Effect of Pressure and Oxygen-Isotope Substitution on Density-Wave Transitions in La$_4$Ni$_3$O$_{10}$

This study utilizes muon-spin rotation and resistivity measurements to reveal that in the trilayer nickelate La$_4$Ni$_3$O$_{10}$, pressure suppresses intertwined spin- and charge-density-wave transitions at a uniform rate while oxygen-isotope substitution selectively enhances the charge-density-wave transition, highlighting a distinct coupling between charge and spin orders that differs from bilayer nickelates.

The Gist

Imagine a bustling city made 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, specifically a three-story building made of lanthanum, nickel, and oxygen called La₄Ni₃O₁₀, the electrons get stuck in traffic jams. They organize themselves into patterns called waves.

This paper is like a detective story where scientists use three different "superpowers" (pressure, changing the weight of the atoms, and tiny magnetic probes) to figure out how these traffic jams form, how they change, and how they might eventually clear the way for a superhighway of electricity (superconductivity).

Here is the breakdown of the investigation:

1. The Setting: A Three-Story Building

Think of the material as a three-story apartment building.

• The Outer Floors (Layers 1 & 3): These are the "loud" floors where the residents (electrons) are very active and form big, organized patterns.
• The Middle Floor (Layer 2): This is the "quiet" floor. The residents here are much calmer and have smaller movements.

In this city, two types of traffic jams happen simultaneously:

• Charge Density Waves (CDW): The electrons arrange themselves in a pattern based on their electric charge (like people standing in a specific formation).
• Spin Density Waves (SDW): The electrons arrange themselves based on their magnetic "spin" (like people all facing North or South).

Usually, scientists thought these two jams were separate events. But in this material, they happen at the exact same time, as if the charge pattern and the magnetic pattern are holding hands and dancing together.

2. The First Clue: The "Freeze" at 132 K

At room temperature, the city is chaotic. But as the scientists cool the building down to about -141°C (132 Kelvin), a sudden "freeze" happens.

• The Sudden Stop: Unlike a slow freeze where things gradually get stiff, this happens all at once. It's like a sudden power outage where everyone freezes instantly. This suggests a "first-order" transition—a dramatic, abrupt change rather than a gentle slide.
• The Two Transitions:
  1. The Big Freeze (132 K): The main traffic jam forms.
  2. The Tilt (80–90 K): As it gets even colder, the residents on the outer floors don't just freeze; they tilt their heads. Their magnetic "spins" which were lying flat on the floor suddenly stand up a bit, pointing toward the ceiling (the c-axis).

3. The Second Clue: Squeezing the Building (Pressure)

The scientists then put the building under a hydraulic press, squeezing it tighter and tighter.

• The Result: In many materials, squeezing makes things more ordered. But here, squeezing destroys the traffic jams.
• The Key Difference: The scientists compared this three-story building to a similar two-story building (La₃Ni₂O₇).
  • In the two-story building, squeezing the building made the charge jam and the magnetic jam split apart and move in opposite directions.
  • In this three-story building, squeezing them together makes both jams disappear at the exact same rate. They are so tightly linked that if you push one, the other collapses with it.
• The Mystery: As the jams disappear, the material starts showing signs of becoming a superconductor (electricity flowing with zero resistance). This suggests that crushing the traffic jams might be the secret recipe to unlocking superconductivity.

4. The Third Clue: The Heavy Backpack (Oxygen Isotope Swap)

To see if the "floorboards" (the oxygen atoms) were helping the electrons organize, the scientists swapped the light oxygen atoms (¹⁶O) with heavy oxygen atoms (¹⁸O). Imagine asking the residents to put on heavy backpacks.

• The Effect on the Jams: When the oxygen atoms got heavier, the temperature at which the traffic jams formed went up. It's like the heavy backpacks made the residents want to organize themselves sooner (at a warmer temperature) to stay stable.
• The Twist: This heavy backpack effect happened to both the charge jam and the magnetic jam only when they were linked together.
• The Exception: When the residents on the middle floor tilted their heads (the second transition at 80 K), the heavy backpacks had no effect. This proves that this specific "tilting" behavior is independent of the oxygen atoms' weight and is driven purely by the electrons themselves.

The Big Picture: Why Does This Matter?

This research is like finding a missing piece of a puzzle for high-temperature superconductors.

• The Connection: It confirms that in this material, magnetism and electricity are deeply intertwined. You can't have one without the other.
• The Path to Superconductivity: The study suggests that to get superconductivity, you need to suppress these "traffic jams." The fact that pressure kills both jams simultaneously in the three-story building (unlike the two-story one) tells us that the coupling between charge and spin is the key.
• The Analogy: Imagine trying to get a crowd to dance (superconductivity). If they are stuck in a rigid formation (the density waves), they can't dance. This paper shows that in this specific material, you have to break the formation of both the charge and the spin at the same time to let the electrons start dancing freely.

In short: The scientists found that in this three-layer nickelate, the magnetic and electric orders are best friends that never separate. Squeezing the material breaks their bond, clearing the path for electricity to flow without resistance, while changing the weight of the atoms proves that the "floor" (lattice) plays a crucial role in keeping these orders together.

Technical Summary

Here is a detailed technical summary of the paper "Effect of Pressure and Oxygen-Isotope Substitution on Density-Wave Transitions in La$_4$Ni$_3$O$_{10}$".

1. Problem Statement

The emergence of high-temperature superconductivity in Ruddlesden-Popper (RP) nickelates under high pressure has sparked intense interest in understanding the interplay between magnetism, charge order, and superconductivity. Specifically, the three-layer RP nickelate La$_4$Ni$_3$O$_{10}$ exhibits intertwined Spin-Density Wave (SDW) and Charge-Density Wave (CDW) orders at ambient pressure.

Key open questions addressed in this study include:

• What is the microscopic nature of the magnetic structure and the sequence of phase transitions in La$_4$Ni$_3$O$_{10}$?
• How do external pressure and lattice vibrations (phonons) influence the stability and coupling of SDW and CDW orders?
• How does the pressure response of the three-layer system (La$_4$Ni$_3$O$_{10}$) compare to the two-layer system (La$_3$Ni$_2$O$_7$), where pressure separates SDW and CDW transitions?
• What is the role of electron-phonon coupling in stabilizing these density-wave states, as probed by oxygen-isotope substitution?

2. Methodology

The authors employed a synergistic approach combining complementary experimental techniques on high-quality polycrystalline samples of pristine, $^{16}$O-substituted, and $^{18}$O-substituted La$_4$Ni$_3$O$_{10}$.

• Sample Characterization:

  • X-ray Diffraction (XRD) & Thermogravimetric Analysis (TGA): Confirmed monoclinic $P2_1/a$ (nearly orthorhombic $Bmab$) structure and near-stoichiometric oxygen content ($\delta \approx 0$).
  • Raman Spectroscopy: Verified successful oxygen isotope exchange ($^{16}$O $\to$ $^{18}$O) by observing systematic downshifts in oxygen-dominated phonon modes.
  • Resistivity Measurements: Performed under ambient and hydrostatic pressure (up to ~2.1 GPa) to detect CDW transitions via anomalies in $R(T)$ and its derivative.

• Muon-Spin Rotation/Relaxation ($\mu$SR):

  • Zero-Field (ZF) and Weak Transverse Field (WTF): Conducted at ambient pressure and under hydrostatic pressure (up to ~2.3 GPa).
  • Function: Probed local internal magnetic fields to identify magnetic ordering temperatures ($T_{SDW}$, $T^*$), determine magnetic volume fractions, and analyze the nature of magnetic order (commensurate vs. incommensurate).
  • Pressure Cells: Used non-magnetic MP35N alloy cells to minimize background signals.

• Theoretical Calculations:

  • DFT+$\mu$: Used to identify muon stopping sites within the crystal lattice.
  • Dipolar Field Simulations: Calculated expected internal field distributions based on proposed magnetic structures to interpret $\mu$SR spectra.

3. Key Contributions and Results

A. Ambient Pressure Magnetic Structure

• Two Magnetic Transitions: The study identified two distinct transitions:
  1. $T_{SDW} \approx 132$ K: The onset of incommensurate SDW order. The transition is first-order-like, evidenced by the abrupt appearance of internal fields and a sharp jump in the magnetic volume fraction.
  2. $T^* \approx 80-90$ K: A spin-reorientation transition where the magnetic field distribution changes from a 2-peak to a 5-peak structure.
• Magnetic Configuration: Dipolar field calculations suggest:
  • Antiferromagnetically coupled SDW order on the outer two Ni layers with large moments.
  • A smaller magnetic moment on the inner Ni layer.
  • Spin Orientation: Above $T^*$, moments lie primarily in the $ab$-plane. Below $T^*$, moments undergo a subtle distortion, acquiring a $c$-axis component.
• Intertwined Orders: The SDW transition coincides with the CDW transition ($T_{SDW} = T_{CDW}$), indicating strong coupling between spin and charge degrees of freedom.

B. Effect of Hydrostatic Pressure

• Suppression of Transitions: Unlike the two-layer La$_3$Ni$_2$O$_7$ (where pressure separates SDW and CDW), pressure in La$_4$Ni$_3$O$_{10}$ suppresses all transition temperatures ($T_{SDW}$, $T^*$, and $T_{CDW}$) at a nearly uniform rate of $\approx -13$ K/GPa.
• Magnetic Fluctuations: The onset of magnetic ordering ($T_{m,onset}$, detected by $\mu$SR as a precursor) decreases more slowly than $T_{SDW}$. This suggests that pressure enhances magnetic fluctuations, which act as precursors to static order.
• Coupling Mechanism: The simultaneous suppression implies that the SDW order is highly dependent on the stability of the CDW order. As pressure suppresses the CDW, the SDW is destabilized.

C. Oxygen-Isotope Effect (OIE)

• Intertwined Regime ($T_{SDW} \approx T_{CDW}$): In the $^{18}$O-substituted sample, both $T_{CDW}$ and $T_{SDW}$ shift to higher temperatures ($\Delta T \approx +2.1$ K). The isotope exponents are nearly identical ($\alpha \approx -0.15$), confirming that lattice vibrations (phonons) play a crucial role in stabilizing the coupled SDW-CDW state.
• Decoupled Regime ($T^*$): The spin-reorientation transition at $T^*$, where the SDW evolves independently of the CDW, shows negligible isotope effect ($\Delta T \approx 0.5$ K). This indicates that the $T^*$ transition is driven primarily by electronic correlations rather than electron-phonon coupling.

4. Significance and Implications

• Distinct Pressure Response: The study highlights a fundamental difference between two-layer (La$_3$Ni$_2$O$_7$) and three-layer (La$_4$Ni$_3$O$_{10}$) nickelates. In the two-layer system, pressure decouples SDW and CDW, allowing SDW to strengthen. In the three-layer system, they remain tightly coupled and mutually suppressed.
• Mechanism for Superconductivity: The results suggest that the suppression of the CDW order by pressure is a critical mechanism that may drive the emergence of high-temperature superconductivity in RP nickelates. Since the SDW is tied to the CDW in La$_4$Ni$_3$O$_{10}$, suppressing the CDW inevitably suppresses the SDW, potentially clearing the path for superconductivity.
• Role of Lattice: The significant isotope effect on the coupled transitions but not on the decoupled spin-reorientation transition provides direct evidence that electron-phonon coupling is essential for the stability of the intertwined density-wave state in La$_4$Ni$_3$O$_{10}$.
• Phase Diagram: The constructed $p-T$ phase diagram provides a comprehensive view of the competition between density-wave orders and the potential superconducting dome, offering a roadmap for future theoretical and experimental investigations into correlated electron systems.

In conclusion, this paper establishes that the magnetic and charge orders in La$_4$Ni$_3$O$_{10}$ are inextricably linked, with their stability heavily reliant on lattice dynamics. This contrasts sharply with the behavior of the two-layer counterpart, suggesting that layer-dependent electronic structures dictate the pathway to superconductivity in nickelates.