Electron microscopic studies of the charge-ordered structures of the bilayered colossal magnetoresistive (CMR) manganite La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7}$

The charge-ordered structures in the bilayered colossal magnetoresistive (CMR) manganite La$_{2-2x}$Sr$_{1+2x}$Mn$_{2}$O$_{7, }$at the electronic doping levels of $x$=0.6 and $x$=0.67 respectively, have been proposed based on the \textit{in-situ} electron microscopic studies. In sharp contrast to the three-dimensional perovskite manganite, evidence of charge-ordered structures with bi-stripe models has been found. A new face-centered charge-ordered superstructure was observed at $x$=0.6. This structure is composed of bi-stripes of Mn$^{3+}$O$_{6}$ and paired Mn$^{4+}$O$_{6}$ rows alternatively stacking along the ordering direction, which is assembled from the building blocks of the charge-ordered phase at $x$=0.5. Taking into account of the systematic absence of reflections with the face-centered symmetry, its modulation vector was deduced as \textbf{q}=(1/10, 1/10, 0). Moreover, for another charge-ordered phase identified at $x$=0.67 with \textbf{q}=(1/6, 1/6, 0), again only the bi-stripe model fits the existing data over the Wigner-crystal model.