Current Noise from a Magnetic Moment in a Helical Edge

We calculate the two-terminal current noise generated by a magnetic moment coupled to a helical edge of a two-dimensional topological insulator. When the system is symmetric with respect to in-plane spin rotation, the noise is dominated by the Nyquist component even in the presence of a voltage bias $V$. The corresponding noise spectrum $S(V,\omega )$ is determined by a modified fluctuation-dissipation theorem with the differential conductance $G(V,\omega )$ in place of the linear one. The differential noise $\partial S/\partial V$, commonly measured in experiments, is strongly dependent on frequency on a small scale ${\tau }_K^{-1}\ll T$ set by the Korringa relaxation rate of the local moment. This is in stark contrast to the case of conventional mesoscopic conductors where $\partial S/\partial V$ is frequency independent and defined by the shot noise. In a helical edge, a violation of the spin-rotation symmetry leads to the shot noise, which becomes important only at a high bias. Uncharacteristically for a fermion system, this noise in the backscattered current is super-Poissonian.

Figure 1

The low-frequency differential noise $\partial S/\partial V$, commonly measured by using bias modulation technique. At moderately low bias $eV\lesssim 4T$ and $\omega \gtrsim {\tau }_K^{-1}$, the main contribution to $\partial S/\partial V$ comes from the equilibrium noise, Eq. (1); the shot noise is smaller by a factor $(\delta J/J{)}^2$. At higher bias, the equilibrium noise contribution falls off exponentially with $eV/T$, and $\partial S/\partial V$ is dominated by the shot noise, Eq. (7). Here $\delta J=0.5J$, and the spin-flip rate ${\tau }_K^{-1}=0.025\mathrm{T}$ at $V=0$.

Figure 2

Super-Poissonian shot noise in the backscattered current $\delta I$. At large bias $eV\gg T$, the local spin spends most of its time in the quasiequilibrium state $\uparrow$, $⟨S_z⟩=1/2$. At random times, on average once per time $\tau \propto 1/J_{--}^2$, the exchange term $\delta J_{--}{S}_{-}{s}_{-}+\mathrm{H}.\mathrm{c}.$ backscatters one electron and flips the localized spin ($\uparrow \to \downarrow$ in figure). After a short time ${\tau }_K\ll \tau$, the local moment relaxes to its quasiequilibrium state ($\downarrow \to \uparrow$). In this process, another electron is scattered back. Since $\tau \gg {\tau }_K$, the time-averaged backscattered current is $2e/\tau$ and the effective backscattered charge is $2e$ when probed at frequencies $\omega \ll 1/{\tau }_K$.