Band-structure and transport calculations in quantum wires using a transfer-matrix technique

he transfer-matrix methodology is used to solve linear systems of differential equations, in situations where the solutions of interest are in the continuous part of the energy spectrum. The technique is actually a generalization in three dimensions of methods used to obtain scattering solutions in one dimension. Using the layer-addition algorithm allows one to control the stability of the computation and describe efficiently periodic repetitions of a basic unit. The paper provides a pedagogical presentation of this technique. It also describes in details how the band structure associated with an infinite periodic medium can be extracted from the transfer matrices characterizing a single basic unit. The method is applied to the calculation of the transmission and band structure of electrons subject to cosine potentials in a cylindrical wire. The simulations show that bound states must be considered because of their impact as sharp resonances in the transmission diagram and to obtain complete band structures. Additional states only improve the completeness of the representation.