J. PRAKASW et al.: Heating Rate Dependence of Maximum Depolarization Current 685

phys. stat. sol. (a) 91, 685 (1985)

Subject classification: 10.2 and 14.4; 22.5.2

Department of Physics, University of Gorakhpurl)

Heating Rate Dependence of Peak and Position of Maximum Depolarization Current in ITC Measurements BY JAI PRAKASH, RAHUL, and A. K. NISHAD

Dedicated to Prof. Dr. Dr. h. c. Dr. E. h. P. GORLICH on the occasion of his 80th birthday

An attempt is made to study the heating rate dependence of the peak and position of the maximum depolarization current in ITC measurements. It is observed that the area under the 1°C curve, where tlepolarization current is plotted as a function of time remains unchanged a t different heating rates. Mathematical equations are developed for obhining the heating rate dependence on the peak and position of the maximum depolarization current. It is observed that the position of' the maximum depolarization current shifts towards higher temperature side with increasing the heating rate. The peak of the maximum depolarization current is also found to increase with the increase in the heating rate. The dependence on the peak and position of the maximum de- polarization current is used to evaluate the dielectric relaxation parameters. The evaluated param- eters are found to be in good agreement with the reported values.

Es wircl versucht, mittels ITC-Messungen dic Abhangiglreit des rvlaximums und der Lngc des maxima.len Depolarisationsstromes von der Heizrate zu untersuchen. Wenn der Depolarisat.ions- strom als Funkt,ion der Zeit aufgetragen wird, wird beobachtet, daB die Flache nnter der ITC- Kurve sich bei unterschiedlichen Heizraten nicht andert. Mathematische Gleichungen werden entwickelt, um die Heizratenabhangigkeit des Maximums und der Lage des maximalen Depolari- sationsstromes zu beschreiben. Es wird beobachtet, daB sich die Lage des maximalen Depolari- sationsstromes zu hoheren Temperaturen versehiebb, wenn die Heitzrate erhoht wird. Die Bb- hiingigkeit des Maximums und der Lage des maximalen Depolarisationsstromes werden benutzt, um die dielektrischen Relaxationsparameter zu bereelmen. Die berechneten Parameter sind in guter Ubereinstimmung mit veroffentlichten Ergebnissen.

1. Introduction In the quest of understanding the inechanisni inherent in dipolar orientation due to non-in teracting dipoles, a number of experiniental techniques have been employed. Ionic thermocurrent (ITC) nieasurement [l] has been found to be the most convenient, sensitive, and accurate method. The ITC technique has contributed substantially to a better understanding of the role of the dipolar imperfections. For systems involving a single relaxation time, one gets a single peak in ITC measurements. Dielectric relaxation parameters can be evaluated considering the whole area of the ITC curve following the method of Bucci et al. [l]. In the cases having several relaxation times, the recorded ITC spectruni is broadened and consists of a number of peaks. However, the ITC technique is sensitive to such an extent that it can effectively resolve rdaxation processes with only slightly different reorientation energies (peaks spaced as close as 0.01 eV). In such cases, different peaks can be scanned following the peak-cleaning technique [l to 31. However, the evaluation of the area enclosed in the ITC spectruni associated with each individual ITC peak involves a fair degree of uncertainty in the peak-cleaning method, leading to an inaccurate

l) Gorakhpur 273001, India.

686 J. PRAKASH, RAHUL, and A. K. KISHAD

value of the dielectric relaxation parameters. The usefulness of the method of Bucci et al. is thus restricted to cases showing multiple relaxation times. Also, the situation will be similar in cases where the correct evaluation of the area is limited due to back- ground current. It has been shown in this article that the ITC spectrum recorded under similar conditions but a t different heating rates provides a convenient method for the accurate determination of the dielectric relaxation parameters. It has been found that the position of the maximum depolarization current shifts towards the higher temperature side with increasing heating rate. The peak of the ITC spectrum has also been found to increase with the increase in the heating rate. However, the area enclosed in the ITC curve where the depolarization current is plotted as a func- tion of time, remains unchanged a t different heating rates. Dielectric relaxation parameters in a number of cases have been evaluated following the suggested method of analysis. The evaluated parameters are found to be in good agreement with the reported values.

2. Suggested Method of Analysis

The depolarization current observed in the ITC measurement is very similar to the therriioluininescence glow curve with mononiolecular kinetics. In ITC measurenients [l, 41, the sample is polarized by applying an electric field (E,) at some suitable polarization temperature ( Tp) for a time large compared to the relaxation time a t T,. With the electric field still on, the sample is rapidly cooled down to a fairly low temperature where the electric field is switched off. A t such a low temperature, the relaxation time is practically infinite and consequently the polarized dipoles are frozen-in in the crystal lattice. The sample is then heated a t a linear heating rate b such that

T = To + b t . (1) The resulting depolarization current is recorded with the help of an electronieter. To is the temperature a t which the depolarization current starts to appear. The depolari- zation current (I) as a function of temperature is expressed as [l]

T Q 1

I ( T ) = - z exp [ --& exp (-$) d Y ] ,

TO where zo is the pre-exponential factor or fundamental relaxation time, E’, the activa- tion energy for the orientation of I V dipoles, k the Boltzmann constant, and t the relaxation time which decreases exponentially with increasing temperature according to the Arrhenius relation as

z = zo exp [E,/kT] . (3) Q in ( 2 ) represents the total charge released in ITC measurements, which depends obviously on the polarizing conditions as

where N , is the number of IV dipoles per unit volurne, ,u their dipole moment, A the effective area of the specimen, and a the geometrical parameter which for freely rotating dipoles has a value of 1/3. The maximum depolarization current occurs a t a temperature T , defined as

Heating Rate Dependence of ;Maximum Depolarization Currenti n ITC 687

where z,,, is the relaxation time a t T,. In ITC measurements, the depolarization current versus temperature curve is called the ITC spectrum. It is obvious from ( 2 ) that the ITC spectrum is an asymmetric band with the maximum depolarization current a t T,,,. The area S enclosed in the ITC spectrum enables one to evaluate the value of Q as

m

Q = J I d t . (6) t( To 1

The higher temperature end of the ITC spectrum can safely be replaced by 00 as done :in (6). With the help of (1) one gets

t(%") T, Lt is obvious from ( 7 ) that S will be proportional to b provided the same specimen is polarized under similar conditions for different runs of b. A number of measurements 15 to 91 are reported in the literature where the ITC spectrum is recorded for different runs of 6. A representative spectrum recorded by Kunze and Muller [ 5 ] in the AgCl : Xi2+ system for heating rates of 0.033,0.050, 0.067, 0.10, and 0.15 Ks-l is plotted in Pig. 1. 'The very look of the figure suggests that probably the different runs of the ITC spectrum are recorded either on different samples of varying concentration or on t,he same sample polarized under different conditions. However, this notion is niis- leading in the light of (7) which suggests a straight line between S and b provided the ineasurenients are recorded on the same sample polarized under similar conditions. Such a% graph is shown in Fig. 2. The slope of the straight line gives the value of Q. The uncertainty due t'o background current in the area enclosed in the ITC spectrum is also naturally averaged out in t.he straight line plot shown in Fig. 2. Consequently, hhe vahe of Q can be determined more accurately following Fig. 2.

Dielectric relaxation parameters ( E , and zo) from ITC spectrum can be evaluated in a number of ways [l, 10 to 131. Out of the different methods of analysis, the method suggest'ed by Bucci et al. (BFG method), proves to be the most accurate one. Howev- er, thce utility of the method is limited when either the dark current is not pre- cisely est,iinat,ed or more than one relaxation mechanism is involved in the system. Subsequently, t'he points corresponding to lower and higher temperature ends of the ITC spectrum deviate from the required straight line fit in the In z versus 1/T graph leading to uncertainty in the evaluated values of E, and zo.

In the quest of developing a suitable method for the analysis of the ITC spectruin (,5) has been used by Kunze and Miiller 151, Hino [GI , Hino et al. 171, and Pissis et al.

Pig. 1. ITC spectrum recorded at different heating rates of 0.033,0.050,0.067,0.10, and 0.15 Ks-'by Kunze and Muller [5 ] in the AgCl:Ni2+ system

688 J. PRAKASH, RAHUL, and A. K. NISHAD

[9]. Rearrangement of ( 5 ) yields

Fig. 2. Heating rate dependence of the area of the ITC spectrum in o AgCl:Ni+2, amorphous semiconductor with SIM sample geometry, amor- phous semiconductor with MSIM sample geometry, and A SiO, films

Thus a straight line results [I41 when In (T:/b) is plotted against l /Tm. The slope and the intercept of the straight line give the values of E, and to, respectively. It is obvious from (8) that T , shifts towards the higher temperature side with the increas- ing value of 6. However, small variations in b do not change the position of T, niar- kedly.Thus, an appreciable shift in T , can be observed for large variations in the heating rate which ultimately develops a thermal gradient in the sample. Obviously, the evaluated parameters Ea and to using (8) will not be much reliable.

To overcome the problem of dark current and to evaluate the values of E , and to accurately, we reconsider (2). The integral appearing in ( 2 ) cannot be solved. How- ever, in the situations, when (Ea /kT) is large (which fortutiously happens to be) approximations can be imposed [ 151 on the integral. Replacing (Ea/kT) by x, one can develop

2 ! 3! 4! k x 5 2 2 3

1 - - + - - - + . . .

If z is large, we can drop terms except the first and hence

wi th the help of (10) one can rearrange (2) as

The maximum depolarization current (I,) can thus be expressed as

(9)

Heating Rate Dependence of Maximum Depolarization Current in ITC

/ I

689

which in conjugation with (5) yields

Q I , = - exp(-1) TTll

--%+ - 7 k Tm . Ea I -

TO

(13)

Elquation (13) can further be manipulated as

In I , = I n - - 1 --. (14) Q E, 7 0 k T m

Thus, one expects to get a straight line when In I , is plotted against l/Tm. The slope of the straight line gives the value of E,. It is obvious from (13) that there will be an exponential change in I , at a slight change in T,. Consequently, a large variation in b is not required, which would ultimately develop a thernial gradient in the sample. Thus (14) shall give a better value of E, in comparison to (8). This value of E, helps in deterniiningt, using ( 5 ) , which in conjugation with (3) gives the value of T ~ . Thus, the dielectric relaxation parameters E, and to can conveniently be evaluated following (14).

Fig. 3. Variation of a) In (T&/b) and b) In I , with (l/Tm) in the AgCl:pi2+ system

44 physic:& ( a ) 91 /2

690 J. PRAKASH, RAHUL, and A. K. NISHAD

T a b l e 1 Dielectric relaxation parameters for different systems

system E , (eV) t o ( s ) ref. _________ ~. -

reported following following reported following following value (8) (14) value (8) (14)

AgCl:Ni2+ 0.339 0.33 0.32 7.54 x 10-14 2.8 x 10-13 6.7 x 10-13 [B] Si0,films 0.19 0.19 0.20 8.2 x 10-3 8.7 x 10-3 6.0 x 10-3 ~ 6 , 71 amorphous 0.60 0.58 0.58 - 5.3 x 10-14 5.0 x 10-14 [81 semiconductor 0.60 0.60 0.60 - 1.8 x 1.6 x [8] ice 0.17 0.17 - 7.67 x 10-7 7.7 x 10-7 - [91

different systems evaluated from the plots similar to Fig. 3 a and b are presented in Table 1. It is obvious from the table that the values of E, and zo evaluated following (8) and (14) are in very good agreement with the reported ones. Further experimental data a t various heating rates are still awaited to substantiate the conclusions we have arrived a t here. In the case of SiO, films [6, 71 a heating rate up to 1.04 Ks-1 does not pose a problem particularly because of the micron thickness of thin films. Hypothetical situations have been considered by Agarwal [a] in the case of amorphous seuii- conductors. The data in the first row correspond to a sample geometry of tnetal-semi- conductor-insulator-metal (MSIM) conibinatlon, whereas the data in the second row are recorded on a sample geometry of semiconductor-insulator-metal (SIM) combination. In the case of ice systems reported by Pissis et al. [9], we could not apply (14) because of the unavailability of ITC spectra a t different heating rates. We could also not verify the statement made by Yissis et al. that the activation energy evaluated from (8) conies out to be less than that reported by the initial rise method [lo]. The data reported in Table 1 are found to be in close agreement with the values evaluated following the BFG method. Thus, the suggested method of analysis has an additional advantage of application where the BFG method cannot be applied with certainty. It is obvious, therefore, that (8) and/or (14) can effectively be used for the deter- mination of the dielectric relaxation parameters in cases where either more than one relaxation mechanisms are involved or the estimation of background current becomes uncertain. It can further be seen that (14) is independent of the type of the heating rate and one can evaluate the value of E, from the slope of the In I m versus (l/Tm) plot whether either a linear heating rate or a reciprocal/hyperbolic heating rate is employed to record the ITC spectrum as shown in the Appendix.

Acknowledgements

The authors are thankful to Prof. N. K. Sanyal for providing the necessary facilities and to Prof. F. Fischer (Munster University, FRG) for valuable discussions. Two of the authors (R and AKN) are thankful to CSIR (New Delhi) for financial assistance.

Appendix

Instead of using a linear heating rate for recording the ITC spectrum one may also use reciprocal heating rate as suggested by Miiller and Teltow [16] and applied by Hickmott [ 171 in SiO, films. In the reciprocal/hyperbolic heating rate (a) the temper- ature is varied according to

- a t . -_- 1 - 1 T To

Heating Rate Dependence of Maximum Depolarization Current in ITC 691

With this value of a onedevelops an expression for the depolarization current as [16] T

This expression represents an asyninietric glow curve with t,he niaxirnum depolariza- tion current at T , according to

where sm is the relaxation time at T,. Coniparison of (5) and (AS) helps in correlating cc and b as

(A4) 2 aTm = b .

Equation (A3) can further be manipulated as

In ( l /a) = In ( y o ) + & __

which suggest,s that In ( l / a ) plotted against (l/Tn,) should result into a straight line with the slope E,/k. Such a plot enables the evaluation of the dielectric relaxation parameters similar to (8).

The integration appearing in (A21 can easily be solved to give

Here, we have replaced To at which the depolarization current starts to appear by 0. Equation (A6) for the maxirriuni depolarization current (I;) can be expressed as

8 zrn azrnEa

I,, = - exp [ -4, which .in conjugation with (A31 yields

Q 1, = - exp [ - I ] . zni

This can further be rearranged as

E, k T m

l n I m = I n - -l--.

Thus, zt plot of In I, against (1/Trn) should result in a straight line similar to (14). It, is obvious that the In I, versus (l/T,) plot will be independent of the type of the heating rate. The suggested method of analysis for the evaluation of the dielectric relaxation parameters can thus be utilized without taking care of the type of heating rate. 44-

692 J. PRAKASH et a]. : Heating Rate Dependence of Maximum Depolarization Current

References

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(Received X a y 14, 1985)