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Self-field reduces critical current density in thick YBCO layers
Physica C 451 (2007) 66–70 www.elsevier.com/locate/physc
Self-field reduces critical current density in thick YBCO layers L. Rostila *, J. Lehtonen, R. Mikkonen Institute of Electromagnetics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland Received 2 June 2006; received in revised form 21 September 2006; accepted 23 October 2006 Available online 1 December 2006
Abstract The engineering current density in YBCO coated conductor applications can be improved in two ways. Either the critical current density should be improved or the superconducting films made thicker. Unfortunately, it has often been observed that the average critical current density decreases when the thickness of films increases. Suggested reasons for this behaviour include e.g. two dimensional pinning properties, microcracks and imperfect crystallographic alignment. However, it is often forgotten that the self-field effect unavoidably reduces the critical current density when the thickness of YBCO films increases and thereby total current rises. In this paper, the influence of self-field on the average critical current density is studied computationally as a function of film thickness. The situation is also scrutinized at different external magnetic fields in order to find ways to distinguish self-field effects from problems related to the manufacturing process. For this purpose, critical current measurements in external field perpendicular to the film surface are proposed. 2006 Elsevier B.V. All rights reserved. PACS: 74.25.Sv* Keywords: YBCO; Critical current
1. Introduction In superconducting power applications the engineering current density, Je, is one of the main parameters which determine the viability of devices [1]. YBCO films provide extremely high local critical current densities, Jc, up to several MA Æ cm2 at 77 K but the engineering current density is drastically lowered because the superconducting layer is very thin compared to the substrate on which it is grown [2]. Obviously Je should be improved if the superconducting films are made thicker [3]. So far, there have been some encouraging examples in which average critical current density of YBCO layer Jca at self-field seems to be quite independent of thickness of YBCO layer d. Such situations have been reported in tapes manufactured with the TFAMOD method [4,5] or with yttrium rich compositions [6]. However, most often it has been found that Jca decreases *
Corresponding author. Tel.: +358 3 31152014; fax: +358 3 3115 2160. E-mail address: lauri.rostila@tut.fi (L. Rostila).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.10.008
if the thickness d of YBCO layer is increased. Several possible reasons have been suggested for this behaviour. According to the collective pinning theory Jca should be proportional to d1/2 when d is smaller than the pinning correlation length [7]. This explanation has been criticized because at sufficiently high magnetic fields Jca can be higher in a thicker superconducting tape [8] and the Jca(d)-dependence does not vanish at low temperatures as predicted [9]. More probable reasons have been found from the deteriorating structure of YBCO material. When d increases regions where the crystallographic c-axis is no more normal to the substrate plane can be created [9,10]. If d exceeds some tenths of micrometers also microcracking usually arises due to the difference between thermal expansion coefficients in the substrate and in the YBCO film [11]. However, recently the film thickness up to the micrometer range has been achieved on sapphire substrates without microcracking [12] but even then Jc has been found to plummet with increasing film thickness [13]. Then the film porosity and roughness increased and thereby the defect structures responsible for flux pinning changed.
L. Rostila et al. / Physica C 451 (2007) 66–70
Jca(d)-variations cannot be explained solely on the basis of material quality. In the analysis, it should be remembered that the self-field effect reduces unavoidably the critical current density when the thickness of YBCO films increases and the total current rises. The self-field effect has been studied both computationally and experimentally in LTS films [14], BSCCO tapes [15–17] and recently a onedimensional computational model is applied on YBCO films [18]. In this paper, the self-field induced reduction of the average critical current density is calculated as a function of film thickness. The situation is also scrutinized at different external magnetic fields in order to find ways to distinguish problems related to the YBCO quality from the effects of self-field. Because mesh generation problems have been encountered in finite element method due to high aspect ratio [19], a self-made algorithm based on finite integration method was chosen here. Furthermore, this approach is simple and fast. 2. Computational model In computations homogeneous YBCO material and DC operation current were assumed. Thus, the variations in the current density distribution were exclusively due to the spatial variations of self-field. A Cartesian coordinate system was used. The x-axis was parallel with the broad face of the tape, the y-axis parallel with the narrow face and current flowed in positive z-direction. The magnetic flux density created by transport current flowing in the tape was denoted as Bself. Furthermore, the tape can be exposed to an external magnetic flux density Bext. The main idea of the algorithm was to determine the current density distribution that fulfils the condition J = Jc(Bext + Bself(J)) inside the film. Here the local Jc(B)dependence was described with the Kim model [20] which was modified to take into account material anisotropy as a eB ; ð1Þ J c ðBÞ ¼ J c0 1 þ B0
Fig. 1. The default mesh with Nx = 16 and Ny = 12. Definitions of h and hloc are also included.
The film cross-section was divided into Nx times Ny rectangular elements in order to calculate the magnetic flux density, Bself(r), created by J(r 0 ). where r = [x y]T and r 0 = [x 0 y 0 ]T. In computations the mesh shown in Fig. 1 with Nx = 16 and Ny = 12, was used as a default. When current densities inside such elements are assumed to be constant the magnetic flux density can be calculated from the two-dimensional Biot–Savart law Z l 0 1 J ðr0 Þðr r0 Þ 0 Bself ðrÞ ¼ 0 ds ; ð3Þ 3=2 4p 1 0 S 0 ðr r0 Þ where l0 is the vacuum permeability and S 0 is the cross-section where the current flows. The flux diagram in Fig. 2 shows the principle of the algorithm starting from constant current density. The magnetic fields are calculated in the centre of elements and a new current density distribution is achieved as Jnew = Jc(Bext + Bself(Jold)), where Jold is the previous current distribution. Then the critical current is computed as an
where anisotropy is denoted with the anisotropic scaling [21] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðhÞ ¼ cos2 ðhloc Þ þ c2 sin2 ðhloc Þ: ð2Þ hloc is the angle between Bext + Bself and c-axis. Here the parameter values were chosen as Jc0 = 3 MA cm2 [18], B0 = 20 mT [18], a = 0.65 [22] and c = 5 [23]. These parameters are aimed to give a typical example about the Jc(B)-dependence of YBCO because only general trends of self-field effect are searched. When this analysis is applied to any particular sample, critical current measurements should be carried out and special attention paid on the derivation of Jc(B) from measured critical currents. The parameters of Jc(B)-dependence are hard to measure accurately due to the presence of self-field [24].
67
Fig. 2. Block diagram about the computation of critical current.
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integral of the new current density over the sample crosssection. The critical current was compared with its value from the previous iteration. If the relative change was below a given tolerance the final current density distribution was obtained. Otherwise the procedure was repeated. 3. Results and discussion The simulations were done with the self-made algorithm implemented to MATLAB [25]. About ten iterations were required to find the final critical current with the relative tolerance of 106. It was found out that acceptable solutions can be achieved with a very coarse mesh. For example, if the self-field critical current in a 4 mm times 1.5 lm tape was computed with Nx = 4, Ny = 3 the result deviated less than 2.95% compared to the mesh Nx = 64 and Ny = 48. The deviation was less than 0.57% with default mesh and it took about 0.3 s to compute one critical current with AMD Athlon 64 3400+. The self-field depends always on the sample geometry and dimensions. Thus, both the sample width and film thickness should influence on the average critical current density Jca. However, Jca is practically constant at a given film thickness if the sample width exceeds one tenth of millimetre as shown in Fig. 3. Thus, in practice the sample width can be neglected and attention can be paid solely to the film thickness. In order to explain this result the magnetic field inside the film was studied. Fig. 4 shows that after the sample width reaches one millimetre the average of By was always practically constant. The average of Bx could increase slightly until the sample width reached one millimetre but in an anisotropic superconductor the critical current density is mainly determined by By [15]. Usually the short sample critical current is measured as a function of magnetic flux density magnitude at constant field orientation which is defined by the angle h between the field and the normal of the film surface. Fig. 5 presents Jca as a function of YBCO film thickness in both parallel
Fig. 3. Normalized average critical current density as a function of YBCO tape width with different film thicknesses: (s) 0.25 lm, (h) 0.5 lm, (·) 1.0 lm, (d) 1.5 lm and (+) 3.0 lm.
Fig. 4. Average Bx (black) and By (gray) inside the YBCO as a function of tape width with different film thicknesses: (s) 0.25 lm, (h) 0.5 lm, (·) 1.0 lm, (d) 1.5 lm and (+) 3.0 lm.
Fig. 5. The computed normalized average critical current density as a function of YBCO film thickness at different external magnetic fields oriented (a) in perpendicular and (b) in parallel with film surface: (s) 0 mT, (h) 10 mT, (·) 20 mT, (d) 30 mT and (+) 40 mT.
and perpendicular external magnetic fields. In self-field Jca(d) resembled closely exponential decay, and in a 3 lm thick film Jca was reduced about 35% if compared to the real zero field critical current density Jc0. The exponential decay has also been fitted to measurements in references [13,26]. For example, the decay measured in reference [26] was probably in large part caused by the self-field effect because no structural problems were found in the samples. Here the function d J ca ðdÞ J c0 0:41984 exp þ 0:57563 1:7929 106 ð4Þ
L. Rostila et al. / Physica C 451 (2007) 66–70
69
fits to the computed self-field Jca values in Fig. 5 with maximum relative error of 0.42%. The parameter values in Eq. (4) were presented with five digits accuracy because the fitting was sensitive to them. However, even much better fit was achieved as J ca ðdÞ J c0 1 þ
d 1:2826 106
0:35509 ;
ð5Þ
which produced an error of 0.016% only. In external magnetic fields the situation changes dramatically. Already in parallel fields the shape of Jca(d) is no longer exponential. In perpendicular fields Jca(d) was quite constant if the thickness was below 0.60, 1.42 and 2.46 lm at By of 10, 20 and 30 mT, respectively. In other words, the thickness has no significant effect on Jca if the external field is higher than the maximum value of self-field the y-component. Therefore, the YBCO thin film samples should be measured in a sufficiently high perpendicular external magnetic field, where the self-field effect is cancelled out and Jca will directly tell about the material quality. Thus, e.g. the measured critical currents in Ref. [8] suggested that the quality of the studied material improved with increasing d. In order to further visualize the self-field effect Fig. 6 presents the average critical current density as a function of external magnetic flux density at different field orientations in a 1.5 lm thick YBCO film. At low parallel fields Jca stays quite constant due to the self-field effect. Fig. 7 illustrates the current density distributions in the YBCO film at different external field. Here the mesh with Nx = 64 and Ny = 48 was used in order to achieve an appropriate resolution for figures about the current density distribution. In self-field B naturally equals zero in the film centre, and therefore the current density is highest there. When the film is exposed to an external magnetic field, the self-field and external field compensate each other in some parts. The region of the highest J shifts to the point
Fig. 7. Current density in YBCO film with thickness of 1.5 lm at different external magnetic fields: (a) self-field, (b) h = 0, B = 10 mT, (c) h = 45, B = 10 mT and (d) h = 90, B = 10 mT.
where the compensation is most effective. Thus, at low external fields just the shape of J-distribution is reformed but the average critical current density does not change a lot. At perpendicular fields also the Jca drops because critical current density decreases drastically even at low values of B. Similar behaviour is often found in high quality BSCCO tapes as well [16,27]. The self-field effect is noticeable in Jca as long as the current density distribution remains inhomogeneous. When the external magnetic field rises, the current distribution is homogenised and the selffield effect can be neglected. The parameter values in Eq. (1) can vary significantly. Furthermore, the Jc(B)-dependence can be very complicated in anisotropic HTS materials and Eq. (1) does not necessarily describe it correctly [28]. Therefore, the sensitivity of presented results was studied with respect to the parameters J0, B0, a and c. The basic definition states that the sensitivity S nN of the quantity N with respect to parameter n is computed as S nN ¼
n dN : N dn
ð6Þ
Here the derivative was approximated numerically as dN N ð1:1nÞ N ð0:9nÞ : dn 0:2n
Fig. 6. Normalized average critical current density as a function of external magnetic flux density at different field orientations in YBCO film with thickness of 1.5 lm: (s) h = 0, (h) 22.5, (·) 45, (d) 67.5 and (+) 90. For comparison normalized critical current density as given by Eq. (1) is shown in gray.
ð7Þ
Eq. (6) gives a rough estimate how much a change of a parameter value effects on the Jca. Table 1 shows the sensitivity of Jca/Jc0. The most sensitive parameter was a, whereas variations in the anisotropy factor c had almost no influence on the results. It is expected that a and c do not vary significantly between samples, but Jc0 and B0 can depend strongly on the sample quality. Therefore, when this kind of analysis is applied on real samples attention should be paid to determine their values. However, small variations of the chosen parameters leave the main aspects in results unaltered.
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Table 1 Sensitivity of normalized average critical current density with respect to parameters in Eq. (1)
Jc0 a B0 c
d1, B1
d2, B1
d3, B1
d1, B2
d2, B2
d3, B2
0.06 0.06 0.06 0.01
0.19 0.25 0.19 0.01
0.25 0.36 0.25 0.01
0.00 0.45 0.33 0.00
0.05 0.45 0.31 0.01
0.15 0.48 0.32 0.01
The sensitivity is studied with thicknesses d1 = 0.25 lm, d2 = 1.5 lm, d3 = 3 lm and external magnetic flux densities B1 = 0 mT, B2 = 20 mT.
4. Conclusions In this paper, the influence of self-field on the critical current was studied computationally as a function of film thickness. The self-field effect unavoidably reduces the critical current density when the thickness of YBCO films increases and total current rises. It was found that in practical YBCO samples the average critical current density does not depend on the sample width but decreases with increasing film thickness in self-field. In order to distinguish self-field effects from problems related to the manufacturing process of real YBCO films it was proposed that the critical current measurements should be carried out in external field perpendicular to the film surface such that external field exceeds the maximum value of the perpendicular component of self-field. References [1] M.N. Wilson, Superconducting Magnets, Oxford University Press, Oxford, 1983. [2] M. Chen, W. Paul, M. Lakner, L. Donzel, M. Hoidis, P. Unternaehrer, R. Weder, M. Mendik, Physica C 372–376 (2002) 1657. [3] H.-W. Neumu¨ller, W. Schmidt, H. Kinder, H.C. Freyhardt, B. Stritzker, R. Wo¨rdenweber, V. Kirchoff, J. Alloy. Compd. 251 (1997) 366. [4] T. Izumi et al., Physica C 412–414 (2004) 885. [5] Y. Tokunaga et al., Cryogenics 44 (2004) 817. [6] R. Feenstra, A. Gapud, F. List, E. Specht, D. Christen, T. Holesinger, D. Feldmann, IEEE T. Appl. Supercon. 15 (2005) 2805. [7] A.I. Larkin, Y. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [8] T. Matsushita et al., Physica C 426–431 (2005) 1096.
[9] T. Matsushita, M. Kiuchi, K. Kimura, S. Miyata, A. Ibi, T. Muroga, Y. Yamada, Y. Shiohara, Supercond. Sci. Tech. 18 (2005) S227. [10] A. Mogro-Campero, L. Turner, E. Hall, N. Lewis, L. Peluso, W. Balz, Supercond. Sci. Tech. 3 (1990) 62. [11] G. Ka¨stner, C. Scha¨fer, S. Senz, D. Heese, M. Lorenz, H. Hochmuth, M. Getta, M.A. Hein, T. Kaiser, G. Mu¨ller, IEEE T. Appl. Supercon. 9 (1999) 2171. [12] K.D. Develos-Bagarinao, H. Yamasaki, Y. Nakagawa, H. Yamada, Physica C 392–396 (2003) 1229. [13] K. Develos-Bagarinao, H. Yamasaki, C. Nie J, Y. Nakagawa, Supercond. Sci. Tech. 18 (2005) 667. [14] J.R. Gavaler, A.T. Santhanam, A.I. Braginski, M. Ashkin, M.A. Janocko, IEEE T. Magn. 17 (1981) 573. [15] N. Shaked et al., Physica C 354 (2001) 237. [16] J. Lehtonen, A. Korpela, W. Nah, J. Kang, P. Kova´cˇ, T. Melisˇek, Physica C 403 (2004) 257. [17] S. Spreafico, L. Gherardi, S. Fleshler, D. Tatelbaum, J. Leone, D. Yu, G. Snitchler, IEEE T. Appl. Supercon. 9 (1999) 2159. [18] A. Brojeny, J. Clem, Supercond. Sci. Tech. 18 (2005) 888. [19] S. Stavrev, F. Grilli, B. Dutoit, S.P. Ashworth, Supercond. Sci. Tech. 18 (2005) 1300. [20] Y.B. Kim, C.F. Hempstead, A.R. Strnad, Rev. Mod. Phys. 36 (1964) 43. [21] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [22] A.A. Gapud, R. Feenstra, D.K. Christen, J.R. Thompson, T.G. Holesinger, IEEE T. Appl. Supercon. 15 (2005) 2578. [23] L. Civale et al., Appl. Phys. Lett. 84 (2003) 2121. [24] M. Masti, J. Lehtonen, R. Pera¨la¨, W. Nah, J. Kang, Physica C 401 (2004) 155. [25] www.mathworks.com. [26] S. Foltyn, P. Tiwari, R. Dye, M. Le, X. Wu, Appl. Phys. Lett. 63 (1993) 1848. [27] J. Lehtonen, R. Pera¨la¨, R. Mikkonen, Physica C 401 (2004) 151. [28] J. Lehtonen, A. Korpela, J. Pitel, P. Kova´cˇ, Physica C 403 (2004) 139.
Self-field reduces critical current density in thick YBCO layers L. Rostila *, J. Lehtonen, R. Mikkonen Institute of Electromagnetics, Tampere University of Technology, P.O. Box 692, FIN-33101 Tampere, Finland Received 2 June 2006; received in revised form 21 September 2006; accepted 23 October 2006 Available online 1 December 2006
Abstract The engineering current density in YBCO coated conductor applications can be improved in two ways. Either the critical current density should be improved or the superconducting films made thicker. Unfortunately, it has often been observed that the average critical current density decreases when the thickness of films increases. Suggested reasons for this behaviour include e.g. two dimensional pinning properties, microcracks and imperfect crystallographic alignment. However, it is often forgotten that the self-field effect unavoidably reduces the critical current density when the thickness of YBCO films increases and thereby total current rises. In this paper, the influence of self-field on the average critical current density is studied computationally as a function of film thickness. The situation is also scrutinized at different external magnetic fields in order to find ways to distinguish self-field effects from problems related to the manufacturing process. For this purpose, critical current measurements in external field perpendicular to the film surface are proposed. 2006 Elsevier B.V. All rights reserved. PACS: 74.25.Sv* Keywords: YBCO; Critical current
1. Introduction In superconducting power applications the engineering current density, Je, is one of the main parameters which determine the viability of devices [1]. YBCO films provide extremely high local critical current densities, Jc, up to several MA Æ cm2 at 77 K but the engineering current density is drastically lowered because the superconducting layer is very thin compared to the substrate on which it is grown [2]. Obviously Je should be improved if the superconducting films are made thicker [3]. So far, there have been some encouraging examples in which average critical current density of YBCO layer Jca at self-field seems to be quite independent of thickness of YBCO layer d. Such situations have been reported in tapes manufactured with the TFAMOD method [4,5] or with yttrium rich compositions [6]. However, most often it has been found that Jca decreases *
Corresponding author. Tel.: +358 3 31152014; fax: +358 3 3115 2160. E-mail address: lauri.rostila@tut.fi (L. Rostila).
0921-4534/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.10.008
if the thickness d of YBCO layer is increased. Several possible reasons have been suggested for this behaviour. According to the collective pinning theory Jca should be proportional to d1/2 when d is smaller than the pinning correlation length [7]. This explanation has been criticized because at sufficiently high magnetic fields Jca can be higher in a thicker superconducting tape [8] and the Jca(d)-dependence does not vanish at low temperatures as predicted [9]. More probable reasons have been found from the deteriorating structure of YBCO material. When d increases regions where the crystallographic c-axis is no more normal to the substrate plane can be created [9,10]. If d exceeds some tenths of micrometers also microcracking usually arises due to the difference between thermal expansion coefficients in the substrate and in the YBCO film [11]. However, recently the film thickness up to the micrometer range has been achieved on sapphire substrates without microcracking [12] but even then Jc has been found to plummet with increasing film thickness [13]. Then the film porosity and roughness increased and thereby the defect structures responsible for flux pinning changed.
L. Rostila et al. / Physica C 451 (2007) 66–70
Jca(d)-variations cannot be explained solely on the basis of material quality. In the analysis, it should be remembered that the self-field effect reduces unavoidably the critical current density when the thickness of YBCO films increases and the total current rises. The self-field effect has been studied both computationally and experimentally in LTS films [14], BSCCO tapes [15–17] and recently a onedimensional computational model is applied on YBCO films [18]. In this paper, the self-field induced reduction of the average critical current density is calculated as a function of film thickness. The situation is also scrutinized at different external magnetic fields in order to find ways to distinguish problems related to the YBCO quality from the effects of self-field. Because mesh generation problems have been encountered in finite element method due to high aspect ratio [19], a self-made algorithm based on finite integration method was chosen here. Furthermore, this approach is simple and fast. 2. Computational model In computations homogeneous YBCO material and DC operation current were assumed. Thus, the variations in the current density distribution were exclusively due to the spatial variations of self-field. A Cartesian coordinate system was used. The x-axis was parallel with the broad face of the tape, the y-axis parallel with the narrow face and current flowed in positive z-direction. The magnetic flux density created by transport current flowing in the tape was denoted as Bself. Furthermore, the tape can be exposed to an external magnetic flux density Bext. The main idea of the algorithm was to determine the current density distribution that fulfils the condition J = Jc(Bext + Bself(J)) inside the film. Here the local Jc(B)dependence was described with the Kim model [20] which was modified to take into account material anisotropy as a eB ; ð1Þ J c ðBÞ ¼ J c0 1 þ B0
Fig. 1. The default mesh with Nx = 16 and Ny = 12. Definitions of h and hloc are also included.
The film cross-section was divided into Nx times Ny rectangular elements in order to calculate the magnetic flux density, Bself(r), created by J(r 0 ). where r = [x y]T and r 0 = [x 0 y 0 ]T. In computations the mesh shown in Fig. 1 with Nx = 16 and Ny = 12, was used as a default. When current densities inside such elements are assumed to be constant the magnetic flux density can be calculated from the two-dimensional Biot–Savart law Z l 0 1 J ðr0 Þðr r0 Þ 0 Bself ðrÞ ¼ 0 ds ; ð3Þ 3=2 4p 1 0 S 0 ðr r0 Þ where l0 is the vacuum permeability and S 0 is the cross-section where the current flows. The flux diagram in Fig. 2 shows the principle of the algorithm starting from constant current density. The magnetic fields are calculated in the centre of elements and a new current density distribution is achieved as Jnew = Jc(Bext + Bself(Jold)), where Jold is the previous current distribution. Then the critical current is computed as an
where anisotropy is denoted with the anisotropic scaling [21] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðhÞ ¼ cos2 ðhloc Þ þ c2 sin2 ðhloc Þ: ð2Þ hloc is the angle between Bext + Bself and c-axis. Here the parameter values were chosen as Jc0 = 3 MA cm2 [18], B0 = 20 mT [18], a = 0.65 [22] and c = 5 [23]. These parameters are aimed to give a typical example about the Jc(B)-dependence of YBCO because only general trends of self-field effect are searched. When this analysis is applied to any particular sample, critical current measurements should be carried out and special attention paid on the derivation of Jc(B) from measured critical currents. The parameters of Jc(B)-dependence are hard to measure accurately due to the presence of self-field [24].
67
Fig. 2. Block diagram about the computation of critical current.
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L. Rostila et al. / Physica C 451 (2007) 66–70
integral of the new current density over the sample crosssection. The critical current was compared with its value from the previous iteration. If the relative change was below a given tolerance the final current density distribution was obtained. Otherwise the procedure was repeated. 3. Results and discussion The simulations were done with the self-made algorithm implemented to MATLAB [25]. About ten iterations were required to find the final critical current with the relative tolerance of 106. It was found out that acceptable solutions can be achieved with a very coarse mesh. For example, if the self-field critical current in a 4 mm times 1.5 lm tape was computed with Nx = 4, Ny = 3 the result deviated less than 2.95% compared to the mesh Nx = 64 and Ny = 48. The deviation was less than 0.57% with default mesh and it took about 0.3 s to compute one critical current with AMD Athlon 64 3400+. The self-field depends always on the sample geometry and dimensions. Thus, both the sample width and film thickness should influence on the average critical current density Jca. However, Jca is practically constant at a given film thickness if the sample width exceeds one tenth of millimetre as shown in Fig. 3. Thus, in practice the sample width can be neglected and attention can be paid solely to the film thickness. In order to explain this result the magnetic field inside the film was studied. Fig. 4 shows that after the sample width reaches one millimetre the average of By was always practically constant. The average of Bx could increase slightly until the sample width reached one millimetre but in an anisotropic superconductor the critical current density is mainly determined by By [15]. Usually the short sample critical current is measured as a function of magnetic flux density magnitude at constant field orientation which is defined by the angle h between the field and the normal of the film surface. Fig. 5 presents Jca as a function of YBCO film thickness in both parallel
Fig. 3. Normalized average critical current density as a function of YBCO tape width with different film thicknesses: (s) 0.25 lm, (h) 0.5 lm, (·) 1.0 lm, (d) 1.5 lm and (+) 3.0 lm.
Fig. 4. Average Bx (black) and By (gray) inside the YBCO as a function of tape width with different film thicknesses: (s) 0.25 lm, (h) 0.5 lm, (·) 1.0 lm, (d) 1.5 lm and (+) 3.0 lm.
Fig. 5. The computed normalized average critical current density as a function of YBCO film thickness at different external magnetic fields oriented (a) in perpendicular and (b) in parallel with film surface: (s) 0 mT, (h) 10 mT, (·) 20 mT, (d) 30 mT and (+) 40 mT.
and perpendicular external magnetic fields. In self-field Jca(d) resembled closely exponential decay, and in a 3 lm thick film Jca was reduced about 35% if compared to the real zero field critical current density Jc0. The exponential decay has also been fitted to measurements in references [13,26]. For example, the decay measured in reference [26] was probably in large part caused by the self-field effect because no structural problems were found in the samples. Here the function d J ca ðdÞ J c0 0:41984 exp þ 0:57563 1:7929 106 ð4Þ
L. Rostila et al. / Physica C 451 (2007) 66–70
69
fits to the computed self-field Jca values in Fig. 5 with maximum relative error of 0.42%. The parameter values in Eq. (4) were presented with five digits accuracy because the fitting was sensitive to them. However, even much better fit was achieved as J ca ðdÞ J c0 1 þ
d 1:2826 106
0:35509 ;
ð5Þ
which produced an error of 0.016% only. In external magnetic fields the situation changes dramatically. Already in parallel fields the shape of Jca(d) is no longer exponential. In perpendicular fields Jca(d) was quite constant if the thickness was below 0.60, 1.42 and 2.46 lm at By of 10, 20 and 30 mT, respectively. In other words, the thickness has no significant effect on Jca if the external field is higher than the maximum value of self-field the y-component. Therefore, the YBCO thin film samples should be measured in a sufficiently high perpendicular external magnetic field, where the self-field effect is cancelled out and Jca will directly tell about the material quality. Thus, e.g. the measured critical currents in Ref. [8] suggested that the quality of the studied material improved with increasing d. In order to further visualize the self-field effect Fig. 6 presents the average critical current density as a function of external magnetic flux density at different field orientations in a 1.5 lm thick YBCO film. At low parallel fields Jca stays quite constant due to the self-field effect. Fig. 7 illustrates the current density distributions in the YBCO film at different external field. Here the mesh with Nx = 64 and Ny = 48 was used in order to achieve an appropriate resolution for figures about the current density distribution. In self-field B naturally equals zero in the film centre, and therefore the current density is highest there. When the film is exposed to an external magnetic field, the self-field and external field compensate each other in some parts. The region of the highest J shifts to the point
Fig. 7. Current density in YBCO film with thickness of 1.5 lm at different external magnetic fields: (a) self-field, (b) h = 0, B = 10 mT, (c) h = 45, B = 10 mT and (d) h = 90, B = 10 mT.
where the compensation is most effective. Thus, at low external fields just the shape of J-distribution is reformed but the average critical current density does not change a lot. At perpendicular fields also the Jca drops because critical current density decreases drastically even at low values of B. Similar behaviour is often found in high quality BSCCO tapes as well [16,27]. The self-field effect is noticeable in Jca as long as the current density distribution remains inhomogeneous. When the external magnetic field rises, the current distribution is homogenised and the selffield effect can be neglected. The parameter values in Eq. (1) can vary significantly. Furthermore, the Jc(B)-dependence can be very complicated in anisotropic HTS materials and Eq. (1) does not necessarily describe it correctly [28]. Therefore, the sensitivity of presented results was studied with respect to the parameters J0, B0, a and c. The basic definition states that the sensitivity S nN of the quantity N with respect to parameter n is computed as S nN ¼
n dN : N dn
ð6Þ
Here the derivative was approximated numerically as dN N ð1:1nÞ N ð0:9nÞ : dn 0:2n
Fig. 6. Normalized average critical current density as a function of external magnetic flux density at different field orientations in YBCO film with thickness of 1.5 lm: (s) h = 0, (h) 22.5, (·) 45, (d) 67.5 and (+) 90. For comparison normalized critical current density as given by Eq. (1) is shown in gray.
ð7Þ
Eq. (6) gives a rough estimate how much a change of a parameter value effects on the Jca. Table 1 shows the sensitivity of Jca/Jc0. The most sensitive parameter was a, whereas variations in the anisotropy factor c had almost no influence on the results. It is expected that a and c do not vary significantly between samples, but Jc0 and B0 can depend strongly on the sample quality. Therefore, when this kind of analysis is applied on real samples attention should be paid to determine their values. However, small variations of the chosen parameters leave the main aspects in results unaltered.
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Table 1 Sensitivity of normalized average critical current density with respect to parameters in Eq. (1)
Jc0 a B0 c
d1, B1
d2, B1
d3, B1
d1, B2
d2, B2
d3, B2
0.06 0.06 0.06 0.01
0.19 0.25 0.19 0.01
0.25 0.36 0.25 0.01
0.00 0.45 0.33 0.00
0.05 0.45 0.31 0.01
0.15 0.48 0.32 0.01
The sensitivity is studied with thicknesses d1 = 0.25 lm, d2 = 1.5 lm, d3 = 3 lm and external magnetic flux densities B1 = 0 mT, B2 = 20 mT.
4. Conclusions In this paper, the influence of self-field on the critical current was studied computationally as a function of film thickness. The self-field effect unavoidably reduces the critical current density when the thickness of YBCO films increases and total current rises. It was found that in practical YBCO samples the average critical current density does not depend on the sample width but decreases with increasing film thickness in self-field. In order to distinguish self-field effects from problems related to the manufacturing process of real YBCO films it was proposed that the critical current measurements should be carried out in external field perpendicular to the film surface such that external field exceeds the maximum value of the perpendicular component of self-field. References [1] M.N. Wilson, Superconducting Magnets, Oxford University Press, Oxford, 1983. [2] M. Chen, W. Paul, M. Lakner, L. Donzel, M. Hoidis, P. Unternaehrer, R. Weder, M. Mendik, Physica C 372–376 (2002) 1657. [3] H.-W. Neumu¨ller, W. Schmidt, H. Kinder, H.C. Freyhardt, B. Stritzker, R. Wo¨rdenweber, V. Kirchoff, J. Alloy. Compd. 251 (1997) 366. [4] T. Izumi et al., Physica C 412–414 (2004) 885. [5] Y. Tokunaga et al., Cryogenics 44 (2004) 817. [6] R. Feenstra, A. Gapud, F. List, E. Specht, D. Christen, T. Holesinger, D. Feldmann, IEEE T. Appl. Supercon. 15 (2005) 2805. [7] A.I. Larkin, Y. Ovchinnikov, J. Low Temp. Phys. 34 (1979) 409. [8] T. Matsushita et al., Physica C 426–431 (2005) 1096.
[9] T. Matsushita, M. Kiuchi, K. Kimura, S. Miyata, A. Ibi, T. Muroga, Y. Yamada, Y. Shiohara, Supercond. Sci. Tech. 18 (2005) S227. [10] A. Mogro-Campero, L. Turner, E. Hall, N. Lewis, L. Peluso, W. Balz, Supercond. Sci. Tech. 3 (1990) 62. [11] G. Ka¨stner, C. Scha¨fer, S. Senz, D. Heese, M. Lorenz, H. Hochmuth, M. Getta, M.A. Hein, T. Kaiser, G. Mu¨ller, IEEE T. Appl. Supercon. 9 (1999) 2171. [12] K.D. Develos-Bagarinao, H. Yamasaki, Y. Nakagawa, H. Yamada, Physica C 392–396 (2003) 1229. [13] K. Develos-Bagarinao, H. Yamasaki, C. Nie J, Y. Nakagawa, Supercond. Sci. Tech. 18 (2005) 667. [14] J.R. Gavaler, A.T. Santhanam, A.I. Braginski, M. Ashkin, M.A. Janocko, IEEE T. Magn. 17 (1981) 573. [15] N. Shaked et al., Physica C 354 (2001) 237. [16] J. Lehtonen, A. Korpela, W. Nah, J. Kang, P. Kova´cˇ, T. Melisˇek, Physica C 403 (2004) 257. [17] S. Spreafico, L. Gherardi, S. Fleshler, D. Tatelbaum, J. Leone, D. Yu, G. Snitchler, IEEE T. Appl. Supercon. 9 (1999) 2159. [18] A. Brojeny, J. Clem, Supercond. Sci. Tech. 18 (2005) 888. [19] S. Stavrev, F. Grilli, B. Dutoit, S.P. Ashworth, Supercond. Sci. Tech. 18 (2005) 1300. [20] Y.B. Kim, C.F. Hempstead, A.R. Strnad, Rev. Mod. Phys. 36 (1964) 43. [21] G. Blatter, M.V. Feigel’man, V.B. Geshkenbein, A.I. Larkin, V.M. Vinokur, Rev. Mod. Phys. 66 (1994) 1125. [22] A.A. Gapud, R. Feenstra, D.K. Christen, J.R. Thompson, T.G. Holesinger, IEEE T. Appl. Supercon. 15 (2005) 2578. [23] L. Civale et al., Appl. Phys. Lett. 84 (2003) 2121. [24] M. Masti, J. Lehtonen, R. Pera¨la¨, W. Nah, J. Kang, Physica C 401 (2004) 155. [25] www.mathworks.com. [26] S. Foltyn, P. Tiwari, R. Dye, M. Le, X. Wu, Appl. Phys. Lett. 63 (1993) 1848. [27] J. Lehtonen, R. Pera¨la¨, R. Mikkonen, Physica C 401 (2004) 151. [28] J. Lehtonen, A. Korpela, J. Pitel, P. Kova´cˇ, Physica C 403 (2004) 139.