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

RAMÍREZ, FRANCISCO; DUNDAS, DANIEL; SÁNCHEZ, CRISTIÁN G.; SCHERLIS, DAMIAN A.; TODOROV, TCHAVDAR N.

JOURNAL OF PHYSICAL CHEMISTRY C

AMER CHEMICAL SOC

Año: 2019 vol. 123 p. 12542 - 12555

1932-7447

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.