Impurity Bands in a Magnetic Field

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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.

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