Three-dimensional spin susceptibility in Ba$_{0.75}$K$_{0.25}$Fe$_{2}$As$_{2}$: Out-of-plane modulation revealed by neutron spectroscopy and theoretical modeling

We present a combined experimental and theoretical investigation of the spin dynamics in the iron-based superconductor Ba${0.75}$K${0.25}$Fe$2$As$2$. Time-of-flight inelastic neutron scattering measurements reveal the three-dimensional (3D) nature of the spin fluctuations, manifested as out-of-plane modulations of the low-energy magnetic intensity. As the energy increases, this 3D-like modulation gradually fades away, leading to a more two-dimensional (2D) profile – a clear signature of a 3D-to-2D crossover in the spin dynamics. By incorporating a realistic 3D electronic band structure derived from density functional theory (DFT), we reproduce the experimentally observed features of the spin susceptibility, including the pronounced out-of-plane modulation at low energies and its gradual evolution into a more 2D character at higher energies. The calculated susceptibility exhibits a peak at the experimental ordering wavevector $\mathbf{q}{\mathrm{AFM}} = (0.5, 0.5, 1)$, demonstrating that the DFT-derived 3D model accurately captures the tendency toward out-of-plane antiferromagnetic (AFM) order. Notably, electronic states away from the Fermi level play a crucial role in shaping the susceptibility peak at $\mathbf{q}{\mathrm{AFM}}$, highlighting the limitations of the Fermi surface nesting picture in explaining the out-of-plane AFM instability. The demonstrated agreement between experiment and theory serves as a benchmark for validating the DFT-derived model as a realistic description of the material-specific electronic structure.

💡 Research Summary

This paper presents a comprehensive study of the spin dynamics in the iron‑based superconductor Ba₀.₇₅K₀.₂₅Fe₂As₂, combining time‑of‑flight inelastic neutron scattering (TOF‑INS) with first‑principles calculations based on density functional theory (DFT) and the random phase approximation (RPA). The experimental work employed the AMA‑TERAS and 4SEASONS spectrometers at J‑PARC, measuring the four‑dimensional scattering function S(Q, ω) over a wide range of incident neutron energies (7.74–125 meV) and crystal orientations. By rotating the co‑aligned 5‑g crystal array in fine steps and also using a continuous‑rotation technique, the authors reconstructed full 4D maps of magnetic intensity. In the low‑energy regime (≈5 meV), the magnetic signal centered at (0.5, 0.5, L) displays a pronounced L‑modulation: odd L values give intensity maxima while even L values give minima. This out‑of‑plane modulation persists up to ≈14 meV but fades rapidly above ≈15 meV, where the intensity becomes essentially L‑independent, indicating a crossover from three‑dimensional (3D) to two‑dimensional (2D) spin behavior. Temperature scans show that the modulation survives well below the Néel temperature (T_N ≈ 90 K) and disappears near T_N, confirming its magnetic origin.

On the theoretical side, the authors performed DFT calculations using Quantum ESPRESSO with the GGA functional, constructing a ten‑orbital tight‑binding model for the crystallographic unit cell (two Fe atoms). This model was unfolded to an effective five‑orbital description in the 1‑Fe Brillouin zone, and the band structure was renormalized by a factor of three to match ARPES data, while K‑doping was simulated by a rigid shift of the Fermi level. The dynamical spin susceptibility χ_s(q, ω) was then evaluated within multiorbital RPA, employing interaction parameters U = 0.41 eV, J = U/8, and the standard relations U′ = U − 2J, J′ = J. Calculations were performed on a dense 128³ k‑mesh at a low temperature (k_BT = 0.03 eV) with a small smearing δ = 0.005 eV.

The RPA results reproduce the experimental observations with remarkable fidelity. The susceptibility exhibits a sharp peak at the ordering wave vector q_AF = (0.5, 0.5, 1), exactly where the neutron data show the strongest odd‑L enhancement. Importantly, analysis of the bare susceptibility χ₀(q, ω) reveals that the dominant contributions to the peak arise from electronic states located several tens of meV away from the Fermi level, rather than from perfect nesting between the hole pockets at Γ and the electron pockets at X. This finding challenges the conventional nesting picture that has often been invoked to explain antiferromagnetism in FeSCs and underscores the role of the full three‑dimensional electronic structure, including kz‑warping and orbital mixing, in driving the out‑of‑plane AFM instability.

By comparing the energy dependence of the calculated χ_s(q, ω) with the INS data, the authors demonstrate that the 3D‑to‑2D crossover is captured naturally: at low energies the susceptibility retains strong L‑dependence, while at higher energies the interlayer modulation is suppressed, yielding a quasi‑2D response. The agreement validates the DFT‑derived band structure as a realistic, material‑specific foundation for modeling spin fluctuations in iron‑based superconductors.

In summary, the work establishes (i) the existence of a pronounced out‑of‑plane modulation of low‑energy spin excitations in Ba₀.₇₅K₀.₂₅Fe₂As₂, (ii) a clear energy‑driven crossover from 3D to 2D spin dynamics, (iii) the necessity of incorporating the full three‑dimensional electronic structure—beyond simple Fermi‑surface nesting—to explain the observed antiferromagnetic correlations, and (iv) the reliability of DFT‑based multiorbital RPA calculations for quantitative comparison with neutron spectroscopy. This combined experimental‑theoretical approach provides a robust benchmark for future investigations of spin‑mediated pairing mechanisms in iron‑based and other unconventional superconductors.