Electrically tunable band gap in silicene

We report calculations of the electronic structure of silicene and the stability of its weakly buckled honeycomb lattice in an external electric field oriented perpendicular to the monolayer of Si atoms. The electric field produces a tunable band gap in the Dirac-type electronic spectrum, the gap being suppressed by a factor of about eight by the high polarizability of the system. At low electric fields, the interplay between this tunable band gap, which is specific to electrons on a honeycomb lattice, and the Kane-Mele spin-orbit coupling induces a transition from a topological to a band insulator, whereas at much higher electric fields silicene becomes a semimetal.

Figure 1

Atomic structure of silicene, together with a sketch of the charge density for the highest occupied valence band in the vicinity of the K point.

Figure 2

DFT-PBE phonon dispersion curves for silicene in zero external field and at $E_z=0.51$ V Å${}^{-1}$. In both cases the calculations were performed using the method of finite displacements, with the atomic displacements being 0.0423 Å, in a supercell consisting of $3\times 3$ primitive cells with a $20\times 20$ $\mathbf{k}$-point grid in the primitive cell.

Figure 3

DFT-PBE band structures for silicene in a cell of length $L_z=26.5$ Å with a plane-wave cutoff energy of 816 eV and a $53\times 53$ $\mathbf{k}$-point grid: (a) in zero external electric field, (b) with $E_z=0.26$ V Å${}^{-1}$, and (c) with $E_z=0.51$ V Å${}^{-1}$ (shown both with and without the relaxation of the atomic coordinates in the electric field). The zero of the external potential is in the center of the silicene layer. The dashed line shows the Fermi energy in each case and the insets show the spectrum near the Fermi level in the vicinity of the K point.

Figure 4

DFT gap against applied electric field $E_z$ for silicene with a plane-wave cutoff energy of 816 eV and a $53\times 53$ $\mathbf{k}$-point grid. Unless otherwise stated, the PBE functional was used. The box length in the $z$ direction was varied from $L_z=13.35$ Å to 26.46 Å. The results have been extrapolated to the limit $L_z\to \infty$ of infinite box length (solid lines) as described in Sec. 5d4. Unscreened band gaps calculated using perturbation theory are also shown. The inset table shows the calculated rate at which the band gap opens.

Figure 5

DFT-PBE and DFT-LDA band structures with and without SO coupling taken into account. The inset shows the bands around the K point, revealing a small band gap induced by SO coupling. The width of the bottom panel corresponds to $1/200$ of the $\Gamma$K line.

Figure 6

Phonon dispersion curves for silicene obtained with castep and vasp using different exchange-correlation functionals and supercell sizes. The results for a $3\times 3$ supercell were obtained with a box length $L_z=13.35$ Å, a $20\times 20$ $\mathbf{k}$-point grid in the primitive cell, and a plane-wave cutoff energy of 816 eV. The matrix of force constants was evaluated using the method of finite displacements, with the displacements being 0.042 Å. The results for a $7\times 7$ supercell were obtained with a box length $L_z=15.0$ Å, a $12\times 12$ $\mathbf{k}$-point grid in the supercell, and a plane-wave cutoff energy of 500 eV. The matrix of force constants was evaluated using the method of finite displacements, with displacements of 0.09 Å.

Figure 7

DFT-PBE gap against plane-wave (PW) cutoff energy for silicene subject to an electric field of 0.257 V/Å in a cell of length $L_z=13.35$ Å with a $15\times 15$ $\mathbf{k}$-point grid including K.

Figure 8

DFT-PBE gap against the reciprocal of the number $N_{\mathbf{k}}^2$ of $\mathbf{k}$ points for silicene subject to an electric field of 0.257 V/Å in a cell of length $L_z=13.35$ Å with a plane-wave cutoff energy of 816 eV.

Figure 9

DFT gap with different exchange-correlation functionals (LDA and PBE) and pseudopotentials [on-the-fly ultrasoft (USP) and norm conserving (NC)[15]] for silicene subject to an electric field of 0.257 V/Å in a cell of length $L_z=13.35$ Å with a $15\times 15$ $\mathbf{k}$-point grid and a plane-wave cutoff energy of 816 eV.

Figure 10

DFT-PBE gap against the reciprocal of the box length $L_z$ for silicene subject to an electric field of 0.257 V/Å with a $15\times 15$ $\mathbf{k}$-point grid and a plane-wave cutoff energy of 816 eV. The solid line is a fitted quadratic in $1/L_z$.