Current-voltage relation for asymmetric ballistic superconducting junctions

We calculate the dc current-voltage (I-V) relation for a ballistic superconductor-normal metal-superconductor (S₁NS₂) junction, where the two superconducting contacts have different energy gaps (Δ₂>~Δ₁>~0). Several new features appear in the I-V relation as it evolves from a NS junction (Δ₁=0) to the symmetric SNS junction (Δ₁=Δ₂). The dc current I₀ flowing at small voltages is I₀=4eΔ₁/h, and is different from the critical current I_c. The differential conductance changes abruptly at eV=Δ₂-Δ₁ due to the loss of a single Andreev reflection from the larger-gap superconductor. Finally, the voltages Δ_n at which subharmonic gap structures appear in the I-V relation for temperatures k_BT≃Δ₂ depend on both energy gaps Δ₁ and Δ₂. We find a systematic way to classify this subgap structure.