Noise in the rf SQUID - Cornell University · of practical SQUID systems, indirectly it establishes the tank circuit noise amplitude and hence strongly affects the overall SQUID performance

Journal of Low Temperature Physics, Vol. 19, Nos. 3/4, 1975

Noise in the rf SQUID

L. D. Jackel and R. A. Buhrman

School o f Applied and Engineering Physics Laboratory o f Atomic and Solid State Physics

and The Materials Science Center Cornell University, Ithaca, New York

( R e c e i v e d O c t o b e r 7, 1974)

This paper presents the results of an extensive study of the sensitivity-limiting noise sources in rf-biased SQUID flux detectors. Our investigation has included both experiments with point contact and thin-film SQUIDs and digital computer simulations. The paper begins with a discussion of rf SQUID operation, with special emphasis on those aspects that determine flux detector sensitivity. We then discuss the origin and consequences of intrinsic device noise in rf SQUID systems. We describe a straightforward technique for determining intrinsic noise amplitude directly from SQUID rf I-V characteristics. Deviations in the measured intrinsic noise of many devices Jbom the Kurkiji~rvi-Webb prediction are attributed to departures from the ideal Josephson sinusoidal current-phase relation in the SQUID weak link. Kurkifi~rvi's SQUID tank circuit noise analysis is reviewed and we show how SQUID sensitivity can be maximized by an optimal coupling between SQUID ring and tank circuit. This optimal coupling depends upon circuit parameters, ring inductance, and weak-link properties. The analysis reveals that variations in the current-phase relation can result in a difference of as much as a factor of ten in the attainable sensitivity of two otherwise identical SQUID systems. We also suggest efforts in circuit design that are most likely to lead to further improvements in flux detector sensitivity. Finally, other factors, including external magnetic noise and underdamping of the SQUID ring, are discussed.

1. INTRODUCTION

Considerable progress has been made in the past several years in under- standing the origin of noise in rf-biased superconducting flux detectors (SQUIDS) which operate with transitions between adjacent fluxoid quantum states. The theoretical work began with the analysis by Kurkij~irvi (K-l) 1 of the effect of thermodynamic fluctuations on fluxoid transitions in a

201

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 1001 t. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher,

202 L.D. Jackel and R. A. Buhrman

superconducting ring closed by a weak link. Subsequent experiments have verified this analysis. Fluctuation effects on rf SQUID operation were then described in detail by Kurkij~irvi and Webb (K-W), 2 who predicted an intrinsic SQUID sensitivity limit. Finally, Kurkijgrvi (K-2) 3 completed the theoretical picture by describing the indirect but controlling effect that intrinsic fluctuations have on noise in SQUID system tank circuits.

In this paper we report results of experiments and computer simulations which (a) have tested the applicability to practical flux detectors of this recent theoretical effort, and (b) have provided a basis for optimizing SQUID sensitivity. We have found that the K-W prediction gives a lower limit for intrinsic noise in rf SQUIDs, but this limit is not always attained in practice. Indeed, while we have achieved the predicted intrinsic noise in some SQUIDs, often the intrinsic noise considerably exceeds the K-W prediction. Our results indicate that most, if not all, of the excess noise is caused by departures in the weak link from the Josephson sinusoidal current-phase relation.

While the intrinsic noise seldom limits directly the realizable sensitivity of practical SQUID systems, indirectly it establishes the tank circuit noise amplitude and hence strongly affects the overall SQUID performance. By governing the intrinsic noise amplitude and the SQUID ring dissipation, the current-phase relation of the weak link is the crucial determinant of the minimum attainable noise of any particular SQUID system.

In Section 2 we outline the basics of rf SQUID operation, with particular emphasis on those aspects that, in the subsequent sections, are shown to affect noise performance.

In Section 3 we discuss the K-W intrinsic noise analysis. We describe an effective technique for directly determining the intrinsic noise amplitude by measurement of the fractional step rise e of the SQUID rf I-V characteristic. The effect of nonsinusoidal current-phase relations upon the intrinsic noise is analyzed.

In Section 4 we report the results of measurements of both the amplitude of the intrinsic noise in practical SQUIDs and the dependence of this noise upon SQUID ring parameters. The rather large variation of intrinsic noise that is observed in apparently identical SQUIDs is also discussed.

In Section 5 we review Kurkij/irvi's SQUID tank circuit noise analysis and show how the sensitivity of a given SQUID system is optimized by a particular choice of coupling between the SQUID ring and the tank circuit.

In Section 6 we point out those areas where efforts are likely to lead to significant improvement in SQUID sensitivity. Numerical examples are given to illustrate these points.

Finally, in Section 7 we discuss other factors which in practice may affect SQUID sensitivity, including flux creep in the SQUID ring and its

Noise in the rf SQUID 203

environment, external magnetic noise that is incident upon the SQUID ring, and underdamping in the SQUID ring.

2. SQUID OPERATION

The most widely used class of if-biased SQUIDs utilizes transitions between adjacent fluxoid states of a superconducting ring closed by a weak link. Our study concerns only this type of SQUID. A number of review articles 4-7 have given detailed and largely correct descriptions of such SQUIDs, and these descriptions will not be repeated in their entirety here. We will only sketch essential features, emphasizing those aspects that determine SQUID sensitivity.

2.1. Fluxoid States of a SQUID Ring

An rf SQUID ring is shown schematically in Fig. 1. A superconducting ring of inductance L s is closed by a superconducting weak link. The weak link is modeled by a superconducting element with current-phase relation i c f (O) shunted by a resistance Rs and capacitance Cs. In practical SQUID rings, L s ,-~ 5 x 10 -1~ H, R s may vary from <0.1 to > 100fL and Cs ~ 0.1 pF. For best operation the weak link critical current is set to i c ~ ~)o/L~,

where ~b o, the flux quantum, is 2.07 x 10 -15 Wb. Fluxoid quantization relates the flux ~ threading the ring to the phase

difference 0 of the superconducting order parameter across the weak link:

o = (1)

Thus, through the current phase relation i J ( O ) , the flux q5 determines the supercurrent circulating around the ring,

i s = i c f ( - 2 ~ ( o / ~ b o ) (2)

where f(0) is an odd function of maximum amplitude _+ 1, and ic is the weak- link critical current.

We now consider the response of the SQUID ring to an applied magnetic flux q~x. The flux ~b threading the ring is related to the applied flux q5 x by

O=c)x + Lsi

Fig. 1. A schematic drawing of an rf SQUID ring. The weak link is modeled by a superconducting element with current- phase relation i,:f(O) shunted by a resistance Rs and capacitance C s. Due to fluxoid quantization 0 = -2~q5/~ o.

(3)

Ls


where i is the total circulating current. In the absence of thermal fluctuations and in the limit where 4;x/4)o << Rs/Ls , i reduces to an equilibrium supercurrent i s. Under these conditions we can combine Eqs. (2) and (3), to obtain

4) = 0~, - Lsicf(Zr~4)/Oo) (4)

In general, Eq. (4) gives a reentrant, snakelike curve in the 4)~-4) plane. An example with icf(O ) = (1.254)o/Ls) sin 0 illustrating some important general features is shown in Fig. 2(a). Kurkij/irvi (K-l) 1 has shown that the segments of such curves having positive slope correspond to regions of local potential minima, while those segments with negative slope correspond to local potential maxima. Thus at fixed 4)~, metastable fluxoid states on the positive slope segments are separated from each other by potential barriers that peak on the negative slope segments. The barriers vanish at values of the applied flux where the curve becomes vertical. If we assume the absence of thermal fluctuations and initially set 4) = 4)~ = 0 and then increase 4)~, 4) will follow along the curve given by Eq. (4) until 4)x = 4)xc and 4) = (Pc. At this point the potential barrier vanishes and the system will make a transition to a lower energy state. The time required to make this transition is on the order of Ls/Rs . If the system is properly overdamped, i.e., (2rcic/4)o)l/2R~Cls/2 <~ 1, it will come to rest at the next adjacent metastable fluxoid state, in our example, where 4) _~ 4)o. Otherwise, if not sufficiently damped, the system may jump through several fluxoid states before stabilizing. As we will see in Section 7, the underdamped situation is a possible source of extra SQUID noise. If we now decrease 4)~ for the overdamped case, 4) will remain near 4)o until 4)~ reaches - (4)~ - 4)o). At this point the system will make a transition to the original fluxoid state near 4) = 0 and then will make a second transition to 4) ,,~ -4)o when 4)~ = -4)~c. Thus if an oscillatory flux with amplitude 4)~c is applied to the SQUID ring, the system will folloW a hysteretic path as indicated in Fig. 2(a). It is these hysteresis loops which provide the signal in an if-biased SQUID.

In real systems 4) fluctuates in the potential wells whose minima define the metastable fluxoid states. Thermal fluctuations drive 4) over the barrier before q~x reaches 4)x~. The transitions are distributed around an applied flux 4)'c where the corresponding enclosed flux q5 is defined as 4)'c. The width of this distribution, i.e., the uncertainty in 4)x at which the transitions occur, ultimately limits SQUID sensitivity. These distributions have been investigated both theoretically 1 and experimentally. 8-~~ Depending upon SQUID ring parameters and 4;x, (4)~ - 4);,~) is between 4)o and 4)0/4. When viewed over a time scale much longer than the fluctuation correlation time (LffR), so as to average out individual fluctuations in the potential wells, the path of the system in the 4)x-~b plane is very nearly piecewise linear. In Fig. 2(b) we show a possible path for the flux in the presence of fluctuations for three


; i

/ / ~ - 2 ~ o-

(a)

r 1 6 2 ~

Jrl

, ,,? ~ ,c '~

(b)

Fig. 2. (a) A plot of Eq. (4) with Lsicf(2r~4)/(oo) = 1.25q~ o sin (2rc~b/qSo). For this case q~xc ~ 1.5~bo and q5 c = 0.25q~o. The thick lines indicate the hysteresis loop followed by the system when an oscillating flux of amplitude ~b= is applied to the SQUID ring. Here we assume the absence of fluctuations. (b) The path followed in the q~-q~ plane by the same SQUID ring in the presence of fluctuations. The fluxoid transitions are distributed about ~b'xc and occur at 4~ ~ ~b;.

sweeps of 4> x w i th a m p l i t u d e 4~xc. As we have shown, the t r a n s i t i o n s a re d i s t r i b u t e d a r o u n d q~ = q~c.

The c u r v e o f q5 vs. ~bx f r o m Eq. (4) d e p e n d s s t r o n g l y on the f o r m of the c u r r e n t - p h a s e r e l a t i o n icf(O ). As w e a k l inks b e c o m e l o n g c o m p a r e d to the s u p e r c o n d u c t i n g c o h e r e n c e length , icf(O) is e x p e c t e d to d e p a r t f r o m the

206 L. D, Jackel and R. A. Buhrman

sinusoidal Josephson relation. ~'12 In such weak links the max imum supercurrent i c is generated when 0 is considerably in excess of n/2. It is even possible to have a mult ivalued cur ren t -phase relation where the phase must wind to a value 0c > ~ in order for i S to reach ic. If 0 is increased beyond 0~, a "phase slip" occurs during which the superconduct ing order parameter in the weak link momentar i ly goes to zero and i s is reduced to the value it had when 0 = 0c - 27r. In Fig. 3(a) we show a hypothet ical mult ivalued cur ren t - phase relation. In Fig. 3(b) we show the ~b vs. q5 x curve that results if a weak link with this cur ren t -phase relation is used to close a S Q U I D ring. We have chosen our example so that we have the same value of ~bxc (l.5q~o) as we had in Fig. 2(a). No te the increased value of q~ in Fig. 3(b), corresponding to the increased phase winding in the weak link; ~bc here is 0.7q~o, whereas for the sinusoidal current--phase relation q~c ~ 4)o/4.

It is also impor tan t to note that the spacing in q~ between adjacent fluxoid states of Fig. 3(b) is reduced from the spacing of Fig. 2(a). Thus, if we sweep ~bx with ampli tude q~,~ in Fig. 3(b), we will trace out smaller area hysteresis loops than we did in Fig. 2(a). In general, for any cur rent -phase relation the area A of the hysteresis loops in the q~-q9 plane is A -- 4)~d?o/7~, where ~ = ~b~/(q~x~ - �89 - ~b~/$~). In the example of Fig. 3, 4'c = 0.7q50 and ~b~ = 1.54)o, so that ~ ~ 2.8. For a ring with a sinusoidal weak-link cur rent -phase relation and qS~, = 1.5, as in Fig. 2, we have ~c _-__ 1.8. General ly

is(8)

I

-2,~ -0o I -,~ ~

~ -i c

(a)

/ i

r

J _.J,c

(b)

Fig. 3. (a) A possible multivalued current-phase relation for a weak link long compared to the superconducting coherence length. Here 0c = 1.4r~. (b) The hysteresis loop in the 4~-4~ plane in the absence of fluctuations for a SQUID ring where the current-phase relation is as shown in (a).


when the critical external flux is relatively low, qSx~ > ~b0, the parameter 7c depends strongly upon the current-phase relation of the weak link, with 7~ increasing in value as the current-phase relation becomes more and more skewed. (We will show below that the magnitude of yc is an essential factor in determining SQUID sensitivity.) When q~x~ >> ~bo and q ~ >> ~b~, the area of the hysteresis loops becomes independent of the current-phase relation as 7c approaches its limiting value of unity. As we will show in Section 6, this fact makes weak links with essentially sinusoidal current-phase relations and with critical current such that qS~ >> q5 0 more desirable than those with "lag" skewed or multivalued f(O) and/or with critical current such that ~bxc ~> ~bo and ~bx~ > qS~.

The transition distributions for quasisinusoidal f(O) can be determined using the method of K-l, which considers fluctuations only ifi the phase 0 of the superconducting order parameter. For highly nonsinusoidal f(O), such as the example of Fig. 3, it is necessary to also consider fluctuations in the amplitude of the order parameter. Calculations of the transition distribution for this phase-winding case have not been done.

2.2. S Q U I D Systems

In further discussion in this section we will assume that all transitions occur at the distribution center q~;,c, where 4~ -- ~b'c. In Section 3 we will discuss the important effects of finite distribution width.

When operated as an rf SQUID, the SQUID ring is inductively coupled to a high-Q tank circuit resonant at oJrf, where usually ~Orf/2~Z ~ 10 MHz, although higher bias frequencies may be used. The tank circuit consists of an inductor Lt and a capacitor Ct shunted by a load resistor Rt (see Fig. 4). The circuit is driven at or near the resonant frequency by an rfcurrent source. The rf voltage across the circuit is amplified by a high-input-impedance, low-noise amplifier and is subsequently detected, providing the S Q U I D system signal output. The rf I-V characteristic of the SQUID system can be mapped out if we record the variation in the peak rf voltage V~f with a slowly increasing drive current amplitude Irf. An idealization of this situation is shown in Fig. 5. Here we have a digital computer simulation of a noise-free SQUID which illustrates some important effects obscured in the operation of real SQUIDs, (Details of this simulation will be given elsewhere,) To better illustrate these effects, we have chosen the coupling between SQUID ring and tank circuit, K = M/(LsL~) t/2, to be 1/3, a much larger value than is desired in practical SQUIDs. For the same reason we have also chosen Q -- 30, an unusually small value for practical devices.

In an rf S Q U I D the applied flux is a sum of a quasi-dc flux q~r which we wish to measure, and the biasing rf flux ~b~ 0 generated by oscillating

208 L .D. Jackel and R. A. Buhrman

0 Cryostat

Icf R :s j .

7

1 I I I I

I I I I I I

Fig. 4. A schematic drawing of an rf SQUID system.

Vr f (~

| |

vo( ~oI --

~ ~step ~i = Iriser ~'~

Trf

Fig. 5. The rf I-V characteristic of a digitally simulated, noise-free SQUID. The characteristic was computed with the rf drive current slowly increasing linearly with time. The rate of increase was such that the time elapsed between point A and point B corresponds to about 500 rf cycles. In this simulation Q = 30, 6xc = 1.7q5o and 7 c = 1.


currents in the tank circuit inductor. The peak rf flux is related to Vrf by

(~(x r f ) = M~rf/~rfLt (5)

We first consider the case where ~b~ d~) = n4~o. If l~r is slowly increased from zero, V~f increases proportionately and the flux oscillates at o~f with slowly increasing amplitude along the path of Fig. 6(a). There are no fluxoid transitions, except possibly a few initial transitions which serve to bring the average value of ~b equal to q~~ This low level of tank circuit excitation corresponds to the initial rise in Fig. 5, where ~f -- I r f ~ r f L t Q . When l~f is increased so that Vrf = V~(n~b0) = ~rfL,~'xc/M, the critical applied flux is reached on either the positive or negative half of the rf cycle and the system undergoes a pair of fluxoid transitions. A dissipative hysteretic path, as shown in Fig. 6(b), is traced out. An energy z~E proportional to the area of the hysteresis loop in the q~-~b~ plane is dissipated by the SQUID ring:

A E = 2c~'x~ao/L~7 ~ (6)

-~IcC

(a)

fl (b)

~XC {c) (d)

Fig. 6. The path followed by a SQUID ring in the ~b~-q~ plane when qS~ r~ is supplied by a tank circuit driven at O)rf2 In this figure ~b~r n~b o. (a) max 1~b~xrnl < 4~xc. Since no fluxoid transitions occur, the path is not hysteretic. (b) The tank circuit is only driven hard enough so that ~b~ rrj just reaches qSxc. On the negative half of the rf cycle - ~bxc is not reached, due to the energy losses from the previous transitions. (c) The tank circuit is driven sufficiently hard so that two sets of flnxoid transitions can occur during one rf cycle. (d) ~b~r < max ]~b]r~J < qS~c + 4, o. As ~b~ ~ increases above ~b~,c, tails are added to the hysteresis loops.

210 L.D. Jaekel and R. A. Bulu'man

where we have now redefined ?r slightly to be

7r = qS;,c/(q~;~ - �89 - qS'~) (7)

The tank circuit voltage is lowered by

A V L = aEC~rfM/r (8)

The recovery rate of the tank circuit from the energy loss and hence the period of the resulting relaxation oscillations depends on Q and I~t. Near point A of Fig. 5 several rf cycles elapse before V~f again reaches V~ where ~bx -- ~b~. As I~f increases, the duration between transitions decreases, until at B the dissipative path of Fig. 6(b) is traversed every rf cycle.

Beyond B every few cycles ]qSxr I is reached on both the positive and negative rf swings, so that a second set of transitions occurs (Fig. 6c). Again the duration between second sets of transitions decreases with increasing I,r until at C there are two sets of transitions every rf cycle. As I~r increases beyond C, V~f grows approximately linearly with irf creating a "riser," as the tails of Fig. 6(d) are added to the hysteresis loops. The riser terminates at D in Fig. 5, where a third set of transitions begins to occur, causing a new step. If Irf is further increased, more hysteresis loops are added each cycle as the staircase rf I - V characteristic for ~ r = n~b0 is generated.

The length of the step l~t~p as measured in rf drive current from A to C in Fig. 5 can be calculated by a method similar to one used by Kurkij/irvi (K-2). 3 The increased dissipation from A to B caused by two sets of fluxoid transitions every cycle is oo~fAE/n. If we make the approximation that the voltage across the step is the constant V~ = (p'x~oorf(Lt)/M, we have

or

VcI~tr = 2OrfAE/2~ (9)

/step = 2d~oM/nLtLsyc (10)

Notice that the step length is sensitive to the form of the current-phase relation, and to qSxc , through the dependence on ?r In fact the step length can indicate the extent of phase winding in the weak link ; the greater the phase winding, the shorter ]'step becomes, particularly for low q~c. We show in Section 5 that I~tep plays an important role in determining the maximum attainable SQUID system sensitivity.

Now suppose q~c} = (n + 1)~o. As Irf slowly increases from zero, Vrf increases as before. The system oscillates at corf with increasing amplitude along a path like Fig. 7(a). This path is similar to Fig. 6(a) except that it is displaced by ~bo/2. But now the drive current Irr need only be increased to a value where Vrf = Vc[(n + �89 = (~'~r - �89 to reach ~'~r and cause a set of transitions (Fig. 7b). Thus a step begins at point E in Fig. 5.

Noise in the r f S Q U I D 211

I

(a) (b)

4~

(c)

~ J

~c ~x

(d)

4,

(e)

Fig. 7. The path followed by a S Q U I D ring in the q5 x ~b plane when ~b~ ~) = (n + �89 The solid dot represents the equilibrium posit ion for the system when qS~ ~ = 0. (a) max kbTf)l < ~bxc. (b) max I~b~~ = ~bxc - ,;bo/2. (c) ~bx~ - ~o/2 < max kb~f~ I < ~bxc + q~o/2. (d) max Iq~f)l = q~xc + ~o/2 on only one of the half-cycles. (e) max I~b~ml = , ~ + ~bo/2 on both positive and negative half-cycles.

As before, the recovery rate between transi t ions increases a long the step, until at poin t F a pair occurs every cycle, and a riser begins. This step is only half as long as the first step for ~b~ c) = nq~ o because at the end of this step only one and not two pairs of t ransi t ions occurs every rf cycle. Along the riser between F and G, tails are added to the hysteresis loops, Fig. 7(c), so that the ampl i tude of ~b~ rf) increases and hence V~f increases. At G in Fig. 5 a second set of t ransi t ions begins to occur, Fig. 7(d), and later at H a third set


is initiated, until at I a full step is completed and a new riser begins as a pattern similar to Fig. 7(e) is traversed every cycle. We have thus generated the limiting forms of the SQUID rf I - V characteristic as a function of 4~ c). If q~c) gradually changes from n~bo to (n + �89 the full steps split, with the left sides moving down and the right sides moving up, until at 4~ ) = (n + �89 0 the second pattern of Fig. 5 is obtained.

Note that in Fig. 5 the average value of V~f increases across a given step. Thus, as pointed out by Kurkij/irvi and Webb (K-W), 2 there is an "inherent" step slope even in the absence of noise effects. The average voltage change along a half step is AVL. This effect is the step slope observed by Simmonds and Parker 1~ in their analog rf SQUID simulation. The voltage separating the rf steps is

A V o --- OgrfL,c~o/m = (Ogrf/K)(LJLs)l/2~po (11)

For practical SQUIDS the coupling K is small enough that AVL << V0. A computer simulation with a "practical" value of K (K -- 0.1) is shown in Fig. 8. The relaxation oscillations and the inherent step slope of Fig. 5 are not observable on the scale of our drawing.

When operated as a flux detector the SQUID is biased on a step by a constant/rf" Changes in ~b~ ~) are detected as changes in Vrf. In the noise-free

Vrf

i i

Irf

Fig. 8. A simulated rf I - u characteristic of a noise-free SQUID under "practical" operating conditions; K = 0.1 and Q = 100,


case the change in V~f, AVe, for a flux change Aq~ (de) at fixed lrf is

AV~ ~ oorfL~Adye)/M (12)

provided Aq~ (d~) < ~b0/2. Thus the voltage signal increases with decreasing M, but obviously M cannot be made arbitrarily small without ultimate loss of signal. In order to remain on a step at constant Irf for all values of ~b(~ ~), we must have/step ~-~ /riser" The riser length/riser is the difference in Irf from points like C and D in Fig. 5:

/riser ~ AVo/~ = (ao/MQ (13)

Therefore the coupling between the ring and tank circuit must be tight enough that

2dpoM/nLtL~7~ >_ (ao/MQ (14)

or

M2Q/L,LsTc -= KZQ/7~ > ~/2 (15)

In order to achieve maximum voltage signal, M should be reduced until the equality in (15) is satisfied. However, in the following sections we will show that due to tank circuit noise this is not necessarily the value of M that maximizes the sensitivity of a given device.

The sensitivity of an if-biased SQUID depends on how accurately the step voltage can be determined. In the absence of external magnetic noise there are three important noise components in a SQUID system that may affect the sensitivity. These are : (1) noise in the rf amplifier, (2) noise arising from dissipation in the rf tank circuit, and (3) intrinsic S Q U I D noise due to the finite transition distribution width. Under the somewhat ideal assump- tion that amplifier noise is independent of source impedance, the amplitude of (1) is independent of (2) and (3). On the other hand, K-2 has shown that tank circuit noise is not independent of intrinsic noise. Rather, the intrinsic noise determines indirectly the amplitude of the tank circuit noise and hence the overall performance of a given S Q U I D system. We proceed to discuss intrinsic noise.

3. INTRINSIC NOISE

In the previous section we assumed that all fluxoid transitions occur at a single value of applied flux ~bx = ~b'xc. In real SQUID rings thermal fluctuations result in a finite state lifetime for any q~x less than ~b~c. The lifetime is long if ~bx is far from ~b~c but goes to zero as q~ approaches ~b~c. Thus fluxoid transitions occur over a distribution in ~b~. The form of the distribution, including its width a and center q~'xc, depend on how q~x is swept and on SQUID ring parameters.

214 L.D. Jackel and R. A. Buhrrnan

Kurkij~irvi (K-I) 1 has calculated transition distributions for a weak link modeled by a Josephson junction with a sinusoidal current-phase relation shunted by a resistance Rs and capacitance C~. Kurkijiirvi only considers the case where q~x/qSo << R I f L e . For constant q~x, K-1 finds

a = aoLjc(2rck~Tj f f )o i~)2/3(3/4x/ /2) 2/3 (16)

where Ts is the temperature of the SQUID ring and i~ is the weak-link critical current. The prefactor ao is of order unity and depends only very weakly on ring parameters and dx. Typically a ~ q50/20 for L, ~ 5 x 10-X~ T~ ~ 4.2 K, and i c ~ 8/~A,

Experimentally the transition distribution has been measured at constant, low sweep rates for rings closed with oxidized point contacts 8'9 and tin microbridges of submicron dimensions formed by electron beam resist techniques.~~ These experiments are in agreement with the K-1 theory with no adjustable parameters. However, we have found that larger tin microbridges give distributions which are more than twice as wide as predicted by the K-1 model. Apparently the K-1 theory represents a lower bound for a not attained by all weak links.

In an if-biased SQUID the applied rf flux ~b(x rf) is swept sinusoidally in time. If the amplitude of q~f) is small compared to qSxc, the system is con- fined to regions in the q~-qS~ plane where fluxoid state lifetimes are always long and thus transitions are very improbable. As the amplitude of qS(~f ) is increased, the system is driven into regions where the lifetime becomes shorter and transitions are more probable. As the amplitude of ~b(~ rf) nears ~b~, the transition probability approaches unity. Kurkijfirvi and Webb (K-W) 2 have found that this amplitude-dependent lifetime has two important effects on rf SQUID performance.

3.1. Amplitude of Intrinsic Noise ~6~b~)t/2

First K-W showed that the statistical nature of energy adsorption from the tank circuit generates random fluctuations in the tank circuit voltage, destroying the simple periodicity of the relaxation oscillations discussed in Section 2. This so-called intrinsic noise can be expressed as an equivalent flux noise (6~p~) I/2 and is the lowest noise that one can ever attain in a given SQUID system. The intrinsic noise is directly proportional to the transition distribution width a. At frequencies much less than o)r~ the spectral density of the intrinsic noise is given by

~q~2) /Hz ~ (~/(Drf)O "2 (17)

K-W found that for the case of the K-1 shunted Josephson junction model and for corf/2r~ < R s x 6 MHz, where R s is in ohms and the ring parameters


is(0) 1 ie

-ir

-tr ~/ 2 t r ,0

Fig. 9. A possible skewed, single-valued current-phase relation.

are as given above,

(~/2>1/2 ,.. ~F2~q 1/2. .1/3F2~kBTs72/3F 3 72/3 ~_ u . , ~ . ~ [ Lszc" / - - , / ~ / (18)

Hz1/2 L rfJ L q~0 ] [_4x/2]

For the SQUID ring parameters given above, Eq. (18) yields

(6~b2)1/2/Hz 1/2 ~ O.0740o(2n/~Orf) */2

This intrinsic noise is rarely observed directly in practical SQUID systems since the amplifier and tank circuit noise contributions usually dominate.

The effect of nonsinusoidal current phase relations on intrinsic noise has been outlined by Kurkij/irvi (K-2), 3 to a first approximation. He finds that (&b2) */2 should vary as [D(4)c)] 1/3, where D(~bc)=-(qS0/2n)2 x cl=f(2rt4)/4)o)/cl4) z evaluated at ~b = q5r For a Josephson junction D(~bc) = [1 - ((Oo/2nLsi~)2], a factor we ignored in writing Eq. (18). A possible example of a skewed but still single-valued current-phase relation is shown in Fig. 9. In this case D(qSc) = 4.3. We see that here the intrinsic noise can be expected to be ,-~ 1.6 times greater than that for a Josephson junction.

If the current-phase relation becomes even more skewed so that it becomes multivalued, as shown, for example, in Fig. 3, then Kurkijfirvi's phase fluctuation analysis is no longer applicable. In this case fluctuations in the amplitude of the order parameter as discussed by Langer and Ambegaokar 14 and by McCumber and Halperin is must also be taken into account. We expect that such fluctuations should lead to increased fluxoid transition uncertainty, but regrettably this theoretical analysis has not been done.

3.2. Thermal-Noise-Induced Step Slope and Fractional Rise =

The second effect of the transition uncertainty is the tilting of the steps in the SQUID rf I - V characteristic and the rounding of the step edges. As

216 L.D. Jackel and R. A, Buhrman

we will show in Section 5, the step slope has a controlling influence on tank circuit noise and therefore on the total noise of the SQUID system. K-W have shown that the fluctuation-induced step slope is directly proportional to the amplitude of the intrinsic noise. Under most practical operating conditions, this slope is much larger than the "inherent" step slope discussed in Section 2.

The increased step slope is explained in the following argument. Near the beginning of a step, the amplitude of ~b~ f) is just large enough so that transitions are only likely to occur every few rf cycles. Near the end of the step, the amplitude of qS~ f) must increase such that transitions are very probable each cycle. Because the tank circuit voltage V~f is directly proportional to ~b(2 f), the result is an rf step slope which reflects directly the width of the transition distribution due to thermal fluctuations.

The rf I -V characteristic measured for a typical SQUID with a post- detection bandwidth of ,-, 1 Hz is shown in Fig. 10. Note that in contrast to the fluctuation-free simulation of Fig. 8, Fig. 10 shows a substantial step slope as well as noise-rounded step corners.

Extension of the K-W analysis yields the crucial result that the step slope of a given SQUID system is directly proportional to the SQUID's intrinsic flux noise, regardless of the form of the transition distribution. Furthermore, as we have verified experimentally, the ratio ~ of the voltage rise along a step AVs to the voltage difference AV0 between steps is inde-

~f

I r f

Fig. 10. The rf I-V characteristic of an actual SQUID. This characteristic was recorded using a post-detection bandwidth of ~ 1 Hz to reduce system noise. The construction used to determine the fractional step rise c~ is shown.


pendent of tank circuit parameters and is given by

Hzl/2 L2~_ I 0.7q~o (19)

While some results of K-W are only applicable to SQUIDs with sinusoidal current-phase relations driven at fairly low bias frequencies, Eq. (19) has a much more general validity. Apart from possible changes of order unity in the constant 0.7, ~ directly reflects the intrinsic noise of a SQUID with any current-phase relation driven at any bias frequency < Rs/L ~. It is difficult to overemphasize the value of ~ as a straightforward measure of intrinsic noise. It permits the quick comparison of the quality of different SQUID rings regardless of variations in the particular tank circuits or amplifiers used. Measurements of~ have shown that all SQUIDs do not have as low an intrinsic noise as given by Eq. (18). In fact, for some devices ~ is more than five times the value predicted by the K-W model. In the next section we report the determination of (602i) 1/2 for practical SQUIDs through measurements of ~.

4. INTRINSIC NOISE: EXPERIMENTAL RESULTS AND DISCUSSION

In this section we report the results of an experimental investigation of intrinsic noise through measurements of the fractional step rise ~ for numerous point contact SQUID rings and a few thin tin film microbridge SQUID rings. We have found major discrepancies between experiments and some of the K-W predictions. Our experiments indicate that these discrepancies are caused by nonsinusoidal current-phase relations in the weak link.

4.1. Point Contact Results

Measurements reported in this section were made at rf drive frequencies ~ 4 MHz, with the only exception being the measurement of the frequency dependence of ~. SQUID rf l - V characteristics from which ~ was obtained were plotted using the following technique (see Fig. 11): The rf drive current was modulated at ~0.01 Hz by a triangle wave, which was also used to drive the X axis of an X - Y plotter. After preamplification, the SQUID rf signal was mixed with a local rf oscillator providing a 50-kHz intermediate frequency (i.f.) signal. The i.f. signal was passed through a 50-kHz tuned filter with Q = 10 and then further amplified to an average level of several volts, allowing good linear detection with a simple diode detector at all but the lowest signal levels. Post-detection bandwidth was reduced to a few Hz so that the rms flux noise was ~ 10-3q50 . This low noise signal was then used


Mocl lator [

Cryostgt

lator I

, . J

Fig. 11. A schematic drawing of the circuit used to record the SQUID rf I V characteristic.

to drive the Y axis of the plotter. Determination of c~ from the I - V characteristic was then made using the construction of Fig. 10.

From Eq. (19) we see that the K-W intrinsic noise prediction of Eq. (18), evaluated for L S = 5 x 10 - l ~ H, T s = 4.2 K, and ic = 2dpo/L,, gives c~ ~ 0.1. In contrast, measurements of ~ made under these conditions range from a low of 0.12 to a high in excess of 0.4. The most commonly observed values were between 0.2 and 0.3.

From Eqs. (18) and (19) we also see that according to the K-W theory c~ should vary as Ls, "c il/3, and _sT 2/3. In general, only the variation of c~ with L s agreed with this prediction.

The inductance dependence of e for a permanently adjusted two-hole point contact S Q U I D ring 16'17 was determined by first measuring the S Q U I D rf I - V characteristic and then remeasuring the I - V with a niobium


rod inserted into one of the S Q U I D ring holes. The change in L s was determined f rom the observed change in the ratio V~(n4o)/z~Vo oc Lj~. Within 5 exper imenta l accuracy, c~, and hence the intrinsic noise, varied linearly with Ls even though c~'s for the var ious S Q U I D s tested varied by over a factor of two at fixed L,.

The measured ic and Ts dependences for poin t contacts were in much less accord with the K - W predictions. These measurements were made with an appa ra tus which al lowed low- tempera ture ad jus tment of the point contact. The var ia t ion of c~ with ic was .de termined by varying the point contac t posit ion at a cons tant t empera tu re of 4.2 K. In some instances a series of adjus tments gave, as predicted, 0c oc i~/3. Such a result somet imes occurred when the critical current of a freshly sharpened and oxidized point contact was increased smooth ly f rom an initial low value as shown in Fig. 12. However , in mos t measurements no regular var ia t ion of c~ with ic was found, a l though contacts with large ic tended to have large ~.

0.4

toO,3

n "

e ~ o ~0.2 r- .o

O. L @

"/'co= L~- = 4.1/.zA

. I I I

(Ic /Tco)

Fig. 12. The results of one series of measurements of the variation of ~ with critical current for an adjustable point contact SQUID. The solid line is a one-parameter fit of the K-W prediction to the data. These data are not typical of our measurements.


The T~ dependence of c~ was studied over a temperature range of 4.2 K to ~ 2 K. As T~ was varied, the contact was readjusted when necessary so that c~ could be measured at constant ic. Despite extensive measurements of this sort, no definite correlation of e with T~ was established, not a very sur- prising result in light of the critical current results. However, measurements of e for some permanently adjusted oxidized niobium point contacts ~6 as a joint function of i c and T~ have yielded the K-W predicted results. Figure 13 shows c~ vs. i~/3T ~/3 for such a device in the temperature range 4.2-2.7 K. It is important to note that it was with such permanently adjusted point contacts that a sinusoidal current-phase relation was measured. 18 Further- more, the transition distribution of SQUID rings incorporating these contacts measured for an applied flux swept at a low constant rate was in agreement with the K-I prediction. 9

We have also investigated the dependence of e on corr. K-W predict to increase only very slowly with increasing drive frequency for (Orf/2~I <~ 1 MHz. However, at higher frequencies c~ is predicted to increase much more rapidly with increasing corr. According to the K-W numerical calculation, this rapid increase should begin to occur for the SQUID ring parameters given above when ~orr/2zc g Rs x 6 MHz, where Rs is expressed in ohms.

Ic(/ZA)

12 I

Ico = i-= = 4. I t.I.A L - S

0 2

i

O

O

O. I ~ / 1 l ~ t l ~ l l ~ l f t l l i

t I I I 0 I 2 3 4 T~ (K)

Fig. 13. The variation of ~ with critical current and SQUID temperature for a permanently adjusted, oxidized point contact SQUID. The solid line is a one-parameter fit to the data of the K-W prediction. The insert shows the temperature dependence on the critical current.


0.4

0.3 ~3

n.-

o. 0.2

-6 t-

O

~0.~ Lt.

I I I I [ I I I I I 0 2 4 6 8 10 t2 14 16 18 20

r f Bies Frequency frf (MHz)

Fig. 14. The frequency dependence of sr for a permanently adjusted, oxidized point contact SQUID.

Of course, once ~orf > R, /L s, the barrier escape analysis of K-W fails completely; the fluxoid transition now occurs at or even beyond the noise- free critical applied flux ~b~c.* Nevertheless, due to thermal fluctuations there remains an uncertainty in the value of applied flux at which the fluxoid transition begins. As long as dissipation caused by transitions between quantum states is the dominant effect the SQUID ring has upon the resonant rf circuit, this uncertainty in the fluxoid transition will result in fluctuations of the rf step voltage and hence in an intrinsic equivalent flux noise and in a fractional step rise c~.

The frequency dependence of c~ was measured for a permanently adjusted, oxidized point contact SQUID ring which at 3 MHz had e ~ 0.17, fairly close to the K-W prediction. As we report later in this section, oxidized niobium point contacts have a resistance of about 1 ~. The results shown in Fig. 14 indicate that, as predicted, above 6 MHz, c~ begins to increase. One should be cautioned against extrapolating the data of Fig. 14 to frequencies

*The concept of "transition between fluxoid states" of course becomes somewhat strained at very high bias frequencies, co,f > Rs/L ~. Even in the absence of fluctuations the system never manages to settle in the minima of the potential wells that define the ftuxoid quantum states. The applied flux 4~ changes so rapidly that the internal flux 4 finds itself pushed by the sizes of the potential well, much like a surfboard on a wave. The potential barriers have long vanished before ~b moves toward a new well. With fluctuations the system appears to wander around the q~x-4~ plane, with the average path for an rf cycle only approximating the sharply cornered path traversed at low frequencies.


j . ~ - 0 0 - 9 6 Niobium C : ~ ~ / Screws

~ I ] Niobiom S pacer - - , ~ . . . ~ ~ Block

Shorting / I ~ Point Contact ContoctJ I I

Fig. 15. A schematic representation of the device used to measure normal state de resistance R, and fractional step rise ~ of adjustable point contact SQUIDs.

much higher than 20 MHz, Preliminary experiments in our laboratory 19 indicate that at 450 MHz, ~ for these SQUID rings increases only to about 0.4, Apparently ~ varies only slowly with frequency after the initial increase.

We supposed that the excess in the measured values of c~ above the prediction of Eqs. (18) and (19) may simply have been due to anomalously small values of the weak-link normal state resistance R S. We expected a large

to be correlated with a small R s. Our measurements, which used the device shown in Fig. 15, gave just the opposite results. The measurement procedure was to first adjust the point contact to an appropriate critical current and plot the dc I -V characteristic determining R s. Then the shorting screw was

I 1 Ico = ~o/Ls = 1.9~A o( )3 0.5

0.4

0.3

0.2

o. IJ'- - [

ee �9

l w

Q

Theoretical Value

L L L I I 2 .5 I0 20 50

R (ohms)

Fig. 16. Normalized fractional step rise ~(ico/ic) 1/3 versus weak-link normal state resistance of adjustable point contact niobium SQUIDs. The division by the K-1 �9 1/3 dependence was found to reduce scattering of the data. I c


adjusted to close the superconducting ring. The device was operated as a SQUID with e)rf/2g = 3 MHz and e was recorded. This procedure was repeated for numerous adjustments of the point contact for both oxidized and unoxidized point contacts. In Fig. 16 we plot the data as o~/i~/3 vs. R~ to adjust for the observed approximate increase of c~ with i~/3. Although there is certainly no regular dependence on R~, there is a definite correlation between c~ and R~. As R~ varies from 1 ~ to greater than 50 f2, c~ varies from a low of 0.14, close to the K-W predicted value for this SQUID ring, to a high of 0.6.

We found that if a freshly oxidized and sharpened point was used to make a weak link, the resulting ~ was usually small. These point contacts also usually had small Rs. As the contact was repeatedly adjusted, it became more difficult to get small enough critical currents. When a satisfactory adjustment could be made, the contact typically had a high R s and a large ~. Visual examination of the point contact-after many adjustments revealed not only a flattened point, but also the destruction of most of the dull gold- or blue-colored oxide layer, leaving a shiny niobium surface.

4.2. Possible Causes of Deviations from K-W Predictions

We interpret these results to mean that the large values of c~ are caused by some nonideal behavior in the niobium point contact which is correlated with its resistance. The current-phase relation of an oxidized niobium point contact has been measured to be almost exactly sinusoidal, but the current- phase relation of clean niobium contacts has not been precisely determined. However, we have found that when the critical current was fairly low, ic >~ (ao/Ls, SQUIDs with large R s and large c~ also have a shorter rf step length Istev than SQUIDs with smaller Rs and smaller c~. This was the case even though all tank circuit parameters were held fixed and the q~'xc values of the two SQUIDs under comparison were essentially identical. Only when ic .~> Oo/L~ did/step become approximately equal for SQUIDs with significantly different e. As discussed in Section 2, the shortened step length found in SQUIDs with a large fractional step rise indicates that 7c, and hence q~'~ in these SQUIDs, is considerably greater than in those SQUIDs having a smaller c~.

Figure 17 shows two point contact SQUID rf I - V characteristics, taken under identical circuit conditions, which illustrate the correlation of /step with c~. The I - V characteristic of SQUID (a), which has a step length ,-~ 14 % shofter than does SQUID (b), also exhibits a fractional step rise c~ that is almost twice as large as that of SQUID (b). The critical applied fluxes of SQUIDs (a) and (b) are approximately the same, ~b'= ~ 2q~o, and from the ratio l~te~(a)/I~t~p(b) we can estimate the difference between ~b~(a) and qS'~(b).


Vr f

(a)

(b)

/i T _ t~ o~ . - - I" J-step - ~.o~

~~, i- I s ~ r b . unit) 'H

! .

Ir t

Fig, 17. Two different measured point contact SQUID rf l-V characteristics illustrating the correlation of/step with Co.

We find q~;(a),,~ 0.86~b;(b)+ 0.28q~o. Depending on the value of q~;(b), which must be ~ ~bo/4 or greater, qS;(a) can be as much as twice ~b;(b). Since the large values of ~b; result from skewed current-phase relations, we conclude that most, if not all, of the excess intrinsic noise as measured by ~ [when measured at small enough COrf that Eq. (18) should be valid] can be attributed to the nonsinusoidal current-phase relations.

4.3. Thin-Film Tin Microbridge Results

In order to further verify this concluMon, some thin-film tin microbridge SQUID rings have been produced. 2~ The dimensions of the microbridges were sufficiently large, ~ 1/~m, that we expected their current-phase relation to be distinctly nonsinusoidal. Preliminary measurements on these rings have fulfilled that expectation. At a temperature slightly below the transition temperature of the ring where the critical current of the bridge is ~ tho/L s, measurements of q~ vs. q~x using the technique of Ref. 16 have shown these samples to have large values of q~c and hence very non-sinusoidal current- phase relations. A typical result of such a measurement is shown in Fig. 18, where the hysteresis pattern produced by several sweeps of ~x is given. Here we find that q~'c ~ 0.8tPo, indicating that at this critical current this par-


,r ~ ~ , t- -- t ~ -- o.S~o

Fig. 18. The measured q~-qS:, hysteresis loop of a tin thin-film microbridge SQUID ring, The phase winding in this ring is such that ~b'c ~ 0.8~bo, indicating that the current-phase relation of the rnicrobridge is actually even more reentrant than the hypothetical current-phase relation shown in Fig. 3.

ticular microbridge has a current-phase relation that is somewhat more skewed than the hypothetical multivalued current-phase relation shown in Fig. 3(a). Using the technique of Ref. 9, we have measured the transition distribution for these samples at a low, constant-4x sweep. The measured distribution widths were 2-3 times larger than those predicted by K-1 for a Josephson junction. Finally, when operated as if-biased SQUIDs the intrinsic noise of these rings as measured by the fractional step rise a was found to be much greater than that predicted by K-W.

4.4. Summary We have found that the intrinsic noise as measured by ~ varied con-

siderably for different SQUID rings with essentially identical values of L~, i c, T~, and ~orf, in contrast to predictions based on the K-W shunted Josephson junction model. The functional dependence of a on SQUID ring parameters generally deviated from the K-W predictions. In our point contact samples we have found a correlation between ~ and the weak-link normal state resistance R~. Most significantly, measurements of the step length/step, which yield 7c, have indicated a strong correlation of ~ with the weak-link current-phase relation. Oxidized point contacts with low R~ and measured sinusoidal current-phase relations exhibit a fractional step rise that is in reasonable agreement with K-W. The importance of the current- phase relation has been further corroborated by measurements on thin-film samples; microbridge SQUID rings with very nonsinusoidal current-phase relations were found to have wide transition widths and large ~'s.

For some weak links, particularly thin-film microbridges, R~ ~< 0.1 f~. Our measurements of the frequency dependence of a indicate that a SQUID


employing such a weak link can be expected to have intrinsic noise which is three or four times greater than given by Eq. (18) even at bias frequencies as low as I MHz and even if the current-phase relation is sinusoidal. Appa- rently the ideal weak link, particularly for high-frequency SQUIDs, is one with both an essentially sinusoidal current-phase relation and R s considerably greater than 1 f~.

5. TANK CIRCUIT NOISE AND OPTIMUM COUPLING

The overall sensitivity of a SQUID system is seldom directly limited by intrinsic noise: noise contributions from the rf amplifier and tank circuit usually dominate. However, the step rise generated by the intrinsic noise- related fluxoid transition uncertainty affects SQUID resolution in two ways. First the step rise reduces the q~c) modulation of the rf I - V characteristic by (1 - c0. Unless e is particularly large, the resultant signal reduction is relatively minor. Second, and most importantly, the step slope governs the amplitude of the tank circuit noise. The rf step slope varies from SQUID ring to SQUID ring with the product eye. Consequently, as we shall see, the current-phase relation of the SQUID ring, by affecting the amplitude of both ~ and 7c, establishes the maximum realizable sensitivity of a given SQUID system.

In this section we briefly review the K-2 tank circuit noise analysis. We show that the tank circuit noise depends on the mutual inductance M between the tank circuit inductor and the SQUID ring. The mutual inductance is the most easily adjusted parameter of a SQUID system ; we show how to choose M to maximize sensitivity in a SQUID system where all other parameters are fixed.

5.1. Tank Circuit Noise

Johnson noise currents in the tank circuit load resistance R t give rise to tank circuit noise. The noise current in the bandwidth of the tank circuit resonance appears as a fluctuating drive current which adds to the drive current Irf supplied by the rf current source. At fixed Irf the noise currents result in a fluctuation in the biasing of the SQUID on the rf 1-V characteristic. Thus there is a voltage fluctuation in the tank circuit (6V~c) which depends on Irf through the effective tank circuit impedance dV~f/dlrf. The tank circuit noise also adds to the rounding of the steps that is caused by transition uncertainty.

On a step V~f is related to l~f by

V~f = 1~f x (step slope) + const (20)


By definition of the fraction step rise ~ we have that

step slope = ~ A Vo/ls,~p o r

Wrf : ( l r f f ' l s t~p ) (~bOe)r fL , /m)~ 27 c o n s t

Following K-2, we define the effective Q on a step by

dV~f onstcp d l r f : - Qef f~OrfL t

SO

(21)

(22)

(23)

Qeff = d?o~x/Mlstep = n~ycLsL,/2M 2 (24)

Now, the noise current 6ITc is given by

(6IZc)/Hz = 4kB T/R, = 4kB T/Qco,.fL, (25)

where the unloaded Q is Q = RjeorfL t. In Eq. (25), T is the effective noise temperature of the tank circuit. For a typical SQUID system with a room- temperature amplifier T ~ 200 K.

Using Eqs. (22), (24), and (25), we can write the tank circuit noise as an 6 2 equivalent flux noise (~bTC),

( ( ~ b 2 c ) ( M / O ) r f L t ) 2 (2) 2 __ 2 2 2 _ M Qeff(b/TC ) VTC) (26) Hz (1 - a)2 Hz (1 - ~x)Z(Hz)

0r [k TL Hz 1/2 - 1 - ~ L e~-f QM~] (27)

or

5.2. Maximization of SQUID Sensitivity Through Adjustment of M

The sensitivity of an if-biased SQUID system in the absence of externally applied magnetic noise is determined by the sum of the tank circuit noise (6~bx2c), the intrinsic flux noise (&b2), and the equivalent flux noise of the rf amplifier (&b2A). If we represent the total equivalent flux noise of a given SQUID system by (b~b~) then we have

o r

(6dp~) ~ I M ]2 (6V2)/Hz rc2.Ey2kBTL s L~L~ Hz ~ ~ -r (1-o02cG~ QM 2

(28)

27 03rf(l __ 00 2 (29)

228 L . D . Jaekel and R. A. Buhrman

where (6 V])t/2/Hz 1/2 is the rms input noise voltage per Hz t/2 of the amplifier. [The extra factor of (1 - e)2 in the intrinsic noise term corrects for the approximation of zero slope made by K-W in calculating (&b2)1/2.]

Of all the SQUID system pa, rameters, M is the most easily adjusted. As we can readily determine from Eq. (29), there is a unique value of M that minimizes (6q~2). However, for most SQUID applications we must have I~t~p > I~i .... or K2Q = (M2/L~Lt)Q _> frye/2. Due to the noise rounding of the steps, the above restriction is usually further constrained to/step ~ 2Im~ o r

K2Q > rcT~ (30)

Because of the constraint imposed by Eq. (30), SQUID systems fall into one of two categories for choice of optimal M. The categorization is determined by the relative amplitude of (6q52c) compared to (6qS]) evaluated when M is adjusted so that K2Q = 7c~z.

Case I ' @~b~) >_ (64~c) when K2Q = 7~rc. In this case, sensitivity is maximized by having the minimum coupling KEQ = MaQ/LtLs = 7c~z. We then find

(6qb 2) ~ ycrcL~ (6V~)/Hz rckBTL,7co~ 2 n~2~b 2

Hz - O)rf(l __ ~)2 QcOrfL t + O.)rf( 1 _ ~)2 + O f f ( 1 __ ~)2 (31a)

In those cases where the tank circuit and intrinsic noise terms can be neglected, this equation simplifies to

Case 2. (6q5~) < (6q5~c) when K2Q = 7on. In this case, sensitivity is maximized by adjusting M so that (c54~c) = (6qS~), resulting in K2Q > Lrt. The optimal mutual inductance M o is given by

2 2 2 2 3 M o F~ kBTL~7~%SrfLT~ 1/'*

= L Q(c~vE4)/Hz J (32)

In terms of optimal coupling K o we can write Eq. (32) as 2 2 2 1/2 M~Q ~ Yzo~ZkBTc~ (33)

K 2 Q - LsL, - L (6V2A)/Hz J

We have experimenta!ly verified the existence of an optimal coupling for one of our SQUID systems by measuring (6q5~) as a function of M. In this SQUID system Q ~ 1500, corf/2n ~ 4MHz, and (6V2A)I/2/Hz 1/2 ~_ 10nV/Hz 1/2. The results are shown in Fig. 19. For this system we found

Noise in the rf S Q U I D 229

- - 2 112 <aq~ 5 -4

oo 40 (Hz)

4

zi

I I I 2

Wrf/2-/T = 4MHz

a =0.15

1 I I I J 1 J I 3 4 5 6 7 8 9 I0

I /M (orb units)

Fig. 19. SQUID system noise as a function of mutual inductance M for a point contact SQUID. The minimum noise is obtained for a value of M where Istop > 2Ir~ .... indicating that for this SQUID system

that KZQ > nTc and hence the minimum coupling was not the optimum coupling.

Using Eqs. (29) and (32), we can write down the minimum noise of a given SQUID system for case 2 as

Hz - (Drf(-~ ~ ) 2 L- Q~O-)rrZ--- t _J + (1 -- ~)20)rf (34a)

or if the intrinsic noise term can be neglected

(#(~)'/2 ..~ I 2~L~a'h ];/2[k.T(6V2A)/Hz];/4 n z = 3 L -j (34b)

In the next section we will discuss Eqs. (31) and (34) from the point of view of determining the most effective methods for improving SQUID sensitivity. However, first we should point out here that our noise analysis results, including Eqs. (31), (33), and (34), are strictly correct only for SQUID systems that employ tank circuits consisting of lumped circuit elements directly coupled to the rf amplifier. At the higher bias frequencies, COrf/2g >> 10 MHz, the SQUID system circuitry is usually somewhat different; the tank circuit may be replaced by a resonant rf cavity and the coupling G between this resonant cavity and the amplifier may be less than unity. This changes the analysis somewhat. For example, (6C~ZA) 1/z is increased by the factor G-1. However, the general conclusions of the noise analYsis remain valid: The noise contribution (6(~c) lie of the resonant circuit still varies as ~, 7c, and M-x ; the optimum coupling between the SQUID


ring and the resonant circuit is either that which causes the circuit noise to be equal to the noise contributed by the amplifier or that which causes i~,~p ~ 27~i .... whichever is greater. Equations (31) and (34), although perhaps in error by multiplicative factors such as G, still give the correct functional dependence of (6q~ 2) 1/2 upon SQUID ring and circuit parameters, provided Qm~fL t is now taken to be the resonant impedance of the rf cavity.

6. SENSITIVITY IMPROVEMENTS FOR OPTIMALLY COUPLED SQUID SYSTEMS

Equations (31) and (34) show how the SQUID system noise @~b~) 1/2 responds to changes in system parameters provided M is kept optimally adjusted for either case 1 or case 2, respectively. We will now discuss how SQUID sensitivity can be improved through changes in these parameters.

6.1. Amplifier and Tank Circuit Effects

In both cases I and 2, ('5q52) depends upon the quantity (~V2A)/QCOrfLt. Furthermore, whether a particular SQUID system falls within case 1 or case 2 also depends strongly on ('SVE)/Q(o~L,. This is important since a SQUID system using a given SQUID ring, rf drive frequency, and effective tank circuit temperature will have lower noise when case 2 applies. Thus when considering ways to improve SQUID sensitivity, the quantity (,5 V 2)/Qoo~eL, should be examined most carefully.

The most common approach to increase SQUID sensitivity has been to go to higher bias frequencies. If the intrinsic noise term can be neglected, as is usually the case, we see that the effectiveness of this approach depends upon the variation (~SV2A)/Q%eL,. In fact, since it is unlikely that state of the art high-frequency (o2rf/2rt >> 10 MHz) amplifiers will have less noise than the best available low-frequency (%f/2n ,,~ 10 MHz) amplifiers, it is actually the tank circuit resonant impedance QcorfL , that determines potential improvements in sensitivity.

If an increase in the resonant circuit impedance can be achieved by operating at a higher frequency, then Eqs. (31) and (34) show that it is

: . , 1 / 2 possible for the SQUID sensitivity to improve even faster than as ~rt , the previously suggested variation of sensitivity with frequency, s In practice such improvements may not develop. Unless considerable effort is made in circuit design, the impedance at higher operating frequencies may be even less than that which can be obtained at usual operating frequencies of

10 MHz. In that case the SQUID sensitivity will vary more slowly than (o~r/2. As an example of the importance of tank circuit impedance in determin-

ing SQUID sensitivity, we can point to the results we have obtained while


operating at 3 MHz. At this frequency tank circuit resonance impedances of greater than 105 ~ have been achieved, more than 20 times that typical of standard 20-MHz SQUID systems. With such impedances and with an amplifier having (6V2A) 1/2 ~ 10nV/Hz 1/2 (which is five times higher than the noise found in the best available low-frequency amplifiers), we have measured values of (q~v)l/2/Hz 1/2 as low as 1.5 x 10-4qgo/Hz 1/2. This sensitivity is essentially identical with the best that has been obtained at 20-30 MHz. Thus, increasing the bias frequency does not automatically improve the SQUID sensitivity.

Of course, an effect of a large tank circuit resonant impedance, or more precisely of a high Q, is that it narrows the signal bandwidth of the SQUID system. In some applications a narrow SQUID bandwidth may be undesir- able. Unless both a suitable bandwidth and a high impedance can be achieved by operating at a sufficiently high frequency, an increased bandwidth can only be purchased at the cost of an increase in the SQUID noise spectral density.

It is seen from Eq. (34) that the dependence of SQUID system performance upon amplifier noise and effective tank circuit noise temperature when case 2 applies is fairly weak, (&b2)l/2 oc (OV~)I/4T1/~. Efforts directed toward decreasing these two quantities will not yield particularly large dividends unless quite substantial changes are made. In a case 1 system the dependence upon amplifier noise is stronger: (3q5~) 1/2 oc (3VA 2) ~/2. Reduc- tion in (6V]) 1/2 until K~Q > ~7c is then much more profitable.

The recent report of Gaerttner 21 concerning the performance of a SQUID biased at 440 MHz is illustrative of how improvements in sensitivity can be made. Gaerttner used a low-temperature amplifier which reduced the effective noise temperature of the tank circuit. The noise voltage of the amplifier was not significantly greater than those used in lower frequency SQUIDs, and by careful circuit design the circuit impedance was maintained high. Thus he was able to obtain an equivalent flux noise (cSq~2)I/2/Hzl/Z "~ 7 x 10-6q~o/Hz 1/2.

6.2. Weak Link Effects

We now discuss further the dependence of the equivalent flux noise (3q5~) 1/2 for optimum M upon weak link properties.

In Section 4 we reported that for SQUID rings with Ls ~ 5 • 10 - l~ H, c~ varies from approximately 0.1 to 0,5, with 0.25 being a fairly typical value. The step length parameter 7c ranges upward from unity, with the precise value depending upon the critical applied flux qS'xc and the current-phase relation of the weak link. We have seen from both Eqs. (31) and (34) that the minimum noise (6~p~) 1/z oc ?~/2. Thus if qS'~ >~ q5 o, ~b;, then y~ >> 1 and the noise performance of the SQUID is degraded considerably. This excess

232 L . D . Jackel and R. A. Buhrman

noise can be avoided by making the critical current, ic = ((0;,c - (0'c)/L~, large enough to ensure that (0'xc >> (0o, (0; and ?c ~ 1. Unfortunately, as we have seen in Section 2, if the weak link has a very nonsinusoidal current- phase relation, then the condition (0'x~ >> (0'~ requires a large critical current and in turn results in an increase in ~. Underdamped SQUID operation is also more likely when the critical current is large. (See Section 7.)

Although larger values of 7~ are of course possible, we may take an upper limit of 3 for 7~ for the purpose of illustrating the variation in noise that one might obtain from different weak links. From Eq. (34) we see that if one SQUID ring has c~ -- 0.5 and 7c = 3 while another, otherwise identical SQUID ring has cc = 0.1 and 7~ = 1, in case 2 systems there will be a variation of approximately a factor of seven in the minimum noise. For case 1 systems the variation is somewhat less, about a factor of 3.5. Thus the properties of the SQUID weak link play a very significant role in determining SQUID sensitivity even when the direct contribution of the intrinsic noise can be neglected. Since large values of c~ are correlated with nonsinusoidal current- phase relations and since 7~ tends to increase with (0'~, if one wishes to obtain maximum sensitivity with a given SQUID system, it is necessary to use a SQUID ring which has a current-phase relation that is sinusoidal or nearly so.

6.3. Ring Inductance Effects

The fractional step rise ~ varies as Ls, and as i~/a and T2s/3 in the K-W case. Since SQUIDs are usually Operated at a critical current ic ~ (0o/L~, the dependence of ~ upon Ls should really be written as ~ oc L 2/3. In a case 2 SQUID system the dependence of the equivalent flux noise upon L s is quite strong; (6(0~) a/2 oc L 5/6 when the contribution of (6(0~ 2) is small. In the unusual case where (6(02) 1/2 is dominated by (6(02) 1/2 we find

[ 2/3 1"1/2 (6(0~) 1/2 oc _s . When case 1 applies (3(0zu) 1/2 oc _~ , a somewhat weaker dependence.

Since the variation of (6(0~) ~/2 with L~ is often so strong, one may be led to think that quite significant gains in flux sensitivity can be obtained by reducing Ls. However, reducing L~ generally reduces the ability to couple signals efficiently into the flux detector. Indeed Claassen 22 has shown that a proper figure of merit f.m. for apparently all SQUID applications is not simply (~q52) ~/2, but rather

f.m. 1/2 2 1/2 1/2 = J L , /((3(0N) /Hz ) (35)

where J is the coupling coefficient between the SQUID ring and a signal coil. In a case 2 system, f.m. oc j/Lls/3. The maximum value of J that one can obtain depends upon the enclosed area of the SQUID and its geometrical design, although not necessarily in a simple manner. For example, in a


cylindrical geometry SQUID with a small inclosed area A, J is likely to be a very strong function of A. As A increases so that the actual SQUID volume occupied by the signal coil windings becomes relatively small, J becomes a much weaker function of Ls. Thus the figure of merit will be maximized at an enclosed area for which J is a moderate function of the SQUID size (d oc L~/3). Although the best size scale for a given ring geometry must be determined more or less empirically, we find that 2-ram-diameter holes in single or double hole rings are fair!y close to the optimum size.

Of course one should avoid features in the ring design which increase Ls but which do not increase signal coupling ability. As an example, Fig. 20 shows two different double hole SQUID rings. In Fig. 20(a) the point contact is symmetrically placed between the two holes; in Fig. 20(b) it is placed as close to one hole as possible. Compared to the symmetric SQUID ring, the asymmetric SQUID ring has ~ 40 ~o less inductance and yet it allows greater signal flux coupling efficiency.

Besides eliminating "parasitic" inductances, there are several techniques by which Ls can be reduced without seriously degrading the ability to couple signals into the SQUID ring. An excellent example of such techniques is the fractional turn SQUID ring design ofZimmerman. 5 By having 12 (sometimes more) superconducting rings in parallel, the total inductance L~ of the SQUID ring is greatly reduced. When used with a coupling coil the maximum J that can be obtained with this device is essentially the same as can be obtained with a one-turn SQUID ring with the same enclosed area as each of the 12 loops in the fractional turn SQUID ring. Compared to single or two hole SQUID rings, the Zimmerman SQUID ring, when used in a case 2 system, yields both reduced flux noise and an increase in the Claassen figure of merit. In a case 1 SQUID system, (g~D2) 1/2 ~ (~(/)2~1/20C L~/2

Fig. 20. Two different designs of the two-hole point contact SQUID. The asymmetric SQUID (b) has ~ 40 ~o less inductance than the symmetric SQUID (a). (b]

iI

234 L.D. Jackei and R. A. Bulu'man

and hence there is no gain in the figure of merit from the fractional turn SQUID ring.

A different technique for reducing L S is the insertion of a hollow, normal metal cylinder into one of the holes of a two-hole point cohtact SQUID ring. ~4 This hollow cylinder acts as a low-pass normal metal filter with a cutoff somewhat below the bias frequency of the SQUID. Its primary purpose has been to shield sensitive samples from the rf flux. It also serves to reduce the effective rf inductance of the SQUID and hence e and the intrinsic noise. The extent of this reduction is dependent upon how tightly the cylinder couples to the SQUID. In our laboratory, reductions in L, of more than 50 % have been effected in this manner. The advantage of the filter is that the low-frequency inductance of the SQUID, and hence the coupling of dc or low-frequency signals to the device, is unaffected by its. presence.

The use of the normal metal filter does have the negative effect of introducing extra magnetic noise into the system. The rms amplitude of the flux noise that the cylinder contributes is

( ~ ) 1 / 2 ~ c(LIkBT)I/2 (36)

where c is the coupling factor between the cylinder and the SQUID and Ly is the inductance of the cylinder. Similarly, if Ry is the resistance around the circumference of the filter, then, since Ry/Ly is the approximate cutoff frequency of the filter, the flux noise per unit bandwidth is

(64~}) 1/2 c[ 4L}k~ T / 1/2 (37) Hz1/2 ~ I Rf }

For a typical filter of ~ 2 mm diameter and ~ 1.5 em length (6~}) 1/2 will be somewhat less than 0. I~o.

The filter-generated magnetic noise which lies within the SQUID signal bandwidth contributes directly to (6~2) 1/2. If the cutoff frequency of the filter is sufficiently high, i.e., fairly close to the bias frequency, then this extra noise component is small enough so that it does not have a significant effect upon SQUID sensitivity. As we will show in Section 7, the higher frequency filter noise acts to tilt the rf step by an a m o u n t (6q~})1/2/~ 0. Since (6r o is less than or of the order of 0.1, the filter-generated tilt can be more than offset by the decrease in e resulting from decreased L~. Thus in some instances increased sensitivity can actually be obtained through use of these filters. We have demonstrated this possibility by measuring c~ for a permanently adjusted SQUID first without a filter and then with one. As shown in Fig. 21, the result was an overall reduction in c~ by use of the filter and hence a possible reduction, albeit a fairly small one, in total system noise.


Vrf

-- a : 0 .27

SQUID without Bross Filter

(a) Irf

v,t

a : O . 2

SQUID with Brass F i l ter

Irf (b)

Fig. 21. (a) The ff I-V characteristic of a permanently adjusted point contact SQUID. (b) the rf I-V characteristic of the same SQUID with a hollow brass cylinder inserted into one of the holes. When the brass filter is inserted into the hole, the rf coupling is reduced between the SQUID ring and the tank circuit inductor. In order to compensate for the above effect, in this figure we have used a different scale for the coordinates in (a) and in (b).

An a l te rna t ive fi l tering technique for bo th reducing Ls and shielding sensitive samples f rom the rf is to shunt the signal coil , which has induc tance L s , with a res is tor R I . If L r and R I are chosen correc t ly and if the signal coil is t ight ly coup led to the S Q U I D ring, results s imi lar to those ob ta ined with the n o r m a l meta l filter can be obta ined .

236 L.D. Jackei and R. A. Buhrman

6.4. Numerical Example and Attainable Sensitivity

Finally, as a numerical example we can discuss the sensitivity that might be possible with a SQUID system biased at 20 MHz. Typically, the best sensitivity of such SQUIDs having an inductance L~ g 5 x 10 -1~ H is quoted as 1 x 10-4tPo/Hz 1/z. This apparently is obtained with SQUID systems where Lt ~ 5 x 10 -7 H, Q ~ 100, T ,~ 200 K, (6V2) 1/2 A ~ 2 nV/ Hz 1/2, and ~ ~ 0.2 or greater. The value of yc is more difficult to determine, but 7c ~ 2 might be a typical value. With these parameters Eq. (33) gives KZQ < ~ty~ and hence the minimum SQUID noise should be given by Eq. (31), which yields (6q~)l / : /Hz 1/2 ~ 8 • 10-s~0/Hz 1/2. Considering the uncertainties in the calculation and the fact that the exact optimum coupling might not have been obtained in practice, this result is in quite good agreement with the experimental value.

However, for ~t ~ 0.2 the intrinsic noise (6tp 2) ~ 3 x 10-Sq~o/Hz 1/2. It would seem that with reasonable efforts in tank circuit design the tank circuit and amplifier noise contribution could be reduced to this level or less, giving a factor of three improvement in sensitivity even without increasing the bias frequency. Of course, if SQUID weak links can be made which when operated at 20 MHz give both a fractional step rise ~ approaching the ideal limit of 0.1 and ~c ~ 1, then a sensitivity approaching 1 x 10-5~bo/Hz at 20 MHz should be attainable. Some efforts along these lines are suggested.

7. ADDITIONAL FACTORS LIMITING SQUID SENSITIVITY

We have seen how the current-phase relation, bydetermining ~ and ~c, affects the realizable flux sensitivity of a SQUID system. However, there are three other phenomena associated with the SQUID ring which may cause (6~b~) 1/2 to be significantly greater than that given by Eqs. (31) and (34). These phenomena are (1) low-frequency flux motion in the SQUID ring or its environment, (2) broad band, externally applied flux noise, and (3) multiple-fluxoid transitions in an underdamped SQUID ring.

7.1. Flux Motion

Some SQUID systems exhibit a noise signal whose spectral density is considerably greater at low frequencies ( f < 10 Hz) than at higher frequencies. The noise appears as an apparent drift in q~O and sometimes has the characteristic of i/f noise. The noise is apparently caused by flux motion in the SQUID ring itself or perhaps in the superconducting shield that almost always surrounds the SQUID ring.

We have sometimes found such low-frequency noise in niobium point contact SQUID rings which have been sealed with helium exchange gas in

238 L.D. Jackei and R. A. Buhrman

thin film at or near the weak link. Apparent!y some thin-film microbridge SQUID rings may be more likely to trap flux and to have subsequent flux motion than bulk niobium SQUID rings. However, there is certa!nly no reason to expect that random flux motion necessarily occurs in all thin-film SQUID rings.

7.2. External Magnetic Flux Noise

It is sometimes necessary to operate SQUIDs in the presence of external magnetic noise. In some cases magnetic noise may also be contributed by a system component, such as the flux transformer filters that are essential for particular SQUID applications mentioned in Section 6.3. We have found that such external noise, besides contributing directly to the total noise of the system, can affect the tank circuit noise amplitude.

We have investigated the effect of magnetic noise upon SQUID performance with computer simulations. The simulations were carried out by adding white magnetic noise of variable bandwidth from dc to some cutoff frequency to an otherwise noise-free SQUID.

We found that the noise component in the frequency range dc to cnff/2rcQ, i.e.. within the signal bandwidth of the SQUID system, is treated as a flux signal. Hence this noise contributes directly to the total flux noise of the SQUID system. This effect is seen in the simulated rf I V characteristic shown in Fig. 22. Notice also that if the rf I-V characteristic in Fig. 22 is

/

Vrf

I

Noise Bond width = O.OlWrf

it

Trf

Fig. 22. The rf I-V characteristic of a simulated SQUID which has low-frequency magnetic noise of rms amplitude 0.kbo incident upon it. The simulation shows that low-frequency noise tends to split the step.


lead plated cans. The cans are used for magnetic and environmental shielding.16 The flux motion apparently originates in the lead shield rather than in the SQUID ring itself. The flux motion seems to be caused by temperature inhomogeneities in the helium bath. This conclusion is reached since the extra noise invariably vanishes abruptly if the bath temperature is lowered below the lambda point. Below Ta we find that to within the limits of the measurements the noise spectrum is white within a bandwidth extending from f < 0.1 Hz up to the cutoff frequency of SQUID system signal response. We have obtained this result in systems biased at ~ 3 MHz with a spectral noise density as low as (6t~2)!/2/Hz1/2 ~,~ 1.5 x 10-4~bo/Hz 1/2. In addition, Gaerttner 21 reports a white noise spectrum in a toroidal point contact SQUID ring system biased at 440 MHz. This measurement of (6~b 2) 1/2/Hz1/2

7 x 10-6~o/Hz 1/2 was made over a frequency range extending from l MHz to l0 Hz, where the lower frequency was set only by the limits of his spectrum analyzer.

We have found that above Tx flux motion noise can also be eliminated or reduced to a negligible level, at least for point contact SQUID rings, if the proper procedures are followed. These procedures include (1) the use of a thick and as uniform as possible lead plating on the exchange gas can, (2) the avoidance insofar as possible of trapping flux in the shield, and (3) either the complete elimination of superconducting solder joints (and large pieces of normal metal) from the inside of the can, or the placement of these items where they do not couple well to the SQUID.

A particularly effective technique for eliminating flux motion is the enclosure of the SQUID ring in a tight superconducting shield machined from bulk niobium. All solder joints are made outside this shield. Both the SQUID ring and the shield are then sealed with helium exchange gas in an unplated copper can. An alternative shielding technique is the use of a toroidal SQUID ring where the SQUID ring body itself serves to shield the active area of the ring from flux motion in the environment.

Flux motion in the bulk niobium SQUID rings themselves apparently occurs only rarely, at least if the SQUID ring temperature is kept reasonably constant. This conclusion is based on numerous measurements of the magnetic response of niobium SQUID rings to external flux. During these measurements we have found: no changes in SQUID ring magnetization which could be attributed to flux motion in the SQUID ring.

On the other hand, in similar experiments with thin-film tin microbridge SQUID rings, changes in the ring magnetization of variable amplitude have often been observed, although only in some rings. In addition, for some microbridges, which have fairly large weak-link dimensions, ,-~2 #m, the critical current has been found to vary irregularly and nonrepeatedly with temperature. This effect is indicative of the movement of trapped flux in the


time-averaged, the full rf steps split. The splitting occurs if ~b(f~ = n~b 0, because both positive and negative charges in magnetic flux cause the rf step voltage to vary in only one direction, down on the first half of the step and up at the second half. A similar splitting occurs if ~b~ c) = (n + �89 0. The amount of splitting that occurs is approximately equal to the rms value of the applied magnetic flux noise. The splitting indicates the presence and amplitude of external noise with frequency <oorf/2rc Q.

The effect of the higher frequency noise, co > mrr/2rcQ, is more indirect. The simulation results shown in Figs. 23(a) and 23(b) indicate that the major effect is the tilting of the rf step. The amount of rf step rise e' that this tilt causes is approximately equal to the rms flux noise, (6qb~)l/2/Jpo. The simulations also indicate that the magnitude of c( is independent of the bandwidth of the magnetic noise, provided this bandwidth is much greater than the bandwidth of the tank circuit, o)rf/Q. The degradation of sensitivity caused by this high-frequency noise depends on the relative amplitudes of c( and fluxoid transition uncertainty fractional step rise ~.

If c( is much less than e, the external magnetic noise plays no role in the determination of S Q U I D sensitivity. If, on the other hand, c~' > c~, then it is the external noise and not the intrinsic S Q U I D fluctuations that sets the amplitude of the tank circuit noise and hence determines the attainable SQUID sensitivity.

7.3. Underdamping

If the parameters of a SQUID ring are such that fi > I, where

fl = (27tic/C~c)1/2 RsC~/2 (38)

then the SQUID ring is marginally damped or undamped rather than well overdamped, fl << 1. When an underdamped SQUID ring makes a fluxoid transition it is likely to pass through several potential minima before coming to rest in a state that is more than one quantum number from the initial state. The exact amount of flux change that occurs in a slightly underdamped S Q U I D ring during a transition is determined stochastically. The random nature of the fluxoid transition may result in increased SQUID system noise. If the S Q U I D ring is very much underdamped, fl >> 1, then it is usually useless as a flux detector. If fl -~ 1, single as well as multiple fluxoid transitions occur and the device may operate satisfactorily, possibly with increased noise. For point contact weak links the shunt capacitance C~ is likely to be ~< 0.1 pF. For a S Q U I D with L s ~ 5 • 10- a o H and i c ,,~ qbo/L s the damping parameter fl < 0.04Rsi~0/2, where R, is in ohms and ico = Lsic/4)o. Obviously, underdamping can be a problem in practical SQUID rings which have fairly high-resistance weak links or large critical currents.

Vrf

(a )


Vrf

•

Noise Bandwidth : OJrf

] ' r f

.__ I

Noise Bandwidth = IO~rt

Trf (b)

Fig. 23. The ff I -V characteristic of a simulated SQUID in the presence of external noise where (a) the noise bandwidth is co a and (b) the noise bandwidth is 10o~rf. Here the principal effect of the noise is to tilt the step.


We have investigated the effect of slight underdamping upon rf SQUID performance with digital computer simulations. The simulation was carried out by introducing a weighted probability for random, multiple fluxoid transitions into an otherwise noise-free simulation of an rf SQUID. Some rf I-V characteristics obtained from these studies are shown in Fig. 24. In Fig. 24(a) there is a 20 ~o probability that when the SQUID ring makes a fluxoid transition the system will go to the second lower fluxoid state rather than the first. In Fig. 24(b) the probability for such a double fluxoid transition is 0.95.

In comparing Fig. 24(a) to the overdamped simulation of Fig. 8, we see that multiple transitions add a small amount of noise to the I-V characteristic while the step slope remains unaffected. However, this noise is negligible in comparison to the intrinsic noise originating with fluxoid transition uncertainty.

When the amount of underdamping is greater, so that, as in Fig. 24(b), consecutive multiple transitions are the rule rather than the exception, the voltage fluctuations of the rf step increase markedly. We have found that the amplitude of this noise relative to AVo varies with the coupling K. In Fig. 24(b) we have KZQ = 1, where Q = 100. For much tighter coupling the steps can be completely obscured.

When the multiple transition probability is high, the amplitude and relative sign of the noise vary along the rf steps so that the time-averaged step voltage shows a tilt. In a real SQUID system this tilt is likely to add to the intrinsic noise step slope. The effect of multiple fluxoid transitions upon an rf SQUID also depends upon the dc flux seen by the SQUID. We find that for q~(~c) = (n + 1)4~ o double fluxoid transitions do not cause very significant fluctuations in the step voltage. For other values of ~b(~ ac), if the probability of multiple transitions is high, the noise on the rf steps is quite appreciable. For a multiple transition probability of ,,~0.8 or greater the voltage fluctuations are comparable to or greater than the intrinsic noise. The simulations show that while a very slightly underdamped ring can operate satisfactorily in a SQUID system, rings with large underdamping incorporated into SQUID systems result in increased system noise.

A point to notice in Fig. 24 is that multiple fluxoid transitions cause the steps to split asymmetrically. The amount of asymmetry in the split steps depends directly upon the percentage of multiple transitions. One can use this asymmetry in the I-V characteristics to determine to what extent multiple fluxoid transitions are occurring in a given operational SQUID. We used this technique to eliminate data from underdamped SQUIDs in our study of the correlation of intrinsic noise with point contact resistance, which was discussed in Section 4.

Vrf

(a)

0.2

I r f

p = 0 . 9 5

Vrf

242 L.D. Jackel and R, A, Bulltman

I r f (b)

Fig. 24. The rf I-V characteristics of a simulated underdamped SQUID. (a) The probability of multiple fluxoid transitions is 0.2. (b) The probability of multiple fluxoid transitions is 0:95.


v,f

j

P

I t f

Fig. 25. The rf I-V characteristic of an actual SQUID which shows the asymmetry in the step splitting that is indicative of an underdamped SQUID.

Slight underdamping in rf SQUIDs, as indicated by asymmetric I-V characteristics, is fairly common. (See, for example, Fig. 2 and the discussion in Appendix A of Ref. 6.) We have often observed asymmetric I-V characteristics in our experiments with point contact SQUID rings, usually when the niobium contact was clean and hence of high resistance. An example of a measured asymmetric characteristic is shown in Fig. 25. A comparison of Fig. 25 with Fig. 24 reveals that the percentage of multiple transitions in this SQUID ring is fairly high. Even though this SQUID appears to operate satisfactorily as a flux detector, such asymmetry should probably be avoided if one is interested in maximum sensitivity.

It is interesting that typical values of Cs and Rs for point contact SQUIDs are such that the SQUIDs are generally only slightly overdamped. For only somewhat higher than usual resistance, R > 10 fl say, or for only moderately high critical currents, multiple fluxoid transitions are fairly likely to occur. In general if high-resistance weak links are to be used in SQUIDs of a given design, some care should be taken to make C~ as small as possible in order to reduce the probability of the system being underdamped.

8. CONCLUSION

A straightforward technique for the determination of the intrinsic flux noise amplitude of operational rf-biased SQUIDs has been presented.


Employing this technique, we have observed that the intrinsic noise of point contact SQUIDs is often in considerable excess of the predicted level. The most probable cause of this excess noise is a departure from a sinusoidal current-phase relation in the point contact weak link.

Utilizing the results of the tank circuit noise calculations by Kurkij~irvi, we have determined the total equivalent flux noise of a SQUID system having a given amplitude of intrinsic flux noise. We have shown that for any particular system there is a unique coupling between tank circuit and SQUID ring that maximizes flux sensitivity. Expressions for the equivalent flux noise of SQUID systems with such optimum coupling have been derived. These expressions have been analyzed to determined the most effective methods for increasing SQUID sensitivity.

The maximum attainable SQUID sensitivity is strongly dependent upon the properties of the superconducting ring and weak link even when the direct contribution of the intrinsic fluctuations to the total noise can be neglected. Although the total equivalent flux noise (6~)2) 1/2 varies quite rapidly with ring inductance, the single strongest influence upon SQUID sensitivity is exerted by the current-phase relation of the weak link. By governing the length and tilt of the rf steps, the current-phase relation determines the amplitude of the tank circuit noise, the minimum amplifier equivalent flux noise, and hence the maximum attainable sensitivity. Differences in (&k 2) 1/2 of greater than a factor of seven between otherwise identical SQUIDs can easily occur if one SQUID has a low critical current and a skewed current- phase relation with ~bc >> ~b0/4 while the other has an essentially sinusoidal current-phase relation and a moderate critical current.

Underdamping and flux motion in the SQUID ring do not significantly affect the optimum coupling condition; rather, these phenomena primarily cause additional terms to be added directly to (t~t~2) 1/2. The presence of underdamping can be readily identified from the rf I-V characteristic and can be avoided by proper choice of weak-link parameters.

Flux motion can cause very significant increases in the low-frequency noise of the SQUID system. The elimination of suchexcess noise may require some combination of improved shielding, temperature regulation, and proper choice of material for the superconducting ring. We have been able in our point contact SQUID systems to reduce flux motion to a negligible level. The noise level of a well-designed SQUID system should approach that given by Eq. (31) or Eq. (34).

The noise analysis that has been presented in this paper is not applicable for SQUIDs with critical current so low that there is only one accessible fluxoid quantum state. (For a sinusoidal current-phase relation this occurs for Lic < ~bo/2n.) Such SQUIDs operate as flux detectors in the so-called "inductive mode" rather than on the basis of fluxoid transitions. While the

Noise in the ff SQUID 245

theory of inductive SQUIDs has been investigated, 23 a noise analysis has not yet been carried out. We can speculate that the intrinsic device noise for these inductive SQUIDs is likely to be lower, approaching the classical limit 5 (0q~2)U2 /Hz 1/2 = (4L2skBT/Rs) 1/2 as i c -~ 0. On the other hand, an examination of the theoretical inductive SQUID I-V characteristics recently published by Hansma 23 reveals that the modulation of the rf voltage for a given dc flux change is considerably less than eorfLJpo/M (the value for a fluxoid transition SQUID with ~ = 0), at least for the circuit parameters chosen. Hence the equivalent flux noise of the rf amplifier will be greater for the inductive SQUID except when compared to fluxoid transition SQUIDs with fractional step rise ct of order 0:9. Alternatively, if the sensitivity is limited only by amplifier noise, we can say that the inductive SQUID signal fs less while the noise is the same as for a fluxoid transition SQUID. The effective Q on the I-V steps of the inductive SQUID, although somewhat dependent upon dc flux, is also greater thaaa is typical for a fluxoid transition SQUID. thus the tank circuit noise amplitude will be greater. Unless the regime can be reached where the intrinsic noise is the direct limiting factor, the overall performance of the inductive SQUID is likely to be inferior to the SQUIDs discussed here. This conclusion is of course very preliminary; detailed noise analysis of the inductive SQUID system will be necessary to ~ompletely clarify this point.

ACKNOWLEDGMENTS

It is a pleasure to thank J. H. Claassen, J. E. Lukens, and especially T. D. Clark, J. Kurkij/irvi, J. M. Warlaumont, and W. W. Webb for many helpful discussions and contributions to the formulation and results presented here. Prof. Webb, J. L. Jackel, and Dr. Clark also made helpful suggestions on the manuscript. We would also like to acknowledge a very useful conversa- tion with Prof. J. Clarke concerning low-frequency SQUID noise. This research is supported by the Office of Naval Research and by the National Science Foundation through the Materials Science Center at Cornell.

REFERENCES

1. J. Kurkij~irvi, Phys. Rev. B6, 832 (1972). 2. J. Kurkij~irvi and W. W. Webb, in Proc. Appl. Superconductivity Conf., Annapolis, IEEE

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Noise in the rf SQUID - Cornell University · of practical SQUID systems, indirectly it establishes the tank circuit noise amplitude and hence strongly affects the overall SQUID performance