not only in one axis direction but also in two axes direction which gives an
orthorhombic phase (space group Amm2, No. 38) or in three axes direction with a
rhombohedral cell of R3 m space group (No. 160) [30].
The most important magnetic species which can be incorporated into perovskite
structure are ions which have incompletely filled d- and f-shells. In case of transition metal cations, the magnetic moment is related to the electron spin quantum
number. At high temperatures the magnetic moments on these cations are disordered and the material is in paramagnetic state. Below a certain temperature, the
moments start to interact what leads to parallel (ferromagnetic) or antiparallel
(antiferromagnetic, ferrimagnetic) alignment [31]. In magnetic oxides, the antiferromagnetic ordering is realized by a process called superexchange [32]. The superexchange is the strong antiferromagnetic coupling between two cations through
non-magnetic anion. If the two next nearest magnetic cations are connected at 90°
to the non-magnetic anion, the interaction can lead to the parallel moment’s
alignment. If the cation–anion–cation bond angle is 180°, then the coupling is the
most frequently antiferromagnetic. Additionally, the Pauli Exclusion Principle
decides that between two magnetic ions with half-filled orbitals the coupling should
be antiferromagnetic, between half-filled and filled orbitals ions will be ferromagnetic and between one ion which has half-occupied or filled orbital and the second
with an empty orbital the coupling may be antiferro- or ferromagnetic but the
second one is preferable. The theory of the superexchange interaction was developed by Anderson in 1950 [33] and then was reviewed by him in 1959 [34].
Briefly, a cation with a spin-up configuration interacts with a spin-down electron in
filled p-orbital of oxygen. The other spin-up electron in the orbital must then induce
spin-down configuration in the other cation. In case of the perovskite transition
metal ions are placed at B-sites, in the middle of oxygen octahedra and cation–
anion–cation geometry is linear, so the superexchange interaction favors the antiferromagnetic alignment of the magnetic moments [35].
9.3.1.1 BiFeO 3
One of the most important and widely studied magnetoelectrics is bismuth ferrite
BiFeO 3 (BFO). This is one of only a few compounds which have electric polarization and magnetic order at room temperature. The ferroelectric Curie temperature
is considerably high and is approximately T C = 1100 K. At room temperature,
bismuth ferrite is antiferromagnetic and is in this magnetically order state up to Néel
temperature T N = 643 K. This makes the material very interesting from an application point of view [1, 36, 37].
Bismuth ferrite at room temperature adopts an R3c rhombohedral (no. 161) [38,
39] structure. The unit cell in the hexagonal coordinate system (Fig. 9.4a) is indeed
distorted perovskite-like crystal structure and belongs to trigonal perovskites, typified by the (3, 3) phase LaAlO 3 at room temperature. The unit cell is the result of a
rotation of [FeO 6 ] octahedra about a trigonal axis normal to a triangular octahedron
face compared to the cubic parent structure (Fig. 9.4d). This is trigonal crystal class,
286
P. Stoch and A. Stoch
orthorhombic phase (space group Amm2, No. 38) or in three axes direction with a
rhombohedral cell of R3 m space group (No. 160) [30].
The most important magnetic species which can be incorporated into perovskite
structure are ions which have incompletely filled d- and f-shells. In case of transition metal cations, the magnetic moment is related to the electron spin quantum
number. At high temperatures the magnetic moments on these cations are disordered and the material is in paramagnetic state. Below a certain temperature, the
moments start to interact what leads to parallel (ferromagnetic) or antiparallel
(antiferromagnetic, ferrimagnetic) alignment [31]. In magnetic oxides, the antiferromagnetic ordering is realized by a process called superexchange [32]. The superexchange is the strong antiferromagnetic coupling between two cations through
non-magnetic anion. If the two next nearest magnetic cations are connected at 90°
to the non-magnetic anion, the interaction can lead to the parallel moment’s
alignment. If the cation–anion–cation bond angle is 180°, then the coupling is the
most frequently antiferromagnetic. Additionally, the Pauli Exclusion Principle
decides that between two magnetic ions with half-filled orbitals the coupling should
be antiferromagnetic, between half-filled and filled orbitals ions will be ferromagnetic and between one ion which has half-occupied or filled orbital and the second
with an empty orbital the coupling may be antiferro- or ferromagnetic but the
second one is preferable. The theory of the superexchange interaction was developed by Anderson in 1950 [33] and then was reviewed by him in 1959 [34].
Briefly, a cation with a spin-up configuration interacts with a spin-down electron in
filled p-orbital of oxygen. The other spin-up electron in the orbital must then induce
spin-down configuration in the other cation. In case of the perovskite transition
metal ions are placed at B-sites, in the middle of oxygen octahedra and cation–
anion–cation geometry is linear, so the superexchange interaction favors the antiferromagnetic alignment of the magnetic moments [35].
9.3.1.1 BiFeO 3
One of the most important and widely studied magnetoelectrics is bismuth ferrite
BiFeO 3 (BFO). This is one of only a few compounds which have electric polarization and magnetic order at room temperature. The ferroelectric Curie temperature
is considerably high and is approximately T C = 1100 K. At room temperature,
bismuth ferrite is antiferromagnetic and is in this magnetically order state up to Néel
temperature T N = 643 K. This makes the material very interesting from an application point of view [1, 36, 37].
Bismuth ferrite at room temperature adopts an R3c rhombohedral (no. 161) [38,
39] structure. The unit cell in the hexagonal coordinate system (Fig. 9.4a) is indeed
distorted perovskite-like crystal structure and belongs to trigonal perovskites, typified by the (3, 3) phase LaAlO 3 at room temperature. The unit cell is the result of a
rotation of [FeO 6 ] octahedra about a trigonal axis normal to a triangular octahedron
face compared to the cubic parent structure (Fig. 9.4d). This is trigonal crystal class,
286
P. Stoch and A. Stoch