# Question on superconductivity theory beyond BCS.

Dear all:
Now I am trying hard to interpret superconductivity in MgB2, and when reading some paper such as J.Kortus et. al, J.M.An et. al on PRL/PRB, I find that their way to weigh transition temperature is different, for instance, in J.Kortus's paper(PRL86,4656(2001)), they use McMillan-Hopfield formulate, while in J.M.An's paer(PRL86, 4366(2001)), they use Allen-Dynes's equation( relatet to "deformation potential due to frozen-in phonon?), and some other papers use Eliasberg theory, here I am a little confused:
1 which is more precise?
2 is there any difference between them?(certainly, but plz outline the differences on a more general aspects, for example, strong electron-phonon coupling, etc)
3 plz refer me to one or two classic book which would include all or part of these theories.
Thank you all.

ZapperZ
Staff Emeritus
leoant said:
Dear all:
Now I am trying hard to interpret superconductivity in MgB2, and when reading some paper such as J.Kortus et. al, J.M.An et. al on PRL/PRB, I find that their way to weigh transition temperature is different, for instance, in J.Kortus's paper(PRL86,4656(2001)), they use McMillan-Hopfield formulate, while in J.M.An's paer(PRL86, 4366(2001)), they use Allen-Dynes's equation( relatet to "deformation potential due to frozen-in phonon?), and some other papers use Eliasberg theory, here I am a little confused:
1 which is more precise?
2 is there any difference between them?(certainly, but plz outline the differences on a more general aspects, for example, strong electron-phonon coupling, etc)
3 plz refer me to one or two classic book which would include all or part of these theories.
Thank you all.

What you are discovering here is what many physicists have seen many times - that often, there are more than one way to skin a cat, and that different approaches can produce amazingly similar results when they start with a physically valid model.

Notice, for instance, that both papers produced amazingly similar band structures. However, the POINT that each paper is trying to make is subtle and differ from each other. The Kortus paper is trying to calculate the phonon spectrum and prove that there is a huge electron-phonon coupling to account for the high Tc of MgB2. The An paper, on the other hand, is trying to show that superconductivity resides primarily in the sigma band. They both reach the same conclusion of which phonon modes are responsible for superconductivity.

Note that at the end of An's paper, there's even a Note Added citing the Kortus "unpublished" paper as being consistent with their results.

Zz.

Dear Zapper:
Yeah, I have read the two papers and those related, and what I want to do is to get an understanding of superconductivity dominated by eletron-phonon interaction following their papers taking MgB2 as an example. And though BCS is important to tell us superconducting mechanism, i think, it's too simple(a few monthes ago I have read the paper following your advice, here I want to thank you again), and it seems like that Eliasberg's theory is better and considers more factors, so I want to have a look at it.
Now I think I've find some useful books such as Grimvall's "electron-phonon interaction in metals" and P.B. Allen's paper in Solid State Physics vol.37, 1982, so do you have some good advice for me to go further on superconductivity theory?
btw: I am very confused to HTSCs in cuprates, what can we do to understand their mechanism using traditional band structure and solid state physics knowledge, especially is electronic structure calculation, for instance, DFT method(which I am desperate to learn), useful?
Anyway, give me some advice on references of ep interaction and Eliasberg theory.
Thank you

ZapperZ
Staff Emeritus
leoant said:
Dear Zapper:
Yeah, I have read the two papers and those related, and what I want to do is to get an understanding of superconductivity dominated by eletron-phonon interaction following their papers taking MgB2 as an example. And though BCS is important to tell us superconducting mechanism, i think, it's too simple(a few monthes ago I have read the paper following your advice, here I want to thank you again), and it seems like that Eliasberg's theory is better and considers more factors, so I want to have a look at it.
Now I think I've find some useful books such as Grimvall's "electron-phonon interaction in metals" and P.B. Allen's paper in Solid State Physics vol.37, 1982, so do you have some good advice for me to go further on superconductivity theory?
btw: I am very confused to HTSCs in cuprates, what can we do to understand their mechanism using traditional band structure and solid state physics knowledge, especially is electronic structure calculation, for instance, DFT method(which I am desperate to learn), useful?
Anyway, give me some advice on references of ep interaction and Eliasberg theory.
Thank you

The Eliashburg theory is an extension of the original BCS theory. Remember that the original BCS is a WEAK COUPLING theory. It accounts for weak coupling between electrons and phonons, and thus, can explain the very low Tc of Hg and stuff. However, it failed when one tries to do that to metals like Pb. The Eliashburg extension of the BCS theory goes into the strong coupling regime. This then can be applied to the conventional superconductors that have higher Tc's.

When MgB2 was discovered, we found out that it too are based on electron-phonon mechanism. With such a high Tc, one tends to automatically assumes that this is a strong-coupling scenario. So the original BCS theory doesn't work, but the extended BCS theory will, and does so far.

As for high-Tc, who knows. The band-structure calculation is a bit dicey because you can't use the conventional calculation - it predicts that the undoped parent compound is a metal (1/2 filling), yet it's an insulator! I don't know how successful DFT has been. You may want to look at the Dynamical Mean Field Theory (DMFT) technique that was popularized by Kotliar et al. That model suffers from its own set of conceptual problems, but it seems to be able to predict at least the metal-insulator "crossover" as a function of doping.

Zz.

P.S. You may want to read this: http://xxx.lanl.gov/abs/cond-mat/0106143

Last edited:
from cond-mat/0106143
"Eliashberg theory goes beyond BCS theory because it includes retardation effects; however, it is still a weak coupling theory, in the sense that the Fermi energy is the dominant energy, and the quasiparticle picture remains intact."
Here I am confused. From the authors it seems like that all mechanism based on Fermi Liquid theory is weak coupling, so what is beyond Fermi Liquid theory?
And there is a theory of conventional superconductors named after McMillan, so what is it? (sorry for so many trivial questions, )

ZapperZ
Staff Emeritus
leoant said:
from cond-mat/0106143
"Eliashberg theory goes beyond BCS theory because it includes retardation effects; however, it is still a weak coupling theory, in the sense that the Fermi energy is the dominant energy, and the quasiparticle picture remains intact."
Here I am confused. From the authors it seems like that all mechanism based on Fermi Liquid theory is weak coupling, so what is beyond Fermi Liquid theory?
And there is a theory of conventional superconductors named after McMillan, so what is it? (sorry for so many trivial questions, )

OK, so we need to be a bit MORE careful as to what is meant by "coupling". The coupling that I mentioned earlier is the electron-phonon coupling. This corresponds to the "strength" of the cooper pairing.

The "coupling" that you are citing (and the inclusion of the Fermi Liquid theory), is the coupling in the electron-electron interaction as QUASIPARTICLES. This is based on the Fermi Liquid theory in which the electron-electron interaction is weak. Thus, such interaction, according to Landau, can be renormalized. This reduces a single many-body problem into many single-body problems - you gain back the independent particle picture, except your particle now has an effective mass different than the bare mass.

Zz.

Thank you very much, dear Zapper.
About McMillan's theory, would you be kind to make a short comment?

ZapperZ
Staff Emeritus
leoant said:
Thank you very much, dear Zapper.
About McMillan's theory, would you be kind to make a short comment?

If by "McMillan's theory" you mean the McMillan-Rowell method, then I can make a quick comment off the top of my head since I'm about to go to bed. So be warned that I'm relying this only on memory.

The M-R method is a way to "invert" a tunneling density of states to arrive at the phonon spectrum of the solid (I'm only familiar with superconductors). What you get, if I remember correctly, is the $$\alpha^2 F$$ spectra of the phonon structure. One usually gets this from the 2nd derivative of the I vs V tunneling spectra.

Don't ask me about the detail of the inversion process, because I don't remember.

Zz.