Ferromagnetic ferroelectricity due to orbital ordering

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 the inversion symmetryI. Therefore, such a phase cannon be obtained by purely magnetic means, as in other multiferroics with complex magnetic order. Here, we show how it can be designed by considering orbital degrees of freedom. The idea can be traced back to a basic principle of interatomic exchange coupling, which states that an alternation of occupied orbitals along a bond (i.e., antiferro orbital order) favors ferromagnetic interactions between the spins. The new aspect of this canonical picture is that the antiferro orbital order breaksI, so that the bond becomes not only ferromagnetic but also ferroelectric. Then, we formulate basic 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 change 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 are iodides with a d2configuration and relatively weak d-p hybridization. The situation is illustrated on the van der Walls compound VI3, which we expect to be ferromagnetic ferroelectric.