Coexisting magnetic structures and spin-reorientation in Er$_{0.5}$Dy$_{0.5}$FeO$_{3}$: Bulk magnetization, neutron scattering, specific heat, and \emph{Ab-initio} studies

The complex magnetic structures, spin-reorientation and correlated exchange interactions have been investigate in Er$_{0.5}$Dy$_{0.5}$FeO$_3$ using bulk magnetization, neutron diffraction, specific heat measurements and density functional theory calculations. The Fe$^{3+}$ spins order as G-type antiferromagnet structure depicted by ${\Gamma}_{4}$($G_{x}$,$A_{y}$,$F_{z}$) irreducible representation below 700K, similar to its end compounds. The bulk magnetization data indicate occurrence of the spin-reorientation and rare-earth magnetic ordering below $\sim$75 K and 10 K, respectively. The neutron diffraction studies confirm an "incomplete" ${\Gamma}_{4}$${\rightarrow}$ ${\Gamma}_{2}$($F_{x}$,$C_{y}$,$G_{z}$) spin-reorientation initiated $\leq$75 K. Although, the relative volume fraction of the two magnetic structures varies with decreasing temperature, both co-exist even at 1.5 K. At 8 K, Er$^{3+}$/Dy$^{3+}$ moments order as $c_{y}^R$ arrangement develop, which gradually increases in intensity with decreasing temperature. At 2 K, magnetic structure associated with $c_{z}^R$ arrangement of Er$^{3+}$/Dy$^{3+}$ moments also appears. At 1.5 K the magnetic structure of Fe$^{3+}$ spins is represented by a combination of ${\Gamma}_{2}$+${\Gamma}_{4}$+${\Gamma}_{1}$, while the rare earth moments coexists as $c_{y}^R$ and $c_{z}^R$ corresponding to ${\Gamma}_{2}$ and ${\Gamma}_{1}$ representation, respectively. The observed Schottky anomaly at 2.5 K suggests that the "rare-earth ordering" is induced by polarization due to Fe$^{3+}$ spins. The Er$^{3+}$-Fe$^{3+}$ and Er$^{3+}$-Dy$^{3+}$ exchange interactions, obtained from first principle calculations, primarily cause the complicated spin-reorientation and $c_{y}^R$ rare-earth ordering, respectively, while the dipolar interactions between rare-earth moments, result in the $c_{z}^R$ type rare-earth ordering at 2 K.