Bibliography

S. I. Pekar, Studies on the Electronic Theory of Crystals. Moscow–Leningrad, Gostekhizdat, 1951, 256 pp., 5000 copies, 10 rub. 50 kopecks.

The present monograph substantially summarizes the studies of S. I. Pekar and his co-workers concerning the theory of semiconductors and dielectrics with an ionic crystal lattice, carried out over a number of recent years. It should first of all be noted that S. I. Pekar considers a one-electron problem, studying in detail the question of the behavior of an excess electron in ionic crystals. The author points out that in ionic crystals an essential role is played by the interaction of the additional electron with lattice vibrations and the polarization of the lattice by the field of the electron itself; on these assumptions he develops a theory of the electrical and optical properties of ionic crystals. It turns out that taking into account the new type of interaction proposed by the author not only changes the quantitative characteristics of phenomena that could be expected on the basis of the generally known band model, where the interaction of the electron with lattice vibrations is usually regarded as weak, but in certain cases also leads to qualitatively different results. From this point of view, the study of effects caused by the interaction of the electron with lattice vibrations is of considerable interest for the electronic theory of ionic crystals and is very promising.

S. I. Pekar’s theory is, to a certain extent, a further development of the very important works of L. D. Landau and Ya. I. Frenkel, in which certain qualitative features of phenomena taking place for an electron in a lattice had already been formulated and which found their development in the works of S. I. Pekar. Thus, in L. D. Landau’s work the important hypothesis was advanced that the electron becomes self-localized in a crystal as a result of the deformation of the surrounding lattice by the field of the electron itself. Ya. I. Frenkel pointed out that the electron deforms the ions nearest to it and that this deformation follows it through the crystal.

The basic idea of S. I. Pekar’s theory, to which the exposition of the entire monograph is devoted, is that the electron polarizes the crystal lattice surrounding it; this polarization, in turn, exerts on it a reciprocal action equivalent to the action of a certain effective potential well. The depth of this well, as S. I. Pekar’s calculations show, proves in a number of crystals to be rather large \((0.5—1.0\ \text{eV})\), so that discrete energy levels can exist in it. The local polarization caused by the electron is associated with the displacement of ions from their mean equilibrium positions. Since the displacements of the ions lag behind the instantaneous states of the electron, the polarization consequently lags behind and forms for it a potential well. Owing to the inertia of the ions, they are acted upon by the mean field of the electron, which may be regarded as the static field of a charge, dis-

defined with density \(e|\psi|^2\), where \(\psi\) is the wave function of the electron in the polarization well. The states of a crystal with a polarization well in which the electron is localized were called polarons by the author.

The considerations set forth are of a “quasiclassical” character. In the transition to a more consistent quantum treatment, these ideas reduce to taking account of the interaction of the electron with the phonon field, in a manner fundamentally analogous to the treatment of the interaction of the electron with the photon field in modern quantum electrodynamics. This analogy is very deep and fruitful. It helps one to understand that what is the object of experimental investigation in ionic crystals is not simply the electron as such, but a more complicated object (usually called a polaron)—an electron plus the lattice vibrations (phonons) interacting with it. It is precisely the properties of this object that determine the course of the so-called “electronic” processes in semiconductors. The situation is exactly the same here as in electrodynamics, where we never deal with a “free” electron, but always investigate a more complicated object—an electron plus the electromagnetic field. In electrodynamics, however, owing to the smallness of the binding energy of the electron with the field, allowance for this circumstance leads in most cases only to small corrections. In crystals, on the other hand, especially ionic ones, the energy of the electron’s coupling with the field may be relatively very large, and for this reason taking this coupling into account becomes essentially necessary. Ordinary band theory regards the interaction of electrons with phonons as weak, and therefore in a number of cases omits from consideration, or incorrectly conveys, important features of the phenomena. It is for this reason that the theory of polarons is of substantial interest for solid-state physics.

In its content the monograph is divided into two parts. The first of them treats the theory of polarons (Chs. II and III), and the second—the theory of \(F\)- and \(F'\)-centers (Chs. IV, V, VI). In the introduction (Ch. I) general questions of polaron theory are discussed and a critical analysis of band theory is given.

S. I. Pekar shows that the energy levels of an electron in a polarization well lie below the bottom of the ordinary electron conduction band and that therefore electrons are mainly in states in which they are accompanied by polarization caused by themselves. Showing further that the electron, together with its polarization well, can move through the crystal, the author comes to the conclusion that in ionic crystals, in electronic conductivity, the principal role is played by polaron states, which may be treated as particles with a charge equal to the charge of the electron and with a certain effective mass, which, depending on the parameters of the crystal, may considerably exceed the mass of the electron. In the second part of the monograph the author applies the methods he has developed to the theory of \(F\)- and \(F'\)-centers. Here the energy spectrum and absorption by \(F\)-centers, the energy of thermal dissociation of \(F\)-centers, and some of their characteristics are calculated. The results of the theory are, as the author asserts, in sufficiently satisfactory agreement with the experimental data.

Turning now to a more detailed consideration of the author’s separate results, it should first of all be noted that all the investigations are carried out by him in the approximation of the effective-mass method and that, if the use of the latter is abandoned, they may undergo a number of substantial changes, not only quantitative but also, what is especially important, qualitative ones.

In Chapters II and III the author studies the behavior of an electron in a crystal lattice when the polarization caused by it is taken into account (this effect apparently plays an essential role only for ionic crystals). After establishing the fact that a polaron can move freely through the cry-

stage, the author proceeds to the calculation of the mobilities of polarons, which is carried out according to the usual scheme of perturbation theory under the assumption that the probabilities of scattering of polaron waves by optical and acoustic vibrations are small. The results thus obtained are, as the author says, in fairly satisfactory agreement with experimental data. Since in this case the polaron levels lie below the zone levels of the electron (i.e., of the state of an electron that has not polarized the crystal), it is obvious that the overwhelming majority of electrons belong to polaron states, and therefore the principal current carriers in such crystals are polarons.

Further, it seems to us, one should dwell in greater detail on two questions: first, on the question of the nature of the energy spectrum as a whole, and second, on the question of the limits of applicability of the theory.

An electron in its polarization well (i.e., a polaron) has discrete energy levels; in addition, the electron together with its well can move through the crystal as a whole. In this connection S. I. Pekar assigns to the eigenvalues of the electron energy the eigenvalues of the energy of the electron in the polarization well, and assigns the energy of translational motion to the energy of the whole system, since the polaron is defined by him as a state of the crystal with a polarization well in which the electron is localized. It seems to us, however, more natural to regard the electron and the well accompanying it as a single whole, since in a number of effects—for example, in electron conductivity—the object of investigation is precisely such a formation, and accordingly to assign to the energy of the polaron both the energy of the electron at a discrete level of the polarization well and the energy of the translational motion of the polaron as a whole. This seems all the more appropriate because, if one uses the formulation of S. I. Pekar, then the motion of a polaron through the crystal will represent nothing other than the displacement through the crystal of a state of the crystal itself with a polarization potential well. Since, however, this question is to a certain extent related to what is to be called a polaron, we shall not dwell on it further. In any case, the energy of the translational motion of a polaron, calculated by the author, is a continuous function of the quasi-wave vector \(k\): \(E(k) = - \frac{\hbar^2 k^2}{2M}\), where \(M\) is some quantity playing the role of the effective mass of the polaron and depending on the parameters of the lattice. The fact that the energy of the translational motion of a polaron is a quadratic function of its quasimomentum is entirely due to the author’s use of the approximation of the effective-mass method. If, in studying the spectrum of the system, the author did not omit the periodic potential (replacing the true electron mass by the corresponding effective one), then the energy of the translational motion of the polaron as a whole would be a periodic function of the quasi-wave vector, i.e., would have a band character. It is precisely in this sense that one may speak of bands in the energy spectrum of polarons. Therefore the objections expressed that taking polarization into account leads to a rejection of the band spectrum are simply incorrect. This circumstance seems to us extremely important for experimentalists in their interpretation of experimental data, for it means that they can continue to use the band scheme. In fact, the parameters of the band scheme—the band widths, the distances between them—are determined, as a rule, not from theoretical formulas but from experiment. In view of this, the difference between the new theoretical formulas and the old ones does not play so large a role for experimentalists in interpreting experimental data. The inclusion of polarization proves essential in calculating temperature effects, but, unfortunately, this question is almost not touched upon in S. I. Pekar’s investigations.

Let us note in this connection that the band form of the spectrum is by no means itself a band theory. The band character of the spectrum is a consequence of a very general property of the system under consideration—the translational symmetry of the lattice—and therefore will occur in any theory that correctly takes this symmetry into account.

The question of the limits of applicability of S. I. Pekar’s theory has several different aspects. First of all, in writing the Hamiltonian it was assumed, as the author indicates, that, on the one hand, the bound electrons of the dielectric adiabatically follow the motion of the conduction electron and, on the other hand, that the conduction electron adiabatically follows the motion of the ions. These conditions, as can be seen, are satisfied for a number of crystals.

Another aspect of the question of the criteria of applicability of the theory is connected with the possibility of using the effective-mass approximation. This method is valid if the de Broglie wavelength \(\lambda\) is considerably larger than the lattice constant. Here the conditions of applicability are fulfilled less well, since it turns out that \(\lambda\) exceeds the lattice constant by only one or two times (in S. I. Pekar’s monograph, evidently through a misunderstanding, the effective size of the polaron, approximately coinciding with \(\lambda\), is compared not with the length of the edge of the elementary cube, but with the distance between the nearest ions, which is half the edge of the cube). This makes the agreement with the experimental data, noted in the monograph, less convincing.

It should also be noted that, apart from this quantitative aspect, the application of the effective-mass method encounters another limitation not noted by S. I. Pekar. Namely, when passing to the effective-mass method the symmetry of the Hamiltonian is raised (for example, in cubic crystals the field becomes spherically symmetric instead of having the symmetry of the lattice). And this leads to an erroneous consideration of all phenomena whose character is determined to one degree or another by the symmetry of the field. For example, all excited levels of an \(F\)-center (except the \(s\)- and \(p\)-levels) in the field of a cubic lattice are not hydrogen-like because of Stark splitting. This makes somewhat doubtful the author’s quantitative results for the calculation of \(F\)-centers. The transition to the effective-mass approximation leads to inaccuracies in other places as well.

Thus, it is easy to show that the “theorem \(1:2:3:4\)” formulated in the monograph, from which the author obtains an important relation between the energies of optical and thermal dissociation, loses its force when the periodic field is taken into exact account.

The second part of the monograph (Chs. IV, V, VI), devoted to the theory of \(F\)- and \(F'\)-centers, begins with an analysis of experimental data. On the basis of this analysis the author shows that the most natural assumption about the nature of \(F\)-centers in alkali-halide crystals is de Boer’s hypothesis that an \(F\)-center is an electron replacing a negative halide ion in the lattice. The author develops the quantitative theory of \(F\)-centers on the basis of the de Boer model analogously to the theory of polarons: taking into account the polarization of the lattice by an electron situated at a halide vacancy of the lattice. It is assumed here that, first, the motion of the localized electron can be treated in the effective-mass approximation and, second, that the polarization radius of the \(F\)-center is sufficiently large for the electrical polarization of the lattice to be calculated macroscopically. Both these assumptions mean, as is easy to see, that the region where the wave function of the electron situated at the halide vacancy is appreciably different from zero must be much larger than the lattice constant. But this condition is fulfilled for \(F\)-centers considerably worse than for

polarons, since the effective radius of the electron cloud turns out to be of the order of the lattice constant or smaller.

In an analogous way, the theory of \(F'\)-centers is constructed; these are regarded as a pair of electrons bound to a single halogen vacancy. Such a model was chosen in accordance with experimental data on the formation of two \(F\)-centers upon dissociation of one \(F'\)-center, and conversely.

On the basis of the assumptions indicated, the monograph calculates the energy of thermal dissociation of \(F\)-centers, the width, shape, and temperature dependence of the \(F\)-absorption band in ionic crystals; it also considers questions of electron recombination at positively charged centers and at \(F\)-centers, investigates the quantum yield of dissociation of \(F\)- and \(F'\)-centers, and proposes an explanation of the metallic type of conductivity of certain semiconductors. One should especially note the author’s commendable tendency to bring the calculation results, despite mathematical difficulties, to a form suitable for comparison with experiment. Unfortunately, however, the comparison of the quantitative results of the theory with experimental data has been made without taking account of the errors introduced by the factors mentioned above, and therefore they must be treated with caution.

In S. I. Pekar’s theory, the principal constants characterizing the properties of crystals and determining the behavior of electrons in these crystals are the dielectric constant and refractive index, the limiting frequency of optical vibrations, and the effective mass of the electron. The first three of these are taken from direct measurements, while the effective mass, entering into all the formulas as an unknown parameter, is chosen so as to satisfy, in the best possible way, the sum of all known experimental facts. A comparison of theoretical data with experimental ones shows that in certain cases the theory gives not only qualitative but also quantitative agreement. This indicates that the basic physical premises used in the theory more or less correctly reflect the physical aspect of the phenomena.

Among the shortcomings of the monograph one should include the fact that it is written in a difficult language; moreover, the difficulty of exposition was not caused by the complexity of the questions considered. Likewise, the shortcomings of the monograph include the weak connection, in the presentation, between the theoretical questions and the experimental data, which are cited in fact only when the final results are being discussed, making the monograph difficult to read.

On the whole it should be said that S. I. Pekar’s monograph is of definite interest to persons engaged with these questions.

S. V. Tyablikov