Electric toroidal dipole order and hidden spin polarization in ferroaxial materials

Abstract

We investigate the role of electric toroidal dipoles in the prototypical ferroaxial materials NiTiO_{3} and K_{2}Zr(PO_{4})_{2}, which undergo ferroaxial structural phase transitions of order-disorder and displacive type, respectively. Using first-principles electronic structure theory, we compute the evolution across the ferroaxial transitions of the local electric toroidal dipole moments, defined in terms of both the vortices formed by local dipoles as well as the cross product of orbital and spin angular momenta. Our calculations confirm that the electric toroidal dipole acts as the order parameter for these ferroaxial transitions and highlight the importance of spin-orbit coupling in generating a nonzero atomic-site electric toroidal dipole moment. We find that, while the ferroaxial phases of NiTiO_{3} and K_{2}Zr(PO_{4})_{2} preserve global inversion symmetry, they contain inversion-symmetry-broken subunits that generate vortices of local electric dipole moments. In addition to causing the net electric toroidal dipole moment, these vortices induce a hidden spin polarization in the band structure.