Effects of carrier concentration on the superfluid density of high-Tccuprates

The absolute values and temperature, T, dependence of the in-plane magnetic penetration depth of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$ and ${\mathrm{HgBa}}_2{\mathrm{CuO}}_{4+\delta }$ 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 ${\rho }_s\mathrm{}\left(0\right)\mathrm{}$ 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 $\left[{\rho }_s\mathrm{}\right(T)/{\rho }_s\mathrm{}(0\left)\mathrm{}\right]$ 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.