Effects of carrier concentration on the superfluid density of high-T c cuprates

The absolute values and temperature, T, dependence of the in-plane magnetic penetration depth of ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CuO}}_{4}$ and ${\mathrm{HgBa}}_{2}{\mathrm{CuO}}_{4+\ensuremath{\delta}}$ have been measured as a function of carrier concentration. We find that the superfluid density ${\ensuremath{\rho}}_{s}$ changes substantially and systematically with doping. The values of ${\ensuremath{\rho}}_{s}(0)$ are closely linked to the available low-energy spectral weight as determined by the electronic entropy just above ${T}_{c},$ and the magnitude of the initial slope of $[{\ensuremath{\rho}}_{s}(T)/{\ensuremath{\rho}}_{s}(0)]$ increases rapidly with carrier concentration. The results are discussed in the context of a possible relationship between ${\ensuremath{\rho}}_{s}$ and the normal-state (or pseudo)energy gap.