Quantum and Boltzmann transport in the quasi-one-dimensional wire with rough edges
We study quantum transport in Q1D wires made of a 2D conductor of width W and length L>>W. Our aim is to compare an impurity-free wire with rough edges with a smooth wire with impurity disorder. We calculate the electron transmission through the wires by the scattering-matrix method, and we find the Landauer conductance for a large ensemble of disordered wires. We study the impurity-free wire whose edges have a roughness correlation length comparable with the Fermi wave length. The mean resistance <ρ> and inverse mean conductance 1/<g> are evaluated in dependence on L. For L -> 0 we observe the quasi-ballistic dependence 1/<g> = <ρ> = 1/N_c + ρ_qb L/W, where 1/N_c is the fundamental contact resistance and ρ_qb is the quasi-ballistic resistivity. As L increases, we observe crossover to the diffusive dependence 1/<g> = <ρ> = 1/N^eff_c + ρ_dif L/W, where ρ_dif is the resistivity and 1/N^eff_c is the effective contact resistance corresponding to the N^eff_c open channels. We find the universal results ρ_qb/ρ_dif = 0.6N_c and N^eff_c = 6 for N_c >> 1. As L exceeds the localization length ξ, the resistance shows onset of localization while the conductance shows the diffusive dependence 1/<g> = 1/N^eff_c + ρ_dif L/W up to L = 2ξ and the localization for L > 2ξ only. On the contrary, for the impurity disorder we find a standard diffusive behavior, namely 1/<g> = <ρ> = 1/N_c + ρ_dif L/W for L < ξ. We also derive the wire conductivity from the semiclassical Boltzmann equation, and we compare the semiclassical electron mean-free path with the mean free path obtained from the quantum resistivity ρ_dif. They coincide for the impurity disorder, however, for the edge roughness they strongly differ, i.e., the diffusive transport is not semiclassical. It becomes semiclassical for the edge roughness with large correlation length.