Influence of gap structures to specific heat in oriented magnetic fields: Application to the orbital dependent superconductor, Sr$_2$RuO$_4$

Abstract

We discuss influence of modulation of gap function and anisotropy of Fermi velocity to field angle dependences of upper critical field, $H_{c2}$, and specific heat, $C$, on the basis of the approximate analytic solution in the quasiclassical formalism. Using 4-fold modulation of the gap function and the Fermi velocity in the single-band model, we demonstrate field and temperature dependence of oscillatory amplitude of $H_{c2}$ and $C$. We apply the method to the effective two-band model to discuss the gap structure of Sr$_2$RuO$_4$, focusing on recent field angle-resolved experiments. It is shown that the gap structures with the intermediate magnitude of minima in $[100]$ direction for $\gamma$ band, and tiny minima of gaps in $[110]$ directions for $\alpha$ and $\beta$ bands give consistent behaviors with experiments. The interplay of the above two gaps also explains the anomalous temperature dependence of in-plane $H_{c2}$ anisotropy, where the opposite contribution from the passive $\alpha\beta$ band is pronounced near $T_c$.