Twist-induced quantum valley Hall states in bilayer germanene

In van der Waals bilayer germanene, we show a twist-induced nontrivial topological transition and find it can host the quantum valley Hall effect (QVHE). It is demonstrated that band inversion is induced at each K of the supercell Brillouin zone (SBZ). The topological phase transition comes from the fact that the valley Chern number, Cv=CK-CK′, switches from 0 to 2, which characterizes the nontrivial QVHE phase. Taking into account spin-orbit coupling (SOC), this band topology of the moiré superlattice originates from both the low-buckled structure within each layer and the coexistence of distinct stacking domains - namely, AA, AB, and BA regions - in the bilayer, particularly for small twist angles. Our findings advance potential applications for device design in topological valleytronics and twistronics.