Computational Physics

124 Solving the Schrödinger equation in periodic solids

6.1.2 Reciprocal lattice

A function which is periodic on a Bravais lattice can be expanded as a Fourier series
with wave vectorsKwhose dot product with every lattice vector of the original
lattice yields an integer times 2π:

R·K= 2 πn, integern. (6.2)

The vectorsKform another Bravais lattice – thereciprocal lattice. The basis vectors
bjof the reciprocal lattice are defined by

ai·bj= 2 πδij (6.3)

and an explicit expression for thebjis

bj= 2 πεjkl

ak×al

a 1 ·(a 2 ×a 3 )

. (6.4)

εjklis the Lévi–Civita tensor, which is+1 forjklan even permutation of(1, 2, 3),
and−1 for odd permutations.
In the reciprocal lattice, the first Brillouin zone is defined as the volume in
reciprocal space consisting of the points that are closer to the origin than to any
other reciprocal lattice point. A general wave vectorqis usually decomposed into
a vectorkof the first Brillouin zone and a vectorKof the reciprocal lattice:

q=k+K. (6.5)

For a finiterectangularlattice of sizeLx×Ly×Lz, the allowed wave vectorsqto
be used for expanding functions defined on the lattice are restricted by the boundary
conditions. A convenient choice is periodic boundary conditions in which functions
are taken periodic within the volume ofLx×Ly×Lz. In that case, vectors in reciprocal
space run over the following values:

q= 2 π

(

nx

Lx

,

ny

Ly

,

nz

Lz

)

(6.6)

with integernx,nyandnz.

6.2 Band structures and Bloch’s theorem

We know that the energy spectra of electrons in atoms are discrete. If we place two
identical atoms at a very large distance from each other, their atomic energy levels
will remain unchanged. Electrons can occupy the atomic levels on either of both
atoms and this results in a double degeneracy. On moving the atoms closer together,
this degeneracy will be lifted and each level splits into two; the closer we move the
atoms together, the stronger this splitting. Suppose we play the same game with