BAND THEORY OF SOLIDS AND THE LIMITS OF ITS APPLICABILITY*)

F. F. Vol’kenshtein

1. INTRODUCTION

The modern theory of solids may be of interest from various points of view. The most diverse branches of physics and technology ultimately run up against the theory of solids.

1. The theory of solids is of interest to the electrical engineer. Indeed, any electrical-engineering problem reduces to a combination of bodies with high and low conductivities. Questions of the electrical conductivity of metallic and nonmetallic crystals, questions of electrical breakdown of dielectrics—as soon as we begin to be interested in the mechanism of the phenomenon—come up against the modern theory of solids.

2. Questions of luminescence likewise lead us to the general problems of the theory of solids. The mechanism of the processes taking place in crystal phosphors under illumination can be understood only on the basis of the modern theory of solids.

3. The theory of solids is of interest to physical chemists concerned with adsorption and heterogeneous catalysis. Crystals (metallic and nonmetallic) are, as a rule, catalysts. Chemical processes developing inside a crystal and on its surface cannot be fully understood if one ignores the modern concepts of the theory of solids. In these processes the solid is not an inert medium or substrate, but an equal participant in the process. The theory of heterogeneous catalysis rests on the theory of solids.

What, then, is the task of the theory of solids?

The task consists in explaining the electrical, photoelectric, magnetic, and, in general, macroscopic properties of a solid from the point of view of the structure of the crystal as a whole. The question is

*) Report at the Seventh All-Union Conference on the Physics of Semiconductors.

of revealing the microscopic mechanism of the macroscopic processes occurring in a solid.

What, then, is the state of affairs in this area?

In describing electronic processes in crystals one usually makes use of the concepts of the so-called band theory of solids. This is a very convenient and visual theory. Experimentalists usually speak in the language of band theory. In this language they interpret their results.

At the same time, many theorists categorically object to band theory, since defects of a fundamental character are built into its very foundation.

The band theory of solids is an approximate theory. Like any approximate theory, it has limits of applicability. Experimentalists often overstep these limits and use the band picture where it is plainly inapplicable. Many theorists, on the contrary, overstep these limits in the opposite direction and, defending the purity of the theory, reject the band model even where it is an entirely acceptable approximation.

In reality, in the theory of solids there exists a whole series of problems that can be solved with a quite satisfactory approximation within the framework of band theory. There are, however, also problems which, by their very nature, go beyond these limits. It is only important to have a clear idea of precisely which group of problems is encompassed by these limits and which group of problems lies outside them.

The present article is devoted mainly to the question of the limits of applicability of band theory.

2. BASIC PREMISES OF BAND THEORY

First of all I should like to dwell on those basic premises on which the band theory of solids is based.

What is a crystal from the physical point of view?

It is a house in which a whole family of electrons lives. This house has a definite architecture, and the electrons inhabiting it obey definite rules of internal order.

The task consists in describing the behavior of these electrons.

The electrons filling the crystal are in interaction with one another. Taking this interaction into account constitutes the main and fundamental difficulty of the theory.

This is a difficulty because quantum mechanics cannot solve exactly the problem of a system of interacting particles. In solving such problems, quantum mechanics is content with approximate methods.

Therefore the theory of the crystal, by its very essence, is an approximate theory. It remains only to see to it that this theory constitutes a sufficiently good approximation.

The problem of the crystal would be an altogether simple problem if it were possible to neglect the interaction between electrons, retaining only the interaction of the electrons with the nuclei. This incorrect path can, of course, be somewhat corrected if the interaction of the electrons is replaced by the introduction of a certain external effective force field, in which each electron moves independently of the others. Such an effective field, calculated as the field created by the smeared-out charge of all the electrons of our system, is called a self-consistent field.

Taking the interaction into account by the self-consistent-field method is a very imperfect way of allowing for the interaction. This method, however, is attractive in that it makes it possible to transform the many-electron problem into a series of mutually independent one-electron problems.

Using this method, we have the right to speak of the behavior of each electron separately, to assign to each electron its own individual wave function and its own individual value of the energy. Within the framework of this method each electron lives its own individual life, as it were “not noticing” the other electrons, as if no other electrons existed at all. The only thing that reminds it of them is the Pauli principle, which forbids it to occupy quantum states already occupied by other electrons.

It is precisely on this method, the self-consistent-field method, that the so-called band theory of solids is based.

Thus, at the foundation of band theory lies the reduction of the many-electron problem to a one-electron problem. Band theory is the theory of one electron.

This is the first, but not the only, characteristic feature of band theory.

The second characteristic feature is a definite assumption about the nature of the force field in which each individual electron moves. This field has a periodic character, reflecting in itself the periodic structure of the lattice. For band theory it is not essential what the concrete form of this field is. What is essential is only that this field is periodic with the period of the lattice.

The characteristic band structure of the energy spectrum of the electron, i.e. the system of allowed energy bands separated, generally speaking, by forbidden regions, is precisely the result of this periodicity. Thus, the band picture of the spectrum is already contained in the very premises of the theory: in the assumption of the periodic character of the potential.

3. SOME SHORTCOMINGS OF BAND THEORY

Let us note some characteristic shortcomings of band theory, some of its features already concealed in the very formulation of the problem.

If electrons are regarded as “not noticing” one another, then the possibility is not excluded that they may accumulate on one and the same atom. It may happen that several electrons at once are concentrated on one and the same atom (or ion), at the expense of the stripping of some other atom or of some other atoms (or ions)1. Such states, characterized by an “overconcentration” of electrons in one place, figure in band theory on an equal footing with the others.

When a lattice atom receives an extra electron (over and above the normal complement), such an atom is transformed into a negative ion. This extra electron is borrowed from another atom, which is thereby transformed into a positive ion.

What is essential here is not so much the fact itself that band theory permits an electron to jump from one lattice atom to any other atom, as the fact that such a jump, from the point of view of band theory, does not require an expenditure of energy.

A similar result is well known to us from the theory of the molecule. In molecular theory there exists a method completely equivalent to Bloch’s method2, which is used in band theory. This is the so-called method of molecular orbitals, or otherwise the Mulliken–Hund method3. Solving by this method, for example, the problem of the hydrogen molecule, we obtain, alongside the so-called nonpolar states, also polar states, i.e. such states in which both electrons are concentrated at one of the two nuclei of the hydrogen molecule, while the other nucleus is thereby bared. The hydrogen molecule thereby acquires a heteropolar character. In the Mulliken–Hund method such heteropolar states have the same statistical weight as the homeopolar states, and correspond to the same value of the energy. In a crystal lattice considered from the point of view of Bloch’s method, we have exactly the same state of affairs.

Such a result is a consequence of the “collectivization” of electrons characteristic of the Mulliken–Hund method, as well as of Bloch’s method (and in general of band theory). Each electron belongs to the same degree both to its own nucleus and to all the other nuclei of the lattice, and moves through the lattice independently of the other electrons. In band theory an electron possesses two qualities: it “does not notice” other electrons, and at the same time it “forgets” the atom to which it originally belonged (together with which it was introduced into the lattice).

4. LIMITS OF APPLICABILITY OF BAND THEORY

Band theory is an approximate theory and, consequently, has its own limits of applicability.

There are problems for which the approximations underlying band theory are acceptable. On the other hand, there are problems for which such approximations are unacceptable.

Let us formulate the conditions that determine the sphere of applicability of band theory. Four such conditions may be indicated.

1) Each atom (or ion) forming part of the crystal lattice possesses its normal complement of electrons. Band theory is obviously inapplicable to these “own” electrons belonging to the atoms or ions of the lattice. The behavior of these electrons cannot be correctly described within the framework of band theory.

However, in addition to its “own” electrons, an atom or ion of the lattice may also possess an “extra” electron. In the case of an atomic lattice, such an “extra” electron, placed on a neutral atom in excess of the complement, makes it a negative ion. In the case of an ionic lattice, such “extra” electrons lead to the appearance of anomalous ions, i.e. ions with an anomalous charge. These extra electrons may be introduced into the lattice from outside, or they may be borrowed from the lattice’s own resources by the transfer of electrons from one normal atom or ion to another normal atom or ion.

If one can speak at all of the applicability of band theory to the electrons of the lattice, then only with respect to these “extra” electrons^1,4.

2) Band theory gives a correct description of the behavior of these “extra” electrons, again not always, but only so long as the concentration of these “extra” electrons is sufficiently small. In other words, for band theory it is necessary that the number of anomalous atoms (ions) be small in comparison with the total number of atoms (ions) in the lattice.

3) The preceding condition is necessary but not sufficient. Band theory is applicable to “extra” electrons only on the condition that the “extra” electron does not change the state of the “own” (inner) electrons of the atom (or ion) on which this “extra” electron is placed^4. This occurs, for example, in the case when the “own” electrons of the atom (or ion) form a closed shell.

4) Even when all the preceding conditions are satisfied, band theory nevertheless ceases to hold in those problems in which the interaction between two or several “extra” electrons is essential. In other words, band theory is inapplicable to those problems in which we are dealing with encounters of two anomalous

atoms (or ions), when these two anomalous atoms (or ions) turn out to be adjacent to one another and enter into interaction with each other.

The conditions formulated above pertain in general to band theory as a method for describing the behavior of electrons in a lattice, regardless of whether one is dealing with a metallic or a nonmetallic lattice.

5. ON THE APPLICABILITY OF BAND THEORY IN THE CASE OF METALS AND IN THE CASE OF NONMETALLIC CRYSTALS

Let us see to what extent the four conditions formulated are satisfied in the case of metals.

Leaving aside the question of whether the first two conditions are fulfilled or not, one may assert that in the case of a metal the third condition is certainly not fulfilled: the proper electrons of a metallic atom do not form a closed shell; the “extra” electron attached to the metallic atom distorts the behavior of its inner electrons. These inner, or, better to say, proper, electrons of the atom feel the appearance of the “extra” electron.

Thus, in the case of metals the application of band theory, strictly speaking, should be regarded as unjustified.

What is the situation in the case of nonmetallic ionic crystals?

An ionic crystal may contain “extra” electrons. It is precisely these electrons (whatever their origin may be) that are the current carriers in a nonmetallic crystal. In other words, among the normal ions there are, in some quantity, anomalous ions with an excess negative or excess positive charge. The former give rise to electronic conductivity; the latter—to hole conductivity. It is the behavior of precisely these anomalous ions that interests us when we speak of the electronic conductivity of the lattice.

The concentration of these anomalous ions in a nonmetallic ionic crystal under ordinary conditions is sufficiently small. It is so small that the ions may be regarded as separated, on average, by sufficiently large distances—distances at which their interaction with one another may quite legitimately be neglected.

Furthermore, the proper electrons of the lattice ions form closed shells; therefore an extra electron attached to an ion beyond the normal complement does not distort, or almost does not distort, the behavior of the proper electrons.

Thus, the first three conditions are approximately fulfilled. The criterion for the applicability of band theory in the case of an ionic crystal is, consequently, the last, fourth condition.

We can use band theory in describing the behavior of conduction electrons so long as these electrons may be regarded as sufficiently dilute. There is a broad class of problems for which this assumption may be adopted.

However, there are problems in which the electrons come substantially closer together and an interaction between them arises as a result. Problems of this kind cannot be described in the language of band theory. They require a different approach.

6. THE BAND PICTURE TRANSLATED INTO MODEL LANGUAGE

Let us note that, in the case of an ionic crystal, the band picture can easily be translated into a visual model language ⁵.

Consider an ideal ionic lattice. For simplicity, let us deal with a crystal of the type \(M_mR\) (where \(M\) is the symbol of a metal, and \(R\) is the symbol of a metalloid), built of positive metallic and negative metalloid ions.

So long as all the ions of the lattice possess their normal charge, electronic conduction in the lattice is altogether impossible. In the band scheme this means that the lower band (the so-called normal band) is completely “filled” with electrons, i.e., contains not a single hole, while the upper band (the conduction band) contains not a single electron.

Let us now imagine that we have removed an electron from one of the negative ions and placed it on some sufficiently distant positive ion. In the band scheme this will mean that we have transferred an electron from the lower band to the upper one.

Thus, the appearance of an electron in the conduction band and the simultaneous appearance of a hole in the normal band mean the creation in the lattice of two anomalous ions with opposite additional charges, separated sufficiently far from one another. The energy expended in such an operation is equal to the distance between the lower and upper bands (more precisely, between the middles of these bands).

The motion of an electron through the conduction band means the displacement through the lattice of a metallic anomalous ion. The motion of a hole through the normal band means the displacement of a metalloid anomalous ion. Here, of course, what is meant is not the displacement of the anomalous ion itself, but the displacement of the anomalous state, transmitted from one ion to a neighboring ion of the same name.

The recombination of two opposite anomalous states, accompanied by the restoration of a pair of normal ions, is described in the language of the band scheme as the falling of an electron from the conduction band into a hole of the normal band.

Let us note that this recombination process lies outside the framework of band theory. It is precisely an example of a problem for whose description

to which band theory is inapplicable. Indeed, the recombination of an electron and a hole is the result of an interaction arising between them. It means that two anomalous ions with opposite additional charges have found themselves in immediate proximity to one another. In this process of recombination, band theory is capable of giving only the initial and final states.

The same applies, of course, to the reverse process—the process of ionization (the formation of two anomalous ions from a pair of normal ions). Here too, band theory describes only the initial and final states.

Between these two states there exist intermediate states which, naturally, do not fit in any way into the band scheme. These are the so-called exciton states. They will be discussed later.

7. THE “NORMAL” BAND IN THE BAND SCHEME OF A NONMETALLIC CRYSTAL

Thus, the behavior of a conduction electron, i.e. an electron in the upper band, can be described by Bloch’s method. This problem lies within the framework of band theory. Here no objections to band theory arise.

Objections to band theory arise, however, when we pass to the electrons of the lower (normal) band, i.e. when we pose the problem of the behavior of the electrons filling (completely or almost completely) the lower band.

Indeed, in describing the behavior of electrons in the conduction band, we remain within the framework of a one-electron problem. However, in describing the behavior of a hole in the normal band, we are dealing, although with a “one-hole” problem, if one may put it that way, but not with a one-electron problem.

This means that the electrons filling the lower band cannot lay claim to being regarded as independent of one another (in other words, they cannot lay claim to individual wave functions).

Such a claim would be illegitimate, for the electrons of the normal band are the proper electrons of the lattice ions. The problem of their behavior is essentially a many-electron problem and cannot be reduced to a series of independent one-electron problems.

The electrons of the lower band form a certain family, whose behavior is described by a collective wave function depending on the coordinates of all the electrons and not decomposing into products of separate individual functions.

This collective function depends, as has already been said, on the coordinates of all the electrons. What is meant here are all the intrinsic electrons of the metalloid ions of the lattice, with the exception of one single electron, whose absence signifies the presence of a hole. Its coordinates are absent from the collective function, and therefore one may say of such a function that it describes the behavior of the hole.

Thus the lower band has a meaning different from that of the upper band. The upper band is a system of electronic levels, whereas the lower band is a system of hole levels.

The relation between them is the same as that between optical terms in a one-electron atom and X-ray terms of a many-electron atom, which correspond to the removal of an electron from one or another electron shell. An X-ray term represents not the energy of an individual electron, but the energy of the whole system as a whole.

For an atom we obtain a whole series of X-ray terms, since the energy of the system will be different depending on which particular electron is missing from this system. This, and only this, is the meaning of the lower band in the usual band scheme of a nonmetallic crystal.

In the figures that appear in textbooks on atomic theory, electronic levels are usually plotted from bottom to top, and hole (X-ray) levels from top to bottom. We have exactly the same thing in the band scheme of a crystal, in which, however, electronic and hole levels are combined in a single drawing.

Let us note that this, the only acceptable interpretation of the normal band, differs from the usual Bloch interpretation. Indeed, in Bloch’s method the “intrinsic” electrons are considered on an equal footing with the “extra” electrons. In Bloch’s theory the electrons of the lower band are just as independent of one another as the electrons of the upper band, and each is described by its own individual function. This is just as incorrect as it would be incorrect to construct a many-electron atom containing \(n\) electrons by taking the system of hydrogen levels and seating all our \(n\) electrons on these levels one after another, taking into account only the Pauli principle.

Objections to the band theory as applied to the electrons of the normal band are, in essence, objections to the Bloch interpretation of this band. Such an understanding of the lower band in the narrowly literal Bloch sense is, of course, incorrect. This is quite obvious.

However, when an experimentalist draws the lower normal band (and cannot do without it in interpreting his experimental results), he by no means ascribes to it a Bloch meaning.

The lower band is not a system of electronic levels filled or almost filled with electrons. It is a system of hole levels, almost not filled with holes. The lower band is “empty” with respect to holes to the same degree as the upper band is with respect to electrons.

When we speak of the lower band, we are speaking not of the behavior of individual electrons independent of one another, but of the behavior of individual holes independent of one another. Here we do not reduce the many-electron problem to a one-electron one. Here we reduce the many-electron problem to a one-hole problem.

Thus, with the aid of band theory one cannot describe the normal state of the system, when there is not a single electron in the upper band and not a single hole in the lower band. With the aid of band theory, however, one can describe the excited states of the system, characterized by the presence of electrons in the upper band and holes in the lower band.

8. SOME APPROXIMATIONS CONTAINED IN THE BAND SCHEME

It is necessary to note that the band scheme, which in its simplest form consists of two energy bands separated by a forbidden region, is connected with certain additional simplifying assumptions.

This scheme applies only to an ideal and, moreover, infinite crystal lattice, i.e. to a lattice in which there are no bounding surfaces and no internal structural defects of any kind. This is the first simplifying assumption.

Thus, all those properties of real crystals which are caused by their deviation from the ideal state, i.e. the entire group of so-called structure-sensitive properties, cannot be interpreted from the point of view of this simple band scheme.

Another approximation contained in this scheme consists in the assumption that the appearance of an anomalous ion in the lattice causes no changes in the positions of the normal ions surrounding it. All ions remain in their places. In other words, the appearance of an electron in the conduction band or of a hole in the normal band is not accompanied by any deformation of the lattice.

Both assumptions (the assumption of the absence of defects in the lattice and the assumption of the absence of deformation around anomalous ions) constitute a known simplification of the actual picture.

Freeing ourselves from the first assumption, we arrive at the notion of “local” levels.

If we free ourselves from the second assumption, this will lead us to the notion of so-called “polarons.”

Let us take both these steps.

9. “DISORDER” IN THE LATTICE AND ITS REFLECTION IN THE BAND SCHEME

Let us pass from the ideal crystal to the real crystal. A real crystal differs from an ideal one by the presence of defects, i.e. local disturbances in the strictly periodic structure of the lattice. Examples of such defects may be empty sites in the lattice, atoms of a foreign impurity, and also the lattice’s own atoms or ions introduced into interstices[^5].

Defects in the lattice have their reflection in the energy spectrum. The presence of a defect usually leads to the appearance of a discrete energy level (the so-called local level) falling within the forbidden region between the bands.

Local levels are characterized by wave functions having an essentially different form than the wave functions corresponding to the levels of an energy band.

Indeed, an electron located in the conduction band is described by a wave function whose modulus is periodic with the period of the lattice. This means that such an electron freely wanders throughout the entire crystal, or, in other words, that such an electron is equally likely to be found at any point of the crystal. One may say of such an electron that it is uniformly smeared over the whole crystal.

An electron sitting on a local level is characterized by a wave function having a sharp maximum near the defect and rapidly decaying as the distance from it increases. This means that such an electron is localized in a definite region of the crystal (namely, in the region of the defect). The probability of its presence is maximal near the defect and decreases with distance from it. Considering the electron as a smeared-out charge, one may say that the density of this charge decreases with distance from the defect.

Let us note that the degree of localization of the electron depends on the position of the local level in the energy spectrum. The closer the electronic local level is situated to the conduction band, the smaller the degree of localization of the electron sitting on this level, the more smeared out is the wave function corresponding to this level. As the local level approaches the lower edge of the conduction band, the localized electron gradually turns into a collectivized one.

electron, and the very concept of a local level gradually loses its meaning[^6].

Defects present in the lattice and giving rise to local levels in the energy spectrum, generally speaking, possess a certain mobility. This mobility is a consequence of the periodic structure of the lattice itself.

A number of facts attest to the wandering of a defect inside a crystal. The very existence of ionic conductivity in a crystal is such evidence.

Defects wandering in the lattice, when approaching one another, enter into interaction with one another. Thus, for example, two empty metallic sites repel one another, tending to move apart in different directions. A metallic empty site and a metallic interstitial ion, on the contrary, are attracted to one another.

When two or several defects combine together, a new defect is formed which, generally speaking, has other properties. Thus, within the lattice we have a kind of “chemistry of defects”[^7].

Among the various reactions into which defects can enter, there also exist reactions that lead, in general, to the disappearance of the reacting “particles” (recombination reactions), as well as reactions leading, on the contrary, to the appearance of defects. In other words, the lattice itself is capable of generating and absorbing defects.

At each given temperature there exist definite equilibrium concentrations for defects of different kinds.

The energy spectrum of a crystal, therefore, should be regarded, generally speaking, as changing with temperature. As heating proceeds, local levels of one kind arise at the expense of the disappearance of local levels of another kind, and the total number of levels changes. In other words, the energy spectrum reflects the chemical reactions in which defects participate[^7],[^8].

The total number of defects in the lattice characterizes what may be called the disorder in the lattice. This disorder has a dual origin. In it one may distinguish two parts: thermal and biographical.

Biographical disorder is the irreversible part of the disorder. It is that share of disorder which is preserved at zero temperature. The degree of biographical disorder is determined by the technology of the specimen, by the entire history of its life—in other words, by its biography. This is the disorder given to the lattice, so to speak, “from birth.”

Superposed upon this biographical disorder is a reversible thermal disorder, having a temperature origin.

The relation between thermal and biographical disorder depends, naturally, on the temperature and on the biography of the specimen. Thus, the transition from an ideal crystal to a real crystal is connected with the superposition of a certain disorder upon the strict order that prevails in an ideal crystal.

Whereas the so-called stable properties of a crystal are determined by the order factor, in other words, by the periodic structure of the lattice, the structure-sensitive properties, on the contrary, are determined by the disorder factor, i.e., by local violations of this periodicity.

The accounting for disorder in the lattice, i.e., the transition from an ideal crystal to a real crystal, is carried out within the framework of band theory. Band theory is not connected in any organic way with the ideality of the crystal.

10. THE CONCEPT OF POLARONS AND BAND THEORY

Let us now turn to another simplifying assumption that underlies ordinary band theory.

This is the neglect of the deformation of the lattice that arises around a conduction electron, i.e., around an “extra” electron introduced into the lattice.

In the case of an ionic lattice, this deformation reduces to dielectric polarization of the lattice by the field of the electron. In ordinary band theory this effect is ignored.

The problem of an electron in an ionic lattice with allowance for the polarization caused by this electron was considered by S. I. Pekar[^9]. An electron surrounded by a polarized lattice was called a polaron.

Such an electron wanders through the crystal together with the deformation surrounding it. Moving through the crystal, it carries with it the polarization surrounding it. Conductivity due to such electrons is called polaron conductivity. The difference between electronic and polaron conductivity reduces to a difference in the magnitude of the effective mass or in the magnitude of the mobility of the current carriers. Polaron conductivity is, in essence, ordinary electronic conductivity with allowance, however, for the polarization of the lattice in the region surrounding the moving electron.

Allowance for polarization leads, however, to a substantial complication in the character of the electron’s motion: the crystal polarized by the electron represents for the electron a potential well within which the electron oscillates. The potential well itself moves through the crystal together with the electron oscillating in it[^10].

From the point of view of quantum mechanics, allowance for polarization means allowance for the interaction of the electron with the phonon field. This interac-

the action may be large or small in comparison with the energy of the electron in the periodic field of an ideal lattice. Accordingly, we speak of “strong” or “weak” coupling[^11].

In the case of “weak coupling,” taking polarization into account, as should be expected, gives only small corrections to the ordinary band picture. The conduction band shifts somewhat downward, which seems quite natural. One may even say that it is obvious a priori. Indeed, a lowering of the electronic levels means a decrease in the energy of the system. But the deformation of the lattice occurs precisely because this is energetically advantageous.

In the case of “strong coupling” (realized in many crystals) the situation proves to be somewhat more complicated. From considerations of translational invariance it is clear that in this case as well the energy spectrum of the system has a band character. However, the structure of these bands is, generally speaking, not at all the same as in the usual Bloch theory. They are characterized by other parameters and have a more complicated form.

Let us note that, along with “electron polarons,” if one may put it this way, it would be possible to consider polarons of another kind, which might be called “hole polarons.” Indeed, a hole, i.e. an anomalous ion with an excess positive charge, like an electron, causes a polarization of the lattice around itself. In crystals possessing hole conductivity, the current is carried by such hole polarons.

Taking this polarization into account causes the entire hole band as a whole to shift upward, which represents a gain in energy due to the deformation (polarization) of the lattice. Thus, taking the polarization effect into account ultimately leads to a narrowing of the forbidden region between the upper and lower bands.

However, from the standpoint of the band scheme with which the experimenter operates, this circumstance is not essential. Indeed, quantities such as the widths of bands or the width of the forbidden region between bands are never taken from theoretical formulas, but are determined on the basis of experimental data.

The width of the forbidden region, i.e. the ionization energy, may be determined, for example, from optical data or from electrical-conductivity data.

Let us note that these two methods of determining the ionization energy usually give different results. The optical ionization energy turns out to be greater than the thermal one.

This circumstance becomes quite understandable from the standpoint of the “polaron” theory.

Indeed, absorption of a quantum of light is an instantaneous process, in which the ions of the lattice do not have time to shift from their equilibrium positions, and, consequently, the polarization of the lattice does not have time to take place.

Measuring the ionization energy optically, we obtain the distance between the bands corresponding to the old Bloch theory.

In the case of the thermal ionization energy, however, we are dealing with a polarized lattice. Taking polarization into account, as has already been said, leads to a bringing together of the lower and upper bands. The forbidden region between the bands proves to be narrowed.

The concept of the polaron remains within the framework of the band theory of a solid. Here we are dealing with the inclusion of a certain effect (the polarization effect), which is neglected in the usual (Bloch) version of band theory. The inclusion of this effect is an important correction to band theory, changing the structure of the band spectrum, but nevertheless not changing in any principled way the positions from which band theory proceeds, and not resolving those difficulties which are characteristic of it.

Indeed, the problem of the polaron is considered as a one-electron problem. In it the interaction between polarons is ignored, just as the interaction between electrons is ignored in ordinary band theory. Therefore this theory ceases to operate in the same place where band theory ceases to operate. They have the same limits of applicability.

The polaron theory developed by S. I. Pekar and his collaborators is an account of a certain important effect which ordinary band theory neglects (the effect of lattice deformation), but it preserves within itself the basic feature of band theory: the neglect of electron interaction.

The problem of several colliding polarons, entering into interaction with one another, presents still greater difficulties than the problem of colliding electrons, which, as has already been indicated, cannot be described in the language of band theory.

11. EXCITONS

Problems of this kind require a different approach.

Band theory is not the only approximate method for solving the problem of electrons in a crystal. Other approximate methods exist.

In conclusion we shall dwell on some problems that go beyond the framework of band theory. A typical example of such a problem is the problem of the exciton.

The concept of the exciton was put forward by Ya. I. Frenkel[^12] as applied to atomic lattices. Let us imagine that among the atoms of a lattice, which are in the usual normal state, one of the atoms is excited. Such an excited state of an individual atom, which may arise, for example, as a result of absorption of a quantum, was called an exciton (exitation — excitation).

The excited state has a definite lifetime. Thus an exciton that has arisen in the lattice inevitably disappears after some time. During its lifetime the exciton can move through the lattice. This means that the state of excitation can be transferred from one atom to another, neighboring atom, and thus can wander through the crystal.

The concept of the exciton was developed by Frenkel, as already stated, for homopolar (metallic) lattices. What corresponds to the Frenkel exciton in the case of an ionic lattice?

In the case of an ionic crystal, the exciton may be represented as a pair of anomalous ions with opposite excess charges, situated in immediate proximity to one another and bound by the forces of Coulomb attraction.

In the case, for example, of an NaCl lattice, an exciton is obtained as a result of the transfer of an electron from a \(Cl^-\) ion to its neighboring \(Na^+\) ion, as a result of which we obtain a neutral sodium atom and, beside it, a neutral chlorine atom. Such a formation can wander through the crystal as a single whole.

The absorption of light by alkali-halide crystals in the region of the intrinsic absorption band, according to Hilsch and Pohl \(^{13}\), is caused precisely by such a transfer of an electron from a negative ion to a neighboring positive ion. In modern language this is the exciton mechanism of light absorption.

An exciton wandering through a crystal does not carry current, for the exciton is an electrically neutral formation. Thus, the formation of an exciton is not associated with the appearance of conductivity.

If the two opposite anomalous ions joined in the exciton are separated sufficiently far from one another, they become independent of each other and from that moment can be described by band theory. Such a rupture of the exciton, or, in other words, its dissociation into an electron and a hole, requires an expenditure of energy. The exciton is therefore energetically more advantageous than the free electron and hole that appear in band theory.

12. DOUBLONS

The problem of the exciton is the problem of an electron and a hole interacting with one another.

Let us now consider another problem, also going beyond the limits of band theory: the problem of the behavior of two interacting electrons \(^{14}\).

We shall approach our problem as a many-electron problem and shall solve it by Heisenberg’s method \(^{15}\). This method is well known in the theory of metals. It is entirely equivalent to the fact that

in the theory of molecules is called the Heitler–London method¹⁶. In this method the problem is solved as a many-electron problem, without reducing it to independent one-electron problems.

Solving this problem, we obtain two types of solutions (two types of states of the system).

First, we obtain solutions of the Bloch type, corresponding to separated (mutually independent) electrons, when the position in the lattice of one electron in no way determines the position of the other.

Second, we obtain solutions of an essentially different type, corresponding to paired electrons, when two electrons sitting on two neighboring ions of the lattice turn out to be rigidly bound to one another. In other words, in this case the wave function of the system has a maximum at the minimum distance between the electrons and decreases as this distance increases.

The normal state of the system is a state of the first type. Among the excited states of the system, however, there are states of both the one and the other type.

The joining of two electrons into a pair is due to taking account of the interaction between them, which, along with the Coulomb interaction, also includes an exchange interaction of exactly the same type as that which occurs in the hydrogen molecule.

Here we have a bond of the chemical type, like the bond which is realized in a diatomic molecule. Indeed, in a lattice constructed, for example, of singly charged ions \(M^+\) and \(R^-\), the presence of two electrons means the presence of two neutral atoms \(M\) and \(M\). Such two neutral states, wandering through the crystal, may turn out to be coupled like two free neutral atoms entering into a chemical bond and forming a diatomic molecule.

It would be convenient to assign a special name to a pair of electrons bound by the exchange interaction. We have called such a pair a “doublon” (double — двойной). Such a pair behaves as a single whole. The doublon, of course, is not fixed in the lattice, but wanders freely through the crystal.

Doublons, unlike electrons, possess a number of specific properties. The effective mass of a doublon is not equal to twice the effective mass of an electron. Doublons, unlike electrons, possess an integral, and not half-integral, spin (the spin of a doublon is 0 or 1). They obey Bose statistics, and not Fermi statistics, as electrons do.

Let us note that band theory does not give, and by the very nature of the approximations contained in it cannot give, doublon states for electrons in the lattice. Such states appear when the crystal is considered as a many-electron problem.

13. Conclusion

In conclusion, let us ask ourselves: what, in the end, is the state of affairs in the modern theory of the solid state, and what are the prospects for its further development?

At the present time we have a sufficiently developed electronic theory of metals. This theory is sufficiently developed from the point of view of its applications, although it cannot be considered sufficiently developed from the point of view of its justification.

The task of the theory of metals consists, perhaps, not only in (or, better to say, not so much in) constructing a good and rigorous theory, as in understanding why an admittedly poor theory (Bloch’s theory, or even Sommerfeld’s theory) gives sufficiently good results.

As for the theory of nonmetallic crystals, here we have a sufficiently developed so-called band theory and, in addition, a number of separate theoretical works lying outside the framework of band theory.

Band theory has until now served in the hands of experimentalists as the working scheme by means of which the various processes occurring in nonmetallic crystals are interpreted.

A narrowly Blochian, entirely incorrect content is often put into this scheme—content which in fact this scheme does not possess.

However, even with its correct interpretation, the band scheme cannot be regarded as either sufficiently perfect or sufficiently all-embracing. Nevertheless, in the theory of nonmetallic crystals there exists a definite and rather extensive range of problems that can be solved within the framework of the band scheme.

Here we have very rich experimental material, far from all of which has yet been incorporated into theory.

The further development of band theory (to which it is fully entitled, when properly understood), its development from the point of view of expanding the experimental material that it embraces and interprets—this is one of the directions in the further development of the theory of nonmetallic crystals.

Another direction, which seems to us more interesting, more fundamental, and more promising, is the development of the theory of the solid state outside the framework of band theory. Very little has yet been done along this path. There exists a whole series of problems that can receive a satisfactory solution only along this second path, and not along the first.

The first path, in which the problem of electrons in a crystal is treated as the problem of a single electron (or a single hole) moving in a periodic field, can provide an explanation—

not many experimental facts that still remain unexplained, but on this path one can hardly expect anything fundamentally new at present.

The second path, in which the problem of electrons in a crystal is solved as a many-electron problem (without reducing it to a one-electron problem and to a periodic field), can not only provide an explanation for various facts, but along this path entirely new and, perhaps, unexpected features in the behavior of electrons may be discovered.

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