# Superconductor surface resistance

If the superconductive state is due to a phase transition from ohmic to a state a zero-resistivity, why the superconductive materials are usually identified by their surface resistance, or by their RRR parameter?

• For superconductors usually you use the resistivity right above the transition, or extrapolate the resistivity to zero temperature. – KF Gauss Feb 9 '17 at 23:51

The problem of evaluation the surface resistance of a superconductor thin, is very important in the Superconductive Resonant Cavity in microwave band.

The response of a superconductor to RF fields as well described by the Two-Fluid Model that consist to identify the

• Cooper pairs as superfluid
• Unpaired electrons as normal fluid, yield conductivity $\sigma_n$.

Studying the response of two fluids to a periodic electric field we can separate the two different contributes:

• Normal Current by Ohm's Law $J_n=\sigma_nE_0\exp(-i\omega t)$
• Supercurrent given by London equations $J_s=i\frac{n_c 2 e^2}{m_e \omega}E_0 \exp(-i\omega t)$

Now we can write the total current as

$J=J_n+J_s = \sigma E_0 \exp(-i\omega t)$

with $\sigma$ is a complex conductivity given by

$\sigma = \sigma_n + i\sigma_s \text{ with } \sigma_s=\frac{2n_ce^2}{m_e\omega}=\frac{1}{\mu_0\lambda_L^2\omega}$

So, is possible to define the surface resistance as: the real part of the the complex surface impedance

$R_{surf} = \Re\left(\frac{1}{\lambda_L(\sigma_n+i\sigma_s)}\right)=\frac{1}{\lambda_L}\frac{\sigma_n}{\sigma_n^2+\sigma_s^2}$

at microwave frequency $\sigma_n^2 \ll \sigma_s^2$ so $\boxed{R_{surf}\approx\frac{\sigma_n}{\lambda_L\sigma_s^2}}$

This model can be refined using an order parameter strictly link to the impurities of material.

Reference:

[1] - Basic Principles of RF Superconductivity and Superconducting Cavities - Peter Schmuser

[2] - Surface Resistance of a Superconductor - H. Safa

Superconductors possess zero resistance when cooled below critical temperature. There is a threshold between zero resistance and close-to-zero resistance.

I'm sending you links to two very useful links, you might want to check them out. Read this one and following Wikipedia article.