Temperature dependence of ac conductivity of ...

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Temperature dependence of a.c. conductivity of chalcogenide glasses S. R. Elliott

a

a

Cavendish Laboratory, Madingley Road, Cambridge, England Available online: 20 Aug 2006

To cite this article: S. R. Elliott (1978): Temperature dependence of a.c. conductivity of chalcogenide glasses, Philosophical Magazine Part B, 37:5, 553-560 To link to this article: http://dx.doi.org/10.1080/01418637808226448

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PHILOSOPHICAL MAGAZINEB, 1978, VOL.37, No. 5, 553-560

Temperature dependence of ax. conductivity of chalcogenide glasses By S. R. ELLIOTT Cavendish Laboratory, Madingley Road, Cambridge, England [Received 20 January 1978 and in present form 28 February 19781

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ABSTRACT A theory based on two electrons hopping between defect sites recently proposed (Elliott 1977, 1978) to account for the frequency dependence of the conductivity of chalcogenide glasses is analysed here with regard to the predicted temperature dependence. To a first approximation, the logarithm of the a.c. conductivity is predicted to be directly proportional t o the temperature, a finding qualitatively horne out by experimental data. A detailed quantitative analysis is made on data taken by Rockstad (1972) on thin films of a-As,Te,. The present theory is obeyed excellently a t low temperatures, and an estimate of 6 x 10-la s is made for the relaxation time involved. At higher temperatures a different mechanism is deduced to predominate; namely, that. of quantum-mechanical tunnelling between localized states a t the valence band edge. A value of 130 A for the spatial extent of the localized wavefunction is estimated for the energy at which tunnelling is believed to take place, namely some 0.03 eV from the mobility edge.

3 1. INTRODUCTION Chalcogenide glasses and films, in common with other amorphous semiconductors, exhibit, in a particular frequency range, an a x . conductivity u(w) whose dependence on frequency w is given approximately by A d , where s 6 1 (see Mott and Davis 1971). This apparently general behaviour can, a t least for the chalcogenides, be subdivided into two categories. In the first, s lies very close to unity and is independent of temperature ; A is also temperatureindependent and has approximately the same magnitude in a wide variety of materials (see Pollak (unpublished),Jonscher 1977, Kocka 1976). I n the second, both s and A are dependent on temperature, the former tending to unity a t low T,and both are related to the band gap of the material (Elliott 1977, 1978). It is materials that fall into the second category with which this paper is concerned. A recent theoretical treatment by the author (Elliott 1977, 1978), based on a model in which two electrons simultaneously hop over a barrier between charged defect centres, is able to explain the dependence of s and A on bandgap. The temperature dependence of s was alluded to in Elliott (1977, 1978) : here the temperature dependence of A , and hence the combined temperature dependence predicted for u(w), is considered. I n order to discuss experimental data, it is necessary to realize that, a t low frequencies and/or high temperatures, a departure from the relation Aw" occurs for a perfectly understandable reason. This is that the magnitude of Aw" falls below the d.c. conductivity, which is ascribed to a quite different mode of conduction, namely, transport by carriers (holes in the case of chalcogenides)

S. R. Elliott


554

excited into the (valence) band. The frequency dependence of this mode of conduction is expected to be very small unless the excited carriers move b y hopping between localized states a t the valence band edge. I n this latter case a dependence on frequency is expected b u t the process should be easily distinguishable from the one that is the main topic of this paper by means of its temperature dependence which, because the Fermi level lies near mid-gap, should be activated with an energy approximately equal to half the bandgap. Examples of experimental data we are attempting to explain are illustrated in figs. 1 ( a ) , ( b ) and (c), and refer to Se (Lakatos and Abkowitz 1971), As,Se, (Ivkin and Kolomiets 1970) and As,Te, (Rockstad 1972), the first two materials being in the form of glass and the last being an evaporated film. The hightemperature (frequency-independent) slopes correspond to activation energies of 0.95 eV, 0.97 eV and 0.39 eV respectively, values that are roughly half the energy gaps and hence correspond to the activated process referred to above. The additional broken lines in fig. 1 (c) will be referred to later in this paper.

IO~I-

._

3

4

5 lO?T (K')

(a)

2.0

3.0 103T (8)

25

(6)

4

6

8

10

1dT ($1

(c)

Temperature dependence of the total measured conductivity uT (solid line) for various chalcogenide glasses, measured at different frequencies. uT is plotted logarithmically versus inverse temperature. (a) Se (Lakatos and Abkowitz 1971); (6) As,Se, (Ivkin and Kolomiets 1970); (c) As,Te, films (Rockstad 1972, taken from Fritzsche 1974). The dash-dotted line in diagram (c) is uac (cT-udC),and the dashed line the prediction from the QMT model (uJ, fitted to the low-temperature data. The inset shows the temperature dependence of the difference between the measured (uac)and the theoretical (ul) value of the a.c. conductivity, i.e. U , = U ~ - U ~ ~ - U ~ .

5 2. THEORETICAL TREATMENTS OF A.C.CONDUCTIVITY Theories proposed in the past for a.c. conduction in amorphous semiconductors (Austin and Mott 1969, Pollak 1971) have mostly assumed that carrier motion occurs through quantum-mechanical tunnelling (QMT) between localized (defect) states near the Fermi level.

Temperature dependence of U . C . conductivity of chalcogenide glasses

555

The expression for uac derived from this model is u B c=AN2(Ep)ar-5e2kTw ln4(1/wro)

(1)

or a, CKw s,

where


s = 1 - 4jln ( l / w r o )

(2)

and N(E,) is the density of states a t the Fermi level ( ~ m - ~ e V - ' )a-' , is the spatial extent of the localized wavefunction, ro is some characteristic relaxation time, and A is a numerical constant for which different values have been obtained b y various authors (7~13(Austin and illott 1969), 7 ~ ~ / 9(Pollak 6 1971), 3.66.rr2/6 (Butcher and Hayden 1977)). Thus, the temperature dependence of uac predicted using this model which assumes uncorrelated tunnelling (Pollak 1971), is o,,ocT. Hence, the temperature dependence of uac in this case is expected to be weaker than that for udc,as observed experimentally. From eqn. ( I ) , written as u = BT, where B is temperature-independent, the gradient for In uaC plotted logarithmically versus inverse temperature should be d(ln uac)/d(1/ T )= - T . Comparing with the experimental data shown in fig. 1, it can be seen immediately that the slope of such a plot does show a tendency to decrease with decreasing temperature. However, the slope is frequency-dependent (see, particularly, fig. 1 (a))-. feature that conflicts with theory. Together with the observation that the QMT model fails to account for the experimentally observed temperature dependence of the exponent s of the frequency dependence of uac for chalcogenide glasses, as well as various other features (Elliott 1977), i t thus appears that this model is inappropriate for the case of these materials. The present author has recently proposed a n alternative theory for a.c. conduction in chalcogenide glasses (Elliott 1977, 1978) that is based on the model for charged defect centres in chalcogenide glasses advanced by Street and Mott (1975), Mott, Davis and Street (1975) and by Kastner, Adler and Fritzsche (1976). This model assumes th at carrier motion occurs by means of hopping over the potential barrier separating two defect centres. For simple activation over a barrier, whose height W is a random variable, the exponent s of the frequency dependence of uac is predicted to be exactly unity (Gevers and Du PrB 1946, Frbhlich 1949, Butcher and Morys 1974, Pollak 1976) ; in the present case, the experimentally observed variation of s with temperature is accounted for by assuming a Coulombic correlation between the charged defect centres (Elliott 1977), resulting in a correlation between W and the intersite separation (Pike 1972), i.e correlated barrier hopping (CBH). The expression for uac derived on this model is, to a first-order approximation (Elliott 1977), (3)

where

~=1-6kT/B

(4) and K is the dielectric constant, B the (optical) bandgap, T~ a characteristic relaxation time, and N the spatial density of defect states ( ~ r n - ~ ) .

S. R. Elliott

556


A better approximation, useful at high temperatures, and for materials with small B, which is correct to second order and a very good approximation to third order (Elliott 1977)) is

It can be seen that there is no explicit temperature dependence contained in eqn. (3), only an implicit dependence through the exponent s. Equation (3) may be rewritten in the form uac=abTITo to make this dependence clearer; here, b = (1/w0) and n stands for all other (temperature-independent) parameters in eqn. (3). The parameter To=B/6k and has a value typically N 3-4 x lo3K. It is of interest to compare the temperature dependence of uac calculated on this model with that of the d.c. conductivity udcfor a material with the same bandgap; the latter can be written as udc=Cexp(-Tl/T), where C is a constant, and T, N BJ2k with a typical value of 2: 1 O4 K. Thus d -(In

dT

In b

uac)=-

z 5 x lW3K-1,

TO

whereas

at room temperature. Hence, in accord with experiment, the temperature dependence of udc is expected to be greater than that for uac for the present model, even at room temperature. For purposes of qualitative comparison with experimental data, as displayed in fig. 1, the quantity

is required.

For the approximate expression (eqn. (3))uac = abTIToand hence, d(ln uac)- - T 2 In 6 . To

--W/T)

This slope decreases as the temperature decreases, in accord with experimental observation. In addition, the slope is seen to have a frequency dependence through the In b term ( =In [ l / W T o ] ) ; as the frequency w decreases, In b, and hence the slope, increases for a given temperature, again in agreement with experimental data. This latter behaviour in fact is an alternative way.of expressing the fact that the exponent s of the frequency dependence is a decreasing function of temperature for these materials. Finally, it should be noted that, for both QMT and CBH models for ax. conduction, an unambiguous estimate for the characteristic relaxation time r,, involved in the hopping process may be made. For materials that are described by the QMT model (most likely to be amorphous germanium or silicon), a value for T~ may be extracted simply from the frequency dependence of uaC,where the . materials that are described by the CBH exponent s = 1- 4/ln ( l / w ~ ~ ) For

Temperature dependence of a.c. conductivity of chalcogenide glasses

557

model (e.g. chalcogenide glasses), then (to a first approximation) a value for T~ may be deduced from a knowledge of the temperature dependence of sac, viz.

assuming that the value for To(= B/6k),and hence the band gap B, is known. Alternatively, T~ may be deduced unambiguously from a combined experimental determination of both the frequency and the temperature dependence of sac, since the frequency exponent s( = 1 - TIT,) includes the factor T o (eqn. (4)).


9 3. APPLICATION OF THE CBH MODEL TO A PARTICULAR EXAMPLE We propose here to apply the CBH model to a specific example, that of amorphous As,Te,, and to determine whether the temperature dependence predicted from this model satisfactorily fits the experimental data. The temperature dependence of a.c. conductivity of thin films of a-As2Te3, as measured by Rockstad (1 972), is shown in fig. 1 ( c )(takenfrom Fritzsche 1974) ; again, urPis plotted logarithmically versus inverse temperature. Qualitatively, the same features are observed as in figs. 1 ( a ) and ( b ) ; namely, a weak temperature dependence of a,, a t low temperature, becoming stronger a t higher temperatures, and a frequency dependence of the slope a t low temperatures, decreasing slightly with increasing frequency. The total measured conductivity uT is shown by the full lines; a t high temperatures, uT tends to the d.c. value ad,, and its (activated) temperature dependence is indicated by the dotted line. Rockstad (1972) analysed his results in terms of the QMT model, t o which the data fitted tolerably well a t low temperatures. However, upon extrapolating the low-temperature fit (ul) on the basis of the QMT model ( a a c ~ Tto) higher temperatures (dashed line), it was observed that the extrapolated values lay below the experimental data shown by the dash-dot curve. Rockstad interpreted this for aaD(= aTbehaviour as indicating that uac consisted of two contributions; the first (uJ, that predicted by the QMT model for tunnelling a t the Fermi level and the second ( a 2 ) ,important only a t higher temperatures, and consequently having a larger temperature dependence, was tentatively ascribed to QMT amongst localized states near the band edges, rather than a t the Fermi level. We intend here to analyse these data in terms of the CBH model, since it has already been shown that QMT a t the Fermi level as a mechanism for a.c. conductivity is inappropriate for chalcogenide glasses (Elliott 1977). Since, on this model, a t least to a fmt approximation (eqn. (3)), lna,, is predicted to be proportional to T (rather than a,,ocT as for QMT), the data for uaC(= uT - udc) measured a t lo5Hz from fig. 1 ( c ) is replotted in this manner in fig. 2 (dashed line). It can be seen that a t low temperatures, the behaviour is indeed accounted for by a dependence In uacocT,but a t hgher temperatures, u a c from experiment increases faster than this. One should really use the full expression (eqn. (5)) for uac derived on the CBH model for this material, since the optical bandgap B (0.83eV) is so small. A best fit to this expression is shown in fig. 2 by the full line, the fit being accomplished by variation of the parameter ln(l,’WTO). Although excellent agreement is attained a t low temperatures, the value predicted for u R Cusing eqn. (5) is still less than the experimental data a t high temperatures.

S. R. Elliott

558

Fig. 2


.

-

0

1

0

50

100

150

200

250

30(

T(K)

Temperature dependence of uaC (uT- u d c ) for a-As,l'e, iiieasured at lo5 Hz, plotted logarithmically versus temperature (experimental data from Rockstad 1972). Filled circles denote experimental data, the dashed line is a plot of the approximate theoretical expression derived from the CBH model (eqn. (3)), and the full line is the more exact expression (eqn. ( 5 ) ) . We now propose, following Rockstad (1972), that uac is indeed composed of two contributions; the first, ul, is given by the CBH model (eqn. (5)) and the other, = (sac - ul, is shown plotted logarithmically versus inverse temperature in fig. 3. This is seen t o be simply activated, with a better straight line than in Rockstad's interpretation-see inset of fig. 1 ( c ) (Fritzsche 1974). The slope of this plot of log u2 versus 1/T in fig. 3 is found t o be 0.36 eV, compared with the activation energy for d.c. conduction of 0.39eV (Rockstad 1972). We therefore believe that u2is caused by carriers undergoing QMT in the localized states a t the band edge, some 0.03eV from the mobility edge, and that this behaviour dominates the CBH behaviour for temperatures greater than 'v 200 K. The best fit t o the experimental data below this temperature using the CBH model and eqn. (5) was accomplished for a value of 12.5 for the parameter In ( ~ / w T ;~for ) the frequency of 105Hz used in the experiment, this gives a value for the characteristic relaxation time T~ N 6 x 10-l2s, a reasonable estimate considering i t to be the inverse of a typical phonon frequency (Mott and Davis 1971). One may also analyse the QMT regime above 200K of u2. For QMT a t an energy E away from the Fermi level, the a.c. conductivity is (Pollak, private communication) 972

u --In

2-12

a-5

(2)N2(E)-e2kTwln4( ~ 16

/ w Texp ~ ) ( - E/kT),

(6)

Temperature dependence of a x . conductivity of chalcogenide gkzsses

559

where N ( E ) is the density of states ( ~ m - ~ e V - 'a) t an energy E from the Fermi level, and all other parameters are as for eqn. (1). Knowing the value of u, at a given frequency and temperature (say 2.3 5 i2-l em-' a t 105Hz and 300K, cf. fig. 3), one may estimate a value for a-l (the radius of the localized wavefunction a t the energy where QMT takes place), assuming values for N ( E ) and T ~ . We estimate N ( E )2 10,' ~ m - ~ e V - la, typical figure a t the band edges, and ~ ~10-'3s, 2 : a typical inverse phonon frequency, and hence deduce a value for a-l= 130& a not unreasonable figure for the radius of the localized wavefunction very close to the mobility edge (some 0.03 eV away) (see Mott and Davis 1971)


Fig. 3

Temperature dependence of u, (uac-ul) for a frequency lo5Hz for a-As,Te,, plotted logarithmically versus inverse temperature (experimental data from Rockstad 1972); (rl is derived from the CBH model (eqn. ( 5 ) ) . The line has a slope of 0.36 eV.

5 4. CONCLUSIONS We have sought to show that, in the case of chalcogenide glasses, the appropriate model for a.c. conduction is one in which two electrons hop between two charged defect centres over a potential barrier, whose height is correlated with the intersite spacing (Elliott 1977). The temperature dependence of a.c. conductivity obtained from this model is, t o a first approximation, In o,,ccT. This dependence is shown to be qualitatively obeyed by various chalcogenide glasses. A quantitative comparison is made for the case of a-As,Te, films, as measured by Rockstad (1972) : agreement is found t o be excellent for temperatures lower than 200K and a value

560

Temperature dependence of a.c. conductivity of chalcogenide glassw

for the characteristic relaxation time T~ N 6 x 10-l2s is deduced. At temperatures higher than 200K, another mechanism for aaCis believed to predominate, that of excitation to the localized states a t the band edges, where conduction takes place instead by quantum-mechanical tunnelling. A value for the extent of the localized wavefunction 01-l N 130 tf is deduced for tunnelling taking place some 0.03 eV from the mobility edge.


ACKNOWLEDGMENTS The author is grateful for useful discussions with Dr. E. A. Davis and Dr. J. Kocka. REFERENCES AUSTIN,I. G., and MOTT,N. F., 1969, Adv. Phys., 18, 41. BUTCHER, P. N., and HAYDEN, K. J., 1977, Proceedings of the 7th International Conference on Amorphous and Liquid Semiconductors, Edinburgh, p. 2. BUTCHER, P. N., and MORYS,P. L., 1974, Proceedings of the 5th International Conference on Amorphous and Liquid Semiconductors, Garmisch-Partenkirchen, p. 153. DAVIS,E. A., and MOTT,N. F., 1970, Phil. Mag., 22, 903. ELLIOTT,S. R., 1977, Phil. Mag., 36, 1291; 1978, Ibid. B, 37, 135. FRITZSCHE, H., 1974, Amorphous and Liquid Semiconductors, edited by J. Taw (Plenum). FROHLICH,H., 1949, Theory of Dielectrics (Oxford: Clarendon Press). GEVERS,M., and DITPRB, F., 1946 a, Discuss. Faraday SOC. A, 42,47; 1946 b, Philips Res. Rep., I, 279. IVKIN, E. B., and KOLOMIETS, B. T., 1970, J . non-crystallineSolids, 3, 41. JONSCHER, A. K., 1977, Nature, Lond., 267, 673. KASTNER, M., ADLER,D., and FRITZSCHE, H., 1976, Phys. Rev. Lett., 32, 1504. KOCKA, J., 1976, Czech. J . Phys. B, 26, 807. LAKATOS, A. I., and ABKOWITZ,M., 1971, Phys. Rev. B, 3, 1791. MOTT,N. F., and DAVIS,E. A., 1971, Electronic Processes in Non-CrystallineMaterials (Oxford: Clarendon Press). MOTT, N. F., DAVIS,E. A., and STREET, R. A., 1975, Phil. Mag., 32, 961. PIKE,G. E., 1972, Phys. Rev. B, 6,1572. POLLAK, M., U.S. National Techn. Int. Service AD-757 097 (unpublished); 1971, Phil. Mag., 23, 519; 1976, Proceedings of the 6th International Conference on Amorphous and Liquid Semiconductors, Leningrad, p. 79. ROCKSTAD, H. K., 1972, J . non-crystallineSolids, 8-10, 621. STREET, R. A., and MOTT,N. F., 1975, Phys. Rev. Lett., 35, 1293.