Doping dependence of anisotropic resistivities in the trilayered superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }$

The doping dependence of the thermopower, in-plane resistivity ${\rho }_{\mathrm{ab}}\mathrm{}\left(T\right),$ out-of-plane resistivity ${\rho }_c\mathrm{}\left(T\right),$ and susceptibility has been systematically measured for high-quality single crystal ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10+\delta }.$ We found that the transition temperature $T_c$ and pseudogap formation temperature $T_{{\rho }_c}^{*},$ below which ${\rho }_c$ shows a typical upturn, do not change from their optimum values in the “overdoped” region, even though doping actually proceeds. This suggests that, in overdoped region, the bulk $T_c$ is determined by the always underdoped inner planes, which have a large superconducting gap, while the carriers are mostly doped in the outer planes, which have a large phase stiffness.