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/${\mathrm{\tau }}_{\mathrm{in}}$).