Büttiker-Landauer conductance formulas in the presence of inelastic scattering

We derive analytic expressions for the two- and four-probe Büttiker-Landauer conductances in one-dimensional quantum-wire structures. The wire contains both elastic barriers and distributed inelastic scatterers and is connected between two contacts or electron reservoirs. It is shown that the transport of electrons in such structures may be described by two local-group-velocity distributions. The concept of group velocities has been exploited to decompose the current density into two oppositely flowing electronic current components. The two current streams are governed by continuity equations in which the local group velocities determine the inelastic-scattering probabilities in the forward and the backward directions. The inelastic-scattering events essentially provide a mechanism through which electrons are exchanged between the two current streams. Solution of the continuity equations for phase-breaking processes yields the two- and four-probe conductances that are expressed as analytic functions of the group velocities and phase-randomizing scattering time.