Magnetic ferroelectric (FE) materials (that is, multiferroics) have received much attention and can have wide technological applications owing to the possibility of controlling their electric properties and their magnetic properties by electric fields. [1, 2] An important class of multiferroic and FE [3] materials is based on perovskites, ABO anion). In most cases, the A cations are small, so that t < 1 and the AÀO bonds are too long to maintain the ideal cubic structure. Consequently, the A cation moves away from the center of the B 8 cube, which is accompanied by the bending of the B-O-B bonds and the rotation of the BO 6 octahedra to form a lower-coordinate AO n polyhedron (n < 12) with short AÀO bonds. The distortion of the ideal cubic perovskite toward a non-centrosymmetric FE structure requires another local instability apart from t < 1, namely, the second-order Jahn-Teller (SOJT) instability [4][5][6] of the A-site and/or B-site cation.As found for PbTiO 3 , [7] PbVO 3 , [8] and BiCoO 3 , [9] moving the A cation toward the center of one B 4 face (Figure 1 a) leads to a tetragonal FE structure with space group P4mm. The B cations move away from the approaching A cations such that the coordinate environment of each B cation is more like a BO 5 square pyramid with shortened axial B À O bond. In PbTiO 3 , PbVO 3 , and BiCoO 3 , the s 2 A-site cations Pb 2+ and Bi 3+ are susceptible to a SOJT distortion, which mixes the empty 6p orbital of A into the filled 6s 2 orbital of A hence forming a lone pair on A.[4] In PbTiO 3 , the B-site cation Ti 4+ (d 0 ) is also susceptible to a SOJT distortion, which mixes the empty Ti 3d orbitals into the filled 2p orbitals of the oxygen atom of the axial Ti À O bond. [5,6] Thus, in PbTiO 3 , the need to make short PbÀO bonds and the SOJT instabilities of both Pb 2+ and Ti 4+ ions cooperate to give rise to the observed FE distortion. [10, 11] Another type of the A-cation displacement from the center of the B 8 cube is the movement toward one corner of the B 8 cube along the body-diagonal direction (that is, the C 3 rotational axis; Figure 1 b) to form a non-centrosymmetric FE trigonal structure with space group R3c, as found for BiFeO 3 .[12] The movement of the A cation toward one corner is equivalent to that toward the opposite corner, and the two energetically-equivalent FE structures are mutually interconverted through the centrosymmetric paraelectric (PE) trigonal structure with space group R3 c (Figure 1 c). [13, 14] For BiFeO 3 , the FE Curie temperature (T E = 1123 K) [15] and the three-dimensional antiferromagnetic ordering temperature (T N = 650 K) [16] are well above room temperature. For with space group R3c. For simplicity, only the two BO 6 octahedra at the two corners of the B 8 cube along the body-diagonal direction and the AO 3 unit lying between the two octahedra are shown. The AO 3 unit merges with the upper BO 6 octahedron to form a distorted AO 6 octahedron, which shares its face with the upper BO 6 octahedron, resulting in the face-sharing octahe...
