A SQUID femtoamperemeter with bulk transformer for bolometer readout
M.Tarasov, A.Kalabukhov, S.Kovtonjuk, I.Lapitskaya, S.Gudoshnikov, M.Kiviranta, O.Snigirev, L.Kuzmin, H.Seppa
Chalmers University of Technology, Sweden
Institute of Radio Engineering and Electronics, Moscow, Russia
M.Lomonosov Moscow State University, Russia
Institute of Terrestrial Magnetism and Radio Wave Propagation, Troitsk, Russia
VTT, Finland

Abstract
We developed a dc SQUID femtoamperemeter with an on-chip intermediate planar transformer and an input bulk transformer with toroid ferromagnetic core.  It was used to read-out current noise from SIN tunnel junction that is temperature sensor in a normal metal hot electron bolometer.  The best measured current noise sensitivity is 35 fA/Hz1/2 in the white noise region.  Optimal matching conditions for SQUID femtoamperemeter are analitically analysed and experimentally studied.  Ferromagnetic bulk transformer allows to achieve requirement to the transformer that its output inductance is large compared to input inductance of SQUID chip and that matching factor for such transformer is close to unity contrary to planar transformers with k=0.50.7.

Introduction
For many practical applications such as bolometers and other direct radiation detectors from microwave to X-ray it is necessary to have read-out sensitivity of the order of 10 fA/Hz1/2.  One of such receivers is a normal metal hot electron microbolometer (NHEB) with capacitive coupling proposed in  and experimentally studied in 2.  The output signal in NHEB is read out from a SIN tunnel junction with normal resistance 11 k and dynamic resistance 0.1-1 M.  Ordinary room-temperature semiconducting amplifiers provide current resolution about 0.5 pA/Hz1/2 at the best.  The obvious solution to this problem in cryogenic receivers is using SQUID as a current meter.  SQUID is supposed to be optimal for measurements with low-ohmic signal source.  For measurements with high-resistance source an input transformer can be used.  On-chip planar transformers 3 can provide SQUID current noise don to 25 fA/Hz1/2.  For better current resolution a so-called cryogenic current comparator (CCC) 4 can provide current noise down to 4 fA/Hz1/2.  The CCC is a bulk wire-wound transformer with two coils enclosed in a superconducting torus whose extremes overlap without being electrically connected.  The inductance of CCC with inner diameter 2 cm is 1 nH and 10000 turns if input coil brings 1 H inductance.  A CCC triple shielding is 10 cm in diameter and 10 cm long.  For arrays of cryogenic detectors it is necessary to develop a small and robust SQUID femtoamperemeter to be able to place tens of them in one cryostat.  

Principle
One can estimate ultimate SQUID fAmeter current sensitivity from the fundamental Johnson noise in resistive shunts of Josephson junctions that produce noise current in squid loop
 
in which kb is Boltsman’s constant, T is temperature, R=4Rd is equivalent series resistance in the SQUID loop.  This noise current corresponds to the flux noise N and equal to the flux produced by input signal
 , 
in which n is the equivalent total turns ratio for multi-transformer design, Lsq is SQUID loop inductance and Isig is input current equal to noise
 
For bare SQUID with 1  resistance it brings 5 pA/Hz1/2 and with 1:1000 equivalent transformer the optimistic estimation for equivalent input noise resolution is 5 fA/Hz1/2.  If we take a more realistic value of flux noise

and matching coefficients k1=k2=0.7 then current noise resolution can be about 1020 fA/Hz1/2.  

Signal current matching
For detail analysis of requirements to SQUID and transformer consider the equivalent circuit for simple double-transformer design (equivalent circuit see in Fig. 1).  The intermediate loop comprising L2 and L3 can be described by following relations 
 
The SQUID loop can be presented for high frequencies and high bias current as series connected loop inductance Lsq and series resistance Rs=4Rd
 
For low signal frequencies and low dc bias the Josephson junction can be presented as connected in parallel inductance of critical current and shunting resistance.  In ordinary case L1 inductance of loop and inductance of critical current are close to equal.  The loop can be presented as connected in parallel dynamic inductance Ld and dynamic resistance Rd  
 
For rough estimation we can skip resistance term and circuit equation is simplified to
 
Assuming Ld=LSQ and M2=LsqL3k2 one can get
 
If we take the flux noise in the SQUID loop n that is equal to the input signal I1  
 
Taking  , LSQ*=LSQ(1-k22) and n*=n1n2 the equation is further simplified to 
 
It is worse to notice, following 5, that intuitive setting for intermediate inductances L2=L3 is not optimal.  According to 3, the loss factor of 2 can be fully recovered for the case L2>>L3 and in their design this ratio is 10.  This statement can be deduced from the signal current transfer coefficient for SQUID with input bulk transformer :
  
that is reduced at =0 to simple relation
 
The requirement L2>>L3 (deduced from invariant n1n2) differs from one for magnetometer in which signal is delivered through a pickup coil and the flux transfer coefficient has maximum for LpL1(1-k12) and L2L3(1-k22).   Actually for our current source there is upper limit for inductances.  If Rsin=L1 the current is reduced by a factor of 2 and this is natural limit for increasing the input inductance.  For 1 k and 10 kHz it brings input inductance only about 1 mH.  
This input loop matching factor is 

Equivalent impedance
From the above estimations a conclusion can be made about equivalent resistance that should be referred to the input of transformer stage.  First analyse the equivalent series circuit with relation of currents
 
 , in which I1M1=U1, now we can estimate impedance
 
It means that the SQUID series resistance is recalculated to the primary circuit with multiple n2 at frequencies comparable to Josephson frequency.  For lower frequencies it can be reduced to

For lower frequencies and parallel equivalent SQUID circuit impedance the impedance in intermediate loop can be presented as series circuit
 ,
or can be naturally presented as parallel circuit with resistance Rp parallel to equivalent Lp 
  and 
Now we can estimate impedance in the input circuit.  

the last equation brings analytic expression for the input circuit impedance 
 
asymptotic for this relation in the low frequency limit is .  
For frequencies over 100 kHz that are necessary for frequency multiplexing of many channels, the impact of impedance of SQUID transformed by both transformers can be so high that switches measurement mode from current-meter to voltmeter.  
Equivalent circuit can also be transferred to parallel circuit of inductance Lt and resistance Rt  

  and 

SQUID design
First experimental results were obtained with SQUIDs designed and fabricated by VTT 8.  In the next generation of SQUIDs we choose the relatively simple layout that allows fabricating samples without tight requirements to technological facilities, photomask, alignment, etc.  It consists of a SQUID loop with octagonal 24 m hole and 5 turnes of input coil in the center, and gradiometric transformer with square holes 230 m and 22 turns input coils to the sides.  Junctions area is 2 m x 2m, critical current 20 A, SQUID normal resistance 10 , resistance of damper in intermediate loop is 6 , shunting capacitor  250x250 m area is 4 pF, SQUID loop inductance 30 pH, SQUID input coil inductance 750 pH, transformer loop inductance 250 pH and input inductance of such gradiometric transformer 250 nH. Microphotograph of the fabricated SQUID chip center part made in reflected light is shown in Fig. 3.

Bulk transformers
At the beginning we used solenoid transformers wound on small rod of amorphous permalloy 6.  Drawbacks of such design are low inductance, low coupling efficiency k and high sensitivity to external magnetic fields.  These disadvantages can be avoided by using torus cores wound by thin tape of amorphous permalloy with insulation of turns by thin powder.  Such cores maintain high >10000 for cryogenic temperatures below 4 K.  We compared different cores from different manufacturers 7 including CRYOPERM and VITROVAC amorphous tape-wound cores 9-E3007-W305 and 6-E3009-W564 with removed case..  Approximate inductance per turn is about 1 H.  Parameters of some of our transformers are presented in Table 1.  

NN	size	turns	Input inductance	Sensitivity SI1/2, 
fA/Hz1/2 , at 1 kHz	
PT1 	Integrated on- chip	1:8:1:16	0,3 mH 	1 pA/Hz1/2 	
FM1 	20x8x11	1:400	0.9 H	34 fA/Hz1/2 	
FM2	11.2x5.1x5.7	1:200	15.1 mH	80 fA/Hz1/2 	
FM3	7.3x3.5x3.8	1:100	3.6 mH	150 fA/Hz1/2 	

Main advantage of ferromagnetic bulk transformer is its high inductance per turn that can easy satisfy the strong requirement of L2>>L3 in our equivalent circuit.  One of problems with ferromagnetic transformers is excess noise that can be caused by external magnetic field.  Remagnetisation of core leads to Barkhausen noise with 1/f frequency dependence.  Such noise can be suppressed down to sufficient level by combined ferromagnetic and superconducting magnetic shields, see Fig. 5. 

Discussion
The SQUID design and performance for different manufacturers achieved comparable level.  To realize SQUID fAmeter sensitivity in a current meter mode the main problem is to develop simple, robust, small matching circuit.  At present there are three competing technologies: planar transformer (PT), cryogenic current comparators (CCC) and superconducting transformers with ferromagnetic core (SFT).  Advantage of PT and CCC is low intrinsic noise.  With SFT it is much easier to achieve high enough input inductance and it is not so sensitive to external magnetic noise. Due to the nonequilibrium nature of 1/f noise it can be suppressed in the transformer down below the noise level of SQUID and  brings possibility to solve problem of SQUID matching to the high resistance signal source.

Acknowledgements
Authors acknowledge VACUUMSCHMELZE Gmbh & Co. KG for samples of their ferromagnetic cores.  This work was carried under support from INTAS project 00-384, KVA, VR, and ISTC.

References
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Figure 1.  Schematic of the SQUID with matching input transformer.