Condensed-Matter Physics & Materials Science Seminar

This talk deals with the calculation of the energy gap (symptomatic of Coper pairing) and of the critical temperature Tc within the BCS theory of superconductivity.



We start with a review. We recall that when the pair scattering potential (in k-space) is constant and attractive (negative) the BCS integral equations yield a constant "energy gap". Tc is obtained by setting this gap=0. But when the scattering potential is "structured", i.e. when it depends on k,k' in a non-trivial way,

the energy gap becomes the "gap function" and becomes a nontrivial function of the momentum and energy. Such a gap function can be constant on a surface of constant energy ("s-wave" gap) or it can have two or more nodes or nodal lines on such a surface ("p-waves", "d-waves" or higher.) The symmetry of the solution is obtained trivially for any "separable" potential function of the form V(k,k')=gv(k)v(k'), as it is determined by v(k).



Then we show a peculiar example of a nonseparable potential V that is nowhere repulsive (i.e. V ≤ 0 for all k,k') for which the BCS equations are exactly solvable. The gap function is s-wave. But for this particular potential there is indeed a surprise: its negative, –V , which is nowhere attractive, also exhibits an s-wave gap function! This establishes that it is not the sign of the potential that determines the pairing but its structure. Purely repulsive potentials (in k-space) can cause pairing!



We then turn to the "screened" potentials, in which the interaction of the two electrons is mediated by the rest of the electron fluid. If the dielectric function epsilon(q,w) has zeroes the effective screened potential will correspondingly be negatively or positively infinite near the zeroes. Nevertheless, as we shall show,Tc remains finite because of the sharply reduced lifetimes of the quasi-particles at the resonances. This also affects the "isotope effect" rendering it nontrivial. To analyze this quant