Thermally Assisted Penetration and Exclusion of Single Vortex in Mesoscopic Superconductors

0.56 MB·English

by  Shi-Zeng Lin

ThermallyAssistedPenetrationandExclusionofSingleVortexinMesoscopicSuperconductors
Shi-Zeng Lin1, Takahiro Nishio2, Lev N. Bulaevskii1, Matthias J. Graf1 and Yukio Hasegawa3
1TheoreticalDivision, LosAlamosNationalLaboratory, LosAlamos, NewMexico87545, USA
2AdvancedDeviceLaboratory,RIKENAdvancedScienceInstitute,2-1Hirosawa,Wako,Saitama351-0198,Japan
3TheInstituteforSolidStatePhysics,TheUniversityofTokyo,Kashiwa-no-ha,Kashiwa277-8581,Japan
(Dated:January25,2012)
Asinglevortexovercomingthesurfacebarrierinamesoscopicsuperconductorwithlateraldimensionsof
severalcoherencelengthsandthicknessofseveralnanometersprovidesanidealplatformtostudythermalac-
tivation of a single vortex. In the presence of thermal fluctuations, there is non-zero probability for vortex
penetrationintoorexclusionfromthesuperconductorevenwhenthesurfacebarrierdoesnotvanish. Wecon-
2 siderthethermalactivationofasinglevortexinamesoscopicsuperconductingdiskofcircularshape.Toobtain
1 statisticsforthepenetrationandexclusionmagneticfields,slowandperiodicmagneticfieldsareappliedtothe
0 superconductor.Wecalculatethedistributionofthepenetrationandexclusionfieldsfromthethermalactivation
2 rate.Thisdistributioncanalsobemeasuredexperimentally,whichallowsforaquantitativecomparison.

n
a PACSnumbers:74.78.-w,74.78.Na
J

4
I. INTRODUCTION Refs. [6–8]. Theimportanceofthermalfluctuationsinthese
2
cases can be estimated using the Ginzburg number G3D =
i
] 1(k T /[H (0)ξ(0)3])2, where k is the Boltzmann constant,
n Thermalactivationofmacroscopicquantumobjectsoveran 2 B c c B
T thecriticaltemperature, H (0)thethermodynamiccritical
o energybarrierisubiquitousincondensedmatterphysics. One c c
field and ξ(0) the coherence length at zero temperature[9].
c well established example is the thermally assisted escape of
r- phase particle in a Josephson junction when the bias current For conventional superconductors, G3iD ≈ 10−10 and ther-
p is close to the critical current corresponding to a vanishing malfluctuationsareweak;whileforhigh-Tcsuperconductors,
u barrier[1]. Combinedwithanalyticalcalculationsandexper- G3iD ≈10−2andthermalfluctuationsbecomeimportant.
s iments, a quantitative description has been achieved in this For a large size superconductor studied in Refs. [4–8],
.
t case. Magnetic vortices as topological excitations in super- many vortices penetrate into the superconductor simultane-
a
m conductors are difficult to excite thermally because of their ouslyoncetheappliedfieldreachesthepenetrationfield. To
largeself-energycomparedtothatofphaseparticleinJoseph- observe single vortex penetration, the lateral size of the su-
-
d sonjunctions. Foralargeofnumberofvorticestheprobabil- perconductorshouldbeofseveralcoherencelengthsonly. In
n ity of thermal activation increases, which allows for an ex- these mesoscopic superconductors, a single vortex enters or
o perimentalobservation. Thermalactivationofbundlesofvor- exists when the applied magnetic field is tuned. Due to the
c tices,suchascreepmotionofvortices[2], hasbeenobserved advancesinmicrofabricationtechniques, penetrationandex-
[
longtimeago. However,nodetailedquantitativeexperimen- clusionofasinglevortexbyadjustingappliedmagneticfield

1 talstudyofthermalactivationofasinglevortexhasbeenre- has been observed repeatedly in many experiments[10–12].
v portedsofarbecausetheprobabilityofthermalactivationfor InRef. [10], thepenetrationandexclusionofvortexismea-
8 asinglevortexisusuallysmall. sured by the Hall probe and in Refs. [11, 12] it is measured
4
byascanningtunnellingmicroscope(STM)tip. However,no
0 Vortices overcoming the surface barrier in finite-size su-
statisticsonthepenetrationandexclusionfieldhasbeencar-
5 perconductors has attracted considerable interest, because of
. the possible observation of thermal activation of a quantum riedouttoinvestigatethethermalactivationinthesestudies.
1
0 object. For a superconductor with finite size in lateral di- For mesoscopic superconductors with several atomic lay-
2 mensions, there exists a surface barrier known as the Bean- ers,thereductionofthedimensionspromotesthethermalfluc-
1 Livingston(BL)barrier[3]. Whentheappliedmagneticfield tuations. Evenformesoscopicsuperconductorsmadeofcon-
v: increases,theBLbarrierdiminishesandfinallydisappearsat ventionalsuperconductors,G2iD = 21(kBTc/[Hc(0)ξ(0)2d])2 ≈
i the penetration field Hp, where vortex enters the supercon- 10−4 with the film thickness d ∼ 1nm, which is greatly
X ductor. The penetration field is larger than H , the ther- enhanced by a factor (ξ/d)2 (cid:29) 1 compared to the bulk
c1
r modynamic lower critical field for type II superconductors value[13]. By solving the Ginzburg-Landau equation and
a
without surface. For the same reason, the exclusion field the Fokker-Planck equation analytically for a superconduct-
H , when vortex leaves the superconductor, is smaller than ing disk, Pogosov found that the thermal activation of vor-
e
H . Thermallyassistedpenetrationofvorticesinathicksu- tex over the BL barrier becomes feasible in mesoscopic
c1
perconductorwasinvestigatedtheoreticallybyPetukhovand superconductors[13]. The effects of thermal fluctuations on
Chechetkindecadesago.[4]Theyfoundthatthethermalacti- thepenetrationofsinglevortexhavealsobeenstudiednumer-
vationispracticallyimpossibleforvortex. Forhigh-T super- icallybysolvingthetime-dependentGinzburg-Landauequa-
c
conductors, Kopylov et. al. found experimentally that the tionswithnoiseterm[14]. Theyobservedthatthenumberof
penetration field decreases when the sample temperature is vorticesinsuperconductorsisfluctuatingduetothermalacti-
increased[5]. Thethermaleffectonthepenetrationandexclu- vationwhenG3Dislargeinsimulations.
i
sionfieldinhigh-T superconductorshasalsobeenstudiedin Inthispaper,weconsiderthethermallyassistedpenetration
c
2
FIG.1. (coloronline). (a): Asuperconductingdiskwithradiusofseveralcoherencelengthsandthicknessofseveralnanometers. ASTM
tipispositionedatthecenterofthedisktocheckthepresenceofavortexbymeasuringthezero-biasconductivity. (b): Profilesofthetime
dependentappliedmagneticfield.Themaximalmagneticfieldislargerthanthemean-fieldpenetrationfieldtoinitializeavortexatthecenter
ofthedisk. Inthebranchwithincreasing H, onemeasuresthepenetrationfieldandinthebranchwithdecreasingfield, onemeasuresthe
exclusion field. (c) and (d): Distribution of the (c) penetration field and (d) exclusion field P(H) at several temperatures. The mean-field
penetrationandexclusionfieldisrepresentedbyadashedline.ThedistributionprofilesareobtainedusingEqs.(19),(28)and(30).
and exclusion of a single vortex in a small superconducting whether the vortex is present or not, by measuring the zero-
disk. Wefindthedistributionofpenetration(exclusion)field bias conductivity[11]. Without thermal fluctuations, the vor-
whentheappliedmagneticfieldincreases(decreases)adiabat- tex penetrates into the disk when the applied field is larger
ically, as a function of disk size and temperature. The setup thanthemean-fieldpenetrationfield. Withthermalactivation,
andmainresultsarepresentedinSec. II.InSec. III,wede- the probability of vortex penetration is non-zero even when
rivetheBLbarrierforasuperconductingdiskbysolvingthe the applied magnetic is smaller than the mean-field penetra-
Londonequation,whileneglectingthermalfluctuations.From tionfield. Tofindthedistributionofpenetrationfield,period-
the BL barrier, we obtain the mean-field penetration and ex- icallyvaryingmagneticfieldsareappliedperpendiculartothe
clusionfieldforasinglevortex. InSec. IV,wecalculatethe disk. Thesetupwithastaticmagneticfieldhasalreadybeen
thermalactivationrateforvortexpenetrationandexclusionby implementedinRefs. [11,12,15]
solvingthecorrespondingFokker-Planckequationforvortex-
as-a-particle and derive the corresponding distribution func-
For the vortex penetration, one changes the applied mag-
tions. ThepaperisclosedbyadiscussioninSecV.
netic field periodically and slowly, see Fig. 1(b). To present
analytical calculations, we choose the sawtooth wave. One
records the magnetic field when vortex enters in the branch
II. PROPOSEDMEASUREMENT withincreasingfield.Byrepeatingthemeasurement,onethen
obtainsadistributionofthepenetrationfieldatagiventemper-
Weconsideracirculardiskofseveralcoherencelengthsin ature. Wecalculatethisdistributionfunctionanalytically[see
lateralsizeandseveralnanometersinthicknessdwithd (cid:28)ξ, Eq. (30)below],whichallowsforaquantitativecomparison.
where the thermally assisted penetration and exclusion of a The measurement of the thermally assisted vortex exclusion
single vortex can be observed experimentally, see Fig. 1(a). is similar. It is determined from the branch with decreasing
An STM tip is positioned at the center of the disk to check field,seeFig. 1(b).
3
III. ENERGYBARRIER Thenwearriveatthetotalenergyofavortexinacirculardisk
ofradiusR
Inthissection,wederivethemean-fieldsurfacebarrierfor Φ2 (cid:34) (cid:32) r2(cid:33) (cid:32)R(cid:33) πR2H (cid:32) r2(cid:33)(cid:35)
vortex penetration and exclusion in a superconducting disk U(r )= 0 ln 1− v +ln − 1− v .
without accounting for thermal fluctuations in a static mag- v 8π2Λ R2 ξ Φ0 R2
(8)
netic field. The thickness d of the disk is much smaller than
theLondonpenetrationdepthd (cid:28) λ. Forsuchthinfilms,the Equation (8) is valid when the vortex is not too close to the
screeningofmagneticfieldsisweakandtheeffectivepenetra- edge. Fora vortexclose tothe edge, R−rv < ξ, one should
tion depth is given by the Pearl length[16] Λ = 2λ2/d. The account for the finite size of the vortex core. When R is of
theorderξ,adifferentapproach,suchasdirectsolutionofthe
coherencelengthisalsoassumedtobemuchsmallerthanthe
penetrationdepthξ (cid:28)λ,andthuswecanusetheLondonap- Ginzburg-Landauequation,isneeded[13]. Thechangeofthe
magnetizationwhenavortexisatthecenterofthediskis
proximation,wheretheamplitudeofthesuperconductingor-
derparameterisuniforminspaceexceptforthevortexcore.
Φ R2
TtehxeismgaigvneentibcyfitehledLdoisntdriobnuteiqounahtizo(nx[,2y])associatedwithavor- Mv ≡−∂HU = 8π0Λ . (9)
2πΛ Thus the penetration and exclusion of a single vortex can be
hz+ c (∇×g)·eˆz =Φ0δ(r−rv), (1) determinedbymeasuringMvusingtheHallprobe. [10]
Avortexcanbetrappedinthediskifthereisalocalmini-
wheregisthesheetcurrentdensity,eˆz aunitvectoralongthe mumatr = 0. Localminimumdevelopswhenthemagnetic
zdirectionandΦ =hc/2eisthequantumflux. Herer isthe v
0 v fieldislargerthanthemean-fieldexclusionfieldH ,
e
position of the vortex and we only consider a single vortex.
ForthesuperconductingPbfilminRef.[15]Λ ≈ 20µm. The Φ
H = 0 . (10)
lateral size of the film is much smaller than the Pearl length e πR2
R (cid:28) Λ. In this case, the screening of the magnetic field is
negligible. Wecanneglectthefirsttermattheleft-handside AboveHethereisabarrierforvortexexclusion. Theposition
of Eq. (1). For convenience, we introduce a scalar stream ofthebarrierisdeterminedbytheconditionwhen∂U/∂ρ=0
functionG,suchthat[17] withρ≡rv/R,whichgives
g=∇×(Geˆ ). (2) H
z ρ2 =1− e. (11)
b H
At the boundary, the component of g normal to the bound-
aryiszero,whichrequiresthatGisconstantattheboundary. Itfollowsthattheheightofthepotentialatρ is
b
Without loss of generality, we takeG = 0 at the edge of the
disk. ThenEq. (1)isreducedtothePoissonequationandthe Φ2 (cid:34) (cid:32) Φ (cid:33) (cid:32)R(cid:33) (cid:35)
U(ρ )= 0 ln 0 +ln −1 . (12)
problemisequivalenttotheoneinelectrostatics, b 8π2Λ πR2H ξ
cΦ
∇2G =− 0δ(r−r ). (3) WhenU(ρ ) = 0,thebarrierforvortexpenetrationvanishes,
2πΛ v becausethebenergyofavortexattheedgeisU(r = R) = 0.
v
ForacirculardiskwithradiusR,wehave[18,19] ThisgivesthepenetrationfieldH
p
G(r)= 4cπΦ2Λ0 ln(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)r−(rR−/rrvv)2rv(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)2 rRv. (4) Hp = πΦeξ0R = eRξHe, (13)
Theenergyofthevortexisthengivenby withe = 2.718. AtfieldsaboveH ,nucleationofvortexinto
p
Φ Φ thediskbecomesfavorable. WhentheradiusRincreases,the
(cid:15)(r )= 0h (r→r )= 0G(r→r ). (5) surfacebarrierdecreasesthusH alsodecreases.WhenU(ρ=
v 8π z v 2c v 0) = 0,avortexatthecenterofpthediskbecomestheground
Theenergy(cid:15) divergesatr = r andtoavoidthedivergence, state,whichgivesanothercharacteristicfieldH
v g
we introduce the standard cutoff ξ near the vortex core, |r−
r |=ξ. Thentheenergyofvortexatr=r reads Φ (cid:32)R(cid:33)
v v H = 0 ln . (14)
Φ2 (cid:34) (cid:32) (cid:18)r (cid:19)2(cid:33) (cid:32)R(cid:33)(cid:35) g πR2 ξ
(cid:15)(r )= 0 ln 1− v +ln . (6)
v 8π2Λ R ξ ForR = 10ξ = 300nm,wehave H ≈ 7mTand H ≈ 26
e p
mT.SeveraltypicalprofilesofU(r )aredisplayedinFig. 2.
Weneedalsotoaddtheinteractionbetweentheappliedfield v
Slightly above H , a local minimum appears at r = 0, and
H andthevortex,U = −M H withthemagneticmomentof e v
z v U(R)<U(0). WhenthefieldisincreasedaboveH ,thelocal
thevortex[20–22] g
minimum becomes the global minimum and U(R) > U(0).
(cid:90)
1 AboveH ,thesurfacebarrierforvortexpenetrationvanishes
M = dr(r×g)·eˆ . (7) p
v 2c z andtheonlystablesolutionisavortexatthecenterofthedisk.
4
of the sample just above T . r ≤ R is the coordinate of the
c v
vortexintheradialdirectionandr˙ istheradialvortexveloc-
2 .4 v
ity. Theequationofmotionforthevortexinthepresenceof
H /H p = 0 .0
1 .8 thermalfluctuationsis
H /H p = 0 .2 1 7
∂L(r ,t)
1 .2 ηr˙ = v +F , (16)
H /H p = 0 .4 0 8 v ∂r n
v
L2] 0 .6 H /H p = 0 .5 4 4 wherewehaveintroducedanoiseforceF toaccountforthe
p n
2/80 0 .0 thermal fluctuations. F has a zero mean value (cid:104)F (t)(cid:105) = 0
n n
FU/[-0 .6 H /H p = 0 .8 1 5 andhasaGaussiancorrelator
-1 .2 H /H p = 0 .9 5 1 (cid:10)Fn(t,rv)Fn(t(cid:48),r(cid:48)v)(cid:11)=2ηkBTδ(t−t(cid:48))δ(rv−r(cid:48)v), (17)
H /H p = 1 .0 8 7 whereT isthetemperature. Whenthechangeoftheapplied
-1 .8
-1 .0 -0 .5 0 .0 0 .5 1 .0 fieldismuchslowerthanthethermalizationtimeofthevortex
r /R (seeSec. V),wecanusetheadiabaticapproximationtosolve
v
Eq. (16)andL(r ,t)canbereplacedby−U(r ). Aschematic
v v
view for thermally assisted vortex exclusion and penetration
FIG.2. (coloronline). Profileoftheenergybarrierforavortexin
a circular disk with radius R/ξ = 10 at several typical fields. The isdepictedinFig. 3.
penetrationfieldis H = 0.074H andtheexclusionfieldis H = Herewehaveconsideredthermallyassistedpenetrationand
p c2 e
0.02H , with H = Φ /(2πξ2) the thermodynamic upper critical exclusion of a single vortex. For two vortices, the barrier is
c2 c2 0
field. twiceaslargeasthatforasinglevortex. Sincetheprobability
for thermal activation is much smaller than that of a single
vortex,thesimultaneouspenetrationandexclusionofmultiple
IV. THERMALACTIVATION vorticescanbeavoided.[23]
InEq. (16),wehaveapproximatedthemotionofthevortex
When H approaches H from below, the barrier decreases in the disk by that of a particle in a one dimensional poten-
p
andthermalfluctuationswillpromotethepenetrationofavor- tial. Duringthepenetrationorexclusionofvortex,themotion
tex. Thus in the presence of fluctuations, a vortex enters at of single vortex is dictated by the force associated with the
magnetic fields below H . For vortex exclusion, the vortex surface barrier. The diffusion of vortex along the azimuthal
p
leavesthesuperconductoratfieldsabovethemean-fieldvalue direction due to thermal fluctuations thus can be safely ne-
H duetothermalactivation. glected.
e
Inthepresenceoftime-dependentmagneticfieldasinFig.
1(b),theenergybarrierchangeswithtimeinthiscase. There-
fore it is more appropriate to describe the vortex motion in A. Thermallyassistedexclusion
termsofaLagrangian. TheLagrangianofthevortexis
LetusfirstcalculatethevortexexclusionratewhenH > H
e
Φ2 (cid:34)πR2H(t)(cid:32) r2(cid:33) (cid:32) r2(cid:33) (cid:32)R(cid:33)(cid:35) usingtheKramersequationfordiffusionoverbarrier[24]
L(r ,t)= 0 1− v −ln 1− v −ln .
v 8π2Λ Φ R2 R2 ξ √
0 −U(cid:48)(cid:48)(r )U(cid:48)(cid:48)(r )
(15) Γ= m b exp[−∆U/k T] (18)
The mass of the vortex is small and neglected. The dissipa- 2πη B
tionfunctionisD = ηr˙2/2,whereη = Φ2/(2πξ2c2R )isthe
v 0 n where U(cid:48)(cid:48)(r ) is the second derivative at the surface barrier
Bardeen-StephendissipationcoefficientwithR theresistance b
n and U(cid:48)(cid:48)(r ) is the second derivative at the energy minimum
m
wherethevortexresidesinitially. Thebarrierheightis∆U ≡
U(r )−U(r ). The Kramers equation is valid when ∆U (cid:29)
b m
k T.
B
UsingEq. (18),weobtaintherateforthevortexexclusion
(cid:113) (cid:32) (cid:34) (cid:32) (cid:33) (cid:35)(cid:33)
(cid:15) 1
Γ = c 2f (f −1)2exp −v ln + f (19)
e πηR2 ef
with f = H/H > 1, (cid:15) = Φ2/(8π2Λ)andv = (cid:15) /(k T). We
e c 0 c B
estimate (cid:15) ≈ 2000K for Λ ≈ 20µm. Hence v (cid:29) 1 in the
c
wholesuperconductingstateT <T ≈7.2K.
c
The characteristic frequency is ω = (cid:15) /(πηR2) ≈ 10GHz
c c
FIG.3. (coloronline). Schematicviewofthermallyassistedvortex forR ≈ 300nm. Theratedecreaseswhenthelateraldimen-
exclusion when He < H < Hg (a) and (b) penetration when Hg < sion of the superconductor is reduced at a given H. More
H<H inapotentialU(r ).
p v importantly, because (cid:15) and v decrease linearly with the film
c
5
thicknessd, thethermalactivationrateincreasesgigantically dominantcontributionisfromtheregionnearthebarrier. We
becausevappearsasanexponent. Notethatthespuriousde- usethesaddlepointapproximation
creaseofΓ ,whenHisveryclosetoH ,isanartifact,because
e e
Eq. (19)becomesinvalidwhen∆U ∼kBT. Inreality,therate U(rv) ≈U(r )+ 1U(cid:48)(cid:48)(r )(r −r )2, (25)
Γ increasesmonotonicallywhenHapproachesH . (cid:15) b 2 b v b
e e c
withU(cid:48)(cid:48)(r ) < 0. Furthermoreweusethefollowingapproxi-
b
mationbecausev(cid:29)1
B. Thermallyassistedpenetration
(cid:115)
2π
Forvortexpenetration, U(cid:48)(cid:48)(rm)isnotdefinedattheedges. exp[vU(cid:48)(cid:48)(rb)(rv−rb)2/2]≈ −vU(cid:48)(cid:48)(r )δ(rv−rb). (26)
The Kramers equation becomes inapplicable, and an explicit b
solutionoftheFokker-Planckequationisneeded.
Thentheratebecomes
We use the Fokker-Planck equation associated with Eq.
(16) (cid:114)
D −vU(cid:48)(cid:48)(r )
Γ = exp[−vU(r )] b . (27)
∂ρ(r ,t)=−∂ J(r ,t), (20) p ξ b 2π
t v rv v
whereρ(r ,t)istheprobabilitydensityoffindingavortexat Let us compare Eq. (27) to the Kramers formula Eq. (18).
rvandJisvtheprobabilitycurrent tThheevseorttwexopeexnpertersastiioonn,stchoeinvocirdteexifisUth(cid:48)(cid:48)e(rrmma)li=ze4dπaktBrT/=ξ2R. aFnodr
v
J(rv,t)=−Dexp[−U/(kBT)]∂rv{exp[U/(kBT)]ρ}, (21) vgoairntesxkpinoestiitcioennoefrgtyhekBoTrd.erBξe,caounseemoafythaessuignnceartnaionstcyilolaftitohne
withthediffusionconstant D = kBT/η. Weusethestandard frequency of the order kBT/ξ2. We then arrive at a consis-
boundaryconditionforthecalculationsoftherate[24],byas- tentdescriptionbasedonthedirectcalculationsoftheFokker-
sumingasourceofvorticesatr =Randasinkatr =0. The PlanckequationandtheKramersequation.
v v
totalprobabilityoffindingavortexintheregionR−ξ ≤r ≤R Finally, for the circular disk, we have the rate for vortex
v
isunitybecausevortexisinitiallythermalizedattheedges, penetration
(cid:90) 2π (cid:90) R D (cid:32)ξef(cid:33)v (cid:114)2vf (f −1)
ρ dφ r dr =1, (22) Γ = , (28)
v v p Rξ R π
0 R−ξ
whichgivesρ(r ≈R,t)=1/(2πRξ).R−ξappearsinthelower with characteristic frequency ω = D/Rξ ≈ 10GHz. Note
v c
boundofintegralinEq. (22)becauseoftheuncertaintyofthe that both Γ and Γ are strongly nonlinear functions of the
e p
vortexcoordinateduetothevortexcoreoffinitesize. Weare radiusRandtheappliedfieldH.
interest in the probability for the vortex penetration, where
initiallynovortexislocatedatthecenterofthedisk. Tofind
thethermalactivationrate,wecanassumeasinkatthecenter C. Results
of the disk and use the boundary condition ρ(r = 0,t) = 0.
v
Thatmeanswerestartthemeasurementoncethevortexenters
Toestimatetheactivationrate,weusetheparametersfrom
intothedisk. experiments [15]. λ = 150nm, ξ = 30nm at 2 K and T =
Here we consider the regime where ∆U (cid:29) kBT. In this 7.2 K. We take R = 300nm, d = 2.5nm and R = 400cΩ.
n
case,theactivationrateforvortexissmall,thus J and∂tρare Thetemperaturedependencesofξ(T)andΛ(T)arecalculated
small[25]. Thentheprobabilitycurrentisalmostindependent
withtheBCStheory. Therateforthevortexpenetrationand
onr ,i.e.,∂ J =0. IntegratingEq. (21)fromr =0tor =R
v rv v v exclusionisshowninFig. 4.
andusingtheboundarycondition,wehave
Experimentally, one measures the distribution of the pen-
(cid:90) R etration or exclusion fields when the magnetic field is swept
Dρ(r ≈R)= J exp[U/(k T)]dr , (23) periodically. The distribution of the penetration field can be
v B v
0 obtainedfromtherateinthefollowingway[26]. Theproba-
bility for a vortex entering the disk after a waiting time t for
where we have used U(r = R) = 0 because the vortex does
v thebranchwithincreasingmagneticfieldinFig. 1(b)is
notexperiencethesurfacebarrierattherim. Thusthethermal
activationrateisgivenby (cid:34) (cid:90) t (cid:35)
W[H(t)]=1−exp − Γ [H(t(cid:48))]dt(cid:48) . (29)
p
D(cid:34)(cid:90) R (cid:35)−1 0
Γ ≡2πRJ = exp[U/(k T)]dr . (24)
p ξ B v Hencetheprobabilitydistributionofthepenetrationfieldis
0
Tvh=e (cid:15)cch/a(rkaBcTte)ri(cid:29)stic1,etnheergiyntesgcraalendodfrtohpesbvaerrryierrapisid(cid:15)lyc.anSdinthcee P(H)= ddWH = ΓpH[˙H]exp(−y), (30)
6
currentexertsaLorentzforceonthevortex, thusitgivesad-
ditional contribution to the surface potential. Thirdly, one
shouldcheckthevalidityoftheadiabaticapproximationused
intheevaluationoftherateinEqs. (19)and(28).
The induced electric field according to the Faraday’s Law
is E(r) = −H˙r/(2c), and the generated Joule heating is Q =
σ πH˙2R4d/(8c2), whereσ istheconductivityofquasiparti-
q q
clesandismuchsmallerthantheconductivityofthenormal
state. Usingtheheatbalanceequation,
Q
dCρ ∂T = +α(T −T) (33)
d t S B
thetemperatureincreaseafterthesystemreachesastationary
stateis∆T =σ H˙2R2d/(8c2α),whereαistheheatexchange
q
ratewithsubstrate. HereC istheheatcapacityandρ isthe
d
massdensityofthesuperconductor. Onecanneglectheating
if ∆T/T (cid:28) 1. For the sawtooth wave in Fig. 1(b) and (c),
FIG.4. (coloronline).Thermalactivationratefor(a)thepenetration H˙ isoforderofω H withω beingtheperiodofthetime-
H p H
ofavortex,(b)theexclusionofavortex.Themean-fieldpenetration
dependent field. Togive an orderof magnitudeestimate, we
and exclusion fields are also shown in the labels. Please note that
takeα≈107W/m2·KandtaketheconductivityofPbfilmin
the mean-field penetration field depends on temperature while the
thenormalstateaboveT ,σ d =0.01Ω−1.Forω (cid:28)1THz,
mean-fieldexclusionfieldistemperatureindependent. c q H
theheatingeffectisnegligiblysmall.
TheinducedcurrentofCooperpairsbythetime-dependent
with magnetic field is accounted for by the last term on the right-
hand side of Eq. (8). The induced quasiparticles J (r) =
y(H)=(cid:90) H Γp[H(cid:48)]dH(cid:48), (31) −σqH˙r/(2c) tilts additionally the potential for vortexqpene-
H˙(cid:48) trationorexclusion. Thechangeofthepotentialis
0
where H˙ is the derivative of H with respect to time. For the (cid:90) R Φ dσ H˙Φ R2
penetrationfieldmeasurement, H˙ > 0. Inasimilarway, one ∆Uq =d Jq c0dr= q4c20 . (34)
0
obtains the distribution of the exclusion field for the branch
withdecreasingmagneticfieldinFig. 1(b) It is required that ∆U /(cid:15) (cid:28) 1. This condition is satisfied
q c
whenω (cid:28)1THz.
Γ [H] (cid:34) (cid:90) +∞ Γ [H(cid:48)] (cid:35) H
P(H)= e exp − e dH(cid:48) . (32) To validate the adiabatic approximation employed in Eqs.
|H˙| H |H˙(cid:48)| (19)and(28),thechangeoftheappliedfieldshouldbemuch
slower than the thermalization time of vortex (or the relax-
Fortheexclusionfieldmeasurement,H˙ < 0. Experimentally,
ation time of a vortex to reach the energy minimum), such
theupperlimitoftheintegralisreplacedbythemaximalmag-
that the vortex remains in equilibrium when the magnetic
netic field used in experiment. The difference is negligible
field changes[27]. From Eq. (16), the thermalization time
becauseΓe ∼exp(−vH/He)≈0whenH (cid:29) He. isη/∂2U ≈1/ω ≈10−10sfortheparametersusedinthepre-
x c
Thedistributionsofthepenetrationfieldandexclusionfield
vious section. The adiabatic approximation thus is justified
asafunctionoftemperatureareshowninFig. 1(e)and(f). In whenω (cid:28)10GHz.
the calculations, we use H˙ = 14mT/s for penetration field H
Let us discuss the detailed form of the applied magnetic
and H˙ = −14mT/s for the exclusion field. As the temper- field. The rate description is valid when ∆U/(k T) (cid:29) 1.
B
ature increases, the distribution profile becomes broad. The The barrier near H can be written as ∆U = (cid:15) (1− H/H ).
p c p
most probable penetration field decreases with temperature, Thisgivesanupperboundfortheappliedfield1−H/H (cid:29)
p
and deviates from the mean-field value. As T increases, the k T/(cid:15) ≈ 1/200atT . Thismeansthattheratedescriptionis
B c c
exclusion field increases and also deviates strongly from the
valid almost in the whole region of the magnetic field below
mean-fieldvalue. Thereforethehystereticregionbetweenthe
H . Ontheotherhand,theperiodoftheappliedfieldshould
p
vortex penetration and exclusion is reduced with increasing
be large enough to count the event of vortex penetration in
temperature.
one period, which gives an upper limit for ω . The upper
H
limitcanbeestimatedasfollows. Theprobabilitytoobserve
vortexpenetrationaccordingtoEq. (29)is,W =1−exp(−y).
V. DISCUSSIONS TheratecanbewrittenasΓ ≈ ω exp(−v(1−H/H )]. Then
p c p
wehave
Let us discuss the effect of time dependent applied field.
(cid:34) (cid:32) (cid:33)(cid:35)
First, the time dependent magnetic field induces an electric y=−ωcHp exp −v 1− H ≈ ωc exp(−10), (35)
fieldandcurrent,whichgeneratesheat.Secondly,theinduced vH˙(cid:48) H vω
p H
7
wherewehavetakenv(1−H/H )=10,suchthattheratede- perconductor. Wederivedtheenergybarrierforavortexasa
p
scriptionisvalid. Inordertohavelargeprobabilityforvortex particleusingtheLondonapproximation,fromwhichweob-
penetration inone period, y > 1, whichgives ω < 10kHz. tainedthemean-fieldpenetration,exclusionfieldandthermo-
H
Forsuchfrequencies,theheatingeffectandtiltofthesurface dynamiccriticalfield. Wethencalculatedthethermalactiva-
barrierarenegligible. tionrateforthevortexpenetrationandexclusionbasedonthe
Fokker-PlanckequationandKramers’escaperateintheadia-
Some clues for the thermally assisted penetration and ex-
baticregionoffieldchange. Basedontheactivationrate,we
clusion of a vortex in mesoscopic superconductors can be
obtainedthedistributionofthepenetrationandexclusionfield
foundbycomparingtwoexperiments[11,12]. Inarecentex-
duetothermalactivation. Finallyweproposedmeasurements
periment by Cren et. al.[12], they measured the penetration
ofthedistributionofpenetrationandexclusionmagneticfields
and exclusion of the vortex in Pb films using an STM. No
totestourmodelofthermallyassistedsingle-vortexmotionin
hysteresis in the penetration and exclusion of the vortex was
mesoscopicsuperconductors.
observed. WhileinasimilarexperimentbyNishioet. al.[11],
hysteresis was clearly observed. These two experiments use
Pb superconducting disks with similar thickness and radius.
VI. ACKNOWLEDGEMENT
R ≈ 2ξ ≈ 60nm in both experiments and d = 2.5nm in
Ref. [11] and d = 5.5nm in Ref. [12]. The temperature in
Work at the Los Alamos National Laboratory was per-
Ref. [12] is 4.3K while in Ref.[11] is 2K with T ≈ 7.2K.
c formed under the auspices of the U.S. DOE contract No.
According to Fig. 1 (e) and (f), increasing the temperature
DE-AC52-06NA25396 through the LDRD program. Work
reducesthehysteresis, whichdisappearseventuallyathigher
at RIKEN (TN) was supported by JSPS Grant-in-Aid for
temperature. The thermally assisted penetration and exclu-
sionofvortexthuscanqualitativelyaccountforthedifference Young Scientists (B) No.22760018 and Special Postdoctoral
ResearchersProgramofRIKEN.WorkatUniversityofTokyo
inRef. [12]andRef. [11].
(YH)wassupportedbyGrand-inAidforScientificResearch
In summary, we have studied the thermally activated pen- (21360018), Ministry of Education, Culture, Sports, Science
etrationandexclusionofasinglevortexinamesoscopicsu- andTechnology(MEXT),Japan.
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