SOME ISSUES IN THE THEORY OF SEMICONDUCTORS AND DIELECTRICS AND PATHS FOR FURTHER DEVELOPMENT OF THE THEORY*)

S. I. Pekar

1. The phenomenological theory of semiconductors, based on the equation of electrical conductivity and diffusion of current carriers, Poisson’s equation, the exponential dependence of the concentration of current carriers on temperature, etc., is in a comparatively satisfactory state in the case of not-too-large fields that do not disturb the thermal equilibrium of the conduction electrons, and also when one may use the concept of local thermal equilibrium of current carriers (a small dependence of the chemical potential of the electrons on the coordinates). This includes the diffusion theory of rectifiers with chemical and contact blocking layers, the theory of the contact of a semiconductor with a metal and of the contact of two semiconductors, the theory of the dependence of the thermoelectron work function from a semiconductor into vacuum on the electric field in vacuum (the latter penetrates into the semiconductor and is gradually screened in it, causing a “bending” of the energy bands).

The phenomenological theory of processes in which the local thermal equilibrium of the conduction electrons is disturbed is in a somewhat worse state: the theory of the internal photoeffect, the consideration of current in a semiconductor with a gradual transition from electronic conductivity to hole conductivity (an \(n—p\) transition), the theory of bipolar diffusion, injection, the theory of conductivity caused by irradiation with fast particles, the theory of nonstationary relaxation processes, etc. The difficulties here are connected with the need to abandon the formula of the equilibrium distribution of electrons over energies and to replace it by kinetic equations containing many

*) A revised and supplemented presentation of a report at the VIII All-Union Conference on Semiconductors. The literature references in no way claim to be complete, but are only examples characterizing the present state of the theory and making it possible briefly to indicate various methods of the theory.

poorly known parameters (for example, the quantum yield of the internal photoeffect, the probabilities of thermal transitions of electrons, the positions of local electron levels in the forbidden band, etc.).

A substantial shortcoming of the phenomenological theory is the large number of parameters entering into it, such as: 1) the mobilities of conduction electrons and holes; 2) their diffusion coefficients; 3) their effective masses; 4) the concentrations and energies of many types of local electron states in the semiconductor; 5) the magnitudes of the heat releases accompanying optical transitions of electrons between different energy levels (these quantities determine the difference between the energies of optical transitions and the activation energies of the corresponding thermal transitions of electrons); 6) the probabilities of thermal transitions of electrons between different energy levels and their substantial temperature dependences (for example, the coefficients of linear and quadratic recombination of current carriers, etc.); 7) the spectra of intrinsic and impurity absorption of light; 8) the probabilities of formation of current carriers under the influence of light and corpuscular radiation; 9) the work function of electrons from a semiconductor into vacuum and from a metal into a semiconductor; 10) other characteristics of the contact between a semiconductor and a metal; the concentrations and energies of surface electron states, etc.

The calculation of most of these parameters is a task of the microtheory of semiconductors. However, the latter is in an unsatisfactory state (see below) and, with rare exceptions, does not yet make it possible to calculate these parameters reliably. It should be borne in mind that only a part of the parameters listed above enters into the theory of each particular phenomenon; nevertheless, the number of parameters is often still too large, and they cannot be determined even by comparing theory with experiment. In these cases, owing to the large number of unknown parameters, the approximating power of the formulas of the theory is too great, and even agreement of the theory with experimental data becomes little convincing. In these cases the role of the phenomenological theory is reduced merely to the schematization and systematization of experimental data.

A second substantial shortcoming of the modern phenomenological theory of semiconductors is its ignoring of the nonideality of the object. For example, in the theory of the contact between a semiconductor and a metal, the surface of the metal is almost always assumed to be ideal (a plane or a sphere). In reality, however, it has a polycrystalline, sharply angular relief, at whose peaks the lines of force of the electric field crowd together. As a result, at the peaks the electric-field strength may exceed the mean field over the cross section by a factor of a hundred. The problem becomes substantially nonuniform in the near-electrode layer, with a thickness of the order of the distance between the peaks. If the potential drop in this layer is substantial, then the influence of the relief

will be reflected in the results. This occurs, for example, in semiconductor rectifiers with a blocking layer adjacent to the metallic electrode and having a thickness of \(10^{-4}\)—\(10^{-5}\) cm. In the blocking direction, the entire potential difference applied to the rectifier falls across the blocking layer, and the whole problem of the theory reduces to considering this layer. This essentially inhomogeneous problem is replaced in the modern phenomenological theory by a one-dimensional one.

There is an attempt to take into account the sharply pointed relief of the metallic electrode in the theory of the contact blocking layer\(^1\), but in most subsequent works on contact theory the influence of the relief is still ignored.

As another example, let us cite the frequently observed substantial dependence of the electrical and photoelectric properties of a semiconductor on its polycrystalline structure. Theory, however, usually ignores the structural factor.

As a consequence of the above-mentioned shortcomings, the modern phenomenological theory of semiconductors (with the exception of rare favorable cases) is not an exact quantitative theory. Its accuracy and content will gradually increase as its numerous parameters become known. The latter, as well as the relations between them, will become known partly through the development of microtheory, and partly through the formulation of special experiments that make it possible to measure these parameters separately in the study of elementary phenomena.

However, even in its present form the phenomenological theory is useful as a qualitative theory, making it possible to understand many phenomena, to establish connections among them, to explain and qualitatively predict the role and influence of various factors, and to calculate orders of magnitude.

Further development of the phenomenological theory of semiconductors in the following directions appears desirable.

A) The theory of semiconductor devices (rectifiers, triodes, thermocells, photocells, photoresistors, photo-, thermo-, and secondary-electron cathodes, etc.). A connection must be established between the technical characteristics of devices and the parameters and structure of semiconductors. The theoretical limits of the capabilities of semiconductor devices, the optimal data for semiconductors used in devices, should be determined; a comparative characterization of the various possible designs of a device should be given, and from all this the paths for possible improvement of devices should be determined.

B) Generalization of the phenomenological theory to the case of large fields and currents, which lead to violation of the thermal equilibrium of conduction electrons, to an increase in their mean thermal energy, to ionization of the atoms of the semiconductor by impacts of fast conduction electrons, etc. These phenomena are essential, for example, in rectifiers under

in current in the blocking direction, when the blocking layer is in the pre-breakdown state. Under these conditions Ohm’s law is inapplicable; the usual equation of electrical conductivity and diffusion is also inapplicable and requires substantial generalization. The studies known up to now of the electrical conductivity of certain semiconductors in large electric fields (Poole’s law \(^{2}\), Frenkel’s law \(^{3}\)) are insufficient, since they refer only to the case of a spatially homogeneous semiconductor and a homogeneous electric field, and in the general case are inapplicable. To be convinced of this, it is enough to consider the case of the absence of current when the electric-field strength and the gradient of the concentration of conduction electrons are nonzero. In this case the concentration of current carriers depends on the field, but according to Boltzmann’s law, and not according to Poole’s or Frenkel’s law. Further it turned out that the increase of electrical conductivity with the electric-field strength is expressed by different formulas in different semiconductors. Such parameters of the phenomenological theory as the recombination coefficient of conduction electrons also become functions of the field and the current. The dependence of the concentration of conduction electrons on temperature changes completely. All these questions have scarcely been investigated.

C) Supplementing the commonly considered problems of electrical conductivity and diffusion of current carriers with problems of heat conduction in connection with Joule heat release, additional heat release in places with predominant recombination of electrons, and absorption of heat in places with predominant dissociation of electrons. Taking into account the positive or negative Peltier heat, which in semiconductors is much greater than in metals. These thermal effects are significant in rectifiers and thermocathodes (especially at large emission in the pulsed regime).

D) Supplementing the phenomenological theory by taking exciton states into account. The latter sometimes play an essential role in optical phenomena, the internal and external photoelectric effect, and also give rise to additional heat conduction, carrying energy and releasing it in the form of heat in that part of the crystal where excitons annihilate without luminescence.

E) Finding and calculating experiments that make it possible to measure separately the parameters of a semiconductor in the study of elementary phenomena.

1. Kinetics of electrons in the conduction band of a semiconductor. The importance of this branch of the theory is obvious, if one takes into account that it is used to calculate the mobility of current carriers, their diffusion coefficient, the thermal conductivity of electrons, the thermoelectric emf, the Hall effect, thermo- and galvanomagnetic phenomena, etc. However, the state of this branch of the theory is generally unsatisfactory: only in rare cases are results obtained that agree quantitatively with experiment (for example, the calculation of the mobility due to

scattering by ionized impurities, and also the Hall effect\(^{4-6}\)). In a number of cases the results agree with experiment only qualitatively (for example, the temperature dependence of the mobility due to scattering by thermal vibrations), and sometimes even contradict experiment qualitatively (for example, the temperature dependence of the thermoelectric power). The good agreement of kinetic calculations with experiment in some cases and the contradiction to experiment in other cases is apparently connected with the fact that the kinetic equation and the assumptions adopted in solving it are justified in the former cases, and inadmissible in the latter. The range of applicability of the kinetic equation and of the additional assumptions has not been sufficiently investigated, and therefore the kinetic equation is often used outside its range of applicability.

The method of the kinetic equation rests essentially on the notion that the interaction of a current carrier with the lattice has the character of rare “collisions,” and that over the course of a mean free path it moves conservatively, like a particle in an external electromagnetic field applied to the crystal. Such a notion could not have raised doubts in the time of Boltzmann and Lorentz; however, it must be reconsidered with the advent of wave mechanics. The latter has shown that the energy of a current carrier is not an entirely precise concept and that its uncertainty is

\[ \Delta E \gg \frac{\hbar}{\tau}, \]

where \(\tau\) is the mean free time. The mean \(\bar{\tau}\) can be related to the mobility of the current carrier

\[ u=\frac{e}{M}\bar{\tau}, \tag{1} \]

where \(M\) is the effective mass of the current carrier.

Thus,

\[ \Delta E \gg \frac{\hbar e}{M u}. \tag{2} \]

If one assumes that \(M\) is equal to the mass of a free electron, and \(u = 50\ \mathrm{cm}^2/\mathrm{sec}\cdot\mathrm{V}\), then one obtains \(\Delta E > 0.023\ \mathrm{eV}\). Consequently, the uncertainty of the energy proves to be greater than the thermal energy of the current carrier at room temperature. Under such conditions, of course, the above-mentioned initial notion on which the kinetic equation is based loses its meaning, and the kinetic equation cannot be used. Meanwhile, the values of \(M\) and \(u\) adopted above are typical. Thus, the very method of the kinetic equation proves to be unjustified for a large group of semiconductors.

The objection considered above against the concept of a mean free path is practically removed for semiconductors for which

\[ \frac{M}{m}u \gg 500\ \frac{\mathrm{cm}^2}{\mathrm{sec}\cdot\mathrm{V}}, \tag{3} \]

\(m\) is the mass of a free electron. This is why, in such semiconductors as germanium, which possess a high mobility of current carriers, the application of the kinetic equation in a number of cases has led to a reasonable result consistent with experiment (mobility, thermoelectric emf).

The second important initial conception of kinetics is the conception of the current carrier, of its internal characteristics, and of the form of the energy of its interaction with the vibrations of the crystal. It is precisely these conceptions that determine the probability of “collision” of the current carrier with the lattice, the time and length of the free path, etc. To show how complicated this question is, let us consider the example of crystals with an ionic lattice. If the current carrier is taken to be an electron in the usual “band” state, calculated for ions fixed at the lattice sites, if this electron is characterized by the effective mass \(\mu\), and if the entire interaction with the polarization vibrations of the crystal is regarded as a small perturbation causing scattering of current carriers, then we arrive at the simplest and most popular conception of the current carrier. On the basis of this conception Fröhlich and Mott\(^{7}\) calculated the mobility of current carriers. In order that, in the case of copper oxide, the calculated mobility coincide with the observed one, they had to take

\[ \frac{\mu}{m}=\frac{1}{4}. \]

Moreover, \(\frac{\mu}{m}u \ll 500\), so that the method of the kinetic equation used by them is not justified. In general, the probability of scattering of the current carrier which they obtained was so large that the interaction with vibrations can no longer be considered a small perturbation. But there also exists another conception of the current carrier, introduced by the author, according to which the latter is a polaron\(^{8}\) (further references are given there). Here the state of the conduction electron is calculated not for ions fixed at the lattice sites, but for ions capable of being displaced. The interaction of the electron with polarization vibrations is not assumed to be a small perturbation, but is introduced into the Hamiltonian already in the zeroth approximation. As a result it turns out that the greater part of the energy of interaction of the electron with the vibrations does not lead to scattering, but forms, together with the electron, a “polaron,” moving conservatively through the crystal with a permanently conserved momentum and current; only a comparatively small part of the energy of interaction with polarization vibrations acts as the cause of scattering of polarons. Therefore the free-path time of a polaron \(\tau\) turns out to be much greater than that of a band electron. At the same time the effective mass of the polaron \(M\) is usually found to be much greater than the mass of the band electron \(\mu\). Relation (1) is valid both for band electrons and for polarons. Therefore, for a given crystal with a given mobility \(u\), the quantum-mechanical uncertainty of the energy \(\Delta E\) will be \(\frac{M}{\mu}\) times smaller for polarons than for band electrons, and the region of applicability

the kinetic equation will be just as many times broader for polarons. In addition, the expression for the probability of scattering of a current carrier in interaction with lattice vibrations is different in the polaron and band theories.

From the example considered it is clear that both the theoretical value of the mass of a current carrier and the probability of its collisions with the lattice depend essentially on the microscopic theory of the states of conduction electrons and will change as the microscopic theory is developed. This assertion, paradoxical at first sight, is in fact a reflection of a general proposition of quantum mechanics, according to which the closer an approximately calculated state of an isolated system is to its exact stationary state, the smaller is the probability of a quantum transition from this state to another.

Further development of the kinetics of conduction electrons appears desirable in the following directions.

A) For semiconductors with small $\dfrac{M}{m}u$, for which the kinetic equation is inapplicable, to develop a method for calculating the $\psi$-function of the system from the time-dependent Schrödinger equation in a nonstationary state in the presence of an external electric field applied to the crystal. The current is then calculated from the formula

\[ I=i\frac{e\hbar}{2m}\left[\psi\nabla\psi^*-\psi^*\nabla\psi\right] \tag{4} \]

without using the kinetic equation.

The solution of this problem would raise the theory of semiconductors to a higher level. This solution would also find application in other areas of theoretical physics.

B) To introduce into kinetics new, more precise concepts of the current carrier (polarons, elementary excitations) and, correspondingly, new expressions for the interaction energy of the current carrier with lattice vibrations and new expressions for the probability of “collisions.”

C) A critical analysis of the theories of mobility, diffusion, thermoelectric power, the Hall effect, thermomagnetic and galvanomagnetic effects; verification of the validity of simplifying assumptions and removal from the theory of unjustified assumptions. Bringing the theory into systematic, at least qualitative, agreement with experiment.

D) Application of kinetics to the theory of semiconductor devices.

1. The theory of thermal (nonradiative) transitions of electrons in semiconductors is necessary for calculating such important quantities as the probability of recombination of current carriers, the probability of their transfer into the conduction band from local states, the quantum yield of the internal photoelectric effect and its temperature dependence, the quantum yield of luminescence and its temperature dependence, etc. This theory is one of the most difficult and least developed branches of the theory of semiconductors.

In the work of Męglikh and Rompe^9, the interaction of an electron with the thermal vibrations of a crystal is considered as a small perturbation that is the cause of thermal transitions of the electron. A small anharmonicity of the vibrations is taken into account. As a result, it turns out that, in the first approximation, only transitions with a change in the electron energy by one vibrational quantum of the lattice, \(\hbar \omega_x\), are allowed; in the second approximation, transitions occur with a change in the electron energy by two vibrational quanta, etc. The thermal transitions usually observed, with a change in the electron energy by thirty to forty vibrational quanta \(\hbar \omega_x\), are allowed in the theory of Męglikh and Rompe only in the thirtieth to fortieth approximation and have a negligible probability (as the transition energy increases by \(\hbar \omega_x\), the transition probability decreases by a factor of \(10^6\)). This theory predicts probabilities for transitions that are far too small in comparison with those observed experimentally. Moreover, it is practically impossible to calculate such high approximations for a real model of a crystal.

In subsequent works on the theory of thermal transitions, by Adirovich^10, Huang and Rhys^11, Kubo^12, Davydov^13, and Krivoglaz^14, thermal transitions are considered on more correct foundations: the interaction of the electron with the vibrations is not assumed to be a small perturbation; instead, it is assumed that the state of the electron adiabatically follows the motion of the atoms; the cause of the transition is the nonadiabatic term, which is assumed to be a small perturbation.

A shortcoming of Adirovich’s work is the overly simplified model of a local electronic center (the electron is considered in a rectangular potential well; the vibrations of the atoms and their influence on the electron are replaced by vibration of the radius of the potential well). A shortcoming of the work of Huang and Rhys is the unjustified replacement of the influence of ion vibrations on the electron by the influence of a certain equivalent uniform electric field. Kubo does not take into account the changes in the equilibrium positions of the atoms as a result of the electron transition, whereas in reality this plays a greater role than the change in the frequencies of the normal vibrations that he takes into account.

Perhaps M. A. Krivoglaz went further than the others. He considered a very general case of a crystal of arbitrary structure and an electron localized in a “center” of any nature. The model of the center is not specified. The initial assumptions of the theory (harmonicity of the vibrations of the atoms, etc.) are fairly close to reality. It is especially important that Krivoglaz showed and used the connection between the parameters of the theory of thermal transitions and the parameters of the theory of the corresponding photo-transitions of the electron (the half-width of optical absorption bands, the Stokes shift). Thus, in a number of cases he obtained the temperature dependence of the probability of a thermal transition without unknown parameters^15. In two cases this temperature dependence was compared with experiment; the theoretically calculated tempera-

temperature dependence of the quantum yield of fluorescence of CaWO$_4$, activated by Pb, and also of MgWO$_4$, proved to be in agreement with Flamm’s measurements$^{16}$.

Since a thermal transition of an electron is most often accompanied by a large local fluctuation of the atomic configuration, the applicability of the approximations used in the above-mentioned theories is questionable, including the harmonic approximation in considering atomic vibrations, the linear dependence of the electron’s potential energy on atomic displacements, etc.

Theories in which the cause of a nonradiative transition of an electron is small nonadiabaticity are applicable only to transitions of an electron between discrete energy levels; in the region of the continuous energy spectrum of the electron the adiabatic approximation is in general inapplicable. In the theoretical consideration of a transition from a discrete level to the conduction band, or conversely, the following ideas have emerged in recent years:

a) If the energy spectrum of the electron in a local “center” becomes infinitely dense as one approaches the bottom of the conduction band, as occurs, for example, when the center abandoned by the electron remains positively charged, then the decisive stage in the process of capture of a conduction electron may be regarded as the transition of the electron from the conduction band to one of the mentioned discrete levels, densely situated just below the very bottom of the band. In such a transition one quantum of lattice vibrations is produced. Therefore the probability of such a transition can be calculated with the aid of ordinary perturbation theory, without resorting to the adiabatic approximation. The captured electron drops out of the number of current carriers. Thus the electron transition considered above can be identified with an act of recombination, provided only that the return of the electron to the conduction band is much less probable than its transitions to the lowest levels of the local center. The latter occurs only under conditions in which the concentration of conduction electrons considerably exceeds their concentration at thermal equilibrium.

b) During a large fluctuation of the configuration of the atoms surrounding the center, the discrete energy levels of the electron in the center, adiabatically following the deformation of the lattice, may all merge into the conduction band, so that in the center at the moment of fluctuation there will be no discrete levels at all. At this moment the electron will leave the center and go into the conduction band, and when the fluctuation has passed, the discrete levels will again separate out from the band, but the electron will no longer be on them. In this way thermal ionization of a local center may occur. Calculation of the probability of such a phenomenon does not require consideration of any quantum transitions of the electron. It is only necessary to determine the configurations of the atoms for which the discrete levels in the local center disappear, and to calculate the probability of such configurations. A similar problem, in the particular case of an ionic crystal,

when the center is a point positive charge, was considered by W. Schottky^17 under a number of simplifying assumptions.

Further work on the theory of thermal transitions is desirable in the following directions.

A) To compare carefully and repeatedly the existing theories of thermal transitions with experiment, to determine their accuracy and the limits of applicability. It should be recommended that parallel experimental studies be made of the absorption and luminescence spectra of the corresponding optical transitions (measurements of the half-width of the bands and of the Stokes shift), as well as of the frequencies of atomic vibrations, which will make it possible to determine the parameters of the theory of thermal transitions in an independent way.

B) Generalization of the theory and its extension to new cases, in particular the development of methods making it possible to consider transfers of electrons into a continuous spectrum and vice versa.

C) Clarification of the question of in which cases the thermal transition of an electron is accompanied by a large local fluctuation of the atomic configuration, requiring that anharmonicity be taken into account, the assumption of small displacements of atoms be abandoned, etc.

D) Introduction into the general theory of thermal transitions of elements borrowed from the microscopic theory of electronic states: calculated energy levels and electron wave functions, transition matrix elements, etc. This will make it possible to calculate a number of parameters of the theory of thermal transitions.

E) Use of the results of the theory of thermal transitions in the phenomenological theory of semiconductors (temperature dependence of the quantum yields of the photoeffect, luminescence, lifetimes of current carriers, and other relaxation times; consideration of the saturation current in the case when, in a semiconductor, electronic conductivity gradually changes into hole conductivity and the current is limited by the frequency of thermal dissociation of electrons and holes).

1. Theory of phototransitions of electrons in semiconductors. Phototransitions of electrons and phenomena associated with them are used in many semiconductor devices: photocells, photoresistors, phosphors, light filters, etc. In addition, the study of the spectra of light absorption and luminescence is one of the principal methods for investigating the composition and structure of semiconductors, the energy spectrum of electrons in semiconductors; and the study of semiconductors under pulsed illumination makes it possible to determine important kinetic parameters (the lifetime of current carriers, the probabilities of thermal transitions of electrons, etc.).

The theory of phototransitions of electrons in an ideal impurity-free crystal, based on the usual band theory, has existed for more than 30 years, has gained great popularity, and is very widespread in the literature. Nevertheless it must be admitted that this theory does not

does not make it possible either to calculate the spectra of absorption and emission of light, or even to predict their form qualitatively. This is the result of: 1) the practical impossibility of calculating the wave functions of an electron in a periodic field and the density of energy levels in the allowed bands (indeed, the periodic field itself is not known exactly, and its introduction for the crystal’s own electrons is justified only in rare cases); 2) the more complex nature of the excitation of the crystal electrons, which cannot be interpreted as a transition from one band to another and in general cannot be reflected by the approximation of band theory.

As an example let us cite the formation of excitons. The theory of excitons, first proposed by Frenkel in 193118 and subsequently developed as applied to ionic crystals by Wannier19, Mott20, and Pekar and Dykman21, and as applied to molecular crystals by Davydov22, explained the absorption of light that does not lead to the photoconductivity of the crystal. At the present time there is no doubt whatever as to the existence of excitons. The experiments of Apker and Taft23, Gross and collaborators24, and others25 convincingly prove the existence of excitons and confront theorists with the task of further developing the theory of excitons.

At present there are two directions in the theory of excitons: the first of them is based on the Heitler—London—Heisenberg method (H. L. H.). Here it is assumed that the energy spectra and wave functions of the electrons of the atoms (or molecules) from which the crystal is built are known; the wave function of the system in the zero approximation is constructed as a product of atomic wave functions. This is permissible if the interaction between atoms (molecules) is small. Then, introducing the interaction as a small perturbation, one can obtain a correction of the first order of smallness, i.e. calculate the small difference between the spectrum of crystals and the spectrum of an isolated atom (molecule). This method is suitable rather for molecular crystals, in which the spectrum of light absorption is similar to the spectrum of absorption by a gas, but this method is little suitable for semiconductors, in which such a similarity is most often absent. Quantitative calculations encounter difficulties similar to those met in band theory and in the calculation of polyatomic molecules.

The second direction in the theory of excitons consists in considering the bound motion of an electron and a hole, to which the corresponding effective masses are assigned, after which the periodic potential is ignored. In this (so far the only) case there exists a quantitative calculation of excitons. However, such a model is good only if the effective radius of the exciton exceeds the lattice constant. Such cases are realized (for example, cuprous oxide, germanium, and in general semiconductors with large $\frac{\varepsilon}{\mu}$). A serious test-

of this direction of the theory will be an attempt at a quantitative calculation of the experimental results of Gross and co-workers, who are studying excitons in cuprous oxide.

Excitons can be formed only if the frequency of the exciting light lies in definite ranges, for example at the red edge of the intrinsic absorption region. At other values of the frequency, photo-transitions are certainly possible in which a free electron and hole are produced, capable of carrying current. Thus, the formation of excitons does not exhaust even the most important kinds of photo-transitions of electrons in a crystal. The question remains open of non-excitonic transitions, of whether their qualitative interpretation is possible solely on the basis of band theory or whether the creation of new supporting concepts of the theory is necessary.

In summary, it should be noted that band theory as yet does not give a qualitative interpretation of the details of the absorption spectrum of a crystal: the positions of the maxima, the ratios of their heights and widths, etc. The theory of excitons makes it possible to interpret these details of the spectra qualitatively, but it applies only to a particular type of photo-transition. In rare favorable cases (for example, large exciton radii), a quantitative calculation of excitons is possible.

The theory of impurity absorption of light and impurity luminescence of semiconductors and dielectrics is in an incomparably better state. It concerns photo-transitions of electrons localized at impurities and defects of the crystal. A very substantial feature of this case, favorable for the theory, is the discreteness of the energy spectrum of the optical electrons, which makes it possible to apply the adiabatic approximation, according to which the state of the electrons follows adiabatically the motion of the atomic nuclei. It is not assumed here that the interaction of the electrons with the vibrations of the atoms is small. On the basis of this approximation a theory was developed of the shape and temperature dependence of the bands of impurity absorption and luminescence, at first for special cases (ionic crystals, particular types of local centers26–31), and then for the very general case of a crystal of arbitrary structure and a center of any nature32–35.

Especially simple and general results are obtained in the case of large heat release in photo-transitions, i.e., when this heat release is considerably greater than the quantum of atomic vibrations \(h\omega\). The shape of the absorption and luminescence bands, their half-width, dependence on temperature, the Stokes shift, the relation between the energy of the photo-transition and the activation energy of the corresponding thermal transition of the electron, etc., have been calculated. In those cases in which the criterion of applicability of the theory is fulfilled, and also when exact experimental data are available, the theory has repeatedly proved to be in quantitative agreement with experiment.

Less general and more complex results are obtained in the case of small heat release. Comparison with experiment here proved to be

although favorable for the theory, is still too small and does not permit final conclusions to be drawn.

Further development of the theory of photo-transitions of electrons in crystals in the following directions appears desirable.

A) Development of the theory of quantum stationary states of the system, since it is precisely its inadequacy that most of all limits the development of the theory of photo-transitions (the theory of stationary states is the subject of the next chapter of the article).

B) Creation of a more exact and consistent theory of the internal field acting on the optical electron from the polarization fields of the light wave of the surrounding atoms of the crystal. The Lorentz expression for the internal field does not take into account the inhomogeneity of this field within the limits of the elementary crystal cell. As a consequence of this inhomogeneity, the internal-field coefficient is transformed from a constant into an oscillating function of the coordinates with the periods of the crystal, and the matrix element of the photo-transition is not reduced to the matrix element of the dipole moment even when the ratio of the wavelength of light to the lattice constant is large. Taking this circumstance into account will lead not only to a quantitative change in the value of the probability of a photo-transition, but will also substantially change the selection rules.

C) Development of new approximate methods that make it possible to consider photo-transitions under strong interaction of electrons with atomic vibrations and considerable heat release, but do not use the adiabatic approximation, for this approximation is applicable only when considering localized electrons (when the electron subsystem is located on a discrete energy level). Abandoning the adiabatic approximation would make it possible to consider intrinsic absorption of light (and luminescence), in which an excited state of the system belonging to a continuous energy spectrum is formed.

D) Further development of qualitative theories that make it possible to establish the correlation and distinction between the spectra of isolated atoms or molecules and the spectra of crystals.

E) Taking into account the change in the frequencies of atomic vibrations during photo-transitions of an electron.

F) Development of methods for summing the probabilities of an infinite number of elementary photo-transitions in which one and the same frequency of light is absorbed.

G) Further development of the theory of impurity absorption and luminescence on the basis of the adiabatic approximation, in particular consideration of the case of intermediate heat release.

1. Methods for calculating the quantum stationary states of a crystal. The calculation of the stationary wave functions of the system and the eigenvalues of its energy is the central task of the theory, since these quantities must be known in order to construct—

... for constructing the theory of optical and thermal transitions, transitions caused by corpuscular radiation, for calculating the elements of the kinetic equation (for example, the mean free paths of charge carriers, their effective masses, etc.), for considering the strength and elastic properties of a solid, for application to systems of thermodynamics and statistics, and so on. Thus, knowledge of the quantum stationary states of a system is the basis for all branches of the theory of semiconductors and dielectrics, except for the phenomenological theory, which leaves many parameters of the crystal unknown.

a) State of the charge carriers

The most popular band theory of electrons in a crystal is based on two simplifying assumptions:

1) the many-electron problem is replaced by the consideration of separate noninteracting electrons moving in an externally prescribed periodic field;

2) the interaction of conduction electrons with thermal vibrations of the atoms is regarded as a small perturbation or is ignored altogether.

The first of these approximations is usually justified by the Hartree—Fock self-consistent field method, or by the assumption that the state of the proper electrons of the dielectric follows adiabatically the comparatively slow conduction electron.^36,37 A critique, partial justification, and limits of applicability of the first assumption are discussed in detail in papers 36—38 and will not be repeated here. From these works it follows that assumption 1) is inapplicable to the proper electrons of a dielectric, but, provided certain inequalities are satisfied, it is applicable to the excess electron—the conduction electron. The conduction band is then the lowest of the allowed bands. It makes no sense to speak of any lower-lying completely filled bands.

The one-electron approximation is capable of describing only large-radius excitons, consisting of an electron and a hole bound by Coulomb attraction and moving as two separate quasiparticles. “Small-radius” excitons, in which a wave of the excited state propagates along the crystal, leaving all molecules of the crystal neutral, cannot be described by that most widespread variety of the one-electron approximation in which each electron is assumed to move in an external periodic field. However, these excitons can be obtained in the more general form of the Hartree—Fock one-electron approximation,^36 in which the self-consistent potential, however, is no longer periodic and which, consequently, lies outside the limits of band theory. Among the popular methods, the most general one, allowing the above-mentioned “small-radius” excitons to be considered, is the T. L. G. method.

Turning to the discussion of assumption 2), it should be noted that in many homopolar crystals it is apparently admissible, but up to now there have been no convincing investigations of this question and no corresponding criteria. In inertially polarizing (ionic) crystals this assumption is justified only in rare cases, for example when the conduction-electron energy is several electron-volts. Such electrons play a role, for example, in the external photoelectric effect or in secondary-electron emission from semiconductors. In the majority of cases, however, assumption 2) for conduction electrons in ionic crystals is not justified. A sufficiently detailed investigation of this question, as is known, led to the concept of the polaron, which is a quasiparticle carrying current \(^{8}\) (further references are given there).

At the present time the theory of polarons of large radii has been developed in detail, since this case is favorable in that it makes it possible to introduce two very substantial simplifications into the theory: 1) by the effective-mass method for the electron \(^{39}\), to exclude from the wave equation the periodic potential of the crystal. This potential is not known with sufficient accuracy, and its presence in the equation creates great mathematical difficulties; 2) since, for a large polaron radius, the electric field polarizing the crystal varies sufficiently smoothly in space, the polarization of the crystal can be calculated by means of macroscopic electrodynamics, using the macroscopic dielectric constant and the refractive index of light.

For brevity we shall call this case of the theory the macro-theory of polarons. Historically, the limiting case of strong coupling of the electron with the polarization vibrations of the ions was considered first by the author (see \(^{8}\); further references are given there). It was assumed that the state of the electron follows adiabatically the relatively slow vibrations of the ions, and the approximate Born–Oppenheimer method was used, applicable when \(\alpha^2 \gg 10\), where

\[ \alpha=\frac{C e^2}{\hbar}\sqrt{\frac{\mu}{2\hbar\omega}} . \tag{5} \]

Here

\[ C=\frac{1}{n^2}-\frac{1}{\varepsilon}, \]

where \(\varepsilon\) is the static dielectric constant, and \(n\) is the refractive index of light; \(\omega\) is the frequency of the longitudinal polarization vibrations of the ions (we neglect frequency dispersion). In the zeroth approximation, for the energy of the ground state of the system \(E_0\) and for the effective mass of the polaron \(M\), the values obtained were

\[ \frac{E_0}{\hbar\omega}=-0.109\alpha^2-\frac{3}{2}, \tag{6} \]

\[ \frac{M}{\mu}=20.8\cdot 10^{-3}\alpha^4 . \tag{7} \]

It should be emphasized that the Born–Oppenheimer method is not an expansion in powers of \(1/\alpha^2\): each approximation of this method—

contains terms with different powers of \(\alpha\). Thus, for example, both terms in (6) constitute the zeroth approximation. Allowance for nonadiabaticity in the following approximations introduces into the expression for \(E_0\) a correction containing \(\alpha\) in the zeroth degree and in even negative powers. The term (6), proportional to \(\alpha^2\), receives no correction in the following approximations and therefore represents the exact value of the energy in the limiting case \(\alpha \to \infty\). (Here we ignore the small error of the numerical determination of this term in \(^{40,41}\).)

To the term with \(\alpha\) in the zeroth degree \(\left(-\dfrac{3}{2}\right)\) there is obtained a correction, when nonadiabaticity is taken into account, already in the first approximation. The interpretation given in \(^{8}\) of the term \(-\dfrac{3}{2}\) as the loss of three vibrational degrees of freedom of the ions is justified only in the case of limiting adiabaticity.

Then, beginning in 1950 with the work of Fröhlich, Pelzer, and Zienau \(^{42}\), the investigation of polarons begins in the limiting case of weak coupling, i.e., the case of small \(\alpha\). Treating the interaction of the electron with the polarization vibrations of the ions as a small perturbation, these authors obtained

\[ \frac{E_0}{\hbar \omega}=-\alpha+\ldots;\quad \frac{M}{\mu}=\frac{1}{1-\dfrac{\alpha}{6}}\simeq 1+\frac{\alpha}{6},\quad \frac{\alpha}{6}\ll 1. \tag{8} \]

G. Höhler \(^{43}\) calculated the energy up to and including the term of order \(\alpha^2\):

\[ \frac{E_0}{\hbar \omega}=-\alpha-0.0157\alpha^2+\ldots \tag{9} \]

The most mathematically complicated case is that of intermediate coupling. At the same time, this case is of practical importance, since in almost all real crystals strong or intermediate coupling occurs. For intermediate coupling there are as yet no systematic approximation methods; there are only attempts to calculate the energy of the ground state of the system by direct variational methods. It is necessary to test many approximations to the wave function of the system and to give preference to those with which the lowest value of \(E_0\) will be obtained. The approximations selected in this way will give a rather accurate value of \(E_0\), but may give the most incorrect values of the effective mass of the polaron \(M\). If, among the published direct variational methods, one chooses those which give approximately coincident and rather correct values of \(E_0\), it turns out that the values of \(M\) obtained thereby differ from one another by a factor of 200! The author explained the reason for this and showed \(^{44}\) that among the approximations giving the exact value of \(E_0\), one must then choose that which gives the largest value of \(M\). This laborious task will still require the work of many scientific groups over a number of years.

Among the direct variational methods proposed up to the present time, one should note the approximations of Gurari \(^{45}\) and of Lee, Low, and Pines \(^{46}\),

which employed a method analogous to Gomonai’s intermediate-coupling method in mesodynamics or to V. A. Fock’s functional method. These authors hoped to cover the region of intermediate coupling, but, unfortunately, obtained results coinciding with the weak-coupling theory and valid only for \(x/6 \ll 1\).

More successful are the approximations of G. Höhler\(^{47}\) and R. Feynman\(^{48}\). In the latter, the expression \(E_0\) for small \(\alpha\) has the form

\[ \frac{E_0}{\hbar\omega}=-\alpha-0.0123\alpha^2-0.00064\alpha^3+\ldots \tag{10} \]

The first two terms of this expression agree approximately with the exact expansion (9). For \(\alpha \to \infty\), Feynman has

\[ \frac{E_0}{\hbar\omega}\to -0.106\alpha^2, \]

which differs by only \(3\%\) from the exact value, \(-0.109\alpha^2\), obtained by Pekar. In the intermediate-coupling region, for \(\alpha \sim 4\text{--}6\), \(E_0\) is obtained lower than in all variational methods known up to now. It is important that, for Feynman, the expression \(M\) at small \(\alpha\) goes over into (8), while at large \(\alpha\) it is only \(4\%\) less than the exact value (7).

In calculating the polaron by the usual direct variational method, it is extremely difficult to take into account the infinite number of excited polaron states of the system. In particular, all the above-mentioned works using the direct variational method consider only the ground state of the system at a temperature equal to absolute zero. This is insufficient for most applications of the theory. Thus, for example, to calculate the thermodynamic functions of the system it is necessary to know the partition function of the system, and for this it is necessary to determine the entire energy spectrum of the system.

Such difficulties have been completely overcome in a new work by the author together with M. A. Krivoglaz\(^{49}\), where a variational method is formulated in which the extremum of the functional determines not the energy, but directly the partition function of the system at an arbitrary temperature. In this case it is the Hamiltonian of the system that has to be approximated, and not the wave functions. The latter need not be determined at all in this method. Through the partition function one can express all thermodynamic functions of the system and, in particular, the average energy, which as \(T \to 0\) goes over into the energy of the ground state \(E_0\). The method makes it possible to obtain not only the effective mass of the polaron \(M\), but also the entire, generally speaking nonquadratic, dependence of the energy on the total momentum of the system.

With an appropriate choice of the approximation to the Hamiltonian, in \(^{49}\) the Feynman value of \(E_0\) is obtained for all values of the coupling constant. At the same time, for \(M\), Feynman values are obtained in the strong- and weak-coupling regimes, and somewhat more accurate values in the intermediate-coupling regime. For the first time in the theory of polarons, the partition function of the system has been obtained at all temperatures. In the limiting case

in the case of weak coupling it tends to its exact value, which is also determined in ^49^. In the limiting case of strong coupling it comes very close to its exact value, obtained in the adiabatic approximation ^8^.

The theory of small-radius polarons is still in an embryonic state. Great complications arise in connection with the necessity of abandoning the electron effective-mass method and the macroscopic calculation of polarization. In addition, a small-radius polaron interacts substantially with short-wavelength polarization oscillations, and these oscillations have a substantial dispersion of their eigenfrequencies, which has not been quantitatively studied for most crystals. What has been said means that, in the theory of small-radius polarons, one must explicitly introduce the poorly known periodic micropotential of the electron in the crystal and base the theory on “atomic” wave functions of the electron. Even merely writing down the energy of interaction of the electron with the short-wavelength polarization oscillations of the ions presents great difficulties and is possible only after a rough modeling of the character of the deformation of the ions during oscillations. For these reasons, in the coming years one may hope only for the construction of a rough qualitative theory.

Some consolation is provided by the circumstance that, in the majority of semiconductors of technical significance, large-radius polarons are realized.

A problem analogous to that of polarons also exists in homopolar crystals, with the sole difference that here the electron interacts not with polarization oscillations but with acoustic oscillations of the atoms. Here the case of weak coupling is most often realized, when the aforementioned interaction may be regarded as a small perturbation. Such a problem has been solved ^49^ for a crystal with isotropic elastic properties, but the solution can easily be generalized to anisotropic cases. It turned out that in a homopolar crystal the interaction of the electron with short-wavelength acoustic oscillations is also always substantial. However, the Hamiltonian used in ^49^ is justified only for long acoustic waves. Therefore the results obtained make it possible only to estimate approximately the binding energy and effective mass of the polaron analogue, which perhaps should be called a condenson.

It appears desirable to develop the theory of stationary quantum states of current carriers in the following directions.

A) Development of general methods for solving the many-electron problem.

B) Formulation of new types of quasiparticles or elementary excitations representing current carriers in a crystal.

C) New forms and justifications of one-electron approximations for a crystal. In particular, more general justifications and definitions of the limits of applicability of the approximation in which the electrons are regarded as independently moving in an external periodic field.

Г) Development of methods for calculating electrons in a periodic field.

Д) Consideration of higher approximations of the effective-mass method for the electron, as well as the development of other approximate methods that make it possible to omit the periodic potential by introducing one or two parameters into the theory.

Е) Obtaining more accurate and easily verifiable criteria for the applicability of band theory to an extra electron in a homopolar crystal.

Ж) Consideration of higher approximations in the theory of polarons, in particular the inclusion of nonadiabaticity corrections in the case of strong coupling of the electron with the polarization vibrations of the ionic lattice. Estimation of the errors of the theory.

З) Development of methods for calculating polarons in the case of intermediate coupling. In particular, the development of approximations not based on the direct variational method, which is of little use for determining the dependence of the energy of the system on the momentum of the polaron and for calculating the effective mass of the polaron.

И) Consideration of small-radius polarons without using the effective-mass method and the dielectric-continuum model. Obtaining polaron energy bands, a large number of which are contained in the energy interval \(kT\).

К) Investigation of the peculiarities that arise when the velocity of the polaron coincides with the speed of sound or with the velocity of polarization waves in the crystal.

Л) Generalization of the theory of polarons to anisotropic crystals and to the case of complex dispersion laws for the frequencies of optical vibrations of ions.

М) Inclusion of acoustic vibrations of the crystal in the theory of polarons.

Н) Further development of the theory of large-radius excitons, in which the coupled motion of an electron and a hole is considered as the motion of two quasiparticles with effective masses. Inclusion of the deformation of the crystal caused by the exciton.

О) Development of the theory of excitons on the basis of H. L. H. and the search for other methods of a many-electron treatment of excitons.

П) Calculation of the free paths of current carriers and excitons *) with respect to thermal vibrations of the crystal, elastic and inelastic collisions with impurities, as well as calculation of their lifetimes.

b) The state of electrons localized on impurities and defects

Here matters are comparatively favorable in the case of large effective radii of the electronic states, when one can use the effective-mass method and ignore the periodic potential; replace the crystal by a dielectric continuum, and polo-

*) Anselm and Firsov have recently calculated the free path of an exciton33.

positively charged “center” at which the electron is localized by one or two point positive charges. Thus, for example, an atomic impurity center of large radius in a homopolar crystal reduces to a hydrogen-like problem, into which an effective mass and a dielectric constant are introduced. The same center in an ionic crystal leads to the solved problem of the \(F\)-center. In the above-mentioned cases the wave function and the energy of the localized electron depend almost not at all on the nature of the introduced impurity atom, but only on the magnitude of the positive charge. Large radii of states are usually realized in crystals with a large ratio \(\frac{\varepsilon}{\mu}\).

At the present time cases have also been considered of two electrons near one positive charge \(^{31}\), and of two electrons near two positive charges \(^{29}\). More complicated centers are also being considered, representing associations of the above-mentioned simple centers.

The state of the theory is far less satisfactory in the case of small radii of the electron \(\psi\)-cloud, when one cannot use either the effective-mass method or the dielectric-continuum model \(^{50-52}\). Here the theory encounters difficulties similar to those met in the case of many-electron atoms and molecules. A radical improvement of the situation here is possible only on the basis of the development of entirely new approximate methods for solving many-electron problems.

Desirable directions for the further development of the theory of local electron centers are points A), B), and D) of the preceding list, and also:

P) Further improvement of the theory of local centers of large radius. Consideration of new cases (new models). Improvement in the account of the strong interaction of electrons with the thermal vibrations of atoms and in the calculation of the equilibrium deformation of the crystal by the localized electron. The latter depends on the quantum state of the electron in the local center; it is necessary to know it in order to make use of the theory of impurity absorption of light and luminescence, and also the theory of thermal transitions.

C) Consideration of centers of small radius on the basis of the H. L. H. approximation and the approximation of strongly bound Bloch electrons, as well as the development of new approximate methods.

T) Consideration of the excitation and ionization of centers by impacts of current carriers, excitons, corpuscular and light radiation.

U) Consideration of cases in which the impurity atoms are rare-earth elements and in which the influence of the surrounding crystal on the state of the optical electron may be regarded as a small perturbation.

F) Development of the theory of spin-electron resonance in local electron centers, which is necessary for the correct interpretation of experimental results and for the possibility, from experimental data, of judging the type and model of the center.

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