Energy conservation and reversibility during thermodynamic changes of state in superconductors: Joule heat vs. magnetocaloric cooling

In the Meissner phase of a superconductor, an external constant magnetic field is shielded by circulating persistent zero-resistance supercurrents that are formed by Cooper pairs. However, a thermodynamic change of state within this phase, such as cooling or heating, inevitably generates normal currents of thermally excited unpaired charge carriers, induced by the time-dependent variations in the local magnetic field. They not only lead to deviations of the magnetic-field distribution from textbook Meissner profiles but also cause dissipative Joule heating. This sharply contradicts the expected reversibility of a truly thermodynamic superconducting state, a fact that has largely been overlooked in the literature. We show that these normal currents also produce a magnetocaloric cooling, which in total instantaneously and precisely compensates for the dissipated heat, thus ensuring overall energy conservation and reversibility. However, the Joule heating and magnetocaloric cooling processes are spatially distinct and should therefore lead to temperature inhomogeneities. We quantify these effects assuming realistic material parameters and conclude that they are challenging to measure with current experimental techniques. Significant temperature gradients are expected only directly at the first-order transition to the superconducting state, where the discontinuous flux expulsion should induce normal currents that are much larger than those deep in the Meissner phase. We also argue that the underlying physics in superconductors is fundamentally identical to that of thermomagnetic generators, where electromechanical work can be extracted from magnetized matter subjected to thermal cycles, and where the magnetocaloric cooling is balanced by the heat supplied from an external thermal reservoir.