Effective microwave surface impedance of superconducting films in the mixed state

The effective microwave surface impedance of multilayer structures made of high-T_c superconducting films in the mixed state, lossy dielectrics, and normal metals are theoretically calculated. The linear response of the superconductor to a microwave field is analyzed within both transmission line theory and the framework of a self-consistent treatment of vortex dynamics reported by Coffey and Clem (1991). The microwave properties are investigated as a function of static field and film thickness for nonresonant structures. The effect of substrate thickness on the resonant phenomenon is carefully studied as well. Numerical results reveal that the substrate resonance in the Meissner state behaves like a parallel lumped-parameter resonator, while in the vortex state it behaves as a series lumped resonant circuit. The basic distinction suggests that care should be taken in microwave applications when using superconducting films in the vortex state