Magnetic and noncentrosymmetric Weyl fermion semimetals in the $\mathit{R}\mathrm{AlGe}$ family of compounds ($\mathit{R}=\mathrm{rare}\mathrm{earth}$)

Weyl semimetals are novel topological conductors that host Weyl fermions as emergent quasiparticles. In this Rapid Communication, we propose a new type of Weyl semimetal state that breaks both time-reversal symmetry and inversion symmetry in the $R\mathrm{AlGe}$ ($R=\mathrm{rare}-\mathrm{earth}$) family. Compared to previous predictions of magnetic Weyl semimetal candidates, the prediction of Weyl nodes in $R\mathrm{AlGe}$ is more robust and less dependent on the details of the magnetism because the Weyl nodes are generated already by the inversion breaking and the ferromagnetism acts as a simple Zeeman coupling that shifts the Weyl nodes in $k$ space. Moreover, $R\mathrm{AlGe}$ offers remarkable tunability, which covers all varieties of Weyl semimetals including type I, type II, inversion breaking, and time-reversal breaking, depending on a suitable choice of the rare-earth elements. Furthermore, the unique noncentrosymmetric and ferromagnetic Weyl semimetal state in $R\mathrm{AlGe}$ enables the generation of spin currents.

Figure 1

Lattice structure, Brillouin zone (BZ), and density of state (DOS) of $R\mathrm{AlGe}$ ($R=\mathrm{La}$, Ce, and Pr). (a) Body-centered tetragonal structure of $R\mathrm{AlGe}$ with space-group $I{4}_{1}md$ (109). The structure consists of stacks of rare-earth elements (RE), Al, and Ge layers, and along the (001) direction each layer consists of only one type of element. (b) The schematic of the RE atomic layer showing the skew axis along the $c$ axis. (c) The bulk and (001) surface BZ. (d)–(f) First-principles DOS of (d) LaAlGe, (e) CeAlGe, and (f) PrAlGe. The partial DOS for spin-up and spin-down states are plotted in red and violet colors, respectively. The DOS from localized $f$ orbitals are drawn in green color.

Figure 2

First-principles band structure of $R\mathrm{AlGe}$ ($R=\mathrm{La}$, Ce, and Pr). (a) and (b) Calculated bulk band structure of LaAlGe without and with the inclusion of spin-orbit coupling. (c) and (d) Bulk band structure of CeAlGe without and with the inclusion of spin-orbit coupling. In (c) the bands of spin-up and spin-down states are plotted in red and violet colors, respectively. (e) and (f) The same as in (c) and (d) but for PrAlGe.

Figure 3

Weyl fermions in LaAlGe, CeAlGe, and PrAlGe. (a) and (c) Weyl nodes (denoted by $W3$) in the first BZ of CeAlGe and PrAlGe with SOC. The arrows indicate the magnetization orientation. The red and blue dots denote the new Weyl nodes ($W4$) generated by the magnetization of the $f$ orbitals of Pr. (b) Projection of the Weyl nodes on the (001) surface Brillouin zone (SBZ) of CeAlGe. The configuration of the other half of the SBZ can be obtained by considering mirror symmetry. (d) and (e) The same as in (b) but for PrAlGe. (f)–(h) Schematics of the band dispersion of the Weyl fermions in LaAlGe, CeAlGe, and PrAlGe, respectively.

Figure 4

A new route for generating magnetic Weyl fermions. (a) and (b) The Dirac node in a Dirac semimetal splits into a pair of Weyl nodes upon the inclusion of a magnetic field. The Weyl nodes are connected by a Fermi arc surface state. (c) and (d) Two pairs of Weyl nodes are present in an inversion-breaking Weyl semimetal. The Weyl nodes are shifted in momentum space upon the inclusion of a magnetic field, generating regions in $\mathbf{k}$ space with a nonzero Chern number.