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Doping Dependence of Gap Inhomogeneities at Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ Surfaces

We study the inhomogeneity of the electronic pairing gap observed by STM near the surface of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+δ}$ to be correlated with interstitial O defects. We treat the problem in a slave boson mean field theory of a disordered $t-t^{\prime}-J$ model, and identify three aspects of the O defects related to the inhomogeneity: (1)the superexchange interaction is locally enhanced in their vicinity, which enhances the local pairing gap and reduces the coherence peak; (2)they donate holes into CuO$_2$ plane, which reduces the spinon density of states of and hence the average gap at large doping; (3)holes are locally attracted to the vicinity of oxygen defects, which causes impurity bound state and further reduces the coherence peak. The interplay of these mechanisms explains simultaneously the locally enhanced pairing gap around oxygen defects, and the reduction of average gap as increasing oxygen concentration.