Classical and quantum transport from generalized Landauer-Büttiker 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-B\"uttiker 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(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$), from a channel at position ${\mathbf{r}}_{\mathit{m}}$ to one at ${\mathbf{r}}_{\mathit{n}}$. A formal solution of the GLBE can be written as an effective transmittance T\ifmmode \tilde{}\else \~{}\fi{}(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$) which satisfies T\ifmmode \tilde{}\else \~{}\fi{}(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$)=T(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$) +Fd${\mathbf{r}}_{\mathit{i}}$ T(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{i}}$)${\mathit{g}}_{\mathit{i}}$T\ifmmode \tilde{}\else \~{}\fi{}(${\mathbf{r}}_{\mathit{i}}$,${\mathbf{r}}_{\mathit{m}}$), where 1/${\mathit{g}}_{\mathit{i}}$=Fd${\mathbf{r}}_{\mathit{j}}$ T(${\mathbf{r}}_{\mathit{j}}$,${\mathbf{r}}_{\mathit{i}}$). 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 ${\mathit{N}}_{0}$, Fermi velocity v, mean free path l, and localization length \ensuremath{\lambda}\ensuremath{\ge}l), the geometrical constraints, the measurement probes, and the electron-phonon interaction (providing the inelastic rate 1/${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$).By using known results for the Green's functions of infinite systems in dimension d, we show that T\ifmmode \tilde{}\else \~{}\fi{} defined in the above equation describes a conductivity of the form \ensuremath{\sigma}${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{d}}$=2${\mathit{e}}^{2}$D${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{d}}$${\mathit{N}}_{0}$. In the ballistic regime (v${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$l) the conductance is limited by inelastic scattering and the diffusion coefficient is D${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{d}}$=${\mathit{v}}^{2}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$/d. In the metallic regime of weak disorder (v${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$\ensuremath{\ll}\ensuremath{\lambda}), we obtain D${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathit{d}}$=${\mathit{D}}_{\mathit{d}}$==vl/d. These results, derived from microscopic principles, formalize an earlier picture of Thouless. Hence, we obtain the weak-localization correction for a quasi-one-dimensional case as a factor [1-(${\mathit{D}}_{1}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$${)}^{1/2}$/\ensuremath{\lambda}] in the diffusion coefficient. For strong localization (\ensuremath{\lambda}\ensuremath{\ll}${\mathit{D}}_{1}$${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$) we get D${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{1}$=${\ensuremath{\lambda}}^{2}$/3${\mathrm{\ensuremath{\tau}}}_{\mathrm{in}}$. The wide range of validity of the whole description gives further support to the GLBE which are then very appropriate to deal with transport not only in meso- scopic systems but also in macroscopic systems in the presence of inelastic processes.