Surface impedance and optimum surface resistance of a superconductor with an imperfect surface

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.

Figure 1

(a) A surface layer of gradually reduced BCS pairing constant $g\left(x\right)$. Inset shows a profile of $g\left(x\right)$. (b) A superconductor covered with a normal layer of thickness $d$. The vertical black line in (b) shows the S-N interface giving rise to the contact resistance $R_B$.

Figure 2

Density of states at the surface calculated for $\mathrm{\Psi }=0.2$ and $\mathrm{\Gamma }=0.01$ (red line). The blue line shows DOS in the bulk.

Figure 3

A contour map of the self-consistent pair potential $\mathrm{\Psi }$ calculated from Eq. (43) at $k_{B}T/\mathrm{\Delta }=0.057$ and $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$. The parameters $\alpha$ and $\beta$ are defined by Eqs. (11) and (12).

Figure 4

Densities of states at (a) N side and (b) S side of the interface, calculated for $\alpha =0.05$, $\mathrm{\Gamma }=0$, $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$, and $\beta =0.1,1,10$. Inset in (b) shows details of the proximity-induced low-energy tail of $N\left(\epsilon \right)$ at S side.

Figure 5

Minigap ${\epsilon }_0$ in the N layer as a function of $\beta$ calculated from Eq. (57).

Figure 6

Densities of states at (a) N side and (b) S side of the interface calculated for $\alpha =0.05$, $\mathrm{\Gamma }=0.05\mathrm{\Delta }$, $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$, and $\beta =0.1,1,10$. Inset in (b) shows $N\left(\epsilon \right)$ at $\epsilon \ll \mathrm{\Delta }$.

Figure 7

Arrhenius plots for $R_s\left(T\right)$ calculated from Eqs. (82) and (83) for $\hslash \omega =0.01\mathrm{\Delta }$ and $\mathrm{\Gamma }/{\mathrm{\Delta }}_0=0.01$, 0.03, and 0.06. Here $R_1={\mu }_0^2{\lambda }^3\omega \mathrm{\Delta }/2\hslash {\rho }_s$, the temperature dependencies of $\mathrm{\Delta }$ and $\lambda$ at $T<T_c/2$ are neglected, and $\mathrm{\Delta }\left(\mathrm{\Gamma }\right)$ is given by Eq. (20).

Figure 8

Minimum in the surface resistance $R_s\left({\mathrm{\Gamma }}_p\right)$ as a function of the spin-flip pair-breaking parameter ${\mathrm{\Gamma }}_p$ calculated from Eqs. (87, 88, 89, 90, 91, 92, 93) at $\hslash \omega =0.005\mathrm{\Delta }$ and $k_{B}T/\mathrm{\Delta }=0.1,0.12,0.15$. Inset shows $N\left(\epsilon \right)=N_{s}coshucosv$ calculated from Eqs. (90, 91, 92) at ${\mathrm{\Gamma }}_p/\mathrm{\Delta }=0.001,0.02,0.05$.

Figure 9

$R_s\left(T\right)$ calculated from Eqs. (94, 95, 96, 97) and (30) for $\mathrm{\Gamma }=0.01\mathrm{\Delta }$, $\lambda =5{\xi }_S$, and $\mathrm{\Psi }=0,0.4$. Here $R_1={\omega }^2{\mu }_0^2{\lambda }^3{\sigma }_s/2$.

Figure 10

Arrhenius plots calculated from Eqs. (79, 80, 81) for $\alpha =0.05$, $\lambda =4{\xi }_S$, $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$, $D_n=D_s/3$, $\beta =0.1,2,4,30$, and (a) $\mathrm{\Gamma }=0.01\mathrm{\Delta }$, (b) $\mathrm{\Gamma }=0.05\mathrm{\Delta }$. Here $R_1={\mu }_0^2{\omega }^2{\xi }_S{\lambda }^2/2{\rho }_s$.

Figure 11

Minima in $R_s\left(\beta \right)$ as a function of the interface barrier parameter $\beta$ calculated at $\mathrm{\Delta }/k_{B}T=4,8,12,20$, $\mathrm{\Gamma }=0.01,0.02,0.03$, $\lambda =4{\xi }_S$, $D_n=0.1D_s$, $\alpha =0.05$, and $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$. Here $R_s(T,\beta ,\mathrm{\Gamma })$ are normalized to their respective values $R_{s0}(T,\mathrm{\Gamma })$ for each $\mathrm{\Gamma }$ in the absence of the N layer.

Figure 12

Behavior of $N\left(\epsilon \right)$ in a narrow energy range $\epsilon \approx \mathrm{\Delta }$ at the S side of the N-S interface calculated for $\beta =0.1,0.5,1$ at $\alpha =0.05$, $\hslash \mathrm{\Omega }=11\mathrm{\Delta }$, and $\mathrm{\Gamma }=0$.