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Short proofs of Josephson's results
Abstract
Upon the consideration of a superconducting ring interrupted by a barrier and the use of Wilson–Sommerfeld quantization rule along with some fundamental facts of electrodynamics and characteristic properties of the superconductive state, we give a short proof of Josephson's voltage-frequency relation and we establish, in a simple manner, the dependence of Josephson's critical current on the magnetic flux. The original insights into the derivation, due to their elementary and intuitive character, as well as the link they provide with the old quantum theory, will allow the reader to understand the peculiarities underlying the Josephson effect.
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Yes, except that the rigorous equations are nonlocal and involve the internal pair coordinates.
DOI:https://doi.org/10.1103/PhysRevLett.7.46
Superconducting systems partitioned by barriers which are thin enough to allow supercurrents to pass through them are discussed. The implications for such systems of superconducting long-range order are considered, and a phenomenological theory developed and applied to a number of topics: penetration of a magnetic field into a barrier, interference of supercurrents in the presence of a magnetic field, the cut-off frequency associated with the propagation of electromagnetic waves along a barrier, a.c. supercurrents, and structure in the d.c. current-voltage characteristics of a barrier resulting from an applied microwave field or the a.c. supercurrent. The phenomenological theory is justified by the use of microscopic theory, and a new approach, based on the Gor'kov method, is described. The results of this approach, which is of more general validity than the tunnelling Hamiltonian method, are shown to be expressible in terms of Feynman diagrams. Attention is drawn to the analogy between supercurrents and other processes involving phase coherence.
The Josephson effect in a superconducting ring, interrupted by a barrier, is shown to be the direct consequence of fundamental principles and, hence, to exactly obey the voltage-frequency relation used for a precise determination of e/h. Supplementary considerations deal with the role of electron pairing and with additional features such as those found in the presence of two parallel junctions. The special case of a thin ring is used to illustrate the treatment of dynamical properties, including the possible occurrence of hysteresis. A more detailed discussion of the effects due to a potential barrier is presented, followed by a rederivation of Josephson's differential equation which is seen to govern the time dependence of the penetrating flux.