we know that atomic orbitals hybridise into bands and we know that the states in these bands behave in a wavelike manner. Now we want to know the E(K) curves, ie the band structure which determines the materials properties.
7.2 Nearly Free Electron Theory
When the wavelength resonates with a periodicity K=np/d , we get an infinity strong back scattering.
Figure 7.1 - Back Scattering in a Crystal
The only solutions to this are standing waves which go nowhere. The Nearly Free Electron model treats the crystal as a uniform potential except for the, about, 10% of the crystal volume, clse to the atoms cores
Figure 7.2 - One Dimensional Nearly Free Electron Model
In resonant conditions, the standing wave is a linear superposition of eKx and e-Kx . Degenerate Perturbation Theory says there must be two orthogonal wavefunctions
ó õy1*(x)y2(x)dx=0
y1 peaks between the atoms whilst y2 peaks at the atoms. The energy of y1 is greater than y2 . The energy difference is the Fourier component of V(x) at a spatial frequency K .
Figure 7.3 - In K Space
In the positive part of the x-axis in figure 7.3 the ??electron?? oscillates more frequently than the lattice period, in the negative region it oscillates less frequently. The bandgaps appear at these resonant K values.
7.3 Reduced Zone Scheme
The Extended Zone Scheme ignores the peridic nature of the reciprocal lattice. In fact each reciprocial lattice cell is equilivent and therefore we can draw all the atomic information (electron states) in one primitive unit cell. Normally we represent all the bands in a First Brilleiun Zone and give each a band index. Thus each band originated from a different atomic orbital. This automatically accounts for the fact that the lattice can supply momentum in units of G (where G is the reciprocal lattic vector).
