Two-magnon Raman scattering in quadratic double-layer antiferromagnets

Two-magnon Raman spectra of the quadratic double-layer antiferromagnet ${\mathrm{K}}_3$ ${\mathrm{Mn}}_2$ ${\mathrm{F}}_7$ ($T_N=58.3\mathrm{}$ K) have been investigated from 2 K up to beyond the ordered regime. Following an analysis of the Raman-scattering Hamiltonian on the basis of symmetry, the spectra are interpreted in terms of a two-dimensional four-sublattice spin-wave theory appropriate to the double layer. The method used is the equation of motion of Green functions. It appears essential to include those quartic terms in the spin-wave Hamiltonian that couple the lower and upper magnon branches, but in the random-phase approximation do not contribute to renormalization of the magnon dispersion. Spectra calculated with first-order decoupling agree well with the spectra observed at 2 K. In second order, the damping of magnons near the zone boundary is expressed in an approximate analytical form. The computed second-order spectra account for the observed temperature dependence of both the peak position and the width of the scattering up to 1.1 $T_N$. Fits to spectra of ${\mathrm{K}}_3$ ${\mathrm{Mn}}_2$ ${\mathrm{F}}_7$, with the anisotropy set at $\Delta =g{\mu }_{\mathrm{B}}\frac{H_A}{5}\mathrm{}\left|J\right|S=0.002\mathrm{}$, yield a nearest-neighbor exchange $J=-7.68\mathrm{}\pm 0.07$ K, in agreement with findings from the sublattice magnetization and susceptibility. From the low-temperature spectrum of the isomorphic ${\mathrm{K}}_3$ ${\mathrm{Ni}}_2$ ${\mathrm{F}}_7$ taken from the literature, we derive $J(1+1.04\mathrm{}\Delta )=-95.4\mathrm{}\pm 0.5$ K.