# Driven Liouville-von Neumann Equation for Quantum Transport and Multiple-Probe Green's Functions

## TL;DR

This paper unifies the Driven Liouville-von Neumann equation with the multiple-probe Green's functions approach, providing a new formal foundation and exploring their performance in simulating quantum transport in various models.

## Contribution

It demonstrates that under certain conditions, the hairy-probes formalism can be expressed in the same algebraic form as the Driven Liouville-von Neumann equation, offering a new theoretical basis.

## Key findings

- The formal equivalence between the two methods is established.
- Performance comparison in ballistic, disordered, and resonant models.
- Insights into the applicability of the Driven Liouville-von Neumann approach without 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 to fix the electrochemical potentials in two regions of an open quantum system. 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 performance 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.