  We study the Kane-Mele-Hubbard model both at half-filling and away from
half-filling using a slave-boson mean-field approach at zero temperature. We
obtain a phase diagram at half-filling and discuss its connection to recent
results from quantum Monte Carlo, slave-rotor, and $Z_2$ mean-field studies. In
particular, we find a small window in parameter space where a spin liquid phase
with gapped spin and charge excitations reside. Upon doping, we show the spin
liquid state becomes a superconducting state by explicitly calculating the
singlet pairing order parameters. Interestingly, we find an "optimal" doping
for such superconductivity. Our work reveals some of the phenomenology
associated with doping an interacting system with strong spin-orbit coupling.
