Microscopic NMR evidence for successive antiferroelectric and antiferromagnetic order in the van der Waals magnet CuCrP$_2$S$_6$

Abstract: We present a comprehensive $^{31}$P and $^{65}$Cu nuclear magnetic resonance (NMR) study of the layered van der Waals magnet CuCrP$2$S$_6$. The compound exhibits a sequence of structural and magnetic phase transitions: a high-temperature paraelectric state, followed by a quasi-antiferroelectric (QAFE) state near 185 K, a long-range antiferroelectric (AFE) phase below 150 K, and finally, antiferromagnetic (AFM) order below $T\mathrm{N}$ = 30 K. The evolution of the NMR spectra, NMR shift, and spin-lattice ($T_1^{-1}$) and spin-spin ($T_2^{-1}$) relaxation rates provide direct microscopic fingerprints of these transitions. The splitting of both the NMR line and $T_1^{-1}$ below the AFE transition demonstrates the emergence of two inequivalent P sites. From $K - χ$ analysis, we extract nearly isotropic transferred hyperfine couplings and show that the NMR shift anisotropy originates primarily from the dipolar contribution, in contrast to Mn$2$P$_2$S$_6$ and Ni$_2$P$_2$S$_6$. We determine the ferromagnetic intralayer exchange $J{intra}\approx$ -4.9 K from the Curie Weiss temperature, consistent with ferromagnetic layers antiferromagnetically stacked along the $c$ axis, and evaluate the Moriya high temperature relaxation rate including cross correlation effects of the P P dimer. Critical divergence of $T_1^{-1}$ near $T_\mathrm{N}$ yields a critical exponent $γ\simeq$ 0.45(4), placing CuCrP$_2$S$_6$ in a three dimensional Heisenberg universality regime.

• The paper demonstrates successive phase transitions from a quasi-antiferroelectric to a long-range antiferroelectric state, followed by antiferromagnetic order at 30 K using detailed NMR spectroscopy.
• It employs angle-dependent NMR shift analysis and relaxation dynamics to quantify hyperfine couplings, exchange interactions, and critical exponents, aligning with 3D Heisenberg universality.
• The study reveals how local symmetry breaking due to Cu ion displacements creates inequivalent P sites, serving as a diagnostic tool for probing spin-lattice coupling in van der Waals magnets.

Microscopic NMR Evidence for Successive Antiferroelectric and Antiferromagnetic Orders in CuCrP$_2$S$_6$

Structural Overview and Phase Sequence

CuCrP$_2$S$_6$, a member of the quasi-2D van der Waals thiophosphate family ($T_2$P$_2$S$_6$), exhibits intricate structural and magnetic ordering phenomena closely coupled via charge, lattice, and spin degrees of freedom. The crystal structure is characterized by Cr$^{3+}$ ions forming a two-dimensional triangular magnetic lattice within a network of [P$_2$S$_6$]$_6$0 units, and Cu$_6$1 ions situated in distorted sulfur octahedra. At room temperature, the system is in a monoclinic $_6$2 space group with Cu ions distributed over partially occupied sites. On cooling, this system undergoes a sequence of transitions: from a high-temperature paraelectric phase to a quasi-antiferroelectric (QAFE) regime near 185 K, followed by a long-range antiferroelectric (AFE) state below 150 K, and finally antiferromagnetic (AFM) order on the Cr$_6$3 sublattice below $_6$4 K.

Figure 1: Crystal structure of CuCrP$_6$5S$_6$6 revealing Cr$_6$7 triangular lattices, P--P dimers, alternately displaced Cu ions, and quasi-2D stacking.

The low-temperature AFE phase is defined by antipolar ordering of Cu$_6$8 ions that occupy alternating up- and down-displaced positions along $_6$9, resulting in cancellation of macroscopic polarization but distinctly inequivalent local environments. The transition to AFM order manifests as ferromagnetic alignment within $_2$0-planes and antiferromagnetic stacking along the $_2$1-axis, forming an A-type structure validated via neutron diffraction and thermodynamic measurements.

NMR Spectroscopy Across Structural and Magnetic Transitions

Comprehensive $_2$2P and $_2$3Cu NMR measurements provide microscopic resolution of these phase transitions. In the high-temperature phase, $_2$4P NMR spectra show a single sharp resonance, confirming crystallographic equivalence of P atoms and the absence of detectable Pake doublet due to broad magnetic linewidths. Angle-dependent measurements of the NMR shift at room temperature reveal moderate anisotropy and nonzero asymmetry parameter ($_2$5), indicating deviation from uniaxial symmetry compared to other thiophosphates.

Upon cooling, spectra evolve significantly: below $_2$6180 K (QAFE regime), line broadening and asymmetry signal loss of local symmetry and precursor fluctuations of Cu displacements. Below $_2$7 K, two distinct $_2$8P resonance lines emerge, evidencing two inequivalent P sites due to symmetry breaking. The progressive growth of line splitting, normalized by the susceptibility, delineates the order parameter evolution associated with the AFE transition and confirms the first-order character.

Hyperfine Coupling and Shift Analysis

NMR shift $_2$9 tracks local spin susceptibility and its evolution through structural transitions. Above $_6$0, $_6$1 and bulk susceptibility $_6$2 are linearly proportional; below $_6$3, two shifts ($_6$4 and $_6$5) reflect site-specific hyperfine field variations.

Shift versus susceptibility ($_6$6--$_6$7) analysis yields nearly isotropic transferred hyperfine coupling constants for both principal field orientations ($_6$8~T/$_6$9, $T_2$0~T/$T_2$1). The orbital shift $T_2$2 remains nearly identical for both sites, signifying that electronic environments of P atoms are unaltered by structural transition. The NMR shift anisotropy arises primarily from the dipolar hyperfine contribution, a stark contrast to Mn$T_2$3P$T_2$4S$T_2$5 and Ni$T_2$6P$T_2$7S$T_2$8 which exhibit stronger, more anisotropic transferred couplings due to distinct electronic configurations and bonding pathways.

Relaxation Dynamics and Criticality

Spin-lattice ($T_2$9) and spin-spin ($_2$0) relaxation rates are direct probes of local magnetic fluctuations and structure-dynamics correlations. At high temperatures, $_2$1 is exchange-narrowed and nearly constant, governed by fast Cr$_2$2 spin fluctuations. Across the AFE transition, $_2$3 splits into two rates associated with inequivalent P sites, quantitatively explained by differing hyperfine couplings, while sharing identical dynamical responses. Below $_2$4, $_2$5 and $_2$6 exhibit critical divergence as the system approaches AFM order, captured by power-law fits yielding a critical exponent $_2$7. This exponent is consistent with 3D Heisenberg universality, ruling out lower-dimensional magnetic criticality.

Spin-spin relaxation reveals oscillatory, Gaussian-modulated echo decays at high temperatures attributable to dipolar coupling in P--P dimers, which are strongly damped and ultimately replaced by pure exponential decay below the AFE transition, reflecting the freezing of dipolar fluctuations and establishment of static order.

Exchange Interactions and Theoretical Implications

Extraction of exchange constants from Curie--Weiss analysis and ESR data yields $_2$8 K (ferromagnetic intralayer) and $_2$9 K (antiferromagnetic interlayer) for Cr$_6$0 triangular lattices. Moriya-type calculations of $_6$1, incorporating cross-correlation effects within the P--P dimer, account for the factor-of-two enhancement in the symmetric relaxation channel compared to single-site theory—a manifestation of coordinated local field fluctuations.

The observed nearly isotropic transferred hyperfine coupling and predominant dipolar shift anisotropy indicate unique Fermi contact interactions restricted by Cr$_6$2 $_6$3 orbital configuration and $_6$4-bonding, in contrast to anisotropic, $_6$5-bonding dominated systems.

Implications and Future Perspectives

The findings underscore CuCrP$_6$6S$_6$7 as a prototypical vdW magnet with intimately coupled electric-dipole and magnetic order, accessible via local probes such as NMR. The confluence of paraelectric, quasi-antiferroelectric, antiferroelectric, and antiferromagnetic phases, as well as the direct spectral and relaxation evidence for symmetry breaking and criticality, provide pivotal benchmarks for understanding multiferroic and low-dimensional magnetic systems.

Figure 1: Crystal structure of CuCrP$_6$8S$_6$9 revealing Cr$^{3+}$0 triangular lattices, P–P dimers, alternately displaced Cu ions, and quasi-2D stacking.

These insights will inform future theoretical modeling of spin-lattice interactions, dimer-fluctuation physics, and electric-magnetic coupling in van der Waals materials. Practically, the NMR fingerprint across transitions offers a diagnostic paradigm for characterizing ferroic and multiferroic materials with potential applications in spintronics and magnon-based information technologies. Further investigation into monolayer or heterostructure variants may elucidate the evolution of these coupled orders at reduced dimensionality and interface-driven environments.

Conclusion

This work provides comprehensive microscopic NMR documentation of successive antiferroelectric and antiferromagnetic ordering phenomena in CuCrP$^{3+}$1S$^{3+}$2, resolving inequivalent P sites and quantifying hyperfine couplings, exchange constants, and critical exponents characteristic of 3D Heisenberg universality. The results establish the unique interplay of structure and magnetism in CuCrP$^{3+}$3S$^{3+}$4 and position NMR as an essential tool in probing coupled order parameters in layered van der Waals systems (2604.02898).