Photoinduced Topological Phase Transition and a Single Dirac-Cone State in Silicene

Silicene (a monolayer of silicon atoms) is a two-dimensional topological insulator (TI) that undergoes a topological phase transition to a band insulator under external electric field $E_z$. We investigate a photoinduced topological phase transition from a TI to another TI by changing its topological class by irradiating circular polarized light at fixed $E_z$. The band structure is modified by photon dressing with a new dispersion, where the topological property is altered. By increasing the intensity of light at $E_z=0$, a photoinduced quantum Hall insulator is realized. Its edge modes are anisotropic chiral, in which the velocities of up and down spins are different. At $E_z>E_{\mathrm{cr}}$ with a certain critical field $E_{\mathrm{cr}}$, a photoinduced spin-polarized quantum Hall insulator emerges. This is a new state of matter, possessing one Chern number and one-half spin-Chern numbers. We newly discover a single Dirac-cone state along a phase boundary. A distinctive hallmark of the state is that one of the two Dirac valleys is closed and the other open.

Figure 1

The band structure of a silicene in the SDC state. The gap is open at the $K$ point with a parabolic dispersion but closed at the $K^{\prime }$ point with a linear dispersion.

Figure 2

Phase diagram in the ($E_z$, ${\mathcal{A}}^2/\Omega$) plane. A circle shows a point where the energy spectrum is calculated and shown in Fig. 3. Heavy lines represent phase boundaries indexed by $K_{\eta }$ and $s_z=\uparrow \downarrow$. A SDC state appears along the line, which is characterized by a single gapless Dirac cone at the $K_{\eta }$ point with spin $s_z$. The topological charges ($C$, $C_s$) are also indicated.

Figure 3

The photon-dressed band structure of a silicene nanoribbon at marked points in the phase diagram (Fig. 2). 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. The spin $s_z$ is practically a good quantum number that we have assigned to the Dirac cones.

Figure 4

Enlarged edge states of silicene nanoribbons for the QSHI, SPM, P-QHI, and PS-QHI. An anisotropic helical (chiral) current flows in the QSHI (P-QHI), where the velocities of up and down spins are different in each edge. A topological flat band appears in SPM. Solid lines marked $T$ represent edge modes propagating on the top edge, while dotted lines marked $B$ represent edge modes propagating on the bottom edge.