Ising-Glauber Spin Cluster Model for Temperature-Dependent Magnetization Noise in SQUIDs

Clusters of interacting two-level-systems, likely due to $\text{Farbe}+(F^{+})$ centers at the metal-insulator interface, are shown to self-consistently lead to $1/f^{\alpha }$ magnetization noise [with $\alpha (T)\lesssim 1$] in SQUIDs. Model calculations, based on a new method of obtaining correlation functions, explains various puzzling experimental features. It is shown why the inductance noise is inherently temperature dependent while the flux noise is not, despite the same underlying microscopics. Magnetic ordering in these systems, established by three-point correlation functions, explains the observed flux- inductance-noise cross correlations. Since long-range ferromagnetic interactions are shown to lead to a more weakly temperature dependent flux noise when compared to short-range interactions, the time reversal symmetry of the clusters is also not likely broken by the same mechanism which mediates surface ferromagnetism in nanoparticles and thin films of the same insulator materials.

Figure 1

Proposed $1/f^{\alpha }$ noise model consisting of interacting and fluctuating TLSs in a cluster. The clusters are assumed to form due to random defects at the SQUID’s metal-insulator interface and 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

(a) Temperature dependent net correlation function for 400 spin clusters, each with 6–9 spins with ferromagnetic($+|J_o|$) RKKY interactions. Note that the overall results are independent of cluster size [48]. (b) Corresponding flux noise spectrum showing $1/f^{\alpha }$ noise below $\sim 0.1\mathrm{Hz}$ and (c) its respective slope $\alpha$, where $\alpha \lesssim 1$ below about 0.1 Hz. (d) Distribution of $k_{F}a$ for each cluster, with mean $⟨k_{F}a⟩$ and standard deviation ($\mathrm{\Sigma }$) as indicated.

Figure 3

(a) Sample flux noise power spectrum $P_{\varphi }$ and (b) its slope for 400 spin clusters with nearest neighbor ferromagnetic interactions that vary randomly for each cluster.

Figure 4

(a) Power spectrum of the inductance noise, $P_L$ (associated noise of $P_{\varphi }$ of Fig. 2) and (b) its respective slope ($\alpha$) for the 400 spin clusters with ferromagnetic ($+|J_o|$) RKKY interactions. The integrated $⟨\alpha ⟩$ between 0.001 and 0.05 Hz is $\{1.569,1.415,1.247,1.242\}$ for the respective temperatures in ascending order.

Figure 5

Normalized sum of all three-point correlation functions, $\sum ⟨s_i(0)s_j(t_1)s_k(t_2)⟩/N^3$, indicating flux-inductance-noise cross correlation for a single cluster of $N=10$ spins with ferromagnetic ($+|J_o|$) RKKY interactions at temperatures of (a) $T=200\mathrm{mK}$ (b) $T=300\mathrm{mK}$, and (c) $T=400\mathrm{mK}$.