Pairing Symmetry and Upward Curvature of Upper Critical Field in Superconducting $Na_{0.35} Co O_2 \cdot y H_2 O$

The origin for the upward curvature of the upper critical field ($H_{c2}$) observed in hydrate cobaltate Na$_{0.35}$CoO$_{2} \cdot y$H$_{2}$O is investigated based on the microscopic gap equation. It is shown that the observed upward curvature results from the transition between two different pairing symmetries that occur on different energy bands. Furthermore, different pairing symmetries involved in the transition results in different upward curvatures. By considering transitions among all lowest possible pairing symmetries, it is found that the transition of the pairing symmetry from s-wave at low fields to $d_{x^2-y^2}+ id_{xy}$ at high fields is the best fit to the experimental data. Our results provide an important clue to the understanding of the superconductivity in hydrate cobaltate.