Rapid enhancement of nodal quasiparticle mass with heavily underdoping in Bi2212

We report substantial advance of our low-energy angle-resolved photoemission study of nodal quasiparticles in Bi₂Sr₂CaCu₂$O_{8+\delta }$. The new data cover the samples from underdoped down to heavily underdoped levels. We also present the nodal Fermi velocities that determined by using an excitation-photon energy of $h\nu =7.0$ eV over a wide doping range. The consistency between the results with $h\nu =8.1$ and 7.0 eV allows us to rule out the effect of photoemission matrix elements. In comparison with the data previously reported, the nodal effective mass increases by a factor of $\sim 1.5$ in going from optimally doped to heavily underdoped levels. We find a rapid enhancement of the nodal quasiparticle mass at low doping levels near the superconductor-to-insulator transition. The effective coupling spectrum, $\lambda \left(\omega \right)$, is extracted directly from the energy derivatives of the quasiparticle dispersion and scattering rate, as a causal function of the mass enhancement factor. A steplike increase in $\mathrm{Re}\lambda \left(\omega \right)$ around $\sim$ 65 meV is demonstrated clearly by the Kramers-Kronig transform of $\mathrm{Im}\lambda \left(\omega \right)$. To extract the low-energy renormalization effect, we calculated a simple model for the electron-boson interaction. This model reveals that the contribution of the renormalization at $\left|\omega \right|\le 15$ meV to the quasiparticle mass is larger than that around 65 meV in underdoped samples.