Quantum Mechanics: Landauer's Formula

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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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.

  • Reading Material: Landauer's Formula
  • Slides: Diffusive vs. Ballistic Transport
  • Slides: Landauer's Formula Derivation
  • Slides: Buttiker Formula Derivation
  • Resonant Tunneling Diode Simulator
  • Homework Assignments for Modeling Resonant Tunneling Diodes
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    Cite this work

    Researchers should cite this work as follows:

    • www.eas.asu.edu/~vasilesk
    • Dragica Vasileska, Gerhard Klimeck (2008), "Quantum Mechanics: Landauer's Formula," http://nanohub.org/resources/4958.

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    In This Series

    1. Resonant Tunneling Diode Simulator

      10 Oct 2005 | Tools | Contributor(s): Michael McLennan

      Simulate 1D resonant tunneling devices and other heterostructures via ballistic quantum transport

    2. Resonant Tunneling Diodes: an Exercise

      06 Jan 2006 | Teaching Materials | Contributor(s): H.-S. Philip Wong

      This homework assignment was created by H.-S. Philip Wong for EE 218 "Introduction to Nanoelectronics and Nanotechnology" (Stanford University). It includes a couple of simple "warm up" exercises and two design problems, intended to teach students the electronic properties of resonant tunneling...

    3. Reading Material: Landauer's formula

      08 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska


    4. Slides: Diffusive vs. ballistic transport

      08 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska


    5. Slides: Landauer's formula derivation

      08 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska


    6. Slides: Buttiker formula derivation

      08 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska