S.E. Savel’ev, J. Mirkovic and K. Kadowaki- The London theory of the crossing-vortex lattice in highly anisotropic layered superconductors

8/3/2019 S.E. Savel’ev, J. Mirkovic and K. Kadowaki- The London theory of the crossing-vortex lattice in highly anisotropic la…

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The London theory of the crossing-vortex lattice in highly anisotropic layered

superconductors

S.E. Savel’ev∗, J. Mirkovic+, and K. KadowakiInstitute of Materials Science, The University of Tsukuba, 1-1-1 Tennodai, Tsukuba 305-8573, Japan, and

CREST, Japan Science and Technology Corporation (JST), Japan

A novel description of Josephson vortices (JVs) crossed by the pancake vortices (PVs) is proposedon the basis of the anisotropic London equations. The field distribution of a JV and its energyhave been calculated for both dense (a < λJ ) and dilute (a > λJ ) PV lattices with distance abetween PVs, and the nonlinear JV core size λJ . It is shown that the “shifted” PV lattice (PVsdisplaced mainly along JVs in the crossing vortex lattice structure), formed in high out-of-planemagnetic fields Bz > Φ0/γ 2s2 [A.E. Koshelev, Phys. Rev. Lett. 83, 187 (1999)], transforms intothe PV lattice “trapped” by the JV sublattice at a certain field, lower than Φ0/γ 2s2, where Φ0 isthe flux quantum, γ is the anisotropy parameter and s is the distance between CuO2 planes. Withfurther decreasing Bz, the free energy of the crossing vortex lattice structure ( PV and JV sublatticescoexist separately) can exceed the free energy of the tilted lattice (common PV-JV vortex structure)in the case of γs < λab with the in-plane penetration depth λab if the low (Bx < γ Φ0/λ2ab) or high(Bx Φ0/γs2) in-plane magnetic field is applied. It means that the crossing vortex structureis realized in the intermediate field orientations, while the tilted vortex lattice can exist if themagnetic field is aligned near the c-axis and the ab-plane as well. In the intermediate in-planefields γ Φ0/λ2ab Bx Φ0/γs2, the crossing vortex structure with the “trapped” PV sublattice

seems to settle in until the lock-in transition occurs since this structure has the lower energy withrespect to the tilted vortex structure in the magnetic field H oriented near the ab-plane. Therecent experimental results concerning the vortex lattice melting transition and transitions in thevortex solid phase in Bi2Sr2CaCu2O8+δ single crystals are discussed in the context of the presentedtheoretical model.

PACS numbers: 74.60.Ge, 74.60.Ec, 74.72.Hs

The mixed state of high temperature superconduc-tors is complex and rich with various vortex phases1,2.Besides the vortex lattice described by 3D anisotropicGinzburg-Landau model, the new types of vortex struc-tures can occur within the large part of the phase diagramof the mixed state where the coherence length along the

c-axis is smaller than the distance between CuO2 planes.In such a case, the magnetic field, aligned with the c-axis, penetrates a superconductor in the form of quasitwo-dimensional pancake vortices (PVs)3 while the fieldapplied parallel to the ab-plane generates Josephson vor-tices (JVs) in the layers between CuO2 planes4,5. In mag-netic fields tilted with respect to the c-axis, PVs and JVscan form a common tilted lattice5 or exist separately asa crossing (combined) lattice6,7. The tilted lattice rep-resents the inclined PVs stacks in fields applied close tothe c-axis while, at higher angles, the pieces of JVs link-ing PVs are developed5,6. The crossing lattice is anotherstructure containing both a PV stack sublattice and a JVsublattice which coexist separately.

The vortex-solid phase diagram in the tilted mag-netic fields was first proposed by Bulaevskii, Ledvij, andKogan6. According to their model, which does not takeinto account the interaction between PV and JV sub-lattices in the crossing lattice structure, the tilted lat-tice is formed for all orientations of the magnetic fielduntil the lock-in transition8 occurs if the in-plane Lon-don penetration depth λab is larger than the Josephsonvortex core with size γs (γ is the anisotropy parame-

ter and s is the distance between CuO2 planes). Inthe opposite limit, γ s > λab, the tilted lattice trans-forms into the crossing lattice (as the magnetic field isinclined away from the c-axis) at a certain angle beforethe lock-in transition happens6. Later, the possibilityof the coexistence of two vortex sublattices with differ-

ent orientations was analyzed numerically by comparingthe free energy of such system with the free energy of mono-oriented tilted vortex lattice at different field ori-entations and different absolute values of the externalmagnetic field for the case of 3D anisotropic (Londonmodel) superconductors9–11 as well as layered (Lawrence-Doniach model12) superconductors13. According to thatanalysis11,13 performed for γ = 50 − 160, the crossinglattice can be energetically preferable in the quite lowmagnetic fields (B =

B2z + B2

x Φ0/λ2ab) in the in-termediate field orientations 0 < θ1 < θ < θ2 < π/2with θ = arctan(Bx/Bz) (Bz and Bx are the field compo-nent along the c-axis and parallel to the ab-plane, respec-tively). However, the interaction of two coexisting vortex

sublattices was not considered in those works9–11,13. Re-cently, Koshelev7 has studied the case of extremely ani-sotropic superconductors γs ≫ λab and has shown thatthe crossing lattice can occupy substantially larger regionof the vortex lattice phase diagram in the oblique fieldsdue to the renormalization of the JV energy E J throughthe interaction of a Josephson vortex and the PV sublat-tice. In addition, such interaction leads to the attractionof PVs to JVs14,7 at low out-of-plane magnetic fields Bz

1

http://arxiv.org/abs/cond-mat/0105370v1























































(the some sort of pinning effect). This pinning may in-duce transitions between different substructures of thecrossing lattice structure. However, there is still no the-oretical investigation how PV sublattice can influence onthe JV lattice in the crossing vortex structure in the caseof moderate anisotropic superconductors with γs < λab.In this regard, the phase diagram6 of the vortex latticefor strongly anisotropic layered superconductors should

be reconsidered (at least for the case of γ s < λab) bytaking into account the renormalization of E J and thepinning of PVs by JVs.

The vortex structures in highly anisotropic layered su-perconductors are usually studied on the basis of thenonlinear discrete Lawrence-Doniach model12, but thismodel is quite complex and the detailed analysis of thevortex system is complicated. On the other hand, thelayerness of superconductors can be ignored on scaleslarger than the size of the nonlinear Josephson vortexcore. Therefore, the linear anisotropic London modelcould be applied for a study of the vortex crossing latticeoutside JV cores. In this paper we introduce the extendedLondon theory which allows to describe the crossing lat-tice as well as to calculate the energy E J and the fielddistribution of JV in the presence of the crossed PV sub-lattice. It is shown that with decreasing the perpendicu-lar magnetic field the pancake sublattice transforms fromthe “shifted” sublattice characterized by one componentPV displacement along JVs to the “trapped” sublatticewhere JVs are occupied by PV rows. The comparison of the free energies of the tilted lattice and the crossing vor-tex structure for the case λab > γs indicates that in low(Bx γ Φ0/λ2ab) and high (Bx Φ0/γs2) in-plane fields,the tilted lattice can exist if the vector B = Bxex + Bzezis directed close to the c-axis as well as near the ab-plane,while the crossing lattice is realized in the fields oriented

far enough from the crystal symmetry axes. Further-more, in the intermediate in-plane fields the tilted vortexlattice exists only at the magnetic field orientations nearthe c-axis whereas crossing vortex structure settles in thewide angular range until the lock-in transition happens.

This paper is organized as follows. The general equa-tions for the magnetic field distribution and the energyof Josephson vortex in the presence of the pancake lat-tice are derived in section I. The dense pancake lat-tice is studied in section II. It is shown that, in thelimit γs ≫ λab and Φ0/(γs)2 ≪ Bz ≪ Φ0γ 2s2/λ4ab, ourmodel reproduces the results which were earlier obtainedby Koshelev7, while the shear deformation of the PV lat-

tice significantly renormalizes the JV energy at the higherout-of-plane fields. Section III is devoted to the dilute PVlattice. It is described how the novel vortex substructurewith the PV lattice “trapped” by the JV lattice can berealized at low Bz. The phase diagram of the vortex-solidphase in the tilted magnetic fields is considered for thecase of λab > γs in section IV while the recent experi-mental results are discussed in section V.

I. JOSEPHSON VORTEX IN THE PRESENCE OFPANCAKE VORTEX LATTICE: GENERAL

EQUATIONS

We consider a Josephson vortex crossed with the pan-cake lattice in the framework of the modified Londonmodel. On scales which are much larger than both thedistance between CuO2 planes and the in-plane coher-

ence length ξab, the pancake vortex stack could be con-sidered as an ordinary vortex line at temperatures signif-icantly lower than the evaporation temperature3,15. Thesame approach can be also used for the description of the Josephson vortex far from the nonlinear core. TheJV current acts on PVs through the Lorentz force causingtheir displacements along JV, which can be interpreted asa local inclination of the PV lines away from the c-axis.In turn, the local tilt of the PV stacks induces an ad-ditional current along the c-axis which redistributes the“bare” JV field. Such physical picture can be describedwith one-component PV displacement u = (u, 0, 0) whichdoes not depend on the x-coordinate (Fig. 1). The free

energy functional F PJ can be written as

F PJ =1

8π

d3R

h2 p + ∇×h p

↔Λ∇×h p

+ hJ 2 + ∇×hJ

↔

Λ∇×hJ + 2h phJ + 2∇×h p↔

Λ∇×hJ

,

(1)

where h p and hJ are the magnetic fields of PV lines

and JV, respectively, and↔

Λ is the penetration-depth ten-sor, ∇ = (∂/∂x,∂/∂y,∂/∂z). In the considered coordi-nate system the tensor has only the diagonal componentsΛxx = Λyy = λ2ab, Λzz = λ2c with anisotropic penetrationdepths λab and λc. The field h p is determined by the dis-

placement u of PVs through the London equation (see,for instance,2)

h p+∇×↔Λ∇×h p

= Φ0

i

dz

×ez +

∂u(Y i, z)ex∂ z

δ(r−Ri(z) − u(Y i, z)ex), (2)

where Ri(z) = (X i, Y i, z) is the equilibrium position of the i-th PV line, r = (x, y, z) while ez and ex are unitvectors along the z and x axes, respectively. (Here wehave accepted that the parametric equation ri(z) (withparameter z) describing the i-th vortex line takes the

form of x(z) = X i + ui(z), y(z) = Y i, z(z) = z.) Thein-plane coordinates of the unshifted lines X i and Y i areexpressed through the distances a and b (see Fig. 1 b)between PVs and PV rows as X i = al/2 + aj and Y i = blwith integer l and j. In our approach, the field of theJosephson vortex also obeys the London equation

hJ − λ2ab∂ 2hJ

∂z2− λ2c

∂ 2hJ

∂y2= Φ0δ(y)δ(z), (3)

2



[

\

]

D

E

-

,

FIG. 1. The JV crossed by PV stacks which are shifted bythe currents induced by JV: a) 3D sketch depicts the defor-mation of the PV lattice in the different CuO2 planes, b) 2Dsketches show the deformation of the PV lattice in a CuO2

plane for the dense (left) and dilute (right) PV lattices. Thefilled circles correspond to the unshifted PVs while the openones represent the PVs shifted due to the interaction with JVcurrents. The shaded area images the nonlinear JV core re-gion. The dashed-dotted lines mark the rows of the unshiftedPVs while the dashed curve shows the deviation of PVs fromthese rows.

where δ-functions should be smoothed on a scale of theJosephson vortex core. The JV core size along the z-axisis fixed by the interlayer distance s, while the core lengthalong the y-direction is limited by the condition that thecurrent along the c-axis can not exceed the maximum

interlayer current jc ∼ cΦ0/(8π2λ2cs). In the presenceof PVs, the current across the layers consists of boththe current of JV itself and the current born by the localinclination of PV lines. Therefore, the core size along they-axis λJ can be renormalized in the presence of PVs andshould be calculated self-consistently (see next section).Furthermore, the space variable y could be replaced byy − y0 in the argument of the δ-function with 0 ≤ y0 ≤ bsince the PV lattice can be arbitrary shifted from thecenter of JV. However, we take y0 = 0 which correspondsto the energetically more preferable position7.

Next, in order to find the distribution of the magneticfield in the vortex system and the energy of JV, we will

minimize the free energy (1) as a functional of the dis-placement u. The fields h p and hJ can be obtained usingequations (2,3) with the displacement u fixed by the min-imization of (1). Then, the energy of JV, E J , defined asthe difference of the free energies (1) with and withoutJV, will be derived. This energy includes the self energyof JV and the change of the free energy of the PV latticeborn by the interaction with JV. We will use the elasticapproximation, i.e., the free energy (1) and the magneticfield of PVs (2) will be expanded up to the second order

in u.Using the integral representation of δ-function, the

equation (2) can be rewritten as

h p+∇×↔Λ∇×h p

= Φ0

i

dz

× d3q

(2π)3

eiqre−iqRi(z)ez(1

−iqxui(z)

−1

2

q2xu2i (z))

+∂ui(z)

∂ z(1 − iqxui(z))ex

. (4)

The field h p of PV lines changes on different space scales.The first scale is determined by the characteristic gradi-ent of the displacement u(y, z) and usually is much largerthan the distance a between PVs. The second scale is de-fined by the discreteness of the PV lattice and it is abouta. To separate the contribution to the free energy fromthese scales, we introduce the Fourier variables u(ky, kz):

u(Y i, z) = π/b−π/b

dky2π

dkz2π u(ky, kz)ei(kyY i+kzz), (5)

and

u(ky, kz) = bY i

dzu(Y i, z)e−i(kyY i+kzz), (6)

where the domain of variation of kz is restrictedby the inequality |kz| 1/s born by the layernessof the system. Substituting (5) into (4) and usingthe well-known equality

i

dz exp

i(k− q)Ri(z)

=

(2π)3(Bz/Φ0)

Qδ(k − q −Q), where Q = (Qx, Qy, 0)

are the vectors of the reciprocal lattice (Qx =2πm/a, Qy = π(2n + m)/b with integer m and n), onegets the expansion of the field of PVs in series with re-spect to the displacement u:

h p = h(0) p + h(1) p [u] + h(2) p [u], n p = n(0) p + n(1) p [u] + n(2) p [u],

h(i) p + ∇×↔Λ∇×h(i) p = n(i) p (7)

where

n (0) p = ezΦ0

i

δ2(r⊥ −R⊥i ),

n (1) p = Bz

Q dkydkz

4π2

× (eziQx + exikz)u(ky, kz)e−iQxxei(ky−Qy)yeikzz,

n (2) p = Bz

Q

dkydkzdk′ydk′z

16π4u(k)u(k′)

×

−1

2Q2

xez − Qxkzex

ei(kz+k′z)ze−iQxxei(ky+k′y−Qy)y.

(8)

3



The term of h p with Q = 0 corresponds to the continuousapproximation and varies on the large scale, while termswith Q = 0 are related to the field components changingon the scale of a.

It is easy to see that only terms with Qx = 0 willgive contribution to the part of the free energy describ-ing the interaction between PVs and JV, because thefield hJ does not depend on x and all terms with Qx = 0

vanish after integration over x. Therefore, it is con-venient to divide h

(1) p and n

(1) p into two components:

n(1) p = n

(Q) p +exn∗ p, h

(1) p = h

(Q) p +exh∗ p, where h

(Q) p and

n(Q) p include summands with Qx = 0 while h∗ p and n∗ p do

not vary with x. Then, the free energy functional F cross,containing only terms dependent on the displacement u,can be introduced as F cross = F PJ − F P − F J , whereF P and F J are the free energies of the unperturbed PVlattice and the “bare” JV, respectively. Using equations(7), we obtain the expression for F cross as

F cross =1

8π d3Rn(Q) p h(Q) p + 2h(0) p n(2) p

+ 18π

d3R

h∗ pn∗ p + 2hJ n

∗ p

. (9)

The first contribution comes from the terms with Qx = 0and depends only on the short-scale variations of h p. Itis determined by the shear deformation and the tilt de-formation. The second part describes the interaction of PVs with JV and with the current generated by PVsalong the y-axis. Using the equations (8), the free en-ergy F cross can be rewritten in term of Fourier variablesu(ky, kz):

F cross =1

2 dkydkz

4π2

(U 66(ky) + U 44(ky, kz)) u(k)u(

−k)

+BzΦ0

4π

Qx=0

dkydkz

4π2ikzu(k)

× f (kz, ky − Qy) − (Bz/2Φ0)ikzu(−k)

1 + λ2abk2z + λ2c(ky − Qy)2, (10)

with the shear energy

U 66 =B2z

4π

Qx=0

Q2

x

1 + λ2abQ2x + λ2ab(ky − Qy)2)

−Q2

x

1 + λ2ab(Q

2x + Q

2y), (11)

and the tilt energy

U 44 =B2z

4π

Qx=0

Q2

x

1 + λ2abk2z + λ2abQ2x + λ2ab(ky − Qy)2

− Q2x

1 + λ2abQ2x + λ2ab(Qy − ky)2

+k2z

1 + λ2abk2z + λ2cQ2x + λ2c(ky − Qy)2

+ (λ2c − λ

2ab)Q

2xk

2z

(1 + λ2abk2z + λ2abQ2x + λ2ab(ky − Qy)2)

× 1

(1 + λ2cQ2x + λ2c(ky − Qy)2 + λ2abk2z)

. (12)

The expressions for U 44 and U 66 represent sums over thereciprocal lattice vectors with Qx = 0, while the sum-mation in the last term of (10) is performed only overthe reciprocal lattice vectors with Qx = 0. The functionf (q) in (10) appears due to smoothing of δ function ineq. (3) and can be evaluated as f (q) ≈ 1 in the rectan-gular region |qz| ≤ 1/s, |qy| ≤ 1/λJ and f ≈ 0 outsidethat area. The summation in the expression (11) for the

shear elastic energy was done by Brandt16 in the limitky ≪ π/b:

U 66 = C 66k2y, (13)

where the shear elastic modulus C 66 is expressed asC 66 = (BzΦ0)/(8πλab)2 for a0 =

Φ0/Bz < λab,

while C 66 =

πλab/(6a0)Φ20/(4πλ2

ab)2 exp(−a0/λab) fora0 > λab. The tilt energy was obtained in2,17:

U 44 =BzΦ0

32π2λ4ab

ln

1 +

k2zλ−2ab + K 20

+k2zλ4ab

λ2

c

lnξ−2ab

K 2

0 + (k2z/γ

2

) + λ−2

c (14)

for kz K 0 = 2π/b, while

U 44 =

3.68

Φ20

(4πλab)4+

BzΦ0 ln(a2/ξ2ab)

32π2λ2c

k2z = C eff 44 k2z

(15)

for kz ≪ K 0. Performing summation over Qy in thesecond term of equation (10) (see Appendix A), we finallyobtain the free energy functional:

F cross =

dky2π

dkz2π

1

2(U 44 + U 66) u(k)u(−k)

+ Bz4π

ikzΨ(kz , ky)

u(k) − ikz Bz2Φ0

u(k)u(−k)

, (16)

where Ψ is defined by the equation

Ψ(kz , ky) =Φ0b

2λc

1 + λ2abk2z

×sinh

1 + λ2abk2zb/λc

cosh

1 + λ2abk2zb/λc

− cos kyb

(17)

4



for ky < min(π/b, 1/λJ ) and kz < 1/s while Ψ ≈ 0 out-side that rectangular area. In the case of small values of wave vector k (ky ≪ π/b and kz ≪ γ/b), the discretenessof PV lattice is irrelevant and the function Ψ coincideswith the Fourrier image of the “bare” JV field, but Ψ ismodified substantially for larger ky or kz.

The minimization of the free energy functional (16)determines the displacement u as

u(k) =Bz

4π

ikzΨ(k)

U 44 + U 66 + (B2zk2z/4πΦ0)Ψ(k)

. (18)

In order to describe the field distribution of a JV inthe crossing lattice, the averaged magnetic induction BJ

along the x-direction generated by both JV and inclinedPV lines can be introduced. By substituting the founddisplacement (18) into equations (7,8), the magnetic in-duction of JV BJ = hJ + h∗ p is rewritten as

BJ =

dqzdqy(2π)2

Φ0eiqyy+iqzz

1 + λ2cq2y + λ2abq2z

+BzQy

dky

dkz

(2π)2ik

zu(k)

1 + λ2c(ky − Qy)2 + k2zλ2abei(ky−Qy)+ikzz,

(19)

where qy and qz are the wave vectors of a “bare” JV(|qy| < 1/λJ , |qz| < 1/s), while the wave vectors k of thePV lattice are restricted also by the first Brillouin zoneof the PV lattice (|ky| < min(1/λJ , π/b), |kz| < 1/s).To get the energy E J it is necessary to add F cross tothe energy of the JV itself. Obviously, the energy of aJV in the presence of PV lines is always lower than oneof a “bare” JV. Indeed, for the displacement of PVs udetermined by equation (18), the energy F cross takes the

minimum value which is smaller than zero, since F cross =0 at u = 0. Finally, the energy of JV in the crossinglattice obeys the equation

E J =Φ20

8π

dqydqz(2π)2

1

1 + λ2cq2y + λ2abq2z

− B2z

32π2

dkydkz(2π)2

k2zΨ(k)Ψ(−k)

U 44 + U 66 + (k2zB2z/4πΦ0)Ψ(k)

. (20)

Equations (18-20) together with the condition that thecurrent density along the c-axis should be smaller thanthe maximum current density jc, determine completelythe behavior of the PV lines and the JV. However, in

further analysis it is convenient to investigate the dense(γs ≫ a) and dilute (γs ≪ a) pancake lattices separately.

II. JOSEPHSON VORTEX IN THE PRESENCEOF DENSE PV LATTICE

For the case of the dense PV lattice, many PV rowsare placed on the nonlinear JV core (Fig. 1b, left sketch).It means that the magnetic field of a bare JV varies on

scales larger than the distance between PV lines evennear the JV core. Thus, the continuous approximation isapplicable in the whole space. In this case |ky | < 1/λJ <π/b and, therefore, the cosine and hyperbolic functions in(17) can be expanded in the series. Hence, the functionΨ can be rewritten as

Ψ =Φ0

1 + λ

2

abk2

z + λ2

ck2

y

. (21)

Substituting this expression for Ψ into (19,20) and omit-ting the difference between k and q, the equations for thedense PV lattice (which determine the field distributionand the energy of JV) are deduced as

BJ =

1/λJ−1/λJ

dqy2π

1/s−1/s

dqz2π

× Φ0eiqyy+iqzz

1 + λ2cq2y + λ2abq2z + q2zB2z/(4π(U 44 + C 66k2y))

, (22)

and

E J =Φ20

8π

1/λJ−1/λJ

dqy2π

1/s−1/s

dqz2π

× 1

1 + λ2cq2y + λ2abq2z + B2zq2z/(4π(U 44 + C 66q2y))

. (23)

The last undefined parameter, λJ , can be obtained fromthe condition |∂ BJ (y ≈ λJ , z = 0)/∂y | ∼ (4π/c) jc:

π

λ2cs≃ λJ

1/λJ−1/λJ

dqy

1/s−1/s

dqz

×

q2y

1 + λ2

cq2

y + λ2

abq2

z + q2

zB2

z/(4π(U 44 + C 66q2

y))

. (24)

The region of integration is shown in Fig. 2a. Therectangular domain of possible wave vectors replaces theusual elliptical one due to a peculiar core structure of JV. In anisotropic London model, the core of an ordi-nary vortex is defined by the elliptical stream line of thepersistent current having the depairing value18. How-ever, in our case the maximum value of qz is determinedby the layerness of the medium while the largest valueof qy is restricted by the Josephson critical current alongthe c-axis. The rectangular domain of wave vectors (Fig.2 a) can be divided into “screened” (|kz | 1/b) and“remote”(

|kz

|< 1/b) subdomains. The first one corre-

sponds to the region where one can roughly neglect theweak logarithmical dependence on kz in eq. (14) to ex-press the tilt energy as:

U 44 ≈ U 44 + C 44k2z , (25)

with U 44 = (BzΦ0/32π2λ4ab)ln(1 + kz2

/(λ−2ab + b−2)),

C 44 = BzΦ0

32π2λ2cln

ξ−2

ab

b−2+(kz2/γ2)+λ−2

c

and kz ∼

1/bs.

5



6

\

6

]

6 F U H H Q H G

U H J L R Q

6 F U H H Q H G

U H J L R Q

5 H P R W H

U H J L R Q

8

- γ -

6

\

6

]

6 F U H H Q H G

U H J L R Q

6 F U H H Q H G

U H J L R Q

5 H P R W H

U H J L R Q

8

-

γ -

λ-

π -

D

E

3 L Q Q L Q J

U H J L R Q

3 L Q Q L Q J

U H J L R Q

1 R Q V F U H H Q H G U H J L R Q

1 R Q V F U H H Q H G U H J L R Q

λ-

π -

FIG. 2. The integration region in equations (19) and (20)for the dense PV lattice (a) and the dilute PV lattice (b). Inthe dense case (a), the region of the available wave vectors kof the PV lattice coincides with the accessible domain of wavevectors q; the continuous approximation (21) for Ψ is alwaysvalid. The approximation (25) is correct in the “screened”domain of wave vectors, while it fails in the “remote” re-gion. In the dilute case (b), the continuous approximationis correct only in the region |qy| ≪ π/b; |qz | ≪ γ/b; the“non-screened” region |qy| > π/b is not accessible for k. Thecontinuous approximation for Ψ(k) is broken in the “pinning”region |ky| < π/b; γ/b |kz| 1/s.

The expression (25) is completely wrong in the “remote”region in which it is necessary to use equation (15). Byusing approximation (25) for the tilt energy, the integralsin equations (23) and (24) are easily evaluated (AppendixB) and we get

λJ ≈ max

λcs

λeff ab

,

λcs

λeff ab

λab

(26)

and

E J ≈ Φ20

16π2λeff ab λc

C 66

U 44λ2ab

λab

λJ + ln

λcut

λJ

, (27)

where

λcut ∼ min(λc, min(γb, max(bλc/λeff ab ,

bλcλab/λeff

ab ))),

while the renormalized penetration depth λ

eff

ab is ex-pressed as

λeff ab =

λ2ab +

B2z

4πU 44. (28)

The physical reason of the renormalization of the in-planepenetration depth is related to the screening of the JVfield by currents born by the local inclination of PV lines.From eq. (26) it is easy to see that the size of the non-linear JV core also decreases due to the interaction of JV and PVs. The similar conclusion was given earlierby Koshelev7, who considered the additional phase vari-ation of the order parameter born by the displacementof PVs. However, the shear contribution to the renor-malization of E J and λJ was neglected in7, which couldbe done only for λJ > λab (see equations (26) and (27)).In the opposite case, i.e., when the London penetrationdepth exceeds the JV core size, the shear deformationbecomes relevant and, as a result, λJ decreases with Bz

slower than it was proposed in7.To understand how the field of JV is distributed in

the real space, we rederive the results considering thefree energy functional of the displacement u defined asa function of the spatial coordinates. In the limit γs ≫a, the JV field varies on scales larger than the distancebetween PVs even near the JV core. This means that

the field h p along the x-axis can be averaged out on thescale larger than a:

h∗ p − λ2ab∂ 2h∗ p∂z2

− λ2c∂ 2h∗ p∂y2

= n∗ = Bz∂u

∂z. (29)

However, the short range variations of the field h p givethe shear energy U 66 and the tilt energy U 44. After ignor-ing the slow logarithmic dependence on k in the expres-sion for U 44, one can conclude that the density of the tiltenergy in the real space is U 44u2(y, z), while the densityof the shear energy is U 66 = C 66(∂u/∂y)2. Thus, the freeenergy functional is expressed as

F cross =1

8π

d3R

4πC 66

∂u

∂y

2

+ 4πU 44u2 + h∗ pBz∂ u

∂z+ 2hJ

∂ u

∂z

. (30)

The first three terms represent the elastic energy (bornby shear, electromagnetic tilt and Josephson coupling tiltrigidity, respectively), but the last term is related to the

6



interaction of the PV lines with the current generated byJV.

In order to get the complete set of equations for thedisplacement u and the averaged magnetic induction BJ ,we have minimized the functional (30) and have addedtogether equations (3) and (29):

−4πC 66

∂ 2u

∂y2

+ 4πU 44u

−2Bz

∂ BJ

∂z

= 0,

BJ − λ2ab∂ 2BJ

∂z2− λ2c

∂ 2BJ

∂y2= Φ0δ(y)δ(z) + Bz

∂u

∂z. (31)

This set of equations is applicable if the continuous ap-proximation is valid (λJ ≫ a) and, strictly speaking, onlywhen the tilt energy U 44(kz) can be replaced by the con-stant U 44. The last condition fails on the distances farfrom JV (z2 + y2/γ 2 > b2). In this “remote” region, the

constant U 44 has to be substituted by −C eff 44 ∂ 2/∂z2. Be-sides, if λab > γ s, the parameter U 44 should be replacedby −C 44∂ 2/∂z2 near the JV core (z < λab/γ ).

Even though we consider only the situation when theset of equation (31) is valid, i.e., the case λab < γs and

the region z2 + y2/γ 2 < b2, the solution of equations (31)seems to be quite complicated. The relation between thedisplacement u and the magnetic induction BJ , which isobtained from the first equation of (31), becomes nonlo-cal due to the shear rigidity of the PV lattice:

u =Bz

8π

C 66U 44

∞−∞

dy∂ BJ (y, z)

∂ze−|y−y|/δ (32)

where δ = λab

C 66/(U 44λ2ab) ∼ λab is the characteris-

tic length of a nonlocality. However, the nonlocality isirrelevant if the space scale of the variation of BJ is sub-stantially large then δ, i.e., if λJ

≫λab. In such a case,

the equations (31) for BJ and u can be decoupled

u =Bz

4πU 44

∂ BJ

∂z,

BJ − (λeff ab )2

∂ 2BJ

∂z2− λ2c

∂ 2BJ

∂y2= Φ0δ(y)δ(z). (33)

The equation (33) for induction BJ is the London equa-tion with the renormalized in-plane penetration depthλeff ab . Therefore, the field distribution BJ , not far from

the center of the Josephson vortex (z2 + y2/γ 2 b2), canbe approximated as

BJ =

Φ0

2πλeff ab λc

K0 z2/(λeff ab )2 + y2/λ2c , (34)

where K0(x) is a modified Bessel function of zero order.Using the free energy functional (30) and equations (31),it is easy to show that the energy of JV is determined bythe field in its center, i.e., E J = Φ0/(8π)BJ (y ≈ λJ , z ≈s):

E J =Φ20

16π2λeff ab λc

ln(λ∗ab/s) (35)

with the length λ∗ab = λeff ab . However, the set of equa-

tions (31) becomes incorrect in the region z2+y2/γ 2 > b2

which cuts off that length as λ∗ab ≈ b. Thus, the expres-sion (35) coincides with the earlier obtained equation (27)in the studied case λJ > λab. The results (34,35) canbe interpreted in terms of the effective anisotropy pa-

rameter γ eff = λc/λeff ab which governs the JV lattice.

Since λeff ab > λab, the effective anisotropy γ eff is re-

duced in the presence of PVs with respect to the “bare”one γ = λc/λab. The similar anisotropy γ eff was earlierintroduced7 as a ratio γ eff = λJ /s, but these two dif-ferent definitions of γ eff give the same value in the caseλJ > λab when the shear deformation is irrelevant.

Here, we discuss how the core size and the JV en-ergy are changed with the magnetic induction Bz if γs > λab. For quite high magnetic inductions Bz B1 =(Φ0/λ2ab) × (γs/λab)2, the size of the nonlinear core λJ issmaller than λab and the shear contribution to the freeenergy is important. The second logarithmic term in theJV energy (27) can be omitted, and the core size obeysthe equation λJ (Bz) ≃ √

γsa. With decreasing of induc-

tion, the core size increases proportionally to B−1/4z andreaches λab at Bz ≈ B1. At low fields, the shear interac-tion between rows is irrelevant, the JV core size becomesλJ = λcs/λeff

ab ≈ γsa/λab, and the energy of JV is deter-mined by the logarithmic term in (27). Below the fieldΦ0/λ2ab, at which the distance a between PVs exceedsλab, the currents generated by PVs practically do not in-fluence on the JV field and, thus, the renormalization of λab, λJ and the E J vanishes. In the case of λab > γsthe physical picture is different from the previous situa-tion. The core size obeys the law λJ ≃ √

γsa at fieldsBz > Φ0/(γs)2. Below this field, the effective value of the

in-plane London penetration depth λeff ab

∼λ2ab/a (28) is

still larger than λab, while the JV core size is saturatedas λJ = γs. This means that JV field shows differentbehavior far from JV (z2 + y2/γ 2 > b2/γ 2), where theredistribution due to the local inclination of PV lines isstill important, and close to the JV core.

III. JOSEPHSON VORTEX IN THE PRESENCEOF DILUTE PV LATTICE.

Far from the JV center, z2 + y2/γ 2 > b2/γ 2, theJV field varies slowly which causes the smooth varia-tion of the displacement u even for the case of the di-

lute PV lattice (a > γ s). In that spatial region, thecontinuous approximation is still valid. On the otherhand, near the JV core (|y| < b), the JV current in-creases quite fast inducing a large displacement of thePV stack placed on the center of JV. In this case, thecontinuous approximation is not applicable. To de-scribe such physical situation, we consider the wave vec-tor area of k divided into two domains (Fig. 2b). Inthe first interval |ky| < π/b and |kz| < γ/b, the func-tion Ψ can be still roughly approximated by equation

7



Ψ ≈ Φ0/(1 + λ2ck2y + λ2abk2z), while Ψ ≈ Φ0b/(2λcλabkz)in the second region |ky| < π/b and γ/b < |kz | < 1/s(“pinning” region in Fig. 2b). Following this approach,the energy of JV is evaluated as

E J (a ≫ γs) ≈ Φ20

8π

π/b−π/b

dqy2π

γ/b−γ/b

dqz2π

×1

1 + λ2cq2y + λ2abq2z + B2zq2z/(4π(U 44 + U 66))

+Φ20

16π2λcλabln

b

γs

− BzΦ30

128π3aλ2cλ2ab

1/sγ/b

dkzU 44 + BzΦ0kz/(8πλcλaba)

. (36)

The first term comes from the spatial region far from thecenter of JV while the second and the third terms arerelated to the vicinity of the JV center. The screeningof the “bare” JV field vanishes near JV (“non-screened”region in k-space) which determines the second term in(36). The last term in (36) represents the energy gain

due to the strong interaction between the PV line placedon the JV core and the JV currents (the energy gainof a PV stack placed on a JV in the limit γs ≫ λab

and Bz → 0 was calculated by Koshelev7). Since thelast term is sensitive to the mutual position of JV andthe nearest PV line, this contribution can be called asthe “crossing lattice pinning”. Using the results of Ap-pendix B and taking into account that the evaluationBzΦ0kz/(8λcλaba) C 44k2z is held in the “pinning re-gion” (kz > γ/b), the energy of JV is finally obtained:

E J ≈ Φ20

16π2λcλeff ab

2µ1C 44γ 2

πU 44λ2ab

λ3abb2λeff

ab

+

µ22C 66

U 44λ2ab

λab

b

+Φ20

16π2λcλeff ab

ln

λcut

b

+

φ2016π2λcλab

ln

b

γs

− µΦ20

4πaλcarctan

b − γs

U 44/C 44sb + γ

C 44/U 44

, (37)

where µ = BzΦ0/(32π2λcλ2ab

C 44U 44) < 1 is the di-

mensionless function depending quite slowly on Bz andthe numerical parameters µ1 and µ2 are about unity.

Next, we will discuss how the renormalization of theJV energy comes in with increasing of the z-componentof the magnetic field. At low fields, Bz

≪Φ0/λ2ab

(a ≫ λab), the first term and the last term in (37) canbe omitted and the expression for the energy of a “bare”JV reported earlier in19,20 is reproduced

E J =Φ20

16π2λcλabln

λc

γs

. (38)

For the case λab > γs, the renormalization of JV en-ergy becomes relevant at Bz ≈ Φ0/λ2ab, i.e., earlier thanthe JV core size starts to decrease which occurs only in

fields Bz > Φ0/(γs)2. The origin of this behavior is thatthe additional current along the c-axis induced by tiltedPV stacks is much smaller than jc near JV core in thefield interval Φ0/λ2ab Bz Φ0/(γs)2, but the inclina-tion of all PV lines can still cause the renormalizationof the JV field on scales larger than a. In the field in-terval Φ0/λ2ab < Bz Φ0/γ 2s2, the main contributionto the Josephson vortex energy (37) comes from the first

term related to the tilt elastic rigidity (born by Josephsoncoupling of PVs) and shear elasticity of the PV lattice.Strictly speaking, from our rough estimation of (20), wecan not conclude how strongly E J is suppressed in thatfield interval, i.e. in the presence of the dilute PV lat-tice. Nevertheless, the pinning energy (last term in (37))could be the same order of magnitude as the first andthe second terms in (37) in fields Bz ∼ Φ0/λ2ab and maydecrease E J substantially.

Another interesting possibility arising due to the“crossing lattice pinning” is the rearrangement of thePV lattice in the presence of the JV sublattice. In thein-plane magnetic fields Bx, JVs form a triangular lat-tice with distances a

J and b

J between JVs (see inset in

Fig. 3a). In general, the PV sublattice and the JV sub-lattice are not commensurate: aJ = pb with integer p.This means that the considered one-component displace-ment of PVs u = (u(y, z), 0, 0) (the “shifted” PV latticeshown in Fig. 3a) does not provide the energy gain com-ing from the “crossing lattice pinning” since the PV rowscan not occupy the centers of JVs. However, the PVs canbe rearranged in order to occupy all JVs (the “trapped”PV lattice shown in Fig. 3b) if the PV lines shift alsoalong the y-direction: u = (ux(x, y, z), uy(x, y, z), 0).The “crossing lattice pinning” decreases the free energyof the “trapped” PV lattice, while the additional sheardeformation acts in the opposite way through increasing

the free energy. For the case Bx < γ Bz, the energy gainrelated to the “trapped” PV lattice is calculated by nor-malizing the last term of equation (37) per unit volume:

E tr = µBxΦ0

4πaλcarctan

b − γs

U 44/C 44sb + γ

C 44/U 44

.

(39)

But, in order to trap the PV lattice, the total displace-ment of PVs along the y-axis between the two nearest JVrows, i.e., on the scale aJ , should be about b. Followingthe simple analysis21, the extra shear deformation (insetin Fig. 3b) is about δb/b

∼b/aJ (δb is the change of

the distance between rows of PVs) and the energy lossE shear can be estimated as:

E shear ≃ νC 66

b

aJ

2≈ νC 66

Bx

γBz(40)

with numerical constant ν 1. For the case γs ≫ λab,the shear elastic energy (40) is strongly suppressed in thefields Bz < Φ0/(γs)2 where the “crossing lattice pinning”is active, since C 66 is exponentially small if a > λab (13).

8



δ -

-

F

D

E

D-

E-

FIG. 3. The different substructures of the crossing lattice:a) the “shifted” PV lattice characterized by one-componentdisplacement along JVs, b) the PV lattice trapped by theJV sublattice for the case when the distance aJ between JVsexceeds the distance b between PV rows, c) the “trapped”PV lattice for aJ < b, i.e., in the case of the field orienta-tions very close to the ab-plane. The dotted lines depict therows of unperturbed lattice (crossings of these lines and filledcircles mark the positions of unshifted PVs) in all sketches.Dashed-dotted lines indicate the rows of the “trapped” PVlattice which are deformed in order to match with JV sublat-tice. The arrows directed from the filled circles to the openones show the two component displacement of PVs for the

cases represented in sketches b) and c). Inset in a): the JVsublattice with lattice parameters aJ and bJ (lines mark theCuO2 planes). Inset in b): the additional deformation of thePV lattice which is required for trapping of PVs by JVs. Theupper sketch is the equilateral triangle of the unperturbed PVlattice which is incommensurate with JV lattice (aJ = pb withinteger p). The lower sketch is the isosceles triangle of the PVlattice matched with the JV sublattice (aJ = p(b + δb)).

Therefore, the “trapped” PV lattice seems to be realizedas soon as a > γ s. In the opposite case, λab γs, thetransformation22 from the “shifted” PV lattice to the“trapped” PV lattice occurs when the energy gain E trexceeds the energy loss E shear. It happens in a certainout-of-plane field between the field Φ0/(γs)2, at whichthe “crossing lattice pinning” is activated, and the fieldBz ∼ Φ0/λ2ab, where the shear elastic energy rapidly de-

creases. Next, we discuss the difference between the con-sidered “trapped” state and the “chain” state proposedfor the crossing lattice7. The “trapped” state is relatedto the rearrangement of PVs on the scale aJ between thenearest rows of JVs. On the other hand, the “chain”state is associated with the creation of an extra PV row(an interstitial in the PV lattice) on a JV, but the influ-ence of the neighbouring JVs is completely ignored. Asa result, the “trapped” and “chain” states have the dif-ferent in-plane field dependence of the out-of-plane tran-sition fields. The out-of-plane transition field23 betweenthe “shifted” and “trapped” PV lattices does not dependon H ab in contrast to the H ab-dependent out-of-planefield7 of the destruction of the “chain” state. Since theanalysis7 is correct only in the case of γs ≫ λab anda ≫ λab, the transformation of the PV lattice discussedhere seems to be more likely in the case λab > γ s.

IV. PHASE DIAGRAM OF VORTEX LATTICE INTILTED MAGNETIC FIELDS

In this section we discuss the vortex lattice structuresformed at different field orientations. The tilted latticeconsists of mono-oriented vortices and transforms contin-uously from the tilted PV stacks in fields near the c-axis(Fig. 4a) to the long JV strings connected by PV kinks

for the field orientations close to the ab-plane (Fig. 4b).On the other hand, the tilted lattice is topologically dif-ferent from the crossing vortex structure (Fig. 4c), andthey replace each other via phase transition6. For theanalysis of the vortex phase diagram in tilted fields, thefree energy of the crossing and tilted vortex structureswill be compared. We concentrate on the case γs < λab,when, according to Bulaevskii et al.

6 and Koshelev20, thetilted lattice is energetically preferable above the lock-in transition8. We will consider a thin superconductingplatelet with the c-axis perpendicular to the plate. Inthis geometry the lock-in transition occurs at very lowfields6 Bz

≈(1

−nz)Φ0/(4πλ2ab)ln(γs/ξab) with demag-

netization factor nz (1 − nz ≪ 1).For the field oriented close enough to the c-axis, tan θ =

Bx/Bz ≪ γ , the free energy of the tilted lattice F t can beevaluated as F t = F 0t + 1

2C tilt44 (k = 0)B2x/B2

z in analogyto the analysis given in ref.7. Here, F 0t represents thefree energy in the absence of the in-plane magnetic field,while the tilt modulus is expressed as C tilt44 (k = 0) =

B2z/4π + C eff 44 with C eff 44 defined in (15) for the case of

Bz Φ0/(4πλ2ab). As a result, we have:

9



D

E

F

+

ab

+

c

+

c

+

ab

+

ab

+

c

FIG. 4. The 3D sketches of the different vortex structuresin the tilted magnetic field with the components H c and H abalong the c-axis and in the ab-plane, respectively: a) the tiltedvortex lattice near the c-axis (TI), when the current betweenCuO2 planes is much smaller than the critical value jc, i. e.,the Josephson strings linking PVs are not developed; b) thetilted vortex lattice far away from the c-axis (TII), when theJV strings are formed; c) the crossing vortex lattice.

F t ≃ B2z

8π+

Φ0Bz

32π2λ2abln

H c2⊥Bz

+B2x

8π

+ 3.68

Φ20

2(4πλab)4

B2x

B2z +

B2xΦ0

64π2λ2cBz ln

H c2⊥

Bz , (41)

where H c2⊥ = Φ0/2πξ2ab. The first two terms form thefree energy for Bx = 0. The third term is the in-planemagnetic energy, the fourth one comes from the electro-magnetic interaction of the inclined PVs, and the lastcontribution is connected with the Josephson coupling of PVs.

The free energy of the crossing lattice F c consists of twocontributions from the PV sublattice and the JV sublat-

tice while the interaction of PVs and JVs is taken intoaccount through the renormalization of the JV energy:

F c ≃ B2z

8π+

Φ0Bz

32π2λ2abln

H c2⊥Bz

+B2x

8π+

Bx

Φ0E J . (42)

The renormalized JV energy, E J , is defined by equa-tion (20) in which the lower limits of integrations are

restricted by the conditions qy, ky 1/aJ and qz, kz 1/bJ .The tilted lattice is energetically preferable in the fields

oriented near the c-axis because F t ∝ B2x, while F c ∝ Bx,

i. e., F t < F c for low Bx. The phase boundary betweenthe tilted lattice and the crossing structure can be ob-tained from the condition F t = F c which is rewritten inthe form:

Bx ≃ E J Φ0

B2z

1.84Φ20/(4πλab)4 + Φ0Bz/(64π2λ2c)ln(H c2⊥/Bz)

.

(43)

The transition from the tilted lattice to the crossing

structure occurs at the field oriented quite close to thec-axis for high anisotropic superconductors due to: 1)the high energy cost of the inclination of PV stacks inthe tilted lattice related to the electromagnetic interac-tion of PVs, and 2) the decrease of the JV energy inthe crossing lattice structure. For the dense PV latticeBz ≫ Φ0/(γs)2 and λab > γ s, equation (43) can be sim-plified:

Bx ≃

C 66

U 44λ2ab

2λ2abλJ λ

eff ab

B2z

γΦ0

4.3π2λ2ab

+ Bz

2γ ln(H c2⊥/Bz).

(44)

Next, we will study the field orientations close to theab-plane, Bx > γBz. Here, the electromagnetic interac-tion between PVs in the tilted lattice is not so importantand the free energy in the low c-axis fields Bz < Φ0/λ2abis reduced6 to:

F t ≃ B2x

8π+

Φ0Bx

32π2λabλcln

Φ0

γs2Bx+

H J Bz

4π, (45)

where H J = Φ0/(4πλ2ab)ln(γs/ξab). The first two terms

are related to the energy of JV strings while the last oneis associated with the energy cost of the formation of PV kinks. In more detail, the PV kink generates the in-

plane current which decreases with the distance r fromthe kink center as 1/r up to a critical radius r0 of theregion with 2D behavior where the current along thec-axis is about the maximum possible current jc

6. Atlarger distances, the in-plane current decays exponen-tially. Simple evaluation gives r0 = γs for the PV kink6.We note, that the tilted vortex lattice in the consideredangular range of the magnetic field orientations seems toexist as a kink-walls substructure, where kinks (belong-ing to different vortices) are collected in separated walls

10



parallel to the yz-plane20. For the kink-wall substruc-ture of the tilted lattice, the contribution to the free en-ergy (45), attributed to the PV kinks, is slightly reducedin the high in-plane magnetic fields Bx > Φ0/γs220,which can be taken into account through renormaliza-tion H J = Φ0/(8πλ2ab)ln(γH c2/Bx).

In the considered field interval, Bx > γBz, Bz ≪Φ0/λ2ab, the renormalization of the JV energy in the

crossing lattice structure vanishes. However, the inter-action of PV and JV sublattices still manifests itself through the “crossing lattice pinning”:

F c =B2x

8π+

Φ0Bx

32π2λabλcln

Φ0

γs2Bx+

H c1⊥Bz

4π

−µBz

BxΦ0

16π2γλ2ab

arctan

1 −

Bx/H 0

H xλ/H 0 +

Bx/H xλ

, (46)

with H c1⊥ = Φ0/(4πλ2ab)ln(λab/ξab) (the critical radius

r0 for the in-plane currents of a PV stack is about λab6),

H 0 = Φ0/γs2 and H xλ = γ Φ0/λ2ab. The third term is theenergy of the unperturbed PV lattice, while the last term

corresponds to the “crossing lattice pinning” contributionwhich can significantly decrease the free energy F c in thein-plane field interval γ Φ0/λ2ab Bx Φ0/γs2. Thedifference between the “crossing lattice pinning” contri-butions to the free energy in the cases of low Bx < γBz

and high Bx > γBz in-plane fields (see equations (39),(46)), emerges because the number of PV lines is suffi-cient to occupy all JVs (Fig. 3b) at Bx < γBz whilesome JV strings do not carry PV rows (Fig. 3c) in theopposite case. By analyzing equations (45) and (46),we can conclude that, at least for Bx γ Φ0/λ2ab andBx Φ0/γs2, the tilted lattice exists near the ab-planesince the condition F t < F c is held due to the inequal-

ity H J < H c1⊥. The tilted lattice is replaced by thecrossing lattice with increasing the out-of-plane magneticfield above Bz = Bx/γ . However, it is difficult to deter-mine the contour of the possible phase line between thecrossing and tilted vortex structures, since it requires themore precise calculations of the free energies F c and F tin the region Bx < γBz. In the intermediate in-planemagnetic fields γ Φ0/λ2ab Bx Φ0/γs2, the “crossinglattice pinning” could make the crossing structure to bemore energetically preferable with respect to the tiltedlattice. In that case, the crossing lattice (CII, Fig. 3b)with a < aJ transforms into the crossing lattice (CIII,Fig. 3c) with the extremely dilute PV sublattice a > aJ

at the angle θ = arctan Bx/Bz ∼ arctan γ .Therefore, we find a complicated picture of phase tran-sitions between the tilted vortex structure and the cross-ing vortex structure in the case γs < λab. The proposedphase diagram24 is shown in figure 5. As it was suggestedearlier7 and according to our calculations by using equa-tion (43), the tilted vortex structure (TI) of inclined PVstacks (see Fig. 4a) can be replaced by the crossing lat-tice quite close to the c-axis (see phase diagram obtainedfor γ = 500, inset in Fig. 5).

+ , -

N2H

7,

7,,

7,,

&,,,

&,,

&, +

.

2 H

+

, -

2 H

+ .

2 H

& ,

7 ,

FIG. 5. The proposed phase diagram of the vortex solidphase in the oblique magnetic fields calculated using equations(39), (40), (43), (45) and (46) with parameters mentioned inthe text, T = 45 K, γ = 100 and ν = 1. The dotted line isthe line Bx = γBz, while the shaded area marks the regioninside which the transition from the crossing lattice C(II) tothe tilted lattice T(II) or the crossing lattice C(III) happens.The arrows from enframed TI and TII are directed toward theregions where these vortex structures are realized. Inset: thepart of the phase diagram close to the c-axis for strongly ani-sotropic superconductors with γ = 500 (γs > λab) and T = 45K; the solid line marking the transition from the tilted lattice(TI) to the crossing lattice is obtained from eq. (43), while

the dashed line, corresponding to the same transition, is cal-culated by using eq. (6) of Ref.7. The parameters are chosento give some insight to the behavior of Bi2Sr2CaCu2O8−δ inthe oblique magnetic fields.

Nevertheless, in the high c-axis magnetic fields, the calcu-lated in-plane magnetic fields of this transition are higherthan ones obtained by using the model7. This differencecomes from the shear contribution to E J which was omit-

11



ted in ref.7. The crossing lattice, which exists in a wideangular range, can have different substructures. At highenough out-of-plane fields Bz > Φ0/γ 2s2, the “shifted”PV sublattice is realized in the crossing lattice structure(CI, Fig. 3a). In this substructure, the JV currents shiftthe PVs mostly along the x-axis. The “shifted” phase cantransform into the “trapped” PV lattice (CII, Fig. 3b),when the energy gain related to the “crossing lattice pin-

ning” exceeds the energy needed for the additional sheardeformation (the dashed line in Fig. 5 separating CI andCII has been obtained from the condition E tr = E shear).Around the line Bx = γBz, the lattice CII can be changedby the tilted lattice (TII) with JV strings linked by PVkinks (Fig. 4b) or by the crossing lattice structure (CIII,Fig. 3c) at which all PV stacks are placed on a few JVs.The domain in the H c − H ab phase diagram with the lat-tice CIII is determined by the condition F c < F t whereF t and F c are defined by the equations (45) and (46),respectively. With increasing temperature, the region of the lattice CIII becomes narrower and disappears at acertain temperature (see Fig. 6). The proposed phasediagram suggests the possibility of the re-entrant tilted-crossing-tilted phase transition as the magnetic field (atleast with low B γ Φ0/λ2ab or high B Φ0/γs2 abso-lute value) is tilted away from the c-axis to the ab-plane.Such possibility for low fields was earlier mentioned inthe works11,13 in which the interaction of crossed sub-lattices was not considered. Moreover, we note that theinstability of the tilted lattice was found numerically byThompson and Moore25 (for γ 100) only at the inter-mediate field orientations 0◦ < θ1 < θ < θ2 < 90◦, whichcould also support the discussed scenario. The parame-ters taken for the H c − H ab phase diagram at T = 45 K(Fig. 5) and the H ab − T phase diagram at the magneticfield orientation Bz = Bx/γ (Fig. 6) were chosen to give

some insight into the behavior of vortex array in BSCCOin the tilted magnetic fields (see further discussion) as

λab = 2000/

1 − T 2/T 2c A, ξab = 30/

1 − T /T c A,s = 15 A, T c = 90 K, γ = 100 (γ = 500 for inset inFig.5 and γ = 150 for inset in Fig. 6).

V. CONCLUSION

This theoretical investigation was partially motivatedby the recent intensive experimental studies of the vor-tex lattice melting transition26–30 as well as transitionsin the vortex solid phase29,31 in Bi2Sr2CaCu2O8+δ sin-

gle crystals. The observed linear decay of the c-axismelting field component H mc with in-plane fields26,27 wasinterpreted7 as an indication of the crossing vortex lat-tice. Thus, the tilted lattice could be replaced by thecrossing vortex structure quite near the c-axis. Accord-ing to our calculations, the angle where such transitionmay occur is about 7◦ at Bz ≈ 100 Oe and γ ≈ 500while that angle reaches 14.5◦ in the higher out-of-planefield Bz = 500 Oe, which correlates well with the disap-

pearance of the hexagonal order along the c-axis foundby neutron measurements32. With further tilting of themagnetic field, the linear dependence of the c-axis melt-ing field component H mc (H ab) abruptly transf orms into aweak dependence27–29, which, as was shown28, can not beexplained in the frame of the model7. Such behavior sug-gests a phase transition in the vortex solid phase in tiltedmagnetic fields in Bi2Sr2CaCu2O8+δ, which was detected

in the recent ac magnetization measurements29,31

.

7 , ,

& , , ,

+ ,

-

N 2 H

7 .

7,,

&,,,

7 .

+ ,

-

N 2 H

FIG. 6. The H ab − T phase diagram of the vor-tex solid phase at the field oriented near the ab-plane(Bx/Bz = γ = 100). The region of the crossing latticephase (CIII) becomes narrower and finally disappears withincreasing temperature. Inset: the same phase diagram forγ = 150. The dotted lines correspond to the experimen-tally found temperature dependence H mab ∝ 1 − T 2/T 2c of thein-plane characteristic fields of the vortex lattice melting tran-sition in Bi2Sr2CaCu2O8+δ in the tilted fields30 (for instance,symbols represented in the inset exhibit the temperature de-pendence of the maximum in-plane field of the vortex latticemelting transition).

12



As was mentioned by Ooi et al.31, the behavior of the new

anomaly of the local magnetization in BSCCO attributedto the phase transition in the vortex solid slightly remindsthe peak effect related to the vortex pinning, which, inturn, could be induced by the vortex trapping by planardefects21. Such analogy, as well as a very weak depen-dence of the c-axis magnetic field component H ∗c of thenovel phase transition on in-plane magnetic fields31 in

a wide angular range, could suggest the tansition fromthe “shifted” PV sublattice (CI) to the “trapped” PVsublattice (CII) in the crossing lattice structure. Our es-timation of the c-axis field H trapc of the transition fromCI to CII (H trapc ≈ 450 Oe for γ = 100 at T = 45 K (seeFig. 5))33 is in a reasonable agreement with experimentalfindings31 H ∗c (T = 45 K) ≈ 430 Oe. Near the ab-plane,the properties of the observed anomaly changes abruptlyand the field H ∗c sharply goes to zero31, which may in-dicate a transformation23 in the JV lattice or a trace of the phase transition from TII to CIII. At higher in-planefields, the found step-wise behavior28 of the vortex lat-tice melting transition may be related to the existence of one more phase transition in the solid phase. Interest-ingly enough, all characteristic in-plane fields of the vor-tex lattice melting transition depend on temperature30

proportionally to 1 − T 2/T 2c , which is similar to the cal-culated temperature dependence of the phase transitionfrom CIII to TII (see Fig. 6).

In summary, we discussed the crossing lattice struc-ture in a strongly anisotropic layered superconductor inthe framework of the extended anisotropic London the-ory. The renormalization of the JV energy in the crossinglattice structure was calculated in the cases of the densePV lattice as well as the dilute PV lattice. It was shown,that the “crossing lattice pinning” can induce the rear-rangement of the PV sublattice in the crossing lattice

structure as soon as the out-of-plane magnetic field be-comes lower than a certain critical value. The free energyanalysis indicates a possibility of the re-entrant tilted-crossing-tilted lattice phase transition with inclination of the magnetic field away from the c-axis to the ab-planein the case of λab > γ s.

APPENDIX A. APPROXIMATE SUMMATION INEQUATION (10)

Our aim is to sum the sequence:

Φ0

∞n=−∞

ikzu(k) f (kz, ky − 2πn/b) − (Bz/2Φ0)ikzu(−k)1 + λ2abk2z + λ2c(ky − 2πn/b)2

=

ikzΨ1(ky, kz)u(k) + (Bz/2Φ0)k2zΨ(ky, kz)u(k)u(−k). (A1)

The equation (17) for Ψ(ky, kz) can be directly obtainedby using the well-known mathematical equality

n

1

r2 + (t + 2πn/α)2=

α

2r

sinh(rα)

cosh(rα) − cos(tα)

with real numbers r, t, and α.Next the sum Ψ1(ky, kz) = Φ0

Qy=2πn/b

f (kz, ky −Qy)/(1 + λ2abk2z + λ2c(ky − Qy)2) needs to be estimated.By using inequailty kz 1/s, we obtain f (kz, ky−Qy) ≈f J (ky − Qy) where f J ≈ 1 for |ky − Qy| 1/λJ and zerootherwise. In the case of the dense PV lattice (b ≪ λJ )we retain only the term with n = 0 in the sum and getthe expression Ψ1(ky > 1/λJ ) = 0 and Ψ1(ky 1/λJ ) =

Φ0/(1+λ2abk2z+λ2ck2y). Taking into account the inequalityky < 1/λJ ≪ 1/b and

1 + λ2abk2zb/λc < b/(γs) ≪ 1,

one can rewrite Ψ1(ky < 1/λJ , b ≪ λJ ) ≃ Ψ(ky, kz).Thus, we come to the equation (16).

For the case of the dilute PV lattice (λJ ≪ b),many terms give contributions to the sum Ψ1 ≃Φ0

N n=−N 1/(1+λ2abk2z+λ2c(ky−2πn/b)2), since inequal-

ity |ky − 2πn/b| < 1/λJ is held until n exceeds N ≫ 1.Thus, the function Ψ1 is estimated as:

Ψ1(ky, kz) ≈ Ψ(ky, kz) − Φ0b

π

∞1/λJ

dx

1 + λ2abk2z + λ2cx2

(A2)

Finally, we obtain Ψ1 = (1 − β (ky, kz))Ψ with

β (ky , kz) ≈ cosh(

1 + λ2abk2zb/λc) − cos(kyb)

sinh(

1 + λ2abk2zb/λc)1 − 2

πarctan

λc

λJ

1 + λ2abk2z

(A3)

At |kz | ≪ 1/s, it is easy to show that β λJ /b ≪ 1 andthe approximation Ψ1 = Ψ used in (16) is excellent. Onlyfor kz ∼ 1/s, the function Ψ1 can differ from the functionΨ by a factor about unity in the case of the dilute PV

lattice (the factor is 1 − β ≈ 0.5 in the framework of ourrough consideration). However, the correct estimation of the value of the factor depends on the type of the smooth-ing function and requires more precise analisys than onein the framework of the London approach. Therefore,we can always assume Ψ1 = Ψ in our semiquantitativeconsideration.

APPENDIX B. EVALUATION OF INTEGRALSIN EQUATIONS (23,24,36)

In this appendix, the integrals in equations (23,24,36)

are evaluated. We start with the dense PV lattice, a ≪λJ . In the region qz < 1/b, the tilt energy is small (15).Therefore, the denominators of the integrands in (23,24)are substantially larger than ones in the case qz > 1/band we can roughly neglect the contribution related tothe region qz < 1/b. Using equation (25) for the tiltenergy in the domain qz > 1/b, the integrals (23,24) canbe rewritten as follows:

13





After taking f (q2y) = λJ q2y in (B5) and by ignor-

ing C 66q2y/U 44 in the case λJ > λab or by neglect-ing unity in the opposite case, we obtain the expres-

sion λJ ≈ max

λcs

λeffab

,

λcs

λeffab

δ

. It coincides with

(26) since δ ≈ λab. The energy of JV (27) is eas-ily derived if one puts f (q2y) = Φ2

0/(32π3) in (B5).Next, we roughly estimate the first integral in equation

(36). The inequality (1 + λ2cq2y + λ2abq2z)C 44 ≪ B2z isstill correct in the domain qz ≪ γ/b, qy ≪ 1/b forBz ≫ Φ0/λ2c . Thus, we can get the estimation (B4)also for the dilute case (a > γs). However, in contraryto the dense PV lattice, the contribution related to thefirst term in (B4) remains important:

I (a ≫ λJ ) = 4C 44γ

U 44b(λeff ab )2

µ1/b1/λc

dqyf (q2y)

+2π

λcλeff ab

µ2/b1/λcut

dqyf (q2y)

qy

1 +

C 66

U 44q2y. (B6)

Here, we have introduced numerical parameters µ1 andµ2, since the upper limits of integration are not well de-fined. The corresponding contribution to the energy isobtained from the last equation for f = Φ2

0/32π3 as pre-sented in the text.

∗ On leave from All Russian Electrical Engineering Institute,111250 Moscow, Russia

+ On leave from Faculty of Sciences, University of Mon-

tenegro, PO Box 211, 81000 Podgorica, Montenegro, Yu-goslavia

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(1990); S. N. Artemenko and A. N. Kruglov, Phys. Lett. A143, 485 (1990); J. R. Clem, Phys. Rev. B 43, 7837 (1991).

4 L. Bulaevskii and J. Clem, Phys. Rev. B 44, 10234 (1991).5 D. Feinberg, Physica 194C, 126 (1992).6 L. N. Bulaevskii, M. Ledvij, and V. G. Kogan, Phys. Rev.

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13 M. Benkraouda and M. Ledvij, Phys. Rev. B 51, 6123(1995).

14 D. A. Huse, Phys. Rev. B 46, 8621 (1992).15 L. N. Bulaevskii, S. V. Meshkov, D. Feinberg, Phys. Rev.

B 43, 3728 (1991).16 E. H. Brandt, J. Low Temp. Phys. 26, 735 (1977); the

expression of C 66 for a0 > λab was presented, for instance,in G. Blatter, V. Geshkenbein, A. Larkin, H. Nordborg,Phys. Rev. B 54, 72 (1996).

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

18 V. G. Kogan and L. J. Campbell, Phys. Rev. Lett. 62, 1552(1989).

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20 A. E. Koshelev, Phys. Rev. B 48, 1180 (1993).21 I. F. Voloshin et al., JETP 84, 1177 (1997).22 One of the possible physical pictures of the replacement

of the “shifted” PV lattice by the “trapped” PV latticeis the sequence of the phase transitions. At each of thosetransitions, the distance L between the nearest JV rows,occupied by PV rows, increases by aJ and takes valuesL = aJ (lattice in Fig. 3b), L = 2aJ (every second JV rowis occupied), etc.

23 The structural transitions between the Josephson vortexlattices with different periods4, which are related to thelayerness of the medium, cause the nonmonotonic depen-dence of aJ on the in-plane field28. Such behavior of theJV sublattice can induce the complicated alternation of the “shifted” and “trapped” states.

24 For simplicity, we have assumed that the demagnetizationfactor nz is very close to unity, i.e., the lock-in transitionand Meissner state are far below the considered region of Bz.

25 A. M. Thompson and M. A. Moore, Phys. Rev. B 55, 3856(1997).

26

S. Ooi, T. Shibauchi, N. Okuda, and T. Tamegai, Phys.Rev. Lett. 82, 4308 (1999).27 J. Mirkovic, E. Sugahara, and K. Kadowaki, Physica

284B-288B, 733 (2000).28 J. Mirkovic, S. E. Savel’ev, E. Sugahara, and K. Kadowaki,

Phys. Rev. Lett. 86, 886 (2001).29 M. Konczykowski, C. J. van der Beek, M. V. Indenbom,

and E. Zeldov, Physica 341C-348C, 1213 (2000).30 J. Mirkovic, S. Savel’ev, E. Sugahara, and K. Kadowaki,

Meeting Abstracts of the Physical Society of Japan, vol.55, issue 2, part 3, p. 504, 55th Annual Meeting, September22-25, 2000; J. Mirkovic, S. Savel’ev, E. Sugahara, and K.Kadowaki, Physica C, to be published.

31 S. Ooi, T. Shibauchi, K. Iteka, N. Okuda, T. Tamegai,

Phys. Rev. B 63, 20501(R) (2001).32 J. Suzuki, N. Metoki, S. Miyata, M. Watahiki, M. Tachiki,K. Kimura, N. Kataoka, and K. Kadowaki, Advances inSuperconductivity XI, proc. of the 11th International Sym-posium on Superconductivity (ISS’98), November 16-19,1998, Fukuoka, Japan, p. 553 (Springer Verlag).

33 Note that the anisotropy parameter γ of BSCCO canchange in quite wide range depending, in particular, onthe oxygen stoichiometry (see for instance G. Balestrino et al., Phys. Rev. B 51, 9100 (1995)).

15

Documents

S.E. Savel’ev, J. Mirkovic and K. Kadowaki- The London theory of the crossing-vortex lattice in highly anisotropic layered superconductors