ELECTRONIC PROCESSES IN REAL CRYSTALS

F. F. Vol'kenshtein

§ 1. Introduction § 2. Ideal and real crystals § 3. Theory of free electrons of Sommerfeld § 4. Band theory § 5. Conductors and insulators from the standpoint of band theory § 6. Electronic levels in a real lattice § 7. Electrical conductivity in nonmetallic crystals § 8. Influence of impurities and temperature on the electrical conductivity of semiconductors § 9. Deformation of energy bands under the influence of an external electric field § 10. Ionizing action of an electric field § 11. Photoelectric processes

§ 1. INTRODUCTION

All macroscopic properties of crystals may be divided into two classes. To one class belong all those properties that are determined by the periodic structure of the lattice and for which allowance for the defects inevitably present in every real lattice plays the role of an insignificant correction. To the other class belong all those properties which, on the contrary, are determined precisely by local disturbances in the periodic structure of the lattice, and for which, consequently, allowance for defects is of fundamental importance. Properties belonging to this second class are usually called structure-sensitive properties.

The structure-sensitive properties include the optical and photoelectric properties of alkali-halide crystals, the electrical conductivity of semiconductors, and others. To this same class of properties belong the activity of crystalline catalysts, as well as the optical properties of phosphors.

All these properties are extremely sensitive to every kind of impurity contained in the crystal. The word “impurity,” however, should not be understood here too literally as the presence in the lattice of foreign atoms. An “impurity” is any disturbance of the periodic structure of the lattice. In particular, in lattices represented by the chemical formula \(M_nR_r\), where \(M\) is the symbol of a metal and \(R\) the symbol of a metalloid, by “impurity” one may understand

any deviation from the stoichiometric ratio. In the formula \(M_mR_r\), such deviations can be represented if the indices \(m\) and \(r\) are regarded not as integers but as fractional numbers, though close to integers.

Thus, under the influence of impurities, the electrical conductivity of semiconductors changes millions of times; the coloration of crystals changes (i.e., the intensity and position of absorption bands). The catalytic activity of catalysts changes (increases or decreases). In the variation of all these diverse properties a certain parallelism is observed.

One may think that all these diverse properties of crystals ultimately have one and the same origin. Indeed, a crystal as a whole constitutes a certain unified electrical system. In this system distinctive electronic processes take place. They determine both the stable and the structure-sensitive properties of crystals. Two factors, contending with one another, govern these processes: the factor of order and the factor of disorder. The first of them predetermines the stable properties of the crystal. The second determines the structure-sensitive properties.

Thus, in the theoretical interpretation of structure-sensitive properties we must deliberately proceed from conceptions of a real crystal lattice, i.e., a lattice whose strictly periodic structure is disturbed. If one bases oneself on an ideal lattice, as is done in most questions of the theory of metals, then all structure-sensitive properties of crystals will necessarily remain outside the scope of the theory.

The theory of the electrical properties of semiconductors, on the one hand, and the theory of the optical and photoelectric properties of alkali-halide crystals, on the other, represent, as it were, two different chapters of one and the same theory of the real crystal in general. Although alkali-halide crystals differ substantially in their macroscopic properties from semiconductors, nevertheless there is no fundamental distinction between these two classes of crystals. In both cases we have crystals of the type \(M_mR_r\) with disturbed stoichiometry.

Both alkali-halide crystals and semiconductor crystals correspond to one and the same scheme of energy levels, and the difference between them reduces merely to a difference in the scale adopted in this scheme. All the optical and photoelectric properties of alkali-halide crystals should be repeated in semiconductors, but must in this case be shifted from the ultraviolet and visible parts of the spectrum into the red and infrared region, poorly accessible to experimental investigation. On the other hand, all the electrical properties of semiconductors, theoretically speaking, should also be observed in alkali-halide crystals, but in

significantly higher temperatures, lying beyond the limits of experimental possibilities.

The theory of the catalytic properties of crystalline (nonmetallic) catalysts is, as it were, yet another chapter of the same general theory of the real crystal. Indeed, in all these cases we are dealing essentially with one and the same object, considered only from different points of view.

§ 2. IDEAL AND REAL CRYSTALS

A real crystal differs from an ideal one by the presence of defects. By the word “defect” we shall understand any disturbance in the periodic structure of the lattice. The defects present in a real lattice may be of two kinds: defects of a macroscopic and defects of a microscopic character.

Fig. 1. Principal types of microdefects.

Macroscopic defects include cracks, colloidal particles of foreign matter, observed, for example, in colored alkali-halide crystals, various kinds of macroscopic inclusions, etc.

As for microscopic defects, let us note here the following principal types of defects of this kind:

1) an empty site, formed as a result of removal from the ideal lattice of an atom or ion;

2) an intrinsic atom or ion of the lattice, situated in an interstice;

3) an ion in a heteropolar lattice, situated in its normal position but carrying an anomalous charge;

4) an impurity atom placed in an interstice;

5) an impurity atom placed in a site, i.e. replacing an intrinsic atom of the lattice.

In Fig. 1 these five types of microdefects are shown schematically. The first two of them may be called mechanical defects.

tions. Such defects, in the case of a homeopolar lattice, do not alter its chemical composition, while in the case of a heteropolar lattice they may lead to one or another violation of the stoichiometric ratio. Defects of the third type may be called electrical defects. They likewise do not affect the chemical formula of the lattice. Finally, the last two types of microdefects may be called chemical defects. They are an “impurity” in the narrow sense of the word, distorting the chemical composition of the crystal.

Chemical defects are introduced into the lattice from outside. Their occurrence is due to one or another treatment of the specimen. Mechanical defects may arise not only by introducing them from outside, but also as a result of heating the lattice. Finally, electrical defects not only may be introduced from outside or created by heat; they may also arise in the lattice under the influence of illumination.

Let us consider in particular heteropolar lattices of the type \(M_mR_r\), possessing mechanical microdefects. Almost all semiconductors belong to this type of lattice.

The structural elements of such a lattice are positive metal ions with charge \(+p\) and negative metalloid ions with charge \(-q\), where

\[ \frac{q}{p}=\frac{m}{r}. \]

Here \(p\) and \(q\) are the absolute magnitude of the charge, expressed in units of the electron charge. In our lattice we have two kinds of sites: metal sites, in which metal ions are placed, and metalloid sites, in which metalloid ions sit.

If all the metal and metalloid sites of the lattice are occupied by the corresponding ions and if, in addition, there are no ions situated outside these sites, then our lattice is an ideal lattice. In such a lattice the stoichiometric ratio is necessarily satisfied.

In practice, however, we always deal with lattices in which the stoichiometric ratio is violated to one degree or another. All the catalytic, electrical, and optical properties of the crystal depend sharply on the character and degree of this violation. Stoichiometric violations indicate the presence of defects in the lattice.

Stoichiometric violations in a crystal may be of two kinds: we may have a lattice with a stoichiometric excess of metal or with a stoichiometric excess of metalloid. An excess of metal may be produced by introducing additional metal ions into the lattice or by removing metalloid ions from it. An excess of metalloid may be produced by introducing metalloid ions or by removing metal ions.

Thus, in the case of a catalyst or a semiconductor we are dealing with a real heteropolar lattice, which differs from the ideal one in the following features. In such a lattice not all sites are occupied by ions: it contains a certain number of vacant metallic and metalloid sites. In addition, in such a lattice not all ions are located at sites: it contains a certain number of interstitial metallic and metalloid ions.

In such a real heteropolar lattice the stoichiometric ratio may be observed, or it may be violated in one direction or another and to one degree or another, depending on the ratio between the numbers of defects of different kinds.

According to the role that these defects play in the lattice, they may be divided into two groups. Interstitial metallic ions and vacant metalloid sites may be called electropositive defects. For an electron introduced into our lattice, these defects serve as centers of attraction. Interstitial metalloid ions and vacant metallic sites may be called electronegative defects. With respect to the electron they play the role of centers of repulsion.

Such a classification of defects (according to their physical nature and according to the role that they play in the lattice) is explained by the scheme shown in Fig. 2.

Fig. 2. Classification of defects (according to their role in the lattice and according to their physical nature).

The total number of electropositive and electronegative defects contained in the lattice characterizes the degree of deviation of our real lattice from the ideal state. The difference between the numbers of electropositive and electronegative defects determines the degree of violation of the stoichiometric ratio. A predominance of electropositive defects indicates a stoichiometric excess of metal. A stoichiometric excess of metalloid is characterized, conversely, by a predominance of electronegative defects.

The degree of deviation of the real lattice from the ideal, as well as the degree of deviation from the stoichiometric ratio, are two independent characteristics. Thus, every real heteropolar lattice, for its description, requires the introduction of two parameters. These parameters determine all the principal properties of the lattice. By varying these parameters, we can obtain different types of temperature curves of electrical conductivity, different types of the internal photoelectric effect (positive or neg—

lative photoeffect or photoelectrically inactive absorption), etc.^1

The values of these parameters are determined by two factors. First of all, they are determined by the biography of the lattice, i.e., they depend on the way in which the specimen was prepared and on the influences to which it was subjected during all its preceding life.

Secondly, they depend on temperature. Indeed, upon heating, the lattice ions dissociate from their sites, forming interstitial ions and empty sites. Temperature thus causes additional disorder in the lattice. As the temperature rises, the degree of deviation of the real lattice from the ideal one increases (but the degree of deviation from the stoichiometric ratio remains unchanged).

Thus, by its origin, disorder in the lattice may be innate (irreversible) or thermal (reversible).

In some cases the thermal disorder may be neglected in comparison with the innate disorder; in other words, defects caused by temperature may be neglected in comparison with defects stored in the lattice from the very beginning and present in it, so to speak, in a frozen state. This occurs in sufficiently spoiled lattices at not too high temperatures. It is with this case that we deal in the theory of semiconductors. In this case the number of defects in the lattice may be regarded as practically independent of temperature. By defects we mean here either empty sites or interstitial ions, depending on the nature of the lattice and on the treatment to which it has been subjected. By calcining a specimen in vacuum or in one atmosphere or another, at one pressure or another, we can change the nature of the defects and their concentration in the lattice.

In other cases, on the contrary, thermal disorder predominates over innate disorder. In this case the number of frozen-in defects may be neglected in comparison with the number of thermal defects. This occurs, apparently, in the case of alkali-halide crystals, as is shown by the well-known experimental fact: the exponential increase in the number of color centers with temperature. In this case the lattice contains both interstitial ions and empty sites. The stoichiometric ratio in this case remains practically undisturbed.

We have seen that the ideal lattice is transformed into the real one as a result of the removal of ions from sites or the introduction of ions into interstices. Let us note that this operation must necessarily be accompanied by the introduction of electrons into the lattice or their removal from it. This second process must accompany the first in order that the electric charge arising as a result of the introduction or removal of ions,

would be compensated, i.e. in order that the electrical neutrality of the lattice be preserved. Instead of speaking of the removal of an electron, we may speak of the introduction of a “hole,” and conversely. The presence of a hole means the absence of an electron. This mode of expression proves to be very convenient.

The creation in the lattice of an electropositive defect requires the introduction of a certain number of electrons. The creation of an electronegative defect must be accompanied by the introduction of a certain number of holes. These electrons and holes, present in any real lattice for the maintenance of electrical balance, are not, generally speaking, obliged to recombine with one another. In some lattices such recombination is energetically favorable, in others it is unfavorable, i.e. it leads to an increase in the total energy of the system. These electrons and holes play a very important role. They constitute the reserve by means of which, under appropriate conditions, the electrical conductivity of the lattice arises. They also determine the optical properties of the lattice.

The total supply of electrons and holes depends on how strongly our lattice differs from the ideal one. The predominance of electrons or of holes in this total supply is expressed the more sharply, the more strongly the stoichiometric ratio is disturbed. In lattices with a stoichiometric excess of metal we have an excess of electrons; in lattices with a stoichiometric excess of metalloid we have a deficiency of electrons, i.e. an excess of holes.

The compensating electrons and holes, introduced into the lattice and distributed over its ions, lead to the formation of anomalous ions, i.e. ions with an anomalous charge. These may be, for example, metallic ions with charge \(+p-1\), or metalloid ions with charge \(-q+1\). Thus a heteropolar lattice possessing mechanical defects must, generally speaking, also contain electrical defects which ensure its electrical neutrality.

Anomalous ions with additional charges of the same sign repel one another, while anomalous ions with additional charges of unlike sign attract one another. Here it would be more correct to say that it is not anomalous ions that repel or attract, but anomalous states of ions. In addition, anomalous ions with a negative additional charge are attracted to electropositive defects, and ions with a positive additional charge—to electronegative defects. Thus electropositive defects serve as traps for electrons, while electronegative defects play the role of traps for holes.

In the normal state all electrons and holes are localized near the corresponding defects. Thus the entire supply of electrons and holes contained in the lattice is conserved. The action—

acting on the lattice by one or another external agent (heating, illumination, etc.), one can remove electrons and holes from the region of action of the defects. These liberated electrons and holes provide the electronic and hole conductivity of the lattice.

§ 3. SOMMERFELD’S THEORY OF FREE ELECTRONS

A crystal lattice is a system of interacting electrons moving in the field of positive nuclei, which, to a first approximation, may be regarded as fixed in certain rigidly fixed positions of equilibrium. The problem consists in determining the behavior of such a system. This problem cannot be solved exactly. Usually, in solving it, one makes use of various simplifications.

The first simplification that we shall make consists in replacing the interaction of the electrons by a certain effective force field, in which each of the electrons will be regarded as moving independently of the others (the method of the “self-consistent” field). In this way our

Fig. 3. Potential energy of an electron in the lattice.

many-electron problem is reduced to a one-electron problem. To each electron there can be assigned its own individual state and its own individual energy. The field in which the electron moves has a periodic character. It is schematically shown in Fig. 3, a. The behavior of an electron in such a periodic field is considered in Bloch’s theory².

The second simplification consists in replacing the periodic potential by a certain mean value of it, constant throughout the whole volume of the crystal. In other words, the potential curve shown in Fig. 3, a, is replaced by the approximating curve shown in Fig. 3, b. Sommerfeld’s theory³ proceeds from such an approximation.

We shall first consider our problem within the framework of Sommerfeld’s theory, and then see how our results change when the periodic potential is taken into account.

In Sommerfeld’s theory we are dealing, in essence, with the same model of the crystal from which the classical theory of metals developed by Drude\(^4\) proceeded. The electrons in the crystal are treated as an electron gas confined in a potential box with a flat bottom.

However, Sommerfeld’s theory differs from Drude’s theory in the following two essential features.

The first distinguishing feature consists in the fact that in Sommerfeld’s theory the behavior of each individual electron obeys the laws of quantum mechanics, whereas in Drude’s theory it was described by the methods of classical mechanics.

This reform has as its consequence the fact that the possible values of the momentum for an electron now form not a continuous, but a discrete series. An electron in a crystal can have only such a momentum \(\mathfrak p\), whose components \(p_1, p_2, p_3\) satisfy the conditions

\[ p_i=\frac{h}{2L}n_i, \tag{1} \]

where \(i=1,2,3\). Here \(L\) is the length of the crystal (we imagine the crystal in the form of a cube), and \(n_i\) is any integer:

\[ n_i=0,\pm 1,\pm 2,\ldots \]

The momentum space shown in Fig. 4 is divided into cells of volume \(\Delta\gamma=h^3/8L^3\). The electron can be located only at those points of this space which correspond to the nodes of the “lattice” depicted in Fig. 4. At the same time the energy spectrum also becomes discrete. For the energy \(W\) of the electron we have, according to (1):

\[ W=\frac{\mathbf p^2}{2m}=\frac{h^2}{8mL^2}(n_1^2+n_2^2+n_3^2), \tag{2} \]

where \(m\) is the mass of the electron. The integer variables \(n_1, n_2, n_3\) play the role of quantum numbers. The dependence of \(p_i\) and \(W\) on \(n_i\) is shown in Fig. 5, \(a\) and 5, \(b\). On the vertical axis of Fig. 5, \(b\) is plotted the system of energy levels of the electron.

Fig. 4. Momentum space.

The second feature distinguishing Sommerfeld’s theory from Drude’s theory consists in the fact that, in describing the behavior of the entire system

electrons as a whole, the classical statistics of Maxwell–Boltzmann is replaced by the quantum statistics of Fermi–Dirac.

Fermi–Dirac statistics, in contrast to Maxwell–Boltzmann statistics, deals with identical particles, i.e., with particles that are in principle indistinguishable. This means that different states of a system which differ by a permutation of particles are regarded as one and the same state.

Moreover, the particles with which Fermi–Dirac statistics is concerned obey the Pauli principle. In the case of noninteracting particles, the Pauli principle reduces to the requirement that all particles of our system be in different individual states. This means that at one and the same point of momentum space, shown in Fig. 4, there cannot simultaneously be more than one electron (if spin is ignored), just as several electrons cannot be located at one and the same point of ordinary coordinate space.

Fig. 5. Possible values of momentum and energy for an electron in a lattice (Sommerfeld theory).

Taking both of these circumstances into account (the identity of particles and the Pauli principle) is reflected in the final formulas. Let us note, however, that all formulas of quantum statistics practically pass over into the classical formulas in the following limiting cases:

a) when the particles have a sufficiently small mass;

b) when the temperature is sufficiently high;

c) when the concentration of particles is sufficiently small.

In the case of an electron gas in a metal, the transition from Fermi statistics to Maxwell statistics occurs at very high temperatures, lying considerably above those temperatures with which the experimenter has to deal. In the case of an electron gas in a semiconductor, owing to the low density of this gas, such a transition takes place, on the contrary, at very low temperatures, lying below the temperatures of practical interest. Therefore the electron gas in a metal must be described by Fermi formulas, whereas the electron gas in a semiconductor can be described by Maxwell formulas.

In Fig. 6 the total energy \(W\) of an electron gas is shown as a function of the temperature \(T\). The curve \(AB\) corresponds to quantum statistics, and the straight line \(OB\) to classical statistics. The slope of the curve \(W = W(T)\) characterizes the heat capacity \(C\) of the electron gas.

Indeed, by definition:

\[ C=\frac{dW}{dT}. \]

The point \(T_0\) in Fig. 6 divides the entire temperature scale into two regions:

1) for \(T \gg T_0\), quantum statistics practically coincides with classical statistics (nondegenerate gas);

2) for \(T \ll T_0\), quantum statistics differs substantially from classical statistics (degenerate gas).

Considering Fig. 6, we can draw two conclusions that are characteristic consequences of applying Fermi statistics to the electron gas:

a) in the region of sufficiently low temperatures (in the region of degeneracy), the heat capacity of the electron gas is practically equal to zero;

b) at absolute zero temperature, the energy of the electron gas is different from zero.

At absolute zero, the distribution of electrons over the cells of phase space (Fig. 4) corresponds to the densest packing. The electrons fill the sphere shown in Fig. 4, whose radius \(p_0\) is determined from the condition

\[ \frac{\frac{4}{3}\pi p_0^3}{h^3}=\frac{N}{2}, \]

where \(N\) is the total number of electrons (per unit volume of the crystal); the factor two on the right-hand side of the equality appears as a result of taking spin into account. Upon heating, the dense packing of the electrons becomes loosened: within the sphere of radius \(p_0\), free places (holes) appear owing to electrons thrown beyond the limits of this sphere. At the same time, however, the symmetry in the distribution of electrons with respect to momenta, characteristic of absolute zero, is preserved. This means that there is no current. The appearance of a current (when an electric field is applied) indicates a breaking of symmetry. Thus, temperature and the electric field are factors causing a redistribution of electrons in phase space.

Fig. 6. Total energy of the electron gas as a function of temperature.

\(OB\)—Maxwell–Boltzmann statistics, \(AB\)—Fermi–Dirac statistics.

§ 4. BAND THEORY

Sommerfeld’s theory has a number of shortcomings. In particular, while remaining within Sommerfeld’s theory, it is impossible to construct an optics of metals. Indeed, the Sommerfeld free electron gas cannot absorb light; its interaction with light waves сафಲ

quanta reduces to the Compton effect. Moreover, from the point of view of Sommerfeld’s theory it is impossible to explain the differences between metallic and nonmetallic crystals.

These shortcomings are due to the crude approximations that underlie the Sommerfeld model. They disappear upon passing to a theory that takes account of the periodic potential of the lattice.

Taking account of the periodic potential is reflected above all in the structure of the energy spectrum. The energy levels shown in Fig. 5, b contract into separate groups, thus forming the so-called energy bands, separated by forbidden regions of greater or lesser width.

The structure of the bands and their position in the spectrum can be clarified if the problem is solved by starting from isolated atoms, as Bloch^2 does, and considering the lattice as the result of bringing together such initially isolated atoms. In Bloch’s method the internal electrons of the atom, together with the nucleus, are regarded as a positively charged atomic core, in whose field the valence electrons move, in the first approximation not interacting with one another. The behavior of the electron inside an individual atom is considered known. The problem is to determine what happens to the electron and to its system of energy levels under the influence of the remaining atoms of the lattice. The action of these latter is treated as a weak perturbation. Strictly speaking, this method is acceptable only in the case of lattices built of monovalent atoms.

Fig. 7. Energy spectrum of a lattice.

In Bloch’s theory each band is the result of the splitting of the corresponding term of an isolated atom. This splitting is caused by the interaction of each given atom with the other atoms of the lattice. Indeed, every atom of the lattice is in the perturbing field of its neighbors. Each term gives rise to its own band, as shown in Fig. 7. Thus, the energy spectrum of the lattice reflects the individuality of the atoms of which the lattice is built.

Let us consider the \(s\)-term of an isolated atom. This term differs from the others in that it is nondegenerate. When atoms combine into a lattice, the \(s\)-term becomes an \(s\)-band. In the case of a cubic lattice the structure of this band, as Bloch showed, is determined

by the formula

\[ W = W_0^s + \alpha^s + 2\beta^s\left(\cos \frac{2\pi a}{L} n_1 + \cos \frac{2\pi a}{L} n_2 + \cos \frac{2\pi a}{L} n_3\right), \tag{3} \]

where \(a\) is the lattice constant; \(n_1, n_2, n_3\) are quantum numbers. These numbers may take the following values:

\[ n_i = 0, \pm 1, \pm 2, \ldots, \pm \frac{L}{2a}. \tag{4} \]

We obtain a whole system of levels corresponding to all possible values of the quantum numbers \(n_1, n_2, n_3\). The totality of these levels forms a band. The triple of quantum numbers \(n_1, n_2, n_3\) may be interpreted as the number of a level within the band. All the levels lie between the upper and lower boundaries of the band.

\[ \begin{aligned} \text{Upper boundary of the band:}\quad & W_{\max} = W_0^s + \alpha^s + 6\beta^s,\\ \text{Middle of the band:}\quad & W_0 = W_0^s + \alpha^s,\\ \text{Lower boundary of the band:}\quad & W_{\min} = W_0^s + \alpha^s - 6\beta^s,\\ \text{Width of the band:}\quad & \Delta W = W_{\max} - W_{\min} = 12\beta^s. \end{aligned} \]

We see that the parameter \(\alpha^s\) in (3) characterizes the shift of the middle of the band relative to the original term \(W_0^s\). The parameter \(\beta^s\) characterizes the width of the band. The value of this parameter depends on how strongly the wave functions of two neighboring atoms overlap. The higher the original term \(W_0^s\) lies and the more closely the atoms are arranged in the lattice, the stronger this overlap, the larger \(\beta^s\), and the wider the band.

To each triple of quantum numbers \(n_1, n_2, n_3\) there corresponds a definite state of the electrons, i.e. a definite wave function describing the behavior of the electron. These wave functions have the following general property. The modulus of each wave function is a periodic function of the coordinates with a period equal to the period of the lattice. This means that the electron can, with equal probability, be found near any atom of the lattice. In other words, all atoms of the lattice participate in the game on equal terms. On the other hand, this means that our electron is completely collectivized. It belongs to all atoms of the lattice to the same degree.

It follows from (4) that the band contains a finite number of states, equal to the number of atoms in the lattice. Since each state can be occupied by only two electrons with opposite spins, the band therefore has a definite capacity with respect to electrons. It can accommodate only a limited number of electrons.

To an electron in the state \(n_1, n_2, n_3\), one may assign a momentum \(\mathbf{p}\,(n_1,n_2,n_3)\). This is the mean value of the momentum corresponding to the group velocity of the wave packet. The momentum \(\mathbf{p}\),

as a function of the quantum numbers \(n_1,\ n_2,\ n_3\), can be calculated if the energy \(W\) is known as a function of \(n_1,\ n_2,\ n_3\). The relation between \(\mathbf p\) and \(W\) is given by the Jones–Zener formula

\[ \mathbf p=\frac{Lm}{h}\operatorname{grad}_{\mathbf n} W . \tag{5} \]

The subscript \(\mathbf n\) on the gradient means that here the gradient is meant not in ordinary coordinate space \(x,\ y,\ z\), but in the space of the quantum numbers \(n_1,\ n_2,\ n_3\), so that the components of the vector \(\mathbf p\) have the form:

\[ p_i=\frac{Lm}{h}\,\partial W/\partial n_i, \]

where \(i=1,\ 2,\ 3\). Substituting (3) here, we shall have:

\[ p_i=\frac{4\pi a m}{h}\,\beta^s \sin \frac{2\pi a}{L}\,n_i . \tag{6} \]

Formulas (6) and (3) replace the corresponding formulas (1) and (2) of the Sommerfeld theory. The dependence of \(p_i\) and \(W\) on \(n_i\) is shown in Fig. 8 (cf. Fig. 5). Comparing (6) and (3), we see that to one and the same energy \(W\) there correspond two states with oppositely directed momenta. This follows from the fact that the energy \(W\) is an even function of \(n_1,\ n_2,\ n_3\), whereas the momenta \(p_i\) are odd functions of these parameters. Thus, on one and the same energy level there may be placed two electrons moving in opposite directions.

Fig. 8. Possible values of the momentum and energy for an electron in a lattice (Bloch theory).

Let us note that near the upper boundary of the band and near its lower boundary the cosines in (3) may be expanded in a series and one may confine oneself to the first terms of the expansion. Then expression (3) takes the following form:

\[ W=W_0^s+\alpha^s\pm 6\beta^s\pm \frac{4\pi^2 a^2}{L^2}\,\beta^s\left(n_1^2+n_2^2+n_3^2\right), \]

where the upper sign corresponds to the expansion near the upper boundary, and the lower sign to the expansion near the lower boundary of the band. Comparing this formula with formula (2) for free electrons, we note that an electron situated at the bottom of the band or raised to the ceiling of the band may be treated as a free electron of the Sommerfeld theory, with the only difference that now it must be assigned an ef-

which effective mass \(m^*\) instead of the true mass \(m\)

\[ m^*=\mp \frac{h^2}{32\pi^2 a^2 \beta^s}. \]

The absolute value of this mass is smaller the wider the band; moreover, this mass is positive at the bottom of the band and negative under the ceiling of the band. This means that under the ceiling of the band the electron behaves in a very peculiar manner: it is accelerated in the direction opposite to the force acting on it.

Let us now consider the bands corresponding to degenerate terms. As an example of such a term, let us consider a \(p\)-term. This term is triply degenerate. It may be regarded as a set of three terms coinciding with one another. A \(p\)-term gives rise to a \(p\)-band, which may be represented as consisting of three bands superposed on one another. In the case of a cubic lattice, the structure of these three bands, as Wilson\({}^{6}\) showed, is determined by the formulae:

\[ \left\{ \begin{aligned} W^1&=W(n_1,n_2,n_3),\\ W^2&=W(n_2,n_1,n_3),\\ W^3&=W(n_3,n_2,n_1), \end{aligned} \right. \tag{7} \]

where

\[ W(n_1,n_2,n_3)= \]

\[ = W_0^p+\alpha^p+2\beta^p \cos \frac{2\pi a}{L}n_1 +2\gamma^p\left(\cos \frac{2\pi a}{L}n_2+\cos \frac{2\pi a}{L}n_3\right), \]

The quantum numbers \(n_1,n_2,n_3\) still take the values (4). We see that all three bands (7) have common boundaries:

\[ W^1_{\max}=W^2_{\max}=W^3_{\max} = W_0^p+\alpha^p+2\beta^p+4\gamma^p, \]

\[ W^1_{\min}=W^2_{\min}=W^3_{\min} = W_0^p+\alpha^p-2\beta^p-4\gamma^p. \]

The bands produced by higher terms (\(d\)-term, \(f\)-term, etc.) have a more complex structure\({}^{7}\). The number of states contained within such a complex band is always equal to the number of atoms in the lattice multiplied by the degeneracy of the corresponding term. The moduli of the wave functions always remain periodic. For any band, the general Jones–Zener formula (5), expressing the relation between the energy \(W\) and the momentum \(\mathbf{p}\) of the electron, remains valid.

The individual bands in the energy spectrum of the lattice are, generally speaking, separated by forbidden regions. If, however, two neighboring terms in the energy spectrum of the isolated atom are situated sufficiently close to one another, and the bands generated by them are sufficiently wide, then the forbidden region between the bands may prove sufficiently narrow. In particular, it reduces to zero when two neighboring bands touch or overlap one another.

In this case neighboring zones cannot be considered separately. They, as it were, interact with one another, and their structure is distorted.

Until now we have regarded our problem as a one-electron one. If the interaction of the electrons is taken into account rigorously, our many-electron problem can no longer be reduced to a one-electron problem. In this case assigning to an individual electron its own system of energy levels loses its meaning. One can now speak only of the energy spectrum of the entire system of electrons as a whole.

An exact solution of the many-electron problem proves impossible. In the theory of metals the interaction of the electrons is usually taken into account approximately, without going beyond the framework of the one-electron problem. This is achieved, as we have already noted, by adding to the periodic potential of the lattice a certain additional so-called self-consistent potential^8. The self-consistent potential is the energy of interaction of the given electron with the smeared-out charge of all the electrons of our system, including its own smeared-out charge. Thus the interaction is replaced by the introduction of a certain effective field in which the electrons move independently of one another. The self-consistent potential is distinguished by an important property: it has the same periodicity as the lattice potential.

The method of the self-consistent potential is a very imperfect method. We can expect satisfactory results from it only in the case of a homeopolar lattice built of monovalent atoms. On passing to a heteropolar lattice, the applicability of this method becomes doubtful.

From the point of view of Bloch’s theory, a heteropolar lattice should be regarded as consisting of a system of positive ions and a family of collectivized electrons composed of the valence electrons of both the metallic and the metalloid atoms. The electron gas with which we are dealing here has a high density: several electrons fall to each crystal cell. Each electron moves in the field of the metallic and metalloid positive ions and in the self-consistent field of the electrons.

It is possible, however, to take another approach to the problem^1, one that is more satisfactory from the standpoint of accounting for the interaction of the electrons. A heteropolar lattice may be pictured as built of positive metallic and negative metalloid ions. In this case the family of collectivized electrons moving in the field of these ions consists of random electrons introduced into the lattice from outside or torn away from individual ions. In this case the density of the electron gas is negligibly small in comparison with the density of the electron gas in Bloch’s method. Therefore the interaction of the electrons can quite legitimately be entirely neglected. For the same reason

one may neglect the distorting influence of anomalous ions present in the lattice. Indeed, the number of such anomalous ions that have given up their electrons for common use cannot exceed the number of collectivized electrons. The field of the lattice, in which the collectivized electrons move, retains the property of periodicity. Therefore, in our approach to the problem the energy spectrum of the lattice retains the band structure characteristic of Bloch theory.

§ 5. CONDUCTORS AND INSULATORS FROM THE POINT OF VIEW OF BAND THEORY

In order for a lattice to possess conductivity, two conditions must be satisfied:

a) the lattice must contain free electrons, i.e. electrons that can move without hindrance throughout the entire volume of the crystal;

b) the external electric field must be capable, to a certain degree, of regulating the disordered motion of the electrons, producing the necessary asymmetry in the distribution of the electrons with respect to velocities.

In classical theory the first of these two conditions served as the criterion for whether we are dealing with a conductor or an insulator. In quantum theory the second condition serves as such a criterion.

The difference between free and bound electrons, in the classical understanding of these terms, can be explained with the aid of Fig. 3, a. This figure shows a periodic field in which an electron moves: potential wells alternate with potential barriers. The total energy \(W\) of the electron is represented in Fig. 3, a by a horizontal level. This level may lie above the tops of the potential barriers (level \(AA'\)) or below them (level \(BB'\)). In the first case we are dealing with a free electron. Such an electron passes without hindrance from atom to atom over the barriers separating them. In the second case we are dealing with a bound electron. From the classical point of view such an electron is compelled always to remain within its own atom; indeed, in order to pass to a neighboring atom, the electron must pass through a forbidden region (shaded in Fig. 3,a), within which its total energy is less than the potential energy, i.e. its kinetic energy is negative. The atoms prove to be separated by walls that are completely impenetrable for a classical electron.

In quantum theory the sharp distinction between free and bound electrons, in the classical understanding of these terms, is erased, since quantum mechanics allows an electron to pass

from atom to atom not only over the barrier, but also through the barrier. If in the classical theory the probability of such a transition could be equal to unity (a free electron) or to zero (a bound electron), now this probability can take all intermediate values. It is the greater, the smaller the area shaded in Fig. 3, a. Thus, from the point of view of quantum theory, all electrons of a crystal are “free” in the sense that they all can, with one or another degree of ease, move through the crystal.

The difference between a conductor and an insulator from the quantum point of view is due to the band structure of the energy spectrum of the lattice.

If the number of collectivized electrons in our lattice is such that, when they are placed on the levels of a band (two electrons in each state), they fill the band completely and if, moreover, our filled band is separated by a forbidden interval from the empty band lying above (see Fig. 9, a), then in this case the lattice constitutes an insulator. Indeed, a dense packing of electrons over the levels of a band corresponds to a symmetric distribution in velocities. An electron with momentum $p$ can always be matched by another electron with momentum $-p$. Redistribution of electrons by momenta within the given band proves impossible. The only thing that can be done is to interchange electrons with one another, but in this way the desired asymmetry in the distribution by momenta cannot be obtained. The external field turns out to be incapable of affecting in any way the motion of the electrons located in the given band and, consequently, the lattice proves insensitive to the action of the external field.

Labels in the figure: Insulator; Semiconductor; Conductor; $W$; a); b); c); d); Lower band filled completely; Band filled partially; Free levels; Occupied levels.

Fig. 9. Difference between a conductor and an insulator (from the point of view of band theory).

Let us now suppose that the total number of collectivized electrons is less than twice the number of states within the band. In this case the electrons do not fill the band completely. With maximal

in close packing, only the lower levels of the band turn out to be occupied. Adjacent to the occupied levels there are empty levels (see Fig. 9, g). By transferring electrons from some levels to others within the given band, one can obtain an asymmetric distribution in momenta, indicating that the number of electrons moving in one direction exceeds the number of electrons moving in the opposite direction. The role of the external field is reduced precisely to such a transfer of electrons, which violates the symmetry in the momentum distribution. In this case our lattice is a conductor.

Let us imagine, finally, that the band is filled with electrons to capacity, but at the same time touches or overlaps with a higher-lying empty band (see Fig. 9, v). In this case the lattice will again be a conductor. Indeed, the upper empty band is in this case a continuation of the lower filled band. Redistribution of the electrons in momentum can be carried out at the expense of the levels of this upper band. Such two touching or overlapping bands may be regarded as one band only partially filled with electrons.

Let us consider the lattice of an insulator. As a result of external influences, a certain conductivity can be artificially created in such a lattice.

From the point of view of classical theory, this can be accomplished as a result of transferring bound electrons into the free state. For this it is necessary to raise an electron from the levels \(BB'\) in Fig. 3, a to the levels \(AA'\), located above the potential barriers. This process, as is seen from Fig. 3, a, requires an expenditure of energy.

From the point of view of quantum theory, this can be accomplished by transferring electrons from the lower band into the upper band. This process is again associated with an expenditure of energy, as is seen in Fig. 9, a.

The lower filled band of an insulator we shall call the normal band. The upper empty band we shall call the conduction band. Electrons that have entered the conduction band fully acquire the ability to carry current. The holes formed in the normal band as a result of removing electrons from it also serve as current carriers. In the lattice of an insulator, in which as a result of external influences some of the electrons have been transferred from the normal band to the conduction band, the current is composed of an electronic and a hole component.

The external factors that produce conductivity in an insulator we shall call ionizing agents.

An example of such an agent may be light. In this case the transfer of an electron into the conduction band occurs at the expense of the energy of a light quantum.

Another example may be temperature. Upon heating, the maximally dense packing of electrons within the normal band,

corresponding to absolute zero temperature, begins to become diffuse: electrons are ejected from it into the conduction band. The number of electrons raised into the conduction band is the greater, the higher the temperature, and, at a given temperature, the narrower the forbidden region between the bands. Crystals with a sufficiently wide forbidden region (Fig. 9,a) remain ideal insulators even at the highest experimentally attainable temperatures. Crystals with a sufficiently narrow forbidden region (Fig. 9,b) exhibit appreciable conductivity already at room temperatures. Such crystals are usually called semiconductors.

We see that, from the standpoint of band theory, the distinction between insulators, semiconductors, and conductors (metals) has a purely quantitative origin. An insulator may be regarded as one limiting case of a semiconductor, while a metal represents the other limiting case.

§ 6. ELECTRON LEVELS IN A REAL LATTICE

The transition from an ideal lattice to a real one has its reflection in the structure of the energy spectrum.

Upon the system of energy bands characteristic of an ideal lattice there is superimposed a system of discrete, so-called local levels. The origin of the energy bands is due to the periodic structure of the lattice. The origin of the local levels is connected with local violations of this periodicity. Every defect in the lattice, being such a violation, gives rise to the appearance of a discrete level. The arrangement of the energy bands reflects the nature of those atoms of which the lattice is built. The arrangement of the local levels in the spectrum reflects the nature of the defects contained in the lattice.

The moduli of the wave functions corresponding to the levels of a band are periodic functions of the coordinates with the period of the lattice. This means that an electron residing within a band is spread out over the entire volume of the crystal, or, in other words, wanders freely through the crystal. Local levels, on the contrary, are characterized by wave functions having sharply pronounced maxima near the sites of damage and falling off rapidly with distance from the damage. This means that an electron that has fallen onto such a level proves to be localized in a more or less narrow region of space.

Local levels may be donors or acceptors. Donors are those local levels which in the normal state are occupied by electrons. Acceptors are free local levels. Local levels, falling into the forbidden region between the normal band and the conduction band of an insulator, play an important role in electronic processes. Donors serve as suppliers

electrons. From these levels conduction electrons can be drawn. Acceptors play the role of traps for electrons: they capture and bind electrons.

Electronic transitions between energy bands determine the stable properties of the crystal. Transitions in which local levels take part determine structure-sensitive properties. The energy spectrum of the crystal lattice is the key to understanding all its principal properties. From an analysis of the energy spectrum there follow, as consequences, all the electrical, photoelectric, and optical properties of the lattice.

Let us consider various types of real lattices from the point of view of their energy spectrum.

The simplest example of a real lattice may be a lattice bounded on one side. Such a lattice was considered by Tamm^9. The presence of a bounding surface is already in itself a violation of the periodic structure of the lattice. Indeed, an ideal lattice, strictly speaking, must be infinite. The breaking off of the lattice leads to the appearance of so-called surface levels. The corresponding wave functions have a maximum at the surface of the crystal and decrease rapidly as one moves away from the surface into the depth of the crystal. This means that electrons are capable of adhering to the surface of the crystal. Let us note that any macroscopic defects in the lattice are characterized by Tamm surface levels. Indeed, all kinds of cracks, pores, and macroscopic inclusions in the crystal testify to a breaking off of the lattice.

As another example of a real lattice, let us consider a homopolar lattice with empty sites. We shall assume that the perturbing potential of the defect falls off rapidly with distance, so that the presence of the defect is felt in practice only by the atoms immediately neighboring the defect. In this case the local level is degenerate, the order of degeneracy being equal to the number of atoms directly surrounding the defect. The position of the local levels in the spectrum may be different, depending on the nature of the lattice. If the local levels fall into the forbidden region between bands, then the corresponding wave functions have maxima the more sharply expressed, the greater the distance between the local level and the band. As this distance decreases, the wave functions become more and more diffuse. When this distance becomes zero, the local level as such disappears. However, the number of states within the band increases correspondingly. In this case all wave functions (more precisely, the moduli of the wave functions) have a periodic character.

As a third example of a real lattice, let us consider a homopolar lattice with foreign atoms embedded in it, located in the interstices. In this case, as calculation shows,

in the energy spectrum, alongside the local levels due to the very fact of the violation of periodicity, there also appear, independent of them, the proper levels of the impurity atoms. Their position in the spectrum is determined both by the nature of the impurity atoms and by the nature of the principal atoms of the lattice.

Finally, let us consider a heteropolar lattice in which the defects are either vacant sites of two kinds (metallic and metalloid), or interstitial ions likewise of two kinds (metallic and metalloid). A special case of such lattices is constituted by lattices with a disturbed stoichiometric ratio. Let us see what the energy spectrum of such lattices looks like. For simplicity, suppose that we are dealing with lattices built of singly charged ions (lattices of alkali-halide crystals). In addition, we shall assume that the concentration of defects practically does not change with temperature.

In the energy spectrum of the lattice we shall have two systems of local levels corresponding to the two kinds of defects present in the lattice: levels due to electropositive defects, and levels due to electronegative defects. These levels are situated in the forbidden region between the normal band and the conduction band. Let, per unit volume of the crystal, we have:

\[ \begin{cases} N_1 \text{— the number of electropositive levels (defects),}\\ N_2 \text{— the number of electronegative levels (defects),} \end{cases} \]

\[ \begin{cases} \text{for a stoichiometric excess of metal } N_1 > N_2,\\ \text{for the stoichiometric ratio being observed } N_1 = N_2,\\ \text{for a stoichiometric excess of metalloid } N_1 < N_2. \end{cases} \]

To ensure electrical neutrality we must introduce into our lattice \(N_1\) electrons and \(N_2\) holes. In the normal state these compensating electrons are localized near the electropositive defects, while the compensating holes are associated with the electronegative defects. Thus the electropositive defects are donors, and the electronegative defects are acceptors.

Let us imagine that the electropositive levels \(W_1\) are situated in the energy spectrum below the electronegative levels \(W_2\), as is shown in Fig. 10, \(a\). In the language of our energy scheme, recombination of an electron with a hole means the transition of an electron from the level \(W_1\) to the level \(W_2\). In Fig. 10, \(a\) such a transition is indicated by an arrow. This transition, as we see, is associated with an expenditure of energy and, consequently, is energetically unfavorable.

Suppose now that the electropositive levels \(W_1\) are situated above the electronegative levels \(W_2\). This case is shown in Fig. 10, \(b\). Recombination of an electron with a hole, as in the preceding case, is expressed by the transition of an electron from the level \(W_1\) to the level \(W_2\). However, now such transitions (shown by arrows—

... in Fig. 10, б) are energetically favorable. In a lattice left to itself and isolated from the action of any exciting agents, the electrons will leave the levels \(W_1\) and pass to the levels \(W_2\), as a result of which the energy diagrams shown in Fig. 10, б must be replaced by the energy diagrams shown in Fig. 10, в. When the stoichiometric ratio is satisfied, all electropositive levels \(W_1\) become acceptors, and all electronegative levels \(W_2\) become donors. With a stoichiometric excess of metal, on the levels \(W_1\) there remain only those electrons for which there was not enough room on the levels \(W_2\). With a stoichiometric excess of metalloid, among the levels \(W_2\) only those remain free for the filling of which there were not enough electrons.

Labels in the figure:
а), б), в).
“Lattice with a stoichiometric excess of metal”; \(N_1>N_2\).
“Lattice with a maintained stoichiometric ratio”; \(N_1=N_2\).
“Lattice with a stoichiometric excess of metalloid”; \(N_1<N_2\).
\(W_1\) — levels of electropositive defects;
\(W_2\) — levels of electronegative defects;
\(-\) — acceptors;
\(\oplus\) — donors;
hatched region — conduction band;
cross-hatched region — normal band.

Fig. 10. Diagram of electron levels (heteropolar lattice with mechanical defects of two kinds).

Fig. 10, а and Fig. 10, в correspond to two different types of lattices. In lattices of the first type (Fig. 10, а), electropositive and electronegative defects in the normal state are neutralized by compensating electrons and holes. In such lattices ionic conductivity is absent. Indeed, in such lattices the defects can move only together with the anomalous ions attached to them, i.e. only as electrically neutral formations. In lattices of the second type (Fig. 10, в), neutralized...

only that part of the defects which determines the deviation from the stoichiometric ratio. Ionic conductivity of the lattice is provided by electropositive and electronegative defects deprived of neutralizing shells.

The energy schemes shown in Fig. 10 and constructed here as applied to alkali-halide crystals can be extended to the general case of heteropolar lattices represented by the formula \(M_m R_r\), i.e., to the case of semiconductors. Semiconductors are described by the same energy scheme as alkali-halide crystals. The difference between them is reduced only to the scale adopted in this scheme. If the energy spectrum of an alkali-halide crystal is compressed, bringing all the levels closer together, then we obtain the spectrum of a semiconductor.

In conclusion, let us note one more type of local levels, caused not by mechanical but by electrical defects in the lattice. Let us imagine a heteropolar lattice devoid of mechanical defects. Suppose that such a lattice contains a certain number of free electrons and holes. The presence of such electrons and holes testifies to the existence in the lattice of ions with anomalous charge. Around such anomalous ions the lattice naturally proves to be deformed. The displacement of the anomalous state from one ion to another must be accompanied by the corresponding displacement of the entire region of deformation. Thus, a free electron (hole) in the course of its motion creates for itself traps in which it may become stuck, since the deformation of the lattice arising around a free electron (hole) is a factor localizing it. Local levels of this origin were considered by Frenkel\(^ {10}\), and also by Landau\(^ {11}\). In real heteropolar lattices possessing mechanical defects (vacant sites, interstitial ions), local levels of this kind apparently play a secondary role.

§ 7. ELECTRICAL CONDUCTIVITY IN NONMETALLIC CRYSTALS

The electrical conductivity of a crystalline lattice, by its nature, may be ionic and electronic. A necessary and sufficient criterion of ionic conductivity is the Faraday effect. A sufficient but not necessary criterion of electronic conductivity is the Hall effect. Depending on which particles serve as current carriers, ionic conductivity may be cationic, anionic, or mixed. Electronic conductivity may be purely electronic, purely hole, or mixed.

Strictly speaking, in electrical conductivity all components are mixed in one proportion or another. However, the ratio between the different components varies depending on the conditions. By external influences on the lattice one can increase the role of one or another of the compo-

influencing it. Usually, one or another of them predominates substantially, so that the others may be neglected. Let us note that the mobility of ions is very small in comparison with the mobility of electrons. Therefore the predominance of the electronic component over the ionic one sets in even when the number of free electrons is still very small in comparison with the number of free ions.

In semiconductors the electronic conductivity substantially predominates over the ionic one. Insulators (for example, alkali-halide crystals), on the contrary, under ordinary conditions possess ionic conductivity. This is connected with the fact that the energy necessary for transferring an electron from the bound state to the free one is, in the case of semiconductors, considerably smaller than in the case of insulators. In the region of strong fields, as the field increases, the ratio between the ionic and electronic components in an insulator changes rapidly in favor of the latter. This occurs because the ionic component practically does not depend on the field; in other words, the ionic component, up to very high fields, obeys Ohm’s law. For the electronic component, however, Ohm’s law, which holds in weak fields, is replaced, on transition to strong fields, by Poole’s law, i.e., by an exponential increase of conductivity with the field. This is explained by

Fig. 11. Electronic and ionic components of conductivity as functions of the field.
$AA'$—electronic component, $BB'$—ionic component.

a) Semiconductors  b) Insulators

the fact that, in Fig. 11, the curve $AA'$ schematically represents the electronic component of the conductivity, and the curve $BB'$ the ionic component as a function of the field. Naturally, the transition from Ohm’s law to Poole’s law in insulators occurs at considerably stronger fields than in semiconductors (the point $E_0$ in Fig. 11). Moreover, in the case of an insulator the transition to Poole’s law testifies to a change in the nature of the conductivity.

Let us examine in more detail the mechanism of ionic conductivity.

As long as the lattice remains ideal, ionic conductivity is impossible. Indeed, the only possible way for ions to move in an ideal lattice without disturbing its ideal...

ness, consists in exchanges of places between like-named ions. However, such exchanges of places cannot produce a current.

In order to make ionic conductivity possible, it is necessary first to disturb the ideality of the lattice by bringing, for example, some number of ions out of the sites into the interstices. Such disturbances are produced, as we know, by temperature. Interstitial ions are able to move within the lattice and thus serve as current carriers. On the other hand, vacant sites are also able to wander through the lattice. They, too, may be regarded as current carriers. Accordingly, we distinguish two mechanisms of ionic conductivity:

1. The Schottky mechanism^12 (the current carriers are vacant sites).

2. The Frenkel mechanism^13 (the current carriers are interstitial ions).

Cathodic conductivity.
Anodic conductivity

a) Schottky mechanism

Cathode
Anode

b) Frenkel mechanism

\[ \left\{ \begin{array}{l} \oplus\ \text{metallic ion}\\ \ominus\ \text{metalloid ion} \end{array} \right. \]

Fig. 12. Mechanisms of ionic conductivity.

These two kinds of carriers, generally speaking, have different mobilities. The number of carriers of both kinds in a real lattice need not be the same. Indeed, a real lattice, from the very beginning, i.e. at absolute zero temperature, may possess some reserve of vacant sites or interstitial ions.

Strictly speaking, when we are dealing with ionic conductivity, both mechanisms act simultaneously: the Frenkel mechanism and the Schottky mechanism. However, in some cases the leading role belongs to the Frenkel mechanism, in others, conversely, to the Schottky mechanism. These two mechanisms are schematically illustrated in Fig. 12.

Let us now turn to electronic conductivity.

First consider an ideal lattice of the type \(M_m R_r\), constructed of metallic ions with charge \(+p\) and metalloid ions with charge \(-q\). In such a lattice electronic conductivity is impossible. To make electronic conductivity possible, the lattice must first be excited. In other words, an anomalous charge must be created on some ions of the lattice. This can be accomplished at the expense of the internal resources of the lattice, by transferring electrons from some ions of the lattice to others. Thus, for example, in the lattice

NaCl, constructed from \(Na^+\) and \(Cl^-\) ions, under the action of illumination, electrons can be transferred from negative chlorine ions to positive sodium ions, as a result of which neutral sodium atoms and neutral chlorine atoms arise in the lattice, serving, respectively, as carriers of electronic and hole conductivity.

Let us note that the presence in the lattice of anomalous ions is a condition necessary, but not sufficient, for the occurrence of electronic (hole) conductivity. Indeed, if anomalous ions of opposite signs are in immediate proximity to one another, then they turn out to be bound to each other. Such a group of anomalous ions, electrically neutral as a whole, constitutes a formation similar to Frenkel’s exciton (this term was first introduced by Frenkel\(^{14}\) as applied to homopolar lattices). The exciton can move through the lattice (this means that the whole group of anomalous states, remaining bound to one another, moves as a single whole), but it cannot give a current. In order to make conductivity possible, the anomalous ions must be separated so far that the interaction between them can be neglected, i.e., so that the behavior of each anomalous ion can be considered independently of the others.

From the point of view of band theory, the creation in the lattice of two free anomalous ions with opposite additional charges means the transfer of an electron from the normal band to the conduction band. The motion of an electron in the conduction band or of a hole in the normal band indicates the displacement of an anomalous state, which is successively transferred from one ion to a neighboring ion of the same name. If we are dealing with an anomalous ion possessing an excess negative charge, then we speak of an electronic current. If, however, the anomalous ion possesses an excess positive charge, then we speak of a hole current. In both cases the displacement of the anomalous state occurs by means of the jumping of an electron from ion to ion. Thus, the difference between electronic and hole conductivity is of a terminological rather than a physical character.

Depending on which ions (metallic or metalloid) serve as carriers of the electron or the hole, we may distinguish two mechanisms of electronic and two mechanisms of hole conductivity. In some cases it is energetically favorable for the electron to travel along metallic ions; in other cases—along metalloid ions. The same may be said of the hole. These possible mechanisms are schematically illustrated in Fig. 13. They may also be represented by the following symbolic formulas.

Electronic conductivity:

\[ \begin{aligned} 1)\quad & M^{+p-1} + M^{+p} \longrightarrow M^{+p} + M^{+p-1},\\ 2)\quad & R^{-q-1} + R^{-q} \longrightarrow R^{-q} + R^{-q-1} \end{aligned} \]

Hole conductivity:

\[ 3)\quad R^{-q+1} + R^{-q} \to R^{-q} + R^{-q+1}, \]

\[ 4)\quad M^{+p+1} + M^{+p} \to M^{+p} + M^{+p+1}. \]

Thus, for example, in the lattice \(CuO = Cu^{++} + O^{--}\) the hole current is carried by anomalous \(O^{--}\) ions (mechanism No. 3), whereas in the lattice \(Cu_2O = Cu^{+} + O^{--}\) it is carried by anomalous \(Cu^{++}\) ions (mechanism No. 4).

Fig. 13. Mechanisms of electronic conductivity.

Let us now turn to a real heteropolar lattice possessing mechanical defects (interstitial ions or empty sites). Such a lattice can from the very beginning already contain a certain number of anomalous ions. However, in the normal state these anomalous ions are localized near the sites of damage. A defect together with the anomalous ion attached to it constitutes an electrically neutral entity, whose displacement through the lattice does not create a current. Thus, defects bind anomalous ions and thereby exclude electronic (hole) conductivity. The anomalous ions, in turn, bind the defects, and thereby the possibility of ionic conductivity is excluded.

Electronic (hole) conductivity in a real lattice can be excited not only as a result of the creation of new free anomalous ions, but also as a result of the transfer of already existing anomalous ions from a bound state to a free one. This second process requires smaller energy expenditures than the first. In the language of the energy diagram of Fig. 10, such a transfer of bound

of anomalous ions into a free state means the transfer of electrons from electropositive local levels into the conduction band, or the transfer of holes from electronegative local levels into the normal band.

§ 8. THE INFLUENCE OF IMPURITIES AND TEMPERATURE ON THE ELECTRICAL CONDUCTIVITY OF SEMICONDUCTORS

In magnitude of electrical conductivity, semiconductors occupy an intermediate position between metals and insulators. The electrical conductivity of a metal is of the order of \(10^5\ \Omega^{-1}\ \mathrm{cm}^{-1}\). The electrical conductivity of insulators under ordinary conditions is of the order of \(10^{-15}\ \Omega^{-1}\ \mathrm{cm}^{-1}\). Semiconductors cover a broad intermediate region. The electrical conductivity of semiconductors ranges from \(10^2\) to \(10^{-10}\ \Omega^{-1}\ \mathrm{cm}^{-1}\). Thus, the group of semiconductors includes bodies with very diverse values of electrical conductivity.

The electrical conductivity of semiconductors depends essentially on external conditions. By acting on a semiconductor with various physical factors, one can often change its conductivity by millions of times. Such factors include:
a) impurities, b) temperature, c) a strong electric field, d) illumination.
Here we shall consider the role of impurities and temperature.

In semiconductors that are compounds of the type \(M_m R_r\), the presence of an “impurity” is usually understood to mean any deviation from the stoichiometric ratio. In this case, an impurity semiconductor is any semiconductor in which the stoichiometric ratio is violated to one degree or another, in contrast to a pure semiconductor, in which the stoichiometric ratio is observed.

In semiconductors with a stoichiometric excess of metal, the conductivity is, as a rule, electronic in nature. In semiconductors with an excess of metalloid, it is of hole character. Let us note that in some cases we are able to make a semiconductor electronic or hole-conducting at will, by producing stoichiometric deviations in one direction or the other.

For all semiconductors one and the same regularity is characteristic: the slightest stoichiometric changes cause sharp changes in the magnitude of the electrical conductivity. The electrical conductivity is the greater, the more strongly the stoichiometric ratio is violated. It is customary to consider that it is precisely the presence of “impurities” to which a semiconductor owes its electrical conductivity. “Pure” semiconductors, with which, incidentally, we never deal in reality, are from this point of view essentially insulators. This idea was first expressed by Gudden1.

A sharp dependence of electrical conductivity on impurity concentration may apparently serve as an explanation of why the measurements of different authors usually diverge so sharply from one another. Different authors, in studying one and the same material, naturally deal with specimens prepared in different ways and, consequently, differing in impurity content. The electrical conductivity of a semiconductor cannot be regarded as a constant of the material, in the way we regard it for metals and insulators.

The dependence of electrical conductivity on temperature is different for different semiconductors. The various types of this dependence are schematically shown in Fig. 14. In the same figure some characteristic representatives of each type are listed.

Fig. 14. Various types of dependence of electrical conductivity on temperature.

schematically shown in Fig. 14. Thus, over a not very wide temperature interval the electrical conductivity \(\sigma\) of a semiconductor can always be expressed by the exponential law:

\[ \sigma = A e^{-\frac{B}{T}} . \tag{8} \]

In particular, here we may have \(B=0\), which corresponds to the metallic type of conductivity. This type of conductivity (Fig. 14, a) occurs in semiconductors with sufficiently large \(\sigma\).

Fig. 15. Temperature course of electrical conductivity for specimens with different impurity content.

The coefficient \(A\) in (8) either does not depend at all on the impurity concentration, or increases with increasing concentration. In the latter case this corresponds to a displacement of the straight lines in Fig. 14 upward parallel to themselves, as shown in Fig. 15, a.

The coefficient \(B\) likewise either does not depend at all on the impurity concentration, or decreases as the concentration increases. In the latter-

ELECTRONIC PROCESSES IN REAL CRYSTALS

In this case this corresponds to a decrease in the slope of the straight lines in Fig. 14. As the concentration increases, the straight lines become flatter, as shown in Fig. 15, b.

On passing from one temperature interval to another, the values of the coefficients \(A\) and \(B\) change, which corresponds to the break in the straight lines in Fig. 14. On passing from low temperatures into the region of high temperatures, the coefficients \(A\) and \(B\) in some semiconductors increase (Fig. 14, c), whereas in others, on the contrary, they decrease (Fig. 14, d).

The various types of temperature dependence are not directly connected with the chemical nature of the semiconductor. Different specimens of one and the same semiconductor may give dependences of different types. Thus, for semiconductors not only the magnitude of the electrical conductivity, but also the temperature course of the electrical conductivity, is not a characteristic of the material, as is the case for metals and insulators.

The theory of the electrical conductivity of semiconductors proposed by Wilson\(^{16}\) is a development of Gudden’s idea, according to which the electrical conductivity of a semiconductor is determined entirely by defects of the crystal lattice. The theory applies to the case in which the lattice contains defects of only one particular kind. It follows from this that, when the stoichiometric ratio is observed, the lattice must be regarded as ideal: defects in the lattice would indicate a violation of stoichiometry.

Wilson starts from the energy scheme shown in Fig. 16. In the forbidden region between the bands we have a system of local levels which are either donors (Fig. 16, a) or acceptors (Fig. 16, b). Fig. 16, a applies to semiconductors with an excess of metal. Fig. 16, b—to semiconductors with an excess of metalloid. In the first case, temperature transfers electrons from the local levels into the conduction band. These electrons, whose number increases exponentially with temperature, determine the electronic electrical conductivity of the semiconductor. The normal band then takes no part at all in the process (this is true only at not too high temperatures). In the second case, temperature transfers electrons from the normal band to the local levels. The holes thereby arising in the normal band provide hole conductivity

Fig. 16. Level scheme for a semiconductor with defects of one kind (Wilson scheme).

a) Semiconductor with donors
b) Semiconductor with acceptors

of a semiconductor. In this case the conduction band is excluded from consideration (this is true, again, only at not too high temperatures).

Wilson’s theory leads to the following formula for the electrical conductivity as a function of the temperature \(T\) and the concentration of defects \(N\):

\[ \sigma = a\sqrt{N} e^{-\frac{u}{2kT}} . \tag{9} \]

Thus, Wilson’s theory gives the correct exponential law (8) and explains the dependence of the constant \(A\) in (8) on the concentration of defects. The parameter \(u\) in (9) has a simple physical meaning: it is the ionization energy, i.e. the energy expended in transferring an electron (hole) from a bound state into a free one (see Fig. 16, \(a\) and \(b\)).

However, Wilson’s theory has the following shortcomings:

1) it does not explain the influence of the concentration of defects on the value of the coefficient \(B\) in the exponential law (8). In Wilson’s theory the coefficient \(B = \dfrac{u}{2k}\) is determined entirely by the nature of the lattice and the nature of the defects;

2) the character of the dependence of the coefficient \(A\) on the concentration of defects, to which Wilson’s theory leads, \(A = a\sqrt{N}\), is by no means always confirmed experimentally;

3) Wilson’s theory does not explain breaks in the temperature curve of electrical conductivity;

4) semiconductors with a metallic course of electrical conductivity remain outside the framework of Wilson’s theory.

A certain generalization of Wilson’s theory is Neuburg’s theory.^17 Neuburg deals with a semiconductor whose energy spectrum simultaneously contains both donor levels and acceptor levels, situated at one and the same horizontal level, as shown in Fig. 17. The lower band is excluded from consideration. For semiconductors of this type, as Neuburg showed, the electrical-conductivity curve has the form shown in Fig. 14, \(g\). On passing from the region of high temperatures to the region of low temperatures, the slope of the curve increases twofold. Such a break in the temperature curve of electrical conductivity, with the slope changing by a factor of two, was found by Yander^18 in ZnO crystals. Neuburg’s formulas have the form:

Fig. 17. Semiconductor with donors and acceptors (Neuburg scheme).

\[ \text{at high temperatures: } \sigma = a\sqrt{N_1} e^{-\frac{u}{2kT}}, \]

\[ \text{at low temperatures: } \sigma = b \frac{N_1}{N_2} e^{-\frac{u}{kT}}, \]

where

$N_1$ — concentration of donors,
$N_2$ — concentration of acceptors,
$u$ — distance between the local levels and the conduction band.

Wilson’s theory and Nĕybur’s theory give plausible results for individual special cases, while being far from encompassing the whole body of experimental facts.

Let us note that, alongside the Wilson and Nĕybur schemes (Figs. 16 and 17), one could also consider more complicated energy schemes, characterized by one or another arrangement of donors and acceptors in the energy spectrum. The consideration of such schemes presents no fundamental difficulties. However, the choice of one scheme or another must be physically motivated.

Let us consider the schemes shown in Fig. 10. These schemes reduce to the Wilson scheme in the special case when $N_1 = 0$ or $N_2 = 0$, and to the Nĕybur scheme in the special case when $W_1 = W_2$. Semiconductors represented by the schemes of Fig. 10 differ from Wilson semiconductors by the presence in the lattice of two kinds of defects rather than one (electropositive and electronegative defects). The number of defects of one kind does not directly depend on the number of defects of the other kind. Thus the lattice is characterized not by one, but by two mutually independent parameters. In other words, the degree of violation of the stoichiometric ratio and the degree of deviation of the lattice from the ideal state are two independent characteristics of the semiconductor. From the point of view of this model, in contrast to the Wilson model, restoration of the stoichiometric ratio in a semiconductor by no means signifies a return of the lattice to the ideal state.

Starting from the energy schemes shown in Fig. 10, one can obtain a general formula for the electrical conductivity $\sigma$, as a function of temperature $T$ and the parameters $N_1$ and $N_2$ [19]. This formula has a rather complicated form. However, in separate temperature intervals, whose boundaries depend on the values of the parameters $N_1$ and $N_2$, this formula reduces to simple exponential laws of type (8). The coefficients entering into this exponential law, while remaining constant within each such interval, change their values on passing from one interval to another, thereby causing breaks in the temperature curve. By varying the values of the parameters $N_1$ and $N_2$, one can obtain a family of theoretical curves. Such a family of curves is shown schematically in Fig. 18. It encompasses all types of experimental dependences of $\sigma$ on $T$.

The transition from one rectilinear segment in Fig. 18 to another rectilinear segment is associated with a change in the mechanism of conductivity,

i.e., it is characterized by the coming into play of some levels and the exclusion of others, so that the supply of electrons to the conduction band (or holes to the normal band) in different temperature regions occurs practically from different reservoirs. The bends in the curves testify not at all to a dependence of the ionization work on temperature, but to the presence of two simultaneously acting sources of ionization (with definite ionization works not depending on temperature), whose relative role changes with temperature. The metallic character of the conductivity, observed in some cases, is obtained from the general formula as a particular case corresponding to the saturation or, more precisely, the depletion of that reservoir from which the conduction electrons (or holes) are drawn.

Fig. 18. Dependence of electrical conductivity on temperature (family of theoretical curves).

In the theoretical formula for $\sigma$, along with parameters characterizing the nature of the lattice, there enter the parameters $N_1$ and $N_2$, which depend on the biography of the specimen. This means that the electrical conductivity of a semiconductor is determined not only by the nature of the material, but also by the technology of each given specimen. Consequently, the electrical conductivity of a semiconductor can in no way be regarded as a constant of the material.

All semiconductors, therefore, can be described by means of one and the same energy model. The qualitative differences between them have a purely quantitative origin. The diversity of the experimental curves is the result of the fact that different experimenters, dealing with specimens differing from one another both in their nature and in their technology, fall on different portions of the theoretical curve.

§ 9. DEFORMATION OF ENERGY BANDS UNDER THE INFLUENCE OF AN EXTERNAL ELECTRIC FIELD

If we impose on the crystal an external electric field of sufficient strength, the behavior of an electron in the lattice becomes very peculiar.

In Fig. 19,a, the curve of the potential energy for an electron in the lattice in the absence of an external field is shown once again (cf. Fig. 3,a). When the field is applied, the curve assumes the form shown in Fig. 19,b (the field is assumed homogeneous and directed along the $x$ axis). We see that the field produces two effects:

a) it raises the potential crater of each atom;

b) in addition, it perturbs each atom, i.e., distorts the shape of this crater.

These two effects have as their consequence a deformation of the energy spectrum. The first effect leads to a tilting of the energy bands. The second effect leads to a broadening of the bands.

Let us first consider the effect of band tilting. This effect (for the one-dimensional case) was investigated by Zener[^20]. The tilted band is shown in Fig. 20. In this figure the region of values of $W$ and $x$ is hatched within which the modulus of the wave function is periodic with the period of the lattice. This region, in accordance with the terminology adopted in § 4, will be called an energy band. Outside the band the wave function rapidly decreases as one moves away from the band boundaries, so that far from the band it may be regarded as practically equal to zero.

Fig. 19. Curve of the potential energy for an electron in a lattice.
a) in the absence of a field, b) in the presence of a field (true curve), c) in the presence of a field (approximated curve).

Thus, for the electron we obtain, strictly speaking, a continuous energy spectrum. All energy values are possible. To each energy value there corresponds a definite wave function, which, however, is practically different from zero only in a limited region of space. The scheme shown in Fig. 20 by no means represents a dependence of the energy $W$ on the coordinate $x$. It should be understood merely as a characteristic of the wave function.

Fig. 20. Tilted band.

An electron with a given energy $W$ proves to be confined inside the band. In other words, it is localized in a more or less narrow region of space bounded by the planes $x=x_A$ and $x=x_B$ (see Fig. 20). This region is the narrower, the stronger the field.

Indeed, if \(d\) is the width of the inclined zone in the “vertical” direction, and \(l\) the width of the zone in the “horizontal” direction, then

\[ l=\frac{d}{eE}, \]

where \(e\) is the absolute value of the electron charge, \(E\) is the intensity of the electric field.

Thus, strange as it may seem at first glance, as the field is strengthened the electron becomes, in a certain sense, more and more bound. Such an electron oscillates between the left and right boundaries of the zone (points \(A\) and \(B\) in Fig. 20). In the right (lower) half of the zone (segment \(OB\) in Fig. 20) the electron has a positive effective mass and, consequently, is accelerated in the direction of the force acting on it, i.e. from right to left in Fig. 20. In the left (upper) half of the zone (segment \(OA\) in Fig. 20) the effective mass of the electron proves to be negative. On entering this region, the electron displays properties paradoxical from the classical point of view: it begins to accelerate in the direction directly opposite to the force acting on it, i.e. from left to right in Fig. 20. Thus, if in the absence of a field the electron was bound to a definite level inside the zone, i.e. could be characterized by the triple of quantum numbers \(n_1, n_2, n_3\), then now the electron begins to travel over the levels of the zone. Its state can no longer be characterized by any definite triple of values \(n_1, n_2, n_3\).

Let us now turn to the effect of zone broadening2. This effect is due to the Stark splitting of the corresponding terms of an isolated atom. Any degenerate term may be regarded as a collection of several terms which, in the absence of a field, coincide with one another. When a field is applied, these coincident terms separate, and the displacement of each of them is, generally speaking, different. As a result we obtain a splitting of the term, as is shown in Fig. 21. The energy zone corresponding to such a degenerate term may be imagined as the result of superposing several zones, the number of which is equal to the multiplicity of the degeneracy. The splitting of the term causes the corresponding zones to move apart, as a result of which the zone as a whole becomes broadened. This effect is illustrated by Fig. 22, which shows two neighboring zones, of which the lower corresponds to a nondegenerate \(s\)-term, and the upper to a \(p\)-term possessing threefold degeneracy.

If the external field is sufficiently weak, so that the potential of the field changes little in passing from one atom to a neighboring one, then the potential within each atom may be regarded as constant. In this case the potential curve shown in Fig. 19, \(b\), may be approximated by the step curve shown in Fig. 19, \(c\):

With such an approximation the effect of band broadening turns out to be lost.

Band broadening has as its consequence a narrowing of the forbidden region between the bands, as is seen from Fig. 22. Thus, allowance for the Stark effect leads to a coming together of the bands that increases as the field increases.

This circumstance should be reflected first of all in the optical spectrum of the crystal, leading to a deformation of the absorption bands. Thus, for example, the band corresponding to electron transitions from

Fig. 21. Stark splitting of terms in an isolated atom.

Fig. 22. Effect of broadening of energy bands.

the normal band to the conduction band (the so-called intrinsic band), upon application of a field, should broaden; moreover, the red edge of this band, as it might seem at first glance, should shift toward the red side of the spectrum. In reality, however, a shift of the red edge toward the red side by no means must always occur and cannot be regarded as a criterion of Stark broadening of bands. One can be convinced of this if one takes into account the selection rules by which optical transfers of electrons from one band to another are governed.

According to these rules, from each given level of a given band a transition is possible only to one quite definite level of another band, so that, if all levels within each band are regarded as renumbered, then an optical transition can occur only between two levels with identical numbers. It is essential here that the numbering of the levels must be carried out for some bands from bottom to top, and for others from top to bottom, depending on the nature of the band. In Fig. 23, a, two neighboring bands (the normal band and the conduction band) are represented with the same order of numbering of the levels, and in Fig. 23, b—with the opposite order of numbering. Arrow No. 1 depicts the transition corresponding to the red edge

absorption band; arrow No. 2 corresponds to the violet boundary. In the case shown in Fig. 23, a, the red boundary is determined by a transition from the lower level of the lower band to the lower level of the upper band. This case occurs, for example, for a combination of an \(s\)-band with a \(p\)-band. In the case shown in Fig. 23, b, the red boundary is due to a transition from the upper level of the lower band to the lower level of the upper band. This case occurs, for example, for a combination of an \(s\)-band with a \(d\)-band.

Fig. 23. Selection rules for optical transitions between bands.
↑ 1 — red boundary of the absorption band in the absence of a field, ↑ 2 — violet boundary in the absence of a field, ↑ 3 — red boundary in the presence of a field.

Let us suppose that, upon application of an external field, only the lower band undergoes broadening (the boundaries of such a broadened band are shown in Fig. 23 by dashed lines). This occurs when the upper band is an \(s\)-band. The dashed arrow in Fig. 23, a and 23, b represents the red boundary of the absorption band in the presence of the field. We see that, when the field is switched on, the red boundary is shifted toward the red in the case shown in Fig. 23, b, whereas in the case shown in Fig. 23, a, it is, on the contrary, shifted toward the violet side, although in both cases the forbidden region between the bands is narrowed when the field is switched on.

§ 10. IONIZING ACTION OF AN ELECTRIC FIELD

In order to produce conductivity in a lattice, it is necessary to act on it with such agents as would supply electrons to the conduction band or empty the normal band. Such ionizing agents are temperature and illumination. An external electric field, if sufficiently strong, can also act as an independent ionizing agent. The ionizing ability of the field becomes possible owing to the deformation that it causes in the energy spectrum of the lattice. This deformation, as we have seen, reduces to an inclination and broadening of the bands.

Owing to the inclination of the bands, an electron acquires the ability to pass from one level to another across the forbidden region separating them by two paths, which are indicated in Fig. 24 by the vertical and horizontal arrows 3 and 4. The vertical transition is associated with an expenditure of energy; this transition is analogous to excitation.

Transitions of this kind are due to the mechanism of impact ionization (ionization by collisions). A horizontal transition occurs without expenditure of energy; it is similar to seepage through a potential barrier. We shall call such transitions electrostatic ionization. The field, by causing the bands to tilt, makes possible transitions of both kinds. In both cases the ionizing capacity of the field is the greater, the narrower the forbidden region that must be overcome by the electron.

Both horizontal and vertical transitions can occur:

1) from local levels to the conduction band (purely electron current);

2) from the normal band to local levels (purely hole current);

3) from the normal band to the conduction band (mixed current).

Horizontal transitions, in addition, can occur:

4) from the cathode to the conduction band (purely electron current);

5) from the normal band to the anode (purely hole current).

Thus, ionization may develop either throughout the entire volume of the crystal (cases 1, 2, 3) or only in its near-electrode layers (cases 4, 5).

Owing to the expansion of the bands, the forbidden region between the bands narrows (both in the horizontal and in the vertical direction). In Fig. 24 the boundaries of such an expanded band are shown by dashed lines. Thus, the Stark broadening serves as a factor facilitating ionization by any of the mechanisms. Taking the Stark effect into account, we thereby make both ionization mechanisms (impact ionization, electrostatic ionization) come into action at fields somewhat weaker than is the case when the Stark effect is neglected. Thus, the narrowing of the forbidden region has the same effect as an additional increase of the field. In addition, the Stark effect strengthens the role of thermal ionization (which may be significant in the case of semiconductors). Indeed, narrowing of the forbidden region increases the concentration of thermal electrons in the conduction band or of holes in the normal band. Here the narrowing of the forbidden region exerts

Fig. 24. Ionizing actions of the field.

causes the same effect as an additional increase in temperature.

Let us consider in more detail the mechanism of impact ionization. Vertical transitions occur as a result of collisions between electrons, so that the energy of one electron is increased at the expense of another. In these processes the conduction band is enriched with electrons, or the normal band is enriched with holes, or both occur simultaneously.

In the first case a free electron (the ionizing electron), moving in the conduction band, collides with an electron sitting on a local level (the ionized electron) and raises it into the conduction band, losing thereby precisely the energy acquired by the ionized electron. The ionizing electron continues to remain in the conduction band, falling, however, from its higher levels to lower ones. Obviously, a free electron can make collisions of this kind only when, while moving through the conduction band, it goes sufficiently far from its lower boundary.

Fig. 25. Mechanism of impact ionization.
a) ionization by impact of an electron, b) ionization by impact of a hole; \(ABCD\) is the path of the ionizing electron (hole), the jump \(BC\) corresponds to the moment of ionization. \(B'C'D'\) is the path of the ionized electron (hole). Evidently: \(B'C' = BC\).

Its distance from the lower boundary must be not less than \(u\) (if counted vertically, see Fig. 25, a), where \(u\) is the width of the forbidden region between the lower boundary of the conduction band and the level on which the ionized electron sits. This occurs only in sufficiently strong fields. Indeed, a free electron, moving toward the anode, descends along the steps of a staircase lying inside the conduction band (see Fig. 25, a); each transition from one step to the next lower one indicates an act of collision with the lattice. The length of a step \(\lambda\) is the mean free path, and the height of a step \(\varepsilon\) is that average portion of energy which the electron loses in each such collision. Whether this stepped path of the electron will enter that region of the conduction band where impact ionization becomes possible, or will lie wholly—

to lie below this region—this depends (for given $\lambda$ and $e$) on how strongly the band is tilted, i.e. on the magnitude of the external field $E$. In the case under consideration, when ionization takes place with the aid of a free electron moving in the conduction band, the energy of the ionizing electron before the collision is, of course, greater than the energy of the electron being ionized. After the collision, the energy of the electron that has produced the ionization may turn out to be either greater or smaller than the energy of the electron that has been ionized.

In the second case, ionization takes place with the aid of a hole moving in the normal band. In reality, two electrons collide here, both belonging to the normal band, as a result of which one of them (the one being ionized) is thrown upward beyond the limits of the normal band onto an empty local level, while the other (the ionizing one), remaining in the normal band, falls to a lower level of it, which is assumed to be free. It is obvious that such a collision can occur only under the condition that the hole is located sufficiently far from the upper boundary of the normal band. Indeed, moving toward the cathode, the hole rises along the steps of the staircase lying within the normal band (Fig. 25, б). Whether this staircase enters the region required for ionization depends on how strongly the band is tilted. In the case under consideration, before the collision between the electrons, the energy of the ionizing electron need not be greater than the energy of the one being ionized. After the collision, however, the energy of the electron that has produced the ionization (i.e. the one remaining in the normal band) is, of course, certainly less than the energy of the ionized electron (i.e. the one thrown out of the band).

Finally, in the third case we have collisions between a free electron of the conduction band and a bound electron of the normal band. In such a collision the bound electron is made free and, in this way, the conduction band, being enriched with electrons, thereby enriches the normal band with holes.

Alongside the process of impact ionization, in which the electron in one way or another makes a vertical transition from below upward, there occurs the reverse process, in which the electron makes the same vertical transition but from above downward, giving its energy to the lattice. This process leads to the binding of a free electron (hole). If this process did not exist, then the current in the crystal would increase with distance like an avalanche. A stationary current, constant throughout the entire volume of the crystal, is established only when both processes (ionization by collision and binding) come into equilibrium. Taking the binding process into account is precisely what distinguishes the modern theories of impact ionization22, 23 from the old theories.

Let us now turn to the mechanism of electrostatic ionization. The horizontal transitions in Fig. 24, i.e. transitions without a change of energy, become possible as a result of the tilting of the bands. In fact—

Indeed, the wave function for an electron with a given energy rapidly decays in the forbidden region; however, its tail extends quite far. This is what accounts for the circumstance that the electron always has some chance of finding itself on the other side of the forbidden region.

The probability of such penetration is extremely sensitive to the length of the forbidden region overcome by the electron. This region is shortened as the bands are tilted, i.e., as the field increases. The probability of a horizontal transition, being practically equal to zero for fields that are not too strong, almost suddenly becomes quite considerable as soon as the bands acquire a sufficient tilt. Thus, in a sufficiently strong field the conduction band can always expect that an electron will appear in it, having penetrated into it from a local level or directly from the cathode or, finally, from the normal band. The normal band, in turn, can always expect that a hole will arise in it due to an electron that has gone through the tunnel to an empty local level or directly to the anode or, finally, to the conduction band.

Thus, the crystal may be enriched with free electrons and holes, on the one hand, at the expense of its own resources, and on the other hand—at the expense of the metallic electrodes (arrows 1 and 2 in Fig. 24). The role played in the crystal by its own electrons and holes is considered in Zener’s work^20. The role that foreign electrons and holes, torn from the surface of the electrodes, may play is considered in Fowler’s work^25. Fowler’s theory has an advantage over Zener’s theory, since it requires overcoming a shorter barrier.

Fig. 26. Electrical conductivity as a function of field.

The curve of electrical conductivity as a function of field is shown in Fig. 26. This is an experimental curve. It is divided into three regions, replacing one another as the field increases:

a) the region of Ohm’s law,
b) the region of Poole’s law,
c) the prebreakdown region.

The exponential increase of conductivity in the Poole region cannot be explained with the aid of the ionization mechanisms considered. Impact ionization leads to an incorrect temperature dependence. Electrostatic ionization gives an incorrect dependence on \(E\) (a more rapid increase of conductivity with the field than actually occurs). As for the prebreakdown region, in this range electrostatic ionization may be regarded as one of the possible mechanisms.

In the mechanism of impact ionization, as also in the mechanism of electrostatic ionization, the electric field acts as an independent ionizing agent. Both these mechanisms become possible, as we have seen, owing to the tilting of the bands. Let us consider still another ionization mechanism, in which the ionizing agent is temperature, while the field acts as an aid. Here the auxiliary role of the field is made possible by the Stark broadening of the bands.

For conductivity caused by thermal ionization, we have:

\[ \sigma = C e^{-\frac{U}{2kT}}, \tag{9} \]

where \(U\) is the width of the forbidden region between the levels supplying electrons (holes) and the levels accepting electrons (holes). Taking the Stark effect into account, the quantity \(U\) must be regarded as a function of the field:

\[ U(E)=U(0)-\Delta U(E). \tag{10} \]

The dependence of \(\Delta U\) on \(E\) can be obtained from the theory of the Stark effect \(^{26}\). It is expressed by the following empirical formula:

\[ \Delta U(E)=aE^{2}+bE. \tag{11} \]

In this formula the parameters \(a\) and \(b\) themselves vary slowly with the field; moreover, as \(E\) increases the ratio \(\frac{a}{b}\) decreases, so that in weak fields the second term in (11) may be neglected in comparison with the first (the region of the quadratic Stark effect), while in strong fields, conversely, the first term may be neglected in comparison with the second (the region of the linear Stark effect).

Substituting (10) into (9) and taking (11) into account, we obtain in the region of strong fields:

\[ \sigma = A e^{\alpha E}. \tag{12} \]

This formula expresses Poole’s law. The coefficient \(\alpha\) has the correct order of magnitude. Moreover, the dependence of \(\alpha\) on the temperature \(T\) also has the correct character. Thus, the Stark effect is evidently one of the possible mechanisms responsible for the increase in conductivity in the Poole region \(^{27}\).

In conclusion, we note that the facilitation of thermal ionization under the influence of the field may occur not only as a result of the Stark effect, but also through a completely different mechanism, considered in Frenkel’s work \(^{28}\). Frenkel’s mechanism is similar to Schottky’s mechanism, which causes a lowering of the work function in the thermal emission of electrons from a metal. From the point of view of Frenkel’s theory, the work \(U\), necessary in order to transfer an electron from one ion of the lattice to another and to separate the ano-

small ions to infinity must, in the presence of an external electric field \(E\), be reduced by the amount

\[ \Delta U = 2\sqrt{\frac{e^3 E}{\varepsilon}}, \]

where \(\varepsilon\) is the dielectric constant of the crystal. Thus, Frenkel’s theory leads to the expression

\[ \sigma = Be^{\beta \sqrt{E}} \tag{13} \]

instead of expression (12), which is based on the mechanism of the Stark effect. Frenkel’s formula (13) differs from Poole’s empirical law by the replacement of \(E\) by \(\sqrt{E}\). This circumstance, however, should not be regarded as an objection to the theory, since it cannot be said that the available experimental data would testify against such a replacement.

§ 11. PHOTOELECTRIC PROCESSES

The absorption spectrum of a real crystal usually contains two systems of bands:

a) the so-called intrinsic absorption bands, lying in the short-wavelength part of the spectrum;

b) the so-called impurity absorption bands in the long-wavelength part of the spectrum.

The bands of these two types possess essentially different properties. Intrinsic bands are fairly stable: they react only to a very slight degree to external influences. Impurity bands, on the other hand, are very fragile formations: their structure is extremely sensitive to external influences. In some specimens these bands may be absent altogether. They can be created artificially and destroyed artificially. Under the action of certain factors they can transform into one another, i.e., the absorption of a crystal may increase in one impurity band at the expense of a weakening of absorption in another impurity band. The absorption coefficient in the impurity bands is, on average, considerably smaller than in the intrinsic bands.

Fig. 27. Absorption spectrum of alkali-halide crystals.

Let us consider certain properties of the absorption bands in alkali-halide crystals. The absorption spectrum for these crystals has the form shown in the schematic Fig. 27. Along the ordinate is plotted the absorption coefficient; along the abscissa—the wavelengths.

The intrinsic bands lie in the far ultraviolet. We have several intrinsic bands superposed upon one another. They thus form a continuous spectrum with separate maxima following one another.

As for impurity bands, in alkali-halide crystals two principal impurity bands can be observed. One of them lies in the ultraviolet part of the spectrum, near the intrinsic bands. This is the so-called \(U\)-band. The other is in the visible part of the spectrum. This band is usually called the \(F\)-band. The color of the crystal is determined by this band. In the absence of an \(F\)-band the crystal is transparent in the visible part of the spectrum.

The \(U\)- and \(F\)-bands differ from one another above all in their photoelectric properties. The \(F\)-band possesses a considerable photoeffect: absorption of light in this band leads to the appearance of conductivity. The \(U\)-band, on the contrary, is photoelectrically insensitive. The position of these bands in the spectrum in different alkali-halide crystals does not depend on the chemical nature of the crystal, but is determined entirely by the lattice constant. We have the following empirical formulas:

\[ \begin{aligned} \text{for the } F\text{-band}\quad & \nu a^{2} = \mathrm{const},\\ \text{for the } U\text{-band}\quad & \nu a = \mathrm{const}, \end{aligned} \]

where \(a\) is the lattice constant, and \(\nu\) is the frequency corresponding to the maximum of the band.

Impurity bands can be transformed into one another. If a crystal containing a \(U\)-band but not containing an \(F\)-band is illuminated with frequencies belonging to the \(U\)-band, then the \(U\)-band will gradually disappear, and at its expense an \(F\)-band will arise. The crystal will gradually become colored. The reverse process occurs spontaneously. It proceeds the faster, the higher the temperature.

The \(F\)-band can also be obtained by other methods. There are three such methods.

1. Illumination of the crystal by X-rays or by ultraviolet light from the long-wavelength part of the intrinsic band. This is the so-called subtractive method of coloration. With prolonged illumination saturation is reached: the \(F\)-band attains its maximum intensity under the given conditions, which does not change upon further illumination. This maximum intensity is the greater, the more defects are contained in the lattice and the higher the temperature at which coloration occurs. Crystals colored by this subtractive method spontaneously bleach. This bleaching can be accelerated by raising the temperature or by applying an external electric field.

2. Heating the crystal in vapors of an alkali metal. This is the so-called additive method of coloration. The \(F\)-band formed in this case is determined by the nature of the crystal, but not by the nature

of that alkali metal in whose vapors the heating takes place. In a crystal colored in this way, when a field is applied, the entire colored region moves toward the anode, leaving the crystal behind it colorless.

1. Finally, a crystal can be colored by means of an electric field. With sharp electrodes and a sufficiently high temperature, coloration appears in the region of the cathode. It seems to emerge from the cathode and gradually spreads into the depth of the crystal. At the same time, together with the \(F\)-band, a \(U\)-band also arises. As the colored block moves in the direction of the anode, its intensity weakens; at the same time the intensity of the \(U\)-band increases. When the direction of the field is reversed, the motion of the colored region changes to the opposite. In this way the \(F\)-band can be driven back into the electrode. The crystal again becomes colorless. However, the \(U\)-band remains stuck in the crystal.

In subtractive coloration, the coloring centers in the crystal are created at the expense of its own resources. In additive coloration they are introduced from outside. In this sense, the creation of an \(F\)-band as a result of illumination with \(U\)-frequencies may be regarded as one kind of subtractive coloration. Coloration by an electric field is one kind of additive coloration.

All these distinctive properties of the absorption bands of alkali-halide crystals have now been studied very well. We have several hundred experimental works. These include the works of Pohl’s school[^29] in Germany, as well as an entire series of works by P. S. Tartakovskii[^30] and his students in the USSR.

As for semiconductors, their absorption spectrum possesses the same structure as the absorption spectrum of alkali-halide crystals. The principal difference is that, on passing to semiconductors, all the bands are shifted toward the red side of the spectrum. The intrinsic bands fall in the visible part. Semiconductor crystals are therefore opaque. The impurity bands turn out to be shifted correspondingly into the red and infrared regions, which are not very convenient for experimental investigation. It is apparently precisely this circumstance that is the reason why the spectra of semiconductors have so far been studied very poorly. Here we have only a few isolated works.

The absorption spectra of real heteropolar crystals can, in general outline, be interpreted from the point of view of the energy schemes presented in Fig. 10. Every absorbed quantum transfers an electron from one level to another, higher-lying level. From the point of view of these schemes, impurity bands correspond to transitions of electrons from local levels into the conduction band, or from the normal band to local levels. Intrinsic bands correspond to transitions from band to band. Such an interpretation leads

lead to the following consequences, which are in qualitative agreement with the experimental data:

1. The intensity of impurity bands from this point of view must be determined not only by the nature of the crystal, but also by the degree of deviation of our real lattice from the ideal state and by the degree of deviation from the stoichiometric ratio. In other words, the intensity of these bands must be in direct dependence on the treatment of the specimen and, thus, may be different in different specimens. The intensity of the intrinsic band, on the contrary, is determined exclusively by the nature of the crystal, which accounts for its stability. These properties of absorption bands are, as we have seen, in accord with reality.

2. The impurity bands must lie in a longer-wave part of the spectrum than the intrinsic bands. This conclusion follows directly from consideration of Fig. 10. In addition, the impurity bands must be considerably less intense than the intrinsic bands. Indeed, in our interpretation the intensity of impurity bands is determined by the number of defects in the lattice, whereas the intensity of intrinsic bands is determined by the number of regular atoms. These properties of the bands also correspond to reality.

3. Irradiation of a crystal with frequencies belonging to some impurity band may in certain cases, as is evident from the analysis of Fig. 10, lead to a transfer of electrons from some local levels to other local levels (for example, through the conduction band). Indeed, electrons torn by light from some levels, after entering the conduction band and wandering in it, may become attached to other levels. As a result, donors become acceptors, and acceptors are transformed into donors. The intensity of our impurity band is weakened owing to the appearance of another impurity band. The crystal is brought into a kind of metastable state, which will gradually dissipate. Processes of this kind, as we have seen, are actually observed.

Let us note, however, that the interpretation of absorption bands from the standpoint of the schemes shown in Fig. 10 at first sight encounters the following difficulty. From the point of view of these schemes, absorption of light in any band leads to the formation of electrons in the conduction band or holes in the normal band, or to both simultaneously. Thus all absorption bands must be photoelectrically active, i.e. absorption of light in all cases must be accompanied by an internal photoelectric effect. In reality this is not so. Some absorption bands possess a considerable photoelectric effect, while in other bands the photoelectric effect is practically entirely absent. This occurs, as we have already noted, in alkali-halide crystals, in which, with a strong photoelectric effect in the \(F\)-band, absorption of light in the \(U\)-band does not produce any noticeable conductivity. Such photoelectricity

chemical inactive absorption is also observed in some semiconductors (for example, in \(V_2O_5\)).

This circumstance served as the basis for Ya. I. Frenkel’s development of the concept of the exciton[^14]. Indeed, such photoelectrically inactive absorption can naturally be compared with nonconducting excited states of the lattice, when the two anomalous ions that arise as a result of absorption of a quantum remain bound to one another and thus cannot participate in the transport of current.

The concept of excitons is not, however, indispensable for explaining photoelectrically inactive absorption. This effect can be interpreted within the framework of the ordinary energy diagrams with which we operate in the theory of semiconductors. What is essential here is only that, in the energy spectrum of the crystal, both donor levels and acceptor levels appear simultaneously as local levels (see Figs. 10, 17). It can be shown[^31] that light can transfer electrons into the conduction band and that, nevertheless, under certain conditions this process may fail to produce any appreciable increase in conductivity. Such a transfer of electrons into a free state, not accompanied by an increase in conductivity, is caused by changes in the conditions of recombination that occur under the action of light.

Let us note in conclusion that, alongside the normal photoelectric effect and the photoelectrically inactive absorption of light, in some semiconductors under certain conditions there occurs an anomalous (so-called negative) photoeffect, when under the influence of illumination the conductivity does not increase, but decreases. Such a negative photoeffect has been observed in selenium, stibnite, molybdenite, and cuprous oxide. The results of the investigations are highly contradictory. What is common, however, is the fact that the negative photoeffect, if it occurs at all, is observed only at not very low temperatures and in sufficiently strong electric fields. When the temperature is lowered, or when the field is reduced (with all other conditions unchanged), the negative effect disappears and turns into a normal effect. The critical temperature at which the photoeffect changes sign is the lower, the greater the field strength.

On the basis of the available experimental data it is still difficult to decide whether the negative photoeffect is due to some independent mechanism or is the result of secondary processes superposed on the normal effect. Let us note, however, that the very fact of a decrease in conductivity under the influence of illumination can be explained from the standpoint of the energy diagrams shown in Fig. 10[^32]. What is essential here is that the light and thermal electrons should differ in their origin, i.e., that the light should draw electrons from levels other than those from which heat draws them. In this

in this case the free electrons acquire the possibility of recombining with holes of two kinds: with holes created by light, and with holes created by heat. The corresponding recombination coefficients need not be the same. For a certain ratio between these coefficients, the conditions for recombination when illumination is switched on may prove to be considerably facilitated, as a result of which the total electrical conductivity, in the presence of illumination, may turn out to be less than in the dark.

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