---
title: "Ginzburg–Landau Theory, Vortices, and Type-II"
module: Superconductivity
moduleNumber: 10
lessonNumber: 3
order: 1003
summary: >
  A complex order parameter and a free-energy expansion turn the superconducting
  transition into a Landau theory. Two lengths emerge — the coherence length and
  the penetration depth — whose ratio kappa sorts superconductors into type I and
  type II. Type-II materials admit flux as an Abrikosov lattice of vortices, each
  threading exactly one quantum h/2e, between a lower and an upper critical field.
topics: [Superconductivity]
draft: false
sources:
  - book: Kittel
    ref: "Ch. 10 — Superconductivity"
  - book: Ashcroft & Mermin
    ref: "Ch. 34 — Superconductivity"
  - book: Hook & Hall
    ref: "Ch. 12 — Superconductivity"
---

[London theory](/condensed-matter/superconductivity/london-theory-and-the-meissner-effect)
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,

$$
f = f_n + \alpha\,|\psi|^2 + \frac{\beta}{2}\,|\psi|^4
  + \frac{1}{2m^\ast}\left|\left(-i\hbar\nabla - q\vec A\right)\psi\right|^2
  + \frac{B^2}{2\mu_0} ,
$$

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 $T_c$,

$$
\alpha(T) = \alpha_0\,\frac{T - T_c}{T_c}, \qquad \alpha_0 > 0 ,
$$

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 $\alpha|\psi|^2 + \tfrac{1}{2}\beta|\psi|^4$ gives

$$
|\psi_0|^2 = -\frac{\alpha}{\beta} \quad (T < T_c),
\qquad
f_s - f_n = -\frac{\alpha^2}{2\beta} .
$$

The condensation-energy density equals $-\tfrac{1}{2}\mu_0 H_c^2$, which fixes the
critical field in terms of the expansion coefficients,
$\mu_0 H_c^2 = \alpha^2/\beta$.

$$
% caption: The Landau free energy as a function of the order parameter. Above T_c
% the single minimum sits at psi = 0; below T_c the curve becomes a double well
% with minima at plus and minus psi_0, the equilibrium condensate amplitude.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-3.0,0) -- (3.0,0) node[below] {order parameter};
  \draw[->, black] (0,-1.4) -- (0,2.6) node[left] {free energy};
  % above Tc: parabola upward, min at 0
  \draw[black, dashed, domain=-1.9:1.9, samples=60, variable=\x]
    plot ({\x},{0.7*\x*\x});
  \node[black, anchor=west, font=\scriptsize] at (1.2,1.7) {$T > T_c$};
  % below Tc: double well f = (x^2-1)^2 - 1 scaled
  \draw[acc, very thick, domain=-2.2:2.2, samples=100, variable=\x]
    plot ({\x},{0.55*((\x*\x/1.6 - 1)*(\x*\x/1.6 - 1)) - 0.55});
  \node[acc, anchor=west, font=\scriptsize] at (1.7,0.4) {$T < T_c$};
  % minima markers
  \fill[acc] (1.265,-0.55) circle (1.8pt);
  \fill[acc] (-1.265,-0.55) circle (1.8pt);
  \draw[black, dashed] (1.265,-0.55) -- (1.265,0);
  \node[anchor=south, font=\scriptsize] at (1.52,0.02) {min};
\end{tikzpicture}
$$

## The two length scales

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

$$
\frac{1}{2m^\ast}\left(-i\hbar\nabla - q\vec A\right)^2\psi
  + \alpha\psi + \beta|\psi|^2\psi = 0 ,
$$

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

$$
\vec J_s = \frac{q}{m^\ast}\,\mathrm{Re}\!\left[\psi^\ast\left(-i\hbar\nabla - q\vec A\right)\psi\right] .
$$

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**

$$
\xi(T) = \frac{\hbar}{\sqrt{2m^\ast|\alpha|}} ,
$$

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

$$
\lambda(T) = \sqrt{\frac{m^\ast\beta}{\mu_0 q^2|\alpha|}}
  = \sqrt{\frac{m^\ast}{\mu_0 q^2|\psi_0|^2}} ,
$$

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**

$$
\kappa = \frac{\lambda}{\xi} .
$$

> **Definition (Coherence length and penetration depth).** $\xi$ is the length
> over which the order parameter $\psi$ varies; $\lambda$ is the length over which
> a magnetic field is screened. A superconductor is characterized by the single
> dimensionless ratio $\kappa = \lambda/\xi$, which is nearly independent of
> temperature.

$$
% caption: A normal-to-superconductor interface. The order parameter rises from
% zero over the coherence length xi; the magnetic field decays into the
% superconductor over the penetration depth lambda. Their relative size sets the
% surface energy and the type.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[black, thick] (0,0) -- (0,3.0);
  \node[anchor=south east, black, font=\scriptsize] at (-0.05,2.7) {normal};
  \node[anchor=south west, acc, font=\scriptsize] at (0.15,2.7) {superconductor};
  \draw[->, black] (-2.4,0) -- (5.6,0) node[below] {$x$};
  \draw[->, black] (0,0) -- (0,3.3);
  % field decays into superconductor (x>0)
  \draw[acc, very thick, domain=0:5.2, samples=70, variable=\x]
    plot ({\x},{2.4*exp(-\x/1.1)});
  \node[acc, anchor=west, font=\scriptsize] at (1.5,1.4) {$B$};
  \draw[black, dashed] (1.1,0) -- (1.1,{2.4*exp(-1)});
  \node[anchor=north west, font=\scriptsize] at (1.1,0) {pen.};
  % order parameter rises over xi (shorter here: type II with lambda>xi)
  \draw[black, very thick, domain=0:5.2, samples=70, variable=\x]
    plot ({\x},{2.2*(1 - exp(-\x/0.45))});
  \node[black, anchor=west, font=\scriptsize] at (2.6,2.35) {order parameter};
  \draw[black, dashed] (0.45,0) -- (0.45,{2.2*(1-exp(-1))});
  \node[anchor=north east, font=\scriptsize] at (0.5,-0.02) {coh.};
\end{tikzpicture}
$$

## 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 $\sim\tfrac{1}{2}\mu_0 H_c^2\,\xi$ per unit area. Over a layer
of width $\sim\lambda$ the field penetrates, saving the field-exclusion energy
$\sim\tfrac{1}{2}\mu_0 H_c^2\,\lambda$. The surface energy per unit area is the
difference,

$$
\sigma_{ns} \sim \tfrac{1}{2}\mu_0 H_c^2\,(\xi - \lambda) .
$$

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,

$$
\hbar\nabla\theta = q\vec A ,
$$

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 $2\pi$,

$$
\oint \nabla\theta\cdot\d\vec l = 2\pi n
\quad\Longrightarrow\quad
q\oint\vec A\cdot\d\vec l = q\,\Phi = 2\pi n\hbar .
$$

The enclosed flux is therefore quantized,

$$
\Phi = n\,\frac{h}{q} = n\,\frac{h}{2e},
\qquad
\Phi_0 = \frac{h}{2e} = 2.07\times10^{-15}\ \text{Wb} .
$$

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.[^fluxq]

[^fluxq]: Flux quantum $\Phi_0 = h/2e$, NIST fundamental constants, <https://physics.nist.gov/cgi-bin/cuu/Value?flxquhz2e>. Kittel, Ch. 10, "flux quantization."

## 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.

$$
% caption: A single Abrikosov vortex. The order parameter is suppressed to zero
% in a core of radius xi; the magnetic field is peaked there and screened over
% lambda by a circulating supercurrent. The tube carries one flux quantum.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (-4.0,0) -- (4.0,0) node[below] {$r$};
  \draw[->, black] (0,0) -- (0,2.8);
  % field B(r): peaked at r=0, decays over lambda both sides
  \draw[acc, very thick, domain=-3.6:3.6, samples=100, variable=\x]
    plot ({\x},{2.2*exp(-abs(\x)/1.3)});
  \node[acc, anchor=west, font=\scriptsize] at (1.4,1.3) {$B(r)$};
  % order parameter |psi|: zero at core, recovers over xi
  \draw[black, very thick, domain=-3.6:3.6, samples=120, variable=\x]
    plot ({\x},{2.0*(1 - exp(-(\x*\x)/0.45))});
  \node[black, anchor=south, font=\scriptsize] at (-2.4,1.9) {order parameter};
  % core width xi
  \draw[black, <->] (-0.5,0.35) -- (0.5,0.35);
  \node[anchor=south, font=\scriptsize] at (0,0.4) {core};
  \draw[black, dashed] (1.3,0) -- (1.3,{2.2*exp(-1)});
  \node[anchor=north, font=\scriptsize] at (1.4,0) {penetration};
\end{tikzpicture}
$$

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

$$
H_{c2} = \frac{\Phi_0}{2\pi\mu_0\xi^2} = \sqrt{2}\,\kappa\,H_c .
$$

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

$$
H_{c1} = \frac{\Phi_0}{4\pi\mu_0\lambda^2}\,\ln\kappa \approx \frac{\ln\kappa}{\sqrt{2}\,\kappa}\,H_c .
$$

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.

$$
% caption: Phase diagram of a type-II superconductor. Below H_c1 the state is a
% flux-free Meissner phase; between H_c1 and H_c2 vortices penetrate as the mixed
% state; above H_c2 the material is normal. The dashed curve is the type-I
% thermodynamic critical field for comparison.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (6.4,0) node[below] {$T$};
  \draw[->, black] (0,0) -- (0,3.6) node[left] {$H$};
  % Hc2 (top curve)
  \draw[acc, very thick, domain=0:5.4, samples=70, variable=\x]
    plot ({\x},{3.2*(1 - (\x/5.4)*(\x/5.4))});
  \node[acc, anchor=west, font=\scriptsize] at (3.0,1.9) {$H_{c2}$};
  % Hc (dashed thermodynamic)
  \draw[black, dashed, domain=0:5.4, samples=70, variable=\x]
    plot ({\x},{1.7*(1 - (\x/5.4)*(\x/5.4))});
  \node[black, anchor=west, font=\scriptsize] at (3.4,0.95) {$H_c$};
  % Hc1 (bottom curve)
  \draw[black, very thick, domain=0:5.4, samples=70, variable=\x]
    plot ({\x},{0.75*(1 - (\x/5.4)*(\x/5.4))});
  \node[black, anchor=west, font=\scriptsize] at (0.25,0.28) {$H_{c1}$};
  \node[acc, anchor=west, font=\scriptsize] at (0.9,3.35) {normal};
  \node[black, anchor=west, font=\scriptsize] at (2.0,1.95) {mixed};
  \node[black, anchor=west, font=\scriptsize] at (1.6,0.5) {Meissner};
  \node[anchor=north, font=\scriptsize] at (5.4,0) {$T_c$};
\end{tikzpicture}
$$

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

$$
% caption: The Abrikosov vortex lattice viewed end-on: vortices on a triangular
% lattice, each a normal core with a circulating supercurrent carrying one flux
% quantum. Denser packing corresponds to a higher field.
\begin{tikzpicture}[scale=1.0, >=stealth, font=\footnotesize]
  \definecolor{acc}{HTML}{4A6FA5}
  \foreach \row in {0,1,2}{
    \pgfmathsetmacro{\off}{mod(\row,2)==0 ? 0 : 0.65}
    \foreach \col in {0,1,2,3}{
      \pgfmathsetmacro{\cx}{\col*1.3 + \off}
      \pgfmathsetmacro{\cy}{\row*1.12}
      \draw[acc, thick] (\cx,\cy) circle (0.22);
      \fill[acc!18] (\cx,\cy) circle (0.22);
      \fill[acc] (\cx,\cy) circle (2pt);
    }
  }
  % circulation on one vortex
  \draw[acc, ->, thick] (0.55,2.24) arc (0:300:0.33);
  \node[acc, anchor=west, font=\scriptsize] at (4.3,2.24) {one quantum};
  \draw[acc, ->] (4.25,2.24) -- (3.6,2.24);
  \node[black, anchor=north, font=\scriptsize] at (2.4,-0.35) {triangular lattice};
\end{tikzpicture}
$$

| Property | Type I | Type II |
| --- | --- | --- |
| GL parameter | $\kappa < 1/\sqrt{2}$ | $\kappa > 1/\sqrt{2}$ |
| Surface energy $\sigma_{ns}$ | positive | negative |
| Flux entry | none until $H_c$, then normal | vortices between $H_{c1}$ and $H_{c2}$ |
| Critical fields | single $H_c$ | $H_{c1} < H_c < H_{c2}$ |
| Typical materials | pure metals (Pb, Sn, Al) | alloys, Nb$_3$Sn, cuprates |

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](/condensed-matter/superconductivity/bcs-theory).