$d+i{d}^{\prime }$ Chiral Superconductivity in Bilayer Silicene

We investigate the structure and physical properties of the undoped bilayer silicene through first-principles calculations and find the system is intrinsically metallic with sizable pocket Fermi surfaces. When realistic electron-electron interaction turns on, the system is identified as a chiral $d+i{d}^{\prime }$ topological superconductor mediated by the strong spin fluctuation on the border of the antiferromagnetic spin density wave order. Moreover, the tunable Fermi pocket area via strain makes it possible to adjust the spin density wave critical interaction strength near the real one and enables a high superconducting critical temperature.

Figure 1

(a) Optimized geometry of the BLS. (b) The corresponding phonon spectrum. In (a), both the top view (left) and side view (right) are shown. The vertical bond length $l_v$, the intralayer nearest neighbor bond length $l_n$, and the angle $\theta$ between them are marked, together with the hopping integrals between each two of the four atoms $A_i$, $B_i$ ($i=1,2$) within a unit cell.

Figure 2

(a) The band structure of BLS corresponding to the optimized lattice structure shown in Fig. 1a. (b) The FS patches around the $K$ point. In (a), the zooming in low energy band (right) is also shown, where the tight-binding (TB) model (red scatters) is compared with the FP results (blue solid line). In (b), the central pocket (red) is electronlike and the outer three identical pockets (green) are holelike, with the total areas of the two kinds of pockets equal.

Figure 3

(a) $\mathbf{k}$ dependence of the free static susceptibility. (b) The SDW ordered spin pattern within a unit cell.

Figure 4

(a) The biaxial strain dependence of Fermi pocket area ratio viz. the ratio of the total area of the electron and hole pockets against the total area of the Brillouin zone and the SDW critical value $U_c$ of the BLS. (b) The interaction strength $U$ dependence of the largest eigenvalue $\lambda$ of the linearized gap function (3), which is related to $T_c$ through $T_c=1.13\hslash {\omega }_D{e}^{-1/\lambda }$. Results for different strain values are compared. (c) $\mathbf{k}$ dependence of the gap function of the $d_{x^2-y^2}$ symmetry for $U=1\mathrm{eV}$, which is antisymmetric about the axes $x=\pm y$ shown in the reciprocal space. Inset: zooming in the vicinity of $K^{\prime }$.