Scaling of the conductance and resistance of square lattices with an exponentially wide spectrum of the resistances of links

Abstract

The conductance G̅ and \(\overline {{G^{ - 1}}} \) resistance average over realizations of disorder have been calculated for various sizes of square lattices L. In contrast with different direction of change in the two quantities at percolation in lattices with the binary spread of conductances of links (g_i = 0 or 1), it has been found that the mean conductance and resistance of lattices decrease simultaneously with an increase in L in the case of an exponential distribution of local conductances g_i = exp(−kx_i), where x_i ∈ [0,1] are random numbers. When L is smaller than the disorder length L₀ = bk^v, G̅(L) and \(\overline {{G^{ - 1}}} \)(L) are proportional to L⁻ⁿ with n = k/5 and k/6, respectively. A similar behavior is characteristic of the distributions of conductances of links, which simulate a transition between the open and tunneling regimes in semiconducting lattices of antidots created in a two-dimensional electron gas.