Two length scales determine magnetic-interface behavior

The GL gradient energy opposes rapid amplitude variation, while the negative quadratic coefficient favors the uniform superconducting amplitude. Balancing these terms defines ξ as the square root of ħ² divided by twice the effective pair mass and |α|. This is the conventional GL coherence parameter; the precise exponential recovery length around a particular uniform solution can include a numerical factor. It should also be distinguished from the microscopic BCS pair-size scale away from the near-transition GL regime.

The same uniform amplitude fixes a penetration depth through the charged current response. Near the mean-field transition, both ξ and λ diverge with the same inverse square-root temperature dependence in the simplest coefficient expansion, leaving their ratio approximately constant. In anisotropic systems these lengths can depend on direction. Multiple order-parameter components can introduce additional healing scales. Treating every real material as if it had one scalar coherence length and one scalar penetration depth can therefore hide important departures from this classification.

In local single-component GL theory the planar normal–superconducting interface has positive energy for κ below 1/√2 and negative energy above it. Positive interface energy favors separating normal and superconducting regions with as little boundary as possible: this is type-I behavior. Negative interface energy allows flux-carrying vortex structures to become favorable: this is type-II behavior. At the boundary the ideal GL interface energy vanishes. This criterion is exact for that model near its validity regime, not a universal classification theorem for all multiband or nonlocal superconductors.

The thermodynamic critical-field scale Bc=μ₀Hc equates magnetic field-energy density to the zero-field condensation energy. In a type-I bulk sample of simple geometry, it marks the equilibrium transition, although demagnetizing fields can produce an intermediate domain state. A type-II sample instead has a lower field Hc1 where vortices become thermodynamically favorable and an upper field Hc2 where bulk superconductivity is lost in the orbital GL description. Surface barriers and pinning can shift observed entry and exit fields away from equilibrium values.

For κ much greater than one, a vortex has a narrow amplitude core and a wider magnetic halo. Its lower critical field contains a logarithm of κ plus a core-energy correction. That large-κ estimate should not be extrapolated to the type boundary. The upper critical field instead follows from the lowest eigenvalue of the charged covariant-gradient operator in a uniform field. A short coherence length raises this orbital scale, but a separate paramagnetic limit or strong fluctuations can prevent a real material from reaching the orbital estimate.

The function lab isolates the gradient-energy penalty behind healing. Reset and set both controls to a=0.5 and x=1, predicting y=0.5. Reset and set a=0.5 and x=2, predicting y=2. Assign x a normalized amplitude gradient and y its quadratic gradient cost. A gradient twice as steep costs four times as much. This lab has no spatial order parameter, magnetic screening, or interface calculation; it cannot decide the sign of interface energy without the GL analysis that also includes condensation and field energies.