Ambegaokar-Baratoff relations of Josephson critical current in heterojunctions with multi-gap superconductors

An extension of the Ambegaokar-Baratoff relation to a superconductor-insulator-superconductor (SIS) Josephson junction with multiple tunneling channels is derived. Appling the resultant relation to a SIS Josephson junction formed by an iron-based (five-band) and a single-band Bardeen-Cooper-Schrieffer (BCS) type superconductors, a theoretical bound of the Josephson critical current ($I_{\mathrm{c}}$) multiplied by the resistance of the junction ($R_{\mathrm{n}}$) is given. We reveal that such a bound is useful for identifying the pairing symmetry of iron-pnictide superconductors. One finds that if a measured value of $I_{\mathrm{c}}{R}_{\mathrm{n}}$ is smaller than the bound then the symmetry is $\pm s$-wave, and otherwise $s$-wave without any sign changes. In addition, we stress that temperature dependence of $I_{\mathrm{c}}{R}_{\mathrm{n}}$ is sensitive to the difference of the gap functions from the BCS type gap formula in the above heterojunction.