Main Question or Discussion Point
Why is the magnetic field of a superconductor normally excluded?
The flow of Cooper pairs.lufc88 said:What causes these screening currents?
It's not really excluded. It's held constant. If you try to change the magnetic field, dB/dt induces an electric field, which cause currents through superconductor that cancel out the change in the magnetic field. This happens in any conductor, but in a normal conductor, there is resistance, so the screening currents die out, and eventually the field changes. In superconductor, these currents keep going, so the field inside never changes.Why is the magnetic field of a superconductor normally excluded?
No, B actually is zero in the bulk of the SC - Meissner effect. This is why perfect conductors differ from superconductors.It's not really excluded. It's held constant. If you try to change the magnetic field, dB/dt induces an electric field, which cause currents through superconductor that cancel out the change in the magnetic field. This happens in any conductor, but in a normal conductor, there is resistance, so the screening currents die out, and eventually the field changes. In superconductor, these currents keep going, so the field inside never changes.
Naturally, the actual physics of how and why is a bit more complex and involves quantum mechanics. If you understand at least the Shroedinger's Equation, I can elaborate a bit more.
I see. I was somewhat confused over how the sueprconducting magnet works, leading me to believe that the field is "frozen in" rather than expelled. I'll have to take a closer look at the QM involved, though, I think I'm starting to see why it actually has to be zero rather than just constant.No, B actually is zero in the bulk of the SC - Meissner effect. This is why perfect conductors differ from superconductors.
Yes, this is the explanation following from the Ginzburg-Landau phenomenological theory.On a more abstract level, the expulsion of the magnetic field from a superconductor is due to the breaking of the U(1) gauge symmetry. Due to the Anderson-Higgs mechanism, the electromagnetic field becomes massive and therefore cannot enter far into the superconductor.
The Anderson Higgs mechanism was just recently confirmed in elementary particle physics with the detection of the Higgs boson. In a superconductor, the Cooper pairs correspond to the Higgs boson.
I wouldn't say so. Anderson derived the Anderson-Higgs mechanism in superconductors using random phase approximation to derive the current response to electromagnetic fields. The BCS hamiltonian is not gauge invariant and hence J=-QA cannot be derived from it. That's why Anderson (and others) was looking for a more elaborate microscopic description.Yes, this is the explanation following from the Ginzburg-Landau phenomenological theory.
Neither do I, sometimes :-)I'm afraid I don't follow your logic.
But, that is the whole point. The flow of a suppercurrent does not excite any excitations, and that is why it is dissipationless. If you go through the calculation, you will see the Q kernel has a Δ2 in the numerator, and must vanish in the normal state of the metal. After the integral over all momenta and the sum over Matsubara frequencies is performed, you see that at zero T, the dependence on Δ cancels, and the Q parameter is proportional to the total density of the electrons. This is understandable, since all the electrons are in a "BCS condensate" at zero temperature. As the temperature is increased, the magnitude of the Q parameter decreases, mainly because the density of electrons in the "condensate" decreases due to thermal excitations and breaking of Cooper pairs.However, the classical BCS explanation of the Meissner effect involves only single particle excitations, the reduced BCS hamiltonian does not even have collective excitations which could be described as a flow of Cooper pairs.
Actually, there is a Coulomb interaction in any metal, regardless whether its SC or normal. But, this interaction is screened and is no longer of infinite range (see Thomas-Fermi screening). Translated into k-space, the potential Coulomb potential energy is no longer:In a real superconductor there is a long range repulsive interaction between the electrons, namely the Coulomb interaction which gets collective excitations out of the gap.
That's why many people didn't believe the explanation of the Meisner effect of BCS.
I don't understand what you mean here. What collective modes? And how is screening supposed to work for "other modes" that it doesn't work for "these modes"?No, the finite energy of the plasmons (irrespective of in the superconductor or normal metal) is due to the long range of the Coulomb forces. Screening does not work for this kind of collective modes.
Yes, I remember reading his Nobel lecture. Let me emphasize once more that:The doubts on the validity of the BCS explanation of the Meissner effect spurred several important developments (I can give you references tomorrow).
For example Nambu wrote in his Nobel lecture ( http://www.nobelprize.org/nobel_prizes/physics/laureates/2008/nambu-lecture.html )
"I will now recall the chain of events which led me to the idea of SSB and
its application to particle physics. One day in 1956 R. Schrieffer gave us a
seminar on what would come to be called the BCS theory  of supercon-
ductivity. I was impressed by the boldness of their ansatz for the state vector,
but at the same time I became worried about the fact that it did not appear
to respect gauge invariance. Soon thereafter Bogoliubov  and Valatin 
independently introduced the concept of quasiparticles as fermionic excita-
tions in the BCS medium. The quasiparticles did not carry a definite charge
as they were a superposition of electron and hole, with their proportion
depending on the momentum. How can one then trust the BCS theory for
discussing the electromagnetic properties like the Meissner effect? It actually
took two years for me to resolve the problem to my satisfaction. There were
a number of people who also addressed the same problem, but I wanted to
understand it in my own way. Essentially it is the presence of a massless col-
lective mode, now known by the generic name of Nambu-Goldstone (NG)
LPN 2008 ekvationer...
boson, that saves charge conservation or gauge invariance."
Let me try to answer this question first: Plasmons are collective modes. They are responsible for the screening in metals and superconductors. However the plasmon cannot screen itself.I don't understand what you mean here. What collective modes? And how is screening supposed to work for "other modes" that it doesn't work for "these modes"?