Finite frequency backscattering current noise at a helical edge

Magnetic impurities with sufficient anisotropy could account for the observed strong deviation of the edge conductance of 2D topological insulators from the anticipated quantized value. In this work we consider such a helical edge coupled to dilute impurities with an arbitrary spin $S$ and a general form of the exchange matrix. We calculate the backscattering current noise at finite frequencies as a function of the temperature and applied voltage bias. We find that, in addition to the Lorentzian resonance at zero frequency, the backscattering current noise features Fano-type resonances at nonzero frequencies. The widths of the resonances are controlled by the spectrum of corresponding Korringa rates. At a fixed frequency the backscattering current noise has nonmonotonic behavior as a function of the bias voltage.

Figure 1

Sketch of the setup: a helical edge of a 2D topological insulator contaminated by dilute spin-$S$ magnetic impurities (see text).

Figure 2

Dependence of the normalized absolute value of backscattering admittance, $V{\mathcal{S}}_{\mathrm{bs}}/\left(2T\right|I_{\mathrm{bs}}\left|\right)$ at $V\ll JT$, on the dimensionless frequency $2\omega /\left(\pi T\right)$. The red solid curve is plotted for ${\mathcal{J}}_{xx}={\mathcal{J}}_{zx}={\mathcal{J}}_{yy}/2={\mathcal{J}}_{zz}$. The green dashed curve corresponds to ${\mathcal{J}}_{xx}={\mathcal{J}}_{zx}={\mathcal{J}}_{yy}/2=2{\mathcal{J}}_{yx}=5{\mathcal{J}}_{zy}={\mathcal{J}}_{zz}$. The blue dotted curve is plotted for ${\mathcal{J}}_{xx}/2={\mathcal{J}}_{zx}={\mathcal{J}}_{yy}={\mathcal{J}}_{zz}$. For all curves ${\mathcal{J}}_{zz}$ is equal to 0.01.

Figure 3

Dependence of the backscattering current noise, $4{\mathcal{S}}_{\mathrm{bs}}/\left(\pi T\right)$, on the dimensionless frequency $2\omega /\left(\pi T\right)$ at different values of $V/T\gg 1$ and $S$. The black lines are guides for an eye, marking the positions of the resonances.

Figure 4

Dependence of the backscattering current noise normalized by its value at zero bias voltage, ${\mathcal{S}}_{\mathrm{bs}}(\omega ,V)/{\mathcal{S}}_{\mathrm{bs}}(\omega ,0)$, on the dimensionless voltage $V/\left(2T\right)$, for $S=1/2,1,3/2,$ and 2. The exchange interaction is the same as in Fig. 3, i.e., ${\mathcal{J}}_{xx}={\mathcal{J}}_{zx}={\mathcal{J}}_{zz}=0.01$, ${\mathcal{J}}_{zy}={\mathcal{J}}_{yx}=0.005$, and ${\mathcal{J}}_{yy}=0.02$. The frequency is equal to $2\omega /\left(\pi T\right)=0.015$.