Valley-dependent multiple quantum states and topological transitions in germanene-based ferromagnetic van der Waals heterostructures

Topological and valleytronic materials are promising for spintronic and quantum applications due to their unique properties. Using first principles calculations, we demonstrate that germanene (Ge)-based ferromagnetic heterostructures can exhibit multiple quantum states such as quantum anomalous Hall effect (QAHE) with Chern numbers of C=-1 or C=-2, quantum valley Hall effect (QVHE) with a valley Chern number of Cᵥ=2, valley-polarized quantum anomalous Hall effect (VP-QAHE) with two Chern numbers of C=-1 and Cᵥ=-1, as well as time-reversal symmetry broken quantum spin Hall effect (T-broken QSHE) with a spin Chern number of Cₛ4pt0ex~4pt0ex1. Furthermore, we find that the transitions between different quantum states can occur by changing the magnetic orientation of ferromagnetic layers through applying a magnetic field. Our discovery provides different routes and material platforms with a unique combination of diverse properties that make it well suitable for applications in electronics, spintronics, and valley electronics.