Low field magnetization curves of cylindrical samples of high-Tc granular superconductors

Physics LettersA 164 (1992) 109—1 14 North-Holland

PHYSICS LETTERS A

Low field magnetization curves of cylindrical samples of high- T~granular superconductors Roberto de Luca Università degli Studi di Salerno, 84081, Baronissi (SA), Italy Received 25 November 1991; revised manuscript received 5 February 1992; accepted for publication 6 February 1992 Communicated by J. Flouquet

Low field magnetization curves of cylindncal samples ofhigh-T. granular superconductors are calculated analytically by means ofa critical state model based on a Josephson junction array description ofthis class of superconductors.

1. Introduction Critical state models have been often applied to interpret the low-field diamagnetic properties ofhigh7’, granular superconductors [1,2]. Analytical calculations of the field and current distributions for this class of superconductors have already been carried out for the most known critical state models [3]. However, it has recently been observed by Ginzburg et al. [4] that the appropriate field (h) and temperature dependence ofthe critical current density J~ for granular samples of YBCO is the following, 2 ( T) / [H02( T) + h2] (1) ‘~c( h, T) /Jc0 (T) = H0 where H 0 is a measure of the spreading of the above bell shaped curve. Moreover, it has been shown that a critical state picture follows immediately from the analysis of the low-field low-temperature diamagnetic properties of sintered granular superconductors modelled by means ofJosephson junction arrays [5,6]. Therefore, in the present paper we propose a critical state model in which, according to the analysis of refs. [5,6], the critical current (Jo) is identified with the maximum value ofthe Josephson current I~ crossing through the junctions of the equivalent network constructed for the superconducting sample. It is briefly noted that, starting from the usual Fraunhofer pattern of the 4 versus h diagrams for small junctions, the phenomenological expression for the ,

local field dependence of the critical current of eq. (1) is compatible with a Josephson junction array model for these systems. Based on the functional relation of eq. (1), the low field magnetization curves of a cylindrical sample of high-i’6 granular superconductor are calculated analytically. Finally, a discussion of the temperature effects on the hysteresis cycles is given.

. -

2. The critical state model A critical state model can be applied only in the temperature range in which the diamagnetic metastable states realized result long lived with respect to the interval oftime in which the measurement is performed. With this restriction in mind we start our analysis by considering the fourth Maxwell equation in cylindrical coordinates for an axial magnetic field applied to a cylindrical sample of high-Ta granular superconductor: dh/dr= ± J~ (h T) (2) where r is the radial distance from the center of the sample and J~, (h, T) is related to the critical current 2, where R as follows: J~, (h, T) = 4 (h, T) / (2Rg) 5 is the grain radius. In eq. (2) the plus sign indicates that the field is locally increasing and, vice versa, the minus sign indicates that h is locally decreasing [7].

0375-9601/92/S 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

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The field dependence of the critical current density is given in eq. (1). In order to derive an expression for H0 ( T) in terms of the characteristics ofthe material, we compare eq. (1) with the field dependence of the maximum J0sephson current in a small junction, namely J~(h,T) J~0(T)

—

I~(h,T)

—

Ijo( T)

—~

sin(x~P~) I

(3) I ‘ 2Rg ) 2 and Y~j= ~PJ( T) = ~hS, ( T) / where 4/ ( the effective flux penetrated area of ~o’ s~ (JjT)=being the junction and ct~the elementary flux quantum. jf we set the low field Taylor series expansion of j. ( h, T) equal to the one for J~, (h, T), we obtain —

(4)

Approximately the same expression can be derived if, instead of eq. (3), one adopts the following averaged expression for the maximum Josephson current density, by considering an array of Njunctions with identical value ofthe superconducting coupling energy,

6 April 1992

sample, and neglecting demagnetization effects for simplicity, we obtain (6) where A~ ~‘ex ( T) = H/H0 ( T) and A~,=h, (r, T) = h (r) /H0 ( T) are normalized fields, R is the radius of the sample and a ( T) = H0 ( T) /J~0( T) is a constant with the dimension of length. In what follows we shall not write the temperature dependence of the various quantities explicitly in order to simplify the notation. By solving the above algebraic equation, we can find the field distribution ~ valid for the range [R A, R], where A is a field-dependent penetration depth, —

~

+ [~~ex(~x+3)—3(Rr)/a]2+

—

[~‘4~ex(’~x + 3)

l]1~/2}~3

3(Rr)/a]2+ 1

]l/2}

I/3

(7)

The field-dependent penetration length A immedi-

J30(T)

—

1

sin(7tfW3)

N

ately follows from condition ~~,1(R—A)=h56, in eq. (5)

where,)~is a form factor assumed to have a uniform distribution about the unitary mean. Eq. (5) has been used to fit experimental data on the magnetic field dependence of the critical current of ceramic YBCO by Paternô et al. [8] consistently with a Josephson junction array superconductors. In any case, wemodel notice for thatgranular it is possible to compare the low field expansion of eq. (1) and those of eqs. (3) and (5), while the critical state pictures consideredinref. [3] do notpresentanequivalentlowfield expansion. Indeed, for the generalized critical state model of ref. [3], the leading term in the low field Taylor series expansion is linear rather than quadratic. 3. Field distributions In the present section we shall derive the magnetic field distributions inside the cylindrical sample for cyclically applied external fields. By solving eq. (2) with the aid of eq. (1) when the external field is raised from zero to H in a virgin 110

(7): +~ ~ , (8) where h561 is a normalized lower threshold field [5]. A normalized full penetration field ,~can be defined as the field at which A = R, so that 4-i. = { ~ [hgc ~ + 3) + 3R/a I ‘ + [~ h~, (,f~,+ 3) + 3R/a j2 + 1]’ /2} 1/3

A(A~ex)a[(A~ex~

+ { ~ ~ (#~, + 3) + 3R/a] [‘[~~(h2 +3)+3R/a12+l]”2}”3.

(9)

The field distribution given in eq. (7) is shown in fig. 1 for values of 4~ less, equal, and larger than ~ respectively. We now turn to the calculation of the field profiles in the sample when the externally applied magnetic field gradually decreases from a maximum normalized value A to a minimum ~a’ The current distribution will adjust to this new perturbation, starting from the external portion of the sample, by locally reversing the critical current direction [7]. This perturbation penetrates the sample from r=R to a distance r_, which depends on the value ofthe applied

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1.2

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R 2 (ha)]. By imposing now the continuity condition on ,f2 at r=r_, we have r_=R—~a[ha(h~+3)—hex(h~x+3)].

h1/H0

(11)

to In ha. fig.Following 2 ~2 is shown, the same for hareasoning ~hf, for decreasing as above, we values ob-

~e~=I.2

Of hex.

We finally give the expression of the field distribution for values ofthe applied field going from ha ~Z~=hf=~8~ex=hf=o82

tam, for ha~h~, ex=06

h3(r) ={~[ha(h~ (R-r)/a

0.0

2+ 1] I/2} 1/3 + [~[ha(h~ ha(h~ +3) +3(R—r)/a] +{~ +3)+3(R—r)/a]

1.0

Fig. 1. Magnetic field distribution in a virgin cylindrical sample for increasing values of the external field (eq. (7) in the text). The normalized lower threshold field value is 0.01, the ratio R/a is 1.0. The values of the normalized externally applied fields are shownon top ofeach curve.

—

[~“~a(’~~ + 3) + 3(Rr)/a]2+ 11 I/2} 1/3

for 0~ r< r~,

field. An implicit expression for the resulting field distribution A2(r) for these values ofthe applied field is obtained by solving the fourth Maxwell equation incylindrical this time with a minus sign coordinates in front off for r_ ~ 6, due to field reversal, and with boundary conditions h2 (R) = hex. Therefore, by algebraically solving for A2 (r), as it was done before for h1 (r), we obtain, for ha~hf,

{~[hex(h~x+3) —3(R—r)/a]

+ U [h~~(h~~+ 3) 3 (R r) /a + { ~[h~(h~ + 3)— 3 (R r) /a]

1

]I/2}I/3

2+

1] 1/2} 1/3

2+

11 l/2} 1/3

—

—

—

U [h~~(h~~+ 3)— 3(R—r)/a]

for r~~ r~R,

(l2b)

where r~is obtained by imposing the continuity condition on A2~

A2(r) = {~ [ha(h~+ 3) + 3)— 3(R—r)/a]

1.5

h

+ [1[ha(h~+3)3(R3)/a]

(1 2a)

h 3(r)

2+

+3) +3(R—r)/a]

2/H0

hex 06

+{~[ha(h~ +3) —3(R—r)/a]

2+1]~2}”3

—

[~ha(h~+3)3(Rr)/a] (l0a)

A2(r)={i[h~~(h~~+3)+3(R—r)/a] for0~r.
0.0

hex~O.6

[~hex(h~x +3) +3(R_r)/a]2+ 1] 1/2} 1/3

+{~[hex(h~x+3)+3(Rr)/a] —

[~hex(h~x +3) +3(R—r)/a]2+ 1] 1/2} 1/3

15 0.0

forr_~r~R. For ha
2 ( r) has the same functional dependence on r as in eqs. (1 Oa), (lOb), except that it goes discontinuously to zero at r= R A (ha) and vanishes in the interval [0, —

(R-r)/a

1.0

(lOb) Fig. 2. Magnetic field distribution for decreasing values of the external fieldThe after first magnetization up to h~= (eqs. (10) in the text). normalized lower threshold field1.2 value is 0.01, the ratio R/a is 1.0. The values ofthe normalized externally applied fields are shown on top of each curve.

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r+=R—~a[ha(h~+3)+hex(h~x+3)].

(13)

m=
3>I~—hex 1’or~~exgoingfromAto~.

(l5c) Again, for ha
In order to present the results of the above integrations in a concise way let us define the following, = (a/2pR)

3 +y~ —2(y~3+y~!3)]

x{(l —2ca/3R) [y~’ ~

(16)

4. Magnetization curves In order to derive the magnetization of the sample, we define the average field in the annular region enclosed between the radii r 1 and r2 as follows, r2

where c= ~hex(h~x+ 3), y~=x±(x2+ 1)1/2, with x = c —3 (R r) /2a. Let now a be the integral when hex is replaced by ha. Furthermore, define —

= (a/2pR) 3+z~3—2(z~(3+z~!3)]

I~=—~Jh(r)rdr,

(14)

where 1/p is the ratio between the total effective intergranular area and the total area of the cross section of the sample [6]. The normalized magnetization is thus given by the following relations,

X{—(l+2ca/3R)[z~ (17) where z+=~±(c~2+l)”2, with ~=c+3(R—r)/2a. Let again a be the integral when hex is replaced by ~ha. In this way we have the expressions

m =
1

>1 ~

hex

for hex going from 0 to ha, (1 5a)

m=

I~—h~~ for hex going from ha to

ha, (l5b)

_________________________

for the recurrent integrals by means of which we express eqs. (15) as follows, (a) For ha I~_2—h~~

hexgoingfrom0toha,

(18a)

1.5 hexl

2

h3/H0

m= a I~i~ + I~_ ~ hex

hexOô

going from ha to

mIr++
0.0~o.~i

ha,

(l8b)

~iR

~

/ I r+

~ex

hexgoingfrom—hatoha. (b) For ha?~A~f

(18c)

hex~O.6

m= I~_~ ~ hexgoingfrom0toha,

_________________________ (R-r)/a 1.0

0.0

Fig. 3. Magnetic field distribution for increasing values ofthe external field after first magnetization down to 4= — 1.2 (eqs. (12) in the text). The normalized lower threshold field value is 0.01,

the ratio R/a is 1.0. The values ofthe normalized externally applied fields are shown on top of each curve. 112

(19a)

m=aI~i+I~—hex ifr_>0,

=

hex

I~~ex ifr_ ~0,

going from ha to

ha,

(1 9b)

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>aI&’~+ I~.—h~ if r~>0,

m=
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that the analysis of the diamagnetic properties of superconducting samples carried only out if bythermal means actiof a critical state picture is correct

= 1 ~ hex if r÷~ 0, hex going from ~ha to ha.

(1 9c)

In eqs. (19) we have made use of negative values of r÷and r.. as an artifice which allows us to group to-

gether the above equations. In fig. 4 we show the graphical representation of the hysteresis loops for ha ~hf (eqs. (19) for two sets of values of the parameters p and a.

vation of flux motion can be neglected. If the appropriate temperature range is considered, then, one

should first normalize the fields with respect to H0 ( T), rewriting eq. (4) according to eq. (20). As a consequence, the length scale a takes the following form, a(T)= 2A5(O)+t H0(0) 225(T)+tJ60(T)

(21)

In this respect, it is obvious from section 3 that the

5. Temperature effects In the present section we shall discuss finite ternperature effects. We can derive the temperature dependence of H0( T) by taking into account the temperature variation of the effective area ofthe junction S, ( T), so that, in the case of a rectangular tunnel junction [9], we may write 22g ( T) + t (20) S~ ( T) S~(O) 2A5(O)+t’ —

—

where t is the thickness of the barrier and Ag is the penetration depth of the grains. We must emphasize

2.0 _____________________________ -.

~

temperature dependence of the field distribution is determined by the following two aspects: the normalization of the fields and the presence of the parameter a ( T) in eq. (6). It goes beyond the scope of the present paper to investigate the exact form of a( T). However, some comments on the hysteresis cycles reported in fig. 4 can still be made. For this purpose, it can be noticed from eq. (21) that, for classical T<< T6, asuperconductors, ( T) is a decreasing J function of T, if, as for 60 ( T) varies much more slowly than 25(T) in this temperature range. Conversely, the normal fraction 1/p can be considered a monotonically increasing function of T for any temperature less than T6, since it can be shown to be proportional to Ag( T). Two main temperature effects can thus be qualitatively detected by means of this simple picture. Indeed, the first effect is a shrinkage of the loop with increasing temperatures in the low-T range where one expects relative temperature variations of Ag( T) to be more significant than relative variations off60( T); the second, essentially due to the increase of 1/p with T, is a decrease of the tilting angle of the entire loop with respect to the field axis. These features have been purposely exaggerated in fig. 4 for an appropriate choice of the parameters for two hysteresis cycles at different temperatures.

-2.0 __________________________ -2.5 ~ex=H/Ho 2.5

6. Conclusion Fig. 4. Hysteresis cycles ofa cylindrical superconducting granular sample for values ofthe external magnetic field~ranging in the interval [—2.5,2.5]. The normalized lower threshold field value is 0.01. The ratio R/a and the normal fraction 1/p are, respectively, 2.0 and 4 for the dashed line loop, and 0.5 and 4 for the full line loop.

We give an analytical expression for the low field magnetization curves of cylindrical samples of highT6 granular superconductors at very low temperatures by means of a critical state picture based on a 113

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Josephson junction array model. This particularcritical state model reproduces the main features of the experimental curves obtained for the most common high-i’6 superconducting oxides at low fields [10— 121. A qualitative discussion on the generalization of the model to finite temperatures is finally given.

Acknowledgement The author would like to thank Professor S. Pace and Professor R.D. Parmentier for the helpful discussions on the subject.

References [1] J.R. Cave, P.R. Critchlow, P. Lambert and B. Champagne, IEEE Trans. Magn. 27 (1991) 1379.

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[2] D.-X. Chen, A. Sanchez, J. Nogues and J.S. Munoz, Phys. Rev. B 41(1990) 9510. [3] M. Xu, D. Shi and R.F. Fox, Phys. Rev. B 42 (1990)10773. [4] S.L. Ginzburg, V.P. Khavronin, G. Yu. Logvinova, I.D. Luzyanin, J. Herrmann, B. Lippold, H. Borner and H. Schmiedel, Physica C 174 (1991)109. [5] R. De Luca, S. Pace and B. Savo, Phys. Lett. A 154 (1991) 185. [6] R. De Luca, S. Pace and G. Raiconi, Critical state in Josephson junction arrays as models of superconducting granular systems, submitted to Physica C (1991). [7] C.P. Bean, Rev. Mod. Phys. 36 (1964) 31. [8] G. Paternô, C. Alvani, S. Casadio, U. Gambardella and L. Maritato, Appl. Phys. Lett. 53 (1988) 609.

[91A. Barone and G.

Paternô, Physics and applications of the Josephson effect (Wiley, New York, 1982). [10] F.S. Razavi, F.P. Koffyberg and B. Mitrovic, Phys. Rev. B 35 (1987) 5323. [Ill D.C. Larbalestier, M. Daeumling, X. Cai, J. Seuntjens, J. McKinnel, D. Hampshire, P. Lee, C. Meingast, T. Willis, H. Muller, R.D. Ray, R.G. Dillenburg, E.E. Hellstrom and R. Joynt, J. AppI. Phys. 62 (1987) 3308. [12] V. Calzona, M.R. Cimberle, C. Federghini, M. Putti and A.S. Siri, Physica C 157 (1989) 425.