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.