A comparative study of the properties of evaporated a-Si films before and after hydrogenation

A comparative study of the properties of evaporated a-Si films before and after hydrogenation

Thin Solid Films, 147 (1987) PREPARATION 213-222 213 AND CHARACTERIZATION A COMPARATIVE STUDY OF THE PROPERTIES OF EVAPORATED a-Si FILMS BEFORE AN...

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Thin Solid Films, 147 (1987) PREPARATION

213-222

213

AND CHARACTERIZATION

A COMPARATIVE STUDY OF THE PROPERTIES OF EVAPORATED a-Si FILMS BEFORE AND AFTER HYDROGENATION K. P. CHICKa,

K. C. KOONa AND B. Y. TONGb

’ Department

of Physics.

b Department

of Physics, University of Western Ontario. London,

(Received

February

2

The Chinese

I,

University of Hong Kong (Hong

1986; revised July 8, 1986; accepted

Kong)

Ontario (Canada)

August 5, 1986)

The electron spin resonance signal, dark conductivity and photoconductivity of evaporated amorphous silicon before and after hydrogenation have been measured as a function of deposition rates and of annealing temperatures. In order to produce films with good photoconductivity a low deposition rate of about 1 8, s- ’ and an annealing temperature above 800 K are recommended. Contamination by residual gases such as oxygen in a high vacuum of lo-’ Torr during deposition can “poison” the film, diminishing the effect of post-hydrogenation. The poisoning effect can be removed by annealing above 800 K. Hydrogenation reduces the dangling bond density and suppresses the hopping contribution to electron conduction but does not seem to affect the band tail width. Hydrogenation also shifts the Fermi level towards the conduction band. This may be due to the formation of an electron accumulation layer in the hydrogen-rich surface region which affects the position of the Fermi level in the whole film.

1.

INTRODUCTION

At present there is tremendous interest in hydrogenated amorphous silicon (a-Si:H) films prepared by glow discharge (GD) deposition. However, the film growth process during GD deposition can be quite complicated and depends on a large number of variables. Hence a slight alteration of deposition conditions may produce films of quite different properties. For example, in undoped a-Si:H, the preexponential factor of the dark electrical conductivity can vary over more than five orders of magnitude’. In contrast, deposition by vacuum evaporation is simple. However, this method of preparation has been largely neglected simply because evaporated silicon films in general have poorer quality than GD-produced films. Recently this method has attracted some new interest. It has been shown that the properties of evaporated films can be improved by post-hydrogenation293 or that a similar improvement can be obtained by reactive evaporation4-6. Apart from the practical application possibility of post-hydrogenated films, post-hydrogenation can be used as a means to study how atomic hydrogen can alter the properties of asdeposited films. 0040-6090/87/$3.50

0

Elsevier Sequoia/Printed

in The Netherlands

214

K . P . CHIK, K. C. KOON, B. Y. TONG

In this paper the results of a detailed study of the electrical properties of evaporated a-Si films prepared with a systematic variation of coating rates and annealing treatment are reported. Film properties before and after hydrogenation were measured and compared. Thereby one may hope to find out how the gap states, band tail, film homogeneity or deep traps are affected. Moreover, one may try to determine an optimum condition so that films with reproducible good quality can be obtained. 2. EXPERIMENT a-Si films were prepared by evaporation of silicon powder through electron beam heating in a conventional high vacuum coating unit at a base pressure of about 10-7 Torr. Corning 7059 glass was used as the substrate. The film thickness lay between 0.3 and 0.5 ~tm. Several sets of samples were prepared with the following two objectives in mind: (1) to study the effect of deposition rates on film properties and (2) to study the effect of annealing temperature on film properties. In the first case, deposition rates between 1 and 6,& s-1 were used while all samples were annealed at 843 K after deposition. In the second case, all samples were deposited at a rate of 1.3 .& s 1 while the annealing temperatures were varied. Annealing was carried out with the films in a pure nitrogen atmosphere for 20 min at a specified temperature. After annealing, the electron spin resonance (ESR) signals of the samples were measured at room temperature in the X-band in a J E O L spectrometer at g = 2.005 with a typical line width of about 6 G. The dark electrical conductivity a D of the samples as function of temperature was then measured. After such measurements, the films were post-hydrogenated in a 0 pinch plasma as described elsewhere 7. To avoid contamination of electrode material during hydrogenation, metal knife-edges acting as electrodes were pressed on the samples during the above ao measurements so that post-hydrogenation was carried out on bare films only. After hydrogenation, ESR signals of the samples were again measured before further electrical measurements were carried out. After hydrogenation, all samples exhibited appreciable photoconductivity ap, which is defined as O'illuminated--O'D.Thus both aD and crp were measured using silver paste electrodes with a 2 m m gap separation. An H e - N e laser was used throughout as the light source and when necessary appropriate filters could be used to reduce the photon flux on the samples. All electrical measurements were carried out at temperatures between 125 and 393 K. 3. EXPERIMENTAL RESULTS

3.1. General It is well known that evaporated a-Si normally shows negligible photoconduction. Thus, before hydrogenation, the electrical data refer almost exclusively to the dark conductivity aD. At high temperatures, aD has a large activation energy between 0.7 and 0.8 eV consistent with published results 8. As the temperature is lowered, aD goes over to a regime with a much smaller activation energy, which in our case lies around 0.16 eV under all preparation conditions we have investigated.

PROPERTIES OF a-Si AND a-Si: H FILMS

215

Thus the temperature dependence of (7Dcan be represented by 0.D = 0.1D°exp~--~-~-)+0.2D°exp -

(1)

At high temperatures (region 1), the conduction mechanism 9 is usually attributed to electron conduction above the electron mobility edge Ec. In the low temperature region (region 2), conduction is mostly interpreted in terms of the variable-range hopping mechanism. However, it has been shown that above 160 K the conduction is better interpreted by the nearest-neighbour hopping mechanism 1o and, below 160 K, 0.D begins to deviate from the behaviour ofeqn. (1). Post-hydrogenation affects the values of alD °, E1D, aZD ° and to a lesser extent E2D. The dark conductivity can still be represented by two exponential terms, namely 0.D(H) = 0.1D°(H)exp{

EID(H)~+0.2D°(H)exp{ kT

.~

E2D(H!~ kT J

(2)

However, aD(H) begins to deviate from the behaviour of eqn. (2) at a higher temperature than 160 K, depending on the free spin density present in the samples. Photoconduction can again for most cases be represented by two exponential functions 11, i.e. 0.p= a,pO e x p//~ _ ~E1P~ _ ) + a2pOexp

(--kZJ E2P~

(3)

Since ap (except in very special cases) can only be observed in hydrogenated samples, we shall drop the designation (H) for all photoconductivity data, keeping in mind that these values refer to hydrogenated samples. Thus the electrical properties of our films are almost completely characterized by the parameters 0"10, 0"20, E 1 and E 2. Because of the large amount of data obtained in the present investigation, we present here the values of these four parameters. We would like to emphasize that the 0.o and E values presented in this paper are obtained by fitting the experimental data with eqns. (1)-(3) to give the best values for 0.o, 0.20, E 1 and E2, as described by Chik et al. 11 Values of E and a ° obtained directly from graphical plots tend to give a lower value in the high temperature regime. This is more serious in areas where a in regime 1 can only be measured over a small temperature range. 3.2. Effect o f deposition rate

In order to see how the deposition rate can affect film properties and posthydrogenation, films were prepared with different deposition rates ranging from 1 to 6 ,~ s- 1. All the samples were annealed at the same temperature of 843 K before any measurements were made. Figures l(a)-l(e) present results showing the dependence of0.1D °, 0.2D0, EID, E2D and the spin density N s respectively on the deposition rate. In addition, Fig. l(f) shows the values OfaD before and after hydrogenation and ap after hydrogenation measured at 160 K whereas Fig. l(g) gives the corresponding values from the same samples measured at a higher temperature of 300 K. There are several features that should be pointed out. First, the deposition rate

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Fig. 1. Properties of a-Si as a function of deposition rates: +, properties before hydrogenation; ©, O, properties after hydrogenation (Q, values measured in the dark; O, values measured under illumination from an He-Ne laser with a photon flux of 2 x l016 cm -2 S 1). has a strong effect on the value o f N s . A higher deposition rate produces films with a higher value of N s. The variations in a2o ° are interrelated. P o s t - h y d r o g e n a t i o n reduces N s with a c o n c o m i t a n t decrease in a2D °. This further supports the suggestion that the magnitude ofa2D ° depends on Ns. However, E2D is practically unaffected by deposition rate. Because of the fluctuation in the values of alp °, one cannot see any correlation between trOD° and the deposition rate. Within the experimental errors, the variation in E~D with deposition rate, if any, should also be very small. Secondly, post-hydrogenation produces a simultaneous shift of a~ DOand E1D to lower values, but the a m o u n t of the shift is uncorrelated with the deposition rate. Thus this shift is typical of the post-hydrogenation process. Thirdly, post-hydrogenated films all b e c o m e photosensitive. In Fig. 1 trD and ap are shown at both 300 K and 160 K. One will notice that the lower deposition rate tends to produce films with a higher trp value. The m a x i m u m value ofap is observed at a deposition rate around 1.3/~ s - 1. One m a y further notice that tro and to a lesser extent also aD(H) can be controlled to a value fluctuating through m u c h less than one order of magnitude. H o w e v e r , ap is m o r e sensitive to post-hydrogenation, s h o w i n g larger fluctuations than aD, which we believe are due m o r e to the hydrogenation process rather than to the film deposition. At high deposition rates ap

PROPERTIES

OF a-Si

AND a-Si:H

FILMS

217

can fluctuate considerably. Such films probably have a more loosely packed structure, and hence they are more sensitive to the actual hydrogenation process. Thus, to produce films with good and reproducible photoconducting properties, a low deposition rate around 1 A s- ~ is advisable.

3.3. Effect of annnealing temperature To study the effect of annealing temperature on film properties, separate sets of samples different from those used in Section 3.2 were used. The samples were all deposited at the same rate of 1.3/~ s - 1. After deposition each sample was annealed at different temperatures between 693 and 921 K. Figures 2(a)-2(e) show the dependence of E1D, trip °, Ns, azo ° and E2o respectively on the annealing temperature. Figures 2(f) and 2(g) show aD and a e after hydrogenation measured at 300 K and 160 K respectively. It has been shown 12 that the spin density Ns depends on the annealing temperature. Our present results are consistent with the former result that N s has a minimum value at around 600-700 K. Above this temperature Ns rises rapidly with TA. Thus we can divide our samples into two groups according to their response to -4

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Fig. 2. Propertiesof a-Si as a function of annealing temperature:symbolsas for Fig. 1.

218

K . P . CHIK, K. C. KOON, B. Y. TONG

post-hydrogenation: group A for TA > 750 K and group B for TA < 750 K. In group B post-hydrogenation has a small effect on the properties of films; there is no detectable reduction in N s and practically no increase in photosensitivity. All these films have a very low dark conductivity (of the order of 10 - 9 ~ - z cm-1 at room temperature) both before and after hydrogenation. This is comparable with the conductivity of good G D a-Si:H samples, and hopping conduction could not be observed down to 250 K, below which trD is not measurable, so that a2 ° and E2 values are not available for group B films. Judging from the absence of hopping conduction in these samples, one may expect such samples to exhibit photoconduction even without hydrogenation. Indeed, such samples are photosensitive but trp only reaches the order of 10- s ~ - , c m - ~ at room temperature even with a photon flux as high as 2×1016 photons cm -2 s -1 (Fig. 3). Post-hydrogenation of such samples cannot improve their photoconductivity and the addition of hydrogen seems to have little effect on the film properties.

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I loyr (K~I Fig. 3. Dark conductivity and photoconductivity of an unhydrogenated a-Si sample annealed at 677 K (photon flux, 2 x 1016 c m - 2 S- 1 from an H e - N e laser): O, trp; O, at).

For group A samples, N s is strongly reduced by post-hydrogenation. There is a corresponding increase in trv and reduction in E~o. However, in contrast with the behaviour with deposition rates, the amount of the shift in E 1Dis not directly related to a proportional decrease in tr~D° where this is more apparent for TA > 850 K. There is another feature which differs from the results of Section 3.2. Both (rD and (rp increase with increasing TA with an almost constant trp/aD ratio. An increase in TA above 800 K does not improve crp very much. Thus to obtain the best trp value TA should be chosen to be around 800 K.

PROPERTIES OF a-Si AND a - S i : H FILMS

.219

4. DISCUSSION 4.1. Preparation condition and photoconduction The effect of deposition conditions and gas contamination on the properties of evaporated a-Si films has been reported by different researchers 13 is. In general, higher coating rates produce films with a larger free spin density N s and a higher electrical conductivity at) consistent with our present results. The presence of oxygen during evaporation can reduce N s and ~rDconsiderably. Bahl et al. ~5 pointed out that in high vacua, even in the absence of the deliberate introduction of an appreciable gas pressure, the properties of a-Si films are affected by contaminants. This is borne out by annealing experiments on a-Si and a-Ge films~ 2.~6.~ 7. According to Chik and Fung 16 both Ns and aD are first reduced with increasing annealing temperature until a minimum is reached. A further increase in the annealing temperature produces a rapid rise in Ns and aD again until crystallization occurs. A higher base pressure during evaporation produces deeper minima 17 of aD and N s where the decrease in ~rt) is consistent with contamination by oxygen during evaporation. The annealing of as-deposited films probably not only removes unstable defects in the film but also promotes the formation of S i - - O bonds, thus reducing the dangling bond density considerably. If the annealing temperature is high enough, the S i - - O complexes are broken up, regenerating dangling bonds. The film becomes less contaminated by oxygen but has gained a higher value of Ns. It should be noted that this phenomenon is observed in films prepared in high vacuum with a base pressure of 10- ~ Torr. The results of the present investigation of the behaviour of the annealed films after post-hydrogenation are in complete agreement with the above interpretation. Films annealed at about 600-700 K have values of at) and Ns near the minimum values. According to the above interpretation, we speculate that most of the dangling bonds will be modified by oxygen. We made no attempt to measure the oxygen content of our films, since its value is expected to be quite small. Apparently, post-hydrogenation cannot remove the oxygen complexes in the film and the S i - - O complexes formed are defects and may act as deep traps. Their presence can only suppress the film's photoconductivity. Furthermore, if the films are annealed at increasing high temperature, more S i - - O defects will be removed. The dangling bonds resulting from such removal can now be modified by hydrogen, leading to an increase in photoconduction. Since the best photoconductivity is obtained by posthydrogenation in films annealed above 800 K, this means that the removal of S i - - O complexes is most effective above this temperature. Thus it is necessary to avoid gaseous contamination, especially by oxygen, during evaporation in order to produce good quality films for post-hydrogenation. 4.2. Dark conductivity and hydrogenation Preparation conditions have a marked effect on the spin densities of the films. A larger N s is accompanied by a larger azD ° but al is uncorrelated with N s. Thus according to eqn.(1) the increase in aD at room temperature (RT) with deposition rate and with annealing temperature is due exclusively to a greater contribution from a2. F r o m Figs. l(b) and l(d) one sees that hydrogenation suppresses a 2 significantly.

220

K . P . CHIK, K. C. KOON, B. Y. TONG

One would then expect a corresponding large decrease in aD(RT ) after hydrogenation according to eqn. (2). In contrast with this expectation, aD(RT ) (see Fig. l(g)) is only reduced by less than one order of magnitude. This is because EID and trtD ° both become smaller after hydrogenation and this effect may be due to the following. Films before hydrogenation can be prepared with a small variation in EtD and alD° which is not directly related to the preparation conditions. However, the variation between a i d ° and E1D is strongly correlated as shown in Fig. 4 which indicates that the dependence can be represented by log O'ID0 = log O'ID 00 -[- AEID

(4)

The plot in Fig. 4 shows a well-defined straight line with a slope A = 21 __+2 e V After hydrogenation the exponential variation behaviour of alD° with E1D is preserved with an almost identical slope A = 18 e V - 1 but with a different intercept. A parallel shift of the log trOD° VS. EtD plot as result of hydrogenation is more apparent if we take the E1D and O'ID0 data from Fig. 1 and plot AE1D = EzD(H)--E1D against log {O'ID0(H)/O'IDO} (Fig. 5). One obtains again a straight line given by AE1D -- B o + B l o g ~ ~ O'ID J

(5)

where B = 20 e V - ~ which is exactly the same value found for the results in Fig. 4. This means that post-hydrogenation does not affect the A value ofeqn. (4) but only the intercept log a~D°° which seems to be characteristic of the particular hydrogenation process employed. One possible explanation is that there exists a space charge layer on the surface of the samples. Fritzsche and G o o d m a n t8"19 have pointed out that in high quality films the presence of a space charge layer on the film surface can affect the position of E F throughout the film. Fritzche's calculation of the effective pre-exponential factors and the Fermi level shift is agrees very well with our data in Fig. 5 if electron accumulation near the surface is assumed. During posthydrogenation from a 0 pinch plasma, hydrogen is introduced into the sample from

:r

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015

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Lo {ev)

~8

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J 0.2 E1DIH ) - E1D (eV)

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Fig. 4. Plot of logO'lD° VS. EID for samples prepared under different conditions: +, values before hydrogenation; C), values after hydrogenation. Fig. 5. The change in alp ° as a function of the change in EID resulting from post-hydrogenation.

PROPERTIES OF a - S i AND a - S i : H

FILMS

221

the top surface. As a result a thin layer exists at the surface with a very high hydrogen concentration as observed by a nuclear profiling experiment 2°. This hydrogen-rich layer probably provides the space charge layer which shifts EF towards E c. The linear relation of log alD ° with EID ofeqn. (4) has also been observed in G D a-Si:H containing different amounts of n-type and p-type dopant 21 where A = 15 eV-1. In undoped G D a-Si:H a similar behaviour is also observed but the data have considerable scatter 1. In both these cases, alD ° varies over five orders of magnitude in comparison with a variation of less than two orders of magnitude reported here. The variation in a i d ° over several orders of magnitude in a-Si is difficult to explain. Several models have been proposed to explain this phenomenon L22-24. Overhof and Beyer 22 have shown that the statistical shift of EF alone can produce a a~D° variation of several orders of magnitude but a large statistical shift o f E v is only possible in a-Si:H samples where the gap state density is small. The variation in trlD ° in our evaporated samples (without hydrogenation) obviously cannot be explained by this model since these samples have a very high density of gap states. Recently Cohen e t al. 24 proposed an electron transport model for amorphous semiconductors in which electron-phonon interaction is taken into account. They showed that in the temperature range where electron-phonon interaction becomes important one can write E~D = E c - E F + k T ~ I o g

(6)

tr* is the pre-exponential factor without electron-phonon interaction and T~ is the temperature above which electron-phonon interaction becomes important, a l p ° now depends on the width of the band tail. If EF is pinned as in the case of our evaporated samples, eqn. (6) will be identical with eqn. (4). This seems to explain well the good fit of our data to eqn. (4). Thus we associate the fluctuation oftrxD° values in our case with a fluctuation in the band tail width which probably comes from the random nature of the evaporation process. One may note that the variation in a~D0 in evaporated films is much smaller in magnitude than that o f G D a-Si:H films. This reflects a simpler film formation process in evaporation which also means that abetter control of film quality reproducibility can be achieved. More quantitative analyses of the experimental results will be carried to test the feasibility of the above interpretation. 5. CONCLUSION

The effect of hydrogenation on evaporated a-Si has been investigated by comparing film properties before and after hydrogenation. The immediate consequence of hydrogenation is the reduction in dangling bond densities, leading to a suppression of the contribution due to hopping conduction and films which become photoconducting. The dangling bond density of evaporated films can also be considerably lowered by annealing at about 650-750 K. However, such films have only a very small photoconductivity which cannot be improved by subsequent hydrogenation. We believe that these films are "poisoned" by gaseous contamin-

222

K . P . CHIK, K. C. KOON, B. Y. TONG

ation from oxygen during deposition even though the films are prepared in a vacuum of 10 -7 Torr. The poisoning effect can be greatly reduced by annealing above 800 K, whereby dangling bonds are regenerated. These dangling bonds can now be removed by hydrogenation. Evaporation can produce films with a high temperature pre-exponential factor aID ° which varies by less than two orders of magnitude which is interpreted as being due to fluctuation in band tail width. Hydrogenation does not seem to affect the band tail width noticeably, but it does shift EF towards Ec. One possible explanation for this is the formation of an electron accumulation layer after hydrogenation in the hydrogen-rich surface layer. Further investigation should be carried out to study the surface effect resulting from hydrogenation. REFERENCES

1 W . E . Spear, D. Allan, P. LeComber and A. Gaith, Philos. Mag. B, 41 (1980) 419. 2 D. Kaplan, N. Sol, G. Velasco and P. A. Thomas, Appl. Phys. Lett., 35 (1978) 440.

3

B.Y. Tong, P.K. John, S.K. W o n g a n d K . P. Chik, AppI. Phys. Lett.,38(1981)789.

4

A . J . Lewis, G. A. N. Connell, W. Paul, J. R. Pawlik and R. J. Temkin, in M. H. Brodsky, S. Kirkpatrick and D. Weaire (eds.), Proc. Int. Conf. on Tetrahedrally Bonded Amorphous Semiconductors, American Institute of Physics, New York, 1974, p. 27. N. Kniftter, W. W. Mi.iller, J. M. Pirrung, N. H/inisch, B. Schr6der and J. Geiger, J. Phys. (Paris), Suppl. 10, 42 (1981) C4-811. M. Shindo, S. Sato, I. Myokan, S. Mano and T. Shibata, J. Non-Cryst. Solids, 59-60 (1983) 747. P . K . John, S. K. Wong, P. K. Gogna, B. Y. Tong and K. P. Chik, J. Phys. (Paris), Suppl. 10, 42 (1981) C4-639. W. Beyer and J. Stuke, Phys. Status Solidi A, 30 (1975) 511. N . F . Mott and E. A. Davis, Electronic Processes in Non-Crystalline Materials, Clarendon, Oxford, 1979.

5 6 7 8 9 10 11 12 13 14 15

16 17 18 19 20 21 22 23 24

K.P. ChikandK. C. Koon, Philos. Mag. B, 53(1986)399. K.P. Chik, P.K. Lim, B.Y. Tong, P.K. John, P.K. GognaandS. K. Wong, Phys. Rev. B, 27(1983) 3562. K . P . Chik, S. Y. Feng and S. K. Poon, Solid State Commun., 33 (1980) 1019.

M.H. Brodsky, R.S. Title, K. WeiserandG. D. Petit, Phys. Rev. B,l(1970)2632. P.G. LeComber, R. J. Loveland, W. E. Spear and R. A. Vaughan, in J. Stuke and W. Brennig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 245. S . K . Bahl, S. M. Bhagat and R. Glosser, Solid State Commun., 13 (1973) 1159. S. K. Bahl, S. M. Bhagat and R. Glosser, in J. Stuke and W. Brennig (eds.), Amorphous and Liquid Semiconductors, Taylor and Francis, London, 1974, p. 69.

K.P. ChikandS. H. Fung, J. Non-Cryst. Solids, 24(1977)431. S. Hasegawa, S. Yazaki and T. Shimizu, Solid State Commun., 26 (1978) 407. H. Fritzsche, Sol. Energy Mater., 3 (1980) 447. N.B. G o o d m a n and H. Fritzsche, Philos. Mag. B, 42 (1980) 149. B.Y. Tong, S. K. Wong, P. K. John and K. P. Chik, unpublished work, 1981. D . E . Carlson and C. R. Wronski, in M. H. Brodsky (ed.), Amorphous Semiconductors, Springer, Berlin, 1979, p. 287. H. Overhof and W. Beyer, Philos. Mag. B, 43 (1981) 433. M. Kikuchi, J. Non-Cryst. Solids, 59-60 (1983) 25. M . H . Cohen, E. N. Economou and C. M. Soukoulis, J. Non-Cryst. Solids, 66 (1984) 285.