Multiple Magnetic Transitions in the Trilayer Nickelate Pr$_4$Ni$_3$O$_{10}$ Revealed by Muon-Spin Rotation

A muon-spin rotation study of the trilayer nickelate Pr$_4$Ni$_3$O$_{10}$ reveals three distinct magnetic transitions at ambient pressure, including a weakly first-order spin-density-wave ordering at 158 K, and demonstrates that hydrostatic pressure linearly suppresses both the ordering temperature and the magnetic moment, indicating a gradual weakening of the magnetic instability.

The Gist

Imagine a microscopic city built out of atoms, where the residents are electrons. In most materials, these electrons flow freely like traffic on a highway. But in a special family of materials called nickelates, the electrons get stuck in patterns, forming "traffic jams" that create magnetism.

Scientists have been trying to figure out how to clear these jams to make the electrons flow without resistance—a state called superconductivity (which allows electricity to move with zero energy loss). To do this, they need to understand the "traffic rules" of these materials before they become superconductors.

This paper is a detailed investigation of a specific nickelate material called Pr4Ni3O10 (a three-layer sandwich of nickel and oxygen). The researchers used a super-sensitive "magnetic camera" called Muon-Spin Rotation (µSR) to take snapshots of the electrons' behavior. Think of muons as tiny, invisible compass needles that get stuck in the material and spin around, telling the scientists exactly what the magnetic fields look like inside.

Here is what they discovered, broken down into simple concepts:

1. The Three-Act Play of Magnetism

When the scientists cooled this material down, they didn't just see one change. They found three distinct "acts" in the story of its magnetism:

• Act I: The Big Freeze (158 K). As the material cooled, the electrons suddenly organized themselves into a rigid, wavy pattern called a Spin-Density Wave (SDW). Imagine a crowd of people suddenly deciding to march in perfect, synchronized waves. This happened at about -115°C. The transition was very sharp, like a light switch flipping on.
• Act II: The Rearrangement (90–100 K). As it got colder, the magnetic pattern didn't disappear, but it subtly shifted. It's like the marching band changing their formation from a straight line to a zigzag. The overall "magnetic volume" didn't change much, but the internal structure got tweaked.
• Act III: The Praseodymium Takeover (25–27 K). At very low temperatures, a different set of atoms (Praseodymium) woke up and joined the dance. This caused a major reconstruction of the magnetic landscape, like a new director stepping in and completely changing the choreography.

2. The "Hysteresis" Mystery

One of the most interesting findings was about the first transition (Act I). The researchers found a tiny "lag" or hysteresis.

• The Analogy: Imagine pushing a heavy door open. It takes a little extra force to get it moving (cooling down), but once it's open, it stays open until you push it back with a slightly different force (warming up).
• What it means: This "lag" suggests the transition isn't a smooth, gentle slide; it's a bit "jumpy" or first-order. The material hesitates before committing to the magnetic state, which tells scientists that the forces holding it together are very delicate and complex.

3. Squeezing the Sponge (Pressure Experiments)

The researchers didn't just watch the material; they squeezed it. They applied hydrostatic pressure (like putting the material in a giant, invisible hydraulic press) up to 2.2 GPa (about 20,000 times atmospheric pressure).

• The Result: Squeezing the material acted like a "magnetism dimmer switch."
  • The temperature at which the magnetic waves started dropped linearly.
  • The strength of the magnetic waves themselves got weaker.
• The Metaphor: Think of the magnetic order as a stiff, rigid gel. When you squeeze the gel, it becomes softer and less organized. The scientists found that pressure makes the magnetic "jams" less stable, which is a crucial step because superconductivity usually appears only after these magnetic jams are suppressed.

Why Does This Matter?

This paper is like a map for a treasure hunt.

• The Treasure: Superconductivity (electricity with zero loss).
• The Obstacle: Magnetic order (the electron traffic jams).
• The Map: This study shows exactly how the magnetic order behaves, how it changes with temperature, and how it crumbles under pressure.

By understanding that the magnetic state in this material is "weakly first-order" (a bit jumpy) and that it weakens predictably under pressure, scientists can better predict where and when to look for superconductivity in these materials. It confirms that to get the "super" electricity, you first have to gently break the "magnetic" order, and this material is a perfect laboratory for learning how to do that.

In short: The scientists used tiny magnetic compasses to watch a nickel-based material change its mind three times as it got cold, and then watched it get "scared" and lose its magnetic strength when squeezed. This helps us understand the secret recipe for turning these materials into superconductors.

Technical Summary

1. Problem Statement

The discovery of pressure-induced superconductivity in layered Ruddlesden–Popper (RP) nickelates has sparked intense interest in understanding the interplay between magnetism, charge-density waves (CDW), and spin-density waves (SDW). While superconductivity in trilayer nickelates (e.g., La$_4$Ni$_3$O$_{10}$ and Pr$_4$Ni$_3$O$_{10}$) emerges only after the suppression of intertwined density-wave orders, the microscopic details of these magnetic states remain unresolved.

Specifically, for Pr$_4$Ni$_3$O$_{10}$, three critical questions were open:

1. What is the precise sequence of magnetic transitions upon cooling?
2. What is the nature of the primary high-temperature SDW transition (continuous vs. first-order)?
3. How robust is the static Ni magnetic order under hydrostatic pressure in the regime preceding superconductivity?

Addressing these is essential for establishing a quantitative magnetic phase diagram and understanding how the suppression of magnetism connects to the emergence of superconductivity.

2. Methodology

The authors employed Muon-Spin Rotation/Relaxation ($\mu$SR), a highly sensitive local probe for detecting internal magnetic fields, to investigate polycrystalline Pr$_4$Ni$_3$O$_{10}$.

• Experimental Conditions:
  • Ambient Pressure: Measurements performed at the GPS spectrometer (low background).
  • High Pressure: Measurements performed up to 2.2 GPa using double-wall piston-cylinder clamp cells (MP35N alloy) at the GPD spectrometer.
• Measurement Modes:
  • Zero-Field (ZF): To detect static internal magnetic fields and characterize magnetic order parameters.
  • Weak Transverse-Field (WTF): To determine the magnetic volume fraction and detect thermal hysteresis.
• Sample Characterization:
  • Synthesized via a modified sol–gel Pechini method.
  • Phase purity confirmed by X-ray and Neutron Powder Diffraction (NPD), confirming a monoclinic structure ($P2_1/a$).
  • Oxygen content determined via thermogravimetric analysis (stoichiometry Pr$_4$Ni$_3$O$_{10.08(2)}$).
• Data Analysis:
  • ZF spectra were analyzed using a sum of cosine oscillations with exponential relaxation to model complex, incommensurate magnetic field distributions.
  • WTF spectra were used to extract the non-magnetic volume fraction ($1-f_m$) to track the onset of magnetic order.

3. Key Contributions and Results

A. Identification of Three Distinct Magnetic Transitions

At ambient pressure, the study revealed a complex magnetic phase diagram with three transitions:

1. High-Temperature SDW Onset ($T_{SDW} \approx 158$ K):

   • Marks the onset of static, incommensurate spin-density-wave order.
   • Characterized by a sharp transition width of $\Delta T_{SDW} = 0.65(4)$ K.
   • Nature of Transition: WTF measurements revealed a finite thermal hysteresis of $0.27(6)$ K ($T_{SDW}^{\text{warming}} > T_{SDW}^{\text{cooling}}$). This indicates weakly first-order-like behavior (metastability), likely driven by the coupling between spin and charge order parameters.
   • The critical exponent $\beta \approx 0.138$ suggests quasi-two-dimensional magnetic correlations, consistent with the trilayer structure.

2. Intermediate Transition ($T^* \approx 90\text{--}100$ K):

   • Involves a subtle reconstruction of the magnetic structure.
   • ZF-$\mu$SR shows the emergence of new internal field components and splitting of existing peaks, but the total magnetic volume fraction remains unchanged. This suggests a reorientation or modification within the existing SDW phase rather than a new phase.

3. Low-Temperature Transition ($T_{SDW}^{Pr} \approx 25\text{--}27$ K):

   • Accompanied by a pronounced reconstruction of the magnetic field distribution.
   • Attributed to the ordering of the Pr sublattice (Pr 4f moments), which couples to the Ni sublattice, enhancing 3D magnetic coherence.

B. Pressure Evolution of Magnetic Order

Under hydrostatic pressure (up to 2.2 GPa):

• Suppression of $T_{SDW}$: The transition temperature decreases linearly with a rate of $dT_{SDW}/dp = -4.9(1)$ K/GPa.
  • Comparison: This suppression is weaker than in La$_4$Ni$_3$O$_{10}$ ($\approx -13$ K/GPa), consistent with Pr$_4$Ni$_3$O$_{10}$ requiring higher pressures ($\sim$20–25 GPa) to achieve superconductivity compared to La-based analogs.
• Reduction of Magnetic Moment: The ordered Ni magnetic moment ($M$) is gradually reduced.
  • The intrinsic pressure dependence of the order parameter amplitude is $d \ln M/dp = -2.0(5) \times 10^{-2}$ GPa$^{-1}$ (approx. 2% reduction per GPa).
  • This confirms that pressure weakens both the thermodynamic stability of the SDW phase and the magnitude of the magnetic order itself.

4. Significance

• Microscopic Phase Diagram: The study provides the first quantitative microscopic magnetic phase diagram for Pr$_4$Ni$_3$O$_{10}$ in the normal state, establishing the baseline from which superconductivity eventually emerges.
• Nature of SDW Instability: The identification of weakly first-order behavior and the specific critical exponent supports the theoretical view that SDW order in these trilayer systems is intimately intertwined with CDW order, leading to complex phase transitions.
• Dimensionality Effects: The results highlight the role of structural anisotropy (strong intra-trilayer coupling vs. weak inter-trilayer coupling) in promoting quasi-2D magnetic behavior.
• Pressure Mechanism: The findings demonstrate that the suppression of magnetism under pressure is intrinsic, affecting both the transition temperature and the order parameter amplitude. This gradual weakening of the SDW instability is a prerequisite for the emergence of superconductivity in trilayer RP nickelates.
• Generalizability: The observation of multiple magnetic transitions (SDW onset, intermediate reconstruction, and rare-earth ordering) suggests a generic feature of trilayer and structurally related RP nickelates.

In conclusion, this work utilizes $\mu$SR to resolve the complex magnetic landscape of Pr$_4$Ni$_3$O$_{10}$, revealing a delicate balance between competing ordered states and providing critical insights into the mechanisms governing the transition from magnetism to superconductivity in high-pressure nickelates.