Electric toroidal dipole order and hidden spin polarization in ferroaxial materials

TL;DR

This study explores how electric toroidal dipoles serve as order parameters in ferroaxial materials, revealing hidden spin polarization linked to local dipole vortices and emphasizing the role of spin-orbit coupling.

Contribution

It demonstrates the first-principles calculation of electric toroidal dipoles as order parameters in ferroaxial phase transitions and uncovers hidden spin polarization effects.

Findings

Electric toroidal dipoles act as order parameters in ferroaxial transitions.

Local dipole vortices generate hidden spin polarization.

Spin-orbit coupling is crucial for non-zero electric toroidal dipoles.

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 both in terms of the vortices formed by local dipoles, as well as 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 non-zero 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 sub-units 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.