Set of supplementary notes.

74 CHAPTER 5. BANDSTRUCTURE OF REAL MATERIALS
Bloch’s theorem again.
The eigenstates are of the form given by Bloch’s theorem
ψ nk (r) = e ik·r u nk (r) (5.3)
where u(r) is periodic on the lattice. Notice that if we make the substitution k → k + G, (5.3)
continues to hold. This tells us that k can always be chosen inside the first Brillouin zone for
convenience, although it is occasionally useful to plot the bands in an extended or repeated
zone scheme as in Fig. 4.2.
Crystal momentum.
The quantity h¯k is the crystal momentum, and enters conservation laws for scattering processes.
For example, if an electron absorbs the momentum of a phonon of wavevector q, the final state
will have a Bloch wavevector k ′ = k + q + G, where G is whatever reciprocal lattice vector
necessary to keep k ′ inside the Brillouin zone. Physical momentum can always be transferred
to the lattice in arbitrary units of h¯G. Notice that depending on the energy conservation,
processes can thus lead to transitions between bands.
Counting states.
We saw that the spacing between k-points in 1D was 2π/L, where L is the linear dimension of
the crystal. This generalises to 3 dimensions: the volume associated with each k is
(∆k) 3 = (2π)3
V
(5.4)
with V the volume of the crystal. Within each primitive unit cell or Brillouin zone of the
reciprocal lattice the number of k states allowed by the periodic boundary conditions is equal
to the number of unit cells in the crystal. In practice N is so big that the bands are continuous
functions of k and we only need to remember density of states to count. Since electrons are
fermions, each k-point can now be occupied by two electrons (double degeneracy for spin). So
if we have a system which contains one electron per unit cell (e.g. a lattice of hydrogen atoms),
half the states will be filled in the first Brillouin zone. From this, we obtain the even number
rule’:
Even number rule.
Allowing for spin, two electrons per real space unit cell fill a Brillouin zone’s worth of k states.
Periodic boundary conditions and volume per k-point
A formal proof of the number of allowed k-points uses Bloch’s theorem, and follows from the imposition
of periodic boundary conditions:
ψ(r + N i a i ) = ψ(r) (5.5)