- Email: [email protected]
Relevance of thermal and quantum fluctuations to the irreversibility line in a melt-textured YBCO sample
PHYSICA® ELSEVIER
Physica C 248 (1995) 353-358
Relevance of thermal and quantum fluctuations to the irreversibility line in a melt-textured YBCO sample O.F. de Lima *, R. Andrade Jr. lnstituto de Fisica "Gleb Wataghin", 13083-970 Campinas, SP, Brazil Received 3 March 1995; revised manuscript received 25 April 1995
Abstract The irreversibility line for a melt-textured Y B a 2 C u 3 0 7_ ~ (YBCO) sample was determined in two orientations of the field H relative to the sample &axis. In both orientations we obtain good fits using a melting equation with quantum correction to describe the Abrikosov lattice melting. The scaling analysis based on a fluctuation theory for 3D systems was applied to our magnetization data allowing the determination to be made of the mean-field critical temperature. Our results are similar to those found in the literature for very clean untwinned samples. We suggest that the Abrikosov lattice melting is weakly affected by quenched disorder, at least in the relatively low fields (H < 50 kOe) and high temperatures (T/Tc > 0.8) probed in this work.
1. Introduction The irreversibility line (IL) separates a magnetically irreversible from a reversible state in the H × T phase diagram of type-II superconductors. This is a very well documented experimental fact for the highTc superconductors, although its physical interpretation has aroused some issues. In order to describe the IL, simple power laws of the type Hm(T) --H0(1 T/Tc) n have been suggested, where H 0 is a modeldependent fitting parameter and T~ is the critical 3 temperature. The exponent n = ~is predicted by a quasi-de Almeida-Thouless behavior [1] as well as by the giant flux-creep model [2,3]. The exponent 4 • n = 3 Is expected from a vortex-glass formation [4-6], as well as from Tinkham's giant flux-creep model [3]. An exponent which is dependent on the * Corresponding author.
pinning strength can be found in a collective fluxcreep model [7]. In a more conventional approach the IL is identified with a depinning line, whose solution comes from an equation describing the constant diffusivity, Do(H, T), of the flux lines [8]. Finally, the interpretation of the IL as the melting transition line of the Abrikosov lattice [9-12] has received enough confirmation, and nowadays the melting hypothesis seems to be a well established matter [13-15]. The basic idea is that the Abrikosov lattice becomes unstable and melts, when the mean displacement amplitude of the flux lines, (u2) 1/2, reaches an appreciable fraction of the lattice parameter, a 0. This latter condition is usually expressed in terms of the Lindemann criterion, (u2) 1/2= CLaO, with the Lindemann number c L varying between 0.1 and 0.3. Recently, Blatter et al. [14] discussed the relevance of quantum fluctuations that, combined with thermal fluctuations [11], produce an effective mean
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 2 8 8 - X
354
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
displacement amplitude of the flux lines which may correct the melting-line position in some cases. For instance, thermal fluctuations are known to be dominant in the strongly anisotropic BiS-CCO compounds, making quantum corrections vanishingly small [14]. However, in the less anisotropic YBa2Cu307_ a (YBCO) compound the correction due to quantum fluctuations becomes important, as has been discussed by Schilling et al. [16] and Blatter et al. [14], using data from single-crystalline samples. In this paper we discuss results that provide further confimation for the relevance of adding quantum correction in the calculation [14] of the melting line for YBCO. The analysis takes in consideration the universal scaling behavior for the magnetization, caused by fluctuations in the amplitude of the superconducting order parameter [17]. Our study was done on a melt-textured YBCO sample, in contrast with the very clean single crystals analyzed before [1417]. Hence, we conclude that the occurrence of quenched disorder (pinning centers) in the melt-textured sample has little influence on the relative weight of quantum and thermal fluctuations that are required to interpret the IL as a melting line. We measured the IL in two angular directions ( a = 0 °, 60 °) of the field H with respect to the e-axis of the textured sample, and found that these results obey the anisotropy-induced angular dependence predicted for the melting line [18,19].
800
.~. 4 0 0 _
2O0
o (d,,o,,,.)
I ...... i 20
25
1
30 35 40 45 20 (degree)
50 55 60
Fig. 1. X-ray diffraction pattern in the 8-axis oriented geometry for the melt-textured YBCO sample. The inset shows a rocking curve for the (005) peak that reveals an angular spread of ~ 2.5° for the grain orientations.
transition temperature measured under a field of H = 5 0 e is 91.0 K, determined at the transition onset, and AT.-~ 1.3 K, between 10% and 90% of the maximum shielding. Each point in the M(T) curves corresponds to an average over two scans for a length of 3 cm. We employed in this work a magnetic and thermal history similar to that reported by Schilling et al. [21], aiming at the IL determination by conventional M versus T measurements. First, a remanent magnetization curve, Mrem, was obtained (Fig. 2) by cooling the sample from 100 K
.............
i .........
80.0 ~
2. Experimental details
~
-2.0
1
-,,o
o
o
i .........
i .........
i .........
o.01. . . . . . . O
80.0
The melt-textured YBCO sample studied here was obtained by the partial-melt-growth method [20]. It was cut into a parallelepiped having dimensions 2.7 × 2.5 × 1.1 mm 3 and contains long crystalline grains, well aligned with the e-axis direction. Fig. 1 presents the X-ray diffractogram for the e-axis oriented geometry, showing essentially the occurrence of (00l) peaks. The inset of Fig. 1 displays a rocking curve for the (005) peak that reveals an angular spread around 2.5 degrees for the grain orientations along the d direction. DC magnetization measurements, M(T), were made using a commercial SQUID magnetometer (Quantum Design, model MPMS5). The magnetic
(005)
oo
40.0
~" 20.0
f,,,
~IYj
= "llO~
i.,~. ~
o
i .........
o
...... r
I
(~o
0.0 -20.0
~ lid
.........
70
H = 20 kOe i .........
75
i .........
80
i .........
85
i .........
90
a .........
95
h*
100
T (K) Fig. 2. Typical magnetization curves Mrem and MFC (see text) of the melt-textured YBCO sample. Two angles, a = 0°, 60°, between the applied field H and the 8-axis direction were employed. The inset displays a magnified view of the reversible region for ot = 60 ° where the temperatures Ti and Tc2 (see text) are indicated.
O.F.
de Lima,
R. Andrade
Jr./Physica
(normal state) to 10 K"under an applied field H = 50 kOe. Then, a measuring field H was set and the sample warmed up to 70 K, from which the Mrem(T) curve was measured by slowly increasing the sample temperature in steps AT = 0.4 K. Next, a field-cooled curve, MFc(T), was obtained by cooling the sample from 100 K to 70 K, in a measuring field H. From that point the MFc(T) curve was measured by slowly increasing the sample temperature in steps AT = 0.4 K, as before. For each chosen field H, the curves Mrem(T) and MFc(T) merge at an irreversibility temperature Ti(H) as shown in the inset of Fig. 2 for H = 30 kOe. A practical criterion was applied to define Ti(H): where the difference A M = M rem MFC becomes less than the standard deviation of the measurements. An experimentally determined upper critical field, He2 =H(Tc2), is associated with the temperature To2 (see the inset of Fig. 2) where the linear extrapolation [17] of the reversible region meets the normal base line of the magnetization curve. It is worth noting that our data did not show a consistent feature like a break in the slope of the magnetization curve at a temperature T * > Ti, as reported by Schilling et al. [21] in their study with YBCO single crystals.
3. Results and discussion
C 248
(1995)
0.0
Fig. 3(a) shows the characteristic fan-shaped magnetization curves [17,23] in the reversible region near Tc(H), well above the irreversibility point, for fields between 10-45 kOe. The occurrence of a common crossing point at a certain temperature has been evaluated as an effect of fluctuations in the vortex line positions (phase fluctuations) for 2D-like superconductors [22,23]. Vortex-line fluctuations are also expected to occur in anisotropic 3D materials like YBCO [22]. However, this subject is not discussed in the present paper, although Fig. 3(a) clearly displays the cross-over phenomenon at T = 90 K. Next, we present a scaling analysis based on fluctuations in the amplitude of the superconducting order parameter, aiming at the determination of the mean-field transition temperature for zero magnetic field, Tco. In high magnetic fields and around T~(H),
.... ' .........
355
' .........
' .........
' .........
'?:z;::l~ziziz: .
.
.
.
-1.0 c
~"
10kOe
20 kOe 30 kOe 45 kOe
-2.0 ~:~
-3.0 88
90
..., .........
, .........
92 94 T (K) , ..........
. ........
, .........
96
98
100
, .......... . ........ , ....
g o.o g8 -0.5
Yco=91"8K
( / - f- b ) -1,0
g. "r"
~-~ -1,51
#
...JJ. ............................................................ -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 (T-Tc(H)I/(T H)z's (10 4 K 'U3/Oe2t3)
Fig. 3. (a) Reversible region of the magnetization curves, for /4 II~, showing the crossing point around T = 90 K. (b) Scaled magnetization assuming a three-dimensional(3D) superconductor (see Section3.1). Tco= 91.8 K is the mean-fieldcritical temperature.
the scaling behavior predicted for the magnetization due to this kind of fluctuations is given by [17,24]: 4'rrM
3.1. Scaling due to fluctuations
353-358
[
-Tc(H)]
(-~)-, = F A T-( TH )'~
,
(1)
where F is the universal scaling function, A is a temperature and field-independent coefficient, and n is ~ or ~, respectively, for a 3D or 2D system. Fig. 3(b) shows a plot of the scaled magnetization, corresponding approximately to the same region shown in Fig. 3(a), assuming a 3D behavior [17]. The only free parameter in this scaling fit is To0, which can be related to Tc(H) and S n - - d H c 2 / d T by the linear relationship [17,23] T~(H)= T~0+ H/S H. Using S H = - 1.9 × 104 O e / K (see Section 3.2) the best scaling fit occurs when T~0 = 91.8 K. This value is shifted 0.8 K above the experimental transition temperature of 91.0 K, taken for H = 5 Oe. This shift in temperature is known to reach values of about 4.0 K in the case of the more anisotropic BISCCO compounds [23], where fluctuation effects are much stronger. 2
1
356
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
Comparing Figs. 3(a) and (b) one sees that the scaled magnetization curves [Fig. 3(b)] almost collapse onto a single curve, as expected from Eq. (1). The observed spread most probably comes from the combination of two factors: (1) the angular spread ( ",, 2.5 °) of the grains orientation in the melt-textured sample, and (2) the angular uncertainty (,-, 1°) of the sample orientation due to slight bendings of the sample holder during the magnetization measurements. Both situations imply some uncertainty in the function T~(H), which could be minimized by using singlecrystalline samples [17] and by taking all M(T) curves in a single-run experiment.
3.2. Melting line As shown by Houghton et al. [11] the melting line due to thermal fluctuations, for H parallel to the d-axis of an anisotropic material, is
t
b 1/2
( 1 - t ) I/2 ( l ~ b )
[ 4(V~-- 1) [
Vri'----~
+
1
2~
=
- ~ - e c 2,
(2) where t=T/T~ and b=B/H~2, with H¢2(t)= Hc2(0X1- t). The Ginzburg number is Gi = 161r3K4(kBTc)2/~30Hc2(O) ~ 1.06 × 10-9Tc2K4/ He2(0), where K is the Ginzburg-Landau factor, 40 is the flux quantum and k B the Boltzmann constant. The sample a n i s ~ described by the effective mass ratio ~ = ~mob/m c , whose value is e = ~ for YBCO [11,14]. Eq. (2) does not admit a closed-form solution for the melting line B(T), therefore it has to be solved in a self-consistent way when fitting the experimental data. However, close to Tc and taking up to linear terms in v~ for the expansion of Eq. (2), one gets
B( T) -~./3thHc2(0) where (1 - t)/t appears as the natural variable [14], instead of the commonly used form ( 1 - t). The numerical constant is fltn = {2~r/[4(v~- - 1) + 1]}2 = 5.6. Fig. 4 shows the best fits to our IL data ( H If ~), for the complete thermal equation [Eq. (2)] and for
40
,,, ........ Eq. (2) lo
.....
eq.(s)
Eq. (4)
i ~,.
dl. ~..~.
~
' ~
o 76 78 80 82 84 86 88 90 92 T (K)
Fig. 4. Magnetic phase diagram of the melt-textured YBCO sample. The data points define the irreversibility line (O) and the experimental Hc~ line (©). The solid, dotted, and dashed lines represent fits of the melting equation, respectively with quantum correction [Eq. (4)], without quantum correction [Eq. (2)], and in the approximated quadratic form [Eq. (3)]. A linear fit of the Hc~ line, Hc*2(T)=Hc2(O)(1-T/Tc'), is represented by the dotdashed line. The thin solid line shifted to the right represents the Hc2 line, corrected for fluctuations of the order parameter (see Section 3.1).
its simplified form [Eq. (3)], both using the Lindemann number c L = 0.20. Eq. (2) gives a reasonable fit (dotted line) only for H < 20 kOe while the quadratic form, Eq. (3), fits only very close to Tc (dashed line), for H < 10 kOe. Fig. 4 also shows the experimentally determined Hc~ line, fitted by the linear form HcE(T) = H e 2 ( 0 ) ( 1 - T/T¢*) with He2(0) = 1.72 x 10 6 0 e and the extrapolated critical temperature T~* = 90.6 K. The thin solid line shifted to the right in Fig. 4 represents the H~2 line, corrected for fluctuations of the order parameter, as explained in the previous section. This line extrapolates to the mean-field transition temperature, T~0 = 91.8 K, which was employed in all three melting equation forms fitted to the IL data (Fig. 4). The lines Hc~ (T) and Hc2(T) have the same slope S n -dHc2/dT-~ -1.9 × 104 Oe/K, comparable to the commonly quoted values for high-quality YBCO crystals [13,16]: Using these quoted figures and the Ginzburg-Landau factor x = 80, which is representative for YBCO [25,15], we get the Ginzburg number Gi = 5.3 X 10 -3. This Gi number is about six orders of magnitude larger than what is found in conventional superconductors, thus stressing the relevance of thermal fluctuations in the high-Tc material [141. According to Blatter et al. [14], the combined
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
effect of thermal and quantum fluctuations leads to the following compact equation for the melting line: 402 b(t) = (4) 1+
.... , .........
50
~
i .........
357 , .........
,.~
, .........
,o
.........
i .........
, .........
, .........
,.
i .........
i .........
i .........
h~
SO*
~- 30
9 +--
i
~
l 10
with the temperature variable 0
C2
/
~th
1),
and the suppression parameter S
=
2.4 # K F + C
Gi '
.a .........
i .........
i .........
i .........
t .........
76 78 80 82 84 86 88 90 92 T (K)
(6)
where K F is the Fermi wave v e c t o r ( K F = 0.15-0.20 £-1 [14]) and v is a fitting parameter, which relates a cutoff frequency g2 with the superconducting energy gap A, v = h O / A . As pointed out by Blatter et al. [14] only part of the suppression parameter S appearing in Eq. (6) can be attributed to the effect of quantum fluctuations, while the larger contribution actually arises from the suppression of the order parameter, ~ , close to the HoE line. Indeed, an increase of the magnetic penetration depth, A A / ( I ~ 12) = A/(1 - b) 1/2, follows from GinzburgLandau theory [26]. Eq. (4) fits very well to our data as shown by the solid line in Fig. 4, producing c L = 0.24 and v = 4 in quantitative agreement with the analysis of Blatter et al. [14] for untwinned single crystals of YBCO. However, the melting line in our case is shifted to lower temperatures, as would be expected in the presence of quenched disorder [27,14]. For instance, at H = 50 kOe we have Ti = 75.5 K, instead of T i = 81.0 K which was observed in untwinned single crystals [28]. These results suggest that the occurrence of quenched disorder in the melt-textured sample has a negligible influence on the determination of c L and v appearing in Eq. (3). This probably means that the melting of the Abrikosov lattice is weakly affected by the quenched disorder occurring in the melt-textured sample, at least in the relatively lowfield and high-temperature region studied here.
3.3. Angular scaling Fig. 5 shows the IL shifted to higher temperatures when the field H makes an angle a = 60 ° ( + 3 °)
Fig. 5. Magnetic phase diagram of the melt-textured YBCO sample, showing the effect of anisotropy on the irreversibility line for a = 0° (O) and a = 60 ° (O). The solid line represents a fit of the melting equation taking into account thermal and quantum fluctuations [Eq. (4)] for a = 0°. The dashed line represents that same fit corrected by the angular scaling rule B(ot, T ) = B(0, T ) / ( ~ 2 sin2a +cos2a) 1/2.
with the ~-axis direction. The dashed line represents the new position of the melting line B(a, t) when the effect of anisotropy is corrected by the following scaling rule [15,18,19]:
B(O, T) B( or, T) = ~/E2 sin2ot + cos2ot ,
(7)
where a is the angle between the d-axis and H. The dashed line in Fig. 5 is obtained using • = ½, the same mass-anisotropy factor employed in the melting equations [Eqs. (2-4)]. Similar to the conclusion of Beck et al. [19], we believe that the verification of the scaling rule given by Eq. (7) corroborates the melting hypothesis.
3.4. Power-law fits Using T~0 = 91.8 K the power law Hm(T) = H0(1
- T/T¢o) n was tested [29] for n = 3 (giant flux-creep model [2,3] and quasi-de Almeida-Thouless line [1]) and n = 34 (vortex glass model [4-6]). For the first case (n = 3) a very good fit was found, as already reported from other experiments [2,14,16,21]. For the second case (n = 4) only a very poor fit was obtained. At first sight it seems that the giant flux-creep model or a quasi-de Almeida-Thouless line could be good explanations for the IL, while the vortex glass model should be discarded. However, following Blatter et al. [14], we could not draw such conclu-
358
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
sions based only on those power-law fits, because they lack a sound theoretical background. Hence, other types of experimental data, like high-resolution I - V curves [30] and long time decay of magnetization [6] would be required in order to allow for a more conclusive analysis based on the relevant critical exponents [6,15].
4. Conclusions The irreversibility line for a high-quality melt-textured YBCO sample was determined in two orientations of the field H relative to the sample &axis. The hypothesis based on the Abrikosov lattice melting is strongly corroborated by the good fit to the data (Fig. 4) of the melting equation with quantum correction, Eq. (4), as well as by the verification of the angular scaling rule (Fig. 5) predicted for the melting line, Eq. (7). We found values for the Lindemann constant (c L -~ 0.24) and other parameters similar to those found for single crystals, although the sample studied here surely contains more quenched disorder (pinning centers). Thus, we conclude that the Abrikosov lattice melting is probably weakly affected by the quenched disorder, at least in the relatively low fields (H < 50 kOe) and high temperatures (t > 0.8) probed in this work.
Acknowledgements This work was partially supported by Funda~o de Amparo a Pesquisa do Estado de Silo Paulo (Fapesp) and Conselho Nacional de Pesquisas (CNPq). We are thankful to Y. Kopelevich and M.A. Avila for many valuable discussions. We also acknowledge S.S. Sugui Jr., for providing the melt-textured sample used in this study.
References [1] K.A. Mi~ller, M. Takashige and J.G. Bednorz, Phys. Rev. Lett. 58 (1987) 1143.
[2] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lctt. 60 (1988) 2202. [3] M. Tinkham, Phys. Rev. Lett. 61 (1988) 1658. [4] M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [5] P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953. [6] D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130. [7] T. Matsushita, T. Fujiyoshi, K. Toko and K. Yamafuji, Appl. Phys. Lett. 56 (1990) 2039. [8] A. Gupta, P. Esquinazi, H.F. Braun and H.-W. Neumiiller, Phys. Rev. Lett. 63 (1989) 1869. [9] D.R. Nelson, Phys. Rev. Lett. 60 (1988) 1973. [10] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak and D.J. Bishop, Phys. Rev. Lett. 61 (1988) 1666. [11] A. Houghton, R.A. Pelcovits and A. Sudb¢, Phys. Rev. B 40 (1989) 6763. [12] E.H. Brandt, Phys. Rev. Lett. 63 (1989) 1106. [13] H. Safar, P.L. Gammel, D.A. Huse, D.J. Bishop, J.P. Rice and D.M. Ginsberg, Phys. Rev. Lett. 69 (1992) 824. [14] G. Blatter and B.I. Ivlev, Phys. Rev. B 50 (1994) 10272. [15] G. Blatter, M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. This is a very useful review paper. [16] A. Schilling, R. Jin, J.D. Guo, H.R. Ott, I. Tanaka and H. Kojima, Physica B 194-196 (1994) 1555. [17] W. Welp, S. Fleshier, W.K. Kwok, R.A. Klemm, V.M. Vinokur, J. Downey, B. Veal and G.W. Crabtree, Phys. Rev. Lett. 67 (1991) 3180. [18] G. Biatter, V.B. Geshkenbein and A.I. Larkin, Phys. Rev. Lett. 68 (1992) 875. [19] R.G. Beck, D.E. Farrell, J.P. Rice, D.M. Ginsberg and V.G. Kogan, Phys. Rev. Lett. 68 (1992) 1594. [20] D. Shi, J.G. Chen, S. Salem-Sugui Jr. and K.C. Goretta, Physica C 185-189 (1991) 2329. [21] A. Schilling, H.R. Ott and Th. Wolf, Phys. Rev. B 46 (1992) 14253. [22] L.N. Bulaevskii, M. Ledvij and V.G. Kogan, Phys. Rev. Lett. 68 (1992) 3773. [23] Q. Li, M. Suenaga, T. Hikata and IC Sato, Phys. Rev. B 46 (1992) 5857. [24] S. Ullah and A.T. Dorsey, Phys. Rev. Lett. 65 (1990) 2066; idem, Phys. Rev. B 44 (1991) 262. [25] M.B. Salomon and J. Bardeen, Phys. Rev. Lett. 59 (1987) 2615. [26] E.H. Brandt, Physica C 195 (1992) 1. [27] W.K. Kwok, J. Fendrich, U. Welp, S. Fleshier, J. Downey and G.W. Crabtree, Phys. Rev. Lett. 72 (1994) 1088. [28] H. Safar, P.L. Gammel, D.A. Huse, D.J. Bishop, W.C. Lee and D.M. Ginsberg, Phys. Rev. Lett. 70 (1993) 3800. [29] R. Andrade Jr., PhD. thesis, Unicamp (1995), unpublished. [30] P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953.
Physica C 248 (1995) 353-358
Relevance of thermal and quantum fluctuations to the irreversibility line in a melt-textured YBCO sample O.F. de Lima *, R. Andrade Jr. lnstituto de Fisica "Gleb Wataghin", 13083-970 Campinas, SP, Brazil Received 3 March 1995; revised manuscript received 25 April 1995
Abstract The irreversibility line for a melt-textured Y B a 2 C u 3 0 7_ ~ (YBCO) sample was determined in two orientations of the field H relative to the sample &axis. In both orientations we obtain good fits using a melting equation with quantum correction to describe the Abrikosov lattice melting. The scaling analysis based on a fluctuation theory for 3D systems was applied to our magnetization data allowing the determination to be made of the mean-field critical temperature. Our results are similar to those found in the literature for very clean untwinned samples. We suggest that the Abrikosov lattice melting is weakly affected by quenched disorder, at least in the relatively low fields (H < 50 kOe) and high temperatures (T/Tc > 0.8) probed in this work.
1. Introduction The irreversibility line (IL) separates a magnetically irreversible from a reversible state in the H × T phase diagram of type-II superconductors. This is a very well documented experimental fact for the highTc superconductors, although its physical interpretation has aroused some issues. In order to describe the IL, simple power laws of the type Hm(T) --H0(1 T/Tc) n have been suggested, where H 0 is a modeldependent fitting parameter and T~ is the critical 3 temperature. The exponent n = ~is predicted by a quasi-de Almeida-Thouless behavior [1] as well as by the giant flux-creep model [2,3]. The exponent 4 • n = 3 Is expected from a vortex-glass formation [4-6], as well as from Tinkham's giant flux-creep model [3]. An exponent which is dependent on the * Corresponding author.
pinning strength can be found in a collective fluxcreep model [7]. In a more conventional approach the IL is identified with a depinning line, whose solution comes from an equation describing the constant diffusivity, Do(H, T), of the flux lines [8]. Finally, the interpretation of the IL as the melting transition line of the Abrikosov lattice [9-12] has received enough confirmation, and nowadays the melting hypothesis seems to be a well established matter [13-15]. The basic idea is that the Abrikosov lattice becomes unstable and melts, when the mean displacement amplitude of the flux lines, (u2) 1/2, reaches an appreciable fraction of the lattice parameter, a 0. This latter condition is usually expressed in terms of the Lindemann criterion, (u2) 1/2= CLaO, with the Lindemann number c L varying between 0.1 and 0.3. Recently, Blatter et al. [14] discussed the relevance of quantum fluctuations that, combined with thermal fluctuations [11], produce an effective mean
0921-4534/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved
SSDI 0 9 2 1 - 4 5 3 4 ( 9 5 ) 0 0 2 8 8 - X
354
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
displacement amplitude of the flux lines which may correct the melting-line position in some cases. For instance, thermal fluctuations are known to be dominant in the strongly anisotropic BiS-CCO compounds, making quantum corrections vanishingly small [14]. However, in the less anisotropic YBa2Cu307_ a (YBCO) compound the correction due to quantum fluctuations becomes important, as has been discussed by Schilling et al. [16] and Blatter et al. [14], using data from single-crystalline samples. In this paper we discuss results that provide further confimation for the relevance of adding quantum correction in the calculation [14] of the melting line for YBCO. The analysis takes in consideration the universal scaling behavior for the magnetization, caused by fluctuations in the amplitude of the superconducting order parameter [17]. Our study was done on a melt-textured YBCO sample, in contrast with the very clean single crystals analyzed before [1417]. Hence, we conclude that the occurrence of quenched disorder (pinning centers) in the melt-textured sample has little influence on the relative weight of quantum and thermal fluctuations that are required to interpret the IL as a melting line. We measured the IL in two angular directions ( a = 0 °, 60 °) of the field H with respect to the e-axis of the textured sample, and found that these results obey the anisotropy-induced angular dependence predicted for the melting line [18,19].
800
.~. 4 0 0 _
2O0
o (d,,o,,,.)
I ...... i 20
25
1
30 35 40 45 20 (degree)
50 55 60
Fig. 1. X-ray diffraction pattern in the 8-axis oriented geometry for the melt-textured YBCO sample. The inset shows a rocking curve for the (005) peak that reveals an angular spread of ~ 2.5° for the grain orientations.
transition temperature measured under a field of H = 5 0 e is 91.0 K, determined at the transition onset, and AT.-~ 1.3 K, between 10% and 90% of the maximum shielding. Each point in the M(T) curves corresponds to an average over two scans for a length of 3 cm. We employed in this work a magnetic and thermal history similar to that reported by Schilling et al. [21], aiming at the IL determination by conventional M versus T measurements. First, a remanent magnetization curve, Mrem, was obtained (Fig. 2) by cooling the sample from 100 K
.............
i .........
80.0 ~
2. Experimental details
~
-2.0
1
-,,o
o
o
i .........
i .........
i .........
o.01. . . . . . . O
80.0
The melt-textured YBCO sample studied here was obtained by the partial-melt-growth method [20]. It was cut into a parallelepiped having dimensions 2.7 × 2.5 × 1.1 mm 3 and contains long crystalline grains, well aligned with the e-axis direction. Fig. 1 presents the X-ray diffractogram for the e-axis oriented geometry, showing essentially the occurrence of (00l) peaks. The inset of Fig. 1 displays a rocking curve for the (005) peak that reveals an angular spread around 2.5 degrees for the grain orientations along the d direction. DC magnetization measurements, M(T), were made using a commercial SQUID magnetometer (Quantum Design, model MPMS5). The magnetic
(005)
oo
40.0
~" 20.0
f,,,
~IYj
= "llO~
i.,~. ~
o
i .........
o
...... r
I
(~o
0.0 -20.0
~ lid
.........
70
H = 20 kOe i .........
75
i .........
80
i .........
85
i .........
90
a .........
95
h*
100
T (K) Fig. 2. Typical magnetization curves Mrem and MFC (see text) of the melt-textured YBCO sample. Two angles, a = 0°, 60°, between the applied field H and the 8-axis direction were employed. The inset displays a magnified view of the reversible region for ot = 60 ° where the temperatures Ti and Tc2 (see text) are indicated.
O.F.
de Lima,
R. Andrade
Jr./Physica
(normal state) to 10 K"under an applied field H = 50 kOe. Then, a measuring field H was set and the sample warmed up to 70 K, from which the Mrem(T) curve was measured by slowly increasing the sample temperature in steps AT = 0.4 K. Next, a field-cooled curve, MFc(T), was obtained by cooling the sample from 100 K to 70 K, in a measuring field H. From that point the MFc(T) curve was measured by slowly increasing the sample temperature in steps AT = 0.4 K, as before. For each chosen field H, the curves Mrem(T) and MFc(T) merge at an irreversibility temperature Ti(H) as shown in the inset of Fig. 2 for H = 30 kOe. A practical criterion was applied to define Ti(H): where the difference A M = M rem MFC becomes less than the standard deviation of the measurements. An experimentally determined upper critical field, He2 =H(Tc2), is associated with the temperature To2 (see the inset of Fig. 2) where the linear extrapolation [17] of the reversible region meets the normal base line of the magnetization curve. It is worth noting that our data did not show a consistent feature like a break in the slope of the magnetization curve at a temperature T * > Ti, as reported by Schilling et al. [21] in their study with YBCO single crystals.
3. Results and discussion
C 248
(1995)
0.0
Fig. 3(a) shows the characteristic fan-shaped magnetization curves [17,23] in the reversible region near Tc(H), well above the irreversibility point, for fields between 10-45 kOe. The occurrence of a common crossing point at a certain temperature has been evaluated as an effect of fluctuations in the vortex line positions (phase fluctuations) for 2D-like superconductors [22,23]. Vortex-line fluctuations are also expected to occur in anisotropic 3D materials like YBCO [22]. However, this subject is not discussed in the present paper, although Fig. 3(a) clearly displays the cross-over phenomenon at T = 90 K. Next, we present a scaling analysis based on fluctuations in the amplitude of the superconducting order parameter, aiming at the determination of the mean-field transition temperature for zero magnetic field, Tco. In high magnetic fields and around T~(H),
.... ' .........
355
' .........
' .........
' .........
'?:z;::l~ziziz: .
.
.
.
-1.0 c
~"
10kOe
20 kOe 30 kOe 45 kOe
-2.0 ~:~
-3.0 88
90
..., .........
, .........
92 94 T (K) , ..........
. ........
, .........
96
98
100
, .......... . ........ , ....
g o.o g8 -0.5
Yco=91"8K
( / - f- b ) -1,0
g. "r"
~-~ -1,51
#
...JJ. ............................................................ -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 (T-Tc(H)I/(T H)z's (10 4 K 'U3/Oe2t3)
Fig. 3. (a) Reversible region of the magnetization curves, for /4 II~, showing the crossing point around T = 90 K. (b) Scaled magnetization assuming a three-dimensional(3D) superconductor (see Section3.1). Tco= 91.8 K is the mean-fieldcritical temperature.
the scaling behavior predicted for the magnetization due to this kind of fluctuations is given by [17,24]: 4'rrM
3.1. Scaling due to fluctuations
353-358
[
-Tc(H)]
(-~)-, = F A T-( TH )'~
,
(1)
where F is the universal scaling function, A is a temperature and field-independent coefficient, and n is ~ or ~, respectively, for a 3D or 2D system. Fig. 3(b) shows a plot of the scaled magnetization, corresponding approximately to the same region shown in Fig. 3(a), assuming a 3D behavior [17]. The only free parameter in this scaling fit is To0, which can be related to Tc(H) and S n - - d H c 2 / d T by the linear relationship [17,23] T~(H)= T~0+ H/S H. Using S H = - 1.9 × 104 O e / K (see Section 3.2) the best scaling fit occurs when T~0 = 91.8 K. This value is shifted 0.8 K above the experimental transition temperature of 91.0 K, taken for H = 5 Oe. This shift in temperature is known to reach values of about 4.0 K in the case of the more anisotropic BISCCO compounds [23], where fluctuation effects are much stronger. 2
1
356
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
Comparing Figs. 3(a) and (b) one sees that the scaled magnetization curves [Fig. 3(b)] almost collapse onto a single curve, as expected from Eq. (1). The observed spread most probably comes from the combination of two factors: (1) the angular spread ( ",, 2.5 °) of the grains orientation in the melt-textured sample, and (2) the angular uncertainty (,-, 1°) of the sample orientation due to slight bendings of the sample holder during the magnetization measurements. Both situations imply some uncertainty in the function T~(H), which could be minimized by using singlecrystalline samples [17] and by taking all M(T) curves in a single-run experiment.
3.2. Melting line As shown by Houghton et al. [11] the melting line due to thermal fluctuations, for H parallel to the d-axis of an anisotropic material, is
t
b 1/2
( 1 - t ) I/2 ( l ~ b )
[ 4(V~-- 1) [
Vri'----~
+
1
2~
=
- ~ - e c 2,
(2) where t=T/T~ and b=B/H~2, with H¢2(t)= Hc2(0X1- t). The Ginzburg number is Gi = 161r3K4(kBTc)2/~30Hc2(O) ~ 1.06 × 10-9Tc2K4/ He2(0), where K is the Ginzburg-Landau factor, 40 is the flux quantum and k B the Boltzmann constant. The sample a n i s ~ described by the effective mass ratio ~ = ~mob/m c , whose value is e = ~ for YBCO [11,14]. Eq. (2) does not admit a closed-form solution for the melting line B(T), therefore it has to be solved in a self-consistent way when fitting the experimental data. However, close to Tc and taking up to linear terms in v~ for the expansion of Eq. (2), one gets
B( T) -~./3thHc2(0) where (1 - t)/t appears as the natural variable [14], instead of the commonly used form ( 1 - t). The numerical constant is fltn = {2~r/[4(v~- - 1) + 1]}2 = 5.6. Fig. 4 shows the best fits to our IL data ( H If ~), for the complete thermal equation [Eq. (2)] and for
40
,,, ........ Eq. (2) lo
.....
eq.(s)
Eq. (4)
i ~,.
dl. ~..~.
~
' ~
o 76 78 80 82 84 86 88 90 92 T (K)
Fig. 4. Magnetic phase diagram of the melt-textured YBCO sample. The data points define the irreversibility line (O) and the experimental Hc~ line (©). The solid, dotted, and dashed lines represent fits of the melting equation, respectively with quantum correction [Eq. (4)], without quantum correction [Eq. (2)], and in the approximated quadratic form [Eq. (3)]. A linear fit of the Hc~ line, Hc*2(T)=Hc2(O)(1-T/Tc'), is represented by the dotdashed line. The thin solid line shifted to the right represents the Hc2 line, corrected for fluctuations of the order parameter (see Section 3.1).
its simplified form [Eq. (3)], both using the Lindemann number c L = 0.20. Eq. (2) gives a reasonable fit (dotted line) only for H < 20 kOe while the quadratic form, Eq. (3), fits only very close to Tc (dashed line), for H < 10 kOe. Fig. 4 also shows the experimentally determined Hc~ line, fitted by the linear form HcE(T) = H e 2 ( 0 ) ( 1 - T/T¢*) with He2(0) = 1.72 x 10 6 0 e and the extrapolated critical temperature T~* = 90.6 K. The thin solid line shifted to the right in Fig. 4 represents the H~2 line, corrected for fluctuations of the order parameter, as explained in the previous section. This line extrapolates to the mean-field transition temperature, T~0 = 91.8 K, which was employed in all three melting equation forms fitted to the IL data (Fig. 4). The lines Hc~ (T) and Hc2(T) have the same slope S n -dHc2/dT-~ -1.9 × 104 Oe/K, comparable to the commonly quoted values for high-quality YBCO crystals [13,16]: Using these quoted figures and the Ginzburg-Landau factor x = 80, which is representative for YBCO [25,15], we get the Ginzburg number Gi = 5.3 X 10 -3. This Gi number is about six orders of magnitude larger than what is found in conventional superconductors, thus stressing the relevance of thermal fluctuations in the high-Tc material [141. According to Blatter et al. [14], the combined
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
effect of thermal and quantum fluctuations leads to the following compact equation for the melting line: 402 b(t) = (4) 1+
.... , .........
50
~
i .........
357 , .........
,.~
, .........
,o
.........
i .........
, .........
, .........
,.
i .........
i .........
i .........
h~
SO*
~- 30
9 +--
i
~
l 10
with the temperature variable 0
C2
/
~th
1),
and the suppression parameter S
=
2.4 # K F + C
Gi '
.a .........
i .........
i .........
i .........
t .........
76 78 80 82 84 86 88 90 92 T (K)
(6)
where K F is the Fermi wave v e c t o r ( K F = 0.15-0.20 £-1 [14]) and v is a fitting parameter, which relates a cutoff frequency g2 with the superconducting energy gap A, v = h O / A . As pointed out by Blatter et al. [14] only part of the suppression parameter S appearing in Eq. (6) can be attributed to the effect of quantum fluctuations, while the larger contribution actually arises from the suppression of the order parameter, ~ , close to the HoE line. Indeed, an increase of the magnetic penetration depth, A A / ( I ~ 12) = A/(1 - b) 1/2, follows from GinzburgLandau theory [26]. Eq. (4) fits very well to our data as shown by the solid line in Fig. 4, producing c L = 0.24 and v = 4 in quantitative agreement with the analysis of Blatter et al. [14] for untwinned single crystals of YBCO. However, the melting line in our case is shifted to lower temperatures, as would be expected in the presence of quenched disorder [27,14]. For instance, at H = 50 kOe we have Ti = 75.5 K, instead of T i = 81.0 K which was observed in untwinned single crystals [28]. These results suggest that the occurrence of quenched disorder in the melt-textured sample has a negligible influence on the determination of c L and v appearing in Eq. (3). This probably means that the melting of the Abrikosov lattice is weakly affected by the quenched disorder occurring in the melt-textured sample, at least in the relatively lowfield and high-temperature region studied here.
3.3. Angular scaling Fig. 5 shows the IL shifted to higher temperatures when the field H makes an angle a = 60 ° ( + 3 °)
Fig. 5. Magnetic phase diagram of the melt-textured YBCO sample, showing the effect of anisotropy on the irreversibility line for a = 0° (O) and a = 60 ° (O). The solid line represents a fit of the melting equation taking into account thermal and quantum fluctuations [Eq. (4)] for a = 0°. The dashed line represents that same fit corrected by the angular scaling rule B(ot, T ) = B(0, T ) / ( ~ 2 sin2a +cos2a) 1/2.
with the ~-axis direction. The dashed line represents the new position of the melting line B(a, t) when the effect of anisotropy is corrected by the following scaling rule [15,18,19]:
B(O, T) B( or, T) = ~/E2 sin2ot + cos2ot ,
(7)
where a is the angle between the d-axis and H. The dashed line in Fig. 5 is obtained using • = ½, the same mass-anisotropy factor employed in the melting equations [Eqs. (2-4)]. Similar to the conclusion of Beck et al. [19], we believe that the verification of the scaling rule given by Eq. (7) corroborates the melting hypothesis.
3.4. Power-law fits Using T~0 = 91.8 K the power law Hm(T) = H0(1
- T/T¢o) n was tested [29] for n = 3 (giant flux-creep model [2,3] and quasi-de Almeida-Thouless line [1]) and n = 34 (vortex glass model [4-6]). For the first case (n = 3) a very good fit was found, as already reported from other experiments [2,14,16,21]. For the second case (n = 4) only a very poor fit was obtained. At first sight it seems that the giant flux-creep model or a quasi-de Almeida-Thouless line could be good explanations for the IL, while the vortex glass model should be discarded. However, following Blatter et al. [14], we could not draw such conclu-
358
O.F. de Lima, R. Andrade Jr./Physica C 248 (1995) 353-358
sions based only on those power-law fits, because they lack a sound theoretical background. Hence, other types of experimental data, like high-resolution I - V curves [30] and long time decay of magnetization [6] would be required in order to allow for a more conclusive analysis based on the relevant critical exponents [6,15].
4. Conclusions The irreversibility line for a high-quality melt-textured YBCO sample was determined in two orientations of the field H relative to the sample &axis. The hypothesis based on the Abrikosov lattice melting is strongly corroborated by the good fit to the data (Fig. 4) of the melting equation with quantum correction, Eq. (4), as well as by the verification of the angular scaling rule (Fig. 5) predicted for the melting line, Eq. (7). We found values for the Lindemann constant (c L -~ 0.24) and other parameters similar to those found for single crystals, although the sample studied here surely contains more quenched disorder (pinning centers). Thus, we conclude that the Abrikosov lattice melting is probably weakly affected by the quenched disorder, at least in the relatively low fields (H < 50 kOe) and high temperatures (t > 0.8) probed in this work.
Acknowledgements This work was partially supported by Funda~o de Amparo a Pesquisa do Estado de Silo Paulo (Fapesp) and Conselho Nacional de Pesquisas (CNPq). We are thankful to Y. Kopelevich and M.A. Avila for many valuable discussions. We also acknowledge S.S. Sugui Jr., for providing the melt-textured sample used in this study.
References [1] K.A. Mi~ller, M. Takashige and J.G. Bednorz, Phys. Rev. Lett. 58 (1987) 1143.
[2] Y. Yeshurun and A.P. Malozemoff, Phys. Rev. Lctt. 60 (1988) 2202. [3] M. Tinkham, Phys. Rev. Lett. 61 (1988) 1658. [4] M.P.A. Fisher, Phys. Rev. Lett. 62 (1989) 1415. [5] P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953. [6] D.S. Fisher, M.P.A. Fisher and D.A. Huse, Phys. Rev. B 43 (1991) 130. [7] T. Matsushita, T. Fujiyoshi, K. Toko and K. Yamafuji, Appl. Phys. Lett. 56 (1990) 2039. [8] A. Gupta, P. Esquinazi, H.F. Braun and H.-W. Neumiiller, Phys. Rev. Lett. 63 (1989) 1869. [9] D.R. Nelson, Phys. Rev. Lett. 60 (1988) 1973. [10] P.L. Gammel, L.F. Schneemeyer, J.V. Waszczak and D.J. Bishop, Phys. Rev. Lett. 61 (1988) 1666. [11] A. Houghton, R.A. Pelcovits and A. Sudb¢, Phys. Rev. B 40 (1989) 6763. [12] E.H. Brandt, Phys. Rev. Lett. 63 (1989) 1106. [13] H. Safar, P.L. Gammel, D.A. Huse, D.J. Bishop, J.P. Rice and D.M. Ginsberg, Phys. Rev. Lett. 69 (1992) 824. [14] G. Blatter and B.I. Ivlev, Phys. Rev. B 50 (1994) 10272. [15] G. Blatter, M.V. Feigel'man, V.B. Geshkenbein, A.I. Larkin and V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. This is a very useful review paper. [16] A. Schilling, R. Jin, J.D. Guo, H.R. Ott, I. Tanaka and H. Kojima, Physica B 194-196 (1994) 1555. [17] W. Welp, S. Fleshier, W.K. Kwok, R.A. Klemm, V.M. Vinokur, J. Downey, B. Veal and G.W. Crabtree, Phys. Rev. Lett. 67 (1991) 3180. [18] G. Biatter, V.B. Geshkenbein and A.I. Larkin, Phys. Rev. Lett. 68 (1992) 875. [19] R.G. Beck, D.E. Farrell, J.P. Rice, D.M. Ginsberg and V.G. Kogan, Phys. Rev. Lett. 68 (1992) 1594. [20] D. Shi, J.G. Chen, S. Salem-Sugui Jr. and K.C. Goretta, Physica C 185-189 (1991) 2329. [21] A. Schilling, H.R. Ott and Th. Wolf, Phys. Rev. B 46 (1992) 14253. [22] L.N. Bulaevskii, M. Ledvij and V.G. Kogan, Phys. Rev. Lett. 68 (1992) 3773. [23] Q. Li, M. Suenaga, T. Hikata and IC Sato, Phys. Rev. B 46 (1992) 5857. [24] S. Ullah and A.T. Dorsey, Phys. Rev. Lett. 65 (1990) 2066; idem, Phys. Rev. B 44 (1991) 262. [25] M.B. Salomon and J. Bardeen, Phys. Rev. Lett. 59 (1987) 2615. [26] E.H. Brandt, Physica C 195 (1992) 1. [27] W.K. Kwok, J. Fendrich, U. Welp, S. Fleshier, J. Downey and G.W. Crabtree, Phys. Rev. Lett. 72 (1994) 1088. [28] H. Safar, P.L. Gammel, D.A. Huse, D.J. Bishop, W.C. Lee and D.M. Ginsberg, Phys. Rev. Lett. 70 (1993) 3800. [29] R. Andrade Jr., PhD. thesis, Unicamp (1995), unpublished. [30] P.L. Gammel, L.F. Schneemeyer and D.J. Bishop, Phys. Rev. Lett. 66 (1991) 953.