This sim shows a small superconducting sample in a magnetic field, in cross-section, and lets you change the temperature T, the applied field B and the material parameter κ. It follows what the field does inside the sample, because that is what separates a superconductor from a wire that merely has no resistance.

What's on screen

• Applied field lines (blue): the field B you set, running down the page.
• Screening skin λ_L (green): a thin band at the sample's surface.
• Flux tubes (rose): drawn only in some states of the sample.
• Operating point (yellow dot): where your T and B sit on the phase diagram, whose curves are the boundaries between states.
• Readout cards: Sample state (Phase, Δ), Boundary fields (B_c, B_c1, B_c2) and Flux in the sample (n).

Press ⌖ on a row to see where that control sits in the sim.

Fixed: T_c = 9.25 K, B_c(0) = 200 mT, sample face A = 1 µm². The state depends only on the three sliders: there is no history and no flux pinning. The sim shows no resistance.

A conductor with zero resistance and nothing more is a perfect conductor. With no resistance, E = 0 inside it, and Faraday's law then gives ∂B/∂t = 0: the field inside cannot change. Whatever was there when it cooled stays frozen in.

A superconductor does more. Below its lowest boundary (B_c for type I, B_c1 for type II) it holds B = 0 inside, whether the field was there before cooling or arrived afterwards. This is the Meissner effect.

• Screening currents J_s flow in the skin at the surface and make a field that cancels the applied one inside. The sim marks them ⊙ and ⊗.
• The sim models only the superconductor. It has no perfect conductor to compare with.

The screening is not perfect at the surface. The field dies away with depth x over a distance λ_L, the London penetration depth: B(x) = B_surface e^−x/λ_L. That layer is the green skin.

A second length, the coherence length ξ, is the distance over which the superconducting state itself can change, for instance from a normal region into a superconducting one. Their ratio is κ.

• Below κ = 1/√2 ≈ 0.707, a wall between normal and superconducting material costs energy (type I). Above it, a wall lowers the energy (type II). Section 05 follows the consequences.
• In the sim ξ is drawn at a fixed size, so the κ slider changes the thickness of the green skin: λ_L = κ ξ. The drawing is a schematic, not to scale.

The thermodynamic critical field B_c falls with temperature and reaches zero at T_c:

• Type I (κ < 0.707): B_c is the only boundary. Below it the sample is Meissner, above it normal.
• Type II (κ > 0.707): B_c splits into B_c1 and B_c2. Between them is the mixed state, superconducting but with field inside. The B_c1 formula is a large-κ estimate, which is why the sim leaves it out for κ ≤ 1.

In the superconducting state the electrons form pairs, and it takes an energy 2Δ to break one. Δ is the energy gap. The sim gives it the zero-field value Δ(0) = 1.764 k_BT_c = 1.406 meV, and an interpolation that reaches zero at T_c:

• Δ barely moves at low T, then falls steeply and reaches 0 at T_c = 9.25 K.
• In the sim Δ depends on T alone. It reads zero as soon as the sample turns normal, whether T or B caused it. A real gap also shrinks as B rises.

In the mixed state the field passes through the sample as separate flux tubes (vortices): thin normal cores in a superconducting background. Each carries exactly one flux quantum Φ₀ = h/2e = 2.0678 × 10⁻¹⁵ Wb, where the 2e is the charge of a pair.

• The sim counts the applied flux through the face A = 1 µm² and rounds down: n = ⌊B·A/Φ₀⌋. One more tube appears every Φ₀/A = 2.068 mT. At 50 mT, B·A/Φ₀ = 24.18, so n = 24; at 100 mT it is 48.
• More field means more tubes, never fatter ones.
• This count is accurate only for B well above B_c1. Just above it a real sample holds far fewer tubes, but the sim's count opens at 14 (at 4.2 K, κ = 1.50).

• MRI and research magnets. They are wound from type II wire, usually niobium–titanium, in liquid helium. Type II keeps superconducting far above B_c1, up to B_c2. Elemental type I metals stop at about 0.1 T or less.
• Magnetic levitation. A magnet floats over a superconductor because the expelled field pushes back on it. The familiar demonstrations use type II ceramics, where pinned flux tubes also hold the magnet in place, which this sim does not model.
• SQUID magnetometers. A superconducting loop can hold only whole multiples of Φ₀. Devices built on this detect changes of a small fraction of Φ₀, which makes them sensitive probes of tiny magnetic fields, such as those of the brain.

Key points

• Zero resistance freezes the field (∂B/∂t = 0). Superconductivity expels it (B = 0) apart from a skin of depth λ_L.
• B_c(T) = B_c(0)[1 − (T/T_c)²]. For κ < 1/√2 it is the one boundary; above, it splits into B_c1 and B_c2 with the mixed state between.
• At B = 0, Δ falls to zero at T_c, where the operating point meets the boundary.
• In the mixed state n = ⌊B·A/Φ₀⌋ tubes, each carrying one Φ₀.

Common mix-ups

• "A superconductor is a perfect conductor." A perfect conductor would freeze the field in. A superconductor pushes it out.
• "A type II sample stops superconducting at B_c1." At B_c1 only the complete exclusion ends. Superconductivity lasts to B_c2.
• "Doubling B makes each tube carry twice the flux." Every tube carries one Φ₀. Doubling B doubles the number of tubes.

Things to try on your own

• Set κ = 0.50 and T = 4.2 K. Raise B until Phase changes, and compare that value with B_c in the Boundary fields card.
• Set κ = 1.50 and T = 4.2 K. Find the smallest B at which n appears, and compare it with B_c1.
• Set B = 0 and warm the sample. Note the temperature at which Δ reaches zero.