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CHAPTER 1
An electron in a periodic crystal lattice
1.1. General properties
It is well known that metals are good conductors of electricity. This is because the outer electronic shells of atoms that make up a metal overlap to a considerable extent. Therefore, the electrons in these shells (called valence electrons) move easily from atom to atom, so that one cannot say to which atom they really belong. This collectivization of outer electrons leads to the generation of the large binding energy of metals and accounts for their specific mechanical properties.
As for the inner electronic shells, because of the small degree of overlap they may be regarded approximately the same as in isolated atoms.
Thus, a metal is a crystalline lattice made up of positive ions, into which are "poured" collectivized electrons of the valence shells. They are also called conduction or "free" electrons. In acutal fact, these electrons strongly interact with one another and with the lattice ions, the potential energy of these interactions being of the order of the kinetic energy of electrons.
The construction of the theory of such a system seems at first sight quite impossible. However, there actually exists at present a sufficiently rigorous description of most of the interesting phenomena that occur in metals. This is associated with two circumstances. First, the behaviour of a system of strongly interacting electrons (or of an electron liquid) is in many respects analogous to that of a system of non-interacting particles (i.e., a gas) in a certain external field, which is the averaged field of the lattice ions and the other electrons. Second, although this field is difficult to calculate exactly, one can deduce much from the fact that the averaged field displays the symmetry properties of the crystal lattice, in particular periodicity. We will start therefore from the study of the auxiliary problem of the behavior of an electron in a periodic field.
Let us consider an electron moving in an external field with a potential energy U(r). The function U(r) is periodic, i.e.,
U(r + an) = U(r), (1.1)
where an is an arbitrary lattice period. As is known, the vector an can always be represented as a linear combination of basis vectors ai:
an = n1a1 + n2a2+n3a3, (1.2)
where ni, are positive or negative integers or zeros.
The Schrödinger equation for an electron is
[MATHEMATICAL EXPRESSION OMITTED] (1.3)
It is not difficult to see that ψ(r + an) is also a solution of this equation, with the same eigenvalue ε. Therefore, if the electron level ε is nondegenerate, i.e. has a single eigenfunction ψ, then we must have
ψ (r + an) = Cψ(r)(1.4)
where C is a constant.
But if the level ε is degenerate, i.e., has several eigenfunctions ψv, we may write
[MATHEMATICAL EXPRESSION OMITTED] (1.5)
Since the functions ψµ form an orthogonal and normalized set, i.e.
[MATHEMATICAL EXPRESSION OMITTED]
[MATHEMATICAL EXPRESSION OMITTED] (1.6)
it follows that by shifting the integration variable r by an and using formula (1.5) we obtain
[MATHEMATICAL EXPRESSION OMITTED] (1.7)
Hence, Cµv is a unitary matrix, i.e.,
C+=C-1 (1.8)
But such a matrix can be diagonalized. In other words, certain linear combinations of the functions ψv exhibit the property (1.4). The normalization condition here gives
|C|2 = 1. (1.9)
Thus, we may write
[MATHEMATICAL EXPRESSION OMITTED] (1.10)
where π is a real function of the displacement an.
Let us now consider two successive displacements: a and a'. In the first displacement the function ψ is multiplied by C(a) and in the second by C(a'). At the same time, the two successive displacements are equivalent to a single displacement by a + a'. Here the function ψ must simply be multiplied by C(a + a'). Hence,
C(a + a') = C(a) C(a'). (1.11)
From this it follows that the function π in formula (1.10) must be a linear function of an:
[MATHEMATICAL EXPRESSION OMITTED] (1.12)
where p is a vector coefficient.
It is easy to see that this vector has been defined ambiguously. Namely, if to p we add the vector hK, which satisfies the condition Kan = 2πm for any lattice period an (where m is an integer), we will obtain the same coefficients C(a"). The equations Kan = 2πm are satisfied by an infinite system of vectors, all of which may be written in the following form:
[MATHEMATICAL EXPRESSION OMITTED] (1.13)
Here qi are integers Ki and are the smallest noncoplanar vectors exhibiting the property Kan = 2πm . From formula (1.2) it follows that this condition must be satisfied for the basis vectors at. It will then be satisfied for any period an. From this it is easy to obtain the vectors Ki:
[MATHEMATICAL EXPRESSION OMITTED] (1.14)
Thus, we see that the vectors Ki are equal to 277 multiplied by the reciprocal heights of the unit cell. Taking K1, K2 and K3 as basis vectors, we can construct the so-called reciprocal lattice. Hence, the reciprocal lattice is wholly determined by the translational properties of the crystal under consideration (by the vectors ai), i.e., by its Bravais lattice, and has the same symmetry properties. But, as is known, there may exist various Bravais lattices with the same symmetry. The relationship between the Bravais lattice and the reciprocal lattice is as follows: if the Bravais lattice is body-centered, then the reciprocal lattice is face-centred, and vice versa; to a base-centered Bravais lattice there corresponds a base-centered reciprocal lattice.
The electron energy ε depends on the vector p. Since p and p + hK are physically equivalent, it follows that the energy ε(p) must evidently be a periodic function with periods hKi. To each value of p there may, generally speaking, correspond several energy levels εl(p) and each of these functions is periodic in the reciprocal lattice. The wave function describing the movement of an electron in the periodic field and having the property
[MATHEMATICAL EXPRESSION OMITTED]
may be represented as
[MATHEMATICAL EXPRESSION OMITTED] (1.15)
where u(r) is a periodic function:
u(r + an) = u(r).
Formula (1.15) is known as the Bloch theorem. The wave function ψ in the form (1.15) resembles a plane wave describing the motion of a free particle, but here the wave is modulated by a periodic function. Therefore, the vector p, which is analogous to the momentum, is not in fact the momentum of a particle in the ordinary sense of the word. It is called the quasimomentum of the electron.
Since the vectors p and p + hK are physically equivalent, for the sake of uniqueness we may consider only one unit cell of the reciprocal lattice. The volume of the region of the unique determination of p is given by
[MATHEMATICAL EXPRESSION OMITTED]
where v = (a1[ a2a3]) is the volume of the unit cell of the principal lattice.
In order to obtain the solution of the Schrödinger equation, one has to know the boundary conditions. However, in an infinitely large volume the successive states will be infinitely close to one another. We are actually interested only in the density of states, i.e., the number of states per energy interval or given volume element in quasimomentum space. The density of states is independent of the particular form of the boundary conditions, and therefore it is easier to determine by assuming the simplest conditions.
Assuming that the metal specimen under consideration has the shape of a rectangular parallelepiped, we specify periodic boundary conditions:
[MATHEMATICAL EXPRESSION OMITTED] (1-16)
Assuming that each of the dimensions L1, L2 and L3 contains an integer number of periods in its direction, we obtain:
[MATHEMATICAL EXPRESSION OMITTED]
from which it follows that
[MATHEMATICAL EXPRESSION OMITTED] (1.17)
where nx, ny and nz are integers.
Thus, the vector p proves to be a discrete variable. But if the lengths L1, L2 and L3 are very large, then the summation over the states may be replaced by an integration. To do this, we have to know the number of states in a given volume of p-space. From eq. (1.17) we find
[MATHEMATICAL EXPRESSION OMITTED]
so that the number of states in the interval d3p = dpx dpy dpz is equal to
[MATHEMATICAL EXPRESSION OMITTED]
where V = L1L2L3 is the volume of the sample. This means that the density of states in p-space is
V/(2πh)3. (1.18)
As has already been pointed out, the region of the unique determination of p is the unit cell of the reciprocal lattice with a volume (2πh)3/v. Therefore, the total number of various values of p is equal to
[MATHEMATICAL EXPRESSION OMITTED]
where N is the number of unit cells in the sample under consideration. It must also be kept in mind that the electron has a spin s=1/2, whose projection on a certain axis may have two values, sz = ±1/2. This doubles the number of states. Thus, it turns out that to each of the functions εl(p) there correspond IN various states.
The functions εl(p) are periodic in the reciprocal lattice and naturally oscillate between the maximal and minimal values. Hence, for each number l we obtain "bands" of allowed energy values. These bands may be separated by "energy gaps" (i.e., energy values unattainable for electrons), but they may also overlap.
Let us consider some general properties of the functions εl(p). The complete Schrödinger equation has the form
[MATHEMATICAL EXPRESSION OMITTED]
We will now turn to the complex-conjugate equation and perform the transformation t -> -t. Here we obtain
[MATHEMATICAL EXPRESSION OMITTED]
that is, the same Schrödinger equation with a Hamiltonian H*. But H is a Hermitian operator, i.e., the eigenfunctions and eigenvalues of the operators H and H* are the same. From this it follows that if ψlp(r, t) = exp[- iεl{(p)t/h]] ψlp(r) is an eigen-function of H, then the function ψ*lp(r,-t) is also an eigenfunction of H. Upon displacement of r by a period a the function ψlp acquires a factor exp(ipa/h) and the function ψ*lp(r,-t) acquires a factor exp(-ipa/h). It then follows that εl(p) = εl (-p).
We have so far used the unit cell of the reciprocal lattice as the region of the unique determination of the quasimomentum p. But it is more convenient to define this region in a different way. Of course, it must have a volume equal to the volume of the unit cell of the reciprocal lattice and, besides, it must not contain points differing by a period of the reciprocal lattice or more. We will define it as follows. Let us draw from some reciprocal lattice point all K-vectors that connect it with the other lattice points. Then, we draw planes perpendicular to each of these vectors and dividing them in half. These planes will cut out a certain figure in the space of the reciprocal lattice which has the shape of a polyhedron. It is not difficult to see that such a polyhedron possesses all the required properties and may therefore be taken as the region of specification of the quasimomentum p. It is called the Brillouin zone. Figure 1 shows examples of Brillouin zones for the face-centred cubic (a) and body-centered cubic (b) lattices.
As a rule, the crystal lattices of metals exhibit high symmetry. This gives rise to certain properties of the function εl(p). Suppose, for example, that the symmetry plane perpendicular to the axis px passes through the point p = 0. If there exist faces of the Brillouin zone perpendicular to the px axis, then εl(p) as functions of px must have extrema on these faces. Indeed, let us single out the points of these faces, p1 and p2, which are symmetric with respect to the symmetry plane (fig. 2). They differ by a reciprocal lattice period (multiplied by h). Therefore, at these points
[MATHEMATICAL EXPRESSION OMITTED]
But by virtue of the symmetry with respect to the px = 0 plane we have
[MATHEMATICAL EXPRESSION OMITTED]
Hence,
[MATHEMATICAL EXPRESSION OMITTED]
In an analogous way we obtain in this case
[MATHEMATICAL EXPRESSION OMITTED]
Thus, we arrive at the conclusion that for symmetrical lattices, as a rule, there are extrema of the functions εl(p) in the center of the Brillouin zone or at its boundaries.
The conclusions concerning the electron energy as a function of the quasimomentum are illustrated in fig. 3, which refers to the one-dimensional case. Evidently, the Brillouin zone here is the segment -πh/a < p < π/a, where a is the period of a linear chain.
1.2. The strong-coupling approximation
For the function εl(p) to be calculated exactly, use is made of rather complicated methods (see ch. 14). To illustrate the general properties of the functions εl(p), we will consider in the following the two simplest techniques, although they are not very efficient for the exact determination of the functions εl(p) in real metals.
We begin with the so-called strong-coupling method and for the sake of simplicity consider first a one-dimensional metal, i.e., a linear chain of atoms. We assume that the electronic shells overlap little and that, in the zeroth approximation, each electron belongs to one atom. The overlap of the shells is regarded as a perturbation.
The potential energy of the electron in the field of all the ions has the form
[MATHEMATICAL EXPRESSION OMITTED] (1.19)
and the Schrödinger equation is
[MATHEMATICAL EXPRESSION OMITTED] (1.20)
Let the following Bloch functions be the exact solutions of eq. (1.20):
[MATHEMATICAL EXPRESSION OMITTED]
and the corresponding eigenvalues be equal to ε(p). We now form so-called Wannier functions from the functions ψp:
[MATHEMATICAL EXPRESSION OMITTED] (1.21)
where N is the number of atoms in the chain, and the sum over p is limited by a one-dimensional Brillouin zone: [MATHEMATICAL EXPRESSION OMITTED]. The inverse transformation has the form
[MATHEMATICAL EXPRESSION OMITTED] (1.22)
Indeed, substituting formula (1.21) into eq. (1.22), we have
[MATHEMATICAL EXPRESSION OMITTED]
(since p and p' are limited by the Brillouin zone). The functions wn(x) with different n are orthogonal. As a matter of fact,
[MATHEMATICAL EXPRESSION OMITTED] (1.23)
where L = Na is the length of the chain.
(Continues…)
Excerpted from "Fundamentals of the Theory of Metals"
by .
Copyright © 1988 A. A. Abrikosov.
Excerpted by permission of Dover Publications, Inc..
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