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3D XY versus 3D LLL revisited in YBa2Cu3O7−δ
Physica C 452 (2007) 1–5 www.elsevier.com/locate/physc
3D XY versus 3D LLL revisited in YBa2Cu3O7d Sebastian Ujevic b
a,*
, E.Z. da Silva b, S. Salem-Sugui Jr.
c
a CIFMC, Universidade de Brası´lia, 70904-970, Brası´lia – DF, Brazil Instituto de Fı´sica, Universidade Estadual de Campinas, 13083-970 Campinas – SP, Brazil c Instituto de Fı´sica – UFRJ CP 68528 Rio de Janeiro, RJ, Brazil
Received 22 August 2006; received in revised form 12 November 2006; accepted 14 November 2006 Available online 11 January 2007
Abstract We analyzed magnetization data from a YBa2Cu3O7d sample with a sharp transition at Tc = 92.3 K. Two models are discussed and compared. The XY model in three dimensions and the Ginzburg–Landau description based in the lowest Landau level approximation. We confirm the validity of the XY model scaling on low applied magnetic fields. For high applied magnetic fields the scaling behavior of both models provides similar results. Ó 2006 Elsevier B.V. All rights reserved. PACS: 74.72.Bk; 74.25.Ha; 74.25.Bt Keywords: High-Tc superconductors; Thermodynamic properties; Scaling phenomena
1. Introduction In the last years, since the discovery of the high-Tc superconductors, the topic of thermal fluctuations near the superconducting transition have been a subject of great interest by the scientific community [1–16]. Several properties of physical quantities near the critical point, like magnetization [2], heat capacity [12], irreversible line [13], anisotropy [13,16] and magnetic torque [14], can be understood in terms of scaling functions. Thermal fluctuations are enhanced in high-Tc superconductors due to their small correlation lengths, highly laminar structure and high transition temperatures. Two models prove to be important in the description of these fluctuations, each one assume to be valid in different ranges of applied magnetic fields (H); the XY model in three dimensions (3D XY) [17] and the Ginzburg–Landau theory in the lowest Landau level (LLL) approximation [18].
*
Corresponding author. Tel.: +55 61 33072571. E-mail addresses: [email protected] (S. Ujevic), zacarias@ifi.unicamp.br (E.Z. da Silva). 0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.11.006
The LLL approximation is based on the phenomenological theory of Ginzburg–Landau which is valid only in the vicinity of the superconducting transition, i.e., close to Tc. However the LLL description requires high applied magnetic fields (HAMF) (above 1 T) to confine the Cooper pairs in the LLL [19,20]. In the other side the 3D XY model is based on renormalization group theories and scaling. Through these properties and universality ideas, scaling forms for physical thermodynamic quantities near the superconducting critical point can be constructed [14,15, 21,22]. Once these quantities are available, previously calculated values of the critical exponents for an specific model can be used. In this work we employed a critical exponent [23] for the 3D XY model valid for low applied magnetic field (LAMF) (below 1 T). Due to their different constructions, the scaling forms for the LLL description and the 3D XY model are formally correct near the critical point and HAMF/LAMF, respectively. In this paper, we are interested to compare the scaling behavior of the magnetic fluctuation, for different ranges of applied magnetic fields, using these two approaches. To accomplished this, we used high-precision magnetic measurements for Hkab-planes in a high-quality twin-aligned
2
S. Ujevic et al. / Physica C 452 (2007) 1–5
single crystal of YBa2Cu3O7d, with a sharp transition at Tc = 92.3 K at zero applied magnetic field. The paper is organized as follows, in Section 2 we present the experimental details of the sample, in Section 3 the scaling functions used by each model are shown, in Section 4 we examine the behavior of the LLL and 3D XY scaling in different ranges of the applied magnetic field. Finally in Section 5 the conclusions are presented.
2. Experimental This work presents precision magnetic measurements for Hkab-planes, in a high-quality twin-aligned single crystal of YBa2Cu3O7d with a sharp transition (which occurs within a DT < 300 mK) at the critical temperature of Tc = 92.3 K at zero applied magnetic field. Magnetic measurements were performed in a commercial SQUID with the magnetic field applied parallel to the ab-planes of the crystal. Scans of 3 cm were used. The sample having a triangular shape with 3 mm height is attached with vacuum grease to a smooth clean plastic surface parallel to H. The background signal due to the sample holder is found to be very small (less than 107 emu for H = 0.05 T) and temperature independent for a fixed field. A previous calibration of the sample position with temperature, and the use of an appropriate routine for data acquisition, allows the sample to be centered during any temperature varying experiment. For the type of geometry considered in the present experiments, a large enough misorientation angle could allow current flowing in the ab-planes [24]. This possibility is discarded, since no transversal magnetic moment (magnetization measured along the c-axis) was observed even in the irreversible region, for H = 5 T measurements. The misorientation of the sample with H is therefore estimated to be smaller than 1°. Zero field-cooled (ZFC) and field-cooled (FC) Magnetization (M) vs temperature (T) curves were obtained for each value of the applied magnetic field. Zero field-cooled measurements were preceded by an oscillating discharge of the magnet from 1 T at T = 130 K. The magnetization was read at intervals of 0.3 K from 80 to 130 K and 1.5 K from 130 to 200 K. Measurements were performed with the magnetic field parallel to the twin boundaries of the crystal. Fig. 1 shows M vs T curves for H 6 1 T in the temperature range of 87–105 K, and in Fig. 2 we present curves for H P 2 T in the range of 87–103 K. All M vs T curves in Fig. 1 exhibit a clear temperature value above which no detectable diamagnetism is observed. These temperature values are associated to Tc(H). Nevertheless, from Fig. 2, a small broadening of the transition around Tc is observed for fields higher than 2 T. Such a behavior above Tc has been attributed to superconducting fluctuation diamagnetism [25]. For T > 150 K (this temperature is smaller for fields H < 5 T), the magnetization is found to be practically temperature independent, and is considered as the normal state magnetization background, which is subtracted.
Fig. 1. Reversible magnetization for Hkab-plane, for H = 0.05 to 1 T in the temperature range of 87–105 K.
Fig. 2. Reversible magnetization for Hkab-plane, for H = 2 to 5 T in the temperature range of 87–103 K.
3. Scaling forms The LLL description predicts a scaling function for the Helmholtz free energy of the form [1,26,27]
S. Ujevic et al. / Physica C 452 (2007) 1–5 4
F ðH ; T Þ ¼ ðHT Þ3 fLLL ðxÞ;
ð3:1Þ
where fLLL(x) is an universal scaling function of a variable x given by x¼
½T T c ðH Þ 2
:
ðHT Þ3
ð3:2Þ
The function Tc(H) in the scaling variable x refers to the transition temperature at each applied magnetic field. Starting from the Helmholtz free energy of Eq. (3.1) we are able to calculate a scaling form for the magnetization function, which is given by 2
MðH ; T Þ ¼ ðHT Þ3 mLLL ðxÞ;
ð3:3Þ
where mLLL(x) is an universal function of x. On the other hand, the 3D XY model foresee a scaling form for the Helmholtz free energy [1,26] d
F ðH ; T Þ ¼ H 2 fXY ðyÞ;
ð3:4Þ
with the scaling variable y given by y¼
½T =T c 1 1
H 2m
:
ð3:5Þ
In Eqs. (3.4) and (3.5) d is the dimensionality of the system, fXY(y) is an universal function and m is the critical exponent associated to the correlation distance in the transition. In principle more sophisticated Helmholtz free energy function for the 3D XY model can be used, with explicit angular applied field or correlation length dependencies, to study for example 2D–3D XY crossover due to high anisotropy [10,14]. However, the expressions considered in this work, Eqs. (3.1) and (3.4), are the simplest for our purposes. Note that in the 3D XY model the scaling variable y depends on a fixed critical temperature Tc and not on a field depended critical temperature Tc(H), like in the LLL scaling variable x. The critical exponent for Eq. (3.5) have already been calculated for the 3D XY model [23] providing the value of 2m1 ¼ 0:747 for the field variable. Using Eq. (3.4) it is possible to deduce an scaling form for the magnetization function [6,26] of the form MðH ; T Þ ¼ H 0:5 mXY ðyÞ;
3
4. Results 4.1. Low applied magnetic field The LAMF region was investigated by considering magnetization data at four applied magnetic fields, 1, 0.5, 0.25 and 0.05 T, as shown in Fig. 1. We have analyzed these data within the XY model, Eqs. (3.5) and (3.6), obtaining Fig. 3. We have considered in the scaling variable of Eq. (3.5) a critical temperature of Tc = 92.25 K. The temperature range in Fig. 3 for the 1 and 0.05 T curves are 78– 97 K and 83–94 K, respectively. The 3D XY model gives an accurate description of the scaling behavior of the magnetization fluctuation in this region. Data lines corresponding to different applied magnetic fields fall perfectly in a single line in the temperature range. The analysis of the LAMF data using the LLL approach, with an scaling variable given by Eq. (3.2), shows no scaling behavior due to the fact that the Cooper pairs are not confined in the LLL. 4.2. High applied magnetic field The HAMF region was investigated by considering magnetization data at four applied magnetic fields values, 2, 3, 4 and 5 T, as shown in Fig. 2. Within the context of the 3D XY model, Eqs. (3.6) and (3.5), we obtained Fig. 4. A critical temperature of Tc = 92.15 K was considered for the plots. In Fig. 4 the temperature range for the 5 and 2 T curves goes from 73 to 109 K and 82 to 100 K, respectively. Note the pronounce scaling behavior with
ð3:6Þ
where the field exponent value of 0.5 arises from considering a 3D dimensional system (d = 3). A similar expression for the magnetization with the same critical exponent was employed by Hubbard et al. [16] to scaled fluctuation magnetization data for a number of oxygen deficient in YBa2Cu3O6+d samples with a good consistency with the behavior predicted by the 3D XY model. In the next section the LLL and the 3D XY approaches (Eqs. (3.3) and (3.6)) are employed to study thermal fluctuations in different ranges of applied magnetic field.
Fig. 3. Magnetization scaling using 3D XY model for low applied magnetic fields.
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S. Ujevic et al. / Physica C 452 (2007) 1–5
used for the LAMF and HAMF scaling within the 3D XY model. Note that the curves from LAMF and HAMF do not scale if we put them together using an unique Tc, evidencing different behaviors. The analysis of the HAMF data using the LLL description give us Fig. 5. An accurate description of the scaling behavior is observed. The values of Tc(H) obtained through this scaling are in agreement with the values obtained using a linear extrapolation of the reversible part of the M vs H curves. The temperature scale in Fig. 5 goes from 81 to 110 K and 86 to 100 K for the 5 and 2 T curves, respectively. 5. Conclusions
Fig. 4. Magnetization scaling using 3D XY model for high applied magnetic fields.
Fig. 5. Magnetization scaling using LLL description for high applied magnetic fields.
the collapse of all data points corresponding to different applied magnetic fields into a single line. We have obtained a difference of 0.10 K between the critical temperatures
In this work we used magnetization data from a YBa2Cu3O7d sample for high and low applied magnetic field to describe the magnetization scaling by two approaches, the LLL approximation and the 3D XY model. Our study corroborated that for LAMF the description of thermal fluctuations based on the 3D XY model is adequate, with a good scaling behavior for all studied magnetic fields values. In the same region of applied fields the LLL description gives us a poor account of the fluctuation behavior. Still today, they are no scaling functions based on the Ginzburg–Landau phenomenological theory that describe correctly the magnetization scaling at low applied magnetic fields values. For the HAMF region we obtained interesting results. Despite of the fact that the 3D XY model is formally correct close near the critical point and LAMF (due to explicit construction), the model could be used in this region to describe the magnetization scaling with good results. This is in accordance with previous investigation of magnetization fluctuation performed in the high applied field region [13,16]. The 3D XY scaling in LAMF and HAMF were achieved using different critical temperatures Tc. We could not found a unique Tc that could fit both data in the same curve, therefore, there is not a simple criterion to determine the value for Tc. This behavior must be taken into account in processes that explore both regions of applied magnetic fields. The results depicted in Figs. 4 and 5 leads us to conclude that the scaling behavior of the magnetization fluctuations in HAMF could be represented either by the LLL formalism or by the 3D XY model. Both scalings, although different, give excellent universal plots. However, we performed a dispersion analysis between all points in Figs. 4 and 5 over the critical temperature Tc. The analysis show that more accurate results can be obtained using the LLL formalism. Ref. [1] claims a perfect scaling in this region using the 3D XY model, even better than LLL scaling, our results do not verify this last one point. Finally, it is important to comment that the LLL formalism have an explicit analytical form for the scaling function of the Helmholtz free energy [5], Eq. (3.1). This analytical function for the XY model is somehow unclear.
S. Ujevic et al. / Physica C 452 (2007) 1–5
However, Hofer et al. [15] obtained (for the XY model) an analytical expression for the derivative of the free energy universal function for important limit values of the scaling variable and tested it in torque magnetometry and magnetization [14] with good results. It would be interesting, due to the accuracy of the LLL scaling, to compared results for both approaches for other physical quantities in the high applied magnetic field region. Also we hope that it might be possible to use our results as guidance in more detailed investigations of the analytical representations of scaling functions for the XY model, and to further increase our understanding of thermal fluctuations near the critical point. Acknowledgments This work was partially supported by ‘‘Fundac¸a˜o de Amparo a` Pesquisa do Estado de Sa˜o Paulo – FAPESP’’ and ‘‘Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico – CNPq’’. Sebastian Ujevic acknowledge financial support from ‘‘Financiadora de Estudos e Projetos – FINEP’’ and ‘‘Ministe´rio da Cieˆncia e Tecnologia – MCT’’. References [1] M. Roulin, A. Junod, E. Walker, Physica C 260 (1996) 257. [2] J.M. Calero, J.C. Granada, E.Z. da Silva, Phys. Rev. B 56 (1997) 6114. [3] E.Z. da Silva, S. Salem-Sugui Jr., Physica C 257 (1995) 173.
5
[4] K. Moloni, M. Friesen, S. Li, V. Souw, P. Metcalf, L. Hou, M. McElfresh, Phys. Rev. Lett. 78 (1997) 3173. [5] N.K. Wilkin, M.A. Moore, Phys. Rev. B 48 (1993) 3464. [6] M.B. Salamon, J. Shi, Phys. Rev. B 47 (1993) 5520. [7] S. Ullah, A.T. Dorsey, Phys. Rev. B 44 (1991) 262. [8] N. Overend, M.A. Howson, I.D. Lawrie, Phys. Rev. B 54 (1996) 9499. [9] Z. Tesanovic, L. Xing, Phys. Rev. Lett. 67 (1991) 2729. [10] T. Schneider, J.M. Singer, Physica C 341–348 (2000) 87. [11] T. Schneider, J.M. Singer, Physica C 313 (1999) 188. [12] N. Overend, M.A. Howson, I.D. Lawrie, Phys. Rev. Lett. 72 (1994) 3238. [13] D. Babic, J.R. Cooper, J.W. Hodby, Chen Changkang, Phys. Rev. B 60 (1999) 698. [14] J. Hofer, T. Schneider, J.M. Singer, M. Willemin, H. Keller, T. Sasagawa, K. Kishio, K. Conder, J. Karpinski, Phys. Rev. B 62 (2000) 631. [15] J. Hofer, T. Schneider, J.M. Singer, M. Willemin, H. Keller, C. Rossel, J. Karpinski, Phys. Rev. B 60 (1999) 1332. [16] M.A. Hubbard, M.B. Salamon, B.W. Veal, Physica C 259 (1996) 309. [17] D.D. Betts, Phase Transitions and Critical Phenomena 3, vol. 569, Academic Press, London, 1974. [18] W.J. Skocpol, M. Tinkham, Rep. Prog. Phys. 38 (1975) 1049. [19] U. Welp, S. Fleshler, W.K. Kwok, R.A. Klemm, V.M. Vinoleur, J. Downey, B. Veal, G.W. Crabtree, Phys. Rev. Lett. 67 (1991) 3180. [20] P.A. Lee, S.R. Shenoy, Phys. Rev. Lett. 28 (1972) 1025. [21] V. Privman, P.C. Hohenberg, A. Aharony, Phase Transitions and Critical Phenomena 14, vol. 1, Academic Press, London, 1991. [22] L.P. Kadanoff, Phase Transit. Crit. Phenom. 5A (1976) 1. [23] J.C. Le Guillou, J. Zinn-Justin, Phys. Rev. B 21 (1980) 3976. [24] L.J. Campbell, M.M. Doria, V.G. Kogan, Phys. Rev. B 38 (1988) 2439. [25] W.C. Lee, R.A. Klemm, D.C. Johnston, Phys. Rev. Lett 63 (1989) 1012. [26] M. Friesen, P. Muzikar, Physica C 302 (1998) 67. [27] I.D. Lawrie, Phys. Rev. B 50 (1994) 9456.
3D XY versus 3D LLL revisited in YBa2Cu3O7d Sebastian Ujevic b
a,*
, E.Z. da Silva b, S. Salem-Sugui Jr.
c
a CIFMC, Universidade de Brası´lia, 70904-970, Brası´lia – DF, Brazil Instituto de Fı´sica, Universidade Estadual de Campinas, 13083-970 Campinas – SP, Brazil c Instituto de Fı´sica – UFRJ CP 68528 Rio de Janeiro, RJ, Brazil
Received 22 August 2006; received in revised form 12 November 2006; accepted 14 November 2006 Available online 11 January 2007
Abstract We analyzed magnetization data from a YBa2Cu3O7d sample with a sharp transition at Tc = 92.3 K. Two models are discussed and compared. The XY model in three dimensions and the Ginzburg–Landau description based in the lowest Landau level approximation. We confirm the validity of the XY model scaling on low applied magnetic fields. For high applied magnetic fields the scaling behavior of both models provides similar results. Ó 2006 Elsevier B.V. All rights reserved. PACS: 74.72.Bk; 74.25.Ha; 74.25.Bt Keywords: High-Tc superconductors; Thermodynamic properties; Scaling phenomena
1. Introduction In the last years, since the discovery of the high-Tc superconductors, the topic of thermal fluctuations near the superconducting transition have been a subject of great interest by the scientific community [1–16]. Several properties of physical quantities near the critical point, like magnetization [2], heat capacity [12], irreversible line [13], anisotropy [13,16] and magnetic torque [14], can be understood in terms of scaling functions. Thermal fluctuations are enhanced in high-Tc superconductors due to their small correlation lengths, highly laminar structure and high transition temperatures. Two models prove to be important in the description of these fluctuations, each one assume to be valid in different ranges of applied magnetic fields (H); the XY model in three dimensions (3D XY) [17] and the Ginzburg–Landau theory in the lowest Landau level (LLL) approximation [18].
*
Corresponding author. Tel.: +55 61 33072571. E-mail addresses: [email protected] (S. Ujevic), zacarias@ifi.unicamp.br (E.Z. da Silva). 0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.11.006
The LLL approximation is based on the phenomenological theory of Ginzburg–Landau which is valid only in the vicinity of the superconducting transition, i.e., close to Tc. However the LLL description requires high applied magnetic fields (HAMF) (above 1 T) to confine the Cooper pairs in the LLL [19,20]. In the other side the 3D XY model is based on renormalization group theories and scaling. Through these properties and universality ideas, scaling forms for physical thermodynamic quantities near the superconducting critical point can be constructed [14,15, 21,22]. Once these quantities are available, previously calculated values of the critical exponents for an specific model can be used. In this work we employed a critical exponent [23] for the 3D XY model valid for low applied magnetic field (LAMF) (below 1 T). Due to their different constructions, the scaling forms for the LLL description and the 3D XY model are formally correct near the critical point and HAMF/LAMF, respectively. In this paper, we are interested to compare the scaling behavior of the magnetic fluctuation, for different ranges of applied magnetic fields, using these two approaches. To accomplished this, we used high-precision magnetic measurements for Hkab-planes in a high-quality twin-aligned
2
S. Ujevic et al. / Physica C 452 (2007) 1–5
single crystal of YBa2Cu3O7d, with a sharp transition at Tc = 92.3 K at zero applied magnetic field. The paper is organized as follows, in Section 2 we present the experimental details of the sample, in Section 3 the scaling functions used by each model are shown, in Section 4 we examine the behavior of the LLL and 3D XY scaling in different ranges of the applied magnetic field. Finally in Section 5 the conclusions are presented.
2. Experimental This work presents precision magnetic measurements for Hkab-planes, in a high-quality twin-aligned single crystal of YBa2Cu3O7d with a sharp transition (which occurs within a DT < 300 mK) at the critical temperature of Tc = 92.3 K at zero applied magnetic field. Magnetic measurements were performed in a commercial SQUID with the magnetic field applied parallel to the ab-planes of the crystal. Scans of 3 cm were used. The sample having a triangular shape with 3 mm height is attached with vacuum grease to a smooth clean plastic surface parallel to H. The background signal due to the sample holder is found to be very small (less than 107 emu for H = 0.05 T) and temperature independent for a fixed field. A previous calibration of the sample position with temperature, and the use of an appropriate routine for data acquisition, allows the sample to be centered during any temperature varying experiment. For the type of geometry considered in the present experiments, a large enough misorientation angle could allow current flowing in the ab-planes [24]. This possibility is discarded, since no transversal magnetic moment (magnetization measured along the c-axis) was observed even in the irreversible region, for H = 5 T measurements. The misorientation of the sample with H is therefore estimated to be smaller than 1°. Zero field-cooled (ZFC) and field-cooled (FC) Magnetization (M) vs temperature (T) curves were obtained for each value of the applied magnetic field. Zero field-cooled measurements were preceded by an oscillating discharge of the magnet from 1 T at T = 130 K. The magnetization was read at intervals of 0.3 K from 80 to 130 K and 1.5 K from 130 to 200 K. Measurements were performed with the magnetic field parallel to the twin boundaries of the crystal. Fig. 1 shows M vs T curves for H 6 1 T in the temperature range of 87–105 K, and in Fig. 2 we present curves for H P 2 T in the range of 87–103 K. All M vs T curves in Fig. 1 exhibit a clear temperature value above which no detectable diamagnetism is observed. These temperature values are associated to Tc(H). Nevertheless, from Fig. 2, a small broadening of the transition around Tc is observed for fields higher than 2 T. Such a behavior above Tc has been attributed to superconducting fluctuation diamagnetism [25]. For T > 150 K (this temperature is smaller for fields H < 5 T), the magnetization is found to be practically temperature independent, and is considered as the normal state magnetization background, which is subtracted.
Fig. 1. Reversible magnetization for Hkab-plane, for H = 0.05 to 1 T in the temperature range of 87–105 K.
Fig. 2. Reversible magnetization for Hkab-plane, for H = 2 to 5 T in the temperature range of 87–103 K.
3. Scaling forms The LLL description predicts a scaling function for the Helmholtz free energy of the form [1,26,27]
S. Ujevic et al. / Physica C 452 (2007) 1–5 4
F ðH ; T Þ ¼ ðHT Þ3 fLLL ðxÞ;
ð3:1Þ
where fLLL(x) is an universal scaling function of a variable x given by x¼
½T T c ðH Þ 2
:
ðHT Þ3
ð3:2Þ
The function Tc(H) in the scaling variable x refers to the transition temperature at each applied magnetic field. Starting from the Helmholtz free energy of Eq. (3.1) we are able to calculate a scaling form for the magnetization function, which is given by 2
MðH ; T Þ ¼ ðHT Þ3 mLLL ðxÞ;
ð3:3Þ
where mLLL(x) is an universal function of x. On the other hand, the 3D XY model foresee a scaling form for the Helmholtz free energy [1,26] d
F ðH ; T Þ ¼ H 2 fXY ðyÞ;
ð3:4Þ
with the scaling variable y given by y¼
½T =T c 1 1
H 2m
:
ð3:5Þ
In Eqs. (3.4) and (3.5) d is the dimensionality of the system, fXY(y) is an universal function and m is the critical exponent associated to the correlation distance in the transition. In principle more sophisticated Helmholtz free energy function for the 3D XY model can be used, with explicit angular applied field or correlation length dependencies, to study for example 2D–3D XY crossover due to high anisotropy [10,14]. However, the expressions considered in this work, Eqs. (3.1) and (3.4), are the simplest for our purposes. Note that in the 3D XY model the scaling variable y depends on a fixed critical temperature Tc and not on a field depended critical temperature Tc(H), like in the LLL scaling variable x. The critical exponent for Eq. (3.5) have already been calculated for the 3D XY model [23] providing the value of 2m1 ¼ 0:747 for the field variable. Using Eq. (3.4) it is possible to deduce an scaling form for the magnetization function [6,26] of the form MðH ; T Þ ¼ H 0:5 mXY ðyÞ;
3
4. Results 4.1. Low applied magnetic field The LAMF region was investigated by considering magnetization data at four applied magnetic fields, 1, 0.5, 0.25 and 0.05 T, as shown in Fig. 1. We have analyzed these data within the XY model, Eqs. (3.5) and (3.6), obtaining Fig. 3. We have considered in the scaling variable of Eq. (3.5) a critical temperature of Tc = 92.25 K. The temperature range in Fig. 3 for the 1 and 0.05 T curves are 78– 97 K and 83–94 K, respectively. The 3D XY model gives an accurate description of the scaling behavior of the magnetization fluctuation in this region. Data lines corresponding to different applied magnetic fields fall perfectly in a single line in the temperature range. The analysis of the LAMF data using the LLL approach, with an scaling variable given by Eq. (3.2), shows no scaling behavior due to the fact that the Cooper pairs are not confined in the LLL. 4.2. High applied magnetic field The HAMF region was investigated by considering magnetization data at four applied magnetic fields values, 2, 3, 4 and 5 T, as shown in Fig. 2. Within the context of the 3D XY model, Eqs. (3.6) and (3.5), we obtained Fig. 4. A critical temperature of Tc = 92.15 K was considered for the plots. In Fig. 4 the temperature range for the 5 and 2 T curves goes from 73 to 109 K and 82 to 100 K, respectively. Note the pronounce scaling behavior with
ð3:6Þ
where the field exponent value of 0.5 arises from considering a 3D dimensional system (d = 3). A similar expression for the magnetization with the same critical exponent was employed by Hubbard et al. [16] to scaled fluctuation magnetization data for a number of oxygen deficient in YBa2Cu3O6+d samples with a good consistency with the behavior predicted by the 3D XY model. In the next section the LLL and the 3D XY approaches (Eqs. (3.3) and (3.6)) are employed to study thermal fluctuations in different ranges of applied magnetic field.
Fig. 3. Magnetization scaling using 3D XY model for low applied magnetic fields.
4
S. Ujevic et al. / Physica C 452 (2007) 1–5
used for the LAMF and HAMF scaling within the 3D XY model. Note that the curves from LAMF and HAMF do not scale if we put them together using an unique Tc, evidencing different behaviors. The analysis of the HAMF data using the LLL description give us Fig. 5. An accurate description of the scaling behavior is observed. The values of Tc(H) obtained through this scaling are in agreement with the values obtained using a linear extrapolation of the reversible part of the M vs H curves. The temperature scale in Fig. 5 goes from 81 to 110 K and 86 to 100 K for the 5 and 2 T curves, respectively. 5. Conclusions
Fig. 4. Magnetization scaling using 3D XY model for high applied magnetic fields.
Fig. 5. Magnetization scaling using LLL description for high applied magnetic fields.
the collapse of all data points corresponding to different applied magnetic fields into a single line. We have obtained a difference of 0.10 K between the critical temperatures
In this work we used magnetization data from a YBa2Cu3O7d sample for high and low applied magnetic field to describe the magnetization scaling by two approaches, the LLL approximation and the 3D XY model. Our study corroborated that for LAMF the description of thermal fluctuations based on the 3D XY model is adequate, with a good scaling behavior for all studied magnetic fields values. In the same region of applied fields the LLL description gives us a poor account of the fluctuation behavior. Still today, they are no scaling functions based on the Ginzburg–Landau phenomenological theory that describe correctly the magnetization scaling at low applied magnetic fields values. For the HAMF region we obtained interesting results. Despite of the fact that the 3D XY model is formally correct close near the critical point and LAMF (due to explicit construction), the model could be used in this region to describe the magnetization scaling with good results. This is in accordance with previous investigation of magnetization fluctuation performed in the high applied field region [13,16]. The 3D XY scaling in LAMF and HAMF were achieved using different critical temperatures Tc. We could not found a unique Tc that could fit both data in the same curve, therefore, there is not a simple criterion to determine the value for Tc. This behavior must be taken into account in processes that explore both regions of applied magnetic fields. The results depicted in Figs. 4 and 5 leads us to conclude that the scaling behavior of the magnetization fluctuations in HAMF could be represented either by the LLL formalism or by the 3D XY model. Both scalings, although different, give excellent universal plots. However, we performed a dispersion analysis between all points in Figs. 4 and 5 over the critical temperature Tc. The analysis show that more accurate results can be obtained using the LLL formalism. Ref. [1] claims a perfect scaling in this region using the 3D XY model, even better than LLL scaling, our results do not verify this last one point. Finally, it is important to comment that the LLL formalism have an explicit analytical form for the scaling function of the Helmholtz free energy [5], Eq. (3.1). This analytical function for the XY model is somehow unclear.
S. Ujevic et al. / Physica C 452 (2007) 1–5
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