If you believe that the novel phenomenon of superconductivity in all materials should be explained from a single theory then BCS is certainly not the right theory. But as you know that BCS theory is fairly successful to explain superconductivity (not everyone will agree to this statement and I have my doubts as well) in metals and alloys. But it fails miserably in many newly discovered superconducting materials, viz. cuprates, Fe-based pnictides and chalcogenides, heavy fermions etc. etc. Here the common wisdom is that the magnons or spin fluctuations, rather than phonons, act as glue for the Cooper pairs. If this is right then obviously phonon mediated BCS theory is not the right theory for these materials. I have given only one alternative mechanism but there are several others in the literature. We are still looking for a genius who will construct a theory of supewrconductivity that will work for all superconductors in a similar way as Newton/Einstein showed that mechanics that works for our ordinary everyday objects is also the same one that can be used for celestial objects like the sun and the planets,stars, glaxies, supernova, black holes etc. None has come forward so far!

BCS theory developed by using second quantization (SQ), undoubtedly unravels certain truth behind superconductivity but one should keep in mind that SQ makes certain compromises to simplify the process of formulation of a theory of many body quantum systems such as an electron fluid which becomes superconducting below T_c. While its conclusion that electrons form Cooper pairs which condense into single particle state of zero momentum of the CM of a pair and these pairs lead to superconductivity represents an important truth, its use of SQ which uses single particle basis (SPB) is its serious weakness. Notably SPB identifies each electron in electron fluid as a free particle with a possibility that it can have any momentum/energy is far from the real state of an electron. The quantum nature of electron demands that the electron surrounded by ions in a solid can not have an energy/momentum lower than what is permitted by the cavity size formed neighboring ions. Consequently, BCS theory achieved only limited success in its objected. I hope that sooner or later this point would be accepted by actively working scholars in the field and we will have a viable theory of superconductivity with a framework that would help in understanding the origin of the phenomenon in widely different systems. There are other equally important reasons for the limited success of BCS theory. However, I feel that the main point answers the question at hand.

Prof. Murat Sertkol,

BCS theory works pretty well for most metallic elements and some conventional superconductors.

Unconventional superconductors continue to be without a unified microscopy theory/mechanism despite all efforts.

Heavy fermions, High Tc, ruthenates, binary compounds, ferromagnetic superconductors, 3He isotopes, just to mention some.

Nodes in their gaps, jumps in thermodynamical and kinetic properties, the influence of non-magnetic impurities, their unitary states at very low temperatures, there are explanations for most compounds but without a microscopic theory.

BCS collapses for the Heike Kamerlingh Onnes’s experiment (1911). In a mercury ring (superconductivity below Tc=4.15 K) we establish a persistent supercurrent. Then we organize temperature cycles (T-cycles) in the cryostat, say from 3 K to 2.9 K and back. According to the BCS theory, the pair density decreases at warming, i.e. a not negligible fraction of pairs annihilates; the same fraction of pairs emerges back at cooling. Annihilated pairs lose their ordered supercurrent momentum on the atom lattice; newly created pairs do not experience any electromotive-force (EMF), since the EMF is no longer available in the ring. Hence, the supercurrent must decrease at every T-cycle and dissipate after a number of T-cycles. However, in all experiments the supercurrent remains constant and, thus, the pair recombination (assumed in BCS) doesn’t take place (note, every cryostat device produces not negligible temperature fluctuations, so every observation of long-lived supercurrents is the experiment with T-cycles).

Do the pairs really recombine in the eternal supercurrent? BCS is out of game for this paradox.