Neither Energy Gap Nor Meissner Effect Imply Superflow

I have read several times in lecture notes, textbooks and online forums that the persistent current in a superconductor of annular geometry is a result of either:

  1. The opening of a superconducting gap at the Fermi surface
  2. The Meissner Effect

This is not correct, actually.

The energy gap at the Fermi surface is neither a sufficient nor necessary condition for the existence of persistent supercurrents in a superconducting ring. It is not sufficient because gaps can occur for all sorts of reasons — semiconductors, Mott insulators, charge density wave systems all exhibit energy gaps separating the occupied states from the unoccupied states. Yet these systems do not exhibit superconductivity.

Superconductivity does not require the existence of a gap either. It is possible to come up with models that exhibit superconductivity yet do not have a gap in the single-particle spectra (see de Gennes Chapter 8 or Rickayzen Chaper 8). Moreover, the cuprate and heavy fermion superconductors possess nodes in their single-particle spectra and still exhibit persistent currents.

Secondly, the Meissner effect is often conflated with superflow in a superconductor, but it is an equilibrium phenomenon, whereas persistent currents are a non-equilibrium phenomenon. Therefore, any conceptual attempts to make a conclusion about persistent currents in a superconducting ring from the Meissner effect is fraught with this inherent obstacle.

So, obviously, I must address the lurking $64k question: why does the current in a superconducting ring not decay within an observable time-frame?

Getting this answer right is much more difficult than pointing out the flaws in the other arguments! The answer has to do with a certain “topological protection” of the current-carrying state in a superconductor. However one chooses to understand the superconducting state (i.e. through broken gauge symmetry, the existence of a macroscopic wavefunction, off-diagonal long-range order, etc.), it is the existence of a particular type of condensate and the ability to adequately define the superfluid velocity that enables superflow:

\textbf{v}_s = \frac{\hbar}{2m} \nabla \phi

where \phi is the phase of the order parameter and the superfluid velocity obeys:

\oint \textbf{v}_s \cdot d\textbf{l} = n\hbar/2m

The details behind these ideas are further discussed in this set of lecture notes, though I have to admit that these notes are quite dense. I still have some pretty major difficulties understanding some of the main ideas in them.

I welcome comments concerning these concepts, especially ones challenging the ideas put forth here.

8 responses to “Neither Energy Gap Nor Meissner Effect Imply Superflow

  1. I am no superconductivity expert, and I don’t have access to the book chapters you refer to.
    I do have a question though: you say a gap is not (sufficient nor) necessary.
    While I agree in view of the nodal gap states where the total DOS (integrated over the full Brillouin zone) is not gapped, I always understood that the pairing by definition creates a gap – otherwise the pairs are not bound, and the superconducting condensate is not stable above 0 K.

    If that is correct, a gap is necessary. Seeing the gap then depends on its structure and the technique used to observe it (e.g. STS vs k-resolved ARPES).

    Could you clarify what you mean by “a gap is not sufficient nor necessary”?


    • Hi DK, thanks for your comment.

      It has been known for a while now that superconductivity can survive without a gap. This is most often seen with time-reversal breaking perturbations. When one applies a magnetic field, for instance, there is a narrow region before superconductivity is destroyed where the superconducting gap can close completely. The sample can still maintain persistent currents. This closing of the gap has been seen for other time-reversal breaking perturbations as well. Adding magnetic impurities or applying a current through a superconducting sample can also cause the gap to close.

      According to Tinkham Chapter 10.2, the closing of the gap (while maintaining superconductivity) was actually theoretically predicted by Abrikosov and Gorkov in the presence of magnetic impurities. You can use Google books’ snippet view to read the relevant page in Tinkham if you don’t have the book handy.

      So what I meant by “a gap is not sufficient nor necessary” is that a gap does not imply superconductivity (charge density waves open up a gap at the Fermi surface as well) and superconductivity does not imply a gap.

      This is why I thought it necessary to think about and seek out an explanation of persistent currents that did not rely on the existence of an energy gap.


      • Okay, just to be clear for me: are you saying that in the examples you give ALL the places on the Fermi surface where a superconducting gap existed have closed this gap?
        Are there still remnants of the coherence peaks in spectra at these locations? I.e. would there be a fraction of the fluid still be gapped while another fraction is not?

        Liked by 1 person

  2. Good question — yes, I’m saying that the gap is closed everywhere. You can take a look a the following references (tunneling and optics):

    Note that there is still a depression in the density of states at the Fermi energy, but that there is no gap. Abrikosov and Gorkov predicted that the superconducting gap would disappear for impurity concentrations higher than about 90% of the critical impurity concentration required to completely destroy superconductivity.


  3. Hm, I don’t think these 2 experimental papers qualify properly to claim that superconductivity without a gap exists; they essentially average the DOS over a large volume, which by necessity of the method (of impurities) contains localized states, which, when distributed “randomly enough” often/normally exhibit a tail in the DOS towards Ef.
    So isn’t the main result of the experiments performed simply a superposition of a gapped superconducting fluid with a DOS from random impurities…?

    (I.e. I’m still not convinced that a 2 fluid system (one gapped and superconducting, and one not gapped – and not superconducting) can not explain any experimental result where superconductivity exists and the total DOS is not gapped – but I think I am overstaying my welcome here, meaning that it is up to me to figure this out more, and then agree or disagree with your post. I do thank you for spending time in formulating educational responses to my questions!)

    Liked by 1 person

    • I asked to be challenged on the ideas presented, so I’m happy that you’ve been asking thought-provoking questions.

      Unfortunately, the only study with STM I could find quickly was this one:

      In this study, it does look like that the regions that don’t have magnetic impurities have a perfect gap. However, there is one major caveat to this study in relation to our conversation, which is that the impurity concentration was not high enough to enter the “gapless superconductivity” phase. But I suppose you could imagine the density of Mn atoms getting high enough such that the density of states does not go to zero anywhere in the sample.

      While perhaps it shouldn’t matter that “gapless superconductivity” was predicted before observation, I feel like this in itself is pretty strong evidence for its existence. Though, as you said, I have not been able to find for you the perfect experiment that can or cannot refute your claim of possible two-fluid behavior.

      Please let me know what you come up with if you decide to look further.


  4. I had some time yesterday night to browse around a bit.

    Looking at Andrei Marouchkine’s book “Room-temperature superconductivity”, page 55, he discusses magnetic impurities in conventional superconductors where the gap decreases faster than Tc. He ends up with a 2 fluid model: remnant Cooper pairs (–>gap and coherence peaks in spectrum) coexist with the pair-broken quaisparticles (–> no gap) coexist, resulting in supercurrent and arbitarily low excitation frequencies to be present at the same time.
    He then remarks this can be done by any pair-breaking perturbation : strong magnetic field, high (near critical) current, magnetic impurities. Even a proximity effect (with a low induced pair density) is essentially gapless superconductivity.
    (Unconventional superconductors are another story; page 229 argues it would only locally destroy the gap – in the end I think that’s not a fundamental difference because in conventional superconductors “locally” is just larger than in unconventional ones simply because the coherence length of unconventional superconductors is on the order of a unit cell.)

    I don’t know him, so I can’t vouch for his statements being right. He is at Cambridge though and seems to have a track record in this field.

    B.S. Alexandrov appears to say something similar in his book “Theory of Superconductivity: From Weak to Strong Coupling” p. 57 (google books).
    And so do Vladimir Kresin and Stuart Wolf in their book “Fundamentals of Superconductivity”, page 38 (google books).

    All of this suggests to me that gapless superconductivity is the result of a superposition of the spectral features of the original superconducting condensate keeping its gap structure (and I see no reason why the gap structure in k-space should change – but for conventional superconductors this may be a moot point anyway), and a gapless quasiparticle liquid, resulting in both a supercurrent and a low energy DOS like a metal.

    These are results confirming what I thought, though they do not prove other ideas are wrong. I.e. it’s a self-serving literature search instead of an attempt to falsify other ideas (which would be more rigorous). Nevertheless, I think other ideas are necessarily more complex, and I generally favor simple ideas over complex ones if they explain observations.

    Thank you for this discussion, and keep up the good work of blogging; it is a way for me to read outside the box (see your recent post “Breadth versus Depth…!).

    Liked by 1 person

    • Thanks, DK. Those resources look good. I hopefully will get a chance to go through them a little more extensively soon, but upon first glance, they look worth spending time on.

      I’ll also keep you posted if I come across a definitive answer on this point of local vs. global gapless superconductivity.


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