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Anisotropy of the electrical resistance in high-Tc granular superconductors under magnetic field
PHYSICA ELSEV1ER
Physica C 341-348 (2000) 1869-1870 www.elsevier.nl/Iocate/physc
Anisotropy of the electrical resistance in high-To granular superconductors under magnetic field D. Daghero, a A. Masoero, b P. Mazzetti a, and A. Stepanescu a aUnit~ I N F M del Politecnico di Torino, Dipartimento di F i s i c a , c.so Duca degli Abruzzi 24-10129 Torino, I t a l y bUnit~ I N F M del Politecnico di Torino, Dipartimento di Scienze e Tecnologie Avanzate dell'Universit~ P.O. " A m e d e o Avogadro", c.so Borsalino 54-10131 Alessandria, Italy The electrical resistance of granular high-To superconductors submitted to a magnetic field and crossed by a supercritical current strongly depends on the angle between magnetic field and current density. In monocrystalline samples, this effect may be simply explained in terms of the angle-dependent coupling between fluxoids and current. In the case of polycrystalline superconductors, an alternative explanation is needed to take into account the effect of the granular structure. In this paper, we propose a quantitative model in which the resistance is mainly determined by the resistive transition of weak links. The electrical anisotropy is related to the inhomogeneous distribution of local magnetic field intensities and local current densities in the weak links, due to the magnetic screening effect of the superconducting grains and to the angular distribution of weak links surfaces. The theoretical results of this model are compared to the experimental ones obtained on a YBCO specimen under different physical conditions. As well known, a granular high-To superconductor in the presence of a magnetic field shows a two-step superconducting transition whose second stage (for T < To) is entirely controlled by the intergrain regions, which can be viewed as Josephson junctions between adjacent grains (weak links). In this second stage, the resistance depends on H, j and on the angle between vectors H and j (see F i g . l a ) . This indicates the existence of a magnetic field-induced electrical anisotropy. In m a n y papers, the resistance anisotropy is explained in terms of the angle-sensitive coupling between current and vortices [1] by treating the material as a uniform medium. Actually, this approximation is correct as long as Ae~ >> a, where a is the mean grain size [2]. We propose here an alternative explanation, in which the effective medium approximation is avoided by essentially relating the resistance at T < Tc to the resistive transition of weak links (WLs) under the effect of both local magnetic field and current density. As long as the grains are in Meissner state, the screening supercurrents flowing on their surfaces create a demagnetising field [3] whose superposition to the external one
gives rise either to flux compression or to flux suppression in the WLs, depending on their angular position with respect to H [4]. The resulting inhomogeneity of the local magnetic field strongly affects the W L transition probability and then creates an anisotropic distribution of resistive WLs. Let us follow a statistical a p p r o a c h and represent the WLs as small flat elements of the surface separating adjacent grains, with normal versor n randomly oriented in space. Let 0 be the angle between n and the external magnetic field H . W h e n the grains are in the Meissner state, the local magnetic field H t in each W L is parallel to the W L ' s surface, and we expect its intensity to depend on 0. In fact we will suppose the local magnetic field intensities H~(0) in the WLs with the same 0 to follow a gaussian distribution f(H~[0) with mean value ( H l ( 0 ) ) e n s = k.H.sin(0), (where k is a constant "flux compression coefficient"), and s t a n d a r d deviation a t [5]. As shown elsewhere [5], when the specimen is macroscopically resistive it can be viewed as a series of superconducting equipotential domains s e p a r a t e d by layers of resistive W L s extended throughout the specimen cross-section. In these
0921-4534/00/$ - see front matter © 2000 ElsevierScience B.V. All rights reserved. PII S0921-4534(00)01224-7
1870
19, Daghero et aL /Physica C 341-348 (2000) 1869-1870
,51
•
,
•
i
i
t
(Ht(O))e~s and (Ht,c(jt))ens. B can be expressed in terms of spherical co-ordinates 0 and ¢ as:
~-j= 2x104 N m 2 2.0[-T = 27 K
cos(B) = cos(P) if H II J (2) cos(fl) = sin(P) • cos(C) if H _Lj (3) Now, let nL be the number of resistive layers separating superconducting domains. The mean resistance of a layer is inversely proportional to its mean surface and then (r) o¢ (I COS~l)res" The number of layers is nL o¢ NT" (I COSfl{)res where NT = f Ptr(O,/~)d~ is the total number of resistive WLs. The specimen resistance is then
!
or
1.5~(
H
a)
c~ 1.0 H Ill
•~
rr
0.5 0.0
.-
,
t
'
I
'
I
0.6 ! j-- 2x104 A/m 2 =27K
R -- (r). nL OCNT" (({ COS ~ l ) r e s ) 2
Q.
~.
0.4
(b)
Q..
II ~" 0.2 i
• • 0.0
'
r
0
20
,
I
40
{ •
experimental
I
theoretical ,
r
60
,
f
80
,
100
Magnetic field (Oe) Figure 1. (a) Electrical resistance vs. the applied magnetic field H measured on a YBCO specimen at T=27 K and with j--2.104A/m 2 in the two cases H {{j and H _Lj. (b) Comparison of the experimental and theoretical ~/ vs. H curves obtained with pH=43.4, at=0.6. (Ht(0))e,s and at=0.3. (Hl,~(jt)) . . . . In both (a) and (b), the small arrows indicate the beginning of the flux penetration into grains. links, the local current density je is parallel to the normal n and then perpendicular to He independently of the relative orientation of the corresponding macroscopic vectors H and j. We suppose a WL to undergo the resistive transition when He is equal to a current-dependent critical value He,c(je) = He,c(j)/cosfl where fl is the angle between n and j [5]. We expect the critical fields Ht,c(j~) to follow a gaussian distribution, with mean value (He,c(jl))ens and standard deviation ac. The transition probability for a given WL is then:
Ptr(0,B) =
/?
Notice that
y(I-Iel0)P(tte >_ I-Ie,~(j~))dtte (1)
Ptr(0,13) depends on B through
(4)
where, according to eqs.(2,3), both NT and (I cos fl])res depend on whether the magnetic field H is parallel or perpendicular to the current density j. Finally, the ratio ~ = RII/R_L = Pll/P± can be numerically calculated. The resulting theoretical ~/ vs. H curve, which is reported in fig.lb, depends on the ratio PH = (He,c(j))ens/k and, to a much minor extent, on the values of at and ac, which are assumed to be proportional to (ne(o))ens and (He,c(jt))ens respectively. Best fit parameters for the comparison to the experimental results [5] are reported in the figure caption. An high-field deviation of the theoretical curve from the experimental points is what is expected when the flux penetration within grains takes place. At low fields, instead, the deviation is due to the fact that equation (4) is only valid far from the percolative threshold (H ~ 10 Oe). An independent determination of k would allow evaluation of (He,c(j))ens, a quantity which is apt to characterize the weak link ensemble for each value of the current density j. REFERENCES 1.
2. 3. 4. 5.
A. Kilic, K. Kilic, S. Senoussi, K. Demir, Physica C 294 (1998) 203, and references therein. S.L. Ginzburg, V.P. Khavronin, I.D. Luzyanin, Supercond. Sci. Technol., 11 (1998) 255. J. Evetts, B.A. Glowacki, Cryogenics 28 (1988) 641. M. Chandran, P. Chaddah, Supercon. Sci. Technol. 8 (1995) 774. D. Daghero, A. Masoero, P. Mazzetti, A. Stepanescu, to be published.
Physica C 341-348 (2000) 1869-1870 www.elsevier.nl/Iocate/physc
Anisotropy of the electrical resistance in high-To granular superconductors under magnetic field D. Daghero, a A. Masoero, b P. Mazzetti a, and A. Stepanescu a aUnit~ I N F M del Politecnico di Torino, Dipartimento di F i s i c a , c.so Duca degli Abruzzi 24-10129 Torino, I t a l y bUnit~ I N F M del Politecnico di Torino, Dipartimento di Scienze e Tecnologie Avanzate dell'Universit~ P.O. " A m e d e o Avogadro", c.so Borsalino 54-10131 Alessandria, Italy The electrical resistance of granular high-To superconductors submitted to a magnetic field and crossed by a supercritical current strongly depends on the angle between magnetic field and current density. In monocrystalline samples, this effect may be simply explained in terms of the angle-dependent coupling between fluxoids and current. In the case of polycrystalline superconductors, an alternative explanation is needed to take into account the effect of the granular structure. In this paper, we propose a quantitative model in which the resistance is mainly determined by the resistive transition of weak links. The electrical anisotropy is related to the inhomogeneous distribution of local magnetic field intensities and local current densities in the weak links, due to the magnetic screening effect of the superconducting grains and to the angular distribution of weak links surfaces. The theoretical results of this model are compared to the experimental ones obtained on a YBCO specimen under different physical conditions. As well known, a granular high-To superconductor in the presence of a magnetic field shows a two-step superconducting transition whose second stage (for T < To) is entirely controlled by the intergrain regions, which can be viewed as Josephson junctions between adjacent grains (weak links). In this second stage, the resistance depends on H, j and on the angle between vectors H and j (see F i g . l a ) . This indicates the existence of a magnetic field-induced electrical anisotropy. In m a n y papers, the resistance anisotropy is explained in terms of the angle-sensitive coupling between current and vortices [1] by treating the material as a uniform medium. Actually, this approximation is correct as long as Ae~ >> a, where a is the mean grain size [2]. We propose here an alternative explanation, in which the effective medium approximation is avoided by essentially relating the resistance at T < Tc to the resistive transition of weak links (WLs) under the effect of both local magnetic field and current density. As long as the grains are in Meissner state, the screening supercurrents flowing on their surfaces create a demagnetising field [3] whose superposition to the external one
gives rise either to flux compression or to flux suppression in the WLs, depending on their angular position with respect to H [4]. The resulting inhomogeneity of the local magnetic field strongly affects the W L transition probability and then creates an anisotropic distribution of resistive WLs. Let us follow a statistical a p p r o a c h and represent the WLs as small flat elements of the surface separating adjacent grains, with normal versor n randomly oriented in space. Let 0 be the angle between n and the external magnetic field H . W h e n the grains are in the Meissner state, the local magnetic field H t in each W L is parallel to the W L ' s surface, and we expect its intensity to depend on 0. In fact we will suppose the local magnetic field intensities H~(0) in the WLs with the same 0 to follow a gaussian distribution f(H~[0) with mean value ( H l ( 0 ) ) e n s = k.H.sin(0), (where k is a constant "flux compression coefficient"), and s t a n d a r d deviation a t [5]. As shown elsewhere [5], when the specimen is macroscopically resistive it can be viewed as a series of superconducting equipotential domains s e p a r a t e d by layers of resistive W L s extended throughout the specimen cross-section. In these
0921-4534/00/$ - see front matter © 2000 ElsevierScience B.V. All rights reserved. PII S0921-4534(00)01224-7
1870
19, Daghero et aL /Physica C 341-348 (2000) 1869-1870
,51
•
,
•
i
i
t
(Ht(O))e~s and (Ht,c(jt))ens. B can be expressed in terms of spherical co-ordinates 0 and ¢ as:
~-j= 2x104 N m 2 2.0[-T = 27 K
cos(B) = cos(P) if H II J (2) cos(fl) = sin(P) • cos(C) if H _Lj (3) Now, let nL be the number of resistive layers separating superconducting domains. The mean resistance of a layer is inversely proportional to its mean surface and then (r) o¢ (I COS~l)res" The number of layers is nL o¢ NT" (I COSfl{)res where NT = f Ptr(O,/~)d~ is the total number of resistive WLs. The specimen resistance is then
!
or
1.5~(
H
a)
c~ 1.0 H Ill
•~
rr
0.5 0.0
.-
,
t
'
I
'
I
0.6 ! j-- 2x104 A/m 2 =27K
R -- (r). nL OCNT" (({ COS ~ l ) r e s ) 2
Q.
~.
0.4
(b)
Q..
II ~" 0.2 i
• • 0.0
'
r
0
20
,
I
40
{ •
experimental
I
theoretical ,
r
60
,
f
80
,
100
Magnetic field (Oe) Figure 1. (a) Electrical resistance vs. the applied magnetic field H measured on a YBCO specimen at T=27 K and with j--2.104A/m 2 in the two cases H {{j and H _Lj. (b) Comparison of the experimental and theoretical ~/ vs. H curves obtained with pH=43.4, at=0.6. (Ht(0))e,s and at=0.3. (Hl,~(jt)) . . . . In both (a) and (b), the small arrows indicate the beginning of the flux penetration into grains. links, the local current density je is parallel to the normal n and then perpendicular to He independently of the relative orientation of the corresponding macroscopic vectors H and j. We suppose a WL to undergo the resistive transition when He is equal to a current-dependent critical value He,c(je) = He,c(j)/cosfl where fl is the angle between n and j [5]. We expect the critical fields Ht,c(j~) to follow a gaussian distribution, with mean value (He,c(jl))ens and standard deviation ac. The transition probability for a given WL is then:
Ptr(0,B) =
/?
Notice that
y(I-Iel0)P(tte >_ I-Ie,~(j~))dtte (1)
Ptr(0,13) depends on B through
(4)
where, according to eqs.(2,3), both NT and (I cos fl])res depend on whether the magnetic field H is parallel or perpendicular to the current density j. Finally, the ratio ~ = RII/R_L = Pll/P± can be numerically calculated. The resulting theoretical ~/ vs. H curve, which is reported in fig.lb, depends on the ratio PH = (He,c(j))ens/k and, to a much minor extent, on the values of at and ac, which are assumed to be proportional to (ne(o))ens and (He,c(jt))ens respectively. Best fit parameters for the comparison to the experimental results [5] are reported in the figure caption. An high-field deviation of the theoretical curve from the experimental points is what is expected when the flux penetration within grains takes place. At low fields, instead, the deviation is due to the fact that equation (4) is only valid far from the percolative threshold (H ~ 10 Oe). An independent determination of k would allow evaluation of (He,c(j))ens, a quantity which is apt to characterize the weak link ensemble for each value of the current density j. REFERENCES 1.
2. 3. 4. 5.
A. Kilic, K. Kilic, S. Senoussi, K. Demir, Physica C 294 (1998) 203, and references therein. S.L. Ginzburg, V.P. Khavronin, I.D. Luzyanin, Supercond. Sci. Technol., 11 (1998) 255. J. Evetts, B.A. Glowacki, Cryogenics 28 (1988) 641. M. Chandran, P. Chaddah, Supercon. Sci. Technol. 8 (1995) 774. D. Daghero, A. Masoero, P. Mazzetti, A. Stepanescu, to be published.