Magnetoelectric effect in the Nd$_3$Ga$_5$SiO$_{14}$ paramagnet

The electric polarization induced by a magnetic field $H$ in Nd$_3$Ga$_5$SiO$_{14}$ noncentrosymmetric trigonal paramagnetic single crystals has been detected. In low fields, it is quadratic in $H$ and its components in the basal $ab$ plane are described by two magnetoelectric susceptibilities $\alpha_1$ and $\alpha_2$. The polarization along the trigonal $c$ axis is manifested only starting from the fourth-order terms $\sim H^4$. In high magnetic fields at low temperatures, when the magnetic moments of Nd$^{3+}$ are saturated, the polarization qualitatively changes its field dependence to nearly quasilinear for all crystallographic directions and increases strongly (up to 250 $\mu$C/m$^2$ at 5 T along the c axis). The effects observed in Nd$_3$Ga$_5$SiO$_{14}$ have been quantitatively described using the spin Hamiltonian of the Nd$^{3+}$ ion in local low-symmetric (C2) sites taking into account the magnetoelectric interaction allowed by symmetry. Magnetoelectric invariants nonlinear in H that make it possible to describe the observed the field, temperature, and angular dependences of the polarization using several magnetoelectric parameters have been constructed in terms of the local susceptibilities of Nd$^{3+}$ ions taking into account the nonequivalence of their magnetization.