Prediction of Dirac semimetals and hourglass surface states in stacked hydrogenated Xenes ($X=\mathrm{Sn}$ and Pb)

Based on density-functional theory calculations and symmetry analysis, topological Dirac semimetals (DSMs) and exotic hourglass surface states are predicted in stacked hydrogenated stanene and plumbene [Xenes ($X$ = Sn and Pb)]. Two types of topologically nontrivial DSMs are predicted in the hydrogenated Xene crystals, with the two most stable stacking patterns, respectively. The Dirac points in SnH are found occurring along the three- or sixfold (screw) rotation axes due to the inversion of the Sn $5s$ ($5p_z$) and Sn $5p_{x,y}$ bands. The unique Fermi arcs are observed on the surfaces parallel to the rotation axes in the crystals. Very specially, the nonsymmorphic symmetry of a glide-mirror plane in the hydrogenated Xene crystals gives rise to exotic hourglass surface states which can lead to a giant value of spin-Hall conductivity and make the crystals be ideal systems applied in prospective dissipationless spintronics. The topological nature of the both types of the DSMs is identified by the calculations of the ${\mathrm{Z}}_2$ indexes. Phase transitions from topologically nontrivial DSM states to normal insulators (NIs) or to other types of DSMs are also investigated. Our work provides an ideal material platform for carrying out DSMs and a comprehensive study for understanding how DSMs evolve from two-dimensional NIs or quantum spin Hall materials.

Figure 1

(a) Top and side views of the monolayer $X\mathrm{H}$ ($X$ = Sn and Pb). The black solid lines indicate the unit cell. (b) Top and side views of the four types of the 3D $X\mathrm{H}$ ($X$ = Sn and Pb) structures studied. For the $\mathit{AA}$ structure, the 2 × 2 × 2 supercell is shown, while for the $\mathit{AB}, \mathit{AB}$′, and $\mathit{AB}$″ structures, the 2 × 2 × 1 supercells are displayed. The green solid line and blue dashed line in the top view of the $\mathit{AB}$ structure indicate a glide mirror plane $\tilde{M}$ and a mirror plane $M$ of the structure, respectively.

Figure 2

Phonon dispersions for the 3D SnH crystals with the $\mathit{AA}$ (a) and $\mathit{AB}$ (b) stacking configurations.

Figure 3

(a) The first BZ of the 3D SnH crystal with the $\mathit{AA}$ stacking configuration. Its projections onto the (001) and $\left(1\sqrt{3}0\right)$ planes are also shown. Panels (b) and (c) are the band structures of the system without and with SOC, respectively. The projected Sn atomic orbitals are also displayed. The sizes of the dots are proportional to the orbital contributions. (d) The amplified band structure along the high-symmetry line Γ-$A$, where the Dirac point is labeled by $D_0$. (e) Schematic diagram of the band evolution from Γ to $A$. Between the two vertical orange dashed lines, the red and green lines denote doubly degenerate bands belonging to different irreducible representations.

Figure 4

The evolution of the Wannier function centers for the $\mathit{AA}$ SnH crystal on the $k_z=0$ (a) and $k_z=\pi$ (b) planes in the BZ. Panels (c) and (d) are the surface band structure and Fermi surface contour for the (001) plane of the $\mathit{AA}$ SnH crystal, respectively. Panels (e) and (f) are the same as (c) and (d), respectively, except for $\left(1\sqrt{3}0\right)$ plane instead. The red dots in (f) indicate the Dirac points. $k_1$ and $k_2$ are the wave-vector units along the $\tilde{\mathrm{\Gamma }}\tilde{S}$ and $\tilde{\mathrm{\Gamma }}\tilde{Z}$ directions, respectively.

Figure 5

(a) The projected atomic orbital band structure of the 3D SnH crystal with the $\mathit{AB}$ stacking configuration. The SOC is turned on. The sizes of the dots are proportional to the orbital contributions. (b) The amplified band structure along the high-symmetry line Γ-$A$. The Dirac point is labeled by $D_1$. Panels (c) and (d) are the surface band structure and Fermi surface contour on the (001) plane of the $\mathit{AB}$ SnH crystal, respectively. Panels (e) and (f) are the same as (c) and (d), respectively, except for the $\left(1\sqrt{3}0\right)$ plane instead. The red dots in (f) indicate the Dirac points. $k_1$ and $k_2$ are the wave-vector units along the $\tilde{\mathrm{\Gamma }}\tilde{S}$ and $\tilde{\mathrm{\Gamma }}\tilde{Z}$ directions, respectively. (g) The schematic diagram of surface bands on the $\left(1\sqrt{3}0\right)$ plane.

Figure 6

The band structures of the $\mathit{AA}$ SnH crystal under the in-plane biaxial strain of −2% (a) and 5% (b). (c) The phase diagram of the $\mathit{AA}$ SnH crystal as a function of the strain applied. Two types of phases are obtained: NI (orange region) and DSM (blue region). Panels (d) and (e) give the band structures of the $\mathit{AB}$ SnH crystal under the biaxial strain of −2% (d) and 1%, respectively. (f) The phase diagram of the $\mathit{AB}$ SnH crystal as a function of the strain. Three types of phases are obtained: NI (orange region), DSM1 (with a pair of Dirac points, blue region), and DSM2 (with two pairs of Dirac points, green region).

Figure 7

The band structures of the 3D PbH crystals with the $\mathit{AA}$ (a) and $\mathit{AB}$ (b) stacking patterns, respectively. Panels (c) and (d) give the surface band structure and Fermi surface contour on the $\left(1\sqrt{3}0\right)$ plane of the 3D $\mathit{AB}$ PbH crystal, respectively. $k_1$ and $k_2$ in (d) are the wave-vector units along the $\tilde{\mathrm{\Gamma }}\tilde{S}$ and $\tilde{\mathrm{\Gamma }}\tilde{Z}$ directions, respectively. The Dirac points in (d) are labeled by the red dots.