# Is there any positive temperature from which superconductivity ceases?

From what I understand about superconductivity, it is due to a coupling between Cooper pairs and phonons. At the absolute 0, there is no phonon, so I assume superconductivity cannot exist at that temperature. But for superconducting materials, is it possible to find a positive temperature below $T_c$ below which superconductivity ceases? If not, why not? If so, was it found for any material yet?

• "At the absolute 0, there is no phonon, so I assume superconductivity cannot exist at that temperature." That is not correct. At absolute zero, there are no thermally activated phonons, but there are still phonons induced by the motion of the electrons themselves. – DanielSank Aug 17 '18 at 1:49

Based on BCS theory, the phonon-mediated interaction is given by $$V(\bf{k},\bf{q})\sim \frac{1}{(\epsilon_{\bf{k}}-\epsilon_{\bf{k}-\bf{q}})^2-(\hbar\omega_{\bf{q}})^2}$$ No matter how small $q$, and hence $\omega_{\bf{q}}$ is, we can always find $\bf{k}$ near the Fermi surface to make the energy difference as small as we like, and in fact the difference can be made strictly 0. As a consequence, the interaction can be made effectively attractive, leading to the Cooper instability.