Orbital magnetization in the hopping regime

The orbital magnetic moment of localized electrons is considered, both in the absence and in the presence of the electron-phonon interaction. This magnetic moment, proportional to the persistent current circulating around groups of at least three localized states, arises from interference of the localized electronic wave functions. It is shown that for the isolated electronic system the magnetic moment has a definite sign which alternates as the location of the Fermi level is varied. When the electrons are coupled to the thermal bath, the orbital magnetic moment decreases due to two effects: (i) the appearance of a current that flows in the opposite direction to that of the isolated system—this countercurrent results from resonant electron-phonon processes, suggested by Holstein; (ii) an overall Debye-Waller factor that diminishes the magnitude of the orbital response to the external magnetic field. A possible experimental realization to observe the countercurrent is discussed.