This occurs when
2 k=−
2 π
a(n′+n) (10.76)Within the first Brillouin zone, this occurs precisely when
k= 0 n′=−n 6 = 0
k=±
π
an′+n=∓ 1 (10.77)At each of those points, the spectrum of the free Hamiltonianis degenerate, since two states occur
with the same energy.
We concentrate on the degeneracies at the edge of the Brillouin zone,k=k+=π/a, where
the levelsn′=−n−1 andnare degenerate. We now turn onany perturbation which mixes the
degenerate levels. We introduce the notation,
〈k+,−n− 1 |V|k+,n〉=V−+ 〈k+,n|V|k+,n〉=V++
〈k+,n|V|k+,n′〉=V+− 〈k+,−n− 1 |V|k+,−n− 1 〉=V−− (10.78)Self-adjointness ofV impliesV−+ =V+∗−, and we require that this mixing matrix element be
non-zero. The energy levels in the presence of the potentialV for sufficiently smallV are then
determined by the equation
det(
E−E 0 −V++ −V+−
−V−+ E−E 0 −V−−)
= 0 (10.79)whereE 0 =Ek+,n=Ek,−n− 1. This gives the following quadratic equation for the two energy levels,
(
E−E 0 −1
2
(V+++V−−)
) 2
=|V+−|^2 +1
4
(V++−V−−)^2 (10.80)
with solutions,
E=E 0 +1
2
(V+++V−−)±
1
2
√
(V++−V−−)^2 + 4|V+−|^2 (10.81)This effect opens an energy gap in the spectrum, and creates a band structure.
10.10Level Crossing
The problem of electronic band formation, discussed in the preceding section, is fundamentally
a problem of level crossing. In fact, the conclusion obtained above indicates that, as soon as
interactions are turned on, levels do not cross one another. This effect is quite general, and we
shall now also prove it more generally.
We shall assume that two levels, with energiesE^0 nandEn^0 ′, are very close to one another, so
thatEn^0 ′−En^0 is small compared to the gaps between eitherEn^0 orEn^0 ′and any other energies in