ON THE PROBLEM OF THERMAL AND ELECTRICAL BREAKDOWN OF SOLID DIELECTRICS

G. I. SKANAVI

Classification of Types of Breakdown. Thermal Breakdown

The study of the mechanism of breakdown of solid dielectrics and of their behavior in strong pre-breakdown fields is highly topical not only because the problem of increasing the electrical strength of materials, if it can be solved, is of enormous practical interest, but also because this study makes it possible to penetrate more or less deeply into the complex processes of motion of electric charges in a solid body and, in particular, into the structure of real solid bodies, which distinguishes them from ideal ones.

In the literature the following classification of the types or forms of breakdown of solid dielectrics has become established: thermal breakdown, electrical or intrinsic (internal intrinsic) electrical breakdown, and the chemical form of breakdown. It should first of all be noted that the chemical form of breakdown cannot claim a separate place in this classification, since it is a complex aggregate of processes of irreversible chemical changes in a solid substance, mainly under the action of ionization of every kind, which most often end in thermal (thermal breakdown) or mechanical destruction of the dielectric.

For a long time it was considered that for electrical breakdown, in contrast to thermal breakdown, what is primarily characteristic is the independence of the breakdown voltage from the temperature and from the duration of application of the electrical load (down to very short durations).

At the same time, one of the main characteristics of a solid dielectric was considered to be a certain critical temperature, below which the breakdown voltage does not depend on it (the region of electrical breakdown) and above which \(E_{\text{br}}\) decreases more or less sharply with increasing temperature (the region of thermal breakdown). At present, however, it is obvious that this so-called critical temperature is by no means connected solely and directly with the peculiarities of the substance or material, but is determined by the combination of one or another type of dependence of the dielectric’s conductivity or dielectric losses on temperature with the conditions of heat transfer in which the particular specimen is found, and with the frequency of the applied voltage. It is easy to see that a specimen of dielectric enclosed in an ideal heat-insulating shell must, generally speaking, undergo thermal breakdown at any small voltage. Such a thermal “adiabatic” breakdown may occur at any initial temperature of the specimen.

It should not be thought, however, that thermal breakdown of dielectrics must be completely excluded from consideration by physicists. A tendency toward this has recently appeared in a number of foreign works. On the contrary, the possibility of thermal processes must above all be taken into account

in setting up any physical experiment in the field of investigating electrical strength.

As is known, the theory of thermal breakdown has long since been developed, and rather fully, in a number of works, according to which thermal breakdown occurs at that voltage at which the dielectric specimen leaves thermal equilibrium because the heat generation due to conductivity or dielectric losses becomes greater than the heat removal.

The criterion of purely thermal breakdown, independently of the variants of its theory, is in essence the combination of two circumstances:

1) The breakdown voltage must decrease with temperature, and moreover in such a way that, in the coordinates \(\log U_{br}; \dfrac{1}{T}\), this decrease is expressed by a straight line with a slope half as large as the slope of the straight line expressing the dependence of the logarithm of the resistance or of the dielectric-loss factor \((\varepsilon \operatorname{tg}\delta)\) on reciprocal temperature.

2) The breakdown voltage must decrease as the duration of the electrical load is increased over a wide range of times; moreover, before breakdown there must be observed a progressive rise in the temperature of the specimen. If both of the indicated circumstances are not present, then the breakdown is not purely thermal, even in the case when the breakdown voltage varies with temperature.

General Characterization of Electrical Breakdown

Electrical breakdown, as follows from the work of the last decade, in contrast to older conceptions, is not characterized by the absence of temperature dependence. Purely electrical breakdown takes place in the case when the influence of thermal processes is excluded (short-time application of voltage or very small conductivities and dielectric losses) and when ionization of the medium, surface discharges and corona, and ionization of air inclusions are absent, which is ensured by a high degree of field uniformity and by macroscopic (at least) homogeneity of the dielectric specimen. The breakdown voltage in electrical breakdown can, to a certain degree, serve as a measure of the electrical strength of a solid dielectric as such.

Characteristic of electrical breakdown is the comparatively high value of the breakdown voltage, which for the enormous majority of solid dielectrics lies within the relatively narrow limits \(1—10\) mv/cm.

A characteristic sign of electrical breakdown may be considered to be the combination of independence of \(E_{br}\) from the duration of voltage application down to \(10^{-7}—10^{-8}\) sec (in a uniform field) and from the thickness of specimens down to \(\sim 10^{-4}\) cm, with a comparatively weak dependence on temperature. In this case the temperature behavior of simple ionic crystals is expressed in a weak, almost linear increase of \(E_{br}\) with increasing temperature (for voltage pulses of short duration \(\leq 10^{-6}\) sec), or has the form of a curve with a maximum at temperatures in any case above \(0^\circ\text{C}\) (\(+50^\circ\text{C}\), \(+200^\circ\text{C}\)).

For other dielectrics the temperature behavior in electrical breakdown has been studied poorly. In some works there is an indication of a weak decrease of \(E_{br}\) with increasing temperature for quartz glass.

There exists a large number of theories of electrical breakdown, among which, in general, preference should be given to quantum theories based on the concept of impact ionization by electrons. The main difficulty encountered in comparing the results of theory with experiment is connected with the fact that in any real case the breakdown of a solid dielectric

consists of at least two stages: the stage of loss of electrical strength, characterized by an increase in current without an increase in the applied voltage, and the stage of destruction of the dielectric, which is usually accompanied by the formation of a conducting channel of finite dimensions. Modern theories consider only the first stage, whereas experiment usually encompasses both stages; precisely for this reason a quantitative comparison of theory with experiment is very difficult.

In light of the above it is appropriate to emphasize that the first basic experimental task at present is the separation of the stages of breakdown, and the creation of such experimental conditions under which it would be possible to observe an increase in current upon loss of electrical strength independently of destruction of the specimen; the first approach to solving this problem may be the measurement and oscillographic recording of the current in the specimen before breakdown and at the instant of breakdown. Incidentally, an almost decisive argument in favor of impact ionization would be the existence of a rectilinear dependence of the logarithm of the breakdown current on the thickness of the specimen under conditions of constant field strength in a homogeneous field. From the slope of these straight lines, if they occur, it would even be possible to estimate the magnitude of the effective coefficient of impact ionization (the Townsend coefficient for the case of gases).

A second important task at present is the study of the influence of cold emission from the cathode and of space charge on the process of electrical breakdown, for a number of experimental data even now apparently cannot be interpreted without invoking concepts of cold emission and space charge (pre-breakdown “noises,” the temperature dependence of $E_{pr}$ of ionic crystals for different durations of electrical loading, etc.).

Theories of electrical breakdown may be divided into two groups: theories that consider the destruction of the dielectric—the second stage of breakdown—and theories that consider the loss of electrical strength—the first stage of breakdown. All modern theories belong to the second group.

Some Classical Theories of Electrical Breakdown

Among the old classical theories of electrical breakdown one should note Rogowski’s theory of rupture of the ionic lattice and Gorowitz’s theory of microcracks. In Rogowski’s theory the field strength is calculated at which the external force acting on the ions is greater than or equal to the maximum counteracting binding force. The calculation is carried out on the assumption that, for large ionic displacements, the restoring force is not quasi-elastic and depends both on the first and on the third power of the displacement along the field, and on the square of the displacements in directions perpendicular to the field. Rogowski calculates the coefficient of elastic binding and the coefficients of anharmonicity from the structure of the NaCl lattice, using a point model and neglecting the polarizability of the ions. As is known, the result of the calculations is that rupture of the NaCl lattice can occur only at a field strength greater than or equal to $\sim 10^8$ V/cm. It should be noted that in Rogowski’s calculations the distinction between the acting (local) and the mean field is not taken into account. At the same time, after calculating by one method or another the field strength that must act on each ion of the lattice in order for its rupture to occur, it is then necessary to find the strength of the mean field, since it is precisely this that is determined experimentally and is called the breakdown strength. If, in a first rough estimate, one assumes that the internal field is equal to the Lorentz field

\(\frac{4\pi I}{3}\) (\(I\) is the electric moment of unit volume), then the mean field is

\[ E_{\mathrm{av}}=\frac{3}{\varepsilon+2}E \simeq \frac{3}{8}E \]

(the dielectric constant of NaCl is \(\varepsilon \simeq 6\)). Moreover, for the large displacements of ions that must arise before rupture of the lattice (according to Rogovskii, the displacement of the ions must reach \(\sim 0.3\) of the nearest distance between ions), the local field must differ still more strongly from the mean field. A simple calculation, taking into account the six nearest neighboring ions, gives that the additional internal field, due to the action of the charges and dipole moments of the displaced ions, is equal to

\[ E_{\mathrm{add}}=\frac{q}{a_0^2}\left[ \frac{4\xi}{(1-\xi^2)^2} - \frac{4\xi}{(1+\xi^2)^{3/2}} \right] + \frac{\alpha_e E}{a_0^3} \left[ \frac{2}{(1-\xi)^3} + \frac{2}{(1+\xi)^3} + \frac{1-4\xi+2\xi^2}{\left[(1-\xi)^2+1\right]^{5/2}} \right], \]

where \(\xi\) is the relative displacement, equal to \(0.3\), \(a_0\) is the nearest distance between ions, \(q\) is the ion charge, \(\alpha_e\) is its electronic polarizability, and \(E\) is the strength of the field acting on the ion.

Therefore the mean field is equal to

\[ E_{\mathrm{av}}=\frac{3}{\varepsilon+2} \left[ E\left(1-\frac{L_e}{a_0^3}B\right) - \frac{q}{a_0^2}A \right], \]

where

\[ A=\frac{4\xi}{(1-\xi^2)^2} - \frac{4\xi}{(1+\xi^2)^2} \quad \text{and} \quad B=\frac{2}{(1+\xi)^3} + \frac{2}{(1-\xi)^3} + \frac{1-4\xi+2\xi^2}{\left[(1-\xi)^2+1\right]^{5/2}}. \]

Substituting numerical values, we obtain

\[ E_{\mathrm{av,\,br}}=\frac{3}{8} \left[ 0.944\,E_{\mathrm{br}}-66.2\cdot 10^6 \right]\ \mathrm{V/cm}. \]

If, according to Rogovskii, \(E_{\mathrm{br}}\) is, for example, \(1\cdot 10^8\ \mathrm{V/cm}\) (for the case when it is directed along the diagonal of the cube), then \(E_{\mathrm{av,\,br}}=1.1\cdot 10^7\ \mathrm{V/cm}\), which is already only one order of magnitude greater than the experimental value. If the crude approximate nature of the calculations is taken into account, this difference should not be considered excessively large. It must further be noted that the coefficients of elastic coupling and anharmonicity can in principle be estimated from experimental data on the dielectric constant, or on the natural frequency of oscillations of the lattice ions and on the temperature coefficient of the dielectric constant, instead of by direct calculation on the basis of the lattice structure, which requires very cumbersome computations and apparently cannot be carried out with sufficient accuracy. Unfortunately, the indicated experimental data are also not sufficiently accurate. In any case, it may be asserted that the field strength rupturing the ionic lattice may be less than \(10^8\ \mathrm{V/cm}\), and that, in connection with this, the possibility of such rupture should perhaps not be rejected altogether in all cases. Of course, the matter cannot be so simple as to say that electrical breakdown of an ionic crystal lattice is rupture in a strong field, but one must consider that attempts at a more rigorous calculation of \(E_{\mathrm{br}}\) in the future are of some interest.

In Gorowitz’s theory, the breakdown voltage of any solid dielectric is calculated on the assumption that electrical breakdown occurs as the rupture of a microcrack on its surface. As a result of the calculations, a very simple expression for \(E_{pr}\) was obtained from the condition that the additional energy of the ellipsoidal surface of the crack, consisting of the field energy and the surface-tension energy, has a maximum value at a certain \(E = E_{pr}\). According to Gorowitz’s formula,

\[ E_{pr}=4\sqrt{\frac{\sigma}{\varepsilon\lambda}}, \]

where \(\sigma\) is the surface tension, \(\varepsilon\) is the dielectric constant, and \(\lambda\) is the crack length. Comparing the calculated value of \(E_{pr}\) for glass, porcelain, and rock salt, the author of the theory came to the conclusion that the calculated values differ noticeably from the experimental ones. Possibly in connection with this, Gorowitz’s theory is now essentially forgotten. On closer examination, however, it turns out that the experimental values taken into account by Gorowitz in 1927 were greatly underestimated.

If one compares more recent experimental data with the results of Gorowitz’s calculations, it is clear that the discrepancy is not very large. Thus, for a crack depth normal to the surface of the dielectric of \(10^{-4}\ \text{cm}\), for glass and porcelain \(E_{pr\,calc} \simeq 1.2\cdot10^6\ \text{V/cm}\) (\(E_{pr\,exp}=(2\text{–}4)\cdot10^6\ \text{V/cm}\) for glass and \(E_{pr\,exp}=(3\text{–}7)\cdot10^5\ \text{V/cm}\) for porcelain); for rock salt \(E_{pr\,calc}=0.6\cdot10^6\ \text{V/cm}\), while the experimental value is \(\sim 10^6\ \text{V/cm}\). Of course, the depth of a crack, if one exists, is not known with sufficient accuracy, nor is the magnitude of the surface tension, so that the indicated agreement cannot be regarded as in any way decisive; nevertheless, it is possible that in the electrical breakdown of imperfect dielectric specimens and in the breakdown of technical materials, the possibility of the Gorowitz mechanism must be taken into account. It would be advisable to carry out the corresponding experimental investigation.

Quantum theories of breakdown based on the concept of ionization by electron impact

The ideas of Ioffe and Smurov on impact ionization in solid dielectrics were fruitfully developed in the works of Hippel, who formulated the principal problems subject to quantitative treatment. Subsequently a number of authors developed Hippel’s concept; somewhat different treatments of the question were proposed in the theory of Fröhlich, in the theory of Adirovich, Chuenkov, and others. All these theories concern the breakdown of ionic (polar) crystals. The breakdown of nonpolar crystals has been theoretically discussed in a smaller number of works, the most important of which are the works of Heller and Franz. The second stage of breakdown was considered qualitatively by Seitz.

In discussing the process of impact ionization in a solid dielectric, it is appropriate first of all to pose the general question why Paschen’s law for gases, in which breakdown is caused by impact ionization, cannot be extended to solid dielectrics. The breakdown voltage of a gas is a function of the product of the pressure \(p\) by the distance between the electrodes \(\delta\), or of the ratio \(\delta/\lambda\), where \(\lambda\) is the mean free path of the electron. In going from a gas to a liquid or a solid, the density increases approximately 2000-fold. When the substance is compressed 2000-fold, the electron mean free path would seem to have to decrease 2000-fold, and since according to Paschen’s law \(E_{pr}\) is approximately proportional to the product \(p\delta\), or to the ratio \(\delta/\lambda\), then, consequently, for a solid or liquid

for a dielectric it should be 2000 times greater than for a gas under normal conditions. The breakdown strength of air, as is known, is \(3\cdot 10^4\ \mathrm{V/cm}\); consequently, \(E_{\mathrm{br}}\) of solid and liquid dielectrics would have to be \(6\cdot 10^7\ \mathrm{V/cm}\). In reality, however, it is smaller by approximately a factor of ten.

The reason for this discrepancy is, in general, as follows. When electrons move in a gas, a considerable part of the energy they have accumulated in the field is spent on inelastic collisions that do not cause ionization, since the atoms and molecules of gases possess excitation levels lying far below the ionization potential; in dense phases, however, the interaction of atoms with one another is so great that the difference between excitation and ionization is almost erased. Thus, the accumulation of energy by an electron in a gas under the action of an electric field is impeded by appreciable pre-ionization losses (to excitation, dissociation, etc.), and the breakdown strength per unit density is higher than for a solid dielectric.

However, the fact that the electrical strength of a solid dielectric is lost not at arbitrarily small field strengths, but at \(E_{\mathrm{br}}\sim 10^6\ \mathrm{V/cm}\), indicates that in a solid dielectric too there are pre-ionization losses of electron energy (if breakdown is due to impact ionization). Since the process of exciton formation (excitation) in a crystal lattice requires an energy close to the ionization energy, the principal factor determining the pre-ionization electrical losses of an electron is the slowing down (scattering) of conduction electrons by lattice vibrations. A comparatively slow electron, colliding with a molecule whose mass is many times larger, cannot transfer to it any appreciable energy by a “mechanical” route. Therefore one can speak only of scattering and slowing down of the electron by lattice vibrations.

In a certain rough approximation it may be assumed that an electron, approaching a molecule and interacting electrically with it, can change the binding forces between the atoms composing the molecule and alter their vibrations. The molecule thereby passes into a new vibrational state of another, higher frequency and, after the electron has gone away, generally remains in this new state. The indicated change in the vibrational state of the molecule occurs at the expense of the energy of the electron, which loses a quantum of energy of comparatively low definite frequency (a phonon). The reverse transition also has a finite, but smaller, probability. In gases the indicated interaction of electrons with atoms in a molecule cannot have an appreciable influence on the breakdown strength, since the electron already acquires too much energy in a single free path, and an inelastic collision of the electron with a molecule leads directly to excitation or ionization. In a solid dielectric, on the contrary, the electron interacts electrically not with a few atoms, but with a large number of atoms or ions forming the crystal lattice. Therefore, in the indicated interaction there arise not only vibrations of two nearest particles relative to one another, but a whole spectrum of elastic vibrations of the lattice. In connection with this, the frequency interval of the excited vibrations becomes theoretically unlimited, and the excitation boundary disappears.

As is known, the elastic vibrations of an ionic crystal lattice are divided into two groups: optical vibrations, for which the phase difference of the displacements of ions of different signs lies within the limits from \(\pi/2\) to \(\pi\), and acoustic vibrations, for which this phase difference lies within the limits from 0 to \(\pi/2\). The frequency of optical vibrations, in the first approximation, does not depend on the wavelength and can be found from optical measurements of the frequency of residual rays. The frequency of acoustic vibrations depends on the wavelength and, for long waves, is inversely proportional to it.

As the calculations of Callen, Fröhlich, Ziger, and Teller have shown, the energy losses of an electron in optical vibrations of the ionic lattice, referred to unit time, depend on the electron energy \(\varepsilon\) and pass through a maximum at \(\varepsilon_m \simeq 4 \frac{h}{2\pi}\), where \(\omega\) is the frequency of the optical vibrations and \(\frac{h}{2\pi}\omega\) is the phonon energy.

If the energy acquired by free electrons from the field is equal to the energy losses, then the electric current is stationary. If, however, the field strength is so large that the energy acquired by the electrons from the field exceeds the energy losses, the stationary state is destroyed and impact ionization arises, leading to electrical breakdown. The whole further difficulty consists in formulating a quantitative criterion for breakdown and calculating theoretically the breakdown field strength, since the electrons possess different energies and different momenta. In a crystal, in the presence of a strong field, there is a complicated distribution of conduction electrons with respect to energy and the angles between the electron momentum and the field.

Hippel, and subsequently Fröhlich, succeeded in formulating a criterion for the occurrence of impact ionization without solving the kinetic equation and without having an expression for the indicated distribution function. According to Hippel’s criterion, impact ionization arises when the energy acquired by an electron from the field is equal to the maximum energy losses, i.e., to the losses of an electron with energy \(\varepsilon_m = 4 \frac{h}{2\pi}\omega\). It is evident that in this case the remaining electrons, losing a smaller amount of energy, will be accelerated by the field. Since the energy acquired by an electron in the field depends on the field strength, Hippel’s criterion makes it possible to calculate the breakdown field strength if the electron energy losses in lattice vibrations are known. The calculation of the latter is the main subject of the works developing Hippel’s concept.

Fröhlich gives another criterion for breakdown, assuming that the Maxwellian velocity distribution is applicable to conduction electrons up to electrons possessing an energy equal to the ionization energy, and that in any field there exist in the crystal electrons with energy close to or equal to the ionization potential. This is the principal difference between Fröhlich’s theory and Hippel’s theory, which assumes that fast electrons are captured by lattice defects and that before the onset of impact ionization, i.e., in the stationary state, there are no fast electrons in the crystal. According to Fröhlich’s criterion, impact ionization arises when the energy acquired by an electron in the field is equal to the energy losses of such an electron, whose energy is equal to the ionization potential. Like Hippel’s criterion, this criterion also makes it possible to calculate the breakdown field strength. Fröhlich’s calculations, as well as those of Hippel—Callen, give values of \(E_{\mathrm{br}}\) of the order of magnitude close to the experimental ones. However, as Chuenkov showed, more accurate calculations, in particular a more detailed account of the difference between the effective charge of an ion and the true charge in optical and acoustic vibrations, lead to the result that the breakdown field strength according to Hippel’s theory proves to be overestimated, and according to Fröhlich’s theory underestimated, which is, generally speaking, quite natural.

The following works are devoted to the solution of the kinetic equation (Franz, Heller, and Chuenkov). In Chuenkov’s theory a physically clear criterion of breakdown is given, and as a result of rather cumbersome calculations results are obtained that are more or less in agreement with experiment.

All the indicated theories, under certain assumptions, give numerical values of \(E_{pr}\) more or less close to the experimental ones, and a weak increase of \(E_{pr}\) with increasing temperature. According to these theories, strengthening should occur in thin layers at thicknesses of about \(10^{-5}\) cm. The time for the development of an electron avalanche, however, turns out to be very small; in this connection one cannot expect strengthening at times of the order of \(10^{-7}\)—\(10^{-8}\) sec. In this respect the theories of impact ionization prove insufficient, since it has been shown experimentally, as already mentioned, that for pulses of \(10^{-7}\)—\(10^{-8}\) sec electrical strengthening, in all probability, does nevertheless occur. It is possible that precisely the second stage of electrical breakdown—the destruction of the dielectric—requires a time of the order of \(10^{-7}\)—\(10^{-8}\) sec. In that case, however, one must assume that the magnitude of the breakdown strength is connected not only with the process of the occurrence of impact ionization, but also with the mechanism of destruction of the dielectric. This circumstance naturally could not be reflected in the indicated theories, which do not touch at all upon the second stage of breakdown.

For the breakdown of gases, as is known, statistical lag is characteristic. Electrical strengthening at small times for solid dielectrics might be connected precisely with this process, and then the theories of impact ionization would be completed within the framework of the single first stage of breakdown. However, statistical lag of breakdown of solid dielectrics in a homogeneous field has not yet been unambiguously established, although interesting attempts in this direction exist (the works of Kawamura and others). The difficulties here consist in the fact that the number of initial conduction electrons in a solid is large and uncertain, and also may depend on external factors, for example, cold emission from the cathode. In any case, detailed experimental elaboration of the question of the dependence of \(E_{pr}\) on time at small times and of the questions connected with it is a current physical problem at the present time.

Not uninteresting semi-quantitative estimates of the second stage of electrical breakdown have been made by Seitz. If the condition for the occurrence of impact ionization is fulfilled, then the number of secondary electrons \(n\), produced by one initial electron, must be proportional to the residence time of the electron inside the crystal and, consequently, will depend on the dimensions of the specimen, the place of appearance of the initial electron, and certain basic properties of the crystal. For a rough estimate one may assume that breakdown of the dielectric will occur when each atom of the substance receives from all the electrons, on the average, an energy equal to the energy of the crystal lattice (\(\sim 10\) ev). In view of the fact that the ionization potential is also close to \(10\) ev for many crystals, the results of the estimate under consideration do not change if one starts from the assumption that, during breakdown, each atom must be ionized, and the dielectric thereby will not be destroyed but will turn into a conductor. If \(E = 10^{6}\) v/cm and the mobility of electrons in strong fields is \(\sim 1\ \text{cm}^{2}/\text{v sec}\), then an electron will travel in the direction of the field a path of \(1\) cm in a time \(\tau = 10^{-6}\) sec, since its drift velocity will be equal to \(10^{6}\) cm/sec. In addition, as a result of thermal motion the electrons will diffuse in the plane perpendicular to the field. The diffusion coefficient may be estimated as \(D = \lambda v/3\), where \(\lambda\) is the mean free path of the electron, \(v\) is its velocity; this gives \(D \simeq 1\ \text{cm}^{2}/\text{sec}\).

Thus, in moving \(1\) cm in the direction of the field, the electron will deviate perpendicular to this direction on average by a distance \(x = \sqrt{D\tau} = 10^{-3}\) cm. Consequently, when traversing a path of \(1\) cm in a time \(10^{-6}\) sec, the average electron will transfer its energy to atoms located in the volume of a cylinder of radius \(10^{-3}\) cm and length \(1\) cm. At \(E = 10^{6}\) v/cm this energy is equal to \(10^{6}\) ev. The number of atoms in the indicated

in a volume of \(\sim 3\cdot 10^{-6}\ \text{cm}^3\) is approximately \(10^{17}\). At breakdown, according to the assumption made, the energy transferred to all these atoms must be equal to \(10\cdot 10^{17}=10^{18}\ \text{eV}\). Such total energy will be possessed by \(10^{18}/10^{6}=10^{12}\) electrons (i.e., for every \(10^{17}/10^{12}=10^{5}\) atoms in the tube under consideration, one primary electron must create one secondary electron).

In reality, only the end of the cylindrical channel will be subject to destruction, but this is not essential for a rough estimate. The number of necessary impact ionizations is determined, obviously, from the equality \(2^n=10^{12}\), i.e. \(n \simeq 40\). If the distance between the electrodes is \(\delta\), then for breakdown it is necessary that the average free electron produce ionization along the path \(L \ll \frac{1}{40}\delta\). Thus, in addition to the criterion for the onset of impact ionization, a definite relation between \(L\) and \(\delta\) is also necessary. It is obvious that \(L\) decreases as the field strength increases. If the thickness \(\delta\) is sufficiently small, then the condition \(L \ll \frac{1}{40}\delta\) will be satisfied only at an increased field strength. Hence follows the strengthening in thin layers which, according to Seitz’s estimate, should occur at thicknesses of the order of \(\sim 10^{-4}\ \text{cm}\). The estimate given does not take into account that the primary electron may be located anywhere in the dielectric, and not at the cathode. However, this circumstance can hardly greatly alter the indicated result (since the probability of the initial electron being found near the cathode is always sufficiently large), but it may be associated with a statistical delay of breakdown. If the field strength is insufficient, \(L\) may be greater than \(\delta\), and then the electron avalanche will not culminate in breakdown.

It is possible that such electron avalanches can be detected in the form of noticeable current pulses before breakdown. In this respect, the works of Kawamura and his collaborators are of interest, in which prebreakdown “noises” occurring in mica were investigated¹. In light of these considerations, the theories of electrical breakdown that have recently been abandoned and that are not connected with the idea of impact ionization acquire new significance: Zener’s theory of “tunnel” breakdown, according to which the growth of current at the moment of breakdown is due to the tilting, in a strong external field, of the energy bands and to the seepage of electrons from the normal band into the conduction band; the theory of breakdown of electrode origin (the seepage of electrons into the dielectric conduction band from the cathode and from the normal band of the dielectric into the anode) (Wolkenstein, Fowler), and Frenkel’s theory of thermal ionization.

Indeed, if electron avalanches are not sufficiently intense to lead to the second stage of breakdown, then, as the field strength increases, the processes indicated may become decisive even before the electron avalanches acquire sufficient intensity.

Attempts to construct such a theory may acquire substantial importance—one that would be based on a combination of the processes of cathode emission, impact ionization, and field distortion by space charge, analogous to those recently made for the case of breakdown of superpure liquids.

In the works of A. A. Vorob’ev and E. K. Zavadovskaya, attention is drawn to the relation between \(E_{pr}\) of alkali-halide crystals and the energy of the crystal lattice \(W\). The approximate proportionality that occurs between \(E_{pr}\) and \(W\) could be explained on the basis of the theory of rupture of the crystalli-

…of the crystal lattice by the field. However, there are no grounds to believe that this theory is justified. It should be emphasized that from the theory of impact ionization there follows a decrease of \(E_{\mathrm{pr}}\) with an increase in the lattice constant, i.e., with a decrease in the energy of the crystal lattice. Thus, the indicated parallelism is not unexpected and does not contradict, in particular, theories of impact ionization.

In concluding the general part of the report, the following circumstances must be emphasized.

1. The process of electrical breakdown of solid dielectrics is so complex that at the present time it would be wrong, without sufficient experimental and theoretical grounds, to become carried away by the construction of any “simple” and pseudo-general qualitative hypotheses or theories, striving, so to speak, to reduce the solution of a system of equations with many unknowns to one simplest equation with one unknown.

2. However, under no circumstances may the opposite conclusion be drawn—that the problem is so confused that there is no hope of bringing clarity to it. It seems to me that the essential role of impact ionization in the process of electrical breakdown at the present time is not in doubt. But the nature of breakdown is still not determined merely by the occurrence of impact ionization; the latter, even under pure experimental conditions, is combined with cold emission from the cathode and is complicated by the influence of space charge and by the process of destruction of the dielectric. Intensive experimental work in the directions indicated above, in close contact with theoretical investigations, will undoubtedly in the near future to a significant degree lift the curtain separating us from one of the most complex and very interesting phenomena of solid-state physics—the electrical breakdown of solid dielectrics.

Passing to the second part of the report, it should be noted that in practice there are often cases of breakdown which at first glance are difficult to classify on the basis of the general propositions set forth above. These cases must be clearly separated from other, physically simpler ones, without posing to nature questions to which it cannot give an answer. This applies first of all to the phenomenon of breakdown of macroscopically inhomogeneous dielectrics.

On the Temperature Dependence of \(E_{\mathrm{pr}}\) in Electrical Breakdown

In the second part of the report I shall take the liberty of presenting individual experimental results obtained in the Laboratory of Dielectric Physics of the Lebedev Physical Institute of the Academy of Sciences of the USSR, which may serve as an illustration of the general propositions set forth above. A separate report by E. A. Konorova is devoted to the important question of the dependence of the electric strength of certain solid dielectrics on pulse duration and on thickness, and I shall not dwell on this question. Another essential question from our point of view is the temperature dependence of the electric strength of alkali-halide single crystals for different durations of voltage application. The corresponding investigation is being carried out in our laboratory by L. A. Sorokina.

At previous conferences, in discussions of the reports by A. A. Vorob'ev and E. K. Zavadovskaya, I pointed out that in the foreign literature there are several works in which a dependence of \(E_{\mathrm{pr}}\) of alkali-halide crystals in the region of electrical breakdown on temperature was obtained, in contradiction to the older works of German and Soviet scientists. The fundamental importance of the phenomenon, and a certain inconsistency of the indicated experi-

experimental results and the negative attitude toward them on the part of the speakers and some other specialists compelled us to undertake a systematic investigation, which has not yet been completed, but already makes it possible to draw sufficiently definite conclusions.

Figure 1 presents experimental data from various authors. It is seen that in almost all cases a dependence of \(E_{pr}\) on temperature is obtained which, at constant voltage, is expressed by a curve with a maximum, while for pulses, as the pulse duration decreases, it turns into a monotonically increasing curve close to a straight line.

Fig. 1. Dependence of \(E_{pr}\) on temperature.

1—constant voltage and \(2 \cdot 10^{-8}\) sec; 2—constant voltage; 3—pulses \(10^{-4}\) sec; 4—pulses \(5 \cdot 10^{-8}\) sec; 5—pulses \(10^{-6}\) sec; 6—pulses \(10^{-4}\) sec; 7—constant voltage and 60 hr; 5, 6, 6a, and 7—Hippel and Alger, KBr [1]; 2, 3, and 4—Inuishi and Suita, KCl [2]; 1—Calderwood and Cooper [3].

In some works the temperature maximum of \(E_{pr}\) was not found even at constant voltage. The absolute value of \(E_{pr}\) under pulses is, for some authors, higher than the maximum \(E_{pr}\) at constant voltage, while for others, on the contrary, it is lower. In individual cases, for a pulse duration of \(10^{-6}\) sec, no temperature dependence of \(E_{pr}\) was observed at all.

In our work, the temperature dependence of \(E_{pr}\) of KBr single crystals was measured at constant voltage and with pulses of duration \(10^{-2}\), \(10^{-4}\), and \(10^{-6}\) sec in the temperature range above \(0^\circ\)C (the low-temperature region is currently being investigated). One of the most su-

essential for obtaining reliable results is uniform heating of the samples, ensuring the absence of cracking of the samples. KBr crystals were grown from the melt, cleaved into plates with an area of \(1\text{–}2\ \mathrm{cm}^2\) and a thickness of \(3\text{–}4\ \mathrm{mm}\), in which cylindrical recesses with a hemispherical bottom were drilled. The thickness of the sample at the thinnest point was thereby brought to \(0.2\text{–}0.3\ \mathrm{mm}\). After such treatment the samples were annealed to relieve the mechanical stresses produced during drilling. Annealing was carried out at a temperature close to the melting temperature (\(t_{\mathrm{m}}=730^\circ\mathrm{C}\), \(t_{\mathrm{ann}}=680\text{–}700^\circ\mathrm{C}\)) for 2–3 hours, after which the samples were slowly cooled. This annealing noticeably reduced the scatter of the values of \(E_{\mathrm{br}}\). After annealing, the surface of the pit and the flat surface of the samples were polished, and the thickness of the sample was brought to \(0.1\text{–}0.2\ \mathrm{mm}\). Before the electrodes were applied, the samples were examined under a microscope to detect visible cracks and scratches on the polished surfaces. The electrodes were applied by vacuum deposition of silver. Breakdown of the samples was produced in a porcelain beaker, in oil or an organosilicon liquid; the high-voltage electrode was lowered into the pit. Heating was carried out by means of a thermostat and was monitored by several thermocouples. The rate of heating was set in such a way that the readings of three thermocouples pressed against the sample and of a thermocouple in the liquid coincided, while the reading of the thermocouple placed inside the metal grounded electrode exceeded the readings of the other thermocouples by no more than \(10^\circ\mathrm{C}\). Under these conditions no cracks appeared in the sample. The heating time chosen in this way was, for a temperature rise from room temperature to \(50^\circ\mathrm{C}\), 1.5–2 hours, and for a temperature rise to \(200^\circ\mathrm{C}\), about 4 hours.

Fig. 2.

The experimental data obtained are presented in Fig. 2. The experimental values of \(E_{\mathrm{br}}\) are averages of 12–15 measurements. The scatter of the experimental data, shown in Fig. 2, is comparatively small and does not obscure the regularities obtained. It may be considered established that, under the experimental conditions described above, the electric strength of KBr crystals under a constant voltage has a pronounced temperature maximum at \(50^\circ\mathrm{C}\); as the pulse duration is reduced, this maximum is smoothed out, and at a pulse duration of \(10^{-6}\ \mathrm{sec}\) there is a monotonic increase of \(E_{\mathrm{br}}\) with increasing temperature. Thus, our results basically confirmed the results obtained for KBr by Hippel and Alger, with the difference, however, that according to our data the temperature maximum of \(E_{\mathrm{br}}\) corresponds to \(+50^\circ\mathrm{C}\), and not to \(+200^\circ\mathrm{C}\), as in Hippel and Alger. It should be noted that in that work not silver but gold electrodes were used. In other works gold electrodes were not used, and the temperature of the maximum of \(E_{\mathrm{br}}\) proved to be close to \(40\text{–}50^\circ\mathrm{C}\). It is significant that the temperature maximum of \(E_{\mathrm{br}}\) of ionic crystals occurs only for a sufficiently long duration of voltage application.

In this connection the applicability of Fröhlich’s second, so-called “high-temperature,” theory is placed in considerable doubt. This theory, however, has a certain general significance, since for the present it is the

apparently the only work in which an attempt is made to take into account the influence of lattice defects on the process of electrical breakdown. At low temperatures the number of electrons on defect levels is small, since the number of defect levels is small. With an increase in temperature the number of defect levels increases strongly, and in the process of electrical breakdown, according to Fröhlich’s assumption, electrons captured by lattice defects begin to participate. At a sufficiently high temperature the probability of collision of a free electron with an electron on a local level may noticeably exceed the probability of scattering by lattice vibrations. In this case, as in the plasma of a gas discharge, electrons in the crystal in the presence of a strong field must possess their own “electron temperature” \(T_e\), exceeding the lattice temperature \(T_0\).

Further, it is assumed that free electrons, having received energy from the field \(U\), interacting with electrons on local levels, transfer energy to the latter, and not directly to the lattice. The electrons on the local levels in turn transfer energy to the lattice. Levels of various depths are considered here, but the principal role is played by shallow levels lying below the conduction band by \(\Delta V\), with \(\Delta V \ll 1\) eV. In contrast to Fröhlich’s first theory and Hippel’s theory, here one determines the energy acquired from the field by all conduction electrons, and the energy given to the lattice by all electrons on local levels. The distribution of electrons between the conduction band and local levels is found with the aid of the Fermi–Dirac function.

Approximate calculations lead to the result that the energy acquired on average by one electron from the field, \(A\), increases at an increasing rate as the electron temperature rises, while the energy given on average by one local electron to the lattice (energy losses), \(B\), increases with increasing electron temperature at a decreasing rate. At the same time the quantity \(A\) naturally also increases when the field strength is increased. The breakdown criterion is determined, as before, by equality of the quantities \(A\) and \(B\), but only at a definite value of the electron temperature \(T_e\), corresponding to tangency of the curves \(A=f(T_e)\) and \(B=f(T_e)\). If the curves intersect, then a stationary state is established; if the curve \(A=f(T_e)\) lies above the curve \(B=f(T_e)\), then a stationary state is impossible. The limiting case is, evidently, the case of tangency of the curves. With the aid of the indicated breakdown criterion Fröhlich estimates the magnitude \(E_{\mathrm{br}}\); it turns out that \(E_{\mathrm{br}}\) decreases with increasing crystal temperature \(T_0\), while the critical electron temperature exceeds the crystal temperature only slightly.

Thus, in the temperature region above a certain definite \(T_c\), \(E_{\mathrm{br}}\) according to Fröhlich should decrease with increasing temperature. At \(T_0=T_c\) the number of collisions of free electrons with lattice vibrations is comparable with the number of their collisions with local electrons. At \(T_0<T_c\) there are few local electrons, as indicated above, and the interaction of free electrons with lattice vibrations (considered by all theories of impact ionization), for which \(E_{\mathrm{br}}\) increases with increasing temperature, acquires the principal role. It follows from this that near \(T_c\) a maximum of \(E_{\mathrm{br}}\) must occur. An increase in the number of structural defects in the transition from a monocrystalline to a polycrystalline and amorphous state of the dielectric should lower the temperature \(T_c\) and, consequently, the temperature of the maximum of \(E_{\mathrm{br}}\). According to Fröhlich’s considerations, in the electrical breakdown of amorphous dielectrics \(E_{\mathrm{br}}\) should decrease with temperature over a wide temperature range. On the other hand, a decrease in the electron mean free path should, generally speaking, increase \(E_{\mathrm{br}}\). Thus, the possibility opens up of estimating the change in the magnitude and temperature dependence

$E_{\mathrm{br}}$ when the structure of the dielectric changes. The experimental data on the temperature dependence of $E_{\mathrm{br}}$ for crystalline and amorphous quartz are in qualitative agreement with the indicated results.

Unfortunately, the change in the temperature dependence of $E_{\mathrm{br}}$ of ionic crystals with a change in the duration of the electrical load, which is confirmed by our experimental data, does not fit within the framework of Fröhlich’s “high-temperature” theory. Although the process of energy transfer from free electrons to local electrons and from the local electrons to the lattice may, and indeed must, require a different (longer) time than the process of direct energy transfer from a free electron to the lattice, and the time required for breakdown to develop may be different, it is nevertheless hardly possible to think that the temperature dependence of $E_{\mathrm{br}}$ at constant voltage differs from the temperature dependence of $E_{\mathrm{br}}$ already for comparatively long pulses $(10^{-2}\ \text{sec})$.

Therefore, the more probable cause of the temperature maximum of $E_{\mathrm{br}}$ should, in agreement with Hippel, be recognized as the influence of space charge, which is sufficiently small only for comparatively short pulses $(\sim 10^{-6}\ \text{sec})$. In this connection it should be borne in mind that space charge may be of two kinds: electronic (in the presence of cathode emission) and ionic—positive, formed during the propagation of an electron avalanche.

A quantitative account of the influence of space charge apparently presents very great difficulties. If the space charge increases the field strength in a sufficiently large volume of the dielectric, then $E_{\mathrm{br}}$ should decrease. If, on the contrary, a space charge of the opposite sign is formed in a thin layer near the electrode, then in the greater part of the dielectric volume the field is weakened and, generally speaking, $E_{\mathrm{br}}$ may increase, provided that the increase of the field at the cathode, which causes growth of cold emission, does not affect the breakdown process (otherwise the result may be the opposite). In one recent work [3] the influence of space charge was investigated by the simultaneous application to the specimen of a high dc voltage, which, in the opinion of the authors, creates space charge, and of an additional pulse voltage of different polarity. It turned out that at low temperatures $(10^\circ\mathrm{C})$ the space charge increases the breakdown voltage, while at high temperatures $(180^\circ\mathrm{C})$ it has no effect at all. Thus, space charge may exert different influences on the magnitude of $E_{\mathrm{br}}$.

Hippel assumes that the temperature maximum of $E_{\mathrm{br}}$ is explained by a decrease of $E_{\mathrm{br}}$ at low temperatures under the influence of an electronic space charge formed by cold emission, and by a decrease of $E_{\mathrm{br}}$ under the influence of a positive space charge at high temperatures. Near the temperature maximum, according to Hippel, compensation of the action of the negative and positive space charges occurs. In connection with the above, this assumption is experimentally insufficiently substantiated. Thus, although at present the experimental data are clearly insufficient for a detailed interpretation of the temperature course of $E_{\mathrm{br}}$ in ionic crystals, it nevertheless seems probable that, for sufficiently long durations of voltage application, the combination of space charge with cold emission and with the process of impact ionization has the most substantial influence on the course of the curve $E_{\mathrm{br}} = f(T)$.

Breakdown of Inhomogeneous Dielectrics (Polycrystalline Titanates)

We shall now turn to another group of phenomena in the field of breakdown of solid dielectrics, one that has substantial practical significance. Above

it was said that such experimental regularities are often observed which, without a sufficiently detailed analysis, cannot be attributed either to thermal or to electrical breakdown. This is especially the case in studies over a wide range of frequencies and temperatures of solid dielectrics of comparatively complex structure. As an example, it is possible to cite some of the experimental results on the investigation of the electric strength of polycrystalline titanates obtained in our laboratory by V. I. Sarafanov [4, 5]. Measurement of \(E_{\mathrm{br}}\) for titanates of magnesium, zinc, calcium, barium, barium tetratitanate, and strontium-bismuth

Fig. 3.
\(1—\mathrm{MgTiO_3}\), \(2—\mathrm{ZnTiO_3}\), \(3—\mathrm{BaO4TiO_3}\)

(BST) as a function of frequency (direct voltage, \(50\) c/s, \(143\) kc/s, \(1\) Mc/s, and \(2.5\) Mc/s) and of temperature (from room temperature to \(300^\circ\mathrm{C}\)) under conditions of an approximately uniform field showed, in general, that \(E_{\mathrm{br}}\) of most titanates decreases with increasing frequency and is almost independent, or only weakly dependent, on temperature over a fairly wide temperature interval (Figs. 3, 4). At first sight it seems that these regularities

Fig. 4.

cannot be attributed either to thermal or to electrical breakdown. At high frequencies (above \(1\) Mc/s), the breakdown of all titanates is thermal, despite the comparatively weak temperature dependence of \(E_{\mathrm{br}}\), associated with the fact that the loss factor \(\varepsilon \tan\delta\) increases comparatively slowly with increasing temperature. This is seen from Fig. 5, which presents the dependence of the specimen temperature on time at different values of the field strength at a frequency of \(2.5\) Mc/s for zinc titanate. For \(E > E_{\mathrm{br}}\), the temperature progressively increases with time, and breakdown occurs. For \(E < E_{\mathrm{br}}\), a steady state is established at a certain po-

constant temperature. At a frequency of 50 cps, the breakdown of all titanates is electrical, the regularities of which are complicated by the inhomogeneity of the material (the practical independence of \(E_{pr}\) from temperature up to \(180^\circ\mathrm{C}\) cannot occur in the case of thermal breakdown).

At constant voltage, the breakdown of strontium titanate, barium titanate, and, in all probability, barium tetratitanate is thermal, while that of the others is electrical.

In the intermediate frequency region (140 kcps–1 Mcps), the breakdown of titanates has a special character, caused by the presence of air pores in polycrystalline titanate ceramics. In the work of Piryatinskii [6], an idea was given of thermoionization, more precisely electrothermomechanical, breakdown of porous ceramics, according to which the liberation of heat in air pores during ionization raises the temperature (more precisely, the temperature gradient) at the pore wall, and therefore the ceramic is mechanically destroyed. This concept is in general applicable to the breakdown of titanates in the frequency interval 100 kcps–1 Mcps. However, it should be noted that the formula obtained by Piryatinskii for \(E_{pr}\) contains certain errors. Piryatinskii estimates the specific heat generation in a pore from the dielectric losses in the material, correctly attributing these losses at high frequency and high voltage mainly to the energy expended on ionization of the pores. Assuming further that, according to experimental data,

Fig. 5. Dependence of the specimen temperature on time for zinc titanate at \(f=2.5\) Mcps, and \(E_t\) equal to: 1—44 kV/cm; 2—34 kV/cm; 3—30 kV/cm; 4—25 kV/cm; 5—21 kV/cm.

\(\operatorname{tg}\delta = k_1 E_{av}\), he equates the specific losses in the material to the specific losses in the pore, which is incorrect. In fact, the specific losses in the pore must be larger; if

\[ P_{mat}=\frac{f\varepsilon\,\operatorname{tg}\delta\,E_{av}^{2}}{k}, \quad \text{then}\quad P_{por}=\frac{f\varepsilon\,\operatorname{tg}\delta\,E_{av}^{2}}{k\Theta}, \quad \text{where } k=4\pi\cdot 9\cdot 10^{11}, \]

and \(\Theta\) is the volume concentration of pores. If \(\Theta=0.01\), then the specific losses in the pore are 100 times greater than those determined by Piryatinskii. In this case they are precisely, as an estimate shows, sufficient to produce such a temperature gradient, and not the magnitude of the temperature, as Piryatinskii believes, at the pore wall, at which mechanical destruction of the ceramic is possible. In addition, in Piryatinskii’s calculations the dielectric permittivity of the material was inadvertently omitted. Thus, this formula must have the following form

\[ E_{pr}=\sqrt[3]{\frac{3\lambda\sigma_{kr}\Theta(1-\nu)}{\alpha\beta a^{3}k_1 f\varepsilon}}, \tag{1} \]

where \(\sigma_{kr}\) is the temporary resistance of the material, \(\nu\) is Poisson’s ratio, \(\alpha\) is the coefficient of linear expansion, \(\beta\) is Young’s modulus, \(a\) is the pore radius, \(\lambda\) is the thermal conductivity of the material, and \(k=4\pi\cdot 9\cdot 10^{11}\).

Piryatinskii’s formula, unlike the one given here, does not contain \(\varepsilon\) and \(\Theta\). The regularities for \(E_{pr}\) of titanates in the intermediate frequency interval (100 kcps–1 Mcps) correspond to formula (1), for \(E_{pr}\) is approximately inversely proportional to \(\sqrt[3]{f}\) and practically does not depend on temperature up to \(180^\circ\mathrm{C}\). At higher temperatures, for some titanates there occurs

purely thermal breakdown. In addition, the decrease of \(E_{\mathrm{br}}\) with increasing \(\varepsilon\) according to formula (1) in the indicated range of frequencies and temperatures is, qualitatively, fulfilled with some exceptions.

The breakdown regularities of strontium bismuth titanate— a dielectric possessing a very high dielectric permittivity (800–1000), but not being a ferroelectric—are of considerable interest. Dielectrics of this type have recently been discovered and studied by us in rather great detail. I shall report on the mechanism of polarization and losses of these dielectrics in my second paper. Here I shall only briefly discuss the breakdown of strontium bismuth titanates.

Figure 6 presents curves of the dependence of \(E_{\mathrm{br}}\) of SBT on temperature at direct voltage, at a frequency of 50 cps, and at a frequency of 140 kcps. It is seen that in the temperature range \(20\text{–}180^\circ\mathrm{C}\), \(E_{\mathrm{br}}\) is practically independent of temperature. At a temperature of \(-180^\circ\mathrm{C}\) there is a distinctly elevated value of \(E_{\mathrm{br}}\). At temperatures above \(200^\circ\mathrm{C}\), in the case of direct voltage, thermal breakdown takes place (\(E_{\mathrm{br}}\) falls with temperature).

Fig. 6. Dependence of \(E_{\mathrm{br}}\) and SBT on temperature.

The mechanism of electrothermomechanical breakdown in the interval \(20\text{–}200^\circ\mathrm{C}\), apparently, does not occur, since, owing to the comparatively low value of \(E_{\mathrm{br}}\) at frequencies of 50 cps and 140 kcps, the field in a small pore is insufficient for its intense ionization and breakdown. Under direct voltage the time constant for charging the pores is greater than the voltage rise time that occurred under the experimental conditions, and it is difficult to expect the pores to be punctured sufficiently often. Under direct voltage and at a frequency of 50 cps, the breakdown of SBT apparently has an electrical character; the reduced value of \(E_{\mathrm{br}}\) may perhaps be attributed to the multiphase structure of the ceramic, although this question is still not sufficiently clear. At a frequency of 140 kcps, despite the absence of a temperature dependence of \(E_{\mathrm{br}}\) in the interval \(20\text{–}280^\circ\mathrm{C}\), the breakdown is thermal breakdown, but of a special kind. Figures 7 and 8 give curves of the dependence of the steady-state temperature of the specimen on time at different field strengths of frequency 140 kcps, and the curve of the dependence of the steady-state temperature on field strength. It is seen that if \(E > E_{\mathrm{br}}\), the rise in temperature slows down, but nevertheless ends in breakdown. From Fig. 7 it is seen that the slope of the curve \(T=f(E)\) sharply increases at a certain \(E\) (\(E=8\text{–}12\ \mathrm{kV/cm}\)), and then again decreases, and only with a new increase of temperature with field strength does breakdown occur. This regularity suggested that near \(190\text{–}200^\circ\mathrm{C}\), at a frequency of 140 kcps, there must be a second additional maximum of \(\tan\delta\) and \(\varepsilon\). Measurements showed that this idea is correct (Fig. 9).

Thus, with a complex dependence of \(\varepsilon\) and \(\tg \delta\) on temperature, the regularities of thermal breakdown also become more complicated. In connection with the foregoing, it should be noted that the problem of experimental and theoretical study of the regularities of thermal breakdown of dielectrics with strongly pronounced relaxation polarization and losses (solid dielectrics containing weakly bound ions in structural defects or dipole radicals) is of unquestionable interest.

Fig. 7.

Fig. 8.

Breakdown of polycrystalline titanates was studied in our laboratory under such experimental conditions in which the influence of thermal processes and other complicating factors is excluded, i.e., with pulses of different duration using electrodes with as homogeneous a field as possible (in flat specimens, spherical dimples were drilled out and polished). The results of measurements carried out by V. V. Krasnopev-

Fig. .

...are presented in Fig. 10 for zinc titanate, which has the highest electric strength among all the titanates, barium titanate, and SVT. It may be thought that for pulse durations of \(10^{-6}\) sec and less the influence of thermal processes is excluded. The practical absence of a dependence of \(E_{\mathrm{br}}\) for zinc titanate, barium titanate, and SVT on time down to \(10^{-6}\) sec confirms the conclusion made above that the breakdown of these dielectrics is electrical in character. The strengthening of SVT in the region of very short pulse durations (less than \(10^{-6}\) sec) may be connected with the presence in the material of different structural phases, one of which (crystalline) is characterized by relaxation polarization and a high dielectric constant.

Fig. 10.

Indeed, the field in intercrystalline layers, which have a low dielectric constant, is much stronger than in crystallites with \(\varepsilon \sim 1000\), and therefore breakdown must begin with the destruction (breakdown) of the intercrystalline layers. Therefore, the higher the \(\varepsilon\) of the crystallites, the lower \(E_{\mathrm{br,\,av}}\) should be. On the other hand, a high \(\varepsilon\) is due to relaxation polarization, whose relaxation time at room temperature is not very small (\(10^{-6}\)—\(10^{-7}\) sec). Consequently, for pulse durations of \(10^{-7}\) sec the relaxation polarization does not have time to develop, the \(\varepsilon\) of the crystallites is appreciably less than 1000, the field in the intercrystalline layers is relatively smaller, and \(E_{\mathrm{br,\,av}}\) is higher. When the temperature is lowered, the relaxation time of the polarization of SVT increases greatly; therefore, if the above assumption is correct, then at low temperatures electrical strengthening should occur beginning with longer pulse durations. This circumstance is currently being checked by us experimentally.

It is more difficult to interpret the electrical strengthening of barium titanate, whose \(\varepsilon\) begins to decrease only at \(10^{10}\) Hz, and the relaxation time of the polarization is, consequently, much smaller than for SVT. It must be assumed, on the basis of the fact that \(E_{\mathrm{br}}\) of \(\mathrm{BaTiO_3}\) under constant voltage, when its breakdown is purely thermal in character, is close to \(E_{\mathrm{br}}\) under pulses, that owing to the high conductivity thermal processes play a role even in breakdown under pulses, and only at \(\tau \leq 10^{-7}\) sec

their influence is weakened. The indicated interpretation, however, does not seem to us entirely satisfactory. The next task is to investigate the electrical strength of single crystals with high dielectric permittivity in the region of electrical breakdown and to elucidate the influence of the internal field on the breakdown process.

References

1. Hippel A and Alger P. S. Phys. Rev. 76, 127, 1949.
2. Inushi J and Suita T. J. Phys. Soc. Japan, 8, No. 4, 567, 1953.
3. Calderwood and Cooper, Proc. Phys Soc, 66, 2, 74, 1953.
4. Sarafanov V. I., ZhETF, 27, No. 5 (11), 590, 1954.
5. Skanavi G. I. and Sarafanov, ZhETF, 27, No. 5 (11), 595, 1954.

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

Errata

1. Quite recently, however, the authors of these works came to the conclusion that the prebreakdown “noises” are not connected with the formation of electron avalanches. ↩