CeAlGe: Magnetic Weyl Semimetal

• CeAlGe is a noncentrosymmetric magnetic semimetal characterized by a polar tetragonal (I4₁md) structure that enables Weyl physics through inversion symmetry breaking.
• It exhibits low-carrier density and antiferromagnetic order near 5 K, with its magnetic properties and topological transport signatures highly sensitive to stoichiometry and external magnetic fields.
• Electronic transport and NMR studies reveal field-induced topological phases and emergent Kondo coherence, positioning CeAlGe as a pivotal platform for exploring magnetic Weyl phenomena.

CeAlGe is a cerium aluminum germanide in the noncentrosymmetric tetragonal $R$AlGe/$Ln$Al$Pn$ family that has been studied as a candidate magnetic Weyl semimetal, including specifically as a proposed type-II Weyl semimetal. Its importance arises from the simultaneous presence of inversion-symmetry breaking in the crystal lattice and low-temperature magnetic order, a combination that provides the symmetry conditions required for Weyl physics. Across the literature, CeAlGe is consistently identified as a low-carrier-density semimetal with magnetic ordering near $5$ K, but its detailed magnetic structure, field response, and topological transport signatures have proven highly sensitive to crystal quality, stoichiometry, pressure, and magnetic-field orientation (Hodovanets et al., 2018, Puphal et al., 2020, Wang et al., 2024).

1. Crystal structure and materials control

CeAlGe crystallizes in the polar tetragonal LaPtSi-type structure with space group $I4_1md$ (No. 109), rather than the centrosymmetric $I4_1/amd$ alternative. This structural assignment is central to its interpretation as a Weyl-semimetal candidate, because the proposed topological state requires inversion symmetry breaking. Single-crystal x-ray diffraction established the $I4_1md$ structure for high-quality single crystals, and the Flack parameter being near zero supported the absolute structure. The ordered $I4_1md$ lattice contains distinct Al and Ge sublattices, whereas the centrosymmetric alternative corresponds to mixed 50:50 Al/Ge occupancy on symmetry-equivalent sites. Several studies also emphasize that powder diffraction alone cannot reliably distinguish $I4_1md$ from $I4_1/amd$ (Hodovanets et al., 2018, Puphal et al., 2019, Singh et al., 2020).

Reported lattice parameters are mutually consistent across studies. One review gives $Ln$0 Å and $Ln$1 Å, with atomic positions Ce at $Ln$2, Al at $Ln$3, and Ge at $Ln$4. Powder refinement on polycrystalline material yielded $Ln$5 Å and $Ln$6 Å. Floating-zone-grown single crystals gave $Ln$7 Å and $Ln$8 Å, whereas flux-grown material showed larger parameters, $Ln$9 Å and $Pn$0 Å, together with Al enrichment (Singh et al., 2021, Puphal et al., 2019, Singh et al., 2020).

The synthesis route is therefore not ancillary. Al self-flux growth produces single crystals suitable for diffraction and bulk-property measurements, but flux-grown crystals tend to be Al-rich. For CeAlGe, flux-grown compositions were reported as $Pn$1 on a cleavage plane and $Pn$2 on the surface. By contrast, floating-zone growth was described as crucial for obtaining nearly stoichiometric crystals, with EDS giving $Pn$3. This materials sensitivity is directly tied to later disagreements over magnetic anisotropy, Hall response, and the visibility of topological transport features (Puphal et al., 2019, Piva et al., 2023).

2. Magnetic ordering and anisotropy

Bulk measurements establish that CeAlGe orders antiferromagnetically below approximately $Pn$4 K. Single-crystal magnetization, specific heat, and transport measurements identified antiferromagnetic order below $Pn$5 K, with inverse susceptibility showing a clear anomaly at the Néel temperature and Curie–Weiss fits above $Pn$6 K giving $Pn$7, consistent with Ce$Pn$8, together with a Weiss temperature of $Pn$9 K. A broader review gives $5$0 K and $5$1, again close to the Ce$5$2 free-ion value. Polycrystalline susceptibility yielded $5$3 and $5$4 K. In each case, the negative Weiss scale was taken as evidence for dominant antiferromagnetic interactions (Hodovanets et al., 2018, Singh et al., 2021, Singh et al., 2020).

The anisotropy is strong and somewhat sample- and probe-dependent in the published record. Early single-crystal work found the easy magnetic axis along $5$5, implying that the ordered moment lies in the tetragonal $5$6-plane below about $5$7 K, and reported that near the ordering temperature $5$8 for $5$9 is about ten times larger than for $I4_1md$0. Floating-zone-grown crystals were likewise described as easy-$I4_1md$1-plane antiferromagnets, with susceptibility maxima at $I4_1md$2 K for $I4_1md$3 and $I4_1md$4 K for $I4_1md$5. By contrast, angular magnetization work reported a four-fold symmetry of the in-plane $I4_1md$6 data with the $I4_1md$7 set of easy directions. That same work linked the discrepancy to a magnetic phase transition between two in-plane magnetic structures and found that the boundary could be tuned by Al deficiency (Hodovanets et al., 2018, Puphal et al., 2019, Hodovanets et al., 2021).

The thermodynamic signatures are consistent with localized Ce $I4_1md$8 moments and anisotropic low-energy spin excitations. After subtraction of the LaAlGe reference, the magnetic contribution to the heat capacity was fitted at low temperature by

$I4_1md$9

yielding a magnon excitation gap of $I4_1/amd$0 K and a magnetic entropy of about $I4_1/amd$1, consistent with a doublet crystal-field ground state for Ce$I4_1/amd$2. A separate review reported a Sommerfeld coefficient $I4_1/amd$3, much smaller than in the more strongly correlated comparison compounds CeNiGe$I4_1/amd$4 and CeGe, and interpreted CeAlGe as a system in which low-temperature physics is governed primarily by RKKY exchange rather than strong Kondo compensation (Hodovanets et al., 2018, Singh et al., 2021).

3. Field-induced phases and microscopic magnetic textures

Magnetic field suppresses or reconstructs the ordered state at comparatively small fields, but the relevant scales depend strongly on field orientation. In the single-crystal study, fields of $I4_1/amd$5 kOe for $I4_1/amd$6 and $I4_1/amd$7–$I4_1/amd$8 kOe for $I4_1/amd$9 were sufficient to suppress signatures of magnetic order. At $I4_1md$0 K and $I4_1md$1, a sharp first-order spin-flop transition with hysteresis was observed below $I4_1md$2 kOe, producing a partially saturated state with moment less than $I4_1md$3. At $I4_1md$4 kOe, the saturation moment remained only about $I4_1md$5 for $I4_1md$6 and a little over $I4_1md$7 for $I4_1md$8, both well below the free-ion Ce$I4_1md$9 value of $I4_1md$0. Floating-zone-grown crystals showed related metamagnetic features near $I4_1md$1 T for $I4_1md$2 and $I4_1md$3 T for $I4_1md$4 (Hodovanets et al., 2018, Puphal et al., 2019).

Neutron scattering later showed that the ordered state is not a simple commensurate antiferromagnet. Powder diffraction and SANS identified an incommensurate propagation vector $I4_1md$5 with $I4_1md$6 r.l.u. in neutron powder diffraction and $I4_1md$7 r.l.u. in SANS, together with the full four-arm propagation star $I4_1md$8 and $I4_1md$9. The preferred zero-field description was a square-coordinated multi-$I4_1md$0 structure in the superspace group $I4_1md$1. In this description, the normalized local moment field contains quadrants with half-integer topological charge density $I4_1md$2, even though the macroscopic $I4_1md$3 averages to zero at zero field. For $I4_1md$4, second-order scattering at $I4_1md$5 appears under field, indicating that the incommensurate multi-$I4_1md$6 state becomes anharmonic rather than simply disappearing (Puphal et al., 2020).

Subsequent neutron diffraction refined this picture using low- and high-$I4_1md$7 magnetic peaks measured in a common normalization framework. A localized-moment model with four Fourier amplitudes $I4_1md$8 reproduced the observed intensities, including the $I4_1md$9 zeroth-order magnetic satellites that are especially sensitive to $I4_1/amd$0. The combined refinement gave $I4_1/amd$1, $I4_1/amd$2, $I4_1/amd$3, and $I4_1/amd$4, and the authors found no unambiguous evidence that an itinerant-electron magnetic contribution was required by the diffraction data. The refined structures support localized winding centers with approximately $I4_1/amd$5, and under an added uniform $I4_1/amd$6-axis component the average topological charge becomes nonzero over an intermediate range, providing a microscopic route to the field-induced topological phase (Pomjakushin et al., 15 Sep 2025).

A related but distinct formulation emerged from the CeAlSi$I4_1/amd$7Ge$I4_1/amd$8 series. In that work, CeAlGe ($I4_1/amd$9) was described as hosting a commensurate ferromagnetic component $Ln$00 together with incommensurate Bragg peaks at $Ln$01 and $Ln$02, with $Ln$03, corresponding to a modulation wavelength of about $Ln$04 nm. The zero-field state was identified as a $Ln$05-$Ln$06 state, evolving under in-plane field into a $Ln$07-$Ln$08 state before the incommensurate component vanished. This literature suggests that CeAlGe supports multiple closely competing magnetic descriptions, all of which emphasize noncollinearity, multi-$Ln$09 order, and strong field tunability (Yao et al., 22 Sep 2025).

4. Electronic transport, semimetallicity, and Hall effects

Transport measurements consistently classify CeAlGe as a semimetal, but with substantial sample dependence in quantitative Hall parameters. In the single-crystal investigation, the residual-resistivity ratio was about $Ln$10, the Hall coefficient was positive and nearly temperature independent up to $Ln$11 K, and a one-band analysis gave a hole density of $Ln$12, two orders of magnitude below copper and comparable to classic semimetals. Magnetoresistance was large and negative over the measured field range, in contrast to LaAlGe, and anomalies in Hall resistivity occurred near the fields where magnetic order was suppressed (Hodovanets et al., 2018).

Later studies showed that the transport response is extraordinarily sensitive to minute stoichiometric changes. Two flux-grown crystals with nearly identical WDS compositions, $Ln$13 and $Ln$14, and similar $Ln$15 values of $Ln$16 and $Ln$17, nevertheless displayed qualitatively different Hall signals. At ambient pressure, one sample showed a clear topological-Hall-like peak near $Ln$18 T at $Ln$19 K, whereas the other lacked a comparable peak in that field range and instead showed only a much smaller feature around $Ln$20 T. Using the high-field Hall slope, that study extracted hole densities of $Ln$21 and $Ln$22 at room temperature, increasing slightly at $Ln$23 K to $Ln$24 and $Ln$25. The authors attributed these differences to Fermi-level shifts induced by tiny stoichiometric or defect variations and to the role of domain walls (Piva et al., 2023).

The Hall response is commonly decomposed as

$Ln$26

Within this framework, CeAlGe exhibits a topological Hall effect for $Ln$27 associated with field-induced magnetic textures. Neutron and transport work identified an intermediate-field topological magnetic phase between low-field and high-field regimes in the $Ln$28 phase diagram, with a finite $Ln$29 correlated with features in $Ln$30. A later pressure study reported a loop-shaped topological Hall effect at $Ln$31 K and $Ln$32, with hysteresis between $Ln$33 and $Ln$34 T, $Ln$35, and $Ln$36, while the Hall curve remained nearly linear to $Ln$37 T and magnetization tended to saturate above $Ln$38 T. That work argued that the signal originates more naturally from chiral domain walls than from a conventional anomalous Hall effect (Puphal et al., 2020, He et al., 2022).

Hydrostatic pressure strengthens the antiferromagnetic ground state but can either induce or split topological Hall features. In one study, $Ln$39 increased approximately linearly with slopes $Ln$40 K/GPa and $Ln$41 K/GPa in two different samples; a single ambient-pressure Hall peak could split already at $Ln$42 GPa into two peaks near $Ln$43 T and $Ln$44 T, while a sample lacking a clear ambient-pressure topological Hall peak developed one around $Ln$45 T at $Ln$46 GPa. Another pressure study found $Ln$47 K/GPa and a pressure-induced splitting of the loop-shaped topological Hall effect into field regions of $Ln$48–$Ln$49 T and $Ln$50–$Ln$51 T at $Ln$52 GPa. Both analyses emphasized magnetoelastic control of domain-wall chirality and the coexistence of antiferromagnetic and field-polarized regions as the operative mechanism (Piva et al., 2023, He et al., 2022).

5. Correlations, Kondo coherence, and local spin dynamics

CeAlGe occupies an intermediate position between simple local-moment antiferromagnetism and a more strongly correlated Kondo lattice. Comparative review work described it as a Ce-based correlated-electron antiferromagnet with weaker correlations than CeNiGe$Ln$53 and CeGe, a comparatively small Sommerfeld coefficient $Ln$54, and no visible Kondo-resistivity signature. In that description, the dominant low-temperature scale is RKKY exchange, represented schematically by $Ln$55, rather than strong Kondo screening (Singh et al., 2021).

Local NMR measurements refined this picture by showing that coherent Kondo coupling does emerge, but that its signatures are masked by the much larger Ce-$Ln$56 spin susceptibility. In $Ln$57Al NMR, the Knight shift was analyzed as

$Ln$58

and the Clogston–Jaccarino plot $Ln$59 versus $Ln$60 was linear for $Ln$61 K with $Ln$62. Below $Ln$63 K, the data deviated from linearity, yielding a Knight-shift anomaly interpreted as the onset of coherent Kondo coupling. The authors argued that this scale is not a crystal-electric-field effect, because the CEF gap is about $Ln$64 meV, far above $Ln$65 K (Wang et al., 2024).

The same NMR study found that dynamic magnetic fluctuations develop well above the bulk ordering temperature. The relaxation rate $Ln$66 shows the expected peak near $Ln$67 K, but also an additional hump near $Ln$68 K. This was connected to topology-stabilized magnetic fluctuations previously associated with an incommensurate wavevector $Ln$69, consistent with nesting between Weyl nodes at roughly $Ln$70. The calculated Al-site hyperfine form factor remains sizable at $Ln$71, about $Ln$72 after normalization, so $Ln$73Al NMR is not blind to fluctuations at the nesting wavevector. A plausible implication is that CeAlGe cannot be classified simply as a weakly correlated magnetic semimetal or simply as a heavy-fermion metal; rather, the relevant low-energy regime combines local-moment magnetism, coherent $Ln$74-$Ln$75 hybridization below $Ln$76, and topology-linked short-range fluctuations above $Ln$77 (Wang et al., 2024).

Polycrystalline AC-susceptibility work adds a further layer by reporting low-field, low-temperature spin-lattice relaxation phenomena below the ordering temperature. Using a modified Cole–Cole analysis, that study extracted a broad distribution of relaxation times with, at $Ln$78 T, $Ln$79, $Ln$80 at $Ln$81 K; $Ln$82, $Ln$83 at $Ln$84 K; and $Ln$85, $Ln$86 at $Ln$87 K. That work interpreted the relaxation and the negative, asymmetric longitudinal magnetoresistance in terms of Rashba–Dresselhaus spin-orbit interaction and Berry-curvature-induced anomalous velocity. This suggests that the low-temperature dynamics sensed by local probes and by transport need not be exhausted by static order parameters alone (Singh et al., 2020).

6. Topological interpretation and comparative position

The topological interpretation of CeAlGe rests on a conjunction of crystallography, magnetism, and semimetallic transport. The noncentrosymmetric $Ln$88 lattice breaks inversion symmetry, magnetic order breaks time-reversal symmetry, and the carrier density is low enough that Weyl-node-derived band topology remains relevant to transport and magnetic interactions. The single-crystal structural result is therefore not merely crystallographic classification: it is the prerequisite for treating CeAlGe as a Weyl-semimetal platform. This is why the literature repeatedly treats stoichiometry control, magnetic anisotropy, and field-tuned phase boundaries as topological issues rather than only materials-science details (Hodovanets et al., 2018, Puphal et al., 2019).

Within the $Ln$89AlGe family, CeAlGe has become a reference system for magnetic Weyl physics with localized rare-earth moments. In a DFT+DMFT comparison of the noncentrosymmetric RGaGe family, CeAlGe and CeAlSi were used as benchmarks for the localized-$Ln$90 limit, where the $Ln$91 electrons mainly generate magnetic moments and the Weyl nodes are formed by itinerant $Ln$92 bands. That study argued that CeGaGe lies closer to the same localized-$Ln$93 limit as CeAlGe but possesses a much more favorable Weyl geometry: one Weyl-point pair has “substantial chiral separation,” “significantly greater than in CeAlSi and CeAlGe,” producing long, well-isolated Fermi arcs on the $Ln$94 surface. By implication, CeAlGe remains topologically important, but its surface-state phenomenology is less favorable for direct ARPES-type observation than that of CeGaGe (Li, 24 Apr 2025).

CeAlGe is also central to compositionally tuned magnetic-topology studies. In Ce$Ln$95Pr$Ln$96AlGe, the Ce-rich end member retains the incommensurate multi-$Ln$97 state with moments predominantly in the $Ln$98-plane, and Pr substitution continuously suppresses the finite field needed to stabilize the topological magnetic phase known from CeAlGe. The crossover near $Ln$99 was identified as the regime where the field-induced topological phase of CeAlGe may become the zero-field ground state. In the complementary CeAlSi$Pn$00Ge$Pn$01 direction, Ge substitution beyond $Pn$02 introduces the incommensurate component and the singular angular magnetoresistance seen in CeAlGe, again tying its band topology to collective magnetism and field-angle-sensitive transport (Puphal et al., 2020, Yao et al., 22 Sep 2025).

Taken together, the literature presents CeAlGe as a noncentrosymmetric magnetic semimetal in which several descriptions coexist without being mutually trivial: a low-carrier-density antiferromagnet, a candidate type-II or magnetic Weyl semimetal, a Kondo Weyl semimetal with $Pn$03 K, and a host of multi-$Pn$04, topologically nontrivial magnetic textures that generate topological Hall and angular-magnetoresistance anomalies. The persistence of stoichiometry sensitivity, competing anisotropy assignments, and multiple field-induced phases is not peripheral to its identity. It instead suggests that CeAlGe occupies a narrow regime where crystal symmetry, $Pn$05-electron magnetism, correlation effects, and Weyl-derived electronic structure are all comparably important (Wang et al., 2024, Piva et al., 2023, Pomjakushin et al., 15 Sep 2025).