Phonons behave like Bosons, so they have a Bose Einstein distribution. If you take the zero temperature limit, the distribution vanishes. So at zero temperature there are basically no phonons (up to quantum fluctuations?). But to build a Cooper Pair in the BCS Theory you need phonons, the electrons get attracted to each other by phonon interaction. So for me, there cant be any cooper pairs at zero T although the Cooper Pair distribution in the BCS groundstate looking almost like a Fermi distribution, says its full of them? Also what escapes my understanding is, that this distribution says that most cooper pair states with small electron energys are occupied, although we assume (or even showed) that only electrons around the Fermi surface can form Cooper pairs? How to resolve my paradoxon? Thanks in advance for answers!
The electrons create their own virtual phonons. As an electron moves through the lattice of positive ions, the ions are atrracted to the electron and move slighly towards it. The electron is moving quite fast (at the Fermi velocity) and the ions are quite heavy, so it takes them a while (on the order of the inverse of the Debye frequency) to relax back to their original positions. Each electron therefore leaves behind it a trail of slightly positive charge that can attract another electron.
This is a classical picture of course, but we describe it in terms the exchange "virtual phonons" just as we describe the classical electrostatic force as being due to exchange of virtual photons. Mathematically, the Feyman propgator of the off-shell virtual phonon or photon is essentially the expression for the attractive potential.