Topological nodal-line semimetals in alkaline-earth stannides, germanides, and silicides

Based on first-principles calculations and an effective Hamiltonian analysis, we systematically investigate the electronic and topological properties of alkaline-earth compounds $A{X}_2$ $(A=\text{Ca}$, Sr, Ba; $X=\text{Si}$, Ge, Sn). Taking ${\mathrm{BaSn}}_2$ as an example, we find that when spin-orbit coupling is ignored, these materials are three-dimensional topological nodal-line semimetals characterized by a snakelike nodal loop in three-dimensional momentum space. Drumheadlike surface states emerge either inside or outside the loop circle on the (001) surface depending on surface termination, while complicated double-drumhead-like surface states appear on the (010) surface. When spin-orbit coupling is included, the nodal line is gapped and the system becomes a topological insulator with ${\mathbb{Z}}_2$ topological invariants (1;001). Since spin-orbit coupling effects are weak in light elements, the nodal-line semimetal phase is expected to be achievable in some alkaline-earth germanides and silicides.

Figure 1

(a), (b) Crystal structure of ${\mathrm{BaSn}}_2$ with $P\overline{3}m1$ symmetry. (c) Brillouin zone of the bulk and the projected surface Brillouin zones of the (001) and (010) surfaces.

Figure 2

(a) Calculated band structure without SOC. The irreducible representation of selected bands at $A$ and along the $A\text{−}H$ line are indicated. Red and green dots in band structures indicate the projection onto the Sn $s$ and $p_z$ orbitals, respectively. (b) Corresponding 3D isoenergy surface at $E=E_F+25$ meV. (c) Top view and (d) side view of the isoenergy surface from the (001) and (010) directions of the Brillouin zone, respectively.

Figure 3

Calculated band touching points (nodal line) in 3D $\mathbf{k}$ space using the effective Hamiltonian of Eqs. (1) and (2). (b) Evolution of Wannier charge center $z(k_x,k_y)$ for the occupied bands.

Figure 4

Calculated (001) surface band structure and Fermi surface without SOC for [(a), (b)] Sn-terminated and [(c), (d)] Ba-terminated surfaces. The Fermi surfaces shown in (b) and (d) are calculated with the chemical potential at 60 meV [green dashed lines in (a) and(c)]. Note that the inner circle in (d) originates from a concave conduction band state, not from the drumheadlike surface state.

Figure 5

(a) Calculated (010) surface band structure and (b) Fermi surface without SOC. The Fermi surfaces shown in (b) are calculated with the chemical potential at 10 meV [green dashed lines in (a)]. The white dashed lines in (b) forming an ellipse and two crescents indicate the projection of the bulk nodal loop on the (010) surface. Surface states exist outside the projected nodal loop and inside the ellipse. (c) Enlargement of the double-drumhead-like (010) surface bands near $\tilde{Z}$. (d), (e) Schematic illustration of the formation of “double-drumhead” surface states due to nodal loop twisting and intersecting.