Proceedings of the Tomsk Polytechnic Institute Named after S. M. Kirov, Holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Volume 247　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1977

On the Electromotive Force of High-Voltage Polarization in Ionic Dielectrics

Yu. M. Annenkov, G. M. Malofienko

(Presented by the scientific seminar of the electrophysical laboratory of the Research Institute of Radiation Physics, Tomsk Polytechnic Institute)

The displacement of charges in solid dielectrics under the influence of an external electric field may be partially or completely impeded by various defects, leading to the formation of a space charge. A. F. Ioffe called the potential drop caused by the reverse field of the space charge the electromotive force of polarization \(P\) [1]. In this case the current flowing through the specimen is equal to

\[ J=\frac{U_0-P}{r_{\mathrm{ist}}}, \tag{1} \]

where \(U_0\) is the voltage applied to the dielectric;
\(P\) is the emf of high-voltage polarization;
\(r_{\mathrm{ist}}\) is the true resistance of the specimen.

On the basis of expression (1), a number of methods were developed for determining the magnitude of \(P\) in many insulating materials [1–9]. It must be borne in mind, however, that formula (1) is not applicable in all cases. The absorption current in a dielectric is formed as a result of the occurrence of various polarization processes, the number of which may be very large. Therefore high-voltage polarization is one of the factors determining the form of the absorption-current function, but by no means in all cases can this factor be predominant. For this reason, the legitimacy of applying one or another method for determining \(P\) must be sufficiently justified.

In the present article, the conditions for the occurrence of the emf of high-voltage polarization as applied to ionic dielectrics and the possibilities of methods for measuring \(P\) are discussed.

Conditions for the Occurrence of the emf \(P\)

The formation of a space charge occurs mainly under the following conditions [1, 2, 10, 11].

1. A clearly expressed inhomogeneity, for example, a two-layer dielectric. Charge accumulates at the interface.

2. As a result of electrolysis or forming, near-electrode layers may be formed whose conductivity differs from the conductivity of the interior of the crystal. Space charge accumulates in these layers.

1. The blocking properties of the “metal–dielectric” contact cause ions to accumulate at the electrodes.

2. The presence of trapping centers in the crystal lattice leads to the formation of a space charge throughout the entire thickness of the dielectric.

In all four cases, the accumulation of space charge will be accompanied by an absorption current. However, in the first two cases the accumulation of space charge may lead to the emergence of an emf \(P\) and to a redistribution of potential across the thickness of the dielectric, as shown in Fig. 1. In the third (and sometimes also in the second) case, the accumulation of charges may occur in very thin near-electrode layers and manifest itself as a double electric layer [12]. On the surface of the dielectric and of the electrode (in the plane of their contact), charges equal in magnitude and opposite in sign are formed. From the condition of electrical neutrality of the complete double electric layer, one should expect that in this case no redistribution of potential across the thickness of the specimen and no formation of emf \(P\) will occur.

Fig. 1. Distribution of potential (b) across the thickness of the dielectric (a) after application of a constant voltage.

With a uniform distribution of space charge in the dielectric, redistribution of the potential is possible only under the condition that the charge carriers are injected from the electrode. In this case, the space-charge density can readily be related to the potential by Poisson’s equation [2]

\[ \frac{\partial^{2}\varphi}{\partial x^{2}}=-\frac{4\pi}{\varepsilon}\rho, \tag{2} \]

where \(x\) is the distance from one of the electrodes,
\(\varepsilon\) is the dielectric permittivity of the dielectric.

If the space-charge density \(\rho=\mathrm{const}\) throughout the entire volume of the dielectric, then from (2) we have

\[ \varphi=\frac{1}{2}Cx^{2}+C_{1}x, \tag{3} \]

where \(C\) and \(C_{1}\) are integration constants.

At \(x=0\), \(\varphi=0\); at \(x=d\) (the specimen thickness),

\[ \varphi=\frac{1}{2}C\cdot d^{2}+C_{1}d=\varphi_{\max}; \]

the emf \(P\) varies continuously over the entire thickness of the dielectric. Now equation (1) takes the form

\[ J=\frac{U_{0}-P(x)}{r}. \tag{4} \]

A uniform displacement of the dielectric’s own weakly bound ions will not affect the average macroscopic distribution of potential. A process will occur analogous to the bulk dielectric polarization of a homogeneous dielectric and obeying the equation [2]

\[ E_{\mathrm{av}}=\frac{4\pi I}{\varepsilon-1}. \tag{5} \]

Here

\[ E_{\mathrm{av}}=\frac{U_{0}}{d}; \]

\(l\) is the electric moment per unit volume of the dielectric;
\(d\) is the thickness of the specimen.

Thus, from a brief consideration of the conditions for the formation of space charge, it is evident that the latter does not always lead to the formation of the emf \(P\).

Absorption Currents in Dielectrics

At present there is no doubt about the fact that the decrease of current with time in solid dielectrics is due not only to the formation of space charge. A definite contribution to the observed current can be made by such phenomena as charging of the geometrical capacitance and various polarization processes [2, 13—16]. Schematically, the general curve of the dependence of current on time can probably be represented as the sum of currents caused by the enumerated processes (Fig. 2). For example, in alkali-halide crystals, under the action of an electric field, complexes of the type “divalent metal ion—cation vacancy” may become oriented. Work in our laboratory [17] has shown that the relaxation time constant of such dipoles at room temperature is \(10^{-2}\) s. Consequently, orientational polarization of impurity complexes should make a substantial contribution to the recorded value of the absorption current (Fig. 2) at times of the order of \(10^{-2}\) s from the moment the voltage is switched on. Divacancies, vacancy coagulates, and other formations also respond to the action of an external electric field. Migration of single point defects (cation and anion vacancies, interstitial ions) can lead to the formation of space charge. However, because of the low mobility of vacancies and the need for their displacement over large distances, this process is rather long in time.

Fig. 2. Schematic representation of the dependence of current on time when a constant voltage is applied to a dielectric. 1—resultant curve; 2—current of charge of the geometrical capacitance; 3—currents due to various polarization processes; 5—through-conduction current.

Of all the phenomena listed, only the formation of space charge leads to the appearance of the emf \(P\) (in accordance with its definition). Hence, in order to measure \(P\), it is necessary, from the entire experimental curve \(I(t)\) (curve 1, Fig. 2), to single out the portion caused by the formation of space charge, for which formula (1) is valid.

Usually, in the experimental determination of \(P\), one strives to measure the initial current \(I_0\), corresponding to the moment the voltage is switched on. However, it is well known [2, 14] that at the moment the electric field is switched on (\(t = 0\)), \(I_0\) is determined by the applied voltage and the resistance of the external circuit and in no way characterizes the properties of the material under study. Thus, the experimenter’s desire to record \(I_0\), or a current close to it in value, without clarifying its nature when measuring the emf \(P\), is unjustified and may lead to erroneous results.

In those cases where, in a homogeneous dielectric, the emf \(P\) does not arise upon application of a constant voltage, the change of current with time is well explained by the change in complex conductivity according to the expression

\[ I(t)=\sigma(t)\cdot U_0 . \tag{6} \]

Here \(\sigma(t)\) is the complex conductivity of the dielectric, directly connected with the characteristics and properties of the insulating material and satisfactorily explaining the physics of the process under consideration [2, 6, 13--15].

On methods for measuring the emf \(P\)

Studies have shown [2--9] that, upon heating alkali-halide crystals, \(P\) rapidly decreases and at \(T = 200\)--\(250^\circ\mathrm{C}\) becomes practically equal to zero. The authors explain this behavior of the dependence \(P=f(T)\) by the dissipation of the space charge with increasing temperature. However, direct measurements of the space charge itself indicate [18--20] that it increases with increasing temperature. For example, M. S. Kosman and V. F. Pisarenko [20] observed appreciable \((10^{-3}\)--\(10^{-4}\ \mathrm{C}/\mathrm{cm}^3)\) space charges in alkali-halide crystals at \(450\)--\(600^\circ\mathrm{C}\). In our opinion, the noted contradiction is due to the insufficient physical justification for the application of methods for measuring \(P\) in certain cases, as was discussed in the preceding paragraphs of the article. In addition, we would like to draw attention to the following.

The reliability of the measured initial current is determined to a large extent by the inertia of the measuring instrument. The faster the process of formation of the space charge proceeds, and the larger the time constant of the instrument, the greater the error introduced into the measured quantity. With increasing temperature, the rate at which polarization processes are established increases. For example, in rock salt the residual current is established at a temperature of \(70^\circ\mathrm{C}\) within several minutes, at \(100^\circ\mathrm{C}\) within several seconds, and at \(140^\circ\mathrm{C}\) within thousandths of a second [21]. A point is reached when the polarization time constant becomes comparable with, and then smaller than, the time constant of the instrument; this leads to a false impression of a weakening of the process under study. In our opinion, this is the first reason that led to the conclusion that, with increasing temperature, the emf \(P\) sharply decreases.

In ionic crystals the temperature dependence of the through-conduction current obeys the law

\[ i_{\mathrm{sc}} = i_{\mathrm{sc}}^{0}\cdot e^{-\frac{w}{KT}} \tag{7} \]

and is not connected with a change in the emf \(P\) [10, 14] (\(W\) is the activation energy of the charge-transport process). In most methods [2--9], the quantity \(P\) is determined by the ratio of the through-conduction current to the initial current

\[ \left(\text{for example, } P = U_0\left(1-\frac{i_{\mathrm{sc}}}{J_0}\right)\right). \]

Naturally, with increasing temperature the quantity \(P\) will decrease for reasons wholly unrelated to a decrease in the space charge. \(i_{\mathrm{sc}}\) increases according to formula (7), while \(I_0\) decreases because of the inertia of the instrument. Obviously, this is the second reason that led to the conclusion that \(P\) decreases with increasing temperature.

Conclusion

The decrease of current with time under a constantly applied voltage characterizes the presence of various polarization processes in the dielectric and, in most cases, cannot be a measure of the magnitude of the emf \(P\). The observed phenomenon, in our opinion, is better described by expression (6).

Formula (1) mathematically describes the change of current with time, but the quantity $P$ acquires a definite physical meaning only in the specific case when the decrease of the current is associated with the formation of a space charge in a bounded region of the dielectric. Insufficient attention to this condition, laid by A. F. Ioffe as the basis of formula (1), led to its application in all cases of current decay with time without sufficient physical justification.

Since the formation of a space charge is not in all cases accompanied by the appearance of an emf $P$, more correct methods should be used to study the laws of its formation; among these one may include the method of thermodepolarization [22]¹.

REFERENCES

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¹ Since polarization in solid dielectrics many times exceeds all polarization values in liquids, it was therefore (and only therefore) given the name high-voltage polarization [6].