A Monte Carlo method for solving the NEGF equations for electron transport

TL;DR

This paper introduces a Monte Carlo scheme for solving NEGF equations in electron transport, proving its correctness and demonstrating its effectiveness in nanowire conductivity calculations, with potential for further speed improvements.

Contribution

The paper develops a Monte Carlo method for NEGF equations, providing formal proof of its accuracy and testing it on nanowire conductivity, highlighting regimes where it is fastest.

Findings

Monte Carlo estimator matches NEGF lead current

Good agreement with traditional approaches in conductivity calculations

Identifies a regime where Monte Carlo is the fastest method

Abstract

We derive, by introducing restrictions to the lesser self energy, a Monte Carlo scheme that solves the NEGF equations for electron transport. In doing so we formally prove that the Monte Carlo estimator has an expectation value equal to the lead current of the NEGF solution, and we provide a simple test of the Monte Carlo scheme by calculating conductivity in nanowires within Buttiker's approximation of scattering. Good agreement between the Monte Carlo simulations and the alternative approaches are obtained, and we also demonstrate the existence of a regime where the Monte Carlo method is the fastest method. In our tests this regime is to extreme to be of practical use. We discuss various ways in which to speed up our prototype and how it can be extended to include more physics. Although the full applicability range of our assumptions about the lesser self energy remains to be better understood, we argue that they should apply as long as the transport process can be considered stationary.