Resonant States in High-Temperature Superconductors with Impurities

A microscopic theory of resonant states for the $\mathrm{Zn}$-doped $\mathrm{CuO}_2$ plane in the superconducting phase is formulated in the effective $t$–$J$ model. In the model derived from the original $p$–$d$ model, $\mathrm{Zn}$ impurities are considered as vacancies for the d states at $\mathrm{Cu}$ sites. In the superconducting phase, in addition to the local static perturbation induced by the vacancy, a dynamical perturbation appears that results in a frequency-dependent perturbation matrix. Using the $T$-matrix formalism for the Green's functions in terms of the Hubbard operators, we calculate the local density of electronic states with $d$, $p$, and $s$ symmetries.