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 ${\mathbf{q}}_{\mathrm{A}\mathrm{F}\mathrm{M}}=(0.5,0.5,1)$, 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 ${\mathbf{q}}_{\mathrm{A}\mathrm{F}\mathrm{M}}$, 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.