Development of the magnetism in the solid solution of the candidate Weyl semimetals Ce$_x$Pr$_{1-x}$AlGe

Abstract: We investigate the macroscopic and microscopic physical properties of the solid solution of Ce${1-x}$Pr${x}$AlGe. The series tunes from CeAlGe with its multi-$\vec{k}$ structure and a major Moment in the ab-plane, to PrAlGe with an easy-c-axis ferromagnetic ground state co-existing with a low density of nanoscale textured magnetic Domain walls. Using AC-, DC-susceptiblity, resistivity, specific heat, muon spin relaxation/rotation and neutron scattering we analyze the magnetic ground state of the series. We provide further evidence supporting our previous claim for spin-glass like properties in pure PrAlGe. With introduction of Pr to CeAlGe the finite magnetic field required to stabilize the topological multi-$\vec{k}$ magnetic phase for $x=0$ becomes suppressed. The crossover between the two end-member ground states occurs in the vicinity of $x=0.3$, a region where we further anticipate the field-induced topological magnetic phase for $x < 0.3$ to become the zero field ground state.

• The paper reveals a sharp crossover near x = 0.3, where competing multi‑k and ferromagnetic orders yield a tunable zero‑field topological state.
• It utilizes a comprehensive suite of techniques, including neutron scattering, muon spin rotation, and thermodynamic measurements, to characterize the microscopic and macroscopic magnetic properties.
• Findings indicate that chemical substitution in Ce₁₋ₓPrₓAlGe allows precise control of topological magnetic states, paving the way for novel quantum device integration.

Magnetism Evolution in the Ce$_{1-x}$Pr$_x$AlGe Solid Solution: A Study of Candidate Magnetic Weyl Semimetals

Introduction

The Ce$_{1-x}$Pr$_x$AlGe series occupies a central role in the exploration of topological quantum materials, notably within the family of polar magnets predicted to host Weyl fermions upon time-reversal and spatial symmetry breaking. The ability to tune from CeAlGe to PrAlGe via chemical substitution provides a platform for controlling both macroscopic magnetic phases and their topological textures, a prerequisite for functional exploitation of Weyl semimetals. This study presents a comprehensive investigation of the microscopic and macroscopic magnetic properties across the full substitution series, with rigorous experimental insight from AC/DC magnetometry, heat capacity, transport, muon spin rotation/relaxation, and neutron scattering methods.

Structural and Compositional Verification

Stoichiometric control across the series was confirmed by high-precision XRF and EDS, demonstrating homogeneity and the absence of the Al/Ge off-stoichiometry issues that had previously confounded flux-grown analogs. Rietveld refinement of neutron diffraction patterns unambiguously resolved the crystallography into the noncentrosymmetric LaPtSi-type I4$_1$md structure throughout, ensuring that any magnetic or topological phenomena can be attributed to the intrinsic physics of the targeted compositions. Structural parameters (lattice constants, $z$-coordinates) showed a Vegard's law evolution consistent with a robust solid solution.

Magnetic Ground State Evolution

End Member Behavior and Order Parameters

CeAlGe manifests an incommensurate multi-$\vec{k}$ spin texture upon cooling below $T_N \approx 4.5$ K, as characterized by polarized neutron data and muon spectroscopy. This topologically non-trivial texture is typified by a periodic array of domains with opposite winding numbers, generically cancelling out at macroscopic scale. Conversely, PrAlGe orders via a robust easy-$c$-axis ferromagnetic transition near $T_C \approx 16$ K, with SANS data indicating an additional population of nanoscale domain walls misaligned from the Ising axis. Prior reports suggested and the present study reinforces the presence of a glassy component in the low-temperature PrAlGe magnetic state (2002.05745).

Crossover Physics and Tunability

Between $x \approx 0$ and $x \approx 1$, the ground state undergoes a marked crossover. For $x \lesssim 0.3$, the CeAlGe-type multi-$\vec{k}$ order survives with only weak admixture of a Pr-derived ferromagnetic component. MuSR data continue to exhibit coherent oscillations characteristic of long-range ordering, and the critical temperature is nearly unchanged. At $x \approx 0.3$, the ground states coexist and interact—a unique region where multi-$\vec{k}$ and ferromagnetic order parameters are simultaneously manifested, as shown by the coexistence of neutron scattering features, thermodynamic anomalies, and magnetotransport signatures.

For $x \gtrsim 0.4$, a sharp transition occurs: all moments, both Pr and Ce, align predominantly along the $c$-axis in a uniform ferromagnetic configuration. The anomalous SANS intensity at low $q$ associated with domain wall texture increases with $x$, and the coercive field and saturation magnetization scale linearly with Pr concentration.

Notably, the field required to stabilize the field-induced topological phase in CeAlGe is rapidly suppressed with Pr substitution, reaching near-zero for $x \gtrsim 0.3$. This implies the capability to realize a zero-field topological phase—a crucial property for practical device integration of topological transport.

Dynamic and Thermodynamic Probes

AC Susceptibility and Domain Dynamics

AC susceptibility elucidates the shift from antiferromagnetic (multi-$\vec{k}$) to ferromagnetic domain-dominated relaxation. For $x \geq 0.3$, Arrhenius law analysis of $m''$ peaks yields increasing energy barriers for domain reversal with Pr, while the relaxation time $\tau_0$ rapidly shortens. The data indicate a substantial change in the domain wall pinning landscape as the system approaches pure PrAlGe.

DC Measurements, Hysteresis, and Metamagnetic Transitions

Field-swept magnetization and susceptibility show that only above $x \sim 0.4$ does a clear hysteresis and robust FM loop develop. Below this, the system remains dominated by non-collinear and glassy responses. For $x = 0.3$, metamagnetic transitions evidenced in magnetoresistance and $M(H)$ shift to substantially lower fields than in CeAlGe, demonstrating enhanced tunability.

Specific Heat and Entropy Release

Heat capacity data show that entropy is primarily released at the ordering transitions, with $S \approx R\ln2$ for CeAlGe and a fractional value for PrAlGe, consistent with reduced ordered moments and persistent glassiness in the latter. Double anomalies for $x=0.3$ precisely correspond to the vanishing of low-$q$ SANS scattering, further confirming the crossover and reentrance region.

Muon Spin Rotation/Relaxation

ZF MuSR differentiates the glassy regime (fast relaxation, incoherent internal fields) in Pr-rich samples from the long-range ordered multi-$\vec{k}$ regime (damped oscillations) in Ce-rich samples. For $x=0.3$, increased damping but preservation of oscillatory components reflects strong competition or phase separation between ground states.

Neutron Scattering and Topological Textures

Powder neutron diffraction and SANS confirm the evolution of the microscopic order parameter: for $x < 0.3$, the magnetic Bragg peak structure matches the multi-$\vec{k}$ solution; for $x > 0.3$, these features collapse to $q=0$ FM peaks with significant diffuse scattering indicative of domain wall textures. Notably, the multi-$\vec{k}$ modulation vector, $\vec{k}$, subtly increases with $x$, and the FM ordered moment size increases monotonically with Pr content, matching the bulk DC results.

SANS further reveals that for intermediate compositions ($x=0.2,0.3$), low-$q$ scattering persists at higher temperatures but disappears at the lowest temperatures, implying a fully aligned state absent domain walls—a phase only accessible due to the fine-tuned competition of orders.

Implications and Outlook

The Ce$_{1-x}$Pr$_x$AlGe solution evidences highly tunable macroscopic and topological magnetic order via isovalent substitution. The robustness of the multi-$\vec{k}$ order up to $x \lesssim 0.3$, and the suppression of the field-induced topological phase to zero field at this composition, opens new avenues for static stabilization of topological spin textures without external fields. Such states are ideal for probing axion electrodynamics, nontrivial Hall responses, and dissipationless transport regimes. The co-existence and competition region revealed near $x = 0.3$ is particularly promising for studies of domain wall dynamics, topological Hall effects, and emergent phases at multicriticality.

Future directions include single-crystal studies of transport and neutron scattering across the crossover region, exploration of control by hydrostatic pressure or uniaxial strain, and the design of devices exploiting the predicted zero-field topological magnetic states. The demonstrated tunability in Ce$_{1-x}$Pr$_x$AlGe underlines the power of bulk chemical substitution for topological phase engineering in quantum materials.

Conclusion

Ce$_{1-x}$Pr$_x$AlGe offers a prototypical example of controlled tuning between distinct correlated and topological magnetic states in a noncentrosymmetric Weyl semimetal context. The sharp crossover near $x = 0.3$—from a field-stabilized multi-$\vec{k}$ spin texture to a uniform ferromagnet with domain wall glassiness—demonstrates the promise of this family for fundamental transport studies and the realization of topological phases functional at liquid helium temperatures. The experimental evidence indicates a narrow window for stabilizing zero-field topological magnetic order, providing a testbed for theoretical and application-oriented advances in quantum matter and potentially informing practical device realization of topological functionalities.