Nodal surface semimetals: Theory and material realization

We theoretically study the three-dimensional topological semimetals with nodal surfaces protected by crystalline symmetries. Different from the well-known nodal-point and nodal-line semimetals, in these materials, the conduction and valence bands cross on closed nodal surfaces in the Brillouin zone. We propose different classes of nodal surfaces, both in the absence and in the presence of spin-orbit coupling (SOC). In the absence of SOC, a class of nodal surfaces can be protected by space-time inversion symmetry and sublattice symmetry and characterized by a ${\mathbb{Z}}_2$ index, while another class of nodal surfaces are guaranteed by a combination of nonsymmorphic twofold screw-rotational symmetry and time-reversal symmetry. We show that the inclusion of SOC will destroy the former class of nodal surfaces but may preserve the latter provided that the inversion symmetry is broken. We further generalize the result to magnetically ordered systems and show that protected nodal surfaces can also exist in magnetic materials without and with SOC, given that certain magnetic group symmetry requirements are satisfied. Several concrete nodal-surface material examples are predicted via the first-principles calculations. The possibility of multi-nodal-surface materials is discussed.

Figure 1

(a) Lattice structure and (b) unit cell of the carbon allotrope QGN(1,2). (a) A $2\times 2\times 2$ supercell, in which the unit cell is marked by the black box. In (b), the red-colored carbon atoms have a $s{p}^2$ hybridization character (which are labeled from site 1 to site 8), whereas the blue-colored carbon atoms have a $s{p}^3$ hybridization character. (c) Corresponding Brillouin zone, where the two nodal surfaces are schematically shown by the shaded surfaces. (d) Calculated band structure for QGN(1,2) without SOC.

Figure 2

(a) Crystal structure of ${\mathrm{K}}_6{\mathrm{YO}}_4$ and (b) the corresponding Brillouin zone. In (b), R and E are the midpoints of the paths A-L and $\mathrm{\Gamma }$-M, respectively. (c) Band structure of ${\mathrm{K}}_6{\mathrm{YO}}_4$ without SOC. The red arrows indicate the band degeneracy along the paths on the nodal surface.

Figure 3

(a) Perspective view and (b) top view of the crystal structure of the $A{\mathrm{Mo}}_3{X}_3$ family materials, where $A=$ (Na, K, Rb, In, Tl) and $X=$ (S, Se, Te). The insert in (b) shows the face-sharing ${\mathrm{Mo}}_6$ octahedra chains. (c) Band structure of ${\mathrm{RbMo}}_3{\mathrm{S}}_3$ without SOC. The red arrows indicate the band degeneracy along the paths on the nodal surface. The Brillouin zone here has the same shape as that in Fig. 2.

Figure 4

Calculated band structures of ${\mathrm{TlMo}}_3{\mathrm{Te}}_3$ (a) without SOC and (b) with SOC. Panel (c) shows the Brillouin zone. R and E are the midpoints of the paths A-L and $\mathrm{\Gamma }$-M, respectively. (d) Enlarged view of the low-energy band dispersion around the R and H points, showing that the original band crossing at the nodal surface is gapped by SOC.

Figure 5

(a) Side view and (b) top view of the crystal structure for ${\mathrm{Ta}}_3{\mathrm{TeI}}_7$. The inset in (b) shows the cluster unit of the triangular lattice. (c) Calculated band structure for ${\mathrm{Ta}}_3{\mathrm{TeI}}_7$ in the presence of SOC. The right panel shows the enlarged view of the band structure near the Fermi level along a generic path E-R-H, where E-R is perpendicular to the $k_z=\pi$ plane and R-H is in the $k_z=\pi$ plane. The Brillouin zone and the labeled $k$ points are the same as in Fig. 2.

Figure 6

(a) Side view and (b) top view of the crystal structure for ${\mathrm{CsCrI}}_3$. (c) Calculated band structure for ${\mathrm{CsCrI}}_3$ in the absence of SOC, showing a ferromagnetic state. The red and blue lines represent spin-up and spin-down bands, respectively. In the absence of SOC, the nodal surface is preserved, and the band structure does not depend on the direction of the magnetic moments.

Figure 7

Calculated band structures for ${\mathrm{CsCrI}}_3$ in the presence of SOC, with magnetic moments along (a) the $\widehat{x}$ direction and (b) the $\widehat{z}$ direction. The nodal surface is preserved in (a) but not in (b).

Figure 8

(a) Crystal structure of ${\mathrm{Cu}}_3{\mathrm{Se}}_2$ and (b) the corresponding Brillouin zone. In (b), ${\mathrm{\Gamma }}_1, {\mathrm{X}}_1, {\mathrm{Y}}_1$, and ${\mathrm{M}}_1$ label the four $k$ points that are above the points $\mathrm{\Gamma }$, X, Y, and M and are located in the $k_z=\pi /2c$ plane. (c) Band structure of ${\mathrm{Cu}}_3{\mathrm{Se}}_2$ (with SOC) along some generic paths. The red arrows indicate the band degeneracy along the paths on the two orthogonal nodal surfaces.

Figure 9

Comparison of band structures of the eight-band tight binding model in Eq. (A1) (red dashed lines) and the DFT result (blue solid lines). In the tight-binding model, the values $t_1=2.95$ eV and $t_2=1.30$ eV are obtained from fitting the low-energy DFT band structure.

Figure 10

(a) Band structure for QGN(1,2) with SOC. The SOC strength is artificially enhanced by 30 times, in order to show that the SOC gaps the original nodal surfaces. (b) Enlarged view around the original band crossing, corresponding to the region as indicated by the red arrow in (a).

Figure 11

(a) Crystal structure of ${\mathrm{Rb}}_2{\mathrm{Se}}_5$ and (b) the corresponding Brillouin zone. The insert in (a) shows the helical ${\mathrm{Se}}_5^{2-}$ chain. In (b), ${\mathrm{\Gamma }}_1$ is the body center of 1/8 Brillouin zone; ${\mathrm{X}}_1, {\mathrm{Y}}_1$, and ${\mathrm{Z}}_1$ are the face centers of the 1/8 Brillouin zone; ${\mathrm{U}}_1, {\mathrm{T}}_1$, and ${\mathrm{S}}_1$ are the midpoints of the paths U-R, T-R, and S-R, respectively. (c) Band structure of ${\mathrm{Rb}}_2{\mathrm{Se}}_5$ with SOC.

Figure 12

Effects of the Hubbard U correction on band structures of (a) ${\mathrm{K}}_6{\mathrm{YO}}_4$ (without SOC) with $U=4.0$ eV, (b) ${\mathrm{RbMo}}_3{\mathrm{S}}_3$ (without SOC) with $U=4.0$ eV, (c) ${\mathrm{TlMo}}_3{\mathrm{Te}}_3$ (SOC included) with $U=4.0$ eV, (d) ${\mathrm{Ta}}_3{\mathrm{TeI}}_7$ (SOC included) with $U=2.0$ eV, (e) ${\mathrm{CsCrI}}_3$ (without SOC) with $U=5.0$ eV, and (f) ${\mathrm{Cu}}_3{\mathrm{Se}}_2$ (SOC included) with $U=5.0$ eV.