Ferromagnetic spin fluctuation and possible triplet superconductivity due to inter-orbital Hund's-rule coupling in Na$_x$CoO$_2 \cdot y$H$_2$O

Electronic structure and superconductivity in Na$_x$CoO$_2 \cdot y$H$_2$O are studied theoretically by using a fluctuation- exchange approximation. This material has partially- filled Co $t_{2g}$ orbitals and LDA calculation shows that its Fermi surface consists of more than two bands. Thus, we expect that the multi-band or multi-orbital contributes to the low- energy electronic state in this material. We employ a multi-orbital Hubbard model which includes the Co $t_{2g}$ orbitals. Tight-binding parameters are determined to reproduce the LDA band dispersions. To analyse this model, we extend the fluctuation-exchange approximation to a triply- degenerate orbital case. We will discuss that several important and interesting aspects appear which are not expected in a single-band model. One of them is a ferromagnetic (FM) spin fluctuation which is enhanced by the inter-orbital Hund's-rule coupling. This FM spin fluctuation leads to triplet pairing mainly on the disconnected hole-pocket Fermi surfaces, in contrast to the naive expectation of RVB superconductivity in a single-band $t$-$J$ model. We will also discuss the obtained results in the light of available experimental findings. On the basis of these analyses, we will point out that Na$_{0.35}$CoO$_2 \cdot$1.3H$_2$O can provide a key material for clarification of roles of orbitals on the superconductivity in the strongly correlated electron systems.