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



Daniel S. Gruss, Kirill Velizhanin, Michael P. Zwolak


Landauer's formula relates the conductance of a region of interest to its transmission probability. It is the standard theoretical tool to examine ballistic transport in nano- and meso-scale junctions and devices. This view of transport as transmission necessitates that any variation of the junction with time, i.e., a molecule fluctuating in solution, must be slow compared to the time required to establish a steady-state and slow compared to the relaxation of the resulting disturbances in the surrounding leads/reservoirs, in addition to its neglect of many-body interactions. Transport through structurally dynamic junctions is, however, increasingly of interest for sensing, harnessing fluctuations, and real-time control. Towards this end, we derive a Landauer-like formula for the steady-state current when relaxation of electrons in the reservoirs is present. We demonstrate that the finite relaxation time gives rise to three regimes of behavior. On the one hand, weak relaxation within a small region adjacent to the junction gives a contact-limited current. Strong relaxation, on the other hand, localizes electrons, distorting their natural dynamics and reducing the current. In an intermediate regime, the standard Landauer view is recovered. This behavior is analogous to Kramers' turnover in chemical reactions. While we focus on the steady-state properties and the correspondence with traditional methods, we also demonstrate that a simple equation of motion emerges which is suitable for investigating transport through structures that rapidly change with time due to their own natural dynamics or due to current-induced changes, as well as the inclusion of many-body interactions.
Scientific Reports


Nanoscale electron transport, computational methods


Gruss, D. , Velizhanin, K. and Zwolak, M. (2016), Landauer's formula with finite-time relaxation: Kramers' crossover in electronic transport, Scientific Reports, [online], (Accessed April 17, 2024)
Created April 20, 2016, Updated June 2, 2021