Advanced Topics in Superconductivity

The Ginzburg—Landau (GL) theory provides a phenomenological description of superconductivity near $T_c$ using a complex order parameter $\psi(\mathbf{r})$ where $|\psi|^2 = n_s$ is the superfluid density.

The GL free energy functional:

$\mathcal{F} = \mathcal{F}_n + \alpha|\psi|^2 + \frac{\beta}{2}|\psi|^4 + \frac{1}{2m^*}\left|\left(-i\hbar\nabla - e^*\mathbf{A}\right)\psi\right|^2 + \frac{|\mathbf{B}|^2}{2\mu_0}$

Where $\alpha = \alpha_0(T - T_c)$ (negative below $T_c$), $\beta > 0$, $m^* = 2m_e$, $e^* = 2e$ (Cooper pair charge), and $\mathbf{A}$ is the vector potential.

Minimising with respect to $\psi^*$ gives the first GL equation:

$\alpha\psi + \beta|\psi|^2\psi + \frac{1}{2m^*}\left(-i\hbar\nabla - e^*\mathbf{A}\right)^2\psi = 0$

Minimising with respect to $\mathbf{A}$ gives the second GL equation (supercurrent):

$\mathbf{J}_s = \frac{e^*\hbar}{m^*}\left(\psi^*\nabla\psi - \psi\nabla\psi^*\right) - \frac{e^{*2}}{m^*}|\psi|^2\mathbf{A}$

Two fundamental length scales emerge from the GL theory:

Coherence length (characterises the spatial variation of $|\psi|$):

$\xi(T) = \sqrt{\frac{\hbar^2}{2m^*|\alpha|}} = \frac{\xi_0}{\sqrt{1 - T/T_c}}$

Penetration depth (characterises the decay of $\mathbf{B}$):

$\lambda(T) = \sqrt{\frac{m^*}{\mu_0 e^{*2}|\psi_\infty|^2}} = \frac{\lambda_0}{\sqrt{1 - T/T_c}}$

Where $|\psi_\infty|^2 = |\alpha|/\beta$ is the bulk equilibrium value.

The ratio of these length scales determines the superconductor type:

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

• $\kappa < 1/\sqrt{2}$: Type I (positive surface energy)
• $\kappa > 1/\sqrt{2}$: Type II (negative surface energy, mixed state favourable)

In the mixed state of a Type II superconductor ($B_{c1} < B < B_{c2}$), magnetic flux penetrates in quantised vortices, each carrying one flux quantum:

$\Phi_0 = \frac{h}{2e} = 2.07 \times 10^{-15}\ \mathrm{Wb}$

The vortex core (radius $\sim\xi$) is in the normal state, while supercurrents circulate around it (decaying over $\sim\lambda$).

The upper critical field from GL theory:

$B_{c2} = \frac{\Phi_0}{2\pi\xi^2}$

The lower critical field:

$B_{c1} = \frac{\Phi_0}{4\pi\lambda^2}\ln\kappa$

The thermodynamic critical field:

$B_c = \frac{\Phi_0}{2\pi\sqrt{2}\xi\lambda}$

These satisfy $B_{c1} < B_c < B_{c2}$ for $\kappa > 1/\sqrt{2}$.

Flux quantisation. The GL order parameter must be single-valued. Integrating the supercurrent around a closed loop enclosing flux $\Phi$:

$\oint \nabla\theta \cdot d\mathbf{l} = \frac{2\pi\Phi}{\Phi_0} = 2\pi n$

Where $\theta$ is the phase of $\psi$ and $n$ is an integer. Hence $\Phi = n\Phi_0$.

DC Josephson effect. For a superconductor—insulator—superconductor (SIS) junction with phase difference $\delta$:

$I = I_c \sin\delta$

Where $I_c$ is the critical current.

AC Josephson effect. Applying a voltage $V$ across the junction causes the phase to evolve as $\dot{\delta} = 2eV/\hbar$. Giving:

$I = I_c\sin\!\left(\delta_0 + \frac{2eV}{\hbar}t\right)$

The oscillation frequency $\nu = 2eV/h$ provides the basis for the Josephson voltage standard: $V = n(h/2e)\nu$.

• Confusing $\xi$ and $\lambda$. The coherence length $\xi$ governs how fast the order parameter $\psi$ varies in space; the penetration depth $\lambda$ governs how fast the magnetic field decays. They are independent length scales that happen to appear in the same theory.
• Type I vs Type II threshold. The criterion $\kappa = 1/\sqrt{2}$ is exact within GL theory. Do not confuse this with $\kappa = 1$ or other round numbers. The $1/\sqrt{2}$ factor arises from comparing the surface energies of normal-superconducting boundaries.
• Flux quantum uses $2e$, not $e$. Each vortex carries $\Phi_0 = h/(2e)$ because the superconducting condensate consists of Cooper pairs with charge $2e$. A common error is to use $h/e$ (the normal-state flux quantum for single electrons).
• GL theory is valid only near $T_c$. The phenomenological expansion in $|\psi|$ assumes $|\psi|$ is small, which holds when $T$ is close to $T_c$. Far below $T_c$, microscopic BCS theory is required.
• AC Josephson frequency is exact. The relation $\nu = 2eV/h$ does not depend on junction geometry or material properties. It is a fundamental quantum relation used to define the volt.

The GL theory is phenomenological — it does not explain why superconductivity occurs. The microscopic BCS theory (Bardeen, Cooper, Schrieffer, 1957) provides this explanation:

• Cooper pairs. An attractive interaction mediated by phonons (lattice vibrations) allows electrons with opposite momenta and spins to form bound pairs. The binding energy is the superconducting gap $\Delta(T)$, which vanishes at $T_c$.
• Gap equation. At $T = 0$, the gap relates to $T_c$ via $2\Delta(0) = 3.53\, k_B T_c$ (weak-coupling limit). This ratio is approximately universal for conventional superconductors.
• Coherence length from BCS. The BCS coherence length $\xi_0 = \hbar v_F / (\pi \Delta_0)$ where $v_F$ is the Fermi velocity. Typical values: $\xi_0 \sim 10$—$1000$ nm for conventional superconductors.
• Penetration depth from BCS. $\lambda_0 = \sqrt{m^*/(\mu_0 n_s e^{*2})}$ where $n_s$ is the superfluid density (all conduction electrons below $T_c$).

The GL parameters $\alpha$ and $\beta$ can be expressed in terms of BCS quantities near $T_c$: $\alpha = -1/(N(0)\xi_0^2)$ and $\beta = 1/(N(0)\Delta_0^2)$, where $N(0)$ is the density of states at the Fermi level. This connection shows that GL theory is the correct Ginzburg—Landau limit of BCS theory when $T \to T_c$.

The GL theory underpins the design of all these devices. For MRI and accelerator magnets, the critical current density $J_c$ (determined by vortex pinning) is the key engineering parameter.