DEGENERACY OF THE ELECTRON GAS IN SEMICONDUCTORS

A. G. Samoilovich and L. L. Korenblit

CONTENTS

§ 1. Introduction. “Metallic” properties of semiconductors and the role of degeneracy of the electron gas . . . . . . . . . . . . . . . . . . . . . . 578

§ 2. Equilibrium of electrons and holes in semiconductors
(2.1). Chemical potential and degeneracy of the electron gas. (2.2). Band model and energy spectrum of electrons in semiconductors. (2.3). Equation for determining the chemical potential. (2.4). Neutrality equation for a semiconductor . . . . . . . . . . . . . . . . . . . . . . 579

§ 3. Determination of the temperature dependence of the chemical potential in semiconductors
(3.1). Graphical methods for determining the chemical potential. (3.2). Approximate analytical methods for determining the chemical potential in semiconductors. (3.3). Temperature dependence of the chemical potential in amphoteric semiconductors. (3.4). Degeneracy temperature . . . . . . . . . . . . . . . . . . . . . . . . . 586

§ 4. Influence of degeneracy on the magnetic properties of the electron gas in semiconductors
(4.1). Formulas for the magnetic susceptibility of an electron gas. (4.2). Magnetic properties of gray tin. (4.3). Magnetic properties of the intermetallic compound Mg₂Sn . . . . . . . . . . . . . . . . . . . . . 595

§ 5. Influence of degeneracy of the electron gas on kinetic coefficients in semiconductors
(5.1). Dependence of the mean free path on energy. (5.2). Summary of formulas for kinetic coefficients at any degree of degeneracy of current carriers. (5.3). Approximate calculation of Fermi integrals. (5.4). Graphical study of kinetic coefficients. (5.5). Case of mixed conductivity . . . . . . . . . . . . . . . . . . . . 602

§ 6. Scattering of electrons by impurity ions in degenerate semiconductors . . . . . . . . . . . . . . . . . . . . 610

§ 7. Comparison of the theory with experimental data
(7.1). Electrical conductivity and the Hall effect in impurity silicon and germanium (case of partial degeneracy). (7.2). Case of strong degeneracy. (7.3). Degeneracy of the electron gas and thermal conductivity of semiconductors. (7.4). Hall effect in germanium at low temperatures. (7.5). Temperature dependence of thermopower in the presence of degeneracy . . . . . . . . . . . . . . . . . 612

§ 8. Some unsolved questions in the theory of degeneracy of the electron gas in semiconductors. (8.1). Electrical conductivity of strongly degenerate semiconductors. (8.2). The question of impurity bands. (8.3). Electrical conductivity of PbS. (8.4). The Hall effect in germanium at low temperatures. (8.5). Kinetic phenomena in the impurity band . . . . 620
§ 9. Brief outline of the development of the theory of degenerate semiconductors . . . . 627
§ 10. Conclusion . . . . 629
Cited literature . . . . 629

§ 1. INTRODUCTION. “METALLIC” PROPERTIES OF SEMICONDUCTORS AND THE ROLE OF DEGENERACY OF THE ELECTRON GAS

The great role of semiconductors in various fields of technology has led to the need to investigate in greater detail the various physical properties of semiconductors. In doing so it has become clear that in a number of cases degeneracy of the electron gas in semiconductors begins to play an essential role. As is known, semiconductors differ from metals chiefly in that the number of current carriers in the former depends substantially on temperature, whereas in the latter it is determined exclusively by the nature of the metal. At the same time the concentration of current carriers in semiconductors is usually so small that the corresponding distribution over energies differs little from the Maxwell—Boltzmann distribution. Later it became clear that in a very large number of cases the application of the Maxwell—Boltzmann distribution to electrons in semiconductors does not lead to correct quantitative results; it was necessary to introduce for the calculation the general formulas of Fermi—Dirac statistics. Thus, for example, it was found that in some semiconductors (PbS^1, SiC^2, GaSb^3, Si^4, Ge^5, etc.) within certain temperature intervals and at certain impurity concentrations the number of current carriers remains constant. Thus, under certain conditions the “metallic” properties of semiconductors begin to appear. This is reflected in almost all electrical properties of semiconductors: in electrical conductivity, thermoelectric power, and thermal conductivity. The appearance of “metallic” properties is observed in semiconductors with low specific resistance. On the other hand, precisely such semiconductors are of technical importance as materials for the construction of semiconductor diodes and triodes.

From the physical point of view, in all these cases there is degeneracy of the electron gas. As is known, by degeneracy of the electron gas is understood such a state of it in which the distribution over energies differs from the Maxwell—Boltzmann one, which may be regarded as the limiting case of the Fermi—Dirac distribution. It should be noted, however, that the degeneracy of the electron gas in semiconductors is of an essentially different character than in metals. In the latter we have the case of very strong de-

generation. In doing so, one can again make use of the well-known simplification of the formulas of quantum statistics, which considerably facilitates the calculations. In semiconductors, as we shall see below, in some cases a strong degeneracy of the electron gas also occurs, and this case is of great theoretical interest, despite the fact that such semiconductors as yet have no special technical application. Most often, however, in semiconductors an intermediate case obtains, when neither one nor the other approximation can be used. This requires the development of special methods of calculation that make it possible to consider the physical properties of semiconductors in this general case.

In view of the importance of this problem as a whole, we have considered it expedient in the present article to give a systematic exposition of the questions pertaining to it.

§ 2. EQUILIBRIUM OF ELECTRONS AND HOLES IN SEMICONDUCTORS

2.1. Chemical potential and degeneracy of the electron gas. The Fermi—Dirac distribution can be written in the form

\[ \bar n_i=\frac{1}{e^{\frac{\varepsilon_i-\mu}{kT}}+1}, \tag{2.1} \]

where \(\bar n_i\) is the mean number of particles in the microstate number \(i\), \(\varepsilon_i\) is the energy of this microstate, and \(\mu\) is the chemical potential. Let \(\varepsilon_0\) be the energy of the lowest state. If \([(\varepsilon_0-\mu)/kT]\gg 0\), then the unity in the denominator of (2.1) may be neglected, which corresponds to the absence of degeneracy of the electron gas,

\[ \bar n_i=e^{\frac{\mu-\varepsilon_i}{kT}} \tag{2.2} \]

(the Maxwell—Boltzmann distribution). One may conventionally take as the boundary between the degenerate and nondegenerate states the value

\[ \mu=\varepsilon_0. \tag{2.3} \]

For \(\mu\ll \varepsilon_0\) we shall speak of a nondegenerate state of the system. Conversely, the system may be regarded as completely degenerate if \(\mu\gg \varepsilon_0\). The intermediate case corresponds to partial degeneracy of the system. Thus, in order to construct a theory of the degeneracy of the electron gas in semiconductors, it is necessary, first of all, to be able to determine the chemical potential of electrons or holes. It is often convenient to normalize the energy so that \(\varepsilon_0=0\). Then the magnitude and sign of the chemical potential serve as the criterion of degeneracy of the system of current carriers. For \(\mu\ll 0\) we have the Maxwell—Boltzmann ...

case; for small absolute values of \(\mu\) it is already necessary to take into account the degeneracy of the electron gas. To determine \(\mu\) it is necessary to specify a semiconductor model.

Fig. 1. Energy diagram of an impurity semiconductor.

2.2. Band model and energy spectrum of electrons in semiconductors. We shall consider semiconductors within the framework of the usual band model. According to the latter, the energy spectrum of valence electrons in a semiconductor consists of the main (valence) band, the conduction band, and local impurity (acceptor and donor) levels (Fig. 1). Let us introduce the following notation:

\(\Delta E\) — the width of the forbidden region in the energy spectrum between the top of the main band and the bottom of the conduction band,

\(\Delta E_a\) — the width of the forbidden region between the top of the main band and the acceptor impurity levels,

\(\Delta E_d\) — the width of the forbidden region between the bottom of the conduction band and the donor impurity levels,

\(N_a,\ N_d\) — the concentrations of impurity acceptor and donor atoms in the semiconductor, respectively.

2.3. Equation for determining the chemical potential. The methods of statistical physics make it possible easily to find, in each particular case, the equation for determining the chemical potential. The essence of the method applied here is as follows[^6]. If the energy of the electron system can be written in the form

\[ E=\sum n_i \varepsilon_i, \tag{2.4} \]

where \(n_i\) are the occupation numbers of the corresponding microstates, then the thermodynamic potential \(\Omega\) is introduced by means of the relation

\[ e^{-\frac{\Omega}{kT}} = \sum_{n_1,n_2,\ldots} e^{\frac{\sum_i n_i(\mu-\varepsilon_i)}{kT}} . \tag{2.5} \]

Since electrons obey the Pauli principle, each \(n_i\) can take only two values: 0, 1. In this case (2.5) is easily transformed into the form

\[ e^{-\frac{\Omega}{kT}} = \prod_i \left(1+e^{\frac{\mu-\varepsilon_i}{kT}}\right), \tag{2.6} \]

or

\[ \Omega=-kT\sum_i \ln \left(1+e^{\frac{\mu-\varepsilon_i}{kT}}\right). \tag{2.7} \]

It is easy to verify, by simply differentiating both sides of (2.7) with respect to \(\mu\), that the total number of electrons is

\[ N=-\left(\frac{\partial \Omega}{\partial \mu}\right)_T . \tag{2.8} \]

Thus, knowing the energy spectrum of the electrons, one can compute \(\Omega\) with the aid of (2.7). Equation (2.8) can then serve to determine the chemical potential as a function of the temperature and of the parameters determining the energy spectrum of the electrons.

2.4. The neutrality equation of a semiconductor

Let us now apply the general theory set forth above specifically to semiconductors. We first consider the case of an electronic impurity semiconductor, when there are only donor local levels. Let \(\varepsilon_i\) denote the energy of electrons in the conduction band, \(\varepsilon'_i\) the energy of an electron in the valence band, and \(E_j\) the energy of an electron on a local level. We assume that at absolute zero the valence band is completely filled and that there is one electron on each local impurity level (the impurity levels are assumed to be nondegenerate), while there are no electrons in the conduction band. Since the interaction of conduction electrons is neglected, \(\Omega\), as it turns out, is composed additively of parts referring to the conduction band \((\Omega_1)\), the valence band \((\Omega_2)\), and the local levels \((\Omega_3)\):

\[ \Omega=\Omega_1+\Omega_2+\Omega_3 . \tag{2.9} \]

We proceed to calculate each of these parts separately.

a. Calculation of \(\Omega_1\). Using the effective-mass method, the energy of a conduction electron can be written in the following form:

\[ \varepsilon_i=f_n\frac{p_i^2}{2m}, \tag{2.10} \]

where

\[ f_n=\frac{m}{m_n^*}, \tag{2.11} \]

\(m_n^*\) is the effective mass of the electron (as the common origin for measuring electron energies in a semiconductor, as usual, the energy corresponding to the bottom of the conduction band is adopted). In formula (2.7) one may pass from summation over microstates to integration

over the energies, and it is necessary to introduce, as is known, the statistical weight

\[ g(\varepsilon)\,d\varepsilon = 2\,\frac{2\pi(2m)^{3/2}}{h^3 f_n^{3/2}}\sqrt{\varepsilon}\,d\varepsilon \tag{2.12} \]

(the volume of the semiconductor is taken to be equal to unity). Then we obtain:

\[ \Omega_1 = -\frac{4\pi kT(2m)^{3/2}}{h^3 f_n^{3/2}} \int_0^\infty \ln\left(1+e^{\frac{\mu-\varepsilon}{kT}}\right) \sqrt{\varepsilon}\,d\varepsilon . \tag{2.13} \]

It is convenient to pass to the dimensionless integration variable \(x=(\varepsilon/kT)\). Denoting also by

\[ \mu^*=\frac{\mu}{kT} \tag{2.14} \]

the reduced chemical potential, we obtain:

\[ \Omega_1 = -\frac{4kT}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_n} \right)^{3/2} \int_0^\infty \ln(1+e^{\mu^*-x})\sqrt{x}\,dx . \tag{2.15} \]

Hence, according to (2.8), for the number of conduction electrons we obtain:

\[ N_n = \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_n} \right)^{3/2} F_{1/2}(\mu^*), \tag{2.16} \]

where

\[ F_m(\mu^*)=\int_0^\infty \frac{x^m\,dx}{e^{x-\mu^*}+1} \tag{2.17} \]

are the so-called generalized Fermi integrals.

1. Calculation of \(\Omega_2\). The energy of the system of electrons located in the ground band is

\[ E'=\sum_i n_i'\varepsilon_i', \tag{2.18} \]

where \(n_i'\) are the occupation numbers of the microstates in the ground band. The summation is carried out over all (both occupied and free) microstates of the ground band. We now introduce, instead of the occupation numbers \(n_i'\), the quantities

\[ m_i=1-n_i', \tag{2.19} \]

which can be interpreted as the occupation numbers of holes, since, like the occupation numbers \(n_i'\), \(m_i\) take two values: 0 or 1, pri-

where \(m_i = 1\) for free states. In addition to \(m_i\), let us also introduce the hole energy \(\varepsilon_i'\), defined by the relation

\[ \varepsilon_i' = -\Delta E - \varepsilon_i'', \tag{2.20} \]

where

\[ \varepsilon_i'' = f_p \frac{p_i^2}{2m}, \tag{2.21} \]

\[ f_p = \frac{m}{m_p^*}, \tag{2.22} \]

\(m_p^*\) is the effective mass of a hole. Formula (2.20) means that we take into account the already chosen origin of energy and that the hole energy, as is customary, is measured from the upper edge of the main band into the main band. Then the energy of the system of electrons located in the main band will take the form

\[ E' = \mathcal{E} + \sum_i m_i \left(\varepsilon_i'' + \Delta E\right), \tag{2.23} \]

where \(\mathcal{E} = \sum_i \varepsilon_i'\) is the energy of the electrons in the main band at absolute zero temperature. Using expression (2.23), we obtain for \(\Omega_e\):

\[ \Omega_e = \mathcal{E} - \mu N_0 - kT \sum_i \ln \left(1 + e^{-\frac{\mu + \Delta E + \varepsilon_i''}{kT}}\right), \tag{2.24} \]

where \(N_0\) is the number of states in the main band. Analogously to how this was done for conduction electrons, we obtain for the number of electrons in the valence band \(N_n'\):

\[ N_n' = -\left(\frac{\partial \Omega}{\partial \mu}\right)_T = N_0 - \frac{4}{\sqrt{\pi}}\left(\frac{2\pi m kT}{f_p h^2}\right)^{3/2} F_{1/2}\left(-\mu^* - \Delta E^*\right), \tag{2.25} \]

where \(\Delta E^* = (\Delta E/kT)\). Taking into account that

\[ N_0 - N_n' = N_p \tag{2.26} \]

is the number of holes in the main band, we have:

\[ N_p = \frac{4}{\sqrt{\pi}}\left(\frac{2\pi m kT}{f_p h^2}\right)^{3/2} F_{1/2}\left(-\mu^* - \Delta E^*\right). \tag{2.27} \]

Let us note that, from comparison of this formula with (2.16), the quantity

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

can be interpreted as the chemical potential of holes.

c. Calculation of \(\Omega_3\). With our choice of the origin for the electron energy scale, an electron located on an impurity level has energy \(E_i=-\Delta E_{\text{д}}\). If the impurity term is an \(s\)-term, then two electrons can fit on it. However, in fact this cannot occur because of the strong repulsion of electrons located on one impurity atom. This circumstance must be taken into account in calculating \(\Omega_3\). Since we neglect the interaction between electrons located on different impurity atoms, instead of (2.6) we obtain the following formula for determining \(\Omega_3\):

\[ e^{-\frac{\Omega_3}{kT}} = \left( \sum_{n_1,n_2} e^{\frac{(n_1+n_2)(\mu+\Delta E_{\text{д}})}{kT}} \right)^{N_{\text{д}}}. \tag{2.28} \]

Here, since two electrons cannot be located simultaneously on one impurity level, when summing over \(n_1\) and \(n_2\) one must exclude the case when both of them are simultaneously equal to unity. Then we easily obtain:

\[ e^{-\frac{\Omega_3}{kT}} = \left( 1+2e^{\frac{\mu+\Delta E_{\text{д}}}{kT}} \right)^{N_{\text{д}}}, \]

whence

\[ \Omega_3 = -N_{\text{д}}kT\ln \left( 1+2e^{\frac{\mu+\Delta E_{\text{д}}}{kT}} \right). \tag{2.29} \]

For the number of electrons on donor levels \(N_{\text{л}}\), in the usual way we obtain:

\[ N_{\text{л}} = \frac{N_{\text{д}}} {\frac{1}{2}e^{-\Delta E_{\text{д}}^{*}-\mu^{*}}+1}. \tag{2.30} \]

Let us note that, in the general case of a \(g\)-fold degenerate impurity level, we have:

\[ N_{\text{л}} = \frac{N_{\text{д}}} {\frac{1}{g}e^{-\Delta E_{\text{д}}^{*}-\mu^{*}}+1}. \tag{2.31} \]

Differentiating (2.9) with respect to \(\mu\) and taking into account (2.15), (2.25), and (2.30), we obtain for the total number of electrons in the main band, in the conduction band, and on local levels:

\[ N=N_n+N'_n+N_{\text{л}}. \tag{2.32} \]

On the other hand, \(N=N_0+N_{\text{д}}\). Therefore we have:

\[ (N_{\text{д}}-N_{\text{л}})+N_p=N_n. \tag{2.33} \]

This equality asserts that the total number of conduction electrons is equal to the total number of holes on the impurity levels \((N_{\text{д}}-N_{\text{л}})\)

and in the valence band \(N_p\). Therefore it is called the neutrality equation of an impurity electronic semiconductor. Substituting the explicit expressions for \(N_n\), \(N_p\), and \(N_l\) into (2.33), we can rewrite (2.33) in the following form:

\[ \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_n} \right)^{3/2} F_{1/2}(\mu^*) = \frac{N_{\mathrm d}}{2e^{\Delta E_{\mathrm d}^*+\mu^*}+1} + \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_p} \right)^{3/2} F_{1/2}(-\mu^*-\Delta E^*). \tag{2.34} \]

In the case of an intrinsic semiconductor without impurities \((N_{\mathrm d}=N_a=0)\), the neutrality equation takes the form

\[ N_p=N_n, \]

or, explicitly,

\[ F_{1/2}(\mu^*)= \left( \frac{f_n}{f_p} \right)^{3/2} F_{1/2}(-\mu^*-\Delta E^*). \tag{2.35} \]

Analogously to the preceding case, for an impurity hole semiconductor \((N_{\mathrm d}=0,\ N_a\ne 0)\), the neutrality equation assumes the following form:

\[ N_p=(N_a-N'_{\mathrm l})+N_n, \tag{2.36} \]

where \(N'_{\mathrm l}\) is the number of holes on acceptor local levels:

\[ N'_{\mathrm l}= \frac{N_a}{2e^{-\Delta E_a^*+\mu^*+\Delta E^*}+1}, \tag{2.37} \]

or, writing out the full expressions for \(N_n\), \(N_p\), \(N'_{\mathrm l}\),

\[ \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_p} \right)^{3/2} F_{1/2}(-\mu^*-\Delta E^*) = \]

\[ = \frac{N_a}{\frac{1}{2}e^{\Delta E_a^*-\Delta E^*-\mu^*}+1} + \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{h^2 f_n} \right)^{3/2} F_{1/2}(\mu^*). \tag{2.38} \]

Finally, in the case of an amphoteric semiconductor \((N_{\mathrm d}\ \text{and}\ N_a\ \text{both nonzero})\), the neutrality equation may be written in the following form:

\[ (N_{\mathrm d}-N_{\mathrm l})+N_p=(N_a-N'_{\mathrm l})+N_n, \tag{2.39} \]

or

\[ \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{f_n h^2} \right)^{3/2} F_{1/2}(\mu^*) + \frac{N_a}{\frac{1}{2}e^{\Delta E_a^*-\Delta E^*-\mu^*}+1} = \]

\[ = \frac{4}{\sqrt{\pi}} \left( \frac{2\pi m kT}{f_p h^2} \right)^{3/2} F_{1/2}(-\mu^*-\Delta E^*) + \frac{N_{\mathrm d}}{2e^{\Delta E_{\mathrm d}^*+\mu^*}+1}. \tag{2.40} \]

§ 3. DETERMINATION OF THE TEMPERATURE DEPENDENCE OF THE CHEMICAL POTENTIAL IN SEMICONDUCTORS

The neutrality equations (2.34), (2.35), (2.38), and (2.40) can be used to determine the chemical potential and its dependence on temperature in a wide variety of specific cases. At the same time it must be noted that these equations contain Fermi integrals of half-integer index \(F_{1/2}\), which, generally speaking, cannot be expressed in terms of elementary functions. Therefore the neutrality equations can be solved analytically for \(\mu\) only in the complete absence of degeneracy (when the replacement of \(F_{1/2}(\mu^*)\) by \(\dfrac{\sqrt{\pi}}{2} e^{\mu^*}\) becomes possible), and even then only in the simplest cases. Finding the temperature dependence of the chemical potential in the general case of an arbitrarily degenerate electron gas therefore requires the development of special approximate methods. These methods can be divided into two groups: a) graphical and b) analytical methods for determining the chemical potential in semiconductors. Below we shall briefly discuss some of them.

3.1. Graphical methods for determining the chemical potential. Graphical methods for determining the temperature dependence of the chemical potential have been proposed by a number of authors \(^{7-9}\). All of them, however, suffer from one common shortcoming, namely: they are either applicable only in certain particular cases, or they do not take into account the possibility of degeneracy of the electron gas. The method presented in \(^{10}\), though quite general in itself, is too cumbersome in its implementation. Here we shall set forth the idea of a simple method largely free of the shortcomings indicated above.

As an example, let us consider the case of a Wilson impurity electronic semiconductor, when only electron transitions between impurity donor levels and the conduction band are taken into account. The neutrality equation according to (2.34) has in this case the following form:

\[ A\left(\frac{T}{f_n}\right)^{3/2} F_{1/2}(\mu^*) = \frac{N_{\mathrm{d}}}{2e^{\Delta E_{\mathrm{d}}^*+\mu^*}+1}, \tag{3.1} \]

where

\[ A=\frac{4}{\sqrt{\pi}}\left(\frac{2\pi m k}{h^2}\right)^{3/2}=\mathrm{const}. \]

We shall regard the right- and left-hand sides of equality (3.1) as functions of \(\mu^*\), parametrically dependent on \(T\), \(f_n\), \(N_{\mathrm{d}}\), and \(\Delta E_{\mathrm{d}}^*\). Then the required quantity \(\mu^*\), for given values of these parameters, is determined as the abscissa of the point of intersection of the graphs

\[ A\left(\frac{T}{f_n}\right)^{3/2} F_{1/2}(\mu^*) \]

and

\[ \frac{N_{\mathrm{d}}}{2e^{\Delta E_{\mathrm{d}}^*+\mu^*}+1}. \]

The essence of the method consists in such a choice of scales along the coordinate axes that the right- and left-

part of (3.1) always, for any values of the parameters \(T\), \(f_n\), \(N_{\mathrm{d}}\), and \(\Delta E_{\mathrm{d}}^{*}\), could be represented by two universal graphs, whose form would not depend on the particular value of these parameters. Only the origins of the coordinate axes on which these universal curves are plotted can depend on the latter. It is easy to see that this can be achieved in the following way. If a logarithmic scale is chosen on the ordinate axis, then the graph of the expression \(A\left(\dfrac{T}{f_n}\right)^{3/2}F_{1/2}(\mu^{*})\), considered as a function of \(\mu^{*}\) for arbitrary \(T\) and \(f_n\), coincides with the graph of the dependence \(AF_{1/2}(\mu^{*})\), up to a shift of the origin along the ordinate axis by the amount \(\dfrac{3}{2}(\ln T-\ln f_n)\). As for the dependence of the right-hand side of (3.1) on \(\mu^{*}\), it likewise (when a logarithmic scale is used on the ordinate axis) coincides, up to a shift of the origin along the ordinate axis by \(N_{\mathrm{d}}\) and along the abscissa axis by \(\ln 2+\Delta E_{\mathrm{d}}^{*}\), with the universal dependence \(\ln(1+y)\) on \(\ln y\). Thus, for any values of the parameters \(T\), \(f_n\), \(N_{\mathrm{d}}\), and \(\Delta E_{\mathrm{d}}^{*}\), it proves possible to find the corresponding values of \(\mu^{*}\) as the abscissas of the points of intersection of two “universal” curves \(AF_{1/2}(\mu^{*})\) and \(\ln(1+y)=f(\ln y)\).

When electron transitions from the principal band are taken into account (equation (2.34)), the construction becomes somewhat more complicated, but the method of finding \(\mu^{*}\) remains the same. In this case \(\mu^{*}\) is found as the abscissa of the point of intersection of the universal curves \(AF_{1/2}(\mu^{*})\) and the curve representing both terms on the right-hand side of (2.34). We note that, with the logarithmic scale along the ordinate axis chosen by us, the dependence of the right-hand side of (2.34) on \(\mu^{*}\), except for a small region of values of \(\mu^{*}\) where \(N_{\mathrm{d}}-N_{\mathrm{a}}\simeq N_p\), coincides alternately either with the dependence of \(N_{\mathrm{d}}-N_{\mathrm{a}}\) on \(\mu^{*}\), or with the dependence of \(N_p\) on \(\mu^{*}\). Thus, outside the region \(N_{\mathrm{d}}-N_{\mathrm{a}}\simeq N_p\), where the corresponding interpolation is required, the right-hand side of (2.34) can be represented either by the universal graph \(\ln(1+y)=f(\ln y)\), or by the graph \(AF_{1/2}(\mu^{*})\), previously turned over with respect to the ordinate axis and shifted along the abscissa axis by \(\Delta E^{*}\). This also solves the problem posed. Quite analogous constructions are applicable in all the remaining cases (acceptor-impurity semiconductors, amphoteric semiconductors, etc.), and we shall not dwell on this here.

The values of \(F_{1/2}(\mu^{*})\) in a sufficiently wide interval of values of \(\mu^{*}\) have been found numerically by a number of authors. Along with other Fermi integrals, the values of \(F_{1/2}(\mu^{*})\) have been tabulated in \(^{12}\) and in \(^{13}\) (see also \(^{9}\)).

The application of the graphical method to the determination of the chemical potential in semiconductors and of its temperature dependence has made it possible to clarify a number of interesting features. The onset of degeneracy of the electron gas is favored by large concentrations of impurity of one sign and small values of the corresponding

activation energies. In intrinsic semiconductors, or in amphoteric semiconductors with \(N_d \approx N_a\) and \(\Delta E_d \approx \Delta E_a\), degeneracy of the current carriers practically does not occur at any temperature, provided only that the difference in the effective masses of electrons and holes does not reach \(\sim 10^2\). In the case of an impurity semiconductor with one type of carrier, the dependence of \(\mu^*\) on temperature has a whole series of special features. The curve of the dependence of \(\mu^*\) on \(T\), if it is considered over a sufficiently wide temperature interval, is always a curve with a maximum (Fig. 2) (with a minimum in the case of a hole semiconductor), and the rise on the low-temperature side is steeper than on the high-temperature side. If \(\mu^*_{\max}\) (the value of \(\mu^*\) at the point of the maximum) \(< -2\), then throughout the whole temperature interval the electron gas may be regarded as nondegenerate, and the ordinary Maxwell–Boltzmann statistics may be used for its description.) For \(-2 < \mu^*_{\max} \leq 0\) there exists a temperature region in which allowance for partial degeneracy of the electron gas may become essential. For \(\mu^*_{\max} > 0\) there is a temperature interval bounded by two degeneracy temperatures \(T_1\) and \(T_2\) (Fig. 2), within which the electron gas is degenerate. In this case the maximum on the curve \(\mu^*=\mu^*(T)\) becomes less sharply pronounced, especially on the high-temperature side. It turns out that as long as the concept of local* impurity centers remains valid, degeneracy of the electron gas never (for any values of the impurity concentration and of the activation energy of the latter) acquires the character of strong degeneracy in the sense in which we encounter this concept in the case of metals, since \(\mu^*_{\max}<2\).

Fig. 2. Temperature dependence of \(\mu^*\) in an impurity electron semiconductor.

The physical mechanism leading to the fact that the curve \(\mu^*(T)\) turns out to be a curve with a maximum is as follows. At low temperatures, transitions of electrons from local impurity levels into the conduction band are hindered—the density of electrons in the conduction band

*) Correspondingly, in the case of a hole semiconductor all relations remain the same; it is only necessary to replace \(\mu^*\) by \(-\mu^*-\Delta E^*\). Thus, for example, the condition for absence of degeneracy in the hole gas has the form
\(\mu^*_{\min} > 2-\Delta E^*\). Without loss of generality, in what follows, for definiteness, we shall speak mainly of the electron gas.

conductivity is very small and the electron gas is nondegenerate. The level of the chemical potential in this case passes somewhere in the middle of the gap \(\Delta E_{\text{д}}\). At higher temperatures, such that \(kT \simeq \Delta E_{\text{д}}\), such transitions are strongly activated; however, the thermal energy of the conduction electrons is insufficient to “scatter” the electrons to a significant extent over the states in the conduction band. The electrons in the conduction band occupy compactly, for the most part, only the lowest states of the conduction band. The level of the chemical potential rises strongly and, under some favorable conditions, may even rise above the bottom of the conduction band. With a further increase in temperature, a noticeable “scattering” of electrons over the states of the conduction band begins, and the level of the chemical potential begins slowly to fall. The decrease of the chemical potential may be promoted to a considerable extent by transitions of electrons from the valence band, if the temperature is high enough to bring the mechanism of intrinsic conductivity into operation.

3.2. Approximate analytical methods for determining the chemical potential in semiconductors

In order to make it possible to describe analytically the temperature dependence of the chemical potential in semiconductors, it is necessary first to interpolate the Fermi integral \(F_{1/2}(\mu^*)\), entering into the neutrality equation, with one or another degree of accuracy and in one or another interval of values of \(\mu^*\), by some simple expression that permits solution of the neutrality equation with respect to \(\mu\). Unfortunately, even in such a simplified formulation of the problem, the analytical determination of the dependence \(\mu^*(T)\) is difficult and can be carried through to the end in by no means all cases. Therefore the analytical methods for determining the chemical potential, despite a whole series of advantages of the latter over graphical methods, are considerably inferior to them in generality. To this it must be added that even in the case when \(\mu^*(T)\) can be determined in analytically closed form, the corresponding formulas often prove too cumbersome for application. Below we shall consider the question of the analytical determination of the temperature dependence of the chemical potential in the case of an impurity electronic semiconductor (equation (2.34)).

Following \(^{14}\), the integral \(F_{1/2}(\mu^*)\) in the interval of variation of \(\mu^*\) that is practically important from the point of view of the theory of semiconductors, \(-\infty < \mu^* \leq 1.5\), can be approximated with a high degree of accuracy by the following expression:

\[ F_{1/2}(\mu^*)=\frac{\sqrt{\pi}}{2}\, \frac{e^{\mu^*}}{1+b e^{\mu^*}+c e^{2\mu^*}}, \tag{3.2} \]

where \(b=0.369,\ c=-0.028\).

In ^9 a number of other, simpler approximating formulas are given. These formulas, however, are applicable in a narrower interval of variation of \(\mu^*\), and we shall not present them here. Substitution of (3.2) into (2.34) leads in the general case to an algebraic equation of high degree with respect to the unknown \(e^{\mu^*}\). Therefore we shall seek the solution in various limiting cases, relying on results obtained by the graphical method.

a. Temperature dependence of the chemical potential at low and intermediate temperatures. By low and intermediate temperatures we shall mean temperatures satisfying, respectively, the inequalities \(T \ll -\dfrac{\Delta E}{k}\) and \(T < -\dfrac{\Delta E}{k}\). In this case, obviously, electron transitions from the valence band to the conduction band may be neglected. Indeed, from \(\Delta E^* \gg 1\) it follows that \(F_{1/2}(-\mu^*-\Delta E^*) \ll F_{1/2}(\mu^*)\), and the hole component in the right-hand side of (2.34) may be neglected. The neutrality equation then takes the following form:

\[ -\frac{z_n e^{\mu^*}}{1+be^{\mu^*}+ce^{2\mu^*}} = -\frac{N_{\mathrm{d}}}{2e^{\Delta E_{\mathrm{d}}^*+\mu^*}+1}, \tag{3.3} \]

where

\[ z_n=2\left(\frac{2\pi m kT}{f_n h^2}\right)^{3/2}, \]

whence

\[ e^{\mu^*}= \frac{ -z_n+N_{\mathrm{d}}b+ \sqrt{(z_n-N_{\mathrm{d}}b)^2+4N_{\mathrm{d}}\left(2z_n e^{\Delta E_{\mathrm{d}}^*}-N_{\mathrm{d}}c\right)} }{ 2\left(2z_n e^{\Delta E_{\mathrm{d}}^*}-N_{\mathrm{d}}c\right) }. \tag{3.4} \]

Formula (3.4) is based on the obvious assumption that \(\Delta E_{\mathrm{d}}^*\) is much smaller than \(\Delta E\). It is easy to verify that the dependence \(\mu^*(T)\) described by (3.4) is a curve with a maximum.

b. The case of high temperatures \(T \lesssim -\dfrac{\Delta E}{k}\). In this case the balance of electronic transitions begins to be strongly affected by intrinsic electron transitions from the valence band to the conduction band. As was said above, in this case the system of conduction electrons is far from degeneracy (provided only that the difference between \(f_n\) and \(f_p\) does not reach the order \(10\)—\(10^2\)). Therefore \(F_{1/2}(\mu^*)\) may in this case simply be replaced by \(\dfrac{\sqrt{\pi}}{2}e^{\mu^*}\), and \(F_{1/2}(-\mu^*-\Delta E^*)\) by \(\dfrac{\sqrt{\pi}}{2}e^{-\mu^*-\Delta E^*}\). In addition, the atoms of the donor impurity may be regarded, in the region of intrinsic conduction, as completely ion-

ionized, so that

\[ \frac{N_{\mathrm{d}}}{2e^{\Delta E_{\mathrm{d}}^{*}+\mu^{*}}+1}\simeq N_{\mathrm{d}}. \]

Therefore the neutrality equation in the high-temperature region can be approximately written in the following form:

\[ z_n e^{\mu^{*}}=N_{\mathrm{d}}+\frac{z_p}{b\,e^{\Delta E^{*}}e^{\mu^{*}}}, \tag{3.5} \]

whence

\[ e^{\mu^{*}}=\frac{N_{\mathrm{d}}}{2z_n} +\sqrt{\left(\frac{N_{\mathrm{d}}}{2z_n}\right)^2 +\left(\frac{f_n}{f_p}\right)^{3/2}e^{-\Delta E^{*}}}. \tag{3.6} \]

Formulas (3.4) and (3.6) contain, as special cases, the usual simplified dependences \(\mu^{*}(T)\), which are ordinarily cited in the literature on semiconductors\({}^{15*}\). Thus, for example, if \(\mu_{\max}^{*}<-2\) and degeneracy does not set in even at low temperatures, we may set the constants \(b\) and \(c\) in formula (3.4) equal to zero. Then we obtain:

\[ e^{\mu^{*}}=\frac{e^{-\Delta E_{\mathrm{d}}^{*}}}{4} \left[-1+\sqrt{1+8\,\frac{N_{\mathrm{d}}}{z_n}\,e^{\Delta E_{\mathrm{d}}^{*}}}\right]. \tag{3.7} \]

Since usually

\[ \frac{N_{\mathrm{d}}}{z_n}e^{\Delta E_{\mathrm{d}}^{*}}\gg 1, \]

we have:

\[ e^{\mu^{*}}=\sqrt{\frac{N_{\mathrm{d}}}{2z_n}}\, e^{-\frac{\Delta E_{\mathrm{d}}^{*}}{2}}. \tag{3.8} \]

From formula (3.6) one can also easily obtain (putting \(N_{\mathrm{d}}=0\)) the usual formula for the chemical potential in the case of semiconductors without impurity,

\[ e^{\mu^{*}}=\left(\frac{f_n}{f_p}\right)^{3/4} e^{-\frac{\Delta E^{*}}{2}}, \tag{3.9} \]

which indicates that, when the effective masses of electrons and holes differ only slightly, the level of the chemical potential in this case lies in the middle of the forbidden energy band.

\[ \text{——————} \]

\({}^{*}\) Let us only note that a certain difference between the formulas given here and those usually accepted (see, for example, \({}^{15}\)) is due to the fact that the density of holes on donor impurity levels was written in the form

\[ \frac{N_{\mathrm{d}}}{\left(e^{\Delta E_{\mathrm{d}}^{*}+\mu^{*}}+1\right)} \]

instead of the correct

\[ \frac{N_{\mathrm{d}}}{\left(2e^{\Delta E_{\mathrm{d}}^{*}+\mu^{*}}+1\right)}. \]

Let us note that the case of an impurity hole semiconductor differs little from the case considered of an electronic impurity semiconductor. It is only necessary to replace \(\mu^*\) by \(-\mu^*-\Delta E^*\), \(N_d\) by \(N_a\), and \(\Delta E_d^*\) by \(\Delta E_a^*\). In this case, however, there is a certain difference connected with the fact that, instead of the term

\[ \frac{N_d}{\left(2e^{\Delta E_d^*+\mu^*}+1\right)} \]

from (2.34), in (2.38) there appears a term of the form

\[ \frac{N_a}{\left(\frac{1}{2}e^{\Delta E_a^*-\mu^*-\Delta E^*}+1\right)}. \]

It is necessary to point out that formula (3.9) correctly describes the behavior of the chemical potential in an intrinsic semiconductor up to the point at which degeneration of the electron or hole gas sets in. If the effective masses of electrons and holes differ greatly from one another, then such degeneration may ultimately occur, and formula (3.9) becomes inapplicable. Let us consider this case in more detail. For definiteness, let the effective mass of the electrons be much smaller than the effective mass of the holes (usually \(m_n^* < m_p^*\)). Then, as is seen from (3.9), the chemical potential tends to increase, approaching the conduction band. Therefore the neutrality equation in this case may be written in the following form:

\[ \frac{z_n e^{\mu^*}}{1+be^{\mu^*}+ce^{2\mu^*}} = \frac{z_p}{e^{\Delta E^*+\mu^*}}, \tag{3.10} \]

whence

\[ e^{\mu^*} = \frac{ z_p b+\sqrt{z_p^2 b^2+4z_p\left(z_n e^{\Delta E^*}-z_p c\right)} }{ 2\left(z_n e^{\Delta E^*}-z_p c\right) }. \tag{3.11} \]

Formula (3.11) supplements relation (3.9) for the case \(\mu^* \leq 1.5\), i.e., when degeneration of the electron gas occurs.

3.3. Temperature dependence of the chemical potential in amphoteric semiconductors.

In amphoteric semiconductors, when both \(N_d\) and \(N_a\) are different from zero, the chemical potential must be found with the aid of equation (2.40). At the same time, it is easy to convince oneself that replacing the Fermi integrals according to formula (3.2) leads to an overly cumbersome algebraic equation with respect to \(e^{\mu^*}\). In this case the following considerations may prove very useful. The presence in the semiconductor simultaneously of both donor and acceptor impurities leads to a redistribution of electrons among the states in the energy spectrum. Indeed, from the condition of minimum energy of the semiconductor it follows that at absolute zero temperature not a single acceptor level can remain unoccupied by electrons if there are still electrons on donor levels or

in the conduction band *). This makes it possible to derive formulas for the temperature dependence of the chemical potential that are valid in strictly definite temperature intervals. In doing so it is necessary to distinguish three cases.

a. \(N_a > N_d\). Naturally, as \(T \to 0\) in this case the conduction band is empty and all donor atoms are ionized. At low temperatures, therefore, only transitions of electrons from the valence band to the remaining unoccupied \(N_a - N_d\) acceptor levels take place. One may use here a formula of the type considered above for the case of an impurity semiconductor at low temperatures, replacing only \(N_a\) by \(N_a - N_d\). The level of the chemical potential then passes through the gap \(\Delta E_a\). At higher temperatures \(\left(T \sim \frac{\Delta E_a}{k}\right)\) saturation of the acceptor levels by electrons sets in, and the concentration of holes in the valence band does not increase up to those temperatures at which electron transitions from acceptor levels to donor levels begin actively. The activation energy of such a process is \(\Delta E - (\Delta E_a + \Delta E_d)\). The chemical potential then shifts into the energy region lying between the donor and acceptor levels. It must be taken into account, however, that transitions of electrons from the valence band to the vacated acceptor levels also occur, as well as transitions from donor levels into the conduction band. An electronic component of the system of current carriers appears.

With a further rise in temperature, up to \(T \sim \frac{\Delta E}{k}\), intrinsic transitions of electrons from the valence band to the conduction band appear, and the corresponding formulas for calculating the chemical potential in intrinsic semiconductors may be used.

In the case when \(\Delta E_d\) or \(\Delta E_a\) is much smaller than \(\Delta E\), one may, for simplicity of calculation, neglect the presence of the acceptor (or donor) impurity already in the region of intermediate, and still more in the region of high, temperatures.

At the same time, the considerations indicated are to a large extent qualitative. A rigorous quantitative treatment of the question is possible only with the aid of the graphical method.

b. \(N_a = N_d\). In the region of low and high temperatures this case differs little from the case of an intrinsic semiconductor.

c. \(N_a < N_d\). As \(T \to 0\) all acceptor levels are occupied by electrons, while a part \(N_d - N_a\) of the donor atoms is not ionized. This case, naturally, is to a large extent equivalent to the case examined—

*) This fact is established not only on the basis of general theoretical considerations, but also has direct experimental confirmation (see, for example, \(^{16}\)), expressed in the fact that the addition of a donor impurity to a hole semiconductor or of an acceptor impurity to an electronic semiconductor leads in both cases to a noticeable increase in the resistance of the indicated semiconductors at low temperatures.

to the case considered above, \(N_a > N_d\); only electrons and holes exchange roles.

3.4. Degeneracy temperature. Using the previously derived dependence of the chemical potential on temperature for the case of an electronic impurity semiconductor (3.4), it is easy to obtain a relation for determining the degeneracy temperature of the electron gas for a given spectrum of impurity energies and impurity concentration for the semiconductor. Here the degeneracy temperature is conventionally understood as the temperature at which the chemical potential reaches the bottom of the conduction band. It is necessary, however, to note that appreciable degeneracy of the electron gas, with a more or less precise quantitative interpretation of the experimental data, can already be detected at \(\mu^* > -2\). This must be borne in mind when estimating the magnitude of the temperature range of degeneracy of the electron system. As was already said, in the case of the type of semiconductor considered here there usually exist two (lower and upper) degeneracy temperatures, \(T_1\) and \(T_2\), which bound the temperature range of degeneracy. In fact, the region of degeneracy of the electron gas is somewhat wider than the interval \((T_1,T_2)\).

The desired formula for determining the degeneracy temperatures can be obtained by putting \(\mu=0\) in (3.4). Then we obtain:

\[ T_{1,2}^{3/2}\left(2e^{\frac{\Delta E_d}{kT_{1,2}}}+1\right) = 0.671\left(\frac{f_n h^2}{2\pi m k}\right)^{3/2}N_d . \tag{3.12} \]

The degeneracy temperatures are the roots of this equation. That there are two roots is easy to verify on the basis of the following considerations. The expression on the left-hand side of (3.12), considered as a function of the degeneracy temperature, is a curve with a minimum. Let \(T_m\) be the value of the parameter \(T\) at which this minimum is attained. If \(T_m\) is such that

\[ T_m^{3/2}\left(2e^{\frac{\Delta E_d}{kT_m}}+1\right) < 0.671\left(\frac{f_n h^2}{2\pi m k}\right)^{3/2}N_d, \]

i.e. the minimum lies below the straight line \(0.671\left(\dfrac{f_n h^2}{2\pi m k}\right)^{3/2}N_d\), then the curve

\[ T^{3/2}\left(2e^{\frac{\Delta E_d}{kT}}+1\right) \]

intersects this straight line twice, and we obtain two values of \(T\) (\(T_1\) and \(T_2\)) for which the equality (3.12) is satisfied, i.e. the entire range of degeneracy. If \(T_m\) lies above the indicated straight line, then degeneracy will not occur at any temperature. It is easy to see that \(T_m\) must satisfy the equation

\[ 2e^{\frac{\Delta E_d}{kT_m}} = \left(\frac{2}{3}\frac{\Delta E_d}{kT_m}-1\right)^{-1}, \tag{3.13} \]

whence

\[ T_m=\frac{1}{1.645}\,\frac{\Delta E_{\text{д}}}{k}. \tag{3.14} \]

Thus, degeneracy can occur in a certain temperature interval \((T_1, T_2)\), determined by the solutions of equation (3.12), only if

\[ N_{\text{д}}\left(\frac{f_n}{\Delta E_{\text{д}}}\right)^{3/2} >2.425\cdot 10^{22}\ \text{cm}^{-3}\ \text{eV}^{-3/2}. \tag{3.15} \]

As is evident from this formula, the onset of degeneracy is favored by high impurity concentrations, small activation energies, and small effective masses of the carriers. The last two factors are the most effective.

§ 4. THE INFLUENCE OF DEGENERACY ON THE MAGNETIC PROPERTIES OF THE ELECTRON GAS IN SEMICONDUCTORS

The equilibrium properties of the electron gas in semiconductors have as yet been little studied. Obviously, the most accessible quantities for study should be the heat capacity and the magnetic susceptibility of the electron gas. In recent years, in particular, the magnetic properties of semiconductors have begun to be studied. The study of the magnetic properties of semiconductors is important in that it makes it possible to establish the nature of the chemical bond in semiconductors and its influence on their electrical and other properties. In the present section we shall briefly examine the question of the influence of degeneracy of the electron gas on the magnetic properties of semiconductors.

4.1. Formulas for the magnetic susceptibility of the electron gas. Let us first consider the question of the magnetic susceptibility of an electronic semiconductor. By methods of statistical physics it is easy to obtain the following expression for the magnetic susceptibility of electrons[^17], valid for any degree of degeneracy of the carrier system:

\[ \chi_n=-\frac{2}{\sqrt{\pi}}\,z_n\,\frac{dF_{1/2}(\mu^*)}{d\mu^*}\cdot \frac{\mu_{\text{B}}^{2}}{kT} \left(1-\frac{f_n^2}{3}\right), \tag{4.1} \]

where \(\mu_{\text{B}}\) is the Bohr magneton. It is known that the concentration of electrons in the conduction band \(N_n\) is given by the following formula:

\[ N_n=\frac{2}{\sqrt{\pi}}\,z_n F_{1/2}(\mu^*). \tag{4.2} \]

If the electron gas is nondegenerate, then

\[ \frac{dF_{1/2}(\mu^*)}{d\mu^*}=F_{1/2}(\mu^*), \]

and we obtain the well-known formula for the susceptibility of a nondegenerate electron gas

\[ \chi_n=\frac{N_n \mu_B^2}{kT}\left(1-\frac{f_n^2}{3}\right). \tag{4.3} \]

Let us note that in formulas (4.1) and (4.2) the second term in the factor \((1-f_n^2/3)\) takes into account the Landau orbital diamagnetism of the electrons. For a sufficiently small effective mass of the electrons the orbital diamagnetism may exceed the spin paramagnetism. The magnetic susceptibility of a hole gas has an analogous expression:

\[ \chi_p=-\frac{2}{\sqrt{\pi}}\, z_p \frac{dF_{1/2}(-\mu^*-\Delta E^*)}{d\mu^*} \cdot \frac{\mu_B^2}{kT} \left(1-\frac{f_p^2}{3}\right). \tag{4.4} \]

When both electrons and holes are present in a semiconductor, the magnetic susceptibility of the current carriers has the form

\[ \chi=\chi_n+\chi_p. \tag{4.5} \]

The preceding formulas were obtained under the assumption that the constant-energy surfaces of both electrons and holes have a spherical form \((\varepsilon=p^2/2m)\). Deviation from sphericity may in certain cases play a large role. If the isoenergetic surface is approximated by an ellipsoid and the anisotropy of individual single crystals is taken into account, then for a polycrystalline specimen we shall have (in the case, for example, of an electronic semiconductor)\(^{18}\):

\[ \chi_n=-\frac{2}{\sqrt{\pi}}\, z_n \frac{dF_{1/2}(\mu^*)}{d\mu^*} \cdot \frac{\mu_B^2}{kT} \cdot \left(1-\frac{\overline{F_n^2}}{3}\right), \tag{4.6} \]

where

\[ \overline{F_n^2}=\overline{f_{xx}^n f_{yy}^n-\left(f_{xy}^n\right)^2}, \tag{4.7} \]

and \(f_{ij}^n\) are the components of the tensor \(m/m_n^*\) (\(m_n^*\) is the tensor of the effective mass of the electron); the bar above denotes averaging over all possible orientations of the principal axes of the microcrystallites relative to the magnetic field.

In measuring the magnetic susceptibility, what is actually taken into account is not only the susceptibility of the electron gas, but also the susceptibility of the lattice of the basic substance and of impurity centers. As regards the susceptibility of the lattice of the basic substance, at small impurity concentrations it may be assumed that it does not change. Assuming that it is known from special measurements made on the pure substance, it can be subtracted from the total susceptibility of the semiconductor. The susceptibility of the impurity centers, however, can be calculated theoretically, making definite assumptions about the model of the impurity center. In the simplest case one deals with so-

called a hydrogen-like impurity, in which case the electron or hole rotates about the impurity center. In this case the magnetic susceptibility of the impurity centers can be calculated by the formula\(^ {18}\)

\[ \chi_{d,a}=N_{d,a}\left(\frac{\mu_B^2}{kT}-\frac{2.42\cdot 10^{-6}}{L}\,\varepsilon_0'^2 f_{n,p}^3\right), \tag{4.8} \]

where \(L\) is Avogadro’s number, and \(\varepsilon_0'\) is the dielectric constant. The first term of this formula is due to the spin paramagnetism of the electron or hole bound to the impurity center; the second term is the Langevin diamagnetism.

By subtracting from the measured susceptibility of the semiconductor the susceptibility of the pure substance and of the impurity centers, one can judge the susceptibility of the electron gas and the influence of degeneracy on it.

4.2. Magnetic properties of gray tin. The magnetic properties of gray tin, both pure and with impurities, were studied in detail by Busch\(^ {18}\). Busch showed that all the experimental data can be reasonably interpreted from the standpoint of the formulas given above, with an appropriate choice of the parameters \(f_n\), \(f_p\), \(\overline{F_n^2}\), \(\overline{F_p^2}\). We shall dwell here briefly on some results obtained by Busch, on the basis of which one may judge the degeneracy of the electron gas in this semiconductor. The values of the parameters that must be chosen in order to reconcile all the experimental data are given in Table I.

Table I

	\(\Delta E\), eV	\(f_n\)	\(f_p\)	\(\overline{F_n^2}\)	\(\overline{F_p^2}\)	\(N_a\), cm\(^{-3}\)	\(N_d\), cm\(^{-3}\)	\(\Delta E_a\), eV	\(\Delta E_d\), eV
\(\alpha\)-Sn (pure) . . . .	0.08	1.5	0.33	41	23	—	—	—	—
\(\alpha\)-Sn with Al impurity .	0.08	1.5	0.33	41	23	\(1.45\cdot 10^{19}\)	—	0.005	—
\(\alpha\)-Sn with Sb impurity .	0.08	\(\chi_{\mathrm{pr}}: 1.5\) \(\chi_d: 1.2\)	0.33	41	23	—	\(1.45\cdot 10^{19}\)	—	0.004

It is interesting to note that, both in the case of pure gray tin and in the presence of an impurity, the parameters retain their values. A slight exception is the Sb impurity, where, for quantitative agreement of the experimental data with the theory, it is necessary to ascribe several different values of the constant \(f_n\) for

conduction electrons and for electrons bound to impurity atoms.

a. Pure gray tin. Figure 3 presents the temperature dependence of the susceptibility of electrons and holes in pure gray tin. The circles denote experimental data. The solid line is the theoretical curve calculated with degeneracy taken into account; the dashed curve is without taking degeneracy into account. As can be seen, for \(T > 125^\circ\mathrm{K}\) the experimental curve fits well the curve calculated with degeneracy taken into account, although the difference is small. It is interesting to note that this degeneracy is completely imperceptible in the study of electrical conductivity, and it sets in despite the fact that here we are dealing with an intrinsic semiconductor, owing to the large difference in the effective masses of electrons and holes, and also owing to the smallness of \(\Delta E\). We note that from measurements of the temperature dependence of the electrical conductivity one obtains the same value of the activation energy of intrinsic conduction, \(\Delta E = 0.08\) eV.

Fig. 3. Temperature dependence of the specific magnetic susceptibility of current carriers in impurity-free gray tin.

b. Gray tin with Sb or Al impurities. Figures 4 and 5 give the temperature dependences of the magnetic susceptibility for \(\alpha\)-Sn with Sb and Al impurities, respectively. The impurity concentrations in the corresponding specimens are given in Table II.

Table II

Alloy No.	\(N_{\text{a,d}},\ \mathrm{cm}^{-3}\)
1	\(4.35\cdot 10^{20}\)
2	\(1.45\cdot 10^{20}\)
3	\(4.35\cdot 10^{19}\)
4	\(1.45\cdot 10^{19}\)
5	\(1.35\cdot 10^{18}\)

Antimony is a pentavalent element and gives rise in tetravalent gray tin to monovalent hydrogen-like impurity donor centers. Trivalent Al, on the contrary, leads to the appearance of monovalent hydrogen-like impurity acceptor centers. We note that the diamagnetism of \(\alpha\)-Sn + Sb alloys increases with increasing concentration of Sb atoms, since, because for electrons \(f_n > 1\), the resulting susceptibility of impurity centers will be diamagnetic. For Al the opposite effect occurs, since \(f_p < 1\), and the resulting susceptibility of impurity centers will be paramagnetic. The theoretical curves calculated for alloy specimens 4 pass close to the

of the experimental ones. Let us note that, while in calculating the susceptibility of alloys with antimony the degeneracy of the electron gas must be substantially taken into account, for Al, on the contrary, this is insignificant. This is explained by the fact that for electrons \(f_n > 1\) and degeneracy sets in sooner than for holes, for which \(f_p < 1\). This is also confirmed by Fig. 6, from which it is seen that the chemical potential of the electrons over the entire temperature interval studied lies near the bottom of the conduction band, whereas the chemical potential of the holes lies near the ceiling of the valence band only at very low temperatures. From Fig. 5 it is also seen that the susceptibility of alloys with Al depends only weakly on temperature at high Al concentrations. Apparently, this is also connected with the onset of degeneracy of the electron gas at high Al concentrations in the alloy; however, this has not been verified theoretically.

Fig. 4. Temperature dependence of the specific magnetic susceptibility in gray tin doped with aluminum.

Fig. 5. Temperature dependence of the specific magnetic susceptibility in gray tin doped with antimony.

Fig. 6. Temperature dependence of the chemical potential of electrons and holes in doped gray tin.

Of great interest is Fig. 7, which presents the dependences of the magnetic susceptibility of ternary alloys \(\alpha\)-Sn—Al—Sb. As is seen from this figure, when the Al concentration is varied, the dependence is closer to the type indicated in Fig. 5;

on the contrary, when the antimony concentration is exceeded—closer to the type indicated in Fig. 4. Of special interest is the case of different concentrations of the alloy \(\alpha\)-Sn + Al + Sb. In this case the temperature dependence of the magnetic susceptibility is very reminiscent of that for pure gray tin (see Fig. 3). This confirms what was said in § 3 about the properties of amphoteric semiconductors.

Fig. 7. Temperature dependence of the specific magnetic susceptibility in ternary alloys \(\alpha\)-Sn + Al + Sb.

4.3. Magnetic properties of the intermetallic compound \(\mathrm{Mg_2Sn}\).

The semiconductor character of the intermetallic compound \(\mathrm{Mg_2Sn}\) was established in the work of\({}^{19}\). Its magnetic properties were investigated in papers\({}^{20}\). The results of the latter authors make it possible to establish the presence of degeneracy of the electron gas when the alloy deviates significantly from the stoichiometric composition. In Fig. 8 are shown the temperature dependences (at different degrees of deviation from the stoichiometric composition) of the measured magnetic susceptibility after subtracting the magnetic susceptibility of the compound—

Fig. 8. Temperature dependence of the magnetic susceptibility of electrons in the intermetallic compound \(\mathrm{Mg_2Sn}\) at different impurity concentrations.

Fig. 9. Temperature dependence of the thermopower in \(\mathrm{Mg_2Sn}\) at different impurity concentrations.

tion very close to the stoichiometric composition (quantitatively, the degree of deviation from the stoichiometric composition was not determined, but from the measurement of the electrical conductivity it could be established that in samples 1, 2, 3 the degree of deviation from stoichiometry increased respectively). In parallel with the measurement of the magnetic susceptibility, thermoelectric-power measurements were carried out. The results are shown in Fig. 9.

On the basis of considerations that will be set forth below, one may suppose that the curves of Fig. 8 give the susceptibility of the electron gas, i.e. that lattice defects contribute practically nothing to the temperature dependence of the magnetic susceptibility. Indeed, let us consider sample 1. The temperature dependence of the thermoelectric power in this case has the form characteristic of nondegenerate semiconductors. As is known, the thermoelectric power is related to the reduced chemical potential by the relation

\[ -\frac{k}{e}\,\alpha = \mu^* + \mathrm{const}, \tag{4.9} \]

where \(k\) is Boltzmann’s constant, \(e\) is the magnitude of the electron charge. On the basis of formulas (4.2) and (4.3) one can obtain, for nondegenerate semiconductors, the relation

\[ \ln \chi - \frac{1}{2}\ln T = \mu^* + \mathrm{const}, \tag{4.10} \]

whence the equality

\[ \ln \chi - \frac{1}{2}\ln T = \frac{k}{e}\,\alpha + \mathrm{const}. \tag{4.11} \]

This relation is indeed well satisfied for sample 1, on the basis of which one may conclude that \(\chi\) is the susceptibility of the electron gas and that the latter in sample 1 is nondegenerate. Let us now turn to sample 3, corresponding to the greatest deviation from stoichiometry. In this case \(\alpha\) is very small and is practically independent of temperature. It follows from this that \(\mu^*\) is practically independent of temperature and that we are dealing with a degenerate electron gas. Then it follows from formula (4.1) that \(\chi_n\) should be proportional to \(\sqrt{T}\), which is indeed the case (see Fig. 8). Evidently, sample 2 corresponds to an intermediate case. Comparison of the experimental values of \(\chi\) and \(\alpha\) with the data of formula (4.11) makes it possible here to determine the value \(f_n = 1.66\).

Thus, the investigation of magnetic phenomena makes it possible, in a number of cases, to establish the presence of degeneracy of the electron gas in semiconductors.

§ 5. THE INFLUENCE OF DEGENERACY OF THE ELECTRON GAS ON THE KINETIC COEFFICIENTS IN SEMICONDUCTORS

A question of the greatest practical importance is that of the influence of degeneracy of the electron gas on electrical conductivity, the Hall effect, thermoelectric power, and other effects connected with the passage of current through a semiconductor. In contrast to magnetic properties, all these effects, being connected with irreversible processes, depend to the highest degree on the mechanism of interaction of electrons with the lattice. In the modern theory of irreversible processes in semiconductors this interaction is characterized by a definite dependence of the mean free path of the current carriers on the energy. Different mechanisms of scattering of electrons and holes by the lattice correspond to different dependences of the mean free path on the energy. Anticipating the results of the investigation, one may say, however, that in most cases the influence of degeneracy of the electron or hole gas manifests itself mainly through the influence of degeneracy on the chemical potential. In the case of not very strong degeneracy the theory can be in good agreement with experimental data. However, in the case of strong degeneracy there are serious difficulties, which will be discussed below. Here we shall confine ourselves only to a summary of the necessary formulas and a discussion of the results, since the derivations of these formulas can be found in the special literature and are of interest only to theorists.

5.1. Dependence of the mean free path on energy. Here we shall restrict ourselves only to the case of scattering of current carriers by thermal vibrations of the lattice. The various scattering mechanisms depend on the nature of the binding forces. The dependence of the mean free path on energy in various cases is given in Table III (p. 604).

5.2. Summary of formulas for kinetic coefficients at an arbitrary degree of degeneracy of current carriers. The calculation formulas for computing the specific electrical conductivity, the Hall coefficient, thermoelectric power, and other kinetic coefficients at any degree of degeneracy of the current carriers can be written, for the case of current carriers of one sign, in the following form ^26:

\[ \sigma=\frac{16\pi m^{*} e^{2} k T l_{0}}{3h^{3}}\,(r+1)F_r(\mu^{*}), \tag{5.1} \]

\[ R=\pm \frac{1}{N_{n,p}e}\,\frac{3}{2}\, \frac{\left(2r+\frac{1}{2}\right)}{(r+1)^2}\, \frac{F_{1/2}(\mu^{*})F_{2r-1/2}(\mu^{*})}{F_r^{2}(\mu^{*})}, \tag{5.2} \]

\[ \alpha=\pm \frac{k}{e} \left\{ \frac{(2+r)F_{r+1}(\mu^{*})}{(1+r)F_r(\mu^{*})} -\mu^{*} \right\}, \tag{5.3} \]

\[ \frac{d\sigma}{\sigma} = \frac{e^{2}l_{0}^{2}}{2m^{*}kT} \left[ \frac{\left(2r+\frac{1}{2}\right)F_{2r-\frac{1}{2}}(\mu^{*})} {(r+1)F_{r}(\mu^{*})} \right]^{2}H^{2}, \tag{5.4} \]

\[ Q= \frac{k l_{0}}{(2m^{*}kT)^{1/2}} \frac{(r+2)\left(2r+\frac{1}{2}\right)F_{r+1}(\mu^{*})F_{2r-\frac{1}{2}}(\mu^{*})} {(r+1)^{2}F_{r}^{2}(\mu^{*})} - \frac{\left(2r+\frac{3}{2}\right)(r+1)F_{r}(\mu^{*})F_{2r+\frac{1}{2}}(\mu^{*})} {(r+1)^{2}F_{r}^{2}(\mu^{*})}, \tag{5.5} \]

\[ P= \frac{3}{2N_{n,p}k} \frac{F_{\frac{1}{2}}(\mu^{*})}{(r+1)F_{r}(\mu^{*})} \times \]

\[ {}\times \frac{(r+1)\left(2r+\frac{3}{2}\right)F_{r}(\mu^{*})F_{2r+\frac{1}{2}}(\mu^{*})} {(r+2)^{2}F_{r+1}^{2}(\mu^{*})-(r+3)(r+1)F_{r+2}(\mu^{*})F_{r}(\mu^{*})} - \]

\[ {}- \frac{(r+2)\left(2r+\frac{1}{2}\right)F_{r+1}(\mu^{*})F_{2r-\frac{1}{2}}(\mu^{*})} {(r+2)^{2}F_{r+1}^{2}(\mu^{*})-(r+3)(r+1)F_{r+2}(\mu^{*})F_{r}(\mu^{*})}, \tag{5.6} \]

\[ S= \pm \frac{l_{0}e}{(2m^{*}kT)^{1/2}(r+1)F_{2}(\mu^{*})} \times \]

\[ {}\times \left[ \frac{2(r+1)(r+2)\left(2r+\frac{3}{2}\right)F_{r}(\mu^{*})F_{r+1}(\mu^{*})F_{2r+\frac{1}{2}}(\mu^{*})} {(r+2)^{2}F_{r+1}^{2}(\mu^{*})-(r+1)(r+3)F_{r}(\mu^{*})F_{r+2}(\mu^{*})} - \right. \]

\[ \left. {}- \frac{(r+1)^{2}\left(2r+\frac{5}{2}\right)F_{r}^{2}(\mu^{*})F_{2r+\frac{3}{2}}(\mu^{*})} {(r+2)^{2}F_{r+1}^{2}(\mu^{*})-(r+1)(r+3)F_{r}(\mu^{*})F_{r+2}(\mu^{*})} - \right. \]

\[ \left. {}- \frac{(r+2)^{2}\left(2r+\frac{1}{2}\right)F_{r+1}^{2}(\mu^{*})F_{2r-\frac{1}{2}}(\mu^{*})} {(r+2)^{2}F_{r+1}^{2}(\mu^{*})-(r+1)(r+3)F_{r}(\mu^{*})F_{r+2}(\mu^{*})} \right], \tag{5.7} \]

where \(r\) is the exponent in the dependence of the mean free path on energy: \(l=l_{0}x^{r}\); \(F_{r}(\mu^{*})\) are Fermi integrals of index \(r\); \(\sigma\) is the electrical conductivity, \(\alpha\) the thermopower, \(R\) the Hall constant, \(\frac{d\sigma}{\sigma}\) the change of conductivity in a magnetic field; \(Q\), \(P\), \(S\) are, respectively, the Nernst, Ettingshausen, and Righi–Leduc coefficients. The formulas have been derived under the assumption of a small magnetic field. The temperatureրաժեշտ

Table III

Run length: \(l=l_0 x^r,\quad x=\dfrac{\varepsilon}{kT}\)

No.	Type of semiconductor	\(r\)	\(l_0\)	Note	Reference
1	Atomic semiconductors	0	\(\displaystyle \frac{9\pi}{4}\,\frac{\hbar^{4}\omega MC^{2}}{kTm^{*2}a^{3}}\)	\(\omega\) — speed of sound, \(M\) — mass of atoms in the cell, \(C\) — Bloch constant	22
2	Atomic semiconductors (deformation potential)	0	\(\displaystyle \frac{\pi\hbar^{4}C_{11}}{m^{*2}\varepsilon_{1n}^{2}kT}\)	\(C_{11}\) — longitudinal modulus of elasticity	23
3	Ionic semiconductors (high \(t^\circ\))	1	\(\displaystyle \frac{a}{2\pi}\,\frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma ze^{2}/a}\right)^2\)	—	24
4	Ionic semiconductors (low \(t^\circ\), small carrier concentrations)	½	\(\displaystyle \frac{3\left(n^2-\varepsilon_0'\!+1\right)}{n^2-\varepsilon_0'}\,\frac{kT}{\sqrt{\hbar\omega_0}}\,\frac{\hbar^2}{m^{*}e^2}\,e^{\hbar\omega_0/kT}\)	\(n\) — refractive index, \(\varepsilon_0'\) — dielectric constant, \(\omega_0\) — frequency of residual rays	25, 15
5	Ionic semiconductors (low \(t^\circ\), large carrier concentrations)	½	\(\displaystyle \frac{a}{2\pi}\,\frac{M}{m}\left(\frac{\hbar\omega_0}{\gamma ze^{2}/a}\right)\frac{kT}{\sqrt{\hbar\omega_0}}\,e^{\hbar\omega_0/kT}(1-n_i)\)	\(n_i\) — Fermi distribution	24
6	Polar semiconductors	2	\(\displaystyle 0.0916\,\frac{e^2ckT}{(\hbar\omega_0)^3}\)	\(\displaystyle c=\frac{1}{n^2}-\frac{1}{\varepsilon_0'}\)	41a

the dependence of the above-mentioned kinetic coefficients is determined either by the temperature entering directly into the corresponding expressions, or through the chemical potential, whose temperature dependence must be determined, on the basis of specific data on the energy spectrum, by the methods set forth in § 3. Most often it is precisely this part of the temperature dependence that proves decisive for the dependence of a given kinetic coefficient on temperature. It is therefore necessary to dwell on the question of the calculation, or proper interpolation, of the Fermi integrals \(F_r(\mu^*)\).

5.3. Approximate calculation of Fermi integrals.

As was said above, the integral \(F_{1/2}(\mu^*)\) is approximated very well by formula (3.2). It can be shown\(^{27}\) that other Fermi integrals with half-integer argument of the type \(F_{m+\frac12}\) can also be approximated, for \(-\infty \leqslant \mu^* \leqslant 1.5\), by formulas of the form

\[ F_{m+\frac12}(\mu^*)= \frac{\sqrt{\pi}}{2}\, \frac{3\cdot 5\ldots(2m+1)}{2^m}\, \frac{e^{\mu^*}}{1+A_m e^{\mu^*}}, \qquad (m \geqslant 1), \tag{5.8} \]

where

\[ A_1=0.0619,\qquad A_2=0.0285,\qquad A_3=0.0144. \tag{5.9} \]

As is seen from this formula, the role of the correction coefficients \(A_m\) decreases as the index \(m\) increases.

As for the Fermi integrals with integer index \(F_n(\mu^*)\), it is easy to show that

\[ F_0(\mu^*)=\ln(1+e^{\mu^*}). \tag{5.10} \]

The integrals \(F_n(\mu^*)\) for \(n>0\) can also be approximated, for \(-\infty \leqslant \mu^* \leqslant 1.5\), by dependences of the form\(^{27}\)

\[ \left. \begin{aligned} F_n(\mu^*)&=n!\,\frac{e^{\mu^*}}{1+B_n e^{\mu^*}}\quad (n>0),\\ B_1&=0.164,\qquad B_2=0.017, \end{aligned} \right\} \tag{5.11} \]

where

As in the case of (5.9), the role of the correction \(B_n\) decreases as the index \(n\) increases.

Tables of certain Fermi integrals and expansions of the latter in series in powers of \(e^{\mu^*}\) or \(e^{-\mu^*}\) are available in \(^{12,13,24,9}\).

5.4. Graphical study of kinetic coefficients.

For convenience, in Fig. 10 (from a to zh) are given graphs of the dependences on \(\mu^*\) of certain combinations of Fermi integrals entering respectively into formulas (5.1)—(5.7).

Fig. 10. Graphs of the dependence of the coefficients \(K_i\) on the reduced chemical potential.

Let us note that the adopted notation is:

\[ \sigma=\frac{K_1(\mu^*)\,16\pi m^* e^2 kT l_0}{3h^2}, \tag{5.12} \]

\[ \alpha=\pm K_2(\mu^*)\frac{k}{e}, \tag{5.13} \]

\[ R=\pm \frac{K_3(\mu^*)}{N_{n,p}e}, \tag{5.14} \]

\[ \frac{\Delta\sigma}{\sigma}=\frac{K_4(\mu^*)e^2 l_0^2}{2m^*kT}\,H^2, \tag{5.15} \]

\[ Q=\frac{K_5(\mu^*)k l_0}{(2m^*kT)^{1/2}}, \tag{5.16} \]

\[ P=\frac{3}{2N_{n,p}k}K_6(\mu^*), \tag{5.17} \]

\[ S=\pm \frac{e l_0}{(2m^*kT)^{1/2}}K_7(\mu^*). \]

The curves denoted by the numbers I, II, III, IV correspond respectively to the cases: a) \(r=0\), b) \(r=-\frac{1}{2}\), c) \(r=\frac{1}{2}\), with the factor \(1-\bar n_i\) included in \(l_0\), and d) \(r=1\). Let us note that in the case when \(l_0\) contains the factor \(1-\bar n_i\) (the case of ionic crystals at low temperatures and high concentrations of current carriers; see Table III), formulas (5.11)—(5.17) change. The replacement of the corresponding combinations of Fermi integrals must be made according to the following table:

\[ \left. \begin{aligned} (r+1)F_r(\mu^*) &\to \frac{3}{4}\left[F_{-\frac12}(\mu^*)+\frac{1}{2}F_{-\frac12}(\mu^*)\right],\\ (r+2)F_{r+1}(\mu^*) &\to \frac{5}{4}\left[F_{\frac32}(\mu^*)+\frac{3}{2}F_{\frac12}(\mu^*)\right],\\ (r+3)F_{r+2}(\mu^*) &\to \frac{7}{4}\left[F_{\frac52}(\mu^*)+\frac{5}{2}F_{\frac32}(\mu^*)\right]. \end{aligned} \right\} \tag{5.18} \]

The graphs in Fig. 10 make it possible to conclude that in the case of electrical conductivity and thermopower, if one confines oneself to the region of variation \(-\infty<\mu^*<1.5\), which is practically important from the point of view of semiconductor theory, the corresponding kinetic coefficients, independently of the value of \(r\), depend monotonically on \(\mu^*\). They differ only in absolute magnitude, and not very much. The same also applies to the dependence of the Hall coefficient on \(\mu^*\) for different \(r\) (apart from the case represented by curve III). Therefore one may state in advance that “anomalies” in the temperature variation of the electrical conductivity, thermopower, and Hall effect

(with the exception of the question of the temperature dependence of the mobility of current carriers, which we shall not touch upon here) can hardly be attributed to differences in the mechanism of electron scattering. Here the decisive importance for interpreting the temperature dependence of these coefficients is acquired by the temperature behavior of the chemical potential, which, depending on the particular structure of the energy spectrum and the degree to which it is filled by electrons, may indeed lead to the most varied and “unexpected” results. This is especially clearly seen in the example of the thermoelectric power (see formula (5.3)). Indeed, the first term in braces,

\[ \frac{(2+r)F_{r+1}(\mu^*)}{(1-r)F_r(\mu^*)}, \]

the only one that is in any way connected with the mechanism of electron scattering, is very inert from the point of view of temperature dependence. Owing to the similar character of the dependence of \(F_{r+1}\) and \(F_r\) on \(\mu^*\), their ratio practically does not depend on \(T\), even when \(\mu^*\) changes strongly with temperature. Therefore, as a rule, to within an almost constant factor, the temperature behavior of the thermoelectric power coincides with the temperature dependence of \(\mu^*\).

The reason why, despite the strong dependence of the mobility of current carriers on the mechanism of their scattering, this usually has almost no effect on the temperature behavior of the electrical conductivity is as follows. Usually the mobility is proportional to some power of the temperature, whereas the formula for the electrical conductivity also contains the concentration of free current carriers, which depends much more strongly on temperature. As a result, the temperature dependence of the mobility is masked and can be revealed only by simultaneously comparing data on the electrical conductivity and the Hall effect. However, this is not always so. In the case of strong degeneracy of the electron gas, or when saturation sets in (complete ionization of impurity atoms), \(\mu^*\) almost ceases to depend on temperature. In this temperature range the contribution of the mobility to the temperature behavior of the electrical conductivity is manifested.

A different situation occurs in the case of the Nernst, Ettingshausen, and Righi–Leduc coefficients. Here a change in the scattering mechanism can lead both to a radical change in the form of the dependence of the corresponding integral on \(\mu^*\), and to a change of sign (the case of \(Q\) and \(P\)). For certain dependences of the mean free path on energy, the Nernst and Ettingshausen coefficients identically vanish or become very small when the expression is removed. Therefore, the study of the temperature dependence and magnitude of these coefficients is, in our opinion, of great interest, since it can provide essentially new information about current mechanisms in semiconductors that escape the attention of experimentalists in the study of the “traditional” Hall effect, thermoelectric power, and electrical conductivity.

5.5. The case of mixed conductivity. In the presence of mixed conductivity, formulas (5.1)—(5.7) must be modified.

are not, since, with the exception of electrical conductivity, the kinetic coefficients in the case of amphoteric semiconductors are usually not a simple sum (or difference) of the corresponding quantities for each of the gases (electrons or holes) separately. Below a method will be indicated for calculating the electrical conductivity, thermoelectric power, and Hall effect in amphoteric semiconductors in the presence of degeneracy. First we make the following remark. As was already indicated in § 3, in amphoteric semiconductors, when the concentrations of donor and acceptor impurities, the effective masses of the carriers, etc. differ only slightly, the system of current carriers is nondegenerate, and, consequently, in this case one may use the usual formulas for the kinetic coefficients in semiconductors with mixed conductivity in the absence of degeneracy of the current carriers (see, for example, \(^{28}\)). If, however, the conditions for degeneracy are present, then, obviously, either the electron gas or the hole gas is degenerate. For definiteness let us assume that the system of electrons is degenerate, while the hole gas may be regarded as nondegenerate. In this case the calculation formula for the electrical conductivity takes the following form:

\[ \begin{aligned} \sigma=\sigma_n+\sigma_p &= \frac{16\pi e^2 kT}{3h^2} \left[ m_n^* l_0^n (r+1)F_r(\mu^*) + m_p^* l_0^p \beta_r e^{-\mu^*-\Delta E^*} \right], \end{aligned} \tag{5.19} \]

where \(\beta_r=1\) for \(r=0\), \(\beta_r=\dfrac{3}{4}\sqrt{\pi}\) for \(r=\dfrac{1}{2}\), and \(\beta=2\) for \(r=1\). The thermoelectric power can be calculated from the following formula:

\[ \alpha = -\frac{k}{e}\, \frac{1}{\sigma_n+\sigma_p} \left\{ (r+2)F_{r+1}(\mu^*)-\mu^*(r+1)F_r(\mu^*) - \left[ \gamma_r+\delta_r(\mu^*+\Delta E^*) \right] e^{-\mu^*-\Delta E^*} \right\}, \tag{5.20} \]

where \(\gamma_r=2\) for \(r=0\), \(\gamma_r=\dfrac{15}{8}\sqrt{\pi}\) for \(r=\dfrac{1}{2}\), and \(\gamma_r=6\) for \(r=1\),

\[ \delta_r=1 \text{ for } r=0,\quad \delta_r=\frac{3}{4}\sqrt{\pi} \text{ for } r=\frac{1}{2},\quad \delta_r=2 \text{ for } r=1. \]

Finally, the Hall constant may be calculated, in the case of an amphoteric semiconductor considered by us, from the formula

\[ R= \frac{R_n\sigma_n^2+R_p\sigma_p^2}{(\sigma_n+\sigma_p)^2}, \tag{5.21} \]

where \(\sigma_n\) and \(\sigma_p\) are the electron and hole components of the electrical conductivity; \(R_n\) and \(R_p\) are the Hall coefficients for electrons and holes separately, calculated from formula (5.2). In this connection, according to what was said, for \(R_p\) in our case one may use the simplified

by a relation of the form

\[ R_p=\frac{1}{N_p e}\,\xi_r, \tag{5.22} \]

where \(\xi_r=\dfrac{3\pi}{8}\) for \(r=0\), \(\xi_r=1\) for \(r=-\dfrac{1}{2}\), and \(\xi_r=\dfrac{15}{128}\pi\) for \(r=1\).

We shall not present here the formulas for calculating the remaining kinetic coefficients in the case of mixed conductivity, since they turn out to be too cumbersome.

§ 6. SCATTERING OF ELECTRONS BY IMPURITY IONS IN DEGENERATE SEMICONDUCTORS

Up to now we have considered only the scattering of electrons by thermal vibrations of the lattice. However, as has become known from detailed studies of the mobility of current carriers in a number of semiconductors, in certain cases scattering by impurity ions plays a role, so that both scattering mechanisms must be taken into account simultaneously; as a result, the mean free path can no longer be represented as a simple power function of the energy. The mean free path due to scattering by impurity ions has the simple form

\[ l_i=l_i^0 x^2, \tag{6.1} \]

where

\[ l_i^0=\frac{4\varepsilon_0^{\prime\,2}(kT)^2}{\pi N_d e^4\ln G},\qquad G=1+\frac{\varepsilon_0^{\prime\,2}m^{*2}d^2}{e^4}\,v^4, \tag{6.2} \]

\(v\) is the electron velocity, and \(d\) is the mean distance between impurity ions. It is assumed here that the ions form a regular cubic lattice. In order to take proper account of both scattering mechanisms, the total mean free path must be found by adding the reciprocals of the lattice and impurity mean free paths. Thus we obtain

\[ \frac{1}{l}=\frac{1}{l_L}+\frac{1}{l_i}, \tag{6.3} \]

whence

\[ l=\frac{l_L l_i}{l_L+l_i}. \tag{6.4} \]

Restricting ourselves to the case of atomic semiconductors, the best studied experimentally (\(l_L\) does not depend on the electron energy), we obtain:

\[ l=\frac{l_L x^2}{x^2+\varkappa^2}, \tag{6.5} \]

where

\[ \varkappa^2=\frac{l_L}{l_i^0}. \tag{6.6} \]

Then the electrical conductivity can be calculated by the formula

\[ \sigma=\frac{16\pi e^2 m^* l_k kT}{3h^3}\int_0^\infty \frac{x^3}{x^2+\varkappa^2}\, \frac{e^{x-\mu^*}}{\left(e^{x-\mu^*}+1\right)^2}\,dx . \tag{6.7} \]

This integral does not reduce to Fermi integrals. However, formulas convenient for calculation can be obtained for any degree of degeneracy. In doing so, the following two assumptions are made: 1. The impurity is completely ionized. 2. Since on the most important part of the range of integration \(\ln G\) changes slowly, it is replaced by a constant value, putting it equal to \(v^2=6kT/m^*\) [29]. Thus,

\[ G=1+\frac{36\varepsilon_0'^2 k^2T^2 d^2}{e^4}. \tag{6.8} \]

Under these assumptions the integral entering formula (6.7) can be represented in the form of rapidly convergent series convenient for computation.

1. \(\mu^*<0\).

\[ \sigma=\sigma_0\left\{\ln\left(1+e^{\mu^*}\right) -\varkappa^2\sum_{n=1}^{\infty}(-1)^n e^{n\mu^*} \left[\operatorname{ci}(n\varkappa)\cos(n\varkappa)+ \operatorname{si}(n\varkappa)\sin(n\varkappa)\right]\right\}, \tag{6.9} \]

where the integral sine and cosine are defined by the formulas

\[ \operatorname{si} y=-\int_y^\infty \frac{\sin t}{t}\,dt,\qquad \operatorname{ci} y=\int_\infty^y \frac{\cos t}{t}\,dt; \tag{6.10} \]

\(\sigma_0\) has the meaning that \(\sigma_0\ln(1+e^{\mu^*})\) is the electrical conductivity of an atomic semiconductor when only lattice scattering is taken into account.

1. \(\mu^*>0\).

\[ \sigma=\sigma_0\left\{\ln\left(1+e^{\mu^*}\right) -\varkappa^2\sum_{n=1}^{\infty}(-1)^n e^{-n\mu^*} \left[\operatorname{ci}(n\varkappa)\cos(n\varkappa)+ \operatorname{si}(n\varkappa)\sin(n\varkappa)\right] +4\pi\varkappa^2\mu^*\sum_{n=0}^{\infty} \frac{\varkappa+(2n+1)\pi}{\left\{[\varkappa+(2n+1)\pi]^2+\mu^{*2}\right\}^2} \right\}. \tag{6.11} \]

In practice, in the formulas given it is sufficient to take into account exactly only the first 2–3 terms of the infinite sums. Analysis of these formulas shows that, as \(\mu^*\) increases in the case \(\mu^*>0\), the influence of the terms due to scattering by impurity ions decreases. This fact has a simple physical meaning. As the degree of degeneracy increases, the average velocity of the electrons increases and, consequently, the Rutherford scattering of electrons by ions, which underlies the derivation of the formula for \(l_i\), decreases.

A. G. SAMOILOVICH AND L. L. KORENBLIT

§ 7. COMPARISON OF THE THEORY WITH EXPERIMENTAL DATA

According to the theory set forth, degeneration of an electron or hole gas may be expected, under suitable conditions, in any semiconductor. In the main, the conditions favoring the occurrence of degeneracy of an electron gas reduce to the presence of large impurity concentrations and small activation energies. Very often these conditions occur in intermetallic compounds, an example of which is the intermetallic compound Mg\(_2\)Sn mentioned above; as was first established in Ref. 19,* it is a semiconductor, but one in which, under suitable conditions, degeneration of the electron gas occurs. Analogous facts were established in Ref. 30 for CdSb. By changing

Fig. 11a. Temperature dependence of the electrical resistivity of silicon with boron impurity of various concentrations.

the ratio between the amount of metal and oxygen in metal oxides of the type TiO\(_2\), ZnO, CdO, etc., by reducing the latter, one can also obtain a degenerate state of the system of charge carriers. Interesting objects of study from this point of view are alloys of silicon and germanium with trivalent or pentavalent elements, which we shall consider in somewhat greater detail.

7.1. Electrical conductivity and the Hall effect in impurity silicon and germanium (case of partial degeneracy). In Figs. 11a and 11b are shown graphs of the dependence

of the electrical resistivity and the concentration of holes on temperature in silicon alloys with boron according to the works\(^4\). In Fig. 12—the data for a silicon alloy with phosphorus\(^ {31}\). In Fig. 13—the data for an alloy of germanium with arsenic\(^5\). The data for the concentration of electrons (or holes) were found from measurements of the Hall effect, and the concentration was calculated by the formula

Fig. 116. Temperature dependence of the concentration of free holes in silicon with a boron impurity.

Sample 2 — \(N_a = 6.7 \cdot 10^{17}\ \mathrm{cm}^{-3}\)
» 3 — \(N_a = 1.3 \cdot 10^{18}\ \mathrm{cm}^{-3}\)
» 4 — \(N_a = 2.7 \cdot 10^{18}\ \mathrm{cm}^{-3}\)
» 5 — \(N_a = 6.7 \cdot 10^{18}\ \mathrm{cm}^{-3}\)
» 7 — \(N_a = 1.3 \cdot 10^{20}\ \mathrm{cm}^{-3}\)
» 8 — \(N_a = 1.3 \cdot 10^{21}\ \mathrm{cm}^{-3}\)

\[ R = \frac{1}{N_{p,n} e}. \tag{7.1} \]

The results for the silicon alloy with boron\(^ {11}\) were subjected to especially careful theoretical study. In this work, using the experimental data relating to the concentration of the boron impurity and the activation energy of impurity holes given in\(^4\), the chemical potential was calculated graphically. The results of the calculation are presented in Fig. 14. We note that, in order to bring the theoretically found values of the hole concentrations in the main band into agreement with the data\(^4\), it was necessary to assume

\(m_p^* = 0.436\,m\), and not \(m_p^* = m\), as was done in \(^{4}\). The reason for the discrepancy is the incorrect expression for the concentration of electrons on acceptor impurity levels, used in \(^{4}\),

\[ \left( N_a/[1+e^{-\Delta E_0+\mu^*+\Delta E^*}] \ \text{instead of} \ N_a\left[1+\frac{1}{2}e^{-\Delta E_a+\mu^*+\Delta E^*}\right]\right). \]

In doing this it was necessary to take into account:

1. The decrease of the activation energy of holes with increasing \(N_a\), according to the empirical formula \(^{4,33}\)

\[ \Delta E_a=\Delta E_a^0-\alpha N_a^{1/3}, \tag{7.2} \]

where \(\Delta E_a^0 = 0.08\) eV, \(\alpha = 4.3\cdot 10^{-8}\) eV/cm.

Fig. 12. a) temperature dependence of the electrical resistivity of silicon with an impurity of phosphorus of various concentrations; b) temperature dependence of the concentration of free electrons in silicon with an impurity of phosphorus.

Sample \(A\) — \(N_d = 4.7\cdot 10^{17}\ \text{cm}^{-3}\)
» \(B\) — \(N_d = 2.7\cdot 10^{18}\ \text{cm}^{-3}\)
» \(C\) — \(N_d = 4.7\cdot 10^{19}\ \text{cm}^{-3}\)
» \(D\) — \(N_d = 4.7\cdot 10^{20}\ \text{cm}^{-3}\)

1. In calculating the electrical conductivity—the scattering by ions according to formulas (6.9) and (6.11).

1. The electrical conductivity was calculated to within a constant, temperature-independent factor.

The results of the calculations show that curves 2, 3, 4 can be obtained quite accurately. It turns out that for samples 3 and 4 it is essential to take into account the partial degeneracy of the electron gas, since \(-\mu^*-\Delta E^*>-2\). Thus, in the case of not very strong degeneracy \((1>-\mu^*-\Delta E^*>-2\) and \(\Delta E_a\ne0)\), the formulas based on the theory developed above make it possible to interpret the experimental data very well. The only unclear point of the theory is the absence of a reliable theoretical justification for formula (7.2) for the dependence of \(\Delta E_a\) on \(N_a\) \({}^{33}\).

Fig. 13. a) temperature dependence of the electrical conductivity of germanium with an arsenic impurity; b) temperature dependence of the concentration of free electrons in germanium doped with arsenic, for various concentrations of arsenic atoms.

Sample 55 — \(N_d=8\cdot10^{13}\ \mathrm{cm}^{-3}\)
» 53 — \(N_d=8\cdot10^{14}\ \mathrm{cm}^{-3}\)
» 64 — \(N_d=1.5\cdot10^{16}\ \mathrm{cm}^{-3}\)
» 54 — \(N_d=7\cdot10^{16}\ \mathrm{cm}^{-3}\)
» 61 — \(N_d=7\cdot10^{17}\ \mathrm{cm}^{-3}\)
» 58 — \(N_d=8\cdot10^{18}\ \mathrm{cm}^{-3}\)

However, the case of strong degeneracy lies outside the scope of this theory. Here one has to abandon the introduced in the previous

notions of local impurity levels. This question requires separate consideration, to which we now turn.

7.2. The case of strong degeneracy. Strong degeneracy of the hole gas is characteristic of samples 5, 7, and 8 Si+B (see Figs. 11a and 11b). First of all we note that the anomalous increase in the concentration of holes with decreasing temperature in sample 5 is apparently connected with an incorrect recalculation of the concentration from data on the temperature dependence of the Hall constant, as will be discussed in more detail in 7.4.

Fig. 14. Temperature dependence of the reduced chemical potential of holes in silicon with various concentrations of the boron impurity. The numbering of the samples is the same as in Fig. 11.

In samples 7 and 8 the concentration of holes does not depend on temperature, similarly to the concentration of electrons in a metal. Since for all samples, beginning with the 5th, the impurity concentration is already so large that \(\Delta E_a = 0\), it may be assumed that the local levels disappear, and we have the same situation as in a metal, when the activation energy of the current carriers is equal to zero, i.e., we have a single band in which, already at absolute zero temperature, all free holes are found. Although from the theoretical point of view such a conception runs into a number of ambiguities, which will be discussed in more detail in § 8, it nevertheless makes it possible to understand the independence of the hole concentration from temperature. It turns out that in the case of strong degeneracy of the hole gas one is already dealing with such concentrations of current carriers that the approximate formulas of Fermi–Dirac statistics corresponding to strong degeneracy are applicable, which are usually used in the theory of metals. This makes it possible easily to calculate the chemical potential from the well-known Sommerfeld formula

\[ \mu_0 = \frac{h^3}{2m^*}\left(\frac{3N_{a,d}}{8\pi}\right)^{2/3} \tag{7.3} \]

or from the Sommerfeld formula with a correction term

\[ \mu = \mu_0\left[1-\frac{\pi^2 k^2 T^2}{12\mu_0^2}\right]. \tag{7.4} \]

Let us note that for \(\mu^* > 3\) these formulas give the chemical potential with an accuracy up to 3%. It is also possible to calculate by the well-known for-

… the degeneration temperature in the theory of metals:

\[ T_0=\frac{h^2}{2\pi k m_e^*}\,N_{a,\partial}^{2/3}. \tag{7.5} \]

In Fig. 116 the region of degeneration, calculated from formula (7.5), is shown by the dotted curve. As is seen from the figure, samples 5, 7, 8 fall within this region.

However, with regard to the calculation of the electrical conductivity in this case there are serious difficulties, which will be discussed in detail in § 8.

7.3. Degeneration of the electron gas and thermal conductivity of semiconductors. An interesting case of degeneration of the electron gas was observed by A. V. Ioffe and A. F. Ioffe[^34]. As is known, the thermal conductivity of a crystal consists of two parts: the thermal conductivity of the lattice and the electronic thermal conductivity:

\[ \chi=\chi_p+\chi_\mathrm{e}. \]

The latter is proportional to the electrical conductivity

\[ \chi_\mathrm{e}=\beta\sigma. \tag{7.6} \]

Fig. 15. Dependence of the thermal conductivity of a semiconductor on electrical conductivity.

The coefficient \(\beta\) (the so-called Wiedemann–Franz ratio) has the value:

\[ \beta= \begin{cases} 2\left(\dfrac{k}{e}\right)^2 T, & \text{for a nondegenerate electron gas,}\\[6pt] \dfrac{\pi^2}{3}\left(\dfrac{k}{e}\right)^2 T, & \text{in the case of a strongly degenerate electron gas.} \end{cases} \tag{7.7} \]

The authors cited investigated the thermal conductivity of a series of samples of one intermetallic compound with different specific resistivities. It turned out that the value of the coefficient \(\beta\) for high-resistance samples is close to the value \(2\left(\dfrac{k}{e}\right)^2T\), while for low-resistance samples the value of \(\beta\) is close to \(\dfrac{\pi^2}{3}\left(\dfrac{k}{e}\right)^2T\), which quite evidently indicates the presence of degeneration of the electron gas in the latter case (Fig. 15).

7.4. The Hall effect in germanium at low temperatures. At low temperatures in germanium there is sometimes observed an anomalous (Fig. 16) smooth decrease of the Hall constant with decreasing temperature. If one uses the simple formula (7.1) for

coefficient, it would follow from this that the concentration of carriers grows as the temperature is decreased, which is physically meaningless. A similar case was considered in work \(^{35}\), where it was shown that, when the degeneracy of the electron gas is taken into account (the coefficient \(K_3(\mu^*)\) in formula (5.14)), the paradox disappears: despite the

Fig. 16a. Temperature dependence of the Hall constant in germanium at low temperatures according to \(^{35}\).

Fig. 16b. Temperature dependence of the concentration of current carriers in germanium at low temperatures \(^{35}\).

fact that the Hall constant falls as the temperature is decreased, the concentration of current carriers also falls or remains constant.

7.5. Temperature dependence of the thermopower in the presence of degeneracy. As has already been said, the temperature dependence of the thermopower, to within an additive almost constant quantity, coincides with the temperature dependence of the reduced chemical potential. Therefore, in the presence of degeneracy of the electron gas one should expect small values of the thermoelectromotive force, almost independent of temperature, with a tendency to grow as the temperature rises. This, as a rule, is also observed experimentally. In Fig. 17 the temperature dependences of the thermopower in germanium with an admixture of aluminum or antimony are presented. In samples 27L and 26Z (Fig. 17a), and also 34F, 33N, and 34E (Fig. 17b), the concentration of current carriers is such that degeneracy sets in; the temperature dependence of the thermopower indeed increases slightly with increasing temperature in the region of degeneracy (the temperature range from \(-150^\circ\text{C}\) to \(+250^\circ\text{C}\)). At the same time, the value of the thermopower of the degenerate samples (\(\sim 200\text{–}250\ \mu\text{V}/\text{deg}\) for the samples Ge + Al and \(\sim 450\ \mu\text{V}/\text{deg}\) in the case of the samples Ge + Sb) is indeed smaller in absolute magnitude and changes much more weakly with temperature than in the case where degeneracy is removed. It is easy to show that the peculiarities of the temperature dependence of the thermopower in nondegenerate samples (the presence of a maximum and a minimum of \(\alpha(T)\)

Fig. 17a. Temperature dependence of the thermopower in germanium doped with aluminum

Sample \(35N\) — \(N_a = 5.6 \cdot 10^{15}\ \mathrm{cm}^{-3}\)
» \(35M\) — \(N_a = 1.5 \cdot 10^{17}\ \mathrm{cm}^{-3}\)
» \(26Z\) — \(N_a = 4.1 \cdot 10^{18}\ \mathrm{cm}^{-3}\)
» \(27L\) — \(N_a = 7.0 \cdot 10^{18}\ \mathrm{cm}^{-3}\)

Fig. 17b. Temperature dependence of the thermopower in germanium doped with antimony

Sample \(26L\) — \(N_{\mathrm{d}} = 3.4 \cdot 10^{15}\ \mathrm{cm}^{-3}\)
» \(33E\) — \(N_{\mathrm{d}} = 9.6 \cdot 10^{16}\ \mathrm{cm}^{-3}\)
» \(34E\) — \(N_{\mathrm{d}} = 5.8 \cdot 10^{17}\ \mathrm{cm}^{-3}\)
» \(34F\) — \(N_{\mathrm{d}} = 7.1 \cdot 10^{17}\ \mathrm{cm}^{-3}\)
» \(33N\) — \(N_{\mathrm{d}} = 7.4 \cdot 10^{17}\ \mathrm{cm}^{-3}\)

In the case of Ge + Al, the character of the temperature dependence of \(\alpha\) in sample \(26L\) (Ge + Sb) can readily be related to the corresponding temperature dependence of \(\mu^*\) in these samples. Indeed, the curves \(\alpha(T)\) in the case of samples \(35N\) and \(35M\) are typical for the temperature dependence of the chemical potential in the presence of mixed conductivity: at low temperatures the conductivity is impurity hole conductivity and the chemical potential is located in the lower half of the forbidden band; at high temperatures it shifts into the upper half owing to the intrinsic conductivity that becomes activated, since the effective mass of the electrons is much smaller than the effective mass of the hole (see formula (3.9)). The mechanism of intrinsic conductivity also accounts for the removal of degeneracy of the hole gas in samples \(27L\) and \(27Z\) at \(t^\circ > 400^\circ\mathrm{C}\) and for the tendency of the corresponding curves \(\alpha(T)\) toward a change of sign. In the case of electron impurity germanium (samples \(26L\), \(33E\), \(34E\), \(33N\), and \(34F\)) in the region of intrinsic conductivity, naturally, there is no such tendency toward a change of sign.

It should be noted that the temperature dependence of the thermoelectric power at very low temperatures is greatly affected by the process of “drag” of electrons by phonons, or the “Gurevich effect”\({}^{36}\). The physical essence of this effect is that, under certain conditions (in the case of nondegenerate semiconductors at low temperatures), in order to obtain the correct result it is necessary to take into account the disturbance of the equilibrium state of the phonon system caused by the interaction of the latter with conduction electrons. As shown in \({}^{37}\), the Gurevich effect in degenerate semiconductors is insignificant at high temperatures; however, at low temperatures the situation changes sharply\({}^{38}\). Since, in the presence of degeneracy of the electron gas, this effect apparently has little influence even in the low-temperature region, we shall not discuss it here.

§ 8. SOME UNSOLVED QUESTIONS IN THE THEORY OF DEGENERACY OF THE ELECTRON GAS IN SEMICONDUCTORS

In this paragraph we shall note some unsolved questions of the theory of semiconductors that arise in connection with the study of the electron gas in them.

8.1. Electrical conductivity of strongly degenerate semiconductors. As was already noted above, the theory of the electrical conductivity of strongly degenerate semiconductors encounters serious difficulties. From Fig. 11a it is seen that in samples 7 and 8 of the Si + B alloy the electrical conductivity is practically independent of temperature. This is usually explained by the theory of Johnson and Lark-Horovitz\({}^{39}\), which is based on the following assumptions: a) since, with decreasing temperature, \(l_L\) increases, then in the region

at low temperatures (when degeneracy is observed) \(l_L \gg l_i\), so that the resultant \(l \simeq l_i\), i.e., the scattering is due exclusively to impurity ions. b) At low temperatures, chiefly electrons with small velocities are scattered. Therefore in expression (6.2) for \(G\) one did not introduce \(kT\) instead of \(m^*v^2\), while in the expression for \(l_i^0\) the function \(\ln G\) was expanded in a series

\[ \ln G \simeq \frac{\varepsilon_0'^2 m^{*2} d^2 v^4}{e^4}, \tag{8.1} \]

so that

\[ x^2 = l_L N_{d,a}\pi d^2 x^2; \tag{8.2} \]

c) In the region of strong degeneracy one may use the approximate formulas of Fermi—Dirac statistics usually employed in the theory of metals. Under these assumptions the result of Johnson and Lark-Horovitz is obtained directly from formula (6.7). Indeed, in order to take the first assumption into account, it is sufficient in the denominator of the integrand to neglect \(x^2\) in comparison with \(\chi^2\), so that

\[ \sigma = \frac{16\pi e^2 m^* l_L kT}{3h^3} \int_0^\infty \frac{x^3 e^{x-\mu^*}}{\chi^2\left(e^{x-\mu^*}+1\right)} \, dx . \tag{8.3} \]

Substituting here \(\chi^2\) from (8.2), we obtain an elementary integral, which gives

\[ \sigma = \frac{16e^2 m^* kT}{3h^3 N_{d,a} d^2} \ln\left(1+e^{\mu^*}\right). \tag{8.4} \]

In the region of strong degeneracy \(\mu^* > 1\) and \(\ln(1+e^{\mu^*}) \simeq \mu^*\), so that

\[ \sigma = \frac{16e^2 m^*\mu}{3h^3 N_{d,a} d^2} \tag{8.5} \]

or, using (7.3),

\[ \sigma = \frac{2e^2}{3hd^2\left(3N_{d,a}\pi^2\right)^{1/3}} . \tag{8.6} \]

This formula coincides with the formula derived by Johnson and Lark-Horovitz, except for the factor 2, which, apparently, they erroneously omitted because they did not take into account the statistical weight of the spin. Thus, according to formula (8.6), the electrical conductivity of degenerate semiconductors, in agreement with experiment, does not depend on temperature. As Johnson and Lark-Horovitz state, formula (8.6) gives the correct order of magnitude of the observed electrical conductivity.

Nevertheless, it is easy to see that the assumptions on the basis of which formula (8.6) was obtained do not correspond to the true state of affairs. The error in the reasoning of Johnson and Lark-Horovitz

consists in the failure to take account of the important circumstance that, as the degree of degeneracy increases, \(l_i\) also increases, although it clearly does not depend on temperature. Indeed, in a degenerate gas, as is known, only electrons whose energy is of the order of the Fermi energy (the chemical potential) can be scattered, and the latter increases with the growth of the degree of degeneracy. On the other hand, the probability of Rutherford scattering decreases strongly with increasing energy, as a result of which \(l_i\) increases. Thus, the assumption that the inequality \(l_L \gg l_i\) is fulfilled has no basis. For the same reasons one cannot assume in the expression for \(G\) a term depending on the electron energy that is small in comparison with unity (in it the energy must be set equal to \(\mu\)), and one cannot use an expansion of the logarithm. The incorrectness of the assumptions of Johnson and Lark-Horovitz can be demonstrated in yet another way. They used the assumption that \(x^2 \ll \chi^2\), whereas in (6.7) the integration extends from \(0\) to \(\infty\). This approximation has meaning if, for \(x \gtrsim \chi\),

\[ \frac{e^{x-\mu^*}}{(e^{x-\mu^*}+1)^2} \ll 1. \]

But this can be only in the case when \(\mu^* \ll \chi\). However, an estimate of these quantities for sample 7 Si B gives the opposite inequality. Let us note that, as was already pointed out above, formula (6.11) also shows that, with increasing degeneracy of the electron gas, the role of scattering by impurity ions is negligibly small.

Thus, the question of the theoretical interpretation of the temperature dependence of the electrical conductivity of strongly degenerate semiconductors remains open. In this connection, however, a whole series of important new questions arises.

8.2. The question of impurity zones. It should be noted that the assumptions on the basis of which formula (6.7) was obtained for the electrical conductivity of semiconductors with allowance for scattering by impurity ions, and from which formulas (6.9) and (9.11) were then obtained for degenerate semiconductors, may also be subjected to criticism. The point is that this formula is based on the assumption of the additivity of the reciprocal mean free paths. Physically this means that the two scattering mechanisms—scattering by thermal vibrations of the lattice and scattering by impurity ions—act independently of one another. Meanwhile, since degeneracy in semiconductors sets in at large impurity concentrations, it is natural to expect a strong interaction of the impurity with the lattice, as a result of which the energy spectrum of the electrons in the lattice may be altered in an essential way, so that no trace of additivity may remain. One form of expression of this idea is the suggestion, repeatedly made recently in the literature (A. F. Ioffe\(^{40}\)), of the existence of impurity zones in semiconductors, i.e., sufficiently narrow energy zones located in the region

...of the forbidden energy band and leading to the delocalization of “impurity” electrons or holes (Fig. 18). It must be emphasized at once that the assumption of the existence of impurity bands is based on the qualitative consideration that, at sufficiently high impurity concentrations, the wave functions of the local states of “impurity” electrons overlap in the same way as the overlap of the wave functions of isolated atoms, when they are brought sufficiently close together, leads to the formation of energy bands in metals. The only difference is that the effective radius of the wave functions of “impurity” electrons is greater than the radius of the wave functions of isolated atoms, since an “impurity” electron moves in a medium with a large dielectric constant (\(\varepsilon'_0 = 12\) in Si and \(\varepsilon'_0 = 16\) in Ge). Therefore the effective “overlap” of the wave functions begins here at large distances, and the energy band obtained is narrow. From the quantitative point of view, these ideas for ionic semiconductors were developed by Pekar and Krivoglaz on the basis of the polaron theory of \(F\)- and \(F'\)-centers. However, for atomic semiconductors the question of the mechanism of formation of impurity bands remains, from the theoretical side, completely undeveloped, although this question is extremely important. There is no doubt that the question of the dependence of the activation energy of current carriers from local levels on the impurity concentration is most closely connected with the question of the formation of impurity bands. Indeed, the appearance of a very narrow band in the statistical calculation of the concentration of current carriers may manifest itself as a decrease in the activation energy of the latter.

Fig. 18. Energy spectrum of an impurity semiconductor with narrow impurity bands.

In this article we are interested chiefly in the question of the influence of impurity bands on the statistical properties of an electron or hole gas. We shall dwell in somewhat more detail on the main experimental facts that are interpreted as evidence for the presence of impurity bands, and consider how justified such an interpretation is.

8.3. Electrical Conductivity of PbS.

Figure 19 shows the temperature dependence of the specific resistance of PbS at different impurity concentrations. As is seen from the figure, at low temperatures (down to \(2^\circ\ \mathrm{K}\)) the dependence \(\rho(T)\) has a metallic character, whereas at high temperatures it is of the typical semiconductor type. The temperature of the transition from metallic conductivity to semiconductor conductivity increases with increasing impurity concentration.

Fig. 19. Temperature dependence of the specific resistance of lead sulfide.

From the point of view of the concept of impurity bands these results are interpreted as follows. At low temperatures the electrons are concentrated in the impurity band, which is not completely filled, and therefore the conductivity has the same character as in a metal. As the temperature is raised, the electrons gradually pass into the conduction band. Since in the impurity band, owing to its small width (\(0.04\ \mathrm{eV}\) in PbS according to the calculations of Pekar and Krivoglaz \(^{416}\)) and, consequently, large effective mass, the mobility of the impurity electrons is much less than the mobility in the conduction band, the transition of electrons into the conduction band will lead to a decrease of the resistance. A change in the character of the conductivity will occur when the current in the conduction band begins to predominate. Naturally, the higher the impurity concentration, i.e., the more electrons there are in the impurity band at low temperatures, the higher the transition temperature will be.

8.4. The Hall Effect in Germanium at Low Temperatures.

A paper by Hung and Gliessman \(^{11}\) has recently been published, in which careful measurements were made of the resistance and Hall effect in germanium at low temperatures. Of particular interest to us are the data relating to the Hall effect. The characteristics of the investigated samples are given in Table IV. The experimental data are presented in Figs. 20 and 21. A characteristic feature of all the samples, except the strongly compensated ones (SB-4, SB-5, B-3), is the presence of a sharply pronounced maximum of the Hall coefficient at low temperatures; moreover, the decrease in the concentration of charge carriers with increasing temperature is so considerable that it cannot be attributed to a change in the coefficient \(K_3(\mu^*)\) in formula (5.14). Analyzing their data, the authors come to the conclusion that their results can be obtained if it is assumed that the electron spectrum in germanium has the form shown in Fig. 18, with the acceptor band at absolute zero being completely filled (this assumption is necessary in order that

Table IV

Sample	Carrier concentration in the region of saturation, \(n_0\) \((\mathrm{cm}^{-3})\)	Concentration of scattering impurity centers \(N_0\) at \(0^\circ\mathrm{K}\) \((\mathrm{cm}^{-3})\)	\(N_A=n_0+\dfrac{1}{2}N_0\) (or \(N_A\) for \(p\)-type) \((\mathrm{cm}^{-3})\)	\(\Delta E_n\), eV	Resistivity at low temperatures \((\Omega\cdot\mathrm{cm})\)	Mobility in the impurity zone \((\mathrm{cm}^2/\mathrm{sec})\)
HP—1 \((n)\)	\(1.2\cdot10^{14}\)	\(10^{14}\)	\(1.7\cdot10^{14}\)	0.0135	\(4\cdot10^8\)	\(1.5\cdot10^{-4}\)
HP—2 \((p)\)	\(2.5\cdot10^{14}\)	\(10^{15}\)	\(7.5\cdot10^{14}\)	0.0123	\(10^7\)	\(3\cdot10^{-3}\)
SB—1 \((n)\)	\(7.0\cdot10^{15}\)	\(1.8\cdot10^{15}\)	\(7.9\cdot10^{15}\)	0.0093	\(2\cdot10^4\)	0.05
SB—2 \((n)\)	\(7.0\cdot10^{15}\)	\(2\cdot10^{15}\)	\(8.0\cdot10^{15}\)	0.0083	\(10^5\)	0.01
SB—3 \((n)\)	\(4.5\cdot10^{16}\)		\(4.5\cdot10^{16}\)	0.0055	\(3\cdot10^2\)	0.5
SB—4 \((n)\)	\(1.5\cdot10^{17}\)		\(1.5\cdot10^{17}\)		0.5	100
SB—5 \((n)\)	\(4.0\cdot10^{18}\)		\(4.0\cdot10^{18}\)		0.0045
B—1 \((p)\)	\(8.5\cdot10^{15}\)	\(8.5\cdot10^{15}\)	\(1.3\cdot10^{16}\)	0.0090	\(3\cdot10^2\)	2.8
B—2 \((p)\)	\(3.3\cdot10^{16}\)	\(3.3\cdot10^{16}\)	\(5\cdot10^{16}\)		3	67
B—3 \((p)\)	\(1.8\cdot10^{17}\)	\(1.8\cdot10^{17}\)	\(2.7\cdot10^{17}\)		0.1	400

explain a certain discrepancy between the concentration of scattering centers and the impurity concentration, which follows from the temperature dependence of the mobility). The presence of a maximum in this scheme is explained qualitatively very simply. At low temperatures the conductivity of the impurity band predominates. Therefore, when electrons begin to pass into the conduction band, this appears as a decrease in the concentration of carriers in the impurity band, which gives an ascending branch for the Hall coefficient.

Fig. 20. Temperature dependence of the Hall constant in germanium at low temperatures according to [11]. The HP samples are of high purity; the SB samples are with an admixture of antimony. All samples are \(n\)-type, with the exception of HP—2.

Conversely, when at higher temperatures the current in the conduction band begins to predominate, this appears as an increase in the concentration of carriers in the latter, which gives a descending branch of \(R(T)\).

It should be noted, however, that the arguments of these authors give rise to certain reservations. First, the maxima appear most sharply in specimens with such small impurity concentrations that, from the standpoint of the considerations developed above, the formation of impurity bands seems highly improbable. Conversely, in strongly degenerate specimens containing large impurity concentrations, where one should have expected the appearance of impurity bands, the maximum on the \(R(T)\) curve disappears. Further, it should be noted that, in order to obtain quantitative results, the authors, when calculating the conductivity and the Hall effect in the impurity band, use the same formulas that are employed for calculating these effects in the conduction band, whereas the validity of this is extremely doubtful.

Fig. 21. Temperature dependence of the Hall constant for germanium specimens subjected to neutron irradiation\(^{11}\). Bombardment changes the sign of the conductivity from \(n\)-type to \(p\)-type.

8.5. Kinetic phenomena in the impurity band. Kinetic phenomena in the impurity band have also been little studied from the theoretical side. However, on qualitative grounds one may expect here an essential difference in the temperature behavior of the kinetic coefficients as compared with what takes place in the conduction band. The point is that when the width of an energy band is of the order of \(kT\), thermal motion distributes the electrons more or less uniformly over all states in the band. Meanwhile it is known that the behavior of an electron in the upper half of the band has essential peculiarities: negative effective mass, etc. Therefore, in calculating kinetic phenomena and the statistical properties of an electron gas in an impurity band, the effective-mass method, which correctly represents the energy spectrum near the lower or upper edge of the band, proves to be absolutely inapplicable and cannot be used in those cases where the spectrum of the band as a whole plays an essential role. A preliminary quantitative development of these ideas is given in work\(^{42}\). The effect of scattering of electrons over all the states of the band in electrical conductivity manifests itself as a decrease in the effective number of current carriers. Naturally, the greater the degree of filling of the impurity band, the more readily the scattering of electrons over the band takes place. All this radically changes the temperature dependence of the electrical conductivity in the impurity band. In thermoelectric phenomena and in the effect

For Hall, when the band is filled by more than half, the kinetic coefficients change sign. The typical course of the chemical potential of an electron gas in a narrow energy band with temperature is shown in Fig. 22. The following feature is present: when the band is filled by less than half, the degree of degeneracy increases as the temperature is lowered; when it is filled by more than half, it increases as the temperature is raised. An estimate shows that

Fig. 22. Dependence of the chemical potential of the electron gas on temperature in a narrow energy band.

the band ceiling has an effect up to \((\Delta/kT) \sim 20\). For wider bands one may already use the usual formulas. In this connection we note that, since the maxima on the curves \(\rho(T)\) for PbS occur in the interval \(400\)—\(600^\circ\text{C}\) \((kT \sim 0.06\ \text{eV})\), while the width of the impurity band according to Pekar and Krivoglaz is \(0.04\ \text{eV}\), it is clear that an analysis of the phenomena in PbS at low temperatures cannot be carried out without taking into account the theory of narrow energy bands. Therefore the qualitative arguments given in 8.3 and 8.4 are not convincing.

§ 9. BRIEF OUTLINE OF THE DEVELOPMENT OF THE THEORY OF DEGENERATE SEMICONDUCTORS

The phenomenon of metallization of a semiconductor at low temperatures was first discovered in the study of the electrical properties of SiC by Kurchatov, Kostina, and Rusinov in 1935[^2]. The authors at that time put forward the hypothesis of an impurity band to explain the then apparently anomalous properties of SiC at low temperatures. Later the same phenomenon was discovered in PbS. However, the explanation of the above-named authors[^2] seemed insufficiently substantiated, since it was impossible to understand the reason for the formation of an impurity band at concentrations of excess lead in PbS of the order of \(10^{18}\ \text{cm}^{-3}\). The physical mechanisms leading to the formation of an impurity band were revealed only in 1952 in the work of Pekar and Krivoglaz[^41] on the basis of polaron theory.

However, PbS belongs, strictly speaking, not to the category of semiconductors, but to semimetals (impurity metals). The latter

characterized by the fact that metallic conductivity appears in them at low temperatures, beginning from absolute zero. The phenomenon of degeneration of the electron gas, discovered in a number of well-conducting semiconductors, differs in that it is removed not only at high but also at low temperatures, so that as \(T \to 0\), \(\sigma \to 0\). It was found that, depending on the impurity concentration, one and the same substance can behave both as a degenerate semiconductor and as a semimetal. It was also found that, long before the onset of purely “metallic” features in semiconductors, the electron gas in them can no longer be regarded as an ordinary “classical” gas obeying Maxwell–Boltzmann statistics.

The idea that the anomalous properties, to one degree or another, of degenerate semiconductors, as well as of semimetals, should be connected with the statistical properties of the system of current carriers in them—in particular, with the temperature dependence of the chemical potential—was first expressed and realized in the works of Shifrin\({}^{24}\) in 1944. He was the first to derive general dependences of various kinetic coefficients in semiconductors and semimetals on the chemical potential for various types of dependence of the mean free path on energy (formulas (5.1)—(5.3)); he investigated the question of the influence of degeneracy on the dependence of the mean free path on energy in the case of ionic semiconductors, the temperature dependence of the chemical potential in the case of a simple impurity semiconductor of the Wilson type, and a number of other questions. Unfortunately, Shifrin did not obtain analytic dependences of the chemical potential on temperature, which considerably hindered the use of his results for interpreting the ever-growing body of experimental material on the electrical properties of semiconductors. The significance of Shifrin’s work was truly appreciated only in the postwar period, in connection with the broad development of semiconductor technology based on well-conducting semiconductors. Many of the effects predicted by Shifrin have indeed been found in careful investigations of the electrical properties of Ge, Si, and SiC.

Among the postwar works that substantially supplement Shifrin’s results, it is necessary to note Busch’s work on the investigation of the behavior of the electron gas in SiC\({}^{14}\). He obtained formula (3.4), which makes it possible to determine the temperature dependence of the chemical potential in impurity semiconductors. Reitz\({}^{26}\), in his work, substantially systematized Shifrin’s results and supplemented them with a number of dependences for kinetic coefficients (formulas (5.4)—(5.7)). Graphical methods were developed for determining the chemical potential in semiconductors with a complex energy spectrum, taking into account the degeneracy of the electron gas (for example, Mooser\({}^{10}\)). In Kontorova’s work\({}^{21}\), the influence was investigated

degeneracy of the electron gas on the thermoelectric properties of semiconductors.

The present article somewhat clarifies and develops the results obtained mainly by Shifrin.

§ 10. CONCLUSION

Summarizing what has been said, the following conclusions may be drawn:

1. Degeneracy of the electron gas in semiconductors really exists and is manifested in many phenomena.

2. Taking into account the degeneracy of the electron gas in semiconductors makes it possible to explain, on the basis of band theory, a larger number of facts.

3. In the case of weak degeneracy of the electron gas, the theoretical formulas are in good agreement with experimental data.

4. In the case of strong degeneracy a number of difficulties arise. These difficulties are mainly connected with the fact that the energy spectrum of the electron in strongly degenerate semiconductors has been insufficiently studied.

5. One of the urgent directions for further research is the development of a theory of the formation of impurity bands in atomic semiconductors and of the properties of semiconductors with an impurity band.

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