Complex Inductance, Excess Noise, and Surface Magnetism in dc SQUIDs

We have characterized the complex inductance of dc SQUIDs cooled to millikelvin temperatures. The SQUID inductance displays a rich, history-dependent structure as a function of temperature, with fluctuations of order 1 fH. At a fixed temperature, the SQUID inductance fluctuates with a $1/f$ power spectrum; the inductance noise is highly correlated with the conventional $1/f$ flux noise. The data are interpreted in terms of the reconfiguration of clusters of surface spins, with correlated fluctuations of effective magnetic moments and relaxation times.

Figure 1

(a) Schematic of asymmetric dc SQUID circuit for investigation of surface magnetism. The ac excitation current $I_{\mathrm{ex}}$ is injected directly into the SQUID loop. The current $I_{\mathrm{null}}$ provides a compensating flux, and the SQUID response to the excitation current is measured with a lock-in amplifier. (b) Layout of the $\mathrm{Nb}\mathrm{\text{−}}{\mathrm{AlO}}_x\mathrm{\text{−}}\mathrm{Nb}$ SQUIDs. The Nb SQUID loops have thickness 80 nm, trace width $2\mu \mathrm{m}$, and loop diameter of 40 and $80\mu \mathrm{m}$, corresponding to measured inductances of 110 and 250 pH, respectively.

Figure 2

Temperature dependence of complex inductance $L=L^{\prime }-i{L}^{\prime \prime }$ of a 110 pH SQUID. (a) $\Delta L^{\prime }$ (upper panel) and $\Delta L^{\prime \prime }$ (lower panel) vs temperature, measured at an excitation frequency $f_0=100\mathrm{Hz}$. The direction and ordering of the temperature sweeps are indicated by the arrows. The increase in $\Delta L^{\prime }$ at higher temperatures is due to the temperature-dependent kinetic inductance of the superconductor. The inset shows a blowup of the fine structure in $\Delta L(T)$, demonstrating strong correlation between fluctuations in $L^{\prime }$ and $L^{\prime \prime }$. (b) $\Delta L^{\prime }(T)$ probed at $f_0=10\mathrm{Hz}$ and 100 Hz. The sharp structure in $\Delta L^{\prime }$ below 2 K is independent of probe frequency. (c) As in (b), but for $\Delta L^{\prime \prime }(T)$; the 10 Hz data have been scaled by a factor 10 to facilitate comparison with the data at 100 Hz. The fluctuations in $L^{\prime \prime }$ scale linearly with probe frequency.

Figure 3

$1/f$ inductance noise in a 250 pH SQUID. (a) $S_L(f;f_0=100\mathrm{Hz})$ for excitation amplitudes $I_{\mathrm{ex}}=30$ and $100\mu \mathrm{A}$, measured at 300 mK. (b) Power spectra $S_{L^{\prime }}(f;f_0)$ and $S_{L^{\prime \prime }}(f;f_0)$ of fluctuations in the real and imaginary parts of the SQUID inductance, for probe frequencies $f_0=150$ and 277 Hz. The data were obtained at 300 mK. (c) $S_L(f;f_0=100\mathrm{Hz})$ measured from 100 mK to 2 K.

Figure 4

Amplitude of the cross spectrum $S_{L^{\prime \prime }\Phi }/(S_{L^{\prime \prime }}{S}_{\Phi }{)}^{1/2}$ of inductance and flux fluctuations of a 250 pH SQUID, measured from 100 mK to 3 K.