Negligible Normal Fluid in Superconducting State of Heavily Overdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ Detected by Ultra-Low Temperature Angle-Resolved Photoemission Spectroscopy

In high temperature cuprate superconductors, it was found that in the overdoped region the superfluid density decreases with the increase of hole doping. One natural question is whether there exists normal fluid in the superconducting state in the overdoped region. In this paper, we have carried out high-resolution ultra-low temperature laser-based angle-resolved photoemission measurements on a heavily overdoped Bi2212 sample with a $T_{\mathrm{c}}$ of 48 K. We find that this heavily overdoped Bi2212 remains in the strong coupling regime with $2 \mathit{\Delta}_0 / k_{\mathrm{B}} T_{\mathrm{c}}=5.8$. The single-particle scattering rate is very small along the nodal direction ($\sim$5 meV) and increases as the momentum moves from the nodal to the antinodal regions. A hard superconducting gap opening is observed near the antinodal region with the spectral weight at the Fermi level fully suppressed to zero. The normal fluid is found to be negligibly small in the superconducting state of this heavily overdoped Bi2212. These results provide key information to understand the high $T_\mathrm{c}$ mechanism in the cuprate superconductors.