Relaxation-limited electronic currents in extended reservoir simulations

TL;DR

This paper analyzes how relaxation strength affects electronic conductance in extended reservoir simulations, deriving explicit formulas for weak and strong relaxation limits and exploring their implications in nanoscale systems.

Contribution

It provides explicit, general expressions for conductance in both weak and strong relaxation regimes, clarifying the role of relaxation in extended reservoir simulations.

Findings

Conductance increases linearly with relaxation strength in the weak limit.

Conductance is inversely proportional to relaxation strength in the strong limit.

Identifies conditions leading to higher order and pseudo-plateau behaviors.

Abstract

Open-system approaches are gaining traction in the simulation of charge transport in nanoscale and molecular electronic devices. In particular, "extended reservoir" simulations, where explicit reservoir degrees of freedom are present, allow for the computation of both real-time and steady-state properties but require relaxation of the extended reservoirs. The strength of this relaxation, $\gamma$, influences the conductance, giving rise to a "turnover" behavior analogous to Kramers' turnover in chemical reaction rates. We derive explicit, general expressions for the weak and strong relaxation limits. For weak relaxation, the conductance increases linearly with $\gamma$ and every electronic state of the total explicit system contributes to the electronic current according to its "reduced" weight in the two extended reservoir regions. Essentially, this represents two conductors in series -- one at each interface with the implicit reservoirs that provide the relaxation. For strong relaxation, a "dual" expression -- one with the same functional form -- results, except now proportional to $1/\gamma$ and dependent on the system of interest's electronic states, reflecting that the strong relaxation is localizing electrons in the extended reservoirs. Higher order behavior (e.g., $\gamma^2$ or $1/\gamma^2$) can occur when there is a gap in the frequency spectrum. Moreover, inhomogeneity in the frequency spacing can give rise to a pseudo-plateau regime. These findings yield a physically motivated approach to diagnosing numerical simulations and understanding the influence of relaxation, and we examine their occurrence in both simple models and a realistic, fluctuating graphene nanoribbon.