NMR Studies of pseudogap and electronic inhomogeneity in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

We report ${}^{17}$O NMR measurements in single crystals of overdoped Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$ with $T_c=82$ K. We measure the full anisotropy of the planar oxygen Knight shift, electric field gradient, and spin lattice relaxation rate tensors, and show that the entire temperature dependence is determined by the suppression of the density of states in the pseudogap below $T^{\ast }\sim 94$ K. The linewidth can be explained by a combination of magnetic and quadrupolar broadening as a result of an inhomogeneous distribution of local hole concentrations that is consistent with scanning tunneling microscopy measurements. This distribution is temperature independent, does not break $C_4$ symmetry, and exhibits no change below $T^{\ast }$ or $T_c$.