Spin dynamical properties and orbital states of the layered perovskite La 2 − 2 x Sr 1 + 2 x Mn 2 O 7 ( 0.3 x 0.5 )

Low-temperature spin dynamics of the double-layered perovskite ${\mathrm{La}}_{2\ensuremath{-}2x}{\mathrm{Sr}}_{1+2x}{\mathrm{Mn}}_{2}{\mathrm{O}}_{7}$ (LSMO327) was systematically studied in a wide hole concentration range $(0.3l~xl0.5).$ The spin-wave dispersion, which is almost perfectly two-dimensional, has two branches due to a coupling between layers within a double-layer. Each branch exhibits a characteristic intensity oscillation along the out-of-plane direction. We found that the in-plane spin stiffness constant and the gap between the two branches strongly depend on x. By fitting to calculated dispersion relations and cross sections assuming a Heisenberg model, we have obtained the in-plane ${(J}_{\ensuremath{\Vert}}),$ intra-bilayer ${(J}_{\ensuremath{\perp}})$ and inter-bilayer ${(J}^{\ensuremath{'}})$ exchange interactions at each x. At $x=0.30,$ ${J}_{\ensuremath{\Vert}}=\ensuremath{-}4\mathrm{meV}$ and ${J}_{\ensuremath{\perp}}=\ensuremath{-}5\mathrm{meV},$ namely almost isotropic and ferromagnetic. Upon increasing x, ${J}_{\ensuremath{\perp}}$ rapidly approaches zero while $|{J}_{\ensuremath{\Vert}}|$ increases slightly, indicating an enhancement of the planar magnetic anisotropy. At $x=0.48,$ ${J}_{\ensuremath{\Vert}}$ reaches $\ensuremath{-}9\mathrm{meV},$ while ${J}_{\ensuremath{\perp}}$ turns to $+1\mathrm{meV},$ indicating an antiferromagnetic interaction. Such a drastic change of the exchange interactions can be ascribed to the change of the relative stability of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ and ${d}_{{3z}^{2}\ensuremath{-}{r}^{2}}$ orbital states upon doping. However, a simple linear combination of the two states results in an orbital state with an orthorhombic symmetry, which is inconsistent with the $I4/mmm$ tetragonal symmetry of the crystal structure. We thus propose that an ``orbital liquid'' state realizes in LSMO327, where the charge distribution symmetry is kept tetragonal around each Mn site. Orbital liquid states are formulated in a theoretical model which takes into account strong electron correlations. The calculated results satisfactorily explain the systematic changes of the exchange interactions in LSMO327 observed in the experiments.