Conductance of a disordered linear chain including inelastic scattering events

A study of the conductance in a disordered linear chain of finite length L including inelastic scattering processes is presented. Inelastic scatterers are introduced as defined by Büttiker and are assumed to be uniformly distributed along the system. This defines an inelastic scattering time ${\mathrm{\tau }}_{\mathrm{in}}$ plus a condition of charge conservation which in turn introduces incoherent electrons. The four-probe conductance of the system is then reduced to a Landauer-like behavior G=2(${\mathit{e}}^2$/h)${\mathit{T}}_{\mathrm{eff}}$/ (1-${\mathit{T}}_{\mathrm{eff}}$), where the effective transmission through the sample, ${\mathit{T}}_{\mathrm{eff}}$, is the sum of two terms, one of which accounts for the phase-coherent electrons which have not suffered any inelastic collision and another for electrons which have suffered at least one inelastic collision in their journey. To show explicitly this point, the conductance of an ordered system is analyzed. Analytical and numerical results are presented for disordered chains, where resonances in the transmission present a width which is associated with the minimum between the escape time and the relaxation time. Because of the denominator in the Landauer formula, strong fluctuations on the conductance are present even in the weak-disordered situation in which the localization length λ>L, but we observed that they become of order ${\mathit{e}}^2$/h when the inelastic scattering length ${\mathit{L}}_{\mathrm{in}}$=L. Further decrease of the inelastic length causes the fluctuations to reduce following similar laws to that of the metallic regime.