Phenomenology of Superconductivity

Hello! Welcome to the tenth module of our course.

In the previous modules, we have journeyed through the electronic and vibrational properties of solids, culminating in a deep dive into the complex world of strongly correlated systems with Dynamical Mean-Field Theory. We are now shifting our focus to an equally profound, but distinct, collective quantum phenomenon: superconductivity. This topic is a cornerstone of modern condensed matter physics, and understanding its foundations is crucial.

This first lesson will establish the phenomenological basis of superconductivity. We will move away from the microscopic Hamiltonians of the last module and instead focus on the striking macroscopic behaviors that define a superconductor. By the end of this lesson, you will be able to describe the three fundamental experimental hallmarks of this state: perfect conductivity (zero resistance), the Meissner effect (perfect diamagnetism), and magnetic flux quantization.

1. The Discovery and Zero Resistance

Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes, who observed that the electrical resistance of mercury vanished abruptly below a critical temperature, $T_c$, of 4.2 K. While this might sound like the simple behavior of a "perfect" conductor, the reality is much more subtle and profound.

To begin, let's explore this foundational property with a lecture from a leading physicist in the field.

As Prof. Kivelson explains, simply measuring a very small resistance isn't enough to claim it's truly zero. The most compelling evidence comes from persistent currents. When a current is induced in a superconducting ring, it flows indefinitely without any measurable decay. Experiments have set lower bounds on the decay time that are longer than the age of the universe. This implies a state of matter with fundamentally zero dissipation for DC currents, unlike a normal metal where scattering from impurities and phonons always leads to resistance.

2. The Meissner Effect: A Defining Characteristic

For over two decades after its discovery, superconductivity was thought of as simply a state of infinite conductivity. The true, defining feature that distinguishes a superconductor from a hypothetical "perfect conductor" was discovered in 1933 by Meissner and Ochsenfeld.

A perfect conductor, according to Faraday's law of induction ($\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$), would maintain whatever magnetic field was present when it became "perfect." Since $\mathbf{E}=\mathbf{j}/\sigma$, infinite conductivity ($\sigma \to \infty$) implies $\mathbf{E}=0$. This leads to $\frac{\partial \mathbf{B}}{\partial t}=0$, meaning the magnetic field inside is frozen in time.

A superconductor does something far more remarkable.

As the reading highlights, a superconductor actively expels magnetic fields from its interior when cooled below $T_c$. This phenomenon, the Meissner effect, demonstrates that the superconducting state is a true thermodynamic phase where the magnetic induction $\mathbf{B}=0$ inside the bulk material. This property is also known as perfect diamagnetism. The magnetic field only penetrates a small distance near the surface, known as the London penetration depth, ${\lambda }_L$.

The London equations, a simple phenomenological model, provide a first theoretical explanation for the Meissner effect. Let's see how this works.

This expulsion is the key to famous demonstrations like magnetic levitation, which we will see shortly.

3. Flux Quantization: Macroscopic Quantum Mechanics

The final hallmark is perhaps the most direct evidence of the quantum mechanical nature of superconductivity on a macroscopic scale. If a superconductor is formed into a ring or any shape with a hole, any magnetic flux ($\Phi$) passing through that hole is quantized.

The flux is not just quantized; it's quantized in units of ${\varphi }_0=h/2e$.

The appearance of Planck's constant, $h$, confirms this is a quantum effect. More strikingly, the charge 2e in the denominator was a profound early clue that the charge carriers responsible for superconductivity are not single electrons, but bound pairs of electrons. These Cooper pairs are the foundation of the microscopic theory of superconductivity (BCS theory), which we will explore later in this module.

The quantization condition arises from the requirement that the macroscopic quantum wavefunction describing all the Cooper pairs must be single-valued. As one traverses a path around the hole within the superconductor, the phase of the wavefunction must return to its original value (modulo $2\pi$), which constrains the enclosed magnetic flux to be an integer multiple of ${\varphi }_0$.

4. A Glimpse Ahead: Type I and Type II Superconductors

The phenomena we've discussed are universal to all superconductors. However, their behavior in a magnetic field allows for a crucial classification into two types. Your experience with experiments in high magnetic fields makes this distinction particularly relevant.

• Type I Superconductors: These exhibit a complete Meissner effect up to a critical magnetic field $H_c(T)$, above which superconductivity is abruptly destroyed and the material becomes normal. Most pure elemental superconductors (e.g., Lead, Tin, Mercury) are Type I.

• Type II Superconductors: These materials have two critical fields, $H_{c1}(T)$ and $H_{c2}(T)$.

  • For $H<H_{c1}$, they behave like Type I superconductors, with a complete Meissner effect.
  • For $H_{c1}<H<H_{c2}$, they enter a mixed state or vortex state. The magnetic field penetrates the material in the form of quantized flux tubes (vortices), each carrying a single flux quantum ${\varphi }_0$. The regions outside the vortex cores remain superconducting.
  • For $H>H_{c2}$, the material becomes fully normal.
  • Most alloys and all high-temperature superconductors are Type II.

This ability of Type II superconductors to remain superconducting in the presence of high magnetic fields is what makes them useful for applications like MRI machines and particle accelerators. It also explains the stable magnetic levitation seen in demonstrations.

Conclusion

In this lesson, we have laid the experimental and phenomenological groundwork for our study of superconductivity. We've seen that it's not just a state of zero resistance but a distinct thermodynamic phase of matter with unique properties.

Key Takeaways:

• Zero Resistance: The DC electrical resistance vanishes completely below $T_c$. This is best demonstrated by persistent currents in superconducting rings, which flow without any observable decay.
• Meissner Effect: A superconductor actively expels magnetic fields from its interior, behaving as a perfect diamagnet ($\mathbf{B}=0$). This is the key property that distinguishes it from a simple "perfect conductor."
• Flux Quantization: Magnetic flux trapped in a hole within a superconductor is quantized in units of the superconducting flux quantum, ${\varphi }_0=h/2e$. The factor of 2e provides crucial evidence for electron pairing.
• Type I vs. Type II: Superconductors are classified by their response to magnetic fields. Type II materials allow partial flux penetration in a "mixed state," which enables them to remain superconducting at much higher fields than Type I materials.

Preview of the Next Lesson

We have described what a superconductor does. In the next lesson, we will begin to explore how to model this behavior. We will introduce Ginzburg-Landau theory, a powerful phenomenological framework. We will use it to apply Ginzburg-Landau theory to define the coherence length and penetration depth, and distinguish Type I and Type II superconductors, putting a more quantitative footing on the concepts we introduced today.