Time domain simulation of Shapiro steps in Josephson junctions

Using the time domain formulation of the theory of the tunnel junction, we investigated Shapiro steps by digital simulation, under the condition of a constant current source given by I_{dc} + I_{rf} \sin \omegat . The integral kernels for the Josephson and the quasiparticle current were computed assuming a nonzero pair breaking parameter \delta=0.1 , and T = 0K. We obtained I rf dependence of the zeroth and first Shapiro steps, I 0 and I 1 , and the frequency dependence of I_{1}^{\MAX} , the maximum of the first Shapiro step as a function of I rf , for a few values of the junction capacitance. we found the following results. (1) I_{1}^{\MAX}/I_{c} (I c :the critical current), showed the Riedel peak at \omega/\omega_{g} \simeq 1 , where ω g is the gap frequency 4 Δ/h. (2) For \omega/\omega_{g} >1\cdot I_{1}^{\MAX}/I_{c} agrees well with that for the constant voltage bias. (3) As the frequency becomes smaller below \omega_{g} \cdot I_{1}^{\MAX}/I_{c} is severely depressed compared to that for the constant voltage bias.