Reason for a Cooper "Pair" Formation

I'm an Australian high school student (Grade 12). We are currently covering superconductors as a topic for our syllabus. Can someone explain (inasmuch as an advanced 18-year-old physics student can understand) why Cooper pairs of electrons (as per the BCS theory) only form in pairs? As I understand it, Cooper pairs form between 2 electrons with opposite spin due to a "phonon exchange", wherein the first electron passing through the metallic lattice induces a distortion of the lattice and hence the creation of a region of higher positive charge density behind it, attracting the second electron with opposite spin (we don't need to know why an opposite spin is required). However, if this second electron also passes through the lattice, and hence creates a region of higher positive charge density behind it, why does it not thence attract another electron and so on? In other words, why do Cooper Pairs form and not Cooper chains? Thanks for your help in advance. *Note: I don't mind what level of understanding your reply requires, I will just try my best to understand more complicated ideas. Thanks

• It is because you can only find cooper pair near the Fermi surface only. The two interacting electrons are just above the fermi surface, all other electrons are below. And non interacting. If you solve the Schrodinger, you will find that they are bound with just below 2* Fermi surface. So the two electrons with momenta k and -k, and opposite spin, floating just below Fermi surface can combine, reducing the total energy. They are floating just above the Fermi surface forever. The lifetime of these quasiparticles increases as they come closer to Fermi surface, and at the surface it is infinity. Commented Jun 5, 2018 at 2:34
• Thanks Arpad. Makes sense that the outcome is to reduce the total energy. Where you said "all other electrons are below", why is there only two electrons above the Fermi surface? (or is that just a definition/ I should read the Wikipedia page on Fermi surfaces?) Thanks again for your response. Commented Jun 5, 2018 at 3:21

You probably know about the Pauli principle: you cannot occupy a single quantum state by more than one electron. This is in fact related to their spin which is 1/2.

In superconductivity, two electrons team up to build a Cooper pair. The spin of the cooper pair can be 0 or 1. It therefore behaves rather like a boson than like a fermion. This is a key ingredient to superconductivity because now, the Cooper pairs can occupy the same macroscopic quantum state!

If we now think about what happens when a third electron joins the party, the spin of the composite state is half integer again and they are not allowed to team up in a macroscopic quantum state.

So what about four electrons then? Here the spin can be bosonic again. At the moment I cannot think of a reason why - in principle - there could not be a composite state made out of these four.

But already at the level of 2 electrons, the formation of Cooper pairs by the mechanism you present is a delicate business that only works at very low temperatures. The probability of 4 four electrons meeting at just the right spot with just the right momentum will be much smaller for sure. While at the same time, the energy barrier to destroy this team of 4 and break it into 2 pairs of 2 will be very low.

• Thanks for your answer. Am I right to hence read that having a Cooper "Pair" of four electrons is possible but very unlikely then? Commented Jun 6, 2018 at 1:10
• Isn't spin 1 magnetic and therefore not superconductive? Commented Aug 9 at 20:13

For the simplest cases, explained by the original BCS theory, all the Cooper pairs form a superconducting condensate. There are no spatially separated pairs in this condensate. They overlap. However, cooling way below the transition temperature of the material is required to convert all the conduction electrons to the condensate of pairs. A transition temperature is a temperature where the condensate becomes non-fragmented.

"Pairing" is a tendency of spins to become opposite, as the conduction electrons lose momentum to the crystal lattice. The more the momenta become similar (lower momentum means less kinetic energy and lower momenta cannot be as different as high momenta), the more the difference must be for spins, in order to follow the Pauli principle, which applies only for these electrons that overlap both spatially and in momenta.

There is usually no reason for Cooper pairs to form any more ordered structures even near absolute zero. It is important to remember: the lattice distortion in BCS theory is a virtual process. It does not form bound states with neither electrons nor pairs of electrons. The lattice still looks the same both below and above the superconducting transition temperature.

• Nice answer (+1) Could you clarify if "the momenta becomes similar" means that the two-particle wavefunction in momentum space $\Psi(p_1, p_2)$ becomes especially concentrated in the $\Psi(p_1 = p_2)$ states? How does this comply with Pauli exclusion, which seems to state that $\Psi(p_1 = p_2) = 0$ due to anti-symmetricity? Commented Aug 10 at 20:39
• No, the momenta of the condensed electrons with the same spin still differ but the interaction with virtual phonons makes it impossible for them to become thermally excited to even higher momentum by an arbitrarily small amount of energy. These phonon modes are frozen out. The absence of superconductivity and Cooper pairs at extremely low temperatures for some metals is due to the competing effect of decreased local screening for singly occupied states. Commented Aug 11 at 12:08
• Ok, thanks. I take it you meant that $\Psi(p_1 = p_2) \neq 0$ does not violate Pauli exclusion as long as the spins of the paired electrons become opposite at the same time? Commented Aug 11 at 12:50
• Yes, that's what I meant. I'm not sure why the two-particle wavefunction in momentum space deserves attention. Can it be useful at finite temperatures? Commented Aug 11 at 18:21
• Thank you, I appreciate the enthusiasm & all the best. Commented Aug 11 at 22:04