Field-dependent nonlinear surface resistance and its optimization by surface nanostructuring in superconductors

We propose a theory of nonlinear surface resistance of a dirty superconductor in a strong radio-frequency (rf) field, taking into account magnetic and nonmagnetic impurities, finite quasiparticle lifetimes, and a thin proximity-coupled normal layer characteristic of the oxide surface of many materials. The Usadel equations were solved to obtain the quasiparticle density of states (DOS) and the low-frequency surface resistance R-s as functions of the rf field amplitude H-0. It is shown that the interplay of the broadening of the DOS peaks and a decrease of a quasiparticle gap caused by the rf currents produces a minimum in R-s(H-0) and an extended rise of the quality factor Q(H-0) with the rf field. Paramagnetic impurities shift the minimum in R-s(H-0) to lower fields and can reduce R-s(H-0) in a wide range of H-0. Subgap states in the DOS can give rise to a residual surface resistance while reducing R-s at higher temperatures. A proximity-coupled normal layer at the surface can shift the minimum in R-s(H-0) to either low and high fields and can reduce R-s below that of an ideal surface. The theory shows that the behavior of R-s(H-0) changes as the temperature and the rf frequency are increased, and the field dependence of Q(H-0) can be very sensitive to the materials processing. Our results suggest that the nonlinear rf losses can be minimized by tuning pair-breaking effects at the surface using impurity management or surface nanostructuring.