Relaxation Dielectric Polarization and the Internal Field in Solid Dielectrics

G. I. Skanavi

General Remarks

For a long time it was believed that relaxation polarization and the dielectric losses associated with it in solid dielectrics occur only in a few cases, when the dielectrics contain dipole groups or radicals. Thus, for example, for oxidized paraffin and for paraffin containing polar impurities, Jackson obtained characteristic temperature-frequency maxima of the loss angle. Kobeko and co-workers observed an analogous phenomenon in sugar.

In 1937 we put forward the hypothesis that, in inorganic glasses, a considerable part of the dielectric losses is associated with relaxation polarization if the glass contains weakly bound ions, primarily alkali-metal ions. In connection with this hypothesis we showed that the theory of thermal ionic polarization and dielectric losses can readily be developed by analogy with Debye’s theory of dipole polarization in liquids.

In a number of works in recent years the hypothesis of thermal ionic polarization in solid dielectrics has been considerably developed. There has also appeared a series of works showing that relaxation dielectric losses occur in a number of inorganic crystalline dielectrics. Thus, K. A. Vodop’yanov showed that in crystals of gypsum containing water of crystallization, and in mica-phlogopite, characteristic temperature-frequency maxima of $\tg \delta$ are observed. G. P. Mikhailov studied relaxation losses in polymers. Also of considerable interest are the works of Breckenridge, who was the first to draw attention to the fact that, in alkali-halide crystals, relaxation temperature maxima of $\tg \delta$ may occur, though they are strongly masked by conduction losses.

In the dielectrics physics laboratory of the P. N. Lebedev Physical Institute of the Academy of Sciences of the USSR, work was carried out over a number of years to elucidate the mechanism of thermal ionic polarization in solid dielectrics, to search for such dielectrics in which it manifests itself especially vividly, and to develop the theory of relaxation polarization and losses. In this, special attention was paid to finding the possibility of combining, in a dielectric, relaxation polarization with an internal field favorable to polarization, which, on the basis of general considerations, must inevitably lead to a very high dielectric permittivity over a wide frequency range, not associated with ferroelectric properties. Four years ago this problem was in principle solved, but it has not yet been reflected in the literature.

In recent years, non-ferroelectric dielectrics with very high dielectric permittivity have been studied in detail in the laboratory, and work has been carried out on their introduction into practice. New results have been obtained.

dielectrics of this class, as well as dielectric semiconductors with a heterogeneous structure, possessing an even higher dielectric permittivity of the same type, to which Ksendzov drew attention in studying chemically pure titanium dioxide. At the same time, attention was devoted to the study of the dielectric properties of dipolar organic crystals, the dielectric properties of solid dielectrics at ultrahigh frequencies, and the investigation was begun of relaxation dielectric losses in alkali-halide single crystals, discovered by Breckenridge, with the aim of checking his results, disputed by some authors, and of working out the question of dielectric losses in the simplest solid dielectrics as a whole. The study of the electret state of matter, undertaken recently, also essentially belongs to this same cycle of investigations.

In the present report I shall take the liberty of setting forth some results of the indicated cycle of investigations of the laboratory which correspond to the title of the report.

The Acting (Local) Field in Relaxation Polarization

One of the most important questions in the theory of polarization and losses of solid dielectrics is the question of the time of establishment of polarization under conditions in which the local (acting) field differs from the average field and is not equal to the Lorentz field, for these conditions are realized in any solid dielectric, except for the case of a roughly approximate treatment of diatomic cubic crystals.

In previous works we limited ourselves to determining the time of establishment of relaxation polarization in the approximation that the acting field is equal to the Lorentz or to the average field. The chief difficulty in solving the indicated problem is that, without knowledge of the structure of the crystal (and all the more of glass), one cannot calculate the structural coefficients of the internal field by the method proposed by us earlier and, consequently, cannot estimate the magnitude of the internal field. Meanwhile, in many interesting and important cases we do not know the structure of the dielectric in sufficient detail. Therefore we approached the solution of the posed problem from the other end. In the presence of relaxation polarization, not masked by other processes, simple measurements of the temperature and frequency dependences of \(\varepsilon'\) and \(\varepsilon''\) (\(\varepsilon^*\) and \(\tan \delta\)) place in the hands of the experimenter abundant material for determining important molecular constants of the dielectric. It was therefore natural to suppose that a general consideration of the question of relaxation polarization with the introduction of unknown coefficients of the internal field would make it possible to find a method for determining these coefficients from experimental data on dielectric permittivity and the loss angle.

As was shown by our work carried out jointly with A. N. Gubkin [1] and E. N. Matveeva, such a possibility does indeed exist. If one introduces generalized coefficients of the internal field in the following natural way:

\[ E = E_{\mathrm{cp}} + \beta_1 I_{\mathrm{rel.}} + \beta_2 I_0, \tag{1} \]

where \(E\) is the acting field, \(E_{\mathrm{cp}}\) the average field, \(I_{\mathrm{rel.}}\) and \(I_0\) the electric moments of unit volume of the dielectric caused, respectively, by relaxation and elastic polarization, and \(\beta_1\) and \(\beta_2\) are generalized coefficients of the internal field. The differential equation characterizing the pro-

process of establishing relaxation polarization can be written in the following simple form

\[ \tau \frac{dI_{\mathrm{rel}.}}{dt} = - I_{\mathrm{rel}.} + B E_{\mathrm{av}.} + \beta_1 B I_{\mathrm{rel}.} + \beta_2 B I_0 . \tag{2} \]

Here \(B=\dfrac{n' q^2 x^2}{12 kT}\) for thermal ionic polarization and \(B=N_0 \dfrac{\overline{m m^*}}{3 kT}\) for dipole polarization (\(n'\) is the number of weakly bound ions in \(1\ \mathrm{cm}^3\), \(x\) is the distance between neighboring local minima of the ion’s potential energy, \(q\) is the ion charge, \(kT\) is the energy of thermal motion, \(N_0\) is the number of unit crystal cells in \(1\ \mathrm{cm}^3\), \(\overline{m m^*}\) is the mean value of the product of the dipole moment of the unit cell, due to relaxation polarization \(m\), and the dipole moment of the sphere \(m^*\) surrounding the given cell when the dipole moment of the cell \(m\) is fixed);

\[ A=\frac{\beta_2 n_3 \alpha_3}{1-\beta_2 n_3 \alpha_3}; \]

(\(n_3\alpha_3\) is the product of the equivalent number of elastically polarizable particles in \(1\ \mathrm{cm}^3\) and the effective elastic polarizability of each particle); \(\tau=\dfrac{e^{U/kT}}{2\nu}\) (\(U\) is the local potential barrier, \(\nu\) is the frequency of elastic vibrations of the particles participating in the relaxation polarization).

In the case of a sinusoidally alternating voltage, equation (2) immediately gives the value of the complex electric moment per unit volume due to relaxation polarization:

\[ I_{\mathrm{rel}.}^{*} = \frac{E_{\mathrm{av}.}B(1+A)\gamma}{\gamma^2+\omega^2\tau^2} - i\,\frac{E_{\mathrm{av}.}\omega\tau B(1+A)}{\gamma^2+\omega^2\tau^2}, \tag{3} \]

where \(\gamma=1-\beta_1B(1+A)\), and \(\omega\) is the angular frequency of the field.

From (3) it is easy to obtain the values of the real and imaginary parts of the dielectric constant in the Debye form

\[ \varepsilon' = \varepsilon_\infty + \frac{\varepsilon_0-\varepsilon_\infty}{1+\omega^2\Theta^2} \quad \text{and} \quad \varepsilon'' = \frac{(\varepsilon_0-\varepsilon_\infty)\omega\Theta}{1+\omega^2\Theta^2}, \tag{4} \]

where the time constant for the establishment of relaxation polarization is \(\Theta=\tau/\gamma\); \(\varepsilon_0\) and \(\varepsilon_\infty\) are the dielectric constants, respectively, at zero frequency and at the frequency at which relaxation polarization is absent. In this case

\[ \varepsilon_0 = 1+\frac{4\pi A}{\beta_2} + \frac{4\pi B(1+A)\left(1+\dfrac{\beta_1}{\beta_2}A\right)}{\gamma}, \qquad \varepsilon_\infty = 1+\frac{4\pi A}{\beta_2}. \tag{4a} \]

Thus, the general Debye formulas remain valid for any internal field, but the time constant \(\Theta\) depends on the form of the internal field.

If the quantities \(\varepsilon_0\) and \(\varepsilon_\infty\), which can be found experimentally, are specified, then it is easy to relate the internal-field coefficients \(\beta_1\) and \(\beta_2\), the coefficient \(\gamma\), and the quantity \(B\) to one another by means of the preceding formulas.

\[ \beta_1 = \frac{4\pi(\varepsilon_0-\varepsilon_\infty)} {B(\varepsilon_0-1)\left[4\pi+(\varepsilon_\infty-1)\beta_2\right]} - \frac{4\pi}{\varepsilon_0-1}; \tag{5a} \]

\[ \gamma = \frac{\beta_1(\varepsilon_\infty-1)} {\beta_2(\varepsilon_0-1)+4\pi}. \tag{5b} \]

But the quantity \(\beta\), as was indicated, is determined through the dipole moments and the temperature and must always be related to \(\varepsilon_0\) and \(\varepsilon_\infty\), for example, by means of Fröhlich’s equation for dipolar crystals. Therefore, for any dielectric with relaxation polarization, having found experimentally only two quantities, \(\varepsilon_0\) and \(\varepsilon_\infty\), one can determine, with the aid of equations (5), two of the three unknown coefficients \(\beta_1, \beta_2\), and \(\gamma\). If, in addition, \(\gamma\) is found from other experimental data, then the problem of finding the coefficients of the internal field from experimental data will be completely solved.

Calculations show that if the activation energy (potential barrier) \(U\) can be found independently of the coefficient \(\gamma\), then the latter is determined through the quantity \(U\) in the following simple way:

\[ \gamma=\frac{D^2 kT_1}{2U}, \tag{7} \]

where \(T_1\) is the temperature corresponding to the maximum of the dielectric permittivity \(\varepsilon'\) at a given frequency \(\omega\), \(k\) is Boltzmann’s constant, and

\[ D=\frac{\varepsilon'_{\max}-\varepsilon_\infty}{\varepsilon''_1},\quad \varepsilon''_1=\varepsilon'_1\cdot \operatorname{tg}\delta_1 \]

is the loss factor (or the imaginary part of the dielectric permittivity) at temperature \(T_1\) and the same frequency \(\omega\), if \(U \gg kT_1\), which is usually the case.

The activation energy \(U\) is usually found from the shift of the temperature maximum of \(\operatorname{tg}\delta\) for a prescribed change in frequency, or, conversely, from the shift of the frequency maximum of \(\operatorname{tg}\delta\) for a prescribed change in temperature. However, in doing so the coefficient \(\gamma\) is taken equal to unity or to

\[ \frac{\varepsilon_\infty+2}{\varepsilon_0+2}, \]

which corresponds to the effective field being equal to the mean field or to the Lorentz field. In reality, however, without knowing the coefficient \(\gamma\), \(U\) cannot be determined by the indicated method. A simple consideration shows that there is also another method of finding \(U\), which does not require knowledge of \(\gamma\). If the temperature of the maximum \(\varepsilon'\), \(T_1\), and the temperature of the maximum of \(\operatorname{tg}\delta\), \(T_2\), are known from experiment, then the activation energy can be calculated. If \(T_1\) and \(T_2\) are close (which is usually the case) and the activation energy is less than \(10\ \mathrm{eV}\), then one may assume that

\[ 1-\frac{T_1}{T_2}\ll \frac{kD^2}{2U}\frac{T_1^2}{T_2}, \]

and for the calculation the formula may be used:

\[ U=\frac{k\ln\!\left(D\sqrt{\varepsilon_{02}/\varepsilon_\infty}\right)} {1(T_2-1)T_1}, \tag{8} \]

where \(\varepsilon_{02}\) is the dielectric permittivity at zero frequency and at the temperature of the maximum of \(\operatorname{tg}\delta\) for that frequency for which the coefficient \(D\) has been calculated. The calculation can also be carried out without the indicated, quite reasonable assumptions, by means of a graphical solution of the equations. It should be noted that the activation energy can be determined independently of the coefficient \(\gamma\) by several other methods as well.

Thus, the problem, substantially important from our point of view, of finding the coefficients of the internal field can be completely solved for a solid dielectric of any unknown structure, provided that it possesses relaxation polarization and losses not masked by other processes. In addition, determination of the energy

of the activation of particles relaxing in an electric field makes it possible to find the natural frequency of their elastic vibrations \(\nu\), proceeding from simple relations, for example, from the condition of the temperature-frequency maximum of \(\operatorname{tg}\delta\)

\[ \nu = \frac{\omega e^{U/kT}}{\gamma}\sqrt{\frac{\varepsilon_{\infty}}{\varepsilon_{0}}}. \tag{9} \]

The frequency of the elastic vibrations of relaxing particles is of special interest if the principal particles of a solid dielectric are relaxing.

Since the internal field is closely connected with the basic structure of the substance, the activation energy—with structural defects, and the frequency of elastic vibrations is the most important characteristic of the basic structure, or of particles in structural defects, or, finally, of impurity particles (this must be analyzed in each particular case), it seems to me that here, in principle, a new path is opened for “dielectric” investigations of the structure of a solid and its peculiarities.

Indeed, in many dielectrics that do not possess relaxation polarization, it can be created by small polar additions. In this case the activation energy and the internal-field coefficient \(\beta_{1}\), found from dielectric measurements, will depend also on the nature of the additions, while the internal-field coefficient \(\beta_{2}\) will be connected practically only with the structure of the basic substance. If, however, the dielectric as such possesses relaxation polarization and losses, then the method indicated will make it possible to investigate certain peculiarities of its structure without introducing any changes into it. It should be noted that the problem of determining the structure of a body from the internal-field coefficients is hardly soluble and, in all probability, presents still greater difficulties than the inverse problem; however, in combination with other methods, the indicated “dielectric” method of investigating structural peculiarities may be sufficiently fruitful.

Up to now I have not yet touched upon the essential question that in any real dielectric with relaxation polarization there is always a more or less broad set of relaxation times or activation energies. According to Cole’s well-known method, it is possible to find the distribution of relaxation times around the most probable one, and the value of the most probable relaxation time, from the same dielectric measurements.

Success along the lines of the possible investigations indicated depends first of all on how wide the interval of frequencies and temperatures is in which \(\varepsilon\) and \(\operatorname{tg}\delta\) of the given particular dielectric can be measured. Preference here should be given to inorganic dielectrics with low conductivity in which relaxation processes take place. A large group of such dielectrics has been synthesized in our laboratory.

Organic dielectrics with low conductivity and relaxation polarization are easier to obtain (dipolar organic crystals, polar polymers), but, unfortunately, their low heat resistance limits the temperature interval from above. This applies especially to organic crystals. However, there are such organic crystals in which relaxation polarization appears especially clearly in the form of simple, I would say classical, regularities. This prompted us to investigate olevaxes—hydrogenated castor oils, in which relaxation polarization was first observed by Walter, Inge, and Trambitskaya [2]. In their work, however, no measurement was made of the frequency dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) over a wide frequency interval, while the temperature dependences of the dielectric properties of olevaxes cannot be determined for a wide temperature interval because of their low melting temperature.

and insufficient frost resistance. In our work jointly with Gubkin [3], measurements were therefore made of the frequency dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) over a wide frequency interval and at two temperatures for oleowaxes of different degrees of hydrogenation, having different contents of the crystalline phase. The results of these measurements for one of the oleowaxes are presented in Fig. 1. Very distinct frequency maxima of \(\operatorname{tg}\delta\), a characteristic decrease of \(\varepsilon\) with frequency, and temperature shifts are visible. Only four curves, given in Fig. 1, proved sufficient to find the activation energy of the dipole groups, the frequency of elastic oscillations of the relaxing dipoles, the most probable relaxation time, the distribution of relaxation times, the internal-field coefficients and the acting field, which differ from the Lorentz field, the change in entropy upon activation (by applying the theory of absolute reaction rates and with the known number of unit crystal cells in \(1\ \mathrm{cm}^3\)), and the effective dipole moment.

Fig. 1.

The same rather few experimental data for oleowaxes with different degrees of hydrogenation made it possible to follow the change of the enumerated parameters with the change in the content of the crystalline phase and to draw the corresponding conclusions. I shall not dwell on the details, since articles containing them have been published comparatively recently. I shall only note that, for those dielectrics in which ionic elastic polarization is absent and the square of the refractive index is small, the determination of the internal-field coefficients in the first approximation is simplified. It is precisely for this reason that, for oleowaxes, it was possible to find these coefficients without measuring the temperature dependences of \(\varepsilon\) and \(\operatorname{tg}\delta\), and, in particular, without observing the temperature maximum of \(\varepsilon\). Indeed, if the electronic polarization is small, then for \(\varepsilon_\infty\) the Clausius–Mossotti equation is applicable in the first approximation, and therefore the internal-field coefficient produced by the elastic polarization \(\beta_2\) may be put equal to \(4\pi/3\). Then the other internal-field coefficient \(\beta_1\) is found from equation 5a, and \(\gamma\) from equation 5b. The quantity \(B\), entering into 5a, for dipole crystals can be determined through \(\varepsilon_0\) and \(\varepsilon_\infty\) with the aid of the Fröhlich equation:

\[ \frac{(\varepsilon_0-\varepsilon_\infty)(2\varepsilon_0+\varepsilon_\infty)}{9\varepsilon_0} = \frac{4\pi}{3}\,N_0\,\frac{\overline{mm^*}}{3kT} = 4\pi B. \tag{10} \]

The values of the coefficient of the internal field created by relaxation polarization, \(\beta_1\), calculated from (5a), turned out to be small and negative, as they should be when \(\beta_2 = \dfrac{4\pi}{3}\); otherwise the notorious \(4\pi\)-catastrophe would occur. The values of the coefficient \(\gamma\), on the contrary, noticeably exceed the Debye values \(\left(\gamma = \dfrac{\varepsilon_\infty + 2}{\varepsilon_0 + 2}\right)\). Figure 2 gives a plot of the values of \(\beta_1\) as a function of \(\varepsilon_0\) for \(\beta_2 = \dfrac{4\pi}{3}\) at the given value of \(\varepsilon_\infty\).

Fig. 2.

Nonsegnetoelectric dielectrics with very high dielectric permittivity

In seeking ways to synthesize nonsegnetoelectric dielectrics with very high dielectric permittivity, as was already indicated, we were guided by the idea of combining relaxation polarization with an internal field favorable to polarization. The latter, as follows from our previous work, is determined by the specific structure of crystals and is characteristic of crystal lattices of the rutile and perovskite types. Thus, it was necessary to solve the problem of creating relaxation (thermal ionic) polarization in dielectrics with the indicated structures. The first result in this direction was obtained in our work carried out with the participation of A. I. Demeshina [4]. In that work, relaxation polarization and high dielectric permittivity were discovered in rutile with small additions of oxides of alkaline-earth metals. However, it turned out that the relaxation polarization of such materials has a comparatively long relaxation time; the expected high values of dielectric permittivity occurred only at low and audio frequencies, and therefore the regularities of relaxation polarization and losses were obscured by conductivity, which, incidentally, was not very small. It may be thought that in polycrystals of the rutile type with small additions, the relaxation polarization is due to the formation of structural defects (and weakly bound ions) in the form of foreign structural formations of titanates of alkaline-earth metals dispersed in the polycrystal and apparently producing a large number of small local electronic levels, and consequently an increased electronic conductivity, which was investigated in our new

in work carried out jointly with A. G. Chrelashvili and A. I. Demeshina [5]. The indicated structural defects at the same time create comparatively large local potential barriers for ions, and relaxation polarization is characterized by a large relaxation time. In this connection we turned to a structure of the perovskite type, in which the crystals contain not one but two kinds of cations and in which solid solutions are formed comparatively easily. We expected that obtaining solid solutions of a perovskite structure of the subtraction type, in which substitution of a considerable fraction of divalent cations by trivalent cations, with the formation of vacant cation sites, may occur, should give a favorable result.

Indeed, a comparatively large number of vacant cation sites should create a considerable number of “defects,” which should not alter the necessary perovskite structure as a whole but should make it possible for the cations to relax, having a small relaxation time.

From this would follow the possibility of obtaining a comparatively rapidly established relaxation of ionic polarization and, consequently, a very high dielectric permittivity not only at low but also at high frequencies.

In addition, the preservation, as a whole, of a homogeneous crystalline structure may perhaps lead only to a small number of shallow local levels for electrons and, consequently, ensure low electronic conductivity.

The indicated general considerations were fully confirmed experimentally. In our work, carried out jointly with E. N. Matveeva, new polycrystalline dielectrics of perovskite structure based on strontium titanate and bismuth polytitanates (strontium—bismuth—titanates—SBT) were synthesized; these possess a very high \(\varepsilon\) (up to 1000) not only at low but also at high frequencies and a very low conductivity (about \(10^{-15}\ \Omega^{-1}\mathrm{cm}^{-1}\)) and do not exhibit ferroelectric properties. At the same time their polarization and losses have a clearly pronounced relaxation character.

In Fig. 3 are given curves of the dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) on temperature at different frequencies for one of the strontium—bismuth—titanates. Processing of these experimental data by the methods described above made it possible to find a number of physical parameters characterizing the process of relaxation polarization (the most probable relaxation time, the distribution of relaxation times, the activation energy of weakly bound ions, the frequency of their natural elastic vibrations, the change in entropy upon activation, etc.), since the values \(\varepsilon_0\) and \(\varepsilon_\infty\) can be estimated rather accurately from the experimental data.

X-ray analysis, carried out by N. A. Porai-Koshits, showed that over a broad interval of concentrations the strontium—bismuth—titanates possess, as we expected, a perovskite structure with a practically unchanged lattice constant, equal to \(3.89\ \text{Å}\). The lattice constant remains unchanged over a fairly broad range of concentrations. This is easily explained if one assumes that bismuth ions, having the same radius as strontium ions (\(1.2\ \text{Å}\)), are situated at the strontium sites. In this case the electroneutrality of the lattice is preserved if, upon partial substitution of divalent strontium ions by trivalent bismuth ions, vacant sites also appear. A subsequent work, in which V. A. Trigubenko participated, was devoted to a detailed study of the dielectric properties of materials of the system \(\mathrm{SrTiO_3}—\mathrm{Bi_2O_3}\cdot 2\mathrm{TiO_2}\) with the ratio of components varying from 0 to 100%. In Fig. 4 is shown the concentra-

tion curve of \(\varepsilon\), \(\operatorname{tg}\delta\), and the sintering temperature. Bismuth polytitanates have \(\varepsilon\) of about 100, increasing with temperature, which, apparently, is due, unlike in other titanates, to considerable ionic polarization. On the basis of the results of X-ray analysis, all materials of the system can be divided into the following groups.

Fig. 3. Dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) on temperature at various frequencies for one of the strontium—bismuth—titanates.

First group: materials containing in the batch from 0 to 25% \(\mathrm{SrTiO_3}\) by weight; the structure of these materials is complex and apparently inherent mainly in bismuth polytitanates.

Second group: materials with a content of \(\mathrm{SrTiO_3}\) in the batch from 25 to 65% by weight, which are a mixture of bismuth polytitanates and strontium titanate (the Debyegrams are the result of superposition of the Debyegrams of the components).

Third group: materials containing in the batch 65—100% \(\mathrm{SrTiO_3}\), perovskite structure with lattice constant 3.89 Å.

The dielectric properties of the materials of the first group are not very far from the properties of bismuth polytitanates.

The dielectric properties of the materials of the second group are the result of an approximately additive superposition of the properties of the components, corresponding to the logarithmic formula. The third group of materials, in terms of dielectric properties, is divided into two subgroups: a) concentration of \(\mathrm{SrTiO_3}\) in the batch 65—76% and b) concentration of \(\mathrm{SrTiO_3}\) in the batch 76%—100%. For group 3-a the following are characteristic: relaxation polarization due to weakly bound ions, \(\varepsilon \sim 800—1000\), low conductivity, very low losses at audio frequencies, and somewhat increased losses at high frequencies (\(\sim 2\) MHz). All materials of this subgroup possess the desired relaxation polarization, and their properties are close to those already described-

properties of the strontium–bismuth titanates first synthesized by us.

Fig. 4.

In the materials of groups 3–6 an interesting combination of electrical properties was found, sharply distinguishing them from the other materials of the system. The dielectric constant of these materials reaches 5000–6000, the loss angle is large; the conductivity also has an increased value and rises sharply with increasing voltage (Fig. 5). The study showed that these anomalous properties are due to the heterogeneity of the structure. In the materials of the group under consideration, during firing—which, incidentally, requires a very high temperature (up to \(1600^\circ\mathrm{C}\))—there occurs a fairly intense partial reduction of the oxides (apparently titanium dioxide) entering the batch. As a result, the sintered materials have a characteristic “steel” coloration. The presence of a semiconducting phase in the materials of this group leads to relaxation of the Maxwell type. It is interesting, however, that the relaxation time of the polarization of the materials considered is comparatively small, and therefore, in contrast to the materials studied by Ksendzov, their extremely high dielectric constant is also observed at high frequencies. However, already in the region of several

megacycles there begins a sharp drop in \(\varepsilon\) and an increase in \(\operatorname{tg}\delta\) with frequency, which had not previously been observed for any of the known dielectrics. In

Fig. 5. Material composition (by batch): SrTiO\(_3\)—93%,
Bi\(_2\)O\(_3\cdot 2\)TiO\(_2\)—7%, sample thickness \(\sim 2\) mm.

Figs. 6 and 7 give plots of the temperature and frequency dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) for some of the dielectrics studied, illustrating what has been said.

Fig. 6. Material composition (by batch)
SrTiO\(_3\)—93%, Bi\(_2\)O\(_3\cdot 2\)TiO\(_2\)—7%.

Fig. 7. Batch composition: SrTiO\(_3\)—85%, Bi\(_2\)O\(_3\)·2TiO\(_2\)—15%.

Dielectric Permittivity and Loss Angle of Polycrystalline Dielectrics at Superhigh Frequencies

The study of relaxation polarization of various types makes work with dielectrics in the region of superhigh frequencies especially topical at the present time, since the widest possible extension of the frequency range, as has already been pointed out, places in the hands of the investigator a large amount of additional experimental data. Moreover, physics in general cannot fail to be interested in this region, which is intermediate between the radio-frequency and infrared ranges of the spectrum of electromagnetic waves. In connection with this, systematic investigations of solid dielectrics in the region of superhigh frequencies at different temperatures have been begun in our laboratory. At present there are many purely methodological difficulties here, on which I shall not dwell. We have succeeded in overcoming some of these difficulties (G. V. Zakhatkin took part in the work, and G. E. Arkhangel’skii, G. A. Lipaeva, L. K. Vodol’yanov, A. D. Zhlud’ko, and others are participating) and in obtaining by now certain reliable results. Some of these results, obtained by G. A. Lipaeva, are presented in Figs. 8, 9, 10, and 11, which show curves of the dependence of \(\varepsilon\) and \(\operatorname{tg}\delta\) on temperature at different frequencies (\(10^{10}\), \(10^6\), and \(10^3\) Hz) and on frequency up to \(10^{10}\) Hz at room temperature, for

![Figure 8 graph: frequency dependence of ε and tg δ for strontium titanate and calcium titanate]

Fig. 8. Frequency dependence of \(\varepsilon\) (—) and \(\operatorname{tg}\delta\) (– – – –) of strontium titanate (○) and calcium titanate (△).

![Figure 9 graph: tg δ and ε versus frequency for ZnTiO3, steatite, MgTiO3, and BaO·4TiO3]

Fig. 9. \(\operatorname{tg}\delta\) (– – – –), \(\varepsilon\) (—);

1 — \(\mathrm{ZnTiO_3}\) (– – – –),
2 — steatite (– – – –),
3 — \(\mathrm{MgTiO_3}\) (—),
4 — \(\mathrm{BaO}\cdot 4\mathrm{TiO_3}\) (– – – –),

1 — \(\mathrm{ZnTiO_3}\) (—),
2 — steatite (—),
3 — \(\mathrm{MgTiO_3}\) (– – – –),
4 — \(\mathrm{BaO}\cdot 4\mathrm{TiO_3}\) (—).

solid dielectrics with dielectric permittivity from 6 to 170. It is seen that in the region of ultrahigh frequencies the loss angle begins to increase with frequency, while the dielectric permittivity remains almost unchanged. The temperature dependence of $\varepsilon$ and $\operatorname{tg}\delta$ at ultrahigh frequencies differs little from that at lower frequencies. It should be emphasized that here we are dealing with such dielectrics which

Fig. 10. $\operatorname{tg}\delta$ (— — —), $\varepsilon$ (—),

1 — CaTiO$_3$ ($f=10^6$) (— — —), 2 — SrTiO$_3$ — ($f \to =10^{10}$),
3 — CaTiO$_3$ ($f=10^3$), 4 — SrTiO$_3$ — ($f=10^3$),
5 — SrTiO$_3$ ($f=10^3,\ 10^6,\ 10^{10}$), 6 — CaTiO$_3$ ($f=10^3,\ 10^7,\ 10^{10}$).

possess predominantly elastic polarization. The increase of dielectric losses in them at ultrahigh frequencies indicates the possible presence of very rapid relaxation processes, which should manifest themselves at still higher frequencies, or, what is less probable, is connected with resonant absorption in the infrared region. Unfortunately, we have not yet completely overcome the methodological difficulties in measuring very high dielectric permittivities ($\varepsilon > 200$) and large $\operatorname{tg}\delta$ ($\operatorname{tg}\delta > 0.05$) at ultrahigh frequencies (especially at different temperatures), and therefore we cannot yet speak about the results of studying the dielectrics considered above with relaxation polarization, as well as ferroelectrics, at ultrahigh frequencies. It is interesting to note that very important investigations of solid dielectrics in the long-wave infrared range ($\lambda \sim 100$–$500\ \mu$) are already beginning to appear in individual works by foreign (German) authors. The first results of these works apparently confirm the assumption that, in dielectrics with increased dielectric permittivity, relaxation of polarization occurs in this region of the spectrum. This first result was obtained for materials of the barium titanate type and, what is especially interesting, for materials rich in rutile and having not very large $\varepsilon$.

With a decrease in wavelength, \(\varepsilon\) decreases in all the cases investigated. For rutile dielectrics this circumstance, if it is definitively established, should lead to some development, and perhaps also modification, of our conception of the causes of the increased \(\varepsilon\). In our laboratory, preparations have begun for the development, assembly, and adjustment of optical apparatus for investigations of this kind.

Fig. 11. \(\operatorname{tg}\delta\) \((— — —)\), \(\varepsilon\) \((— \mathrm{O} — \mathrm{O} —)\).

1 — steatite, 2 — \(\mathrm{MgTiO_3}\), 3 — \(\mathrm{BaO}\cdot4\mathrm{TiO_3}\),
4 — \(\mathrm{ZnTiO_3}\) at \(f = 10^{10}\) cycles.

Dielectric Relaxation in Simple Ionic Crystals

Up to now the discussion has concerned relaxation phenomena in polycrystalline dielectrics of more or less complex structure. It is obvious that the detection and investigation of these phenomena in the simplest crystals is of fundamental importance. As was indicated in the works of Breckenridge, weakly expressed relaxation maxima of \(\operatorname{tg}\delta\) were found in alkali-halide crystals. At the same time, for a long period after the work of N. P. Bogoroditskii and V. I. Malyshev it was considered that in these crystals dielectric losses are due only to conductivity. Recently, work has been carried out in our laboratory (B. N. Matsanashvili) on the study of the nature of dielectric losses in “pure” alkali-halide crystals without impurities introduced during the growth of the single crystals, and with specially introduced impurities. This work encountered many methodological difficulties in connection with the need to obtain specimens of single crystals of sufficiently large area and in connection with the hygroscopicity of many crystals, the influence of the type of electrodes on small changes in dielectric losses, and, finally, in connection with the small value of \(\operatorname{tg}\delta\) at moderate temperatures. A considerable part of these difficul-

has been overcome, and at present one may apparently regard as reliable the results obtained for loss angles greater than \(3 \cdot 10^{-4}\). The corresponding curves are shown in Figs. 12, 13, and 14. The LiF and KCl crystals, the data for which are given in Figs. 12, 13, and 14, have comparatively low hygroscopicity; therefore the results for these crystals obtained in atmosphere and in vacuum are close.

Fig. 12. KCl (in atmosphere)

\(1\)—1 kc/s, \(2\)—450 kc/s, \(3\)—3 Mc/s, \(4\)—9 Mc/s.

1. The straight lines give the dependence \(\lg \tan \delta\) at a given \(f\), calculated from the conductivity measured at direct voltage.

2. The dashed line indicates parts of the curves corresponding to the sensitivity limit of the apparatus.

In connection with the hygroscopicity of many crystals, the measurements were carried out for the most part in vacuum, with electrodes in the form of disks of chemically pure iron having optical contact with the surface of the single-crystal specimen. The single crystals were grown in porcelain glazed, corundum, fireclay, and quartz crucibles. The amount of impurities in the single crystals was determined by polarographic and optical methods. It was established that in all the crystals investigated there are weakly expressed relaxation maxima of \(\tan \delta\),

and the dielectric losses are not completely determined by the conductivity. In this connection, the relaxation maxima of \(\tg \delta\), in contrast to Breckenridge’s indications, are most characteristic of crystals that have not been subjected to any treatments after their growth. It is significant that in impurity-free

Fig. 13. LiF (in air): 1 — 1 kc, 2 — 1 Mc,
3 — 1 Mc, 4 — 4.75 Mc.

Dots — forward run; \(x\)—\(x\)—\(x\) — reverse run.

1. The straight lines give the dependence of \(\lg \tg \delta\), at the given \(f\), calculated from the conductivity measured at constant voltage.

crystals, as a rule, one maximum of \(\tg \delta\) is found, whereas in crystals with introduced impurities there are up to four maxima. All temperature maxima of \(\tg \delta\) shift regularly when the frequency is changed, which makes it possible to calculate the activation energy of weakly bound ions; its value practically does not depend on the frequency interval chosen for the calculation and is about \(0.7\)–\(1\) eV (in different cases). Measurements of the frequency behavior of \(\tg \delta\) for the same specimens at different temperatures show that, as was to be expected, there are frequency maxima of \(\tg \delta\), shifting with increasing temperature into the high-frequency region. The activation energy calculated from these data agrees quite well with the activation energy calculated from the frequency shifts of the temperature maxima of \(\tg \delta\). Having no possibility of

Without dwelling in greater or lesser detail on the results of the work, I shall only point out that considerable temperature changes of $\varepsilon$ have also been found in the low-frequency region. The totality of the results makes it possible to consider definitively established the fact of the existence of relaxation losses,

Fig. 14. LiF (in vacuum).

caused by defects in the crystal lattice of the simplest ionic single crystals. This conclusion, it seems to us, is of fundamental importance.

On the question of new (inorganic) electrets

In conclusion I shall allow myself to dwell on experimental results concerning the electret state of matter, obtained by us quite recently.

In 1924 Eguchi discovered that certain organic dielectrics (carnauba wax, etc.), cooled in a sufficiently strong constant field from a temperature close to the melting or softening temperature down to room temperature, acquire a residual bound charge of both signs, which is retained for a long time, especially if the specimens are stored wrapped in metal foil. Thereafter, in many works, these so-called elec-

thirds, obtained from organic dielectrics. It was found that the charge of electrets is a combination of (a) the so-called heterocharge, having signs opposite to the signs of the electrodes, which create the external polarizing field during manufacture of the electret, and (b) the so-called homocharge, having signs that coincide with the polarity. The heterocharge is apparently due to frozen-in polarization, and the homocharge to the entry of charges into the dielectric from outside. In weak fields \((E \sim 5\ \mathrm{kV/cm})\) only the heterocharge, characteristic of polar organic dielectrics, is formed; in stronger fields \((E \sim 10 \div 20\ \mathrm{kV/cm})\), in addition to the heterocharge, a homocharge also arises.

Fig. 15 (a). Electret state of magnesium titanate.

In recent works (Swan et al.) a phenomenological theory of electrets has been given, according to which the potential difference of their surfaces is due to the finite and different rate of decrease of homo- and heterocharges with time, at which compensation of these charges by the charges of adhering air ions does not have time to occur. The electret state of a substance is thus thermodynamically nonstationary, but possesses a very large time constant (up to several years). A satisfactory molecular theory of the electret state is still lacking. However, there is no doubt that the formation and properties of electrets are closely connected with the nature of polarization and conductivity of dielectrics, and also, possibly, with the properties of the contact layer. In the literature there is still no information on inorganic electrets and, in particular, on electrets obtained from inorganic dielectrics with increased and high dielectric permittivity. The exception is a very recent brief report on the electret state of barium titanate. It is evident that, in the study of the polarization of these dielectrics, one cannot pass over the study of the electret state. Already a year ago we obtained electrets from certain titanates.

By the present time we, jointly with A. N. Gubkin and R. Kh. Mukhamedieva, have shown that the electret state of matter is characteristic of all polycrystalline titanates and even of ordinary ceramic materials (steatite).

It is essential here that at small intensities of the polarizing field \((E \sim 5\ \mathrm{kV/cm})\) a heterocharge is obtained, while at larger fields a homocharge is obtained. The stability of the new electrets is high (Fig. 15). The charge

practically does not decrease even over the course of a year. Our immediate task is to clarify the physical causes responsible for the occurrence of the electret state of a substance, in connection with the mechanism

Fig. 15 (b). Schematic circuit for measuring the charge of an electret by the induction method.

1. Electret specimen. 2. Electrostatic voltmeter.

of polarization and conductivity; further investigations in the field of obtaining new electrets and their practical application; and the construction of a molecular theory of electrets. Work in this direction is under way.

REFERENCES

1. G. I. Skanavi and A. N. Gubkin, ZhETF, 28, No. 1, 85, 1955.
2. A. F. Walter, L. D. Inge, and T. A. Trambitskaya, ZhTF, 10, 1970, 1940.
3. G. I. Skanavi and A. N. Gubkin, ZhETF, 27, No. 6, 742, 1954.
4. G. I. Skanavi and A. I. Demeshina, ZhETF, 19, No. 1, 1949.
5. G. I. Skanavi, A. I. Demeshina, and A. G. Chrelashvili, DAN SSSR 27, 673, 1951.

Moscow, P. N. Lebedev Physical Institute,
Academy of Sciences of the USSR.

A. A. Vorob’ev draws attention to the discrepancy in the results obtained at FIAN (presented in the report) and by Breckenridge.

B. N. Matsonashvili replies that Breckenridge found relaxation maxima at low frequency and not in vacuum. Breckenridge did not prepare the crystals himself and did not investigate their purity. We grew and prepared the crystals ourselves and ensured clean experiments. The relaxation maxima were found in vacuum at high frequency. The presence of maxima is firmly established.