The theory that, in crystals, electrons fall into allowed energy bands, between which lie forbidden bands. Although free-electron theory can explain the electrical properties of metals, to understand fully the nature of electrical conduction, free-electron theory must be modified to include the effect of the crystal lattice in which the electrons move. Band theory modifies the free-electron treatment by including a regular periodic potential resulting from the positive ions in the lattice. Although the presence of the lattice alone does not give rise to electron scattering, except under special conditions, the periodic potential generated by the lattice does change the distribution of electron states given by the simple free-electron model.
In principle the original particle-in-a-box problem is modified by the addition of an extra potential term V(x, y, x), where V(x, y, x) has the same periodicity of the lattice it represents. The Schrödinger equation (see also quantum mechanics) that must be satisfied by these electron matter waves, ψ, is then:
− (ħ/2m)2(∂2ψ/∂x2 + ∂2ψ/∂y2 + ∂2ψ/∂z2) + V(x,y,z)ψ = Eψ
where E is the energy of the state associated with ψ. This equation is often called the Bloch equation; it is difficult to solve even using approximate methods.
As with the free-electron model, the Bloch equation leads to a relationship between the momenta that characterize an electron state, and the energy of that state. However, this relationship is no longer the simple parabolic function:
E = p2/2m,
where p is the electron momentum that characterizes the state, and m is the mass of the electron. Instead the function is multivalued and there are certain forbidden bands of energy, i.e. no permitted states exist for them. The relationship between E and p for a one-dimensional crystal is shown in diagram (1).
The main feature is that all the curves repeat themselves over an interval in p of h/a, where h is the Planck constant and a is called the lattice constant. The electron states on the lower curve indicated by the points S, S′, and S″, are identical states except for their momentum values. The upper curves show that for each momentum value there are several permitted energies; one within each energy band (e.g. P, P′, P″). In order to avoid the confusion that would arise if the complete multivalued set of curves were used, it is conventional to choose certain sections of them in which as p increases so does energy E. The smallest values, starting from p = 0, are reserved for the lowest energy band – the lowest curve in diagram (1). The next set from p = h/2a to h/a and −h/2a to −h/a is reserved for the second energy band (the middle curve in diagram (1), and so on. This zone structure is illustrated in diagram (2) and is called the extended zone scheme. It is merely a useful convention; any sets of p values that cover a range of h/a could be used to span all the available energy states.
A comparison of the extended zone scheme and the results of the free-electron model may easily be made by considering the relationship between E and p in the two models. This is shown in diagram (2) with the broken line corresponding to the free-electron parabolic relationship, E≅p2.