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Ginzburg-Landau theory of superconductivity organizes a continuous superconducting transition through a slowly varying complex order parameter. In the simplest isotropic single-component SI model,Here $\beta >0$, $m_{\ast }>0$, $\alpha$ changes sign at $T_c$, and $q=-2e$ fixes a charge/phase convention. The phenomenological normalization of $\psi$ determines the coefficient values; it should not be confused without qualification with a literal density of independently identifiable pairs.

Microscopic derivation of Ginzburg-Landau theory expands the BCS response near $T_c$ in small gap amplitude and slow spatial gradients. It identifies the complex order parameter with a scaled pair amplitude, supplies the coefficients from the electronic microscopic parameters, and explains the pair charge in the gauge coupling. The agreement is a controlled near-transition limit, not a statement that the truncated local functional remains exact at all temperatures. Lev Gor'kov established this connection explicitly in 1959.^[ref]

Below the transition, minimization gives $|{\psi }_0{|}^2=-\alpha /\beta$ and condensation-energy density ${\alpha }^2/(2\beta )$. A variation with respect to ${\psi }^{\ast }$ givesVariation with respect to $\mathbf{A}$ gives the supercurrentcoupled to the magnetostatic Ampère law. For nearly uniform amplitude this reduces to London screening and the Meissner effect. The theory supplies two different length scales:The amplitude can vary over the superconducting coherence length, whereas the magnetic field is screened over the London penetration depth.

The surface-energy distinction at $\kappa =1/\sqrt{2}$ explains why type-I and type-II behavior are possible within the same framework. In the type-II regime, quantized winding produces an Abrikosov vortex with a finite core and circulating current. Periodic solutions yield an Abrikosov vortex lattice. Linearization about the normal state gives $B_{c2}={\Phi }_0/(2\pi {\xi }^2)$, while core and field energetics determine the lower entry field. These predictions connect a spatial field theory to measurable vortex patterns and critical fields, beyond a spatially uniform BCS gap equation. Thin-film critical-field observations helped motivate the large-$\kappa$ regime; magnetic-decoration experiments subsequently made vortex positions visible.^[ref]

These phenomena also enable applications. Type-II conductors support high-field magnets, but flux pinning is needed to limit vortex-motion losses. Phase coherence and fluxoid quantization permit persistent-current loops and interference devices. A weak link supplies Josephson coupling energy, allowing SQUID magnetometry and nonlinear quantum circuits; a junction's detailed current–phase law requires a weak-link model, rather than following automatically from the spatially uniform bulk functional. Microstructure can raise a useful critical current without greatly changing the equilibrium upper field, illustrating the difference between thermodynamic stability and transport performance.

The simple functional has identifiable limits. Far below $T_c$, the small-gap expansion is uncontrolled; rapidly varying fields can require nonlocal microscopic electrodynamics. Very near a transition in a fluctuating low-dimensional or strongly anisotropic material, mean-field minimization can miss important thermal or quantum fluctuations. Unconventional or multiband pairing can require several order-parameter components and additional gradient couplings. Strong fields can introduce spin-polarization pair breaking omitted by the orbital model. Finally, the static functional predicts equilibrium extrema, not quasiparticle spectra, dissipation, vortex velocities or a unique amplitude-mode frequency without further dynamical input. These are reasons to choose an appropriate extension or a microscopic calculation, not reasons to discard the successful near-transition predictions.

Take a homogeneous material exhibiting superconductivity in the half-space $z\ge 0$, bounded by vacuum at $z=0$. Fix the temperature below the superconducting transition temperature, take a constant positive London penetration depth ${\lambda }_L$, and assume a local, linear London equation description in a Meissner state. The applied magnetic field is weak enough that it neither nucleates a quantum vortex nor appreciably suppresses the material's superconductivity. Assume translation invariance in $x,y$, no imposed transport electric current, and a tangential boundary magnetic field $\mathbf{B}(0)=B_0\hat{\mathbf{x}}$. Here $B_0$ is the local surface magnetic flux density, equal to the imposed vacuum magnetic field in this infinite planar geometry; it is not automatically the far-away applied value for a finite object.

For a bounded, decaying interior solution, the London equations, magnetostatic Ampère law and Gauss's law for magnetism giveThe solution isAt one London penetration depth the magnetic field is $e^{-1}\simeq 0.368$ of its surface value; at five it is $e^{-5}\simeq 0.00674$. There is no finite depth where this exponential is exactly zero. The differential equation excludes the growing exponential through the deep-interior boundary condition. A constant normal component cannot decay in this geometry: Gauss's law for magnetism would require ${\partial }_z{B}_z=0$. A perpendicular applied magnetic field on a finite plate therefore requires a different spatial problem.^[ref]

In a local London equation description, the measured London penetration depth satisfies ${\lambda }_L^2=m/({\mu }_0{n}_s{e}^2)$ when $n_s$ counts superconducting electrons and $m$ is their effective mass. Thus screening measures $n_s/m$. If instead one counts Cooper pairs, use pair mass $2m$ and pair electric charge $-2e$, giving pair density $n_s/2$ for the same screening length. Identifying this superconducting carrier density with the total conduction-electron density requires the additional assumption that the relevant carriers all participate in screening at the measurement temperature.

The gradient and quadratic terms of the stated model balance at the amplitude-variation scale $\xi$. It is a superconducting length scale, distinct from optical coherence length. The weak-field London penetration depth in the same normalization obeysA core suppresses the pairing amplitude over a scale set by $\xi$; screening electric currents and the magnetic field involve $\lambda$. Neither scale is a universal particle size.

Superconductivity is a phase of matter that supports steady electric current with zero bulk electrical resistivity and exhibits the Meissner effect: equilibrium screening of a sufficiently weak, static magnetic field from the interior. It occurs below a material-dependent superconducting transition temperature, provided the electric current and applied magnetic field are also sufficiently small. ^[ref] Bulk expulsion means that a sample much thicker than its London penetration depth has an almost field-free interior; the magnetic field need not vanish at its surface. Conventional BCS theory explains superconductivity through correlated electrons forming Cooper pairs. ^[ref]

The two observations answer different questions. A resistance-versus-temperature experiment for superconductivity asks whether a steady electric current requires a measurable electric potential difference. A Field-cooled Meissner experiment asks whether a magnetic field already present in the material is expelled when it enters superconductivity. An ideal perfect conductor with zero electrical resistivity can preserve that pre-existing magnetic field; zero electrical resistance alone therefore does not establish the Meissner effect.

The usefulness follows directly from Joule heating, $P=I^{2}R$: maintaining a steady electric current through a segment with zero electrical resistance produces no resistive heating in that segment. This makes persistent electric currents attractive for producing steady magnetic fields. The beauty is that one phase has both a transport property and an equilibrium magnetic response, and the second cannot be inferred just by making an ordinary electrical resistance very small. The London equations encode this extra magnetic response. ^[ref]