Low-field hopping conductivity in granular metals

The hopping conductivity of granular metals is known to be of the form sigma varies as exp (-(T0/T)12/) in the temperature range 20K<or=T<280K. A theoretical derivation of this relation using the critical percolation conductance method as developed in the context of the variable range hopping mechanism of conduction in amorphous systems, is discussed. Use is made of the empirical fact that the ratio of intergrain separation and grain size is nearly constant for macroscopically homogeneous samples. It is also assumed that the hopping of electrons is predominantly between nearest neighbours. Extension of the treatment to thin films and comparison with previous theoretical work is briefly discussed.