Pressure-Invariant Isotope Effect as Evidence for Electronically Driven Intertwined Order in Pr$_4$Ni$_3$O$_{10}$

Muon-spin rotation measurements on Pr$_4$Ni$_3$O$_{10}$ reveal that the oxygen-isotope effect on the spin-density wave transition remains invariant under hydrostatic pressure, providing evidence that the intertwined order in this trilayer nickelate is predominantly driven by electronic interactions rather than electron-phonon coupling.

The Gist

Imagine a crowded dance floor where two groups of dancers, Spin and Charge, are trying to move in perfect synchronization. In certain special materials called "nickelates," these dancers don't just dance together; they get so intertwined that they form a single, rigid pattern called a Spin-Density Wave (SDW). This pattern is like a frozen wave of order that stops the material from conducting electricity freely.

Scientists have long debated: Who is the choreographer? Is the dance floor (the crystal lattice made of atoms) forcing them to dance this way, or are the dancers themselves (the electrons) deciding to sync up on their own?

To find the answer, the researchers in this paper used a clever trick involving oxygen isotopes.

The "Heavy Shoes" Experiment

Think of the oxygen atoms in the material as the shoes the dancers wear.

• Normal Oxygen ($^{16}$O): Light, nimble shoes.
• Heavy Oxygen ($^{18}$O): Heavy, clunky boots.

The researchers made two identical samples of the material, $Pr_4Ni_3O_{10}$. In one, the dancers wore light shoes. In the other, they wore heavy boots. They didn't change the music (the electrons) or the room layout (the crystal symmetry); they just changed the weight of the shoes.

The Logic: If the dance pattern (the SDW transition) depends heavily on the floor or the shoes (lattice vibrations), the dancers wearing heavy boots should struggle to form the pattern at the same temperature as the dancers in light shoes. The "heavy shoe" group should need more heat to break the pattern, or less heat to form it.

The Result: They found a difference! The dancers with heavy boots formed their synchronized wave at a slightly different temperature (about 1.8 degrees higher) than the light-shoe dancers. This proved that the shoes do matter; the lattice is involved in the dance.

The Pressure Test: Squeezing the Dance Floor

Here is where the story gets interesting. The researchers then put both groups of dancers under pressure (squeezing the room).

In many other materials (like high-temperature superconductors), when you squeeze the room, the dancers get more confused, and the "shoe weight" becomes the most important factor. The heavy shoes would cause a massive delay in the dance compared to the light shoes. The "isotope effect" would get huge.

But in this material, something strange happened:
As they squeezed the room harder and harder, the dancers in both groups slowed down their dance at the exact same rate.

• The light-shoe dancers slowed down by 4.93 degrees per unit of pressure.
• The heavy-shoe dancers slowed down by 4.90 degrees per unit of pressure.

The gap between them (the isotope effect) stayed exactly the same. It didn't grow, it didn't shrink. It was "pressure-invariant."

The Big Conclusion: The Dancers Lead the Floor

This result is like a detective story with a twist.

1. The Clue: The fact that the heavy shoes changed the temperature at all means the floor (lattice) is part of the story.
2. The Twist: The fact that squeezing the room didn't make the shoes matter more means the floor isn't the boss.

If the dance were driven by the floor (lattice vibrations), squeezing the room would have made the floor's influence explode, and the heavy shoes would have caused a massive delay. Since the delay stayed constant, it tells us that the dancers (electrons) are the ones leading the dance.

The electrons are so strongly connected to each other that they decide to form this wave pattern on their own. The floor (the oxygen atoms) just kind of "goes along for the ride," participating but not controlling the show.

Why Does This Matter?

This discovery is a huge deal for understanding superconductivity (electricity flowing with zero resistance).

• In older theories, scientists thought the "floor" (lattice) was the main engine driving these exotic states.
• This paper suggests that in these specific nickelate materials, the "engine" is purely electronic. The electrons are talking to each other so loudly that they create this order without needing the floor to push them.

In a nutshell: The researchers squeezed a material with heavy and light oxygen "shoes" and found that the shoes didn't change how the material reacted to the squeeze. This proves that the material's strange magnetic order is driven by the electrons themselves, not by the vibrations of the atoms they live in. It's a victory for the "electrons lead, lattice follows" theory.

Technical Summary

1. Problem Statement and Scientific Context

The study addresses the fundamental nature of the intertwined charge-density wave (CDW) and spin-density wave (SDW) order in trilayer Ruddlesden-Popper (RP) nickelates, specifically $\text{Pr}_4\text{Ni}_3\text{O}_{10}$.

• The Core Question: Is the mechanism driving the SDW transition primarily electronic (spin correlations) or lattice-driven (electron-phonon coupling)?
• Background: In correlated electron systems, the oxygen-isotope effect (OIE) serves as a probe for lattice involvement. A finite OIE suggests significant electron-phonon coupling. Previous studies on bilayer nickelates ($\text{La}_3\text{Ni}_2\text{O}_7$) showed that the SDW transition has no isotope effect, while the CDW does. Conversely, in trilayer $\text{La}_4\text{Ni}_3\text{O}_{10}$, the CDW and SDW are strongly intertwined (occurring at the same temperature) and exhibit a finite OIE.
• The Hypothesis: If the intertwined state in trilayer nickelates is analogous to cuprate superconductors, where suppressing magnetic order (via doping or pressure) enhances lattice participation, one would expect the isotope effect on the SDW transition to increase as pressure suppresses the magnetic order. The authors aim to test if pressure enhances the lattice contribution to the SDW mechanism in $\text{Pr}_4\text{Ni}_3\text{O}_{10}$.

2. Methodology

The researchers employed a combination of advanced experimental techniques on high-quality single crystals of $\text{Pr}_4\text{Ni}_3\text{O}_{10}$ with two oxygen isotopic compositions: $^{16}\text{O}$ and $^{18}\text{O}$ (enriched to 87%).

• Muon-Spin Rotation/Relaxation ($\mu$SR):
  • Technique: Weak-transverse-field (WTF) $\mu$SR measurements were performed to determine the magnetic volume fraction ($f_m$) and the SDW transition temperature ($T_{\text{SDW}}$).
  • Conditions: Measurements were conducted at ambient pressure and under hydrostatic pressure (up to ~2.26 GPa).
  • Analysis: The time spectra were fitted to extract the paramagnetic volume fraction. The transition temperature $T_{\text{SDW}}$ was determined by fitting the temperature dependence of $f_m$ using a phenomenological Fermi-function form.
• Raman Spectroscopy:
  • Used to characterize oxygen-related vibrational modes and quantify the oxygen participation parameter ($f_O$) in phonon modes. This confirmed the efficiency of the isotope substitution and identified which modes were dominated by oxygen motion.
• Pressure Application:
  • Hydrostatic pressure was applied using a piston-cylinder cell. The analysis accounted for background signals from the pressure cell walls, which stop a significant fraction of muons.

3. Key Results

A. Ambient Pressure Observations

• Finite Isotope Shift: At ambient pressure, a clear isotope shift was observed in the SDW transition temperature:
  • $^{16}T_{\text{SDW}} = 158.04(5)$ K
  • $^{18}T_{\text{SDW}} = 159.81(6)$ K
  • Shift: $\Delta T_{\text{SDW}} = 1.77(8)$ K.
• Interpretation: This confirms that lattice dynamics (oxygen mass) play a role in stabilizing the intertwined CDW/SDW state, consistent with previous findings in $\text{La}_4\text{Ni}_3\text{O}_{10}$.

B. Pressure Dependence

• Suppression of $T_{\text{SDW}}$: Under hydrostatic pressure, $T_{\text{SDW}}$ decreases linearly for both isotopes.
  • Slope for $^{16}\text{O}$: $d(^{16}T_{\text{SDW}})/dp = -4.93(5)$ K/GPa.
  • Slope for $^{18}\text{O}$: $d(^{18}T_{\text{SDW}})/dp = -4.90(7)$ K/GPa.
• Pressure-Invariant Isotope Effect: Crucially, the difference between the transition temperatures of the two isotopes remains constant across the entire pressure range. The isotope shift does not grow as the magnetic order is suppressed.
• Transition Width: The transition width ($\Delta T_{\text{SDW}}$) broadens slightly with pressure, likely due to non-hydrostatic conditions or intrinsic effects, but this broadening is identical for both isotopes.

C. Raman Spectroscopy Findings

• Most phonon modes showed expected downshifts with $^{18}\text{O}$ substitution, confirming effective isotope exchange.
• No anomalous phonon softening or linewidth broadening was observed near the transition, suggesting the lattice does not undergo a critical instability at $T_{\text{SDW}}$.

4. Key Contributions and Conclusions

• Electronic Origin of SDW: The primary finding is that the isotope effect on $T_{\text{SDW}}$ is pressure-invariant. This contradicts the behavior seen in cuprates, where suppressing magnetic order enhances the isotope effect (indicating increased spin-lattice coupling).
• Mechanism of Intertwined Order: The results suggest that the intertwined CDW/SDW instability in trilayer RP nickelates is predominantly electronic in origin, driven by strong spin correlations. While the lattice participates (evidenced by the finite, non-zero isotope shift at ambient pressure), it plays a secondary, cooperative role rather than being the primary driver.
• Distinction from Cuprates: The study highlights a fundamental difference between trilayer nickelates and cuprate superconductors. In cuprates, the suppression of antiferromagnetism leads to a stronger lattice contribution to the ordering mechanism. In $\text{Pr}_4\text{Ni}_3\text{O}_{10}$, the lattice contribution remains static even as the system is tuned toward the superconducting regime by pressure.
• Consistency with Other Probes: These findings align with recent inelastic X-ray scattering results (Jia et al.) which found no phonon softening at the ordering wave vector in trilayer nickelates, further supporting the dominance of electronic susceptibility over lattice coupling.

5. Significance

This work provides critical constraints on microscopic models of density-wave formation and the mechanism of superconductivity in Ruddlesden-Popper nickelates. By demonstrating that the lattice does not become more dominant as the magnetic order weakens, the paper rules out models where superconductivity in these materials emerges from a strong enhancement of electron-phonon coupling driven by the suppression of magnetism. Instead, it points toward a scenario where electronic correlations are the primary engine for both the intertwined order and the subsequent emergence of superconductivity in trilayer nickelates.