Stability of Weyl points in magnetic half-metallic Heusler compounds

We employ ab initio fully relativistic electronic structure calculations to study the stability of the Weyl points in the momentum space within the class of the half-metallic ferromagnetic full Heusler materials, by focusing on ${\mathrm{Co}}_2\mathrm{TiAl}$ as a well-established prototype compound. Here we show that both the number of the Weyl points together with their $k$-space coordinates can be controlled by the orientation of the magnetization. This alternative degree of freedom, which is absent in other topological materials (e.g., in Weyl semimetals), introduces functionalities that are specific for the class of half-metallic ferromagnets. Of special interest are crossing points which are preserved irrespective of any arbitrary rotation of the magnetization axis.

Figure 1

Calculated electronic band structure of ${\mathrm{Co}}_2\mathrm{TiAl}$ using the scalar-relativistic formalism. The zero energy in the vertical axis is the Fermi level. We have denoted the six crossing points of interest with the symbols A–F. Note that all high-symmetry points are within the $k_z=0$ plane.

Figure 2

Orbital character of the bands which cross at the six points along the $K-\mathrm{\Gamma }-W$ direction shown in Fig. 1.

Figure 3

(a) Calculated electronic band structure of ${\mathrm{Co}}_2\mathrm{TiAl}$ using the fully relativistic formalism and setting the quantization axis to be the [100]. (b) Same as (a), with [001] as the quantization axis. In both cases, the zero energy in the vertical axis is the Fermi level. We have encircled with red colors the crossing points where the degeneracy is lifted and crossing no longer occurs, and with green color the ones where the crossing is preserved.

Figure 4

Upper panel: Fully relativistic band structures of ${\mathrm{Co}}_2\mathrm{TiAl}$ computed along the $K\text{−}\mathrm{\Gamma }\text{−}{K}^{\prime }k$ path ($\mathrm{\Gamma }\text{−}K$ and $\mathrm{\Gamma }\text{−}{K}^{\prime }$ directions are denoted explicitly) for three orientations of the magnetization $\mathbf{M}$, i.e., [001], [111], and [110]. ($E,k$) points of interest (of A–D type) are emphasized by green or red circles depending on their crossing or anticrossing character, respectively. Minority- and majority-spin states are colored as red and blue, respectively. Lower panels: The corresponding nonequivalent points of each type, plotting the spectral weight on $\mathrm{\Gamma }$-centered spheres in 3D $k$ space (green letters mark crossing character and black letters mark anticrossing character). Red and blue lines and surfaces mark minority- and majority-spin states as in the upper panel; note that at some parts of the spheres, there is superposition of the two different spin states and thus of the two colors used to denote them.

Figure 5

Calculated electronic band structure of ${\mathrm{Co}}_2$(${\mathrm{Ti}}_{0.8}{\mathrm{V}}_{0.2}$)Al using the scalar-relativistic formalism within the virtual crystal approximation for a strained lattice (see text for details). Note that now the B points are located exactly at the Fermi level which is the zero energy in the vertical axis.