Giant Spin Rotation in the Junction between a Normal Metal and a Quantum Spin Hall System

We study theoretically the reflection problem in the junction between a normal metal and an insulator characterized by a parameter $M$, which is a usual insulator for $M>0$ or a quantum-spin-Hall system for $M<0$. The spin rotation angle $\alpha$ at the reflection is obtained as a function of $M$ and the incident angle $\theta$ measured from the normal to the interface. The $\alpha$ shows rich structures around the quantum critical point $M=0$ and $\theta =0$; i.e., $\alpha$ can be as large as $\sim \pi$ at an incident angle in the quantum spin Hall case $M<0$ because the helical edge modes resonantly enhance the spin rotation, which can be used to map the energy dispersion of the helical edge modes. As an experimentally relevant system, we also study the spin rotation effect in quantum-spin-Hall–normal-metal–quantum-spin-Hall trilayer junction.

Figure 1

$N/\mathrm{QSH}$ junction and corresponding band structures (below). Dotted lines represent helical edge modes.

Figure 2

Spin rotation angle $\alpha$ as a function of injection angle and $M$. (a) $C=-0.08\mathrm{eV}$, (b) $C=-0.1\mathrm{eV}$, and (c) $C=-1\mathrm{eV}$. Note that $\alpha =\pi$ and $\alpha =-\pi$ are equivalent and the continuous increase of $\alpha$ in the clockwise direction is separated into several sheets in (b) and (c). Note that $\alpha (-\theta )=-\alpha (\theta )$ is satisfied. Finite barrier potential at the interface is taken into account in (d) and (e). (d) $C=-0.1\mathrm{eV}$ and $Z=10$. (e) $\alpha$ as a function of $Z$ at $M=-0.01\mathrm{eV}$ and $\theta =0.01\pi$.

Figure 3

Expectation values of ${\sigma }_x$ and ${\sigma }_y$ as a function of $L/w$ for $C=-1\mathrm{eV}$ and $M=-0.01\mathrm{eV}$. Inset shows the model.