We have investigated effects of Zn and Ni on the Cu-spin dynamics and superconductivity from the zero-field muon-spin-relaxation (ZF-muSR) and magnetic-susceptibility, chi, measurements for La_2-x_Sr_x_Cu_1-y_(Zn,Ni)_y_O_4_ with x=0.15-0.20, changing y up to 0.10 in fine step. In the optimally doped x=0.15, it has been concluded that the formation of a magnetic order requires a larger amount of Ni than that of Zn, which is similar to our previous results of x=0.13. From the estimation of volume fractions of superconducting (SC) and magnetic regions, it has been found for x=0.15 that the SC region is in rough correspondence to the region where Cu spins fluctuate fast beyond the muSR frequency window for both Zn- and Ni-substituted samples. According to the stripe model, it follows that, even for x=0.15, the dynamical stripe correlations of spins and holes are pinned and localized around Zn and Ni, leading to the formation of the static stripe order and the suppression of superconductivity. These may indicate an importance of the dynamical stripe in the appearance of the high-T_c_ superconductivity in the hole-doped cuprates. In the overdoped regime of x=0.18 and 0.20, on the other hand, the SC region seems to be in rough correspondence to the region where Cu spins fluctuate fast beyond the muSR frequency window, though it appears that the Cu-spin dynamics and superconductivity are affected by the phase separation into SC and normal-state regions.
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