Hopping transport in granular metals and superconductors

Valerii Vinokur

Materials Science Division, Argonne National Laboratory, Argonne, USA

Arrays of metallic granules, granular metals, are in the insulating phase at low temperatures if the coupling between granules is weak and exhibit stretched exponential conductivity behavior exp[-(T0/T)1/2] resembling Efros-Shklovskii hopping conductivity in semiconductors. This temperature dependence observed not only in numerous experiments on disordered granular metals but, strikingly, also in perfectly periodic arrays of metallic granules and periodic arrays of semiconductor quantum dots without visible traces of disorder posed a fundamental problem that resisted almost three decade long intense attacks of theorists. We explain the observed dependence in terms of the Mott-Efros-Shklovskii-like variable range hopping mechanism. Our approach includes two key ingredients: (i) electrostatic disorder induced on the granules by the charged centers imbedded in dielectric matrix of granular conductors and (ii) the theory of coherent co-tunneling through the chain of granules providing mechanism for the electron hopping over several granules.