Spectral properties and localization of an electron in a two-dimensional system with point scatterers in a magnetic field

Electron spectral properties and localization in a two-dimensional system with point potentials subject to a perpendicular magnetic field are studied. A brief review of the known results concerning electron dynamics in such systems is presented. For a set of periodic point potentials, exact dispersion laws and energy-flux diagram (Hofstadter-type butterfly) are obtained. It is shown that, in the case of one-dimensional disorder, the electron localization in a strong magnetic field is described by the random Harper equation. Energy-flux diagram for the localization length is presented and the fractal structure of the localization length is demonstrated. Near the Landau levels an exact formula for the localization length as a function of energy and disorder is obtained. The corresponding critical exponent is equal to unity which is reminiscent of one-dimensional characteristics.