Noiseless manipulation of helical edge state transport by a quantum magnet

The current through a helical edge state of a quantum spin Hall insulator may be fully transmitted through a magnetically gapped region due to a combination of spin-transfer torque and spin pumping [Meng et al., Phys. Rev. B 90, 205403 (2014)]. Using a scattering approach, we here argue that in such a system the current is effectively carried by electrons with energies below the magnet-induced gap and well below the Fermi energy. This has striking consequences, such as the absence of shot noise, an exponential suppression of thermal noise, and an obstruction of thermal transport. For two helical edges covered by the same quantum magnet, the device can act as a robust noiseless current splitter.

Figure 1

Schematic drawing of the geometry we consider: One helical edge (a) or both edges (b) of a two-dimensional topological insulator (gray) exchange coupled to a magnetic insulator.

Figure 2

Schematic illustration of reflection and transmission as a function of energy, for the special case that the exchange coupling $h\left(x\right)$ is a smooth function of $x$ and the chemical potential $\mu$ lies inside the spectral gap induced by the coupling to the ferromagnet. The vertical axis refers to the electrons' kinetic energy, which changes by an amount $\hslash \omega =eV$ upon reflection. Electrons carrying the actual current are shown by red dashed lines. Although details change if the coupling function $h\left(x\right)$ is not smooth, the conclusion that the current is carried effectively by electrons far away from the Fermi level remains true as long as the chemical potential is inside the gap [20].

Figure 3

Illustration of distribution functions for incoming (top) and outgoing (bottom) electrons, for the case $T_{\mathrm{L}}/eV=0.2, T_{\mathrm{R}}/eV=0.02, {\left|r\left(\varepsilon \right)\right|}^2=1-{\left|t\left(\varepsilon \right)\right|}^2=0.8$. The location of the “step” in the distribution function for the outgoing electrons corresponds to that of the opposite reservoir, consistent with the perfect transmission of the incident current. For this example, the width of the step is, however, predominantly that of the same-side reservoir, consistent with the fact that the device acts as a “thermal insulator.”

Figure 4

Noiseless partitioning of the current between the two edges of the topological insulator (TI) covered by the same magnet. The current is injected through the left contact of the upper edge 1 with chemical potential ${\mu }_{1\mathrm{L}}={\mu }_{1\mathrm{R}}+eV$. In the steady state the magnetic moment precesses at a frequency $\hslash \omega =eV/2$. In this case half of the injected current is reflected back with the energies ${\mu }_{1\mathrm{R}}+eV/2>\varepsilon >{\mu }_{1\mathrm{R}}$. In the upper right contact uncompensated right movers have energies ${\mu }_{1\mathrm{R}}+eV/2>\varepsilon >{\mu }_{1\mathrm{R}}$. Even though half of the current is reflected, there is no shot noise, since every single electron state is always either fully occupied, or fully empty. Electrons which are physically transmitted from left to right in edge 1 (shown in red) have energies well below the Fermi energy, $\varepsilon \approx -{\varepsilon }_{1\mathrm{gap}}$ (cf. Fig. 2). Electrons from the lower edge 2 also have to change their spin and energy upon reflection. This results in the noiseless current $I_1=e^{2}V/2h=-I_2$.