Electron-phonon driven superconductivity: do we really have a microscopic understanding of it?

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100 years after the discovery of superconductivity in elemental Mercury, we have innumerable families of superconducting materials, each of which seems to have little in common with one another. The phenomenon of superconductivity is all about the movement of charge, and even our understanding of the mechanism for this remarkable state of matter, at least for so-called conventional superconductors, is about pairing through charge fluctuations. In contrast, for newer superconductors, many of the current ideas about the mechanism involved center around `spin fluctuations' as the `pairing glue'. But to understand how electrons can both carry the current and provide the pairing mechanism, we propose to go back and gain a deeper understanding of how electrons pair in the conventional mechanism. The historical theory (Eliashberg) explains what happens to the electrons, but never really explains simultaneously what happens to the phonons, as they interact with the electrons. In this sense it remains a phenomenology, and we propose to provide a truly microscopic description of the superconducting state, through advanced theoretical techniques. In this way, not only will we gain insights into how the conventional superconductor works, but we should be better equipped to deal with the more difficult problem of understanding an unconventional superconductor where the carriers and the pairing glue appear to originate from the same electrons. The goal is that a deeper understanding of the mechanism of superconductivity will aid material scientists in the search for higher temperature superconductivity.