  In order to understand the experimentally proposed phase diagrams of
NaxCoO2.yH2O, we theoretically study the CoO2-layer-thickness dependence of
magnetic and superconducting (SC) properties by analyzing a multiorbital
Hubbard model using the random phase approximation. When the Co valence (s) is
+3.4, we show that the magnetic fluctuation exhibits strong layer-thickness
dependence where it is enhanced at finite (zero) momentum in the thicker
(thinner) layer system. A magnetic order phase appears sandwiched by two SC
phases, consistent with the experiments. These two SC phases have different
pairing states where one is the singlet extended s-wave state and the other is
the triplet p-wave state. On the other hand, only a triplet p-wave SC phase
with dome-shaped behavior of Tc is predicted when s=+3.5, which is also
consistent with the experiments. Controversial experimental results on the
magnetic properties are also discussed.
