Ginzburg-Landau Theory: Coherence, Penetration, and Superconductor Types

Hello! Welcome back to our study of superconductivity.

In our last lesson, we explored the defining phenomenological hallmarks of superconductivity: zero resistance, the Meissner effect, and flux quantization. We also introduced the crucial classification of superconductors into Type I and Type II based on their response to an external magnetic field.

Today, we will build a powerful theoretical framework to describe this behavior quantitatively. We will delve into Ginzburg-Landau (GL) theory, a cornerstone of condensed matter physics that, despite being phenomenological, provides profound insights into the nature of the superconducting state. Our goal is to apply this theory to formally define the two fundamental length scales of superconductivity—the coherence length ($\xi$) and the penetration depth ($\lambda$)—and understand how their ratio distinguishes between Type I and Type II materials.

1. The Ginzburg-Landau Free Energy

At its heart, GL theory is an extension of Landau's general theory of second-order phase transitions, which you've likely encountered in statistical mechanics. The key idea is to expand the free energy of a system as a power series in an order parameter, a quantity that is zero in the disordered (high-temperature) phase and non-zero in the ordered (low-temperature) phase.

For a superconductor, the order parameter is a complex function, $\Psi (\mathbf{r})$, which can be thought of as the macroscopic wavefunction of the entire condensate of Cooper pairs. The local density of superconducting carriers is then given by $n_s(\mathbf{r})=|\Psi (\mathbf{r}){|}^2$.

Let's start by seeing how the free energy is constructed.

As the video explains, for a homogeneous system in zero field, the difference in free energy density between the superconducting ($f_S$) and normal ($f_N$) states is expanded as:

$f_S-f_N=\alpha |\Psi {|}^2+\frac{\beta }{2}|\Psi {|}^4$

Here, $\beta >0$ is a constant, and $\alpha ={\alpha }_0(T-T_c)$ with ${\alpha }_0>0$. This temperature dependence of $\alpha$ ensures that for $T>T_c$, the free energy is minimized at $\Psi =0$ (normal state), while for $T<T_c$, it's minimized at a finite value $|{\Psi }_{\infty }{|}^2=-\alpha /\beta =({\alpha }_0/\beta )(T_c-T)$ (superconducting state).

Ginzburg and Landau's crucial insight was to generalize this to include spatial variations of $\Psi (\mathbf{r})$ and the presence of a magnetic field. The total free energy $F_S$ becomes a functional of $\Psi (\mathbf{r})$ and the vector potential $\mathbf{A}(\mathbf{r})$:

$F_S[\Psi ,\mathbf{A}]=F_N+\int d^{3}r\left[\alpha |\Psi {|}^2+\frac{\beta }{2}|\Psi {|}^4+\frac{1}{2m^{\ast }}{\left|\left(-i\hslash \nabla -q^{\ast }\mathbf{A}\right)\Psi \right|}^2+\frac{|\mathbf{B}{|}^2}{2{\mu }_0}\right]$

Let's break down the two new terms in the integral:

1. Kinetic Energy Term: $\frac{1}{2m^{\ast }}{\left|\left(-i\hslash \nabla -q^{\ast }\mathbf{A}\right)\Psi \right|}^2$. This is the quantum mechanical kinetic energy density. The term $-i\hslash \nabla$ is the standard momentum operator, and the coupling to the magnetic field is included via minimal coupling, $\mathbf{p}\to \mathbf{p}-q^{\ast }\mathbf{A}$, a concept central to gauge theories and one you're familiar with from your QFT studies. Here, $q^{\ast }=2e$ and $m^{\ast }=2m_e$ are the charge and effective mass of a Cooper pair. This term penalizes spatial variations in the order parameter.
2. Magnetic Field Energy: $\frac{|\mathbf{B}{|}^2}{2{\mu }_0}$, where $\mathbf{B}=\nabla \times \mathbf{A}$. This is simply the energy density stored in the magnetic field itself.

The equilibrium state of the superconductor is the one that minimizes this total free energy.

2. The Ginzburg-Landau Equations

To find the functions $\Psi (\mathbf{r})$ and $\mathbf{A}(\mathbf{r})$ that minimize the free energy functional, we use the calculus of variations. By requiring that the variation of $F_S$ with respect to ${\Psi }^{\ast }$ and $\mathbf{A}$ is zero ($\delta F_S=0$), we obtain two coupled differential equations.

The minimization procedure yields the two celebrated Ginzburg-Landau equations:

1. First GL Equation (from $\delta F_S/\delta {\Psi }^{\ast }=0$):
   $\alpha \Psi +\beta |\Psi {|}^2\Psi +\frac{1}{2m^{\ast }}{\left(-i\hslash \nabla -q^{\ast }\mathbf{A}\right)}^2\Psi =0$
   This equation determines the spatial profile of the order parameter $\Psi (\mathbf{r})$. It has the form of a non-linear Schrödinger equation.

2. Second GL Equation (from $\delta F_S/\delta \mathbf{A}=0$):
   ${\mathbf{j}}_s=\frac{q^{\ast }}{2m^{\ast }}\left[{\Psi }^{\ast }(-i\hslash \nabla -q^{\ast }\mathbf{A})\Psi +c.c.\right]$
   This equation gives an expression for the dissipationless supercurrent density ${\mathbf{j}}_s$. When combined with Maxwell's equation ($\nabla \times \mathbf{B}={\mu }_0{\mathbf{j}}_s$), it determines the magnetic field distribution.

These two equations form the complete basis for describing a superconductor's behavior within this phenomenological theory.

3. Characteristic Length Scales: $\xi$ and $\lambda$

Now we will use the GL equations to derive the two fundamental length scales that govern superconductivity. We'll do this by analyzing the equations in simplified physical limits.

Let's summarize and formalize those results.

Coherence Length, $\xi$

Consider a boundary between a superconductor ($x>0$) and a vacuum/insulator ($x<0$) with no magnetic field ($\mathbf{A}=0$). The order parameter must vanish for $x<0$ and recover to its bulk value ${\Psi }_{\infty }$ deep inside the superconductor. The first GL equation describes this recovery:
$-\frac{{\hslash }^2}{2m^{\ast }}\frac{d^2\Psi }{d{x}^2}+\alpha \Psi +\beta |\Psi {|}^2\Psi =0$
By rearranging this equation (for $T<T_c$, where $\alpha <0$) and normalizing $\Psi$ by its bulk value, one finds a natural length scale emerges:
${\xi }^2(T)=\frac{{\hslash }^2}{2m^{\ast }|\alpha |}=\frac{{\hslash }^2}{2m^{\ast }{\alpha }_0(T_c-T)}$
This is the Ginzburg-Landau coherence length.

• Physical Meaning: $\xi$ is the characteristic length scale over which the superconducting order parameter $\Psi$ can vary. It represents the minimum distance required for $\Psi$ to "heal" back to its bulk value after a perturbation (like an interface). Microscopically, it's related to the size of a Cooper pair.
• Temperature Dependence: Near $T_c$, the coherence length diverges as $\xi (T)\propto (T_c-T{)}^{-1/2}$.

Penetration Depth, $\lambda$

Now consider a weak magnetic field applied to the bulk of a superconductor, where we can assume $\Psi \approx {\Psi }_{\infty }$ is constant. The second GL equation for the supercurrent simplifies to the London equation:
${\mathbf{j}}_s=-\frac{(q^{\ast }{)}^2|{\Psi }_{\infty }{|}^2}{m^{\ast }}\mathbf{A}$
Combining this with Ampere's law ($\nabla \times \mathbf{B}={\mu }_0{\mathbf{j}}_s$) and $\mathbf{B}=\nabla \times \mathbf{A}$, and using the vector identity $\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}$, we arrive at (in the London gauge where $\nabla \cdot \mathbf{A}=0$):
${\nabla }^2\mathbf{B}=\frac{{\mu }_0(q^{\ast }{)}^2|{\Psi }_{\infty }{|}^2}{m^{\ast }}\mathbf{B}$
This shows the magnetic field decays exponentially, and we can identify the decay length:
${\lambda }^2(T)=\frac{m^{\ast }}{{\mu }_0(q^{\ast }{)}^2|{\Psi }_{\infty }{|}^2}=\frac{m^{\ast }\beta }{{\mu }_0(q^{\ast }{)}^2|\alpha |}=\frac{m^{\ast }\beta }{{\mu }_0(q^{\ast }{)}^2{\alpha }_0(T_c-T)}$
This is the magnetic penetration depth.

• Physical Meaning: $\lambda$ is the characteristic length over which an external magnetic field is screened and decays to zero inside the superconductor.
• Temperature Dependence: Near $T_c$, the penetration depth also diverges as $\lambda (T)\propto (T_c-T{)}^{-1/2}$.

4. Distinguishing Type I and Type II Superconductors

The distinction between Type I and Type II superconductors, which we introduced phenomenologically in the last lesson, can be understood elegantly through the competition between $\xi$ and $\lambda$. This competition determines the energy cost of creating an interface between a normal and a superconducting region.

As Prof. Frolov beautifully explains, the sign of the surface energy ${\sigma }_{ns}$ at a normal-superconductor interface is key:

• Positive Surface Energy (${\sigma }_{ns}>0$): If $\xi \gg \lambda$, the system loses superconducting condensation energy over a wide region ($\xi$) while only gaining magnetic energy over a narrow region ($\lambda$). The net effect is a positive surface energy. The system will try to minimize the number of such interfaces. This leads to the complete Meissner effect until the field is strong enough to destroy superconductivity entirely. This is a Type I superconductor.

• Negative Surface Energy (${\sigma }_{ns}<0$): If $\lambda \gg \xi$, the system gains more magnetic energy than it loses in condensation energy at the interface. It becomes energetically favorable to create normal-superconducting boundaries. The system allows magnetic flux to penetrate in the form of quantized flux tubes (vortices), creating a mixed state. This is a Type II superconductor.

The behavior is governed by the dimensionless Ginzburg-Landau parameter:
$\kappa =\frac{\lambda (T)}{\xi (T)}$
Since both $\lambda$ and $\xi$ have the same temperature dependence near $T_c$, $\kappa$ is approximately temperature-independent. A more detailed calculation by Abrikosov showed that the sign of the surface energy changes precisely at:
$\kappa =\frac{1}{\sqrt{2}}\approx 0.707$

• $\kappa <1/\sqrt{2}⟹$ Type I
• $\kappa >1/\sqrt{2}⟹$ Type II

This parameter provides a clear, quantitative criterion for classifying any superconductor. Your experimental work in high magnetic fields directly probes the consequences of this classification, as the high critical fields required for many applications are only possible in Type II materials.

Test your understanding!

A certain elemental superconductor is Type I in its pure form. An experimentalist adds impurities to the material. Impurities reduce the electron mean free path, which is known to decrease the coherence length $\xi$ significantly while having a smaller effect on the penetration depth $\lambda$. What might happen to the classification of this material if enough impurities are added? Explain your reasoning in terms of the Ginzburg-Landau parameter $\kappa$.

Show answer

Adding impurities decreases $\xi$. Since the Ginzburg-Landau parameter is defined as $\kappa =\lambda /\xi$, a decrease in $\xi$ will cause $\kappa$ to increase. If enough impurities are added, it is possible for $\kappa$ to increase from a value below $1/\sqrt{2}$ to a value above $1/\sqrt{2}$. This would cause the material to transition from being a Type I superconductor to a Type II superconductor. This is a known phenomenon; for example, pure niobium is a borderline Type I/II superconductor, but adding impurities reliably makes it Type II.

Conclusion

In this lesson, we have moved from a qualitative description of superconductivity to a powerful quantitative framework. Ginzburg-Landau theory, though phenomenological, successfully captures the essential physics of the superconducting state, including its response to magnetic fields and the existence of two distinct types of behavior.

Key Takeaways:

• Ginzburg-Landau theory is based on a free energy functional expanded in terms of a complex order parameter $\Psi (\mathbf{r})$, where $|\Psi {|}^2$ represents the density of superconducting carriers.
• Minimizing this functional yields the two Ginzburg-Landau equations, which govern the spatial behavior of the order parameter and the supercurrent.
• The theory naturally defines two fundamental length scales: the coherence length $\xi$, over which $\Psi$ can vary, and the penetration depth $\lambda$, over which magnetic fields are screened.
• The ratio of these lengths, the Ginzburg-Landau parameter $\kappa =\lambda /\xi$, determines the sign of the surface energy at a normal-superconductor interface.
• This criterion distinguishes between Type I ($\kappa <1/\sqrt{2}$) and Type II ($\kappa >1/\sqrt{2}$) superconductors, explaining the formation of the mixed state in the latter.

Preview of the Next Lesson

Ginzburg-Landau theory provides an incredibly successful "how" model, but it doesn't explain the microscopic "why". Why do electrons form pairs in the first place? What is the origin of the attractive interaction that overcomes their Coulomb repulsion? In our next lesson, we will begin to answer these questions by exploring the microscopic origins of superconductivity. We will start by demonstrating the Cooper instability, which shows that an arbitrarily weak attractive interaction between electrons near the Fermi surface can lead to the formation of a bound state. This is the first step towards the celebrated BCS theory.