On the Statistical Time Lag of Breakdown of Solid Dielectrics

E. A. Konorova

Since at the present time the most generally accepted theories of the electrical breakdown of solid dielectrics are theories of impact ionization by electrons, it may be supposed that certain regularities already established for gases are also applicable to solid dielectrics. Therefore the question of the statistical time lag of electrical breakdown naturally arose. If experimenters were able to prove the existence of a statistical time lag of breakdown in solid dielectrics, this would be one of the decisive proofs of the correctness of the theory of impact ionization. As is known, many experimenters have established that if a statistical time lag does exist, then in any case it is of the order of no more than \(10^{-8}\) sec. Since the time required for the destruction of the dielectric may have the same order of magnitude, the question of statistical time lag remains experimentally open to this day.

Attempts have been made to estimate theoretically the time required for the development of an avalanche. Whitehead [1] calculated this time, proceeding from the breakdown criterion proposed by Fröhlich, as the time required for an electron to acquire a velocity sufficient for ionization.

As a result of the calculation, a value of the order of \(10^{-10}\) sec was obtained. Chuenkov [2], on the basis of the breakdown criterion proposed by him, likewise determined the time of acceleration of an electron to the ionization energy. It proved to be equal to \(10^{-13}\) sec.

For crystals with a disturbed structure and for amorphous bodies, Fröhlich [3] proposed that the field energy is transferred to conduction electrons not directly by the lattice, but by the excitation of electrons located on defect levels. With such a mechanism of energy exchange one might expect a delay in the electron’s acquisition of sufficient energy. However, the corresponding calculation, carried out by Simpson [1], showed that the time required for breakdown lies between \(10^{-10}\) and \(10^{-12}\) sec. Thus, all the calculations cited show that the time of avalanche formation is considerably less than \(10^{-8}\) sec, the time at which experimenters have detected strengthening.

In 1949 Seitz [4], adhering mainly to Hippel’s views, pointed out the possibility of deviations of the electron’s behavior from the average. Seitz’s reasoning reduces to the following. A conduction electron in an electric field has a certain probability of traversing, without collisions, the distance necessary to acquire the ionization energy. This probability determines the interval of time during which the electron can once be accelerated to the ionization energy. This same probability, on the other hand, determines the path over which the electron undergoes one collision accompanied by an ionization act. Thus, one may introduce the concept of the “free path” length of an electron, \(\lambda\), which depends—

depends on the magnitude of the applied field. According to Seitz, this mean free path is determined by the formula:

\[ \lambda = \frac{E\varepsilon'}{eE_{\mathrm{H}}^{2}}\exp(\alpha E_{\mathrm{H}}/E), \]

where \(E_{\mathrm{H}}\) is the breakdown field, calculated according to the Hippel criterion;
\(E\) is the applied field;
\(\varepsilon'\) is the energy transferred by the electron to the lattice in one collision;
\(\alpha\) is a coefficient which, according to Seitz’s calculations, is equal to 3.8 for nonpolar crystals and is close to this value for polar ones. For KBr, \(E_{\mathrm{H}} = 0.85 \cdot 10^{6}\) V/cm and \(\varepsilon' = \varepsilon_{\varphi} = 0.021\) eV. In this case \(\lambda\) varies from \(10^{-6}\) cm at \(E = E_{\mathrm{H}}\) to \(10^{-5}\) at \(E = 0.6E_{\mathrm{H}}\).

The value of \(\lambda\) given above for a field equal to \(0.6E_{\mathrm{H}}\) lies within reasonable limits for specimens ordinarily used in breakdown, and therefore one may expect that the value of the actual breakdown strength will be lower than the critical strength according to Hippel.

Since the avalanche that destroys the dielectric is formed as the result of a series of successive ionizations, one may think that a breakdown strength equal to \(0.6E_{\mathrm{H}}\) corresponds to a specimen thickness of the order of \(10^{-4}\)—\(10^{-3}\) cm and increases as the thickness decreases. That is, in this range of thicknesses a dependence of breakdown strength on thickness should be observed. As the thickness increases, the dependence becomes less noticeable.

Applying to a solid dielectric the calculation of the probability of formation of an avalanche with number of electrons \(n\), one can determine the mean statistical time lag of breakdown [8]. It is determined by the formula:

\[ \tau = \frac{1}{V_0}\exp\left(\frac{n_0}{\bar n}\right), \]

where \(V_0\) is the number of electrons formed in the dielectric per sec;
\(n_0\) is the number of electrons in the avalanche necessary for destruction of the material;
\(\bar n\) is the most probable number of electrons in the avalanche, corresponding to a definite mean free path \(\lambda\) and specimen thickness \(d\). This number of electrons is determined by the formula:

\[ \bar n = \exp\left(\frac{d}{\lambda}\right). \]

It follows from the formula given that the mean statistical time lag depends strongly on the thickness of the specimens if \(\bar n\) is less than \(n_0\) and \(\exp\left(\frac{n_0}{\bar n}\right)\) differs from unity. For thick specimens the exponential term tends to unity and \(\tau = \frac{1}{V_0}\). Thus, the dependence of the breakdown time of dielectrics on breakdown strength should be more sharply expressed and more easily observed for thin specimens.

In 1950–1953 a number of works by Japanese researchers on the electrical breakdown of solid dielectrics were published [5, 6, 7]. Dependences were obtained of the breakdown strength of mica and KCl on the time of voltage application and on thickness. The thickness of the mica specimens was of the order of \(10^{-4}\) cm, and of KCl, \(10^{-3}\) cm. For mica, a dependence was obtained of the breakdown time lag on the magnitude of the breakdown voltage, shown in Fig. 1, and a dependence of the breakdown voltage on thickness—Fig. 2. As is seen from the curve in Fig. 1, the dependence of the breakdown voltage—

dependence on time begins already at \(10^{-4}\) sec, which clearly indicates the presence of a statistical time lag of breakdown. The curve in Fig. 2 shows a strong dependence on thickness. For KCl single crystals, the same authors did not obtain dependences on thickness and on time.

On the basis of the results obtained, the authors of the work conclude that the breakdown mechanism proposed by Seitz (“single-avalanche breakdown”) occurs in mica and is absent in KCl crystals and in glass. The latter circumstance is explained by the presence of interaction between the avalanche electrons, when the density of the latter is large, and by the inhibition of further development of the avalanche. This entails the necessity of increasing the voltage for breakdown, which weakens the significance of fluctuations. The interaction between the avalanche electrons should affect the behavior of an electron in the case where the energy of this interaction is comparable with the energy of interaction of the electron with the lattice. Since in mica the latter is greater than in KCl, the interaction between the electrons is not of decisive importance.

Fig. 1. Dependence of the breakdown strength of mica-biotite on the time of application of the voltage (Kawamura, Okura, and Kikuchi).

Fig. 2. Dependence of the breakdown strength of mica-biotite on the thickness of the dielectric (Ryu and Kawamura).

The correctness of such an explanation must, of course, be confirmed by further experiments. Here it should only be noted that the absence of a dependence on thickness and on time for KCl still does not exclude the possibility of a Seitz breakdown mechanism in it. First, for KCl and glass the measurements were carried out on specimens whose thickness was an order of magnitude greater than for mica; second, we still have no data on the magnitude \(V_0\)—the number of electrons formed per unit time in the gap. Therefore the experimental conditions for KCl may have proved unfavorable for revealing the dependence.

Since the experimental data obtained are of considerable interest from the point of view of the theory of electrical breakdown, we also carried out measurements of the breakdown strength of muscovite mica and glass as a function of thickness and of the duration of the applied voltage. The measurements were made on specimens with thicknesses from 2 to 10 microns. Since, in measuring breakdown strength, an error can very easily arise because of field nonuniformity, we paid great attention to the selection of the appropriate electrode shape and of the medium into which the specimen was immersed. In the final measurements, for the anode we used the electrode system shown in Fig. 3. A mica plate was coated—

silver electrodes in the form of disks, made by evaporation in vacuum; the diameter of the smaller disk was 1.5 mm. The specimens were placed between a sphere–plane electrode system and flooded with a mixture of glycerin and alcohol. The highly conducting medium largely smoothed out the influence of the edge effect. The high values obtained for the electric strength, as well as the fact that in 50% of the cases breakdown occurred at the center of the smaller disk, indicate that the chosen electrode system did not introduce any substantial distortion of the field.

Fig. 3. 1—specimen, 2—upper electrode, 3—silver, 4—lower electrode.

Fig. 4. — — — according to Kawamura.

Glass breakdown was carried out on flat films placed between sphere–plane electrodes in a mixture of glycerin and alcohol. The dependence of breakdown strength on the time of application of the voltage was obtained over the range of times from \(5\cdot 10^{-8}\) to \(5\cdot 10^{-6}\) sec for specimens of thickness \(4 \div 5\cdot 10^{-4}\) cm. Pulses of exponential form were applied to the specimens, with an amplitude somewhat greater than the breakdown voltage of the specimen, so that breakdown always occurred on the front. The breakdown voltage was measured directly from the oscillogram. The thickness of the specimens was measured with a capacitance micrometer, with an accuracy of 5% for the thinnest specimens.

The results obtained are presented in Fig. 4 in the form of a curve, from which it is seen that no dependence of breakdown strength on time was observed. In addition, the values of breakdown strength obtained by us for muscovite mica at pulses of \(5\times 10^{-6}\) sec exceed the values of breakdown strength obtained by the Japanese investigators. It is therefore difficult to assume the presence of error in our measurements.

Fig. 5. The indicated thickness of the specimens corresponds to the mean value of the thickness in the interval 1 micron — — — Kawamura’s data.

As regards the dependence on thickness, here the results obtained by us and in Japan essentially coincide. The dependence on thickness was measured by us with pulses of \(5 \times 10^{-6}\) sec and is presented in Fig. 5.

The development of electrical breakdown in solid dielectrics is evidently strongly influenced by such factors as cold emission from the cathode and the formation of space charges. These factors and the degree of their influence on breakdown have not been studied at all. At the same time, the formation of space charges, especially in thin layers, where the voltages applied to the specimen are small, may be the cause of a change in breakdown strength with time. It is possible that this is the reason for the discrepancy in the experimental data in the present case, since identical experimental conditions for cold emission are difficult to reproduce.

In conclusion it should be said that the existence of a statistical time lag in the breakdown of mica cannot be regarded as an experimentally proven fact.

In further investigations of this question, attention should be paid to the influence of such factors as cathode emission and the formation of space charges, especially since the study of these phenomena is of independent interest to experimentalists.

References

1. S. Whitehead. “Dielectric breakdown of solids.”
2. Chuenkov V. A. Dissertation, FIAN, 1954.
3. H. Fröhlich and J. O’Dwyer. Proc. Phys. Soc. Sec. A, 63 part 2, No. 362A.
4. Seitz. Phys. Rev. 76, pp. 1376–1393.
5. Kawamura. Onuki and Ohkura.
   I J. Phys. Soc. Japan U 7 No. 5, 1952.
   II J. Phys. Soc. Japan V 8 No. 6, 1953.
6. Kawamura, Ohkura and Kikushi. J. Phys. Soc. Japan U 9 No. 4, 1954.
7. Ruy and Kawamura. J. Phys. Soc. Japan V 9, pp. 438–439, 1954.
8. Furry. Phys. Rev. 52, 569, 1937.

Moscow, P. N. Lebedev Physical Institute
of the Academy of Sciences of the USSR