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Do we have any tunneling current for $0 < V \leq V_c$? Yes. If yes, then why don't we show it in the diagram? It is in the diagram, you just have to understand how that diagram was measured. Junction basics The Josephson junction is governed by two equations \begin{align} I &= I_c \sin(\delta) \\ V &= (\Phi_0 / 2\pi) \dot{\delta} ...


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I think there is still a current flowing. I found this picture on this website. The gist is that above a certain voltage $V_c=2E_g/e$ (twice the energy gap divided by elementary charge), the voltage large enough to overcome the band gap and Cooper pairs can flow. EDIT: Also look at this excellent post.



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