Successive spin reorientation and rare earth ordering in Nd$_{0.5}$Dy$_{0.5}$FeO$_{3}$: Experimental and $Ab$-$initio$ investigations

Abstract: In present study, the magnetic structure and spin reorientation of mixed doped orthoferrite Nd${0.5}$Dy${0.5}$FeO$3$ have been investigated. Similar to both parent compounds (NdFeO$_3$ and DyFeO$_3$), the magnetic structure of Fe$^{3+}$ belongs to ${\Gamma}{4}$ irreducible representation (G${x}$, F${z}$) at room temperature. The experimental measurements confirmed the spin reorientation where magnetic structure of Fe$^{3+}$ changes from ${\Gamma}{4}$ to ${\Gamma}{2}$(F${x}$, G${z}$) between 75 and 20 \,K while maintaining G-type configuration. Such a gradual spin reorientation is unusual since the large single ion anisotropy of Dy$^{3+}$ ions causes an abrupt ${\Gamma}{4}$${\rightarrow}$ ${\Gamma}{1}$(G${y}$) spin reorientation in DyFeO$_3$. Between 20 and 10 \,K, the Fe$^{3+}$ magnetic structure is represented by ${\Gamma}{2}$ (F${x}$, G${z}$). Unexpectedly, magnetic structure of Fe$^{3+}$ with ${\Gamma}{4}$ representation re-emerges below 10\,K which also coincides with the development of rare-earth (Nd$^{3+}$/Dy$^{3+}$) magnetic ordering having C${y}$ configuration with magnetic moment of 1.8 ${\mu}{B}$. The absence of any signature of second order phase transition in the specific heat confirms the role of $R$(Nd$^{3+}$/Dy$^{3+}$)-Fe$^{3+}$ exchange interaction in the observed "rare-earth ordering" unlike DyFeO$_3$ where Dy$^{3+}$ ordering takes place independently to the magnetic ordering of Fe$^{3+}$ magnetic structure. Our (DFT+U+SO) calculations show that the C-type arrangement of rare-earth ions (Nd$^{3+}$/Dy$^{3+}$) with ${\Gamma}{2}$ configuration for Fe$^{3+}$ moments is the ground state whereas ${\Gamma}_{4}$ phase is energetically very close. Nd-Fe and Nd-Dy exchange interactions, estimated from DFT, are observed to have significant roles in the rare earth ordering and Fe spin reorientation corroborating our experimental results.