Room-Temperature Ferrimagnet with Frustrated Antiferroelectricity: Promising Candidate Toward Multiple State Memory

On the basis of first-principles calculations we show that the M-type hexaferrite BaFe12O19 exhibits frustrated antiferroelectricity associated with its trigonal bipyramidal Fe3+ sites. The ferroelectric (FE) state of BaFe12O19, reachable by applying an external electric field to the antiferroelectric (AFE) state, can be made stable at room temperature by appropriate element substitution or strain engineering. Thus M-type hexaferrite, as a new type of multiferoic with coexistence of antiferroelectricity and ferrimagnetism, provide a basis for studying the phenomenon of frustrated antiferroelectricity and realizing multiple state memory devices.

The phenomenon of frustration, typically observed in the field of magnetism, is found in solids [1] and soft materials [2]. Prototype examples of geometrical spin frustration are provided by magnetic systems consisting of triangular or pyrochlore spin lattice with nearest-neighbor antiferromagnetic spin exchange. Frustrated magnetic systems can give rise to exotic phenomena such as quantum spin liquid [3] and spin-order induced ferroelectricity [ 4,5]. An important question in the field of ferroelectricity is whether geometrically frustrated antiferroelectricity exists or not. Currently, all well-known multiferroics (e.g., TbMnO3, BiFeO3) possess simultaneously ferroelectricity and antiferromagnetism. It is not clear whether antiferroelectricity can coexist with ferromagnetism.
Hexaferrite contains triangular lattices of Fe 3+ ions, which has been found to display intriguing magnetoelectric (ME) effects at room temperature and low magnetic fields (~ 0.01 T) [6,7,8,9]. The room-temperature insulating ferrimagnetic undoped M-type hexaferrite, AFe12O19 (A = Ca, Sr, Ba, Pb, etc.), was widely believed [10] to crystallize in the magnetoplumbite-type centrosymmetric structure (space group P63/mmc) with high-spin Fe 3+  TBP site has been controversial. In the centrosymmetric structure, the TBP Fe 3+ ion lies on the equatorial plane (i.e., the local mirror plane) of the TBP FeO5. The x-ray-diffraction study at room temperature [11] suggested that the Fe 3+ ion is displaced out of the equatorial mirror plane by about 0.16 Å, which was supported by a Mössbauer study [12]. Collomb et al. [13] carried out neutron-diffraction studies for a number of hexagonal ferrites at room temperature and 4.2 K.
At room temperature, they found an even greater displacement (i.e., 0.26 Å) of the TBP site ions.
However, their structure refinement at 4.2 K suggested a freezing of the Fe 3+ ion at the mirror-2 plane site, and this was supported by empirical rigid-ion model calculations [14]. In this work, we carry out a comprehensive first-principles study to resolve this controversy and reveal that the M-type hexaferrite BaFe12O19 exhibits frustrated antiferroelectricity, giving rise to both roomtemperature polar order and strong ferrimagnetic order. Our work predicts that BaFe12O19 represents a first example of a new type of multiferroic possessing both antiferroelectricity and ferrimagnetism.
The presence of a structural instability in BaFe12O19 can be examined by computing its phonon dispersion within the density functional theory (see [15] for details). In BaFe12O19, the high-spin Fe 3+ ions (S = 5/2) order ferrimagnetically below 450 °C with 16 up-spin and eight down-spin Fe 3+ ions per unit cell, resulting in a net magnetization 20 μB per unit cell [10] (see Fig. S1). We first show that this ferrimagnetic state is the magnetic ground state by calculating the spin exchange parameters of BaFe12O19 (see SM), and then carry out phonon calculations for the ferrimagnetic spin ground state. Contrary to the conclusion of the empirical rigid-ion model [14], our calculations show the presence of two unstable modes in the whole Brillouin zone of the phonon dispersion [ Fig. 1(b)], providing a clear evidence for the structural instability in BaFe12O19. Both unstable phonon modes at Γ are contributed mainly by the displacement of the Fe 3+ ions at the TBP sites along the c axis [ Fig. 1(c)]. The lower (higher) frequency mode is associated with the in-phase (out-of-phase) vibration of the two TBP Fe 3+ ions in the unit cell.
The eigenvectors of the two modes can be used to generate one FE and one AFE structure along the c axis. After performing structural relaxations, the FE and AFE structures become more stable than the centrosymmetric paraelectric (PE) structure by 4.3 meV/f.u. and 0.1 meV/f.u., respectively. This further evidences the structural instability of the TBP Fe 3+ ions at the local 3 mirror-plane sites. In the FE structure, the TBP Fe 3+ ion moves out of the mirror plane by 0.19 Å, in good agreement with the experimental result (0.16 Å) [11].
We now investigate the interaction between the local dipoles caused by the displacements of the TBP Fe 3+ ions by considering the five different dipole arrangements ( After structural relaxations, we obtain the relative energies of these five states summarized in where ij e  is a unit vector parallel to the line joining the centers of the two dipoles, r is the distance between two dipoles, i p  and j p  , and C is a constant related to the dielectric constant. 4 It can be easily seen that the two ferroelectrically aligned dipoles along the c axis has a lower DDI energy than that of an antiparallel dipole pair, while two dipoles with the dipole direction perpendicular to the distance vector tend to be antiparallel to each other (see the inset of Fig. 2). To find out the ground state configuration of the dipole arrangement, we adopt two approaches.
One is to enumerate all the symmetrically nonequivalent configurations with the total number of dipoles no more than 12 in each supercell [17]. We find that the 2 1 × AFEab-FEc state has the lowest DDI energy. The other approach is to perform parallel tempering Monte Carlo (MC) [18] simulations, which confirm that the 2 1 × AFEab-FEc state (space group: Pnma) is the ground state and is consistent with the DFT result that it has the lowest energy among all five considered states. The ground state of the NN antiferromagnetic (AFM) Ising model on a triangular lattice is known to have a macroscopic degeneracy. The 6-fold degenerate 2 1 × AFEab-FEc state has the lowest energy due to the long range nature of the DDI. As a matter of fact, a similar 2 1 × chainlike AFM state is the ground state of the Ising model on a triangular lattice with AFM NN and 5 AFM next NN interactions [19]. Although the AFEab-FEc state has the lowest energy, there are many low-lying excited states, and this affects the thermodynamic properties of BaFe12O19.
We perform parallel tempering MC simulations on a 10 10 4 × × lattice using the DDI model to determine the thermodynamic properties of BaFe12O19. The effect of vibrational free energy is discussed in SM. Our calculations reveal that there is a sharp peak at around 3 K in the specific heat curve, indicating a long-range order of the dipoles (see Fig. 3 . This is so because the distance (5.8 Å) between the NN dipoles in the plane is much shorter than that (11.5 Å) between the NN dipoles along the c axis, thus the in-plane DDI is much stronger. The inplane correlation i j ab p p 〈 ⋅ 〉   saturates to 1 3 − at the transition temperature, which is the smallest value that can be achieved in a 2D Ising triangular system. However, this does not mean that the system fully orders in the ground state below the transition temperature because there are many low-lying excited states with the same i j ab p p 〈 ⋅ 〉   and i j c p p 〈 ⋅ 〉   as does the ground state. FEc ground state, the order parameter is 1. Fig. 3 shows that this parameter starts to become nonzero only when the temperature is below the transition point, suggesting that the low temperature phase is the AFEab-FEc state.
We now compare our theoretical results with previous experiments. We find that the TBP Fe 3+ ion is displaced out of the equatorial mirror plane, which agrees with the x-ray-diffraction study [11] and the Mössbauer study [12]. The neutron-diffraction study by Collomb et al. [13] found a large displacement of the TBP Fe 3+ ion at room temperature, but suggested that the TBP Fe 3+ ion freezes at the mirror-plane site at 4.2 K. This is contrary to the usual phenomenon that symmetry lowers when temperature decreases as predicted by Landau's theory. The puzzling neutron-diffraction results may be due to the frustrated nature of the TBP Fe 3+ related dipoles and the fact that the TBP Fe 3+ ion is at the mirror-plane site on average in the AFE state. A future neutron-diffraction study at very low temperature (e.g., 1 K) may confirm the AFE ground state predicted in this work.  Fig. 4(a) shows that the FE state becomes more stable by replacing the TBP Fe 3+ ions with Al 3+ or Ga 3+ ions, but becomes less stable when the TBP Fe 3+ ions are replaced with Sc 3+ and In 3+ ions. This is the PE-like arrangement becomes more stable. The ionic radii [21] increase in the order Al 3+ < Ga 3+ < Fe 3+ < Sc 3+ < In 3+ , suggesting that the stability of the FE state follows the relationship Al 3+ >Ga 3+ >Fe 3+ >Sc 3+ >In 3+ , in good agreement with our first-principles results [see Fig. 4(a)].
Thus, the FE state of BaFe12O19 can be made more stable at a higher temperature if the TBP Fe 3+ ions can be selectively replaced with smaller cations Al 3+ and Ga 3+ . 8 The other way of stabilizing the FE state at room temperature is to apply compressive epitaxial strain [ Fig. 4(b)]. An in-plane compressive epitaxial strain makes the TBP FeO5 elongated along the c axis, so the distance between the Fe ion and the apical O ion becomes longer than the sum of the ionic radius, which enhances the FE distortion of the TBP Fe 3+ site.
The stability of the FE state increases with decreasing the in-plane lattice constant. For example, if BaFe12O19 is grown on CaFe12O19 (which introduces a 2% compressive strain), the FE state is more stable than the PE state by about 26 meV/dipole, close to the room temperature energy scale. Our molecular dynamics [22] simulation shows that the FE state at a 5% compressive strain is stable at least up to room temperature: The TBP Fe 3+ ions stay at the original positions of an initial FE state after a 5 ps simulation at 300 K. As can also be seen from Fig. S7 of SM, the FE state becomes even more stable than the AFE state when the compressive strain is larger than 4% possibility due to the strain-polarization coupling. We also calculate the energy barrier from the FE state to the AFE state by using the climbing image nudged elastic band method [23].
As shown in Fig. 5 In summary, the M-type hexaferrite BaFe12O19 exhibits frustrated antiferroelectricity due to the local dipole moments arising from its TBP Fe 3+ ion sites, and is a novel multiferroic with both ferrimagnetic order and antiferroelectric order. The M-type hexaferrites are expected to provide a basis for realizing room-temperature multiple state memory devices.
We thank Professor M.-H. Whangbo for invaluable discussions. Work was supported by