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Numerical evidence for Lorentz-force independent dissipation in layered superconductors
Physica C 412–414 (2004) 422–424 www.elsevier.com/locate/physc
Numerical evidence for Lorentz-force independent dissipation in layered superconductors Qing-Hu Chen a
a,b,*
, Meng-Bo Luo
a,b
, Qing-Miao Nie
a,b
, Xiao Hu
a
Computer Materials Science Center, National Institute for Material Science, Tsukuba 305-0047, Japan b Department of Physics, Zhejiang University, Hangzhou 310027, China Received 29 October 2003; accepted 26 January 2004 Available online 10 May 2004
Abstract We study numerically the dynamics of Josephson vortex systems in high-Tc superconductors. A uniform external current is applied in the CuO2 plane, which is either along or perpendicular to the magnetic field. For high magnetic field and/or high anisotropy, we find Lorentz-force independent dissipations, which is consistent with previous experimental results in BSCCO. The IV characteristics show nice power-law behaviors at intermediate temperatures, which clearly reveals Kosterlitz–Thouless property of the system. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.50.+r; 74.40.+k Keywords: Superconductivity; Josephson vortex; Dynamics
1. Introduction It is generally accepted that in high-Tc cuprates superconductivity occurs mainly in the CuO2 layers intervened by layers of charge reservoir. When a magnetic field is applied parallel to the CuO2 plane, Josephson vortices are induced, which are located at the insulating layers in between the CuO2 planes. The CuO2 planes play intrinsic pinning effects on the movement of the Josephson vortices in the direction perpendicular to the planes. A puzzling phenomenon has been observed *
Corresponding author. Tel.: +86-571-8795-2068; fax: +86571-8795-1328. E-mail address: [email protected] (Q.-H. Chen).
in Bi2 Sr2 CaCu2 O8þy that the resistivity does not depend on the angle between the magnetic field and current when they are both parallel to the CuO2 layer [1]. Later, power-law, non-Ohmic dissipations have been found by several groups [2,3]. Blatter et al. [4] proposed a novel Kosterlitz– Thouless (KT) [5] scenario at high magnetic fields, characterizing the behavior by a smectic state where the interlayer shear modulus vanishes. Recently, Hu and Tachiki performed a large-scale Monte Carlo (MC) simulation and mapped out a phase diagram with three-dimensional lattice phase, two-dimensional (2D) KT phase and liquid phase and a multicritical point [6]. This work presents results by large-scale dynamical simulations.
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.064
Q.-H. Chen et al. / Physica C 412–414 (2004) 422–424
423
2. Model Josephson vortices in high-Tc superconductors can be described by the three-dimensional anisotropic XY model on a simple cubic lattice [7,8] X H ¼ Jij cosð/i /j Aij Þ; ð1Þ hiji
where /i is the phase of the superconducting order parameter on site i, Jij ¼ J except for bonds along the c axis where Jij ¼ J =c2 and c ¼ kc =kab is the anisotropy constant. The y direction is along the external magnetic field and ^x ? ^c ? ^y , A ¼ ð0; 0; 2pfxÞ. The system is of size Lx Ly Lc ¼ 128d 128d 20d with d the separation between CuO2 layers. We fix the strength of magnetic field at f ¼ 1=32 and anisotropy constant at c ¼ 20, which is above the multicritical point pffiffiffi f c ¼ 1=2 3 [6]. The resistivity-shunted-junction dynamics is incorporated in simulations, which can be described as [9,10] X rh X _ oH ð/i /_ j Þ ¼ þ Jext;i gij ; ð2Þ 2e j o/i j where Jext;i is the external current which vanishes except for the boundary sites. The gij is the thermal noise current with zero mean and a correlator hgij ðtÞgij ðt0 Þi ¼ 2rkB T dðt t0 Þ. The fluctuating twist boundary condition [9] is applied in all directions to maintain the external current. Dynamics of thePfluctuating twist variable D is ðsÞ given by D_ a ¼ L13 hijia ½Ji!j þ gij Ia , (a ¼ x; y; z). The voltage drop is Va ¼ LD_ a . Successive averages of the voltage over time intervals ½2n1 ; 2n ðn ¼ 1; 2; . . .Þ are recorded and the final voltage is obtained when the steady state is archived. Units are taken of 2e ¼ J0 ¼ h ¼ r ¼ kB ¼ 1. 3. Results and discussion In order to elucidate the possible effect of the Lorentz force on the transport properties, the external current is applied either parallel or perpendicular to the magnetic field. According to an equilibrium MC simulation, the transition temperature for these parameters is around T ¼ 0:93
Fig. 1. Voltage–current (IV ) curves at temperatures T ¼ 0:8 and 0.9 for currents parallel (Iy ) and perpendicular (Ix ) to the magnetic field. The IV exponents aðT Þ at various temperatures are shown in the inset.
[6]. In Fig. 1, we plot the voltage–current curves at T ¼ 0:8 and 0.9. It is striking to observe that below the transition temperature, the current–voltage characteristics are independent of the current direction, which clearly shows that the Lorentz force plays no role in dissipations at the macroscopic level. Detailed creeping processes are to be revealed in the future. The IV curves show a nice power-law behavior at both temperatures, similar to the KT ones originally observed in 2D systems. This observation is consistent with the results of MC simulation [6], in which decoupling of Josephson vortex in the c-axis is revealed by the calculated structure factors. Therefore, the system is basically of quasi-2D nature. In the inset of Fig. 1, we present the temperature dependence of the exponent aðT Þ defined from the IV curves, V I aðT Þ . A jump roughly from 3 to 1 of IV exponent at around T ¼ 0:93 is clearly shown, providing another evidence of the KT phase transition. The isotropic exponents aðT Þ agree with the isotropic helicity moduli obtained in an equilibrium study [6], providing the relation a ¼ 1 þ p=kB T still holds. In summary, by dynamical simulations we show that a novel KT intermediate phase appears when a strong magnetic field is applied parallel to the CuO2 plane in high-Tc superconductors, which just accounts for Lorentz-force independent
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Q.-H. Chen et al. / Physica C 412–414 (2004) 422–424
dissipation observed experimentally. The detailed results will be presented elsewhere. Acknowledgements The present simulations were performed on the Numerical Materials Simulator (SX-5) at NIMS, Japan. This work was partially supported by Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) No. 15540355). One of us (QHC) is supported in part by the National Nature Science Foundation of China under Grant No. 10274067. References [1] Y. Iye, S. Nakamura, T. Tamegai, Physica C 159 (1989) 433.
[2] Y. Ando, N. Motohira, K. Kitazawa, J.I. Takeya, S. Akita, Phys. Rev. Lett. 67 (1991) 2737. [3] A.K. Pradhan, S.J. Hazell, J.W. Hodby, C. Chen, Y. Hu, B.M. Wanklyn, Phys. Rev. B 47 (1993) 11374. [4] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Phys. Rev. Lett. 66 (1991) 2392. [5] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [6] X. Hu and M. Tachiki,; X. Hu, J. Low Temp. Phys. 131 (2003) 979. [7] X. Hu, S. Miyashita, M. Tachiki, Phys. Rev. Lett. 79 (1997) 3498; P. Olsson, S. Teitel, Phys. Rev. Lett. 82 (1999) 2183; R.E. Hetzel, A. Sudbo, D.A. Huse, Phys. Rev. Lett. 69 (1992) 518. [8] X. Hu, M. Tachiki, Phys. Rev. Lett. 85 (2000) 2577; X. Hu, M. Tachiki, Phys. Rev. Lett. 80 (1998) 4044. [9] Q.H. Chen, L.H. Tang, P.Q. Tong, Phys. Rev. Lett. 87 (2001) 067001; L.H. Tang, Q.H. Chen, Phys. Rev. B 67 (2003) 024508. [10] Q.H. Chen, X. Hu, Phys. Rev. Lett. 90 (2003) 117005.
Numerical evidence for Lorentz-force independent dissipation in layered superconductors Qing-Hu Chen a
a,b,*
, Meng-Bo Luo
a,b
, Qing-Miao Nie
a,b
, Xiao Hu
a
Computer Materials Science Center, National Institute for Material Science, Tsukuba 305-0047, Japan b Department of Physics, Zhejiang University, Hangzhou 310027, China Received 29 October 2003; accepted 26 January 2004 Available online 10 May 2004
Abstract We study numerically the dynamics of Josephson vortex systems in high-Tc superconductors. A uniform external current is applied in the CuO2 plane, which is either along or perpendicular to the magnetic field. For high magnetic field and/or high anisotropy, we find Lorentz-force independent dissipations, which is consistent with previous experimental results in BSCCO. The IV characteristics show nice power-law behaviors at intermediate temperatures, which clearly reveals Kosterlitz–Thouless property of the system. Ó 2004 Elsevier B.V. All rights reserved. PACS: 74.60.Ge; 74.50.+r; 74.40.+k Keywords: Superconductivity; Josephson vortex; Dynamics
1. Introduction It is generally accepted that in high-Tc cuprates superconductivity occurs mainly in the CuO2 layers intervened by layers of charge reservoir. When a magnetic field is applied parallel to the CuO2 plane, Josephson vortices are induced, which are located at the insulating layers in between the CuO2 planes. The CuO2 planes play intrinsic pinning effects on the movement of the Josephson vortices in the direction perpendicular to the planes. A puzzling phenomenon has been observed *
Corresponding author. Tel.: +86-571-8795-2068; fax: +86571-8795-1328. E-mail address: [email protected] (Q.-H. Chen).
in Bi2 Sr2 CaCu2 O8þy that the resistivity does not depend on the angle between the magnetic field and current when they are both parallel to the CuO2 layer [1]. Later, power-law, non-Ohmic dissipations have been found by several groups [2,3]. Blatter et al. [4] proposed a novel Kosterlitz– Thouless (KT) [5] scenario at high magnetic fields, characterizing the behavior by a smectic state where the interlayer shear modulus vanishes. Recently, Hu and Tachiki performed a large-scale Monte Carlo (MC) simulation and mapped out a phase diagram with three-dimensional lattice phase, two-dimensional (2D) KT phase and liquid phase and a multicritical point [6]. This work presents results by large-scale dynamical simulations.
0921-4534/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2004.01.064
Q.-H. Chen et al. / Physica C 412–414 (2004) 422–424
423
2. Model Josephson vortices in high-Tc superconductors can be described by the three-dimensional anisotropic XY model on a simple cubic lattice [7,8] X H ¼ Jij cosð/i /j Aij Þ; ð1Þ hiji
where /i is the phase of the superconducting order parameter on site i, Jij ¼ J except for bonds along the c axis where Jij ¼ J =c2 and c ¼ kc =kab is the anisotropy constant. The y direction is along the external magnetic field and ^x ? ^c ? ^y , A ¼ ð0; 0; 2pfxÞ. The system is of size Lx Ly Lc ¼ 128d 128d 20d with d the separation between CuO2 layers. We fix the strength of magnetic field at f ¼ 1=32 and anisotropy constant at c ¼ 20, which is above the multicritical point pffiffiffi f c ¼ 1=2 3 [6]. The resistivity-shunted-junction dynamics is incorporated in simulations, which can be described as [9,10] X rh X _ oH ð/i /_ j Þ ¼ þ Jext;i gij ; ð2Þ 2e j o/i j where Jext;i is the external current which vanishes except for the boundary sites. The gij is the thermal noise current with zero mean and a correlator hgij ðtÞgij ðt0 Þi ¼ 2rkB T dðt t0 Þ. The fluctuating twist boundary condition [9] is applied in all directions to maintain the external current. Dynamics of thePfluctuating twist variable D is ðsÞ given by D_ a ¼ L13 hijia ½Ji!j þ gij Ia , (a ¼ x; y; z). The voltage drop is Va ¼ LD_ a . Successive averages of the voltage over time intervals ½2n1 ; 2n ðn ¼ 1; 2; . . .Þ are recorded and the final voltage is obtained when the steady state is archived. Units are taken of 2e ¼ J0 ¼ h ¼ r ¼ kB ¼ 1. 3. Results and discussion In order to elucidate the possible effect of the Lorentz force on the transport properties, the external current is applied either parallel or perpendicular to the magnetic field. According to an equilibrium MC simulation, the transition temperature for these parameters is around T ¼ 0:93
Fig. 1. Voltage–current (IV ) curves at temperatures T ¼ 0:8 and 0.9 for currents parallel (Iy ) and perpendicular (Ix ) to the magnetic field. The IV exponents aðT Þ at various temperatures are shown in the inset.
[6]. In Fig. 1, we plot the voltage–current curves at T ¼ 0:8 and 0.9. It is striking to observe that below the transition temperature, the current–voltage characteristics are independent of the current direction, which clearly shows that the Lorentz force plays no role in dissipations at the macroscopic level. Detailed creeping processes are to be revealed in the future. The IV curves show a nice power-law behavior at both temperatures, similar to the KT ones originally observed in 2D systems. This observation is consistent with the results of MC simulation [6], in which decoupling of Josephson vortex in the c-axis is revealed by the calculated structure factors. Therefore, the system is basically of quasi-2D nature. In the inset of Fig. 1, we present the temperature dependence of the exponent aðT Þ defined from the IV curves, V I aðT Þ . A jump roughly from 3 to 1 of IV exponent at around T ¼ 0:93 is clearly shown, providing another evidence of the KT phase transition. The isotropic exponents aðT Þ agree with the isotropic helicity moduli obtained in an equilibrium study [6], providing the relation a ¼ 1 þ p=kB T still holds. In summary, by dynamical simulations we show that a novel KT intermediate phase appears when a strong magnetic field is applied parallel to the CuO2 plane in high-Tc superconductors, which just accounts for Lorentz-force independent
424
Q.-H. Chen et al. / Physica C 412–414 (2004) 422–424
dissipation observed experimentally. The detailed results will be presented elsewhere. Acknowledgements The present simulations were performed on the Numerical Materials Simulator (SX-5) at NIMS, Japan. This work was partially supported by Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) No. 15540355). One of us (QHC) is supported in part by the National Nature Science Foundation of China under Grant No. 10274067. References [1] Y. Iye, S. Nakamura, T. Tamegai, Physica C 159 (1989) 433.
[2] Y. Ando, N. Motohira, K. Kitazawa, J.I. Takeya, S. Akita, Phys. Rev. Lett. 67 (1991) 2737. [3] A.K. Pradhan, S.J. Hazell, J.W. Hodby, C. Chen, Y. Hu, B.M. Wanklyn, Phys. Rev. B 47 (1993) 11374. [4] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Phys. Rev. Lett. 66 (1991) 2392. [5] J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181. [6] X. Hu and M. Tachiki,