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Theoretical upper critical field Hc2 for inhomogeneous high temperature superconductors
Physica C 408–410 (2004) 348–349 www.elsevier.com/locate/physc
Theoretical upper critical field Hc2 for inhomogeneous high temperature superconductors E.S. Caixeiro *, J.L. Gonz alez, E.V.L. de Mello Departamento de Fısica, Universidade Federal Fluminense, av. Litor^ania s/n, Niteroi, RJ 24210-340, Brazil
Abstract We present the theoretical upper critical field Hc2 ðT Þ of the high temperature superconductors (HTSC), calculated through a linearized Ginzburg–Landau equation modified to consider the intrinsic inhomogeneity of the HTSC. The unusual behavior of Hc2 ðT Þ for these compounds, and other properties like the Meissner and Nernst effects detected at temperatures much higher than the critical temperature Tc of the sample, are explained by the approach. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.20.)z; 74.81.)g Keywords: Percolation; Inhomogeneity
1. Introduction The H–T phase diagram of the HTSC exhibit, in certain cases, an unusual behavior: a positive curvature for the upper critical field Hc2 ðT Þ, with no evidence of saturation at low temperatures [1]. Another interesting feature of the HTSC is the inhomogeneity of the distribution of charge carriers and superconducting gaps in the CuO2 planes, as demonstrated by STMS/S experiments [2,3]. Based on this experiments we have developed a new theory to explain the phase diagram of the HTSC [4]. In our theory, below the pseudogap temperature T ( Tc ) [5] some localized regions become superconducting. This is consistent with the Meissner effect above Tc measured by Iguchi et al. [6] and the drift of magnetic vortices measured in HTSC at temperatures above Tc through the Nernst effect [7]. Each one of these localized regions has a local superconducting temperature Tc ðqðrÞÞ, and local density qðrÞ. The differences in the local charge densities yield insulator and metallic regions, being the metallic ones corresponding to su-
*
Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.S. Caixeiro).
perconducting regions. This inhomogeneous medium is modeled by a probability distribution P ðqðrÞÞ [4] of charge densities. As the temperature is lowered, the size of these regions grow. At the critical temperature Tc of the sample these superconducting regions percolate, and the system becomes able to hold a dissipationless current. Therefore, this percolating approach together with a GL theory, will be considered in order to compute the contribution of the superconducting regions to the upper critical field Hc2 of the sample.
2. The upper critical field To develop Hc2 ðT Þ in a GL theory we consider only the case of an applied magnetic field parallel to the cdirection, i.e. perpendicular to the CuO2 planes (abdirection) [8]. Therefore, the coherence length may be written as [8] h2 Tc ¼ n2ab ð0Þ ðT < Tc Þ; ð1Þ n2ab ðT Þ ¼ 2mab aðT Þ Tc T h2 =2mab aTc is the extrapolated coherence where n2ab ð0Þ ¼ length, mab is the part of the mass tensor for the ab plane and a is a constant [8]. Substituting Eq.(1) in the GL
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.100
E.S. Caixeiro et al. / Physica C 408–410 (2004) 348–349
349
equation for Hc2 ðT Þ parallel to the c-direction [8] we have U0 Tc T ðT < Tc Þ: ð2Þ Hc2 ðT Þ ¼ Tc 2pn2ab ð0Þ As was already mentioned, below T isolated superconducting regions starts to develop. Since a given local superconducting region ‘‘i’’ has a local temperature Tc ðiÞ and probability Pi , it will contribute to the upper critical i field with a local linear upper critical field Hc2 ðT Þ near Tc ðiÞ. Therefore, the total contribution of the local superconducting regions to the upper critical field is the i ðT Þs. Thus, applying Eq. (2), the Hc2 sum of all the Hc2 for an entire sample is N U0 1 X Tc ðiÞ T P Hc2 ðT Þ ¼ i Tc ðiÞ 2pn2ab ð0Þ W i¼1 ¼
N 1 X i Pi Hc2 ðT Þ ðT < Tc ðiÞ 6 Tc Þ; W i¼1
ð3Þ
where N is the number of superconducting regions, or superconducting islands each with its local Tc ðiÞ 6 Tc and P W ¼ Ni¼1 Pi is the sum of all the Pi s. Increasing the applied field some regions become normal and eventually when the regions with Tc ðiÞ Tc turn to the normal phase, the system is about to have a nonvanishing resistivity. This value of the applied field is taken as the Hc2 in our theory, and it is the physical meaning of Eq. (3).
Fig. 1. In (a) the theoretical result (solid lines) of Hc2 for the LSCO hqi ¼ 0:15 with the distribution of Ref. [4] compared with the experimental data of Ref. [1]. The result for a linear normalized distribution (dotted line) is also shown. In (b) the theoretical result (dot-dashed line) of Hc2 for the LSCO hqi ¼ 0:20 together with the Nersnt signal measurements curve of Ref. [7] (solid line). The dashed line in both is a GL fitting of Eq. (2).
which controls the onset of the flux-flow dissipation and vanishes at a temperature close to Tc . In Fig. 1 are the theoretical results compared with the experimental data for these two magnetic fields.
4. Conclusion Using a distribution of local critical temperatures Tc ðiÞs of different superconducting regions in GL expression, we have been able to fit the Hc2 curves derived by two different experimental procedures, namely the resistive magnetic fields and the Nernst signal.
3. Comparison with experiment In this section we compare the results of Eq.(3) with the experimental upper critical field data of Refs. [1,7]. The experimental Hc2 of Ref. [1] is obtained from resistivity measurements, defined as the field relative to a fraction of the ‘‘normal-state’’ resistivity. Once there is no conclusion in what fraction may lead to the correct Hc2 , we identified Honset as the upper critical field for the hqi ¼ 0:15 compound of the LSCO series since its definition in Ref. [1] is the same as our for Hc2 . In the same way, H of Ref. [7] for the hqi ¼ 0:20 may be considered the upper critical field since it represents an intrinsic field
References [1] [2] [3] [4] [5] [6] [7] [8]
Y. Ando et al., Phys. Rev. B 60 (1999) 12475. K.M. Lang et al., Nature 415 (2002) 412. S.H. Pan et al., Nature 413 (2001) 282, cond-mat/0107347. E.V.L. de Mello, E.S. Caixeiro, J.L. Gonzalez, Phys. Rev. B 67 (2003) 024502. T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. I. Iguchi et al., Nature (London) 412 (2001) 420. Y. Wang et al., cond-mat/0205299. M. Cyrot, D. Pavuna, Introduction to Superconductivity and High-Tc Materials, World Scientific, 1992.
Theoretical upper critical field Hc2 for inhomogeneous high temperature superconductors E.S. Caixeiro *, J.L. Gonz alez, E.V.L. de Mello Departamento de Fısica, Universidade Federal Fluminense, av. Litor^ania s/n, Niteroi, RJ 24210-340, Brazil
Abstract We present the theoretical upper critical field Hc2 ðT Þ of the high temperature superconductors (HTSC), calculated through a linearized Ginzburg–Landau equation modified to consider the intrinsic inhomogeneity of the HTSC. The unusual behavior of Hc2 ðT Þ for these compounds, and other properties like the Meissner and Nernst effects detected at temperatures much higher than the critical temperature Tc of the sample, are explained by the approach. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.72.)h; 74.20.)z; 74.81.)g Keywords: Percolation; Inhomogeneity
1. Introduction The H–T phase diagram of the HTSC exhibit, in certain cases, an unusual behavior: a positive curvature for the upper critical field Hc2 ðT Þ, with no evidence of saturation at low temperatures [1]. Another interesting feature of the HTSC is the inhomogeneity of the distribution of charge carriers and superconducting gaps in the CuO2 planes, as demonstrated by STMS/S experiments [2,3]. Based on this experiments we have developed a new theory to explain the phase diagram of the HTSC [4]. In our theory, below the pseudogap temperature T ( Tc ) [5] some localized regions become superconducting. This is consistent with the Meissner effect above Tc measured by Iguchi et al. [6] and the drift of magnetic vortices measured in HTSC at temperatures above Tc through the Nernst effect [7]. Each one of these localized regions has a local superconducting temperature Tc ðqðrÞÞ, and local density qðrÞ. The differences in the local charge densities yield insulator and metallic regions, being the metallic ones corresponding to su-
*
Corresponding author. Tel.: +55-21-620-7729; fax: +55-21620-3881. E-mail address: [email protected]ff.br (E.S. Caixeiro).
perconducting regions. This inhomogeneous medium is modeled by a probability distribution P ðqðrÞÞ [4] of charge densities. As the temperature is lowered, the size of these regions grow. At the critical temperature Tc of the sample these superconducting regions percolate, and the system becomes able to hold a dissipationless current. Therefore, this percolating approach together with a GL theory, will be considered in order to compute the contribution of the superconducting regions to the upper critical field Hc2 of the sample.
2. The upper critical field To develop Hc2 ðT Þ in a GL theory we consider only the case of an applied magnetic field parallel to the cdirection, i.e. perpendicular to the CuO2 planes (abdirection) [8]. Therefore, the coherence length may be written as [8] h2 Tc ¼ n2ab ð0Þ ðT < Tc Þ; ð1Þ n2ab ðT Þ ¼ 2mab aðT Þ Tc T h2 =2mab aTc is the extrapolated coherence where n2ab ð0Þ ¼ length, mab is the part of the mass tensor for the ab plane and a is a constant [8]. Substituting Eq.(1) in the GL
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.02.100
E.S. Caixeiro et al. / Physica C 408–410 (2004) 348–349
349
equation for Hc2 ðT Þ parallel to the c-direction [8] we have U0 Tc T ðT < Tc Þ: ð2Þ Hc2 ðT Þ ¼ Tc 2pn2ab ð0Þ As was already mentioned, below T isolated superconducting regions starts to develop. Since a given local superconducting region ‘‘i’’ has a local temperature Tc ðiÞ and probability Pi , it will contribute to the upper critical i field with a local linear upper critical field Hc2 ðT Þ near Tc ðiÞ. Therefore, the total contribution of the local superconducting regions to the upper critical field is the i ðT Þs. Thus, applying Eq. (2), the Hc2 sum of all the Hc2 for an entire sample is N U0 1 X Tc ðiÞ T P Hc2 ðT Þ ¼ i Tc ðiÞ 2pn2ab ð0Þ W i¼1 ¼
N 1 X i Pi Hc2 ðT Þ ðT < Tc ðiÞ 6 Tc Þ; W i¼1
ð3Þ
where N is the number of superconducting regions, or superconducting islands each with its local Tc ðiÞ 6 Tc and P W ¼ Ni¼1 Pi is the sum of all the Pi s. Increasing the applied field some regions become normal and eventually when the regions with Tc ðiÞ Tc turn to the normal phase, the system is about to have a nonvanishing resistivity. This value of the applied field is taken as the Hc2 in our theory, and it is the physical meaning of Eq. (3).
Fig. 1. In (a) the theoretical result (solid lines) of Hc2 for the LSCO hqi ¼ 0:15 with the distribution of Ref. [4] compared with the experimental data of Ref. [1]. The result for a linear normalized distribution (dotted line) is also shown. In (b) the theoretical result (dot-dashed line) of Hc2 for the LSCO hqi ¼ 0:20 together with the Nersnt signal measurements curve of Ref. [7] (solid line). The dashed line in both is a GL fitting of Eq. (2).
which controls the onset of the flux-flow dissipation and vanishes at a temperature close to Tc . In Fig. 1 are the theoretical results compared with the experimental data for these two magnetic fields.
4. Conclusion Using a distribution of local critical temperatures Tc ðiÞs of different superconducting regions in GL expression, we have been able to fit the Hc2 curves derived by two different experimental procedures, namely the resistive magnetic fields and the Nernst signal.
3. Comparison with experiment In this section we compare the results of Eq.(3) with the experimental upper critical field data of Refs. [1,7]. The experimental Hc2 of Ref. [1] is obtained from resistivity measurements, defined as the field relative to a fraction of the ‘‘normal-state’’ resistivity. Once there is no conclusion in what fraction may lead to the correct Hc2 , we identified Honset as the upper critical field for the hqi ¼ 0:15 compound of the LSCO series since its definition in Ref. [1] is the same as our for Hc2 . In the same way, H of Ref. [7] for the hqi ¼ 0:20 may be considered the upper critical field since it represents an intrinsic field
References [1] [2] [3] [4] [5] [6] [7] [8]
Y. Ando et al., Phys. Rev. B 60 (1999) 12475. K.M. Lang et al., Nature 415 (2002) 412. S.H. Pan et al., Nature 413 (2001) 282, cond-mat/0107347. E.V.L. de Mello, E.S. Caixeiro, J.L. Gonzalez, Phys. Rev. B 67 (2003) 024502. T. Timusk, B. Statt, Rep. Prog. Phys. 62 (1999) 61. I. Iguchi et al., Nature (London) 412 (2001) 420. Y. Wang et al., cond-mat/0205299. M. Cyrot, D. Pavuna, Introduction to Superconductivity and High-Tc Materials, World Scientific, 1992.