$5\times 5$ patch. As $\alpha$ increases, the Andreev resonance moves to lower energy and sharpens up, and the resonance energy $\Omega_0$ can approach zero when $\alpha$ becomes greater than 1. This means that the order parameter actually changes its phase by $\pi$ on an atomic scale. The question relevant to the present study is how these Andreev states depend on the local gap patch in which they are positioned. In order to answer this, we have studied the LDOS near phase impurities with $\alpha=1.5$ in the center of gap patches of size $5\times 5$. Assuming that Zn causes the same local changes of the gap irrespective of the local gap environment (i.e. same $\alpha$ for all local gap values of the $5\times 5$ patch), then one would expect sharper resonances in small-gap regions since there the phase impurity is effectively stronger. Indeed, the special case of a phase impurity of strength $\alpha$ in a large-gap region with increased local gap also of size $\alpha$ has effectively no phase impurity at all, and hence no low-energy resonance.

In Fig. \[fig7\] we show the LDOS at the center of the phase impurity as a function of the local gap change $\delta \Delta$ 