Trends in spin exchange interactions and ferroelectric polarization in the orthorhombic $R$MnO$_{3}$ series

Recently, magnetic ferroelectricity induced by Heisenberg-type interactions has been theoretically predicted in E-type antiferromagnetic (AFM) HoMnO$_{3}$ \footnote{S. Picozzi et al., Phys. Rev. Lett. (in press)}. In order to fully clarify this unconventional microscopic mechanism, we have studied the structural, magnetic and ferroelectric properties for the entire family of orthorhombic $R$MnO$_{3}$ ($R$= rare earth ions), based on first-principles density functional calculations. The ferromagnetic exchange interaction between nearest-neighbor Mn sites decreases with the ionic radius of $R$ (concomitantly with the in-plane Mn-O-Mn bond angle), whereas the anti-ferromagnetic next-nearest neighbor interaction stays rather constant in the series. The competition of these exchange interactions results in a complicated magnetic phase diagram. The decrease in the Mn-O-Mn angle also affects the hopping integrals between Mn ions (as determined from Wannier functions), so that the calculated electric polarization in E-type AFM $R$MnO$_{3}$ is remarkably reduced throughout the rare earth series.