Possible presence of a charge-orbital density wave in layered manganites ${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_{1+x}{\mathrm{MnO}}_4$

We have studied the structural modifications related to the charge-orbital ordering for single-layered manganites ${\mathrm{Nd}}_{1-x}{\mathrm{Ca}}_{1+x}{\mathrm{MnO}}_4$ with both commensurate and incommensurate doping levels $(0.55\mathrm{}\leqq x\leqq 0.75).$ High-resolution electron microscopy (HREM) revealed transverse and sinusoidal structural modulations with a doping-level-dependent period of $d_s=a/\left(1\mathrm{}-x\right)$ in the charge-orbital ordered phases. We discuss the possibility of an orbital-density wave as well as a charge-density wave based on the HREM observation of the single-layered manganites.