Universal Crossover between Efros-Shklovskii and Mott Variable-Range-Hopping Regimes.

A universal scaling function, describing the crossover between the Mott and the Efros-Shklovskii hopping regimes, is derived, using the percolation picture of transport in strongly localized systems. This function is agrees very well with experimental data. Quantitative comparison with experiment allows for the possible determination of the role played by polarons in the transport. 72.20.-i, 71.55.Jw, 05.60+w, 71.38+i Typeset using REVTEX 1 Electronic interactions are known to play an important role in the strongly localized regime. More than two decades ago, Pollak [1], and Efros and Shklovskii [2] pointed out that the long-range nature of the interactions leads to a dip in the single-particle density of states, ρ(ǫ), at the Fermi energy. Using a constraint on the single-particle excitations, Efros and Shklovskii [2] argued that this soft gap is of the form ρ(ǫ) ∼ ǫ, where ǫ is the energy measured from the Fermi energy and d the space dimension. The gap in the density of states was indeed observed by tunneling [3,4] and photoemission [5] measurements. Moreover, it modifies the Mott variable-range hopping law, logR ∼ 1/T , from x = d + 1 to x = 1/2. The modified exponent has been observed in many experiments [6]. The Efros-Shklovskii hopping law is expected to be relevant at low temperatures (compared to the size of the gap) [2,7], and in the last few years there have been a multitude of experiments aimed at exploring the crossover between the Mott to the Efros-Shklovskii hopping regimes as a function of temperature [8]. A significant step was taken more recently, when Aharony and coworkers [9,10] have argued that this crossover is described by a universal scaling function, and obtained this scaling function phenomenologically, using energy additivity. It is the aim of this paper to demonstrate that the “microscopic” percolation picture [11], describing the transport in the strongly localized regime, indeed leads to such a universal crossover function, and to derive an equation for this function. The results agree excellently with existing experimental data [4], and allow quantitative comparison between features in the density of the states and the transport data, taken on the same physical system. The starting point of this calculation is the mapping of the resistance problem into a percolation criterion [11] of an equivalent random resistor network [12], consisting of randomly placed sites. Without interactions, the activation energy from site i to site j is given by ǫj−ǫi, where ǫi are the energy of site i, which in this case is distributed uniformly. The resistance between each pair of sites is given by [12] Rij = exp {(|ǫi|+ |ǫj |+ |ǫi − ǫj |)|/2T + 2rij/ξ}, where all resistances are measured in some unit of resistance. Since the resistances vary exponentially, the overall resistance will be determined by the weakest link, which is the largest 2 resistance, R, such that the cluster formed by all resistances (bonds), satisfying R > Rij, percolates. Clearly, all states participating in the percolating network (defined as occupied sites) must satisfy R > exp(|ǫi|/2T ). Following [11], the percolation criterion employed here is the following [13] — given such an occupied site , the number of bonds attached to it has to be higher than a critical threshold, Zc, for the system to percolate, Zc = 1 2 ∫ T log R −T log R ρ0dǫ ∫ 2T logR −2T logR ρ0 dǫ1 ∫