Chiral spin-wave excitations of the spin-$\frac{5}{2}$ trimers in the langasite compound Ba${}_3$NbFe${}_3$Si${}_2$O${}_{14}$

The inelastic scattering of neutrons from magnetic excitations in the antiferromagnetic phase of the langasite compound Ba${}_3$NbFe${}_3$Si${}_2$O${}_{14}$ is analyzed theoretically. In the calculations presented, the strongly coupled spin-$\frac{5}{2}$ Fe triangles are accounted for as trimerized units. The weaker interactions between the trimers are included within the mean field and random phase approximations. The theory is compared with linear spin-wave theory, and a model is developed that leads to good agreement with the published results from unpolarized and polarized neutron-scattering experiments.

Figure 1

Fe triangles are stacked on top of each other along the $c$ axis in the Ba${}_3$NbFe${}_3$Si${}_2$O${}_{14}$ crystal. The line parallel to the $c$ axis through the center of the triangles is a threefold symmetry axis, and the three altitudes of one triangle are twofold symmetry axes. The oxygen ions are placed in different positions along the paths determining the interplanar exchange constants $J_3$, $J_4$, and $J_5$. The shortest super-superexchange path (and bond angles closest to $\pi$) defines (Ref. 1) the structural chirality ${\epsilon }_T^{}=\pm 1$. If this path is the one determining $J_5$ as defined in the figure, the corresponding chirality would be ${\epsilon }_T^{}=-1$, whereas ${\epsilon }_T^{}=+1$ if this path is the one leading to the exchange interaction $J_3$. The right part of the figure shows the ordered spins in one of the triangles, where the moments rotate the angle $\gamma ={\epsilon }_{\gamma }^{}(2\pi /3)$ for increasing site index. The upper triangle shows the case where ${\epsilon }_{\gamma }^{}=+1$, whereas the orientation of the ordered spins in the lower triangle corresponds to ${\epsilon }_{\gamma }^{}=-1$.

Figure 2

The triangles within an $ab$ plane with a definition of the exchange parameter $J_2$.

Figure 3

The calculated susceptibility components as functions of temperature compared with the experimental results of Marty et al. (Ref. 8). The red line and the red crosses denote, respectively, calculated and experimental results, when the field is applied along the $c$ axis. The green symbols denote the results obtained when the field is applied perpendicular to the $c$ axis. The dashed line shows the MF behavior for the corresponding simple $S=5/2$ system. The difference between the dashed and solid lines indicates the importance of correlation between the three spins in the triangle clusters.

Figure 4

The calculated spin-wave dispersion relations along the $c$ axis compared with the experimental results obtained at 2.5 K by Stock et al. (Ref. 3) along $\left(20q\right)$. The calculated energies of the two $w$ modes and the two branches of the $c$ mode are shown by, respectively, the blue and orange lines, and the spectrum includes only the two times two branches observed in the experiments. The solid lines show the results obtained when introducing the calculated values of the energy splittings and the matrix elements in Eqs. (14) and (16), whereas the dashed lines are the results obtained using the linear spin-wave expressions (18) and (19).

Figure 5

Logarithmic contour plots of the calculated spin-wave scattering intensities along different scattering vectors in reciprocal lattice units. The black squares indicate the experimental energies determined at $T=2.5$ K by Stock et al. (Ref. 3). The logarithmic scale shown to the right is common for all three cases shown here and also applies to Fig. 7 and to the left figure in Fig. 8. The use of the logarithmic scale may be slightly misleading in the sense that the low-intensity branches (light green and light blue), which are easily identified in these plots, are probably not visible under realistic circumstances.

Figure 6

Detail comparisons between the experimental (Ref. 3) and calculated cross sections for a selection of the scattering vectors considered in Fig. 5.

Figure 7

The logarithmic scattering intensities calculated along $\left(00q\right)$ and $\left(40q\right)/\left(04q\right)$. The experimental results included in the $\left(40q\right)/\left(04q\right)$ figure are the same as those appearing in Fig. 5 and obtained by Stock et al. (Ref. 3), but the results have been translated by wave vectors $(00\ell \pm Q)$.

Figure 8

Contour plots of the calculated spin-wave scattering intensities along $\mathbf{k}=\left(01q\right)$. The left figure shows the logarithmic intensities $S(\mathbf{k},\omega )$ derived for unpolarized neutron scattering. Notice that the intensities of the branches starting out from zero energy at $(01\ell -Q)$ are much higher than the intensities of the $(01\ell +Q)$ branches. The figure in the middle shows the calculated polarization function $C(\mathbf{k},\omega )$ using a linear scale. The calculated intensities are compared with the experimental results obtained by Loire et al. (Ref. 4), which are shown by the green and blue squares lying in the interval between 1 and 6 meV. The figure to the right shows the calculated dynamic scattering ratio $C(\mathbf{k},\omega )/S(\mathbf{k},\omega )$.

Figure 9

Comparison between theory and experiments for two of the scattering vectors considered in Fig. 8. The experimental results are obtained by Loire et al. (Ref. 4) from inelastic scattering of polarized neutrons. The intensity scaling factor and the resolution width are kept constant in all the calculated results.