Surface impedance and reflectivity of superconductors

Abstract

A complete solution to the Mattis–Bardeen equations of the anomalous skin effect in superconductors [Phys. Rev. 111, 412 (1958)] is presented in the case of plane, bulk conductors. This solution shows good agreement with existing solutions in the microwave region, and for the first time, it correctly describes measurements in the far-infrared region. It turns out that the solution to the Mattis–Bardeen equations for the extreme anomalous limit cannot be used for a correct description of experimental results. In addition, our exact solution is also applicable to strong-coupling superconductors.

Superconducting Microstrip Line Model Studies at Millimetre and Sub-Millimetre Waves

“…For the magnetic and electric field penetration depths, λ and δ r respectively, the latter is also known as a skin depth, we can write the following expressions [18]: In Figure 2, we present the plots of the complex conductivity components and the penetration depths for E and H field of the electromagnetic wave into Nb material with given properties. The sharp change in the frequency dependence of σ 1 and connected to that the magnetic field penetration depth λ is clearly seen around Nb gap frequency (for chosen Nb material parameters) at about 650 GHz.…”

“…The applicability of the Mattis -Bardeen theory to specific superconducting materials is discussed in [17], [18] and integral form equations for calculating  1 and  2 are presented in [17].…”

“…The actual magnetic field penetration depth,  has a noticeable frequency dependence for frequencies even well below the gap frequency and, as the frequency approaches the gap frequency of the superconducting material,  becomes highly frequency dependent [14] - [16]. The microstrip line with the superconducting electrodes becomes material-dispersive and, above the gap frequency has additionally an increased conducting loss in the strip and ground electrodes due to breaking of Cooper pairs [17], [18].…”

“…The specific surface impedance Z s per square (unit of area) of a superconducting film with thickness t is given in reference [18] as:…”

Superconductor–insulator–superconductor tunnel strip line: Features and applications

“…However, at frequencies near the gap frequency F g ϭeV g /h of the superconducting material, a microstrip line with superconducting electrodes is dispersive because 0 is frequency dependent, 12,13 and above the gap frequency the line has increased losses. 14,15 The Mattis-Bardeen theory 16 of the anomalous skin effect describes the behavior of a superconductor in terms of complex conductivity ,…”