Peak effect at the weak- to strong pinning crossover

0.25 MB·English

by  J. A. G. Koopmann

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J Peak effect attheweak-to strongpinning crossover

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J.A.G.Koopmann,V.B. Geshkenbein,andG.Blatter
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n Theoretische Physik, ETH-H¨onggerberg,CH-8093 Zu¨rich,Switzerland
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Abstract
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Intype-IIsuperconductors,themagneticfieldentersintheformofvortices;theirflowunderapplicationofacurrent
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t introduces dissipation and thus destroys the defining property of a superconductor. Vortices get immobilized by
a
m pinningthroughmaterial defects, thusresurrecting thesupercurrent.Inweak collective pinning,defectscompete
and only fluctuations in the defect density produce pinning. On the contrary, strong pins deform the lattice and
-
d inducemetastabilities.Here,wefocusonthecrossoverfromweak-tostrongbulkpinning,whichistriggeredeither
n by increasing the strength fp of the defect potential or by decreasing the effective elasticity of the lattice (which
o is parametrized by theLabusch force fLab). With an appropriate Landau expansion of the free energy we obtain
c apeak effect with a sharp risein thecritical current densityjc ∼j0(a0ξ2np)(ξ2/a20)(fp/fLab−1)2.
[

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Keywords: superconductivity, vortex pinning,peak effect
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Pinningofvorticesbymaterialdefectsiscrucial havebeenrelatedinarecentwork,wherewehave
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inestablishingthedissipation-freeflowofasuper- studiedthe pinning diagramexhibiting crossovers
0
4 current.Inrecentyears,thefocushasbeenonweak between various regimes when varying the defect
0 collective pinning theory [1] describing the action density n measured with respect to the vortex
p
/ ofmanycompetingdefects;pinningthenisdueto density 1/a2 and the pinning force f measured
t 0 p
a fluctuations.Thecriticalcurrentisdeterminedby with respect to the elasticity, cf. Fig. 1. At low
m
thestatisticalsummationofthecompetingpins[2] defectdensity, vortexlattice (bulk) pinning is rel-
- andischaracterizedbyaquadraticdependenceon evant; for weak defect strength pinning is collec-
d
n the defect density.Onthe contrary,firstattempts tive, whereas defects with a force larger than the
o todescribevortexpinninggobacktoLabusch[3], LabuschforcefLab(seebelow)pinthelatticeindi-
c who studied the action of strong individual pins. vidually.Athigherdefectdensitieswehaveestab-
:
v Strongpinsdeformthelattice[4,5,6,7]andinduce lished two regimes where pinning acts on individ-
i metastabilities; the pinning energy landscape be- ualvortexlines.
X
comesmulti-valued,producinganon-zeroaverage In this work, we focus on the crossover from
r
a ofthepinningforceandacriticalcurrentwhichis weak-tostrongbulkpinning,whichistriggeredei-
linearinthe defectdensity.Thesetwo approaches therbyincreasingthe defectstrengthκ∼f /ξ (ξ
p
is the coherence length of the superconductor) or
by decreasing the effective elasticity C¯ of the lat-
Email address: [email protected]
tice;thecriticaldefectforceatthecrossoveristhe
(J.A.G. Koopmann).
Preprintsubmitted toElsevier Science 2February2008
Labusch force fLab = C¯ξ. The crossover is struc- 2ξ 1D cp
0
turally relatedto the Landautheory offirst order a
p
phasetransitionsandcanbeanalyzedwithanap- n 3D wcp
propriateLandauexpansionofthefreeenergy.As 1 1D sp
aresult,weobtainastrongincreaseinthecritical
currentdensity atthe crossover,
j ∼j (a ξ2n )ξ2 fp −1 2, (1) u 3D sp
c 0 0 p a2 (cid:18)f (cid:19) R
0 Lab
wherej0 isthedepairingcurrentdensityanda0 = 0 1 fp fLab
(Φ /B)1/2 is the mean vortex spacing (Φ is the
0 0
superconducting flux unit); see below for the pre- Fig.1.Pinningdiagramdelineatingthecrossoversbetween
variouspinningregimesinvolvingcollectiveversusindivid-
cise definition of f and f . The crossover from
p Lab ual pinning and 1D-line versus 3D-bulk pinning: 3D wcp
weak-tostrongpinningcanbetriggeredbyreduc-
–bulkweakcollectivepinning,1Dcp–collectivelinepin-
ing the effective elasticity C¯ ∝ fLab, producing a ning, 1D sp – strong line pinning, 3D sp – bulk strong
peak inthe currentdensity[1]. pinning. The crossover fromweak- to strong bulkpinning
Below,we willdescribe the strongpinning situ- occurs at the Labusch forcefLab.
ationin more detail anduse a Landautype of ex-
withr =(R ,z)andf =−∇ e (u)thepinning
pansionto find the jumps in the free energy land- ν ν p u p
force of the defect. In the last equation, we have
scape.Thesejumpsarethenusedtoderivethere-
assumed a defect of range much smaller than a
0
sult(1)forthe criticalcurrentdensity.
pinning at most one vortex; we further have cho-
In order to derive a quantitative criterion for senR asthedistancetothevortexclosesttothe
d
strong pinning, we consider the effect of a single defect. Evaluating (3) for r = (R ,0), we arrive
(strong)defectonthevortexlattice.Weassumea ν d
atthe self-consistencyequation
defectlocatedattheoriginwithpinningpotential
ep(r)producingthe pinning contribution uα(R,0)=Gαβ(r=0)fβ[R+u(R,0),0]
p
E (r,u)= e (r)δ2(R−R −u(R ,z)) =C¯−1fα[R+u(R,0),0] (4)
p p ν ν p
Xν
withthe effective elasticconstant
to the free energy density of the vortex system;
r=(R,z)andthevorticesarepositionedatRν− d3k Gαβ(k)=C¯−1δαβ. (5)
u(R ,z), with R the equilibrium positions and Z (2π)3
ν ν
u the displacement field. The elastic part of the
For a < λ the effective elasticity is C¯ ∼ ε /a
vortexlattice free energyreads[2] 0 0 0
with ε = (Φ /4πλ)2 the vortex line energy. The
0 0
F = d3k uα(−k) Gαβ(k) −1uβ(k), (2) solutionat (R,0) allows for the calculationof the
el Z (2π)3 wholedisplacementfieldu(R ,z),
(cid:2) (cid:3) ν
with Gαβ(k) the Fourier transform of the elastic u(R ,z)=Gαβ(R −R,z)C¯u(R,0), (6)
ν ν
Green function Gαβ(r); Greek indices denote the
and we can calculate the total free energy
in-planecomponents(x,y)andwesumoverdouble
e (u,R)(containingboth,thecontributionfrom
indices. pin
The displacement field u is obtained from vari- elasticity and disorder) of the vortex system as a
functionofu(R,0),
ationofthe totalenergy(elasticandpinning),
uα(rν)=−Z d3r′ Gαβ(rν −r′)[∂uβEp](r′,u′) (3) epin= 12Z(2dπ3k)3 Gαγ(−k) Gαβ(k) −1Gβδ(k)
(cid:2) (cid:2) (cid:3)
=Gαβ(R −R ,z)fβ[R +u(R ,0),0], ×uγ(R,0)uδ(R,0) +e (R+u(R,0))
ν d p d d p
(cid:3)
2
e ∆e
pin pin This (Labusch) criterion[3] for strongpinning in-
volves the maximal negative curvature above the
∆e inflection point; it tests an individual isolatedpin
pin
andclassifiesitasweakorstrong.
In a next step, we calculate the average pin-
R ning force in the strong pinning case. Using the
x x a~ d
Cu=f + − self-consistency equation (4), we calculate the
pin derivative of the pinning energy e (x,y) =
pin
e (u(R),R) with respect to the drag parameter
pin
weak / strong pinning
x,
∂ e (x,y)=C¯u∂ u−f (R+u)[xˆ+∂ u]
x pin x p x
R =−fx(R+u); (9)
d p
this equation relates the force along x exerted by
the pin at the vortex position R+u(R,0) to the
freeenergylandscapecontainingboth, the contri-
Fig. 2. The pinning energy epin and the force fpin are bution from pinning and elasticity. The pinning
sketched as a function of the drag Rd between the vortex forcehastobeaveragedoverdefectlocationsand,
systemandthedefect.Inthecaseofweakpinning(dashed
due to the bistabilities, depends on the prepara-
line),thetwoquantities aresingle-valued(hence theaver-
age over the pinning force is zero), whereas in the strong tion of the system; here, we focus on the critical
pinning case (solid line, the dotted parts denote unstable currentdensity andthereforesearchforthe maxi-
branches) the average over the force is determined by the malforceagainstdrag.Averagingtheaccumulated
jump ∆epin in the pinning energy. The thick lines show dragforceoverthe‘impactparameter’y,weobtain
the choice of the two possible branches that occurs when
dxrdagirgeicntgiotnh.e lattice through the defect along the positive Lx Ly fx(R+u)
p
hf i= dx dy (10)
= 1C¯u2+e (R+u). (7) pin Z Z LxLy
2 p 0 0
Lx Ly
−∂ e (x,y)
x pin
For weak pinning, the displacement u is small = dx dy ;
and the solution u ≈ f (R)/C¯ of (4) is unique; Z Z LxLy
p 0 0
the pinning energy e (R) = e (u(R),R) then
pin pin the integraloverxcanequallywellbe interpreted
is a single-valued function. Strong pinning, how-
asanintegrateddragforceorasanaverageoverpin
ever,produces multi-valuedsolutions u(R,0) and
locations (maximized with respect to the bistable
e (R),ascanbeseeninFig.2.Thedisplacement
pin branches).Weexpresstheaveragealongthex-axis
turns multi-valued when ∂ u → ∞. Assuming a
R throughthe jump ∆e (y)inthe pinning energy,
defect symmetric in the plane (e (R) = e (R)) pin
p p
anddraggingthelattice throughthedefectcenter a0/2
alongthe x-afxp′i(sx(R+du)=(x,0)),we find hfpini=−−aZ0/2 day0 ∆ea˜p(iyn)(y) =−ta⊥20∆epin, (11)
∂ u(x)=
x C¯−fp′(x+u) where we have assumed a maximal trapping dis-
(notethatx>0impliesu<0).Thedisplacement tancet⊥alongtheyaxisandrequirethepinnotto
collapses when the (negative) curvature f′ of the overdragthe vortex;thentheperiodicityofepin is
p thesameasthelatticeperiodicityanda˜=a .The
defect potential overcompensates the elasticity of 0
remaining task is the determination of the jump
the lattice,
∆e ;aswewillseebelow,itdoesnotdependon
pin
∂ f (x+u)=−∂2e (x+u)=C¯. (8) the impactparametery.
x p x p
3
0
e (u,x)
pin
ξ x+ u+
u / unpinned pinned −x−
u−
−1
z
−2 u −x+
x x− x
0 ∆e
pin
e
+
u+ u− u
0
e
e /pin e− Fofigt.h4e.Sdcishpelmacaetmicepnltotuoffotrhedipffinerneinntg(enneegrgaytivaes)avfaulunecstioonf
the drag x. The sketch shows the trapping process of an
unpinned vortex at the point of instability −x+ and the
−1 appearance of the jump∆epin.
0 1 2 x /ξ 4
thedefectwithanegativedragparameterx,there
Fig.3.Displacementuandenergyepinversusdragxofthe is only one solution u+ corresponding to the un-
vortex lattice relative to a defect producing a Lorentzian
pinned vortex system, cf. Figs. 2 and 4. With the
pinning potential ep,L = −e0/(1+u2/ξ2) (we measure vortex approaching the defect a second minimum
unitsoflengthinξandunitsofenergyine0).Theevolution
of the bistability appearing above the Labusch criterium appears, but is separated from the first one by a
is shown for pinning centers of increasing strength with barrier.Therelevantsolutionthereforeremainsthe
C¯/κ = 0.85,0.75,0.65,0.55; dashed lines denote unstable unpinned one, although at some point the pinned
branches. The insetshows thegeometry defining the drag
solutionbecomeslowerinenergy.Forx=−x the
+
parameter xand the displacement u.
unpinned solution becomes unstable and the vor-
Inafirststep,wediscussthesolutionsoftheself- texgetstrapped,asshowninFig.4.Itremainsso
consistencyequation(4)whendraggingthelattice until at x = x− the pinned solution u− turns un-
through the defect center along the positive x di- stableandthedefectreleasesthe vortex.Thecor-
rection and calculate the displacement u numeri- responding jumps in the free energy are shown in
callyfor aLorentziandefect potential Fig.2.
In order to attain a more quantitative descrip-
e
e (u)=− 0 . tionofthedifferentbranchesandtheircorrespond-
p,L 1+u2/ξ2
ing jumps, we perform a Landau type expansion
of the free energy close to the Labusch condition
The resultisshownin Fig.3,wherewe haveplot-
κ=C¯andusetheanalogytotheLandautheoryof
tedthedisplacementuandthepinningenergye
pin
fordifferentvaluesofC¯/κ∼f /f ,where−κ= first order phase transitions. We denote the loca-
Lab p
−e /2ξ2 is the maximalnegative curvature of the tion of the maximal negative curvature −κ by uκ
0
andexpandthepinningpotentiale (x+u)around
defect potential (realized at u = u = ξ), which p
κ
this point,
is the relevant curvature in the Labusch criterion
(8).Thebranchu−(u+)correspondstothepinned
e (x+u)≈−ǫ+ν(x+u−u ) (12)
(unpinned) vortex and becomes unstable at x− p κ
κ α
(x+);the dashedlines denoteunstable branches. − (x+u−uκ)2+ (x+u−uκ)4.
2 24
The solutions u(x) of the self-consistency equa-
tion(4)correspondtotheminimaofthefunctional FortheLorentzianpotentialtheexpansionparam-
e [u,x] for fixed x, cf. Fig. 4. Starting far from etersreadu =ξ,ǫ=e /2,ν =e /2ξ,κ=e /2ξ2,
pin κ 0 0 0
4
andα=3e /ξ4.Weinsertthisexpansionintothe Goingbeyondthisone-dimensionalanalysis,we
0
pinningenergye =Cu2/2+e (R+u);thelat- havetoaccountforafinite‘impactparameter’yof
pin p
ter maps to the free energy of a one-component thevortexwithrespecttothedefect(R =(x,y))
d
(Ising) magnet in a magnetic field [8] if we define anddeterminethetransversetrappingdistancet⊥
the order parameter φ = x+u−u , the reduced for the vortex to get trapped when draggedalong
κ
‘temperature’τ =C¯−κ, and the ‘magnetic field’ x,cf. Fig.5.Makinguse ofthe rotationalsymme-
h=C¯(x−uκ−ν/C¯), try of the problem, we find circles with radius x±
τ α limitingthebistableregions.Thejumpsthenoccur
epin[u,x]=emag[φ,h]= φ2+ φ4−hφ, (13) when a solution is dragged through its instability
2 24
line: The unpinned vortex gets pinned when x =
where we have dropped the constant term. Con- −(x2−y2)1/2;thisonlyhappensforimpactparam-
+
sider the zero field case first: In the paramagnetic
eters |y|< x , which leads to a trapping distance
+
prahmaseeteartφhigvha-ntiesmhepse,rawthuirleesφτ >∝ 0±t|hτe|1/o2rdienr tphae- t⊥ =2x+(note,thatt⊥ ≈uκ+ν/C¯ remainsfinite
atthetransitiontoweakpinning).Thepinnedvor-
low temperature (τ < 0) ferromagnetic phase. tex is releasedwhen x = (x2 −y2)1/2. Note, that
−
The two ferromagnetic states (φ > 0 and φ < 0)
thecorrespondingmagnitudesofthejumpsdonot
are separated by an energy barrier increasing as
dependony because ofthe planarsymmetry.
∆e ∝τ2. Inthe low temperature phase τ <0,
mag
a field h induces a first order transition across
uncompensated
h = 0 between the two ferromagnetic states. y
pinned
Metastable/hysteretic behavior appears within
the field regime |h| < h⋆, with h⋆ ∝ |τ|3/2 follow-
ing from comparing the barrier ∆e and the
mag
fieldenergyhφ.
e
The above considerations translate to the pin- bl
a
ning problemin the following way:the high ‘tem- st x+ x- x
bi pinned
perature’phaseτ >0describesweakpinning.The
two low temperature states stand for the pinned
(φ < 0) and unpinned (φ > 0) configurations,
unpinned
which transform into one another via the first or-
dertransition.Thehystereticregimetranslatesto vortex trajectory
the bistable pinning domainbounded by
Fig.5.Trappingareaaroundthedefectforacircularsym-
x± =uκ+(ν∓h⋆)/C¯, metric situation with pinned, unpinned, and bistable re-
gions;thedarkuncompensated areaproducesthenetpin-
with h⋆ = (2/3C¯) 2/α|τ|3/2. At the instability ningforce.Avortextrajectoryforfiniteimpactparameter
x = x− (h = h⋆p) the order parameter jumps y<0isshown.
from the pinned solution u− = φ− +uκ −x− to
the unpinned state u+ = φ+ + uκ − x−, where Weinsertthevalueforthejump∆epininto(11),
φ− = − 2/α|τ|1/2 and φ+ = 2 2/α|τ|1/2. As uset⊥ =2x+andobtainthepinningforcehfpini=
a result,pwe find the jump ∆e =pepin(u−,x−)− −2x+∆epin/a20. Adding the contributions of inde-
epin(u+,x−)=(9/2α)τ2inthefreeenergy;ajump pendent strong defects with density np, the total
ofequalmagnitudeisfoundatx+.Thetotaljump pinning force density reads Fpin = nphfpini and
∆epin inthe pinning energythenreads the critical current density is obtained as jc ∼
−cF /B. Close to the Labusch condition, we fi-
∆e =2∆e=(9/α)(C¯−κ)2. pin
pin nallyobtainthe criticalcurrentdensity
WithinthisLandaudescription,theLabuschcon-
ditionC¯ =κcorrespondstothe criticalend-point ξ2 f 2
j ∼j a ξ2n p −1 , (14)
T =Tcterminatingalineoffirstordertransitions. c 0 0 pa20 (cid:18)fLab (cid:19)
5
wherewehaveuseda defectsizeξ,implying t⊥ ∼ [8] L.D. Landau and E.M. Lifschitz, Statistical Physics,
ξ;wedefinetherelevantdefectstrengthf =ξκ= Vol. 5(Pergamon Press,London/Paris, 1958).
p
′
ξmaxu[fp(u)]intermsofthemaximalcurvatureof [9] A.B.Pippard, Phil.Mag.19,217 (1969).
thedefectpotential.TheLabuschforcethentakes
[10]M. Marchevsky, M.J. Higgins, and S. Bhattacharya,
the form fLab = ξC¯ ∼ ε0ξ/a0. Interpolating with Nature 409,591 (2001).
the weakcollectivepinning result[2]
[11]M. Avraham, B. Khaykovich, Y. Myasoedov, M.
ξ2 f 4 Rappaport,H.Shtrikman,D.E.Feldman,T.Tamegai,
jwcp ∼j (a ξ2n )2 p P.H. Kes, M. Li, M. Konczykowski, K. van der Beek,
c 0 0 p λ2 (cid:18)fLab(cid:19) and E.Zeldov, Nature411,451 (2001).
(we assume pinned bundles of size larger than
λ), we observe a sharp rise in the critical current
density once the strong pinning force overcomes
the weak pinning result; this peak effect [1] oc-
curs close to the Labusch condition at f /f =
p Lab
1+(a /λ) a ξ2n .
0 0 p
Thecrospsoverfromweak-tostrongpinningcan
berealizedinexperiments:increasingthemagnetic
field and approaching the upper critical field H
c2
leads to a marked softening of the elastic moduli,
therebyreducingf ∝C¯;theresultingcrossover
Lab
intothestrongpinningregimethenisobservedas
apeak[1,9]inthecurrentdensity.Whethersucha
crossoverfromweak-tostrongpinningcanexplain
therecentlyobservedpeakeffects(see,e.g.,[10,11]
andRefs.therein)remainsto be seen.
We acknowledge discussions with Anatoly
Larkin and financial support from the Swiss Na-
tionalFoundation.
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