LETTERS TO THE EDITOR

ON S. V. TYABLIKOV’S REVIEW OF S. I. PEKAR’S BOOK STUDIES IN THE ELECTRON THEORY OF CRYSTALS

In the third issue of volume 48 of your journal (p. 447) there appeared S. V. Tyablikov’s review of S. I. Pekar’s monograph Studies in the Electron Theory of Crystals, Gostekhizdat, Moscow–Leningrad, 1951.

This review seems to us not objective and, in a number of places, simply erroneous.

Being students of S. I. Pekar, we are far from the thought of giving a sufficiently objective assessment of the above-mentioned book, all the more since a detailed review by A. I. Anselm has appeared in print, one that raises no objections on our part (Soviet Book, No. 11, 1952). However, we consider it necessary, for the proper information of the readers of Uspekhi Fizicheskikh Nauk, to point out a number of factual errors made in S. V. Tyablikov’s review.

1. In discussing the question of the energy spectrum of an electron in a crystal, the reviewer particularly insists on its continuous and band character, in contrast to the consideration, carried out in the monograph, according to which one may speak of the discrete spectrum of an electron in a polarization well.

It should be emphasized that only in the adiabatic approximation does it make any sense at all to speak of the energy levels of a single electron. In this approximation the (discrete) energy of the electron proves to be a function of the coordinates of all the ions. In the second stage of the adiabatic approximation (the solution of the equations for the vibrations of the ions), the total energy of the entire system reduces to the energy of the ions, with the electron’s proper energy entering as a term in the potential energy of the ions. This total energy has a continuous spectrum.

Here it is quite appropriate to recall the analogy with a moving hydrogen atom, in which the electron’s energy spectrum has a discrete character, whereas the translational motion of the whole system has a continuous energy spectrum.

The presence of discrete levels for a single electron should manifest itself (as distinct from the former band theory of the solid) in the possible absorption of light by current carriers (i.e., by polarons). Only in this sense (i.e., in the adiabatic approximation) can one speak (as is emphasized in the monograph) of the discreteness of the energy spectrum of a single electron.

The question is not, “... what should be called a polaron...,” whether “it is more natural to regard the electron and the well accompanying it as a single whole,” as S. V. Tyablikov writes, but rather to what extent the adiabatic approximation is applicable to the problem of studying the motion of an electron in a crystal and what its degree of accuracy is. If, as is the case in the monograph in question, the criteria for the adiabatic approximation are fulfilled sufficiently well, the terminology adopted by S. I. Pekar should not give rise to misunderstandings.

1. The reviewer’s assertion is erroneous that the expression for the energy of a moving polaron

\[ \frac{\hbar^2 k^2}{2M} \]

(\(k\) is the wave vector) “... is entirely due to the author’s use of an approximate method of effective mass (in the further discussion, for brevity, we shall be using the abbreviation MEM). If, in investigating the spectrum of the system, the author had not omitted the periodic potential (with the corresponding replacement of the true electron mass by the effective one), then the energy of 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.”

Indeed, S. I. Pekar considered the motion of electrons with comparatively small energies, lying near the lower edge of the conduction band (in the sense of the old band theory). In this case the MEM proves to be sufficiently accurate and in fact takes account of the existence of the periodic potential. Usually the width of the allowed band is assumed to be of the order of several ev; at the same time, as has been calculated, the kinetic energy of the electron in a polarization well reaches no more than \(0.2\)–\(0.4\) ev, while the energy of the translational motion of the polaron

\[ \frac{\hbar^2 k^2}{2M} \]

amounts to hundredths of an ev \(\left(\sim \frac{3}{2} kT\right)\).

The upper region of the energy spectrum of the translational motion of the polaron was not investigated at all by S. I. Pekar, in view of a number of specific difficulties arising when the criterion of “smallness” of the polaron velocity (indicated in the monograph) is violated. However, these difficulties have nothing in common with the existence of the periodic potential, since they arise already in that region of polaron energies where the MEM is obviously applicable.

Consequently, a simple reference to the existence of the periodic potential in the crystal is still wholly insufficient for asserting the band character of the polaron energy spectrum.

The following remark by the reviewer seems to us incorrect: “Therefore the statements which have been made to the effect that allowance for polarization leads to rejection of the band spectrum are simply wrong. This circumstance seems to us extremely important for experimenters in interpreting their experimental data, for it means that they may still use the band scheme. Indeed, the parameters of the band scheme—the widths of the bands, the distances between them—are determined, as a rule, not from theoretical formulas, but from experiment....” It incorrectly orients experimenters, since it completely ignores the essential difference between the previous band theory and the theory of polarons. It is necessary to emphasize that the polaron theory makes it possible to predict a number of quantities which previously could be taken only from experiment, for example, the energies of thermal and optical dissociation, the coefficient of recombination of polarons at vacancies and \(F\)-centers, the effective masses of the carrier of current, its mobility, and so on, when only a single parameter, taken from experiment, is available.

Therefore the difference between the new theoretical formulas and the old ones must necessarily be taken into account in interpreting experimental results.

1. Complete bewilderment is caused by S. V. Tyablikov’s remark at the end of p. 449 that allowance for polarization in the calculation of temperature effects “... has hardly been touched upon in S. I. Pekar’s investigations.” The monograph considers the following temperature effects: the temperature dependence of the mobility and concentration of current carriers, as well as the concentration of various impurity centers, the distinction between the optical and thermal energies of dissociation of \(F\)-centers, the temperature dependence of the coefficient of recombination of polarons at vacancies and \(F\)-centers, the temperature dependence of the widths of impurity absorption bands, and

locations of their maxima, the temperature dependence of the conductivity of a semiconducting metal of the titanium type.

All these numerous effects are most closely connected with the phenomenon of polarization of the lattice by the electron field. How, after this, is one to understand the statement that “the question has not been touched upon”? Perhaps the reviewer has in mind the dependence of the effective mass of the current carrier on temperature? This effect, in principle, obviously exists, but, being a correction of higher orders of approximation in the theory, should have little effect on the quantitative results of the theory.

1. The reviewer’s assertion that the use of the EMM raises the symmetry of the Hamiltonian and that “... this leads to an erroneous consideration of all phenomena whose character is conditioned, to one degree or another, by the symmetry of the fields” (p. 450) is completely mistaken.

If the criterion for applicability of the EMM is satisfied*), then the solutions of the auxiliary equation with effective mass can be rigorously classified as \(s\)-, \(p\)-, \(d\)-, ... states. However, it by no means follows from this that the true wave function of the electron possesses the same symmetry.

Therefore, no “loss of the symmetry of the solutions” can, as the reviewer believes, make several “... of the author’s quantitative results on the calculation of \(F\)-centers” questionable. The cause of quantitative inaccuracies here can only be an insufficiently precise fulfillment of the criterion for applicability of the EMM.

The considerations just given also apply fully to the so-called “theorem 1:2:3:4,” which may lose force if the criteria for applicability of the EMM and of the macroscopic approximation are violated, and not at all when the periodic field is taken into account, as the reviewer supposes.

1. S. V. Tyablikov’s remark at the end of the review about “... the slight connection, in the exposition, between theoretical questions and experimental data...” looks somewhat strange. One may, of course, dispute the advisability of placing the comparison of theory with experiment in a separate chapter (incidentally, a rather large one), but it should be noted that all the numerical results presented were compared with experiments. The number of such comparisons was limited only by the insufficiency of experimental data.

These remarks should be borne in mind when reading S. V. Tyablikov’s review, in order to obtain a correct orientation in questions of polaron theory.

M. F. Deigen, K. B. Tolpygo