PHYSICAL IDEAS OF THE METHOD OF ELEMENTARY EXCITATIONS

(The Many-Electron Problem in Solid-State Theory)

V. L. Bonch-Bruevich

§ 1. INTRODUCTION. DIFFICULTIES OF THE “ONE-ELECTRON” THEORY OF METALS

The problem of studying systems consisting of many interacting particles occupies one of the central places in modern condensed-matter physics. As we shall see, it arises with particular acuteness in the theory of the metallic state (and also in the theory of liquids); however, even in the theory of semiconductors, where it would seem that the interaction of conduction electrons with one another may be neglected because their concentration is small, there are problems in which allowance for the electron–electron interaction is essential. It is enough to recall at least the following standard argument: owing, for example, to a nonuniform distribution of an impurity in the lattice, the concentration of electrons in it also turns out to be different at different places, and the redistribution of electrons continues until the resulting space charge creates a field that prevents further displacement of the electrons. Such arguments are constantly encountered in the theory of surface states, the photo-emf, etc., and, as is clearly seen, they are entirely based on the existence of interaction of the electrons with one another. At the same time, a correct solution of the quantum-mechanical (and equally of the classical) problem of the behavior of a system of many interacting particles presents considerable difficulties. In the present article we should like, without going into the details of calculations, to set forth the essence of one of the methods for the approximate solution of the many-body problem, which seems to us the most promising and which has been developing most intensively in recent times. The exposition does not claim exhaustive completeness;

the task consists merely in acquainting the non-theoretical reader with the substance of the matter.

Since the principal “sphere of application” of methods for solving the many-body problem at the present time is solid-state physics, it is appropriate to begin by considering some difficulties of the modern theory of metals. The latter, despite the presence of an actually strong interaction between electrons, until very recently developed almost exclusively as a “one-electron”) theory, in which correlations between electrons either were not taken into account at all (the simplest model of an “electron gas”¹), or were taken into account only very approximately by the self-consistent-field method. Nevertheless, a number of qualitative conclusions of the “one-electron” theory (even in its most primitive form) are in good agreement with experiment. These include, for example, the temperature dependences of the electrical conductivity and electronic heat capacity of metals, the theory of the paramagnetism of alkali metals, etc. At first sight this circumstance seems paradoxical: a physically untenable theory leads to correct results; obviously², the task of the theory of metals consists above all in understanding the causes of this paradox). Let us note, however, that the successes of the “one-electron” model should not be overestimated (and such an overestimation of the band theory of metals undoubtedly occurs in a number of works**); especially characteristic in this respect are the books by Mott and Jones and by Seitz cited above).

A number of phenomena receive no explanation at all within the framework of the “one-electron” theory (among them, apparently, is, for example, superconductivity), while other phenomena are explained only, we would say, in a purely formal way. Perhaps the most vivid example

*) The only exception is the theory of ferromagnetism, in which the necessity of an essentially “many-electron” treatment was recognized long ago.

**) An attempt to solve the question in a trivial way, by considering the interaction energy between electrons to be small in comparison with their kinetic energy, proves untenable. Indeed, the Fermi energy for an electron gas, as is known, is given by the formula

\[ E_F=(3\pi^2)^{2/3}\frac{\hbar^2}{2m}\,n^{2/3}, \]

where \(n\) is the number of electrons per unit volume; \(m\) is the electron mass; the average energy of the Coulomb interaction, obviously, will be of the order of

\[ e^2 n^{1/3}. \]

The ratio of these energies is of the order of \(4\cdot 10^7 n^{-1/3}\), which for reasonable (\(\sim 10^{22}\ \mathrm{cm}^{-3}\)) values of \(n\) is a quantity of order unity. This circumstance, it seems to us, deprives the very notion of a “Fermi sphere” for electrons in a metal of meaning.

***) The band theory of metals has repeatedly been subjected to justified criticism in the works of Soviet scientists²–⁶.

of such a formal explanation is encountered in the theory of the magnetic properties of metals at low temperatures. As is known, at low temperatures the magnetic susceptibility \(\chi\) of a number of metals depends periodically on the strength of the magnetic field \(H\). In qualitative form this effect would seem to be excellently explained by the “one-electron” theory\(^{7-9}\), which indeed gives the required periodic dependence. With a certain choice of the parameters entering into the theoretical formula (the effective mass*) and the concentration of conduction electrons), in most cases quantitative agreement between theory and experiment is also obtained. However, a more careful consideration of the question\(^{10}\) shows that this explanation is to a considerable extent illusory. Namely, for the parameters entering into the theoretical formula, implausibly small values are obtained (and ones not agreeing with the data of other measurements). Thus, for example, for zinc the number of conduction electrons per atom turned out to be of the order of \(0.8\cdot 10^{-6}\), which, as indicated in\(^{11}\), is only one thousandth of the number needed to explain the experimentally observed values of the electronic heat capacity of this metal. The situation is similar for other metals (beryllium and bismuth). Thus it turns out that the “one-electron” theory correctly conveys only the qualitative form of the dependence, but by no means the quantitative side of the matter. However, the form of the \(\chi(H)\) curve is determined to a considerable degree only by the statistical properties of the system. With the concrete values of the parameters that depend on the nature of the system (for example, on the masses, charges, and concentration of the particles) there are connected only the numerical characteristics of the curve.

In this connection it should be noted that, in order to obtain, for example, the qualitative picture of the temperature course of the electronic heat capacity, it is also necessary to know only the statistics of the electrons; likewise, only the statistical properties of the electrons and of the thermal vibrations of the lattice determine the form of the temperature dependence of the electrical conductivity of metals.

Thus, in the one-electron theory by no means all relations are obtained correctly, but only those that are due mainly to the statistical properties of the system—the fact that the electrons obey Fermi statistics; the same regularities in which the concrete form of the energy spectrum and the concrete values of the parameters determining it are essential are, as a rule, not conveyed by the “one-electron” theory of metals.

*) It would be more correct to say “effective masses,” since, owing to the anisotropy of the crystal lattices of the metals under consideration, the electrons in them are characterized not by one, but by three effective masses (corresponding to the three principal axes of the crystal).

To clarify the “paradox” indicated above, and also to investigate as yet unsolved problems in the theory of metals, a “many-electron” approach to the problem appears necessary. We thus arrive at the problem of investigating the properties of a system of many particles strongly interacting with one another.

§ 2. THE HYPOTHESIS OF “ELEMENTARY EXCITATIONS”

The difficulties in solving the problem posed are as follows:

a) The wave equation for a system of many \((\sim 10^{23})\) interacting particles is extraordinarily complicated; its exact solution by the means of modern mathematics, even with machines, is scarcely possible.

b) Even if it were possible to determine exactly the possible energy levels of the many-body system under consideration, in order to find a number of experimentally observable quantities (heat capacity, magnetic susceptibility, etc.) one would still have to calculate the statistical sum

\[ Z=\sum_n e^{-\frac{E_n}{kT}} \tag{2.1} \]

(\(E_n\) are the possible energy levels, numbered by the index \(n\)). Indeed, knowing \(Z\), one can, as is well known, find the free energy and, consequently, all the thermodynamic properties of the system. This problem is, generally speaking, not much simpler than the first one (let us recall that the classical analogue of (2.1) is an integral of multiplicity \(\sim 10^{23}\), and the integrand, generally speaking, cannot be represented as a product of factors each depending on a small number of variables).

The difficulties named are extremely serious, and in the present state of mathematics it scarcely makes sense to try to solve them “head-on.” From consideration of the expression for the statistical sum (2.1), however, it follows that in practice such a general formulation of the problem is not necessary. Indeed, only energy levels sufficiently close to the ground state play a noticeable role in (2.1). Therefore one may restrict oneself to investigating only such “weakly excited” states of the system. This circumstance, as we shall see, simplifies the problem to an extraordinary degree.

In several concrete cases the problem of weakly excited states of a system of many particles has long since been solved. We shall recall here two well-known examples and consider them (from a purely qualitative point of view), with the aim of revealing certain features of the behavior of such systems which, as will later turn out, have a quite general character.

A. Vibrations of the Crystal Lattice

The problem of thermal vibrations of a crystal lattice is, evidently, historically the first example of the study of the collective behavior of many interacting particles. Obviously, the state of “complete order,” when all the atoms (or ions) of the lattice are uniformly (and periodically) distributed in space,* is here the state of lowest energy. Excitation of the system consists in the appearance of small vibrations of the atoms about their equilibrium positions; the spatial distribution of the atoms will then, naturally, be slightly nonuniform. In other words, excitation of the system in the present case consists in the appearance of certain “special states”—local changes of density; the latter do not “freeze” in one place, but propagate in wave-like fashion throughout the entire lattice (ordinary sound vibrations are a particular case of these waves).

For small amplitudes of vibration (in comparison with interatomic distances)—this is precisely the condition of “smallness of the excitation”—the principle of superposition is valid for elastic waves, i.e. they propagate independently of one another, and the energy of the system is additively composed of the energies of the individual waves.

In a quantum-mechanical treatment of the problem[^12-14], these waves are, naturally, associated with discrete formations—sound quanta** (phonons). Interaction between the latter is absent, since the principle of superposition is valid for the corresponding waves (the “harmonic” approximation).

Thus, from the energetic point of view, weakly excited states of a crystal lattice may be regarded as an “ideal gas” of certain “quasiparticles”—phonons. We emphasize that these “quasiparticles” have nothing in common with the atoms composing the given system, but represent only the corpuscular aspect of the collective vibrational motion of the latter.

The state of a phonon (in a simple lattice) is specified by its polarization (a longitudinal or transverse wave) and by three components of a certain vector (in many respects analogous to momentum), which determine the energy. Since the possible values in principle of the velocity of sound in the lattice are not bounded,*** in one and the same

* We are reasoning in a purely “classical” form, abstracting from the existence of zero-point vibrations, the presence of which is inessential for our discussion.

** The concept of sound quanta was first introduced by I. E. Tamm[^12].

*** Of course, at a sufficiently high intensity of sound waves they can no longer be regarded as independent. This circumstance, however, is inessential for our discussion (although because of it the corresponding intensities are far greater than those at which the “Bose” character of phonon statistics becomes manifest).

in the states there may be any number of “quasiparticles”; consequently, they obey Bose—Einstein statistics (independently of the type of statistics obeyed by the atoms themselves that make up the lattice). As is well known, on the basis of the concept of phonons one can construct the entire thermodynamics of the crystal lattice, and also consider a number of kinetic processes in it.

B. Spin waves in a ferromagnet

The second example that we wish to consider concerns weakly excited states of a ferromagnet. As is known,\(^{15}\) the fundamental energy level of a ferromagnet corresponds to the state of “complete magnetization,” when the magnetic moments of all atoms of the lattice are oriented in the same way) (the spins of all “magnetic”*) electrons have one and the same component along some axis).

The excitation of the system consists in a change in the directions of the magnetic moments of certain atoms (in a “reversal” of the spins of some of the “magnetic” electrons), i.e., as in the first example, in the appearance of certain special states (in the present case, spins of one direction in the “medium” of spins of the opposite direction), propagating in a wave-like manner throughout the entire crystal lattice.

Indeed, owing to the physical equivalence of the different lattice sites it is obvious that a state with a “reversed” spin cannot “get stuck” at some single atom***), but, because of the interaction of the electrons with one another, will propagate through the lattice. In the stationary case a “reversed” spin may be found with equal probability at any atom of the lattice (if it is simple). These states are called spin waves. So long as the number of “reversed” spins is small in comparison with the total number of “magnetic” electrons (the condition of small excitation) and, consequently, the probability of their meeting in the lattice is small, one may assume that the spin waves propagate independently of one another and that each of them is characterized by a definite energy. The energy of the system of electrons

*) When the weak magnetic interaction of the electrons is taken into account, this assertion requires some clarification, which, however, changes nothing essential in our reasoning.

**) By “magnetic” we mean here electrons whose spins (in the absence of a magnetic field) can be oriented in any direction (i.e., for example, electrons located in unfilled atomic shells).

***) Here we are speaking only of an ideal lattice, containing no structural defects that break the translational invariance of the system.

in this case (to within an inessential additive constant) is composed of the energies of the individual spin waves. Naturally, with the latter one can associate certain “quasiparticles” (sometimes called “ferromagnons,” since they are characteristic of ferromagnets), and we, as in the first example, arrive at the representation of an ideal gas of certain “quasiparticles” depicting weakly excited states of a system of strongly interacting particles (in the present case—electrons in a ferromagnet). Let us emphasize again that these quasiparticles have nothing in common with the electrons themselves*), but characterize only the corpuscular aspect of their collective motion.

The representation in terms of spin waves proved to be extremely fruitful in the theory of ferromagnetism, making it possible theoretically to derive the dependence of the spontaneous magnetization near saturation on temperature^16 and on the magnitude of the external field^17, and also to construct a quantum theory of magnetic anisotropy^18 and magnetostriction^19.

We see that in both of the cases considered the energy corresponding to weakly excited states of the system is represented in the form of a sum of the energies of independent “quasiparticles,” and the study of the properties of the system in these states reduces to the study of the behavior of a “gas” of quasiparticles (i.e., to a well-known problem that is solved without difficulty). Since the systems we have considered are physically quite different, it is natural to suppose that this state of affairs is characteristic also of any quantum system of many interacting particles: excitation of the system always reduces to the appearance of certain special states—“elementary excitations”—which (by virtue of translational invariance) propagate through the system in a wave-like manner; weakly excited states of any quantum system of many interacting particles can be represented as an ideal gas of certain “quasiparticles” associated with these waves**).

The possible values of the energy of the quasiparticles, of their momentum, and of other similar quantities, as well as the statistics obeyed by the elementary excitations, give a complete characterization of the weakly excited states of the system. (Depending on the statistics of the elementary excitations, one speaks of spectra “of Fermi type” and “of Bose type”^14; the terms “Fermi” and “Bose” branch are also sometimes used.) The quasiparticles in question, generally speaking, have nothing in common with those

*) Let us note in this connection that, as is known^15, spin waves obey Bose statistics (and not Fermi statistics, as electrons do).

**) As far as we know, this idea was first expressed by L. D. Landau. In recent years it has been the subject of investigations by N. N. Bogolyubov, S. V. Vonsovskii, and a number of other Soviet scientists.

particles of which the given system consists, but represent only a certain aspect of their collective motion*).

It is quite obvious that the representation in terms of elementary excitations at once resolves both of the difficulties of the many-body theory noted above. Indeed, since the energy values corresponding to weakly excited states of the system are expressed in the form

\[ E=\sum_k w(k)n(k), \tag{2.2} \]

where \(n(k)\) is the number of elementary excitations characterized by the set of “quantum numbers” \(k\) (for example, momentum, spin, etc.), the problem is reduced to computing the energy of a single quasiparticle, \(w(k)\). One may expect this to be considerably simpler than solving the many-body problem in its general formulation. And indeed, in a number of cases the energy spectrum of elementary excitations can be computed quite effectively. Different weakly excited levels of the system correspond, obviously, to different sets of the numbers \(n(k)\), i.e. to different distributions of elementary excitations over their quantum states.

Further, the question of computing the statistical sum in this case generally loses all urgency, since we are dealing with an ideal gas, for which the equilibrium distribution function of particles over energies is well known.

Finally, it is clear that the representation in terms of elementary excitations makes it possible, without any special difficulty, to consider nonequilibrium problems as well. Indeed, since definite values of energy and also, possibly, charge, momentum, etc., correspond to quasiparticles, one can speak of their transporting the corresponding quantities. Thus problems concerning transport processes in condensed media are reduced to analogous problems in the kinetic theory of an ideal gas. For example, the consideration of thermal conductivity due to the lattice itself is reduced to the study of energy transport by a flux of phonons. Questions of the establishment of statistical equilibrium in the system can likewise be considered without particular difficulty by the method of elementary excitations. For this it is only necessary to introduce (as the next approximation) a weak interaction between quasiparticles, leading to the establishment of an equilibrium Fermi or Bose distribution of them over states.

* From what has been said it follows that, for example, it would be quite meaningless to attempt to “assemble” quasiparticles (for example, phonons) “into a box.” They exist only insofar as there exists a system of interacting particles undergoing collective motion, represented by the quasiparticle picture; with the destruction of the system (for example, by the melting of a crystal), the corresponding elementary excitations naturally also disappear.

Indeed, from the point of view of the representation of elementary excitations, the establishment of thermodynamic equilibrium in a system of many interacting particles reduces to the establishment of an equilibrium distribution in the gas of “quasiparticles.”

The effectiveness of the method of elementary excitations was demonstrated in a number of works devoted to the solution of specific equilibrium and nonequilibrium problems. Among these works one should first of all mention the theory of superfluidity of liquid helium II20–24, first successfully developed by L. D. Landau exclusively on the basis of the representation of elementary excitations*). In the work of N. N. Bogolyubov21, in which the spectrum of elementary excitations in a gas of weakly interacting Bose particles was theoretically calculated, these representations received a microscopic foundation. An example of the successful application of the method of elementary excitations is also the theory, developed by I. Ya. Pomeranchuk25, of the thermal conductivity of paramagnetic dielectrics. In these substances there are specific excitations associated with the presence of exchange interaction of the electrons of paramagnetic atoms. Namely, as in a ferromagnet, the excited states of the system in the present case may differ from the ground state by another distribution of magnetic moments (the difference in comparison with the ferromagnetic case consists in the fact that now the ground state does not correspond to complete magnetization. To indicate the exact distribution of magnetic moments in the ground state of a paramagnet in the absence of an external field is difficult; for our purposes, however, it is sufficient to know only that some such distribution exists). Naturally, deviations from the ground-state distribution of magnetic moments are not localized on individual atoms, but, owing to the interaction of the electrons with one another, propagate in wave-like fashion throughout the whole lattice. These elementary excitations are called magnons. (For weakly excited states, when the number of magnons is small in comparison with the total number of atoms in the lattice, their interaction energy with one another may be neglected, and, consequently, the excitation energy is the sum of the energies of the individual magnons.) Magnons (which apparently obey Fermi statistics) interact with phonons, thereby affecting their free path, and also themselves take part in the transport of heat. As shown in25, this leads to specific features in the temperature dependence of the thermal conductivity \(\chi\) at low temperatures (the dependence \(\chi(T)\) proves to be nonmonotonic).

In addition, the method of elementary excitations was successfully applied in the study of the approach to the state of equilibrium in ferro- and paramagnets26, 27. It was also used in attempts

*) These investigations are not considered here in detail, since there is a detailed review in the literature65, to which we refer the reader.

construction of a many-electron theory of metals and semiconductors, which will be discussed in the next paragraph.

Let us note, finally, that in all other cases the successive (even if approximate) consideration of the many-body problem “by itself” leads to the concept of elementary excitations. Thus, for example, this is the situation in the theory of antiferromagnetism, which is being developed on the basis of a somewhat generalized notion of spin waves \(^{28-31}\); weakly excited states of molecular crystals can be described with the aid of the concept of excitons \(^{32-35}\)—“quasiparticles” whose motion characterizes the transfer of excitation energy (obtained, for example, from light) from one lattice site to another.

This notion of the exciton naturally carries over also to the case of any homopolar crystal. If one of the atoms of the lattice has in one way or another acquired some excess energy, then it is clear that, as a result of interatomic interaction, this energy will be transferred also to other atoms (in the end, on the average being distributed uniformly among them). The wave-like displacement of the excited state may be regarded as the motion of a “quasiparticle”—an exciton*).

Finally, the concept of elementary excitations is in fact widely used also in the ordinary theory of semiconductors (see, for example, \(^{36}\)). Indeed, “holes” in semiconductors are typical quasiparticles, describing states in which, on certain atoms, there is an incomplete set of electrons. (This example illustrates especially well both the essentially “collective” character of elementary excitations and the correspondence of this concept to physical reality. Indeed, hardly anyone would deny the reality of the existence of “holes” in semiconductors, just as it would probably occur to no one to try to collect them in some vessel.)

Returning now to the difficulties of the theory of metals considered above, it is easy to see that, in principle, they are at once resolved by the idea of elementary excitations. Indeed, from this point of view the well-known success of the “one-electron” model is entirely understandable: a system of many interacting electrons in a metal, like any system of many interacting particles, is characterized by certain excitations, and what in the Sommerfeld and Bloch theories was called an electron is in fact not an electron, but a “quasiparticle”: the “electron gas” of the primitive theory of metals is in fact a “gas of elementary

*) The concept of the exciton, in a somewhat different form, can also be extended to the case of ionic crystals. Consideration of this question, however, does not fall within our task.

...“excitations” of a many-electron system obeying Fermi statistics. In this sense one may say that in the theory of metals people have always spoken the language of elementary excitations, without knowing it. It is therefore not surprising that regularities determined, in the main, only by statistics are correctly conveyed by the “one-electron” theory (they are simply not connected with its simplifying assumptions); it is equally clear why the “one-electron” model fails in the analysis of those characteristics of the system for the study of which more concrete information about its energy spectrum is necessary*).

It should be emphasized, however, that what has been said must in no way be regarded as a final solution of the difficulties of the modern theory of metals. Here only a possible path of solution has been indicated—a path which, it seems to us, is correct and very promising, but by no means yet traversed. In order that the solution of the question set forth above may be considered satisfactory, it is necessary, first of all, to prove that the energy of the electron system in a metal is indeed expressed in the form (2.2), and to determine the form of the function \(w(k)\), as well as the statistics of the elementary excitations**). At the same time, of course, it must be clarified precisely which excitations are possible in one or another system (it is obvious that in a given concrete system not all types of elementary excitations can, in general, be realized. Spin waves, for example, arise in ferromagnets, but in a metal of the type of, say, beryllium, they apparently do not). In other words, the question arises of developing methods for investigating the spectra of elementary excitations. We would like especially to emphasize the importance of this problem. The point is that the ease of operating with elementary excitations (once their statistics and the form of the function \(w(k)\) are known) can easily lead one onto the false path of simply “fitting” one or another excitation spectrum to experimental data without proper theoretical justification. Thereby a certain semblance of understanding and explanation of phenomena would be created, while in fact both would be absent, and the concept of elementary excitations would lose its meaning and interest***).

*) From what has been said it is clear how fruitless are attempts at quantitative refinement and improvement of the methods of the “one-electron” theory, still being undertaken in some works. These attempts, to be sure, are not wholly “harmful,” for they do not distort the “Fermi” character of the spectrum, but they are useless to the same degree.

**) In what follows we shall see that, as a rule, the spectrum proves to be mixed, i.e., there are excitations of both Fermi and Bose type.

***) What has been said should not be understood perversely. We, of course, do not object to determining, for example, the effective mass of a quasiparticle from experimental data, once it has been proved that excitations of the given type can indeed arise in the given system. Equally, one should...

From what has been said it is clear how important is the problem of the actual determination of the weakly excited levels of a system of many interacting particles. In the following paragraph we shall consider the modern ideas concerning the spectrum of elementary excitations of a system of electrons in a solid.

§ 3. ELEMENTARY EXCITATIONS AND THE ELECTRON THEORY OF SOLIDS

As we saw in the preceding paragraph, a number of difficulties in the electron theory of metals could, in all likelihood, be resolved if it were possible to represent the energies of the weakly excited states of a many-electron system in the form of a set of elementary excitations. It should only be borne in mind that in substances possessing metallic properties, at least some (if not all) excitations must be characterized by two special features:

a) their motion in the lattice must be accompanied by the transport of charge (otherwise they are not at all carriers of current),

b) their formation must not require the expenditure of finite energy.

Indeed, otherwise the number of elementary excitations of the given type (i.e. the number of current carriers) at low temperatures would decrease exponentially as the temperature is lowered, which would also lead to a corresponding behavior of the electrical conductivity. In fact, however, this apparently does not occur (although, experimentally, the question of the temperature dependence of the electrical conductivity of metals at low temperatures is still not entirely clear). Moreover, in our opinion, the elementary excitations of a many-electron system that are characteristic specifically of the metallic state of a substance must obey Fermi statistics*). Indeed, it is known from experiment that the electronic heat capacity of a metal depends linearly on temperature.

recognize as very important the “inverse problem” of the theory of elementary excitations—the determination of the form of the spectrum from experimental data (very substantial results in this direction have been obtained by I. M. Lifshitz and his collaborators⁶³,⁶⁷). We wished only to caution against possible attempts simply to postulate the existence of one or another spectrum without investigating whether its occurrence is possible in reality. It should be pointed out that in all the works cited above such “fitting” did not occur; the form of the elementary excitations was established either on the basis of a direct calculation or with the aid of theoretical considerations of a qualitative character.

*) This does not mean that there is no “Bose” branch of excitations in metals. It undoubtedly exists (at least in some metals). We merely wish to say that, apparently, a “Fermi” branch of the energy spectrum must also be present.

This dependence is easily obtained theoretically if the elementary excitations of a many-electron system form a degenerate Fermi gas; in the case of Bose-type excitations, however, such a dependence can be obtained only under special assumptions concerning the density of energy levels. In fact, the total energy \(E\) of a gas of elementary excitations is given by the well-known relation \(^*)\)

\[ E=\int_{\varepsilon_{\min}}^{\infty} \frac{\rho(\varepsilon)\,d\varepsilon} {\exp\left\{\frac{\varepsilon-\mu}{kT}\right\}\pm 1}, \tag{3.1} \]

where \(\mu\) is the chemical potential, \(\varepsilon\) is the energy of an individual excitation, \(\rho(\varepsilon)\,d\varepsilon\) is the number of states in the energy interval \((\varepsilon,\varepsilon+d\varepsilon)\), and the signs “\(+\)” and “\(-\)” correspond respectively to Fermi and Bose statistics.

For a degenerate Fermi gas \(\mu>0\) and \(\mu/kT\gg 1\); the asymptotic expansion in \(kT/\mu\) gives, as is well known (see, e.g., \(^{14}\)):

\[ E\simeq \int_{\varepsilon_{\min}}^{\mu}\varepsilon\rho(\varepsilon)\,d\varepsilon + \frac{\pi^2}{6}(kT)^2 \left\{ \rho(\mu)+\mu\left.\frac{d\rho(\varepsilon)}{d\varepsilon}\right|_{\varepsilon=\mu} \right\}, \tag{3.2} \]

whence—whatever the form of \(\rho(\varepsilon)\)—a linear temperature dependence of the heat capacity is obtained.

For a gas obeying Bose statistics, on the other hand, as is known \(^{14}\), the chemical potential is always negative (and small in absolute value if the gas is degenerate); therefore an expansion of the type (3.2) is not possible, and the dependence of \(E\) on \(T\) is determined by the specific form of the function \(\rho(\varepsilon)\) \(^ {**}\).

In the present paragraph we propose, without going into the computational side of the matter, to consider the current state of the question of elementary excitations of the many-electron system in metals and semiconductors.

Some types of excitations are already known to us—these are the excitons and spin waves considered in the preceding paragraph.

\(^*)\) If there are several types of excitations, one must take the sum of expressions of the type (3.1). At very large values of \(\varepsilon\) the very notion of elementary excitations becomes inaccurate, but this region makes practically no contribution to the integral.

\(^ {**}\) In this case \(\rho(\varepsilon)\) usually turns out to be such that the heat capacity is proportional to a power of the temperature higher than the first and therefore is very small at low temperatures. Thus, for example, for phonons in a simple lattice, as is known, the heat capacity is proportional to \(T^3\).

It is clear, however, that they are by no means characteristic of a metal. Indeed, an exciton is a neutral formation, for whose occurrence, in addition, a finite energy is required. Consequently, it has no relation to the property most characteristic of metals—high electrical conductivity. Likewise, under a simple displacement of a state with “reversed” spin (a spin wave), electric charge is by no means displaced through the lattice*), for the average number of electrons at each individual atom remains unchanged^37,38.

Thus, the model of spin waves considered earlier (sometimes called the “exchange” model*) describes, strictly speaking, not a metal, but a ferromagnetic dielectric. This is understandable, since in the “exchange” theory of spin waves a very important feature of the metallic state of matter is not taken into account—the collectivization of part of the electrons among all atoms of the lattice (it is precisely thanks to this process that electrons acquire the ability to move freely through the lattice, forming what in the phenomenological theory of electricity is called “free charges”). In order that the theory of spin waves could be used in the investigation of ferromagnetic metals, it is necessary to generalize it somewhat, taking into account the inevitable “collectivization” of at least part of the electrons among all atoms of the metal. Such a generalization has been carried out in two directions. First, one should bear in mind that in real metals the electrons of incompletely filled \(d\)-shells are apparently responsible for ferromagnetism; in electrical conductivity, however, the main role is evidently played by the “peripheral” electrons, which belonged (before the formation of the metal crystal lattice) to the outer atomic shells and are “collectivized” among all atoms of the metal. S. V. Vonsovskii^39,40 proposed that the \(d\)-electrons be treated according to the many-electron exchange model, taking into account only additionally their exchange interaction with the “collectivized” electrons; the interaction of the latter with one another is not taken into account. The energy spectrum of the whole system, in this way, consists of two “branches”—a set of spin waves (Bose-type excitations) and the sum of the energies of the outer electrons (which may be regarded as Fermi-type excitations). The presence of an interaction between the inner and outer electrons is manifested in the dependence of the effective mass of the latter on the total spin of the former, i.e. on the magnetization of the specimen.

In such an “\(s\)–\(d\) exchange” model**) both the ferromagnetic and the electrical properties are reflected simultaneously, and (thanks to taking into account the exchange interaction between the inner and outer electrons) their interrelation can also be investigated.

*) See, however, the footnote on p. 69.
**) The name is connected with the fact that the outer electrons are considered to have originally been (before the formation of the metal) in \(s\)-states.

It is obvious, however, that the \(s-d\) model does not fully solve the problem of determining the energy spectrum of the system of electrons in a metal, since there are no grounds at all for neglecting the interaction of the “generalized” electrons with one another (this circumstance is also noted in the papers themselves \(^{39,40}\)). Moreover, the very possibility of dividing the single system of electrons into two “parts” (“inner” and “outer” electrons) is by no means obvious; in a more exact theory, evidently, one should dispense with such superfluous model representations.

More consistent with respect to taking interelectronic interaction into account is the second possible generalization of the spin-wave theory—the so-called “polar” model \(^{3,4,5,28,38,41,42}\). According to this model, in the ground state of a metal the electrons are, on the average, uniformly distributed over all atoms; an excitation of the system consists in a deviation of the charge distribution from the uniform (more precisely, periodic) one, i.e.—in pictorial language—in the transfer of some electrons to “foreign” atoms, as a result of which atoms with an excess and a deficiency of electrons appear in equal number in the lattice (the corresponding states of the atoms are called “polar”—this is also connected with the name of the model itself). Naturally, by virtue of the translational invariance of the system, polar states in an ideal crystal are not localized on any definite atoms, but with equal probability may be found at any place in the lattice. Translated into the language of elementary excitations, the polar states correspond to quasiparticles—“doublons” and “holes,”—whose motion characterizes, respectively, the wave-like propagation of states with “excess” and missing electrons*). It is obvious that the transfer of electric current is associated with the displacement of a doublon or a hole. The statistics obeyed by excitations of this type may be either Fermi or Bose statistics. Thus, for example, if in the normal state the valence shell of an atom contains an odd number of electrons and the excess electron appears in the same valence shell, then the doublons and holes have integral spin and, consequently, obey Bose statistics. On the other hand, it may happen that in the ground state the atoms have integral spin; then the spins of the doublon and the hole are half-integral and their statistics is Fermi. The first case was considered in the above-cited papers of S. V. Vonsovskii; the second (on the particular example of a beryllium-type metal) was briefly discussed in the author’s paper \(^{43}\).

*) In the mathematically most perfect version of the polar model, proposed by N. N. Bogolyubov and S. V. Tyablikov, “doublons” and “holes” are not introduced explicitly, but taking polar states into account leads to the fact that spin waves turn out to be connected with the transfer of electric charge.

In works 44 and 45 the polar model was used to investigate the electrical conductivity of a metal and its magnetic properties. In accordance with the basic idea of the elementary-excitation method, in both cases the problem was reduced to the study of the corresponding properties of an ideal gas of quasiparticles—doublons and holes. The statistics obeyed by the elementary excitations was not established in work 45. In work 44 excitations of the Bose type were considered. This circumstance led to a specific dependence of the electrical conductivity \(\rho\) on temperature

\[ \left( \rho = \frac{\gamma}{T} + \frac{\beta}{T^2}, \text{ where } \gamma \text{ and } \beta \text{ are constants} \right), \]

a dependence which, apparently, is indeed observed experimentally at low temperatures in such metals as cesium and certain others. It is essential that it cannot in any way be obtained within the framework of the one-electron theory of metals,^1 since the decisive role here is played precisely by the type of statistics obeyed by the current carriers (in the “one-electron” theory the current carriers are free electrons, and Fermi statistics for them leads to the well-known law: \(\rho \sim T^{-5}\)).

In work 46 the electrical conductivity of a metal was studied within the framework of a somewhat different variant of the polar model: it was assumed that in the ground state the distribution of electron density is maximally nonuniform (almost all lattice sites are occupied either by doublons or by holes), while excitation of the system is associated with a decrease of the nonuniformity in the charge distribution (partial “depolarization” of the crystal), i.e., with a decrease in the number of doublons and holes. The corresponding elementary excitations also obey Bose statistics, and the temperature dependence of the electrical conductivity turns out to be the same as in 45. In essence, in 46 the metal is regarded as something like an ionic crystal. We do not find such an approach to the problem convincing (it is not very clear, for example, what the situation will be with X-ray diffraction); however, the methodological value of the cited work is beyond doubt.

The known success of the polar model, however, must not obscure its serious defects, which are organically connected with its initial assumptions. From what has been said above it is clear that, in its present form, the polar model unquestionably cannot be applied to “good” metals containing a large number of current carriers. Indeed, the appearance of current carriers in the polar model is necessarily connected with excitation of the system (in the ground state current carriers are absent). Consequently, in weakly excited states (where alone the quasiparticle method is applicable) there will be few current carriers, and we obtain a substance with poor conductivity.

Moreover, in a number of cases a finite energy proves necessary for the formation of a “doublon” and a “hole,” which must

would lead to an exponential dependence of the electrical conductivity and other quantities on temperature. Finally, it is not very clear how, in the polar model (with a Bose-type spectrum), matters will stand with the electronic heat capacity of a metal. We encounter similar difficulties also in the further generalization of the polar model in the so-called “polar-exciton” model,³ in which the presence of three types of elementary excitations at once is taken into account—“pairs,” “holes,” and excitons.

The impression is created that, in general, in its present form the polar-exciton model of a solid describes not a metal, but a semiconductor with an atomic lattice. Indeed, for a semiconductor the exponential dependence of the number of current carriers on temperature is precisely characteristic. Let us note in this connection that the polar-exciton model of a crystal has in fact also been used to investigate the magnetic⁴⁷ and electrical⁴⁸ properties of semiconductors. In the latter work there was given for the first time a “many-electron” justification of a number of assertions of the “ordinary” (based on the one-electron approximation) theory of semiconductors. It was specifically an atomic crystal that was considered, in which the “pairs” and “holes” of the polar model obey Bose statistics. It was shown that the behavior of these excitations corresponds to what we expect from the “ordinary” conduction electrons and holes of the theory of semiconductors (the energy spectrum is band-like, a finite energy is required for the occurrence of excitations, etc.). The difference in the types of statistics to which the current carriers are subject according to the one- and many-electron theories is, however, of a fundamental (and in some cases also practically essential) character. In the region of degeneracy this circumstance will naturally lead to sharply different predictions concerning the electrical and magnetic properties of a semiconductor. The corresponding experimental investigations would be of considerable interest for the theory of the solid state. It should be borne in mind, however, that in semiconductors of the germanium or silicon type the spectrum of the “pairs” and “holes” of the polar model proves to be Fermi-like. This case was considered in work⁶⁸. As was to be expected, it turned out that in such a semiconductor the “pairs” behave analogously to the conduction electrons of the one-electron theory, thereby justifying the qualitative conclusions of the latter (this applies both to an ideal lattice and to a lattice with defects; to a certain extent this is also true in the presence of external electric and magnetic fields). Let us note, however, that this justification in no way pertains to the computational methods of the one-electron theory. All constants characterizing the form of the energy spectrum (the width of the forbidden band, the effective mass, etc.) are calculated in the many-electron theory quite differently than in the one-electron theory. Only qualitative conclusions receive justification.

representations of the “zone” model (which, incidentally, are chiefly what interest the experimentalist).

It does not follow from what has been said that one must altogether abandon the representations of the polar model of a metal, restricting the sphere of its application only to semiconductors. Apparently, excitations of the types considered above do nevertheless exist in metals as well, but they do not exhaust the entire energy spectrum of the latter*). These excitations, taken together, form what might be called the “semiconductor spectrum” of a metal. They are equally possible also in nonmetallic crystals and constitute that common feature which is present in the electronic energy spectra of all crystals with atomic lattices.

In metals**), however, there apparently exist excitations of another type, which do not require a finite energy for their formation and therefore are present in large numbers even at low temperatures. (They probably obey Fermi statistics.) The problem of investigating this specifically metallic branch of the energy spectrum still remains open (here we encounter very great mathematical difficulties). In this connection it becomes of interest to study at least the simplest cases of spectra of this type—Fermi spectra and those without an energy gap, while the system must already contain current carriers in its ground state. Apparently the easiest system to consider is that of weakly interacting conduction electrons in a crystal. In this case, in order to determine the energy spectrum, one may make use of perturbation theory (the small parameter being the ratio of the concentration of conduction electrons to the number of lattice sites per unit volume)***). The corresponding method was developed in work 49 and used in calculating the electrical conductivity of metals according to the many-electron model in 50 and 51. In the last two papers it was shown that, as was to be expected, the “statistical” results of the one-electron theory (the temperature law of electrical conductivity) remain valid also when the current carriers are elementary excitations of the Fermi type. It should be remembered, however, that the quantitative results of 49 (and, consequently, of 50) have only limited significance and, in our

*) In this connection it should be noted that the methods so far proposed for calculating the spectra of elementary excitations practically do not yield a complete system of eigenfunctions of the Hamiltonian of the many-electron problem.

**) It would be more correct to say that it is precisely those substances in which these specific excitations occur that are metals.

***) Such a formulation of the problem is natural for the nondegenerate (or weakly degenerate) case. In the presence of strong degeneracy, the role of the small parameter may be played by the ratio of the mean interaction energy to the Fermi energy (see 68).

in our opinion, cannot be applied to real metals, in which the concentration of conduction electrons is by no means small. It would be more correct to regard these results as pertaining to semiconductors, where the conditions for the applicability of the given method of calculation are indeed fulfilled.

Recently the idea of local changes in density as elementary excitations of a many-electron system has received very broad development from a point of view somewhat different from that of the usual polar model \(^{52-59}\). In these works) an idea, long ago expressed by Bloch \(^{60}\), was developed in detail: that the elementary excitations of a many-electron system are nothing other than sound waves propagating in it (i.e., deviations from a spatially homogeneous electron distribution—oscillations of the plasma type \(^{61-62}\)). The corresponding “quasiparticles”—phonons—naturally obey Bose statistics*).

In works \(^{53}\), and especially \(^{57}\), however, it was shown that, for the three-dimensional case of real physical interest, sound oscillations by no means exhaust all the excitations of a many-electron system.

Alongside them there also exists a Fermi branch of the spectrum, which is evidently of particular interest for our purposes. It is also essential that, unlike ordinary sound waves propagating in neutral and not charged systems, excitation of plasma oscillations requires the expenditure of a finite energy \(E_0\), and this quantity is by no means small:

\[ E_0=\hbar \sqrt{\frac{4\pi}{m} n e^2}, \]

where \(n\) is the number of electrons per unit volume, and \(m\) and \(e\) are the mass and charge of the electron. For \(n\sim 10^{22}\ \mathrm{cm}^{-3}\), \(E_0\) turns out to be \(\sim 8.6\ \mathrm{eV}\); therefore at ordinary temperatures “plasma” phonons are practically absent—there exist only zero-point oscillations of the plasma. It is precisely—

*) In accordance with the main purpose of the present article, we touch only on the physical content of these works, without entering into a comparative evaluation of the calculational methods developed in them. We shall merely note that the most complete and rigorous consideration of the question is given, in our opinion, in \(^{57}\).

**) From the conceptual side these ideas are very close to the polar model. However, density fluctuations had hitherto in fact been investigated either by a method that did not allow one to reveal excitations of the Fermi type, or with the periodic character of the field in the crystal lattice neglected (the positive charge of the system was assumed to be uniformly distributed in space, and its role consisted only in compensating the total negative charge of the electrons). The polar model in its modern form is essentially connected with the idea of the correct periodic arrangement of atoms in the crystal lattice. It is possible that in the future—when the results of \(^{57}\) are generalized to the case of the presence of a periodic lattice field—the polar model and the method of density fluctuations will turn out to be, to some extent, equivalent to each other, representing merely two different approaches to one and the same problem.

they, as well as the Fermi branch, are of interest for the theory of the metal as such.

Let us note, however, that the emission of “plasma” phonons may play an essential role in the braking of fast charged particles moving through a metal*) (the energy of the particle may be expended on the excitation of “plasma” oscillations).

The study of the Fermi branch has not yet been carried out in this method of calculation with due thoroughness. In our opinion, it requires taking into account the atomic structure of the crystal, in particular the periodic field of the lattice (on “plasma” oscillations this field apparently has no noticeable effect, since the wavelengths involved there are sufficiently large that the discrete structure of the crystal is “smeared out”).

Summarizing all that has been said, it should be recognized that at present the notion of elementary excitations (which began to develop comparatively recently) has already led to serious successes and has firmly acquired “citizenship rights” in the physics of condensed systems. In application to the theory of the solid state it may be considered firmly established that the energy spectrum of a system of many interacting electrons in a crystal has a “mixed” character—containing both a Bose and a Fermi branch. The first of these may be regarded as having been studied to some extent; it is apparently exhausted by spin waves (in ferromagnets), excitons, and fluctuations of the electron density (in one form or another). As for the second branch, so far only its “semiconductor” part has been successfully studied. The creation of new methods of calculation that would make it possible theoretically to study excitations of the “metallic” type appears to us to be one of the most urgent tasks of solid-state physics.

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