Can Cooper Pairs Have Different Orientations in a Superconductor?

[Davephaelon]

Can Cooper pairs have any spatial orientation within a superconductor? In most graphical depictions of a Cooper pair they are both on the same axis (moving through the same channel within the lattice), and parallel with the current direction. Is there any reason the pair cannot be at a right angle to the current flow, with the individual electrons of the Cooper pair moving in different channels of the metal crystal?

 

[DrDu]

In most ordinary superconductors, the wave function of the Cooper pairs is spatially isotropic, i.e. there is no preferred direction for the relative motion of the electrons forming the pair. There are special superconductors, where the Cooper pair wave function has some angular dependence, in particular that of a d-orbital.

 

[f95toli]

Just to add to what DrDu has already stated. Most common high-Tc superconductors (e.g. YBCO) has a d-wave symmetry meaning the transport is not isotropic with respect to the a-b plane of the lattice. This can e.g. be used to create devices where the Cooper pairs pick up an extra phase of pi when they are transported from a "plus" lobe to a "minus" lobe (this is known as a pi-junction and has some interesting properties).
There are also p-wave superconductors and possibly also superconductors with mixed symmetries such as d+s. d+is (i is the imaginary unit) etc

 

[Davephaelon]

I was just looking at d-wave symmetry for a pair of electrons in atomic orbit, and it shows a 4 leaf clover pattern. Does this mean that the pattern traced out by a Cooper pair, as they trace out their motion within the lattice is like a 4-leaf clover, even though the scale of the pattern could be hundreds of times larger than in the atomic case?

 

[f95toli]

Not quite, because the d-wave pattern is in k-space; not real space.
I am not sure there is a simple way of thinking about the actual motion, I've never come across a "simple" mental image that was accurate.

 

[radium]

d wave symmetry refers to the symmetries of the cooper pair order parameter in Fourier space. There are actually several examples of types of pairing in superconductors which you can see from mean field solutions for the order parameter. In addition to this, in the high Tc hole doped cuprates, there are a lot of different interesting phases with different orders that occur on the phase diagram (some in the non superconducting part where there are not yet Cooper pairs) depending on the hole doping and temperature. Many of these have been seen in experiments. A Cooper pair density wave in the superconducting phase is when the cooper pair is at finite momentum.

Theoretically you can have p wave pairing although I believe there is no known example.
In 2d a spinless p+ip paired sc mode has topologically protected Majorana fermion modes on the edge. You can however get a similar state if you put an s wave superconductor on top of a topological insulator.

So in summary these different types of pairings are all very interesting.

 

[ZapperZ]

Theoretically you can have p wave pairing although I believe there is no known example.

The ruthenates, such as Sr₂RuO₄, are often cited as having such paring symmetry.

Zz.

 

[Davephaelon]

In this pdf file http://www.phys.ufl.edu/~pjh/teaching/phz7427/7427notes/ch5.pdf, devoted to superconductivity, it states under section 5.5.2 [Is the BCS order parameter general?], "and 2) insisting on a spin singlet state because the phonon mechanism leads to electron attraction when the elec- 16 trons are at the same spatial position (because it is retarded in time!),"... This is at the bottom of page 16 and top of page 17.

I was puzzled what they meant by "...retarded in time!" Does it mean something is time reversed?

 

[DrDu]

No, it means that the interaction between the two electrons is not instantaneous, as is the Coulombic repulsion, but that the phonons emitted by one electron need a finite time to reach the other electron. As photons travel much faster than phonons, this effect can usually be ignored in solid state physics.

 

[Davephaelon]

DrDu, thank you for the reply and explanation. But I have another question. Cooper pairs have a slight binding energy attraction of milli-volts, mediated by lattice vibrations (phonons). From what I've read here, and elsewhere, Cooper pairs, along with their lattice phonon interactions, are treated as a quasiparticle, which I assume means that a single wavefunction can be used to describe the whole system. Is there a term in that wavefunction that denotes this tiny electrical attraction?

I'm a long way from being fluent in the mathematical language of solid state physics/QM, so I'm kind of jumping ahead of myself here. But I have a number of books on the subjects, and am working on learning these fields beyond the pretty much basic level that I'm at now.

 

[DrDu]

I didn't find it easy to find a decent book on superconductivity. I have a row on the shelf, but I think one of the best is still J. Schrieffer, "Theory of superconductivity". I think you will find answers to many of your questions in this book.