The coexistence of electric and magnetic orders in multiferroics imposes certain restrictions on the symmetry of the magnetic and crystalline structures of substances. However, many of them, having a non-centrosymmetric crystal structure, can exhibit magnetoelectric properties even in the paramagnetic state when a magnetic field is applied. Here we report the detection of electric polarization induced by the magnetic field H in single crystals of a trigonal non-centrosymmetric paramagnetic Nd₃Ga₅SiO₁₄ with a langasite structure. In small fields, the polarization is quadratic in H and for components in the basic ab plane it is described by two magnetoelectric susceptibilities (α₁ and α₂), while along the trigonal axis c it appears only starting from the fourth-order terms, ~H⁴. In strong magnetic fields at low temperatures, when the Nd³⁺ magnetic moments are saturated, the dependence of polarization on the field changes qualitatively and approaches to quasi-linear one for all crystallographic directions, and its magnitude increases greatly (up to 250 µC/m² along the c axis at 5 T, see Figure). A quantitative description of the effects observed in Nd₃Ga₅SiO₁₄ based on the spin-Hamiltonian of the Nd³⁺ ion in local low-symmetric (C₂) positions is presented, taking into account the symmetry-allowed magnetoelectric interactions. Nonlinear magnetoelectric invariants based on the local susceptibilities of Nd³⁺ ions are constructed, taking into account the nonequivalence of their magnetization and allowing us to describe the observed field, temperature and orientation dependences of polarization using few magnetoelectric parameters.

Figure. (a) The angular dependence of the electric polarization P_c along c axis, measured at 4.2 K at H =1.3 T when the magnetic field rotates in the ac plane. (b) The dependences of P_c  on a magnetic field applied in ac plane at an angle of 60^o from c axis at different temperatures (the inserts in (a) and (b) explain the geometry of the experiment). (c) The dependence of the dP_c/dH⁴ derivative on 1/T³, which characterizes the temperature dependence of the fourth order magnetoelectric susceptibility α₃. Symbols are an experiment, lines are a theory.

A.A. Mukhin , V.Yu. Ivanov, A.M. Kuzmenko, A.Yu. Tikhonovsky, B.V. Mill
 JETP Letters 121, issue 4 (2025)