Is there any explanation of Josephson effect based on Schrodinger equation?

[zhanhai]

All explanations of Josephson effect I have read so far are based on Ginzburg–Landau theory. There seems no explanation based on Schrödinger equation. Why?

While an explanation of Josephson frequency of 2eV/h seems not difficult to envisage, the major problem, I guess, should be with electron pairing. Josephson effect takes place with a driving voltage of less than 1mV, but a typical superconducting energy gap should be of 10meV or greater (especially for HTS), which should prevent electrons below the gap from being excited by phonons. Thus, how would an electron be driven to transit by a voltage of less than 1mV while it is prevented from transition by phonons of over 10meV?

Do I miss or mistaken anything?

 

[f95toli]

Yes, you can derive it based on some very general principles using the SE. See for example Feynman's lectures (probably volume 3).

In order to derive the Josephson equations you have to assume that there is some sort of coupling between the wavefunctions on each side of the barrier and you also have to assume that the current carrying "particle" has a charge 2e; but that is it.

Note that the Josephson effect is very general and can also be observed in e.g. Bose-Einstein condensates. Hence, it does not rely on any details of the underlying mechanism for superconductivity.

Edit: Also, the Josephson effect in SC deals with Cooper pairs, not electrons.

 

[zhanhai]

Yes, you can derive it based on some very general principles using the SE. See for example Feynman's lectures (probably volume 3).

In order to derive the Josephson equations you have to assume that there is some sort of coupling between the wavefunctions on each side of the barrier and you also have to assume that the current carrying "particle" has a charge 2e; but that is it.

Note that the Josephson effect is very general and can also be observed in e.g. Bose-Einstein condensates. Hence, it does not rely on any details of the underlying mechanism for superconductivity.

Edit: Also, the Josephson effect in SC deals with Cooper pairs, not electrons.

Thanks for your comments. I found Feynman's lectures on Josephson effect on this webpage:
http://www.feynmanlectures.caltech.edu/III_21.html#Ch21-S9

Frankly, I am not sure whether the use of common wave function is justified, and no electronic state structure is discussed.

I have thought the question again after my previous post, and come to realized it might not be so much due to treatment of electrons in pairing; rather, the lower states of the structure seemingly suitable for eV transitions are occupied. This should be the true reason that prevents an explanation based on eV transitions; such an explanation thus could not be established...unless some related notion is modified.

 

[DrDu]

I am not sure whether I understand your question, but the Ginzburg Landau equations can be derived from the microscopic hamiltonian describing the electrons. This has been done by Gorkov: L. P. Gor'kov, Sov. Phys. JETP, 9, 1364(1959).

 

[f95toli]

You don't need to deal with electron structure to describe the Josephson effect. You can -as I wrote above- observe the Josephson effect in Bose-Einstein condensates (no electrons involved) which illustrates that is a very general phenomenon. You can obviously derive them starting from electron structure (although it is rarely done) but it is sort of missing the point a bit; doing it that way would be a bit like trying to explain the quantum Hall effect by studying the electron structure of GaAs (too many trees and all of that).

Also -on a more practical note- we don't actually have a microscopic theory that works for all the materials that exhibit the Josephson effect; the high-Tc superconductors are a case in point. You can get reasonably good agreement with experiments using models that take the Andreev states into account, but the symmetry of the wavefunction (nearly) always assumed to have a specific form (.e.g d or d+is) in these calculations.
That said, for a specific configuration you can always get perfect agreement with the Josephson equations as long as you use phenomenological parameters for the critical current etc (and in the d-wave case include higher harmonics for the current-phase relation).

 

[zhanhai]

I have made a model, in which the double frequency 2ω can be resulted from non-linear relationship. Why ω=eV/ħ seems more complicated; first, greater ω corresponds to smaller current needed to maintain the system; second, there seems some effect concerning quality factor, and when ω=eV/ħ the quality factor could be the best.

A related result is a microscopic explanation of flux quantum, which seems to indicate that carrier electrons can be "deep electrons", accordingly superconducting electrons may like water flowing in a trench, and a tentative mechanism could be constructed for it.