BCS Theory

BCS theory explains the occurrence of superconductivity quantum mechanically through the interaction of electrons and phonons [14][15]. One electron pulls in nearby positive ions, generating a local excess positive charge which then draws a second electron towards the first. This attraction leads to the formation of Cooper pairs. These are bound states composed of two electrons whose momenta each exceed the Fermi momentum p_F while their combined energy, both potential and kinetic, has fallen to less than 2E_F. Although these two electrons continually scatter each other to new individual momenta, the total momentum of each Cooper pair is unchanging and identical. The net energy of a Cooper pair is lowest when it possesses zero momentum and the electrons comprising it have opposite spins. The BCS cutoff stipulates that the attraction needed to create Cooper pairs transpires only between electrons within a Debye energy $k_B\Theta_D$ of the Fermi level E_F. For this reason the number of scattering processes allowed to the electrons of a Cooper pair, and consequently the amount by which their total energy decreases, is sharply maximal when their centre of mass is stationary. The existence of more Cooper pairs leaves fewer momentum states available for scattering into, diminishing the negative potential energy associated with the attractive interaction. The BCS ground state contains as many Cooper pairs as can form with a negative potential energy of greater magnitude than the requisite kinetic energy increment. This achieves the lowest possible total energy of all the electrons. Exciting a superconductor to higher states necessitates one or more Cooper pairs breaking up.

The dissociation of a Cooper pair yields two quasiparticles [14]. These are electrons no longer restrained to occupy states of equal and opposite momenta. BCS theory proposes that splitting up a Cooper pair needs at least an energy $2\Delta$. This added energy supplies the binding energy of the Cooper pair and lifts the total energy of all the electrons. Consequently no quasiparticle energy levels exist within a BCS energy gap $\Delta$ of the Fermi energy E_F. The states absent from this energy region reside at its upper bound $E_F + \Delta$, creating there a peak in the density of states N(E) [16]. As the temperature Trises, the BCS energy gap $\Delta(T)$ shrinks in all k-space directions [17] and thermally excited quasiparticles become more numerous.

The BCS energy gap $\Delta(T)$ is an important superconducting parameter. A spherically symmetric gap $\Delta$ is termed s-wave pairing, while one with the symmetry of the crystal is known as anisotropic s-wave pairing. Unconventional pairing results from interactions other than that between electrons and phonons. This produces gap $\Delta$ symmetries lower than that of the crystal [16], for example an energy gap $\Delta(T)$ with nodal lines. Through the relation [18]

$$\xi_{BCS} = \frac{\hbar v_F}{\pi\Delta (0)}$$ (4.9)

where v_F is the Fermi velocity, the BCS energy gap $\Delta$ provides an estimate of the scale $\xi_{BCS}$ of the spatial correlation of the superelectrons. This characteristic range $\xi_{BCS}$ is called the BCS coherence length. The BCS energy gap $\Delta$ turns out to be directly proportional to the order parameter $\psi$ of the Ginzburg-Landau theory.