Classical and quantum transport from generalized Landauer-Büttiker equations

HORACIO M. PASTAWSKI

PHYSICAL REVIEW B - SOLID STATE

American Physical Society

Lugar: Ridge N.Y.; Año: 1991 vol. 44 p. 6329 - 6339

0556-2805

Phys. Rev. B 44, 6329 - 6339 (1991) 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(rn,rm), from a channel at position rm to one at rn. A formal solution of the GLBE can be written as an effective transmittance T̃(rn,rm) which satisfies T̃(rn,rm)=T(rn,rm) +Fdri T(rn,ri)giT̃(ri,rm), where 1/gi=Fdrj T(rj,ri). 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 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/τin). 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 σ̃d=2e2D̃dN0. In the ballistic regime (vτin<l) the conductance is limited by inelastic scattering and the diffusion coefficient is D̃d=v2τin/d. In the metallic regime of weak disorder (vτin≪λ), we obtain D̃d=Dd==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-(D1τin)1/2/λ] in the diffusion coefficient. For strong localization (λ≪D1τin) we get D̃1=λ2/3τin. 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.