CLASSICAL AND QUANTUM TRANSPORT FROM GENERALIZED LANDAUER-BUTTIKER EQUATIONS

The electronic transport in a finite-size sample under the presence of inelastic processes, such as electron-phonon interaction, can be described with the generalized Landauer-Buttiker equations (GLBE). These use the equivalence between the inelastic channels and a continuous distribution of voltage probes to establish a current balance. The essential parameters in the GLBE are the transmission probabilities, T(r(n),r(m)), from a channel at position r(m) to one at r(n). A formal solution of the GLBE can be written as an effective transmittance T approximately (r(n),r(m)) which satisfies T approximately (r(n),r(m)) = T(r(n),r(m)) + integral dr(i) T(r(n),r(i))g(i)T approximately (r(i),r(m)), where 1/g(i) = integral dr(j) T(r(j),r(i)). The T's are obtained from the Green's functions of a Hamiltonian which models the electronic structure of the sample (with density of states N0, Fermi velocity upsilon, mean free path l, and localization length lambda greater-than-or-equal-to l), the geometrical constraints, the measurement probes, and the electron-phonon interaction (providing the inelastic rate 1/tau-in). By using known results for the Green's functions of infinite systems in dimension d, we show that T approximately defined in the above equation describes a conductivity of the form sigma approximately d = 2e2D approximately (d)N0. In the ballistic regime (upsilon-tau-in < l) the conductance is limited by inelastic scattering and the diffusion coefficient is D approximately d = upsilon-2-tau-in/d. In the metallic regime of weak disorder (upsilon-tau-in << lambda), we obtain D approximately d = D(d) = upsilon-l/d. These results, derived from microscopic principles, formalize an earlier picture of Thouless. Hence, we obtain the weak-localization correction for a quasi-one-dimensional case as a factor [1-(D1-tau-in)1/2/lambda] in the diffusion coefficient. For strong localization (lambda << D1-tau-in) we get D approximately 1 = lambda-2/3-tau-in. The wide range of validity of the whole discription gives further support to the GLBE which are then very appropriate to deal with transport not only in mesoscopic systems but also in macroscopic systems in the presence of inelastic processes.