Kronig-Penney model

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Potential U (x) of the Kronig-Penney model

The Kronig-Penney model (after Ralph Kronig and William Penney ) is a simple model of solid-state physics that explains the behavior of valence electrons in crystalline solids . This results in a band structure of energy similar to that found in nature, for example in metals and semiconductors .

Explanation

The one-dimensional model is based on a series of potential wells of depth  U 0 , each a wide and separated by a barrier of width  b . The pots represent the area near the atomic nucleus where the electrons are attracted by the nuclear charge . The barriers correspond to the areas between the nuclei in which the charge of the nuclei is shielded by that of the electrons near the nucleus. The grid is infinitely extended, and according to Bloch's theorem spatially periodic solutions of the Schrödinger equation result . Applying these prerequisites results in energy bands that are separated by forbidden energy areas. There can be no electrons with energies from the forbidden areas in the crystal, only with energies from the energy bands.

At the boundary between permitted and prohibited energy ranges, standing waves form due to Bragg reflection . Those standing waves in which the electrons are preferentially located in the vicinity of atomic nuclei are energetically more favorable than those in which the electrons are preferentially located between atomic nuclei. The Bragg reflection at the periodic potential can therefore be seen as the cause of the forbidden energy ranges.

For the explicit band gap calculation as well as the level splitting for the one-dimensional Penney-Kronig model see p. Müller-Kirsten.

See also

literature

  • R. de L. Kronig, WG Penney: Quantum Mechanics of Electrons in Crystal Lattices . In: Proceedings of the Royal Society of London. Series A . tape 130 , no. 814 , March 2, 1931, p. 499-513 , doi : 10.1098 / rspa.1931.0019 .

Individual evidence

  1. ^ NF Mott, H. Jones: The Theory of the Properties of Metals and Alloys (Clarendon Press, 1936; Dover Publ.Reprint, 1958), p. 63.
  2. ↑ HJW Müller-Kirsten: Introduction to Quantum Mechanics: Schrödinger Equation and Integral Path. 2nd ed. (World Scientific, 2012), ISBN 978-981-4397-74-2 , pp. 458-477 and 325-329.