Creating Weyl nodes and controlling their energy by magnetization rotation

As they do not rely on the presence of any crystal symmetry, Weyl nodes are robust topological features of an electronic structure that can occur at any momentum and energy. Acting as sinks and sources of Berry curvature, Weyl nodes have been predicted to strongly affect the transverse electronic response, like in the anomalous Hall or Nernst effects. However, to observe large anomalous effects the Weyl nodes need to be close to or at the Fermi level, which implies the band structure must be tuned by an external parameter, e.g., chemical doping. Here we show that in a ferromagnetic metal tuning of the Weyl node energy and momentum can be achieved by rotation of the magnetization. First, taking as example the elementary magnet hcp-Co, we use electronic structure calculations based on density-functional theory to show that by canting the magnetization away from the easy axis, Weyl nodes can be driven exactly to the Fermi surface. Second, we show that the same phenomenology applies to the kagome ferromagnet Co 3 Sn 2 S 2 , in which we additionally show how the dynamics in energy and momentum of the Weyl nodes affects the calculated anomalous Hall and Nernst conductivities. Our results highlight how the intrinsic magnetic anisotropy can be used to engineer Weyl physics.

As they do not rely on the presence of any crystal symmetry, Weyl nodes are robust topological features of an electronic structure that can occur at any momentum and energy. Acting as sinks and sources of Berry curvature, Weyl nodes have been predicted to strongly affect the transverse electronic response, like in the anomalous Hall or Nernst effects. However, to observe large anomalous effects the Weyl nodes need to be close to or at the Fermi level, which implies the band structure must be tuned by an external parameter, e.g., chemical doping. Here we show that in a ferromagnetic metal tuning of the Weyl node energy and momentum can be achieved by rotation of the magnetization. First, taking as example the elementary magnet hcp-Co, we use electronic structure calculations based on density-functional theory to show that by canting the magnetization away from the easy axis, Weyl nodes can be driven exactly to the Fermi surface. Second, we show that the same phenomenology applies to the kagome ferromagnet Co 3 Sn 2 S 2 , in which we additionally show how the dynamics in energy and momentum of the Weyl nodes affects the calculated anomalous Hall and Nernst conductivities. Our results highlight how the intrinsic magnetic anisotropy can be used to engineer Weyl physics. DOI: 10.1103/PhysRevResearch. 1.032044 Materials hosting unconventional quasiparticles, such as Weyl semimetals, constitute a framework with potential for novel electronic devices. One ground for such expectation is the possibility of enhancing the response to external fields by engineering topological properties of the electronic states. For a material to specifically host Weyl fermions, the spin degeneracy of the electronic bands has to be removed by breaking either time-reversal symmetry ( ) or inversion. Karplus and Luttinger [1] first noticed that in a -broken system the spin-orbit coupling can introduce in the manifold of Bloch states a left-right asymmetry which in turn, in the presence of an electric field, causes a Hall current at zero magnetic field. This scattering-independent mechanism originates in the so-called anomalous velocity of the wave packets, which can be written in terms of the Berry curvature of the Bloch states in momentum space. Weyl nodes are monopoles of Berry curvature, which implies first that they can only be created and annihilated in pairs of opposite monopole charge and second that wave packets made out of Weyl fermions can have a large anomalous velocity. As this velocity is perpendicular to the This effect has been argued to be at work in different materials in which the anomalous velocity contribution intrinsic to the band structure is at the heart of enhanced electric and thermoelectric performance in the regimes of both linear [2][3][4][5] and nonlinear [6,7] response. Still, a central problem for optimizing Berry-curvature-based effects is the energy of the Weyl nodes, which currently is not a controlled variable from a material design point of view. Indeed, as the only symmetry restriction is to break inversion or , Weyl nodes can occur at any momentum and energy [8][9][10]. The interplay with the symmetry enables interesting effects of external magnetic fields as, e.g., using the field to split symmetry-protected fourfold crossing to create Weyl nodes [11][12][13][14][15][16]. In this work, we analyze a different interplay between magnetism and topology, one that does not rely on the existence of such kind of crossings: We explore the effects of changing the orientation of the magnetization (m) on the electronic structure topology of ferromagnets. Rotation of m can be achieved by changing thermodynamic variables such as external magnetic fields [17], temperature [18,19], hydrostatic pressure [20], and uniaxial strain [21].
Here, we show that the magnetization reorientation can be used as a strategy to tune Weyl nodes to the Fermi surface. We consider two examples, the elementary magnet hcp-Co and the half-metal Co 3 Sn 2 S 2 , a ferromagnet recently found to be a Weyl semimetal and to display a large anomalous Hall effect (AHE) [22][23][24][25]. Based on density functional theory (DFT), we show that canting m away from the easy axis leads to the creation of new Weyl nodes and to large displacements of the Weyl points in energy and in momentum space such that, at specific orientations, they can be placed exactly at the Fermi surface. Furthermore, we analyze how this tuning affects the anomalous Hall conductivity (AHC) and its thermal counterpart, the anomalous Nernst conductivity (ANC). Calculations as a function of the canting angle reveal that the ANC in particular is highly susceptible to the Weyl dynamics: It displays sharp peaks when asymmetric hole and electron pockets exist nearby the band crossing that reaches the Fermi energy.
Controlling Weyl nodes in an elementary magnet. We start our discussion with hcp-Co (space group 164), a ferromagnet with a magnetic anisotropy energy (MAE) of 65 μeV per Co in favor of the easy-axis configuration [26,27]. The magnetic field required to put the magnetic moments in plane is less than 2 Tesla [28].
We performed DFT calculations of the ferromagnetic phase with m in the xz plane at an angle θ fromẑ [29]. Figure 1(b) presents the band structure for the cases θ = 0, π/4. The rotation of m induces appreciable changes in the splitting between some of the bands, and even makes some band crossings be avoided and vice versa. We searched for Weyl nodes in a low-energy window [−40, 40] meV following their trajectories as a function of θ [30]. Remarkably, at specific angles, Weyl nodes are exactly at the Fermi surface It is interesting to note that while different exchange and correlation functionals yield significantly different MAE [the generalized gradient approximation (GGA) results in 8 μeV while GGA plus the orbital polarization correction gives 150 μeV], calculations based on both functionals result in similar Weyl-node energy trajectories [31]. This shows that the energy change of the Weyl nodes need not scale proportionally to the MAE and, therefore, that analogous tuning of the Weyl nodes to the Fermi surface can occur in magnets with significantly different MAE. It is naturally of interest to analyze if such tuning is also possible in materials with a low-energy band structure simpler than that of hcp-Co. Therefore, we focus now on the Weyl semimetal Co 3 Sn 2 S 2 .
The half-metal Co 3 Sn 2 S 2 . This system presents a ferromagnetic phase of Curie temperature T C = 172 K [32]. The crystal structure has space group R-3m (no. 166) with lattice parameters a = 5.37 Å and c = 13.18 Å and consists of quasitwo-dimensional (quasi-2D) Co 3 Sn layers separated by S 2 Sn layers with the particularity that the stack of cobalt atoms forms a kagome lattice; see Fig. 2(a). Recently, different groups [22,23] have reported in the magnetic ground state six Weyl nodes lying 60 meV above the Fermi energy. Similar to hcp-Co, the rotation of m induces appreciable changes in electronic structure [see Fig. 2(b)]. The ground state corresponds to θ = 0, with a magnetic moment of 0.33 μ B , which agrees well with magnetization measurements, 0.29 μ B [33] or 0.31 μ B [32]. The calculated MAE is ≈0.19 meV per Co atom. The field required to rotate the moment into the plane at zero temperature is estimated to be ≈26 Tesla [28,31].
Interestingly, the number of Weyl points in the energy range [−100, 150] meV increases from 6 to 26 as θ changes from 0 to π/2. We find two regimes in the evolution of the band structure as m goes from out of plane to in plane. The first occurs at small angles and consists in the splitting in energy of the original six Weyl points due to the breaking of the C 3 symmetry. Namely, the nonmagnetic point symmetry group is D 3d (b) [31] and a finite m breaks and affects some of the unitary symmetry operations. In total, for m ẑ, there are 12 point symmetry operations and the application of these to a Weyl node that lies in the k y = 0 plane leads to six degenerated Weyl nodes. Upon canting m at an angle θ towardsx, the only remaining symmetries-in addition to the identity and to inversion-are · C 2 (y) and · M(y). This forces the six Weyl nodes to split in energy as a doublet (formed by those with k y = 0) and a quartet. Figure 3(a) shows the calculated energy of the Weyl nodes as a function of θ . For θ < π/4, this splitting is smaller than 4 meV.
Further increasing θ , the second regime is signaled by large changes in the Weyl-node structure. In particular, the Weyl-node quartet goes from above to below the Fermi level and new Weyl nodes are created. Notice that in this angle range, the symmetry does not change. Therefore, the Weylnode creation is not due to the splitting of a Dirac point as in Ref. [11][12][13][14][15][16]. The vertical lines in Fig. 3(a) indicate the angles (labeled s, t, . . ., w) at which a Weyl node crosses the Fermi surface. The remarkable changes in energy go hand in hand with large displacements in momentum space. Figure 3(b) shows the projection of the trajectories on the semiplane k z = 0 with k y 0, illustrating the rich dynamics controlled by θ . As shown in Figs. 3(s)-3(w), the energy dispersion close to the nodes reaching the Fermi energy can differ significantly among the different cases [34].
Experimental consequences. We analyze next observables that can make evident the rich Weyl dynamics induced by rotating m. We focus on response functions that originate in the Berry curvature of the Bloch states through the anomalous velocity [35]. We compute the Hall response to an electric field E or to a temperature gradient ∇T at zero external magnetic field. In the linear response, the Hall current is j a = σ ab E b − α ab ∇ b T , where σ ab and α ab read Here, abc is the Levi-Civita tensor, E nk is the energy dispersion of the nth band, n,c is the c component of the Berry curvature, and n (k is the entropy density, and μ the chemical potential [37][38][39][40][41][42][43][44][45][46]. As T → 0 K, the entropy vanishes and so does α ab , while at finite T , s(E nk ) is maximum at μ and decreases exponentially away from the Fermi surface. Joint measurements of electric and thermoelectric coefficients can determine σ ab and α ab [47]. Figures 4(a) and 4(b) show the calculated AHCs as a function of θ [48]. For θ = 0, only σ xy is nonzero and its very large value 1200 ( cm) −1 is in good agreement with values reported of 1310 ( cm) −1 [23] and 1100 ( cm) −1 [22]. When m acquires a finite projection onx, the only symmetry involving a mirror plane is M(y) and both σ xy and σ yz are symmetry allowed. Namely, under mirror symmetry, behaves as a pseudovector so that M(y) y (k) = y (k x , −k y , k z ) while M(y) (k) = − (k x , −k y , k z ), where refers to the components of alongx andẑ (parallel to the mirror plane). Therefore, if M(y) is a symmetry, the Berry curvature satisfies y (k) = − y (−k x , k y , −k z ) and (k) = (−k x , k y , −k z ) and these constraints only make σ xz vanish. As θ increases, σ xy and σ yz follow opposite trends and at θ = π/2, σ yz ∼ 200 ( cm) −1 is much larger than σ xy . At this angle, the xy component does not vanish as it is still symmetry allowed. The smallness of its value is still related to the effects of the symmetries on the Berry curvature flux. In this compound, the large AHC arises mainly from the nodal lines that become gapped when spin-orbit coupling is included. When m points alongẑ, the combination of C 3 and the mirror planes M(x), M(y) forces the different patches of the nodal lines to contribute additively to the flux of z [31]. When m acquires a component alongx, the reduction in the symmetries removes this constraint and large cancelations occur due to the opposite flux of z at different points (k x , k y ). This geometrical effect controls the overall evolution of σ xy which is, therefore, quite smooth and monotonous. σ yz lacks this geometrical effect and, as a result, is more sensitive to detailed changes in the band structure and Weyl dynamics, as shown in Fig. 4(b). For instance, the red arrow in the plot indicates an angle at which the creation of Weyl nodes leads to a steplike increase of σ yz . As θ increases further, a set of Weyl nodes reaches the Fermi energy at θ = s and σ yz exhibits a small plateau. The energy dispersion of these Weyl fermions is shown in Fig. 3(s) and makes clear that this node is type I, and therefore the observed plateau is consistent with the general prediction for such Weyl nodes when their energy approaches the Fermi level [49].
Admittedly, the resulting changes in the AHC related to the Weyl nodes are not large. It turns out that the ANC shows much larger changes when Weyl nodes approach the Fermi energy. Figures 4(c) and 4(d) show the ANCs as a function of θ for different values of T . Since thermal fluctuations of the Co magnetic moments are not considered in Eq. (1), we restrict our analysis to a range of T where the magnetization is essentially saturated (T < 100 K) [32]. Remarkably, the dependence on θ is nonmonotonous and includes sharp peaks at specific angles. The peaks are centered at some of the angles in which a Weyl node crosses the Fermi energy (specifically, θ = t and v) and become broader as T increases. The nonmonotonicity remains, however, clearly visible for all temperatures. The magnitude of the enhancements is worth noting: At 91 K, starting from ≈0.1 A(mK) −1 at π/4, |α yz | reaches a maximum 1.8 A(mK) −1 . This is five times the maximum obtained in Mn 3 Sn [47,50] and half of the giant ANC in Co 2 MnGa [51].
Entropy flow and Weyl fermions. To understand why and when a peak in the ANC occurs, it proves useful to analyze how different carriers contribute to the ANC. Since in a fermionic system the available entropy is restricted to states of energy ≈k B T around μ, we are interested in resolving the contribution of different electronic states within this thermal energy window. We recast the ANC as (see Ref. [31]). This equation has the overall shape of the Mott relation-involving the derivative with respect to energy of the AHC and the entropy-but holds for larger temperature and motivates us to define the entropy-flow density per energy, s f ,ab (ε) = ∂σ ab ∂ε s(ε). The entropy flow s f ,ab (ε) measures the contribution to α ab of carriers of different energy and, therefore, its calculation allows us to compare the contribution of different sets of Weyl points.
We focus on α yz and first analyze θ = s. Figure 4(e) presents s f ,yz (ε) for θ = s and for two other angles close to it, s ± δ with δ ∼ 4 • . For each angle, the entropy flow presents a peak as a function of energy which highlights the energy of the carriers that contribute the most to the ANC. Notice that this peak does not need to be centered at the energy of maximum entropy (ε = μ). For instance, at θ = s − δ the largest contribution is found ≈10 meV above μ. As θ increases, this peak achieves a larger value and its position shifts to smaller energy in a rather monotonous way. In particular, at θ = s the contribution of carriers at the Fermi surface is markedly smaller than that of the states conforming the entropy-flow peak. This explicitly shows that the set of Weyl nodes crossing the Fermi energy at this angle does not contribute significantly to the ANC.
The situation is different at angles θ = t and v. Figure 4(f) shows s f ,yz (ε) for θ = v and θ = v ± δ and makes clear that the entropy-flow peak is at the chemical potential when the Weyl nodes reach the Fermi surface (θ = v). This indicates that in this case the Weyl points indeed dominate the ANC.
The different contribution of different Weyl nodes to the ANC can be traced to the band structure close to the Weyl nodes [see Figs. 3(s), 3(t) and 3(v)]. While at θ = s the Weyl point at the Fermi surface is type I with only a small tilt, at angles θ = t and v the dispersion of one of the bands involved in the crossing at the Fermi level is such that it produces asymmetric electron and hole pockets close to the node. To illustrate this, Fig. 4(g) shows that the density of states (DOS) in a sphere of radius π/(5a) centered at the corresponding Weyl node at the Fermi surface is larger and more electronhole asymmetric for θ = v than for s. In an analagous way, for θ = v, the more asymmetric distribution of states close to the nodes at the Fermi surface contributes to a larger ∂σ yz /∂ε and hence a larger entropy flow. We thus associate the enhanced ANC at some of the angles at which a Weyl node reaches the Fermi surface with having asymmetric hole and electron pockets close to the Weyl point at the Fermi energy.
In summary, we have shown that in ferromagnets a rich dynamics of Weyl points results from a change in the direction of the magnetization: Weyl nodes can be created, annihilated, and shifted in energy-momentum space over large distances. This dynamics can be used to place Weyl nodes exactly at the Fermi surface, which can lead to sharp enhancements in the anomalous Nernst conductivity that survive up to relatively high temperatures. Whereas we established the effect of rotation of the magnetization on Weyl nodes in the elementary magnet hcp-Co and in the semimetal Co 3 Sn 2 S 2 , the finding that the energy change of the Weyl nodes does not need to scale proportionally to the magnetic anisotropy energy suggests that similar tuning of Weyl nodes to the Fermi surface can be expected in a wide set of ferromagnets. This provides a pathway to experimentally probe, manipulate, and control Weyl-fermion transport properties in magnetic Weyl semimetals.