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Surface Curvature andVortex Stability P. Voll, N. apRoberts-Warren, and R.J. Zieve Physics Department, University of California at Davis Weexaminethestabilityofapinnedsuperfluidheliumvortexlinebymeasuringitspersistenceatelevated 6 temperatures. Each vortex terminates at the surface of the container, at either a rounded bump or a conical 0 indentation. We find that pinning with the bump termination is much easier to overcome thermally. This 0 behaviorwouldnotbeexpectedfromconsiderationsofvortexlinetensionalone. Wetaketheobservationsas 2 evidence of anadditional contributiontothepinning energeticsarisingfromtheinteraction of thesuperfluid n order parameter singularity with the curvature of the container’s surface. By favoring pinning at points of a negativeGaussiancurvature,thesurfaceinteractionmakesthebumpalessadvantageouspinsite. J 8 1 Topologicaldefectsplayimportantrolesinavarietyofsys- pressurethatattractsthevortextothisposition. tems. Cosmic strings, defects created in the early universe, The present experimenttests the pinning of a vortex by a ] mayprovideanexperimentallyaccessibletestofstringtheo- bump. Since a roughly hemispherical bump on a flat back- n o ries[1]. Domainwallsaffectthebehaviorofmagneticrecord- ground has positive Gaussian curvature everywhere except c ing heads and other magnetic sensors [2], and flux line mo- near the rim where the bump meets the flat surface, the pre- - tion in superconductors introduces dissipation. Topological dictedcurvatureinteraction[5,6]wouldfavorvortexpinning r p defects may also governproteinfolding mechanisms[4]. In aroundtheedgeofthebump,ratherthanatitspeak. u somesituations,suchasliquidcrystalcoatingsofsmallparti- Weuseastraightvibratingwiretotrapasinglevortexinsu- s cles,defectsmustalwaysexist[3]. . perfluid4He. Asdescribedelsewhere[12],wedetectthevor- t Recent work has explored how topological defects inter- a texthroughachangeinthebeatfrequencyofthewire’slow- m act with surface curvature [5, 6], a problem relevant to sys- estnormalmodes. Themeasurementsaredoneonapumped - tems involving coated particles or flexible membranes. The 3He cryostat, which we rotate to create vorticity. All mea- d energeticsgoverningthe variation of the orderparameterfa- surements,however,takeplacewiththecryostatstationary.If n vorsiteswithnegativeGaussiancurvature.Thuspointdefects avortexbecomestrappedalongtheentirelengthofthewire, o confinedtoatwo-dimensionalcurvedsurfacetendtoposition an especially stable configuration ensues. Unless we delib- c themselves at saddle points. The same arguments extend to [ erately disturb the vortex in some way, it usually remains in a three-dimensional system with line defects that terminate placeuntilthecryostatwarmstonearorabovethesuperfluid 1 alongacurvedboundary. transition temperature. This behavior was the basis for the v Vortices in superfluid helium are precisely such defects. original demonstrations of quantized circulation in superflu- 5 9 Each vortex core must either close on itself, forming a vor- ids 4He [13, 14] and 3He [15]. With mechanicalor thermal 3 texring,orterminateatasurfaceofthehelium. Thissurface perturbation,thevortexcandislodgefromthewire,leadingto 1 canbeafreesurface,ifoneexists, orawallofthecontainer circulationvalues intermediateto the expectedquantumlev- 0 holding the superfluid. In the latter case, experimental and els.Acommonconfigurationafterthevortexcomesfreeisfor 6 computationalworkshowsthattheendofthevortexcanbe- oneendofthe vortexto leave thewire andprogressthrough 0 comepinnedinplace. thecell,terminatingonthecylindricalwall[12,16]. Notsur- / t Earlymeasurementsof torsionaloscillatordampingfound prisingly,suchapartiallyattachedvortexhasanintermediate a m acomplicatedfrequencydependencethatcanbeexplainedby effectonthenormalmodes. Witha50mmwire,wecande- waves along pinned vortices [7]. Measurements of thermal tectthepositionoftheattachmentpointinthisconfiguration - d counterflow[8] androtationalacceleration[9] also gaveevi- tobetterthan10µmprecisionforavortexdetachingnearthe n dence for pinningof vorticeson roughwalls. A moredirect middleofthewire. o experimentalverification [10] tracked the motionof a single Our measurements focus on the thermal stability of a c vortexalongthecellwall,includingoccasionalpinningevents trapped vortex line. We assume that as a vortex comes free : v whenthemotionceased. Insomecasesthevortexworkedits fromthewire,itmovesalongtheendofthecelluntilitreaches Xi wayfree,whileinothersitremainedpinneduntilthehelium the cylindrical cell wall. If the cell endcap is flat, as in the leftitssuperfluidphase. leftmostinsetofFigure1,thenthevortexlengthincreaseslin- r a On the computational side, Schwarz uses a single vortex earlyasitleavesthewire.Furthermore,sincethewireismuch terminatingonahalf-infiniteplaneandexposedtoaconstant largerthan the free vortexcore, the energyper length of the external flow velocity parallel to the plane [11]. For suffi- vortexislargerwhenitisnottrappedonthewire. Ifthevor- cientlylowflowrates,iftheplanehasahemisphericalbump texendisafixeddistancefromthewire,minimizingthetotal nearenoughtothevortex’spath,thevortexwillspiralontothe vortexenergyshowsthatthedetachedportionmakesanangle bump and remain pinned there. The simulations correspond θwiththewire,wherecosθ = EW. HereE andE arethe EF F W welltoanaivepictureofvortexpinning: byterminatingatop energiesperlengthofafreevortexandofavortextrappedon the hemisphere, the vortex has less length and correspond- thewire, respectively. Using a freevortexcoreradiusof1.3 ingly less kinetic energy. In an alternative formulation, the A˚, a wire radiusof 8 µm, and a cell radiusof 1.5 mm gives fluidvelocityishighestatopthebump,causingareductionin θ = 71◦. With this value, we can then calculate the change 2 40 sponds in Figure 1 to considering the energy change for the dashedcurvebetween1mmand1.5mm,ratherthanfrom0 to1.5mm. Thebumpsare formedfromStycast1266. We beginwith J)30 a stycast surface cut flat with an end mill. To get an aspect 6 −1 ratio near 1 for the bump, we add a droplet of Stycast 1266 0 1 thatisalreadypartlysetandisfairlyviscous. Oncethebump ( e has dried, we measure its dimensions and drill a small hole g an20 through it for our wire. After the wire is in place, we add h c additionalstycasttocovertheholethatitemergesfrom. y g The measurements described here come from three cells. r e n Cell 1, of radius 1.5 mm, has one conical and one flat end. E10 Cells 2 and 3, of radius 3.5 mm, each have a bump at one end with the other end flat. The larger radius in cells 2 and 3,whichwasusedtoprovidespaceforthebump,shouldalso increasetheenergybarrierforavortexinthesecellstodepin. 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Incell2thebumphasheight1.9mmandradius1.9mm. Its Distancefromwire(mm) cross-sectionalongthecellendisquiteregularaswell,mak- ing it close to hemispherical. The bump in cell 3 has height FIG.1: Additionalenergyasvortexendmovesawayfromwire, if 1.2 mm and radius 1.8 mm, with an irregular cross-section. theendofthecellisflat(solid),conical(dotted),oraroundedbump Forbothcells2and3theStycastbumpsarenotperfectlycen- (dashed).Insetsshowthesedifferentgeometriesforthecellend. tered. Their closest approachto the cell wall is between 0.5 and 1.0mm. The wire of cell 1 is centeredmuchbetter, en- teringthe cellthrougha hole 0.44mm in diameterat the tip oftheconicalsurface.Thuspartofeachbumpisclosertothe inenergyasthevortexendmovesawayfromthewire,shown cellwallthanisthewireofcell1. asthesolidblackcurveinFigure1. Oncethisenergybarrier To test stability, we first trap a vortexso thatit appearsto is overcomeand the vortex has reached the cylindricalwall, cover the entire wire. We then heat the superfluid to test at its motiondownthe wall decreasesthe length ofthe trapped what temperature the vortex comes free. Since the damping vortex(andthecorrespondingenergy)withoutfurtherchange on the wire is too high for reliable measurementsabove600 totheaveragefreevortexlength. mK, we cool down after a few minutes to check whether an Changing the endcap geometry affects the energy consid- endof thevortexhasdislodged. If not, we raise the temper- erations. Iftheendcapisdrilledout,itresemblesthemiddle insetofFigure1. Thehalf-angleis59◦,theangleofthetipof atureagain,usuallyto a slightlyhighervalue,andrepeatthe cycling until the vortex does leave the wire. In some cases astandardtwistdrill. Thisgeometryreducestheenergyprice thevortexremainstrappeduntilthecryostatexhaustsits3He for the vortex to move away from the wire, since the vortex supplyandwarmsaboveT . Figure2showsatypicalheating length increases less than for a flat endcap. The dotted blue λ curveinFigure1showsthisnewenergy. Athirdgeometryforthecellend,sketchedintherightmost insetofFigure1,isaroughlyhemisphericalbump. Herethe geometry enhancesthe energyas the vortex leaves the wire. 1 ThedashedredcurveinFigure1plotstheenergyforahemi- 1.2 sphericalbumpwithradius1mm. In a conventional picture of vortex pinning, considering K) only the line energyof the vortex, one would expectthe en- )4 e ( ergybarriertothevortexleavingthewiretobelargestwhen /m atur tnhaetevsoortnexatceornmicinaaltseusrfoancea.bHuomwpe,vaenrd,tshmeaplrleedsticwtehdengeiotmteermtriic- (iκh0.75 0.8mper h e contributiontovortexenergyfromsurfacecurvaturealtersthe T situationforcellsthatendwithabump.Ifthegeometricterm is strong enough, then the pinned vortex will not follow the wire until it reaches the bump. Rather, the vortex will leave thewireandterminateinthenegativeGaussiancurvaturere- gionaroundtheedgeofthebump.Fromthere,thevortexneed 0.50 0.2 0.4 0.6 0.80.4 onlytraversetheremainingflatportionoftheendcaptoreach Time (hours) the cylindrical wall. Since the distance involved is shorter than if the wire began in the center of the endcap, the addi- FIG.2:Typicaltemperaturesequence(black,rightaxis)andcircula- tional energy required to reach the cylindrical wall is lower tionbehavior(magenta,leftaxis)forcellwithbumpatoneend. than for either of the other two end geometries. This corre- 3 sequence, along with the signalwe observe from the vortex. Athighertemperatures,the dampingissolargethatwecan- The temperature is raised successively to 600 mK, 800 mK, notextract the circulationat all, so the circulation curvehas 1 K, and 1.2 K. (The temperature briefly overshoots its tar- gapsduringthehigh-temperatureregions. geteachtime,sincethecryostat’stemperaturecontrolisopti- We repeatedthisprocedurewithmanytrappedvorticeson mizedforrapidsettlingatanewtemperature.Theovershoots thethreewires. Theresultsaresummarizedinthehistograms arerepeatablefromonethermalcycleto thenext.) On cool- ofFigure3. Thefilleddiamondsrepresentcell1,theopencir- ingbackto500mK,thevortexclearlyremainsinplaceafter clescell2,andthex’scell3.Theuppergraphshowsthehigh- thefirstthreeanneals,butnotafterthelast. Weoftenfindvor- esttemperaturereachedwithoutdislodgingthe vortex,while texprecessionsignaturesoncethecirculationlevelfallsbelow thelowergraphshowsthetemperaturesatwhichvorticesdid hκi = 1, whichmeansthatthevortexisdislodgingfromthe comefree. Thetwographsdonotrepresentexactlythesame wireatoneend. Theextranoiseinthecirculationdataat600 vortices. Some vortices on the upper graph never dislodged mKcomesfromtheincreaseddampingofthevibratingwire. before the cryostat warmed up; in this case there is no cor- respondingpointon the lower graph, and the temperaturein the upper graph may be artificially low. Conversely, some vorticesonthelowergraphdislodgedontheirfirstannealing. Thatannealmay havebeenabovethe minimumtemperature 8 s needed to dislodge the vortex, so some points on the lower e c graphmaybeartificiallyhigh. Thekeyobservationisthatin n re6 the absence of the bump no vortex ever depinned below 1.6 r u K(lowergraph),whilewithabumpnovortexeverremained c c pinned above 1 K (upper graph). This dramatic difference o f 4 clearlyindicatesthatvorticesalongthewirearelessstablein o r thepresenceofabump. e b Inonerespect,thehistogramsdonotadequatelydisplaythe m 2 contrastbetween the cells. Each trapped vortex figures only u N once in each graph of Figure 3. In cell 1, a vortex is often extremely stable even at the highest annealing temperatures, 0 lastingthroughnotjustonethermalcycleto1.8Kbuttwenty 0.4 0.8 1.2 1.6 2 ormore. Thusthedifferenceinthenumberofthermalcycles Highest stable T (K) survivedbyvorticesisfargreaterthansuggestedbyFigure3. Inthepresentexperiment,wecannotdirectlydetectwhich endofthevortexdislodges.Weexpecteachcelltohavealess 8 stable end at which the vortex generally works free. Table I shows the different possibilities for where the vortex depins s e in each cell. Our measurementshows that the vortexdepins c n6 more easily in cells 2 and 3 than in cell 1; the implication e r fortherelativeenergybarriersisgiveninthethirdcolumnof r u c TableI. Notethatinthefirsttwocases,themeasurementssug- c o4 gestthata vortexdepinsmoreeasilyfromabumpthanfrom of a conicalend. Thesecondtwo cases giveno informationon r the relative stability of the bump and cone; but both lead to e b theunphysicalconclusionthatavortexdepinsfromaflatsur- m2 u facemoreeasily ina largercell, whereitmusttravelfarther. N Thusweconcludethatincells2and3thevortexdoesdepin 0 0.8 1.2 1.6 2 Depinning T (K) TABLEI:Possiblerelationshipsamongenergybarrierstodislodging avortex. Thefirsttwocolumnscompare theenergybarriersatthe FIG.3: Highest annealingtemperaturesthatfailstodepinavortex twoendsofeachcell.Thethirdcolumnincorporatestheexperimen- trappedathκi=1(upper)andtemperaturesthatdodepinsuchavor- talresultthatthebarrierishighestincell1. Termsinparenthesesin tex(lower). Insomecasesthecryostatwarmedupbeforethevortex thethirdcolumn simplyrepeat aninequality fromone of theother depinned; thisexplains some low temperatures intheupper graph. columns. Similarly, in some cases the first anneal was at ahigh temperature Cell1 Cells2and3 Measurementimplication and the vortex immediately depinned; this accounts for some high cone<flat1 bump<flat2 bump<cone temperatures in the lower graph. Data for three wires are shown; flat1<cone bump<flat2 bump<flat1(<cone) cells2(blackcircles)and3(redx’s)terminatedinbumpswhilecell 1(greenfilleddiamonds)didnot. cone<flat1 flat2<bump flat2<cone(<flat1) flat1<cone flat2<bump flat2<flat1 4 fromthebump,andthattheenergybarrierforthisprocessis thisvaluewithN=1.Wecannotdistinguishwhetherthisvalue smallerthanfordepinningfromtheconicalendofcell1. correspondstoavortexcoveringtheentirewireoronly98% Our stability result is the opposite of what would be ex- of it. We would find a directsignalonlyif the vortexsome- pectedfromthesimpleenergyconsiderationsofFigure1. As timescoveredtheentirewireandsometimesdetachedashort suggested above, an interaction between the vortex and the distancefromtheend. Wewouldthenobservemorethanone surfacecurvaturemayprovidetheexplanation. Althoughthe fairlystablecirculationlevelinthevicinityofN=1.Anappro- existing calculations consider a two-dimensionalsystem [5], priatelyshapedbump,withseveralmetastablepositionsfora wenotethattheinteractionenergy ρsκ2V isatleastcompara- vortextoterminate,mayallowthis. 4π bleinmagnitudetothelineenergiesinourgeometry.Hereρ For future measurements, we can determine at which end s and κ are the superfluid density and quantum of circulation, thevortexcomesfreebyusingacellwithadiameterchangein respectively,andV isacurvature-dependentfactoroforder1. themiddle. Afteravortexworksfree,itssubsequentpreces- Since our proposedscenario has the vortex pinningto the sion,whichdependsstronglyonthelocalcelldiameter[16], edge rather than the top of the bump, we address the possi- willidentifywhichhalfofthecellcontainsthedetachedpor- bility of a more direct measurement of vortex pinning near tion.Withthisinformationwecanbettercomparethepinning a bump. If a vortextrapped along the wire terminates at the strengthofdifferentendcapgeometries.Anotherplanforfur- edgeofthebump,thenthevortexmustleavethewireshortly therworkistopositionabumponthecylindricalwallmidway beforethe wire entersthe topof the bump. Theshortlength along the cell, where the measurementsensitivity is highest. ofwirewithnocirculationarounditwillslightlyaltertheob- We have previously seen oscillation signatures for a vortex served beat frequency. However, the accuracy of our mea- pinningonwallroughness[10],anditwouldbeinterestingto surementis muchlower than its precision, since an absolute seehowavortexbehavesonencounteringalargerandbetter- calibration depends on the mass per length of the wire. Al- characterizedobstruction. Quantitative measurementsof the though we know this density approximately, it varies by up pinstrengthmayalsobeeasierwiththeimprovedsensitivity. to 10% among wires. Each wire is a single strand of NbTi, Ourpresentresultsalreadydemonstratetheneedforanaddi- approximately16µmin diameter,cutfrommultifilamentary tionalcontributionto the energeticsof a pinnedvortex, such superconductingmagnetwire. Furthermore,thesensitivityis asaninteractionpotentialbetweenthevortexandthesurface muchlowerneartheendofthewire,sincetheinfluenceofthe curvature[5,6]. vortexonthewiredisappearsatavibrationalnode. Intypical We thankD. NelsonandV. Vitelliforhelpfuldiscussions. measurementsonevalueofthebeatfrequencynearwherewe ThisworkwassupportedbytheNationalScienceFoundation expectN=1isfarmorestablethananyothers,andweidentify underPHY-0243904andbyUCDavis. [1] T.W.B.Kibble,“CosmicStringsReborn,”astro-ph/0410073. [10] L.A.K.Donev, L.Hough, andR.J.Zieve,“Depinningofasu- [2] N.Spaldin,MagneticMaterials–FundamentalsandDeviceAp- perfluidvortexlinebyKelvinwaves,”Phys.Rev.B64180512 plications,(CambridgeUniversityPress,Cambridge,2003). (2001)andcond-mat/0010240. 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