Fragile Magnetic Ground State in Half-Doped Manganite LaSr$_2$Mn$_2$O$_7$: Orbital Instability

Recently, a careful doping control study, however, reported that the La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ material has an exotic phase diagram very near the half doping with extremely narrow antiferromagnetic phase boundaries at x$\simeq$ 0.5 $\pm$ 0.005 -- the $CE$-type within the boundaries but the $A$-type outside.\footnote{Q. Li $et~al.$, Phys. Rev. Lett. 98, 167201 (2007).} To understand a complexity on this material's phase diagram, we investigated the orbital and antiferromagnetic ordering behaviors of the half-doped bilayer manganite La$_{2-2x}$Sr$_{1+2x}$Mn$_2$O$_7$ (x $\simeq$ 0.5) by using Mn $L_{2,3}$-edge resonant soft x-ray scattering. We confirmed the predicted $CE$-type antiferromagnetic order for the true half-doped (x = 0.5) case. Moreover, we found that such a narrow phase boundary is due to the close competition of the two antiferromagnetic ordering phases via $3d$ Mn $e_g$ orbital instability. Our study reveals the spin and orbital orders of electrons in the sample as well as information about their ground states.\footnote{J.-S. Lee $et~al.$, Phys. Rev. Lett. 107, 037206 (2011).}