ABSTRACT
A convenient approach to the electronic properties of a solid consists of using the one-electron model, where the core electrons are treated as almost completely localized, and the energy eigenvalue problem for the valence electron system, Eq.(1.77), reduces to solving the Schrodinger wave equation (1.79) for each valence electron as it moves in the spatially dependent potential energy U(r). If forces on the valence electrons can be ignored, U(r) does not depend on r such that Eq.(1.79), if we drop U (?), reduces to the free-electron problem:
— V2yA(r) = £y/(r) 2 me
Although this free-electron approximation is a drastic assumption, it provides an excellent model for the electronic properties of metals. Solutions of this wave equation are subject to appropriate boundary conditions. As we have seen in Section 1.4, when assuming that y/(r) must vanish over a closed surface in real space, the allowed solutions are standing waves. The same density of states, but more physical insight, results if we use the periodic boundary conditions, as for the lattice vibrational modes, which immediately lead to allowed plane wave solutions of the form:
¥ ( r ) = - L e i[f (5.1) V v
which are normalized with respect to the volume V of the crystal. The corresponding energy eigenvalues have the form:
£ = (5.2) 2 me
and the momentum eigenvalues of free electrons are given by:
p y ( r ) = — v i_ L _ £ '* r \ = hky/(r) or p - f i k (5.3) i IV v
If we assume the volume V to be a finite parallelepiped of sides Pa,Qb and Sc, the periodic boundary conditions require that the solutions are periodic over the distances Pa, Qb and Sc respectively, namely:
y/(r 4-Pa) = y/(r + Qb) = y/(r + Sc) = \j/(r)
Substituting Eq.(5.1) yields:
P k - a - 2 n p , Qk • b = 2icq, Sk -c - 2tis
where p, q and s are arbitrary integers. Comparing these equations with the definitions of the reciprocal base set, Eq.(3.7), it is clear that k can be considered to be a vector in k-space:
p p _** q p* sk = — a -l b H-c P Q S
Note that, for a general vector in reciprocal space, the components p/P, q/Q, and s/S can be fractions of an integer. If they are integers h = p/P, k = q/Q and l = s/S, then k
becomes a reciprocal lattice vector, as defined by Eq.(3.5) and denoted by K. In view of the definition of the first Brillouin zone, Eq.(3.9), which can also be written as k 2 = (k + K)2, it follows that the propagation vector may always be decomposed into
the sum of a reciprocal lattice vector K and a vector k lying in the first Brillouin zone. We note that the lattice vibration properties have been discussed in terms of k values in the first zone only, but there is no such restriction for electrons, where two or more zones will often be considered. Hence, Eq.(5.1) can be rewritten as:
¥ k ( r ) = - j = e ‘k ':eltr (5.4)
where elRr is a function with the periodicity of the lattice, and the corresponding eigenvalue, Eq.(5.2), becomes a function of the reduced wavevectork :
£t-= — (k + K)2 (5.5) * 2 me
In this representation, each possible k - state outside the first zone is translated by a specific Khkl and remapped into the zone. The electronic energy calculations are usually presented for wave numbers along symmetry lines of the crystal structure.
