Chemical potential shift in lightly doped to overdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{1-x}{R}_x{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$ $(R=\mathrm{Pr},$ Er)

We have studied the chemical potential shift in the high-temperature superconductor ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_{1-x}{R}_x{\mathrm{Cu}}_2{\mathrm{O}}_{8+y}$ $(R=\mathrm{Pr},$ Er), where the hole concentration is varied from 0.025 to 0.17 per Cu, by precise measurements of core-level photoemission spectra. The result shows that the shift is depressed in the underdoped region as in the case of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ (LSCO) but the depression is much weaker than in LSCO. The observed shift in the present system can be relatively well explained by numerical results on the doped two-dimensional Hubbard model and suggests that the change of the electronic structure induced by hole doping is less influenced by stripe fluctuations than in LSCO.