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Effect of two-gap structure on flux pinning in MgB2
Physica C 445–448 (2006) 462–465 www.elsevier.com/locate/physc
Effect of two-gap structure on flux pinning in MgB2 J. Wang a, Z.X. Shi b
a,*
, H. Lv a, T. Tamegai
b
a Department of Physics, Southeast University, Sipailou No. 2, Nanjing 210096, China Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Available online 11 July 2006
Abstract The magnetic hysteresis loops of MgB2 polycrystalline bulk sample were measured at different temperatures and calculated numerically with different models. The influence of grain orientation, grain boundary pinning, surface pinning and flux creep on the magnetic hysteresis loops of polycrystalline MgB2 was investigated carefully. A two-exponential model taking into account the grain orientation and the flux creep is found to describe the experimental data well, which may be related to the two energy gaps in MgB2. 2006 Elsevier B.V. All rights reserved. PACS: 74.70.Ad; 74.25.Qt; 74.25.Op; 74.25.Sv Keywords: MgB2 polycrystalline bulk; Two-exponential pinning model; Two-gap structure
1. Introduction
2. Experimental and numerical method
The discovery of superconductivity in MgB2 [1] with a transition temperature Tc 39 K, has attracted great scientific interest. Characteristics of the superconducting mechanism: two-gap structure and anisotropic properties of MgB2 are focuses of recent research. Although extensive studies have been performed, the vortex pinning mechanism in MgB2 is still unclear. Some groups have reported their data on polycrystalline samples such as Kramer scaling behavior of flux-pinning force [2] and exponential field dependence of critical current density [3]. However, when the influence of the anisotropy of MgB2 with layered structure is considered, these relationships may not be accurate enough for discussing the pinning mechanism. In this paper, we describe the magnetization curves in polycrystalline MgB2 by numerical calculation considering both grain orientations and flux creep. The two-exponential field dependence of critical current density is found to agree with experimental data nicely, which is believed to be an intrinsic property of MgB2 and may be related to the two-gap structure.
A polycrystalline sample of MgB2 was synthesized under high pressure at high temperature [4], and cut into rectangular shape with 430 · 400 · 225 lm3 and Tc is about 38 K. The magnetization curves were measured by using a SQUID magnetometer. Details of experiments were described in Ref. [5]. We considered a two-dimensional slab with infinite length along the y-axis and thickness of 2d along the x-axis. The magnetic field is applied along the y-axis, and calculated the magnetic hysteresis loops along the x-axis for simplicity. To simulate the effect of random orientation of the polycrystalline sample, a model sample containing 80 grains with 5 lm in size, and with grain randomly oriented (designated by the angle h between the c-axis and the xaxis) as shown in Fig. 1 was considered. The upper critical field Hc2(h) of each grain can be determined by the anisotropic Ginzburg–Landau model Hc2(h) = Hc2(0)/(cos2(h) + sin2(h)/c2)1/2. The anisotropy parameter c depends on temperature as c(T) = c* + k(1 T/Tc), where c* = 1.87 is the effective mass anisotropy and k = 6 [6]. In order to account for the discontinuous critical current density Jc caused by the different Hc2(h) of neighboring grains, a numerical calculation is necessary to simulate the hysteresis loop. From
*
Corresponding author. Tel.: +86 25 83795083; fax: +86 25 83792868. E-mail address: [email protected] (Z.X. Shi).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.037
J. Wang et al. / Physica C 445–448 (2006) 462–465
463
Fig. 1. Sketch for different grain orientations and " is the c-axis orientation of grain.
Fig. 2. Profile of the induction distribution inside the sample calculated numerically.
the diffusion equation, dBðxÞ ¼ 4pJ c ðBðxÞÞ, the induction dx profile inside the sample was calculated as shown in Fig. 2. The average magnetization can be obtained through integration over the whole sample, given by 4pM = hB(x)i H at every applied field H. 3. Results and discussion Before fitting the experimental data, the influence of grain boundary pinning was investigated. The critical current density was chosen as Jc(B(x)) / c1Jc1(B) + c2Jc2(B), where Jc1(B) is the intra-grain critical current density supposed as a two-exponential model and Jc2(B) / exp (jB(x)/DHc2j) is the inter-grain critical current density determined by the interface pinning [7], and DHc2 is the difference of upper critical field of adjacent grains. Considering a strong–weak alternative pinning model and the critical current density at the grain boundaries is two-order magnitude higher than that in the grain, the magnetic hysteresis loops (MHLs) were calculated for model samples with grain sizes of 50 lm, 5 lm, 0.5 lm and 50 nm, shown in Fig. 3. The MHL changes very small when the grain size is larger than 5 lm. The grain size effect is not obvious until the grain size is smaller than 1 lm, such as nano-size. In our case (grain sizes >1 lm), intra-grain flux pinning dominates the induction distribution in the whole sample and the effect of inter-grain flux pinning is negligible. However, the influence of surface pinning and flux creep on the MHL should be considered. Surface pinning affects dH=dt the effective magnetic field as H eff ¼ H jdH=dtj DH , spe-
Fig. 3. Magnetization curves for polycrystalline MgB2 with different grain sizes at T = 30 K calculated in consideration of inter-grain flux pinning.
cially at lower fields, which may change the peak position of the MHL. Flux creep affects the shape of the MHL due the field-dependent pinning energy U0(T, B). In calculation, the critical current density is taken as J c ðB; T ; tÞ ¼ t J c0 1 U 0kT ln 1 þ , where the pinning energy was ðB;T Þ t0 selected as U0(T, B) / (1 T/Tc)1.65B2 following a related experimental paper published [8] and t = 10 s for the measurement time of the SQUID magnetometer. To describe the experimental results precisely, several models have been tried to fit the obtained MHLs, as shown in Fig. 4(a). The field dependence of critical current density is Jc(B(x)) = J0 exp(cjB(x)/DHc2(h)j) for exponential model, Jc(B(x)) = J0jB(x)/Hc2(h)j0.5(1 jB(x)/Hc2(h)j)2 for Kramer model [9], and Jc(B(x)) = J01 exp(10jB(x)/Hc2(h)j) + J02 exp(2.2jB(x)/Hc2(h)j) for the two-exponential model. From the reduced field dependence of the pinning force density Fp calculated by the Bean model from the MHLs as shown in Fig. 4(b), it is obvious that the two-exponential model is the best among the three models. All MHLs measured at different temperatures are fitted by the two-exponential model nicely by considering the grain orientations, the flux creep, and the surface pinning. The scaling behavior of the flux pinning force is shown in Fig. 5(a) and the fitting parameters are shown in Fig. 5(b). Hc2(h) is deduced from our previous paper [6] and the Heff is determined by the peak position of the MHL. Compared with the theoretical curve (dashed line in Fig. 5(a)), the non-scaling behavior is obviously caused
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J. Wang et al. / Physica C 445–448 (2006) 462–465
Fig. 4. (a) Calculated magnetization curves by different models compared to experimental data. (b) The reduced field dependence of flux pinning force deduced from (a).
Fig. 5. (a) The reduced field dependence of the flux pinning force deduced from both experimental magnetization curves and calculated M–H loops of a two-exponential model. Dashed line shows the ideal two-exponential model without grain orientation and flux creep. (b) Temperature dependence of the fitting parameters.
by the dispersion of the grain orientation and the flux creep. The dispersion of the grain orientations causes rapid decrease of critical current density at mediated fields from H cc2 to H ab c2 and moves the peak of flux pinning force to lower fields. While the pinning force is normalized by the peak position (Hpeak, Fp,peak), the grain orientations result in higher normalized pinning force Fp/Fp,peak at higher reduced fields, especially at lower temperatures due to the larger anisotropy parameter. Flux creep results in smaller normalized pinning force Fp/Fp,peak and smaller irreversibility fields. Consequently both the flux creep and the grain orientations destroy the scaling behavior of flux pinning force in polycrystalline MgB2 and distort the field dependence of critical current density, which further affects the judgement of pinning mechanism in polycrystalline MgB2. The two-exponential model agrees with the experimental results reported by another group [10], which is independent of samples and is the intrinsic characteristic of
MgB2. Each component of the two exponential model controls the magnetic behavior at lower and higher magnetic fields, respectively. It is possible that the two-exponential model may be related to the two energy gaps in MgB2 residing on r band and p band [11]. At higher fields, the order parameter of the p band is suppressed rapidly due to smaller energy gap and strong pair breaking. So the critical current density is determined by two characteristic fields. Details will be discussed elsewhere. In summary, the MHLs of a polycrystalline MgB2 at different temperatures have been fitted nicely by the two-exponential model taking into account the grain orientation, the flux creep and the surface pinning. The non-scaling behavior of the flux pinning force is explained by the dispersion of the grain orientations and the fielddependent flux creep. The two-exponential field dependence of the critical current density may be a common and an intrinsic characteristic of MgB2, and may be a manifestation of the effect of the two energy gaps on the flux pinning.
J. Wang et al. / Physica C 445–448 (2006) 462–465
Acknowledgements This work is partly supported by the project sponsored by the SRF for ROCS, SEM, and the project sponsored by the SRF of Southeast University and the project sponsored by Jiangsu Province Government. References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature 410 (2001) 63. [2] D.C. Larbalestier, M.O. Rikel, L.D. Cooley, A.A. Polyanskii, J.Y. Jiang, S. Patnaik, X.Y. Cai, D.M. Feldmann, A. Gurevich, A.A. Squitieri, M.T. Naus, C.B. Eom, E.E. Hellstrom, R.J. Cava, K.A. Regan, N. Rogado, M.A. Hayward, T. He, J.S. Slusky, P. Khalifah, K. Inumaru, M. Haas, Nature 410 (2001) 186.
465
[3] Mun-Seog Kim, C.U. Jung, Min-Seok Park, S.Y. Lee, Kijoon H.P. Kim, W.N. Kang, Sung-Ik Lee, Phys. Rev. Lett. 87 (2001) 087002. [4] Y. Takano, H. Takeya, H. Fujii, H. Kumakura, T. Hatano, K. Togano, H. Kito, H. Ihara, Appl. Phys. Lett. 78 (2001) 2914. [5] Z.X. Shi, A.K. Pradhan, M. Tokunaga, K. Yamazaki, T. Tamegai, Y. Takano, K. Togano, H. Kito, H. Ihara, Phys. Rev. B 68 (2003) 104514. [6] Z.X. Shi, M. Tokunaga, T. Tamegai, Y. Takano, K. Togano, H. Kito, H. Ihara, Phys. Rev. B 68 (2003) 104513. [7] E.J. Kramer, G.S. Knapp, J. Appl. Phys. 46 (1975) 4595. [8] Y.Z. Zhang, R. Deltour, H.H. Wen, Z.X. Zhao, Physica C 388–389 (2003) 173. [9] Edward J. Kramer, J. Appl. Phys. 44 (1973) 1360. [10] J. Hovat, S. Soltanian, A.V. Pan, X.L. Wang, J. Appl. Phys. 96 (2004) 4342. [11] Eun Mi Choi, Heon-Jung Kim, S.K. Gupta, P. Chowdhury, Kijoon H.P. Kim, Sung-Ik Lee, W.N. Kang, Hyeong-Jin Kim, Myung-Hwa Jung, Soo-Hyun Park, Phys. Rev. B 69 (2004) 224510.
Effect of two-gap structure on flux pinning in MgB2 J. Wang a, Z.X. Shi b
a,*
, H. Lv a, T. Tamegai
b
a Department of Physics, Southeast University, Sipailou No. 2, Nanjing 210096, China Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Available online 11 July 2006
Abstract The magnetic hysteresis loops of MgB2 polycrystalline bulk sample were measured at different temperatures and calculated numerically with different models. The influence of grain orientation, grain boundary pinning, surface pinning and flux creep on the magnetic hysteresis loops of polycrystalline MgB2 was investigated carefully. A two-exponential model taking into account the grain orientation and the flux creep is found to describe the experimental data well, which may be related to the two energy gaps in MgB2. 2006 Elsevier B.V. All rights reserved. PACS: 74.70.Ad; 74.25.Qt; 74.25.Op; 74.25.Sv Keywords: MgB2 polycrystalline bulk; Two-exponential pinning model; Two-gap structure
1. Introduction
2. Experimental and numerical method
The discovery of superconductivity in MgB2 [1] with a transition temperature Tc 39 K, has attracted great scientific interest. Characteristics of the superconducting mechanism: two-gap structure and anisotropic properties of MgB2 are focuses of recent research. Although extensive studies have been performed, the vortex pinning mechanism in MgB2 is still unclear. Some groups have reported their data on polycrystalline samples such as Kramer scaling behavior of flux-pinning force [2] and exponential field dependence of critical current density [3]. However, when the influence of the anisotropy of MgB2 with layered structure is considered, these relationships may not be accurate enough for discussing the pinning mechanism. In this paper, we describe the magnetization curves in polycrystalline MgB2 by numerical calculation considering both grain orientations and flux creep. The two-exponential field dependence of critical current density is found to agree with experimental data nicely, which is believed to be an intrinsic property of MgB2 and may be related to the two-gap structure.
A polycrystalline sample of MgB2 was synthesized under high pressure at high temperature [4], and cut into rectangular shape with 430 · 400 · 225 lm3 and Tc is about 38 K. The magnetization curves were measured by using a SQUID magnetometer. Details of experiments were described in Ref. [5]. We considered a two-dimensional slab with infinite length along the y-axis and thickness of 2d along the x-axis. The magnetic field is applied along the y-axis, and calculated the magnetic hysteresis loops along the x-axis for simplicity. To simulate the effect of random orientation of the polycrystalline sample, a model sample containing 80 grains with 5 lm in size, and with grain randomly oriented (designated by the angle h between the c-axis and the xaxis) as shown in Fig. 1 was considered. The upper critical field Hc2(h) of each grain can be determined by the anisotropic Ginzburg–Landau model Hc2(h) = Hc2(0)/(cos2(h) + sin2(h)/c2)1/2. The anisotropy parameter c depends on temperature as c(T) = c* + k(1 T/Tc), where c* = 1.87 is the effective mass anisotropy and k = 6 [6]. In order to account for the discontinuous critical current density Jc caused by the different Hc2(h) of neighboring grains, a numerical calculation is necessary to simulate the hysteresis loop. From
*
Corresponding author. Tel.: +86 25 83795083; fax: +86 25 83792868. E-mail address: [email protected] (Z.X. Shi).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.04.037
J. Wang et al. / Physica C 445–448 (2006) 462–465
463
Fig. 1. Sketch for different grain orientations and " is the c-axis orientation of grain.
Fig. 2. Profile of the induction distribution inside the sample calculated numerically.
the diffusion equation, dBðxÞ ¼ 4pJ c ðBðxÞÞ, the induction dx profile inside the sample was calculated as shown in Fig. 2. The average magnetization can be obtained through integration over the whole sample, given by 4pM = hB(x)i H at every applied field H. 3. Results and discussion Before fitting the experimental data, the influence of grain boundary pinning was investigated. The critical current density was chosen as Jc(B(x)) / c1Jc1(B) + c2Jc2(B), where Jc1(B) is the intra-grain critical current density supposed as a two-exponential model and Jc2(B) / exp (jB(x)/DHc2j) is the inter-grain critical current density determined by the interface pinning [7], and DHc2 is the difference of upper critical field of adjacent grains. Considering a strong–weak alternative pinning model and the critical current density at the grain boundaries is two-order magnitude higher than that in the grain, the magnetic hysteresis loops (MHLs) were calculated for model samples with grain sizes of 50 lm, 5 lm, 0.5 lm and 50 nm, shown in Fig. 3. The MHL changes very small when the grain size is larger than 5 lm. The grain size effect is not obvious until the grain size is smaller than 1 lm, such as nano-size. In our case (grain sizes >1 lm), intra-grain flux pinning dominates the induction distribution in the whole sample and the effect of inter-grain flux pinning is negligible. However, the influence of surface pinning and flux creep on the MHL should be considered. Surface pinning affects dH=dt the effective magnetic field as H eff ¼ H jdH=dtj DH , spe-
Fig. 3. Magnetization curves for polycrystalline MgB2 with different grain sizes at T = 30 K calculated in consideration of inter-grain flux pinning.
cially at lower fields, which may change the peak position of the MHL. Flux creep affects the shape of the MHL due the field-dependent pinning energy U0(T, B). In calculation, the critical current density is taken as J c ðB; T ; tÞ ¼ t J c0 1 U 0kT ln 1 þ , where the pinning energy was ðB;T Þ t0 selected as U0(T, B) / (1 T/Tc)1.65B2 following a related experimental paper published [8] and t = 10 s for the measurement time of the SQUID magnetometer. To describe the experimental results precisely, several models have been tried to fit the obtained MHLs, as shown in Fig. 4(a). The field dependence of critical current density is Jc(B(x)) = J0 exp(cjB(x)/DHc2(h)j) for exponential model, Jc(B(x)) = J0jB(x)/Hc2(h)j0.5(1 jB(x)/Hc2(h)j)2 for Kramer model [9], and Jc(B(x)) = J01 exp(10jB(x)/Hc2(h)j) + J02 exp(2.2jB(x)/Hc2(h)j) for the two-exponential model. From the reduced field dependence of the pinning force density Fp calculated by the Bean model from the MHLs as shown in Fig. 4(b), it is obvious that the two-exponential model is the best among the three models. All MHLs measured at different temperatures are fitted by the two-exponential model nicely by considering the grain orientations, the flux creep, and the surface pinning. The scaling behavior of the flux pinning force is shown in Fig. 5(a) and the fitting parameters are shown in Fig. 5(b). Hc2(h) is deduced from our previous paper [6] and the Heff is determined by the peak position of the MHL. Compared with the theoretical curve (dashed line in Fig. 5(a)), the non-scaling behavior is obviously caused
464
J. Wang et al. / Physica C 445–448 (2006) 462–465
Fig. 4. (a) Calculated magnetization curves by different models compared to experimental data. (b) The reduced field dependence of flux pinning force deduced from (a).
Fig. 5. (a) The reduced field dependence of the flux pinning force deduced from both experimental magnetization curves and calculated M–H loops of a two-exponential model. Dashed line shows the ideal two-exponential model without grain orientation and flux creep. (b) Temperature dependence of the fitting parameters.
by the dispersion of the grain orientation and the flux creep. The dispersion of the grain orientations causes rapid decrease of critical current density at mediated fields from H cc2 to H ab c2 and moves the peak of flux pinning force to lower fields. While the pinning force is normalized by the peak position (Hpeak, Fp,peak), the grain orientations result in higher normalized pinning force Fp/Fp,peak at higher reduced fields, especially at lower temperatures due to the larger anisotropy parameter. Flux creep results in smaller normalized pinning force Fp/Fp,peak and smaller irreversibility fields. Consequently both the flux creep and the grain orientations destroy the scaling behavior of flux pinning force in polycrystalline MgB2 and distort the field dependence of critical current density, which further affects the judgement of pinning mechanism in polycrystalline MgB2. The two-exponential model agrees with the experimental results reported by another group [10], which is independent of samples and is the intrinsic characteristic of
MgB2. Each component of the two exponential model controls the magnetic behavior at lower and higher magnetic fields, respectively. It is possible that the two-exponential model may be related to the two energy gaps in MgB2 residing on r band and p band [11]. At higher fields, the order parameter of the p band is suppressed rapidly due to smaller energy gap and strong pair breaking. So the critical current density is determined by two characteristic fields. Details will be discussed elsewhere. In summary, the MHLs of a polycrystalline MgB2 at different temperatures have been fitted nicely by the two-exponential model taking into account the grain orientation, the flux creep and the surface pinning. The non-scaling behavior of the flux pinning force is explained by the dispersion of the grain orientations and the fielddependent flux creep. The two-exponential field dependence of the critical current density may be a common and an intrinsic characteristic of MgB2, and may be a manifestation of the effect of the two energy gaps on the flux pinning.
J. Wang et al. / Physica C 445–448 (2006) 462–465
Acknowledgements This work is partly supported by the project sponsored by the SRF for ROCS, SEM, and the project sponsored by the SRF of Southeast University and the project sponsored by Jiangsu Province Government. References [1] J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, J. Akimitsu, Nature 410 (2001) 63. [2] D.C. Larbalestier, M.O. Rikel, L.D. Cooley, A.A. Polyanskii, J.Y. Jiang, S. Patnaik, X.Y. Cai, D.M. Feldmann, A. Gurevich, A.A. Squitieri, M.T. Naus, C.B. Eom, E.E. Hellstrom, R.J. Cava, K.A. Regan, N. Rogado, M.A. Hayward, T. He, J.S. Slusky, P. Khalifah, K. Inumaru, M. Haas, Nature 410 (2001) 186.
465
[3] Mun-Seog Kim, C.U. Jung, Min-Seok Park, S.Y. Lee, Kijoon H.P. Kim, W.N. Kang, Sung-Ik Lee, Phys. Rev. Lett. 87 (2001) 087002. [4] Y. Takano, H. Takeya, H. Fujii, H. Kumakura, T. Hatano, K. Togano, H. Kito, H. Ihara, Appl. Phys. Lett. 78 (2001) 2914. [5] Z.X. Shi, A.K. Pradhan, M. Tokunaga, K. Yamazaki, T. Tamegai, Y. Takano, K. Togano, H. Kito, H. Ihara, Phys. Rev. B 68 (2003) 104514. [6] Z.X. Shi, M. Tokunaga, T. Tamegai, Y. Takano, K. Togano, H. Kito, H. Ihara, Phys. Rev. B 68 (2003) 104513. [7] E.J. Kramer, G.S. Knapp, J. Appl. Phys. 46 (1975) 4595. [8] Y.Z. Zhang, R. Deltour, H.H. Wen, Z.X. Zhao, Physica C 388–389 (2003) 173. [9] Edward J. Kramer, J. Appl. Phys. 44 (1973) 1360. [10] J. Hovat, S. Soltanian, A.V. Pan, X.L. Wang, J. Appl. Phys. 96 (2004) 4342. [11] Eun Mi Choi, Heon-Jung Kim, S.K. Gupta, P. Chowdhury, Kijoon H.P. Kim, Sung-Ik Lee, W.N. Kang, Hyeong-Jin Kim, Myung-Hwa Jung, Soo-Hyun Park, Phys. Rev. B 69 (2004) 224510.