Thank you, Dr Armando. However, I would appreciate if you may kindly provide me the reference which makes such a statement or makes such a conclusion based on a mathematical analysis, since a number of physicists that I have interacted with strongly believed that these are only the electrons falling around $k=k_F$ make Cooper pairs.

Thanks Dr Armando

I guess in superconducting states, the normal electrons above Fermi level may fall into the bosonic zone transfomed from band gap so that zero resistrance occurs in the view point of band structure scheme as follows,

In a superconductor, all electrons form Cooper pairs (at T=0K), but the energy and occupation probability of electronic states near the Fermi level change remarkably in the vicinity of the Fermi level; for other electronic states, changes in energy and occupation probability do not show very significant changes.

As already mentioned by some of the other contributors, at T=0 ALL electrons will be bound in Cooper pairs, while for 0 < T < Tc Cooper pairs and electrons coexist:

The attached figure shows the quasiparticle ("= electron") density of states below Tc normalized to the density of states in the normal state. Below Tc, an energy gap starts forming, the ground state of a BCS superconductor becomes unstable and electrons of opposite spins start forming Cooper pairs. For sufficiently low temperatures, the thermal energy of the lattice is smaller than this energy gap and no interaction between the Cooper pairs and the lattice can take place. Consequently, the condensate becomes a superfluid and all traces of electric resistance vanish.

@Armando Gama Goicochea:

"However, all current is carried by those forming pairs, therefore the state is superconducting at any temperature below Tc."

In general, quasiparticles definitely also carry charge (and spin and energy) current above T=0. This comes in addition to the supercurrent carried by the Cooper pairs. The quasiparticle current is transformed into supercurrent for quasiparticles with energies inside the gap. This is called "gap scattering" and happens on the scale of the superconducting correlation length.

Figure taken from my PhD thesis (http://ntnu.diva-portal.org/smash/record.jsf?pid=diva2:621656)

Electrons can be paired because they alter their direction or velocity after running into atoms (electron-phonon interaction).  This does not happen if the electrons are deep inside the Fermi sea (Pauli principle).

Thanks a lot Dr. Zheng for your valued answer to my question that I asked some time in October 2013.

Over this long period I had been working hard to get answers to several of my questions on BCS theory and related aspects. I also critically examined the basic foundations of BCS theory and and the basic ideas that I used for developing the simple theory of interacting fermions which not only explains superconductivity and

related aspects but also explains superfluidity of systems like liquid $^3$He. You may find that my theory not only explains the observation of superconductivity at high temperatures but also explains all phenomena (listed by Hirsch Physica Scripta, Vol. 80 (2009), 035702 (11pp)) that BCS theory does not explain even for metallic supercondutors.

However, all research groups working in the field (excluding the group of Hirsch) of superconductivity considers BCS theory as the basic model to analyse their experiments on classical superconductors and to develop the theory of HTS for the

reasons known to them.  They totally ignored my theory because it seems to have a  conflict with their interests, -not for a scientific reason. This is matter of chance that, in whatever way it concludes, the basic conclusions of BCS theory that electrons form (q,-q) bound pairs and there is an energy gap between superconducting and normal states of the conductor are right. However, it is not correct that  superconductivity arises only due to electron-phonon interaction pairs are formed they way it concludes. In this context,  I would simply state that as established in the paper (attached hear), it should be clear, beyond any doubt, that the basic

premises of BCS theory have errors.   Naturally, no effort that uses BCS idea as its foundation would render a viable theory of the phenomenon.  Interestingly, this is evident from the fact that no theory of superconductivity could be developed over the last three decades after the discovery of HTS in 1986.

For the benefit of other interested readers I would like to add what my theory concludes about the origin of superconductivity.

Superconductivity (or superfluidity of a system like liquid $^3$He) is basically a property of the ground state ($T=0$ state) of a system of interacting fermions (SIF). The state remains stable even at non zero $T < T_c$ just because of energy gap

between the normal and super states of every such SIF.

Fermions in this state get organised in position space as well as in phase space. In the position space they assume localised positions with an arrangement where they have identically equal nearest neighbour separation. Further, because of its localisation, the state of each electron is represented by the superposition of two plane waves of equal and opposite momenta, the state is identified with a state of a pair of electrons moving with equal and opposite momenta (q,-q).

In the phase space their positions are separated by $\Delta\phi = 2n\pi$ phase difference which ensures their coherent motion.

They cease to have any relative motion; they can move only in the order of their positions. Naturally, they do not have collisions and as a result of this they do not find any resistance for their flow through narrow channels in the lattice.

As such each electron in the superconducting state can be identified as a part or a representative of a bound pair of electrons which move with (q,-q) momenta with respect to the CM of the pair, while the CM can move with an momentum K. While q is restricted to have only $\pi/d$ momentum for the localisation of electrons in the

channel of diameter $d$ through which it moves, K can have any value between o and $K_F$ (Fermi momentum). Interestingly, momentum K gets detached from the electron due to its localisation and represents a gas of non-interacting fermions. We conclude that gap is contributed by all interactions between two electrons and all possible interactions between electron and lattice.

This picture is concluded by using first quantisation approach without using any major approximation. Consequently, it has potential to explain every aspect of superconductivity. On RESEARCH GATE, I have placed several articles which explain these aspects separately.

I invite intersted researchers to have their critical reading and help in improving the quality of their discussion. I assure every body who extends his help that your help would be duly acknowledged. I would be agreeable even to write joint papers.

The main difference between pairs on the Fermi surface (FS) and under FS:

The pairs on FS are stable (permanent), whereas under FS the pairs can be created and annihilated, what destroys a supercurrent.

Indeed, if the pairing occurs under FS, then there are electrons with energy higher than the energy of electrons at the pairing level. These higher energy electrons can fall under FS to the pairing level and, thus, form new pairs, replacing the “old” ones. If the pairs emerge on FS, then there are no electrons above FS, no electrons can fall to FS, so new pairs don’t arise and the “old” ones become permanent, keeping the supercurrent and zero resistivity.