Single pair of Weyl fermions in the half-metallic semimetal $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$

Materials with the ideal case of a single pair of Weyl points (WPs) are highly desirable for elucidating the unique properties of Weyl fermions. $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ is an antiferromagnetic topological insulator or Dirac semimetal depending on the different magnetic configurations. Using first-principles band-structure calculations, we show that inducing ferromagnetism in $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ can generate a single pair of WPs from splitting the single pair of antiferromagnetic Dirac points due to its half-metallic nature. Analysis with a low-energy effective Hamiltonian shows that a single pair of WPs is obtained in $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ because the Dirac points are very close to the zone center and the ferromagnetic exchange splitting is large enough to push one pair of WPs to merge and annihilate at Γ while the other pair survives. Furthermore, we predict that alloying with Ba at the Eu site can stabilize the ferromagnetic configuration and generate a single pair of Weyl points without application of a magnetic field.

Figure 1

(a) Band structure of $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ in PBE+$U$+SOC for $A$-type AFM with magnetic moment along $c$ axis (AFMAc). (b) crystal structure of $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ in space group $164\left(P\overline{3}m1\right)$. Band structure along Γ$A$ for (c) AFMAc, (d) moments in plane along major axis (AFMAa) breaking threefold rotational symmetry, and (e) FM moment along $c$ (FMc) breaking the combined $S=\mathrm{\Theta }{\tau }_{1/2}$ symmetry. The green shade over the band structure stands for the projection of As $p$ orbitals. (f) Berry curvature on the $k_x\text{−}{k}_z$ plane showing chirality for the single pair of Weyl points in (e).

Figure 2

Fermi arcs at $E_F$ on the (a) top and (b) bottom (110) surfaces of $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ in FMc with a single pair of Weyl points; and those at $E_F+0.048$ eV on the (c) top and (d) bottom (110) surfaces. The color scheme of the intensity from high to low is red, white, and blue.

Figure 3

Bulk band structure of $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ along Γ$A$ direction for nonmagnetic (NM) without Eu $4f$ in (a) PBE and (b) PBE+SOC; AFM in (c) PBE+$U$ and (d) PBE+$U$+SOC; and FM in (e) PBE+$U$ and (f) PBE+$U$+SOC. The NM and FM have the primitive 5-atom unit cell. The AFM has the 10-atom unit cell by doubling along $c$ direction. The green shade shows the projection on As $p$ orbitals. Each band in (c) has a different color to show the orbital degeneracy in the valence band.

Figure 4

Band structure near Γ along Γ$A$ of the 4-band low-energy Hamiltonian based on $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$ with different values of exchange-splitting parameters $h_1$ and $h_2$ in meV.

Figure 5

Band structure of 50% Ba (gray) substituted $\mathrm{EuC}{\mathrm{d}}_2\mathrm{A}{\mathrm{s}}_2$, or $\mathrm{EuBaC}{\mathrm{d}}_4\mathrm{A}{\mathrm{s}}_4$, for (a) AFMAc and (b) FMc with moment along $c$. Unit cells are fully relaxed and FMc is preferred. The green shade shows the projection on As $p$ orbitals.

Figure 6

Phonon dispersion of $\mathrm{EuBaC}{\mathrm{d}}_4\mathrm{A}{\mathrm{s}}_4$ calculated in a (2 × 2 × 2) supercell of the FM configuration.