Once Again on the Temperature Dependence of Electric Strength in Electrical Breakdown

A. A. Vorob’ev and E. K. Zavadovskaya

Through the investigations of Soviet scientists—Academician Semenov, Academician Ioffe, Academician Fock, Corresponding Member of the Academy of Sciences of the USSR Vul, Corresponding Member of the Academy of Sciences of the USSR Aleksandrov [1], and others—the basic concepts of the breakdown of dielectrics in an electric field were developed. Two forms of breakdown have been established: thermal and electrical. For the thermal form of breakdown of solid dielectrics, an exponential decrease in the value of electric strength with increasing temperature is characteristic. For the electrical form of breakdown, the characteristic feature is the independence of the value of electric strength from temperature over the practically interesting temperature interval. These basic concepts of dielectric breakdown have in the past repeatedly been subjected to criticism by foreign scientists (Moon and Nordcross [2], Hippel [3], and others).

The value of the electric strength of dielectrics calculated according to the theories of electrical breakdown of dielectrics by Hippel, Froehlich, Zener is found to be approximately the same. The temperature dependence of electric strength given by the various theories is different. Therefore, it seemed possible to establish the validity of a theory of dielectric breakdown by studying the temperature dependence of electric strength.

At the present time, the question of the temperature dependence of electric strength in the electrical breakdown of solid dielectrics has again been raised. In a large number of works by Froehlich [4], Whitehead [5], Hippel, and others, a dependence of electric strength in the electrical form of breakdown on temperature is proved theoretically. In this connection it seems of interest to reconsider this question, since the accumulated materials of experimental investigations, which possess objective value, are being interpreted incorrectly.

We have shown that there is a linear dependence between the value of electric strength and the lattice energy [6].

Temperature has little effect on the value of the energy of the crystal lattice [7]. In accordance with such a small change in the lattice energy, one should expect that the change in the electric strength of a dielectric with a change in temperature will likewise be small. It has been shown [8] that the value of the electric strength of a dielectric can be calculated from the condition of the minimum mobility of electrons. Taking into account collisions of the electron with lattice vibrations, a formula was obtained for calculating the value of electric strength in the form:

\[ L_{\text{pr}}=\frac{U_0\sqrt{m V_{\text{zv}}}}{e a_0 \sqrt{kT}}, \]

where \(U_0\) is the magnitude of the potential barrier for electrons in the dielectric,
\(m\) and \(e\) are the mass and charge of the electron,
\(a_0\) is the lattice constant,
\(V_{\text{zv}}\) is the speed of sound in the dielectric.

Studies show that the quantity \(V_{\mathrm{s}}/\sqrt{T}\) decreases slowly with increasing temperature.

It is known that the measured electric strength varies considerably from specimen to specimen. Therefore it is hardly possible to establish experimentally a small change in it when the temperature changes.

We believe that in the region of the electrical form of breakdown the value of the electric strength should not depend appreciably on temperature.

In Fröhlich’s theory of breakdown [4], an essential role is played by the critical temperature \(T_k\), at which a substance passes from the crystalline state into the amorphous state. The electric strength of dielectrics depends on temperature in different ways. At a temperature below \(T_k\), the electric strength of a dielectric should increase with increasing temperature according to the condition:

\[ E(T)=E(0)\left[1+2\left(e^{h\nu/kT}-1\right)^{-1}\right]^{1/2}, \tag{1} \]

where \(E(T)\) is the electric strength at temperature \(T\), and \(\nu\) is the frequency of the residual radiation.

The electric strength \(E_{\mathrm{br}}\sim(\varepsilon-\varepsilon_0)\), where \((\varepsilon-\varepsilon_0)\) is the magnitude of the ionic polarization due to displacement of the ions as a whole, which depends little on temperature. In some temperature interval, changes in the dielectric constant may be significant for dielectrics with a dipole structure. Above the temperature \(T_k\), when the substance becomes amorphous, Fröhlich gives, for describing the change in its electric strength, the condition

\[ \ln E=\mathrm{const}+\frac{\Delta_{\mu}W}{2kT}, \tag{2} \]

where \(\Delta_{\mu}W\) is the average energy difference required to transfer electrons located on excited levels of inhomogeneities into the conduction band.

Disruption of the crystalline structure lowers \(T_k\). At the same time, disruption of the structure, by decreasing the free path length, should increase the electric strength of the dielectric. This phenomenon is observed in solid solutions of inorganic and organic dielectrics, for example, in chlorinated polyethylene.

According to Franz’s impact-ionization theory [5], which takes into account collisions of electrons with acoustic lattice vibrations, the formula obtained is

\[ E_{\mathrm{br}}=\frac{30}{a_0^{3}}\cdot\frac{30}{A}\cdot\frac{4\cdot10^{5}}{V_{\mathrm{s}}}\sqrt{\frac{T}{300}}, \tag{3} \]

where \(A\) is the atomic weight, \(T\) is the temperature in \({}^{\circ}K\), and \(V_{\mathrm{s}}\) is the speed of sound in cm/sec.

This formula is derived for a region of not too low temperatures.

For diatomic ionic lattices, \(A\) takes the value

\[ A=\frac{2}{\dfrac{1}{A_1}+\dfrac{1}{A_2}}. \]

For rock salt at room temperature, at \(T=300^{\circ}K\), when \(V_{\mathrm{s}}=4\cdot10^{5}\) cm/sec, formula (3) takes the form

\[ E_{\mathrm{br}}=30\,a_0^{-3}. \]

According to the impact-ionization theory that takes into account collisions of electrons with optical vibrations, developed by Kollen [5], the dependence of electric strength on temperature proves to be complicated. In the region of low temperatures, the electric strength of alkali-halide compounds with po-

with an increase in temperature, it should grow linearly at a rate of approximately \(1\ \mathrm{kV/cm}\) per degree.

According to Zener’s theory of electrostatic ionization by a field [5], the electric strength of a dielectric may be represented by the formula:

\[ E_{\mathrm{br}} = 0.1\,\frac{a_0(\Delta W)^2}{n}, \]

where \(E_{\mathrm{br}}\) is in \(\mathrm{MV/cm}\), \(\Delta W\) is the width of the forbidden energy gap in the energy spectrum of the crystal, overcome by means of the tunneling effect. It is expressed in electron-volts; \(a_0\) is the lattice constant in angstroms; and \(n\) is the ordinal number for the electron waves before leaving the bound state, that is, the number of electron wavelengths that fit into the length \(2a_0\).

According to this formula, the electric strength does not depend on temperature.

Oakes studied in detail the electric strength of certain synthetic polymers: polyethylene, polystyrene, polyisobutylene, polymethyl methacrylate, chlorinated polyethylene, and oxidized polyethylene over the temperature interval from \(-200\) to \(110^\circ\mathrm{C}\). It turned out that at low temperatures the electric strength of the nonpolar polymers polyethylene, polystyrene, and polyisobutylene retains an almost unchanged value. Above a certain definite temperature, the electric strength drops rapidly with increasing temperature. For polyethylene this temperature is about \(40^\circ\mathrm{C}\), for polystyrene about \(90^\circ\mathrm{C}\), and for polyisobutylene about \(50^\circ\mathrm{C}\).

The electric strength of the polar polymer polymethyl methacrylate at low temperatures is considerably higher than the strength of nonpolar polymers and decreases with increasing temperature. An increase is observed in the rate of change of electric strength with temperature in the region above \(20^\circ\mathrm{C}\). Weak chlorination of polyethylene leads to an increase in its electric strength at low temperature and to a decrease in it at high temperatures. At low temperatures the electric strength of oxidized polyethylene is greater than the electric strength of non-oxidized polyethylene.

Fig. 1. Dependence of the electric strength of polyethylene on temperature according to the data of Oakes (O) and Austen and Pelzer (X).

In Fig. 1 the change in the electric strength of polyethylene is presented over the temperature interval from \(-196\) to \(130^\circ\mathrm{C}\). The average molecular weight of the polyethylene was 15000. Measurements were carried out on specimens with hemispherical depressions, which were formed by pressing at

at a temperature of 120–170°C. Measurements were made on specimens with a bottom thickness of 0.001–0.02 cm. Aquadag electrodes were used. Breakdown was carried out under constant voltage. During the experiment the temperature was maintained constant to an accuracy of 0.5°C.

The results presented in Fig. 1 show that the electric strength of polyethylene is almost independent of temperature over a considerable interval. This result contradicts Fröhlich’s theory. The decrease in strength at temperatures above 30°C is associated with the occurrence of thermal breakdown.

Fig. 2. Dependence of the electric strength of polystyrene on temperature (Oakes).

A similar behavior is observed for polystyrene (Fig. 2) with an average molecular weight of about 100,000.

Figure 3 presents the temperature dependence of electric strength for polyisobutylene. In the region of low temperatures a noticeable decrease in strength is observed, which is in contradiction with Fröhlich’s theory, as given by formula (2).

Fig. 3. Dependence of electric strength on temperature for polyisobutylene according to Oakes.

Figure 4 presents the temperature dependence of electric strength for polymethyl methacrylate (Plexiglas) with an average molecular weight of approximately 800,000. The very high electric strength, \(\sim 10^7\) V/cm, of this polar polymer in the low-temperature region is noteworthy. By chlorinating polyethylene under various conditions, thermoplastic polymers are obtained. An increase in chlorine content leads to...

correspond to changes in the mechanical properties of the material at room temperature, from the rubber-like state to the glassy state.

Figure 5 shows the dependence of electric strength on temperature for polyethylene chlorinated to different degrees. An increase in the content of chlorine atoms in polyethylene, which have a high electron affinity, is accompanied by an increase in the electric strength of the material [9].

Fig. 4. Dependence of the electric strength of polymethyl methacrylate (Plexiglas) on temperature.

Thus, for highly polymerized synthetic dielectrics, various types of dependence of electric strength on temperature are observed. In the low-temperature region, the electric strength of polymers, as the temperature changes, may retain an unchanged value, may decrease, or may increase.

Fig. 5. Dependence of electric strength on temperature for chlorinated polyethylene: ○○ 8% chlorine—cold chlorination; X-X-X—8% chlorine—hot chlorination; --- polyethylene from Fig. 1.

It should also be noted that special studies of polyethylene specimens by X-rays at temperatures of 96, 20, and 45°C showed their crystalline structure.

An increase in the electric strength of solid solutions of alkali-halide salts was observed, for example, in the system NaCl + AgCl [10]. The value of the residual-radiation frequency \(\nu_{\text{res}}\) is large for all dielectrics and changes little with temperature. For alkali-halide salts it is relatively smaller than for other dielectrics, and for them \(T_k\) lies in the range of experimentally accessible low temperatures.

According to Fröhlich [1], for crystals of alkali-halide salts in the low-temperature region one should expect an increase in electric strength with temperature. Such phenomena have been observed experimentally. Figure 6 shows the change in the value of the electric strength with temperature for \(KBr\) crystals [11]. As can be seen, as the temperature is lowered below 300°K the electric strength of \(KBr\) decreases markedly. A decrease in electric strength is also observed when the temperature is raised above 370°K. It is probably connected with thermal breakdown.

Fig. 6. Dependence of electric strength on temperature for \(KBr\) crystals: 1—calculated according to Fröhlich’s theory, 2—measured by Beusle and Hippel, and 3—measured by Austen and Whitehead.

The experiments of Beusle and Hippel [11] on \(KBr\), and of Austen and Whitehead in the range from 200 to 300°K, showed a large increase in electric strength with increasing temperature. This increase in electric strength differs from that required by the theory of impact ionization, but it was nevertheless regarded as a qualitative confirmation of the theory.

These data are in contradiction with the measurements of Walter and Inge [11], who in 1931, for rock salt, obtained independence of breakdown voltage from temperature over a wide temperature interval.

The increasing temperature dependence of electric strength seemed to confirm the validity of the theory of impact ionization and the incorrectness of the theory of electrostatic ionization. Moreover, the value of the electric strength calculated according to Zener’s theory of electrostatic ionization was much too large. Subsequent measurements of the temperature dependence of the electric strength of \(KBr\), carried out by Hippel and Alger [11], showed that the measurement results depend significantly on the form of the electric field, the method of applying the electrodes, and the duration of the action of the voltage.

The observed change in the temperature dependence of the electric strength of \(KBr\) with changes in the conditions indicated above is well explained by taking into account the formation and dissipation of space charge in the dielectric. Measurement of the temperature dependence of the electric strength of \(KBr\) under the action of a pulsed voltage of duration \(\sim 10^{-6}\) sec showed that there is a weak linear dependence between these quantities, amounting to 0.4 kV/cm per degree. This is two times lower than is obtained from Collen’s theory.

Considering the considerable scientific interest of the temperature dependence of electric strength, new measurements should be carried out with due regard for all methodological comments concerning the measurements performed.

In our experiments we also observed a sharp decrease in the electric strength of rock salt and potassium chloride crystals in the region of low temperatures. Breakdown was carried out in a Dewar vessel while cooling the specimens with liquid air, or in transformer oil, gasoline, and other media while cooling in a cryostat with liquid air.

In some experiments the specimens were provided with coatings. The electrodes were applied to them by evaporating metal in vacuum. In other experiments the specimens were taken in the form of plates of various thicknesses, which were placed between flat and hemispherical electrodes.

In our experiments at low temperatures a very large scatter of the measured values of breakdown voltages was observed. However, as the temperature was lowered, the breakdown voltage of the specimens decreased considerably. Sometimes at low temperatures values of electric strength were observed that were the same as at room temperature. This led us to the idea that the lowering of the breakdown voltage observed in our experiments is connected with cracking of the crystals as a result of the nonuniform change of their dimensions along different crystallographic directions when the temperature is lowered.

The measures taken to make the cooling of the specimens more uniform, for which purpose the cooling rate was changed, did not alter the results of the experiments. To check for the existence of irreversible changes in the specimens after cooling, the specimens were cooled to a low temperature and then their temperature was slowly raised to room temperature. The breakdown voltage of such specimens, previously cooled and then punctured at room temperature, as a rule was likewise found to be low. These experiments show that the lowering of the breakdown strength of specimens after cooling to a low temperature is caused by irreversible changes in the crystals in the process of their cooling and subsequent heating. Such changes may be microdisruptions of the structure and cracking. These phenomena and their influence on electric strength were studied by us. It was shown that internal porosity considerably lowers the magnitude of the breakdown voltage.

From the standpoint set forth, the electric strength of dielectrics with poor thermal stability may decrease as the temperature is lowered as a result of a violation of their continuity. Dielectrics that are mechanically more stable under temperature fluctuations should not exhibit a decrease in electric strength as the temperature is lowered.

In the graphs given below, borrowed from the works of various scientists, we have confirmation of the point of view expressed.

Figure 7 gives the dependence of electric strength on temperature for ice according to the data of Prikhodko [12].

Experiments on the study of the dependence of the magnitude of electric strength on temperature for porcelain (Fig. 8), carried out by him, showed that the strength of porcelain remains constant. The inhomogeneous dielectric porcelain has considerable porosity and good temperature stability; therefore, for it no dependence of electric strength on temperature is found even for many cycles of cooling and heating.

Figure 9 presents the dependence of electric strength on temperature for mica: muscovite and phlogopite.

As follows from these data [13], in the region of low temperatures the electric strength retains a constant value. In the region of high temperatures (480°K) a decrease in electric strength is observed, associated with thermal breakdown.

Fig. 7. Dependence of the magnitude of electric strength on temperature for ice and water under impulse voltages (after Prikhodko).

Fig. 8. Dependence of the magnitude of electric strength on temperature for porcelain (after Prikhodko).

Fig. 9. Dependence of the magnitude of electric strength on temperature for mica: 1—calculated for \(\lambda_{\mathrm{rem}} = 20\,\mu\); 2—calculated for \(\lambda_{\mathrm{rem}} = 10\,\mu\); 3—for muscovite, 4—for phlogopite.

Fig. 10 presents the dependence of electric strength on temperature [5], 1—for crystalline quartz and 2—for fused quartz. With decreasing temperature, a noticeable decrease in the electric strength of crystalline quartz is observed. For fused quartz no such dependence is observed. It is known that fused quartz has a small coefficient of thermal expansion and high thermal stability.

Fig. 11 presents the dependence of electric strength on temperature for compound Э—3 according to the data of Vorob'ev and Snegirev [7]. The region of a sharp drop in electric strength corresponds to cracking of the material. The cracking temperature of the compound in this region was determined by special experiments. Studies were carried out on the properties of electrical apparatus and insulating materials at low temperatures. Methods were developed for cooling, cryostating, and studying the properties of materials and apparatus in large and small volumes. The results of these investigations were published [14].

Fig. 10. Dependence of the magnitude of electric strength on temperature for 1—crystalline quartz, 2—for fused quartz.

Fig. 11. Dependence of electric strength on temperature for compound Э—3.

For some viscous materials, for example, viscous compounds, transformer oil, rubber, impregnated paper, and others, an increase in electric strength was observed when the temperature was lowered.

Our theoretical ideas about the breakdown of solid dielectrics, experimental investigations, and also analysis of the results of investigations obtained by other authors allow one to conclude that the magnitude of the electric strength of crystalline dielectrics is not noticeably dependent on temperature. The point of view being developed by Soviet researchers—that electric strength in the electrical form of breakdown does not depend on temperature—has repeatedly been denied by foreign scientists. New investigations by foreign authors are now cited as proof of the correctness of Fröhlich’s theory. It seems to us that these results are explained more simply and convincingly if the secondary phenomena arising in a dielectric at low temperatures are taken into account.

Geller [15], who developed the kinetico-statistical theory of electrical breakdown in nonpolar crystals, came to the conclusion that at temperatures below the characteristic temperature (Debye temperature)

\[ \Theta^\circ = \frac{h\nu_{\text{osc}}}{k}, \]

there should be no noticeable dependence on temperature. For diamond, for example, this should hold up to room temperatures.

Keller [16] published the results of investigations of the electric strength of glasses over a wide temperature range. To exclude thermal breakdown, the experiments were carried out with thin specimens up to 50 microns thick. The duration of the applied voltage was varied from \(6 \cdot 10^{-6}\) to 20 sec. The experiments showed that, when heating of the specimen by the current flowing through the dielectric is excluded, the electric strength of glasses in a certain range does not depend on temperature.

Figure 12 presents the dependence of the electric strength of glasses on temperature for different durations of the applied voltage.

Fig. 12. Dependence of the electric strength of glass on temperature for different durations of the applied voltage (after Keller)

Thus, we arrive at the conclusion that, in the practically interesting temperature range where electrical breakdown takes place, the magnitude of the electric strength probably depends only weakly on temperature. Sometimes the observed dependence of electric strength on temperature is explained by irreversible changes in the dielectric, for example, by a disruption of the continuity of the material or by other secondary phenomena.

REFERENCES

1. Physics of Dielectrics. GTTI, 1932.

2. Walter and Inge. ZhTF, 8, 749, 1932.
   Moon and Norcross. AIEE. Jan. 1930.

3. A. Hippel. Appl. Phys. 8, 815, 1937.
   R. Buehl and A. Hippel. Phys. Rev. 56, 941, 1939.

4. H. Fröhlich. Proc. Roy. Soc. A 160, 230, 1937.

5. S. Whitehead. Dielectric Breakdown of Solids, Oxford, Clarendon Press, 1951.
   W. Franz. Zs. f. Phys. 113, 607, 1939. H. B. Callen. Phys. Rev. 76, 1394, 1949.
   C. Zener. Proc. Roy. Soc. A 145, 523, 1934.

6. A. A. Vorob’ev and E. K. Zavadovskaya. DAN, XXXI, 375, 1951.

7. A. A. Vorob’ev. Proceedings of the Tomsk Polytechnic Institute, 29, 169, 1941.

8. E. K. Zavadovskaya. DAN, XXXII, 565, 1952;
   A. A. Vorob’ev and E. K. Zavadovskaya (in this issue).

9. W. G. Oakes. Proc. I.E.E. 96, 56, 1948.

10. A. Hippel. Zs. f. Phys. 68, 309, 1931; 75, 145, 1932; 88, 358, 1934.

11. A. E. W. Austen and S. Whitehead. Proc. Roy. Soc. A. 176, 33, 1940.
    R. C. Buehl and A. Hippel. Phys. Rev. 56, 941, 1939.
    A. Hippel and R. S. Alger. Phys. Rev. 76, 127, 1949.

12. N. A. Prikhod’ko. Proceedings of the Siberian Physico-Technical Institute 6, 114, 1942.

13. Chakkett and Thomas. I.I.E.E. 88, 295, 1941.

14. E. K. Zavadovskaya. Proceedings of the Siberian Physico-Technical Institute, 6, 120, 1942; 6, 8, 1942.

15. Heller. Phys. Rev. 84, 1131, 1951.

16. K. I. Keller. Physica 14, 15, 1949.