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