Coexistence of fully spin-polarized Weyl nodal loop, nodal surface, and Dirac point in a family of quasi-one-dimensional half-metals

Recent investigation has connected topology with magnetization, which has been attracting great research interest. Here, based on symmetry analysis and first-principle calculations, we reveal rich band crossings in ferromagnetic quasi-one-dimensional compounds ${XYZ}_3$ ($X=\mathrm{Cs},\mathrm{Rb}$, $Y=\mathrm{Cr},\mathrm{Cu}$, $Z=\mathrm{Cl},\mathrm{I}$). We show that these materials host hourglass Weyl loops, nodal surfaces, and Dirac points in the absence of spin-orbital coupling. Notably, these topological phases only occur in a single spin channel, leading to fully spin-polarized phases. Based on symmetry analysis, we present that the hourglass-like Weyl loop is enforced by nonsymmorphic symmetries, which give rise to drumhead surface states. Particularly, the Weyl loop can persist when a magnetic direction is normal to the plane where it lies on. Additionally, we also find the nodal surfaces protected by the combination of screw rotation and time reversal symmetry. Besides, there exists a Dirac point enabled by nonsymmorphic symmetry and time reversal symmetry. Notably, this Dirac point shows a linear dispersion in one direction and quadratic dispersion in the normal plane, showing two Fermi arcs in its surface energy spectrum. By tuning the direction of magnetization, the Dirac point can be reduced to a pair of Weyl points. Our work suggests a concrete platform for exploring the fascinating physics associated with nonsymmorphic band crossings in quasi-one-dimensional ferromagnetic systems.

Figure 1

(a) Side and (b) top views of the unit cell of ${\mathrm{CsCrCl}}_3$ compound. (c) The bulk Brillouin zone with the corresponding surface Brillouin zone of ${\mathrm{CsCrCl}}_3$ compound.

Figure 2

Electronic band structure and the projected density of states of ${\mathrm{CsCrCl}}_3$ in the (a) spin-up channel and (b) spin-down channel. In (a), the band crossing points in the $k$ paths $M\text{−}L$ and $L\text{−}\mathrm{\Gamma }$ are labeled as P1 and P2. The arrow indicates the nodal surface in the $k_z$ = $\pi$ plane. In (b), the size of band gap is labeled.

Figure 3

(a) Orbital-projected band structure of ${\mathrm{CsCrCl}}_3$ for paths near P1 and P2 points. (b) Illustration of the Weyl nodal loop in the (100) plane. (c) The 3D plot of band dispersions of the Weyl nodal loop. (d) The shape of Weyl nodal loop in the Brillouin zone.

Figure 4

(a) The comparison of band structures from the Wannier model and DFT. (b) Schematic illustration of the surface state in the projection surface. (c) Calculated surface band structure with the drumhead surface states pointed by the black arrows. (d) The constant energy slice of the surface states near the Fermi level.

Figure 5

(a) Electronic band structure along the $M\text{−}L\text{−}\mathrm{\Gamma }$ path under SOC with magnetization along the [010] direction. (b), (d) Schematic figures for the magnetization along [010] and [100] directions, respectively. (c) Electric band structure along the $M\text{−}L\text{−}\mathrm{\Gamma }$ path under SOC with magnetization along the [100] direction.

Figure 6

(a) The enlarged view for the band structure along $M\text{−}A\text{−}L\text{−}H\text{−}A$ path. The inset image in (a) shows the Fermi surface of ${\mathrm{CsCrCl}}_3$. (b) Enlarged view for the band structure along a generic path E-R [marked in Brillouin zone (d)]. (c) 3D plotting of the bands in the $k_z$ = $\pi$(left) and $k_z$ = $0.9\pi$ (right) planes. (d) The position of the nodal surface in the Brillouin zone (denoted as green planes). R and E are the centers of the triangles $A\text{−}L\text{−}H$ and $\mathrm{\Gamma }\text{−}M\text{−}K$, respectively.

Figure 7

Band structure along the $A\text{−}L\text{−}H\text{−}A$ path with SOC. Three magnetization directions have been considered: (a) [010], (b) [100], and (c) [001].

Figure 8

(a) The enlarged view of the band structure in the $M\text{−}A\text{−}\mathrm{\Gamma }$ paths. (b) Calculated surface band structure and (d) the corresponding constant energy slices near the Fermi level. Band structure along the $M\text{−}A\text{−}\mathrm{\Gamma }$ paths with SOC included in (c) [010] and (d) [001] magnetization directions.

Figure 9

The calculated electronic band structures for (a) ${\mathrm{RbCuCl}}_3$, (b) ${\mathrm{CsCrI}}_3$, and (c) ${\mathrm{CsCuCl}}_3$ along the $M\text{−}L\text{−}\mathrm{\Gamma }\text{−}A\text{−}L\text{−}H\text{−}A$ path. The size of band gaps in spin-down channels is labeled.