THEORY OF THE METALLIC STATE

L. Nordheim

IV. MODEL THEORY*

§ 1. The Model of a Crystal in Wave Mechanics

Up to now we have carried out almost all calculations in the zero approximation, assuming that the electrons in the metal are completely free. To obtain the basic results it was sufficient to take into account only the Pauli principle and the quantization connected with the finite volume of the metal. The interaction with the metal ions was taken into account in a summary way, by introducing a jump of the potential at the boundary and a definite probability of quantum transitions. The interaction between the electrons, however, was not taken into account at all (more precisely—only in the Pauli principle). The task of the present chapter consists in a further refinement of the theory. For this purpose, on the one hand, it is necessary to investigate to what degree and in what cases the assumptions made will correspond to reality. Secondly, it is necessary to clarify the question of the transition probability; this should in particular explain the dependence of conductivity on temperature. To analyze all these questions it is necessary to use the methods of quantum mechanics. Since the calculations here sometimes turn out to be very complicated, in certain places we shall confine ourselves only to presenting and explaining the results.**

To construct the theory it is necessary to take as its basis a definite model of the crystal as a whole. In doing so it is advisable first to start from special limiting cases as the zero approximation. For the choice of such cases the following possibilities present themselves:

1. First, isolated atoms are considered, situated at a large distance from one another and, moreover, in the non-ionized state. Then the nuclei are mentally brought to their places in the cry—

* See Uspekhi fizicheskikh nauk, 15, 570, 675, 779, 1935, translated by S. G. Kalashnikov.

** We therefore give already here the most important literature. The fundamental works concerning the foundations of the theory belong to Bloch¹ and Peierls²; see also the work of Nordheim³. A very detailed survey exposition has been given by Brillouin⁴. These works are cited below as B, P, respectively, without calculation details, by Bloch⁵ and Nordheim⁶.

... metallic lattice, and the interaction arising in this case between neighboring atoms is regarded as a small perturbation. In this way one can take into account both the Coulomb interaction and the exchange forces. This model was used by Heisenberg in the theory of ferromagnetism. It is obvious that this approximation will be insufficient in the case where the atomic spheres overlap one another; for good conductors (for example, the alkali metals), with their large atomic volume, this is precisely what should occur.

In Fig. 1 the distribution of the potential is given schematically for the case of two ions (\(C\)-ionization potential). If there were no interaction (solid curves), then each of the optical electrons would belong only to its own atom. The true course of the potential, when the interaction is taken into account, is determined by the superposition of these curves (dotted line), as a result of which the potential barrier between the ions will be lowered. If this lowering is of the same order as the ionization potential of the isolated atom, then it is no longer possible, obviously, in constructing the model, to start from the neutral atom. Such a case is most likely to be expected, again, for good conductors, for which the ionization potential is especially small.

Fig. 1. Superposition of potential curves for two ions

It should be noted, however, that the Heisenberg model has not been systematically applied in the theory of electrical conductivity*. There is reason to think that this model may also lead to metallic conductivity, and therefore it would be extremely important to investigate the applicability of this model, especially for superconductors. One may expect that for superconductors an important role will be played by the interaction between electrons (exchange effects), which up to now could be taken into account only in the Heisenberg model. In addition, one must also take into account the circumstance that superconductors can in no case be assigned to the class of ordinary good conductors.

II. The second possibility is as follows. One may regard the crystal from the very beginning as an integral system. Then any electron of the metal must be regarded as being in the resultant field of all ions and all electrons. This field must have, on the average, the same periodicity as the crystal itself, i.e., it must be triply periodic in space. Further, an individual atom can, as is well known, be described with sufficient approximation by means of Hartree’s “self-consistent” field (at least in the case of many-electron atoms), i.e., of such a field as would give the energy levels of all electrons**.

* Attempts in this direction were made by Bloch \(^{7}\).

** With the aid of such a field we take into account (at least on the average) the action of all electrons and positive nuclei on any one electron.

Introducing such a field, we can consider the individual electrons as independent (just as in the one-electron problem) and take into account only the Pauli principle. It is therefore expedient to regard the field of the entire crystal as a whole as self-consistent. One may think that the Hartree method, which has led to good results in application to individual atoms, gives here too a sufficiently good approximation. In such a treatment the interaction between electrons is again taken into account only in the aggregate, with the aid of the Pauli principle and of the self-consistent field; exchange forces are not taken into account here at all. The introduction of such a self-consistent field is the starting point of Bloch’s theory. It turns out here that the most essential point is the periodicity of the field, which must correspond to the periodicity of the crystal structure; knowledge of the exact form of the field plays no special role.

III. The last possibility consists in considering, in the zero approximation, truly free electrons, and regarding the interaction with the ions as a small perturbation. The development of the theory in this direction was given by Peierls (l. c.). Here, as before, exchange forces are not taken into account at all, and the interaction with the electrons—only to the extent to which it is included in the Pauli principle.

The three models analyzed cover the transition from isolated atoms (I) to very dense packing (III). Model (II) occupies an intermediate position.

It is necessary to note that, despite the apparent difference between these three modes of description, the final results should not differ greatly from one another. Indeed, a continuous transition is possible between models (II) and (III), and this in itself shows that these two pictures must have many features in common. The connection between (I) and (II) is best clarified by comparison with the theory of molecules. In the theory of the molecule one may, on the one hand, take as a basis the Heitler and London model, in which the starting point is the isolated atom; it corresponds exactly to the Heisenberg model of the crystal, which is an extension of the Heitler–London theory to a very large number of atoms. On the other hand, in the theory of the molecule one may take as a starting point the problem of two (or many) bodies. Then the investigation of the proper functions and transitions in the system of terms (Hund) makes it possible to draw definite conclusions about real molecules. It turns out that both pictures, at least in those cases where the question is not one of exact numerical values, have considerably greater applicability than could have been expected a priori. In many respects both pictures give not too different results, and it often even happens that, for one and the same molecule, it is expedient to apply different models in order to investigate different phenomena.*

* Quantum mechanics provides the basis for such a possibility of different modes of description, which are not identical from the point of view of classical mechanics. The state of a system is described by a definite Schrödinger function. Different pictures give complete systems of functions, in terms of which the proper functions could be expanded. The choice of some particular complete system of functions (picture) is expedient when the proper function can be well represented by a linear combination of only a small number of terms. Therefore there is no contradiction in the fact that, in the investigation of some properties, one system proves preferable, while in the investigation of other properties another does.

We therefore should not be surprised when we find the same state of affairs also in the more complicated case of a solid, where for the investigation of ferromagnetism (and, probably, superconductivity) the Heisenberg model proves necessary, while for the ordinary processes of electrical conductivity—the Bloch–Peierls–Sommerfeld model. Here too, just as in the theory of molecules, one may expect that study of the transition between (I) and (II) will lead to valuable results*. We shall return to this question later.

It follows from the most general considerations that models (II) and (III) will be applicable the better, the greater the conductivity of the metal. For metals with high electrical conductivity and with magnetic properties independent of temperature (paramagnetism or diamagnetism), these models should give a sufficiently good approximation.

§ 2. Electrons in a Periodic Field

General Results

We shall start from picture II (Bloch) and investigate the properties of the eigenfunction of an electron in a triply periodic potential field \(U(x,y,z)\). We shall regard the latter as the self-consistent field of the crystal lattice.

In what follows we shall restrict ourselves to the case of a cubic lattice (the extension of the results to an arbitrary lattice involves no fundamental difficulties). Let the lattice constant be \(a\). In consequence of the periodicity of the field

\[ U(x,y,z)=U(x+n_1a,y+n_2a,z+n_3a), \tag{1} \]

where \(n_1,n_2,n_3\) are integers. We shall represent the potential as expanded in a triple Fourier series

\[ U=\sum_{\mathbf n} U_{\mathbf n} e^{\frac{2\pi i(\mathbf r\mathbf n)}{a}} =\sum_{n_1n_2n_3} U_{n_1n_2n_3} e^{\frac{2\pi i}{a}(xn_1+yn_2+zn_3)} . \tag{2} \]

We shall consider a finite crystal, corresponding to the potential box Vol. I, § 6, with edge length

\[ K=aG\left(-\frac{K}{2}<x,y,z<+\frac{K}{2}\right). \tag{3} \]

The eigenfunctions must in this case be cyclic, with period \(K\), so that

\[ \psi_k(x)=\psi_k(x+K)\ \text{etc.} \tag{4} \]

They are determined by the Schrödinger equation

\[ \Delta\psi+\varkappa^2(E-U)\psi=0;\quad \varkappa^2=\frac{8\pi^2m}{h^2}. \]

On the basis of the periodicity conditions (1) and (4), one can already make certain statements about the properties of the eigenfunction without resorting to a solution of Schrödinger’s equation. Since here we have three degrees of freedom, each eigenfunction will depend on three quan-

* This question was discussed by Wilson\(^8\) and Goudsmit\(^9\).

tic numbers \(k_x, k_y, k_z\) (vectorially \(\mathbf{k}\)). Let us choose some definite eigenfunction \(\psi_{\mathbf{k}}(x,y,z)\) and form from it a new function \(\psi_{\mathbf{k}}(x+a,y,z)\). The new function satisfies condition (4) and, owing to condition (1), is also a solution of the Schrödinger equation, and moreover it belongs to the same eigenvalue as the original one. These two functions can therefore differ only by a constant factor *:

\[ \psi(x+a,y,z)=\lambda\psi(x,y,z). \tag{5} \]

Hence we obtain

\[ \psi(x+na)=\lambda^n\psi(x). \]

Using further the boundary condition (4), we may write

\[ \psi(x+Ga)=\psi(x+K)=\lambda^G\psi(x)=\psi(x), \]

whence we find:

\[ \lambda^G=1;\quad \text{i.e.}\quad \lambda=e^{\frac{2\pi i k_x}{G}}. \tag{7} \]

Here \(k_x\) is some integer. We therefore obtain

\[ e^{-\frac{2\pi i k_x}{Ga}(x+a)}\psi(x+a) = e^{-\frac{2\pi i k_x x}{Ga}}\psi(x) \]

and, consequently,

\[ u(x)=e^{-2\pi i\frac{k_x x}{K}}\psi(x) \]

must be a periodic function with period \(a\) (the dependence on \(y\) and \(z\) still remains arbitrary). Thus

\[ \psi(x)=u(x)e^{2\pi i\frac{k_x x}{K}}. \]

In exactly the same way we find that

\[ \psi(y)=u(y)e^{2\pi i\frac{k_y y}{K}} \]

\[ \psi(z)=u(z)e^{2\pi i\frac{k_z z}{K}}. \]

Hence we conclude that \(\psi\) must have the form

\[ \psi_{\mathbf{k}}(x,y,z) = u_{\mathbf{k}}(\mathbf{r})e^{\frac{2\pi i}{K}(\mathbf{k}\mathbf{r})} \equiv u_{k_x k_y k_z}(x,y,z) e^{\frac{2\pi i}{K}(k_xx+k_yy+k_zz)}, \tag{8} \]

where \(u_{\mathbf{k}}\) is a triply periodic function with period \(a\):

\[ u_{\mathbf{k}}(\mathbf{r}+a\mathbf{n})=u_{\mathbf{k}}(\mathbf{r}), \tag{9} \]

\[ u_{\mathbf{k}}=\sum_{\mathbf{n}}u_{\mathbf{k}\mathbf{n}}e^{\frac{2\pi i}{a}(\mathbf{r}\mathbf{n})}. \]

\[ \overline{\phantom{xxxxxxxxxxxxxxxx}} \]

* In the case of degeneracy one would have had to write the more general relation

\[ \psi_r(x+a)=\sum_s \lambda_{rs}(x)\psi_s(x), \tag{6} \]

where the summation extends over all eigenfunctions belonging to the same characteristic number. By means of a suitable linear combination [transformation to the principal axes of the matrix \(\lambda_{rs}(x)\)] this expression can again be brought to the form (5), and indeed for the \(x\)- and \(y\)-displacements simultaneously, since the corresponding matrices can be interchanged. The latter follows from the circumstance that the result cannot depend on the order of the sequence of displacements.

L. NORDHEIM

The result obtained will be valid quite exactly for any periodic potential. For normalization we require, as usual,

\[ \int \psi_k \overline{\psi}_k\, dV = \int_{K^3} u_k \overline{u}_k\, dV = 1, \tag{10a} \]

i.e., so that on integration over the elementary cell \(a^3\)

\[ \int_{a^3} u\overline{u}\, dV = \frac{1}{G^3}. \tag{10b} \]

We see that even the rigorous expression for the eigenfunction for any periodic potential has the form of plane waves, only now these plane waves are modulated by the periodic function \(u_k\), determined by the crystal lattice (these functions are in general different for different \(k\)). For the case of a constant \(V\) (when only a constant term is present in the Fourier series), the expression obtained passes into the ordinary de Broglie plane waves (Part 1, § 6), which are thus a special case of (8)

\[ \left(u_k=\frac{1}{\sqrt{K^3}}=\mathrm{const}\right). \]

The Bloch waves considered here possess properties very similar to those of the plane waves of free electrons. First, they give a definite electron current, not equal to zero.

This can be shown most simply in the following way. The complete expression for the eigenfunction with the time factor is

\[ \psi_k(t) = u_k e^{-\frac{2\pi i}{h}\left(\varepsilon_k t-h\frac{kr}{K}\right)}. \tag{11} \]

Let us now construct a packet from such waves with very close values of \(k\). Proceeding in the same way as in the case of plane waves, we obtain the group velocity equal to *

\[ v_x = \lim_{k'\to k} \frac{\frac{\varepsilon_{k'}-\varepsilon_k}{h}} {\frac{K}{K}\left(k'_x-k_x\right)} = \frac{K}{h}\frac{\partial \varepsilon_k}{\partial k_x}, \tag{12} \]

in complete agreement with de Broglie’s relation, Part 1, § 6 (8) and (10).

\[ \text{* For more detail see Sommerfeld, Wave Mechanics. Let our packet be} \]

\[ \psi(x,t) = \int_{k}^{k+\Delta k} c_k u_k e^{-\frac{2\pi i}{h}\left(\varepsilon_k t-\frac{hkr}{K}\right)}\, dk = \]

\[ = u_k^{*} \int_{k}^{k+\Delta k} c_k e^{-\frac{2\pi i}{h}\left(\varepsilon_k t-\frac{h}{K}kr\right)}\, dk. \]

Here the second mean-value theorem has been used, where \(u_k^{*}\) denotes the mean value of the function \(u_k(r)\) in the interval \(\Delta k\). Further, we note that the function \(u_k^{*}\) has the small period \(a\). Therefore the essential role in the behavior of the packet as a whole will be played only by the second factor, which here is completely analogous to the corresponding expression for a plane wave.

(For the latter we obtain

\[ \varepsilon=\frac{h^2 k^2}{2mK^2}, \qquad v_x=\frac{h k_x}{mK}=\frac{p_x}{m}. \tag{12a} \]

The expression obtained earlier (12a) thus proves to be valid only in the limiting case.) For the current \(S_x\) (the mean transport of electricity) we find*

\[ S_x=e v_x=-\frac{eK}{h}\frac{d\varepsilon_k}{dk_x}. \tag{13} \]

Expression (13) shows that Bloch waves corresponding to a periodic potential give a definite finite current.

Exactly as in the case of free electrons, this current can be increased by the action of an external electric field (the electron can be accelerated). To describe this process one must again make use of a wave packet. The calculation can be carried out in the following way. Let the packet be given in the form

\[ \psi=\sum_{\mathbf{k}} c_k(t)\psi_k. \]

Since all \(\psi_k\) are normalized and orthogonal, then

\[ c_k=\int \psi\,\overline{\psi}_k\,dV. \tag{14} \]

The quantity

\[ |c_k|^2=f_k \]

determines the probability of finding the electron in state \(k\); in statistical calculations this quantity would determine the degree of participation of the electron under consideration in filling the cell \(k\).** If there is now also an electric field \(F\) in the direction \(x\), i.e., the perturbing potential

\[ H_1=-eFx, \]

then by means of the usual method of perturbation theory one can obtain the relation

\[ \frac{dc_k}{dt} = -\frac{2\pi i}{h}\sum_l c_l \int H_1\psi_l\overline{\psi}_k\,dV = \frac{2\pi i}{h}eF\int \psi x\overline{\psi}_k\,dV. \tag{15} \]

By definition [cf. (14)]

\[ \frac{\partial c_k}{\partial k_x} = \lim_{k_x'\to k_x}\int \psi\,\frac{\overline{\psi}_{k'}-\overline{\psi}_k}{k_x'-k_x}\,dV = \int \psi\,\frac{\partial \overline{\psi}_k}{\partial k_x}\,dV. \]

Further, according to (8)

\[ \frac{\partial \overline{\psi}_k}{\partial k_x} = -\frac{2\pi i}{K}\,\overline{\psi}_k + e^{-\frac{2\pi i(k_x r)}{K}} \frac{\partial \overline{u}_k}{\partial k_x}. \]

As the magnitude of the volume within which the electrons are enclosed increases, the second term decreases without bound, since the successive values

* A proof of formula (13) by calculating the quantum-mechanical expression for the current density was given by Peierls.¹⁰

** See the work of Kikuchi and Nordheim.¹¹

\(u_k\) converge; therefore we may neglect the second term in comparison with the first, after which, on the basis of (15), we obtain

\[ \frac{\partial c_{\mathbf{k}}}{\partial k_x} = -\frac{2\pi i}{K}\int \psi x\overline{\psi}_k\,dV = -\frac{h}{eFK}\,\frac{dc_{\mathbf{k}}}{dt}. \]

Thus

\[ \frac{dc_{\mathbf{k}}}{dt} = -\frac{KeF}{h}\,\frac{\partial c_{\mathbf{k}}}{\partial k_x}. \]

Taking into account that the same relation will hold for \(\overline{c}_{\mathbf{k}}\), we finally find

\[ \frac{\partial |c_{\mathbf{k}}|^2}{\partial t} = c_{\mathbf{k}}\dot{\overline{c}}_{\mathbf{k}} + \overline{c}_{\mathbf{k}}\dot c_{\mathbf{k}} = -\frac{eFK}{h} \left( c_{\mathbf{k}}\frac{\partial \overline{c}_{\mathbf{k}}}{\partial k_x} + \overline{c}\frac{\partial c_{\mathbf{k}}}{\partial k_x} \right) = -\frac{eFK}{h}\frac{\partial |c_{\mathbf{k}}|^2}{\partial k_x} \]

or

\[ \frac{\partial f_{\mathbf{k}}}{\partial t} = -\frac{eFK}{h}\,\frac{\partial f_{\mathbf{k}}}{\partial k_x}. \tag{16} \]

The expression obtained gives the change of the distribution function caused by the action of the external field; it is completely analogous to the relation *

\[ \frac{\partial f}{\partial t} = \frac{eF}{m}\frac{\partial f}{\partial \xi}, \tag{16a} \]

obtained by us in Part III, § 3 (2).

We thus arrive at the following fundamental results. Electrons situated in a strictly periodic field behave, in many respects, in the same way as free electrons considered in Part I, § 6. Only their eigenfunctions will be modulated waves, given by expression (8); they are still characterized by the vector quantum number \(\mathbf{k}\) (triplet). Such electrons produce a definite current (13) and can be accelerated by an external field. Finally, we have shown that the action of an external field, by causing a flow of electrons, thereby changes also the statistical distribution in \(k\)-space (which appears here instead of the former momentum space).

These completely general propositions already make it possible to draw essential conclusions about the mechanism of electrical conductivity. First of all it is clear that an ideal metallic crystal** would have to possess infinitely large electrical conductivity; the electronic waves in such a crystal would experience no scattering and would pass through the crystal without any change in intensity. From this point of view, any ohmic resistance can arise only in the presence of irregularities in the lattice and, conversely—

\[ \frac{df}{dt} = \frac{eH}{c} \left(\frac{K}{h}\right)^2 \left( \frac{\partial \varepsilon}{\partial k_x}\frac{\partial f}{\partial k_y} - \frac{\partial \varepsilon}{\partial k_y}\frac{\partial f}{\partial k_x} \right). \tag{17} \]

* In this way the action of a magnetic field on the wave packet can also be taken into account. In this case one obtains (the magnetic field is directed along \(z\))

For more details see Kronig’s dissertation \(^{12}\) and Peierls’ paper \(^{13}\).

** On the distinction between conductors and insulators in this respect see § 6.

...growth, any perturbation, by causing scattering of electron waves, must necessarily also create resistance. The principal cause of such perturbations must be the thermal vibrations of the crystal. From this the fact of the strong temperature dependence of the free path of electrons becomes understandable.

We see that the new ideas differ fundamentally from the classical picture of the reflection of electrons from elastic spheres. The very presence of ions by no means yet causes the appearance of resistance, which arises only when the regularity of their spatial arrangement is disturbed. Such a result can be explained vividly by means of an analogy with the propagation of light in a transparent crystal. While each individual atom is a scattering center, a plane wave can pass through the crystal without attenuation. The mechanism of the process, as is well known, is the following. Each atom, acting as a scattering center, creates, under the action of the primary wave, a secondary spherical wave. This would cause an attenuation of the primary wave, owing to the transfer of energy into scattered radiation. However, when the scattering centers are arranged regularly, the individual secondary waves interfere, according to Huygens’ principle, in such a way that they again give a properly propagating wave. And only when the lattice is disordered does diffuse scattering appear in all directions (the Tyndall effect). As for the velocity of propagation, it proves to be different inside the crystal than in vacuum. We find the analogy to this in the motion of electrons in the fact that the group velocity of the electron waves inside the crystal will likewise be different [(12) instead of (12a)].

On the basis of these ideas it becomes understandable how electrons can have the large free path found in Part III, § 5; there are likewise no grounds at all for supposing that electron waves could not propagate inside a crystal over distances of the order of hundreds of atomic spacings. On the contrary, in the classical picture the free path should always be of the same order as the lattice constant.

§ 3. Perturbation Method for Low Terms

For static calculations it is necessary to know the distribution of eigenvalues, i.e. the energy values corresponding to definite wave numbers; the law of distribution of the eigenvalues directly determines the magnitude of the current [§ 2 (13)]. The solution of this problem in the general form is associated with great difficulties, and we shall therefore try to elucidate the basic features of the solution by means of suitable approximations*.

* The whole theory has been analyzed in detail by Brillouin¹⁴. There a more exact investigation of the one-dimensional case is also given, with an indication of the mathematical literature. For special forms of the potential the problem has been analyzed by Morse¹⁵ and by Kronig and Penney¹⁶; see also Bethe’s work¹⁷.

The one-dimensional problem can be solved comparatively simply. Taking for the potential the expression

\[ U=U_0+U_1\cos\frac{2\pi x}{a}, \]

we shall find that the Schrödinger equation turns into the well-known Mathieu equation. In the case of an arbitrary periodic potential we arrive at the general Hill problem. Here, for the most general case, it can be shown that all eigenvalues must lie within bands of finite width; moreover, for small energies these bands must be very narrow, and for large energies—wide. In Fig. 2 these zones are shown by hatching, and the course of the periodic potential to which these zones correspond is also depicted. All zones, even for high values of the energies, are separated by intervals of finite width. In each region there are contained \(G\) eigenvalues

\[ \left(-\frac{G}{2}<k\leq+\frac{G}{2}\right). \]

Fig. 2. Allowed and forbidden regions of eigenvalues of a periodic potential

Such a splitting into finite zones also takes place for the three-dimensional case, only here it occurs in a more complicated way. We shall now consider approximate solutions of the problem under certain special conditions.

A solution can be obtained comparatively easily for two limiting cases—very low and very high terms. For low terms, Bloch adopts as the initial solution the eigenfunctions and eigenvalues of one isolated potential well. A periodic field can always be described by means of a potential of the form

\[ V=\sum_{\mathbf n}F=\sum_{\mathbf n}F(\mathbf r-a\mathbf n);\quad (\mathbf n=n_x,\ n_y,\ n_z), \tag{1} \]

created by a certain lattice; \(F(\mathbf r)\) gives the potential created by one ion at the point \(\mathbf n=0\). If this potential term is sufficiently low, then a whole series of eigenvalues will correspond to it. They can be found from the equation

\[ \Delta\varphi_{\mathbf n}+\chi^2(\varepsilon-F_{\mathbf n})\varphi_{\mathbf n}=0, \]

where the index \(\mathbf n\) denotes that the electron is located precisely at the \(\mathbf n\)-th lattice point. If \(F\) decreases so rapidly that already at the neighboring lattice point it is considerably reduced, then \(\varphi_{\mathbf n}\) will already represent an approximate solution of the problem for the full potential. This solution will, however, be degenerate, and indeed \(G^3\)-fold degenerate, since there are altogether \(G^3\) such potential wells, and it is entirely immaterial in which of them the electron is located. When the interaction is taken into account as a small perturbation, these terms split and give \(G^3\) different states. According to the general principles of perturbation theory, in this case the eigenfunction can be found in first approximation as a linear combination of the degenerate \(\varphi_{\mathbf n}\).

We shall therefore set

\[ \psi=\sum_{\mathbf n} a_{\mathbf n}\varphi_{\mathbf n}+\varphi_1, \tag{2a} \]

\[ \varepsilon=\varepsilon_0+\varepsilon_1, \tag{2b} \]

where \(\varphi_1\) and \(\varepsilon_1\) will be regarded as small quantities of the first order. Substitution of these expressions into the Schrödinger equation gives

\[ \Delta\varphi_1+\chi^2(\varepsilon_0-U)\varphi_1 = -\chi^2\sum_{\mathbf n} a_{\mathbf n}\varphi_{\mathbf n}(\varepsilon_1-U+F_{\mathbf n}). \tag{3} \]

The inhomogeneous equation obtained, as is known, has a solution only when its right-hand side is orthogonal to all solutions of the homogeneous equation. These solutions are, it is true, unknown to us; however, they will differ the less from the \(\varphi_{\mathbf n}\) themselves, the less the potentials of the individual ions overlap one another. Therefore, within the framework of our approximations we may use the \(\varphi_{\mathbf n}\) directly for the orthogonality condition. It should be noted that this device is entirely analogous to that used in the theory of the molecule by Heitler and London. Thus we may assume that the \(\varphi_{\mathbf n}\) are quasi-orthogonal, i.e. that the relation

\[ \int \varphi_{\mathbf n}\bar{\varphi}_{\mathbf m}\,dV=\delta_{\mathbf{nm}}, \tag{4} \]

holds, where \(\delta_{\mathbf{nm}}=1\) for \(\mathbf n=\mathbf m\), and for \(\mathbf n\ne\mathbf m\) is so small that it may be neglected.

We therefore require that

\[ \sum_{\mathbf n} a_{\mathbf n}\int(\varepsilon_1-U_{\mathbf n}^{*})\varphi_{\mathbf n}\bar{\varphi}_{\mathbf m}\,dV=0 \tag{5} \]

for all \(\mathbf m\),

\[ \left(\text{i.e. for }-\frac{G}{2}\le m_x,\ m_y,\ m_z\le +\frac{G}{2}\right). \]

The quantity entering here,

\[ U_{\mathbf n}^{*}=U-F_{\mathbf n}, \tag{6} \]

gives the lattice potential from which the node \(\mathbf n\) has been removed.

Since we assume that \(\varphi_{\mathbf n}\) decreases sufficiently rapidly (exponentially) in the neighborhood of the point \(\mathbf n\), the integral of \(U_n\) will have an appreciable value only when \(\mathbf m\) coincides with \(\mathbf n\) or at least \(\mathbf n\) is adjacent to \(\mathbf m\) (for example, \(m_x=n_x+1;\ m_y=n_y;\ m_z=n_z\), etc.); in all other cases we may neglect it altogether. We shall therefore put

\[ \int U_{\mathbf n}^{*}\varphi_{\mathbf n}\bar{\varphi}_{\mathbf m}\,dV= \begin{cases} \alpha & \text{for } \mathbf n=\mathbf m,\\ \beta & \text{for } \mathbf n \text{ and } \mathbf m \text{ adjacent},\\ 0 & \text{in all other cases}. \end{cases} \tag{7} \]

The “resonance” integrals written here correspond exactly to the exchange integrals of Heitler and London. For strongly bound electrons \(\alpha\) and \(\beta\) are always positive.

With the aid of (7) and the quasi-orthogonality conditions (4), we obtain from (5) equations for determining \(a_{\mathbf n}\) and \(\varepsilon_1\):

\[ (\varepsilon_1-\alpha)a_{m_xm_ym_z} +\beta\bigl(a_{m_x+1,m_y,m_z} +a_{m_x-1,m_y,m_z} +a_{m_x,m_y+1,m_z}+\ldots\bigr)=0 \]

\[ \left(-\frac{G}{2}\le m_x,\ m_y,\ m_z\le +\frac{G}{2}\right). \tag{8} \]

On the other hand, the periodicity conditions give

\[ a_{m_x m_y m_z}=a_{m_x \pm G,\,m_y m_z}, \tag{9} \]

and so on.

We shall satisfy these requirements if we set

\[ a_{m_x m_y m_z}^{k_x k_y k_z} = e^{\frac{2\pi i}{G}(k_x m_x+k_y m_y+k_z m_z)}, \tag{10} \]

where \(\mathbf{k}\,(k_x k_y k_z)\) determines a triple of integers. To each such triplet there corresponds its own special solution, and their total number is exactly \(G^3\). Condition (8) will indeed be satisfied if the energy parameter has the value

\[ \varepsilon_{\mathbf{k}} = \alpha - 2\beta \left( \cos \frac{2\pi k_x}{G} + \cos \frac{2\pi k_y}{G} + \cos \frac{2\pi k_z}{G} \right). \tag{11} \]

We thus find the required change of energy under the perturbation. The corresponding eigenfunction of the zeroth approximation will be

\[ \psi_k = \sum_{m_{x,y,z}=-\frac{G}{2}}^{\frac{G}{2}} e^{\frac{2\pi i}{G}(k_x m_x+k_y m_y+k_z m_z)} \varphi_{m_x m_y m_z}, \tag{12} \]

and it is obtained precisely in the form* required by the general theory [§ 2 (8)].

After the energy has been determined as a function of the wave numbers, the magnitude of the current can be obtained from § 2 (13):

\[ S_x = \frac{eK}{h}\frac{\partial \varepsilon_k}{\partial k_x} = \frac{4\pi e\beta K}{hG}\sin\frac{2\pi k_x}{G} = e\frac{4\pi a\beta}{h}\sin\frac{2\pi k_x}{G}. \tag{13} \]

Thus the problem in the limiting case under consideration is completely solved. To each eigenvalue of an individual potential well there correspond \(G^3\) eigenvalues of the crystal which, according to (11), lie within a band of width \(12\beta\) (in the one-dimensional problem \(4\beta\)). Thus in the case examined the three-dimensional problem also proves soluble without particular difficulties.

Since for any band a shift of the numbering \(k\) by the amount \(lG\) (\(l\) an integer) plays no role, we could renumber all states; for example, for the eigenfunction of the lowest eigenvalue we could put

\[ -\frac{G}{2}<k\leq +\frac{G}{2}, \]

and for the eigenfunction of the next eigenvalue take

\[ \frac{G}{2}<k\leq \frac{3G}{2}, \]

and so on.

\[ \text{* In this case, of course, the condition of § 2 is also fulfilled} \]

\[ \psi_k(\mathbf{r}+\mathbf{n}a) = e^{\frac{2\pi i}{G}(\mathbf{n}\mathbf{k})}\psi(\mathbf{r}). \]

Expression (10) gives a solution also when the influence of more distant potential wells is taken into account (see Brillouin, loc. cit.). However, in this case additional terms enter the expression for \(\varepsilon_k\), having the form of products

\[ \cos\frac{2\pi k_{x,y,z}}{G} \]

and so on. Hence it is clear that the separate dependence on each of the directions in (11) is only a first approximation.

We see that for the case of low terms there is obtained a very substantial deviation from the properties of free electrons, for which we had

\[ \varepsilon_k=\frac{h^2|\mathbf{k}|^2}{2mK^2}; \qquad v_x=\frac{hk_x}{mK}. \]

The dependence of \(\varepsilon_k\) on \(\mathbf{k}\) no longer has spherical symmetry. We obtain it only in the case when \(|\mathbf{k}|\) is close to 0 or when

\[ k_{x,y,z}\sim \pm \frac{G}{2}. \]

When \(|\mathbf{k}|\ll \frac{G}{2}\), we can expand \(\sin\) and \(\cos\) in a series; then

\[ \varepsilon_k=\alpha-6\beta+\beta\,\frac{4\pi^2}{G^2}\left(k_x^2+k_y^2+k_z^2\right), \tag{14a} \]

\[ S_x=ev_x=\frac{8\pi^2 e\beta K}{hG^2}\,k_x =\frac{8\pi^2ea^2\beta}{h^2}\,\frac{hk_x}{K}. \tag{14b} \]

An electron in such a state behaves in the same way as a free one, only it must be assigned the apparent mass

\[ m^*=\frac{h^2}{8\pi^2a^2\beta}. \tag{15} \]

In other words: an electron in a periodic field moves as if, moving in the absence of a field, it possessed the mass \(m^*\).

In the neighborhood of the corners of the cube \(G^3\) in \(k\)-space we can, conversely, carry out an expansion in a series in \(k_x^*=\pm\left(\frac{G}{2}-k_x\right)\), and so on. Then we obtain

\[ \left. \begin{aligned} \varepsilon_{k^*} &=\alpha+2\beta\left(\cos\frac{2\pi k_x^*}{G}+\cdots\right) \simeq \alpha+6\beta \\ &\quad -\frac{4\pi^2\beta}{G^2}\left(k_x^{*2}+k_y^{*2}+k_z^{*2}\right), \end{aligned} \right\} \tag{16a} \]

\[ S_x=-\frac{eK}{h}\,\frac{\partial \varepsilon_{k^*}}{\partial k^*} =\frac{8\pi^2ea^2\beta}{h^2}\,\frac{hk_x^*}{K} =\frac{4\pi^2ea\beta}{h}-\frac{8\pi^2ea^2\beta}{h^2}\,\frac{hk_x}{K}. \tag{16b} \]

An electron in such a state behaves as if it possessed the negative mass \(m^*\).

Figure 3 shows the course of the change of the energy and of the current as a function of the wave number for the one-dimensional case (negative \(k\) denotes motion to the left). From the figure it is seen that for the highest values of the energy within the band the current turns to zero (in this the role of the negative mass is manifested). The use of the approximate expression (14) means replacing the cosine by a segment of a parabola. In this case the apparent mass turns out to be the larger (and, correspondingly, the current the smaller), the smaller the interaction of neighboring potential wells, i.e. the smaller the value of the resonance integral \(\beta\).

In passing from the one-dimensional problem to the case of several dimensions, the results become considerably more complicated. For two dimensions a visual interpretation is still possible. In this case we can lay off the components of the wave vector \(k_x\) and \(k_y\) in two mutually perpendicular directions (we consider for

for simplicity only the case of a cubic lattice), and the values of the energy—perpendicular to the plane \(k_z-k_y\). We shall then obtain an energy surface which will be the analogue of the curve in Fig. 3. In the plane of the drawing we obtain lines of constant energy. To any vector with integral components \(k_x\) and \(k_y\), drawn in the plane from the origin, there corresponds a definite possible stationary state. For small \(|k|\) the energy surfaces will, according to (14), be paraboloids of revolution

Fig. 3. Dependence of the energy and current on the wave number. Strongly bound electrons (one-dimensional problem)

Fig. 4. Curves of constant energy as a function of the wave number (two-dimensional problem)

\[ \varepsilon_k=\operatorname{const}+\frac{4\pi^2\beta}{G^2}(k_x^2+k_y^2), \]

and the curves of constant energy will be circles with center at the origin. For large energies these will, according to (16), be circles with centers at the corners of the square \(G^2\); the form of the lines of constant energy is shown in Fig. 4.

Fig. 5. Surface of constant energy in the three-dimensional problem

It is very difficult to give a geometrical interpretation for the three-dimensional problem. Fig. 5 is an attempt to depict a surface of constant energy for three-dimensional \(k\)-space.

In the preceding arguments we have everywhere assumed that the terms due to the individual potential wells are nondegenerate. This will correspond to reality if we consider, for example, the \(p\)-, \(d\)-states of ions, etc. In this case a special investigation is necessary.*

§ 4. Perturbation method for high terms

As another limiting case we shall consider very high terms, for which the energy levels lie above the maximum rise

* Calculations for the \(p\)-terms of a cubic lattice were carried out by Wilson \({}^{18}\).

of the potential. Here the periodic potential itself may be regarded as a small perturbation, and a plane wave may be chosen as the initial function

\[ \psi_{\mathbf{k}}^{0}=\frac{1}{\sqrt{K^{3}}}\,e^{\frac{2\pi i}{K}(\mathbf{k}\mathbf{r})}, \tag{1} \]

where

\[ \varepsilon_{\mathbf{k}}^{0}=\frac{h^{2}}{2mK^{2}}\left(k_{x}^{2}+k_{y}^{2}+k_{z}^{2}\right);\quad v_{x}^{0}=\frac{h}{mK}\,k_{x}. \tag{2} \]

We shall represent the perturbing potential expanded in a Fourier series

\[ U=\sum_{\mathbf{n}} U_{\mathbf{n}} e^{\frac{2\pi i}{a}(\mathbf{n}\mathbf{r})}. \tag{3} \]

We first make use of the usual method of perturbation theory, according to which one may write

\[ \varepsilon_{\mathbf{k}}=\varepsilon_{\mathbf{k}}^{0}+\varepsilon_{\mathbf{k}}^{1};\quad \varepsilon_{\mathbf{k}}^{1}=U_{\mathbf{k}\mathbf{k}}, \tag{4a} \]

\[ \psi_{\mathbf{k}}=\psi_{\mathbf{k}}^{0}+\psi_{\mathbf{k}}';\quad \psi_{\mathbf{k}}'=\sum_{\mathbf{k}'}'\frac{\psi_{\mathbf{k}'}^{0}U_{\mathbf{k}\mathbf{k}'}}{\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}'}}. \tag{4b} \]

The symbol \(\sum_{\mathbf{k}'}'\) denotes summation over all \(\mathbf{k}'\) with the exception of \(\mathbf{k}'=\mathbf{k}\).

The matrix elements \(U_{\mathbf{k}\mathbf{k}'}\) entering expressions (4) have the following value:

\[ \left. \begin{aligned} U_{\mathbf{k}\mathbf{k}'}&=\int \bar{\psi}_{\mathbf{k}'}^{0}U\psi_{\mathbf{k}}^{0}\,dV=\\ &=\frac{1}{K^{3}}\int \sum_{\mathbf{n}}U_{\mathbf{n}}e^{\frac{2\pi i}{K}(\mathbf{n}G+\mathbf{k}-\mathbf{k}')\mathbf{r}}\,dV. \end{aligned} \right\} \tag{5} \]

Only those of them for which the relations

\[ \mathbf{k}'-\mathbf{k}=\mathbf{n}G;\quad U_{\mathbf{k}\mathbf{k}'}=U_{\mathbf{k},\,\mathbf{k}+\mathbf{n}G}=U_{\mathbf{n}}, \tag{6} \]

are satisfied are nonzero; in this case

\[ \varepsilon_{\mathbf{k}}=\varepsilon_{\mathbf{k}}^{0}+U_{0} \tag{7} \]

and the whole result reduces only to a change in the origin from which the energy is measured. In this case the arrangement of the energy levels proves to be entirely undisturbed.

The eigenfunction is obtained as

\[ \left. \begin{aligned} \psi_{\mathbf{k}}&=\psi_{\mathbf{k}}^{0}+\sum_{\mathbf{n}}'\frac{\psi_{\mathbf{k}+\mathbf{n}G}^{0}U_{\mathbf{k},\,\mathbf{k}+\mathbf{n}G}}{\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}+\mathbf{n}G}}=\\ &=e^{\frac{2\pi i}{K}(\mathbf{k}\mathbf{r})}\frac{1}{\sqrt{K^{3}}}\left(1+\sum_{\mathbf{n}}'\frac{e^{\frac{2\pi i}{a}(\mathbf{n}\mathbf{r})}U_{\mathbf{n}}}{\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}+\mathbf{n}G}}\right). \end{aligned} \right\} \tag{8} \]

Since in the brackets there stands a periodic function with period \(a\) (a Fourier series), we obtain the eigenfunction still in the form indicated by general considerations [§ 2 (9)]. When the coefficients \(U_{\mathbf{n}}\) are small, the eigenfunction will also differ little from a plane wave.

The method applied loses its validity when one of the denominators \((\varepsilon_{\mathbf{k}}-\varepsilon_{\mathbf{k}+\mathbf{n}G})\) becomes very small and resonance occurs. This case is characterized by the fact that, in addition to condition (6), the additional relation

\[ |\mathbf{k}|^2 \simeq |\mathbf{k}'|^2 . \tag{9} \]

will be satisfied.

It is not difficult to clarify the physical meaning of this condition. Indeed, the vector \(\mathbf{k}\) determines the wave numbers of the cube \(K^3\). The corresponding de Broglie wavelength is

\[ \lambda=\frac{K}{\sqrt{|\mathbf{k}|^2}} =\frac{K}{\sqrt{k_x^2+k_y^2+k_z^2}}, \]

and therefore condition (9) denotes equality of the wavelengths \(\lambda\) and \(\lambda'\). Further, \(k/G\) gives the number of waves fitting into a length equal to the lattice constant. Therefore condition (6) expresses the requirement that in the elementary cube \(a^3\) the differences of the wave numbers be integers; and this coincides exactly with Laue’s interference conditions. Indeed, introducing instead of the vector \(\mathbf{k}\) its absolute value \(k\) and the direction cosines of the wave-front normal \(\alpha,\beta,\gamma\), we obtain

\[ k=k', \]

\[ k_x=\alpha k;\qquad k'_x=\alpha' k\quad \text{etc.} \]

\[ \frac{k}{K}=\frac{1}{\lambda}=\frac{1}{\lambda'}. \]

After this, (6) gives the well-known Laue conditions in the usual form

\[ a(\alpha'-\alpha)=n n_x \tag{10} \]

and so on. Relations (10) are equivalent to the Bragg conditions for selective reflection from the \((n_x n_y n_z)\)-face of the crystal [if the numbers \(n_x,n_y,n_z\) are not mutually prime, but can be represented in the form \(n_i=l m_i\), then reflection of the \(l\)-th order from the face \((m_x m_y m_z)\) will occur].

Thus, if the electron waves undergo selective reflection inside the crystal, solution (8) loses its validity. However, quantum mechanics here too provides a method for solving the problem. In this case, as the zeroth approximation, one must choose a linear combination of both degenerate solutions and exclude from the expansion (8) smaller terms. We shall examine this case in more detail. If there is a pair of close wave vectors \(\mathbf{k}\) and \(\mathbf{k}'\), then simultaneously with condition (6) one may also put

\[ \varepsilon^0_{\mathbf{k}'}=\varepsilon^0_{\mathbf{k}}+\delta, \]

where \(\delta\) is a small quantity. For definiteness we shall assume that \(|\mathbf{k}'|>|\mathbf{k}|\), so that \(\delta>0\). In addition, we shall assume that there is only one* interfering wave \(\mathbf{k}'\).

We shall seek the solution in the form

\[ \psi=a_1\psi^0_{\mathbf{k}}+a_2\psi^0_{\mathbf{k}'}+\psi_1, \tag{11a} \]

\[ \varepsilon=\frac{1}{2}\left(\varepsilon^0_{\mathbf{k}}+\varepsilon^0_{\mathbf{k}'}\right)+\varepsilon_1, \tag{11b} \]

where \(\psi_1\) and \(\varepsilon_1\) are small of first order [we consider the perturbing potential (3) to be of the same order of smallness]. Since \(\psi^0_{\mathbf{k}}\) and \(\psi^0_{\mathbf{k}'}\) satisfy the equation

\[ \Delta\psi^0_{\mathbf{k}}+\chi^2\varepsilon^0_{\mathbf{k}}\psi^0_{\mathbf{k}}=0, \]

* If it turned out that there are several interfering waves (which can occur only in exceptional cases), then one would have to take a linear combination of all these waves.

then, substituting (11) into the equation

\[ \Delta \psi+\varkappa^2(\varepsilon-U)\psi=V, \]

we obtain, accurate up to small higher-order terms,

\[ \Delta \psi_1+\varkappa^2\frac{\varepsilon_k+\varepsilon_{k'}}{2}\psi_1 =-\varkappa^2\left\{a_1\psi_k^0(\varepsilon-\varepsilon_k^0-U)+ a_2\psi_{k'}^0(\varepsilon-\varepsilon_{k'}^0-U)\right\}. \]

In order that the inhomogeneous equation be solvable, it is necessary, as before, that the right-hand side be orthogonal to the solutions of the homogeneous equation. In our approximation the latter are simply \(\psi_k^0,\ \psi_{k'}^0\), and we obtain the system of equations

\[ \left. \begin{aligned} a_1(\varepsilon-\varepsilon_k^0)-a_2U_{k'k}&=0,\\ a_2(\varepsilon-\varepsilon_{k'}^0)-a_1U_{kk'}&=0. \end{aligned} \right\} \tag{12} \]

Equating the determinant to zero,

\[ (\varepsilon-\varepsilon_k^0)(\varepsilon-\varepsilon_{k'}^0)-|U_{kk'}|^2=0;\quad (U_{kk'}=\overline{U}_{k'k}), \]

we find two solutions

\[ \varepsilon=\frac12\left(\varepsilon_k^0+\varepsilon_{k'}^0\right) \pm \sqrt{\frac14\left(\varepsilon_k^0-\varepsilon_{k'}^0\right)^2+|U_{kk'}|^2}, \]

comparing which with (11), we finally obtain

\[ \varepsilon_1=\pm \sqrt{\frac14\left(\varepsilon_k^0+\varepsilon_{k'}^0\right)^2+|U_{kk'}|^2}. \tag{13} \]

The ratio of the coefficients \(a_1\) and \(a_2\) is found to be

\[ \left. \frac{a_1}{a_2} =\frac{\varepsilon-\varepsilon_{k'}^0}{U_{kk'}} = -\frac12\frac{\varepsilon_{k'}^0-\varepsilon_k^0}{U_{kk'}} \left( 1\mp \sqrt{1+4\left|\frac{U_{kk'}}{\varepsilon_k^0-\varepsilon_{k'}^0}\right|^2} \right) = \frac{U_{kk'}}{\varepsilon-\varepsilon_k^0}. \right\} \tag{14} \]

The absolute value of the coefficients can be determined from the normalization condition

\[ |a_1|^2+|a_2|^2=1. \tag{15} \]

We see that in this case a perturbation of the energy is obtained already in the first approximation, in contrast to the case when there is no degeneracy.

For a better elucidation of the results found, it is useful to compare them with the general expressions of § 2. For this purpose we shall write the expression for \(\psi\) in the general form [§ 2 (9)], and here we obtain two functions, since for \(a_1\) and \(a_2\) two values are obtained. We shall introduce, corresponding to the two signs in (14), the designations \(a_1^+, a_2^+\) and \(a_1^-, a_2^-\). Then, in the zero approximation according to (11), (14), (1), and (6), the expressions for \(\psi^+\) and \(\psi^-\) may be written in the following form:

\[ \left. \begin{aligned} \psi^\pm &=a_1^\pm\left(\psi_k^0+\frac{a_2^\pm}{a_1^\pm}\psi_{k'}^0\right)=\\ &=e^{\frac{2\pi i}{K}(kr)} \frac{a_1^\pm}{\sqrt{K^3}} \left(1+\frac{a_2^\pm}{a_1^\pm}e^{\frac{2\pi i}{a}(nr)}\right), \end{aligned} \right\} \tag{16a} \]

or, with equal right,

\[ \left. \begin{aligned} \psi^{\pm} &= a_2^{\mp}\left(\psi_{k'}^0+\frac{a_1^{\pm}}{a_2^{\pm}}\psi_k^0\right)= \\ &= e^{\frac{2\pi i}{K}(k'r)}\,\frac{a_2^{\pm}}{\sqrt{K^3}}\left(1+\frac{a_1^{\pm}}{a_2^{\pm}}e^{-\frac{2\pi i}{a}(nr)}\right). \end{aligned} \right\} \tag{16b} \]

Both expressions agree with the general form of § 2 (9); however, they contain different wave vectors (\(k\) and \(k'\)), which gives rise to a certain ambiguity of the function \(\psi\). We can eliminate this ambiguity in the following way. We perform an adiabatic transition \(U_{kk'} \to 0\) (the coefficient of Fourier \(U_n \to 0\)). It then turns out that one of the functions \(\psi\) goes over into \(\psi_k^0\), the other into \(\psi_{k'}^0\). Indeed, as is easy to show, under such a transition

\[ \left|\frac{a_1^+}{a_2^+}\right|\to\infty;\qquad \left|\frac{a_1^-}{a_2^-}\right|\to\frac{1}{\infty}, \]

and this leads precisely to the indicated result. But then we can denote the two functions \(\psi\) obtained above as \(\psi_k\) and \(\psi_{k'}\), and write them already quite unambiguously in the following form:

\[ \left. \begin{aligned} \psi_k=\psi^+&=e^{\frac{2\pi i}{K}(kr)}u_k;\quad u_k=\frac{a_1^+}{\sqrt{K^3}}\left(1+\frac{a_2^+}{a_1^+}e^{\frac{2\pi i}{a}(nr)}\right)\\ \psi_{k'}=\psi^-&=e^{\frac{2\pi i}{K}(k'r)}u_{k'};\quad u_{k'}=\frac{a_2^-}{\sqrt{K^3}}\left(1+\frac{a_1^-}{a_2^-}e^{-\frac{2\pi i}{a}(nr)}\right) \end{aligned} \right\} \tag{17} \]

The corresponding eigenvalues will be:

\[ \left. \begin{aligned} \varepsilon_k&=\frac{1}{2}\left(\varepsilon_k^0+\varepsilon_{k'}^0\right) -\sqrt{\frac{1}{4}\left(\varepsilon_k^0-\varepsilon_{k'}^0\right)^2+\left|U_{kk'}\right|^2}\\ \varepsilon_{k'}&=\frac{1}{2}\left(\varepsilon_k^0+\varepsilon_{k'}^0\right) +\sqrt{\frac{1}{4}\left(\varepsilon_k^0-\varepsilon_{k'}^0\right)^2+\left|U_{kk'}\right|^2} \end{aligned} \right\} \tag{18} \]

Thus definite wave vectors can be unambiguously associated also with the perturbed eigenfunction, and the energy also proves to be an unambiguous function of the real wave numbers.

We can now give a visual description of the energy anomaly (13). Eliminating \(k'\) from (6) and (9), we obtain the equation of a family of planes

\[ (kn)+n^2\frac{G}{2}=0;\quad (n^2=n_x^2+n_y^2+n_z^2) \tag{19} \]

in \(k\)-space. The expressions (18) which we have obtained show that whenever the vector \(k'\) passes through one of these planes, the energy [which previously was a continuous function (2)] undergoes a jump. Taking into account that on the plane (19) the difference \(\varepsilon_k^0-\varepsilon_{k'}^0\) vanishes, we find the magnitude of the jump equal to

\[ 2|U_n|=2|U_{k,k+nG}|. \]

On each of these planes the derivative with respect to the normal also vanishes,

\[ \frac{\partial \varepsilon}{\partial k_\nu}, \]

and this leads, according to § 2 (13), to the vanishing of the corresponding component of the current.

For the one-dimensional problem the results obtained admit a simple graphical representation (it may also correspond to the three-dimensional problem for the special case \(k_x=k,\ k_y=k_z=0\)). The parabola in Fig. 6 gives the unperturbed energy; the heavy line shows the actual distribution of energy described by formulas (18). The points of discontinuity are given by (19) and lie at

\[ k=n\frac{G}{2};\quad (n=\pm 1,\ \pm 2,\ldots). \]

Fig. 6. Dependence of the energy and the current on the wave number for weakly bound electrons (one-dimensional problem)

Fig. 7. Lines of discontinuity of the energy (two-dimensional problem)

We see that the energy curve is broken and bent, and in this example it is particularly clear how the band structure in the energy distribution appears: according to (18), smaller values of the energy \(\varepsilon_k\) are lowered still further, while larger \(\varepsilon_k\) are raised still higher. At each point of discontinuity the energy curve has a horizontal tangent; at this point the magnitude of the current becomes zero. It is easy to see that the type of distribution of the eigenvalues obtained here is exactly the same as in the case of strongly bound electrons (see Fig. 3, § 3). Figure 7 shows the first lines of discontinuity for the two-dimensional problem, calculated from (19). On each of these lines the energy undergoes a certain jump.

Let us briefly formulate the results obtained. In those cases when, according to the classical laws, electrons can pass through periodic potential barriers, the corresponding elec-

electron waves change little. The latter, however, does not occur if the length of the electron wave is chosen so that the Bragg–Laue conditions (6) and (9) are satisfied. In this case phenomena occur that are analogous to the diffraction of X-rays. If, for any one of the waves, such a condition is fulfilled, then it can no longer propagate stationarily by itself. Such a wave will be in equilibrium with the reflected one [according to (11) and (14)], as a result of which the stationary state will be described by a standing wave. In this case the transport of electricity will be equal to zero. The last result follows from § 2 (13), since

\[ \frac{\partial \varepsilon_k}{\partial k}=0, \]

which is, however, obvious directly as well. The condition

\[ \frac{\partial \varepsilon_k}{\partial k}=0 \]

shows that in this case the energy surface has a horizontal tangent, as indeed we found (Fig. 6).

§ 5. General survey of the distribution of eigenvalues

The results obtained in the analysis of the limiting cases of high and low terms already make it possible to form a general idea of the complete energy spectrum of the crystal. Therefore we shall not enter into the details of a more general calculation.

The general picture of the distribution of eigenvalues is as follows. For the very lowest terms no difficulties are encountered. The terms of the crystal adjoin the corresponding terms of a single potential well, and the individual zones prove to be far removed from one another. However, on passing to higher energy values the width of the individual zones increases, and, finally, neighboring zones may begin to overlap one another. In this case the method set forth in § 3 loses its force*.

We shall now proceed from the other limiting case (§ 4). Here, each time the wave vector passes through the plane of discontinuity [§ 4 (19)], there is a jump of the energy. Let us consider the first cube

\[ \left(-\frac{G}{2}\leq k_x,k_y,k_z\leq +\frac{G}{2}\right). \]

The lateral surfaces of this cube are by no means surfaces of constant energy (Fig. 4, § 3). Thus, for example, the corners of the cube correspond (if the perturbation is neglected) to an energy 3 times greater than the points of intersection of the planes of the cube with the axes.

If the difference of the corresponding energy values is less than

* If only a small number of terms (multiplet) adjoin one another, while the remaining terms are sufficiently remote, then the method of § 3, modified accordingly, may be applied here as well; in this case it is only necessary additionally to take into account the mutual perturbation of these terms.

THEORY OF THE METALLIC STATE

If there is a discontinuity at the side surfaces, then none of the energy surfaces will intersect the side planes. All these points inside the cube (for them $|k|<\dfrac{G}{2}$) correspond to smaller values of the energy than points lying outside the cube, for which $|k|>\dfrac{G}{2}$. Accordingly, the distribution of eigenvalues over zones separated by forbidden energy regions will also be preserved in this case. Conversely, if the jump in energy is smaller than the indicated magnitude, the constant-energy surfaces will pass out of the cube into the exterior region, and the individual zones will overlap one another. In this case no forbidden energy values are obtained at all, and only for certain energy values are some directions of the wave vector excluded.

In the general case of transition from the first case to the second, we shall find that deep zones will be separated by broad regions, which will decrease with increasing energy; at some value of the energy, approximately equal to the maximum value of the rise of the potential, overlap of the individual zones will set in. This general picture is illustrated by Fig. 8, which gives the number of stationary states $A(\varepsilon)d\varepsilon$ in the energy interval $d\varepsilon$ as a function of the energy. The dotted curve gives the distribution for the case of a constant field.

It is especially noteworthy that the indicated structure of the distribution is preserved also in the case when the eigenvalues lie above the maximum rise of the potential; according to classical laws we ought here to expect complete continuity.

The case of forbidden states (definite energy values for definite directions) can be realized if the crystal is subjected from outside to bombardment by electrons of a suitable velocity. According to § 4, the electrons in this case must be reflected by the crystal, and indeed precisely according to Bragg’s law. Such selective reflection maxima are in fact observed in experiments on electron diffraction by Davisson and Germer, Thomson, Rupp, and others. The theory of electron diffraction is entirely analogous to the calculations given above^15, ^16, ^17. In this connection it turns out to be possible, from measurements of the differences of the diffraction maxima, to draw definite conclusions about the magnitude of the potential coefficients in the Fourier series.

Interesting evidence for the existence of a band distribution of eigenvalues at high energies is provided by the study of the fine structure of X-ray absorption edges in crystals^19. X-ray absorption, as is well known, is connected with the ejection of an electron from a sharp term of one of the inner electron shells of the atom. However, the final states inside the crystal do not coincide with the continuous states of an isolated atom, but are determined by the high terms of the crystal itself. Therefore, if for some energy value the eigenvalues inside the crystal are less dense, the probability of transferring an electron

in such states will also be diminished. Near the absorption edge there must arise certain intensity minima which do not occur in the case of free atoms. Such a structure of the edge is indeed found experimentally. The observed regularities (position of the bands, influence of temperature) are in very good agreement with the theory.

The modulation of electron waves in a crystal must also have an essential influence on optical effects. In Part II of the present article, when discussing the photoelectric effect, it was already pointed out that a free electron cannot take part in absorption, and that for this a certain coupling of the electron with the lattice is necessary. Such a coupling, in addition to the surface jump of the potential, is effected by the periodic field of the crystal. Substituting the Bloch waves of § 2 (8), (9) into the expression for the matrix element, Part II, § 5 (1), we find for the spatial factor of the matrix element (instead of Part II, § 5 (1a)) the expression

\[ M_{\mathbf{k}\mathbf{k}'} \sim \int \sum_{nn'} u_{\mathbf{k}'n}\, u_{\mathbf{k}n}\, e^{\frac{2\pi i}{K}(\mathbf{k}'-\mathbf{k}+n'\mathbf{G}-n\mathbf{G})}, \tag{1} \]

which is not equal to zero if the interference condition is fulfilled

\[ \mathbf{k}'-\mathbf{k}=n\mathbf{G}. \tag{2} \]

Thus certain transitions prove possible in which the electron is thrown from one zone into another*; transitions of the electron within one and the same zone prove to be forbidden.

§ 6. Conductors and Insulators

Up to now we have considered a single electron in the periodic field of a crystal. However, in order to construct a theory of electrical conductivity it is necessary to consider the entire crystal as a whole, with all its electrons. Here we can proceed analogously to the way this is done in the Bohr theory of the periodic system, and imagine that the electronic terms of the crystal are filled by electrons in the order of their energy sequence. In this case, to each term in translational motion (definite values of the triple of components of the wave vector \(\mathbf{k}\)) there will correspond two electrons, since here we have degeneracy due to the presence of spin; these two electrons will have oppositely directed spins. At low temperatures, therefore, the deepest levels of the crystal will be filled until all the electrons have been exhausted. The higher levels will remain completely free. The electrons filling the levels of the crystal are nothing other than the electrons of the individual atoms, and therefore it is natural to pose the question of in what

* A more detailed discussion may be found in the works by Tamm and Shubin, as well as by Kronig and Fujioka, cited in Part II, § 5.

correspond to the terms of an individual atom with the terms of the whole crystal. In this formulation of the question we have a complete analogy with Hund’s investigations concerning diatomic molecules, where a relation is likewise established between the terms of the isolated atom and the terms of the molecule (to which, in our case, the whole crystal corresponds).

Let us first consider the innermost atomic electrons, belonging to the \(K\)-shell. Here we have the case of low terms analyzed in § 3. The \(2N\) electrons completely fill the very lowest band with \(N\) different states of translational motion. However, such closed atomic shells do not take part in the phenomena of electrical conductivity. An electron can be accelerated only when, in its neighborhood, there are unoccupied states with higher energy; in this case a wave packet with a charge nonuniformly distributed in space can in fact be formed from it (§ 2). Considering now the filled band, one can see that, although a definite current corresponds to the individual terms, the resulting current will be equal to zero, since for any electron with a given direction of motion one can always find a paired one moving oppositely.

We shall obtain exactly the same result for any other closed shell, at least insofar as the corresponding crystal terms may be regarded as “low” in the sense of § 3. Each completed shell of the atom will give a filled band of the crystal, and therefore in the following arguments we may disregard completed shells altogether.

It is obvious that only the very highest electrons are essential for electrical conductivity; these must have arisen from the valence electrons of the isolated atom. We note that in numbering the states of these electrons it is expedient to begin with the wave number equal to unity.

If the temperature of the crystal is not equal to zero, the sharp boundary between occupied and unoccupied states disappears; in this case the Fermi–Dirac distribution will hold

\[ f_k=\frac{1}{e^{(\varepsilon_k-\varepsilon_0)/kT}+1}, \]

from which it is seen that an increase in temperature causes a loosening of the highest electrons of the bands.

Now we are in a position to decide whether a crystal with given properties will be a conductor or an insulator. In order that the crystal possess metallic conductivity, it is evidently necessary that in the region where the very highest electrons are situated (i.e., near \(\varepsilon_0\)) there be free places. Otherwise, the ground state of the crystal with all its electrons must belong to a continuum*. If the electrons completely fill the corresponding

\[ \text{* Strictly speaking, a true continuum is possible only in the case of an infinitely large crystal. However, the distance between neighboring terms within one and the same band is so small that practically one may speak here also of a continuum.} \]

zone, and the neighboring free zone is separated from it by a finite energy value, then the crystal under consideration cannot conduct; we obtain a model of an insulator. Conversely, if the energy level \(\varepsilon_0\) falls inside some zone, or if mutual overlap of separate zones has already begun, our crystal will be a metallic conductor.

Let us consider in more detail the simplest cases. If there is an element with a single valence electron, for example an alkali metal, then the number of conduction electrons present is equal to \(N = G^3\) (the number of atoms). However, the innermost cube in \(k\)-space contains \(2N\) states, since here, as before, we have an expression due to the presence of electron spin. In this case half the places within the zone will remain free, and the crystal will be a conductor. Hence it is clear that monovalent elements with an atomic lattice must be metallic conductors (in solid hydrogen, which is an insulator, the lattice is not atomic but molecular). In the case under discussion of monovalent elements, the energy anomaly considered in § 4 plays no essential role. Indeed, if the periodic terms of the potential are small, then in the first approximation the surfaces of constant energy in \(k\)-space will be spherical. Since a sphere of volume

\[ \frac{G^3}{3} \]

has radius

\[ r=\left(\frac{3}{\pi}\right)^{\frac{1}{3}}\frac{G}{2}, \]

the surface of constant energy will lie entirely inside the cube under consideration (whose edge is \(G\)), and here there will be no anomalies associated with Bragg reflection at all. (This need not necessarily be so if the perturbation in the distribution of eigenvalues is taken into account; however, in any case its influence must be small.)

In the case of elements with two valence electrons (for example the alkaline-earth metals) there are \(2N\) electrons, which fill the zone completely; in this case the substance could be an insulator. However, in this case as well we can obtain conductivity if the above-mentioned overlap of zones takes place. A priori no substantial assertions can be made here, and, on the contrary, on the basis of experimental data one must conclude that for the alkaline-earth metals precisely the latter occurs. However, for elements with many valence electrons Bragg reflection will always play a known role, and therefore we must conclude that their electrical conductivity will always be less than that of monovalent elements (see the interesting summary in Brillouin’s book).

A complete subgroup in an atom does not always give a filled zone in a crystal. Nor can it be said conversely that a filled zone of a crystal is formed from some one complete subgroup. Thus, for example, in crystals with symmetry lower than

cubic, as a result of the strong splitting of the \(p\)-terms by the field of the crystal (see the note at the end of § 3), a filled band could be formed from each separate \(p\)-term. Whether this will actually occur or not depends on the concrete quantitative relations and cannot be predicted in advance in general form. Therefore it is difficult to determine in advance whether a given element will give insulating crystals or conducting ones. Nevertheless, it proves possible to make here as well a number of qualitative statements \(^{20}\) and, to a certain extent, to explain the available experimental material.

In the case of atoms of noble gases or molecules without free valence (a completed outer shell), the basic term of the free atoms does not split upon their approach (this follows from Heitler and London’s theory of valence). Such systems can give only insulators, which, moreover, must possess a low melting temperature (since only van der Waals forces act here). The continuum of the higher terms of the crystal must be imagined as arising from the terms of rarefied atoms, of which at least one is in an excited state. For the alkaline-earth metals, which, to be sure, also have the ground state \(^{1}S\), the conditions change, since in the separate atom the states \(^{3}P\) are very close to the ground state. Therefore one can no longer neglect such states in solving the problem of excitation (for example, in § 3), even if the \(S — P\) distances do not exceed the interaction energies significantly. This means that in the immediate vicinity of the \(2G^{3}\) terms of the crystal formed by the \(s\)-electrons of the isolated atoms, there will also be terms corresponding to the \(p\)-electrons (strictly speaking, such a division is approximate, and here one should take the corresponding linear combination of all the near-lying terms). Such a situation corresponds to the overlapping of individual bands considered in § 5.

If the atoms from which the lattice is built are such that not too great an energy is required to convert them into positive or negative ions, then the energy released upon the approach of the ions by the forces of electrostatic attraction can compensate the losses due to ionization. In this case we obtain an ionic lattice. If the ions have a completed outer shell, then no splitting of the terms will occur, and we obtain an insulator.

In these cases (van der Waals or ionic lattices), for the basic term a sufficiently good approximation is already obtained if it is calculated from the combination of the proper functions of the individual constituent parts. Let us note that for metals such localization of the electrons would be impossible.

Thus one can easily explain, though qualitatively, the origin of both metallic and ionic lattices, and also of the low-melting insulators. Refractory insulators, a typical representative of which is diamond, require special consideration. Since the question of refractory insulators already lies outside the domain of the theory of metals, we shall confine ourselves only to presenting the principal results.

The splitting of the whole variety of terms into two separate regions, of which the lower one, as it turns out, is everywhere filled, occurs for any lattice, provided only the condition

\[ n = r = t. \tag{1} \]

is fulfilled.

Here \(n\) is the number of valence electrons in the atom, \(r\) is the number of centers in the elementary cell, \(t\) is the number of proper functions of an individual atom (the degree of degeneration).

Under the condition that relation (1) is fulfilled, the bond can be expressed by means of valence strokes. In this case, in forming a linear combination from the proper functions of the atoms, one may first form

a suitable combination of the eigenfunctions of isolated atoms (potential wells), and only then pass to Bloch combinations § 3, (12) (this constitutes one of the essential features of Hund’s theory).

For diamond condition (1) is fulfilled. C has four valence electrons, and each atom in the diamond lattice is surrounded by four other atoms (a tetrahedron). Further, the valence electron of a C atom may be either in the orbit \(2s\), or in one of the three unusually closely situated orbits \(2p\); therefore the degree of degeneracy of the initial eigenfunction of the electron will be precisely \(1+3=4\). The circumstance that the higher groups C (Si, Ge, and Sn) are metals is explained by the considerably greater difference in the \(s\)- and \(p\)-terms. Let us note that in this case the number \(t\) is in general difficult to determine.

Although the principal term of a nonconductor lies isolated, the higher excited states will, according to the general scheme of § 5, give a continuum. If an electron somehow falls into one of these unoccupied states, then it can already move freely there. Such a case occurs, for example, in the internal photoeffect. If the electron excited in this way is spatially removed from its initial position, then it can no longer return to the original state, since all the lower-lying terms are occupied. Let us note that the photoelectric work function of the electron here will not coincide with the difference of the energies of two bands, but will always be larger; this follows from the selection principle of § 5 (2), according to which in an ideal crystal transitions are possible only between corresponding places of two bands.

Optical absorption in an insulator must set in at a definite frequency corresponding to the first possible electronic transition; at lower frequencies the insulator must be completely transparent. Conversely, at higher frequencies the absorption band must be continuous. The presence of discrete lines in alkali-halide compounds must therefore be ascribed to the excited terms of an individual ion, but not to the terms of the whole crystal.

§ 7. Refinement of the Theory of Electrical Conductivity.
Anomalous Hall Effect

We shall now attempt to refine the theory of electrical conductivity of part III by means of the new results obtained. Here, first of all, the distribution of eigenvalues near the Fermi surface must be changed, since only the electrons of this region participate in electrical conductivity. These same electrons also determine the thermal properties of the metal (in particular, its heat capacity) and the magnitude of the current [§ 2 (13)]. Further, the magnitude of the kinetic part of the electronic energy, i.e. the difference between the mean value of the periodic potential (for the critical zone) and the energy \(\varepsilon_0\) of the Fermi distribution, must be subjected to review. The latter quantity coincides with the energy of the absolute zero in the sense in which this is given by measurements by means of electron diffraction.

It will now no longer coincide exactly with the width of the filled region of the upper band at \(T=0\) (from this it is clear that one cannot expect complete agreement of the numerical values discussed in Part II, § 3). Therefore the distribution of eigenvalues for small energies will not play any special role, and, consequently, quantities of the type of the “total number of free electrons,” which cannot be rationally defined, will likewise play no role. However, in the case of a homogeneous field (or a weakly modulated one) the number of valence electrons does indeed establish a definite connection between the energy of the absolute zero and the number of free electrons in the form in which this was set forth in Parts I–III. In this case both these quantities are, in principle, quite measurable: the energy of the absolute zero (in the sense indicated above) from electron diffraction, and the density of the distribution of eigenvalues from the magnitude of the heat capacity due to the electrons. However, the experimental data needed for this are still too inaccurate for such a calculation actually to be carried out.

Fig. 8. Distribution of eigenvalues upon overlap of higher zones

Statistical calculations for equilibrium states will not differ from those analyzed in Part I. Here the discrete set of energy levels will be determined by the existing periodic field, while the statistical method itself will remain unchanged. The filling of the cells here too will be determined by the Fermi–Dirac distribution [Part III, § 3 (12)]

\[ f_0=\frac{1}{e^{(\varepsilon-\varepsilon_0)/kT}+1}. \tag{1} \]

However, the number of states \(A(\varepsilon)\,d\varepsilon\) in the energy interval \(d\varepsilon\) will now be different, as a result of which the value of the parameter \(\varepsilon_0\) will also change. The quantity \(\varepsilon_0\) will still be determined from the relation

\[ N=\int A(\varepsilon)f_0(\varepsilon)\,d\varepsilon. \tag{2} \]

If the distribution of eigenvalues has the character shown in Fig. 8, § 5, then the influence of the electrons on the heat capacity of the crystal should also be noticeable.* We should then obtain the following obvious course of phenomena. If the first zone is entirely

* The question was first analyzed by Bloch (loc. cit.), and then more precisely by Morse and Stueckelberg \(^{21}\). Here, however, so far only qualitative results have been obtained.

filled (for example, in an insulator), then at first there should be no influence of the electrons at all. Only when \(kT \cong \varepsilon_2-\varepsilon_1\) (\(\varepsilon_2-\varepsilon_1\) is the energy difference for two neighboring zones) can a slow increase of the heat capacity be expected. If the critical zone is not completely filled (a normal metal), at first there should be a linear increase in accordance with Sommerfeld’s theory, and the rate of increase may even be somewhat greater than in Sommerfeld’s theory if the energy levels are more closely spaced than for free electrons. At higher temperatures a decrease of the heat capacity is again possible, which should begin from the moment when the first region in \(k\)-space is completely filled.

In passing to the kinetic theory of electrical conductivity, a great difficulty immediately arises, since now the energy turns out to depend in a very complicated way on the three wave numbers (which here replace the components of the velocity). In particular, the energy function no longer possesses spherical symmetry with respect to \(\mathbf{k}\). As a consequence, the fundamental equation of Part III, § 4 (2), for the distribution function can no longer be solved by a substitution of the type of Part III, § 4 (6), and further calculations under any general assumptions become very difficult. Therefore, in almost all works now available the hypothesis is made that here too a function of spherical symmetry can approximately be used, so that one may put

\[ \varepsilon(\mathbf{k})=\omega\left(k_x^2+k_y^2+k_z^2\right)=\omega k^2 . \tag{3} \]

\(\omega\) is a constant which, for convenience of agreement with the formulas of the theory of free electrons, we may set equal to [cf. § 4 (2)]

\[ \omega=\frac{h^2}{2m^*K^2}. \tag{4} \]

Thus the whole influence of the periodic potential is taken into account only by introducing an “apparent mass” in the sense of §§ 3–5*.

Let us note that the occurrence of spherical symmetry for microcrystalline systems could be regarded as a certain statistical effect.

Accepting relation (3), we find from § 2 (16) the value of the current \(S_x\), equal to

\[ S_x=e\xi=\frac{eK}{h}\frac{\partial \varepsilon}{\partial k_x} =e\frac{2\omega K}{h}k_x =\frac{e}{m^*}\frac{h}{K}k_x . \tag{5} \]

Assuming that the electric field acts along the \(x\)-axis \((F=F_x)\), we obtain from § 2 (16) the change of the distribution function caused by the external field in the form

\[ -\frac{\partial^{(1)} f}{\partial t} =\frac{eFK}{h}\frac{\partial f_0}{\partial k_x} =\frac{eFK}{h}\frac{\partial f_0}{\partial \varepsilon}\frac{\partial \varepsilon}{\partial k_x} =e\frac{Fhk_x}{m^*K}\frac{\partial f_0}{\partial \varepsilon} =eF\xi\frac{\partial f_0}{\partial \varepsilon}. \tag{6a} \]

* Of course, the introduction of the same apparent mass for all electrons is impossible. However, we have seen that only electrons of low energy, close to \(\varepsilon_0\), play an essential role. Therefore it is natural to take the value \(m^*\) precisely for these electrons.

In exactly the same way as in Part III, § 3, in order to obtain the basic equation it is also necessary to take into account the influence of diffusion of wave packets, which here as well will be determined by the magnitude of the group velocity

\[ -\frac{\partial^{(2)} f}{\partial t} = \xi \frac{\partial f_0}{\partial x} = \frac{K}{h}\frac{\partial \varepsilon}{\partial k_x}\frac{\partial f_0}{\partial x} = -\frac{h}{m^{*}K}\,k_x\frac{\partial f_0}{\partial x}. \tag{6b} \]

We see that all the relations of Part III, § 3 remain valid; only, instead of the true electron mass \(m\), one must introduce the apparent mass \(m^{*}\) (let us note that all these formulas could have been retained even without assumption (3), if only \(\xi\) were replaced by \(\dfrac{K}{h}\dfrac{\partial \varepsilon}{\partial k_x}\)).

In the expression for the transition probability it is expedient to introduce, instead of the component of the group velocity \(\xi\), the wave number proportional to it,

\[ k_x=\frac{m^{*}K}{h}\,\xi = m^{*}\frac{K^2}{h^2}\frac{\partial \varepsilon}{\partial k_x}. \tag{7} \]

After this we obtain the basic equation [cf. Part III, § 4 (2), (5) and (14)] in the following form:

\[ \frac{\partial f}{\partial t} = \frac{h}{m^{*}K}\,k_x\frac{\partial f_0}{\partial \varepsilon} \left( eF+kT\frac{\partial \alpha}{\partial x} -\frac{\varepsilon}{T}\frac{\partial T}{\partial x} \right) = \frac{K}{h}\frac{\partial \varepsilon}{\partial k_x}\frac{\partial f_0}{\partial \varepsilon} \]

\[ \left( eF+kT\frac{\partial \alpha}{\partial x} -\frac{\varepsilon}{T}\frac{\partial T}{\partial x} \right) = b-a = -\iiint \left\{ V_{\mathbf{k}'}^{\mathbf{k}} f(1-f') - V_{\mathbf{k}}^{\mathbf{k}'} f'(1-f) \right\} \,dk'_x\,dk'_y\,dk'_z . \tag{8} \]

In the presence of an external magnetic field in the direction \(z\), an additional term would enter the left-hand side of this equation,

\[ \frac{eH}{c}\left(\frac{K}{h}\right)^2 \left\{ \frac{\partial \varepsilon}{\partial k_x}\frac{\partial f}{\partial k_y} - \frac{\partial \varepsilon}{\partial k_y}\frac{\partial f}{\partial k_x} \right\} = \frac{eH}{m^{*}c} \left\{ k_x\frac{\partial f}{\partial k_y} - k_y\frac{\partial f}{\partial k_x} \right\}, \tag{9} \]

analogous to the expression in Part III, § 7 (1). The fact that all the former formulas change only very insignificantly is a consequence of the great similarity between Bloch waves and ordinary plane waves.

Under the assumptions made, all the principal consequences of the former theory prove to be valid here as well (only the new quantity \(m^{*}\) will enter the formulas everywhere). Only the old expression for the absolute zero energy [Part III, § 5 (2) (3)] proves to be unreliable,

\[ \varepsilon_0 = \frac{h^2}{2m^{*}} \left( \frac{3n}{4\pi G} \right)^{\frac{2}{3}}, \tag{10} \]

since now it can no longer be assumed that there will be one and the same value of \(m^{*}\) for the whole zone. For the calculation of the quantities \(K_n\) [Part III, § 4 (13)] this circumstance is immaterial, and therefore the whole former theory of thermoelectric effects* at high temperatures remains—

* A formal solution of equation (8) under very general assumptions (spherical symmetry \(\varepsilon(\mathbf{k})\) is not assumed, and energy exchange is not neglected) was given by Uehling\({}^{22}\). In this way the most general formulas for thermoelectric phenomena, including the Bridgman effect, were obtained.

is introduced. The Wiedemann–Franz law [Part III, § 5 (6)] likewise remains fully valid; the energy of the absolute zero does not enter into it at all.

The approximation considered here must be the better, the closer the real electrons are, in their properties, to free ones. For good conductors the energy of the absolute zero is certainly of the same order as that given by expression (10), under the assumption that the apparent mass \(m^*\) is equal to the actual mass. This permits the conclusion that the approximation found is not too bad.

In those cases where the level \(\varepsilon_0\) lies near the anomalous region, Bragg reflection and the related features in the distribution of eigenvalues may play an essential role. For monovalent metals, according to § 6, this is difficult to expect; however, for elements with several valence electrons, especially if the individual potential wells are deep (in poor conductors), the latter may occur.

The anomalies associated with Bragg reflection will be especially sharp if the electron zone is almost completely occupied, so that only a few electrons are lacking for its filling. In this case the apparent mass (see § 4) may change its sign*, and then one may speak of “defect” conductivity. The phenomena that occur in this case can be most easily understood if, instead of the motion of electrons, one considers the displacement of the empty places in the electron zone (“holes”). The trajectories of motion of such holes under the action of an electromagnetic field will be exactly the same as those of the corresponding electrons. According to the picture adopted by us, we regarded the electrons, in their motion, as independent of one another, and represented the common motion of all the electrons by a certain flow in phase space. The hole under consideration will participate in this motion just as an air bubble participates in the motion of a fluid flow. However, the magnitude of the current of the hole will be related as if there were a positive charge in its place. We saw above that the resultant current of a completed zone is equal to zero, since to each of its electrons another electron with oppo-

* In a real metal an overlap of the critical zone must occur. In this case part of the conduction electrons will have a negative apparent mass, part a positive one, and the resulting effect in the Hall phenomenon will depend on which of these electrons predominates. Bloch and Nordheim** carried out recently a calculation for a suitably generalized model under the assumption of a cubic (and not central) symmetry of the distribution of eigenvalues. In doing so it proved possible to give an exact criterion for the anomalous sign of the Hall effect, and also for the anomalous Thomson effect (in Part III, § 6 it is already indicated that the Thomson effect with abnormal sign is observed in many cases; it should be noted, however, that the conditions for the occurrence of the anomalous Thomson effect do not coincide with those for the Hall effect). By means of such a generalization it was also possible to obtain the correct order of magnitude for the change of resistance in a magnetic field and the correct dependence of this effect on direction.

in the opposite direction of motion. If one electron is removed from such a filled zone, then a current arises, produced by its uncompensated partner. Since only the ratio \(e/m\) enters the equations of motion, the indicated circumstances will lead to identifying such a hole in the anomalous region (i.e., with negative apparent mass) with an electron of positive mass and positive charge. We shall then find a normal electrical conductivity, which will appear to be caused by positive particles. The latter should also lead to a reversal of the sign in the Hall effect. Thus the anomalous Hall effect, which had previously seemed quite mysterious, receives in the new theory a simple and natural explanation.*

For a detailed calculation of the effect one must take into account that the “holes” obey Fermi–Dirac statistics, and that they therefore may be operated with in the same way as with real electrons. Within the zone under consideration, by assumption, there will be only a very small number of unoccupied places, and the degree of filling of the cells (by holes) will be very small; this permits the limiting Lorentz case to be used in the calculation (Part III, § 4; see also the following paragraph).

The anomalous Hall effect is observed only in a small number of materials. If for some substance the normal sign of the effect is observed and if it agrees, in order of magnitude, with the effect calculated according to Part III, § 7 (12), then one may be sure that all the assumptions of Part III are also well justified for the given substance. However, the character of the dependence of the resistance on the magnetic field (Part III, § 7) leads to the conclusion that here too the indicated anomaly is present. Since here the question is one of the disappearance or non-disappearance of the zero approximation (which has a predominant significance in comparison with the subsequent terms), very small anomalies are already manifested here, which may be imperceptible in other effects.

In connection with the questions under consideration it should be especially noted that experiments of the Tolman type, in which the inertial properties of the electron are manifested, will give the true mass, and not the apparent one, and therefore will always lead to the correct value of \(e/m\). The essential feature of such experiments consists in the fact that the conductor is accelerated. Owing to inertia the electrons lag behind the motion of the conductor and thereby create a certain potential difference, which is measured by compensation with an external field. If this process is considered in a coordinate system connected with the moving conductor, then the acceleration will be equivalent to the appearance of an apparent additional force \(mb\). Compensation by the external field \(F\) will occur when the relation

\[ eF = mb, \tag{11} \]

is fulfilled, quite independently of how the electrons react to the external field.**

* See Peierls’ work.\(^{10}\) A pictorial explanation by means of holes belongs to Heisenberg.\(^{24}\)

** What has been said will be valid in any case as long as the crystal lattice may be regarded as rigid during the acceleration; under the conditions of Tolman’s experiments this is always the case.

§ 8. Semiconductors

Up to now we have considered two limiting cases: insulators (widely separated electron bands, of which one is completely filled) and metals (either an unfilled band, or an overlap of separate bands). Let us now consider the intermediate case, when there are discrete bands, of which one, the lower one, is filled at \(T=0\), but the interval between this band and the neighboring one is extremely small. Such a crystal will not conduct at low temperatures.

At higher temperatures, thermal excitation (the quantity \(kT\)) may become comparable with the energy difference of neighboring bands; then some of the electrons will be transferred into the higher-lying band, which will lead to the appearance of a definite electrical conductivity. At the same time “holes” are also formed in the lower band. Such a model will correspond to a semiconductor*; its resistance tends to infinity at low temperatures. At sufficiently high temperatures the resistance of such a semiconductor may again increase owing to disturbances connected with the thermal disordering of the lattice. However, depending on the concrete relations between the parameters of the system, such an increase of the resistance may also not be observed.

For the magnitude of the electrical conductivity, what is essential is the number of thermally excited electrons (the number of electrons in the upper band) or the equal number of free places in the lower band. It can still be calculated from the Fermi distribution [§ 7 (1)], only the parameter \(\varepsilon_0\) must now be determined differently. To determine \(\varepsilon_0\) we shall, as always, have the relation § 7 (2); however, we must take into account that here the number of eigenvalues \(A(\varepsilon)d\varepsilon\) will not be the same as in Part I, § 6 (11) for free electrons.

It is natural to make the assumption that, under thermal excitation, electrons are removed mainly from the region immediately adjacent to the upper edge of the first band (\(I\)). We shall denote the energy level of this edge by \(\varepsilon_1\) (Fig. 9). For the states of this region we may use the approximation § 3 (16) and put

\[ \varepsilon(I)=\varepsilon_1-\varepsilon_1^* =\varepsilon_1-\beta_1\frac{4\pi^2}{G_1^2}\left(k_x^{*2}+k_y^{*2}+k_z^{*2}\right); \tag{1} \]

\[ (k^*_{x,y,z}=\pm 1,\pm 2,\ldots;\ \varepsilon<\varepsilon_1,\ \varepsilon_1^*>0). \]

Each of these states may correspond to \(G_1\) electro-

* The model was proposed and calculated by Wilson\(^{25}\). However, Wilson’s calculations are in some places very complicated. We therefore use here a more heuristic method, which leads exactly to the same results and which seems to us more direct. All of Wilson’s formulas are obtained from ours if one puts \(G_1=2\), \(G_2=6\) and

\[ m_{1,2}=\frac{h^3}{8\pi^2 a^2\beta_{1,2}}. \]

us*. The number of states calculated for an interval of wave numbers will therefore be

\[ A(k_x^*,\,k_y^*,\,k_z^*)\,dk_x^*\,dk_y^*\,dk_z^* = G_1\,dk_x^*\,dk_y^*\,dk_z^*, \tag{2a} \]

or, if it is recalculated for an energy interval,

\[ A_1(\varepsilon)\,d\varepsilon = \frac{VG_1}{h^3}\,2\pi(2m_1)^{3/2}\varepsilon^{1/2}\,d\varepsilon = \frac{G_1G^3\varepsilon^{1/2}\,d\varepsilon}{4\pi^2\beta_1^{3/2}}; \quad (\varepsilon<\varepsilon_1), \]

\[ m_1=\frac{h^2}{8\pi^2a^2\beta_1}. \tag{2b} \]

(It should be borne in mind that the quantities \(m_1\) and \(m_2\) occurring here and below are always positive and therefore may differ in sign from the apparent mass \(m^*\) [§ 3 (15)].)

Fig. 9. Electron zones and the energy of the absolute zero for semiconductors

For the upper zone (II), in making the calculation we must start only from the lower edge \(\varepsilon_2\) (here, by assumption, \(\varepsilon_2>\varepsilon_1\)). Here the mass will have the normal sign, and we may put

\[ \varepsilon(\mathrm{II}) = \varepsilon_2+\varepsilon_2^* = \varepsilon_2+\beta_2\frac{4\pi^2}{G^2} \left(k_x^2+k_y^2+k_z^2\right), \tag{3} \]

and correspondingly we obtain

\[ A_{II}(\varepsilon)\,d\varepsilon = \frac{VG_2}{h^3}\,2\pi(2m_2)^{3/2}\varepsilon^{1/2}\,d\varepsilon; \quad (\varepsilon>\varepsilon_2). \tag{4} \]

Relation § 7 (2) for determining \(\varepsilon_0\) requires that

\[ N = \int_I \frac{A_1(\varepsilon)\,d\varepsilon} {e^{(\varepsilon-\varepsilon_0)/kT}+1} + \int_{II} \frac{A_{II}(\varepsilon)\,d\varepsilon} {e^{(\varepsilon-\varepsilon_0)/kT}+1}. \tag{5} \]

* Since the designations of the weights \(G_1, G_2\) are supplied with indices, there is hardly any danger of confusing these quantities with the number of atoms \(G\) on an edge of the crystal. Usually \(G_1=2\), corresponding to the presence of spin in the electron. However, the weight may also be expressed by a larger number if not \(s\)-terms but, for example, \(p\)-terms take part in the formation of the crystal. In the following, precisely the case is considered in which the lower zone is formed from \(s\)-terms (\(G_1=2\)), and the upper from \(p\)-terms (\(G_2=6\)).

On the other hand, from the assumption that at \(T=0\) zone I is completely filled, it follows that

\[ N=\int_{\mathrm{I}} A_{\mathrm{I}}(\varepsilon)\,d\varepsilon \tag{6} \]

(i.e., that the number of electrons is exactly equal to the number of vacancies in zone I). Therefore equation (5) gives

\[ \left. \begin{aligned} \int_{\mathrm{I}}\left(1-\frac{1}{e^{(\varepsilon-\varepsilon_0)/kT}+1}\right)A_{\mathrm{I}}(\varepsilon)\,d\varepsilon &= \int_{\mathrm{I}}\frac{A_{\mathrm{I}}(\varepsilon)\,d\varepsilon}{1+e^{(\varepsilon_0-\varepsilon)/kT}} \\ &= \int_{\mathrm{II}}\frac{A_{\mathrm{II}}(\varepsilon)\,d\varepsilon}{e^{(\varepsilon-\varepsilon_0)/kT}+1}. \end{aligned} \right\} \tag{7} \]

After this, from (1), (2b), (3), and (4) we find the equation for determining \(\varepsilon_0\) in the following form:

\[ \left. \begin{aligned} G_1(m_1)^{\frac{3}{2}} \int_0^\infty \frac{\varepsilon_1^{*\,\frac12}\,d\varepsilon_1^*} {1+e^{(\varepsilon_1^*+\varepsilon_0-\varepsilon_1)/kT}} &= \\ &= G_2(m_2)^{\frac{3}{2}} \int_0^\infty \frac{\varepsilon_2^{*\,\frac12}\,d\varepsilon_2^*} {e^{(\varepsilon_2^*+\varepsilon_2-\varepsilon_0)/kT}+1}. \end{aligned} \right\} \tag{8} \]

Replacing in this expression the exactly unknown upper limit by \(\infty\), we do not make a large error, since for large \(\varepsilon_1^*\) and \(\varepsilon_2^*\) the integrand rapidly tends to zero. Equation (8) cannot be solved exactly. However, the integrals entering it correspond to the types \(J_{1/2}\) in Part I, § 8 (1), and may be evaluated approximately. Assuming that \(\varepsilon_1<\varepsilon_0<\varepsilon_2\) (the correctness of this we shall now confirm), we shall find that the exponents will always be positive, and therefore for not too high temperatures we can use the first approximation [Part I, § 8 (3a)]:

\[ G_1(m_1)^{\frac{3}{2}}e^{-(\varepsilon_0-\varepsilon_1)/kT} = G_2(m_2)^{\frac{3}{2}}e^{-(\varepsilon_2-\varepsilon_0)/kT}; \tag{9} \]

whence we obtain

\[ \varepsilon_0=\frac{1}{2}(\varepsilon_1+\varepsilon_2) -\frac{1}{2}kT\lg\frac{G_2}{G_1}\left(\frac{m_2}{m_1}\right)^{\frac{3}{2}}. \tag{10} \]

Since the expression standing under the sign \(\lg\) is of order 1, the last term may be altogether neglected, at any rate so long as \(kT\ll \varepsilon_2-\varepsilon_1\).

We obtain in (10) a remarkable result: it turns out that the level \(\varepsilon_0\) lies exactly in the middle of the forbidden region (Fig. 9). It is obvious that the level \(\varepsilon_0\) must therefore be regarded as only virtual. A clear explanation of the reason for so strange a result may be given in the following way. If zone I were not

completely filled, the level \(\varepsilon_0\) would lie inside this very band. If there were electrons in band II even at \(T=0\), this level would lie inside band II. In passing from one case to the other, \(\varepsilon_0\) must undergo a jump, and therefore in our example an intermediate value for \(\varepsilon_0\) must have been obtained (exactly analogous to what happens when a discontinuous function is expanded in a Fourier series). A very essential circumstance is that the excitation energy turns out to be equal not to the full difference \(\varepsilon_2-\varepsilon_1\), but only to one half of it,

\[ \frac{\varepsilon_2-\varepsilon_1}{2}=k\theta_H, \tag{11} \]

where \(\theta_H\) is the characteristic temperature for the given semiconductor. For each of the bands the distribution functions will have the following form:

\[ f_I=\frac{1}{e^{(\varepsilon-\varepsilon_0)/kT}+1} =\frac{1}{e^{-(k\theta_H+\varepsilon_1^*)/kT}+1}, \tag{12a} \]

\[ f_{II}=\frac{1}{e^{(k\theta_H+\varepsilon_2^*)/kT}+1}. \tag{12b} \]

The distribution function for the free places (holes) in band I will be

\[ f_L=1-f_I=\frac{1}{e^{(k\theta_H+\varepsilon_1^*)/kT}+1}; \tag{12c} \]

it has exactly the same form as \(f_{II}\). Equation (8) thereby acquires the following simple meaning: it merely expresses the equality of the number of holes \(n_L\) in band I (left-hand side) and the number of excited electrons \(n_{II}\) (right-hand side).

The total number of excited electrons per unit volume is obtained as

\[ n_{II}=n_L=\int_{II} A_{II} f_{II}\,d\varepsilon = \left\{ \begin{aligned} &=\frac{G_2}{h^3}\,2\pi(2m_2)^{\frac{3}{2}} \int_0^\infty \frac{\varepsilon_2^{*\,\frac{1}{2}}\,d\varepsilon_2^*} {e^{(k\theta_H+\varepsilon_2^*)/kT}+1}. \end{aligned} \right. \tag{13a} \]

If here, too, in calculating the integral we restrict ourselves only to the first term of the expansion [Part I, § 8 (3a)], then the following dependence of the number of excited electrons on temperature is obtained:

\[ n_{II}=n_L=\frac{G_2}{h^3}(2\pi m_2 kT)^{\frac{3}{2}} e^{-\frac{\theta_H}{T}}; \tag{13b} \]

this dependence agrees well with experimental data.

The electrical conductivity* of such a semiconductor can be calcul—

\[ \text{* The quantitative theory of the remaining effects (for example, paramagnetism, heat capacity, etc.) can be found in Wilson’s work.} \]

will be determined in the following way. The total electrical conductivity will be composed of two parts: the conductivity of band I and the conductivity of band II. For band I there is a typical defect conductivity. Here the calculation can be carried out by operating with holes in band I (see the note at the end of § 7), which will be similar to positive electrons, with positive energy \(\varepsilon_1^*\) and positive mass \(m_1\). According to (12c), the degree of filling of the cells by such holes will be \(\ll 1\), and we shall have the case of weak degeneracy. We can directly transfer here all formulas of the Lorentz theory, Part III, in which it will only be necessary to assign a positive sign to the electron charge \(e\) and to replace the actual mass \(m\) by the apparent \(m_1 \ne m\). Using for the number of electrons (holes) \(n_L\) expression (13b), we obtain the required part of the electrical conductivity \(x_1\) [Part III, § 5 (1)] in the following form:

\[ \left. \begin{aligned} x_1=x_L&=e^2\,\frac{4}{3}\,\frac{n_L l_1}{\sqrt{2\pi m_1 kT}}=\\ &=\frac{4}{3}\,\frac{G_2 e^2 l_1}{h^3}\left(\frac{m_2}{m_1}\right)^{\frac12}2\pi m_2 kT e^{-\frac{\Theta_H}{T}}, \end{aligned} \right\} \tag{14a} \]

where \(l_1\) is the length of the free path of the holes under consideration (the possible dependence of the free path on the velocity is neglected here).

For the electrical conductivity of the upper band II an analogous expression is obtained, only now the normal sign must be taken for \(e\). Here too we must use the Lorentz theory (and not Sommerfeld’s), whence we find

\[ x_{\mathrm{II}}=e^2\,\frac{4}{3}\,\frac{n_{\mathrm{II}} l_{\mathrm{II}}}{\sqrt{2\pi m_2 kT}} =\frac{4}{3}\,\frac{G_2 e^2 l_{\mathrm{II}}}{h^3}\,2\pi m_2 kT e^{-\frac{\Theta_H}{T}} . \tag{14b} \]

The total electrical conductivity will be

\[ \left. \begin{aligned} x&=e^2\,\frac{4}{3}\,\frac{n_{\mathrm{II}}}{\sqrt{2\pi kT}} \left(\frac{l_1}{m_1^{1/2}}+\frac{l_{\mathrm{II}}}{m_2^{1/2}}\right)=\\ &=\frac{4}{3}\,\frac{e^2 G_2}{h^3}\,2\pi m_2 kT \left\{l_1\left(\frac{m_2}{m_1}\right)^{\frac12}+l_{\mathrm{II}}\right\} e^{-\frac{\Theta_H}{T}} . \end{aligned} \right\} \tag{15a} \]

The most essential feature in the expression obtained is the temperature dependence. As will be shown in the next paragraph, for pure metals at not very low \(T\) (for \(T>\Theta_D\), where \(\Theta_D\) is the Debye characteristic temperature) the model theory leads to proportionality between \(l\) and \(1/T\). The same dependence is naturally to be expected in our case also, since the basic conclusions of the theory must be valid here as well. Empirical data for pure metals also lead to analogous results. Therefore from (15a) it fol-

the following dependence of the electrical conductivity on temperature:

\[ \chi = \chi_0 e^{-\frac{\Theta_H}{T}}, \tag{15b} \]

where \(\chi_0\) no longer depends on \(T\). The ideal semiconductor under consideration must change its electrical conductivity according to an exponential law in the temperature range \(\Theta_D < T < \Theta_H\), and its resistance must tend to infinity at low temperatures (transition to an insulator).

For determining the nature of the electrical conductivity, the Hall effect plays an especially important role, since with its aid one can distinguish defective conductivity from normal conductivity. The numerical value of the Hall effect will be expressed by the sum of two terms of the type in Part III, § 7 (12) L., in one of which, corresponding to defective conductivity, it will be necessary to choose a positive sign for the charge. Since for an ideal semiconductor the number of holes must be exactly equal to the number of excited electrons, we arrive at the important conclusion that the Hall effect for an ideal semiconductor must vanish.

If \(T > \Theta_H\), then the assumption that the number of excited electrons is small will no longer be valid, and the results obtained will cease to be true. In this case the discontinuity in the distribution of eigenvalues will no longer play an essential role, and the semiconductor, in its properties, must approach a metal; in particular, at sufficiently large \(T\), owing to the increasing influence of the thermal disorder of the lattice, its resistance must increase with temperature according to a linear law. For a sufficiently small value of \(\Theta_H\), such a semiconductor should exhibit a minimum of resistance, lying approximately at \(T \simeq \Theta_H\).

The conditions for a crystal to be a semiconductor in the indicated sense depend on rather special numerical ratios between the parameters; first of all, \(\Theta_H\), which determines the width of the forbidden region, must have a suitable value. Since in real specimens there is almost always a number of perturbing influences, it is rather difficult to decide whether a given substance fits the model analyzed or not. In particular, for polycrystalline substances the transition resistance of the individual microcrystallites may be such that it is bridged only at high temperatures, when the characteristic properties of the semiconductor may be lost. In addition, various inclusions in the crystal lattice play an especially large role. At present it is hardly possible to point to even a single pure substance that would be an ideal semiconductor; usually various perturbing factors are always present.

The best-studied semiconductor is \(\mathrm{Cu_2O}\), which has found wide technical application in dry rectifiers and valve photoelements \(^{26}\). In this case the electrical conductivity apparently arises entirely at the expense of impurities (probably \(^{27}\) from

(residual 0), since Cu₂O conducts the more poorly, the purer the specimen. Just as in Cu₂O, a noticeable Hall effect is observed. To explain all these facts, Wilson proposed the following scheme.

A foreign atom produces inside the crystal a localized perturbation that violates the periodicity of the lattice. At such places there may appear electronic states that do not belong to the system of terms of the crystal itself, and which therefore may also be located inside the forbidden region (Fig. 10, level \(A\)). In this case the transition of electrons into the higher-lying continuum of band II can occur already from level \(A\), so that the required excitation energy will be considerably reduced*. The number of electrons that can be excited in this case will be considerably smaller than in the preceding case (the number of perturbing atoms is small in comparison with the number of valence electrons); however, this circumstance may be more than compensated by the magnitude of the exponential factor. The magnitude of the electrical conductivity will now be determined only by the number of electrons in band II. If all the perturbing atoms are identical, then the perturbing levels will lie at the same height. In this case the electrical conductivity can be calculated simply.

Fig. 10. Electronic bands of a semiconductor in the presence of a perturbing atom.

Let the energy of the perturbing levels, which at \(T=0\) contain one electron each, be \(\varepsilon_1\). Let the number of levels (the number of impurity atoms in \(1\ \mathrm{cm}^3\)) be \(n_0\). Further, let the energy \(\varepsilon_2\) correspond to the beginning of the continuum of band II. If the upper edge of the filled band I lies considerably below \(\varepsilon_1\), then we need not take into account at all the electrons of this band. According to the Fermi distribution, the number of electrons on the additional levels of interest to us for \(T\ne0\) is

\[ \frac{n_0}{e^{(\varepsilon_1-\varepsilon_0)/kT}+1}. \tag{16} \]

We shall find the energy of absolute zero \(\varepsilon_0\) by the former method. Instead of equation (8) we obtain

\[ n_0=\frac{n_0}{e^{(\varepsilon_1-\varepsilon_0)/kT}+1}+n_{\mathrm{II}}, \]

* If level \(A\) falls into one of the bands, then its influence will be insignificant. In this case the perturbing atom will cause only a certain scattering of the electronic waves, creating an additional resistance of the crystal (see § 11).

** Wagner\(^{28}\) put forward the supposition that the electrical conductivity of Cu₂O is due to holes in band I, which are formed upon thermal excitation of electrons. Such purely defective conductivity is theoretically quite possible. In this case, however, the sign of the Hall effect should be anomalous, which is not observed for Cu₂O.

which, with the aid of (13a), we can write in the form

\[ \frac{n^0}{1+e^{(\varepsilon_0-\varepsilon_1)/kT}} = \frac{G_2}{h^3}\,2\pi(2m_2)^{3/2} \int_0^\infty \frac{\varepsilon_2^{*\,1/2}\,d\varepsilon_2^*} {e^{(\varepsilon_2^*+\varepsilon_2-\varepsilon_0)/kT}+1}. \tag{17} \]

Under the same assumptions as before, i.e., neglecting 1 in the left-hand side and using the expansion of Part I, § 8 (3a), we find*

\[ n_0 e^{-(\varepsilon_0-\varepsilon_1)/kT} = n_{\mathrm{II}} = \frac{G_2}{h^3}(2\pi m_2 kT)^{3/2} e^{-(\varepsilon_2-\varepsilon_0)/kT}, \tag{18a} \]

whence we obtain

\[ \varepsilon_0 = \frac{\varepsilon_1+\varepsilon_2}{2} -\frac{kT}{2}\lg \frac{G_2}{h^3 n_0} (2\pi m_2 kT)^{3/2}. \tag{18b} \]

The exact expression for \(n_{\mathrm{II}}\) will be

\[ n_{\mathrm{II}} = \left\{ \frac{n_0G_2}{h^3}(2\pi m_2 kT)^{3/2} \right\}^{1/2} e^{-\,1/2(\varepsilon_2-\varepsilon_1)/kT}, \tag{19} \]

where, as before, we shall henceforth denote

\[ k\theta_H=\frac{1}{2}(\varepsilon_2-\varepsilon_1). \tag{20} \]

The characteristic constant of the semiconductor \(\theta_H\) thus also turns out here to be equal to half the energy difference between the boundary of the upper continuum and the perturbing level. Relation (19) is quite unexpected; according to it, the number of excited electrons is proportional to the square root of the number of perturbing atoms; nevertheless formula (19) can raise no doubts.

The magnitude of the electrical conductivity can be found in the same way as before, only now one must take into account exclusively the electrons of the upper continuum, since the tightly bound perturbing atoms cannot participate in the formation of the current. According to (14b) and (19) we obtain:

\[ \chi=\chi_{\mathrm{II}} = e^2\frac{4}{3} \frac{n_{\mathrm{II}}l}{\sqrt{2\pi m_2 kT}} = \frac{4}{3}e^2l \left(\frac{n_0G_2}{h^3}\right)^{1/2} (2\pi m_2 kT)^{1/4} e^{-\theta_H/T}. \tag{21} \]

We find a result entirely analogous to (15-a), except that now, instead of the factor \(T\), \(T^{1/4}\) enters; such a difference, owing to the strong influence of the exponential function, should practically not show up in experiment at all. The mean free path \(l\) here will no longer be proportional to \(1/T\); it may rather be regarded as not depending on temperature at all, if the principal part of the resistance of the semiconductor is produced by the perturbing atoms themselves (see § 11).

* Neglecting 1 corresponds to the transition to the limiting case of Boltzmann statistics. We could, consequently, have obtained these results from the very beginning by using classical statistics.

As a result of the circumstance that the lower zone does not participate at all in the formation of the current, we obtain here a definite Hall effect, not equal to zero. Assuming that the electrons of the upper zone, in their properties, approach free electrons (which seems quite justified), we obtain for the Hall constant, according to Part III, § 7 (12) L., the following expression:

\[ R=\frac{3\pi}{8}\frac{1}{ecn_I} =\frac{3\pi}{8ec}\left\{\frac{h^3}{n_0G_2}(2\pi m_2 kT)^{-3/2}\right\}^{1/2}e^{\theta_H/T}, \tag{22} \]

which also makes it possible to estimate the number of perturbing atoms \(n_0\). Fort found, for the Cu\(_2\)O specimens he studied, a value of \(n_0\) of the order \(10^{17}\), and \(k\theta_H\) about \(0.3\) V (therefore, \(\varepsilon_2-\varepsilon_1 \simeq 0.6\) V).

Such an arrangement of the additional levels must be recognized as, to some extent, accidental. However, any other assumptions about these levels would not introduce many changes into the qualitative results. The calculation could be performed in a completely analogous way; moreover, in the first place here, too, there would be the calculation of \(\varepsilon_0\) from the equations of § 7 (2). After determining \(\varepsilon_0\), all the other quantities are computed simply. In doing so it would be inappropriate to assume that the additional levels are distributed throughout the whole forbidden region. In that case they would have to be located also in the immediate neighborhood of the edge of the \(\varepsilon_2\) zone, which would lead to a corresponding decrease in the magnitude of \(\theta_H\). Since, however, no exact statements can be made here, a more detailed investigation of the various possible hypotheses about the location of the levels would hardly be expedient.

The theory of semiconductors finds an important application in detection phenomena; however, here we are forced to confine ourselves only to citing some of the literature.\(^{28}\)

§ 9. Causes of Electrical Resistance

The preceding considerations constitute a formal theory of electrical processes in metals. In order actually to elucidate the mechanism of electrical conductivity, it is necessary to examine the question of the origin of the mean free path from the standpoint of model theory. We have already mentioned more than once that crystals with an ideal periodicity of the lattice, provided only that they belong at all to the class of conductors, must possess infinitely large electrical conductivity. Any deviations from the regular structure of the lattice will cause scattering of electron waves, and electrons accelerated by the action of an external field will experience braking. Therefore every violation of the regularity of the lattice will create resistance and, conversely, the presence of resistance always unambiguously indicates deviations from periodicity inside the crystal.

The following causes of violation of periodicity may be indicated.

1. Thermal motion of the lattice. Since thermal vibrations must depend essentially on temperature, the resistance caused by them must also possess a strong temperature dependence. This cause must play the predominant role for pure metals.

II. Irregularities in the structure of the crystal

These include:

a) distortions of the lattice caused by irregularities in individual microcrystallites; elastic deformations, etc.

b) inclusion of impurities in the lattice and the formation of alloys. If inclusions of foreign atoms are present, then such inclusions, first, produce a deformation of the lattice; secondly, they will cause around themselves local changes in the distribution of the potential even in those cases when they are uniformly distributed inside the lattice and form a solid solution. Disturbances of type II, having a purely static character, should give a resistance almost independent of temperature. The magnitude of the total resistance must be composed of these two parts.

In order to take quantitatively into account* effect I (to which alone we shall confine ourselves in the present paragraph), it is necessary to supplement the equations describing the state of the crystal by taking account of thermal vibrations. Since at not very high temperatures the principal role in the phenomena under consideration will be played by slow vibrations, we may use the basic ideas of Debye’s theory of heat capacity.** According to this latter theory, thermal vibrations may be regarded as a superposition of independent elastic normal vibrations of the crystal, the number of which is determined by the total number of degrees of freedom and is therefore equal to \(3G^3\).

Each normal vibration of the crystal may be represented in the following form:

\[ \mathbf{u}_{fj}(xyzt)=a_{fj}\mathbf{n}_{fj}e^{2\pi i\left(\frac{\mathbf{f}\mathbf{r}}{K}-\nu_f t\right)}; \qquad (\mathbf{f}\mathbf{r}=xf_x+yf_y+zf_z). \tag{1} \]

In accordance with the cyclic character of the boundary conditions for the electrons, it is expedient here to choose as a normal vibration of the crystal a traveling wave; \(\mathbf{f}\) is the wave vector of the acoustic wave. The index \(j\) indicates a definite polarization. To each value of the wave vector there correspond three waves: one longitudinal and two transverse. \(\mathbf{n}_{fj}\) denotes the unit vector in the direction of displacement (for longitudinal waves—perpendicular to the plane of the front \((\mathbf{f}\mathbf{r})=\mathrm{const}\), for transverse waves—lying in this plane). Finally, \(a_{fj}\) denote the amplitudes of the individual waves. These latter may be regarded as quantum-mechanical variables of our system. The energy of the oscillators is quantized, so that oscillator \(f\) may have only the energy

\[ \varepsilon_f=(N_f+1/2)h\nu_f, \]

* The theory was developed by Bloch (loc. cit.) and extended by Peierls (loc. cit.). The character of our exposition corresponds to Nordheim’s work (loc. cit.). The idea that thermal vibrations of ions are responsible for electrical resistance is not new and was first expressed already in the classical theory by Wien.²⁹ This idea was used in Sommerfeld’s theory by Houston.⁵⁰ The latter attempted to transfer to the theory of metals the basic formulas of the theory of X-ray scattering, which are replaced in the works cited above by others, consistently derived from quantum mechanics.

** See, for example, Müller-Pouillet, vol. III, 2, ch. VIII, § 3.

where \(N_i\) is an integer. In a state of thermodynamic equilibrium the relation

\[ N_i=\frac{1}{e^{\frac{h\nu_i}{kT}}-1}, \tag{2} \]

is fulfilled, the wave number and the frequency being connected by the relation

\[ \nu_i=\frac{c}{\lambda_i}=\frac{c}{k}\sqrt{f_x^2+f_y^2+f_z^2}. \tag{3} \]

Furthermore, the maximum value of \(\nu_i\) coincides with the Debye limiting frequency \(\nu_m\), which is equal to

\[ h\nu_m=k\theta_D, \tag{4} \]

where \(\theta_D\) is the characteristic temperature of the metal. If one neglects the dispersion of the acoustic waves under consideration (which has no meaning in view of the other approximations of the theory), then, as is known,

\[ k\theta_D=hc\left(\frac{3n}{4\pi}\right)^{1/3}, \tag{5} \]

where \(n\) is the number of atoms (lattice sites) in \(1\ \mathrm{cm}^3\) of the metal.

The calculation of the influence of thermal vibrations on the motion of electrons is rather complicated, and we shall therefore confine ourselves to a visual explanation of the results. A separate longitudinal wave produces in the crystal a change of density*. In the periodic structure thus formed, Bragg reflection of electron waves will occur. This process is quite analogous to the Compton effect, which we may represent as the scattering of electrons by light waves**. In terms of the corpuscular theory we could describe such a process as a collision of electrons with acoustic quanta. As a result of such a collision the state of the oscillator may also change. It is easy to show that the latter either loses or gains one vibrational quantum. In such processes the law of conservation of energy must be satisfied, i.e. it must be

\[ \varepsilon_k-\varepsilon_{k_1}=\pm h\nu. \tag{6} \]

In addition, the Bragg condition*** will also be satisfied

\[ \mathbf{k}-\mathbf{k}' + \mathbf{f}=0. \tag{7} \]

* Since transverse waves cause only the appearance of stresses, but not a change of density, it is evident that as a first approximation one should take into account only the interaction of longitudinal waves with electrons. An exact calculation fully confirms this conclusion.

** See, for example, Schrödinger’s work.\(^{31}\)

*** Formula (7) will be valid only if plane waves are chosen as the electron’s proper functions. If, however, the modulation of these waves in the crystal, i.e. the factor \(u_k\) § 2, (8), is taken into account, then such transitions also become possible for which

\[ \mathbf{k}-\mathbf{k}' + \mathbf{f}=l\mathbf{G}\quad (l\equiv(l_x,l_y,l_z)=0\pm1,\pm2\ldots). \tag{7a} \]

Paierls drew attention to this circumstance. Transitions with \(l\ne0\) are called “Umklapp processes,” since the condition \(l\ne0\), together with (6), denotes that the direction of the electrons changes almost by

Equation (6) and (7), for given initial and final states, completely determine both the wave vector and the frequency of the elastic vibration participating in the given quantum transition. Since \(h\nu_m = k\theta_D\) is much smaller than the energy of the absolute zero of the electron gas, the vectors \(\mathbf{k}\) and \(\mathbf{k}'\) will differ very little in magnitude (although the difference in direction may also be considerable). Nevertheless, in processes of this type we shall always have a definite, although very small, energy exchange between the electrons and the lattice.

To calculate the probability of such processes it is necessary to make a definite assumption about the character of the influence of thermal vibrations on the motion of electrons. It is obvious that this influence reduces to the displacement of ions by an elastic wave and, thereby, to a change in the potential field in which the electrons move. The unperturbed (by thermal vibrations) potential we expressed in the form § 3 (1)

\[ U=\sum_{\mathbf n} F_{\mathbf n}(\mathbf r-a\mathbf n). \tag{8} \]

The thermal displacement of any ion can be defined [see (1)] by the vector

\[ \mathbf u=\sum_{fj}\mathbf u_{fj}. \tag{9} \]

Therefore the perturbed potential can be represented in the following form:

\[ U=\sum_{\mathbf n} F_{\mathbf n}(\mathbf r-a\mathbf n-\mathbf u_{\mathbf n}), \tag{10} \]

where \(\mathbf u_{\mathbf n}\) is determined by the value of expression (9) at the position of the corresponding ion.* Since in the preceding calculations we took (8) as the unperturbed (thermal vibrations not taken into account) potential, the perturbing potential of our problem will be represented by the difference between (10) and (8). Assuming that the amplitude of thermal vibrations is small in comparison with the distance between

directly opposite. At high temperatures transfer processes will occur more rarely than normal ones (for which \(l=0\)), and therefore their significance for the temperature-dependent resistance is small. At low temperatures, however, they may play a significant role (cf. § 10).

* In expressing the potential by relation (8), we compose it from fields created by the individual ions. In adopting expression (10) for the perturbed potential, we make the hypothesis that in thermal motion the ions are displaced as wholes; the change of the resultant field may then be represented simply as the displacement of the fields of the individual ions (Nordheim). It would be possible, however, to introduce the idea of deformed ions and to consider that elastic waves produce some continuous perturbation of the resultant potential. Then, instead of (10), we would have

\[ U=\sum_{\mathbf n} F[\mathbf r-a\mathbf n-\mathbf u(\mathbf r)], \tag{10a} \]

where \(\mathbf u(\mathbf r)\) is a continuous function (Bloch). A priori it is difficult to make the correct choice between these two hypotheses; probably in reality, as is always the case, there is an intermediate situation. In what follows it will be seen that the choice of either hypothesis is not essential for the general conclusion, since both lead to identical qualitative results. Nevertheless, it still seems to the author that the formulas given in the text should better correspond to reality, since the principal part of the lattice field must coincide with the purely electrostatic field of the ions.

atoms, we can expand (10) in a Taylor series; then for the interaction energy (omitting the factor \(e\)) we find

\[ U=\sum_n \{F_n(\mathbf r-a\mathbf n-\mathbf u_n)-F(\mathbf r-a\mathbf n)\}\cong \sum_n [\mathbf u_n \operatorname{grad} F(\mathbf r-a\mathbf n)]. \tag{11} \]

After this everything else is readily calculated. The transition probabilities will still depend on the state of the oscillators (on the number of acoustic quanta, determined by formula (7)). Namely, the probability \(W^+\) of a process in which emission of an acoustic quantum takes place will be proportional to \(N+1\), while the probability \(W^-\) of an absorption process will be proportional to \(N\). The ratio of these probabilities will be equal to

\[ \frac{W^+}{W^-}=\frac{N+1}{N}=e^{\frac{h\nu}{kT}}=e^{(\varepsilon-\varepsilon')/kT} \tag{12} \]

in full agreement with the relation of Part III, § 3 (13). At sufficiently high temperature (\(kT>h\nu_m=k\theta_D\)) this ratio will be close to unity and \(W^+\cong W^-\). In this case energy exchange may be disregarded altogether, since the mean energy of an oscillator will be considerably greater than an individual quantum, and the act of absorption or emission will not noticeably affect the state of the oscillator. Under this condition we again arrive at the formal theory of Part III. All general conclusions of this theory remain completely valid; in particular, the correct Wiedemann–Franz law remains, and the electrical conductivity is found proportional to \(T^{-1}\), in agreement with experimental data. We note that the method of Part III loses force when \(T\leq \theta_D\).

The exact expression for the magnitude of the free path is obtained in the following form (see Nordheim, loc. cit.):

\[ \frac{1}{l}=\frac{\pi^3}{h^4c^6de^2}\,kT\int_0^{k\theta_D}G^6|H|^2x^3\,dx. \tag{13} \]

From this we find, using Part III, § 5 (1), (3), the value of the resistance equal to

\[ \rho=\frac{3\pi^2}{K16e^2h_{fl}*c^6de_0^3}\,kT\int_0^{k\theta_D}G^6|H|^2x^3\,dx. \tag{14} \]

Here \(c\) is the velocity of sound, \(d\) the density of the metal, \(\varepsilon_0\) the zero-point energy. \(\theta_D\) denotes, as before, the characteristic Debye temperature, \(x=h\nu\) the energy of a lattice vibration. \(G\) is equal to the number of atoms fitting on the edge of the crystal. In order to obtain the specific resistance, one must put \(K=1\ \mathrm{cm}\). Finally, \(H\) is the matrix element, the exact expression for which is

\[ H_{kk'}=F_{kk'}+\frac{Ch^2}{4\pi^2m}, \tag{15} \]

where

\[ F_{kk'}=\int_{K^3}\bar{\psi}_{k'}F\psi_k\,dV, \tag{16a} \]

\[ C=\int_{a^3}\left|\frac{\partial u}{\partial x}\right|^2\,dV. \tag{16b} \]

\(m\) already denotes the true mass of the electron. In the expression \(F\) (16a) there enters the complete eigenfunction corresponding to the potential of an individual ion; in the expression for \(C\) (16b), however, only the modulating factor \(u\) of the eigenfunction enters. The integration in (16b) extends over the elementary cell \(a^3\). Expression (13) shows that the magnitude of the mean free path depends essentially on the energy of the electrons, namely, is proportional to \(\varepsilon^2\).

The most important result of the calculation is the explanation of the proportionality of \(\rho\) and \(T\). One might, however, try to estimate, with the aid of (14), (15), and (16), also the absolute magnitude of the resistance. For this it would be necessary to make more detailed assumptions about the character of the potential field [for the calculation of the matrix element (15)]. However, in the present state of the theory such an estimate can be made only very roughly; therefore its results cannot be regarded as very conclusive, although the order of magnitude of the resistance does come out correctly*.

If one chooses for \(F\) the expression for the potential of a screened ion**

\[ F = \frac{z e^2}{r} e^{-\frac{r}{b}}, \tag{17} \]

and considers plane waves as eigenfunctions (moreover putting \(m^*\) equal to the true mass of the electron), then from the empirical value of the resistance the screening radius \(b\) can be determined; this quantity alone enters as the unknown in equation (16). For the alkali metals \(b\) turns out to be approximately equal to half the distance between atoms in the lattice. Thus the order of magnitude proves quite correct, and this exhausts all that can be demanded of the theory. We note that precisely for the alkalis one may most probably expect that our assumption of almost free electrons will correspond to reality.

For other good conductors, for Cu, Ag, Au, the calculation under the same assumptions leads to too large a value of the electrical conductivity, approximately 10 times greater than the true one. Since the principal simplifications of the calculation amounted to neglecting the bonds of the electrons inside the metal, such a result is not unexpected. A quantitative estimate shows that even a very small modulation of the eigenfunctions [in the sense of § 2 (8)] is sufficient to introduce a factor of 10. It is quite understandable that for the alkalis such a modulation could be neglected. For other metals, however, its influence is noticeably manifested in a lowering of the electrical conductivity.

* For a more detailed discussion of the question see Nordheim (loc. cit.). The earlier data of Houston, which at one time seemed more accurate, must now be recognized as unreliable.

** \(ze\) is the charge of the ion, with \(z\) approximately equal to the number of valence electrons per atom; \(b\) is the screening radius, by means of which we take into account the screening influence of the smeared-out charge of the conduction electrons.

§ 10. Low Temperatures

Whereas for high temperatures the theory can be developed comparatively simply, in the case of low temperatures extraordinarily great difficulties arise. One of the principal difficulties consists in the fact that one can no longer put \(W^{+}=W^{-}\) (i.e., neglect energy exchange). In this case it becomes necessary to use the full expression of Part III, § 3 (14), for the number of collisions, and the integral equation for the distribution function can no longer be reduced to the ordinary one.

In exactly the same way, the law of distribution of the energy of elastic vibrations [§ 9 (2)] here ceases to be valid. This circumstance was pointed out by Peierls. Indeed, in applying the law § 9 (2), we assume that the vibrations of the lattice are at all times in a state of thermal equilibrium. But the establishment of an equilibrium state is due to two principal processes. First, to collisions with electrons, which cause definite quantum transitions in the lattice and thereby ultimately bring the lattice into an equilibrium state. Secondly, the presence of a definite coupling between the individual elastic waves, which we have not taken into account at all, may play the same role; such a coupling may arise, for example, as a consequence of the anharmonicity of the waves and may occur both in a metal and in an insulator, determining a definite thermal resistance of the latter. Bloch’s hypothesis, which we tacitly used in the preceding paragraph, reduces to the assumption that such equalizing processes occur more rapidly than the electronic processes, or, in other words, that the law of distribution of the energy of thermal vibrations is not substantially changed by the presence of an electric current.

At high temperatures this hypothesis gives no particular grounds for doubt. Moreover, since the Wiedemann–Franz law is obtained only by introducing this hypothesis, the latter may be regarded as experimentally confirmed. However, Peierls, proceeding from regularities relating to the thermal conductivity of insulators, showed that there must exist a temperature region in which the energy exchange between thermal vibrations is extraordinarily slowed down, and the condition indicated above changes into the directly opposite one. At present it is impossible to determine theoretically with any accuracy the magnitude of this temperature; however, according to Peierls it should be of the same order as the characteristic Debye temperature.

A rigorous theory could be constructed in the following way. In addition to the equation for the distribution function of the electronic states [Part III, § 3 (7)], one would have to introduce another one, determining the corresponding distribution law for the vibrations of the lattice, and start from a system of two coupled integro-differential equations. In doing this, in addition to the processes examined in § 9, one would also have to take into account the processes arising as a consequence of the coupling between thermal vibrations. The character of the pro-

processes of the latter type could be determined, at least in principle, from data on the thermal conductivity of insulators.^32 Such a theory, however, will be hopelessly complicated and therefore will not be able to yield any concrete results. At present only some limiting cases have been analyzed, and even for them the calculations prove to be very complicated. We shall confine ourselves to a brief presentation of some results.

Let us first consider the consequences following from the initial hypothesis of automatic equalization of the distribution. Bloch succeeded in obtaining for this case an approximate solution*, which leads to the following expression for the electrical resistance:

\[ \rho = \frac{3\pi^2}{K\,16\,e^2 h m^* c\,d\varepsilon_0^3}\, \frac{1}{kT} \int_0^{k\theta_D} \frac{G^6/H^2 x^5\,dx} {(e^{x/kT}-1)(1-e^{-x/kT})}. \tag{1} \]

Here the same notation is used as in the preceding paragraph. Expression (1) for high temperatures goes over into the formula § 9 (14), as indeed it should; for low temperatures the resistance proves to be proportional to \(T^5\). If it is assumed that the matrix element \(H\) does not depend on \(x\) (approximately this is actually so), then expression (1) takes the simple form

\[ \rho = C(kT)^5 \int_0^{\theta_D/T} \frac{z^5\,dz}{(e^z-1)\cdot(1-e^{-z})}, \tag{2} \]

where \(C\) is a material constant. According to Grüneisen’s measurements^33 relation (2) is well justified experimentally, giving for pure metals at low temperatures precisely a proportionality of the resistance to the fifth power of the absolute temperature.

The calculation of the thermal conductivity and of the thermoelectric effects proves to be considerably more complicated and has not so far been carried through to numerical results. However, Peierls and Brillouin nevertheless succeeded in establishing that the thermal conductivity (due to the electrons) must be proportional to \(T^{-2}\), which agrees very well with the experimental data. As for the Wiedemann–Franz law, the theory shows that at low temperatures this law must in no case be fulfilled. Thus both from the theoretical point of view and on the basis of experimental data the applicability of the Wiedemann–Franz law is limited only to the region of sufficiently high temperatures \(T>\theta_D\).

The results to which Bloch’s theory leads are in good agreement with experiment. However, Peierls pointed out that the basic conceptions

* See Bloch’s paper (l. c.) (II), and also Nordheim’s papers (l. c.). The integral equation is analyzed in great detail by Brillouin (l. c.). It is also shown there that expression (2) gives an approximation valid for any temperatures, whereas Bloch himself found it at first only for the two limiting cases—very low and very high temperatures.

of this theory must be regarded as more than doubtful. Therefore Peierls investigated another limiting case, under the assumption that energy exchange due to the coupling between thermal vibrations may be altogether neglected and that only the processes described in § 9 are included. In this case a curious result is obtained. If one restricts oneself only to normal collisions of § 9 [for which (7) with \(l=0\) is valid], then it turns out that the resulting wave number

\[ J=\sum k+\sum f \tag{3} \]

of the system (of the electrons and the lattice) remains constant all the time (we note that, for free electrons, this would mean conservation of momentum). Since energy exchange between the thermal vibrations is neglected, relation (3) is one of the invariants of the equation of motion, on which the distribution function must depend in a definite way.* If we now imagine the presence of an electric field which accelerates the electrons and thereby increases their wave numbers, it will turn out that the energy accumulated by the electrons cannot be transferred to the remaining parts of the system (the lattice). In this case a stationary state would be altogether impossible. Such a stationary state becomes possible again, however, if umklapp processes are taken into account. Within the framework of the Peierls theory under consideration, these processes must, consequently, necessarily be taken into account even in the case when their probability is negligibly small in comparison with normal collisions. The calculation proves in this case to be very complicated and cannot be carried out with the same precision as in Bloch’s theory. Nevertheless, Peierls was still able to show that this hypothesis (which neglects the direct interaction of the elastic vibrations) again leads, for low temperatures, to a resistance proportional to \(T^5\).

For high temperatures Bloch’s limiting case will correspond more closely to reality. The circumstance that the hypotheses considered lead, at low temperatures, to one and the same temperature dependence is extremely important, since it explains why relation (2) is in agreement with experiment. However, we can no longer expect formula (2) to have universal applicability for all metals, as follows from Lox’s theory and as is indeed observed experimentally for certain good conductors.

It may not be superfluous to explain vividly for what reasons such a rapid decrease of the resistance occurs at low temperatures.

The weakening of the current occurs as a result of the transfer of energy from the electrons to the crystal lattice. It is immediately clear that the processes of absorption of energy by electrons (absorption of an acoustic quantum) must decrease very rapidly with temperature, since the energy of the elastic waves then rapidly decreases. The energy of the absolute zero of the lattice plays no role here, since it cannot be transferred to the electrons. At first sight it might seem that the reverse processes (emission of an acoustic quantum)

* Namely, in statistical equilibrium the distribution function will be:

\[ f=f_0+\alpha k\,\frac{\partial f_0}{\partial \varepsilon};\quad N=N_0+\alpha f\,\frac{\partial N_0}{\partial (h\nu)}, \tag{4} \]

where \(f_0\) is the Fermi distribution, \(N_0\) the Debye distribution, and \(\alpha\) an arbitrary constant.

should not thereby be weakened at all, since the energy supplied to the electrons (by the external field) does not depend on the temperature. In reality, however, these processes too will be extremely weakened owing to the influence of the Fermi factor in the expression for the number of collisions (in Part III, § 3, this factor played no role, since we neglected energy exchange). Indeed, if, for example, at \(T=0\) there is no current, the electrons fill a certain region in phase space with the maximum possible density, determined by the Pauli principle. When an external field is applied, this distribution (in phase space) will begin to move as a whole, and the density of the distribution will remain constant at all times. Let us note that this result is valid both in classical mechanics (Liouville’s theorem) and in quantum mechanics [§ 2 (16)]. Thus in this case there will be no free places at all which the electrons could occupy after emitting an acoustic quantum, and, consequently, the electrical conductivity at \(T=0\) becomes infinitely large. This result is of great generality and does not depend at all on any special hypotheses.* To obtain the exact form of the law (for example \(T^5\)), a quantitative analysis of the transition processes is, of course, necessary.

Despite the fact that the theory gives the correct law for the decrease of resistance with temperature and leads at \(T=0\) to an infinite electrical conductivity of pure metals, it is completely unable to explain the phenomenon of superconductivity. Despite various refinements in the calculations, with the chosen initial assumptions it proves impossible to explain the presence of a sharp jump in electrical conductivity and the absence of the so-called residual resistance (§ 11). However, although the mechanism of superconductivity is at present entirely unclear, there are nevertheless some grounds for believing that this phenomenon is due to the joint action of a large number of electrons. The presence of a perfectly definite transition temperature must be regarded as a certain collective effect caused by the complete configuration of all the electrons. For a correct understanding of this effect, the interaction between electrons, which we have throughout systematically neglected, must evidently be of essential importance in the first place. Therefore the impotence of the theory in the present question should not seem especially surprising.

§ 11. Matthiessen’s Rule. Alloys

We shall now consider the influence of irregularities in the structure of the crystal on the resistance** (cause II, § 9).

* This result will not be valid, of course, for semiconductors, since in them [§ 8 (15)] at \(T=0\) there will be no conduction electrons at all.

** More exact calculations may be found in Nordheim’s work.

In the expression for the potential, § 9 (8), we can take into account the presence of an irregularity in the structure of the lattice by shifting each of the components of the potential \(F(\mathbf r-a\mathbf n)\) by some vector \(\Delta_{\mathbf n}\) (components \(\delta x_n,\delta y_n,\delta z_n\)). If there are foreign inclusions, or if the substance is an alloy, different ions will be located at different lattice sites, and the potential \(F\) will itself be a function of the number of the lattice site. To take this circumstance into account we shall assign the index \(n\) to \(F\). Thus, instead of § 9 (10) we shall start from the following expression for the perturbing potential:

\[ U=\sum_{\mathbf n}\left\{F(\mathbf r-a\mathbf n-\mathbf u_{\mathbf n}-\Delta_{\mathbf n}) -\widetilde F(\mathbf r-a\mathbf n)\right\}, \tag{1} \]

where \(\widetilde F\) denotes the averaged periodic potential. If the displacements \(\Delta_{\mathbf n}\) are small compared with the lattice constant \(a\), we may use expansion in a series, and the preceding expression may be represented in the form

\[ U=\sum_{\mathbf n}\left\{[F_n-\widetilde F_n-\Delta_{\mathbf n}\operatorname{grad}F_n] -\mathbf u_{\mathbf n}\operatorname{grad}F_n\right\}, \tag{2} \]

where in each term of the sum \(F\) is a function of the argument \((\mathbf r-a\mathbf n)\). The difference of this expression from the formula § 9 (11) consists in the fact that here there appears an additive term, marked by square brackets*.

In this case it turns out that the perturbation energy splits into two parts, one of which does not depend at all on the amplitude of the thermal vibrations, \(q\). This shows that in processes associated with irregularity of structure, the influence of thermal oscillators does not appear. In terms of quantum mechanics this result reduces to the assertion that the transition probabilities corresponding to the two terms of the sum (2) are independent of one another.

The presence of thermal vibrations gives rise to processes which are always accompanied by energy exchange. Conversely, irregularities in the structure of the lattice lead only to purely elastic scattering, in which only the direction of motion of the electrons changes, but not their energy. The influence of such processes will show itself in the appearance of an additional additive term in the basic equation, which will have the simple form of Part III, § 3 (15); this follows from the circumstance that in scattering processes the energy of the electrons remains constant.

This means that also for collisions of the type under consideration one may introduce, after the manner of Part III, § 4 (8), a definite mean free path \(l_{\mathrm{II}}\), and for the resulting mean free path one obtains the simple relation

\[ \frac{1}{l}=\frac{1}{l_{\mathrm I}}+\frac{1}{l_{\mathrm{II}}}. \tag{3} \]

Since the resistance of a conductor [Part III, § 5 (11)] is proportional to \(\dfrac{1}{l}\), the last result shows that the resulting

\[ \text{* Strictly speaking, the second term in expression (2) does not coincide with § 9 (8),} \]
since for different \(n\) the function \(F_n\) will be different. However, it can be shown that the influence of this circumstance on the final results may be neglected (see Nordheim, loc. cit., § 9). The definition of the averaged potential \(\widetilde F\) will be clear from what follows.

resistance is additively composed of the resistance caused by thermal oscillations and the resistance due to irregularities of the lattice. Taking into account that the mean free path \(l_{II}\) does not depend on temperature, we arrive at a theoretical justification of Matthiessen’s rule, according to which there is always superposed on the ordinary resistance of a pure metal some additional constant resistance, whose magnitude is different for different metals and different specimens. This rule will be valid so long as one may confine oneself to the first term of the expansion in (2).

Expression (2) depends on the distribution of the displacements \(\Delta_n\) and on the arrangement of foreign inclusions. Therefore, for further calculation it is necessary to prescribe some special form of this distribution. We shall make the natural assumption that the distribution of the scattering centers is completely random. Indeed, this distribution will vary from place to place; however, in macroscopic measurements with a large crystal we shall always find the result of some average distribution.

This average value can be easily found if the individual fluctuating quantities are statistically independent, i.e., if the probability of a definite value for the site \(n+1\) does not depend at all on the corresponding value for the site \(n\).

In what follows we shall restrict ourselves to the case of the inclusion of foreign atoms (for example, in the formation of an alloy), and in the perturbation energy we shall take into account only the terms

\[ \sum_{n}\left[F_n(r-an)-\widetilde{F}(r-an)\right]. \]

The sum written extends over all lattice sites, and each of its terms gives the deviation of the true value of the potential at some site from its averaged value \(\widetilde{F}\). According to the assumption of randomness of the distribution, the values of the potential at different lattice sites are independent of one another, and therefore the separate sites may be regarded by us as independent scattering centers.

In the presence of a single partial potential field \(F(r)\), the probability of scattering of an electron (transition from the state \(k\) to the state \(k'\)) is proportional to the square of the matrix element

\[ |F_{kk'}|^2=\left|\int F(r)\psi_k\psi_{k'}^{*}\,dV\right|^2 . \tag{4} \]

If the matrix element corresponding to \(F(r-an)\) is denoted simply by \(F_n\), then the total probability of transition may be represented in the form

\[ W_{kk'}\sim \sum_{n}\left|F_n-\widetilde{F}\right|^2, \tag{5} \]

Now we shall take into account that, in the scattering of a plane wave, the result is completely independent of the place where the scattering center is located. This means that the functions \(F_n\) do not depend on the coordinates, but are determined only by the properties (kind) of the ion sitting at the site. This circumstance greatly simplifies the summation (5).

We shall assume that our substance contains \(l\) different kinds of atoms with relative concentrations \(p_1, p_2,\ldots\). The total number of atoms of kind \(s\) will therefore be \(N_s=G^3p_s\) (\(G^3=N\) is the total number of all atoms), with

\[ \sum_s p_s=1. \tag{6} \]

The mean value of any quantity \(X\) will be

\[ \widetilde{X}=\sum_s p_s X_s, \tag{7} \]

where \(X_s\) is its value for atoms of species \(s\). Denoting the corresponding matrix element by \(F_s\), we find, instead of (5):

\[ W_{\mathbf{k}\mathbf{k}'} \sim \sum_s N_s \left|F_s-\widetilde{F}\right|^2 =G^3\sum_s p_s\left|F_s-\widetilde{F}\right|^2 . \]

Taking into account the definition of the mean value of any quantity, we can write the last expression also in the following form:

\[ W_{\mathbf{k}\mathbf{k}'} \sim G^3\,\overline{\left|F_s-\widetilde{F}\right|^2} = G^3\left\{\overline{|F|^2}+|\widetilde{F}|^2-2|\widetilde{F}|^2\right\} = \]

\[ =G^3\left\{\overline{|F|^2}-|\widetilde{F}|^2\right\}, \tag{8} \]

i.e. in the form of the usual fluctuation formula, which occurs, for example, in the Rayleigh theory of light scattering for fluctuations of the gas density. Finally applying relations (6) and (7), we find ultimately*:

\[ |W_{\mathbf{k}\mathbf{k}'}|L = G^3\,\frac{1}{2}\sum_{st}p_s p_t\,|F_s-F_t|^2, \tag{9} \]

where

\[ F_s-F_t=\int (F_s-F_t)\,\psi_{\mathbf{k}}\overline{\psi}_{\mathbf{k}'}\,dV \tag{10} \]

is the matrix element corresponding to the difference of the potentials \(F_s\) and \(F_t\) for one and the same lattice site.

The results obtained admit a clear interpretation. Relation (9) shows that the action of the impurity will be as if, at each lattice site (in all \(G^3\) sites), there were scattering centers with scattering power

\[ \frac{1}{2}\sum_{s,t} p_s p_t\,|F_s-F_t|^2 . \]

We can characterize an individual scattering center by a certain effective cross section \(Q\), which is connected by the simple relation of the kinetic theory of gases with the mean free path \(l\):

\[ \frac{1}{l}=G^3Q. \tag{11} \]

For the effective cross section, the expansion will evidently also be valid

\[ Q=\frac{1}{2}\sum p_s p_t Q_{st}, \tag{12} \]

\[ \text{* Namely:} \]

\[ \frac{1}{2}\sum_{s,t}p_s p_t|F_s-F_t|^2 = \frac{1}{2}\sum p_s p_t \left\{|F_s|^2+|F_t|^2-F_s\overline{F}_t-\overline{F}_sF_t\right\} = \]

\[ =\sum_s p_s|F_s|^2-\sum_{s,t}p_s p_t F_s\overline{F}_t = \overline{|F|^2}-\widetilde{F}\,\overline{\widetilde{F}} . \]

where \(Q_{st}\) denotes the effective cross section for the case when there is a single impurity atom of species \(t\) in a lattice of type \(s\), or conversely.

THEORY OF THE METALLIC STATE

For the magnitude of the additional resistance caused by impurity atoms, we find, according to Part III, § 5 (3), the expression

\[ \rho_L=\frac{3h^3}{16\pi e^2 m \varepsilon_0}\,G^3\,\frac{1}{2}\sum_{s,t} p_s p_t Q_{st} = \]

\[ =\frac{h}{2e^3 z^{2/3}}\left(\frac{3n}{\pi}\right)^{1/3}\frac{1}{2}\sum p_s p_t Q_{st}, \tag{13} \]

in which, instead of \(\varepsilon_0\), its value from Part III, § 5 (2) has been substituted \((G^3=n;\ z\) is the number of free electrons per 1 atom). The quantum-mechanical calculation of \(Q_{st}\) can be carried out easily; it leads to the formula

\[ Q_{st}=\frac{32\pi^3 m^2}{h^4}K^6\int |F_s-F_t|^2(1-\cos\theta)\sin\theta\,d\theta, \tag{14} \]

from which it would be possible to calculate also the absolute magnitude \(Q_{st}\), if the matrix elements \(F\) were known. However, even without exact knowledge of \(Q_{st}\), formula (13) permits certain conclusions to be drawn about the dependence of the additional resistance on the impurity concentration and admits experimental verification.

Fig. 11. Dependence of the residual resistance of a solution of two metals on the concentration of the components.

Let us consider the simplest case of a mixture of two components with concentrations \(p_1=p\) and \(p_2=1-p\). In this case we find that the additional resistance is equal to

\[ \rho_L=\mathrm{const}\cdot\frac{1}{2}\,p(1-p). \tag{15} \]

It is symmetric with respect to both components (\(p=1\) corresponds to pure component 1, \(p=0\) to pure component 2). The magnitude of the additional resistance varies according to a parabolic law with the change in impurity concentration, and for small concentrations an approximately linear law is obtained, as was to be expected (Fig. 11).

The formulas derived will be valid only in the case when there is indeed a completely disordered distribution of impurity atoms and when the crystal lattice is in no other respect distorted.* For the case of solid solutions we have a good approximation to such conditions. In this sense the best example is an alloy of silver and gold. In this case the lattice constant of both metals is indeed the same, just as both kinds of atoms are, in their structure, homologues and easily replace one another; this is indicated, for example,

* If the two metals had different lattice constants, or if their atoms had different numbers of valence electrons, then deviations from formula (15) should be expected.

complete miscibility of both components at all concentrations. Likewise, as a result of the equality of the lattice constants for the object under consideration, no distortion of the lattice is to be expected. The magnitude of the additional resistance \(\rho_L\) can be simply determined from measurements at low temperatures. In this case the ordinary resistance of the metals is extremely small, so that the “residual resistance” can be simply identified with the quantity \(\rho_L\) and used directly for comparison with the theory.

For the case Ag—Au, no better agreement between theory and experiment could be desired. This is seen from the table below, which contains data* on the dependence of the additional resistance of Ag—Au on concentration:

\(p\)	\(0.01\)	\(0.025\)	\(0.316\)	\(0.629\)
\(\rho_L\) (exper.) . . . .	\(0.35\)	\(0.86\)	\(7.3\)	\(8.2\)
\(\rho_L\) (theor.) . . . .	\(0.35\)	\(0.88\)	\(7.6\)	\(8.2\)

\(p\) is the ratio of the number of silver atoms to the total number of atoms. The quantities \(\rho_L\) (theor.) were calculated from the value at \(p = 0.01\) with the aid of (15). The small discrepancies between the theoretical and experimental values are wholly covered by the unavoidable errors, which here are rather large, since in determining the residual resistance one has to extrapolate to \(T = 0\). In addition, it is rather difficult to eliminate definitively the influence of preliminary treatment, impurities, etc.

It should be noted that for pure Au the ordinary resistance at \(T = 273^\circ\mathrm{C}\) is \(\rho = 2.14 \cdot 10^{-6}\ \Omega\); for equal amounts of the components it turns out to be considerably smaller than the additional resistance. From the point of view of the classical electron theory such an effect proves to be completely incomprehensible. On the contrary, from the point of view of our interference ideas, one obtains not only the correct dependence on \(p\), but also the correct order of magnitude. Starting from the measured values of \(\rho_L\), one can calculate from (13) the effective cross section \(Q_{12}\). For the silver–gold system \((n = 5.9 \cdot 10^{22};\ z = 1)\) one obtains the value \(1.27 \cdot 10^{-16}\ \mathrm{cm}^2\), which in order of magnitude coincides with the cross section of an atom. If now the same quantity is estimated theoretically by means of (14) [calculating, for simplicity, \(F_s - F_t\) as a matrix element of the potential § 9, (16)], then one obtains a value precisely of order \(10^{-16}\ \mathrm{cm}^2\).

In those cases in which the incorporation of foreign atoms is accompanied by a distortion of the basic lattice, the parabolic curve (15)

* The experimental figures are borrowed from Grüneisen’s paper (Handb. d. Phys. 13, ch. 1). The resistance is given in \(10^{-6}\ \Omega\).

will be deformed. However, in this case as well, at small concentrations, when the impurity atoms are at large distances from one another, the increase of the resistance will follow a linear law.

At higher temperatures the ordinary resistance will begin to play a noticeable role. Since the latter will be different for the individual components, and since the presence of an impurity substantially affects the magnitude of the ordinary temperature resistance, the curve of the dependence of resistance on concentration will become asymmetric. The form of the curve for this case is shown in Fig. 12; the initial and final ordinates correspond to the resistance of the pure metals. In exactly the same way, the parabolic law (15) will be distorted under various kinds of preliminary treatment. In all these cases only the typical form of the curve is preserved; its analytic representation then becomes very difficult.

Fig. 12. Change of the resistance curve upon raising the temperature of the mixture.

Deviations of another kind occur in the case when the mixture is not ideal. With the aid of interference theory it is easy, qualitatively here as well, to explain the observed phenomena. If, instead of a solid solution, there is a mixture of crystallites of the pure components, and if, moreover, the sizes of the crystallites are larger than the mean free path of the electrons, then the resulting resistance is obtained by adding the partial resistances according to the ordinary rule of mixtures. In the general case the dependence will be as follows. At small concentrations the impurity atoms can still remain uniformly distributed over the main lattice. Under these conditions the increase of resistance with concentration will follow a parabolic law. At a certain concentration, however, crystallization of the impurity will begin, and the whole specimen will turn out to be a mixture of crystallites which will contain the maximum possible number of foreign atoms. In this case the parabolic law will pass into a straight-line dependence for a mechanical mixture,* as is shown in Fig. 11 (dotted line). The points of intersection of the straight line and the parabola determine the maximum concentrations of both components at which the existence of a solid solution is still possible.

The character of the change in resistance of a mixture of metals that are different (in their properties) can also be easily explained qualitatively. For this case there exists an empirical rule according to which the atomic increase of resistance is the greater, the smaller

* We do not consider those complications which arise as a consequence of the nonuniform distribution of current in an inhomogeneous conductor. The latter circumstance leads to deviations from the straight-line law.

affinity between the impurity and the base metal. Such a result follows directly from the interference theory as well, since the greater the difference in ionic potentials, the greater the scattering should be and, consequently, the resistance. In this same sense the distortion of the lattice will also act. For illustration let us give some data. When one atomic percent of the metals listed below is admixed, an increase in the specific resistance is observed:

\[ \begin{aligned} \mathrm{Ag}\ \text{in}\ \mathrm{Cu} &— 0.22 \cdot 10^{-6}\ \Omega,\\ \mathrm{Ag}\ \text{in}\ \mathrm{Au} &— 0.35 \cdot 10^{-6},\\ \mathrm{Au}\ \text{in}\ \mathrm{Cu} &— 0.62 \cdot 10^{-6}. \end{aligned} \]

As was to be expected, Au in Cu gives a larger additional resistance than Ag in Cu and Ag in Au.

Finally, let us note one more effect which may occur when the components of a mixture enter in a stoichiometric ratio. In this case the formation of a chemical compound is possible. When a compound is formed, an ordered arrangement of atoms is obtained, and we may expect that the specimen, in its electrical properties, will again approach the pure metals. And indeed, it has repeatedly been observed experimentally that upon the formation of such compounds the additional resistance drops sharply; moreover, a clearly expressed temperature dependence of the resistance always appears at the same time. In Fig. 13 the course of the resistance for this case is shown schematically.

Fig. 13. Course of the change in residual resistance upon formation of a chemical compound.

At

\[ p = \frac{1}{2} \]

there is a chemical compound; moreover, for the interval of values of \(p\) from 0 to \(\frac{1}{2}\) the presence of a mixture of microcrystallites is also assumed. Let us note that the magnitude of the temperature resistance of such a compound will by no means be determined by the mixing rule, and therefore the corresponding point for it will not fall on the straight line of the mixture even in the ideal case.

The experimental facts set forth here receive a natural explanation in the new theory of electrical resistance. From the point of view of the classical conceptions they turn out to be completely incomprehensible, since here we have typical interference effects. Therefore the phenomena analyzed prove to be especially good confirmation of the correctness of the new conceptions.

References

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