Advanced Solid State Physics

Figure 26: Dispersion relationship for a fcc for the empty lattice approximation

7.2.2 Plane Wave Method - Central Equations

Now we want to talk about the Schrödinger equation with a periodic potential (in the free electron
model the potential was zero), i.e. electrons moving through a crystal lattice. Such calculations are
called band structure calculations (there are bands and bandgaps). Now the plane wave method is
used (simple method, numerically not very efficient). We will end up with something that’s called the
central equations.
We start with the Schrödinger equation, which is a linear differential equation with periodic coeffi-
cients.

−~^2

2 m

∇^2 Ψ +U(r)Ψ =EΨ

When the functionU(r)is periodic, it can be written as a Fourier- series (potential as a sum of plane
waves,Gas a reciprocal lattice vector). ForΨa former solution is assumed, which is also periodic,
but in the scale of the crystal (i.e. 1 cm). So the boundary conditions don’t matter.

U(r) =

∑

G

UGeiGr

Ψ =

∑

k

Ckeikr

These are put in the Schrödinger equation to get an algebraic equation out of a differential equation:

∑

k

~^2 k^2

2 m

Ckeikr+

∑

G

∑

k

UGCkei(G+k)·r=E

∑

k

Ckeikr (48)

We know~

(^2) k 2
2 m and we know whichGvectors correspond to the potential. The coefficientsCkare
unknown, but there are linear equations for the C’s. It can be written in a matrix - there is a different