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

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is a superconducting phase characterized by broken translational-symmetry, where Cooper pairs form with non-zero momentum between Zeeman-split Fermi surfaces. This state is highly sensitive to band structure and pairing symmetry. In multiband superconductors, the FFLO state can significantly deviate from its standard form, but experimental verification has remained challenging. Here, we present $^{75}$As nuclear magnetic resonance (NMR) measurements on the multiband superconductor KFe$_2$As$2$. In the low-temperature, high-magnetic-field region above the upper critical field $B{c2}$, we observe a clear increase in the second moment of the NMR spectrum, along with a strong enhancement in the spin-lattice relaxation rate divided by temperature 1/$T_1$$T$. These results indicate an emergence of superconducting spin smecticity and Andreev bound states from the spatially modulation of the superconducting gap, providing microscopic evidence for the FFLO state. The obtained phase diagram reveals a distinct boundary line between the FFLO and homogenous superconducting (HSC) states with a low critical temperature of the FFLO state $T^\ast \approx 0.2 T_c$, which can be attributed to the multiband effects in KFe$_2$As$_2$. Our results show that the iron-based superconductors are a good material platform for studying the FFLO state and highlight the importance of the multiband effects on this exotic phase.

💡 Research Summary

**
In this work the authors present a comprehensive microscopic investigation of the Fulde‑Ferrell‑Larkin‑Ovchinnikov (FFLO) state in the multiband iron‑based superconductor KFe₂As₂ using ^75As nuclear magnetic resonance (NMR). High‑quality single crystals (RRR ≈ 3000, Tc = 3.8 K) were studied in a dilution refrigerator down to 0.2 K with magnetic fields applied precisely parallel to the ab‑plane. The key observations are twofold. First, the second moment of the central NMR line, σ²¹/², which reflects the distribution of local spin polarization, remains constant in the homogeneous superconducting (HSC) state but shows a pronounced increase in a narrow field‑temperature window (B ≈ 5.3 T, T < 0.8 K). This broadening, rather than a clear two‑horn splitting, is attributed to spin smecticity – the spatial modulation of the superconducting order parameter characteristic of an FFLO phase – and is enhanced by quadrupolar broadening of the As nucleus. Second, the spin‑lattice relaxation rate divided by temperature, 1/T₁T, exhibits sharp peaks at the same fields (5.3 T and 5.7 T) where σ²¹/² rises. These peaks are interpreted as signatures of Andreev bound states localized at the nodal planes of the FFLO modulation, which increase low‑energy quasiparticle density and thus the relaxation rate. The coincidence of the two independent NMR signatures provides compelling microscopic evidence for an FFLO state.

The authors further demonstrate the extreme sensitivity of this high‑field state to the field orientation: tilting the field by only 1.7° eliminates both the line broadening and the 1/T₁T peaks, confirming that the observed phase is not a vortex‑related phenomenon but a genuine FFLO state stabilized only for fields strictly parallel to the layers. By combining NMR data with AC susceptibility measurements, they construct a detailed B–T phase diagram. The transition line from HSC to FFLO (denoted B* or T*) moves to lower fields as temperature decreases, opposite to the behavior expected for a single‑band superconductor. The authors attribute this unusual curvature to multiband effects: KFe₂As₂ possesses several Fermi‑surface sheets (α, β, γ, ε) with markedly different gap magnitudes and anisotropies. The γ band has a very small gap, while β and ε exhibit line nodes and a flat dispersion that favor nesting. Interband coupling between the weakly gapped γ band and the more robust bands can lower the critical field for FFLO formation at low temperature, producing the observed ascending HSC–FFLO boundary.

The measured Maki parameter α_M ≈ 3 is modest compared with heavy‑fermion (α_M ≈ 10) and organic superconductors (α_M ≈ 8), where orbital pair‑breaking is negligible and the FFLO region extends up to T* ≈ 0.4–0.6 Tc. In KFe₂As₂, orbital effects are non‑trivial, which together with the moderate α_M suppresses the FFLO critical temperature to T* ≈ 0.2 Tc. This low T* is consistent with similar observations in FeSe, where α_M is of comparable magnitude.

By simultaneously detecting spin‑smectic broadening and enhanced relaxation, the study resolves a long‑standing experimental gap: previous FFLO candidates (e.g., CeCoIn₅, β’’‑(ET)₂SF₅CH₂CF₂SO₃, Sr₂RuO₄) displayed only one of these signatures. The work thus establishes KFe₂As₂ as the first iron‑based superconductor with unequivocal microscopic FFLO evidence and highlights how multiband physics reshapes the stability and phase boundaries of this exotic superconducting state. The findings open new avenues for exploring FFLO physics in other multiband systems and for testing theoretical models that incorporate interband coupling and orbital pair‑breaking.