Electrical Breakdown and Electron Mobility in Dielectrics and Semiconductors

E. K. Zavadovskaya

(Presented by the scientific seminar on dielectrics of TPI)

In a dielectric, under a high field intensity, complex physical processes take place, culminating in its breakdown. The destruction of the solid structure of the dielectric probably occurs because the electrons participating in the transport of current transfer to the lattice part of their energy, acquired from the external field. Such transfer may occur both through direct collisions of electrons with lattice sites and through the interaction of electrons with quanta of acoustic and optical vibrations.

It is obvious that the process of energy transfer from an electron to lattice particles cannot proceed effectively through their direct collision. Owing to the excessively large difference between the masses of the electron and the ion, the latter will receive little energy from the electron. Energy transfer will be more effective in interaction with a phonon. When their masses are comparable in magnitude, the electron will transfer to lattice vibrations a significant fraction of its energy. When the energy transferred from the electron current to the lattice sites becomes comparable with the energy of the lattice, destruction of the bonds between the sites occurs—the breakdown of the dielectric.

The breakdown process is characterized by an intensified transfer of energy from electrons to lattice sites and must be accompanied by a decrease in the velocity of the electrons. A decrease in the velocity of electron motion occurs in an increasing electric field; therefore their mobility must decrease. Fröhlich calculated the energy accumulated by electrons in an electric field. He defined the breakdown condition by requiring that the power accumulated by an electron be equal to the power transferred by the electron to the lattice. The power received by the lattice goes into intensifying the thermal vibrations of its sites. In this case the energy of the electrons must be equal to the ionization energy. The intensification of thermal vibrations leads to heating and melting of the lattice. Thus Fröhlich explains the process of electrical breakdown. Fröhlich considers established such a view of the process of electrical breakdown as the melting of the crystalline lattice.

We consider it possible that electrical breakdown ends with mechanical destruction of the crystal. The breakdown site represents mechanical destruction.

We believe that the energy accumulated by electrons in the electric field is transferred to the lattice and is expended in breaking the bonds between its particles. The transfer of energy from electrons to the lattice occurs especially intensely when the energy of the electrons is of the order of magnitude of the bonding energy between the particles of the lattice.

In this case one may write the condition of proportionality between the magnitude of the energy accumulated by the electrons in the electric field and the energy of the lattice.

An electron accumulates energy in an electric field of strength \(E\) over a free path \(l\), equal to

\[ elE. \tag{1} \]

The power accumulated by the electron may be written as

\[ evE. \tag{2} \]

We assume the energy transferred by the electron to the lattice during the time \(\Delta t\) (the breakdown time) to be proportional to the energy of the lattice:

\[ evE\Delta t = AU, \]

\[ euE^{2}\Delta t = AU, \tag{3} \]

where \(U\) is the energy of the lattice, and \(A\) is the proportionality coefficient.

Experimental data indicate that for crystals of alkali-halide salts there is a linear dependence between the breakdown field strength and the lattice energy [1]. According to formula (3), a linear dependence between the breakdown field strength and the lattice energy is obtained under the condition that the electron mobility varies inversely proportionally to the field strength, i.e.,

\[ u \sim \frac{1}{E}. \]

Such a decrease in the mobility of electrons with changing field strength (according to the condition \(v = uE\)) corresponds to a constant velocity of the electrons.

One may make another assumption: that the electron transfers to the lattice the energy \(\Delta W\) which it accumulates during the free-flight time \(\tau\).

\[ \Delta W = evE\tau = euE^{2}\tau = \frac{e^{2}E^{2}}{m}\tau^{2}. \tag{4} \]

Let us suppose that all the energy accumulated by the electron over the free path is transferred to the particles of the lattice and is expended in breaking the bond between them. In this case it is likewise possible to write a condition of proportionality between the magnitude of the energy accumulated by the electron in the electric field over the free path and the energy of the lattice:

\[ \frac{e^{2}E^{2}}{m}\tau^{2} = BU. \tag{5} \]

In accordance with the experimental linear dependence between the electric strength of crystals of alkali-halide salts and the lattice energy, according to (5) the free-flight time must vary with the field strength according to the law

\[ \tau \sim E^{-\frac{1}{2}}. \tag{6} \]

And consequently the mobility must vary according to the same law,

\[ u \sim E^{-\frac{1}{2}}. \tag{7} \]

Apparently, in weaker fields the mobility varies according to the law

\[ u \sim E^{-\frac{1}{2}}, \]

when the electrons transfer to the lattice or to impurities the energy accumulated during the free-flight time. In stronger fields the elec-

electrons transfer to the lattice the energy that they accumulate over a time corresponding to several mean free paths. In this case, at the end of each free path the electron transfers only part of its energy to the lattice and continues to accumulate it until, in interaction with phonons, it transfers a significant part of all the accumulated energy to the lattice.

It may be assumed that, in the case where the dielectric contains a larger quantity of impurities, the electrons, interacting with them, transfer their energy at the end of the free path. The electrical breakdown of such a dielectric will be preceded by a change in the mobility of the electrons according to the law \(u \sim E^{-1/2}\), and relation (5) will correspond to the breakdown mechanism of a dielectric with impurities.

An electron that has accumulated energy in an electric field transfers it to the vibrations of the lattice. Thus, the increase in the energy of the lattice vibrations is also caused by the electric field. The strengthening of the interaction of the electron with the lattice in strong fields is accompanied by a decrease in its kinetic energy and in its velocity of motion. One may say that the electric field acting on the dielectric, on the one hand, accelerates the electrons, and on the other, by strengthening the interaction of the electron with the lattice, retards its motion. Under certain conditions an equilibrium may be established and a stationary current through the dielectric may arise.

Here we have considered the mobility of electrons when they are swept along in an electric field.

Three types of electron mobility in dielectrics and semiconductors are distinguished.

1. Mobility when electrons are carried (drift) in an electric field, \(u_E\). This mobility of electrons differs from the true mobility. When carried in an electric field, the electrons experience time delays near various electron-trapping centers. As a result, the average velocity of electron motion in the electric field over the observation interval will be less than the average velocity over the mean free path.

2. Mobility of electrons in the conduction band, or the true microscopic mobility, \(u_m\).

3. Mobility when electrons are carried in a magnetic field, \(u_H = R\sigma\). Calculations show that \(u_m = u_H\).

At present there is still no established point of view on the dependence of the mobility of electrons in a crystal lattice on the field strength.

Some authors believe that the mobility of electrons in dielectrics remains a constant quantity, independent of the field strength. Thus, Ya. I. Frenkel [2], in his latest article devoted to the question of the pre-breakdown state of dielectrics, assumes that the mobility of electrons in dielectrics does not change with the field strength.

Regarding the character of the dependence of the mobility of electrons in semiconductors when the field strength changes, there exist different points of view. Thus, for example, Weibel explained the increase of electrical conductivity in semiconductors with increasing field strength by an increase in the mobility of electrons.

A. F. Ioffe and A. V. Ioffe [3] investigated the influence of the field on the electrical conductivity of certain semiconductors and expressed the view that electron mobility does not play an essential role in the increase of the electrical conductivity of semiconductors with increasing field strength.

But in their experiments A. F. Ioffe and A. V. Ioffe limited themselves to field strengths up to \(5 \cdot 10^4\ \mathrm{V/cm}\) and showed that in the region below \(5 \cdot 10^4\ \mathrm{V/cm}\)

An increase of the electrical conductivity with the field occurs as a consequence of an increase in the electron concentration. This same point of view concerning electron mobility is supported by F. F. Vol’kenshtein [4]. In his book Electrical Conductivity of Semiconductors he points out: “Joffe’s experiments completely rule out the supposition that the growth of conductivity is due to an increase in mobility. Also, from the theoretical point of view, there are no grounds for expecting the mobility to increase in any substantial way with the field.”*

Theoretical investigations by Soviet scientists are known in which the dependence of electron mobility on field strength in semiconductors is considered. Such investigations were carried out by L. Landau and A. Kompaneets [5], V. I. Davydov and I. M. Shmushkevich [6], N. L. Pisarenko [7], and others.

L. Landau and A. Kompaneets calculated the energy of electrons in strong electric fields in semiconductors with an atomic structure, taking into account collisions of electrons with acoustic vibrations of the lattice. In this case it was found that the mobility of electrons in semiconductors with an atomic lattice decreases with increasing field strength according to the law \(u \sim E^{-1/2}\).

V. I. Davydov and I. M. Shmushkevich likewise obtained, for semiconductors with an atomic lattice, a decrease in electron mobility with increasing field strength according to the law \(u \sim E^{-1/2}\). For semiconductors with an ionic lattice they obtained an increase in electron mobility in strong electric fields.

N. L. Pisarenko investigated the motion of electrons in semiconductors with an atomic lattice at high electric-field strength, taking into account ionization by collision and trapping. He found that the electron mobility decreases with increasing field strength according to the law \(u \sim E^{-1/2}\).

The mobility of electrons in a strong field may decrease as a result of an increase in the random velocity and, consequently, a difficulty in the drift of electrons in the electric field.

A theoretical investigation of the mobility of electrons in an \(n\)-type germanium semiconductor was carried out by Shockley [8]. He pointed out that an approximate theory of electron mobility can be constructed. For this, one should take as a basis the proposition that in strong fields the electron transfers to the lattice a power equal to that which the electron acquires in the electric field.

The energy transferred to the crystal lattice is given mainly by those electrons whose energy is several times greater than \(kT\).

The power transferred by the electrons to the crystal lattice is expressed by the formula

\[ P_A=\left(4mc^2\frac{v_1}{l}\right)\left[\left(\frac{v_1^2}{2v_T^2}\right)-1\right], \tag{8} \]

where \(v_T^2=\dfrac{2kT}{m}\); \(l\) is the mean free path.

The power acquired by the electrons is

\[ P_E=euE^2. \tag{9} \]

From further calculations Shockley finds an expression for the velocity of the electrons in the crystal lattice

\[ v=uE=\left(\frac{32}{3\pi}\right)^{\frac12}(C_uTE)^{\frac12}. \tag{10} \]

It follows from formula (10) that the electron mobility in germanium \(u\) should decrease with increasing field strength according to the law \(u \sim E^{-1/2}\) in strong fields, in which the power transferred by the electron to the lattice is equal to the power acquired by the electron.

Of interest is the theoretical investigation of electron mobility in solids carried out by E. M. Conwell [9], since she considers the mobility of electrons in strong electric fields. Conwell points out that the formal theory of conductivity can also be applied to strong fields, assuming approximately that the processes causing electron scattering are elastic. This condition is fulfilled in a crystal lattice for the range of temperatures and field strengths in which collisions of electrons with acoustic lattice vibrations and impurities mainly occur. As experiments show, for germanium this is observed up to a field strength of \(4000\ \text{V/cm}\) at room temperature. This condition is also fulfilled for electrons in a gas, so long as electron collisions do not cause ionization.

E. M. Conwell denotes the electron distribution function in the presence of an electric field by \(f(k,E)\), where \(k\) is the electron wave vector, \(E\) is the field strength, and writes the distribution function approximately in the form

\[ f(k,E)=f_0(k,E)-f_1(k,E)\cos\Theta, \]

where \(\Theta\) is the angle between the wave vector “\(k\)” and the direction of the field. The first term \(f_0(k,E)\) represents the electron distribution function in weak fields or in a field equal to zero. The second term \(f_1(k,E)\cos\Theta\) shows the change in the electron distribution function caused by the action of the external electric field. The function \(f_1\) is much smaller than the function \(f_0\).

In the steady state, the rate of change of the electron distribution function \(f\) due to the field is balanced by the rate of change of the electron distribution function caused by collisions of the electrons.

Further, E. M. Conwell points out that the probability of electron scattering by lattice vibrations or by traps (irregularities of the lattice or impurities) does not depend on the electric-field strength up to fields of the order of \(6\cdot 10^5\ \text{V/cm}\). This remark by E. M. Conwell indicates that electrons can accumulate energy in semiconductors as the field increases up to such high values. For the current density in a solid, E. M. Conwell obtained the expression

\[ j=\frac{e^2}{3m}En\left[\frac{1}{k^2}-\frac{d}{dk}\left(k^3\tau\right)\right], \tag{12} \]

where \(n\) is the number of electrons per unit volume.

It follows from this that the electron mobility can be written in the following form:

\[ u=\frac{e}{m}\left[\frac{1}{k^2}-\frac{d}{dk}\left(k^3\tau\right)\right]. \tag{13} \]

Such an expression for the electron mobility is indeed valid in weak fields in a solid. It may also be valid for those strong fields in which electron scattering is elastic.

In strong fields the electron distribution function changes with the field, and consequently the electron mobility calculated on its basis must also depend on the field strength.

The formula obtained by E. M. Conwell for the mobility of electrons in strong fields is analogous to the known expression for mobility in semiconductors when the electrons are distributed according to the Maxwell–Boltzmann function.

In E. M. Conwell’s work, the essential point is the indication that in strong fields, when the electron distribution function changes with the field, the electron mobility must also change with the field.

In addition, the observation is of interest that in a certain range of field strengths (up to fields of \(6 \cdot 10^5\ \mathrm{V/cm}\)) the probability of collisions of electrons with the lattice or with impurities does not increase with the field, and the electrons acquire the possibility of accumulating energy.

We [10] obtained an expression for the mobility of electrons in dielectrics. Investigating the dependence of electron mobility on field strength, we obtained a decrease in electron mobility with increasing field. We believe that the electron mobility reaches its smallest value at the most intense transfer of energy by the electrons to the lattice. From the condition for the minimum of the electron mobility we found an expression for determining the electric strength of dielectrics in the form

\[ E_{\mathrm{pr}}=\frac{\varepsilon W}{ea}, \tag{14} \]

where \(\varepsilon\) is the dielectric constant, \(W\) is the electron energy, \(e\) is the electron charge, and \(a\) is the lattice constant.

The value of the electric strength of crystals of alkali-halide salts calculated by this formula is close to the experimental values of the breakdown field strengths for these dielectrics.

The dependences of the electric strength on the dielectric constant, on the residual-radiation energy, if one sets \(W=h\nu\), and also on the lattice constant, which follow from this formula, are well confirmed by experimental data.

Thus, the dependence of electron mobility on field strength obtained by us is in agreement with experimental data. The decrease of electron mobility with field strength appears to be a necessary phenomenon accompanying the transfer of energy by electrons to the lattice and the subsequent breakdown.

Thus, theoretical investigations of electron mobility in semiconductors have shown that the electron mobility changes with the field strength.

In semiconductors with an atomic structure the electron mobility decreases with the field strength. In semiconductors with an ionic structure the electron mobility may increase. We have shown that the mobility of electrons in dielectrics must decrease with field strength according to the law \(u \sim \frac{1}{\sqrt{E}}\), and in stronger fields according to the law \(u \sim \frac{1}{E}\). Then, at a field strength corresponding to breakdown, the electron mobility reaches a minimum value. In this case the electrons transfer to the lattice the energy they have accumulated, sufficient to destroy the dielectric. Breakdown of the dielectric occurs.

REFERENCES

1. Vorob’ev A. A. and Zavadovskaya E. K. On the theory of electrical breakdown of solid dielectrics. DAN, LXXI, 3, 375, 1951.

2. Frenkel Ya. I. On the pre-breakdown state of dielectrics. ZhETF, 23, 6, 619, 1952.

1. A. F. Ioffe and A. V. Ioffe. Semiconductors in strong electric fields. The influence of the field on the electrical conductivity of Cu₂O, Tl₂S, V₂O₅, Se, Sb₂S₃, MoS₂, WO₃. ZhETF, 9, 12, 1423, 1451, 1939.

2. F. F. Vol’kenshtein. Electrical Conductivity of Semiconductors. Gostekhizdat, 1947.

3. L. Landau and A. Kompaneets. On deviations of semiconductors from Ohm’s law in strong electric fields. ZhETF, 5, 3, 270, 1935.

4. B. I. Davydov and I. M. Shmushkevich. Theory of electronic semiconductors. UFN, XXIV, 21, 1940.

5. N. L. Tsiparenko. The behavior of semiconductors in strong electric fields. Izv. Acad. Sci. USSR, Phys. Ser., 5–6, 631, 1938.

6. W. Shockley. Mobility of electrons in high electric fields: Theory. Phys. Rev., 82, 2, 330, 1951.

7. E. M. Conwell. Mobility of electrons in high electric fields. Phys. Rev., 88, 6, 1379, 1952.

8. A. A. Vorob’ev and E. K. Zavalovskaya. Determination of the magnitude of the electric strength of dielectrics from the condition of minimum electron mobility. Proceedings of the Tomsk Polytechnic Institute, 73, 3, 1952.