Ginzburg-Landau Theory

Ginzburg-Landau Free Energy Functional

First GL Equation (Order Parameter Equation)

This is a nonlinear Schrodinger equation for the order parameter. The nonlinear term $\beta|\psi|^2\psi$ arises from the self-interaction of the condensate and prevents the amplitude from diverging.

Second GL Equation (Supercurrent Equation)

This is the gauge-invariant supercurrent density. Writing $\psi = |\psi|e^{i\theta}$, the supercurrent takes the familiar London form $\mathbf{J}_s = \frac{e^*|\psi|^2}{m^*}(\hbar\nabla\theta - e^*\mathbf{A})$, showing that the supercurrent is driven by gradients of the phase.

GL Coherence Length

The coherence length $\xi$ characterizes the distance over which the order parameter can vary. It diverges as $\xi \propto (T_c - T)^{-1/2}$ approaching the transition. Near a surface or defect, $\psi$ recovers to its bulk value over a distance $\sim \xi$.

London Penetration Depth

The penetration depth $\lambda$ characterizes the distance over which an external magnetic field is screened inside the superconductor. Like $\xi$, it diverges as $(T_c - T)^{-1/2}$ near the transition.

Ginzburg-Landau Parameter

Crucially, $\kappa$ is temperature-independent within GL theory, since both$\lambda$ and $\xi$ diverge with the same exponent. The value of $\kappa$ determines the magnetic behavior of the superconductor.

Flux Quantization

The flux quantum $\Phi_0 = h/(2e)$ reflects the Cooper-pair charge $e^* = 2e$. Its experimental observation by Deaver & Fairbank and by Doll & Nabauer in 1961 provided direct evidence for pairing in superconductors before the GL theory was microscopically justified.

Derivation of the First GL Equation

Starting from the GL free energy functional, consider a variation $\psi^* \to \psi^* + \delta\psi^*$. The kinetic energy term is:

Under the variation $\delta\psi^*$, the first-order change in $F_{\text{kin}}$ is:

Integrating by parts (discarding the surface term for a bulk superconductor):

Combining with the potential terms $\delta(\alpha|\psi|^2 + \frac{\beta}{2}|\psi|^4) = (\alpha\psi + \beta|\psi|^2\psi)\delta\psi^*$, setting $\delta F / \delta\psi^* = 0$ gives:

Derivation of the Second GL Equation

Varying with respect to $\mathbf{A}$: from the kinetic term, we use$(-i\hbar\nabla - e^*\mathbf{A})\psi = -i\hbar\nabla\psi - e^*\mathbf{A}\psi$. The variation $\mathbf{A} \to \mathbf{A} + \delta\mathbf{A}$ gives:

This simplifies to $\delta F_{\text{kin}} = -\int d^3r\, \mathbf{J}_s \cdot \delta\mathbf{A}$, where the supercurrent density is:

From the field energy term $B^2/(2\mu_0) = |\nabla\times\mathbf{A}|^2/(2\mu_0)$, the variation gives (after integration by parts) $(\nabla\times\mathbf{B})/\mu_0 \cdot \delta\mathbf{A}$. Setting the total variation to zero yields Ampere's law:

Writing $\psi = |\psi|e^{i\theta}$, the supercurrent becomes$\mathbf{J}_s = (e^*|\psi|^2/m^*)(\hbar\nabla\theta - e^*\mathbf{A})$, the gauge-invariant form.

Derivation: London Equation from GL Theory

In the London limit, $|\psi|$ is uniform and equal to its equilibrium value$\psi_0 = \sqrt{|\alpha|/\beta}$. The supercurrent simplifies to:

(in the London gauge $\nabla\cdot\mathbf{A} = 0$ and $\mathbf{A}\cdot\hat{n} = 0$at the surface). Taking the curl of both sides and using $\nabla\times\mathbf{J}_s = (\nabla\times\nabla\times\mathbf{B})/\mu_0 = -\nabla^2\mathbf{B}/\mu_0$:

For a semi-infinite superconductor with $B(0) = B_0$, the solution is $B(x) = B_0 e^{-x/\lambda}$ — exponential screening of the applied field. The associated supercurrent is $J_s(x) = -(B_0/\mu_0\lambda)e^{-x/\lambda}$, flowing parallel to the surface.

Derivation: Upper Critical Field $H_{c2}$

At $H_{c2}$, the order parameter is infinitesimally small. Linearizing the first GL equation ($\beta|\psi|^2 \to 0$):

This is the Schrodinger equation for a charged particle in a uniform magnetic field — the Landau level problem. With $\mathbf{A} = (-By, 0, 0)$ (Landau gauge), the eigenvalues are:

The onset of superconductivity occurs when the lowest Landau level ($n = 0$) equals $|\alpha|$:

Solving for $B_{c2}$:

using $\Phi_0 = h/e^* = 2\pi\hbar/e^*$. This is a remarkable result: the upper critical field is determined entirely by the coherence length, which in turn sets the minimum size of a vortex core.

Step 1: The Free Energy Functional

We begin with the full GL free energy functional:

We seek the equilibrium configuration by requiring that $F$ be stationary under independent variations of $\psi^*$ and $\mathbf{A}$.

Step 2: First GL Equation from $\delta F/\delta\psi^* = 0$

Let $\psi^* \to \psi^* + \delta\psi^*$ with $\psi$ held fixed. The variation of the potential terms is straightforward:

For the kinetic term, define the covariant derivative $\mathbf{D} = -i\hbar\nabla - e^*\mathbf{A}$and its conjugate $\mathbf{D}^{\dagger} = i\hbar\nabla - e^*\mathbf{A}$. Then:

The variation with respect to $\psi^*$ gives:

We integrate by parts. Writing out the components and using the identity$(i\hbar\nabla_j - e^*A_j)\delta\psi^* \cdot (-i\hbar\nabla_j - e^*A_j)\psi$, we shift the derivatives from $\delta\psi^*$ to $\psi$:

Setting $\delta F = 0$ for arbitrary $\delta\psi^*$ in the bulk yields the first Ginzburg-Landau equation:

Step 3: Supercurrent Equation from $\delta F/\delta\mathbf{A} = 0$

Now vary $\mathbf{A} \to \mathbf{A} + \delta\mathbf{A}$. The kinetic energy term$|\mathbf{D}\psi|^2 = |(-i\hbar\nabla - e^*\mathbf{A})\psi|^2$ yields:

This can be written as $\delta F_{\text{kin}} = -\int d^3r\, \mathbf{J}_s\cdot\delta\mathbf{A}$, where:

The field energy term $|\nabla\times\mathbf{A}|^2/(2\mu_0)$ varies as:

(after integration by parts). Setting the total variation to zero gives $\nabla\times\mathbf{B} = \mu_0\mathbf{J}_s$, the second GL equation (Ampere's law with the supercurrent as source).

Step 4: Natural Boundary Condition

The surface term from the integration by parts in Step 2 must vanish independently:

This is the natural boundary condition of GL theory. It states that the normal component of the gauge-invariant momentum of the superconducting condensate vanishes at the surface — there is no supercurrent flowing through the boundary. For a superconductor-vacuum interface, this ensures that the order parameter smoothly adjusts at the surface without any flux of Cooper pairs leaving the material.

Step 1: Phase Representation of the Order Parameter

Write the order parameter in amplitude-phase form:

Substituting into the gauge-invariant supercurrent expression:

This shows that the supercurrent is driven by the gauge-invariant combination of the phase gradient and the vector potential.

Step 2: Contour Deep Inside the Superconductor

Consider a closed contour $\mathcal{C}$ lying deep inside the bulk of a multiply-connected superconductor (e.g., a ring or cylinder), far from any surface. Deep in the interior, the Meissner effect ensures $\mathbf{B} = 0$ and therefore$\mathbf{J}_s = 0$. From the supercurrent expression:

Step 3: Integration Around the Loop

Integrate both sides around the closed contour $\mathcal{C}$:

The right side is the magnetic flux through the loop by Stokes' theorem:$\oint \mathbf{A}\cdot d\mathbf{l} = \int_S \mathbf{B}\cdot d\mathbf{S} = \Phi$. For the left side, single-valuedness of $\psi = |\psi|e^{i\phi}$ requires that$\phi$ change by an integer multiple of $2\pi$ around any closed loop:

Step 4: The Flux Quantum

Combining these results:

With the microscopic identification $e^* = 2e$ (the charge of a Cooper pair), the fundamental flux quantum is:

This was experimentally confirmed by Deaver and Fairbank, and independently by Doll and Nabauer, in 1961. The observation of $h/(2e)$ rather than $h/e$ provided direct evidence for Cooper pairing before the BCS theory had been fully accepted.

The Order Parameter Identification

Gor'kov started from the BCS anomalous Green's function (the Gor'kov function) $\mathcal{F}(\mathbf{r}, \mathbf{r'})$, which describes the pairing correlations. Near $T_c$, the self-consistency equation for the gap function$\Delta(\mathbf{r})$ can be expanded in powers of $\Delta$ and its gradients. The GL order parameter is directly proportional to the gap:

The proportionality constant is fixed by convention. The key insight is that$|\psi|^2$ gives the superfluid density, consistent with the GL interpretation.

Microscopic Expressions for GL Coefficients

By expanding the BCS free energy near $T_c$ in powers of $\Delta$, Gor'kov obtained the following identifications for the GL coefficients in terms of microscopic quantities:

Coherence Length from BCS

The GL coherence length $\xi(T)$, which sets the scale of spatial variations of the order parameter, is related to the BCS coherence length $\xi_0$ by:

Here $v_F$ is the Fermi velocity and $\Delta_0$ is the BCS gap at$T = 0$. The divergence $\xi \to \infty$ as $T \to T_c$reflects the critical softening of the order parameter — spatial variations cost progressively less energy as the transition is approached. This is a universal feature of continuous phase transitions, connecting GL theory to the broader framework of critical phenomena.

Domain Wall Setup

Consider a planar interface (domain wall) between a normal region ($x \to -\infty$, where $\psi = 0$ and $B = B_c$) and a superconducting region ($x \to +\infty$, where $|\psi| = \psi_0$ and $B = 0$). Working in dimensionless units (lengths in $\lambda$, fields in $B_c = \mu_0 H_c$, order parameter in $\psi_0 = \sqrt{|\alpha|/\beta}$), the GL free energy density relative to the normal state in a field $H_c$ is:

where $f = |\psi|/\psi_0$, $a = A/(\lambda B_c)$, and $b = B/B_c$.

Competing Length Scales

The surface energy arises from the competition between two effects at the domain wall:

The net surface energy is approximately:

The Critical Value $\kappa = 1/\sqrt{2}$

An exact analysis of the GL equations for the domain wall profile (first performed by Abrikosov) reveals that the precise boundary between positive and negative surface energy occurs at $\kappa = 1/\sqrt{2}$:

Type-I ($\kappa < 1/\sqrt{2}$): Positive surface energy means that creating N-S interfaces costs energy. The superconductor avoids domain formation and exhibits a first-order transition with the intermediate state (macroscopic normal and superconducting laminae). Examples: Pb ($\kappa \approx 0.48$), Sn ($\kappa \approx 0.15$), Al ($\kappa \approx 0.01$).

Type-II ($\kappa > 1/\sqrt{2}$): Negative surface energy means that interfaces are energetically favorable. The superconductor spontaneously subdivides into as many domains as possible, leading to the Abrikosov vortex lattice in the mixed state between$H_{c1}$ and $H_{c2}$. Examples: Nb ($\kappa \approx 1.05$), NbTi ($\kappa \approx 70$), YBCO ($\kappa \approx 95$).

Domain Wall Profile from GL Equations

In dimensionless variables, the coupled GL equations for the domain wall are:

At the special value $\kappa = 1/\sqrt{2}$, these equations admit the first integral:

which together with $b = 1 - f^2$ (the Bogomolny equation) can be solved exactly, confirming $\sigma_{ns} = 0$ at the critical $\kappa$. This self-dual point marks the exact boundary between type-I and type-II behavior.