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, $\lambda_{ab}$, of La_{2-x}Sr_xCuO_4 and HgBa_2CuO_{4+\delta} have been measured as a function of carrier concentration. We find that the superfluid density, $\rho_s$, changes substantially and systematically with doping. The values of $\rho_{s}(0)$ are closely linked to the available low energy spectral weight as determined by the electronic entropy just above T_c and the initial slope of $\rho_{s}(T)/\rho_{s}(0)$ increases rapidly with carrier concentration. The results are discussed in the context of a possible relationship between $\rho_s$ and the normal-state (or pseudo) energy gap.