Some symmetry aspects of ferroics and single phase multiferroics

The symmetry conditions for the occurrence in a same phase of one or more of the four primary ferroic properties, i.e., ferroelectricity, ferromagnetism, ferrotoroidicity and ferroelasticity, are discussed. Analogous conditions are outlined for the admission of so-called secondary and tertiary ferroic effects, such as magnetoelectric, piezoelectric, piezomagnetic, piezotoroidic, etc. Formerly postulated `magnetotoroidic' and 'electrotoroidic' effects are found to be describable as tertiary ferroic magnetoelectric effects. For understanding ferroic and multiferroic domains and their possibilities of switching, knowledge of the pairs of prototype point group/ferroic phase point group (so-called 'Aizu species') is decisive. A classification into ensembles of species with common properties, recently extended to ferrotoroidic crystals, allows distinguishing between full, partial or no coupling between order parameters and understanding domain patterns and poling procedures. The switching by reorientation with angles other than 180 degrees of ferromagnetic, antiferromagnetic and ferroelectric domains by magnetic fields, electric fields or by stress requires the ferroic phase to be ferroelastic. For ferromagnetic/ ferrotoroidic and antiferromagnetic/ ferrotoroidic phases, the ferrotoroidic domains are found to be identical with the ferromagnetic and antiferromagnetic ones, respectively. As a consequence and depending on symmetry, ferrotoroidic domains can be switched by crossed electric and magnetic fields, by collinear electric and magnetic fields or by a magnetic field alone. Examples of ferrotoroidic domains are discussed for Fe2-xGaxO3, Co3B7O13Br and LiCoPO4. Recent new results on symmetry and domains of the antiferromagnetic incommensurate phase of BiFeO3 are also discussed.