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The effects of a single non-magnetic impurity on superconducting states inthe Penson-Kolb-Hubbard model have been analyzed. The investigations have beenperformed within two steps: (i) the homogeneous system is analysed using theBogoliubov transformation, whereas (ii) the inhomogeneous system isinvestigated self-consistent Bogoliubova-de Gennes equations (by exactdiagonalization and the kernel polynomial method). We analysed both signs ofpair hopping, which correspond to s-wave and $\eta$-wave superconductivity. Ourresults show that an enhancement of the local superconducting gap at theimpurity-site occurs for both cases. We obtained that Cooper pairs arescattered (at the impurity site) into the states which are from theneighborhoods of the states, which are commensurate ones with the crystallattice. Additionally, in the $\eta$-phase there are peaks in the local-energygap (in momentum space), which are connected with long-range oscillations inthe spatial distribution of the energy gap, superconducting order parameter aswell as effective pairing potential. Our results can be contrasted with theexperiment and predicts how to experimentally differentiate these two differentsymmetries of superconducting order parameter by scanning tunneling microscopytechnique.

Probe-type of superconductivity by impurity in materials with short coherence length: the s-wave and $\eta$-wave phases study