Unraveling Orbital Ordering in ${\mathrm{L}\mathrm{a}}_{0.5}{\mathrm{S}\mathrm{r}}_{1.5}{\mathrm{M}\mathrm{n}\mathrm{O}}_4$

Orbital ordering (OO) in the layered perovskite ${\mathrm{L}\mathrm{a}}_{0.5}{\mathrm{S}\mathrm{r}}_{1.5}{\mathrm{M}\mathrm{n}\mathrm{O}}_4$ has been investigated using the enhanced sensitivity of soft x-ray resonant diffraction at the Mn $L$ edges. The energy dependence of an OO diffraction peak over the $L_{2,3}$ edges is compared to ligand-field calculations allowing a distinction between the influences of Jahn-Teller distortions and spin correlations. The energy dependence of the diffraction peak at the Mn $L_1$ edge is remarkably different from that observed at the Mn $K$ edge.

Figure 1

(a) Charge, orbital, and spin ordering in the ${\mathrm{M}\mathrm{n}\mathrm{O}}_2$ planes of ${\mathrm{L}\mathrm{a}}_{0.5}{\mathrm{S}\mathrm{r}}_{1.5}{\mathrm{M}\mathrm{n}\mathrm{O}}_4$; the arrows represent the magnetic moments. The primitive $I4/mmm$ unit cell is shown by the solid line ($a=b=3.864\AA$) and the OO unit cell by the dashed line. (b) The energy dependence (solid circles with line) of the ($\frac{1}{4},\frac{1}{4},0$) peak recorded over the Mn $L_{2,3}$ edges at 134 K and the calculated spectrum in $D_{2h}$ symmetry for large (red line) and small (green line) JTD. The inset shows a ($\frac{1}{4}+\Delta q,\frac{1}{4}+\Delta q,0$) scan of the forbidden reflection arising from OO at a photon energy of 643.8 eV and at $T=134\mathrm{K}$ (open circles). The solid line is a Lorentzian fit to the peak. The experimental geometry is also shown.

Figure 2

The relative intensity changes with temperature of the peaks shown in Fig. 1b. The symbols correspond to the features marked with the same symbols in Fig. 1b; the broken lines are guides to the eye.

Figure 3

Intensity of the ($\frac{1}{4},\frac{1}{4},0$) diffraction as a function of temperature normalized to the intensity at 134 K and recorded at a photon energy of 643.8 eV (solid circles). The error in the measurement is smaller than the data points. The correlation length, $\xi$, calculated from the half width at half maximum of the OO peak is also shown (open squares with line). The inset shows the natural log of the peak intensity against inverse temperature at the onset of OO (solid circles) along with the fit to the data (solid line).

Figure 4

The energy dependence (solid circles with line) of the OO peak recorded across the Mn $L_1$ edge at 134 K. The inset shows a ($\frac{1}{4}+\Delta q,\frac{1}{4}+\Delta q,0$) scan of the peak measured at a photon energy of 785 eV (open circles) with a Lorentzian fit (solid line). For comparison the OO peak across the Mn $K$ edge [6] is shown on the same relative energy scale with the peak positioned $\sim 14\mathrm{e}\mathrm{V}$ above threshold (open squares with dashed line). The estimated position of the threshold is marked by the arrow.