Unusually thin Dayem bridges asQ-band mixers

P.K.D. Froome and A.H. Beck

Indexing terms: Mixers (circuits), Solid-state microwave circuits, Superconducting function devices

Abstract: Josephson junctions using Dayem bridges made from unusually thin films have been operated asmixers. The thickness of the film was made small to increase the r.f. impedance of the device and so toimprove r.f. matching, but such thin films often fail to act as Josephson junctions because of surface rough-ness. In the device reported here, attempts were made to obtain specular reflection of the electron wave func-tions from the surface. Our devices behaved as good Josephson junctions and gave an internal mixing efficiencyof 4-5% at 32 GHz.

1 Introduction

Mixers based on the Josephson effect may show conversiongain and operate up to the far infra-red.16 In most of thesuperconducting weak links that show Josephson behaviour,however, the two essentials for widespread use, mechanicalstability and a reasonable impedance, seem to be mutuallyexclusive. We have been investigating the Dayem bridge, aform of Josephson junction where the weak link is a smallbridge of superconductor connecting two relatively largeareas of superconducting film — typically lOOnm thick. Amicrobridge must not be much bigger than a micrometre inany direction if it is to be capable of showing the Josephsoneffect. The short length usually leads to a low resistance —typically 50 m£2. On the other hand, the small cross-sectionleads to the low capacitances that are necessary for goodhigh-frequency operation. The construction is mechanicallyrobust because the device is supported on a substrate. Inthis paper, we report on attempts to increase the deviceresistance to a few ohms, by making a very thin film, with-out destroying the weak-link properties.

A convenient theory to study is the one formulated byBaratoff, Blackburn and Schwartz by solving the Ginzburg-Landau equations for a simplified, 1-dimensional model.1

They find that a microbridge will behave as a Josephsonjunction provided that the temperature-dependent coher-ence lengthas of the order of the bridge length, and pro-vided that the electron mean free path in the bridge region(lb) is significantly reduced below that in the bulk of thefilm (la) by the collisions between the electrons and thesides of the bridge. Formally, the condition may be written

(1)

where £a and £b are the coherence lengths in the tworegions, and L is the length of the bridge. Thus, as themicrobridge is warmed up from a temperature well belowthe transition temperature (T<€.TC), weak-link behaviourwill begin when 2%b —L, and continue until 2£& /£a — L.

The impedance of a microbridge cannot be increased in-definitely by increasing its length because it rapidly be-comes impossible to fulfil the criteria of eqn. 1, andalthough narrowing the bridge gives a general improvementin performance, there is obviously a limit to which this canbe taken - in practice this is about 0-3 jum. Making the film

Paper T59S, received 21st March 1977Dr. Froome and Prof. Beck are with the University EngineeringDepartment, Trumpington Street, Cambridge CB2 1PZ, England

SOIJD-STA TE AND ELECTRON DEVICES, JUL Y 1977, Vol. 1, No. 4

thinner gives rise to difficulties if the electron free path be-comes so reduced by diffuse reflections with the entire sur-face that the additional reduction due to the width of thebridge becomes negligible, and, for the same reason, it isnot possible to increase the resistivity by increasing the con-centration of impurities and imperfections; i.e. making thefilm 'dirty'.

Suppose, however, that a bridge was made in a film witha surface such that many of the collisions with it werespecular (mirror-like). In such a film, the effective meanfree path of the electrons is independent of the thickness andso the resistance can be increased by making the film thin-ner without interfering with the weak-link behaviour. Thepreparation of metal surfaces that show some specularscattering of the conduction electrons is certainly possible,as has been demonstrated with both polycrystalline andepitaxial gold and silver films2~s and also in tin films6' 7. Itis probably an oversimplification to assume that such be-haviour implies an atomically smooth surface, although thesurface should certainly be as smooth as possible,8 and anangularly dependent specularity parameter, such as pro-posed by Parrott,9 is more likely. In this model, the scat-tering changed from nonspecular (specularity parameterp = 0) to specular (p = 1) when the angle of incidence ofthe electrons to the surface normal, 6, exceeds some criticalvalue, d0. The model predicts that ordering of the trajec-tories of the conduction electrons will occur at low tem-peratures, because, if surface collisions predominate, oncean electron starts moving on a path which meets the surfaceat an angle greater than 0o it will tend to continue on thispath, and so the value of p for these collisions will rapidlyincrease. Therefore the conduction electrons becomeordered in the bulk film into paths roughly parallel to thesurface. However, since the drift velocity due to an appliedelectric field is always small compared with the randommotion of the electrons, there is no preferred direction inthe plane of the film, and so electrons entering the bridgeregion will be doing so from all angles. Thus, the scatteringfrom the sides of the bridge will have p = 0; and further-more, since these nonspecular collisons will tend to deflectelectrons towards the film surface, the value of p for sur-face collisons in the bridge region will be reduced as well,tending to zero when the mean free path exceeds the bridgewidth. Hence, in a film where 0O <90°, it is possible toarrange at low temperatures for p ^ 1 in the bulk film butp ^ 0 in the bridge; in this case,the mean free path in thebridge region is determined by the film thickness ratherthan the bridge width, and so the bridge provides a moreeffective geometrical constriction than one of the same sizein a thicker film.

117

With this theory in mind, we decided to make some'clean' films about 50 nm thick with surfaces as smooth aspossible. Bridges made in such films should have a largeratio of /<,//& at low temperatures, and, hence, behave asideal Josephson junctions over a finite temperature rangewhile having much higher resistance than usual.

2 Preparation of the films

Indium was chosen for the experiments because it wetsglass, and glass should form a good smooth surface. Indiumhas a high adatom mobility on soft glass substrates at roomtemperature, when evaporated in the vacuum of 10~7 torr.This causes the formation of disconnected islands up to5/im in diameter rather than a continuous film. To reducethis adatom mobility the substrates were cooled by a suit-able holder1 s which could have its temperature controlledto within 2K anywhere between 110K and room tempera-ture.

Although cooling down to 140 K produced films whichwere thinner when they first started to coalesce sufficientlyto conduct, their resistance increased rapidly as they werewarmed. At room temperature they appeared as grey filmsformed by the coalescence of islands about 1 jum in dia-meter. At 140K the films would coalesce when they wereno more than 50 nm thick, but this network structure wastoo coarse for reproducible microbridges.

To achieve a more homogeneous film, we tried 'freezing'the structure in the low-temperature form by admittingoxygen to a pressure of about 10~5 torr for a minute or twoafter the deposition. This treatment, which has been usedsuccessfully before on tin films,10 oxidises the surface andmakes agglomeration more difficult. Films made this waywere shiny, but could be seen through if held up to thelight and examination using a scanning electron microscopeshowed that the islands were now separated by narrow thinregions about 500 nm apart and 300 nm long. We decidedto make some bridges in these films and see how theybehaved.

3 Manufacture of the microbridges

We found that a reliable method of making Dayem bridgesis the etching-and-scribing method developed by Gregers-Hansen.11 In this method, a glass substrate is scratched witha single-edged razor blade and etched for a short time inhydrofluoric acid to produce a fairly deep, but narrow,groove. Next, a thin film of superconductor is evaporatedover the etched groove, and then a fine scratch is madethrough the film, perpendicular to the etched groove, withanother razor blade; this scratch separates the film into twoexcept where it has gone down into the groove, where themicrobridge is formed. The second scratch can be madefiner than 1 /nm if the razor blade is freely mounted butheld in such a way that it cannot twist or skate over thesurface. Details of the simple mount used can be found inReference 15. Bridges made in this way were, typically,800 nm square.

4 D.C. measurements

According to the theory for Josephson tunnel junctions,

IT =2eR.

(2)

the superconductor, and Rn is the normal-state resistanceof the junction.12 Experimentally, it is found that eqn. 2holds not only for tunnel junctions but also for all othertypes of weak link. For reduced temperatures (t = T/Tc)greater than about 0-9, A, and hence / j , is proportional to1 — t, and this is a convenient test for Josephson behaviour.Dayem bridges have such an Ij against t curve, for tempera-tures where eqn. 1 is satisfied. Song and Rochlin13 havefound that the lower temperature limit of Josephson behav-iour is marked by a sudden change in slope of this curve asIj tends towards lower bulk-like values. Fig. 1 shows such aplot for one of our bridges, and it shows two discontinuities,a lower temperature one, similar to that observed by Songand Rochlin for their best bridges, although our films weremuch thinner than their successful films, and also a hightemperature one, which, presumably, represents the pointat which the other inequality given in eqn. 1 ceases to hold.

where Ij is the maximum supercurrent that a junction cancarry, 2A(r) is the temperature-dependent energy gap of

118

0-76 078 060 0-82 0-84 0-86 088 0-90 0-92 094 0-96 098 10T/T.

Fig. 1 Ij against TJTC for bridge E 12/2

The device behaves like an ideal Josephson junction over the moresteeply sloping part of the curve

Eqn. 2 also relates Ij and /?„ at any particular tempera-ture. Table 1 gives the measured values of Rn for fourbridges and the values calculated from the observed Ijs onthe steeply sloping portions of the Ij against t plots. Thecalculated values are systematically very low, but this is tobe expected because the theory assumes a uniform junction,and so the calculated Rn is the value that would occur werethe film ideally flat. We corrected for the uneven film asfollows: the parameters of each film that are most readilyobtained are the average thickness (measured using either aquartz-crystal monitor during deposition, or an interfer-ometer) and the electrical resistance. The presence of short,but very thin, regions in a newly coalesced film will increaseits resistance above the value it would be were the materialevenly distributed, and so if the average thickness and theresistance are combined to give a value f' for the resistivityof the film the number obtained is larger than the true resis-tivity f according to the relationship

f = S[T(r)]$ (3)

where S is a complicated function depending upon thedetails of the position-dependent thickness r(r) and, hencevarying from specimen to specimen. The temperature-dependent part of f' can be calculated from the resistancesat room temperature and 77 K, and, assuming a bulk valuefor f, S can be found for each film. S is the ratio of theactual resistance to that worked out assuming a flat film;thus, the calculated values for Rn should be multiplied by Sto give the correct theoretical estimate, and this is done inthe fourth column of Table 1. The agreement betweentheory and experiment is now very satisfactory, especially

SOLID-STATE AND ELECTRON DEVICES, JUL Y 1977, Vol. 1, No. 4

as the value of S used is an average over the whole film,whereas the bridge occupies such a small area that somelocal variation is inevitable; it suggests that all four bridgesare behaving as weak links on the more steeply slopingportion of the h against t plot although the film thicknessis very much less than the bridge width, and, hence, thatthe desired surface properties have been achieved.

Table 1 Normal-state resistance of four microbridges

Bridge

0 2/3£6/3£12/1£12/2

Thickness of film

nm60566075

Measured

n15

21-31-7

Calculated

f*n

Q.6-70-460-690-67

CalculatedX S

D.122-21-21-6

i.f. output

standardmixer mount

wg 22 waveguide

glasssubstrate

plug-indevice holder

Fig. 2 Schematic of mixer mount

The d.c. bias leads have been omitted for clarity. The microbridge isvertical, with the plane of the film perpendicular to the plane of thediagram

bias current

Fig. 3 Current-voltage characteristic and i.f. output characteristicfor bridge H 8/1

Upper curve is the junction voltage: scale 0-5 mV/division Bias cur-rent: scale 0-5 mA/division Lower curve is the i.f. outputTaken at T/Tc = 0-75. This bridge had Rn = 2-6SI, and was made ina film 50 nm thick. The local oscillator and signal source were at32 GHz. The mixing efficiency, defined in this case as (Vh/Rdyn)l(I%Rn), is 4-5% at this temperature where Rdyn = 0-317, but sinceVif = IgRfiyn, the efficiency should increase greatly at temperaturesnearer Tc, where Rdyn > ^n

5 Mixing at Q-band

We carried out the mixing experiments using a standardQ-band mixer mount modified by replacing the crystal witha Dayem bridge, as illustrated in Fig. 2, and immersing themount directly in the liquid helium bath. This simplemethod has two drawbacks. First, the r.f. power is absorbedonly by the very small microbridge region where the deviceis not superconducting. Secondly, the temperature fluctu-ations in the bath cause variations in the critical current andthis produces noise in the i.f.

Fig. 3 shows a typical variation in the output at the30 MHz i.f. as the bias current is varied but the 32 GHzsignal and local oscillator are kept constant in power. Thecurrent-voltage characteristic of the bridge is also given.

Although it is difficult to measure the signal (or l.o.) cur-rent directly, it can be deduced from the value of the maxi-mum supercurrent in the presence of the signal. Accordingto theory,14 as the signal power increases from zero, thefirst complete suppression of the critical current (which hasan amplitude that varies with r.f. power in a manner resem-bling a zero-order Bessel function) occurs when the signalcurrent is equal to 1-1 Ij\ this enables the r.f. sources to becalibrated in terms of the induced currents. For the con-ditions of Fig. 3, the l.o. power present in the device was600 nW, which is about 2% of the estimated available powerfrom the generator; the signal power in the device was 20nW, with a similar coupling efficiency.

We also calculated the i.f. output indirectly, using thefact that, as the signal power increases, the i.f. output dueto the modulation of an I-V curve with well defined stepswill saturate when Vif equals the separation of the constant-voltage portions of the steps, as is explained below. In thiscase, the saturation is at 32juV since the steps are separatedby about 1 /uV for every 500 MHz of l.o. frequency. Fig. 3was taken just as this saturation was occurring, and, hence,the i.f. peaks have an amplitude of 32 jitV. Since the coup-ling was so poor we thought it most informative to workout the conversion efficiency in terms of the i.f. and signalpowers actually present in the device, and, in this case, itwas 4-5%.

Josephson-effect mixing takes place as follows: if a weaklink is exposed to microwave radiation, steps appear in thecurrent-voltage characteristic at voltages corresponding tothe frequency of the incident radiation and its harmonics,the amplitude depending in an oscillatory way on theintensity of the radiation. Small-area weak links respond toa small applied r.f. signal and a local oscillator as if the l.o.was amplitude modulated by the signal;14 the resulting in-tensity fluctuation, at the difference frequency, varies thecurrent amplitude of the constant-voltage steps producedby the l.o., and, if, as in our case, the device is currentbiased, an i.f. voltage is produced with an amplitude pro-portional to Rdyn, the dynamic resistance at the biaspoint.17' 18 It is clear from Fig. 3 that the i.f. peaks at cur-rents corresponding to regions of maximum Rdyn< ratherthan at points of maximum curvature in the I-V plot aswould occur if classical mixing was taking place.

In our experimental arrangement, the temperature fluc-tuations in the helium bath made it impossible to bias thedevice between steps unless they were adjusted, by loweringthe temperature below the ideal regime, to be very rounded.At temperatures nearer Tc values for Rdyn of several ohmswere observed. If the devices had, for example, been oper-ated in an evacuated chamber where temperature fluctu-ations were much less, it would have been possible to

SOLID-STATE AND ELECTRON DEVICES, JULY 1977, Vol. 1, No. 4 119

bias the device on such steep steps. This would have led to ahigher efficiency and possibly even conversion gain.

6 Conclusions

Microwave mixers, using very thin films of indium con-structed as Josephson-junction Dayem bridges, have beentested. Up to 4-5% internal conversion efficiency has beenfound. With more stable temperature surroundings, micro-wave resistances of a few ohms, and even higher conversionefficiencies, could have been obtained. The films are un-usually thin for Dayem-bridge Josephson junctions. Theirsuccess is attributed to specular reflection of the electronwave functions from the surface of the indium films, possiblywith an angular dependent scattering parameter afterParrot.9

7 References

1 BARATOFF, A., BLACKBURN, J.A., and SCHWARTZ, B.B.:'Current-phase relationship in short superconducting weak links',Phys. Rev. Lett., 1970, 25, pp. 1096-1099

2 G1LLIIAM, E.J., PRESTON, J.S., and WILLIAMS, B.E.: 'Astudy of transparent, highly conducting gold Films', Philos. Mag.,1955,46, pp. 1051-1068

3 CHOPRA, K.L.: 'Size effects in the longitudinal magnetoresist-ance of thin silver films', Phys. Rev., 1967, 155, pp. 660-662

4 ABELES, F., and THEYE, M-L.: 'Simultaneous determination ofthe scattering parameter and the mean free path of conductionelectrons in thin films', Phys. Lett., 1963, 4, pp. 348-349

5 LARSON, D.C., and BOIKO, B.T.: 'Electrical resistivity of thinepitaxially grown silver films', Appl. Phys. Lett., 1964, 5, pp.155-156

6 LEARN, J.L., and SPRIGGS, R.S.: 'Behaviour of film conduc-tance during vacuum deposition', J. Appl. Phys., 1963, 34, pp.3012-3021

7 NIEBUHR, J.: 'Der elektrische Widerstand dunner Zinnschichtenmit Gitterstoerungen', 2. Physik, 1952, 132, pp. 468-481

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10 CASWELL, H.L., and BUDO, Y.: 'Influence of oxygen on thesurface mobility of tin atoms in thin films',/. Appl. Phys., 1964,35,644-647

11 GREGERS-HANSEN, P.E., and LEVINSEN, M.T.: 'Normal-stateresistance as determining parameter in behaviour of Dayembridges with sinusoidal current-phase relations', Phys. Rev. Lett.,1971, 27, pp. 847-849

12 AMBEGAOKAR, V., and BARATOFF, A.: Tunnelling betweensuperconductors', ibid, 1963, 10, pp. 486-489, and 1963, 11,p. 104 (erratum)

13 SONG, Y., and ROCHLIN, G.I.: 'Transition from bulklike be-haviour to Josephson-junction-like behaviour in superconductingmicrobridges', ibid., 1972, 29, pp. 416-419

14 RUSSER, P.: 'Influence of microwave radiation on current-voltage characteristics of superconducting weak links', /. Appl.Phys., 1972, 43, pp. 2008-2010

15 FROOME, P.K.D.: 'Microwave applications of superconductingmicrobridges', Ph.D. thesis, University of Cambridge

16 MCDONALD, D.G., LOSE, V.E., EVENSON, K.M., WELLS, J.S.,and CUPP, J.D.: 'Harmonic generation and submillimeter wavemixing with the Josephson effect', Appl. Phys. Lett., 1969, 15,pp. 121-122

17 GRIMES, C.C., and SHAPIRO, S.: 'Millimeter-wave mixing withJosephson junctions', Phys. Rev., 1968, 169, pp. 397-406

18 TAUR, Y.,CLAASSEN, J.H., and RICHARDS, P.L.: 'Conversiongain in a Josephson effect mixer', Appl. Phys. Lett., 1974, 24,pp. 101-103

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