ELECTRONIC CONDUCTIVITY OF NONMETALS*)

G. Busch

CONTENTS

1. Ionic and electronic conductivity in solids . . . . . . . . . . 258
2. Types of semiconductors . . . . . . . . . . . . . . . . . . . 259
       2.1. Intrinsic semiconductors . . . . . . . . . . . . . . . . . 259
       2.2. Electron and hole semiconductors . . . . . . . . . . . . . 263
       2.3. The case of degeneracy . . . . . . . . . . . . . . . . . . 266
       2.4. On the activation energy. Nature of impurity levels . . . . 268
3. Electrical conductivity and galvanomagnetic phenomena . . . . . 276
       3.1. Theoretical principles . . . . . . . . . . . . . . . . . . 276
       3.2. Experimental part . . . . . . . . . . . . . . . . . . . . 282
       3.3. Summary of experimental data . . . . . . . . . . . . . . 285
4. Thermoelectromotive force in semiconductors . . . . . . . . . . 312
       4.1. Formula for the differential thermopower . . . . . . . . . 312
       4.2. Experimental data . . . . . . . . . . . . . . . . . . . . 314
5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 317

1. IONIC AND ELECTRONIC CONDUCTIVITY IN SOLIDS

The electrical conductivity of a solid is due to the transport of ions or electrons under the action of an electric field.

Whereas ionic conductivity is observed essentially only in heteropolar crystals and glasses, electronic conductivity can in principle occur in any solid, regardless of the nature of the bond.

In turn, electronic conductors are subdivided into two distinct groups: metals and semiconductors.

Metals are characterized by high electrical conductivity, which lies approximately in the range \(5.9 \cdot 10^3\)—\(6.7 \cdot 10^5\ \Omega^{-1}\cdot\mathrm{cm}^{-1}\). The electrical conductivity of metals increases with decreasing temperature, and in a number of metals and their compounds superconductivity sets in near absolute zero temperature**)—

*) G. Busch, Zeitschr. angew. Math. und Physik, 1, 3, 1950 and 2, 81, 1950. Translated from the German by P. P. Konorov, edited by Acad. A. F. Ioffe.
**) See, for example, 5.

state in which the electrical conductivity becomes of the order of \(10^{30}\ \mathrm{ohm}^{-1}\cdot \mathrm{cm}^{-1}\).

According to present-day views, the onset of superconductivity unambiguously characterizes the metallic state. As for the large magnitude of the conductivity and its negative temperature coefficient, these are not yet sufficient signs of the metallic state.

Various elements, such as Si, Ge, Te, were formerly regarded as metals; from the modern point of view, however, on the basis of their electrical conductivity they must be regarded as semiconductors.

Semiconductors, in the modern sense of the word, are electronic conductors whose electrical conductivity becomes vanishingly small near absolute zero. With increasing temperature the electrical conductivity of semiconductors increases. However, there may be temperature regions in which their temperature coefficient is negative. In general, one may suppose that for every semiconductor there exists a certain critical temperature above which the temperature coefficient of conductivity becomes negative, as in metals.

A positive temperature coefficient of electrical conductivity is not an unambiguous sign of a semiconductor, since the electrical conductivity of solids with ionic conduction in most cases has the same temperature dependence. Moreover, ionic and electronic conductivities are often observed simultaneously.

A clear separation of semiconductors and substances with ionic conductivity is much more difficult than is commonly assumed, and requires at least an accurate measurement of transport numbers. The well-known case of \(\alpha\)-\(\mathrm{Ag_2S}\) gives a clear idea of the difficulties of such an analysis of the mechanism of electrical conductivity*).

The present review concerns electrical conductivity, galvanomagnetic, and thermoelectric phenomena in semiconductors. Substances with ionic conductivity and metals are not considered in this review. Nor are rectification phenomena considered, which are characteristic of the boundary between a metal and a semiconductor, nor questions of photoconductivity and the photo-emf.

2. TYPES OF SEMICONDUCTORS

2.1. INTRINSIC SEMICONDUCTORS

A substance crystallizing in a regular lattice contains, in some volume \(R\), a large number \(N\) of identical atoms or ions. On the basis of the electron theory of solids\(^{1,2,7,8,9}\) it is known that the energy spectrum of the totality of all electrons situated in the volume \(R\) is a discrete series of

*) See, for example, \(^{3}\).

of discrete energy levels (see Fig. 1). The “allowed” energy bands, bounded by the energy values \(E_0 \div E_1\), \(E_2 \div E_3\), \(E_4 \div E_5\) (in Fig. 1 shaded by horizontal lines), alternate with forbidden energy bands \(E_1 \div E_2\), \(E_3 \div E_4,\ldots\). Each energy band contains \(N\) possible levels, at each of which no more than two electrons may be placed, provided that their spins have opposite directions. Consequently, each energy band can contain no more than \(2N\) electrons. In elements with an even number of electrons the energy bands are completely filled. Conversely, incomplete filling of bands is characteristic of elements with an odd number of electrons.

Fig. 1. Energy bands in a crystal lattice.

The metallic state is characterized by the fact that the electrons incompletely fill the highest allowed band (alkali metals), or else the filled band overlaps with the neighboring unfilled band (alkaline-earth metals). On the contrary, for the nonmetallic state it is typical that an allowed band is completely filled with \(2N\) electrons at absolute zero temperature.

In practically attainable electric fields, electrons located in a completely filled band cannot acquire additional momentum and, consequently, cannot participate in the creation of an electric current. Consequently, near absolute zero nonmetals are insulators. However, at finite temperatures the situation is different. As the temperature rises, the probability increases that electrons from the upper levels of the filled band will reach the levels of the neighboring higher-lying free band. These electrons, freed by thermal motion, can move more or less freely through the lattice and, under the action of an external electric field, create an electric current. The number \(n_E\) of electrons that have entered the conduction band will be exactly equal to the number \(n_L\) of remaining empty places or “holes” in the filled band, which under the action of the field will create a “hole” current. While electrons, having a negative charge, will be accelerated against the field, the “holes,” which may be regarded as positive charges, will move in the direction of the field. Thus, motion of charges is possible in both bands.

Nonconducting at absolute zero, a crystal becomes conducting at higher temperatures.

The number \(N_E\) of electrons in the conduction band can be calculated as follows \(^{129}\): the number of energy states (eigenvalues) in the energy interval \(dE\), i.e. the so-called density of states \(D(E)\), near the lower edge \(E_2\) of the conduction band is

\[ D_2(E)\,dE=\frac{R}{4\pi^2}\left(\frac{2m_2^*}{\hbar^2}\right)^{\frac{3}{2}}(E-E_2)^{\frac{1}{2}}\,dE, \tag{1} \]

where \(R\) is the volume of a certain region of the crystal, \(m_2^*\) is the effective mass of an electron in the conduction band, \(\hbar=\dfrac{h}{2\pi}=1.01\cdot 10^{-27}\,\mathrm{erg\cdot sec}\) (Planck’s constant).

For the upper edge of the filled band

\[ D_1(E)\,dE=\frac{R}{4\pi^2}\left(\frac{2m_1^*}{\hbar^2}\right)^{\frac{3}{2}}(E_1-E)^{\frac{1}{2}}\,dE, \tag{2} \]

where \(m_1^*\) is the effective mass of an electron in the filled band.

The effective electron masses \(m_1^*\) and \(m_2^*\) are in general not identical with the rest mass of the electron \(m_0\). They are introduced into the discussion in order to reflect the fact that an electron inside a crystal lattice moves not in a field with a constant potential, but in a field whose potential has a spatial periodicity with the period of the crystal lattice.

For the number of electrons \(N(E)\) with energies lying in the interval \(E,\,E+dE\), we have:

\[ N(E)\,dE=2D(E)W(E)\,dE, \tag{3} \]

where \(W(E)\) is the Fermi–Dirac distribution function, equal to

\[ W(E)=\frac{1}{e^{\frac{E-\zeta}{kT}}+1}, \tag{4} \]

where \(\zeta\) is an energy parameter equal, at \(T=0\), to the limiting energy of the Fermi–Dirac distribution, \(k\) is Boltzmann’s constant, and \(T\) is the absolute temperature.

In the stationary case the total number of electrons in the conduction band must be equal to the number of “holes” in the filled band. From this and taking (3) into account, we have:

\[ N_E=N_L=2\int_{\text{conduction band}} D_2(E)W(E)\,dE =2\int_{\text{filled band}} D_1(E)[1-W(E)]\,dE. \tag{5} \]

Taking into account expressions (1), (2), and (4) and substituting the limits of integration, we obtain:

\[ m_2^{*\,\frac{3}{2}}\int_{E_2}^{\infty} \frac{(E-E_2)^{\frac{1}{2}}\,dE}{e^{(E-\zeta)/kT}+1} = m_1^{*\,\frac{3}{2}}\int_{-\infty}^{E_1} \frac{(E_1-E)^{\frac{1}{2}}\,dE}{e^{-(E-\zeta)/kT}+1}. \tag{6} \]

The limiting Fermi energy \(\zeta\) lies between the band boundaries \(E_1\) and \(E_2\) (see Fig. 2). In the left-hand side of equality (6) the integration is carried out over energies \(E\geq E_2>\zeta\). For the integral in the right-hand side of the equality \(E\leq E_1<\zeta\).

Fig. 2. Energy diagram of an intrinsic semiconductor.

If the width of the forbidden gap is

\[ \Delta E_1 = E_2 - E_1 \tag{7} \]

not very small, then at sufficiently low temperatures the unit in the denominator is much smaller than the exponential and may be neglected. This means that the energy levels of the conduction band prove to be only partially occupied. The concentration of electrons in the conduction band, like that of “holes” in the filled band, is also very small. As a consequence, the electron gas proves to be nondegenerate and may be described rather accurately by Maxwell–Boltzmann statistics. The problem of degeneracy will be considered in the following paragraph.

In the absence of degeneracy, equality (6) is simplified and takes the form:

\[ m_2^{*\,\frac{3}{2}}\int_{E_2}^{\infty} (E-E_2)^{\frac{1}{2}} e^{-\frac{E-\zeta}{kT}}\,dE = m_1^{*\,\frac{3}{2}}\int_{-\infty}^{E_1} (E_1-E)^{\frac{1}{2}} e^{\frac{E-\zeta}{kT}}\,dE, \]

which leads to the following expression for the limiting energy \(\zeta\):

\[ \zeta=\frac{E_1+E_2}{2}+\frac{kT}{2}\ln\left(\frac{m_1^*}{m_2^*}\right)^{-\frac{3}{2}}. \tag{8} \]

Starting from formulas (1), (4), (5), and (7), and substituting the expression found for \(\zeta\) into formulas (5) and (7), with \(n=\frac{N}{R}\), we obtain an expression for the concentration of electrons in the conduction band and of “holes” in the filled band:

\[ n_E=n_L=\frac{1}{4}\left(\frac{2m_2^* kT}{\pi\hbar^2}\right)^{\frac{2}{3}}\cdot \left(\frac{m_1^*}{m_2^*}\right)^{\frac{3}{4}}e^{-\frac{\Delta E}{2kT}}. \tag{9} \]

It is evident from this expression that the concentration of current carriers increases very rapidly with increasing temperature, depends on the effective masses of the carriers \(m_1^*\) and \(m_2^*\), and, moreover, depends substantially on the magnitude \(\Delta E\)—the so-called activation energy. Each of these quantities depends on the geometry of the lattice, as well as on the manner in which atoms or ions are joined in the lattice.

Substances whose conductivity is brought about by the mechanism described above are called intrinsic semiconductors.

Since the conductivity is proportional to the number of current carriers, on the basis of formula (9), in a given crystal lattice it should depend only on temperature and, at a given temperature, should be a material constant for each given substance.

This, however, is in contradiction with a large part of the experiments carried out so far. It has been established that electrical conductivity depends substantially on the degree of deviation of the composition of a substance from stoichiometry, as well as on the type and concentration of impurities foreign to the lattice.

2. 2. ELECTRON AND HOLE SEMICONDUCTORS

Any disturbances of an ideal crystal lattice, such as, for example, vacancies in the lattice or impurities of foreign atoms, lead to the appearance in the system of new additional energy levels for electrons. If the width of the energy gap between any of these levels, filled at low temperatures by electrons, and the lower edge of the conduction band is small, then such a level acts as a supplier of electrons (a donor level). With increasing temperature, electrons will pass from such levels into the conduction band.

and create electronic conductivity there (Fig. 3). If these levels at low temperatures are free of electrons and are located near the upper edge of the filled band, then they have the opposite effect and can capture electrons (acceptor levels). As the temperature is raised, part of the acceptor levels is filled by electrons from the filled band. In this process “holes” are formed in the filled band, and this leads to hole conductivity of the crystal (Fig. 4). The former semiconductors are called electronic, or semiconductors of the \(n\)-type, and the latter—hole, or semiconductors of the \(p\)-type. Taken together they are called impurity semiconductors.

Fig. 3. Scheme of the energy levels of an electronic semiconductor.

Fig. 4. Scheme of the energy levels of a hole semiconductor.

The concentration of electrons \(n_E\), as well as the concentration of “holes” \(n_L\), if they are small, can be calculated in the same way as was done for intrinsic semiconductors. In the state of thermal equilibrium the number of electrons in the conduction band must be equal to the number of ionized donor levels, i.e.

\[ N_E = 2 \int_{E_2}^{\infty} D_2(E) W(E)\, dE = N_D \left[1 - W(E'_1)\right], \tag{10} \]

where \(N_D\) is the number of donor levels, \(E'_1\) is the energy level of the impurity containing excess electrons. Substituting here from formula (1) the expression for the eigenvalue of the energy,

and also introducing an explicit expression for the distribution function and the concentration of donor levels \(n_D = \dfrac{N_D}{R}\), we have:

\[ \frac{1}{2\pi^2}\left(\frac{2m_2^*}{\hbar^2}\right)^{\frac{3}{2}} \int_{E_2}^{\infty} \frac{(E-E_2)^{\frac{1}{2}}\,dE} {e^{\frac{E-\zeta}{kT}}+1} = \frac{n_D} {e^{\frac{E_1-\zeta}{kT}}+1}. \tag{11} \]

This expression is simplified if the activation energy \(\Delta E_D = E_2 - E_1'\) (Fig. 3) is sufficiently large and the temperature is low:

\[ \frac{1}{2\pi^2}\left(\frac{2m_2^*}{\hbar^2}\right)^{\frac{3}{2}} \int_{E_2}^{\infty} (E-E_2)^{\frac{1}{2}} e^{-\frac{E-\zeta}{kT}} = n_D e^{-\frac{E_1'-\zeta}{kT}} . \tag{12} \]

The integral is easily evaluated, and for \(\zeta\) the following expression is obtained:

\[ \zeta = \frac{E_2+E_1'}{2} + kT \left[ \ln n_D^{\frac{1}{2}} - \ln \left( \frac{m_2^* kT}{2\pi\hbar^2} \right)^{\frac{3}{4}} \right]. \tag{13} \]

Hence, taking into account the right-hand side of equality (12), we obtain:

\[ n_E = \frac{1}{2} \left( \frac{2m_2^* kT}{\pi\hbar^2} \right)^{\frac{3}{4}} n_D^{\frac{1}{2}} e^{-\frac{\Delta E_D}{2kT}}, \tag{14} \]

where \(n_D\) is the number of donor levels in \(1\ \mathrm{cm}^3\), \(m_2^*\) is the effective mass of an electron in the conduction band, and \(\Delta E_D\) is the activation energy of donor impurities.

A similar expression is also obtained for hole semiconductors:

\[ n_L = \frac{1}{2} \left( \frac{2m_1^* kT}{\pi\hbar^2} \right)^{\frac{3}{4}} n_A^{\frac{1}{2}} e^{-\frac{\Delta E_A}{2kT}}, \tag{15} \]

where \(n_A\) is the number of acceptor levels in \(1\ \mathrm{cm}^3\), \(m_1^*\) is the effective mass of an electron in the filled band, and \(\Delta E_A\) is the activation energy of acceptor impurities.

In both cases the dependence of the concentration on temperature is the same as for intrinsic semiconductors, with the only difference that the exponent of the temperature entering the pre-exponential factor is smaller. In addition, the concentrations of donor and acceptor levels \(n_D\) and \(n_A\) appear here as new quantities. Thus the theory formally takes into account—

confirms the experimental fact that the electrical conductivity of all known semiconductors can vary over wide limits under the influence of such chemical processes as, for example, oxidation and reduction, or the introduction of foreign impurity atoms.

In every semiconductor both donor and acceptor levels may be present simultaneously. It should be borne in mind that donor and acceptor impurities occurring together have different activation energies. In addition, one must take into account that every impurity semiconductor, at a sufficiently high temperature which does not, however, substantially alter the crystal lattice, must exhibit intrinsic conductivity.

From all that has been set forth above it follows that, in order to explain conductivity phenomena in impurity semiconductors, very careful experimental investigations and a detailed analysis of the data obtained are needed.

2. 3. THE CASE OF DEGENERACY

The expressions given in the preceding paragraph for the carrier concentration as a function of temperature are valid only so long as the concentration is considerably smaller than the critical value, which is obtained from the degeneracy criterion:

\[ n_{\text{crit}}=\frac{1}{3\pi^2}\left(\frac{2mk}{\hbar^2}\right)^{\frac{3}{2}}T^{\frac{3}{2}} =3.6\cdot10^{16}T^{\frac{3}{2}}. \tag{16} \]

The temperature at which degeneracy of the electron gas in a semiconductor sets in depends on the manner in which the electron concentration \(n\) varies with temperature.

From the expression for the dependence of the concentration on temperature

\[ n\sim e^{-\frac{\Delta E}{2kT}} \]

it is seen that the concentration falls very rapidly as the temperature decreases, and, as the temperature increases, tends toward a constant value. Consequently, degeneracy will occur neither at very low nor at very high temperatures; rather, it should be expected at certain intermediate temperatures.

At concentrations higher than \(n_{\text{crit}}\), the approximations made in deriving formulas (9), (14), and (15), which are equivalent to using Maxwell–Boltzmann statistics, are inadmissible.

On the other hand, the concentration of electrons or “holes,” even in well-conducting semiconductors, does not reach such large values as are found in metals, where complete degeneracy of the electron gas is observed. Therefore, in semi-

...in conductors one should expect only weak degeneracy. This region of weak degeneracy, important for semiconductors, has been analyzed in detail by Shifrin^121.

For electronic semiconductors this problem can be solved rigorously with the aid of equality (11). Denoting

\[ x=\frac{E-E_2}{kT},\quad \mu=\frac{\zeta-E_2}{kT}\quad \text{and}\quad \varepsilon=\frac{E_2-E'_1}{kT}=\frac{\Delta E_D}{kT}, \]

from equality (11) we obtain:

\[ \frac{A^*}{e^{\varepsilon+\mu}+1} = \frac{1}{2}\int_0^\infty \frac{x^{1/2}\,dx}{e^{x-\mu}+1} = f_{1/2}(\mu), \tag{17} \]

where

\[ A^*=\pi^2 n_D\left(\frac{\hbar^2}{2m_2^*kT}\right)^{3/2}. \]

Shifrin showed that the parameter \(\mu\) in these formulas expresses the degree of degeneracy of the electron gas. The condition \(\mu=0\) is identical with the degeneracy criterion (16) and means that the boundary energy \(\zeta\) exactly coincides with the lower edge of the conduction band.

In the case when \(\mu \ll -2\), i.e., while the boundary energy lies below the lower edge of the conduction band by more than \(2kT\), the electron gas may be considered nondegenerate. Conversely, for values \(\mu \gg +1\) complete degeneracy already occurs. In this case the boundary energy \(\zeta\) lies above the lower edge of the conduction band.

From the equation \(\mu=0\), equivalent to formula (16), one can easily obtain the relation between the critical value of the activation energy \(\Delta E_D\) and the concentration of donor impurities \(n_D\) as functions of the temperature at which degeneracy sets in. Solving the problem again requires determining \(\zeta\) or \(\mu\) from equality (17) as a function of temperature. Shifrin used a graphical method for this purpose. On the other hand, Bush and Labhart^36 obtained for \(f_{1/2}(\mu)\) an explicit expression in the form of a function valid for all values \(\mu \ll +1.5\), with satisfactory accuracy \(\pm 3.5\%\), which made it possible to apply formula (17) for analytical calculations*). For values \(\mu \gg +1.5\) there exists a known expression in Sommerfeld’s electron theory of metals.

) As Putley^108 noted, the function \(f_{1/2}(\mu)\) was computed and tabulated very accurately by McDougall and Stoner^62. (Editor’s note.*)

Degeneracy phenomena play an essential role for good conducting semiconductors, such as PbS, PbSe, SiC, and for semiconductor elements of particular current interest, such as Si and Ge.^{81, 106, 108}

2.4. ON THE ACTIVATION ENERGY. NATURE OF IMPURITY LEVELS

The activation energy \(\Delta E_1\) of an intrinsic semiconductor is the energy gap between the conduction band and the first lower-lying filled band (Fig. 2). The theoretical determination of this quantity requires exact knowledge of the eigenfunctions and eigenvalues of the electron energies for each lattice. Such calculations were carried out, for example, by Kimball^84 and Hund^76 for diamond, and by Mulleneer^102 and Golmstock^75 for crystalline silicon, which has a diamond-type lattice.

Unfortunately, for semiconductors with small energy gaps, these calculations have so far been made too inaccurately; therefore experimental determinations of the activation energy are usually used.

For impurity semiconductors the activation energy can sometimes be calculated, if one makes use of model representations corresponding to the character of the impurities.

2.41. Ionic crystals

According to modern views, based on the experimental work chiefly of Wagner and Schottky*), heteropolar crystals consist of positive metallic \((M^+)\) and negative metalloid \((X^-)\) ions, with corresponding deviations from the ideal crystalline structure of the lattice in the form of various violations of periodicity. These violations may cause the onset of electronic conductivity.

There are two typical cases of violation of periodicity:

A. A neutral metal atom \(M\) is introduced into an interstice of the crystal lattice (Fig. 5). Since, generally speaking, in the space between the nodes of the crystal lattice there is too little room for a neutral atom, it will be located there in the form of an ion \(M^+\), from which an electron has been torn away in the force field;

B. Individual sites of ions \((X^-)\) in the crystal lattice are unoccupied (Fig. 6).

*) See.^4

For preservation of the neutrality of the lattice, in place of each missing ion \((X^-)\) there is an electron which, by chemically combining with one of the neighboring ions \(M^+\), converts it into a neutral atom \(M\). Since the electron can with equal probability attach itself to any of the neighboring ions \(M^+\), the “hole” \(\square\) (the empty place left after the negative charge) may be regarded as a positive charge in whose force field the electron moves.

In both cases the crystal lattice contains an excess of metal, which can be produced either by chemical reduction or by heating the crystal in the vapors of the given metal.

Fig. 5. Sites of distortions in an ionic lattice with an excess of metal. A neutral metal atom is located in an interstice of the crystal lattice. Electronic semiconductor.

Fig. 6. Distortions of the ionic lattice by an excess of metal. In place of negative ions there are empty sites. Electronic semiconductor.

According to Mott\(^6\), impurities of the above-described kind, irrespective of the type of introduced or missing ions, may be regarded as hydrogen-like atoms, in which the electron moves in a field with Coulomb potential

\[ V(r)=-\frac{e}{r^2}\frac{1}{K_{\mathrm{eff}}}, \tag{18} \]

where \(K_{\mathrm{eff}}\) is the effective dielectric constant, which depends on the polarization of the medium surrounding the site of the disturbance (impurity atom), and whose value lies between the static and optical dielectric constants of the crystal.

The radius of the orbit of such an electron, owing to the weakening of the field, is considerably larger than in the hydrogen atom and may amount to several atomic distances.

For the energy states of the electron in this case one obtains an expression similar to the expression for the hydrogen atom,

\[ E_n=-\frac{e^4 m^*}{2\hbar^2 n^2}\frac{1}{K_{\mathrm{eff}}^2} \tag{19} \]

with quantum number \(n=1, 2, 3\ldots\).

The ionization energy \(E_j\) of some impurity atom will therefore be \(\frac{1}{K_{\mathrm{eff}}^2}\) times smaller than for the hydrogen atom, i.e.

\[ E_{jE}=\frac{e^4 m_E^*}{2\hbar^2}\frac{1}{K_{\mathrm{eff}}^2} =\frac{13.53}{K_{\mathrm{eff}}^2}\frac{m_E^*}{m_0}\ \text{eV}, \tag{20} \]

where \(m_E^*\) is the effective mass of the electron, \(m_0\) the mass of an electron at rest, and \(E_{jE}\) is nothing other than the work necessary in order to free the electron located at the impurity level, i.e. to transfer it into the conduction band. The ground state of the impurity atom also lies within the limits \(E_{jE}\) below the lower edge of the conduction band, i.e. \(E_{jE}\) is identical with the activation energy \(\Delta E_D\) of an electronic semiconductor (Fig. 3).

The quantity \(\Delta E_{jE}\) can be estimated by using formula (20). This estimate shows that one may expect smaller values for the activation energy the stronger the polarization of the crystal lattice.

Fig. 7. Disturbance of the ionic lattice by a deficiency of metal. In place of positive ions there are vacant sites. Hole semiconductor.

Semiconductors with impurities of type \(A\) and \(B\) are electronic semiconductors, since these impurity levels act as donors. Their concentration increases with an increase in the excess of metal, i.e. during chemical reduction of the semiconductor. Thus, semiconductors with an excess of metal and semiconductors obtained by chemical reduction are identical.

One of the typical and most thoroughly investigated representatives of this group of semiconductors is ZnO. Other examples of such semiconductors will be given in Section 3.3.

B. Disturbances of type \(A\) may correspond to semiconductors with excess atoms of metalloid \(X\) in the interstices of the lattice. However, this kind of disturbance of the order in the lattice cannot be realized because of the large ionic radii of the metalloids.

C. Individual sites of \(M^+\)-ions in the lattice are not occupied (Fig. 7). In order to preserve the electrical neutrality of the lattice, for each

ELECTRONIC CONDUCTIVITY OF NONMETALS

one electron must be removed from the lattice from the missing metallic ion. This electron may be detached from one of the neighboring \(X^{-}\)-ions and thus form a neutral atom \(X\). However, with equal success this electron may be detached from an \(M^{+}\)-ion, after which an \(M^{++}\)-ion is formed. Which of these two cases is actually realized depends on whether the electrons in the very highest filled band belong to the \(X^{-}\)-ions or to the \(M^{+}\)-ions.

The missing electron behaves as a positively charged “hole,” which moves in the same way as, in case B, the negative charge that forms an empty site (a missing positive charge) moves in the force field.

Mott’s explanation can be extended to this case as well, since here too we have a system similar to the hydrogen atom, differing only in the signs of the charges.

For the ionization of such impurities, in other words, for the creation of hole conductivity, the same energy \(E_{jL}\) is required, determined by formula (20), taking into account the difference in the effective masses \(m^{*}\). But removal of a positively charged “hole” is physically equivalent to the attachment of an electron. Therefore, in order that a freely moving “hole” be formed in the filled band, an additional reserve of energy, equal to \(E_{jL}\), must be supplied to the electron in the filled band. This energy, corresponding to the acceptor energy level, is equal to

\[ E_{jL}=\frac{13.53}{K_{\mathrm{eff}}^{2}}\,\frac{m_{L}^{*}}{m_{0}}\ \text{eV}, \tag{21} \]

and the acceptor level lies above the upper edge of the filled band (Fig. 4). Here \(m_{L}^{*}\) is the effective mass of the “hole.”

Thus, \(E_{jL}\) is equal to the activation energy \(\Delta E_{A}\) for a hole semiconductor. In both cases B and C there is a deficiency of metal atoms in the lattice. The concentration of acceptors increases when the amount of metal is reduced relative to the stoichiometric ratio, i.e., upon oxidation.

One of the most thoroughly investigated hole semiconductors is \(\mathrm{Cu}_{2}\mathrm{O}\). Other hole semiconductors will be indicated in Section 3.3.

Several compounds are also known that can possess both hole and electronic conductivity.

Studies by Bauser \(^{31}\), Eckart and Reitel \(^{48}\), Eisenmann \(^{50}\), Ginterberger \(^{70}\), and Morton \(^{101}\) of the compounds PbS and PbSe, as well as studies by Meyer and Neldel \(^{96}\) of \(\mathrm{UO}_{2}\), showed that these substances, with an excess of metal, exhibit electronic conductivity, and with an excess of metalloid, hole conductivity. Such semiconductors we shall call amphoteric.

2.42. Valence Crystals

It is clear that in semiconductors (or insulators) whose lattices are built of neutral atoms, such as, for example, diamond, Si, Ge, SiC, with completely filled energy bands, there must be a different mechanism of conduction than in ionic crystals.

On the basis of extensive experimental work*) one may form the following picture of the conductivity of these substances. Silicon and germanium are elements of group IV of the periodic system. It has been established that an impurity, in a small concentration, of trivalent elements, such as, for example, B and Al, causes

Fig. 8. Distortion of the lattice of a valence crystal. A trivalent boron atom substitutes for a silicon atom. Hole conduction.

Fig. 9. Distortions in the lattice of a valence crystal. A pentavalent phosphorus atom substitutes for a silicon atom.

hole conduction, while an impurity of pentavalent elements, such as P, Sb, As, on the contrary, causes electron conduction. The resulting disturbances of the lattice periodicity, according to the data of Pearson and Bardeen \(^{106}\), have the character of substitution, i.e. regular sites in the crystal lattice, normally occupied by silicon atoms, are replaced by foreign atoms P or B (Figs. 8 and 9).

In an ideal silicon lattice (a diamond-type lattice) each of the silicon atoms is located at the vertices of a regular tetrahedron. If now one of the silicon atoms is replaced by a pentavalent phosphorus atom, then one of the valence electrons of the phosphorus atom will not participate in the chemical bond. It will behave like an electron captured by an empty site in the lattice

*) See, for example, \(^{10}\).

of an ionic crystal, i.e., it will move around the introduced atom P in a hydrogen-like orbit.

The quantitative calculation here is exactly the same as for an ionic lattice. It shows that in this case electronic conduction occurs.

If, as an impurity, a trivalent boron atom is introduced, then, on the contrary, it lacks one electron, compared with the surrounding atoms, for the saturation of the valence bonds, and, as is easy to see, this leads to hole conduction.

On the basis of these ideas one should expect that semiconductors with a homeopolar character of bonding are mainly amphoteric. Indeed, in Si, Ge, and SiC one can observe both hole and electronic conduction.

In this interpretation, however, some unclear points still remain. The introduction, for example, as an impurity, of Sn atoms, which also belong to group IV of the periodic system, likewise produces impurity conduction, which is difficult to explain from the point of view described above.

2.43. Semiconductors with an unfilled $3d$ shell

A special position among semiconductors is occupied by compounds of the elements Se, Ti, V, Cr, Mn, Fe, Co, and Ni. In these elements the $3d$ shell is not completely filled, whereas the $4s$ shell is either entirely, or at least partially, filled. Chemical compounds of these elements, such as, for example, MnO, CoO, NiO, Fe$_2$O$_3$, etc., also possess an incompletely filled band of $3d$ levels. According to the band theory of electrical conductivity, metallic conductivity should be expected in these substances. In reality, however, their electrical conductivity at room temperature has a magnitude of the order of $10^{-7}$—$10^{-8}$ ohm$^{-1}\cdot$cm$^{-1}$, and the dependence of electrical conductivity on temperature corresponds to typical semiconductors. These circumstances were first pointed out by de Boer and Verwey$^{44}$. De Boer and Verwey’s view of the mechanism of conduction in semiconductors of the transition elements is as follows. Band theory is inapplicable to these compounds, since, owing to weak interaction, the eigenfunctions of the $3d$ electrons of neighboring atoms or ions in the lattice of these compounds do not overlap. In other words, the $3d$ electrons belong to a definite atom or ion and cannot move freely through the lattice, as is assumed for electrons in a metal. Thus, there are potential “hills” between the individual layers of the crystal lattice, and the $3d$ electrons do not participate in the mechanism of formation of the chemical bond.

The $3d$ band of a positive doubly charged ion lies energetically deep (the 3rd ionization potential) and, owing to the negligible interaction between the $3d$ electrons of neighboring

of atoms, its width is small. Under these conditions the mechanism of electrical conductivity in compounds with an unfilled \(3d\)-shell is very similar to the mechanism of electronic conductivity in ionic conductors. Owing to thermal excitation and the tunneling effect, \(3d\)-electrons can move to neighboring equivalent sites of the crystal lattice. If the surrounding nodes are normally occupied, then the electrons will immediately return back.

However, if nearby there is an ion of the same element that lacks one electron, then the residence time of the electron at this site will be so large that during this time an electron from some other site of the crystal lattice will jump to the vacant site. If an electric field is applied to the crystal, then in this case it creates a preferred direction for such an exchange of sites, so that as a result we obtain a directed displacement of charges.

For such a mechanism of electrical conductivity to occur, it is necessary that in equivalent sites of the crystal lattice there be ions of one and the same element with different charges. This can arise for various reasons. First, the reason may be a deviation from the correct stoichiometric ratio of the substance. An example of this may be \( \mathrm{NiO} \) with a small excess of oxygen. According to the investigations of de Boer and Verwey, in this case vacant sites of \( \mathrm{Ni}^{++} \) ions remain in the lattice. To preserve the neutrality of the lattice, two \( \mathrm{Ni}^{++} \) ions adjacent to this defect must each give up one electron.

The \( \mathrm{Ni}^{+++} \) ions that arise in this way, while only slightly affecting the position of the \(3d\)-zones, also give rise to the above-mentioned mechanism of conductivity.

Secondly, such conductivity can be produced by a crystal lattice in which ions of one and the same element, but with different charges, may be situated at equivalent sites. Such a ratio occurs, for example, in a lattice of the spinel type. This lattice is characteristic of compounds belonging to the groups described here, such as, for example, \( \mathrm{Fe}_3\mathrm{O}_4 \), \( \mathrm{FeOCr}_2\mathrm{O}_3 \), \( \mathrm{MnOFe}_2\mathrm{O}_3 \), etc.

The spinel lattice has the closest, essentially cubic, packing of oxygen ions (Fig. 10). If in such a lattice one selects an elementary cubic cell containing 32 oxygen ions, then between these oxygen ions there will be 32 vacant sites surrounded octahedrally by \( \mathrm{O}^{--} \) ions, and 64 vacant sites surrounded tetrahedrally by \( \mathrm{O}^{--} \) ions. Of all these vacant sites, 16 octahedral and 64 tetrahedral ones are filled, respectively, by divalent or trivalent metal ions, so that cubic holohedral symmetry is preserved.

In many substances all trivalent ions occupy octahedral, and all divalent ions—tetrahedral, vacancies. Thus, in this case, at equivalent lattice sites everywhere there are ions with identical charges, which makes such a lattice nonconducting. Bart and Pozdniak\({}^{119}\), however, showed that in many substances with the spinel lattice eight tetrahedral vacancies are occupied by trivalent ions, so that 16 octahedral vacancies turn out to be occupied by divalent and trivalent ions in equal numbers. On this basis Verwey, Haaijman, and Romeijn\({}^{124}\) constructed their theory of the mechanism of electrical conductivity in oxides with the spinel structure.

Fig. 10. Unit cell of the spinel lattice.

The essence of this theory is as follows: if the divalent and trivalent ions occupying the octahedral vacancies of the lattice belong to one and the same element and are statistically distributed, then the transfer of an electron from a divalent to a trivalent ion will not be accompanied by changes in the lattice, and on the average the lattice energy will not change. When a field is applied, an electric current will flow.

The existence of such a peculiar spinel lattice in \(\mathrm{Fe}_3\mathrm{O}_4\) can explain the high electrical conductivity of this compound. In \(\mathrm{Fe}_3\mathrm{O}_4\), \(\mathrm{Fe}^{++}\) ions are readily replaced by other divalent ions, such as, for example, \(\mathrm{Ni}^{++}\), \(\mathrm{Zn}^{++}\), etc., and in such a crystal a statistical distribution of \(\mathrm{Ni}^{++}\) and \(\mathrm{Fe}^{++}\) ions over octahedral vacancies is observed.

The transfer of an electron from \(\mathrm{Ni}^{++}\) to \(\mathrm{Fe}^{+++}\) requires an expenditure of energy and is less probable\({}^{*}\). Therefore the introduction of such ions reduces the electrical conductivity of the substance, and at large differences in energy the substance practically becomes an insulator.

\({}^{*}\) If in this case energy is gained, then nickel will play the role of the trivalent ion, and iron that of the divalent ion. (Editor’s note.)

3. ELECTRICAL CONDUCTIVITY AND GALVANOMAGNETIC PHENOMENA

3.1. THEORETICAL CONSIDERATIONS

3.11. Electrical Conductivity

The electrical conductivity of semiconductors in which both electron and hole conductivity exist simultaneously is determined by the following expression:

\[ \sigma = en_E b_E + en_L b_L . \tag{22} \]

Here \(n_E\) and \(n_L\) denote, respectively, the concentrations of electrons and “holes,” and are expressed by formulas (9), (10), and (11); \(b_E\) and \(b_L\) are the mobilities of the corresponding charge carriers.

Mobility is the mean translational velocity of the charge carriers acquired by them under the action of a field equal to \(1\ \mathrm{V/cm}\).

In the case of a nondegenerate electron gas, the mobility can be calculated from the formula given by the classical Lorentz theory,

\[ b_{E,L}=\frac{4}{3}\frac{e l_{E,L}}{(2\pi m^*_{E,L} kT)^{1/2}}, \tag{23} \]

where \(l_{E,L}\) denotes the mean free path, and \(m^*_{E,L}\) are the effective masses, respectively, for electrons and “holes.”

The principal difficulty for the theory of the electrical conductivity of semiconductors lies precisely in determining the magnitude of the mean free path. The question of the dependence of the mean free path on temperature in many cases proves to be much simpler. Since the question of the mean free path plays an essential role in the theory of semiconductors, we shall dwell on it here in somewhat greater detail.

The mean free path of an electron is inversely proportional to the probability of scattering of the electron within the crystal lattice. The measure of the probability of electron scattering is the degree of deviation from the strict periodicity of an ideal crystal lattice. These deviations are caused, on the one hand, by the thermal motions of atoms or ions in the lattice, and, on the other hand, by distortions of the lattice consisting in the absence of atoms or ions in interstitial positions or in the presence of impurities of foreign atoms or ions.

As in the case of alloys, the mean free path \(l\) may be represented as consisting of two parts. This, of course, will be valid only in the simplest case, namely, under the assumption of the independence of these two additive terms, as occurs in alloys:

\[ \frac{1}{l}=\frac{1}{l_g}+\frac{1}{l_s}. \tag{24} \]

ELECTRONIC CONDUCTIVITY OF NONMETALS

In this case \(l_g\) depends on the thermal vibrations of the lattice, while \(l_s\) depends on disturbances of the crystal lattice, i.e., on the introduction of foreign atoms.

The temperature dependence of the part \(l_g\), determined by lattice vibrations, can be understood on the basis of elementary considerations. The probability of scattering of an electron caused by the thermal motion of particles in the lattice is proportional to the effective scattering cross section, i.e., proportional to the square of the amplitude of the thermal motion, and therefore at sufficiently high temperatures is proportional to the absolute temperature \(T\). The mean free path \(l_g\) is inversely proportional to the scattering probability and, consequently, inversely proportional to the temperature, i.e.,

\[ l_g \sim \frac{1}{T}. \tag{25} \]

Wilson \({}^{129}\) and Bronstein \({}^{31}\) theoretically considered the quantitative side of this problem. As a supplement to Bloch’s theory \({}^{23}\), they took into account the interaction of electrons in a semiconductor with sound vibrations of the lattice. The difference from the theory of metals here consists in the fact that the electrons are treated as a nondegenerate gas, i.e., as particles possessing thermal motions. This gives, not only for high temperatures but also for all practically attainable temperatures, i.e., beginning with \(T > 1^\circ \mathrm{K}\), the same temperature dependence as that given by formula (25).

A more complete theory of the mean free path of electrons in ionic crystals was given by Fröhlich and Mott \({}^{57,58}\). In this theory the polarity of the crystal lattice is taken into account. The principal cause of electron scattering here is the optical branch of lattice vibrations. Using Maxwell–Boltzmann statistics, these authors found that for temperatures below the characteristic temperature of the lattice \(\theta\), i.e., for \(T < \theta\), the following dependence holds:

\[ l_g = \frac{6}{\sqrt{\pi}}\, \frac{\varepsilon-\varepsilon_0+1}{\varepsilon-\varepsilon_0}\, a_0 \left(\frac{T}{\theta}\right)^{\frac{1}{2}} \left(e^{\frac{\theta}{T}}-1\right), \tag{26} \]

where the constants \(\varepsilon\) and \(\varepsilon_0\) are the static and, respectively, optical dielectric constants,

\[ a_0=\frac{\hbar^2}{m^* e^2}, \]

\(m^*\) is the effective mass of the electron and \(e\) is its charge.

The corresponding theory for a valence crystal was developed by Seitz \({}^{120}\). In this case, owing to the nonpolar character of the valence bond, electric polarization is manifested neither in the acoustic nor in the optical branches of lattice vibrations. This model corresponds to the previously considered model of Wilson and

Bronstein, and therefore one should expect similar results. Indeed, Seitz, on the basis of his more rigorous theory, obtained for the mean free path an inverse dependence on temperature, namely:

\[ l_g=\frac{9\pi}{4}\,\frac{\hbar^2 c^3 n_0 M}{C^2 k m^{*2}}\,\frac{1}{T}; \tag{27} \]

where \(c\) is the velocity of propagation of sound vibrations, \(n_0\) is the number of atoms in \(1\ \mathrm{cm}^3\), \(M\) is the mass of an atom, \(k\) is Boltzmann’s constant, and \(C\) is a constant which is approximately proportional to the characteristic temperature of the lattice and serves as a measure of the interaction of the electrons with the lattice. Unfortunately, this quantity is determined directly very inaccurately and can only be roughly estimated.

In Seitz’s theory, the interactions of electrons with energies of the order of \(kT\) are considered only with acoustic vibrations of the lattice, since optical vibrations, owing to the high characteristic temperatures of most valence crystals, may be disregarded at practically attainable temperatures.

The influence of impurities in the lattice on the mean free path or, correspondingly, on the mobility \(b_s\) was investigated by Conwell and Weisskopf\({}^{42}\), under the assumption of elastic scattering of electrons of thermal velocities by foreign ions statistically distributed in the lattice. They obtained:

\[ \frac{1}{b_s} = \frac{n_s \pi^2 e^3 m^{\frac{1}{2}}} {2^{\frac{7}{2}}\varepsilon^2 (kT)^{\frac{3}{2}}} \ln(1+\chi^2), \tag{28} \]

where

\[ \chi=\frac{\varepsilon d kT}{e^2}, \]

and the remaining constants have the following meaning: \(n_s\) is the concentration of foreign ions, \(\varepsilon\) is the dielectric constant of the crystal, and \(2d\) is the mean distance between neighboring foreign ions.

For nondegenerate electronic and hole semiconductors with statistically distributed foreign ions, the following picture is obtained on the basis of theoretical calculations: whereas the term \(l_g\), depending on lattice vibrations, according to formula (26) or (27), decreases with increasing temperature, the term \(l_s\), due to the presence of impurities, according to formula (28), should increase with increasing temperature. This makes it possible, for valence crystals with an admixture of foreign

ions, compute the true course of the dependence of mobility on temperature. Starting from formulas (23) and (28) and assuming additivity of the mean free path, expressed by formula (24), we obtain:

\[ \frac{1}{b}=\beta_g T^{\frac{3}{2}}+\beta_s T^{-\frac{3}{2}} . \tag{29} \]

The first term of the expression on the right is due to lattice vibrations, and the second to scattering by impurities.

For ionic crystals, according to formula (26), one should expect a different dependence on temperature. A comparison with experiment is given in Section 3.3.

On the basis of the foregoing, it follows that simple and clear relations can be obtained only under special assumptions concerning the scattering mechanism. Restricting ourselves to valence crystals and to the region of sufficiently high temperatures, in this case the dependence of electrical conductivity on temperature can be rather simply visualized.

In this case, scattering of conduction electrons occurs only on thermal vibrations of the lattice, so that in formula (29) only the first term remains. On the other hand, if sufficiently low temperatures are taken, at which there is no appreciable intrinsic conductivity, then from formula (22), with \(n_L=0\), and formulas (29) and (14), the following regularity is obtained for the conductivity of a purely electronic semiconductor:

\[ \sigma=a_D T^{-\frac{3}{4}} e^{-\frac{\Delta E_D}{2kT}}, \tag{30} \]

where

\[ a_D=\frac{e}{2\beta_0}\left(\frac{2m_{2k}^{*}}{\pi \hbar^2}\right)^{\frac{3}{4}} n_D^{\frac{1}{2}} . \tag{30a} \]

In this expression all quantities not depending on temperature enter into the constant \(a_D\). This, in fact, is one of the formulas for electrical conductivity derived on the basis of Wilson’s work.

However, formula (30) does not give a general regularity, and it is not surprising that, in general, only a few experimental results correspond to it.

Degenerate semiconductors naturally require a different solution of the problem of electrical conductivity. For the case of complete degeneracy an expression has been found \(^{81}\), similar to the expression for the conductivity of metals. As for incomplete degeneracy (the transition region of weak degeneracy), these questions are still at the stage of study \(^{36,121}\).

3.12. Hall Effect

The study of the Hall effect is of exceptionally great importance for elucidating the mechanism of electrical conductivity in semiconductors. If in a conductor located in a magnetic field \(H_z\), directed along the \(z\)-axis, an electric current \(j_x\) flows in the direction of the \(x\)-axis, then in such a conductor there arises an electric field \(E_y\), perpendicular to the directions of \(j_x\) and \(H_z\), and equal in magnitude to

\[ E_y = R j_x H_z, \tag{31} \]

where \(R\) is the so-called Hall constant.

For semiconductors with simultaneous electron and hole conductivity, the electron theory gives the following expression for the Hall constant:

\[ R = -\frac{3\pi}{8e}\cdot \frac{n_E b_E^2 - n_L b_L^2}{(n_E b_E + n_L b_L)^2}. \tag{32} \]

Hence, in the case of intrinsic conductivity \((n_E = n_L)\), the following expression is obtained:

\[ R_y = -\frac{3\pi}{8}\frac{1}{ne}\frac{b_E - b_L}{b_E + b_L}. \tag{33} \]

For a purely electronic semiconductor \((n_L = 0)\):

\[ R_E = -\frac{3\pi}{8}\frac{1}{e n_E}. \tag{34} \]

For a purely hole semiconductor \((n_E = 0)\):

\[ R_L = +\frac{3\pi}{8}\frac{1}{e n_L}. \tag{35} \]

In formulas (32)—(35), \(e\) denotes the absolute value of the electron charge. A negative value of the Hall constant corresponds to the negative sign of the current carriers, i.e. electrons; a positive value corresponds to the positive sign of the current carriers, i.e. “holes.”

Thus, from the sign of the Hall constant one can judge the nature of the conductivity of the semiconductor—whether it is hole or electronic.

With mixed conductivity, the Hall constant according to formula (32), under known circumstances, may vanish (if \(n_E = n_L\) and \(b_E = b_L\)).

The quantitative determination of \(R\) also makes it possible to determine the concentration of carriers \(n_E\) or \(n_L\), knowledge of which is of substantial importance.

Comparing the Hall constant \(R\) with the electrical conductivity \(\sigma\), one can obtain the following remarkable result. Multiplying the expression for \(\sigma\), corresponding to a purely electronic semicon-

water ($n_L=0$) and determined by formula (22), into the corresponding expression for $R$ given by formula (34), we obtain

\[ (R\sigma)_E=-\frac{3\pi}{8}\,b_E, \tag{36} \]

and, correspondingly, for a purely hole semiconductor,

\[ (R\sigma)_L=\frac{3\pi}{8}\,b_L. \tag{37} \]

By measuring the dependence of $\sigma$ and $R$ on temperature and substituting their values into formula (36) or (37), we obtain the dependence of the mobility $b$, and consequently also of the mean free path $l$, on temperature.

Thus, to elucidate the mechanism of conduction in semiconductors, simultaneous measurements of conductivity and of the Hall effect are necessary.

The influence of degeneracy on the Hall effect is very small. In this case only the factor in formulas (32)—(35) changes, which under complete degeneracy, instead of

\[ \frac{3\pi}{8}=1.175 \]

becomes exactly equal to 1.

3.13. Change of electrical conductivity in a magnetic field

The change of conductivity or, correspondingly, of the specific resistance $\rho$ in a magnetic field is closely connected with the Hall effect. The theory gives the following expression for the relative change of conductivity in a magnetic field $H$:

\[ \frac{\sigma_0-\sigma_H}{\sigma_0}=B(T)H^2. \tag{38} \]

Such a quadratic dependence is valid for fields with strength $H\ll H_0$, where $H_0$ is the critical strength, which at room temperature is a quantity of order $\sim 500\,000\, a\cdot cm^{-1}$.

For the temperature dependence of $B(T)$ the following formula has been found:

\[ B(T)=\frac{4-\pi}{\pi}(R\sigma_0)^2. \tag{39} \]

The quantity $(R\sigma_0)^2$, according to formulas (36) or (37), for purely electronic or purely hole semiconductors coincides, to within a constant factor, with the square of the mobility $b$. Accordingly, neglecting in formula (29) the dependence of the pre-exponential part on temperature, for $B(T)$ one may expect the following temperature dependence:

\[ B(T)\sim \frac{1}{T^3}. \tag{40} \]

Consequently, the conductivity for semiconductors in a constant magnetic field increases rapidly as the temperature decreases, which, as is known, is characteristic of metals.

For strong fields, i.e., for \(H \gg H_0\), on the contrary, one should expect the attainment of a limiting value for \(\sigma\), independent of the material and temperature,

\[ \frac{\sigma_0-\sigma_\infty}{\sigma_0}=0.117 . \tag{41} \]

Thus, from measurements of the dependence of the change in \(\sigma\) in a magnetic field on temperature, one can also draw conclusions about the course of the change of mobility with temperature, as from measurements of electrical conductivity and the Hall effect.

3. 2. EXPERIMENTAL PART

3. 21. Experimental data

The theoretical considerations set forth above are valid only for homogeneous conductors. In reality, however, no semiconductor can be considered homogeneous in the strict sense of the word, since distributed impurities are always present in the lattice. The condition of homogeneity is almost equivalent to the requirement that in each region whose dimensions are of the order of the mean free path there be a constant number of identical impurity atoms. In practice this condition is fulfilled only in a very limited number of cases and depends strongly on the circumstances under which crystallization takes place. Therefore, the preparation of suitable specimens for investigation is one of the principal problems for experimenters.

The large amount of experimental material available on the properties of semiconductors has been obtained on specimens prepared by the most diverse methods, namely, on:

1. natural single crystals (minerals);
2. synthesized single crystals obtained, for example, by sublimation or crystallization from the melt with or without the addition of mineralizers;
3. dense polycrystalline bodies obtained, for example, from a melt or by oxidation of a metal (from metallic foil);
4. porous polycrystalline bodies, for example, pressed and sintered specimens;
5. condensed polycrystalline layers obtained by vacuum deposition.

Of all the possibilities listed here, the second is the most expedient. By the sublimation method one can obtain the necessary single crystals with a quite definite stoichiometric ratio. The difficulty consists only in obtaining single crystals of sufficiently large size.

Frithschem[^55] obtained ZnO single crystals by sublimation. In this case the largest crystals were obtained in the form of needles up to 5 mm long and several tenths of a millimeter thick. Recent studies carried out on ZnO single crystals are reported in the works of Hahn, Russell, and Miller[^65]. However, very much work still remains to be done in this direction.

Crystallization from the melt makes it possible to obtain fairly large single crystals; however, it must be borne in mind that the unavoidable presence in this case of the material of the crucible or, in any event, the added mineralizers may react with the substance being prepared, and this circumstance is not subject to control.

Measurements of the electrical conductivity of very pure tellurium single crystals were carried out by Cartwright[^38], [^40], [^41], Kaiser, Temmerman, and Kolebunder[^45], and also by Bott[^27]. The investigations by the method of Schmidt and Wassermann[^112] were carried out mainly on single crystals prepared artificially.

Mention should also be made of the investigations of Busch and Labhart[^36], carried out on carefully prepared synthetic SiC crystals. It should be noted that the commercially available finished product, with an unknown degree of purity, is usually unsuitable for solving many important problems.

A noteworthy method for preparing cuprous oxide specimens was proposed by Engelhardt[^49]. It consists in the fact that, upon many hours of heating a copper single crystal in a current of air, the latter is transformed into a homogeneous single crystal of cuprous oxide (\(\mathrm{Cu_2O}\)).

Solid polycrystalline specimens can be obtained from the melt only in rare cases, since most semiconductor compounds have a very high melting point. Exceptions in this respect are silicon and germanium. Recent work on silicon and germanium[^10] was carried out on specimens obtained from the melt. In this case, however, the matter is complicated by the difficulty of reproducing the conditions for preparing the specimens.

The next method, which makes it possible to obtain fairly dense specimens, amounts to oxidizing a very pure metallic foil in a current of oxygen. Recently, Wright and Andrews[^30], using this method, investigated the electrical conductivity of NiO.

Unfortunately, the predominant number of measurements of semiconductor conductivity has been made on pressed and sintered specimens. The homogeneity of specimens obtained in this way is for the most part highly doubtful, since the stoichiometric ratio at the outer part of a specimen may be different from that inside. Owing to such inhomogeneity, various contact phenomena may arise between the individual crystallites of the specimen, which, as is known, lead to a distortion of the results.

This circumstance can, for example, explain the fact that investigations by various authors of one and the same substance, if not qualitatively, then at least quantitatively, have never yielded coincident results.

The situation is no better in this respect for semiconductor layers obtained by evaporation in vacuum followed by oxidation. The advantages of this method consist in the fact that the processes of oxidation and reduction in this case are easily carried out. However, the difficulty here lies in determining the absolute values of the electrical conductivity, since the thickness and density of the layer can be determined only approximately.

Measurements on thin films were carried out, for example, by Bauer \({}^{20}\) on oxides of Cd, Tl, and Sn, by Güntherberger \({}^{70}\) on PbS, and by Fritsche \({}^{56}\) on ZnO.

3. 22. Methods of measurement

Electrical conductivity in the simplest case can be measured by measuring the voltage drop across a specimen of regular geometrical shape, the voltage being measured statically or by a compensation method between two probes. Measurements without potential probes, because of the generally known properties of contacts at the metal–semiconductor boundary, are less reliable.

Since the substances under investigation are usually prepared in the form of a fine-grained powder, other methods of studying conductivity are also of interest. Felkel \({}^{125}\) and Goller \({}^{62}\) placed the semiconductor under investigation, prepared in the form of a powder, in a medium free from dielectric losses—for example, vaseline or quartz powder—and used the resulting mass as a dielectric for filling a capacitor. Together with an inductance coil, the capacitor filled with this mixture of semiconductor and dielectric formed an oscillatory circuit. Measurement of the decrement of damping of this circuit made it possible to determine the conductivity of the given semiconductor. A disadvantage of this method is that sufficiently good measurement results can be obtained only on specimens with conductivities from \(10^{-5}\) to \(10^{-7}\ \Omega^{-1}\cdot \mathrm{cm}^{-1}\).

An analogous method, suitable for measuring substances with higher conductivity, was proposed by Kurchatov, Kostin, and Rusinov \({}^{86}\). In semiconductor particles, eddy currents were induced by means of a high-frequency magnetic field. By measuring the resulting rise in temperature, it was possible to judge the electrical conductivity of the given semiconductor. However, in this case, for accurate measurements it is necessary to know the shape and dimensions of the particles.

Finally, Ferbrote^51 developed a method for measuring conductivity that is suitable at high temperatures. This method consists in the following: the substance whose conductivity is to be measured is deposited as a dense layer on a tungsten wire, which can be heated by a current. This system is placed in a gas-discharge tube. The electrical conductivity of the semiconducting layer is determined from the current–voltage characteristic, one of the electrodes being the tungsten filament and the other the plasma of the gas discharge. It should be noted that this method, too, is not free from shortcomings.

All the methods listed, with the exception of the first, have limited applicability*). For measuring the Hall effect, naturally, one has to use only the first method.

3.3. SUMMARY OF EXPERIMENTAL DATA

3.31. Semiconductor substances

The number of semiconductors investigated is very large. Here we shall confine ourselves only to listing the substances investigated, and only in individual cases shall we touch upon the mechanism of conduction.

In classifying semiconductor substances we shall follow the method proposed by Meier^93. Compounds whose type of conduction is not reliable are placed in parentheses. Literature references are given on the basis of the most recent works.

Electronic semiconductors:
Al₂O₃^66, TiO₂, V₂O₅, Fe₂O₃, (CuO), Cu₂O₃^131, ZnO, MoO₃, (ScN), (Nb₂O₅), CdO, CdS, CdSe, SnO₂, SnSe^43, Cs₂S, Cs₂Se, BaO, BaTiO₃^34, Ta₂O₅, WO₃, (Au₂O₃), Hg₂S red. (Hg₂S black), Tl₂O₃, PbCrO₄^60,88, Bi₂Se₃^43, U₃O₈ (UO₃).

Hole semiconductors:
Cr₂O₃, (MnO), CoO, (Co₃O₄), NiO, CuJ, Cu₂O, Cu₂S, (Cu₂Se), (Cu₂Te), (GeO), MnO₂^71, Ag₂O, (SnO), SnS^12, (Sb₂S₃), (Tl₂O), (Tl₂S)^71, (Bi₂O₃), (Bi₂S₃), (BiSe₃), (BiTe₃).

) An original method for measuring the conductivity of well-conducting semiconductors in the solid and liquid states was developed by A. R. Regel. By this method, measurements have recently been made on a whole series of semiconductors. See A. R. Regel, “Measurement of the electrical conductivity of metals and alloys in a rotating magnetic field,” ZhTF 18, 1511 (1948), and A. I. Blum and A. R. Regel, Electrical properties of solid solutions of mercury selenide and selenium, ZhTF 21, 316 (1951). (Ed. note*)

Amphoteric semiconductors:

Si¹⁰⁶, SiC³⁶, (Cr₂O₃), (Mn₂O₃), (Mn₃O₄), (Co₃O₄), Ge⁸⁷, ¹⁰⁷, ¹⁰⁸, ¹⁰⁹, RnO₂, (Os₂S₂), IrO₂, (PbO), PbS, PbSe, UO₂.

In the following paragraph some special cases of semiconductor conductivity will be examined in detail, since their investigation has been carried out with particular thoroughness.

3.32. Special examples

3.321. Zinc oxide ZnO.

Zinc oxide has been studied very extensively. It was investigated by Baχ¹⁷ on natural single crystals of unknown purity, then by Sommerwil¹²³, Jander and Stamm⁷⁵, Friedrich⁵³ and Hüllery⁶³ on pressed specimens, and, more reliably, by Skaupy¹²³, Hüllery⁶⁴, Meyer⁹⁴, Baumbach and Wagner²² on sintered specimens.

Measurements on a synthetic single crystal of pure ZnO were first carried out by Fritzsche⁶⁶. The results of his measurements are given in Fig. 11.

Fig. 11. Electrical conductivity of ZnO at low temperatures.

The dashed curves 1 and 3 show the course of the conductivity for two ZnO single crystals. Their conductivity at room temperature is equal to 3.1 and 1.4 ohm⁻¹·cm⁻¹, while the activation energy is respectively equal to 0.012 and 0.009 eV.

Upon heating in an atmosphere of hydrogen for 30 hours at 900° C, the conductivity at room temperature fell to \(4 \cdot 10^{-6}\) ohm⁻¹·cm⁻¹. At the same time the activation energy increased to 0.4 (see curves 24, 28 and 30). Between 300 and 80° K in all cases the dependence of \(\ln \sigma\) on \(\frac{1}{T}\) is represented by a straight line, as was to be expected according to formula (30) for an electron or hole semiconductor.

According to the measurements of Baumbach and Wagner, the conductivity of ZnO at high temperatures is a single-valued function of the oxygen pressure (Fig. 12) and, at constant temperature, is expressed by the empirical relation:

\[ \sigma = \text{const}\cdot p^{-\frac{1}{4.3}}. \]

As can be seen, the conductivity decreases with increasing oxygen pressure, i.e., with a decrease in the excess amount of Zn relative to the stoichiometric ratio. ZnO is, consequently, an electronic semiconductor. This is directly confirmed by measurements of the Hall effect carried out by Fritsch. Unfortunately, Fritsch did not continue the Hall-effect measurements on the same single crystals, but confined himself only to measurements on

Fig. 12. Dependence of the electrical conductivity of ZnO on oxygen pressure.

sintered specimens. For the Hall constant \(R\), values between \(-7.5\) and \(-380\ \dfrac{\text{cm}^3}{\text{A}\cdot\text{sec}}\) were obtained, with the sign corresponding to the negative sign of the carriers.

The mobility \(b_E\) lies between 7 and 30 \(\dfrac{\text{cm}^2}{\text{V}\cdot\text{sec}}\) and shows a surprisingly weak increase with decreasing temperature.

The latest measurements by Hahn, Russell, and Miller\(^{66}\) were made on single crystals and sintered specimens. In this case, the single crystals showed a higher mobility, namely: \(b_E \sim 100—200\ \dfrac{\text{cm}^2}{\text{V}\cdot\text{sec}}\), whereas for sintered specimens, on the contrary, only \(5—60\ \dfrac{\text{cm}^2}{\text{V}\cdot\text{sec}}\), i.e., values of the same order of magnitude as Fritsch’s data. The smaller mobilities for polycrystalline specimens may be explained by the influence of transition layers between grains. There is reason to think that in ZnO crystals, which possess a hexagonal lattice, there exists

a clearly expressed anisotropy of mobility, which is smoothed out in sintered specimens.

The question of the decrease in activation energy with increasing concentration of excess Zn atoms remains, up to the present time, entirely unresolved. According to Mott,^6 this can be explained by the fact that, at high concentrations, the emerging mutual influence of the electron orbits of impurity atoms reduces the activation energy.

Fig. 13. Electrical conductivity of ZnO at high temperatures.

Above room temperature these relationships are somewhat simplified. According to Baumbach and Wagner, up to a temperature of about \(900^\circ\mathrm{K}\) the conductivity increases exponentially with increasing

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

At higher temperatures the increase in electrical conductivity becomes slower. If in this case “saturation” really occurs, as Miller^97 believes, then this question remains unresolved. Miller found, on sintered specimens above \(600^\circ\mathrm{K}\), a similar course of conductivity to that observed by Baumbach and Wagner. In doing so he obtained for the activation energy a value equal to \(0.71\ \text{eV}\) (Fig. 13). The measurements were carried out so rapidly that equilibrium with the oxygen of the air did not have time to be established. The number of distortions in the lattice in this case remained constant, and the course of the conductivity was quite reversible. Between 300 and \(400^\circ\mathrm{K}\), in some specimens, a large decrease in conductivity and an increase in mobility with increasing temperature were observed. It is regrettable that Miller, whose research results for the most part confirmed earlier work, does not provide curves of the dependence of \(\sigma\) on \(T\) for the entire temperature range investigated.

In general it is evident that, despite numerous experimental works, a quantitative understanding of the mechanism of conductivity in ZnO has not been achieved. Therefore further careful investigations on well-formed single crystals are necessary.

3.322. Cuprous oxide Cu₂O.

From the works of Auwers1, Dönvalda and Wagner2, Gudden3, Gundermann, Gauffe and Wagner4, Le Blanc and Sachs5, and Waibel it follows that the conductivity of cuprous oxide increases with increasing excess oxygen, i.e., with a decrease in the relative amount of copper.

Engelhardt[^49] investigated the conductivity and the Hall effect below room temperature on samples of cuprous oxide obtained by oxidation of copper. By heating in an oxygen atmosphere or in vacuum between 500 and 1000°C, it was possible to vary the concentration of oxygen. In Figs. 14 and 15 the principal results of his measurements are presented.

Figs. 14 and 15. Hall constant and specific resistance of Cu₂O at low temperatures.

The temperature dependence of the conductivity of Cu₂O can be expressed in the usual form:

\[ \sigma = A e^{-\frac{\Delta E}{2kT}} . \]

The Hall constant has a positive sign, which shows that Cu₂O is a hole semiconductor. For the activation energy, values between 0.232 and 0.390 eV are obtained. Generally speaking, oxidation gives larger, and reduction smaller, values for the activation energy. The pre-exponential factor \(A\) depends especially strongly on the prior treatment of the material.

The acceptor concentration, calculated for one of the samples, had the value \(n_A = 1.4\cdot 10^{18}\ \mathrm{cm}^{-3}\). When heated in an oxygen atmosphere it increased to \(3.4\cdot 10^{19}\), while when heated in vacuum it fell to \(5.4\cdot 10^{17}\). These data are in agreement with theoretical notions.

From measurements of the conductivity \(\sigma\) and the Hall constant \(R\), one can obtain the dependence of mobility on temperature, which is presented in Fig. 16. According to Fröhlich and Mott \(^{58}\), the experimental curve coincides with the theoretical one constructed on the basis of formula (26), if it is assumed that the characteristic temperature of cuprous oxide is \(\theta = 280^\circ\), and the effective electron mass is \(m^* = 0.25\,m_0\).

Fig. 16. Dependence of the mobility of “holes” in Cu\(_2\)O on temperature.

Donald and Wagner, as well as Juse and Kurchatov \(^{83}\), investigated the temperature region between 100 and 1000° C. According to the data of Juse and Kurchatov, the conductivity of cuprous oxide can be well represented by a two-term formula of the form:

\[ \sigma = A_1 e^{-\frac{\Delta E_1}{2kT}} + A_2 e^{-\frac{\Delta E_2}{2kT}} . \]

The constants \(A_1\) and \(\Delta E_1\) depend little on the oxygen concentration and have the values:

\[ A_1 = 44\ \mathrm{ohm}^{-1}\cdot \mathrm{cm}^{-1}; \quad \Delta E_1 = 0.72\ \mathrm{eV}. \]

\(\Delta E_2\) has values from 0.129 to 0.134 eV, while \(A_2\) increases with increasing oxygen concentration from \(4\cdot 10^{-3}\) to \(0.4\ \mathrm{ohm}^{-1}\cdot \mathrm{cm}^{-1}\). The authors conclude that at low temperatures we have a hole semiconductor, while at high temperatures intrinsic conductivity sets in. Correspondingly, one should expect that with increasing temperature, owing to the fact that the concentration of electrons grows faster than the concentration of “holes,” the Hall constant becomes a small positive quantity and, finally, changes sign. Indeed, this circumstance was confirmed by Schottky and Waibel \(^{113}\). However, in one of their subsequent works it was refuted by them \(^{114}\).

More recent investigations are due to Angelo \(^{14}\) and Feldman \(^{52}\). Angelo’s measurements were confined to only a narrow temperature region between \(-40\) and \(+100^\circ\)C and, naturally, did not allow—

...make broad generalizations. Fig. 17 is taken from Feldman’s work. Above \(400^\circ\text{C}\) the course of the curves agrees with the data of Juse and Kurchatov. The slope of the rectilinear part of the curve gives, for all samples, independently of the treatment, the value \(\Delta E = 0.78\) ev, which agrees satisfactorily with the data of previous measurements. However, according to Feldman, this value is not identical with the activation energy \(\Delta E_A\) of a hole semiconductor and, consequently, a different interpretation must be given here.

As acceptors in \(\mathrm{Cu}_2\mathrm{O}\) there appear vacant sites in the partial lattice of copper atoms. According to Dünwald and Wagner, the diffusion of copper ions proceeds so rapidly that, in contrast to \(\mathrm{ZnO}\), the degree of disorder corresponds to a state of thermodynamic equilibrium. In this case the concentration of acceptors increases with increasing temperature and oxygen pressure and, according to Wagner\(^{127}\), obeys the formula:

Fig. 17. Conductivity (curve \(A\)) and Hall constant (curve \(B\)) for \(\mathrm{Cu}_2\mathrm{O}\) at high temperatures.

\[ n_A = \mathrm{const}\, T^{-\frac{5}{8}} p^{\frac{1}{4}} e^{-\frac{E_f}{kT}}, \tag{42} \]

where \(E_f\) is the energy of formation of an impurity level, i.e., the energy which must be expended in order to place an oxygen atom in the \(\mathrm{Cu}_2\mathrm{O}\) lattice and form two vacant sites in the lattice of copper ions.

According to formula (15), the concentration of “holes” is expressed by the formula:

\[ n_L = \mathrm{const}\, T^{\frac{3}{4}} n_A^{\frac{1}{2}} e^{-\frac{\Delta E_A}{2kT}}, \tag{43} \]

and therefore, taking formula (42) into account, we obtain the fol—

following expression for the conductivity:

\[ \sigma=\mathrm{const}\, p^{\frac18} T^{-\frac1{16}} l e^{-\frac{(E_f+\Delta E_A)}{2kT}}, \tag{44} \]

where \(l\), as usual, denotes the mean free path.

Thus, from the slope of the rectilinear part of curve \(A\) in Fig. 17 we obtain not the activation energy \(\Delta E_A\), but the sum \(E_f+\Delta E_A\). Hence the above value for the activation energy, equal to \(0.78\) eV, is obtained. The true value of the activation energy may therefore be considerably smaller.

Labels in the figure:
\( \ln \sigma \); ○ measurements under pressure; ● measurements without pressure; \(1000^\circ\mathrm{C}\), \(900^\circ\mathrm{C}\), \(800^\circ\mathrm{C}\); \(\ln P_{O_2}\) (mm Hg).

Fig. 18. Dependence of the electrical conductivity of \(\mathrm{Cu_2O}\) on oxygen pressure.

The dependence of the conductivity on oxygen pressure expected according to formula (44) at temperatures above \(400^\circ\mathrm{C}\) is confirmed by the experiments of Dünwald and Wagner, as is seen from Fig. 18.

Thus, one may conclude that, contrary to the views set forth above, \(\mathrm{Cu_2O}\) at high temperatures possesses no intrinsic conductivity. The most recent Hall-effect measurements, carried out by Feldman, seem to have fully confirmed this conclusion, and no change in the sign of the Hall constant was detected. Unfortunately, above \(400^\circ\mathrm{C}\) Hall-effect measurements were not carried out (Fig. 17, curve \(B\)), whereas, in the opinion of the above-mentioned authors, the Hall constant should have changed sign at a temperature not lower than \(500^\circ\mathrm{C}\).

The mobility of “holes” \(b_L\) changes extremely strongly with temperature; below \(150^\circ\mathrm{C}\), \(b_L \sim T^{-5}\), while above \(150^\circ\mathrm{C}\), \(b_L \sim T^{-7}\).

Below \(300^\circ\mathrm{C}\) the impurity concentration is “frozen in,” i.e., the number of acceptors \(n_L\) becomes independent of temperature,

and depends only on the initial state (“prehistory of the sample”), as Engelhardt also observed.

The measurements of Engelhardt and Feldman were limited mainly to the region of room temperatures. Unfortunately, up to the present time there are no reliable measurements that have been carried out on the same samples both at high and at low temperatures. Such measurements would make it possible to explain more fully the mechanism of conduction in Cu₂O.

3. 323. Silicon carbide. Attempts to measure the conductivity of silicon carbide were undertaken by Seemann¹¹⁹, Sears and Becker¹¹⁷, Felkl¹²⁵, Hollerý⁶², Kurchatov, Kostina and Rusinov⁸⁶, Osterberg¹⁰⁵, Henninger⁶⁸, Bose and Castrum²⁵ and, more recently, by Busch, and also by Busch and Labhart³⁶. The previous measurements extended only to the region of very low temperatures, and the results of the measurements were strongly distorted by the presence of transition layers with large resistance at the boundary between individual grains of the substance. Busch investigated the conductivity on a large number of single crystals by the probe method in the temperature range between 77 and 1400°K.

SiC is a valence crystal with the wurtzite structure, where each silicon atom, together with four carbon atoms, forms a regular tetrahedron.

Technically produced crystals, because of the presence of foreign atoms, have the most varied coloration. Pure SiC is colorless⁷⁷; with increasing contamination, yellowish, green, and finally black crystals are obtained. Spectral analysis shows that the yellowish crystals correspond to a content of 0.01% Fe and Mg; the green ones—to as much as 0.05% Fe and Mg; and the black crystals correspond to a content of 0.2% Al and Ca and 0.1% Mg.

The dependence of conductivity on temperature is presented in Figs. 19 and 20. For not too low temperatures, the conductivity can be represented by the following two-term formula:

\[ \sigma = A_1 e^{-\frac{\Delta E_1}{2kT}} + A_2 e^{-\frac{\Delta E_2}{2kT}} . \]

Measurements of the Hall effect by Busch and Labhart showed that yellow and green SiC correspond to the \(n\)-type, i.e., are electron semiconductors, while black corresponds to the \(p\)-type, i.e., is a hole semiconductor.

For \(n\)-SiC the conductivity at 0°C varies from about \(10^{-12}\ \Omega^{-1}\cdot\text{cm}^{-1}\) for almost colorless crystals to about \(5\ \Omega^{-1}\cdot\text{cm}^{-1}\) for dark-green crystals. For the activation energies \(\Delta E_1\) and \(\Delta E_2\), values are obtained lying respectively between \(1.57 \pm 0.157\) eV and \(0.191 \pm 0.051\) eV. For the pre-exponential factors \(A_1\) and \(A_2\), values are obtained respectively of \(6.3\)—\(260\ \Omega^{-1}\cdot\text{cm}^{-1}\) and \(0.3\)—\(10\ \Omega^{-1}\cdot\text{cm}^{-1}\). The corresponding range of values for

$p$-SiC is considerably narrower. For 50°C the values lie between $6.2 \cdot 10^{-1}$ and $1.8\ \Omega^{-1}\cdot\mathrm{cm}^{-1}$. The ranges of values for the remaining quantities are:

\[ \Delta E_1:\ 0.226—0.281\ \text{eV}; \qquad \Delta E_2:\ 0.038—0.052\ \text{eV}; \]

\[ A_1:\ 160—350\ \Omega^{-1}\cdot\mathrm{cm}^{-1}; \qquad A_2:\ 0.18—0.95\ \Omega^{-1}\cdot\mathrm{cm}^{-1}. \]

At high temperatures the conductivity has a sharply pronounced maximum, which for $n$-SiC lies between 480 and 1100°K, and for $p$-SiC between 950 and 1020°K. Above these temperatures the conductivity decreases up to the very highest temperatures at which the investigations were carried out (1400°K). In all cases complete reproducibility of the results was obtained for one and the same crystal. Prolonged heating in air up to 1000°C had no effect on the measurement results, so that disturbances in the lattice may undoubtedly be regarded as being in a state of “frozen” equilibrium.

Fig. 19. Dependence of the electrical conductivity of $n$-SiC on temperature.

Fig. 20. Dependence of the electrical conductivity of $p$-SiC on temperature.

Since SiC has a homopolar bond, its conductivity must be expressed by formula (30), which is based

ELECTRONIC CONDUCTIVITY OF NONMETALS

in the dependence of mobility on temperature, given by formula (29), i.e. \(b \sim T^{-3/2}\). Formula (30), for a known value of the activation energy \(\Delta E\), i.e. one determined experimentally, makes it possible to determine \(T_{\max}\), at which a maximum of the conductivity should be expected. However, the temperature determined in this way turns out to be 5–10 times greater than it actually is. The contradiction can be removed if one assumes a considerably stronger dependence of the mobility on temperature. Experiment shows, however, that such an assumption is incorrect.

Comparison of the data on conductivity and the Hall effect gives the temperature dependence for \(P_g\) shown in Figs. 21 and 22. In Fig. 23 the dependence of the product \(R\sigma\) on \(T^{-3/2}\) is given, which at high temperatures agrees with the dependence later theoretically justified by Seitz.

Fig. 21. Dependence of the electron mobility on temperature for \(n\)-SiC.

Fig. 22. Dependence of the “hole” mobility on temperature for \(p\)-SiC.

The lower value of the temperature at which the conductivity reaches a maximum cannot be due to a stronger dependence of the mobility on temperature, as was assumed above, but must be due to a slight increase in the concentration of electrons with increasing temperature, which was to be expected from Maxwell–Boltzmann statistics.

Indeed, at high temperatures in \(n\)-SiC one obtains an electron concentration of the order of \(10^{18}\ \mathrm{cm}^{-3}\), and in \(p\)-SiC—of the order of \(10^{19}\ \mathrm{cm}^{-3}\). Such a concentration of electrons is already so great

high, which must be taken into account in the following expression. The derivation, which is given in Section 2.3, gives the following expression for

Fig. 23. The \(T^{-3/2}\) law for mobility in SiC.

the conductivity of SiC at medium and high temperatures in the case of a weakly expressed electron gas:

\[ \sigma = Q^{*} \frac{T^{-\frac{3}{2}}} { 1+ \left[ 1+ \left(\frac{4D^{*}}{nD}\right) T^{-\frac{3}{2}} e^{\frac{\Delta E}{kT}} \right]^{\frac{1}{2}} }, \tag{45} \]

where:

\[ Q^{*}=2\frac{e^{2}}{m^{*}}\tau_{0}n_{D}, \qquad \tau_{0}=\bar{\tau}T^{\frac{3}{2}}, \]

\[ D^{*} = \left( \frac{m^{*}k}{2^{\frac{3}{3}}\pi\hbar^{2}} \right)^{\frac{3}{2}}; \]

\(\bar{\tau}\) is the mean relaxation time and \(n_{D}\) is the concentration of impurity centers.

It is easy to see that, for low temperatures, expression (45) is identical with formula (30), derived by Wilson. For very high temperatures, however, we have a decrease in conductivity with increasing

temperature, namely:

\[ \sigma \sim T^{-\frac{3}{2}}. \]

Möglikh \(^{48}\) arrived at the same temperature dependence on the basis of Saha’s formula. As can be seen from Fig. 24, the experimental—

Fig. 24. Measured and calculated conductivity according to formula (45) in the region of weak degeneracy for SiC.

data for high temperatures agree well with formula (45).

At low temperatures the curves of the dependence of conductivity on temperature have, especially for \(p\)-SiC, characteristic sections with a gentle slope, corresponding to a small activation energy.

Simultaneously with the lowering of temperature the mobility decreases rapidly (Figs. 21 and 22). According to Busch and Labhart this circumstance can be explained by the fact that SiC has impurities of two kinds, located at levels with different activation energies. The strong decrease in mobility with decreasing temperature is due to the fact that impurities of one kind, owing to their

of high concentration (from \(10^{18}\) to \(10^{20}\) per \(1\ \mathrm{cm}^3\)), interact with one another and can exchange electrons. Therefore, in the additional impurity conduction band the charge carriers possess a very large effective mass and, accordingly, a small mobility.

Fig. 25. Dependence of the resistivity on \(\dfrac{1}{T}\) for Si—B alloys.

However, from the modern point of view it is more probable that the decrease in mobility at low temperatures is caused by an increase in the probability of scattering by impurities in the SiC lattice. Proceeding from this, it is possible, on the basis of equation (29), to calculate the dependence of mobility on temperature.

On the basis of the results set forth above, SiC is an amphoteric semiconductor. It is quite possible that electron and hole conduction are caused by the same mechanism as in Si and Ge. An impurity of elements whose valences are less than four (Ca, Al) creates hole conduction (\(p\)-type); contamination by iron leads, as in silicon, to electron conduction (\(n\)-type).

3.324. Silicon Si. The mechanism of the high conductivity of silicon for a long time remained unclear. Against the metallic character of the conductivity spoke measurements of the distribution of intensities in the emission spectra of long-wavelength X-rays, made by O’Bryan and Skinner \(^{104}\). From this it could be concluded that the valence bonds are completely saturated, and therefore there should be no metallic conductivity. This conclusion agrees with the first measurements of Königsberger and Schilling \(^{85}\), which showed that Si possesses nonmetallic conductivity.

However, in contradiction to this were the investigations of Seemann \(^{118}\), Schulze \(^{115}\), and Luster \(^{89}\). These authors found, at temperatures between \(77^\circ\) and \(1000^\circ\ \mathrm{K}\), on single crystals of silicon a weak decrease of conductivity with increasing temperature and made

from which follows the conclusion about the metallic character of the conductivity of silicon.

In contrast to this, Meissner and Fort^93 firmly established that the single crystal which they investigated at \(1.2^\circ\mathrm{K}\) possesses no metallic temperature coefficient.

A complete explanation was given in investigations among which the works of Pearson and Shockley^107 and Pearson and Bardeen^106 should especially be noted. Pearson and Bardeen used, as the starting material, very pure Si with an impurity content of only about \(0.03\%\), which was obtained in the form of long needle-like crystals by reducing \(\mathrm{SiCl}_4\) with Zn. This material was doped with additions from \(0.0005\) to \(1\) percent by weight of boron and from \(0.001\) to \(1\) percent by weight of phosphorus. In accordance with the ideas set forth in Section 2.42, an impurity of trivalent boron creates hole conductivity, and an impurity of pentavalent phosphorus—electron conductivity. The temperature dependence of the specific resistance

\[ \rho = \frac{1}{\sigma} \]

is presented in Figs. 25 and 26.

Fig. 26. Dependence of the specific resistance on
\[ \frac{1}{T} \]
for Si—P alloys.

As is seen from these figures, at high temperatures the temperature course of the resistance of all samples is represented by one

and the same straight line. This straight line corresponds to the intrinsic conductivity of silicon with activation energy \(\Delta E_1 = 1.12\ \text{eV}\). At low temperatures the conductivity increases in a characteristic way with increasing impurity concentration.

From the measurement of the Hall constant \(R\), one can, on the basis of equalities (32), (34), and (35), determine the carrier concentration \(n\).

The results are presented in Figs. 27 and 28. From these figures it is seen that at small concentrations of impurities \(B\) or \(P\) the concentration of “holes” or, respectively, electrons increases with rising temperature approximately according to an exponential law. At larger impurity concentrations the carrier concentration is very high (up to \(5 \cdot 10^{20}\ \text{cm}^{-3}\)) and is practically independent of temperature. The dotted curve \(T_0\) in both figures represents, in accordance with expression (16) for the degeneracy criterion, the dependence of the degeneracy temperature on the carrier concentration. For samples 7 and 8 in Fig. 25 and \(C\) and \(D\) in Fig. 26, degeneracy is present practically throughout the entire temperature interval.

Fig. 27. Dependence of the carrier concentration on \(\dfrac{1}{T}\) for Si—B alloys; \(T_0\) is the degeneracy temperature.

Calculating the products \(R\sigma\), one can find the dependence of the carrier mobility on temperature, which is presented in Figs. 29 and 30. Here, just as for SiC, the theoretically substantiated proportionality at high temperatures between the mobility and \(T^{-3/2}\) is very well confirmed (see Section 3.11). Below \(100^\circ\text{K}\) the mobility again begins to decrease with decreasing temperature, which corresponds to an increase in carrier scattering by impurities in the lattice (at B or P ions).

In the region of impurity conduction, as for other semiconductors, a characteristic decrease in the activation energy is observed with increasing impurity concentration. For very pure Si (sample No. 1) with acceptor concentration \(n_A = 1.5 \cdot 10^{16}\ \text{cm}^{-3}\), the activation energy \(\Delta E_A\) is found to be about \(0.075\ \text{eV}\).

The dielectric constant of silicon is equal to 13. The activation energy calculated from equation (20) (Section 2.4), \(\Delta E_A = 0.08\) eV, approximately coincides with the experimental value. For the highest concentrations observed experimentally, \(n_A = 5.3 \cdot 10^{18}\), the activation energy is practically equal to zero. Figure 31 (see p. 303) presents the dependence of the activation energy on the concentration of impurity levels.

Fig. 28. Dependence of carrier concentration on \(\dfrac{1}{T}\) for Si—P alloys; \(T_0\) is the degeneracy temperature.

According to Pearson and Bardeen, such a dependence can be interpreted as follows. When an isolated atom is ionized, the force of attraction between the remaining ion and the detached electron becomes equal to zero. However, in a semiconductor the ionized donors or acceptors are usually partially screened by conduction electrons. Therefore, in the ionized state there arises an attractive force, which will be the greater, the smaller the average distance between impurity atoms. This distance is proportional to \(n_A^{-1/3}\), where \(n_A\) denotes the concentration

Fig. 29. Dependence of the mobility of “holes” on \(\dfrac{1}{T}\) for Si—B alloys.

Fig. 30. Dependence of electron mobility on \(\dfrac{1}{T}\) for Si—P alloys.

acceptors. The activation energy will therefore be smaller the greater the concentration of acceptors, and can be represented in the following form:

\[ \Delta E_A=\Delta E_A^0-a n_A^{\frac{1}{3}}, \]

where \(\Delta E_A^0\) denotes the activation energy at an infinitely small concentration of acceptors and, on the basis of experiment, is approximately \(0.08\) ev. Substituting this value and the value \(a=4.3\cdot10^{-8}\), found empirically, into the above formula, we obtain the curve shown in Fig. 31, which well represents the observed course of the change in activation energy with changing acceptor concentration.

Fig. 31. Dependence of activation energy on impurity concentration for Si—B and Si—P alloys.

On the basis of a quantitative estimate, a value of about \(3\cdot10^{-8}\) was found for \(a\). Such agreement in order of magnitude is quite satisfactory and confirms this explanation.

For very high impurity concentrations a different relation is obtained.

Silicon with a concentration of impurity atoms B or P greater than \(10^{19}\) per \(1\ \mathrm{cm}^3\) has an electrical conductivity which is practically independent of temperature (curves 6—8 in Fig. 25 and curves \(C\) and \(D\) in Fig. 26). Pearson and Shockley\({}^{107}\) found no change in specific resistance even down to temperatures of about \(10^\circ\) K. The activation energy for such alloys is practically equal to zero, so that the concentration of charge carriers, independently of temperature, is equal to the concentration of donors or acceptors. The electron gas in this case is strongly degenerate, and the conductivity depends only on the concentration of carriers and on their mobility.

For low temperatures the mobility is limited only by scattering on impurities. Johnson and Lark-Horovitz\({}^{81,82}\) calculated this

scattering for a degenerate electron gas and found its complete independence of temperature. Thus, in this case the specific resistance $\rho$ depends only on the density of impurity levels, which is identical with the number of carriers.

For the mobility the following regularity is obtained:

$$ \frac{1}{b}=\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\frac{\pi \hbar}{4e} n^{\frac{2}{3}}. $$

Whence for the specific resistance $\rho=\frac{1}{enb}$ it follows that

$$ \rho=\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\frac{\pi \hbar}{4e^2} n^{-\frac{1}{3}}. $$

Or, substituting numerical values:

$$ \rho=6270\, n^{-\frac{1}{3}}\ \text{ohm}\cdot\text{cm}, \tag{46} $$

which agrees, in order of magnitude, with the observed values.

The properties of silicon as a semiconductor are thus fully explained, and the experimental data are in good agreement with the latest theoretical concepts.

Fig. 32. Conductivity of mixed crystals Fe$_3$O$_4$MgCr$_2$O$_4$ at different contents of MgCr$_2$O$_4$.

3.325. Oxides with the spinel structure. In Section 2.43 some elements were set forth of the theory developed by Verwey, Haman, and Romeijn[^124] concerning the electrical conductivity

oxides with the spinel structure. The theory set forth was experimentally confirmed in the same work by these authors in the study of \(Fe_3O_4\) in the form of mixed crystals with \(MgCr_2O_4\) or \(ZnCr_2O_4\). At the same time, not only divalent but also trivalent ions could be introduced into the lattice.

In preparing the specimens, \(Fe_3O_3\), \(MgO\), and \(Cr_2O_3\), or \(Fe_2O_3\), \(ZnO\), and \(Cr_2O_3\), were mixed in the proper proportion. The specimens were pressed and annealed in the corresponding atmosphere at \(1200\)—\(1400^\circ C\). In Figs. 32 and 33 the curves are given for the dependence of conductivity on temperature, obtained on these polycrystalline specimens. As was to be expected, with an increase in the content of \(MgCr_2O_4\) or \(ZnCr_2O_4\), the conductivity decreased,

Fig. 33. Conductivity of mixed crystals
\(Fe_3O_4 — ZnCr_2O_4\) at various contents of
\(ZnCr_2O_4\).

while the activation energy, on the contrary, increased. The fact deserves attention that the magnitude of the conductivity and the activation energy can vary within wide limits. The electrical properties of these semiconductors are not very sensitive to deviations from the stoichiometric ratio, and therefore the preparation of specimens with identical properties presents no difficulty.

3.33. The Meyer–Neldel Rule

The conductivity of pure electronic and hole semiconductors at a sufficiently low impurity concentration, i.e., in the absence of degeneracy, can with sufficient accuracy be represented by the following expression:

\[ \sigma = A e^{-\frac{\Delta E}{2kT}} . \tag{47} \]

The weak temperature dependence of the preexponential factor \(A\) may be neglected in comparison with the exponential dependence on temperature.

According to Meyer and Neldel\(^{36}\), for a number of semiconductors whose conductivity at room temperature is less than \(10^{-2}\ \Omega^{-1}\cdot\text{cm}^{-1}\), there exists a remarkably simple relation between the preexponential factor \(A\) and the activation energy \(\Delta E\), namely:

\[ \ln A=\alpha+\beta\Delta E . \tag{48} \]

Fig. 34. Dependence between the activation energy \(\Delta E\) and the constant \(A\). Meyer–Neldel rule: ◯ — ZnO; ⊖ — TiO\(_2\); ● — Fe\(_2\)O\(_3\); ⊙ — UO\(_2\); ● — SiC; ⊕ — BaTiO\(_3\).

In Fig. 34 the data for ZnO, TiO\(_2\), Fe\(_2\)O\(_3\), and WO\(_3\) according to Meyer and Neldel, for SiC according to Busch\(^{32}\), and for BaTiO\(_3\) according to Busch and Flüri\(^{34}\) are compared. As is evident from this figure, the experimental points lie on straight lines, the position and slope of which are characteristic for each given substance.

The constant \(A\), as is seen from formulas (30) and (30a), is proportional to the square root of the impurity concentration \(n_D\) or \(n_A\). Thus, as follows from formula (48), the activation energy for poorly conducting semiconductors increases with increasing impurity concentration.

For semiconductors whose conductivity at room temperature is greater than \(10^{-2}\ \Omega^{-1}\cdot\text{cm}^{-1}\), the opposite dependence is observed, i.e., a decrease of \(\Delta E\) with increasing \(A\). The explanation of this effect, apparently, should be sought in the interaction of impurity atoms, similar to what was noted in Section 3.324 for the case of silicon.

The Meyer–Neldel rule may also be interpreted in another way, for example as was done by Busch\(^{33}\). As long as a semiconducting compound, for example an oxide, is in thermodynamic equilibrium with the oxygen atmosphere surrounding it, the impurity concentration, according to Wagner\(^{137}\) [formula

(42), Section 3.322] has the following temperature dependence:

\[ n_D = Ce^{-\frac{E_f}{kT}}. \]

The factor \(C\) depends, among other things, on the oxygen pressure and, in comparison with the exponential, depends only weakly on temperature. \(E_f\) is the energy that must be expended in order to remove an atom or ion from a lattice site (“disordering energy”). The conductivity of semiconductors was usually measured at such low temperatures (for example, ZnO) that the impurity concentration remained practically constant. This evidently means that disturbances in the ideal crystal lattice (“disorder”) are “frozen in” below a certain temperature \(\theta\), characteristic for each given semiconductor. Below \(\theta\) the stable impurity concentration is determined by the “freezing” temperature, i.e.,

\[ n_D = Ce^{-\frac{E_f}{k\theta}}. \tag{49} \]

For impurity semiconductors the conductivity is proportional to \(n_D^{\frac{1}{2}}\), i.e., neglecting quantities that vary little with temperature, we obtain:

\[ \sigma = an_D^{\frac{1}{2}} e^{-\frac{\Delta E_D}{2kT}}, \tag{50} \]

or, using formula (49),

\[ \sigma = a'e^{-\frac{E_f}{2k\theta}} e^{-\frac{\Delta E}{2kT}}. \tag{51} \]

Replacing \(E_f+\Delta E\) by \(U\), where \(U\) is the energy characteristic of each given lattice and necessary in order to ionize an atom or ion in the undistorted lattice, i.e., to transfer it into the conduction band, and substituting the expression for \(E_f\) into formula (51), we obtain:

\[ \sigma = a'e^{\frac{(\Delta E-U)}{2k\theta}} e^{-\frac{\Delta E}{2kT}}. \tag{52} \]

Comparing equalities (52) and (47), we obtain the following expression for the pre-exponential factor:

\[ A = a'e^{\frac{\Delta E-U}{2kT}}, \]

or, taking logarithms, we obtain:

\[ \ln A=\alpha+\beta \Delta E, \]

which agrees with the empirical Meyer–Neldel rule (48). Whereas \(\alpha\) depends in a complicated way on \(U\), the gas pressure, the mobility of the charge carriers, etc., for \(\beta\) one obtains a simple expression of the following form:

\[ \beta=\frac{1}{2k\theta}. \tag{53} \]

The dependence \(\Delta E=f(\ln \theta)\) is represented graphically by straight lines (Fig. 34), whose slope is the greater, the higher the “freezing” temperature \(\theta\) of the substance under consideration. The “freezing” temperatures calculated from the slope of these straight lines have the following values:

Semiconductor	\(\theta\), calculated in °K	\(\theta\), observed in °K
\(\mathrm{UO_2}\)	\((340)\ *)\)	1000?
\(\mathrm{Fe_2O_3}\)	430	400
\(\mathrm{ZnO}\)	533	600
\(\mathrm{TiO_2}\)	685	700
\(\mathrm{BaTiO_3}\)	710	950
\(\mathrm{SiC}\)	\(1350\ **)\)	\(>2000\ ***)\)

) Value unreliable: the straight line was drawn through two points.
) Corrected value.
**) According to Bozard’s observations. ETH diploma thesis, 1947. Unpublished.

In the third column, for comparison, are given the temperatures at which, upon heating in vacuum, noticeable dissociation begins and, together with it, annealing of lattice defects. These temperatures are naturally determined only very approximately; however, the correlation with the calculated values is unquestionable.

The Meyer–Neldel rule is fulfilled for the most diverse chemical compounds, so that the possibility is not excluded that it has quite general applicability. According to Busch’s considerations set forth above, however, this rule should be limited mainly to semiconductors with “frozen-in” lattice defects.

Gisolf^(59) gave an entirely different explanation of the Meyer–Neldel rule. It is based on very special assumptions about the simultaneous existence of donor and acceptor levels, and also on the existence of a definite relation between the mobilities and concentrations of impurity levels; this, however, is to a considerable extent inconsistent with the views set forth earlier and in this section. For a more complete solution of this question, additional experimental data are required.

3.34. Relation between the dielectric constant and the activation energy

The activation energy of impurity semiconductors, according to Mott^(6) [equation (20) in Section 2.4], depends on the effective dielectric constant \(K_{\mathrm{eff}}\) of the semiconductor and may be represented as follows:

\[ \Delta E \sim \frac{1}{K_{\mathrm{eff}}^{2}} . \tag{54} \]

According to this, a high dielectric constant, i.e., a large polarizability of the lattice region closest to the impurity center, leads to a small activation energy. Although, on the basis of this formula, one can directly obtain values of the activation energy that are correct in order of magnitude, systematic studies in this direction would nevertheless be very important. Unfortunately, such studies at present scarcely exist. In Henniger’s brief review^(69), only isolated experimental data are presented concerning the dependence of the activation energy on impurity concentration and dielectric constant. However, precise data on the substances studied, as well as numerical values, are absent there.

In order to find at least qualitative confirmation of relation (54), Busch, Flury, and Merz^(35) undertook a study of \(\mathrm{BaTiO_3}\) single crystals. This substance is an electronic semiconductor and, owing to its special properties, is known as a ferroelectric. \(\mathrm{BaTiO_3}\) exhibits a very strong dependence of the static dielectric constant on temperature. Near the Curie temperature \(T_k \simeq 120^\circ \mathrm{C}\), the dielectric constant has a very sharp maximum and reaches values as high as 10,000.

Unlike all ferroelectrics presently known, in this compound at the Curie point there is also a clearly expressed maximum of the refractive index in the visible region. At room temperature \(n = 2.40\), and at \(120^\circ \mathrm{C}\) it reaches a value of about 2.46.

On the basis of these observations one might expect that the activation energy associated with the dielectric constant \(K_{\mathrm{eff}}\) will have a similar dependence on temperature. According to formula (54), at the maximum value of \(K_{\mathrm{eff}}\) the activation energy \(\Delta E\) will have a minimum, which should correspond to the minimum slope of the straight line representing the dependence

\[ \ln \sigma = f\!\left(\frac{1}{T}\right). \]

This proposition is confirmed experimentally.

For high and low temperatures the activation energy has, for example, the value \(\Delta E = 1.75\) eV, while near the Curie point it passes through a minimum equal to \(1.15\) eV.

Since the question of the dependence of the activation energy on the dielectric constant is very essential for the theory of semiconductors, it would be of interest to carry out such comparisons between \(\Delta E\) and \(K_{\mathrm{eff}}\) for other compounds investigated—if not quantitatively, then at least qualitatively.

3.35. The influence of hydrostatic pressure on electrical conductivity

The change in the electrical conductivity of metals under the influence of hydrostatic pressure was investigated by Bridgman\(^{28}\). For metals, with increasing pressure, both a decrease and an increase in conductivity were observed.

For semiconductors, very few such investigations have been carried out up to the present time.

Holmes and Allen\(^{74}\) observed in selenium a strong decrease in specific resistance with increasing pressure, as compared with metals, and found that the change in resistance obeys the following formula:

\[ \frac{1}{R_0}\,\frac{\Delta R}{\Delta P} = -3.1 \cdot 10^{4}\ \frac{\mathrm{cm}^{3}}{\mathrm{kg}} . \]

An exceptionally strong change in resistance was observed by Bridgman\(^{29}\) for tellurium. At a pressure of about \(30\,000\ \dfrac{\mathrm{kg}}{\mathrm{cm}^{3}}\), the conductivity of tellurium increased by a factor of 400 or 600, compared with the conductivity at atmospheric pressure, depending on the orientation of the crystal in the electric field.

Similar data were also obtained by Cartwright\(^{39}\). The work of Cartwright and Gaberfeld\(^{41}\), as well as the more recent investigations of Kehlon and Lark-Horovitz\(^{111}\), Bottom\(^{27}\), and Johnson\(^{79}\), established that tellurium is a semiconductor in which, already at temperatures somewhat above room temperature, intrinsic conductivity with activation energy \(\Delta E_1 = 0.38\) eV is detected. At lower temperatures hole conductivity is observed.

Bridgman suggested that, with increasing pressure, the metallic character of tellurium becomes ever stronger. Indeed, at a pressure of \(45\,000\ \dfrac{\mathrm{kg}}{\mathrm{cm}^2}\) tellurium undergoes a transformation\(^{30}\).

For the change in resistivity at low pressures, Bardeen\(^{18}\) gave a very convincing explanation. With increasing pressure the lattice constant of tellurium decreases, and one may assume that the energy gap between the highest filled band and the conduction band thereby decreases; consequently, the activation energy \(\Delta E\), which determines the intrinsic conductivity, also decreases. From Bridgman’s measurements Bardeen determined the activation energy \(\Delta E\) as a function of pressure. His calculation

Fig. 35. Activation energy \(\Delta E_1\) for tellurium, calculated from the temperature dependence of the resistance at different pressures.

is based on the assumption that the conductivity varies with temperature according to the law:

\[ \sigma = Ae^{-\frac{\Delta E_1}{2kT}}. \]

Therefore, to determine \(\Delta E_1\) it is sufficient to have conductivity data at two different temperatures. The results of these calculations are shown graphically in Fig. 35.

For a pressure of about \(30\,000\ \dfrac{\mathrm{kg}}{\mathrm{cm}^2}\), the activation energy is indeed very small, and at a somewhat higher pressure a transition to a truly metallic phase probably takes place owing to the overlap of the energy bands. The fact that at pressure \(P=0\) values less than \(0.38\ \mathrm{eV}\) are obtained for \(\Delta E_1\) can be readily explained by the tellurium studied by Bridgman still having impurity conductivity.

4. THERMOELECTROMOTIVE FORCE IN SEMICONDUCTORS

4.1. FORMULA FOR THE DIFFERENTIAL THERMO-EMF

Between the integral thermo-emf \(V_{\mathrm{MP}}\) of a metal–semiconductor–metal element and the Thomson coefficients \(\mu_{\mathrm{M}}\) and \(\mu_{\mathrm{P}}\) for the metal and the semiconductor there exists the thermodynamic relation:

\[ \frac{dV_{\mathrm{MP}}}{dT} = \int \frac{\mu_{\mathrm{M}}-\mu_{\mathrm{P}}}{T}\,dT . \tag{55} \]

For the Thomson coefficients, the electron theory*) gives the following expression:

\[ \mu = -\frac{T}{e}\, \frac{\partial}{\partial T} \left[ \frac{K_2}{K_1 T} - \frac{\zeta}{T} \right]. \tag{56} \]

Here \(\zeta\) denotes the limiting energy of the Fermi–Dirac distribution function, while \(K_1\) and \(K_2\) are expressed through the integrals:

\[ K_n = \frac{1}{2\pi^2} \left( \frac{2m^*}{\hbar^2} \right)^{\frac{3}{2}} \int_0^\infty l E^n \frac{\partial w_0}{\partial E}\,dE, \tag{57} \]

where \(E\) is the total energy of the electron, \(m^*\) is the effective mass of the electron, \(l\) is the mean free path, and \(w_0\) is the unperturbed Fermi–Dirac distribution function.

Under the assumption that the mean free path does not depend on energy, we easily obtain that

\[ \frac{K_2}{K_1T}=2k, \tag{58} \]

where \(k\) is Boltzmann’s constant, i.e.

\[ \mu = -\frac{T}{e}\, \frac{\partial}{\partial T} \left[ 2k-\frac{\zeta}{T} \right]. \tag{59} \]

Owing to the very weak temperature dependence of the limiting energy in metals, the Thomson coefficient \(\mu_{\mathrm{P}}\) for semiconductors is, in order of magnitude, much larger than \(\mu_{\mathrm{M}}\). Taking this into account, from formulas (55) and (59) we obtain the following expression for the differential thermo-emf:

\[ \frac{dV_{\mathrm{MP}}}{dT} = \pm \frac{k}{e} \left[ 2-\frac{\zeta}{kT} \right]. \tag{60} \]

Substituting the expressions for \(\zeta\), we immediately obtain the differential thermo-emf as a function of temperature. For example, for electronic semiconductors, applying formula (13) of section 2.2 and taking into account the negative sign of the electron charge,

*) See, for example, 8.

we obtain:

\[ \frac{dV_{\mathrm{MP}}}{dT} = -\frac{k}{e} \left[ 2-\ln C n_D^{\frac12} +\frac{\Delta E_D}{2kT} \right], \tag{61} \]

where

\[ C= \left( \frac{2\pi \hbar^2}{m_e^* kT} \right)^{\frac34}, \]

\(n_D\) is the donor concentration and \(\Delta E_D\) is the activation energy.

The absolute value of the thermoelectric emf increases, at constant temperature, with increasing activation energy and decreasing impurity concentration.

For electron semiconductors the thermoelectric emf has a negative sign, and for hole semiconductors a positive one. In mixed semiconductors the thermoelectric emf may vanish altogether.

The dependence of the thermoelectric emf on temperature may be represented in the following form:

\[ \frac{dV_{\mathrm{MP}}}{dT} = F(T)-\frac{\Delta E_D}{2eT}. \tag{62} \]

The first term \(F(T)\), in comparison with the second, changes little with temperature, and therefore, on the basis of the given theory, the thermoelectric emf should increase as the temperature decreases.

This result is in explicit contradiction with the basic thermodynamic Nernst equation, according to which the thermoelectric emf must vanish at absolute zero. Therefore equality (62) may be regarded as quite valid only for high temperatures; on the contrary, for low temperatures it is incorrect. The basis for this is, on the one hand, the fact that the approximations made in deriving formula (56) become obviously invalid at low temperatures.^2 On the other hand, in differentiating expression (56) it was assumed that the thermal conductivity, just as in metals, is mainly due to electrons. This is, of course, meaningless for semiconductors at low temperatures, since, with the decrease in electron concentration then occurring, the lattice thermal conductivity begins to predominate. To this one should also add the considerations of Herring,^67 according to which, in order to find the correct value of the thermoelectric emf, it is necessary to take into account contact phenomena between individual grains of an inhomogeneous semiconductor.*

* The considerations to which the author refers here are in fact incorrect, since they follow from Herring’s erroneous work. See the article by B. I. Boltaksa, “On the dependence of the thermoelectric-power coefficient in semiconductors on the temperature difference of the alloy,” ZhTF 20, 1039 (1950). (Editor’s note.)

In general it should be noted that agreement of theory with experiment is observed only in exceptional cases.

For degenerate semiconductors, relations are obtained similar to those for metals. Johnson and Lark-Horovitz\(^{80}\) gave a corresponding formula, which agrees well with observations on germanium.\(^{87}\)

4.2. EXPERIMENTAL DATA

Data from measurements of thermoelectric power in semiconductors, as compared with metals, are set forth in the following works:

Electron semiconductors:

TiO\(^{67}\); V\(_2\)O\(_5^{54,71}\); CuO\(^{54}\); ZnO\(^{22,56}\); CdO\(^{15,22,54,72,73}\); WO\(_3^{71}\).

Hole semiconductors:

Cr\(_2\)O\(_3^{54}\); CoO\(^{71}\); Co\(_3\)O\(_4^{54}\); NiO\(^{22,54,130}\); CuJ\(^{103}\);

Cu\(_2\)O\(^{47,98,110,116,126}\); MoS\(_2^{71}\); SnS\(^{12,13}\); Bi\(_2\)S\(_3^{71}\).

Amphoteric semiconductors:

Si\(^{53}\); SiC\(^{37,71}\); Ge\(^{87}\); Se\(^{24,71}\); Tl\(_2\)S\(^{71}\); PbS\(^{46,50,91,101}\); PbSe\(^{21}\).

In most cases the integral thermoelectric power has been measured for a single temperature, as a result of which only the order of magnitude and the sign of the thermoelectric power can be regarded as firmly established. Determination of the sign of the carriers by this method gives results that coincide with data obtained from measurements of the Hall effect.

From both the practical and the theoretical point of view, measurements of the differential thermoelectric power over a wide temperature interval are of considerably greater interest. Unfortunately, the number of such investigations is very small. Schweikert\(^{116}\) and Rode\(^{110}\) investigated the element Cu—Cu\(_2\)O—Cu between \(-80\) and \(+70^\circ\)C. In this temperature region a decrease of the differential thermoelectric power with increasing temperature and an increase of it with increasing activation energy were observed. This is in qualitative agreement with the theoretical conclusions. However, proportionality to the quantity reciprocal to the temperature appears not to hold. In addition, a strange dependence of the differential thermoelectric power on the temperature difference is found, which has so far received no explanation.\(^{100}\)*)

Good confirmation of the proportional dependence between the differential thermoelectric power and \(\dfrac{1}{T}\) was obtained from the measurements of Wright and Andrews\(^{130}\) on NiO. Nickel oxide, despite its unfilled \(3d\)-shell, is not a metal, but a hole semiconductor. Wright and Andrews, by oxidizing a nickel ribbon, obtained a dense—

) On the question of the dependence of the thermoelectric-power coefficient on the temperature difference of junctions, see \(^{42}\) in the supplementary bibliography. (Editor’s note.*)

... layers of NiO and, along with the conductivity, also investigated the Hall effect and the differential thermoelectric power up to \(700^\circ\text{C}\). The temperature difference \(\Delta T\) between the hot and cold ends was \(40—50^\circ\), and they found no dependence of the thermoelectric power on the difference \(\Delta T\). The question of whether this is also true for considerably smaller temperature differences remains open. The results of the measurements of Wright and Andrews are presented in Fig. 36. Between 370 and \(640^\circ\text{C}\), proportionality

Fig. 36. Dependence of the differential thermoelectric power of the Pt—NiO—Pt element on reciprocal temperature.

between the differential thermoelectric power and \(\dfrac{1}{T}\) is in fact satisfied very well; in this case, for the activation energy \(\Delta E_A\), values of about \(1.67 \div 1.94\ \text{eV}\) are obtained, which agree satisfactorily with the values obtained from conductivity measurements.

The theoretically expected dependence of the thermoelectric power on the impurity concentration \(n_A\) was not found. The reason evidently lies in the fact that all the NiO specimens investigated had the same oxygen content. The concentration of donor impurities fluctuated slightly about an average value equal to \(n_A = 5.37 \cdot 10^{20}\ \text{cm}^{-3}\).

Hogarth’s investigations \(^{72,73}\), carried out on CdO between 240 and \(570^\circ\text{C}\), gave quite similar results. In particular, the theoretically predicted temperature dependence was confirmed. In addition, these measurements made it possible to draw an important conclusion concerning the influence of impurity concentration on the differential thermoelectric power. According to formula (42) of Section 3.322, con-

centration of impurities in oxides depends on the oxygen pressure in the following way:

\[ n_D=\mathrm{const}\, P^{\frac{1}{n}} . \tag{63} \]

Substituting this expression into formula (62) of Section 4.1, we obtain (at constant temperature) the following relation:

\[ \frac{dV_{\mathrm{MP}}}{dT}\sim \ln P . \tag{64} \]

This was confirmed by Hogarth’s measurements on CdO, Cu\(_2\)O, NiO, and ZnO. For the quantity \(n\) entering the exponent, values are obtained that agree well with the values derived on the basis of thermodynamic considerations by Baumbach and Wagner \(^{23}\), and also by Djunvald and Wagner \(^{47}\).

Fig. 37. Dependence of the differential thermo-emf of the element Cu—SiC—Cu on temperature.

The temperature behavior of the differential thermoelectric emf, which deviates completely from that predicted by modern theory, was obtained by Busch, Schmidt, and Spenlich \(^{37}\) for SiC. Fig. 37 presents the results of their experiments for temperatures lying in the interval from 80 to 300°K. The observational data of Gochberg and Sominskii \(^{71}\) concerning the sign of the thermo-emf for SiC agree with the data of the work cited above, and also with the data of Busch and Labgart obtained from Hall-effect measurements.

In the studies of Gochberg and Sominskii the thermo-emf was measured on single-crystal SiC specimens relative to copper. No dependence of the thermo-emf on the temperature difference was detected. It was found that the differential thermo-emf decreases as the temperature is lowered, which is especially clearly expressed for black \(p\)-SiC. At 100°K the thermo-emf disappeared completely and at lower temperatures remained equal to zero, which indicated the absence of a change in the sign of the thermo-emf. An explanation of this phenomenon from the standpoint of electron theory is still lacking.

5. CONCLUSION

For more than 40 years, physicists and physical chemists have been more or less intensively engaged with the problem of electrical conductivity in nonmetals. A large number of experimental facts have been collected, and, at least in broad outline, a theoretical explanation has been found for the processes occurring in solids. There is nothing surprising in the fact that, up to the present, theory agrees with experiment only in a small number of cases, and even then only qualitatively. On the one hand, our knowledge of the actual structure of solids is still very insufficient; on the other hand, our preparative methods have not yet attained such perfection as to yield substances corresponding to our simple models. The difficulties encountered along this path are clearly seen in the example of selenium. Despite the large number of works devoted to the electrical properties of this element, there is still no explanation of the regularities observed in this substance. Therefore, despite their technical importance, such semiconductors were not considered in the present review. Recently our knowledge in the field of semiconductor theory has increased considerably, and there is no doubt that the difficulties that have arisen can be overcome by the further accumulation of new experimental material.

In any case, for the investigator interested in this very attractive field, there is the broadest possible scope for activity.

EDITOR’S ADDENDA

The article by the Swiss physicist G. Busch, “Electronic Conductivity of Nonmetals,” is a review of experimental data on the electrical, thermoelectric, and galvanomagnetic properties of semiconductor materials investigated up to 1950.

The theory of semiconductors at present constitutes an extensive field of solid-state physics, which is finding ever wider application in the most varied branches of industry and technology. A compilation of experimental data on the properties of semiconductors and their interpretation within the framework of the so-called “band theory” are of considerable interest to persons working in the field of semiconductor physics and technology.

Since, however, two years have passed since the article appeared, it requires certain additions. Moreover, the review omits a number of earlier important experimental and theoretical investigations, chiefly by Soviet physicists. It seems to us that the value of the article will be enhanced if the most substantial omissions are corrected within the framework set by the author.

Even in the prewar years, we in the Soviet Union investigated the influence of a strong electric field on the electrical conductivity of semiconductors[^1][^2][^3]. It was shown that, beginning with a certain critical value of the applied field (characteristic for each semiconductor), Ohm’s law ceases to hold in semiconductors and an exponential increase of current with voltage begins. At that same time Soviet physicists proposed and developed theories of these phenomena[^4][^5][^6]. Much later, about 10 years afterward, similar investigations were carried out in England and the USA.

Let us note in passing that the conversion of hole semiconductors into electronic ones and vice versa was carried out in the Soviet Union on the example of Tl₂S[^7] and PbS[^8] long before the foreign works described by the author.

Bush’s review also omits the question of a new class of semiconductor materials—intermetallic compounds of constant composition. Investigations by Soviet physicists showed that these compounds (discovered more than 40 years ago by N. S. Kurnakov and his collaborators[^9][^10]) possess the properties of impurity semiconductors, whose electrical conductivity can be varied over a wide range of values—from metallic to semiconducting[^11][^12][^13][^14][^15].

Not included in the review is the fundamentally important question of the electrical conductivity of liquid semiconductors at temperatures above the melting point. Investigations by A. R. Regel’ and others[^34][^35] showed that the band structure of the energy spectrum is preserved in semiconductors even after melting, and consequently the discrete structure of the energy spectrum is determined not so much by the strictly periodic arrangement of atoms or ions in the crystal as by their so-called “short-range order,” which is preserved after melting as well. Moreover, depending on whether melting is accompanied by a transition of the substance to a denser or to a less dense packing, the specific electrical conductivity at the melting point either increases abruptly or decreases abruptly. These investigations are of substantial importance for understanding the mechanism of conductivity and the place occupied by the band theory of semiconductors in the theory of the solid state.

A significant number of investigations by Soviet authors on the electrical properties of semiconductors1[^17][^18][^19], carried out and published by them both before 1950 and later, likewise did not enter Bush’s review.

Perhaps the most substantial gap in the review is the absence of data on germanium. In recent years, an extensive series of works connected with the properties of germanium and silicon as materials for solid-state amplifiers has been published in the foreign press. The electrical properties of germanium have been studied as a function of the conditions of preparation and treatment of specimens[^20][^21][^22] (the introduction ...

impurities, irradiation, etc.). The question of the motion of holes introduced into electron-type germanium, and of electrons in hole-type germanium, has been studied in detail \(^{23,24,25}\). In 1949 Suhl \(^{26}\) described the magneto-concentration effect discovered by him—the local change in the conductivity of germanium caused by the lateral deflection, by a magnetic field, of current carriers introduced into germanium. All these investigations constituted a new chapter in the physics of homeopolar-type semiconductors with an atomic lattice, whereas earlier attention had been directed toward compounds of the polar type. Here the predictions of the theory were most fully confirmed.

The author completely ignores Pekar’s theory of polarons \(^{27}\), which gives a new interpretation of the processes of conduction in ionic crystals, Adirovich’s theory of radiationless transitions \(^{28}\), and the works of Soviet authors on the electronic theory of crystals \(^{29,30}\).

The survey also lacks an analysis of a number of theoretical investigations by foreign authors devoted to the mobility of current carriers in ionic and atomic semiconductors \(^{31}\), the Hall constant in semiconductors with impurities \(^{32,47}\), the mechanism of impurity-band conduction \(^{45}\), etc.

We have noted only the principal questions that did not enter Busch’s survey. Despite these shortcomings, however, Busch’s survey is the best of the existing summaries on the electrical conductivity of semiconductors. The gaps in the article are partly made up for by the bibliographical references to the original works given at the end. This index also includes works published after 1950.

In conclusion, let us note that in the pages of the Journal of Technical Physics a discussion has unfolded on certain problems of the electronic theory of crystals and, in particular, on questions of the band theory of semiconductors \(^{65,66}\). This discussion goes considerably beyond the stage of band theory that is set forth in Busch’s article.

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1. L. Brillouin, Quantenstatistik (Springer, Berlin 1932).
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4. W. Jost, Diffusion und chemische Reaktion in festen Stoffen (Steinkopf, Dresden und Leipzig 1937).
5. E. Justi, Leitfähigkeit und Leitungsmechanismus fester Stoffe (Vandenhoeck & Ruprecht, Göttingen 1948).
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38. E. Engelhard, Ann. der Physik 17, 501 (1933).
39. L. Engelhard, Ann. der Physik 38, 121 (1940).
40. J. A. V. Fairbrother, Proc. Roy. Soc. 157, 50 (1936).
41. W. Feldmann, Phys. Rev. 64, 113 (1943).
42. F. Fischer und K. Dehn, Ges. Abh. Kenntn. Kohle 12, 526 (1937).

ELECTRONIC CONDUCTIVITY OF NONMETALS

1. F. Fischer, K. Dehn und H. Sustmann, Ann. der Physik 15, 109 (1932).
2. E. Friedrich, Zeits. f. Physik 31, 813 (1925).
3. O. Fritsch, Ann. der Physik 22, 375 (1935).
4. H. Fröhlich, Proc. Roy. Soc. 160, 230 (1937).
5. H. Fröhlich and N. F. Mott, Proc. Roy. Soc. 171, 496 (1939).
6. J. H. Gisolf, Ann. der Physik 1, 3 (1947).
7. J. Goldmann and A. W. Lawson, Phys. Rev. 64, 11 (1934).
8. B. Gudden, Ber. phys.-med. Soc. Erlangen 62, 287 (1930).
9. P. Guillery, Ann. der Physik 14, 216 (1932).
10. P. Guillery, Phys. Zeits. 32, 891 (1931).
11. J. Gundermann, H. Hauffe und C. Wagner, Zeits. f. phys. Chem. 37, 148, 155 (1931).
12. E. E. Hahn, B. R. Russel and P. H. Miller, Phys. Rev. 75, 1631 (1949).
13. W. Hartmann, Zeits. f. Physik 102, 709 (1936).
14. H. K. Henisch, El. Comm. стр. 163 (1948).
15. F. P. Henninger, Ann. der Physik 28, 245 (1937).
16. F. P. Henninger, Phys. Zeits. 39, 911 (1938).
17. H. Hintenberger, Zeits. f. Phys. 119, 1 (1942).
18. B. M. Гохберг и M. Я. Соминский, Phys. Zeits. Sow. Union 13, 198 (1938).
19. C. A. Hogarth, Nature 161, 60 (1948).
20. C. A. Hogarth, Phil. Mag. 40, 273 (1949).
21. R. M. Holmes and H. W. Allen, Phys. Rev. 55, 593 (1939).
22. D. K. Holmes, Phys. Rev. 76, 179 (1949).
23. F. Hund, Phys. Zeits. 36, 888 (1935).
24. R. Iley and H. L. Riley, Nature 160, 468 (1947).
25. W. Jander und W. Stamm, Zeits. anorg. Chem. 199, 265 (1931).
26. V. A. Johnson, Phys. Rev. 74, 1255 (1948).
27. V. A. Johnson and K. Lark-Horovitz, Phys. Rev. 69, 259 (1946).
28. V. A. Johnson and K. Lark-Horovitz, Phys. Rev. 71, 374, 909 (1947).
29. V. A. Johnson and K. Lark-Horovitz, Phys. Rev. 72, 531 (1947).
30. B. Жузе и Б. Курчатов, ЖЭТФ 2, 309 (1932).
31. G. E. Kimball, J. Chem. Phys. 3, 560 (1937).
32. J. Königsberger und K. Schilling, Ann. der Physik 32, 129 (1910).
33. И. В. Курчатов, Т. З. Костина и В. Л. Русинов, Phys. Zeits. Sow. Union 7, 129 (1935).
34. K. Lark-Horovitz, A. E. Middleton, E. P. Miller and J. Walerstein, Phys. Rev. 69, 258 (1946).
35. T. W. Lashof, J. Chem. Phys. 11, 196 (1943).
36. F. Lauster, Zeits. f. Physik 59, 83 (1929); 61, 578 (1930).
37. K. Legrand, Zeits. f. Physik 124, 219 (1948).
38. M. Le Blanc und H. Sachse, Ann. der Physik 11, 727 (1931).
39. J. McDougall and E. C. Stoner, Phil. Trans. Roy. Soc. London 771, 67 (1938).
40. W. Meissner und B. Voigt, Ann. der Physik 7, 892 (1930).
41. M. Meyer, Zeits. f. Physik 85, 278 (1933).
42. W. Meyer, Zeits. El. Chem. 50, 279 (1944).
43. W. Meyer und H. Neldel, Zeits. techn. Phys. 18, 588 (1937).
44. P. H. Miller, Phys. Rev. 60, 890 (1941).
45. F. Möglich, Zeits. f. Physik 109, 503 (1938).

1. G. Mönch, Zeits. techn. Phys. 16, 361 (1935); Ann. der Physik 26, 481 (1936).
2. G. Mönch, Ann. der Physik 34, 265 (1939).
3. M. C. Morton, Trans. Faraday Soc. 43, 194 (1947).
4. J. F. Mullaney, Phys. Rev. 66, 326 (1944).
5. K. Nagel und C. Wagner, Zeits. f. phys. Chem. 25, 71 (1934).
6. H. M. O’Bryan and H. W. Skinner, Phys. Rev. 45, 370 (1934).
7. H. Osterberg, Phys. Rev. 50, 1187 (1936).
8. G. L. Pearson and J. Bardeen, Phys. Rev. 75, 865 (1949).
9. G. L. Pearson and W. Shockley, Phys. Rev. 71, 142 (1947).
10. E. H. Putley, Proc. Phys. Soc. 62, 284 (1949).
11. W. Ringer und H. Welker, Zeits. Naturf. 3a, 20 (1948).
12. W. Rohde, Ann. der Physik 34, 259 (1939).
13. W. Scanlon and K. Lark-Horovitz, Phys. Rev. 72, 530 (1947).
14. E. Schmid und G. Wassermann, Zeits. f. Physik 46, 653 (1928).
15. W. Schottky und F. Waibel, Phys. Zeits. 34, 858 (1933).
16. W. Schottky und F. Waibel, Phys. Zeits. 36, 912 (1935).
17. A. Schulze, Phys. Zeits. 31, 1062 (1930); Zeits. techn. Phys. 61, 578 (1930).
18. H. Schweickert, Ann. der Physik 34, 250 (1939).
19. R. W. Sears and J. A. Becker, Phys. Rev. 40, 1055 (1933).
20. H. J. Seemann, Phys. Zeits. 28, 765 (1927); 29, 94 (1928); Zeits. f. Phys. 61, 576 (1930).
21. H. J. Seemann, Phys. Zeits. 30, 143 (1929).
22. F. Seitz, Phys. Rev. 73, 549 (1948).
23. K. Shifrin, ZhETF 8, 242 (1944).
24. F. Skaupy, Zeits. f. Phys. 1, 259 (1920).
25. P. Sommerville, Phys. Rev. 34, 311 (1912).
26. E. J. W. Verwey, P. W. Haayman and F. C. Romeijn, J. Chem. Phys. 15, 181 (1947).
27. A. Voelkl, Ann. der Physik 14, 193 (1932).
28. W. Vogt, Ann. der Physik 7, 183 (1930).
29. C. Wagner, Trans. Faraday Soc. 34, 851 (1938).
30. F. Waibel, Wiss. Veröff. Siemens-Konzern 10, 65 (1931).
31. A. H. Wilson, Proc. Roy. Soc. 133, 458 (1931); 134, 277 (1932).
32. R. W. Wright und J. R. Andrews, Proc. Phys. Soc. 62, 446 (1949).
33. B. Zhusé and Ya. Starchenko, ZhETF 10, 331 (1940).

ADDITIONAL BIBLIOGRAPHY

I. Works published before 1950

1. A. V. and A. F. Ioffe, ZhETF 9, 1428 (1939).
2. A. V. and A. F. Ioffe, ZhETF 9, 1451 (1939).
3. Ya. I. Frenkel, ZhETF 8, 1292 (1938).
   3a. Pruzhinina-Granovskaya, ZhTF 19, 100 (1949).
4. N. L. Pisarenko, Izv. AN SSSR, 631 (1938).
5. F. F. Volkenstein, Electrical Conductivity of Semiconductors, Gostekhizdat, 1947.
6. R. Bray, Phys. Rev. 76, 1, 1949; E. Ryder and W. Shockley, Phys. Rev. 75, 2, 310 (1949).
7. W. Brattain and J. Bardeen, Phys. Rev. 74, 231 (1948).
8. J. Bardeen and W. H. Brattain, Phys. Rev. 74, 230 (1948).
9. N. S. Kurnakov and N. I. Stepanov, ZhRFKhO 37, 568 (1905).
10. N. I. Stepanov, Proceedings of the Mining Institute III, nos. 3–4, 208 (1911).

1. B. I. Boltaks and V. P. Zhuze, ZhTF XVIII, No. 12, 1459 (1948).
2. I. V. Mochan, ZhTF XVIII, No. 12, 1485 (1948).
3. V. P. Zhuze, I. V. Mochan, S. M. Ryvkin, ZhTF XVIII, No. 12, 1494 (1948).
4. T. A. Kontorova, ZhTF XVIII, No. 12, 1478 (1948).
5. B. I. Boltaks, DAN SSSR LXIV, No. 4, 487 (1949); DAN SSSR LXIV, No. 5, 653 (1949).
6. Yu. A. Dunaev and Yu. P. Maslakovets, ZhETF 17, No. 10, 901 (1947).
7. N. D. Morgulis, P. G. Borzyak, B. I. Dyatlovitskaya, Izv. AN SSSR, ser. fizich. XII, No. 2, 125 (1948).
8. B. I. Volfson and V. N. Rozhdestvenskii, ZhETF 3, 447 (1933).
9. A. N. Arsen’eva and B. V. Kurchatov, ZhETF 4, 456 (1934).
10. W. Ringer u. N. Welner, Zeits. Naturforsch. 39, No. 1, 20 (1948).
11. A. Gerritsch, Physica 15, No. 3—4, 427 (1949).
12. K. Lark-Horovitz, Electr. Eng. 68, No. 12, 1047 (1949).
13. G. L. Pearson, Phys. Rev. 76, 179 (1949).
14. W. Shockley, Bell Syst. Tech. J. 28, 435 (1949).
15. J. R. Haynes and W. Shockley, Phys. Rev. 75, 691 (1949); The Bell Syst. Tech. J. XXVIII, No. 3, 344 (1943).
16. H. Suhl and W. Shockley, Phys. Rev. 75, 1617 (1949).
17. S. I. Pekar, ZhETF 16, No. 4, 335 (1946); ZhETF 16, No. 4, 341 (1946); ZhETF 18, No. 2, 105 (1948); ZhETF 19, No. 9, 796 (1949).
18. E. I. Adirovich, DAN SSSR 63, 111, 635 (1948); Izv. AN SSSR, ser. fizich. 13, 101 (1949).
19. B. I. Davydov and I. M. Shmushkevich, UFN 24, 21 (1940).
20. I. E. Tamm, Sow. Phys. 1, 733 (1932); Zeits. Physik 76, 849 (1932).
21. F. Seitz, Phys. Rev. 73, 549 (1948); H. Fröhlich and N. F. Mott, Proc. Cambr. Phil. Soc. 34, 568 (1938).
22. K. Lark-Horovitz and V. A. Johnson, Bull. Amer. Phys. Soc. 21, 18 (1946).
23. A. R. Regel, ZhTF 18, 1511 (1948).

II. Works published in 1950—1951*)

1. A. I. Blum and A. R. Regel, ZhTF 21, 316 (1951).
2. A. I. Blum, N. P. Mokrovskii, A. R. Regel, Reports at the VII Conference on Semiconductors in Kiev, 1950.
3. N. A. Goryunova, ZhTF XXI, 814 (1951).
4. B. I. Boltaks, ZhTF XX, 180 (1950).
5. Ya. M. Ksendzov, ZhTF XX, 117 (1950).
6. B. I. Boltaks, F. I. Vasenin and A. E. Salunina, ZhTF XXI, 532 (1951).
7. V. I. Lyashchenko and A. F. Skubenko, Nauk. Zap. Kievsk. univer., vol. IX, issue II, 77 (1950).
8. S. G. Kalashnikov, ZhTF 21, 129 (1951).
9. B. I. Boltaks, ZhTF XX, 9, 1039 (1950).
10. V. E. Lashkarev and V. I. Lyashchenko, Collection dedicated to the seventieth anniversary of Academician A. F. Ioffe, 535 (1950).
11. C. S. Hung, Phys. Rev. 79, No. 4, 327 (1950).
12. C. Erginsoy, Phys. Rev. 80, No. 6, 1104 (1951).
13. C. S. Hung and Gliessman, Phys. Rev. 79, No. 4, 726 (1950).

*) For a detailed list of the works of foreign physicists during this period, see “Abstract Collection on Semiconductors.”

1. V. A. Johnson and K. Lark-Horovitz, Phys. Rev. 79, No. 1, 176 (1950).
2. V. A. Johnson and K. Lark-Horovitz, Phys. Rev. 79, No. 2, 409 (1950).
3. W. C. Dunlap, Phys. Rev. 79, No. 2, 286 (1950).
4. G. L. Pearson, J. D. Struthers and H. C. Theurer, Phys. Rev. 77, 809 (1950).
5. C. S. Hung and V. A. Johnson, Phys. Rev. 79, No. 3, 535 (1950).
6. G. L. Pearson, J. R. Haynes, W. Shockley, Phys. Rev. 78, No. 3, 295 (1950).
7. R. Wyrick and H. Levinstein, Phys. Rev. 78, No. 3, 304 (1950).
8. C. Erginsoy, Phys. Rev. 79, No. 6, 1013 (1950).
9. J. Dunlap, Phys. Rev. 77, 759 (1950).
10. E. J. Ryder and W. Shockley, Phys. Rev. 81, No. 1, 133 (1951).
11. F. Stöckmann, Zeits. f. Physik 127, 563 (1950).
12. Stönelin and G. Busch, Helv. Phys. Acta. XXIII, 530 (1950).
13. E. E. Hahn, Journ. Appl. Phys. 22, No. 7, 855 (1951).
14. E. J. Morin, Phys. Rev. 83, No. 5, 1005 (1951).
15. E. Loebner, Phys. Rev. 84, No. 1, 153 (1951).
16. F. F. Volkenshtein and V. L. Bonch-Bruevich, ZhETF 20, 624 (1950).
17. E. I. Adirovich, “Some Problems of the Theory of Luminescence of Crystals,” Gostekhizdat, 1951.
18. S. I. Pekar, “Investigations on the Electron Theory of Crystals,” Gostekhizdat, 1951.
19. A. I. Anselm, ZhTF XXI, No. 4, 489 (1951).
20. F. F. Volkenshtein, ZhTF XXI, 1544 (1951).