Mesoscopic electron and phonon transport through a curved wire

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. An interesting 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.

Figure 1

(a) An example of the type of two-dimensional curved wire geometry considered in this paper. The shape of the wire is used to define local orthogonal coordinates $X$ and $Y$, with $X$ the arclength along one of the edges. The width of the wire is $b$. (b) Scattering problem in the $XY$ frame. Here the wire appears straight, but the curvature induces an effective potential $V$ that causes scattering.

Figure 2

Dispersion relation for unperturbed scattering states, for both spinless electrons and scalar phonons. $\alpha$ is equal to $2mϵ∕{\hslash }^2$ in the case of electrons, or ${ϵ}^2∕{\hslash }^2{v}^2$ in the case of phonons. $\sigma$ is 1 on the right half of the figure, and $-1$ on the left. $b$ is the wire width.

Figure 3

Individual electron transmission probabilities $\mid t_{n{n}^{\prime }}{\mid }^2$ as a function of energy for a circular right-angle bend with $R∕b=1.20$. Here ${\Delta }_{\mathrm{el}}\equiv {\pi }^2{\hslash }^2∕2m{b}^2$.

Figure 4

Total electron transmission probability ${\mathsf{T}}_{\mathrm{el}}$. System parameters are the same as in Fig. 3.

Figure 5

Individual phonon transmission probabilities $\mid t_{n{n}^{\prime }}{\mid }^2$ as a function of energy for a circular right-angle bend with $R=2b$. Here ${\Delta }_{\mathrm{ph}}\equiv \pi \hslash v∕b$.

Figure 6

Individual phonon transmission probabilities $\mid t_{n{n}^{\prime }}{\mid }^2$ as a function of energy for a circular right-angle bend with $R=1.001b$.

Figure 7

Dimensionless thermal conductance $G_{\mathrm{th}}∕G_q$ as a function of temperature, for a $100\mathrm{nm}$ curved Si-like quantum wire, with outer radius $R=1.001b$. Here $G_q\equiv \pi k_B^{2}T∕6\hslash$, which is itself linearly proportional to temperature.

Figure 8

The solid curve is the same as Fig. 7. Dashed curve is the dimensionless thermal conductance for a straight Si-like wire with $b=100\mathrm{nm}$. Thermal transport is hardly suppressed by the bending.

Figure 9

Individual phonon transmission probabilities $\mid t_{n{n}^{\prime }}{\mid }^2$ as a function of energy for a circular right-angle bend with $R=1.00001b$.