Spin-wave analysis in the two-dimensional antiferromagnetK2FeF4. I. Neutron scattering

In this and the following papers, spin waves in the quadratic-layer basal-plane antiferromagnet ${\mathrm{K}}_2$Fe${\mathrm{F}}_4$ are examined in detail. Here elastic and inelastic neutron scattering are employed to determine the magnetic structure, the spin-wave dispersion, and the sublattice magnetization versus temperature. The spin-wave analysis is based on the spin Hamiltonian for ${\mathrm{Fe}}^{2+}$ in a tetragonally distorted cubic crystal field, which appears to contain, except Heisenberg nearest-neighbor exchange $-J{\overrightarrow{\mathrm{S}}}_1·{\overrightarrow{\mathrm{S}}}_2$, anisotropies of the form $DS_z^{}{}_{}{}^2$ and $e(S_{+}^4+S_{-}^4)$. The anisotropies favor spin ordering along the in-layer magnetic axes, which is confirmed by experiment; the stacking of the layers appears to be unique. Holstein-Primakoff spin waves are expanded in $\frac{1}{2S}$, and first-order corrections to the leading-order theory after Oguchi are included. To ensure sufficient convergence of the expansion, the quartic anisotropy is partially decoupled within the random-phase approximation in spin space, converting it to an in-layer anisotropy of the form $E(S_x^2-S_y^2)$, with $E$ temperature dependent according to $〈S_x^2-S_y^2〉$. The spin-wave description accounts for the dispersion and sublattice magnetization up to $\frac{3}{4}{T}_N$, with for the first-order corrected theory $J=-15.7\mathrm{}\pm 0.3$ K, $D=5.7\mathrm{}\pm 0.1$ K, and $E(T=0\mathrm{})=-0.49\mathrm{}\pm 0.03$ K. Both $D$ and $E$ are in agreement with the crystal-field estimates. Additionally, the critical behavior is studied. There is a well-defined phase transition at $T_N=63.0\mathrm{}\pm 0.3$ K, in contrast to Mössbauer findings, suggesting some sort of local semistatic order to persist up to 70 K. The critical exponents $\beta$, $\gamma$, and $\nu$ compare with those of other quadratic-layer systems.