Intertwined Charge and Spin Density Waves in Trilayer Nickelate La$_4$Ni$_3$O$_{10}$ Revealed by $^{139}$La NQR

Using $^{139}$La NQR, this study reveals that La$_4$Ni$_3$O$_{10}$ undergoes a first-order-like phase transition at approximately 133 K driven by an intricate interplay between incommensurate charge and spin density waves, providing critical microscopic insights into the relationship between density wave orders and superconductivity in nickelates.

The Gist

The Big Picture: A Dance of Electrons

Imagine a crowded dance floor where the dancers are electrons. In most materials, these dancers move around somewhat randomly. But in special materials called nickelates (specifically one called La4Ni3O10), something fascinating happens when the temperature drops.

The electrons stop dancing randomly and start organizing themselves into patterns. Sometimes they line up in waves of charge (where they bunch up in some spots and leave gaps in others). Other times, they line up in waves of spin (where their magnetic "directions" align in a specific rhythm).

Scientists call these patterns Density Waves (DW). The big question this paper answers is: How do these two types of waves behave, and do they dance together or separately?

The Tool: Listening to the "Heartbeat"

To figure this out, the researchers used a technique called NQR (Nuclear Quadrupole Resonance).

• The Analogy: Imagine trying to hear a specific instrument in a noisy orchestra. The researchers tuned their radio to listen specifically to the "heartbeat" of the Lanthanum (La) atoms inside the material.
• The Setup: They tested two types of samples:
  1. Polycrystalline: Like a pile of broken puzzle pieces glued together (many tiny crystals with different orientations).
  2. Single-Crystal: Like one perfect, giant crystal (all the atoms are perfectly aligned).
• Why it matters: The single-crystal sample is like a high-definition photo, while the polycrystalline sample is a blurry snapshot. The high-quality sample revealed details the blurry one missed.

The Discovery: A Sudden "Snap"

As they cooled the material down, they watched what happened to the Lanthanum atoms' "heartbeat" around 133 Kelvin (about -140°C).

1. The "Snap" (First-Order Transition):
   In the perfect single-crystal sample, the signal didn't change slowly. It snapped instantly.

   • Analogy: Think of water freezing into ice. Usually, it takes time to freeze, but here, it's like the water instantly turning into a block of ice the moment it hit the freezing point. This suggests a very sharp, sudden change in the material's state.
   • Note: In the "blurry" polycrystalline sample, this snap looked like a slow slide because the tiny crystals weren't all freezing at the exact same moment.

2. The "Messy" Pattern (Incommensurate Waves):
   When the transition happened, the signal lines got very wide and fuzzy.

   • Analogy: Imagine a marching band. If they march in perfect lockstep (commensurate), you see a clean, sharp line. If they are marching to slightly different rhythms that don't quite match the size of the stadium (incommensurate), the line looks blurry and messy.
   • The Finding: The waves in this material are "messy" (incommensurate). They don't fit perfectly into the crystal grid.

3. The "Double Trouble" (Intertwined Charge and Spin):
   The researchers noticed that the signal changed in a way that couldn't be explained by just charge waves OR just spin waves. It needed both.

   • The Analogy: It's like a couple dancing a tango. You can't explain the movement by looking at just the man's feet (charge) or just the woman's feet (spin). They are moving together in a complex, intertwined way.
   • The Conclusion: The material has both charge density waves and spin density waves happening at the same time, and they are influencing each other.

The "Heat" of the Moment (Spin Fluctuations)

The researchers also measured how fast the atoms relaxed after being excited (called spin-lattice relaxation).

• The Finding: Right at the moment the "snap" happened (133 K), the atoms got very "excited" or "hot" in terms of magnetic fluctuations.
• The Paradox: Usually, if a change happens suddenly (like a first-order snap), the excitement (fluctuations) should be low. But here, the excitement was huge.
• The Explanation: The paper suggests that the Charge Waves caused the sudden snap, but the Spin Waves were causing the huge excitement. They are so tightly linked that even though the charge changed abruptly, the spins were still raging with activity.

Why This Matters

This material (La4Ni3O10) is a cousin to other nickelates that become superconductors (conduct electricity with zero resistance) when squeezed under high pressure.

• The Takeaway: Before these materials can become superconductors, they have to deal with these "Density Waves." This paper shows us that the waves are complex, messy, and intertwined.
• The Metaphor: If you want to understand how a car drives (superconductivity), you first need to understand how the engine parts (density waves) are moving and interacting. This paper gives us a clear map of how those engine parts are moving in this specific nickelate.

Summary

• What they did: Listened to the atomic "heartbeat" of a nickelate crystal as it cooled down.
• What they found: At 133 K, the material suddenly changed state.
• The Nature of the Change: It was a sharp "snap" (first-order) caused by charge waves, but it involved messy, non-matching (incommensurate) waves of both charge and spin.
• The Key Insight: Charge and spin are dancing together in a complex, intertwined tango, creating a state that competes with superconductivity.

Technical Summary

Technical Summary: Intertwined Charge and Spin Density Waves in Trilayer Nickelate La$_4$Ni$_3$O$_{10}$ Revealed by $^{139}$La NQR

Problem Statement
The recent discovery of superconductivity in pressurized Ruddlesden-Popper (RP) phase nickelates, specifically La$_3$Ni$_2$O$_7$ and La$_4$Ni$_3$O$_{10}$, has renewed interest in the microscopic mechanisms governing high-$T_c$ superconductivity. In these materials, superconductivity emerges upon the suppression of density wave (DW) orders. While previous studies using X-ray diffraction, neutron scattering, and $\mu$SR have identified the presence of charge density wave (CDW) and spin density wave (SDW) orders in La$_4$Ni$_3$O$_{10}$ near 140 K, several critical questions remain unresolved. Specifically, the nature of the phase transition (first-order vs. second-order), the commensurability of the DW orders, and the precise interplay between CDW and SDW fluctuations are not fully understood. Furthermore, discrepancies exist between different measurement techniques regarding the order of the transition and the existence of multiple SDW phases, necessitating a local probe to clarify the microscopic mechanisms.

Methodology
This study employs $^{139}$La nuclear quadrupole resonance (NQR) spectroscopy to investigate the local electronic and magnetic environment in La$_4$Ni$_3$O$_{10}$ under ambient pressure. The research utilizes both high-quality single-crystal and polycrystalline samples synthesized via flux growth and sol-gel methods, respectively.

• Spectroscopy: NQR spectra were acquired by sweeping frequencies and integrating spin-echo intensities in zero external magnetic field. The study focuses on the La(2) sites (located outside the NiO$_2$ trilayers), which exhibit distinct quadrupole frequencies compared to La(1) sites.
• Relaxation Measurements: The spin-lattice relaxation rate ($1/T_1$) was measured using the saturation-recovery method to probe spin fluctuations. The data was fitted using a stretched-exponential formula to extract $T_1$ and the stretching factor $\beta$.
• Simulation: Theoretical simulations were performed to model the NQR line shapes, incorporating contributions from both charge modulation (affecting the electric field gradient) and internal magnetic fields (arising from SDW) to distinguish between commensurate and incommensurate orders.

Key Results

1. Nature of the Phase Transition:

   • In single-crystal samples, the NQR linewidth and frequency of the La(2) site exhibit an abrupt change at $T_{DW} \approx 133$ K. This sudden broadening and frequency shift, particularly in the $\pm 5/2 \leftrightarrow \pm 7/2$ transition, provide compelling evidence for a first-order-like phase transition.
   • In contrast, polycrystalline samples show a more gradual change below $T_{DW}$, attributed to sample disorder and oxygen vacancy distributions.
   • This finding contrasts with previous NQR results in bilayer La$_3$Ni$_2$O$_7$, which indicated a second-order transition, suggesting distinct behaviors between the two nickelate systems.

2. Incommensurate and Intertwined Orders:

   • The NQR spectra in the DW state show significant broadening without distinct line splitting. This absence of splitting indicates that the DW order is incommensurate, differing from the commensurate order observed in La$_3$Ni$_2$O$_7$.
   • Analysis of the line broadening and frequency shifts reveals that the changes cannot be explained solely by charge modulation (quadrupole broadening) or solely by an internal magnetic field aligned along the $c$-axis.
   • The data is best explained by a model where the internal magnetic field ($B_{int}$) at the La(2) site is perpendicular to the $c$-axis. This implies that the Ni magnetic moments on the outer layers are aligned along the $c$-axis, consistent with specific theoretical proposals.
   • The simultaneous observation of spectral broadening (attributed to CDW) and significant line shifts/fluctuations (attributed to SDW) confirms the intertwined nature of the CDW and SDW orders.

3. Spin Fluctuations and Relaxation:

   • The spin-lattice relaxation rate divided by temperature ($1/T_1T$) exhibits a strong enhancement at $T_{DW}$ for single-crystal samples.
   • Despite the first-order nature of the transition (which typically suppresses critical fluctuations), the strong enhancement in $1/T_1T$ indicates the presence of strong spin fluctuations originating from the SDW order above the transition.
   • The stretching factor $\beta$ decreases rapidly below $T_{DW}$, consistent with the development of spin and charge modulations.
   • At low temperatures, $1/T_1T$ in La$_4$Ni$_3$O$_{10}$ retains a substantial residual value (~20% of the high-temperature limit), suggesting that a significant portion of the Fermi surface remains ungapped. This contrasts with La$_3$Ni$_2$O$_7$, where $1/T_1T$ becomes negligible, implying a fully gapped Fermi surface.

Significance and Claims
The authors claim that these findings elucidate the microscopic mechanisms of the DW state in La$_4$Ni$_3$O$_{10}$. Specifically, the study establishes that:

• The DW transition in La$_4$Ni$_3$O$_{10}$ is a first-order-like transition driven primarily by the CDW, while the strong spin fluctuations are driven by the SDW.
• The DW orders are incommensurate and intertwined, likely arising from Fermi surface nesting that prevents complete gapping of the Fermi surface.
• The orientation of Ni magnetic moments on the outer layers is aligned along the $c$-axis, generating an internal magnetic field perpendicular to $c$ at the La(2) sites.

The paper concludes that these results provide a crucial framework for understanding the competition and interplay between DW and superconducting phases in nickelate superconductors, highlighting the complexity of the electronic orders in the trilayer system compared to its bilayer counterpart.