Hourglass Fermion in Two-Dimensional Material

The hourglass fermion, as an exotic quasiparticle protected by nonsymmorphic symmetry, has excited great research interest recently. However, its bulk counterpart in two-dimensional (2D) solid-state materials has seldom been studied. In this Letter, we propose a 2D rectangular lattice made of $p_x$ and $p_y$ orbitals with glide mirror symmetry but without inversion symmetry to realize the hourglass fermion. The glide mirror symmetry guarantees a Dirac nodal line, while the Rashba spin-orbital coupling splits it into two Weyl nodal lines and generates two pairs of hourglass fermion located at the glide mirror plane. Furthermore, based on first principles calculations, we predict a surface-supported 2D material $\mathrm{Bi}/\mathrm{Cl}\text{−}\mathrm{SiC}(111)$ to realize our proposal, making a huge-bandwidth hourglass cone. Moreover, the hourglass fermion exhibits a spin-momentum locking spin texture and also sustains a giant spin Hall conductivity. Our results demonstrate a general routine for designing an hourglass fermion in 2D materials, which will be easily extended to other surfaces with different adatoms and lattice symmetries.

Figure 1

Schematic designing principle of the hourglass fermion. Left panel: 2D rectangular lattice with two lattice sites ($A$ and $B$) per unit cell, and each site has two orbitals ($p_x$ and $p_y$). Middle panel: first BZ with high-symmetric $k$ points. Right panel: schematic hourglass fermion along the $\mathrm{\Gamma }\text{−}Y$ and $X\text{−}M$ directions, characterized by a hourglass cone with four degenerate points at boundary (red dot) and one degenerate point at center (blue dot). The numbers in the bracket denote the eigenvalue of ${\tilde{M}}_x$ at the degenerate point.

Figure 2

(a) and (b) With the inversion symmetry, the TB band structure without and with SOC, respectively. The parameters are $V_1^{\sigma }=1.5\mathrm{eV}$, $V_2^{\sigma }=-0.1\mathrm{eV}$, $V_3^{\sigma }=0.1\mathrm{eV}$, $V_4^{\sigma }=-0.05\mathrm{eV}$, $V_1^{\pi }=-0.25\mathrm{eV}$, $V_2^{\pi }=0.1\mathrm{eV}$, $V_3^{\pi }=0.1\mathrm{eV}$, $V_4^{\pi }=-0.05\mathrm{eV}$, ${ϵ}_{p_x}={ϵ}_{p_y}=0.0\mathrm{eV}$, ${ϵ}_{p_z}=-6.0\mathrm{eV}$, and $\lambda =0.5\mathrm{eV}$. (c) Without the inversion symmetry, the TB band structure with SOC. The hopping energy between the $(p_x,p_y)$ and $p_z$ orbital is described by ${\gamma }_1=-0.3\mathrm{eV}$, ${\gamma }_2=0.1\mathrm{eV}$, ${\gamma }_3=-0.1\mathrm{eV}$, and ${\gamma }_4=0.05\mathrm{eV}$. The other parameters are the same as (a) and (b). (d) is the same as (c), but ${\gamma }_1=-0.7\mathrm{eV}$. The inset in (a) is the same as (d), but without SOC.

Figure 3

(a)–(e) Energy comparison for different Cl adsorption patterns on SiC(111) surface at $1/2\mathrm{ML}$ coverage in the $4\times 2\sqrt{3}$ superlattice. The energy for pattern (a) is set as the reference energy. The labels $T$, $H$, and $T_4$ in (a) denote different adsorption sites. The blue dashed line and red dashed line in (a) denote the $2\times \sqrt{3}$ superlattice and $1\times 1$ lattice, respectively. (f) Charge transfer between SiC(111) and Cl in pattern (a). The red and blue colors denote negative and positive value, respectively.

Figure 4

(a) Top and side view of atomic structure of $\mathrm{Bi}/\mathrm{Cl}\text{−}\mathrm{SiC}(111)$. For clarity, only the top two-layer SiC(111) is shown in the side view. (b) and (c) Band structure and PDOS of $\mathrm{Bi}/\mathrm{Cl}\text{−}\mathrm{SiC}(111)$ without SOC. (d) $p_x+p_y$ orbital of Bi projected band structure of $\mathrm{Bi}/\mathrm{Cl}\text{−}\mathrm{SiC}(111)$ with SOC. The black and cyan colors denote the maximum and zero value, respectively. (e) 3D band plotting of the hourglass fermion. (f) Spin texture of the hourglass fermion along the $\mathrm{\Gamma }\text{−}Y\text{−}\mathrm{\Gamma }$ and $X\text{−}M\text{−}X$ directions. The red and blue colors denote the positive and negative values of $S_x$, respectively. The dashed lines are used to guide the bandwidth of the hourglass cone.

Figure 5

(a) Band structure comparison between Wannier fitted and first-principles calculated hourglass fermion. (b) Energy-dependent intrinsic spin Hall conductivity, showing a giant spin Hall effect.