BCS Theory Magnetic Flux Question

[Johnleprekan]

In superconductors, electrons form Cooper pairs that move at the same speed. Also, magnetic flux becomes quantized. Can someone explain in layman's terms why and how the magnetic flux becomes quantized? Especially in contrast to the magnetic flux in a conductor. Does it have something to do with the cooper pairs and their formation?

I thought that perhaps it might have something to do with the overlapping of the cooper pairs. Wikipedia stated that the cooper pairs at a sufficiently low temperature and form something like a Bose-Einstein condensate where the pairs form a single entity. Could someone explain how a Bose-Einstein condensate works in layman's terms as well? Such as how Cooper pairs could overlap, what is binding them together into a condensate? In a regular Bose-Einstein condensate as well, not just Cooper pairs and why in order to break one pair, you have to break all the pairs. Wikipedia said it had something to do with the energy.

Here is the article:

http://en.wikipedia.org/wiki/BCS_theory

 

[Jolb]

BCS theory is a very difficult theory and I'm not surprised there aren't many people jumping to answer this. I am admittedly no expert on BCS, but here's a simple explanation I've heard about flux quantization.

A true superconductor actually expels 100% of the magnetic flux through it. This is called the Meissner effect, and it explains why a magnet can never come in contact with a superconductor (in fact a magnet will levitate above a superconductor). So a pure superconductor has no flux quanta.

Now suppose you really try shoving some flux through the superconductor. What will happen is that some energy will go into making a small region nonsuperconducting (it will heat up above the critical temperature in that little region), and then a flux quantum, or a few flux quanta, will "thread" the "hole" in the superconductor. The holes tend to be very small because to make a bigger hole requires more energy since you need to heat more of the superconductor above the critical temperature--so the hole a flux quantum threads tends to shrink to as small as it can possibly be. But the hole cannot go away completely as long as a flux quantum is threading it.Now why do the flux quanta only come in certain discrete sizes? I'm not entirely sure whether this applies to BCS theory, but in the quantum hall effect you also see flux quantization and it has to do with the fact that solutions for the electrons' wavefunction are only possible for certain values of the flux, similar to Landau Levels.

A Bose-Einstein condensate basically is condensation of bosonic particles so that all of them go into the ground state. This is usually achieved by cooling things down. Cooper pairs of fermions are actually themselves bosons, so they can all share the same bosonic ground state. This explains why all the Cooper pairs move in unison.

What binds repulsive electrons together into a cooper pair? One way I've heard this can be visualized is that an electron traveling through a lattice actually causes the positive charges in the lattice around it to be slightly drawn in toward the electron, so a positive charge builds up around the electron. The other electron is actually attracted to this small accumulation of positive charge, and the two electrons can thus form a bound state--the Cooper pair. As one heats the system, thermal fluctuation in the positions of the positive charges in the lattice tends to wash out the small accumulation the electron creates, so the critical temperature has to do a lot with when the fluctuations wash out the positive charge accumulation.

Yes, it sounds like magic when explained this way, so if you don't believe it's possible, go ahead and work through the formalism. It's probably not as colorful as what I've said. A similarly impossible sounding phenomenon is the binding of electrons with one another in the quantum hall regime--for this look up "Haldane Pseudopotential".

Hopefully that doesn't butcher the holy theory of Bardeen, Cooper, and Schrieffer. Please, anyone, correct me if I've gone wrong.

 

[Dotini]

Can someone explain in layman's terms why and how the magnetic flux becomes quantized?

Individual atoms have a quality called spin which is quantized, or in discrete states, usually called up or down. Magnetism arises from the spin of atoms when they all line up. In a recent experiment, a Swiss team observed atoms spontaneously self-organize into non-random spin patterns.
http://www.huffingtonpost.com/2013/...n_3339243.html?utm_hp_ref=mostpopular,physics

 

[Johnleprekan]

So the flux gets quantized because all the spins align?

I know that electrons are in specific orbits because of their wave function (I believe I am using this correctly). So they can only occupy certain states. They cancel out in other states; De Broglie. What is it about atoms that causes electrons to do this? Is it just the attraction it has? Why doesn't this happen in normal conductors and only in superconductors? I ask this to better clarify my earlier question as to why magnetic flux gets quantized.

 

[Jolb]

Thanks guys.

So the flux gets quantized because all the spins align?

No. What Dotini is saying is an example of the fact that magnetic moments are quantized. But I think he's misleading you; for example a superconducting magnet can create a magnetic field without any "spin" effects going on--a superconducting magnet has more to do with the fact that charges in motion create magnetic fields. I think Dotini is thinking of the Landau mean field theory of ferromagnets, which really has little to do with your question. (In fact BCS is a newer and better theory than the Ginzburg-Landau mean field theory of superconductors because it is not a mean field theory.)

 

[Johnleprekan]

From Wikipedia:

"An electron moving through a conductor will attract nearby positive charges in the lattice. This deformation of the lattice causes another electron, with opposite spin, to move into the region of higher positive charge density. The two electrons then become correlated. Because there are a lot of such electron pairs in a superconductor, these pairs overlap very strongly and form a highly collective condensate. In this "condensed" state, the breaking of one pair will change the energy of the entire condensate - not just a single electron, or a single pair. Thus, the energy required to break any single pair is related to the energy required to break all of the pairs (or more than just two electrons). Because the pairing increases this energy barrier, kicks from oscillating atoms in the conductor (which are small at sufficiently low temperatures) are not enough to affect the condensate as a whole, or any individual "member pair" within the condensate. Thus the electrons stay paired together and resist all kicks, and the electron flow as a whole (the current through the superconductor) will not experience resistance. Thus, the collective behavior of the condensate is a crucial ingredient necessary for superconductivity."

Can someone explain how exactly the Cooper pairs overlap each other? What are they doing that prevents one pair from being broken?

 

[Jolb]

Okay, it seems that Wikipedia agrees with what I said about the formation of Cooper pairs, except that wikipedia is written in a much more confusing and jargony way. Please remember that Wikipedia is not a physics textbook; the quality of articles is extremely variable.

When they say the Cooper pairs "overlap" one another, they are using terminology that comes from quantum mechanics--they probably mean that the wavefunctions have a nonzero (or non-negligable) inner product. But that terminology is a little vague so stick to the basics: The cooper pairs are bosons so they can all share the same ground state (boson wavefunction). They all condense into this ground state once they get cool enough, so they're all described by one relatively simple wavefunction. So the wavefunction, which to physicists is more fundamental than the individual pairs themselves, describes all of the pairs together--we can think of them as one big object. Maybe you could think of this as the pairs "overlapping".

----Edit:
It might be enlightening for you to look at the analogous phenomenon of superfluidity. For example, Helium 4 atoms (which are bosons) will condense into a superfluid phase which flows with zero viscosity for the same basic reasons that the condensed cooper pairs move in a superconductor with zero resistance. Helium superfluid is also a Bose-Einstein condensate. Helium 3 is a fermion but it can also go into superfluid phase (that requires lower temperature than Helium 4) by forming atom pairs.
-----

The Cooper pairs can be broken, it's just that it requires energy to break the pair since the two electrons are in a bound state. As wikipedia says, there are collective effects in what the binding energy is, so there's a sort of mutual reinforcement of cooper pairs staying bound. The nontrivial part is why the electrons attract, since only attractive things can be in a bound state, and I think on that part what I said is pretty much the same as what Wiki says.

 

[ZombieFeynman]

When they say the Cooper pairs "overlap" one another, they are using terminology that comes from quantum mechanics--they probably mean that the wavefunctions have a nonzero (or non-negligable) inner product. But that terminology is a little vague so stick to the basics: The cooper pairs are bosons so they can all share the same ground state (boson wavefunction). They all condense into this ground state once they get cool enough, so they're all described by one relatively simple wavefunction. So the wavefunction, which to physicists is more fundamental than the individual pairs themselves, describes all of the pairs together--we can think of them as one big object. Maybe you could think of this as the pairs "overlapping".

I don't think some of what you wrote is quite right, or atleast it is rather vague. The size of the cooper pairs is essentially the coherence length of the order parameter. The inter cooper pair spacing is found to be smaller than this length in many superconductors. Cooper pairs are rather large. They are formed from electrons on opposite edges of the fermi sea, so the electrons have equal but opposite momenta. One can easily show that such electrons will form cooper pairs outside the fermi sea for arbitrarily weak attractive force between them. This (phonon mediated) interaction can often be hundreds of nanometers, while the interelectron spacing is much less than a nanometer. This means that although not all conduction electrons are unstable to pair formation, many cooper pairs will often occupy the same volume. It is in this sense that cooper pairs overlap.

 

[Jolb]

Thanks ZombieFeynman! That's an insightful point that really does clear up the "overlap" terminology.

The part I'm curious about is what you say about the "coherence length of the order parameter." I am a newbie to BCS theory so I would appreciate any basic references you could give me on the order parameter you speak of!

 

[ZombieFeynman]

Thanks ZombieFeynman! That's an insightful point that really does clear up the "overlap" terminology.

The part I'm curious about is what you say about the "coherence length of the order parameter." I am a newbie to BCS theory so I would appreciate any basic references you could give me on the order parameter you speak of!

By order parameter i refer to the Ginzburg Landau order parameter (which is very analogous to the ground state wavefunction).

If you minimize the Ginzburg Landau free energy functional for an inhomogenous region (say for x less than zero the order parameter is zero), you will find that for x greater than zero the order parameter becomes proportional to a hyberbolic tangent of x over some parameter with unit of length. This parameter is (to a factor of sqrt2) the order parameter.

If you would like to know more I suggest you look at Annetts book (sort of easy) or Tinkhams (hard)