4.4 Multiferroics
4.4.1 Type-I Multiferroics
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Figure 4.1: Geometric frustration - (a) Ising antiferromagnets on the triangle lattice. The remain-ing one spin can no longer point in a direction oppsite to both two spins. (b) Scheme of water ice molecules where hollow circles are oxygen atoms and filles circles are hydrogen atoms. For each oxygen atom, two of the neighboring must reside in the far position and two of them in the near position.
the atom. The condition for ferroelectricity requires the spontaneous breaking of spatial reversal symmetry.
4.3 Geometric Frustration
The concept of geometric frustration [25, 26] is important in some system. It means that there is no single ground state in the system. A nonzero entropy remains in the system even at zero temperature. It can be traced back to 1950 [27]. An Ising antiferromagnet on the triangle lattice, when two of three spins are pointed in oppsite directions to satisfy their antiferromagnetic interaction, the remaining one can no longer point in a direction oppsite to both two spins. That is to say, it is impossible to simultaneously minimize the energy of all interactions. The geometric frustration is not a phenomenon occurs unusually. In fact, it occurs in the ordinary ice which we contact almost everyday [28]. Four oxygen atoms form a tetrahedral structure, and the hydrogen atoms locate between two oxygen atoms being closer to one of the two. Every oxygen atom is surrounded by four hydrogen atoms. For each oxygen atom, two of the neighboring must reside in the far position and two of them in the near position, so-called ’Ice rules’.
4.4 Multiferroics
4.4.1 Type-I Multiferroics
The sources of ferromagnetism and ferroelectricity of type-I multiferroic materials are dif-ferent, and weak or no magnetoelectric effect appears in such a system. The critical temperature of appearance of ferroelectricity is often higher than magnetism, and the spontaneous electric polarization P (of order 10 - 100 µC/cm2) is often rather large. Four different subclasses
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Figure 4.2: Perovskite Structure - The general stoichiometry ABX3, where A (blue particles located at the corners of the cube) and B (orange particle located in the centre of the cube) are cations and X (red particles located in the face-centred positions of the cube) are anions.
pending on the mechanism of ferroelectricity will be discussed as follows.
Perovskites
Firstly, we introduce the perovskite structure which has the general chemical formula ABX3, named originating from the perovskite mineral (CaTiO3), where A and B are cations and X is an anion (often the oxygen atom). In an ideal perovskite structure, the A cations (blue) is located at the corners of the cube, the B cations (orange) in the centre and the X anions (red) in the face-centred positions. There are many magnetic and ferroelectric materials with the perovskite structure. What scientists want to find is a material simultaneously has the ferromagnetism and ferroelectricity with strong cross coupling. But, early studies represent that the conditions to cause the ferromagnetism and ferroelectricity usually interfere each other. For example, a con-dition for causing ferroelectricity requires empty d-orbitals, while for ferromagnetism requires partially filled d-orbitals. It is considered that the coexistence of ferromagnetism and ferro-electricity is impossible. However, out of speculation, some materials with perovskite structure do simultaneously have the ferromagnetism and ferroelectricity [29, 30], for example, RMnO3 (R=Tb, Dy, Ho). This is the so-called d0-dnproblem. A possible way to explain this problem is to make ”mixed” perovskite with d0 and dn ions. It means that the source of the ferromag-netism and ferroelectricity originate from different ions. The d0transition metal ions locate on B-cites, the empty d-orbitals hybridized with p-orbitals of the surrounding one or three oxy-gen atoms forming the strong covalent bond and causing the off-centered shifts which break the spatial inversion symmetry, i.e. causing the spontaneous electric polarization. And the dn ions locate on A-cites contributing to the ferromagnetism. There seens to be no contradiction in this way. Unfortunately, this kind of multiferroic materials have weak or no cross coupling between ferromagnetism and ferroelectricity since the sources are different.
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Figure 4.3: YMnO3- (a) The centrosymmetric structure above the critical temperature. (b) The ferroelectric structure below the critical temperature where practically rigid MnO5tilt breaking the spatial inversion symmetry.
Lone Pairs
There is another way oppsite to the above one to explain the coexistence of ferromagnetism and ferroelectricity, where ferroelectricity is caused by the A-cite cations, and the B-cite cations contribute to the ferromagnetism. This way usually occurs on those materials with perovskite structure having active ns2 electrons, called lone-pair, on the cations located on A-cite. Take BiFeO3for example, the outer Bi-6s2lone-pair cause the empty Bi-6p-orbital to come closer in energy to the O-2p-orbital. This lead to the hybridization of Bi-6p and O-2p orbitals and drive the off-centered shifts resulting in the ferroelectricity. And the dn transition metal ions locate on B-cites contributing to the ferromagnetism. No contradiction occurs. However, the magneto-electric effect in this kind of system is none or weak, due to different sources of ferromagnetism and ferroelectricity.
Structured Distortion
The mentioned two above are related to bondings that lead to the off-centered shifts of atoms.
Here we introduce a mechanism of ferroelectricity caused by the relatively complicated defor-mation of the crystal structure. Take YMnO3 [32]for example, a tilting of the practically rigid MnO5 in the crystal occurs as belowing the critical temperature. The asymmetric change of distance of Y-O bonds in the system lead to the ferroelectricity. And the ferromagnetism comes from the magnetic Mn3+ions.
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In the typical ferroelectric materials, the main source of ferroelectricity is the deformation of cations and anions in the crystal as above. Recently, it is found that the ferroelectricity can occur due to the charge ordering [31]. In some strong correlation systems, due to the strong interaction between electrons, charges are localized on different sites leading to a disproportion and an ordered superlattice breaking the spatial inversion symmetry. This is often observed in transition metal oxides, especially those formally containing transition metal ions with different valence. For example, the magnetite Fe3O4is a mixed-valence oxide where the iron atoms have a statistical distribution of Fe3+and Fe2+above the critical temperature. Below the critical tem-perature, the combination of Fe2+ and Fe3+ species arrange themselves in an ordering pattern, causing the ferroelectricity.