Comment on "Quantum transport with electronic relaxation in electrodes: Landauer-type formulas derived from the driven Liouville-von Neumann approach" [The Journal of Chemical Physics 153, 044103 (2020)]

TL;DR

This paper provides a unified analytical framework for steady-state current in electronic junctions using the driven Liouville-von Neumann approach, clarifying convergence to Landauer limits and explaining interference effects.

Contribution

It derives a simple, general analytical expression for steady-state current in identical reservoir-connected junctions within the DLvN framework, extending previous results.

Findings

Analytical formula for steady-state current in identical reservoirs

Convergence to Landauer and Meir-Wingreen results explained

Destructive interference causes slow convergence in certain cases

Abstract

In a recent article, Chiang and Hsu [The Journal of Chemical Physics 153, 044103 (2020)] examine one and two site electronic junctions identically connected to finite reservoirs. For these two examples, they derive analytical solutions, as well as provide asymptotic analyses, for the steady-state current from the driven Liouville-von Neumann (DLvN) equation - an open system approach to transport in non-interacting systems where relaxation maintains a bias. The two site junction they examine has destructive interference, which they show leads to slow convergence of the DLvN to the Landauer limit with respect to reservoir size and relaxation. We previously derived the general solution for the steady-state current in both the DLvN and its many-body analog [Gruss et al., Scientific Reports 6, 24514 (2016)]. The many-body analog is a Lindblad master equation, which, when restricted to non-interacting systems, is exactly the DLvN. Here, we demonstrate that applying the more general expression to identical left and right reservoirs (i.e., finite reservoirs with the same density of states and coupling to the system) and Markovian relaxation provides a simple analytic form that applies to arbitrary, but identically connected, junctions. Moreover, we briefly discuss the convergence of the current to the Landauer and Meir-Wingreen result for non-interacting and interacting systems, respectively. Convergence occurs as the reservoirs' lesser Green's functions begin conforming to the fluctuation-dissipation theorem. Our approach sheds light on the behavior Chiang and Hsu observe for destructive interference. Finally, we show that the analytical results yield the asymptotic formulas derived in Gruss et al.