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Quantum Mechanics: Periodic Potentials and Kronig-Penney Model

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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Abstract

The Kronig-Penney model is a simple approximation of a solid. The potential consists of a periodic arrangement of delta functions, square well or Coulomb well potentials. By means of epitaxial growth techniques artificial semiconductor superlattices can be realized, which behave very similar to the assumptions of the Kronig-Penny model. To get better understanding of the Kroinig-Penney model, we provide written text, slides, homework assignments and access to the Periodic Potential Lab and the Bandstructure Tool that can be used to solve the homework assignments and investigate the particle dispersion relation as a function of the well geometry and well to well interactions.

  • Derivation of the Kronig-Penney model for delta-function potentials
  • Slides: Kronig-Penney model
  • Piece-wise constant potential barrier tool
  • From 1 well to 2 wells to 5 wells to periodic potentials
  • Bandstructure Lab
  • Exercise: Periodic potential and bandstructure
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    NSF

    Cite this work

    Researchers should cite this work as follows:

    • www.eas.asu.edu/~vasilesk
    • Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: Periodic Potentials and Kronig-Penney Model," http://nanohub.org/resources/4962.

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    In This Series

    1. Piece-Wise Constant Potential Barriers Tool

      30 Jun 2008 | Tools | Contributor(s): Xufeng Wang, Samarth Agarwal, Gerhard Klimeck, Dragica Vasileska, Mathieu Luisier, Jean Michel D Sellier

      Transmission and the reflection coefficient of a five, seven, nine, eleven and 2n-segment piece-wise constant potential energy profile

    2. Band Structure Lab

      19 May 2006 | Tools | Contributor(s): Samik Mukherjee, Abhijeet Paul, Neophytos Neophytou, Raseong Kim, Junzhe Geng, Michael Povolotskyi, Tillmann Christoph Kubis, Arvind Ajoy, Bozidar Novakovic, Sebastian Steiger, Michael McLennan, Mark Lundstrom, Gerhard Klimeck

      Computes the electronic and phonon structure of various materials in the spatial configuration of bulk , quantum wells, and wires

    3. Slides: Kronig-Penney Model Explained

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

      www.eas.asu.edu/~vasileskNSF

    4. Periodic Potentials and the Kronig-Penney Model

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

      This material describes the derivation of the Kronig-Penney model for delta-function periodic potentials.

    5. Periodic Potentials and Bandstructure: an Exercise

      02 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

      This exercise teaches the students that in the case of strong coupling between the neighboring wells in square and Coulomb periodic potential wells electrons start to behave as free electrons and the gaps that open at the Brillouin zone boundaries become smaller and smaller (thus recovering the …

    6. From 1 well to 2 wells to 5 wells to periodic potentials: an Exercise

      02 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

      This exercise demonstrates that the interaction between the wells lifts the degeneracy of the quasi-bound states and if in the limit we have infinite periodic potential it leads to formation of energy bands. Notice that when the interaction is less strong the energy levels are more sharp and the …

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