Meissner Effect

Definition

When a superconductor is cooled below its critical temperature T_c, it actively expels essentially all magnetic flux from its interior. This is the Meissner effect, and it is the true defining signature of superconductivity.

The crucial word is active. The material does not simply trap whatever field was present when it became superconducting — it pushes the field out, even a field that was already there. The bulk interior settles to:

B_interior = 0    (deep inside, in the bulk)

Equivalently, a superconductor is a perfect diamagnet with magnetic susceptibility χ = −1: its induced magnetization exactly opposes any applied field, driving the net interior field to zero.

Why it is NOT just zero resistance

This is the single most common misconception, so it is worth being precise. Imagine a hypothetical perfect conductor — a material with exactly zero resistance but no other special property. Maxwell's equations tell us that in a perfect conductor the field cannot change: any attempt to change the enclosed flux drives a current (Lenz's law) that perfectly opposes the change. So:

• Perfect conductor, cooled in zero field, then field applied: the field is excluded. (Looks like Meissner.)
• Perfect conductor, cooled in a field: the field is frozen in place and stays there. The interior field is whatever it was at the moment of transition.

A real superconductor behaves differently in the second case. Cool it in a steady field and the field is still expelled — the final state is independent of history. That is impossible for a mere perfect conductor and proves superconductivity is a distinct thermodynamic phase. Walther Meissner and Robert Ochsenfeld measured exactly this in 1933 with tin and lead cylinders, and the distinction has carried their name ever since.

How it works — screening currents and the London depth

The expulsion is not magic; it is paid for by real, persistent currents. When the field is expelled, dissipationless screening currents spontaneously flow in a thin skin at the surface. By Ampère's law these currents create a magnetic field inside the bulk that is exactly equal and opposite to the applied field. The two cancel, and the interior B is zero.

But the cancellation cannot be infinitely sharp. The screening currents — and therefore the residual field — live in a surface layer of finite thickness. The field decays into the material exponentially:

B(x) = B₀ · e^(−x / λ_L)

where x is depth below the surface and λ_L is the London penetration depth. The two London brothers, Fritz and Heinz London, derived this in 1935 from a phenomenological modification of Ohm's law for the superelectrons. The penetration depth is set by the density n_s of superconducting carriers:

λ_L = √( m / (μ₀ · n_s · e²) )

The denser the superfluid of paired electrons, the thinner the skin and the more completely the field is expelled. For typical elemental superconductors λ_L ≈ 20–100 nm — a few hundred atomic spacings. After about five penetration depths (≈ 100–500 nm), e^(−5) ≈ 0.0067, so less than 1% of the surface field survives. In a millimetre-thick sample the interior is field-free to extraordinary precision.

Worked example — field at depth in niobium

Take niobium, the workhorse of superconducting magnets and RF cavities, with λ_L ≈ 39 nm at low temperature. Suppose the surface sees an applied field of B₀ = 50 mT (just below niobium's lower critical field). How deep must we go before the field drops to 1% of its surface value?

B(x) / B₀ = e^(−x / λ_L) = 0.01
−x / λ_L = ln(0.01) = −4.605
x = 4.605 × λ_L = 4.605 × 39 nm ≈ 180 nm

So at a depth of about 180 nm — less than one five-thousandth of a millimetre — the 50 mT surface field has collapsed to 0.5 mT. At one penetration depth (39 nm) the field is already down to B₀/e ≈ 18.4 mT, a 63% drop. The screening is brutally fast. This is why a superconducting RF cavity can sustain enormous surface fields while keeping its bulk perfectly field-free, and why thin films thinner than λ_L behave very differently from bulk — they cannot fully expel the field at all.

Depth x	x / λ_L	B(x) / B₀	B(x) at B₀ = 50 mT
0 nm	0	1.000	50.0 mT
39 nm	1	0.368	18.4 mT
78 nm	2	0.135	6.8 mT
117 nm	3	0.050	2.5 mT
156 nm	4	0.018	0.9 mT
195 nm	5	0.0067	0.34 mT

The critical field H_c — when expulsion fails

Expelling the field is not free. Keeping a field B out of a volume costs magnetic energy density B²/2μ₀. Superconductivity, meanwhile, only buys you a fixed condensation energy — the energy advantage of the paired superconducting state over the normal state. When the applied field is small, condensation energy wins and the field is expelled. When the field is large enough that B²/2μ₀ exceeds the condensation energy, superconductivity can no longer pay the bill and collapses, letting the field back in.

The threshold is the thermodynamic critical field H_c:

½ μ₀ H_c²  =  condensation energy density (per unit volume)

H_c is temperature dependent, vanishing at T_c and rising as the sample is cooled, following the empirical relation:

H_c(T) = H_c(0) · [ 1 − (T / T_c)² ]

So flux is fully expelled only below H_c. Above it the Meissner state is destroyed. For type-I metals H_c(0) is small — lead is about 80 mT, tin about 30 mT, aluminium about 10 mT — which is why a simple bar magnet can punch straight through a chunk of pure lead.

Variants — type-I, type-II, and the vortex state

Whether the breakdown is abrupt or gradual depends on the ratio κ = λ_L / ξ of the penetration depth to the coherence length ξ (the size of a Cooper pair). This single number splits all superconductors into two families.

Property	Type-I	Type-II
Ginzburg–Landau κ	κ < 1/√2	κ > 1/√2
Meissner state	Complete up to H_c	Complete up to H_c1 only
Above first threshold	Abruptly normal	Mixed (vortex) state up to H_c2
Flux penetration	None until H_c	Quantized vortices, Φ₀ = h/2e ≈ 2.07×10⁻¹⁵ Wb each
Typical examples	Pb, Sn, Al, Hg	Nb, NbTi, Nb₃Sn, cuprates
Upper critical field	≈ 10–80 mT	up to tens of teslas (Nb₃Sn ≈ 30 T)

In a type-II material between H_c1 and H_c2 the field does not stay fully out: it threads through in discrete flux vortices, each a tiny normal-cored tube carrying exactly one flux quantum Φ₀. The bulk between vortices remains superconducting. Crucially, these vortices can be pinned by defects, which both enables high-field magnets (the vortices don't move and dissipate) and produces the rock-stable levitation seen in demonstrations — a pinned magnet can even hang underneath a superconductor, which pure Meissner repulsion could never do.

Where the Meissner effect shows up

• Magnetic levitation and Maglev. Flux expulsion plus flux pinning gives stable, contactless levitation. Quantum-locking demos and superconducting Maglev train prototypes both ride on this. See magnetic levitation.
• MRI and high-field magnets. Type-II NbTi and Nb₃Sn coils carry persistent currents with no dissipation, producing the steady multi-tesla fields inside MRI scanners and particle accelerators.
• SQUID magnetometers. The combination of flux quantization and the Meissner state lets SQUIDs measure magnetic fields down to femtotesla levels — sensitive enough to map brain and heart currents.
• RF cavities. Niobium accelerator cavities exploit the thin λ_L skin to confine huge surface fields while keeping the bulk field-free and lossless.
• Magnetic shielding. A superconducting can excludes external fields almost perfectly, far better than any ferromagnetic shield.
• Identifying superconductivity. Because zero resistance alone can be faked by experimental artefacts, a clean Meissner signal (χ → −1) is the gold-standard test for a genuine superconductor.

Common pitfalls and misconceptions

• "It's just zero resistance." No. Zero resistance freezes flux in; the Meissner effect expels it. A field-cooled perfect conductor keeps its field; a field-cooled superconductor does not. This is the whole reason the effect is named.
• "The field is exactly zero right at the surface." It isn't. The field penetrates a skin of order λ_L (tens of nanometres) and only becomes negligible several λ_L deep. Films thinner than λ_L never fully expel the field.
• "Any magnet will levitate over any superconductor." Only below H_c. A strong enough magnet drives the surface past the critical field and quenches the superconducting state locally, so levitation collapses.
• "Type-II superconductors expel all the field." Only below H_c1. Between H_c1 and H_c2 they admit quantized vortices while remaining superconducting overall.
• "Levitation requires pinning." Pure Meissner repulsion can levitate a magnet, but the equilibrium is unstable (Earnshaw's theorem); the stable, hands-off levitation in demos comes from flux pinning in type-II materials.
• "χ = −1 means it's a strong magnet." The opposite — it means the material perfectly opposes the field. It is the strongest possible diamagnet, not a ferromagnet.

Derivation analysis — the London equation

The exponential decay falls straight out of the London equation. The Londons proposed that in a superconductor the current density obeys

∇ × J = − (n_s e² / m) · B          (the second London equation)

Combining this with Ampère's law (∇ × B = μ₀ J, ignoring displacement current) and the identity ∇ × (∇ × B) = −∇²B for a divergence-free field gives

∇²B = (μ₀ n_s e² / m) · B = B / λ_L²

For a flat surface at x = 0 with the field along it, this one-dimensional equation ∂²B/∂x² = B/λ_L² has the physically sensible (decaying) solution

B(x) = B₀ · e^(−x / λ_L),     λ_L = √( m / (μ₀ n_s e²) )

The same equation governs J, so the screening currents share the identical λ_L skin. Plugging in electron mass, charge, and a carrier density n_s ~ 10²⁸–10²⁹ m⁻³ reproduces λ_L ≈ 20–100 nm without any free parameters — a striking confirmation that the field is screened by a real, dense superfluid of charge carriers, the Cooper pairs of BCS theory.