Equivalence of wave function matching and Green's functions methods for quantum transport: generalized Fisher-Lee relation

TL;DR

This paper proves the exact equivalence between wave function matching and Green's functions methods in quantum transport, enabling flexible approach selection and deriving a generalized Fisher-Lee relation for phonon mode transmission analysis.

Contribution

It establishes a rigorous proof of equivalence between two key quantum transport methods and derives a generalized Fisher-Lee relation for phonon transmission.

Findings

Proved the exact equivalence of wave function matching and Green's functions methods.

Derived a generalized Fisher-Lee formula for phonon mode transmission.

Validated the approach with a simple model demonstrating accuracy and simplicity.

Abstract

We present a proof of an exact equivalence of the two approaches that are most used in computing conductance in quantum electron and phonon transport: the wave function matching and Green's functions methods. We can obtain all the quantities defined in one method starting from those obtained in the other. This completes and illuminates the work started Ando[Ando T 1991 Phys. Rev. B 44 8017] and continued later by Komyakov et al.[Khomyakov P A, Brocks G, Karpan V, Zwierzycki M and Kelly P J 2005 Phys. Rev. B 72 035450]. The aim is to allow for solving the transport problem with whichever approach fits most the system at hand. One major corollary of the proven equivalence is our derivation of a generalized Fisher-Lee formula for resolving the transmission function into individual phonon mode contributions. As an illustration, we applied our method to a simple model to highlight its accuracy and simplicity.