$T_c$ suppression and resistivity in cuprates with out-of-plane defects

The suppression of the critical temperature due to isotropic impurity scattering in a $d$-wave superconductor is expected to be described by the Abrikosov-Gor'kov formula. However recent experiments on cuprate superconductors with out-of-plane defects show a nearly linear decrease of the critical temperature as a function of the residual resistivity in contradiction to the Abrikosov-Gor'kov result. Other experiments suggest further that the $T_c$ suppression is stronger for out-of-plane than for in-plane impurities. Both results can be explained assuming that elastic forward scattering is the dominant scattering process for out-of-plane disorder. We present a simple model of forward scattering allowing an analytical solution that already includes the key features of $T_c$ suppression for out-of-plane defects. We also verify the intuitive result of this simple model by comparing it to numerical calculations assuming randomly distributed out- of-plane defects and a realistic band structure appropriate for YBCO.