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We propose a theory of nonlinear surface resistance of a dirty superconductorin a strong radio-frequency (RF) field, taking into account magnetic andnonmagnetic impurities, finite quasiparticle lifetimes, and a thinproximity-coupled normal layer characteristic of the oxide surface of manymaterials. The Usadel equations were solved to obtain the quasiparticle densityof states (DOS) and the low-frequency surface resistance $R_s$ as functions ofthe RF field amplitude $H_0$. It is shown that the interplay of the broadeningof the DOS peaks and a decrease of a quasiparticle gap caused by the RFcurrents produces a minimum in $R_s(H_0)$ and an extended rise of the qualityfactor $Q(H_0)$ 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 whilereducing $R_s$ at higher temperatures. A proximity-coupled normal layer at thesurface can shift the minimum in $R_s(H_0)$ to either low and high fields andcan reduce $R_s$ below that of an ideal surface. The theory shows that thebehavior of $R_s(H_0)$ changes as the temperature and the RF frequency areincreased, and the field dependence of $Q(H_0)$ can be very sensitive to thematerials processing. Our results suggest that the nonlinear RF losses can beminimized by tuning pairbreaking effects at the surface using impuritymanagement or surface nanostructuring.

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