Quantum and Boltzmann transport in a quasi-one-dimensional wire with rough edges

Abstract

We study electron transport in quasi-one-dimensional metallic wires. 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 first study the impurity-free wire whose edges have roughness with a correlation length comparable with the Fermi wavelength. Simulating wires with the number of the conducting channels (N c ) as large as 34-347, we observe the roughness-mediated effects which are not observable for small N c (∼ 3-9) used in previous works. First, we observe the crossover from the quasi-ballistic transport to the diffusive one, where the ratio of the quasi-ballistic resistivity to the diffusive resistivity is ∼N c independent of the parameters of roughness. Second, we find that transport in the diffusive regime is carried by a small effective number of open channels, equal to ∼6. This number is universal-independent of N c and of the parameters of roughness. Third, we see that the inverse mean conductance rises linearly with the wire length (a sign of the diffusive regime) up to the length twice larger than the electron localization length. We develop a theory based on the weak-scattering limit and semiclassical Boltzmann equation, and we explain the first and second observations analytically. For the impurity disorder we find a standard diffusive behavior. Finally, we derive from the Boltzmann equation the semiclassical electron mean free path and we compare it with the quantum mean free path obtained from the Landauer conductance. They coincide for the impurity disorder; however, for the edge roughness they strongly differ, i.e., the diffusive transport in the wire with rough edges is not semiclassical. It becomes semiclassical only for roughness with a large correlation length. The conductance then behaves like the conductance of the wire with impurities, also showing the conductance fluctuations of the same size.

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