Abstract
Even if Weyl semimetals are characterized by quasiparticles with welldefined chirality, exploiting this experimentally is severely hampered by Weyl lattice fermions coming in pairs with opposite chirality, typically causing the net chirality picked up by experimental probes to vanish. Here, we show this issue can be circumvented in a controlled manner when both timereversal and inversion symmetry are broken. To this end, we investigate chirality disbalance in the carbide family RMC_{2} (R a rareearth and M a transition metal), showing several members to be Weyl semimetals. Using the noncentrosymmetric ferromagnet NdRhC_{2} as an illustrating example, we show that an odd number of Weyl nodes can be stabilized at its Fermi surface by properly tilting its magnetization. The chiral configuration endows a topological phase transition as the Weyl node transitions across the Fermi sheets, which triggers interesting chiral electromagnetic responses. Further, the tilt direction determines the sign of the resulting net chirality, opening up a simple route to control its sign and strength.
Introduction
Since their experimental discovery in the TaAs family, Weyl semimetals continue to gain interest. The nontrivial topology of this electronic phase follows from the geometrical properties^{1} associated with electronic bands. Specifically, for band crossing points, a topological invariant can be defined as the flux of Berry curvature through a surface enclosing the point. The lowenergy effective theory around such a point corresponds to the Weyl equation^{2}, identifying the topological invariant with the corresponding Weyl fermion chirality. In lattice systems, however, Weyl nodes must come in pairs of opposite chirality^{3}. Each pair having zero net chirality severely hampers experimental probes sensitive to Brillouin zone integrated quantities from picking up the Weyl node chirality: for this to work, one should create an overall chirality imbalance.
The essential condition to enable the existence of Weyl nodes in the first place is broken spin degeneracy of Bloch states at a generic crystal momentum. This requires timereversal symmetry (Θ) or inversion symmetry (\({{{\mathcal{I}}}}\)) to be broken. While the first experimental confirmations of topological semimetals were achieved in Θsymmetric compounds^{4,5,6,7,8,9,10,11}, magnetic compounds are naturally appealing due to the broad prospects that the interplay between the electronic structure and external magnetic fields can offer^{12,13,14,15,16,17,18,19,20,21,22}. Still, experimentally confirmed magnetic topological semimetals are rather scarce. Two recently investigated cases are the nodalline semimetal Co_{2}MnGa^{23,24} and the Weyl semimetal Co_{3}Sn_{2}S_{2}^{25,26,27,28}, both centrosymmetric ferromagnets with nontrivial topology. In EuCd_{2}As_{2}^{29} and EuCd_{2}Sb_{2}^{30}, on the other hand, a Weyl semimetallic phase is realized for the fully spinpolarized state induced by an external magnetic field.
Here we focus on the rareearth carbides RMC_{2}, with R a rareearth metal and M a transition metal. This broad family of compounds exhibits a diversity of Θbreaking or \({{{\mathcal{I}}}}\)breaking phenomena^{31,32,33,34,35,36,37,38,39,40,41,42,43,44,45} both in longrange ordered magnetic compounds^{31,32,33,34,35,36,37} as well as in superconducting phases^{38,39,40,41} and a complex interplay between these phases and charge density waves (CDW)^{42,43,44,45,46,47,48,49,50,51}. We show by consideration of available experimental information and own densityfunctional calculations (DFT) that RMC_{2} compounds can be categorized in four classes: (I) Θsymmetric and \({{{\mathcal{I}}}}\)broken semimetals (YCoC_{2} and LuCoC_{2}); (II) Θbroken and \({{{\mathcal{I}}}}\)symmetric metals (GdRuC_{2}); (III) both Θ and \({{{\mathcal{I}}}}\)broken semimetals (PrRhC_{2}, NdRhC_{2}, GdCoC_{2} and GdNiC_{2}); and (IV) insulators (LaRhC_{2}), the latter being of secondary interest for this work. We find that all the mentioned semimetals possess Weyl nodes close to the Fermi energy, as summarized in Table 1, where they also tend to have relatively simple and uncluttered band structures. Figure 1 illustrates the two observed structure types of the (semi)metallic compounds.
Systems belonging to class III are of particular interest as they allow for specific demonstrations of the unique interplay between topology and magnetism offered by \({{{\mathcal{I}}}}\)broken symmetry. As a proof of principle, we show for NdRhC_{2} and GdCoC_{2} how in ferromagnetic (FM) noncentrosymmetric phases tilting of the magnetization (m) along a lowsymmetry direction produces a disbalance in the number of opposite chirality Weyl fermions near the Fermi surface: of all Weyl nodes the degeneracy is lifted. Noteworthy, the direction in which the magnetization is canted controls the sign of the chirality disbalance, allowing therefore to switch the dominant lowenergy chirality of the electrons. Such chirality imbalanced configuration naturally leads to a variety of physical consequences, some simply related with the chiral symmetry induced on the electronic structure, others with the possibility—unique of chiral systems—of topological phase transitions in which a single Weyl node crosses the Fermi level and moves between regions enclosed by different Fermi surface sheets. We discuss the ramifications of chirality imbalance and the related topological transitions for existence and observation of chiral and gyrotropic magnetic effects (GME) in these compounds.
Results and discussions
From Y to Lu
For compounds based on different R elements, we first present the main structural, electronic and magnetic properties of interest for this work. Our DFT results were obtained using the generalized gradient approximation (GGA)^{52}, as implemented in FPLO18^{53} (https://www.fplo.de/) and considering for the treatment of the occupied 4fstates both the opencore approximation (OC) with Rspecific 4f spin moment but spherical orbital occupation^{54} and the GGA + U method with the fulllocalized limit for the double counting correction^{55} and parameters U = 7 eV and J = 1 eV. We study compounds which have already been synthesized, including R = Y, La, Nd, Pr, Gd or Lu, and M = Co, Rh, Ni or Ru^{31,32,34,43}, and use the lattice parameters from The Materials Project^{56} (see the Supplementary Note I for further details). Table 1 includes the available experimental data. For most of the compounds, we find that the DFT calculations agree with the reported magnetic properties. In order to analyze the electronic properties of the whole family on the same footing, we will fix in the following the quantization axis along the direction [001].
We find that the compounds involving rare earths with empty or completely filled fshells, R = Y, La or Lu, result in Θsymmetric nonmagnetic (NM) states, in agreement with experiment^{32,43,57}. Among these, LaRhC_{2} is the only system that crystallizes in the noncentrosymmetric tetragonal space group P4_{1}^{32} and displays an insulating electronic structure (see Supplementary Note III). Opposed to this, YCoC_{2} and LuCoC_{2} grow in the noncentrosymmetric orthorhombic space group Amm2^{43,57} and exhibit a semimetallic band structure (see Fig. 2a and Supplementary Note III). Four bands dominate the energy spectrum of LuCoC_{2} near the Fermi energy. These arise from hybridized Co3d, C2p and R5d states (referred to as pd states from now on). The same characteristic four bands are present at low energy for other compounds with M = Co or Rh (Fig. 2b, c, e). On the other hand, compounds based on transition metals not isoelectronic to Co, namely, Ru or Ni, exhibit a shift of the Fermi energy of roughly ±1 eV (Fig. 2d). This shift naturally changes the nature of the bands around the Fermi energy and, as we will see, their electronic properties.
Members of the family based on magnetic R ions are characterized as a lattice of localized moments on the R4f states coupled with the pd states via an onsite (Kondo) exchange. Ruderman–Kittel–Kasuya–Yosida exchange interactions between the 4f moments can lead to longrange magnetic order including the induced moments on the pd states. Consequently, these compounds are an interesting platform to explore how external magnetic fields, which couple primarily with the R4f states, can tune Weyl node properties of the lowenergy electronic structure associated with the pd states.
Among the magnetic compounds, those obtained by replacing Lu by magnetic rare earths while retaining the space group Amm2 (Fig. 1a, c) belong to Class III. The crystal structure in this case can be regarded as a stack of rareearth planes, responsible for the breaking of Θ at low temperatures, and planes of MC_{2}, responsible for the breaking of \({{{\mathcal{I}}}}\).
For these compounds, different magnetic states have been reported. GdCoC_{2} was first described as antiferromagnetic (AFM) with inplane moments canted at 48° from the aaxis and T_{N} = 15.6 K^{34}. However, a more recent study by Meng et al. finds a FM state below ~15 K^{36}. GdNiC_{2}, on the other hand, as most of the Nibased members in the family^{35}, presents an AFM ground state^{34}. For PrRhC_{2} and NdRhC_{2}, experimental data is limited to hightemperature susceptibility measurements^{32}. These indicate small Curie–Weiss temperatures (θ_{CW} ~ −13 K and ~0 K, respectively) and magnetic moments in good agreement with the values expected for the trivalent R ions. GdRuC_{2} exhibits an interesting contrast to the above, since it presents a transition to a FM phase but the \({{{\mathcal{I}}}}\)breaking distortion in the MC_{2} layers is absent and the space group is Cmcm (Fig. 1b), thus belonging to Class II^{33}.
Our DFT results correctly capture the NM states in YCoC_{2} and LuCoC_{2} and the FM states in GdCoC_{2} and GdRuC_{2}. For NdRhC_{2}, we find that both GGA+OC and GGA + U predict an FM ground state. Results for PrRhC_{2} are somewhat more complex, as GGA+OC and GGA + U predict as ground state AFM and FM order, respectively. The only clear deviation from experiment is found in GdNiC_{2}, where our calculations predict the ground state to be FM. One should, however, keep in mind that in the Nibased carbides, a strong interplay between the magnetic state and a CDW (not explored in our calculations) has been established. In fact, there are indications that the CDW tends to compete against the FM phase^{42,46,48,50}. Specifically in GdNiC_{2}, different metamagnetic transitions have been observed under moderate external magnetic fields, yielding an interesting and complex phase diagram^{42,45,51}.
Weyl nodes
We now turn our attention to the topological properties of the electronic structure. For this, we will focus on the GGA + OC calculations. As a common reference for the following, we define for each compound the number of valence electrons as N and search for Weyl points in a relevant lowenergy window [−120, 120] meV, using the Pyfplo module of the FPLO package ().
Figure 2g shows the energy of the identified Weyl nodes for all the compounds considered in this work, ordered as a function of their unit cell volume (V). Table 1 includes the position and energy of the respective lowest energy Weyl nodes (for a complete list, see Supplementary Note III). Weyl nodes at low energy are found both for Θsymmetric and Θbreaking cases and, among the latter, both for compounds with FM or AFM orderings. YCoC_{2} has been previously studied in ref. ^{57} and our results agree with the addition that the search for crossings between bands N − 1 and N discloses Weyl nodes at even lower energy, indicated in Fig. 2g.
Relations between the Weyl node structure of different compounds can also be established. For instance, an inspection of the node coordinates reveals that the fourfold degenerate Weyl nodes of lowest energy in YCoC_{2}, which lie in the k_{z} = 0 plane, are also present in LuCoC_{2} but are higher in energy. It further suggests that on replacing Y by Gd, the large exchange field induced by the Gd4f spins moves these Weyl nodes away from the k_{z} = 0 plane and to a lower energy. To confirm this, we carried out explicit computation of Weyl points in YCoC_{2} under external magnetic field acting on the spin degrees of freedom. Indeed, we find that the field endows the nodes a finite k_{z} component.
Regarding GdNiC_{2} and GdRuC_{2}, the Fermi energy shift associated with their different number of valence electrons with respect to GdCoC_{2} is naturally in opposite directions and therefore has different consequences. On GdRuC_{2}, it increases the density of states and the complexity of the lowenergy band structure. While it is interesting that it presents Weyl nodes close to the Fermi surface, it should not be considered as a semimetal. On the other hand, the upward shift of Fermi energy makes the Nibased compound of strong interest. Indeed, upon this shift, a single pair of Weyl nodes—the minimum possible in a periodic system—lies ~100 meV above the Fermi energy. Our calculations performed on different compounds neatly explain the origin of these Weyl nodes: the isostructural but NM LuCoC_{2} features twofold degenerate bands along the line XT forming a massive Dirac cone ~1 eV above the Fermi level (Fig. 2a). The band degeneracy is protected by a reflection symmetry and is lifted in the isolectronic FM compounds (Fig. 2c, e), where the Dirac cone is split into Weyl cones. Substituting Co by Ni shifts these Weyl nodes closer to the Fermi level, while keeping them along XT (Fig. 2d). While the FM phase in GdNiC_{2} has only been stabilized with an external magnetic field^{42,45}, it could be interesting to study this phenomenology in SmNiC_{2}, where the competing CDW is suppressed^{40} leading to a FM ground state^{35}.
The rather large volume change caused by the substitutions Gd → Nd or Pr and Co → Rh naturally induces sizable changes in the electronic structure which remains, however, semimetallic in NdRhC_{2} and PrRhC_{2}. Also, lowenergy Weyl nodes are present in these compounds (Fig. 2g). While there are already examples of noncollinear AFM Weyl semimetals^{58}, should the predicted magnetic structure of PrRhC_{2} be confirmed, this compound would be (to the best of our knowledge) the first realization of a collinear AFM Weyl semimetal. In addition, it would allow for the presence of the recently uncovered crystal Hall effect^{59}. This effect is characterized by a finite anomalous Hall conductivity (AHC) in a collinear antiferromagnet due to the asymmetric local environment of the magnetic sublattices. Our AHC calculations (see Table 1) show that this effect is indeed present in PrRhC_{2}.
Pumping chirality to and through the Fermi surface
Due to the NielsenNinomiya “nogo” theorem for chiral lattice fermions^{3,60}, Weyl nodes come in pairs of opposite chirality^{61} (which can be broken up by very strong electron–electron interactions^{62,63}). In a crystal, Weyl nodes occur in multiplets, their degeneracy being dictated by the Shubnikov group of the material which relates Weyl nodes of same or different chirality. Depending on the specific Shubnikov group, the Weyl node degeneracy need not be even. If the degeneracy is odd, pairs of opposite chirality are necessarily energysplit. The energy splitting between nodes of opposite chirality, a key quantity for the magnitude of different electromagnetic responses sensitive to the Weyl node chirality, actually depends on the material and external conditions. Low symmetry, in particular the absence of inversion and mirror symmetries is of the essence. Here we build on the idea of using the magnetic degrees of freedom in a \({{{\mathcal{I}}}}\)broken material to reduce the symmetry such that the Shubnikov group contains only the identity.
As a proof of principle, we consider NdRhC_{2}, although the physics we discuss can be readily extended to other noncentrosymmetric magnetic compounds. In the FM ground state, m points along the [001] direction and the Shubnikov group contains {E, m(x)Θ, m(y)Θ, C_{2}(z)}, where m and C_{2} represent the mirror symmetries and the twofold rotation, respectively. Therefore, for each Weyl node away from highsymmetry lines, there are three degenerate symmetryrelated partners. Any component of m along a lowsymmetry direction does not only break the rotation C_{2}(z) but also the symmetries involving mirrors. Thus, starting from m along [001], a perturbation that cants m toward, e.g., [111] leaves {E} as the only symmetry element, removing all degeneracies among the Weyl nodes. The natural question is how large this effect is.
Figure 3a shows the energy of Weyl nodes in NdRhC_{2} as m is canted at an angle θ toward [−111] (on the left) or [111] (right). At θ = 0, there are two sets of fourfold degenerate Weyl nodes in the energy range [−120, 80] meV. Remarkably, even a moderate canting of m is enough to produce experimentally meaningful energy splittings between Weyl nodes of opposite chirality, of the order of tens of meV. We have verified that similar results are obtained using either the experimentally reported crystal structures^{32} or crystal structures relaxed keeping the ratio of the lattice constants as in the experimental structure (see Supplementary Note II).
The tuning of Weyl nodes to the Fermi surface at specific angles resembles the prediction for Co_{3}Sn_{2}S_{2}^{17}, with the difference that in the latter case the crystal structure is centrosymmetric. This difference is crucial as \({{{\mathcal{I}}}}\) always connects Weyl nodes of opposite chirality and, therefore, enforces a vanishing total chirality of the Weyl nodes at the Fermi surface. While mirror symmetries can be reversibly broken by external magnetic fields, breaking \({{{\mathcal{I}}}}\) is the structural prerequisite for tuning an odd number of Weyl nodes to the Fermi surface.
The two chosen canting directions are related by a crystal mirror symmetry (Fig. 1d), and therefore, the chirality of the Weyl nodes reaching the Fermi surface at a certain θ is opposite for [−111] and [111]. Thus, the canting direction provides an experimentally viable way of discarding effects not originated in the chirality disbalance. It is worth noting that this chirality valve is effective not only at the specific angle at which a Weyl node crosses the Fermi surface. Indeed, due to the particularly large energy trajectory of one of the nodes below the Fermi energy, in a wide range of angles for θ ≳ 20°, NdRhC_{2} exhibits two Weyl fermions of negative (positive) and only one of positive (negative) chirality in a transportrelevant energy window of ±40 meV around the Fermi energy for θ toward [111] ([−111]).
Magnetisminduced bulk topological phase transitions
The dynamics of a single Weyl node going from above to below the Fermi level is a topological phase transition which can be described by the flux of the Berry curvature through the different Fermi surface sheets:
where α denotes different sheets of the Fermi surface associated with the band n. C_{nα} can be computed based on the momentum position of the Weyl nodes connecting the bands n and n + 1 and the bands n and n − 1^{64}. The latter are found to play no role in the energy and angle range analyzed here. In the following we will focus on only one band so we drop the n index. In NdRhC_{2}, two electronlike and two holelike bands conform the Fermi surface, shown in Fig. 3b. For θ < θ_{c1} ~ 20°, the four Weyl nodes above the Fermi energy at θ = 0 lie inside the outer hole pocket which has, therefore, C_{hole} = 0 (Fig. 3c). At θ = θ_{c1}, one Weyl node sits exactly on the Fermi surface. This is a typeII node connecting the outer hole pocket with an electron Fermi surface (Fig. 3d). Further increasing θ, this Weyl node goes below the Fermi level and leaves the hole pocket. In this situation, ∣C_{hole}∣ = 1. This holds for an angle range between θ_{c1} and θ_{c2} ~ 27°, since at θ_{c2} another Weyl node of opposite chirality to the previous one transitions out of the hole pocket. As a result, in the parameter space of canting angle θ and chemical potential, the system presents a series of topological phase transitions in which the topological invariant C_{hole} changes (Fig. 3e).
Magnetic anisotropy and Weyl node dynamics
A relevant question is how large is the magnetic field required to cant the magnetization in order to enable the Weyl node dynamics described above. Within the OC approximation employed here, the contribution of the 4f states to the magnetic anisotropy energy (MAE) is not considered. Our estimates for the MAE arising from the remaining valence electrons suggest that the first topological phase transition is accessible at ~5 T. An order of magnitude estimate for the 4f contributions can be obtained from the experimental anisotropy field of the wellknown permanent magnet compound Nd_{2}Fe_{14}B in comparison with the anisotropy field of Y_{2}Fe_{14}B. The former has an anisotropy field of about ~7 T at room temperature^{65}, the latter somewhat <3 T^{66}, implying similar magnitude of the Nd4f and the non4f contributions.
Figure 4 shows the energy of the Weyl nodes closest to the Fermi energy in GdCoC_{2} as the magnetization is canted. As in the Ndbased compound, the induced energy splitting between Weyl nodes can reach tens of meV. This is interesting since in this compound the 4fcontribution to the MAE vanishes to firstorder in the spinorbit coupling. These results are in line with previous work where it has been shown that the energy scale associated with the change in energy of the Weyl nodes due to canting of the magnetization in general does not scale with the MAE^{17}. Last, notice that the larger magnetic moment of Gd^{3+} contributes to lowering the required magnetic field to cant the magnetization.
Discussion
The predicted Weyl nodes dynamics has consequences on a number of physical observables. Some of them are related to the effects that canting of the magnetization has on the symmetry of the electronic structure, others follow from the topological phase transitions associated with having a single Weyl node going through the Fermi surface. We now briefly outline these open possibilities.
The bulk Fermi surface topological phase transitions lead to visible consequences on the surface electronic structure. To illustrate this, we consider the (100)surface of NdRhC_{2} terminated on a RhC_{2} plane. In addition to the corresponding projection of the bulk Fermi surface, the surface Fermi surface exhibits clear Fermi arcs connecting the projection of opposite chirality Weyl nodes, as shown in Fig. 5. For θ = 0, Fig. 5b shows the Fermi arcs fixing the Fermi energy at the Weyl node energy (ε_{F} = 0.043 meV), while Fig. 5c corresponds to the chargeneutrality chemical potential (ε_{F} = 0). As one of the Weyl nodes moves out from the hole pocket, the upper part of the Fermi arc, which at θ = 0 starts in the hole pocket (Fig. 5d), at θ = 25°, starts in the electron Fermi surface (Fig. 5d). We notice that similar results are obtained for an Ndterminated surface.
The GME accounts for the generation of an electrical current by a timedependent magnetic field and can be regarded as the lowfrequency limit of the natural optical activity in inversionbroken materials^{67,68}:
The tensor \({\alpha }_{ij}^{\,{{\mathrm{GME}}}\,}\) is determined by the angular momentum of the Bloch states at the Fermi surface, and the chiral symmetry is a sufficient condition for it to be nonzero^{67}. Thus, as explained earlier, canting of the magnetization along a lowsymmetry direction, which breaks all the mirror symmetries, together with the inversionbroken lattice structure enable the GME in the carbides family. When evaluated for simplified models for Weyl semimetals, the GME is found to scale linearly with the energy splitting between Weyl nodes^{67}. While the question of how such results are modified when having more realistic band structures deserves further investigation, the sizable and controllable energy splitting here predicted both for NdRhC_{2} and GdCoC_{2} makes them interesting platforms to explore the GME.
The chiral magnetic effect (CME) also accounts for the generation of an electrical current under applied magnetic field but is fundamentally different to the GME in that it is driven by topological properties of the Fermi surface^{69}. Specifically, the relevant topological invariants are the Chern numbers associated with the Fermi surfaces (Eq. (1)). In the presence of electric and magnetic fields having a common projection E ⋅ B, the C_{α} invariants break the conservation of the electron density n_{i} of individual Fermi surface sheets. Specifically, the continuity equation describing the conservation of charge reads:
As shown in ref. ^{69}, a natural consequence of nonzero C_{α} invariants is the CME. Specifically, turning on a E ⋅ B for a finite time causing a difference Δμ between the chemical potential of sheets having opposite Chern number C leads to a finite current j = CΔμB/4π^{2}. Remarkably, as indicated by Fig. 3e, canting of the magnetization in noncentrosymmetric magnets can provide a way to turn on/off the CME. In particular, when a typeII Weyl node crosses the Fermi energy, as predicted here for NdRhC_{2} and GdCoC_{2}, the switching on/off of the CME directly exposes the underlying Weyl node dynamics.
Last, we emphasize that although both the GME and the CME rely on the system having a chiral Shubnikov group, the requirements for the CME are stronger as also a finite Berry curvature flux trough the Fermi surfaces sheets is necessary. This distinction suggests that the dependence of the GME and of the CME on the canting angle may be qualitatively different, the latter including rather abrupt changes according to the phase diagram in Fig. 3e.
To summarize, realization of a chirality imbalance configuration in real materials is of paramount importance for exploiting the most essential property of Weyl fermions, their chirality. Our findings not only show that the family of carbides with broken \({{{\mathcal{I}}}}\) and broken Θ are natural candidates to that end, but also that canting of the magnetization toward a lowsymmetry direction in inversionbroken magnets can be generally considered as a knob to measure and control various chiral electromagnetic responses, particularly of gyrotropic and chiral magnetic type.
These effects were recently shown to harbor particular application potential: as our asymmetric Weyl semimetals from the RMC_{2} carbide family naturally have a chirality dependent Fermi velocity of Weyl cones, timedependent pumping of electrons from a nonchiral external source can be used to generate a nonvanishing chiral chemical potential. This again generates, via the chiral anomaly, a current along the direction of an applied magnetic field even in the absence of an external electric field so that the material acts as a rectifying element^{70}. Moreover, chirality imbalanced has been predicted to trigger an interesting photoresponse. The simultaneously broken particlehole and spatial inversion symmetry can generate giant photocurrents, suggesting a potential application of these asymmetric Weyl semimetals for creating tuneable THz photosensors^{71}.
Methods
Band structure calculations
DFT calculations were carried out using the Perdew–Burke–Ernzerhof implementation^{52} of the GGA using the fullpotential localorbital (FPLO) code^{53}, version 18.0057 (https://www.fplo.de). A kmesh with 12 × 12 × 12 subdivisions was used for numerical integration in the Brillouin zone along with a linear tetrahedron method. Spinorbit effects were included in the selfconsistent calculations via the 4spinor formalism implemented in the FPLO code.
The description of the 4f elements within the framework of DFT is still a subtle problem^{54}. In this work, we considered both of the widelyused approaches to treat the 4fshell: the OC approximation, and the GGA + U scheme. In the first approach, we set the spin for R^{3+} ions according to the Hund’s first rule while maintaining a spherically averaged distribution of electrons in the 4f shell^{54}. The trivalent configuration of the rareearth atoms is found in the related Nibased carbides RNiC_{2} (R = Pr, Nd, Gd)^{34,72} and GdCoC_{2}^{36}. In the second approach, to circumvent the usual problem of multiple metastable solutions, we used the occupation matrix control^{73,74}, whereby several different initial 4f density matrices, corresponding to +3 valence of R ions, were considered to explore the energy landscape. We used the fulllocalized limit of the double counting scheme with F^{0} = 7.0 eV, F^{2} = 11.92 eV, F^{4} = 7.96 eV, and F^{6} = 5.89 eV, leading to U = 7 eV and J = 1 eV. For most of the compounds, both GGA + OC and GGA + U favors the magnetic moments to be in the rareearth planes. Therefore, we consider m∥[001] for all the compounds for brevity and comparison. While we can anticipate that the existence of Weyl nodes at low energies is a finding robust to these explored choices, we will discuss to what extent details such as position in energy and momentum can be affected in a forthcoming publication.
Topological electronic properties
To study the topological properties, we constructed a tightbinding model based on maximally projected Wannier functions. In the basis set, states lying in the energy window −9.5–10 eV were considered. The Wannier basis set typically included the 5d and 6s states for R (respectively, 4d and 5s for Y), valence d and s states for M, and 2s and 2p states for C. The accuracy of the resulting tightbinding models was typically ≲15 meV compared to the selfconsistent band structures.
Magnetic configurations
To study the relative stability of different collinear longrange ordered states (applicable to materials belonging to classes II and III), different lowersymmetry AFM configurations were generated:

AF1, with AFM interlayer and ferromagnetic (FM) intralayer couplings.

AF2, with FM interlayer and AFM intralayer interactions.

AF3, with AFM interlayer as well as AFM intralayer interactions.
Figure 6 shows these AFM configurations for compounds belonging to classes II and III.
Data availability
The data supporting the present work are available from the corresponding author(s) upon request.
Code availability
The data presented in the manuscript were obtained using the FullPotential LocalOrbital (FPLO) code. For details, please see: https://www.fplo.de/. The raw data were subject to basic postprocessing (e.g. sorting) using bash/python scripts. All relevant data points were included.
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Acknowledgements
We thank Ulrike Nitzsche for technical assistance and Klaus Koepernik for discussion. We acknowledge financial support from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) via SFB1143 Project No. A5 and under Germany’s Excellence Strategy through WürzburgDresden Cluster of Excellence on Complexity and Topology in Quantum Matter—ct.qmat (EXC 2147, Project No. 390858490). R.R. and M.R. acknowledge partial financial support from the European Union (ERDF) and the Free State of Saxony via the ESF Projects No. 100231947 and No. 100339533 (Young Investigators Group Computer Simulations for Materials Design—CoSiMa) during the early stages of the project. J.I.F. acknowledges the support from the Alexander von Humboldt foundation.
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R.R. carried out the DFT calculations. B.S. computed and analyzed the topological properties under the guidance of J.I.F. M.R. and J.v.d.B. were responsible for project planning. All the authors contributed to the analysis of results and preparation of the manuscript.
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Ray, R., Sadhukhan, B., Richter, M. et al. Tunable chirality of noncentrosymmetric magnetic Weyl semimetals in rareearth carbides. npj Quantum Mater. 7, 19 (2022). https://doi.org/10.1038/s4153502200423z
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DOI: https://doi.org/10.1038/s4153502200423z