ON THE STATE OF SEMICONDUCTOR THEORY*)

A. I. Anselm

In my report I attempt to answer the following questions, which, from my point of view, are of interest to experimentalists: 1) why the study of semiconductors plays such a large role in the study of the solid state, 2) which propositions and results of semiconductor theory may be regarded as sufficiently well founded, and 3) what problems semiconductor theory poses for experiment.

The electronic theory of crystals began to develop with its application to metals. Semiconductor theory arose in the early 1930s, after the establishment of the laws of quantum mechanics and quantum statistics. In the prewar period, semiconductor theory received fundamental development in the works of A. Wilson, Ya. I. Frenkel, L. D. Landau, H. Fröhlich, N. Mott, B. I. Davydov, W. Schottky, and others. These works considered questions of the chemical potential in semiconductors, the motion of an electron (hole) in atomic and ionic crystals, current rectification at the contact between semiconductors and metals, exciton (currentless) excitation of a crystal, the effect of a strong electric field, and others. In the postwar period one should note the interesting works of S. I. Pekar on the theory of polarons with strong coupling, and the important works of W. Shockley and his collaborators on the theory of electron-hole transitions.

WHY THE STUDY OF SEMICONDUCTORS PLAYS SUCH A LARGE ROLE IN THE STUDY OF THE SOLID STATE

I do not intend to dwell on the question of why modern physics devotes such great attention to the study of semiconductors. The answer to this question is obvious—it is connected with the large and ever increasing role of semiconductors in modern technology and,

*) Report at the plenary session of the All-Union Conference on Semiconductors in Leningrad, November 14–20, 1955.

in particular, radio engineering. I wish to dwell only on the question posed in the title of this section.

First, let us note that the features of the zonal structure of the energy spectrum of an electron in a crystal manifest themselves much more concretely and diversely in semiconductors than, for example, in metals. Only in semiconductors do we distinguish with complete clarity two kinds of current carriers (electrons and holes), anisotropy of the effective mass (cyclotron resonance), localized energy levels (donors, acceptors, and surface levels), and the superposition of degenerate energy bands (tellurium, etc.).

Second, vibrations of the crystal lattice play a particularly active role in semiconductors. Thus, electrons in ionic crystals interact not only with the acoustic branch but, chiefly, with the optical one. Taking such interaction into account, we obtain an entirely different dependence of the electron mean free path on its energy and on temperature. In some cases the interaction between an electron and lattice vibrations cannot be considered small. The development of these ideas for ionic crystals led to the theory of polarons with strong coupling (S. I. Pekar) and to the theory of nonradiative transitions (E. I. Adirovich, S. I. Pekar, M. A. Krivoglaz, K. Huang, and A. Rhys). Finally, in semiconductors the intensity of lattice vibrations (temperature) determines such an important quantity as the concentration of current carriers.

Third, when the temperature of a semiconductor is changed, the concentration of current carriers and the ratio of the number of electrons to the number of holes change greatly. By varying the temperature in germanium from that of liquid helium to several hundred degrees Celsius, one can, under appropriate circumstances, change the concentration of current carriers by more than ten orders of magnitude. Such a wide change in concentration and the presence of two kinds of current carriers make it possible to investigate various kinetic processes much more diversely and profoundly: electrical conductivity, thermal conductivity, thermomagnetic and galvanomagnetic phenomena, etc. Thus, for example, the study of the Wiedemann–Franz law at different electron concentrations makes it possible to follow the change in the coefficient appearing in the formula from 2 (nondegenerate gas) to \(\frac{\pi^2}{3}\) (degenerate gas).

Fourth, the possibility of varying the concentration, the presence of two kinds of current carriers, and the two types of semiconductor have led to the creation of a new field of semiconductor physics and technology: rectification and amplification at electron-hole transitions. The development of these fields of solid-state electronics has played an outstanding role not only in radio engineering, but also in the study of physical processes in semiconductors (the determination of drift mobility and of the lifetime of current carriers, etc.).

Fifthly, a whole series of physical phenomena in general could be discovered only in semiconductors or dielectrics; among them are: the internal photoelectric effect, cyclotron resonance, phenomena associated with \(F\)- and \(F'\)-centers, the effect of a strong electric field, etc.

Finally, sixthly, I would like to note the deep connection that exists in semiconductors between electronic processes and all kinds of defects of the crystal. Quite apart from the fact that the very existence of impurity semiconductors is due to the presence of foreign atoms or lattice defects, the role of the latter is also great in the kinetics of electronic processes. Scattering of electrons by impurity ions is manifested in some cases at temperatures close to room temperature. An especially large role is played by the scattering of current carriers by compensated impurity ions (a high concentration of donors and acceptors compensating one another). Apparently, lattice defects can play a significant role in electron scattering. This makes it possible to think of a deeper connection that exists between the electronic and mechanical properties of crystals.

One could point to still a number of features of semiconductors that make their study so fruitful for the problem of investigating the properties of the solid state; however, we shall confine ourselves to what has been said above. Let us proceed to consideration of the second question.

WHICH PROPOSITIONS AND CONCLUSIONS OF THE THEORY OF SEMICONDUCTORS MAY BE REGARDED AS SUFFICIENTLY WELL FOUNDED

Let us first consider the very important question of the applicability of the one-electron approximation in crystals.

This question seems to me very important, since rejection of the one-electron approximation (band theory) deprives us not only of the customary scheme of energy bands, but also of the possibility of using Fermi statistics and the kinetic equation. One may encounter statements that the one-electron approximation in crystals is not justified and has exhausted its possibilities, so that its further development is inexpedient. Thus, for example, S. V. Vonsovskii writes^1: “At the present time the quantum theory of crystals is experiencing a certain crisis in connection with the fact that the most widespread one-electron (so-called ‘band’) model has already exhausted its possibilities and its further application to the explanation of the real properties of crystals leads to difficulties of a fundamental character. The cause of these difficulties lies mainly in the simplifying assumption of this model—the neglect of the interaction between electrons in the crystal.” Still more categorically writes V. L. Bonch-Bruevich^2 (note on p. 65): “From what has been said it is clear how fruitless are attempts at quantitative refinement and improvement of the methods of the ‘one-electron’ theory, hitherto still pre-

adopted in some works. These attempts, to be sure, are quite “harmless,” since they do not distort the “Fermian” character of the spectrum, but they are equally useless. Let us consider the question in more detail.

First of all, let us note that the frequently encountered assertion that in the self-consistent-field method (the one-electron approximation) the interaction between electrons is neglected is incorrect. In the self-consistent-field method we replace the action of all the other electrons on a given one by the field of their averaged position, so that we neglect only the correlation in the motion of the electrons. The successes of applying the self-consistent-field method to many-electron atoms are well known. Calculations of the electron density and of energy terms for many-electron atoms give not only good qualitative but also quantitative results. Finally, it should be borne in mind that the very scheme for filling the electron shells of atoms in the periodic system of elements is based on the one-electron approximation. There is no reason to think that the one-electron approximation is inapplicable to molecules. The Mulliken—Hund method also uses one-electron wave functions. All this makes it possible to think that the one-electron approximation is to a large extent applicable also to electrons in a crystal, since their average concentration in a solid is of the same order as in atoms and molecules. The conditions for applying the one-electron approximation are especially favorable in the case of electronic semiconductors.

At concentrations of conduction electrons of the order of \(10^{12}—10^{18}\ \text{cm}^{-3}\), the average distance between them is of the order of \(10^{-4}—10^{-6}\ \text{cm}\), i.e., sufficiently large for screening to take place.

In Pekar’s work \(^{3}\), devoted to the question under discussion here, it is shown that the Bloch one-electron functions can be justified as solutions of the Hartree—Fock self-consistent-field equations in the case of a conduction electron that weakly perturbs the electron shells of the atoms of the crystal. In other cases, as the author himself emphasizes, “it is not excluded, however, that the Bloch one-electron functions may be justified in some other sense.” Thus, for example, as Pekar showed in another work \(^{4}\), if the polarization of the electron shells of the atoms of the crystal adiabatically follows the motion of the “extra” electron, then the problem is likewise reduced to considering the motion of one electron in an external periodic field. Thus, band theory is in any case applicable to electronic semiconductors. However, for the reasons set forth above, I believe that the limits of its applicability are much broader. Although metals are not the immediate topic of my report, one may cite some arguments in favor of the legitimacy of applying the one-electron approximation to metals as well. As was noted by Gurevich \(^{5}\), the effective collision cross section, calculated for all conduction electrons of a metal, must be reduced

with respect to \(\left(\dfrac{kT}{\varepsilon_0}\right)^2\), where \(\varepsilon_0\) is the maximum Fermi energy. This circumstance is connected with the fact that colliding electrons can change their quantum state only in the case when their energies lie in the region where the Fermi distribution is smeared out. Against this it was objected that the Fermi distribution itself is based on the one-electron approximation and therefore cannot serve as its justification. However, we cannot regard this objection as well founded, since in theoretical physics we almost always justify the initial assumptions not a priori, but a posteriori (for example, in perturbation theory).

It may be worth noting that the one-nucleon approximation in the theory of the atomic nucleus, the so-called shell theory, was justified by Fermi\({}^{6}\) by the same considerations connected with the application of the Pauli principle to the nucleons in the nucleus. It should be emphasized that, for understandable reasons noted by Bohr\({}^{7}\), the one-nucleon approximation in the atomic nucleus is much less well justified than the one-electron approximation for atomic systems.

The first significant success of the many-electron model of crystals is connected with the quantum theory of ferromagnetism of Frenkel—Heisenberg. For the quantitative development of the theory Heisenberg made use of the Heitler—London model, according to which, in the zeroth approximation, near each lattice site there moves one electron in an \(s\)-state, unperturbed by the action of the other lattice sites. Consideration of the excited states of the atoms of the crystal allowed Frenkel to introduce currentless excitations of a dielectric—the so-called excitons. It should be noted that the phenomenon of ferromagnetism and the currentless excitations of a crystal do not fit within the framework of the one-electron approximation. The polar model of Shubin—Wonsowsky, in which the possibility is taken into account of an electron passing from one lattice site to another with the formation of a hole and a doublet, imparted to the Heitler—London—Heisenberg model the property of current conductivity—electronic and hole conductivity.

Every many-electron theory of a crystal has the advantage over a one-electron theory that in it the interaction of electrons (correlation in the motion of the electrons) is taken into account more accurately. However, it should be emphasized that concrete results in the many-electron theory are obtained only under certain simplifying assumptions.

First of all, it should be noted that the interaction of the conduction electrons of a metal with the electrons of the closed shells of the ions is, in the many-electron model, treated in the one-electron approximation. At the same time, the state of the electrons in the closed shells of the ions cannot differ essentially from the state of conduction electrons with energy \(\varepsilon\), if \((\varepsilon_0-\varepsilon)\gg kT\), where \(\varepsilon_0\) is the limiting Fermi energy.

Secondly, the polar model (as the author himself notes[^1]), by virtue of the approximations made in it, is applicable only to semiconductors and not to metals, i.e., in the case when the one-electron approximation is sufficiently justified.

Thirdly, the concrete conclusions of the many-electron theory are obtained as a result of using, for the expansion of the wave function, an incomplete system of atomic functions. It is very difficult to estimate the error thereby introduced.

Fourthly, the usual many-electron model, in which near each lattice site there is one electron, does not satisfy the condition of spin saturation characteristic of the closed shells of the atoms of a semiconductor. For this reason the quasiparticles of such a model obey Bose statistics, in contrast to the electrons and holes of band theory. Nor can one regard as justified, as has been done, the application of such a model to the explanation of the magnetic properties of a germanium-type semiconductor. Of course, in principle one can consider a model with two valence electrons at each site; however, it is unclear what results, comparable with experiment, can then be obtained.

Does it follow from this that we should altogether abandon the development of a many-electron theory of crystals? Of course not. First, such a theory can always lead to results of a fundamental character, both in the sense of discovering new effects and in the sense of interpreting known facts.

Secondly, there exist phenomena such as, for example, ferromagnetism, exciton excitation, the electrical properties of elements of the transition group, and possibly superconductivity, which in general cannot be explained on the basis of the one-electron approximation.

Thirdly, the schematic character and crude approximations inherent in the many-electron theory today may be overcome tomorrow.

Finally, fourthly, the many-electron theory must give us exact criteria for the applicability of the one-electron approximation.

However, we do not believe that band theory has exhausted its possibilities, that its further development is useless, and that all the difficulties it encounters in comparison with experiment are connected with an inaccurate allowance for the interaction between electrons in the self-consistent-field method. Moreover, we believe that band theory is quite well founded when applied to semiconductors. If one takes into account the successes of the application of the Hartree—Fock method to atomic systems, the features of the interaction of electrons associated with the Pauli principle, the results of the one-electron theory of metals and the conclusions of the many-electron theory that coincide with them to a considerable degree, then one should think that band theory remains applicable, to some extent, also to metals. Unfortunately, we cannot give a quantitative criterion for the applicability of the one-electron approxima-

tion, since the latter must follow from the exact many-electron theory.

The estimate sometimes used for the applicability of the one-electron method, based on comparing the total energy of an electron with the potential energy of its interaction with another electron located at the mean distance from it, cannot be regarded as convincing, since in the self-consistent-field method we do not ignore the interactions between electrons. As A. Wilson writes[^8]: “However, in doing so we do not discard the other electrons altogether. We simply replace their action by the action of a smeared-out field, which represents their field averaged in such a way that the correlation between the positions of the electrons is neglected.”

We do not consider it necessary to dwell in detail on the well-known successes of band theory. The fact that some results of band theory can be obtained in the many-electron approximation testifies, it seems to us, to a sufficiently sound justification of the one-electron approximation. Among the most notable recent successes of the band theory of crystals, we would like to mention the quantitative theory of cyclotron resonance[^9], based on a deep and concrete analysis of the structure of the energy bands of electrons and holes in germanium and silicon.

It should be emphasized, however, that, despite a number of successes of band theory, there is a fairly large number of observations that are not in agreement with it. Thus, for example, the effective mass of current carriers, determined from different phenomena observed in a semiconductor, turns out to be different; the temperature dependence of mobility does not satisfy the laws obtained either for atomic or for ionic crystals; the theory does not satisfy the temperature dependence of the thermoelectromotive force in semiconductors, etc.

However, we do not think that this discrepancy is explained by an inaccurate account of the interaction between electrons in band theory. We would like to point out a number of circumstances which, in various cases, are probably responsible for the fact that the one-electron theory is not in agreement with experiment.

First, the properties of the semiconductor samples studied by the experimenter are sometimes very far from those ideal crystals that figure in the theory. We have in mind here the existence of poorly conducting interlayers, various cases of dependence of the number and type of crystal defects on temperature, the presence of cracks, dislocations, etc. Taking these circumstances into account in the theory is difficult because of the great arbitrariness in the choice of the defining parameters.

Secondly, the possibility of degeneracy of the current carriers is not always taken into account in the proper way. Small effective masses of the current carriers promote the onset of degeneracy at unusually low concentrations. This circumstance can be taken into account in the theory comparatively simply.

Thirdly, only very recently has due attention been paid to taking account of the anisotropy of the effective mass of electrons in a crystal. In a recently published work by Herring \({}^{10}\) it was shown that, under certain simplifying assumptions, the general form of the formulas for various kinetic phenomena remains the same even when the anisotropy of the effective mass of the current carriers is taken into account. However, the components of the tensor of the reciprocal effective mass enter in different ways into different effects, which can in principle explain the nonconstancy of the values of the “average” effective mass measured by experimenters.

Fourthly, we must take into account the possibility that very different mechanisms of scattering of current carriers may exist. Besides the usual scattering by acoustic and optical vibrations and by impurity ions, in some cases it is necessary to take into account scattering by optical branches of the atomic crystal, by neutral impurity atoms, by piezoelectric vibrations, by transverse waves of vibrations, as well as scattering caused by interaction with two phonons, etc. Of course, not all these mechanisms will act simultaneously, but under certain conditions each of them may prove essential.

Fifthly, we must allow for the possibility of the existence in ionic crystals of polarons with strong coupling. In this case the formulas for kinetic effects undergo a considerable change.

Sixthly, we must reckon not only with the anisotropy of the effective mass of the current carriers in a crystal, but also with the possibility of a complex structure of the energy band. The existence of such bands with a complex structure has been proved theoretically and experimentally in the cases of germanium and silicon. There are many grounds for assuming a complex band structure in tellurium, etc. Calculations of kinetic effects then become very difficult, since in this case we are forced to abandon the effective-mass method.

Seventhly, we have still studied very little the influence of the form of the vibrational spectrum of a crystal on the scattering of current carriers. Meanwhile the true form of the vibrational spectrum, even in the simplest case of an NaCl crystal, differs substantially from Debye’s schematic assumptions.

Finally, eighthly, we must remember the limits of applicability of perturbation theory in calculating the probability of transition of an electron when it collides with phonons and the applicability of kine-

…kinetic equation. Both applicability criteria coincide and reduce to the requirement \(\lambda/l \ll 1\), where \(\lambda\) is the de Broglie wavelength of the electron, and \(l\) is its mean free path. Unfortunately, quite often, especially for ionic crystals, this inequality is not satisfied, and in this case we have no grounds for demanding agreement between theory and experiment. This limitation of the existing theory seems to us the most severe and the most difficult to overcome.

The questions noted above do not in any way claim to be a program for the further development of semiconductor theory. They merely indicate those features of band theory which, within the framework of the one-electron approximation, may fail to correspond to the real properties of semiconductors. It seems advisable to us to draw the attention of experimentalists to this aspect of the question, which is essential for comparing theory with experiment.

WHAT TASKS DOES THE THEORY OF SEMICONDUCTORS SET BEFORE EXPERIMENT?

I must again stipulate that I in no way claim to put forward a general program of experimental research in semiconductor physics. Quite apart from the fact that such a task is beyond my powers, it is obvious that such a program is determined mainly by the needs of the technical utilization of semiconductors, and not by the requirements of theory.

I would like to draw attention only to those directions and tasks of the experimental study of semiconductors which, being especially essential for the further development of the theory, have not received proper development in our country. Since I am not an experimentalist, I shall confine myself only to the most general considerations, without going into details.

I think that in a number of cases the discrepancy between theory and experiment is due to the fact that the experimentalist cannot precisely establish with what substance he has worked. Therefore, from the point of view of theory, the broad development of work on obtaining ultrapure substances with precisely controlled impurities appears very important.

It is also necessary to apply old methods of structural analysis more widely, and to develop new methods, for the study of defects in crystals. The main point here is to increase the sensitivity of the methods, which should make it possible to detect defects in crystals (impurities, interlayers, cracks, etc.) at concentrations of the order of \(10^{18}\ \text{cm}^{-3}\) and lower.

Insufficient attention is being paid in our country to the investigation of the influence of crystal anisotropy on various kinetic phenomena. Such studies are essential, since the theory is developed mainly for single crystals.

Until recently, experimenters have mainly measured three quantities pertaining to semiconductors: the electrical conductivity $\sigma$, the Hall constant $R$, and the thermoelectromotive force $\alpha$. Under certain conditions this is sufficient for determining three quantities: the concentration of current carriers $n$, their mobility $u$, and their effective mass $m^*$. However, for testing the theory it is desirable to study a large number of kinetic effects (thermal conductivity, changes in resistance in a magnetic field, the Nernst effect, etc.), since only in this case can one be convinced of the internal consistency or inconsistency of the theoretical assumptions.

The latter case is especially important, since it compels us to examine more deeply the phenomena occurring in a semiconductor. For testing the theory it is also essential to study kinetic effects over a broader range of temperatures and of magnetic- and electric-field strengths. Particularly interesting are investigations of semiconductors at helium temperatures and in strong magnetic fields, $uH/c \gg 1$.

A study of thin films at low temperatures would be of great interest; apparently, it could provide a direct method for measuring the effective mass.

Research in solid-state spectroscopy is being carried out insufficiently, in particular infrared spectroscopy connected with electronic transitions, and X-ray spectroscopy in the region of soft X-rays.

An unacceptably small amount of experimental research is being conducted on paramagnetic resonance in solids, discovered in our country by E. K. Zavoiskii,^11 and no work at all is being done on cyclotron resonance, discovered by Ya. G. Dorfman^12 in our country and by Dingle^13 abroad.

In general, work on the radiospectroscopy of the solid state must be broadly developed; it is a powerful source for the study of energy levels and the properties of crystals.

Research on the magnetic susceptibility of semiconductors is being carried out insufficiently. Meanwhile, as Prof. A. G. Samoilovich emphasizes, the study of equilibrium states of a semiconductor has a number of advantages in comparison with the investigation of kinetic processes.

Finally, work on the study of atomic-diffusion processes in semiconductors, and on the influence of external mechanical actions (pressure, uniaxial tension along different crystallographic axes, etc.) on the electrical properties of semiconductors, is being carried out quite insufficiently.

I would like to confine myself to consideration of the above-mentioned directions of experimental study of semiconductors, which seem to me essential for the further development of the theory.

References Cited

1. S. V. Vonsovskii, UFN 48, 289 (1952).
2. V. L. Bonch-Bruevich, UFN 56, 55 (1955).
3. S. I. Pekar, ZhETF 18, 525 (1948).
4. S. I. Pekar, Studies in the Electron Theory of Crystals, Gostekhizdat, 1951, p. 21.
5. L. E. Gurevich, DAN SSSR 20, 355 (1938).
6. E. Fermi, Lectures on Atomic Physics, IL, 1952, p. 72.
7. N. Bohr, Nature 137, 344 (1936).
8. A. Wilson, The Quantum Theory of Metals, Gostekhizdat, 1941, p. 25.
9. G. Dresselhaus, A. F. Kip and C. Kittel, Phys. Rev. 98, 368 (1955).
10. C. Hering, Bell Syst. Techn. Journ. 34, 237 (1955).
11. E. K. Zavoiskii, ZhETF 15, 253, 344 (1945).
12. Ya. G. Dorfman, DAN SSSR 81, 765 (1951).
13. R. B. Dingle, Proc. of the Internat. Conf. on Very Low Temperat., edit. by R. Bowes (Oxford, England, Aug. 1951), p. 165.