Structural and electronic origin of the magnetic structures in hexagonal ${\mathrm{LuFeO}}_3$

Using combined theoretical and experimental approaches, we studied the structural and electronic origin of the magnetic structure in hexagonal ${\mathrm{LuFeO}}_3$. Besides showing the strong exchange coupling that is consistent with the high magnetic ordering temperature, the previously observed spin reorientation transition is explained by the theoretically calculated magnetic phase diagram. The structural origin of this spin reorientation that is responsible for the appearance of spontaneous magnetization, is identified by theory and verified by x-ray diffraction and absorption experiments.

Figure 1

(a) Displacement patterns of the ${\mathrm{FeO}}_5$ local environment (trigonal bipyramid) in the three phonon modes that freeze in the $P{6}_3/mmc\to P{6}_{3}cm$ structural transition in hexagonal ferrites (h-${\mathrm{RFeO}}_3)$. The arrows indicate the relative displacement of the atoms. $\vec{K}$ is the wave vector of the modes in the reciprocal space of the $P{6}_3/mmc$ structure. (b) Four independent spin structures $(A_1$, $A_2$, $B_1$, and $B_2)$ of the 120-degree magnetic orders viewed along the $c$ axis. The arrows indicate the spins $\left({\vec{S}}_i^{Z_{\text{Fe}}}\right)$ on the Fe sites. The Fe sites shown in the polyhedra are in the $Z_{\text{Fe}}=0$ layer while all the other Fe sites are in the $Z_{\text{Fe}}=c/2$ layers. In the $B$ $\left(A\right)$ phase, ${\vec{S}}_1^0$ is parallel (antiparallel) to ${\vec{S}}_1^{c/2}$.

Figure 2

(a) Representative $d_{xy}$, $d_{z^2}$, and $d_{xz}$-like LWFs viewed from [001] direction. (b) Illustrations of two independent SSE paths $J_1^c$ and $J_2^c$ between ${\mathrm{Fe}}_0$ at $z/c=0$ and three neighboring iron ions ${\mathrm{Fe}}_1$, ${\mathrm{Fe}}_2$, and ${\mathrm{Fe}}_3$ at $z/c=1/2$ (c) Atomic displacements of the $K_1$ phonon mode. (d) Atomic displacements of the $K_3$ phonon mode, viewed from [001] direction.

Figure 3

(a) $\Delta J\left(\delta Q\right)=J\left(Q\right)-J\left(Q_0\right)$ for each individual $K_1$, $K_3$, and ${\Gamma }_2^{-}$ phonon mode, where $\delta Q=Q-Q_0$ and $Q_0$ is the value at 300 K, while the other two phonon displacements are kept as zero. Inset: theoretical phase diagram as functions of mode amplitudes of $K_1$ and $K_3$. (b) $J_1^c$ and $J_2^c$ as functions of $Q_{K_1}$, while $Q_{K_3}$ and $Q_{{\Gamma }_2^{-}}$ are fixed at the experimental values [25].

Figure 4

Structural changes indicated by the XRD and XAS. (a) The XRD intensity ratio between the (104) and (004) peaks (normalized to the 30 K value) and the change of lattice constant $c$ (with respect to the 30 K value) as functions of the temperature (the representative error bars are shown). Inset: the simulated intensity ratio between the (104) and (004) peaks as functions of phonon displacements. (b) The change of Fe-$3d$ crystal field levels (relative to the 300 K values) as functions of the temperature (the error bars for $e^{\prime }$ levels are shown as examples); the bold line is a guide to the eye to highlight the common peak-like feature. (c) Simulated change of Fe-$3d$ levels as functions of phonon displacements [25].