Chebyshev polynomial method to Landauer–Büttiker formula of quantum transport in nanostructures

The Landauer–Buttiker formula describes the electronic quantum transport 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, the Chebyshev polynomial method has attracted intense interest 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. Much progress has 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–Buttiker formula for the conductance of quantum transport in nanostructures. We first describe the full algorithm based on the standard bath kernel polynomial method (KPM). Then, we present two simple but efficient improvements. One of them has time consumption remarkably less than that of the direct matrix calculation without KPM. Some typical examples are also presented to illustrate the numerical effectiveness.