THEORY OF SEMICONDUCTORS AT THE EIGHTH ALL-UNION CONFERENCE ON SEMICONDUCTORS

V. L. Bonch-Bruevich

The present article should not be regarded as a review of the current state of semiconductor theory. It is a review of specific works reported at the sessions of the theoretical section of the Eighth All-Union Conference on Semiconductors. In this connection, two reports on the general problems of semiconductor theory, delivered by S. I. Pekar and A. I. Anselm at the plenary session of the conference, are almost not considered here.*)

At the five sessions of the section on the theory of semiconductors, thirteen reports were heard, which may be divided into two classes, namely, those relating to the theory of stationary states and those relating to the kinetics of electronic processes in semiconductors.**)

I. THEORY OF STATIONARY STATES OF ELECTRONS IN CRYSTALLINE LATTICES

Most of the works in this section, in one form or another, were connected with the consistent allowance for the interaction of electrons with the lattice in the investigation not only of the kinetics of phenomena, but also of the equilibrium states of the system. In this respect, the work of S. I. Pekar and M. A. Krivoglaz, “The Spur Method in the Electron Theory of Crystals,” aroused great interest.

In a generalization of Feynman’s well-known work², the authors developed a new (and, apparently, very effective) mathematical apparatus,

) The reports mentioned are being published in this same issue of UFN*; see also¹.

**) Naturally, this division (like any other) is not absolute, if only because the study of relaxation processes presupposes the availability of some information about the stationary states of the system. Therefore, in almost all reports of the second section, problems relating to the first were considered to some extent as well; however, taking into account the orientation of one or another work, it is easy to assign it to one of these two classes.

permitting one to find the sum of states for a particle in a quantum field without solving the wave equation (specifically, the interaction of electrons with vibrations of the crystal lattice in ionic and homopolar crystals was under discussion). This enabled the authors consistently to carry out the program of polaron theory,^3,4 in which the interaction of electrons with lattice vibrations (at least in its main part) is taken into account already in determining the possible values of the energy of the system, and is not treated simply as a cause of scattering of “free” electrons. Cases of both weak and strong coupling were considered; moreover, what is especially important, the theory is valid for arbitrary temperatures (let us recall that in earlier studies on polaron theory by methods of strong or adiabatic coupling^3,4 it was necessary to restrict oneself only to sufficiently low temperatures). The results of the calculations were applied by the authors to determining the ground-state energy, the effective mass of the current carrier*), the equilibrium concentration of conduction electrons in a semiconductor, the electronic heat capacity, and the thermoelectric current (we emphasize that in the present case not only the first two, but also all the remaining questions are by no means trivial, since in the presence of interaction with lattice vibrations the Maxwellian distribution for electrons, generally speaking, no longer applies).

The report by K. B. Tolpygo was devoted to presenting a number of works by the author and his collaborators on the microtheory of electronic states in a crystal. Proceeding from the exact Hamiltonian of the problem, the author used a unified system of approximations in considering various groups of phenomena; thereby the constants characterizing the most diverse phenomena proved to be interconnected. In particular, the interrelation between the electronic and vibrational degrees of freedom in a solid was investigated in detail. The main results amount to the following:

a) A calculation was made of the natural frequencies of lattice vibrations and of the heat capacity (for ionic crystals) with allowance for the finite sizes of the ions and their polarizability.

b) In homopolar crystals of the diamond type, in addition to three acoustic and three optical branches, two more branches of vibrations were discovered (jointly by K. B. Tolpygo and V. S. Mashkevich), called by the authors “light” branches and describing the propagation of long electromagnetic waves in the crystal. It is significant that in the latter case the displacements of the lattice atoms from their equilibrium positions are also different from zero; on the other hand, owing to the coupling of the electronic and vibrational degrees of freedom, the displacements

*) The effective mass of an electron not interacting with phonons (taking into account only the periodic field of an ideal lattice) was, naturally, assumed to be known.

atoms (in a homopolar lattice) can, under certain conditions, be associated with oscillations of dipole moments. As a result, direct absorption of light of a definite frequency by an ideal lattice of a homopolar crystal proves possible (in this case only one quantum of lattice vibrations is excited).

c) The possibility of the formation of polarons in homopolar crystals was investigated (the inertial polarization necessary for the formation of a polaron is caused by a change in the dipole moment of the elementary cell both for optical and for acoustic vibrations). The result again proved negative: in homopolar crystals, polarons of large (in comparison with the lattice constant) radius turn out to be unstable.

d) By a somewhat generalized Heitler–London method, the motion of an “excess” electron or hole in a homopolar crystal (in particular, also in an imperfect one*) was considered. Of special interest here, in the opinion of the author of the review, was the indication of the specific character of the interaction of a hole with optical vibrations of the lattice in homopolar crystals: since the absence of one of the valence electrons weakens the bond strength between atoms, the latter are, as it were, “stretched apart” when a hole appears, owing to which optical vibrations may arise. This effect, apparently, is by no means small. (The mobility of a hole associated with scattering by optical phonons proves proportional to \(T^{-5/2}\) for \(kT \gg \hbar\omega_0\) and to \(\left(\exp \frac{\hbar\omega_0}{kT} - 1\right)^{-1}\) for \(kT \ll \hbar\omega_0\); \(\omega_0\) is the frequency of the longitudinal optical vibrations.)

M. F. Deigen, in the report “Local States of Electrons in Homopolar Crystals,” presented the results of his investigations on the interaction of electrons with acoustic vibrations of the lattice in homopolar crystals. Taking the named interaction into account in calculating the energy of the system, the author arrived at the following visual picture: an “excess” electron, by its electric field, causes a deformation of the lattice (an increase in density), which, in turn, leads to a local increase of the dielectric permittivity**). As a result, an additional field arises which acts on the electron; the latter, in this way,

*) By a somewhat different method, an analogous problem was considered by the author of the present review⁵. As is known, the consideration of a hole constitutes, in principle, a many-electron problem; the many-electron approach to the problem of the “excess” electron, as was shown by Pekar, is very important in constructing a microscopic theory of \(F\)-centers.

**) Such a deformation of the lattice, naturally, may be represented in the form of a superposition of a number of acoustic vibrations, which also makes it possible to regard the whole effect described as the result of the interaction of the electron with the field of acoustic phonons.

Thus, it moves through the lattice together with the potential well (region of compaction) that it itself has created. This formation—“electron plus region of compaction”—was called by the author a “condenson.” According to the views of M. F. Deigen, it is precisely condensons that are the current carriers in such crystals. Experimentally, the formation of condensons apparently manifests itself in a noticeable (even at extremely low temperatures) broadening of the absorption band of light by the crystal. According to the speaker’s estimate, in crystals of the sulfur type this effect may be quite appreciable; in germanium and silicon, on the contrary, it proves to be small.

Two papers were devoted to the theory of the exciton. E. I. Rashba, in the work “Interaction of Excitons with Lattice Vibrations,” investigated the properties of currentless excited states in molecular crystals) while allowing for the possible deformation of the lattice by excitons (when the system of electrons is excited, the forces of bonding between molecules change, and this leads to deformation of the crystal). It had previously been shown by A. S. Davydov⁶ that, depending on the nature of the substance, in molecular crystals one of two types of excitons may exist, which he called “free” and “localized” (in E. I. Rashba’s terminology, these are “light” and “heavy” excitons). In the first case, the excitation energy passes from one lattice site to another in a time short in comparison with the period of molecular vibrations, so that the lattice practically has no time to “rearrange itself” (thanks to which the exciton may be regarded as moving more or less freely); in the second case the situation is the opposite, and consequently the lattice has time to deform completely before the excitation moves to another site. As a result, migration of the excitation energy is greatly hindered—the exciton is practically “localized”*). Generalizing these results, the speaker established that, alongside free excitons (and simultaneously with them), under certain conditions excitons of a third type also arise—the so-called “deforming” excitons. As in the case of “heavy” excitons, a deformation of the lattice takes place here under electronic excitation, which, in turn, affects the energy of the electrons; as a result, self-consistent excitation states are formed, propagating through the lattice together with the deformation they have caused. Unlike

*) As the speaker noted, in molecular crystals it is very convenient to study the fundamental absorption of light, which in this case has a purely excitonic character, whereas in atomic and ionic lattices fundamental absorption with the formation of current carriers is possible.

**) This expression should not be understood literally. From the translational invariance of the problem it follows that in this case too the exciton will move: “waves of excitation” are formed in the lattice; however, their velocity is much smaller than for free excitons.

of “heavy” excitons consists in the fact that, in the given case, the exciton nevertheless moves sufficiently rapidly in comparison with the vibrations of the lattice (in typical cases its effective mass, according to the speaker’s estimate, is \(10^2—10^3 m_0\), whereas the effective mass of a heavy exciton is of the order of \(10^5 m_0\); \(m_0\) is the mass of a free electron). Experimentally, the presence of deforming excitons is apparently confirmed by the existence in molecular crystals of broad absorption bands at low temperatures (the formation of “free” excitons corresponds, as can be shown, to narrow absorption lines).

The report by L. L. Korenblit was devoted to a new derivation of the Schrödinger equation for an exciton. Considering (within the framework of the “polar” model of S. I. Shubin and S. V. Vonsovskii\(^7\)) “extra” electrons and holes as elementary excitations of a many-electron system, the author obtained for them a “Schrödinger equation” in the configuration space of the excitation coordinates. In this, owing to the influence of the “background” of all the electrons of the crystal, the interaction potential of the “extra” electron with the hole proved to be Coulombic only asymptotically—at sufficiently large distances between them. At small distances, corrections of exchange origin appear.

In the report by Yu. N. Obraztsov, “The Chemical Potential and the Pressure of Phonons in a Solid,” a number of paradoxes were considered that arise when the usual thermodynamic relations are carelessly applied to a phonon or electron gas in semiconductors:

\[ \left(\frac{\partial \mu}{\partial T}\right)_p=-s;\quad \mu N=E+pV-TS \tag{1} \]

(\(\mu\) is the chemical potential, \(s\) is the entropy calculated per particle, \(S\) is the total entropy, \(E\) is the internal energy, \(N\) is the number of electrons, \(p\) is the pressure, \(V\) is the volume of the system). Thus, for example, it follows from (1) that the entropy of the phonon gas does not tend to zero as \(T\to0\). The speaker showed that the cause of the paradoxes lies in the nonadditivity of the thermodynamic functions of the phonon and electron gases when they are considered as separate (“independent”) systems. Indeed, in a solid, when the volume changes, the total number of atoms \((N_a)\) remains unchanged, while the number of atoms per unit volume (and, consequently, the number of electronic and phonon states per unit volume determined by it) changes. This means that, for example, the free energy of the system \(F\) depends not on two parameters, \(\dfrac{V}{N}\) and \(T\) (as in a gas), but on three: \(\dfrac{V}{N}\), \(T\), \(\dfrac{V}{N_a}\).

As was shown in the report, it follows from this that relations (1) are incorrect in the present case and must be replaced by the following:

\[ \left(\frac{\partial \mu}{\partial T}\right)_{p,N} = - s - N\left(\frac{\partial s}{\partial N}\right)_{p,T}, \tag{2} \]

\[ \mu N = E+pV-TS - N_a\left(\frac{\partial F}{\partial N_a}\right)_{N,V,T}. \]

In this case all paradoxes disappear.

II. KINETICS OF ELECTRONIC PROCESSES IN SEMICONDUCTORS

The works belonging to this section, in turn, may be subdivided into three groups:

a) the theory of scattering of current carriers in semiconductors;
b) the phenomenological kinetics of electronic processes;
c) the physical theory of recombination of current carriers*).

The first group includes the reports of L. E. Gurevich (“On the question of the scattering of electrons by lattice vibrations in semiconductors”), A. I. Gubanov (“The mean free path of electrons in liquid and amorphous semiconductors”), A. G. Samoilovich and M. I. Klinger (“Properties of semiconductors with a narrow impurity band”), and G. E. Pikus (“Thermo- and galvanomagnetic effects in intrinsic semiconductors”).

L. E. Gurevich noted that at sufficiently low temperatures, when \(kT \ll mc^2\) (\(m\) is the effective mass of the electron, \(c\) the velocity of sound), the velocities of the overwhelming majority of electrons are less than \(c\). In this case one-phonon processes (emission and absorption of one phonon by an electron) are forbidden by the conservation laws of energy and quasimomentum—in complete analogy with the fact that in electrodynamics the emission and absorption of light by a uniformly moving electron are impossible if the velocity of the latter is less than the phase velocity of light. Under these conditions the interaction of the current carrier with the lattice (which causes scattering) can occur only in processes involving no fewer than two phonons (the absorption or emission of both phonons is, of course, impossible, but a process of the Compton-effect type may occur—with the absorption of one phonon and the emission of another). The corresponding relaxation time turns out to be proportional to \(T^{-8}\), i.e. the mobility increases with decreasing temperature much more rapidly than in one-phonon processes (where the calculation gives the law \(T^{-3/2}\)). It should be borne in mind, however, that whereas in electrodynamics the case considered is typical, in the theory of the solid state it is a very “exotic” one, since the critical temperature \(T_0=\dfrac{mc^2}{k}\sim 1^\circ\mathrm{K}\). As was note-

*) By this we mean a theory that attempts to elucidate the mechanism of the recombination process.

established by the author of the paper, in fact in this temperature region scattering by impurities plays the decisive role; therefore everything presented may be of experimental interest only for the case of an extremely pure semiconductor.

A. I. Gubanov considered the specific processes of scattering of current carriers within the framework of the model of liquid and amorphous semiconductors that he had developed earlier1. In addition to the usual scattering processes by phonons, two more proved possible, arising from the peculiarities of the systems under consideration. First, even in the absence of phonons, scattering occurs owing to the violation of the strict periodicity of the potential—the absence of long-range order (in the author’s terminology, “liquid scattering”). Second, for the same reason—because of the absence of long-range order—when an electron is scattered by a phonon the law of conservation of quasimomentum need not be satisfied, which leads to additional scattering of electrons (“phonon-liquid scattering”).

In the work of A. G. Samoilovich and M. I. Klinger, the equilibrium and kinetic properties of a semiconductor with a narrow impurity band were studied in detail. A model example of an ideally regular arrangement of impurity atoms was considered; it was assumed that they form a simple cubic lattice. Although highly idealized, this example nevertheless makes it possible to understand some important features of a semiconductor with an impurity band, especially since in this case all calculations can be carried through to the end. The main results of this work have already been published2.

G. E. Pikus considered the kinetics of a number of processes in semiconductors with allowance for surface recombination of current carriers.

The second group includes the work of É. I. Adirovich and G. M. Guro, “Characteristic Times of Electronic Processes in Semiconductors.” In recent years, in connection with a number of physical and technical problems, the study of the kinetics of nonequilibrium recombination processes has acquired exceptional importance. As a rule, experiments measuring the “lifetimes” of current carriers have until now been interpreted on the basis of the statistical theory of Shockley and Read3, constructed on the assumption of stationarity of the process. As the authors of the paper noted, this assumption is by no means always justified by the actual experimental situation, and therefore it becomes necessary to consider the problem on the basis of the kinetic equations of recombination. This was the subject of the investigation in the work of É. I. Adirovich and G. M. Guro. Under the assumption that the cross sections of the elementary acts of capture of current carriers by recombination centers do not depend on the carrier concentration, the authors succeeded in studying in detail the time dependence of the concentration of electrons and holes in a number of standard experiments for determining lifetimes. Under certain conditions the results differ sharply from those given by the “stationary” theory3. In particular, the widely accepted view proved to be incorrect

the widespread opinion that the rate of a process is always determined by the lifetime of the minority carriers. In a number of cases the situation is just the reverse: what is characteristic (and is determined from experiment) is the lifetime of the majority carriers.

A number of reports delivered in the section on semiconductor electronic devices also pertain to the phenomenological theory of semiconductors (K. B. Tolpygo and V. A. Fomenko, “Rectification at a point contact in germanium,” E. I. Adirovich, V. G. Kolotilova, and B. V. Malin, “Transient processes in semiconductor devices,” and others).

The physical theory of recombination was the subject of the reports by L. N. Sosnovskii*) (Institute of Physics of the Polish Academy of Sciences, Warsaw), “On recombination upon collision of current carriers in semiconductors,” Yu. E. Perlin, “Recombination of conduction electrons by impurity centers in ionic crystals,” and V. L. Bonch-Bruevich, “Toward a theory of recombination of current carriers in heavily doped semiconductors.” In all three works the mechanism of capture of carriers by impurity centers was investigated, but for different cases (and, accordingly, in different models).

Yu. E. Perlin considered the capture of a current carrier by an impurity center in an ionic crystal. It was assumed (as usual) that the energy released upon capture is transferred to lattice vibrations; however (in contrast to a number of works by other authors) it was taken into account that the current carrier in an ionic crystal is in fact not an electron (or a hole), but a polaron. This circumstance proved very important for a correct estimate of the order of magnitude of the capture cross section.

L. N. Sosnovskii, drawing attention to the difficulties involved in transferring energy to phonons when carriers are captured at deep local levels, pointed to another mechanism for its removal: the energy released upon capture of a current carrier may be transferred to another carrier remaining in the conduction band. Being associated with a “collision” of two electrons, this mechanism received the name “impact” mechanism. Evidently, it should lead to a quadratic dependence of the reciprocal lifetime on the carrier concentration, which, apparently, was observed in the works \(^{11,13}\)**). The approximate theory developed in work \(^{15}\) is, in the opinion of the speaker, in qualitative agreement with experiment. We note, however, that a more exact calculation, carried out in work \(^{16}\) within the framework of the “hydrogenic” model of an impurity center, apparently leads to the conclusion that the effective capture cross sections according to this mechanism turn out to be somewhat smaller than the observed ones.

) This report was read at a plenary session of the conference.
*) See also \(^{14}\).

appealing to the known successes of band theory (in particular, to investigations of cyclotron resonance). It seems to the author of the review, however, that in the theory of solids the self-consistent-field method is far from being as effective as in the theory of the atom, for the fundamental physical and mathematical difficulties that have to be faced in its use are well known\(^{18,22}\). As for the successes of the one-electron theory, in our view they are in fact successes not of the one-electron, but of the “one-particle” approximation, based on the concept of elementary excitations) (whose spectrum, in particular, is studied in experiments on cyclotron resonance). Indeed, it can be shown\(^{5}\) that in semiconductors the behavior of Fermi-type excitations (“excess electrons” and “holes”) is qualitatively analogous to the behavior of the “noninteracting electrons” of the band scheme*). In particular, in determining the excitation spectrum all the symmetry considerations are valid which play so essential a role in the band scheme. Let us note that what has been said applies even to such cases (with the participation of inner electrons) where the one-electron approximation, understood literally, can no longer be applied at all. It should be emphasized, however, that so long as we are interested only in the analogy in the behavior of excitations and “noninteracting electrons,” the difference between them remains purely verbal. If the role of the many-electron theory were reduced only to establishing this analogy (i.e., only to justifying a number of “band” assertions), then it would be no more enviable than the role of the Moor who departs after having done his duty. The true fruitfulness of the “many-electron” approach can be revealed only where the difference between the two points of view begins, and the many-electron theory is applied to concrete problems. Such a case is found, for example, in the theory of an electron plasma, where the Coulomb interaction leads to screening of the external field acting on the current carriers, and to the appearance of a Bose branch in the excitation spectrum—oscillations of the “plasma” type. Another example of a successful “many-electron” approach is provided by the work of K. B. Tolpygo considered above (see also \(^{24}\)). In addition, in modern solid-state physics problems requiring an essentially many-electron treatment are playing an increasingly important role. A classical example is furnished by the theory of the exciton, which has been developing very intensively in recent

*) For more detail see \(^{21}\).

**) Similar results were obtained in \(^{23}\) for charged excitations of the Bose type. We agree, however, with A. I. Anselm that this circumstance is not a convincing argument in favor of the many-electron point of view, since a) in a number of problems there exists a statistical model to which the excitations are subordinate, and b) it is not clear to what extent the model adopted in \(^{23}\) can correctly describe the behavior of a semiconductor in a magnetic field. At the same time, we emphasize that neither of these objections applies to the work \(^{5}\).

The author of the present review proposed a new capture mechanism, associated with the transfer of energy not directly to the lattice, but to “plasma”-type oscillations, caused by the collective Coulomb interaction of electrons (and holes) with one another (in connection with this, an approximate theory of the electron plasma in semiconductors was developed in the work). It turned out that this mechanism apparently plays an important (if not decisive) role, provided it comes into action; the latter, however, can occur only at a sufficiently large (\(>10^{18}\ \mathrm{cm}^{-3}\)) concentration of current carriers (otherwise the transfer of the energy of the captured carrier into plasma oscillations becomes very difficult, since the corresponding vibrational quanta are too small).

III. SOME CONCLUSIONS

Analyzing the results of the section’s work, it seems to us possible to draw certain conclusions about the main directions in the development of the modern theory of semiconductors*). Apparently, four such directions may be distinguished (the point is not concrete important problems, of which there are of course far more, but precisely tendencies in the development of the theory).

a) The consideration of electrons and lattice vibrations as a single system. This means that the interaction of conduction electrons with phonons is regarded not simply as a cause of scattering (which determines the electrical resistance of the substance), but is included in the Hamiltonian of the zeroth approximation, i.e., is taken into account already in determining the stationary states of the system. The current carrier then proves to be a certain “conglomerate” of an electron and phonons. This point of view arose naturally in the theory of polarons in ionic crystals \(^{3,4}\), when the coupling of electrons with lattice vibrations is, as a rule, not weak; subsequently it became clear that it gives by no means trivial results also when applied to molecular and homeopolar lattices (reports by K. B. Tolpygo, M. F. Deigen, E. I. Rashba; see also \(^{17}\)).

b) A consistently many-electron approach to the solution of a number of concrete problems. The question of the possibilities and limits of applicability of the one-electron approximation in the theory of semiconductors has repeatedly been discussed in recent years \(^{7,18-21}\). However, apparently, complete agreement has not yet been reached, and, for example, A. I. Anselm in his report at the Leningrad Conference quite definitely defended the one-electron scheme, pointing out that to a certain extent it too takes account of the interaction of electrons (by the method of the self-consistent field), and also

*) Let us emphasize that what follows is a presentation of the point of view of the author of the review.

time in connection with a whole series of diverse problems25–28. Therefore the further development of the many-electron theory of semiconductors appears to us to be highly timely.

c) Abandonment of a number of idealizations in constructing a phenomenological theory. It has already been pointed out more than once (see, for example,1) that phenomenological theories (of rectification and other phenomena) based on the use of microscopic equations of kinetics, Poisson’s equation, etc., are strongly limited in their applicability because of a number of poorly justified idealizations (restriction only to the stationary case, neglect of the geometrical features of the given structure, etc.). Recently the aforementioned defects have gradually begun to be overcome, and the works mentioned earlier are very indicative in this respect. Besides the three indicated directions, broadly represented in the papers considered, one more should be indicated, although at the conference, unfortunately, much less attention was paid to it than it deserves. We have in mind the problem, posed long ago by Acad. A. F. Ioffe,21 of taking account of the nature of the chemical bond in the theory of semiconductors. One of the concrete expressions of this is

d) Taking account of the specific features of the energy spectrum in calculating kinetic coefficients.

Until recently all calculations of mobility, the Hall constant, and other kinetic coefficients related exclusively to an extremely simplified model of a semiconductor with nondegenerate bands and spherically symmetric isoenergetic surfaces. At the same time, direct experimental data on cyclotron resonance29 clearly show that the actual situation in a number of cases (in particular, in germanium and silicon) is much more complicated. This leads to the necessity of revising the existing theory of kinetic coefficients, also taking into account the inevitable modification of the law of interaction of current carriers with phonons. (Apparently, the features of the band structure have the most substantial effect on certain effects involving a magnetic field—changes of resistance in a magnetic field30, 31 and others.) Recently a whole series of works devoted to this problem32–37 has appeared, but the question is still far from being exhausted.

Such are some results of the work of the theoretical section of the VIII All-Union Conference on Semiconductors.

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