6.2 London equations

The London equations provide a phenomenological description of how superconductors expel magnetic fields and carry current without resistance. Developed in 1935, they translate the two hallmark properties of superconductors (zero resistance and the Meissner effect) into a precise mathematical framework built on classical electromagnetism. Understanding these equations is essential before moving to more complete theories like Ginzburg-Landau or BCS.

Foundations of London equations

The London equations describe the macroscopic electromagnetic behavior of superconductors without requiring a full microscopic theory. They connect supercurrent density to the electromagnetic field, capturing the essential physics of the superconducting state.

Superconductivity basics

Below a material-specific critical temperature $T_c$, a superconductor exhibits two defining properties:

• Zero electrical resistance: Current flows indefinitely with no voltage drop.
• Perfect diamagnetism (Meissner effect): The material actively expels magnetic flux from its interior, not just trapping whatever flux was present when it cooled.

That second point is what separates a superconductor from a hypothetical "perfect conductor." A perfect conductor would freeze in whatever field existed when resistance vanished. A superconductor expels the field regardless of history.

Two-fluid model

The London equations rest on the two-fluid model, which treats the electron system as two interpenetrating fluids:

• Normal electrons scatter off impurities and phonons, contributing to resistivity just like in an ordinary metal.
• Superconducting electrons form Cooper pairs and flow without dissipation, carrying the supercurrent.

The fraction of electrons in the superconducting condensate grows as temperature drops below $T_c$, reaching unity at $T = 0$. This model explains why thermal and electromagnetic properties change continuously below $T_c$.

London brothers' contribution

Fritz and Heinz London proposed their equations in 1935 as a phenomenological extension of Maxwell's equations. Rather than deriving superconductivity from first principles, they postulated a specific relationship between supercurrent and electromagnetic fields that would reproduce the observed Meissner effect and zero resistance. Their key insight was treating the superconducting electrons as a quantum-mechanical fluid whose collective behavior could be captured by a simple constitutive relation.

Mathematical formulation

First London equation

The first London equation relates the time derivative of the supercurrent density to the electric field:

$\frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m} \mathbf{E}$

This says that an electric field accelerates superconducting electrons without any frictional (resistive) term. In a normal metal, friction balances the electric force and you get Ohm's law. Here, there's no friction, so the current accelerates freely.

In the London gauge (where $\nabla \cdot \mathbf{A} = 0$ and appropriate boundary conditions hold), this can be integrated to give the equivalent form:

$\mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{A}$

where:

• $n_s$ is the superconducting electron density
• $e$ is the elementary charge
• $m$ is the electron mass
• $\mathbf{A}$ is the magnetic vector potential

Second London equation

Taking the curl of the first London equation (in its integrated form) and using $\nabla \times \mathbf{A} = \mathbf{B}$:

$\nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B}$

This is the second London equation, and it's the one that directly explains the Meissner effect. It constrains the relationship between supercurrent circulation and the local magnetic field. Combined with the Maxwell equation $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s$ (neglecting displacement current), it leads to an equation showing that magnetic fields decay exponentially inside a superconductor.

Deriving the screening equation

Combining the second London equation with Ampère's law gives a differential equation for $\mathbf{B}$ inside the superconductor:

1. Start with $\nabla \times \mathbf{B} = \mu_0 \mathbf{J}_s$.
2. Take the curl: $\nabla \times (\nabla \times \mathbf{B}) = \mu_0 (\nabla \times \mathbf{J}_s)$.
3. Use the vector identity $\nabla \times (\nabla \times \mathbf{B}) = -\nabla^2 \mathbf{B}$ (since $\nabla \cdot \mathbf{B} = 0$).
4. Substitute the second London equation for $\nabla \times \mathbf{J}_s$:

$\nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2} \mathbf{B}$

This tells you that $\mathbf{B}$ decays exponentially from the surface into the bulk, with characteristic length $\lambda_L$.

Physical implications

Meissner effect explanation

The screening equation derived above has solutions of the form $B(x) = B_0 \, e^{-x/\lambda_L}$ for a semi-infinite superconductor with a flat surface. The field at the surface equals the applied field, but it drops to nearly zero within a few penetration depths.

The supercurrents that flow in this thin surface layer generate a magnetic field that exactly cancels the applied field in the interior. This expulsion occurs spontaneously when the material is cooled below $T_c$, even if the field was applied first. That spontaneous expulsion is what distinguishes the Meissner effect from simple persistent currents in a perfect conductor.

Magnetic field penetration depth

The London penetration depth sets the length scale over which magnetic fields decay:

$\lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}}$

where $\mu_0$ is the vacuum permeability.

• Typical values range from about 10 to 100 nm for conventional superconductors (e.g., $\lambda_L \approx 50$ nm for aluminum).
• $\lambda_L$ increases with temperature as $n_s$ decreases, diverging at $T_c$.
• For thin films with thickness comparable to or less than $\lambda_L$, the field penetrates throughout the sample, and the effective penetration depth is modified (the Pearl length $\Lambda = 2\lambda_L^2/d$ becomes relevant for a film of thickness $d$).

Flux quantization

When supercurrent flows around a closed loop, the single-valuedness of the Cooper pair wavefunction requires that the enclosed magnetic flux be quantized:

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

The factor of $2e$ (rather than $e$) reflects the fact that the charge carriers are Cooper pairs. Flux quantization is directly observable in superconducting rings and provides strong evidence for the paired nature of superconducting electrons. It also underlies the operation of SQUIDs (Superconducting Quantum Interference Devices).

Limitations and extensions

Validity in the local limit

The London equations assume a local relationship: the current at a point depends only on the field at that same point. This is valid when the coherence length $\xi$ (the spatial extent of a Cooper pair) is much smaller than the penetration depth $\lambda_L$.

• Works well for type-II superconductors (where $\kappa = \lambda_L/\xi > 1/\sqrt{2}$) and for "dirty" type-I superconductors where impurity scattering shortens the effective coherence length.
• Breaks down in clean type-I superconductors at low temperatures, where $\xi$ can be much larger than $\lambda_L$.

Pippard's non-local generalization

Brian Pippard (1953) extended the London theory to handle clean superconductors by introducing a non-local relationship: the current at a point depends on the vector potential averaged over a region of size $\sim \xi$. This is analogous to the anomalous skin effect in normal metals. Pippard's theory bridges the gap between the phenomenological London approach and the microscopic BCS theory, and it correctly predicts the modified penetration depth in clean materials.

Ginzburg-Landau theory vs London theory

Ginzburg-Landau (GL) theory is more general than the London equations in several ways:

• It introduces a complex order parameter $\psi(\mathbf{r})$ whose magnitude describes the local superconducting electron density and whose phase determines the supercurrent.
• It handles spatial variations of the superconducting state, including interfaces, vortices, and the mixed state of type-II superconductors.
• It applies near $T_c$ and in the presence of strong fields where the London equations (which assume uniform $n_s$) break down.
• In the limit of uniform $|\psi|$, GL theory reduces to the London equations.

Experimental verification

Magnetic field expulsion measurements

• Sensitive magnetometers detect the sudden onset of diamagnetism below $T_c$.
• Magnetic susceptibility measurements yield $\chi = -1$ (in SI, for full Meissner state), confirming complete flux expulsion.
• Levitation of a magnet above a superconductor provides a visual demonstration.
• Scanning SQUID microscopy maps the spatial distribution of magnetic field near the superconductor surface with sub-micron resolution.

Penetration depth experiments

Several techniques measure $\lambda_L$ and its temperature dependence:

• Microwave cavity perturbation: A superconducting sample changes the resonant frequency of a cavity; the shift is proportional to $\lambda_L$.
• Muon spin rotation ($\mu$SR): Implanted muons precess in the local field, mapping the field profile inside a vortex lattice and extracting $\lambda_L$.
• The temperature dependence of $\lambda_L(T)$ reveals information about the gap symmetry (exponential for s-wave, power-law for d-wave).

Flux quantization observations

• Deaver and Fairbank (1961) measured quantized flux in superconducting cylinders, confirming $\Phi_0 = h/2e$.
• Magnetic decoration techniques (Bitter decoration) image individual vortices in type-II superconductors, each carrying one flux quantum.
• SQUID-based measurements detect single flux quanta in superconducting loops with extraordinary sensitivity.

Applications in superconductivity

Josephson junctions

A Josephson junction consists of two superconductors separated by a thin barrier (insulator, normal metal, or weak link). Cooper pairs tunnel coherently across the barrier, producing:

• DC Josephson effect: A supercurrent flows with zero voltage, determined by the phase difference across the junction.
• AC Josephson effect: A DC voltage $V$ produces an oscillating current at frequency $f = 2eV/h$.

These junctions are the building blocks of SQUIDs, superconducting digital logic, and superconducting qubits for quantum computing.

SQUID devices

SQUIDs combine one or two Josephson junctions in a superconducting loop. The critical current of the loop oscillates as a function of the enclosed flux with period $\Phi_0$, making SQUIDs the most sensitive magnetometers available (sensitivity below $10^{-15}$ T). Applications include:

• Biomagnetism (magnetoencephalography, magnetocardiography)
• Geophysical surveying
• Non-destructive materials testing

Superconducting magnets

Type-II superconductors like NbTi and $\text{Nb}_3\text{Sn}$ carry large currents in high magnetic fields without resistance, making them ideal for:

• MRI machines (fields of 1.5 to 7 T in clinical use)
• Particle accelerators (the LHC uses over 1,200 dipole magnets at 8.3 T)
• Fusion reactor confinement magnets

These magnets require cryogenic cooling (liquid helium at 4.2 K for low-$T_c$ materials), but once energized, they maintain their field with negligible power input.

London equations in different geometries

Thin films

When the film thickness $d$ is comparable to or smaller than $\lambda_L$, the magnetic field penetrates the entire film and the standard exponential decay picture breaks down. The effective screening is weaker, and the relevant length scale becomes the Pearl length $\Lambda = 2\lambda_L^2/d$, which can be much larger than $\lambda_L$. This regime is critical for superconducting electronics, single-photon detectors, and qubit design.

Cylindrical geometries

For superconducting wires and cables, the London equations are solved in cylindrical coordinates. The field and current distributions depend on the wire radius relative to $\lambda_L$ and on the orientation of the applied field. Understanding these distributions is important for calculating AC losses in superconducting power transmission lines, where time-varying fields drive resistive losses in the normal-electron component.

Spherical superconductors

Solving the London equations for a superconducting sphere gives the field distribution around and inside a spherical grain or nanoparticle. As the particle size approaches $\lambda_L$, the Meissner effect becomes incomplete and the critical field is enhanced relative to the bulk value. These size effects are relevant for granular superconductors and superconducting nanoparticle research.

Connection to microscopic theory

BCS theory relationship

The London equations emerge naturally from BCS theory in the long-wavelength, low-frequency limit. BCS theory provides microscopic expressions for the parameters in the London equations:

• The penetration depth $\lambda_L$ is related to the energy gap $\Delta$, Fermi velocity $v_F$, and density of states at the Fermi level.
• The temperature dependence of $\lambda_L(T)$ follows from the temperature dependence of $\Delta(T)$ and the thermal depletion of the condensate.

This connection validates the London equations as the correct macroscopic limit of the full quantum theory.

Quasiparticle excitations

The London equations describe only the superfluid component. At finite temperature, thermally excited quasiparticles (broken Cooper pairs) also respond to electromagnetic fields and contribute a dissipative (normal) current. The two-fluid model captures this by writing the total current as $\mathbf{J} = \mathbf{J}_s + \mathbf{J}_n$, where $\mathbf{J}_n$ obeys Ohm's law. Quasiparticle contributions become significant for AC response and thermal transport.

Cooper pairs and London equations

The London equations describe the collective, coherent motion of the Cooper pair condensate. The macroscopic wavefunction $\psi = \sqrt{n_s} \, e^{i\theta}$ has a well-defined phase $\theta$, and the supercurrent is proportional to the gradient of that phase. Flux quantization and the Josephson effect both follow directly from the single-valuedness and coherence of this macroscopic phase.

Numerical methods

Finite element analysis

For realistic device geometries (not just infinite slabs or cylinders), the London equations are solved numerically using finite element methods. The mesh must resolve the penetration depth, which at tens of nanometers can be orders of magnitude smaller than the device dimensions. Boundary conditions at superconductor-vacuum and superconductor-normal interfaces must be handled carefully.

Computational challenges

• Multiscale problems: $\lambda_L$ is nanoscale while devices can be millimeters or larger, requiring adaptive meshing.
• Nonlinearity: Near the critical current or critical field, the superfluid density is no longer constant and the London equations must be replaced by GL or more complete models.
• Coupled physics: Practical simulations often couple electromagnetic, thermal, and mechanical equations (e.g., quench propagation in a magnet).

Simulation of superconducting systems

Modern simulation tools combine the London (or GL) equations with full Maxwell solvers to design superconducting magnets, RF cavities for particle accelerators, and quantum circuits. These simulations incorporate models for flux pinning and critical current density, enabling virtual prototyping that reduces the cost and time of experimental development.

London equations in modern research

High-temperature superconductors

High-$T_c$ cuprates and iron-based superconductors are layered, anisotropic materials. The London penetration depth becomes a tensor, with different values for currents flowing parallel and perpendicular to the layers. The anisotropy ratio $\gamma = \lambda_c / \lambda_{ab}$ can exceed 100 in highly anisotropic cuprates like $\text{Bi}_2\text{Sr}_2\text{CaCu}_2\text{O}_8$. Strong thermal fluctuations and short coherence lengths in these materials push the London framework to its limits.

Unconventional superconductivity

For superconductors with non-s-wave pairing symmetry (d-wave in cuprates, p-wave candidates like $\text{Sr}_2\text{RuO}_4$), the London equations must be adapted to account for the angular dependence of the gap. Nodes in the gap function lead to power-law (rather than exponential) temperature dependence of $\lambda_L$, and exotic vortex core structures can appear.

Topological superconductors

In topological superconductors, the electromagnetic response can acquire additional terms related to Berry curvature and topological invariants. Modified London-type equations predict phenomena such as the quantized magnetoelectric effect. These materials are of intense interest for their potential to host Majorana bound states, which could serve as building blocks for topological quantum computing.