AC susceptibilities of grain-textured superconductors

Physica C 468 (2008) 1431–1434

Contents lists available at ScienceDirect

Physica C journal homepage: www.elsevier.com/locate/physc

AC susceptibilities of grain-textured superconductors N. Sakamoto a,*, Y. Fukuda a, M. Koga a, T. Akune a, H.R. Khan b, K. Lüders c a b c

Department of Electrical Engineering, Kyushu Sangyo University, 2-3-1 Matsukadai, 813-8503 Fukuoka, Japan Institut von Ionenstrahl und Vakuum Technologie, 73728 Esslingen, Germany Freie Universität Berlin, Arnimallee, Fac.Physik, D-14195 Berlin, Germany

a r t i c l e

i n f o

Article history: Available online 28 May 2008 PACS: 74.72 74.25.Ha 74.80.Bj Keywords: Grained Bean model AC susceptibility Pinning penetration field

a b s t r a c t In-phase vn0 and out-phase vn00 components of nth harmonics of AC susceptibility with measuring parameters of a DC magnetic field Bdc, an amplitude Ba and a frequency f of the superimposed AC magnetic fields give substantial information of the superconducting properties. In low-Tc metallic superconductors, v10 shows smooth transition and v100 does single peak. High-Tc oxide superconductors with anisotropic and grain-textured structures show deformed complex characteristics. Double peaks in v100 and shoulders in v10 appear in AC susceptibility of Hg-1223 superconductors. Instead of simple Bean model, a grained model, where the superconducting grains are immersed in weak superconducting matrix, are proposed. The susceptibilities numerically analyzed using the model show varied and deformed curves and are successfully compared with the measured results. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction

2. Grained Bean model

High-Tc ceramics is textures of grains and interconnecting links as shown in Fig. 1 [1]. Sintered high-Tc superconductors are considered to possess two type of properties. One is intrinsic to the superconducting grains and another is characteristic due to the coupling between the grains. In such materials, the coupling component supports supercurrents and has its own effective critical temperature Tc‘, critical current density Jc‘ and pinning penetration depth Bp‘. The situation is less certain, but lack of stoichiometry at the grain boundaries and microbridges between the grains give rise to a proximity-effect coupling. The coupling region shows a weak superconductivity and the field penetrates more freely through them compared to the grain regions. Then the surface field of the grains is determined by the coupling matrix superconducting regions. The intrinsic superconductors with high pinning penetration field Bpg are immersed in the weak matrix with low Bp‘. Field distribution and magnetization in the multi-phases are calculated by Bean model [2]. Fourier integration of magnetization is carried out numerically and gives rise to the real part v0 and the imaginary part v00 of AC susceptibilities. Measured results of Hg1223 superconductors [1,3] are compared with the simulated results to give the link characteristics.

Matrix superconductor with the penetration field Bp‘ determines the magnetic fields at the grain surface with the penetration field Bpg as shown in Fig. 1. Field distribution Boi outside the ith grain at a grain position {i is given by Bp‘, using Bean model as [4]

* Corresponding author. Tel.: +81 92 673 5636; fax: +81 92 673 5091. E-mail address: [email protected] (N. Sakamoto). 0921-4534/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2008.05.134

Boi ¼ Bo  BP ‘

  Xi : outside the grain; D

ð1Þ

where D is one half the thickness of the superconductor. The field distribution Bi inside the ith grain with the grain size dgi is given by Bpg

Bi ¼ Boi  Bpg



x dgi



: inside the grain;

ð2Þ

where x is a position inside the grain from the each grain surface as shown in the inset of Fig. 1. Average of the magnetic flux density in the increasing period for the superconductor which consist of the matrix and ng grains with grain interval dvi is given by

hBi ¼ Bo 

 ng  Bp X dgi mi : þ 2 dvi i¼1

ð3Þ

Grain magnetization mi is easily computed following magnetization process of each grains. In the case of uniform grain structure with constant grain size dg (=dgi for all i) and intergrain distance dv (=dvi for all i), where fg (=dg/dv) is the grain content factor. Then

1432

N. Sakamoto et al. / Physica C 468 (2008) 1431–1434

B a = 0.02 T cl / T cg = 1 B pg(0) = 1 f g = 0. 5

χ ''

0. 2

0. 1

0

Bpl(0 )

χ'

0.03

—0 .5

1.0

—1

Fig. 1. Field distribution Boi outside the grain and distribution Bi inside the grain. Penetration fields are Bp‘ and Bpg for matrix and grain, respectively. The grain diameter is 2dg and the distance between grains is 2dv.

the magnetization M of the superconductor at the field Bo for each magnetization processes is given by

0.3 0.5

0.05 0.1

0.6

0.8

1

T / Tcg Fig. 3. AC susceptibilities v0 and v00 as a function of temperature T reduced by the grain critical temperature Tcg. Parameters are the same in Fig. 2 and Ba/Bpg(0) = 0.02 and grain content of fg = 0.5. Temperature the penetration fields n dependenceonof g;‘ Bp‘(T) and Bpg(T) is given as Bp‘;g ¼ Bp‘;g ð0Þ 1  ðT=T c‘;g Þ2 with ng = 2 and n‘ = 6.

n

M ¼ hBi  Bo ¼ 

g X Bp mi : þ fg 2 i¼1

ð4Þ

3. Results and discussion The equations for the AC susceptibilities v0 and v00 under an AC field Ba cos xt are derived from Fourier integrals of the magnetization. The fundamental Fourier components v10 and v100 are denoted as v0 and v00 hereafter, and expressed as [5]

v ¼ v0 þ iv00 ¼

1 pBa

Z p

M expðixtÞdxt:

ð5Þ

p

Numerical integration of Eq. (5) is carried out and the results are shown in Fig. 2, where the imaginary part v00 and the real part v’ are plotted as a function of the amplitude of the superposed AC field Ba normalized by a grain penetration field at 0 K, Bpg(0). When the link penetration field at 0 K, Bp‘(0) is low value of Bp‘(0)/ Bpg(0) = 0.01, the typical double peak characteristics appear in the

imaginary part and double transition in the real part. With increasing Bp‘(0), lower peak begins to move higher field and unite to a single peak. The peak of v00 due to the link component shifts smoothly to that of the high grain one and v0 shows smooth transition at Bp‘(0) = 1.0. Temperature dependence of v0 and v00 is obtained by introducing the temperature T variation of the penetration fields of Bp‘(T) and Bpg(T). If the fields Bp‘,g’s have a usual parabolic dependence n ong;‘ of the form as, Bp‘;g ¼ Bp‘;g ð0Þ 1  ðT=T c‘;g Þ2 , where over the temperatures Tc‘ and Tcg the pinning effect for the fluxoid motion disappears in the link and the grain region, respectively. At a fixed grain content of fg = 0.5, the AC susceptibilities are numerically computed and plotted in Fig. 3, where the parameters are the same in Fig. 2 and AC field amplitude Ba/Bpg(0) = 0.02 and ng = 2, n‘ = 6. With decreasing Bp‘(0) from 1 to 0.03, the peak of v00 decreases and the additional peak is born at low temperatures. This means that the grain penetration field Bpg approaches Ba at rather high temperature than the matrix penetration field Bp‘. This type of

B pg(0) = 1

0.2

0.2 χ ''

χ ''

fg = 0.5 0.1

B pg (0) = 1.0 f g = 0.3

0.1

0

0 B pl(0)

0.01

0.01

0.03 0.1

0.03

χ'

χ'

B pl(0)

0.3 1.0

0.1

0.3 1.0

—1

—1 10

—2

10

0

10

2

B a / B pg(0) Fig. 2. Computed AC susceptibilities as a function of AC field Ba normalized by Bp‘(0). With increasing Bp‘, low field peak gradually shifts to high temperature and unites to single peak.

0.6

0.8

1

T / T cg Fig. 4. AC susceptibilities v0 and v00 as a function of temperature T reduced by the grain critical temperature Tcg. Parameters are the same in Fig. 3 except grain content of fg = 0.3.

1433

N. Sakamoto et al. / Physica C 468 (2008) 1431–1434

χ ''

0.2

double peaks is often observed in high-Tc materials [6]. Characteristics for low grain content factor of fg = 0.3 in Fig. 4 reveals the peak position shift to higher temperature owing to large n‘ values. The effect of Bpg is shown in Fig. 5. The imaginary peaks decrease with increasing Bpg(0) and new peaks due to the grains appear at higher temperature and coexist with low temperature peaks. The double peak structure in the v00 curve is specifically indicated to originate in the cooperative function of the grain and the interconnecting link.

Ba = 0.02 Bpl(0) = 0.1 Tcl / Tcg = 1

0.1

fg = 0.5

0

B pg (0)

χ'

0.1

1.0 3.0

0.2 0.3

–0.5

4. Grain and link characteristics in Hg-1223 superconductors

10

0.5

–1 0.6

0.8 T / T cg

1

Fig. 5. AC susceptibilities v0 and v00 as a function of temperature T reduced by the grain critical temperature Tcg. Parameters are the same in Fig. 3 and Ba/Bp‘(0) = 0.02 at a grain content of fg = 0.5. New peak appears with higher Bp‘.

The shoulder characteristics of HgBa2Ca2Cu3Re0.15O8+d (Re-15c) [1,3] sample are shown in Fig. 6(a), where temperatures are normalized by the critical temperature Tc(Bdc) at the applied DC magnetic field Bdc of 2–14 T. The computed results by the grain model are plotted in Fig. 6(b) and parameters Bpg(0) and Bp‘(0) in the inset. The decrease of the penetration field Bp‘(T) with increasing Bdc induces the bulges below the peaks as observed data in (a). At the lower fields below 1 T, typical double peak characteristics were observed as reported in the preceding report [3]. These features are quite well demonstrated in Fig. 3. The link region shows a rapid

0.003 Hg —Re —15 c

χ ''

χ ''

0.04

Ag—03—03

0.002 0.001

0.02

0 0

B dc (T)

2 3 6

—0.4 —0.6

χ'

χ'

—0.2

B a = 0.5 (mT)

—0 .02

f = 100 (H z) B a = 0.5 (mT)

10 12 14

f = 100 (H z)

—0 .04 —0 .06 0

0

0.5

0.5 T / Tcg (B dc)

1

T / T c (B dc )

0.003 12, 14 (T)

—0 .02

10

2

0

0.2

—0 .04

10

—2

10

—3

0.2 0.4 0.6 0.8 Bdc(T)

—0 .06

0

T / T cg (B dc )

1

Fig. 6. (a) Temperature dependence of AC susceptibilities of Re-15c sample and (b) Numerically computed AC susceptibilities as a function of T/Tcg. Higher Bdc induces shoulder parts in v00 .

1.0

B pg(0) B pl (0)

—3

Bdc (T)

0.8, 1.0 (T )

1.0

Penetration field (T)

—2

14

χ '

Penetoration field (T)

χ '

Bpg (0) Bpl (0)

01

= 0.2, 0.4, 0.6,

0

2

0

10

1

0.001

0.02

—0.4

dc

0.002

14

—0.2

B

χ ''

χ ''

Bdc =2 , 3, 6, 10 0.04

B dc (T ) 0.2 0.4 0.6 0.8 1.0

0

1

0.5 T / T cg (B dc )

0.2 1

Fig. 7. (a) Temperature dependence of AC susceptibilities of Ag-03 sample. (b) Numerically computed AC susceptibilities as a function of T/Tcg. The grain penetration field Bpg(0) decreased with increasing Bdc.

1434

N. Sakamoto et al. / Physica C 468 (2008) 1431–1434

temperature change with indices n‘ = 6 compared with that with ng = 2 in the grains. Observed curves in an excess Ag added sample Ag0.3(HgBa1.9Bi0.1Ca2Cu3O8+d) (Ag-03) are shown in Fig. 7(a). At low temperatures, weak second peaks due to the interconnecting regions were observed. Fitting results with the grained Bean model indicate that Bp‘(0) is nearly constant with applying Bdc and the Bpg(0) tends to decrease with Bdc as can be seen in the inset of Fig. 7(b). High addition of Ag is considered to result in boundary improvement but deterioration of grain properties.

the imaginary part v00 of AC susceptibility and shoulder transition in v0 , appears in the computed results. Emergence of double peak and its disappearance is successfully demonstrated by the grained Bean model. Effect of addition of Re and Ag into Hg-based superconductors are discussed with the grained Bean model. Obtained values of the pinning penetration fields indicate that addition of Re into Hg-1223 superconductor improves the grain properties and Ag strengthens the interconnecting link regions.

5. Conclusions

[1] M. Kubo, T. Akune, N. Sakamoto, Physica C 463–465 (2007) 478. [2] C.P. Bean, Phys. Rev. Lett. 8 (1962) 250. [3] N. Yamada, T. Akune, N. Sakamoto, Y. Matsumoto, Physica C 412–414 (2004) 425. [4] W. Yumoto, T. Akune, N. Sakamoto, Ann. Rep. RISS 4 (2007) 86 (in Japanese). [5] T. Matsushita, E.S. Otabe, B. Ni, Physica C 182 (1991) 95. [6] L. Fàbrega, A. Sin, A. Calleja, J. Fontcuberta, Phys. Rev. B 61 (2000) 9793.

Textures of grains and interconnecting links in high-Tc superconductors are simulated by the grained Bean model, where the superconducting regions are divided into two parts; grains and interconnecting links. A variety of characteristics, double peak in

References