Orbital order in La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$: Beyond a common local Jahn-Teller picture

The standard way to find the orbital occupation of Jahn-Teller (JT) ions is to use structural data, with the assumption of a one-to-one correspondence between the orbital occupation and the associated JT distortion, e.g., in O${}_6$ octahedron. We show, however, that this approach can be incorrect, e.g., for layered systems. Using the layered manganite La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$ as an example, we found from our x-ray-absorption measurements and electronic structure calculations that the type of orbital ordering strongly contradicts the standard local distortion approach for the Mn${}^{3+}$O${}_6$ octahedra, and that the usually ignored long-range crystal-field effect and anisotropic hopping integrals are actually crucial for determining the orbital occupation.

Figure 1

Polarization-dependent Mn-$L_{2,3}$ XAS spectra of La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$ for $\mathbf{E}\parallel \mathbf{c}$ (red curve) and $\mathbf{E}\perp \mathbf{c}$ (black curve) taken at 150 K. (b) The corresponding linear dichroism (LD) spectrum. Theoretical LD calculated for orbital scenarios with (c) Mn${}^{3+}$ $3x^2-r^2/3y^2-r^2$ or (d) Mn${}^{3+}$ $x^2-z^2/y^2-z^2$ orbital occupation, each with the Mn${}^{4+}$ $t_{2g}^3$.

Figure 2

(a) PDOS of the Mn${}^{3+}$ and Mn${}^{4+}$ ions in the CE-type antiferromagnetic state calculated by LSDA $+$ $U$. (b) Charge density contour plot (0.05–0.4 $e$/Å${}^3$) of the Mn${}^{3+}$-O-Mn${}^{4+}$ network in the $xy$ plane for the occupied $e_g$ states within 1 eV below Fermi level. The ferromagnetic zigzag chain is marked by the dashed line. (c) Charge density contour plot of the Mn${}^{3+}$-O-Mn${}^{4+}$ chain in the $xz$ and $yz$ planes. (b) and (c) clearly show the occupied $3x^2-r^2/3y^2-r^2$orbitals of the Mn${}^{3+}$ ions.

Figure 3

(a) Sketch of the surrounding of a Mn${}^{3+}$ ion in La${}_{0.5}$Sr${}_{1.5}$MnO${}_4$ illustrates a different contribution of the further neighbors to the crystal field. The local Mn${}^{3+}$-O bond lengths are also shown. (b) The Mn${}^{3+}$ $e_g$ PDOS calculated by LSDA, projected onto the ($3x^2-r^2,y^2-z^2$) basis giving eigenstates (upper panel), and onto the ($x^2-z^2,3y^2-r^2$) basis, with mixed occupation of both orbitals (lower panel).

Figure 4

Variation of the Mn${}^{3+}$ $e_g^1$ orbital state with respect to the $c$-axis Mn-O bond length $R^z$, in terms of the orbital mixing angle $\theta$ calculated by LSDA $+$ $U$ ($U=5$ eV). The error bar is $\pm$4${}^{\circ }$ for $U=0\text{--}8$ eV. The insets show the definition of the orbital states, and the real Mn${}^{3+}$-O bond lengths with $R^z=2.00$ Å, at which the OO state is practically the pure $3x^2-r^2/3y^2-r^2$ type as discussed in the main text.