Determination of the Magnitude of the Electrical Strength of Dielectrics from the Condition of Minimum Electron Mobility

A. A. VOROB'EV, E. K. ZAVADOVSKAYA

A hypothesis is advanced that the mobility of electrons in a dielectric decreases before breakdown. From the condition for the minimum of the function describing the dependence of mobility on field strength, formulas are obtained for calculating the electrical strength of crystals. The electrical strength is expressed through lattice characteristics and the energy of the electron. The calculated values of the electrical strength for crystals of alkali-halide salts are close to the measured ones.

Introduction

In a dielectric at high field strength, complex physical processes occur that end in its breakdown. The destruction of the solid structure of a dielectric probably occurs because the electrons participating in current transfer transmit to the lattice part of their energy, acquired from the external field. When the energy transmitted from the electron flow to the lattice sites becomes comparable with the energy of the lattice, the bonds between the sites are destroyed—the breakdown of the dielectric occurs.

The breakdown process is characterized by enhanced transfer of energy from electrons to the lattice and must be accompanied by a decrease in their velocity. Since the decrease in the velocity of electron motion occurs with an increasing electric field, their mobility consequently decreases [1].

Thus, before breakdown there should be a decrease in the mobility of electrons in the dielectric. The mobility of electrons in solid dielectrics must depend on the field strength. This is the case in gases. The magnitude of the breakdown strength of the electric field can be found from the condition for the minimum of the function determining the dependence of electron mobility on field strength.

Davydov [2] showed that in semiconductors the mobility of electrons decreases inversely proportional to the square root of the field strength. From the measurements of Pruzhinina-Granovskaya [3] it follows that in mica, in the field-strength interval \(10^5\)—\(10^6\) V/cm, the mobility of electrons decreases inversely proportional to the field strength.

Recently [4], experiments were described which showed that the mobility of electrons in germanium at high fields decreases inversely proportional to the field strength. We shall give some simple calculations showing that the mobility of charged particles in a lattice can depend on the field strength. A decrease in the mobility \(u\) of current carriers according to the condition \(u \sim E^{-1}\) will lead to a saturation current, since \(i = uenE\). Such a phenomenon was observed by Hippel [5] in the photoconductivity of colored KCl crystals. As is seen in Fig. 1, in these crystals at vo-

at field strengths of the order of 600 kV/cm and above, a saturation current was observed. Breakdown of these crystals occurs at a field strength of about 1000 kV/cm.

The mean free path \(\delta\) of an electron in a crystal depends on the composition of the crystal, the temperature, the concentration of traps, and the field strength \(E\). Measurements of the electron mean free path in crystals of alkali-halide salts and in diamond were carried out at various temperatures and field strengths of the order of several thousand V/cm.

Fig. 1. Saturation current in colored KCl crystals under photoconductivity

In this range of measurements it has been established for many ionic crystals that the mean free path of electrons increases linearly with increasing field: \(\delta = cE\). According to Hecht, who determined \(\delta\) by the light-probe method, for AgCl crystals in a field of \(104\text{--}312\ \text{V/cm}\) the value \(c = 2.5 \cdot 10^{-3}\); in a field of \(5 \cdot 10^{3}\ \text{V/cm}\) and at a temperature \(t = 196^\circ\text{C}\), \(c = 2.2 \cdot 10^{3}\). According to Lefeld, for AgCl at \(t = 170^\circ\text{C}\) and a field varying from 20 to 300 V/cm, the value \(c = 0.04\). For TlBr at \(-70^\circ\text{C}\) and a field of 20–300 V/cm, \(c = 0.022\) was obtained. In a field of \(5 \cdot 10^{3}\ \text{V/cm}\), at a temperature of \(-196^\circ\text{C}\), for TlBr \(c = 1.24 \cdot 10^{-3}\). It is difficult to compare the values obtained for the proportionality coefficient \(c\), obtained at different temperatures and for crystals with different impurity concentrations. For LiF crystals, according to these measurements, the mean free path of secondary electrons at a temperature of \(300^\circ\text{K}\) in a field where

\[ E = 5 \cdot 10^{3}\ \text{V/cm} \]

is

\[ \delta = cE = 0.11 \cdot 5 \cdot 10^{3} = 5.5 \cdot 10^{2}\ \text{cm}. \]

Assuming that, at breakdown, electrons are transferred from the \(F\) band, which is separated from the conduction band in LiF crystals by \(4.95\ \text{eV}\), we obtain that in the field \(E_{\text{br}} = 3.1\ \text{MV/cm}\) the electrons must be displaced by an amount

\[ \delta = \frac{4.95}{eE} = \frac{4.95}{3.1\,10^{6}} = 1.65 \cdot 10^{-6}\ \text{cm}. \]

Such a considerable decrease in the mean free path of electrons at a field strength equal to the breakdown strength may be caused either by a decrease in electron mobility as a result of the strengthening of their interaction with the lattice, or by an increase in trap concentration, or by the simultaneous action of both causes.

Let us consider the transition of an electron from one equilibrium state to another at absolute-zero temperature. Two neighboring equilibrium positions for an electron are located at a distance \(2a\) from one another and are separated by a potential barrier of height \(u_0\).

The probability of transition from one equilibrium position to another at each oscillation is, for an electron, \(e^{-u_0/w_k}\), or \( \nu e^{-u_0/w_k}\) per second, where \(\nu\) is the frequency of the electron’s natural oscillations and \(w_k\) is the energy of the electron. If there is an electric field of strength \(E\) in the dielectric, then the potential barrier in the direction of action of the accelerating ...

the field is reduced relative to the equilibrium position by the amount \(E'ea\), where \(a\) is the distance between the equilibrium position and the maximum of the potential barrier, and \(e\) is the charge of the electron or hole, and

\[ E' = E \varepsilon_0,\quad \varepsilon_0 = n^2, \]

where \(E'\) is the average calculated value of the field strength and \(\varepsilon_0\) is the optical part of the dielectric coefficient.

Fig. 2. Change in the periodic distribution of the potential energy for an electron in a crystal lattice under the action of a constant electric field.

The probability of an electron transition in the direction of the accelerating field during one oscillation will be

\[ e^{-\frac{u_0 - E' \cdot e \cdot a}{w_k}} . \]

The frequency of electron transitions per second is

\[ \nu_1 = \nu e^{-\frac{u_0 - E' \cdot e \cdot a}{w_k}}, \tag{1} \]

where \(\nu\) is the frequency of oscillations of the electron at the crest of the potential barrier.

In the opposite direction, the transition frequency for electrons is

\[ \nu_2 = \nu e^{-\frac{u_0 + E' \cdot e \cdot a}{w_k}} . \tag{2} \]

The total number of electron transitions in the direction of the accelerating field and in the opposite direction is

\[ \Delta \nu = \nu \left( e^{-\frac{u_0 - E'ea}{w_k}} - e^{-\frac{u_0 + E'ea}{w_k}} \right). \tag{3} \]

The current density is determined by the expression: \(i = n \cdot e \cdot \Delta \nu \cdot \delta\), where \(n\) is the number of electrons per unit volume. We have

\[ i = ne\delta\nu \left( e^{-\frac{u_0 - E'ea}{w_k}} - e^{-\frac{u_0 + E'ea}{w_k}} \right). \tag{4} \]

We define the electron mobility \(u\) from the condition \(i = ne\delta\Delta\nu = neuE'\), where \(\delta\) is the distance by which the electron is displaced in the dielectric; whence

\[ u = \frac{\delta\Delta\nu}{E'} = \frac{\delta\nu}{E'} \left( e^{-\frac{u_0 - E'ea}{w_k}} - e^{-\frac{u_0 + E'ea}{w_k}} \right). \tag{5} \]

Let us examine the change of the function \(u\) as a function of \(E'\) for certain particular values of \(w_k\).

Change in the Mobility of Electrons in an Electric Field

At our request A. M. Gaidamovich investigated equation (5) for several special cases (Figs. 3 and 4). If \(w=\mathrm{const}\), then the function represented by equation (5) has a minimum at \(E=0\) (Fig. 3). The minimum value of the mobility is

Fig. 3. Change in the mobility of electrons as a function of the field at constant electron energy.

\[ OM=\frac{2\delta \nu ae}{w_k}\cdot e^{-\frac{u_0}{w}} . \]

Assuming that the energy of the electrons increases linearly with increase of the field, that is, \(w=mE'\), we obtain:

\[ u=\frac{\delta \nu}{E'}\cdot e^{-\frac{u_0}{mE'}} \left(e^{\frac{ae}{m}}-e^{-\frac{ae}{m}}\right) = \frac{B}{E'}\cdot e^{-\frac{u_0}{mE'}}, \]

where

\[ B=\delta \nu\left(e^{\frac{ae}{m}}-e^{-\frac{ae}{m}}\right). \]

In Fig. 4 the change of the function \(u\) with variation of the field strength \(E'\) is shown schematically.

The curve located in the third quadrant shows the decrease in mobility with increasing retarding action of the electric field.

The curve located in the first quadrant shows the complex character of the dependence of the electron mobility on the electric-field strength. At first, with increasing electric-field strength, the electron mobility increases. The mobility reaches its greatest value at the field strength determined by the condition: \(E^*=\dfrac{u_0}{m}\). Po-

the subsequent increase in the field strength is accompanied by a decrease in the mobility of the electrons.

Let us consider particular solutions of equation (5) under certain additional conditions.

Fig. 4. Change in electron mobility as a function of the field, under the condition of a linear increase of electron energy in the field.

1. Suppose that the energy of an electron is equal to the thermal energy and, in the direction of the action of the field, is \(w_k = KT = \text{const}\); then we write equation (5) in the form

\[ u = A \frac{\operatorname{sh} bE'}{E'}, \tag{6} \]

where

\[ A = 2\gamma e^{-\frac{u_0}{KT}} \quad \text{and} \quad b = \frac{a_0 e}{KT}. \]

Let us consider the variation of the derivative of the function \(u\). In this case, assuming that \(e^{-bE'} \sim 0\), we obtain

\[ u' = \left(\frac{b}{E'} - \frac{1}{E'^2}\right)\frac{Ae^{bE'}}{2}. \tag{7} \]

The second derivative is

\[ u'' = \left(\frac{b^2}{E'} - \frac{2b}{E'^2} + \frac{2}{E'^3}\right)\frac{Ae^{bE'}}{2}. \]

It follows from these expressions that the function \(u\), at the value \(E' = 1/b\), has a minimum value.

From (7) we determine

\[ E' = \frac{1}{b} = \frac{KT}{ea}. \tag{8} \]

When using this formula for calculating the electrical strength, it is necessary to take into account that the magnitude of the field strength in the dielectric which we have calculated is, owing to polarization, smaller than the average calculated ...

field strength by a factor of $\varepsilon$. Introducing this correction for determining the electrical strength of a dielectric, we obtain the formula

\[ E_{pr}=\frac{KT\varepsilon_0}{ea}, \tag{9} \]

where $\varepsilon_0$ is the part of the dielectric coefficient due to the displacement of electrons. In crystals of the rock-salt type, the motion of electrons occurs predominantly along the direction of the face diagonal. Along this crystallographic direction, along which positive or negative ions are arranged, the potential energy of the electrons retains its smallest value. In this direction the distance between two neighboring like ions will be $\sqrt{2}\,a$. With the corrections indicated, for calculating the magnitude of the electrical strength of rock salt we obtain the formula

\[ E_{pr}=\frac{KT\varepsilon_0}{ea\sqrt{2}} . \tag{10} \]

From this formula for rock salt we obtain

\[ E_{pr}=\frac{1.4\cdot 10^{-16}\cdot 300\cdot 2.25\cdot 300} {4.8\cdot 10^{-10}\cdot 2.81\cdot 10^{-8}\cdot 1.41} =1.5\cdot 10^{6}\ \text{V/cm}, \]

i.e., a value in satisfactory agreement with that measured experimentally. In a strong electric field the electrons can accumulate an energy $\overline{\omega}_k$ considerably exceeding their mean thermal energy $3/2\,KT$. In general, as a result of this, the electron distribution function will differ greatly from the equilibrium Maxwellian one. This must be taken into account in a more exact treatment of the problem.

1. It is of interest to compare the proposed hypothesis with others advanced earlier. Hippel [5] considered the question of the interaction of an electron with lattice vibrations. He was interested in the accumulation by an electron of the energy necessary to produce impact ionization in the lattice. For this it was necessary to find the conditions under which the accumulation of energy by an electron becomes possible as it moves in an electric field. Hippel calculated the electron velocity in an accelerating field, at which the least loss of energy by the electron occurs, as follows. The transfer of energy to lattice vibrations is most probable when the energy of the moving charge is close to the energy of the natural vibrations of the lattice ions. It is known that the energy of the natural vibrations of the lattice ions is determined from the residual radiation and is

\[ h\nu_{\text{res}}\sim 1-3,10^{-2}. \]

At this value of the electron energy their interaction with lattice vibrations will be greatest. According to Hippel’s hypothesis, as long as an electron, over the path $a$, accumulates in the electric field an energy smaller than that lost by it to excitation of lattice vibrations, i.e. $h\nu_{\text{res}}$, where $\nu_{\text{res}}$ is the frequency of the residual radiation, it cannot be accelerated. If, however, over a segment of path equal to the lattice constant, the electron accumulates an energy greater than $h\nu_{\text{res}}$, then the amount of energy transferred by it to the lattice is sharply reduced. In this case the probability of collisions of the electron with ions also decreases.

Under such conditions, accumulation by the electron of the energy necessary for subsequently producing impact ionization becomes possible.

An energy of the order of \(10\ \mathrm{eV}\), necessary for producing impact ionization in a rock-salt crystal, is accumulated by an electron in a field \(E \simeq 10^{6}\ \mathrm{V/cm}\) over a path equal to several hundred interionic distances.

The explanation of breakdown that we propose differs substantially from the one given above. It seems to us that the onset of breakdown is determined by the condition of maximum interaction of the electron with the lattice when it passes from one equilibrium position to another, and by the magnitude of its energy.

Neglecting the second term in equation (5) increases the value of the mobility by a factor \(e^{-1}\). This shows how approximate our arguments are.

Let us assume that \(\omega_k \simeq h\nu_{\mathrm{res}}\).

Under this condition equation (9) is written in the form

\[ E_{\mathrm{br}}=\frac{h\nu_{\mathrm{res}}\varepsilon_0}{ea}. \tag{11} \]

The table presents the results of calculations of the electrical strength for crystals of alkali-halide salts. The values of the electrical strength of alkali-halide crystals obtained by calculation from formula (11) and those measured agree satisfactorily.

Table

Electrical strength of crystals of alkali-halide salts

Figures 5, 6, and 7 present the dependences of the magnitude of the electrical strength on the energy of the residual radiation, on the lattice constant \(a\), and on the fraction of the dielectric coefficient \(\varepsilon_0\) due to the displacement of electrons, according to experimental data. As is seen, between the magnitude of the electrical strength \(E_{\mathrm{br}}\), on the one hand, and \(h\nu_{\mathrm{res}}\), \(a^{-1}\), and \(\varepsilon_0\), on the other, there is a satisfactory linear dependence. This is in agreement with the results of our calculations, represented by formula (11).

It is interesting to note that in his semiempirical theory of breakdown, Hippel used the formula

\[ E_{\mathrm{br}}=C\,\frac{h\nu_{\mathrm{res}}}{ea\cdot 2}, \]

where \(C\) is a coefficient of the order of 1.7, that is, two. In the calculation given by us, the formula constructed by Hippel was obtained as a result of the adopted

breakdown hypothesis. The coefficient \(C\), determined empirically by Hippel, acquired the physical meaning of the dielectric coefficient and logically fell into place.

Fig. 5. Dependence of the electrical strength of alkali-halide salt crystals on the lattice constant \(a\), according to experimental data.

Dependence of the electrical strength \(E_{\mathrm{br}}\) on \(a_0^{-1}\)

Equation of the straight line:

\[ E_{\mathrm{br}} = 11.8\,(a_0^{-1} - 0.232)\ \text{MV/cm} \]

Electrical strength and the quantum of residual-radiation energy

Quantum of residual-radiation energy in \(10^{-2}\ \mathrm{eV}\) (according to measurements)

Fig. 6. Dependence of the value of the electrical strength of alkali-halide salt crystals on the quantum of residual-radiation energy, according to experimental data.

Dependence between the electrical strength and the dielectric coefficient for alkali-halide crystals

Optical part of the dielectric coefficient \(\varepsilon_0\)

Fig. 7. Dependence of the value of the electrical strength of alkali-halide salt crystals on the value of the dielectric coefficient due to electron displacement, according to experimental data.

1. Let us consider the case when \(w_k=f(E)\), for example, \(w_k=mE\); then

\[ u\sim E^{-1} e^{-\frac{u_0-aeE}{w_k}} =E^{-1}e^{-nE}\cdot e^{-\frac{ae}{m}} =AE^{-1}e^{-nE}, \tag{12} \]

where

\[ n=\frac{u_0}{m}\quad \text{and}\quad A=e^{-\frac{ae}{m}}. \]

Let us investigate the function

\[ u^{-1}\sim Ee^{nE}. \tag{13} \]

It is obvious that the maximum value of the function (13) will correspond to the minimum of the function (12). We have

\[ (u^{-1})'=(Ee^{nE})'=e^{nE}\left(1-\frac{n}{E}\right). \]

Therefore, from the condition \((u^{-1})'=0\), we have \(E=n\),

\[ (u^{-1})''=\frac{n}{E^2}e^{nE}>0. \tag{14} \]

Consequently, the function \(u^{-1}\) at \(E=n\) has a minimum, and the function of interest to us, \(u\), at \(E=n\) has a maximum.

This result coincides with that obtained above in considering equation (5).

1. Let us take into account the change in the kinetic energy of an electron moving in a lattice due to collisions with phonons.

For an electron moving in an atomic lattice [6, 7] and undergoing collisions with lattice vibrations, the energy \(w_k\) is expressed by the condition:

\[ w_k=\frac{eEa}{v_{\mathrm{sound}}}\sqrt{\frac{KT}{m}}=eEa n, \tag{15} \]

where \(v_{\mathrm{sound}}\) is the speed of sound in the dielectric,

\(m\) is the electron mass.

Owing to the proportionality of the electron energy to the field strength, this case reduces to the preceding one.

The calculations presented for the value of the electric strength of dielectrics show that this quantity may depend on temperature. The dependence of electric strength on temperature is determined by the change in the electron energy in a strong electric field, the lattice constant \(a\), and the potential barrier for electrons \(u_0\). These quantities depend only weakly on temperature. Thus, one should not expect a significant temperature dependence of the electric strength in electrical breakdown.

1. Let us assume that the mobility of electrons is determined by the expression

\[ u=\frac{\partial v}{E}\,e^{-\frac{u_0-E'e a}{w_k}}. \tag{16} \]

At the moment preceding breakdown, the mobility has a minimum value and varies according to the condition \(u\sim E^{-1}\). Therefore one may write that

\[ e^{-\frac{u_0-ea_0E}{w_k}}=1. \tag{17} \]

Consequently,

\[ u_0-ea_0E=0, \tag{18} \]

whence

\[ E_{\mathrm{br}}=\frac{u_0\varepsilon_0}{ea}. \tag{19} \]

Assuming for rock salt that the height of the potential barrier separating two neighboring equilibrium positions for electrons is equal to \(1\ \mathrm{eV}\), we obtain

\[ E_{\mathrm{br}}=76\cdot 10^6\ \mathrm{V/cm}. \]

The calculated value of \(E_{\mathrm{br}}\) exceeds by 40–50 times the maximum value of the electrical strength determined from experiment.

The very high value of the electrical strength obtained in this case is probably a consequence of a purely electrostatic treatment of the question of the liberation and motion of the electron.

1. For the energy of electrons in semiconductors with an ionic lattice in strong fields, Davydov and Shmushkevich [6] give a limiting expression for the mean energy \(w_k=1.5\,w_i\), where \(w_i\) is the energy of liberation (ionization) of the electron. In that case

\[ E_{\mathrm{br}}=\frac{w_i\varepsilon_0}{5ea\sqrt{2}}. \tag{20} \]

It was shown by us that, for crystals of the alkali-halide series, there is a linear dependence between the value of the electrical strength and the liberation energy of the most weakly bound electrons of the \(F\) band [8] (Fig. 8). The formula (20) obtained is in qualitative agreement with these results. For rock salt \(w_i=\Delta w_F=2.65\ \mathrm{eV}\). According to formula (20), the value of the electrical strength is obtained as

\[ E_{\mathrm{br}}=0.29\cdot 10^8\ \mathrm{V/cm} =2.9\cdot 10^7\ \mathrm{V/cm}. \]

This value is 20 times higher than that determined experimentally and calculated by formula (19). Possibly this resulted because the calculation used a high average value of the energy of electrons formed as a result of ionization processes. It is known [9] that calculations of electrical strength carried out by other authors on the basis of ideas about ionization also led to a high value of the electrical strength.

Fig. 8. Dependence of the value of the electrical strength of alkali-halide salt crystals on the value of the binding energy of electrons in the \(F\) band.

Thus, consideration of the phenomenon of breakdown leads to interesting values of the electron energy, of the order of the lattice energy. Consequently, in breakdown the participating electrons are not fast electrons having energy of the order of the ionization energy, but slower ones, having energy of the order of the energy of the natural vibrations of the lattice ions.

Neglect of the interaction of electrons with the lattice and an electrostatic treatment of the question of ionization and motion of electrons in the lattice lead to very high values of the electrical strength. Allowance

interaction of electrons with lattice vibrations during breakdown leads to a value of the electric strength that agrees satisfactorily with that determined experimentally.

1. Let us calculate the average increase in the energy of an electron moving in the conduction band of an ionic crystal [10]. The increment of the electron energy per second will be \(\Delta w=eEv\), where \(v\) is the group velocity. The probability of collision and deflection of an electron per unit time may be written in the form \(1/\tau(w)\). The function \(\tau(w)\), representing the time of motion of an electron before deflection or collision, may be called the electron mean free time.

The mobility of electrons located in the conduction band is determined as:

\[ u(w)=\frac{e}{m}\tau(w). \tag{21} \]

The drift velocity \(v\) of electrons in the field \(E\) is written in the form

\[ v=uE=\frac{e}{m}\tau(w)E. \tag{22} \]

Thus,

\[ \Delta w=evE=\frac{e^2E^2}{m}\tau(w). \tag{23} \]

Let us assume that all the energy received by the electrons from the electric field, the rate of accumulation of which is represented by formula (23), is expended in destroying the lattice during breakdown. The magnitude of the energy expended on breakdown of the dielectric is determined by the condition

\[ w_k=\Delta w\,\Delta t=\frac{e^2E^2}{m}\tau(w)\Delta t, \tag{24} \]

where \(\Delta t\) is the breakdown time of the dielectric.

The quantity \(\tau\) in the breakdown region depends on the field strength, \(\tau=qE^{-1}\), where \(q\) is a proportionality coefficient. Therefore the energy of the electrons participating in energy transfer during breakdown is determined by the condition:

\[ w_k=\frac{e^2}{m}qE\Delta t=pE, \]

where

\[ p=\frac{e^2}{m}q\Delta t. \]

Thus, this case reduces to the one already considered earlier.

In the case considered, we obtain the formula for calculating the magnitude of the electric strength of a dielectric in the form

\[ E_{\text{br}}=\frac{u_0}{p}=\frac{u_0m\varepsilon_0}{e^2\cdot q\cdot \Delta t}. \tag{25} \]

CONCLUSION

1. The mobility of electrons in solid dielectrics depends on the field strength.

2. From the condition of the minimum of the function describing the dependence of mobility on field strength, formulas can be obtained for calculating the magnitude of the electric strength of a dielectric.

1. The magnitude of the electric strength of a dielectric decreases with increasing lattice constant and depends on the energy of the electron in the lattice and on the dielectric constant.

2. Neglect of the interaction of electrons with lattice vibrations, i.e., a purely electrostatic treatment of the problem of ionization and of the motion of electrons in the lattice, leads to implausibly high values of the electric strength.

3. Breakdown develops when the electron energy is of the order of the energy of the natural vibrations of the lattice ions.

We express our gratitude to A. M. Gaidamovich for useful advice and discussion of individual questions.

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