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Europhys. Lett., 67 (

B. Plac¸ais 1, N. Lu¨tke-Entrup 1, J. Bellessa 1, P. Mathieu 1, Y. Simon 1 and E. B. Sonin 2 1 Laboratoire Pierre Aigrain, D´epartement de Physique de l?Ecole Normale Sup´erieure 24 rue Lhomond, 75005 Paris, France 2 Racah Institute of Physics, Hebrew University of Jerusalem - Jerusalem 91904, Israel (received 5 March 2004; accepted in final form 7 June 2004) PACS. 74.25.Qt ? Vortex lattices, flux pinning, flux creep.

PACS. 64.60.Cn ? Order-isorder transformations;statistical mechanics of model systems.

A peak in the critical current vs. magnetic field plot, the peak effect (PE), is observed in superconductors close to the transition line where the critical current vanishes. From the very first studies [1?4] it was supposed that PE originates from softening of the vortex lattice (VL) by disorder near the transition. This results in a more effective vortex pinning, which corresponds to a higher critical current. The phenomenon is directly connected with a fundamental problem of condensed-matter physics: the competition between elasticity and disorder. Numerous scenarios of PE have been discussed, but all of them dealt with the competition between vortex elasticity and bulk pinning. Here we present an essentially different scenario of PE: it is surface pinning of vortices which interplays with bulk vortex elasticity.

A controlled surface roughness ?(r) is obtained by etching the Nb sample surfaces with 500 eV Ar+ ions (fig. 1). The sputtering of Nb atoms by low-energy ions is a stochastic process. It gives rise to a white corrugation spectrum S? (k) = dr e?ikr ?(r + R)?(R) R a3? z/? for |k| < 1/a, where a = 0.26 nm is the Nb lattice parameter and ? z the average sputtering depth. In our experiment ? z ? 10 µ m for a 90 min exposure to a 1.5 mA/cm2 Ar+ flux so that S? ? 50 nm4 and the total roughness ?? = ?(r)2 < (a? z)1/2 = 50 nm. Atomic force microscopy (AFM) in fig. 2 confirms the above estimates with S? (k) 40 nm4 for k 40 µ m?1. Unfortunately, the finite AFM-tip radius masks the large k spectrum, sothat we can only bracket the upper cut-off kc of S? (k) in the range 10?2 < kca < 1. This entails a large uncertainty in ?? = 0.5?50 nm. Importantly for our scenario, AFM data indicate the presence of roughness at small scale, with wave numbers k ? a0?1 (50 nm)?1, where a0 = ?B/?0 is the VL reciprocal unit and ?0 = h/2e is the flux quantum. Note that a roughness is smaller than vortex spacing so that the weak surface pinning approximation used below is justified.

c EDP Sciences

The peak effect is generally observed in the critical current data Ic(B) o rIc(T ). We see it in our samples as well, but since Ic 20 A are quite large, we prefer torely on the AC linear surface impedance Z(?) = ?i?µ 0?AC which is a more accurate and non-destructive probe of the vortex state, especially in the vicinity of a transition. According to [5, 6], the AC penetration depth ?AC in thick samples is given by

1 ?AC = 1 LS + 1 ?2 + i?µ 0?f

C 1/2 ,

LS = lSB µ 0? .

Here ?f is the flux-flow resistivity, ??0 is the vortex-line tension, and ?C the Campbell depth
for bulk pinning [7]. Expression (

If vortices do not interact, the slippage length lS does not depend on vortex density and is
on the order of a curvature radius of the surface profile (individual pinning). But, in general,
(

Bpk 103 f (Hz) 106

6 -1 µ()m l/S 1 0.2

B(T) 0.3 0.2

B (T) 0.3 lS may depend on vortex density, i.e. on magnetic field (see fig. 4). If vortices strongly interact the theory of collective pinning [4] assumes that within the so-called Larkin-Ovchinnikov domain of size Lc the vortices move mostly coherently without essential deformation of the vortex lattice. But then because of the random directions of pinning forces on every vortex, the total force on vortices in the domain is proportional to ?Nc and not to Nc = L2/a20, c the number of vortices in the domain. Correspondingly, the pinning force per vortex must be smaller by the factor ?Nc = Lc/a0, i.e. 1/lS = 1/l0?Nc = a0/l0Lc. Lc is usually derived from the balance between the elastic and pinning energy. Pinning is collective as long as Nc 1. The condition Nc ? 1 (o rlS ? l0) determines the crossover from the collective to the individual pinning. Later in the paper we shall derive lS without these heuristic arguments.

Figure 3 shows the PE in the inverse surface-pinning length. The sample, with dimensions
25 × 10.1 × 0.87 mm3, was annealed in ultra-high vacuum which gives a low residual resistivity
?n = 11 n?cm (resistivity ratio ? 1300) and an upper critical field Bc2 = 0.29 T at 4.2 K [10].
Data points are obtained by fitting the penetration depth spectra (inset of fig. 3) with eq. (

We quantitatively separate the bulk and surface pinning contributions, ?C and LS, by
fitting the full 1 kHz?1 MHz spectrum ?AC(f ) with eq. (

Using the Abrikosov expression [11], µ 0? (Bc2 ? B)/2.32?2 with ? = ?/? = 1.3 (? is the
coherence length and ? is the London penetration depth), we deduce from eq. (

The first step of our analysis addresses the response of the semi-infinite VL to a Fourier component f (r) = f (k)eikr of the surface force on vortices. The force produces vortex displacements in the sample bulk (z < 0) in the form u(r, z) = kz U (k, kz)eikr+ikzz. We look for the elastic constant C(k) = f (k)/u(k), connecting the Fourier components of the surface force f (k) tothe surface displacement u(k) = kz U (k, kz). The force is assumed to be transverse, [f (k), U (k, kz) ? k], since VL compressibility is quite low and the response to the longitudinal force is weak. The possible values of out-of-plane wave vector component kz must be found from the equation of the elasticity theory:

C66k2 + C44(k, kz)kz2 U (k, kz) = 0,
(

C44(k, kz) =

B2

1
µ 0 1 + ?2(k2 + kz2)
+ ?B
is the tilt-modulus, which takes into account nonlocal effects due to long-range vortex-vortex
interaction. The general solution of eq. (

C66 (1 + ?2k2µ 0?/B) µ 0 1 + ?2k2 .

At large ?k, eq. (

The second step consists in calculating the deformations produced by surface pinning from the corrugation profile ?(r). The random force on the vortices is f (ri) = ???0??(ri), where ri is the 2D position vector of the i-th vortex. The Fourier component of the force is f (k) = ??B Q i[Q ? k?(k? · Q)] dr e?i(k+Q)r?(r), where the factor in brackets separates the transverse component of the force and the summation over the reciprocal VL vector Q appears because the force is applied in discrete sites of the VL. Collecting contributions from all Fourier components u(k) = f (k)/C(k), we obtain the mean-square-root shear deformation at the surface: (?u)2 = ?ux + ?uy 2 ?y ?x = ?2?20 4?2 k2 dk C(k)2

Q

Q2 ? Here the integration over k is fulfilled over the VL Brillouin zone. In the following we shall approximate the surface corrugation spectrum by S?(k) = 2???2rd2e?krd , where the corrugation correlation radius rd is determined by the spectrum cut-off kc, if kc? < 1: rd ? kc?1. But since the vortex cannot probe corrugation on scales less than its ?size? ?, rd ? ? if kc? > 1. Approximating the sum over Q by an integral, we obtain for k ? 1/a0 Q ? 1/rd (?u)2 = ?2?20a20 8? 2/a0 k3 dk 0

C(k)2 0 ?

?B 3r2d2 .

S? (Q)Q3 dQ ? C66 l0 Here we used the expression C(k) ? k?0(C66?/B)1/2 for large k, which is a good approximation when ? a0. Note that since C(k) ? k at small k, the integral for the mean-square-root displacement u2 is divergent. This means that even a weak disorder destroys the long-range order near the surface, as was revealed in ref. [15]. However, our analysis shows that destruction of long-range order near the surface is not essential for the peak effect, which is governed by the mean-square-root deformation, but not by the mean-square-root displacement.

In the third step we derive the boundary condition eq. (

Since ?u is proportional to u, we arrive at the boundary condition eq. (

Q dk |k + Q|2 Q2 ?

Comparing with the expression lS = l0Lc/a0, we see that the size of the Larkin-Ovchinnikov
domain is Lc ? l0a0 C66/??0. In deriving eq. (

For low magnetic fields B Bc2, one has ? ? (?0/µ 0?2) ln(Bc2/B), C66 ? ?0B/µ 0?2
and, according to eq. (

Still, this crossover cannot explain a fully developed PE. Despite lS ? ?Bc2 ? B
decreases at B approaching Bc2, according to eq. (

The close relation between PE and vanishing of the shear modulus of VL was suggested in the early studies of PE [3, 4]. The new feature of our scenario is that at B < Bpk the shear modulus vanishes only at distances on the order of the deformation penetration depth 1/p ? 1/ C?66 from the surface. Our scenario agrees with STM imaging of the vortex array by Troyanovski et al. [16]. They revealed that PE is accompanied by the disorder onset on the surface of a 2H-NbSe2 sample, but they related it with bulk pinning. In order to discriminate two scenarios, it would be useful to supplement the STM probing of the vortex array at the surface by probing vortex arrangements in the bulk.

In conclusion, we presented the experiment and the theory, which support a new scenario for the peak effect based on competition between vortex-lattice shear rigidity and weak surface disorder. The peak is accompanied by a crossover from collective to individual vortex pinning and from a weakly disordered crystal to a glass state at the sample surface. Beside its experimental relevance, this mechanism offers an interesting paradigm for elastic systems at the upper critical dimension for disorder.

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We thank F. R. Ladan and E. Lacaze for the ion-beam etching and AFM measurements. We acknowledge discussions with B. Horowitz, T. Natterman and T. Giamarchi. This work was funded by the French-Israel program Keshet and by the Israel Academy of Sciences and Humanities. The Laboratoire Pierre Aigrain is ?unit´e mixte de recherche? (UMR8551) of the Ecole Normale, the CNRS, and the universities Paris 6 and Paris 7.