This theory, developed by Bardeen, Cooper and Schrieffer, states that electrons experience an attractive interaction through the lattice that overcomes their normal repulsive interaction, forming Cooper pairs. At low temperatures, these pairs move without resistance through the lattice, causing the material to become a superconductor. The electron-lattice-electron interaction must be stronger than the direct electron-electron interaction for superconductivity to occur.

1. BCS THEORY :
This theory was developed by Bardeen, Cooper and Schrieffer.
This theory states that the electron experience special kind of attraction
Interaction, overcoming the coulomb forces off repulsion between them, as a
result Cooper pairs.
At low temperature, these pairs move without scattering i.e. without
any resistance through the lattice points and the material become
superconductor.
Here the electron – lattice – electron interaction should be stronger than
electron – electron interaction.
Electron –Lattice –Electron Interaction :
When an electron (1st) moves through the lattice, it will be attracted by
the core (+ ve charge) of the lattice. Due to this attraction, ion core is disturbed
and it is called as lattice distortion. The lattice vibrations are quantized in terms
of phonons.
The deformation product a region of increased positive charge. Thus if
another electron (2nd) moves through this region as shown in fig. It will
be attracted by the greater concentration of positive charge and hence the
energy of the 2nd electron is lowered.
Hence the two electrons interact through the lattice (or) the
phonons field resulting in lowering of energy of the electron. This lowering of
energy implies that The force between the two electrons are attractive. This
type of interaction is called Electrons – lattice – electron interaction. The
interaction is strong only when the two electrons have equal and opposite
momentum and spins.
The Nobel Prize in Physics 1972was awardedjointly to John Bardeen, Leon Neil Cooper and
John Robert Schrieffer"for their jointly developedtheory of superconductivity, usually called
the BCS-theory".

2. Explanation
Consider the 1st electron with wave vector k distorts the lattice,
there by emitting a phonons of wave vector q. This result in the wave
vector k - q for the 1st electron. Now if the 2nd electron with wave
vector k’, seeks the lattice, it takes Up the energy from the lattice and its
wave vector changes to k + q as shown in fig. Two electrons with wave
vector k – q and k’ + q from a pair of electrons known as Cooper pairs.
Cooper pairs
The pairs of electrons formed due to electron - lattice – electron
(phonons), electron - electron interaction (forces of attraction) by overcoming
the electron – electron interaction (force of repulsion) with equal and
opposite momentum and spins i.e., with wave vector k - q and k + q are
called cooper pairs.
Coherence length
In the electron - lattice - electron interactions, the electrons will not be
fixed, they move in opposite directions and their co-relations may persist over
lengths of maximum of 10 -6 m. This length is called coherence length.