Multiple ferroic orders and toroidal magnetoelectricity in the chiral magnet BaCoSiO4

The investigation of ferroic orders and their associated domain structures is essential for understanding the fundamental science of ordering phenomena and engineering the relevant devices. Here, the authors show experimentally that the BaCoSiO${}_{4}$ single crystal carries all four types of ferroic orders - polarization, magnetization, ferrorotation, and toroidal moment. The toroidal moment induces an off-diagonal linear magnetoelectricity, and coupling between magnetization and toroidal moment enables the control of toroidal linear magnetoelectricity with external magnetic fields.