Mesoscopic Electron and Phonon Transport through a Curved Wire

TL;DR

This paper develops a theoretical method to analyze wave propagation in curved nanowires, revealing that curvature minimally affects phonon thermal transport but can influence electron transport.

Contribution

It introduces a general technique for modeling wave scattering in curved wires, including a novel analysis of phonon transport and curvature effects.

Findings

Curvature causes negligible suppression of phonon thermal transport.

The method applies to both electron and phonon wave equations with different boundary conditions.

Reflection probability for phonons vanishes at long wavelengths, simplifying low-energy analysis.

Abstract

There is great interest in the development of novel nanomachines that use charge, spin, or energy transport, to enable new sensors with unprecedented measurement capabilities. Electrical and thermal transport in these mesoscopic systems typically involves wave propagation through a nanoscale geometry such as a quantum wire. In this paper we present a general theoretical technique to describe wave propagation through a curved wire of uniform cross-section and lying in a plane, but of otherwise arbitrary shape. The method consists of (i) introducing a local orthogonal coordinate system, the arclength and two locally perpendicular coordinate axes, dictated by the shape of the wire; (ii) rewriting the wave equation of interest in this system; (iii) identifying an effective scattering potential caused by the local curvature; and (iv), solving the associated Lippmann-Schwinger equation for the scattering matrix. We carry out this procedure in detail for the scalar Helmholtz equation with both hard-wall and stress-free boundary conditions, appropriate for the mesoscopic transport of electrons and (scalar) phonons. A novel aspect of the phonon case is that the reflection probability always vanishes in the long-wavelength limit, allowing a simple perturbative (Born approximation) treatment at low energies. Our results show that, in contrast to charge transport, curvature only barely suppresses thermal transport, even for sharply bent wires, at least within the two-dimensional scalar phonon model considered. Applications to experiments are also discussed.