A Femtoamperemeter Based on a Superconducting Quantum Interferometer and a Volume Transformer

M. A. Tarasov, A. S. Kalabukhov, S. A. Kovtonyuk, I. L. Lapitskaya, S. A. Gudoshnikov, M. Kiviranta, O. V. Snigirev, L. S. Kuz'min, and H. Seppa

@Abstract@---A femtoamperemeter based on a superconducting quantum interferometer device (SQUID) with a planar gradiometric transformer on a substrate and an additional volume cryogenic transformer on a toroidal ferromagnetic core is developed, fabricated, and experimentally studied. The conditions for the optimum matching of the SQUID to the signal source are analyzed. It is demonstrated that the conditions for the optimum matching and high coupling coefficient are satisfied in the transformer with a ferromagnetic core. The excess noise introduced by the core is studied experimentally. It is shown that double shielding using superconducting and ferromagnetic shields makes it possible to lower the cutoff frequency of the excess noise to less than 1 kHz.

Received February 14, 2003

1. INTRODUCTION
High-sensitivity bolometers and direct detectors for wavelengths ranging from microwaves to X rays must be equipped with readout devices whose sensitivity is no worse than 10 fA/Hz@1/2. One of such receivers is the normal metal hot-electron bolometer (NHEB) with capacitive coupling proposed in [1] and experimentally studied in [2]. The output signal is measured in proportion to the temperature by a superconductor--insulator--normal metal (SIN) junction with a normal resistance of 1--10 k@[Omega] and a differential resistance of 0.1--1 M[Omega]. At room temperature, the current resolution of semiconductor amplifiers is normally about 0.5 pA/Hz@1/2. A natural solution for a cryogenic receiver is the application of a superconducting quantum interferometer device (SQUID) in the case of a low-resistance signal source, such as the bolometer based on a transition from the superconducting to normal state. A matching transformer is needed in the case of a high-resistance source. A planar transformer integrated with the to a SQUID on a single chip [3] exhibits a current resolution of up to 250 fA/Hz@1/2. One can obtain an even better resolution of 4 fA/Hz@1/2 with cryogenic current comparator (CCC) [4]. The comparator represents a volume transformer with two coils wound using a superconducting wire and placed in a superconducting toroid with overlapping ends without electric contact. The inductance of such a superconducting stocking with an inner diameter of 20 mm is 1 nH, and the inductance of the input coil with 10 000 turns is 1 H. The triple CCC shield determines the overall dimensions of the device: both the diameter and the length are 10 cm. An array of bolometers requires a large number of readout channels. Therefore, the shield size imposes strict limitations on the possible number of channels.

2. MATCHING PRINCIPLES FOR THE SQUID FEMTOAMPEREMETER 
The ultimate parameters of the SQUID femtoamperemeter can be estimated based on the intrinsic thermal noises in resistive shunts of the Josephson junctions that induce a noise current @I@n[roman] in the SQUID loop:
@,
Key: 1. @n[roman]; 2. @B[roman]
where @k@B[roman] is the Boltzmann constant, @T is temperature, and @R = 4@R@d[roman] is the equivalent series resistance in the SQUID loop. If the signal-to-noise ratio equals unity, this noise current corresponds to the noise flux @[Phi]@n[roman] and equals the flux induced by the input signal:
@,
Key: 1. @n[roman]; 2. @S[roman]; 3. @B[roman]; 4. @sign[roman]
where @n is the total equivalent current transformation coefficient for the device with a few transformers, @L@S[roman] is the inductance of the SQUID loop, and @I@sign[roman] is the input current equal to the noise and represented as
@.
Key: 1. @sign[roman]; 2. @B[roman]
For a SQUID with a normal resistance of 10 @[Omega], this corresponds to a noise current of 5 pA/Hz@1/2. In the presence of a 1:1000 transformer, an optimistic estimate of the equivalent input current resolution is 5 fA/Hz@1/2. For more realistic flux noises given by
@
Key: 1. @[Phi]|@[roman]; 2. @B[roman]; 3. @d[roman]
and the coupling coefficients @k@ = @ = 0.7, the current resolution can reach 10--20 fA/Hz@1/2.

3. MATCHING THE SIGNAL CURRENT
To accurately analyze the requirements for the SQUID and the transformer, we consider an equivalent circuit for a simple device with two transformers (Fig. 1). Coils @L@2 and @L@3 are connected in the intermediate loop and the currents satisfy the following condition:
@.
Key: 1. @S[roman]
For high frequencies and strong bias currents, the SQUID loop can be represented as a series connection of loop inductance @L@S[roman] and resistance of the two junctions @R@S[roman] = 4@R@d[roman], so that
@.
Key: 1. @S[roman]
At low frequencies and small bias currents, the Josephson junction can be represented as a parallel connection of the critical current inductance and the shunt resistance. Normally, @[beta]@L[roman] @ 1 and the loop inductance is close to the critical current inductance. The SQUID loop can be represented as a parallel connection of dynamic inductance @L@d[roman] and dynamic resistance @R@d[roman]:
@.
Key: 1. @d[roman]
For rough estimates, we can neglect the resistive term, simplify the circuit equation, and represent it as
@.
Assuming that @L@d[roman] = @L@S[roman] and @M@ = @L@S[roman]@, we find that
@.
Let the flux noise in the SQUID loop @[Phi]@n[roman] be equal to the flux induced by input signal @I@1:
@.
Key: 1. @n[roman]; 2. @S[roman]
Using the notation @k@ = @ and @n@ = @, we can represent the above relationship as
@.
Key: 1. @n[roman]; 2. @S[roman]
Note that, as was mentioned in [5], the intuitive desire to equate inductances in the intermediate circuit (@L@2= @L@3) does not represent the optimum for the amperemeter. It was demonstrated in [3] that a mismatch coefficient of about 2 can be eliminated completely if @L@2 @ @L@3 (in [3], the corresponding ratio equals 10). This statement can easily be deduced from the expression for the signal transformation coefficient of the SQUID with an input transformer represented as
@.
Key: 1. @S[roman]; 2. @d[roman]
For @[omega] = 0, this expression can be simplified to
@.
Key: 1. @d[roman]
Condition @L@2 @ @L@3 (obtained in the case of invariance of @n@1@n@2) differs from the conventional condition for magnetometers, in which the signal comes from receiving coil @L@r[roman] and the flux transformation coefficient exhibits maximum at @L@r[roman] @ @L@1@ and @L@2 @ @L@. In fact, there is an upper bound on the inductance of the resistive current sourc. This bound is related to the cutoff frequency and the time constant of the circuit. If the resistance of the SIN junction is @R@SIN[roman] = @, the current decreases by a factor of two relative to the maximum value, which yields the natural limit for the increase in the inductance. For 1 k@[Omega] and 10 kHz, the effective input inductance @L@ is no greater than 1 mH.
For an arbitrary frequency, the matching coefficient of the input circuit is given by
@.
Key: 1. @SIN[roman]

4. EQUIVALENT IMPEDANCE
Based on the above calculations, we can infer the equivalent input impedance of the femtoamperemeter. First, we analyze the equivalent series circuit with the following relationships for the currents:
@.
Key: 1. @S[roman]; 2. @d[roman]; 3. where
The impedance can be estimated as
@.
Key: 1. @S[roman]; 2. @d[roman]
It is seen from this relationship that the SQUID series resistance is recalculated into the input circuit with a factor of @n@2 at frequencies comparable to the Josephson frequency. For low frequencies, the above expression can be simplified and represented as
@.
Key: 1. @d[roman]; 2. @S[roman]
For low frequencies and the parallel equivalent circuit of the SQUID impedance in the intermediate loop, the impedance can be represented using a series circuit as
@
Key: 1. @d[roman]; 2. @S[roman]
or a parallel circuit with resistance @R@p[roman] and equivalent inductance @L@p[roman] 
@.
Key: 1. @p[roman]; 2. @S[roman]; 3. and; 4. @d[roman]
Then, we can estimate the impedance in the input circuit as
@,
Key: 1. @p[roman]
@.
Key: 1. @p[roman]; 2. @SIN[roman]
The expression for the desired impedance can be derived from the last formula:
@.
Key: 1. @p[roman]; 2. @SIN[roman]
In the low-frequency limit, this expression can be written in simplified form as 
@.
Key: 1. @SIN[roman]; 2. @d[roman]
For frequencies higher than 100 kHz (required for frequency multiplexing), the contribution from the impedance of the SQUID with a transformer can be so high that a change occurs from current measurements to voltage measurements.
The equivalent circuit can also be represented as a parallel circuit with inductance @L@t[roman] and resistance @R@t[roman]:
@.
Key: 1. @t[roman]; 2. @SIN[roman]; 3. @p[roman]; 4. and

5. SQUID DESIGN
The first experimental results were obtained using SQUIDs produced by VTT [8, 9]. For the next generation of SQUIDs, we have chosen a relatively simple topology that makes it possible to produce samples without imposing strict requirements on the technology, photomasks, alignment, etc. Figure 2 shows a structure consisting of an octagonal SQUID loop with a 24 @[mu]m opening and 5 turns of the input coil. The same substrate contains a gradiometric transformer with 230 @[mu]m square openings and 22 turns of the input coil on each half. The size of the Josephson junctions is 2@2 @[mu]m@2, the critical current is 20 @[mu]A (Fig. 3), the normal SQUID resistance is 10 @[Omega], the resistance of the damper in the intermediate loop is 6 @[Omega], the capacitance of the shunting capacitor with dimensions of 250@250 @[mu]m@2 is 4 pF, the inductance of the SQUID loop is 30 pH, the inductance of the SQUID input coil is 750 pH, the inductance of the transformer loop is 250 pH, and the input inductance of the gradiometric transformer is 250 nH. Figure 4 shows a microphotograph of the central part of the SQUID under consideration.

6. VOLUME TRANSFORMERS
At the first stage, we used transformers in the form of solenoids wound on a thin rod made of amorphous permalloy [6]. The disadvantages of such a transformer are low inductance, a low coupling coefficient, and high sensitivity to the magnetic fields of external noise. To eliminate these disadvantages, one can apply toroidal cores wound with ribbon amorphous permalloy with a fine-powder insulation of the turns. Such cores retain high permeability, @[mu] > 10 000, at cryogenic temperatures down to at least 4 K. We studied various cores from various manufacturers [7], including CRYOPERM and VITROVAC materials representing cores of 9-E3007-W305 and 6-E3009-W564 standard sizes with the sheath removed. The inductance per turn is about 1 @[mu]H. The table shows the parameters of some of our transformers.
The main advantage of the ferromagnetic volume transformer is its high inductance per turn. In combination with a high coupling coefficient, this makes it possible to satisfy condition @L@ in the equivalent circuit under consideration. One of the problems of the ferromagnetic transformer is the excess noise that can be excited by external magnetic fields.
The core magnetization switching gives rise to Barkhausen noise with the 1/@f@2 spectrum. This noise can be substantially suppressed using a shield with a combination of ferromagnetic and superconducting properties, which is proved by the results of the measurements for an FM1 core (Fig. 5). The reason for this is the nonequilibrium nature of the 1/@f@2 noise (the noise is absent in the absence of the external switching flux). The equilibrium thermal noise of magnetic domains with a uniform spectrum decreases with lowering the transformer temperature (similarly to the thermal noise of resistors).

7. CONCLUSIONS
At present, the limit parameters of SQUIDs from various manufacturers are comparable. The main problem in fabrication of the SQUID femtoamperemeter is related to having to manufacture a simple, reliable, and compact matching device. There are three competing technologies: planar transformers (PTs), CCCs, and superconducting transformers with ferromagnetic cores (SFTs). The advantages of PTs and CCCs are the ultimately low intrinsic noises. The advantages of SFTs are easily reachable high inductance and relatively low susceptibility to external magnetic noises. Double shielding of the transformer enables one to make the transformer noises lower than the SQUID intrinsic noises. As a result, we obtain a current resolution of 34 fA/Hz@1/2 in the range of the white noise.

ACKNOWLEDGMENTS
We are grateful to VACUUMSCHMELZE GmbH & Co. KG for providing the samples of the cores. This work was supported by INTAS (grant nos. 00-384 and 01-686), KVA, VR, and the International Science and Technology Center (grant no. 1239).

REFERENCES
1. Kuzmin, L., @Physica B: Condens. Matter,@ 2000, vols. 284-288, p. 2129.
2. Tarasov, M., Fominskii, M., Kalabukhov, A., and Kuz'min, L., @Pis'ma Zh. Eksp. Teor. Fiz.,@ 2002, vol. 76, no. 8, p. 588.
3. Polushkin, V., Glowacka, E.Gu., Goldie, D., and Lumley, J., @Physica C@ (Amsterdam), 2002, vol. 367, p. 280.
4. Gay, F., Piquemal, F., and Geneves, G., @Rev. Sci. Instrum.,@ 2000, vol. 71, no. 12, p. 4592.
5. Ketchen, M., @IEEE Trans. Magn.,@ 1987, vol. 23, no. 2, p. 1650.
6. Tarasov, M., Gudoshnikov, S., Kalabukhov, A., @et al., Physica C@ (Amsterdam), 2002, vol. 368, p. 161.
7. www.vacuumschmelze.de, www.cmr.uk.com/lttdesc.html.
8. Seppa, H., Kiviranta, M., Satrapinski, A., @et al., IEEE Trans. Appl. Supercond.,@ 1993, vol. 3, no. 1, p. 1816.

9. Kiviranta, M. and Seppa, H., @IEEE Trans. Appl. Supercond.,@ 1995, vol. 5, no. 2, p. 2146.

TABLE
Parameters of cryogenic transformers
Key: 1. Dimensions, mm@3; 2. Turns; 3. Input inductance, mH; 4. sensitivity @S@, fA/Hz@1/2

FIGURE CAPTIONS
Fig. 1E circuit of the SQUID with a matching input transformer.
Key: 1. @SIN[roman]; 2. @S[roman]
Fig. 2. Topology of the SQUID with an integral planar gradiometric transformer. A 24 @[mu]m octagonal SQUID loop with 5 turns of the input coil is located in the center. In the lower part, there are two planar gradiometric transformers, each with a 230-@[mu]m-wide opening and 22 turns of the input coil.
Fig. 3. @I--@V characteristics of the SQUID for two input magnetic fluxes differing from each other by half of the flux quantum (curves @1 and @2).
Key: 1. @[mu]V; 2. @[mu]A
Fig. 4. Microphotograph of the central part of the SQUID with the SQUID loop, Josephson junctions, and circuit for damping spurious oscillations.
Fig. 5. Flux noises of the SQUID with an input ferromagnetic transformer and different shielding levels: (@1) in the absence of the transformer, (@2) in the presence of the transformer with a lead shield, and (@3) in the absence of the superconducting shield.
Key: 1. @S@[Phi], @[mu]F/Hz@1/2; 2. Hz

Translated by A. Chikishev
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