In rare-earth ($\mathit{R}$) ferroborates, R${\mathrm{Fe}}_3$(${\mathrm{BO}}_3$)${}_4$ with R = Eu, Gd, and Tb, the magnetoelectric (ME) responses appear to stem from both the antiferromagnetic order of the iron (Fe) spins and the magnetic moments on the $\mathit{R}$ ions. We measured the electric polarization ($P$) along the $a$ axis while rotating a magnetic field ($H$) around the $a$ axis and found that the target compounds show mutually distinctive $H$-direction dependencies. ${\mathrm{EuFe}}_3$(${\mathrm{BO}}_3$)${}_4$ (R = Eu) shows an almost constant spontaneous $P$ with a slight modulation when $H$ is slanted from the $c$ axis. The $H$-angle (${\theta }_H$) dependence of the $P$ can be described by a formula $P=P_0-\Lambda {sin}^2{\theta }_H$. As for ${\mathrm{GdFe}}_3$(${\mathrm{BO}}_3$)${}_4$ and ${\mathrm{TbFe}}_3$(${\mathrm{BO}}_3$)${}_4$, they show highly anisotropic ${\theta }_H$ dependence of $P$, which characterizes the respective ME responses from their $\mathit{R}$ magnetic moments. In certain regions of ${\theta }_H$, the $P$ can be described by $P=P_0-Ksin2{\theta }_H$ and $P=P_0\mp \Gamma sin{\theta }_H$ for R = Gd and Tb, respectively. We devised a theory for the ME response of the individual magnetic ions in a R${\mathrm{Fe}}_3$(${\mathrm{BO}}_3$)${}_4$ crystal and applied it to these compounds focusing on their local symmetry and their ground-state multiplet structures. The above formulas successfully reproduce the observed results as the summation of $P$ from each magnetic subsystem, which in turn enables us to assign the first and second terms to the spontaneous $P$ due to a collinear antiferromagnetic ordering of the Fe spins and the ME response of the $\mathit{R}$ ion under $H$, respectively. The thermal and $H$-induced evolutions of the magnetic-ion resolved $P$ quantitatively agree with the theoretical predictions, ensuring the relevant microscopic ME mechanism for each magnetic ion. The measurement of angular dependence of $P$ is particularly useful to decompose the overlapped ME responses into the respective origins in the system with multiple magnetic subsystems.

• Received 1 April 2014

DOI:https://doi.org/10.1103/PhysRevB.89.195126