Granular electronic systems

Granular metals are arrays of metallic particles of a size ranging usually from a few to hundreds of nanometers embedded into an insulating matrix. Metallic granules are often viewed as artificial atoms. Accordingly, granular arrays can be treated as artificial solids with programmable electronic properties. The ease of adjusting electronic properties of granular metals assures them an important role for nanotechnological applications and makes them most suitable for fundamental studies of disordered solids. This review discusses recent theoretical advances in the study of granular metals, emphasizing the interplay of disorder, quantum effects, fluctuations, and effects of confinement. These key elements are quantified by the tunneling conductance between granules $g$, the charging energy of a single granule ${E}_{c}$, the mean level spacing within a granule $\ensuremath{\delta}$, and the mean electronic lifetime within the granule $\ensuremath{\hbar}∕g\ensuremath{\delta}$. By tuning the coupling between granules the system can be made either a good metal for $gg{g}_{c}=(1∕2\ensuremath{\pi}d)\mathrm{ln}({E}_{c}∕\ensuremath{\delta})$ ($d$ is the system dimensionality), or an insulator for $gl{g}_{c}$. The metallic phase in its turn is governed by the characteristic energy $\ensuremath{\Gamma}=g\ensuremath{\delta}$: at high temperatures $Tg\ensuremath{\Gamma}$ the resistivity exhibits universal logarithmic temperature behavior specific to granular materials, while at $Tl\ensuremath{\Gamma}$ the transport properties are those generic for all disordered metals. In the insulator phase the transport exhibits a variety of activation behaviors including the long-puzzling $\ensuremath{\sigma}\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}({T}_{0}∕T{)}^{1∕2}]$ hopping conductivity. Superconductivity adds to the richness of the observed phases via one more energy parameter $\ensuremath{\Delta}$. Using a wide range of recently developed theoretical approaches, it is possible to obtain a detailed understanding of the electronic transport and thermodynamic properties of granular materials, as is required for their applications.