Time domain simulation of Shapiro steps in Josephson junctions

Abstract

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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{dc} + I_{rf} \sin \omegat</tex> . The integral kernels for the Josephson and the quasiparticle current were computed assuming a nonzero pair breaking parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\delta=0.1</tex> , and T = 0K. We obtained I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rf</inf> dependence of the zeroth and first Shapiro steps, I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> and I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , and the frequency dependence of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{1}^{\MAX}</tex> , the maximum of the first Shapiro step as a function of I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">rf</inf> , for a few values of the junction capacitance. we found the following results. (1) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">I_{1}^{\MAX}/I_{c}</tex> (I <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> :the critical current), showed the Riedel peak at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega/\omega_{g} \simeq 1</tex> , where ω <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</inf> is the gap frequency 4 Δ/h. (2) For <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega/\omega_{g} &gt;1\cdot I_{1}^{\MAX}/I_{c}</tex> agrees well with that for the constant voltage bias. (3) As the frequency becomes smaller below <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\omega_{g} \cdot I_{1}^{\MAX}/I_{c}</tex> is severely depressed compared to that for the constant voltage bias.