4.4 Multiferroics
4.4.2 Type-II Multiferroics
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.
4.4.2 Type-II Multiferroics
The theme of multiferroics which gets the most attention recently is the type-II multiferroics.
The discovery of strong magnetoelectric effect in the materials excite scientists to devote rather considerable effort to search for new compounds and theoretical models. They can be separated into two groups with different mechanisms of multiferroics, the spiral and collinear magnetic structures in the materials as follows.
Spiral Magnetic Structure
The first group is the spiral magnetic structure in the material which appears below the critical temperature along with the ferroelectricity. Most of the type-II multiferroics known to date belong to this group. Take the pioneering work on TbMnO3 for example [24]. Below TN1, the magnetic structure is a sinusoidal spin-density wave, and every spins point to the same direction but different magnitude of local moment. The magnitude of these moments varies periodically as shown in Figure 4.4 [33]. This structure is centrosymmetic and consequently not ferroelectric. As the temperature decreases below TN2, the cycloidal spiral with the wave vector Q = Qx and spins rotating in the xz-plane appears. The spontaneous electric polarization along the z-axis appears since the spiral magnetic structure break the spatial inversion symmetry. By applying the strong enough magnetic field along the y-axis, the direction of electric polarization will change from z-direction to x-direction. The strong magnetoelectric effect inspire scientists to find more new compounds and study the theoretical models. A microscopic approach [34]
and a phenomenological approach [35] are proposed to describe the mechanism of the electric polarization. The former, which is the so-called inverse effect of (relativistic) Dzyaloshinskii-Moriya interaction, gives a relation between the ferroelectricity and the arrangement of spins,
P ∝ ri j× [Si× Sj], (4.1)
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Figure 4.4: Different types of spin structures relevant for TbMnO3 - (a) TN2 < T < TN1. Si-nusoidal spin density wave. Every spins point to the same direction but different magnitude of local moment. The magnitude of these moments vary periodically. This structure is centrosym-metic and consequently not ferroelectric. (b) T < TN2. The cycloidal spiral magnetic structure with the wave vector Q = Qxand spins rotating in the xz-plane. This structure breaks the spa-tial inversion symmetry so that the spontaneous electric polarization appears. (c) In a so-called
”proper screw” the spins rotate in a plane perpendicular to Q. Here the inversion symmetry is broken, but most often it does not produce polarization.
where ri j is the vector pointing from spins Si to Sj. The magnitude of the polarization is pro-portional to the spin-orbit coupling constant. In a particular case, the spiral magnetic structure as shown in Figure 4.4(c) breaks the spatial inversion symmetry but appears no ferroelectricity most often, although in certain cases it might. And the later also gives a relation obtained by analysing the symmetries of the ferroelectricity and the magnetism of the system,
P ∝ [(M · ∇)M − M(∇ · M)]. (4.2)
Since the ferroelectricity occurs relating to the magnetism, it is not surprised that the ferroelec-tricity is strongly affected by the magnetic field.
Collinear Magnetic Structure
In the second group, the collinear magnetic structure in the material strongly affects the fer-roelectricity. All spin moments aligned along the same direction, and the spin-orbit coupling in the system is neglected. Since the strength of exchange striction of ferromagnetic and antifer-romagnetic arrangements of two neighbor spins (↑↑ and ↓↓) are different, the crystal distortion may occur breaking the spatial inversion symmetry, i.e. lead to the spontaneous electric po-larization. When the magnetic field is applied in the certain direction, direction of the electric polarization changed relating to the transformation of spin arrangement. Take Ca3CoMnO6 for example, the crystal has the inversion symmetry at high temperature. However, below the
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Figure 4.5: Ising chain composed of Co2+and Mn4+- Ising chains with the up-up-down-down spin order and alternating ionic order, in which electric polarization is induced through sym-metric exchange striction. The two possible magnetic configurations leading to the opposite polarizations are shown. The atomic positions in the undistorted chains are shown with dashed circles.
ical temperature, Co2+ and Mn4+ ions alternating along the simplified Ising chains exhibit an up-up-down-down (↑↑↓↓) magnetic order. Two possible crystal distortion due to the exchange striction are shown in Figure 4.5. The two possible magnetic configurations leading to the oppo-site polarizations are shown. In this case, valences of the two transition metal ions are different (Co2+ and Mn4+), and thus lead to the different strength of exchange striction. Interestingly, the same effect occur even for same magnetic ions despite the fact that the exchange interac-tion in transiinterac-tion metal oxides usually occurs via intermediate oxygens and depends on both the distance between the metal ions and the metal-oxygen-metal bond angle.