Anisotropy and dimensional crossover of the vortex state in ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_{8+\mathrm{\delta }}$ crystals

We report muon-spin-rotation studies of the magnetic-field distribution n(B) in the vortex state in both underdoped and strongly overdoped ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_{8+\mathrm{\delta }}$ single crystals. With increasing magnetic field we observe sudden changes in n(B) that are indicative of a reduction in the dimensionality of the vortex system. The crossover field ${\mathit{B}}^{\mathrm{*}}$ depends strongly on the doping state and the corresponding anisotropy γ of the system. From magnetization measurements we find evidence that the second peak in the M-H loops is related to the crossover in the vortex structure in ${\mathit{B}}^{\mathrm{*}}$.