Electronic Structure of a Single-Walled Carbon Nanotube in Tight-Binding Wannier Representation

This Demonstration shows an alternative way to represent the reciprocal space zone-folding (ZF) method for computing the tight-binding (TB) electronic structure (right plot) of a single-walled carbon nanotube (SWNT) with given chirality. The TB Hamiltonian is constructed in real space representation or Wannier representation and the electronic energy dispersion relation is obtained from the eigenvalues of the corresponding Hamiltonian matrix (left plot). The diagonal matrix elements are given by the on-site energy parameter , while the off-diagonal matrix elements are given by the hopping parameter . To find which of these matrix elements are nonzero, one has to consider the whole set of the atomic coordinates in one SWNT unit cell and the hopping of an electron from a given site with coordinates to each of its first three nearest neighbors with coordinates . Hence, for , where is the carbon-carbon bond length. Periodic boundary conditions along the SWNT axis (for with the axial period of the SWNT) can be expressed by multiplying the hopping parameter by the complex exponential phase factor . By changing the phase in the range , the whole 1D Brillouin zone can be sampled. This approach lets you sample a finite number of Brillouin zone -points by choosing a finite lattice model with sites; hence the term small crystal approach. In order to show the full equivalence of this method to the reciprocal space zone-folding method, the eigenvalues obtained from diagonalization of the Wannier Hamiltonian for a given are superimposed on the plot of the ZF band structure.