$1/f$ flux noise in low-$T_c$ SQUIDs due to superparamagnetic phase transitions in defect clusters

It is shown here that $1/f^{\alpha }$ flux noise in conventional low-$T_c$ SQUIDs is a result of low temperature superparamagnetic phase transitions in small clusters of strongly correlated color center defects. The spins in each cluster interact via long-range ferromagnetic interactions. Due to its small size, the cluster behaves like a random-telegraphic macrospin when transitioning to the superparamagnetic phase. This results in $1/f^{\alpha }$ noise when ensemble averaged over a random distribution of clusters. This model is self-consistent and explains all related experimental results which includes $\alpha \sim 0.8$ independent of system size. The experimental flux-inductance-noise spectrum is explained through three-point correlation calculations and time-reversal symmetry-breaking arguments. Also, unlike the flux noise, it is shown why the second-spectrum inductance noise is inherently temperature dependent due to the fluctuation-dissipation theorem. A correlation-function calculation methodology using Ising-Glauber dynamics was key for obtaining these results.

Figure 1

$1/f^{\alpha }$ noise model consisting of interacting and fluctuating TLSs in a cluster. The clusters are assumed to form at the metal-insulator interface or on the surface. They are sufficiently far apart so that only spins within a single cluster interact. Number of TLSs within a cluster and the lattice constant $a$ vary.

Figure 2

Comparison between calculated net correlation functions and fits to $\sum C_{\nu }exp(-{\mathrm{\Gamma }}_{\nu }t)$ at various temperatures. The fits are in perfect agreement with the calculations done for 50 spin clusters, each with seven spins.

Figure 3

(a) Net correlation function for 400 spin clusters, each with 6–9 spins. (b) The corresponding power spectrum for the flux noise $P_{\varphi }$ and (c) the slope of the flux noise. (d) The distribution of $k_{F}a$ for each spin cluster. The mean and standard deviation ($\mathrm{\Sigma }$) are as indicated.

Figure 4

Power spectrum comparison for different cluster sizes. (a) Power spectrum and (b) corresponding noise slope for 75 cluster with 6–9 spins. (c) Power spectrum and (d) corresponding noise slope for 20 clusters with six spins.

Figure 5

Calculations for spin clusters with nearest neighbor ferromagnetic interactions showing the (a) net correlation function. (b) The corresponding power spectrum for the flux noise and (c) the slope of the flux noise. (d) The distribution of $k_{F}a$ for each spin cluster. The mean and standard deviation ($\mathrm{\Sigma }$) are as indicated. The calculations are for 400 clusters with 6–9 spins.

Figure 6

Sum of all three-point autocorrelation and cross-correlation functions $C_{\text{3pt}}$, for $N=9$ spins with ferromagnetic RKKY interactions at (a) $T=200$ mK, (b) $T=300$ mK, and (c) $T=400$ mK where we note that the $z$ axis is ${10}^3$ times smaller. (d) The corresponding normalized power spectrum.

Figure 7

Superparamagnetic phase transitions for a single TLS cluster as a function of temperature. (a) Sum of three-point correlations $C_{\text{3pt}}$, (b) susceptibility $\chi$ shown for $N=6$ with varying $k_{F}a$, (c) $C_{\text{3pt}}$, and (d) $\chi$ for varying $N$ with $k_{F}a=1$. All values are obtained at initial times ($t_1=t_2=0$).

Figure 8

Relaxation times ${\tau }_r$ as a function of inverse temperature, leading up to $T_f$, shown for (a) $N=6$ and (b) $N=7$ for various $k_{F}a=1$. The lines are fits to an Arrhenius equation ${\tau }_{0}exp(E_b/T)$.

Figure 9

Sum of two-point correlations as a function of normalized temperature at initial times ($t=0$) shown for (a) $N=6$ with varying $k_{F}a$ and (b) varying the number of spins with $k_{F}a=1$.

Figure 10

Left axis shows the ensemble averaged sum of three-point correlations ($C_{3\mathrm{pt}}$) as a function of temperature (at $t_1=t_2=0$ initial time). The ensemble average is over the exact same 75 random $k_{F}a$ and $N$, which gives the noise exponent $\alpha$ on the right axis.

Figure 11

(a) Comparison of the noise exponent $\alpha$ against the specific heat ${\mathcal{C}}_v$ for 75 spin clusters, each with 6–9 spins. ${\mathcal{C}}_v$ is calculated using the Monte Carlo method. This is the same distribution of spin clusters that gives the $1/f$ noise shown in Fig. 4 with $\langle k_{F}a\rangle =0.63$. (b) A close up of $\alpha$.

Figure 12

Critical temperature $T_c^{\prime }$ from Monte Carlo simulations for the spin clusters with RKKY interactions as a function of cluster size $N$ for various normalized lattice spacing $k_{F}a$.

Figure 13

Inductance noise plots for different $\mathrm{\Delta }\nu ={\omega }_b-{\omega }_a$ showing the (a) Power spectrum and (b) its respective slope for ${\omega }_b=10$ Hz and ${\omega }_a={10}^{-3}$ Hz. (c) Power spectrum and (d) its respective slope for ${\omega }_b=10$ Hz and ${\omega }_a={10}^{-2}$ Hz. (e) Power spectrum and (f) its respective slope for ${\omega }_b=10$ Hz and ${\omega }_a={10}^{-1}$ Hz. Calculations are for 400 cluster with 6–9 spins.

Figure 14

The errors signifying how good the Gaussian approximation is for (a) ferromagnetic (FM) RKKY interaction and (b) antiferromagnetic (AFM) RKKY interactions. Here $t_1=t_2=t_3$ for $N=6$ and $k_{F}a=0.75$. The results are of the same order for larger $N$ and randomly sampled $t_1$ and $t_2$. (c) Corresponding specific heat for $t_1=0$.