Impurities effects in Type II SCs

[Shiricon]

Hello,

I have a question about how non-magnetic impurities affect the transition temperature in type II superconductors.

I know that the impurities introduce additional scattering and therefore act to reduce Tc, but is there a simple formula that would relate the effect on Tc due to particles of a particular size in relation to xi or lambda? Also, what about particle shape, as a sphere would cause scattering different from that of a long needle-like impurity?

Most of the papers I have found present experimental results showing the decrease of Tc, but I haven't found any that cite where I could find a theoretical estimation for the reduction of Tc with a specific impurity. I have found some very lengthy and complex formulas in a few papers, but it was not clear to me how the size or shape would factor into these formulas.

If you know a source or if you can guide me in the right direction, that would be helpful. Thank you!

 

[f95toli]

"Particles of a particular size" doesn't really make much sense in this context, when we talk about impurities in SC we generally refer to atoms (well, ions); not particles. It is a microscopic effect. Also, "geometric size" is usually not very important, what matters is how well an impurities scatters electrons/pairs which in turn depends on its moment.

Particles will of course act affect vortex dynamics etc, but that won't affect Tc directly.
If you have a large number of particles dispersed in the solid one could of course imagine that this affects the phonon spectrum etc which might affect Tc somewhat; but that effect would presumably be very small.

 

[Shiricon]

Thank you for your input. I am trying to get the sense for this...

In a type II superconductor in the vortex state, impurities can act as pinning centers. However, with the addition of impurities/defects you reduce the mean free path by introducing additional scattering. As the Cooper pair is of the order of xi, I would naively think that a particle below the order of xi would have much less effect on scattering than a particle >xi. By particle I could mean ion; I am using the term generically to mean "something" of finite size, whether it be a single atom or collection of atoms.

But this is what I am trying to find out... "We" like to add impurities to help pin vortices, but what effect are we having on Tc (experimentally I know that the Tc will be reduced).

For magnetic impurities, the Abrikosov-Gorkov theory provides a pair breaking term alpha which takes into account the exchange energy between the electron spin and the magnetic impurity. This alpha is in terms of the concentration and contains a critical concentration for which superconductivity can not exist for any temperature. In this way, I can use the magnetic properties of the impurity with the concentration to calculate the effect on lowering Tc.

I would like a similar formula for a non-magnetic particle/impurity/ion. I would like a formula for which I can take the properties and concentration (and potentially the particle size) of the non-magnetic particle and roughly estimate the effect on Tc, whether that shift be .01, 1, or 10 degrees.

I suppose I should mention that I am interested in doing calculations for bulk samples, and not thin films or wires.

The point... If I am making a bulk superconductor, say like a pellet of MgB2, and I want to mix in a powder of non-magnetic impurities to increase pinning in the sample, but I also want to know how that would affect Tc. If I'm using a powder, what effect do I expect my choice of powder size (nano-powders versus micro-powders) to have on the reduction of Tc? I can look at the experimental results and see the Tc shifts will typically be small, but I assume there is a formula somewhere that estimates the relative shift in Tc, and that is what I am looking for.

 

[Shiricon]

I thought I would post this in case anyone comes across this post in the future and is looking for a similar result.

I found a paper that answers my question sufficiently.
Markowitz, D. and Kadanoff, L. P. (1963). Effects of impurities upon critical temperature of anisotropic superconductors. Physical Review, 131(2):563.

There is an appendix at the end of the article that contains a formulation for dT_c if the scattering cross section and anisotropy interaction are known.