Spin-Orbit Coupling and Ion Displacements in MultiferroicTbMnO3

The magnetic and ferroelectric (FE) properties of ${\mathrm{TbMnO}}_3$ are investigated on the basis of relativistic density functional theory calculations. We show that, due to spin-orbit coupling, the spin-spiral plane of ${\mathrm{TbMnO}}_3$ can be either the $bc$ or $ab$ plane, but not the $ac$ plane. As for the mechanism of FE polarization, our work reveals that the “pure electronic” model by Katsura, Nagaosa, and Balatsky is inadequate in predicting the absolute direction of FE polarization. Our work indicates that to determine the magnitude and the absolute direction of FE polarization in spin-spiral states, it is crucial to consider the displacements of the ions from their centrosymmetric positions.