9.1 Meissner effect

Discovery of Meissner effect

Walther Meissner and Robert Ochsenfeld discovered this effect in 1933 while studying the magnetic properties of superconducting tin and lead. They observed that a superconductor expels magnetic fields from its interior when cooled below its critical temperature ($T_c$). This wasn't just a consequence of zero resistance; it was a distinct thermodynamic property.

The distinction matters. A "perfect conductor" (hypothetically, zero resistance but not a superconductor) would simply trap whatever magnetic flux was already inside it when resistance vanished. A true superconductor actively expels the field, regardless of whether the field was applied before or after cooling. That active expulsion is what makes the Meissner effect a defining feature of superconductivity.

Experimental setup for demonstrating Meissner effect

A typical demonstration works like this:

1. Place a superconducting sample in an external magnetic field at a temperature above $T_c$ (the field penetrates the sample normally).
2. Cool the sample below $T_c$.
3. As the material enters the superconducting state, it expels the magnetic field from its interior.

You can detect this expulsion by measuring the magnetic flux density around the sample with a magnetometer, or more dramatically, by watching a small permanent magnet levitate above a cooled superconductor. The levitation occurs because the expelled field creates a repulsive force between the magnet and the superconductor's surface.

Magnetic field expulsion in superconductors

When a material enters the superconducting state, persistent surface currents (called screening currents) spontaneously arise. These currents produce a magnetic field that exactly cancels the applied field inside the bulk of the superconductor. The external field lines bend around the sample rather than passing through it.

From a thermodynamic perspective, this expulsion minimizes the free energy of the superconducting state. The system "prefers" to spend energy generating surface currents rather than allowing magnetic flux inside, because the condensation energy gained by remaining superconducting outweighs the kinetic energy cost of those currents.

Critical magnetic field for destroying superconductivity

The Meissner effect only persists up to a critical magnetic field ($H_c$). Above this field, the energy cost of expelling the field exceeds the condensation energy, and superconductivity is destroyed.

• Type-I superconductors exhibit a complete Meissner effect for all fields below a single $H_c$. At $H_c$, superconductivity vanishes abruptly.
• Type-II superconductors have two critical fields. Below the lower critical field $H_{c1}$, the Meissner effect is complete. Between $H_{c1}$ and the upper critical field $H_{c2}$, magnetic flux partially penetrates in quantized vortices (the mixed state). Above $H_{c2}$, superconductivity is fully destroyed.

Temperature dependence of critical magnetic field

The critical field $H_c$ decreases as temperature increases, reaching zero at $T_c$. A useful empirical approximation is:

$H_c(T) = H_c(0)\left[1 - \left(\frac{T}{T_c}\right)^2\right]$

Here $H_c(0)$ is the critical field at absolute zero. This parabolic relationship means that a superconductor operating close to $T_c$ can tolerate only weak fields, while one cooled well below $T_c$ can withstand much stronger fields. Device designers must account for this trade-off.

Microscopic origin of Meissner effect

The Meissner effect originates from Cooper pairs. These are pairs of electrons bound together through an attractive interaction mediated by lattice vibrations (phonons). One electron slightly deforms the positive ion lattice, and a second electron is attracted to that deformation. The resulting bound pair has a lower energy than two separate electrons at the Fermi surface.

Cooper pairs are bosons (they have integer spin), so they can all occupy the same quantum state. This collective condensation into a single macroscopic quantum state is what gives rise to both zero resistance and the Meissner effect.

Role of Cooper pairs in magnetic field expulsion

Cooper pairs carry a net charge of $2e$ and respond collectively to electromagnetic fields. When an external magnetic field is applied, the pairs accelerate and establish supercurrents that flow along the surface of the material. These supercurrents generate a magnetization that precisely cancels the applied field in the interior.

Because all Cooper pairs share the same macroscopic wavefunction, their response is coherent. Any attempt by the magnetic field to penetrate the bulk would require breaking this coherence, which costs energy. As long as the field stays below $H_c$, the system maintains full expulsion.

London equations for describing Meissner effect

Fritz and Heinz London proposed their phenomenological equations in 1935 to describe the electrodynamics of superconductors. The two key equations are:

• First London equation: Relates the time derivative of the supercurrent density $\mathbf{J}_s$ to the electric field $\mathbf{E}$. It captures the fact that supercurrents accelerate freely (no resistance).
• Second London equation: Relates the curl of the supercurrent density to the magnetic field. Combined with Maxwell's equations, it predicts that magnetic fields decay exponentially inside a superconductor.

The characteristic decay length is the London penetration depth $\lambda_L$, typically on the order of 10–100 nm. The field isn't expelled perfectly sharply at the surface; it decays as $B(x) \propto e^{-x/\lambda_L}$ from the surface inward.

Thermodynamics of Meissner effect

The Meissner effect can be understood as a consequence of minimizing the Gibbs free energy in the presence of an applied magnetic field. The superconducting state is thermodynamically favored whenever the energy gained from Cooper pair condensation exceeds the energy cost of expelling the magnetic field.

Gibbs free energy considerations

The Gibbs free energy of a superconductor in a field $H$ has two competing contributions:

• Condensation energy: The energy gained by forming the superconducting state (this favors superconductivity).
• Magnetic energy: The energy cost of expelling the applied field (this opposes superconductivity).

Below $H_c$, the condensation energy wins, and the superconducting state has lower Gibbs free energy than the normal state. At exactly $H_c$, the two energies balance, and the two states have equal free energy. Above $H_c$, the magnetic energy cost dominates, and the normal state becomes favorable.

Entropy change during superconducting transition

The transition into the superconducting state is accompanied by a decrease in entropy. This makes physical sense: Cooper pairs represent a more ordered state than unpaired electrons, with fewer available electronic excitations.

At $H = 0$, the transition at $T_c$ is second-order (no latent heat, but a discontinuity in specific heat). In the presence of an applied field, the transition at $H_c(T)$ becomes first-order, with a latent heat related to the entropy difference through the Clausius-Clapeyron relation:

$\frac{dH_c}{dT} = -\frac{\Delta S}{\Delta M}$

where $\Delta S$ is the entropy change and $\Delta M$ is the magnetization change at the transition.

Meissner effect vs perfect diamagnetism

The Meissner effect is often called "perfect diamagnetism," and while the two are closely related, there's an important conceptual distinction.

• Perfect diamagnetism simply means $B = 0$ inside the material, with a magnetic susceptibility of $\chi = -1$. This describes the result but not the mechanism.
• The Meissner effect specifically refers to the spontaneous expulsion of flux when a material is cooled through $T_c$ in an applied field. A hypothetical "perfect conductor" would maintain $B = 0$ if cooled in zero field and then exposed to a field (by Lenz's law), but it would trap flux if the field were applied first. A superconductor expels the field either way.

This history-independence is what proves the Meissner effect is a true thermodynamic equilibrium property, not just a consequence of zero resistance. It was this observation that led the Londons to develop their equations and that ultimately pointed toward superconductivity being a distinct phase of matter.

Type-I vs type-II superconductors

Superconductors fall into two categories based on how they respond to magnetic fields:

• Type-I (e.g., mercury, lead, tin): Complete Meissner effect below a single critical field $H_c$. At $H_c$, superconductivity is destroyed in a first-order transition. These tend to be pure elemental metals with relatively low $H_c$ values.
• Type-II (e.g., niobium, vanadium, high-temperature cuprate superconductors): Complete Meissner effect below $H_{c1}$. Between $H_{c1}$ and $H_{c2}$, they enter a mixed state where magnetic flux partially penetrates. Above $H_{c2}$, superconductivity is fully destroyed.

The distinction depends on the Ginzburg-Landau parameter $\kappa = \lambda_L / \xi$, where $\xi$ is the coherence length. Type-I superconductors have $\kappa < 1/\sqrt{2}$, and type-II have $\kappa > 1/\sqrt{2}$.

Differences in magnetic properties

Type-II superconductors are far more useful in practice because their upper critical fields can be extremely high, allowing them to carry large currents in strong magnetic fields.

Abrikosov vortex lattice in type-II superconductors

In the mixed state (between $H_{c1}$ and $H_{c2}$), magnetic flux enters a type-II superconductor not uniformly but in discrete tubes called Abrikosov vortices. Each vortex carries exactly one flux quantum:

$\Phi_0 = \frac{h}{2e} \approx 2.07 \times 10^{-15} \text{ Wb}$

Each vortex has a normal-state core (radius on the order of the coherence length $\xi$) surrounded by circulating supercurrents that decay over a distance $\lambda_L$. The vortices repel each other and, in a clean system, arrange into a regular triangular (hexagonal) lattice to minimize the total energy. Alexei Abrikosov predicted this structure theoretically in 1957, and it was later confirmed experimentally.

Pinning these vortices at defect sites is critical for applications, because moving vortices dissipate energy and destroy the zero-resistance property.

Applications exploiting Meissner effect

The Meissner effect enables several technologies that rely on magnetic field expulsion, flux trapping, or frictionless magnetic support.

Magnetic levitation using superconductors

The repulsive force from the Meissner effect can stably levitate a permanent magnet above a superconductor (or vice versa). This principle underlies:

• Maglev trains: Superconducting magnets on the vehicle interact with guideway coils or tracks, providing contactless, nearly frictionless propulsion. Japanese SCMaglev trains use this approach and have reached speeds above 600 km/h.
• Low-gravity research: Superconducting levitation can suspend small samples in a stable, vibration-free environment for studying material behavior.

Superconducting bearings and flywheels

Superconducting bearings use flux trapping (particularly in type-II superconductors) to create contactless, extremely low-friction support for rotating shafts. A magnet is levitated above or within a superconductor, and the trapped flux provides a strong restoring force that keeps the rotor centered.

These bearings are used in:

• Flywheel energy storage systems: The low friction allows flywheels to spin for very long periods with minimal energy loss, making them useful for grid-scale energy storage.
• High-speed centrifuges and precision instruments: Where mechanical bearings would introduce unacceptable friction or vibration.

Limitations of Meissner effect

Thickness dependence of magnetic field penetration

The magnetic field isn't expelled with a perfectly sharp boundary. It decays exponentially over the London penetration depth $\lambda_L$, which is typically 20–200 nm depending on the material.

For bulk samples (thickness $\gg \lambda_L$), this is negligible. But for thin films with thickness comparable to or less than $\lambda_L$, the field penetrates significantly through the sample, weakening the Meissner effect. This must be accounted for when designing thin-film superconducting devices such as microwave resonators or SQUID sensors.

Demagnetization effects in non-ellipsoidal samples

The Meissner effect works most cleanly in ellipsoidal geometries, where the demagnetizing field is uniform throughout the sample. In non-ellipsoidal shapes (rectangular slabs, thin disks, irregular geometries), the magnetic field distribution becomes non-uniform.

Near sharp corners and edges, the local field can be significantly enhanced above the average applied field. If this local field exceeds $H_c$ (or $H_{c1}$ for type-II), superconductivity breaks down in those regions even though the applied field is nominally below the critical value. Careful sample geometry and orientation relative to the applied field are necessary to avoid premature flux entry in practical devices.