Charge Transport in Voltage-Biased Superconducting Single-Electron Transistors

Charge is transported through superconducting SSS single-electron transistors at finite bias voltages by a combination of coherent Cooper-pair tunneling and quasiparticle tunneling. At low transport voltages the effect of an ``odd'' quasiparticle in the island leads to a $2e$-periodic dependence of the current on the gate charge. We evaluate the $I-V$ characteristic in the framework of a model which accounts for these effects as well as for the influence of the electromagnetic environment. The good agreement between our model calculation and experimental results demonstrates the importance of coherent Cooper-pair tunneling and parity effects.