Explain the formation of Cooper pair in a superconductor. Obtain an expression for the energy gap in a superconductor at absolute zero temperature.

In the quantum world of superconductors, electrons usually repel each other due to their negative charges. But in superconductors, there's a fascinating phenomenon where these electrons actually attract each other! How does this happen? The answer lies in 'phonon mediation'.
Phonons are quantized vibrations of atoms in the crystal lattice of a material. As an electron moves through the lattice, it distorts the nearby positive ions, creating an area of positive charge around it. This distortion essentially 'echoes' through the lattice and attracts another electron.
Here’s the key point: even though the electrons never come close to each other directly, the interaction through the lattice creates an attractive force between them. This attraction is strong enough to overcome their natural repulsion at low temperatures, leading to the formation of Cooper pairs.

One of the defining features of a superconductor is the 'energy gap'. This gap is an energy difference that exists at absolute zero temperature and it plays a crucial role in superconductivity.
Imagine the energy levels in a superconductor like rungs on a ladder. At absolute zero, all the electrons in the superconductor are snugly paired up and sitting on the lowest rung (ground state). The energy gap represents the minimum energy required to break one of these Cooper pairs apart.
Breaking a Cooper pair promotes the electrons to higher energy states, disrupting the superconducting state. According to BCS Theory, the energy gap \( \Delta(0) \) at absolute zero is given by the formula:
\[\Delta(0) = 1.76 k_B T_c \]
Here, \( k_B \) is the Boltzmann constant, and \( T_c \) is the critical temperature below which the material becomes superconducting. This energy gap ensures that at low temperatures, thermal excitations aren’t strong enough to break the Cooper pairs, maintaining superconductivity.

To fully understand the marvel of superconductivity, we must dive into the BCS Theory, named after John Bardeen, Leon Cooper, and Robert Schrieffer who proposed it in 1957. This theory explains how and why superconductivity occurs in most materials.
According to BCS Theory, superconductivity is all about Cooper pairs. When phonon-mediated interactions lead to the formation of these pairs, they occupy a collective quantum state that spans the whole crystal. These pairs can move through the superconductor without any resistance.
In this paired state, even if an electron encounters an impurity or other normal scattering event, the entire Cooper pair moves smoothly, ensuring there's no loss of energy. The collective behavior of Cooper pairs under BCS Theory is what makes superconductors so unique, allowing them to conduct electric current without resistance and expel magnetic fields, a property known as the Meissner effect.
The BCS Theory doesn't just explain superconductivity; it also provides the crucial formula for the energy gap \( \Delta(0) \), solidifying our understanding of why and how superconductors function.