Topological domain walls and quantum valley Hall effects in silicene

Silicene is a two-dimensional honeycomb lattice made of silicon atoms, which is considered to be a new Dirac fermion system. Based on first-principles calculations, we examine the possibility of the formation of solitonlike topological domain walls (DWs) in silicene. We show that the DWs between regions of distinct ground states of the buckled geometry should bind electrons when a uniform electric field is applied in the perpendicular direction to the sheet. The topological origin of the electron confinement is demonstrated based on numerical calculations of the valley-specific Hall conductivities, and possible experimental signatures of the quantum valley Hall effects are discussed using simulated scanning tunneling microscopy images. Our results strongly suggest that silicene could be an ideal host for the quantum valley Hall effect, thus providing a pathway to the valleytronics in silicon-based technology.

Figure 1

(a) The geometry of the low-buckled silicene. (b) A domain wall (DW) geometry between $\alpha$ and $\beta$ low-buckled states. (c) Top view of a supercell geometry with zigzag DWs. A pair of solitonlike and antisolitonlike domain walls (S and $\overline{\mathrm{S}}$) are separated by 10.3 nm in the supercell. The magnified views of the detailed atomic structure of the S and $\overline{\mathrm{S}}$ regions are also illustrated. (d) Schematic configuration of the junction geometry under applied electric fields. The arrows indicate the direction of the applied electric field.

Figure 2

Atomic geometry of the elbowed junction and its total energy curve as a function of the elbow angle $\theta$.

Figure 3

Total energy of the junction geometry as a function of the width of the transition region.

Figure 4

Position-dependent buckling order parameter $\mu \left(x\right)$ for soliton (S) and antisoliton ($\overline{\mathrm{S}}$).

Figure 5

(a) Band structure of the supercell geometry. The solid gray lines in the inset correspond to bulk states, and the dotted and dashed lines indicate the kink states at $\mathrm{S}$ and $\overline{\mathrm{S}}$, respectively. The blue and red color schemes are used to represent $K$ and $K^{\prime }$, respectively. The dots (a–d) correspond to the states at 10 meV above the Fermi level. (b) Probability distributions of the QVH kink states at a–d. (c) Schematic illustration of kink states. Electrons at different valleys propagate in opposite directions on the DWs. $\mathrm{S}$ supports an upward $K$ current and a downward $K^{\prime }$ current and vice versa for $\overline{\mathrm{S}}$. The senses of the arrows perpendicular to the page represent the sign of the valley Hall conductivities associated with each valley.

Figure 6

Probability distributions of the wave functions at $E=10$ meV from the Fermi energy. The top panel corresponds to a (or equally to d) state, and the bottom panel corresponds to b (or equally to c) state in Fig. 5. By increasing the applied electric field, states become more concentrated on the DW.

Figure 7

Changes in the probability distribution near a domain wall. The green and red (overlapped) curves represent probability distributions obtained with and without the DW, respectively, in the absence of the applied electric field. The blue (lower) curve represents the changes in the probability distributions due to the DW.

Figure 8

Berry curvatures for the $\alpha$ state (left panel) and the $\beta$ state (right panel) under the applied electric field of 0.5 V/Å using the color code scale in units of $e^2/h$.

Figure 9

(a)–(d) Simulated scanning tunneling microscopy (STM) images for soliton [(a) and (c)] and antisoliton [(b) and (d)] DWs at different sample biases. The rhombus in (a) represents the $1\times 1$ unit cell. Sample biases are chosen to be $-20$ mV in (a) and (b), and $-1.4$ V in (c) and (d). An external field of 0.5 V/Å is applied perpendicular to the sheet in (a) and (b). Bright spots correspond to the upper buckled silicon atoms, and DWs reside along the center of the images.