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.