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

Cooper pair heuristic diagram.

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:

\[ \hat{H}_{\text{BCS}} = \sum_{k, \ \sigma}\xi_k \hat{a}^{\dagger}_{k\sigma}\hat{a}_{k\sigma} - \frac{g}{V}\sum_{k,k',q}\hat{a}^{\dagger}_{k+q, \uparrow}\hat{a}^{\dagger}_{-k,\downarrow}\hat{a}_{-k'+q,\downarrow} \hat{a}_{k', \uparrow} \]
where the summation is over all wave vectors \(k\) in the Brillouin zone, and over all spin states \(\sigma\) (up or down along some axis); \(\epsilon_k\) is then energy at some wave vector \(k\); and \(V\) is the volume of the system. The coupling constant \(g\), is assumed to be positive (for the case with a negative sign in front) and only to be non-zero in a width above of below the Debye frequency \(\omega_{D}\) from the Fermi energy \(\epsilon_F\).

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

\[ \underbrace{- \frac{g}{V}\sum_{k,k',q}\hat{a}^{\dagger}_{k+q, \uparrow}\hat{a}^{\dagger}_{-k,\downarrow}\hat{a}_{-k'+q,\downarrow} \hat{a}_{k', \uparrow}}_{\text{Encourages electron 'pairing'}} \]
This can be shown explicity by considering the behaviour of the two-particle correlation function as temperature is varied.

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.

\[ C(\vec{q},\tau) =\frac{1}{V^2}\sum_{k,k'}\langle \bar{\psi}_{k+q,\uparrow}(\tau) \bar{\psi}_{-k,\downarrow}(\tau) \psi_{-k'+q,\downarrow}(\tau) \psi_{k',\uparrow}(0)\rangle \]
It's form can be shown (via the Bethe-Salpeter equation) to depend on a vertex function \(\Gamma\):
\[ \Gamma_q = \frac{g}{1-\frac{gk_B T}{V}\sum_p G_{p+q}G_{-p}} \]
Making a few assumptions (and applying the expanded correlation function derived in the first section of this article) this can be shown to have the approximate form:
\[ \Gamma_{(0,0)} = \frac{g}{1-g\nu \ln (\frac{\omega_D}{T})} \]
which has singular behaviour at the critical temperature \(T_c = \omega_D e^{-\frac{1}{g\nu}}\). Due to the vertex's relation to the two-particle correlation function, this also implies a singularity in it's behaviour as well; indicating a phase transition of some sort, in which 'Cooper pairs' have a relatively strong bond or tendency to condensate.

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:

\begin{align} \Delta &= \frac{g}{V}\sum_k \langle \Omega_s\vert \hat{a}_{-k,\downarrow} \hat{a}_{k,\uparrow}\vert\Omega_s\rangle\\ \\ \bar{\Delta} &= \frac{g}{V}\sum_k \langle \Omega_s\vert \hat{a}_{k,\uparrow}^{\dagger} \hat{a}_{-k,\downarrow}^{\dagger}\vert\Omega_s\rangle\\ \end{align}
where \(\vert\Omega_s\rangle\) is introduced as the new ground state (of the new regime). As the order parameters of the transition, these functions are only non-zero in the range \(T < T_c \).

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

\[ \sum_k \hat{c}_{-k,\downarrow}\hat{c}_{k,\uparrow} = \frac{V}{q}\Delta +(\underbrace{\sum_k \hat{c}_{-k,\downarrow}\hat{c}_{k,\uparrow} - \frac{V}{q}\Delta}_{=\delta}) \]
and assuming \(\delta\) to be small enough that we may ignore it's second order terms, we may expand as below to yield our new mean-field Hamiltonian:
\begin{align} \hat{H}_{\text{BCS}}&= \sum_{k,\sigma}\Big[\epsilon_{k} \hat{a}^{\dagger}_{k\sigma}\hat{a}_{k\sigma}-\big(\bar{\Delta}\hat{a}_{-k,\downarrow}\hat{a}_{k,\uparrow}+\Delta\hat{a}^{\dagger}_{k,\uparrow}\hat{a}^{\dagger}_{-k,\downarrow}+\frac{v}{g}\vert\Delta\vert^2+\mathcal{O}(\delta^2)\big)\Big]\\ \hat{H}_{\text{BDG}}=\big(\hat{H}_{\text{BCS}}\big)_{\text{MF}} & =\sum_{k,\sigma}\Big[\epsilon_{k} \hat{a}^{\dagger}_{k\sigma}\hat{a}_{k\sigma}-\big(\bar{\Delta}\hat{a}_{-k,\downarrow}\hat{a}_{k,\uparrow}+\Delta\hat{a}^{\dagger}_{k,\uparrow}\hat{a}^{\dagger}_{-k,\downarrow}+\frac{v}{g}\vert\Delta\vert^2\big)\Big] \end{align}
this is sometimes termed the 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.

\[ \Psi_k = \begin{bmatrix} \hat{a}_{k,\uparrow} \\ \hat{a}^{\dagger}_{-k,\downarrow}\end{bmatrix} \\ \]
Now, this spinor may be used to transform to another basis of creation and annihilation operators via any unitary transformation. This allows us to perform a so-called Bogoliubov Transformation, of the following form:
\[ \Phi_k = U \Psi_k \]
Where we now have a new set of ladder operators that are linear combinations of the previous set.

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

\[ \hat{H}_{\text{BDG}}-\mu\hat{N} = \sum_k \Psi^{\dagger}_k\begin{bmatrix} \xi_k & -\Delta \\ -\bar{\Delta} & -\xi_{-k}\end{bmatrix}\Psi_k + \sum_k \xi_k +\frac{V}{g}\vert \Delta \vert^2 \]
where \(\xi_{k} = \epsilon_k - \mu\) is defined as the difference between the energy \(\epsilon_k\) at some wave vector \(k\) and the chemical potential \(\mu\). We now perform the Bogoliubov transformation below, to diagonalize this Hamiltonian.
\[ \begin{bmatrix} \hat{c}_{k,\uparrow} \\ \hat{c}^{\dagger}_{-k,\downarrow}\end{bmatrix} = \begin{bmatrix} \cos\theta_k & \sin\theta_k\\ \sin \theta_k &-\cos\theta_k\end{bmatrix} \begin{bmatrix} \hat{a}_{k,\uparrow} \\ \hat{a}^{\dagger}_{-k,\downarrow}\end{bmatrix} \\ \] \[ \Rightarrow \ \ \ \hat{H}_{\text{BDG}}-\mu\hat{N} = \sum_{k,\sigma}(\pm\lambda_k)\hat{c}^{\dagger}_{k\sigma}\hat{c}_{k\sigma}+\sum_k(\xi_k-\lambda_k)+\frac{\Delta^2 V}{g} \]
where we now can read off the dispersion relation as \(\lambda_k\), defined as below:
\[ \lambda_k = \sqrt{\Delta^2+\xi_k^2} \]
Note that particle number here is conserved again, but now the Hamiltonian is written in terms of creation and annihilation operators for different particles entirely (linear combinations of operators for electrons and holes), these new 'particles' are termed 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.