Anomalous Hopping Exponents of Ultrathin Films of Metals

and x are constantswhich depend on the disorder, the details of the inter-actions and the dimensionality of the system. Simpleactivated hopping over a constant barrier results in theArrhenius form with x = 1. For noninteracting electrons,when the average hopping distance depends on temper-ature due to the compromise between hopping to siteswhich are close in energy, but farther away, Mott variablerange hopping [1] is expected, with x = 1/(d+1), where dis the dimension. Efros and Shklovskii (ES) showed thatincluding Coulomb interactions between electrons resultsin a soft gap in the density of states at the Fermi energy,which changes the variable range hopping exponent tox = 1/2 in all dimensions [2].Hopping conduction has been investigated in a widevariety of materials, such as doped semiconductors [3,4],semiconducting heterostructures [5], amorphous metals[6–10], magnetic materials [11] and superconductors [12].Both the Mott and the ES forms of variable range hop-ping have been observed, as well as a crossover betweenthe two regimes [5,6]. It should be emphasized, how-ever, that it is often hard to distinguish between Mottand ES hopping, particularly in experiments in which theresistance changes only by one or two orders of magni-tude. The unambiguous identification of the Mott or EShopping can be further complicated by factors which areusually neglected, such as the granularity of the system,possible temperature dependence of R