Investigation of Helium-Cooled Planar Transformer-Coupled SQUID Magnetometer

We investigated helium-cooled planar transformer-coupled SQUID magnetometers with regard to their field resolution δB by varying the SQUID loop inductance Ls, input coils and the pick-up loop Lp. It was found that the pick-up-loop area Ap is the most important parameter for δB of transformer-coupled magnetometers. δB with Ap = 10 × 10 mm2 reached about 3 fTA/□Hz, even using direct readout scheme without any feedback circuitries.


Introduction
Our transformer-coupled SQUID magnetometer consists of three parts: a dual-washer SQUID with two series-opposite integrated input coils L in connected to an on-chip pickup loop L p .
An important figure of merit for a magnetometer is the magnetic field resolution δB, which is a product of the SQUID system flux noise δФ and the flux-to-field transfer coefficient ∂B/∂Ф, i.e., δB = δФ × (∂B/∂Ф). δФ consists of two parts, the SQUID intrinsic noise δФ s and the preamplifier noise contribution δФ preamp , i.e., δФ 2 = δФ s 2 + δФ preamp 2 . Minimizing δФ of SQUIDs was analyzed in detail in many previous studies, e.g., in reference [1]. For transformer-coupled magnetometers, ∂B/∂Ф can be represented as ∂B/∂Ф / , , / , , [2], whereby L p and A p,eff denote the inductance and the effective area of the pick-up loop, L in,eff and L s,eff the effective inductances of the input coil and the SQUID loop, and k the corresponding coupling coefficient. The value of L in,eff should be designed to equal L p in order to achieve a minimum of ∂B/∂Ф for a given L s,eff . Generally, larger L s leads to a larger δФ, but a smaller (i.e., better) value of ∂B/∂Ф. In practice, most magnetometers are designed with L s > 100 pH to meet the high field resolution requirements [3]. We experimentally studied the properties of magnetometers employing SQUIDs with large β c ≈ 3 to find the combination of inductances L s , L in and L p that is optimum for minimizing δB.

SQUID magnetometer and readout electronics
In our experiments, mostly SQUID magnetometers with Steward-McCumber parameter β c ≈ 3 were employed. In our earlier work we have shown that large β c leads to a large flux-to-voltage transfer coefficient ∂V/∂Φ > 300 µV/Φ 0 at L s = 350 pH [4], thus reducing δΦ preamp = V n /(∂V/∂Φ), whereby the preamplifier voltage noise V n is 0.9 nV/√Hz (AD797). In this case, the SQUIDs can be connected to the preamplifier directly without any feedback circuitries in order to construct a simple SQUID system with an acceptable δΦ [5], measured in flux-locked loop (FLL) inside a niobium shielding tube. The general layout of our SQUID magnetometer is shown in reference [6]. In this layout, we used the dual-washer gradiometric SQUID to increase the coupling between L s and L in .

Different input coil types
We designed and fabricated three types of input coils, with the input coil consisting of 4.5 × 2 turns (see figure 1(a)) at L s = 350 pH and A p = 5 × 5 mm²: (i) whole coil placed on the SQUID washer of "Ketchen-type" [7]; (ii) whole coil located inside the hole of the SQUID washer; (iii) a part of coil is overlaying the SQUID washer and the other part is inside the hole. Subsequently, we measured δΦ and ∂B/∂Φ to determine δB. We found that δΦ ≈ 5 µΦ 0 /√Hz for all types of magnetometers investigated, while ∂B/∂Φ strongly depended on the positions of the input coils. The transfer function ∂B/∂Φ increased from 1.5 (for type (i)), and 1.6 (for type (iii)) to 2.7 nT/Φ 0 (for type (ii)). Of course, the "Ketchen-type" magnetometer provides the best coupling method for the transformer-coupled SQUID magnetometer. However, it is surprising that placing all of the coil turns inside the SQUID hole (in the case of type (ii)) reduced the effective area of the magnetometer only to about 50 %, because a much smaller value of the effective area was speculated. In the work described below, only input coils of "Ketchen-type" were utilized

SQUID effective inductance
In a transformer-coupled magnetometer, the design (geometrical) value of the SQUID inductance L s is reduced due to the screening effect when the input coils are connected to the pick-up loop. Therefore, we introduce the SQUID effective inductance L s,eff , because it (and not the geometrical inductance L s ) is responsible for δB of magnetometers. The value of L s,eff can be determined by the SQUID screening parameter β L = 2I 0 L s,eff /Φ 0 , where I 0 is the critical current of one junction and Φ 0 is the flux quantum. The value of β L is determined by the ratio of I cmin /I cmax . The value of I cmax is 2I 0 in the above expression of β L , when two junctions are identical. The dependence of β L on I cmin /I cmax shown in the inset in figure  1(b) is reproduced from [1]. The screening parameter β L increases monotonously with increasing

Field resolution δB
To investigate δB of the magnetometer, we varied the SQUID layout parameter, such as L s , or the turn number of the input coil. Two SQUID pick-up areas of A p = 5 × 5 mm 2 (table 1) and 10 × 10 mm 2 (table 2) were employed. In table 1, three important parameters of the magnetometers, L s,eff , δΦ and ∂B/∂Ф, are listed. The value of L s,eff , for example, at L s = 350 pH, obviously increases with decreasing the turn number of L in , thus increasing δΦ, but concurrently improving ∂B/∂Ф. In fact, for a given L s of 350pH, L in,eff slightly changes with the turns of L in varying in a certain range shown in table I and II, since L in,eff depends on both the turns of L in and L s,eff [2], which leads to L in,eff ≈ 14 nH to match L p ≈ 15nH [6]. Consequently, δB was almost constant for such transformer-coupled magnetometers, even when varying L s from 350 up to 620 pH. For A p of 10 × 10 mm 2 (table 2), we increased the turn number of L in , e.g., to 7.5 × 2 at L s = 350 pH, to improve the matching between L in and L p , thus leading to the reduction of ∂B/∂Ф ≈ 0.55 nT/Φ 0 , although L s,eff further reduced to 120 pH. Indeed, making A p larger is very useful for the improvement of δB. For different L s , δB ≈ 3 fT/√Hz of such magnetometers was achieved. Note that δФ of the present SQUID magnetometers in table 2 was larger than that listed in table 1, due to β c > 3. Larger β c led to a higher δΦ s , which dominated δΦ [5].

Conclusion
We studied different layouts of transformer-coupled SQUID magnetometers with varying L s , L in and L p . An input coil fully overlaying the SQUID washer ("Ketchen type") provides the best coupling between L s and L in , whereas placing the input coil fully inside the SQUID hole will reduce the pick-up area by about 50%. The effective SQUID inductance L s,eff obviously increases with decreasing turn number of L in , thus increasing δΦ, but at the same time reducing ∂B/∂Ф. Consequently, δB is almost constant for a certain pick-up-loop area A p . In other words, L s , L in and the matching between L in and L p will not influence greatly δB. Only by increasing A p will the field resolution improve.