Field-induced electric polarization and elastic softening caused by parity-mixed d-p hybridized states with electric multipoles in Ba₂CuGe₂O₇

We performed high-magnetic-field magnetization, polarization, and ultrasonic measurements in Ba₂CuGe₂O₇ to investigate field-induced multiferroic properties arising from a cross-correlation between electric dipoles and electric quadrupoles in addition to cross-correlation between magnetic dipoles and electric dipoles. Magnetization M shows saturation behavior above 20 T for several magnetic field directions, however, electric polarization Pc exhibits an increase, and elastic constants show a softening above 20 T. Based on quantum states with a crystalline electric field for the D₂₃ point group and d-p hybridization between Cu-3d and O-2p electrons, we confirmed that the matrix of an electric dipole Pᵦ was proportional to that of an electric quadrupole Oₗₘ. Furthermore, considering the spin-orbit coupling of 3d electrons and the Zeeman effect, we showed that Pᵦ and Oₗₘ simultaneously exhibited field-induced responses. These findings indicate that the orbital degrees of freedom, in addition to the spin degrees of freedom, contribute to the high-field multiferroicity in Ba₂CuGe₂O₇.