Spin valleytronics in silicene: Quantum spin Hall–quantum anomalous Hall insulators and single-valley semimetals

Valley-based electronics, known as valleytronics, is one of the keys to breaking through to a new stage of electronics. The valley degree of freedom is ubiquitous in the honeycomb lattice system. The honeycomb lattice structure of silicon, called silicene, is a fascinating playground of valleytronics. We investigate topological phases of silicene by introducing different exchange fields on the $A$ and $B$ sites. There emerges a rich variety of topologically protected states, each of which has a characteristic spin-valley structure. The single Dirac-cone semimetal is such a state where one gap is closed while the other three gaps are open, evading the Nielsen-Ninomiya fermion-doubling problem. We have newly discovered a hybrid topological insulator named the quantum spin–quantum anomalous Hall insulator, where the quantum anomalous Hall effect occurs at one valley and the quantum spin Hall effect occurs at the other valley. Along its phase boundary, single-valley semimetals emerge, where only one of the two valleys is gapless with degenerated spins. These semimetals are also topologically protected because they appear in the interface of different topological insulators. Such a spin-valley-dependent physics will be observed by optical absorption or edge modes.

Figure 1

Phase diagram in the $\ell E_z$-$\Delta M$ plane. Heavy lines represent phase boundaries where the band gap closes. There are three types of topological insulators as indicated by QSH with $(0,1)$ and SQAH with $(\pm 1,\frac{1}{2})$. There are two types of trivial band insulators as indicated by AF and CDW. There emerge the SDC semimetal and the SV semimetal in the phase boundary. A circle shows a point where the energy spectrum is shown in Fig. 2. The gap closes at $E_z=\pm 17$ meV/Å on the $E_z$ axis and at $\Delta M=\pm {\lambda }_{\text{SO}}$ on the $\Delta M$ axis.

Figure 2

The band structures of zigzag silicene nanoribbons at the points indicated in the phase diagram (Fig. 1). The vertical axis is the energy in units of $t$, and the horizontal axis is the momentum. We can clearly see the Dirac cones representing the energy spectrum of the bulk. Lines connecting the two Dirac cones are edge modes.

Figure 3

Phase diagram in the $\overline{M}$-$\Delta M$ plane. Heavy lines represent phase boundaries. There are five types of topological insulators as indicated by QAH with $(\pm 2,0)$, QSQAH with $(\pm 1,1/2)$, and QSH with $(0,1)$. There is one trivial band insulator, as indicated by AF. There emerge the mQAH metal and the TM in the phase boundary. A circle shows a point where the energy spectrum is calculated and shown in Fig. 4. The gap closes at $\overline{M}=\pm {\lambda }_{\text{SO}}$ on the $\overline{M}$ axis and at $\Delta M=\pm {\lambda }_{\text{SO}}$ on the $\Delta M$ axis.

Figure 4

The band structures of silicene nanoribbons at marked points in the phase diagram (Fig. 3). The vertical axis is the energy in unit of $t$, and the horizontal axis is the momentum. Lines connecting the two Dirac cones are edge modes.