Cosine and other terms in the Josephson tunneling current

To second order in perturbation theory the tunneling current between two superconductors can be expressed as follows: I(V,T)=IJ1(V,T)sinφ+IJ2(V,T)cosφ+Iqp(V,T), where V is a constant voltage across the tunneling barrier, T is the temperature, and φ=-2eVt+φ₀ is the difference in the phases of the wave functions of the superconductors on each side of the barrier. Numerical evaluations of each of the terms are presented as functions of voltage for several temperatures. For the second term we find a different sign from that found in previous numerical work. When the superconductors on each side of the tunneling barrier are different, structure occurs at a voltage corresponding to the difference in the energy gaps. For the first two terms this structure was previously unrecognized. In addition, it is shown that the term in cosφ has no effect upon rf-induced steps in the time-averaged current-voltage curve for a tunneling junction biased by a voltage source. Finally a relation is discussed between tunneling and other experiments such as far-infrared absorption and acoustic attenuation in superconductors. It is shown that tunneling can be thought of in terms of a slight generalization of the coherence effects which dominate the other kinds of experiments.