MODERN THEORY OF ELECTRICAL BREAKDOWN OF SOLID DIELECTRICS (Review)

Yu. A. Starikin

Introduction

The theory of electrical breakdown of solid dielectrics proceeds from the premise that the current carriers in the electrical breakdown of solid dielectrics are electrons. The idea of the electronic nature of the breakdown of solid dielectrics was first put forward by Smurov in 1927 and was subsequently developed by Hippel, who gave experimental confirmation of the electronic character of breakdown. The essence of Hippel’s experiment amounted to the following: he subjected to electrical breakdown a crystal in which copper ions had been diffused; the coloration of the copper ions that appeared after breakdown meant that free electrons had appeared during breakdown.

From the point of view of the theory of electrical breakdown, it is necessary that free electrons appear in avalanche-like fashion in a solid dielectric, which, under the action of an external electric field, would form a sufficiently strong current leading to destruction of the dielectric material.

Hippel and Zener proposed two different mechanisms of electrical breakdown of solid dielectrics. Both mechanisms proceeded from the conception of the band structure of the energy spectrum of electrons in a solid. According to this conception, a solid dielectric is characterized by the presence of two different energy bands: the valence band and the conduction band, separated from one another by a band of forbidden energy, or forbidden band. At absolute zero all the energy levels of the valence band are occupied by electrons, while all the energy levels of the conduction band are free. Therefore, in order to obtain conductivity in a dielectric it is necessary to transfer some of the electrons from the lower, filled valence band into the upper, empty conduction band. Such a transition of electrons is associated with an expenditure of energy and may be accomplished under the influence of external actions, for example, upon absorption of light, upon heating, or under the action of an external electric field. Electrons that have entered the conduction band can, in the presence of an external electric field, create an electric current, and therefore they are called conduction electrons.

The first mechanism of breakdown of a solid dielectric, proposed by Hippel [1], in 1936 received the name “the theory of electronic impact ionization.”

Let there be at least one conduction electron in the conduction band. Placing the dielectric in an external electric field, we note that the electron is acted upon, on the one hand, by the external electric field, accelerating its motion, and, on the other hand, as a result of the interaction of the electron with the vibrations of the crystal lattice, it loses part of its energy and is therefore retarded. In order for the electron to be able to acquire sufficient energy for ionization, it is necessary that the accelerating action

the external electric field was stronger than the retarding action of the crystal lattice. An electron moving with acceleration can, by impact ionization, liberate one of the bound electrons of the valence band and transfer it to the conduction band. Then an avalanche-like multiplication (proliferation) of conduction electrons begins, which leads to a sharp increase in the number of conduction electrons in the dielectric and, ultimately, to breakdown of the dielectric.

Therefore, as the breakdown criterion, Hippel considered that the action of the field on the electron must be stronger than its retardation. To overcome the retardation of an electron it is necessary to apply a certain electric field, whose strength \(G(E)\) will depend on the electron energy and is called the equilibrium field strength. According to Hippel, the strength of the external field at breakdown must be greater than the maximum equilibrium field strength, i.e.,

\[ F > G_m(E). \tag{1} \]

A quantitative calculation for determining the field strength at breakdown from the standpoint of Hippel’s theory was given by Fröhlich in 1937 and by Franz in 1939.

Fröhlich [2], in solving the problem, restricted himself exclusively to ionic crystals and proposed that the retardation of electrons is a consequence of the excitation of optical vibrations of the lattice. Optical vibrations of the crystal lattice possess, according to Fröhlich, a definite frequency \(\nu_0\) and require for their excitation a definite amount of energy \(h\nu_0\). If an electron has an energy of several \(h\nu_0\), it is strongly retarded, giving up its energy to the excitation of lattice vibrations; fast electrons, on the contrary, experience moderate retardation. For discussion of the theory of impact ionization, what is of interest is not the retardation itself, but the field strength necessary to overcome the retardation, or the equilibrium field strength as a function of the energy of conduction electrons. The course of this function is shown graphically in Fig. 1. It is seen from the graph that the maximum of the equilibrium field strength \(G_m(E)\) corresponds to an electron energy

\[ E_1 = h\nu_0 \simeq \frac{1}{30}\ \text{eV}, \]

and for higher energies the equilibrium field strength decreases.

Fig. 1. Equilibrium field strength as a function of energy.

\(F_H\) — field strength for breakdown according to Hippel,
\(F_{Fr}\) — field strength for breakdown according to Fröhlich.

Hippel’s criterion indicates that the breakdown field strength must be greater than \(G_m(E)\). Then all electrons from zero energy up to the ionization energy \(J\) (i.e., up to the energy necessary for impact ionization of bound electrons in the valence band) will be accelerated and, upon reaching the ionization energy, can liberate bound electrons in the valence band, transferring them to the conduction band. The process of increase in conduction electrons, or their multiplication, proceeds avalanche-like, which leads to breakdown of the dielectric.

Fröhlich proposed another breakdown criterion: he held that breakdown will occur when conduction electrons with energy \(J\), overcoming braking, begin to accelerate, i.e., according to Fröhlich,

\[ F > G(J). \tag{2} \]

In contrast to Hippel’s criterion [1], called the “low-energy criterion,” Fröhlich’s criterion [2] is called the “high-energy criterion.”

Fröhlich’s theory did not take into account the braking action of acoustic vibrations of the crystal lattice and, consequently, could not be applied to nonpolar crystals.

Fig. 2. Equilibrium field strength as a function of energy.

— with allowance for optical vibrations of the lattice,
-- with allowance for acoustic vibrations of the lattice.

For nonpolar crystals there are no optical vibrations; therefore the braking action of the crystal lattice must be due to acoustic vibrations of the lattice. The main difference between “optical” vibrations and “acoustic” ones consists in a different energy dependence of the braking action (Fig. 2): the braking action of acoustic vibrations increases up to the energy \(E_2 \simeq 1\ \mathrm{eV}\) and then decreases. The maximum of this curve, \(G(E_2)\), determines, according to Hippel, the breakdown field strength, i.e., \(F > G(E_2)\).

Both breakdown criteria (Hippel’s criterion as well as Fröhlich’s criterion) were analyzed in a number of works and proved untenable.

Thus, in 1950 Seitz showed that, because of the statistical distribution of conduction electrons, for breakdown \(1/5\) of the Hippel criterion for the breakdown field strength is sufficient, i.e.,

\[ F > \frac{1}{5} G_m(E). \]

Zener and Teller, Fröhlich, Franz, and Kallén came out against Fröhlich’s criterion, proving its untenability.

The untenability of these breakdown criteria, among other things, is also explained by the fact that both Hippel and Fröhlich completely failed to take into account the process of electron recombination.

The second mechanism of dielectric breakdown is the so-called “theory of internal cold emission,” proposed by Zener in 1934 and further developed by Franz in 1939.

Zener [3] showed that, in the presence of a strong electric field, electrons can pass from the valence band into the conduction band. The possibility of such a transition is explained by the fact that, from the point of view of quantum physics, the imposition of an electric field on a dielectric changes the energy levels of the electrons and, consequently, changes the energy bands: if the external field is directed, for example, to the right, then the energy bands acquire a slope likewise to the right (Fig. 3).

Fig. 3. Distribution of energy bands in a solid dielectric in the presence of an external field.

Therefore an electron having energy \(E_1\) will move along a horizontal line, crossing the valence band, where it was initially located, and passing into the conduction band through the forbidden band. This seepage of the electron through the forbidden band is a tunnel effect through the potential barrier \(ABC\).

Since the probability of transition through an energy barrier depends on the width and height of this barrier, in order to obtain the tunnel effect it is necessary to apply a sufficiently strong external electric field, since the slope of the bands depends on the field strength.

Zener showed that, at an electric-field strength of the order of \(10^6\ \text{V/cm}\), an electron from the valence band can move by several hundred atomic distances and enter the conduction band. With time (the duration of breakdown according to the theory of cold emission, as also according to the theory of impact ionization, is \(10^{-12}\ \text{sec}\)), a sufficient number of conduction electrons appears, which leads to breakdown.

A major shortcoming of Hippel’s theory of impact ionization and Zener’s theory of internal cold emission is that, despite a sufficiently good substantiation of the individual processes, in determining the breakdown strength one has to use unreliable criteria. In addition, both theories require a great idealization of the external conditions (homogeneity of the crystal, absence of disturbances in the crystalline structure), which in reality cannot be obtained.

The modern theory of electrical breakdown, developed by W. Franz [4, 5, 6], gives several new propositions for explaining breakdown in an inhomogeneous crystal. Unlike the previous theories, it takes into account the possibility of disturbances in the crystalline structure and considers all the basic processes—the process of electron braking in the crystal lattice, the process of impact ionization and recombination in their dynamics, and the process of internal cold emission—and then gives an analysis of the breakdown conditions.

According to Franz, breakdown and then destruction of the material occur when the stable state of the substance is disturbed, when the dielectric begins to melt or else turns into a semiconductor (and, as a consequence of this, also soon melts). Such a change in the state of the substance occurs under the influence of the heat released during the passage of electric current through the dielectric.

Therefore it is important to determine how the current strength changes with time at different electric-field intensities and, consequently, how the heat released in the dielectric changes. For this it is necessary first of all to determine the conditions under which electron multiplication occurs, since the change in the number of conduction electrons ultimately determines the change of current in the dielectric.

Therefore, we shall first consider the states of electrons in the dielectric and their changes due to various processes, and then determine the conditions for electron multiplication; after this we shall proceed to consideration of the change in the thermal state of the dielectric and determine the conditions under which dielectric breakdown occurs.

Electron States in the Conduction Band

To resolve the question of changes in the state of electrons in the conduction band, we shall use the concept of the flux of conduction electrons across a certain energy boundary in energy space. On the basis of the laws of statistics, conduction electrons will have different energies. A schematic representation of the distribution function of conduction electrons may be given in the form of the curve shown in Fig. 4. (A more exact form of this function will be obtained later.) Here $\chi(E)\,dE$ is the density of conduction electrons whose energy lies in the energy interval from $E$ to $E + dE$.

Fig. 4. Distribution function of electrons $\chi(E)$ (schematic representation).

Here we note that the number of conduction electrons, generally speaking, changes with time, and therefore the expression $\int_0^\infty \chi(E)\,dE = N$ (where $N$ is the density of conduction electrons) applies to some fixed instant of time.

On the energy axis we choose an energy boundary and its neighborhood $(E - \varepsilon, E + \varepsilon)$. Electrons can cross this boundary as a result of their interaction with lattice vibrations and with the external electric field, and also as a result of the processes of impact ionization and recombination of electrons.

Let us consider in more detail the flux of electrons across the energy boundary $E$.

A. Passage of Electrons across an Energy Boundary as a Result of Their Interaction with Vibrations of the Crystal Lattice

The interaction of conduction electrons with vibrations of the crystal lattice may be regarded as a process in which electrons absorb and emit phonons. If an electron lying in the energy region between \(E-\varepsilon\) and \(E\) absorbs a phonon whose energy is equal to or greater than \(\varepsilon\), then the electron will cross the energy boundary \(E\) into the region of higher energies. On the other hand, if an electron lying in the energy region between \(E\) and \(E+\varepsilon\) emits a phonon whose energy is equal to or greater than \(\varepsilon\), then it will cross the energy boundary \(E\) into the region of lower energies. Assuming that the probability for an electron to absorb or emit a phonon depends only weakly on the electron energy, and neglecting this dependence, we obtain for the flux of electrons through the energy boundary \(E\) an expression that depends linearly on \(\chi(E)\) and \(\dfrac{d\chi(E)}{dE}\):

\[ S_{\Phi}(E)=A_{1}\chi(E)+B_{1}\frac{d\chi(E)}{dE}, \tag{3} \]

where \(A_{1}\) and \(B_{1}\) are certain coefficients, and the symbol \(\Phi\) indicates that the flux \(S(E)\) is calculated as a result of the interaction of electrons with phonons.

B. Passage of Electrons across an Energy Boundary as a Result of the Action of an External Electric Field on the Electrons

As a result of the action of an electric field on an electron, its momentum changes by the amount \(\Delta \vec p=e\vec F\tau\), where \(\tau\) is the time between two successive collisions of the electron. Consequently, as a result of the action of the electric field on the electron, the electron energy will also change. We shall assume that, owing to the small density of conduction electrons, the interaction between electrons is weak and may be neglected. Then the dependence of the electron energy on its momentum may be written in the form \(E=\dfrac{p^{2}}{2m}\), i.e., the electron energy depends on the absolute value of the momentum \(|\vec p|\). But the absolute value of the electron momentum under the action of the electric field will either increase or decrease depending on the mutual orientation of the direction of the field \(\vec F\) and the direction of the momentum \(\vec p\), namely: if the angle between the direction of the field and the direction of the momentum is acute, then the absolute value of the momentum increases, and the electron energy also increases, i.e., the electron can cross the energy boundary \(E\) toward higher energies; if, however, this angle is obtuse, then the absolute value of the momentum decreases, and the electron energy decreases; the electron crosses the energy boundary into the region of lower energies.

Taking this action of the electric field on the electron into account, one can calculate the flux of electrons through the energy boundary \(E\) as a result of the action of the field. Then we find an expression likewise depending linearly on \(\chi(E)\) and \(\dfrac{d\chi(E)}{dE}\):

\[ S_{n}(E)=A_{2}\chi(E)+B_{2}\frac{d\chi(E)}{dE}, \tag{4} \]

where \(A_{2}\) and \(B_{2}\) are coefficients depending on the field strength \(F\).

The resulting electron flux through the energy boundary \(E\), due to the action on the electrons of lattice vibrations and of the external electric field, is the sum of two fluxes, \(S(E)=S_{cp}(E)+S_n(E)\), and will be represented by an expression that depends linearly on \(\chi(E)\) and \(\dfrac{d\chi(E)}{dE}\):

\[ S(E)=A\chi(E)+B\frac{d\chi(E)}{dE}. \tag{5} \]

B. Influence of ionization and recombination processes on the electron flux

The process of ionization and recombination of electrons can change the number of electrons located in the energy region in the vicinity of the energy boundary.

Indeed, if a conduction electron has sufficient energy, then, upon collision with a bound electron, it can cause the bound electron to pass from the valence band into the conduction band; then the number of conduction electrons increases. Obviously, in order for the newly appearing electron to enter the energy region between \(E\) and \(E+dE\), the ionizing electron must be in the energy region between \(J+2E\) and \(J+2E+d(2E)\), where \(J\) is the ionization energy of the bound electron; the remaining part of the electron energy is distributed equally between the ionizing electron itself and the newly appearing electron. Since both electrons after ionization will have energy in the interval from \(E\) to \(E+dE\), two new electrons will appear in the energy region between \(E\) and \(E+dE\) as a result of such ionization.

On the other hand, an electron located in the energy region between \(E\) and \(E+dE\), during ionization, gives up part of its energy and, consequently, leaves this region; in this case the energy region must lie beyond the energy boundary \(J\), otherwise ionization will not occur. Thus, in the process of ionization the electron flux through the energy boundary can both increase and decrease.

In addition to the ionization process, a recombination process occurs simultaneously, in which conduction electrons pass into the valence band; this process will reduce the electron flux through the energy boundary \(E\).

This change in the flux is determined by the relation

\[ \frac{dS(E)}{dE}=4w_i\chi(J+2E)-(w_i+w_r)\chi(E), \tag{6} \]

where \(w_i\) is the probability of ionization, and \(w_r\) is the probability of recombination.

If \(\dfrac{dS(E)}{dE}>0\), this means an increase in the number of electrons with energy \(E\), or their multiplication; whereas if \(\dfrac{dS(E)}{dE}<0\), this means a decrease in the number of conduction electrons having energy \(E\). Moreover, if this decrease in electrons occurs due to recombination, then in the conduction band there is a reduction in the number of electrons; but if the decrease in electrons with energy \(E\) occurs due to ionization, then this leads to an increase in the number of electrons in the conduction band, or to their multiplication. The latter circumstance is especially important for the theory of breakdown.

Taking into account both the interaction of conduction electrons with lattice vibrations and with the external electric field, and the processes of ionization and recombination, we obtain two equations relating the distribution function \(\chi(E)\) and the electron flux \(S(E)\), namely:

\[ S(E)=\int_E^\infty (w_i+w_r)\chi(E')\,dE' -4\int_E^\infty w_i\chi(J+2E')\,dE'; \tag{7} \]

\[ \chi(E)=\frac{3m}{2E}\int_E^\infty \frac{S(E')\,dE'}{\tau e^2(F^2+H^2)} \cdot \exp\left\{\int_E^{E'}\frac{dE''}{E''}\cdot \frac{3G^2-2H^2-3F^2}{2(H^2+F^2)}\right\}, \tag{8} \]

where
\[ \begin{aligned} E&\text{ — an arbitrary energy boundary;}\\ \tau&\text{ — the time between two successive collisions of an electron;}\\ F&\text{ — the strength of the external electric field;}\\ G&\text{ — the equilibrium field strength;}\\ H&\text{ — a certain computational quantity;}\\ w_i&\text{ — the probability of electron ionization;}\\ w_r&\text{ — the probability of electron recombination;}\\ J&\text{ — the ionization energy;}\\ m\ \text{and}\ e&\text{ — the mass and charge of the electron.} \end{aligned} \]

Considering expression (8) for \(\chi(E)\), we see that, in a first approximation, the function \(\chi(E)\) is determined by the exponential term, whose magnitude depends substantially on the function \(G(E)\).

The qualitative behavior of the function \(\chi(E)\) can be obtained by considering the function \(G(E)\)—the equilibrium field strength. Suppose that an external electric field is applied to the dielectric, whose strength \(F\) is less than the maximum equilibrium field strength (Fig. 5). Then electrons with energy \(E_1\), at which the field strength is equal to the equilibrium field strength, do not change their energy. Slower electrons are accelerated by the field, moving into the vicinity of the point \(E_1\) in energy space. Electrons with energy in the interval from \(E_1\) to \(E_2\) will be decelerated more strongly by lattice vibrations than they are accelerated by the external field, and therefore will also move into the vicinity of the point \(E_1\) in energy space. Electrons with energy greater than \(E_2\) are again accelerated by the field and, upon reaching the energy \(J\), become capable of producing impact ionization. Thus, in the vicinity of the point \(E_1\) we shall have a maximum of the function \(\chi(E)\) and a sharp decrease of it at energy values greater than the ionization energy \(J\). The behavior of the function \(\chi(E)\) is shown graphically in Fig. 6.

From equation (7) it is easy to find approximate values for \(S(E)\), under the condition that for \(E>J\), \(w_r\ll w_i\):

\[ S(E)=S(J)-S(J+2E)+\int_E^\infty w_r\chi(E')\,dE' \qquad \text{for } E\leq J \tag{7'} \]

\[ S(E)=S(J)\cdot \exp\left\{-\sqrt{\frac{3mw_i}{2J\tau}}\frac{E-J}{eF}\right\}. \tag{7''} \]

The function \(S(E)\) is shown graphically in Fig. 7.

Fig. 5. Effect of an external field on conduction electrons.
\(F\)—strength of the external electric field, \(G(E)\)—equilibrium field strength. (Electrons with energy \(0 < E < E_1\) and \(E > E_2\) are accelerated, while electrons with energy \(E_1 < E < E_2\) are decelerated.)

Fig. 6. Electron distribution function \(\chi(E)\).

Fig. 7. Electron flux \(S(E)\) in energy space.

As already indicated, an increase of the function \(S(E)\) means an increase in the number of conduction electrons, while a decrease of the function \(S(E)\) means a decrease in the number of conduction electrons. From the graph it is seen that there are two decreases of \(S(E)\): in the vicinity of \(E_1\) and for \(E>J\). Since we assume that for \(E<J\), \(w_i=0\), while for \(E>J\), \(w_i\simeq 0\), the decrease in the vicinity of \(E_1\) consequently means a decrease in the conduction electrons due to recombination of electrons, whereas the decrease for \(E>J\) means a decrease of conduction electrons in this region due to ionization. But impact ionization leads to multiplication of electrons in the region of lower energies (which also explains the increase of \(S(E)\) in the region of small energies). For breakdown theory it is precisely the fall of the curve in the region \(E>J\) that plays an important role, for the steepness of this fall determines the intensity of electron multiplication. The decrease of \(S(E)\) in the region \(E>J\) follows an exponential law \((7'')\), and the expression appearing in the exponent depends in a complicated way on the field strength \(F\). Therefore the steepness of the curve will depend on the field strength and, consequently, the intensity of multiplication also depends on the field strength.

Thus, investigation of the change in the state of the electrons has led to the question of electron multiplication.

Stationary distribution and multiplication of conduction electrons

Suppose that we have a solid dielectric in which, along with the valence band and the conduction band, there are also \(f\)-local levels (or an \(f\)-band), caused by disturbances in the crystalline structure of the lattice (absence of ions at lattice sites or the presence of foreign impurities). Schematically we may depict the arrangement of the bands in the form shown in Fig. 8.

Fig. 8. Distribution of electrons in a solid dielectric over separate energy bands.

Let there be \(N\) electrons in a unit volume. Of these, \(N_e\) are in the conduction band, \(N_f\) are in the \(f\)-band, and in the valence band there are \(N_e+N_f\) unoccupied levels, or “holes.”

Usually, instead of the absolute number of electrons, one takes the relative density, taking the density of all electrons as unity. Then in the conduction band the electron density will be

\[ n_e=\frac{N_e}{N}, \]

in the \(f\)-band the electron density is equal to

\[ n_f=\frac{N_f}{N}, \]

and in the valence band the density of “holes” will be

\[ n_e+n_f=\frac{N_e+N_f}{N}. \]

The transition of electrons from one band to another occurs under the action of various causes, namely:

1) impact ionization can transfer an electron from the valence band into the conduction band. The transition of electrons from the \(f\)-band into the conduction band is negligibly small because of the small number of electrons in the \(f\)-band, and therefore we shall neglect this process;

2) thermal vibrations can transfer an electron from the valence band into the \(f\)-band or the conduction band, and also from the \(f\)-band into the conduction band;

3) the external electric field, as a result of internal cold emission, can transfer an electron from the valence band or the \(f\)-band into the conduction band; the transition of an electron from the valence band into the \(f\)-band by means of cold emission is also small because of the small number of free places in the \(f\)-band;

4) external radiation can transfer an electron from the valence band into the conduction band; the transition of electrons from the \(f\)-band into the conduction band and from the valence band into the \(f\)-band is small, since in the first case the number of electrons \(n_f\) is small, and in the second case the number of free places \(f-n_f\) is small; therefore we shall neglect these transitions;

5) spontaneous recombination, or the capture process, can transfer an electron from the conduction band into the \(f\)-band and the valence band, and also from the \(f\)-band into the valence band.

All these processes listed above can change the density of electrons in the conduction band and in the \(f\)-band, and consequently also the density of “holes” in the valence band. Therefore, taking into account the Pauli exclusion principle, one may write the following equations for the change in the density of electrons in the conduction band and the \(f\)-band:

\[ \frac{dn_e}{dt}=n_e w_i+w_{ve}+n_f w_{fe}-n_e(n_e+n_f)w_{ev}-n_e(f-n_f)w_{ef}; \tag{9} \]

\[ \frac{dn_f}{dt}=(f-n_f)w_{vf}+n_e(f-n_f)w_{ef}-n_f(n_e+n_f)w_{fv}-n_f w_{fe}, \tag{10} \]

where: \(w_{ik}\) is the probability of transition from the \(i\)-th band into the \(k\)-th band.

Here it has been taken into account that \(n_e\) and \(n_f\) are small in comparison with unity and, consequently, the number of free places in the conduction band and the number of occupied places in the valence band are close to unity.

This system of equations makes it possible to determine the density of electrons in the conduction band as a function of time, \(n_e=f(t)\).

It should be noted that, for purely electrical breakdown of solid dielectrics, only the processes of impact ionization, internal cold emission, and spontaneous recombination play a decisive role in the transition of electrons from one band to another. These are the processes taken into account in calculating the transition probabilities. Transitions of electrons under the action of thermal vibrations of the lattice and of external radiation are of secondary importance for purely electrical breakdown, and they are usually neglected.

In the absence of an external electric field, the density of electrons in the conduction band is very small. Therefore, when an external electric field is applied, at the initial instant of time the processes of impact ionization and internal cold emission will predominate over the process of spontaneous recombination. An increase in the density of electrons in the conduction band occurs (multiplication of electrons).

As the density of conduction electrons increases, spontaneous recombination of electrons increases. Finally, a stationary state may occur, when the multiplication of electrons is compensated by their recombination; therefore the density of electrons in the conduction band

and in the \(f\)-zone becomes constant. We shall denote the stationary distribution in the form: \(n_e(\infty)\) and \(n_f(\infty)\).

In calculating the multiplication of electrons, one should consider separately its initial stage and the final stage of multiplication, when a stationary distribution of electrons is established. In this, substantially different results are obtained for the case of pure impact ionization and for the case of pure internal cold emission. Thus, in the initial stage, with pure impact ionization, the growth of conduction electrons with time follows an exponential law, whereas with pure internal cold emission the growth of conduction electrons has a linear dependence on time. In the final stage, as already indicated, the distribution becomes stationary and the quantities \(n_e\) and \(n_f\) become equal to \(n_e(\infty)\) and \(n_f(\infty)\).

Graphically this process of growth of conduction electrons is shown in Fig. 9.

Fig. 9. Schematic course of multiplication of conduction electrons.
— for the case of pure impact ionization,
--- for the case of pure internal cold emission.

Change in the Thermal State of the Dielectric

As already indicated above, in order to solve the question of the breakdown conditions of a solid dielectric it is necessary to determine how, under the action of an electric current, the thermal state of the dielectric changes with time.

If, at some initial instant of time, a sufficiently strong electric field is switched on, then multiplication of conduction electrons begins, as a result of which the current strength in the dielectric increases. The current density is determined by the relation:

\[ j = eN_e bF, \tag{11} \]

where \(N_e\) is the number of conduction electrons per unit volume,
\(b\) is the mobility of the electrons,
\(F\) is the field strength.

In our case the number of conduction electrons is determined through the relative density, namely, since \(N = \frac{\nu}{\Omega_0}\) (where \(\nu\) is the number of electrons per elementary cell of the lattice, \(\Omega_0\) is the volume of the elementary cell), then \(N_e = n_e \cdot \frac{\nu}{\Omega_0}\), and therefore the current density may be written in the form

\[ j = en_e \cdot \frac{\nu}{\Omega_0}\, bF, \tag{12} \]

i.e., the current density is proportional to \(n_e\). Therefore the increase of the density

of the current with time is determined by the increase in the density of conduction electrons (Fig. 9)¹).

Thus, at the initial moment we shall have a current that increases either according to an exponential law (in the case of pure impact ionization), or according to a linear law (in the case of internal cold emission). The increasing electric current will raise the temperature of the dielectric relative to the temperature of the surrounding medium. As the temperature of the dielectric rises, the heat transfer to the surrounding medium will also increase. It is possible that in this case thermal equilibrium will be established, when, owing to strong recombination of the multiplication electrons, the conduction-electron multiplication will cease, the increase of the current will likewise cease, and all the energy released by the current will be carried away into the surrounding medium through heat transfer.

If, however, there is a sufficiently strong current, the temperature of the dielectric may reach the melting temperature, when a strong ionic current arises and the material is destroyed (breakdown occurs). We shall call this temperature the critical temperature \(T_{\mathrm{cr}}\).

The question of the rise in the temperature of the dielectric is resolved as follows.

If by \(L(t)\) we denote the power released by the electric current in a unit volume of the dielectric, \(Q(T)\) the amount of energy given up to the surrounding medium by a unit volume of the dielectric in 1 sec,

\[ \rho \chi \frac{dT}{dt} \]

the amount of heat used to heat a unit volume of the dielectric in 1 sec (\(\rho\) is the density of the substance, \(\chi\) its specific heat), then on the basis of the law of conservation of energy one may write the following relation:

\[ L(t)=Q(T)+\rho \chi \frac{dT}{dt} \]

or

\[ \rho \chi \frac{dT}{dt}=L(t)-Q(T); \tag{13} \]

the power released by the current is determined by the relation:

\[ L(t)=jF=en_e\frac{\nu}{\Omega_0}\, bF^2. \tag{14} \]

For the initial moment of time one may put \(Q(T)=0\), since the temperature difference between the dielectric and the surrounding medium is insignificant. Then from (13) and (14) we have:

\[ T-T_0=\int_0^t \frac{L(t')\,dt'}{\rho \chi} = \frac{e\nu b}{\rho \chi \Omega_0}F^2 \int_0^t n_e\,dt'. \tag{15} \]

Consequently, the rise in temperature at the initial moment is determined by the increase of \(n_e\) during the same time.

\[ T-T_0 \sim \int_0^t n_e\,dt. \]

Since, for the case of pure impact ionization, \(n_e\)

¹) It should be noted that in calculating the current density, along with conduction electrons, it is necessary to take into account “holes” in the valence band, which in an electric field behave similarly to conduction electrons and for which one can obtain an expression for the current density analogous to expression (12), but instead of \(n_e\) one should take \(n_e+n_f\). The resulting current density is in this case determined by the sum of these two current densities.

varies according to an exponential law; consequently, in this case the temperature rise at the initial moment will also proceed according to an exponential law. In the case of pure internal cold emission, at the initial moment \(n_e\) varies according to a linear law, and therefore the temperature rise will proceed according to a quadratic law.

Fig. 10. Graph of the temperature rise of the dielectric.

— for the case of pure impact ionization (at the initial moment—exponential course of the rise),

• • • for the case of pure internal cold emission (at the initial moment—quadratic course of the rise)

As electron multiplication proceeds, their number in the conduction band increases and reaches a certain stationary value \(n_e(\infty)\). If, for this moment, the quantity of heat transfer \(Q(T)\) is neglected (which may be done if the process of establishment of the stationary current occurs so rapidly that the dielectric does not have time to heat up), then the temperature rise will proceed according to a linear law, since from (15) we have

\[ T - T_0 = \frac{e \nu b}{\rho \chi \Omega_0} F^2 \int_0^t n_e(\infty)\,dt' = \frac{e \nu b}{\rho \chi \Omega_0} F^2 n_e(\infty)t, \tag{16} \]

finally, when thermal equilibrium sets in and \(dT=0\), we obtain \(L(\infty)=Q(T)\), i.e. the temperature becomes constant and will be determined by the relation:

\[ \frac{e \nu b}{\Omega_0} F^2 n_e(\infty)=Q(T). \tag{17} \]

Fig. 11. Graph of the dependence of the temperature rise of the dielectric on the intensity of the external field \(F_2>F_1\).

The course of the temperature rise is shown in Fig. 10. The temperature rise will also depend on the intensity of the external electric field, \(T - T_0 \sim F^2\). Therefore, as the field intensity increases, the temperature rise will occur more strongly, and the final stationary temperature will be higher. This can be seen from the schematic drawing in Fig. 11.

Breakdown conditions

It was indicated above that breakdown occurs when the material can no longer remain in a stable state and begins to melt or to transform into a semiconductor. Such a change takes place at a certain critical temperature \(T_{\mathrm{cr}}\). Consequently, the necessary condition for the onset of breakdown is

\[ T(t)=T_{\mathrm{cr}}. \tag{16} \]

In the preceding section we found how the temperature of a dielectric changes with time for different field strengths. Let us consider the case of the 4 fields shown in Fig. 12.

From the graph the following conclusions may be drawn:

1) at the electric-field strength \(F_1\), the temperature of the dielectric does not reach the critical value \(T_{\mathrm{cr}}\), and therefore breakdown does not occur;

2) at the field strength \(F_2\), the temperature of the dielectric reaches the critical value \(T_{\mathrm{cr}}\) after a long time and, consequently, breakdown can occur only after a long wait;

Fig. 12. Graph of the rise of temperature for different stages of the process: I—the initial stage, II—the intermediate stage, III—the final stage \((F_1<F_2<F_3<F_4)\).

3) at the field strength \(F_3\), the temperature of the dielectric reaches the critical value \(T_{\mathrm{cr}}\) in the intermediate stage, when we have a stationary current. Consequently, breakdown occurs upon establishment of the stationary distribution of conduction electrons, although the final temperature has not yet been reached;

4) at the field strength \(F_4\), the temperature of the dielectric reaches the critical value \(T_{\mathrm{cr}}\) in the initial stage, when multiplication of conduction electrons takes place.

Using relation (13), one can find the values for \(F_2\), \(F_3\), and \(F_4\):

1) for \(F_2\), when \(L(\infty)=Q(T_{\mathrm{cr}})\), i.e.

\[ \frac{e \nu b}{\Omega_0} F^2 n_e(\infty)=Q(T_{\mathrm{cr}}), \]

and hence

\[ F_2=\sqrt{\frac{\Omega_0 Q(T_{\mathrm{cr}})}{e \nu b n_e(\infty)}} ; \tag{18} \]

2) for \(F_3\), when the stationary current sets in:

\[ \frac{e \nu b}{\rho \chi \Omega_0} F^2 n_e(\infty)t=T_{\mathrm{cr}}-T_0, \]

and hence

\[ F_3=\sqrt{\frac{\rho \chi \Omega_0 (T_{\mathrm{cr}}-T_0)} {e \nu b\, n_e(\infty)\, t}}\,; \tag{19} \]

3) for \(F_4\), when multiplication of electrons occurs:

\[ \frac{e \nu b}{\rho \chi \Omega_0}F^2 \int_0^t n_e\,dt' = T_{\mathrm{cr}}-T_0 \]

or

\[ F_4=\sqrt{\frac{\rho \chi \Omega (T_{\mathrm{cr}}-T_0)} {e \nu b\int_0^t n_e\,dt'}}\,. \tag{20} \]

The first case is practically not observed, since in order to obtain breakdown at \(F_2\) an infinitely long observation time is required.

Fig. 13. Dependence of the breakdown field strength on the lattice constant for alkali-halide crystals.

○ — experimental values according to Hippel.
● — theoretical calculations of Fröhlich.

As regards the second case (breakdown upon the onset of a steady-state current in the dielectric), owing to the thermal effect it is impossible to observe purely electrical breakdown.

Fig. 14. Temperature dependence of the breakdown field strength for KBr.

— experimental data according to Hippel and Alger;
— theoretical calculations of Fröhlich.

Therefore only the last case is of interest, when breakdown occurs in the initial stage in the presence of multiplication of electrons.

In the formulas obtained by Fröhlich, the breakdown field strength depends (weakly) on the temperature of the dielectric and on the lattice constant \((\Omega_0=a^3)\). Comparison with the experimental data is in good agreement with Fröhlich’s theoretical calculations (Figs. 13 and 14).

Thus, the theory of electrical breakdown of solid dielectrics considered here is not limited to the examination of the initial stage of breakdown, but investigates in detail the entire breakdown process up to the moment of destruction of the crystal lattice.

This theory cannot claim to be complete, since it does not explain the dependence of the breakdown voltage on the crystallographic direction.

REFERENCES

1. Hippel, A.: Frg. exakt. Naturwiss. 14, 79, 1935.
2. Fröhlich, H.: Proc. Roy. Soc. (Lond) A 160, 230, 1937.
   Fröhlich, H.: Proc. Roy. Soc. (Lond) A 172, 94, 1939.
3. Zener, Cl.: Proc. Roy. Soc. (Lond) A 145, 523, 1934.
4. Franz, W.: Erg. exakt. Naturwiss. 27, 1, 1953.
5. Franz, W.: Z. angew. Physik. 3, 72, 1952.
6. Franz, W.: Z. Physik 132, 285, 1952.

Novosibirsk, West Siberian
Branch of the USSR Academy of Sciences.