Maximal persistent current in a type-II superconductor with an artificial pinning array
at the matching magnetic field
B. Rosenstein,1I. Shapiro,2and B. Ya. Shapiro2
1Department of Electrohysics, National Chiao Tung University, Hsinchu, Taiwan, Republic of China 2Department of Physics, Institute of Superconductivity, Bar-Ilan University, 52900 Ramat-Gan, Israel
共Received 17 November 2009; revised manuscript received 20 January 2010; published 17 February 2010兲 The current carrying steady state of the pinned flux line lattice created by magnetic field is described. We calculate analytically the critical current for the case of the matching field共when the number of vortices is equal to that of the pinning centers兲 using a simple variational method in the framework of Ginzburg-Landau equations. The vortex cores are deformed and displaced in the current carrying state. Displacement of the centers of the vortices with respect to pinning centers and structure of these states are determined.
DOI:10.1103/PhysRevB.81.064507 PACS number共s兲: 74.20.De, 74.25.Uv, 74.78.Fk
I. INTRODUCTION
The great interest in the problem of magnetic flux pinning in type-II superconductors is associated with its relevance to technological applications of superconductivity. An impor-tant challenge in applications of type-II superconductors is achieving optimal critical currents under given magnetic fields. This requires preventing depinning of Abrikosov vor-tices during formation of the resistive state under the applied current. Random pointlike pinning centers naturally appear due to imperfections of lattice structure or chemical disorder. However, in technologically important materials critical cur-rent due to intrinsic pinning is not strong enough, especially at high magnetic fields. One of the main reasons is destruc-tive competition of pinning centers, as demonstrated by the collective pinning theory.1It was predicted theoretically2and confirmed experimentally3–5 that when pinning centers are
arranged into a periodic array commensurate with the Abri-kosov lattice, the critical current increases dramatically. The effect is maximized when the filling fraction f 共defined as a ratio between number of vortices to that of the pinning cen-ters兲 is 1, when one pinning center traps a single vortex. Additional vortices are “interstitial” and can be depinned easily, thus significantly reducing the critical current.6
Re-cently there has been an advance in the fabrication method of the periodic arrays of pinning sites.7The arrays with
trian-gular, square, and rectangular geometries have been fabri-cated using either microholes or blind holes,3magnetic dots,4
and columnar defects.5
Theoretically these systems were studied, using mostly numerical methods, within a model of interacting two-dimensional共2D兲 points representing vortices subject to pin-ning potential.8,9 This approach is appropriate to describe
weak magnetic fields and sparse pinning arrays, so that the structure of the vortex core can be ignored. Recently, how-ever, the arrays are fabricated on the nanometer scale, and the range of fields applied continuously increases. Therefore the distribution of the order parameter becomes of impor-tance and one has to resort to a more fundamental approach. Since microscopic approach is not practical, the only avail-able tool is the Ginzburg-Landau 共GL兲 phenomenological approach.1 Within this approach the periodic pinning
prob-lem was tackled numerically by Priour and Fertig.10 They
demonstrated that the pinning centers deform the vortex core and, moreover, that the current carrying state for a large square-shaped pinning center displaces a vortex in the direc-tion perpendicular to the persistent current. Unfortunately, only one vortex and one pinning center were simulated on the square sample with area carrying just one unit of flux ⌽0=hc2e, while the rest of the vortex-pinning center pairs were represented by periodic boundary conditions duplicating the “squares.” It is well known that in an isotropic supercon-ductor the intervortex repulsion 共which is rather strong at elevated fields兲 forces them to form a hexagonal vortex lattice.1The square lattice will therefore be in conflict with
these forces and the question is whether this is an important factor in the pinning problem. In addition, it is clear that in order to maximize the critical current the pinning center ar-ray should be hexagonal with one pinning center per vortex 共the matching field兲 and this is the situation we consider in the present analytical calculation.
In this paper we employ the GL equations for the order parameter⌿ in order to determine the persistent current and to describe the structure of the pinned vortex matter in su-perconducting films at matching field 共f =1兲. The sample is considered to be infinite in directions perpendicular to ap-plied magnetic field, so that the vortex-vortex interactions are fully accounted for. Current carrying states of the flux lattice at matching field and hexagonal array of pinning cen-ters are characterized by displacement of the vortices with respect to that of the pins and by deformations of the vortex cores. A variational method exploiting the two lowest Lan-dau levels共LLLs兲 is used to determine the displacement and to calculate the critical current at the matching field for rect-angular pinning centers of arbitrary aspect ratios. The depen-dence of the persistent current on the displacement is linear at small displacements and approaches its maximum 共the critical current兲 at about half of the intervortex distance. The coefficient in the linear part, the Labusch constant, is calcu-lated.
II. GINZBURG-LANDAU EQUATIONS WITH A PERIODIC PINNING ARRAY
Let us consider a type-II superconducting film of width s under constant magnetic field H perpendicular to the film.
Static magnetic properties of the superconductor are de-scribed by the GL Gibbs energy1 as a function of the order
parameter⌿ and vector potential A, F关⌿,A兴 = s
冕
dr冋
ប 2 2mⴱ兩D⌿兩 2− a⬘
共r兲兩⌿兩2 +b⬘
2兩⌿兩 4+ 1 8共B − H兲 2册
. 共1兲Here D⬅+i共2/⌽0兲A denotes the covariant derivative, where⌽0= hc/eⴱ, eⴱ= 2兩e兩 is the unit of flux, B=⫻A is the magnetic induction, and mⴱ is the effective mass. Assuming that the ratio ef f⬅ef f/Ⰷ1, where ef f= 22/s is the
ef-fective penetration depth andis the coherence length, mag-netization is by a factor 1/2smaller than the field. Conse-quently for magnetic fields few times larger than Hc1, one
uses B⬇H. The vector potentials are chosen in the symmet-ric gauge, Ax= − 1 2By, Ay= 1 2Bx. 共2兲
The pinning centers are located at points ra 关2D vectors r =共x,y兲 will be denoted by bold letters兴 共see Fig.1兲. When
pinning is absent, the coefficient a
⬘
共r兲=␣共Tc− T兲 in Eq. 共1兲 isuniform. The static free energy is minimized by a hexagonal Abrikosov lattice of vortices with cores, with primitive vec-tors of hexagonal lattice
a1= a⌬
冉
1 2,冑
32
冊
, a2= a⌬共1,0兲, 共3兲 where the lattice spacing is a⌬= 21/23−1/4冑
⌽0/B. Pinning is represented by an inhomogeneous coefficienta
⬘
共r兲 =␣共Tc− T兲 − Tc兺
aU共r − ra兲, 共4兲
where U are “potentials” around pinning centers ra. As
dis-cussed above, an interesting configuration corresponds to a hexagonal periodic array located at
ra= n1a1+ n2a2 共5兲
commensurate with the static Abrikosov lattice at matching field.
The superconducting current density has a form
J =ie
ⴱប 2mⴱ共⌿
ⴱD⌿ − ⌿D⌿ⴱ兲. 共6兲
In the pinned state considered in the present paper electric field is absent outside of very narrow shelf near the boundaries1共of the width of order of兲. Therefore there is no
normal current present in the bulk of the sample, so that J represents both the persistent and the diamagnetic 共namely, the one circling around the vortex cores兲 components of the supercurrent density. The persistent current, which originates from the normal electron’s current in the leads, is seen as an imbalance between the currents on two sides of vortices共see Fig.2兲. We will be mostly interested in the sample average of the current density关assumed to be along the y direction 共see Fig.1兲兴,
J = 1 LxLy
冕
d2rJy共r兲 ⬅ 具Jy典. 共7兲
Only the persistent current component contributes to it. Below the coherence length =ប/共2mⴱ␣Tc兲1/2 will be
used as a unit of length r→r/and Hc2=⌽0/22as a unit of magnetic field, h = B/Hc2. The scaled order parameter
= 2−1/2⌿/⌿0, where 兩⌿0兩=共␣Tc/b
⬘
兲1/2, so that thedimen-sionless energy can be written in the following form:
fGL=
冕
d2r共f2+ f4+ fp兲, 共8兲 f2=ⴱ冋
− D2 2 − 1 − t 2册
, 共9兲FIG. 1. 共Color online兲 A 2D array of pinning centers commen-surate with hexagonal Abrikosov lattice 共superfluid density is shown兲 at matching field. All the currents are diamagnetic, no per-sistent current present.
j
FIG. 2. 共Color online兲 A persistent current carrying state. Vorti-ces are displaced with respect to the pinning centers. The current density J, in addition to the diamagnetic component, has a rela-tively small persistent current component. A variational order pa-rameter configuration includes two lowest Landau levels.
f4=2共 兲 , 共10兲 fp=
兺
a
V共r − ra兲ⴱ, 共11兲
where t = T/Tcand V共r兲=U共r/兲/␣is dimensionless pinning
potential. It is convenient to present the persistent current in the units of the depairing current Jd= cHc2/22.The
di-mensionless of the current density is defined by
J = Jdj 共12兲
with
j = i
2关
ⴱD−共D兲ⴱ兴. 共13兲 Below the energy Eq. 共8兲 is minimized with fixed average current 关Eq. 共7兲兴 first by using a simple variational proce-dure.
III. VARIATIONAL METHOD FOR PERSISTENT CURRENT CARRYING PINNED STATES A. Qualitative description and symmetry considerations
Neglecting the vortex creep due to thermal fluctuations on the mesoscopic scale, the vortex matter in the presence of pinning can be in one of the two stationary states, either a pinned共static兲 vortex lattice or a flux flow. Both states gen-erally carry the electric current; however the nature of con-ductivity is totally different. In the pinned state the transport is dissipatedness due to the “persistent” superconducting cur-rent and electric field vanishes inside the sample. When the persistent current in the pinned state approaches a critical magnitude Jc, vortices are depinned and the flux flow ensues.
In the flux flow regime the electric field does penetrate the bulk of the superconductor and Ohmic 共Bardeen-Stephen兲 dissipation arises.
In the absence of persistent current at the matching field the Abrikosov lattice coincides共“matches”兲 with the pinning array. In particular vortex centers共zeros of the order param-eter兲 will coincide with the pinning centers at ra共see Fig.1兲.
Persistent current not only displaces the “vortices” with re-spect to the pinning centers 共by the Lorentz force兲 but also significantly deforms their shape10共see Fig.2兲. At small
per-sistent current densities the displacement of vortices should be small enough, so that linear elasticity theory applies. For increasing current densities nonlinear effects appear and grow. Eventually at a critical current density the static dis-torted Abrikosov lattice becomes unstable to depinning. The critical current and the order parameter configuration at given current density depend both on the strength and on the shape of the pinning center.
Free energy of a pinned system in which there is one vortex per pinning site共f =1兲 关Eq. 共8兲兴 has a hexagonal lat-tice translational symmetry
r→ r + n1a1+ n2a2, 共14兲 where the lattice vectors were defined in Eq.共3兲. In addition it has a rotation by /6 symmetry. In the presence of the
persistent current共see Fig.2兲, the configuration of the order parameter in principle might not be symmetric 共namely, a spontaneous symmetry breaking takes place兲. However nu-merical simulations we made demonstrate that in the present case spontaneous symmetry breaking of the translational symmetry关Eq. 共14兲兴 does not occur. This greatly constrains possible solutions and allows a simple variational procedure. The rotational hexagonal symmetry is broken spontaneously down to reflection symmetry with respect to the supercurrent direction.
B. Why a lowest Landau level order parameter configuration cannot carry net supercurrent in a pinned state
Let us consider a simple case of short-range pinning po-tential and then generalize to arbitrary shape of the pinning center. It is clear that with such a choice of pinning potential a properly normalized LLL state,
共r兲 =
冑
ah A 0共r兲, 0共r兲 = 31/8冑
h兺
i eil2/2exp再
i冋
− h 2xy + 共2l + 1兲 a冉
x − a 4冊
册
−h 2冋
y − 共2l + 1兲 ah册
2冎
, 共15兲where a = a⌬/, ah=共1−t−h兲/2, is still an approximate
Abri-kosov solution of the GL equation for zero net supercurrent. It seems natural to try to look for periodic configurations of the order parameter describing the current carrying states among the other LLL states. It is well known that an LLL is uniquely defined by locations of its zeros, so that possible candidates are displaced configurations,0共r+u兲. It turns out that this naive assumption fails since generally these states have vanishing persistent current. Indeed it can be shown8
that in any LLL state the current density is proportional to curl of superfluid density
Ji⬀ ijj兩兩2, 共16兲
so that an integral over unit cell vanishes due to periodicity. Hereijis an antisymmetric tensor in two dimensions.
Con-sequently to describe states carrying the persistent supercur-rent, higher Landau levels are necessary.
C. Trial functions
The simplest such states involve, in addition to the LLL, just a shifted first Landau level,
=
冑
ahA
关c00共r + u兲 + ic11共r + u兲兴, 共17兲 where
1共r兲 = 21/231/8h
兺
l冋
y −共2l + 1兲 ah册
e il2/2 exp再
i冋
−h 2xy +共2l + 1兲 a冉
x − a 4冊
册
− h 2冋
y − 共2l + 1兲 ah册
2冎
. 共18兲 FunctionsNare normalized by具兩0共r兲兩2典=1. Variationalco-efficients c0and c1are assumed to be real for the following reasons. First the overall phase does not influence gauge in-variant quantities and second the supercurrent is directed along the y axis only when the relative phase is /2. Sub-stituting Eq.共17兲 into the current density defined in Eq. 共13兲 and using Dy0=
冉
y+ i h 2x冊
0= −冑
h 21, 共19兲 one obtains 具jy典 = ah A冑
2hc1c0. 共20兲This means that the persistent current appears due to a mix-ture of LLL and the first LL 共see Fig. 2兲. Similarly when higher LLs are present the current will get contributions due to mixture of NandN+1only关“a selection rule” 共see Ref.
8 for description兲兴. Therefore higher Landau levels N⬎1 will contribute very small corrections because, as mentioned above, the Abrikosov LLL solution is dominant. We there-fore should minimize the free energy 关Eq. 共8兲兴 for a set of variational parameters c0, c1, and u.
IV. STRUCTURE OF THE PINNED STATE AND CRITICAL CURRENT FOR VARIOUS PINNING CENTERS
A. Minimization of energy
The variational energy 共averaged over the unit cell兲 has the quadratic in order parameter, quartic and the pinning con-tributions defined in Eq. 共8兲,
具f2典 = ah A 关共h − ah兲c1 2 − ahc0 2兴, 共21兲 具f4典 = ah 2 2A 2具关c02兩0兩2+ c12兩1兩2+ ic0c1共0ⴱ1−01ⴱ兲兴2典 = ah A
冋
ahc04 2 + ah1 2A c14+ 2ah12 A c02c12册
, 共22兲 具fp典 = ah A 关c0 2 0+ c1 2 1+ 2c0c112兴, 共23兲 where具¯典 is an average over sample 共equivalently over the unit cell due to translational symmetry discussed above兲. The last contribution involves integrals over the pinning potential 共obviously assuming nonoverlapping pinning potentials兲,0=具V共r兲兩0共u + r兲兩2典; 1=具V共r兲兩1共u + r兲兩2典, 共24兲
12= i 2具V共r兲0
ⴱ共u + r兲1共u + r兲典 + c.c. 共25兲 The constants in Eq.共22兲 are
1=
冕
dxdy兩1兩4= 2.1126,12=
冕
dxdy兩0兩2兩1兩2=A/2 = 0.58. 共26兲The minimization equations with respect to variational pa-rameters of the trial functions c0 and c1 are
dfGL dc0 ⬀ ah共− 1 + c0 2 + c12兲c0+ c00+ c112= 0, 共27兲 dfGL dc1 ⬀
冋
共h − ah兲 + ah1 A c12+ ahc02册
c1+ c11+ c012= 0. 共28兲 The displacement of vortices u determines the transport cur-rent, so that to find the critical current density, one has to maximize the current when u runs over the unit cell共see Fig. 3兲. This set of equations is solved numerically in two cases: the␦potential and rectangular pins. In particular it is instruc-tive to solve the equations using perturbation theory in small displacement u 共linear elasticity theory兲. Since the current will be flowing along the y direction, the displacement vector should be oriented in the x direction.B. Location of vortices for the␦-pinning array: Linear elasticity theory
For simplicity let us consider first an array of identical pinning centers each of which is described by the potential
V共r兲 = U
兺
a
␦共r − ra兲. 共29兲
This is appropriate when the size 共radius兲 of the pin w is smaller than the coherence length. The dimensionless pin-ning strength parameter can be estimated by U =w2/Tc,
where⬃Tc− Tc0is potential well energy. In this case
inte-grals in Eqs.共24兲 and 共25兲 are simply
0= U兩0共− u兲兩2, 1= U兩1共− u兲兩2,
12= i 2U关0
ⴱ共− u兲1共− u兲 −0共− u兲 1
ⴱ共− u兲兴. 共30兲 We start from derivation of the linear elasticity of the pinned vortex lattice in which the displacement 共which is along the x direction兲 is assumed to be small.
There is only one stable solution for ux= u = 0:
c0= 1 , c1= 0 共there is another unstable solution with c0= 0兲. This is just the Abrikosov lattice configuration with zero critical current 共higher order corrections in ah involve 6th,
12th, etc., Landau levels, which are very small and well be-yond our variational procedure兲. Expanding functions 共15兲
and 共18兲 in small displacement u around this equilibrium solution one obtains for densities Eq. 共30兲 appearing in the pinning energy, 0= U兩0共− u兲兩2⯝ U 2h兩1共0兲兩 2u2+ O共u3兲, 1⯝ U兩1共0兲兩2+ O共u兲, 12= iU 2 关0
ⴱ共− u兲1共− u兲 −0共− u兲 1 ⴱ共− u兲兴
= − U
2共2h兲1/2兩1共0兲兩2u + O共u2兲, 共31兲 where兩1共0兲兩2= 3.77h.
To the leading nontrivial order therefore one obtains c0= 1 +0u2, c1=1u. Substituting this into the minimization equation关Eq. 共27兲兴, one obtains to order u2
ah共20u2+1 2
u2兲 +0+1u12= 0. 共32兲 Using Eq.共31兲, one obtains
ah共20+1
2兲 +U兩1共0兲兩
2h −1
U兩1共0兲兩
2
冑
2h = 0. 共33兲 Similarly the second minimization equation关Eq. 共28兲兴 to the first order in u readsh1+ U兩1共0兲兩21−1 2
冑
1
2hU兩1共0兲兩
2= 0, 共34兲 determining the correction
1=1 2
冑
1 2h U兩1共0兲兩2 U兩1共0兲兩2+ h. 共35兲 Consequently the average persistent current density respon-sible for the pinning force isj = ah A
冑
2hc1= ah 2A U兩1共0兲兩2 U兩1共0兲兩2+ hu. 共36兲 The Lorentz force on one vortex depends linearly on dis-placement j = Ku, where the Labusch parameter isK = ah 2A
3.77U
3.77U + 1. 共37兲
This parameter enters various phenomenologically important quantities such as surface impedance of the microwave absorption.11
C. Beyond the elasticity theory: Critical current
Let us now turn to the calculation of the critical current. It is natural to assume that above certain current density the static solution loses its stability. The linear elasticity, which breaks down well below this critical current density, is reached共within a harmonic well depinning is actually impos-sible and formally the critical current is infinite兲. To estimate the critical current, the minimization equations therefore should be solved numerically. In Fig. 3共a兲 we show depen-dence of the current on the displacement of vortices in the x direction for different values of pinning strength U for h = 0.5 and t = 0共that is, for ah= 0.25兲. For small displacement
u the persistent current density rises linearly consistently with perturbation theory. In addition it is clear that the cur-rent vanishes when vortex stays right in the middle between the pinning centers, that is, for u = a⌬/2. Therefore the maxi-mal persistent current jcshould exist at certain displacement
ucin between. The displacement ucweakly depends on
pin-ning strength decreasing as the pinpin-ning strength rises. In the critical current one observes that at U⬍0.1 the critical cur-rent is well approximated by
jc=
ah A
U. 共38兲
For stronger pinning the vortex is hold tightly共confined兲 by the pining center and jc diverges as in the linear elasticity
theory. For pinning centers of the order or larger than coher-ent lengththe model should be generalized. In addition the shape of large pinning center might become important.
1 2 3 4 5 0.2 0.4 0.6 0.8 1.0
u
0.01 0.02 0.03 0.04j
0.6 0.7 0.8 0.9 1.0h
0.01 0.02 0.03 0.04Jc
(b) (a)FIG. 3.共Color online兲 共a兲 Dependence of the dimensionless per-sistent current on displacement for the delta pinning for different pinning strengths 共curve 1 U=0.05, curve 2 U=0.1, curve 3 U = 0.2, curve 4 U = 0.4, and curve 5 U = 1兲. 共b兲 Dependence of the maximal persistent current on the magnetic field for the delta pinning for different values of pinning strengths U = 0.05, 0.1, 0.2, 0.4, 0.6, 0.8共from bottom to top兲.
D. Rectangular pinning center array
A commensurate array of rectangular artificial pinning centers can be modeled using the following single pin poten-tial:
V共r兲
=
再
V0 for − wx/2 ⬍ x ⬍ wx/2 and − wy/2 ⬍ y ⬍ wy/20 otherwise.
冎
共39兲 In this case the integrals appearing in the pining term关Eqs. 共24兲 and 共25兲兴, 0=
冕
−wx/2 wx/2 dx冕
−wy/2 wy/2 dy兩0共u + r兲兩2, 1=冕
−wx/2 wx/2 dx冕
−wy/2 wy/2 dy兩1共u + r兲兩2, 12= i 2冕
−wx/2 wx/2 dx冕
−wy/2 wy/2 dy0ⴱ共u + r兲1共u + r兲 + c.c., 共40兲 were performed numerically. The dependence of the current on displacement for different aspect ratios r = wx/wy, pinningareas S = wxwy, and the potential strength U is given in Fig.4.
Magnetic field and temperature are fixed as before at h = 0.5, t = 0, so that ah= 0.25. The results are discussed
below.
V. DISCUSSION AND CONCLUSIONS
The model of the vortex crystal pinned by the periodic array of inclusions at the matching magnetic field considered here is remarkably simple and useful for theoretical analysis. It allows us to study, from first principles, various general properties of the vortex matter including critical current, elasticity, and destruction of the vortex crystal. Dependence of the dimensionless persistent current on the displacement of the vortex lattice with respect to pins for the ␦ pinning was calculated analytically 关see Fig. 3共a兲兴. In order to ex-press the results in physical units the dimensionless current density must be multiplied by the depairing current density Jd= cHc2/22. At small pinning strength U the critical
cur-rent increases rapidly and saturates at large U关see Fig.3共b兲兴. The shape of the pinning center also affects the critical cur-rent when both the size of the pinning center共on the scale of coherence length兲 and its strength are sufficiently large 共see Fig.4兲. The critical current is always largest for a more sym-metric square pin关lines numbered 1 on Figs.4共a兲and4共b兲兴. It is followed by a rectangular center with the long side
par-1 2 3 0.2 0.4 0.6 0.8
u
0.01 0.02 0.03 0.04j
square wxwy14 wxwy4 0.1 0.2 0.3 0.4 0.5 0.6 0.7u
0.005 0.010 0.015 0.020 0.025 0.030j
square wxwy14 wxwy4 0.2 0.4 0.6 0.8u
0.005 0.010 0.015 0.020 0.025 0.030 0.035j
square wxwy14 wxwy4 0.2 0.4 0.6 0.8 1.0u
0.005 0.010 0.015 0.020j
(b) (a) (c) (d)FIG. 4. 共Color online兲 共a兲 Dependence of the persistent current on displacement for the rectangular pinning centers for small area of the pinning center and three different pinning strengths 共curve 1 U=3, curve2 U=1, and curve 3 U=0.33兲. The straight lines correspond to elasticity theory关Eq. 共37兲兴. 共b兲 Dependence of the persistent current on displacement for various shapes of pinning centers for fixed pinning potential V0= 3 and area wxwy= 0.5.共c兲 Dependence of the persistent current on displacement for various shapes of pinning centers for fixed
pinning potential V0= 1 and area wxwy= 0.5.共d兲 Dependence of the persistent current on displacement for various shapes of pinning centers for fixed pinning potential V0= 0.33 and area wxwy= 0.5.
allel to the current关lines numbered 2 on Figs.4共a兲and4共b兲兴. The rectangular pinning centers with the long side perpen-dicular to the current 关lines numbered 3 on Fig. 4共a兲兴 have always the lowest critical current. At small pinning potential the difference is insignificant, as can be learnt from Figs.4共c兲 and4共d兲. The results for the smaller square pinning centers are consistent with the␦-function approximation, as can be seen from the linear part of the dependence of the current on displacement in Figs. 4共b兲 and 4共d兲. The lines follow a simple formula for the Labusch parameter 关Eq. 共38兲兴. The dependence of the largest persistent current, Jc, on the
mag-netic field H for the␦ pinning for various pinning strengths is presented in Fig.3共b兲. It vanishes as共1−H/Hc2兲3/2 in the
limit H→Hc2. A similar dependence was obtained for the
longitudinal 共parallel to the magnetic field direction兲 critical current in Ref. 12.
The physical picture of the persistent current in the pinned state can be considered from two complementary angles. From one point of view, the nonzero persistent current ap-pears in the pinned state due to shift and deformation of the vortex core, while from another point of view, the deforma-tion is due to the pinning force creating the persistent cur-rent. Vortex cores are no longer circular, as was observed in numerical simulations by Priour and Fertig.10 The
deforma-tion of the vortex cores is observable by currently existing scanning tunnel microscopy techniques.13
Most favorable conditions to look for the phenomena de-scribed in this paper are the following. Thermal fluctuations on the mesoscopic scale 共not included in the calculation兲
ought to be minimized since they lead to thermal depinning of vortices at elevated temperatures. This means that for strongly fluctuating materials such as high Tc cuprates the
temperature should be lower than the depinning temperature. Pinning should be strong enough similar to the one achieved in an array of artificial magnetic dots.
It is well known that a small deviation from the matching magnetic field leads to a sharp decrease of the critical current roughly to the level of an equivalent random pinning array.4
Therefore there exists a sharp peak in the critical current of a small width⌬B. This is due to the fact that even for a small deviation from the matching condition, interstitial vortices 共or vortex vacancies兲 appear and determine the reduced criti-cal current. The current is still larger than that in the random pinning array with the same number of pinning sites because the interstitial vortices continue to move in the periodically modulated environment created by the pinned set of the vor-tex “channels.”14,15
ACKNOWLEDGMENTS
We thank T. Maniv, V. Metlushko, R. F. Hong, D. Berco, and M. Lewkowicz for discussions. We acknowledge support from the Israel Scientific Foundation 共Grant No. 499/07兲. Work of B.R. was supported by NSC of R.O.C. Grant No. 982112M009014MY3 and MOE ATU program. B.R. ac-knowledges the hospitality and support at Physics Depart-ment of Bar Ilan University during sabbatical leave.
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