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I have two questions one practical and one theoretical. Even though I have a decent understanding of superconductivity both phenomenological as well as theoretical (i.e. BCS), some things just slipped through the cracks.

First question: Say you have a decent quality superconducting ring completely isolated from external magnetic fields. Then a supercurrent is supposed to flow (with no voltage applied) which can can last for years without much dissipation. Will this current just automatically set in the moment I cool my sample below $T_c$? Or do I have to apply an external perturbation for a short duration, remove the perturbation, and let the current go around the ring indefinitely? If the latter is true then what is this perturbation? It can't be voltage (can it?) since no voltage is required to make the current flow. If it is voltage (or any other perturbation) then how do I apply it?

Now, the second question is a bit theoretical. It even (I think) superficially contradicts what I described above. Since cooper pairs (in at least $s$-wave pairing) are formed by $\psi_{\mathbf{k},\uparrow}$ and $\psi_{-\mathbf{k},\downarrow}$ their net momentum is zero. How do they give rise to a supercurrent then? According to BCS they're not supposed to move, right?

Can someone please help me reconcile this?

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"The have two questions one practical and one theoretical." Then please post two questions. That's how a Q&A site is supposed to work. –  dmckee Mar 22 '13 at 0:14
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@dmckee: I know how a Q&A site works. The questions are intimately related to one thing: "supercurrent." –  PhHEP Mar 22 '13 at 1:11
    
For the answer to your first question, you can refer to physics.stackexchange.com/questions/69222/… –  Martin J.H. Jun 26 '13 at 13:08
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1 Answer

First question: Say you have a decent quality superconducting ring completely isolated from external magnetic fields. Then a supercurrent is supposed to flow (with no voltage applied) which can can last for years without much dissipation. Will this current just automatically set in the moment I cool my sample below Tc? Or do I have to apply an external perturbation for a short duration, remove the perturbation, and let the current go around the ring indefinitely? If the latter is true then what is this perturbation? It can't be voltage (can it?) since no voltage is required to make the current flow. If it is voltage (or any other perturbation) then how do I apply it?

You need to perturb your system of course. It is sufficient to apply a magnetic flux inside the loop for instance. Note that some funny loops interrupted with peculiar Josephson junctions are able to produce their own current. They are called $\pi$-phase Josephson junction. See http://en.wikipedia.org/wiki/Pi_Josephson_junction for more informations about $\pi$-phase Josephson system.

Now, the second question is a bit theoretical. It even (I think) superficially contradicts what I described above. Since copper pairs (in at least s-wave pairing) are formed by $\Psi_{\mathbf{-k} \uparrow}$ and $\Psi_{\mathbf{k} \downarrow}$ their net momentum is zero. How do they give rise to a supercurrent then? According to BCS they're not supposed to move, right?

You're perfectly right regarding the momentum of the Cooper pair, but that's not how we describe current in superconductors :-). To describe the current propagation, or any electromagnetic property, you need to generate inhomogeneous correlations, say between $\Psi_{\mathbf{-k} \uparrow}$ and $\Psi_{\mathbf{k}+\mathbf{q} \downarrow}$.

To conclude, an historical remark.

When BCS paper appears, people immediately realised that this kind of correlation would break gauge invariance. You indeed have something like the current $ \mathbf{j}\propto \left<\Psi_{\mathbf{-k} \uparrow} \left(\mathbf{k}+\mathbf{q}\right) \Psi_{\mathbf{k}+\mathbf{q} \downarrow} \right> \propto \mathbf{A}$ proportional to the vector potential in order to explain the Meissner effect (if I remember correctly). The resolution of this problem ends up with the explanation by Anderson (Nambu was also important in understanding this point, as well as Bogoliubov, since Landau didn't believe the BCS theory was right until the gauge invariance can be proven, and he asked Bogoliubov to work on this problem) that the gauge invariance can be kept at the expense that the plasmon field becomes massive. This mechanism is known as the Anderson-Higgs mechanism in condensed matter. Since Anderson was working in non-relativistic field theory, high-energy physicists usually refer to this mechanism as the Higgs mechanism (sometimes the Higgs-Englert-Brout-Guralnik-Hagen-Kibble mechanism for all the people having done the same discovery independently), without acknowledging Anderson. A good reference about the Nambu-Anderson-Bogoliubov calculations is the book by Schrieffer, Superconductivity, Benjamin Inc. (1964). A discussion about the Higgs mechanism is a talk by Weinberg: http://cerncourier.com/cws/article/cern/32522.

Schrieffer is also an excellent introductory book, still really interesting despite its age.

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