Loss Mechanisms and Quasiparticle Dynamics in Superconducting Microwave Resonators Made of Thin-Film Granular Aluminum

Superconducting high kinetic inductance elements constitute a valuable resource for quantum circuit design and millimeter-wave detection. Granular aluminum (grAl) in the superconducting regime is a particularly interesting material since it has already shown a kinetic inductance in the range of $\mathrm{nH}/\square$ and its deposition is compatible with conventional $\mathrm{Al}/\mathrm{AlOx}/\mathrm{Al}$ Josephson junction fabrication. We characterize microwave resonators fabricated from grAl with a room temperature resistivity of $4\times {10}^3\mu \mathrm{\Omega }\mathrm{cm}$, which is a factor of 3 below the superconductor to insulator transition, showing a kinetic inductance fraction close to unity. The measured internal quality factors are on the order of $Q_i={10}^5$ in the single photon regime, and we demonstrate that nonequilibrium quasiparticles (QPs) constitute the dominant loss mechanism. We extract QP relaxation times in the range of 1 s and we observe QP bursts every $\sim 20\mathrm{s}$. The current level of coherence of grAl resonators makes them attractive for integration in quantum devices, while it also evidences the need to reduce the density of nonequilibrium QPs.

Figure 1

Optical images of the grAl resonators and the 3D waveguide sample holder. The copper waveguide provides a low-loss environment [51, 52] for three grAl resonators patterned by $e$-beam lift-off lithography on a $10\times 15{\mathrm{mm}}^2$ $c$-plane sapphire chip. The resonator dimensions and resonant frequencies $f$ are the following: $A$—$400\times 5.4\mu {\mathrm{m}}^2$, $f=6.994\mathrm{GHz}$; $B$—$1000\times 40\mu {\mathrm{m}}^2$, $f=6.025\mathrm{GHz}$; and $C$—$600\times 10\mu {\mathrm{m}}^2$, $f=6.287\mathrm{GHz}$. The 20 nm thick grAl film has a resistivity $\rho =4\times {10}^3\mu \mathrm{\Omega }\mathrm{cm}$, corresponding to a sheet resistance $R_s=2\mathrm{k}\mathrm{\Omega }/\square$. By comparing the measured resonant frequencies with a finite elements method (FEM) simulation we extract a kinetic inductance $L_{\text{kinetic}}=2\text{nH}/\square$.

Figure 2

Measurement of radio frequency loss mechanisms in grAl films. (a) Measurement of the relative shift of the resonant frequency $\delta f/f=[f(T)-f(0.02)]/f(0.02)$ as a function of temperature. From FEM simulations we expect the kinetic inductance to be three orders of magnitude larger than the geometric inductance. The data can be fitted using a Bardeen-Cooper-Schrieffer (BCS) model [59, 60], $\delta f(T)/f=-\frac{\alpha }{2}\sqrt{\pi {\mathrm{\Delta }}_0/(2k_{B}T)}\exp (-{\mathrm{\Delta }}_0/k_{B}T)$, which is expected to approximately describe the temperature dependence of the frequency, but which does not take into account corrections to the prefactor which can arise either as temperature changes from below to above $hf/k_B$ or due to deviations of grAl from standard BCS theory [32]. The black line shows the fit for the kinetic inductance fraction $\alpha =1$. We extract a value for the grAl superconducting gap ${\mathrm{\Delta }}_0=(288\pm 4)\mu \mathrm{eV}$, in agreement, within 15%, with the measured gap from THz spectroscopy [32, 49], and previously reported values [33]. Notice that the grAl gap is $\sim 1.6$ times larger than that of thin film aluminum. When using $\alpha$ as a free fit parameter, we obtain similar values ($\alpha =0.92\pm 0.06$, ${\mathrm{\Delta }}_0=(282\pm 4)\mu \mathrm{eV}$). (b) Measured internal quality factors $Q_i$ as a function of the average circulating photon number $\overline{n}$. The solid lines represent a fit to the QP activation model of Eq. (1), discussed in the main text. (c) Comparison between measured $Q_i$ in the single photon regime as a function of the metal-substrate participation ratio $p_{\mathrm{MS}}$ for different resonator geometries (see Supplemental Material [54]). These results suggest that high kinetic inductance grAl resonators $A–C$ are limited by excess QPs, not by surface dielectric loss. grAl resonators in CPW geometry, with $p_{\mathrm{MS}}>{10}^{-3}$, are limited by a surface dielectric loss tangent $\tan (\delta )=2.4\times {10}^{-3}$, similar to aluminum qubits [61]. For a detailed discussion see the main text.

Figure 3

Measurement of QP generating events. (a) Typical plot of a continuous monitoring of a resonator’s phase signal at one frequency point. Multiple time traces are consecutively recorded, covering a total time of about 45 min (for clarity only partially shown). The measurement reveals discrete jumps of the resonant frequency to a lower value followed by a relaxation over seconds approximately every 20 s. (b) Plot of the time trace indicated by the arrow in panel (a), where the phase response is converted into a frequency shift, showing an instantaneous drop followed by a slow relaxation. (c) By recording multiple events and averaging them (see Supplemental Material [54]), an exponential tail can be seen. The characteristic relaxation time, ${\tau }_{\mathrm{ss}}$, depends on the average circulating photons $\overline{n}$.

Figure 4

Steady state QP relaxation constant ${\tau }_{\mathrm{ss}}$ as a function of $\overline{n}$, and its correlation with $Q_i$. (a) Resonator $A$ (orange triangles) shows a decrease of ${\tau }_{\mathrm{ss}}$ by approximately an order of magnitude whereas ${\tau }_{\mathrm{ss}}$ of $B$ and $C$ stays constant, within a factor of 2. The solid line is a fit to the phenomenological model described by Eq. (2). All fit parameters are given in the inset of panel (b). Error bars, where not plotted, are approximately the size of the marker and represent the statistical error of the fit (see Supplemental Material [54]). (b) Correlation between ${\tau }_{\mathrm{ss}}$ and $Q_i$ for resonators $A$ and $B$. We obtain ${\tau }_{\mathrm{ss}}$ and $Q_i$ in separate measurements and plot $Q_i({\tau }_{\mathrm{ss}})$ for values measured at the same photon number. The orange line is traced using the fits for the measured ${\tau }_{\mathrm{ss}}$ [panel (a)] and $Q_i$ [Fig. 2].