Impurity Bands in a Magnetic Field

Abstract

The formation of an impurity band for independent electrons moving under actions of a static magnetic field and short-range potentials due to impurity centers is discussed. The theoretical basis is the disordered-lattice Green's function formalism of Yonezawa and Matsubara. When the impurity potential is attractive and the magnetic field H is strong enough, there is at least one bound level associated with each center, which is shown to display a spread to form a band because of the presence of many impurities. The critical condition of this impurity band merging into the main band is expressed in a form of an “equation of states”. The trace of the disordered-lattice Green's function, Z ( E ), defined so that it satisfies a self-consistent relation between the Green's function and its self-energy is shown to be endowed with some satisfactory analytic properties which assures the sum rule for the impurity-band density of states and other characteristics. While the imaginary part of Z ( E ) represents the density of allowed energy bands, the real part of Z ( E ) is shown to express the degree of localization of the scattered amplitude due to individual centers. On these bases a thermal distribution of electrons over both bands and the high-field conductivities are calculated.