Conductance calculations for quantum wires and interfaces: mode matching and Green functions

TL;DR

This paper generalizes Ando's mode matching technique for calculating quantum conductance, linking it to Green function methods, and demonstrates its application to simple models and first-principles Fe|vacuum|Fe junctions.

Contribution

It extends mode matching techniques to any tight-binding Hamiltonian and connects them with Green function approaches for conductance calculations.

Findings

Derived a compact expression for transmission matrix elements.

Showed equivalence of mode matching and Green function results.

Applied methods to analytical models and first-principles Fe junctions.

Abstract

Landauer's formula relates the conductance of a quantum wire or interface to transmission probabilities. Total transmission probabilities are frequently calculated using Green function techniques and an expression first derived by Caroli. Alternatively, partial transmission probabilities can be calculated from the scattering wave functions that are obtained by matching the wave functions in the scattering region to the Bloch modes of ideal bulk leads. An elegant technique for doing this, formulated originally by Ando, is here generalized to any Hamiltonian that can be represented in tight-binding form. A more compact expression for the transmission matrix elements is derived and it is shown how all the Green function results can be derived from the mode matching technique. We illustrate this for a simple model which can be studied analytically, and for an Fe|vacuum|Fe tunnel junction which we study using first-principles calculations.