Microwave response of superconductors that obey local electrodynamics

Surface impedance of type-II superconductors is determined by their local optical conductivity $\sigma \left(\omega \right)$. Standard BCS-like theoretical descriptions of $\sigma \left(\omega \right)$, due to Mattis and Bardeen [Phys. Rev. 111, 412 (1958)] or Zimmermann et al. [Physica C 183, 99 (1991)], do not take pair-breaking processes into account. Therefore they do not provide a quantitative explanation of the microwave response, and in particular, they cannot predict the magnitude of the coherence peak. Here, based on the recently developed concept of Dynes superconductors, which does take also the pair-breaking scattering into account, we provide a simple but complete description of the microwave response of type-II superconductors, concentrating on the unexpected properties of clean superconductors which are often used as cavity materials.

Figure 1

Temperature dependence of the $\omega \to 0$ limit of ${\sigma }_1\left(T\right)/{\sigma }_N$ as a function of $T/T_c$ for several values of $\gamma$ and ${\gamma }_s$. Note that the same peak height can be reached for different combinations of $\gamma$ and ${\gamma }_s$.

Figure 2

Height of the coherence peak $\kappa ={\sigma }_{1,\max }/{\sigma }_N-1$ (magnitude indicated by the black labels) as a function of the dimensionless scattering rates ${\gamma }_s$ and $\gamma$.

Figure 3

Dependence of the slopes $\gamma /{\gamma }_s=f\left(\kappa \right)$ of the equal-$\kappa$ lines in the small-scattering-rate corner of Fig. 2 on the coherence-peak height $\kappa$. The inset shows an analogous plot of the slopes $\gamma /{\gamma }_s$ of the equal-$T_{\max }/T_c$ lines in the small-scattering-rate corner of Fig. 4 as a function of $T_{\max }/T_c$.

Figure 4

Position $T_{\max }/T_c$ (indicated by the black labels) of the coherence peak of ${\sigma }_1/{\sigma }_N$ as a function of ${\gamma }_s$ and $\gamma$.

Figure 5

Temperature dependence of the real part of the microwave conductivity ${\sigma }_1\left(T\right)/{\sigma }_N$ for three choices of $\gamma$ and ${\gamma }_s$ with a fixed ratio $\gamma /{\gamma }_s$.

Figure 6

Temperature dependence of the imaginary part of the microwave conductivity ${\sigma }_2\left(T\right)/{\sigma }_2\left(0\right)$ for several choices of $\gamma$ and ${\gamma }_s$. Note that when ${\gamma }_s$ decreases and $\gamma$ increases, the curves are pushed slightly downwards with respect to Eq. (14), which is shown by the blue line.

Figure 7

Coefficient $A$ describing the $T$ dependence of the imaginary part of the microwave conductivity close to $T_c, {\sigma }_2\left(T\right)/{\sigma }_2\left(0\right)=A(1-T/T_c)$, plotted as a function of ${\gamma }_s$ for several values of $\gamma$ and $x$. Note that $A$ depends only very weakly on $\gamma$ in the limit of small pair breaking.

Figure 8

Evolution of the normalized conductivities ${\sigma }_1\left(T\right)/{\sigma }_N$ (main panel) and ${\sigma }_2\left(T\right)/{\sigma }_2\left(0\right)$ (inset) with the strong-coupling rescaling parameter $x$. In both figures we have used ${\gamma }_s=0.89$ and $\gamma =0.012$.

Figure 9

Real and imaginary parts of the conductivities of the two samples from Fig. 4 in [18] (symbols), together with their fits by the theory of Dynes superconductors with the strong-coupling corrections described for both samples by $x=1.145$ (lines).