Ferromagnetic Ferroelectricity due to Orbital Ordering

TL;DR

The article proposes a route to ferromagnetic ferroelectricity by activating orbital degrees of freedom to realize antiferro orbital order that both strengthens ferromagnetic exchange and breaks inversion symmetry. Building on Goodenough-Kanamori-Anderson and Kugel-Khomskii frameworks, the authors show how orbital flexibility, enhanced by Hund’s second rule, can overcome the obstacles that prevent FM order from inducing polarization. The honeycomb lattice VI$_3$ emerges as a promising d$^{2}$ system where Kugel-Khomskii-type orbital ordering can drive a FM-FE state, with Hartree-Fock analyses indicating that inversion symmetry breaking via orbital order is essential for stabilizing ferromagnetism and enabling magnetoelectric coupling. Magnetic-field control of polarization is predicted through spin-orbit coupling that mixes orbital states, yielding sizable polarization changes and hysteresis, which could enable practical magnetoelectric devices. Overall, the work outlines a principled design strategy for FM-FE materials and highlights Hund’s second-rule effects as a new lever in strongly correlated oxides and van der Waals magnets.

Abstract

Realization of ferromagnetic ferroelectricity, combining two ferroic orders in a single phase, is the longstanding problem of great practical importance. One of the difficulties is that ferromagnetism alone cannot break inversion symmetry $\mathcal{I}$. Therefore, such a phase cannon be obtained by purely magnetic means. Here, we show how it can be designed by making orbital degrees of freedom active. The idea can be traced back to a basic principle of interatomic exchange, which states that an alternation of occupied orbitals along a bond (i.e., antiferro orbital order) favors ferromagnetic coupling. Moreover, the antiferro orbital order breaks $\mathcal{I}$, so that the bond becomes not simply ferromagnetic but also ferroelectric. Then, we formulate main principles governing the realization of such a state in solids, namely: (i) The magnetic atoms should not be located in inversion centers, as in the honeycomb lattice; (ii) The orbitals should be flexible enough to adjust they shape and minimize the energy of exchange interactions; (iii) This flexibility can be achieved by intraatomic interactions, which are responsible for Hund's second rule and compete with the crystal field splitting; (iv) For octahedrally coordinated transition-metal compounds, the most promising candidates appear to be iodides with a $d^{2}$ configuration and relatively weak $d$-$p$ hybridization. Such a situation is realized in the van der Walls compound VI$_3$, which we expect to be ferromagnetic ferroelectric.

Figures (11)

• Figure 1: Total and partial densities of states of cubic BaTiO$_3$ and KNbO$_3$ in the local density approximation. The Fermi level is in the middle of the band gap (shown by dot-dashed line).
• Figure 2: Spin spirals and inversion symmetry breaking. (a) Conical and (b) cycloidal magnetic order. $\boldsymbol{q}$ is the propagation vector. (c) Illustration of inversion symmetry breaking by noncollinear spins in centrosymmetric bond. $\times$ is the inversion center. The noncollinear spin configuration can be decomposed in ferromagnetic and antiferromagnetic counterparts. The former is invariant under $\mathcal{I}$, while the latter is invariant under $\mathcal{IT}$. Since $\mathcal{I}$ cannot coexist with $\mathcal{IT}$, the inversion symmetry is broken (from MDPI2025).
• Figure 3: Examples of ferro and antiferro orbital order around the inversion center. Ferro orbital order tends to stabilize AFM coupling and preserve the inversion symmetry in the bond. Antiferro orbital order stabilizes FM coupling and breaks the inversion symmetry.
• Figure 4: Illustration of superexchange interactions: (a) Example of occupied and unoccupied orbitals in the bond; (b) Transfer integrals between occupied and empty states in the case of FM and AFM alignment. The splitting is $\Delta^{\uparrow \uparrow} = U - J_{\rm H}/2 \equiv \tilde{U}$ and $\Delta^{\uparrow \downarrow} = U + J_{\rm H}/2$.
• Figure 5: Example of the antiferro orbital order realized in perovskite (a) and honeycomb (b) lattice. In the perovskite lattice, the directions of polarization induced in each bond are shown by arrows. The inversion centers are shown by circles (which coincide with the positions of oxygen atoms in undistorted perovskite structure).