Effects of the interaction between R and Fe modes of the magnetic resonance in RFe$_3$(BO$_3)_4$ rare-earth iron borates

Resonance modes that are due to magnetic excitations in the exchange-coupled subsystems of rare-earth ions (R = Nd$^{3+}$, Sm$^{3+}$, Gd$^{3+}$) and Fe$^{3+}$ ions have been detected in submillimeter transmission spectra (0.1–0.6 THz) of RFe$_3$(BO$_3)_4$ iron borate-multiferroic single crystals. The strong interaction between spin oscillations of the Fe and R subsystems has been revealed, which determines the behavior of the modes depending on the anisotropy of the exchange splitting of the ground doublet of the R ion. It has been shown that the intensities of coupled modes (contributions to the magnetic permeability) depend strongly on the difference between the $g$ factors of Fe and R ions. This dependence makes it possible to determine the sign of the latter $g$ factor. In particular, a noticeable intensity of exchange Nd modes in NdFe$_3$(BO$_3)_4$ is due to an increase in their contribution at $g^{\rm Nd}_{\perp,\parallel}<0$, while in GdFe$_3$(BO$_3)_4$ with $g_{\rm Gd}\approx g_{\rm Fe}\approx2$, the Fe and Gd contributions compensate each other and the exchange (Gd) mode is not observed. In spite of the weak interaction of Sm ions with the magnetic field, SmFe$_3$(BO$_3)_4$ exhibits resonance modes, which are attributed to the excitation of Sm ions through the Fe subsystem.