Limits of Meissner screening 2026-10-14
For a bulk type-II superconductor with negligible demagnetizing field, raising the applied magnetic field intensity past the lower critical value favors a superconducting mixed state containing quantum vortex lines; superconductivity survives until the upper critical value . The magnetic field extends around each vortex over a penetration scale, not solely inside its core. Thus a nonzero average magnetic flux density can coexist with superconductivity. Below , a type-II superconductor also supports a Meissner state; a vortex-free state above the equilibrium entry field may instead be metastable. [ref]
9.8 Superconductivity. §9.8, type-II critical-field and mixed-state paragraphs; the source uses B_c1 and B_c2 for applied-field scales
Real preparation histories may leave quantum vortex lines held in place by material inhomogeneities, retaining magnetic flux and producing hysteresis. This flux pinning restricts access to equilibrium: it is different from the arbitrary frozen interior magnetic field of the ideal perfect conductor model. Harvard's vortex-imaging work illustrates strong pinning to bulk disorder in a type-II superconductor. [ref]
Scanning Tunneling Microscopy Research by Hoffman Laboratory. Vortex Imaging in Iron Pnictide Superconductor, Fig. 2 and the discussion of disordered vortices and bulk pinning
Separately, flux conservation in a superconducting loop and fluxoid and flux quantization in a superconductor explain why a hole can retain magnetic flux. Neither bore retention, vortex penetration nor weak screening in a thin sample should be silently identified with the equilibrium Planar London screening calculation. Meissner effect and flux history 2026-10-14
Compare two ideal models for a stationary, solid, simply connected sample. In the perfect conductor flux-history model the interior electric field is constrained to zero from the moment perfect conduction starts. Faraday's law of induction then givesConsequently the interior magnetic field retains its value at that moment. Cooling in an already applied magnetic field freezes a nonzero interior value; starting perfect conduction in zero magnetic field and then applying a magnetic field preserves zero. The zero initial interior magnetic field often assumed in perfect conductor wave-reflection problems is one particular preparation, not a general prediction of expulsion. [ref]
9.8 Superconductivity. §9.8, Eq. (9.38) and the argument that zero resistivity conserves existing flux rather than expelling it
The Meissner state instead selects the screened equilibrium after either preparation, provided that equilibrium is accessible and the applied magnetic field lies in the Meissner state regime. The first London equation alone illustrates the missing information: at fixed , combining it with Faraday's law of induction givesThis permits an arbitrary time-independent integration field. The second London equation imposes the additional zero value of that bracket in the vortex-free local model.[ref]
Lecture notes on the electrodynamics of metals and superconductors. §2, pp. 12–13, Eqs. (64), (66)–(68): time-independent freedom and the second London equation
Establishing superconductivity during cooling is not the fixed-parameter, zero-electric field model: transient electric fields accompany changing magnetic flux, and itself depends on temperature. The equilibrium London equations predict final spatial screening, not the cooling rate or the time-resolved expulsion pulse. Meissner state 2026-10-14
A Meissner state is an equilibrium state of superconductivity without quantum vortex penetration, with screening electric currents responding to the applied magnetic field. In a solid, simply connected sample much thicker than the London penetration depth, the deep interior magnetic field is approximately zero. This is a statement about a phase and a geometry, not a rule that every sample exhibiting superconductivity always has . In particular, the Meissner state exists for sufficiently weak magnetic fields in both type-I superconductors and type-II superconductors. [ref]
9.8 Superconductivity. §9.8, Properties of Superconductors, paragraphs distinguishing type I, type II and their two critical fields
