Vortex structures in anisotropic superconductors at low inductions

PHYSICA

Physica C 201 ( 1992 ) 369-378 North-Holland

Vortex structures in anisotropic superconductors at low inductions A. S u d b o A T& T Bell Laboratories, Murray Hill, NJ 07974, USA

Received 4 August 1992

The stability of a simple unpinned, possibly distorted, hexagonal flux-line lattice (FLL) is examined in anisotropic superconductors at oblique angles 0 z (B, ~) in the low induction regime b = B / B c2<< 1. The normal modes of the FLL are strongly dispersive due to the long-range nature of the vortex-vortex interaction. For 0¢ 0, they may even become soft away from the zonecenter, signalling a structural instability of a distorted hexagonal FLL. Exact total energy calculations on FLLs with a basis are performed and compared with the hexagonal FLL. The simple hexagonal FLL does not always represent the lowest energy state. In particular, vortex-chains embedded in an Abrikosov lattice, which may be considered as a FLL with a basis, have a lower energy than a simple hexagonal FLL at large angles 0 and low enough inductions. Anisotropic London-theory is thus shown to provide a useful framework for discussing novel types of vortex-structures in type-II superconductors.

1. Introduction In the context o f high-To superconductivity, the properties o f the flux-line lattice ( F L L ) in type-II superconductors have received renewed attention. Due to extreme m a t e r i a l p a r a m e t e r s a n d pronounced anisotropy o f these materials, novel and unexpected properties o f the F L L s are found. Since the discovery, by Bitter p a t t e r n d e c o r a t i o n experiments, o f apparently thermally decorrelated flux lines in the high-To superconductors YBa2Cu307 ( Y B C O ) a n d Bi2.2Sr2Cao.sCu2Os ( B S C C O ) even at quite low t e m p e r a t u r e s [ 1 ], a central a n d theoretically interesting issue has been whether or not a F L L m a y melt [2 ]. Several groups have tackled this p r o b l e m [ 2 4 ]. It is well established that the F L L in anisotropic extreme type-II superconductors is intrinsically soft due to the long range o f v o r t e x - v o r t e x interactions [ 3-5 ]. As a consequence, estimates based on a Lind e m a n n criterion show that t h e r m a l decorrelation o f flux line positions m a y be expected over a large portion o f the (H, T) p h a s e - d i a g r a m [ 3,4 ]. M o r e recent d e c o r a t i o n experiments [ 6 ] at low T revealed fascinating, exotic vortex arrangements (vortex chains e m b e d d e d in a vortex lattice) when the a p p l i e d field was oriented at oblique angles with respect to the crystal ~-axes. This observation was not

expected from earlier work on anisotropic superconductors, which for very small B p r e d i c t e d vortex chains without a F L L in between [ 7 ]. This conclusion was reached by c o m p a r i n g the energy o f an isolated vortex with the energy per vortex o f a vortexchain structure. Due to anisotropy, the field-component surrounding one single vortex m a y change sign along the vortex-direction in a certain direction ( p e r p e n d i c u l a r to the v o r t e x ) at a given distance from the flux line ( d e t e r m i n e d by mass-anisotropy and a m o u n t o f tilting). This could give rise to an attractive v o r t e x - v o r t e x interaction along this direction p e r p e n d i c u l a r to the vortices, thereby stabilizing vortex-chains c o m p a r e d to an "ideal gas" o f r a n d o m l y d i s t r i b u t e d vortices. The observations in ref. [ 6] raise the m o r e interesting question o f whether a " s i m p l e " F L L o f interacting vortices m a y be unstable due to the anisotropic character [8,9] o f the long-range v o r t e x vortex interaction. This is a very different p r o b l e m from that discussed in ref. [ 7 ]. An i m p o r t a n t issue is the identification o f a m i n i m a l theoretical framework within which well-defined quantitative calculations can be done, and which will account for novel types o f vortex-structures, for instance such as observed in ref. [ 6 ]. Most treatments o f the elasticity o f the F L L in an-

0921-4534/92/$05.00 © 1992 Elsevier Science Publishers B.V. All fights reserved.

370

A. Sudbo / Vortex structures at low inductions

isotropic superconductors applied to thermal fluctuations of the FLL, treat the situation where the induction B is oriented parallel to the £-axis of the uniaxial compound. In this case, the form of the elastic energy of the FLL, which is used to determine the phonon-propagator, is particularly simple. At oblique orientations of the induction, the situation is considerably more complicated. The elastic energy has a more complicated form and contains more terms. Furthermore, the definition and calculation of the elastic moduli is more difficult. Recently [ 8 ], progress has been made in the large induction regime, where the angular dependence of the thermal and pinning-caused fluctuations has been derived from scaling arguments. In this paper, we address the issue of the stability of a class of hexagonal FLLs so far believed to represent the (unpinned) vortex ground state in an anisotropic type-II superconductor. We will demonstrate the instability of these proposed ground states at sufficiently low inductions in very anisotropic superconductors at large angles 0 = / (B, £), provided that the nonlocal character of the anisotropic vortex-vortex interaction is accounted for properly. We emphasize that the full anisotropic interaction between vortices is accounted for exactly. Throughout this paper we use nonlocal, anisotropic London theory, which is valid when b=B/Be2 < 0.25 and K>> 1. Here, K is the Ginzburg-Landau parameter and Be2 the upper critical field. The region of very low reduced induction b=B/B¢2 << 1 is of particular importance in high-T~ superconductors due to their large values of Bc2. Even the apparently simple case of an ensemble of rigid, parallel flux-lines inclined with respect to the ~-axis of the superconductor is a highly nontrivial one. Within this framework it will be demonstrated that at sufficiently low inductions, a FLL with a basis may be energetically favourable compared to a simple hexagonal lattice.

F= ~

~ f ~ dr~dr~V~(r,-rj) .

Here Vap(r) is the nonlocal interaction potential between any two infinitesimal vortex segments in the system, ( a , fl)e (x, y, z) and q~o=2.07× 10 -7 G c m 2 is the flux-quantum. Expanding this energy to second order in the displacements u (k) of the flux lines from their equilibrium positions in the ground state, we find the excess energy due to these displacements, (" d3k

~=

½j ~

us( -k)~p(k)up(k),

(2)

where now (or, fl)e (x, y) (flux lines along z). The elastic matrix O~p(k) is given by [9] B2

O~,(k) = ~ ~Q [f~(k+Q) -f~,(Q) ] , f~,(k ) = k 2 f'~,(k) + k~k a Vzz(k ) - 2kzk, f'z~(k) .

(3) The sum runs over the reciprocal lattice vectors of the FLL in its appropriate ground state. The last term of the tensor f~a(k) in eq. (3), first considered by Sardella [ 9 ], vanishes for all k when B I[~; then only one shear modulus of the FLL exists. At oblique angles, however, this term gives rise to an additional anisotropy in the shear stiffness if one goes beyond the local limit k ~ 0 discussed in re£ [ 11 ], but does not influence the tilt moduli of ref. [ 9 ]. In this paper, we will consider the simple, but as it turns out highly nontrivial, case of rigid vortices, in which case kz= 0 andrea(k) takes the simple form

f,~,(k) =k,~k, Pz~(k±),

(4)

where k± = (k~, ky). The above expressions are quite general and make no reference to specific values of the reciprocal lattice vectors, which only should belong to an equilibrium FLL. For a uniaxially anisotropic London superconductors the Fourier-transformed interaction tensor V~a(k) is given by [9,10]

2. Soft shear moduli and normal modes

V~p(k) The total energy of an arbitrarily distorted system of unpinned vortices may be derived from the anisotropic London equations [ 10 ],

(1)

exp(-2g) (~_ l+A,k2

A2q~qp

.'~, (5)

1+ A l ~ E q 2 ]

where A 1 :'~ab, 2 2 A2 =)'c2 --'~ab, q = k × ~ , 2ab and 2~ are the magnetic penetration depths parallel and perpendicular to the basal ab-plane, respectively, and

A. Sudbo / Vortexstructuresat low inductions ~ab and ~ are the corresponding coherence lengths. One has 2c/~tab=~ab/~c=N//--M--~, where M J M is the mass-anisotropy. The cutoff factor in eq. (4) contains g = ~2bq 2 + ~2 (k. £) 2, which means that the lattice sum is elliptically cut off at Q-vectors of the order of the core-size. In the isotropic case this cutoff was shown to reproduce the results of GinzburgLandau theory [ 5 ]. For uniaxial anisotropy and not too small b >> 1/2~¢2, the equilibrium FLL predicted by [ 11 ] has reciprocal lattice vectors Qm, = nQl + mQ2, where QI = 2n [ x / ~ + y2~]/x/~ay, Q2=anTYc/x/~a, a2=2Oo/x/~B, and y4=cos20+M/Mz sin20. Here, 0 is the angle of the induction B away from the ~-axis of the uniaxial superconductor. The various elastic moduli of the FLL, derivable from eqs. (2) and (3), are central quantities in the statistical mechanics and dynamics of the FLL and hence deserve special attention. In particular, the tilt modulus c44(k) and shear moduli are crucial in theories of thermal fluctuations [2-4] and collective pinning of the FLL [12]. The bulk modulus clt(k) normally plays a less prominent role, entering the theory of collective flux creep at T > 0 [ 13 ] but not the collective pinning theory at T = 0 [ 12 ]: the thermally activated "jumping volume" depends on the bulk modulus, whereas the static "correlated volume" does not. A detailed discussion of dispersive tilt-moduli of the FLL in anisotropic superconductors has been given elsewhere [8,9]. Their main term originates from the overlap of the vortex fields, has lorentzian dispersion, and is positive at all k. Hence, nonuniform tilt waves do not induce a structural instability of the FLL. This is a general result, independent of the specific FLL structure. In most considerations of the elasticity of the FLL, it is tacitly assumed that the shear modulus of the FLL is essentially dispersionless in the dominating part of the BZ. Given the importance of the dispersion of the tilt moduli and the large contribution of k-values close to the BZ boundary to the thermal fluctuations of the FLL [ 3 ], it is fair to check to what extent the equally important shear moduli really are dispersionless. These moduli depend on the structure of the FLL, and yield important information about its stability. For an isotropic superconductor, the shear modulus and its geometric dispersion was

371

calculated numerically in ref. [5 ] from the elastic matrix ~,,p(k), using the definitions c66 (k) = ~ ( 0 , k, 0 ) / k 2 and c66(k ) = t2byy(k, 0, 0 ) / k 2, which are obvious from comparing eq. (3) with the expression for the elastic matrix in continuum theory, • otfl(k) =

[C,l (k) -c66(k)]k~k, -bJaa[c66(k)(k2x+k2)+c44(k)k 2] .

In

(6)

isotropic

superconductors, the assumption is indeed quite reasonable for k < 0.3kBz, where kBz is the radius of the circularized Brillouin-zone (BZ), k~z=4Bn/~o. For the case 0= 0 this finding remains true even in uniaxially anisotropic superconductors. For 0 # 0 there are two shear moduli, c~g~ and c~h~ (easy and hard) [ 11 ]. These two moduli may be extracted from the elastic matrix ~ p ( k ) as

c66(k)~c66(0)

c~g)(k)= ~xx(O, k, 0) k2

c~(k)_

qbyy(k, 0, 0) k2

(7)

This may again be seen by comparing eq. (3) with the Fourier-transform of the general elastic energy of a FLL of rigid vortices (since kz=0 for pure shear) when also a torsion term describing the coupling of the FLL to the underlying uniaxial crystal is included [ 11 ]. The shear moduli c ~ ( O ) and c ~ ( O ) as computed from eq. (6) at the zone-center k = 0, agree with the results of ref. [ 11 ], where a different cutoff scheme to truncate the lattice sum at the vortex core was used. The results for M J M = 2 , 5 and 25, b=0.0001 and x = 5 0 are shown in the inset of fig. 1. For k = 0 , both c~g~(k) and c~h6~(k) remain positive for all 0~[0, n / 2 ] . The small value of c~g~(O) for k = 0 and 0,~,n/2 is responsible for the existence of a whole class of nearly-degenerate hexagonal vortex structures with one vortex per unit cell [14,15]. We next consider the shear moduli at finite values of k. A surprising feature that we find is the pronounced dispersion of c~g)(k) when O#0. This is shown in fig. 1 for parameters x = 5 0 , b=B/ B¢2= 0.0001 and M J M = 3600. Note that in this case c~g ) (k) changes sign at kr7/Q~o ~. 0.62, when 0= 3n/

A. Sudbo / Vortex structures at low inductions

372

I

'

'

'

'

i

i

i

i

i

[

=

i

i

~

i

i

i

i

i

I

tices (kz = O) in the limit of small but finite k. Using eq. (6), we obtain to leading order in k

I

1.0

c~g)(k)_c~g)(O) ~ k 2 ~ Q2

-

--:t/8

-

c ~ ) ( k ) _ c ~ ) ( O ) ~ k 2 • Q2 04V=(Q) e

0.9_ v

O

OQ~----~-r ,

(8)

and

0.5

g

0417=(Q)

0Q 4

(9)

Explicitly, the tensor-component V=(Q) is given by 0.0 - 4

i "

l

Vlz

~- = 2 s ~ : 5

,

1

;

',

-

I

V = ( Q ) - l+2a2k 2 1

h

',

-

--

2 2 2 ' 2 (2zc--22ab)Q~xsinZO] 2 2 2 2 l+2cQx+AoQy J,

(10)

2 2 • 2 0+2cCOS 2 2 0. Thus, the anisotropic 2o=2abSln part of V=(Q) will depend more strongly on Qy than on Ox at large Q when 0 ~ = / 2 and M=/M>> 1, and the correction to the isotropic result is larger for cag)(k) than for ca6h)(k) in very anisotropic superconductors. The latter essentially retains its weak dispersion at 0=0, which is the same as for the isotropic case. Given the above results, we further investigate the stability of the FLL by considering its strongly dispersive normal modes. Regardless of the symmetry of the FLL or the orientation of the induction B, the FLL always has precisely two normal modes since only displacements perpendicular to the flux lines have physical significance. The normal modes g2-+(k) of the FLL are found by diagonalizing the matrix • ,#(k). At 0=0, the mode g2-(k) corresponds to a transverse phonon mode, which in the continuum limit k . << Kto=4n/v/3a is given by with

-0.5

--

0

I

I

F - ~ . ~

I I I i i 0.0

0.2

L

i

I

i

I

i

I

0.4

I

i

I

0.6

i

i

J

t

I i 0.8

ky

Fig. 1. Dispersion ofc~g)(k)/c~g)(k=O) at 0=0, ~/8, ~/4 and 3~/8, with x=50, MJM=3600 and b=B/B¢2=O.O001 (rigid flux-lines, k~= 0 ). Notechangeof sign in c~) (k) for Icy~Qm~ 0.62 at 0= 3n/8, signalling a structural instability of the FLL. Inset shows c~)(0) and c~)( 0) at k = 0, normalized to their values at 0=0, for three mass-anisotropies MJM=2, 5 and 25, at b=0.0001 and x=50. 8. Using the same parameters as above, but M~/ M = 2 5 , no change of sing in ckg)(k) is seen at any angle. The hard modulus c~6h) (k) is only weakly dispersive and remains positive down to the lowest fields (b=0.00001) that we have considered. The reciprocal lattice vectors Qm, which we use in calculating the matrix ~ p ( k ) and structure dependent elastic moduli, given below eq. (4), represent the global minimum of a class of variational solutions obtained by uniform deformation of trial lattice structures [ 14,15 ]. This is the reason why the structure dependent elastic moduli at k = 0 are found to be positive at all angles. As we have seen, c ~ ) (k) may become negative at finite k when 0 and M J M are sufficiently large and b is sufficiently small. This finding hints at a structural instability with respect to periodic shear of the FLL defined by the above given Qm,. The origin of the strong and weak dispersion of c ~ ) (k) and c~h) (k) respectively, when 0 # 0, may be understood qualitatively by considering rigid vor-

~Q- ( k ) = c 6 6 ( k ) k 2 + c 4 4 ( / ) k z

2 .

(1 1 )

The hard mode Q+ (k) in this geometry corresponds to a longitudinal mode which, however, goes soft at k = 0. This is due to screening of the magnetic field by supercurrents, and a resulting exponentially small interaction potential at large enough distances. However, at finite k it always remains hard, even at 0~ 0 (see below); it will not be considered further, the transverse mode is the dominant mode of fluctuations of the FLL at 0= 0 and in this case also remains hard in the entire BZ away from the zone-center. This is expected from our discussion of c~g)(k) at 0=0, which demonstrated that the FLL is robust against uniform as well as nonuniform shear deformations

A. Sudbo / Vortex structures at low inductions

in this geometry, as is also the case for tilt and compressional deformations [ 9 ]. We now turn to the more interesting situation where 0 # 0. From our consideration of the modulus c~g>(k) we are led to consider t 2 - ( k ) at 0=3n/8; 12÷ (k) remains hard, and will not be considered further. In fig. 2, we show 12- (k) at b=0.0001, x = 5 0 for Mz/M= 25 and 3600. The most important feature is that this mode becomes completely soft at finite k-vectors in the BriUouin-zone when 311=/ M = 3600, but not for Mz/M= 25. Further, in fig. 3 we show 12-(k) for the parameters M=/M=3600, x = 50 and 0= 3n/8 for variaous values of be [0.0001, 0.0004]. While the mode is soft at finite k for b=0.0001 and b=0.0002, this is not the case for the largest inductions. These results show that the instabilities of a hexagonal FLL only develop at very low inductions in anisotropic superconductors provided also that the vortices are tilted well away from the iaxis. It is not necessary to invoke surface effects or boundary conditions to see these instabilities.

~_

kyI

I 3 ~

I-

Y

b : o.oool

/

1

j

\\

,

/ 1\

o.oo

I

F

.J

X

Fig. 2. Normal mode 12- (k)/No along three symmetry directions in the BZ for rigid flux lines ( k z = 0 ) and mass anisotropies M J M = 2 5 and 3600 at 0=3n/8, with No=B2/4x. We have used b= B/Bc2 = 0.0001, and x = 50. The normal mode is plotted with kx in units of Klor and ky in units of Klo/7, with 7= (cos20+M/ M:sin 2 O) 1/4 , and Klo= 4n/x/~a.

373

0.06 0

=

_

_

4 2

Mz 36oo

0.04

/-b

= 0.0004 0.0003

I I ^ \~./.

III \kV

0.0002

0.02

0.00

-0.02 r

I J

I X

F

Fig. 3. Same as fig. 2, for one value of M=/M= 3600 with reduced induction b=B/B¢2=O.O001, 0.0002, 0.0003 and 0.0004. Only at the two lowest inductions is t2- (k) soft for finite k.

The value o f t 2 - (k)/(B2/4~) is controlled by the vortex-field overlap-parameter 70 =2=bK~o2 z = 4xbx2/x/~. Structural instabilities of the FLL at too large values of 70 are not possible, since substantial overlap of vortex-fields render the effective intervortex interaction uniformly repulsive. In this regime of larger inductions 0.0002 << b < 0.2 we therefore expect a stable hexagonal FLL at all angles 0. This is consistent with recent results ofref. [ 8 ], which in this field regime treated the nonlocal elasticity at oblique angles of the induction by scaling transformations of the isotropic case. The instability we find, at finite k and b << 1, implies a vortex ground state of a different nature than those discussed in refs. [ 14] and [15], which considered ground states which were stable to uniform deformations. A particular possibility is one where the vortices form a FLL with a basis, which we now investigate.

3. Ground state energies The exact ground state energy of a simple FLL of rigid flux lines per unit length and per vortex is given by (in the thermodynamic limit)

A. S u d b o / Vortex structures at low inductions

374

F

B 2 K"

LNv - 8re ~

V=:(Q)

'

(12)

where the relevant component of the interaction-tensor is given explicitly in eq. (10), Nv is the total number of vortices in the system, and L is the thickness of the sample. For a brief discussion of this expression, it is instructive to rewrite ~'=(Q) slightly, to the form

will be small compared to the previous two. A detailed discussion of this last term was given in ref. [ 14] (see also references therein), for the case where vortices in a simple FLL (one vortex per unit cell) were inclined to lie almost in the ab-plane. For a FLL with a basis, the above expression is modified to F

B2

L N~ Nb

--

8n

5"

v~ ~ ( Q

'

)f(Q

'

)

(14)

.

1

v:--(Q)= 1 +2oQy+2~,Qx 2 2 2 2 x (sin20+cos20 i1~q_~Q2 .]"

(13)

Since Q2~ B, it is seen that in hard superconductors x>> 1 at inductions b ~ 0.1, the leading term in this sum is the Qm.=0-term (when the sum is cut off at the vortex-core), in which case the summand is V=(Q) = 1. Irrespective of mass-anisotropy and orientation of the FLL, the energy per vortex per unit length (in units of B2/4rc) is therefore always approximately 0.5. The details of the FLL structure gives corrections of order ~ 1/x2b, which will be enhanced at lower inductions, but are O( 10 -2) when x = 100 and b=0.01. The distinction between various FLL structures is in general thus a subtle one. Moreover, V~=(Q) written in the above form reveals an important fact which is not apparent from eq. (10), namely that the interaction in Q-space is always positive, regardless of vortex-orientation and mass-anisotropy. From eq. (12), the total energy per vortex of the system is thus guaranteed to be positive, in compliance with the Meissner-effect. Through the work of refs. [ 11-14-15 ] it is now established that the energy in eq. (12) is minimized by a simple (possibly distorted) hexagonal FLL at all angles 0, defined by the reciprocal lattice vectors given below eq. (5). The energy eq. (12) may in general be decomposed into three distinct contributions. ( 1 ) A term B2/8~ associated with the magnetic energy density of the average induction in the superconductor; (2) a term given by the self-energy of each vortex multiplied by the areal density of vortices, the "ideal vortex-gas" contribution; and (3) a term arising from the anisotropy of the vortexoverlap due to FLL structure, and which in most cases

Here, Nuc is the number of unit cells and Nb is the number of vortices in the basis. The Q's run over the reciprocal lattice vectors of the superlattice being used, while f ( Q ) is a structure function determined by the basis of the FLL, f ( Q ) =Z'j exp (iQ'ri), where ri represent the positions of the vortices in the basis. We consider three cases: (1) simple distorted hexagonal FLL found in ref. ( 15 ), in which case eq. (14) reverts back to eq. (12), (2) simple distorted hexagonal FLL with one extra vortex embedded between every vortex in every Nth row in the rows running parallel to the ab-component of the tilted field (y-direction), (3) same as case (2), but with rows running perpendicular to the ab-component of the tilted field (xdirection). Case (2) is a FLL with a N + 1-point basis, case (3) a FLL with a 2 N + 1-point basis. Cases 2 and 3 are illustrated in figs. 4 (a) and (b). For case ( 2 ), we find for the structure factor

f(Qmn) = e i~"+N =e

inn

e2in'n/N=1

,

e2ir~m/N # 1 ,

(15)

whereas for case (3), we have

f(Qmn)=ei~n+N[l+e = e inn

i,~m]

e2ir~n/N=l , e2inn/N=/~: 1 .

(16)

In fig. 5, we show the ground state energy at 0=0, 3•/8, and M d M = 25 as a function of induction. We have chosen the value of N = 6 in rough correspondence to the experimentally observed interchain distance at 0 ~ 7 0 ° (however, see comments below). Moreover, the induction used in each case is scaled by a factor o f x / N / ( N + 1 ) in case (2) and a factor of x/2N/(2N+ 1 ) in case (3), as compared to the

A. Sudbo / Vortexstructures at low inductions (a)

?(:~x§

Y

= § x (;,x §) (b)

X

y y=Bx(cxB)

Fig. 4. Two cases ofa FLL with a basis. Figure 4(a) shows a hexagonal FLL with chains running parallel to the projection of the tilted magnetic field onto the ab-plane. Figure-4(b) shows the situation where the chains run perpendicular to the direction of the projection of the magnetic field onto the ab-plane. The former represents a FLL with an N+ 1-point basis, whereas the latter forms a FLL with a 2N+ l-point basis. case o f a simple FLL, in order to preserve the overall density o f vortices in all cases. We find that case (2) has a lower energy than case ( 1 ) at low inductions, but not case (3). Further, since this result applies for M J M = 25, while the normal modes were hard for this mass-ratio in the entire Brillouin-zone at this induction, a barrier must exist for reconstruction at this value of Mz/M. The simple hexagonal FLL is found to be stable at 0 = 0. This explicitly demonstrates that vortex structures qualitatively similar to the ones observed in ref. [ 6 ] may be stabilized at low inductions within the framework o f bulk anisotropic L o n d o n theory. This is due to in-

375

terference effects between the factors f(Qmn) and ~=(Qm,) in eq. (10), which alter the b-dependence, and in fact also the 0-dependence, o f the lattice sum compared to the simple hexagonal FLL. The angular dependence o f the energy is shown in fig. 6. It is seen that F/LNv in units of B2/4x has a m i n i m u m at 0 = x / 2 due to the softening o f the interaction potential evident from eq. (10). However, a simiale physical argument for why this more complicated vortex-structure is (slightly) lower in energy than the simple FLL, seems to be elusive. The effect is clearly a subtle one, arising only at low inductions, where the interference effects from the structure-dependent terms in the lattice sum become appreciable. That this is the case is not surprising, since even the distinction between simple FLL-structure (such as rectangular and hexagonal) is a subtle one. We have not yet addressed the issue of how to determine the parameter N (interchain distance) from our approach. However, some insights may be gained by considering the exact real-space many-body potential with which the tilted, rigid vortices interact. This potential is the Fourier-transform o f eq. (10), alternatively eq. (13). The contribution to this quantity originating in the isotropic part o f eq. (10) is calculated easily, yielding a zeroth order modified Bessel function Ko(r). The integrations involved in the Fourier-transformation of the anisotropic part are apparently difficult to perform explicitly, but may be reduced to an auxiliary integral over a finite region through the use of Feynman's relation [ 16 ]

I F, F2 -

! l

du

[ F , u + F 2 ( 1 - u ) ] 2"

(17)

Using eq. (10), we then obtain the following expression for the anisotropic vortex-vortex interaction f d 3k/( 2n ) 3 Vzz(k) exp ( ik- r ) at arbitrary inclination o f the vortices with respect to the &axis:

V=(x, y, O, M J M ) = ~

1

Z~Z~b

2

2

2

"

[Ko(r)

2

_ ~.ab(,~c--2aO)Sln 0 G(X, y, O, M z / M ) ] 2 1

du a ( x , y, O, M J M ) = f a 3 ( u ~ ( u ) 0

376

A. S u d b o I Vortex structures at low inductions

2.0

(a)

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I I III1[

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iii

(b)

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t

K=50

= 25

t .i~

N=6

.J

Mz/M

0 = 0

SIMPLE ,5 FLL CASE 2 .......... CASE 3

K=50 N=6

---

~" 1.5

1.5 -iI -I il

2

0=3

Mz/M=25

%

~_ 4 2

----

SIMPLE ,5 FLL CASE 2 .......... CASE 3

/

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I I I[Pl

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5

/J ..~1

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/~

./'/

..f,> ...."S''" • 0.5 10

i 100

1000

I 10000

0.5

I [IkJ

10

100

1000

- (c)

10000

(B/Bc2) "1

(B/Bc2) -1

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///:."

M z, M = 2 5

1.6 ~-

K = 50

/.S-"

N = 6

----

%

/.~...."

SIMPLE A FLL CASE 2

.......... CASE 3

,z///''" / ~£ ", 1/f//

1.4

2

115111111

g z

ii.., ...

ill

ii. ... 14.,.. 14.... /i...... I....

1.2

1.0 2000

i

i

5000 (B/Sc2) "1 X2

Fig. 5. Ground state energy E/LN,cNb (in units of B2/4n) as a function of Bc2/Bfor three types of vortex-structures described in the text at two angles 0=0, 3~/8 for MJM=25 as a function of induction. At O=3n/8, the hexagonal FLL is unstable, for low enough induction, where the energy for case (3) (dotted line) crosses the energy for the simpled hexagonal FLL (solid line). (a) 0=0, (b)0= 3~/8, (c) 0= 3x/8 showing a blowup for lower inductions, to resolve the details of the energy-crossings. Case (2) and (3) refer to the tWOcases of different basises of the FLL which are being used; details are explained in the text. The value of N was chosen to be 6 in all cases. a2(

2 2 U) =~c2 "~-U(2ab--2c ),

b2(u)=22

Here, Ko and KI are m o d i f i e d Bessel functions o f zeroth and first order, respectively, and we have introduced

x2 y2 r2_--7~- + - - ,

;ta~

X2 p

2

a(u) 2

22b y2 4 - - -

b ( u ) 2'

+u(22ab-22)

.

(19)

N o t e that although V = ( Q ) is positive at all Q, regardless o f vortex-orientation, this is not the case for the real-space expression. At 0 = 0 the potential is isotropic and repulsive. On the other hand, for instance at 0 = 3 n / 8 and Mz/M=25, long ridges o f large V=(x, y, O, M . / M ) have developed along the axis around w h i c h the field has been tilted (x-axis). In contrast, m i n i m a develop along the y-direction,

A. Sudb~ / Vortex structures at low inductions

_--- ~ .~'~.;¢.~.~. t

I

i

t

m

m

I

t

m

t

t

I i

2.0

~ -

~

.

'

Mz/M = 25

K = 50 N=6 b = 0.000345

" ~

x~. ~x SIMPLE A FLL - - - CASE 2 .......... CASE 3

1.5

2

N~.~x N~x "~\

Z 1,0

0.5

L

0.0

i

t

i

[

0.5

i

t

t

o

t

I

1.0

I

t

I

t

I

1.5

Fig. 6. Angular dependence of the energy E/LNucNbin units of B2/4~ at B/Bc2=O.O00345,which is the largest induction where one sees that the FLL with a basis has lower energy than simple FLL for Mz/M=25. Note that a critical angle 0 is necessaryfor the stabilization of the vortex structure in case ( 3 ) (dotted line ) tO o c c u r .

where Vz=may even become negative and hence give rise to vortex-vortex attraction. Considering the case where the chains run parallel to the projection of the tilted field on the ab-plane (which seems qualitatively to correspond to what is experimentally observed), the value of N is presumably determined as a result of a tradeoff between the energy gained by aligning vortices along the soft direction of the interaction potential and the energy lost by the relatively hard inter-chain interaction. This alignment (binding of vortices into chain-structures) may occur for inductions which correspond to intervortexdistances which are smaller than the intervortexspacing which leads to an attractive interaction arising from V=(x, y, O, Mz/M) in eq. (18). The point is that this new type of vortex-structure is bound with respect to the conventional Abrikosov-lattice of interacting vortices, but not necessarily with respect to the irrelevant case of an ensemble of non-interacting vortices. An attraction is therefore no required for binding of vortices into chains, as was argued in ref. [7]. For vortex-chain formation to occur in the background of a conventional Abrikosov-lattice of interacting vortices, it is only required that there ex-

377

ists a direction in which the vortex-vortex interaction is relatively soft. This is in fact a general feature of the potential in eq. (18 ) at any finite angle 0. In principle~ chain formation should therefore occur]n an unpinned FLL at any angle, and at any mass-'anisotropy, although they may be impossible to observe due to their dilution. For very low inductions, it is also clear that pinning will intervene seriously with the above description due to the (exponentially) small shear-stiffness of the FLL. More work is required to understand the details by which the system picks a specific field independent length scale (N) in the direction parallel to the axis around which the field is tilted, and not perpendicular, like one naively would have guessed. Anisotropic bulk London-theory even for the simple case of rigid flux lines appears to provide a well-defined quantitative framework for such studies. Finally, we mention the possibility, of thermally driven reconstruction of a FLL, at given Mz/M, 0, and b = B/Bc2. The lattice sums in eqs. (12) and (14) depend on induction and x only through a specific combination 4nx2b/v/3, which is denoted as the vortex-field overlap parameter [ 5 ]. Lowering the induction to produce the instability of a hexagonal FLL seen in fig. 5, is equivalent to fixing the induction and lowering the temperature, assuming only that x is roughly temperature independent in the relevant temperature regime, and Bc2(T) is a decreasing function of the temperature T. Decoration experiments performed at different temperatures, but otherwise identical parameters, may reveal such a reconstruction, provided that B (napplie d ) is appropriately chosen to prevent melting of the FLL to intervene.

4. Conclusion

In summary, we have discussed the structural instability of a class of hexagonal FLLs in uniaxial superconductors at oblique orientations of B in the extreme low-induction regime. This instability is induced by nonuniform shear deformations of the FLL. The ground state energy calculations performed here indeed show that simple hexagonal vortex structures may be less stable than more complex ones, although we have not performed a stability

378

A. Sudbo / Vortex structures at low ~nductions

analysis o f the n o v e l types o f structures we h a v e proposed. Hence, we do n o t yet k n o w what the correctly r e c o n s t r u c t e d g r o u n d state really is. A m o r e direct a p p r o a c h to the precise d e t e r m i n a t i o n o f such a global g r o u n d state at o b l i q u e angles, c u r r e n t l y u n d e r investigation, is r e p r e s e n t e d by the u t i l i z a t i o n o f m o lecular d y n a m i c s m e t h o d s a p p l i e d to r a n d o m i n i t i a l pin-free vortex c o n f i g u r a t i o n s i n t e r a c t i n g with the real-space v e r s i o n o f the a n i s o t r o p i c L o n d o n p o t e n tial. H o w e v e r , the m a i n c o n c l u s i o n o f this p a p e r is that a n effective-mass theory like a n i s o t r o p i c Lond o n - t h e o r y e v i d e n t l y does p r o v i d e a f r a m e w o r k for discussing flux-line lattices with exotic vortex structures, n o t o n l y c o n v e n t i o n a l Abrikosov-lattices.

Acknowledgements T h e a u t h o r acknowledges useful discussions with E.H. B r a n d t , P.L. G a m m e l , H.F. Hess, P.C. H o h enberg, D.A. Huse, M.F. N e e d e l s a n d C.M. R o l a n d .

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[ 3 ] E.H. Brandt, Phys. Rev. Len. 63 ( 1989 ) 1106; A. Houghton, R.A. Pelcovits and A. Sudbo, Phys. Rev. B 40 (1989) 6763. [4 ] M.V. Feigel'man, V.B. Geshkenbein and A.I. Larkin, Physics C 167 (1990) 177. [ 5] E.H. Brandt, J. Low Temp. Phys. 26 (1977) 709; ibid., 26 (1977) 735; ibid., 28 (1977) 263; ibid., 28 (1977) 291. [6] C.A. Bolle, P.L. GAmmel, D.G. Grier, C.A. Murray, D.J. Bishop and A. Kapitulnik, Phys. Rev. Lett. 66 (1990) 112. [7] A.M. Grishin, A.Yu. Martynovich and S.V. Yampol'skii, Sh. Eksp. Teor. Fiz. 97 (1990) 1930; A.I. Buzdin and A.Yu. Simonov, Soy. Phys. JETP Lett. 51 (1990) 191. [8] G. Blatter, V.B. Geshkenbein and A.I. Larkin, Phys. Rev. Lett. 68 (1992) 875. [9] A. Sudbo and E.H. Brandt, Phys. Rev. Lett. 66 ( 1991 ) 1781; E. Sardella, Phys. Rev. B 45 ( 1991 ) 3141. [ 10 ] W. Barford and J.M.F. Gunn, Physica C 165 ( 1988 ) 515; E.H. Brandt, Physica B 165-166 (1990) 1129. [ 11 ] V.G. Kogan and L.J. CampbellSPhys. Rev. Lett. 62 (1989) 1552. See also ref. [ 15]. [ 12 ] A.I. Larkin and Yu.N. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [ 13 ] M.V. Feigel'man, V.B. Geshkenbein, AJ., Larkin and V.M. Vinokur, Phys. Rev. Lett. 63 (1989) 2303. [14] B.I. Ivlev, N.B. Kopnin and M.M. Salomaa, Phys. Rev. B 43 (1990) 2896. [ 15] L.J. Campbell, M.M. Doria and V.G. Kogan, Phys. Rev. B 38 (1988) 2439; K.G. Petzinger and G.A. Warren, Phys. Rev. B 42 (1990) 2023. [ 16] R.P. Feynman, Phys. Rev. 76 (1949) 769.