# The BCS Model of Superconductivity: The Origin of Cooper Pairs.

The Bardeen-Cooper-Schrieffer (BCS) model of superconductivity was the first microscopic theory explaining the origin of superconducting states in conventional superconductors. This model is presented in brief below.

Famously, the first superconducting state was discovered experimentally in the early 1900s; but
a microscopic theory explaining such phenomena eluded researchers until the late 1950s, when Bardeen-Cooper-Schrieffer published
this article on the *Theory of Superconductivity*. Preceding the BCS model, phenomenological
models (that is, macroscopic models developed from trends in data) were developed, namely the London equations, and
later, the Landau-Ginzburg free energy.

The intuition for the model is that as electrons travel through the underlying lattice, at low enough temperatures, the ions behind the moving electrons form squeezed channels, that result in a higher local concentration of positive charge, attracting other electrons. This happens such that the electrons tend to form 'Cooper pairs' that condensate below a certain temperature and then contribute to the superconducting state.

#### History of Superconductors

In 1911, Heike Kamerlingh Onnes discovered the superconducting state of super-cooled Mercury at a
temperature of 4.19 Kelvin. He published this finding in a paper of the same year titled *The Superconductivity of Mercury*.

Largely for this work, he recieved the 1913 Nobel prize in physics "for his investigations on the properties of matter at low temperatures which led, inter alia, to the production of liquid helium".

### What we're looking for...

Before showing the results, it's important to give some intuition for the model considered. And, atleast two concepts should be explicitly considered
to motivate what follows: the resistivity of superconductors drops to zero discontinuously at a certain temperature (as opposed to some smooth
transition), which hints at a *phase transition* that should be characterizable by some order parameter; and, furthermore, the behaviour of these states
depends on the underlying ionic lattice (shown experimentally).

The order parameter we're looking for turns out to be a so-called *gap function* which corresponds to a certain singularity in the two-particle
correlation function. And the lattice dependence points towards a phonon-mediated effect on the electrons.

## BCS Hamiltonian

The BCS Hamiltonian essentially gives a mechanism by which electrons may pair and form a new set of quasi-particles that experience no resistance but still may transport charge (and more fundamentally, screen out entirely any external magnetic field).

Below, we start with an *effective* electron Hamiltonian, termed as such
since it's additional term arises from an underlying electron-phonon interaction (which we neglect to treat in detail here). The derivation of a similar effective Hamiltonian,
as an averaging over phonon modes can be seen in the second section of this article (as well as the derivation of a
result for the correlation function's form, which we will use later).

The effective BCS Hamiltonian (for electrons) can be given in the second-quantized form:

The second term above, shown again below, actually encourages electrons to 'pair up'.

#### 4-Point Correlation Function & The Cooper Instability

The 4-point or 2-particle correlation function \(C(\vec{q},\tau)\) describes the scattering amplitude of pairs of electrons with related momenta.

### Mean-Field Transition

This phase transition hints at a new normal or 'mean-field' behaviour in the new regime (that is, below the critical temperature). To explore this transition, we
define the *order parameters* (we will not derive them here) below:

In our new regime, we now expect fluctuations about the order parameter to be small. That is, we make the substitution:

*Bogoliubov de-Gennes (BDG) Hamiltonian*. The benefit of this form is that now the Hamiltonian is again quadratic in operators, which guarantees we'll be able to diagonalize it in some basis to extract, analytically, information about it's behaviour.

Also note that this effective mean-field Hamiltonian is Hermitian (necessarily), but doesn't conserve the number of particles!

#### Nambu Spinors & The Bogoliubov Transformation

We may package the annihilation and creation operators of the up and down spin states of particles into a spinor \(\Psi_k\), termed a Nambu Spinor, which is defined as below.

*Bogoliubov Transformation*, of the following form:

We now rewrite the BDG Hamiltonian in terms of Nambu Spinors, below. And rewrite the

*Bogoliubons*or Bogoliubov quasi-particles.

## Further Resources

The approaches taken here are really just to give a rough heuristic for the behaviour of typical superconductors. A more refined treatement may utilize the functional field integral approach.

Much of the discussion here is covered (arguably better, and in more detail) in Altland and Simmons *Condensed Matter Field
Theory* in section 6.5; as well as in this set of
lecture notes.