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

This study utilizes comprehensive $^{31}$P and $^{65}$Cu nuclear magnetic resonance (NMR) measurements to provide direct microscopic evidence of successive quasi-antiferroelectric, antiferroelectric, and antiferromagnetic phase transitions in the van der Waals magnet CuCrP$_2$S$_6$, while characterizing its magnetic exchange interactions and identifying its critical behavior within the three-dimensional Heisenberg universality class.

The Gist

Imagine a microscopic city built from layers of flat, sticky pancakes (these are the "van der Waals" layers). Inside this city live two types of residents: Copper ions and Chromium ions. The Copper ions are like shy, quiet neighbors who sometimes move around, while the Chromium ions are the energetic, magnetic "sports fans" who love to cheer in unison.

This paper is a detective story about how this city changes its behavior as the temperature drops, turning from a chaotic summer party into a highly organized winter village. The scientists used a special tool called NMR (Nuclear Magnetic Resonance)—think of it as a super-sensitive "ear" that can listen to the tiny whispers of the atoms inside the material.

Here is the story of what they found, broken down into simple steps:

1. The Hot Summer Party (High Temperature)

When the material is hot (above 185 K), the Copper residents are restless. They are jumping between different spots on the "pancake" layers, and the Chromium fans are cheering randomly.

• The NMR Ear: When the scientists listened, they heard one single, clear voice. This meant that, from the perspective of the Phosphorus atoms (the "streetlights" of the city), everything looked the same. The city was symmetrical and chaotic.

2. The First Chill: The "Almost" Order (185 K to 150 K)

As the temperature drops to about 185 K, the Copper residents start to get a little shy. They stop jumping randomly and start clustering in small groups, but they haven't decided on a final rule yet.

• The NMR Ear: The single voice starts to sound fuzzy and distorted. It's like a crowd starting to murmur in different directions. The scientists call this the Quasi-Antiferroelectric state. It's a "practice run" for order, where the Copper ions are trying to decide which way to face, but they haven't fully committed.

3. The Big Freeze: The Perfect Mirror (150 K)

At 150 K, the Copper residents finally make a decision. They split into two distinct groups: half of them decide to stand "up," and the other half stand "down." They arrange themselves in a perfect, alternating pattern (Up-Down-Up-Down).

• The NMR Ear: Suddenly, the single fuzzy voice splits into two distinct voices. One voice is high-pitched, and the other is low-pitched. This is the "smoking gun" evidence that the city has changed its architecture. The streetlights (Phosphorus atoms) now see two different neighborhoods: one next to the "Up" Copper and one next to the "Down" Copper. This is the Antiferroelectric state. The city has become a perfect mirror image.

4. The Final Freeze: The Silent Stadium (30 K)

When it gets really cold (below 30 K), the Chromium "sports fans" finally stop their random cheering. They organize themselves too! Inside each layer, they all cheer in the same direction (Ferromagnetic), but the layer above them cheers in the opposite direction (Antiferromagnetic).

• The NMR Ear: The voices split even further and get quieter. The material has now entered a Magnetic state. The scientists measured how fast the atoms "relax" (calm down) after being disturbed. They found that the way the atoms calm down follows the exact mathematical rules of a 3D Heisenberg Antiferromagnet. It's like finding out that this tiny city follows the same physics laws as a giant, 3D magnet, proving it's not just a flat 2D toy, but a robust 3D system.

The Big Discoveries (The "Aha!" Moments)

• The "Dipole" Mystery: In similar cities (like those made of Nickel or Manganese), the way the atoms talk to each other is complicated and depends heavily on direction. But in this Copper-Chromium city, the scientists found that the "magnetic conversation" is surprisingly isotropic (the same in all directions). It's as if the Copper-Chromium residents speak a language that doesn't care if you are facing North or East; the message is the same. This is because of how their electrons are arranged (like a specific type of handshake).
• The "Twin" Effect: The Phosphorus atoms come in pairs (dimers). Usually, scientists treat them as two separate people. But the scientists realized that because these two atoms are so close and connected to the same neighbors, they "hear" the same magnetic fluctuations at the same time. It's like two twins standing next to each other; if one hears a noise, the other hears it too. This "cross-correlation" makes the material relax twice as fast as expected. It's a team effort!

Why Does This Matter?

This material is special because it combines electric order (the Copper ions lining up) and magnetic order (the Chromium ions lining up) in the same place at the same time.

Think of it as a smart material that could be the basis for future computers. If you can control the "electric" switch (the Copper), you might be able to flip the "magnetic" switch (the Chromium) without using electricity, making devices that are faster and use less power.

In summary: The scientists used a microscopic ear to listen to a layered material as it cooled down. They watched it go from a chaotic mess, to a practice run, to a perfectly mirrored electric city, and finally to a synchronized magnetic stadium. Along the way, they discovered that the atoms in this city talk to each other in a unique, direction-independent way, offering a new blueprint for building next-generation electronic devices.

Technical Summary

1. Problem Statement

Layered van der Waals (vdW) magnets, particularly the transition-metal thiophosphate family ($T_2P_2S_6$), are promising platforms for studying low-dimensional magnetism and magnetoelectric coupling. While bulk properties of CuCrP$_2$S$_6$ are known to exhibit a sequence of phase transitions—paraelectric $\to$ quasi-antiferroelectric (QAFE) $\to$ long-range antiferroelectric (AFE) $\to$ antiferromagnetic (AFM)—the microscopic evolution of local electronic and magnetic environments across these transitions remains unclear.

Specifically, open questions include:

• How do local structural distortions (Cu ion displacements) affect the local symmetry and hyperfine fields at the phosphorus sites?
• What is the nature of the hyperfine coupling mechanism in CuCrP$_2$S$_6$ compared to related compounds like Mn$_2$P$_2$S$_6$ and Ni$_2$P$_2$S$_6$?
• How do the critical dynamics near the magnetic ordering temperature ($T_N$) behave, and do they follow 2D or 3D universality classes?

2. Methodology

The authors performed a comprehensive Nuclear Magnetic Resonance (NMR) study on high-quality single crystals of CuCrP$_2$S$_6$.

• Probes: They utilized $^{31}$P ($I=1/2$, non-quadrupolar, clean probe) and $^{65}$Cu ($I=3/2$) nuclei.
• Techniques:
  • Temperature- and Angle-Dependent Spectra: Measured from room temperature down to low temperatures ($<30$ K) under magnetic fields of 2.5 T, 4 T, and 7 T applied along different crystallographic axes ($b$ and $c^*$).
  • Relaxation Measurements:
    • Spin-Lattice Relaxation ($T_1$): Measured via inversion recovery to probe spin fluctuations and critical slowing down.
    • Spin-Spin Relaxation ($T_2$): Measured via spin-echo decay to probe local field inhomogeneities and dipolar couplings.
  • Analysis:
    • K-$\chi$ Analysis: Correlating NMR shift ($K$) with bulk magnetic susceptibility ($\chi$) to extract transferred hyperfine coupling constants ($A_{hf}$).
    • Moriya Theory: Applied to high-temperature relaxation rates, explicitly incorporating cross-correlation effects within the P–P dimers.
    • Critical Exponent Extraction: Fitting the divergence of relaxation rates near $T_N$ to power laws ($T^{-1} \propto \epsilon^{-\gamma}$) to determine the universality class.

3. Key Contributions and Results

A. Microscopic Evidence of Phase Transitions

The NMR data provided direct, site-resolved evidence for the sequence of transitions:

1. Paraelectric State ($T > 185$ K): A single, sharp $^{31}$P resonance line indicates crystallographically equivalent P sites in the high-symmetry $C2/c$ structure.
2. Quasi-Antiferroelectric (QAFE) Regime ($150 < T < 185$ K): The NMR line broadens and becomes asymmetric. This indicates the onset of local Cu displacements and short-range correlations without long-range order. The spin-spin relaxation ($T_2$) shows damping of oscillatory components, signaling the freezing of dipole fluctuations.
3. Long-Range Antiferroelectric (AFE) State ($T < 150$ K):
   • Spectral Splitting: The single NMR line splits into two distinct peaks of equal intensity, confirming the emergence of two inequivalent P sites due to symmetry lowering (transition to $Pc$ space group).
   • Relaxation Splitting: The spin-lattice relaxation rate ($1/T_1$) also splits into two distinct values corresponding to the two P sites. The ratio of relaxation rates matches the square of the ratio of hyperfine couplings, confirming that the sites experience the same magnetic fluctuations but with different coupling strengths.
   • Order Parameter: The line splitting ($\Delta\nu$) grows continuously from zero at 180 K but shows a discontinuity (first-order character) at 150 K, consistent with calorimetric data.
4. Antiferromagnetic (AFM) State ($T < 30$ K):
   • Further splitting of the NMR lines occurs due to static internal fields from ordered Cr$^{3+}$ moments.
   • A sharp peak in $1/T_1$ at $T_N = 30$ K signals critical slowing down of spin fluctuations.

B. Hyperfine Coupling Mechanism

• Isotropic Transferred Coupling: By analyzing the $K$-$\chi$ relationship, the authors extracted the transferred hyperfine coupling constants ($A_{tra}$). Unlike Mn$_2$P$_2$S$_6$ and Ni$_2$P$_2$S$_6$, where transferred coupling is highly anisotropic, CuCrP$_2$S$_6$ exhibits nearly isotropic transferred coupling.
• Origin of Anisotropy: The observed NMR shift anisotropy in CuCrP$_2$S$_6$ is attributed primarily to the dipolar contribution, not the transferred term. This is attributed to the Cr$^{3+}$ ($3d^3$) electronic configuration, where unpaired electrons in $t_{2g}$ orbitals interact via $\pi$-bonding, leading to isotropic Fermi contact interaction, unlike the $\sigma$-bonding in $e_g$ orbitals of Mn/Ni analogues.

C. Exchange Interactions and Critical Dynamics

• Exchange Constants: Using the Curie-Weiss temperature and ESR data, the intralayer exchange was determined to be ferromagnetic ($J_{intra} \approx -4.9$ K), while interlayer coupling is antiferromagnetic ($J_{inter} \approx +2.8$ K), consistent with A-type AFM order.
• P-P Dimer Cross-Correlation: The experimentally observed high-temperature $1/T_1$ was found to be roughly twice the value predicted by standard single-site Moriya theory. The authors successfully explained this enhancement by including cross-correlation effects between the two P nuclei in the P–P dimer, which share the same fluctuating Cr spin bath.
• Universality Class: The critical divergence of $1/T_1$ and $1/T_2$ near $T_N$ yielded a critical exponent $\gamma \approx 0.45(4)$. This value is close to the theoretical prediction for a 3D Heisenberg antiferromagnet ($\gamma = 1/2$ or $1/3$) and significantly different from 2D systems, indicating that despite the layered structure, the critical behavior is three-dimensional.

4. Significance

• Microscopic Validation: This work provides the first microscopic confirmation of the structural symmetry breaking and site inequivalence in the AFE phase of CuCrP$_2$S$_6$, resolving ambiguities from bulk measurements.
• Unique Electronic Environment: It highlights a distinct spin-polarization mechanism in CuCrP$_2$S$_6$ compared to its family members, driven by the specific orbital occupancy of Cr$^{3+}$.
• Methodological Advance: The study demonstrates the necessity of accounting for nuclear dimer cross-correlations in relaxation theories for systems with strong intralayer coupling, offering a corrected framework for interpreting NMR data in similar vdW materials.
• Dimensionality Insight: The determination of the 3D Heisenberg universality class clarifies the nature of magnetic ordering in this quasi-2D material, showing that interlayer coupling is sufficient to drive 3D criticality near the transition.

In conclusion, the paper establishes CuCrP$_2$S$_6$ as a rare example where electric-dipole order and quasi-2D magnetism coexist and interact, with NMR serving as a powerful tool to disentangle the complex interplay of structural and magnetic degrees of freedom.