Band Theory of Solids
One of the most basic features of solid materials is that at the microscopic level, the atoms are characteristically lined up in an orderly fashion. When there is no order, materials are called amorphous, but even in a great many materials we do not think of as crystalline, there exist small domains within which the atoms are orderly but the adjacent domains have no systematic order with respect to one another. The fundamental effect of the order within a domain or within a larger crystal is to produce a periodic potential well instead of the isolated atomic potentials we have considered previously. This chapter will examine the effects of this periodicity in an idealized way. If one first considers a series of atomic potentials, each at a fixed spacing
a from one another, the superposition of the overlapping Coulomb potentials will produce a net potential as pictured in Figure 8.1. At the edge of the material, the potential will of course rise to zero, but the overlapping potentials inside will produce a lowered maximum potential and far from the edge the potential will be periodic. In our idealized case, we will basically ignore the deviations from periodicity at the edge and take the potential to be exactly periodic. In addition to the periodicity of the well due to the overlapping poten-
tials, the proximity of the nearest neighbors causes the energy states to be perturbed. Since the potential barriers are not infinite, there is tunneling between atoms, and we need to examine the effect of coupled wells. Considering two wells only with a barrier between them, as in Figure 8.2, the three regions are characterized by the left, center, and right wave functions
ψ`(x) = Aeiαx +Be−iαx , −a ≤ x ≤ −b , (8.1) ψc(x) = Ceβx +De−βx , −b ≤ x ≤ b , (8.2) ψr(x) = F eiαx +Ge−iαx , b ≤ x ≤ a , (8.3)
where E < V so that
α = √ 2mE/~ ,
β = √ 2m(V − E)/~ .