Functionalized germanene as a prototype of large-gap two-dimensional topological insulators

We propose two-dimensional (2D) topological insulators (TIs) in functionalized germanenes (GeX, X$=$H, F, Cl, Br, or I) using first-principles calculations. We find GeI is a 2D TI with a bulk gap of about 0.3 eV, while GeH, GeF, GeCl, and GeBr can be transformed into TIs with sizable gaps under achievable tensile strains. A unique mechanism is revealed to be responsible for the large topologically nontrivial gap obtained: due to the functionalization, the $\sigma$ orbitals with stronger spin-orbit coupling (SOC) dominate the states around the Fermi level, instead of original $\pi$ orbitals with weaker SOC. Thereinto, the coupling of the $p_{xy}$ orbitals of Ge and heavy halogens in forming the $\sigma$ orbitals also plays a key role in the further enlargement of the gaps in halogenated germanenes. Our results suggest a realistic possibility for the utilization of topological effects at room temperature.

Figure 1

Top (a) and side (b) views of an optimized structure of GeI displaying the primitive cell with Bravais lattice vectors a${}_1$ and a${}_2$ and the buckling of Ge plane $h$. Green and magenta balls represent Ge and I atoms, respectively.

Figure 2

(a, b) Band structures for GeI without SOC (blue line) and with SOC (black line) with zooming in the energy dispersion near the Fermi level. The red circles and green squares represent the weights of the Ge-$s$ and Ge-$p_{xy}$ character, respectively. (c) The parities of eleven occupied bands at $\Gamma$ and three M points for GeI. The product of the parities at each $k$ point is given in brackets on the right.

Figure 3

(a) Electronic structure for the armchair GeI nanoribbon with a width of 9.3 nm. The helical edge states (red lines) can be clearly seen around the $\Gamma$ point dispersing in the bulk gap. (b) Real-space charge distribution of edge states at $\Gamma$.

Figure 4

(a, b, e) Band structures with SOC for unstrained GeH, 12% strained GeH, and germanene, respectively. Splitting of the $\sigma$ orbital at $\Gamma$ under SOC (${\xi }_{\sigma }$), $\approx$$0.2$ eV. (c, d) Schematic diagram of the evolution of energy levels at $\Gamma$ for GeH. Without strain, $p_{xy}^{+}$ is lower than $s^{-}$ (c). After applying enough large strain, $p_{xy}^{+}$ and $s^{-}$ are inverted (d). Under SOC, $p_{xy}^{+}$ is split into $|p,\pm 3/2\rangle$ and $|p,\pm 1/2\rangle$ states.