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

  • Sergey Gredeskul
  • Masha Zusman
  • Yshai Avishai
  • Mark Ya. Azbel
Part of the The IMA Volumes in Mathematics and its Applications book series (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.

Keywords

GaAs Tral Dinates Clarification Univer 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Sergey Gredeskul
    • 1
  • Masha Zusman
    • 1
  • Yshai Avishai
    • 1
  • Mark Ya. Azbel
    • 2
  1. 1.Department of PhysicsBen Gurion University of the NegevBeer-ShevaIsrael
  2. 2.Raymond and Beverly Sackler School of Physics and AstronomyTel-Aviv UniversityRamat GanIsrael

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