WOLFRAM|DEMONSTRATIONS PROJECT

Brillouin Zone Sampling of a Periodic Chain with N Sites

​
sites
2
a
1.
ϵ
-1.
t
-3.
This Demonstration shows the sampling of the
k
-points in the first Brillouin zone (BZ) of a virtually infinite linear crystal as a function of the number of sites in the unit cell. By choosing a periodic chain with
N
sites one can sample
N
k
-points in the reciprocal space of the first BZ, whose spacing is inversely proportional to
N
and the lattice parameter
a
. Then
k=2πκ/(Na)
​
, where
κ
is the allowed quantum number for the chain (
κ=0,1,2,…,N-1,
or equivalently,
κ=-N/2+1,…,N/2+1
). There is also cyclic periodicity in
±N
. The
k
-points thus obtained are mapped onto the analytical form of the tight-binding electronic dispersion relation
E(k)
for the chain. Diagonalizing the associated Bloch Hamiltonian gives the electronic energy eigenvalues
E(k)=ϵ+2tcos(ka)
. These are calculated and plotted as a function of the tight-binding hopping parameter
t
and the on-site energy parameter
ϵ
expressed in electron volts.