Pushing the limit of quantum transport simulations

Abstract

Simulations of quantum transport in coherent conductors have evolved into mature techniques that are used in fields of physics ranging from electrical engineering to quantum nanoelectronics and material science. The most efficient general-purpose algorithms have a computational cost that scales as L^{6,⋯,7} in three dimensions (L: length of the device), which on the one hand is a substantial improvement over older algorithms, but on the other hand still severely restricts the size of the simulation domain, limiting the usefulness of simulations through strong finite-size effects. Here we present a class of algorithms that, for certain systems, allows us to directly access the thermodynamic limit. Our approach, based on the Green's function formalism for discrete models, targets systems that are mostly invariant by translation, i.e., invariant by translation up to a finite number of orbitals and/or quasi-one-dimensional electrodes and/or the presence of edges or surfaces. Our approach is based on an automatic calculation of the poles and residues of series expansions of the Green's function in momentum space. This expansion allows us to integrate analytically in one momentum variable. We illustrate our algorithms with several applications: devices with graphene electrodes that consist of half an infinite sheet, Friedel oscillation calculations of infinite two-dimensional systems in the presence of an impurity, quantum spin Hall physics in the presence of an edge, and the surface of a Weyl semimetal in the presence of impurities and electrodes connected to the surface. In this last example, we study the conduction through the Fermi arcs of the topological material and its resilience to the presence of disorder. Our approach provides a practical route for simulating three-dimensional bulk systems or surfaces as well as other setups that have so far remained elusive.