Coulomb effects and hopping transport in granular metals

We investigate effects of Coulomb interaction and hopping transport in the insulator phase of granular metals and quantum dot arrays considering both spatially periodic as well as irregular grain or dot arraignments. We study the Mott transition between the insulating and metallic phases in a strictly periodic system and find the dependence of the Mott gap on the intergranular coupling. In this case the conductivity of the insulating state has the activation form with the Mott gap entering the exponent. In the irregular arrays the electrostatic disorder induces the finite density of states near the Fermi level giving rise to the variable range hopping conductivity. We derive the transport properties of the irregular array in the dielectric, low coupling limit and show that the conductivity follows the Efros-Shklovskii law. We develop a theory of tunneling through a chain of grains and discuss in detail both elastic and inelastic cotunneling mechanisms; the former dominates at very low temperatures and very low applied electric fields, while the inelastic mechanism controls tunneling at high temperature or fields. Our results are obtained within the framework of the technique based on the mapping of the quantum electronic problem onto the classical gas of Coulomb charges. The processes of quantum tunneling of real electrons are represented in this technique as trajectories (world lines) of charged classical particles in d+1 dimensions. The Mott gap is related to the dielectric susceptibility of the Coulomb gas in the direction of the imaginary time axis.