Phenomenological Superconductivity: Meissner Effect, London & Ginzburg-Landau

1. Discovery & The Superconducting State

In 1911, Heike Kamerlingh Onnes discovered that when pure mercury ($\text{Hg}$) is cooled below a critical temperature $T_c = 4.15\text{ K}$, its electrical resistivity drops abruptly to zero:

$$\rho(T) = 0 \quad \text{for } T < T_c$$

Modern experimental measurements on closed superconducting rings place an upper bound on the resistivity of $\rho < 10^{-26}\ \Omega\cdot\text{m}$ (over 18 orders of magnitude lower than high-purity copper). Persistent electrical currents induced in superconducting rings have circulated without measurable decay for decades, implying a carrier lifetime $\tau_{\text{decay}} > 10^5\text{ years}$.

2. Empirical Critical Field Law

The superconducting state is destroyed and normal electrical resistance is restored when an external magnetic field exceeds a critical value $H_c(T)$. Experimentally, the critical field follows an accurate parabolic temperature dependence:

$$H_c(T) = H_0 \left[ 1 - \left( \frac{T}{T_c} \right)^2 \right]$$

where $H_0 = H_c(0)$ is the critical field extrapolated to absolute zero ($T = 0\text{ K}$).

Similarly, passing an electrical current through a superconductor creates a self-magnetic field. According to Silsbee's Rule, superconductivity is destroyed when the total magnetic field at the specimen surface (from external sources and transport currents combined) reaches $H_c(T)$. For a cylindrical wire of radius $r$, the critical current is:

$$I_c(T) = 2\pi r H_c(T)$$

1. The Meissner-Ochsenfeld Effect

In 1933, Walther Meissner and Robert Ochsenfeld discovered that when a superconductor is cooled below $T_c$ in the presence of a constant external magnetic field $\vec{B}$, the magnetic flux is actively expelled from the interior of the specimen:

$$\vec{B}_{\text{interior}} = 0 \quad (\text{Meissner Effect})$$

Inside the bulk material, the magnetic induction is identically zero:

$$\vec{B} = \mu_0 (\vec{H} + \vec{M}) = 0 \implies \vec{M} = -\vec{H}$$

The volume magnetic susceptibility of a bulk superconductor is:

$$\chi_v = \frac{M}{H} = -1 \quad (\text{Perfect Diamagnetism})$$

2. Distinction Between a Perfect Conductor and a Superconductor

Consider a hypothetical classical perfect conductor with zero resistivity ($\rho = 0$). By Ohm's law, $\vec{E} = \rho \vec{j} = 0$. Faraday's law of induction requires:

$$\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t} = 0 \implies \frac{\partial\vec{B}}{\partial t} = 0 \implies \vec{B}(t) = \vec{B}(0) = \text{constant}$$

A perfect conductor traps whatever flux was present when resistance vanished:

• If cooled in zero field, it excludes subsequent fields ($B = 0$).
• If cooled in an applied field $B_0$, it traps $B_0$ inside indefinitely ($\vec{B} = B_0 \ne 0$). The final state depends on thermodynamic history!

In contrast, a true superconductor always expels the magnetic field upon cooling through $T_c$, regardless of whether the field was applied before or after cooling:

$$\vec{B} = 0 \quad \text{always for } T < T_c$$

The Meissner effect proves that superconductivity is not merely infinite conductivity, but a true thermodynamic equilibrium phase governed by an equation of state $\vec{B} = 0$.

1. Free Energy & Condensation Energy Density

Because superconductivity is an equilibrium thermodynamic state, we apply the Gibbs free energy per unit volume:

$$dG = -S dT - \mu_0 M dH$$

In the normal state ($M_n \approx 0$):

$$G_n(T, H) = G_n(T, 0)$$

In the superconducting state ($M_s = -H$):

$$G_s(T, H) = G_s(T, 0) - \mu_0 \int_0^H (-H') dH' = G_s(T, 0) + \frac{\mu_0 H^2}{2}$$

At the phase boundary $H = H_c(T)$, the normal and superconducting phases are in thermodynamic equilibrium:

$$G_s(T, H_c) = G_n(T, H_c) \implies G_s(T, 0) + \frac{\mu_0 H_c^2(T)}{2} = G_n(T, 0)$$

Rearranging gives the superconducting condensation energy density:

$$\Delta f_{\text{cond}} \equiv G_n(T, 0) - G_s(T, 0) = \frac{\mu_0 H_c^2(T)}{2}$$

The superconducting state has a lower free energy than the normal state by exactly the magnetic energy density required to expel the critical field.

2. Entropy & Discontinuous Specific Heat Jump

Differentiating with respect to temperature:

$$S_n - S_s = -\frac{d}{dT}[G_n(T, 0) - G_s(T, 0)] = -\mu_0 H_c(T) \frac{dH_c}{dT}$$

• At $T = T_c$, $H_c(T_c) = 0$, so $S_s(T_c) = S_n(T_c)$. The entropy is continuous: there is zero latent heat in zero magnetic field ($Q = T_c \Delta S = 0$).
• Differentiating entropy to obtain the specific heat $C = T \frac{dS}{dT}$: $$C_s - C_n = T \frac{d}{dT}(S_s - S_n) = \mu_0 T \left[ \left(\frac{dH_c}{dT}\right)^2 + H_c(T) \frac{d^2H_c}{dT^2} \right]$$ At $T = T_c$, where $H_c(T_c) = 0$: $$\Delta C \equiv C_s(T_c) - C_n(T_c) = \mu_0 T_c \left( \left.\frac{dH_c}{dT}\right|_{T_c} \right)^2 > 0$$

Using the parabolic law $H_c(T) = H_0[1 - (T/T_c)^2]$, the slope at $T_c$ is $\left.\frac{dH_c}{dT}\right|_{T_c} = -\frac{2H_0}{T_c}$:

$$\Delta C = \mu_0 T_c \left(-\frac{2H_0}{T_c}\right)^2 = \frac{4\mu_0 H_0^2}{T_c}$$

This finite discontinuity in specific heat with zero latent heat rigorously classifies the superconducting transition in zero field as a second-order phase transition.

1. The Two London Equations

In 1935, Fritz and Heinz London proposed a phenomenological two-fluid model where the electron density divides into a normal component $n_n$ and a superconducting component $n_s$ ($n = n_n + n_s$).

Superconducting electrons accelerate without friction under an electric field:

$$m \frac{d\vec{v}_s}{dt} = -e\vec{E} \implies \frac{\partial \vec{j}_s}{\partial t} = \frac{n_s e^2}{m} \vec{E} \quad (\text{First London Equation})$$

Taking the curl of both sides and invoking Faraday's law $\nabla \times \vec{E} = -\frac{\partial\vec{B}}{\partial t}$:

$$\frac{\partial}{\partial t} \left( \nabla \times \vec{j}_s + \frac{n_s e^2}{m} \vec{B} \right) = 0$$

To encompass the Meissner effect (where $\vec{B} = 0$ is the unique equilibrium state, not merely a conserved constant), the London brothers boldly set the quantity in parentheses identically to zero:

$$\nabla \times \vec{j}_s = -\frac{n_s e^2}{m} \vec{B} \quad (\text{Second London Equation})$$

In terms of the magnetic vector potential $\vec{A}$ in the London gauge ($\nabla \cdot \vec{A} = 0$):

$$\vec{j}_s = -\frac{n_s e^2}{m} \vec{A}$$

2. The London Penetration Depth $\lambda_L$

Taking the curl of Maxwell's Ampère law $\nabla \times \vec{B} = \mu_0 \vec{j}_s$ (neglecting displacement current):

$$\nabla \times (\nabla \times \vec{B}) = \mu_0 (\nabla \times \vec{j}_s)$$

Using the vector identity $\nabla \times (\nabla \times \vec{B}) = \nabla(\nabla \cdot \vec{B}) - \nabla^2 \vec{B} = -\nabla^2 \vec{B}$ (since $\nabla \cdot \vec{B} = 0$):

$$-\nabla^2 \vec{B} = -\mu_0 \frac{n_s e^2}{m} \vec{B} \implies \nabla^2 \vec{B} = \frac{1}{\lambda_L^2} \vec{B}$$

where $\lambda_L$ is the London penetration depth:

$$\lambda_L \equiv \sqrt{\frac{m}{\mu_0 n_s e^2}}$$

For a semi-infinite superconductor filling $x \ge 0$ with an external field $B_0 \hat{z}$ at $x = 0$:

$$B(x) = B_0 \exp\left( -\frac{x}{\lambda_L} \right)$$

The magnetic field does not vanish discontinuously at the surface, but penetrates exponentially into a thin surface sheath of thickness $\lambda_L \sim 30-100\text{ nm}$, supported by screening supercurrents $\vec{j}_s(x) = \frac{B_0}{\mu_0 \lambda_L} e^{-x/\lambda_L} \hat{y}$.

As $T \to T_c$, $n_s(T) \to 0$, causing the penetration depth to diverge:

$$\lambda_L(T) = \lambda_0 \left[ 1 - \left( \frac{T}{T_c} \right)^4 \right]^{-1/2}$$

1. Classification: Type-I vs Type-II Superconductors

Superconductors are categorized into two fundamentally distinct classes based on their magnetic response:

• Type-I Superconductors (e.g. Pb, Sn, In, Al, Hg): Exhibit complete Meissner expulsion ($\vec{B} = 0$) up to a single sharp thermodynamic critical field $H_c(T)$. When $H > H_c$, superconductivity is destroyed abruptly, and the material reverts to the normal state in a first-order phase transition. Values of $H_c$ are typically low ($< 0.1\text{ T}$), rendering pure Type-I materials useless for high-field electromagnets.
• Type-II Superconductors (e.g. Nb, Nb-Ti, $\text{Nb}_3\text{Sn}$, YBCO, BSCCO): Possess two distinct critical magnetic fields: the lower critical field $H_{c1}(T)$ and the upper critical field $H_{c2}(T)$.
  • For $H < H_{c1}$: Complete Meissner effect ($B = 0$).
  • For $H_{c1} < H < H_{c2}$: The Mixed State (or Shubnikov phase / Vortex state). Magnetic flux penetrates the bulk in the form of quantized microscopic flux tubes (vortices), while the intervening matrix remains superconducting. Zero electrical resistance persists!
  • For $H > H_{c2}$: The bulk completely transitions to the normal state. $H_{c2}$ can be exceptionally large ($15\text{ T}$ in Nb-Ti, $30\text{ T}$ in $\text{Nb}_3\text{Sn}$, $>100\text{ T}$ in cuprates), enabling modern MRI scanners and particle accelerator magnets.

2. The Surface Energy of a Normal-Superconducting Interface

Consider a planar interface between a normal domain ($B = H_c$) and a superconducting domain ($B = 0$):

• The magnetic field penetrates a distance $\lambda$ into the superconductor, releasing magnetic expulsion energy $\frac{1}{2}\mu_0 H_c^2 \lambda > 0$ (energy gain).
• The superconducting order parameter $|\psi|^2$ requires a characteristic distance $\xi$ (the coherence length) to recover its bulk value, losing condensation energy $\frac{1}{2}\mu_0 H_c^2 \xi < 0$ (energy cost).

The net surface energy per unit area of the interface is approximately:

$$\sigma_{\text{ns}} \approx \frac{1}{2}\mu_0 H_c^2 (\xi - \lambda)$$

• Positive Surface Energy ($\xi > \lambda$): The interface costs energy. The system minimizes interface area by remaining either fully superconducting or fully normal $\implies$ Type-I Superconductor.
• Negative Surface Energy ($\xi < \lambda$): Creating interfaces is energetically favorable! The material spontaneously maximizes normal-superconductor boundaries by shredding into a dense array of microscopic magnetic flux tubes $\implies$ Type-II Superconductor.

1. The Ginzburg-Landau Order Parameter & Free Energy

In 1950, Vitaly Ginzburg and Lev Landau formulated a general phenomenological theory of superconductivity based on Landau's theory of second-order phase transitions. They introduced a complex macroscopic pseudo-wavefunction $\psi(\vec{r}) = |\psi|e^{i\theta}$ as an order parameter, whose local density represents the superconducting Cooper pair density:

$$|\psi(\vec{r})|^2 = \frac{1}{2} n_s(\vec{r})$$

Near $T_c$, where $|\psi|^2$ is small, the Gibbs free energy density is expanded in powers of $|\psi|^2$ and its spatial gradients:

$$f_s = f_n + \alpha(T) |\psi|^2 + \frac{\beta}{2} |\psi|^4 + \frac{1}{2m^*} |(-i\hbar\nabla - q^*\vec{A})\psi|^2 + \frac{B^2}{2\mu_0}$$

where $m^* = 2m$ and $q^* = -2e$ are the mass and charge of a Cooper pair.

• $\beta > 0$ ensures stability.
• $\alpha(T) = \alpha_0 (T - T_c)$:
  • For $T > T_c$: $\alpha > 0$, minimized at $|\psi| = 0$ (normal state).
  • For $T < T_c$: $\alpha < 0$, minimized at $|\psi_\infty|^2 = -\frac{\alpha}{\beta} = \frac{|\alpha|}{\beta}$.

2. The Ginzburg-Landau Equations & Coherence Length $\xi(T)$

Minimizing the total free energy functional $\mathcal{F} = \int f_s d^3r$ with respect to $\psi^*$ and $\vec{A}$ yields the two Ginzburg-Landau equations:

$$\frac{1}{2m^*} (-i\hbar\nabla - q^*\vec{A})^2 \psi + \alpha \psi + \beta |\psi|^2 \psi = 0 \quad (\text{First GL Equation})$$ $$\vec{j}_s = \frac{q^*\hbar}{2m^* i} (\psi^* \nabla\psi - \psi \nabla\psi^*) - \frac{(q^*)^2}{m^*} |\psi|^2 \vec{A} = \frac{1}{\mu_0} \nabla \times \vec{B} \quad (\text{Second GL Equation})$$

In zero magnetic field, the first equation in 1D becomes:

$$-\frac{\hbar^2}{2m^*|\alpha|} \frac{d^2\psi}{dx^2} - \psi + \frac{\beta}{|\alpha|} \psi^3 = 0$$

Defining the Ginzburg-Landau coherence length $\xi(T)$:

$$\xi(T) \equiv \sqrt{\frac{\hbar^2}{2m^* |\alpha(T)|}} \propto \left(1 - \frac{T}{T_c}\right)^{-1/2}$$

$\xi(T)$ represents the characteristic spatial scale over which the superconducting order parameter can vary without excessive gradient energy penalty.

We define the dimensionless Ginzburg-Landau parameter $\kappa$:

$$\kappa \equiv \frac{\lambda(T)}{\xi(T)}$$

The exact boundary derived from GL theory is:

$$\kappa < \frac{1}{\sqrt{2}} \approx 0.707 \implies \text{Type-I Superconductor}$$ $$\kappa > \frac{1}{\sqrt{2}} \approx 0.707 \implies \text{Type-II Superconductor}$$

1. Derivation of Magnetic Flux Quantization

Consider a thick superconducting cylinder containing a hole or hollow core. Inside the superconducting bulk far from surfaces, $\vec{j}_s = 0$. Writing $\psi(\vec{r}) = |\psi| e^{i\theta(\vec{r})}$:

$$\vec{j}_s = \frac{q^*}{m^*} |\psi|^2 (\hbar\nabla\theta - q^*\vec{A}) = 0 \implies \hbar\nabla\theta = q^*\vec{A} = -2e\vec{A}$$

Integrating around a closed path $C$ enclosing the hole deep inside the bulk:

$$\hbar \oint_C \nabla\theta \cdot d\vec{l} = -2e \oint_C \vec{A} \cdot d\vec{l}$$

Because $\psi(\vec{r})$ must be single-valued, the phase change around any closed loop must be an integer multiple of $2\pi$: $\oint_C \nabla\theta \cdot d\vec{l} = 2\pi n, n \in \mathbb{Z}$.

By Stokes' theorem, $\oint_C \vec{A} \cdot d\vec{l} = \iint (\nabla \times \vec{A}) \cdot d\vec{a} = \iint \vec{B} \cdot d\vec{a} = \Phi$:

$$2\pi n \hbar = -2e \Phi \implies |\Phi| = n \frac{h}{2e} \equiv n \Phi_0$$

The magnetic flux enclosed by a superconducting ring is strictly quantized in discrete integer units of the flux quantum $\Phi_0$:

$$\Phi_0 \equiv \frac{h}{2e} \approx 2.0678 \times 10^{-15}\text{ Wb} = 2.0678 \times 10^{-7}\text{ G}\cdot\text{cm}^2$$

The factor of $2e$ provided the first direct experimental proof (Deaver & Fairbank; Doll & Näbauer, 1961) that the fundamental charge carriers in superconductors are electron pairs (Cooper pairs).

2. The Abrikosov Mixed Vortex State

In 1957, Alexei Abrikosov discovered periodic solutions to the GL equations for $\kappa > 1/\sqrt{2}$. In the mixed state ($H_{c1} < H < H_{c2}$):

• Magnetic field penetrates as an array of microscopic cylindrical filaments called vortices (or fluxons).
• Each vortex has a normal core of radius $\sim \xi(T)$ where $|\psi| \to 0$.
• Surrounding the core, circulating supercurrents decay over radius $\lambda(T)$, screening the magnetic field.
• Each individual vortex carries exactly one single flux quantum $\Phi_0 = h/2e$.
• Mutual repulsive forces between circulating supercurrents cause vortices to arrange into a regular Abrikosov triangular flux line lattice with lattice parameter $a_{\triangle} = \left(\frac{2\Phi_0}{\sqrt{3}B}\right)^{1/2}$.
• The upper critical field occurs when vortex cores overlap: $$B_{c2} = \frac{\Phi_0}{2\pi \xi^2}$$