APS/123-QED

Peak effect in vortex systems with strong pinning

X. B. Xu,1, ∗ H. Fangohr,2 X. N. Xu,1 M. Gu,1 Z . H. Wang,1 S. M. Ji,1 S . Y. Ding,1 D. Q. Shi,3 and S. X. Dou3

1National Laboratory of Solid State Microstructures, Department of physics,

Nanjing University, Nanjing 210093, P. R. China2School of Engineering Sciences, University of Southampton, Southampton SO17 1BJ, United Kingdom

3 Institute for Superconducting and Electronic Materials,

University of Wollongong, NSW 2522, Australia

(Dated: December 7, 2008)

We have performed 2D-Langevin simulations studying the peak effect (PE) of the critical cur-rent taking into account the temperature dependence of the competing forces. We observe andreport that the occurrence of the PE results from the competition of vortex-vortex interactions andvortex-pin interactions and thermal fluctuations which have different temperature dependencies.The simulations reveal that the PE can only take place for certain pinning strengths, densities ofpinning centers and driving forces, which is in good agreement with experiments. No apparentvortex order-disorder transition is observed across the PE regime. Besides, the PE is a dynamicalphenomenon and thermal fluctuations can speed up the process for the formation of the PE.

PACS numbers: 74.25.Fy, 74.25.Qt

One pronounced phenomenon in type-II superconduc-tors is the so called peak effect (PE), which is the ap-pearance of a peak in the critical current density Jcbefore decreasing to 0 with increasing temperature orfield[1]. The PE has widely been observed in a varietyof low and high temperature superconductors by differ-ent experimental techniques[2], such as transport[3–5],magnetization[6, 7], and ac-susceptibility[8–10]. It wasproposed long ago that the PE originated from softeningof the elastic moduli of vortex lattice[1], which causedby the competitions between elastic energy Eel, pinningenergy Epin of vortex lattice and the energy of thermalfluctuations Eth[11]. Considering the competition be-tween the strength of vortex lattice and the pinning forceof isolated pinning center, Labusch showed that Jc > 0only if the pinning force dominates over the strength ofvortex lattice, otherwise Jc = 0[12]. According to theLabusch criterion the weak pinning centers in supercon-ductors cannot pin the vortex lattice. Larkin and Ovchin-nikov showed that weak pinning centers can act collec-tively in the correlation volume Vc (=LcR

2c , where Lc and

Rc are the longitudinal and transverse correlation lengthsrespectively), where the vortices are of long-range order,reducing much the Labusch criterion[13]. In the collec-tive pinning theory the PE was interpreted as resultingfrom the abrupt decrease of Vc due to the reduction inelastic interaction[14]. The PE was recently shown toappear naturally at the crossover from weak collectiveto strong pinning [15]. Further calculations showed thatthe ”Bragg glass” phases exist to be a quasi-long-rangeordered vortex lattice on a length scale r ≫ Rc[2]. Fur-thermore, the occurrence of PE was explained as evi-dence for a Bragg glass transition or an order-disorder(OD) transition[2, 5, 9, 16]. The OD transition was sug-gested to be a thermodynamic phase transition inducedby thermal fluctuations or pinning centers[2], which has

been confirmed experimentally by the direct structuralobservation of the vortex lattice, such as small angle neu-tron scattering (SANS)[9], and muon spin relaxation[17].Also, the investigation of the reversibility of the OD tran-sition provided strong support for its thermodynamicalnature[18].

However, the exact nature of the PE phenomenon re-mains controversial[6, 7, 10, 19–21]. Recently, acrossthe PE regime no noticeable change in the order of thevortex lattice was observed by SANS[19], and by Bitterdecoration[20]. It was also reported that the PE is situ-ated on a boundary separating the strong pinning regimefrom the thermal fluctuations dominated regime, whilethe weak-strong pinning crossover is located far from thePE regime[7]. On the other hand, for some supercon-ductors with strong pinning centers, it has been foundthat the occurrence of the PE depends strongly on thestrength and amount of the pinning centers[3, 6, 22, 23].Especially, the PE can be adjusted by changing the pin-ning strength of the twin boundaries (changing the anglebetween field and twin boundaries), which is difficult toexplain through the OD mechanism[3].

In this work we investigated the PE of a vortex systemwith strong pinning by using Langevin dynamics simu-lations. The random pinning centers act independently.The temperature dependence of pinning force fpv(T) isdifferent from that for the elastic forces between vorticesfvv(T). Thus, the competition between fpv(T) and fvv(T)takes place as temperature changes. The PE was clearlyseen for certain pinning strengths and densities of pin-ning. No marked change in the order of the vorticeswas observed. In addition, the PE is a dynamical phe-nomenon, and thermal fluctuations are not crucial forthe occurrence of the PE but can speed up its dynamicalprocess.

We use the overdamped Langevin equation of motion

2

for a vortex at position ri is[24]

Fi =

Nv∑

j 6=i

Fvv(ri−rj)+

Np∑

k

Fvp(ri−r

pk)+F

L+FTi = ηdridt

where Fi is the total force acting on vortex i, Fvv and

Fvp are the forces due to vortex-vortex and vortex-pin in-

teractions, respectively, FL is the driving force due to thecurrent J (FL ∝ J × ẑ) and FT is the thermal stochas-tic force, η is the Bardeen-Stephen friction coefficient:η ∝ φ0Bc2/ρn, Nv the number of vortices, Np the numberof pinning centers and rpk the position of the kth pinningcenter. We choose Fvv(ri − rj)=(φ

20s)(2πµ0λ

2)−1(ri −rj)(|ri − rj |)

−2 = fvv(ri − rj)(|ri − rj |)−2, where φ0

is the flux quantum, s the length of the vortex, µ0the vacuum permeability. We employ periodic bound-ary conditions and cut off the logarithmic vortex-vortexrepulsion potential smoothly[25]. Pinning centers ex-ert an attractive force on the vortices: Fvp(ri − r

pk) =

−fpv(rik/rp) exp(−(rik/rp)2)r̂ik, where fpv tunes the

strength of this force and rp determines its range[26]. Weassume rp = 0.2λ and fpv ∝ B

2c2(1−B/Bc2)ξ

2/κ2 as corepinning is considered[27], where κ = λ/ξ, Bc2 dependson the temperature via Bc2 = Bc2(0)(1 − (T/Tc)

2) (Theform for Bc2(T ) (also for λ(T ) and ξ(T ) in the followedcontext) is correct in the Ginzburg-Landau theory whenT approaches Tc, while providing a good fit to the BCSform over the whole temperature range[28]). The thermalforce is taken as FTi = fth

∑j δ(t−tj)Γ(tj)Θ(p−qj), where

fth represents the intensity of thermal force[29], Γ(tj) is arandom number chosen from a Gaussian distribution ofmean 0 and width 1, p represents the frequency of theaction to the vortex by thermal noise, and qj is a randomnumber uniformly distributed between 0 and 1. To rep-resent the temperature dependence of Fvv, Fvp, κ andF

T, we use λ(T )/λ(0) = (1−T/Tc)−1/2 and ξ(T )/ξ(0) =

(1 − T/Tc)−1/2 [28]. The average x component of the

velocities of the vortices is 〈Vx〉 =1

Nv

∑Nvi vxi which

is proportional to the resulting voltage. We normalizelengths by λ0 = λ(0), forces by f0 = (φ

20s)(2πµ0λ

30)

−1

and time by τ0 = λ0η(0)/f0. All quantities shown in thefollowing figures are expressed in simulation units.

For a fixed magnetic field, we perform simulations witha Lorentz driving force along the x-axis while coolingfrom 0.99Tc to 0.1Tc. The total number of vortices Nv =676 is used in the calculations presented here. For largersystems, similar results are observed. We employ B =0.015Bc20, λ0 = 690Å, s = 12Å, and η0 = 1.4×10

−17kg/sand, unless specified otherwise, the pinning strength atzero temperature is fpv0 = 9f0, the rate of change oftemperature is dT/dt = −0.02Tc/t0, the driving forceFL = 2f0, the number of pinning centers Np = 0.2Nv,and fth = 1.

Fig. 1(a) shows a typical plot of the average veloc-ity of the vortices in the x-direction, Vx, against the re-duced temperature T/Tc (filled triangles). The figure

0.0 0.5 1.00

7

14

0.0 0.5 1.00

1

2 fvv=fvv0(1-T/Tc)

fpv=fpv0(1-(T/Tc)2)(1+T/Tc)

fpv0

3

randomly distributed in the simulation cell at T1, asshown in Fig. 1(b). When decreasing the temperature,the pinning force becomes more important than both thethermal fluctuations and the elastic force (fpv > fvv, seealso the inset in Fig. 1(a)). Thus, more and more vor-tices are trapped by the random pinning centers. Closeto T =ps, almost all of vortices are pinned.

In the PE region (Tpe ≤ T ≤Tps), the vortices arepinned and disordered. It is clear that the high Jc isdue to fpv(T ) dominating over either fvv(T ) or ther-mal fluctuations. Besides, because the vortex pinning isstrong enough, several vortices are trapped by one pin-ning site, see for example the insert of Fig. 1(c). Thisis consistent with the recent experimental and numericalobservations[30].

At low temperatures (T < Tpe), fvv(T ) becomes moreimportant and thus more and more vortices are depinned,see for example the insert of Fig. 1(e). The vortex pathsform channel flows as shown in Fig. 1(d). These chan-nels will be destroyed by the quickly enhancing elasticforce at very low temperatures, shown in Fig. 1(e). Notethat no ordered vortex structure is observed even in lowtemperature T ∼ 0.

Our calculations clearly show that there exists the PEfor such a 2D disordered vortex system, where pinningis strong and the random pinning centers act indepen-dently. Furthermore, the vortices are always in disor-dered states across the PE regime. In other words, no ap-parent change in the vortex order can be observed whenthe PE takes place. The PE can be simply explainedby the competition between pinning and elastic forces ofthe vortices, which results from their different tempera-ture dependencies. It is evident that our interpretationon the PE is different from that based on the OD transi-tion or on the abrupt reduction in the correlation volume.This is the central result of this work.

Fig. 2 demonstrates the Vx-T/Tc curves for differentpinning strengths fpv0 and densities np(= Np/Nv). Itcan be seen that the PE appears only for certain fpv0 ata given density np and Lorentz force. For small fpv0, thevortices cannot be effectively pinned, resulting in a verylow Jc and no observable PE. While for very large fpv0,the vortices are always pinned until to very low temper-atures, see the data for fpv0 = 90f0 shown in Fig. 2.Therefore, the PE can be numerically realized by chang-ing pinning strength, which is in good agreement withtransport experiment by Kwok and co-workers who suc-cessfully observed the PE by adjusted pinning strengthof two twin boundaries in YBCO[3].

The insert of Fig. 2 shows Vx-T/Tc curves at severalnp at given fpv0. The PE is hardly seen if np is eithertoo large or too small. That is, the situation for lowdensity is similar to that for low pinning strength. Forthe presence of the PE, therefore, these results implythat the vortex-pin interactions depending on the prop-erties of the defects in superconducting materials should

0.3 0.6 0.90

3

6

0.3 0.6 0.9

0

3

6Np/Nv

0.05 0.1 0.2 0.5

T/Tc

T/Tc

fpv0 down

FIG. 2: Effect of pinning strength f0 on the PE at a fixeddensity of pinning centers. The Lorentz force fpv0 varies from4.5 to 90 f0 (4.5, 6, 6.6, 7.5, 9, 10.5, 90) f0. Inset: Effect ofdensity of pinning center on the PE at fixed pinning strengthand Lorentz force.

0.3 0.6 0.90

1

2

3

0.3 0.6 0.90

2

4

T/Tc

FL

3 2.5 2 1.5 1

T/Tc

dT/dt up

FIG. 3: Effect of temperature ramping rates dTc/dt0 on thePE, dTc/dt0 ranging from -0.002 to 0.1 Tc/t0 (-0.002, -0.02,-0.04, -0.06, -0.08, -0.1) Tc/t0. Inset: Effect of the drivingforce on the PE.

be comparable with the vortex-vortex interactions beingan intrinsic property of vortex matter. Our simulatedresults are supported by the experimental investigationabout the dependence of the PE on the pin density, whichwas adjusted by radiation or annealing treatments[6, 31].

We now study the effects of the experimental speed onthe PE. Fig. 3 shows the Vx-T/Tc curves for several tem-perature ramping rates dT/dt at fixed pinning and driv-ing forces. For high dT/dt, the moving vortices cannotbe pinned due to the (relatively slow) vortex relaxation.Thus the PE will progressively disappear with increasingdT/dt. At low temperature ramping rates, the PE is pro-gressively visible because the vortices have enough timeto move into the pinning wells. These simulated results

4

0.5 1.00

5

10

0.5 1.00

2

4fth=0dT/dt(Tc/t0)

-0.002 -0.02

T/Tc

T/Tc

fth 1 0.8 0.6 0

dT/dt=-0.02Tc/t0

FIG. 4: Vx-T/Tc characteristics for various thermal fluctua-tion forces. Inset: Vx-T/Tc characteristics without the ther-mal fluctuations for different temperature ramping rates.

indicate that the PE is a metastable phenomenon.It is expected that an increment of pinning force at

fixed driving current is equivalent to a decrement of ap-plied current at fixed pinning force. We therefore studiedthe effect of external current on the PE. In the insert ofFig. 3 we show the Vx-T/Tc curves for various drivingforces at fixed pinning force. For large FL (= 3f0), noPE could be observed, just as the case for small pinningforce. Decreasing the driving force FL (= 2.5f0) a dip inVx-T/Tc curve is progressively more pronounced, just asseen here, see Fig. 3 in Ref.[32]. As the driving force isfurther decreased (FL = 2f0), the pinning force becomesdominant, leading to a broadening of the PE[4].

Finally, in order to clarify the role of thermalfluctuations[7], we studied the effects of thermal fluctu-ations on the PE. Shown in Fig. 4 are Vx-T/Tc charac-teristics for various fth. It can be seen that the PE isgrowing with enhanced thermal fluctuations. The reasonis that the ”shaking” effect of thermal fluctuation caneffectively reduce the relaxation time within which thevortices diffuse into the potential wells. To further con-firm this we study the effects of the dT/dt on the PEunder fth = 0, which is shown in the inset of Fig. 4. It isfound that the PE occurs for low dT/dt (= −0.002Tc/t0)but disappears for large dT/dt (= −0.02Tc/t0). This is astrong evidence that thermal fluctuations are not crucialfor the occurrence of the PE in our 2D vortex system,contrasting with the reported result[7].

In summary, we performed 2D-Langevin simulationsstudying the PE for a vortex system with random strongpinning at various temperatures. The pinning centersact independently when they pin the disordered vortices.Due to different temperature dependencies the forces ofvortex-vortex, vortex-pin and thermal fluctuations com-pete with each other as temperature changes. The PE

is ubiquitously observed for certain strengths and den-sities of pinning sites and the Lorentz force, which is ingood agreement with the experiments. No vortex order-disorder transition or collective-individual vortex pinningcross is observed. The PE can be simply explained bythe competition between pinning and elastic forces due totheir different temperature dependencies. The relaxationrelating to the PE is observed by changing the temper-ature ramping rate or thermal fluctuation force, demon-strating its dynamical characteristics. In addition, ther-mal fluctuations are not crucial to the PE in the stronglypinned system but it can speed up the process for theformation of the PE.

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