On the Electronic Theory of Metals*

About five years ago the electronic theory of metals finally emerged from the contradictions that had stifled it and that, even quite recently, had seemed irremediable. It is noteworthy that the sudden resolution of these contradictions was due not to any new physical conceptions of the nature of the metal (the physical picture of the metal remained the same), but merely to changed forms of calculation.

One may distinguish four stages in the development of our ideas about the metal:

1. Drude was the first to put forward the hypothesis of an electron gas. Richardson ascribed to this “gas” a Maxwellian velocity distribution. The development of this point of view led to a brilliant agreement between theory and experiment with regard to thermionic emission (the Richardson effect) and the Wiedemann–Franz law. However, difficulties immediately arose concerning the heat capacity of metals.

2. Subsequent works, aimed at overcoming these and certain other difficulties, all developed in approximately one and the same direction—

* Kronig and Penny, Quantum Mechanics of Electron in Crystal Lattices. Proceed. of Royal Soc., A. 130, p. 499, 1931.

fluence: in the direction of improving or even completely changing Drude’s model.

Let us note here the interesting work of J. J. Thomson, who considered a metal as a kind of dielectric which, when placed in an electric field, constitutes a system of oriented dipoles. The difference from a dielectric, according to this point of view, is only quantitative, not qualitative. Whereas in a dielectric the interaction between the two poles of a dipole (the electron and the positive ion of the corresponding atom) is manifested more strongly than between separate dipoles, in a metal just the opposite is observed: the electron forming the negative pole of a dipole is pulled over by the positive pole of a neighboring dipole. A chain of such stepwise moving electrons, according to Thomson, accounts for the electric current.

Another very interesting work of the same period, belonging to J. J. Utterman, transfers to the metal the hypothesis of the chemical dissociation of the atom into ion and electron and, studying the equilibrium conditions of such a “solution,” attempts to explain the temperature dependence of electrical conductivity.

There is no possibility here of enumerating all the models proposed at that time, associated with the names of Wien, Verweyde, Borelius, Bridgman, and others. Let us note only the common feature of all these works. All these works, to one degree or another, explained some one or, at best, several particular questions, proving powerless to explain their entire totality.

1. The way out of the situation consisted in returning to Drude’s model—modified, to be sure, but not with respect to its physical content, rather with respect to the mathematical methods of describing it. The free electrons of the metal were still treated as an ideal gas; however, this attractive analogy had to be broken off earlier than had been done in the initial development of Drude’s ideas. Instead of Maxwell–Boltzmann statistics, the new Fermi–Dirac statistics was applied to the electron gas.

This reform may be briefly characterized as follows.

Let us consider an electron gas. Let \(f(\varepsilon)\) be the distribution function, defined, as is known, by the relation

\[ \frac{N}{\delta n}=f(\varepsilon)\,d\varepsilon, \]

where \(\delta n\) is the number of electrons actually possessing energy \(\varepsilon\) (or, more precisely: an energy contained in the interval from \(\varepsilon\) to \(\varepsilon+d\varepsilon\)), and \(N\) is the total number of electrons capable of possessing such energy. According to classical statistics, this number \(N\) is the total number of electrons forming the gas. Indeed, according to classical conceptions, the number of gas particles corresponding to a given energy \(\varepsilon\), prin-

cipally by nothing except the total number of particles contained in the gas. In the expression

\[ \delta n = N f(\varepsilon)\, d\varepsilon \tag{1} \]

Fermi statistics replaced the Maxwellian distribution function (we consider electrons moving only in one direction)

\[ f(\varepsilon)=\frac{1}{\frac{1}{A} e^{\frac{\varepsilon}{kT}}} \]

by the expression

\[ f(\varepsilon)=\frac{1}{\frac{1}{A} e^{\frac{\varepsilon}{kT}}+1}, \]

which becomes Maxwellian in the case when \(A\) is small. The Pauli principle changed, in equality (1), the factor \(N\), which until then had the meaning of the total number of electrons constituting the gas and thus did not depend on \(\varepsilon\), and now made it a function of \(\varepsilon\). The Pauli principle is a restrictive principle, allowing not any number of electrons of our gas to be assigned to a given energy, but only a strictly definite number of them, the larger the greater the energy.

All the difficulties that the original Drude model encountered on its way now miraculously disappeared before the restored, but essentially still the same, model of the electron gas.

1. It is possible, of course, to regard the free electrons of a metal as an ideal electron gas only in the first approximation. However fruitful such an interpretation may be, it is still only a crude scheme. The further development of the theory curtailed still more the analogy between free electrons and a gas. It was necessary to take into account the ionic metallic lattice, that material “framework” which until then had been almost neglected, but which in fact is immersed in the electron gas and must, of course, influence its properties.

The fundamental work here belongs to Bloch. Bloch’s ideas are developed and made concrete by Kronig and Penney in a very interesting paper, to which the present abstract is devoted.

In order to make Bloch’s idea clear and natural, let us first consider an isolated atom of a metal: for simplicity let it be a monovalent metal, i.e., in the first approximation, a hydrogen-like atom. The potential energy of the valence electron depends on its distance from the nucleus; it increases as this distance increases (i.e., being negative, it decreases in absolute value, following, as is not hard to see, a hyperbolic law). The atom, therefore, may be represented as a “potential crater” formed by two hyperbolas (Fig. 1), having as asymptotes the positive and negative semiaxes \(OX\) and the negative semiaxis \(OU\). The position of the nucleus corresponds to \(x=0\). Immersed at the bottom of this “crater” are

atomic electrons, which can be raised only at the expense of energy absorbed from outside. In its normal state such an atom constitutes a stable system, and one may be sure that no electron, once it has descended to the bottom of the crater, will leave it again, at least until the atom is subjected to an external influence.

Fig. 1.

Let us now imagine an entire chain of atoms completely similar to one another, situated at a distance \(a\) from one another. Such a chain will be for us nothing other than a one-dimensional model of a crystal lattice. Fig. 1 must now be replaced by Fig. 2. The positions of the nuclei of the individual atoms making up the chain correspond to the points \(x = 0, \pm a, \pm 2a, \pm 3a, \ldots\). From Fig. 2 it is evident that neighboring atoms are separated from one another by potential barriers, whose height, generally speaking, is the smaller the more closely the atoms in our chain are arranged, i.e., the smaller the lattice constant \(a\).

Fig. 2.

In order that an electron belonging to some one of the atoms should pass into a neighboring atom, according to classical views it is necessary that its energy exceed the height of the potential barrier. According to wave ideas, however, a transition into a neighboring atom is possible not only over the barrier, but also through the barrier; i.e., this transition, or, more precisely, “leakage,” is also possible for an electron possessing an energy smaller than the height of the barrier. Thus the bringing together of atoms leads to the “socialization” of electrons that up to this point belonged inseparably to each of them. Here one already feels a step in the direction of the hypothesis of an electron gas. However, in considering “free” electrons as those which leak from one atom to a neighboring one and thus wander inside the metal, perhaps forever having lost their native atom, we are still very far from the analogy with a gas on which Drude’s model is built. Indeed, Bloch electrons move not in free space, like Drude electrons, but in a space divided by potential partitions.

Bloch’s idea consists in the fact that he takes into account the influence of the crystal lattice on the motion of “free” electrons by replacing the lattice with a field that varies periodically in space. Thus, the potential energy of an electron moving, for example, along the \(OX\) axis (Fig. 2) is a periodic function of the distance \(x\) with period \(a\), as is immediately clear from Fig. 2.

Our task is to describe the behavior of such an electron, whose potential energy is a periodic function of \(x\) (we confine ourselves to a one-dimensional model). In the language of wave mechanics, the fruitful spirit of which permeates all modern metallophysics, this means: to find the eigenfunctions \(\psi_k(x)\) (Eigenfunktionen) and the eigenvalues \(W_k\) (Eigenwerte).

The first step toward determining \(\psi(x)\) can be made on the basis of purely physical considerations. Assuming that all the atoms making up the lattice are completely identical, we must suppose that the probability of finding the electron at one or another place in the lattice, while generally different, is nevertheless the same at distances that are multiples of \(a\) (one-dimensional model!). But since this probability is proportional to the product \(\psi\psi^*\), where the asterisk denotes the complex conjugate quantity, we obtain the condition:

\[ \psi(x)\psi^*(x)=\psi(x+na)\psi^*(x+na), \tag{2} \]

where \(n\) is any positive or negative integer. Let us note here that from the periodicity of the product \(\psi\psi^*\), of course, it is still by no means possible to draw a conclusion about the periodicity of \(\psi\).

Condition (2) can be satisfied if we put

\[ \psi(x)=u(x)e^{i\alpha x}, \tag{3} \]

where \(\alpha\) is an arbitrary real parameter, and \(u(x)\) is some periodic function with period \(a\):

\[ u(x)=u(x+na). \]

We shall not dwell on the proof that solution (3) is not only possible but also the unique one satisfying condition (2). As for the parameter \(\alpha\), which will play an essential role for us, its arbitrariness is somewhat restricted if, following Bloch, we consider a metal of length \(L\) (\(L=Ga\), where \(G\) is some integer, and \(a\) is still the lattice constant) and, in addition to condition (2), introduce the further condition:

\[ \psi(x)=\psi(x+L), \tag{4} \]

which expresses the equality of the wave function \(\psi\) at the boundaries of the metal, which is natural. Substituting (3) into (4), we obtain:

\[ e^{i\alpha x}=e^{i\alpha(x+L)}, \]

i.e.

\[ \alpha L=2\pi k, \]

whence

\[ \alpha=\frac{2\pi k}{ga}. \]

Thus \(\alpha\) proves to be a function of the integer variable \(k\), and consequently the possible values of \(\alpha\) form a discrete series. To each \(\alpha\) there corresponds its own \(\psi_k\) and \(u_k\). We shall not undertake to prove that the integer variable \(k\) is bounded by the limits

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

and that, accordingly, \(\alpha\) assumes in all only \(G\) different values. Only in the case of an unbounded (infinitely long) crystal does \(\alpha\) acquire the possibility of running through an innumerable series of values.

However, we cannot, of course, be satisfied with solution (3), which contains a certain function \(u(x)\) about which nothing is known except that this function is periodic. Substituting (3) into the Schrödinger equation

\[ \frac{d^2\psi}{dx^2}+\frac{8\pi^2m}{h^2}\,[\,W-V(x)\,]\psi=0 \tag{6} \]

[\(V(x)\) is a periodic function with period \(a\)], which, of course, (3) must satisfy, we obtain a differential equation for \(u(x)\), the solution of which, however, is associated with insurmountable mathematical difficulties.

Fig. 3.

Here we come close to the work of Kronig and Penney.

Kronig and Penney make use of an artificial method, usually adopted in solving the Schrödinger equation when its direct solution proves too difficult—the method of “potential jumps.”

Let us replace the continuous periodic function \(V(x)\) by the saw-toothed discontinuous curve shown in Fig. 3.

Then equation (6) must be replaced by two equations:

\[ \frac{d^2\psi}{dx^2}+\frac{8\pi^2m}{h^2}\,W\psi=0, \tag{6a} \]

for

\[ n(a+b)\leq x\leq n(a+b)+a, \]

where \(n\) is any integer, and

\[ \frac{d^2\psi}{dx^2}+\frac{8\pi^2m}{h^2}\,(W-V_0)\psi=0 \tag{6b} \]

for

\[ n(a+b)+a\leq x\leq (n+1)(a+b); \]

substituting \(\psi\) from (3) into (6b), we obtain for \(u(x)\):

\[ u^{(1)}=Ae^{(-i\alpha+\gamma)x}+Be^{(-i\alpha-\gamma)x}, \tag{7a} \]

and, substituting the same expression (3) into (6a), we obtain:

\[ u^{(2)}=Ce^{i(-\alpha+\beta)x}+De^{i(-\alpha-\beta)x}, \tag{7b} \]

where:

\[ \beta^2=\frac{8\pi^2 m}{h^2}W \quad\text{and}\quad \gamma^2=\frac{8\pi^2 m}{h^2}(V_o-W). \]

However, since the function \(u(x)\) must necessarily be continuous, we have to assume that at the points of discontinuity \(x=0\) and \(x=a\) (we restrict ourselves to one period) not only the functions \(u^{(1)}\) and \(u^{(2)}\) are equal, but also their first derivatives:

\[ \left. \begin{aligned} u^{(1)}_{x=0}&=u^{(2)}_{x=0}\\ u^{(1)}_{x=a}&=u^{(2)}_{x=a}\\ \left(\frac{du^{(1)}}{dx}\right)_{x=0} &=\left(\frac{du^{(2)}}{dx}\right)_{x=0}\\ \left(\frac{du^{(1)}}{dx}\right)_{x=a} &=\left(\frac{du^{(2)}}{dx}\right)_{x=a}. \end{aligned} \right\} \tag{8} \]

These four equations constitute a system from which the constants of integration \(A\), \(B\), \(C\), and \(D\) can be determined. Carrying out some simple calculations, one can verify that the condition for the compatibility of these equations is given by the expression:

\[ \frac{\gamma^2-\beta^2}{2\beta\gamma}\,\operatorname{Snh}\gamma b\,\sin\beta a+\operatorname{Csh}\gamma b\,\cos\beta a =\cos\alpha(a+b). \tag{9} \]

The whole ingenuity of the work of Kronig and Penney consists in the fact that, having reached this point by the same generally accepted paths which we too have followed schematically, they pass to the limit, assuming that the potential barriers by which the entire crystal is partitioned become infinitely high \((V_o\to\infty)\), but at the same time infinitely thin \((b\to0)\), and in such a way that the area \(bV_o\) under this “deformation” remains unchanged. Denoting by the letter \(P\) a quantity proportional to this area and characteristic for each metal,

\[ P=\lim_{\substack{b\to0\\ V_o\to\infty}}\frac{\gamma^2ab}{2} \]

and passing to the limit, we obtain instead of (9):

\[ \frac{P}{\beta a}\sin\beta a+\cos\beta a=\cos\alpha a. \tag{9a} \]

It is not difficult to see that the quantity \(P\) characterizes the degree of “transparency” of the barriers with respect to the electrons, or, in other words, the degree of binding of the electrons inside the atom. We have two limiting cases: if \(P=0\), then before us is a completely free electron gas, the kind with which Drude originally dealt; in this case

potential barriers are smoothed out, their role reduced to zero; the other limiting case \(P=\infty\) corresponds to complete impenetrability of the barriers; electrons are thus locked in separate cells of the lattice, and their motions are confined within narrow limits.

We can speak of the “height” and “thickness” of the barriers so long as we are dealing with a one-dimensional model. However, this geometric visualization, unfortunately, disappears as soon as we pass to the three-dimensional case: for the continuation of the same geometric interpretation we no longer have enough three dimensions here. We must imagine the metal as partitioned by “potential” walls—something resembling a honeycomb—walls that are permeable to electrons to one degree or another.

Fig. 4.

Let us turn to equation (9a). Its left-hand side is shown in Fig. 4 as a function of \(\beta\). From an analysis of this damped curve we can obtain the physical characteristic of the metal. The values of \(\beta\) satisfying equation (9a) will be obtained as projections onto the \(a\beta\) axis of the points of intersection of our damped curve with straight lines parallel to the \(a\beta\) axis and drawn at a distance equal to \(\cos \alpha a\) from it. There will be as many such straight lines as there are possible values of \(\alpha\), i.e., an infinite number in the case of an unbounded crystal and \(G\) in the case of a crystal of length \(L=Ga\). In both cases the possible straight lines will lie within the limits \(+1\) and \(-1\), and since the maxima and minima of our damped curve lie outside these limits, the admissible values of \(\beta\) will form separate “zones” (indicated in Fig. 4 by bold segments), separated from one another by “gaps” (dotted lines). Within each zone the values of \(\beta\) of interest to us form a discrete series, the denser the longer the crystal, and a continuous infinity if the crystal is unbounded. The totality of all these admissible \(\beta\)’s [i.e., satisfying equation (9a)] constitutes nothing other than the energy spectrum of the metal. The width of the individual zones will increase with increasing \(\beta\), while the width of the gaps will correspondingly—

decrease substantially. Within each individual zone, however, the possible values of \(\beta\) will not be distributed uniformly, but will be crowded near its edges, which is evident from the diagram (Fig. 4), if one recalls that \(\alpha\) varies uniformly (i.e., discontinuously, but always by one and the same amount).

Metals differing from one another in the “mobility” (more precisely, the “boundness”) of their electrons will have different energy spectra, since the curve of Fig. 4, which determines the admissible energies, while remaining in character one and the same for all metals, will nevertheless differ, in the case of different metals, by the different values of the parameter \(P\) entering into it. When \(P=0\) (a free electron gas), our damped curve degenerates into an ordinary cosine with extrema equal to \(\pm 1\); consequently, the energy “zones” merge with one another, the gaps between them disappear altogether, and the energy spectrum of the metal, having become uniform, though still remaining discrete, approaches the continuous all the more closely the greater the length \(L\) of the crystal in comparison with the magnitude \(a\) of the elementary crystal cell. We thus arrive at the same result as that to which the theory of the free electron gas leads directly. If we now put \(P=\infty\) (absolutely bound electrons, corresponding to an isolated atom), we shall see that the zones are no longer arranged at all but, on the contrary, contract into separate points. Instead of each zone we obtain a single value of \(\beta\). The energy spectrum will be formed by separate levels, remote from one another. Bringing atoms together into a crystal lattice, we shall again find a splitting of these levels.

Under ordinary conditions, in the case of a monovalent metal, the energies of the electrons correspond to the first zone. Thus only the first, lowest zone (or half of it, if spin is taken into account) proves to be “occupied,” while the remaining zones remain empty. Indeed, the first zone (as, incidentally, every other one) contains \(G\) possible energy states; if the metal is monovalent, then the number of free electrons is also equal to \(G\); restricted by the Pauli principle, they are distributed among the \(G\) states, completely filling the first zone.

However, by means of an external action the electrons may be raised to the upper zones, and, returning from them back to their old place, they emit a certain frequency \(\nu\). Kronig and Penney calculated the probability of such jumps between separate zones as a function of \(\beta\). It turned out that only jumps between the corresponding \(\beta\)’s are probable, i.e., between such two values of \(\beta\) (of which the first lies within one zone and the second within another) that correspond to identical values of \(\cos \alpha\), or, more precisely, to identical (or differing by \(\pi\)) values of \(a\alpha\).

From these calculations of Kronig and Penney it follows that to each pair of zones one can assign a definite frequency \(\nu\), and not a multitude of different frequencies, as would be the case if there were possible ...

all transitions are possible: from any place in one zone to any place in another. True, owing to the fact that the individual zones have different widths (and therefore the density of “possible” $\beta$’s within each separate zone is the smaller, the wider the zone), the frequency $\nu$ is somewhat smeared out and turns into an entire band, bounded by $\nu_{max}$, corresponding to the transition between the right-hand boundaries of the zones under consideration, on the one hand, and by $\nu_{min}$, corresponding to the transition between their left-hand boundaries, on the other (see Fig. 4).

Thus, in the geometrical properties of Fig. 4 we see the physical nature of the metal. Here, as often happens in physics, it is more convenient for us to use not directly the language of the phenomenon itself, but the language of its geometrical interpretation.

The paper under review is very characteristic of the present state of the electron theory of the metal, which, following Bloch’s example, increasingly constrains the free motion of electrons, so that the term “electron gas” is already beginning not to help, and may even hinder, the understanding of what actually occurs in a metal.

F. Vol’kenshtein, Moscow