Ginzburg–Landau Theory, Vortices, and Type-II

London theory treats the superconducting electron density $n_s$ as a fixed number and cannot describe a boundary, a partially suppressed condensate, or the entry of flux. Ginzburg and Landau (1950) promoted $n_s$ to a spatially varying complex field and wrote the free energy as an expansion in it. The theory predates the microscopic mechanism yet predicts the two characteristic lengths, the existence of two classes of superconductor, the vortex lattice, and the quantization of flux — all before BCS.

The order parameter and the free-energy expansion

Introduce a complex order parameter $\psi (\vec{r})$ whose squared magnitude is the local density of superconducting carriers, $|\psi {|}^2=n_s$. It is zero in the normal phase and nonzero below $T_c$. Near the transition $\psi$ is small, and by Landau's hypothesis the free-energy density is analytic in $\psi$ and, by gauge invariance, depends only on $|\psi {|}^2$ and on gradients. Keeping terms through $|\psi {|}^4$ and one gradient term,

with pairing charge $q=-2e$ and mass $m^{\ast }=2m$. Stability requires $\beta >0$. The coefficient $\alpha$ carries the temperature dependence and changes sign at $T_c$,

so $\alpha >0$ above $T_c$ (the minimum sits at $\psi =0$, the normal state) and $\alpha <0$ below (a nonzero $\psi$ is favored).

For a uniform sample in zero field the gradient and field terms vanish, and minimizing $\alpha |\psi {|}^2+\frac{1}{2}\beta |\psi {|}^4$ gives

The condensation-energy density equals $-\frac{1}{2}{\mu }_0{H}_c^2$, which fixes the critical field in terms of the expansion coefficients, ${\mu }_0{H}_c^2={\alpha }^2/\beta$.

The two length scales

Varying the free energy with respect to ${\psi }^{\ast }$ gives the first Ginzburg–Landau equation,

and varying with respect to $\vec{A}$ gives the second, a current expression identical in form to the London supercurrent,

Two lengths fall out. In zero field, linearizing the first equation about ${\psi }_0$ by writing $\psi ={\psi }_0+\delta \psi$ shows a perturbation relaxes back over the coherence length

the distance over which $\psi$ heals from any disturbance, such as a surface or a normal region. The field term supplies the penetration depth

the same screening length as London's, now with the equilibrium condensate density $|{\psi }_0{|}^2$. Both diverge as $|\alpha {|}^{-1/2}\propto (T_c-T{)}^{-1/2}$ near $T_c$, so their ratio is temperature-independent. That ratio is the Ginzburg–Landau parameter

Surface energy and the two types

Consider the boundary between a normal and a superconducting region in a field at $H_c$. Over a layer of width $\sim \xi$ the order parameter is suppressed, costing condensation energy $\sim \frac{1}{2}{\mu }_0{H}_c^2\xi$ per unit area. Over a layer of width $\sim \lambda$ the field penetrates, saving the field-exclusion energy $\sim \frac{1}{2}{\mu }_0{H}_c^2\lambda$. The surface energy per unit area is the difference,

The sign flips at $\xi =\lambda$, and the exact calculation places the boundary at $\kappa =1/\sqrt{2}$:

• Type I ($\kappa <1/\sqrt{2}$): the surface energy is positive. The system minimizes interface area, expels flux completely, and reverts to normal in one step at $H_c$.
• Type II ($\kappa >1/\sqrt{2}$): the surface energy is negative. The system gains energy by making interfaces, so flux enters as finely divided tubes as soon as it is energetically allowed.

Most pure elemental metals are type I; alloys and compounds, with their shorter mean free path and hence shorter $\xi$, are type II. All useful high-field magnet materials are type II.

Flux quantization

Take a superconducting ring, or a path deep inside a type-II material where the supercurrent has died away, ${\vec{J}}_s=0$. The second GL equation with ${\vec{J}}_s=0$ forces the canonical momentum to vanish,

where $\theta$ is the phase of $\psi =|\psi |e^{i\theta }$. Integrate around a closed loop enclosing the hole. Single-valuedness of $\psi$ requires the phase to change by an integer multiple of $2\pi$,

The enclosed flux is therefore quantized,

The measured quantum is $h/2e$, not $h/e$: the carriers have charge $2e$. Flux quantization is thus a macroscopic manifestation of electron pairing, and its value is a fundamental constant.¹

The mixed state and the vortex lattice

In a type-II material at fields above a lower critical field $H_{c1}$, it becomes favorable for flux to enter, but it can only do so in units of ${\Phi }_0$. Each quantum of flux forms an Abrikosov vortex: a normal core of radius $\sim \xi$ where $\psi \to 0$, surrounded by a circulating supercurrent that screens the field over $\sim \lambda$. The field is peaked at the core and decays outward.

Abrikosov showed in 1957 that the vortices repel and settle into a triangular lattice, the arrangement of lowest energy at a given flux density. Raising the field packs the vortices closer, since the areal density of vortices is $B/{\Phi }_0$, until at the upper critical field the cores overlap and superconductivity is destroyed. Setting the intervortex spacing to $\xi$ gives

The lower critical field, at which the first vortex enters, is set by the energy of a single vortex line,

For $\kappa \gg 1$, $H_{c2}=\sqrt{2}\kappa H_c$ can exceed $H_c$ by two orders of magnitude, reaching tens of tesla in Nb${}_3$Sn or the cuprates. This is why type-II superconductors, not type-I, build high-field magnets.

The vortices themselves form a lattice visible in decoration and imaging experiments, each site carrying one flux quantum.

Ginzburg–Landau theory delivers the coherence length, the penetration depth, their ratio, the vortex state, and flux quantization from a single free-energy expansion, with $\alpha$ and $\beta$ left as phenomenological constants. Deriving those constants — and the pairing that gives $q=2e$ — is the task of the microscopic theory.