Driven Liouville–von Neumann Equation for Quantum Transport and Multiple-Probe Green’s Functions

Abstract

The so-called driven Liouville–von Neumann equation is a dynamical formulation to simulate a voltage bias across a molecular system and to model a time-dependent current in a grand-canonical framework. This approach introduces a damping term in the equation of motion that drives the charge to a reference, out of equilibrium density. Originally proposed by Horsfield and co-workers, further work on this scheme has led to different coexisting versions of this equation. On the other hand, the multiple-probe scheme devised by Todorov and collaborators, known as the hairy-probes method, is a formal treatment based on Green’s functions that allows the electrochemical potentials in two regions of an open quantum system to be fixed. In this article, the equations of motion of the hairy-probes formalism are rewritten to show that, under certain conditions, they can assume the same algebraic structure as the driven Liouville–von Neumann equation in the form proposed by Morzan et al. (J. Chem. Phys.2017, 146, 044110). In this way, a new formal ground is provided for the latter, identifying the origin of every term. The performances of the different methods are explored using tight-binding time-dependent simulations in three trial structures, designated as ballistic, disordered, and resonant models. In the context of first-principles Hamiltonians, the driven Liouville–von Neumann approach is of special interest, because it does not require the calculation of Green’s functions. Hence, the effects of replacing the reference density based on the Green’s function by one obtained from an applied field are investigated, to gain a deeper understanding of the limitations and the range of applicability of the driven Liouville–von Neumann equation.