## Mind the angle

Interplay between real- and momentum-space properties of materials can lead to exotic phenomena. Suzuki *et al.* studied electrical transport in the presence of a magnetic field in cerium-aluminum-germanium, a Weyl semimetal that also harbors magnetism (see the Perspective by Hassinger and Meng). As they varied the orientation of the applied field, they noticed spikes of resistivity sharply centered around the high symmetry axes of the material. The spikes were a consequence of the small overlap of Fermi surfaces—which “live” in momentum space—on either side of magnetic domain walls, which occur in real space. This extreme angular sensitivity may be useful in practical applications.

## Abstract

Transport coefficients of correlated electron systems are often useful for mapping hidden phases with distinct symmetries. Here we report a transport signature of spontaneous symmetry breaking in the magnetic Weyl semimetal cerium-aluminum-germanium (CeAlGe) system in the form of singular angular magnetoresistance (SAMR). This angular response exceeding 1000% per radian is confined along the high-symmetry axes with a full width at half maximum reaching less than 1° and is tunable via isoelectronic partial substitution of silicon for germanium. The SAMR phenomena is explained theoretically as a consequence of controllable high-resistance domain walls, arising from the breaking of magnetic point group symmetry strongly coupled to a nearly nodal electronic structure. This study indicates ingredients for engineering magnetic materials with high angular sensitivity by lattice and site symmetries.

The past decade has revealed an unexpected depth of phenomena in topological aspects of electronic solids and in strongly spin-orbit–coupled materials (*1*–*9*). A key recent advance is the discovery of Weyl and Dirac fermions, which occur at topologically nontrivial points, known as “nodes,” at which bulk bands cross (*10*–*14*). The band crossings themselves, and the surface Fermi arcs they engender, have been observed through angle-resolved photoemission (*15*, *16*). Transport studies detect pronounced longitudinal negative magnetoresistance, which may be attributed to the chiral anomaly of Weyl fermions (*17*, *18*).

It is known that Weyl fermions require breaking of either inversion or time-reversal symmetry. The former option is realized structurally in TaAs (*19*) and related materials. Magnetic Weyl semimetals, which evince the second possibility, have been harder to identify but have recently been discussed in systems such as HgCr_{2}Se_{4} (*20*), *R*AlGe (where *R* is a rare-earth element) (*21*), SrMnSb_{2} (*22*), and Mn_{3}*Z* (where *Z* is Ga, Sn, or Ge) (*23*, *24*). Magnetic Weyl semimetals could host controllable topological defects in the form of domain walls (DWs) that can be moved, created, and annihilated in situ in a single sample. It is intriguing to contemplate the interactions between these real-space topological defects and the Weyl points, which are momentum-space topological defects (*11*). More generally, substantial coupling of magnetic configurations to semimetallic electronic structure may afford technologically useful behavior for sensors and spintronic devices (*25*).

Here we identify a sharp signature of this interaction, singular angular magnetoresistance (SAMR) in CeAlGe_{1−}* _{x}*Si

*(crystal structure shown in Fig. 1A). SAMR is a pronounced enhancement of resistance occurring over a narrow angular range [down to full width at half maximum (FWHM) < 1°] leading to a derivative of resistance with respect to angle exceeding 1000% per radian, under particular conditions of applied magnetic fields (Fig. 1B). This is in clear contrast to conventional angular magnetoresistance (AMR), which describes a smooth dependence of resistivity ρ on the orientation of magnetization*

_{x}**M**and electrical current

**I**in magnetic conductors (

*26*). As depicted in Fig. 1C, we argue that this arises from momentum-space mismatch across real-space magnetic domains.

CeAlGe (*x* = 0) crystallizes in a nonsymmorphic tetragonal structure with space group *I*4_{1}*md* (Fig. 1A) (*27*, *28*). The magnetic Ce ions (located at the Wyckoff 4*a* positions) comprise two interpenetrating body-centered–tetragonal sublattices (A and B) offset by 0, *a*/2, and *c*/4 along the [100], [010], and [001] direction, respectively. The lattice is tetragonal, but the Ce site has a lower orthorhombic symmetry 2*mm*. This orthorhombicity makes the local in-plane axes of each Ce ion inequivalent, resulting in an anisotropic *g*-tensor *x* and *y* parallel to the <100> tetragonal axes). Furthermore, the structure contains 4_{1} (and 4_{3}) screw axis operations involving 90° rotations of the in-plane axes that interchange the Ce sublattices (*29*), thus making the *g*-tensor components *g*_{100} along the global [100] direction sublattice-dependent [*S ^{i}* (defined to have unit magnitude) and magnetic moments

_{B}is the Bohr magneton and

*i*,

*j*= {

*x*,

*y*,

*z*}). As we describe below, this allows for an unusual ferrimagnetic magnetic moment

**m**ordering in a conventional antiferromagnetic spin

**S**network, with acute sensitivity to the magnetic point group. Although a range of magnetic ground states has been previously reported for CeAlGe attributed to the sensitivity of these materials to growth conditions (

*27*,

*30*–

*32*), we find that this magnetic state is stabilized using chemical pressure with isoelectronic substitution of Ge by Si (CeAlGe

_{1−}

*Si*

_{x}*) (*

_{x}*29*).

The most notable consequence of the above structure is the AMR observed in the longitudinal resistivity ρ* _{xx}*(

*H*, θ) below the magnetic ordering temperature

*T*

_{N}= 5.6 K. This is shown for CeAlGe

_{0.72}Si

_{0.28}at

*T*= 2 K in Fig. 1B, where the current

**I**is applied along [010] and μ

_{0}

*H*= 2.2 T rotated in the tetragonal plane (θ is measured from [100] as shown in Fig. 1A; where μ

_{0}is the vacuum permeability and

*H*is the magnetic field). A sharp enhancement is observed when θ is near the crystallographic high-symmetry directions <100>. This can be contrasted with the behavior at

*T*= 6.5 K, which shows a smoothly varying profile captured by conventional AMR and has the symmetry of the underlying nonmagnetic group symmetry (

*29*). This comparison is also shown in a polar plot in the inset of Fig. 1B where the sharpness of the features along all four tetragonal directions is apparent. Neutron scattering experiments show no sign of a structural transition at

*T*

_{N}(

*29*), indicating instead that the SAMR is purely of magnetic origin.

A more detailed profile of ρ* _{xx}*(θ,

*H*) is shown in Fig. 2A. With

**H**aligned precisely along the [100] direction (θ = 0°), a prominent peak is observed in ρ

*(*

_{xx}*H*), reaching a maximum near μ

_{0}

*H*= 2 T. At higher

*H*, this peak is suppressed and conventional magnetoresistance is seen. More notably, the peak signal rapidly disappears with increasing θ, whereas the remainder of ρ

*(*

_{xx}*H*) remains unchanged: for θ = 0.7° the peak is largely suppressed and appears to completely vanish by θ = 1.5°. The field evolution of the magnetization

*M*(

*H*) at the same

*T*= 2 K is shown in Fig. 2B (θ = 0°). At a low field μ

_{0}

*H*

_{0}≡ 0.02 T, a sharp rise in

*M*(

*H*) is observed (inset) followed by a gentle increase toward a saturation moment of 0.97 μ

_{B}/Ce. From the associated real part of the ac magnetic susceptibility χ′ and its field derivative

*d*χ′/

*dH*, two further field scales μ

_{0}

*H*

_{1}≡ 1.86 T and μ

_{0}

*H*

_{2}≡ 3.28 T can be identified, which bracket the ρ

*(*

_{xx}*H*) peak associated with SAMR (marked by arrows in Fig. 2B). An alternate view of the SAMR regime with changing θ is shown in Fig. 2C, where outside of

*26*,

*29*). Focusing on the SAMR region, the sharpest response is seen as a peak in ρ

*(θ) for μ*

_{xx}_{0}

*H*= 2.2 T with FWHM Δθ = 0.7°. The largest

_{0}is the zero-field resistivity) is two orders of magnitude larger than in permalloy (a conventional AMR material) (

*26*), whereas the small Δθ is reminiscent of the response in Sr

_{3}Ru

_{2}O

_{7}(

*33*,

*34*) but is more than an order of magnitude narrower here.

We now argue that the observed angular magnetoresistance phenomena signify magnetic quantum phase transitions and broken magnetic point group symmetry in the intermediate *H*_{1} < *H* < *H*_{2} state. We consider the energy function that governs ordering of the Ce 4*f* spins, which are represented by fixed-length (set to 1 here) vectors **S**_{α} = (*S ^{x}*

_{α},

*S*

^{y}_{α},

*S*

^{z}_{α}) that should be interpreted as the expectation value of the pseudospin describing the Kramers doublet ground state on the Ce sublattice α = A, B. Assuming ferromagnetic intrasublattice interactions, all spins within a single sublattice polarize identically, and the minimal model for the energy per unit cell is then

*g*-tensors

*29*). Within this model, for

*g*-tensor, the system has a nonzero net (ferrimagnetic) moment, consistent with the observed powder neutron diffraction (

*29*). An infinitesimal field selects a direction for the spins via Zeeman coupling; the ordering pattern of

**m**in small

*m'*

_{010}[i.e., the combination of the (010) plane mirror and time reversal], and the magnetic state is invariant under this transformation as well. Figure 3B summarizes the configuration of

**m**on sublattice A and B with increasing

*H*. For fields along <100>, the spins remain strictly antialigned, along

**H**, up to a critical field (which we identify with

*H*

_{1}) where they acquire a nonzero in-plane component perpendicular to the field. With further increasing field, the magnetization increases until both spins are aligned along the field (at a field strength identified as

*H*

_{2}), where the aforementioned symmetry is recovered and a single domain exists. The intermediate “canted” phase (between

*H*

_{1}and

*H*

_{2}) spontaneously breaks

*m*′

_{010}, and contains two types of magnetic domains: for

*S*

^{y}_{A}> 0 or

*S*

^{y}_{A}< 0.

We explain the resistivity anomaly and SAMR as follows. Without special preparation of the system, and for *m'*_{010} symmetry is not explicitly broken (it is microscopically present) and hence can be broken spontaneously only for *m'*_{010} symmetry explicitly and selects a single preferred domain. The consequent removal of DW resistance generates the observed SAMR signal. As shown in Fig. 2C, in a field μ_{0}*H* = 2.2 T this occurs at a tilt angle θ_{t} ≈ 0.6°, corresponding to a DW motion field scale *H*sinθ_{t} ≈ 230 Oe, comparable to that observed in conventional magnets (*25*).

The above theoretical interpretation requires a high DW resistance for a large SAMR effect. We carried out ab initio density functional theory calculations using the full noncolinear magnetic structure and found a semimetallic band structure with Fermi surfaces consisting of small pockets proximate to Weyl nodes (*29*). The electronic structure is similar to that observed in LaAlGe (*35*), supporting the presence of Weyl points throughout the *R*AlGe family. We argue that a high DW resistance is germane to such strongly spin-orbit–coupled magnetic semimetals, because the strongly anisotropic interactions and overall antiferromagnetic nature of the ordering favor narrow DWs that strongly scatter the low-density electrons (*36*). Phase space alone places a lower bound on the DW resistance due to Fermi surface mismatch (Fig. 1C and figs. S13B and S22). Following a Landauer-Büttiker approach, electrons transmit across the wall and contribute to conduction only if they can preserve their transverse momentum at the Fermi energy; this requires overlap of the Fermi surfaces on either side. The overlap is minimized for small Fermi surfaces that are strongly displaced by magnetic order—exactly the conditions pertaining to magnetic nodal semimetals. We estimate the DW contribution to the macroscopic resistivity enhancement [see sections 13 and 14D of (*29*) for a general discussion and a band calculation, respectively] as*L*_{w} is the typical domain size, and *29*). It is particularly desirable to apply real-space (*37*) and momentum-space (*36*) techniques along with dynamical analysis to probe the structure of the magnetic domains or, alternatively, to examine crystals with dimensions comparable to the domain size to directly probe *L*_{w}.

We next examine the phase space for SAMR. SAMR onsets below *T*_{N}, which itself is marked by a kink in ρ* _{xx}*(

*T*) and a peak of

*dM*/

*dT*(

*29*). For

*T*<

*T*

_{N}features in

*M*(

*H*) and

*d*χ′/

*dH*associated with

*H*

_{0},

*H*

_{1}, and

*H*

_{2}develop as summarized in the

*H*-

*T*phase diagram in Fig. 3C with the SAMR response δρ

*= ρ*

_{xx}*(*

_{xx}*T*,

*H*, θ = 0°) − ρ

*(*

_{xx}*T*,

*H*, θ = 1.5°) shown as the color scale (

*29*). Compared with the relatively subtle features in the thermodynamic measurements, the transport response sharply identifies the region

*H*

_{1}<

*H*<

*H*

_{2}in which

*m*′

_{010}is spontaneously broken. We have also observed an enhancement in the transverse resistivity ρ

*, which similarly highlights the SAMR phase (*

_{yx}*29*).

We note several features of this symmetry-breaking SAMR region. First, we observe a tendency toward separation between the SAMR state and the zero-field critical point in Fig. 3C. Theoretical analyses using both symmetry-based Landau theory and a microscopic lattice model confirm that the canted phase must be disconnected from *T*_{N} (*29*). Furthermore, a substantial broadening is observed for SAMR for rotations out of the (001) plane (Fig. 4A). We calculate the corresponding phase diagram for **H** (Fig. 4B); the canted phase persists when the field is rotated within the (100) or (010) planes. Finally, we determine the zero-field ground state using the full model (*29*), including all weak anisotropies beyond those in Eq. 1, and find a specific noncolinear state with in-plane magnetizations, which collapses rapidly to the colinear one in a small field (which we associate with *H*_{0}).

Finally, we suggest key ingredients for realizing SAMR materials. Magnetically, the phenomenon relies upon strong anisotropy so that crystalline symmetries are fully communicated to the spins, as well as a relatively low exchange scale so that a state change can be affected by manageable fields. The response of the magnetism of CeAlGe to chemical substitution is useful in realizing such a ground state, which may serve as guide for further material realizations (*29*). Electronically, the Fermi surface mismatch leading to high-resistance DWs requires a semimetal with a small Fermi surface in close proximity to a node, and narrow bandwidth and strong coupling to the spins are beneficial. Note that specifically Weyl points are not essential, and Dirac or other nodal structures should suffice. The SAMR effect itself requires a magnetic point group operation (here *m*′_{010}), which remains a symmetry in a magnetic field along some specific axes, but not for other orientations, and that this symmetry is spontaneously broken in some parameter regime. Less abstractly, a deviation of the field from the prescribed axis must couple linearly to the order parameter of the canted phase. This condition requires that the canted phase must not break translational symmetry. Therefore, structures with multiple magnetic sites in the crystalline primitive unit cell are desired in the case of spin antiferromagnets. From Eq. 2, we see that the strength of the response Δρ/ρ is enhanced in cleaner systems with larger ℓ but smaller *L*_{w}. Optimized growth conditions may reduce the defect density to control the former (*29*), whereas device fabrication may allow for control of the energy and microstructure of magnetic domains responsible for determining the latter. As in transition metal–based magnets, the applied **H** is expected to affect the DW orientation and, in turn, modulate Δρ/ρ (*38*)—this is consistent with the observed breaking of <100> degeneracy for SAMR in Fig. 1B. More generally, the angular sharpness of Δθ is limited by the pinning of the magnetic DWs. Reducing the pinning potential would allow SAMR to be narrowed toward the δ function response expected from theory. Engineering the SAMR behavior in materials with reduced electronic and magnetic defects for improved sensitivity or with magnetic 3d transition elements for higher *T* operation may provide pathways to devices capable of vector magnetic sensing with singular angular sensitivity (*39*).

## Supplementary Materials

science.sciencemag.org/content/365/6451/377/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S22

Tables S1 to S3

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We acknowledge discussions with T. Kurumaji and J. P. Paglione and technical support from D. Graf, Y. Zhao, and Z. Xu.

**Funding:**This work was supported by in part by the Gordon and Betty Moore Foundation EPiQS Initiative, grant GBMF3848 (J.G.C.), and a postdoctoral fellowship grant GBMF4303 (L.S.). L.B. was supported by the NSF under grant DMR-1818533. J.-P.L. was supported by ARO grant W911-NF-14-1-0379. L.S. is also grateful for the hospitality and funding of the KITP, where part of this work was carried out under NSF grant PHY-1748958. We acknowledge the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the BT-7 neutron facility used in this work.

**Author contributions:**T.S. synthesized the materials and performed the electrical and thermodynamic measurements. L.S. and J.-P.L. performed theoretical calculations. T.S. and J.W.L. performed neutron scattering experiments. All authors contributed to writing the manuscript. L.B. and J.G.C. supervised the project.

**Competing interests:**The authors declare no competing interests.

**Data and materials availability:**The data in the manuscript are available from the Harvard Dataverse (

*40*).