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 (Delta(2) greater than or equal to Delta(1) greater than or equal to 0) Several new features appear in the I-V relation as it evolves from a NS junction (Delta(1)=0) to the symmetric SNS junction (Delta(1)=Delta(2)). The de current I-0 flowing at small voltages is I-0=4e Delta(1)/h, and is different from the critical current I-c. The differential conductance changes abruptly at eV=Delta(2)-Delta(1) due to the loss of a single Andreev reflection from the larger-gap superconductor. Finally, the voltages V-n at which subharmonic gap structures appear in the I-V relation for temperatures k(B)T similar or equal to Delta(2) depend on both energy gaps Delta(1) and Delta(2) We find a systematic way to classify this subgap structure.