Weyl nodal-line surface half-metal in ${\mathrm{CaFeO}}_3$

Manipulating the spin degrees of freedom of electrons affords an excellent platform for exploring novel quantum states in condensed-matter physics and material science. Based on first-principles calculations and analysis of crystal symmetries, we propose a fully spin-polarized composite semimetal state, which is combined with the 1D nodal lines and 2D nodal surfaces, in the half-metal material ${\mathrm{CaFeO}}_3$. In the nodal line-surface half-metal, the Baguenaudier-like nodal lines feature six rings linked together, which are protected by the three independent symmetry operations: $\mathcal{PT}, {\mathcal{M}}_y$, and ${\tilde{\mathcal{M}}}_z$. Near the Fermi level, 2D nodal surfaces with fully spin-polarized are guaranteed by the joint operation $\mathcal{T}{\mathcal{S}}_{2i}$ in the $k_{i(i=x,y,z)}=\pi$ plane. Furthermore, high-quality ${\mathrm{CaFeO}}_3$ harbors ultraclean energy dispersion, which is rather robust against strong hydrostatic compressional strain and correlation effect. The realization of the Weyl nodal line-surface half-metal presents great potential for spintronics applications with high-speed and low-power consumption.

Figure 1

(a) The crystal structure and (b) the 3D magnetic charge density of ${\mathrm{CaFeO}}_3$. Note: Gold, pink, and light blue colors stand for iron, oxygen, and calcium atoms, respectively.

Figure 2

Spin-resolved band structure and the density of state of (a) ${\mathrm{CaFeO}}_3$. Orbital-projected band structure of spin-up channel for (b) ${\mathrm{CaFeO}}_3$ contributed by O-2$p$ and Fe-3$d$ ($d{}_{xz}$ and $d{}_{x^2-y^2}$) orbitals. Note: the red and blue arrows represent spin-up and spin-down channels, respectively.

Figure 3

The 3D profile of the nodal lines lie on the (a) $k{}_z$ = 0 plane and the (c) $k{}_y$ = 0 plane, respectively. Band structures of spin-up channel for ${\mathrm{CaFeO}}_3$ along the X-$\mathrm{\Gamma }$-Y path and X-$\mathrm{\Gamma }$-Z path. The eigenvalues of ${\tilde{\mathcal{M}}}_z$ and ${\mathcal{M}}_y$ symmetry operators for each band are labeled $+$ and $-$ in the enlarged band structures (b) and (d). The green dashed lines labeled $L{}_1$ and $L{}_2$ represent two respective loops along which the Berry phase is calculated. The topological protection of nodal lines is further checked by directly calculating the nontrivial $\pi$ Berry phase along two small loops ($L{}_1$ and $L{}_2$) enclosing the nodal lines.

Figure 4

(a) The 3D view of the snakelike nodal lines in the BZ. The calculated Berry phase of an arbitrary green dashed ring (i.e., $L{}_3$) intersecting a snakelike nodal line. (b) The projected surface states for the (010) plane. (c) The xoz perspective of the nodal line profiles in the BZ. (d) The isoenergy ($E{}_f$ = 0.1 eV) surface around the $\overline{\mathrm{\Gamma }}$ point for the (010) plane.

Figure 5

(a) Spin-resolved band structure of ${\mathrm{CaFeO}}_3$. (b) The corresponding BZ. Note: M is the middle point of the $\mathrm{\Gamma }$-R path and ${\mathrm{N}}_1$ (${\mathrm{N}}_2$ and ${\mathrm{N}}_3$) is the middle point of the X-R (Y-R and Z-R) path. Enlarged band dispersion along the M and ${\mathrm{N}}_1$ (c) ${\mathrm{[N}}_2$ (d) and ${\mathrm{N}}_3$ (e)] path showing the linear crossing between two bands along the $k{}_x$ ($k{}_y$ and $k{}_z$) direction.

Figure 6

Schematics of four different magnetic structures (a) FM, (b) AFM1, (c) AFM2, and (d) AFM3 for ${\mathrm{CaFeO}}_3$. The red atoms stand for the spin-up direction and the blue atoms stand for the spin-down direction.

Figure 7

(a) Schematic of the nodal surface. The bands in $k_{i(i=x,y,z)}=\pi$ planes, highlighted in cyan, are double degenerate. (b) The 3D band structure with ${\mathrm{t}}_1=0.5$ and ${\mathrm{t}}_2=0.4$. Here only the two bands near the Fermi level are shown. There is a nodal line, highlighted by the yellow line, centering on the $Y$ point.

Figure 8

Spin-resolved band structures of ${\mathrm{CaFeO}}_3$ in the absence of SOC effects for (a) $U=3$ eV, (b) $U=5$ eV, (c) $U=6$ eV, and (d) $U=7$ eV. Red and blue lines represent spin-up and spin-down channels, respectively.

Figure 9

Spin-resolved band structures of ${\mathrm{CaFeO}}_3$ ($U=4$ eV) with different hydrostatic compressional strains for (a) $-$1%, (b) −2%, (c) −3%, (d) −4%, (e) −5%, and (f) −6% are calculated.

Figure 10

(a) Calculated band structure of ${\mathrm{CaFeO}}_3$ ($U$ = 4 eV) with SOC. (b) Band structure of ${\mathrm{CaFeO}}_3$ after artificially increasing the intrinsic SOC by 10 times.

Figure 11

The projected surface states of ${\mathrm{CaFeO}}_3$ for (a) the (100) and (b) the (001) planes.