Bilayer stacking ferrovalley materials without breaking time-reversal and spatial-inversion symmetry

Ferrovalley, which refers to the valley polarization being nonvolatile and switchable, is highly desired for valleytronics applications but remains challenging due to rare candidate materials. Here we propose a strategy to realize ferrovalley with bilayer stacking (BSFV) in many candidate systems. As a special case of BSFV, sliding ferrovalley corresponds to the bilayers obtained by a direct AA stacking and subsequent in-plane sliding. Different from previous approaches, the BSFV strategy not only maintains time-reversal symmetry, but also keeps spatial-inversion symmetry in many cases. Importantly, switching of the valley polarization can be easily achieved by interlayer sliding. Group theory analysis is systematically performed over all kinds of lattices to identify those that can host BSFV. High-throughput screening is carried out and leads to 14 BSFV candidates with direct bandgap and 338 with indirect bandgap. First-principles verification of BSFV indicates that the valley polarization can be realized in, e.g., (i) the hexagonal $\mathrm{RhC}{\mathrm{l}}_3$ bilayer with a threefold rotation symmetry and 39 meV energy difference among valleys, and (ii) the square-latticed InI bilayer with a fourfold rotation symmetry and 326 meV energy difference among valleys. The presently proposed BSFV strategy offers a highly convenient approach for the realization of polarizers and the advancement of valleytronics applications.

Figure 1

Illustration of realizing valley polarization through BSFV theory. (a) Schematics of a bilayer with $C_{nz}$ rotational symmetry and (b) breaking of $C_{nz}$ with interlayer sliding $\tau$. (c) Brillouin zone of a hexagonal lattice and degenerate valleys connected by $C_{3z}$ symmetry. (d) Valley polarization realized by breaking $C_{3z}$ with sliding. Red and blue dots represent the positions of degenerate and polarized valley, respectively. Panels (e) and (f) are similar to (c) and (d), respectively, but for square lattice with $C_{4z}$ symmetry.

Figure 2

Realizing BSFV in hexagonal $\mathrm{RhC}{\mathrm{l}}_3$. (a) Geometry of bilayer $\mathrm{RhC}{\mathrm{l}}_3$ with $G$ and $L_1$ stackings. The left and right panels show the top and side views, respectively. (b) Energy landscape for different bilayer stackings. The white dotted parallelogram denotes the unit cell of monolayer. Points of $G, L_{1/2/3}$, and $D_{1,2}$ represent stackings with fractional translations of (0,0), (1/3,0), (0,1/3), (2/3,2/3), (1/3,2/3), and (2/3,1/3), respectively. The ground states locate at $D_1$ and $D_2$ stackings, while $L_{1/2/3}$ stackings are local minima. Panels (c) and (d) show band structures for $G$ and $L_1$ stackings, respectively. The spin-orbit coupling is not considered for its negligible strength. The insets show high-symmetry points of the Brillouin zone, as well as the corresponding stackings.

Figure 3

Realizing BSFV in square InI. (a) Geometry of bilayer InI with $G$ and $N_1$ stackings. The left and right panels show the top and side views, respectively. (b) Energy landscape for different bilayer stackings. Points of $G$ and $N_{1,2}$ represent stackings with fractional translations of (0,0), (1/2,0), and (0,1/2), respectively. The ground states locate at $N_1$ and $N_2$ stackings, while $G$ stacking is a local minimum. Panels (c) and (d) show band structures for $G$ and $N_1$ stackings, respectively. The spin-orbit coupling effect is not negligible, thus is considered here. The insets show high-symmetry points of the Brillouin zone, as well as the corresponding stackings.