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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

Naoki Murai, Katsuhiro Suzuki, Masamichi Nakajima, Maiko Kofu, Seiko Ohira-Kawamura, Yasuhiro Inamura, Ryoichi Kajimoto

Abstract

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.

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

Abstract

We present a combined experimental and theoretical investigation of the spin dynamics in the iron-based superconductor BaKFeAs. 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 , 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 , 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.

Paper Structure

This paper contains 6 sections, 2 equations, 5 figures.

Table of Contents

  1. INTRODUCTION
  2. EXPERIMENTAL METHODS
  3. THEORETICAL METHODS
  4. RESULTS AND DISCUSSION
  5. SUMMARY
  6. ACKNOWLEDGMENTS

Figures (5)

  • Figure 1: (a)--(c) 3D and cross-sectional views of the DFT-derived Fermi surface for BaFe$_2$As$_2$, with the Fermi level shifted to account for $25\%$ K-doping. Panel (a) shows a 3D view, (b) a top view from the (0 0 1) direction, and (c) a side view from the (1 $\bar{1}$ 0) direction. The color code represents the orbital character projected onto the Fermi surface. Panels (d) and (e) show the theoretical and experimental spin susceptibilities at $\omega = 10$ meV, respectively, both peaking at $\mathbf{q} = (0.5, 0.5)$. The data in panel (e) were obtained from a fixed-geometry scan ($\varphi = 0^\circ$), where the sample $c$-axis was aligned parallel to the incident neutron beam. The measurements were performed using $E_i = 31.3$ meV at $T = 30$ K. The intensities in panels (d) and (e) are shown on different scales.
  • Figure 2: (a--c) Low-energy spin excitations measured with $E_i = 15.15$ meV, focusing on the out-of-plane ($L$) dependence of the magnetic response. (a) 3D density map of the magnetic scattering intensity in $(H,K,L)$ space at $\omega = 5 \pm 2$ meV. (b) Horizontal slice of panel (a) in the $(H, H, L)$ plane. (c) False-color energy spectrum along $(0.5, 0.5, L)$. Panels (a)--(c) reveal a pronounced periodic intensity modulation along $L$, with maxima at odd $L$ and minima at even $L$, thereby establishing the 3D character of the low-energy spin excitations. (d--g) Spin excitations measured with a higher incident energy of $E_i = 55.6$ meV. (d--f) Constant-energy slices in the $(H, H, L)$ plane at $\omega = 10 \pm 2$, $14 \pm 2$, and $20 \pm 2$ meV. While the odd-$L$ modulation remains clearly visible at 10 and 14 meV, it becomes strongly suppressed at 20 meV, where the intensity approaches a nearly uniform distribution along $L$. (g) False-color energy spectrum along $(0.5,0.5,L)$, showing that the $L$-dependent modulation progressively weakens above $\omega \sim 15\,\mathrm{meV}$. (h) Representative constant-energy cuts along $(0.5,0.5,L)$ at $\omega = 4 \pm 1$, $7 \pm 1$ meV ($E_i = 15.15$ meV) and $\omega = 13 \pm 1$, $16 \pm 1$, $19 \pm 1$, and $22 \pm 1$ meV ($E_i = 55.6$ meV), highlighting the crossover from a strongly $L$-modulated 3D response at low energies to a nearly $L$-independent 2D profile at higher energies. All data were collected at $T < 10$ K.
  • Figure 3: Temperature dependence of magnetic scattering in the $(H, H, L)$ plane at (a) $\omega = 0 \, \mathbin{\pm} \, 0.075$ meV and (b) $\omega = 4 \, \mathbin{\pm} \, 1$ meV, respectively, collected with $E_i = 7.74$ meV. At finite energy transfer in panel (b), magnetic scattering peaks appear at odd $L$ positions and persist above $T_\mathrm{N}$, indicating robust out-of-plane AFM correlations.
  • Figure 4: (a) Constant-energy map of the RPA spin susceptibility at $\omega = 5$ meV in the $(H, H, L)$ plane, calculated using a DFT-derived 3D band structure. The intensity exhibits a periodic modulation along $L$, with maxima at odd $L$ and minima at even $L$ positions. (b) Corresponding map at $\omega = 50$ meV, showing a nearly uniform intensity distribution along $L$. (c) Energy dependence of the RPA spin susceptibility along the $(0.5, 0.5, L)$ direction. (d) Constant-energy cuts along $(0.5, 0.5, L)$ at $\omega = 5$, 13, 20, 30, and 50 meV, highlighting the strong $L$-dependent intensity variation at low energies and its gradual suppression at higher energies.
  • Figure 5: (a) Orbital-resolved intraorbital components of the RPA spin susceptibility, showing a common momentum dependence with a peak at $L = 1$ across all orbitals. (b) Schematic illustration of the imperfect nesting between hole and electron pockets. Red dotted lines indicate the hole Fermi surfaces translated by the AFM wavevector $\mathbf{q}_{\mathrm{AFM}} = (0.5, 0.5, 1)$ (blue arrow). (c) The spin susceptibility along the $(0.5, 0.5, L)$ direction, computed using only electronic states within $\pm5$ meV of the Fermi level, thereby isolating the contribution from the Fermi surface geometry. (d) The spin susceptibility at $\omega = 5$ meV, obtained using the full DFT band structure without any energy cutoff, thereby including contributions from a broad energy range. Black arrows indicate the peak positions along $L$: 0.19, 1, and 1.81 in (c), and 1 in (d).