Evidence for temperature-dependent spin diffusion as a mechanism of intrinsic flux noise in SQUIDs

The intrinsic flux noise observed in superconducting quantum interference devices (SQUIDs) is thought to be due to the fluctuation of electron-spin impurities, but the frequency and temperature dependence observed in experiments do not agree with the usual $1/f$ models. We present theoretical calculations and experimental measurements of flux noise in rf SQUID flux qubits that show how these observations, and previous reported measurements, can be interpreted in terms of a spin-diffusion constant that increases with temperature. We fit measurements of flux noise in 16 devices, taken in the $20\text{--}80$ mK temperature range, to the spin-diffusion model. This allows us to extract the spin-diffusion constant and its temperature dependence, suggesting that the spin system is close to a spin-glass phase transition.

Figure 1

Theoretical calculation of flux noise due to spin diffusion in a rectangular SQUID of sides $L_{\parallel }$ and $L_{\perp }$, with wire width $W$ (we used $L_{\parallel }/L_{\perp }=350/3$ and $L_{\perp }/W=3$ to compare to the experiments below). The solid curves show ${\tilde{S}}_{\Phi }\left(f\right)$ for a range of $D$. The axes are normalized to $f_0$ and $S_0\equiv {\tilde{S}}_{\Phi }\left(f_0\right)$, where $f_0$ is chosen to be the point where the curves appear to cross. We also normalize $D$ by the quantity $D_0=2\pi W^2{f}_0$. The dashed line shows the fitting function used in Eq. (12). Inset: A zoom into the crossing band region at $f_0$. The arrows show the behavior for increasing $D$. At frequencies lower than $f_0$, the noise spectral density decreases with increasing $D$; at frequencies higher than $f_0$ the opposite behavior takes place.

Figure 2

Typical measurements of low-frequency noise. We plot ${\tilde{S}}_{\Phi }\left(f\right)$ vs $f$ for one of the qubits ($q2$) at $T=20$ and 60 mK. The solid curves show fits to Eq. (11). We observe a crossing at $0.1\text{--}1$ Hz across the measured temperature range, roughly consistent with Ref. [10].

Figure 3

Fit values of $\sigma \chi /{\chi }_0$ vs temperature for 16 devices. $\sigma \chi /{\chi }_0$ was obtained by fitting the measured power spectral density to Eq. (11). Within the experimental error bars, $\sigma \chi /{\chi }_0$ seems to be independent of temperature and device. For $\chi ={\chi }_0$, this suggests the spin areal density, $\sigma$, is constant.

Figure 4

Fit value of the spin-diffusion constant $D$ vs temperature for 16 devices. $D$ was obtained by fitting the measured power spectral density to Eq. (11). In the low-temperature range, $D$ shows a clear trend to increase with temperature. In the higher-temperature range the increasing contribution of the white-noise background $w_n$ increases the error bars, making this trend less clear.

Figure 5

${\tilde{S}}_{\Phi }\left(1\mathrm{Hz}\right)$ vs $T$ for all 16 devices. We used the fit parameters shown in Figs. 3 and 4 and Eq. (12) to calculate ${\tilde{S}}_{\Phi }\left(1\mathrm{Hz}\right)$. Note that ${\tilde{S}}_{\Phi }\left(1\mathrm{Hz}\right)$ decreases with increasing $T$.

Figure 6

An illustration of the geometry of the rf SQUID flux qubit wiring used in our theoretical calculations and experimental measurements. Our SQUIDs had wire width $W=1$ $\mu$m, wire thickness $b=0.2$ $\mu$m, and lateral dimensions $L_{\parallel }=350\mu$m and $L_{\perp }=3\mu$m. For simplicity we include a single Josephson junction in the illustration. The measurements reported herein were taken with compound-compound Josephson-junction devices described in detail elsewhere [18].

Figure 7

Calculations of the modulus of the flux vector $\mathit{F}\left(\mathit{r}\right)$ as a function of spin position $\mathit{r}$ for the rf SQUID flux qubit geometry of Fig. 6. Here $x$ is the spin's coordinate along the SC wire width (the wire edges are at $x=\pm W/2$), and $z$ is the spin coordinate perpendicular to the wire ($z=0$ is at the wire surface). Note that at the wire surface $\left|\mathit{F}\right|$ is within 20% of $F_0=4$ n${\Phi }_0$.

Figure 8

Plots of $\sqrt{w_n}$ vs refrigerator thermometry $T$ for 16 qubits. The data scale as expected from Eq. (B9), direct evidence that the qubit temperature matches the refrigerator temperature. Note in particular that $\sqrt{w_n}$ does not saturate at low $T$, demonstrating that the qubit temperature and refrigerator thermometry are in agreement even at the lowest temperatures.