Spin-Orbital Driven Ferroelectricity

Proposing a general formulism in terms of local coordinates representing the tilting of the ligands' octahedra, we evaluate the electric polarization in a chain of transition metal ions with unpolar octahedron rotation. We find the orbital ordering produced by the ligands's rotation and the spin order, together, determine the polarization features, manifesting that nonvannishing polarization appears in collinear spin order and the direction of polarization is no more restricted in the plane of spin rotation in cycloidal ordering.