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.

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DETAILS

C. Kittel, Introduction to Solid State Physics, 7th ed., Hoboken, New Jersey: J. Wiley and Sons, 1996.
S. L. Altmann, Band Theory of Solids: An Introduction from the Point of View of Symmetry, Oxford: Clarendon Press, 1991.
J. Alfonsi, Small Crystal Models for the Electronic Properties of Carbon Nanotubes, PhD thesis, University of Padova, 2009, Chap. 4 and references therein.
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