Millimeter-wave surface impedance of optimally-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ single crystals

Precision measurements of active and reactive components of in-plane microwave surface impedance were performed in single crystals of optimally-doped Fe-based superconductor Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ ($x=0.074$, $T_c=22.8$ K). Measurements in a millimeter wavelength range ($K_a$ band, 35–40 GHz) were performed using whispering gallery mode excitations in the ultrahigh quality factor quasioptical sapphire disk resonator with YBa${}_2$Cu${}_2$O${}_7$ superconducting ($T_c=90$ K) end plates. The temperature variation of the London penetration depth is best described by a power-law function, $\Delta \lambda \left(T\right)\sim T^n$, $n=2.8$, in reasonable agreement with radio-frequency measurements on crystals of the same batch. This power-law dependence is characteristic of a nodeless superconducting gap in the extended $s$-wave pairing scenario with a strong pair-breaking scattering. The quasiparticle conductivity of the samples, ${\sigma }_1\left(T\right)$, gradually increases with the decrease of temperature, showing no peak below or at $T_c$, in notable contrast with the behavior found in the cuprates. The temperature-dependent quasiparticle scattering rate was analyzed in a two-fluid model, assuming the validity of the Drude description of conductivity and generalized expression for the scattering rate. This analysis allows us to estimate the range of the values of a residual surface resistance from 3 to 6 $\mathrm{m}\Omega$.

Figure 1

Schematics of the slotted sapphire disk resonator experiment. The sapphire disk with a single crystal of Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ placed in the slot is sandwiched between superconducting YBa${}_2$Cu${}_3$O${}_7$ film end plates. Whispering gallery mode excitation at a $K_a$ band frequency (35 to 40 GHz) produces an electric field $E$ parallel to the conducting plane of the sample, enabling measurements of in-plane surface impedance.

Figure 2

Temperature-dependent (a) quality factor and (b) resonant frequency shift of the resonator with the single crystal Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$ (curves 1) and of the empty resonator (curves 2). Inset in panel (a) shows $Q\left(T\right)$ of the empty resonator in the temperature interval up to the superconducting transition of YBCO film end plates ($T_c=90$ K).

Figure 3

Temperature-dependent surface impe-dance of the single crystal of optimally-doped Ba(Fe${}_{1-x}$ Co${}_x$)${}_2$As${}_2$, $x=0.074$. Inset shows $R_s\left(T\right)$ in the low-temperature range.

Figure 4

The London penetration depth $\lambda \left(T\right)$ in single crystal of optimally-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, $x=0.074$, at $T<T_c/2$. The solid line is the power-law fit, $\sim T^{2.8}$, the dashed and the dot-dashed lines correspond to the experimental data of Refs. 11 and 12, respectively. The open circles represent experimental data. Inset shows $\lambda \left(T\right)$ in a broader temperature range.

Figure 5

The temperature-dependent superfluid density in a single crystal of optimally-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, $x=0.074$. The solid lines correspond to the power-law $\Delta \lambda \sim T^{2.8}$. A dashed line corresponds to a single-gap isotropic $s$-wave BCS superconductor with $\Delta =1.76k_B{T}_c$. Inset shows $n_s\left(T\right)$ calculated with $\Delta \lambda \left(T\right)\sim T^{2.8}$ (solid line) and the best exponential fit resulted in $\Delta =0.75k_B{T}_c$ (dash-dotted line).

Figure 6

Temperature-dependent quasiparticle conductivity, ${\sigma }_1\left(T\right)$, in a single crystal of optimally-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, $x=0.074$, calculated for different values of $R_{\mathrm{res}}$.

Figure 7

The temperature-dependent quasiparticle scattering rate ${\tau }^{-1}$ in a single crystal of optimally-doped Ba(Fe${}_{1-x}$Co${}_x$)${}_2$As${}_2$, $x=0.074$, calculated for different values of $R_{\mathrm{res}}$.