We have derived the effective low-energy Hamiltonian for Zn- or Ni-substituted high-$T_c$ cuprates from microscopic three-band models consisting of the most relevant Cu or impurity $3d$ and O $2p$ orbitals. We find that both the scattering potential and hopping integral induced by impurities have a finite range but decay very fast with distance from the impurity. The Zn scattering potential is very strong and attractive for electrons. The Ni scattering potential is much weaker than the Zn case, resulting from the hybridization between Ni ions and O holes. This profound difference is due to neither the electric charge nor d-level location, but rather because of the interplay between the valence state of the impurity and the strong correlation background. It gives a natural account of the unusual effect of Ni and Zn on the reduction of the superconducting transition temperature. The interlayer hopping of electrons is highly anisotropic and nonlocal, determined by the in-plane electronic structure. This leads to a quantum interference of states from different sites and affects strongly the scanning tunneling spectrum perpendicular to ${\mathrm{CuO}}_2$ planes.

• Received 31 July 2002

DOI:https://doi.org/10.1103/PhysRevB.66.174504