Magnetic order, spontaneous polarization, and magnetoelectric effect in rare earth iron borates: Ho$_{1-x}$Nd$_x$Fe$_3$(BO$_3$)$_4$

Comprehensive results are presented for the thermodynamic, magnetic, dielectric, and magnetoelectric properties of HoFe$_3$(BO$_3$)$_4$ and the solid solution Ho$_{1-x}$Nd$_x$Fe$_3$(BO$_3$)$_4$ (x = 0.5 and 0.75). All compounds undergo a Ne\'{e}l order at T$_N>$ 30 K and a spin reorientation at T$_{SR}<$ 10 K. HoFe$_3$(BO$_3$)$_4$ shows a spontaneous electrical polarization below T$_N$ which decreases below T$_{SR}$ and in external magnetic fields. Ho$_{1-x}$Nd$_x$Fe$_3$(BO$_3$)$_4$ exhibits both, a spontaneous polarization and a large positive magnetoelectric effect. The superposition of spontaneous polarization induced by the internal magnetic field and the magnetoelectric polarization in external fields results in a complex behavior of the electrical polarization as function of temperature and/or magnetic fields. The magnetic order of HoFe$_3$(BO$_3$)$_4$ is further explored by neutron scattering experiments in external magnetic fields.