Spectral and transmission properties of periodic 1d tight-binding lattices with a generic unit cell: an analysis within the transfer matrix approach

Abstract

Wereport on the electronic structure, density of states and transmission properties of the periodic one-dimensional tight-binding (TB) lattice with a single orbital per site and nearest-neighbor hoppings, with a generic unit cell of u sites. The determination of the eigenvalues is equivalent to the diagonalization of a real tridiagonal symmetricu-Toeplitz matrix with (cyclic boundaries) or without (fixed boundaries) perturbed upper right and lower left corners.Wesolve the TB equations via the Transfer Matrix Method, producing analytical solutions and recursive relations for its eigenvalues, closely related to the Chebyshev polynomials.Weexamine the density of states and provide relevant analytical relations.Weattach semi-infinite leads, determine and discuss the transmissioncoefficient at zero bias and investigate the peaks number and position, and the effect of the coupling strength and asymmetry as well as of the lead properties on the transmission profiles.Weintroduce a generic optimal coupling condition and demonstrate its physical meaning.