Elastic properties of the flux line lattice in high-Tc superconductors

Pmlgll

Physica C 213 (1993) 43-50 North-Holland

Elastic properties of the flux line lattice in high-To superconductors S. N i e b e r a n d H. K r o n m i i l l e r Max-Planck-lnstitut J~r Metallforschung, Institut f~r Physik, Heisenbergstrafle I, 1)-7000 Stuttgart 80, Germany Received 14 February 1993

The elastic matrix, governingthe non-local elastic properties of the flux line lattice, is derived within the anisotropic London theory for an arbitrary arrangement of straight or curved vortices in uniaxial high-To superconductors. The non-local elastic moduli for compression, shear and tilt of the flux line lattice in YBa2Cu3OT_6and BizSr2CaCu2Os+6are numerically calculated for magnetic fields of various strengths applied parallel and perpendicular to the c-axis. The conventional hexagonal flux line lattice becomes soft for certain compression and shear modes with wave vectors near the boundary of the BriUouin zone. A comparison of the numerical results with the isotropic approximation is made and analytical expressions for the line tension of single flux lines oriented along one of the main axes are given.

1. Introduction

An essential feature of the high-To superconductors (HTSC's) is a pronounced anisotropy induced by their fundamental building blocks, the CuO-multilayers [ 16,24 ]. This anisotropy is characterized by the ratio between the effective masses for electron motion parallel to the c-axis (m ~*) and within the abplane ( m a b ) . I ' 2 - - m c / m a b , and is adequately accounted for by the anisotropic London theory [2,9,12]. Strictly speaking, the London theory is valid for magnetic inductions B much smaller than the upper critical field He2 only. In practice, however, this constraint imposes no limitation since H~2, due to the small coherence lengths of HTSC's, is typically of the order of 102 T. Moreover, by taking the temperature dependence of the London penetration depth 2L into account, the London theory, in contrast to the Ginzburg-Landau theory, may be applied to the whole temperature range between absolute zero and the critical temperature T¢. The London equations describe the spatial variation of the magnetic induction not too close to the flux line core and may therefore only be applied to problems with relevant length scales larger than the extension of the flux line centre which in continuum Ginzburg-Landau theory is given by the anisotropic coherence length. All phenomena associated with the

flUX line core, such as core pinning of the flux line, are beyond the scope of the London theory. Nonetheless, it provides a valuable tool for studying the interaction between flux lines which even in isotropic superconductors leads to non-local elastic behaviour of the flux line lattice (FLL) as the distance between neighbouring flux lines, depending on the magnetic induction applied, may be smaller than the range 2L of the magnetic intervortex forces [6]. Compared to conventional isotropic London theory, the FLL in the anisotropic HTSC's features some novel properties: the magnetization is in general not parallel to the applied magnetic field [ 8,10 ], its free energy depends on the orientation relative to the underlying crystal host lattice [ 9,10,23 ] and, for certain orientations away from the c-axis, parallel flux lines may attract each other, leading to the appearance of flux line chains in the plane spanned by the c-axis and the flux line direction [21,22,25 ]. In order to study the effects of anisotropy on the elastic properties of the FLL in HTSC's, the elastic matrix for arbitrarily oriented straight or curved vortices will be derived within anisotropic London theory. For this elastic matrix, the exact form of which has been under discussion [28,29], the three non-local bulk moduli for tilt, compression and shear in YBa2Cu307_6 and Bi2Sr2CaCu2Os+ 6 will be determined numerically for different values of the mag-

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S. Nieber,H. Kronmiiller/ Elasticpropertiesof FLL

44

netic field applied parallel and perpendicular to the c-axis. In the latter case, the elastic moduli for certain compression and shear modes with wave vectors close to the boundary of the BriUouin zone (BZ) display anomalous behaviour indicating a softening of the conventional triangular FLL.

with the summation convention valid in the last two equations. In Fourier representation, which is connected to the real space representation by r

dJk3 k

fop(r) = J (2n)3 f~p(k)e

ikr

,

(6)

the tensor function takes the form 1 2. Elastic matrix for arbitrarily oriented flux lines Within anisotropic London theory, the free energy for a uniaxial superconductor is given by the volume integral over the contributions from the field energy density and the kinetic energy density of the screening currents: 1

F= ~ f d3r[h2(r)+ ( V × h ( r ) ) A ( V x h ( r ) ) ]

(1) Here h(r) denotes the mesoscopic magnetic induction; the tensor A has the components

A,p=Al Jap + A2cacp , Ai =~2b, (2)

where c, are the cartesian components of the unit vector ~ along the c-axis. Variation of the free energy ( 1 ) yields the London equation:

h(r)+Vx [A(Vxh(r))l

(3) with the summation running over all vortices. ~o, J ( r ) are the flux quantum and the two-dimensional delta function, respectively, and the line integral is taken along the ith vortex with the parameter representation: r~(z ) = ( xi( z ), yi( z ), z ). Introducing a tensor functionfap(r) via the ansatz (4)

the free energy may be written as a double sum of the tensorial interaction between all line elements:

F=-8-~ ~ f dr~ ~ dr~f~,(ri-rj),

A2qaqp 1+ ~ 2 q ~ J

(7)

where q is equal to the vector q=k×~ [20,29]. Within linear elastic theory, the difference in free energy between the deformed and the equilibrium FLL may be written as a bilinear non-local functional:

AF[si(z) ] =F[ri(z) ] -F[R,(z) ]

m,n=l,2,

(8)

where CDm,(Z,z') is the symmetric 2 × 2 elastic matrix giving the free energy of a certain displacement mode per unit volume and per square of the displacement. The elastic matrix completely determines the elastic properties of arbitrary flux line arrangements [ 3,6,28,29 ]. The displacement field s, (z) is a two-dimensional vector field defined by the difference in position between the deformed and undeformed, i.e. equilibrium, flux line positions:

si( z ) =ri( z ) -Ri( z ) •

=ePo~ f driO(r-ri),

ha(')=~o ~ f d r ~ f ~ p ( r - , i ) ,

-

=½ ~ ~ dz ~ dz' ~m,(z,z')s'~(z)s](z'),

.

Az=2~--2~b, a , f l = 1 , 2 , 3 ,

[

(5)

(9)

By convention, the z-axis of a cartesian coordinate system is taken along the direction of the macroscopic magnetic induction. Introducing the Fourier transform of the displacements as

s,(z)=

f

d3 k

~s(k)exp[ikR,(z)]

,

(10)

BZ

with the inversion s(k)= ~

fdzs,(z)exp[-ikR,(zl],

(11)

where n = B~ q~ogives the density of flux lines and the integration is taken over the first Brillouin zone together with all wave vectors - oo < k3 < ~ , the free energy of the deformed FLL from eq. (8) can be written as

S. Nieber,H. Kronmfiller/ElasticpropertiesofFLL 1 r

Fourier transforming the right-hand side ofeq. (16), using the identity

d3k

AF= ~ J (-T~n)~ ~mn(k)S~( - k ) s ~ ( k ) , *mn(k)~-~mn(--k)

.

(12)

With all the necessary formulae now at hand, the elastic matrix ~m~ may be evaluated by expanding the free energy difference between the deformed and the equilibrium FLL,

AF=F[Ri(z) +$i(z) ] -F[Ri(z) ] -~b~ [f dr~ f (13)

up to quadratic terms in the displacements,

1

.

+ ~(s~(z)-sAz )

X (si(z) -sj(z')

and comparing with eq. (12) finally gives the elastic matrix which is symmetric, periodic in k-space and vanishes for a homogeneous translation of the FLL (¢,(0) =0):

)60r 0~f~p I,=Ra~)_Rjtz,) ,

[Fm~(kWQ)-Fm~(~)]..

F,.~(k)=k2f~(k)+kmk~f33(k) -kmk3f3n(k)-k~knfm3(k) •

1

)r

(14)

which, after noting that

dr~(z) = t 7 dz

(17)

' ,,

(18)

The summation in eq. (18) runs over ali the reciprocal lattice vectors Q= rQ1+sQ2 within an ellipsis with the orientation dependent half axes

f~a(ri(z) -rj(z' ) ) =f~.p(Ri(z) -Rj(z' ) ) + (si(z) -s~(z') )r Orfaa(r)I,=R,(z)-~jtz')

(ds~

~ dzei~'z)=(2~)3n~(kz) ~ '~(q-Q)'

~,n,~(k)= -'~x ~

--I dz l dz' f33(R,-R~)],

45

for a = 1,2, (15) for a = 3,

1

where r, s are whole numbers, Qt, Q2 are the basis vectors of the reciprocal lattice and ~b denotes the Ginzburg-Landau coherence length for a flux line parallel to the c-axis. The parameter ~=7(O) is given by sin~O 74(0) -- ~ F 2 +cos20,

yields

O=/_ (L ~), (19)

xrds~(z) ds~(z' ) L ~

dz' f,~,a(R,(z)-Rj(z'))

+ ds~(z) (si(z)-sj(z'))rO,fa3(r)l,=s,~,)_Rj~z, ) dz +

~7(z') dz' (s~(z)-sAz'))%A..(r)l.=R,z~_~tz.) 1

+ ~(s~(z) -sAz' ) V(sAz) -sAz' ) )6 × 0r 06A3 (r) I,=~,(~)-aj(=.)]/ •

(16)

This cutoff in the lattice sum is due tO the limitation of the London theory to length scales larger than the extension of the flux line core in real space which is just the reciprocal of the elliptical cutoff in k-space. Moreover, without this cutoff, the self-energy and the elastic moduli would diverge logarithmically[ 6,29 ]~. The sharp cutoff may be modified to a smooth one by introducing a gaussian in the" tensorial fuiaction f,p(k) with an appropriate extension of the summation area [ 31 ]. ' For intermediate fields He1 <
46

Q:

S. Nieber, H. Kronmiiller /Elastic properties of FLL

¢

4~7

=

L-~--~s2,

(20)

with the characteristic length La = ~ and the unit vectorsfb fz in k-space. The first BZ of this lattice covers the area 4n2n and may be approximated by a circle with radius kaz = 4q/-~n. A continuum approximation, which is exactly valid near the centre of the first BZ only, i.e. for small values of the wave vectors (kl, k2<
B2 1 c,,(k,, k2)= 4x l+A~b(k~+kZ~)

(22)

gives a good description of the compression modulus for almost all wave vectors in the first BZ. For an external field applied perpendicular to the c-axis, the moduli for pure "soft" compression (in the ab-plane) and "hard" compression (parallel to the c-axis) in this approximation are given by B2 1 c~, (k,) = 4n 1 +2~Zbk~ ' B2 c]l(kz)= 4n 1

1 2 2, +2ck2

(23)

while for the mixed compressional mode

n

+~mn[C66(k)(k~+k~)+c44(k)k2]) .

so-called "isotropic approximation", which consists of keeping only the term Q=O in the lattice sum (18) [ 6 ] and which has nothing to do with the anisotropy of the uniaxial materials themselves. Thus, for an external field applied parallel to the c-axis,

(21)

B2 Clll, k l , k 2 ) "= 4 ~

3. Elastic moduli for compression, shear and tilt

With the elastic matrix according to eq. (18) and its relation (21) to the bulk moduli, the non-local elastic moduli for compression, shear and tilt may be determined, which play a prominent role in the theories of collective flux creep [ 15 ], collective pinning [ 7] and thermal fluctuations of the FLL [ 11 ]. This was done numerically for different values and orientations of the applied magnetic field with the moderately anisotropic YBa2Cu307_6 ( F 2 = 25, X~b= 102 ) as well as the highly anisotropic Bi2Sr2CaCu2Os+a ( / ' 2 = 3 × 103, Xab=85) [13,14,17,19], using Ewald's method to decrease the computational effort [ 1 ], for "pure" deformation modes, i.e. by considering the dependence on a single component of the wave vector only, while setting the other two components of k equal to zero.

3.1. Compression moduli For magnetic inductions n c l < B < 0.1H¢2 the nonlocal compression moduli are well described by the

1 2 2+/].c2k22" + 2abk l

(24)

The two compression moduli according to eq. (23) coincide only for homogeneous deformations (k= 0), while at the border of the circularized first BZ at kl.2 = kaz their ratio is given by Chl(k) =F 2 c l l ( k ) k=k~z

(25)

Although this value is found in the numerically evaluated lattice sums for external fields much larger than the lower London critical field, at smaller fields the modulus for the soft compression displays additional local minima with small values of C]l(k2) near the boundary of the circularized first BZ as shown for YBa2Cu307_a in fig. 1, indicating a softening of the conventional triangular FLL for compression modes of large wave length.

3.2. Shear moduli As was found in the isotropic case, the dispersion of the shear moduli in the uniaxial HTSC's is much smaller than for the compression moduli. Therefore, in most of the BZ and for a wide range of magnetic inductions He1 < B < 0.1H¢2, the numerical results

S. Nieber, H. Kronmfiller / Elastic properties of FLL

c°---~61b'l = F 4 ,

¥Bo2Cu307 103

..g 101

mo

c

10"I. Hcl (a)

.+0-3

.,(2

1o-I

lO0

k2/kBz 104

I.Hc2

YBa2Cu307

102

_

_

a / ~

100 c

3-

10-2

(b)

10-3

10-2 k2 / ksz

1~

;0° "

Fig. 1. Dispersion of the compression modulus cH(k2) for the flux line lattice in an external magnetic field parallel (a) and perpendicular (b) to the c-axis (soft mode) at varying magnetic inductions in YBa2Cu307 _ a ( F2 = 25, x,~ = 102). Note the softening of the FLL at low magnetic inductions and wave vectors of the displacement field close to the boundary of the circularized first Brillouin zone for the magnetic field perpendicular to the caxis.

agree with the values obtained from local elasticity theory [ 18 ] which, for external fields parallel to the c-axis, yields •

C66 --~-C~S6,

B~o C~s6-- 64/t222b •

(26)

B~o .1-,_7/3

c~6lo_-~-

c~6lo=~

10-I'Hc2

102

d-

47

649222

where 2 is the geometric mean of the London penetration depths along the main axes. However, the modulus for the hard shear features a pronounced dispersion at external fields several orders of magnitude larger than the London lower critical field, where, for the orientations parallel to the ab-plane and wave vectors close to the boundary of the circularized first BZ, the shear modulus decreases strongly, c.f. fig. 2. This phenomenon is more pronounced in the strongly anisotropic B i2Sr2CaCu2Os+a and serves as an indication that the conventional triangular FLL according to eq. (20) proves exceptionally soft for shear deformations of this wave length. Thus, other flux line arrangements, such as flux line chains [ 21,22,25 ] may appear, as there are energetically more favourable flux line configurations, the determination of which from the free energy according to eq. (1) remains a non-trivial problem. It should be pointed out that in all calculations of elastic moduli for straight flux lines, i.e. for the compressional and shear moduli but not for the tilt moduli considered here, the elastic matrix contains not only the lattice sum of eq. (18) but also a corresponding 2D-integral with negative sign over the reciprocal lattice points of density 1/ [n(2~)2]. While I0-3Hc2 I

101

3.100 Bi2Sr2CaCu208

c: 100

,0-1

10-2

For an external field applied in the ab-plane, the difference in the moduli for soft and hard shear is even more pronounced than for the corresponding compression moduli, with their ratio not too close to the boundary of the first BZ taking the value from local elasticity theory:

(27) '

100

k2/kBz Fig. 2. Dispersion o£ the hard shear modulus c~ (k2) for the FLL in an external magnetic field perpendicular to the c-axis at different values of the magnetic induction in Bi2Sr2CaCu=Os+~ Near the boundary of the first Brillouin zone, the strongly reduced values for the modulus c~ indicate a softening of the FLL.

48

S. Nieber, H. Kronrniiller /Elastic properties of FLL

this integral contributes only weakly to the compression moduli, it dominates the shear moduli, more than compensating for the negative values o f the lattice sum [6 ].

C

B~o

3.3. Tilt moduli

1 1

F2x~b

c44(k3) le=o= (4~-~b)~Lf~ n 1+2~bk32

For external fields applied along the c-axis, the numerically obtained tilt moduli coincide with the values from the isotropic approximation: B2

1

4n 1 +2abk3 2 2 +2~(k,z +k~)

C44(k) =

flux line, which follows from eq. (18) by replacing the lattice sum with the corresponding integral, yielding as an excellent analytic approximate expression for the line tension:

(28)

only for small values of the wave vector k3, c.f. fig. 3(a). At larger k3, however, the numerical results tend more closely to the tilt modulus o f an isolated

+

2 ( 1 "t-2abk3) 2 2 ] 1 Nab "'-TT-~'z m22abk3 .-2-5 ~lqab+2abk3 ] • ,

(29)

For an external field applied perpendicular to the caxis, there are two tilt moduli which are well described by the isotropic approximation for not too small fields Hcl < B < 0.1He2 and not too large wave vectors only:

104

c

103

2

YBa2Cu307

YBa2Cu307

]

10 3

,

102

J_~

10 2

101

B /

I01

100

100

10-1.Hc1 10-I (a)

........

,

10-3

.......

10-2

. . . . . . .

,

i0 -I

ab

k3

104 - - ~ . ~ I H c

(b)

]

i0 0

~o-3

........

: ~o-~

.......

: ,o-~

...... ~oo

"

2

YBe2Cu307

103 ¢.1 102 1 -I

101

............... ".................

J 100

(C!

10-3

........

I

10-2

........

t

i0-I

.......

100

Fig. 3. Dispersion of the tilt modulus in YBa2eu30~_a at varying magnetic fields applied parallel (a) and perpendicular (b,c) to the caxis (soft and hard mode). For larger wave vectors and not too low magnetic inductions, tile tilt modulus of flu× lines in the FLL tends to the results for an isolated flux line indicated by the dotted curves (...).

S. Nieber, H. Kronmiiller/ Elastic properties of FLL ch4( k ) =

parallel and perpendicular to the c-axis is given by

B2 1 4 n l +Xab(k2 2 2 + k 32) '

Hlcll

(30)

As noted before, for larger wave vectors k3, the flux lines in the FLL behave as if they were isolated [27,30]. This behaviour is displayed in figs. 3(b) and (c) for YBa2Cu307_& The line tension, which in anisotropic superconductors, just as for dislocations in crystals [ 5 ], is different from the line energy [ 29 ], may be roughly approximated for the hard tilt out of the ab-plane by

..[.

Bqb0

1 ~(21n 2

(FKab) 2

2

2

l~ln F (X~b+'~Zabk3)~ ( r 2=bk3-- , l+F222bk 2 ] ' 2 2

2

(31)

while for the soft tilt in the ab-plane a very good approximation is given by 1 c,~4(k3) = ~3-c4h4(k3) . 1-

(32)

4. Anisotropy of the lower critical field For completeness, it should be noted that, while the upper critical field may readily be obtained from anisotropic Ginzburg-Landau theory [4 ], the lower London critical field follows from the London free energy of a single straight flux line:/-/el = ( 4 n / ~ o )¢L, which may be obtained from the tensor function f33 (k) according to ~ f d2k ~L = - ~ (-~)2f33(kl, k2),

(33)

and yields for the orientations parallel and perpendicular to the c-axis in the limit of large GinzburgLandau parameters (X=b>> l ) 2

_(

*o ~21nF__~,a, F "

F

H A - 1 + (lnF/lnXab) '

B2 1 c,~4(k)= 4rt 1 2 2 2• +2=bkl2 +2ck3

8~4 (k3) = ~

49

(34)

e~ -\4~t,,b)

Thus, while the ratio between the lower critical fields

(35)

the respective value for the line tension of straight flux lines is c44(k3) I o=o

ch44(k3)

_1[1 1 1 k3=O-- ~ - ~ + 21nI'x=b "

(36)

5. Summary For the uniaxial HTSC's, the elastic matrix, which completely determines the non-local elastic properties of flux line arrangements, has been derived within anisotropic London theory for an arbitrarily oriented arrangement of straight or curved vortices. From this elastic matrix, the non-local bulk moduli have been numerically determined for the material parameters of YBa2Cu307_6 and Bi2Sr2CaCu2Oc+6 at different magnetic fields applied parallel and perpendicular to the c-axis. The numerically found compression moduli for the conventional triangular FLL agree at all inductions Hcl < B < 0 . 1 H c 2 with the values of the isotropic approximation, apart from the region near the boundary of the circularized first BZ at low inductions for magnetic fields applied perpendicular to the c-axis, where the modulus associated with the soft compression mode displays anomalous dips. The shear moduli, which in most of the BZ show only weak dispersion and coincide with the results from local elasticity theory, offered another indication for a softening of the conventional FLL for magnetic fields in the ab-plane, as the hard shear modulus at higher inductions becomes very small for wave vectors of the displacement field near the boundary of the first BZ. The numerically obtained tilt moduli agree with the values of the isotropic approximation only at higher magnetic inductions or at small wave vectors. For larger wave vectors, the flux lines of the FLL behave as if they were completely isolated and the exact line tension, for which approximate analytical expressions in external fields parallel to the main axes have been given, deviates strongly from the isotropic approximation.

50

S. Nieber, H. Kronmfiller /Elastic properties of FLL

Acknowledgement Thanks are due to E.H. Brandt for valuable discussions.

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