There are several formulas to describe critical current in Josepshon S-N-S junction mainly based on Eilenberger and Usadel equations for quasi-classical Green's functions. The starting point is the Nambu-Gorkov equation and then some simplifications. Lately, I encountered scattering matrix approach to this problem (Beenakker) and I am not quiet sure if this method can also handle the quasiclassical Green's function. I really don't understand in what regimes I am working.

While searching the net I have found the statement that the Green's function method do not include the phase coherent Andreev reflection (I can't find it now).

What kind of Josephson junction regimes exist: classical, semi-classical and mesoscopic ? Can we have long junction limit in the mesoscopic regime ?

This question arises because I am working on Josephson supercurrent through topological insulators. I am not sure which formalism should I use to describe the critical current. I have seen both of them in literature, but I think it depends on some parameters (length scales etc.).

Thank in advance for Your help.


EDIT: I come back to this answer, since I've seen this review this morning: Scattering matrix approach to the description of quantum electron transport by Lesovik and Sadovskyy, published in Physiks Uspekhi recently (the link is to arXiv actually). It seems to me Lesovik and Sadovskyy made a really nice introduction to the Andreev and Josephson problems.

You did not describe regimes, you described methods. I understand this is confusing since the different methods are usually used to describe different regimes indeed.

So in short, the different methods to describe superconductivity (I focus on heterostructures, not bulk problems, which are an other story):

  • Bogoliubov-de Gennes (BdG): this is a generalised Schrödinger equation, so you can treat any problems with it. It describes the evolution of the excitations of a superconductor. You resolve the wave equation, and then you discuss heterostructures as usual. The method gives tractable results in the short-junction, ballistic limit at zero-temperature... and that's almost all. But it allows to describe complicated band-structure and spin-effect with great accuracy (not necessarily tractable, though).

  • Slightly different: the Andreev approximation of the BdG: in this limit (also called quasi-classic) you simplify the Schrödinger-like BdG-equations to a WKB-like equation. Namely, you have something like $\left(p^{2}-p_{F}^{2}\right)\Psi$ in the BdG equations (the kinetic energy), with $p_{F}$ the Fermi momentum. You transform to $\Psi=e^{\mathbf{i}k_{F}x}\Psi_{0}$ and you use the approximation $\left(\hat{p}^{2}-p_{F}^{2}\right)\Psi\approx e^{\mathbf{i}k_{F}x}p_{F}\hat{p}\Psi_{0}$ which is a simpler equation since it is first-order. IMPORTANT: what to you miss with this approximation? Well, the really microscopic details, appearing at the Fermi length-scale. In normal metal, the Fermi wave-length is about the Angstrom, so you don't really miss a lot, but for semi-conductors ... well you have to discuss case by case, but there are not a lot of effects hidden in the Andreev limit of the BdG, and most of the papers resolving the BdG problem implicitly solve only the Andreev approximation. For really small Fermi energy, this limit is not supposed to work. This method can be used in short/long and ballistic junctions. It gives tractable result in the ballistic limit only, at zero-temperature.

  • Green's functions (GF): you can in principle describe everything using the Green's functions, but this is usually hard in heterostructures, since the Green functions are two-point correlations functions, so people prefer to discuss quasi-classical Green function. The point to remember is that, contrary to the BdG method, you here discuss the original quasi-particles (say, electrons), not the excitation quasi-particles (sometimes called Bogolons) ; of course there is a canonical transformation between them, but this is fair point to mention here.

  • quasi-classic Green functions (QCGF): you transform the Green function in the real space to some correlators in the phase-space using a Wigner transform. Then your function now depends on one point in the real space (and no more two), and one point in the momentum space. You can nevertheless apply a kind of Andreev approximation and suppose all the relevant physics is at $p\approx p_{F}$ and so you end-up with a function depending on one point only. This method is in principle the most general: it allows to describe ballistic to diffusive limits, temperature effects, it doesn't care about the size of the junction (short/long-limit), but to obtain tractable/compact results is sometimes pretty tricky. When the band structure is complicated it can be absolutely impossible to handle.

So in short you have 2 really microscopic methods: BdG and GF, and 2 quasi-classic approximation: Andreev and QCGF. What you miss from the quasi-classic approximation are the details at Fermi wave-length (as well as energy, ...)

Now for each of these methods there are subtutlies really long to discuss. The best is to ask to your supervisor I guess, or some colleagues.

If you want to describe a Josephson junction, the QCGF allow an easy calculation of the thermodynamic current, whereas you have to sum all contributions from above and below the gap in the case of BdG method.

One last word about the method, and their difference:

  • GF are known to allow systematic treatment of perturbation, which can be more involved in BdG.

  • To add disorder (Born approximation) and temperature is straightforward with Green function method, and can be really tricky (or simply impossible) in a wave-function approach (BdG)

  • The wave-function approach (BdG) is transparent: you really see the particles moving from here to there, say. The GF approach is less transparent, only at the end of (a long) calculation you can discuss the physics of the problem.

And finally, a personal remark: I feel people prefer the wave-function approach because it is easy to deal with: if you know how to resolve the Schrödinger equation, there is no much effort to do to understand the BdG scattering theory (see the review by Beenakker for instance, and reference therein). GF (especially QCGF) are more tricky, and less documented in literature, especially for your researchers I fear. Also, QCGF are really tricky in the case of spin-orbit effect, and people have troubles dealing with in topological situations. Things may change in the future, perhaps if one follows gauge-covariant approaches (warning, auto-promotion (well, intended to be a decent introduction to the QCGF approach though, despite giving no example, since those are coming :-) and friends-promotion).

Finally, a remark about topological insulator (TI): if you believe the band structure of the TI is a Dirac dispersion, then there is no difference between the BdG and Andreev approximation... so all my long speech to give you a comment-like answer... sorry for that.

  • $\begingroup$ Wow! This is really perfect answer ! Thank You very much! $\endgroup$ – WoofDoggy Jul 27 '14 at 14:12

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