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Submitted on 1 Jan 1970

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Superconducting magnetometersJ.E. Mercereau

To cite this version:J.E. Mercereau. Superconducting magnetometers. Revue de Physique Appliquee, 1970, 5 (1), pp.13-20. �10.1051/rphysap:019700050101300�. �jpa-00243347�

13.

SUPERCONDUCTING MAGNETOMETERS

By J. E. MERCEREAU, California Institute of Technology, Alfred. P. Sloan Laboratory of Mathematics and Physics,

Pasadena, California (U.S.A.).

Abstract. 2014 Under the proper circumstances a current carrying superconductor cangenerate voltage. This voltage is produced by supercurrent oscillations at frequency

03C9 = 2e/0127 V 2014 a phenomena similar to the Josephson effect in superconducting tunnel junctions.Various structures have been fabricated in which this quantum relation dominates the electricalbehavior. These thin film structures have been developed into reliable sensor elements whichhave been incorporated into various electronic instruments 2014 particularly magnetometers.These instruments include a differential magnetometer with a one second response time andsensitivity greater than 10-10 gauss and a digital magnetometer counting in increments of1/4h/2e at a rate up to 104 per second. The fundamental principles of these instruments willbe discussed as well as their present operational limitations.

REVUE DE PHYSIQUE APPLIQUÉE TOME 5, FÉVRIER 1970, PAGE

Zero electrical resistance has been the principalhallmark of the phenomena of superconductivity. It

was, in fact, this effect which alerted KamerlinghOnnes that he has uncovered a new physical pheno-menon. This electrical aspect of superconductivity,the ability to carry finite current at zero voltage hasreceived major attention from experimentalists forsome time. This is primarily because there is at leastan adequate phenomenological description of this

aspect of superconductivity and also, in a practicalsense, because of the vast economic implications oflow cost power transmission and very high field super-conducting magnets.However, there is another facet of superconductivity

which in the past seems to have received somewhatless attention. Despite the apparent contradiction interms, this kind of superconductivity can probably bestbe characterized as the resistive superconducting state- that is to say, superconductivity at finite voltages.In contrast to zero voltage superconductivity thereseems to be little theory to guide the experimenterinto this intrinsically time dependent problem. The

purpose of this paper is to indicate our use of super-conductivity at finite voltages for instrumentation pur-poses and, in the absence of a more complete theory,to attempt to describe our results in physical terms.

In order to sustain a voltage supporting superconduc-ting state it is usually necessary that the super-electrondensity be inhomogeneous. Kim [1] and Gieaver [2]have studied this state in situations where the inhomo-

geneities are vortex lines produced by an external field.I want to concentrate here on the "field free" casewhere the inhomogeneity is determined by the materialitself. The extreme of this inhomogeneity is the

Josephson [3] tunnel junction, with an insulating bar-rier. However this structure has already had extensivetreatment and since it is relatively delicate is not uni-quely suitable for instrumentation. In what is to

follow I would like to specifically exclude this particularinhomogeneity and consider only electron density varia-tions resulting from current through the Dayem [4]bridge or a nonhomogeneous metal [5].

Voltage-supporting superconductivity. - In figure 1,curve A shows a typical current-voltage characteristicacross such a superconducting structure driven by acurrent source. For currents less than some tempera-ture-dependent critical current, I,,,, current can flow

FIG. 1. - Voltage developed across a Dayem bridgedriven from a current source. Curve A shows a cri-tical current, I.1, of about .2 milliamperes. Curve Bis the same structure irradiated with microwaves atabout 35 GHz.

without developing any voltage; while above thiscritical current, voltage ( ) exists. Voltage inducedabove, ICI’ indicates that power is being delivered tothe superconductor from the current source. Howe-

ver, as I shall indicate, because of the peculiar processof voltage generation in a superconductor only partofthis power is transformed into heat while the remain-der can escape as high frequency radiation, principallyat frequencies v = N(2e/h) V = Ncpo-1 V.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:019700050101300

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These structures are deliberately fabricated in sucha way that there is a definite, localized region of highcurrent density. Within this region the electron driftvelocity is considerably higher than in the surroundingsuperconductor. This usually means that the dimen-sions are small enough, relative to the penetrationdepth, that in the high density region the current canbe assumed to be uniform. Consequently, in whatfollows we will use the concept of current and currentdensity interchangeably. Current less than the cri-tical value can, by definition, be accommodated withoutdeveloping a voltage. However if the current suppliedto the superconductor is greater than critical someother current transfer mechanism is required. WhatI shall do is to assume a two-fluid description for

superconductivity and construct a model which givesrise to a possible mechanism.On a two-fluid model we presume that for currents

greater than critical, part of the current flows as

supercurrent and part as normal current. The normalcurrent requires voltage which is related to the normalcurrent by the resistance, V = I. R. Because of the

geometry, R is predominantly the resistance of thehigh current density region. If, above the criticalcurrent level, the superfluid component of the totalcurrent always remains at its maximum value 1c, thecurrent-voltage characteristic would be as shown infigure 2, curve A; where we presume the voltage is

developed only by the normal component of thecurrent, V = (1- le) R.

FIG. 2. - Idealized current-voltage characteristicfor a Dayem bridge.

However, in the superconductor the relationship [6]between current density j and electron drift velocity, v,is j cc v(1-v2/v2c). As a consequence, the maximumcurrent density ic does not occur at the maximumelectron velocity, Vc. At low velocities, as long ascurrent increases with increasing electron velocity, aflux change will induce current to offset the changeand the resultant Faraday induction will help to keepthe magnetic field from collapsing and penetratinginto the superconductor. On the other hand at highvelocities v > vc/V3 current decreases with increasingelectron velocity and an inverse Faraday induction [7]will occur, spontaneously driving the supercurrent andsuperelectron density to zero in a second order phasetransition as the superelectron velocity reaches v,,.

Therefore the previously described situation of thesimultaneous flow of super and normal currents ( fig. 2,curve A) must not be a stable situation. The electric

field, E, required to drive the normal current must alsoaccelerate the superelectrons. On a free electron

model, v == !.- E. Thus the supercurrent cannot re-

main at Ic in the presence of this electric field. Thefield will accelerate the superelectrons and drive thesupercurrent to zero at a second order phase transition.In the process the voltage will have to increase in orderto sustain the same total current.

If this sequence of events occurs, the current-voltagecharacteristic of the superconductor would be as shownon figure 2, curve C. As the current is slowly increasedfrom zero it eventually exceeds I, and a spontaneoustransition will occur, driving the supercurrent compo-nent to zero, leaving the voltage to become V = IR.This voltage must be due to charge at the interfacesof the region of high current density. Since the regionof high current density is bounded by equipotentialsurfaces (superconductor) these voltages must arisefrom an electrostatic potential across the bridge,creating a "linear capacitor".However this situation is not stable either. If the

system can maintain thermal equilibrium it will nowrecondense at zero velocity, 3 since the electric fielditself is not sufficient to quench superconductivity.The acceleration process will then repeat, part of thecurrent becoming supercurrent, "discharging" theelectric field until the electron drift velocity againreaches vc, etc. If we assume that the inverse Faradayinduction process occurs very rapidly we can charac-terize the period (ï) at which the process repeats asthe time for the velocity to reach critical.

On a free electron model E = mv, and thus :e

However, the definition of coherence length 03BE issuch that :

The coherence length by definition is the shortestdistance over which the superelectron density canchange and thus the smallest distance over which thefield can exist. Thus if the geometry of the situationis such that the electric field exists only across a dimen-sion equivalent to the coherence length the product( E ) § must equal the voltage V, or T N Jîj2e V.Actually this approximation must be exact since it isequivalent to the Josephson voltage-frequency relation-ship which must hold for superconductors in general.In this model i-1 is the oscillation frequency of thesupercurrent. Since the superconductor is drivenfrom a current source, total current must be keptconstant. Therefore the normal current componentmay also oscillate. And to the extent that 7g and I.are not exactly of opposite time phase, there may alsobe displacement currents induced which represent pos-sible radiative processes. As a result of these current"relaxation oscillations", voltage is generated due tothe growth and decay of charge across the region ofhighest current density. Thus, this voltage also oscil-lates at frequency (ù = T-1, but has an average valueequal to V = ~03C9/2e.

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This average voltage thus depends on the timeaverage of the supercurrent - ranging from curve A( fig. 2) when 7g = Ic to curve C when 7g = 0.Experimentally a superconductor ofthis structure oftenshows a current-voltage characteristic such as thatillustrated in figure 2, curve B. On the basis of thismodel the interpretation of curve B is that above levoltage appears, both to represent a decrease in I, andalso the appearance of radiated power. As the fre-

quency of the supercurrent oscillation is increased thetime average of the supercurrent component of thetotal current eventually decreases, probably becauseof some intrinsic relaxation time for supercurrentdecay and recondensation. When finally the periodof the relaxation oscillation is shorter than the intrinsicrelaxation time, the supercurrent remains zero andwe are back to curve C.

Curve B in figure 1 gives some additional experi-mental justification for this model. In this circums-tance the superconductor is being radiated by micro-waves at frequency - 35 GHz. The zero-voltagecurrent is depressed since the threshold current levelmust now accommodate not only the bias current butalso the high frequency current induced by the radia-tion. Above the threshold level any coupling what-ever between the supercurrents and the microwaveswould tend to make the supercurrent move at the mi-crowave frequency. Thus the superconductor wouldtend to have certain prefered oscillation frequenciesor voltages. These data show oscillation at the drivingfrequency and also, because of the non-linearity in thissystem, oscillation at frequencies up to the 5th harmo-nic. The nth harmonic corresponds to n relaxationoscillations per cycle of microwave frequency. Datasuch as this have been observed up to voltages of a fewmillivolts which corresponds to oscillation frequencieson the order 1012 cycles per second.

Structure such as this has also been observed in I- Vcharacteristics for tiny contacts between normal and su-perconducting metals [8], which thus cannot be interpre-ted in terms of the relative behavior of two superconduc-tors, as in the usual Josephson effect. However this

concept of supercurrent oscillation near the boundrycan be used to explain these effects at a normal-superinterface.

In summary then : in structures such as these, abovesome lower critical current Ic1, the supercurrent beginsto undergo relaxation oscillations. The rate of super-current oscillation is determined by the average voltageand the average voltage in turn is determined by therelative time spent, during each cycle, in the super-conducting or normal states. As the current is increa-

sed, the voltage or frequency increases, decreasing theamount of time spent in the superconducting state andthe average voltage gradually increases with increasingcurrent. This process appears to persist to frequenciesat least as high as the energy gap in superconductorwith no apparent degradation in the quantum rela-tionship ùoe = 2eh. Thus the two-fluid model seems

entirely adequate; the normal and supercurrents seemto enter only in an additive way with no interferenceeffects.There must exist an upper critical current le2 at

which these dynamic superconducting processes cease.This may occur because of heating by the normalcurrent or, more fundamentally, because of some fre-quency limitation such as the energy gap or other

relaxation process. However in what follows we willconcentrate only on the low frequency behavior anddiscuss devices incorporating this kind of superconduc-tivity into a superconducting circuit.

Superconducting circuits. - It had been noticedsometime ago that under the right conditions a thinfilm superconducting ring is effectively transparent tomagnetic flux [9]. If such a superconducting ring isplaced in an axial field it will shield flux changes fromits interior, but only up to some critical current Ic1, inthe ring. For flux changes larger than this, currentin the ring remains essentially constant while fluxpasses thru. Under these conditions the ring is inthe previously described voltage-supporting super-conducting state. In order to control this voltage,which is the basis for the instrumentation 1 will discuss,the ring itself is usually made non-homogeneous bymeans of a Dayem [4] bridge or a microscopic materialinhomogeneity [5]. These configurations localize theflux flow region and allow us to experimentally controlthe flux motion.

Driving these rings inductively is equivalent to pro-viding a voltage source rather than a current source,as in the previous discussion. When the inducedcurrent reaches critical in the bridge, it must never-theless spontaneously decay, as before, and in the

process create displacement current, E, in the bridge.After the spontaneous decay the bridge can againcondense but now in a finite electric field producedby the displacement current - this field then "dis-charges" and accelerates the current again to nearlythe critical value. By arguments similar to thoseused before this process can be related to flux loss bythe ring.

Since v .

= e E = e A in the proper gauge, then :

Thus the spontaneous decay of current, which chan-ges the velocity by roughly vc/2 must be accomplished

by a change in potential 8A N mvc 2e. Again substitutingfor MV, h we get 03BE 8A - h/2e. And if the super-

conductor actually manages to limit the current decayto the smallest possible distance 03BE, the flux change,8p = 03BE 8A produced in the ring will be one quantum,Yo = hl2e. This model thus envisages a situation inwhich individual flux quanta escape from the ring ina process which interchanges real and displacementcurrents. The final current, If, in the ring after thetransition must be sufficient to sustain the original flux,minus yo. Thus 1 f = le - ~0/L, where L is theinductance of the ring.The amplitude of the voltage puise, 03B4V, accompa-

nying this transition depends on the relaxation timer,_for the spontaneous decay, 8F = qpo Tï1. As an esti-mate for this time we draw upon Kim’s results forcurrent driven flux motion in superconductors [1].He finds that flux moves perpendicular to the currentat a rate v, = j~0/~. If we assume that this visco-

sity [10] (YJ) limited velocity can be related to a maxi-mum rate of change for the order parameter then :

16

At the critical current this reduces to :

17, - (P-0 7?") ==%/Awhere 6 and À are respectively the normal state con-ductivity and the superconducting penetration depthand 1Y is the energy gap. For most of our circuits

Tl 10-n s.This type of operation will occur only for bridge

widths small enough that a flux vortex cannot be formedwithin the bridge -- i.e. width less than a coherencedistance. For larger bridges, current density is notuniform and is free to adjust internally and permit theformation of flux vortices within the bridge. If theseflux quanta are subsequently removed by the current,via the Lorentz force, a quasi-continuous flux transfercan still occur giving rise to voltage by induction [4].Operation of the wider bridges has been observed butin general is much more erratic that for small bridges,03C9 03BE(T).The magnetic flux enclosed by one of these super-

conducting rings is shown in figure 3 A. This figureshows the magnetic flux enclosed by a superconductingring as a function of the applied flux. This is the fluxin the ring, presuming that the applied flux has startedfrom zero and has been slowly increased. Up to somecritical level, determined by the critical current of thebridge, the flux enclosed by the ring is held constant.Once this critical level is exceeded, flux begins to enterthe ring as quanta. When a quantum enters a ringthe flux shielded by the ring must also decrease by aquantum and the persistent current around the ring

FIG. 3.

A) Magnetie flux enclosed by a thin film superconduc-ting ring as a function of applied flux. Above y, fluxenters as quanta.

B) Emf generated by flux transfer into the ring as afunction of applied flux. Pulse amplitude is determinedby flux transit time.

decreases by yo L-1, where L is the inductance of thering. Thus the external flux must again be increasedby an amount (po to once more reach the critical condi-tion at which the process repeats. Since the relaxationtime is much shorter than any other characteristic timeof our circuit, the emf produced by this flux change isshown in figure 3 B. The time average of this emf isthe voltage to which I referred in the previous discussionof the voltage supporting superconducting state. Inthe previous discussion the total current was held fixedand above some critical level the supercurrent sustainedrelaxation oscillations at a rate determined by thevoltage. In the case of the inductively driven super-conducting ring however the current is not fixed andis free to adjust to flux changes within the ring. Theflux change in this case is not continuous since it is

triggered only when the current reaches its criticalvalue. As a consequence, this inductively driven ringacts as a "flux valve" producing an emf composed ofa series of pulses of amplitude y,,ri 1 spaced by incre-ments, cpo, in external flux.

In the experiments which I will describe, this emfis examined by observing the response of a tunedcircuit driven by the superconducting ring. To illus-trate this response we can Fourier analyze the pulsetrain using flux as the expansion variable and for thetime being ignore the starting transients. To accountfor a possible finite relaxation time in the flux changewe assume the emf to be a series of square pulses.However for convenience we convert this presumedshape dependence to time intervals where Tl is theflux transit time and T is the time between pulses.Then :

In this approximation the superconducting ringgenerates an emf whose amplitude is fixed by therelaxation time but whose frequency depends perio-dically on the applied flux. In the limit 1"1 Tthe emf becomes independent of 1"1 and equals :

depending only on the time between pulses. If the

applied flux oscillates in time, cp == cpr f sin mt, T is no

longer a constant but can be approximated by :

The only other change in the analysis is to introducea phase factor to take account of possible variationsin response to positive or negative fluxes. This varia-tion corresponds to the possibility of persistent currentfrom some steady ambient field. The resultant emffrom a superconducting ring, driven into flux flow byan oscillating field, of amplitude cprf and frequency m,in a steady field ({)dc can be characterized by [11] :

where we display only a component at the fundamentalfrequency. Thus the amplitude ofthis voltage dependsperiodically on the amplitude of both the steady andalternating components of magnetic flux. This vol-

tage has the same functional dependence as though the

17

ring were actually interrupted by a Josephson tunnelbarrier. However it is important to realize that theonly physical requirement here is flux quantization- not the Josephson tunneling phenomena. Thesestructures are not "weakly coupled" in the sense ofthe Josephson tunnel junction but are dynamicallyuncoupled by the critical current.

Experimental results by Nisenoff on this inducedvoltage are shown in figure 4. These data show the

W G. 4.

A) Amplitude of the superconducting emf as a func-tion of steady flux, ~dc. Pump amplitude, cpf, is heldconstant.

B) Amplitude of the superconducting emf as a func-tion of pump amplitude, cprf. Steady field, CfJdc’ is heldconstant. (Pa, and ~dc are on different scales - eachis periodic in h j2e.

amplitude of an rf signal induced in a tank circuitinductively coupled to a superconducting thin filmring. Usually, because of interactions with the reso-nant tank circuit, feedback effects also connect the

superconducting emf with y,, and act to limit the signalamplitude. If this feedback is phased properly thesystem can be made to oscillate. In these data the

experimental variable is the applied magnetic flux,however the scale of the absciss differs for curves Aand B. Figure 4 A illustrates the variation of thesignal amplitude at constant cprf ("pump amplitude")corresponding to changes in the steady flux ~dc; wherethe elementary derivation also leads us to expect asinusoidal variation. And, figure 4 B shows the signalamplitude as a function of pump amplitude when C[)dc isheld fixed. In this case as long as 03C41 ~ T the elemen-tary derivation would lead us to expect a Bessel func-tion type of oscillation. As the drive amplitude isincreased the time between pulses decreases as

T - ~0(03C9~rf) -1. Ultimately the signal cuts off whenthere is no time between pulses to extract the quantumfrom the circuit or when mt - C[)O/C[)rf.

These data were taken at a frequency of 30 MHzon a thin film superconducting tin ring 300 A thick.This ring was formed by evaporation on a quartz rod1 mm in diameter, the resulting ring being about amillimeter long. Into this ring was photo-etched aDayem bridge about a micron wide. Data similar tothis have been obtained from a number of super-conducting configurations. The materials we haveexamined so far have been tin, lead, aluminum, nio-bium and various alloys of tin-lead and indium-tin.The width and thickness of the Dayem bridge largelydetermines the temperature of operation of the device.For a 300 A tin film and a bridge width of about amicron the device will operate from the transition

temperature (,....., 3.8 OK) down to at least 1.2 oR. Overthis temperature range the critical current and thusthe bias level increases with decreasing temperature.However at the lowest temperatures the device still

operates even though the necessary bias level may beseveral thousand times greater than the signal. As the

bridge width increases the temperature range of opera-tion decreases, probably because of the requirementthat w 03BE(T). For a 300 A tin film the operatingrange is reduced to about 15 millidegrees for a 200 mi-cron bridge.

This degree of experimental confirmation of theconcepts of the operation of these superconductingdevices is about all we can expect due to the highlysimplified model and the neglect of higher harmonics.The periodicity of the signal must be accurately repre-sented by this approximation, however the amplitudedependence surely is not. The periodicity in steadyflux has been examined to a fraction of a percent, butthe periodicity in alternating flux has been confirmedto only a few percent. To this accuracy the periodis h/2e. The absolute amplitude of the signal dependson feedback effects arising from the coupling of classicaland quantum circuits, and has also been confirmedexperimentally. As it stands these superconductingelements form a reliable basis for the development ofa unique kind of instrumentation.

If the superconducting circuit contains two Dayemtype bridges it becomes meaningful to consider currentthrough the device, as well as the circulating current.This structure (see fig. 5) has many characteristicssimilar to the dc superconducting interferometer [12].When the device is driven from a direct current source,above some critical current, voltage is producedroughly proportional to the excess current abovecritical :

In addition, the sum of h and 12 is determined bythe current source, I1 + 12 = I and the average fluxin the device is quantized :

where L is the inductance and y,,.t is flux from someexternal source. Making the indicated substitutionsthe voltage becomes :

The quantum number N will always adjust to keep1 N Cf> 0 - Cf>ext 1 as small as possible, leading to a voltagewhich is periodic in flux, with period y.. The apparentdiscontinuous behavior at cpegt = (N + 1/2) cpo will

18

FIG. 5. - Schematic of a two bridge circuit where Dayembridges at 1 and 2 are connected by superconductinglinks. This device is driven by a current source

from A to B.

always be thermally smeared [8] in practice, leadingto a dc voltage whose amplitude is roughly sinusoidalin de magnetic flux. Most of the "point contact"devices [13] probably also operate on principles similarto those discussed here.

Noise. - In this mode of operation the superconduc-ting ring carries only supercurrent, except for the timeinterval of the transition, ’t’1. However, during thistime, voltage appears and normal current with itsattendant resistance must be developed. This appea-rance of resistance leads to an effective noise in the

operation of the device.During the transition we assume the superconduc-

ting circuit contains resistance R(t). At constant

resistance for this circuit, composed of an inductor anda resistor, the current noise is :

This noise is spread over a frequency band-widthroughly from zero to RIL and has a total magnitude of :

The total flux noise (cpn) 2 from this current in a singleturn is cpn = L2 I2 = LkT and is independent of R.Thus, as the resistance drops and the ring tends towardcomplete superconductivity after the transition, thisentire noise is confined to a decreasing bandwidth(RIL), until as R - 0, LkT represents the uncertaintyin the flux "trapped" by the ring at R = 0. And Lk Talso represents the spread expected in the magnitudeof trapped flux in the ring if the trapping process isrepeated many times.

Actually, since the flux trapped in a superconductingring (inductance Lg) is quantized, the trapped fluxitself will always be quantized. This trapped flux isusually as close to the ambient value as can be achieved

within the quantum rules. Adjacent values differingby NC?o are possible but are unlikely because of theextra energy involved. The result [8] is that the

trapped flux is a periodic function of the ambient fluxwith period (po.However in the presence of this noise the ambient

value itselfis uncertain by (L, k T) 1/2. Thus the averagetrapped flux must be determined by averaging theideal periodic behavior over a flux width of LkT.Clearly when LkT > y2 0 the periodicity disappearsand quantum effects vanish.The importance of this to the operation of these

superconducting devices is to put a limit on the induc-tance of any device expected to show quantum behavior

at Ls É kq>; or Ls É 10- 7 h at 4 oK. For a cylin-drical ring, area A and length 1, inductance is given

by L = {J-1. ° Thus for l - 10-3 m, ’ A 5 10-4 m2 ’

or a maximum diameter of one cm.As magnetometers, the important parameter is equi-

valent field noise, 03B4B2 = ~2n/A2. Thus maximum sen-

sitivity will be achieved at the largest size for whichoperation is possible. At this size the noise, cpn, willbe y2 0 so that in the above circuit 03B4Bmin ~ 10-7 gspread over the current band-width. If this signal isexamined with a time constant i’ there will be aneffective decrease in noise by a factor (-r’ QI Q) 1/2 whereSu is the resonance frequency. For most of our cir-cuits 03A9/Q ~ 105, thus a one second time constant

implies a field noise of about 8B N 3 X 10-1o g.Actually this can be improved somewhat by properdesign of the circuit to increase its area while holdingthe inductance down. Our best sensitivity so far,with a one second time constant, is 7 X lO-11 g.

Another source of flux noise at the superconductingcircuit is the input circuit itself. The equivalent inputcircuit for our amplifier is :

Current noise in this circuit is centered at the reso-nance frequency 03A9 _ (LC)-1/2 and is limited bythe Q,of the circuit to a frequency width roughly £2 Q- 1.Most of the noise is generated in the room temperatureresistor R and is coupled to the superconducting circuitthrough mutual inductance M. The superconductingcircuit will respond to all of the noise in the bandwidth

of the input circuit or cpn = M2 f: 1n dúl. These

noise signals are amplified and summed in a detectorto give an equivalent input flux noise at the super-conducting circuit of :

Hère T’ is the température of the résistance R.

For most of our circuits the coupling M 2 is a fewpercent and Q,2 R is about one. Thus this flux

19

noise in a circuit 1 mm in diameter (Ls""’" 10-9) isabout 1/10 quantum.

It turns out that as long as this noise is relativelysmall ( po), the response of the superconductor tothe noise depends strongly on the drive amplitude.Since the signal is quasi harmonic in CPrf there are

many signal maxima as a function of y,, (see f ig. 4 B).And, when the superconductor is biased at such a

maximum, to first order the signal is insensitive tofluctuations in cpr f. Thus, biased to a point of maximumsignal, the effect of noise is minimized. Conversely,by the same reasoning, at minimum signal the noiseis greatest. ,

Figure 6 shows these effects. The variation of the

superconducting emf with C?dc is shown with cprf as aparameter. Note that at 2y,f - (2n -1 ) hl2e the

FIG. 6. - Amplitude of the superconducting emf as afunction of steady magnetic field, Cl>dc’ with pumpamplitude as a parameter.

signal is positive maximum, at 2cprf""’" 2nn/2e the

signal is negative maximum, 3 and in both cases the

noise is small. While at 2prf ’" 2n+1 2 ~/2e the signalis zero and the noise is greatest. Note also the perio-dicity of both the signal and noise with ~dc. Actuallyit has been possible by proper choice of coupling andfeedback to quiet the noise of the tank circuit. At the

proper bias and coupling it is possible to achieve aninduced signal from the tank circuit with considerablyless noise than that generated by the tank itself.

Instrumentation - In our laboratory we have uti-lized these superconducting devices to produce highlysensitive ac and dc magnetometers, voltmeters andradiation detectors. 1 will concentrate here on thedc magnetometer applications. For this purpose the

superconducting circuit is biased to some appropriatelevel such that the response to a change in de field isas shown in figure 4 A. Two types of magnetometerhave been developed, an "analogue" and a "digital"instrument.The "analogue" instrument [14] employs a feedback

circuit which controls current to a solenoid surroun-

ding (or through) the superconducting device. The

magnetically controlled signal from the device is heldconstant (usually zero) by sensing any impendingchange in the signal and compensating for it by anappropriate current in the feedback solenoid. A deter-mination of the field change is then made by measuringthe control current for the feedback loop. The sensi-

tivity of this instrument is determined by the field

periodicity of the superconducting element and noisein the feedback loop. Devices with periodicity as lowas 10-7 gauss have been used in feedback circuits ableto control - 10-3 of a period. The lowest noise deviceso far has produced an instrument whose sensitivity isabout 7 X 10-11 gauss with a one second time constant.The practical bandwidth for these feedback circuits isabout 100 cycle s-1. This simple technique gives us arelatively sensitive magnetometer whose ultimate limithas not yet been achieved; the present limit is appa-rently set by noise in the feedback circuit.One drawback of this previous technique is that the

intrinsic periodicity of the signal (its quantum pro-perties) are not utilized. And also that large fieldchanges may go unobserved if they cause the instru-ment to "break lock" by exceeding the capabilities ofthe feedback loop and skip many periods. As des-

cribed, the instrument cannot tell one signal zero fromanother. To overcome these difficulties a second typeof instrument has been built to count flux periods [15].By differentiating the signal ( fig. 4 A) twice and usingthese derivative signals to gate a counting circuit, boththe sense and number of flux periods can be determined.This technique has been refined [16] so that now wehave an instrument which will count in units of 1 /4po ata rate of 104 per s. Both of these magnetometers havebeen made into absolute instruments by the simple ex-pedient of inverting the superconducting element [17]- the difference in readings being twice the absolutefield.

These magnetometers can be converted to de volt-meters by converting voltage to a known flux. Thiscan be done by impressing the dc voltage on a R-L series

FIG. 7. - Complete instrument package. Superconduc-ting circuit is contained within the tank coil at theend of the coaxial cable.

20

circuit containing N turns. If R is much larger thanthe impedence of the voltage source, the current Ithough this circuit is V/R. This current in turn gene-

rates a flux y = LIN-1 - L V ° The present magne-

tometer noise limit is about 10-3 y, giving an apparent

voltage sensitivity of V - N R 10-18 V. However this

circuit will develop a Johnson noise of its own where(cpn) = LkTN-2. This circuit noise will usually limitthe flux sensitivity so that :

Thus a micro-henry and a micro-ohm at 4.2 OK willgive a sensitivity of about lO-14 V and a time constantof one second.

Conclusion. - By the techniques which 1 have dis-cussed in this paper we have made operating instru-

ments which we are using in further laboratory experi-ments. The present sensitivity and response time ofthese instruments make them appropriate researchtools for a diversity of experiments. Our use has

ranged from determination of magnetic susceptibi-lity [18] and measurements of fundamental cons-

tants [19] to a search for magnetic monopoles [20].There seems no doubt that devices of this type canmake further useful contributions to many other tech-nical and scientific problems.Our present "state of the art" in this cryogenic

instrumentation is represented in figure 7. This

package contains all the electronics required to driveand detect the superconducting thin film circuit I havediscussed. Of course here the tank circuit and super-conducting film at the end of the coaxial cable is theonly element actually cooled to low temperatures.Despite repeated cooling to helium temperature thesesuperconducting circuits nevertheless form reliablestructures - some of our circuits have been success-

fully cooled hundreds of times over what is now a threeyear period.

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