Evolution of the CO-OO and AF ordering in the single-layer manganite $\rm Pr_{1-x}Ca_{1+x}MnO_4$ near half doping

Manganese oxides have attracted considerable attention due to the CMR effect observed in the perovskite manganite $\rm A_{1-x}A'_{x}MnO_3$ near $x=0.3$. A peculiar charge/orbital (CO-OO) accompanied by antiferromagnetic (AF) order occurs when the carrier concentration is close to half doping (x=0.5). To understand the interplay between the charge, lattice and spin degrees of freedom in such insulating state, we used elastic neutron scattering to study the evolution of the CO-OO as well as the AF correlations in the single-layer manganite $\rm Pr_{1-x}Ca_{1+x}MnO_4$ ($x=0.40$, 0.45 and 0.50). Upon cooling, all three samples exhibit long-range CO-OO near 300~K. However, only the $x=0.50$ system displays long-range AF order at low temperatures. The CE-type AF correlations are quickly suppressed and become short-ranged as more $e_g$ electrons are introduced to the $\rm MnO_2$ plane. More interestingly, the CO-OO and AF order associated with Mn$^{3+}$ ions appears at incommensurate positions while the AF order associated with Mn$^{4+}$ ions remains commensurate. Our observations indicate that the orbital physics play an important role in the understanding of the electronic and magnetic properties of doped manganites.