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-Büttiker 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 T̃(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$) which satisfies T̃(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$)=T(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{m}}$) +Fd${\mathbf{r}}_{\mathit{i}}$ T(${\mathbf{r}}_{\mathit{n}}$,${\mathbf{r}}_{\mathit{i}}$)${\mathit{g}}_{\mathit{i}}$T̃(${\mathbf{r}}_{\mathit{i}}$,${\mathbf{r}}_{\mathit{m}}$), 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 λ≥l), the geometrical constraints, the measurement probes, and the electron-phonon interaction (providing the inelastic rate 1/${\mathrm{\tau }}_{\mathrm{in}}$).

By using known results for the Green’s functions of infinite systems in dimension d, we show that T̃ defined in the above equation describes a conductivity of the form σ${\mathrm{̃}}_{\mathit{d}}$=2${\mathit{e}}^2$D${\mathrm{̃}}_{\mathit{d}}$${\mathit{N}}_0$. In the ballistic regime (v${\mathrm{\tau }}_{\mathrm{in}}$<l) the conductance is limited by inelastic scattering and the diffusion coefficient is D${\mathrm{̃}}_{\mathit{d}}$=${\mathit{v}}^2$${\mathrm{\tau }}_{\mathrm{in}}$/d. In the metallic regime of weak disorder (v${\mathrm{\tau }}_{\mathrm{in}}$≪λ), we obtain D${\mathrm{̃}}_{\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$${\mathrm{\tau }}_{\mathrm{in}}$${)}^{1/2}$/λ] in the diffusion coefficient. For strong localization (λ≪${\mathit{D}}_1$${\mathrm{\tau }}_{\mathrm{in}}$) we get D${\mathrm{̃}}_1$=${\lambda }^2$/3${\mathrm{\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.