Making superconducting transition temperature higher in Bi$_{2}$Sr$_{2}$Ca$_{2}$Cu$_{3}$O$_{10+\delta }$

We report an experimental finding of \textit{Tc} enhancement in optimally doped Bi$_{2}$Sr$_{2}$Ca$_{2}$Cu$_{3}$O$_{10+\delta }$. We found that the generally observed pressure effect on \textit{Tc}, $i.e.$, \textit{Tc} first increases with pressure and then decreases after passing a maximum at an optimal pressure, is only held below a critical pressure around 24 GPa in this multilayer material. After that \textit{Tc} enhances remarkably upon further compression, considerably surpassing the first maximum. The critical pressure was then considered as the crossover from antiferromagnetism to superconductivity in the inner CuO$_{2}$ plane. The afterwards \textit{Tc} enhancement was suggested through the optimization of two competing energy scales (pairing and phase ordering) of different CuO$_{2}$ planes. The results have important implications for engineering superconductors with much higher \textit{Tc}'s at ambient conditions.