  We study the inhomogeneity of the electronic pairing gap observed by STM near
the surface of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta}$ 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.
