Microwave response of vortices in superconducting thin films of Re and Al

Vortices in superconductors driven at microwave frequencies exhibit a response related to the interplay between the vortex viscosity, pinning strength, and flux creep effects. At the same time, the trapping of vortices in superconducting microwave resonant circuits contributes excess loss and can result in substantial reductions in the quality factor. Thus, understanding the microwave vortex response in superconducting thin films is important for the design of such circuits, including superconducting qubits and photon detectors, which are typically operated in small, but nonzero, magnetic fields. By cooling in fields on the order of $100\mu \text{T}$ and below, we have characterized the magnetic field and frequency dependence of the microwave response of a small density of vortices in resonators fabricated from thin films of Re and Al, which are common materials used in superconducting microwave circuits. Above a certain threshold cooling field, which is different for the Re and Al films, vortices become trapped in the resonators. Vortices in the Al resonators contribute greater loss and are influenced more strongly by flux creep effects than in the Re resonators. This different behavior can be described in the framework of a general vortex dynamics model.

Figure 1

(Color online) (a) Chip layout showing common feedline and four resonators. (b) Atomic force microscope (AFM) image of portion of Al chip. (c) Schematic of measurement setup including cold attenuators with values listed in dB.

Figure 2

(Color online) (a) Dips in magnitude of $S_{21}$ for different microwave drive power for the Re resonator near 1.8 GHz, $B=92.5\mu \text{T}$. (b) Magnitude and phase of $S_{21}$ for Re resonator near 1.8 GHz cooled in $B=56.4\mu \text{T}$ (symbols) along with the corresponding fit (solid lines) as described in text; $Q_{\text{fit}}=10040$, $f_0=1.758499\text{GHz}$.

Figure 3

(Color online) Magnitude of $S_{21}$ for different cooling fields $B$ for resonator near 1.8 GHz on (a) Re chip with $B$ from 0 to $149.6\mu \text{T}$ and (b) Al chip with $B$ from 0 to $94.5\mu \text{T}$.

Figure 4

(Color online) Fractional frequency shift $\delta f/f_0\left(B\right)$ for (a) Re and (b) Al. Excess loss due to vortices $1/Q_v\left(B\right)$, as defined by Eq. (2) for (c) Re and (d) Al.

Figure 5

(Color online) $r\left(B\right)$ for (a) Re and (b) Al films for four different resonator lengths.

Figure 6

(Color online) $r\left(f\right)$ for Re with $B=-130.9\mu \text{T}$ (closed circles) and Al with $B=-75.5\mu \text{T}$ (open circles) along with fits as described in the text. Fit parameters are $f_d=22.6\text{GHz}$ (4.2 GHz) and $ϵ=0.0039$ (0.15) for Re (Al).

Figure 7

(Color online) $B$ dependence of parameters from fits to $r\left(f\right)$ at each $B$: $f_d$ for (a) Re and (b) Al; $ϵ$ for (c) Re and (d) Al films. Note the different scale factors on $ϵ$ between (c) and (d).

Figure 8

(Color online) Plot of $q\left(f_0/f_d\right)$, computed based on the definition in the text, which is proportional to $\partial \left(1/Q_v\right)/\partial B$ with $ϵ$ values from the text corresponding to Re (solid red line) and Al (dashed purple line).

Figure 9

(Color online) $1/Q_v\left(B\right)$ for $B\ge 0$ for Re and Al together for (a) lowest-frequency resonator and (b) highest-frequency resonator. Solid lines are linear fits to the $B$ dependence well beyond the shoulder region as described in text. $B_{\text{th}}$ corresponds to intercept of fit line with $1/Q_v=0$.

Figure 10

(Color online) (a) Calculated current-density distribution normalized by the average current density, $j\left(x\right)=J_s{\left(x\right)}^2/{⟨J_s⟩}^2$, for CPW geometry with parameters for Al resonator: $w=11.5\mu \text{m}$ (indicated by blue dashed lines) and $6.4\mu \text{m}$ gap between the center conductor and the ground plane (indicated by red dashed-dotted lines). Predicted vortex configurations in the absence of pinning disorder based on Ref. 25 for (b) $B_{\text{th}}<B<2.48B_{\text{th}}$ and (c) $B>2.48B_{\text{th}}$.