Multiferroic crossover in perovskite oxides

The coexistence of ferroelectricity and magnetism in $AB{\mathrm{O}}_3$ perovskite oxides is rare, a phenomenon that has become known as the ferroelectric “$d^0$ rule.” Recently, the perovskite ${\mathrm{BiCoO}}_3$ has been shown experimentally to be isostructural with ${\mathrm{PbTiO}}_3$, while simultaneously the $d^6{\mathrm{Co}}^{3+}$ ion has a high-spin ground state with $C$-type antiferromagnetic ordering. It has been suggested that the hybridization of Bi $6s$ states with the O $2p$ valence band stabilizes the polar phase, however, we have recently demonstrated that ${\mathrm{Co}}^{3+}$ ions in the perovskite structure can facilitate a ferroelectric distortion via the Co $3d$–O $2p$ covalent interaction [L. Weston, et al., Phys. Rev. Lett. 114, 247601 (2015)]. In this paper, using accurate hybrid density functional calculations, we investigate the atomic, electronic, and magnetic structure of ${\mathrm{BiCoO}}_3$ to elucidate the origin of the multiferroic state. To begin with, we perform a more general first-principles investigation of the role of $d$ electrons in affecting the tendency for perovskite materials to exhibit a ferroelectric distortion; this is achieved via a qualitative trend study in artificial cubic and tetragonal $\mathrm{La}B{\mathrm{O}}_3$ perovskites. We choose La as the $A$ cation so as to remove the effects of Bi $6s$ hybridization. The lattice instability is identified by the softening of phonon modes in the cubic phase, as well as by the energy lowering associated with a ferroelectric distortion. For the $\mathrm{La}B{\mathrm{O}}_3$ series, where $B$ is a $d^0–d^8$ cation from the $3d$ block, the trend study reveals that increasing the $d$ orbital occupation initially removes the tendency for a polar distortion, as expected. However, for high-spin $d^5–d^7$ and $d^8$ cations a strong ferroelectric instability is recovered. This effect is explained in terms of increased pseudo-Jahn-Teller (PJT) $p-d$ vibronic coupling. The PJT effect is described by the competition between a stabilizing force ($K_0$) that favors the cubic phase, and a vibronic term that drives the ferroelectric state ($K_v$). The recovery of the lattice instability for high-spin $d^5–d^7$ and $d^8$ cations is due to (i) a reduction in $K_0$ due to a significant volume increase arising from population of the $\sigma$-bonded axial $d{e}_g$ orbitals, and (ii) an increase in the $K_v$ contribution arising from increased $p-d$ hybridization; our calculations suggest that the former mechanism is dominant. Surprisingly, we are able to show that, in some cases unpaired electron spins actually drive ferroelectricity, rather than inhibit it, which represents a shift in the understanding of how ferroelectricity and magnetism interact in perovskite oxides. It follows, that for the case of ${\mathrm{BiCoO}}_3$, the ${\mathrm{Co}}^{3+}$ ion plays a major role in the ferroelectric lattice instability. Importantly, the ferroelectric polarization is greatly enhanced when the ${\mathrm{Co}}^{3+}$ ion is in the high-spin state, when compared to the nonmagnetic, low-spin state, and a large coupling of the electric and magnetic polarization is present. Generally, for $d^5–d^7 B$ cations in $AB{\mathrm{O}}_3$ perovskites, an inherent and remarkably strong magnetoelectric coupling exists via the multiferroic crossover effect, whereby switching the spin state strongly affects the ferroelectric polarization and, potentially, manipulation of the polarization with an externally applied electric field could induce a spin-state transition. This novel effect is demonstrated for ${\mathrm{BiCoO}}_3$, for which the ground spin state is switched by reducing the internal ferroelectric polarization. These results provide a deeper insight into perovskite ferroelectrics and multiferroics.

Figure 1

(a) The unit cell of an $AB{\mathrm{O}}_3$ perovskite oxide with a cubic crystal structure. The $B$ cation is at the center of the unit cell and is sixfold coordinated to oxygen. (b) The same system with a tetragonal ferroelectric distortion. The distortion is characterized by an elongation of the $c$ axis compared to the in-plane lattice vectors, as well as a shift of the $B$ cation with respect to the surrounding O-ion octahedron; the internal distortions break inversion symmetry, leading to a net macroscopic electrical polarization. The $A$ cation of the perovskite structure is represented by a green sphere, the $B$ cation is blue, and O ions are red.

Figure 2

A molecular orbital diagram for an $AB{\mathrm{O}}_3$ perovskite. The bonding states ($\pi$ and $\sigma$) have predominant O $2p$ character, the antibonding states (${\pi }^{*}$ and ${\sigma }^{*}$) have predominant $B$-cation $d$ character. For simplicity, only the valence states are included, and the $A$-cation orbitals have been neglected.

Figure 3

In (a), the projected partial density of states is plotted for low-spin (LS) (top) and high-spin (HS) (bottom) ${\mathrm{LaFeO}}_3$. The dotted red line represents O $2p$ states, the solid blue and purple lines represent, respectively, the $t_{2g}$ ($d_{xy}, d_{xz}, d_{yz}$) and $e_g$ ($d_{z^2}, d_{x^2-y^2}$) orbitals of the Fe $3d$ band. In (b) and (c), the $d$-orbital configurations are shown for LS and HS ${\mathrm{Fe}}^{3+}$.

Figure 4

Crystal structure of ${\mathrm{LaFeO}}_3$ (LFO) in the low-spin (a) and high-spin (b) states. In (c), the total energy of ${\mathrm{LaFeO}}_3$ is calculated as a function of the Fe ion position in the unit cell, presented as a fraction of the $c$-axis lattice vector. Sr atoms are represented by green spheres, Fe are brown, O are red.

Figure 5

(a) On the left the ground-state orbital configuration (${\mathrm{\Psi }}_0$) is shown for high-spin ${\mathrm{Fe}}^{3+}$ in ${\mathrm{LaFeO}}_3$. On the right, the same orbital configuration is shown, however, an electron has been promoted from the O $2p$ band into the $e_g$ band of ${\mathrm{Fe}}^{3+}$, the label ${\mathrm{\Psi }}_1^{e_g}$ indicates that this is the lowest-energy excited state involving promotion of an electron into the $e_g$ band. As can be seen, ${\mathrm{\Psi }}_0$ and ${\mathrm{\Psi }}_1^{e_g}$ have the same spin $S=\frac{5}{2}$, and so ${\mathrm{\Psi }}_1^{e_g}$ contributes to the summation in Eq. (5). (b) The energy difference between the O $2p$ and $B$-cation $3d$ band centers is plotted for the $\mathrm{La}B{\mathrm{O}}_3$ series. The position of the band center is approximated [45], and this plot should only be considered as a qualitative guide.

Figure 6

The charge density isosurface is shown for the $e_g$ band of low-spin (LS) (left) and high-spin (HS) (right) ${\mathrm{LaFeO}}_3$. The isosurface is set to 10% of the maximum value. La ions are represented by green spheres, Fe are brown, and O are red.

Figure 7

The total energy (per formula unit) of cubic high-spin (HS) ${\mathrm{BiCoO}}_3$ and cubic HS ${\mathrm{LaCoO}}_3$ is plotted as a function of the $A$-cation displacement from the high-symmetry site.

Figure 8

The calculated crystal structure for ${\mathrm{BiCoO}}_3$ in the low-spin (a) and high-spin (b) states. In (c), the ground-state $C$-type antiferromagnetic ordering of the high-spin Co ions is shown. Bi atoms are represented by purple spheres, Co are blue, and O are red.

Figure 9

The partial density of states (DOS) for low-spin (LS) (top) and high-spin (HS) (bottom) ${\mathrm{BiCoO}}_3$. The LS state of ${\mathrm{BiCoO}}_3$ is nonmagnetic, whereas the HS state has $C$-type antiferromagnetic ordering. The DOS has been shifted so that the Fermi level is at zero.

Figure 10

The total energy (per formula unit) of low-spin (LS) and high-spin (HS) ${\mathrm{BiCoO}}_3$ is plotted as a function of the generalized coordinate. To model the dependence of the spin-state energetics on the polarization, the internal coordinates are moved continuously from those of the relaxed high-spin structure (generalized coordinate = 1) to a paraelectric structure (generalized coordinate = 0).