VARIABLE-RANGE-HOPPING MAGNETORESISTANCE

The hopping magnetoresistance R of a two-dimensional insulator with metallic impurities is considered. In sufficiently weak magnetic fields it increases or decreases depending on the impurity density n: It decreases if n is low and increases if n is high. In high magnetic fields B, it always exponentially increases with square-root B. Such fields yield a one-dimensional temperature dependence: lnR proportional 1/square-root T. The calculation provides an accurate leading approximation for small impurities with one eigenstate in their potential well. In the limit of infinitesimally small impurities, an impurity potential is described by a generalized function. This function, similar to a delta function, is localized at a point, but, contrary to a delta function in the dimensionality above 1, it has finite eigenenergies. Such functions may be helpful in the study of scattering and localization of any waves.