Z2 Invariance of Germanene on MoS2 from First Principles

We present a low energy Hamiltonian generalized to describe how the energy bands of germanene ((Ge) over bar) 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 bar vertical bar MoS2 bilayers and MoS2 vertical bar(Ge) over bar vertical bar MoS2 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 MoS2 layer(s). Topologically nontrivial gaps for bilayers and trilayers can be almost as large as for a freestanding germanene layer.