Pressure Effect on the Spin Density Wave Transition in La$_2$PrNi$_2$O$_{6.96}$

Using muon-spin rotation under high pressure, this study demonstrates that the magnetic properties of the double-layer Ruddlesden-Popper nickelate La$_2$PrNi$_2$O$_{6.96}$, characterized by a pressure-enhanced Néel temperature and stable critical exponents, remain largely unaffected by Pr-to-La substitution, suggesting similar magnetic behavior in the related superconducting compound La$_3$Ni$_2$O$_7$.

The Gist

Imagine you have a brand new, super-advanced material called a "nickelate." Scientists are incredibly excited about it because, when you squeeze it really hard (apply high pressure), it starts conducting electricity with zero resistance. This is superconductivity, the "holy grail" of physics that could revolutionize power grids, maglev trains, and computers.

However, there's a catch. In the original version of this material (called La₃Ni₂O₇), the superconductivity is a bit of a "ghost." It only happens in tiny, thread-like filaments rather than the whole block of material. It's like trying to power a city with a single, flickering candle instead of a massive power plant. Scientists are trying to figure out why this happens and how to fix it.

One theory is that if you swap some of the atoms in the material (replacing Lanthanum with Praseodymium), you might get a "cleaner" version that superconducts throughout the entire block. This new version is called La₂PrNi₂O₆.₉₆.

The Big Question:
Does this "cleaner" version behave differently magnetically? Since magnetism and superconductivity are often dance partners in these materials, understanding the magnetic behavior is key to unlocking the superconductivity.

The Experiment: The "Muon" Flashlight
To see what's happening inside the atoms, the researchers used a technique called Muon-Spin Rotation (µSR).

• The Analogy: Imagine you want to see how a crowd of people (the electrons) is moving in a dark room. You can't see them directly, so you throw in thousands of tiny, glowing flashlights (muons) that spin as they fly. By watching how these flashlights spin and wobble as they get stuck in the material, you can map out the magnetic "wind" inside the room.
• The Setup: They put a chunk of this new material inside a special pressure chamber (like a high-tech nutcracker) and squeezed it with up to 2.3 Gigapascals of pressure (that's about 23,000 times the atmospheric pressure at sea level!).

What They Found:

1. The Magnet Gets Stronger (But Not Too Much): As they squeezed the material, the temperature at which it becomes magnetic (called the Néel temperature) went up slightly, from about 161°C to 170°C. Think of it like a magnet that gets slightly more "stubborn" and holds its magnetic shape better when you squeeze it.
2. The "Dance" Stays the Same: The way the magnetic atoms interact with each other didn't change its fundamental style. Whether the material was at normal pressure or under heavy pressure, the "dance steps" (mathematical rules describing the magnetism) remained identical.
3. The "Ghost" vs. The "Bulk": The most important finding is this: The magnetic properties of this new "Praseodymium-swapped" material are almost identical to the original "Lanthanum-only" material.

The Conclusion: A Simple Takeaway
The researchers expected that swapping the atoms would completely change the magnetic landscape, perhaps explaining why the new material superconducts better.

Instead, they found that the swap didn't change the magnetic personality of the material at all. The "ghostly" filamentary superconductivity in the original material and the "bulk" superconductivity in the new material seem to be happening despite the magnetic similarities, not because of a magnetic difference.

In Everyday Terms:
Imagine two identical twins (the two materials). One twin can run a marathon (superconduct) easily, while the other can only jog in short bursts. You thought that maybe the jogging twin had a different heart structure (magnetism). But after a medical scan (the muon experiment), you realize their hearts are beating in the exact same rhythm.

This tells scientists that the secret to making the "bulk" superconductor isn't just about fixing the magnetism. The Praseodymium swap helps the material form a better structure (like fixing the skeleton), but the magnetic "heart" remains the same. This is a huge clue: it suggests that the reason for the difference in superconductivity lies in the material's structure and shape, not in how its internal magnets are behaving.

Why does this matter?
It narrows down the search. Scientists can stop looking for a "magnetic miracle" caused by the atom swap and focus on how the atoms are stacked and arranged. It's a step closer to understanding how to build a room-temperature superconductor that works everywhere, not just in tiny threads.

Technical Summary

1. Problem Statement and Motivation

The discovery of high-temperature superconductivity ($T_c \approx 80$ K) in double-layer Ruddlesden-Popper (RP) nickelates, specifically $La_3Ni_2O_7$ under high pressure, has sparked intense research. However, significant challenges remain in understanding the intrinsic nature of this superconductivity:

• Filamentary vs. Bulk Superconductivity: In non-substituted $La_3Ni_2O_7$, superconductivity often appears filamentary with a low volume fraction, raising questions about whether the bulk material is truly superconducting or if the effect is confined to phase interfaces.
• Structural Polymorphism: The material exhibits distinct structural phases (e.g., "2222" and "1313" polymorphs) depending on synthesis conditions, complicating the correlation between structure and physical properties.
• Role of Magnetic Substitution: Recent studies showed that partial substitution of La with Pr ($La_{3-x}Pr_xNi_2O_7$) enhances the purity of the bilayer structure, increases the superconducting volume fraction, and slightly raises $T_c$.
• The Core Question: It is essential to determine if the magnetic properties (specifically the Spin Density Wave or SDW order) that coexist with or precede superconductivity are fundamentally altered by Pr substitution. Understanding the magnetic response of the Pr-substituted system ($La_2PrNi_2O_{6.96}$) under pressure is crucial for clarifying the electronic mechanisms driving superconductivity in these nickelates.

2. Methodology

The authors employed Muon-Spin Rotation/Relaxation ($\mu$SR) spectroscopy, a highly sensitive local magnetic probe, to investigate the magnetic properties of $La_2PrNi_2O_{6.96}$ under hydrostatic pressure.

• Sample: A high-purity $La_2PrNi_2O_{6.96}$ sample (oxygen content verified as $\delta \approx 0.04$) from a batch previously studied at ambient pressure.
• Experimental Setup:
  • Experiments were conducted at the Paul Scherrer Institute (PSI) using the dedicated GPD muon spectrometer.
  • Pressures up to 2.3 GPa were generated using a non-magnetic MP35N alloy double-wall pressure cell.
• Measurement Techniques:
  • Zero-Field (ZF) $\mu$SR: To detect internal magnetic fields arising from ordered magnetic moments.
  • Weak Transverse Field (WTF) $\mu$SR: To measure the magnetic volume fraction and transition temperatures.
• Data Analysis:
  • Signals were separated into sample and pressure cell (background) contributions.
  • The sample polarization function was decomposed into magnetic and non-magnetic components.
  • The internal magnetic field ($B_{int}$) and its temperature dependence were fitted to a power-law relation: $B_{int}(T) = B_{int}(0)[1 - (T/T_N)^\alpha]^\beta$.

3. Key Results

The study yielded four primary observations regarding the magnetic behavior of $La_2PrNi_2O_{6.96}$ under pressure:

• Magnetic Order Type: The magnetic field distribution is well-described by Lorentzian peaks, indicating a commensurate magnetic order (Spin Density Wave). This contrasts with the incommensurate order found in the triple-layer $La_4Ni_3O_{10}$ but aligns with other double-layer nickelates.
• Pressure Dependence of Transition Temperature ($T_N$):
  • The Néel temperature ($T_N$) increases linearly with pressure.
  • At ambient pressure ($p=0$), $T_N \approx 161$ K.
  • At $p = 2.3$ GPa, $T_N$ rises to approximately 170 K.
  • The pressure coefficient is $dT_N/dp \approx 3.7$ K/GPa, which is comparable to values reported for non-substituted $La_3Ni_2O_7$.
• Robustness of Magnetic Order Parameters:
  • The temperature dependence of the internal magnetic field ($B_{int}$) follows a power law with exponents $\alpha \approx 1.95$ and $\beta \approx 0.35$.
  • These exponents remain unchanged across the pressure range, suggesting the magnetic order remains close to a 3D magnet (consistent with 3D Heisenberg, Ising, or XY models where $\beta \approx 1/3$) and is not fundamentally altered by pressure up to 2.3 GPa.
• Ordered Magnetic Moments:
  • The magnitude of the internal field at low temperatures ($B_{int}$ at 10 K) is independent of pressure (approx. 141–143 mT).
  • Since $B_{int}$ is proportional to the ordered Ni magnetic moments ($m_{Ni}$), this implies that the ordered magnetic moments remain constant under pressure.

4. Key Contributions

• Characterization of Pr-Substituted System: This work provides the first detailed high-pressure magnetic characterization of the Pr-substituted double-layer nickelate $La_2PrNi_2O_{6.96}$, establishing it as a robust magnetic system.
• Comparison with Non-Substituted Analog: The study demonstrates that the magnetic properties of $La_2PrNi_2O_{6.96}$ are broadly unaffected by the substitution of La with Pr when compared to $La_3Ni_2O_7$. The primary difference is a slightly higher ambient-pressure ordering temperature ($162$ K vs. $152-157$ K).
• Decoupling of Magnetism and Pressure Effects: The results show that while pressure enhances the magnetic ordering temperature, it does not alter the magnitude of the ordered moments or the critical exponents of the phase transition.
• Implications for Superconductivity: By confirming that Pr substitution does not introduce significant magnetic anomalies, the authors suggest that if superconducting pairing is driven by magnetic interactions, the enhanced superconducting properties (higher $T_c$ and bulk volume fraction) observed in Pr-substituted samples are likely due to structural improvements (phase purity) rather than a fundamental change in the magnetic pairing mechanism.

5. Significance

This paper is significant for the field of high-temperature superconductivity in nickelates for several reasons:

1. Validation of Bulk Superconductivity: By showing that the magnetic properties of the Pr-substituted sample are consistent with the non-substituted system, it supports the hypothesis that the enhanced superconductivity in Pr-substituted samples is a true bulk phenomenon resulting from improved structural homogeneity, rather than an artifact of phase separation.
2. Mechanism Insight: The finding that magnetic moments and critical exponents are pressure-invariant suggests that the mechanism driving the SDW transition is robust. This places constraints on theoretical models of superconductivity in these materials, implying that the pairing glue (potentially magnetic fluctuations) is not drastically modified by moderate pressure or Pr substitution.
3. Benchmark for Future Studies: The establishment of a clear baseline for the magnetic behavior of $La_2PrNi_2O_{6.96}$ under pressure serves as a critical reference for distinguishing between intrinsic electronic effects and extrinsic structural defects in future investigations of RP nickelates.

In conclusion, the study confirms that the partial substitution of La with Pr in double-layer RP nickelates primarily serves to stabilize the desired crystal structure and increase the superconducting volume fraction, without fundamentally altering the underlying magnetic order or its response to pressure.