THEORY OF ELECTRONIC SEMICONDUCTORS. I

B. I. Davydov and I. M. Shmushkevich

1. INTRODUCTION

In constructing an electronic theory of the solid state, one usually starts from the following model: each of the electrons moves, independently of the others, in a force field consisting of the field of all ions or atoms located at the nodes of the crystal lattice and of the field of all the remaining electrons (apart from the given one), averaged over their motion. It should be said that a justification of this method of treatment, called the Hartree method, still does not exist. In its defense one may point to the successful application of this method to atoms. There is also no doubt that a very large number of conclusions obtained with its aid for crystals are likewise in fairly good agreement with experimental data. There are, however, facts which have not yet found a satisfactory explanation in the theory. Chief among them, undoubtedly, is superconductivity. It is possible that the absence of an explanation of this phenomenon is connected with the indicated imperfection of the theory.

The essence of the Hartree method is that it makes it possible to reduce the many-electron problem to a one-electron problem.

Owing to the periodicity of the crystal lattice, the potential energy of the electron is also a triply periodic function of its coordinates. This fact proves sufficient for determining the general form of the electron wave function \(\psi^1\). Let \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\) be three vectors characterizing the elementary cell of the crystal (the lengths of its edges being respectively \(a_1, a_2, a_3\)). In that case the periodicity property of the electron potential energy \(V\) may be expressed as follows:

\[ V(\mathbf{r}+n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3)=V(\mathbf{r}), \tag{1,1} \]

where \(n_1, n_2, n_3\) are three arbitrary integers.

The probability density for finding the electron at some point, \(|\psi(\mathbf{r})|^2\), must evidently also possess this periodicity property:

\[ |\psi(\mathbf{r}+n_1\mathbf{a}_1+n_2\mathbf{a}_2+n_3\mathbf{a}_3)|^2=|\psi(\mathbf{r})|^2. \tag{1,2} \]

On the function \(\psi\) one usually imposes the cyclicity condition

\[ \psi(\mathbf{r}+\mathbf{G}\mathbf{a}_i)=\psi(\mathbf{r}), \tag{1,3} \]

where \(G\) is a very large number, so that the length \(L=Ga\) has macroscopic dimensions. We shall call the part of the crystal having the form of a parallelepiped with edges \(Ga_1,\ Ga_2,\ Ga_3\) the fundamental region.

It is physically clear that, if the crystal is sufficiently large, the choice of boundary conditions for the function \(\psi\) is immaterial. From equation (1,2) we conclude that

\[ \begin{aligned} \psi(\mathbf r+\mathbf a_1)&=\mu_1\psi(\mathbf r),\\ \psi(\mathbf r+\mathbf a_2)&=\mu_2\psi(\mathbf r),\\ \psi(\mathbf r+\mathbf a_3)&=\mu_3\psi(\mathbf r). \end{aligned} \tag{1,4} \]

The multipliers \(\mu_1,\mu_2,\mu_3\) are equal to unity in modulus; their phase is determined by equations (1,3).

From these equations it follows that

\[ \mu_1^G=\mu_2^G=\mu_3^G=1 . \tag{1,5} \]

Thus,

\[ \mu_\alpha=e^{\,i\frac{2\pi z_\alpha}{G}} \qquad (\alpha=1,2,3). \tag{1,6} \]

\(z_1,z_2,z_3\) are three integers, each of which may take \(G\) values, for example from \(0\) to \(G\), or from \(-G/2\) to \(+G/2\) (the two limiting values are equivalent, since \(e^{2\pi i}=1\)). It is easy to see that the function

\[ \psi_{\mathbf k}=e^{i\mathbf k\mathbf r}\,U_{\mathbf k}(\mathbf r)\,e^{-i\frac{\varepsilon_{\mathbf k}t}{\hbar}} \tag{1,7} \]

satisfies equations (1,4) with the coefficients \(\mu_1,\mu_2,\mu_3\), determined by formulas (1,6), if \(U_{\mathbf k}(\mathbf r)\) is a function possessing the same periodicity as the function \(V(\mathbf r)\), and the wave vector

\[ \mathbf k=\frac{2\pi}{b}\left(z_1\mathbf b_1+z_2\mathbf b_2+z_3\mathbf b_3\right), \tag{1,8} \]

where

\[ \mathbf b_1=\frac{[\mathbf a_2,\mathbf a_3]}{\mathbf a_1[\mathbf a_2,\mathbf a_3]}, \qquad \mathbf b_2=\frac{[\mathbf a_3,\mathbf a_1]}{\mathbf a_1[\mathbf a_2,\mathbf a_3]}, \qquad \mathbf b_3=\frac{[\mathbf a_1,\mathbf a_2]}{\mathbf a_1[\mathbf a_2,\mathbf a_3]}, \]

are the so-called scale vectors of the reciprocal lattice. The vector \(\mathbf k\), obviously, may take \(G^3=N\) different values. \(N\) is the number of atoms in the fundamental region.

The wave function of the electron is a modulated plane wave, i.e. a plane wave whose amplitude changes periodically.

Such a wave, without being scattered, could pass through the entire crystal if the lattice of the latter were ideal. The electrical resistance of the crystal would then be equal to zero. Thus, we arrive at the conclusion that the resistance of a crystal is caused by the deviation of the force field in which the electron moves from ideal periodicity. We shall speak of this in more detail later, when we quantitatively determine the electrical conductivity.

If one constructs a wave packet from the functions (1.7), then it can be shown that its group velocity is

\[ \mathbf{v}=\frac{1}{\hbar}\frac{\partial \varepsilon}{\partial \mathbf{k}} . \tag{1.9} \]

We see, therefore, that the behavior of an electron in a periodic field strongly resembles the behavior of a free electron. The quantity \(\mathbf{p}=\hbar\mathbf{k}\), which plays the role of momentum, is called the quasimomentum. The relation between the quasimomentum and the group velocity of an electron in a crystal [equation (1.9)] is the same as that between the momentum and velocity of a free electron. The quasimomentum, however, is not a uniquely defined quantity. It is easy to see that adding to the wave vector \(\mathbf{k}\) the vector \(\mathbf{g}=2\pi(n_1\mathbf{b}_1+n_2\mathbf{b}_2+n_3\mathbf{b}_3)\) does not change the form of the electron wave function (1.7), since \(e^{i\mathbf{g}\mathbf{r}}\) also has the periodicity of the lattice. To eliminate this periodicity, one usually uses the reduced wave vector, whose components along the axes \(\mathbf{b}_1,\mathbf{b}_2,\mathbf{b}_3\) are minimal in absolute value; i.e., \(z_1,z_2,z_3\) in formula (1.8) may take the values

\[ -\frac{G}{2},\quad -\frac{G}{2}+1,\quad -\frac{G}{2}+2,\ldots,\quad +\frac{G}{2}. \]

In a simple cubic lattice with lattice constant \(a\), the components of the quasimomentum are

\[ k_x=\frac{2\pi z_x}{ba},\qquad k_y=\frac{2\pi z_y}{ba},\qquad k_z=\frac{2\pi z_z}{ba}. \]

To determine the character of the energy spectrum of an electron in a crystal, i.e., to determine the totality of possible values of the electron energy in a crystal, we shall start from the behavior of an electron in an isolated atom. Figure 1 shows the dependence of the potential energy of an outer (valence) electron on the distance to the atomic core corresponding to this electron (origin of coordinates in Fig. 1).

Fig. 1

As is known from quantum mechanics, the energy spectrum of an electron in an atom consists of a discrete series of separate levels (the horizontal straight lines in Fig. 1), which merge into a continuous background when the total energy of the electron becomes positive (the potential energy at infinity is taken to be zero).

If there is a system of \(N\) atoms located at large distances from one another, then in place of each atomic level there will be \(N\) identical levels of the whole system. In other words, there will be an \(N\)-fold degeneracy of each level. When the atoms that form the crystal lattice are brought closer together, owing to the perturbation caused by the electrical interaction of the atoms, a splitting of the levels occurs. From each atomic level there will result

a whole band of densely spaced levels. The number of levels in such a band is equal to the number of atoms of the lattice, or, what is practically the same thing, to the number of atoms in the fundamental region. If the lattice consists of different atoms, for example ZnS, then the number of levels in the band is equal to the number of unit cells in the crystal. In this case the crystal contains bands that originate both from the atomic levels of Zn and from the atomic levels of S. Thus, the number of levels in a band is exactly equal to the number of possible values of the wave vector. To each level in the band one may assign a definite wave vector k. The state of the electron is therefore completely determined (if, for the time being, we do not speak of its spin) when its quasimomentum and the energy band to which its energy belongs are specified.

Energy bands, or, as they are often called, zones, are separated from one another by so-called forbidden bands, corresponding to such energy values that the electron cannot possess in the lattice. The higher the atomic level is situated, i.e. the farther the electron is from the corresponding nucleus, the less strongly it is bound to the atom, the more the atomic level is split in the crystalline lattice, and the more easily the electron passes from one atom to another. The ability of an electron to penetrate through the potential barriers between atoms, even if its energy is less than the height of these barriers, is a purely quantum effect. Connected with this effect is the fact that the wave function of the electron is a plane wave capable of passing through the entire crystal. Figure 2 shows the general form of the energy spectrum of electrons in a crystal. The allowed bands (hatched strips) alternate with forbidden ones. The width of the allowed bands, as a rule, increases, while the width of the forbidden regions decreases with increasing energy. The widths of both are of the order of atomic energy, i.e. of the order of one or several electron-volts.

Fig. 2

In the general case, the electron energy \(\varepsilon\) is a certain complicated function of the quasimomentum, which is very difficult to determine. It is possible, however, to indicate the dependence of the energy on the components of the quasimomentum when the electron energy is near the lower or upper edge of an allowed band.

In the first case we expand the energy in a series in the components of the quasimomentum near the lower edge of the allowed band, denoting its energy by \(\varepsilon_1\). Then

\[ \varepsilon=\varepsilon_1+\frac{\hbar^2}{2}\sum_{\alpha,\beta} m_{\alpha\beta}^{-1} k_\alpha k_\beta . \tag{1.10} \]

The linear terms in this expansion are absent, since we are considering the region near the energy minimum. The symmetric tensor \(m_{\alpha\beta}^{-1}\) is called the reciprocal-mass tensor.

In a cubic lattice it, in the simplest case, becomes a scalar. The energy of the electron in this case is

\[ \varepsilon=\varepsilon_1+\frac{\hbar^2 k^2}{2m_-}. \tag{1,11} \]

\(m_-\) is called the effective mass by analogy with the corresponding formula for free electrons. The velocity of the electrons according to formula (1,9) is

\[ \mathbf{v}=\frac{\hbar\mathbf{k}}{m_-}. \tag{1,12} \]

However, in a cubic lattice three coalescing bands may also be obtained, for which the principal axes of the inverse-mass tensor are rotated relative to one another by \(\pi/2\).

To determine the dependence of the energy on the quasi-momentum near the upper edge of the allowed band, we introduce the vector \(\mathbf{k}'\), whose components are defined as follows:

\[ k'_\alpha= \begin{cases} -k_\alpha+k_\alpha^{\max} & \text{for } k_\alpha>0 \quad (\text{then } k'_\alpha>0),\\ -k_\alpha+k_\alpha^{\min}=-k_\alpha-k_\alpha^{\max} & \text{for } k_\alpha<0 \quad (\text{then } k'_\alpha<0). \end{cases} \tag{1,13} \]

Then

\[ \varepsilon=\varepsilon_2-\frac{\hbar^2}{2}\sum_{\alpha,\beta} m_{\alpha\beta}^{-1} k'_\alpha k'_\beta, \tag{1,14} \]

where \(\varepsilon_2\) is the energy of the upper edge of the allowed band. For electrons with a pronounced inverse-mass tensor,

\[ \varepsilon=\varepsilon_2-\frac{\hbar^2 k'^2}{2m_+}. \tag{1,15} \]

The velocity of the electron in this case is

\[ \mathbf{v}=-\frac{\hbar\mathbf{k}'}{m_+}, \tag{1,16} \]

or

\[ v_\alpha=-\frac{\hbar\left(k_\alpha^{\max}-k_\alpha\right)}{m_+} \tag{1,16'} \]

(for \(k_\alpha>0\)).

Thus, for states whose energy is situated close to the upper edge of the allowed band, the velocity of the electron decreases with increasing quasi-momentum [formula (1,16′)], while the energy decreases with increasing velocity [see formula (1,15)]. This seemingly paradoxical fact should not be surprising, since the electrons are in fact not free, but move in a periodic field, and the similarity between the motion of a free electron and the motion of an electron in a periodic field should not be extended arbitrarily far.

It can be shown that, owing to the fact that the wave function of an electron in a crystal is a plane modulated wave, the motion of a wave packet composed of functions (1,7), under

under the action of an external electric field obeys the laws of classical mechanics (Ehrenfest’s theorem for electrons in a periodic field)

\[ \frac{d\mathbf p}{dt}=\hbar \frac{d\mathbf k}{dt}=-e\mathbf E. \tag{1,17} \]

Here and in what follows the charge of the electron is denoted by \(-e\), so that \(e\) is a positive quantity.

With the aid of (1,12), for electrons whose energy is situated close to the lower edge of the band, we obtain the usual relation

\[ m_- \frac{d\mathbf v}{dt}=-e\mathbf E. \tag{1,18} \]

For states situated close to the upper edge of the band, equation (1,17) gives

\[ -m_+ \frac{d\mathbf v}{dt}=-e\mathbf E, \tag{1,18'} \]

i.e. electrons in such states behave in an external field like particles with negative charge and negative mass. Energy levels free of such electrons are equivalent to the presence of particles with positive charge and positive mass.

We have clarified the question of the character of the electron wave function in a crystal, of its energy spectrum, and of its motion in an external electric field.

Let us turn to the question of the distribution of electrons over energy levels. At absolute zero temperature, any system tends toward a minimum of energy. As a consequence of this, it would seem that all electrons at absolute zero should be in one and the same state, corresponding to their minimum possible energy. According to the Pauli principle, however, only one electron can be in each state. Therefore, in a state with a definite energy and quasimomentum there can be at most two electrons, whose states differ by the opposite direction of the projection of the spin onto some axis. At absolute zero, consequently, the number of the lowest occupied levels is equal to the number of electrons in the crystal. In this case there may be two cases: either the very highest of the energy levels occupied at absolute zero are immediately adjoined by other levels (Fig. 3), or the electrons completely fill a certain number of the lowest bands, and the next band above them is entirely free (Fig. 4).

If an electric field is applied to a crystal belonging to the first type, then the electrons, accelerating in it, will be able to pass to energetically higher levels. Since more electrons will move in the direction of the field than in the opposite direction, an electric current will arise as a result.

From what we have said about the motion of electrons in an ideal crystal lattice, it follows that the electrical resistance

is caused by the scattering of electron waves arising from deviations in the force field of the lattice from ideal periodicity. There are two kinds of causes leading to such a deviation. First, these may be distortions caused by mechanical damage to the crystal, by the absence, at some lattice sites, of the corresponding atoms, by the inclusion in the lattice of atoms of foreign substances—impurities, etc. Second, deviations from periodicity are caused by the thermal vibrations of atoms. In an ideal single crystal, obviously, only the latter cause operates. But as the temperature is lowered, the thermal motion diminishes.

Fig. 3

Fig. 4

Therefore, in pure single crystals in which there is an incompletely filled allowed band, the resistance decreases as the temperature is lowered.

If, however, a field is applied to a crystal of the second type (in which the distribution of electrons over energy levels at absolute zero is shown in Fig. 4), then at absolute zero it will produce no current.

Indeed, in order for an electron to be able to increase its energy in an electric field, there must be free energy levels higher than those on which the electrons are located. There are none in the filled zone. The nearest free levels are separated from the electrons of the filled band by a portion of the forbidden zone \(\Delta E\), of width on the order of several electron-volts. In order for an external field, by means of a tunnel transition, to throw electrons from the filled zone into the free one in any appreciable number, it must be very large, on the order of \(10^6\ \mathrm{V/cm}\).

The fields used in practice are much smaller. Therefore crystals in which the completely filled zone is separated by a large interval from the completely free zone are, according to the scheme set forth here, perfect insulators at absolute zero.

At a temperature different from absolute zero, thermal motion throws a small number of electrons from the filled band into the empty one. As the temperature is raised, the number of electrons in the upper zone, i.e. conduction electrons, increases. At the same time, the liberation of an equal number of levels in the filled zone

also causes a certain conductivity. With an increase in the concentration of conduction electrons, as the temperature is raised, the electrical conductivity increases according to an exponential law.

Thus we arrive at the existence of two types of crystals, which differ fundamentally in the dependence of their electrical resistance on temperature. In some, which have unfilled energy bands, the resistance at absolute zero is equal to zero; with increasing temperature it grows. Such substances are metals.

In others, at absolute zero all the lowest bands are completely filled, and the next band after them is entirely free. Their resistance is then equal to infinity. As the temperature is raised, a small electronic conductivity appears, and the resistance, in any case at low temperatures, decreases exponentially. These substances are called electronic semiconductors.

Such a principle of dividing substances into metals and semiconductors according to the dependence of their resistance on temperature (at low temperatures) is at present the only theoretically reasonable one.

The reference to low temperatures is connected with the fact that at high temperatures the character of the change in the resistance of semiconductors, as will be seen below, may in some cases change.

Semiconductors or dielectrics in which ionic electrical conductivity plays the principal role will not be considered in this review.

The electrical properties of semiconductors depend strongly on external conditions. Thus, for example, by illuminating a semiconductor with light, one can transfer a certain number of electrons from the filled band to the conduction band and thereby create an additional conductivity—photoconductivity. The strong influence of temperature (the exponential growth of electrical conductivity) has already been mentioned. In connection with the small number of conduction electrons (in comparison with metals), all thermoelectric effects in semiconductors are considerably larger than in metals. This occurs because in metals the Fermi energy, i.e., the energy below which at absolute zero all states are occupied and above which they are all free, is of the order of several electron-volts, i.e., considerably greater than \(kT\). Therefore, for most metals, up to the melting temperature of the lattice, the electron distribution function, i.e., the distribution of electrons over energy levels, differs very little from the distribution function at absolute zero. In semiconductors, however, the distribution of electrons over the various levels in the conduction band depends very strongly on temperature.

As has already been indicated, impurities, by distorting the lattice, cause scattering of electron waves and thereby create an additional electrical resistance of the substance. In metals at low temperatures this is manifested in the form of the so-called residual resistance (the thermal motion of the atoms and the corresponding emu-

… the resistance is already very small). In semiconductors the role of impurities is not limited to their effect only on the scattering of electrons and thereby on the reduction of the mean free path.

Foreign atoms possess a system of terms (energy levels) different from the terms of the crystal. Since, however, impurity atoms are located at large distances from one another, these additional energy levels are localized. This means that electrons can be on an impurity level only at the place where the impurity atom corresponding to this level is located, and cannot move through the crystal.

An approximate picture of the levels, when foreign atoms are present in the lattice, is shown in Fig. 5. The atoms of which the lattice is built are denoted by the letter \(a\), the impurity atom by the letter \(b\). This figure is, of course, simplified, since owing to the perturbation introduced by atom \(b\), the system of terms in the atoms \(a\) immediately adjoining the \(b\)-atom is somewhat distorted. For understanding the essential point, this inaccuracy is immaterial.

Fig. 5

Impurity levels may fall in the region of the forbidden band between the completely filled and the completely empty bands. A certain number of electrons may be present on these levels. Naturally, it is easier for thermal motion to transfer electrons into the conduction band from this impurity level than from the filled band. Thus, being sources of conduction electrons, impurities exert a strong influence on the electrical conductivity of semiconductors. This model of an impurity semiconductor was first proposed and calculated by Wilson \(^{2}\). An elementary exposition of the theory is available in a number of review articles \(^{3}\).

Impurities may also play the role of levels of electron attachment. In this case they contribute to conductivity because electrons pass onto them from the filled band and thereby free in it a certain number of levels.

When the whole band is completely filled, it contains an equal number of electrons moving in opposite directions. Therefore the resultant current of all electrons of the filled band is equal to zero. If the band is partially filled, then in the absence of a field the current is likewise equal to zero. Under the influence of an external electric field the distribution of electrons becomes asymmetric. The greater part of the electrons moves in the direction opposite to the field (the electron has negative charge). The density of the electric current due to the electrons of one band is

\[ \mathbf{i}=-e\sum_i n_i\mathbf{v}_i . \tag{1,19} \]

Here \(n_i\) is the number of electrons in the \(i\)-th quantum state (according to the Pauli principle it is equal to zero or unity). The summation is carried out over all quantum states of the band.

Formula (1.19) is conveniently transformed, in the case of an almost filled band, as follows:

\[ \mathbf i=-e\sum_i \mathbf v_i+e\sum_i(1-n_i)\mathbf v_i =+e\sum_i(1-n_i)\mathbf v_i . \tag{1.20} \]

The first term \(e\sum_i \mathbf v_i\) gives the current of an entirely filled band; therefore it is equal to zero. Since the electrons move predominantly against the field, among the unfilled levels there are more of those which correspond to a velocity directed along the field.

Formula (1.20) may therefore be interpreted as follows: in the \(i\)-th quantum state there are \(1-n_i\) “holes,” moving with velocity \(\mathbf v_i\) (directed predominantly along the field) and with charge \(+e\).

Analogously to the electric current, one can transform the energy flux of the electrons of an almost filled band,

\[ \mathbf S=-\sum_i n_i(\varepsilon_i-e\varphi)\mathbf v_i, \tag{1.21} \]

where \(e\varphi\) is the potential energy of the electrons in the external field. Further,

\[ \mathbf S=-\sum_i(\varepsilon_i-e\varphi)\mathbf v_i -\sum_i(1-n_i)(\varepsilon_i-e\varphi)\mathbf v_i= \]

\[ =-\sum_i\varepsilon_i\mathbf v_i -e\varphi\sum_i\mathbf v_i -\sum_i(1-n_i)(\varepsilon_i-e\varphi)\mathbf v_i= \]

\[ =-\sum_i(1-n_i)(\varepsilon_i-e\varphi)\mathbf v_i . \tag{1.22} \]

For states situated close to the upper edge of an allowed band, according to formula (1.15) \(\varepsilon_i=\varepsilon_2-\varepsilon'\), where \(\varepsilon'=\dfrac{\hbar^2 k^2}{2m_+}\). Therefore

\[ \mathbf S=\sum_i(1-n_i)(\varepsilon'-\varepsilon_2+e\varphi)\mathbf v_i . \tag{1.23} \]

This formula can be interpreted analogously to formula (1.20). In doing so one must regard \(\varepsilon'\) as the “kinetic energy” of the holes, and \(-\varepsilon_2+e\varphi\) as their potential energy.

Thus, equations (1.15), (1.18), (1.20), and (1.23) show that the free levels in an almost filled band (“holes”) may be regarded as particles to which one should ascribe: positive charge \(+e\), effective mass \(m_+\), quasimomentum \(\hbar\mathbf k\), kinetic energy \(\varepsilon'=\dfrac{\hbar^2 k^2}{2m_+}\), and potential energy \(+e\varphi-\varepsilon_2\).

In semiconductors the number of electrons in the conduction band, or of holes in an almost filled band, is very small, so that the overwhelming majority of them lie in an energy region of order \(kT\) near the corresponding band edge.

Since \(kT\) is much smaller than the band width, we may always use formulas (1.11) and (1.15) for the energies of electrons and holes, for the fraction of those electrons or holes for which these formulas are invalid is negligibly small.

In the first part of this survey, in addition to the introduction and the thermal equilibrium of electrons in the lattice, the following will also be considered: electrical conductivity in a weak field, thermoelectric, thermomagnetic and galvanomagnetic effects in semiconductors with atomic and ionic lattices, as well as the electrical conductivity of semiconductors in strong fields. In the second part, diffusion phenomena and the rectification effects, photo-emf, and also contact resistances associated with them will be analyzed.

The sections of the first part devoted to semiconductors with an ionic lattice, as well as the entire second part, are a presentation of work carried out over the past several years.

2. THERMAL EQUILIBRIUM OF ELECTRONS IN THE LATTICE

The number of atoms in a unit volume of a solid is of the order of \(10^{23}\). The number of levels in each band is of the same order. Therefore, since the distance between neighboring levels is very small, it may be assumed that the levels continuously fill the entire band. To each triple of numbers \(z_1, z_2, z_3\) [see formula (1,8)] there corresponds a definite state of motion of the electron. Consequently, the number of states in an element of volume of \(z\)-space \(dz_1dz_2dz_3\) is

\[ d\Phi = 2dz_1dz_2dz_3. \tag{2,1} \]

The factor two in this formula is present because in each state of motion there may be two electrons having different spin orientations. From formula (1,8) it follows that

\[ dk_\alpha=\frac{2\pi}{G}\left(b_{1\alpha}dz_1+b_{2\alpha}dz_2+b_{3\alpha}dz_3\right). \tag{2,2} \]

Therefore

\[ dk_xdk_ydk_z=\left(\frac{2\pi}{G}\right)^3 Ddz_1dz_2dz_3, \tag{2,3} \]

where \(D\) is the Jacobian of the transformation from the variables \(z_1, z_2, z_3\) to the variables \(k_x, k_y, k_z\). It is easy to see that it is equal to \(\frac{1}{\Omega_0}\), where \(\Omega_0\) is the volume of the elementary cell of the crystal. Since \(\mathbf{p}=\hbar\mathbf{k}\), formulas (2,1) and (2,3) give

\[ d\Phi=\frac{2}{h^3}\Omega\,dp_xdp_ydp_z, \tag{2,4} \]

where \(h=2\pi\hbar\), and \(\Omega\) is the volume of the crystal.

The number of electrons in a unit volume and in the volume element of quasimomentum space \(dp_xdp_ydp_z\) is equal to the number of states \(d\Phi\), multiplied by the distribution function and divided by the volume \(\Omega\),

\[ dn=f(p_x,p_y,p_z)\frac{2}{h^3}\,dp_xdp_ydp_z. \tag{2,5} \]

As is known, electrons obey Fermi statistics. The Fermi distribution function is

\[ f=\frac{1}{e^{\frac{\varepsilon-\zeta}{kT}}+1}. \tag{2,6} \]

The constant \(\mu\), called the chemical potential (in the presence of an external field the chemical potential is \(\mu-u\), where \(u\) is the potential energy of an electron in the external field), is determined from the normalization condition:

\[ \frac{2}{h^3}\int_I \frac{dp_x dp_y dp_z}{e^{\frac{\varepsilon-\mu}{kT}}+1} + \frac{2}{h^3}\int_{II}\frac{dp_x dp_y dp_z}{e^{\frac{\varepsilon-\mu}{kT}}+1} + \sum_i \frac{1}{e^{\frac{\varepsilon_i-\mu}{kT}}+1} = n . \tag{2,7} \]

The first integral gives the number of electrons located in the conduction band (the upper, almost empty band), the second integral gives the number of electrons in the lower, almost filled band of levels. The sum \(\sum_i\) gives the number of electrons located at local levels of impurities; \(n\) is the total concentration of electrons at all these levels.

We shall now consider several simple cases.

1. There is a pure semiconductor without impurities. The band structure of the levels is shown in Fig. 4. The normalization condition (2,7) in this case reduces to the requirement that the number of electrons in the conduction band be equal to the number of holes in the almost filled band. The distribution function of holes is

\[ f' = 1-f=\frac{1}{e^{\frac{\mu-\varepsilon}{kT}}+1}. \tag{2,8} \]

If the energy is measured from the lower edge of the conduction band, then \(\varepsilon=\Delta E-\varepsilon'\). Therefore

\[ f'=\frac{1}{e^{\frac{\varepsilon'+\Delta E+\mu}{kT}}+1}. \tag{2,9} \]

Thus there is complete symmetry between electrons and holes. If one also introduces the chemical potential of holes

\[ \mu'=-\Delta E-\mu, \]

then the functions (2,6) and (2,9) will have the same form. It is more convenient, however, not to do this. Thus, we have:

\[ n_- = n_+, \tag{2,10} \]

or

\[ \int_I f\, dp_x dp_y dp_z = \int_{II} f'\, dp'_x dp'_y dp'_z . \tag{2,11} \]

\(n_-\) and \(n_+\) are the concentrations of electrons and holes, respectively.

Since the number of electrons and holes in the conduction band and of holes in the almost filled band is small, one may expect that for each of them separately a Maxwellian distribution holds. Therefore:

\[ -\frac{\mu}{kT}\gg 1 \quad \text{and} \quad \frac{\Delta E+\mu}{kT}\gg 1. \tag{2,12} \]

In this case the unity in the denominator of formulas (2,6) and (2,9) may

neglect, and we obtain:

\[ n_-=\frac{2}{h^3}e^{\frac{\mu}{kT}} \int\limits_{-\infty}^{+\infty}\!\!\int\!\!\int e^{-\frac{p_x^2+p_y^2+p_z^2}{2m_-kT}}\, dp_x\,dp_y\,dp_z = \frac{2}{h^3}(2\pi m_-kT)^{3/2}e^{\frac{\mu}{kT}}, \tag{2.13} \]

\[ \begin{aligned} n_+&=\frac{2}{h^3}e^{-\frac{\Delta E+\mu}{kT}} \int\limits_{-\infty}^{+\infty}\!\!\int\!\!\int e^{-\frac{p_x^{\prime 2}+p_y^{\prime 2}+p_z^{\prime 2}}{2m_+kT}}\, dp'_x\,dp'_y\,dp'_z \\ &=\frac{2}{h^3}(2\pi m_+kT)^{3/2}e^{-\frac{\Delta E+\mu}{kT}}. \end{aligned} \tag{2.14} \]

From condition (2.10) we find

\[ \mu=-\frac{\Delta E}{2}-\frac{1}{2}kT\lg\left(\frac{m_-}{m_+}\right)^{3/2}. \tag{2.15} \]

If \(kT\ll \Delta E\), then equation (2.15) means that the level of the chemical potential lies almost exactly in the middle of the forbidden band. This justifies assumption (2.12). The concentration of electrons and holes is

\[ n_-=n_+=\frac{2}{h^3}\left(2\pi\sqrt{m_-m_+}\,kT\right)^{3/2}e^{-\frac{\Delta E}{2kT}}. \tag{2.16} \]

In formulas (2.15) and (2.16), \(\Delta E\) is the width of the forbidden band.

1. In the forbidden band there are \(n_0\) local levels (because of the presence of impurities in the crystal), which at absolute zero are all occupied by electrons. We shall assume, first, that the energies of all these levels are the same and, second, that they are situated much closer to the conduction band than to the filled band, and that the forbidden band is so wide that the electrons of the filled band practically take no part in thermal equilibrium [the factor \(e^{-\frac{\Delta E+\mu}{kT}}\) occurring in the second integral of formula (2.7) is in this case extremely small]. This means that the concentration of holes may be neglected. The corresponding level scheme is given in Fig. 6. The dashed line in it gives the position of the local impurity levels.

Fig. 6

If we denote by \(\Delta E'\) the difference in energy between the lower edge of the conduction band and the impurity level, then from (2.7) we obtain the following equation for determining the chemical potential:

\[ n_0= \frac{n_0}{e^{\frac{-\Delta E'-\mu}{kT}}+1} + \frac{2}{h^3}\int \frac{d\tau_{\mathbf p}}{e^{\frac{\varepsilon-\mu}{kT}}+1}. \tag{2.17} \]

Here \(d\tau_{\mathbf p}=dp_xdp_ydp_z\). Since the electrons in the conduction band

\(^{1}\) Here and in what follows we restrict ourselves to the isotropic case, when the inverse-mass tensors of the electrons and holes reduce to scalars.

in this case is also small, then, obviously, \(-\dfrac{\mu}{kT}\gg 1\) (as we have already seen in the preceding example, this inequality is equivalent to the absence of degeneracy of the electron gas, in the present case the “gas” of conduction electrons). In such a case, from (2,17) we have

\[ n_0= \frac{n_0}{e^{-\frac{\Delta E'-\mu}{kT}}+1} + \frac{2}{h^3}(2\pi m_- kT)^{3/2}e^{\frac{\mu}{kT}}. \tag{2,18} \]

Hence

\[ \mu=-\Delta E' + kT\lg \frac{1}{2} \left[ \sqrt{ 1+\frac{2n_0h^3 e^{\frac{\Delta E'}{kT}}}{(2\pi m_- kT)^{3/2}} } -1 \right]. \tag{2,19} \]

This expression is not very transparent; let us therefore consider two limiting cases:

\[ \text{a) }\quad \frac{2n_0h^3 e^{\frac{\Delta E'}{kT}}}{(2\pi m_- kT)^{3/2}}\gg 1, \]

then

\[ \mu=-\frac{\Delta E'}{2} +kT\lg \frac{n_0^{1/2}h^{3/2}}{2^{1/2}(2\pi m_- kT)^{3/4}}. \tag{2,20} \]

The concentration of conduction electrons is

\[ n= \frac{2^{1/2}n_0^{1/2}(2\pi m_- kT)^{3/4}}{h^{3/2}} e^{-\frac{\Delta E'}{2kT}}. \tag{2,21} \]

The distribution function of electrons in the conduction band is

\[ f= \frac{n_0^{1/2}h^{3/4}}{2^{1/2}(2\pi m_- kT)^{3/4}} e^{-\frac{\frac{\Delta E'}{2}+\varepsilon}{kT}}. \tag{2,22} \]

The number of impurity levels per unit volume is approximately the same as the number of impurity atoms. When the concentration of the latter is \(\simeq 10^{17}\) per \(1\ \mathrm{cm}^3\) and at \(T\simeq 300^\circ\mathrm{K}\), the quantity

\[ \frac{n_0h^3}{(2\pi m_- kT)^{3/2}}\simeq 10^{-2}, \]

and therefore the logarithmic term in equation (2,20) is very small. Consequently, the level of the chemical potential passes halfway between the level of the impurities serving as sources of electrons and the lower edge of the conduction band.

\[ \text{b) }\quad \frac{n_0h^3 e^{\frac{\Delta E'}{kT}}}{(2\pi m_- kT)^{3/2}}\ll 1. \]

In this case

\[ \mu=kT\lg\frac{n_0h^3}{2(2\pi m_- kT)^{3/2}}. \tag{2,23} \]

The concentration in the conduction band is \(n=n_0\). Physically this case means that the temperature is so high that all electrons from impurity levels pass into the conduction band. Further increase

the number of electrons ceases to increase with temperature; saturation sets in. More often, apparently, case “a” is realized, when saturation is still far off.

Everything that was said in this example about electrons can be repeated literally for holes if, instead of levels that are sources of electrons (“donors”), there are capture levels (“acceptors”) situated close to the upper edge of the filled band.

1. If impurity levels fall in the conduction band or above it, then in it (in this band) there will always be a definite number of electrons independent of temperature. In this case one obtains what is called an “impurity metal.” If in this case the concentration of “impurity” electrons is so small that there is no degeneracy, then their distribution function has the form:

\[ f=(2\pi m_- kT)^{-3/2} n_0 e^{\frac{\varepsilon}{kT}}, \tag{2,24} \]

where \(n_0\) is the concentration of electrons in the conduction band (independent of temperature).

1. Naiburg\(^4\) has recently considered the following case: in a unit volume, in the forbidden band, there are \(n_0\) impurity levels situated at the same height, near the lower edge of the conduction band. Of these levels only a small fraction \(\nu\) \((\nu \ll n_0)\) is filled with electrons at absolute zero. The normalization condition (2,7) in this case gives

\[ \nu= \frac{n_0}{e^{\frac{\Delta E+\mu}{kT}}+1} + \frac{2}{h^3}\int \frac{d\mathbf{p}}{e^{\frac{\varepsilon-\mu}{kT}}+1}. \tag{2,25} \]

Assuming again that \(-\dfrac{\mu}{kT}\gg 1\), we obtain for \(\mu\) a quadratic equation. Its solution is:

\[ \mu=-\Delta E' + kT\lg \frac{1}{2} \left\{ \sqrt{ \left[ 1+\frac{(n_0-\nu)h^3 e^{\frac{\Delta E}{kT}}} {2(2\pi m_- kT)^{3/2}} \right]^2 + \frac{2\nu h^3 e^{\frac{\Delta E}{kT}}} {(2\pi m_- kT)^{3/2}} } - \left[ 1+\frac{(n_0-\nu)h^3 e^{\frac{\Delta E}{kT}}} {2(2\pi m_- kT)^{3/2}} \right] \right\}. \tag{2,26} \]

In the range of temperatures far from saturation,

\[ \frac{\nu h^3 e^{\frac{\Delta E'}{kT}}}{(2\pi m_- kT)^{3/2}}\gg 1 \tag{2,27} \]

and

\[ \mu=-\Delta E+kT\lg\frac{n_0+\nu}{2(n_0-\nu)}, \tag{2,28} \]

i.e. the level of the chemical potential passes close to the impurity level. This is quite understandable, since for the electrons there are many

of free states at the energetically lowest level—the impurity level. The number of conduction electrons is

\[ n=\frac{n_0+\nu}{n_0-\nu}\frac{(2\pi m_0 kT)^{3/2}}{h^3}\,e^{-\frac{\Delta E}{kT}}. \tag{2,29} \]

We have examined several of the simplest cases, which, of course, do not exhaust all possibilities. It hardly makes sense to consider more complicated schemes here without special need, especially since the analysis of these schemes presents no difficulties.

Qualitatively one may say that, if there are several levels in the forbidden band, then at low temperatures the role is played by those levels which are situated closest to the allowed bands (for the formation of holes it is important that there be trapping levels situated close to the filled band). When, as the temperature is raised, \(kT\) becomes of the order of \(\Delta E_i\), where \(\Delta E_i\) is the distance from the nearest impurity level to the edge of the allowed band, saturation will occur, associated with the transition of almost all electrons from the first impurity level into the allowed band. With a further increase in temperature, electrons from lower levels will come into play. At very high temperature it is already necessary to take into account transitions from the filled band into the conduction band. Incidentally, if, in addition to levels supplying their electrons—sources—there are also trapping levels situated close to the lower band, then transitions from this band to the trapping levels will also begin at a lower temperature.

In one way or another it is easy to indicate simultaneously the temperature dependence of the concentrations of electrons and holes, if it is assumed that the level of the chemical potential passes sufficiently far both from the lower edge of the conduction band and from the upper edge of the almost filled band (which is almost always the case).

If, by \(\Delta E\), as at the beginning of this paragraph, we denote the width of the forbidden band, then according to equations (2,13) and (2,14) the concentrations of electrons and holes will be the following:

\[ n_-=\frac{2}{h^3}(2\pi m_- kT)^{3/2} e^{-\frac{|\mu|}{kT}}, \tag{2,30} \]

\[ n_+=\frac{2}{h^3}(2\pi m_+ kT)^{3/2} e^{-\frac{\Delta E-|\mu|}{kT}}. \tag{2,31} \]

For complete knowledge of the dependence of \(n\) on \(T\) it is still necessary to know \(\mu\) as a function of \(T\). In many cases, however, the relative change of the chemical potential with temperature is very small. Therefore equations (2,30) and (2,31) can give the correct dependence of \(n\) on \(T\) over fairly broad temperature ranges, if \(\mu\) is approximately regarded as independent of temperature.

If, owing to a large number of impurities, the concentration of conduction electrons or holes becomes of the order of \(10^{19}\)—\(10^{20}\ \mathrm{cm}^{-3}\), then degeneracy will set in and all the relations will have a more complicated character.

3. ELECTRICAL CONDUCTIVITY IN A WEAK FIELD, THERMOELECTRIC, GALVANOMAGNETIC, AND THERMOMAGNETIC EFFECTS ^{5,6}

In considering the kinetics of electronic processes, the main role is played by the concept of the mean free time of electrons \(\tau\): this is the time during which the electron on the average loses the component of its quasimomentum in a given direction. More precisely, the change of the component of the quasimomentum in a given direction per unit time, due to scattering,

\[ -\frac{dp_x}{dt}=\frac{1}{\tau}p_x . \tag{3,1} \]

\(\tau\) may depend on the velocity. The mean free path is \(l=\tau v\).

The study of semiconductors is usually carried out at such high temperatures that the scattering of electrons occurs mainly because of the thermal vibrations of atoms, and not because of the presence of impurities. Our task therefore is to determine the probability of scattering of an electron caused by the vibrations of the lattice atoms.

In the first approximation the forces returning the atoms to their equilibrium positions are quasielastic. The vibrations therefore have a harmonic character. If the crystal contains \(N\) elementary cells and in each cell \(s\) atoms, then the number of degrees of freedom of the whole system is \(3Ns\). There will be the same number of independent normal vibrations with different frequencies \(\omega\). A normal vibration of the crystal corresponding to frequency \(\omega\) has the form

\[ \mathbf{u}^{\,l}_{k}=\mathbf{e}_k\cos(\mathbf{q}\mathbf{r}^{\,l}_{k}-\omega t+\beta), \tag{3,2} \]

i.e. it corresponds to a wave running through the entire crystal. In this formula \(\mathbf{r}^{\,l}_{k}\) is the radius vector of the atom of the \(k\)-th kind in the \(l\)-th elementary cell, \(\mathbf{u}^{\,l}_{k}\) is the displacement of this atom from its equilibrium position, \(\mathbf{q}\) is the wave vector of the elastic wave \(\left(\mathbf{q}=\frac{2\pi}{\lambda}\mathbf{n},\ \lambda\text{ is the wavelength, }\mathbf{n}\text{ is the unit vector of propagation of the wave}\right)\), and \(\mathbf{e}_k\) is the polarization vector.

The spectrum of elastic vibrations consists of \(3s\) branches. To each wavelength and direction of its propagation there correspond \(3s\) vibrations differing in their frequency and polarization. Of these \(3s\) branches only the first three (corresponding to the three polarizations) are acoustic. The vibration frequency of infinitely long waves of acoustic vibrations is zero, and therefore for long waves the frequency \(\omega\) is proportional to the wave vector:

\[ \omega=\mathcal{w}q, \tag{3,3} \]

where \(\mathcal{w}\) is the velocity of sound. All the remaining vibrations corresponding to the given wave vector have higher frequencies. They are called optical vibrations. Long-wave optical vibrations have frequencies which do not vanish at \(q=0\),

\(\omega_j^0\) \((j=4,5,\ldots,3s)\). These limiting frequencies, in order of magnitude, are close to the Debye frequency. Experiments with residual rays show that the quantity \(\theta = \dfrac{\hbar \omega_0}{k}\) in various substances ranges from 100 to 1500° K.

In a diatomic lattice (there are two atoms in one elementary cell), for acoustic long-wavelength vibrations neighboring atoms oscillate approximately in the same phase, whereas in optical vibrations with a large wavelength the phases of neighboring atoms or ions are shifted by approximately half a period. In this case the long-wavelength optical vibrations may be pictured as vibrations of a lattice composed of atoms of only one kind (located at the corresponding sites) with respect to the entire aggregate of atoms of the other kind. As the wavelength decreases, the phase difference of the vibrations of neighboring atoms decreases. Therefore, as \(q\) increases, the frequency of the optical vibrations decreases (since the force returning the atoms to the equilibrium position decreases). The characteristic form of the dependence of the frequency \(\omega\) on the wave vector \(q\) is shown in Fig. 7. The lower curve corresponds to the acoustic branch, the upper to the optical branch. In a cubic lattice and for small \(q\), this dependence (for the optical branch) can be determined by expanding \(\omega\) in a series in the components of the wave vector \(q_x, q_y, q_z\). Namely:

Fig. 7

\[ \omega = \omega_0 + \sum_i a_i q_i + \sum_{ik} a_{ik} q_i q_k . \tag{3,4} \]

By virtue of the cubic symmetry of the lattice, the material vector with components \(a_i\) is equal to zero, while the second-rank tensor \(a_{ik}\) reduces to a scalar. Therefore (3,4) may be rewritten in the following form:

\[ \omega = \omega_0 (1 - \alpha a^2 q^2), \tag{3,5} \]

where \(a\) is the distance between neighboring atoms, and \(\alpha\) is a dimensionless coefficient which, in order of magnitude, is equal to \(\dfrac{\Delta \omega}{\omega_0}\); \(\Delta \omega\) is the width of the optical branch of vibrations.

An arbitrary displacement of any particle of the lattice can be represented as a superposition of all the normal vibrations with different amplitudes:

\[ u_k^l = \frac{1}{\sqrt{N}} \sum_{q,j} \mathbf{e}_{k,qj} \left( a_{qj} e^{i\mathbf{q}\mathbf{r}_k^l} + a_{qj}^{*} e^{-i\mathbf{q}\mathbf{r}_k^l} \right). \tag{3,6} \]

For convenience we have passed to the complex form of notation; for the same reasons the normalizing factor \(\dfrac{1}{\sqrt{N}}\) has been introduced. The time-dependent multipliers—

multipliers \(e^{-i\omega t}\) and \(e^{i\omega t}\) are included in the amplitudes \(a_{qj}\) and \(a^{*}_{qj}\), respectively.

In this case the polarization vectors satisfy the orthogonality relation

\[ \sum_k m_k e_{k,j} e_{k,j'} = 0 \quad \text{for } j \ne j'. \tag{3,7} \]

They can also be normalized by setting

\[ \sum_k m_k e^2_{k,j} = 1. \tag{3,8} \]

The total energy of the crystal vibrations, i.e., the sum of the potential and kinetic energies, is expressed in the following way through the amplitudes \(a_{qj}\) and \(a^{*}_{qj}\):

\[ H = 2 \sum_{q,j} \omega^2_{qj} a_{qj} a^{*}_{qj}. \tag{3,9} \]

Putting

\[ a_{qj} = \frac{1}{2}\left(x_{qj} + i \frac{p_{qj}}{\omega_{qj}}\right), \]

we obtain

\[ H = \sum_{q,j} H_{qj}, \tag{3,10} \]

where

\[ H_{qj} = \frac{1}{2}\left(p^2_{qj} + \omega^2_{qj} x^2_{qj}\right). \tag{3,11} \]

\(H_{qj}\) has the form of the Hamiltonian function of an oscillator with natural frequency \(\omega_{qj}\) and mass 1. In passing to a quantum-mechanical description, \(x_{qj}\) and \(p_{qj}\) are treated as the coordinate and momentum operators of the corresponding “elastic oscillator.” As is known from quantum mechanics, the energy of such an oscillator is

\[ E_{qj} = \hbar \omega_{qj}\left(N_{qj} + \frac{1}{2}\right), \tag{3,12} \]

where \(N_{qj}\) is the vibrational quantum number. If the oscillator \(qj\) is in the \(N\)-th quantum state, then it is customary to say that there are \(N\) phonons of type \(qj\). To each phonon, in addition to the energy \(\hbar \omega_{qj}\), one may also assign the momentum \(\hbar q\). Interacting with elastic vibrations, an electron may absorb or emit a phonon. The probability of such a transition is determined according to perturbation theory and is equal to

\[ W = \frac{2\pi}{\hbar} |u|^2 \delta\left(\varepsilon_{\mathbf{p}'} - \varepsilon_{\mathbf{p}} - \hbar \omega_{qj}\right), \tag{3,13} \]

where \(|u|\) is the matrix element of the energy of interaction of the electron with the elastic vibrations. It is nonzero only in the case when

\[ \mathbf{p}' = \mathbf{p} \pm \hbar \mathbf{q}; \tag{3,14} \]

the function \(\hbar c\,\delta\) has a very sharp maximum when its argument is equal to zero, and away from this point is almost equal to zero. Therefore

\[ \varepsilon_{\mathbf{p}'} = \varepsilon_{\mathbf{p}} \pm \hbar \omega_{qj}. \tag{3,15} \]

Thus, in the processes of collision of electrons with phonons, the conservation laws of quasimomentum (3.14) and energy (3.15) hold. The upper signs before \(\hbar\omega_q\) and \(\hbar\mathbf q\) in formulas (3.13), (3.14), and (3.15) refer to absorption of a phonon, the lower signs to emission.

In accordance with what has been set forth above, we shall consider separately semiconductors with a simple atomic lattice (one atom per elementary cell), possessing only acoustic vibrations, and semiconductors with an ionic lattice.

A. Semiconductors with an Atomic Lattice

From the conservation laws (3.14) and (3.15), with the aid of the relations \(\varepsilon=\dfrac{p^2}{2m}\) and \(\omega=wq\), we obtain:

\[ \hbar q=\pm 2mw \mp 2p\cos\vartheta . \tag{3.16} \]

The upper signs refer to absorption of a phonon, the lower to emission. \(\vartheta\) is the angle between the vectors \(\mathbf p\) and \(\mathbf q\). Since the velocity of sound \(w\) is much smaller than the mean velocity of the electrons \(v=\dfrac{p}{m}\), the first term on the right-hand side of (3.16) may be neglected for the overwhelming majority of electrons. In this case, in order of magnitude,

\[ \hbar\omega=\hbar wq \approx mvw . \tag{3.17} \]

Indeed, the mean energy of the electrons in the conduction band is \(\bar{\varepsilon}\approx kT\), and, consequently, the mean quasimomentum \(\bar p\approx \sqrt{mkT}\). Therefore

\[ \frac{\bar p}{mw}\approx \frac{\sqrt{mkT}}{mw} =\sqrt{\frac{kT}{mw^2}} . \tag{3.18} \]

If the velocity of sound is taken to be \(2\cdot 10^5\ \text{cm/sec}\), then at room temperatures

\[ \frac{kT}{mw^2}\approx 0.001 . \]

From equation (3.16) we also conclude that, in the interaction with electrons, the chief role is played by those phonons whose momentum \(\simeq \sqrt{mkT}\). The wavelength of an elastic vibration corresponding to this momentum is

\[ \lambda=\frac{2\pi}{q}\approx \frac{2\pi\hbar}{\sqrt{mkT}} \tag{3.19} \]

and is much larger than the lattice constant. But for such long waves one can not only estimate the order of magnitude of the matrix element \(|u|\) entering into the transition probability (3.13), but also determine its dependence on \(q\). Without giving the calculation, we state the result at once: the transition probability turns out to be proportional to the wave vector \(q\), and for emission of a phonon by an electron it is equal to

\[ W_-=\frac{1}{N}W_0q\,(N_q+1)\,\delta(\varepsilon'-\varepsilon+\hbar\omega_q). \tag{3.20} \]

For absorption of a phonon

\[ W_{-}=\frac{1}{N}\,W_0 q N_q\,\delta(\varepsilon'-\varepsilon-\hbar\omega_q). \tag{3,21} \]

In order of magnitude

\[ W_0 \simeq \frac{V_a^2}{M\omega}, \tag{3,22} \]

where \(V_a\) is a quantity of the order of the atomic energy, \(M\) is the mass of an atom, and \(N\) in the denominator of formulas (3,20) and (3,21) is the number of atoms in the crystal (not to be confused with \(N_q\), the number of phonons). Thus, the probability of phonon absorption is proportional to the number of available phonons of the given type, while the probability of emission is composed of the probability of spontaneous emission and the probability of induced emission, the latter likewise being proportional to \(N_q\).

Since we regard the lattice as being in a state of thermal equilibrium, \(N_q\) is the Planck function

\[ N_q=\frac{1}{e^{\frac{\hbar\omega}{kT}}-1}. \tag{3,23} \]

We can now proceed to the kinetic equation for electrons in the conduction band. It has the form:

\[ \frac{\partial f}{\partial t}+\mathbf v\cdot\nabla f+ \left(e\nabla\varphi-\frac{e}{c}[\mathbf v,\mathbf H]\right)\nabla_p f =(b-a)f. \tag{3,24} \]

Here \(\varphi\) is the electric potential of the external field, \(\mathbf H\) is the magnetic field, \(\nabla\) denotes the gradient in ordinary space, and \(\nabla_p\) denotes the gradient in quasimomentum space. The terms \(\mathbf v\cdot\nabla f+\left(e\nabla\varphi-\frac{e}{c}[\mathbf v,\mathbf H]\right)\nabla_p f\) give the decrease in the number of electrons in a unit volume of phase space under the influence of the external field and of the concentration gradient. \((b-a)f\) denotes the excess of the number of electrons entering, as a result of collisions with phonons, a unit volume of phase space over the electrons leaving this volume for the same reasons.

By analogy, the number of vibrations in the element of volume \(dq_x dq_y dq_z\) is equal to [cf. formula (2,4)]

\[ \frac{\Omega}{(2\pi)^3}\,dq_x\,dq_y\,dq_z. \]

Therefore

\[ \begin{aligned} (b-a)f={}&\frac{\Omega}{N(2\pi)^3} \Bigg\{ \int W_0 q\,[f(\mathbf p+\hbar\mathbf q)(N_q+1)-f(\mathbf p)N_q] \\ &\qquad\qquad\times \delta(\varepsilon'-\varepsilon-\hbar\omega_q)\,dq_x\,dq_y\,dq_z \\ &\quad+ \int W_0 q\,[f(\mathbf p-\hbar\mathbf q)N_q-f(\mathbf p)(N_q+1)] \,\delta(\varepsilon-\varepsilon'-\hbar\omega_q)\,dq_x\,dq_y\,dq_z \Bigg\}. \end{aligned} \tag{3,25} \]

In writing this equation we have assumed that the concentration of electrons in the conduction band is so small that one may pre-

neglect the Pauli principle. Therefore it is easy to see that, in the stationary case \(\left(\dfrac{\partial f}{\partial t}=0\right)\) and in the absence of an external perturbation [the left-hand side of equation (3.24) is zero], the Maxwellian function

\[ f_0=e^{\frac{\mu-\varepsilon}{kT}} \]

reduces \((b-a)f\) to zero. In this paragraph we shall assume that the external electric field, as well as the concentration gradient, are so small that the distribution function differs little from the equilibrium one. Then one may put

\[ f=f_0+\chi, \]

where \(\chi\) is small in comparison with \(f_0\).

Since the terms \(\mathbf v\cdot\nabla f+e\mathbf E_0\cdot\nabla_p f\) on the left-hand side of equation (3.24) do not vanish when \(f_0\) is substituted for \(f\), in them we may neglect the correction function \(\chi\). Conversely, on the right-hand side of equation (3.24), and in that term of the left-hand side which contains the magnetic field \(\mathbf H\), we must retain the function \(\chi\), since \((b-a)f_0=0\) and \([\mathbf v,\mathbf H]\nabla_p f_0=0\). Thus the magnetic field causes a change in the distribution function only insofar as the latter differs from the equilibrium distribution. Therefore, in what follows, it will be possible to obtain results valid not only for a weak magnetic field, but also for a strong one.

Thus, in the stationary case:

\[ \mathbf v\cdot\nabla f_0+e\mathbf E_0\cdot\nabla_p f_0-\frac{e}{c}[\mathbf v,\mathbf H]\nabla_p\chi=(b-a)\chi. \tag{3.26} \]

For the function \(\chi\) we shall make the assumption that it has the form

\[ \chi=\frac{df_0}{d\varepsilon}\,\mathbf p\cdot\mathbf j_-, \tag{3.27} \]

where \(\mathbf j_-\) is a vector depending only on the electron energy, but not on the direction of its quasimomentum. Passing to spherical coordinates in \(q\)-space, with the \(Z\)-axis coinciding with the direction of the vector \(\mathbf p\), we have:

\[ \begin{aligned} (b-a)\chi &=\frac{\Omega_0 W_0}{(2\pi)^3}\frac{df_0}{d\varepsilon} \Bigg\{ \int \big[ \mathbf j_-(\varepsilon+\hbar\omega)(p_j+\hbar q_j) -\mathbf j_-(\varepsilon)p_j \big]N_q \times \\ &\qquad\qquad\times \delta\left( \frac{\hbar^2q^2}{2m_-} +\frac{\hbar pq}{m_-}\cos\vartheta -\hbar\omega q \right) q^3\,dq\,\sin\vartheta\,d\vartheta\,d\varphi \\ &\qquad+ \int \big[ \mathbf j_-(\varepsilon-\hbar\omega)(p_j-\hbar q_j) -\mathbf j(\varepsilon)\cdot p_j \big](N_q+1)\times \\ &\qquad\qquad\times \delta\left( -\frac{\hbar^2q^2}{2m_-} +\frac{\hbar pq}{m_-}\cos\vartheta -\hbar\omega q \right) q^3\,dq\,\sin\vartheta\,d\vartheta\,d\varphi \Bigg\}, \tag{3.28} \end{aligned} \]

where, as before, \(\Omega_0\) is the volume of an elementary cell of the crystal. The integration over the angles is performed easily. Indeed:

\[ \int \delta\left( \pm\frac{\hbar^2q^2}{2m_-} +\frac{\hbar pq}{m_-}\cos\vartheta -\hbar\omega q \right) \sin\vartheta\,d\vartheta\,d\varphi = \frac{2\pi m_-}{\hbar pq}. \tag{3.29} \]

Further,

\[ q_j=q(\cos\vartheta \cos\vartheta' + \sin\vartheta \sin\vartheta' \cos\varphi), \]

where \(\vartheta'\) is the angle between the vectors \(\mathbf p\) and \(\mathbf j\). Since \(\cos\varphi=0\), we have

\[ \begin{aligned} \int q_j \partial \left(\pm \frac{\hbar^2 q^2}{2m}+\frac{\hbar p q}{m}\cos\vartheta-\hbar\omega q\right) \sin\vartheta\,d\vartheta\,d\varphi &= q\cos\vartheta'\,\frac{2\pi m}{\hbar p q}\cdot \frac{\pm \hbar q+m\omega}{2p} \\ &= q\,\frac{p_j}{p}\,\frac{2\pi m}{\hbar p q}\cdot \frac{\pm \hbar q+m\omega}{2p}. \end{aligned} \tag{3.30} \]

All this is substituted into (3.28). In this case the limits of integration are determined from equation (3.16)

\[ q_{\min}=0,\qquad q_{\max}=\frac{2p\pm 2m w}{\hbar}\simeq \frac{2p}{\hbar}. \]

Thus,

\[ (b-a)\chi= \frac{\Omega_0 W_0}{(2\pi)^3}\cdot \frac{df_0}{d\varepsilon}\cdot \frac{2\pi m}{\hbar p}\times \]

\[ \times p_j\left\{ \int\limits_0^{\frac{2p}{\hbar}} \left[ j(\varepsilon+\hbar\omega) \left(1-\frac{\hbar q}{p}\cdot\frac{\hbar q-mw}{2p}\right) -j(\varepsilon) \right] \frac{q^2\,dq}{e^{\frac{\hbar\omega q}{kT}}-1} +\right. \]

\[ \left. +\int\limits_0^{\frac{2p}{\hbar}} \left[ j(\varepsilon-\hbar\omega) \left(1-\frac{\hbar q}{p}\cdot\frac{\hbar q+mw}{2p}\right) -j(\varepsilon) \right] \left(\frac{1}{e^{\frac{\hbar\omega q}{kT}}-1}+1\right) q^2\,dq \right\}. \tag{3.31} \]

In the integrand one may neglect \(mw\) in comparison with \(\hbar q\), for the latter is \(\simeq \sqrt{mkT}\). Further, since \(\hbar\omega \simeq w\sqrt{mkT}\) [see (3.17)], one may, first, neglect \(\hbar\omega\) in comparison with \(\varepsilon\) in the argument of the functions \(j(\varepsilon+\hbar\omega)\) and \(j(\varepsilon-\hbar\omega)\), and, secondly, expand the Planck function in powers of \(\dfrac{\hbar\omega q}{kT}\), retaining the first nonzero term.

After these simplifications the integration is carried out elementarily and gives:

\[ (b-a)\left(\frac{df_0}{d\varepsilon}\mathbf p\cdot \mathbf j\right) = -\frac{\Omega_0 W_0 m kT}{7\pi^2 w\hbar^4}\, p\,\frac{df_0}{d\varepsilon} = -\frac{1}{\tau}\frac{df_0}{d\varepsilon}\mathbf p\cdot \mathbf j. \tag{3.32} \]

It is easy to show that \(\tau\) coincides with the mean free time defined at the beginning of this paragraph. What is essential for what follows is that it is inversely proportional to the temperature and to the electron velocity. The mean free path therefore does not depend on the energy.

\[ l=\tau v= \frac{7\pi w\hbar^4}{\Omega_0 W_0 m^2 kT} \simeq \pi^2\frac{M w^2}{V_a}\cdot \frac{\frac{\hbar^2}{m a^2}}{V_a}\cdot \frac{\frac{\hbar^2}{m a^2}}{kT}\cdot a. \tag{3.33} \]

\[ \frac{M w^2}{V_a} \quad\text{and}\quad \frac{\frac{\hbar^2}{m a^2}}{V_a} \]

are of order unity, while

\[ \frac{\frac{\hbar^2}{m a^2}}{kT}\gg 1 \]

(\(\frac{\hbar^2}{m a^2}\) is of the order of atom-

energy); therefore the mean free path is much greater than the lattice constant.

The function \(f_0=e^{\frac{\mu-\varepsilon}{kT}}\), consequently,

\[ \mathbf v\cdot \nabla f_0+e\nabla\varphi\cdot \nabla_p f_0 = -\frac{\mathbf p}{m_-} \left[ -\frac{\mu-\varepsilon}{T}\nabla T+\nabla(\mu-e\varphi) \right]\frac{df_0}{d\varepsilon}, \tag{3,34} \]

\[ [\mathbf v,\mathbf H]\nabla_p\chi = \frac{\mathbf p}{m_-}\frac{df_0}{d\varepsilon}[\mathbf H,\mathbf j_-]. \tag{3,35} \]

Upon substituting (3,32), (3,33), (3,34), and (3,35) into equation (3,27), the vector \(\mathbf p\) cancels. This justifies the assumption (3,27) about the form of the function \(\chi\). The equation remaining after cancellation of \(\mathbf p\), for determining the vector \(\mathbf j\), has the form:

\[ \frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi) +\frac{e}{c}[\mathbf H,\mathbf j_-] = \frac{p}{l_-}\mathbf j_- . \tag{3,36} \]

Obviously, there will be an analogous equation for holes in an almost filled band. The only difference is that the sign of the charge of holes is different. Just as with the mass, we shall mark the mean free path of holes and the vector \(\mathbf j\) with the sign \(+\), in contrast to electrons, for which these same quantities are marked with the sign \(-\). Recalling that the chemical potential of holes in the absence of an external field is equal to \(-\Delta E-\mu\), we have

\[ \frac{\varepsilon'+(\Delta E+\mu)}{T}\nabla T+\nabla(-\mu+e\varphi) -\frac{e}{c}[\mathbf H,\mathbf j_+] = \frac{p'}{l_+}\mathbf j_+ . \tag{3,37} \]

With the aid of equations (3,36) and (3,37) one can calculate all the effects of interest to us.

a. Electrical conductivity

The magnetic field \(\mathbf H=0\), and the temperature and concentration are the same everywhere. The semiconductor is in a uniform electric field \(\mathbf E=-\nabla\varphi\), whose direction is taken as the \(X\)-axis. With the aid of equations (3,36) and (3,37) we find:

\[ f=f_0+\frac{df_0}{d\varepsilon}\,eEl_-\,\frac{p_x}{p}, \tag{3,38} \]

\[ f'=f_0'-\frac{df_0'}{d\varepsilon'}\,eEl_+\,\frac{p_x'}{p'}. \tag{3,39} \]

The density of the electric current is

\[ \mathbf i = -e\int f\,\frac{\mathbf p}{m_-}\,\frac{2}{h^3}\,d\tau_p + e\int f'\,\frac{\mathbf p'}{m_+}\,\frac{2}{h^3}\,d\tau_{p'} = \]

\[ = \frac{4\pi}{3}e^2\mathbf E \left\{ -\frac{l_-}{m_-}\frac{2}{h^3}\int_0^\infty \frac{df_0}{d\varepsilon}p^3\,dp - \frac{l_+}{m_+}\frac{2}{h^3}\int_0^\infty \frac{df_0'}{d\varepsilon'}p'^3\,dp' \right\}. \tag{3,40} \]

Taking into account that

\[ \frac{2}{h^3}\int_0^\infty f_0\cdot 4\pi p^2\,dp=n_- \quad\text{and}\quad \frac{2}{h^3}\int_0^\infty f'_0\cdot 4\pi p'^2\,dp'=n_+, \]

we obtain

\[ \mathbf{i}=(n_-u_-+n_+u_+)e\mathbf{E}; \tag{3.41} \]

hence the electrical conductivity is

\[ \sigma=e(n_-u_-+n_+u_+). \tag{3.41'} \]

The mobility of electrons and holes is

\[ u_-=\frac{4el}{3\sqrt{2m_-kT}} \quad\text{and}\quad u_+=\frac{4el}{3\sqrt{2m_+kT}}. \tag{3.42} \]

If the mean free path of electrons and holes is taken to be proportional to the temperature, then the mobility \(u\simeq \frac{1}{T^{3/2}}\). Therefore the main dependence of the electrical conductivity on temperature is contained in the factors \(n_-\) and \(n_+\) (provided the given semiconductor is not an impurity metal, and provided we are in a region of temperatures far from saturation). As was established in the second section, the concentrations of electrons and holes increase exponentially with temperature; namely,

\[ n_-\simeq e^{-\frac{|\mu_-|}{kT}} \quad\text{and}\quad n_+\simeq e^{-\frac{\Delta F-|\mu_-|}{kT}}. \]

Since usually the concentration of charge carriers of one sign is much greater than the concentration of carriers of the other sign, the electrical conductivity also increases with temperature according to the same exponential law. If \(1/T\) is plotted along the abscissa axis, and \(\lg\sigma\) along the ordinate axis, then the graph of the dependence of the electrical conductivity on temperature will be a straight line. The tangent of the angle of inclination of this straight line to the \(X\)-axis (multiplied by \(k\)) is equal to the chemical potential of the carriers that determine the conductivity of the semiconductor (electrons or holes). At low temperatures this chemical potential is approximately equal to one half of the distance from the corresponding allowed band to the impurity level (let us assume, for simplicity, that in the forbidden band all local levels lie at the same height). At high temperatures it is equal to one half of the width of the entire forbidden band. Therefore the graph of the dependence of \(\lg\sigma\) on \(1/T\) has the characteristic form of curve \(II\) in Fig. 8. Roughly speaking, this curve consists of two straight lines passing smoothly one into the other.

Fig. 8

The straight line \(I\), which is an asymptote for curve \(II\) at high temperatures, gives the dependence of \(\lg\sigma\) on \(1/T\) for pure semiconductors devoid of impurities. Similar curves were obtained in experiments

Juse and Kurchatov,^9 who studied the influence of impurities and temperature on the electrical conductivity of cuprous oxide \((\mathrm{Cu}_2\mathrm{O})\). According to their measurements, the width of the forbidden band in this semiconductor is \(1.44\ \mathrm{eV}\).

6. Thermoelectromotive force

We still take the magnetic field to be equal to zero. In that case

\[ \mathbf{j}_{-}=-\frac{l_{-}}{\rho}\left[\frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right], \tag{3,43} \]

\[ \mathbf{j}_{+}=-\frac{l_{+}}{\rho}\left[\frac{\varepsilon'+\Delta E+\mu}{T}\nabla T-\nabla(\mu-e\varphi)\right]. \tag{3,44} \]

The current density is

\[ \mathbf{i}=-\frac{e l_{+}}{3m_{+}}\int_{0}^{\infty}p'\left[\frac{\varepsilon'+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right]\cdot \frac{2}{h^{3}}4\pi p'^{\,2}\,dp' \]
\[ -\frac{e l_{-}}{3m_{-}}\int_{0}^{\infty}p\left[\frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right]\cdot \frac{2}{h^{3}}4\pi p^{2}\,dp . \tag{3,45} \]

Integration gives:

\[ \mathbf{i}=n_{+}u_{+}\left[\nabla(\mu-e\varphi)-\frac{(\Delta E+\mu)+2kT}{T}\nabla T\right]+ \]
\[ +n_{-}u_{-}\left[\nabla(\mu-e\varphi)-\frac{\mu-2kT}{T}\nabla T\right]. \tag{3,46} \]

The thermoelectromotive force is measured at zero current. Under this condition we find:

\[ \nabla\left(\varphi-\frac{\mu}{e}\right) = \frac{1}{\sigma} \left\{ n_{-}u_{-}\frac{2kT-\mu}{T} - n_{+}u_{+}\frac{2kT+(\Delta E+\mu)}{T} \right\}\nabla T . \tag{3,47} \]

Let our conductor be between two metallic electrodes made of the same material, and let the temperature distribution be as indicated in Fig. 9. The segment \(bc\) corresponds to the semiconductor; \(ab\) and \(cd\) are the metallic electrodes, whose extreme points \(a\) and \(d\) are at the same temperature. At the points of contact of the metal with the semiconductors there are contact potential differences, as a result of which at these points \(\varphi\) undergoes a jump; however, \(\varphi-\frac{\mu}{e}\) is continuous.

Fig. 9

Since, on the other hand, the metal at points \(a\) and \(d\) is in the same state, \(\mu_a=\mu_d\). Therefore, in order to obtain the total electromotive force \(\theta\), it is necessary to integrate (3,47)

along the entire circuit from \(a\) to \(d\). The thermo-emf introduced by the metal is very small; it may be neglected, and then

\[ \theta = \int_{T_1}^{T_2} \frac{ n-u-\dfrac{2kT-\mu}{T} -n+u_{+}+\dfrac{2kT+(\Delta E+\mu)}{T} }{\sigma}\,dT = \]

\[ = \int_{T_1}^{T_2} \frac{k}{e} \left[ n-u_{-} \left( 2+\lg\frac{2(2\pi m_{-}kT)^{3/2}}{h^3 n_{-}} \right) - n+u_{+} \left( 2+\lg\frac{2(2\pi m_{+}kT)^{3/2}}{h^3 n_{+}} \right) \right]dT . \tag{3,48} \]

If the concentration of carriers of one sign is much greater than the concentration of carriers of the other sign, then formula (3,48) is greatly simplified. In this case

\[ \theta = \pm \frac{k}{e} \int_{T_1}^{T_2} \left[ 2+\lg\frac{2(2\pi m_{\mp}kT)^{3/2}}{h^3 n_{\mp}} \right]dT , \tag{3,49} \]

If approximately \(\mu=-\dfrac{\Delta E}{2}\gg kT\) (where \(\Delta E\) is the width of the forbidden band, or the distance from the edge of the allowed band to an impurity level), then

\[ \theta \approx \pm \frac{\Delta E}{2e}\lg\frac{T_2}{T_1}. \tag{3,50} \]

In formulas (3,49) and (3,50) the upper signs refer to electrons, the lower to holes. From formula (3,48) it is seen that holes partly compensate the thermo-emf produced by electrons, and may even change its sign. This is understandable, since both electrons and holes diffuse from the hot end of the semiconductor to the cold one.

We already know that the introduction of impurities changes the concentration of electrons and holes. Recently Yu. P. Maslakovets (LPTI) has succeeded, by changing the concentration of sulfur in lead sulfide, in changing the character of the conductivity from electronic to hole conductivity. In this case the sign of the thermo-emf changes.

c. Thomson and Peltier Effects

For \(\mathbf H=0\) we also find the vector of the energy-flux density \(\mathbf S\)

\[ \mathbf S = \int f(\mathbf p)(\varepsilon-e\varphi)\frac{\mathbf p}{m_{-}}\cdot\frac{2}{h^3}\,d\tau_{\mathbf p} + \]

\[ + \int f'(\mathbf p')(\varepsilon'+\Delta E+e\varphi)\frac{\mathbf p'}{m_{+}}\cdot\frac{2}{h^3}\,d\tau_{\mathbf p'} -\chi_0\nabla T . \tag{3,51} \]

To the energy flux carried by the electrons we must add the heat carried by the lattice itself; \(\chi_0\) is the coefficient of its thermal conductivity. With the aid of formulas (3,26), (3,28), (3,36), (3,37), (3,43)

from (3.44) we find:

\[ \begin{aligned} \mathbf S={}&\varphi\mathbf i-\frac{2kT}{e}n_{-}u_{-} -\left[\nabla(\mu-e\varphi)-\frac{\mu-3kT}{T}\nabla T\right] \\ &+\frac{2kT}{e}n_{+}u_{+} \left[\nabla(\mu-e\varphi)-\frac{(\Delta E+\mu)+3kT}{T}\nabla T\right]+ \\ &+\frac{\Delta E}{e}n_{+}u_{+} \left[\nabla(\mu-e\varphi)-\frac{(\Delta E+\mu)+2kT}{T}\nabla T\right]. \end{aligned} \tag{3.52} \]

Substituting into this formula \(\nabla(\mu-e\varphi)\) from (3.46), we obtain

\[ \mathbf S=\left(\varphi-\frac{\mu}{e}\right)\mathbf i-\Pi\mathbf i-\chi\nabla T, \tag{3.53} \]

where

\[ \Pi=\frac{n_{-}u_{-}[2kT-\mu]-n_{+}u_{+}[2kT+(\Delta E+\mu)]}{\sigma}= \]

\[ = \frac{ n_{-}u_{-}kT\left[2+\lg\frac{2(2\pi m_{-}kT)^{3/2}}{h^{3}n_{-}}\right] - n_{+}u_{+}kT\left[2+\lg\frac{2(2\pi m_{+}kT)^{3/2}}{h^{3}n_{+}}\right] }{\sigma} \tag{3.54} \]

and

\[ \chi=\chi_{0}+n_{+}u_{+}\frac{2k^{2}T}{e}+n_{-}u_{-}\frac{2k^{2}T}{e} +\frac{n_{+}u_{+}\,n_{-}u_{-}}{n_{+}u_{+}+n_{-}u_{-}}\cdot \frac{(\Delta E+4kT)^{2}}{eT}. \tag{3.55} \]

\(\chi\) is the total thermal conductivity of the crystal.

In the stationary state \(\operatorname{div}\mathbf S=0\) and \(\operatorname{div}\mathbf i=0\). Therefore, in passing from one body to another, the normal components of the vectors \(\mathbf S\) and \(\mathbf i\) are continuous. If \(\varphi-\frac{\mu}{e}\) is also continuous, then from (3.53) it follows that

\[ -\chi_{1}(\nabla T)_{n,1}+\chi_{2}(\nabla T)_{n,2}=(\Pi_{1}-\Pi_{2})i. \tag{3.56} \]

This equality means that \((\Pi_{1}-\Pi_{2})i\) is the heat liberated in 1 sec. at the contact point (Peltier heat), which is removed by thermal conduction [the left-hand side of equation (3.56)]. If the contact between a semiconductor and a metal is considered, then the Peltier coefficient for the metal \(\Pi_{m}\) may be neglected, since it is considerably smaller than the Peltier coefficient for the semiconductor \(\Pi_{n}\).

In an analogous manner we have:

\[ \operatorname{div}\mathbf S =\mathbf i\nabla\left(\varphi-\frac{\mu}{e}\right) -\mathbf i\nabla\Pi-\operatorname{div}(\chi\nabla T). \tag{3.57} \]

Substituting \(\nabla\left(\varphi-\frac{\mu}{e}\right)\) from (3.46), we have

\[ \operatorname{div}(-\chi\nabla T) =\frac{i^{2}}{\sigma}+T\frac{d}{dT}\left(\frac{\Pi}{T}\right)\mathbf i\cdot\nabla T. \tag{3.58} \]

Thus, in the presence of a temperature gradient, in a unit volume, in addition to the Joule heat \(\frac{i^{2}}{\sigma}\), there is also liberated the Thomson heat \(a\mathbf i\cdot\nabla T\). The quantity

\[ a=T\frac{d}{dT}\left(\frac{\Pi}{T}\right) \tag{3.59} \]

is called the Thomson coefficient. Formulas (3,48), (3,54) and (3,59) are in agreement with the thermodynamic Onsager–Thomson relations. As for the thermoelectric emf, the presence, in addition to electrons, of holes diminishes the Thomson and Peltier effects. If there are only electrons, the formulas simplify:

\[ \Pi=\frac{2kT-\mu}{e}, \tag{3,60} \]

\[ \alpha=\frac{1}{e}\left(\frac{\mu}{T}-\frac{d\mu}{dT}\right). \tag{3,61} \]

In the presence of holes alone,

\[ \Pi=-\frac{\Delta E+\mu+2kT}{e}, \tag{3,62} \]

\[ \alpha=-\frac{1}{e}\left(\frac{\Delta E+\mu}{T}+\frac{d\mu}{dT}\right). \tag{3,63} \]

In the same approximation in which formula (3,50) is valid, we have

\[ \Pi=\pm \frac{\Delta E}{2e}, \qquad \alpha=\mp \frac{\Delta E}{2eT}; \tag{3,64} \]

the upper signs refer to electronic (in the narrow sense of the word), the lower signs to hole semiconductors.

d. The Hall effect and the Nernst effect

In the presence of a magnetic field \(\mathbf H\), equations 3,36 and (3,37) have the following solutions:

\[ \mathbf j_-= \frac{l_-}{p}\, \frac{ \frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi) +\frac{e l_-}{cp}\left[\mathbf H,\frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right] }{ 1+\left(\frac{e l_- H}{cp}\right)^2 } + \]

\[ + \frac{ \left(\frac{e l_-}{cp}\right)^2 \left(\mathbf H,\frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right)\mathbf H }{ 1+\left(\frac{e l_- H}{cp}\right)^2 }, \tag{3,65} \]

\[ \mathbf j_+= \frac{l_+}{p'}\, \frac{ \frac{\varepsilon'+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi) }{ 1+\left(\frac{e l_+ H}{cp'}\right)^2 } - \]

\[ - \frac{ \frac{e l_+}{cp'} \left[\mathbf H,\frac{\varepsilon'+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right] }{ 1+\left(\frac{e l_+ H}{cp'}\right)^2 } + \]

\[ + \frac{ \left(\frac{e l_+}{cp'}\right)^2 \left(\mathbf H,\frac{\varepsilon+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right)\mathbf H }{ 1+\left(\frac{e l_+ H}{cp'}\right)^2 }. \tag{3,66} \]

In the case of a weak magnetic field, when quadratic terms with respect to \(\mathbf H\) may be neglected, we have:

\[ \mathbf j_-=\frac{l_-}{p}\left\{ \frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)+ \frac{el_-}{cp}\left[\mathbf H,\frac{\varepsilon-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right]\right\}, \tag{3,67} \]

\[ \mathbf j_+=\frac{l_+}{p'}\left\{ \frac{\varepsilon'+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)- \frac{el_+}{cp'}\left[\mathbf H,\frac{\varepsilon+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right]\right\}. \tag{3,68} \]

Current density

\[ \mathbf i=n_+u_+\left[\nabla(\mu-e\varphi)- \frac{(\Delta E+\mu)+2kT}{T}\nabla T\right]+ \]

\[ +\,n_-u_-\left[\nabla(\mu-e\varphi)+ \frac{2kT-\mu}{T}\nabla T\right]+ \]

\[ +\frac{3\pi}{8c}\left[ n_+u_+^2\frac{\frac{3}{2}kT+(\Delta E+\mu)}{T} +n_-u_-^2\frac{\frac{3}{2}kT-\mu}{T} \right][\mathbf H,\nabla T]+ \]

\[ +\frac{3\pi}{8c}(n_-u_-^2-n_+u_+^2)[\mathbf H,\nabla(\mu-e\varphi)]. \tag{3,69} \]

Energy flux density

\[ \mathbf S=\varphi\mathbf i-\frac{2kT}{e}n_-u_- \left[\nabla(\mu-e\varphi)+\frac{3kT-\mu}{T}\nabla T\right]+ \]

\[ +\frac{2kT}{e}n_+u_+ \left[\nabla(\mu-e\varphi)-\frac{3kT+(\Delta E+\mu)}{T}\nabla T\right]+ \]

\[ +\frac{\Delta E}{e}n_+u_+ \left[\nabla(\mu-e\varphi)-\frac{2kT+(\Delta E+\mu)}{T}\nabla T\right]+ \]

\[ +\frac{3\pi}{8c}\cdot\frac{\Delta E}{e}n_+u_+^2 \left[\mathbf H,\frac{\frac{3}{2}kT+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right]- \]

\[ -\frac{9\pi kT}{16\,ce}\left[ \mathbf H,\left\{ n_-u_-^2\left(\frac{\frac{5}{2}kT-\mu}{T}\nabla T+\nabla(\mu-e\varphi)\right)- \right.\right. \]

\[ \left.\left. -\,n_+u_+^2\left(\frac{\frac{5}{2}kT+(\Delta E+\mu)}{T}\nabla T-\nabla(\mu-e\varphi)\right) \right\}\right]. \tag{3,70} \]

From equation (3,69), in the same approximation (neglecting higher powers of \(H\)), we find the following expression for \(\nabla\left(\frac{\mu}{e}-\varphi\right)\):

\[ \nabla\left(\frac{\mu}{e}-\varphi\right) =\frac{\mathbf i}{\sigma}-\frac{d\theta}{dT}\nabla T -R[\mathbf i,\mathbf H]-Q[\mathbf H,\nabla T], \tag{3,71} \]

where

\[ R=\frac{3\pi}{8ce}\, \frac{n_+u_+^2-n_-u_-^2}{(n_+u_+ + n_-u_-)^2}, \tag{3,72} \]

\[ Q=-\frac{3\pi}{8ce}\left[ \frac{n_{+}u_{+}\cdot n_{-}u_{-}(u_{+}+u_{-})}{(n_{+}u_{+}+n_{-}u_{-})^{2}}\cdot \frac{\Delta E+\frac{7}{2}kT}{T} -\frac{k}{2}\, \frac{n_{+}^{2}u_{+}^{3}+n_{-}^{2}u_{-}^{3}}{(n_{+}u_{+}+n_{-}u_{-})^{2}} \right]. \tag{3,73} \]

\(\dfrac{d\theta}{dT}\) is the thermo-emf per one degree of temperature difference [see (3,48)].

If in a semiconductor a current flows in the direction of the \(X\)-axis, and a magnetic field is switched on perpendicular to it along the \(Z\)-axis, then an electric field arises in the direction of the \(Y\)-axis (a transverse Hall field).

\[ E_y=-\frac{\partial \varphi}{\partial y}=RH_z i_x . \tag{3,74} \]

The semiconductor is then assumed to be uniformly heated—\(\nabla T=0\). This phenomenon is called the Hall effect, and the quantity \(R\) is the Hall constant [see (3,72)]. In the case of one kind of carrier

\[ R=\pm \frac{3\pi}{8c}\frac{1}{en_{\pm}} \tag{3,75} \]

measurements of the Hall effect make it possible to determine the concentration of electrons or holes [in formula (3,75) \(+\) refers to holes, \(-\) to electrons].

In contrast to the isothermal Hall effect, which we have just considered, in experiment, owing to the thermal nonconductivity of the side walls, conditions of the so-called adiabatic Hall effect may sometimes be realized, for which \(S_y=0\).

If instead of an electric current there is a temperature gradient in the direction of the \(X\)-axis, then perpendicular to it and to the magnetic field (directed along the \(Z\)-axis) an electric field also arises

\[ E_y=-\frac{\partial \varphi}{\partial y}=-QH_z\frac{dT}{dx}. \tag{3,76} \]

This phenomenon is called the Nernst effect, and the quantity \(Q\) is the Nernst constant [see (3,73)].

d. Change of resistance in a weak magnetic field

The resistivity \(\rho\) is the ratio of the component of the electric-field strength in the direction of the current-density vector to the magnitude of the current density:

\[ \rho=\frac{\mathbf{i}\cdot\mathbf{E}}{i^2}. \tag{3,77} \]

From formula (3,69), for \(\nabla T=0\), we have:

\[ \rho=\frac{\mathbf{i}\cdot\left(\frac{\mathbf{i}}{\sigma}-R[\mathbf{i},\mathbf{H}]\right)}{i^2} =\frac{1}{\sigma}=\rho_0, \tag{3,78} \]

i.e., there is no change in the resistance. To obtain it, one must, in expanding expressions (3.65) and (3.66) in powers of \(H\), retain the following terms of the expansion as well. Keeping the quadratic and cubic terms with respect to \(H\) and assuming \(\nabla T=0\), we have:

\[ \mathbf{j}_{-}= \frac{el_{-}}{p} \left\{ \mathbf{E} +\frac{el_{-}}{cp}[\mathbf{H},\mathbf{E}] +\left(\frac{el_{-}}{cp}\right)^2(\mathbf{H},\mathbf{E})\mathbf{H} -\left(\frac{el_{-}H}{cp}\right)^2\mathbf{E} -\left(\frac{el_{-}}{cp}\right)^3H^2[\mathbf{H},\mathbf{E}] \right\}, \tag{3.79} \]

\[ \mathbf{j}_{+}= \frac{el_{+}}{p'} \left\{ \mathbf{E} -\frac{el_{+}}{cp'}[\mathbf{H},\mathbf{E}] +\left(\frac{el_{+}}{cp'}\right)^2(\mathbf{H},\mathbf{E})\mathbf{H} -\left(\frac{el_{+}H}{cp'}\right)^2\mathbf{E} +\left(\frac{el_{+}}{cp'}\right)^3H^2[\mathbf{H},\mathbf{E}] \right\}. \tag{3.80} \]

The current density

\[ \mathbf{i}=\sigma\mathbf{E} +\frac{3\pi e}{8c}\left(n_{-}u_{-}^{2}-n_{+}u_{+}^{2}\right)[\mathbf{H},\mathbf{E}] + \]

\[ +\frac{9\pi e}{16c^{2}}\left(n_{-}u_{-}^{3}+n_{+}u_{+}^{3}\right)[\mathbf{H},[\mathbf{H},\mathbf{E}]] - \]

\[ -\frac{27\pi^{2}e}{64c^{3}}\left(n_{-}u_{-}^{4}-n_{+}u_{+}^{4}\right)H^{2}[\mathbf{H},\mathbf{E}]. \tag{3.81} \]

Hence, with the same degree of accuracy, we find:

\[ \mathbf{E}= \frac{3\pi}{8ce}\cdot \frac{n_{+}u_{+}^{2}-n_{-}u_{-}^{2}}{(n_{+}u_{+}+n_{-}u_{-})^{2}} \left\{ 1+ \right. \]

\[ +\left[ \frac{8\pi}{9c^{2}}\cdot \frac{ n_{+}u_{+}^{2}\cdot n_{-}u_{-}^{2}(u_{+}-u_{-}) +n_{+}u_{+}\cdot n_{-}u_{-}(u_{+}^{3}-u_{-}^{3}) }{ (n_{+}u_{+}+n_{-}u_{-})(n_{-}u_{-}^{2}+n_{+}u_{+}^{2}) } - \]

\[ \left. -\frac{9\pi^{2}}{64c^{2}}\cdot \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})^{2}}{(n_{-}u_{-}+n_{+}u_{+})^{2}} \right] H^{2} \Bigg\} [\mathbf{H},\mathbf{i}] + \]

\[ +\frac{1}{\sigma} \left\{ 1+ \left[ \frac{9\pi}{16c^{2}}\cdot \frac{n_{-}u_{-}^{3}+n_{+}u_{+}^{3}}{n_{-}u_{-}+n_{+}u_{+}} - \frac{9\pi^{2}}{64c^{2}}\cdot \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})}{(n_{-}u_{-}+n_{+}u_{+})^{2}} \right] H^{2} \right\}\mathbf{i} - \]

\[ -\frac{1}{\sigma} \left[ \frac{9\pi}{16c^{2}}\cdot \frac{n_{-}u_{-}^{3}+n_{+}u_{+}^{3}}{n_{-}u_{-}+n_{+}u_{+}} - \frac{9\pi^{2}}{16c^{2}}\cdot \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})}{(n_{-}u_{-}+n_{+}u_{+})^{2}} \right] (\mathbf{i},\mathbf{H})\mathbf{H}. \tag{3.82} \]

Consequently,

\[ \rho=\frac{\mathbf{i}\cdot\mathbf{E}}{i^{2}} = \frac{1}{\sigma} \left\{ 1+ \left[ \frac{9\pi}{16c^{2}}\cdot \frac{n_{-}u_{-}^{3}+n_{+}u_{+}^{3}}{n_{-}u_{-}+n_{+}u_{+}} - \frac{9\pi^{2}}{64c^{2}} \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})^{2}}{(n_{-}u_{-}+n_{+}u_{+})^{2}} \right] \left( H^{2}-\frac{(\mathbf{i},\mathbf{H})^{2}}{i^{2}} \right) \right\}, \tag{3.83} \]

and if the magnetic field is perpendicular to the current, then

\[ \frac{\Delta\rho}{\rho} = \frac{\rho-\rho_{0}}{\rho_{0}} = \frac{9\pi}{16c^{2}}\frac{H^{2}}{\sigma} \left[ \left(n_{+}u_{+}^{3}+n_{-}u_{-}^{3}\right) -\frac{\pi}{4}\, \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})^{2}}{n_{-}u_{-}+n_{+}u_{+}} \right]. \tag{3.84} \]

THEORY OF ELECTRONIC SEMICONDUCTORS

A magnetic field parallel to the current does not cause a change in the resistance, as is easily seen from formula (3.83). From equation (3.82) we also find that the Hall constant is

\[ R=\frac{3\pi}{8ce}\cdot \frac{n_{+}u_{+}^{2}-n_{-}u_{-}^{2}} {(n_{+}u_{+}+n_{-}u_{-})^{2}} \left\{ 1+\frac{9\pi eH^{2}}{8c^{2}\sigma} \left[ \frac{n_{+}u_{+}\cdot n_{-}u_{-}(u_{+}+u_{-})^{2}(u_{+}-u_{-})} {n_{-}u_{-}^{2}-n_{+}u_{+}^{2}} -\frac{\pi}{4}\cdot \frac{(n_{-}u_{-}^{2}-n_{+}u_{+}^{2})^{2}} {n_{-}u_{-}+n_{+}u_{+}} \right] \right\}. \tag{3.85} \]

e. Resistance and the Hall effect in a strong magnetic field

If, on the contrary, the magnetic field is very large, then expressions (3.65) and (3.66) can be expanded in powers of \(1/H\). Then

\[ \mathbf{j}_{-}= \frac{c^{2}}{H^{2}}\frac{p}{el_{-}} \left\{ \mathbf{E} +\frac{el_{-}}{cp}[\mathbf{H},\mathbf{E}] +\left[ \left(\frac{eHl_{-}}{cp}\right)^{2}-1 \right] \frac{(\mathbf{H},\mathbf{E})\mathbf{H}}{H^{2}} \right\}, \tag{3.86} \]

\[ \mathbf{j}_{+}= -\frac{c^{2}}{H^{2}}\frac{p'}{el_{+}} \left\{ \mathbf{E} -\frac{el_{+}}{cp'}[\mathbf{H},\mathbf{E}] +\left[ \left(\frac{eHl_{+}}{cp'}\right)^{2}-1 \right] \frac{(\mathbf{H},\mathbf{E})\mathbf{H}}{H^{2}} \right\}. \tag{3.87} \]

The current density is

\[ \mathbf{i}=\frac{c^{2}}{H^{2}} \left\{ \frac{32}{9\pi} \left( \frac{n_{+}e}{u_{+}}+\frac{n_{-}e}{u_{-}} \right)\mathbf{E} +\frac{e}{c}(n_{+}-n_{-})[\mathbf{E},\mathbf{H}] + \left[ \frac{H^{2}}{c^{2}}e(n_{+}u_{+}+n_{-}u_{-}) -\frac{32}{9\pi} \left( \frac{n_{+}e}{u_{+}}+\frac{n_{-}e}{u_{-}} \right) \right] \frac{(\mathbf{H},\mathbf{E})\mathbf{H}}{H^{2}} \right\}. \tag{3.88} \]

Solving this equation with respect to \(\mathbf{E}\), we have

\[ \mathbf{E}= \frac{ \displaystyle \frac{9\pi}{32c}\cdot \frac{n_{+}-n_{-}} {\left(\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}\right)} [\mathbf{H},\mathbf{i}] } { \displaystyle \frac{32}{9\pi} \left(\frac{n_{+}e}{u_{+}}+\frac{n_{-}e}{u_{-}}\right) \left[ \left( \frac{9\pi(n_{+}-n_{-})H} {32c\left(\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}\right)} \right)^{2}+1 \right] } - \tag{3.89} \]

\[ - \frac{ \displaystyle \left\{ 1-\frac{32}{9\pi}\cdot \frac{\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}} {n_{+}u_{+}+n_{-}u_{-}}\cdot \frac{c^{2}}{H^{2}} \left[ 1+ \left( \frac{9\pi}{32c}\cdot \frac{n_{+}-n_{-}} {\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}} H \right)^{2} \right] \right\} \frac{(\mathbf{i},\mathbf{H})\mathbf{H}}{H^{2}}+\mathbf{i} } { \displaystyle \frac{32}{9\pi} \left(\frac{n_{+}e}{u_{+}}+\frac{n_{-}e}{u_{-}}\right) \left[ \left( \frac{9\pi(n_{+}-n_{-})H} {32c\left(\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}\right)} \right)^{2}+1 \right] }. \]

Hence the Hall constant is

\[ R=\frac{1}{ce(n_{+}-n_{-})}. \tag{3.90} \]

The resistance in a field \(H\) perpendicular to the current is

\[ \rho_{\infty}= \frac{32}{9\pi}\cdot \frac{\frac{n_{+}}{u_{+}}+\frac{n_{-}}{u_{-}}} {e(n_{+}-n_{-})^{2}}. \tag{3.91} \]

The longitudinal effect is absent, i.e., a magnetic field parallel to the current does not cause a change in the resistance.

Physically, the difference between the motion of electrons in a weak magnetic field and in a strong one reduces to the fact that in a weak field the trajectory of an electron during its free path is only slightly curved, whereas in a strong field the electron manages during this time to describe several turns of the inclined helical line along which it moves.

It should be said that the calculations given above are not entirely general, since galvanomagnetic effects depend strongly on the anisotropy present in the crystal. We have restricted ourselves to consideration of a cubic lattice, in which the inverse-mass tensor \(m_{ik}^{-1}\) reduces to a single scalar.

In conclusion we note that most of the formulas given in this paragraph for semiconductors with an atomic lattice, in the presence of carriers of both signs, were derived earlier by N. L. Pisarenko (LPTI).

B. SEMICONDUCTORS WITH AN IONIC LATTICE

As the simplest case, let us consider a cubic diatomic lattice, at the sites of which there are ions with masses \(M_{+}\) and \(M_{-}\) and charges \(Ze\) and \(-Ze\). We shall denote the distance between neighboring ions by \(a\). The volume of the elementary cell of such a crystal is \(\Omega_0=2a^3\). As was already said at the beginning of this section, the interaction of electrons with optical vibrations is greater than with acoustic ones. The latter may therefore be completely neglected.

The dipole moment arising in optical vibrations of the lattice, per unit volume near the \(l\)-th cell, is

\[ \mathbf{P}=\gamma Ze\,\frac{\mathbf{u}_{+}^{\,l}-\mathbf{u}_{-}^{\,l}}{\Omega_0}. \tag{3,92} \]

The deviation of \(\gamma\) from unity is determined by the deformation of the electronic shells during vibrations of the ions. For not very strongly pronounced heteropolar compounds \(\gamma\), probably, may be considerably smaller than unity. Instead of \(\mathbf{u}_{+}^{\,l}\) and \(\mathbf{u}_{-}^{\,l}\) one should substitute their expansions in the normal vibrations of the lattice (3,6). Since in what follows only long-wave vibrations are essential, \(\mathbf{P}\) may be regarded as a continuous function of the coordinates. In this case the additional potential \(\Phi\) arising during the vibrations satisfies Poisson’s equation

\[ \nabla^{2}\Phi=4\pi \operatorname{div}\mathbf{P}. \tag{3,93} \]

Solving this equation, we find the perturbation energy

\[ U=-e\Phi, \tag{3,94} \]

the matrix element of which determines the transition probability. The transition probabilities calculated with the aid of (3,94) are equal: for absorption of a phonon by an electron

\[ W^{+}=wN_q\delta(\varepsilon_{p'}-\varepsilon_p-\hbar\omega_q) \tag{3,95} \]

and for emission of a phonon

\[ W^- = w(N_q+1)\delta(\varepsilon_{\mu'}-\varepsilon_\rho+\hbar\omega_q), \tag{3,96} \]

where

\[ w=\frac{(2\pi)^3\gamma^2Z^2e^4}{\Omega_0N\cdot Ma^3\omega q^2} \quad \text{and} \quad M=\frac{M_+\cdot M_-}{M_+ + M_-}. \]

It then turns out that only longitudinal vibrations interact with the electrons. Since in diatomic lattices there are altogether only three optical branches (corresponding to the three polarizations), we shall have to deal with only one branch. We shall not give here the detailed solution of the kinetic equation, as was done for an atomic lattice, but shall confine ourselves to indicating the existing differences. The change in the electron energy in the absorption or emission of an acoustic phonon is very small in comparison with the electron energy. For optical vibrations, however, the quantity \(\hbar\omega_0\) may, in different crystals and at different temperatures, be either smaller or larger than the average electron energy. \(\omega_0\) is the limiting frequency of the longitudinal optical vibrations.

If \(kT \gg \hbar\omega_0\), then, for the overwhelming majority of electrons, the change in energy upon absorption or emission of a phonon is insignificant. If, however, \(kT \ll \hbar\omega_0\), then almost all electrons can only absorb phonons, thereby passing into an energy interval from \(\hbar\omega_0\) to \(2\hbar\omega_0\), i.e. changing their energy very strongly. By emitting phonons, electrons from the second interval will pass back into the first. An electron, however, passes from the second interval into the first with a much greater probability than conversely, since the probability of emission contains the factor \(N_q+1\), which at low temperatures is much greater than the corresponding factor \(N_q\) in the probability of emission:

\[ \frac{N_q+1}{N_q}=e^{\frac{\hbar\omega_0}{kT}}. \]

Therefore the residence time of an electron in the second interval is so small that we may assume that the electron, immediately after absorbing a phonon, emits another phonon of almost the same energy (but not exactly the same, since \(\omega\) depends on \(q\); this is essential for establishing the stationary distribution of the electrons, although it does not explicitly enter the calculations in the case of a weak electric field). As a result of such a double transition, the electron falls into a state in which its energy differs hardly at all from the initial one, but the direction of the quasimomentum may change strongly. Thus, one may still regard the electron as being scattered as a result of interaction with lattice vibrations, almost without changing its energy. Owing to this simplification, the kinetic equation can be solved approximately in both limiting cases of low and high temperatures relative to the temperature

\[ \theta=\frac{\hbar\omega_0}{k}. \]

As before, we shall set the distribution function \(f\) equal to \(f_0+\gamma\), where \(f_0=e^{\frac{\mu-\varepsilon}{kT}}\) for electrons and \(f'_0=e^{-\frac{(\Delta E+\mu)+\varepsilon'}{kT}}\) for holes,

and \(\chi=\dfrac{dt_0}{d\varepsilon}\mathbf p\cdot\mathbf j\). The vectors \(\mathbf j_+\) and \(\mathbf j_-\) satisfy equations (3.36) and (3.37). We shall give, without derivation, the mean free path appearing in the right-hand sides of these equations.

For \(kT\ll \hbar\omega_0\) \(^{10}\)

\[ l=\frac{a}{2\pi}\frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma Ze^2/a}\right)^2 e^{\frac{\hbar\omega_0}{kT}}\sqrt{\frac{\varepsilon}{\hbar\omega_0}}. \tag{3.97} \]

For \(kT\gg \hbar\omega_0\) \(^{11}\)

\[ l=\frac{a}{2\pi}\frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma Ze^2/a}\right)^2 \frac{\varepsilon}{\hbar\omega_0\left(N_q+\dfrac12\right)} \simeq \frac{a}{2\pi}\frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma Ze^2/a}\right)^2 \frac{\varepsilon}{kT}, \tag{3.98} \]

since \(N_q+\dfrac12\simeq \dfrac{kT}{\hbar\omega_0}\) for \(kT\gg \hbar\omega_0\).

All the effects considered for an atomic lattice can also easily be calculated for ionic lattices in the two limiting cases indicated. In the intermediate region all the quantities of interest to us can be found by interpolation. The calculations in this case are analogous to those carried out for the atomic lattice, and we give the results directly. The quantity

\[ \frac{a}{2\pi}\cdot \frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma Ze^2/a}\right)^2, \]

which appears here in formulas (3.97) and (3.98), has the dimension of length and is independent of temperature; for convenience we shall denote it by \(l_0\).

a. Electrical conductivity

\[ \sigma=e\left(n_+u_+ + n_-u_-\right). \tag{3.99} \]

For \(kT\ll \hbar\omega_0\) the mobility is

\[ u_{\pm}=\frac{el_0^{\pm}e^{\frac{\hbar\omega_0}{kT}}}{\sqrt{2m_{\pm}\hbar\omega_0}}, \tag{3.100} \]

for \(kT\gg \hbar\omega_0\)

\[ u_{\pm}=\frac{8el_0^{\pm}}{3\sqrt{2\pi m_{\pm}kT}}. \tag{3.101} \]

b. Thermoelectromotive force

For \(kT\ll \hbar\omega_0\)

\[ \theta=\int_{T_1}^{T_2} \frac{ n_-u_-\dfrac{\dfrac{5}{2}kT-\mu}{T} - n_+u_+\dfrac{\dfrac{5}{2}kT+(\Delta E+\mu)}{T} }{\sigma}\,dT. \tag{3.102} \]

For \(kT\gg \hbar\omega_0\)

\[ \theta=\int_{T_1}^{T_2} \frac{ n_-u_-\dfrac{3kT-\mu}{T} - n_+u_+\dfrac{3kT+(\Delta E+\mu)}{T} }{\sigma}\,dT. \tag{3.103} \]

c. Thomson and Peltier Effects

For \(kT \ll \hbar \omega_0\), the Peltier coefficient is

\[ \Pi= \frac{ n_-u_-\left(\frac{5}{2}kT-\mu\right) - n_+u_+\left(\frac{5}{2}kT+\Delta E+\mu\right) }{e}. \tag{3,104} \]

For \(kT \gg \hbar \omega_0\),

\[ \Pi= \frac{ n_-u_-(3kT-\mu) - n_+u_+(3kT+\Delta E+\mu) }{e}. \tag{3,105} \]

As for the Thomson coefficient, for it it is better to use the thermodynamic relation

\[ \alpha = T \frac{d\Pi}{dT} \tag{3,106} \]

together with formulas (3,104) and (3,105).

d. Thermal Conductivity

For \(kT \ll \hbar \omega_0\), the thermal-conductivity coefficient of the whole crystal is

\[ \chi=\chi_0+ \frac{n_+u_+\cdot n_-u_-}{n_+u_+ + n_-u_-} \frac{\left(\Delta E+\frac{5}{2}kT\right)^2}{eT} + \]

\[ +\frac{5}{2}n_+u_+\frac{k^2T}{e} + \frac{5}{2}n_-u_-\frac{k^2T}{e}. \tag{3,107} \]

For \(kT \gg \hbar \omega_0\),

\[ \chi=\chi_0+ \frac{n_+u_+\cdot n_-u_-}{n_+u_+ + n_-u_-} \frac{(\Delta E+6kT)^2}{eT} + \]

\[ +3n_+u_+\frac{k^2T}{e} + 3n_-u_-\frac{k^2T}{e}. \tag{3,108} \]

The thermal conductivity of the lattice itself, \(\chi_0\), is usually much greater than the thermal conductivity of the electrons. We have given these formulas because, if both electrons and holes are present in comparable quantities, then the second term in formulas (3,107) and (3,108), associated with transport of the electron dissociation energy \(\Delta E\), is \(\left(\frac{\Delta E}{kT}\right)^2\) times greater than the purely electronic or hole thermal conductivity. Perhaps in some cases it may be significant.

e. Hall Effect

At low temperatures \(kT \ll \hbar \omega_0\), owing to the fact that the mean free time \(\tau\) does not depend on the energy [see formula (3,97)], one may give a formula suitable both for weak and for strong fields:

\[ R= \frac{1}{ce} \frac{ n_+u_+^2 - n_-u_-^2 + (n_+-n_-)\dfrac{u_+^2u_-^2H^2}{c^2} }{ (n_+u_+ + n_-u_-)^2 + (n_+-n_-)^2\dfrac{u_+^2u_-^2H^2}{c^2} }. \tag{3,109} \]

The dependence on the field here exists only insofar as holes and electrons are simultaneously present in the semiconductor.

In a weak magnetic field \(\dfrac{u_{\pm}H}{c}\ll 1\), and

\[ R=-\frac{1}{ce}\cdot \frac{n_{+}u_{+}^{2}-n_{-}u_{-}^{2}} {(n_{+}u_{+}+n_{-}u_{-})^{2}} . \tag{3,110} \]

In a strong field \(\dfrac{u_{\pm}H}{c}\gg 1\),

\[ R=\frac{1}{ce}\cdot\frac{1}{n_{+}-n_{-}} . \tag{3,111} \]

At high temperatures \(kT\gg \hbar\omega_{0}\), and in a weak magnetic field,

\[ R=-\frac{15}{16}\cdot\frac{3\pi}{8ce}\cdot \frac{n_{+}u_{+}^{2}-n_{-}u_{-}^{2}} {(n_{+}u_{+}+n_{-}u_{-})^{2}} . \tag{3,112} \]

In a strong field (also for \(kT\gg \hbar\omega_{0}\)),

\[ R=\frac{1}{ce}\frac{1}{n_{+}-n_{-}} . \tag{3,113} \]

e. Thermomagnetic Nernst effect

For \(kT\ll \hbar\omega_{0}\), the Nernst coefficient is

\[ Q=\frac{1}{ce}\cdot \frac{n_{+}u_{+}\cdot n_{-}u_{-}(u_{+}+u_{-})} {(n_{+}u_{+}+n_{-}u_{-})^{2}} \cdot \frac{\Delta E+5kT}{T}. \tag{3,114} \]

For \(kT\gg \hbar\omega_{0}\),

\[ Q=\frac{15}{16}\cdot\frac{3\pi}{8ce}\cdot \frac{ n_{+}u_{+}\cdot n_{-}u_{-}(u_{+}+u_{-}) \dfrac{\frac{13}{2}kT+\Delta E}{T} +\frac{k}{2}(n_{+}^{2}u_{+}^{3}+n_{-}^{2}u_{-}^{3}) } {(n_{+}u_{+}+n_{-}u_{-})^{2}} . \tag{3,115} \]

ж. Change of resistance in a magnetic field

The longitudinal effect is absent both in weak and in strong fields, and both at low and at high temperatures. This effect does not vanish in the presence of anisotropy\({}^{12}\).

The resistance in a transverse magnetic field for \(kT\ll \hbar\omega_{0}\) is

\[ \rho=\rho_{0}\left\{ 1+ \frac{ n_{+}n_{-}(u_{+}+u_{-})^{2}\dfrac{u_{+}u_{-}}{c^{2}}H^{2} } { (n_{+}u_{+}+n_{-}u_{-})^{2} +(n_{+}-n_{-})^{2}\dfrac{u_{+}^{2}u_{-}^{2}}{c^{2}}H^{2} } \right\}. \tag{3,116} \]

As for the Hall effect, the dependence on the field is present here if the semiconductor contains carriers of both signs.

In a weak field,

\[ \rho=\rho_{0}\left\{ 1+ \frac{n_{+}n_{-}(u_{+}+u_{-})^{2}} {(n_{+}u_{+}+n_{-}u_{-})^{2}} \cdot \frac{u_{+}u_{-}}{c^{2}}H^{2} \right\}. \tag{3,117} \]

At high temperatures \(kT \gg \hbar\omega_0\) and in a weak magnetic field

\[ \rho=\rho_0\left\{1+\frac{27\pi}{64}\cdot\frac{eH^2}{c^2} \left[ (n_+u_+^3+n_-u_-^3) -\frac{75\pi}{256}\frac{(n_+u_+^2-n_-u_-^2)^2}{(n_+u_+ + n_-u_-)} \right]\right\}. \tag{3,118} \]

In a strong field at \(kT \gg \hbar\omega_0\)

\[ \rho_\infty=\frac{32}{9\pi}\cdot \frac{\dfrac{n_+}{u_+}+\dfrac{n_-}{u_-}} {e\,(n_+-n_-)^2}. \tag{3,119} \]

Of course, in all formulas for low and high temperatures one must substitute different expressions for the mobilities. In the first case one must use formula (3,100), and in the second, formula (3,101).

The phenomena considered do not exhaust all possible effects of this type in semiconductors. Thus, for example, the question of the adiabatic Hall effect and the related Ettingshausen effect—the formation of a transverse temperature difference—and some other effects has remained unconsidered. We have considered the most essential phenomena; others, if necessary, can easily be calculated by the same methods.

In outward form the formulas for an ionic lattice differ from those for an atomic lattice only by inessential coefficients. However, the temperature dependence of the mobilities of electrons and holes in an ionic lattice is different from that in an atomic one. In an atomic lattice the mobility \(u \simeq T^{-3/2}\), while in an ionic lattice at low temperature \(u \simeq e^{\frac{\hbar\omega_0}{kT}}\), and at high temperatures \(u \simeq T^{-1/2}\). Since the mobilities enter into all formulas for electrical conductivity, thermoelectric emf, etc., the temperature dependence of these quantities in an ionic lattice is different from that in an atomic one. In addition, since electrons interact more strongly with optical vibrations than with acoustic ones, the mean free path, and together with it the mobility, in an ionic lattice must be smaller than in an atomic one.

In connection with this one should make one fundamental remark about the applicability of all calculations carried out in this paragraph. For the probability of collision of an electron with a phonon we used the formulas of perturbation theory. These formulas are valid if the perturbation is small. For electrons the condition that the perturbation be small is expressed by the fact that their de Broglie wavelength \(\dfrac{\hbar}{mv}\) must be considerably larger than the mean free path. In many cases, however, we are apparently at the boundary of applicability of the whole method, and then the results obtained should be regarded only as rough qualitative estimates.

As for impurity metals, in order to transfer to them all the formulas derived in this paragraph, the concentration of those carriers which are present in these metals should be regarded as independent of temperature, and the concentration of carriers of the other sign should be set equal to zero. The formulas are, of course, suitable only if the electron gas is nondegenerate.

4. ELECTRICAL CONDUCTIVITY IN STRONG FIELDS

In contrast to metals, for which Ohm’s law is valid over the entire range of practically attainable fields, the electrical conductivity of semiconductors depends on the field strength. Empirically this dependence is usually expressed by the so-called increase of conductivity with the field

\[ \sigma = \sigma_0 e^{bE}. \tag{4,1} \]

At some critical field, of the order of \(10^5\)—\(10^6\ \mathrm{V\cdot cm^{-1}}\), the smooth increase of the electrical conductivity is replaced by a sharp increase, with transition to a nonstationary regime. This is electrical breakdown—a phenomenon for which no complete theory yet exists. We shall not consider this question in detail, and only at the end shall say a few words about it.

As for the increase of electrical conductivity, the Gudden school generally considers that the electrical conductivity of semiconductors does not depend on the field up to breakdown. Thus, Heninger \(^{13}\), who studied the behavior of semiconductors in a strong field, explains the increase of electrical conductivity with the field by breakdown and by the failure of poorly conducting interlayers.

The elimination of interlayers leads, of course, to an increase in electrical conductivity. Moreover, in the second part of this review it will be shown that phenomena occurring at the boundary between microcrystals can lead to an increase of conductivity with the field.

Careful measurements by A. V. Ioffe and A. F. Ioffe \(^{14}\) showed, however, that deviations from Ohm’s law can by no means always be reduced to the influence of poorly conducting interlayers. Even where the influence of these interlayers is excluded, for example in single crystals, nevertheless, after a more or less wide ohmic region (independence of \(\sigma\) on \(E\)), an increase of electrical conductivity with the field sets in.

On the other hand, theoretically there are no grounds for assuming that in a strong electric field semiconductors must obey Ohm’s law. On the contrary, we shall now show that in strong fields deviations from Ohm’s law must occur.

Let us first consider semiconductors with an atomic lattice \(^{15,16}\). As we saw in the first part of the third section, at each collision with a phonon an electron loses or gains only a small part of its energy. Therefore, in a strong field the electrons may accumulate energy considerably exceeding their average thermal energy \({}^{3}/_{2} kT\). Owing to this, the distribution function will differ strongly from the equilibrium Maxwellian. One may therefore expect that in a strong field not only the average energy of the electrons but also their mobility will depend on the field. Let us confirm this quantitatively.

The energy acquired or lost by an electron at each collision is \(\hbar\omega = \hbar q \overline{u}\). For electrons for which \(\varepsilon \gg kT\), one may estimate in the following way the energy lost by them on average at each

THEORY OF ELECTRONIC SEMICONDUCTORS

the collision energy \(\Delta \varepsilon\): the ratio of the probability of phonon emission to the probability of its absorption is equal to \(\dfrac{N_q+1}{N_q}\), where

\[ N_q=\frac{1}{e^{\frac{\hbar\omega}{kT}}-1}; \]

therefore

\[ \Delta \varepsilon \simeq \frac{(N_q+1)-N_q}{(N_q+1)+N_q}\,\hbar\omega_q = \frac{\hbar\omega_q}{2N_q+1}. \tag{4,2} \]

Since the overwhelming majority of electrons interact with phonons for which \(\hbar\omega \ll kT\), we have \(2N_q+1 \simeq 2\dfrac{kT}{\hbar\omega}\). On the other hand, from the conservation laws in the collision of an electron with a phonon it follows that \(\hbar\omega=\hbar qv \simeq mv\cdot v\), where \(v\) is the electron velocity [see (3,17)]. Therefore

\[ \Delta \varepsilon \simeq \frac{mv^2}{kT}\,\varepsilon . \tag{4,3} \]

Since \(\dfrac{mv^2}{kT}\ll 1\), this relation confirms what we said about the small loss of electron energy in collisions with phonons. In the stationary state this energy loss must be compensated by the action of the electric field. Consequently,

\[ \frac{mv^2}{kT}\,\varepsilon \simeq eEu\,\frac{l}{v}. \tag{4,4} \]

Here \(u\) is the mean translational velocity of the electrons in the direction of the field, and \(\dfrac{l}{v}\) is their free-flight time. In Section 3 it was shown that \(l\) does not depend on the energy.

Analogously to the condition for the energy (4,4), we can write the condition for the momentum. Namely,

\[ mu \simeq eE\,\frac{l}{v}. \tag{4,5} \]

This relation expresses the circumstance that, in the stationary state, the momentum acquired by an electron in the field during the free flight is, on the average, completely lost at each collision.

Relations (4,4) and (4,5) give

\[ \varepsilon \simeq \sqrt{\frac{kT}{mv^2}}\,eEl,\qquad u \simeq \sqrt{\frac{mv^2}{kT}}\,v. \tag{4,6} \]

Thus, already at fields

\[ E \simeq \sqrt{\frac{mv^2 kT}{el}} \simeq 10^3\ \mathrm{V/cm} \]

the mean energy of the electrons begins to increase appreciably with increasing field strength.

The mobility \(u=\dfrac{el}{mv}\). Using the first of relations (4,6), we obtain

\[ u \simeq u_0\left(\frac{\sqrt{mv^2 kT}}{eEl}\right)^{1/2}, \tag{4,7} \]

where \(u_0\) is the mobility in a weak field.

B. I. Davydov and I. M. Shmushkevich

(4.7) shows that in a strong field the mobility of the electrons decreases as \(\dfrac{1}{\sqrt{E}}\). The electrical conductivity is \(\sigma = neu\); therefore one might have thought that, despite the decrease in mobility, the electrical conductivity nevertheless increases owing to the growth of the concentration. In fact, the mean energy of the electrons in a strong field is much greater than the thermal energy; therefore the character of the equilibrium of the electron gas with the lattice changes. For example, the probability of electrons adhering to impurity atoms may decrease, and because of this the concentration of conduction electrons may increase. Since, however, the mean energy \(\bar{\varepsilon}\) is considerably smaller than atomic energies, the probability of recombination of an electron with a positive hole may be regarded as independent of \(\bar{\varepsilon}\) and, consequently, of the field. As Fröhlich and Mott \(^{10}\) indicate, this corresponds to the \(1/v\) law for the capture cross section of slow neutrons by nuclei.

Thus, the concentration of conduction electrons should not change in fields that are not too strong, and, consequently, the electrical conductivity should decrease. It must be said, however, that a decrease of electrical conductivity in strong fields has so far never been observed.

A few words about the solution of the kinetic equation for electrons in a strong electric field. The second of relations (4.6) shows that the antisymmetric part of the distribution function (which alone creates the mean translational velocity) is much smaller than the symmetric part. Mathematically this leads to the fact that, also in a strong field, the distribution function, with accuracy up to terms of higher order relative to \(\dfrac{mv^{2}}{kT}\), can be represented in the form

\[ f(\mathbf{p}) = f_{0}(\varepsilon) + f_{1}(\varepsilon)\frac{p_{x}}{p}, \tag{4.8} \]

i.e., in a form similar to the distribution function in a weak field [see equation (3.38); \(f_{1}(\varepsilon)\) corresponds to the function \(\dfrac{df_{0}}{d\varepsilon}\,\mathbf{j}\cdot\mathbf{p}\) in that equation]. However, \(f_{0}(\varepsilon)\) is now no longer the Maxwellian distribution function. In calculating \((b-a)f_{0}\), we may expand all functions except the \(\delta\)-functions entering under the integral sign in powers of \(\dfrac{\hbar\omega}{\varepsilon}\), restricting ourselves to quadratic terms. For example,

\[ f_{0}(\varepsilon \pm \hbar\omega_{0}) = f_{0}(\varepsilon) \pm \hbar\omega\,\frac{df_{0}}{d\varepsilon} + \frac{1}{2}(\hbar\omega)^{2}\frac{d^{2}f_{0}}{d\varepsilon^{2}} . \]

In this case the first term of the expansion \((b-a)f_{0}\) is equal to zero. By this device the integral expression \((b-a)f_{0}\) is transformed into a differential one. In calculating \((b-a)\left(f_{1}\dfrac{p_{x}}{p}\right)\), one may everywhere neglect—

the quantity \(\hbar\omega\) in comparison with \(\varepsilon\), since here, in contrast to \(f_0\), already the first term (not depending on \(\frac{\hbar\omega}{\varepsilon}\)) is not equal to zero (as also in the solution of the kinetic equation in a weak field). As a result, for \(f_0\) and \(f_1\) the following differential equations are obtained:

\[ \frac{\partial f_0}{\partial t} + \frac{1}{\sqrt{2m\varepsilon}}\, \frac{\partial}{\partial \varepsilon} \left[ -\frac{4m\omega^2\varepsilon^2}{lkT} \left( f_0+kT\frac{\partial f_0}{\partial \varepsilon} \right) -\frac{2eE}{3}\,\varepsilon f_1 \right] =0, \tag{4,9} \]

\[ \frac{\partial f_1}{\partial t} - eEv\frac{\partial f_0}{\partial \varepsilon} + \frac{v}{l}f_1 =0. \tag{4,10} \]

Equation (4,9) may be regarded as a continuity equation, and the quantity standing under the sign of the derivative \(\frac{\partial}{\partial \varepsilon}\) is equal to the electron flux \(j_\varepsilon\) in energy space:

\[ j_\varepsilon = -\frac{4m\omega^2\varepsilon^2}{lkT} \left( f_0+kT\frac{\partial f_0}{\partial \varepsilon} \right) -\frac{2eE}{3}\,\varepsilon f_1, \tag{4,11} \]

provided that \(f_0\) is normalized so that

\[ \int_{0}^{\infty} f_0(\varepsilon)\sqrt{2m\varepsilon}\,d\varepsilon=n, \tag{4,12} \]

where \(n\) is the concentration of electrons (or holes).

In the stationary state

\[ \frac{\partial f_0}{\partial t}=0,\quad \frac{\partial f_1}{\partial t}=0 \quad\text{and}\quad j_\varepsilon=\mathrm{const}. \]

At infinity in energy space there should be no electron flux. We must therefore put \(j_\varepsilon=0\). This gives the following equation for \(f_0\), if \(f_1\), with the aid of (4,10), is expressed through \(\frac{df_0}{d\varepsilon}\):

\[ kT \left\{ 1+\frac{1}{6}\frac{(eEl)^2}{m\omega^2\varepsilon} \right\} \cdot \frac{df_0}{d\varepsilon} + f_0 =0. \tag{4,13} \]

In the limiting case of strong fields, the unity in the square brackets may be neglected. Then:

\[ f_0 = C e^{-\frac{3m\omega^2}{kT(eEl)^2}\varepsilon^2}. \tag{4,14} \]

The constant \(C\) is determined from the normalization condition (4,12). The mean energy

\[ \bar{\varepsilon} = \frac{1}{n} \int_{0}^{\infty} f_0(\varepsilon)\,\varepsilon\sqrt{2m\varepsilon}\,d\varepsilon = \frac{2}{\pi} \sqrt{\frac{2}{3}}\, \Gamma^2\!\left(\frac{5}{4}\right) \sqrt{\frac{kT}{m\omega^2}}\, eEl. \tag{4,15} \]

Current density

\[ i=-\frac{2e}{3}\int_0^\infty f_1(\varepsilon)\varepsilon\,d\varepsilon =ne\,\frac{4}{3^{3/4}\pi^{1/2}}\,\Gamma\!\left(\frac54\right) \left(\frac{e^2 l^2 w^2}{mkTE^2}\right)^{1/4}. \tag{4,16} \]

Hence the mobility is

\[ u=2^{1/2}\cdot 3^{1/4}\Gamma\!\left(\frac54\right)u_0\cdot \left(\frac{\sqrt{mw^2 kT}}{eEl}\right)^{1/2}. \tag{4,17} \]

Equations (4,15) and (4,17) differ from the approximate relations (4,7) and the first of relations (4,6) by a numerical factor of order unity \(\left(\Gamma\!\left(\frac54\right)\simeq 0.91\right)\).

Let us pass to semiconductors with an ionic lattice\(^ {17}\). Here two limiting cases should be distinguished: a) the mean electron energy \(\bar{\varepsilon}\ll \hbar\omega_0\), and b) \(\bar{\varepsilon}\gg \hbar\omega_0\). In the first case, evidently, \(kT\ll \hbar\omega_0\), and, moreover, the field is not too large, so that \(\bar{\varepsilon}\ll \hbar\omega_0\). The establishment of a stationary electron distribution occurs here owing to double transitions described at the beginning of the second part of the third section. In this case the dependence of the frequency of optical vibrations \(\omega\) on the wave vector is essential [see equation (3,5)]. The second case, \(\bar{\varepsilon}\gg \hbar\omega_0\), is realized either at high temperature \((kT\gg \hbar\omega_0)\) and any field, or at low temperature and a very strong field. In this case the dependence of \(\omega\) on \(q\) may be neglected and it may be assumed that all phonons have the same energy \(\hbar\omega_0\).

a) \(\bar{\varepsilon}\ll \hbar\omega_0\).

The continuity equation in energy space in the stationary case has the form:

\[ \frac{1}{\sqrt{2m\varepsilon}}\cdot\frac{d}{d\varepsilon} \left\{ -\frac{4\lambda^2}{3\tau}\sqrt{2m}\,\hbar\omega_0\,\varepsilon^{3/2} \left(\frac{df_0}{d\varepsilon}+\frac{1}{kT}\cdot f_0\right) -\frac{2eE}{3}\,\varepsilon f_1 \right\}. \tag{4,18} \]

According to (3,97), the free-path time is

\[ \tau=\frac{l}{v} =\frac{Ma^3(\hbar\omega_0)^{3/2}\cdot e^{\frac{\hbar\omega_0}{kT}}}{2\pi\sqrt{2m}\,\gamma^2 Z^2e^4} =\frac{l_0}{\sqrt{\dfrac{2\hbar\omega_0}{m}}}\, e^{\frac{\hbar\omega_0}{kT}}. \tag{4,19} \]

Here it is more convenient to use not the length but the free-path time, since it does not depend on energy. In (4,18) it is denoted:

\[ \lambda=a\,\frac{\hbar\omega_0}{\dfrac{\hbar^2}{2ma^2}}, \tag{4,20} \]

where \(a\) is the coefficient in formula (3,5). The relation between \(f_1\) and \(f_0\) is given by the usual formula:

\[ f_1=eEl\,\frac{df_0}{d\varepsilon} =\sqrt{\frac{2\varepsilon}{m}}\,\tau eE\,\frac{df_0}{d\varepsilon}. \tag{4,21} \]

Substituting (4.21) into (4.18) and setting the flux in energy space \(j_\varepsilon\) equal to zero, we obtain an equation whose solution is:

\[ f_0 = Ce^{-\frac{\varepsilon}{k\theta}} . \tag{4.22} \]

Thus, a Maxwellian distribution is obtained with a temperature differing from the lattice temperature,

\[ k\theta = kT\left[1+\left(\frac{eE\tau}{\lambda\sqrt{2m\hbar\omega_0}}\right)^2\right] . \tag{4.23} \]

The mean energy is

\[ \overline{\varepsilon} = \frac{3}{2}kT \left[ 1+ \left( \frac{eE\tau}{\lambda\sqrt{2m\hbar\omega_0}} \right)^2 \right] . \tag{4.24} \]

The mobility is

\[ u=\frac{e\tau}{m}. \tag{4.25} \]

The independence of the mobility, and with it also of the electrical conductivity, from the field (despite the increase in the mean energy) is, of course, a consequence of the fact that the free-flight time \(\tau\) does not depend on energy.

The limits of applicability of formulas (4.24) and (4.25) are established from the requirement that \(\overline{\varepsilon}\) be considerably smaller than \(\hbar\omega_0\). It is therefore necessary that the inequality

\[ \frac{eE\tau}{\lambda\sqrt{2m\hbar\omega_0}} \ll \frac{\hbar\omega_0}{kT} \tag{4.26} \]

be satisfied, or

\[ E \ll \lambda \frac{\hbar\omega_0}{kT} \cdot \frac{\sqrt{2m\hbar\omega_0}}{e\tau} \simeq 1000\ \mathrm{V/cm}, \tag{4.27} \]

i.e., up to these fields Ohm’s law is satisfied [formula (4.25)].

b) \(\overline{\varepsilon}\gg \hbar\omega_0\).

The basic equations in this case are as follows:

\[ \frac{1}{\sqrt{2m\varepsilon}}\frac{dj_\varepsilon}{d\varepsilon} = \frac{1}{\sqrt{2m\varepsilon}}\frac{d}{d\varepsilon} \left\{ - \frac{2\pi m\gamma^2 Z^2 e^4}{Ma^3} \lg\frac{4\varepsilon}{\hbar\omega_0} \left[ f_0+\hbar\omega_0\left(N_q+\frac{1}{2}\right)\frac{df_0}{d\varepsilon} \right] - \frac{2eE}{3}\varepsilon f_1 \right\} =0, \tag{4.28} \]

\[ eE\sqrt{\frac{2\varepsilon}{m}}\frac{df_0}{d\varepsilon} = - \frac{4\pi m\gamma^2 Z^2 e^4\left(N_q+\frac{1}{2}\right)} {Ma^3\hbar\omega_0\sqrt{2m\varepsilon}} \,f_1(\varepsilon). \tag{4.29} \]

If \(E=0\), then for \(kT\gg \hbar\omega_0\), \(\hbar\omega_0\left(N_q+\frac{1}{2}\right)\simeq kT\), and from equation (4.28) a Maxwellian distribution is obtained for \(f_0\).

If, however, \(kT\ll\hbar\omega_0\), then \(\hbar\omega_0\left(N_q+\frac{1}{2}\right)\simeq \frac{1}{2}\hbar\omega_0\), but then equation (4.28) is not valid.

B. I. Davydov and I. M. Shmushkevich

In the case under consideration, \(\bar{\varepsilon}\gg \hbar\omega_0\), we cannot set \(j_\varepsilon\) equal to zero, since the solution of the equation \(j_\varepsilon=0\) gives such a function \(f_0(\varepsilon)\) that cannot be normalized, for the integral

\[ \int_0^\infty f_0(\varepsilon)\sqrt{2m\varepsilon}\,d\varepsilon \]

diverges at the upper limit. This circumstance is connected with the increase in the mean free path of electrons as their energy increases. According to formula (3,98), \(l\) is proportional to \(\varepsilon\). Therefore, even in weak fields—of course, at high temperatures \(kT\gg \hbar\omega_0\)—a strong change occurs in the electron distribution function in the region of high energies. The braking of fast electrons due to their collisions with phonons is insignificant. As a result, it is necessary to take into account ionization by electrons of lattice atoms, or else of impurity atoms if their ionization energy \(\varepsilon_i\) is considerably smaller than that for the lattice atoms. Together with ionization it is then also necessary to take into account the attachment of electrons.

We cannot enter here into the details of the calculations and shall confine ourselves only to presenting the results. It turns out that noticeable deviations from Ohm’s law, namely the growth of the electrical conductivity with the field, set in at fields of the order of

\[ \sqrt{6}\,\pi\,\frac{m}{M}\,\frac{\gamma Ze^2/a}{\hbar\omega_0}\cdot \frac{\gamma Ze^2}{a^2} \simeq 10^5\ \mathrm{V/cm}. \]

Thus, we have the following picture of the dependence of the electrical conductivity on the field for semiconductors with an ionic lattice. At low temperatures \(kT\ll \hbar\omega_0\), the mobility up to fields of the order of \(1000\ \mathrm{V/cm}\) does not depend on the field, i.e. Ohm’s law holds, despite the fact that the mean energy of the electrons may become considerably greater than \({}^{3}/_{2}kT\). The mobility in this case decreases exponentially with temperature [see (3,100)]. Further, an increase of the electrical conductivity begins, which becomes significant at fields of the order of \(10^5\ \mathrm{V/cm}\). In these fields the mobility ceases to depend on the temperature \(\left(\text{for } kT\ll \hbar\omega_0,\ N_q+\frac12\simeq \frac12\right)\).

For \(kT\gg \hbar\omega_0\), the electrical conductivity in fields smaller than \(10^5\ \mathrm{V/cm}\) does not depend on the field. The mobility is proportional to

\[ \frac{1}{\sqrt{T}} \]

[see formula (3,101)]. In fields greater than \(10^5\ \mathrm{V/cm}\), the electrical conductivity increases with the field both because of the increase in mobility and because of the increase in the concentration of electrons owing to ionization. At the same time the mean energy of the electrons also increases. As for the resulting concentration of electrons in a strong field, in order to determine it, besides the probability of ionization it is necessary to know also the probability of recombination or attachment of electrons. This latter probability, and its dependence on the concentration itself, depend on the arrangement and filling of the local energy levels. A similar calculation for semiconductors with an atomic lattice was made by N. L. Pisarenko\(^{18}\).

It may happen that it is easier for electrons with high energy to spend their energy, for example, on excitation of atoms, than on their

ionization. In that case the concentration of electrons will not differ from the equilibrium concentration. The increase in electrical conductivity in a strong field will then be due only to the increase in the mobility of the electrons.

We see that the behavior of semiconductors with an ionic lattice in a strong field differs considerably from the behavior of semiconductors with an atomic lattice, and seems to be in better agreement with the experimental facts.

All the arguments of this section, of course, are equally applicable both to electrons in the upper, almost free band of levels and to positive holes in the almost filled lower band.

Phenomena associated with electrical breakdown are considerably more complex and should probably be considered taking into account the formation of space charges and the very strong nonuniform fields associated with them, which arise owing to the propagation of the avalanche^19.

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