# Picoammeters Based on Gradiometric Superconducting Quantum Interference Devices
## 2 Design, Fabrication and Experimental Setup  
SQUID sensors based on gradiometric design allow for a high current sensitivity to be obtained thanks to high mutual inductance M between the SQUID and input coil, at the same time keeping a high rejection factor to the ambient noise. There are different gradiometric designs: parallel or series configurations of first and higher order and combinations of series and parallel configurations [7,10]. The parallel configuration can ensure relatively low inductance, while the series one has the advantage to completely cancel the screening current everywhere in common mode excitation [12]. The design of both gradiometers in parallel and series configurations is based on a double-washer-shaped SQUID ring inductively coupled to the underlying multiturn input coils [13]. In Figure 1, the pictures of the realized device are reported; in the parallel configuration, the two washer sections are connected in a parallel way, leading to a total inductance \(L_{p}=L_{w}/2\), while in the series connection, the total inductance is \(L_{s}=2L_{w}\), where \(L_{w}\) is the inductance of a single washer. Both the input coils are wound so that the input current circulates in the same direction, generating two concordant magnetic fluxes. A key factor for a SQUID current sensor consists of transfer factor between the input current and the magnetic flux (\(I_{\Phi}\)), often known as current sensitivity and given by the reciprocal of the mutual inductance M as defined above; it represents the current flowing in the signal coil that induces one flux quantum \(\Phi_{0}\) into the SQUID loop. We chose to design the SQUID sensor with a similar flux-to-current transfer factor, taking into account that too high an inductance value could led to performance degradation in the series configuration [14]. In a single-washer-shaped SQUID, its geometrical inductance, neglecting the parasitic inductance of the interconnection of the washer to the junctions (typically a few pH) [15], can be obtained by summing the inductances of the washer and the slit [12]:  
\[L_{h}=1.25\mu_{0}d;\ L_{sl}=0.3\left[\frac{\mathrm{pH}}{\mathrm{\upmu m}} \right]\cdot b_{T}\]  
where \(d\) is the washer hole inner dimension and \(b_{T}\) is the slit length. Referring to a single washer, in the case of parallel configuration, the hole dimension measures 100 \(\upmu\)m while the slit length is 404 \(\upmu\)m; and in the series case, the hole and slit length are 35 \(\upmu\)m and 290 \(\upmu\)m, respectively. Both input coils consist of a series of 2 multiturn windings having \(N_{P}=40\) and \(N_{S}=29\) turns and being magnetically coupled to the SQUID in double-washer configuration. Their inductances can be calculated as [12]:  
\[L_{i}=2\left[N^{2}\left(L_{h}+\frac{L_{sl}}{3}\right)+\ell L_{strip}^{\prime}\right]\]  
where \(N\) is the turn number of input coil; while \(L_{h}\) and \(L_{sl}\) are the inductance of the washer hole and slit, respectively. \(L_{strip}^{\prime}\) represents the stripline inductance for unit length and \(\ell\) is the total length of the input coil winding. \(L_{strip}^{\prime}\) depends on the stripline geometry and the London penetration depth. In this case, this inductance contribution affects the total inductance for a few nH. The mutual inductance between the input coil and the double-washer SQUID can be computed by using the following equations:  
\[M_{i,P}=N\left(L_{h}+\frac{L_{sl}}{2}\right)\quad M_{i,S}=2N\left(L_{h}+\frac {L_{sl}}{2}\right)\]  
In the Table 1, the main geometric features of the devices are summarized and electrical parameters are computed. Figure 1: Picture of SQUID-based current gradiometric sensors. (**a**) Parallel configuration, (**b**) series configuration. In the respective insets, a magnification of area including Josephson junctions, damping and shunt resistors of both devices are shown and indicated by an arrow.