Microscopic evidence for Fulde-Ferrel-Larkin-Ovchinnikov state and multiband effects in KFe$_2$As$_2$

This study provides microscopic evidence for the Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state in the multiband superconductor KFe$_2$As$_2$ through $^{75}$As NMR measurements, revealing a distinct low-temperature phase boundary driven by multiband effects.

The Gist

Imagine a dance floor where pairs of dancers (called Cooper pairs) move in perfect unison. In a normal superconductor, these pairs glide across the floor with zero momentum, creating a smooth, uniform flow. But what happens if you turn up the magnetic "volume" on this dance floor? Eventually, the magnetic force tries to tear the dancers apart.

In most cases, the dance stops, and the material loses its superconducting magic. However, physicists predicted a special, exotic state called the FFLO state (named after Fulde, Ferrell, Larkin, and Ovchinnikov). In this state, instead of giving up, the dancers adapt. They form pairs that move with a specific, non-zero momentum, creating a pattern where the dance floor is no longer uniform. Instead, it becomes a patchwork of "superconducting" zones and "normal" zones, like a striped rug or a layered cake.

This paper reports on a successful hunt for this exotic "striped" state in a specific material called KFe₂As₂ (a type of iron-based superconductor). Here is the breakdown of their findings using simple analogies:

1. The Challenge: Finding a Ghost

The FFLO state is notoriously difficult to find. It's like trying to spot a specific type of cloud formation that only happens when the wind is just right and the air is perfectly clean.

• The Problem: If the material has too many impurities (like dust on the dance floor), the pattern gets ruined.
• The Solution: The researchers used a very pure crystal of KFe₂As₂. Think of this as a pristine, high-end dance floor with almost no dust. They also used a powerful tool called NMR (Nuclear Magnetic Resonance), which acts like a high-resolution camera that can "see" the magnetic spins of the atoms inside the material.

2. The Evidence: Two Clues in the Same Spot

To prove the "striped" FFLO state exists, the team looked for two specific things happening at the same time, in the same cold, high-magnetic-field region:

• Clue A: The "Fuzzy" Line (Spin Smecticity)
  Normally, the NMR signal looks like a sharp, clear line. In the FFLO state, because the superconducting and normal regions are alternating like stripes, the signal gets "smeared" or broadened.

  • Analogy: Imagine looking at a sharp pencil line. If you shake the paper back and forth rapidly, the line looks blurry. The researchers saw this "blur" (an increase in the "second moment" of the spectrum) only at very low temperatures and high magnetic fields. This blur indicates the material has developed that striped, layered structure.

• Clue B: The "Hotspot" Peak (Andreev Bound States)
  Where the "stripes" of the superconductor meet the "stripes" of the normal metal, special energy states form. These act like little traps for particles, causing the material to relax energy faster.

  • Analogy: Imagine a highway where traffic usually flows smoothly. But at the border between two different road types, cars get stuck and honk (release energy). The researchers measured a sudden spike (a peak) in how fast the atoms relaxed their energy.
  • The Smoking Gun: Crucially, they found that the "blurry line" (Clue A) and the "energy spike" (Clue B) appeared at the exact same temperature and magnetic field. This simultaneous occurrence is strong proof that the FFLO state is real.

3. The Twist: Why This Material is Special

The paper highlights two unique features of this discovery that differ from what we see in other materials:

• The "Multiband" Effect:
  Most superconductors are like a single-lane highway. KFe₂As₂ is like a multi-lane highway where different lanes (called "bands") have different rules. Some lanes are wide and open (isotropic), while others are narrow and winding (anisotropic).

  • The Result: The researchers found that the FFLO state in this material is stabilized by the interaction between these different lanes. Specifically, the "winding" lanes help the pattern form, while the "wide" lanes might actually make it harder to form. This complex interaction creates a unique boundary line between the normal superconducting state and the FFLO state.

• The "Low Temperature" Surprise:
  In other materials where FFLO has been suspected, this state usually appears at a relatively high temperature (relative to the material's limit). Here, the FFLO state only appears at a very low temperature (about 20% of the material's maximum superconducting temperature).

  • The Reason: The researchers suggest this is because the "magnetic wind" (orbital effects) in this material is strong enough to push the FFLO state down to lower temperatures, and the specific mix of the multi-lane highway (multiband effects) plays a role in keeping it stable only in that narrow, cold window.

4. The "Angle" Test

To be absolutely sure they weren't just seeing a different phenomenon (like a vortex state, which is another type of magnetic pattern), they tilted the material slightly.

• The Test: They rotated the crystal by a tiny amount (1.7 degrees).
• The Result: The "blurry line" and the "energy spike" disappeared immediately.
• The Meaning: This proves the state is extremely sensitive to the direction of the magnetic field, which is a hallmark of the FFLO state in this type of layered material.

Summary

In short, the researchers used a high-precision "magnetic camera" to look at a very clean iron-based crystal. They found that under extreme cold and strong magnetic fields, the material spontaneously organizes itself into a striped pattern of superconducting and normal regions. They confirmed this by seeing two distinct signals (a broadened signal and an energy spike) appear together. This provides the first microscopic proof of the FFLO state in this class of materials and shows how the material's complex, multi-lane structure (multiband effects) shapes this exotic state.

Technical Summary

Technical Summary: Microscopic Evidence for Fulde-Ferrel-Larkin-Ovchinnikov State and Multiband Effects in KFe$_2$As$_2$

Problem Statement
The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is a superconducting phase characterized by broken translational symmetry, where Cooper pairs form with non-zero center-of-mass momentum between Zeeman-split Fermi surfaces. While theoretically predicted to exist in materials with strong Pauli paramagnetic effects and low disorder, direct experimental verification has remained elusive. A primary challenge is the lack of simultaneous microscopic evidence for the two defining features of the FFLO state: superconducting spin smecticity (manifesting as spectral line splitting or broadening) and the formation of Andreev bound states (manifesting as enhanced spin-lattice relaxation). Furthermore, while iron-based superconductors are considered ideal platforms due to their quasi-two-dimensional structure and multiband nature, no microscopic evidence for the FFLO state had been established in this family of materials prior to this work. Specifically, the impact of multiband effects on the stability and boundaries of the FFLO phase in KFe$_2$As$_2$ required direct investigation.

Methodology
The authors employed $^{75}$As nuclear magnetic resonance (NMR) spectroscopy on high-quality single crystals of the multiband superconductor KFe$_2$As$_2$. The samples, grown via the self-flux method, exhibited a high superconducting critical temperature ($T_c \approx 3.8$ K) and a high residual resistivity ratio (up to 3000), indicating low disorder.

• Experimental Conditions: Measurements were conducted in the low-temperature ($T < 1.5$ K) and high-magnetic-field regime using a $^3$He-$^4$He dilution refrigerator. The external magnetic field was applied parallel to the $ab$-plane ($B \parallel ab$) with precise angular control.
• Key Observables:
  1. NMR Spectral Line Shape: The authors analyzed the central transition line of the $^{75}$As-NMR spectra, specifically calculating the square root of the second moment ($\sigma^2)^{1/2}$ to quantify line broadening, which serves as a proxy for spin polarization and spatial modulation.
  2. Spin-Lattice Relaxation Rate: The temperature dependence of the spin-lattice relaxation rate divided by temperature ($1/T_1T$) was measured using the saturation-recovery method to detect anomalies associated with Andreev bound states.
  3. Angular Dependence: The sensitivity of the observed state to field orientation was tested by rotating the sample by 1.7$^\circ$ from the $ab$-plane.

Key Results

1. Spectral Broadening (Spin Smecticity): In the normal state and the homogeneous superconducting (HSC) state, the NMR spectra remained sharp. However, in a specific low-temperature, high-field region (e.g., $T < 0.6$ K and $4.8 \text{ T} < B < 6.1 \text{ T}$), a clear increase in the second moment $(\sigma^2)^{1/2}$ was observed. This indicates line broadening rather than the distinct two-horn splitting seen in other materials (like Sr$_2$RuO$_4$), attributed to the relatively large intrinsic linewidth from second-order quadrupolar broadening in KFe$_2$As$_2$. This broadening occurs within the superconducting state, even above the Pauli limit field ($B_{Pauli} \approx 4.8$ T), and is distinct from vortex state effects.
2. Enhanced Relaxation (Andreev Bound States): Coinciding with the onset of spectral broadening, the authors observed a distinct peak in $1/T_1T$ at the same magnetic fields and temperatures. This enhancement suggests the presence of polarized quasiparticles localized at the nodes of the order parameter (Andreev bound states), a hallmark of the spatially modulated gap in the FFLO state.
3. Angular Sensitivity: When the magnetic field was tilted by just 1.7$^\circ$ away from the $ab$-plane, both the spectral broadening and the $1/T_1T$ peak disappeared. This confirms the high sensitivity of the observed state to field orientation, consistent with the FFLO state in quasi-two-dimensional systems.
4. Phase Diagram and Multiband Effects: The authors constructed a phase diagram revealing a distinct boundary between the HSC and FFLO states. Two critical features were identified:
   • Low Critical Temperature: The FFLO state emerges at a low critical temperature $T^* \approx 0.2 T_c$.
   • Ascending Boundary: The transition field from HSC to FFLO decreases as temperature decreases, creating an ascending boundary line in the phase diagram.

Significance and Claims
The paper claims to provide the first microscopic experimental evidence for the FFLO state in iron-based superconductors. By simultaneously observing superconducting spin smecticity (via line broadening) and Andreev bound states (via $1/T_1T$ enhancement), the study overcomes previous limitations where only one of these signatures was detected.

The authors attribute the unique features of the observed phase diagram—specifically the low $T^*$ and the ascending HSC-FFLO boundary—to multiband effects inherent to KFe$_2$As$_2$. They argue that:

• The low $T^*$ is likely due to significant orbital pair-breaking (given the Maki parameter $\alpha_M \sim 3$) and the destabilizing influence of isotropic Fermi surfaces ($\alpha$ and $\gamma$ bands) on the FFLO state.
• The ascending boundary line is explained by interband coupling between an "active" band (anisotropic, nodal gaps on $\beta$ and $\varepsilon$ surfaces) and a "passive" band (negligible gap on the $\gamma$ surface). This coupling stabilizes the FFLO state at lower fields as temperature decreases.

The study concludes that iron-based superconductors serve as a robust platform for investigating the FFLO state and highlights the critical role of multiband physics in determining the stability and boundaries of this exotic phase.