Coherence length dependent flux-lattice melting temperature in high-temperature superconductors

Solid State Communications, Vol. 77, No. 3, pp. 225-228, 1991. Printed in Great Britain.

0038-1098/91 $3.00 + .00 Pergamon Press plc

COHERENCE LENGTH DEPENDENT FLUX-LATTICE MELTING TEMPERATURE IN H I G H - T E M P E R A T U R E S U P E R C O N D U C T O R S S. Koka and Keshav N. Shrivastava School of Physics, University of Hyderabad, P.O. Central University, Hyderabad 500134, India

(Received 25 September 1990 by D. Van Dyck) The fluctuation in the distance between flux lines is calculated within the frame work of a Ginzburg-Landau model. It is found that the melting temperature has a term which is linearly proportional to the coherence length in addition to the term depending on the shear modulus of the flux lattice and the superfluid density. The effect arises due to finite wave vector cutoff along the chain direction perpendicular to the ab-plane.

1. I N T R O D U C T I O N RECENTLY, Moore [1] has considered the GinzburgLandau phenomenological free-energy density functional F =

1

r l ~ l z + ½ul~kl4 + ~mmI ( - i h V + (B -

H)2/8~,

2eA/c)~k)z (1)

the solution of which has the periodicity of a triangular lattice. Moore found that the square of the thermal fluctuation is given by d2(T ) =

ka T

4n(ps C66)1/2'

(2)

where Ps is the superfluid density and c66 is the shear modulus of the flux lattice. This expression is independent of the wave vector cutoff and hence the coherence length does not appear in it. Brandt [2] and Nelson and Seung [3] have calculated the thermal fluctuation in the vortex state. Houghton et al. [4] incorporate both an anisotropic tilt modulus and nonlocal elastic constants into the Lindemann criterion. However, their expressions can be evaluated only in some extreme approximations. One of the solutions can be obtained for b >> 1/2 where b = B/Bc2(T) which means that the average induction is larger than half o f the upper critical field. Another solution requires that b ,~ 1/2 which means that the induction is very small. Both the solutions suffer from the infinite limits on the wave vector along the z-direction. The double integrals of Houghton et al. cannot be integrated unless either the wave vector dependence o f the elastic constants is ignored or the same are assumed to be isotropic. It is reported [5] that there are finite phase

fluctuations and long-range order is present in a threedimensional flux-line lattice. Soulen and Wolf [6] have developed the full phase diagram showing the melting region as distinct from the region of depinning. It is important that the approximations made apply to the melting region. In this paper, we use a finite cutoff for the wave vector of the thermal excitations so that there occurs the coherence length in the thermal fluctuation. We find that the fluctuation has one more term than is given by (2). There is a logarithmic contribution to the fluctuations so that the application of the Lindemann criterion leads to a flux lattice melting temperature proportional to the coherence length in addition to what is found from (2). 2. T H E R M A L F L U C T U A T I O N IN A SUPERCONDUCTOR The minimization of the free energy of a superconductor in a magnetic field with pinning forces gives rise to a triangular Abrikosov lattice o f flux lines of lattice constant I and periods rl =

(x~,yj) =

(1,0),

r,, = (xl,, yH) = (1, ~ ) 1 ,

(3)

so that the area Q = 2rrP of the fundamental cell is given by the flux-quantization condition, x,,

/~,

(4)

where ~b0 = he/2e and/~ is the spatial average of the magnetic induction. Moore [1] has calculated the

225

226

COHERENCE LENGTH DEPENDENT FLUX-LATTICE

Vol. 77, No. 3

2/P,

circular Brillouin zone of radius A where A2 = we find,

_

d2(T )

k~ TP ~21e~,1~ I kdk, 4n(pse66) m 0

(12)

the solution of which is given by d2(T) -q- plane

Fig. 1. The contour at R --* ~ and the poles for the integral in equation (1 1) with q± = + i(c66P:'k4[ps) in the complex q-plane. mean square fluctuation

2 ((u~ + uy)),

(5)

to study the stability of the flux lattice against thermal fluctuations. It was found that d2(T)

p2f~

=

d2k i2n)2

koTk 2 (psq2 + c66P2k4),

(6)

where q is the wave vector in the z-direction while Ikl occurs only in the xy-plane and is restricted by the Bfillouin zone. The superfluid density p~ is given by p~ =

4~0(4~r)-'H~(1

-

~/H0~),

(7)

0.48(8nf2)-'H22(2x2 - 1)(1 - BlH¢z,)2 (8)

with f

=

(9)

where H~2 is the upper critical field of the type-II superconductor, x = 2/~ is the Ginzburg parameter, the ratio of the London penetration depth to the coherence length, and /3 is a numerical constant, ~ 1.1 6. The double integral in (6) may be written as, d2(T) -

p2 kn T ~

(10)

We consider the dq part of the integral which along the large semicircle in the upper-half plane is zero as long as c66P2k4/p~ ~ q. There are two poles, q = +_i(c66p2k4[p~)'/z but only one is inside the contour in the upper half plane as shown in Fig. 1. Therefore, dq

~ qZ + (c66PZk,~/p~)

~

\c66,/

Substituting (11) in (10) and taking dk integral over a

\¢66// ..]"

(14)

Using the abbreviation

(15)

aZ = ~-~ (P---Ls~112,

\¢66,,/ it is seen that the integral in (15) reduces into the form, d2(T) =

PkB TI

(2/t) 2(PsC66)112'

(16)

where

(2/p)1/2. f

0

( )~-~ a2

k tan-'

dk.

(17)

Integrating by parts the above integral gives, I

=

½k2 t a n - '

~

+ a2

o

( k4 + a4) ' (18)

the solution of which is,

dq

f q2 + (c66p2k4/ps) f k2d2k.

f d2k tan i [ p ~ (P__2.s~l[2~

(2/0 3(ps C66)I/2

I = 1 + (2x 2 - l)fl,

PkBT

d2(T) -

and the shear modulus [7] of the flux lattice is c66 =

(13)

as found by Moore [1]. However, it is obtained by using infinite boundary conditions for q as in (11) and a circular zone for k. The infinite limit for q introduces unphysical divergence in the ground state energy. Therefore we make an effort to obtain a more accurate result than (I 1). Instead of using infinite limits as in (1 1) we obtain the q-space integral with a finite cutoff, qm" The problem is then slightly different from that of (12) as (10) becomes

q_

dZ(T) =

kB T 4n(p~c~6) '/2'

I

=

]

~ tan-

\e66/ d

qemp' q2ps '

qm / P ~ " 2 In 4c66 +

+ ~ \C66//

(19)

so that the fluctuation becomes, d2(T) _

kBT

(2~)2(PsC66)ll2 + ~

lean_, {_~ (p~']'/2 ~ \C66,] )

(Pm) '12In {\ 4c66qTp q- qZPs'~ \c6v jj.

(20)

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COHERENCE LENGTH DEPENDENT FLUX-LATTICE

Taking the value of the angle arising from the t a n - ' term to be re, the first term o f (20) exactly agrees with (13) but then the second term o f (20) provides correction to (13). Thus the result (20) is more accurate than (13). The wave vector along the z-direction varies from zero to qm with the restriction, qm = (2b)tl2/~z, (21) on the maximum value. The average value of the field and the upper critical field determine b

=

(22)

B/B~,,(T),

and ~: is the coherence length along the z-direction.

[l qm(Ps/e66)112] "~ 7I

and the logarithmic term is small compared with the previous term. Then using (7) and (8) we find, (23)

kBTH~23/2(1 - B/H~2)-3/2f3n(2~z)u2 0.69c~/2(2x2 -- 1)~/2

C a s e II. For a small angle t a n - ' x = x with logarithmic term smaller than tan-J x term, the result is different from that found above. This approxima1,7 ~/2 "~ ,~/2 tion is equivalent to ~-"/mPs ~66 In this case (20) becomes, •

d2(T) =

kB Tqm

l )~/3He2( 1 -

x (1 - B [ H ¢ E ) ( k B T ) - 2 / 3 f

~ =

T~M/ T~)

c-4/3l -4/3,

(24)

in which we apply the flux-quantization condition l 2 = q~o/B so that (24) becomes (0.69)2j3q~01/32 1/3~Z-I/3(2K2 _ l)lJaHc2(1 -- TvM/Tc) x (1 - B / H ~ 2 ) ( k B T ) 2/3f-,

=

e-4/3~)~213]~2/3.

(25) This

result

is

equivalent

to

(1 - T v u / T ~ ) ""

(27)

Using the flux-quantization condition the above expression reduces to 0.96(2r02(2X2 =

81zkBTFMq m f 2 I)(1 --B/H¢2)2H~2(I -- TFu/T~) 2

B2 ,

(28)

which means that 1 -

(29)

(TFM/Tc) "~ c - ~ B ,

apart from the constants, so that the flux-melting temperature measured from the critical temperature becomes a linear function of the magnetic field. However, the cutoff wave vector along the z-direction introduces the average field so that (28) becomes, TFM/T¢)2¢zBt¢~:

8 n f 2 kB TFM(2B) ~¢2

Using the Lindemann melting criterion, d 2 = c2l 2, the critical behavior of the upper critical field H¢2 = He2(1 -- T / T ¢ ) we find

~-- c2l 2.

0.96(2r02H~2(2x2 -- 1)(1 -- B/H¢2) 2 -

0.96(2n)2(2x 2 - )(1 -- B/H¢2)2H2¢2(1 -

(0.69) 2/3~b0~/32- ~/3n -~/3(2 x 2 -

(26)

2(2r02c66'

8r~kBTFMqmf2

The Lindemann's criterion is that when the fluctuation becomes equal to about one tenth of the lattice spacing then the lattice melts. We use this rule on the fluctuation given by (20) along with the lattice spacing flux quantized according to (4). The flux lattice melting temperature, TFM, and hence the Hc2(TFM) line depends upon the relative importance of the two terms of (20). Introduction of the critical behavior opens up the possibility of a normal region above T¢(H¢2). Thus there is a normal region and two superconducting regions one with a flux lattice and the other with a flux liquid.

d2(T) ~_

10-5c-4/3B2/3, and its exponent is consistent with early experiments on the high-temperature superconductors [8]. Indeed, the pairing of electrons at random sites leads to higher critical temperature than does the B.C.S. theory [9].

Using (8), and the Lindemann criterion we find

3. A P P L I C A T I O N O F L I N D E M A N N ' S CRITERION

C a s e I. We assume that tan -l

227

=

c-2~bo2B 2,

(30)

so the critical line behaves as 1 -

(TFM/T¢) "~ C ' B 5j',

(31)

which has much larger exponent than (25). However, (31) applies only when the maximum wave vector along the z-direction is small, q,, ,~ 2(c66/Ps) t/2. C a s e III. For a large logarithic term in (20) compared with the arctan term, the result is similar to (30) as the logarithmic term has only small variations for large changes in its argument. In the high-temperature superconductors, there is a vortex transition below the transition temperature. The flux lattice melting temperature as a function of the upper critical field is determined from the fluctuations of the flux-lattice spacing. In another paper, we

228

COHERENCE LENGTH DEPENDENT FLUX-LATTICE REFERENCES

have treated [10] the fluxons as bosons to calculate the flux-lattice melting temperature. Vinokur et al. [11] find a crossover temperature from the dependence of magnetoresistance above which the magnetoresistivity varies linearly with temperature and below which it is thermally activated, exponentially [12].

4.

4. CONCLUSIONS

5.

We have obtained an analytical expression for the thermal fluctuation of the vortex position. We introduce an upper cutoff in the k-integral. This leads to the correct result for large x for the isotropic superfluid density. There are two important contributions to the fluctuation, one having an arctan term and the other a logarithmic term, the latter of which is found here for the first time.

6.

Acknowledgements - One of the authors (S. Koka) has been awarded Dr K.S. Krishnan D.A.E. Fellowship. We thank the Department of Atomic Energy of the Government of India for financial support.

Vol. 77, No. 3

1. 2. 3.

7. 8. 9. 10. 11. 12.

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