Analysis of the magnetic structure and spin exchange interactions of multiferroic YBaCuFeO$_{5}$ by first principles DFT calculations

In the layered perovskites RBaCuFeO$_{5}$ (R = Y, Lu, Tm), the CuFeO$_{9}$ dumbbells made up of apex-sharing CuO$_{5}$ and FeO$_{5}$ square pyramids share their basal corners to form perovskite layers, and the resulting CuFeO$_{5}$ slabs are stacked along the c-direction. Recently, these compounds were found to exhibit ferroelectric polarization when a modulated magnetic component is superposed on their antiferromagnetic structure. To help understand this finding, we examined the spin exchange interactions between the Fe$^{3+}$ (d$^{5})$ ions, between the Cu$^{2+}$ (d$^{9})$ ions, and between the Fe$^{3+}$ and Cu$^{2+}$ ions on the basis of DFT+U and DFT+U+SOC calculations for YBaCuFeO$_{5}$. The ferroelectric polarization of YBaCuFeO$_{5}$ was also calculated for several modulated magnetic structures that were constructed based on the cone-model.