Z_{2} Invariance of Germanene on MoS_{2} from First Principles

We present a low energy Hamiltonian generalized to describe how the energy bands of germanene (Ge[over ¯]) are modified by interaction with a substrate or a capping layer. The parameters that enter the Hamiltonian are determined from first-principles relativistic calculations for Ge[over ¯]|MoS_{2} bilayers and MoS_{2}|Ge[over ¯]|MoS_{2} trilayers and are used to determine the topological nature of the system. For the lowest energy, buckled germanene structure, the gap depends strongly on how germanene is oriented with respect to the MoS_{2} layer(s). Topologically nontrivial gaps for bilayers and trilayers can be almost as large as for a freestanding germanene layer.