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Wave Propagation in Complex Media pp 95–142Cite as

Electron in two-dimensional system with point scatterers and magnetic field

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 96))

Abstract

This paper is a survey of the electron spectral properties (eigenenergies and wave functions) in a two dimensional system containing point scatterers and subject to a perpendicular magnetic field. Point potentials scatter only s-waves and do not influence waves with higher orbital momentum. Therefore they lift the infinite degeneracy of the Landau levels only partially. As a result the spectrum can be divided into two parts: the set of discrete Landau levels and the set of intervals between these levels. The states on the Landau levels exist in a strong enough magnetic field (the flux per a scatterer has to be larger than a flux quantum). They are regular functions of the spatial coordinates and vanish at the sites where the point scatterers are located. A new approach, based on the theory of entire functions, is proposed for studying of these states, and some of them are explicitly constructed.

States outside the Landau levels exist for an arbitrary magnetic field and have logarithmic singularities at all points where the scatterers are placed. In the ordered case (identical scatterers, placed on the sites of a square lattice) and for some rational values of a magnetic field, dispersion laws are numerically calculated and the Hofstadter-type butterfly is constructed. It is shown that the dispersive subbands in a square lattice of identical point scatterers in the strong field limit are described by the Harper equation. The problem of electron localization in such a system with one dimensional disorder reduces in the strong field limit to the random Harper equation. An explicit formula describing the fractal structure of the localization length is obtained. This structure is influenced by the amplitudes of the Bloch states in the corresponding ordered system.

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Authors and Affiliations

  1. Department of Physics, Ben Gurion University of the Negev, PO Box 653, 84105, Beer-Sheva, Israel

    Sergey Gredeskul, Masha Zusman & Yshai Avishai

  2. Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, 69978, Ramat Gan, Israel

    Mark Ya. Azbel

Authors
  1. Sergey Gredeskul
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  2. Masha Zusman
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  3. Yshai Avishai
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  4. Mark Ya. Azbel
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Editors and Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    George Papanicolaou

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Gredeskul, S., Zusman, M., Avishai, Y., Azbel, M.Y. (1998). Electron in two-dimensional system with point scatterers and magnetic field. In: Papanicolaou, G. (eds) Wave Propagation in Complex Media. The IMA Volumes in Mathematics and its Applications, vol 96. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1678-0_6

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  • DOI: https://doi.org/10.1007/978-1-4612-1678-0_6

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