History of ferroelectricity in Magnetite

Despite the controversies about the existence of charge ordering, magnetite has been proposed as one of the charge- order- based multiferroics. Indeed experimental indications of ferroelectric polarization [102, 124] and the anomalous dielectric properties were already reported in the early 1980’s [125]. Even before that, magnetoelectric effect was measured below the Verwey transition and interpreted by a model with spontaneous polarization along the b-axis [101, 126]. Despite these experimental reports about ferroelectricity neither detailed ferroelectric properties and its mechanism, nor a clear proof of ferroelectricity were available. In 2006 Khomskii had proposed a theoretical model for charge order based ferroelectricity in magnetite by the co-existence of both site and bond centered charge ordering, which break the inversion symmetry and shows the net dipole moment [62, 95]. Besides the site-centered charge ordering, the distances between the charge ordered Fe²⁺_B and Fe³⁺_B sites are strongly modulated along the monoclinic b-direction in which the polarization is observed. In addition to alternation of Fe²⁺_B and Fe³⁺_B ions, there is an alternation of short and long Fe-Fe bonds and is supported by the low temperature structural refinement by Wright etal [123] where they observed the strong modulation between the two charge ordered Fe sites along the b-direction from 2.86 ˚A to 3.05 ˚A. Also his model, based on Wright et al [123], assuming the iron ions on the octahedrally coordinated site form a network of iron tetrahedra, proposed each tetrahedron to show 3:1 charge order arrangement (three Fe²⁺ and one Fe³⁺). However this model is in contrast to Anderson’s criterion, where each tetrahedron has a 2:2 pattern.

But the low temperature refinement by Wrightet al was performed using a small sub cell a_c/√

2×a_c/√

2×2a_c instead of the large a_c√

2×a_c√

2×2a_c Cc cell [114].

Subsequent to Khomskii’s model, a theoretical calculation(DFT) done by Yamauchi et al proposed the ferroelectricity in magnetite to be induced by a noncentrosymmetric charge ordering with the polarization primarily being induced by the localised charged shifts [67]. The calculated P2/c structure as paraelectric shows 3:1 charge configuration, but cancels out the total polarization due to the structure shows inversion symmetry.

Whereas the calculations done considering the Cc lattice symmetry is ferroelectrically active due to the lack of center of symmetry and shows mixed pattern of 3:1 and 2:2 charge ordered configuration (25 % of 2:2 and 75 % of 3:1 tetrahedra). In the figure below, the charge shift can be understood assuming the shift of B12⁰ from B14⁰ site and B12

Figure 4.3: Schematic representation of the possible origin of ferroelectricity in magnetite, from ref [62]. Emphasized are the Fe chains in the B-site running along the [110] directions(in the cubic setting).In th xy chain, in addition to the alternation of Fe²⁺ and Fe³⁺, there is also an alternation of short and long Fe-Fe bonds. The black arrow indicates the shift of the Fe ions and the red arrows indicate the resulting net polarization.

to B14 site of the cell. Each charge shift produces two 2:2 charge ordered tetrahedra, so as to form four 2:2 tetrahedra. The resulting charge ordered pattern lacks inversion symmetry and allows for ferroelectric polarization.

The polarization values calculated from DFT on a monoclinic Cc structure (from Berry phase model [127, 128] , P_a = 4.41µC/cm², P_c = 4.12µC/cm²) is fairly in good agreement with the recently reported experimental value of real-time ferroelectric switching in magnetite expitaxial thin film (P ∼ 11µC/ cm²) [103] as well as with earlier reports on single crystals (P_a = 4.8µC/cm² and P_c = 1.5µC/cm²) [102]. But the recent polarization measurement on a single crystals [97] showed P = 0.5µC/cm² which is very less compare to the value obtained by thin film. However, point-charge model calculation from Senn et al on recently solved low temperature charge ordered structure provided the polarization P_a = 0.118 C/cm² and P_c = 0.405 C/cm², which is several order higher than the previous calculations. Their calculation also indicated that the more than 80% of the polarization in magnetite is induced by charge order and three site distortion. However, there was no microscopic experimental proof of supporting the CO ferroelctricity in magnetite. Even though for a ferroelectric material even if the

4.1 Magnetite

Figure 4.4: Ionic structure model of Fe octahedral sites in Cc cell. Fe²⁺ and Fe³⁺ ions are represented by orange and blue balls respectively. Yellow and green color planes indicates the 2:2 and 3:1 charge ordered pattern of Fe₄ tetrahedra. Red arrow indicates the charge shift with will induce the electric dipole moment, taken from reference [67].

polar structure is established, the switching of this polar structure needs to be shown.

Recently, Schrettle and co-workers proposed that magnetite is a relaxor ferroelctric below 40 K rather than a normal ferroelectric, based on their observation of strongly frequency dependent of dielectric properties and a continuous slowing down of its polar dynamics, dominated by tunnelling at low temperatures [97]. As shown in the figure 4.5, magnetite shows a broad peak in the real part of the dielectric permittivity as a function of temperature and this peak decreases in magnitude and is shift to higher temperature with increasing frequency. This is the typical behavior of relaxor ferroelectrics [129]. But compared to the dielectric properties of classical relaxor, e.g., figure 4.11 magnetite exhibits larger step like feature, which is of extrinsic origin. One of the reason could be the presence of residual conductivity in the ferroelectric phase of magnetite, which is very well visible in the P(E) curves shown in the figure 5.3.

Figure 4.5: Temperature dependent of ⁰of magnetite for various frequencies taken from reference [97]. Symbols are with silver contact and dotteds line indicates the measurements repeated with the gold contacts. The intrinsic features, indicated by the dashed line representing Curie-Weiss behavior, are better visible in the measurements done with gold contacts.

4.2 Synthesis and effect of non-stoichiometry on the