$\mathbb{Z}_2$ invariance of Germanene on MoS$_2$ from first principles

TL;DR

This study develops a low-energy Hamiltonian to analyze how germanene's electronic properties are affected by interaction with MoS2 layers, revealing orientation-dependent topological gaps that can rival free-standing germanene.

Contribution

The paper introduces a generalized Hamiltonian derived from first-principles calculations to describe germanene on MoS2, highlighting the impact of orientation on topological properties.

Findings

Topologically non-trivial gaps can be nearly as large as free-standing germanene.

The energy gap depends strongly on germanene's orientation relative to MoS2.

The Hamiltonian accurately captures substrate effects on germanene's electronic structure.

Abstract

We present a low energy Hamiltonian generalized to describe how the energy bands of germanene ($\rm \overline{Ge}$) 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 $\rm \overline{Ge}|$MoS$_2$ bilayers and MoS$_2|\rm \overline{Ge} |$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 non-trivial gaps for bilayers and trilayers can be almost as large as for a free-standing germanene layer.