

E* kV)’

Fig. 1. The product of the refractive index n and thickness versus the square of the photon energy E’.

d

173

A.M. Elkorashy I Optical constants of a GeS single crystal

measured reflectance R, the one-surface reflectance R, corrected for back-reflection, calculated from eq. (6), and the normalized reflectance R, for a sample in which interference was not observed. The value of d obtained by replacing R, in eq. (6) by R,, which made the normalized reflectance fit to the measured one at the high energy side, clearly disagrees with the true thickness of the sample. We assumed that only an effective thickness deff < d,,,, satisfies eqs. (l), (2), (4) and (5). Moreover, the values of the refractive index obtained from fig. 1 by substituting the effective thickness, not the true one, agrees excellently with those values obtained from the normalized reflectance by eq. (7). Table I shows the values of the effective thickness from interference in transmittance d,,,(T) and from interference in reflectance d,,,(R) as compared to the true thickness d,,,, for several samples. It was found that deff is the same for both the a- and b-axes. The refractive index was also directly measured by the prism method. For this purpose samples in the form of thin prisms with plane surfaces were used. One face of the prism was natural (cleaved) and the other was ground and

where (Yis the absorption coefficient and d the true thickness of the sample. Simultaneous solution of eqs. (4) and (5) gives

R Rs = 1 + T exp(-ad)

.

From eq. (3b) it can be seen that the refractive index is a monotonically increasing function of E. Since the reflectance is directly related to the refractive index, we expect the reflectance to be also a monotonically increasing function of E. Unfortunately, R,, calculated from eq. (6), does not satisfy this requirement. Values of R, were normalized to fit smoothly with the measured values of R at the high energy side by an Eindependent factor (less than one). The refractive index was then calculated from the normalized reflectance by the relation

where values of the extinction coefficient k were neglected in the transparency region where the interference was observed. Fig. 2 shows the WAVELENGTH

0 5

2.0 I""'""'

"

A(prn) 0.5

1.0

1.5 ' '

'

I

1.5 PHOTON

2.0 ENERGY

2.5

3.0

E(eV)

Fig. 2. The reflectance spectral response: the measured reflectance R; ---back-reflection, calculated from eq. (6) and -.the normalized reflectance R,.

the one-surface reflectance R, corrected for

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174

Elkorashy

I Optical constants of a GeS single crystal

Table I Comparison between the true thickness in reflectance for 5 thin samples where

and the effective thickness in transmittance interference was taking place

Sample no.

Effective thickness in transmittance

Effective thickness in reflectance

d,,,(T)

d,,,(R)

1 2 3 4 5

True thickness d,,,, (pm)

25 25 23 20 19

optically polished. Three prisms with the two faces intersecting along the b-axis at angles about 9, 10 and 11” were prepared. A precision optical goniometer was used with the beam always entering the polished face of the prism and emerging from the natural surface, with the prism aligned so that this surface was strictly perpendicular to the emerging beam. Fig. 3(a) shows the spectral response of the refractive index for the a- and b-axes. The squares are the values of the refractive index calculated from the normalized reflectance by eq. (7) and the triangles are the values directly measured by the prism method. Fig. 3(b) shows the spectral response of the refractive index obtained from interference by dividing nd, from fig. 1, by the corresponding value of the effective thickness, d,rr. From figs. 3(a) and 3(b) it is clear that the values of the refractive index obtained by the three independent methods are in excellent agreement. The experimental results were fitted to the Cauchy-Sellmaier expansion, eq. (3), and the constants IZ~and the a’s were obtained. Table II summarizes the constants of eq. (3) for the two crystallographic axes considered. From these results it can be seen that the GeS single crystal exhibits birefringence. The solid lines in fig. 3 are the least-squares fits to eq. (3), and the dashed lines are extensions of these fits above the energy gap. Fig. 4 shows the spectral response of the total absorption coefficient (Ywhich was directly calculated from the measured transmittance and reflectance [12]. This response exhibits a featureless long tail at the low energy side for the two directions of polarization. We extrapolated the

(km)

6.1 6.0 5.5 4.9 4.6

and

(Pm)

5.9 5.8 5.3 4.8 4.4

WAVELENGTH 105

c o z

3

Jlprn) 1

2

0.9

0.8

0.7

3.6

3.2 I

311 1.0

0.5

0

PHOTON

ENERGY

WAVELENGTH 105

38

3

2.0

1.5 EleV)

_I(pm) 1

2

0.9

0.8

0.7

3.2 3 11

u

u.5

1.0 PHOTON

ENERGY

1.5

2.0

E(N)

Fig. 3. The spectral response of the refractive index. (a) The squares are the values calculated from R, by eq. (7) and the triangles are the values measured by the prism method. (b) The circles are the values of the refractive index obtained from interference by dividing nd, from fig. 1, by d,,,. The solid lines are the least-squares fits to eq. (3) and the dashed lines are their extension above the enerev eao.

A.M.

Elkorashy

Table II Summary of the constants of eq. (3)

0.80

Constant

a-axis

b-axis

n, (dimensionless)

3.176 0.166
3.358 0.165
a, a, a3 a4 a; a; a;

ai

(pm’) (pm”) (pm? (pm8) (eV’) (eV-‘) (eV”) (eV-“)

175

I Optical constants of a GeS single crystal

WAVELENGTH 0.75

JX(rm) 0.70

low energy part of the response and subtracted it from the total absorption coefficient to obtain the interband absorption coefficient (Y~.Fig. 5 shows the spectral variation of the interband absorption coefficient (Y~. The extinction coefficient k was calculated from the relation

k= $

(yi = 2

= 9.78 X lo-6 (Yi(cm-‘) E(eV) ’

(8) 1.6

1.5 1.2

WAVELENGTH 1.0 0.8

PHOTON

l(prn)

tl + 5 3 ii tJ_ IO2

fi I= k 5:

m __-____---______----

0 t-

E(eV)

We thus have both n and k known in the energy range of measurements. Fig. 6 shows the spectral response of both 12 and k in the photon energy range from zero to 2 eV for the two directions of polarization. In fig. 6, II is obtained by substituting the numerical values of the constants of eq. (3) from table II. Fig. 7 shows the spectral response of the real and imaginary parts (E, and Ed) of the complex dielectric constant in the photon energy range from zero to 2 eV for both the a- and b-axes. E, and ci were calculated from the values of the refractive index and the extinction coefficient by the relations

W

alOl 2 +

ENERGY

1

1.8

Fig. 5. The spectral response of the interband absorption coefficient ai which is obtained from fig. 4 by extrapolating the low energy tail and subtracting it from the total absorption coefficient a.

Ti 0E ,03

s

1.7

I

1.0

1.5

2.0

PHOTON ENERGY E(d) Fig. 4. The spectral response of the total absorption coefficient a which was directly calculated from the measured transmittance and reflectance, according to ref. [12].

E, =

n2

-

k2

(9)

and ci=2nk.

(10)

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176 WAVELENGTH

Elkorashy

I Optical constants of a GeS single crystal

_Iltml

Fig. 8 shows the spectral variation of the reflectance and the phase change in the same photon energy range as figs. 6 and 7. By this way three complete sets of optical constants (n, k), (E,, Ei), and (R, CD) are given for a GeS single crystal in the transparency region. The complex dielectric constant E* is the optical constant accessible to physical interpretation. Our results of refractive index dispersion below the interband absorption edge correspond to the fundamental electronic excitation spectrum. Wemple and Di Domenico [15] have analyzed more than 100 widely different solids and liquids using a single effective-oscillator fit of the form F &r(E)

PHOTON

ENERGY

Fig. 6. The spectral response extinction coefficient k.

of both

refractive

index n and

R = (n - 1)’ + k2

(11)

(n+Q2+k2

and -2k

1053 III!

(12)

n* + k* - 1 ’ WAVELENGTH

17

2

1+

&

_

E2)

(13)

2

E[eVI

The reflectance R and the phase change @ associated with the reflection were calculated from the values of n and k by the following equations

@=tan-’

=

where the two parameters E, and F are related straightforwardly to the electric dipole strengths and the corresponding transition frequencies of all oscillators. By a special combination of parameters, Wemple and Di Domenico [15] defined a parameter E, as Ed=;.

Eqs. (9) and (13), neglecting values of k in the transparency region, now give

_iipml 1

(14)

0

0

O.LO 0.9

0.8

0.7

10-l

P

GeS

w

-10

@a

K 0.35

/ -20

/ /

% ?

8 -30

y

-LO

w

z

9

$0.30 :

Y -50

0.25

: 0

parts

aI

-60 0.5

1.0 PHOTON

Fig. 7. The spectral response of the real and imaginary of the complex dielectric constant.

: 2

ENERGY

1.5

20

El@‘)

Fig. 8. The spectral response of the reflectance and the phase change. The open circles are the normalized reflectance below the energy gap and the solid circles are the measured reflectance above the energy gap.

A.M.

n*(E)= 1+

E,(E) =

E$.!;2

Elkorashy

177

I Optical constants of a GeS single crystal

(15)

.

0

Table III The single effective oscillator parameters

M-1

(dimensionless)

They found empirically that

(16) where N, is the coordination number of the cation nearest neighbour to the anion, Z, is the formal chemical valency of the anion, and N, is the effective number of valence electrons per anion. We obtained the values of the parameters E,, and E, by plotting (n’ - 1))’ versus E* and fitting it to a straight line, as shown in fig. 9. On the basis of the above-mentioned model, the single oscillator parameters E, and E,, are connected to the imaginary part ci of the complex dielectric constant, and the -1 and -3 moments of the q(E) optical spectrum, as defined in ref. [15], can be derived from the relations (17)

3.49 3.65

a-axis b-axis

31.57 37.38

9.04 10.24

0.74 0.77

In applying eq. (16) to the GeS single crystal, there is some uncertainty about the coordination number of the anion which may be thought to be 6 (deformed NaCl structure) or 3 (taking account of the layer structure). The other parameters in eq. (16) are well defined, namely Z, = 2 and N, = 4 + 6 = 10. The resulting p’s for values of N, between 3 and 6 are given in table IV for the a- and b-axes. It may be instructive to compare the electronic contribution to the static dielectric constant, E: = lim,,, n*(E) = ni, in terms of electronic polarizability. Provided no local-field corrections are taken into account, one can express the average electronic polarizability p per atom by means of the relation

and

M’,

ET,==.

(18)

Table III gives the values of the single effective oscillator parameters E,, Ed, M_, , and M_, for the a- and b-crystallographic axes.

p=

&I)_1 m 47r p’

(19)

where rii is the mean atomic mass and p is the density. On the other hand, with local-field approximation applied, one obtains the ClausiusMossotti relation

0.12 GeS 0.11

h

/

\

0 interference

maximum

interference

minimum

l

written here in the form valid for isotropic and cubic symmetry media. In the case of the orthorhombic GeS, we average the anisotropy by taking the mean of the

-7

‘,

0.10 I

“c

0.09

0.08

0.07) 0

0.5

Table IV Values of the parameter 0 in eq. (16) for NC= 3,4,5, 1.0

1.5

2.0

SQUARE OF PHOTON ENERGY Fig. 9. (n’-

1)-l versus E’.

2.5 E2 (eV)2

NC

3.0 P (eV)

a-axis b-axis

and 6

3

4

5

6

0.53 0.62

0.39 0.47

0.32 0.37

0.26 0.31

178

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Elkorashy

I Optical constants of a GeS single crystal

principal permittivity components and simplify matters by assuming the cubic local field correction to be applicable. We could not carry out optical measurements with the plane of polarization parallel to the c-axis, and we may take the mean of the only available two permittivity components and assume that it is not far from the mean of the three components. Substituting E: = 10.68 (from table II), Ci = 86.86 g [16] and p = 4.01 g/cm3 [17], we obtain an average electronic polarizability of 16.69 x 1O-24 cm3 with no local-field correction, eq. (19), and 3.95 x 1O-24 cm3 with local-field correction, eq. (20).

4. Discussion of the results and conclusion The reflectance spectral response showed a pronounced rise near the fundamental absorption edge, which seems to be common to most layer-type materials. A similar rise was observed for GeSe [18] and for GaS and GaSe single crystals [19]. These crystals also belong to the group of layer-type semiconductors. This rise can be explained as partially due to back surface reflection resulting in multiple reflections within the sample. The agreement between the presently measured reflectance and of Gregora et al. [5] is striking. From fig. 2, it can be seen that multiple reflections reduce the rise in reflectance from 42% to 4.3% for the u-axis and from 36% to 3.5% for the b-axis, but does not remove it completely. Gregora et al. [5] calculated a theoretical value of reflectance for eq. (4) by letting cr+O and normalized the measured reflectance in the transparency region by a factor greater than unity and applied the same factor to normalize the high energy side of the reflectance response. Our procedure of normalization, on the basis of the effective thickness hypothesis, seems to have a more physical basis as it takes account of the layer structure of the material, while in ref. [5] only the back reflection correction was considered. Moreover, it seems to us to have no physical basis to normalize the measured reflectance above the energy gap where the absorption starts to be of significant value, such

that the internal photon scattering by the layer structure of the material is very much reduced. The decrease in reflectance above the fundamental absorption edge at a photon energy of about 2.1 eV for the a-axis can be interpreted as due to the increase in absorption by an allowed transition [5]. No similar decrease was observed for the b-axis. Similar results, namely a decrease in reflectance above the fundamental absorption edge for the u-axis and not for the b-axis, were observed with GeSe [18,20]. The low energy tail exhibited in the absorption spectrum seems to be a common feature of most layer-type crystals. Similar results were reported for GeSe [20], SnS [21], and SnSe [22] and for SnS, and SnSe, [23]. These crystals have layertype structure and the first three are isomorphic in structure with GeS. This tail can be attributed to internal photon scattering by the layer structure of the material [24]. The interpretation of our results of the refractive index, calculated for the sequential interference maxima or minima in transmittance and reflectance, is rather difficult. The main difficulty of this method lies in the fact that the interference order is not unambiguously known or readily determined. When the true thickness of the sample was substituted for d, we obtained values of the refractive index which were unreasonably small and significantly disagreed with the values obtained from the normalized reflectance calculated by eq. (7). This drastic contradiction led us to the conclusion that the whole thickness of the sample could not take place in a simple interference which is expressed by the standard eqs. (1) and (2). We assumed that there exists an effective thinkness deff for which these two equations are satisfied. The effective thickness arises straightforwardly from internal photon scattering by the layer structure of the material. In the actual case, complex equations apply for interference, for reflection, and for transmission. In the equivalent case where the sample is considered to be homogeneous without a layer structure and having the same values of n and k, eqs. (1) and (2) apply for simple interference and eqs. (4) and (5) for reflectance and transmittance, respectively, as shown in fig. 10. In the

A.M.

Layerstrudure (n

Homogeneous

8 k1

complex

some expressions

Elkorashy

for

:

the

I Optical constants of a GeS single crystal

_ no layer

n 8 k followmg

eqs.

1nterterence

eqs(11 F,(Z)

for

reflection.

eq.(LI

reflection,

and

transmission.

Fig. 10. Demonstration

str

for

eq 15) for

apply

:

Interference and

transmlsslon

of the effective thickness hypothesis.

equivalent case the effective thickness deff is substituted. The reproducibility of the results of the effective thickness, as seen from table I, may give a physical basis for the justification of our hypothesis. The effective thickness hypothesis could also be supported by the excellent agreement of the results of the refractive index obtained by applying this hypothesis and dividing nd from fig. 1 by deff with those obtained from the normalized reflectance and those measured by the prism method. The value of deff is the same for the a- and b-axis, but for interference in reflection it is slightly smaller than for interference in transmission (about 3% smaller). A similar contradiction concerning the true and effective thickness was also observed in GeSe [25]. Taking into consideration that in ref. [5] the results of the refractive index obtained by the prism method differ by 2 0.02 for different prisms, it can be seen that they agree excellently with the present results obtained from interference on the bases of the effective thickness hypothesis. The absolute values of the refractive indices obey the relation nb > n, in the fundamental electronic excitation range E < E, (the energy range of the measurements). Similar results were obtained with GeSe [25] and more general results were found to hold for CdSb [26] and ZnSb [27], namely nb > IZ, > IZ, for E < E, with b < a < c. These materials have a crystallographic structure similar to that of GeS. Wemple and Di Domenico [15] have found empirically that the coefficient p in eq. (16) takes the values pi = (0.26 -+ 0.04) eV for ionic

179

binding and p, = (0.37 + 0.05) eV for covalent binding and varies between these two limits for a large number of different materials. By considering these limiting values, pi and /I,, table IV shows that the binding in a GeS crystal is partly ionic and partly covalent, On this basis we come to the conclusion that our results on optical constants fit satisfactorily to the Wemple-Di Domenico model of a single effective oscillator. Applying eq. (20) separately to crystalline Ge and S we can derive the mean atomic polarizability as Pm_ = b(p,, + Ps). Using the corresponding permittivity and density data of Ge and a-S, as reported, for example, in [17,28], we arrive at a value of P,,,,,, = 3.72 X 10ez4 cm3, which is comparable with our results for a GeS single crystal.

References PI N.Kh. Abrikosov, V.F. Bankiana, L.V. Poretskaya, L.E. Shehmova and E.V. Skudnova, Semiconducting II-VI, IV-VI, and V-VI Compounds (Plenum, New York, 1969) ch. II. PI Yu.1. Ravich, B.A. Efimova and I.A. Smirov, Semiconducting Lead Chalcogenides (Plenum, New York, 1970). [31 W.H. Zachariasen, Phys. Rev. 40 (1932) 917. 141 G. Bissert and K.F. Hesse, Acta Crystallogr. B 34 (1978) 1122. PI I. Gregora, B. Velicky and M. Zavetova, J. Phys. Chem. Solids 37 (1976) 785. Fl J.D. Wiley, W.J. Buckel, W. Braun, G.W. Fehrenbach, F.J. Himpsel and E.E. Koch, Phys. Rev. B 14 (1976) 697. 171 L. StouraE, A. Abraham, I. Gregora, B. Velicky and M. Z&&ova, in: Proc. 13th Int. Conf. Phys. Semicond., F.G. Fumi, ed. (Tipografia Marves, Rome, 1977) p. 423. PI R. Emard and A. Otto, Phys. Rev. B 16 (1977) 1616. 191 F. Luke& E. Schmidt and A. Lacina, Solid State Commun. 39 (1981) 921. [lOI A.M. Elkorashy, Egyptian J. Solids 7 (1985) 13. WI A.M. Elkorashy, Egyptian J. Phys. 17 (1986) 29. PI A.M. Elkorashy, J. Phys. C 21 (1988) 2595. (Verlag [I31 H. Schaefer, Chemische Transportreaktionen Chemie, Weinheim, 1962). 1141 A. Hruby, Czech. J. Phys. B 25 (1975) 1413. [151 S.H. Wemple and W. Di Domenico, Phys. Rev. B 3 (1971) 1338. WA J.A. Dean, ed., Lange’s Handbook of Chemistry, 12th ed. (1979) sec. 4, p. 54, 55.

A.M.

180 [17] D.E.

[18]

[19] [20] [21]

Elkorashy

I Optical constants of a GeS single crystal

Gray, ed., American Institute of Physics Handbook, 2nd ed. (1963) sec. 9, p. 27. A.M. Elkorashy, in: Proc. 14th Int. Conf. Phys. Semicond., Edinburgh, 1978, B.L.H. Wilson, ed. (The Institute of Physics, London) p. 817. G.A. Akhundov, N.A. Gasanova and .M.A. Nizametdinova, Phys. Stat. Sol. (b) 15 (1966) K109. A.M. Elkorashy, Phys. Stat. Sol. (b) 135 (1986) 707. A.M. Elkorashy, Chemtronics 1 (1986) 76.

[22] A.M. Elkorashy, J. Phys. Chem. Solids 47 (1986) 497. [23] G. Domingo, R.S. Itoga and CR. Kannewurf, Phys. Rev. 143 (1966) 536. [24] A.M. Elkorashy, to be published. [25] A.M. Elkorashy, Egyptian J. Solids 3 (1981) 219. [26] M. Z&&ova, Czech. J. Phys. B 14 (1964) 271. [27] M. Zavetova, Phys. Stat. Sol. 5 (1964) K19. [28] L.J. Giacoletto, ed., Electronic Designers’ Handbook, 2nd ed. (1977) sec. 2, p. 130.