Characteristic Length of Conductivity in a 3D Miller–Abrahams Resistor Network as a Function of Cell Size

Abstract

Conductivity in the hopping regime is usually described using a simplified  Miller–Abrahams  resistor network model. A solution for the conductivity of  the  infinite Miller–Abrahams resistor network was found in the framework of percolation theory. This solution shows an excellent agreement for hopping distances that are large compared to the localization radius of electrons,  a.  However, the tests were only targeting the argument of the exponential function, which depends on the concentration of localization sites and the localization radius of electrons. In this context, the experiment is carried out as a function of the average impurity separation d₀ or the localization radius. The prefactor of the exponential function was not tested independently. The prefactor is difficult to analyze based on resistivity measurement alone, because the interpretation is ambiguous with the exponential function in the background. In the percolation approach, the prefactor is parameterized as the product of the correlation radius L₀ of the critical subnetwork and an unknown constant A of order of unity. In principle, the prefactor can be investigated independently by making measurements as a function of sample size, near the transition from the microscopic to the macroscopic regime. In this work, the prefactor was investigated by simulating the resistivity of Miller–Abrahams resistor network as a function of cell size. To assist the interpretation, the same cells were additionally solved using the percolation method directly, to find the critical percolation distance  r_c  of each cell. The product  A L₀  was calculated as a function  of  d₀/a.  A  reasonable agreement was found with the prediction of the percolation approach,  L₀ ~ (d₀/a)^ν. Despite this agreement, the intermediate results often show disagreement with the purely percolation-based view, as well as with initial assumptions. Among these, the variation of conductivity  σ(L)  with cell size is the opposite of what was expected; it shows no crossover to a macroscopic regime and exhibit no correlation with the product  A L₀.  There is also an unexpected, universal behavior of  σ(L)  for large  L.  It  is  proposed that, in all simulated regimes, there exists an additional mean-field component of the conductance, which operates in parallel with the conductance by the network of critical resistors.