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We investigate the doping dependent low energy, low temperature ($T$ = 5 K)properties of nodal quasiparticles in the d-wave superconductorBi$_{2.1}$Sr$_{1.9}$CaCu$_2$O$_{8+\delta}$ (Bi2212). By utilizing ultrahighresolution laser-excited angle-resolved photoemission spectroscopy, we obtainprecise band dispersions near $E_{F}$, mean free paths and scattering rates($\Gamma$) of quasiparticles. For optimally and overdoped, we obtain very sharpquasiparticle peaks of 8 meV and 6 meV full-width at half-maximum,respectively, in accord with terahertz conductivity. For all doping levels, wefind the energy-dependence of $\Gamma \sim |\omega |$, while $\Gamma$($\omega=0$) shows a monotonic increase from overdoping to underdoping. The dopingdependence suggests the role of electronic inhomogeneity on the nodalquasiparticle scattering at low temperature (5 K $\lsim 0.07T_{\rm c}$),pronounced in the underdoped region.

Doping-dependence of nodal quasiparticle properties in high-$T_{\rm c}$ cuprates studied by laser-excited angle-resolved photoemission spectroscopy