Chebyshev Polynomial Method to Landauer-B\"uttiker Formula of Quantum Transport in Nanostructures

TL;DR

This paper introduces an efficient Chebyshev polynomial method to compute the Landauer-Büttiker formula for quantum transport in nanostructures, reducing computational costs and enabling scalable simulations.

Contribution

It adapts the Chebyshev polynomial approach to quantum transport calculations, offering a faster and more efficient alternative to traditional matrix inversion methods.

Findings

Significant reduction in computational time compared to direct matrix calculations.

Successful application to typical quantum transport examples.

Enhanced numerical efficiency with proposed algorithm improvements.

Abstract

Landauer-B\"uttiker formula describes the electronic quantum transports in nanostructures and molecules. It will be numerically demanding for simulations of complex or large size systems due to, for example, matrix inversion calculations. Recently, Chebyshev polynomial method has attracted intense interests in numerical simulations of quantum systems due to the high efficiency in parallelization, because the only matrix operation it involves is just the product of sparse matrices and vectors. Many progresses have been made on the Chebyshev polynomial representations of physical quantities for isolated or bulk quantum structures. Here we present the Chebyshev polynomial method to the typical electronic scattering problem, the Landauer-B\"uttiker formula for the conductance of quantum transports in nanostructures. We first describe the full algorithm based on the standard bath kernel polynomial method (KPM). Then, we present two simple butefficient improvements. One of them has a time consumption remarkably less than the direct matrix calculation without KPM. Some typical examples are also presented to illustrate the numerical effectiveness.