Structural modulation, hole distribution, and hole-ordered structure of the incommensurate composite crystal $({\mathrm{Sr}}_2{\mathrm{Cu}}_2{\mathrm{O}}_3{)}_{0.70}{\mathrm{CuO}}_2$

Modulated structure of incommensurate composite crystal $({\mathrm{Sr}}_2{\mathrm{Cu}}_2{\mathrm{O}}_3{)}_{0.70}{\mathrm{CuO}}_2,$ ${\mathrm{“}\mathrm{Sr}}_{14}{\mathrm{Cu}}_{24}{\mathrm{O}}_{41},”$ has been investigated by single-crystal x-ray-diffraction method using centrosymmetric $\mathrm{}(3+1)\mathrm{}$-dimensional superspace group. In $({\mathrm{Sr}}_2{\mathrm{Cu}}_2{\mathrm{O}}_3{)}_{0.70}{\mathrm{CuO}}_2,$ displacive modulation of O atom in the ${\mathrm{CuO}}_2$ chain is fairly large. Considering the modulation of bond angles, it has been found that the Cu-O bond in the ${\mathrm{CuO}}_2$ chain is tilting toward the ${\mathrm{Cu}}_2{\mathrm{O}}_3$ ladder in order that the O atom in the chain plays as apical oxygen for the ${\mathrm{CuO}}_4$ square in the ladder. The bond-valence sum (BVS) method has been applied to investigate the hole distribution in the modulated structure of $({\mathrm{Sr}}_2{\mathrm{Cu}}_2{\mathrm{O}}_3{)}_{0.70}{\mathrm{CuO}}_2.$ It is indicated that the valence of Cu atom in the ${\mathrm{Cu}}_2{\mathrm{O}}_3$ ladder is +2.04, where about 0.03 holes are certainly transferred from the ${\mathrm{CuO}}_2$ chain through the modulated O atom in the ${\mathrm{CuO}}_2.$ The BVS calculation has demonstrated that almost all of the holes are prepared in the ${\mathrm{CuO}}_2$ chain by the large modulation of the Cu-O bond. Cu atoms in the modulated ${\mathrm{CuO}}_2$ chain have been proved to form hole-ordered structure with next-nearest-neighbor ${\mathrm{Cu}}^{2+}$ ions separated by ${\mathrm{Cu}}^{3+}$ ion on the Zhang-Rice singlet site. The periodicity of the hole-ordered structure is five times of the average ${\mathrm{CuO}}_2$ lattice along the crystallographic c axis, which is compatible with the spin-dimerized state at low temperature. The new model of the two-dimensional hole-ordered structure in the ${\mathrm{CuO}}_2$ plane has been obtained by the BVS calculation. Furthermore, the two-dimensional configuration of the spin dimers has been successfully derived from the hole-ordered structure in the ${\mathrm{CuO}}_2$ plane. It has been concluded that the valences of Cu atoms both in the ${\mathrm{Cu}}_2{\mathrm{O}}_3$ ladder and in the ${\mathrm{CuO}}_2$ chain are well controlled by the modulated O atom in the ${\mathrm{CuO}}_2$ chain.