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motivated by another question, i wonder if there are special properties of superconductivity when restricted on 2D or very thin layers related to the effective permittivity in function of the frequency $\epsilon ( \omega )$ near the first-order transitions between the superconducting and normal phases , any references about the main state of the art would be appreciated

Edit: I've noticed that most research papers on superconductivity report on the current transport properties, but they usually don't talk much about the permittivity. Is this because its harder to measure?

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The obvious case for superconductivity superflow in 2D is, well, superconductivity superflow in a 2DEG (2D electron gas). One of the many ways this manifests is in the quantum hall effect. There, one observes a step-like behavior for the resistivity of a 2D sample w.r.t externally controllable parameters such as the magnetic field or carrier concentration. The resistivity is quantized in integer or fractional units as shown in the plot below (courtesy of D.R. Leadley, Warwick University 1997).

Quantized resistance

When the magnetic field is such that the system lies on the "plateaus", in between the quantized resistance values, the 2DEG in the bulk of the sample is in a superconductivity state exhibiting dissipationless transport. This is also depicted by the green line in the plot which measures the longitudinal resistance $\rho_{xx}$ of the sample. We see that this quantity vanishes precisely when the system is on a plateau indicating the presence of superflow in the longitudinal direction.

Edit: As pointed out by @wsc and @4tnmele in the comments below, it is not quite accurate to describe the plateaus as being in a superconducting state. However on the plateaus the hall strip does exhibit dissipationless flow in the longitudinal direction - even though this is primarily due to edge currents. The bulk remains insulating. So while it might be correct to describe this state as exhibiting "superflow" it is incorrect to call it a "superconductor". I have modified the language in my answer to reflect this change. I am not deleting my answer because I feel that it is still relevant in the context of the OP's question.

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+1, i've noticed that most papers about superconductivity don't care too much about measuring the surface permittivity and just focus in the current transport characteristics – lurscher Mar 11 '11 at 17:20
-1. This is incredibly wrong -- there is no superflow in QHE! Superconductivity is much more than vanishing $\rho_{xx}$ anyway, and it is also trivially seen that the dc conductivity $\sigma_{xx} = 0$ in quantum Hall systems as well (this is because resistivity is the inverse of the conductivity tensor). – wsc Mar 11 '11 at 21:42
Dear @wsc, quoting from Jain's "Composite Fermions", pg. 19 "What is remarkable about the quantization of the Hall resistance is that ... concurrent with the quantized plateaus is a “superflow,” i.e., a dissipationless current flow in the limit of zero temperature." I call it superflow, Dr. Jain calls it superflow. You can call it whatever you like. – user346 Mar 12 '11 at 5:59
@wsc you might be right in that the state of the electron fluid on the plateaus is not exactly like that of a superconductor in that it does not exhibit cooper pairing. However, the fact is, the best description for the dissipationless flow observed at the plateaus is as "superflow". Likewise the mechanism for superconductivity in high T_c is different from low T_c, but we still call it superconductivity. – user346 Mar 12 '11 at 6:05
@Deepak Vaid Isn't superconductivity defined as perfect conductor + perfect diamagnet? Not sure FQHE fulfils this...? Although there are a few concepts from superconductivity that arises in FQHE, I am pretty sure that experts don't consider FQH systems as 2d superconductors. – Heidar Mar 12 '11 at 9:11

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