MECHANISM AND THEORY OF BREAKDOWN IN SOLID DIELECTRICS

A. S. Zingerman

1. INTRODUCTION

In 1802 the Russian academician Vasilii Vladimirovich Petrov constructed “an enormous battery, sometimes consisting of 4200 copper and zinc disks,”[^1] which produced an enormous voltage for that time. At that time no physics laboratory in the world possessed such a voltage, except the laboratory of Academician V. V. Petrov. The experiments carried out by Academician V. V. Petrov with this voltage led to a number of discoveries of world importance. In one such series of experiments Petrov discovered the phenomenon of dielectric breakdown. Into a glass tube filled with olive “or some other transparent oil,” metallic or carbon electrodes were inserted. The distance between the electrodes was 2–4 mm. After connecting the electrodes to the poles of the battery, Petrov observed “between the cap of one wire and the piece of charcoal or small metallic cone located at the end of the other wire, light in the form of sparks of various size and brightness.”[^2]

Academician V. V. Petrov carried out the next series of experiments in air. When he began to bring together two carbon electrodes connected to the poles of the battery, “a very bright light” flashed between them. Thus the discovery of the electric arc was accompanied at the same time by the discovery of the phenomenon of electrical breakdown. Already at that time Academician V. V. Petrov knew that the onset of this phenomenon is facilitated by a pointed form of the electrodes and requires an “enormous battery”; moreover, the greater the distance between the electrodes, the larger the battery must be, i.e., the voltage. Thus the priority in the discovery of the phenomenon of dielectric breakdown belongs to Russia and to the Russian academician Vasilii Vladimirovich Petrov.

The problem of electrical breakdown is of extraordinarily great practical importance, especially in our country with its widely spread network of installations and enormous length of lines

high voltage. The great construction projects of communism—the Kuibyshev, Stalingrad, and Kakhovka hydroelectric power stations, being built on the initiative of Comrade Stalin—require the transition of electric power-transmission lines to extra-high voltages. Such a transition lends still greater urgency to the problem of dielectric breakdown.

The especially favorable conditions created in the Soviet country for the development of science have led to Soviet scientists playing the leading role in the study of the breakdown of solid dielectrics.

From the proposed survey of views on the nature of the breakdown of solid dielectrics, it is clear that the original and most fruitful ideas in this field were first proposed by Soviet scientists: A. F. Ioffe, A. A. Smurov, Ya. I. Frenkel, and others. These ideas were developed and experimentally verified by a whole series of Soviet researchers. The leading role here belongs to the school of the Physico-Technical Institute of the Academy of Sciences of the USSR: A. F. Ioffe, N. N. Semyonov, N. V. Kurchatov, B. M. Vul, A. P. Aleksandrov, K. D. Sinelnikov, L. D. Inge, B. V. Gorelik, I. M. Goldman, and others. One should also note the work of A. A. Vorob’ev (Tomsk Physico-Technical Institute), and the work on thermal breakdown by V. A. Fock, G. A. Grinberg, A. M. Zaleskii, and others.

The numerous experimental investigations of the breakdown of solid dielectrics are, in the overwhelming majority of cases, devoted to the study of macroscopic dependences. Investigation of the elementary microprocesses that take place during breakdown is extremely difficult. This largely explains the absence of a generally accepted point of view on the mechanism of breakdown. Although this phenomenon has been studied for a comparatively long time, not one of the proposed mechanisms can be regarded not only as established and proven, but even as developed in sufficient detail. The hypotheses proposed by various authors, as experimental facts accumulated, either fell away or underwent substantial changes. For the success of further work in this field it is extremely useful to make a critical survey of all attempts to solve this question. However, it is first necessary to outline the principal characteristic features of the phenomenon itself that have been firmly established experimentally.

These principal features are as follows.

1. The breakdown of a dielectric does not occur throughout its entire volume, but only at individual places. The transverse cross section (perpendicular to the field) of the punctured region is very small.

2. The process leading to breakdown does not develop simultaneously throughout the entire thickness of the dielectric, but begins at one place and gradually (with some velocity) propagates in the body of the dielectric in the direction toward both electrodes. Often breakdown is observed through only part of the thickness of the dielectric (partial breakdown).

1. The punctured place of the dielectric is characterized either by mechanical, or by thermal (burn-through, melting), or by chemical destruction, and more often by both at the same time.

2. The dependence between the current in a solid dielectric and the voltage applied to it is shown in Fig. 1. At small voltages the current increases in proportion to the voltage (the region of Ohm’s law). A further increase in voltage is accompanied by a more rapid growth of the current (the region of Poole’s law), which assumes an extremely sharp form when the voltage reaches pre-breakdown values.

At point \(A\) the current continues to increase at a constant value of the voltage, and its growth continues even in the case when the voltage decreases. Thus, point \(A\) is characterized by unstable processes in the dielectric, leading to an extremely strong increase in the conductivity of the dielectric. Subsequently, the process of increasing conductivity of the dielectric continues uninterruptedly to develop on its own and leads to a catastrophe, i.e., to destruction of the dielectric. Point \(A\) characterizes breakdown or the beginning of breakdown of the dielectric.

Fig. 1.

B. M. Vul \({}^{3}\) characterizes dielectric breakdown in the following way:

“The breakdown usually observed in an experiment is the final stage of destruction of the dielectric. It is preceded by the loss by the dielectric of its electrical strength. The release of energy resulting from this causes the visible phenomenon of breakdown: the formation of spark discharges in the dielectric, mechanical destruction when a solid dielectric is subjected to breakdown. The dielectric proves to be punctured precisely because, at the limiting field strength, it loses its electrical strength, ceases to be a dielectric. Usually both stages—dielectric destruction and breakdown—immediately follow one another and are regarded as one phenomenon.”

Up to point \(A\) the dielectric possesses electrical strength, although its conductivity increases continuously. The increase in the conductivity of the dielectric occurs at the expense of the transfer to the dielectric by the electric field of a definite amount of energy.

The gradual quantitative change of conductivity up to point \(A\) is replaced by a jump-like one at point \(A\). After point \(A\) the material not only possesses greater conductivity, but also a new regularity of behavior. For example, up to point \(A\) a decrease in voltage causes a decrease in conductivity, which is characteristic of nonconducting materials; after point \(A\) a decrease in voltage causes an increase in conductivity, which is characteristic of conducti

kov materials. Up to point \(A\), the electrical energy supplied to the material is converted mainly into the potential energy of polarization, into a reversible form of energy. After point \(A\), the supplied electrical energy is converted mainly into an irreversible form of energy—heat. Thus, the energy transmitted up to point \(A\) is converted into one form, after point \(A\)—into another. Therefore point \(A\) may be characterized as the point at which certain properties of the material change. The phenomenon under consideration well illustrates the well-known dialectical law of the transition of quantity into quality[^4].

The transition of the material at point \(A\) from the class of nonconductors to the class of conductors is characterized by the loss of electrical strength. In the new state of the material there occurs a much more intense absorption of electrical energy, which causes charring, melting, mechanical, and other destruction of the material. If the energy supplied to the material is limited, the indicated consequences may not occur, and destruction of the material will not take place. This can be observed when testing, for example, some porcelain insulators (without glaze) with a very low-power transformer. When the voltage is raised, at point \(A\) there is a sharp increase in current with a sharp drop in voltage—the insulator loses all the characteristics of a nonconductor. However, no destruction of the material occurs. Such short-term tests of one and the same specimen may be repeated. But an increase in the power of the transformer, or a considerable increase in the duration of the test, causes breakdown with visible destruction of the material. After such a test it is impossible to repeat the experiment with the same specimen; the specimen no longer possesses electrical strength. Under ordinary conditions there is no limitation in the supply of energy, and breakdown follows the loss of electrical strength. Breakdown occurs after point \(A\), in the new state of the material, when it becomes a conductor.

What has been said is illustrated by the two voltage oscillograms shown in Fig. 2. The first oscillogram (a) was taken with a pulse generator having a low resistance. After the insulating material (mica, cemented with a compound mass) loses electrical strength, its destruction occurs. The voltage after breakdown is not restored. The second oscillogram (b) was taken on the same specimen with the same generator, but with an additionally connected large resistance. From the oscillogram it is evident that after the loss of electrical strength, which is characterized by a sharp drop in voltage, it is immediately restored. This indicates that destruction of the insulation did not occur. Destruction takes place only after a number, sometimes several tens, of sharp voltage drops; moreover, the magnitude of the voltage at which the subsequent drops occur is sometimes greater than at the preceding ones.

1. The duration of the processes of loss of electrical strength and breakdown is usually very small. The time required for complete breakdown depends on the thickness of the dielectric, on the magnitude of the field strength, and on other factors. In some solid dielectrics of thickness 1 mm, the time during which the breakdown process develops is of the order of \(10^{-6}\) sec. When short voltage pulses are applied to a dielectric, the breakdown voltage increases as the pulse duration decreases.

2. The breakdown of solid dielectrics has a statistical character. Both the breakdown time and the voltage that causes breakdown, for the same dielectrics, vary over a more or less wide range. The scatter of the voltage may attain a value of more than \(\pm 50\%\) of the mean value.

Fig. 2.

1. The mean field strength that causes breakdown, for most solid dielectrics, has a value of the order of \(10^5\)—\(10^6\) V/cm.

For alkali-halide crystals, the breakdown field strength is the higher, the greater the lattice energy. This dependence has an approximately linear character.

1. There are three kinds of breakdown: electrical, thermal, and chemical, when the increase in conductivity is due to electrical, thermal, or chemical factors. The last kind of breakdown is not considered in this work.

The breakdown field strength depends on a number of factors, including temperature. In thermal breakdown, the breakdown field strength decreases with increasing temperature. In electrical breakdown, the influence of temperature on the breakdown field strength has not yet been definitively established. The available experimental data are contradictory. The most recent experiments indicate an increase in the breakdown strength for some dielectrics with increasing temperature.

2. A. F. Ioffe’s Theory of Impact Ionization

The first theory of breakdown of solid dielectrics regarded breakdown as the result of mechanical rupture of the lattice of a body by an electric field (Rogowski’s theory \(^{5}\)). This theory led to sharp contradictions with experimental data. Attempts to improve this theory (by Rogowski himself \(^{5}\) and by Horowitz \(^{6}\)—the theory of cracks) were based on certain arbitrary quantities, which made it possible to remove some contradictions. However, the artificial character of the assumptions underlying the latter theories, and the inability of mechanical theories to explain the main features of the phenomenon, make them unacceptable.

An entirely different view of the nature of breakdown of solid dielectrics was developed by A. F. Ioffe \(^{7}\). A. F. Ioffe treats breakdown as a catastrophically rapid increase in conductivity, the consequence of which is mechanical or thermal destruction of the dielectric. Thus, the sign that breakdown has occurred is an increase of the electric-current density to a certain critical value. The field strength that causes such an increase of current is the breakdown strength.

The mechanism of breakdown itself is represented as follows. Let the dielectric be in a homogeneous electric field directed along the \(X\)-axis. Suppose that in the interelectrode space, per unit volume and per unit time, some non-impact mechanism produces \(n_{0}\) “primary” ions. If \(l\) is the distance traversed by an ion along the field between two ionizing collisions, then the number of ionizations produced by one ion per unit length of path (in the direction of the field) is equal to \(\frac{1}{l}\). It is further assumed that every collision of an ion accelerated by the electric field leads to ionization. If, into a layer \(dx\) of unit cross section situated at a distance \(x\) from the first electrode, \(n\) ions enter per unit time, then the number of ions \(dn\) formed in the layer is equal to

\[ dn=\frac{n\,dx}{l} \]

and

\[ n=n_{0}e^{\frac{x}{l}}. \]

The density of the ion current at the second electrode will be

\[ i=e\int_{0}^{d} n\,dx=en_{0}l\left(e^{\frac{d}{l}}-1\right), \]

where \(e\) is the charge of the ion, and \(d\) is the distance between the electrodes.

In the absence of the multiplication process the density of the ion current is

\[ i_{0}=en_{0}d, \]

and therefore,

\[ i=\frac{i_0 l}{d}\left(e^{\frac{d}{l}}-1\right). \tag{2,1} \]

Neglecting the distortion of the field due to the presence of ions, one may regard the following relation as valid:

\[ \frac{l}{d}=\frac{U_i}{U}, \]

where \(U_i\) is the ionization potential, and \(U\) is the voltage applied to the dielectric. The density of the current flowing through the dielectric can then be represented in the form:

\[ i=i_0\frac{U_i}{U}\left(e^{\frac{U}{U_i}}-1\right). \]

It is further assumed that breakdown occurs when the current reaches some sufficiently large value, i.e., under the condition that the ratio

\[ \frac{i}{i_0}=\frac{U_i}{U_k}\left(e^{\frac{U_k}{U_i}}-1\right)=K \]

is equal to or exceeds some definite number. From the last relation \(\frac{U_k}{U_i}\) is determined, and hence the critical value of the field strength,

\[ E_k=\frac{U_k}{d}, \tag{2,2} \]

which causes breakdown.

Thus, in the theory of breakdown proposed by Ioffe, what is essential is the presence of some continuously acting mechanism for the reproduction of “primary” ions. Moreover, in order to ensure a sustained self-maintained current, such a mechanism must not be of external origin (for example, photoionization from an extraneous source), but must be due to processes occurring in the dielectric itself. However, Ioffe’s theory does not provide for such a mechanism.

It is also necessary to note that in Ioffe’s theory it remained unclear whether ionization by impact should be understood as the tearing of ions of a definite sign out of the sites of the lattice, or whether it is the ionization of neutral molecules or atoms. In the first case, “ionization” is accompanied by destruction of the lattice, intensified by each passing avalanche of ions. But then it is unclear why ions of only one sign will be torn out or “ionized.” If the latter does indeed take place, then after the passage of an avalanche there will remain “unionized” ions of the other sign, and a space charge will gradually accumulate, which will substantially affect the development of the process. In the case of ionization of neutral particles, the product

of ionization there will be charged particles of both signs. However, the theory completely ignores particles of the other sign, although their role, if these are electrons, is much more active.

Experimental verification showed that the conclusions of the theory are in contradiction with the experimental data.

The small breakdown delay times obtained from experiment do not agree with the theory, in which the main role is played by heavy particles.

According to (2.1), it follows from Ioffe’s theory that, at a constant field strength, the ionization current should increase exponentially as a function of the thickness of the dielectric. Experiments by the author of the theory showed that, for thicknesses greater than \(0.1\) mm, such a dependence does not occur.

Finally, it follows from (2.2) that the breakdown field strength increases as the dielectric thickness decreases. Experiments by Aleksandrov and Ioffe\(^8\) showed that when the dielectric thickness is decreased to \(0.6 \cdot 10^{-4}\) cm, the breakdown field strength does not change.

Experiments by Austen and Whitehead\(^9\) on thinner mica layers showed that, when the thickness is decreased from \(0.6 \cdot 10^{-4}\) cm to \(0.2 \cdot 10^{-4}\) cm, i.e. by a factor of three, the mean breakdown field strength increases only slightly—by approximately 36%.

Despite the fact that experimental verification of Ioffe’s theory showed its inadequacy, the theory nevertheless played a significant role in the development of views on the nature of breakdown in solid dielectrics. Instead of mechanical theories, the idea of impact ionization was proposed. True, in this form this idea was still in contradiction with the observed facts, but subsequently, after undergoing certain substantial changes and being more fully developed by other authors, it proved to be very useful and fruitful.

3. IONIZATION THEORIES OF A. A. SMUROV, A. A. VOROB'EV AND E. K. ZAVADOVSKAYA

a) Theory of A. A. Smurov

The theory of ionization received further development in the works of A. A. Smurov\(^10\). The mechanism of breakdown of solid dielectrics according to Smurov’s views is as follows.

Under the influence of an electric field, electrons are torn away from atoms—electrostatic ionization. Such tearing away of electrons is quite possible, in Smurov’s opinion, in the case of negative ions, and also in complex molecules in which electrons move in large orbits, and is considerably facilitated when the atom is located at the interface between the dielectric and the electrode. There may be a sufficient number of these detached electrons—

precisely for producing short current pulses. When a voltage is applied for a long time, the mechanism of impact ionization also operates, and it is electrons, not ions, that ionize. The latter, because of their much shorter mean free path and very large mass, are not capable of ionizing.

Electrons freed from atoms as a result of electrostatic ionization move under the action of the electric field toward the anode. In the head part of such a moving electron cloud an intense local electric field is created. This field makes possible further electrostatic ionization (especially of negative ions), which precedes the motion of the electron cloud and increases the cloud itself. Within the cloud, electrons and positive ions, formed during the passage of the head part of the cloud, are found intermixed. Therefore, within the cloud it is difficult to expect the appearance of large potential gradients needed for electrostatic ionization. Within the cloud only impact ionization by electrons accelerated by the electric field will occur. In order that the electrons of the cloud not be attracted by neutral atoms, it is necessary that the potential gradient of the electric field have a sufficiently large value. The rate of advance of the cloud from the cathode to the anode is very high, since it corresponds to the velocity of motion of the electrons. After the first cloud reaches the anode, a column of positive ions is formed, in which impact ionization by electrons, continuously produced as a result of electrostatic ionization, continues to occur.

The field strength at which electrostatic ionization can occur is determined by Smirov in the following way. Smirov calculates the trajectory of motion of a valence electron in an atom when the atom is in an electric field. The trajectory is a complex spiral line. As the field strength is increased, the trajectory shifts and reaches a certain limiting position. On this limiting trajectory the electron is in equilibrium, which at some points of the trajectory is unstable. The slightest jolt received by the electron from outside (for example, in a thermal collision) can take it out of the state of equilibrium, and then the electron will move under the action of the field, continuously receding from the atom. The magnitude of the field strength causing the electron to pass onto the limiting trajectory depends on the orientation of the initial electron orbit with respect to the field direction. This field strength will be smallest when the electric force lies in the plane of the initial trajectory. For this case

\[ E = 0.086\,\frac{e}{\rho^{2}}\,\frac{Z}{2}\left(1+\frac{1}{\sqrt{Z}}\right), \]

where \(\rho\) is the radius of the electron orbit, and \(Z\) is the number of valence electrons. For a hydrogen-like atom \(Z=1\), \(\rho=0.5\cdot 10^{-8}\ \mathrm{cm}\), and

\[ E \simeq 5\cdot 10^{8}\ \mathrm{V/cm} \]

at absolute-zero temperature.

If the temperature of the dielectric is above zero, then the field strength required for ionization will be smaller, since part of the work expended on ionization is performed at the expense of the electron’s thermal energy. Since the thermal energy of the electron is small in comparison with the work of ionization, the reduction in the critical field strength at which electrostatic ionization occurs is small, and the order of magnitude does not change.

Another factor which, in Smurov’s opinion, can change the critical field strength is the magnetic field of moving electrons. This field can arise as a result of a change in the electron orbits under the action of an external electric field, even in those atoms in which the magnetic fields of the moving electrons mutually compensate one another in the normal state. The dependence of the breakdown voltage on the magnetic field is explained by the influence of the latter on the motion of the electron. Depending on its orientation, the magnetic field may cause either an increase or a decrease in the breakdown voltage. The relative change in the breakdown field strength, according to Smurov’s calculations, is equal to

\[ 1 \pm \frac{2.5\sqrt{\rho^{3}}}{c\sqrt{mZ}}\,H, \]

where \(H\) is the magnetic-field strength, \(m\) is the mass of the electron, and \(c\) is the speed of light. For a hydrogen-like atom the correction to unity is \(10^{-9}H\). Thus the influence of the magnetic field is small; it does not change the order of magnitude of the breakdown field strength.

Finally, Smurov also points to a third factor lowering the breakdown field strength: the influence of the electric field of neighboring atoms, molecules, and ions, i.e. the influence of the field of the crystal lattice. Although Smurov indicates that this influence may be large, he does not determine it quantitatively. This influence was taken into account by F. F. Vol’kenshtein using the methods of quantum mechanics (see below). It proved to be very significant—capable of lowering the breakdown field strength by several orders of magnitude. Without taking it into account, the potential gradient at which electrostatic ionization can occur is extremely large, approximately three orders of magnitude greater than that obtained from experiment.

Despite the fact that Smurov did not carry the calculations through to the end, as a result of which the results he obtained proved unacceptable, the ideas underlying his theory played a significant role. Smurov was the first to propose the idea of electrostatic ionization,

which ten years later was worked out in greater detail by the methods of quantum mechanics. The idea of impact ionization in solid dielectrics was also modernized by Smurov; the action of light particles—electrons, and not heavy ions—was taken as the basis of this process. The validity of this assumption was confirmed by Hippel’s experiments \(^{11}\). Further, Smurov drew attention to the necessity of the continuous formation of initial ionizing particles. Smurov’s ideas are, in general, clearer than those developed before him; however, they are still extremely imperfect. The question of the mechanism and causes of the destruction of the dielectric is not touched upon at all by the author. In connection with this, the mechanism of breakdown growth has not been worked out. The mechanism of the conductivity of the dielectric during the process of breakdown and after its completion is insufficiently developed and in this form is unacceptable. The influence of various factors on the breakdown process has not been clarified. Finally, there is no rigorous mathematical theory either of the individual processes or of the entire phenomenon as a whole.

b) The theory of A. A. Vorob’ev and E. K. Zavadovskaya

The ionization theory of breakdown proposed by Ioffe possessed, among other shortcomings, two substantial defects which made it unacceptable. As was already noted above, in Ioffe’s theory the role of the ionizing agent was assigned to ions, which contradicted experimental data. Particles with such a large mass as ions are not capable of providing the observed rate of development of the process. This difficulty was eliminated by Smurov, who ascribed the ionizing action not to heavy particles—ions—but to light ones—electrons.

The second major defect in Ioffe’s theory consists in the fact that it leads to an excessively large field strength required for the onset of breakdown. If by ionization one understands the tearing away of an electron from a neutral atom or negative ion as the result of the impact of a particle, then the replacement of the ion as the ionizing particle by an electron considerably reduces the necessary field strength. However, if it is assumed that the electron acquires the energy required for ionization over the course of one mean free path, the field strength must be:

\[ E = \frac{U_i}{l}. \]

In the best case, if the motion of the electron coincides with the direction of the field and the length of the mean free path is equal to the distance between neighboring atoms in the dielectric (of the order of \(10^{-8}\) cm), then, taking into account the value of the ionization potential (of the order of several eV), the smallest field strength is obtained of the order of \(10^8\) V/cm, i.e. approximately two orders of magnitude greater than that observed.

The latter difficulty is what A. A. Vorob’ev and E. K. Zavadovskaya attempt to eliminate in their theory1. Vorob’ev and Zavadovskaya believe that breakdown of alkali-halide crystals occurs as a result of destruction of the lattice by electron impacts. Denoting by \(\tau\) the time from the beginning of the electron’s acceleration to the moment of its collision with the lattice, at which the latter is destroyed, the authors determine the drift velocity, i.e. the mean velocity of the electron in the direction of the electric field \(v\). According to the authors’ definition,

\[ v=\frac{e}{m}E\tau, \]

and the energy acquired by the electron from the electric field is

\[ W=\frac{eE^2}{m}\tau^3. \]

The authors assume that this energy is expended in destroying the lattice, and take it to be proportional to the lattice energy \(U\):

\[ \frac{eE^2}{m}\tau^3=bU. \]

Since neither the value \(b\) and \(\tau\), nor the dependence of \(\tau\) on \(E\), is determined by the authors, there is no possibility of establishing agreement between the theory and the experimental results. However, an estimate of the theory can be made indirectly. Denoting by \(l\) the distance that the electron traverses between two collisions that destroy the lattice, we obtain

\[ l=v\tau=\frac{bU}{eE}. \]

Using the experimental data for the breakdown voltages and lattice energies of alkali-halide crystals given in the work of Vorob’ev and Zavadovskaya, we obtain the values for \(l\), taking for \(b\) the smallest possible value, equal to unity. These values, as well as the ratio of \(l\) to the lattice constant \(a\), are given in Table 1.

Table 1

Crystal	\(U\) (eV)	\(E\) (V/cm)	\(l\) (cm)	\(a\) (cm)	\(l/a\)
KJ	6.4	\(0.57\cdot10^6\)	\(10.8\cdot10^{-6}\)	\(3.5\cdot10^{-8}\)	310
KBr	6.8	\(0.70\cdot10^6\)	\(9.6\cdot10^{-6}\)	\(3.3\cdot10^{-8}\)	290
NaJ	7.05	\(0.69\cdot10^6\)	\(10.3\cdot10^{-6}\)	\(3.2\cdot10^{-8}\)	320
KCl	7.2	\(1.00\cdot10^6\)	\(7.2\cdot10^{-6}\)	\(3.1\cdot10^{-8}\)	230
NaBr	7.55	\(0.81\cdot10^6\)	\(9.5\cdot10^{-6}\)	\(3.0\cdot10^{-8}\)	320
NaCl	7.8	\(1.5\cdot10^6\)	\(4.9\cdot10^{-6}\)	\(2.8\cdot10^{-8}\)	170
KF	8.4	\(1.8\cdot10^6\)	\(4.66\cdot10^{-6}\)	\(2.7\cdot10^{-8}\)	170
NaF	9.3	\(2.4\cdot10^6\)	\(3.7\cdot10^{-6}\)	\(2.3\cdot10^{-8}\)	160
LiF	10.4	\(3.1\cdot10^6\)	\(3.35\cdot10^{-6}\)	\(2.0\cdot10^{-8}\)	170

It is evident from the table that this ratio fluctuates from 160 to 320. In reality this ratio should be still larger, since in the calculation it was assumed that the electron transfers all the accumulated energy upon impact and that its motion occurs only in the direction of the field.

Thus, the theory of Vorob’ev and Zavadovskaya is based on the assumption that the electron travels in the crystal, without collisions, a distance several hundred times greater than the distance between lattice sites. Such an assumption is unacceptable. Between two destructive impacts the electron will undergo many collisions, in which it will transfer to the lattice part of the energy accumulated in the field. This circumstance is not taken into account in the theory of Vorob’ev and Zavadovskaya. The second principal shortcoming of the theory of Vorob’ev and Zavadovskaya is that this theory cannot explain the sharp increase in current observed at breakdown.

4. THE THEORY OF IONIZATION BY SLOW ELECTRONS OF A. HIPPEL

Hippel’s theory\(^{13}\) is based on two assumptions.

The first assumption is that the electrons acquire the energy necessary for ionization not in a single event, but over the course of several runs.

The second assumption is that the energy acquired by the electrons in the electric field is, under certain conditions, greater than the energy lost by them in collisions with the lattice. The field strength at which these conditions begin to be satisfied was taken by Hippel as the breakdown strength.

Since the energy of the electrons at which the second condition begins to be satisfied is small—of the order of \(0.1\)–\(0.2\) eV—Hippel’s theory was called the theory of ionization by slow electrons.

The essence of Hippel’s theory\(^{13}\) is as follows. In every dielectric there is a certain number of free electrons that can be accelerated by the electric field applied to the dielectric. Collisions of electrons with atoms and molecules are accompanied by a loss of energy for the excitation of vibrational motion of the particles (elastic collisions) and for optical excitation or ionization (inelastic collisions). The probability \(P\) of exciting vibration of the particles and the amount of energy \(Q\) expended on this depend on the kinetic energy of the electron and are schematically represented by the curve shown in Fig. 3. The curve must have a maximum. Indeed, electrons with very small and very large energy, when colliding with a particle, transfer little energy to it: in the first case, because of the small amount of energy possessed by the electron; in the second case, because of the short duration of the time of interaction between the electron and the particle.

If, between two collisions, an electron acquires in a field an energy less than the value \(W_B\), corresponding to the maximum of the curve in Fig. 3, then the electron can expend the energy acquired in the field only on the excitation of an atom. As the field strength increases, the average energy acquired by the electron between collisions increases and exceeds \(W_B\); at the same time the energy expended by the electron on excitation of the atom decreases. The possibility of energy accumulation appears. Thus, if over the length of one free path \(l\) an electron acquires on the average an energy less than \(W_B\), then ionization is impossible. But if the average energy acquired by the electron during one free path, although less than \(W_i\)—the ionization energy—is greater than \(W_B\), then ionization is possible. Consequently, in Hippel’s opinion, the critical field strength capable of causing ionization and, as will be shown below, breakdown, is the field in which the electron during one free path acquires an average energy equal to \(W_B\), corresponding to the maximum of the curve of the probability of energy loss for excitation of an atom. In reality the field strength may be somewhat smaller.

Fig. 3.

During ionization, an electron tears away from a negative ion or from a neutral atom an electron which, in turn, together with the first, is accelerated and ionizes. An avalanche of electrons is gradually formed, which moves from the cathode to the anode and leaves behind a positive space charge. As a result of this, the bond between the structural elements of the dielectric (atoms or ions) that existed before ionization is disrupted, and a mechanical stress arises between them. If the structure of the dielectric is not sufficiently elastic, as is the case at low temperatures, slipping occurs and a crack appears. As the electron avalanche advances toward the anode, the crack grows after it. The growth of the crack occurs not continuously, but in jumps, which occur when the stress causing the slipping reaches a certain value. The latter depends on the magnitude and distribution of the space charge.

From the proposed mechanism one may draw the following conclusions:

1. The critical breakdown field strength is approximately equal to

\[ E=\frac{W_B}{l}, \]

the sole condition for the occurrence of breakdown is the onset of ionization.

1. In dielectrics with a crystalline lattice there must exist preferential directions of breakdown, namely those directions along which the bonding force between particles is smaller and, consequently, the probability of energy loss to excitation is smaller.

2. The current accompanying breakdown must grow according to the exponential law characteristic of the process of avalanche formation.

3. For breakdown to begin, a limited number of initial free electrons is necessary. Reproduction of the initial electrons is not required.

4. The principal active agents of ionization are slow electrons.

Hippel sees the proof of the validity of his theorem in the following.

In a solid or liquid dielectric the particles are situated sufficiently close to one another. The electron is under the action not only of the field of its own atom, but also of the field of neighboring atoms. The ionization potential is much smaller than in the case of an isolated atom. It is approximately the same as the excitation potential. The onset of luminescence of a dielectric may be taken, to a first approximation, as the onset of ionization. Therefore, for dielectrics with a crystalline structure, the electric-field strength at which the luminescence of the dielectric begins, multiplied by the lattice constant, must be proportional to the frequency of the radiation. This regularity is confirmed by the author’s experiments.

At the basis of this argument lies the author’s assertion that an electron with energy below \(W_B\), corresponding to the maximum of losses to excitation, cannot excite. This assertion is not obvious and requires more detailed consideration.

Let us plot along the ordinate axis the electric-field strength, and along the abscissa axis the energy \(W\) which the electron acquires in the field during one free path (Fig. 4). The relation between these quantities is represented by a straight line, denoted in Fig. 4 by the letter \(E\). In the same figure is given the relation between the energy \(Q\), lost by the electron to excitation of the atom at each collision, and the electron energy \(W\). For \(Q\) and \(W\) the same scale has been chosen. In addition, let us draw in the figure the straight line \(A\) at an angle of \(45^\circ\).

Suppose that the dielectric is in a field of strength \(E_1\), and that during each free path the electron acquires the energy \(W_1\). After the first collision the electron will have the energy \(W'_1\), equal to the length of the segment \(B_1A_1\). After the second path the electron’s energy

will be \(W_2 = W_1 + W'_1\), and after the second collision—\(W'_2\), measured by the length of the segment \(B_2A_2\). After the third free path the electron will possess the energy \(W_3 = W_1 + W'_2\), and after the third collision—\(W'_3 = B_3A_3\). It is not difficult to see that after each successive collision the electron energy will be either greater or less than \(W'\) and will tend toward the latter value. \(W'\) is determined by the intersection of the curve \(Q\) with a straight line parallel to the abscissa axis and passing through the point \(A_1\). When the electron energy becomes equal to \(W'\), the electron will be in an equilibrium state: on the average, at each collision the electron will lose to the excitation of the atom as much energy as it gains on the average during one free path.

Fig. 4.

With increasing field strength, the energy which the electron gains during one free path increases, and consequently so does the energy of the equilibrium state. At a certain value of the field strength the equilibrium state becomes impossible. This will evidently occur when the residual energy after each collision increases continuously.

The greatest energy \(Q_m\) that the electron can transfer to an atom in a collision is equal to \(CB\)—the \(y\)-ordinate of the maximum of the curve \(Q\). The field strength \(E_m\) at which the electron acquires such an energy during one free path is the limiting one at which an equilibrium state is still possible. However, this equilibrium state will not be stable. All equilibrium states for which the mean energy of the electron is less than \(Q_m\) are stable. Consequently, all values of the field strength

of field strength below \(E_m\) there corresponds an equilibrium state in which the electron cannot acquire a large energy sufficient for excitation. At a value \(E > E_m\) there can be no equilibrium state; after each run the electron energy will increase continuously. Thus the critical field strength at which excitation begins will be equal to \(E_m\), and \(aE_m = h\nu\). In this formula \(a\) is the lattice constant, \(h\) Planck’s constant, and \(\nu\) the frequency of the initial light. This dependence, as was indicated, is confirmed by Hippel’s experiments.

If the curve \(Q\) represents the total losses for rocking and excitation, then by analogous reasoning it can be shown that, at the field strength corresponding to the maximum, ionization processes will begin. This field strength will be greater than the strength corresponding to the onset of excitation. One may therefore agree with the author’s arguments that the experiments confirm the theory of the onset of ionization. However, this cannot serve as confirmation that the onset of ionization necessarily leads to breakdown. The latter is confirmed by other indirect experimental data. The conception of ionization as the sole condition for the occurrence of breakdown leads to the conclusion that the breakdown field strength is independent of the thickness of the dielectric. This proposition is approximately confirmed by experiment.

As a second argument in favor of the validity of his theory, Hippel regards the experimental data on the existence of preferred directions of breakdown in dielectrics with a crystal lattice. In a crystal of rock salt the nearest like-named ion is situated along the diagonal of a square (the distance between them is \(d=\sqrt{2}a\)), then along the diagonal of a cube (\(d=\sqrt{3}a\)) and along the edge of a cube (\(d=2a\)). The closer the particles are situated, the lower the potential barrier between them, the smaller the binding force and the lower the maximum of the losses for rocking, and consequently (see Fig. 4) the smaller \(E_m\). Therefore, in crystals of rock salt one should expect first the appearance of breakdowns in the direction of the plane diagonal. With increasing voltage, breakdowns should occur in the direction of the cube diagonal and then, at still higher voltage, in all directions. These expectations are justified by Hippel’s experiments.

However, the experiments of Walter and Inge\(^{14}\) do not confirm Hippel’s data. According to these authors, the direction of breakdown propagation in crystals depends on the character of the field and on temperature. In crystals of rock salt at low temperatures and positive constant voltage, breakdown propagates predominantly along the cube diagonal.

It should also be noted that the preference for breakdown in the direction of the nearest particle also follows from other ionization-

... theories and is not an exclusive indication of the validity of the theory of ionization by slow electrons.

The breakdown mechanism proposed by Hippel was not developed mathematically by him. A mathematical theory as applied to alkali-halide crystals was proposed by Seeger and Teller[^15]. Proceeding from Hippel’s mechanism, the authors assume that, for breakdown to arise, the onset of ionization is necessary. The conditions under which the loss of energy by an electron in setting atoms into vibration reaches its maximum value are critical for the onset of ionization. From the force of interaction of an electron with an ion, the authors determine the maximum energy that the electron transfers to the ions in traversing unit path length. Equating this energy to the energy acquired by the electron in the field during the time of traversing unit path length, the authors determine the field strength corresponding to the onset of ionization. In the calculations the authors are compelled to make certain assumptions and approximations and obtain the following final formula:

\[ E=\frac{\eta\pi^{2}me\nu}{h}\left(\frac{1}{\varepsilon'}-\frac{1}{\varepsilon}\right), \tag{4,1} \]

where \(\eta=0.74\) if the Bohr approximation is used, and \(\eta=1.82\) if the Born approximation is used. The latter approximation, in the authors’ opinion, is less justified. In (4.1) \(m\) is the effective mass of the electron, \(e\) is the charge of the electron, \(\nu\) is the frequency of the residual rays, \(h\) is Planck’s constant, \(\varepsilon\) is the dielectric constant of the real crystal, and \(\varepsilon'\) is the dielectric constant of the crystal under the assumption of immobile ions. The authors take \(\varepsilon'\) to be equal to the square root of the coefficient of refraction of rays with a frequency lying at the lower boundary of the infrared region.

Since calculation of the effective mass of the electron is very difficult, Seeger and Teller substitute the ordinary value \(m_0\) into the formula and determine the breakdown field strength, using the Bohr approximation, although they consider it less justified. Comparing the calculated value of the field strength with that obtained from Hippel’s experiments, the authors find the ratio between the effective and ordinary masses of the electron. This ratio lies within the limits from 1.2 to 3.3. Seeger and Teller believe that such a ratio lies within reasonable limits for alkali-halide crystals. The authors themselves consider the theory not very accurate and in need of revision. Fröhlich[^16] pointed out certain errors made by Seeger and Teller in their derivation, which render this theory unacceptable.

A completely different formula was proposed by Fröhlich[^17] for determining the field strength at which there occurs an equ...

equilibrium state. This formula was derived by the author in the following way. First, Franck found the distribution of electron momenta by directions, taking into account the action of the electric field and collisions. Then, using this distribution, he determines the mean increase in the electron energy and equates it to the mean energy loss in collisions. As a result, after certain approximations, the author obtains the following formula for the field strength in the equilibrium state:

\[ E_p=\frac{2\pi^2 W\sqrt{3mkT}}{Mva}, \tag{4,2} \]

where \(W\) is the electron energy in ev, at which the losses to excitation are greatest, \(M\) and \(v\) are the mass and velocity of the atoms of the dielectric, \(k\) is Boltzmann’s constant, \(T\) is the temperature, and the remaining notation is as before. Since the electron energy corresponding to the greatest excitation losses is unknown, the formula cannot be used directly to test Hippel’s theory.

In deriving the last formula Franck assumed that the energy acquired by an electron in an electric field is completely expended in collisions. Therefore (4,2) is valid only for equilibrium states. The latter occur for electron energies \(W<Q_m\) (Fig. 3). At electron energy \(W>Q_m\) an equilibrium state cannot exist. Therefore formula (4,2) must have a limit of applicability, which, however, is not established by the author. Neither the derivation of the formula nor the formula itself makes it possible to establish this limit. It can be established, as will be seen from the further exposition, by comparing (4,2) with another formula derived by Franck. The results following from this comparison cast doubt on the validity of the formula itself.

In addition to the above, Franck proposed a second method for determining the breakdown field strength, based on the idea of energy accumulation. This method consists in the following\({}^{17}\).

Between two ionizations an electron may undergo several elastic impacts. The probability that the electron will acquire the energy necessary for ionization without undergoing a single elastic collision is equal to

\[ e^{-\theta}, \tag{4,3} \]

where

\[ \theta=\int_0^{R_i}\chi\,dt, \]

represents the most probable number of collisions during acceleration; \(\chi\) is the probability of collision per unit time,

\[ R_i=\frac{2\pi P_i}{h}, \]

\(P_i\) is the electron momentum corresponding to the ionization energy, and \(h\) is Planck’s constant. Since

\[ \frac{dR}{dt}=\frac{2\pi eE}{h}, \]

then

\[ E=\frac{h}{2\pi e\theta}\int_{0}^{R_i}\chi\,dR. \]

An approximate evaluation of the integral leads to the following final formula:

\[ E=\frac{\pi^3 W k T}{\theta M v^2 a}. \tag{4.4} \]

According to Frantz, \(\theta\) denotes the probable number of electron collisions before ionization and varies within the range from 3 to 10. An exact determination of the value of \(\theta\) is very difficult, and the author takes \(\theta=5\). Substituting the values of the constants \(M=5\cdot10^{-23}\), \(v=4\cdot10^5\), \(a=3\cdot10^{-8}\), and expressing \(W\) in eV, the author obtains the following expression for \(E\):

\[ E \simeq 2\cdot10^4 W\,\frac{T}{\theta}. \]

Taking \(\theta=5\), he obtains for rock salt at ordinary temperatures \((T=300)\) a breakdown field strength equal to

\[ E \simeq W\cdot10^6\ \text{V/cm}. \]

This value is several times greater than the experimental one.

It was shown above (Fig. 4) that the critical value of the electron energy \(W\), at which the onset of ionization is already possible, is equal to \(Q_m\). The electron acquires this critical energy in a field with strength \(E_m\). Consequently, the region of field strengths in which formula (4.2) is valid is characterized by the value

\[ E_p \gg E_m. \]

Since

\[ E_m < E_i, \]

then

\[ E_p < E_i, \tag{4.5} \]

where \(E_i\) is the field strength causing ionization. Formula (4.4) is valid for all values of \(E\). Putting in (4.4) \(W=W_i\), the ionization energy, we obtain

\[ E_i=\frac{\varkappa^3 W_i^2 kT}{\theta M v^2 a}. \tag{4.6} \]

Substituting (4.2) and (4.6) into (4.5), Franck obtains the relation

\[ kT>12\theta^2 m v^2, \tag{4.7} \]

which characterizes the region of applicability of (4.2). Hence the author concludes that for \(kT<12\theta^2 m v^2\) the breakdown field strength is determined by (4.4), and for \(kT>12\theta^2 m v^2\) by (4.2). It is difficult to agree with this conclusion of the author. When substituting (4.2) into inequality (4.5), one cannot put \(W=W_i\) in the right- and left-hand sides of the inequality, as Franck did. In (4.2) it is necessary to put \(W=Q_m\); then

\[ kT>12\theta^2 m v^2\left(\frac{Q_m}{W_i}\right)=\theta, \tag{4.8} \]

with \(Q_m<W_i\).

Furthermore, inequalities (4.7) or (4.8) do not at all give the conditions under which (4.2) or (4.4) should be used to determine the breakdown field strength. They determine the conditions under which an equilibrium state is possible or impossible. Therefore, in all cases the breakdown field strength can be determined from (4.4), if in it one takes \(W=W_i\). For \(kT>\theta\) the breakdown field strength can also be determined from (4.2), if in it one takes \(W=Q_m\).

Even if one uses Franck’s inequality (4.5) and takes, as he does, \(\theta=5\), then for rock salt we obtain \(T>325^\circ\text{ C}\), i.e. an equilibrium state is possible only at a temperature above normal; a somewhat unexpected conclusion. Some objections are also raised by the derivation of (4.4). If \(\theta\) denotes the mean number of collisions between two ionizations, then formula (4.3) is incorrect and \(\varkappa\) cannot have the meaning which the author assigns to it—the probability of collisions per unit time. \(\varkappa\) must then denote the number of collisions per unit time. If, however, \(\varkappa\) has the meaning which the author assigns to it, then formula (4.3) is correct, but then \(\theta\) has another meaning and denotes the probability of a collision during the time of acceleration of the electron up to ionization, and cannot be greater than unity. In the latter case, for rock salt \(T>13^\circ\text{ C}\), which causes no perplexity, but the breakdown field strength will be 5 times greater than that calculated earlier and an order of magnitude greater than the experimental one.

A mathematical theory for ionic crystals, based on Gippel’s assumptions, was also proposed by Callenow\(^{18}\).

According to the theory of Born and Karman, Callen describes the vibrational motions of ions by waves. The vibrations of the ions lie in two regions: “optical” and “acoustic,” and in each region the vibrations are subdivided into two groups of transverse waves and one group of longitudinal waves. Thus, there are in all \(6N\) waves, where \(N\) is the number of cells in the crystal (the cell volume is \(2a^3\), \(a\) is the lattice constant).

The author regards the displacement of ions in the process of vibration from the equilibrium position as a polarization with the corresponding charge density. Only for longitudinal vibrations is this density different from zero.

Since the “optical” vibrations are of essential importance, the author neglects the “acoustic” ones and, in addition, assumes that all \(N\) longitudinal “optical” vibrations have a single frequency \(\nu\), equal to the frequency of the longest wave.

Assuming that the relative displacement of unlike neighboring ions is equal to the difference of the amplitudes of their vibrations, the author determines the electric-field strength associated with this polarization.

Since there are very few electrons in the conduction band, Callen neglects the interaction of the electrons with one another and determines the interaction energy of an electron with the lattice as the mutual energy of the field of the electron with the polarization field. Callen treats this interaction as a perturbation and determines, by quantum-mechanical methods, the probability of transfer of energy equal to \(h\nu\) in one second. If the energy of the electron is less than \(h\nu\), then transfer of energy from the electron to the lattice is impossible; only the reverse process is possible. If the energy of the electron is greater than \(h\nu\), then the probability that the electron will transfer the energy \(h\nu\) to the lattice is greater than the probability of its receiving it.

The quantity of energy \(B\) which an electron with energy \(W\) transfers to the lattice per unit time turns out to be equal to

\[ B = C \sqrt{\frac{h\nu}{W}} \left[ \frac{e^{\frac{h\nu}{kT}}}{e^{\frac{h\nu}{kT}}-1} \ln \frac{1+\sqrt{1-\frac{h\nu}{W}}}{1-\sqrt{1-\frac{h\nu}{W}}} - \frac{1}{e^{\frac{h\nu}{kT}}-1} \ln \frac{\sqrt{1+\frac{h\nu}{W}}+1}{\sqrt{1+\frac{h\nu}{W}}-1} \right], \]

where

\[ C=\frac{\pi\sqrt{2m}\,(ee^*)^2}{Ma^3\sqrt{h\nu}}. \]

Here \(e^*\) is the effective charge of the ion, smaller than its actual charge owing to the displacement of the electron orbits and the electronic polarization, caused in turn by displacement of the ions; \(m\) is the effective mass of the electron, \(M=\dfrac{M_+M_-}{M_+ + M_-}\) is the reduced mass of the ions.

Thus \(\dfrac{B}{C}\) is only a function of \(\dfrac{h\nu}{W}\) and \(\dfrac{h\nu}{kT}\), and has flat maxima as a function of \(\dfrac{h\nu}{W}\) at different values of \(\dfrac{h\nu}{kT}\).

Next Callen determines the energy acquired by an electron in an electric field per unit time. This energy is equal to

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

where \(\tau\) is the relaxation time; it depends on the field strength and on the electron energy.

The relaxation time is defined by Callen as follows:

\[ \tau=\frac{v}{\dfrac{dv}{dt}}, \]

where \(v\) is the mean velocity of the electron directed along the field, and \(\dfrac{dv}{dt}\) is the mean directed acceleration.

This mean acceleration the author defines as the difference between the increment of the directed velocity per unit time under the action of the field and the retardation due to collisions with the lattice. This difference is averaged by the author over all possible directions of the velocity and possible deflections of the electron after a collision, taking into account the probability of the corresponding velocities and deflections.

For \(\dfrac{1}{\tau}\) the following dependence is obtained:

\[ \frac{1}{\tau} = \frac{C}{h\nu} \left( \frac{1}{2}\frac{B}{C}\frac{h\nu}{W} + \sqrt{\frac{h\nu}{W}} \right) \times \left[ \frac{e^{\frac{h\nu}{kT}}}{e^{\frac{h\nu}{kT}}-1} \sqrt{1-\frac{h\nu}{W}} + \frac{1}{e^{\frac{h\nu}{kT}}-1} \sqrt{1+\frac{h\nu}{W}} \right]. \]

From this expression it is seen that \(\dfrac{h\nu}{C\tau}\) is also a function only of \(\dfrac{h\nu}{W}\) and \(\dfrac{h\nu}{kT}\).

Thus, the energy acquired by an electron in an electric field per unit time depends on the electron energy, on the temperature, and, quadratically, on the field strength.

The energy lost by an electron per unit time in collisions with the lattice also depends on the temperature and the electron energy, but does not depend on the field strength. Consequently, there exists an equilibrium field strength at which the energy lost is equal to the energy gained.

Equating \(A\) to \(B\), Callen finds the equilibrium field strength

\[ E = E_0 \sqrt{\frac{h\nu}{C\tau} - \frac{B}{C}}, \]

where

\[ E_0 = \frac{\sqrt{2\pi m e e^{*2}}}{M a^3 h\nu}. \]

The ratio \(\dfrac{E}{E_0}\) also depends only on \(\dfrac{h\nu}{W}\) and \(\dfrac{h\nu}{kT}\). The dependences of \(\dfrac{E}{E_0}\) on \(\dfrac{h\nu}{W}\) for different values of \(\dfrac{h\nu}{kT}\) have maxima (Fig. 5).

Fig. 5.

Let us denote the value of \(E\) corresponding to the maximum of \(\dfrac{E}{E_0}\) at a given temperature \(T\) by \(E_{nT}\). Let the electron energy be \(W_1\), and let the corresponding equilibrium field strength be \(E_1\). If the actual field strength \(E_2\) is greater than \(E_1\), then the energy balance for the electron is positive; its energy will increase to the value \(W_2\) corresponding to the field strength \(E_2\). If \(E_2\) is greater than \(E_{nT}\), then equilibrium is impossible and the electron will continuously gain energy and may ionize. Callen, in accordance with Hippel’s assumption, takes the field strength \(E_{nT}\), corresponding to the maximum of the curve \(\dfrac{E}{E_0}\), to be equal to the breakdown field.

At \(T = 0\) the ratio \(\dfrac{E_{nT}}{E_0}\) proves to be equal to unity; consequently, \(E_0\) is the breakdown field strength at absolute-zero temperature.

Using these formulas, Callen, taking \(m = m_0\), calculated the breakdown field strengths for ionic crystals. These data, first published in Hippel’s work\({}^{19}\), are given in Table II.

Table II

Crystal	Breakdown field strength \((\text{in }10^6\ \text{V/cm})\), calculated by Callen according to Hippel’s theory: Hippel’s data	Breakdown field strength \((\text{in }10^6\ \text{V/cm})\), calculated by Callen according to Hippel’s theory: Callen’s data	Breakdown field strength \((\text{in }10^6\ \text{V/cm})\), calculated by Callen according to Fröhlich’s theory	Experimental data
RbI	1.64	0.52	—	0.49
RbBr	2.00	0.73	0.19	0.63
KI	2.19	0.57	—	0.57
RbCl	2.14	1.12	0.24	0.83
KBr	2.45	0.85	0.23	0.70
KCl	2.9	1.20	0.33	1.00
NaBr	3.34	0.98	0.47	0.81
NaCl	3.60	1.57	0.57	1.50
NaF	5.48	3.57	—	2.40
KF	—	2.96	—	1.60
LiF	—	8.8	—	2.8

For most crystals, Hippel’s theory, according to these calculations, gives values exceeding the experimental ones by a factor of 2–4. Hippel explains such disagreement between experiment and theory by a number of reasons. First of all, by the circumstance that Callen used the perturbation method in a region where its applicability is not legitimate. Further, Callen assumed the presence of a perfect ionic lattice, whereas in reality the heteropolar character of ionic crystals decreases very rapidly with increasing ion size. Therefore the polar disturbance of the potential and the coupling between the electron and the lattice are overestimated. In addition, the maximum in the loss curve cannot be identified with the breakdown field strength. Secondary electrons receive some energy from the ionizing electrons, and there is an appreciable probability of energy transfer from the lattice to the electrons. Therefore breakdown may occur at a field strength somewhat below the maximum. However, the uncertainty due to the electrons having some initial energy cannot be appreciable because of the very flat shape of the curve near the maximum, as Callen showed. On the other hand, Seitz’s calculations\({}^{20}\) for a nonpolar lattice showed that,

that the maximum lies at appreciably higher energies and has a magnitude that cannot be neglected. Therefore, in Hippel’s opinion, given the present state of knowledge, one cannot expect good agreement between the calculated values and the experimental ones, especially taking into account the complexity of the phenomena and the series of simplifications that have to be made in the calculations.

Later Callen\(^{18}\) published other data, given in the same Table II. The latter values are much smaller than those published by Hippel. Callen’s new data for the breakdown field strength of ionic crystals, if LiF is not counted, as is seen from Table II, exceed the experimental data by no more than 60%; for the majority of the listed crystals, by no more than 30%. The very sharp discrepancy between the theoretical and experimental values of the breakdown field strength for LiF can be explained, as the author believes, by the presence in LiF of a large number of anomalous homopolar bonds not taken into account by the theory. Taking these bonds into account should bring the theoretical value closer to the experimental one.

Considering a number of simplifications and approximations adopted by Callen in the derivations, it should be acknowledged that the data of his theory agree rather satisfactorily with the experimental data. Nevertheless, the breakdown field strengths predicted by his theory are always greater than the experimental ones. The author believes that the excess of the theoretical values over the experimental ones should be explained by the presence of fluctuations in the electron energies not taken into account by the theory.

The criterion for the breakdown field strength adopted by Callen, proposed by Hippel, means that breakdown occurs when the average energy of the electron corresponds to the maximum of the curve. In reality, however, because of the statistical distribution of electrons over energies, some of them will have an energy greater, and some less, than the specified one. Thus, for breakdown to occur it is not necessary that all electrons possess an energy not less than the specified one. At a field strength smaller than the maximum, there will also be electrons possessing an energy not less than the specified one and, consequently, capable of ionizing. Seitz\(^{20}\) showed that, at a field strength 80% smaller than the theoretical breakdown value, there is at least one electron capable of ionizing. Seitz showed that if an electron traverses in the crystal, in the direction of the field at the breakdown field strength, a distance equal to 1 cm, then it produces an avalanche of electrons amounting to \(10^5\). For destruction of the dielectric, approximately \(10^{13}\) such avalanches are necessary. Such a number of electrons with energy not less than \(W_B\) (Fig. 3) can be provided depending on the electron distribution function even at an average electron energy less than \(W_B\). The excess of the magnitude of the breakdown field strength obtained from Callen’s theory over the experimental one shows that

criteria. Hippel’s criterion ensures a number of ionizing electrons greater than is necessary. Moreover, the different amount of excess for different crystals indicates the purely formal character of the criterion. Indeed, Hippel’s criterion, properly speaking, indicates the certainty of the presence of electrons capable of ionizing at a field strength corresponding to the maximum of the curve. However, this still does not mean that in this case the ionization will be sufficiently effective to ensure breakdown. It is possible that a less effective ionization is sufficient for this, or, conversely, that a more effective ionization is required. The effectiveness of ionization depends on the magnitude of the avalanche, i.e., on the thickness of the dielectric and its nature, and on the number of ionizing initial electrons, i.e., on the function of the

Fig. 6.

energy distribution of the electrons. These circumstances are not taken into account by the theory.

The formal character of Hippel’s criterion is especially clearly seen when considering the dependence of the breakdown field strength on temperature. According to the Callen–Hippel theory, the breakdown field strength for ionic crystals increases with temperature. The available experimental data are contradictory. Thus, experiments by Malmow[^21] showed the independence of the breakdown field strength for KBr from temperature. Experiments by Austen and Whitehead[^9] showed that the breakdown field strength for KBr increases with temperature, at first slowly, then at a temperature of about \(20^\circ\mathrm{C}\) rises sharply, and finally again increases more slowly. Hippel[^19] tested KBr with constant and 60-cycle voltage and with constant voltage with a rise rate over a time of \(10^{-3}\), \(10^{-4}\), and \(10^{-6}\) sec. up to breakdown (Fig. 6). In the last case a linear dependence increasing with temperature was obtained; in all other cases, brea-

breakdown field strength first increases with temperature and then falls. Hippel obtained the same curve with a maximum for an NaCl sample at constant voltage.

One may agree with Hippel’s assumptions that, when the voltage is increased slowly, high-voltage polarization or other similar phenomena appear, which do not permit the true field strength to be determined correctly. When the voltage is raised rapidly, in a time of \(10^{-6}\) sec., the purest conditions are obtained and the dependence of the breakdown field strength on temperature is linear, as required by Callen’s theory. However, the experimental straight line for this case lies considerably below the theoretical one and, most importantly, it is more gently sloping. The steeper character of the theoretical straight line illustrates the formalism of the Hippel criterion, which does not take into account the increase, with temperature, in the number of electrons capable of ionizing.

5. THE THEORY OF IONIZATION BY FAST ELECTRONS OF H. FRÖHLICH

Fröhlich \(^{32}\) starts from the same premises as Hippel. The difference between these two theories lies only in the criterion. Whereas Hippel assumes that electrons with an energy corresponding to the maximum of the loss curve (of the order of \(0.1\)–\(0.2\ \mathrm{eV}\)) possess ionizing ability, Fröhlich assumes that electrons whose energy is not lower than the ionization energy (of the order of several \(\mathrm{eV}\)) can ionize. Therefore Fröhlich’s theory is usually called the theory of ionization by fast electrons.

The essence of Fröhlich’s theory is as follows. When a dielectric is placed in an electric field, the free electrons present in the dielectric acquire energy. The average amount of this energy is calculated by Fröhlich in the following way. The current density, referred to one electron, is equal to

\[ i = ev, \]

where \(v\) is the average electron velocity in the direction of the field. This velocity is determined through the relaxation time \(\tau\)—the time during which the component of the electron momentum decreases by a factor of \(e\). Then the current density is equal to

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

and the energy acquired by an electron per second will be

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

where \(m\) is the mass of the electron.

The relaxation time \(\tau\) is determined by Fröhlich by methods of quantum mechanics, and for alkali-halide crystals it is proportional-

... of the electron’s energy to the power \( \frac{3}{2} \). As a result,

\[ A = K_1 E^2 W^{\frac{3}{2}}, \]

where \(K_1\) is a constant.

On the other hand, the electrons lose energy in setting into oscillation the atoms and ions of the lattice when colliding with them. The amount of energy transferred to the lattice per unit time is equal to

\[ B = h \nu \bigl(\Phi_e(w) - \Phi_a(w)\bigr), \]

where \(\nu\) is the frequency of oscillation of the lattice elements, \(\Phi_e(w)\) is the probability of transfer of energy \(h\nu\) to the lattice by an electron possessing energy \(W\), and \(\Phi_a(w)\) is the probability of transfer of energy to the electron by the lattice.

Frohlich calculates \(\Phi\) for alkali-halide crystals by methods of quantum mechanics and finds the final expression for \(B\):

\[ B = \mathrm{const}\, W^{-\frac{1}{2}}. \tag{5,1} \]

The condition for an equilibrium state will be

\[ A = B, \tag{5,2} \]

whence one can obtain the value of the electron energy in the equilibrium state,

\[ W_p = \frac{\mathrm{const}}{E}. \tag{5,3} \]

If the ionization energy is denoted by \(W_i\) and in (5,3) one puts

\[ W_p = W_i, \tag{5,4} \]

then the breakdown field strength can be determined by the relation

\[ E_{\mathrm{br}} = \frac{\mathrm{const}}{W_i}. \tag{5,5} \]

It follows from (5,3) that the equilibrium condition is possible in any field. However, Frohlich attempts, by general arguments, to prove that an equilibrium state can occur not at any value of the electron energy \(W_p\), but only when \(W_p > W_i\). Then

\[ E < E_{\mathrm{br}}. \]

If, however, \(W_p \leqslant W_i\), then

\[ E \geqslant E_{\mathrm{br}}, \]

the equilibrium state is impossible, and breakdown occurs.

Frohlich’s proofs reduce to the following:

Suppose that in some field \(E\) an electron has, in the equilibrium state, an average energy \(W_p\). Possessing this energy, the electron is additionally accelerated by the field and accumulates energy. The energy accumulated in excess of \(W_p\) must be transferred by the electron to the lattice, since this is the condition for the equilibrium state.

Let us consider the first case: the field strength is small, i.e., \(E \ll E_{\mathrm{br}}\). The energy of the electron in the equilibrium state is \(W_p > W_i\). The principal collisions will be inelastic, since their probability is 100 times greater than the probability of elastic ones. After ionization the ionizing or liberated electron has an energy \(W_p - W_i > 0\), which is transferred to the lattice, and thereby the conditions of the equilibrium state are satisfied.

The second case: the field strength is greater than \(E_{\mathrm{br}}\); the energy of the electron in the equilibrium state is \(W_p < W_i\). Since the field strength is large, the total accumulated energy of the electron will be \(W_i\), and the electron is in a state to ionize. After ionization the energy of the electron will be \(W \simeq 0\). The lattice receives no energy, and the conditions of the equilibrium state are not satisfied.

Consequently, according to Fröhlich a nonequilibrium state occurs only when \(W_p < W_i\). \(W_p = W_i\) characterizes the critical conditions for the onset of breakdown, and (5.5) gives the critical breakdown strength.

The above arguments of Fröhlich are not flawless. In the second case Fröhlich assumes that the electron accumulates in the field the energy \(W_i - W_p\), which is, in fact, unproven. It is possible that the accumulated energy will be less than this value; then ionization is impossible, energy will be transferred to the lattice, and the conditions of the equilibrium state will be satisfied. However, let us suppose that the energy accumulated by the electron is equal to \(W_i - W_p\); in this case an equilibrium state is also possible if the energy lost by the electron in ionization is equal to the accumulated energy. It should be noted that the question of the existence of an energetic equilibrium state in the presence of ionization processes is in general not essential, since only ionizing processes are necessary for the development of breakdown. The latter condition is necessary, but not yet sufficient. If during ionization an equilibrium electronic state exists, i.e., if the electrons captured are compensated by those liberated by ionization, then breakdown cannot develop. For breakdown to occur, the number of free electrons must grow continuously, i.e., the electronic state of the dielectric must be nonequilibrium. Fröhlich has not proved the presence of precisely this essential condition.

In the first case Fröhlich showed the presence of ionization processes; therefore the question of the energetic state has no significance at all. To prove the impossibility of breakdown, it was necessary to establish the presence of an equilibrium electronic state. However, Fröhlich makes no attempt to prove this; he confines himself to the remark that if ionizing processes exist in this case, then, apparently, reverse processes must also exist.

Thus, Froehlich’s propositions on the existence of a nonequilibrium energy state only when \(W_0 \leq W_i\), and that this is a condition for breakdown, cannot be considered proven.

A comparison of Froehlich’s formula (5.3), which expresses the dependence of the equilibrium energy of an electron on the field strength, with the corresponding formulas of Franz (4.2) and (4.4) shows their sharp contradiction. Whereas according to Froehlich’s formula (5.3) the equilibrium energy of an electron is inversely proportional to the field strength, according to Franz’s formulas (4.2) and (4.4) this dependence is direct. This discrepancy arises because Franz takes into account the action of slow electrons and assumes that the relationship between the electron energy \(W\) and the energy loss \(B\) is represented by the left branch of the curve shown in Fig. 2. Froehlich, however, takes into account the action of fast electrons, and in his treatment the dependence between \(W\) and \(B\) is represented by the right branch, as follows from (5.1). The maximum of the curve is the limiting equilibrium state, and the energy loss on excitation, calculated by Froehlich from (5.1), is less than the energy accumulated by the electron in the field, i.e. \(B < A\). Froehlich’s condition (5.2) is incompatible with (5.4). Froehlich’s final formula (5.5) also supports this conclusion. According to this formula, the breakdown field strength should be the smaller the greater the ionization energy of the atoms of the dielectric—a result that is quite unexpected and contradicts experimental data. Table II gives the breakdown field strengths calculated by Callen on the basis of Froehlich’s ideas. As can be seen from the table, these values are smaller than the experimental ones by a factor of 2 to 8.

6. MECHANISM OF CONDUCTION OF A SOLID FROM THE POINT OF VIEW OF QUANTUM MECHANICS

The ionization theories of electrical breakdown of a solid dielectric considered above proceeded from the assumption that the onset of ionization is at the same time the beginning of the development of the process leading to breakdown. It has already been noted above that the onset of ionization cannot be identified with the beginning of breakdown. For breakdown to occur it is also necessary that the ionization process become unstable and self-sustaining. The second condition may, at first glance, be replaced by another approximate condition, which is as follows.

The development of ionization processes in a dielectric leads to an increase in the number of free electrons in it. These, in turn, cause an increase in the current. An unstable ionization process developing at an accelerating rate must cause a sharp increase in the strength of the current. If, with an insignificant increase in the field strength, the current density in the dielectric increases sharply by several orders of magnitude (from a value,

characteristic of the ordinary conductivity of a dielectric at low field strengths, up to the value occurring at breakdown), it may be recognized that these conditions are almost adequate to the breakdown conditions. The gradients at which a sharp increase of the current by several orders of magnitude occurs may be taken as breakdown gradients.

In fact, such an approach to the estimation of breakdown conditions was already adopted in Joffe’s works. Such an estimate of breakdown conditions leads in fact to a new interpretation of the nature of breakdown. Indeed, breakdown can then be regarded as a process of a sharp increase in the conductivity of a dielectric under the action of an electric field. Breakdown is the result of the limiting increase in the conductivity of a dielectric in a strong electric field. This view of the nature of electrical breakdown opens the possibility of developing quantum-mechanical theories of electrical breakdown.

Before setting forth the quantum-mechanical theories of breakdown, it is necessary, at least briefly, to explain the mechanism of conductivity of a solid from the point of view of quantum mechanics. For this it is necessary to consider the behavior of charged particles in a solid. The electric current in a dielectric always consists of two components: ionic and electronic,

\[ i = i_{\text{i}} + i_{\text{e}} . \]

The dependences of these components on the temperature and on the strength of the electric field are diametrically opposite. The ionic current increases very strongly with increasing temperature and comparatively slowly with increasing electric-field strength. The electronic current, on the contrary, increases only barely noticeably with heating and rises sharply with increasing field strength. Therefore, in the first instance we shall be interested in the nature of the electronic current.

Every dielectric has a regular or irregular crystalline structure. The state of each electron is influenced by all the atoms, ions, or molecules situated at the lattice sites, and by all the other electrons. Let us first consider the state of an electron in an isolated atom.

An electron in the system of an isolated atom possesses kinetic and potential energy. The latter is equal to

\[ V(x_i)=\frac{e^2}{r}\qquad (i=1,2,3), \]

where \(r\) is the distance from the electron to the center of the nucleus.

For a nucleus situated at the origin of coordinates, this potential function is shown in Fig. 7,a.

The state of the electron is described by the Schrödinger equation

\[ -\frac{1}{g}\frac{d^2\varphi}{dx^2}+\left[V(x)-W_0\right]\varphi=0, \]

where

\[ g=\frac{8\pi^2 m}{h^2}, \]

and \(W_0\) is the total energy of the electron. This equation has solutions for discrete values of the energy

\[ W_{0_1},\; W_{0_2},\; W_{0_3},\ldots,\; W_{0_n}, \]

to which there correspond the functions

\[ \varphi_1,\; \varphi_2,\; \varphi_3,\ldots,\; \varphi_n, \]

where the functions \(\varphi\) have the following physical meaning: \(|\varphi_k(x)|^2\) is the probability that an electron with energy \(W_{0_k}\) is located at the point \(x\). For simplicity we have restricted ourselves to a one-dimensional model. An electron possessing the energy \(W_{0_n}\) or \(W_{0_m}<0\) is compelled always to remain inside the atom. The maximum distance to which the electron can move away is \(x_n\) or \(x_m\), since at a greater—

Fig. 7.

—distance its kinetic energy becomes negative, which is impossible. To remove an electron beyond the limits of the atom it is necessary to impart to it an energy \(\Delta W\), where \(\Delta W\) must satisfy the expression \(W_0+\Delta W>0\).

Let us now consider the state of an electron in a crystal. One of the methods by which the state of an electron in a crystalline body is determined is Bloch’s “self-consistent field” method. This method proceeds from the following representation. The valence electrons are torn away from the atoms, collectivized, and belong to all the atoms. They form a single common, so-called “self-consistent” field, whose potential is a periodic function of the coordinates with period \(a\), equal to the distance between the lattice sites of the crystal. This potential is the potential of an equivalent field with energy equal to the energy of interaction of all the electrons with one another. The atoms remaining without valence electrons form a positive periodic field. Thus, an individual electron moves in a common periodic field (the field of the atoms without valence electrons in the “self-consistent” field), independently of the other electrons. The potential energy of the electron \(V\) is the sum of the potential—

energies of an electron in the positive field of each nucleus and the negative “self-consistent” field of the electrons. At a sufficient distance from the boundaries of the crystal, the potential energy \(V\) is a periodic function of the coordinates with a period equal to \(a\), as is seen from Fig. 7,b, which corresponds to a one-dimensional model of the crystal. In practice one may assume that the violation of periodicity will occur only at the outermost nuclei. The potential energy of an electron in the crystal is less than in an isolated atom. As a consequence, electrons that have the former energy \(W_{0m}\) can move freely throughout the entire crystal.

Indeed, at any point in the crystal these electrons possess positive kinetic energy. In classical mechanics such electrons are called free. Electrons possessing the energy \(W_{0n}\), from the standpoint of classical mechanics, still cannot leave the confines of the atom. Classical mechanics calls these electrons bound, since in order to pass into a neighboring atom the electron must overcome a potential barrier. For this the electron must be given an additional energy \(\Delta W\).

According to quantum mechanics, electrons with energy \(W_{0n}\), lying below the potential barrier, have the possibility of leaking through the barrier and entering a neighboring atom. Such a process of an electron leaking through a potential barrier is known as the “tunnel effect.” The probability of such an event depends on the width and height of the barrier, i.e., on the shaded area in Fig. 7,b. The smaller this area, the greater the probability of electron leakage. Thus quantum mechanics does not divide (in the sense indicated above) electrons into bound and free. All electrons possess relative freedom to varying degrees.

Electrons have the possibility, while remaining at the same energy level, of leaking through the potential barrier from one atom to another. The location of the electrons is immaterial and, generally speaking, unknown. One may imagine that around one atom, at different levels, there are several valence electrons (according to the Pauli principle there can be no more than two electrons at one level), while at the same time other atoms will be stripped. The probability of electrons leaking through barriers is the same in all directions. Therefore such leakage cannot create a directed motion of electrons, which constitutes an electric current, and leads to a uniform distribution of electrons over all atoms. An electric current is obtained with an asymmetric probability function, which depends on the distribution of electrons over energy levels.

The influence of the crystal lattice on the state of the electron is not limited to what has been said above. Moreover, this influence is reflected in the energy levels of the electrons, which is much more significant.

In the Schrödinger equation, \(V\) is now a periodic function of the coordinates. To take into account the boundary conditions, an additional condition is imposed on the solution of the Schrödinger equation (the Born–Karman condition), requiring that the solution \(\varphi(x)\) be a periodic function of the coordinates with period equal to \(L\), where \(L\) is the size of the crystal.

The solution of this Schrödinger equation, taking into account the periodicity of \(V(x)\) (with period \(a\)) and \(\varphi(x)\) (with period \(L\)), which has physical meaning, leads to a change in the energy levels of the electron. If, in an isolated atom, the electron had discrete energy levels \(W_{0s}\), then in the lattice each level rises and splits into \(N\) discrete levels forming a band, where \(N\) is the number of nuclei in the crystal: \(N=\dfrac{L}{a}\) (in the one-dimensional model).

\[ W_s = W_{0s} + \alpha_s + 2\beta_s \cos \frac{2\pi}{N} j, \]

where \(s\) is the number of the band, \(j\) is the number of the level in the band \(\left(j=0;\ \pm 1;\ \pm 2;\ldots;\pm \dfrac{N}{2}\right)\), \(\alpha\) is the height by which the corresponding level is raised, and \(\beta\) is a quantity characterizing the width of the band. The change in the energy levels is shown in Fig. 8. The width of the band is equal to

\[ W_{\max s} - W_{\min s} = 4\beta_s . \]

For a spatial lattice,

\[ W_s = W_{0s} + \alpha_s + 2\beta_s \left( \cos \frac{2\pi}{N} j_1 + \cos \frac{2\pi}{N} j_2 + \cos \frac{2\pi}{N} j_3 \right) \]

\[ j_i = 0;\ \pm 1;\ \pm 2;\ldots;\pm \frac{N}{2}. \]

The upper boundary of the band is

\[ W_{\max} = W_{0s} + \alpha_s + 6\beta_s; \]

the lower boundary of the band is

\[ W_{\min} = W_{0s} + \alpha_s - 6\beta_s; \]

the width of the band is

\[ W_{\max} - W_{\min} = 12\beta_s. \]

The splitting of each level into sublevels is due to the presence of nodes in the crystal lattice, while the number of sublevels is due to the number of nodes or to the boundary conditions. The grouping of levels into bands is due to the periodicity of the potential function; the width of the band depends on the degree of interaction of two neighboring atoms.

Thus the zones with energy levels on which electrons can be found (zones with “allowed” levels) are separated from one another by bands called “forbidden,” i.e., regions with such energy values as the electrons cannot have. In each zone there are \(N\) levels.

According to the Pauli principle, at each level there can be no more than two electrons, taking their spin into account. Thus, in each zone there can be no more than \(2N\) electrons. We are interested in the valence electrons, since they determine the conductivity. Three cases are possible.

Fig. 8.

Fig. 9.

The first case is that the number of electrons in the zone is less than \(2N\). The electrons occupy only part of the levels. At absolute-zero temperature the electrons occupy the lowest levels. The upper levels in the zone remain free. To each energy level

\[ W = W(j_1, j_2, j_3) \]

there corresponds a momentum

\[ \mathbf{p} = \mathbf{p}(j_1, j_2, j_3). \]

For an energy distribution symmetric with respect to \(j\), the momenta will also be arranged symmetrically. This distribution is shown in Fig. 9,a.

When an electric field is applied, the electrons are redistributed among the levels in such a way that the number of electrons with positive values of \(j\) exceeds the number of electrons with negative ones.

values, or conversely. The distribution of electron velocities with respect to directions is asymmetric. More electrons move in one direction than in the other. The role of the field is reduced to the fact that it makes the electrons occupy some free levels and vacate others. Such a distribution is shown in Fig. 9,b.

The second case—the band is completely filled with electrons, but the empty band above is adjacent to this band. When a field is applied, electrons from the lower band pass over and occupy part of the levels in the upper band.

The third case—the band is completely filled with electrons, but the empty band above is separated from the lower one by a forbidden region. The application of an electric field cannot produce asymmetry in the distribution of electrons over the levels and therefore cannot produce a directed velocity of the electrons.

Conductor          Dielectric

Occupied levels are indicated by hatching.

Fig. 10.

The first two cases are characteristic of a conductor, the third of a dielectric. This difference between a conductor and a dielectric is illustrated by Fig. 10.

In order for a dielectric to become conducting, it is necessary, from the point of view of classical mechanics, to raise the bound electron from a level lying in a potential well (the level \(W_{0n}\), Fig. 7,b) to a level lying above the potential barriers (to the level \(W_{0m}\), Fig. 7,b), and to make it free. For this it is necessary to expend some energy. From the point of view of quantum mechanics, in order to create conductivity in a dielectric it is necessary to transfer an electron from the lower filled normal band to the upper empty conduction band. This process is associated with an expenditure of energy. Electrons that have entered the empty band may acquire an asymmetric distribution. As a result of the freeing of some levels in the lower band, an asymmetric distribution is now also possible in it. The number of freed levels in the lower band*) is exactly equal to the number of occupied levels in the upper band. Both bands may lie below the potential barriers, since, according to the views of quantum mechanics, as has already been indicated, conductivity is determined only by an asymmetric distribution.

*) Unfilled places in the lower band are often called holes. Hence the terms—hole current, hole conductivity. It seems to us that these terms can hardly be considered successful.

Barriers do not constitute an absolute obstacle to the motion of electrons, since there is always a probability that an electron will leak through the barrier (the tunnel effect). The current is composed of the electronic current of the conduction band and the electronic current of the normal band. The conductivity will be the greater, the more electrons have passed from the normal filled band into the empty one.

The transfer of electrons from the normal band to the conduction band may be accomplished by the action of light, at the expense of the energy of a light quantum, or by heating, at the expense of thermal energy.

7. QUANTUM-MECHANICAL THEORIES OF ELECTROSTATIC IONIZATION

a) Theory of electrodeless breakdown

The potential function inside a crystal is a periodic function with period \(a\). Since, from the point of view of quantum mechanics, the conductivity of a dielectric is determined not by the leakage of electrons through a potential barrier, but by the transition of electrons from the normal band to the conduction band,

Fig. 11.

a) Absence of a field  b) Presence of a field

but not by the leakage of electrons through a potential barrier, the periodicity of the potential function is subsequently immaterial. This periodicity was used to determine the grouping of levels into bands. For further arguments one may therefore replace the periodic function by its mean value, constant throughout the entire volume of the crystal (Fig. 11,a).

The application of an electric field to a dielectric decreases the potential energy of an electron by the amount \(Eex\). In the Schrödinger equation one more term is added:

\[ -\frac{1}{g}\frac{d^{2}\varphi}{dx^{2}}+\left[V(x)-Eex-W\right]\varphi=0. \]

If one assumes that the term \(Eex\) changes little over the distance \(a\), then this equation will have a solution for the value

\[ W = Eex + W_0 + \alpha + 2\beta \cos \frac{2\pi}{N} j . \]

The energy levels are now a linear function of the coordinate and are represented by inclined lines, as shown in Fig. 11,b. The greater the field strength, the greater the inclination of the levels, and consequently of the zones. This does not mean, of course, that the total energy of the electron is now a function of the coordinate; the energy of the electron remains constant even in the presence of the field. It means only that an electron with a definite energy can be located only in a limited region of the crystal.

Fig. 12.

An electron, for example, with energy \(W_1\), can be located throughout the entire region of the crystal in the absence of a field; in the presence of a field, however, only in the regions \(0—x_1\) and \(x_2—L\). In the language of wave mechanics this means that the modulus of the wave function \(\varphi_1\), corresponding to the energy \(W_1\), remains periodic not throughout the whole region of the crystal, but only in the regions \(0—x_1\) and \(x_2—L\); in the region \(x_1—x_2\) it is practically equal to zero. The energy \(W\) in the presence of a field ceases to be uniquely associated with a definite zone; it may correspond to different zones. If the lower zone is the normal (filled) zone, and the upper one is the conduction zone (empty), then in the presence of a field a transition of an electron from the first to the second zone is possible without expenditure of energy (along the straight line \(x_1—x_2\)).

However, in such a transition the electron must overcome the forbidden region \(x_1 - x_2\). The width of this region is

\[ x_2 - x_1 = \frac{U}{Ee}, \]

where \(U\) is the width of the forbidden region, equal to \(W_{\min\, n+1} - W_{\max\, n}\). The greater the field strength \(E\), the smaller the width of the forbidden region.

The probability of an electron tunneling through the forbidden region is the greater, the narrower the region. This probability is equal to\({}^{23}\)

\[ P = e^{-\frac{\pi^2 ma}{h^2}\frac{U^2}{eE}}. \]

Within a band, the electrons execute oscillatory motions in the direction of the field over a length equal to \(\frac{1}{3}l\) (Fig. 12), with frequency

\[ \nu = \frac{eEa}{h}. \]

If it is assumed that the dielectric consists of divalent atoms, then the number of valence electrons per unit volume is

\[ Z = \frac{2}{a^3}, \]

where \(a\) is the lattice constant, equal to the distance between atoms. In each oscillation, the number of electrons tunneling through \(1\ \text{cm}^2\) of surface from the normal band into the conduction band is equal to \(ZP\). In one second this number is

\[ n = Z\nu P. \tag{7,1} \]

The expression

\[ \nu P = \frac{eaE}{h}\, e^{-\frac{\pi^2 ma}{h^2}\frac{U^2}{eE}} \tag{7,2} \]

may be interpreted as the probability of electrostatic ionization per unit time.

The current density is equal to

\[ i = eZ\nu P = \frac{2e^3E}{ha^2}\, e^{-\frac{\pi^2 ma}{h^2}\frac{U^2}{eE}}. \tag{7,3} \]

Taking \(a = 3 \cdot 10^{-8}\ \text{cm}\) and \(U = 4\ \text{eV}\), we obtain the current density in amperes

\[ i = 10^{11}E\,10^{-\frac{8\cdot 10^7}{E}}\ a. \tag{7,4} \]

For

\[ E = 4\cdot 10^6 \quad 5\cdot 10^6 \quad 6\cdot 10^6\ \text{V/cm}, \]

\[ i = 4\cdot 10^{-3} \quad 5\cdot 10 \quad 3\cdot 10^4\ a. \]

When the field strength changes from \(4\cdot 10^6\) to \(6\cdot 10^6\ \text{V/cm}\), the current density increases by \(7\cdot 10^6\) times, reaching values of thousands of amperes per \(1\ \text{cm}^3\). The growth of the current is extremely sharp, and it reaches values that occur during breakdown.

The formula (7.4), obtained by Zener, is, in Cernushi’s opinion\(^{24}\), inaccurate, because Zener made an incorrect assumption in its derivation. Zener assumed that, in the presence of a field, the dependence of the electron wave function on \((W-Eex)\) is the same as on \(W\) in the absence of a field, which, in Cernushi’s opinion, is incorrect. Cernushi proceeded differently: he replaced the smooth distribution of the potential of the external field by a stepwise one. The formulas obtained by Cernushi, already at the beginning of the derivation, require complicated numerical calculations for the determination of the constants. Therefore it is possible to compare the results obtained by him with Zener’s results only for \(U=5\) eV. For this value of \(U\), the currents according to Cernushi’s formula are, approximately, 1.5 times smaller than according to Zener’s formula; the order of magnitude obtained is the same.

Formula (7.2) was also obtained by Franz\(^{17}\) under the condition that the ionization potential (or the width of the band) does not exceed 3–4 eV.

The formula \(n=ZvP\) is valid only under two assumptions: 1) the concentration of electrons in the normal zone remains constant. This occurs under the condition of the immediate arrival of electrons from the electrode into the normal zone in place of those electrons which have tunneled from this zone into the conduction zone; and 2) the electrons which have tunneled into the conduction zone immediately go on to the second electrode, so that the conduction zone remains empty all the time. The mechanism of such conduction may be represented by the following scheme. In the absence of a field, an electron is able to move through the crystal along the zone in any direction, and, owing to the chaotic nature of this motion, there is no electric current. When an electric field is applied to the crystal, the electron is locked by the field in a certain region of the crystal of width \(\frac{1}{3}l\) (Fig. 12). In this region the electron performs, along the field and consequently also along the crystal, an oscillatory motion. At each oscillation the electron has a probability \(P\) of tunneling through the forbidden interval into the conduction zone. If it succeeds, then in its place, from the upper level of the same zone, an electron falls, in whose place an electron enters from a still higher level, and so on. Ultimately, an electron from the electrode enters the vacated place in the normal zone. The electron which has tunneled into the conduction zone can pass, freely falling from level to level, ultimately to the second electrode, since most of the levels of this zone are free.

If the surface of the crystal is not in contact on one side with such a medium in which there is a sufficiently large number of “free” electrons, from which they can enter in unlimited quantity into the normal zone of the crystal, and on the

on the other hand, with the medium into which electrons can freely enter in an unlimited number from the conduction band, formula (7.1) is incorrect. Indeed, suppose that electrons do not enter the conduction band from the electrode at all. Then, during the first oscillation from the normal band through a surface of \(1\ \mathrm{cm}^2\), there will seep into the conduction band

\[ n_1 = ZP \]

electrons. The concentration of electrons in the normal band will decrease and will be equal to

\[ Z - n_1 = Z(1 - P). \]

During the second oscillation, a smaller number of electrons, equal to

\[ n_2 = Z(1 - P)P, \]

will already seep from the normal band. If no reverse seepage of electrons from the conduction band into the normal band occurred, then the normal band would gradually become depleted of electrons. In the latter case the number of electrons entering per second through a surface of \(1\ \mathrm{cm}^2\) into the upper band would be

\[ n' = Z\left[1 - (1 - P)^\nu\right]. \]

In the absence of an electric field, the electrons are only in the normal band. When an electric field is applied, the electrons enter the conduction band only by seeping through the forbidden region. If it is assumed that an electron can be found only in one of these two bands, then the probability of its presence in the conduction band is equal to the probability of seepage through the forbidden region, i.e. \(P\). Consequently, the probability of an electron seeping through the forbidden region from the conduction band into the normal band is equal to the probability of its presence in the normal band, i.e. \(1 - P\). Then the number of electrons seeping from the conduction band into the normal band during the second oscillation is equal to

\[ n_{\mathrm{p}\to\mathrm{n}} = n_1(1 - P) = ZP(1 - P). \]

Thus, after the first oscillation, in the absence of electrodes an equilibrium state is established with concentrations in the normal band

\[ Z_{\mathrm{n}} = Z(1 - P), \]

and in the conduction band

\[ Z_{\mathrm{p}} = ZP. \]

At each subsequent oscillation an equal number of electrons will seep in both directions, equal to

\[ n_{\mathrm{obm}} = Z(1 - P)P. \]

Although the composition of electrons in the conduction band will be continuously renewed, nevertheless there will be no current. The current whose density

which is determined by formula (7.3), will exist only at the moment of application of the electric field for a time

\[ t=\frac{1}{\nu}=\frac{h}{aeE}\cong 10^{-14}\ \text{sec}. \]

Such a short-lived current pulse cannot cause breakdown. The theory of breakdown of intracrystalline origin can explain breakdown only in the presence of electrodes. At the same time, experiment shows that partial breakdown often occurs, the breakdown of some layer inside the dielectric that is not in direct contact with the electrodes. Thus, in the absence of electrodes, the mechanism considered can only ensure the presence of a certain number of electrons in the conduction band when a field is present, i.e., the formation by the field of “free” electrons.

b) Theory of breakdown of electrode origin

When a dielectric is clamped between two electrodes, electrons may leak directly from the negative electrode into the conduction band of the dielectric and from the normal band of the dielectric into the positive electrode.

Fig. 13.

The upper level of the electrode band, filled with electrons \(W_\mathrm{e}\), lies, approximately, midway between the lower level of the conduction band \(W_{\min.\mathrm{p}}\) and the upper level of the normal band \(W_{\max.\mathrm{n}}\). Let us denote the difference of these levels, or the width of the forbidden region, by \(U\)

\[ U=W_{\min.\mathrm{p}}-W_{\max.\mathrm{n}}. \]

From the cathode directly into the conduction band of the dielectric there may pass electrons possessing an energy greater than \(W_{\text{э}}-\dfrac{1}{2U}\) (Fig. 13); from the normal band directly into the anode there may pass electrons with energies lying in the interval \(W_{\text{э}}\) and \(W_{\text{э}}+\dfrac{1}{2U}\).

An electron with energy \(W_{\text{э}}\) must tunnel through a barrier twice as short as the barrier through which an electron with energy \(W_{\text{э}}+\dfrac{1}{2U}\) tunnels (Fig. 13). The height of the forbidden region for the first electron is equal to \(\dfrac{1}{2U}\), and for the second to \(U\). The probability of tunneling for the first electron will be

\[ P_1=e^{-\frac{\pi^2ma}{4h^2}\frac{U^2}{eE}}, \]

and for the second

\[ P_2=e^{-\frac{\pi^2ma}{h^2}\frac{U^2}{eE}}=P_1^{\frac14}. \]

The number of electrons in the metal which can tunnel per second through \(1\ \text{cm}^2\) of the electrode surface is equal to \({}^{25}\)

\[ n \simeq \frac{e^2E^2}{8\pi h\delta}, \]

where \(\delta\) is the height of the potential barrier through which the electron tunnels. \(\delta\) is equal to the difference between the minimum level of the conduction band and the energy level on which the electron is situated in the metal.

For the first electrons \(\delta=\dfrac{U}{2}\), for the second \(\delta=U\). Consequently, the greatest current density is equal to

\[ i=enP_1=\frac{e^3E^2}{4\pi hU}\,e^{-\frac{\pi^2ma}{4h^2}\frac{U^2}{eE}}. \]

Expressing \(U\) in ev, \(E\) in v/cm and \(i\) in a/cm\(^2\), we obtain

\[ i=3.1\cdot 10^{-6}\frac{E^2}{U}\,10^{-1.25\cdot 10^5\frac{U^2}{E}}\ \text{a/cm}^2 \]

At \(U=2\) ev	At \(U=2\) ev	At \(U=4\) ev	At \(U=4\) ev	At \(U=6\) ev	At \(U=6\) ev
\(E\) (v/cm)	\(i\) (a/cm\(^2\))	\(E\) (v/cm)	\(i\) (a/cm\(^2\))	\(E\) (v/cm)	\(i\) (a/cm\(^2\))
\(6\cdot 10^5\)	\(2\cdot 10^{-3}\)	\(2\cdot 10^6\)	\(6\cdot 10^{-4}\)	\(5\cdot 10^6\)	\(10^{-2}\)
\(2\cdot 10^6\)	\(2\cdot 10^4\)	\(4\cdot 10^6\)	\(10^2\)	\(10^7\)	\(10^3\)

A change in the field strength by a factor of two leads to a sharp increase in the current by several orders of magnitude. The current reaches values of thousands of a/cm\(^2\) at field strengths of the order of \(10^6\)—\(10^7\) v/cm.

The possibility of breakdown of electronic origin was considered by Fowler \(^{26}\). Fowler regarded the penetration of electrons from the cathode into the dielectric as the result of cold emission. Fowler derived a formula for the number of electrons pulled out by the field, without taking into account the periodic potential of the lattice. According to this formula, the cold-emission current is very small and does not exceed \(10^{-5}\) a/cm\(^2\) at \(E = 10^7\) v/cm. This led Fowler to believe that cold emission cannot serve as the principal process in the breakdown of solid dielectrics. Cold emission can only supply the electrons that begin to develop the ionization processes leading to breakdown.

However, Volkenstein \(^{27}\) showed that neglecting the periodic field of the lattice leads to a substantially smaller probability of electron leakage; as a result Fowler obtained very small currents.

The theory of breakdown of electronic origin presented above assumed a temperature equal to absolute zero (an increase in temperature is not reflected in the order of magnitude) and did not take into account distortion of the energy spectrum by the boundary surface (the so-called Tamm surface levels).

The mechanism of electrostatic ionization can, as is evident, explain the large increase in current when the field strength is increased approximately twofold. However, this increase in current is not sharp enough to justify the use of the approximate breakdown criterion given above. Moreover, the magnitude of the gradient, although it has the order of magnitude obtained from experiment, is nevertheless several times larger than it.

8. QUANTUM-MECHANICAL THEORIES OF THERMAL IONIZATION

a) The theory of breakdown as a result of the Stark effect, by F. F. Volkenstein

In deriving the influence of an electric field applied to a dielectric on the energy levels, it was assumed that within the atom, i.e., at distances \(a\), the field varies insignificantly and it may be regarded as constant. This causes the crater of each atom to rise in proportion to its distance from the electrode and leads to an inclination of the energy zones. In reality, the electric field not only raises but also distorts the shape of the crater, which causes each level to shift by an amount \(\Delta W\) and the splitting of some levels.

If the upper level of the normal band has shifted by an amount \(\Delta W_n\), and the lower level of the conduction band by an amount \(\Delta W_p\), with \(\Delta W_n > \Delta W_p\), then the height of the forbidden region will decrease

by the amount \(\Delta U=\Delta W_{\mathrm{n}}-\Delta W'_{\mathrm{n}}\). \(\Delta U\) depends on the field strength. The width of the forbidden region will be

\[ U(E)=U_0-\Delta U(E), \]

where \(U_0\) is the width of the forbidden region in the absence of an electric field.

When \(\Delta U\) becomes equal to \(U_0\), the two bands come into contact. The field strength at which this occurs may be taken as the breakdown strength.

From the available empirical dependence of \(\Delta U\) on \(E\), one can estimate the magnitude of the breakdown field strength. It reaches values of \(4\cdot 10^6\)—\(4\cdot 10^7\ \mathrm{V/cm}\). This value is one or two orders of magnitude greater than the experimental one. Therefore Vol'kenshtein\(^{28}\), who developed this theory, in more detailed calculations comes to the conclusion that this effect does not play an essential role in breakdown; it only facilitates electrostatic ionization.

b) Thermal theory of ionization of Ya. I. Frenkel

The theories of electrostatic ionization set out above proceed from the idea that each electron is surrounded by positive ions. Frenkel\(^{29}\) proceeds from a different idea. He assumes that each electron is surrounded by neutral atoms. The behavior of the electron must be considered as in an isolated atom situated in a medium with dielectric permittivity \(\varepsilon\). Therefore, in the absence of an external electric field, the electron cannot tunnel through the potential barrier, which is infinite, and is bound to its atom.

Fig. 14.

When an external electric field is applied, the potential function changes, as is shown in Fig. 14 by the dashed lines. The electron can now tunnel through the potential barrier beyond the limits of the atom. The probability of this depends on the width of the barrier \(A'B'\) and its height \(BC'\). The lower and narrower the barrier, the greater the probability of tunneling of the electron.

In the absence of an external field, the probability of detachment of an electron having thermal energy \(kT\) is equal to

\[ P_0=e^{-\frac{W_i}{2kT}}, \]

where \(W_i\) is the ionization energy, equal to \(BC\). In the presence of a field, the energy

the ionization is reduced by the amount \(CC'\) and will be equal to

\[ \frac{e^3}{\varepsilon r_0}+eEr_0. \]

\(r_0\) can be determined from the condition

\[ \left.\frac{d}{dr}\left(\frac{e^3}{\varepsilon r}+eEr\right)\right|_{r=r_0}=0. \]

Hence

\[ r_0=\sqrt{\frac{e}{\varepsilon E}} \]

and

\[ cc'=\Delta W=2e\sqrt{\frac{eE}{\varepsilon}}. \]

Consequently, the probability of electron leakage or of ionization of an atom in the presence of a field will be

\[ P=e^{-\frac{W_i-\Delta W}{2kT}} =e^{-\frac{W_i-2e\sqrt{eE/\varepsilon}}{2kT}}. \tag{8,1} \]

Substituting (8,1) into (7,1), we obtain the current density

\[ i=\frac{2e^2E}{ha^2}\, e^{-\frac{W_i-2e\sqrt{eE/\varepsilon}}{2kT}}. \tag{8,2} \]

Taking \(a=3\cdot10^{-8}\,\text{cm}\), \(W_i=4\) eV, \(T=300^\circ\text{C}\), \(\varepsilon=6\), and expressing \(E\) in V/cm, we obtain for the current density in A/cm\(^2\)

\[ i\simeq 10^{11}E\,10^{-34.4+2.6\cdot10^{-3}\sqrt{E}}. \]

For \(E=\quad 4\qquad 5\qquad 6\cdot10^6\) V/cm,

\[ i=\quad 2.5\cdot10^{-12}\quad 1.6\cdot10^{-11}\quad 6\cdot10^{-11}\ \text{A/cm}^2. \]

Frankel’s formula gives an extremely insignificant current density and only a weak increase in it.

9. QUANTUM-MECHANICAL THEORIES OF IMPACT IONIZATION

With respect to all theories of electrostatic ionization it may be noted that none of them can explain the development of breakdown in individual places of a dielectric and its propagation from one electrode to the other. According to these theories, breakdown should develop simultaneously throughout the whole thickness of the dielectric. If, moreover, the cases of electrodeless breakdowns are taken into account, one must admit that apparently there is another mechanism, acting more effectively than the mechanism of electrostatic ionization.

According to the views of quantum mechanics, electrons can move through a crystal by leaking through the potential barriers separating atoms from one another, while remaining at the same energy level. During these movements there are possible

collisions between electrons and the transfer of energy by one electron to another. The electron transferring energy descends to a lower free energy level, while the electron receiving energy passes to a higher free energy level. If the electron giving up energy is, before the collision, in the conduction band and remains in it after the collision, while the electron receiving energy is, before the collision, in the normal band and after the collision passes into the conduction band, then as a result of such a collision the number of “free” electrons increases by one. This phenomenon is analogous to the process of impact ionization in classical mechanics and bears the same name. After such events the conduction band is enriched with electrons, and the normal band with free levels. Both these circumstances contribute to an increase in conductivity.

Ionization is possible in the case when the ionizing electron possesses an energy not less than \(W_{\min,\mathrm{p}} + U\), where \(W_{\min,\mathrm{p}}\) is the lowest energy level in the conduction band, and \(U\) is the width of the forbidden region.

In the absence of an electric field, such phenomena practically do not occur in a dielectric, since the number of electrons in the conduction band at ordinary temperature is negligibly small. Electrons can pass from the normal band into the conduction band either as a result of irradiation of the dielectric or as a result of its heating. However, since the width of the forbidden region in dielectrics is usually large, of the order of several electron-volts, in order to transfer an appreciable number of electrons from the lower to the upper band the dielectric must be heated to the melting temperature.

The presence of an electric field facilitates this process. The electric field inclines the energy bands. This gives rise to three causes that assist the process under consideration.

1. The inclination of the bands makes it possible, as was clarified in the preceding paragraph, for a certain number of electrons to seep from the normal band into the conduction band. These electrons can initiate the process of ionization.

2. An electron situated at one of the very lowest energy levels of the conduction band (position 1 in Fig. 15) cannot ionize. Being at some level, the electron executes oscillations whose amplitude is equal to \(\frac{1}{3} l\); in this case, as a result of a collision,

the electron can fall to a lower free level (denoted in Fig. 15 by the numeral 2), located to the right. From this level the electron can fall to an still lower level—3, located still farther to the right. In some positions on the last level the energy of the electron exceeds \(W_{\min,\mathrm{p}}\) by an amount greater than \(U\), and ionization is possible. The ionizing electron will then pass to a lower level 4, but will remain in the conduction band.

The ionized electron from position B will pass to level 5, into the conduction band.

1. Electrons that are in the normal band and cannot ionize in the absence of a field, in the presence of a field are also capable of ionization. This may occur in the following way. An electron from level 7 may, upon collision, fall into the free level 6, situated in the upper part of the normal band, after which level 7 is vacated. Onto this level falls an electron from level 8, and onto 8 one from level 9. The last level is already sufficiently low in the normal band. If now two electrons collide, located, for example, on levels 10 and 11, then one of them will fall to level 9, transferring part of its energy to the other electron, which will rise, at the expense of this energy, into the conduction band, to level 12.

Fig. 15.

Ionization by electrons located both in the conduction band and in the normal band is the more probable, the greater the inclination of the bands and the smaller \(U\). Such ionization is possible only when either the width of the normal band \(U_1\), or the width of the conduction band \(U_2\), is greater than \(U\). In addition, it is necessary that the electron be on the upper levels of the conduction band, or that one of the lower levels in the normal band be freed. With inclined bands there is a possibility for electrons located on the lower levels of the conduction band to make their way stepwise to the upper level. In an analogous manner, in the normal band a lower level can be freed*).

*) In these cases it is often said that a hole travels in the band and ionizes, since a free level is analogous to a hole from which an electron has departed.

The mechanism of impact ionization considered above applies to absolutely ideal crystals. Not only the presence of impurities, but also a disturbance of the lattice structure is of essential importance. If, at the lattice sites, instead of one element there is one or several elements of another kind (for example, an atom instead of an ion, or conversely), then the periodicity of the field is disturbed. Since the periodicity of the field, as was indicated above, determines the band character of the levels, a disturbance of periodicity causes a change in the normal distribution of the bands. In the forbidden region local levels appear, as is shown schematically in Fig. 16. Upon impact an electron may fall onto a local level, for which it must be given an energy \(\Delta W\), smaller than \(U\). From the local level, by a second impact, the electron passes to a higher level or into the conduction band. Since the probability of ionization depends to a very high degree (exponentially) on \(\Delta W\), and this probability is the greater the smaller \(\Delta W\) is, the presence of local levels requires a much smaller field gradient for ionization.

Fig. 16.

Structural disturbances of the lattice or the presence of impurities are entirely random. Therefore the presence of one or another local level is also random. This may explain the development of breakdown at individual places in the dielectric and the statistical scatter of the breakdown field strength.

10. THERMAL BREAKDOWN

When a voltage smaller than the breakdown voltage is applied to a dielectric, an electric current flows through the dielectric and energy is released in the dielectric. The amount of energy released per unit time in a unit volume of the dielectric is equal to

\[ W_1 = \gamma E^3, \]

where

\[ \gamma = A e^{-\frac{B}{T}}; \tag{10,1} \]

\(\gamma\) is the conductivity of the dielectric, \(B\) is a constant coefficient, and \(A\) is a quantity that depends little on temperature.

The release of energy in the dielectric causes an increase in temperature and, as a consequence, an increase in its conductivity. The increase in conductivity leads to an increase in the energy released in the dielectric, which in turn increases the conductivity still further, and so on. Such a process causes a continuous increase in tem—

temperature, if, in parallel with it, there is no reverse process of the dielectric giving up heat to the surrounding medium. The amount of energy \(W_2\) given up by the dielectric also increases with increasing temperature,

\[ W_2 = C(T - T_0), \]

where \(T_0\) is the temperature of the surrounding medium, equal to the initial temperature of the dielectric, and \(C\) is a constant coefficient depending on the conditions of heat removal. Figure 17 shows the dependences of \(W_1\) and \(W_2\) on temperature at a constant field strength. The straight line \(W_2\) may intersect the curve \(W_1\) at two points, as is seen from Fig. 17. The points of intersection correspond to equilibrium states.

Fig. 17.

The lower point \(A\) characterizes stable equilibrium, whereas the upper point \(B\) corresponds to unstable equilibrium. Indeed, suppose that the field strength is equal to \(E_1\) and the initial temperature is equal to \(T_0\). When a voltage is applied to the dielectric, the energy released in it is greater than the energy given up; the dielectric will gradually heat up to the temperature \(T_A\). At the temperature \(T_A\) an equilibrium state is reached, and further heating of the dielectric ceases. If, for some accidental reason, the temperature of the dielectric becomes higher than \(T_A\), then the energy given up by the dielectric becomes greater than that released, and the dielectric will begin to cool down to the temperature \(T_A\). Suppose now that the temperature of the dielectric is \(T_B\). Accidental deviations of the temperature in either direction from \(T_B\) cause a continuous increase of these deviations and a departure of the state from point \(B\). An accidental decrease in temperature causes further cooling of the dielectric down to the temperature \(T_A\); an accidental increase in temperature causes the energy released in the dielectric to exceed that lost, and an unbounded rise of temperature. When point \(B\) is sufficiently far from \(A\), the state \(B\) naturally cannot occur. An increase in the electric-field strength \(E\) or in the initial temperature \(T_0\) leads to a convergence of the points \(A\) and \(B\). For certain values of \(E\) and \(T_0\), the points \(A\) and \(B\) coincide—the point \(C\) in Fig. 17. It is not difficult to see that point \(C\) is a critical point. With an accidental slight increase in temperature above \(T_k\), an unstable state occurs in which the temperature of the dielectric can increase without bound and cause its thermal destruc-

... (melting, annealing). If the straight line \(W_2\) and the curve \(W_1\) do not have a single common point (for definite values of \(T_0\) and \(E\)), an equilibrium state is impossible; the energy liberated in the dielectric is always greater than that removed, and the temperature of the dielectric increases without bound.

Thus the conditions for tangency of the straight line \(W_2\) to the curve \(W_1\) are critical for the onset of thermal breakdown. These conditions are partly expressed by the following differential equation, which characterizes the energy state inside the dielectric:

\[ -\operatorname{div}(\lambda \operatorname{grad} T)+\gamma(\operatorname{grad}\varphi)^2=0, \tag{10,2} \]

where \(\lambda\) is the coefficient of thermal conductivity.

The first term of the equation gives the amount of energy removed from a unit volume per unit time; the second term gives the amount of energy liberated.

However, this equation alone is not sufficient, since an equilibrium state occurs not only at tangency, but also when the curve \(W_1\) intersects the straight line \(W_2\). The position of the straight line \(W_2\) depends on the initial temperature \(T_0\). Taking the initial temperature of the dielectric equal to the temperature of the surrounding medium, the initial conditions can be reduced to boundary conditions.

The boundary conditions under which (10,2) no longer has a solution give the desired critical conditions. The relation between the field strength \(E\) and the boundary temperature \(T_0\) satisfying the critical conditions gives the connection between the breakdown field strength and the initial temperature of the dielectric.

A solution of (10,2) is possible only by numerical integration, which was carried out by V. A. Fock\({}^{30}\). If, however, instead of (10,1) for the dependence between the conductivity of the dielectric and the temperature, one adopts the expression

\[ \gamma=\gamma_0 e^{bT}, \]

then in some simple cases (10,2) can be solved completely.

The theory of thermal breakdown presented is the most complete and rigorous. It was developed by V. A. Fock\({}^{30}\). In cases where heat removal takes place only through the thickness of the dielectric to the electrodes, the breakdown voltage is equal to

\[ U_{\mathrm{br}}=\sqrt{\frac{33.6\,\lambda}{\gamma_0 b}}\, f e^{-\frac{b}{2}T_0}, \]

where \(f\) is a known function depending on the ratio of the thicknesses and thermal conductivities of the electrodes and the dielectric.

In the last formula all quantities are expressed in practical units. From this formula it is evident that thermal breakdown is possible only in the case when the conductivity increases...

with increasing temperature. In the opposite case, an equilibrium state is possible at any initial temperature. If the conductivity of the electrode is large in comparison with the conductivity of the dielectric, and the thickness of the electrode is small in comparison with the thickness of the dielectric, then \(f \simeq 1\) and the voltage at thermal breakdown does not depend on the thickness of the dielectric. For small dielectric thicknesses \(d\), the relation between the breakdown voltage and the dielectric thickness is quite well expressed by a parabola

\[ U_{\mathrm{br}} \sim \sqrt{d}. \]

In Fock’s theory it was assumed that all the heat is removed in the direction toward the electrodes. This occurs only in ideally homogeneous dielectrics. In practice, dielectrics always have more or less sharply expressed inhomogeneities. In the case of a threadlike impurity with a conductivity greater than the conductivity of the dielectric, intensive release of energy will occur in the impurity. Heat removal will take place parallel to the electrodes. An exact solution of this problem encounters considerable mathematical difficulties; therefore only approximate solutions are available. The most accurate solution was given by Rogowski\(^{31}\) and Dreyfus\(^{32}\). The dependence of the breakdown voltage on the dielectric thickness, obtained by Rogowski, for an impurity radius considerably smaller than the dielectric thickness, has the following form

\[ U_{\mathrm{br}} = K d^{3/4}, \]

where \(K\) is a coefficient depending on the thermal conductivity of the dielectric and on the radius of the impurity. The larger the radius of the impurity, the smaller \(K\).

Thus, in thermal breakdown the breakdown voltage is proportional to the dielectric thickness raised to the power \(n\)

\[ U_{\mathrm{br}} = K d^{n}, \]

where \(n\) lies within the limits from \(n = 0.5\) to \(n \simeq 1\), depending on the degree and character of the inhomogeneity of the dielectric.

In the theories of Fock, Rogowski, and Dreyfus it is assumed that breakdown occurs only when the equilibrium state of the dielectric becomes unstable. Some dielectrics may be destroyed at temperatures below those at which the indicated critical state could occur, i.e., at \(T_{\mathrm{p}} < T_{\mathrm{k}}\) (Fig. 17). In these cases the breakdown field strength is that field strength at which \(T_{\mathrm{a}} = T_{\mathrm{p}}\).

Thermal processes are slow. Therefore thermal breakdown requires a comparatively long time. To determine this time it is necessary to add to (10.2) a third term expressing

the energy expended in heating the dielectric,

\[ -\operatorname{div}(\lambda \operatorname{grad} T)+\gamma(\operatorname{grad}\varphi)^2-C\frac{dT}{dt}=0, \tag{10,3} \]

where \(C\) is the heat capacity of the dielectric.

Only approximate solutions of this equation are available. Several methods of approximate solution have been developed by G. A. Grinberg, M. I. Kontorovich, and N. P. Lebedev\({}^{33}\).

The theory of thermal breakdown set forth above proceeded from the assumption that points \(A\) and \(B\) (Fig. 17) must coincide for breakdown to occur. However, even when these points are situated sufficiently close to one another, owing to random fluctuations the temperature of individual places may become somewhat greater than \(T_B\). Thus the onset of an unstable state has a certain probability already upon approach to the critical conditions. The closer the conditions are to the critical ones, the greater the probability of breakdown. Under critical conditions it is close to unity. The very inhomogeneity of the dielectric and its character (magnitude, shape, position, nature) affect the magnitude of the slope of the straight line \(W_2\) and, consequently, the positions of points \(A\) and \(B\). Since this influence depends on random causes, the critical conditions themselves may vary within certain limits for one and the same dielectric. These two circumstances lead to the fact that the breakdown voltage in thermal breakdown has a certain scatter.

Thermal breakdown of a solid dielectric, as is clear from the foregoing, depends on particular conditions (shape, initial temperature, etc.). The theory of thermal breakdown, verified in particular cases, agrees well with experimental data\({}^{34,35}\).

11. CONCLUSION

The analysis carried out above of all the proposed mechanisms of breakdown of solid dielectrics makes it possible to draw the following conclusion.

There is no doubt that the occurrence of breakdown is preceded by a very strong increase in the electric current flowing through the dielectric as a result of the action of the electric field. Three mechanisms may be indicated that contribute to an increase in the conductivity of a dielectric situated in an electric field: thermal, electrostatic, and impact ionization.

Figure 18 gives the dependences of the current density in a solid dielectric, due to various mechanisms of conductivity, on the electric-field strength. It is evident from the figure that the mechanism of thermal ionization comes into action before all others. This mechanism plays the principal role up to gradients somewhat below \(10^6\ \mathrm{V/cm}\). It is interesting to note that, up to several kilovolts per cm, the current of thermal ionization practically obeys Ohm’s law. Thermal ionization may lead to currents preceding—

than breakdown, at gradients of the order of \(10^8\ \mathrm{V/cm}\). At gradients somewhat lower than \(10^6\ \mathrm{V/cm}\), the second mechanism begins to acquire predominant importance—electrostatic ionization. This mechanism can also lead to strong currents preceding breakdown at gradients of the order of \(10^7\ \mathrm{V/cm}\). However, breakdown occurs at smaller gradients, of the order of \(10^6\ \mathrm{V/cm}\), and the currents preceding breakdown are caused by the mechanism of impact ionization, which comes into action almost immediately after the mechanism of electrostatic ionization. That the increase in conductivity in the region of gradients lying between the region of Ohm’s law and the breakdown gradients is not caused by the mechanism of impact ionization follows from the following fact. If, from the current in a dielectric under illumination, one subtracts the dark current, the result is a current due only to photoelectrons. This current strictly obeys Ohm’s law even in such strong fields in which the dark current deviates from Ohm’s law. From this it may be concluded that the deviations of the dark current from Ohm’s law are not caused by impact ionization, but by some other mechanism, since otherwise it would be unclear why photoelectrons cannot participate in impact ionization on an equal footing with dark electrons.

Fig. 18.

For the beginning of the process of impact ionization, the presence of “free” electrons or electrons in the conduction band is necessary. The presence of such initial electrons is ensured by the mechanism of thermal or electrostatic ionization, whose action precedes impact ionization.

Let us consider in greater detail the process of impact ionization from the classical point of view. An available free electron is accelerated by the electric field and, upon collision with a neutral particle, tears off an electron. The detached electron, together with the ionizing ...

accelerate, ionize again, etc. As a result of a series of such successive acts the number of free electrons increases according to an exponential law. These electrons form an avalanche moving toward the anode. If there is one initial electron, one avalanche is formed. One avalanche gives one current pulse. For the continuous passage of electric current it is necessary that successive avalanches be formed, and for this it is necessary, after or during the passage of each avalanche, that a new initial electron or new initial electrons be formed. The continuous passage of current requires the continuous formation of initial electrons. In gas breakdown the initial electrons are formed by photoionization by the radiation accompanying the formation of the avalanche. In a solid dielectric the continuous formation of initial electrons is ensured by thermal or electrostatic ionization. The intensity of such a process must be small, several orders of magnitude less than the intensity of the process of impact ionization.

At the beginning of the development of the avalanche the process has an unstable character. However, as it develops, factors begin to form and intensify which hinder the further development of the process. After the passage of the electron avalanche there remains a wake of positively charged particles*), the density of which increases almost likewise according to an exponential law. This positive space charge retards the motion of some of the electrons.

The retarding action of the positive space charge increases with the growth of the avalanche and acts in a stabilizing manner on the development of the ionization process. Stabilization here is understood in the sense that, starting from some moment, the density of the positive charge in the wake or the ionization intensity becomes constant.

The further processes, relating to the last stage of breakdown, are still not completely clear. Concerning these processes a number of assumptions may be made. Electrons retarded by the positive space charge are trapped and form either negative ions, or else neutralize the positive ions in ionic crystals. In both the first and the second cases the normal distribution of forces between the particles is disturbed. This may produce, at separate places in the dielectric, such large local stresses that a displacement of a group of particles and the formation of microscopic cracks occur.

Another assumption is possible. The retarded electrons cannot ionize in collisions with heavy particles, but are capable of transmitting to them energy which sets them into vibrational motion. Thus, separate places in the dielectric may heat up

*) This is also true for the case of ionic crystals. In this case ionization is expressed in the detachment of an electron from a negative ion. After the passage of the electron avalanche there remains a wake of uncompensated positive lattice ions.

to the high temperature at which melting or burning-through of these places occurs.

In ionic crystals, the neutralization of positive ions leads to the formation of metal atoms, and this may lead to the formation of metallic bridges that short-circuit the gap between the electrodes and are melted by the short-circuit current. Other assumptions are also possible.

From the point of view of quantum mechanics, the breakdown process may be interpreted as follows. For ionization to begin, the presence of a certain number of electrons in the conduction band is also necessary. Ionization occurs when an electron in the conduction band collides with an electron in the normal band. As a result of this, the electron from the normal band passes into the conduction band. The normal band is gradually depleted of electrons, while the conduction band, on the contrary, is replenished. This process, initially unstable in character, is adequate to the process of avalanche formation in the classical interpretation.

As electrons accumulate in the conduction band, the reverse process begins—the transition of electrons from the conduction band into the normal band. This reverse transition is a stabilizing process. Here stabilization should be understood as the establishment of a constant distribution of electrons over the bands between which electron exchange takes place. The transition of an electron from the conduction band to the normal band is possible when energy is transferred by an electron in the conduction band to a heavy particle.

As a result of such energy transfer, the dielectric is heated to melting or burning-through. Since such a process is facilitated and most probable where local levels exist, it develops in places with inhomogeneities. The difference between the classical and the quantum-mechanical interpretation of the process consists in the fact that, from the point of view of quantum mechanics, the transition of an electron from one band to another is not accompanied by a spatial displacement of the electron. Therefore, the accumulation of electrons in the conduction band does not mean a concentration of electrons in some definite place, and in this sense it is not adequate to the formation of an avalanche. Therefore quantum mechanics cannot explain the mechanical destruction of a dielectric by the displacement of particles. For continuity of the process, from the point of view of quantum mechanics, a continuous supply of initial electrons is not required. They are needed only for the start of the process.

The views on the nature of the electrical breakdown of a solid dielectric developed here are, in essence, a deeper interpretation of the idea first proposed by Smurov. In its time this idea did not receive due attention. However, the dialectical development of the doctrine of the nature of electrical breakdown has again led to this idea; moreover, at a new level the idea has, naturally, undergone a fuller and deeper elaboration.

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1. Reference number as printed on the page. ↩