AC TRANSPORT AND STRUCTURAL RELAXATIONS IN DISORDERED SOLIDS

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AC TRANSPORT AND STRUCTURAL RELAXATIONS IN DISORDERED SOLIDS

W. Schmidt, K. Breitschwerdt

To cite this version:

W. Schmidt, K. Breitschwerdt. AC TRANSPORT AND STRUCTURAL RELAXATIONS IN DISORDERED SOLIDS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-171-C4-174.

�10.1051/jphyscol:1981435�. �jpa-00220892�

AC TRANSPORT AND STRUCTURAL R E L A X A T I O N S I N DISORDERED S O L I D S

W.W. Schmidt and K.G. Breitschwerdt

I n s t i t u t fiir Angewandte Physik, Universitdt HeideZberg, Heidelberg, F

.

^R

.

^G

.

Abstract.

-

P.leasurements of the dielectric properties and the a.c. conductivity of glassy As2Se3 are relsorted. A model whereby an initial dielectric response is coupled to structural relaxa- tions describes the available data between 1 kHz and 1 THz.

Introduction.

-

The existing hopping models (1-3) do not adequately describe the experimental data on a.c. conductivity and dielectric properties of glassy As2Se3 (4). The predicted tem~erature dependence is not conpatible with exnerinental data (K.G. Breitschwerdt, J. Haf- ner, and 7J.W. Schmidt, to be published) and they are only applicable in a limited frequency range. A completely different model has been proposed for the explanation of a conductivity proportional to w2 (5).

TJe ascribe the a.c. trans?ort to a two-step orocess whereby an initial dielectric response is coupled to relaxing structural modes.

Experimental results.

-

The dielectric properties of AsZSe3 have been measured between 1 kHz and 30 GHz with resonance techniques (6). The results may be described in terms of a conductivity o(w) = o0+5'(w), where og is the d.c. contribution, or in terms of a com?lex dielectric susceptibility ~ ( w ) = X' (w)

+

ix"(w)

,

where ~ " ( w ) = a ' (w)/w and

w = 2~rf is the angular frequency.

At 300 R the a.c. conductivity is proportional to wn between 1 kHz and

1 G I I z , w'th n

3

5 1 (fig. 1). Above 10 GHz the a.c. conductivity increa-

ses as w

.

This behavior is consistent with results of far infrared measurements in the frequency range up to 3 THz (5). The conductivity contribution of four phonon modes between 2 and 20 TBz (7) also shows a frequency dependence proportional to w2 on the low-frequency side

(fig. 1 ) . However, below 1 THz the magnitude of this contribution is 25 per cent of the measured losses at most. The d.c. conductivity u o * 10-l3 l/Qcm is negligible in comparison with the a.c. conducti- vlty above 1 kXz.

Phenomenological descrintion.

-

A two-step dielectric response may be established by the following causal relationship between the electric field F and the polarization P (8):

where T and D are positive constants and the expression in the brackets represents a memory function. The initial response requires a short- time memory contribution which for reasons of mathematical simplicity is approxinated by a 6-function. B(t) accounts for the coupling to ad-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981435

JOURNAL DE PHYSIQUE

ditional relaxations. The dielectric susceptibility associated with eq. (1) is

c . 7

where B(w) = .&D(t) exp (iwt) dt. The observed a.c. conductivity pro- portional to w", where n & 1, is described by

where r is a constant and 8 > 0 in order to renove singularities for t a 0. An analysis of the associated susceptibility shows, that for 0 6 r < 1 the thermodynamic requirements (9) X' (0) > 0 and X" (0) > 9 for o > 3 are fulfilled. For r A 1 and r<< 8 each of the frequency dependent terms on the right hand side of eq. (2) dominates in a se- parate frequency range. Thus one obtains in good approximation

-

1

D I - f o r o < w < < e - I . (4) x(w) =

D-I [I -iw~] for w > > w 1' ^{( 5 )} where wl = [( 1 -n) / 8 ~ ]

'I2.

Furthernore, a truncated series expansion of B(w) yields an analytical expression for ~ ( w ) at frequencies above w2 = [2(1-r) sin (nn/2)/r (n)] l/(l-n)e-l

.

The corresponding a. c. con- ductivity is

([8"-'/or(n)] cos(nn/2)wn for w2 < < u < <

e-l ,

^{( 6 )}

a' (w) =

2 2

~ " w ~ T / ( l + w i ) for w > > wl.

Thus at high frequencies the initial response gives rise to Debye-like losses while in a frequency range below 1/8 an a.c. conductivity pro- portional to wn is obtained as a consequence of the slowly decaying function B (t)

.

The measured losses in AsZSe3 between 1 kHz and 1 THz ar well des- cribed by a single set of parameters: n = 0.997. 0 = 10-'l sec. D =

30, i = 10-13 sec, and r = 0.9 (figs. 1 and 2). In fig. 2 the dielec- tric function E ' (w)

+

icn(w) = 1

+

^{4n(x' (w)}

+

ixn(w)) is plotted ver- sus frequency. The Kramers-Kronig conjugated real part of the dielec- tric function is shown, added to the dominating contributions due to phonon modes and electronic polarizability (7). The total €'(a) is also in reasonable agreement with the exnerimental results.

Model.

-

Depending on the type of material this two-step response may be electronic, ionic, dipolar, and structural in nature. In all cases the coupling between the entities is essential. As a model system we consider a charge q which can occupy either of two sites separated by a distance R (1). The differences of site energies and site occupation probabilities are denoted by E and q, respectively, where -1

<

q 6 1. If E(t) varies within a time scale considerably

larger than the relaxation time T (I), the actual occupation proba- bilities are well approximated by probabilities corresponding to an instantaneous establishment of the thermal equilibrium between Eft)

~ ( t ) = tanh (BE(t)/2), ( 8 ) where

B =

l/kT. The site energy difference may be written

The first term on the right-hand side is the energy difference with- out electric field at thermal equilibrium,the second term represents the energy due to the electric field, and the third term arises from the coupling between the dipole systen and additional relaxations.The function G(t) associates previous deviations of Q from the equilibrium value

no

without electric field with the instantaneous contribution to the site energy difference. The solution of eqs. (8) and (9) for

~ ( t ) yields the dipole moment qRq(t)/2 as a function of the electric field. For small amplitudes of a field P(t) a exp (-iwt) the corres- ponding polarizability a ( w ) is

x This work

.

^{e . 0 Ref. 5}

- - - - Vibr.. Ref. 7

10'

-

T h e o r y ^{F I R}

Fig. 1 : A.C. Con- ductivity versus frequency.

18- l 6 - 1L- 12

10- 8 6 L

'

rb.

^{' r b 6 ' i o 8 ' io10 '}

iolf,izl

E " " ' ~ ' ~ " - '!

As2Se, 300K

:: ::

x x x x This work

;:

....

^Ref.5

^{r ':}

- -

-

-

^{V~br. and} electr.. ~ e f . 7 dd

:

8 - i d

-

^Theory

- - - - T O ~ O I E'

X X I X X~ I X x p ] l o

i i

-^, l

-

-

- - - - - - -

. -.--

- - - -

.-

- - -

-.

- - - - -

--. 71--7-:

-

- -->.-I

- -

-

⁸

^-

- /

0 1

I I 8 1

-

, - - - x - x 4 *

!i '

-loo

Fig. 2: Real part E '

and imaginary part E "

of

tion the versus dielectric frequency. func-

10-2

lo3

JOURNAL DE PHYSIQUE

a(w) =

[B

( q ~ ) 2/4] [cosh2 ( B T V / ~ )

-

B G ( ~ ) /2]-' (1 0) where G(w) = G(t) exp(iwt) dt. In a first approximation a number

0

of independent, essentially identical, and randomly oriented dipole systems per unit volume may be assumed. One obtains then the dielectric susceptilibity

Thus, losses at frequencies far below 1/8 are due to the coupling to additional relaxations as described by the imaginary part of G(w).

A comparison of eqs. (4) and (11) establishes the following correspon- dence between the phenomenological description end the model

1/D, according to eq. (12), is identical with the susceptility eq. (Il), provided that the dipoles are decoupled from the additional relaxations.

On the other hand, eq. (13) attributes B(w) and, consequently, the a.c. conductivity proportional to wn, to the pro~erties of the relaxa- tion processes coupled to the dipole system.

In the case of As Se3 the relaxation processes may be due to the coupl- ing of a polariza&Lon response of the bulk material, probably enhanced by charged structural defects (lo), with highly unharmonic structural modes widely observed in amorphous materials of this type (11). Since the latter involve cooperative readjustments of many atoms (12), a structural rearrangement may take place over longer periods of time.

This could give rise to the long-time memory effects in the dielectric response. Using eq. (13) and the numerical values for the narameters n, 8 and r in B (t) (eq.3), the function G(t) and, hence,G (w) can be calculated.

1 Plott N . 5 . and Davis E.A., Electronic Processes in Non-Crystalline Elaterials (Clarendon Press, 2nd ed., Oxford 1979).

2 Pollak P I . and Pike G.E., Phys.Fev.Lett.

28

(1972) 1449.

3 Elliott S.R., Phil.Mag.

36

(1977) 1291.

4 Le Cleac'h X. and Palmier J.T., in: Amorphous and Liquid Semiconduc- tors, ed. Spear 1V.E. (Centre for Industrial Consultancy and Liaison, University of Edinburgh, 1977).

5 Stron U. and Taylor P.C., Phys.Rev. B16 (1977) 5512.

6 Breitschwerdt X.G. and Hafner J., J.Non-cryst.Solids 35-36 (1980) 993.

7 Vasko A. and Rieslich R . , in Ref. 4.

8 Schmidt %?.>I. and Breitsch~ierdt K.G.

,

J.Physigue 41-C8 (1 980) 12.

9 Landau L.D. and Lifschitz E . X . , Electrodynamics of Continuous Media (Pergamon Press, London, 1960).

10 Street R.A. and Mott :J.P., Phys.Rev.Lett. 35 (1975) 1293.

1 1 Hunklinger S. and Arnold W., in: Physical

caustics,

eds. rlason V.P.

and Thurston R.N., Vol. XI1 (Academic Press, Yew York, 1976)

12 Anderson 3.?7., Iialperin B.I., and Varna C.11., Phil.Plag. 25 (1972) 1.