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.

💡 Research Summary

The paper investigates a subtle but fundamental aspect of superconductors in the Meissner state: when a superconductor is held in a constant external magnetic field and its temperature is slowly varied (either cooled or heated), normal quasiparticles that are thermally excited generate an induced electric field via Faraday’s law. This induced field drives a normal current Jₙ = σE, where σ is the finite conductivity of the normal component. Because σ > 0, the normal current dissipates power as Joule heat (P_J = Jₙ²/σ), apparently violating the expected reversibility of a thermodynamic superconducting process.

The authors demonstrate that the same induced current also produces a magnetocaloric cooling effect. The time‑dependent magnetic field B(r,t) creates an additional “induced” magnetic field B_ind, and the power density associated with the work done by the induced currents on the magnetic field is P_ind = B_ind·∂B/∂t. By Lenz’s law this term is always negative, meaning that energy is extracted from the material and the local temperature drops. This is precisely the magnetocaloric effect familiar from adiabatic demagnetization, but here it is driven not by an externally varying field but by the temperature‑dependent magnetization of the superconductor itself.

Using the time‑dependent London equation (1b) together with Maxwell’s equations and Ohm’s law, the authors derive explicit expressions for the spatial distributions of Jₙ, P_J, and P_ind. In a cylindrical geometry they show that Joule heating is concentrated near the surface, whereas the magnetocaloric cooling is distributed throughout the bulk. Integrating over the whole volume yields ∫(P_J + P_ind) dV = 0 at every instant, establishing exact global energy conservation and overall reversibility despite the presence of local dissipation.

Numerical estimates are performed for a type‑I superconductor with material parameters similar to niobium (Tc ≈ 9 K, λ₀ ≈ 47 nm, Hc(0) ≈ 0.2 T, normal‑state conductivity σ_n ≈ 1.8 × 10⁹ (Ω·m)⁻¹). The temperature dependence of the penetration depth λ(T) follows a two‑fluid law, and σ(T) is assumed to scale with the normal‑fluid density. For a realistic cooling rate of –0.05 K s⁻¹, the magnetic‑field profile B(r) obtained from the full time‑dependent solution is virtually indistinguishable from the static London solution, indicating that the induced currents are small except very close to the critical temperature. Consequently, measurable temperature gradients are expected only near the first‑order transition where the flux expulsion is discontinuous and normal currents become large.

The paper also draws a parallel with thermomagnetic generators. In such devices a material with a strong temperature‑dependent magnetic susceptibility is cyclically heated and cooled in a constant field; induced currents generate electrical power while a magnetocaloric cooling is compensated by heat supplied from an external reservoir. In the superconducting case, the same two processes occur internally, and they cancel each other, allowing a truly reversible thermodynamic cycle.

In summary, the authors resolve an apparent paradox: normal currents induced by temperature changes in a Meissner‑state superconductor do generate irreversible Joule heat, but an exactly compensating magnetocaloric cooling is produced by the same currents. The net effect preserves energy conservation and thermodynamic reversibility, although spatial separation of heating and cooling leads to tiny, likely unobservable temperature inhomogeneities except near the superconducting transition. This insight deepens our understanding of the interplay between electrodynamics and thermodynamics in superconductors and highlights the fundamental equivalence with magnetocaloric energy conversion in thermomagnetic generators.