# Band-structure and electronic transport calculations in cylindrical wires : the issue of bound states in transfer-matrix calculations

## TL;DR

This paper discusses the transfer-matrix method for calculating band structures and electronic transport in cylindrical wires, emphasizing the importance of considering bound states to accurately model transmission and avoid unphysical results.

## Contribution

It provides a detailed pedagogical update on using transfer matrices to compute band structures in periodic systems, highlighting the role of bound states in electronic transport.

## Key findings

- Bound states cause sharp resonances in transmission probabilities.
- Including additional states enhances the completeness of the model.
- Bound states are essential to avoid unphysical discontinuities in band structures.

## Abstract

The transfer-matrix methodology is used to solve linear systems of differential equations, such as those that arise when solving Schr\"odinger's equation, 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 to describe efficiently periodic repetitions of a basic unit. This paper, which is an update of an article originally published in Physical and Chemical News 16, 46-53 (2004), provides a pedagogical presentation of this technique. It describes in details how the band structure associated with an infinite periodic medium can be extracted from the transfer matrices that characterize 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 probabilities and to remove unphysical discontinuities in the band structure. Additional states only improve the completeness of the representation.