Contribution of Quasiparticles in the Subgap States to the Surface Impedance of Superconductors

The surface impedance of a superconductor has been calculated using the extended Mattis-Bardeen theory, in which the complex gap energy are taken into account. It is found that the surface resistance of the superconductor increases with increasing magnitude of the imaginary part of the gap energy. It is also found that the surface resistance of an NbN film calculated by the extended Mattis-Bardeen equation quantitatively agrees well with the measured one. It is shown that temperature dependence of not only values but also residual numbers of quasiparticles of superconducting resonators agree well with those calculated by the extended Mattis-Bardeen equation. It is also shown that phonon-assisted gradual relaxation of excess quasiparticles passing through the intragap states is dominant rather than the direct recombination into Cooper pairs.