DISCUSSION OF THE REPORTS BY G. I. SKANAVI, Yu. A. STARIKIN, AND V. A. CHUENKOV

A. A. Vorob’ev says:

Physical ideas put forward by researchers—works on the basis of which attempts are made to explain individual aspects of phenomena—prove to be very useful; they must be valued and welcomed. In this respect the work of V. A. Chuenkov is of interest, since it expresses new thoughts deserving attention. One cannot demand from a theory under development a complete explanation of all aspects of a phenomenon, but the theory must explain the basic regularities.

The usual scheme of theories of electrical breakdown is as follows: one seeks the law of the growth of conductivity as a function of field strength. It is tacitly assumed that breakdown will follow the growth of conductivity. Is this how the process actually develops? What is the mechanism of formation of free electrons? It is indeed necessary to find the mechanism of growth of the electron avalanche, but then it is necessary to connect it with the mechanism of destruction of the substance of the crystal lattice itself. Abstract searches for electronic current in a dielectric in general are, of course, useful, but are far from sufficient for solving the question of breakdown. Many researchers work with ionic crystals or with other definite groups of dielectric materials. In studying their properties, one must seek ways of producing materials with high electric strength.

In solving the problem of electrical breakdown, a detailed treatment of the process of energy transfer from electrons to the crystal lattice is necessary. Without clarifying the connection between the electric current and the behavior of the structural elements of the lattice, it is impossible to solve the question of electrical breakdown.

In G. I. Skanavi’s report the electrical properties of certain materials are described. It is possible that the destruction of ceramic materials occurred not because of disruption of the amorphous phase itself, but owing to the formation of cracks and pores. The role of pores has been firmly established by the works of Zavadovskaya, Gutin, Zakgeim, and others. At the branch of the Academy of Sciences of the USSR in Novosibirsk, pressed specimens of dielectrics have been investigated; in this connection a regular change was found in a number of characteristics (conductivity, lattice constant, dielectric constant $\varepsilon$, $\tg \delta$) in relation to the chemical properties and energetic characteristics of the substance. One of the possible directions of investigation is to find the connection between electric strength and the physicochemical properties of dielectrics. These connections must be studied in fields of high intensity, i.e., close to $10^6$ V/cm.

V. I. Odelevskii expresses doubt that, in the ionization processes developing in NaCl, atomic chlorine is formed. In his opinion, neither atomic chlorine nor atomic sodium under impact

ionization is not produced. The breakdown process, despite the considerable currents present in the second stage of breakdown, is not accompanied by electrolysis. It is possible that the currents are associated with the entry of electrons from the cathode.

R. L. Muller says that A. F. Ioffe, studying the electrolysis of NaCl, observed the growth of a dendrite from one electrode to the other, and it was difficult to get rid of the formation of this metallic bridge: special measures had to be taken to prevent the formation of bridges. Breakdown must be accompanied by the liberation of atomic Na and Cl. Microscopic analysis should reveal metallic Na in the crystal (in the channel where breakdown occurred), while chemical analysis should reveal the presence of chlorine in the surrounding medium.

Yu. A. Starikin says the following on the question of the mobility of electrons in solid dielectrics:

In studying the behavior of conduction electrons in solid dielectrics in strong electric fields, the question of the dependence of the electron mobility on the strength of the external electric field is of known interest. This question has been poorly covered in the literature and concerns mainly nonpolar crystals.

However, by a comparatively simple method one can obtain an expression for the mobility of electrons by considering the behavior of an average electron.

By definition, the mobility of an electron is measured by the ratio of the drift velocity \(u\) to the strength of the external electric field \(E\), i.e.

\[ b=\frac{u}{E}. \tag{1} \]

The drift velocity can be expressed in terms of the electron velocity \(v\) from the relation:

\[ \Delta p=m^*u=eE\tau=\frac{eEl}{v}, \tag{2} \]

where \(l\) is the mean free path of the electron.

Therefore expression (1) will be written

\[ b=\frac{el}{\sqrt{2m^*\varepsilon}}, \tag{3} \]

where \(m^*\) is the effective mass of the electron,
\(\varepsilon\) is the kinetic energy of the electron.

In weak fields the energy of the electron will be of the order of \(kT\).

Therefore the mobility of an electron for weak fields is written

\[ b=\frac{el}{\sqrt{2m^*kT}}. \tag{4} \]

In strong electric fields the energy of an electron may differ greatly from the average thermal energy. Let us determine an expression for the energy of an electron in strong fields. To this end we shall take into account that, in the stationary state, the loss of energy by an electron as a result of interaction with lattice vibrations must be compensated by the acquisition of energy from the electric field.

According to quantum theory, the interaction of an electron with lattice vibrations may be regarded as a process of absorption or emission of phonons.

Let \(N_q\) be the mean number of phonons with energy \(\dfrac{h}{2\pi}\omega\); then the probability of phonon absorption in a single act of interaction will be proportional to \(N_q\),

\[ W_a=aN_q, \]

and the probability of phonon emission will be proportional to \(N_q+1\),

\[ W_e=a(N_q+1). \]

Taking into account that \(W_e+W_a=1\), we obtain the following expressions:

1) the probability of phonon absorption per unit time:

\[ W_a=\frac{1}{\tau}\frac{N_q}{2N_q+1}; \]

2) the probability of phonon emission per unit time:

\[ W_e=\frac{1}{\tau}\frac{N_q+1}{2N_q+1}. \]

Therefore the loss of energy by an electron per unit time will be:

\[ \Delta_{\phi}\varepsilon = W_e\frac{h}{2\pi}\omega - W_a\frac{h}{2\pi}\omega = \frac{\dfrac{h}{2\pi}}{\tau(2N_q+1)}. \]

The energy acquired by the electron per unit time is numerically equal to the work done by the field:

\[ \Delta_{\phi}\varepsilon=eEu. \]

In the stationary state we have \(\Delta_{\phi}\varepsilon=\Delta_E\varepsilon\), i.e.

\[ \frac{\dfrac{h}{2\pi}\omega}{\tau(2N_q+1)}=eEu. \]

Substituting the value of \(u\) from (2) and taking into account that \(\tau=\dfrac{l}{v}\), we obtain:

\[ \varepsilon=\frac{(eEl)^2(2N_q+1)}{2h\omega}. \tag{5} \]

Let us consider separately the electron mobility in strong fields for a nonpolar and an ionic crystal.

a) A crystal with an atomic lattice.

For a nonpolar crystal, the main role in the interaction of the electron with lattice vibrations is played by the so-called acoustic vibrations, for which we have:

\[ \left(\frac{h}{2\pi}\omega\right)^2=2mv_3^2\varepsilon, \]

where \(v_3\) is the speed of sound.

Moreover, we always have \(kT\gg \dfrac{h}{2\pi}\omega\), and consequently,

\[ 2N_q+1\simeq \frac{2kT}{\dfrac{h}{2\pi}\omega}. \]

Then from relations (3) and (5) we obtain:

\[ b=\frac{el}{\sqrt{2m^{*}kT}}\left(\frac{\sqrt{2m v_{0}^{2}kT}}{eEl}\right)^{1/2} \tag{6} \]

b) A crystal with an ionic lattice.

For an ionic crystal, the principal role in the interaction of the electron with the lattice vibrations is played by the so-called optical vibrations, for which \(\dfrac{h}{2\pi}\omega=\dfrac{h}{2\pi}\omega_{0}\).

Here two cases are possible:

1st case—the mean energy of the electron is much less than the photon energy: \(\bar{\varepsilon}\ll \dfrac{h}{2\pi}\omega_{0}\). This case is possible only at low temperatures and weak fields. The electron mobility is determined by formula (4).

2nd case—the electron energy is much greater than the photon energy: \(\bar{\varepsilon}\gg \dfrac{h}{2\pi}\omega_{0}\).

It is possible either at low temperatures and a strong field, or at high temperatures and any field.

For a strong electric field we shall have:

1) \(\bar{\varepsilon}\ll \dfrac{h}{2\pi}\omega_{0}\) and \(kT\ll \dfrac{h}{2\pi}\omega_{0}\).

Under these conditions \(2N_{q}+1\simeq 1\). Therefore from (3) and (5) we have:

\[ b=\frac{\dfrac{h}{2\pi}\omega_{0}}{\sqrt{2mh\omega_{0}}}\cdot\frac{1}{E}, \tag{7} \]

2) \(\bar{\varepsilon}\gg \dfrac{h}{2\pi}\omega_{0}\) and \(kT\gg \dfrac{h}{2\pi}\omega_{0}\).

Under these conditions \(2N_{q}+1\simeq \dfrac{2kT}{\dfrac{h}{2\pi}\omega_{0}}\). Therefore from (3) and (5) we have:

\[ b=\frac{\dfrac{h}{2\pi}\omega_{0}}{\sqrt{2mkT}}\cdot\frac{1}{E}. \tag{8} \]

Thus, we see that for nonpolar crystals the electron mobility in strong fields is inversely proportional to the square root of the field strength, while for ionic crystals the electron mobility is inversely proportional to the field strength.

In addition, formulas (4), (6), (7), and (8) make it possible easily to determine also the temperature dependence of the mobility.

G. I. Skanavi says:

When studying polycrystalline substances, it is impossible to test any theory of electrical breakdown, since breakdown theories have in view dielectrics of regular structure, whereas the structure of polycrystals and the bonds in them are complex. There is no point in comparing the electric strength of polycrystalline dielectrics and the lattice energy for these materials. Electrical breakdown and the electric strength of a polycrystal do not allow one to judge the strength of the crystal lattice and the lattice energy.

Of great interest is the dependence of electric strength on temperature, but, unfortunately, all the speakers did not discuss the question of the temperature dependence of \(E_{\mathrm{pr}}\).