# BCS theory of superconductivity: Cooper pairs

The microscopic theory put forward by Bardeen. Cooper and Schrieffer or BCS in 1957, provides the better quantum explanation of superconductivity and account very well for all the properties exhibited by the superconductors. This theory based on the electron phonon-electron interactions, Cooper pairs, the existence of energy gap, Coherence length, and flux quantization.

## 1) electron-phonon-electron interaction:

Suppose an electron approaches a positive ion core. It suffers from an attractive Coulomb interaction. Due to this attraction, the ion core is set in motion and consequently distorts the lattice. Smaller the mass of positive ion core, greater will be the distortion. Suppose towards that side another electron comes and sees this distorted lattice. Then the interaction between the two the electron and the destroyed lattice occurs. Which in its effect lowers the energy of the second electron.

Thus we interpret that the electron interacts via the lattice distortion. Or the phonon field resulting in the lowering of energy for the electrons.The lowering of electron energy implies that the force between the two electrons attractive. Thus the type of is called electron-phonon-electron interaction. This interaction is strongest when the two electrons have equal and opposite momenta and Spins.

### Additional theory

Since the oscillatory distortion of the lattice, as pointed out earlier, is quantized in terms of phonons and therefore above interaction can also be interpreted as the electron-electron interaction through phonons the mediator suppose, as shown in fig. An electron of wave vector K emits a virtual phonon q which is absorbed by an electron K’. K is thus scattered as (K-q) and K’ as K’+q. The process being a virtual one, energy needs out conserved (the phonons involved are called virtual phonons because, as a consequence of uncertainty principle. And their very short lifetime renders it unnecessary to conserve the energy in the process). In fact, the nature of the resulting electron interaction depends on the relative magnitude of the electronic energy change and the phonon energy ~~h~~ωq.

If this phonon energy exceeds electronic energy, the interaction is attractive. Hence the fundamental postulate of BCS theory is that superconductivity when occurs such an attractive interaction between two electrons, by means of phonon exchange, dominates the usual repulsive Coulomb interaction.

#### 2) Cooper pair:

The fundamental postulate of BCS theory is that the superconductivity occurs when such an attractive interaction, mentioned above, between two electrons, by means of a phonon exchange, dominate the usual repulsive Coulomb interaction. Two such electrons which interact attractively in the phonon field are called a cooper-pair. These cooper pairs have certain aspects of single particles.

The energy of the pair of electrons in a bound state is less than the energy of the in the free state (electron separated). The difference of the energy of the two states is the binding energy of the cooper pair and should, therefore, be supplied if the pair is broken. At temperatures less than the critical temperature, therefore, be supplied if the pair is broken. At the temperature less than the critical temperature, electron-phonon-electron interaction is stronger than electron-electron Coulomb interaction, and so the valence electrons tend to pair up. Pairing is complete at T=0K and is completely broken at the critical temperature.

##### 3) The existence of energy gap:

The energy difference between the free state of the electron (i.e. energy of individual electron-a case of normal state) and the paired state (the energy of paired electron-a case of superconducting state) appears as the energy gap at the Fermi surface. The normal electron states are above the energy gap. And superconducting electron states are above the energy gap and superconducting electron states are below the energy gap at the Fermi surface. The energy gap is a function of temperature unlike to case of the constant energy gap in semiconductors and insulaters.

Since pairing is complete at 0K, the difference in energy of free and paired electron states (i.e normal and superconducting electron states) is maximum or in other words energy gap is maximum at absolute zero. At T=Tc, pairing is dissolved and the energy gap reduces to zero. Across the energy gap, there are many excited states for the superconducting Cooper pairs. BCS theory thus predicts many electron ground states, as well as excited for the superconductor in the range 0 to Tc and in these states, coopers pairs are supposed to be in the condensed state with a definite phase coherence. At critical temperature. This coherence disappears and the pairs are broken resulting in the transition of the superconducting state to a normal state.

###### 4) Coherence length:

(Cooper pairs) i.e. the paired electrons are not scattered. This is because of their peculiar property of smoothly riding over the lattice imperfections without even exchanging energy with them. Consequently, they can maintain their coupled motion up to a certain distance called coherence length. The latter is found to be of the order of 10-⁶ m.

###### 5) Main achievements of BCS theory:

(a) Critical temperature Tc = 1.14 ~~h~~(ωD exp(1/N(0)V))/KB

where ~~h~~ω is phonon energy, N(0) is the density of electronic levels for the occupied of single spin. And V is the parameter of the electron-phonon-electron interaction of BCS theory.

(b) Energy gap Eg= 2Δ(0)= 4 ~~h~~ωD exp(1/N(0)V)

(c) Energy gap Eg= 2Δ(0)= 3.5 KB Tc