Electronic Band Structures

Abstract

In deriving (4.24), we did not have to assume that V(r) was small, and in principle this set of equations enables us to find the energies ε _n (k) and the wave-functions ψ _nk (r) for an electron moving in any periodic potential V(r) But in practice, the equations can be solved at all easily only if most of the coefficients V _G are small (compared with ε_F, say). In the NFE approximation we assume that they are all small, so that the solution wave-functions contain only a few non-zero Fourier coefficients c _K . In reality, this will not be so; V(r) varies rapidly with r, as shown in Fig. 5.1, and consequently the coefficients V _G are large, and only decrease in magnitude slowly as |G| increases. Equation (4.24) then becomes a formidably large set of coupled equations. The solution wave-functions contain many Fourier components and, within the ion core at least, they look nothing like plane waves. Physically, this is just what we should expect: within the ion cores, ψ(r) must look something like an atomic wave-function, and such a function will certainly need many Fourier components to represent it adequately.