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The electrical behavior of superconducting thin-filmmicrobridges self-heating and superconducting quantum

processesW.J. Skocpol, M. R. Beasley, M. Tinkham

To cite this version:W.J. Skocpol, M. R. Beasley, M. Tinkham. The electrical behavior of superconducting thin-filmmicrobridges self-heating and superconducting quantum processes. Revue de Physique Appliquee,1974, 9 (1), pp.19-23. �10.1051/rphysap:019740090101900�. �jpa-00243739�

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THE ELECTRICAL BEHAVIOR OF SUPERCONDUCTINGTHIN-FILM MICROBRIDGES

SELF-HEATING AND SUPERCONDUCTING QUANTUM PROCESSES (*)

W. J. SKOCPOL~, M. R. BEASLEY and M. TINKHAM

Department of Physics and Division of Engineering and Applied PhysicsHarvard University, Cambridge, Massachusetts 02138, USA

Résumé. 2014 Nous présentons les mesures des caractéristiques I-V de microponts supraconduc-teurs en couches minces et de ces mesures nous déduisons quelques nouveaux résultats concernantle comportement électrique de ces microponts. A basse température, nous trouvons que leurcomportement est dominé par les effets d’échauffement. Pour les faibles valeurs de la tension etprès de Tc où l’échauffement n’est pas important, nous trouvons que le comportement observéest très bien décrit par un modèle de glissement de phase et que les courants normaux qui circulentpendant le processus de glissement de phase ont une extension spatiale déterminée par la longueurde diffusion des quasi-particules. Même près de Tc l’échauffement devient important lorsque lestensions sont élevées; il est probable que ceci limite la tension maximale (et de là la fréquenceJosephson) jusqu’à laquelle le comportement de type Josephson peut être obtenu.

Abstract. 2014 Measurements of the I-V curves of superconducting thin-film microbridges are

reported and used to infer some new results regarding the electrical behavior of these devices. Attemperatures well below Tc their behavior is found to be dominated by the effects of self-heating.At low voltages near Tc where heating is not important, it is found that the observed behavior isbest understood in terms of a phase-slip mechanism and that the normal currents which flow duringthe phase-slip process extend over a range determined by the quasiparticle diffusion length. Evennear Tc heating becomes important at high voltages and probably puts a limit on the maximumvoltage (hence Josephson frequency) up to which useful Josephson-like behavior can be obtained.

REVUE DE PHYSIQUE APPLIQUÉE TOME 9, JANVIER 1974, PAGE

1. Introduction. - Recent experimental [1]-[3] andtheoretical [4]-[5] work has shown that superconduct-ing thin-film microbridges which are small comparedto the Ginzburg-Landau (GL) coherence length 03BE(T)(i. e., very near Tc) apparently exhibit classical Joseph-son behavior : a sinusoidal current-phase relationship,a linear temperature dependence of the critical current,and an ac Josephson effect with a quasi-Bessel functiondependence on microwave power. By comparison,considerably less attention has been paid recently tothe regime further below 7c? where for conveniencein making practical devices one would hope usefulJosephson behavior might be obtained, even if it isnot of the classical form. We have recently undertakena thorough study of the current-voltage (I- V) charac-teristics of a variety of microbridges (both long andshort) over a wide range of temperatures. In this paperwe attempt to summarize our major results and whatthey reveal about the underlying physical processesgoverning the electrical behavior of these devices.

In brief, our results show that well below Tc the

(*) Work supported in part by the Office of Naval Research,the National Science Foundation, and the Joint Services Elec-tronics Program.

t Danforth Fellow.

electrical behavior of these microbridges is largelydominated by the effects of self-heating. Only at lowvoltage (hence low power) levels near 7c is the beha-vior found to be solely governed by superconductingquantum processes. Moreover, in this intrinsic super-conducting regime, the observed I V curves can best beunderstood in terms of a phase-slip model like thatintroduced originally by Notarys and Mercereau [6]to describe the behavior of proximity effect bridges,rather than the popular shunted-Josephson-junctionmodel [7], which has been used frequently in the pastto interpret the behavior of superconducting micro-bridges. However, in contrast to the theory of thephase-slip process recently developed by Rieger,Scalapino and Mercereau (RSM) [8], we find that inthe tin microbridges we have studied, the normalcurrent flow during the phase-slip process extends overa length determined by the quasiparticle diffusion

length of Pippard, Shepherd and Tindall [9], and notthe Ginzburg-Landau coherence length as found byRSM. In what follows we present the evidence forthese conclusions and discuss their implications forapplications of these devices. In particular, our resultsindicate that self-heating probably plays a role in

limiting the maximum voltage (hence Josephson fre-quency) up to which these devices can exhibit usefulJosephson-like behavior.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:019740090101900

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2. Self-heating effects. - Figure 1 shows the 1- Vcurves for typical long and short tin microbridges atvarious helium bath temperatures. The evidence for

strong self-heating effects is immediately evident. Inboth cases the I- V curves show pronounced anomaliesdue to the onset of bubble nucleation at the samplesurface and a substantial change in the I V curve uponcooling through the superfluid transition. Both changesresult in improved heat transfer. The identification ofthe bubbling anomalies is confirmed by their absenceboth below the superfluid transition and in a vacuumenvironment. Further evidence for heating is obtainedfrom the generally good fit to the observed I V curvesobtained from a simple theory of self-heating (dottedcurves) in which resistance is assumed to arise from a

FIG. 1. -1- V characteristics for a long (40 pm by 3.0 ym by0.1 p) microbridge and a short (- 0.5 pm by - 0.5 pm by0.1 pm) microbridge directly immersed in liquid helium for aseries of bath temperatures. The solid and dashed curves areexperimental data ; the dashed curves are the dc average of acircuit-controlled relaxation oscillation at low bias voltages.The arrows along the current axis indicate the critical currents Ic.The effects of heating are obvious from the anomalies due tobubbling at the sample surface, and the superfluid transition,both of which increase the heat transfer. The dotted curves aretheoretical results obtained from a simple model of a self-heatingnormal hotspot (described in the text). The inserts show therelative size of a typical hotspot compared to that of the bridge.

fully normal hot-spot maintained above T, by Jouleheating. The deviations at low voltages are due toresistive superconducting behavior and are discussedin section 3. The dashed part of the measured I V cur-ves indicates the dc average of circuit-controlled rela-xation oscillations observed for voltage biasing in thisregion. When current biased the bridges show hystere-tic switching at Ic and at the minimum current reachedon each curve.

In the self-heated hostpot theory, which will bediscussed in detail in a subsequent paper [10], heat isgenerated in a fully normal region (see inserts in

figure 1) by an applied current I and is carried awayboth through conduction and heat transfer across theboundaries with the helium bath and sample substrate.The size of the normal region and hence the voltagefor the given current is determined self-consistently bythe condition that the resulting temperature distribu-tion has T = 7c at the superconducting/normal inter-face. For long bridges it is sufficient to assume thatthe temperature is maintained at the bath temperatureat the ends of the bridge. For short bridges it is essen-tial to at least approximate the spread of the normalregion into the wide film at the ends of the bridge. Theresults of our analysis are analytic expressions fromwhich the theoretical I-V curves of figure 1 were

obtained. In calculating these curves, most of the

parameters (e. g., bridge dimensions, the normal

resistivity p, and 7c) were independently determined ;only the thermal conductivity Il and the heat transfercoefficient oc were unknown and treated as adjustableparameters. The fitted values of x are consistent withthose found from p using the Wiedemann-Franz law.The fitted values ofoc fall in the range 1.0-3.5 W/cm2/Kfor transfer to the substrate and/or substrate plusnormal helium bath, and 7-9 W jcm2/K for transfer tothe substrate plus superfluid helium ; this is in goodagreement with published values. The calculatedcurves for short bridges are relatively insensitive to abecause most of the heat is carried away from thenormal region by conduction to the strong coolingarea provided by the wide part of the film at the endsof the bridge. For the same reason, they are substan-tially better cooled than the long bridges. While thedetails of both the observed 1 v curves (e. g., the

region of negative differential resistance) and theirinterpretation as a result of heating are quite interest-ing, our objective here is only to establish the impor-tance of self-heating in these devices and to demons-trate that at low temperatures and high voltages theheating effects can be satisfactorily understood with asimple model of a normal hotspot. A complete discus-sion of these effects and the theory will appear in oursubsequent paper.

3. Superconducting quantum processes. - As isevident in figure 1, at low voltages the data systemati-cally deviate from the predictions of the normal

hotspot model. In this regime superconducting resis-

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tive processes play a role in determining the I V curve.As T - 7c the importance of self-heating diminishesand the behavior becomes governed by these intrinsicsuperconducting processes. This diminution of the

importance of self-heating as T ~ 7c can be seen fromthe fact that the minimum current necessary to main-

tain a fully normal hotspot Im n (i. e., the current atthe point of infinite differential resistance in the theo-retical curves of figure 1) decreases only as (Tc- T)1/2,whereas the critical current, following the GL mean-field behavior, decreases as (Tc - T)312 Thus at

some temperature near Tc, Ic becomes less than Im nand superconducting resistive behavior unaffected byheating has an opportunity to develop. This relation-ship between Te and Im p is clearly evident in figure 1.The low voltage behavior near T, of the short bridge

of figure 1 is shown more fully and enlarged in figure 2.As is evident from this figure, at low voltage the I- Vcurves are very simple : as T - Tc, they are characte-rized by a temperature-independent differential resis-tance d V/dI and an « excess » average supercurrent of~ 1 2 Ic, (At lower bath temperatures the excess super-current is apparently less than 1 2 Ic.) At high voltagesthe I Y characteristics gradually curve toward the

voltage axis and smoothly join the heating dominatedpart of the characteristic.

FIG. 2. -1 V characteristics near Tc for the short bridge offigure 1, showing the heating effects at high voltages and thephase-slip behavior at low voltages. The large dots indicate

the voltage 2 A(T)le.

The form of the 1 V curves in figure 2 is very similarto that observed in proximity effect junctions, and canbe qualitatively understood in terms of a phase-slipmodel originally introduced by Notarys and Merce-reau [6]. In this model the phase slippage which isrequired at finite voltages by the Josephson equationV = (h/2 e) d({J/dt arises from a relaxation oscillationof the order parameter. A qualitative picture of thismodel is as follows. Because for I > I, some of thecurrent is carried by the normal electrons, on the ave-rage there must be a potential gradient along thebridge. An important consequence of this potential

gradient is that the superfluid velocity vs, which is

assumed to behave nearly as in a one-dimensionalsuperconductor, cannot be constant in time but mustaccelerate. However, once the supercurrent densityJ, = 2 e 1 VJ 12 v. reaches Jc, the order parameterbecomes unstable and rapidly collapses to zero, forcingthe supercurrent (but not vs) to zero also and necessi-tating a shift of all the current into the normal channel.The end result of this instability is a region (presuma-bly - 03BE(T) in length) in which # = 0, Js ~ 0, butacross which a phase différence Aç = 2 7r exists. Atthis point the system is assumed to « snap back »to a state where 0394~ has decreased by exactly 2 nand now VJ, Js, and Aç all are approximately equalto zero. At this point superconductivity can reformand the supercurrent reaccelerate. The resultingbehavior is a periodic increase and decrease of thesupercurrent density (at the Josephson frequency)between J., = 0 and J, = Jc coupled with a growthand decay of the amplitude of the order parameter,all of which culminates in an abrupt phase slip of 2 nat the end of each cycle. The dc I- V curve that resultsfrom this process is given by the expression

where Rn is the normal resistance of the region overwhich the normal currents flow during the phase slipprocess. The 1 Y curves shown in figure 2 are justof this form where Is(t) ~ 1 2 Ic and Rn is temperatureindependent. Moreover, as seen from the figure,once started the phase-slip process can persist evenfor I Ic.The details of the phase-slip process are obviously

quite subtle, involving not only complicated super-fluid dynamics, but also delicate issues such as nonequilibrium between the normal and superfluids aswell. Two relatively simple attempts to provide amore detailed theoretical description of this processusing time-dependent Ginzburg-Landau theory to

describe the superfluid dynamics have been under-taken : one in the paper by RSM and the secondmore recently by Tinkham [11]. In both of thesecalculations IS(t) is found to be N i Ic in good agree-ment with our results. However, these two theoriesdiffer substantially in their assumptions regarding thecharacteristic healing length for departures from

mutual equilibrium of the normal and superfluidcomponents of the superconductor, and consequentlylead to different predictions for Rn. We will returnto this issue below.We have found that at low voltages near Tc long

microbridges also exhibit phase slippage at localizedvoltage centers and that this provides the basis of anexplanation of the well-known voltage steps observedin the 1 Y characteristics of long superconductingmicrobridges and whisker crystals [12], [13]. Manysteps occur because the length of the bridges is largecompared to the size of a phase-slip center. By meansof voltage tabs located along the length of some of

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our long microbridges, we have directly establishedthat the observed steps arise from spatially localizedvoltage centers and that the onset current for a givenstep is primarily determined by local variations of Tc,and hence Ic, along the bridge. These variations in Tcwere established by measuring Ic(T) for the varioussegments of the bridge defined by the voltage tabs.Figure 3 shows the individual I V curves associatedwith the first three voltage steps (each correspondingto a different segment of the bridge) in the total I- Vcharacteristic of a 140 03BC long tin microbridge. Thecurves shown in figure 3 were chosen so as to corres-pond approximately to the same temperature differencefrom the local Tc’s. The curves obtained from thethree voltage units are virtually identical and beara striking resemblance to those observed with theshort bridge of figure 2. It seems almost certainthat the spatially localized voltage centers observedin the long bridges are phase-slip centers basicallylike those found in short bridges. Firm confirmationof this assertion lacks only demonstration of theac Josephson effect on one of the voltage plateausin the I v characteristic of a long bridge or whisker.

FIG. 3. - Low voltage behavior of the steps near Tc in the1 Y curves of a long (140 gm by 3 gm by 0.1 1 pm) microbridgewith voltage tabs along its length. The steps arise from phase-slip centers spatially localized in different regions of the bridge.The voltage plotted is the local voltage drop across each of thefirst three steps that appear. The curves presented were selectedto represent approximately equal temperature differences fromthe local Tc’s. The curves show a temperature independent slopeand average supercurrent of ’" tIc like those shown in figure 2,indicating that each voltage center of a long bridge acts just like

a phase-slip center in a short bridge.

Note added in proof. - We have now observed theac Josephson effect in phase-slip centers in a 120 03BClong, geometrically uniform bridge, as evidenced

by microwave-induced steps at the voltage corres-

ponding to the 10 GHz microwave frequency. As inshort bridges, subharmonic steps as well as the funda-mental step are observed, consistent with the anhar-monic current waveform predicted by the phase-slip model. When two phase-slip centers are present,the induced steps occur when the local voltage dropacross either center corresponds to the microwave

frequency ; no induced step is seen when the total

voltage across both centers corresponds to the micro-wave frequency.The importance of the results on long bridges

(in addition to providing an explanation of the observedvoltage steps) lies in the fact that they allow a precisedetermination of the length of the nonequilibriumregion over which normal currents flow during a

phase slip. If this length is denoted by Ln, then forlong microbridges the resistance Rn in (1) is givenby the simple expression

where LT and RT are the total length and normalresistance of the microbridge. For short bridges Rndepends only logarithmically on Ln.Using the data on the bridge shown in figure 3

(RT = 2.4 Q and LT = 140 03BCm), we find that Ln = 14 limand is temperature independent. Data on other longbridges yields values in the range 8-16 ym with thelonger values generally corresponding to the bridgeswith the longer mean free paths 1. Data from short

bridges are consistent with these values. Using thedata of Meyer and v. Minnegerode [13] for voltagesteps in tin whisker crystals, we find Ln lies in the

range 48-60 ym for these very clean (1 ~ 2 ym)systems. These measured values are not consistentwith the result Ln ~ 2 03BE(T) expected from the theoryof RSM. First of all, the observed L.’s are tempera-ture independent, and second, for the samples wehave considered, they are larger than 03BE(T) in the

temperature range under consideration.Since Ln is temperature independent near Tc it

seems natural to suppose that it is somehow relatedto the normal excitations, and not the superfluid.In fact our data are remarkably consistent with theidentification Ln - 2 A, where 039B = (lVF 03C42)1/2 isthe quasiparticle diffusion length introduced byPippard et al. [9], to account for the excess resistance.observed at SNS interfaces. Here VF is the Fermi

velocity and i2 is the inelastic scattering time (includingrecombination) for the quasiparticles. In calculating Afor our bridges, we have used VF = 108 cm/s,z2 - 4 x 10-10 s [14], and 1 = 0.1 ym and 2 )lm(as estimated from p) in microbridges and whiskersrespectively. Physically 039B represents tlle distance

quasiparticles can random walk before inelastic

scattering. A phenomenological model of the phase-slip process in which the healing length for departuresfrom equilibrium between the quasiparticles and thesuperfluid is given by 039B has been developed by Tink-ham [11 ] and is in good agreement with our results.It should be noted, however, that in Tinkham’s

theory it has been assumed that 039B 03BE(T). In theopposite limit A « 03BE(T), the theory of RSM may becorrect.

4. Conclusions. - The results presented in this

paper, along with the previous work on very smallbridges near Tc, allow us to draw the following ten-tative picture of the complete electrical behavior ofsuperconducting thin-film microbridges. Very near

Tc where 03BE(T) is large compared to the length Lof the bridge, classical Josephson behavior is appa-rently obtained. However, as the temperature is

decreased, or if the bridge is not short enough, so

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that 03BE(T) L, behavior like that observed in theseexperiments is obtained ; the bridge acts much more« bulk-like », having a current-phase relation

Iq ex. 1 VJ 12 ~~ and a critical current following themean-field behavior Ic oc (T,, - T) 3/2 , but stillexhibits Josephson-like behavior through the phase-slip mechanism. Unfortunately a completely satis-

factory theory of this phase-slip process is not yetavailable, although some progress has been made

through our work on long bridges, which has identifiedthe important role of the quasiparticle diffusion length039B in the nonequilibrium aspects of the process. Howlong the bridge can be before single-unit Josephson-like behavior is lost is not clear, but a plausible guesswould be L ~ 2 A when A > 03BE, rather that L N 2 03BE(T)as has been suggested previously, since as analyzedby Tinkham [11] ] the presence of one phase-slipcenter tends to stabilize adjacent regions against theformation of additional centers over a range 2 A.Since A can be quite long, this might be useful infabricating microbridge weak links.For temperatures far below Tc, the behavior of

these bridges at finite voltages appears to be almostcompletely dominated by the effects of self-heating,and in this region resistance is produced by a fullynormal hot spot rather than by phase slippage. Evennear 7c, where at low voltages self-heating is not

important, our results suggest that as the bias voltageis increased, self-heating does become important evenat voltages less than 2 A(T)/e. (See figure 2 where the

voltage 2 A(T)/e is indicated by the circles.) Asheating becomes important, the temperature of thebridge begins to rise and Ic and hence IS(t) are decreased.At higher voltages and correspondingly higher heatinputs, phase coherence is ultimately lost and a

hotspot forms. The curvature of the 1- V curves

toward the voltage axis in figure 2 appears to beindicative of just this process.

If this interpretation is correct, then clearly self-

heating plays a role in limiting the maximum voltageand hence Josephson frequency up to which thesedevices can operate. Moreover, preliminary resultsof an analysis of this aspect of the heating problemsuggest that self-heating should be important at

voltages less than but comparable to 2 A(T)le forsmall microbridges generally, regardless of the parti-cular material used. A comparison of the importanceof self-heating in various types of weak links suggeststhat point contacts, where substantial material is

available to carry away the heat in three dimensions,should suffer much less limitation due to heatingthan microbridges in high frequency applications.Proximity effect bridges, on the other hand, maysuffer the most severe limitations, since as they arepresently constructed the junction is effectively placedat the center of a long bridge and therefore is notcooled as efficiently as short microbridges. These

conclusions are clearly speculative, however, anddirect experimental confirmation of this limitationwould be highly desirable.

References

[1] FULTON, T. A. and DYNES, R. C., Phys. Rev. Lett. 25

(1970) 794.[2] GREGERS-HANSEN, P. E. and LEVINSEN, M. T., Phys. Rev.

Lett. 27 (1971) 847.

[3] SONG, Y. and ROCHLIN, G. I., Phys. Rev. Lett. 29 (1972) 416.

[4] BARATOFF, A., BLACKBURN, J. A. and SCHWARTZ, B. B.,Phys. Rev. Lett. 25 (1970) 1096.

[5] GREGERS-HANSEN, P. E., LEVINSEN, M. T. and FOG PEDER-SEN, G., J. Low Temp. Phys. 7 (1972) 99.

[6] NOTARYS, H. A. and MERCEREAU, J. E., Physica 55 (1971)424.

[7] MCCUMBER, D. E., J. Appl. Phys. 39 (1968) 3113 ; STEWART,W. C., Appl. Phys. Lett. 12 (1968) 277.

[8] RIEGER, T. J., SCALAPINO, D. J. and MERCEREAU, J. E.,Phys. Rev. B 6 (1972) 1734.

[9] PIPPARD, A. B., SHEPHERD, J. G. and TINDALL, D. A.,Proc. R. Soc. A 324 (1971) 17.

[10] SKOCPOL, W. J., BEASLEY, M. R. and TINKHAM, M., Bull.Am. Phys. Soc. 18 (1973) 302, and to be published.

[11] TINKHAM, M., to be published.[12] WEBB, W. W. and WARBURTON, R. J., Phys. Rev. Lett.

20 (1968) 461; also see WARBURTON, R. J. and WEBB,W. W., Bull. Am. Phys. Soc. 13 (1968) 379.

[13] MEYER, J. and v. MINNEGERODE, G., Phys. Lett. A 38 (1972)529; Proc. 13th Int. Conf. on Low Temp. Phys. (Boul-der) 1972 to be published.

[14] TINKHAM, M., Phys. Rev. B 6 (1972) 1747.