Two- and three-dimensional incommensurate modulation in optimally-doped Bi₂Sr₂CaCu₂O_8+δ

X-ray scattering measurements on optimally doped single crystal samples of the high-temperature superconductor Bi₂Sr₂CaCu₂O_8+δ reveal the presence of three distinct incommensurate charge modulations, each involving a roughly fivefold increase in the unit cell dimension along the b direction. The strongest scattering comes from the well known (H, K±0.21, L) modulation and its harmonics. However, we also observe broad diffraction which peak up at the L values complementary to those which characterize the known modulated structure. These diffraction features correspond to correlation lengths of roughly a unit cell dimension, ξ_c∼20 Å in the c direction, and of ξ_b∼185 Å parallel to the incommensurate wave vector. We interpret these features as arising from three-dimensional incommensurate domains and the interfaces between them, respectively. In addition we investigate the recently discovered incommensurate modulations which peak up at (1∕2, K±0.21, L) and related wave vectors. Here we explicitly study the L dependence of this scattering and see that these charge modulations are two dimensional in nature with weak correlations on the scale of a bilayer thickness, and that they correspond to short-range, isotropic correlation lengths within the basal plane. We relate these new incommensurate modulations to the electronic nanostructure observed in Bi₂Sr₂CaCu₂O_8+δ using STM topography.