# Picoammeters Based on Gradiometric Superconducting Quantum Interference Devices
## 3 Results and Discussion
Figure 2: Voltage vs. magnetic flux characteristics of the SQUID current sensors in parallel (**a**) and series configuration (**b**), measured at T = 4.2 K. The bias currents are 26 and 28 \(\upmu\)A for the parallel and series configuration, respectively. The crosses indicate the working point for the noise measurements corresponding for both configurations to about 0.15 \(\Phi_{0}\) with respect a minimum (\(\upmu\Phi_{0}\)). A quite-high-value capacitor (10 \(\upmu\)F) connected to the ground (Figure 3) acts as a filtering stage to reduce the effect of the radiofrequency disturbance on the SQUID voltage swing. The measured current sensitivities for the parallel and series configurations are I\({}_{\upvarphi,\mathrm{p}}\) = 0.25 \(\upmu\)A/\(\Phi_{0}\) and I\({}_{\upvarphi,\mathrm{s}}\) = 0.36 \(\upmu\)A/\(\Phi_{0}\), respectively. These values are quite close, as in the starting aim, and the small mismatch is due to the requirement to keep a limited inductance in the series case to avoid an excessive increase in \(\beta_{\mathrm{L}}\) value. Figure 4 reports the magnetic flux noise spectral density, as usual, for both devices. The measurements were performed at T = 4.2 K in small signal mode, biasing the sensor with a dc magnetic flux at its maximum responsivity point and using a spectrum analyzer (Hewlett Packard 3566A). The magnetic flux biases corresponding to the maximum responsivity are \(\Phi_{\mathrm{B}}\cong\) 0.15 \(\Phi_{0}\) (Figure 2) for both configurations. The sensors show a comparable level of magnetic flux noise, which in the white region results in \(\sqrt{\mathrm{S}_{\upvarphi,\mathrm{p}}}\) = 3.8 \(\upmu\Phi_{0}\)/ \(\upmu\)Hz and \(\sqrt{\mathrm{S}_{\upvarphi,\mathrm{S}}}\) = 5.3 \(\upmu\Phi_{0}\)/ \(\upmu\)Hz. Note that both sensors are not affected by an excess of flicker noise since the 1/f corner starts at quite a low frequency, as is evident from the linear spectra with higher resolution in that range. Figure 4: Spectral densities of magnetic flux noise of the current sensors in the parallel (**a**) and series configuration (**b**). In the insets, the same measurements in the linear scale and in the low-frequency range are reported. Figure 3: Measurement scheme for evaluation of the current\(-\)to\(-\)magnetic\(-\)flux transfer factor (I\(\Phi\)) of both SQUID sensors. It is worth noting that the amplifier voltage noise does not significantly contribute to the overall noise. In fact, with the amplifier voltage noise being equal to \(\mathrm{S_{\mathrm{\varphi}E}}^{1/2}=0.7~{}\mathrm{nV}/\mathrm{Hz}^{1/2}\), the contribution to the magnetic flux noise is \(\mathrm{S_{\mathrm{\varphi}E}}^{1/2}=\mathrm{S_{\mathrm{\varphi}E}}^{1/2}/ \mathrm{V_{\mathrm{\varphi}}}=2.0~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{1/2}\) for the parallel configuration and \(2.4~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{1/2}\) for the series one. Since the overall magnetic flux noises reported are \(\mathrm{S_{\mathrm{\varphi}E}}^{1/2}=3.8~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{ 1/2}\) and \(5.3~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{1/2}\) for the parallel and series configuration, respectively, the intrinsic magnetic noise is \(\mathrm{S_{\mathrm{\varphi}1}}^{1/2}=(\mathrm{S_{\mathrm{\varphi}E}}-\mathrm{ S_{\mathrm{\varphi}E}})^{1/2}=3.2~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{1/2}\) for the parallel configuration and \(4.7~{}\mathrm{\mu}\mathrm{\Phi_{0}/Hz}^{1/2}\) for the series one. The current noise spectral density is proportional to the SQUID magnetic flux noise by the current sensitivity, following a simple expression:  
\[S_{I}^{1/2}=I_{\mathrm{\Phi}}\cdot S_{\mathrm{\Phi}}^{1/2}= \begin{cases}1.0 \frac{PA}{\sqrt{\mathrm{Hz}}}&(parallel)\\ 1.9\frac{PA}{\sqrt{\mathrm{Hz}}}&(series)\end{cases}