Current State of the Theory of Electrical Breakdown of Solid Dielectrics

V. A. Tsyenkov

The electrification of our country and the problem of high-voltage transmission of electrical energy over large distances make the question of studying the mechanism of breakdown of solid dielectrics a highly urgent one.

The phenomenon of dielectric breakdown was first observed by Academician Vasilii Vladimirovich Petrov in 1802[^1][^2]. Intensive and comprehensive study of the phenomenon of breakdown of solid dielectrics (henceforth only such dielectrics will be discussed) began in the twenties of our century in connection with the problem of transmitting electrical energy over large distances. The leading role in the study of breakdown of solid dielectrics belongs to Soviet scientists—A. F. Ioffe, A. A. Smurov, A. P. Aleksandrov, B. M. Vul, V. A. Fock, Ya. I. Frenkel, N. V. Kurchatov, K. D. Sineľnikov, A. A. Vorob’ev, and others.

B. M. Vul[^3] showed that the phenomenon of breakdown of solid dielectrics consists of two stages: 1) loss by the dielectric of its electrical strength and 2) destruction of the dielectric (mechanical or thermal), accompanied by the formation of a narrow channel piercing the entire thickness of the material from one electrode to the other.

Electrical strength, i.e., the ability of a dielectric to maintain a small stationary value of electrical conductivity in strong fields, is the most important property of electrical insulating materials. Experience shows that, up to the loss of electrical strength, electrical conductivity slowly increases with increasing field. The quantitative characteristic of the electrical strength of dielectrics is the value of the external field intensity \(E_{\mathrm{br}}\) (breakdown strength) at which the current flowing through the dielectric increases discontinuously to very large values.

During the first stage of breakdown, the properties of the dielectric change reversibly. The second stage of breakdown of solid

dielectrics is accompanied by irreversible changes, as a result of which, after breakdown, the electrical strength of the material is not restored. Thus, by breakdown of a solid dielectric one should understand the phenomenon of loss of electrical strength with subsequent destruction of the material.

Depending on the form of energy into which the energy of the external electric field is converted inside the dielectric, the following forms of breakdown are distinguished: thermal, chemical, and electrical.

The mechanism of thermal breakdown is as follows. A weak electric current passing through the dielectric heats it, which, owing to the presence of a negative temperature coefficient of resistance, leads to an increase in conductivity. This causes further heating of the dielectric. If the conditions of heat removal do not ensure the removal of the released power, the heating ends in melting of the specimen*). The breakdown voltage in thermal breakdown depends substantially on the ambient temperature, the conditions of heat removal, the frequency of the external field, the thickness of the specimen, the time during which the dielectric is under voltage, and also on the material, shape, and dimensions of the electrodes. The thermal form of breakdown has been studied experimentally in detail\({}^{4,5}\). There is also a rigorous theory of this phenomenon, due to Academician V. A. Fock\({}^{6}\), which explains its main regularities.

The chemical form of breakdown includes phenomena in which the passage of an electric current through a dielectric is accompanied by chemical transformations leading to loss of the electrical strength of the specimen\({}^{4,5}\). This form of breakdown occurs more often in organic dielectrics, but is also observed in inorganic materials (growth of dendrites).

By appropriate choice of material and external conditions one can prevent the occurrence of the thermal and chemical forms of breakdown of dielectrics. The possibility of the chemical form of breakdown will be eliminated if a material is selected that practically does not change during the passage of current. Thermal breakdown is eliminated by improving the conditions of heat removal, and also by changing to a pulsed mode of voltage application. In addition, it is necessary to eliminate the possibility of edge discharges, which may cause local irreversible and progressive changes.

When all these necessary conditions are fulfilled, breakdown of dielectrics nevertheless occurs, and at much higher electric fields \((E_{\mathrm{br}}\sim 10^{6}\ \mathrm{V/cm}\) instead of \(10^{4}—10^{5}\ \mathrm{V/cm})\).

*) In thermally unstable materials melting may also occur before the thermal equilibrium is disturbed (the so-called thermal breakdown of the second kind).

The corresponding form of breakdown is called electrical. In the case of electrical breakdown in a uniform field, the electric strength is a direct characteristic of the dielectric material, closely connected with its microscopic structure and practically independent of external conditions. The high value of the quantity \(E_{\mathrm{br}}\) corresponding to electrical breakdown, as well as its close connection with the internal properties of the dielectric material, make the problem of studying the electrical form of breakdown very important both for practice and for the theory of dielectrics.

As a result of numerous investigations of electrical breakdown, the following empirical regularities may be regarded as firmly established:

1. Breakdown is preceded by a more rapid than linear increase of current with voltage (Poole’s law). When a certain field strength \(E_{\mathrm{br}}\) is reached, the electric strength is destroyed, accompanied by a jump-like increase of current and usually by subsequent destruction of the dielectric.

2. Breakdown ends with the formation of a narrow channel, with a cross section of the order of tenths of a millimeter, passing through the entire thickness of the dielectric.

3. The breakdown field strength for various solid dielectrics (crystals, amorphous organic and inorganic bodies) is of one and the same order (\(\sim 10^6\) V/cm; see Table 1).

Table 1

Breakdown field strength of some materials in \(10^5\) V/cm
at room temperature\(^5\)

Substance	\(E_{\mathrm{br}}\)	Substance	\(E_{\mathrm{br}}\)
NaF	2.4	Quartz	4.7–6.7
NaCl	1.5	Mica	10–11
NaBr	1.0–0.8	Paraffin	2–2.4
NaJ	0.8	Alkali glass	2.0
KCl	1.0	Window glass	1.7
KBr	0.8–0.83	Bitumen	1.0
KJ	0.6	Shellac	3.5
RbCl	1.35	Resin	1.0–2.15
RbBr	0.5	Cellulose	1.2–3.2
RbJ	0.5	Nitrogen at a pressure of 90 atm	1.5

The breakdown field strength for compressed gases is of the same order of magnitude.\(^7\)

1. In a uniform electric field, the breakdown field strength does not depend on the thickness of the dielectric up to \(10^{-4}\)–\(10^{-5}\) cm.

With a further decrease in the thickness of the specimen, the breakdown field strength increases (Fig. 1)*.

1. The transition from tests under constant-voltage conditions to the pulsed regime does not change the character of the dependence of \(E_{\mathrm{br}}\) on the thickness of the specimen, nor the numerical values of \(E_{\mathrm{br}}\), down to a pulse duration of the order of \(10^{-7}\)—\(10^{-8}\) sec.

Fig. 1. Average value of \(E_{\mathrm{br}}\) for one of the types of mica.

1. For a pulse duration shorter than \(10^{-7}\)—\(10^{-8}\) sec, a number of investigators observed an increase in the breakdown field strength \(^{8,9,10}\).

2. In the case of a nonuniform electric field, the mean value \(E_{\mathrm{br}} = \dfrac{V}{d}\) does not remain constant in thick specimens, but decreases with increasing thickness of the dielectric.

3. The breakdown field strength does not depend on the medium in which the dielectric is placed, nor on the material of the electrodes \(^{4,11}\).

4. Experiments on the influence of the internal photoeffect on the breakdown of dielectrics showed that the breakdown field strength does not depend on the initial concentration of electrons \(^{12}\).

5. The phenomenon of incomplete breakdown \(^{4}\) and, in particular, the phenomenon of successive breakdown discovered by B. M. Vul \(^{13}\) show that the discharge begins at a definite point of the dielectric, from which it propagates through the thickness of the specimen with a finite velocity (of the order of \(10^{7}\)—\(10^{8}\) cm/sec).

6. There is considerable divergence in the experimental data relating to the temperature dependence of the breakdown field strength. Experiments by some authors \(^{14,15,16,17}\) and others showed that, in the case of electrical breakdown, the breakdown field strength does not depend on temperature. According to data of other authors \(^{18,19,20,21}\), for alkali-halide crystals there exists a temperature dependence of \(E_{\mathrm{br}}\) in the region of electrical breakdown (Figs. 2 and 3; curves 1, 2, and 3 in Fig. 2 were obtained for breakdown at constant voltage).

7. The increase in electrical conductivity with increasing field occurs at the expense of an increase in the number of free current carriers, and not at the expense of an increase in their mobility \(^{22}\). In this connection, as was po-

shown by Pruzhinina-Granovskaya for mica,^23 the overwhelming majority of current carriers at high fields are electrons.

1. There are data indicating that the breakdown strength of alkali-halide compounds is proportional to the lattice energy.^24

Fig. 2. Dependence of the breakdown strength of KBr on temperature; curves \(1^{18}\), \(2^{19}\), \(3^{21}\) were obtained experimentally, curve 4 was calculated according to Fröhlich’s theory.

Fig. 3. Dependence of the breakdown strength of NaCl on temperature.

Electrical breakdown of solid dielectrics is a complex set of physical (electrical, mechanical, thermal, and optical) phenomena. The beginning of the development of breakdown is the moment at which the dielectric loses its electrical strength.

The value \(E_{\mathrm{br}}\) corresponding to these conditions is a physical characteristic of the substance; at \(E = E_{\mathrm{br}}\) the dielectric passes into another physical state. In the case when various kinds of secondary factors are excluded (inhomogeneities of the field, of the material, etc.), which distort the development of breakdown, the breakdown field strength proves, within wide limits, not to depend on the experimental conditions.

The loss of electrical strength determines all the remaining processes which, taken together, form the complete picture of electrical breakdown (spark, destruction of the material, etc.). It is obvious that the first and fundamental task of the theory of electrical breakdown is to clarify the conditions under which electrical strength is lost.

All existing theories of the electrical breakdown of solid dielectrics can be divided into two principal groups. The first of these includes theories that ignore the stage of loss of electrical strength of the dielectric and identify breakdown directly with the mechanical destruction of the material.

The first theory of this type was Rogowski’s theory \(^{25}\), according to which breakdown is the mechanical destruction of the crystal lattice by electrostatic forces. It is obvious that, according to Rogowski, the condition for breakdown must be equality of \(E_{\mathrm{br}}\) to the internal electrostatic forces in the crystal lattice, i.e.

\[ E_{\mathrm{br}} \approx \frac{e}{\chi a^2} \approx 10^8 \ \mathrm{V/cm} \]

(\(e\) is the ion charge, \(a\) is the lattice constant), which is approximately 100 times greater than the experimental values of \(E_{\mathrm{br}}\).

It also follows from Rogowski’s ideas that the breakdown field strength should depend substantially on the dielectric permittivity of the material \(\left(E_{\mathrm{br}} \sim \frac{1}{\chi}\right)\), which is not observed in practice \(^{26}\). Moreover, Rogowski’s theory applies only to ionic structures, whereas, according to experimental data, the order of magnitude of \(E_{\mathrm{br}}\) (\(\sim 10^6\ \mathrm{V/cm}\)) is the same for the most diverse solid dielectrics (ionic and valence crystals, amorphous bodies, etc.; see Table I), as well as for compressed gases. This circumstance points to a common mechanism of breakdown in condensed media of different structure.

The sharp discrepancy between the theory and experiment forced Rogowski to suppose that the overestimated values of \(E_{\mathrm{br}}\) are obtained as a result of ignoring mechanical defects (cracks) in the dielectric. The presumed influence of cracks was reduced to their continuous growth as a consequence of destruction caused by ions entering the cracks and accelerated there by the electric field. Thus, instead of rupture of the lattice by electrostatic forces, the idea was advanced of impact destruction of the lattice—

by fast ions. For a crack length \(\lambda \approx 10^{-5}\ \mathrm{cm}\) and an ion binding energy in the lattice \(W = 2.13 \cdot 10^{-11}\ \mathrm{erg}\), Rogovskii obtained satisfactory values of the breakdown strength

\[ E_{\mathrm{br}}=\frac{W}{e\lambda}=1.34\cdot 10^{6}\ \mathrm{V/cm}, \tag{1} \]

where \(e\) is the ion charge.

This modified theory of Rogovskii’s also gives rise to serious objections (see \(^{4}\)). First, the gas filling the crack will break down at lower fields than the solid dielectric. But in that case the field in the cracks will be weak and appreciable acceleration of ions in them is impossible. Appreciable acceleration of ions is also impossible because of the considerable energy losses in collisions with the atoms of the gas filling the crack (the masses of ions and gas atoms are comparable in order of magnitude). Secondly, it has been experimentally proved that \(E_{\mathrm{br}}\) in dielectrics containing no cracks (glass) is also of the order of \(10^{6}\ \mathrm{V/cm}\), and not \(10^{8}\ \mathrm{V/cm}\). Thirdly, the choice of the value \(\lambda \sim 10^{-5}\ \mathrm{cm}\) is entirely arbitrary.

The first of the objections mentioned above was an attempt to remove by Gorovits \(^{27}\). He assumed that the elongation of cracks is caused by electrostatic forces produced by charges induced on their internal surface. The cause of the appearance of these charges, according to Gorovits, consists in the ionization of the gas in the crack, occurring before the breakdown of the solid dielectric. Gorovits’ calculation for rock salt gave:

\[ E_{\mathrm{br}}=\sqrt{\frac{16\sigma}{\lambda x}}=2\cdot 10^{6}\ \mathrm{V/cm}, \tag{2} \]

where \(\sigma\) is the coefficient of surface tension, equal to \(150\ \mathrm{g/sec^{2}}\), \(x\) is the dielectric constant, equal to 5, and \(\lambda\) is the crack length, equal to \(10^{-5}\ \mathrm{cm}\). Numerical agreement with experiment was achieved, however, by a special choice of the value of \(\lambda\). For \(\lambda < 10^{-5}\ \mathrm{cm}\) and in the absence of cracks Gorovits’ formula disagrees with experiment. Moreover, it gives an erroneous dependence of \(E_{\mathrm{br}}\) on \(x\) (see the experiments of B. M. Vul et al. \(^{26}\) with titanates, where \(E_{\mathrm{br}}=\mathrm{const}\) when \(x\) is varied by two orders of magnitude).

To this same group of theories, which take as the basis of the phenomenon of electrical breakdown the stage of dielectric destruction, should also be assigned the ideas of A. A. Vorob’ev and E. K. Zavadovskaya, developed in recent years (\(^{28,29,30,31}\)). The basis of these ideas is the assumption that ions or atoms are knocked out by direct electronic impact from their equilibrium positions in the solid. A. A. Vorob’ev and E. K. Zavadovskaya see experimental confirmation of their ideas in the fact of proportionality of \(E_{\mathrm{br}}\) to the lattice energy, established

for alkali-halide crystals. Determining \(E_{\mathrm{br}}\) from the condition of proportionality between the lattice energy and the kinetic energy of the electron acquired by it in the field—in one case over the empirically established dielectric-breakdown time, and in the other over the mean free time—A. A. Vorob’ev and E. K. Zavadovskaya arrive at two formulas*):

\[ \frac{e^2 E_{\mathrm{br}}^2}{m}\,\tau\,\Delta t = \alpha W, \tag{3} \]

\[ \frac{e^2 E_{\mathrm{br}}^2}{m}\,\tau^2 = \beta W, \tag{4} \]

where \(\tau\) is the mean free time of the electron, \(\Delta t \sim 10^{-8}\ \mathrm{sec}\) is the empirical dielectric-breakdown time, \(W\) is the lattice energy if the destruction is caused by the knocking out of ions, or the dissociation energy if, during breakdown, the lattice is destroyed into neutral atoms, \(e\) is the electron charge, \(m\) is the electron mass, and \(\alpha\) and \(\beta\) are proportionality coefficients.

Making certain assumptions about the dependence of \(\tau\) on \(E\)

\[ \left[ \text{in case (3) } \tau \sim \frac{1}{E},\quad \text{in case (4) } \tau \sim \frac{1}{\sqrt{E}} \right], \]

the authors obtain a linear dependence of \(E_{\mathrm{br}}\) on the lattice energy, in which they see confirmation of their hypothesis.

Without dwelling on a detailed consideration of a number of inconsistencies in the authors’ reasoning**), let us note the main defect of their theory. It is known that the energy \(\Delta \varepsilon\) transferred by an electron in a collision with an ion or atom of mass \(M\) is equal to only a small fraction of its total energy \(\varepsilon\), namely:

\[ \Delta \varepsilon \approx \frac{m}{M}\,\varepsilon. \]

*) A. A. Vorob’ev and E. K. Zavadovskaya do not indicate what relation these formulas have to one another, nor in which case one or the other of them should be applied.

**) For example, in paper \(^{29}\) \(E_{\mathrm{br}}\) is determined from the condition of a minimum of mobility, the calculation being carried out on the basis of the approximate formula

\[ \mu = \mathrm{const}\cdot \frac{e^{bE}}{E} \]

(\(\mu\) is mobility, \(b=\mathrm{const}\)), by means of which the author approximates the formula

\[ \mu = \mathrm{const}\cdot \frac{\operatorname{sh} bE}{E}. \]

But according to the latter formula \(\mu\) has no minimum at all.

THEORY OF ELECTRICAL BREAKDOWN

When the binding energy of an ion (or atom) in the lattice is of the order of 10 eV, direct destruction of the lattice by electron impact is possible provided that the electron possesses an energy of the order of \(10^5\) eV; for this, in a field of \(10^6\) V/cm, it would have to traverse without collisions a distance of \(10^6\)—\(10^7\) lattice constants.

The possibility of the appearance in a dielectric of electrons of such energies under ordinary breakdown fields is practically excluded, above all owing to the high ionization losses arising at considerably lower energies (20—30 eV). The latter circumstance indicates that, much earlier than the possibility arises of direct mechanical destruction of the lattice by electrons, breakdown of the dielectric must occur, caused by the formation of an electron avalanche as a result of impact ionization.

All the other theories, differing in many details, are similar in that the problem of electrical breakdown is treated in them as the problem of the destruction by the field of the electrical strength of the material. These theories do not consider the entire picture of the development of breakdown, but restrict themselves to its initial stage—the stage of loss of electrical strength.

The first theory of electrical breakdown as a phenomenon of avalanche-like increase in the conductivity of dielectrics was the theory of A. F. Ioffe \(^{32, 33}\), proposed in 1928. A. F. Ioffe was the first to express the idea that impact ionization takes place in a solid when a strong electric field is applied. At the same time, in accordance with the views of that time on the conductivity of dielectrics, A. F. Ioffe considered the current carriers to be free ions, and interpreted the increase in the electrical conductivity of dielectrics in strong fields as the result of ions being knocked out of lattice sites by free ions accelerated by the electric field.

In general outline, the proposed breakdown mechanism was as follows. For impact ionization to occur, a free ion must acquire in the field a certain minimum energy \(eP_0\), equal to the binding energy of the ion in the lattice. Owing to non-ionizing collisions, the ion loses part of the energy received from the field; therefore the potential difference \(P\) over the length \(\lambda\), over which it acquires the energy \(eP_0\), must be greater than \(P_0\). \(P\) is determined from the condition that the quantity \(e(P - P_0)\) be equal to the losses of ion energy along the path \(\lambda\). At a density \(n_0\) of primary free ions, produced in the dielectric itself by some non-impact mechanism, the concentration of ions at the second electrode after the application of the field is equal to

\[ n = n_0 \frac{P}{V}\left(e^{\frac{V}{P}} - 1\right), \tag{5} \]

where \(\dfrac{V}{P}=\dfrac{d}{\lambda}=Z\) is the total number of ionizations produced by one ion in the dielectric at a potential difference \(V\).

The condition for increasing the density of mobile ions in the dielectric is, obviously, the requirement

\[ Z=\frac{d}{\lambda}=\frac{V}{P}\gg 1, \]

where it was assumed that for \(Z\lessgtr 10\) \(\left(\dfrac{n}{n_0}\sim 10^3\right)\) ionization entails only deviations from Ohm’s law, and only at sufficiently large \(Z\) \((Z\gtrsim 20)\) does breakdown of the dielectric occur.

Assuming that \(\lambda\) is of the order of those distances \((10^{-4}\ \text{cm})\) over which a uniform drift of ions in an electric field is established (i.e., when \(v=\mu E\)), A. F. Ioffe comes to the conclusion that for \(Z\gg 1\) (thick samples) the condition for breakdown is the occurrence of impact ionization, since in this case [see (5)] the condition for obtaining the critical concentration of ions at the second electrode will certainly be fulfilled. It follows from this that at large \(d\) the breakdown voltage does not depend on the thickness of the sample.

For \(d\sim\lambda\), the occurrence of impact ionization is no longer sufficient to produce breakdown, since even if it is present the ion concentration at the second electrode may not reach the critical value. Therefore, the condition for breakdown of thin samples is the attainment of the critical concentration of ions at the second electrode, according to formula (5), which is equivalent to the condition \(V_{\text{br}}=\text{const}\) (for \(P\) weakly dependent on \(V\)). Consequently, in the breakdown of thin samples \((d\sim\lambda)\), an increase in

\[ E_{\text{br}}=\frac{V}{d} \]

will take place as \(d\) decreases.

The ideas expressed by A. F. Ioffe made it possible to explain qualitatively certain features of the phenomenon of electrical breakdown of dielectrics: the increase of current before breakdown, the independence of the breakdown voltage from the dielectric thickness in thick samples, and the rise of the breakdown voltage with decreasing \(d\) in the breakdown of thin samples, the independence of \(E_{\text{br}}\) from the surrounding medium and from the electrode material (ionization occurs in the dielectric itself).

However, further experimental and theoretical study of the electrical properties of solids led to the necessity of abandoning the notion of the ionic nature of the conductivity of dielectrics in strong fields. Because of the large losses in ion–ion or ion–atom collisions, the possibility of accelerating an ion in a solid to energies of the order of \(10\ \text{eV}\) is practically excluded\(^{34}\). Direct experiments showed that the increase of elec-

conductivity of ionic crystals at high fields occurs through an increase in the concentration of electrons, not ions[^23]. Theoretical consideration has shown[^35] that, for the ionic character of conduction, Ohm’s law should be obeyed up to breakdown fields, which is not observed experimentally. The breakdown time observed in experiment ($\lesssim 10^{-8}$ sec) does not agree with the notion of ionization by heavy particles.

Following A. F. Ioffe, the idea of impact ionization was developed by A. A. Smurov[^36],[^37]. What is essentially new in Smurov’s theory is the assumption of the electronic character of the electrical conductivity of dielectrics and of the ionization of dielectrics by electron impact in strong electric fields. According to Smurov, the breakdown phenomenon proceeds as follows. When an external electric field is applied, negative charges on the cathode and positive charges on the side of the dielectric adjacent to the cathode form a double electric layer, within which large potential gradients arise, capable of causing electrostatic ionization of the atoms of the dielectric. The free electrons thus formed will move under the influence of the external field from cathode to anode. At the same time, in the head part of the electron cloud (at distances on the order of $10^{-7}$—$10^{-8}$ cm from it), fields on the order of $10^8$ V/cm are formed, leading to additional electrostatic ionization. Along with electrostatic ionization, as the electron cloud moves, ionization by electron impact also occurs; moreover, inside the cloud, where the fields are small, exclusively impact ionization takes place. The positive space charge remaining after the electron cloud has gone to the anode causes a redistribution of the potential and the appearance of internal fields, which contribute to an even stronger increase in the concentration of free electrons in the dielectric as a result of electrostatic and impact ionization. This mechanism, Smurov concludes, must ultimately lead to those values of the currents that occur when the electric strength of a dielectric is disrupted.

The ideas expressed by A. F. Ioffe and A. A. Smurov (ionization by electron impact, electrostatic ionization, the influence of space charges on the breakdown of dielectrics) were preserved in subsequent theories of breakdown and determined the paths of their development. However, in their original form these concepts were insufficient for interpreting the phenomenon of electrical breakdown of solid dielectrics. They were qualitative in character and did not contain a breakdown criterion, i.e., they did not define the conditions under which a qualitative change in the properties of a dielectric occurs abruptly—the loss of electric strength.

The development of quantum mechanics contributed to the development of theories explaining the loss of electric strength of dielectrics ...

from the standpoint of quantum-mechanical ideas about electrons, atoms, and solids. In doing so, some authors (Zener, F. F. Vol’kenshtein, and others) made the idea of electrostatic ionization the basis of their theories, ignoring the mechanism of ionization by electron impact. Another direction in the theory of loss of electric strength (Hippel, Fröhlich, A. Akhiezer, and I. Lifshitz, and others) proceeds from the idea of impact ionization.

Zener \(^{38}\) assumed that the sharp increase in current when the electric strength of a dielectric is disrupted can be explained by the leakage of electrons from the valence band into the conduction band under the influence of a strong electric field (the tunnel effect). The probability of the tunnel effect in solids was calculated by a number of authors \(^{(34, 38, 39, 40)}\). The most rigorous is Houston’s calculation \(^{40}\), which gives the following expression for the probability of an electron passing through a potential barrier per unit time:

\[ p=\frac{eEa}{h}\frac{e^{-\alpha}}{(1-e^{-\alpha})^{2}}, \tag{6} \]

where

\[ \alpha=\frac{\pi^{2}mau^{2}}{eh^{2}E}, \]

where \(E\) is the field strength, \(a\) is the lattice constant, \(e\) is the electron charge, \(h\) is Planck’s constant, \(m\) is the electron mass, and \(u\) is the width of the forbidden interval.

Zener’s formula

\[ p=\frac{eEa}{h}e^{-\alpha} \tag{7} \]

differs little from formula (6) up to fields of the order of \(10^{9}\ \mathrm{V/cm}\). It should be pointed out that all the authors mentioned, in order to determine \(p\), solved, in one approximation or another, the Schrödinger equation

\[ -\frac{\hbar^{2}}{2m}\nabla^{2}\psi(\mathbf r,\mathbf E)+V(\mathbf r)\cdot\psi(\mathbf r,\mathbf E)-e\mathbf E\mathbf r\cdot\psi(\mathbf r,\mathbf E)=\varepsilon\cdot\psi(\mathbf r,\mathbf E), \tag{8} \]

assuming that the potential of the external field changes little over the length of the lattice constant, which is indeed the case.

Zener takes as the criterion for loss of electric strength a relative increase of \(p\) by a factor of 100. According to formula (7), this will occur when \(E\) changes from \(1\cdot 10^{6}\ \mathrm{V/cm}\) to \(1.13\cdot 10^{6}\ \mathrm{V/cm}\), if \(u=2\ \mathrm{eV}\), \(a=3\cdot 10^{-8}\ \mathrm{cm}\). Hence the conclusion is drawn that the theory agrees with experiment (the experimental value \(E_{\mathrm{pr}}\approx 10^{6}\ \mathrm{V/cm}\)). However, upon more detailed consideration, Zener’s theory gives rise to a number of serious objections, which essentially reduce to the following.

The criterion for the loss of electrical strength in Zener’s theory is chosen quite arbitrarily. Why the loss of electrical strength occurs for a relative increase of \(p\) by a factor of 100, corresponding to a change of \(E\) from \(1\cdot 10^6\ \mathrm{V/cm}\) to \(1.13\cdot 10^6\ \mathrm{V/cm}\) (for \(u=2\ \mathrm{eV}\), \(a=3\cdot 10^{-8}\ \mathrm{cm}\)), and not for an increase by a factor of \(10^{15}\), corresponding to a change of \(E\) from \(1\cdot 10^6\ \mathrm{V/cm}\) to \(1.1\cdot 10^6\ \mathrm{V/cm}\) [see (7)], remains unclear. This could be justified if, at \(E \approx 10^6\ \mathrm{V/cm}\), the increase in conductivity caused by the tunnel effect became comparable with the intrinsic conductivity of the dielectric, while at \(E < 10^6\ \mathrm{V/cm}\) it remained much smaller. But, as will be shown below, such a relation between the intrinsic and the additional conductivity does not occur. Thus, the condition for the loss of electrical strength in Zener’s theory does not follow from the theory itself, but is artificially introduced into it from outside. In essence, Zener’s theory contains no criterion for the loss of electrical strength.

Let us determine, according to Zener’s theory, what the current density \(i=en v_{\mathrm{dr}}\) is for a field strength equal to \(E\). The drift velocity of the electrons is \(v_{\mathrm{dr}}=\mu E\), and the mobility \(\mu\) in the general case\(^{41,42}\) depends on \(E\). The concentration of free electrons is determined from the condition of balance of the processes of electron transitions between the valence band and the conduction band:

\[ Np = An^2 . \tag{9} \]

Here \(N\) is the density of electrons in the valence band, \(n\) is the concentration of free electrons, equal to the concentration of holes, \(p\) is the probability of liberation of an electron, and \(A\) is the probability of recombination (\(p\) and \(A\) are referred to unit time).

From (9) it follows that

\[ n=\sqrt{\frac{Np}{A}} . \tag{10} \]

Substituting the value of \(p\) from (7), we find that\(^*\)

\[ i=e\sqrt{\frac{eNa}{hA}}\,\mu E^{3/2} e^{-\frac{\pi^2 m a u^2}{2 e h^2 E}} . \tag{11} \]

\(^*\) Zingerman\(^{43}\), expounding the theories of Zener and Ya. I. Frenkel, writes the current density in the form

\[ i=\frac{2e^2E}{ha^2}\,e^{-\alpha} \]

[see (6) and (7) in Zingerman]. It is easy to see that the formula given by Zingerman is erroneous; its right- and left-hand sides have different dimensions:

\[ [i]=\mathrm{A/cm^2}, \qquad \left[\frac{e^2E}{ha^2}\right]=\mathrm{A/cm^3}. \]

For \(N=10^{22}\ \mathrm{cm}^{-3}\), \(a=3\cdot 10^{-8}\ \mathrm{cm}\), \(u=9.6\ \mathrm{eV}\) (for NaCl), \(E=10^{6}\ \mathrm{V/cm}\), \(\mu=10\ \mathrm{cm}^{2}/\mathrm{V\cdot sec}\), \(A=S_{p}\cdot v\simeq 10^{-15}\cdot 10^{7}=10^{-8}\ \mathrm{cm}^{3}/\mathrm{sec}\) (\(S_{p}\) is the effective recombination cross section)

\[ i=4.3\cdot 10^{-187}\ \mathrm{A/cm}^{2}. \]

This current is vanishingly small not only in comparison with the breakdown currents observed at \(E\simeq 10^{6}\ \mathrm{V/cm}\) (\(i\simeq 10^{-9}\ \mathrm{A/cm}^{2}\) for NaCl\(^{44}\)), but also in comparison with the conduction currents in dielectrics at small fields (\(i\simeq 10^{-13}\ \mathrm{A/cm}^{2}\) at \(E\simeq 10^{4}\ \mathrm{V/cm}\)). Consequently, Zener’s theory does not agree with experiment and does not correspond to the true mechanism of electrical breakdown.

Attempts to improve Zener’s theory\(^{34,39,40,45,46,47}\) were not successful.

F. F. Volkenstein\(^{45}\) solved equation (8), taking into account the change in the potential of the external field over the extent of the lattice constant. He represented the wave function describing the behavior of an electron in a crystal when a field is applied in the form of a linear combination of the electronic wave functions in an isolated atom. The problem was solved by the perturbation method. As a result, F. F. Volkenstein came to the conclusion that, owing to the Stark effect, the allowed energy bands in a solid must broaden when an external field is applied. Taking into account the narrowing of the forbidden band caused by this effect and the corresponding increase in the transparency of the potential barrier, F. F. Volkenstein arrived at the following formula for the probability of passage of an electron from the filled band into the conduction band per unit time:

\[ p=\frac{eEa}{h}\,e^{-\alpha\left(1-\frac{\Delta u}{u}\right)^{2}}, \tag{12} \]

where \(\Delta u(E)\) is the change in the width of the forbidden region, depending on the magnitude of the field strength; the remaining notation is the same as in formula (6). F. F. Volkenstein determines the dependence \(\Delta u(E)\) on the basis of data on the Stark effect in isolated atoms. The criterion for the violation of electrical strength is the fulfillment of the equality

\[ \Delta u(E_{\mathrm{br}})=u, \tag{13} \]

which expresses the condition for the merging of the valence and free bands.

According to F. F. Volkenstein’s estimates\(^{45}\), the value of \(E_{\mathrm{br}}\) under this assumption lies within the limits from \(10^{5}\ \mathrm{V/cm}\) to \(10^{7}\ \mathrm{V/cm}\). One may think, however, that these estimates are much too low and that the assumed effect could occur only at considerably larger fields.

It follows from theoretical considerations that no appreciable broadening of the energy bands in a solid will occur at fields of the order of \(10^{6}\ \mathrm{V/cm}\) (breakdown fields), since each atom of the dielectric, even before the application of an external field, is already subject to the action of stronger fields (\(\sim 10^{8}\ \mathrm{V/cm}\)) from the surrounding atoms.

This circumstance was first pointed out by A. A. Vorob'ev^48. The author of the theory himself also showed in later works^49 that for fields of \(10^6\ \text{V/cm}\), \(\Delta u = 10^{-1}—10^{-2}\ \text{eV}\), whereas the value of \(u\) for alkali-halide crystals is \(\sim 10\ \text{eV}\).

A. A. Vorob'ev's experiments^48 on determining the long-wavelength limit of ultraviolet absorption in alkali-halide crystals placed in a strong field confirmed the validity of the assertion that the width of the forbidden band is independent of the strength of the external field at \(E \simeq 10^6\ \text{V/cm}\). Attempts to relate the breakdown of the electrical strength of dielectrics to electrode effects (leakage of electrons from the cathode into the dielectric as a result of the tunnel effect; see Fowler^47, F. F. Volkenshtein^46) likewise were not successful. The most serious objection to these theories is the fact that the breakdown voltage is independent of the electrode material^11.

Ya. I. Frenkel attempted to explain the phenomenon of the breakdown of the electrical strength of dielectrics and electronic semiconductors by means of a combination of the mechanism of thermal ionization of atoms and the action of an external electric field^50. In this case the field was assigned the role of an auxiliary factor facilitating thermal ionization. Noting the fact that in amorphous dielectrics the breakdown of electrical strength occurs at the same field strengths as in crystals, Ya. I. Frenkel concludes that the specific conditions of electron motion in the crystal, which arise because of the periodicity of the lattice field, play a secondary role in the phenomenon of breakdown. Hence follows the author's assertion that a crystal and an amorphous body may be treated as an aggregate of individual atoms placed in a continuous medium with dielectric constant \(\varkappa\).

Fig. 4. On Ya. I. Frenkel's theory.

Fig. 4 shows the course of the potential energy of an electron as a func-

... (Fig. 5), it is seen that the losses reach a maximum value at a quite definite energy \(\varepsilon_m\). According to the calculations of Seeger and Teller\({}^{35}\), and also Callen\({}^{56}\),

\[ \varepsilon_m \simeq 4\hbar\omega . \]

Since for the optical branch of lattice vibrations the frequencies lie in a narrow interval near \(\omega_{\max}\), and the damping on acoustic vibrations in the case of ionic crystals is insignificant\({}^{57}\) (for small \(\varepsilon\)), the largest energy losses will occur for electrons with energy

\[ \varepsilon_m \simeq 4\hbar\omega_{\max}. \]

Fig. 5. Energy losses of an electron over \(1\ \mathrm{cm}\) of path as a function of its energy.

If, at some value of \(E\), the energy losses of electrons with energy \(\varepsilon_m\) become smaller than, or at least equal to, the energy received by these electrons from the field, then for other \(\varepsilon\) this condition will be all the more satisfied. Consequently, in such a field all electrons will be accelerated, a stationary regime of electrical conduction is impossible, and, as a result of the acceleration of the electrons, impact ionization of the atoms of the dielectric will occur. In this process an electron avalanche is formed, moving from the cathode to the anode. It leaves behind a positive space charge, which distorts the field, weakens the structure, and promotes an increase in the intensity of impact ionization. Ultimately this mechanism will lead to the destruction of the dielectric. No further explanations concerning the final stage of breakdown are given in Hippel’s works.

Such is Hippel’s point of view on the mechanism of electrical breakdown of solid dielectrics. We shall return to a critical analysis of the ideas and results of this theory after considering the works devoted to its quantitative development.

Most rigorously this was done by Callen^56. The theories of Gippel and Callen apply directly to ionic crystals of the alkali-halide type, as the most typical and well studied experimentally and theoretically solid dielectrics. However, the essence of the matter remains exactly the same for other types of solid insulators.

The essence of Callen’s theory is as follows.

According to the dynamical theory of lattices, the normal vibrations of diatomic crystals can be represented in the form of plane waves whose wave vector \(\mathbf{L}\), for which \(\left(|\mathbf{L}|=\dfrac{1}{\lambda}\right)\), assumes \(N\) discrete values, where \(N\) is the number of elementary cells in the crystal^57. To each value of \(\mathbf{L}\) there correspond six kinds of vibrations, three of which are optical and have a relatively high frequency, and three acoustic, having a relatively low frequency. In the limiting case of long waves, neighboring ions of opposite sign in optical vibrations undergo displacements with opposite phases, and in acoustic vibrations with identical phases. One of the three kinds of vibrations of each type corresponds to longitudinal waves, and the other two to transverse waves. It is also known that in diatomic crystals of the ionic type, electrons experience the greatest scattering on longitudinal optical vibrations. Taking this circumstance into account and treating an electron located in the conduction band in the unperturbed state as a free particle with effective mass \(m^*\), Callen, using Fröhlich’s results^58, calculates for a conduction electron the probability of absorption

\[ \Phi_L^a = \left(\frac{e e^*}{a^3 L}\right)^2 \frac{1} {N M \hbar \omega \left(e^{\frac{\hbar \omega}{kT}}-1\right)} \frac{\sin \xi t}{\xi} \delta(\mathbf{K}-\mathbf{K'} \pm \mathbf{L}) \tag{18} \]

and the probability of emission

\[ \Phi_L^e = \left(\frac{e e^*}{a^3 L}\right)^2 \frac{e^{\frac{\hbar \omega}{kT}}} {N M \hbar \omega \left(e^{\frac{\hbar \omega}{kT}}-1\right)} \frac{\sin \xi t}{\xi} \delta(\mathbf{K}-\mathbf{K'} \pm \mathbf{L}) \tag{19} \]

of a phonon having energy \(\hbar\omega\) and momentum \(h\mathbf{L}\). In formulas (18) and (19), \(M\) is the reduced mass of the ions constituting the lattice, \(a\) is the lattice constant, \(\omega\) is the frequency of the longitudinal long-wavelength optical vibrations, \(e^*\) is the effective charge of the ion, differing from the true charge \(e\) owing to the influence, on the electronic polarization of the crystal, of the internal field produced as a result of displacements of the ions in long-wavelength longitudinal optical vibrations; \(\hbar\xi=\varepsilon' - \varepsilon \pm \hbar\omega\), where \(\varepsilon\) and \(\varepsilon'\) are the electron energies respectively before and after the interaction. If \(\mathbf{K}\) and \(\mathbf{K'}\) are the wave—

if the electron’s numbers before and after the interaction, respectively, then from the laws of conservation of energy and momentum

\[ \varepsilon'=\varepsilon \pm \hbar \omega + \hbar \xi, \tag{20} \]

\[ \mathbf{K}'=\mathbf{K}+\mathbf{L} \tag{21} \]

one obtains the following expression for the cosine of the angle \(\alpha\) between the vectors \(\mathbf{K}\) and \(\mathbf{L}\):

\[ \cos\alpha=-\frac{L}{2K}\pm \frac{m^*\omega}{2\pi\hbar KL}+\frac{\xi}{2\pi vL} \begin{cases} + \text{ for absorption,}\\ - \text{ for emission.} \end{cases} \tag{22} \]

Taking into account that \(|\cos\alpha|\leqslant 1\), we obtain limiting values for \(L\) for a given \(K\), namely:

\[ \begin{aligned} L_{\max}^{a}&=K\left(1+\sqrt{1+\frac{\hbar\omega}{\varepsilon}}\right)\\ L_{\min}^{a}&=K\left(-1+\sqrt{1+\frac{\hbar\omega}{\varepsilon}}\right) \end{aligned} \left\{ \begin{array}{l} \text{in the case of absorption} \end{array} \right. \tag{23} \]

\[ \begin{aligned} L_{\max}^{e}&=K\left(1+\sqrt{1-\frac{\hbar\omega}{\varepsilon}}\right),\\ L_{\min}^{e}&=K\left(1-\sqrt{1-\frac{\hbar\omega}{\varepsilon}}\right) \end{aligned} \left\{ \begin{array}{l} \text{in the case}\\ \text{of emission.} \end{array} \right. \]

From relations (23) it is seen that slow electrons interact with longer-wavelength oscillations than fast ones.

Let us derive formulas determining the effective charge of the ion \(e^*\) and the frequency \(\omega\) of long-wavelength longitudinal optical oscillations through experimentally observed quantities. Let \(U_{+L}\) and \(U_{-L}\) be the displacements of positive and negative ions in a long-wavelength longitudinal optical oscillation with wave number \(\mathbf{L}\). Then the relative displacement of the ions and the polarization associated with it, respectively, have the values

\[ \mathbf{X}_L=\mathbf{U}_{+L}-\mathbf{U}_{-L}, \tag{24} \]

\[ \mathbf{P}_L=\frac{e^*}{2a^3}\mathbf{X}_L. \tag{25} \]

The strength of the internal field is equal to

\[ \mathbf{E}=-4\pi\mathbf{P}_L=-\frac{2\pi e^*}{a^3}\mathbf{X}_L, \tag{26} \]

and the corresponding density of electrostatic energy is

\[ w'_L=\frac{\chi_0}{8\pi}\left(\frac{2\pi e^*}{a^3}\mathbf{X}_L\right)^2. \tag{27} \]

Here \(\chi_0\) is the square of the refractive index of the crystal.

Since the internal electric field in the lattice arises only in optical longitudinal oscillations, the volume density

the energy in this case is greater by the amount \(w'_L\) than in the case of transverse vibrations, i.e.

\[ \frac{1}{2}\frac{M\omega^2}{2a^3}\mathbf{X}_L^2 = \frac{1}{2}\frac{M\omega_t^2}{2a^3}\mathbf{X}_L^2 + \frac{\chi_0}{8\pi} \left( \frac{2\pi e^*}{a^3}\mathbf{X}_L \right)^2 . \tag{28} \]

From relation (28) we find the connection between the frequencies \(\omega\) and \(\omega_t\):

\[ \omega^2=\omega_t^2+\frac{2\pi\chi_0 e^{*2}}{Ma^3}. \tag{29} \]

To obtain the second relation connecting \(e^*\) and \(\omega\) with experimentally observed quantities, let us consider the behavior of the crystal in an external field \(\mathbf{D}\). The values of the polarization, field strength, and energy density in the crystal are expressed by the formulas:

\[ \mathbf{P}=\frac{e^*}{2a^3}\mathbf{X}+\frac{\chi_0-1}{4\pi\chi_0}\mathbf{D}, \tag{30} \]

\[ \mathbf{E}=-\frac{2\pi e^*}{a^3}\mathbf{X}+\frac{\mathbf{D}}{\chi_0}, \tag{31} \]

\[ w'=\frac{\chi_0}{8\pi} \left( \frac{\mathbf{D}}{\chi_0} - \frac{2\pi e^*}{a^3}\mathbf{X} \right)^2, \tag{32} \]

\[ w=\frac{M}{4a^3}\omega_t^2\mathbf{X}^2 + \frac{\chi_0}{8\pi} \left( \frac{\mathbf{D}}{\chi_0} - \frac{2\pi e^*}{a^3}\mathbf{X} \right)^2 . \tag{33} \]

Finding from the condition of the minimum of \(w\) the equilibrium value of the displacement \(\mathbf{X}\) and substituting it into (30), we obtain the equilibrium value of the polarization in the external field \(\mathbf{D}\):

\[ \mathbf{P} = \frac{1}{2} \frac{e^{*2}}{Ma^3\omega_t^2+2\pi\chi_0 e^{*2}}\mathbf{D} + \frac{\chi_0-1}{4\pi\chi_0}\mathbf{D}. \tag{34} \]

But

\[ \mathbf{D}=\mathbf{E}+4\pi\mathbf{P} = \frac{\mathbf{D}}{\chi}+4\pi\mathbf{P} \tag{35} \]

and, consequently,

\[ \mathbf{P}=\frac{\chi-1}{4\pi\chi}\mathbf{D}. \tag{36} \]

From (34) and (36) it follows that

\[ \omega_t^2=\frac{2e^{*2}}{Ma^3}\frac{\chi_0^2}{\chi-\chi_0}. \tag{37} \]

Combining (29) and (37), we obtain:

\[ \omega^2=\frac{2\pi e^{*2}}{Ma^3}\frac{\chi_0\chi}{\chi-\chi_0}, \tag{38} \]

or

\[ \frac{\omega^2}{\omega_t^2}=\frac{\chi}{\chi_0}. \tag{39} \]

From (38) and (39), through \(\omega\), \(x_0\), and \(x\), one can easily express \(\omega\) and \(e^*\). Knowing \(\omega\) and \(e^*\), Kallen is able to write expressions for the energy losses and for the energy acquired by the electron from the field in terms of experimentally measurable quantities, and then to compare the theory with experiment. Using expressions (18) and (19), Kallen calculates, for an electron with energy \(\varepsilon\), the mean value of the energy losses per unit time due to interaction with lattice vibrations:

\[ \begin{aligned} B &= C\left(\frac{\hbar \omega}{\varepsilon}\right)^{1/2} \left( \frac{e^{\frac{\hbar \omega}{kT}}}{e^{\frac{\hbar \omega}{kT}}-1} \int_{L_{\min}^{e}}^{L_{\max}^{e}} \frac{dL}{L} - \frac{1}{e^{\frac{\hbar \omega}{kT}}-1} \int_{L_{\min}^{a}}^{L_{\max}^{a}} \frac{dL}{L} \right) \\ &= C\left(\frac{\hbar \omega}{\varepsilon}\right)^{1/2} \left( \frac{e^{\frac{\hbar \omega}{kT}}}{e^{\frac{\hbar \omega}{kT}}-1} \ln \frac{1+\sqrt{1-\frac{\hbar \omega}{\varepsilon}}} {1-\sqrt{1-\frac{\hbar \omega}{\varepsilon}}} - \frac{1}{e^{\frac{\hbar \omega}{kT}}-1} \ln \frac{\sqrt{1+\frac{\hbar \omega}{\varepsilon}}+1} {\sqrt{1+\frac{\hbar \omega}{\varepsilon}}-1} \right), \end{aligned} \tag{40} \]

where

\[ C=\frac{\pi \sqrt{2m^*}\,(ee^*)^2}{Ma^3\sqrt{\hbar\omega}} . \tag{41} \]

The energy acquired by the electron per unit time from the field is equal to

\[ A=\frac{e^2E^2}{m^*}\,\tau, \tag{42} \]

where \(\tau\) is the mean free time for an electron with energy \(\varepsilon\); it can also be calculated by means of formulas (18) and (19):

\[ \begin{aligned} \frac{1}{\tau} &= \frac{\pi(ee^*)^2}{Ma^3K^2v\,\hbar\omega} \left( \frac{1}{e^{\frac{\hbar\omega}{kT}}-1} \int_{L_{\min}^{a}}^{L_{\max}^{a}} L\,dL + \frac{e^{\frac{\hbar\omega}{kT}}}{e^{\frac{\hbar\omega}{kT}}-1} \int_{L_{\min}^{e}}^{L_{\max}^{e}} L\,dL \right) \\ &\quad+ \frac{m^*(ee^*)^2}{2\pi Ma^3\hbar^2K^2v} \left( \frac{e^{\frac{\hbar\omega}{kT}}}{e^{\frac{\hbar\omega}{kT}}-1} \int_{L_{\min}^{e}}^{L_{\max}^{e}} \frac{dL}{L} - \frac{1}{e^{\frac{\hbar\omega}{kT}}-1} \int_{L_{\min}^{a}}^{L_{\max}^{a}} \frac{dL}{L} \right) = \end{aligned} \]

$$ = \frac{B}{2\varepsilon}+\frac{C}{\hbar\omega}\left(\frac{\hbar\omega}{\varepsilon}\right)^{1/2}\cdot \left( \frac{e^{\frac{\hbar\omega}{kT}}}{e^{\frac{\hbar\omega}{kT}}-1} \sqrt{1-\frac{\hbar\omega}{\varepsilon}} + \frac{1}{e^{\frac{\hbar\omega}{kT}}-1} \sqrt{1+\frac{\hbar\omega}{\varepsilon}} \right). \tag{43} $$

The field strength \(E\) at which, for electrons with the given energy \(\varepsilon\), the energy losses for excitation of lattice vibrations are on the average equal to the energy acquired by these electrons from the field is determined from the condition

$$ A=B. \tag{44} $$

Substituting into (44) the values of \(A\) and \(B\) calculated above, Callen finds that

$$ E=E_0\left(\frac{\hbar\omega B}{C^2\tau}\right)^{1/4}, \tag{45} $$

where

$$ E_0=\frac{\sqrt{2}\,\pi m^* e e^{*\,2}}{M a^3\hbar\omega}. \tag{46} $$

As Hippel did, Callen assumes that the loss of electrical strength occurs when condition (44) is fulfilled for electrons with energy \(\varepsilon_m\), whose braking by lattice vibrations is maximal. Consequently, according to Callen,

$$ E_{\mathrm{br}}=E_0\left(\frac{\hbar\omega\cdot B_{\max}}{C^2\tau}\right)^{1/4}. \tag{47} $$

At \(T=0\), \(E_{\mathrm{br}}=E_0\). Consequently, \(E_0\) is the breakdown field strength at absolute zero.

Formulas (33) and (39) make it possible to write the expression for \(E_0\) in terms of \(\omega_t\), \(x\), and \(x_0\):

$$ E_0=\frac{2^{3/4}\pi^2 m^* e}{h^2}\,\hbar\omega_t\, \frac{x-x_0}{(x x_0^3)^{1/2}}. \tag{48} $$

The values of \(E_{\mathrm{br}}\) calculated by Callen from formulas (47) and (48) are in satisfactory agreement with experiment (Table II) for the majority of alkali-halide compounds. However, this agreement is destroyed and significant discrepancies with experiment arise if Callen’s calculation is considered more rigorously and certain essential circumstances not taken into account by Callen are included.

The results presented above are valid under two assumptions: 1) \(\omega\) and \(e^*\) are related to \(\omega_t\), \(x\), and \(x_0\) by formulas (38) and (39);

Table II

Theoretical (according to Callen) and experimental values of the breakdown field strength for ionic crystals
in V/cm · \(10^{-6}\)

Crystals	\(x\)	\(x_0\)	\(\hbar \omega_t\), in eV	\(a\), in Å	\(\hbar \omega\), in eV	\(e^* \cdot 10^{10}\)	\(E_0\), \(m^* = m\) (theory)	\(E_{\mathrm{br}}\), \(m^* = m\), \(T = 288^\circ\) K (theory)	\(E_{\mathrm{br}}\), \(T = 288^\circ\) K (experiment)	\(\dfrac{E_{\mathrm{br}}^{\mathrm{theor}}}{E_{\mathrm{br}}^{\mathrm{exper}}}\), \(T = 288^\circ\) K
LiF	9,27	1,92	0,072	2,07	0,158	4,53	8,75	8,80	3,1	2,8
NaF	6,0	1,74	0,033	2,31	0,061	2,59	3,34	3,57	2,4	1,5
NaCl	5,62	2,25	0,024	2,81	0,038	2,09	1,35	1,57	1,5	1,1
NaBr	5,99	2,62	0,018	2,97	0,027	1,65	0,78	0,98	0,81	1,2
KF	6,05	1,85	0,03	2,66	0,054	3,25	2,72	2,96	1,9	1,6
KCl	4,68	2,13	0,019	3,14	0,028	2,07	0,965	1,20	1,0	1,2
KBr	4,78	2,33	0,015	3,29	0,021	1,86	0,63	0,85	0,70	1,2
KJ	4,94	2,69	0,013	3,53	0,018	1,82	0,40	0,57	0,57	1,0
RbCl	5,0	2,19	0,017	3,27	0,026	2,33	0,88	1,12	0,83	1,3
RbBr	5,0	2,33	0,011	3,42	0,016	1,99	0,49	0,73	0,63	1,2
RbJ	5,0	2,63	0,01	3,66	0,014	1,86	0,33	0,53	0,49	1,1

2) \(\omega\) does not depend on \(L\). The second of these assumptions allows Callen to take outside the integral sign expressions depending on \(\omega\) (see formulas (40) and (43)).

The assumption of the independence of the frequency from the wave number may be considered justified, since the frequency interval of optical vibrations is narrow (see \(^{37}\)). On the contrary, the application of formulas (38) and (39) at those wavelengths in the lattice at which electron scattering actually occurs raises objections. Indeed, only for long longitudinal waves \((\lambda \gg a)\) does the electric field caused by the intrinsic vibrations of the lattice not vanish upon averaging over a volume containing many elementary cells. But according to Callen’s calculations

\[ \varepsilon_m = 4\hbar\omega, \tag{49} \]

which for NaCl, for example, amounts to \(2.44\cdot 10^{-13}\) erg \(= 0.152\) ev (see (38) and Table II). The wave number of an electron at such an energy is

\[ K = 3.18\cdot 10^6\ \text{cm}^{-1}, \]

and

\[ L_{\max}^{a} \simeq L_{\max}^{e} = 6.36\cdot 10^6\ \text{cm}^{-1} \]

(see (23)); correspondingly

\[ \lambda = 1.57\cdot 10^{-7}\ \text{cm}, \]

i.e. \(5\)—\(6\) lattice constants (for NaCl). Consequently, for an electron with energy \(\varepsilon_m\) (see (42), (43)) the energy gain is determined mainly by interaction with such longitudinal polarization waves for which the condition \(\lambda \gg a\) is not satisfied.

In the case of short waves the volume-averaged value of \(\mathbf{X}_L\) is equal to zero. Therefore (see (26) and (27))

\[ \mathbf{E}_L = 0,\quad w'_L = 0 \]

and from (28), instead of (29), it follows that \(\omega=\omega_t\). In addition, in the case of short waves there are no effects leading to the appearance of an effective ion charge \(e^*\) instead of the true charge \(e\).

For

\[ \lambda \simeq (5\div 6)a \]

one should expect that the values of \(\omega\) and \(e^*\) will rather satisfy the equalities

\[ \omega=\omega_t,\qquad e^*=e, \tag{50} \]

than formulas (38), (39).

Dividing the region of integration in (40) and (43) into two intervals \((L_{\min}, L')\) and \((L', L_{\max})\), where \(L'=\dfrac{1}{10a}\), and assuming that only polarization waves lead to the appearance of the internal field

oscillations with wavelength \(\lambda \gg 10a\), we obtain:

\[ \begin{aligned} B &= \frac{\pi \sqrt{2m^{*}}}{Ma^{3}}\, \frac{e^{4}}{\sqrt{\hbar\omega_{t}}} \left(\frac{\hbar\omega_{t}}{\varepsilon}\right)^{1/2} \left[ \frac{e^{\frac{\hbar\omega_{t}}{kT}}} {e^{\frac{\hbar\omega_{t}}{kT}}-1} \ln \frac{L^{e}_{\max}}{L'} - \frac{1} {e^{\frac{\hbar\omega_{t}}{kT}}-1} \ln \frac{L^{a}_{\max}}{L'} \right] \\ &\quad + \frac{\pi \sqrt{2m^{*}}}{Ma^{3}}\, \frac{(ee^{*})^{2}}{\sqrt{\hbar\omega}} \left(\frac{\hbar\omega}{\varepsilon}\right)^{1/2} \times \left[ \frac{e^{\frac{\hbar\omega}{kT}}} {e^{\frac{\hbar\omega}{kT}}-1} \ln \frac{L'}{L^{e}_{\min}} - \frac{1} {e^{\frac{\hbar\omega}{kT}}-1} \ln \frac{L'}{L^{a}_{\min}} \right], \end{aligned} \tag{51} \]

\[ \frac{1}{\tau} = \frac{B}{2\varepsilon} + \frac{1}{2}\, \frac{\pi \sqrt{2m^{*}}\, e^{4}} {Ma^{3}\hbar\omega_{t}\sqrt{\varepsilon}} \left[ \frac{e^{\frac{\hbar\omega_{t}}{kT}}} {e^{\frac{\hbar\omega_{t}}{kT}}-1} \left(L^{e}_{\max}\right)^{2} + \frac{1} {e^{\frac{\hbar\omega_{t}}{kT}}-1} \left(L^{a}_{\max}\right)^{2} \right]. \tag{52} \]

Applying the same breakdown criterion as Callen,

\[ \left( B_{\max}=\frac{e^{2}E_{\mathrm{br}}^{2}}{m^{*}}\,\tau_{\min} \right), \]

we find that the theoretical values of \(E_{\mathrm{br}}\) for alkali-halide crystals are several times (5–6 times) higher than the experimental ones. Thus, Callen’s theory, when the calculation is carried out consistently, gives overestimated values of \(E_{\mathrm{br}}\).

In conclusion, let us note that Callen’s theory implies the temperature dependence of \(E_{\mathrm{br}}\) shown in Fig. 6.

The theory of the disruption of electrical strength, based on Hippel’s ideas, was developed before Callen by Zener and Teller\(^{55}\), and also by Fröhlich\(^{54}\). These theories add nothing new to Callen’s theory and were developed less rigorously. Therefore we shall not consider them in detail.

Fröhlich\(^{58,59,60}\), proceeding essentially from the same assumptions as Hippel, believes, however, in contrast to Hippel

and Callen, that the breakdown of electrical strength is determined by the behavior not of slow, but of fast electrons.

Fig. 6. Temperature dependence of \(E_{\mathrm{pr}}\) for alkali-halide crystals according to Callen’s theory.

For the energy \(A\), obtained by an electron from the external field, and the energy \(B\), given up by the electron to the lattice per unit time, Fröhlich obtains the following expressions:

\[ A=\frac{e^{2}E^{2}}{m^{*}}\tau = \frac{16\sqrt{2}\,Ma^{7}\omega_{t}\varepsilon^{3/2}E^{2}} {2^{1/3}\chi^{3}m^{*1/2}e^{2}\hbar} \int_{0}^{L^{0}_{\max}} L\,dL = \]

\[ = \frac{4\sqrt{2}}{2^{2/3}\chi^{3}}\, \frac{Ma^{5}\omega_{t}\varepsilon^{3/2}E^{2}} {m^{*1/2}e^{2}\hbar} \left(1+\frac{2}{e^{\hbar\omega_{t}/kT}-1}\right)^{-1}, \tag{53} \]

\[ B= \frac{2^{1/2}\pi e^{4}m^{*1/2}} {Ma^{3}\varepsilon^{1/2}} \int_{L_{\min}}^{L^{0}_{\max}}\frac{dL}{L} = \frac{2^{1/2}\pi e^{4}m^{*1/2}} {Ma^{3}\varepsilon^{1/2}} \ln \frac{2\pi(2\varepsilon)^{1/2}} {2^{2/3}m^{*1/2}\omega_{t}a}, \tag{54} \]

where

\[ L_{\min}=\frac{m^{*}\omega_{t}}{2\pi hK}. \tag{55} \]

is determined from (23), while

\[ L^0_{\max}=\frac{1}{2^{1/3}a} \tag{56} \]

is the wave number corresponding to the oscillation with the smallest wavelength for the given lattice; the remaining notation is as before.

The energy \(\varepsilon'\) of those electrons for which, in the field \(E\), the losses and the gain in energy are balanced on the average is determined from the condition \(A=B\), and from (53) and (54) it follows that

\[ \varepsilon'= \frac{2^{1/3}\pi e^3}{4Ma^4} \left[ \frac{2\pi m^*h}{\omega_t} \ln \frac{2\pi(2\varepsilon)^{1/2}} {2^{1/3}m^{*\,1/2}\omega_t a} \right]^{1/2} \times \]

\[ \times \left( 1+ \frac{2}{ \dfrac{\hbar\omega_t^*}{e^{\hbar\omega_t^*/kT}-1} } \right)^{1/2} \frac{1}{E} = \frac{C}{E}. \tag{57} \]

The further course of the argument is as follows. If \(\varepsilon'>I\), where \(I\) is the ionization potential, then the field will accelerate only those electrons whose energy \(\varepsilon>\varepsilon'>I\).

After ionization the electrons will have energy \(\varepsilon_2<I<\varepsilon'\)* and, consequently, will be slowed down, giving up entirely to the lattice the energy which they received from the field before ionization. Hence, Fröhlich concludes, in this case a stationary state obtains and breakdown of the electrical strength cannot occur. If, however, \(\varepsilon'<I\), then, along with electrons whose energy exceeds the ionization potential, electrons with energy \(\varepsilon\) satisfying the inequality \(\varepsilon'<\varepsilon<I\) will also be accelerated. These latter electrons, accelerated by the field up to the ionization energy and beyond, cause ionization of the dielectric; after this their energy becomes negligible, and they are braked by the lattice. Now the energy \(I-\varepsilon\), obtained by the electron from the field, is no longer transferred to the lattice, but is spent on ionization. A portion of the slow electrons is again, as a result of fluctuations, thrown into the energy region \(\varepsilon'<\varepsilon<I^{59}\), which leads to new acts of ionization, and so on. As a consequence, a stationary state for \(\varepsilon'<I\) is impossible, and loss of electrical strength must occur.

Starting from these ideas, Fröhlich considers the condition for breakdown to be the equality

\[ A=B \quad \text{for } \varepsilon'=I, \tag{58} \]

\[ \text{*) According to Fröhlich}^{59}\text{ ionization occurs at } \varepsilon \text{ only slightly exceeding } I. \]

whence (see (53) and (54)) it follows that

\[ E_{\mathrm{br}}=\frac{2^{4/3}\pi e^3}{4Ma^4l} \left[ \frac{2\pi m^*h}{\omega_t} \ln \frac{2\pi(2l)^{1/2}}{2^{2/3}m^{*\,1/2}\omega_t a} \right]^{1/2} \left(1+\frac{2}{e^{\hbar\omega_t/kT}-1}\right)^{1/2}. \tag{59} \]

A comparison of the experimental values of \(E_{\mathrm{br}}\) with the theoretical values according to Fröhlich is given in Table III.

Table III

Theoretical (according to Fröhlich) and experimental values of breakdown fields in crystals of the alkali-halide type (in \(10^5\ \mathrm{V/cm}\))

Crystals	\(E_{\mathrm{br}}\) theor. at \(T=0^\circ\mathrm{K}\)	\(E_{\mathrm{br}}\) theor. at \(T=300^\circ\mathrm{K}\)	\(E_{\mathrm{br}}\) exp. at \(T=300^\circ\mathrm{K}\)
NaCl	6.9	10.7	15
NaBr	6.1	10.6	8.1
NaJ	4.9	9.3	8
KCl	3.8	6.6	10
KBr	3.0	5.6	7
KJ	2.5	5.1	6
RbCl	2.7	4.8	8.3
RbBr	2.0	4.2	6
RbJ	1.4	3.1	5

The temperature dependence of \(E_{\mathrm{br}}\) for KBr, calculated from (59), was given in Fig. 2 (p. 139). From Fig. 2 it is seen that the temperature dependence of \(E_{\mathrm{br}}\) predicted by Fröhlich’s theory is in qualitative agreement with the experimental data of some authors\({}^{19}\), but there is no quantitative agreement. As was noted above, in the experiments of a number of other authors even the form of the temperature dependence is not confirmed.

Let us now compare the conditions for the loss of electrical strength in the theories of Hippel–Callen and Fröhlich. According to Hippel and Callen, the loss of electrical strength occurs when \(A=B\) at \(\varepsilon=\varepsilon_m\); according to Fröhlich, when \(A=B\) at \(\varepsilon=l\). Since

\[ A=\frac{e^2E^2}{m^*}\tau, \]

from the condition \(A=B\) it follows that

\[ E_{\mathrm{br}}=\frac{1}{e}\sqrt{\frac{B\cdot m^*}{\tau}}, \tag{60} \]

where in one case in (60) one should substitute \(B_{\varepsilon_m}\) and \(\tau_{\varepsilon_m}\), and in the other \(B_l\) and \(\tau_l\). Since \(B_{\varepsilon_m}\gg B_l\) and \(\tau_l\gg\tau_{\varepsilon_m}\) (see (51), (52), (53), (54)), the ratio of the breakdown field strength determined from Fröhlich’s criterion to the breakdown field strength determined

from Kallén’s criterion, is equal to

\[ \frac{E_{\mathrm{pr}}^{\Phi}}{E_{\mathrm{pr}}^{K}} = \sqrt{\frac{B_l \tau_{\varepsilon_m}}{B_{\varepsilon_m}\tau_l}} < \frac{1}{10}. \tag{61} \]

Meanwhile, as is evident from Tables II and III, both theories lead to identical values \(E_{\mathrm{pr}}\sim 10^6\ \mathrm{V/cm}\), in agreement with experiment. It was shown above that, when the calculation is carried out more rigorously, Kallén’s theory gives overestimated values of \(E_{\mathrm{pr}}\). An analysis of Fröhlich’s calculations shows that the latter also allows an error; when it is removed, the theory gives values of \(E_{\mathrm{pr}}\) underestimated in comparison with experiment. In calculating the losses by formula (54), Fröhlich assumes the frequency of the longitudinal optical oscillations to be equal to the frequency of the transverse oscillations, and \(e\) to be equal to the true charge of the ion. But this is not valid over the entire interval of integration with respect to wave numbers. Dividing the region of integration into two intervals \((L_{\min}, L')\) and \((L', L^0_{\max})\), where \(L'=\dfrac{1}{10a}\), and assuming that the internal-field phenomenon is due only to longitudinal polarization oscillations with wavelength \(\lambda \gg 10a\), we obtain (see (54), (55), (56)):

\[ \begin{aligned} B &= \sqrt{\frac{2m^*}{\varepsilon}}\, \frac{\pi(ee^*)^2}{Ma^3}\, \ln \frac{L'}{L_{\min}} + \sqrt{\frac{2m^*}{\varepsilon}}\, \frac{\pi e^4}{Ma^3}\, \ln \frac{L^0_{\max}}{L'} \\ &= \sqrt{\frac{2m^*}{\varepsilon}}\, \frac{\pi(ee^*)^2}{Ma^3}\, \ln \frac{\pi\sqrt{2e}}{5a\omega m^{*1/2}} + \sqrt{\frac{2m^*}{\varepsilon}}\, \frac{\pi e^4}{Ma^3}\, \ln \left(10^3\sqrt{\frac{3}{4\pi}}\right). \tag{62} \end{aligned} \]

Using, for the determination of the energy losses, expression (62), which takes into account the polarization effects due to long-wavelength oscillations, and, for the determination of \(\tau\), expression (54)\(^*\), from formula (60) we obtain that the values of the breakdown field strength according to Fröhlich’s theory are several times smaller than the experimental values. As was to be expected, these values are more than an order of magnitude smaller than the values of \(E_{\mathrm{pr}}\) in Kallén’s theory (see (61)).

The Gippel–Kallén theory of the breakdown of the electrical strength of a dielectric requires that the field accelerate the electron with any energy, including with energy \(\varepsilon_m\), at which the strongest braking occurs. Obviously, this requirement is sufficient for breakdown to take place, but not necessary. It is not excluded that the growth in the number of ionizations and the disruption of the stationary regime may also occur at smaller fields. One may think that this is the physical reason why overestimated values of the breakdown field strength are obtained in the Gippel–Kallén theory.

\(^*\) In determining \(\tau\), polarization effects play no role, since \(L^0_{\max} \gg L^0\).

In Fröhlich’s theory the breakdown condition reduces to acceleration by the field of electrons with energy \(\varepsilon > \varepsilon' \simeq I\). But for \(\varepsilon < \varepsilon'\) the electrons are, on the average, decelerated. Consequently, replenishment of the number of electrons in the region \(\varepsilon' < \varepsilon < I\) (and only these electrons can, as a result of their acceleration by the field, produce ionization) can occur only by a fluctuation mechanism. The obtaining of an understated value of \(E_{\mathrm{br}}\) in Fröhlich’s theory is apparently connected with an underestimation of the fact that the probability of such processes is very small.

As in the Hippel–Callen theory, so also in Fröhlich’s theory, the breakdown criterion is introduced formally and does not follow directly from the theory itself*). The ionization process, which in essence determines the loss of electrical strength of dielectrics, is not considered explicitly. As A. Akhiezer and I. Lifshitz correctly pointed out\(^{61}\), a theory of the loss of electrical strength must be based on the solution of the kinetic equation for electrons in a dielectric. In this case, terms describing impact ionization must be explicitly introduced into the kinetic equation, and the breakdown criterion must follow directly from the solution.

Fig. 7. Diagram of the energy levels of a dielectric in the case \(T > T_{\mathrm{cr}}\) (according to Fröhlich).

Before proceeding to the formulation of the problem in this form, let us, for completeness of the survey, consider the second theory of Fröhlich\(^{62}\), proposed by him in 1947 to explain the decrease of \(E_{\mathrm{br}}\) at elevated temperatures, observed by some authors. Fröhlich assumed that at a certain critical temperature \(T_{\mathrm{cr}}\), characteristic of each dielectric, the laws of electrical breakdown change. He proceeded from the energy diagram of a dielectric shown in Fig. 7.

At relatively low temperatures, when the total number of electrons in the conduction band and on the local levels (apart from

*) This circumstance is probably connected with that result of Fröhlich’s, which raises considerable doubts, namely that the breakdown field strength is inversely proportional to the ionization potential (see formula (59)).

(main) level is small, then, in Fröhlich’s opinion, one should take into account only the interaction of the conduction electrons with the field and with the lattice. Conversely, at sufficiently high temperatures, when the concentration of electrons with energy \(\varepsilon>0\) is large (Fig. 7), Fröhlich believes that the probability of collision of a free electron with a local one is much greater than the probability of scattering by lattice vibrations. In this case, similarly to the plasma of a gas discharge\({}^{63}\), the electrons in a crystal must have, in an electric field, their own temperature \(T\), greater than the lattice temperature \(T_0\). The temperature \(T\) is determined from the equilibrium condition:

\[ A=B(T,T_0), \tag{63} \]

where \(A\) is the energy received from the field per unit time by all electrons (conduction electrons plus local electrons) contained in \(1\ \mathrm{cm}^3\), and \(B(T,T_0)\) is the energy given up per unit time by the electrons to the lattice. The quantities \(A\) and \(B\) are determined as follows. Assuming that the Fermi level \(\zeta\) lies midway between the main and the first local levels, i.e.

\[ \zeta=-(V-\Delta V), \tag{64} \]

and

\[ -\zeta \gg kT,\qquad \Delta V \gg kT, \tag{65} \]

we obtain for the electron energy distribution function the following expression:

\[ f(\varepsilon)= \frac{g(\varepsilon)} {e^{\frac{\varepsilon+(V-\Delta V)}{kT}}+1} \simeq g(\varepsilon)e^{-\frac{\varepsilon+(V-\Delta V)}{kT}} . \tag{66} \]

The density of levels in the conduction band is

\[ g(\varepsilon)=\frac{4\pi(2m^*)^{3/2}}{h^3}\sqrt{\varepsilon}, \tag{67} \]

and the mean density of local levels \(\bar g_1\) is assumed sufficiently large that transitions between them with absorption or emission of a quantum \(\hbar\omega\) are possible. Accordingly, the concentrations of conduction electrons and local electrons are equal to

\[ n_i=\int_{\Delta V}^{\infty} f(\varepsilon)d\varepsilon = \frac{4\pi(2m^*)^{3/2}}{h^3}e^{-\frac{V}{kT}} \int_{\Delta V}^{\infty}\sqrt{\varepsilon}\,e^{-\frac{\varepsilon-\Delta V}{kT}}d\varepsilon = \]

\[ = \frac{4\pi(2m^*kT)^{3/2}}{h^3}e^{-\frac{V}{kT}} \int_{0}^{\infty}\sqrt{\frac{\Delta V}{kT}+x}\,e^{-x}dx \simeq \]

\[ \simeq 4\pi\left(\frac{2m^*kT}{h^2}\right)^{3/2} \sqrt{\frac{\Delta V}{kT}}\,e^{-\frac{V}{kT}}, \tag{68} \]

$$ n_2=\sum_{\varepsilon=0}^{\Delta V} g_1(\varepsilon)e^{-\frac{\varepsilon+(V-\Delta V)}{kT}} =e^{-\frac{V-\Delta V}{kT}}\cdot \sum_{\varepsilon=0}^{\Delta V} g_1(\varepsilon)e^{-\frac{\varepsilon}{kT}}\simeq $$

$$ \simeq e^{-\frac{V-\Delta V}{kT}}\overline{g_1}\int_0^{\Delta V} e^{-\frac{\varepsilon}{kT}}\,d\varepsilon \simeq \overline{g_1}\cdot kT e^{-\frac{V-\Delta V}{kT}}. \tag{69} $$

Fröhlich assumes that the number of localized electrons is much greater than the number of conduction electrons, i.e.

$$ n_2 \gg n_1 . \tag{70} $$

Since localized electrons cannot be accelerated by the field, the energy

$$ A=\frac{e^2E^2}{m^*}\tau n_1 \tag{71} $$

is obtained from the field only by the conduction electrons. Conversely, owing to (70), the transfer of energy to the lattice is carried out mainly by the localized electrons, to which the conduction electrons transfer the energy received from the field. Assuming that all lattice oscillations occur with one frequency \(\omega\), Fröhlich obtained for the magnitude of the energy losses per unit time the expression

$$ B(T,T_0)=-\frac{\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \left(e^{\frac{\hbar\omega}{kT_0}-\frac{\hbar\omega}{kT}}-1\right) \sum_{\varepsilon=0}^{\Delta V}\Phi(\varepsilon)\cdot f\left(\frac{\varepsilon}{kT}\right), \tag{72} $$

where \(\Phi(\varepsilon)\) is the probability of an electron transition from one localized state to another as a result of interaction with the lattice.

Substituting expressions (71) and (72) into (63), then dividing both parts of the resulting equation by \(n=n_2+n_1\simeq n_2\) and denoting the mean value of \(\Phi(\varepsilon)\) by \(1/\tau_2\):

$$ \frac{1}{n_2}\sum_{\varepsilon=0}^{\Delta V}\Phi(\varepsilon)\cdot f(\varepsilon)=\frac{1}{\tau_2}, \tag{73} $$

where \(\tau_2\), a function of temperature varying slowly in comparison with the exponential, we arrive at the equation

$$ \frac{e^2\tau}{\pi^2 g_1\hbar^3}\sqrt{2m^*\Delta V}\cdot E^2\cdot e^{-\frac{\Delta V}{kT}} = \frac{\hbar\omega}{\tau_2\left(e^{\frac{\hbar\omega}{kT_0}}-1\right)} \left(e^{\frac{\hbar\omega}{kT_0}-\frac{\hbar\omega}{kT}}-1\right). \tag{74} $$

The left-hand side of this equation expresses the energy received on average from the field by one electron per unit time; the right-hand ...

respectively, the energy given up by one electron to the lattice. The essential dependence on \(T\) is practically contained only in the exponents. In Fig. 8 the left-hand (dashed) and right-hand (solid line) parts of equation (74)*) are shown graphically.

At \(E=E_1\), the losses in the temperature range \(T_1<T<T_2\) exceed the energy obtained from the field. Therefore, when the field is applied,

Fig. 8. On the determination of \(E_{\mathrm{pr}}\) according to Fröhlich for \(T>T_{\mathrm{pr}}\).

a stationary state with electronic temperature \(T=T_1\) will be established. At \(E=E_2\), establishment of a stationary state is impossible, since at any temperature

\[ \frac{A}{n}>\frac{B}{n}. \]

Fröhlich determines the breakdown field strength from the condition of tangency of the curves \(\frac{A}{n}\) and \(\frac{B}{n}\), which corresponds to the transition from a stationary to a nonstationary regime, i.e., to loss of electric strength.

Equating the ordinates and the derivatives of the loss and energy-increase curves, we find \(E_{\mathrm{pr}}\) and \(T\):

\[ T \approx T_0+\frac{kT_0^2}{\Delta V}, \tag{75} \]

\[ E_{\mathrm{pr}} = \frac{\pi \hbar^2 \omega}{e\,\Delta V} \sqrt[4]{\frac{\Delta V}{2m^*}}\, \sqrt{\frac{g_1\hbar}{\tau\tau_2\left(e^{\frac{\hbar\omega}{kT_0}}-1\right)}}\, e^{\frac{\Delta V}{2kT_0}}. \tag{76} \]

*) The different course of the curves is connected with the circumstance that

\[ \frac{\hbar\omega}{2}<kT<\frac{\Delta V}{2}. \]

From formula (75) it is seen that the electron temperature at the moment of breakdown of the electric strength exceeds the lattice temperature only slightly (see (65)). As for \(E_{\mathrm{br}}\), Fröhlich considers it impossible to give a theoretical estimate of this quantity and sees the possibility of comparison with experiment only in the temperature dependence\(^{62}\)*).

Let us note, however, that the very fact of a decrease of \(E_{\mathrm{br}}\) at sufficiently high temperatures cannot serve as an argument in favor of the theory presented, since it is not excluded that at these temperatures thermal rather than electrical breakdown is already taking place. Since \(T_{\mathrm{cr}}\) is introduced into Fröhlich’s theory only formally and does not receive a quantitative estimate, it is also impossible to compare with the theory the value of the temperature in the region of the maximum of the curve \(E_{\mathrm{br}}(T)\), corresponding to the change in character of the temperature dependence of the breakdown field strength (see Figs. 2 and 3).

Fröhlich’s idea of the necessity of taking into account the interaction between electrons undoubtedly deserves attention in the case of large electron densities. However, the formal character of this theory and the impossibility of directly comparing it with experiment compel one to regard it merely as one of the possible hypotheses concerning the mechanism of electrical breakdown of dielectrics in the high-temperature region.

As is clear from the foregoing, Fröhlich’s last theory does not apply to the region of relatively low temperatures, where the phenomenon of electrical breakdown appears in its pure, undistorted form.

The consideration of the experimental data carried out above and the critical analysis of existing theories make it possible to assert that the phenomenon of electrical breakdown of solid dielectrics is based on ionization of the atoms (ions) of the dielectric by electron impact. The braking of electrons, which prevents their accumulation of the energy necessary for impact ionization, is due to the scattering of electrons by lattice vibrations. The onset of breakdown, and at the same time the cause determining the course of all subsequent processes, usually ending in destruction of the dielectric, is the loss by the dielectric of electric strength, i.e. the disturbance of the stationary regime of electrical conductivity. Existing theories of electrical breakdown of solid dielectrics do not give a satisfactory criterion for disrup—

*) Fröhlich does not reveal the meaning of the pre-exponential factor and writes simply:

\[ E_{\mathrm{br}} = C e^{\frac{\Delta V}{2kT_0}}. \]

The impossibility he asserts of a theoretical estimate of \(E_{\mathrm{br}}\) is apparently connected with the fact that \(C\) contains the quantities \(\tau_2\), \(g_1\), and \(\Delta V\), whose values are unknown.

…of electrical strength, following from the theory itself. A rigorous criterion for the violation of electrical strength can be obtained from the solution of the kinetic equation for electrons in a dielectric; moreover, in this equation not only elastic but also ionizing collisions must be taken explicitly into account.

Recently, E. I. Adirovich and V. A. Chuenkov made an attempt to carry out the solution of the problem in this direction. The authors considered the electronic processes occurring in a dielectric under the application of a strong electric field by solving the kinetic equation with allowance for elastic and ionizing collisions.

The kinetic equation for the stationary distribution of conduction electrons in a dielectric may be written in the following form:

\[ eE \frac{\partial f}{\partial p_z} = \varphi^{\mathrm{расс}} + \varphi^{\mathrm{возб}} + \varphi^{\mathrm{иониз}} + \varphi^{\mathrm{рек}}, \tag{77} \]

where the terms on the right correspond to changes in the distribution function \(f(\varepsilon,\theta)\) occurring per unit time as a result of elastic (scattering) and inelastic (excitation, ionization, recombination) collisions. Since the excitation energy in crystals usually differs only slightly from the ionization potential \(I\), while the probability of excitation is small in comparison with the probability of ionization\(^{54–64}\), the term \(\varphi^{\mathrm{возб}}\) may be neglected.

Let us divide the energy zone of the conduction electrons into four regions:

\[ 0 \leq \varepsilon \leq 10\hbar\omega_{\mathrm{деб}} \quad (I), \qquad \varepsilon'' \leq \varepsilon \leq I \quad (III^*), \]

\[ 10\hbar\omega_{\mathrm{деб}} \leq \varepsilon \leq \varepsilon'' \quad (II), \qquad \varepsilon \geq I \quad (IV). \]

It follows from general theoretical considerations that the probability of recombination is different from zero only for slow electrons\(^{65}\). Therefore everywhere, except for region (I), \(\varphi^{\mathrm{рек}}=0\).

In region (II) the probability of scattering by lattice vibrations is sufficiently large to ensure redistribution over the angles of the electrons accelerated by the field. Here the distribution function may be approximately represented in the form of the first two terms

\[ {}^{*)}\qquad \varepsilon'' = \left(\frac{3}{4\pi}\right)^{1/3} \frac{h^2}{8m^*a^2} = \frac{h^2 L^{0\,2}_{\max}}{8m^*}, \]

where \(hL^0_{\max}\) is equal to the maximum possible phonon momentum for the given lattice, \(m^*\) is the effective mass of the electron, and \(a\) is the lattice constant.

expansion in a series in Legendre polynomials \(P_n(\cos\theta)\):

\[ f(\varepsilon,\theta)=f^0(\varepsilon)+f^1(\varepsilon)\cos\theta, \]

and

\[ \begin{aligned} \varphi^{\mathrm{scatt}}(\varepsilon,\theta) &= \frac{2\pi e^2}{Ma^3 v}\,\frac{d f^0}{d\varepsilon} \int_{L_{\min}}^{L_{\max}} e_L^2\,\frac{dL}{L} \\ &\quad+ \frac{\pi e^2\hbar}{Ma^3 v}\,\frac{d^2 f^0}{d\varepsilon^2} \int_{L_{\min}}^{L_{\max}} e_L^2\omega_L\, \frac{e^{\hbar\omega_L/kT}+1}{e^{\hbar\omega_L/kT}-1}\, \frac{dL}{L} \\ &\quad- \frac{2\pi^2 e^2\hbar}{Ma^3m^2v^3}\,f^1(\varepsilon)\cos\theta \int_{L_{\min}}^{L_{\max}} \frac{e_L^2}{\omega_L}\, \frac{e^{\hbar\omega_L/kT}+1}{e^{\hbar\omega_L/kT}-1}\, \frac{dL}{L}. \end{aligned} \tag{78} \]

In expression (78), written for an ionic lattice, account is taken, besides optical vibrations, also of acoustic vibrations affecting the scattering of fast electrons, as well as of polarization effects caused by long-wavelength optical vibrations of the lattice

\[ \left( \text{for } \lambda=\frac{1}{L}\gg a,\quad e_L=e^*=\omega\sqrt{\frac{Ma^3(\chi-\chi_0)}{2\chi_0\chi}}, \quad \omega_L=\omega;\quad \text{for } \lambda\sim a,\quad e_L=e,\quad \omega_L=\omega_t \right)^{*}. \]

The ionization term in this region

\[ \varphi^{\mathrm{ioniz}}(\varepsilon) = \beta N_1 Q(I+2\varepsilon) \sqrt{\frac{2(I+2\varepsilon)}{m^*}} \int_0^\pi f(I+2\varepsilon,\theta)\sin\theta\,d\theta \tag{79} \]

expresses the influx of electrons arising in the process of ionization (\(N_1\) is the number of atoms per unit volume, \(\beta\) is the number of electrons in the outer shell of the ionized atom or ion, \(Q(\varepsilon)\) is the effective ionization cross section). It is assumed: 1) that \(Q(\varepsilon)\) is isotropic both with respect to the direction of motion of the ejected electron and with respect to the direction of motion of the ionizing electron after the ionization event; 2) that both the ejected and the ionizing electrons have the same velocity after ionization. The solution of the kinetic equation in this region makes it possible to determine \(f^0(\varepsilon)\) and \(f^1(\varepsilon)\).

In region (III) the change in the distribution function is caused mainly by the electric field. Here there occurs a sharp decrease—

\[ \text{——} \]

*) \(M\) is the reduced mass of the ions, \(e\) is the electron charge, \(e^*\) is the effective ion charge, \(\omega_t\) is the frequency of transverse vibrations of the ions, \(\omega\) is the frequency of long-wavelength longitudinal optical vibrations, \(\chi\) is the static dielectric permittivity, \(\chi_0\) is the square of the refractive index of light.

a decrease in the relative number of electrons moving at large angles to the direction \(e\mathbf E\) (orientation of the electrons by the field), for comparatively small changes of the distribution function caused by scattering by lattice vibrations and by the arrival of electrons produced as a result of ionization.

At energies exceeding the ionization potential (region (IV)), the distribution function is determined by the action of the electric field and by ionization collisions; here

\[ \varphi^{\mathrm{ioniz}}(\varepsilon) = -\beta N_1 Q(\varepsilon)\sqrt{\frac{2\varepsilon}{m^*}}\,f(\varepsilon,\theta), \tag{80} \]

since the arrival of electrons produced after ionization may be neglected in this region. The solution of the kinetic equation, carried out under the assumption that \(Q(\varepsilon)=S(\varepsilon-I)\), where \(S=\mathrm{const}\), leads here to a distribution function consisting of two factors. The first factor is proportional to the probability of finding an electron with energy \(\varepsilon\) at an angle \(\theta\) to the direction \(\mathbf E\), determined only by the orienting action of the field. The second factor is equal to the conditional probability that an electron which was moving at \(\varepsilon=I\) at the angle \(\theta_1=\arcsin\sqrt{\dfrac{\varepsilon}{I}}\sin\theta\) acquires in the field the additional energy \(\varepsilon-I\) without undergoing ionization collisions.

After the continuity requirements for the solution at the boundaries of the energy regions have been satisfied, the distribution function of the conduction electrons contains two constants of integration, one of which is the normalization constant. The second constant is determined from the condition that the flux of electrons through the sphere \(\varepsilon=\varepsilon''\) in phase space be equal to the number of ionizations occurring per unit time. This ensures the stationarity of the solution of the kinetic equation in the region \(\varepsilon \gg I\), since in the intermediate region \(\varepsilon'' \leqslant \varepsilon \leqslant I\) the scattering of electrons by lattice vibrations and the arrival of electrons produced after ionization may be neglected. However, the stationarity requirement in the region \(\varepsilon \leqslant \varepsilon''\) must also be fulfilled. Since after each act of ionization occurring in the dielectric two electrons appear in this energy region, stationarity here can be maintained only under the condition that the number of recombinations occurring in one second in region (I) is equal to the number of ionizations. The latter requirement is fulfilled at any field so long as \(\varepsilon_p\)—the value of the energy at which, on the average, the losses and gains of energy by the electrons are balanced—is greater than \(\varepsilon_{1/2}\), the value of the energy above and below which an equal number of electrons arrive after ionization. If this condition is fulfilled, an ionization act leads on average to the appearance of fewer than one electron capable of causing

new ionization. In this case the principal agent creating electrons capable of being accelerated by the field up to the ionization potential and higher is thermal motion. The number of ionizations will be determined by the number of fluctuation thermal ejections of electrons into the energy region exceeding \(\varepsilon_p\), i.e., it will remain small and constant for a given value of the electric-field strength. On the contrary, for \(\varepsilon_p < \varepsilon_{1/2}\), ionization processes will lead to a continuous increase in the number of electrons that can receive energy from the field and cause new acts of ionization. The process of ionization and the formation of free electrons in the dielectric acquires an avalanche character and becomes independent of the concentration of fast electrons arising as a result of thermal motion. Beginning with that value of the field strength at which this condition is realized, the number of ionizations will continuously increase at a constant value of \(E\), that is, a stationary regime of electrical conduction becomes impossible.

The breakdown field strength \(E_{\mathrm{br}}\), at which the limit of the electric strength of the dielectric is reached and electrical breakdown occurs, is determined from the equation

\[ \varepsilon_{1/2} = \varepsilon_p, \tag{81} \]

corresponding to the multiplication coefficient of fast \((\hat{\varepsilon} > \varepsilon_p)\) electrons becoming equal to unity\(^*\).

The solution of the kinetic equation makes it possible to express \(\varepsilon_{1/2}\) and \(\varepsilon_p\) as functions of the electric-field strength:

\[ \left(1 + \frac{3}{2}\frac{m^*}{e^2}\frac{br}{E^2 \varepsilon_p^2}\right) \left[ 1 - \frac{y(\varepsilon'')}{y(\varepsilon_p)} \times \right. \]

\[ \left. \times \left( 1 - \frac{e}{3}\sqrt{\frac{2}{m^*}} - \frac{a_1 E \varepsilon''} {br + \frac{2}{3}\frac{e^2}{m^*}E^2 \varepsilon''^{2}} \right) \right] = 1, \tag{82} \]

\[ \varepsilon_{1/2} = 0.589 \sqrt{\frac{eE}{\beta N_1 S}}, \tag{83} \]

\(^*\) The preceding consideration shows that, in order to find \(E_{\mathrm{br}}\), there is no need to solve the kinetic equation for \(\varepsilon \simeq \hbar\omega_{\mathrm{Deb}}\) (region (I), where \(\varphi^{\mathrm{rek}}(\varepsilon,\theta) \ne 0\)). The numerical value of the probability of a recombination event affects only the magnitude of the stationary density of slow electrons, at which recombination processes compensate the difference between the number of electrons entering the region \(\varepsilon < \varepsilon_p\) after ionization and the number of electrons thermally ejected into the region \(\varepsilon > \varepsilon_p\). As long as \(\varepsilon_{1/2} < \varepsilon_p\), a stationary regime is always established in region (I). Consequently, it is not this region that limits the possibility of maintaining electrical strength. We note that from this follows the practical independence (or weak dependence) of the breakdown field strength on the effective cross-section of recombination centers:

where

\[ b=\frac{\pi^{2} e^{2}\sqrt{m^{*}}\hbar}{\sqrt{2}\,Ma^{3}} \left( e^{*2}\omega\, \frac{e^{\frac{\hbar\omega}{kT}}+1}{e^{\frac{\hbar\omega}{kT}}-1} \ln \frac{\pi\sqrt{2\varepsilon}}{5a\omega\sqrt{m^{*}}} + e^{2}\omega_{t}\, \frac{e^{\frac{\hbar\omega_{t}}{kT}}+1}{e^{\frac{\hbar\omega_{t}}{kT}}-1} \ln \frac{20a\sqrt{2m^{*}\varepsilon}}{\hbar} \right), \]

\[ a_{1}=\frac{\pi\sqrt{2m^{*}}e^{2}}{Ma^{3}} \left( e^{*2}\ln \frac{\pi\sqrt{2\varepsilon}}{5a\omega\sqrt{m^{*}}} + e^{2}\ln \frac{20a\sqrt{2m^{*}\varepsilon}}{\hbar} \right), \qquad p=\frac{2}{3}\frac{e^{2}}{m^{*}r}, \tag{84} \]

\[ r=\frac{\pi\sqrt{2m^{*}}e^{4}}{Ma^{3}\hbar\omega_{t}}\, \frac{e^{\frac{\hbar\omega_{t}}{kT}}+1}{e^{\frac{\hbar\omega_{t}}{kT}}-1}, \]

\[ y=\exp\left[ -\frac{a_{1}}{b}\sqrt{\frac{b}{p}}\,\frac{1}{E} \operatorname{arctg}\left(\sqrt{\frac{p}{b}}\,E\varepsilon\right) \right]. \]

Substituting expressions (82) and (83) into equation (81), we find \(E_{\mathrm{br}}\).

Let us apply the results obtained to the calculation of \(E_{\mathrm{br}}\) for alkali-halide crystals. Apart from the ionization constant \(S\), all the lattice parameters entering (82) and (83) (\(a\), \(\omega_t\), \(M\), etc.) are known from experiment\(^{56}\). Assuming that \(Q_{\max}\) of the halide ions in the lattice is of the order of \(10^{-16}\,\text{cm}^{2}\), and \(\varepsilon_{\max}\simeq 5I\), as is the case in gases\(^{66\,*}\), and calculating \(S\) in the approximate formula for \(Q\) from the relation

\[ S=\frac{Q_{\max}}{\varepsilon_{\max}-I}, \]

we find that \(S\simeq 10^{-6}\,\text{sec}^{2}/\text{g}\). For \(S=10^{-6}\,\text{sec}^{2}/\text{g}\), for NaCl \(E_{\mathrm{br}}^{\mathrm{theor}}=1.52\cdot 10^{6}\ \text{V}/\text{cm}\), whereas \(E_{\mathrm{br}}^{\mathrm{exp}}=1.5\cdot 10^{6}\ \text{V}/\text{cm}\). Assuming that \(S_{\mathrm{NaCl}}=10^{-6}\,\text{sec}^{2}/\text{g}\) and that \(S\) is proportional to the square of the ionic radius of the halogen, we obtain theoretical curves for the dependence of \(E_{\mathrm{br}}\) on temperature for RbJ, KBr, NaCl, LiF, shown in Fig. 9. Crosses indicate the experimental values of \(E_{\mathrm{br}}\) for these crystals at room temperature. Fig. 9 also shows the experimen-

*) Ionization cross sections in solids are not known.

tal curve of the temperature dependence of \(E_{\mathrm{br}}\) for KBr, obtained with pulses of duration \(10^{-6}\) sec, when electrical breakdown is not complicated by side effects (heating, high-voltage polarization)\(^{21}\). As is seen from Fig. 9, the theory is in good agreement with experiment both with respect to the temperature dependence of \(E_{\mathrm{br}}\) for KBr and with respect to the sequence in which \(E_{\mathrm{br}}\) increases in alkali-halide crystals of different composition (RbJ, KBr, NaCl, LiF). This agreement practically does not depend on the exact choice of the value of \(S\), since changing \(S\) by an order of magnitude leads to a change of \(E_{\mathrm{br}}\) by approximately a factor of two \((E_{\mathrm{br}}=3\cdot 10^{6}\ \mathrm{V/cm}\) for NaCl at \(S'=10^{-5}\ \mathrm{sec}^{3}/\mathrm{g})\).

Fig. 9. Theoretical curves of the temperature dependence of \(E_{\mathrm{br}}\) for several alkali-halide crystals. For comparison, the dependence of \(E_{\mathrm{br}}\) on temperature for KBr, obtained in tests with pulses of duration \(10^{-6}\) sec, is shown.

Experimentally this is confirmed by the fact that the most diverse dielectrics, consisting of atoms with different effective ionization cross sections, have the same order of magnitude of \(E_{\mathrm{br}}\) (see Table I). Let us also note that, according to the theory developed, the breakdown field strength increases with increasing effective ionization cross section. Physically this is understandable. действи-

Consequently, the larger the effective ionization cross section, the more sharply the distribution function of conduction electrons decreases with increasing \(\varepsilon\) in the region \(\varepsilon>I\), the smaller the average energy of the electrons after ionization, and the more difficult it is to accelerate again the electrons produced as a result of ionization.

It should be noted that all calculations have been made under the assumption that scattering of electrons by lattice vibrations in the region \(I\gg \varepsilon \gg \varepsilon''\) may be neglected.

To justify this assumption, let us extend to the region \(\varepsilon''\leqslant \varepsilon \leqslant I\) the method of calculating the distribution function that was used in the region \(\varepsilon \leqslant \varepsilon''\). From formulas (23) and (56) it follows that an electron with energy \(\varepsilon=\varepsilon''\) (see the note on p. 220) can emit or absorb even such phonons as have the maximum possible momentum for the given lattice. Consequently, for all \(\varepsilon\gg\varepsilon''\)

\[ L^e_{\max}\left(\simeq L^a_{\max}\right)=L^0_{\max}=\frac{1}{2^{1/3}a}. \]

If it is assumed that the crystal is a continuous medium (in this case we shall have a spectrum of elastic vibrations with wavelengths from \(0\) to \(\infty\)), then, when electron scattering by lattice vibrations is taken into account, one should take into account the increase of \(L^e_{\max}\) and \(L^a_{\max}\) with increasing \(\varepsilon\) not only in the region \(\varepsilon\leqslant \varepsilon''\), but also in the region \(\varepsilon\gg\varepsilon''\) (see formulas (23)). If one assumes that scattering by lattice vibrations plays no role only for \(\varepsilon\gg I\), which is certainly permissible (ionization collisions occur much more often than elastic scattering by lattice vibrations*), while for \(\varepsilon''\leqslant \varepsilon \leqslant I\) the effect of scattering is taken into account only in the manner just described (the point is that instead of

\[ \varepsilon''=\frac{h^3 L^0_{\max} c}{8m^*} \]

one should take

\[ \varepsilon'''=I \]

), then we overestimate the influence of interaction with the lattice on the distribution function of conduction electrons. Thus we have the possibility of determining the limits of the values of \(E_{\mathrm{pr}}\) between which the true value of \(E_{\mathrm{pr}}\) must lie.

Substituting \(\varepsilon''=I\) into formulas (82) and (83), we obtain for NaCl at \(T=300^\circ\mathrm{K}\) and \(S=10^{-6}\ \mathrm{sec}^2/\mathrm{g}\) the value \(E_{\mathrm{pr}}=2.5\cdot 10^6\ \mathrm{V/cm}\).

Thus, underestimating the effect of scattering by lattice vibrations leads to the value \(E_{\mathrm{pr}}=1.52\cdot 10^6\ \mathrm{V/cm}\), while overestimating it gives \(2.5\cdot 10^6\ \mathrm{V/cm}\). Consequently, the assumption of the smallness of the effect

*) The number of ionization collisions per unit time is equal to
\(\beta N_1 v Q \simeq 6\cdot 10^{15}\) for \(\beta=6\), \(N_1=6\cdot 10^{22}\ \mathrm{cm}^{-3}\), \(v\simeq 10^8\ \mathrm{cm/sec}\), \(Q=10^{-15}\ \mathrm{cm}^2\), whereas the number of collisions with lattice vibrations for \(\varepsilon \approx I\) is equal to
\[ \frac{1}{\tau}\approx 5\cdot 10^{13} \]
(see formula (53)).

scattering of electrons by lattice vibrations in the range \(\varepsilon'' \ll \varepsilon < I\) at \(E \sim 10^6\ \mathrm{V/cm}\) is entirely justified.

Let us estimate the times and distances necessary for the development of an electron avalanche, i.e., for the occurrence of electrical breakdown of dielectrics. The time during which an electron is accelerated by a field \(E \simeq 10^6\ \mathrm{V/cm}\) from the energy \(\varepsilon_p\) to \(I\) is equal to \(10^{-13}\ \mathrm{s}\). At an average electron velocity of the order of \(10^8\ \mathrm{cm/s}\), the corresponding length is \(10^{-5}\ \mathrm{cm}\). Since the development of an electron avalanche requires several stages of successive ionization of the dielectric by fast electrons arising directly as a result of the act of ionization, the estimate obtained agrees with the experimental data on electrical hardening at thicknesses of the order of \(10^{-4}\)—\(10^{-5}\ \mathrm{cm}^{5}\).

There are data in the literature indicating that the value \(E_{\mathrm{br}}\) for electrical breakdown in a uniform field, which over a wide range of values of \(t\) does not depend on the pulse duration, increases at \(t \lesssim 10^{-8}\ \mathrm{s}^{10,8}\), i.e., at times appreciably longer than the time calculated above for the development of an electron avalanche. This suggests that, unlike electrical hardening at small thicknesses, hardening of the dielectric at sufficiently short pulse times \((t \lesssim 10^{-8}\ \mathrm{s})\) is associated with the conditions for the development of the second stage of breakdown—mechanical destruction of the dielectric—which naturally requires longer times than the process of development of an electron avalanche.

CONCLUSION

The above review of the present state of knowledge concerning the phenomena of electrical breakdown shows that the existing theories can be divided into two groups: 1) theories of loss of electrical strength, 2) theories of mechanical destruction of the dielectric. Neither the former nor the latter give a complete description of the entire picture of breakdown development: loss of electrical strength constitutes the initial stage of electrical breakdown, while destruction of the material completes the breakdown process.

At present, the theory of the first stage of electrical breakdown—the loss of electrical strength—has been developed in greatest detail. It may be thought that the most correct path in constructing a theory of the loss of electrical strength is connected with formulating the problem in the form of a kinetic equation for electrons in a solid.

Owing to a number of difficulties that arise in carrying out such a program of theoretical investigation of breakdown (the impossibility of applying the usual approximate methods of calculation for \(\varepsilon \lesssim \hbar \omega_{\mathrm{Deb}}\), the absence of data on effective cross sections for ionization and recombination in solid dielectrics, etc.), modern

the theory rests on a large number of assumptions, not always sufficiently well substantiated. The most urgent immediate tasks of the theory are: calculation of the effective ionization cross section in a solid in the case of small energies of the ionizing electron \((\varepsilon \gtrsim I)\), calculation of the probability of scattering of slow electrons \((\varepsilon \sim \hbar \omega_{\mathrm{Deb}})\) by lattice vibrations, and calculation of the effective recombination cross section. Consequently, the further development of the theory of electrical breakdown requires the solution of a number of general problems in solid-state theory.

The theory of the second stage of electrical breakdown is almost entirely undeveloped. The destruction of a dielectric during breakdown is considered, as a rule, in isolation from the first stage and is not connected with the conditions of avalanche passage in the solid. Such an approach seems to us incorrect, since it thereby ignores the qualitative feature of a dielectric in breakdown fields—its rapid transition into a highly conducting state, accompanied by the release of large amounts of energy in small volumes and over short times. One may think that the true cause of the destruction of the dielectric is connected precisely with these processes, which lead to the emergence of a shock wave, analogous to what occurs in the breakdown of gases at high pressures. Therefore, in our opinion, a fruitful direction for the theory is the treatment of the processes of dielectric destruction in inseparable connection with the preceding processes of loss of electrical strength.

The author takes this opportunity to express his deep gratitude to É. I. Adirovich for his constant attention to the work and for a number of valuable suggestions.

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PUBLISHER’S NOTE

In formula (18) and in the subsequent expressions, \(h\omega\) should be regarded as factors; for technical reasons the printing house was unable to place them on a single line.