A. E. Koshelev- Josephson vortices and solitons inside pancake vortex lattice in layered...

8/3/2019 A. E. Koshelev- Josephson vortices and solitons inside pancake vortex lattice in layered superconductors

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arXiv:cond-mat/0309021v1

31Aug2003

Josephson vortices and solitons inside pancake vortex lattice in layered

superconductors

A. E. KoshelevMaterials Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne , IL 60439

(Dated: February 2, 2008)

In very anisotropic layered superconductors a tilted magnetic field generates crossing vortex lat-tices of pancake and Josephson vortices (JVs). We study the properties of an isolated JV in thelattice of pancake vortices. JV induces deformations in the pancake vortex crystal, which, in turn,substantially modify the JV structure. The phase field of the JV is composed of two types of phasedeformations: the regular phase and vortex phase. The phase deformations with smaller stiffnessdominate. The contribution from the vortex phase smoothly takes over with increasing magneticfield. We find that the structure of the cores experiences a smooth yet qualitative evolution withdecrease of the anisotropy. At large anisotropies pancakes have only small deformations with respectto position of the ideal crystal while at smaller anisotropies the pancake stacks in the central rowsmoothly transfer between the neighboring lattice positions forming a solitonlike structure. We alsofind that even at high anisotropies pancake vortices strongly pin JVs and strongly increase theirviscous friction.

PACS numbers: 74.25.Qt, 74.25.Op, 74.20.De,

I. INTRODUCTION

The vortex state in layered superconductors has avery rich phase diagram in the multidimensional spaceof temperature-field-anisotropy-field orientation. Espe-cially interesting subject is the vortex phases in lay-ered superconductors with very high anisotropy suchas Bi2Sr2CaCu2Ox (BSCCO). Relatively simple vortexstructures are formed when magnetic field is appliedalong one of the principal axes of the layered structure.A magnetic field applied perpendicular to the layers pen-etrates inside the superconductor in the form of pan-cake vortices (PVs).1 PVs in different layers are coupled

weakly via the Josephson and magnetic interactions andform aligned stacks at low fields and temperatures (PVstacks). These stacks are disintegrated at the meltingpoint. In another simple case of the magnetic field ap-plied parallel to the layers the vortex structure is com-pletely different. Such a field penetrates inside the super-conductor in the form of Josephson vortices (JVs).2,3,4

The JVs do not have normal cores, but have rather widenonlinear cores, of the order of the Josephson length, lo-cated between two central layers. At small in-plane fieldsJVs form the triangular lattice, strongly stretched alongthe direction of the layers, so that JVs form stacks alignedalong the c direction and separated by a large distancein the in-plane direction.

A rich variety of vortex structures were theoreticallypredicted for the case of tilted magnetic field, such as thekinked lattice,4,5,7, tilted vortex chains8, coexisting lat-tices with different orientation.9 A very special situationexists in highly anisotropic superconductors, in which themagnetic coupling between the pancake vortices in dif-ferent layers is stronger than the Josephson coupling. Insuch superconductors a tilted magnetic field creates aunique vortex state consisting of two qualitatively differ-ent interpenetrating sublattices.5,6 This set of crossing

lattices (or combined lattice5) contains a sublattice ofJosephson vortices generated by the component of thefield parallel to the layers, coexisting with a sublattice ofstacks of pancake vortices generated by the componentof the field perpendicular to the layers. A basic reasonfor such an exotic ground state, as opposed to a simpletilted vortex lattice, is that magnetic coupling energy isminimal when pancake stacks are perfectly aligned alongthe c axis. A homogeneous tilt of pancake lattice coststoo much magnetic coupling energy, while formation ofJosephson vortices only weakly disturbs the alignment ofpancake stacks.

Even at high anisotropies JVs and pancake stacks havesignificant attractive coupling.6 The strong mutual in-teraction between the two sublattices leads to a veryrich phase diagram with many nontrivial lattice struc-tures separated by phase transitions. At sufficientlysmall c-axis fields (10-50 gauss) a phase separated stateis formed: density of the pancake stacks located at JVsbecomes larger than the stack density outside JVs.6,14

This leads to formation of dense stack chains separatedby regions of dilute triangular lattice in between (mixedchain+lattice state). Such structures have been observedin early decoration experiments15,16 and, more recently,by scanning Hall probe17, Lorentz microscopy18, andmagnetooptical technique.19,20 At very small c-axis fields(

several gauss) the regions of triangular lattice vanish

leaving only chains of stacks.17 Moreover, there are ex-perimental indications17 and theoretical reasoning21 infavor of the phase transition from the crossing config-uration of pancake-stack chains and JVs into chains oftilted vortices. JVs also modify the interaction betweenpancake stacks leading to an attractive interaction be-tween the stacks at large distances.22 As a consequence,one can expect clustering of the pancake stacks at smallconcentrations.

Many unexpected observable effects can be naturally
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interpreted within the crossing lattices picture. The un-derlying JV lattice modifies the free energy of the vortexcrystal state. An observable consequence of this changeis a shift of the melting temperature. Strong support forthe crossing-lattices ground state is the linear dependenceof the c-axis melting field on the in-plane field observedwithin a finite range of in-plane fields.6,12,13,23,24 In anextended field range several melting regimes have been

observed23,24

indicating several distinct ground states ofvortex matter in tilted fields. Transitions between differ-ent ground state configurations have also been detectedby the features in the irreversible magnetization.25,26

In this paper we consider in detail the properties of anisolated JV in the pancake lattice. We mainly focus onthe regime of a dense pancake lattice, when many rowsof the pancakes fit into the JV core. The pancake latticeforms an effective medium for JVs which determines theirproperties. The dense pancake lattice substantially mod-ifies the JV structure. In general, the phase field of theJV is built up from the continuous regular phase and thephase perturbations created by pancake displacements.Such a JV has a smaller core size and smaller energy ascompared to the ordinary JV.6 The pancake lattice alsostrongly modifies the field and current distribution faraway from the core region.11

The key parameter which determines the structure ofthe JV core in the dense pancake lattice is the ratio = /s, where is the in-plane London penetrationdepth, is the anisotropy ratio, and s is the period of lay-ered structure. The core structure experiences a smoothyet qualitative evolution with decrease of this parame-ter. When is small (large anisotropies) pancakes haveonly small displacements with respect to positions of theideal crystal and the JV core occupies several pancakesrows. In this situation the renormalization of the JV

core by the pancake vortices can be described in terms ofthe continuous vortex phase which is characterized by itsown phase stiffness (effective phase stiffness approach).6

At large (small anisotropies) the core shrinks to scalessmaller than the distance between pancake vortices. Inthis case pancake stacks in the central row form soliton-like structure smoothly transferring between the neigh-boring lattice position.

We consider dynamic properties of JVs in the case ofsmall : the critical pinning force which sticks the JVto the PV lattice and the viscosity of the moving JVdue to the traveling displacement field in the PV lattice.The pinning force has a nonmonotonic dependence on

the c-axis magnetic field, Bz, reaching maximum whenroughly one pancake row fits inside the JV core region.At higher fields the pinning force decays exponentially exp(

Bz/B0). We study JV motion through the

PV lattice and find that the lattice strongly hinders themobility of JVs.

The paper is organized as follows. Section II is devotedto the static structure of an isolated JV inside the PVlattice. In this section we

consider small c-axis fields and calculate the cross-

ing energy of JV and PV stack (IIB);

consider large c-axis fields and introduce the ef-fective phase stiffness approximation, which al-lows for simple description of JV structure insidethe dense PV lattice in the case of large anisotropy(II C);

investigate a large-scale behavior and JV magnetic

field (II D);

analyze the JV core quantitatively using numericminimization of the total energy and find crossoverfrom the JV core structure to the soliton core struc-ture with decrease of anisotropy (II E);

formulate a simple model which describes the soli-ton core structure for small anisotropies (II F).

In Section III we consider pinning of JV by the pancakelattice and calculate the field dependence of the criticalcurrent at which the JV detaches from the PV lattice.In Section IV we consider possible JV dynamic regimes:dragging the pancake lattice by JVs and motion of JVsthrough the pancake lattice. For the second case we cal-culate the effective JV viscosity.

II. STRUCTURE AND ENERGY OFJOSEPHSON VORTEX IN PANCAKE LATTICE

A. General relations

Our calculations are based on the Lawrence-Doniachfree energy in the London approximation, which dependson the in-plane phases n(r) and vector-potential A(r)

F =n

d2r

J2

n 2

0A

2

+EJ

1 cos

n+1 n 2s

0Az

+

d3r

B2

8, (1)

where

J s20

(4)2 s0

and EJ

20

s (4c)2 (2)

are the phase stiffness and the Josephson coupling en-

ergy, ab and c are the components of the Londonpenetration depth and s is the interlayer periodicity. Weuse the London gauge, divA = 0. We assume that theaverage magnetic induction B inside the superconductoris fixed.10 The c component of the field fixes the concen-tration of the pancake vortices nv Bz/0 inside onelayer. The in-plane phases n have singularities at thepositions of pancake vortices Rin inside the layers,

[ n]z = 2i

(rRin) .


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The major obstacle, preventing a full analytical consid-eration of the problem, is the nonlinearity coming fromthe Josephson term. A useful approach for superconduc-tors with weak Josephson coupling is to split the phaseand vector-potential into the vortex and regular contri-butions, n = vn + rn and A = Av + Ar. The vortexcontributions minimize the energy for fixed positions ofpancake vortices at EJ = 0 and give magnetic interac-

tion energy for the pancake vortices. One can express thispart of energy via the vortex coordinates Rn,i. In gen-

eral, the regular contributions may include phases andvector-potentials of the Josephson vortices. The totalenergy naturally splits into the regular part Fr , the en-ergy of magnetic interactions between pancakes FM, andthe Josephson energy FJ, which couples the regular andvortex degrees of freedom,

F = Fr + FM + FJ (3)

with

Fr [rn,Ar] =n

d2r

J

2

rn 2

0Ar

2+

d3r

B2r

8, (4)

FM [Rn,i] =1

2

n,m,i,j

UM(Rn,i Rm,j, n m), (5)

FJ [rn,Ar,Rn,i] =nd

2rEJ1 cosn+1 n

2s

0Az , (6)

and

UM(R, n) =J

2

dk

dqexp[ikR+ iqn)]

k2[1 + 2(k2 + 2(1 cos q)/s2)1]

2J

lnL

R

n s

2exp

s|n|

+

s

4u

r

,

s|n|

(7)

is the magnetic interaction between pancakes1 where

u (r, z) exp(z)E1 (r z) + exp(z)E1 (r + z) , (8)

E1(u) =u (exp(v)/v) dv is the integral exponent

(E1(u) E ln u + u at u 1 with E 0.577),r

R2 + (ns)2, and L is a cutoff length.In this paper we focus on the crystal state. If the

pancake coordinates have only small deviations from the

positions R(0)i of the ideal triangular lattice, Rni =

R(0)i +uni then Fv reduces to the energy of an ideal crys-

tal Fcr plus the magnetic elastic energy FMel consistingof the shear and compression parts,

FMel =d3k

(2)3

Ut(k)

2|ut(k)|2 + Ul(k)

2|ul(k)|2 , (9)

where

uni

d3k

(2)3exp

ikR(0)i +ikzsn

[etut(k)+elul(k)]

(el (et) is the unit vector parallel (orthogonal) to k),

Ut(k) = C66k2 + U44(k),

Ul(k) = U11(k) + U44(k),

U11 is the compression stiffness, C66 is the shear modulus.In particular, at high fields, Bz 0/2, we have

U11 C11(k)k2 B2

z42

1 k2

16nv

,

C66 = nv0/4

The magnetic tilt stiffness,29 U44(k), is given by interpo-lation formula, which takes into account softening due topancake fluctuations,

U44(k) C44(kz)k2z=

Bz02(4)24

ln

1 +

r2cutk2z + r2w

(10)

with r2w = (un+1 un)2

, rcut at a > andrcut a/4.5 at a < ,27 with a =2/(3nv) being thelattice constant. At finite Josephson energy minimiza-tion of the energy with respect to the phases at fixedpancake positions leads to the Josephson term in the tiltstiffness energy.30

The major focus of this paper is the structure of JVcore. This requires analysis of the pancake displacementsand regular phase at distances r c and z a, from the vortex center. At these distances the main con-tribution to the energy is coming from the kinetic en-


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ergy of supercurrents and magnetic screening can be ne-glected. The structure of energy is significantly simpli-

fied: one can neglect the field contributions in the regularand Josephson energy terms, i.e., drop Ar:

Fr [rn,Ar]Fr [rn] =n

d2rJ

2(rn)2 , (11)

FJ [rn,Ar,Rn,i]FJ [rn,Rn,i] =n

d2rEJ (1 cos(n+1 n)) , (12)

and use asymptotics rcutkz 1 in the tilt stiffness (10).Behavior at large distances r c and z a, is im-portant for accurate evaluation of the cutoff in the loga-rithmically diverging energy of the Josephson vortex. Inthis range the Josephson term can be linearized and onecan use the anisotropic London theory.11

B. Small c axis field: Crossing energy

At small fields and high anisotropy factor pancake vor-tices do not influence much structure of JVs. However,there is a finite interaction energy between pancake stackand JV (crossing energy) which causes spectacular ob-servable effects, including formation of the mixed chain-lattice state.6,14,15,16,18 We consider a JV located betweenthe layers 0 and 1 and directed along x axis with centerat y = 0 and a pancake stack located at x = 0 and atdistance y from the JV center. We will calculate struc-ture of the pancake stack and the crossing energy. TheJV core structure is defined by the phases n(y) obeyingthe following equation

d2ndy2

+ sin(n+1 n) sin(n n1) = 0 (13)

with y = y/J0. An accurate numerical solution of thisequation has been obtained in Ref. 4. It is described bythe approximate interpolation formula28

n(y) arctan n 1/2y

+0.35(n 1/2)y

((n 1/2)2 + y2 + 0.38)2

+8.81(n 1/2)y y

2 (n 1/2)2 + 2.77((n 1/2)2 + y2 + 2.02)4

(14)

Interaction between the pancake stack and JV appearsdue to the pancake displacements un under the actionof the JV in-plane currents jn(y) (see Fig. 1). In theregime of very weak interlayer coupling the energy of thedeformed pancake stack is given by

E(y) =

dkz2

UM(kz)

2|u(kz)|2

n

s0c

jn(y)un (15)

un

xz

y

y

(a)(b)

FIG. 1: Configuration of the pancake stack crossing theJosephson vortex:(a) the stack located in the center of JVcore and (b) the stack located at a finite distance y from JVcenter.

where UM(kz) =20

2(4)24 ln

1 + 2

k2z +r2w

is the mag-

netic tilt stiffness of the pancake stack,

jn(y) 2c0(4)

2 spn

y

swith pn(y) dn(y)/dy being the reduced superfluid mo-mentum, and the JV phase n(y) is given by approximateformula (14). In particular, pn(0) = Cn/(n 1/2) withCn 1 at large n. Using the precise numerical phasesn(y) we obtain the interpolation formula

Cn 1 0.265/((n 0.835)2 + 0.566),giving C1 0.55 and C2 0.86. At large distances fromthe core, n, y 1, pn(y) is given by

pn(y) = n 1/2(n 1/2)2 + y2 .

Displacements in the core region have typical wavevectors kz /s. In this range one can neglect kz-dependence of UM(kz),

UM 20

(4)24ln

rw,

and rewrite Eq. (15) as

E(y) =n

sUM

2u2n

s0c

jn(y)un

. (16)


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We neglected weak logarithmic dependence of the tiltstiffness on displacements and the parameter rw in UM is

just a typical value of|un+1un|. Minimizing this energywith respect to un, we obtain the pancake displacements

un(y) =0c

jn(y)

UM 2

2

s ln (/un(y))pn

y

s

with vn(y) |un(y) un1(y)| and crossing energy at fi-nite distance y between the crossing point and the centerof JV core

E(y) = s2UM

n=

0jn(y)

c

2

20

422s ln(/u1(y))A

y

s

(17)

with

A(y) =

n=1[pn (y)]

2=

n=(1 cos(n+1 n)) ,

where the second identity can be derived from Eq. (13).In particular, for the pancake stack located at the JVcenter, A(0) = 2 (exact value) and

E(0) 20

222s ln(3.5s/). (18)

The maximum pancake displacement in the core is givenby

u1(0) 2.22

s ln(2s/). (19)

At large distances, y s, using asymptotics A(y) /4y, we obtain

E(y) 20

16y lny2

s

, with 1. (20)Using numerical calculations, we also obtain the follow-ing approximate interpolation formula for the functionA(y), valid for the whole range of y,

A(y) /4

y2 + y20

1 +

a

y2 + y20

(21)

y0 0.93, a 1.23Eqs. (17) and (21) determine the crossing energy at finitedistance y between the crossing point and the center of

the JV core. Below we will use this result to calculate thepinning force which binds the JV to the dilute pancakelattice.

C. Large c axis field. Approximation of theeffective phase stiffness

At high c axis fields pancakes substantially modifythe JV structure. Precise analysis of the JV core in

the pancake lattice for the general case requires tediousconsideration of many energy contributions (see SectionII E below). The situation simplifies considerably in theregime of very high anisotropy /s. In this caseone can conveniently describe the JV structure in termsof the effective phase stiffness, which allows us to re-duce the problem of a JV in the pancake lattice to theproblem of an ordinary JV at Bz = 0. In Ref. 6 thisapproach has been used to derive JV structures at highfields Bz 0/42. The approach is based on theobservation that smooth transverse lattice deformationsutn(r) produce large-scale phase variations vn(r) withvn = 2nvez utn. This allows us to express thetransverse part of the elastic energy, Fvt, in terms ofvn(r):

Fvt =

dk

(2)3Jv(Bz,k)

2sk2 |v(k)|2 , (22)

with the effective phase stiffness Jv(Bz,k),

Jv(Bz,k) =s

C66k2 + U44

(2nv)2 (23)

J8nv2

2k22

+ ln1 +r2cut

k2z

+ r2w .

Replacing the discrete lattice displacements by thesmooth phase distribution is justified at fields Bz >0/(s)2. The structure of the JV core is determinedby phase deformations with the typical wave vectorsk 1/s < 2/ and kz /s > 1/rw. In this rangethe vortex phase stiffness is k-independent, similar to theusual phase stiffness,

Jv(Bz) JBBz

, B 042

lnrcutrw

. (24)

The phase stiffness energy (22) has to be supplementedby the Josephson energy. In the core region we can ne-glect the vector-potentials and write the total energy interms of rn and v as

F =n

d2r

J

2(rn)2 + Jv

2(vn)2 + EJ (1 cos(n+1 n))

, (25)


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where, again, n rn + vn is the total phase.We now investigate the core structure on the basis of

the energy (25). Eliminating the regular phase, rn =n vn, and varying the energy with respect to vn atfixed n, we obtain the equation

Jn (Jv + J) vn = 0,which gives

vn =J

Jv + Jn. (26)

Substituting this relation back into the energy (25), weexpress it in terms of the total phase

F=n

d2r

Jeff

2(n)2 + EJ (1 cos(n+1 n))

(27)

with the effective phase stiffness Jeff,

J1eff = J1 + J1v or Jeff =

J

1 + Bz/B

. (28)

Note that the smallest stiffness from J and Jv dominatesin Jeff.31 From Eq. (27) we obtain equation for the equi-librium phase

Jeff2yn + EJ [sin(n+1 n) sin(n n1)] = 0,(29)

which has the same form as at zero c axis field, exceptthat the bare phase stiffness is replaced with the effectivephase stiffness Jeff. For the Josephson vortex located be-tween the layers 0 and 1 the phase satisfies the conditions

1 0 0, y 2, y (30)Far away from the nonlinear core the phase has the usualform for the vortex in anisotropic superconductor

n(y) arctan J(n 1/2)y

(31)

where the effective Josephson length

J =

Jeff/EJ = J0/

1 + Bz/B

determines the size of the nonlinear core. Therefore, atlow temperatures the JV core shrinks in the presence of

the c axis magnetic field due to softening of the in-planephase deformations. A number of pancake rows withinthe JV core can be estimated as

Nrows J0

ln(rcut/rw)

4

Bz

B + Bz. (32)

At Bz > B it is almost independent on the field. Anapproximate solution of Eq. (29) is given by Eq. (14),where the bare Josephson length J0 has to be replacedby the renormalized length J.

The JV energy per unit length, EJV , is given by

EJV =

EJJeff lnL

s, (33)

where L is the cutoff length, which is determined byscreening at large distances and will be considered be-low, in Section IID. From Eqs. (26) and (28) we obtainthat the partial contribution of the vortex phase in the

total phase,

vn =Bz

Bz + Bn, (34)

continuously grows from 0 at Bz B to 1 at Bz B. From the last equation one can estimate pancakedisplacements

ux,n(y) =0/2

Bz + Byn

0/2Bz + B

J(n 1/2)y2 + (J(n 1/2))2

. (35)

The maximum displacement in the core can be estimatedas

ux,0(0) 2.22

J0 ln(rcut/rw)

1 + Bz/B. (36)

At Bz B this equation can be rewritten in the formux,0(0)

a 0.58

J0

ln(a/rw), (37)

which shows that condition for applicability of the linearelasticity, ux,0(0) 0.2a, is satisfied if 3/s. Eqs.(31) and (33) describe smooth evolution of the JV struc-ture with increase of concentration of pancakes starting

from the usual vortex at Bz = 0. It is quantitativelyvalid only at very high anisotropies /s and atlow temperatures, when one can neglect fluctuation sup-pression of the Josephson energy. Thermal motion ofthe PVs at finite temperatures induces the fluctuatingphase n,n+1 and suppresses the effective Josephson en-

ergy, EJ CEJ where C

cos n,n+1

. This leads to

reduction of the JV energy and thermal expansion of itscore.

In the range 0/(s)2 < Bz < B the crossing en-ergy regime of Section II B overlaps with the applica-bility range of the effective phase stiffness approxima-tion. To check the consistency of these approximationswe calculate correction to the JV energy at small fieldssumming up the crossing energies and compare the resultwith prediction of the effective phase stiffness approx-imation. The correction to the JV energy is given by

EJV = 1a

Mm=M

E(mb)

0Bz8ln(/u1(0))

lnLys

, (38)


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where Ly = M b is the long-range cutoff length. On theother hand, Eqs. (28) and (33) give at Bz B

EJV =

EJJBz

2Bln

L

s. (39)

This result is identical to Eq. (38) except for expressionsunder the logarithms, which are approximate in bothcases.

D. Large-scale behavior. Screening lengths

In this Section we consider the JV structure at largedistances from the core, n 1, y J. At largedistance screening of supercurrents becomes importantand one can not neglect the vector-potential any more.At these scales the phase changes slowly from layer tolayer so that one can expand the Josephson energy inEq. (6) with respect to the phase difference and usethe continuous approximation, n+1 n 2s0 Az

sz 20 Az,

FJ [n,A] FJ [,A] =

d3rsEJ

2

z 2

0Az

2.

(40)This reduces the Lawrence-Doniach model defined byEqs. (3), (4), (5), and (6) to the anisotropic London

model. Within this model the JV structure outside thecore region has been investigated in detail by Savelev et.al.11. In this section we reproduce JV structure at largedistances using the effective phase stiffness approach. Forthe vortex energy one still can use Eq. (22) with the fullk-dependent phase stiffness (23). Within these approxi-mations the energy (3) is replaced by

F[r, v,A] = Fr[r ,A] + Fvt[v ] + FJ[r + v, Az].(41)

Varying the energy with respect to A, we obtain

02

r A + 22A = 0, (42a)02

z Az + 2c2Az = 0. (42b)

It is convenient to perform the analysis of the large-scalebehavior in k-space. Solving linear equations (42a) and(42b) using Fourier transform,

A = 02

r1 + 2k2

, (43a)

Az =02

z1 + 2ck

2, (43b)

and excluding A, we express the energy in terms ofphases

F = d3k

(2)3 J

2s

2k2

1 + 2k2(r)2 + Jv(k)

2s(v)2 + sEJ

2

2ck2

1 + 2ck2

(z)2 .Following the procedure of the previous Section, we eliminate r, minimize the energy with respect to v, and obtainthe energy in terms of the total phase

F =

d3k

(2)3

Jeff(k)

2s()2 + sEJ(k)

2(z)2

. (44)

where the effective phase stiffness, Jeff(k), and the effective Josephson energy are given by

J1eff (k) = J1 1 +

2k2

2k2+ J1v (k) (45a)

EJ(k) = EJ2ck

2

1 + 2ck2(45b)

In the case of the JV, minimization with respect to the phases has to be done with the topological constrain,zy yz = 2(y)(z), which gives

z = 2ikyJeff(k)Jeff(k)k2y + s

2EJ(k)k2z=

2iky

1 + 2ck2

k2

1 + 2ck2y +

2k2z(1 + w(k) , (46a)

y = 2ikzs2EJ(k)

Jeff(k)k2y + s2EJ(k)k2z

= 2ikz

1 + 2k2

k2

1 + 2ck2y +

2k2z(1 + w(k)) , (46b)


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8

and

EJV = 12

d2k

sEJ(k)Jeff(k)

Jeff(k)k2y + s2EJ(k)k2z

=J

2s

d2k

2 + 2k2y + k2z(1 + w(k))

(47)

with

w(k) = J/Jv(k) =2h

2k2y/2 + ln(1 + k2zr

2cut)

(48)

and h 4nv2. The integration has to be cut at kz /s. In addition, integration with respect to ky is typicallydetermined by ky kz/ so that one can neglect in w(k) the term 2k2y/2 2k2z/22, coming from the shear energy,in comparison with the tilt energy term ln

1 + k2zr

2cut

and the JV energy reduces to

EJV J2s

d2k

2 + 2k2y + k

2z +

2hk2zln (1 + k2zr

2cut)

1

=J

s

kc0

dkz

2 + k2z 1 + 2h/ ln 1 + r2cut/ k2z + r2w. (49)

A similar formula has been derived in Ref. 11. This for-mula shows that the small-kz logarithmic divergence inthe integral cuts off at kz = max(

1, a1). To repro-duce JV energy at Bz = 0 the upper cutoff has to bechosen as kc = 2.36/s.

Lets consider in more detail the case of a large c axisfield h 1, where JV structure is strongly renormalizedby the dense pancake lattice. Formula for the JV energysimplifies in this limit to

EJV Js

2h

kc0

ln

1 + 0.05a2/

k2z + r2w

dkzkz.

To estimate this integral, we split the integration regioninto two intervals, 1/rw kz /s and /a kz 1/rw, and obtain

EJV Js

h

ln

0.2a

rw

ln

2.4rws

+2

3

ln

0.2a

rw

3/2.

Note that the long-range contribution to the energy

scales as a logarithm to the power 3/2.

1. Magnetic field of Josephson vortex

Using Eqs. (46) and (43) we obtain for the JV magnetic field (see also Ref. 11)

Bx(k) =0

1 + 2ck2y +

2k2z(1 + w(k)), (50)

where w(k) is given by Eq. (48). Let us consider the case of large magnetic fields B > B. In a wide region,

0/(s)2. The structure of JV core iscompletely determined by the displacements of pancakevortices and phase distribution. The equilibrium pancakedisplacements depend only on the layer index and on co-ordinate, perpendicular to the direction of the vortex (seeFig. 2). Therefore, the energy can be expressed in terms

FIG. 2: Displacements of the pancake rows in the JV core.

of the displacements of the vortex rows un,i. Different

representations for the magnetic interaction between thevortex rows UMr (un,i um,j, n m) are considered inAppendix A. We will operate with the phase pertur-bation n(r) with respect to equilibrium phase distribu-tion of the perfectly aligned pancake crystal. We splitthis phase into the contribution, averaged over the JVdirection (x axis), n(y), and the oscillating in the x

direction contribution, n(x, y). Pancake displacementsinduce jumps of the average phase at the coordinates ofthe vortex rows Yi, n(Yi + 0) n(Yi 0) = 2un,i/a.The oscillating phase induced by the row displacementsbecomes negligible already at the neighboring row. Thisallows us to separate the local contribution to the Joseph-

son energy coming from n(x, y) (see Appendix B) andreduce initially three-dimensional problem to the two-dimensional problem of finding the average phase androw displacements. Further on we operate only with theaveraged phase and skip the accent in the notationn(y). We again split the total phase into the continu-ous regular phase, rn(y), and the vortex phase, vn(y),n(y) = rn(y) + vn(y; un,i). The vortex phase is com-posed of jumps at the row positions Yi,

vn(y; un,i) = 2a

i

un,i (Yi y) , (55)

where (y) is the step-function ( (y) = 1 (0) at y > 0(y < 0)). In the effective phase stiffness approach of Sec-tion II C we used a coarse-grained continuous approxima-tion for this phase. We neglect x-dependent contributionin the regular phase, which is small at B > 0/(s)2.Collecting relevant energy contributions, we now writethe energy per unit length in terms of the regular phase,


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10

rn(y), and the row displacements, un,i,

EJ=n

dy

J

2

drn

dy

2+ EJ (1 cos(n+1 n))

+1

2

n,m,i,j

UMr(un,i um,j, Yi,j, n m)

+n,i EJosc(un+1,i un,i, n+1,i n,i) (56)

where

n(y) rn(y) + vn(y; un,i) is the total phase; UMr (xn,ixm,j, Yi,j, nm) is the magnetic interac-

tion between the vortex rows separated by distanceYi,j = Yi Yj = b(i j) (see Appendix A),

UMr (un,i um,j, . . .) UMr (Xij + un,i um,j, . . .) UMr (Xij , . . .).

is the variation of this interaction caused by pan-cake row displacements, Xi = 0 for even i andXi = a/2 for odd i.UMr (x, y, n) is periodic with respect to x,

UMr (x + a, y, n) = UMr (x, y, n);

EJosc(un+1,iun,i, n+1,in,i) is the local Joseph-son energy due to the oscillating component of thephase difference (see Appendix B) with

n,i rn(Yi) + un,ia

2a

j>i

un,j (Yj y)

being the external phase at i-th rows and n-th layer.

The energy (56) describes JV structure at distances r c and z a, from its center.

To facilitate calculations we introduce the reduced co-ordinates

y =y

s, vni =

unia

,

and represent the energies in the scaling form. Magneticinteraction between the rows we represent as

UMr(x, y, n) =J a

2VMr ( x

a,

y

a, n),

where

VMr(x, y, n) = 2

a2

n s

2exp

s|n|

ln[1 2cos2x exp(2|y|) + exp (4|y|)]

+s

2a2

m=u

y2 + (x m)2

/a,

s|n|

and u(r, z) is defined by Eq. (8). Note that at x, y 0 VMr (x, y, n) remains finite for n = 0 because, logarithmicdivergency in the first term is compensated by logarithmic divergency of the m = 0 term in the sum. At 2 x2 + y2 0, using asymptotics u(, z) exp(z)

E ln

2

2z

+ exp(z)E1 (2z) with E = 0.5772, we obtain the

limiting value of VMr (x, y, n) at n = 0

VMr(0, 0, n) = s2a2

exp

s|n|

ln

82s|n|

a2

E

+ exp

s|n|

E1

2s|n|

+ 2

m=1

u

am

,

s|n|

.

The local Josephson energy can be represented as

EJosc(u, ) = 2EJa cos()J(u/a),where

J(v) is dimensionless function,

J(v)

4

v2 ln 0.39v at v

1 (see Appendix B). In reduced units the total

energy takes the form

EJ/J0 =n

dy

1

2

drn

dy

2+ 1 cos(n+1 n)

+sa

22

n,m,i,j

VMr (vn,i vm,j, Yi,ja

, n m)

+2a

s

n,i

J(vn+1,i vn,i) cos(n+1,i n,i) (57)


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11

where J0 EJs J/s is the JV energy scale, n(y) = rn(y) 2

i vn,i (Yi y) and a a/s. Varying theenergy, we obtain equations for vn,i and n

yrn(Yi) + sa22

m,j

FMr (vn,i vm,j, Yi,ja

, n m) + as

=1

cos(n,i n+,i)FJ(vn,i vn+,i) = 0, (58a)

d2rndy2

+ sin (n+1 n) sin(n n1) = 0. (58b)

Here FMr (x, y, n) xVMr (x, y, n) is the magnetic interaction force between the vortex rowsFMr (x, y, n) = 2

2

a2

n s

2exp

s|n|

sin2x

cosh2|y| cos2x

+s

a2

m

x m(x m)2 + y2 exp

(x m)2 + y2 + z2/a

.

and FJ(v, ) = J(v)/v (/2)v ln(0.235/v) at v 1. The derivative of the regular phase has jumps at thepositions of the rows

drndy

(Yi + 0) drndy

(Yi 0) = 2as

=1

J(vn+,i vn,i)sin(n+,i n,i). (59)

To find JV structure at low temperatures one has to solveEqs. (58a), (58b), and (59) with condition (30).

Let us consider in more detail magnetic interactions be-tween vortex rows, i.e., the term with FMr in Eq. (58a).Firstly, one can observe that the dominating contribu-tions to the sum over the layer index m and row index jcome from rows in the same layer, m = n, and rows inthe same of stacks, j = i. The former sum determinesthe shear stiffness, while the latter one determines themagnetic tilt stiffness.

m,j

FMr (vn,ivm,j,Yi,ja

, nm) fshear [vn,i]+ftilt [vn,i](60)

with

fshear [vn,i] =j=i

FMr (vn,i vn,j, Yi,ja

, 0) (61)

ftilt [vn,i] =m=n

FMr (vn,i vm,i, 0, n m) (62)

The sum over the rows in fshear [vn,i] converges very fastand effectively is determined by the first two neighbor-ing rows. Note that skipping the terms with m

= n in

fshear [vn,i] is completely justified in the limit a < butleads to overestimation of the shear energy in the limita > . However in this limit the shear energy has al-

ready a very weak influence on JV properties. The sumover the layers in ftilt [vn,i] is determined by large num-ber of the layers of the order of a/s or /s. If we considerlayer n close to the JV core, then the interaction forcewith row in the layer m, in the same stack with n m a/s, /s is given by FMr (vn,i vm,i, 0, n m) (vn,i vm,i)/2(m n). Interactions with remote lay-ers give large contributions even if displacements in theselayers are small, vm,i vn,i. A useful trick to treat thissituation is to separate interaction of a given pancake rowwith the aligned stack pancake rows:

ftilt [vn,i] =m=n

FMr (vn,i vm,i, 0, n m) + fcage (vn,i)

with

FMr (vn,i vm,i, 0, n m)= FMr (vn,i vm,i, 0, n m) FMr (vn,i, 0, n m)

and

fcage (v) =m=0

FMr (v, 0, m)

is the interaction force of a chosen pancake row with thealigned stack of rows (cage force), for which we derivea useful representation

fcage (v) = l=1

4 sin(2lv)(a/)2 + (2l)2

(a/)2 + (2l)2 + 2l

.


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12

At v 1 /a this equation gives fcage (v) v ln(0.433/v). FMr (vn,i vm,i, 0, n m) behaves as vm,i/2(m n)at large m and decays with increase of m much faster than FMr (vn,i vm,i, 0, n m). The same splitting can bemade in the magnetic coupling energy:

1

2

m=n

VMr (vn vm, n m) =

|m|>|n|

(VMr(vn vm, n m) VMr (vn, n m) VMr (vm, n m))

+n

vcage (vn)

with

vcage (v) =n=0

VMr (v, n)

=

l=1

2 (1 cos(2lv))l

(a/)2 + (2l)2

(a/)2 + (2l)

2+ 2l

This function has simple asymptotics at v 1 /a,vcage (v)

(v2/2) ln(0.713/v).

We demonstrate now that in the limit /s Eqs.(58a) and (58b) reproduce the JV structure obtainedwithin the effective phase stiffness approximation. Onecan show that in this limit the local Josephson energyinfluences weakly the JV structure. We calculate correc-tion to the JV energy due to this term in the Appendix C.The dominating contribution to the magnetic interaction

between the pancake rows,

m,j FMr (vn,ivm,j, Yi,ja , nm), comes from the tilt force (62), which with good ac-curacy can be described by the cage force fcage (vn,i)in the limit of small vn,i, fcage (v) v ln(C/v) withC 0.433. Further estimate shows, that the term

m=n

FMr (vn,i

vm,i, 0, n

m) in Eq. (62) amounts to

the replacement of numerical constant C under the loga-rithm by slowly changing function of the order of unity.Because of slow space variations, the discrete row dis-placements vn,i can be replaced by the continuous dis-placement field vn(y). Within these approximations Eq.(58a) reduces to

yrn sa22

vn(y) lnC

vn(y)

Replacing v(y) by the vortex phase vn(y) obtained bycoarse-graining of Eq. (55), vn(y) = (b/2s) yvn(y),we obtain

yrn BBz

yvn

Note that we replaced vn(y) in ln[C/vn(y)] by its typ-ical value and absorbed the logarithmic factor into thedefinition of B (24). From the last equation we obtainrn = (B/Bz)vnand

n =

1 +

BzB

rn

Therefore Eq. (58b) reduces to

11 + BzB

d2ndy2

+ sin (n n+1) + sin (n n1) = 0,

which is just dimensionless version of Eq. (29).

1. Numerical calculations of JV core structure. Crossoverto solitonlike cores.

To explore the JV core structure, we solved Eqs. (58a)and (58b) numerically for different ratios /s and dif-

ferent magnetic fields. We used a relaxation techniqueto find the equilibrium displacements of the pancakerows and the continuous regular phase. Typically, wesolved equations for 20 layers and the in-plane region0 < y < 20.

To test the effective phase stiffness model and to cal-culate uncertain numerical factors, we start from the caseof large anisotropies, /s. Figure 3 shows the greylevel plots of the cosine of the phase difference betweentwo central layers of JV, cos , 1 0 21, for = 0.2s. Fig. 4 shows y dependence of the total phasedifference and the contribution to this phase comingfrom the regular phase for the same parameters. As onecan see, at B 0/(s)2 the core region covers sev-eral pancake rows. At high fields the core size shrinksso that the number of rows in the core does not change,in agreement with the effective phase stiffness model.From Fig. 4 one can see that the fraction of the regularphase in the total phase progressively decreases with in-crease of magnetic field. For the field 80/(s)2 we alsoplotted (y) dependence from the effective phase stiff-ness model, assuming B = 2.10/42. One can seethat the numerically calculated dependence is reasonablywell described by this model.


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13

FIG. 3: Grey level plots of cosine of the phase difference be-tween two central layers of JV, cos , for = 0.2s and sev-eral magnetic fields (dark regions correspond to cos 1

and white regions correspond to cos 1). The total size ofdisplayed region in the horizontal direction is 6s. One cansee that in this regime several pancake rows fit inside the coreregion. At high fields the size of the core shrinks so that thenumber of pancake rows inside the core remains constant.

We now extend study of the core structure to moder-ate anisotropies /s. At Fig. 5 we plot the maximumpancake displacement umax in the core region normalizedto the lattice constant a as function of magnetic fieldfor different /s. The maximum displacement approx-imately saturates at a finite fraction of lattice constantat high field (at high /s one can actually observe aslight decrease ofumax/a with field). Fig. 6 shows depen-dence ofumax/a on /s for fixed field B = 100/(s)2.Dashed line shows prediction of the effective phase stiff-ness model given by Eq. (37). One can see that thisequation correctly predicts maximum displacement for/s < 0.35. Important qualitative change occurs at/s > 0.35, where the maximum displacement umax(0)exceeds a/4. This means that the pancakes initially be-

0

1

2

3

B = 0/(s)

2

regular phase

total phase

0

1

2

3

B = 20/(s)

2

0

1

2

3

B = 40/(s)

2

0

1

2

3

0 1 2 3 4

B = 80/(s)

2

Effective phase stiffness

y

FIG. 4: Coordinates dependence of the phase difference b e-tween two central layers, = 1 0, for /s = 0.2 anddifferent magnetic fields. Circles connected by dotted linesrepresent total phase difference, solid lines show contributionsfrom the regular phase. Jumps of the total phase differenceat the positions of pancake rows are caused by pancake dis-placement and represent the vortex phases. In the lower plot

dashed line represents prediction of the effective phase stiff-ness model with B = 2.10/4

2.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1 2 3 4 5 6 7 8

Maximum pancake displacement in the core

umax

/a

B (in units 0/42)

0.1

0.2

0.3

0.4

0.5

FIG. 5: Field dependencies of the maximum pancake dis-placement in the core at different ratios /s (the curves arelabelled by this ratio).


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14

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3B = 10 0/4

2

/s

umax

/a

max

max

ln 0.582

=

u a

a u s

FIG. 6: Dependence of the maximum pancake displacementin the core on the ratio /s at B = 100/4

2. Dashed linerepresents prediction of the effective phase stiffness model(37).

longing to the neighboring stacks become closer thanthe pancakes belonging to the same stack. This can beviewed as switching of the vortex lines in the central layerof JV. This switching is clearly observed in Fig. 7, whichshows pancake displacements in the central row of pan-cake stacks and its neighboring row for two values of theratio /s, 0.3 and 0.5, and several fields. For /s = 0.5configuration of the pancake rows in the central stack isvery similar to the classical soliton (kink) of the sta-tionary sine-Gordon equation: the stacks smoothly trans-fer between the two ideal lattice position in the region ofthe core. Simplified approximate description of such so-litionlike structure in the case s < is presented belowin Sec. IIF. Fig. 8 shows distribution of cosine of inter-layer phase difference between two central layers, cos .As one can see, at /s = 0.3 there are still extended

regions of large phase mismatch in the JV core (darkregions), while for /s = 0.5 these regions are almosteliminated by large pancake displacements in the core.

F. Simple model for solitonlike cores at moderateanisotropies

In this Section we consider the structure of the JVcore for moderate anisotropies s < 0.5 and high fieldsB > 0/42. Estimate (37) and numerical calculationsshow that at sufficiently small anisotropy, < 0.5/s,the maximum displacement in the core region exceeds aquarter of the lattice spacing. This means that distancebetween displaced pancakes belonging to the same vortexline, 2ux,0(0), becomes larger than the distance betweenpancakes initially belonging to the neighboring lines,a 2ux,0(0) . This can be viewed as switching of the vor-tex lines in the central layer of JV. At lower anisotropiespancake stacks in the central row acquire structure, simi-lar to the soliton of the stationary sine-Gordon model: inthe core region they displace smoothly between the twoideal lattice position (see Fig. 7 for /s = 0.5). In such

configuration a strong phase mismatch between the twocentral layers is eliminated, which saves the Josephsonenergy in the core region. On the other hand, large pan-cake displacements lead to greater loss of the magneticcoupling energy. To describe this soliton structure, weconsider a simplified model, in which we (i) keep onlydisplacements in the central row vn,0 vn, (ii) use thecage approximation for magnetic interactions and (iii)

neglect the shear energy. All these approximations arevalid close to the JV center. At n > 0 we redefine dis-placements as vn 1 + vn. The redefined displacementsdepend smoothly on the layer index, vn+1 vn 1, sothat one can replace the layer index n by continuous vari-able z = ns, un u(z), and use elastic approximationfor the Josephson tilt energy:

Ecore

dz

EJsLJ

2a

du

dz

2+

J a

s2vcageu

a

,

(63)where the logarithmic factor LJ is estimated as LJ ln 0.39(a/s) |du/dz|

1 andvcage(v)

l=1

1 cos (2lv)(2)2l3

is the magnetic cage. For estimates, we replace LJ by aconstant substituting a typical value for du/dz under thelogarithm. The equilibrium reduced displacement v =u/a is determined by equation

d2v

dz2+

2

LJ2 vcage(v) = 0. (64)

Its solution is implicitly determined by the integral rela-

tion v1/2

dvvcage(v)

=Az

with A =

2LJ

. Therefore a typical size of the soliton is

given by

zs /and the applicability condition of this approach zs sis equivalent to s . In fact, an accurate numericalcalculations of the previous section show that the core

acquires the solitonlike structure already at s

2.The core energy is given by

Ecore a

2JEJLJ

10

dv

vcage(v)

At a numerical evaluation of the integral gives10

dv

vcage(v) 1.018/2 and we obtain

Ecore

JEJLJ2

a

.


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15

FIG. 7: Structure of the pancake-stacks row in the center of JV (big circles) and its neighboring row (small circles) for /s = 0.3and 0.5 and several values of the magnetic field. At /s = 0.5 pancakes in the central row form lines smoothly transferringbetween two ideal lattice position (solitonlike structure).

FIG. 8: Gray-level plots of cosine of interlayer phase difference between the two central layers of JV for /s = 0.3 and 0.5 anddifferent fields. For /s = 0.3 the JV core covers roughly three pancake rows, while for /s = 0.5 it shrinks to one pancakerows. In the second case the regions of suppressed Josephson energy are practically eliminated.

We also estimate the shear contribution to the periodicpotential vcage(v)

vshear(v) 22

a22 (1 cos2v)exp 3

1 + exp32

0.017 2

a2(1 cos2v) .

It occurs to be numerically small and only has to be takeninto account when a becomes significantly smaller than

.

It is important to note that the described model doesnot provide precise soliton structure. At distances z / displacements in other pancake rows become compa-rable with displacements in the central row. In fact, atdistances z / the displacements should cross overto the regime, described by the effective phase stiffnessmodel (35). Precise description of the soliton structure israther complicated and beyond the scope of this paper.


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16

x

y

a0F

j

FIG. 9: Pinning of a Josephson vortex by a single pancake-stack row.

III. PINNING OF JOSEPHSON VORTEX BYPANCAKE VORTICES

Dynamic properties of JVs can be probed either byapplying transport current along the c direction or bystudying ac susceptibility for magnetic field, polarizedalong the layers. Although there are numerous experi-mental indications that pancake vortices strongly impedemotion of JVs,32,33,34 no quantitative study (theoreticalor experimental) has been done yet. In this section weconsider a pinning force, which is necessary to apply tothe JV to detach it from the pancake vortex crystal. Weconsider the simplest case of a dilute pancake lattice,a > , which allows us to neglect the influence of pan-cakes on the JV core, and small concentration of JVs, so

that we can neglect influence of the JV lattice on the pan-cake crystal (e.g., formation of phase-separated states).

A. Pinning by a single pancake-stack row (a0 > s)

We consider first the simplest case of an isolatedpancake-stack row crossing JV (see Fig. 9) and estimatethe force necessary to detach JV from this stack, assum-ing that its position is fixed. The consideration is basedon the crossing energy of JV and isolated pancake stack,calculated in Sec. IIB. Let us calculated first the forcenecessary to separate JV from an isolated pancake stack.Using Eq. (17) we obtain for the force acting on JV fromthe pancake stack located at distance y from the centerof JV core

F = ddy

E(y) (65)

20

423s2 ln(/u1(y))A

y

s

with A (y) dA (y) /dy. For the maximum force weobtain

Fmax Cf20

423s2 ln(/sf)

with Cf = maxy

A(y)

and f 1. Numerical cal-culation gives Cf 1.4 and the maximum is located atyf 0.52s. The critical current which detaches JV fromthe row of pancake stacks with the period a0 is given by

jJp = jJCf

2

sa0 ln(/sf). (66)

where jJ = c0/(822cs) is the Josephson current. This

expression is valid as long as the lattice period a0 is muchlarger than the Josephson length s. Otherwise inter-action with several pancake rows has to be considered,which will be done in the next section.

B. Pinning by dilute pancake lattice ( < a0 < s)

When several pancake-stack rows fit inside the JV core(but still a0 > ) interaction energy of JV per unit lengthwith the pancake lattice, EJl(y), can be calculated as asum of crossing energies (17),

EJl(y) =1

a0

n

E (y nb0) (67)

20

422sa0 ln(s/)

m=

A

y mb0

s

where b0 =

3a0/2 is the distance between the PV rows.

This expression has a logarithmic accuracy, i.e., we ne-glected a weak y-dependence under the logarithm andreplaced the dimensionless function under the logarithmby its typical value. Using Fourier transform of E(y),E(p) =

dy exp(ipy)E(y), we also represent EJl(y)

as

EJl (y) = nv

s=

E

2s

b0

exp

i

2m

b0y

In the case b0 s we can keep only m = 0, 1 termsin this sum,

EJl(y) EJl(0) + 2nvE2b0

cos

2yb0

Using interpolation formula (21) for A (y), we obtainthe asymptotics for b0 < s

EJl(y) EJl(0) 0.35 nv20

ln(s/)

s

b0(68)

cos

2y

b0

exp

5.82 s

b0


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17

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2

-c

B [in units 0/(s)2]

FIG. 10: The field dependence of the dimensionless pinningforce of JV by dilute pancake vortex crystal

To compute the field dependence of the critical current,we represent the force acting on JV from the pancakelattice in the scaling form,

FJpl(y)

3s20(4)2(s)4 ln(/s)

F

y

s,

b0s

,

with

F(y, b0) 2b0

d

dy

j

A(y jb0).

The critical pinning current is given by

jJpl = jJ

32

2(s)2 ln(/s)Fc(b0) (69)

with

Fc(b0) = maxy

[F(y, b0)]

Numerically calculated dependence ofFc versus reducedfield (s)2B/0

3/2b20. Maximum Fcmax 1.15 is

achieved at B 0.260/(s)2 (b0 = 2.1s). Thereforethe maximum pinning current can be estimated as

jJpmax jJ 2

(s)2 ln(/s)(70)

For typical parameters of BSCCO this current is only 5-10 times smaller than the maximum Josephson current,i.e., it is actually rather large.

An exponential decay of the pinning energy (68) andforce holds until the row separation b0 reaches . Atlarger fields one have to take into account shrinking ofthe vortex core. Because the number of pancake rows inthe core is almost constant at high field, the exponentialdecay will saturate at a finite value exp(Cs/).

IV. PANCAKE VORTICES AND VISCOSITY OFJOSEPHSON VORTEX

A sufficiently large c axis transport current will driveJVs. Two dynamic regimes are possible, depending on

the relation between the JV-pancake interaction anddisorder-induced pancake pinning. Moving JVs eithercan drag the pancake lattice or they can move throughthe static pancake lattice. Slow dragging of pancakestacks by JVs at small c axis fields have been observedexperimentally.17 As the JV-pancake interaction force de-cays exponentially at high c axis fields in the case < s,one can expect that JVs will always move through the

static pancake lattice at sufficiently high fields.

A. Dragging pancake lattice by Josephson vortices

When moving JVs drag the pancake lattice, one canobtain simple universal formulas for the JV viscosity co-efficient and JV flux-flow resistivity. The effective vis-cosity coefficient per single JV is connected by a simplerelation with the viscosity coefficient of pancake stack perunit length, v,

BxJV = Bzv.

Therefore, the JV flux-flow resistivity,

c

ff(B

) =0Bx/(c2JV ), is given by

cff(B) =0B2x

c2vBz. (71)

As a consequence, we also obtain a simple relation be-tween cff(B) and

abff(Bz)

cff(B) = abff(Bz)

B2xB2z

.

B. Josephson vortex moving through pancakelattice

Consider slow JV motion through the static pancakelattice. JV motion along the y direction with velocity Vyinduces traveling pancake displacement field un(y Vyt).For slow motion, un(y) is just the static displacementfield around JV. Contribution to the energy dissipationcaused by these displacements is given by

W pnvn

dru2n

= p

nvn

dr (yun)2

V2y

where p is the pancake viscosity coefficient. Thereforethe JV viscosity per unit length is given by

JV = pnvn

dr (yun)2 (72)

Using relation between the vortex phase and displace-ment field yvn = 2nvun we obtain an estimate

nvn

dy (yun)2 2.7

3nv(s)3

Cln (rcut/rw)

3/2


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19

at

2 + (2/a)

2|y| 1. Therefore the interaction between the pancake rows decays exponentially at y, sn > , aInteraction of pancake row with stack of rows is given by

UMs(x, y) = 2

n=1

UMr (x, y, n)

=J

aln 1 2 cos(

2x

a)exp

2y

a + exp4y

a +

22J

a

l

exp

i2l

ax

2 +2la

2y

(a/)2 + (2l)2

In particular, the potential created by pancakes belonging to the same row stack (cage potential) is given by

Ucage UMs(x, 0) = 42J

a

l=1

cos

2l

ax

12l

1(a/)2 + (2l)

2

APPENDIX B: LOCAL CONTRIBUTION TOJOSEPHSON ENERGY DUE TO MISMATCH OFPANCAKE ROWS IN NEIGHBORING LAYERS

We consider two pancake rows in neighboring layerswith period a s shifted at distance v with respectto each other in the direction of row ( x axis). In zeroorder with respect to the Josephson coupling these rowsproduce the phase mismatch v(x, y) between the layers,

v(x, y) =m

arctan

xm+v/2y

arctan xmv/2y

.

We measure all distances in units of the lattice constanta. The x-averaged phase mismatch is given by

v(y) = v sgn(y).

The total phase approaches v(y) at y 1/2.

Separating v(x, y) from the total phase, we can rep-resent the Josephson energy as

FJ = EJa2

d2r [1 cos(v(x, y) + (x, y))]

= EJa2

d2r [1 cos(v(y) + (x, y))] + LxEJosc(v, )

where (x, y) is the smooth external phase and the local Josephson energy EJosc(v, ) per unit length is defined as

EJosc(v, ) = EJa2

Lx

dxdy [cos( + v(x, y)) cos( + v(y))] (B1)

= 2EJa cos() Jwhere

J(v) 1/21/2

dx

0

dy [cos(v(y)) cos(v(x, y))] . (B2)

In EJosc(v, ) we can neglect weak coordinate dependence of the external phase and replace it by a constant . At|v| < 1/2 the ground state for fixed v corresponds to = 0, while at 1/2 < |v| < 1 the ground state corresponds to = . EJosc(v, ) has a symmetry property EJosc(1v, ) = EJosc(v, ). The integral over y in J(v) is converges aty 1/2. This allows us to consider a single row separately from other rows and neglect the coordinate dependenceof the external phase .


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20

Using the complex variable z = x + iy one can derive a useful expression for v(x, y):

v(z) = Im

lnm

z m v/2z m + v/2

= Im

ln

sin((z v/2))sin((z + v/2))

.

Going back to the (x, y) representation, we obtain

v(x, y) = arctan

tan((x + v/2))

tanh y arctantan((x

v/2))

tanh y

and

cos v(x, y) =cosh2y cos v cos2x

(cosh2y cos2x cos v)2 (sin2x sin v)2 .

Integral (B2) can now be represented as

J(v) =1/21/2

dx

0

dy

cos(v) cosh2y cos v cos2x

(cosh 2y cos2x cos v)2 (sin2x sin v)2

We obtain an approximate analytical result at small v, v 1. Simple expansion with respect to v produces loga-rithmically diverging integral. To handle this problem we introduce the intermediate scale y0, v y0 1, and splitintegral J into contribution coming from y > y0 (J>) and y < y0 (J y0 we use small-v expansion andobtain

J> = (v)2y0

dy

1/20

dx

1 + sinh

2 2y

(cosh 2y cos2x)2

=(v)

2

2

y0

dyexp(2y)

sinh2y

v2

4ln

1

4y0.

In region y < y0 we can expand all trigonometric functions and obtain the integral

J< 2y00

dy

1/20

dx

1 4(x

2 + y2) v2(4 (x2 + y2) + v2)2 (4xv)2

=v2

2

2y0/v0

dy

1/v0

dx

1 x

2 + y2 1(x2 + y2 1)2 + 4y2

with y = 2y/v and x = 2x/v. Because we only interestedin the main logarithmic term, we can extend integrationover x up to . The obtained integral can be evaluatedas

J< 4

v2

ln y0v

+ 1.58

Adding J> and J


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21

FIG. 11: Dimensionless function J(v) which determines the local Josephson energy. Dashed line is small-v asymptotics (B3).

APPENDIX C: CONTRIBUTION TO JVENERGY COMING FROM LOCAL JOSEPHSON

TERM AT /s

In the limit of very weak coupling the correction toreduced JV energy (57) coming from the local Josephson

energy is given by

EJV

3

4

1 + Bz/B

n=

dy(vn+1 vn)2 ln 0.39|vn+1 vn| cos(n+1 n) (C1)

with y = y/J =

1 + Bz/By/s. We will focus only on the regime Bz B, where this correction can benoticeable. In this regime reduced row displacements are connected with phase gradient by relation

vn(y)

b

2J

yn

and the correction reduces to

EJV

Bz/B

8nv (s)2

n

dyyn+1 yn

2ln

2.83 sB/Bz

ayn+1 yn

cos(n+1 n). (C2)

Using numerical estimates

n=

dy cos(n+1 n)yn+1 yn

2 2.4

n=

dy cos(n+1 n)

yn+1 yn2

ln

1yn+1 yn 4.7obtained with the JV phase n(y) (14), we obtain

EJV

BBz

0.382

(s)2

ln (a/rw)ln

0.1 s

ln

a

rw

. (C3)

As we can see, the correction is smaller than the reduced JV energy at Bz B, EJV

B/Bz ln(a/s), by thefactor 2/(s)2.


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22

1 A. I. Buzdin and D. Feinberg, J. Phys. (Paris) 51, 1971(1990); S. N. Artemenko and A. N. Kruglov, Phys. Lett. A143 485 (1990); J. R. Clem, Phys. Rev. B 43, 7837 (1991).

2 L. N. Bulaevskii, Zh. Eksp. Teor. Fiz. 64, 2241 (1973) [Sov.Phys. JETP 37, 1133 (1973)].

3 J. R. Clem and M. W. Coffey, Phys. Rev. B, 42, 6209

(1990); J. R. Clem, M. W. Coffey, and Z. Hao, Phys. Rev.B 44, 2732 (1991).

4 A. E. Koshelev, Phys. Rev. B 48, 1180 (1993).5 L. N. Bulaevskii, M. Ledvij, and V. G. Kogan, Phys. Rev.

B 46, 366 (1992); M. Benkraouda and M. Ledvij, Phys.Rev. B 51, 6123 (1995).

6 A. E. Koshelev, Phys. Rev. Lett. 83, 187 (1999).7 D. Feinberg and A. M. Ettouhami, Int. J. Mod. Phys. B

7, 2085 (1993).8 A. I. Buzdin and A. Yu. Simonov, JETP Lett. 51, 191

(1990); A. M. Grishin, A. Y. Martynovich, and S. V. Yam-polskii, Sov. Phys. JETP 70, 1089 (1990); B. I. Ivlev andN. B. Kopnin, Phys. Rev. B 44, 2747 (1991); W. A. M.Morgado, M. M. Doria, and G. Carneiro, Physica C, 349,196 (2001).

9 L. L. Daemen, L. J. Campbell, A. Yu. Simonov, and V. G.Kogan Phys. Rev. Lett. 70, 2948 (1993); A. Sudb, E. H.Brandt, and D. A. Huse Phys. Rev. Lett. 71, 1451 (1993);E. Sardella, Physica C 257, 231 (1997).

10 Direction of magnetic induction B almost never coincideswith direction of external magnetic field Hext. The relationbetween B and Hext depends on the shape of sample andconstitutes a separate problem.

11 S. E. Savelev, J. Mirkovic, K. Kadowaki, Phys. Rev. B 64,094521 (2001).

12 B. Schmidt,M. Konczykowski, N. Morozov, and E. Zeldov,Phys. Rev. B 55, R8705 (1997).

13 S. Ooi, T.Shibauchi, N.Okuda, and T.Tamegai, Phys. Rev.Lett. 82, 4308 (1999).

14

D. A. Huse, Phys. Rev. B 46,8621 (1992).15 C. A. Bolle, P. L. Gammel, D. G. Grier, C. A. Murray,D. J. Bishop, D. B. Mitzi, and A. Kapitulnik , Phys. Rev.Lett. 66, 112 (1991).

16 I. V. Grigorieva, J. W. Steeds, G. Balakrishnan, and D. M.Paul, Phys. Rev. B 51, 3765 (1995).

17 A. Grigorenko, S. Bending, T. Tamegai, S. Ooi, and M.Henini, Nature 414, 728 (2001).

18 T. Matsuda, O. Kamimura, H. Kasai, K. Harada, T.Yoshida, T. Akashi, A. Tonomura, Y. Nakayama, J. Shi-

moyama, K. Kishio, T. Hanaguri, and K. Kitazawa, Sci-ence, 294, 2136(2001).

19 V. K. Vlasko-Vlasov, A. E. Koshelev, U. Welp,G. W. Crab-tree, and K. Kadowaki, Phys. Rev. B 66, 014523 (2002).

20 M. Tokunaga, M. Kobayashi, Y. Tokunaga, and T.Tamegai, Phys. Rev. B 66, 060507 (2002).

21

M. J. W. Dodgson, Phys. Rev. B 66, 014509 (2002).22 A. Buzdin and I. Baladie, Phys. Rev. Lett. 88, 147002(2002).

23 M. Konczykowski, C. J. van der Beek, M. V. Indenbom,E.Zeldov, Physica C, 341, 1213 (2000).

24 J. Mirkovic, S. E. Savelev, E. Sugahara, and K. Kadowaki,Phys. Rev. Lett. 86, 886 (2001).

25 S. Ooi, T. Shibauchi, K. Itaka, N. Okuda, and T. Tamegai,Phys. Rev. B 63, 020501(R) (2001)

26 M. Tokunaga, M. Kishi, N. Kameda, K. Itaka, and T.Tamegai, Phys. Rev B 66, 220501(R) (2002).

27 The more quantitative expression for U44(k), which ac-counts for thermal softening, can be obtained using theself-consistent harmonic approximation. At high fields and

kz

> r

1

w

it gives U44

(k)

nv0

22

ln

0.5 +

0.13a2

r2w

, see A. E.Koshelev and L. N. Bulaevskii Physica C, 341-348, 1503(2000).

28 The formula (B12) for the JV phase in Ref. 4 for phase dis-tribution n was written incorrectly. Eq. (14) gives correctphase distribution, which was actually used in all numeri-cal calculations and is plotted in Fig. 8 of Ref. 4.

29 A. E. Koshelev and P. H. Kes, Phys. Rev. B 48, 6539(1993).

30 T. R. Goldin and B. Horovitz, Phys. Rev. B 58, 9524(1998).

31 The identical factor 1 + Bz/B appears also in the renor-malization of defect energy by the pancake vortex crystaland have precisely the same physical origin, see M. J. W.Dodgson, V. B. Geshkenbein, and G. Blatter, Phys. Rev.

Lett. 83, 5358 (1999).32 Yu. Latyshev and A. Volkov, Physica C, 182, 47 (1991).33 H. Enriquez, N. Bontemps, P. Fournier, A. Kapitulnik, A.

Maignan, and A. Ruyter, Phys. Rev. B, 53, R14757 (1996)34 G. Hechtfischer, R. Kleiner, K. Schlenga, W. Walkenhorst,

P. Muller, and H. L. Johnson, Phys. Rev. B 55, 14638(1997).

35 A. E. Koshelev, Phys. Rev. B 62, R3616 (2000).

A. E. Koshelev- Josephson vortices and solitons inside pancake vortex lattice in layered...

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