We calculate a low-frequency surface impedance of a dirty, $s$-wave superconductor with an imperfect surface incorporating either a thin layer with a reduced pairing constant or a thin, proximity-coupled normal layer. Such structures model realistic surfaces of superconducting materials which can contain oxide layers, absorbed impurities, or nonstoichiometric composition. We solved the Usadel equations self-consistently and obtained spatial distributions of the order parameter and the quasiparticle density of states which then were used to calculate a low-frequency surface resistance $R_s\left(T\right)$ and the magnetic penetration depth $\lambda \left(T\right)$ as functions of temperature in the limit of local London electrodynamics. It is shown that the imperfect surface in a single-band $s$-wave superconductor results in a nonexponential temperature dependence of $Z\left(T\right)$ at $T\ll T_c$ which can mimic the behavior of multiband or $d$-wave superconductors. The imperfect surface and the broadening of the gap peaks in the quasiparticle density of states $N\left(\epsilon \right)$ in the bulk give rise to a weakly temperature-dependent residual surface resistance. We show that the surface resistance can be optimized and even reduced below its value for an ideal surface by engineering $N\left(\epsilon \right)$ at the surface using pair-breaking mechanisms, particularly by incorporating a small density of magnetic impurities or by tuning the thickness and conductivity of the normal layer and its contact resistance. The results of this work address the limit of $R_s$ in superconductors at $T\ll T_c$, and the ways of engineering the optimal density of states by surface nanostructuring and impurities to reduce losses in superconducting microresonators, thin-film strip lines, and radio-frequency cavities for particle accelerators.

中文翻译：

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表面不完美的超导体的表面阻抗和最佳表面电阻

我们计算出一个肮脏的低频表面阻抗， $s$具有不完美表面的电磁波超导体，其中包含具有减小的配对常数的薄层或包含接近耦合的法线薄层。这种结构对超导材料的真实表面进行建模，这些表面可能包含氧化物层，吸收的杂质或非化学计量组成。我们自洽地求解了Usadel方程，获得了阶数参数和状态的准粒子密度的空间分布，然后将其用于计算低频表面电阻${\mathrm{[R}}_s（Ť）$ 和磁穿透深度 $\lambda （Ť）$作为温度的函数，在局部伦敦电动力学的极限范围内。结果表明，单波段表面不完美$s$波超导体导致温度的非指数依赖性 $ž（Ť）$ 在 $Ť\ll {Ť}_C$ 可以模仿多频段或 $d$波超导体。准粒子密度状态下表面的不完美和间隙峰的加宽$ñ（\epsilon ）$松散的体积会引起与温度有关的残余表面电阻。我们证明，通过工程设计，可以优化表面​​电阻，甚至可以将其降低到理想表面以下的值以下$ñ（\epsilon ）$使用成对断裂机制在表面上进行腐蚀，特别是通过掺入低密度的磁性杂质或通过调整正常层的厚度和电导率及其接触电阻来实现。这项工作的结果解决了${\mathrm{[R}}_s$ 在超导体 $Ť\ll {Ť}_C$，以及通过表面纳米结构和杂质工程化最佳状态密度的方法，以减少超导微谐振器，薄膜带状线和用于粒子加速器的射频腔中的损耗。