Superconductors in a Magnetic Field

Why can a magnetic field enter one superconductor before it becomes normal? A type-II superconductor can admit flux in vortices while superconductivity survives between their cores. The first vortices enter near the lower critical field; only near the upper critical field does bulk superconductivity disappear. A type-I material has a single bulk transition field.

Compare two synthetic materials near their transition temperature. Raise the field to move from magnetic exclusion to a vortex state or the normal state, then follow the corresponding critical-field curves.

Lower critical field $\text{Loading}$

Upper / thermodynamic field $\text{Loading}$

Applied field $\text{Loading}$

Material state $\text{Loading}$

Change the physical setting

Synthetic material

Temperature relative to transition

0.9

T/T_c

Applied magnetic induction

12

B_a/\mathrm{mT}

Drag the source dial to change the field.

Model and controls

This near-transition Ginzburg–Landau teaching model uses synthetic penetration-depth and coherence-length amplitudes. The type-II sample has a ratio of six; its lower field uses a large-ratio London approximation. The type-I sample has a ratio of one quarter and uses the thermodynamic critical field. Applied induction is compared with bulk critical fields, with demagnetization neglected. The slab, flux markers and vortex lattice are qualitative schematics. The source coil indicates an imposed uniform field; its geometry is not used to calculate that field. Vortex density uses a disclosed interpolated induction proxy and is not a self-consistent GL solution. Pinning, transport current, hysteresis, finite geometry, thermal fluctuations and cuprate pairing mechanisms are omitted.

The focused scene control also supports arrow keys and Home / End.

Observe, then compare

Synthetic model data

Field curves and an ideal Meissner reference, shown separately.

Near-transition critical fields

Magnetic screening profile

Data, units and reference limits

Critical-field curves use synthetic GL amplitudes: cyan is the upper field for type II or the thermodynamic field for type I; blue is the lower field for type II. Amber marks the current setting.

One-sentence intuition: Vortices allow flux through a type-II material while superconductivity survives outside their cores.

Core mathematical model

Characteristic lengths near transition

\lambda=\lambda_* (1-T/T_c)^{-1/2},\quad\xi=\xi_* (1-T/T_c)^{-1/2}

The starred quantities are synthetic GL amplitude parameters, not extrapolated zero-temperature measurements.

Upper critical induction

B_{c2}=\frac{\Phi_0}{2\pi\xi^2}

It decreases linearly with the distance below the transition temperature in this approximation.

Lower field for the large-ratio sample

B_{c1}\simeq\frac{\Phi_0}{4\pi\lambda^2}(\ln\kappa+0.5),\quad\kappa=\lambda/\xi

This London approximation is used only for the synthetic type-II sample with a length ratio of six.

Common difficulties

Reading the model correctly

Typical misconceptionThe plotted critical-field curves predict an arbitrary real superconducting material.

Better mental modelThe amplitudes were selected for teaching. Geometry, pinning and microscopic material physics require additional models and measurements.

Run the experiment

01
Admit the first vortices
Compare Exclude the field and Vortices enter.
What to observe: The lower critical field marks the first thermodynamically allowed flux penetration in the ideal bulk picture.
02
Destroy the remaining order
Select Destroy superconductivity.
What to observe: Above the upper field, the sample is normal throughout.
03
Change material type
Select Type-I comparison and raise Applied magnetic induction.
What to observe: The bulk model switches directly from exclusion to normal at the thermodynamic field.

Superconductors in a Magnetic Field

Observe, then compare

Two lengths set the magnetic response

Core mathematical model

Characteristic lengths near transition

Upper critical induction

Lower field for the large-ratio sample

Common difficulties

Reading the model correctly

Run the experiment

Admit the first vortices

Destroy the remaining order

Change material type

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