Superconducting Gap in Superconductors

Question

Please explain "Exponential behavior usually signals the presence of a gap in the energy spectrum" How The presence of a gap explains the exponential decrease of the specific heat and the thermal conductivity below TC.

Answer 1

At low temperature number of excited electrons above the gap will be proportional to $\exp\{-\Delta/kT\}$ where $\Delta$ is the gap. This decreases very rapidly as $T$ becomes small so will be far fewer excitation than without the gap. Consequently the electronic contribution to the specific heat will be $\exp\{-\Delta/kT\}$ smaller. Gapless superconductors, on the other hand, have specific heats that go to zero as $T^a$ for some power $a$ that depends on the dimension (line or point) of the gapless region on the Fermi surface.