Shot noise in Aharonov-Bohm interferometers: Comparison of helical and conventional setups

TL;DR

This work develops a comprehensive scattering-theory framework for shot noise in Aharonov-Bohm interferometers built from conventional spinless wires and helical edge states of a 2D topological insulator. By deriving energy-averaged expressions for conductance and the Fano factor, it reveals how quantum interference reshapes noise patterns as a function of the AB flux, tunneling strength, and defect-induced backscattering. A key finding is that in the helical interferometer, AB peaks in the Fano factor scale with the backscattering defect strength, enabling detection of topological-protection violations, whereas conventional interferometers exhibit AB-induced noise features even in the ballistic limit. The results, valid at relatively high temperatures where $\Delta \ll T$, provide practical guidelines for extracting defect information from simultaneous conductance and noise measurements in mesoscopic rings.

Abstract

We study tunneling transport through quantum Aharonov-Bohm (AB) interferometers and demonstrate that interference effects strongly modify shot noise of the current. We discuss in detail two simplest setups: conventional single-channel spinless interferometer and interferometer formed by helical edge states of two-dimensional topological insulator. We demonstrate that both in the conventional and the helical case the interference dramatically changes the Fano factor and its magnetic field dependence. For weak tunneling coupling, the Fano factor of both setups exhibits a periodic series of sharp AB peaks depending on the magnetic flux piercing the system. Our key finding is that the Fano factor in the helical interferometer provides information about the presence of backscattering defects violating topological protection. In particular, the amplitude of AB peaks in the helical setup is proportional to the strength of the defect in contrast to conventional setup, where peaks have finite amplitude even in the ballistic case.

Figures (10)

• Figure 1: Aharonov-Bohm interferometers based on conventional spinless single-channel quantum wire (a) and on helical edge channels of 2D topological insulator (b). The leads (shown by gray color) are modeled by conventional single-channel [spinless (a) and spinful (b)] wires. The black dot in panel (b) represents a backscattering impurity. Gray regions represent contacts described by $3\times 3$ and $4\times 4$$S-$matrix for CI and HI, respectively [see Eqs. \ref{['eq:SCI']}, \ref{['S-matrix']} and Figs. \ref{['Fig-conv-lead']}, \ref{['fig:lead']}]. Homogeneous magnetic field, $B,$ is perpendicular to the picture plane. Corresponding magnetic flux, $\Phi=B S,$ is proportional to the area, $S,$ of the region encompassed by one-dimensional channels
• Figure 2: Scattering amplitudes entering scattering matrix Eq. \ref{['eq:SCI']} for CI. The lead is modeled by spinless 1D wire. For tunneling contact, $|t_{\rm r}| \approx 1,$ while the case of metallic contact corresponds to $t_{\rm r}=0.$ Contact region with area $S_0$ is shown by gray color. We assume that $B S_0 \ll \Phi_0,$ so that the magnetic field does not affect S-matrix.
• Figure 3: Schematic illustration of different types of leads (contact area is shown by gray color): panels (a) and (b) shows two different types of tunneling coupling, corresponding to $\gamma=0$ and $\gamma=\infty,$ respectively; panel (c) illustrates "metallic" contact, $\gamma \approx 1.$ Arrows in panels (a) and (b) show processes with non-zero amplitudes (compare with Fig. \ref{['Fig-conv-lead']}). Panel (c) corresponds to a "metallic contact", so that all processes in Fig. \ref{['Fig-conv-lead']} are allowed. Scattering matrices, describing cases (a), (b), and (c) are given by Eq. \ref{['eq:3cases']}.
• Figure 4: The scattering amplitudes entering scattering matrix of contacts, Eq. \ref{['S-matrix']}, which are modeled as a single-channel spinful wire. It is assumed that there is no spin flip at the contact and the two spin polarizations at the contact (shown by red and blue color) are completely separated. The $t\to 1$ case corresponds to a tunnel contact, and the $t\to 0$ case models a metal contact. Contact region with area $S_0$ is shown by gray color. We assume that $B S_0 \ll \Phi_0,$ so that the field does not affect S-matrix.
• Figure 5: The dependence of Fano factor, $\mathcal{F}^\mathrm{CI}(\phi,t_\mathrm{b})$, on the amplitude of backscattering on the contact, $t_\mathrm{b}$, and the magnetic flux, $\phi$, as described by Eqs. \ref{['eq:SCI']}, \ref{['TCIsym']}, \ref{['FFCIsym']}.