# BCS theory energy gap

I am studying BCS theory of superconductivity and have understood everything except this part: According to HyperPhysics "The electron pairs have a slightly lower energy and leave an energy gap above them". What does this mean exactly?Help appreciated

• That the pairing has a finite (albeit) small energy difference, leaving a small band gap between the paired state and the unpaired states. There is not a continuum of states from unpaired to paired. Feb 18 at 19:57

Great question! Because there is an attractive interaction between the electrons that form Cooper pairs, their energy is lowered relative to the Fermi Level. Recall that the Fermi Level is the energy of those electrons that sit at the top of the conduction band at T = 0K.

The attractive energy between the Cooper pairs is very small. For example, for aluminum it is roughly 3.4 x $$10^{-4}$$eV. This means, though, that these electrons have an energy below the Fermi Level. Since Cooper pairs are formed from those electrons that sit at the Fermi Level, there is a gap formed at the Fermi Level. This gap is temperature dependent. It grows larger as the temperature is lowered: $$\Delta(T)=3.2kT_c(1-\frac{T}{T_c})^{0.5}$$ where $$T_c$$ is the superconducting transition temperature and k is Boltzmann's constant.

Therefore, compared to a normal conductor, where there is always an empty state an infinitesimal amount of energy above those electrons at the Fermi Level, that is not true in a superconductor. The empty states are separated from the bound states (and the single particle states deeper in the Fermi sphere) by an energy gap equal to the binding energy of the Cooper pairs.

You can think of this like the gap in a semiconductor, only it is much smaller.

• So if I understand correctly the energy of the electrons of the Cooper pair are below the conduction band which in turn doesn't make the electrons bump to the positive ions of the lattice.
– user288883
Feb 24 at 19:35
• Hi @user288883 . No, the conduction band in a metal is usually a few eV wide, while the gap is of the order $10^{-4}$eV. So the gap energy is much smaller. The size of the gap does not have anything to do with the zero resistance of the superconductor. That is related to the fact that all of the Cooper pairs are in the exact same state. So once a current gets flowing to slow down one Cooper pair means you have to slow them ALL down. That is not possible by the usual mechanisms of resistance, which are lattice vibrations or scattering off of impurities.
– CGS
Feb 28 at 12:29