Finite-frequency admittance and noise of a helical edge coupled to a magnet

TL;DR

This work addresses finite-frequency transport and current fluctuations in a helical edge of a quantum spin-Hall insulator exchange-coupled to an easy-plane magnet. Using a scattering framework, it derives the finite-frequency admittance $G_0(ω)$ and equilibrium noise $S_0(ω)$, showing how the magnet’s slow precession fluctuations restore Nyquist-like noise at $ω o0$ while suppressing noise for $|ω| au o ext{large}$, and verifies the fluctuation-dissipation relation for $|ω| oxdot ω_{Th}$. It further shows that under a DC bias $V$, the admittance and noise maps to the zero-bias results with a bias-dependent relaxation time $τ_V$, yielding bias independence except for $τ_V$, consistent with the absence of shot noise. The paper highlights a Josephson-like relation $Ω_V = eV/ħ$ between magnet precession and applied voltage, discusses conditions on magnet size and damping, and envisions practical uses such as frequency-selective noise suppression and low-pass filtering in quantum spin-Hall devices.

Abstract

The exchange coupling of the helical edge state of a quantum spin-Hall insulator with an easy-plane magnet has no effect on its DC electrical conductance if the magnet's anisotropy axis is aligned with the spin quantization axis of the helical edge state [Meng et al., Phys. Rev. B 90, 205403 (2014)]. We here calculate the AC conductance $G_V(ω)$ and the noise power $S_V(ω)$ in the presence of a DC bias $V$. While both take the universal values $G_V({ω= 0}) = e^2/h$ and $S_V(ω= 0) = 4 e^2 k_{\rm B} T/h$ in the zero-frequency limit, $G_V(ω)$ and $S_V(ω)$ are quickly suppressed for finite $ω$, so that low-frequency transport is effectively noiseless.

Figures (2)

• Figure 1: Schematic illustration of a magnet coupled to the helical edge states of a quantum spin Hall insulator (panel a). The exchange coupling to the magnet opens a gap in the spectrum of the helical edge, causing backscattering of electrons accompanied by a spin-flip. This scattering transfers angular momentum to the magnet, driving it out of the easy plane and, hence, inducing a precession of its magnetization, which, in turn, pumps a spin current into the helical edge. Thermal fluctuations in the current carried by incoming electrons are transferred to the magnet and cause small fluctuations in the out-of-plane canting angle and, hence, in the precession frequency $\Omega$. These small fluctuations cause a pumped current temporally correlated with the fluctuations of the incident electron current. Due to the magnet's finite response time $\tau$, high-frequency components of the noise are suppressed. This frequency filtering is illustrated in the two current traces in the bottom panel b). The bottom left panel shows a current trace with correlations $\langle I(\omega)^2\rangle \propto \hbar \omega (e^{\hbar \omega/k_{\rm B}T}-1)^{-1}$, corresponding to Eq. (\ref{['eq:fdt']}) with $G_0(\omega) = e^2/h$ after removing the zero-point fluctuations, whereas the bottom right panel includes the additional low-pass factor $(1-i\omega\tau)^{-1}$ in the presence of a magnet, see Eq. (\ref{['eq:Ginfinity']}).
• Figure 2: AC conductance $G_0(\omega)$ vs. frequency $\omega$ for the case that the magnet-induced gap in the spectrum of the helical edge is of size $2 \varepsilon_{\rm gap}$, with the Fermi energy in the gap center (see text). The solid curves show $G_0(\omega)$ for $k_{\rm B} T/\varepsilon_{\rm gap} = 0.1$, $1$, and $10$ (see legend). The dashed lines indicate the limit $\tau \to \infty$ of Eq. (\ref{['eq:tauinf']}). The dotted curve indicates the limit $\varepsilon_{\rm gap}/k_{\rm B} T \to \infty$ of Eq. (\ref{['eq:Ginfinity']}). For temperature $k_{\rm B} T \ll \varepsilon_{\rm gap}$, the AC conductance and the thermal noise are strongly suppressed in the frequency range $\tau^{-1} \ll \omega \ll \varepsilon_{\rm gap}/\hbar$.