Mutual phase-locking in Josephson junction arrays

PHYSICS REPORTS (Review Section of Physics Letters) 109, No. 6 (1984) 309—426. North-Holland, Amsterdam

MUTUAL PHAS&LOCKING IN JOSEPHSON JUNCTION ARRAYS A.K. JAIN, K.K. LIKHAREV5, i.E. LUKENS and J.E. SAUVAGEAU Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, U.S.A. Received January 1984

Contents: 1. Introduction 2. Coupling mechanisms 2.1. Non-interacting junctions 2.2. Order-parameter coupling 2.3. Quasiparticle coupling 2.4. High-frequency electromagnetic coupling 2.5. Low-frequency electromagnetic coupling 3. Mathematical tools 3.1. General approach 3.2. SVA method for a Josephson junction 3.3. The simplest applications of the reduced equations 3.4. Junction equivalent circuits 4. Two-junction cell-theory 4.1. High-frequency coupling 4.2. Influence of low-frequency conductivity 4.3. Superconducting low-frequency coupling 5. Experimental results on two coupled junctions 5.1. Early works 5.2. Instrumentation 5.3. Single junction properties 5.4. Experimental results on 2 junction cells 5.5. Superconducting low-frequency coupling 6. One-dimensional multi-junction arrays — General analysis 6.1. General equations

a Permanent

311 313 313 314 315 316 317 318 318 320 324 327 329 329 336 341 347 347 348 352 356 370 372 372

6.2. High-frequency coupling 6.3. Phase stability and the junction interaction range 6.4. Influence of the finite low-frequency conductivity 6.5. Superconducting low-frequency coupling 7. Design and performance of practical multi-junction onedimensional arrays 7.1. Basic design considerations 7.2. Low-frequency bias circuits 7.3. High-frequency coupling: voltage locking 7.4. Microwave generation by linear arrays 8. Applications of linear arrays 8.1. Introduction 8.2. Applications of arrays as generators 8.3. Travelling wave arrays 8.4. Application to microwave detection 8.5. Application to study of the non-equilibrium effects 9. Summary and conclusion Appendix A: Locking of non-identical junctions and harmonic locking Appendix B: Phase locking with resonant coupling Appendix C: Fabrication techniques Appendix D: DC interaction between the junctions References

375 377 381 384 386 386 387 390 394 399 399 400 406 410 413 414 416 418 419 423 424

address, Department of Physics, Moscow State University, Moscow 117234, U.S.S.R.

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MUTUAL PHASE-LOCKING IN JOSEPHSON JUNCTION ARRAYS

A.K. JAIN, K.K. LIKHAREV, J.E. LUKENS and J.E. SAUVAGEAU Department of Physics, State University of New York at Stony Brook, Stony Brook, NY 11794, US.A.

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A.K. Jam et a!., Mutual phase-locking in Josephson junction arrays

311

Abstract: We discuss mutual phase locking of Josephson oscillations in an array of Josephson junctions. The discussion is focussed on phase locking due to electromagnetic coupling. The theoretical analysis is based on a secular-term free perturbative solution of the resistively shunted junction model. Detailed experimental results on phase locking of two coupled micro-bridges are presented. The perturbation analysis provides a complete qualitative description of the experimental results, including the effects of fluctuations on phase locking and the oscillation linewidth. Measurements 2) and the decrease in the radiation have also been made on linear arrays of up to 100 junctions. The observed increase in the radiated power (as N linewidth (as N1) are all in agre~ementwith the theory. Optimized arrays suitable for use as either microwave and millimeter wave radiation sources or as mixers are discussed. The properties of the arrays are found to solve various problems associated with the use of single junctions as mixers, such as excess noise temperature, limited dynamic range and low sensitivity for narrow band signals.

1. Introduction A reasonably complete understanding of the dynamics of a single Josephson junction has been developed during the last decade (see monographs [1—3]and reviews [4-61).However, it is only quite recently that a comparable understanding of the dynamics of systems consisting of several (or many) similar (or nearly similar) Josephson junctions has begun to emerge. Since the initial studies of these multi-junction systems (“arrays”) in the late ‘60’s it has been clear that the most interesting and technologically useful phenomena would involve the mutual phase locking (synchronization) of the Josephson junctions. To understand the meaning of this mutual phase locking we recall that when a junction (the kth junction) is biased in its “resistive” state, i.e. with a current ‘k greater than its critical current, there is a nonvanishing dc voltage Vk across the junctions. Associated with this dc voltage is the Josephson oscillation of the junction supercurrent which has an average frequency given by =

d~/dt= 21TVk/~o

(1.1)

where Cko = hI2e 2 x 1015Wb = (0.5 GHzIp~V)1is the flux quantum, and 4k the phase difference across the junction (we will use the symbol (-) to denote long time averages). When referring to the mutual locking of the Josephson phases 4k we mean not only that the average frequencies iik of all the junctions are equal but that the fluctuating frequencies Wk of the oscillations are also equal over a short time. That is, (O1—0J2--”—WN.

(1.2)

The Wk are averages over some suitably short time as will be discussed more precisely in section 3; thus eq. (1.2) does not imply that all d4k/dt are equal at each instant. This phase locking, which we refer to as the coherent state of the array, can be maintained even in the presence of (inevitable) perturbations, such as random variation of the junction parameters or fluctuations. In spite of the evident importance of this phase-locking phenomena, and the relatively long history of its experimental [7—36]and theoretical [37—61] study, it is only quite recently that a clear picture of the dynamics of phase locking in arrays has emerged. An important step in developing this picture has been to isolate and identify among the various mechanisms through which the junctions can interact to achieve the coherent state. It has been, in large part, the ability to isolate a particular, well characterized mechanisn~(both theoretically and experimentally) which has led to the detailed understanding of mutual phase locking. A few years ago it was realized that the most powerful and controllable mechanism for interaction

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A.K. fain et a!.. Mutual phase-locking in Josephson junction arrays

between junctions is high-frequency electromagnetic coupling. Even the initial experiments [26,27] which were designed for this type of coupling produced much stronger phase locking than had been obtained previously. Moreover, these experiments, carried out for the simplest system (two-junction cells), provided data which could be well explained using a very simple theoretical model [48,51, 53]. The good agreement between theory and experiment in two junction cells has encouraged us to extend the research to more complex multi-junction arrays [30,31, 33, 51, 61] including circuits where additional low-frequency electromagnetic coupling is present. In addition a clear picture of electromagnetic coupling has helped provide a reasonable model for the features of other possible coupling mechanisms, e.g. interaction via the order parameter and interaction via quasiparticle injection (see review [41]).

As a result of the understanding which has been achieved, it has been possible to design and fabricate complex arrays consisting of as many as 100 junctions and to demonstrate their coherent operation [33,36] also with results in good agreement with the theory [51,61]. This has paved the way to the synthesis of even more complex “optimum” arrays which should be valuable for both fundamental research in low temperature physics and for technical applications particularly for radiation generation and detection in the centimeter and millimeter wavebands [52]. During the past several years, extensive research on mutual phase locking has been carried out, in particular, at the State University of New York at Stony Brook and at Moscow State University. Short reports on this work have appeared elsewhere [26, 27, 30, 31, 33, 38, 48, 51—54] but most of the recent results still have not been published. The purpose of this paper is to give a comprehensive picture of the understanding which has been achieved in this rapidly developing field, including the most essential of our recent experimental and theoretical results. We will start our description of mutual phase locking with the discussion (in section 2) of the various mechanisms of interaction among Josephson junctions. Electromagnetic coupling will be shown to be the most powerful and controllable of these mechanisms, thus the remainder of the paper will deal mainly with the phase locking due to this type of coupling. In section 3 we describe the mathematical tools needed for the description of the locking phenomena, and show that a secular-term-free perturbation theory (the “SVA method”) can be used for this purpose. Some simple examples of the application of this method, and its comparison with other possible mathematical techniques are also given in this section. The SVA method is used in section 4 for a theoretical analysis of mutual phase locking in the simplest system the two-junction cell. We develop here the basic features of two kinds of electromagnetic coupling those due to high-frequency and low-frequency currents as well as the results of possible superposition of the two mechanisms. Section 5 presents the results of detailed experimental studies of the two junction cell with both high- and low-frequency coupling along with a description of the experimental techniques and the properties of single junctions of the type used for the study of the phase locking. These data are compared with the theoretical results of section 4 obtained using our analysis based on the resistively shunted junction model. The good agreement between theory and experiment gives us confidence to analyze the dynamics of more complex multi-junction arrays using the same methods (section 6). Here we concentrate attention on the search for conditions under which the phase locking is the strongest and consequently the coherent state of arrays is the most stable with respect to noise and junction parameter variation. The first experiments with multi-junction arrays, described in section 7, have given encouraging results, again in good agreement with the theory. —

—

—

—

—

A.K. fain et al., Mutual phase-locking in Josephson junction arrays

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Possible applications of coherent arrays for research and technology are discussed in section 8. We have given particular emphasis to the specific types of arrays which this work has shown to be most promising. (We also discuss the performance limits of these devices imposed by constraints such as junction quality and lithographic techniques.) In the final section of the paper we summarize our results and discuss the prospects for the future development of coherent arrays. 2. Coupling mechanisms

2.1. Non-interacting junctions Before discussing the possible mechanisms for Josephson junction interaction, it is necessary to discuss why the junctions are usually independent (non-interacting) even if they are located quite close together. There is a common intuitive reaction that two junctions connected by a superconductor as shown in fIg. 1 should be coherent since they are both coupled to the coherent phase of the connecting superconductor. To understand why such junctions in general do not interact consider a representative situation, where the two Josephson junctions 1 and 2 connect three superconducting electrodes A, B and C with the spacing between the junctions larger than several micrometers as shown in fig. 1. According to the Josephson equations, the dynamics of the two junctions are determined by the time evolution of the phase difference & and *~2 across the two junctions, where cA1XBXA

(2.1)

4~2XCXB

(2.2)

and XA,

XB,

Xc are the phases of the order parameter of the three electrodes.

i~I~t~~III (a)

(b)

I

: ~/c:s~ 1(t)

2::

Fig. 1. Series connection of two Josephson junctions 1 and 2 (a) and the equivalent circuit of the system (b).

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A.K. fain et a!., Mutual phase-locking in Josephson junction arrays

The time evolution of the phase difference 4 across a junction is determined by the current through that junction. (A specific example of this can be seen in the discussion of junctions described by the resistively shunted junction model discussed in section 3.) The current could consist for example of the dc bias current and high frequency currents perhaps due to the junctions themselves. If we now connect a current source to the junctions which has an impedance which is large compared to the junction impedance (fig. 1) the junction currents are just equal to the current 1(t) fixed by the external source. Thus, the dynamics of each of the phases 4)i and 4)2 are determined by the current 1(t) and also by parameters of the particular junction but not by the time evolution of the neighboring junction. Of course for identical junctions without fluctuations the solutions for ~ and 4)2 could be identical. The point is, however, that the system has no restoring force coupling 4~ and 4)2 against the inevitable perturbations. This independence has been clearly observed in a number of experiments with arrays of series-connected Josephson junctions [14—16,24]. In these experiments, proximity-effect-type or Dayem-type microbridges have been located as close as 10 to 30 micrometers, with no evidence of a perturbation in the dc voltage. We have constrained the junctions to have a reasonably large separation which preclude the possibility that one junction might effect the parameter of the other (e.g. heating in junction 1 might reduce the critical current of junction 2). Note that if such an array is irradiated by an external ac (microwave) source, the Josephson frequencies ~0k can be simultaneously synchronized by the external source and, thus, be made equal to each other. This external phase locking is not, however, the desired mutual phase locking of the Josephson junctions; the former locking vanishes in the absence of external radiation and would not be useful for most applications. Practical current sources, of course, have some finite impedance Z(w), and thus the current 1(t) can depend on the functions 4) 1,2(t). However, in the usual experimental situations, the Josephson frequencies are quite high (typically from 10 to iO~GHz). For these frequencies, the impedance of standard current sources is also high, typically ~102 fl (which is usually much larger than the normal resistance RN of a Josephson junction). Under these conditions, the junction’s influence upon the current 1(t) is negligibly small, and the junctions can be treated as independent if they are connected in series (fig. 1). Since the series connection is the most interesting one from several points of view, we will now discuss the possible mechanisms for interaction of junctions connected in this way. 2.2. Order-parameter coupling In principle, the Josephson effect does not necessarily result in perturbations of the equilibrium state of the superconducting electrodes; that is, both the modulus i~ and the phase x of the order parameter may remain constant in the electrodes. For example, the well-known theory of the Josephson effect in tunnel junctions [62—65], which is in 2), very good agreement for junctions of moderate assumes that zl andwith x areexperiment constant within each electrode. critical current density (J~ ~ 10~ A/cm In Josephson junctions with a large dc conductivity (“weak links”, such as point contacts, thin-film microbridges, etc. [6,67]) however, the modulus of the order parameter can change noticeably in the electrodes. A qualitative picture of this time and space dependence of zi is shown in fig. 2. The weak link somewhat suppresses the order parameter in the electrodes, the amplitude of the suppression being dependent on the phase difference 4)(t). If there is a voltage (ac Josephson effect) across the junction, the phase difference changes in time according to the Josephson equation, (2.3)

A.K. fain eta!., Mutual phase.locking in Josephson junction arrays

315

L (a) I>Ic_

______

~

2XQ

(a)

I ~(x,t)~

tt2

~.

(b)

i

(b)

Fig. 2. Order-parameter coupling (schematically): weak link of length L (a) and variation of ~1with time in the weak link and electrodes (b).

S~/

Fig. 3. Quasiparticle coupling (schematically): current I through the weak link (a) redistributes periodically between superconducting (Ii) and quasiparticle (4) components, which results in the periodic injection of quasiparticles into the electrodes (b).

and ~ oscillates with the Josephson frequency. These oscillations penetrate into the electrodes to a depth of a coherence length ~, typically of the order of 0.1 p~m.If another Josephson junction is located within this distance, an interaction between the junctions results which could lead to mutual phase locking. Calculations [37, 39, 40, 42, 45] confirm the above qualitative picture. Some evidence for this mechanism has been found in experiments with closely spaced microbridges [18,21] although its strength is usually much smaller than that of the quasiparticle coupling mechanism which we discuss below.

2.3. Quasiparticle coupling Josephson oscillations in a weak link result not only in an order-parameter variation at the Josephson frequency, but also in a quasiparticle current injection to the electrodes, oscillating with the same frequency. Since the ac Josephson effect is essentially the oscillation of the junction supercurrent at the Josephson frequency a fixed bias current (constant in time) implies that part of the junction current must be carried by “normal” charge carriers (quasiparticles) counter-oscillating with the same frequency (fig. 3). These quasiparticles propagate (diffuse) into the electrodes, decaying due to branch relaxation in a time ‘r 0, which is of the order of iO~° s for most practical superconductors. The corresponding decay length A0 depends on the frequency [66—68]and is typically around several micrometers. If two Josephson junctions are located within a distance of the order of AQ, they will interact, one of the possible consequences of the situation being mutual phase locking. A very elementary model for this interaction is as follows [18]: Quasiparticles generated by one of the junctions penetrate through the other junction, inducing a quasiparticle current ‘q’ However, due to the conservation of the net current, a supercurrent I~of the same amplitude must flow in the opposite direction. Since the

316

A.K. fain eta!., Mutual phase-locking in Josephsonjunction arrays

supercurrent is directly related to the Josephson phase difference across the junction, the quasiparticles generated by one junction induce a variation of 4) across the second junction, acting as a coupling force, and vice versa. Phase locking due to quasiparticle injection has been clearly observed in experiments with Dayemtype microbridges separated by 2 to 10 micrometers [18—24,35]. The locking range dependence on the Josephson oscillation frequency Cuk and junction spacing d is in qualitative agreement with simple

theoretical arguments [21,24,32]. A quantitative theory of the quasiparticle interaction has, so far, only been developed for the very special case of proximity-effect microbridges with small critical current [43], and not verified experimentally as yet. 2.4. High-frequency electromagnetic coupling Both interaction mechanisms described above vanish if the junction spacing is larger than several

micrometers. Moreover, neither of these mechanisms is inherent in the Josephson effect itself, but is due to some secondary effects. In Josephson junctions with the best high-frequency properties (tunnel junctions, variable-thickness microbridges, etc. [6]) these secondary effects are small and thus the mechanisms described produce only weak coupling. For this reason another coupling mechanism, electromagnetic coupling, has recently received much attention. The basis of this coupling mechanism is extremely direct: If a changing phase difference across one Josephson junction can change the current through another junction, and vice versa, processes in these junctions will be interdependent as mentioned above (section 2.1). For further discussion, it is very useful to distinguish between two kinds of electromagnetic

interactions those due to high-frequency currents (near the Josephson frequency) and those due to low-frequency currents. Beginning with the high-frequency interaction, consider the circuit shown in fig. —

4. This circuit is very similar to that of fig. 1, but has a linear shunt with finite impedance Ze((U) connecting the junctions. The shunt impedance should be of the order of the junction impedance Z at the frequency of the Josephson oscillations of the junction, i.e.,

lZe(W~

(2.4)

RN.

In this case, oscillations of the voltage across the junctions will produce an appreciable ac current Ie

containing components with frequencies w1 and w2. This current, flowing through both junctions, tends to lock the oscillation phases, thus acting as a coupling force. In the case of the very high frequencies, the external coupling circuit can have dimensions

111 Ze(w)I I

11J

i;-:’~ V1(t)

(L\

V2(t)

Fig 4 High frequency electromagnetic coupling Josephson voltage oscillations across each of the Junctions induce a current I of the same frequency flowing through the coupling circuit Ze and both junctions, This current can then lock the phases of the two junctions.

AK. fain eta!., Mutual phase-locking in Josephsonjunction arrays

317

comparable with the Josephson radiation wavelength, and thus should be treated as distributed rather than lumped. Here, ac coupling can be qualitatively discussed in the following terms: Josephson oscillations of a junction are partly radiated to the surrounding space and part of this radiation induces oscillations in another junction. For a quantitative description of this process, the equivalent circuit shown in fig. 4 can still be used if an impedance Ze appropriate to the radiative coupling is used. It is evident from fig. 4, that ac coupling does not depend on the junction spacing provided that impedance Ze is fixed. In practice for lumped circuits, Ze tends to increase with increasing spacing due to increasing inductive component ZL = jwL, where L is the inductance of the coupling loop. The resulting decrease of the coupling amplitude is, however, much slower than for the nonequilibrium mechanisms discussed above. For example, phase locking has been observed for the junctions separated

by —‘1 mm [33], and estimates show that such locking is possible at spacings as large as several centimeters. Another important advantage of electromagnetic coupling is its controllability: one can change its value by changing the environment but not Josephson junctions themselves. Finally, it is also important

that the electromagnetic coupling amplitude depends directly on the Josephson effect in the junction, and not the parameters of some secondary effects. 2.5. Low-frequency electromagnetic coupling

Low-frequency currents can also contribute to the junction coupling. The clearest example is shown in fig. 5, where two Josephson junctions are connected by a closed superconducting path. Since no dc voltage drop can exist across the superconductors, the dc voltages across the junctions are equal: V1 = V2, and consequently the average Josephson oscillation frequencies are also equal. This occurs since a loop current ‘e is automatically established to compensate any difference of external bias

currents or junction parameters. This circuit is equivalent to that of the well-known “two-junction superconducting interferometer” or “dc SQUID” widely used for magnetometry, digital circuits and other applications of the Josephson effect [1—6]. Even for this ubiquitous circuit the nature of the phase locking is not universally understood, a frequent misconception being that the junctions of a dc SQUID are always phase locked since they are coupled by superconducting loops and thus have the same average frequency. However, as will be discussed in detail in section 4, the inevitable fluctuations can destroy the phase coherence in this circuit in the absence of additional high-frequency interactions.

A

fl::

i

lB L

2

I ‘BC

C Fig. 5. Low-frequency electromagnetic coupling: the current 4 through the superconducting loop automatically establishes equal average voltages V1 and V2 across the junctions.

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AK. fain et al., Mutual phase-locking in Josephson junction arrays

Moreover, even for coherent oscillations, the superconducting ioop can provide an undesirable shift of the junction phases related to the external magnetic field through the loop. In multi-junction arrays, these phase shifts are not readily controllable, and can tend to randomize the phases of the oscillation

thus degrading many of the useful properties of coherent arrays. Thus, low-frequency coupling is just one of the possible mechanisms of the junction interaction, and can even be harmful in efforts to produce in-phase coherent oscillations of the junctions in arrays.

3. Mathematical tools 3.1. General approach

The theoretical description of Josephson junction arrays with electromagnetic interactions requires at least two equations for each junction: 1) A “material” equation which describes the dynamics of the phase of the junction 4) due to a

current 1(t). 2) An “electrodynamic” equation which gives us the currents Ie(t) flowing through the external circuit (and hence through the junction) due to the voltages V(t)=~4)

(3.1)

across the junctions.

The material equation is, generally speaking, different for each type of Josephson junction, and should in principle be deduced from a microscopic analysis. In practice, most of the results concerning

Josephson junction dynamics have been obtained within the framework of the Resistively Shunted Junction (RSJ) model [69,70]. According to the RSJ model, the equivalent circuit of the Josephson junction consists of the parallel connection of four circuit elements (fig. 6), representing the supercurrent I~= I~ sin 4), the quasiparticle current ‘q = V/RN, the displacement current C d VIdt and the fluctuation current F(t). I~,is the critical current of the junction, RN is its normal resistance, and C its intrinsic capacitance. The random function F(t) describes the intrinsic noise current in the junction, and (within the RSJ model) is taken to be the Johnson—Nyquist thermal noise with a constant spectral density given by SF(w)=2I~T.

(3.2)

Within this model the phase dynamics is given by

w~24)+w~14)+sinçb= i(t)+f(t)

(3.3)

where i(t) and f(t) are the junction terminal and intrinsic fluctuation currents respectively in units of I,~, i.e. I(t)=i(t)I~,

F(t)=f(t)I~.

(3.4)

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

319

___p__ I~sin~~

C ~

F(t)~

~

Fig. 6. Equivalent circuit of a Josephson junction within a framework of the Resistively Shunted Junction (RSJ) model.

Also w~,and cu~are the “plasma frequency” and the “characteristic frequency”, -~

°~L~C’

L

35

-

~

COC~L,

(.)

The RSJ model was first introduced as a phenomenological model, however, its validity for superconducting weak links for T T~was soon proved [71];in fact, near T~the RSJ model is valid for tunnel junctions as well. During the last decade, noticeable deviations of real junction behavior from the model have been observed (see e.g. the review [6]).More exact equations for weak links are, however, presently available only in a very complex form [72,73], and are impractical for use in a dynamical analysis. For this reason, the RSJ model is almost universally accepted as a “first approximation”. In contrast with the fundamentally nonlinear material equation (3.3), the electrodynamic equations of —

the external circuit are linear and can be written in the general form 1(t) = ~‘~[V1(t),V2(t), . . . , VN(t)]

(3.6)

,

where N is the total number of the junctions in the array, and is a linear operator. For the simplest case of the two-junction cell (fig. 4), the operator has only one argument: .~

Ie(t)

.~e[Vi(t)+

V2(t)] ,

(3.7)

and is simply related to the complex impedance Ze of external circuit: —

(3.8)

Ye(tu)

For any particular array with known parameters, the self-consistent solution of eqs. (3.2)—(3.6) can be found numerically without great difficulty (or using analog simulation), especially if the fluctuations F(t) are believed to be negligibly small. Such calculations have been carried out by several authors [17, 41, 44, 49, 56, 59]. Their results are usually presented as the dc “I—V curves” Vk = Vk(I1,. . . IN). Within some range of the dc bias currents, dc voltages Vk across two (or several) junctions are equal, thus showing the coherent state described by eq. (1.2). This numerical approach, while straightforward, has however a serious disadvantage. Namely, the ,

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A.K. fain et a!., Mutual phase-locking in Josephson junction arrays

number of parameters for a multi-junction array (and even for a two-junction cell) is quite large; it is therefore very difficult ~o obtain a real understanding of the dependence of the phase-locking characteristics on the array parameters using numerical results obtained for several particular sets of parameter values. In situations like this, analytical results are much more useful, even if their validity is limited by some, but not very severe constraints. For the problem under consideration, such results can be obtained using a perturbation theory in one of its modifications, known as the method of Slowly Varying Amplitudes (SVA) (other names for the same method are sometimes encountered, such as the “Van-der-Pol Method”, or the “Rotating-Wave Approximation”, etc.). 3.2. SVA method for a Josephson junction A perturbation analysis has already been developed [79]for a Josephson junction, approximated by the RSJ model. In this reference the perturbation due to a small high-frequency current I, applied to the junctions, was analyzed and equations developed for the resulting change in the junction characteristics, in particular the low- and high-frequency components of the voltage across the junction. The real utility of this perturbation analysis is that it need only be done once for the isolated junction. The more complicated perturbations which we will be interested in (i.e. the loading effects of external circuits, interactions with other junctions, noise, etc.) can then all be dealt with by a self-consistent determination of the current I which they cause to flow through the junction in question. Thus the introduction of a new set of “electrodynamic” equations representing different perturbations does not require the problem to be completely resolved each time. The single junction analysis will remain valid as long as I is small in a sense which we will make more precise later. We will thus begin our discussion of perturbation analysis by reviewing the results obtained in ref. [79] for a single junction subject to a small current I. The main goals here will be to develop the notation and results needed for the rest of this paper. The actual techniques used will only be outlined, since as noted above, it will never be necessary to repeat these calculations for the systems of junctions we will be discussing in later sections of this paper. The interested reader is referred to ref. [79]for the details of the method. We define an “unperturbed” (or ~‘autonomous”)junction as one fed by only a dc bias current I. The —

variables referring to such unperturbed junctions are denoted by the superscript “u”. Note that the time evolution of the phase 4) in such an unperturbed junction can be written as: 4)

=

4)U(~U 0),

0

(3.9)

= cü”t

and is, in the general case, highly nonlinear (fig. 7), so that the voltage acrossthe junction contains not only the dc component V” and the Josephson frequency component, but higher harmonics (with amplitude s~”~) of this frequency as well: Vu(I,

t)=

“(I)+ ~ r~”~cos(n0), (3.10)

nl

cu”(I) = 2irV”(I)/~o. For the case of junctions with negligibly small intrinsic capacitance

(w~~tu~), which

will concern us in

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

321

this paper, the standard results for the unperturbed RSJ model are [3]: 4)

=

v \ tafl~)] fO\1 2 arctan If j-~~:—;)

—

IT

(3.11) E

c [(1 +

—

2v v2)”2 + v]~‘

Cs)

Cu~V,

where we employ the notation,

v

=

(i2

—

1)1~’2,

i

=

I/It,,

V~=

(3.12)

ICRN.

The main idea of the method of Slowly Varying Amplitudes is to express the solution 4)of the

junction equation (3.3) in the presence of a weak perturbation due to a high-frequency current I in the form 4) = 4)u(~0)+ 4)”~(w,O)+...

(3.13)

where

tu is some instantaneous frequency of Josephson oscillation, which in the general case can slowly vary in time under the influence of the perturbation. As a result, the “linearized” phase 0 is quasilinear

rather than a strictly linear function of time as in eq. (3.9): 0JCu(t)dt,

i.e.w0.

(3.14)

The change in cu is, as we will see below, to a first approximation proportional to the perturbation .1 and is thus assumed to be small: (3.15)

1~W~W.

4~~

I

•

I

u~0.1

0.3 1.0

________

I

Fig. 7. Dependence of the phase çf of an unperturbed Josephson junction on the “linearized phase” 0 oscillation frequency w~ = vw~ (RSJ model, small capacitance).

•

=

wt at various values of the Josephson

322

AK. Jam eta!., Mutual phase-locking in Josephson junction arrays

Equation (3. 13) is then substituted into the junction equation (3.3), and terms of the same order (with respect to I) are set equal. After some transformations, this gives us expressions (so-called “reduced” or “shortened” equations) for tu and for other variables of interest (eqs. (3.17—3.19) below). To our knowledge, such an equation for w was first obtained independently by Volkov [74] and Stephen [75]. In both of these papers, only the “high-frequency” limit was considered, i.e. the unperturbed phase evolution in time was assumed to be a linear function of 0 4)u~0+const

V~’=V”.

(3.16)

This is valid either for high-oscillation frequencies (tu ~ Cue), or high-junction capacitances (w a~ Cur). In his analysis, Stephen also took noise into account, however, in both of these papers [74,75] only the low-frequency properties of the junction were discussed. Likharev and Semenov [76] (see also the review [4]) extended this analysis to obtain the correction to the high-frequency part V of the junction voltage in the same limit, eq. (3.16). Later the same equations (but without fluctuations) were discussed by Forder [77],who also presented the corresponding equivalent-circuit description of the perturbed junction. The conditions which are most important for our analysis of coupled junctions, namely low capacitance and frequencies of the order of or less than the characteristic frequency (tu ~ C5)~)require the use of eq. (3.11) for the phase 4)” rather than the simplified expression in eq. (3.16) to obtain the correct results. Such an approach was first used by Aslamazov and Larkin [71] to find the height of the Josephson current steps. Thompson [78]generalized this approach to the case when the external signal

frequency does not necessarily coincide with the Josephson frequency. In both of these papers [71,78], however, the fluctuations were not considered, and the perturbation of the high-frequency voltage was not analyzed. Eventually, Likharev and Kuzmin [79] (see also monograph [3] chapters 4, 6) obtained the

reduced equations for both low-frequency and high-frequency perturbations of the junction voltage; we will use the results of this work in the following analysis. To formulate these results, we should note that there are two very distinct (as I 0) frequency ranges of interest: frequencies much less than Cu, and frequencies of the order of (or higher than) the Josephson frequency. Here and later we will use the symbol circumflex (~)to denote the low-frequency -

—~

(slowly-varying) part of the variables, including the average (dc) components. The remainder of the variable, which contains high-frequency (rapidly-varying) components, will be denoted by the symbol tilde (.)~Thus, for any variable we can write X(t)=.~t(t)+~(t),

X=)~’(t).

Likharev and Kuzmin [79]obtained the following “reduced” equation for w: Cs)

=

Cu”(I + IM),

‘M =

—

a(21 cos 0).

(3.17)

According to eq. (3.14), tu = O, so that eq. (3.17) gives a first-order differential equation for the “linearized phase” 0. If this equation is solved, and 0(t) is found, both the slowly- and the rapidlyvarying parts of the junction voltage can be calculated as (3.18)

A.K. Jam eta!., Mutual phase-locking in Josephson junction arrays

1= ~ e~”~cos(n0)+~[J(t)]

323

(3.19)

where ~, is a linear operator representing the junction impedance. Equation (3.18) follows from the Josephson equation (3.1) if we take the low-frequency components of the latter, and eq. (3.19) is a generalization of eq. (3.10) for the unperturbed junction. Equations (3.17)—(3.19) show that the high-frequency current I through the junction has two major effects: 1) The Josephson oscillation frequency Cu = 0 is modulated in time in accordance with eq. (3.17). This modulation can be interpreted as the result of mixing of the high-frequency current I with the Josephson oscillations of the junction, resulting in the appearance of an effective low-frequency current component ‘M, through the junction, in addition to the “real” low-frequency current I. The effects of I on the low frequency (.)properties of the junction are identical to those which would be produced by passing an effective bias current ‘M through the junction in addition to the applied low-frequency current I. This current then modulates the Josephson frequency Cu in accordance with the unperturbed junction characteristic.

2) An additional ac voltage drop (represented by the operator ~1’,)appears across the junction, corresponding to some finite junction impedance Z~for the high-frequency current. We will not need the exact expression for this operator, except that at high frequencies the junction impedance is resistive

and equal to the normal resistance of the junction: RN 1(t),

for cv ~

Cu~

(3.20)

i.e. ZI—~RN.

To conclude this section, we note that the real range of validity of eqs. (3.17)—(3.19) is believed to be substantially wider than that of the RSJ model, eq. (3.3). In fact, both experiment and theory show [6] that the main reason for deviations of the RSJ model from reality is its failure to describe dispersion (frequency dependence) of the supercurrent amplitude I~and the quasiparticle conductivity R ~. Careful examination of derivation of eqs. (3.17}-(3.19) shows, however, that the possible dispersion should merely change the parameters tue, ~(n) a and SF rather than the structure of the reduced equations; Within the framework of the RSJ model, the quantities e~”~ and cu” are given by eq. (3.11), while a = 2(1

+

v2)1”2 ~

(3.21)

Thus the results following from the reduced equations are believed to be valid for a wide range of Josephson junctions if the parameters above are understood to be the measured values at a given frequency. The function w”(I) is given by the I—V curve of the unperturbed junction since cv” = (2ir/~’o)V”, amplitudes s~”~ can be deduced from the measurements of the Josephson radiation spectrum, a from the size of the first Josephson step and SF from the linewidth of the Josephson

radiation (see below). Note that eqs. (3.17)—(3.21) are also valid even if the effects of intrinsic noise in the junctions are included, i.e. I(t)-+I(t)+F(t) everywhere. To further simplify the notation we will denote the dominant first harmonic of the junction voltage by e, i.e. e 0).

324

A.K. fain et al., Mutual phase-locking in Josephson junction arrays

3.3. The simplest applications of the reduced equations

Equations (3.17)—(3.19) will be used throughout this paper for discussion of phase locking. To demonstrate the advantages of these equations compared with the initial equation (3.3), and to give a better understanding of their features, let us consider three examples of applications of these equations. Example 1. Let us find the shape of the junction I—V curve in the vicinity of the first Josephson current step. Such a step is formed in the I—V curve due to external microwave irradiation which

induces a high-frequency current

I

Ae C05 Wet

(3.22)

to flow through the junction. Neglecting the fluctuations for the time being (F = 0), and assuming the current I to be the only perturbation of the junction, we can use eq. (3.17) to obtain -

=

—

2cr A~cos 8 cos Wet =

—

a Ae [COS(Wet + 8)+ COS(Wet

—

8)j.

Since we wish to consider only the slowly-varying component, the first term averages to zero and we obtain

a A~cos(co~t 0).

IM

(3.23)

—

The reduced equation, to first order in

‘M,

is given by

dcu”tI~

_IM.

W=w”(I+IM)=w”(I)+

dI

Substituting eq. (3.23) into this equation yields cv

=

w” (I)

—

cos(cv~t 0)

a A~Rd

(3.24)

—

where Rd is the differential resistance of an unperturbed junction Rd

=

dX~1

dI

Q~

(3.25)

2irdl

taken at the point where the first current step appears, V =

(~o/2IT) We.

Substituting eq. (3.24) into eq. (3.18), we obtain the differential equation for 0,

O + a A~cos(w~t 0) = 2~ cv”(T). —

(3.26)

AK. fain et aL, Mutual phase-locking in Josephson junction arrays

325

This can be written in the standard form —~fr+Isin~fr=~I

(3.27)

where: hI~”OWet+~,

R = Rd,

~

I—I~,

‘L

aAe

with I,. defined by: cvU

(I)

=

Equation (3.27) is a phase-locking equation, which describes the synchronization of the Josephson junction by the external signal in the region where cv -~w~.Properties of such an equation are well known, not only from the theory of phase-locked loops, but also from the theory of the Josephson effect. Iffact, eq. (3.27) coincides with the equatioti for the phase 4) of a Josephson junction with a small capacitance in the RSJ model, the critical current of the junction being ‘L and its normal resistance Rd. Equation (3.27) shows that the phase of the Josephson junction is locked to that of the external signal for a range of the dc current [71] I~—IL
(3.28)

Throughout this range, cli is constant in time and hence the frequency cv = O of the junction oscillation is exactly equal to the external frequency ~e. Due to the Josephson frequency voltage relation, the dc voltage across the junction is also constant within this range eq. (3.28). This implies that a current step of height 21,.. occurs in the junction I—V curve, with its center at the current 1~,i.e., the current such that the unperturbed I—V curve gives (3.29)

Cue.

Note that this fact gives a convenient way of experimentally measuring the down-conversion factor a: a =

dIL/dAe.

(3.30)

This derivative should be taken at A~= 0, since the perturbation-theory results are only valid if current amplitude A~is small enough. Analysis shows [3,79] that in the RSJ model A~should satisfy the following condition Ae
I,~

i

fr’3,

l.v,

for v~1, for v 1, ~‘

so that the limitation is rather stringent at low frequencies (cv ~

(3.31)

326

A.K. lain eta!., Mutual phase-locking in Josephson junction arrays

Example 2. We wish to find the linewidthof the Josephson oscillations of the junction when it is not appreciably shunted by an external circuit (I = 0, I = I). Taking into account the junction fluctuations F, we get for the current components of J(t) I(t)+ F(t)

J=i+P,

J=i~.

(3.32)

Assuming fluctuations to be small, we can make successive approximations with respect to F. In the 0th approximation, we neglect F, and from eqs. (3.17), (3.18) get 2zr wo~- V”(I). -

8= Oowot,

-

(3.33)

In the 1st approximation, we take 0 = 0~in eq. (3.17) and find (3.34)

where ~ is an “effective” low-frequency current noise -

F

—

a

[2F cos(wot)].

(3.35)

The fluctuations ~ consist not only of the “real” low-frequency current fluctuations 1~but also of the high-frequency noise F down-converted due to its mixing with Josephson oscillations. The spectral density of the term in brackets is equal to 2SF (w0), and thus the spectral density of the effective noise current ~ is equal [80]to 2 SF(wo). S~(0) SF(0) + 2a

(3.36)

According to eqs. (3.17), (3.18), these current fluctuations modulate the low-frequency voltage across the junction and hence the oscillation frequency cv: V V+ RdS, S~(0)’R~S~(0)

(3.37)

and

cv

=

wo +

Rd ~,

S~(0)=

(~.)2

R~S~(0).

(3.38)

The linewidth ~wof the oscillations modulated by white noise is equal [81]to =

IT S~(0) =

IT

(27TRI~o)2Sv(0),

(3.39)

and thus for this case (single, unperturbed junction) we obtain the result [80] =

IT

(2ir/(I)o)2 R~S~~(0)

(3.40)

A.K. fain et al., Mutual phase-locking in Josephson junction arrays

327

with S~(0)given by eq. (3.36). In the RSJ model, SF(w) = const, eq. (3.2), and a ~ ~, eq. (3.21), so that the contribution of high-frequency noise oscillation linewidth does not exceed 50% of that from low-frequency noise. Equation (3.40) gives a recipe for measurement of the effective low-frequency noise which is valid even if the deviations of the junction fluctuations from the RSJ model are substantial.

Note that our basic result is only valid if noise is relatively small: (3.41)

This condition coincides with the condition that the dc bias point should not be located on the parts of the junction I—V curve which are significantly modified (“smeared out”) by the fluctuations [3,4, 79]. 3.4. Junction equivalent circuits

For some applications, an equivalent-circuit representation [77]of the perturbation-theory results can be more convenient to use than the analytical representation, eqs. (3.17)—(3.19). Figure 8a shows this equivalent circuit. Since the junction properties are different for low-frequency and high-frequency currents, the circuit consists of two (indirectly related) parts. The low-frequency part contains a nonlinear resistance which has the I—V curve of the unperturbed junction. The left current generator represents the low-frequency junction noise F with spectral density SF(O), while the right generator represents the effective current ‘M, eq. (3.17), resulting from mixing of the high-frequency current with the Josephson oscillations. The high-frequency part of the circuit contains a voltage generator e(0) representing the emf of the LOW FREQUENCIES

HIGH

A

I

FREQUENCIES

I

(a)

~4 (b)

~IM+

~? 0

HZe

VU(

J

J

Fig. 8. Two equivalent circuitsof a Josephson junction resulting from the perturbation theory (SVA method): a)with a perturbing current! applied for the junction. The high- and low-frequency noise sources P and F are shown separately. b) The two noise components in a) are replaced by a single effective low-frequency noise source, ~.

328

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

junction oscillations eq. (3.19). Note that the time behavior of the linearized phase 0 depends on the low-frequency voltage eq. (3.18) and hence on the processes in the low-frequency part of equivalent circuit. The impedance Z, is equal to RN at high frequencies, eq. (3.20). Finally, the current generator F represents the junction’s high-frequency noise with the spectral density SF(Cu). If the high-frequency admittance of the external circuit is not too large, we can assume the high-frequency fluctuation current through Z due to F is equal to F In this case, we can simplify the equivalent circuit and represent it in the form shown in fig. 8b. Here we include the high-frequency noise in the effective low-frequency noise ~ in accordance with eq. (3.35). Note that in this case the noise current should be excluded from eq. (3.17), so that effective current ‘M resulting from mixing is equal to IM=—a(2IcosO).

(3.42)

The low-frequency reduced equation (3.17) now reads 2

-

0=

Cu”

[I +

~ +

IM]

w~’(I)+

~

~

—

a (21 cos

0)].

(3.43)

To demonstrate the usefulness of the equivalent-circuit approach, consider one more simple example. Example 3. Let us find the change of the junction I—V curve resulting from its connection to an external microwave circuit with impedance Ze(W). First, we find the high-frequency current which the junction induces in the external circuit. Connecting the right-hand terminals of the equivalent circuit to the impedance Ze (dashed lines in fig. 8b), we obtain (for F = 0): =

I = Re ~

(n) Ze(nw)+ Z,(nw) ~

Now, substituting eq. (3.44) in expression (3.42) for the current IM =

-

ar

(3.44) 1’M,

we obtain [82,83]

Re Ze(W) + Z

1(cv) <0.

(3.45)

According to the left-hand side of fig. 8b, the dc current through the junction at some fixed voltage decreases by an amount IM~dependent on the impedance of the external circuit at the corresponding Josephson frequency. Note that the result (3.45) is only valid if the amplitude A~of the high-frequency current I is not too large, eq. (3.31). Comparing eqs. (3.31) and (3.44), we find that Ze can have any value at high frequencies; at low frequencies, however, Z~should be large enough that 2, for cv S w~. (3.46) IZI + Zel ? RNV If the (rather complex) general expression [76]for Z is substituted into eq. (3.46), one finds that for the perturbation theory to be valid, ZCI should be at least of order RNV3 and hence much greater than I Z~ RNV~. For this reason we have not written the general expression for Z,, and will later use only the expression (3.20), valid in the high-frequency limit. -

—

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

329

4. Two-junction cell-theory

4.1. High-frequency coupling

The simplest system where mutual phase locking can occur is a two-junction cell, discussed briefly in section 1 (see fig. 4). For the sake of generality, we will consider the circuits shown in fig. 9, where each junction can be independently biased by a separate dc current source. From the discussion of section 2, it is clear that mutual phase locking takes place for V1 = ± V2, since the Josephson frequencies of the junctions coincide for these conditions. For convenience, we will refer to them as to “series” (fig. 9a) and “parallel” (fig. 9b) biasing. To discuss the high-frequency coupling we first assume that the external coupling circuit Ze has nonzero admittance only at high frequencies, but zero admittance (infinite impedance) for dc and low-frequency currents (discussion of the finite low-frequency conductivity case will be the subject of section 4.2). Moreover, we will consider here the most interesting case of nearly identical junctions, and will discuss the opposite case in the appendix A (“harmonic” locking is also discussed in this appendix). For series biasing (fig. 9a), the coupling current Ic is Ic =

=

12=

—

Re[s Y(&°’+ &92)],

(series)

(4.1)

where Y(w) is the total complex admittance of the coupling loop at the Josephson frequency: Y~=Z~+2Z1.

(4.2)

For parallel biasing (fig. 9b) it is convenient to take the opposite convention for the voltage V2, so that

___________

4

(a)

V1

Ze

ii 1j

‘AB

B

V2

t

1~

series bias

C

(b)

A

:

V1

7 e

B

[.J

J

t~IAB parallel

bias

—

V2

Fig. 9. Two-junction cell with series biasing (a) and parallel biasing voltages V1~.

~

‘BC

C

Q,). Signs ±show not only the dc voltagepolarities but also the convention for the

330

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

V2 and V1 are always positive. In this case, for high-frequency currents we obtain: Ic =

Ii =

—

12 =

—

Re[e Y(e~°~ e~~)], (parallel)

(4.3)

—

instead of eq. (4.1).

Substitution of eqs. (4.1) and (4.3) into the low-frequency reduced equations (3.43) for each of the two junctions gives: =

w~(I1)+ 2lraRdr [Re Y (1 ±cos ci’) ±Im

Y sin ci’] +

(4.4)

~

2IT~~~8[Re Y(1±cost/i)~Im Ysin~]+~i

02=

w~(l2)+

2.

In the above equation, and in the rest of this chapter, the upper sign corresponds to series biasing and the lower sign to parallel biasing. In eq. (4.4), (4.5)

1/1=01—02,

and lTd ?71,2 =

n,

(4.6)

~1,2,

so that the range of bias currents where mutual locking occurs corresponds to

1/’ =

const. To find this

range, we subtract eqs. (4.4) one from another, and obtain an equation similar to eq. (3.26): ILsint/1=~I+~

R=Rd

(4.7)

where the parameters U and ‘L are now: ~iI= ~ IL

2ar Im y

(4.8) (4.9)

and

(4.10) Note that fluctuations ~ and ~2 of the junctions are independent, so that the spectral density of is equal to the sum of those of the junctions: S,..= S~,+S~=2S,

~_

(4.11)

AK. fain eta!., Mutual phase.!ocking in Josephson junction arrays

331

where the final equality comes from our assumption of equal junctions. The high frequency noise E~ can generate a small noise current in the coupling loop which is common to both junctions. Since the junction oscillations can be coherent the down conversion of this common noise current can produce a small coherent component in .~i and ~ Since this will be the product of two small effects we will neglect it. The fact that we have again obtained eq. (4.7), the well-known RSJ equation, enables us to simply write down all the results ofinterest. Note, however, that the sign of the sin 1/’ termin eq. (4.7) can be either + or depending on the bias polarity and the character of the coupling. This indicates that the stable region can be either 1/’ = 0 or 1/i = IT depending on conditions. We discuss this more fully below. 1) If fluctuations are negligibly small, phase locking takes place in a range of bias current 61 with —

UI

~

ILl

=

2as Im YI

(4.12)

of the bias current. In fact, for identical junctions eq. (4.8) gives (4.13)

611112.

More generally, if there is a small difference in the junctions parameters, experimentally U can be defined in terms of the change in I~and 12 from their values at the center of the locking range. Thus 61= 811—612,

(4.14)

where 6I~= 612 = 0 in the center of locking range. The most interesting feature of formula (4.12) is that the “locking strength” ‘L is proportional to reactive part of the loop admittance Y, so that for equal junctions locking does not take place if the loop is purely resistive. A similar effect is known for the mutual phase locking of two Van-der-Pol oscillators

but in the latter case the locking range vanishes for purely reactive coupling admittance (see, for example, ref. [84]).

Equation (4.12) also provides an estimate of the maximum possible locking strength. Suppose we optimize the impedance Ze of the coupling circuit; from eqs. (4.2) and (4.12) we get max[IL]

=

2 IZ,

(4.15)

which occurs for ReZ~=0,

Im Z~= 2(±ReZ1—Im Zj.

(4.16)

Using the RSJ-model results for the parameters a, e and Re Z1, if the frequency is not too low, we get maxIILI I~. .~‘ , for v 1. (4.17) 2z(i+v)

For v

1, where eq. (4.17) is still qualitatively valid, this gives a locking strength as large as

maxIILI/IC= 15%.

(4.18)

332

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

This estimate shows that mutual phase-locking can be a very pronounced effect for the proper choice of parameters in the coupling circuit. 2) The phase difference inside the locking range depends on the sign of the imaginary part of the coupling loop admittance Y

sin1/=~~1sign(Im Y), cos 1/’ =

~

(4.19)

[1—(6I/IL?]h/2 sign(Im Y),

where the upper and the lower signs correspond again to series and parallel biasing, respectively. In particular, in the middle of the locking range (U = 0) 0,

Im Y<04 series biasing

IT,

Im Y>0,J ~

(4.20) parallel biasing.

In practice, the impedance of the locking ioop is usually inductive in character, i.e., Im Y <0. 3) Note that although the phases change their relative polarity with the change of dc bias polarity (4.20), the voltage convention of V2 also changes (fig. 9). As a result of this double change, certain variables do not change with the bias polarity. An important example of this is the Josephson oscillation amplitude across the entire cell, i.e. between terminals A, C. Assuming for simplicity that IZel ~ Iz~l, from eq. (3.19) we obtain VAc = ,~,e~”~ [cos(nOi)±cos n(0i —

i/r)]

so that for the amplitude V~C of the first voltage harmonic we get 2 (1 ±cos 1/’). =

(4.21)

(4.22)

2r

Comparing this equation with eq. (4.20) we see that for inductive coupling V~is always a maximum and equal to 2e in the middle of the locking range. In contrast, for capacitive coupling the voltage oscillations would be in anti-phase, and the net voltage amplitude would be zero in the middle of the locking range. 4) After 1/i is found from eq. (4.7), we can substitute it back into eqs. (4.4) and thus find the time evolution of ~ The time averages of these quantities give us the dc voltages across the junctions, i.e., the shape of the cell I—V curves: 111,2(11, 12) =

~

O 1,2.

(4.23)

According to eqs. L4.23)~the dc voltages across the junctions are equal inside the locking range, where 1/’ = 0 and thus 01 = 02. This implies that mutual locking results in a change of the junction I—V

A.K. fain et a!., Mutual phase-locking in Josephson junction arrays

333

curves near the point where V~= V~(fig. 10). Note that these changes are not completely localized inside the locking range (4.12) but extend over a range of bias currents,

6II~IILI.

(4.24)

In this region, eq. (4.7) has a periodic solution oscillating with the “beat” frequency ö—çfr=01—02.

(4.25)

In fact, since eq. (4.7) has the same form as the equation for 41 of a single junction, we immediately see that SV( = (PoI2ir)ö) vs. 61 has the same shape as the single junction I—V curve and 1/i(t) has the same waveform as 4i(t) (see fig. 7) with an appropriate change of timescale. As the bias difference 61 approaches the boundary of the locking region, the time evolution of i/i is highly nonlinear (similar to evolution of 41 as v -+0, fig. 7) and as a result t~is a very steep function of 61: as U-+±IL.

(4.26)

This implies that in the absence of fluctuations the “I—V curves” of the cell have sharp angles at the edges of the locking range, and the corresponding derivatives 0111,2/311,2 are discontinuous (fig. 10, solid lines). We will discuss the shape of I—V curves in more detail in section 4.2. 5) Nonvanishing fluctuations modify the behavior discussed above in two important ways. If the fluctuations are small enough, their only effect is to give rise to a finite linewidth of the Josephson oscillations both outside and inside the locking range. We can analyze this effect in the small fluctuation limit by supposing t/1-+çfr+6tfr,

where 61/i and 6w

w-t.cv+Sw are

(4.27)

small variations in phase difference and the coherent oscillation frequency

(cv = 01 = 02) due to the fluctuations. Substitution of eqs. (4.27) into eqs. (4.4) and linearization of the V LOCK ING

0

ICYAB

Fig. 10. Schematic view of the “I—V curves” of the two-junction cell in the absence (solid line) and in the presence (dashed line) of fluctuations.

334

A.K. fain et a!., Mutual phase-locking in Josephson junction arrays

latter with respect to 61/’, 6w give the linear equations [~IL(±cos 1/’ ~ rsin 1/,) 61/i +

6w = ~

6w =

2lrRdl [~IL(+cos tfr + r sin -

-

,~

,t,o

1/’)

(4.28)

,~]

6* + ~2]

where r = Re Y/Im Y, from which 6w and 6i/i can easily be found. For the frequency fluctuations we obtain from eq. (4.28) 6w

=

!~j~ [~+ S~Ir tan cli]’ +

where ~+

= ~

+ ~2

(4.29)

so that the spectral density of the frequency fluctuations is (see eq. (3.39)) 2 ~[1 + r2 tan2 i/i] S~(0). S~(0)= ~(2IT/~o)R

(4.30)

Using the universal relations (3.39) and expression (3.40) for the linewidth of the unperturbed junction, we obtain for the linewidth i~w~ of the coherent oscillations t~wc= ~1kw” [1 + r2 tan2 i/i]

(4.31)

so that the Iinewidth is minimum in the middle of the locking range (1/, = 0): ~ series

min[~wd] =

(1)

Aw” ~ & {parallel.

(4.32)

This important result has a very simple interpretation: phase locking makes the low-frequency voltages across the junctions equal, just as if the junctions were connected in parallel for low-frequency currents. Thus, the two junctions form a single Josephson junction, with doubled noise current density, and halved differential resistance. However, the linewidth of Josephson oscillations is always proportional to Sv(O) = R~S,(0), and thus is only half that of a single junction. 6) If the fluctuations are large enough, they lead to large fluctuations of the phase difference 1/’ and hence to the breaking of the junction coherence. This effect shows up in the I—V curves as a “smearing out” of the sharp edges of the locking range (fig. 10, dashed lines). For even larger fluctuations, the entire locking range is smeared out, and coherence is not achieved for any values of the bias currents. To find this “critical” amplitude of the fluctuations which will destroy locking we refer to eq. (4.7) for the phase difference, and find the dependence of 1/’ on SI, taking fluctuations ~ into account. This mathematical problem has been solved in different ways by several authors [3,81, 85]. The effect of fluctuations on the SI vs. S V curve can be expressed through a parameter y which measures the mean square phase noise: =

(8i/i~).

(4.33)

A.K. fain eta!., Mutual phase-locking in Josephsonjunction arrays

335

be shown from eq. (4.7) that 1/’ only responds to fluctuations ~ below a certain cutoff frequency. Thus the system has an effective noise bandwidth CuN. This can be expressed in terms of the locking strength, WL measured in frequency units It can

CuL =

2lTILRd~o,

(4.34)

as (4.35)

CuN~1TWL.

For the assumed white noise source = (&/1~)=

S~(0)(ON,

~_,

y can thus be written

~ 4 1.

(4.36)

Since S1/i(0) = ~_(0)/IL we have (4.37)

To get a better physical feel for y it is also worth noting that y

=

(I~)/I~ = L~wIwL,

y41

(4.38)

where (I2N) = S~..(0)~N is the effective mean square noise current. The last equality is obtained using eq. (3.40) for the radiation linewidth of the unperturbed junction. Thus, if y 4 1, the fluctuations of 1/’ inside the locking range are small, and only the sharp edges of the 1/’ vs. SI curve will be somewhat smoothed. For y 1, the phase fluctuations are large, and phase locking is broken for any SI, so that the value y = 1 can be taken as the condition for the critical intensity of fluctuations. Note that formulas (4.31), (4.32) for the coherent oscillation Iinewidth are obtained using a linear approximation with respect St/i; in this approximation the fluctuations do not change 1/i. This implies that these formulas are valid only for the parts of the locking range not smeared out by fluctuations, and cannot be applied at all if y ? 1. For large ‘y it is necessary to use the complete Fokker—Plank equations [60].We discuss this further in section 4.2. To finish the discussion of purely high-frequency locking, we should note that in obtaining all the results above, we have assumed Y to be a fixed parameter. This is only valid if the change of Y(cv) inside the locking range is relatively small: ~

(4.39)

This condition is well satisfied for the usual wide-band coupling circuits like resistive and inductive shunts [25—27, 30, 31, 35] which are of practical interest for obtaining tunable coherent arrays. If one uses, however, a series type resonator with a large enough quality factor Q as a coupling circuit [29], condition (4.36) will not be fulfilled, and some of the above results should be modified (see appendix B).

336

A.K. fain eta!., Mutual phase-locking in Josephsonjunction arrays

4.2. Influence of low-frequency conductivity Now let us consider a more complex but a more realistic case, when the coupling circuit (fig. 9) has a nonvanishing conductivity at frequencies much lower than that of the Josephson oscillations: Ze(0) = R~ ~.

(4.40)

To describe this case, we can again use eqs. (4.4). The junction low-frequency currents however, are no longer equal to their respective bias currents ‘AB and ‘BC- The new relations between the currents ‘1,2 which we need for eq. (4.4) and the bias and noise currents are easily seen from fig. 9 to be IlIAB(%~’l±V’

2)/ReFe,

12IBC+(%’l±%~’2)/Re+Fe,

(4.41)

where signs are used as before and Fe denotes the current fluctuations due to resistance Re, with the Nyquist spectral density SF,(w)

=

(4.42)

2~T~

Making the necessary substitution in eqs. (4.4), we obtain the equations V0 %1~ . . 01~w1(IAB__+_)_g(01±o2_ Re R~ + 2(OL[± sin t/i +

r (1 ±cos i/i)]

V0_V~

-

.

2irV02irV0 +

çbo

where

111,2

[1

sin 1/’ + r (1±cos i/i)] +

)

2ITV02lTV0

-.

+

ReRe

~CuL

)

+ ~11 —

82=w2(IBc__+_)+g(o1+o2_ +

41°

4’o 112

4,o

fle

(4.43)

given by eq. (4.6) represent the junction fluctuations and the fluctuations due to R~are

~1e = 2~rRd Fe.

(4.44)

The normalized dc conductivity of the coupling circuit is g = Rd/Re.

(4.45)

The term V0, which does not appear in eqs. (4.4), has been included since it may not be correct to expand to. around the bias current. For example for series bias with large g only a small fraction of the bias current passes through the junction. The precise value of V~,which is unimportant here, is discussed following eqs. (4.55) and (4.56). Solving the eqs. (4.43) for 01.2, one again obtains the same equation (4.7) for the phase difference. Moreover, for the parameters of this equation we obtain the same values (4.8)—(4.11) if biasing is series.

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

337

For parallel biasing, however, some of the parameters are changed due to the finite external conductivity. For parallel bias the effective noise current is changed to ~_~i22Fe,

(4.46)

R = Rd(1 + 2g).

(4.47)

and

This gives us the following important results: 1) The conductivity g does not change the width of the locking range, ‘L, when fluctuations are small. 2) The shunt conductivity does not change the phase difference i/i inside the locking range. 3) In the series case, the effective value y of the fluctuations is not affected by g and again is given by eq. (4.37). In the parallel case, however, y is changed:

=

(1 +2g)~OIL[S,(0) + 2SF~(0)].

(4.48)

Here the effect of g 0 is to decrease the effective noise bandwidth, eqs. (4.34) and (4.47), which tends to reduce y. This is somewhat offset by the additional noise source SFe which is dominant in phase breaking for parallel bias. When the junction noise is comparable to SFe~however, as is the case in the experiments discussed in section 5, y is significantly smaller than for series bias resulting in a larger (less smeared) apparent locking range for parallel biasing. 4) The effect of the shunt conductance on the coherent oscillation linewidth for series and parallel bias is just the reverse of that discussed above. In fact, carrying out the same analysis as in section 4.1, for the frequency fluctuations Sw we get for series biasing

6w = 2(1 + 2g) [(~~

—

211e) +

~-

r tan 4,]

(4.49)

and for parallel biasing 6w

~{(n+) + (fl.

—

2i~)r tan 4,}

(4.50)

where ‘1±= 111±112. As a consequence, the oscillation linewidth in the middle of the locking range (tan 4, = 0) can be strongly reduced for series biasing, =

2(1 +~g~ ~

=

2(2lT/f1~)

where

+ 2~~

2kB TRe

(series)

(4.51)

(4.52)

338

A.K. fain eta!., Mutual phase.!ocking in Josephson junction arrays

is the linewidth a single junction would have due to a voltage noise equal to that generated by the shunt resistance. As g ~ the shunt fluctuation became dominant and -+

-+

as g

(series).

-+

(4.53)

For parallel bias the linewidth remains the same for 4, = 0 as with g = 0: = ~~

(4.54)

(parallel).

The reason for the difference is very simple. The two independent internal noise sources in the two

Josephson junctions of a cell can always be represented as a set of independent components: in-phase and anti-phase, shown in fig. 11. The coupling loop shunts out the in-phase fluctuations (fig. ha), but does not affect the anti-phase fluctuations (fig. lib). Note now that these components play different roles depending on the biasing. For series biasing (fig. 9a), the in-phase component changes both low-frequency voltages V1,2 in the same direction, thus causing a common voltage (and frequency) modulation, i.e. it determines the Josephson oscillation line-width. Thus an external low-frequency conductivity g, shunting out the in-phase fluctuations, reduces the linewidth (4.51). In contrast, the anti-phase component acts in different directions upon V1 and V2, tending to separate these voltages rather than shift both of them in a common direction. Hence, this component is responsible for fluctuations Si/i of the phase difference, but not for the linewidth. Shunting does not affect this component and thus in series the effective value of y remains the same. For parallel biasing the roles of noise components are reversed, e.g. the in-phase component produces phase breaking so now the linewidth is just the same as for g = 0, but y changes. Next, let us discuss the influence of the low-frequency conductivity g on the “I—V curves” V1,2(IAB.BC) of the two-junction cell. Averaging eqs. (4.43), we obtain: 1) Series biasing: —

V= Vo+~1~2g[6I+ILr(1+cos*)1+~[6I+ILsin*]~

(4.55)

1 “2

VO+~1± 2g[SI+ILT(1+CO5*)1~~[SI+ILSiI1*]• (a)

(

~Re

______ ~

(b) I çRe

~H2

~ ~

Fig. 11. In-phase (a) and anti-phase (b) components of the junction noise currents due to sources ~i and circuit, if the circuits low-frequency conductivity is high enough, i.e. if g = RdIR~—’~.

~2is effectively shunted out by the external

A.K. fain et aL, Mutual phase-locking in Josephson junction arrays

339

2) Parallel biasing: ~

~i

(4.56) 1

V2= V0+~[6I+ ILr(h—cos*)]~~1~ 2g[6I~ILsin*].

Here, V0 is the voltage the junction would have for a given interaction, if biased at the center of the locking range: = °

j V~(‘BC 1. V~

—

‘AB

in the absence of a high-frequency

series parallel

Vo/Re)

(‘BC)

(4.57)

and sin4, and cos 1/’ can be found from eq. (4.7) (see ref. [3]chapter 6) and are universal functions of 61/’L and y shown in fig. 12. If fluctuations are small (y4 h), ~iii7/and cos i/i inside the locking range are described by formula (4.19), while outside the range sin4,

= ±sign(6I) {[(SI/IL)2

—

1]hI’2

—

SI/IL}

(4.58) cos9!i=0,

forISIIIL.

Note that in fig. 12 a nonzero value of cost// can occur outside of the locking range for sufficiently large fluctuations. This will result in a voltage V (eq. (4.22)) which can be larger outside of the locking range than the incoherent sum of the junction voltages. This effect was in fact observed several years ago [27]and has recently been discussed in detail by Ambegaokar and Arai [60]. Formulas (4.55), (4.56) show that if the noise is small and if the coupling is purely inductive (Re Y = 0, i.e., r = 0), 1/~and V 2 are linear functions of the bias current (say, I~)inside the locking range (where V1 = V2 = V), with a constant slope Rdh 2 1 + 2g’

for series biasing (4.59)

Rd

-i-,

for parallel biasing.

This slope is always less than the differential resistance Rd of an unperturbed junction. If the resistive component of the coupling admittance is significant (r ~ 1), the V(I) dependences are curved inside the locking range due to the term r (1+ cos 4,)— see fig. 28 for example. This term describes the change of the junction I—V curve due to dissipation of Josephson oscillations in the coupling loop see eq. (3.45); the dissipation increases inside the locking range if the coupling is inductive (IL <0) and decreases if the coupling is capacitive (IL> 0). As a result of such a “curving”, the derivatives 8V/811,2 can pass through zero and become negative inside the locking range. Note, that if the low-frequency conductivity of the coupling circuit is small (g 4 1), the V vs.I curves —

—

340

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

(a)

o.O~~O

~ SI/IL

.0

I

(b)

I~-~ y~0

0.5

0.01 0.03 0.1 0.3

0.0

1.0 3.0

.0.0 8111L

Fig. 12. Averages of sin ~iand cos ~iusedin eqs. (4.55) and (4.56)as functions of 81/IL for variousvalues ofparameter y from eqs. (4.33). Note that sin ~l,is an odd and cos i/i an even function of hI.

are similar for series and parallel biasing. For large conductivities, however, the dc voltage inside the locking range is almost constant for series biasing: 8VIÔIAB

hg

=

RjR~-+0

for Re 4 Rd

(4.60)

but changes substantially for parallel biasing. Finally, we should note that according to eqs. (4.55) and (4.56), formulas (4.31) and (4.51) for the Josephson oscillation linewidth i~WCcan be rewritten in the form (3.39), I~CuC

= IT(21T/tko)2

S~(0)

(4.61)

where the low-frequency spectral density S~(0)of the voltage fluctuations can be found as S~(0)= S~

2+ S~(0)( 2. (4.62) 1(0)(3V/8La.~) t9V/t9IBC)2 + S~~(0) (t9V/OIe) In the last term, Ic is the dc current through the coupling circuit; this last term is equal zero for the parallel biasing. Equation (4.62) expresses the fact that the low-frequency voltage fluctuations (which result in random frequency modulation of Josephson oscillations) result from a change of the voltage

Mi

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

due to random fluctuations of the currents through the junctions and the coupling circuit. This form is particularly useful for comparison with experiments since the partial derivatives in eq. (4.62) can be measured independently. 4.3. Superconducting low-frequency coupling From the analysis of section 4.2, one could conclude that the only effect of the low-frequency conductivity of the coupling loop is to change some of the parameters of coupled junctions, but not to

produce any new coupling mechanism. This is not true, however, for the important case when the junctions are connected by a superconducting loop forming a “dc SQUID” a two-junction superconducting quantum interferometer (fig. 5). Our previous analysis is not valid for this case, because g ce Note that the polarity of the dc bias in the SQUID (fig. 5) always corresponds to what we earlier called parallel biasing (fig. 9b). For the sake of clarity, let us first analyze the case of a pure low-frequency interaction, where one can neglect the high-frequency currents through the coupling circuit: —

-+

(4.63) This assumption is valid either for high Josephson oscillation frequencies, to ~ w,~,for a large ac impedance of the inductive loop, or for Josephson junctions with high capacitances (to w~)which shunt out the high-frequency currents. The low-frequency coupling current ‘e can be found from the basic equation of the flux quantization in SQUIDs [1—6], ~‘

(4.64) where L is the loop inductance, and ~ke is the flux of an external magnetic field applied to the loop. Averaging eq. (4.64) over the high frequencies, we obtain (4.65) since & yields

—

01 =

412=

01—

w~[IAJI

—

02=

4, and

~

(1/i

41~ =

2irP~/Po.Substitution of eq. (4.65) into eq. (3.43) for each junction

41e)] + 111,

=

w~[IBc+ ~(çfr

—

41~)]+ 112

(4.66)

with 111.2 given by eq. (4.6). Note that I~ = I~and 12= ‘BC + I~. In the absence of fluctuations this set of equations has a stable static solution 4, = const. for any parameters of the junctions; for similar junctions —

4, = qS~+41’

(4.67)

342

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

where (I~~IBC).

41, =

—

This solution, of course, corresponds to coherent oscillations in the cell:

O

1=

=

w. Equation (4.67)

shows that the phase difference i/i is proportional to the external magnetic field, since the second term

41,

just represents a constant magnetic flux due to dc supply currents, which does not change with 41e. Thus, phase coupling due to a low-frequency interaction differs drastically from that due to high-frequency interaction:

1) There is not any definite locking range, and coherent oscillations take place for any parameters, if only fluctuations are small. 2) The oscillation phase difference 4, is determined by the flux of the external magnetic field, rather than taking some particular value as is the case for high-frequency coupling. This has important implications for the design of junction arrays (see sections 6, 7) since it is not practical to control the values of the flux in each cell and hence the corresponding phase differences. Taking these features into account, one can assert that the low-frequency coupling does not result in mutual phase locking of the junctions in the usual sense of the word, although this can lead to the coherence of the junction oscillations. In particular there is no frequency pulling as with the hi~hfrequency interaction i.e. w = w” for all ~Pe. Thus variations in ~ cannot induce a loop current (i.e. Ic 15 independent of ~) or cause a modulation of the Josephson frequency. Thus V1,2 are independent of Pe. This implies that the usual effects seen in dc SQUIDs are crucially dependent on the existence of a high-frequency interaction.

To analyze the influence of fluctuations on the low-frequency coupling, we should linearize eqs. (4.66) with respect to small deviations Si/i and Sw, eq. (4.27), induced by noise sources ~i and ~ For 6i/i we get 1~o

~Z~

;

‘

0Si,fr or t. 1) 1) —~w1t1I1~—1~d

hlTl’(

7TL

(4.68)

which is close in structure to eq. (4.7), but linear. Due to this simplification we can find exactly the parameter = (54~2)=

J

S4,(w) dw

(4.69)

which is similar in meaning to the parameter y for high-frequency coupling. From (4.69) we get, assuming ,~ito be frequency independent, =

.~

‘2

(.~)

~2

Rd LS,(0).

(4.70)

~1~0

If S, is equal to

the RSJ model value (see eqs. (3.21), (3.36))

S~(w)= 2 kBT (i. + 2(1 + v2))

(4.71)

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

343

we can rewrite (4.70) in the form:

j~(1

=

+

2(1 +

V2))

~-

(4.72)

~-

where L~is the so-called “critical” or maximum value of inductance [3], L, =

(~)2

(4.73)

~

Thus, thermal fluctuations lead to large fluctuations of the phase difference, if L is larger than the critical value Lf which is about 2 x h0~H at 4K. For the fluctuations of the oscillation frequency, the linearization of eqs. (4.66) give a simple result 6w =

(4.74)

RdS~+

~-

so that the oscillation linewidth is equal to =

~L~w~’,

(4.75)

which coincides with the result of eq. (4.32) for parallel bias. This is just what we would expect according to the discussion in section 4.2 since all arguments concerning noise in the parallel-biased resistive cell (fig. 9b) are completely applicable to the cell with a superconducting coupling loop (fig. 5). Summarizing: the fluctuation stability of a cell with pure super-conducting low-frequency coupling is the same as that of the parallel-biased cell with high-frequency coupling, if the latter cell has a locking strength 1L.

(4.76)

irL

Now we shall consider the more general and practical case when high-frequency coupling is also present in the SQUID-type cell (fig. 5). Repeating the usual calculations, we obtain the following equations 01 = Ci)

[lAB

—

2~L(çli

41~)]—

~ [sin4, + r (1— cos 4,)] + (4.77)

=

cL~’[IBC

+

(*_ 41c)] + ~

[sin4,— r (1— cos i/i)]

+

112

which are, evidently, “superposition” of eqs. (4.4) and (4.66). For the phase difference we now have the equation =

w’i [IAB—~~(~_

41e)]

—

w~[IBc+~~

(ci’

—

41c)] —

wLsin 4,+

~-

(4.78)

344

AK. fain eta!., Mutual phase-locking in Josephson/unction arrays

with 11- =

11i~~ 112-

We are mainly interested in the dynamics of the phase

i/s

near equilibrium (4,-~’0). In

this region the difference of w ~ and w ~should be small, so we can linearize these functions and get ~0.

2ITRd

i/i+ILsin4,=5I——-~.(I/J—41~)+,~i..

o

a

(4.79)

~J~/‘ ~\J’ ~i ~

0

~

~

I

0

C

Fig. 13. Phase plane [i/i, i/i] plot for high-frequency coupling (a), low-frequency coupling (b), and their superposition (c). Black dots show stable and white dots unstable static states.

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

345

To understand the properties of this equation in detail, it is convenient to present it as a formula for the trajectory in the phase plane [4,,4,] (fig. 13), neglecting fluctuations for the time being. For pure high-frequency coupling described by eq. (4.6), i/i is a sum of a constant value (SI) and a sinusoidal part with amplitude proportional to the coupling range ‘L fig. 13a. As a result, we have a stable static solution (4, = 0) and hence a coherent regime, only if the 6.1 is sufficiently small. For pure low-frequency coupling, described by eq. (4.66), i/s gradually decreases with 4,, so that there is always a single static solution eq. (4.68)— fig. 13b. Finally, for a superposition of the two coupling mechanisms we have a linear superposition of the corresponding i/is fig. h3c. From the latter picture, the following conclusions can be made: 1) With joint high frequency and superconducting low-frequency coupling, we always have at least one stable solution, so there is no limited locking range in this case. 2) The properties of the cell in the coherent mode (including stable values of phase difference and stability with respect to fluctuations) depend on the ratio of strengths of two coupling mechanisms. The relation can be characterized by the ratio of high-frequency locking strength ‘L to the low-frequency equivalent given by eq. (4.76): —

—

K =

irLIi.Jt~Po

(4.80)

which is essentially the ratio of high-frequency contribution to the slope di/s/di/i (at a typical point 4, = 0) to the low-frequency contribution to the slope. Figure 14a shows the static value of i/s as a function of external magnetic field for various values of the parameter K. For K 4 1, the phase difference changes proportionally to the external field, see eq. (4.67). If, however, K becomes more than unity, i/c tends to be close to the value predicted by the theory of high-frequency coupling, i.e. ir for Im Y <0. Figure 14b shows the corresponding dependence of the persistent current ‘e~circulating in the superconducting loop. Note that when K decreases, the circulating current is also decreased for all 41~, showing that the junctions behave like unperturbed junctions in the absence of high-frequency coupling. Finally, fig. h4c shows the magnetic-field dependence of the amplitude V~C of the fundamental (first) harmonic of the Josephson oscillations across the cell (fig. 5). Note that for K > 1 all variables shown in fig. 14 are multi-valued functions of 41e. Thus, all properties of the cell depend on whether parameter K is less or greater than unity. For the important special case when all high-frequency currents are flowing through the superconducting coil L (fig. 5), we have Y(w) = (jwL + 2Z~)~,

(4.81)

which (in RSJ model) gives 2 i+

awL(wL+2ImZ1) 2 + (2 Re Z 2] v [(wL+2 Im Z1) 1)

(482)

Thus, for all frequencies, high-frequency coupling is weaker than low-frequency coupling, and we cannot achieve “real” phase locking in the cell. However, we can consider the more general case, when the coupling circuit is not solely the superconducting loop L. An example of such a circuit is shown in fig. 15: the superconducting

346

A.K. lain eta!., Mutual phase-locking in Josephson/unction arrays

2w

________________________ ~‘-~-~-------

__________________

~

0.3

1.0 30

+

I

-w

I

I

I

0

‘S.

—

S

2.

I

I

I

______

_________

a

2.

= ~

4

_ ~~i;;— 2irL ‘e “

K3 K~5~

r4 _~55I,~I

/~ K~o.3

I-ET1 w191 323 m204 323 lSBT

/1 titi

__

2

/

[~uj 2

II

_______________________

_________________ C

-7,

0

w

i_ ~—T.

2w

3w

Fig. 14. Dependence of (a) the phase difference ~s,(b) the persistent dc current I~ in the superconducting loop and (c) the Josephson oscillation amplitude V~in the two junction cell, on the external magnetic field, for various ratios of high-frequency to low-frequency coupling. Note that all variables are 2ir—periodic functions of ~, and that for K > 1 all of these functions are multi-valued.

AK. lain el a!., Mutual phase-locking in Josephson junction arrays

(1~

+

~

347

‘AB

Ze +

2

‘BC

Fig. 15. Two-junction cell with superconducting coupling loop L and additional coupling circuit Z~(w).

inductance is shorted by some other circuit with nonvanishing resistance for dc currents, but with a low impedance Z~at high frequencies: IZ~I4 wL. Such an additional shunt does not affect the low-frequency properties of the cell, but increases the high-frequency conductivity Y enhancing the factor K in eq. (4.80). With the addition of circuits of this type, the factor K can thus be made greater than unity. In this case, as was noted above, the cell can have several stable states, corresponding to different values of 4, and other variables. In this situation, the role of fluctuations is very important. If fluctuations are very small (more exactly, if y defined by eq. (4.35a) is much less than unity), the initial conditions will determine the state of the cell. If, however, the fluctuations are somewhat larger,

they can produce a finite probability of thermally activated transitions between states. In other words, the metastable, states have finite lifetimes (see [3]Chapters 6, 8): CuL

exp(—21y).

(4.83)

Within a time of the order of ‘TL, the cell would come to the state with the lowest energy, which is essentially the state with the lowest value of sin 4,~,the lowest persistent current Ic and the largest oscillation amplitude V~2~ (fig. 14). Thus, in the SQUID-type cell with dominant high-frequency coupling (K ~ 1), small but noticeable fluctuations should lead to the establishment of a coherent state with the junctions oscillating almost in phase (sin 4, 0) forany valueof the external magneticflux. In this senseone can saythat the negative effect of superconducting loop on the phase locking can be overcome by an additional high-frequency coupling circuit Ze (fig. 15). To end this section, let us note that all our results are valid only if the high-frequency currents are sufficiently small (see eq. (3.31)). In particular, for the usual SQUID-type loop (without additional high-frequency conductivity) this implies the following limitation: ‘—

(w/w~)Lw ~ RN.

(4.84)

In this limit the dc voltage dependence on the external magnetic flux is small (as one can show from eqs. (4.77)). As a result, for the dc SQUIDs used as the magnetic flux detectors, lower operating frequencies are necessary, where a differential analytical approach [87],or numerical calculations [88]should be used. 5. Experimental results on two coupled junctions 5.1. Early works

A number of experiments performed during the past decade have demonstrated coherent behavior of two interacting Josephson junctions. The first of these by Finnegan and Wahlsten [10]used tunnel

348

A.K. Jam et a!., Mutual phase-locking in Josephson junction arrays

junctions interacting by radiative coupling in a resonant cavity. Later experiments involved both different types of junctions and different mechanisms of interaction [12—16, 18—24]. Most of these early experiments were aimed primarily at demonstrating the existence of coherence

since neither an adequate theory nor sufficiently controllable measurements were available to develop a very quantitative understanding. The goal of the data presented in this section is to provide just such a quantitative test of the model developed in section 4 for the two junction cell. Our own initial measurements of both quasiparticle [21,90] and electromagnetic [27] coupling were made using planar microbridges. In order to reduce heating effects to make possible a detailed comparison with the theory in section 4, the present measurements have been made using variable thickness indium bridges [91].These bridges, whose properties are described in detail below, have microwave properties rather close to those of the RSJ. However, we must still adopt the philosophy discussed in section 3, that it is proper to use the measured low frequency parameters, e.g. Rd, even if these differ considerably from those of the RSJ model. We begin our discussion of these experiments with a description of the measurement techniques. 5.2. Instrumentation

The measurements on most of the arrays to be discussed consisted of investigation of the microwave properties and the low-frequency characteristics, i.e. I—V curves and (d VIdI)—I curves. a) Microwave measurements: To study the microwave behavior of these devices in detail, a broadband coupling between the Josephson circuit and the receiver was required. To achieve this, the junctions were fabricated in the middle of a 50 fl microstripline on a 1/2” square sapphire substrate, 25 mils thick. The substrate was mounted in a microwave holder equipped with microstrip to coaxial cable transitions, fig. 16a. These were connected to 50 fl stainless steel coaxial cables running out of the dewar. One of the advantages of such an arrangement was that it allowed for both transmission and reflection measurements of the coaxial cable to be made, which was helpful in the determination of microwave discontinuities in the microwave circuit. For the measurement of the power from the junctions, one of the cables was terminated with a 50 [1 resistive termination; the other cable was connected to the input of the receiver. With such an arrangement, assuming the impedance of the Josephson circuit to be small, the power detected by the receiver (for no microwave discontinuities) is given by P=

C(%7

1)=~~~~(171)2

(5.1)

where i7~is the amplitude of the rf voltage across the Josephson circuit, and Z0 is the characteristic impedance (= 50 ~)with which the coaxial lines are terminated. The receiver, shown schematically in fig. 16b was a single sideband swept superheterodyne receiver and operated in the range of 2—18 GH.z. Four low-noise preamplifiers were used to cover this frequency range. These amplifiers, with gains and noise figures of about 30 dB and 6 dB respectively essentially determined the noise figure of the receiver. The amplified signal went through a tracking preselector to a double balanced mixer. The local oscillator (LO) power for this mixer was provided by a microwave sweeper, which could be swept through in the 2—18 GHz range. The preselector was driven by a dc output proportional to the frequency obtained from the sweeper; the center frequency of the preselector was offset from that of the local oscillator by the intermediate frequency (IF). The IF chain

AK. fain eta!., Mutualphase-locking in Josephson junction arrays

349

5011 MICROSTRIP SMA CONNECTOR

(a)

~APPHIR~~

GROUND PLANE MICROSTRIP TO COAX TRANSITION

-1 1810 unction

rl‘PinDiode’ Modulator, L

Preamp

-~

Post Detection Filter Amp

RF Mixer

Termination 5011 sweep Sweep Generator Freq Ref

LO

IF

IF Amplifier

,

Tee Bias;

YIG Preselector Ext Ref

Detector

I

IF Bandpass Filter 1

; L(Ck i’~I Amplifier L___ReLJ

~

I Signal ___________ AveraQerjX

Xv Ij

Recorder]

(b) Fig. 16. (a) Sketch of the array in a microstrip. The encircled area contains the array. (b) Schematic of the 2—18 GHz superheterodyne receiver.

(frequency of IF = 300 MHz) had a gain of 70 dB, and the bandwidth of the IF bandpass filter could be changed from 0.5 MHz to 20 MHz. The gain and the bandwidth of the post detection filter amplifier could also be varied. For narrower spectra where a smaller IF bandwidth was required, we used a Tekronix spectrum analyzer as a fixed frequency IF receiver. Josephson junctions are very susceptible to external microwave signals, to ensure that very little LO power reached the junctions, isolators were used in front of the preamplifiers. Using isolators had the added advantage of improving the input VSWR of the receiver. The isolation of the double balanced mixer and the isolator along with the large off peak attenuation of the preselector reduced the LO power at the junction to a negligible amount (= ~ watt).

350

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

The receiver was usually operated in the swept mode in which the LO was swept with the junction biased with a constant current. In this mode, the receiver was used in conjunction with a Nicolet 1072 signal averager. Apart from the improvement in the signal-to-noise ratio obtained this way, the data storage and arithmetic manipulation capabilities of the signal averager were necessary to obtain the spectrum and the power of the radiation from the junction. The power spectral density was measured for a range of frequencies, with the sample biased at a constant voltage, and stored in the signal

averager; the background signal measured with zero junction current, due primarily to the receiver noise was then subtracted from the stored values to give the spectrum of the Josephson oscillations. The spectrum was then numerically integrated to obtain the integrated power. The calibration of the receiver was performed using a similar procedure with the receiver connected to a solid state noise tube with a calibrated excess noise ratio. The receiver as described has a sensitivity close to the theoretical value for large post detection bandwidths [92].However, for low-level signals (when a small post detection bandwidth was required), the sensitivity was considerably less than the theoretical value due to low-frequency gain fluctuations; this resulted in drifts in the baseline after signal averaging. To reduce the baseline drift in such cases, the receiver was reconfigured into a Dicke receiver [92]; the input to the receiver was chopped with a pin diode modulator and then detected (phase sensitive detection) with a lock-in amplifier. Alternatively, the input to the receiver was switched between the probe and a 50 f~termination using an electromechanical RF switch on alternate sweeps of the LO. At the same time, the signal averager was

operated to respectively add or subtract the input to stored values on alternate sweeps. With such a configuration, we regularly achieved a noise equivalent power within a factor of 2 of the theoretical value [92]. b) DC and low-frequency measurements: The basic dc and low-frequency biasing circuitry for the two-junction cell is shown in fig. 17. The circuitry for other samples were similar or simple modifications of this circuit. The low-frequency connections to the sample were made by bonding gold wires between

the wires running out of the dewar and gold pads on the substrate. These gold pads were connected to the junctions by thin film indium strips, fig. 18. With eight leads connected to the sample, independent four terminal measurements could be performed on either of the two junctions; of the eight leads, four (A, B, C, D) were used for current biasing the array, and the other four (A’, B’, C’, D’) were used for

the voltage measurements. Low pass filters (with a cutoff frequency of about 15 kHz) were used on all the leads to reduce the room temperature noise and transients experienced by the junctions. The currents for biasing the junctions were obtained from a battery of mercury cells connected to a variable voltage divider. These power supplies were connected to the sample terminals A, B, C, D through current limiting resistors (R1 and R2), current sensing resistors (R3 and R4), a current switching matrix and the low pass filters. The switching matrix and the polarities of the power supplies determined the mode of dc biasing of the array. For the measurement of the differential resistances, a small ac current (f— 1 kHz, Iac 0.1 ~iA, coupled via a transformer from the reference channel of a lock-in amplifier) was also passed through the array. Another set of switches determined the ac biasing of the junctions. The dc currents through the junctions were obtained from the voltages across the resistors R3 and R4. The terminals A’, B’, C’, D’, used for the voltage measurements, were connected through low pass filters and through another set of switches to either the lock-in amplifier or a nanovoitmeter, permitting measurements across any pair of terminals. (An impedance matching transformer was used at the input of the lock-in amplifier to obtain a better noise figure.) With this set-up, the differential resistances discussed in the previous section (e.g. d V1/dIAB, d Vj/dIBc, d V,JdI8~,etc.) could be measured.

A.K. fain et aL, Mutual phase.locking in Josephson junction arrays

_______~_____

____

Ref

_______________

~

r

Input

Lock — in Amplifier Output

I _

Input

Nonovoltmeter Output

________________________________________________________________

_

X

_____________

351

XYY~ Recorder

Fig. 17. Schematic of the low-frequency biasing and measurement circuitry.

The presence of biasing leads on the substrate, however, leads to discontinuities in the microwave impedance. For the two-junction cell and other circuits where more than one current had to be varied, the use of biasing leads on the substrate was unavoidable. The resulting microwave discontinuities made accurate determination of the microwave properties very difficult. For those circuits where only a single

bias current was needed, the bias current was fed through the two coaxial cables and the microstips using commercial microwave bias tees.

A B,D

A’ ______________________

B’~D’ C

~

C’

Fig. 18. An optical photograph of a typical two-junction cell. The eight rectangles are the gold pads used for current biasing and voltage measurements. The tapered rectangles at the top and bottom of this photograph connect to 50 0 microstrip lines.

352

A.K. lain er a!., Mutual phase-locking in Josephson junction arrays

The array was located inside a copper can filled with a few millimeters of helium exchange gas. This

can was isolated from the helium bath inside a larger vacuum can. Using a manostat, the temperature of the bath was regulated to within a few m°K.The temperature in the inner can was regulated to within 5 p~°K by using electronic feedback temperature control [93]. The experiments were done inside a screened room, and instruments that were sources of noise (e.g.

the signal averager with its internal clock) were placed outside the room and connected through low pass filters. The cryostat was shielded with two layers of mu metal to attenuate static and low-frequency magnetic fields to about 1 mG at the sample. An aluminum cylinder around the cryostat provided electrostatic shielding. 5.3. Single junction properties

According to the discussion of the previous section, the external coupling impedance Ze, along with the single junction parameters tu”, s, a, Z1 and SF determine the behavior of Josephson junction circuits. Because of non-equilibrium effects, many of the properties of microbridges differ considerably from the RSJ model. Of all the parameters mentioned above, w” is the simplest to measure experimentally because it is given by the I—V curve of the unperturbed junction (w” 2irV”/~o).For low critical currents the I—V characteristics of microbridges are similar to those predicted by the RSJ model. For high critical currents, however, the I—V curves deviate considerably from the RSJ model, e.g. the gap sub-harmonic structure and the “foot” appear in the characteristic [67,94]. These structures are more apparent in the Rd—I curve as can be seen in the typical example shown in fig. 19; this figure shows d V1/dI~vs.

‘AB for

V1 (MV) 424—U

9

?

318

IAB(mA) Fig. 19. dVj/dIAB vs. JAB for the two-junction cell. The voltage across junction 1, V1 is indicated at the top of the figure. was biased at a high voltage.)

(For this figure, junction 2

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

353

the two-junction sample with junction 2 biased at a high voltage. This curve is representative of the R~I characteristic of single junctions; the deviations from RSJ model are obvious. Values of e (the amplitude of the fundamental oscillation at the Josephson frequency) can be obtained from microwave power measurement using eq. (5.1). For the RSJ model, power is related to the dc voltage by P = C {2 V~ii [(1+ z52)”2

(5.2)

—

Thus the power emitted by the RSJ is dependent both on the frequency and critical current. Using the biasing technique discussed in section 5.2 (so that C is given by eq. (5.1) and does not vary appreciably with frequency), the power from an indium VTB was recently measured in the frequency range 2—18 GHz [95]. The results, shown in fig. 20, are consistent with eq. (5.2), though the absolute magnitude of the power is about a factor of 2 smaller. These results are consistent with the earlier observations of Lindelof et al. [96]who determined that the dependence of the power on critical current at 9, 35 and (possibly) 69 GHz is correctly described by eq. (5.2). These results are quite surprising, especially since the I—V curves deviate considerably from the RSJ model. There are very few experiments in which the conversion coefficient a has been determined. This 0

III

liii

x xx

x

X

Xx

C” >

I

X

C.’ N > +

‘C

0.1

-

‘C il

2

4 I,

ii

68101418 [GHz]

Fig. 20. Microwave power detected from a single variable thickness microbridge in the frequency range 2—18 GHz, for I~= 705 isA; RN = 134 mO. The solid straight line is the least square fit to the data and obeys P = v[(v2 + 1)10 — v]~°5 (0.0092). The numerical factor in brackets should have been (0.02) for the RSJ model.

354

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

coefficient, defined by eq. (3.30) is given by eq. (3.21) for the RSJ model. The difficulty in the determination of a is similar for the absolute power; it requires an absolute calibration of the

microwave current passing through the junction; this is difficult to determine with high accuracy. Experiments to determine a in indium VTB’s are in progress in our laboratory [97];some of the results at 2.5, 8 and 13 GHz are shown in fig. 21. For low critical currents, a is close to the prediction of the RSJ model, whereas for high critical currents it becomes significantly smaller. The critical current at which a is a maximum increases with increasing frequency. Though different samples show similar behavior, the dependence of a on critical current and frequency is not understood, and may possibly also be a function of bridge dimensions and thin film properties. These data though incomplete make it clear that one cannot assume the RSJ value for a. The spectral density of the current noise, S~,can be determined by the measurement of the linewidth of Josephson oscillations. For a voltage spectral density which is flat till frequencies much

larger than the linewidth, the lineshape is Lorentzian and the linewidth is given by eq. (3.40) [3,98]. An example of the lineshape is shown in fig. 22. This figure shows the radiation from one of the junctions in the two-junction cell (fig. 9), with the second junction in the V = 0 state. For this curve, the receiver was

tuned to a fixed frequency and the bias current was swept. Since the differential resistance was constant over the sweep range of the current, this procedure correctly reproduces the lineshape of the radiation. Also shown in fig. 22 is a Lorentzian fit to the experimental lineshape. For this fit, the zero for the vertical axis was taken as the receiver output for no junction current. The only free fitting parameter is the linewidth of the radiation. As can be seen in fig. 22, the fit to a Lorentzian lineshape is good. Since any significant excess low-frequency noise (mainly due to external sources [3]) or variation in the spectral density at higher frequencies would have resulted in deviations from a Lorentzian fit [98],this data shows that the spectral density of the noise voltage is relatively flat, at least up to 80 MHz and that

there is no significant influence from external sources. The current noise spectral density as a function of the biasing voltage is shown for two different critical currents in fig. 23. These results are for a single indium VTB and were obtained from linewidth +

2

2.5GHz

or8GHz ~rI3GHz 0

~

0

100

200

300

Critical

I

I

400

500

600

Current (~A)

Fig. 21. The downconversion coefficient a vs. critical current for 2.5, 8 and 13 0Hz. The solid lines show the variation of a according to the RSJ model.

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

70

355

3

60 ~C 50 2 0

f 40’ 36MHz

C

Ii

30

0

20 x20

C

~

‘,

--

-

_______

96Hz Frequency

Fig. 22. (a) The lineshape of the Josephson oscillations in junction 1 with junction 2 in the superconducting state. (b) Same as (a), except that the gain for the vertical axis has been increased twenty times. Also shown in (b) is the calculated Lorentzian lineshape.

-

0

~o0000

10

20

Junction

30

Voltage (~V)

Fig. 23. The current noise spectral density vs. junction voltage for a typical microbridge. Data shown for I~ increase in noise with critical current, and the decrease with increasing junction voltage.

108 ~A and 4 = 705 ~iA.Note

the

A.K. fain et al., Mutual phase-!ocking in Josephson junction arrays

356

measurements. The noise increases with increasing critical current and decreases with increasing bias voltages. Further, it is larger than can be attributed to the Johnson noise of the normal resistance; the mechanism for noise generation is not presently understood. These results on the noise are similar to the experimental observation of Lee [99], who measured the low-frequency noise from microbridges directly using a cryogenic transformer. These results again show that it is essential to individually measure the junction parameters and not to rely on the RSJ model.

No quantitative measurements of the microwave impedance of microbridges have been made. However, experiments of this type have been performed for proximity effect bridges [100],and the results are in qualitative (but not quantitative) agreement with the RSJ model. The above discussion makes clear that there are significant differences between the properties of microbridges and the RSJ model. In comparing the experimental results of mutual phase locking to the

theory, one thus cannot assume the RSJ values for the junction parameters. In this section we shall generally adopt the procedure of treating these parameters as fitting parameters in order to test the functional form of the theory and comparing the parameters obtained from these fits with independent

measurements where possible. 5.4. Experimental results on 2 junction cells A micrograph of a typical two-junction cell coupled with a loop composed of an inductor and resistor in fig. 24. The junctions used in these samples were indium VTBs with bridge and bank thicknesses of 700 A and 7000 A respectively. The nominal dimensions of the bridges were 2000 by is shown

A

3000 A (length x width). The coupling loop consisted of a gold resistor, 1500 A thick and 6 p. x 6 p. square, and the separation between the junctions was 10 p.. (The details of fabrication are discussed in appendix C.) The magnetic inductance of the loop formed by the bridges and the coupling resistor was estimated to be about 10 pH [101,1021. The most easily observable effects of the interaction of two junctions are the distortions in the junctions’ I—V curves which this interaction produces. These effects are clearly seen in fig. 25 which shows the voltage variation of the two junctions of the cell (fig. 9) as the current i~ is varied with ‘Bc held constant (with junction 2 biased at a finite voltage). Figures 25 (a) and (b) correspond to parallel bias and series bias, respectively. The interaction between the junctions due to the coupling, which has a finite low-frequency conductivity, causes many of the features present in fig. 25. The most prominent of these is due to voltage locking; the voltages V1 and V2 across the two junctions lock to a common value,

and this locking persists for a range of I~. In addition to this fundamental voltage locking (V1 = V2), distortions in the I—V curves also occur due to harmonic locking between the junctions when V1 and V2 are integral multiples of each other. Because of the dc conductivity of the coupling loop, the currents passing through junctions 1 and 2 are not in general equal to I~cbut are given by eq. (4.41). As a result of this, as is increased, the current through junction 2 is decreased for series bias and is increased for parallel bias. This results in a decrease of 1/2 with an increase in l~ for series bias and an increase in V2 with an ‘AB,

increase in

‘AC

‘AB

for parallel bias. Also because 11.2 ~ ‘AB,Bc for V2 = 0, the apparent critical current of

junction 1 is increased for series bias and is decreased for parallel bias. (Also see appendix D.) Having seen qualitatively the effect of the interaction on the shape of the I—V curves, we now examine the region near the fundamental locking in detail. The result of section 4, which is central to this discussion is eq. (4.7), which describes the behavior of the difference phase near the region of t/i

AK. fain et a!., Mutua! phase-locking in Josephson junction arrays

357

‘-U----

Fig. 24. Micrograph of two junctions coupled by an inductor. The light rectangle is the gold shunt resistor, which forms a part of the coupling loop. The equivalent circuit is shown in fig. 9.

fundamental locking. The beat frequency, which is the difference in the frequencies w~and of the two junctions, is just the linear component of and is determined (for identical junctions) by eq. (4.7) with R = Rd for series bias and R = Rd(1 + 2g) for parallel bias. We will first discuss the parallel bias case and will come back to the series bias case later. For parallel bias the voltage corresponding to ~1iis just the low-frequency voltage across the array, VAC = ~oçt’/2ir. According to eq. (4.7) the ~I—VAC characteristic should be familiar hyperbola predicted for a single junction by the RSJ model (where = ~(IAB Inc); the currents I~and are assumed to be adjusted such that U 0 for ~ti= 0). This behavior is observed [103],as shown by the solid line in fig. 26. For this curve, U was varied by passing an additional dc current from A to C. To compare the data curve in fig. 26 with that predicted from the single-junction parameters we must allow for the difference in the properties of the two junctions by using the more general form of eq. C02

~(i

—

‘BC

‘AC

358

AK. fain eta!., Mutual phase-locking in fosephson junction arrays

000

500 I~ ( /~A)

~

2000

~

Fig. 25. The voltages across junctions 1 and 2 vs. I~,with ‘BC held constant. (a) parallel bias, Q~)series bias. Note the fundamental locking between the junctions when V 1 = V2 also note harmonic locking when mVi = nV2 (‘BC is not the same for (a) and (b).) For %~2= 0, I~= 1170 ~A.

(4.7) (see appendix A). For non-identical junctions, the +

Rdl+R~)=

VAC—IAC

characteristics is determined by

(Rdl + R~)IAC + WLI sin(tI’ + f3) + ~7eff +c

(5.3)

where is the locking range in frequency units, fleff is an effective noise current, c is a constant and /3 is a phase shift dependent on the non-uniformity of the junctions and on the nature of the coupling loop. All the above parameters are defined in terms of the parameters of the two junctions in appendix A. The two junctions have critical currents of ~ = 120 p.A and 1c2 = 200 p.A and differential resistance Rd, = 212 mQ and Rd, = 232 mEl at VL = 49 p.V, which was the voltage of an individual junction in the locking range. _The resistance Re of the coupling resistor is 92 mEl. Thus, even for non-identical junctions, the VAC-IAC characteristic is the usual RSJ hyperbola but with an asymptotic resistance (L)L

Rasym

(Rd, + Rd2)Re

ERdl+Rd2+ Re]

54

( ) .

and a locking range or “critical current” given by L

—

~1~O (RdlWL + Rd2) ~

55

AK. fain ci aL, Mutual phase-lacking in Josephson junction arrays

I

359

__

0

20

Fig. 26. The “VAC—IAC” characteristic of the two-junction cell, for no external rf current (solid lines) and with external rf current of frequency 400 MHz (dashed lines). This figure is for parallel bias.

The experimental curve in fig. 26 shows considerable noise rounding near the “critical current”; thus to compare the data with the I—V curve predicted by eq. (5.3), the effects ofnoise have to be included. The data was well fit by calculated I—V curves, based on the Ambeogaokar Halperin theory (AH) [85,81, 3] (which takes the effects of noise into account). The noise parameter y (eq. (4.33)) and the locking range ‘L were determined from the fit of the theory to the data. As a further check of the validity of the perturbation theory, we determined the response of the two almost locked junctions to low frequency rf currents. If a current ‘e sin Wet (W~4 w1) is added to the dc current ‘AC, then eq. (5.3) predicts constant voltage difference steps in the VAC—IAC characteristic when the beat frequency between the junctions is equal to a multiple of We. The first and second constant voltage difference steps are distinctly visible in the characteristic shown as the dashed curve in fig. 26; for this data, the external rf current had a frequency of 400 MI{z. The presence of the biasing leads, however, made it impossible to determine the absolute value of this low-frequency rf current passing through the junctions; also because of these leads the amplitudes of the rf currents flowing through the two junctions may not have been identical. The half heights of the n = 0, 1 and 2 steps, corrected for noise rounding, are shown as a function of I~(in arbitrary units) in fig. 27. As can be seen, all three curves are closely fit by the corresponding Bessel functions, just as for a single current biased RSJ in the high-frequency limit [3]. The scale factors in fig. 27, which were the same for all three steps, were obtained by fitting to the maximum and first zero of the n = 0 step. The “characteristic frequency” corresponding to the I—V curve shown in fig. 26 is 100 MH.z. Since the external rf current had a frequency four times the characteristic frequency, the use of the high-frequency limit is appropriate [3].

360

AK. Jam eta!., Mutual phase-locking in Josephson junction arrays

1rf (arbitrary units) Fig. 27. The half height of the zeroth, first and second if induced constant voltage difference steps as a function of I~.The heights of the steps are normalized to the height of the zeroth step for I,s = 0. The solid lines show the Bessel function dependence, f,(2irI,sR,.~,,.,,/Øowi).

The noise-free heights of the various steps were determined by fitting the I—V characteristic in the neighborhood of the steps to the AH theory. This procedure is also justified because of the large normalized frequency of the microwave current [3]. The values of the spectral density of the noise current flen obtained from these fits agree with each other to within 10%. The detailed relation of the effective noise current ~1eff (or the noise parameter y) to the individual noise sources will be discussed below. The data shown in figs. 26 and 27 were obtained for parallel bias. Essentially the same results are obtained for series bias. In this case the voltage corresponding to (~‘oI21T)tlJis VAB— VBC and had to be obtained from independent measurement of the two voltages. For series bias, the current “AC” was fed

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

361

through B and extractedsymmetrically from terminals A and C. The rfcurrent was fed through terminals A and C, as in the parallel bias case. Note forthis rIbias, for identical junctions, eq. (4.7) predictsno constant voltage difference steps. Because of non-identical junctions and inbalance in the rf currents through the junctions, constant voltage difference steps were also observed for the series biasing. According to the perturbation analysis using the measured values of Rdl, Rd2 and Re, the asymptotic resistances of the VAC—IAC characteristic should be 76.2 mEl for parallel biasing and 222 mEl for series biasing. The measured values were 75 mEl and 215 mEl respectively. The locking range is predicted to be about 4 p.A using the RSJ values for e and a; the measured values are both 2.8 p.A. This disagreement in the locking range is related to the use of the RSJ values for the parameters r and a and will be discussed later in this section. The good agreement between the theoretical and experimental VAC—IAC curves, the presence of constant voltage difference steps and the agreement of the heights of these steps with theory all attest to the validity of the perturbation method. In particular note that these characteristics are very close to the RSJ model even though those of the individual junction are not. ‘L

—

V

1 V2

(a)

V2 I~Vt 7.5~A

v>,___~

V12

‘AB

(b)

V2

Fig. 28. The voltages Via vs. I~ near fundamental locking for parallel bias. (a) experiment, (b) theory. The voltage in the middle of the locking range is 22 p~V.The dashed line in (b) indicates the calculated voltages across the two junctions in the absence of noise.

362

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

We next discuss the shape of the I—V curves (given by eqs. (4.55) and (4.56)) of the individual junctions when the junctions are locked or are close to locking. The results of parallel biasing, are shown in fig. 28. For these results ‘Bc was kept constant and I~ was varied. The locking voltage was approximately 22 p~Vand corresponded to a normalized frequency of 0.5. The sample for which the data are shown had almost identical junctions; RN 0.15 fl and for the data shown ‘ci = 190 ~A and 1c2 = 220 ~tAand Rdl = Rd2 = 145 mEl. The hump in the I~—V1,2 curves shown in fig. 28a is characteristic ofinductive—resistive coupling, as discussed earlier. The differential resistance in the middle of the locking range was 78 mEl, which is close to, though somewhat higher than half the differential resistance of the unperturbed junction. Fig. 28b shows the theory corresponding to fig. 28a, calculated on the basis of eq. (4.55). The calculations are shown for the case of no noise (dashed lines) and with noise present (solid lines). The effect of the noise is to smear out all the sharp corners of these I—V curves. The calculated results (with noise) display most of the features shown by the data in fig. 28a. There are small differences between the theory and experiment in the voltages V~and V2 of the two junctions.

(a)

:

1AS

O.5~V

l0,~A

Fig. 29. ~o*2irvs. I,.~.(a) Parallel bias, (b) series bias. The solid lines are the calculated curves (with noise), the dashed lines (no noise). The solid circles are the experimental data points. Note the different scales for parts (a) and (b). The experimental data is for locking voltage of 22 ~V.

A.K. Jam et al., Mutual phase.locking in Josephson junction arrays

363

The agreement between the theory and experiment is best at lower critical currents and at higher frequencies. This suggests that one of the reasons for the discrepancy between experiment and the theory could be the presence of harmonics, which has been omitted in the theoretical analysis. As discussed earlier in this section, the agreement between the theory and experiment of the difference voltage (V1 V2) is quite close; this comparison is shown in fig. 29a for the case of parallel bias and fig. 29b for series bias. The voltages V1,2 for series biasing are shown in fig. 30. These curves were taken at approximately —

the same locking voltage as the data shown for the parallel bias case. Note that because of the large conductivity of the coupling loop (g = 3.4), the voltages of the two junctions, fig. 30a, do not change much in the locked region as compared to the case of parallel biasing. The differential resistance in the

middle of the locking range is about 10 mEl, close to the value of 9.3 mEl predicted by eq. (4.59). Fig. 30b shows the calculated curves for the case of series biasing, again with and without noise. Here, the agreement between the theory and experiment is apparently even better than the parallel

VI

/

(a) VI

v2

4 l~V1

V12

7.5~A>

1AB

Fig. 30. The voltages Via vs. I~near fundamental locking for series. (a) experiment, ~) theory. The experimental data is for the same locking voltage as fig. 28. The dashed line in (b) shows the calculated voltages in the absence of noise.

364

AK. fain eta)., Mutual phase-locking in Josephson

junction arrays

data. This is because most of the variation of the voltage in the locked region is due to the cos cu term in the reduced eq. (4.56). For the case of series bias, the contribution of this term to 01,2 is decreased by a factor of 1/(1 + 2g). The rounding of the 1—V curves due to noise provides an easy though indirect way of measuring the spectral density of the noise sources. Inverting eq. (4.48), for parallel bias we get S~(0)+ 2SF~(0)=

(5.6)

ypar(1+2g)~OIL

and for series bias (using eq. (4.37)) C

(A\

1~OIL Yser~ I z..IT2~1’d

—

The values of (S, + 2SFC) and S,, calculated using eqs. (5.6) and (5.7) respectively for the two biasing schemes, are shown in fig. 31 as a function of the locking voltage. Also shown in fig. 31 are the values of S~+ SF~,obtained from the measurement of the linewidth of radiation from junction 1, with the second junction in the superconducting state. The spectral densities obtained from the series locking data are qualitatively similar to those shown in fig. 23; S~increases with increasing critical current and, as shown in fig. 31, it decreases for increasing junction voltage. Values of S~can also be obtained from the parallel locking and the linewidth data

N

r

2’

\

E

‘C’,’

a

0

N

‘C a

9

‘C

‘~“..

£

‘C

..‘.... ~

‘_lC_ ~

a

a

‘C

* X

—

a

a C

a, —

o

0 0’s,

t

—‘a

0

I

I

I

I

I

10

20

30

40

50

60

Locking Voltage ( ~V) Fig. 31. The spectral densities of the current noise sources obtained from the voltage locking data; (open circles) series bias; (x) parallel bias. The dashed guiding line passing through the series bias data is shown displaced by 2SF, as the dashed line on the top. Also shown is the spectral density obtained using linewidth measurement (A).

A.K. fain era!., Mutual phase-locking in Josephson junction arrays

365

shown in fig. 31 by subtracting the shunt resistors contribution (assuming it to be Johnson noise). The values of S~so obtained closely agree with those obtained from the series locking data; those obtained from the linewidth measurements are somewhat higher. Though the data shown in figs. 28 and 30 were measured at the same locking voltage, the apparent locking range, i.e. the range of current for which difr/dI~is zero (which implies complete locking), is larger for the parallel bias case than for series bias. However, as seen from figs. 29a and 29b, the noise-free locking ranges for the two cases are about equal. The difference in the apparent locking, results from the differing effects of the shunt conductance on the fluctuations which determines the rounding of the I~—V~,2curves as discussed in section 4. For the data shown, 2Sf, S~,thus the effect of the shunt resistor is to reduce y for the parallel case, resulting in less smearing and a larger range of complete locking for parallel bias. This difference in the effect of noise on the apparent locking strength for series and parallel bias explains the puzzling results in the initial observation of voltage locking [18]: namely, that while complete locking was observed for parallel bias it could not be obtained when the bridges were biased in series. The variation with the locking voltage of the locking strength (corrected for noise) is shown for two critical currents in fig. 32. As predicted by the theory the maximum locking strength I~ increases with increasing critical current; the voltage V(I~*) at which this maximum occurs also increases with increasing critical current. The maximum locking strength, however, is always smaller than that predicted by the RSJ model. The dependence of ‘L on the locking voltage VL is compared with the theory in fig. 32. For I~= 58 pA per bridge, the agreement between the theory and experiment is reasonable over a large range of locking voltages for L = 13 pH and Z~= RN, if s is assumed to be 1.4 times smaller than the RSJ value. This is consistent with the value of ~ obtained by power measure-

15 ~‘\

,_ /

\

/

/ / \

I

4 :1.

x

I

/

~l0, 0’ C

I I

x~

\ ‘C

x

X

\\

*

5

Locking

Voltage (MV)

Fig. 32. The locking strength vs. locking voltage for average critical current of 58 isA (solid circles) and 200 isA (x). The solid and the dashed lines show the calculated values using the RSJ model but assuming a to be smaller, as discussed in the text.

A.K. fain ef al., Mutual phase-locking in Josephson junction arrays

single microbridge, as discussed in section 5.3. Also as discussed in section 5.3, a is close to the RSJ value, eq. (3.21) for low critical currents. For I~= 200 pA/bridge, is smaller than given by the theory, even for the smaller value of s. The theoretical curves, however, are calculated for the same inductance (L = 13 pH). Since the coupling loop has significant temperature dependent kinetic inductance, the inductance at this higher critical current could be smaller than 13 pH. Because of the uncertainties in the variation of L and the other parameters, we have not attempted to fit the data to the theory for high critical currents. Qualitatively, the disagreement with the calculated ‘L at voltages lower than V(I~) for this critical current can be explained by the observed discrepancy in the conversion coefficient a which is less than the RSJ value as discussed in section 5.3. The decrease in for voltages beyond V(I~~) is faster than given by the theory, and suggest the existence of an additional cutoff due to relaxation processes present in microbridges but not included in the RSJ model. The linewidth (full width at half maximum) and the integrated power of the radiation from the two-junction cell near fundamental locking is shown in fig. 33 (for the case of parallel bias). For this data, I~= 500 ~A, Rd = 150 mEl and the locking voltage VL = 28 p.V. The linewidth and the power predicted on the basis of eq. (4.61) and eq. (4.22) are also shown in fig. 33. The experiment was performed at a low normalized voltage and as discussed earlier in this section, there is disagreement between the calculated and experimental V1,2 curves which become significant at low normalized ments of a

‘L

‘L

‘AB~

I O/.L A II I

I

I

I

-3

I

I

-2

I

I

l0~A

‘C

*

a,

I

1

I

I

I

I

-l

0

I

2

3

81 Fig. 33. Linewidth andpowervs. flu nearfundamental lockingforparallel bias(bi

flAB/IL). (a)Linewidth: (x)experiment;solidlineshows thelinewidth calculated using measured differential resistances. (b) Power: (x) experiment; solid line shows the calculated power.

AK. fain eta!., Mutualphase-locking in Josephson junction arrays

367

voltages. Also, at this high critical current the unperturbed differential resistance exhibits both “foot” and subharmonic gap structure. For these reasons, the experimentally measured differential resistances were used (see eq. (4.62)) for the calculation of the linewidth. The spectral density of the intrinsic

junction noise, S~,was obtained from the fitting of the VAC—~IABcurve to the AR theory for series locking at this locking voltage, eq. (4.37). The shunt resistor noise, SF~,was assumed to be Johnson noise, eq. (4.42). For large ~i (6i > 2.5) the radiation consists of two distinct peaks at frequencies vi = V1Rk~and -

= V2/k0. Also, sidebands are observed, separated from the main peaks by multiples of v- (an vi the power in ~hesidebands decrease rapidly on increasing ~i. For &i <2.5, the two spectral lines at v~ and i’2 overlap, and the sidebands overlap with the peaks at v~and i.’2. With a decrease of ~i, a single very broad line is observed. On further decrease of ~i, the junctions phase-lock. In order to compare the linewidth data with theory, we must first review the range of validity of the various approximations used. Equation (4.4) should provide the correct spectral density of the radiation except, as noted in section 3, for regions where noise effects the shape of the I— V curve, e.g. the regions — i.’2);

of high differential resistance near the edges of the locking range. Here a more general description of

the effects of noise on the dynamics of the coupled junctions is provided by the complete Fokker—Plank equations; such an analysis has been carried out by Ambegaokar and Arai [60].(Their calculations, however, were done for the case of series bias with zero low frequency conductivity of the coupling

loop, and as we shall see, some of the interesting phenomenon are more pronounced for high shunt conductivity and for parallel bias.) In the region further outside the locking range where eqs. (4.4) are valid it is clear that eq. (4.61) still does not correctly describe the data. The reason for this is that eq. (4.61) represents a further approximation in that it neglects the effects of the beat frequency and its harmonics which cause

additional modulation of the Josephson radiation in this region. While this information is all contained in eq. (4.4) the complete solution including noise is somewhat cumbersome in the region. The essential results have however been demonstrated in computer simulations by Jillie [59].We present below a semiquantitative discussion of these effects which allows one to estimate the regions where significant deviations from eq. (4.61) are tobe expected. Apart from the dc voltages V1 and 1/2 across the junctions, there is also an ac voltage (difr/dt) at the beat frequency v_ and harmonics of v; for large shunt conductivity, the total voltages for parallel bias are given by V1= Vo+~[~I(1+i,~2g)]_~ILrcosl/f (5.8) V2= Vo+~[~I(1_i,~2g)]+~ILrC051fr

where V0 is given by eq. (4.57). Outside the locked region, evolves in the time according to the RSJ equation, eq. (4.7). For large shunt conductivity, the linewidth of the ~troscillations is quite narrow (=3.5 MHz) compared to the linewidth of the Josephson oscillations. The presence of this ac voltage results in periodic FM modulation of the Josephson oscillations at the beat frequency, thus leading to the formation of sidebands at frequencies v1 ±nv_ and i’22~oP_)) ±rn’_. is Theabout effectsunity of the become or modulation larger [98].In the significant when the modulation index m = (~ RdILr/( experiment, the frequency v_ corresponding to a modulation index of one is about 250 MHz, which i/i

368

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

corresponds to Si 2. This explains the existence of the sidebands and the rapid decrease of the power in the sidebands for Si > 2.0, since this power decreases [98]as (J1(m ))2. The modulation also accounts for the rapid increase in the linewidth outside the locked region. Because of the large linewidth of the Josephson oscillations (=70 MHz), the fundamentals at v1 and v2 and the sidebands all overlap for v_ less than a few times the Josephson linewidth, thus leading to large linewidths at these frequencies. For large modulation index, the linewidth is approximately [98]given by 2mv_ = RdIL/’l’o ( 500MHz), which is close to the maximum linewidth observed. We next discuss the deviations of the calculated linewidth inside the locked region. The noise parameter y corresponding to the locking region is about 0.05. Though deviations from the calculated linewidths are to be expected towards the edges of the locking range where the I—V curve is rounded by noise (as discussed in section 4), the experimental linewidths are smaller than the calculated values even for small SI where noise rounding is not significant. One possible reason for this discrepancy is outlined below. The linewidth in the locked region is given by eq. (4.31) as long as the spectral density of the voltage is flat to frequencies larger than ~ (assuming the junctions to be completely locked). A linearized analysis of eqs. (4.43) shows that the voltage noise spectral density is not flat, but is given by 2tan2ç1i)S~(co)+r2tan2lfrSF,} S~(w)=~R~{(1+ r (5.9) tu

~R~Ss(w)

where

—

~=

cos ib’ 2ir (1+2g)cPo

ILRd

The reason for this spectral variation in S~(w)is that the difference phase (which obeys the RSJ equation, eq. (4.7) with R = Rd/(1 + 2g)) does not respond to ~ (tu) for frequencies above the cutoff frequency tue,. However, the large variation of the measured differential resistances (which are used to calculate i~vand are measured at frequencies ‘~w~) inside the locked region for parallel bias (see e.g. fig. 28) are due to the low frequency response of Thus, using the measured differential resistances to i/i

cu’.

obtain S~(w)at frequencies larger than

tu.~,

overestimates the high-frequency voltage noise spectral

density. In the experiment, ILRdJ(l + 2g) is about 0.3 p~Vwhich is close to the voltage corresponding to the coherent linewidth of the two junctions. A result of the effect discussed above is that away from the center of the locked region the linewidth should increase less than the theoretical prediction, eqs. (4.61)

and (4.62), as observed experimentally. The power in the locked region as well as outside the locked region, fig. 33b, is consistent with the theory. The large scatter in the data is due to the various microwave discontinuities which makes C

frequency dependent (eq. (5.3)); it is difficult to correct the measured powers to include the effect of the discontinuities. The power and the linewidth for the case of series biasing are shown in fig. 34. Note the dramatic narrowing of the linewidth inside the locked region due to the shunting effect of the coupling resistor. Because of the shunting of the fluctuations the linewidth is reduced to about 3.5 MHz in the center of the locked region. The linewidth of the autonomous junction would have been about 220 MHz. For the data shown in fig. 34, I,~ 1000 p.A, Rd = 150 mEl, Re = 30 mEl and the locking voltage is 28 p~V.The dotted lines in fig. 34 show the theoretical predictions. For the same reasons as discussed for

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

III\L~J’

369

1111

l0~sA I

I

I

I

I

I

I

I

I

I

‘AG Fig. 34. Linewidth and power near fundamental locking for series bias. (a) Linewidth: (x) experiment; solid line shows linewidth calculated using measured differential resistances. (b) Power: (solid circles) experiment; solid line shows the calculated power.

the parallel bias case, the experimentally measured differential resistances were used to calculate the linewidths. The spectral density of the intrinsic junction noise was determined from the rounding of the (~Poi1’i/2rr)4I~ curves, while the shunt resistor noise was again assumed to have the Nyquist value. Unlike the parallel biased case, the agreement between the theory and experiment is quite good, fig. 34a. The large broadening of the linewidth outside the locking range due to modulation at the beat frequency is not seen for series bias. This is because the linewidth of the beat frequency oscillations, which is given by eq. (3.39) and eq. (4.11) is quite large (= 400 MHz). Also, from eq. (4.55), the modulation index m is equal to one for v_ <500 MHz. Because of the large linewidth of the beat frequency oscillations, the modulation signal is essentially random for v_ <500 MHz, and leads to only

a slight increase in the linewidth. The calculated linewidths around the centerof the locked region are also in reasonable agreement with those measured. As discussed in section 4, there are differences towards the edges of the locking range where the differential resistances are smeared significantly due to fluctuations. Note that Wq, for series bias is (1+ 2g) times larger than for parallel bias, and for the experiment is about 1 GH.z. Hence the measured differential resistances give the high-frequency voltage noise and thus the linewidth accurately over most of the locking range. The power in the locked region, shown in fig. 34b, is again in approximate agreement with the

AK. fain era!., Mutual phase-locking in Josephson junction arrays

370

theory. Becuase of the much smaller linewidth inside the locked region, the microwave discontinuities affect the measured power more than for parallel bias. 5.5. Superconducting low-frequency coupling

In this section, we discuss the experimental results on “dc SQUIDs” (fig. 5) and “shunted dc SQUIDs” (fig. 15); our goal in the following is mainly to compare the behavior of the standard SQUID to that of the shunted dc SQUID. The standard (unshunted) dc SQUID consisted of two planar microbridges having dimensions

dc of

0.3 p. X 0.3 p., thickness of 1000 A, and spaced 6 p. apart. These microbridges had a normal resistance of

200 mEl. The superconducting loop coupling the two microbridges was a thin film inductor; from the modulation of the critical current, the inductance was determined to be about 50 pH. The variation of the integrated power in the fundamental at 4.5 0Hz (for I~ = 190 p.A) is shown in fig. 35. For this data, the current through the SQUID was kept constant, and the external flux, was varied. The power and the SQUID voltage both varied periodically with P~with a period consistent with the flux quantum, The SQUID voltage was maximum when the power detected was maximum. This is as expected according to the discussion following eq. (4.59), since both results imply = The predicted variation of the powerwith ~ is given by eq. (4.22) and eq. (4.79), and is shown by the ~,

cu~

~.

External Flux

0 28

(b)

K~0.75,r I

I

I

= 0.28 I

I

I

I

I

0 External Flux Fig. 35. Linewidth and power vs. ~, for the dcSQUID. (a) Power: the experimental values areshown by the solid circles. The solidlines show the resultsof the calculations for ,c = 0,0.1,0.5,0.75. (b)Linewidth: solid circles—experiment; solid lines show calculated variation of linewidth for a = 0.1, r = 1,5,8. The dotted line shows the calculated Iinewidth for a = 0.75, r = 0.28.

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

371

solid lines in fig. 35a for several values of K. Agreement between theory and experiment is reasonable for K 0.1; this value of K is much smaller than the value (K = 0.73) given by eq. (4.80) assuming RSJ parameter values and Im Z~= 0, Re .Z~= RN. The noise parameter y (which for the case of small K is given by eq. (4.70)) for the data shown was about 0.15; and thus justifies the comparison of the theoretical results for no noise with the experiment. The linewidth of the radiation, shown in fig. 35b, also varied periodically with the external flux. The theoretical linewidth (for small noise) is easily obtained by linearizing eq. (4.77), and is given by ~v~S~~{1+

(5.10)

(i~:~~~)2}

where ifr is determined from eq. (4.79). The linewidth calculated using eq. (5.10) is shown in fig. 35b for K 0.1 and several values of r an Re Y/Im Y. Reasonable agreement with the experiment is obtained for r = 6; this value of r is much larger than that obtained assuming Re Z1 = 2RN, Im Z, = 0 (r = 0.28). We conclude that the qualitative flux dependence of the Josephson radiation linewidth and power from an unshunted SQUID agree with the model discussed in section 4. However, the junction parameters K and r required to obtain precise quantitative agreement between experiment and theory generally lie outside of plausible bounds. Since these data were obtained several years ago [30](before the theory was worked out) using nonideal samples, it would clearly be desirable to have further measurements to test the details of the theory. We now discuss the shunted dc SQUID, shown schematically in fig. 15. The shunted dc SQUID was similar in construction to the unshunted dc SQUID. The planar microbridges were 0.3 p. x 0.3 p. in size, and had a thickness of 1200 A. The bridges were separated by 12 p.. The shunt was a thin film gold resistor (R~= 50 mEl). The inductance of the loop formed by this shunt and the junctions was about 20 pH. The magnetic inductance of the superconducting SQUID loop was 0.5 nH. A shunted two junction cell without a superconducting ioop was also fabricated on the same substrate next to the shunted SQUID and served as a reference for evaluating the performance of the shunted SQUID. Similar to the dc SQUID, the power and the linewidth varied periodically with the external flux, 4’e. Fig. 36 shows the dependence of the power and the linewidth on q~(at 4.5 0Hz, I~ = 190 p.A). In marked contrast to the dc SQUID (fig. 35), the dependence of the power on 4’e is decreased considerably. The maximum power and the minimum linewidth of the radiation from the shunted SQUID were close to those obtained for the two junction cell (without the SQUID loop) biased in parallel in the middle of the locking range. The weak dependence of the quantities on ~ demonstrates that the additional high-frequency coupling due to the resistive-inductive shunt is effective in stabilizing the phase around ifr = iT independent of flux. The power and the linewidth according to the theory are again given by eq. (4.22) and eq. (5.10) (neglecting the noise contribution due to the shunt resistor). The calculated values of the power and linewidth are shown for several values of K and r in fig. 36; agreement with the experimental results are reasonable for K 2.5, r 0.7. These values can be compared to those expected using the parameters determined from the two junction cell without the SQUID loop for the same critical current and locking voltage. The measured locking strength is = 6 p.A which, using eq. (4.80), implies K = 4.7. Assuming Im Z 1 = 0 and Re Z, = RN gives r = 0.7. The noise parameter y, defined in this case by eq. (4.37) was about 0.4; hence the results of the small noise approximation discussed above may be somewhat modified. In view of this, the quantitative agreement between the experimental results and the theory is quite reasonable. This improvement over the agreement obtained for the unshunted SQUID is ‘L

372

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

130

!.

65~~

External Flux

External Flux

Fig. 36. Linewidth and power vs. ,~, for the shunted dc SQUID. The experimental results are shown by the solid circles. (a) Power, (b) Linewidth. The calculated variation of power is shown in (a) by the solid lines for a = 1.5, 2.5, 3 and 4. The calculated linewidth in (b) is shown for a = 2.5, = 0.5, 0.7 and 0.9.

undoubtedly due in part to improved sample quality as well as to a greater use of measured (vs. RSJ) parameters (e.g. IL) in the calculations. As noted above, however, more complete measurements would clearly be desirable. 6. One-dimensional multi-junction arrays — General analysis 6.1. General equations

The general equivalent circuit of a multi-junction array is shown in fig. 37. According to the discussion of section 3, the coupling circuit can be assumed to be linear and hence its response can be described by some linear operator: Iek(t) an Ik(t) =

~

[V1(t), V2(t),

which relates the currents

‘k

..

.

,

Vk(t), . .

.

,

V~(t)] ,

(6.1)

flowing through the kth junction to the voltages across the junctions.

Our approach to the analysis of multi-junction arrays will remain the same for the two-junction cells (section 4): first, we will consider the mutual phase locking, which is primarily due to the high-frequency

coupling mechanism and then we will analyze the effect of the low-frequency conductance of the coupling circuit. For the description of the Josephson junctions, we will again use eqs. (3.19) and (3.43) obtained using the perturbation theory.

For the pure high-frequency coupling, the only component of the junction current ‘k which we need to make eq. (3.43) self-consistent, is the first harmonic of the Josephson oscillations frequency ut. We find this harmonic from eqs. (3.19) and (6.1). In doing this, we should remember that, in principle, each

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

(a)

I

~(Z~l

373

(b)

I ‘f’

I

e

VN

(Z

)N

Fig. 37. (a) General scheme of the coupling in a multi-junction Josephson array, and (b) schematic of a simple rule to find the coefficients

Y

55..

junction of an array can have its own instantaneous frequency of the Josephson oscillation. Thus, the high-frequency current and voltage should be represented as sums over the individual junctions: °k

6k=)exp(jOk”),

1k

=

=

k”=1

k”=i

(6.2)

Vk(

where the complex amplitudes (Ok”) and Vk (Ok”) characterize the current and voltage at junction k due to oscillations in junction k”. Substituting eq. (6.2) into eq. (6.1) and using the linearity of the operator we obtain the connection between the amplitudes: ‘k

2~C,

1k(Ok”) =

where

(6.3)

Y~.Vk’(Ok=),

Y~k’(Ok=)are

the coefficients describing the coupling circuit:

Y~k’exp(J Ok”) =

~

[9,9,. .

,

exp(jO~..)~. ..,

If the Josephson oscillation frequencies

Ok

o].

(6.4)

of the junctions are equal or almost equal, (6.5)

and the coupling circuit is relatively wideband, the coefficients Y~k.do not depend on k”. We will assume this to be the case.

AK. Jam eta!., Mutual phase-locking in Josephson junction arrays

374

(a)

_______

(b)

I T

( 2Z~+z 2 ‘1 \

k=k’

0/

2 2

YkKI— ‘I’

Z5

~

__________

1=

hk

NZ1 +Ze

T

N

—

t

k r1Z~

~M

Ze

Ykk Z0

2T

~-~--

~0=~

(

~

Z1

\I~kI

I

kk

(ze+~/z:+4zize)

(C) (Z÷Z2÷2Z0)

kk

~‘kk(Z+Z2~o+Z,

I

)I k-k’~

k

z0=~(~,’~z+Z22+4zz+z2~Z+z2)

Fig. 38. Some examples of uniform one-dimensional arrays with different interaction ranges K: (a) long-range interaction, (b) and (c) short and medium range interaction, with K dependent on the ratio of the external impedances to Z1.

Now, substituting eqs. (6.2), (6.3) into eq. (3.44) written for the kth junction, we obtain for the first harmonic of the currents:

1k Re~ek.Ykk.exp(jOk.) =

(6.6)

where Ykk’ are the coupling coefficients including the Josephson junction impedance. They are connected with coefficients Y~k’by the relation: =

k”=1

Ykk.’ x [(ZI)kY~k”+ Ôk’k”]

(6.7)

which is more easily presented in a matrix form: )IYH = )Iy1 Xlii + z1Yeir.l,

(6.8)

where 1Z111 is the diagonal matrix of the junction’s high-frequency impedance. Equation (6.6) gives a simple physical meaning to the coefficients Ykk’ and it also provides a more convenient way to determine these coefficients compared to eqs. (6.7), (6.8). Let us replace all the Josephson junctions with their high-frequency impedances Z1, except for the k’th junction, which is

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

375

replaced by its impedance together with a unit-amplitude e.m.f. of frequency w. In this case, according to eq. (6.6), the complex amplitude of the current through the kth junction is just equal to Ykk’. Using this rule, the factors Ykk’ can be easily found for any coupling circuit of interest, using the well-known methods of linear circuit theory. Fig. 38 shows some of the simplest arrays and their corresponding factors Ykk’. Now, we can substitute eq. (6.6) into eq. (3.43) and obtain a self-consistent set of equations for the junction phases °k ‘k

Ok =

w~(Ik)+ aeRd~2iT/~o) Re ~

Ykk. exp~j(Ok’— Ok)] +

~7k(t)

(6.9) (6.10)

Equations (6.9) are valid for any kind of array (assuming only that the conductivities Ykk’ are not too large, eq. (3.46), so that the perturbation theory is valid). In the remaining part of this section, we will discuss the simplest (and seemingly most important) array: one-dimensional arrays. Some possible configurations for these arrays are shown in fig. 38. 6.2. High-frequency coupling

In contrast with the case of two-junction arrays, it is not possible to obtain analytical expressions for the characteristics of an N-junction array with arbitrary coupling coefficients Ykk.. Let us remember, however, that we are most interested not in an arbitrary solution of the set of equations (6.9), but rather in a particular “uniform” solution =

0 = tut + const.,

(6.11)

which describes (in the absence of fluctuations) the result of “perfect” phase locking of all the junctions of the array with a common phase of oscillation. We saw that in a two-junction cell such a solution exists in the middle of a locking range, and could always be achieved by the proper choice of dc bias currents. Thus, we will start our discussions of multi-junction arrays with an analysis of the conditions under which the uniform solution (6.11) exists and is stable. Equation (6.9) shows that this uniform phase solution only exists if the combination (L)~(Ik)+ aeRd

(~) Re

~

Ykk. = cv = const.

(6.12)

does not depend on the junction number, k. Then, one can say that the uniform solution (6.11) again exists in the middle of the locking range now in the multi-dimensional space of the array parameters. However, for large arrays individually tuning each junction by trimming its dc bias current seems impractical, thus we should analyze condition (6.12) more carefully. This condition can be satisfied, first ofall, in infinite uniform arrays, consisting of identical junctions (w~= cv” = const.). In such an array, the coupling coefficients Ykk’ can only be dependent on the “distance” n between the interacting junctions: Ykk’Y(n),

nk—k’,

(6.13)

A.K. Jam et aI., Mutual phase-locking in Josephson junction arrays

376

where we suppose that the junctions are numbered in correspondence with their position in an array (fig. 37). For such arrays, condition (6.12) is satisfied, since Ykk’ =

Y(n) = Y const.

~

(6.14)

If, however, a uniform array has a finite number of junctions, condition (6.12) would not be satisfied near the ends of the array. Two possible ways to maintain ~ Y~,.= const. and thus provide a uniform solution (6.11) in a finite array are as follows: k 1) One can change the parameters of junctions (or their dc bias currents) near the end of the array in a self-consistent way in accordance with eq. (6.12). This approach is rather complicated from a technical point of view; thus it would be desirable to avoid it. 2) Alternatively, the ends of a one-dimensional array can be connected, thus forming a uniform ioop array. In this case, eq. (6.13) is valid, so that condition (6.12) is satisfied. The circuit shown in fig. 38a is a simple example of a loop array; the particular location of the “external” impedance Z~does not affect the uniformity as long as the array’s length is much less than the radiation wavelength as we have assumed (see section 8). The concept of a uniform loop array has proved to be so fruitful from both fundamental and applied points of view that nearly all recent research work has been concentrated on the study of this type of multi-junction array. Thus we will restrict our analysis to these uniform (or nearly uniform) loop arrays in the remaining part of section 6. To analyze the stability of the uniform solution (6.11), we should linearize eq. (6.9) with respect to 0 resulting from small perturbations deviations 60k = —

(6.15)

~k71k+6Wk

due to both junction parameter variations (6.16)

~

and junction fluctuations cvrRd

77k.

The resulting set of linear equations

(~-~) ~ Im Ykk’ (60k 0

—

60k’) +

~k

(6.17)

k’=l

can be solved exactly for a uniform ioop array, where eq. (6.13) is valid. To do this, we expand 60k, Im Ykk’ and ~k in a spatial Fourier series: 60k

~ 0(q)~

~k = ~

~(q)~ik~

Im Y(n) = ~ Im Y(q) e~

(6.18)

where q are the wave numbers m integer,

O~m
(6.19)

AK. Jam et al., Mutual phase-locking in Josephson junction arrays

377

Substitution of eq. (6.18) into eq. (6.17) yields O(q)

—

aeRd

(~)

N[Im Y(0) Im Y(q)] 0(q) = ~(q).

To analyze the stability of the uniform solution (6.11), we put solution 0(q)

(6.20)

—

exp[A(q)tJ

e(q)=

0 and look for the partial

(6.21)

of the resulting homogeneous equation. Solution (6.11) is stable, when all A (q) have negative or vanishing real parts, i.e., when Im Y(0) Im Y(q) ~ 0, —

for all

q.

(6.22)

All the wideband coupling circuits which seem realistic at present satisfy two conditions: 1) They can be represented as consisting of only resistances and inductances, so that Im Ykk <0 and Im Y(q)<0. 2) The coupling coefficient modulus I Y(n )l remains constant or falls off with increasing distance n between interesting junctions. In this case, the Fourier coefficients decrease gradually with increasing q. For these arrays, condition (6.22) is satisfied and the uniform solution (6.11) is stable. Before

proceeding further, let us get some idea of what happens with an array if condition (6.22) is not satisfied (for example, if the coupling is capacitive). In this case, eq. (6.11) without perturbations has a stable solution =

tut +

(— 1)”ir + const.

(6.23)

Solution (6.23) describes an array where each junction oscillates 180°out of phase with its nearest neighbors. Thus, one can say that inductive coupling acts as a mutual attraction of the junction oscillation phases and leads to the in-phase regime of eq. (6.11), while capacitive coupling acts as a mutual repulsion of the phases and leads to the anti-phase regime of eq. (6.23). Note, however, that if the polarity of dc bias alternates along the array with each junction, it is solution (6.23) which is stable for inductive coupling. In complete analogy with the two-junction cell (see eq. (4.20)) this leads to the conclusion that inductive coupling produces in-phase addition of the rf voltages along the array independent of the bias polarity. 6.3. Phase stability and the junction interaction range

Coming back to formula (6.20), we note that, provided the uniform solution is stable, this equation gives us the (small) phase deviations from the uniform solution due to parameter variations 6cv~and noise ilk. In particular, the equation for q = 0:

~ m~(°)

(6.24)

A.K. fain et a!., Mutual phase-locking in Josephson junction arrays

378

gives us the time evolution of the net low-frequency voltage across the array, since according to eq. (6.18) 0(0)=~~6Ok,

O(0)=~~~.ôVk=~6V,

(6.25)

and we obtain N

~

A

U

(6.26)

i~w =—~-.

This important result for the coherent oscillation linewidth is an evident generalization of eq. (4.32) for the two-junction cell, and leads to a similar explanation: mutual phase locking couples the junctions as if they were connected in parallel for low-frequency currents. This connection produces an N-fold decrease of the differential resistance Rd and an N-fold increase of the noise-current spectral density SF, thus decreasing the oscillation linewidth which is always proportional to R~SF. Equations (6.20) with q 0 show that the amplitude of the phase deviations depends drastically on what can be called the interaction range K. This quantity can be defined as the “distance” n = k k’ between two junctions, at which the coupling coefficients Im Y(n) falls off (fig. 39a). Let us analyze the two opposite limits. —

(a)

ImY

(b) ImY

K << N

(c)

ImY

N 2

0

(d)

ImY

Nn

N

2

2

0

No 2

Fig. 39. Dependence of the coupling coefficient Y on the distance n = k—k’ between the interacting Josephson junctions (schematically) (a) infinite one-dimensional array, ~,—d)uniform loop arrays with (b) short-range, (c) medium-range and (d) long.range interactions.

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

379

In an array with K N, each junction is coupled with all the other junctions of the array (fig. 39d); an example of such an array with a long-range interaction is given by fig. 38a, where K is formally infinite, so that from eq. (6.14) ~‘

Im Ykk’ =

Im V = const.

(6.27)

For such arrays, all Fourier coefficients Im Y(q) with q Im Y(0) =

~ Im Ykk’ =

0 are much smaller than the zeroth coefficient

Im Y

(6.28)

and eq. (6.20) gives the following simple result for the time-independent phase deviations: 0(q) = asRd(21r/~o)(—ImY)’

q

0.

(6.29)

The denominator does not depend on q in this limit, so that we can perform the inverse Fourier transform and obtain ~

°kO2~

I

U

~1T cvL=2arRd-~-ImY.

UWk

(6.30)

‘~~0

This formula shows that the deviation of a junction’s phase from the uniform solution is due to deviations in the parameters of that junction only from the average value cv” given by eq. (6.16). Moreover, eq. (6.30) gives us the locking strength, which can be defined as the maximum variation of the junction parameters such that the phase difference remains within reasonable limits: I 0, 0~s 1. In fact, since eq. (6.30) is only valid for small deviations, this approach only determines the locking strength to within a factor of the order of unity (one can show that for the case K ~ N, this factor is exactly unity). As is clear from eq. (6.30) the locking strength is equal to 1wJ21 or ~Ii.. in current units with the expressions for WL and ‘L being the same as those for the two-junction cell in eqs. (4.9) and (4.34). Note that the locking strength for the two-junction cell is also equal wJ2 rather than if it is defined as w~—w”,rather than w~—w~. Thus, the locking strength for a loop array with a long-range interaction is just equal to that of the two-junction cell, provided that lIm YJ is the same. Since Y is the admittance of the loop per junction one might expect it to be roughly independent of array size. For example, for the array shown in fig. 38a, —

0L,

Y = [Z,+ ZJN]’ If Ze is

selected to maximize

Z”

=

(6.31)

-

‘L as in eq.

(4.16) we have, assuming Z~ = RN, (6.32)

jNZ~.

Thus V is explicitly independent of the number of junctions as is

‘L-

380

A.K. lain et al., Mutual phase-locking in Josephson junction arrays

In order to calculate the stability of the locking in the presence of fluctuations, that is, to determine the equivalent of y in eqs. (4.36)—(4.38) for the two-junction cell, we begin with the Fourier series for 60k in eq. (6.18). From eq. (6.20) we see that fluctuations ~(q) with a frequency cv~—WL~~ cv” cause a response in 0(q) given by: O(q)=.

q

,

JWWL12

0.

(6.33)

Obtaining i~(q) from the inverse Fourier transform:

~(q)

(6.34)

~7k ~

gives

=

q

~

1

k’

JWWL12

)

‘h’ ~i~(k_k’)

(6.35)

or

=

jw —w

5J2

(i

[flk

—

+

N

(6.36)

k’~kOk’].

Calculating the mean square phase deviation (~0~) as in section 4, we obtain for the effective noise parameter y

=

N-1~w” N

_____

(6.37)

—

(UL

or 2

N-i

Rd

IT

~ S~(0) ~

,.

(6.38)

.

IN

This formula shows the gradual transition from the value for the two-junction cell to half that value as N ~. The difference results from the fact that for N 1, the noise of a junction acts mainly on the junction itself, while in the arrays with a small number of junctions, its action upon the neighbors is also significant. In the opposite limit, K 4 lV~each junction interacts with only a few neighboring junctions, thus we have an array with a short-range interaction. Possible examples of such arrays are shown in figs. 38b and 38c. It follows from the inverse Fourier transform, that —~

Im Y(q) ~ ~ Im Ykk’.

~‘

(6.39)

A.K. fain eta!., Mutua! phase-locking in Josephson junction arrays

381

In such arrays the first few Fourier coefficients are nearly equal: Im Y(q)= Im Y(0)[1— Cq2~,

C-~K24N2,

for q—~2IT/N42IT.

(6.40)

Consequently, the first Fourier coefficients 0(q) which give the long-range phase deviations would be extremely large in these arrays, 0( )=

O(’~ (641) q (ULN[1 Im Y(q)IIm V(0)] alL \~Kq) with the first coefficient (q = 2ir/N) being especially large. This implies that arrays with short-range interactions fail to stabilize the junction phases against a long-range random walk. In particular, the first spatial harmonics of the random perturbations (due to both parameter variations and the junction noise) induce very large sinusoidal phase variations ~

—

°k

60k -~ a sin[~ (k

—

ko)]~

a

-~~-

(6.42)

(N)2

where ; is the amplitude of the first harmonic of the perturbation 6co”, and k

0 is some (random)

junction number. As a result, both the effective locking rangeinand critical of noise in arrays 2 less than arrays withintensity long-range interactions. In with othershort-range words, the interactions are a factor of (N/2irK) stability of the coherent regime with respect to fluctuations is much worse for the short-range junction interaction, while for the long-range interaction, it is of the same order as (or even larger than) that in two-junction cells.

6.4.

Influence of the finite low-frequency conductivity

Now we should discuss the influence of the dc-bias circuitry on mutual phase locking. Fig. 40 shows some of the possibilities for providing all junctions in a one-dimensional array with equal (or almost equal) dc voltages; one can distinguish “series” biasing (figs. 40a—c) from “parallel” biasing (fig. 40d). Another classification can be made according to the effective dc circuit resistance R~as seen by a junction: “current biasing” for Re ~‘ Rd and “voltage biasing” for Re 4 Rd. Note that each type of dc biasing can, in principle, be used together with any type of high-frequency coupling circuits (fig. 38). In practical arrays, however, it is simpler to employ the same elements (resistors and inductances) to provide both dc biasing and high-frequency coupling; for example, an array can have high-frequency equivalent circuit shown in fig. 38a and low-frequency equivalent circuit shown in fig. 40b, where the same circuit has a high-frequency impedance Ze and a low-frequency resistance Re. If the biasing circuit has non-vanishing low-frequency conductivity, one can use the same methods of analysis as in section 4.2. Namely, the general set of equations (6.9) should be complimented with the following set of equations for the low-frequency currents (Ie)k, similar to eq. (4.41): r

N

~1e)klk

—r

T’~T

~

kk’ Vkr~k, k’-’ 1

Vk~Uk, IT

382

AK. Jam et a!., Mutual phase-locking in Josephson junction arrays

j~

(a)

(b)

Re

I_ Cc)

L

Cd)

Re

N~X~

Fig. 40. Various types of dc-bias circuits for multi-junctions arrays: (a) series current biasing, (b) the same with additional common shunting, (c) series voltage biasing, (d) parallel biasing. Possible additional high-frequency coupling circuits are not shown.

where the constants and Gkk’ describe a particular dc biasing circuit. Diagonal elements of the conductivity matrix Gkk’ determine the spectral densities of the bias circuit fluctuation currents Fk: ‘k

SF,,

=

21~BTGkk..

(6.44)

Note, however, that these fluctuations can be partly correlated. It is not possible, of course, to analyze the influence of the low-frequency circuit on the array dynamics in an arbitrary case. Thus, we will give an analysis for two particular cases of resistive dc biasing which seem most practical. The first possibility is shown in fig. 40b where a single resistance Re shunts the array. An alternative method would be to shunt each junction with a small resistance Re as in fig. 40c.

A.K. fain et al., Mutual phase-locking in Josephson junction

arrays

383

Let us compare the main characteristics of the coherent regime for these two cases. In both cases, we will assume that we have used the most effective high-frequency interaction in the array, i.e. the long range interaction with Im Ykk = Im YIN. For the array shown in fig. 40b, and Gkk’ do not depend on the junction numbers and are given by = I, Gkk’ = 1/Re. Thus eq. (6.9) reduces to ‘k

Ok

(~)

~

,~,

r=Re Y/IIm YI, (6.45)

where the spectral density of the shunt noise Oe ~S (6.46)

~

where =2 (2ITRPO)2kBTRe. For the case where each junction is separately shunted as in fig. 40c 4 obtains Ok

=

j~ O~

w~(I)—

—~

Re

~

(r + j) exp[j(Ok.— Ok)] + Ok +

=

(~)

(fle)k

I, G,~’= t5kk’IRC, and one

(6.47)

where all random functions (fle)k have the same spectral densities as Oe but are independent. Applying the methods developed in sections 6.2 and 6.3, to eqs. (6.45) and (6.47), we obtain the following results: 1) Finite low-frequency conductivities do not change the locking range expressed in deviations 61 of currents flowing through the junctions. In particular, this means that the junction critical current tolerance ~ doesnot changewith the array shunting (see section 4.2).This conclusion remainsvalid for any type of shuntwith finite resistance. Infact, terms like Gkk’, Vk’, eq. (6.43), are all proportional to and can change only the dynamics of the phases, but not the range of the static solution, describing the mutual phase locking. 2) The linewidth of the coherent oscillations (in the middle of the locking range) is equal to ‘L,

°k,

~ CL) cU(t~dI1~e)21~1)e N[1+NRdRe]2 —

648 (.

for the common shunting (fig. 40b) and to c=

L~W”+

(RdRe? l~We

(649)

cv N[1+R~JR~J2 for the “individual” shunting (fig. 40c). Note that in the first case, eq. (6.48), the linewidth is

AK.

384

fain et a!., Mutual phase-locking in Josephson junction arrays

proportional to N2 for small R~, =

~ale/N2,

for Re 4 NRd,

(6.50)

while in the second case, eq. (6.49), Aw’~decreases only as N1. This result should be compared with eq. (4.51) for the two-junction cell. The common shunt does not change the effective intensity of fluctuations y (6.38), while individual shunting does, giving: (N-i)

IT2

Rd ~

5]

(6.51)

N(1+g)~oIL

The second term in the brackets is usually small (see the discussion of eq. (4.48)), so that shunting increases the phase locking stability with respect to fluctuations. Thus, the dc bias circuit shown in fig. 40b has the advantage of strongly decreasing the coherent oscillation linewidth, while that shown in fig. 40c is better in the sense of increasing the fluctuation stability of the coherent mode. 6.5. Superconducting low-frequency coupling

A superconducting dc bias circuit (see fig. 40d for example) can provide coupling of the junctions in an array even in the absence of a high-frequency interaction. Let us again (see section 4.3) first analyze this low-frequency coupling alone assuming there is no high-frequency interaction. For an arbitrary superconducting circuit, we can write the following linear equation: ~

(6.52) ITk,l

kk’

which is an evident generalization of eq. (4.65). For example, in the circuit shown in fig. 40d,

~

=

—

2/L,

k’=k,

i/L,

k’ = k ±1,

(6.53)

k’~k,k±1.

0,

The constants ‘k can depend on the fluxes of the external magnetic field applied to the loops of the circuit. Substitution of eq. (6.52) into the general eq. (6.9) with Ykk’ = 0, i.e. no high-frequency interaction, gives a set of equations OkCd(Ik)~Rd

~ k’=l

~

(6.54) kk~

Limiting ourselves again to the case of loop (or infinite) uniform arrays: Lkk’=L(n),

n=k—k’,

(6.55)

A.K.

fain et aL, Mutual phase-locking in Josephson junction arrays

385

we obtain from eq. (6.54) that the uniform solution (6.11) formally exists and is stable in the absence of fluctuations and parameter variations. The latter factor, however, can strongly affect the low-frequency coupling, if the spacial range K of this coupling is small (K 4 N). For the Fourier coefficients 0(q) of the phase deviations 8O~from the uniform solution, eq. (6.54) gives (6.56) Equation (6.56) with q

0 coincides with that for high-frequency locking, eq. (6.24), and shows that the oscillation linewidth is. again given by eq. (6.26). Equations (6.56) with q = 1 show that for arrays with short-range—low-frequency interactions, the first harmonic of the phase deviation has the maximum amplitude. For example, for the case (6.53) where the interaction range is very small (K = 1), the Fourier-components of inverse inductance are equal =

(6.57)

For the largest (first) harmonic of the phase variations due to a random distribution of parameters, we get L (N)2 a~,

0(1)

(6.58)

so the amplitude ar,. of the first spatial harmonics of the parameter variation should be extremely small for large N. We obtain a similar result for the effective noise intensity: ~cv°LfN\2 =

(6.59)

Rd

which increases in proportion to N2. Combining the results (6.58) and (6.59), we find that for typical oscillation frequencies (cv w~)the number N of junctions in a parallel array should not exceed —V

Nmax

2ir x min[(Lt/L)L’2, (LJL)v2

c51/2]

(6.60)

if one wants the phase deviations to be in reasonably small (ö ~ 1). In eq. (6.60), L~stands for the internal inductance of a Josephson junction (6.61) and S = Bcv”/w” for the relative variation of the junction parameters. Note that the estimate (6.60) is only valid if the optimum “uniform’s or “lateral” biasing of the array is used [104];for the more common single-edge biasing the factor S would be absent in the estimate, so that the maximum possible number of junctions would be even less.

AK. lain et a!., Mutual phase-locking in Josephson junction arrays

386

A simple analysis shows that the estimate (6.60) is also valid for parallel arrays with small cell inductances, LILc s 1 (these arrays cannot be analyzed with perturbation theory but can be treated as a quasiuniform long Josephson junctions). For a typical array with L~-~3 pH (I~ 0.1 mA), L -~ 10 pH and S -= 10%, eq. (5.60) shows that N is limited by junction parameter variations at a very low value: Nmax

10.

(6.62)

Thus, in contrast with long-range high-frequency interactions, the short-range low-frequency coupling turns out to be unable to provide a reliable coherent regime in a long one-dimensional array for realistic parameter values. However, dc biasing with a superconducting loop of the type shown in fig. 40d can be practically useful together with long-range high-frequency coupling. Carrying out the usual calculations with both low-frequency and high-frequency interactions taken into account, we come to a conclusion, similar to the one obtained in section 4 for the two-junction cell: if the loop inductance are large enough, K ~ 1, eq. (4.80), the superconducting bias circuit does not change the characteristics of the array except that the locking range becomes formally infinite. Thus, such circuits can help to provide mutual phase-locking in the case when a large random scattering of the junction parameters is the main problem.

7.

Design and performance of practical multi-junction one-dimensional arrays

7.1. Basic design considerations

We have seen clearly in the previous sections that in order to obtain coherent operation in an array, it is necessary to have strong phase and voltage locking of the junctions against fluctuations and the ever present scatter in junction parameters. The initial evidence for coherence in large linear arrays was obtained by Palmer and Mercereau [14]who reported an N2 power increase in the MHz range in an array of N junctions. The coupling in their arrays was, however, presumably due to the weak short-range quasi-particle interaction (section 2.3) resulting in weak locking at low frequencies and the disappearance of coherence at microwave and higher frequencies, where arrays would have the most potential for technical applications. Our own initial attempts [30]to obtain coherence in multi-junction arrays using short-range electromagnetic coupling also had only limited success. As we have seen in section 6.3 short-range coupling is inherently unstable against long-range phase deviations, thus successful arrays require a long-range interaction, i.e. K fig. 39. This section will be devoted to the discussion of recent experiments to develop practical methods for producing this strong long-range coupling and to examine in more detail the performance of the resulting arrays in light of the analysis of section 6. Principal considerations in the design of the arrays are the low-frequency coupling used to provide the bias currents to the junctions (see e.g. fig. 40) and the high-frequency coupling used to achieve phase locking as shown in fig. 38. In order to give a clearer picture of the constraints which the various technical and fabrication problems have imposed on the design of the array, we will present this discussion from a more historical perspective reversing the order of section 6 to begin our discussion with the design of the low-frequency biasing circuit. The characteristics of the low-frequency coupling were an important factor in determining the design~of the array needed to provide the required high-frequency interaction. —~ ~,

AK. Jam 7.2. Low-frequency

eta!., Mutual phase-locking in Josephson junction arrays

387

bias circuits

Since, for large arrays, it is impractical to have bias currents which can be individually varied for each junction as was done with the two-junction cell, a single bias current must be applied to the array in such a way that the bias I~thru the ith junction gives cv~(i~) within the locking range for all junctions. The simplest bias arrangement, shown in figs. 40a, b is for all I~= i.e. to pass the bias current through the junctions in series. As we saw (eq. (4.18)) the theoretical maximum variation in I~which can be tolerated with this type of biasing is about 30% - In practice, data obtained for planar microbridges from the 2 junction cells indicate that, due to the bridge’s non-RSJ properties, their maximum locking range is about 10%. Since their I~variation is about 20% it is clear that a series biased array of these bridges would not lock. Since planar bridges are by far the easiest junctions to fabricate, we nevertheless decided to continue using them for this work but to use a biasing scheme such as that in fig. 40d which compensates for variation in junction parameters. A possible layout for such a circuit is shown in fig. 41. This is equivalent, at low frequencies, to the parallel connection of the junctions shown in fig. 40d. If the connecting leads are superconductors, the bias current must divide up such that the average voltage is the same across each junction (see section 6.5). The practical realization of such a circuit in which some degree of coherence was achieved was reported by Lindelof et al. [28]. The circuit, as shown in fig. 41a, has a number of serious defects. Since the rf voltages add in series ‘BIAS,

‘B

(a)

_____

‘B

1~

“I, [X

‘~‘‘Bt

K ~JX~X~

[KR

‘I’ ‘B

-~-

K

K

+

+‘B

K

K

(b)

X~

)(

K

K

< ‘B

(c)

Fig. 41. Circuits for providing bias current I~to coupled junctions in parallel. (a) Common SQUID loops tending to short out Josephson current. (b) Separate SQUID loop, presenting high impedance to Josephson currents. (c) Bias current supplied through coaxial cable instead of separate leads.

388

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

along the array the bias circuit will tend to short out the rf. While a sufficiently large inductance in the bias leads could prevent this, one must remember that all of the inductances of the connecting leads

tend to connect in parallel. Thus as the array size and impedance increases the shunt impedance of the bias leads drops rapidly, making this circuit useless for large arrays. A more complex version of the parallel bias circuit, which we have developed [31]and used in the arrays discussed here, is shown in fig. 41b. In this circuit each pair of junctions is shunted by a distinct SQUID loop. While the circuit is still equivalent to that in fig. 40d at low frequencies, the rf now sees all of the SQUID inductances in series, thus the total shunt impedance of the SQUID loops across the array increases along with the array impedance as the array’s size increases. To ensure this series addition of SQUID inductances the mutual inductance between the SQUIDs must be small. An example of the bias circuit above can be seen in fig. 42, which shows a micrograph of a 10 bridge array. The schematic of the array is shown in fig. 42b. For this array, the magnetic inductance of the SQUID loops L,, is 0.5 nH and the mutual inductance between ioops about 10 pH. Since these loops are made of indium films, as are the junctions, they also have a temperature dependence kinetic inductance which becomes significant near T~of the junctions. In addition, the narrowness of the 1200 A thick films comprising the loops gives them a critical current which is only several times that of a junction, thus the bias currents must be fed to each loop separately from a common bus. The connections to these current busses are made through resistors to reduce the rf shunting discussed above. In the absence of high-frequency coupling the essentially random flux through each SQUID loop will determine the relative phase of oscillation of the associated junctions; eq. (6.58) shows that even a small degree of randomness will rapidly destroy coherence. However as we saw in section 6, strong high-frequency coupling (large K, eq. (4.80)) overcomes this randomness in phase just as it overcomes variations in co~without SQUID loops; thus the effect of the phase variation is roughly equivalent to a variation of I~in uncoupled junctions of 40/L,, which is 4 p.A for this array. Since the array operates with an I~of about 1 mA, the effective parameter scatter to be overcome by high-frequency coupling is now less than 1%, which is well within the 10% locking strength observed in the two-junction cell. Since the locking strength decreases as one attempts to operate the array over a wider frequency range, this “excess” locking strength translates directly into a broader band of coherent operation for the array (see section 7.3). In principle L. could be made large enough that 4s0/L~would be less than the thermal fluctuation currents; thus parameter scatter would be negligible. In practice, when the SQUID length approaches a wavelength, resonances would be important as would the coupling of the loops to the ground plane. While these problems could be solved with proper microwave design we have chosen to ignore them for now and limit the SQUID length to 600 p.m which will permit at least a 20 GHz operating frequency. The design of the 10 junction array required that bias current be supplied individually to the SQUIDs both due to the low I~of the leads and in order to conduct the tests of the high-frequency coupling circuits to be discussed below. The disadvantage of this arrangement, however, is that it is very difficult to avoid discontinuities in the microwave impedance due to the bias leads. The close spacing of the leads and the broad range of operating frequencies make proper filtering of these leads quite complex. In order to prevent this problem and obtain a smooth microwave response a different bias scheme was developed, illustrated in fig. 41c, in which the array is supplied with bias current through the microstrip thus avoiding any bias leads to the array which could perturb the microwave impedance. Note that this bias arrangement requires an odd number of junctions to avoid having the SQUID loops short out the bias current. This scheme, which was tested on a 99 junction array, has the disadvantage that the SQUID loops near the ends must carry a current equal to the sum of all of the junction

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

389

Fig. 42. Micrograph (a) of ten junction coherent array and (b) schematic of array showing two sections, A and B, which can be biased at separate voltages. The high-frequency coupling circuit is not shown in (b).

390

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

currents and thus have an I~of almost 100 mA for the 99 junction array. To achieve this these loops were made of 7500 A thick PbIn alloy. The fabrication is discussed in detail in appendix C. 7.3. High-frequency coupling: voltage locking The most straightforward method of achieving the long range, inductive, high-frequency coupling which is required to make the junctions oscillate in phase and to reduce the dependence of the relative phases on the flux through the SQUID loops is shown schematically in fig. 38a, where the external impedance would be Z~= jLetu + Re and could be fabricated as a thin film feedback loop on the same chip as the array. In order to study the effects of this type of coupling a split 10 junctions array, fig. 42, was designed. It was divided into two sections, one of 4 junctions (section A) and one of 6 junctions (section B), by omitting the SQUID loop between the 6th and 7th junctions. Thus, the bias currents through the two sections could be varied independently. This permitted a direct measurement of the effects of the junction interactions by a measurement of the locking of the two sections analogous to the measurement on the two junction cell discussed in section 4. The high frequency coupling circuit for the 10 bridge array can be seen in fig. 42a. Here a lumped inductor is connected across the array, directly to one end and through a low-impedance stripline to the other. This stripline consists of the junctions and their pads separated by a 5000 A silicon monoxide dielectric from the 7000 A thick 20 p. wide PbIn film which forms the top leg of the microstnp. This design of the coupling inductor has the advantage of reducing the inductance of the pads between the junctions; this results in a larger rf voltage between the ends of the array since the inductive voltage drop due to the pads is decreased. The inductance of the thin film loop, L~,was chosen so that Im Y = Re Y at about 3 GHz. The junction resistance was 0.1 fl, the estimated value of Le was 60 pH and Re0.05fl. Figure 43 shows the voltage variation of the two sections as (fig. 42) is varied, keeping ‘B fixed. As with the two-junction cell, there is a range of current, 1~IA(= IL) for which the two sections lock to a ‘A

350

400

450 ‘A per

500

550

600

Junction (PA)

Fig. 43. “I—V” curves of ten junctions array showing phase locking between sections A and B for I,. near 500 ~i.Aper junction.

AK. lain et aL, Mutual phase-locking in Josephson junction arrays

391

common voltage. Similar arrays, but without the coupling inductor, showed no such locking between the two sections. The existence of locking thus demonstrates that the coupling inductor does, in fact, produce coherence in the array. To analyze the locking behavior of the two sections we will approximate each section by an equivalent single junction so that the results of sections 4 and 5 will be applicable. Since we are interested in the effects of an applied perturbation (i.e. the difference in bias currents) which is large compared to the scatter in junction parameters, the approximation will be adequate. For simplicity we will also assume that the two sections of the array are equal. With these approximations, the results of eq. (4.9) apply leading to the following expression for the locking strength of the two sections: L —

~Po Nww~ [(NR)2Le(~) + (L~w)2]2 [cv+ (w2 + w~)h/2](w2 + w~)~2

(7.1)

This is similar to eq. (4.9), where s and Re Z have been increased by a factor of N/2 = 5 from those for the two-junction cell. Note that the characteristic frequency w~remains unchanged from that of a single junction. ‘L is the maximum bias current variation per junction between the sections of the array for which the sections will remain locked. We neglect Re since it is small compared to NR, its only purpose being to break the superconducting path between the ends of the array so that a voltage difference can be established between sections A and B. We also see from eqs. (6.12) and (6.30) that the locking strength defined by eq. (7.1) is essentially the same as that which defines the parameter scatter or fluctuation intensity required to decouple a single junction from the coherent state. Thus the measurement of the locking strength between two halves of an array gives a very general indicatiOn of how well the junctions are coupled by their high-frequency

interaction. As with the two junction cell, eq. (7.1) exhibits two characteristic frequencies; one, tu,, is determined by the time constant of the coupling loop, LeINR, the other, cot, by the properties of the junctions (also wj2ir and ii, w,/21T). We see from eq. (7.1) that ‘L decreases rapidly outside of the range given by these frequencies. The limiting behavior at high- and low-frequency being:

fcv2,

~. L~1

tcv

—2 ,

w4min(cv,,w~) cv max(w,, w~) ~‘

(7.2)

with

‘L being roughly constant in between. Thus cv, and w~ serve as a conservative measure of the limits of the locking range. However locking will in general persist outside of this range until ‘L has decreased to roughly the level of the fluctuations or parameter scatter of the individual junctions. Rewriting eq. (7.1) in terms of ~ = wiw, and ~ = w/w~gives

1-

(

~,

~1+

~)\f~(1+ ~2)1/2[~~_c+ (1 + ~2)1/2])~

(7.3)

where 4.. is the locking strength measured in units of I~,the critical current of an individual junction.

The maximum value for 4~,which is obtained for ~ = 1 and ~ 0.78, is about 15%, thus the maximum permissible scatter in the junction parameters would be 30%, the same as was found for the two-junction cell, eq. (4.18). Note that eqs. (7.1)—(7.3) have assumed that the properties of the individual junctions are given by

392

AK. lain et a!., Mutual phase-locking in Josephson junction arrays

the RSJ model over a wide range of frequencies. Since the I—V curves for microbridges are not well described by the RSJ model (see section 5.3), we would not expect our measurements to be in quantitative agreement with these equations. One should also be cautious of the results obtained from applying perturbation theory for w ~ w~,since the limits of its validity are quite narrow, eq. (3.31), in

this region. The measured dependence of the locking strength between sections A and B on frequency is shown in fig. 44. Data are presented for two different temperatures with corresponding junction characteristic frequencies v~= 12 GHz and v~2 34 GHz. The characteristic frequency of the coupling loop is = 3 GI-Iz. These data are not in quantitative agreement with eq. (7.1); this is not surprising in view of the reservations discussed above. The main qualitative features of eq. (7.1) are, however, clearly observed. The maximum locking strength (measured in p.A) increases with i~ and occurs at a voltage which also increases with and is slightly less than v~.These trends continue as the temperature is

reduced until the junctions’ critical currents become high enough that the junctions become hysteretic. As we will see (section 8.2), it is likely that in arrays optimized for various applications the junctions would have low-inductance shunt resistors and would thus be well described by the RSJ model even for w ~- w~,where w~would be determined by the junction critical current and the resistance of the shunt. Thus it makes sense to examine the quantitative predictions of eqs. (7.1)—(7.3) for this successful locking scheme. In particular, since one of the superior features of Josephson arrays as radiation sources is their

wide potential tuning range, it is worth examining the frequency range over which the array can operate

30

I

—

x*,cx ‘C

‘C

-

I

‘C ‘C

‘~

Locking Frequency a’ (GHz) Fig. 44. Locking strength between sections A and B of ten junctions array as function of the lockingvoltage determined by the constant bias current ‘B. Solid dots are for a critical current per junction of I~ = 250 ~Aand crosses for I~ = 680 p.A.

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

393

if a certain minimum locking strength i~11’must be maintained. From eq. (7.3) we obtain cv4w,, c&i~

(7.4) w,(Uc

cv~’w,,w~.

Thus the ratio of the maximum operating frequency, cv>, to the minimum, w<, will be cv>

_____

w<

\/~j~Ifl

75

In this case cv, and cu~should be chosen such that (7.6)

w,w~V2w> w<.

Thus, for example, if = 0.01 is required, as is estimated for the 10 bridge arrays being discussed, coherent operation would ideally be expected over about two decades in frequency e.g. from 0.3 GHz to 30 GHz. Further if cv, w~the locking strength for such an array would be strongly peaked reaching the maximum value of 4. = 15% around 3 GHz. This behavior is illustrated in fig. 45a. An analysis of the opposite extreme of design where cv, and w~are widely separated and at the edges of the operating range, i.e. cv> max [cv,,cv~]and cv< = mm [cv,,we], gives nearly the same frequency range as eq. (7.5). This case is illustrated in fig. 45b with cv, ~ w~.As can be seen, the major difference between the case where cv, ~ w~and that where cv, cv~is that for the former 1L is roughly constant (at a low value) throughout the locking range in contrast to the strong maximum obtained for the latter case. ~

-.——

~

~

(a)

\\~.\

0.001

0.01

0.1

I

Frequency (cvc)

10

~

I

—.-...

,—_—_

__‘s’s

/\\

(b)

~

0.1

I

10

100

1000

Frequency (UC)

Fig. 45. Calculated locking strength 4. (solid line) vs. frequency in units of the junction characteristic frequency w~,showing separate contributions of the coupling loop parameters (short dash) and the junction parameters (long dashes). In (a) the coupling loop characteristic frequency w~ w~giving maximum locking strength and in (b) ~UI5’ w~giving maximum usable locking range above w~.

AK. fain et a!.. Mutual phase-locking in Josephson junction arrays

394

a

b

19111

Fig. 46. Micrograph of (a) 51-junction array and (b) blowup of feedback mop and resistor R~.

As will be discussed in section 8.2, it would appear that in arrays used for radiation generation the most desirable range of operation is for cv 5 cv~.With this criterion it can be seen from fig. 45 that the useful operating range (cv> w<)/cv~obtained for cv, cv~is approximately twice as many decades as obtained when w~ I’m. Thus the former design may in some cases be best in spite of its much lower maximum locking strength. —

~‘

7.4. Microwave generation by linear arrays

The radiation generated by an array at the Josephson frequency serves as a further probe of the degree of coherence and thus the effectiveness of the coupling circuit. We can easily write down this coherent power. Since the impedance of the arrays discussed in this section is quite low compared to the total load impedance 2ZL (see fig. 16), it is sufficient to treat this array as a voltage source. To calculate this voltage we must take into account the distribution of the inductance in the coupling loop. Note that for the assumed lumped circuit approximation the locking strength only depends on the total Le. If Le ~5 split into two parts, Le = Lei + Le 2 as shown in fig. 47, by the contacts to the microstrip then the magnitude of the voltage across the array VA, including the loading effects of the high-frequency coupling ioop, will be 2]~2 VA

(7.7)

~NLei~/[(Letu?+(NR)

and the power delivered to one of the load resistors ZL is PL= V~J8ZL;

ZL~’NR.

(7.8)

AK. Jam et aL, Mutual phase-locking in Josephson junction arrays

1xx

x x x

395

Lea

Fig. 47. Distribution coupling loop inductance into components in series (L~ 2)and in parallel (Lei) with the array.

2 for mn-phase coherence and would be reduced by a factor N for incoherent operation. It So PL ccalso N be remembered that eqs. (7.7) and (7.8) give the maximum coherent power. If the junctions should are not all in phase the power will be less than this upper limit. The coherent state of the 10-junction array, implied by the locking measurements, is further demonstrated by the measurements of microwave radiation from the array. These measurements were made in the same manner as for the two-junction cell. In these measurements, the total integrated power was observed to vary with ‘A across the locking range, with the maximum power obtained in the region of locking being the coherent sum of the powers from the two sections individually (i.e., PA+B = (P~2+P~2)2).This observation by itself, however, is not enough to conclusively demonstrate complete in-phase coherence, since locking of the sections may change the powers from the individual sections if all of the junctions in a section were not in-phase. The in-phase coherent power can, however, be independently measured: it is possible to injection lock all of the junctions to a small external rf current (I~~ = 20 p.A) passing through the array in series. The Josephson oscillations of each junction are then phase locked to the external rf current, which provides a common reference phase for all the junctions. For small injection levels, the power produced by the array when biased on the radiation step is predominantly due to the synchronized oscillations of the junctions. The power measured from the array when locked by the circulating rf current in the coupling inductor was more than 80% of the power produced by injection locking, again indicating that the coupling inductor produces strong in-phase coherence. The power from a similar array without the coupling inductor was less than 10% of the power detected when this array was injection locked. Also, in the latter case, there was a large modulation of the power by a magnetic field which varied the flux through the SQUID loops. The modulation of the power with flux in the array with the coupling inductor was less than 10%. For the coherent array, the minimum linewidth of the radiation with the two sections locked was around 20 MHz, which is about one tenth of the linewidth of the radiation from a single junction, in agreement with the results of eq. (6.26).

Once the effectiveness of the high-frequency coupling scheme had been demonstrated through the locking measurements above, the same design was incorporated, with a scaled up inductance, in an array of 99 junctions [36]made without separate bias leads as discussed in section 7.2. For this configuration the evidence for coherence came from the properties of the radiation since locking measurements were impossible. The array did indeed generate coherent radiation which, for a critical current of 400 p.A per bridge, had 5 nW of power with a linewidth of 1.6 MHz. Unfortunately, the operating range of this array was anamolously narrow: from 9 to 10 0Hz. According to the considerations discussed above the array should have phase locked from at least 5 GHz to 20 GHz. A larger operating range was also indicated by the results from the 10-junction array. Room temperature measurements on a scaled up (50 X) copper model of the array showed that the propagation velocity in the feedback stripline was about five times

396

AK. fain et a!., Mutual phase-locking in Josephson junction arrays

smaller than what it should have been for a TEM mode. This was explained by the relatively large

capacitance between the two layers comprising the stripline, combined with the large inductance of the lower layer due to the constrictions (the microbridges). Because of this reduced propagation velocity, the length of the array (1.5 mm) became comparable to an EM wavelength at unexpectedly low frequencies.

Since the feedback stripline was shorted at one end and effectively open at the other end (due to the high impedance of the lumped inductor), a quarter wavelength resonance was produced in the stripline at about 8 0Hz. It was this resonant behavior of the feedback loop which limited the operating range of the array. While the predicted N2 power increase and 1/N linewidth decrease were achieved with this array, the narrow operating range was clearly undesirable. Using the scale model mentioned above, it was found that this resonance could be shifted to much higher frequencies by decreasing the width of the PbIn

feedback loop, thus decreasing the capacitance and increasing the propagation velocity. Also, in this case, the desired inductance for the feedback loop could be obtained from the distributed inductance of the strip, thus eliminating the need for a lumped inductor. This further increased the frequency of the resonance, since the first resonance would now occur at a half-wavelength, since both ends of the PbIn stripline were shorted. This improved design was incorporated in an array of 51 junctions shown in

fig. 46. This array also was biased through the stripline. A section of the 2 p. wide inductance loop and the

resistor Re can be seen in fig. 46b. The loop had a calculated inductance of 150 pH. The lower, indium section of the loop containing the bridges also had a temperature-dependent kinetic inductance which was estimated to be of the same order as the magnetic inductance near T~of the bridges. The precise value was not determined since the current paths are quite complex. The resonant frequency of the microstrip formed by this loop was calculated to be about 80 GHz, well above the designed operating range of the array. To further improve the array’s performance indium VTBs were used in place of planar junctions. The VTBs were made using a self-aligned masking technique [95]. The thickness of the banks, however, had to be limited to 3000 A to ensure proper coverage by subsequent layers (see appendix C). The resulting bridges had a resistance of R = 0.1 fl. While these bridges performed better than planar bridges, the full improvement expected from true VTBs could still not be achieved due to the limited thickness of the banks. The onset of in-phase coherence at w< is clearly evident in the measured microwave power as shown

in fig. 48. The powers P

1 and P2 in the first and second harmonics respectively of the Josephson frequency are plotted as the frequency is increased by increasing the bias current at constant temperature. Below the lower threshold, cv< = 9 0Hz, P1 is quite small (—1 pW); at cv<, P1 increases dramatically — by almost three orders of magnitude — over a small-frequency range. The effects of coherence can also be seen in P2 which decreases from about 80 pW to less than 1 pW at cv<. A reduction in P2 is expected at the onset of coherence since the alternating bias polarity of the junctions tend to cause the cancellation of even harmonics, as well as the dc voltage, across the array. The high level of P2 as well as the unexpectedly low level of P1 for cv
AK. fain eta!., Mutual phase-locking in Josephson junction arrays

397

20

1.2

‘—moo

‘C ___—5~__ x.—___--___ ‘C

-.

.,I ~80 o

//

‘u—i

60

0.8~ 0..

‘Cl

/

I

/

/

0.6

I1~

C ‘

~

x

~

/

0,

~ o

,~_.

1.0

~ E I

40

0.4 /

‘C

/

c\J 20

0.2

/

L.__~__u0

2

4

6

8

10

12

14

16

18

Frequency (GHz) Fig. 48. Power in 1st (crosses) and 2nd (dots) harmonics of Josephson frequency from 51-junction array as frequency is varied by sweeping the bias current. Note sharp onset of coherence at 9 0Hz.

steps in contrast to the RSJ prediction, it is possible that the currents at the harmonics of the Josephson frequency may cause mutual locking. However, we have not analyzed this possibility further. The coaxial connection did not permit a quantitative measure of the power above 180Hz; however large, though uncalibrated, power levels were detected using an HP 435A power meter up to bias voltages in excess of 100 p.V per junction. Since these levels were of the same order as that at 180Hz and did not show any abrupt change, such as would be expected from switching out of the coherent state, it is clear that the array remains coherent to at least 50 0Hz and probably to higher frequencies. This approaches the range where the lumped circuit approximation used to analyze the arrays is no longer valid. The power obtained at high I~at 160Hz was 1.6 nW which can be compared with eq. (78) where ZL = 50 11. If we assume Lei = L~2for the microstrip coupling loop, eq. (7.8) predicts 3.1 nW of power compared to the measured 1.6 nW. Most of the discrepancy is due to the lower than predicted power from an individual junction (section 5.3). It would be relatively easy to modify the design used here to obtain an order of magnitude increase even with these junctions. In particular if Le2 ~ Lei and one end of the array were ground instead of both ends being terminated in 50 fl loads, the power should increase by a factor of 16. The variation of the threshold frequency cv< with the critical current is shown in fig. 49. (A similar effect can be seen in the locking range for the 10-junction sample, fig. 44.) A detailed theory of this transition into coherence has not been worked out, and here we will only discuss it qualitatively. Theoretically, at the frequency cv< the locking range approximately equals the effective dispersion in the bias currents, which is about çbo/L. (—4 p.A). A better estimate of this minimum locking range I~’ required for coherence can be obtained experimentally by determining the minimum microwave injection power needed to phase lock the array at frequencies below cv<. Such experiments show that It” is about 15 p.A for .1~= 500 p.A per bridge. The limiting frequencies calculated from eq. (7.1) for

398

AK. Jam et a!., Mutual phase-locking in Josephson junction arrays 12

I

N I

x. x

0

~,Io

x x

C.

x

2In

x

x x

B) I—

X

x 4

I

0

I

200

400

600

Junction Critical Current ( ~A) Fig. 49. Dependence of the threshold frequency for coherent radiation from 51-junction array as a function of the junction critical current.

I~ = 500 p.A and ir = 15 p.A are ~‘< = 2 GHz and v> = 25 0Hz, which are in rough agreement with, though both somewhat lower than, those measured with the array. The large variation of cv< with critical current (and thus with cti~)is not expected as the basis of eq. (7.1). In fact for cv 4 cv,~eq. (7.1) shows ‘L to be independent of cv~.There are, however, two effects not included in this analysis which would tend to give the observed dependence of cv< on i~the inductance Le decreases with increasing I~ due to the changing kinetic inductance of the indium film. In addition, the anomalous noise of the junctions increases substantially (see section 5.3) with I~also tending to increase ir. For these reasons plus the limitations of the theory (eq. (3.31)) for cv 4 cv~,we do not feel the data on cv< are in substantial disagreement with the model used to analyze the system. As mentioned above, accurate measurements of cv> were not possible since it lay outside the range of our instrumentation. 200

500~A ‘C p. I ‘C

Iioo ‘C ‘C

5

‘C

‘C

X~)(~

‘CXX

x

‘C

‘C

Frequency (6Hz) Fig.

50.

Reduction in radiation

linewidth at the onset of coherence in the 51-junction array.

AK. fain

eta!., Mutual phase-locking in Josephsonjunction arrays

399

P

to.

/ Ax’ freq~I0GHz .~,,/ 5011

\

,/4.

\\

0.I

\\\~

-~

/\\

-

• a o

0.

/

/1+ 0.01

/

00

/

I

.Io~

\ “

a

/

,‘

~ /

_1

I

I 0.001

_________________________________________ N I to moo Number of Junctions Fig. 51. Power (+) into a 500 load and radiation linewidth (x) from arrays of 1 to 99 junctions. All arrays had a junction resistance of about 0.1 0 2 power dependence and a total array impedance much less that the load impedance. The lines have the theoretical slopes of 2 (solid line) for the N and — 1 (dashed line) for the 1/N linewidth dependence.

The transition from the incoherent to the coherent state was also evident in the linewidth of the radiation which is shown as a function of frequency in fig. 50. The minimum linewidth observed was 5 M1-Iz; this is nearly 1/N times (N = 51) the linewidth of radiation from single junctions (200—300 MHz) as predicted by eq. (6.26). In addition very large fluctuations both AM and FM in the power were observed in the transition region at w<. The microwave radiation data from a wide range of coherent arrays are summarized in fig. 51. These results show that P cc N2 and ~ v cc 1/N as predicted in the coherent state for the conditions of the measurements. We conclude from these data that our basic understanding of coherence in Josephson arrays is correct and that the coherent state is, with effort, experimentally realizable. In the following section we will extend these ideas to analyze the expected performance and problems of more elaborate arrays suitable for technical and research applications. —

—

8. Applications of linear arrays 8.1. Introduction

Much of the motivation for the work in the preceding sections comes from the possibility that Josephson arrays will far be superior to conventional technology (and certainly to single junctions) for high-frequency devices operating between the microwave and far infrared. In this section we will

400

A.K. Jam et a!., Mutual phase-locking in Josephson junction arrays

consider a number of promising applications of linear arrays to determine the sort of performance to be expected from the types of arrays we have developed and to analyze advanced designs which could have even better performance. We have already seen in sections 6 and 7 that arrays are greatly superior to single junctions as generators of microwave radiation, so we will begin our discussion with an analysis of arrays optimized for radiation generation, which is perhaps their most straightforward application. 8.2. Applications of arrays as generators One of the features of Josephson junctions which makes them so interesting as potential generators is their wide tuning range. As we saw in section 7.3, it will probably be necessary to limit this range to one or two decades in arrays in order to achieve coherence. However, even this range, coupled with the extremely rapid tuning rates which are possible, would be an improvement over current technology for

some applications in the microwave range and would be dramatically superior to competing devices at higher frequencies. We begin our discussion with a review of the properties and problems of single-junction radiation sources. 1) The maximum available power from a single junction, Pr~, is rather low. Theory shows

[3,4, 1061 that (within the RSJ model) the maximum power radiated by a junction to a matched wideband transmission line is 1/8, =

I~RNX

for V~V~

2\/3

-

for

(8.1)

V
Thus for frequencies below the critical frequency (V < V~) Prnax.....04JV

(8.2)

We should also recognize that the amplitude I~of the supercurrent is limited by self-inductance effects, so that for real junctions it is difficult to achieve an L~of more than 3 mA. For example, for the long-wave millimeter band (~ = 40 GHz, V = 80 p.V), we obtain an estimate that prnax 100 nW, which is too small for many applications. 2) The microwave power actually obtained from a Josephson junction has been little more than several nanowatts, i.e., much less than the above estimate, due to a second drawback: the small microwave impedance Z1 of the Josephson junction. Theory gives the following expression for Z1, i.e., for the value of the load resistance RL, corresponding to maximum power in eq. (8.1) RL

— 11, for V>V~ _Zi_RN.~1.V—~~ for V~
(83)

For the example considered above, we obtain Z1=V3V/I~——0.05fL

(8.4)

A microwave source with such a small impedance is seriously mismatched with typical microwave

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

transmission lines (say wave guides or coaxial cables), which have an impedance of RL

401

102 fl In

principle a series-type resonator can be used for coupling [3—6], but this immediately sharply limits the range of frequency tuning. 3) Finally, the linewidth of Josephson radiation is relatively large. According to eqs. (3.2), (3.40), the

thermal noise produces a linewidth of at least =

~RNkBT,

(8.5)

equal to 160 MHz at 4 K for each ohm of the junction’s normal resistance. This is generally increased by other possible noise sources (e.g. 1/f noise). The Iinewidth can be decreased by using a narrow-band resonator [3,29] or a low-resistance external shunt [4,25]. We have already noted the disadvantages of using a resonator; an external shunt is somewhat more acceptable but has its own drawbacks for many

applications. To see how coherent arrays can solve the problems discussed above we will examine the optimization of the type of short (or lumped) arrays discussed in sections 6 and 7 (see e.g. fig. 38) to achieve the maximum power per junction into a 50 load. The linewidth reduction will be discussed separately since in general it can be varied using external shunts and is thus not fixed through the power optimization. Note that the load impedance is assumed to be fixed, thus for the present we are not considering the possible use of impedance transformers to increase the available power. We feel this conservative approach is wise, especially if it is desired to achieve a wide tuning range, since large transformer ratios can then be quite difficult to achieve. We will discuss the possible uses of transformers later, along with a discussion of the limitations which fabrications technology imposes on the results obtained below. The maximum power per junction from the array (eq. (8.1)) will be obtained when the array impedance is equal to the load impedance. This impedance match is achieved by optimizing the number of junctions, giving from eqs. (8.3) ~

ICRL

R

-

V3V

(8.6)

RL/RN,

V>VC.

The optimized power, which is just P°2t= N0~~ Pr”, is then, using eq. (8.1), 1’L,

(112,) A — UI~RL, popt — J 81 c

V

>

Vc

II’ Ti %< V~.

(8.7)

We see the optimized power is always greatest for the highest possible junction critical current. We can therefore take I,. to be fixed by the maximum critical current achievable from the junctions being used (— a few mA for simple single junctions). The junction resistance can, however, in general be varied through the use of low-inductance shunts, the upper limit being set by the intrinsic I~RNproduct of the junction. Figure 52 shows the dependence of ~ and ~ on the junction resistance for a fixed I, and RL. As can be seen, there is very little dependence of P~on RN. However, there is a number of

402

AK. lain et a!., Mutual phase-locking in Josephson junction arrays

I>

‘~1

~

_

a.

.lo~

\

I

I

I

0.1

I

10

a

RN/(V/Ic) Fig. 52. Dependence of optimized power (solid line) and number of junctions required to achieve the power (dotted solid line) on the resistance of individual junctions. Dashed sections are extrapolations between high- and low-resistance limits.

arguments which indicate that in general it is best to fix RN such that ICRNRP0 equals the lowest desired operating frequency of the array, i.e. RN =

Vminhlc.

(8.8)

For larger values of RN, the radiation would have a large harmonic content as well as a power level which varies rapidly with frequency (note that N0~~ from eq. (8.6) is frequency dependent for large RN). On the other hand, for RN smaller than that of eq. (8.8), N0~~ increases rapidly without any increase in power. Furthermore, any heating problems would be compounded by a_reduction in RN. We will thus take RN to be set by eq. (8.8). For these conditions (I~= 3 mA, RN = V/Ia, ZL = 50 fl), N0~~ depends only on frequency and is given by 5/v[GHz]

(8.9)

N0~~10

with RN

i0~ v[GHz]

fl.

(8.10)

For these parameters the array would deliver about PA = 0.1 mW in a 50 fi load at any frequency above that for which it was optimized. The design values of ~ and RN are certainly reasonable with current technology. The required uniformity (see section 7.3) has been reported in junctions designed for digital applications and presumably can also be achieved with some further work in junctions suitable for generation. We will discuss possible methods for overcoming the power limitation imposed by I~ below. First, however, it is necessary to analyze another limitation imposed by the fabrication technology. We have considered above and in section 7.3 the effects of limitations on junction uniformity and critical current. A third important constraint is that of the minimum junction spacing which heating and

A.K.

Jam et a!., Mutual phase-locking in Josephson

junction arrays

403

fabrication techniques will permit. All of the designs for arrays considered so far have treated the array as a lumped element, shorter than the electromagnetic wavelength A. If we limit the array length to a quarter wavelength for example, then in order to achieve the optimized power output in eq. (8.7) the maximum junction spacing S which can be tolerated is S~—=

(8.11)

B~1

4~~,

where B v>/i’< sets the bandwidth over which the array is designed to operate and where ë is the propagation velocity in the array. For arrays operating over a frequency range of an octave or less B 1, thus (8.12)

S=~j-~--=1p..

Note that the maximum permissible spacing does not depend on the operating frequency. A spacing of

the order of 1 p.m is feasible using electron beam fabrication but is clearly near the present limit of the technology. It is possible, however, that as lithography with 100 A resolution continues to develop, 0.1 p.m junction spacing will become feasible. Note that max[S] cc I~’,thus in lumped arrays it will be necessary to decrease S below 1 p.m to make full use of critical currents greater than a few milliamps. For higher critical currents or large operating bands it seems clear that it will not always be possible to fabricate arrays where the maximum number of junction Nm~permitted by the minimum possible spacing Smjn is equal to N0~1.If VmB), defines the highest operating frequency, then Nmax~

~°

_

4 VmaxSminB

(8.13)

-

Figure 53 illustrates the dependence of the power per junction and the total power to the load as a function of N for non-optimized arrays. Note that the maximum power which can be delivered to a given load is max[PAJ

=

2ICRL.

This power, which is 4P~,is achieved when N N0~1- Since the junction spacing already poses a problem for PA = P°A~ and power increases rather slowly N for N > N0~1,we will take as a 2 for N 4with N~,: practical upper limit. Note that PA decreases as N ~‘

j~O~,t

iI~RL,

P~t=min

Pë

2

1

(4s~~B) ~

(8.14) NNmax4Nopt.

For many potential applications, especially those above the microwave range, a tuning band of an octave or less is entirely sufficient. For these applications it can be feasible to transform the load impedance. To achieve the maximum power from such arrays the maximum possible number of

404

AK. fain et a!.. Mutual phase-locking in Josephson junction arrays

—

/

cc NO

•~

0.01

~

,.~ /

0.1

/

/

/

/ / /

/

/

/

N

Z N

C’J N

/

N

I

/

0,11)

0.00I

0~I

N/(N

0pt)

Fig. 53. Total power from array (solid line) and power per junction (dashed line) as a function of the ratio of the number of junctions to the optimum number of junctions.

junctions Nmax should be used. The power should then be maximized by optimizing the transformed load impedance RT so that Nm~= N0~~ from eq. (8.6), where RN is still fixed by eq. (8.8). Thus, taking the essential constraints on arrays to be I~and Smin, we find O1CC

pm

A ~)

(-

)

~Jmin

where we have set B = 1 in eq. (8.13) since only relatively narrow operating bands are being considered. Using I~ = 3 mA, Smjn = 1 p.m and ë = 1.50 X 10~ rn/s gives P~Z”C 0.03 mW. To achieve this the transformed load impedance RT should be RT = 41S

(8.16)

or RT 25 Cl for the parameters above. This is reasonably close to practical values of RL so that large transformer ratios would not be required. For applications of lumped arrays where impedance transformation is practical it would clearly be desirable to have I~,greater than 3 mA. It is in principle possible for very short, wide junctions (e.g. edge aligned junctions) to have much higher critical currents, if they are placed above ground planes so the bias current is fed in uniformly over the width of the junction [6].The high damping required for generators would also tend to damp fluxoid propagations in the junctions to maintain the required constant phase across the junction. However, data do not yet exist to show just how high it is possible to push I~ in a single uniform junction. An alternate approach to higher values of I~would be to use two-dimensional linear arrays. By two-dimensional we mean for example that each junction would be replaced by a parallel string of junctions (that is, the high-frequency voltages would add in parallel). Alternately, one-dimensional

A.K. fain eta!., Mutual phase.locking in Josephson junction arrays

405

chains of junctions can be combined in parallel. Figure 54 illustrates such configurations. We characterize these arrays as linear to indicate that the phase of the Josephson oscillations across the array, i.e. transverse to the flow of rf current to the load, should be constant. For example, we are not considering true two-dimensional arrays which have junctions in both directions such as those originally studied by Clark [7,11] and by Clark and Lindelof [17].While evidence of interactions has been reported for such arrays, true coherent operation with radiation emission seems to be almost universally absent. In retrospect this is not surprising in view of the stability criteria discussed in section 6. For linear arrays, the equations above are still valid if the single junction critical current ‘C is replaced by MI. where M is the number of junctions across the array. One of the very simplest configurations for such arrays, shown in fig. 54a, was studied by Mooij et al. [13]without finding evidence of coherent radiation. Unfortunately, we now know (see section 6.5) that the maintaining a constant phase across such arrays with superconducting loops is quite difficult. The other simple configuration shown in fig. 54b may also be unworkable since the stable state of such an array has alternating rf voltage polarities on adjacent rows of junction since the coupling between rows is inductive. Thus no rf power is coupled to the load. Fortunately, the analysis of section 6 also indicates the solution to this problem, which is shown schematically in fig. 54c. Here the coupling in each series element is inductive resulting in a stable state with in-phase or series addition of the rf voltages as with one-dimensional arrays. However, the coupling between the parallel rows of series elements is capacitive with the result that the stable configuration is with the rf voltages of the series elements adding in parallel across the array even though they are biased in series. The lumped circuit approximation would limit the width of the array to about A/4. Thus a 100 GHz array (N~~1000) with 1 p.m junction spacing would have an available power 1000 times greater than a one-dimensional array, i.e. P°~ = 30 mW. The low impedance of the array, about 0.1 Cl, would require the use of a large ratio transformer to couple this power to a typical load. We will see in section 8.3 however that more advanced designs are possible which can relax the constraints on junction spacing to achieve high power and even a wide tuning range without transformers. A further advantage of arrays as radiation sources is their reduced linewidth. Here we shall see that series biased arrays have a clear advantage. An examination of the radiation linewidths ~ given by eqs. (6.26), (6.48), (6.49), (6.50) for the various types of lumped arrays, i.e., series or parallel, biased with

I

~

I

~

‘bias

c)

LbIQ. +—.-

M

—+

Fig. 54. Various schemes for combining junctions in parallel (at high frequencies) to achieve a higher effective critical current. The stable configuration of (a) and (b) tends to cancel the rf power to the load, while in (c) the stable state is with rf current of all junctions flowing in-phase to the load.

406

AK. lain et a!., Mutual phase-locking in Josephson junction arrays Re

Le

~

Le<> R s L~

R~

Fig. 55. Schematic of low resistance (Re) high inductance (L,) shunt to narrow radiation linewidths. Coupling loop for coherence is comprised of L

11

and Re.

or without low-frequency shunts reveals that in all cases the linewidth of the radiation can be expressed in terms dc bias power Pb1~along with the temperature and frequency as 2/Pbj~, (8.17) AWC 2kBT tu which does not depend explicitly on the number or resistance of the junctions. For example an array optimized for operation at 100 GHz and designed to operate with one watt of bias power would have a linewidth of about 10 Hz. The superiority of series biased arrays derives from the fact that the bias current required for this power is a factor N less than that required for parallel bias. The 100 GHz array discussed above would require a bias current of about 5 A for a series array or 5 kA for a parallel array (since N~,, 1000 at 100 GH.z). A 5 kA bias current is clearly impractical. A power dissipation of 1 W could be a serious problem if it were necessary to dissipate this power on the chip near the array. We can see that this need not be done by examining the design of a series array with a low-frequency shunt resistance Re as shown in fig. 55. For Re 4 NRN we see from eq. (6.51) that the coherent linewidth is ~c cc Awe/N2. In order for this resistance to effectively shunt the lowfrequency noise which determines the linewidth, the shunt impedance must be dominated by Re up to frequencies of the order of A ~ This sets a limit for the maximum permissible shunt inductance and hence determines how close the shunt must be to the array. For a single junction this maximum shunt inductance is Lf 2 x 10~H for a 4 K shunt as discussed in section 4, eq. (4.73). For a multi-junction array, however, this constraint is greatly relaxed and max[L,]

LfN3.

(8.18)

For our 100 GHz example with N = 1000, max[L~] 1 H. Thus the shunt could be in a separate, high inductance loop, as shown in fig. 55, not only placed off of the chip but even at room temperature or in a separate liquid nitrogen temperature reservoir. Of course the linewidth will be determined by the temperature of the shunt but for many applications a factor 10 or 100 increase in the extremely narrow linewidth predicted for a shunt at 4 K could easily be tolerated. This implies the power which must be dissipated on the chip is only about 1 mW regardless of the operating frequency. 8.3. Travelling wave arrays

The discussions of section 8 2 make it clear that the requirements on junction spacing imposed by the lumped array design can be a serious limitation especially if higher power levels are desired in arrays

A.K. fain eta!., Mutual phase-locking in Josephson junction arrays

407

with wide tuning bands. Thus it would be very desirable to be able to spread the junctions over many wavelengths while coupling them coherently. The essential problem in this, as we have seen in previous sections, is that to achieve in-phase locking the coupling current induced by the junctions must lag the

junction oscillations. In a broad band array, properly terminated at both ends, however, waves in general propagate in both directions making it impossible to maintain the proper phase relationship at all points. One solution to this problem, explored by Davidson [34],is to use mismatched terminations so that the array is a resonant structure. Junctions can then be placed at the points of correct phase relationship. Such a structure of course is not continuously tunable. We discuss below another approach to this problem which should permit a distributed travelling wave array which is continuously tunable. A schematic of such a travelling wave array is shown in fig. 56. Here Josephson junctions are inserted in series in a microwave transmission line, for example, a superconducting stripline. Their dc bias

voltage, equal in magnitude but with alternating polarity, is provided through additional superconducting lines in a manner similar to that discussed in sections 6 and 7: these lines form superconducting loops of large inductance L L0, connecting each pair of junctions. The dc bias currents are injected ~‘

symmetrically into each loop so that they produce no magnetic flux ~e through the loops. An equal external flux is produced in all of the loops by an additional current I~,injected at the ends of the array. This current produces an equal phase difference between each pair of the junctions: 0k+10k~e,

(8.19)

(/3e~~IeL

2

3....

N ~T’~

I~ L/2

~ti

L/2

____

29~

3

RL—p—(~)

(b)

N~I

Fig. 56. Possible one-dimensional, travelling wave array; (a) high-frequency equivalent circuits and (b,c) low-frequency equivalent circuits.

A.K. fain et aL, Mutual phase-locking in Josephson junction arrays

408

and is chosen in such a way that

(8.20)

jK~~—q

OelT+K,

where q is the phase shift of the wave between adjacent junctions in the transmission line q=wlën.

(8.21)

Here n is the number of junctions per unit length, and ë is the wave phase velocity. Realistic values of the parameters give 10~s~1,

c

1010

cm/s,

n

—

i0~cm~1,

(8.22)

so q is of the order of 0.1. The operation of this array is very close to those discussed in sections 6, 7, with the important exception that the phase gradient of the travelling wave is matched by the shift of the junction

oscillation phase if k I q. In this case, all the junctions give almost equal contributions to the wave travelling in one direction (which is determined by the sign of K), and give a nearly vanishing net contribution to the wave travelling in the opposite direction. As a result, an intense electromagnetic wave at the Josephson frequency appears in the transmission line travelling from one (“idle”) end to the other (“load”) end. This situation is very close to the well-known resonant wave propagation in long Josephson junctions [1—3].This analogy becomes even more evident if one presents the low-frequency equivalent circuit of

the array under discussion in the form shown in fig. 56b, c which coincides with the equivalent circuit of a long Josephson junction. Note, however, that in contrast with the usual (tunnel) Josephson junctions, high-frequency currents in our array are flowing mostly along a special transmission line, not shown in fig. 56c (see fig. 56a). For this current the Josephson oscillators are connected in series. As a result, a travelling wave is induced in the high impedance (p = VL 0/C0 50 Cl) line. In contrast, in a long tunnel junction the wave is induced in the very low-impedance (p -~ 10~Cl) stripline formed by the junction. To analyze the dynamics of the travelling-wave array, we can again use eq. (6.9) for the linearized

phase =

w~(Ik)+~Rd[Ik+asRe~

ykk,

e~~°~’ Ok)]+ flk

(8.23)

where we have separated the additional low-frequency current ‘k’ flowing through the junction due to the external magnetic flux ~. From the equivalent circuit (fig. 56b, c) we obtain the equations for ‘k, 20k,

L 1k

=

fork

1, N,

Ok_1+Ok+1 02 01— ~e,

for k

°N-i°N~e,

fork=N.

=

1,

(8.24)

To find the coefficients Ykk’ of the high-frequency coupling between the junctions, we can use the recipe described in the beginning of section 6. Namely, we replace all junctions with their internal

A.K. Jam

et aL,

Mutual phase-locking in Josephson junction arrays

41)9

impedances Z, (at high frequencies discussed we can take Z~ = RN), except junction k’ which should be replaced by a resistance RN in series with a voltage generator with unit amplitude at the Josephson frequency. The coefficient Ykk’ is found in this case as the complex amplitude of the current flowing through the kth junction. Neglecting the small admittance of the dc-bias circuit, one obtains Ykk,=(_1)kY(m),

(8.25)

m=k—k’

where Y(m) satisfies a usual wave equation ~

[q2+ 2jqS + j~ t9(wt)] ~‘

=

—

Sjfl,1J,

(8.26)

where S is the wave damping due to energy loss in the Josephson junction S = RN/2p.

(8.27)

For the parameters RN = 0.1 Cl, p = 50 Cl, S is of the order of ~ i.e. is much smaller than both q and unity. Equations (8.23)—(8.26) describe a travelling wave at the Josephson frequency, with a current amplitude A growing from zero at the idle end of the array to a saturation value A—

~g—K —

828

S

achieved in all junctions with numbers k larger than approximately N0 = S~= 2p/RN.

(8.29)

For all of the junctions (with the exception of those close to the idle end k ~ N0), the linearized phase changes linearly with k = — ict + irk + const.

(8.30)

in the absence of fluctuations and junction parameter deviations, so that all junctions give equal contributions to the travelling wave I = Re[A exp{j(wt — id + irk

+

const.)}].

(8.31)

In return, this travelling wave produces a narrow current peak in the I—V curves of the junctions: (8.32)

which is analogous to the “Eck peak” in tunnel junctions [1—3].

410

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

At the top of the resonant peak (zl

0), the wave power is a maximum

p~ax= A~~~p/2 ~

(8.33)

Thus both the maximum power level and the number of junctions required to achieve this power are roughly the same as those given by eqs. (8.6) and (8.7) for lumped arrays. However, the constraint of the junction spacing has now been effectively removed. Combining this travelling-wave array with some of the two-dimensional geometries discussed in section 8.2 for increasing the effective L therefore has the potential for achieving high power levels (of the order of a watt) in widely tunable Josephson array generators. The tuning in this case would be achieved by the simultaneous variation of the bias voltage

and the current ‘e which creates the phase gradient in the array. 8.4. Application to microwave detection During the past two decades there has been a large and diverse effort to develop single junction Josephson effect devices as high frequency detectors. (See for example monograph [3] and reviews [107—109].) The results of this work have generally been disappointing with the exception of the quasiparticle mixer [110]which does not really use the Josephson effect. This is the case even though theoretical analysis [111—114] have shown the potential sensitivity of these devices to be quite high. The reasons for the limited success of single junction devices are now rather well understood, and as we will discuss below, the problems are precisely those which can be overcome using coherent arrays. For Josephson mixers the problems are threefold. For mixers using external local oscillators the combination of the LO frequency and the Josephson frequency mixes numerous noise components into the IF band resulting in a noise temperature significantly above the physical temperature [107,114]. On the other hand, mixers using the Josephson oscillations as an intrinsic LO as shown in fig. 57a have a minimum predetection bandwidth which is limited to the intrinsic linewidth of the Josephson oscillations. Further, the power from a narrow signal is broadened at the IF to cover a band equal to the Josephson linewidth. (Note that external shunt resistors cannot be used with mixers to narrow the linewidth [114].)Since this linewidth is at least 160 MHz per ohm of junction resistance the sensitivity of these devices is severely reduced for narrow band signals. A third limitation on single junction mixers is their low saturation power [107]and limited dynamic range. While the saturation power depends on just how the mixer is used, we can say very roughly that

(a) Wi

o

(b)

w

W

W

~

[i”~1

CL)

W

Fig. 57. Frequency diagrams of microwave receiving devices in which coherent arrays can be used: (a) intrinsic LI) mixer and (b) self-selective quadratic detector. Arrows denote the microwave input signal (~) and the low-frequency output signal (fl. Shaded peaks show the Josephson oscillation lines with width ~v; dashed-line rectangles show the frequency bands for array matching to external circuits.

A.K. fain et aL, Mutual phase-locking in Josephson junction arrays

4i I

saturation will occur when the total external ri current through the junction is of the order of the junction critical current. Since the ICRN product is limited to a few millivolts, high resistance junctions have very low values of I~.While this problem can be circumvented to some degree using transformers, high transformer ratios sharply limit the tuning range of the mixer. In addition, the signal current level for a given power is increased in low impedance junction again limiting the dynamic range. As we have seen in previous sections, coherent arrays permit matching low impedance loads without using impedance transformation to increase the source current, thus giving a much greater dynamic range. Further the intrinsic LO mode can now be used for narrow band signals since the array linewidth is reduced both due to coherence and to the small intrinsic linewidths of the low resistance junctions in the array. For example, a 10 GHz one-dimensional array mixer would have a minimum predetection bandwidth of about 200 Hz. The minimum noise temperature of such a coherent array is the same as for a single junction [52] (TN)min =

T

for

w

0.3wC.

(8.34)

However, with arrays unlike single junctions it should be possible to achieve noise temperatures close to this minimum value even for narrow-band signals while maintaining a broad tuning band and a high dynamic range. The precise design of array mixers depends on the desired operating characteristics. In general achieving a low TN requires operating well below the junction characteristic frequency (e.g. w O.3wC) with the array heavily loaded by the source. For these conditions it is best to use methods of analysis other than the perturbation approach used in this paper. Thus we will point out a few of the main requirements and refer the reader to other papers [108,114] for the details. Two possible mixer circuits are shown in fig. 58. The first of these (fig. 58a) could be directly coupled to a broadband coaxial line while the second (fig. 58b) which is inductively coupled would be suitable for a waveguide system for example. A principal consideration in both of these designs is that the IF voltages add in phase between the terminals AB so the arrays have a high IF impedance making it

A

X

x

x

x

x

(a)

RL

(b)

Fig. 58. Schematics of possible arrays for use as microwave receivers. Microwave input is from RL and low-frequency output is across AE.

412

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

easier to match them to a low noise IF amplifier. The phase of the IF which is just the phase of i,li 61— 62 was analyzed in section 4 and is determined by the relative polarities of the Josephson oscillations, the signal current, and the bias voltage. The important point is that, as we have seen, by varying the feedback coupling and bias circuits virtually any relative combination of signal and IF polarities and thus of signal and IF impedance can be achieved. An example of an array with an extremely low IF impedance (and thus not likely to be useful for a mixer) would be the parallel arrays discussed in section 7 as shown in fig. 40d. The minimum noise temperature is limited to the array temperature since the junction noise at the signal frequency (assumed to Nyguist noise) is down-converted in the same manner as the signal. In designing mixers it is important this noise not be dominated by the low (IF) frequency junction and source noise. This dictates that w 4 w~,and also that the signal line has a high pass filter. It is important to consider the effects of this filter on the mixer characteristics. Such a practical system has been analyzed [114]and has a noise temperature given by eq. (8.34) along with a conversion efficiency (signal to IF) substantially greater than unity. If the mixer is operational in the quantum limit (hi.’> kT) the down-converted noise will be increased by quantum fluctuation making it relatively easier to avoid the effects of the direct low-frequency noise. This allows one to hope that system noise temperatures as low as -~10 K can be achieved at frequencies up to the short-wave edge of millimeter band. Another promising microwave receiving device, the self-selective quadratic detector, is very close in its dynamics to the self-pumped mixer discussed above. The Josephson oscillation frequency w again should be close to the signal frequency w~but now the change in the dc voltage V across the junction plays the role of the output signal (fig. 5Th). For small signal power P5, the device behaves as a quadratic detector, (8.35)

with responsivity i~exhibiting a sharp odd-resonant peak at ut cog. An analysis shows [113,114], that the noise-equivalent power (NEP) of such a quadratic detector is closely related to the noise temperature TN of a self-pumped mixer: 2 (8.36) NEP = 2Vir kB TN (A vAv)’~’ where A p is the input bandwidth, and A v~is the post-detector band-width. This result remains valid for coherent arrays. Both arrays shown in fig. 58 can be used, with the output signal picked up from the terminals AB. Use of such arrays should allow one to come close in device sensitivity to the ultimate value of eq. (8.36) with (TN)mjn T. For a typical case At’ = 10 MHz this implies that a very high sensitivity, NEP/(A v~4”2 2 x

10~W/Hz”2,

(8.37)

can be achieved throughout the millimeter band. To summarize, Josephson effect receiving devices, which have been somewhat obscured for a few years by the quick progress of other cryogenic microwave receivers (semiconductor and superconductor bolometers and SIS mixers) have a chance to overcome these competing devices in sensitivity, at the same time being much simpler and thus more suitable for microwave intergrated-circuit systems.

A.K fain eta!., Mutual phase-locking in Josephson junction arrays

413

8.5. Application to study of the non-equilibrium effects

As an example of possible applications of arrays in solid state physics, let us analyze their use for the study of the non-equilibrium distributions of quasiparticles in conducting materials. Consider an array of the type shown in fig. 40b, with the sample to be studied (say, a thin-film strip) playing the role of the shunt resistance Re. As we saw in section 6.4, if this resistance is small enough, the linewidth Awc of the coherent oscillations of the array will be determined by fluctuations of the voltage across the shunt resistance rather than the intrinsic noise of the junctions (eq. (6.50)): forR~4NRd.

(8.38)

If the current carriers (quasiparticles) in the sample are in thermal equilibrium with the heat bath of temperature T, Sve follows the Nyquist theorem: (8.39)

k~iRe.

Sve =

If, however, the quasiparticles are out of equilibrium for some reason, the intensity of the fluctuations will deviate from the value (8.39). For example, if the dc electric field E in the sample exceeds a value E0 (8.40)

eEOAO—~kBT,

where A0 is a quasiparticle diffusion length, the quasiparticles would become “overheated” by the field, even if the crystal lattice of the material is still at the heat bath temperature. In this case, an approximate expression for S.~can be obtained from eq. (8.39) by replacing T by the effective temperature TE eEOAO/2kB, so that 2

S~~~eEA0R~,for E>E0.

(8.41)

This estimate is correct, only if the sample length L (in the direction of the field E0) is larger than A0 in the opposite limit (L 4 A0) one should replace A0 by L12 in eq. (8.41). The resulting expression SF~=

-~-

eELR~, for E> kBTie,

(8.42)

can be rewritten as Sve~e~Re,

i.e. sFC=—er.

(8.43)

Equation (8.43) coincides with the usual expression for shot noise, and thus shows the close relation between the effects in long and short samples. The quasiparticle diffusion length A0 is of the order of

414

AK. fain eta!., Mutual phase-locking in Josephson junction arrays

several micrometers, thus it seems possible to follow the crossover between the two cases changing the length L of the sample in this range. Similar non-equilibrium effects can be caused by a microwave field, laser irradiation, etc. Of course, the “overheating” of the quasiparticles can be traced by other experimental methods, say by observation of their influence on the sample resistance. Estimates show, however, that the latter quantity has a very weak dependence on the effective temperature TE until it becomes as large as 1 K (at a bath temperature T 4 K). Some preliminary observations of the field-induced quasiparticle overheating have been done [105] for thin bismuth films, where the effect is especially large due to low concentration of carriers. Using coherent arrays of Josephson junction rather than relatively primitive technique of fluctuation measurement used in the work cited above, seemingly should enable one to observe the effect in “real” metals as well. 9. Summary and conclusion Over the past several years, stable coherent oscillation of large numbers of Josephson junctions coupled in arrays has been experimentally demonstrated, and the basic conditions for obtaining the coherent state have been ascertained. The stable coherent state is the result of a mutual phase locking of Josephson junctions united in the array; this locking can be caused by several different physical mechanisms. However, only one of these mechanisms the high-frequency electromagnetic interaction — is strong enough to produce stable phase locking in real arrays, where the junction parameter spread and time-dependent fluctuations tend to destroy the coherent state. This high-frequency electromagnetic interaction of the junctions can be qualitatively described as follows: the Josephson oscillations of a given junction induce currents of the same frequency in other junctions, and vice versa. The junction nonlinearity gives rise to the possibility of phase locking (synchronization) of the junction oscillations through these “external” currents. To obtain strong phase locking using this mechanism, an array should have a special coupling circuit which permits the oscillation currents induced by one junction to flow through the others. One of the requirements for this coupling circuit is that it possess a predominately reactive admittance. For providing in-phase coherent oscillations this admittance should have an “inductive” character. In multi-junction arrays, a further requirement for the coupling circuit exists which is essential for the stability of the coherent state: the circuit should provide a “long-range” interaction, with any pair of junctions interacting (almost) independently of their mutual position in the array. If these conditions are satisfied, the coherent state can be quite stable with respect to both timeindependent perturbations (parameter scatter) and time-dependent perturbations (fluctuations in the junctions and other elements of arrays). For example, critical current variations among the junctions as —

large as —.30% can be tolerated.

For one-dimensional arrays, the conditions listed above can be easily fulfilled by using a uniform ioop array, where all junctions are connected in series by a loop with a properly-chosen high-frequency impedance. In the array, the junctions can be connected in an independent manner (in parallel, for example) by low-frequency circuits to provide the dc bias necessary for the Josephson oscillations. Using these principles, coherent microwave oscillations of arrays consisting of up to 100 Josephson junctions (thin-film microbridges) have been obtained, and relatively large values of radiated power (up to 5 nW) were observed. These experiments have shown generally good agreement with the results of

AK. fain et a!., Mutual phase.!ocking in Josephson junction arrays

415

our theoretical analysis of mutual phase locking, allowing for the deviations of the microbridge properties from the simple RSJ model. Based on the detailed experimental and theoretical studies described, we have found it possible to carry out reliable, quantitative analyses of the performance of arrays. These analyses show that practical coherent arrays can be designed which will offer significant improvements over the present technology for a number of applications both in solid state physics and microwave engineering. For example, coherent arrays can be used for the study of current-induced non-equilibrium of quasiparticles in normal metals, semimetals and alloys by providing a very sensitive method to make high-frequency noise measurements. For device applications relatively simple lumped (with dimensions less than a wavelength) one-dimensional arrays are feasible as radiation generators through the submillimeter band where they can have an output power approaching 0.1 mW, linewidths of a few Hz and a multi-octave tuning range. For higher powers, more complex travelling-wave arrays should be feasible. Their dynamics, however, requires more detailed study, both experimentally and theoretically. In the field of microwave detection, coherent arrays can be used in a number of receiving devices, based on the Josephson effect, such as self-pumped mixers and self-selective quadratic detectors. These devices make the most complete use of the unique electrodynamic properties of the Josephson junctions, and do not require an external pumping source (local oscillator) for their operation. They have not been extensively studied until recently, however, because the contradictory requirements for the Josephson junction resistance limit their performance if single junctions are used. Utilization of coherent arrays can avoid this contradiction, and enables one to design very simple receivers with electronic tuning of the input frequency band. Estimates show that such receiving devices can have respectively, noise temperatures and noise-equivalent powers as low as —10K and ~-~..10’9W/Hz112 throughout the millimeter band. We feel the advent of coherent Josephson arrays provides a real hope that a method now exists to realize the great promise which has long been held for the use of the Josephson effect in high-frequency analog applications. Attempts over the years to realize this promise using single junctions have been the source of much frustration, related in large measure to the junction’s low impedance and low saturation power. There should be other possibilities for the use of arrays beyond those which we have discussed and we are hopeful that this review will stimulate new ideas leading to a wide range of array applications in research and technology.

Acknowledgments One of us (K.K.L.) would like to thank A.M. Klushin, Sd. Borovitiskii and V.V. Migulin for useful discussions and to especially acknowledge the contributions of L.S. Kuzmin and G.A. Ovsyannikov to some of the work discussed here; and, in addition, to thank the State University of New York at Stony Brook for the hospitality and support provided during the preparation of this manuscript. The remaining authors would like to acknowledge many helpful discussions with A.M. Feingold during the initial phases of our work on coupled junctions as well as those with V. Ambegaokar over a number of years and to note the important contributions of C. Varmazis, R.D. Sandell and P.M. Mankiewich to the progress of the experimental work presented here. We would also like to thank S. Chakravarty for some useful criticism of the manuscript. Our work was supported in part by the Office of Naval Research.

416

AK. Jam eta!., Mutual phase~!ockingin Josephson junction arrays

Appendix A: Locking of non-identical junctions and harmonic locking If the parameters of the junctions in a two-junction cell are significantly different, their interaction can be analyzed using the same approach as in section 4, but taking into account the difference in the values of Rd, r and a for each junction. Thus, instead of eqs. (4.4) one should write w’j’(11)+ a, Rdl

=

[e~Re Y ± e2 Re Y cos i~i±r2 Im Y sin

q.’] + ~1i (A.1)

O2 where

co~(I2)+a2Rd2~[r2Re Y±e~Re Ycos ~

I,

e1 Im Ysin tli]+ 712

and 12 are again given by the relations, eq. (4.41), for coupling circuits with finite dc

conductivity.

These two equations are simple to solve, though they lead to rather cumbersome expressions. Only the equation for the phase difference i/i will be presented here. The phase difference is again described by an RSJ type equation:

R0]~WLSIfl(l/J+ f3) B2w~—B,w~+C + ~1eff

ç1[1 +

(A.2)

where

2 + {Im Y (ai Rd

[{ReY (ai Rdlr2 B2 a2 Rd2e1 B1)}

=

c

=

~ [{alRd,r

1B2

tan

/3

=

—

a2Rd2E2Bl} Re

a1RO1E2B2—a2Rd2e,Bl ______________ a, Rd1e2 B2 + a2 Rd2e, B,

R0

1+(Rdl+Rd2)/Re

B1

1+2Rdl/Re

Y

2] (2ir/cko)2

1E2B2 + a2 Rd2e, B1)}

—

+

(Rd,

—

Rd2)( V0 ± V0)]

Re Y Im Y ___

series bias B2 = 1+ 2Rd2/R~

B, = 1

parallel bias B2 = 1 u_ 1 = U—

=

5AB

u~y

—

~- ~‘ ~fJo

0/ 1’e

W1 ~

UIT

series _‘~1/ /t~ ~ V OIPe

bias

A.K. fain eta!., Mutual phase-locking in Josephsonjunction arrays an

w~(I,~) parallel bias

=

(U

2

—

17eff

417

(U2(IBC)

=

{Rdl~lB2— R~~.232B, — Fe[Rd,82

=

{R~1~, — R~32 — FC[Rd,

+

—

Rd2B,J}

R~]}

series bias

parallel bias.

An analysis of eq. (A.1) produced the following distinctive new features, compared with locking of nearly identical junctions (section 4): (1) Mutual locking is possible even for purely resistive coupling Im V = 0, with a locking range (Ui

—

~Ima~=

~—

IaiR~iE2 — a2Rd2 e,IRe Y,

~A.3)

equal, in the RSJ model, to =

w~IR,— R2~Re Y

(here we assume the characteristic frequencies ~C of the junctions to be equal). For such a resistive coupling, ~1c = ± ir/2 in the middle of the locking range. Further, the I—V curves V(I,,2) are linear inside the locking range, with derivatives 9Ik = RJ(Rd)k/(RJ — Rk), j, k = 1, 2; j k (A.4) 8VIt which can be larger than (Rd)k (for some numerical results, see refs. [49, 56, 59]). Note that aV/~9I,and 3V/c9I

2 have opposite signs. (2) If the resistances of the junction differ significantly, say R2 ~ R, the locking term in eq. (A.1) for 02 becomes much larger than that in the equation for 01. Thus one can say that the low-resistance junction is always “stronger” than its high-resistance neighbor. As a result, one can neglect the locking terms in the equation for 0,, so that the phase locking is one-directional (rather than mutual): the current with the amplitude A = e,~Y~induced in the coupling ioop by the low-resistance junction 1 locks the phase of the high-resistance junction 2. If the coupling is purely high-frequency (no dc conductivity of the coupling circuit), this onedirectional locking leads to appearance of almost vertical (3 VI~I2 Rdl 4 R~)step on the I—V curve of the junction 2, with the height -

~12~’2a2riIYl.

(A.5)

At the same time, the I—V curve of the junction 1 shows an even greater locking range (A.6)

418

AK. fain et al., Mutual phase-locking in Josephson junction arrays

having a slope very close to Rd,, so the change of its I—V curve is minimal. Note, however, that even if the coupling circuit has a high dc conductivity Re 4Rdl (see section 4.2), for series biasing (as well as for parallel biasing), both derivatives a VI a1~and 0 V13i2 inside the locking range are still given by eq. (A.3); R~is not effective in reducing the differential resistances inside the locking range for series biasing; this is related to the opposite signs of the voltage derivatives, eq. (A.4). The locking of unequal junctions is similar to “harmonic” locking, when a higher harmonic of one of the junctions (say, the junction 1) locks the phase of the junction 2, i.e. w2nw1,

V2~nV1,

n>1,

(A.7)

inside the locking range. Again, the basic equation (3.43) shows that only the equation for 02 has any locking terms, while the equation for 0, does not. For identical junctions, the reduced equations are el= (U~(Il)+alRdl.~— e~’ReY(uL)+

711(t),

02 = w~(I2)+a2Rd2~ [s~’~Re Y(w2)±g~n)IY(w2)I sin ifr] + ii2(t), /1=

nO1— 02+const.,

n>1.

(A.8)

If the absence of external shunting (or large fluctuations) a vertical current step of height &12

=

a2r~,”~ Y(o2)I

(A.9)

appears at the I—V curve of junction 2 at a voltage (A.7). The locking range for the I—V curve of junction 1 is again given by eq. (A.6), again with the result that there is almost no difference in the slope of the I—V curve from that in the unperturbed regions. In the presence of strong external low-frequency shunting (R~4 Rdl, R~)the locking ranges for currents 11,2 remain the same (A.6), (A.9), but the slopes 3Vi 31,,2 inside the locking range decrease as R~.For harmonic locking, however, this decrease occurs for both series and parallel biasing. Appendix B: Phase locking with resonant coupling condition eq. (4.39) is not satisfied, several characteristics of mutual phase locking still coincide with those in the wide-band systems. For example, inside the locking range (for small fluctuations, y 4 1), the junctions oscillate with the single frequency ô = 01 = 02= ~ Thus the results obtained in sections 4 and 6 are valid if Y is assumed to be a function of this frequency: If

Y=Y(O).

(Bd)

Some calculations of this type have been carried out [57, 58] using a method which is only valid for high frequencies (tu 5~ op). The authors of these papers have considered only the noiseless case. The most distinctive features of the resonant phase-locking are however connected with fluctuations.

AK. fain et aL, Mutual phase-locking in Josephson junction arrays

419

Repeating the calculations of the sections 4 and 6 using eq. (B.1), one obtains the expression for the linewidth of coherent oscillations of N junctions as, (B.2)

Awc=~Awt.

This expression is similar to eq. (6.26), but with the unperturbed linewidth

Awu

changed to Awr, the

linewidth of a single junction in the same resonator [3] =

2(2ir/~o)2R~~

(B.3)

where Rr is the differential resistance corresponding to the junction I—V curve modified by the resonator’s influence (see eq. (3.45)), Rr = dV/dIr, tr = I + ‘M = I as Re[Z~(w)+ NZ —

1(w)]’,

(B.4)

and S~ris the effective low-frequency noise, also modified by the resonator: 2 ~NZj+Zj

[SF(o)+ SFe((O)].

S~r SF(0)+2a

(B.5)

If the resonator is kept at a low temperature, its additional contribution to S~rdescribed by eq. (B.5) cannot be large, S~, SF(0). This means that the most important factor is the reduction of Awr and hence Aa~due to decrease of (Rd)r: (Rd)r

Rd

Q’,

Awe ~ Q~2,

(B.6)

if compared with the nonresonant case. All these formulas are only valid for Awc ~ wiQ. Finally, we should mention some specific effects in resonant systems arising from the possibility of “non-Josephson” oscillations in such systems. A discussion of such effects in arrays can be found in ref. [47].Note, however, that “non-Josephson” oscillations occur at a frequency not connected by any strict relation with the Josephson oscillation frequency, and hence, these oscillations themselves do not produce mutual phase locking of the junctions. Appendix C: Fabrication techniques Cl. General techniques

Optical micrographs of typical arrays shown earlier (see e.g. fig. 18 and fig. 24) make it clear that these samples contain relatively iarge features (e.g. the stripline and the wire bonding pads having dimensions of 10 to 100 micrometers) and very small features (e.g. the microbridges, with a dimension of about 3000 A). These samples were made using lithographic masks; the coarse patterning was done using photolithography, whereas for the given features, electron beam lithography (E.B.L.) was used. The principles of lithography have been reviewed in a number of articles, and will not be discussed in

AK. fain et aL, Mutual phase-locking in Josephson junction arrays

420

detail here. The two basic techniques used for pattern transfer from the lithographic mask to the substrate are shown schematically and explained in fig. 59. The lift-off process, shown in fig. 59b requires undercut in the developed resist profile to prevent tearing along the pattern edges in the lift-off process. This was no problem for E.B.L. since electron scattering results in undercut resist profiles. For photolithography, the scheme shown in fig. 59b was used. Note that the lift-off process results in a reversal of the polarity between the mask and the final pattern, whereas the milling techniques does not. The thickness of the evaporated layer was limited to less than the thickness of the resist in the lift-off process; in the ion milling technique this was in addition limited by the milling rates of the resist and the deposited material.

For photolithography, contact printing was used; the jig used for this was a simplified version of the jig described by Smith [117].The substrate coated with photoresist was placed in a square hole in a brass block. The chrome master mask was held by suction in intimate contact over the substrate. Deep uv, obtained from a mecrury lamp was used to expose the photoresist. The E.B.L. system, based on a modified scanning electron microscope interfaced to a minicomputer, had 16 bit x 16 bit resolution in a 2 mm field. This was convenient because it allowed high resolution work to be done at the same magnification as relatively coarse work. However, the slow speed of the system made it necessary to change the beam current and resolution for doing coarse work. The system also had provision for manually registering the substrate to within 1500 A; registration was necessary in the fabrication of the multilayer samples.

Electron Beam

Electron Beam

/develoPed PMMA mask~

______

I

I

r

~

-I I

Evaporation

F—

‘

‘

______

I

‘ ~ _____________

Ion Beam

I

____________________

evaporated film ‘—substrate Fig. 59. Pattern transfer techniques. (a) Lift-off technique. Resist is spun on to the substrate, exposed and developed, than the desired film is deposited. The unexposed resist along with the film on top of it is removed, leaving the patterned film on the substrate. (b) Ion-milling technique. The desired film is deposited on the substrate, then coated with resist, exposed and developed. The underlying film in the exposed region is then

etched away using an ion beam.

A.K.

fain eta!., Mutual phase-locking in Josephson junction arrays

421

C2. Fabrication of two junction arrays

The steps involved in the fabrication of a sample were as follows: a) Substrate cleaning: The substrates were cleaned by 1) scrubbing with a cotton swab in acetone, 2) ultrasonic in hot detergent (10 mm.), 3) rinse in distilled water, 4) ultrasonic in hot distilled water (10 mm.), 5) ultrasonic in isopropyl alcohol (10 mm.). The clean substrates were stored in isopropyl alcohol. b) The coarse features (which includes the microstrips, registration marks for subsequent E.B.L. steps, and the gold wire bonding pads used for connecting the leads to bias the array) were patterned using photolithography and the scheme shown in fig. 59b. Gold chrome was evaporated on the clean substrates (75 A of chrome, 1000 A of gold), the substrates were then spin-coated with 3000 A of polymethyl—methacrylate (PMMA), and baked at 180°C.The PMMA was contact exposed with deep uv through a chrome mask on a conformable sapphire substrate. After developmentof the resist in 1:3 methyl isobutyl ketone and isoprophyl alcohol, the substrate was ion milled with 1 keV argon ions for about three minutes to remove the gold chrome on the exposed regions. The resist left on the unexposed parts was then washed away in acetone. The chrome master mask was made in a similar way a conformable sapphire substrate with 1000 A thick chrome film on it was coated with 1 ~imof AZ 1350 photo resist and baked at 90°C. The resist was exposed through a photographic negative having the desired pattern. After development of the resist, the chrome in the exposed area was etched away using a solution of cerium ammonium nitrate. The resist left on the exposed area was then dissolved in acetone. c) If the Josephson circuitry involved any resistors, the fabrication of the resistor was the next step. This was done using E.B.L. and the lift-off scheme shown in fig. 59a. The substrate was spin-coated with PMMA (3000 A) baked at 180°C, after which 75 A of aluminum was deposited on the substrate. (This was to prevent charge buildup during the electron beam exposure.) The substrate was manually registered in the E.B.L. system using the registration marks made in (a). After the electron beam exposure, the aluminum was dissolved in sodium hydroxide solution. After the pattern was developed, 1000—1500 A of gold was evaporated on the substrate after cleaning the surface of the substrate in an oxygen glow discharge. The lift-off of the gold in the unexposed regions was effected by leaving the substrate in acetone for about 112 hour. d) The mask for the Josephson junctions (microbridges) and associated biasing leads was next fabricated in a similar E.B.L. process. The biasing leads, which required lesser resolution than the junctions were exposed at a higher beam current (2 nA). The indium evaporation (1000—1500 A) was done at a pressure of 3 x i0~torr with the substrate cooled to 77 K. Prior to the evaporation, the substrate was cleaned in an oxygen glow discharge to remove any residues left behind in the previous lithographic steps. After evaporation and lift-off, the sample was ready to be mounted on the probe. Though the undercut resist profiles needed for the lift-off process are produced as a result of electron scattering, scattering also causes the proximity effect, which makes the exposure at a given point dependent on the exposure of neighboring points within a radius of several microns [118].A variety of techniques using software have been described to correct for the proximity effect in E.B.L. system with large computers [118],but these were difficult to implement on our system. As a result of this effect, the dimensions of the bridges in the developed mask were different from the programmed dimensions and —

422

AK. Jam et a!., Mutu4l phase-locking in Josephson junction arrays

x4~47Z

Fig. 60. Schematic of the shadowing technique used for fabricating variable thickness microbridges.

had to be compensated for in the computer program. The exposure due to proximity effect also limited the resolution of the pattern to about 50% of the resist thickness [25]. As mentioned earlier, we have also fabricated arrays with variable thickness microbndges (VTB; s). Fig. 60 illustrates the technique used for making such bridges. The PMMA mask for these arrays was thick (6000 A—b 000 A). Indium was evaporated in the direction A perpendicular to the surface of the substrate and in the ~,yplane (—~700A) followed immediately by indium in the direction B, also in the xy plane (5000—7000 A). The material evaporated in the direction A formed the bridges and part of the banks, but because of shadowing, the material evaporated in the direction B formed the banks only. The condition for this to happen was that the shadow in the direction B of the mask defining the bridge fall on the resist wall on the opposite side and not on the substrate. However, the effects of electron scattering were much more pronounced for these thick resist layers, and resulted in severely undercut resist profiles, especially in the region which defined the bridges. For resist thickness greater than 1 j.i~m, the undercut was so great that these regions were suspended, and on evaporation two junctions were obtained instead of one. The undercut in this region was decreased to some extent by decreasing the electron beam exposure for the bridge and the banks. C3. Fabrication of larger arrays

In the fabrication of the larger arrays, the problems encountered in the construction of the 2 bridge sample were accentuated. These arrays consisted of five layers of thin films and hence, high reliability was required at each step. Since soft metals like indium, gold and lead were used, diffusion between layers presented a potentially serious problem. For this reason, standard fabrication procedures could not be used for many of the steps, and new techniques had to be developed as outlined in this section. The layers involved in the array are shown in fig. 61 and the fabrication steps are as follows: a) Fabrication of microstrips: As with the 2 junction cells, the arrays were made in the gap of a 5011

A.K. fain et aL, Mutual phase.locking in Josephson junction arrays

423

PbIn SiO

~)~J3~lJ!J~J’ 3, ;;~,)3’Y~3 J5~3~!JUL— Indium Substrate Fig. 61. Cross-section of the 51-junction array showing the different layers.

microstrip (fig. 18) on a 1/2” square sapphire substrates 0.025” thick. After cleaning the substrates, the 5011 gold chrome microstrip and registration marks for subsequent layers were patterned using photolithography and ion beam milling, as described in section C2. b) The second layer (fig. 61) consisted of the junctions and the initial layers of the SQUID loops. The mask for this layer was made with E.B.L. using a 1500 A thick PMMA layer baked at 180°C.(The processing steps involved in E.B.L. are discussed in section C2.) The section of this mask containing the bridges and the banks was exposed using a low electron beam current (30 pA) for high resolution; the SQUID loops, which required lesser resolution, were made using a beam current of (200pA) to reduce the beam writing time. In spite of this, the writing time for the largest arrays was long enough for significant drift to occur in the E.B.L. system. The effects of this had to be compensated by reregistering the sample a number of times during the exposure. The indium film (1500 A) was deposited from a thermally heated boat onto the substrate which was cooled to 77 K. c) Fabrication of the dielectric for the feedback stripline: After lift-off of the indium, the mask for the SiO layer (fig. 61) was made using a 6000 A thick layerof unbaked PMMA. The resist for this layerwas left unbaked to prevent the formation of holes and hillocks in the indium film. A low beam current was required for exposing this mask in order to prevent the bridges from blowing out. The silicon monoxide evaporation (5000 A) was carried out in an oxygen atmosphere at a pressure of 1 X iO~torr. d) Fabrication of the feedback resistor: The resistors required in the feedback loop to prevent the bias current from being shorted out was made on top of the SiO using lift-off from a 1000 A Au film evaporated thru a mask made using E.B.L. in unbaked PMMA. e) Fabrication of PbIn SQUID loops and feedback loop: The mask for the fifth and final layer for the PbIn portions of the SQUID loops and the feedback loop, was made in a 1 p~mthick PMMA layer (fig. 61). This thick PMMA film required some baking to prevent it from cracking during development. It was found that a temperature of 120°Cwas sufficient for this and at the same time minimized damage to the In film. The PbIn evaporation was performed in an ion mill without breaking vacuum after cleaning the sample with 1 keV argon ions for 30 seconds. Indium (1900 A) followed by 5100 A of lead was evaporated at a pressure of 1 x 10_6 torr. The ion milling was necessary to obtain good superconducting contact between the two indium layers. The thicknesses of these two layers were chosen such that the two materials formed a stable PbIn alloy, thus minimizing diffusion to the pure In film. Using the shadow evaporation technique described in section 5, 0arrays with variable thickness bridges have also been fabricated. The bridges in these arrays were 700 A thick, with banks 3000 A thick. To obtain good coverage, the SiO layer was made to be 7000 A thick. The final thickness of the PbIn SQUID loops for these arrays was increased to 10 000 A. Appendix D: DC interaction between the junctions

The dc interaction between the junctions provides a convenient way of determining the conductivity of the coupling loop. For V2 0 the apparent critical current of junction 1, I~is related to the actual

424

AK. lain et a!., Mutual phase-locking in Josephson junction arrays

critical current by ‘Ci

1ci~V~/R~+ii

where we have added a correction term ~, which does not depend on the relative direction of the bias currents. This symmetric term includes the effects of heating and other symmetric interaction~[21]. Thus the difference in the apparent critical currents for series and parallel biasing determines Re. Such measurements show R~to be independent of voltage, though Re does depend on the temperature. The sum of the apparent critical currents for the two biasing schemes given an estimate of the temperature rise of junction 1 due to heat generated in by junction 2 and the shunt resistor. The amount of heat generated is dominated by the contribution from the shunt resistor for high voltages, and results in a temperature rise at junction 1 of about 25 uK/nW, which in turn results in a decrease in the critical current of 0.1 uA/nW. References [1] 1.0. Kulik and 1K. Yanson, Josephson Effect in Superconductive Tunneling Structures (Israel Program for Scientific Translations, Jerusalem, 1972). [2] L. Solymar, Superconductive Tunneling and Applications (Chapman and Hall, London 1972); T. Van Dozer and c.w. Turner, P inciples of Superconductive Devices and Circuits (Elsevier, New York, 1981). [3]K.K. Likharev and B.T. Ulrich, Systems with Josephson Junctions: Basic Theory (Moscow University Publications, Moscow, 1978) Chapters 4, 10, 12 (in Russian). [4] AN. Vystavkin, V.N. Gubankov, L.S. Kuzmin, K.K. Likharev, V.V. Migulin and V.K. Semenov, Rev. Phys. AppI. 9 (1974) 76. [5] JR. Waldram, Rep. Prog. Phys. 39 (1976) 75. [6] K.K. Likharev, Rev. Mod. Phys. 51(1979)101. [7] T.D. Clark, Phys. Lett. 27A (1968) 585. [8] T.D. Clark, Physica 55 (suppl.) (1971)432; also in: Superconductivity (North-Holland, Amsterdam, 1971) p. 432. [9] DJ. Recipi, L. Leopold, W.D. Gregory, M. Behravesh and T. Thompson, Proc. 1972 Appl. Supercond. Conf. (IEEE, New York, 1972) p. 701. [101T.F. Finnegan and S. Wahlsten, AppI. Phys. Lett. 21(1972) 541. [11] ID. Clark, Phys. Rev. B8 (1973)137. [12]T.F. Finnegan, J. Wilson and J. Toots, Rev. Phys. AppI. 9 (1974) 199. [13]J.E. Mooij, CA. Gorter and J.E. Noordam, Rev. Phys. AppI. 9 (1974)173. [14]D.W. Palmer and i.E. Mercereau, Appi. Phys. Lett. 25 (1974) 467. [15] D.W. Jillie, i.E. Lukens and Y.H. Kao, IEEE Trans. Magn. MAG-li (1975) 671. [16] D.W. Palmer and J.E. Mercereau, IEEE Trans Magn. MAG-il (1975) 667. [17] T.D. Clark and P.E. Lindelof, AppI. Phys. Lett. 29 (1976) 751. [18]D.W. Jillie, J.E. Lukens, Y.H. Kao and G.J. Dolan, Phys. Lett. 55A (1976) 381. [19] D.W. Palmer and I.E. Mercereau, Phys. Lett. 61A (1977)135. [20]D.W. Julie, J.E. Lukens and Y.H. Kao, IEEE Trans. Magn. MAG-13 (1977) 578. [21] D.W. Jillie, J.E. Lukens and Y.H. Kao, Phys. Rev. Lett. 38 (1977) 915. [22] G.M. Daalmans, TM. Klapwijk and J.E. Mooij, IEEE Trans. Magn. MAG-13 (1977) 719. [23]P.E. Lindelof, J. Bindslev Hansen, J. Mygind, N.F. Pederson and OH. Soerensen, Phys. Lett. 60A (1977) 451. [24] P.E. Lindelof and J. Bindslev Hansen, J. Low Temp. Phys. 29 (1977) 369. [25] J.E. Lukens, R.D. Sandell and C. Varmazis, Future Trends in Superconductive Electronics, AlP Conf. Proc. No. 44 (1978) p. 298. [26] R.D. Sandell, C. Varmazis, AK. lain and I.E. Lukens, Future Trends in Superconductive Electronics, AlP Conf. Proc. No. 44 (1978) p. 327. [27]C. Varmazis, R.D. Sandell, AK. lain and i.E. Lukens, AppI. Phys. Lett. 33 (1978) 357. [28]P.E. Lindelof, J. Bindslev Hansen and P. Jespersen, Future Trends in Superconductive Electronics, AlP Conf. Proc. No. 44 (1978) p. 322. [29]M. Stern, W. Howard and Y.H. Kao, J. de Phys. 39 (1978) 577. [30]R.D. Sandell, C. Varmazis, A.K. lain and i.E. Lukens, IEEE Trans. Magn. MAG-15 (1979) 462. [31] AK. Jam, P. Mankiewich and J.E. Lukens, Appi. Phys. Lett. 36 (1980) 774. [32] D.W. Jillie, M.A.M. Nerenberg and J.A. Blackburn, Phys. Rev. B21 (1980) 125. [33] A.K. Jam, P.M. Mankiewich, AM. Kadin, RH. Ono and J.E. Lukens, IEEE Trans. Magn. MAG-17 (1981) 99. [34] A. Davidson, IEEE Trans. Magn. MAG.17 (1981) 103.

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