Landauer's formula with finite-time relaxation: Kramers' crossover in electronic transport

TL;DR

This paper extends Landauer's formula to include finite-time relaxation effects, revealing three transport regimes and providing an efficient equation of motion for simulating time-dependent electronic transport.

Contribution

It introduces a modified Landauer framework accounting for reservoir relaxation, identifying distinct transport regimes and deriving a practical equation of motion for dynamic systems.

Findings

Identifies three regimes: contact-limited, localized electrons, and Landauer recovery.

Shows relaxation strength critically influences current behavior.

Provides an efficient simulation method for time-dependent transport.

Abstract

Landauer's formula is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions, but it necessitates that any variation of the junction with time must be slow compared to characteristic times of the system, e.g., the relaxation time of local excitations. Transport through structurally dynamic junctions is, however, increasingly of interest for sensing, harnessing fluctuations, and real-time control. Here, we calculate the steady-state current when relaxation of electrons in the reservoirs is present and demonstrate that it gives rise to three regimes of behavior: weak relaxation gives a contact-limited current; strong relaxation localizes electrons, distorting their natural dynamics and reducing the current; and in an intermediate regime the Landauer view of the system only is recovered. We also demonstrate that a simple equation of motion emerges, which is suitable for efficiently simulating time-dependent transport.