In the lead up to the full formulation of BCS theory, the derivation of Bardeen-Pines interaction played a prominent role. The Bardeen-Pines interaction demonstrated that a net attractive interaction between electrons in an electron gas/liquid can result in the presence of phonons.
The way that Bardeen and Pines derived this result can be understood by reading this paper. The result is actually quite simple to derive using a random-phase-like approximation or second-order perturbation theory. Regardless, the important result from this paper is that the effective interaction between two electrons is given by:
The crucial aspect of this equation is that for frequencies less than the phonon frequency (i.e. for ), the effective interaction becomes negative (i.e. attractive).
It was also shown by Kohn and Luttinger in 1965 that, in principle, one could also obtain superconductivity in the absence of phonons. The attraction would occur using the phenomenon of Friedel oscillations whereby the effective potential can also become negative. This was quite a remarkable result: it showed that a purely electronic form of superconductivity was indeed theoretically possible.
What makes the effective interaction become attractive in these two models? In the Bardeen-Pines case, the phonons screen the electrons leading to a net attraction, while in the Kohn-Luttinger case, Fermi surface effects can again lead to a net attraction. It is important to note that in both papers, the pre-eminent quantity calculated was the dielectric function.
This is because the effective potential, , is equal to the following:
In the aforementioned cases, net attraction resulted when .
This raises an interesting question: is it possible to still form Cooper pairs even when ? It is possible that this question has been asked and answered in the literature previously, unbeknownst to me. I do think it is an important point to try to address especially in the context of high temperature superconductivity.
I welcome comments regarding this question.
Update: In light of my previous post about spin fluctuations, it seems like is not a necessary condition to form Cooper pairs. In the s-wave channel, it seems like, barring some pathology, that would have to be less than , but in the d-wave case, this need not be so. I just hadn’t put two and two together when initially drafting this post.