IEEE Transactions on Electrical Insulation Vol. EI-21 No.3, June 1986

NON-DEBYE BEHAVIOR OF DIPOLAR RELAXATION IN SYSTEMS

WITH DIPOLAR INTERACTION

A. Torres, J. Jimenez, B. Vega

and

J. A. de Saja

Laboratorio de F.'sica del Estado S6olidoFacultad de Ciencias

Valladolid, Spain

ABSTRACT

The study of dipolar relaxation in hydrated cerium sulphatesingle crystals by TSDC reveals a non-Debye behavior. Therelaxation time does not follow an Arrhenius law. This factis ascribed without ambiguities to dipolar interaction occur-ring during the relaxation process. This interaction arisesfrom both the high density of structural dipoles existing inthe crystals and their weak bonding energy. By means of thefractional emptying technique we have gradually reduced thedipole density and therefore the dipolar interaction. Thestudy of the relaxation time related to each TSDC peak leadsto relations between the activation energy of the interactingprocess and the strength of this interaction.

INTRODUCTION

TSDC (Thermally Stimulated Depolarization Currents)measurements in Ce2(S04)3-9H20 single crystals give a

rather complex spectrum, its resolution by the experi-mental methods of Bucci et al. [1] and Creswell et al.[2] gives three different peaks [3]. The shape of thesepeaks is different from that expected of a Debye re-

laxation process [4]. In fact, the plot of the relaxa-tion time as a function of the temperature does notfollow a classical Arrhenius law [5] for these peaks.In other previous work dealing with these specimens, we

ascribed the TSDC peaks to dipolar relaxation phenomena[3,5]. The electric dipole unities accounting for theobserved TSDC peaks were shown to be associated to hy-drogen bonds existing in the structure of the singlecrystals [6].

The non-Debye shape of the peaks can be ascribed tothe high dipolar density relaxation, which leads to a

non-independent dipolar reordening yielding the corres-

ponding delay in the relaxation time. The dependenceon the dipolar density of the relaxation time has beendeduced in an analytical way on the basis of the satis-factory fitting found between the detailed balance equa-tions for general-order kinetics [7,8] and our experi-mental TSDC peaks. Thus, we have empirically defined a

relaxation time obeying the following relationship:

T = T0p(PO/P)q- exp[E/kT] (1)

where -co is the preexponential factor, which has timedimensions, E is the activation energy, k is the

Boltzmann constant, PO is the total polarization asso-ciated to a TSCD peak, P is the remanent polarizationat a temperature T and q is a fitting parameter, whichwe have defined as the interaction parameter, and clas-sically represents the kinetic order in a detailed bal-ance equation. The relaxation time defined by means ofEq. 1 includes a factor (Po/P)q-', which gives a delayin the relaxation time. This ;factor is dependent onboth P and q, in fact the greater q, the longer the de-lay. The physical description of Eq. 1 is the following:during the polarization process under an external biasat high temperature, a density PO of dipoles is driveninto a metastable configuration, which is frozen-in bya fast cooling of the specimen, while the electric fieldis applied. Then the electric field is removed and thesample is heated under short-circuit conditions. As thetemperature increases the dipoles begin to relax, at lowtemperature the density of dipoles that relax is prac-tically negligible, P=Po, the factor (Po/P)q-1z1, and nodelay in the relaxation time is observed. As the re-laxation involves more and more dipoles (increasing tem-perature) (P0/P)>1 and the relaxation time increases.It should be noted that the dipolar interaction idea isstrongly related to the relaxation process itself, inother words the dipole-dipole interaction takes placeduring the reorientation of these dipoles. In fact,during the relaxation the dipoles are in less definedenergetic positions, which favor strong interaction. Onthe other hand, high dipole density in the "intermediate"relaxing state must give an increase of the interaction,so that the interaction parameter q is dependent onboth the relaxing dipolar density and the bonding energyof the relaxing state. The experimental results obtained

0018-9367/86/0600-0395$01.00 @ 1986 IEEE

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IEEE Transactions on Electrical Insulation Vol. EI-21 No.3, June 1986

in Ce2(SO4)3.9H20 single crystals support in a reliableway these assertions [5,9]. In this way, it is note-worthy to say that E and q exhibit a similar behavior.A high value of E is accompanied by a high value of q.This is in agreement with the fact that the activationenergy includes an interacting energy contribution. Inorder to corroborate this idea, we have carried out someexperiments dealing with the behavior of E and q fordifferent polarization conditions and therefore differ-ent dipolar densities. These conditions are obtained bypolarizing the specimen at variable temperatures.

EXPERIMENTAL PROCEDURE

The specimens used in our experiments wereCe2(SO4)3-9H20 single crystals, which are colorless andcrystallize in the hexagonal system, spatial group C2h.The growth method has been described elsewhere [10,11]. The samples were cut either parallel or perpendi-cular to the C6 axis. The preparation of the specimensand the experimental procedure were described in pre-vious papers [3,12]. In order to obtain a good repro-ducibility of the experiments an MISIM (Metal-Insulator-Sample-Insulator-Metal)contact configuration was used[12]. The polarizing electric field was applied in theC6 axis direction. The TSDC spectrum obtained in thesepolarization conditions is shown in Fig. 1. The resolu-tion of this spectrum reveals the existence of threedifferent TSDC peaks, which are represented by thedotted line.

150 200 T(K)Fig. 1: TSDC spectrum of the Ce2(SO4).9H20 single

crystaZ for a poZarizing bias paraZZeZ to the C6symmetry axis. The dotted Zine represents thethree different peaks obtained after experimentalresolution of the spectrum.

Our study about the behavior of q and E as a functionof the dipolar density has been carried out with thepeaks 1 (TMI=162 K) and 3 (TM3=200 K). The variationof the dipolar density is produced by varying the polar-ization temperature Tp, keeping unchanged the otherpolarization conditions (polarization time tp, and po-larizing bias Ep). The lower the Tp, the smaller isthe density of dipoles in the metastable stage; this isobserved as a reduction of the area enclosed by theTSDC curve. This procedure is easily applied for peak1: by polarizing below Tp=170 K, no spurious effectsdue to the other peak is observed. The resolution of 3is more difficult because it is necessary to erase thecontribution from peaks 1 and 2 by means of fractionalemptying. The polarization temperatures for both peaksare represented by arrows in Figs. 2 and 3 respectively.It can be observed in these Figures that both peaks donot present the typical shape expected from a Debye re-laxation mechanism.

150 170 T(K)Fig. 2: ExperimentaZ TSDC peak (TM1 = 162 K) (o o o).

The fulZ Zine is the theoreticaZ pZot obtained fromEq. 3. The arrows indicate the polarization tem-peratures, which aZZow one to vary the dipoZar den-sity. (Ep = 1.4 kV/cm; heating rate 0.25 K/s;samrpZe thickness 0. 4 mm).

The electrical current density is defined as the re-lation of the polarization to the relaxation time:

dP p" 7dt T (2)

By substituting T, as defined in Eq. 1, in (2) the elec-trical current density, after integration, becomes

(3)

J(T) =- exp[-E/kT](1 + ff f'(T') exp[-K/kT']dT') q

where f'(T) is the derivative of the function t=f(T) .

This equation fits satisfactorily our experimentalTSDC currents. The fit, associated with the relaxation

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Torres et al.: Non-debye behavior of dipolar relaxation in systems with interaction

-Q5[

180 200 T(K)Fig. 3: TSDC peak (TM3 = 200 K). (o o o) experimentaZ

vaZues; ( -) theoreticaZ pZot (Eq. 3).

time definition (Eq. 1) and the condition of the maxi-mum of the TSDC peak [4] allows us to study the para-meters, E, Toy and q, which define the interacting di-pole relaxation.

DISCUSSION

The fitting of Eq. 3 to the experimental TSDC peaksconfirms the necessity of a third parameter q, in addi-tion to To and E, in order to account for the experi-mental results. This is an obvious consequence of thenon-Debye dipolar relaxation. We study the dependenceof E and q with the dipolar density. In a non-inter-acting Debye process the area enclosed by a TSDC peakis proportional to the number of dipoles activated bythe polarizing bias [13]. This assertion is not exact-ly true in a non-Debye process, because the delay inthe relaxation time gives an enhancement of the areaand therefore an overestimation of the dipolar density.However, in a first approximation a rough proportional-ity between the area and the dipolar density must exist.In this way, in our experiments the activated dipolardensity (dipoles in metastable state) decrease when pis lowered, so that we can consider the area enclosedunder the TSDC peak as a rough measure of the activateddipole density.

Within this approximation the corresponding E and qvalues have been calculated; they are reported as afunction of the area enclosed under the TSDC peak ob-tained for every polarization temperature of peaks1 (TM1 = 162 K) (Fig. 4) and 3 (TM3 = 200 K) (Fig. 5)respectively. As the area (dipolar density) is reduced,a reduction in both E and q is observed. This behaviorshows the influence of the interaction on the dipolarrelaxation; the activation energy includes an inter-action energy term which is a function of the dipolardensity. The value of q tends to unity (Debye limit)(Figs. 4 and 5), whereas the activation energy tends toa value Eo,=0.5 eV for peak 1 and E03-0.4 eV for peak 3in the experimentally accessible limit of A/A00.1.This suggests that these values are approximately equalto the interactionless activation energies corresponding

31-

1

0 0.5

A/AoFig. 4: Peak 1 (TM1 = 162 K): evolution of theactivation energy E (eV), and the interactionparamneter q, as a function of the dipoZar density,which is represented as the area A encZosed underthe peak obtained for each polarization temperatureT (indicated by arrows in Fig. 2), normaZized tot7e area AO encZosed under the saturated peak(Tp>>TM1).

q(A)

3

2

0 0

@00

0@ 0

A~~~~~~~A A A

A A

A AA

1

0 0.5 1

A/A0Fig. 5: Peak 3 (T = 200 K): evotution of E (eV)

and q as a function of the dipolar density. A andAO denote the scone as in Fig. 4. The polarizationtemperatures Tp are indicated by arrows in Fig. 3.

to the microscopic potential barrier separating themetastable state from the equilibrium state. The factthat Eo1 is higher than E03 may be understood on thebasis of the interaction process itself. In fact, dueto the structural nature of the dipoles (high density)the Debye limit is not experimentally accessible, sothat the additive activation energy due to the

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IEEE Transactions on Electrical Insulation Vol. EI-21 No.3, June 1986

interaction cannot be neglected. On the other hand,some experimental difficulties could contribute to thisobservation. In fact, as the area under peak 1 is re-duced by lowering Tp, the spurious effects due to peak2, which is very close to peak 1, cannot be neglectedand this could account for the overestimation of theasymptotic activation energy calculated for peak 1.

As it has been stated prior to this study, the dipolesare associated with different configurations of the hy-drogen bonds in the structure of these crystals. In ouropinion a study of the sample under different dipolaractivation conditions should provide interesting data onthe interaction process itself. Therefore, Raman spec-troscopy studies under these experimental conditions arecurrently in progress. This study would allow one tofactorize the relaxation time in terms of an interactionenergy.

CONCLUS IONS

Interacting dipolar relaxation in Ce2(S04)3.9H20single crystals can be studied by means of a relaxationtime (Eq. 1) which is a function of the relaxing dipoledensity, the Variation of which causes substantialchanges in the relaxation parameters. In particular, areduction of this density is accompanied by a decreaseof the activation energy. Our empirical method has beenshown to be very useful for studying thermally stimulatednon-Debye dipolar relaxation [5,9]. In our opinion theclassical exponential relaxation time must be factorizedby a term including a dipolar interaction contribution.

The dipolar interaction observed, which has beenpehnomenologically described by Eqs. 1 and 2, is a parti-cular case of the cooperative phenomenon describingother relaxation processes in solids, which do not behaveas a single Debye mechanism [14,15].

REFERENCES

[1] C. Bucci, R. Fieschi and G. Guidi, "Ionic Thermo-currents", Phys. Rev. Vol. 148, pp. 816-823, 1966.

[2] R. A. Creswell and M. M. Perlmann, "Thermal Cur-rents", J. Appl. Phys. Vol. 41, pp. 2365-2375,1970.

[3] A. Torres, J. Jimenez, V. Carbayo and J. A. de Saja,"Courants de depolarisation", Phys. Stat. Sol. (a)Vol. 78, pp. 671-677, 1983.

[4] H. Fr6hlich, Theory of Dielectrics, Oxford Univeri-sity Press 1958, Chapter 3.

[5] A. Torres, J. Jimenez, J. C. Merino, F. Sobronand J. A. Saja, "Etude du Comportement non-Debye",J. Phys. Chem. Sol. Vol. 46, pp. 665-674, 1985.

[6] A. Torres, F. Rull and J. A. de Saja, "PolarizedRaman", Spectrochimica Acta, Vol. 36A, pp. 425-431,1980.

[7] R. Chen and Y. Kirsh, Analysis of ThermallyStimulated Processes, Pergamon Press Oxford 1980,Chapter 1.

[8] C. Bucci and R. Fieschi, "Ionic Thermoconductivity",Phys. Rev. Lett. Vol. 12, pp. 16-19, 1964.

[9] A. Torres, J. Jimenez and J. A. de Saja, "Etude dela Relaxation Dipolaire", J. Phys. Chem. Solids,Vol. 46, pp. 733-741, 1985.

[10] R. Alonso, V. Luu Dang and J. A. de Saja, "Growthand Some Properties", Kristall und Technik, Vol. 8,pp. 457-461, 1973.

[11] A. de Saja, J. M. Pastor, F. Rull and J. A. Saja,"Growth and Some Properties", Kristall und Technik,Vol. 13, pp. 909-914, 1978.

[12] A. Torres, J. Jimenez and J. A. de Saja, "On theInfluence of Contact Configurations", Ferroelec-trics, Vol. 56, pp. 157-160, 1984.

[13] P. Braunlich, Topics in Applied Physics, Vol. 37,Thermally Stimulated Relaxation in Solids, Springer-Verlag Heidelberg 1979, Chapter 4.

[14] A. K. Jonscher, Dielectric Relaxation in Solids,Chelsea Dielectric Press, London 1983, Chapter 8.

[15] R. G. Palmer, D. L. Stein, E. Abrahams and P. W.Anderson, "Models of Hierarchically ConstrainedDynamics", Phys. Rev. Lett. Vol. 53, pp. 958-961,1984.

Manuscript was received 6 September 1985.

This paper was presented at the 5th InternationaZSymposium on EZectrets, HeideZberg, Germany, 4-6September 1985.

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