Time-Reversal-Breaking Weyl Fermions in Magnetic Heusler Alloys

Weyl fermions have recently been observed in several time-reversal-invariant semimetals and photonics materials with broken inversion symmetry. These systems are expected to have exotic transport properties such as the chiral anomaly. However, most discovered Weyl materials possess a substantial number of Weyl nodes close to the Fermi level that give rise to complicated transport properties. Here we predict, for the first time, a new family of Weyl systems defined by broken time-reversal symmetry, namely, Co-based magnetic Heusler materials $X{\mathrm{Co}}_{2}Z$ ($X=\mathrm{IVB}$ or VB; $Z=\mathrm{IVA}$ or IIIA). To search for Weyl fermions in the centrosymmetric magnetic systems, we recall an easy and practical inversion invariant, which has been calculated to be $-1$, guaranteeing the existence of an odd number of pairs of Weyl fermions. These materials exhibit, when alloyed, only two Weyl nodes at the Fermi level—the minimum number possible in a condensed matter system. The Weyl nodes are protected by the rotational symmetry along the magnetic axis and separated by a large distance (of order $2\pi$) in the Brillouin zone. The corresponding Fermi arcs have been calculated as well. This discovery provides a realistic and promising platform for manipulating and studying the magnetic Weyl physics in experiments.

Figure 1

(a) Rocksalt crystal structure of ${\mathrm{ZrCo}}_2\mathrm{Sn}$ with $Fm\overline{3}m$ space group. Co, Zr, and Sn atoms are shown in blue, gray, and green, respectively. (b) Brillouin zone (BZ) of the rocksalt structure. It has three independent time-reversal-invariant points, $\mathrm{\Gamma }(0,0,0)$, $L[(\pi /a),(\pi /a),(\pi /a)]$, and $X[(2\pi /a),0,0]$. (c) Bulk band structure of ${\mathrm{ZrCo}}_2\mathrm{Sn}$ along high-symmetry lines without spin-orbit coupling (SOC). The majority and minority spin bands are indicated by solid black and dashed red lines, respectively. (d) Total and partial density of states without SOC. The positive (negative) lines represent the majority (minority) spin channel.

Figure 2

(a) The nodal line in the $xy$ plane has a big dispersion. (b) Three nodal lines in three planes in three-dimensional (3D) $k$ space. (c) The SOC band structure of ${\mathrm{ZrCo}}_2\mathrm{Sn}$ with the [110] magnetism, opening a small gap in the $\mathrm{\Gamma }\mathrm{W}$ direction in the inset. (d) Weyl points emerging with SOC. The independent $W$, ${\mathrm{W}}_1$, and ${\mathrm{W}}_2$ points are clearly indicated in the figure (top view)—the remaining ones can be obtained by symmetry. $W$ and ${\mathrm{W}}_1$ are in the $xy$ plane, while ${\mathrm{W}}_2$ is out of the plane. The Chern numbers can be calculated with the Wilson-loop method applied on a sphere (illustrated as dashed circles) enclosing a Weyl point. The filled (unfilled) symbols indicate Chern number $+1$ ($-1$). Furthermore, the 001-surface lattice vectors are also given as ${\overrightarrow{k}}_1[(2\pi /a),-(2\pi /a)]$ and ${\overrightarrow{k}}_2[(2\pi /a),(2\pi /a)]$, and the corresponding surface BZ is shown in green.

Figure 3

(a) The flow chart of the average position of Wannier centers obtained by the Wilson-loop method applied on a sphere that encloses a node [57]. The average Wannier center shifts downwards, corresponding to Chern number $-1$ for $W$ and ${\mathrm{W}}_2$, while it shifts upwards, suggesting the Chern number of ${\mathrm{W}}_1$ is $+1$. (b) Calculated Bloch spectral function of ${\mathrm{Zr}}_{0.725}{\mathrm{Nb}}_{0.275}{\mathrm{Co}}_2\mathrm{Sn}$ along high-symmetry lines.

Figure 4

Bloch spectral function of the (001) surface at 0.5 eV above the Fermi level for ${\mathrm{ZrCo}}_2\mathrm{Sn}$. In the (001)-surface Brillouin zone, the surface k points are represented by $x_1{\overrightarrow{k}}_1+x_2{\overrightarrow{k}}_2$. The surface lattice vectors (${\overrightarrow{k}}_1$ and ${\overrightarrow{k}}_2$) are illustrated, and the corresponding surface BZ is shown in green in Fig. 2 (notice the $\pi /2$ rotation of the surface BZ). Only the bulk projections of $W$ (yellow) are separated from the Fermi surface projections. The bulk projections of ${\mathrm{W}}_1$ and ${\mathrm{W}}_2$ (black) sit inside the projection of the bulk Fermi surfaces. The Fermi arcs connecting to the bulk projections of $W$ are shown. The large, square surface state is of a trivial nature. The color code represents $\log (\rho )$.