Magnetic excitations and exchange interactions in the substituted multiferroics $(\mathrm{Nd},\mathrm{Tb}){\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ revealed by inelastic neutron scattering

Inelastic neutron scattering spectra in the antiferromagnetic ferroborates ${\mathrm{Nd}}_{1-x}{\mathrm{Tb}}_x{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ ($x$ = 0, 0.1, 0.2, and 1) reveal various magnetic excitations of the interacting iron and rare-earth subsystems. We observe an evolution of the magnetic system from “easy-plane”, in the Tb-free ($x$ = 0) case, to “easy-axis” anisotropy for samples substituted with Tb. The spectra show hybridized Fe and Nd branches, which are determined by the Fe-Nd exchange splitting of the ground-state ${\mathrm{Nd}}^{3+}$ doublet. In the easy-plane configuration, near the Brillouin zone center, there are two different pairs of anticrossing quasiacoustic Fe and Nd modes in contrast to the easy-axis state, where the two corresponding pairs of the branches are degenerated. The high-energy (exchange) branches are similar in both spin configurations. The Ising-type anisotropy of the Tb ion prevents the magnetic moment from precession. The increasing of the Tb content changes the effective magnetic anisotropy and stabilizes the easy-axis state. The spin-wave dispersion in the substituted and pure $\mathrm{Tb}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ compounds, which have the same, easy-axis magnetic structure, but different crystal symmetry, strongly differ. The observed spectra were analysed in the frame of linear spin-wave theory and the exchange parameters were determined.

Figure 1

Magnetic structures of (a) $\mathrm{Nd}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$, (b) ${\mathrm{Nd}}_{0.9}{\mathrm{Tb}}_{0.1}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$, (c) ${\mathrm{Nd}}_{0.8}{\mathrm{Tb}}_{0.2}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$, and (d) $\mathrm{Tb}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$. The magnetic moments of the rare-earth and Fe ions are shown in green and red color, respectively. The magnetic unit cell, which is twice the chemical one, is shown.

Figure 2

Calculated [(a)–(d)] and observed [(e)–(h)] maps of spin-wave dispersion in the substituted compounds ${\mathrm{Nd}}_{1-x}{\mathrm{Tb}}_x{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ and ${\mathrm{NdFe}}_3{\left({\mathrm{BO}}_3\right)}_4$. [(a), (b)] and [(e), (f)] Maps of ${\mathrm{Nd}}_{0.8}{\mathrm{Tb}}_{0.2}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$, measured along [$\xi$ 0 1.5] and [0 0 $\xi$ ] directions, respectively. Maps of ${\mathrm{Nd}}_{0.9}{\mathrm{Tb}}_{0.1}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ measured along the [$\xi$ 0 1.5] direction [(c), (g)] and maps of $\mathrm{Nd}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ measured along the [$\xi$ 0 1.5] direction [(d), (h)] [22] are shown for comparison. The calculated dispersion curves $\hslash \omega =E\left(\vec{Q}\right)$ are shown as well in the observed maps of spin-wave dispersion, panels [(e)–(h)]. White open circles correspond to the magnon positions, obtained after deconvolution of the measured energy scans for Voigtians. Only branches with nonzero inelastic neutron-scattering cross section are shown, apart from the panel (b), where a branch with zero inelastic neutron-scattering cross section is shown by a white line (see text). The spin-wave dispersion in ${\mathrm{Nd}}_{0.8}{\mathrm{Tb}}_{0.2}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ (e) was measured with a step size of $\mathrm{\Delta }Q=0.05$ reciprocal lattice units (r.l.u), while in the map (f) $\mathrm{\Delta }Q=0.1$ r.l.u. The weak dispersion-less branch in the panel (h) around 1.4 meV is associated with the transition inside the ${\mathrm{Nd}}^{3+}$ doublet [22]. Scales are logarithmic.

Figure 3

Calculated [(a), (b)] and observed [(c), (d)] maps of spin-wave dispersion for ${\mathrm{Nd}}_{0.8}{\mathrm{Tb}}_{0.2}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ along [$\xi$ 0 1.5] and [0 0 $\xi$] directions in an applied magnetic field of 10 T at a temperature of 1.5 K. The calculated dispersion curves $\hslash \omega =E\left(\vec{Q}\right)$ are shown as well in the observed maps of spin-wave dispersion, panels (c) and (d). White open circles correspond to the magnon positions, obtained after deconvolution of the measured energy scans for Voigtians. Scales are logarithmic.

Figure 4

Calculated [(a), (c)] and observed [(b), (d)] spin-wave spectra for $\mathrm{Tb}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ measured at 1.5 K with const-$Q$ scans in the directions [$\xi$ 0 0.5] and [2 0 $\xi$] around the magnetic reflection (2 0 0.5). The calculated dispersion curves $\hslash \omega =E\left(\vec{Q}\right)$ are shown as well in the observed maps of spin-wave dispersion. White open circles correspond to the magnon positions, obtained after deconvolution of the measured energy scans for Voigtians. Red triangles are experimental data from THz-spectroscopy experiments [23, 42]. (e) The profiles of measured and calculated energy scans at $Q$ = [2 0 0.3]. Vertical bars indicate the magnon positions. Scales are logarithmic.

Figure 5

The energy scan measured at Q = [0 0 1.2] - for ${\mathrm{Nd}}_{0.8}{\mathrm{Tb}}_{0.2}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$. The peaks are approximated by the Voigtian line-shape with the same FWHM. Vertical bars indicate the magnon positions at 3.34(1) meV and 3.78(1) meV.

Figure 6

Dependence of the refined absolute value of the exchange parameters on the distance for ($\mathrm{Nd},\mathrm{Tb}){\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ (in red) and $\mathrm{Tb}{\mathrm{Fe}}_3{\left({\mathrm{BO}}_3\right)}_4$ (in blue). The lines are guide-to-the-eyes and correspond to a fit with exponential decay. The error does not exceed the symbol size.