The study of magnetoelectric effects originated with the conjecture of Pierre Curie in 1894 that materials may exist that can be polarized electrically by means of a magnetic field and magnetized by means of an electric field. The different kinds of such magnetoelectric effects known today can be classified on a thermodynamic and symmetry basis. This chapter surveys the thermodynamically reversible induced effects, i.e., the linear and bilinear magnetoelectric effects; the piezomagnetoelectric and thermodynamically irreversible spontaneous effects, i.e., electric field, magnetic field and stress-induced switching or reorientation of domains with spontaneous polarization, spontaneous magnetization, spontaneous toroidal moment and spontaneous deformation. Spontaneous toroidal moments of ferrotoroidics are now recognized as measurable contributions to the linear magnetoelectric effect. Electro-, magneto- and piezotoroidic effects have been postulated. Examples of insulating materials displaying different kinds of magnetoelectric effect are discussed. Useful scientific applications of the effects are the determination of magnetic symmetry, representing a powerful complementary tool for magnetic structure determination by neutron diffraction; the study of magnetic phase transitions, magnetic phase diagrams, toroidal moments and toroidal effects; poling of antiferromagnetic domains; magnetoelectric spin chirality control; linear magneto-optic effect and magnetoelectric spectroscopy at optical frequencies. Technologically promising are composites, for example, to fabricate nonreciprocal magnetic photonic crystals with magnetoelectric symmetry.
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