Orbital Hierarchy Inversion and Magnetic Transition in Mn Doped Sr$_3$Ru$_2$O$_7$

Ruthenium oxides are a very important class of compounds that are extremely sensitive to impurities. Here we present a study of the Mn doped Sr$_3$Ru$_2$O$_7$ by X-ray Absorption Spectroscopy (XAS) and a combination of density functional theory (LSDA) and cluster calculations. We find that delocalized RuO matrix forces the Mn impurities to behave as a negative charge transfer system and donate a hole. As a result Mn impurities behave as Mn3+ acceptors. Our XAS data clearly shows that the occupied $e_g$ orbitals have an in-plane alignment: a very surprising result in a tetragonally distorted system elongated along the $c$-axis. LSDA calculations reveal very strong anisotropy in oxygen bandwidth leading to a reversal of the $e_g$ crystal field hierarchy. This work further establishes that the Mn atoms are sitting in a bath of a uniaxial exchange field that gradually goes out-of-plane to in-plane as the Mn doping level is increased. Overall, the behavior has some similarities with Mn doped GaAs and may be relevant to the physics of dilute magnetic semiconductors.