Orbital-dependent phase control in ${\mathrm{Ca}}_{2-x}{\mathrm{Sr}}_x{\mathrm{RuO}}_4$ $(0\mathrm{}<~x<~0.5)$

We present first-principles studies on the orbital states of the layered perovskites ${\mathrm{Ca}}_{2-x}{\mathrm{Sr}}_x{\mathrm{RuO}}_4.$ The crossover from antiferromagnetic (AF) Mott insulator for $x<\mathrm{}0.2\mathrm{}$ to nearly ferromagnetic (FM) metal at $x=0.5\mathrm{}$ is characterized by the systematic change of the $\mathrm{xy}$ orbital occupation. For the AF side $\mathrm{}(x<\mathrm{}0.2\mathrm{}),$ we present firm evidence for the $\mathrm{xy}$ ferro-orbital ordering. It is found that the degeneracy of $t_{2g}$ (or $e_g)$ states is lifted robustly due to the two-dimensional crystal structure, even without the Jahn-Teller distortion of ${\mathrm{RuO}}_6.$ This effect dominates, and the cooperative occupation of $\mathrm{xy}$ orbital is concluded. In contrast to recent proposals, the resulting electronic structure explains well both the observed x-ray-absorption spectra and the double-peak structure of optical conductivity. For the FM side $\mathrm{}(x=0.5\mathrm{}),$ however, the $\mathrm{xy}$ orbital with half filling opens a pseudo-gap in the FM state and contributes to the spin $S=1/2\mathrm{}$ moment (rather than $S=1\mathrm{}$ for $x=0.0\mathrm{}$ case) dominantly, while $yz,zx$ states are itinerant with very small spin polarization, explaining the recent neutron data consistently.