## Quantum Mechanics: Landauer's Formula

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#### Abstract

When a metallic nanojunction between two macroscopic electrodes is connected to a battery, electrical current flows across it. The battery provides, and maintains, the charge imbalance between the electrode surfaces needed to sustain steady-state conduction in the junction. This static non-equilibrium problem is usually described according to the Landauer picture. In this picture, the junction is connected to a pair of defect-free metallic leads, each of which is connected to its own distant infinite heat-particle reservoir. The pair of reservoirs represents the battery. Each reservoir injects electrons into its respective lead with the electrochemical potential appropriate to the bulk of that reservoir. Each injected electron then travels undisturbed down the respective lead to the junction, where it is scattered and is transmitted, with a finite probability, into the other lead. From there it flows, without further disturbance, into the other reservoir. The reservoirs are conceptual constructs which allow us to map the transport problem onto a truly stationary scattering one, in which the time derivative of the total current, and of all other local physical properties of the system, is zero. By doing so, however, we arbitrarily enforce a specific steady state whose microscopic nature is not, in reality, known a priori. The Landauer construct is highly plausible in the case of non-interacting lectrons.

In the material provided below, we first discuss the concepts of diffusive vs. ballistic transport, then we show the derivation of the Landauer and Landauer-Buttiker formulas, we give a link to a resonant tunneling diode solver and we also provide homework assignments regarding simulation of resonant tunneling diodes.

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- www.eas.asu.edu/~vasilesk