It is a little bit surprising to me that it seems hard to find a detailed, microscopic derivation of the Josephson effect. Generally, you see a hand-waving derivation based on the macroscopic coherence theory (or assumption).
So, could anyone point a relevant paper to me?
I know the paper by Cohen, Falicov, and Phillips, but I cannot make sense of it.
As far as I remember, the paper you cite (let me give the full reference)
is about tunneling effect between two superconductors. The Josephson effect is not discussed in this paper, for the simple reason Josephson didn't published his papers at the time of the above paper ...
Perhaps the Josephson effect was first microscopically calculated by Josephson himself ... What about
Josephson, B. D. (1962). Possible new effects in superconductive tunnelling. Phys. Lett., 1, 251–253.
Josephson, B. D. (1964). Coupled Superconductors. Reviews of Modern Physics, 36, 216–220.
Josephson, B. D. (1965). Supercurrents through barriers. Advances in Physics, 14, 419–451.
Josephson, B. D. (1968). Weakly coupled superconductors. In Superconductivity (in two volumes), vol. 1 (pp. 423–448). Ed. R.D Parks, Marcel Dekker, Inc.
What do you call microscopic ? Calculating from the tunneling Hamiltonian, or from conservation of the current and overlapping Green's function isn't sufficient for you ?
If you're looking for a textbook, then the tunneling calculation, originated in
can be find in
Other important papers, of historical impact, are
Andreev, A. F. (1964). The thermal conductivity of the intermediate state in superconductors. Sov. Phys. JETP, 19, 1228–1231. PDF link
Andreev, A. (1966). Electron spectrum of the intermediate state of superconductors. Sov. Phys. JETP, 22, 18–23. PDF link
Tinkham, M., Blonder, G., & Klapwijk, T. M. (1982). Transition from metallic to tunneling regimes in superconducting microconstrictions: Excess current, charge imbalance, and supercurrent conversion. Physical Review B, 25, 4515–4532.
which start the modern aspects of mesoscopic superconductivity, and quite recent reviews are
Beenakker, C. W. J. (1997). Random-matrix theory of quantum transport. Reviews of Modern Physics, 69, 731–808.
Buzdin, A. I. (2005). Proximity effects in superconductor-ferromagnet heterostructures. Reviews of Modern Physics, 77, 935–976.
Bergeret, F. S., Volkov, A. F., & Efetov, K. B. (2005). Odd triplet superconductivity and related phenomena in superconductor-ferromagnet structures. Reviews of Modern Physics, 77, 1321–1373.
Eschrig, M. (2011). Spin-polarized supercurrents for spintronics. Physics Today, 64, 43–49.
Linder, J., & Robinson, J. W. A. (2015). Superconducting spintronics. Nature Physics, 11, 307–315.
