Spin and charge orderings in single layered cobaltate La$_{2-x}$Ca$_{x}$CoO$_{4}$ (0.3$<x<$0.8)

Neutron scattering experiments were performed on single crystals of layered cobalt-oxides La$_{2-x}$Ca$_{x}$CoO$_{4}$ (LCCO) to characterize the charge and spin orders in a wide hole-doping range of 0.3$\mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} $x$\mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle=}\vphantom{_x}}$}} $0.8. For a commensurate value of x=0.5 in (H,0,L) plane, two types of superlattice reflections concomitantly appear at low temperature; one corresponds to a checkerboard charge ordered pattern of Co$^{2+}$/Co$^{3+}$ ions and the other is magnetic in origin. Further, the latter magnetic-superlattice peaks show two types of symmetry in the reflections, suggesting antiferromagnetic-stacking (AF-S) and ferromagnetic-stacking (F-S) patterns of spins along the c direction. From the hole-doping dependence, the in-plane correlation lengths of both charge and spin orders are found to give a maximum at x=0.5. These features are the same with those of x=0.5 in La$_{1-x}$Sr$_{1+x}$MnO$_{4}$ (LSMO), a typical checkerboard and spin ordered compound.