- Email: [email protected]
Ag tapes
Physica C 309 Ž1998. 197–202
Critical current of high Tc superconducting Bi2223rAg tapes Y.K. Huang ) , B. ten Haken, H.H.J. ten Kate Low Temperature DiÕision, Faculty of Applied Physics, UniÕersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received 7 July 1998; revised 13 October 1998; accepted 16 October 1998
Abstract The magnetic field dependence of the critical current of various high Tc superconducting Bi2223rAg tapes indicates that the transport current is carried through two paths: one is through weakly-linked grain boundaries ŽJosephson junctions.; another is through well-connected grains. The critical current flowing through the weakly-linked grain boundaries, ICw , can be characterized by the Fraunhofer-pattern, taking into account the distribution of junction dimension in the tapes. The critical current along the path of well-connected grains, ICs , is a complex exponential function of the applied field depending on the pinning strength in the grains. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Critical current; Weak links; Flux pinning
1. Introduction The critical current of high-Tc superconducting Bi2223rAg tapes has increased continuously during the last few years due to the improvement of fabrication techniques. The magnetic field dependence of the critical current of the tapes has been studied extensively for applications and for understanding of the pinning mechanism in this material. The critical current density JC of Bi2223rAg tapes is observed to drop significantly in small fields perpendicular to the tape Ž B 5 c-axis.. With further increase of the field, JC decrease, is less pronounced. The power law dependence, JC Ž B . A Bya , is usually used to describe the JC Ž B . curve in low-field region Že.g., 10 - B - 100 mT at 77 K., while in higher fields the exponential dependence, JC Ž B . A expŽyBrB0 ., can be applied w1,2x. These two kinds of field dependen)
Corresponding author. Tel.: q31-53-489-3140; Fax: q31-53489-1099; E-mail: [email protected]
cies can be attributed to different dissipation mechanisms. The Bi2223 compound in a tape is a highly anisotropic polycrystalline material. Due to the small coherence length of the material, defects along grain boundaries will deteriorate the grain connectivity and lead to weak links. The small coherence length and large penetration depth as well as the 2D characteristics of the vortices’ structure result in rather low pinning potential of the material, leading to the occurrence of flux creep and flow at relatively high temperatures. Therefore, two important dissipation mechanisms limiting the current-carrying capacity can be distinguished, the weak-link current suppression and the flux motion. These two mechanisms respond differently to the presence of a magnetic field. In Bi2223rAg tapes, the grains are aligned with the c-axes approximately normal to the broad face of the tape through mechanical deformation and heat treatments, which improves the grain to grain con-
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 6 0 8 - X
198
Y.K. Huang et al.r Physica C 309 (1998) 197–202
nectivity significantly. Various kinds of grain boundaries, e.g., the Ž001. twist boundaries and the resulting colony structure; edge-on c-axis tilt grain boundaries, have been found in the filaments of the tapes. Among these, the low-angle ab-axis grain boundary is considered to be a strongly-linked one for transporting current w3,4,15x and the others may form a weakly-linked Josephson junction network. The current flows simultaneously through these two different paths in the tape. The JC Ž B . curves of a tape reflects the combination of the two dissipation mechanisms mentioned above in these two current-carrying paths, respectively. In this paper we present an analysis of the magnetic field dependence of the critical currents in various Bi2223rAg tapes based on a proposed model and discuss the hysteresis in IC Ž B . curves and pinning force in the tape.
2. Experimental The Bi2223 tapes used in the experiments originated from various research groups in universities and companies. These not-twisted multi-filament tapes are fabricated using the oxide powder-in-tube ŽOPIT. method w5–7x. The number of filaments in the tapes is 19, 30, 37, 55 or 85. The DC voltage– current characteristics Ž V–I curves. of the tapes are measured at 77 K using a four-point configuration. DC magnetic fields up to 0.2 T are applied perpendicular to the tape face. The critical currents are determined using the voltage criterion of 10y4 Vrm.
The ICw and ICs can be distinguished because of their different magnetic field dependencies. For a Josephson junction, the Josephson current between adjacent grains in a local field has a characteristic of the Fraunhofer-pattern w8,9x: IJ s Ž B, B0 . s I0
sin p Ž B q Bm . rB0
p Ž B q Bm . rB0
Ž 2.
where B is the applied magnetic field and Bm is a field representing the conservation of magnetic history in the sample and the effect of self-field due to the transport current. B0 is the characteristic field, B0 s F 0rŽ2 l q d . w, where F 0 is the flux quantum, l the magnetic penetration depth, d and w the thickness and width of the junction. The critical current along the weak-links path Ž ICw . is the sum of the Josephson currents in the tape. Bi2223 tapes contain a large number of grain boundaries, which must have a statistical distribution of their thickness d and width w. Therefore, the characteristic field B0 for a Josephson junction, which is directly related to the junction dimension, should follow a certain distribution function, GŽ B0 .. In our analysis, we assume that this distribution function is a log-normal one: G Ž B0 . s
1
'2p ln s
exp y
Ž ln B0 y ln Bw . 2 ln2s
2
,
Ž 3. where Bw and s are parameters characterizing the distribution function. With this distribution function the ICw can be expressed as
3. Results and discussion
b
Ha I Ž B, B . G Ž B . d B J
3.1. Field dependence of the critical current in the two current-carrying paths Based on the measured IC Ž B . curves and the microstructures of various Bi2223rAg tapes, it is reasonable to assume that the critical current of a tape is flowing in two different paths: one is through weakly-linked grain boundaries ŽJosephson junctions., ICw ; another is through well-connected grains, ICs , i.e., IC s ICw q ICs .
,
Ž 1.
ICw Ž B . s
0
0
b
Ha G Ž B . d B 0
0
.
Ž 4.
0
When the thickness of a grain boundary becomes smaller than the so-called Josephson critical thickness w10,11x, a Josephson junction can be formed. While further reducing the thickness of a grain boundary to be smaller than the coherence length, a strong link is formed. The upper and lower limits of the integration in Eq. Ž3. correspond to these two cases.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
199
The critical current carried by the strongly-linked grains Ž ICs . is controlled by the flux pinning mechanism in the grains. The collective pinning theory w12x predicts an exponential dependence of JC Ž B .. We assume that this can be expressed by a general form as proposed in Ref. w14x: ICs Ž B . s I0 s exp y Ž BrBs .
a
,
Ž 5.
where Bs and a are related to the pinning characteristics of a tape. Usually a has a value between 1 and 2 in the Bi2223rAg tapes measured. 3.2. Comparison to experimental data The IC Ž B . curves of many different Bi2223rAg tapes measured are analyzed using the above described two current-carrying paths model. Three typical examples that represent most of the IC Ž B . characteristics of various tapes measured, are listed in Table 1 and illustrated in Figs. 1–3. The results show that all IC Ž B . curves can be reproduced very well in the entire range of fields measured Ž B - 0.2 T., including the very low-field region where the self-field and remnant field are comparable to the applied field Žsee the insert in Fig. 1.. The critical current contributed by the well-connected grains, ICs , accounts for 20% to 40% of the total critical current at zero applied field. The ICs decreases as a
Fig. 1. Critical current as a function of applied magnetic field of tape T1 Ž85 filaments.. Filled marks: measured IC in increasing field; Open marks: measured IC in decreasing field; Lines: calculated IC Ž B . curve and the critical current components ICs and ICw according to Eqs. Ž1. – Ž5.. The ICs and ICw , attributed to the well-connected current path and weak-link current path, are shown as the dotted line and dot–dash line, respectively. Insert: very low-field part of the IC Ž B . curves.
complex exponential function of B and becomes dominant in higher fields. The critical current contributed by the weakly linked grains, ICw , decreases
Table 1 Sample description and parameters for describing IC Ž B . curves using two-current-path model Sample
Number of filaments Size Žmm2 . SCrAg ratio Ž%. IC ŽA. I0 ŽA. Bw ŽmT. ln s Bm ŽmT. I0s ŽA. Bs ŽT. a
T1
T2
T3
85 4.2=0.25
55 4.3=0.22
30 3.2=0.17
25 31.2 20.73 Ž20.80. 15.10 Ž17.57. 0.952 Ž0.752. y0.45 Žy0.94. 10.60 0.252 1.00
20 20.8 12.81 Ž11.76. 8.90 Ž12.72. 1.082 Ž0.833. y0.83 Žy0.57. 8.99 0.240 1.41
20 8.0 6.86 Ž6.16. 4.22 Ž5.26. 0.981 Ž0.932. y0.46 Žy0.47. 1.67 0.115 2.00
Parameters for decreasing field are shown in parentheses when they are different from those for increasing field.
Fig. 2. Critical current as a function of applied magnetic field of tape T2 Ž55 filaments.. Filled marks: measured IC in increasing field; Lines: calculated IC Ž B . curve and the critical current components ICs Ždot line. and ICw Ždot–dash line. according to Eqs. Ž1. – Ž5..
200
Y.K. Huang et al.r Physica C 309 (1998) 197–202
distribution of favored weak links and supercurrent paths within the weak-link network. The change of local field when the applied field is reduced is related to the flux trapped by persistent currents both within the grains and in the survived current path. We found that the following formula can represent the ICw component rather well and the calculation for ICw is simplified:
½
ICw Ž B . s I1r 1 q Ž B q B1 . rB2
Fig. 3. Critical current as a function of applied magnetic field of tape T3 Ž30 filaments.. Filled marks: measured IC in increasing field; Lines: calculated IC Ž B . curve and the critical current components ICs Ždotted line. and ICw Ždot–dash line. according to Eqs. Ž1. – Ž5..
strongly in low-field range and becomes inversely proportional to the field in higher fields. The decrease of ICw may be slower than that of ICs after the applied field is higher than a certain value ŽFig. 3.. This could provide one of the explanations to the slower decrease of IC Ž B . in high field as reported in several papers Že.g., Refs. w1,3,14x., where JC is lower than 10 6 Arm2 . In some tapes the IC Ž B . curve in increasing field is different from the one in decreasing field. This hysteresis is probably associated only with the change of weak-link current path. The IC Ž B . curve in decreasing field can also be calculated using Eqs. Ž1. – Ž5.. An example is shown in Fig. 1. The parameters associated with the weak-link current path in decreasing field are slightly different from those in increasing field Žsee Table 1, numbers in brackets.. It is found that the B0 distribution function in decreasing field is narrower and shifted to higher Bw compared to the one in increasing field ŽFig. 4.. The change of the B0 distribution function can be understood by considering the change in both the magnitude and direction of the local magnetic field within the weak-link network w13x. Any change in the local field will selectively suppress the Josephson critical current of some weak links and restore the critical current of the others, thereby changing the spatial
2 1r2
5
,
Ž 6.
where I1 , B1 and B2 are fitting parameters. How these parameters relate to the parameters of I0 , Bm , Bw and s in Eqs. Ž2. – Ž4. is not yet clear. Usually, it is observed that I1 f I0 , and B1 is very small Ž- 1 mT.. The B2 increases with the IC and may be related to the self-field of the tape. The current path formed by well-connected grains is probably along low-angle grain boundaries with a large tilt component w15,16x. The flux pinning at the low-angle tilt boundaries is reported to be stronger than the intra-grain pinning w17x. Therefore, the critical current ICs may be controlled by intra-grain pinning. An exponential field dependence of critical current is predicted by the collective pinning theory w12x. However, a deviation from the simple exponential dependence has been observed in various samples. In these cases the more general form of field
Fig. 4. Distribution of the characteristic fields B0 of the Josephson junctions in tape T1 according to Eq. Ž3.. Full line: GŽ B0 . in increasing applied field; Dashed line: GŽ B0 . in decreasing applied field.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
dependence as Eq. Ž5. describes the experimental curves better. We attribute this deviation from the simple exponential dependence to the inhomogeneous distribution of the pinning strength in the tape and the arbitrary choice of the critical current criterion. In a certain applied field, the flux lines may creep or flow, depending on the strength and direction of the Lorentz force locally. The dissipations due to flux creep and flux flow have different field dependence. The contributions by flux creep and flux flow, respectively, to the total dissipation are related to the distribution of the pinning strength and vary with the applied field and the chosen criterion. Therefore, the critical current determined using a certain criterion Že.g., 10y4 Vrm. represents the total contribution to the dissipation originating from flux creep as well as Žlocal. flux flow. In this sense, the index a in Eq. Ž5. may reflect the relative contribution of flux creep and flux flow under the chosen criterion. In fact, a depends on the tape fabrication process and may be closely related to the grain alignment and the micro-crack density in the tape w14,18x. The pinning force density Fp is usually calculated with the following formula Fp s JC B s IC BrS,
Ž 7.
where S is the cross-section area of the superconducting core. However, as we discussed above, the
201
Fig. 6. The correlation between Bp , the field where the pinning force is maximum, and Bw , the most popular characteristic field of Josephson junctions, in various tapes analyzed.
pinning strength is reflected only by the ICs component in a tape. Therefore, we should use the ICs and the effective cross-section area, Se , in Eq. Ž7. instead of the total IC and S, i.e., Fp s ICs BrSe s I0 s B exp y Ž BrBs .
a
rSe .
Ž 8.
The uncertainty of the effective cross-section for carrying the current ICs can be eliminated by normalizing Fp to the maximum pinning force Fpmax :
½
FprFpmax s Ž BrBp . exp Ž 1ra . 1 y Ž BrBp .
a
5, Ž 9.
Fig. 5. Normalized pinning force density as a function of applied magnetic field for the three tapes, T1, T2 and T3.
where Bp s a Žy1r a . Bs is the field where the maximum pinning force appears. The FprFpmax as a function of B for the selected tapes T1, T2 and T3 is shown in Fig. 5. According to the Kramer model w19x for conventional type-II superconducting materials, the macroscopic pinning force is determined by two factors: one is the pinning strength for individual flux lines and the other is the shear strength of the flux line lattice. Flux motion occurs by flux-line depinning from pinning centers in the low-field region Ž B - Bp . and by synchronous shear of the flux-line lattice in the high-field region Ž B ) Bp .. A maximum of pinning force appears where these two factors are approximately equal. For a relatively strong flux-line pinning, the Fpmax appears at a lower reduced field.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
202
The Kramer model at a given temperature can be expressed as q
FprFpmax s Ab p Ž 1 y b . ,
Ž 10 .
where A, p and q are numerical parameters, b s BrBirr is the reduced field, Birr is the irreversibility field. Assuming the Kramer model is still valid for Bi2223rAg tapes, we find, by comparing Eq. Ž9. to Eq. Ž10., that Birr is proportional to the Bs and p f 1 for all tapes. However, q varies from 2 to a few hundred, depending on the index a of the tape. It seems that a higher a Žcorrespondingly a lower q . may reflect the weaker elastic properties of the fluxline lattice, which is related to the anisotropic layered structure of the material w20x. If we plot Bp , the field where the pinning force is maximal, vs. Bw , the most probable characteristic field of the Josephson junction, for various tapes ŽFig. 6., we find that the Bp and Bw are correlated. Higher Bp corresponds to higher Bw . In fact, the grain size in different tapes is about the same. The higher Bw implies that the thickness of the Josephson junction is smaller, i.e., the grains are more compacted. The results presented in this paper are concentrated on the case when the magnetic field is applied perpendicular to the broad face of a tape. However, the analytical process applied is also valid for other field directions if the misalignment of grains in the tape is considered in the calculation of the critical current along the well-connected grains.
4. Conclusion Detailed analysis of the IC Ž B . curves of various Bi2223rAg tapes reveals that the transport current of the tape is carried through the weak-linked Josephson junction network as well as through a chain of well-connected grains. The critical current along the weak-linked path, ICw , is related to the total Josephson current of the active junctions and the distribution of the junction dimensions in the tape. The hysteresis in the critical current in decreasing field is attributed to the change of the active Josephson junctions due to the change of local field, which is related to the flux trapped by persistent
currents both within the grains and in the survived current path. The critical current along the well-connected grains, ICs , is controlled by the pinning properties of the grains, which is governed by the tape fabrication process and is closely related to the grain alignment, micro-cracks and other defects in the grains. The pinning force of the Bi2223rAg tape should be calculated using the ICs instead of the total IC .
References w1x Y. Mawatari, H. Yamasaki, S. Kosaka, M. Umeda, Cryogenics 35 Ž1995. 161. w2x C.M. Friend, D.P. Hampshire, Physica C 252 Ž1995. 107. w3x M. Dhalle, ´ M. Cuthbert, M.D. Johnston, J. Everett, R. Flukiger, S.X. Dou, W. Goldacker, T. Beales, A.D. Caplin, ¨ Supercond. Sci. Technol. 10 Ž1997. 21. w4x A.E. Pashitski, A. Polyanskii, A. Gurevich, J.A. Parrell, D.C. Larbalestier, Physica C 246 Ž1995. 133. w5x G. Grasso, A. Jeremie, R. Flukiger, Supercond. Sci. Technol. 8 Ž1995. 827. w6x W. Goldacker, E. Mossang, M. Quilitz, M. Rikel, IEEE Trans. Appl. Supercond. 7 Ž1997. 1407. w7x Q. Li, G.N. Riley Jr., R.D. Parrella, S. Fleshler, M.W. Rupich, W.L. Carter, J.O. Willis, J.Y. Coulter, J.F. Bingert, V.K. Sikka, J.A. Parrell, D.C. Larbalestier, IEEE Trans. Appl. Supercond. 7 Ž1997. 2026. w8x T. van Duzer, C.W. Turner, Principles of Superconductive Devices and Circuits, Elsevier, North-Holland, New York, 1981. w9x R.L. Peterson, J.W. Ekin, Phys. Rev. B 37 Ž1988. 9848. w10x Y. Zhang, H. Liu, X. Cao, Chin. J. Low Temp. Phys. 7 Ž1985. 12. w11x Q.Y. Hu, R.M. Schalk, H.K. Liu, R.K. Wang, C. Czurda, S.X. Dou, J. Appl. Phys. 78 Ž1995. 1123. w12x M.V. Feigel’man, V.M. Vinokur, Phys. Rev. B 41 Ž1990. 8986. w13x J.E. Evetts, B.A. Glowacki, Cryogenics 28 Ž1988. 641. w14x J. Horvat, S.X. Dou, H.K. Liu, R. Bhasale, Physica C 271 Ž1996. 51. w15x Y.H. Li, J.A. Kilner, M. Dhalle, A.D. Caplin, G. Grasso, R. Flukiger, Supercond. Sci. Technol. 8 Ž1995. 764. ¨ w16x B. Hensel, G. Grasso, R. Flukiger, J. Electron. Mater. 24 ¨ Ž1995. 1877. w17x N. Nakamura, G.D. Gu, K. Takamuku, M. Murakami, N. Koshizuka, Appl. Phys. Lett. 61 Ž1992. 3044. w18x Y. Fang, S. Danyluk, M.T. Lanagan, Cryogenics 36 Ž1996. 957. w19x E.J. Kramer, J. Appl. Phys. 44 Ž1973. 1360. w20x P.H. Kes, J. Aarts, V.M. Vinokur, C.J. van der Beek, Phys. Rev. Lett. 64 Ž1990. 1063.
Critical current of high Tc superconducting Bi2223rAg tapes Y.K. Huang ) , B. ten Haken, H.H.J. ten Kate Low Temperature DiÕision, Faculty of Applied Physics, UniÕersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received 7 July 1998; revised 13 October 1998; accepted 16 October 1998
Abstract The magnetic field dependence of the critical current of various high Tc superconducting Bi2223rAg tapes indicates that the transport current is carried through two paths: one is through weakly-linked grain boundaries ŽJosephson junctions.; another is through well-connected grains. The critical current flowing through the weakly-linked grain boundaries, ICw , can be characterized by the Fraunhofer-pattern, taking into account the distribution of junction dimension in the tapes. The critical current along the path of well-connected grains, ICs , is a complex exponential function of the applied field depending on the pinning strength in the grains. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Critical current; Weak links; Flux pinning
1. Introduction The critical current of high-Tc superconducting Bi2223rAg tapes has increased continuously during the last few years due to the improvement of fabrication techniques. The magnetic field dependence of the critical current of the tapes has been studied extensively for applications and for understanding of the pinning mechanism in this material. The critical current density JC of Bi2223rAg tapes is observed to drop significantly in small fields perpendicular to the tape Ž B 5 c-axis.. With further increase of the field, JC decrease, is less pronounced. The power law dependence, JC Ž B . A Bya , is usually used to describe the JC Ž B . curve in low-field region Že.g., 10 - B - 100 mT at 77 K., while in higher fields the exponential dependence, JC Ž B . A expŽyBrB0 ., can be applied w1,2x. These two kinds of field dependen)
Corresponding author. Tel.: q31-53-489-3140; Fax: q31-53489-1099; E-mail: [email protected]
cies can be attributed to different dissipation mechanisms. The Bi2223 compound in a tape is a highly anisotropic polycrystalline material. Due to the small coherence length of the material, defects along grain boundaries will deteriorate the grain connectivity and lead to weak links. The small coherence length and large penetration depth as well as the 2D characteristics of the vortices’ structure result in rather low pinning potential of the material, leading to the occurrence of flux creep and flow at relatively high temperatures. Therefore, two important dissipation mechanisms limiting the current-carrying capacity can be distinguished, the weak-link current suppression and the flux motion. These two mechanisms respond differently to the presence of a magnetic field. In Bi2223rAg tapes, the grains are aligned with the c-axes approximately normal to the broad face of the tape through mechanical deformation and heat treatments, which improves the grain to grain con-
0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 6 0 8 - X
198
Y.K. Huang et al.r Physica C 309 (1998) 197–202
nectivity significantly. Various kinds of grain boundaries, e.g., the Ž001. twist boundaries and the resulting colony structure; edge-on c-axis tilt grain boundaries, have been found in the filaments of the tapes. Among these, the low-angle ab-axis grain boundary is considered to be a strongly-linked one for transporting current w3,4,15x and the others may form a weakly-linked Josephson junction network. The current flows simultaneously through these two different paths in the tape. The JC Ž B . curves of a tape reflects the combination of the two dissipation mechanisms mentioned above in these two current-carrying paths, respectively. In this paper we present an analysis of the magnetic field dependence of the critical currents in various Bi2223rAg tapes based on a proposed model and discuss the hysteresis in IC Ž B . curves and pinning force in the tape.
2. Experimental The Bi2223 tapes used in the experiments originated from various research groups in universities and companies. These not-twisted multi-filament tapes are fabricated using the oxide powder-in-tube ŽOPIT. method w5–7x. The number of filaments in the tapes is 19, 30, 37, 55 or 85. The DC voltage– current characteristics Ž V–I curves. of the tapes are measured at 77 K using a four-point configuration. DC magnetic fields up to 0.2 T are applied perpendicular to the tape face. The critical currents are determined using the voltage criterion of 10y4 Vrm.
The ICw and ICs can be distinguished because of their different magnetic field dependencies. For a Josephson junction, the Josephson current between adjacent grains in a local field has a characteristic of the Fraunhofer-pattern w8,9x: IJ s Ž B, B0 . s I0
sin p Ž B q Bm . rB0
p Ž B q Bm . rB0
Ž 2.
where B is the applied magnetic field and Bm is a field representing the conservation of magnetic history in the sample and the effect of self-field due to the transport current. B0 is the characteristic field, B0 s F 0rŽ2 l q d . w, where F 0 is the flux quantum, l the magnetic penetration depth, d and w the thickness and width of the junction. The critical current along the weak-links path Ž ICw . is the sum of the Josephson currents in the tape. Bi2223 tapes contain a large number of grain boundaries, which must have a statistical distribution of their thickness d and width w. Therefore, the characteristic field B0 for a Josephson junction, which is directly related to the junction dimension, should follow a certain distribution function, GŽ B0 .. In our analysis, we assume that this distribution function is a log-normal one: G Ž B0 . s
1
'2p ln s
exp y
Ž ln B0 y ln Bw . 2 ln2s
2
,
Ž 3. where Bw and s are parameters characterizing the distribution function. With this distribution function the ICw can be expressed as
3. Results and discussion
b
Ha I Ž B, B . G Ž B . d B J
3.1. Field dependence of the critical current in the two current-carrying paths Based on the measured IC Ž B . curves and the microstructures of various Bi2223rAg tapes, it is reasonable to assume that the critical current of a tape is flowing in two different paths: one is through weakly-linked grain boundaries ŽJosephson junctions., ICw ; another is through well-connected grains, ICs , i.e., IC s ICw q ICs .
,
Ž 1.
ICw Ž B . s
0
0
b
Ha G Ž B . d B 0
0
.
Ž 4.
0
When the thickness of a grain boundary becomes smaller than the so-called Josephson critical thickness w10,11x, a Josephson junction can be formed. While further reducing the thickness of a grain boundary to be smaller than the coherence length, a strong link is formed. The upper and lower limits of the integration in Eq. Ž3. correspond to these two cases.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
199
The critical current carried by the strongly-linked grains Ž ICs . is controlled by the flux pinning mechanism in the grains. The collective pinning theory w12x predicts an exponential dependence of JC Ž B .. We assume that this can be expressed by a general form as proposed in Ref. w14x: ICs Ž B . s I0 s exp y Ž BrBs .
a
,
Ž 5.
where Bs and a are related to the pinning characteristics of a tape. Usually a has a value between 1 and 2 in the Bi2223rAg tapes measured. 3.2. Comparison to experimental data The IC Ž B . curves of many different Bi2223rAg tapes measured are analyzed using the above described two current-carrying paths model. Three typical examples that represent most of the IC Ž B . characteristics of various tapes measured, are listed in Table 1 and illustrated in Figs. 1–3. The results show that all IC Ž B . curves can be reproduced very well in the entire range of fields measured Ž B - 0.2 T., including the very low-field region where the self-field and remnant field are comparable to the applied field Žsee the insert in Fig. 1.. The critical current contributed by the well-connected grains, ICs , accounts for 20% to 40% of the total critical current at zero applied field. The ICs decreases as a
Fig. 1. Critical current as a function of applied magnetic field of tape T1 Ž85 filaments.. Filled marks: measured IC in increasing field; Open marks: measured IC in decreasing field; Lines: calculated IC Ž B . curve and the critical current components ICs and ICw according to Eqs. Ž1. – Ž5.. The ICs and ICw , attributed to the well-connected current path and weak-link current path, are shown as the dotted line and dot–dash line, respectively. Insert: very low-field part of the IC Ž B . curves.
complex exponential function of B and becomes dominant in higher fields. The critical current contributed by the weakly linked grains, ICw , decreases
Table 1 Sample description and parameters for describing IC Ž B . curves using two-current-path model Sample
Number of filaments Size Žmm2 . SCrAg ratio Ž%. IC ŽA. I0 ŽA. Bw ŽmT. ln s Bm ŽmT. I0s ŽA. Bs ŽT. a
T1
T2
T3
85 4.2=0.25
55 4.3=0.22
30 3.2=0.17
25 31.2 20.73 Ž20.80. 15.10 Ž17.57. 0.952 Ž0.752. y0.45 Žy0.94. 10.60 0.252 1.00
20 20.8 12.81 Ž11.76. 8.90 Ž12.72. 1.082 Ž0.833. y0.83 Žy0.57. 8.99 0.240 1.41
20 8.0 6.86 Ž6.16. 4.22 Ž5.26. 0.981 Ž0.932. y0.46 Žy0.47. 1.67 0.115 2.00
Parameters for decreasing field are shown in parentheses when they are different from those for increasing field.
Fig. 2. Critical current as a function of applied magnetic field of tape T2 Ž55 filaments.. Filled marks: measured IC in increasing field; Lines: calculated IC Ž B . curve and the critical current components ICs Ždot line. and ICw Ždot–dash line. according to Eqs. Ž1. – Ž5..
200
Y.K. Huang et al.r Physica C 309 (1998) 197–202
distribution of favored weak links and supercurrent paths within the weak-link network. The change of local field when the applied field is reduced is related to the flux trapped by persistent currents both within the grains and in the survived current path. We found that the following formula can represent the ICw component rather well and the calculation for ICw is simplified:
½
ICw Ž B . s I1r 1 q Ž B q B1 . rB2
Fig. 3. Critical current as a function of applied magnetic field of tape T3 Ž30 filaments.. Filled marks: measured IC in increasing field; Lines: calculated IC Ž B . curve and the critical current components ICs Ždotted line. and ICw Ždot–dash line. according to Eqs. Ž1. – Ž5..
strongly in low-field range and becomes inversely proportional to the field in higher fields. The decrease of ICw may be slower than that of ICs after the applied field is higher than a certain value ŽFig. 3.. This could provide one of the explanations to the slower decrease of IC Ž B . in high field as reported in several papers Že.g., Refs. w1,3,14x., where JC is lower than 10 6 Arm2 . In some tapes the IC Ž B . curve in increasing field is different from the one in decreasing field. This hysteresis is probably associated only with the change of weak-link current path. The IC Ž B . curve in decreasing field can also be calculated using Eqs. Ž1. – Ž5.. An example is shown in Fig. 1. The parameters associated with the weak-link current path in decreasing field are slightly different from those in increasing field Žsee Table 1, numbers in brackets.. It is found that the B0 distribution function in decreasing field is narrower and shifted to higher Bw compared to the one in increasing field ŽFig. 4.. The change of the B0 distribution function can be understood by considering the change in both the magnitude and direction of the local magnetic field within the weak-link network w13x. Any change in the local field will selectively suppress the Josephson critical current of some weak links and restore the critical current of the others, thereby changing the spatial
2 1r2
5
,
Ž 6.
where I1 , B1 and B2 are fitting parameters. How these parameters relate to the parameters of I0 , Bm , Bw and s in Eqs. Ž2. – Ž4. is not yet clear. Usually, it is observed that I1 f I0 , and B1 is very small Ž- 1 mT.. The B2 increases with the IC and may be related to the self-field of the tape. The current path formed by well-connected grains is probably along low-angle grain boundaries with a large tilt component w15,16x. The flux pinning at the low-angle tilt boundaries is reported to be stronger than the intra-grain pinning w17x. Therefore, the critical current ICs may be controlled by intra-grain pinning. An exponential field dependence of critical current is predicted by the collective pinning theory w12x. However, a deviation from the simple exponential dependence has been observed in various samples. In these cases the more general form of field
Fig. 4. Distribution of the characteristic fields B0 of the Josephson junctions in tape T1 according to Eq. Ž3.. Full line: GŽ B0 . in increasing applied field; Dashed line: GŽ B0 . in decreasing applied field.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
dependence as Eq. Ž5. describes the experimental curves better. We attribute this deviation from the simple exponential dependence to the inhomogeneous distribution of the pinning strength in the tape and the arbitrary choice of the critical current criterion. In a certain applied field, the flux lines may creep or flow, depending on the strength and direction of the Lorentz force locally. The dissipations due to flux creep and flux flow have different field dependence. The contributions by flux creep and flux flow, respectively, to the total dissipation are related to the distribution of the pinning strength and vary with the applied field and the chosen criterion. Therefore, the critical current determined using a certain criterion Že.g., 10y4 Vrm. represents the total contribution to the dissipation originating from flux creep as well as Žlocal. flux flow. In this sense, the index a in Eq. Ž5. may reflect the relative contribution of flux creep and flux flow under the chosen criterion. In fact, a depends on the tape fabrication process and may be closely related to the grain alignment and the micro-crack density in the tape w14,18x. The pinning force density Fp is usually calculated with the following formula Fp s JC B s IC BrS,
Ž 7.
where S is the cross-section area of the superconducting core. However, as we discussed above, the
201
Fig. 6. The correlation between Bp , the field where the pinning force is maximum, and Bw , the most popular characteristic field of Josephson junctions, in various tapes analyzed.
pinning strength is reflected only by the ICs component in a tape. Therefore, we should use the ICs and the effective cross-section area, Se , in Eq. Ž7. instead of the total IC and S, i.e., Fp s ICs BrSe s I0 s B exp y Ž BrBs .
a
rSe .
Ž 8.
The uncertainty of the effective cross-section for carrying the current ICs can be eliminated by normalizing Fp to the maximum pinning force Fpmax :
½
FprFpmax s Ž BrBp . exp Ž 1ra . 1 y Ž BrBp .
a
5, Ž 9.
Fig. 5. Normalized pinning force density as a function of applied magnetic field for the three tapes, T1, T2 and T3.
where Bp s a Žy1r a . Bs is the field where the maximum pinning force appears. The FprFpmax as a function of B for the selected tapes T1, T2 and T3 is shown in Fig. 5. According to the Kramer model w19x for conventional type-II superconducting materials, the macroscopic pinning force is determined by two factors: one is the pinning strength for individual flux lines and the other is the shear strength of the flux line lattice. Flux motion occurs by flux-line depinning from pinning centers in the low-field region Ž B - Bp . and by synchronous shear of the flux-line lattice in the high-field region Ž B ) Bp .. A maximum of pinning force appears where these two factors are approximately equal. For a relatively strong flux-line pinning, the Fpmax appears at a lower reduced field.
Y.K. Huang et al.r Physica C 309 (1998) 197–202
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The Kramer model at a given temperature can be expressed as q
FprFpmax s Ab p Ž 1 y b . ,
Ž 10 .
where A, p and q are numerical parameters, b s BrBirr is the reduced field, Birr is the irreversibility field. Assuming the Kramer model is still valid for Bi2223rAg tapes, we find, by comparing Eq. Ž9. to Eq. Ž10., that Birr is proportional to the Bs and p f 1 for all tapes. However, q varies from 2 to a few hundred, depending on the index a of the tape. It seems that a higher a Žcorrespondingly a lower q . may reflect the weaker elastic properties of the fluxline lattice, which is related to the anisotropic layered structure of the material w20x. If we plot Bp , the field where the pinning force is maximal, vs. Bw , the most probable characteristic field of the Josephson junction, for various tapes ŽFig. 6., we find that the Bp and Bw are correlated. Higher Bp corresponds to higher Bw . In fact, the grain size in different tapes is about the same. The higher Bw implies that the thickness of the Josephson junction is smaller, i.e., the grains are more compacted. The results presented in this paper are concentrated on the case when the magnetic field is applied perpendicular to the broad face of a tape. However, the analytical process applied is also valid for other field directions if the misalignment of grains in the tape is considered in the calculation of the critical current along the well-connected grains.
4. Conclusion Detailed analysis of the IC Ž B . curves of various Bi2223rAg tapes reveals that the transport current of the tape is carried through the weak-linked Josephson junction network as well as through a chain of well-connected grains. The critical current along the weak-linked path, ICw , is related to the total Josephson current of the active junctions and the distribution of the junction dimensions in the tape. The hysteresis in the critical current in decreasing field is attributed to the change of the active Josephson junctions due to the change of local field, which is related to the flux trapped by persistent
currents both within the grains and in the survived current path. The critical current along the well-connected grains, ICs , is controlled by the pinning properties of the grains, which is governed by the tape fabrication process and is closely related to the grain alignment, micro-cracks and other defects in the grains. The pinning force of the Bi2223rAg tape should be calculated using the ICs instead of the total IC .
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