Successive spin reorientations and rare earth ordering in ${\mathrm{Nd}}_{0.5}{\mathrm{Dy}}_{0.5}\mathrm{Fe}{\mathrm{O}}_3$: Experimental and ab initio investigations

In the present paper, the magnetic structure and spin reorientation of mixed rare-earth orthoferrite ${\mathrm{Nd}}_{0.5}{\mathrm{Dy}}_{0.5}\mathrm{Fe}{\mathrm{O}}_3$ have been investigated. At room temperature, our neutron-diffraction measurements reveal that the magnetic structure of ${\mathrm{Fe}}^{3+}$ spins in ${\mathrm{Nd}}_{0.5}{\mathrm{Dy}}_{0.5}\mathrm{Fe}{\mathrm{O}}_3$ belongs to ${\mathrm{\Gamma }}_4$ irreducible representation ($G_x, F_z$) as observed in both parent compounds ($\mathrm{Nd}\mathrm{Fe}{\mathrm{O}}_3$ and $\mathrm{Dy}\mathrm{Fe}{\mathrm{O}}_3$). The neutron-diffraction study also confirms the presence of a spin-reorientation transition where the magnetic structure of ${\mathrm{Fe}}^{3+}$ spins changes from ${\mathrm{\Gamma }}_4$ to ${\mathrm{\Gamma }}_2(F_x, G_z$) representation between 75 and 20 K while maintaining a G-type antiferromagnetic configuration. Such a gradual spin reorientation is unusual since the large single ion anisotropy of ${\mathrm{Dy}}^{3+}$ ions is expected to cause an abrupt ${\mathrm{\Gamma }}_4\to {\mathrm{\Gamma }}_1(G_y$) rotation of the ${\mathrm{Fe}}^{3+}$ spins. At 10 K, the ${\mathrm{Fe}}^{3+}$ magnetic structure is represented by ${\mathrm{\Gamma }}_2$ ($F_x, G_z$). Unexpectedly, the ${\mathrm{\Gamma }}_4$ structure of ${\mathrm{Fe}}^{3+}$ spins re-emerges below 10 K, which also coincides with the development of rare-earth (${\mathrm{Nd}}^{3+}/{\mathrm{Dy}}^{3+}$) magnetic ordering having $c_y^R$ configuration. Such re-emergence of a magnetic structure has been a rare phenomenon in orthoferrites. The absence of a second-order phase transition in rare-earth ordering, interpreted from heat capacity data, suggests the prominent role of ${\mathrm{Nd}}^{3+}-{\mathrm{Fe}}^{3+}$ and ${\mathrm{Nd}}^{3+}-{\mathrm{Dy}}^{3+}$ exchange interactions. These interactions suppress the independent rare-earth magnetic ordering observed in both parent compounds due to ${\mathrm{Nd}}^{3+}/{\mathrm{Dy}}^{3+}-{\mathrm{Nd}}^{3+}/{\mathrm{Dy}}^{3+}$ exchange interactions. Our density-functional-theory calculations including Coulomb correlation and spin-orbit interaction effects ($\mathrm{DFT}+U+\mathrm{SO}$) reveal that the C-type arrangement of rare-earth ions (${\mathrm{Nd}}^{3+}/{\mathrm{Dy}}^{3+}$), with ${\mathrm{\Gamma }}_2$ ($F_x, G_z$) configuration for ${\mathrm{Fe}}^{3+}$ moments, is energetically very close to a phase with the same rare-earth magnetic ordering but ${\mathrm{\Gamma }}_4$ ($G_x, F_z$) configuration of ${\mathrm{Fe}}^{3+}$ spins. Further, the ${\mathrm{Nd}}^{3+}-{\mathrm{Fe}}^{3+}$ and ${\mathrm{Nd}}^{3+}-{\mathrm{Dy}}^{3+}$ exchange interactions are observed to play significant roles in the complex ${\mathrm{Fe}}^{3+}$ spin reorientation with the re-emergence of ${\mathrm{\Gamma }}_4$ at low temperature. Consistent with the experimental observations, our calculations established the mixed phase (${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_4$) to be the magnetic ground state of ${\mathrm{Fe}}^{3+}$ moments.

Figure 1

Observed and simulated x-ray diffraction pattern of NDFO at 300 K, refined using $Pbnm$ space group.

Figure 2

Temperature variation of lattice parameters and unit cell volume of NDFO. Solid lines are guide to the eye.

Figure 3

ZFC-FC plots of NDFO at (a) 100 Oe and (b) 1000 and 5000 Oe.

Figure 4

M-H plots of NDFO at 200 K, 75 K, 45 K, 30 K, and 2 K.

Figure 5

Evolution of the neutron diffraction data along with refinement as a function of temperature.

Figure 6

Temperature variation of magnetic moments of ${\mathrm{Fe}}^{3+}$ and ${\mathrm{Nd}}^{3+}/{\mathrm{Dy}}^{3+}$ spins from 1.5 K to 300 K for the various representations.

Figure 7

Schematic representation of NDFO magnetic structure at (a) 300 K: ${\mathrm{\Gamma }}_4$ (b) reorientation region at 45 K: ${\mathrm{\Gamma }}_{42}$ (c) at 20 K : ${\mathrm{\Gamma }}_2$ and (d) 1.5 K second reorientation : ${\mathrm{\Gamma }}_{24}$ and ordering of Nd/Dy as $C_y$.

Figure 8

(a) The molar heat capacity of NDFO measured at 0 T over 2–220 K. (b) The molar heat capacity of NDFO and nonmagnetic analog $\mathrm{La}\mathrm{Ga}{\mathrm{O}}_3$ measured at 0 T over 0.4–15 K. Solid red line shows calculated lattice contribution to the molar heat capacity for $\mathrm{La}\mathrm{Ga}{\mathrm{O}}_3$. Dashed blue line: Renormalized lattice contribution for NDFO. Open triangles show molar heat capacity of NDFO at 5 T over 0.4–15 K. (c) Molar heat capacity for NDFO at 0 and 5 T after subtracting the lattice contribution.

Figure 9

Unit cells of NDFO displaying two possible arrangements of Nd and Dy ions considered in our calculations: (a) alternate, (b) layered.

Figure 10

Relative energies (in meV per unit cell) calculated from noncollinear calculations within $\mathrm{GGA}+U+\mathrm{SO}$ for ${\mathrm{Fe}}^{3+}$ spins along the three crystallographic directions for 300 K (red) and 1.5 K (black) crystal structures.

Figure 11

Magnetic structures of NDFO in (a) ${\mathrm{\Gamma }}_2$ phase and (b) mixed phase of ${\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_4$ with Fe spins canted in the $ac$ plane considered in our calculations. As shown in (b), the mixed phase is observed to be the ground state.

Figure 12

Temperature dependence of Fe sublattice magnetization under molecular exchange field provided by Fe-Fe exchange (gray), Nd/Dy-Fe exchange (red), and Nd-Dy exchange (blue). Next-nearest-neighbor Fe-Fe exchange of $-0.143$ meV is included in addition to the near-neighbor exchange interactions given in Table 3. The Nd-Dy exchange field was considered along $b$ direction only as Nd/Dy order in $c_y$ configuration.