9.2 London equations

London equations for superconductors

The London equations are a phenomenological model that describes how supercurrents respond to electromagnetic fields inside a superconductor. They capture the two defining macroscopic properties of the superconducting state: perfect conductivity (zero resistance) and perfect diamagnetism (complete magnetic field expulsion). The equations grow out of the two-fluid model, which treats the electron system as a mixture of normal electrons and a superfluid condensate with macroscopic quantum coherence.

First London equation: perfect conductivity

The first London equation relates the time derivative of the supercurrent density $\vec{J}_s$ to the electric field $\vec{E}$ inside the superconductor:

$\frac{\partial \vec{J}_s}{\partial t} = \frac{n_s e^2}{m_e} \vec{E}$

Here $n_s$ is the superconducting electron density, $e$ is the electron charge, and $m_e$ is the electron mass.

The physical content is straightforward: an electric field accelerates superconducting electrons without any scattering or dissipation. In a normal metal, collisions balance the acceleration and you get Ohm's law. Here there's no friction term, so the current just keeps growing as long as $\vec{E} \neq 0$.

That means in the steady state, $\vec{E}$ must be zero inside the superconductor. If it weren't, the current would increase without bound. Zero electric field with a finite, persistent current is exactly what "perfect conductivity" means.

Second London equation: perfect diamagnetism

The second London equation connects the curl of the supercurrent density to the magnetic field:

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

This equation is the one that goes beyond perfect conductivity. A merely perfect conductor would freeze in whatever magnetic flux it had when resistance vanished. The second London equation is stronger: it demands that $\vec{B}$ be actively expelled from the bulk, regardless of the field history. That active expulsion is the Meissner effect.

Combined with Ampère's law ($\nabla \times \vec{B} = \mu_0 \vec{J}_s$ in the static case), this equation leads directly to the exponential decay of $\vec{B}$ inside the superconductor and defines the London penetration depth.

Relation between current density and vector potential

The two London equations can be combined into a single compact form using the magnetic vector potential $\vec{A}$ (with $\vec{B} = \nabla \times \vec{A}$):

$\vec{J}_s = -\frac{n_s e^2}{m_e} \vec{A}$

This holds in the London gauge (Coulomb gauge, $\nabla \cdot \vec{A} = 0$, with appropriate boundary conditions). It's a convenient starting point for calculations because it directly gives you the current distribution once you know $\vec{A}$, and it makes the connection to the quantum-mechanical phase of the condensate wavefunction more transparent.

Consequences of the London equations

• Zero electrical resistance and persistent currents in the superconducting state.
• The Meissner effect: complete expulsion of magnetic flux from the bulk.
• An exponential decay of $\vec{B}$ and $\vec{J}_s$ from the surface, characterized by the London penetration depth $\lambda_L$.
• A framework for understanding the distinct magnetic behaviors of type I and type II superconductors.
• Quantitative predictions (penetration depth, flux expulsion) that have been confirmed experimentally through susceptibility measurements, muon spin rotation, and other techniques.

Meissner effect in superconductors

The Meissner effect is the complete expulsion of magnetic flux from the interior of a superconductor when it's cooled below its critical temperature $T_c$. This happens even if the field was already present before the transition. That "even if" is the key distinction from a hypothetical perfect conductor, which would merely trap existing flux rather than expel it.

Expulsion of magnetic fields

When a material enters the superconducting state in a weak applied field, screening supercurrents spontaneously appear at the surface. These currents generate a magnetization that exactly cancels the applied field in the bulk. The field doesn't just fail to penetrate; it's actively pushed out.

This expulsion is a thermodynamic equilibrium property. The superconducting state with $\vec{B} = 0$ in the interior has lower free energy than a state with trapped flux. That thermodynamic character is what separates the Meissner effect from simple perfect conductivity.

Critical magnetic field strength

The Meissner effect holds only below a critical field $H_c$. Above $H_c$, the energy cost of expelling the field exceeds the condensation energy gained by being superconducting, and the material reverts to the normal state.

The critical field decreases with temperature following the approximate relation:

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

At $T = 0$, the critical field is at its maximum $H_c(0)$. At $T = T_c$, it drops to zero.

Type I vs. type II superconductors

Superconductors split into two categories based on how they handle magnetic fields:

• Type I (e.g., Pb, Hg, Al): These show a complete Meissner effect up to a single critical field $H_c$, then abruptly switch to the normal state. The transition is first-order.
• Type II (e.g., Nb-Ti, YBCO, $\text{MgB}_2$): These have two critical fields. Below the lower critical field $H_{c1}$, the Meissner effect is complete. Between $H_{c1}$ and the upper critical field $H_{c2}$, the material enters a mixed state (also called the vortex state), where magnetic flux penetrates in the form of quantized flux tubes called vortices, each carrying one flux quantum $\Phi_0 = h/2e$. Above $H_{c2}$, superconductivity is destroyed.

Type II materials are far more useful technologically because $H_{c2}$ can be very large (tens of tesla), allowing them to carry high currents in strong magnetic fields.

Penetration depth in superconductors

The London penetration depth $\lambda_L$ is the characteristic length scale over which an applied magnetic field decays exponentially from the surface into the bulk of a superconductor. It arises directly from the second London equation combined with Maxwell's equations.

Definition of London penetration depth

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

where $m_e$ is the electron mass, $\mu_0$ is the vacuum permeability, $n_s$ is the superconducting electron density, and $e$ is the electron charge.

Inside a superconductor with a flat surface, the magnetic field profile looks like:

$B(x) = B_0 \, e^{-x/\lambda_L}$

where $x$ is the depth from the surface. So $\lambda_L$ tells you how far the field "leaks" in. Typical values range from about 20 nm (for clean elemental superconductors like Al) to several hundred nm (for high-$T_c$ cuprates and dirty alloys).

Temperature dependence of penetration depth

The penetration depth grows with temperature and diverges as $T \to T_c$:

$\lambda_L(T) = \frac{\lambda_L(0)}{\left[1 - \left(\frac{T}{T_c}\right)^4\right]^{1/2}}$

The divergence makes physical sense: as temperature rises toward $T_c$, thermal energy breaks Cooper pairs, $n_s$ drops, and the remaining superconducting electrons become less effective at screening the field. Right at $T_c$, $n_s \to 0$ and the field penetrates completely.

Relation to superconducting electron density

Since $\lambda_L \propto 1/\sqrt{n_s}$, measuring the penetration depth gives you a direct window into the superconducting electron density. A shorter $\lambda_L$ means more superconducting electrons and stronger screening. Changes in $n_s$ caused by temperature, impurities, or applied fields all show up as changes in $\lambda_L$, making penetration depth measurements one of the most informative experimental probes of the superconducting state.

Coherence length in superconductors

The coherence length $\xi$ is the second fundamental length scale in superconductivity. While $\lambda_L$ describes how fields decay at the surface, $\xi$ describes the spatial extent of the Cooper pair wavefunction and the minimum distance over which the superconducting order parameter can change appreciably.

Definition of coherence length

The BCS coherence length (also called the Pippard coherence length) is:

$\xi_0 = \frac{\hbar v_F}{\pi \Delta}$

where $v_F$ is the Fermi velocity and $\Delta$ is the superconducting energy gap at $T = 0$.

You can think of $\xi_0$ as the "size" of a Cooper pair. For conventional superconductors like Al, $\xi_0$ can be on the order of a micrometer. For high-$T_c$ cuprates, it's only a few nanometers because the gap is much larger relative to the Fermi energy.

Relation to superconducting energy gap

$\xi_0$ is inversely proportional to $\Delta$: a larger gap means a more tightly bound pair and a shorter coherence length. As temperature increases toward $T_c$, the gap shrinks and $\xi$ grows (in the Ginzburg-Landau framework, $\xi(T)$ actually diverges at $T_c$). The coherence length therefore encodes information about the strength of the pairing interaction.

Comparison with London penetration depth

The ratio of these two length scales defines the Ginzburg-Landau parameter:

$\kappa = \frac{\lambda_L}{\xi}$

This single dimensionless number determines the type of superconductor:

Most elemental superconductors are type I ($\kappa$ small), while alloys, compounds, and all high-$T_c$ materials are type II ($\kappa$ large).

Limitations of London theory

The London equations are remarkably successful for a phenomenological model, but they rest on simplifying assumptions that break down in several important situations.

Inability to explain microscopic origin

The London theory takes superconductivity as a given and describes its electromagnetic consequences. It says nothing about why electrons pair up, what mediates the attractive interaction, or what determines $T_c$. For those questions you need the microscopic BCS theory (1957), which explains Cooper pairing via phonon-mediated electron-electron attraction and predicts the energy gap, the isotope effect, and the coherence length from first principles.

Assumption of local electrodynamics

The London equations assume the supercurrent at a point depends only on the fields at that same point. This local approximation works well when $\xi \ll \lambda_L$ (the type II limit), because the order parameter doesn't vary much over the scale of field penetration.

When $\xi \gtrsim \lambda_L$ (as in many clean type I superconductors), the response becomes nonlocal: the current at a point depends on the field averaged over a region of size $\sim \xi$. Pippard's nonlocal generalization of the London equation handles this regime and gives a corrected, effective penetration depth that's larger than the London value.

Failure at high frequencies and short wavelengths

The London theory assumes a quasistatic response. At frequencies approaching the gap frequency $\omega \sim 2\Delta/\hbar$ (typically in the microwave to far-infrared range), the photon energy becomes large enough to break Cooper pairs, and the simple London picture no longer applies. The Mattis-Bardeen theory and the more general Eliashberg formalism are needed to describe the electrodynamic response in this regime, accounting for the frequency-dependent complex conductivity of the superconductor.

Experimental verification of London equations

The predictions of the London theory have been confirmed by a wide range of experiments, establishing the equations as a reliable phenomenological foundation.

Magnetic flux expulsion experiments

Direct measurements of the magnetic susceptibility of superconductors show $\chi = -1$ (perfect diamagnetism) below $T_c$ and below $H_c$, exactly as the London theory predicts. Magnetization curves for type I superconductors show a sharp transition at $H_c$, while type II materials show the expected onset of flux penetration at $H_{c1}$ and the gradual increase of internal flux through the mixed state up to $H_{c2}$.

Measurement of penetration depth

Several experimental techniques probe $\lambda_L$ directly:

• Muon spin rotation ($\mu$SR): Implanted muons precess in the local magnetic field, mapping out the field profile inside the superconductor.
• Microwave cavity perturbation: Changes in the resonant frequency and quality factor of a cavity containing a superconducting sample yield the surface impedance, from which $\lambda_L$ is extracted.
• Low-energy neutron scattering: Sensitive to the field distribution in the vortex lattice of type II superconductors.

These measurements confirm the predicted magnitude of $\lambda_L$, its $1/\sqrt{n_s}$ dependence, and its divergence as $T \to T_c$.

Observation of the Meissner effect

The most vivid demonstration is magnetic levitation: a superconductor cooled below $T_c$ repels a permanent magnet strongly enough to float it in midair. This works because the expelled flux creates a repulsive force. The stability and height of levitation depend on the material's $\lambda_L$, $H_c$ (or $H_{c1}$ and $H_{c2}$), and geometry. Beyond demonstrations, the Meissner effect is the operating principle behind superconducting magnetic shielding and is central to the design of superconducting magnets used in MRI machines and particle accelerators.