Noncentrosymmetric compensated half-metal hosting pure spin Weyl nodes, triple nodal points, nodal loops, and nexus fermions

Materials containing multiple topological characteristics become more exotic when combined with noncentrosymmetric crystal structures and unusual magnetic phases such as the compensated half-metal state, which is gapped in one spin direction and conducting in the other. First-principles calculations reveal these multiple topological features in the compensated half-metal ${\mathrm{Cr}}_2\mathrm{CoAl}$ having neither time-reversal nor inversion symmetries. In the absence of (minor) spin-orbit coupling (SOC), there are (1) a total of 12 pairs of magnetic Weyl points, (2) three distinct sets of triple nodal points near the Fermi level that are (3) interconnected with six symmetry-related nodal lines. This combination gives rise to fully spin polarized nexus fermions, in a system with broken time-reversal symmetry but negligible macroscopic magnetic field. The observed high Curie temperature of 750 K and calculated SOC hybridization mixing of several meV should make these nexus fermions readily measurable. Unlike topological features discussed for other Heuslers which emphasize their strong ferromagnetism, this compensated half-metal is impervious to typical magnetic fields, thus providing a complementary set of experimental phenomena. Making use of the soft calculated magnetic state, large magnetic fields can be used to rotate the direction of magnetism, during which certain topological features will evolve. Our results suggest that these features may be common in inverse-Heusler systems, particularly the isostructural and isovalent Ga and In analogs.

Figure 1

(a) Structure of the inverse-Heusler ${\mathrm{Cr}}_2\mathrm{CoAl}$ with the sequence of Cr1-Cr2-Co-Al along the diagonal direction. (b) Bulk and (001) surface Brillouin zones (BZs) with high-symmetry points of the inverse Heusler systems. The dots in the bulk BZ indicate Weyl points (WPs) with positive (red) and negative (blue) chiralities. There are 12 pairs of WPs protected by the three twofold rotational axes along the $\langle 100\rangle$ directions, and six mirrors (${\mathcal{M}}_{\pm xy}, {\mathcal{M}}_{\pm yz}, {\mathcal{M}}_{\pm zx}$). (c) Total and atom-resolved densities of states (DOSs) per formula unit of ${\mathrm{Cr}}_2\mathrm{CoAl}$, showing the half-metallic character. The spin-down DOS, with a gap at the Fermi level $E_F$, is plotted downward.

Figure 2

(a) GGA spin-up band structure of ${\mathrm{Cr}}_2\mathrm{CoAl}$ near $E_F=0$ in the absence of SOC. The spin-down bands (not shown) are gapped at $E_F$. The blue square indicates nodal points. (b),(c) Enlarged spin-up plots around the Weyl points lying around $-0.07$ eV along the $X-W$ line, and along a line parallel to the $\left[\overline{1}10\right]$ direction, respectively. ${\mathrm{W}}_1$ (${\mathrm{W}}_2$) denotes a WP with positive (negative) chirality at $(0.4132,0,0.887)\frac{2\pi }{a}\left[(0,0.4132,0.887)\frac{2\pi }{a}\right]$. (d) Bands enlarged around the TNPs, lying just 10 meV above $E_F$ along the [001] direction (SOC neglected). The red dots denote a pair of symmetry-related TNPs, with the ${\mathrm{\Delta }}_1$ band having essentially zero velocity at the crossing. (e) Effect of SOC on the TNP, showing a hybridization splitting of 10 meV, for spin along the (001) direction. (f) Plot of the $\pi$-Berry phase for the nodal loops, indicating topologically nontrivial loops. For the trivial nodal lines, the Berry phase is zero. (g) Plot of the nodal lines: a trivial along the $\mathrm{\Gamma }-X$ rotational axis, meeting with topologically nontrivial nodal loops centered on $X$. Only the loops oriented along the $\widehat{z}$ direction are shown. Nexus fermions appear midway along $\mathrm{\Gamma }-X$, marked by blue dots. TNPs (blue dots) occur at crossing points of two nodal lines lying on two perpendicular mirror planes.

Figure 3

Several views of surface spectra neglecting SOC, with bright yellow indicating high surface intensity. Top and bottom rows of panels are for the Cr1-Co and Cr2-Al surface terminations, respectively. (a),(b) The (001) surface spectral functions of the spin-up channel along symmetry lines. These are followed by isoenergy spectral densities of the surface states at energy lying at the WPs (middle column) and TNPs (right column). The WPs in (c) and (d) are denoted as green and red circles, whereas the light-blue squares of (e) and (f) indicate TNPs. The $\overline{X}$ and $\overline{Y}$ points in (a) and (b) lie at each midpoint of adjacent faces of the surface BZ, outlined by white lines in panels (c)–(f). The high-symmetry points relative to bulk are provided in Fig. 1.