Remarks on von Hippel’s Theory of Electrical Breakdown of Solid Dielectrics

E. K. Zavadovskaya

The semi-empirical explanation of the electrical breakdown of solid dielectrics proposed by von Hippel [1] is distinguished by a deep physical treatment of the question and by considerable experimental and theoretical material brought to bear on the explanation of the phenomenon. According to von Hippel, breakdown develops as a result of impact ionization caused by conduction electrons. In the ionization theory, the electric strength of a dielectric is defined as the equilibrium electric-field strength at which the energy accumulated by an electron in the electric field becomes equal to the energy transferred by the electron to the vibrations of the lattice. Slow electrons, moving in an ionic crystal lattice, interact with its sites as a consequence of polarization. The electrons lose part of their energy in exciting lattice vibrations. Acceleration of the electrons will be possible if the increase in the electrons’ energy in the external electric field compensates for the loss of their energy expended on exciting the lattice sites.

The magnitude of the energy transferred by electrons to the excitation of the lattice depends on the velocity of the electrons. Slow electrons transfer little energy to the lattice, since their energy is small. Fast electrons transfer little energy to the lattice sites, because the time of interaction of an electron with a lattice site is small. Thus, the energy transferred by an electron to the lattice, considered as a function of the magnitude of the electron’s energy, has a maximum. Correspondingly, the strength of the external field through whose energy the accumulation of electron energy takes place also passes through a maximum, as shown in Fig. 1.

Fig. 1. Equilibrium field strength \(E(w)\) as a function of the energy of conduction electrons

The magnitude of this maximum, according to von Hippel, corresponds to the energy of the natural vibrations of the lattice ions, calculated from the residual-radiation frequency, or is a multiple of the number of energy quanta of the natural vibrations of the lattice.

Hippel considered the conditions under which any electron, moving in an electric field from one lattice site to another, accumulates more energy than it transfers to the excitation of vibrations.

The condition for breakdown according to Hippel is the possibility that an electron accumulates energy in the field and that its velocity increases up to the production of ionization. The determining factor for breakdown is ionization and the growth of the electronic current, and not the interaction of the electrons with the lattice.

The equilibrium field strength \(E_m(W)\) corresponds in this case to the energy of sufficiently slow electrons \(W_m\). This breakdown condition has received in the literature the name of the breakdown criterion with “low electron energy.”

The breakdown condition for ionic crystals according to Hippel is written in the form:

\[ e E_{np} a_0 = c h \nu_{oct}, \tag{1} \]

where \(e\) is the charge of the electron, \(E_{np}\) is the electric strength,

\(a_0\) is the lattice constant,

\(\nu_{oct}\) is the frequency of vibrations of the residual rays, and

\(c\) is a constant of order \(1.7\).

Another breakdown criterion was proposed by Fröhlich [2]. He takes into account that electrons, in their collisions, experience collisions with lattice phonons; therefore the increase in their energy will be proportional to the square of the field strength, as is the case in the theory of conductivity. Accumulation of energy takes place only for fast electrons.

Impact ionization will begin when the electrons have accumulated the necessary energy \(W_i\). At first an equilibrium will be established between ionization and recombination. When the field is increased above the equilibrium field strength \(E(W_i) = E(W)\), breakdown will occur. Thus, according to Fröhlich, breakdown occurs at a high value of the electron energy, equal to the ionization energy. The electric strength of dielectrics determined according to Fröhlich’s criterion is lower than the electric strength determined according to Hippel’s criterion (Fig. 1).

Franz [3] believes that breakdown develops when the density of the electronic current through the dielectric reaches a value of about \(1\ \text{a}/\text{cm}^2\).

It has been proposed [4] to calculate the value of the electric strength of a dielectric from the condition of the minimum of the function determining the dependence of the electron mobility on the field strength.

Using this last breakdown criterion, let us calculate the value of the electric strength of crystals of the alkali-halide series, assuming that the energy of the electrons is equal to the energy of the natural vibrations of the lattice ions. Let us consider the transition of an electron from one equilibrium state near a positive ion to another, neighboring one, at absolute-zero temperature. The transition takes place in the direction of the cube edge with an unchanged store of electron energy. Two neighboring equilibrium positions for the electron are located at a distance \(2a_0\) (\(a_0\) is the lattice constant) from one another and are separated by a potential barrier of height \(U_0\) near the negative ion. The probability of the electron passing through the barrier at each vibration will be \(e^{-U_0/W_k}\), or \(\nu e^{-U_0/W_k}\) per second, where \(\nu\) is the frequency of the electron’s natural vibrations, and \(W_k\) is its energy.

If there is an electric field \(E\) in the dielectric, then the potential barrier for electrons in the direction of the accelerating field is lowered by the amount

\[ \frac{E e a_0}{\varepsilon_0}, \]

where

\(a_0\) is the distance from the maximum of the potential barrier to its minimum, and

\(\varepsilon_0 = n^2\), where \(n\) is the refractive index.

The probability of an electron transition during one oscillation is \(e^{-\frac{U_0-E' e a_0}{W_k}}\), or per second

\[ \nu_1=\nu \, e^{-\frac{U_0-E' e a_0}{W_k}}, \quad \text{where } E'=\frac{E}{\varepsilon_0}. \]

The current density is determined from the condition: \(i=n e \delta \nu_1\), where \(\delta\) is the magnitude of the displacement of the electrons in the dielectric, or from the condition: \(i=n e U E'\), where \(U\) is the mobility of the electrons. Hence we obtain:

\[ U=\frac{\delta \nu}{E'} \cdot e^{-\frac{U_0-E' a_0 e}{W_k}}. \]

From the condition for the minimum of the mobility we obtain an expression for determining the electric strength in the form:

\[ E_{\text{br}}=\frac{W_k \varepsilon_0}{e a_0}. \tag{2} \]

For \(W_k=h\nu_{\text{res}}\) we obtain:

\[ E_{\text{br}}=\frac{h\nu_{\text{res}}\varepsilon_0}{e a_0}. \tag{3} \]

Thus, a formula has been obtained theoretically for calculating the electric strength, having the same form as the formula obtained by Hippel. The experimentally found constant \(C\) in this case has the physical meaning \(C=\varepsilon_0\).

The table gives the results of calculations by formula (3), substituting into it experimentally determined values of the energy quantum of the residual radiation, the lattice constant, and the dielectric coefficient. The experimentally measured value of the electric strength is given according to the literature data [5]. These data agree closely with those obtained by us. In the worst case, the discrepancy between the measured electric-strength values and those calculated from the experimental data by means of formula (3) does not exceed 55%. Also of interest is the obtained dependence of the electric-strength value on the characteristics of the dielectric \((a_0,\ \varepsilon_0,\ \text{and } h\nu_{\text{res}})\), as determined by formula (3), and its agreement with the experimental data presented in the table.

Table

In the above-cited work of Franz, for calculating the magnitude of the electric strength of dielectrics at breakdown due to field ionization, the formula obtained is

\[ E_{\mathrm{br}} \sim \frac{m a_0 W i^2}{240 e \hbar^2}. \tag{4} \]

The quadratic dependence of the electric strength on the magnitude of the ionization energy, determined from the edge of intrinsic absorption in the ultraviolet part of the spectrum, has been observed previously [7].

I express my gratitude to Prof. A. A. Vorob'ev for discussion of the questions raised in the article.

CITED LITERATURE

1. A. Hippel, Appl. Phys. 8, 815, 1937.
2. H. Fröhlich, Proc. Roy. Soc. A 160, 230, 1937.
3. W. Franz, Naturwissen 39, 1, 18, 1952.
4. E. K. Zavadovskaya, DAN 82, 4, 565, 1952.
5. H. Mott and R. Gurney, Electronic Processes in Ionic Crystals, IL, 1950, p. 226.
6. E. K. Zavadovskaya, DAN (in press).
7. A. A. Vorob'ev, High-Voltage Engineering, GEI, 1945.