In summary, the BCS theory explains how electrons, which are fermions with spin 1/2, can indirectly couple with each other through interactions with the lattice phonon. This results in the formation of a "particle" with spin 1 or 0, which follows Bose-Einstein statistics and exhibits superconductivity. This coupling is visualized as a vibrating lattice site pulling away from its electron and attracting the electron of the neighboring site, effectively bringing the two electrons together. Overall, the BCS theory sheds light on the behavior of fermions and bosons in the presence of lattice vibrations.
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I just don't get it, would anyone care to simplify the BCS theory for me? Please?
electrons are fermions (spin 1/2), they don't like to be in the same state as each other. Boson (spin 1,2...) on the other hand, like to be with each other. The whole point of BCS is that electrons couple with each other indirectly through coupling with the lattice phonon (vibration of lattice). What you have are (effectively) two electron trying to "attract" each other even though they have like charges. Since each has 1/2 spin, the coupled particle has spin 1 (or 0). Now the new e-e "particle" obeys Bose-Einstein statistic, and flow like an ideal fluid, hence superconductivity.
I'd like to imagine this coupling as following: When you don't have much lattice kinetic energy (hence low temp), a vibrating lattice site (positive ion) pulls away from its electron, forming a dipole, and the positive end of this dipole attracts the electron of the next side, which effectively couple the two electrons together.