${\mathbb{Z}}_2$ Invariance of Germanene on ${\mathrm{MoS}}_2$ from First Principles

We present a low energy Hamiltonian generalized to describe how the energy bands of germanene ($\overline{\mathrm{Ge}}$) 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 $\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ bilayers and ${\mathrm{MoS}}_2|\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ 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 ${\mathrm{MoS}}_2$ layer(s). Topologically nontrivial gaps for bilayers and trilayers can be almost as large as for a freestanding germanene layer.

Figure 1

Band structures of (a) AS $p\text{−}\overline{\mathrm{C}}|{\mathrm{MoS}}_2$ and (b) AS $b\text{−}\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ bilayers close to the $\mathbf{K}$ point. The yellow dots are the results of first-principles calculations, the black lines result from the model (1) with parameters from Eq. (4).

Figure 2

Stereographic projection of the phase space of the Hamiltonian (1). Black lines represent boundaries between regions where the gap vanishes; phases on either side of the dashed black lines are the same. The scaling of the ${\lambda }_{\mathrm{so}}$ and ${\lambda }_m$ variables with $R\text{−}Z$ is explained in the text. When germanene is rotated with respect to ${\mathrm{MoS}}_2$, a trajectory is traced out in parameter space which is shown in red for a $\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ bilayer and in blue for a ${\mathrm{MoS}}_2|\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ trilayer, where the two ${\mathrm{MoS}}_2$ layers are rotated with respect to one another by ${\theta }_1-{\theta }_2=15°$.

Figure 3

$\lambda$ parameters as a function of the angle $\theta$ for a fixed height of germanene above ${\mathrm{MoS}}_2$ that minimizes the energy for $\theta =24.8°$ for $b\text{−}\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$. The dashed lines are fits to expressions with appropriate angle symmetries. Details of the calculations and the parameters extracted for both planar and buckled $\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ can be found in Ref. [26].

Figure 4

Dependence of the band gap on the angles ${\theta }_1$ and ${\theta }_2$ that a germanene layer makes with two ${\mathrm{MoS}}_2$ layers in a ${\mathrm{MoS}}_2|\overline{\mathrm{Ge}}|{\mathrm{MoS}}_2$ trilayer with threefold rotation symmetry. The unshaded region is NI, the shaded region TI.