Simulation of the intergranular magnetization of (Bi,Pb)2Sr2Ca2Cu3O10+x superconductors by using Josephson junction arrays

PHYSICA ELSEVIER

Physica C 259 (1996) 10-26

Simulation of the intergranular magnetization of (Bi,Pb)2Sr2Ca2CU3Olo+x superconductors by using Josephson junction arrays Jaakko Paasi *, Antti Tuohimaa, Jarl-Thure Eriksson Laboratory of Electricity and Magnetism, Tampere University of Technology, PO Box 692, 33101 Tampere, Finland

Received 11 August 1995; revised manuscript received 24 November 1995

Abstract

In this paper we present a numerical simulation study of the intergranular magnetization of (Bi,Pb)2Sr2Ca2Cu3Olo+x superconductors. The intergranular-current system was modelled as a two-dimensional Josephson junction array, which consists of superconducting grains connected via overdamped, narrow Josephson weak links. The simulations cover magnetic-flux distribution and magnetic-hysteresis studies, including relaxation effects due to thermal fluctuations in the junctions. Special attention was paid to the influence of intergranular defects on the array magnetization and intergranular flux pinning, with the motivation to find model parameters which fit to the experimental behavior of (Bi,Pb)zSr2Ca2Cu3010+x. A correspondence with the experimental results [Paasi et al., Physica C 259 (1996) 1) (previous paper)] required that there was an intergranular defect of the I~m order located in the middle of the four grains of an array mesh. Such a defect could act as a strong pinning center for intergranular flux.

1. I n t r o d u c t i o n

Magnetization of ceramic superconductors is strongly affected by their granularity. Josephson junctions (JJ's) are believed to occur at each grain boundary. Many of them are weak, resulting in a limitation of current flow from grain to grain. Granular (Bi,Pb)2Sr2Ca2Cu3010+x (BSCCO-2223) superconductors often have a very thin layer of secondphase material at the boundary between adjacent grains. The thickness of the layer is in the nm scale and it consists of either amorphous or 2212 phase

* Corresponding author. Fax: +358 31 316 2160.

B S C C O material [1,2]. Further, the crystal axes of adjacent grains or adjacent grain colonies are usually more or less misoriented. Both the misorientation and the second-phase layers give rise to the formation of such kinds of grain-boundary junctions which reduce the intergranular current flow. In an early theory for the magnetization of granular high-T¢ superconductors, proposed by Clem [3], the material was simply modelled as an array of superconducting grains coupled by Josephson junctions. The free energy of the array was obtained by accounting for only the condensation energy, Eg, of the grains and the Josephson coupling energy, Ej, of the junctions. For a single grain the condensation energy is given by Eg = t z o H 2 V g / 2 , where Hcg is the thermodynamic critical field and Vg the volume

0921-4534/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0921-4534(96)00040-8

J. Paasiet al./Physica C 259 (1996) 10-26

of a grain. The Josephson coupling energy of a single junction is equal to Ej = Ij~b0/27r, where 1j is the maximum Josephson current of the junction and ~b0 is the flux quantum. If E j / 2 E g << 1, the junction is weak and intergranular effects become very important. This is the typical case of granular high-T~ superconductors. When a sample consisting of such JJ's is exposed to a magnetic field, two kinds of current systems will be induced in the superconductor, intergranular and intragranular currents. Both systems affect the magnetization of the superconductor. Despite the fact that BSCCO superconductors do not suffer as drastically from the weak links as granular YBa2Cu307_ x samples, intergranular critical current densities of high-quality silver-sheathed (Bi,Pb)2Sr2Ca2Cu3010÷x tapes have still been found to be much lower than their intragranular Jc's [4]. In fact, intergranular and intragranular current systems of (Bi,Pb) 2Sr 2Ca2Cu 3° 10+x have been found to have not only their own Jc's but also their own dynamics [5,6]. All these indicate that the understanding of the electromagnetic behavior of the intergranular current system is of vital importance. This paper contains a simulation study of the intergranular magnetization of BSCCO superconductors. The aim is to extend theoretical knowledge of the magnetization including a question of intergranular flux pinning. To accomplish this we have done numerical simulations with model specimens, where different kinds of Josephson junction arrays have worked as the models for the intergrain system. The simulations cover the following magnetic properties: penetration of magnetic field into zero-field cooled (ZFC) specimens, hysteretic magnetization versus magnetic field, and relaxation of induced screening currents. Special attention is paid to achieve model parameters which give simulation results comparable to the experimental results of the intergranular magnetization dynamics of (Bi,Pb)2Sr2Ca2Cu3Ol0+x recently done by Paasi et al. [6]. A granular sample was modelled as a system of grains located in the nodes of a two-dimensional square lattice oriented perpendicular to the external field. The grains were characterized by a superconducting order parameter, which is constant in each grain. Adjacent grains were coupled via overdamped Josephson junctions. A main objective was to provide a model as realistic as possible, e.g. accounting

11

for the magnetic-field dependence of the maximum Josephson current, thermal fluctuations in the Josephson junctions. The thermal effects were taken into account by a procedure specially developed for the present study. The paper is arranged as follows: In Section 2 we shall shortly review the theoretical background of some special cases related to the present study, including the behavior of a single intergranular JJ in a magnetic field, the magnetization of a JJ array with negligible mesh inductances, and the magnetization of a superconducting loop closed by a JJ. The array model built for numerical simulations is described in Section 3. Central results of the simulations are given in Section 4. The simulations were performed for arrays including both negligible and considerable mesh inductances. Special attention was paid to the influence of spatial variations of array parameters on the overall magnetic behavior. Finally, concluding remarks are given in Section 5.

2. Background 2.1. Intergranular Josephson junctions

Intergranular JJ's are distinguished into two categories: narrow junctions with a width w < 4A j, and wide junctions with w > 4)tj [7]. Aj gives a measure of the width within which Josephson currents are confined at the edges of the junction. For this reason it is called the Josephson penetration depth. It is determined by [8] As =

2

0J,0(2AL + d) '

(1)

where ~b0 = h / 2 e is the flux quantum, JJ0 is the maximum Josephson current density in the absence of external magnetic field, AL is London penetration depth of adjacent grains, and d is the barrier thickness between the grains. Parameters JJo = 1 × 105 A / c m 2 and (2A L + d) = 0.3 ixm give As --- 1 ~zm. This means that junctions with w < 4 Ixm behave as narrow junctions. A current density of JJ0 = I × 10 4 A / c m 2 will increase the characteristic width to 12 ~m. An external magnetic field gives rise to the modulation of the maximum Josephson current. If w ~< Aj

12

J. Paasi et al./Physica C 259 (1996) 10-26

and the junction is homogeneous, the field penetrates uniformly into the junction and the average Jj of the junction is given as [9]

Josephson vortices. An extensive study o f the pinning phenomena and critical currents in disordered wide JJ's is given in Ref. [13].

2H ° ~x(j2 --jl)AL ] JJ = JJo ~r(j2 _ jl)AL sin 2~/o '

2.2. Magnetization of a JJ array with negligible inductances

(2)

where Jl and J2 are the surface current densities of the adjacent grains, and H 0 is given by H 0 = ~b0/2~0ALW. When the grains are in the Meissner state, the surface current densities are given by J2 = --Jl----H/AL, where H is the local field, and the equation reduces to the familiar Fraunhofer-like J j ( H ) dependence of a uniform narrow JJ [8]. Then J j ( H ) diminishes to zero at fields corresponding to the penetration of an integer number of q~0. However, if the junction is disordered, J j ( H ) remains at a finite level also at these field values. Therefore, the J j ( H ) dependence of a disordered narrow JJ can be approximated by neglecting the sinus term in Eq. (2). The disorder can be just dislocations at the grain boundary due to misorientation of the grains. If the grain boundary JJ is wide, its magnetic-field behavior has some similarities to the magnetization of type-II superconductors. When H is below the Josephson lower critical field of the junction, Hc0, junction currents flow to a depth of Aj and the interior is shielded from the field. In this region the junction acts similar to superconductors in the Meissner state. When /-/co is exceeded, the field enters the junction in the form of coreless Josephson vortices, each carrying one flux quantum. The H field affecting the junction can be either due to an applied field or due to the junction current. In the latter case Hc0 is equal to the level caused by current flowing with a density JJo in a surface layer of thickness /~j around the periphery of the junction. The resulting vortex state above Hc0 is dissipative in the presence of transport current unless the motion of the vortices is impeded. In a uniform junction only the boundaries of the junction can impede the vortex motion. Therefore, a wide and uniform JJ cannot carry a critical current density in the vortex state. Fortunately, grain-boundary JJ's of high-Tc materials have been found to be non-uniform consisting of regions of good and poor Josephson coupling [10]. The spatial variation of Ej can occur even in the nm scale [11,12], and it gives rise to the pinning of

The first theory for the intergranular magnetization of high-To superconductors, proposed by Clem [3], considered the behavior of Josephson vortices in the intergranular junctions. A granular sample was described as a positionally disordered system of superconducting grains weakly coupled via Josephson junctions. The free energy of the system was obtained by accounting for only the intragranular condensation energy, Eg, and the intergranular Josephson coupling energy, Ej. If Ej << Eg, the intergranular system of the sample is modelled as a JJ array. Because the magnetic energy of the mesh current loops of the array is not included in the theory, we can therefore speak of a JJ array with negligible mesh inductances. According to Clem, application of an external field somewhat larger than the Josephson lower critical field causes an array of Josephson vortices to penetrate into the granular specimen. The net internal magnetic field can be thought of as a linear superposition of individual vortex contributions (flux quantum localized, roughly speaking, within area ~A2). If Aj is of the order a few p~m, the Josephson vortex contributions will strongly overlap. The depth of the intergranular vortex penetration depends upon the extent to which these vortices are pinned. Pinning can arise from either discreteness of the JJ array [14] or inhomogeneity of the junction coupling strengths. This results in the existence of macroscopic fluxdensity gradients according to the critical-state model: for a cylindrical geometry in an axially applied field we have OB O r = ~°S~'

(3)

where J~ is the intergranular (transport) critical current density. The Jc cannot locally exceed the maximum Josephson current density of a single junction given by Eq. (2). In general J~ is much less than Jj. Furthermore, thermal energy kBT can excite intergranular vortices from a pinning site over an energy

J. Paasi et al./Physica C 259 (1996) 10-26 barrier into an adjacent one resulting in intergranular flux creep and magnetic relaxation. The energy bartier preventing the flux motion is proportional to Ij. Despite the success of Clem's theory in distinguishing the intergranular and intragranular electromagnetic effects, we have the following reasons to believe that the theory is inadequate in describing the magnetic behavior of BSCCO samples: (1) The intergranular magnetic-flux density gradients measured in Ref. [6] are relatively high. That requires strong pinning of intergranular vortices. However, the pinning due to the array discreteness is weak, if the JJ array has negligible mesh inductances [14,15]. The remaining explanation, that the fluxdensity gradients are due to intergranular vortex pinning in disordered wide JJ's, is doubtful since a great part of the grain boundary junctions in the reference was narrow and narrow junctions have a uniform magnetic-field distribution; (2) There are usually numerous large defects in (Bi,Pb)2Sr2CazCu3010+x samples including non-superconducting particles and voids on the 1-10 Ixm scale. For instance, Ag regions of this size have been found in BSCCO layers carrying current densities of 1.1 × 105 A / c m 2 at 77 K [16]. The importance of this size of defects have been pointed out by several groups [ 17-19]. When such a non-supereonducting region is affected by an external flux, shielding currents start to flow in the grains around the region forming a superconducting loop closed by intergranular JJ's. On the defect scale the loop inductance is no more negligible and the magnetic energy, E m, of the current loop can be even higher than the total Josephson coupling energy, Ej, of the loop. Accordingly both energies should be accounted for the free-energy function of the system; (3) A recent study done by the magneto-optical imaging technique [20] indicates that there are actually two kinds of intergranular magnetization currents in high-quality BSCCO: currents flowing in the entire sample scale with a density equal to the transport Je, and currents flowing only in a much smaller scale in well connected grain blocks, see Fig. 1. The block currents form intergranular current loops extending over few grains, adjacent loops coupled via Josephson weak links. The resulting JJ array can no longer be described without accounting for the magnetic energy of the current loops.

13 Jct

Job ::~;::~: ~:~i~i;~i;~;~::s:~i~i~:: ~:~is*~:~:~:~:~:~:~:~*~:~:~:s;s;~:;:~:: ~:i:i:i:i:i:i:;i:~ii~i¼)ii~i

Jcg

Fig. 1. Currentsystemsin a granularsuperconductorin an applied magnetic field: intragrain screening currents Jog, intergranular block currents Jeb, intergrain screeningcurrents Jet flowingwith a density equal to transport Jc, (Jog >Jcb > Jet).

2.3. Magnetization o f a superconducting loop closed byaJJ The Clem model for the intergranular system can be improved by adding the magnetic-energy component to the total free energy of the system. Let us first review the basic equations for the magnetic-field behavior of a superconducting loop closed by a narrow resistively shunted junction (RSJ), where an ideal Josephson junction is shunted by a small capacitance, C, and a resistance, R [8]. We restrict ourselves to overdamped junctions, where movement of a flux quantum into or out of the loop is related to 2~r change of the superconducting order parameter across the junction. The energy potential function of the JJ loop is

[2

4, (4)

where 4, is the total magnetic flux and 4,e is the applied flux threading the loop. L is the loop inductance. The first term on the right-hand side describes the magnetic energy created by the loop and the second term the Josephson energy of the junction. The loop is characterized by the SQUID paramete r 2"rrLIj( B, T)

/3 =

4'0

(5)

When/3 > 1 the modulation of the parabolic magnetic-energy function by the Josephson energy results in local minima in the energy potential func, tion, U(4,, 4,e). As a result metastable quantum states are created wherein magnetic flux can be trapped.

J. Paasi et al./Physica C 259 (1996) 10-26

14

The magnetization of the JJ loop is best understood by following its trajectories in the U(~b, $~) space. Fig. 2 shows the potential surface in normalized units for the case LIj = 6~b0. Let us first consider a ZFC system in the absence of thermal fluctuations. The system is constrained by a potential barrier A U = U(~bB)-U(~bA). When an external magnetic field is increased from zero, the barrier, AU, preventing transitions decreases and finally vanishes at a point P~ causing a system transition to a next lower metastable state, point C. The high damping coefficient assures that the system will be retrapped in this quantum state. Let this critical flux value be ~bec. Physically the process means that the external field induces a shielding current which perfectly shields the loop interior up to ~be = ~bec. Then a flux quantum irreversibly penetrates the loop. A further increase of the external field causes additional penetration of flux quanta into the loop. When the external field is cycled, the system follows the trajectory indicated by the m o w e d thick line. As the current and the flux of the JJ loop are coupled together by the inductance, a potential barrier AU preventing transitions between adjacent metastable states can be given as a function of current I flowing through the junction [21]:

AU(I)

=e,

[( ))1j2 sin')] 2 l- (;,

(6) When exposed to the thermal energy of an average level kBT ( k B T < A U ) , the system will oscillate around a minimum of a potential well in U(~b, qbe), say point C, and ultimately turn to a lower metastable state, point D. This means there is penetration of a new flux quantum into the loop. The oscillation is due to a thermal noise current in the junction. The average lifetime ~- associated with the flux remaining in a specific quantum state is given by [19,22] 1

-- = tooe-av(1)/%r,

(7)

T

where the amplitude factor (characteristic frequency) to o is of the order R / L . As a result, the screening current I in the loop decreases during each period of

8

::::::::

::::::::::::

÷,/¢, !.

U/Ej 0

-1

0

1

2

3

4

/~o Fig. 2. Potential surface U(~b, ~be) for a JJ loop in the case

LIj = 6~bo. For the legend see the text.

time ~- on an average by factor A I I I = - q b 0 / L . According to Paasi and Eriksson the rate of the current change is obtained from [19] dl dt

~b0too - e -nv(t)/k"T L

(8)

If the shape of the potential barrier is simplified by a linear approximation, AU(1) = Ej(1 - I/Ic), the loop current can be analytically solved

l(t)=l(O)

1--

n 1+--

,

(9)

T0

where T0 = ( LAkBTlc)/( q~otooEj), and I(0) = Ic is the current at t = 0. I~ ~
J. Paasi et al,/ Physica C 259 (1996) 10-26

loop analysis to the network size will be done in the next sections.

15

vector potential, and the integral is taken across the junction. The magnetic flux 49ij through a single mesh is coupled with the mesh current according to (12)

49i,j = 49e -- LIi,j"

3. Model As a model for the intergranular current system subjected to an extemal magnetic field, we consider a system of grains placed at the sites of a two-dimensional square lattice oriented perpendicular to the external field, see Fig. 3. In the absence of thermal fluctuations and field-dependent maximum Josephson currents the basic array model is similar to the one by Majhofer et al. [18]. The grains are characterized by a superconducting order parameter, considered to be constant in each grain. Adjacent grains are coupled via overdamped Josephson junctions, which can be characterized by an ideal junction shunted by a resistance. Oij and O~j describe the "horizontal" and "vertical" gauge-invariant phase differences of the superconducting order parameter across the junction, respectively, and a is the lattice constant. In the absence of extemal currents, the current through a particular junction is given by the superposition of elementary mesh currents I u (i.e. currents circulating around each elementary cell):

where L is the self-inductance of the lattice mesh. Since our two-dimensional array describes a thin slice taken from an infinitely long sample, the influence of mutual inductances on 49ij can be neglected. Consequently, we suppose that any spatial variation in the array would extend over all layers parallel to the one studied. Under more realistic conditions, Eq. (12) still remains valid, but L describes the effective inductance of the mesh [18]. The JJ's of the model can be either small-angle grain boundaries with strong coupling or high-angle grain boundaries with weak coupling. In either case the junctions are expected to be narrow and behave according to overdamped dynamics. Eq. (10) can be rearranged as a system of ordinary differential equations for Ogj and Oij. Accounting for the coupling between the current and the flux, we obtain dOi,j

R

2,r49i_ 1,i

2~49i,j

dt

L

490

490

/3ij sin

(13a)

490 dOi,j

ll,j--li_l,j=Iji,j

sin O i j + 27rR 490

li,j_ 1 --Ii,j=Iji,j sin 0~d+ 2aiR

dt '

(10a)

dOi,j

dt '

2-rr

fA. all,

~i,j sin Oi,j),

(11)

where fli,j is the parameter given by Eq. (5). Because of the flux quantization, 49ij is coupled with Oij and 0 u. For a ZFC system we then have

I = Ij sinv

Vi-l,~i*l.j I~ C[

27r49i,j-1 490

(13b)

and the equivalent expression for O~,j. Here A q~i,j is the difference in phases of the superconducting order parameter in adjacent grains, A is the magnetic

- R

dOi, j = R [ 2a'r49i,j dt L ~ 490

(10b)

with

Oi'J = A gi'J -- 490

Oi,j),

2~R

dt

I

Fig. 3. Schematic view of a JJ array used as a model for the intergranular current system.

2"n'~i,j 490 -- ~i,j -[- Oi,j+ l -- ~ q i + l , j - Oi,j"

(14)

The magnetic behavior of the array can be calculated by solving Eqs. (13) and (14). Required boundary conditions are obtained from the assumption lij = 0 outside the system. Then we have for the left (i = 1) and the bottom ( j = 1) boundary dO1,j R ( 2~r49e d---~ = L x 49o

O i ' J - 01'j+ 1 + ~92,j k

+ 01, j - fll.j sin O l j ) , ]

(15a)

16

J. Paasi et al./Physica C 259 (1996) 10-26

dOi i R [ " = ~ dt L

2 xrthe --+Oil+Oi,2-Oi+l,L

~bo

"

/

0,.,t -/3,.l sin 0iA ) ,

(15b)

Analogous conditions hold for the other two boundaries. The set of equations was solved by means of the standard fourth-order Runge-Kutta method. Each stationary solution for a single mesh corresponds to a local minimum of the energy potential function U(~b, the) given by Eq. (4). The solutions depend only on the value of /3ij and, via the boundary conditions, on the external magnetic flux, the = Bea 2. Uniform arrays are thus described by setting /3ij as constant, /3ij =/3. In real samples there are variations in junction parameters between junctions. This has been modelled either by letting /3ij vary spatially with a random variation around an average value or by giving specific values for each /3ij. Finally, once ~bij has been calculated for each mesh, the average magnetization of the array, M, can be obtained from

1 N N ~ i , 1 - (~e M = - )-". E ' - ~ - - - , bL0N2 i=1 j=l

ITi,j ~- Koi,j( t)lJi,j sin Oi,j"

(17)

The coefficient Koij(t) is given by

Koi.j ( t) = Koi,j ( t -- At) At(2,rrw0e-aV/kB r )

4~i,~+ 4'i- ~j

2a 2

(18)

Eqs. (17) and (18) are for vertical junctions. Analogous equations hold for horizontal junctions, too. The Ij(B) model is similar to Kim's critical-state model [26]. With the choice of A = 2'rrALW/q~o it gives a magnetic-field dependence close to that given by Eq. (2) if the sine term is neglected. This approximation is valid for strongly disordered junctions. Finally, Eqs. (17) and (18) are used in the array model for the determination of /3i,j(~).

(20)

where At is the integration step, AU is the energy barrier preventing transitions, and K o i , j ( t - A t ) is the current coefficient of the previous iteration round. Eq. (20) has been derived in Appendix A. In the array model Eq. (13a) will now be replaced by

dOi, j = R ( 2qT~i_ l,j

dt where Ijo is the maximum current at zero field, A is a sample-dependent parameter, and Bjii is the magnetic-flux density affecting the junction. Bji j is approximated by

Bji,j =

( ! 9)

--I /3i,y(6) sin(Oi,j)tffit_at 1' (16)

where N 2 is the number of meshes in the array. To account for the magnetic-field dependence of Ij, we used a simple model [25]: 1 lJij = IJ0 1 + ABji j '

At non-zero temperatures thermal noise currents will be present in the junctions. As shown in the single-loop analysis these currents have a decreasing influence on the screening currents of the system, the rate of change given by Eq. (8). In the case of a two-dimensional JJ array the behavior is not as straightforward for the following two reasons. First, in an array a mesh current is influenced by changes in neighboring loops. Therefore, it can either increase or decrease due to a thermally activated flux motion. Secondly, a flux quantum may not necessarily be localized in a single mesh but could be spread over several meshes. However, the actual tunneling current is in any case some portion of the tunneling current without thermal fluctuations. For a vertical junction we obtain

L ~

q~o

2'IT~i,j

qbo

- - K o i , j ( t ) / 3 i , j ( ~ ) sin(Oi,j) ) .

(21)

Furthermore, analogous equations will replace Eq. (13b) for the horizontal junctions and Eq. (15) for the boundaries. The resulting set of equations together with Eq. (14) forms the JJ array model in the presence of thermal fluctuations. In the computation the relaxation of each junction current is first computed after a small time step At. Then a new magnetic-flux distribution due to thermally activated changes is computed and currents are corrected to be consistent with the new flux distribution. The procedure is repeated after each iteration step until the desired relaxation time is reached.

J. Paasi et al./Physica C 259 (1996) 10-26

4. Simulations In the simulations we examine a 1 Ixm thick square cross-sectional slice of an infinitely long conductor (a long square conductor with spatial variations across its cross-section, but each cross-section being identical), which is placed in a magnetic field parallel to the conductor (and thus perpendicular to the cross-sectional slice). The plate is described by a N X N JJ array, where N = 50 unless otherwise stated. To assure that all junctions of the array are narrow, we set a constant junction width w = 3 Ixm. The non-superconducting region in the middle of the four adjacent superconducting grains is supposed to be a D × D square, whereby the lattice constant is given by a = D + 3 i~m. The grains are in the Meissner state, and the magnetic field penetrates only to a depth of A L = 0.15 i.Lm. Most of the parameters used in the simulations are given in the text. Additional computation parameters are tabled in Appendix B.

4.1. Influence of/3 on the magnetization Since the magnetic behavior of the model depends strongly on the value of /3, we start the simulations by studying the influence of /3 on magnetic-flux distribution and hysteresis. First we consider a very dense and pure system where D = 0.05 Ixm. The mesh inductance L was calculated from Weber's equation for a rectangular current loop [27]; here L = 9.8 X 10 - 1 4 H. The system behavior is studied for two different maximum Josephson currents, Ij = 0.3 m A ( J j = 104 A / c m 2) and Ij = 3 mA ( J j = 105 A / c m 2) resulting in / 3 = 0 . 1 and / 3 = 1, respectively. Here we suppose the simplest case, I j ( B ) = constant. However, in order to account for spatial variations in junction parameters, /3u was permitted to vary randomly around its average value /3, the range of variation being ___20%. The spatial variations are accounted for because in uniform arrays, where I j ( B ) = constant, the phases of neighboring JJ's can be locked resulting in strong fluctuations in the magnetization, which are not observed in real, non-uniform specimens. The magnetization of the system flu = / 3 -+ 20% = 0.1 _ 20% is shown in Figs. 4(a) and (b), where (a) shows the magnetic-flux distribution at B e = 0.4

17

mT and (b) the virgin and second cycle tzoM(B e) hysteresis curves of amplitude B m = 0.4 mT. External flux penetrates the array almost uniformly. Because the overall current density of the array is related to the flux density gradients by Ampere's law, V × B =/.% J , the array should have very low overall Jc, provided the classical critical-state concept [28] can be applied. The calculation gives an array current density of Je ~ 10 A / c m 2 which is three orders of magnitude smaller than the maximum Josephson current density of individual junctions ( J j = 10 4 A / c m 2 ) . T h i s indicates very low flux pinning in the array, which is also seen from the hysteresis curve. The oscillation of the M(B~) curve does not describe the physics of a real array; it is merely caused by model simplifications [29], and should not be paid too much attention. Since the array junctions are narrow, the hysteresis in the magnetization (and thus also the pinning) is solely due to the discreteness of the array. However, the flux pinning d u e to the discreteness is not very effective, because the/3 of a mesh is small causing a penetrated flux quantum to spread over several meshes (practically over the whole array). It is clear that this kind of array with negligible mesh inductances cannot cause the high intergranular flux density gradients measured in BSCCO bulk [5,6] as well as in B S C C O / A g tape samples [30]. The behavior of the /3u = 1 _+ 20% system is given in Figs. 4(c) and (d) for the flux distribution and the hysteresis curves, respectively. Here, the array magnetization is stronger than in the /3 = 0.1 case. A clear flux-density gradient can be observed from the flux distribution plot. The overall array Jc calculated from the B u gradients is Jc ~-" 400 A / c m 2. The value is higher than the Je of the/3 = 0.1 array, but it is still essentfally smaller than the Jj of individual junctions (here Jj = 105 A / c m 2 ) . Because the mesh inductances o f both / 3 = 0.1 and /3 = 1 arrays are equal, we can conclude from the simulations, that arrays with negligible mesh inductances (actually negligible magnetic energy) seem not to be able to have high transport J~ (this, of course, concerns only arrays consisting of narrow junctions). To study the behavior of arrays with higher/3 and L values we increase the intergranular defect dimensions to 5 ~ m × 5 Ixm. This results in a mesh

18

J. Paasi et al./Physica C 259 (1996) 10-26

inductance of L = 1.2 × 10 -11 H. Now the maximum Josephson currents 6 = 0 . 3 mA (Jj = 1 0 4 A / c m 2) and Ij = 3 mA (Jj = 105 A / c m 2) lead to /3 values of 10 and 100, respectively. A flux-distribution plot of the /3;j = 10 ___20% system at B e = 1.2 mT is given in Fig. 4(e), and the respective magnetic-hysteresis curve in Fig. 4(f). For the /3ii = 100 + 20% case the flux distribution is presented in Fig. 4(g), corresponding to an external flux density of B e = 12 mT, and the I.,oM(B e) hysteresis curves in Fig. 4(h). The array magnetization is strong in both systems, and it is the stronger the higher /3 is. The magnetic-flux density falls linearly with distance from the boundary as in the magnetization of hard type-II superconductors according to B e a n ' s critical-state model [28]. The overall Jc of the/3 = 10 array calculated from the Bij gradients is J~ = 370 A / c m 2. It is about equal to the J¢ of the /3 = 1 array. In the comparison we must, however, remember that the available cross-sectional area for the current here is only 38%, in contrast to the 98% for the /3 = 1 array. Furthermore, the junctions of the /3 = 10 array have an order of magnitude lower Jj with respect to the /3 = 1 system. Therefore the result actually means improved flux pinning in the /3 = 10 system, which can also be seen when comparing the /x0M(Be)-curves of the systems. The strongest pinning, however, is in the /3 = 100 array, where the calculated overall Jc is 4600 A / c m 2. It is still much lower than the theoretical maximum, Jc,max = wJj//a = 38 000 A / c m 2. The ratio Jc//Jc,max = 0.12 is, however, essentially higher than in the /3= 1 system, where JJJc,m~x=(400 A / c m 2 ) / (98 000 A / c m 2) = 0.004. The randomness we used in the /3 parameters does not essentially alter the magnetic behavior of the non-uniform arrays. The ixoM(B ~) curves are about independent of the particular configuration of /3,.j due to the averaging done over the array. However, the level of the magnetization is slightly smaller than the average level of the magnetization in uniform arrays, the difference depending on the range of the variation. In the flux distribution the randomness results in local disturbances the place of which depends on the particular /3ij configuration. The longrange gradient of B~j, however, will be independent of the randomness used (supposing a constant range of the variation).

The special case of the magnetization of an array including grain blocks is shown in Figs. 4(i) and (j). Here, the grain block size and Ij(B) are constant. The flux distribution has been studied in a 70 × 70 array, where each seventh fl,.j = 10 and elsewhere /3ij = 40, i.e. the array consists of 7 × 7 grain blocks where /3ij = 40 (intra-block value), neighboring blocks separated by /3o. = 10 junctions (inter-block value). Such a /3ij = 4 0 / 1 0 array could present the following sample: letting D = 5 txm and w = 3 p,m, the sample has square grain blocks consisting of 7 × 7 grains where each intergranular JJ is a low-angle grain boundary with 1j = 1.2 mA; at the boundary between neighboring blocks the JJ's (inter-block junctions) are high-angle grain-boundary junctions with Ij = 0.3 mA. A flux distribution plot of the /3~j = 4 0 / 1 0 system at B e = 2.0 mT is presented in Fig. 4(i). The figure shows that there are actually two different systems in the array: a macroscopic system with long flux density slopes at each seventh vertical and horizontal junction array, and a local system with the "sandpile" formation within the 7 × 7 blocks. It is easy to see that the macroscopic system is related to the high-angle grain boundaries (/3;j = 10) which limit the macroscopic intergranular current density; the gradient of B35.j gives an overall Jc of 410 A / c m 2. Within the blocks the array current density (and flux pinning) is higher, which leads to sub-systems with stronger local magnetization. Magnetic hysteresis of the /3o. = 4 0 / 1 0 system is given in Fig. 4(,0 for the c a s e B m = 3.5 mT. In order to compare the behavior to that of the /3ij = 10 + 20% system (Fig. 4(0), the tzoM(Be) curve was calculated for a 50 × 50 array with the remainder of the parameters as in Fig. 4(i). From the curves we can see that the magnetization of the /3~j= 4 0 / 1 0 array is about 1.5 times of that of the 13 = 10 array. The M enhancement is mainly due to block screening currents in the 7 × 7 sub-arrays, because whole array screening currents do not seem to be very different in these /3i; = 4 0 / 1 0 and /3 = 10 arrays. The simulation results are consistent with JJ array magnetization studies done by other groups [18,31]. Furthermore, the behavior of the arrays with the higher mesh inductance (the 13 = 10 and /3= 100 systems) is in good qualitative accordance with the measured intergranular magnetization of BSCCO

19

J. Paasi et al./Physica C 259 (1996) 10-26

(~)

(b)

fl=0.1

,oM (inTo!J

.'~...~

-0.6

Be

-0,04 (d)

(~

#oM (mT~.2

(naT)

fl = 1

!

o.~llltllllltllllllUtl~llllltll " ~

f ~

Be

-o.~ (e)~~ ~ o2~ 1~ .

.

fl = los0

.

(f)

#oM(mT~.4~,

(mT)

fl = 10

-4 -2 -o.2

/'t°O~°Ot¢IdOOejt~

0

0

-0,4

fl = 100

10~

2,5

:~ _ o ~

-~

/./or/zo ~

""~/,/o~

-',etioo,~

(i).

~0 0

5o

"~,.,,,d6o ~s ,,,lof~o~,~ ~'~

~

fl = 40/10

o , q ~ ~ p , o

"~/dOOch~. ' ~s

(mT)

"4e'x~Sx.x~"

-~

-.~ (j)

#oM (mT)0.6, ~

Be

(roT)

= 4o/lo

~ o~

~ ~ ~ 4 B ~ (roT) -0.6

J. Paasi et al./ Physica C 259 (1996) 10-26

20 I

I

'

I

~

I

1.5

~

I

'

I

1.25 (b)

'

i

'

'

i

'

i

'

I

'

\ ~",/

, I . I . I . . I . . , -2.0 -1-5 -LO -0-5 0.0 0.5 110 115 2.0

B e (mT)

LO "~'~0.75 ,d ~ 0.5 0.25 0.0

I 3.0

I

-0.5

0.0

3.5

•

0.0

-o.25

0.2

I

0.5

'

I

'

I

Fig. 6. Magnetic-hysteresis curves of 5 0 X 50 arrays with the Ij(B) dependence: ( ) txoM(Be) of a uniform array where /3ij(0)= 10; ( . . . . . . ) IxoM(Be) of an array consisting of 5 X 5 grain blocks with intra-/inter-block parameters of flij(O) = 4 0 / 1 0 , respectively; ( . . . . . ) /xoM (B e) of an array consisting of 5 X 5 blocks where /3~j(0) = 100/10.

(c)

~'~ 2-5

.~ zo

~

1-5 1.0 0.5 0.0 0

25

50

junctions Fig. 5. Penelration of magnetic field into ZFC arrays w h e n the magnetic-field dependence of lj is taken into account: (a) flu(0) = 10, (b) /3ij(0)= 20, (c) /3ij(0) = 100~ The profiles are taken at an array (i, j ) = (25, j).

samples [5,6,30]. Therefore, when studying the special contribution of the present work, the influence of Ij(B) dependence and magnetic relaxation on the JJ array magnetization, we omit arrays with negligible mesh inductance and concentrate only on arrays where /3(0) >> 1.

4.2. Is(B) dependence The connection between Iju(B) dependence and array magnetization was simulated according to Eq. (17). The parameter A was set to 14000. First, field penetration was simulated for uniform arrays com-

prising zero-field parameters of ~ij(O)= 10, 20, and 100, respectively; see Figs. 5(a-c). The flux-density profiles for/3u(0) = 10 are almost linear, despite the strong field dependence of Ij, which is seen from the corresponding ~oM(Be) hysteresis curve in Fig. 6. From the profiles in Figs. 5(b) and (c) we can conclude, that increasing /3(0) results in an increasing curvature, especially in the range of partial flux penetration. However, when B e is sufficiently high, the array magnetization becomes small and the flux profiles are almost linear. This is due to the fact that in high fields Ijij(B) is low and only weakly dependent on B e. The simulations with field-dependent arrays consisting of weakly coupled 5 X 5 grain blocks reveal interesting features. Flux penetration into a /3u(0)= 4 0 / 1 0 array is shown in Figs. 7(a) and (b), wherein (a) shows flux-density profiles of an inter-block array, (i, j ) = (25, j), /325.j(0)= 10, and (b) shows profiles of a next nearest array, (i, j ) = ( 2 3 , j), where /323,j(0) is either 40 or 10 depending on j. Corresponding profiles for /3ij(0)= 100/10 are shown in Figs. 7(c) and (d). When comparing the systems, one can see that the inter-block profiles,

Fig. 4. Magnetic-flux distribution and magnetic hysteresis curves of JJ arrays: (a, b) an array where Bij= 0.1 +__20%; (c, d) a /3ij = 1 + 20% array; (e, f) a flij = 10 + 20% array; (g, h) a flij = 100 + 20% array; (i) flux distribution of a 70 X 70 array, where each seventh /3ij = 10 and elsewhere flij = 40; (j) hysteresis curve of a 50 × 50 array with grain blocks as in (i). The flux-distribution plots have been cut in the center for a better visualization and only the second half is shown.

J. Paasi et aL/Physica C 259 (1996) 10-26

M(B e) curves of high-quality B S C C O - 2 2 2 3 / A g

B25,j, are almost similar, and, surprisingly, they are also similar to the profiles o f the uniform/3ij(0) -- 10 array. The most interesting feature is seen in the (23,j) arrays, wherein flux distribution is strongly dependent on the position: there exist intra-block meshes w h e r e BE3,j = 0 , although the flux has penetrated to the center of the m a y , (i, j) = (25, 25). This is especially characteristic for the profiles of B e = 0.6-0.7 mT. When B e is increased, the flux penetrates into all loops and the flux-density difference between inter- and intra-block meshes decreases slowly. We also computed tzoM(Be) curves for the /3o.(0) = 4 0 / 1 0 and = 1 0 0 / 1 0 arrays and compared them with the curve of the uniform /3o.(0)--10 array. From these curves, Fig. 6, one can see that the intra-block currents have an increasing influence on the average magnetization level. However, even though 80% of the junctions of the flij(O) = 1 0 0 / 1 0 array have 10 times higher 1j(0) as compared to the /3ij.(0) = 10 array, the magnetization is increased only by a factor of two. This underlines the importance of inter-block screening currents in determining the overall behavior of the arrays. A comparison between simulated M(B e) hysteresis curves and measured curves of BSCCO-2223 bulk samples [6] shows that the computed field dependence is stronger than that of the measured one. However, if the simulations are compared to

tapes [4], the qualitative correspondence is much better. The better correlation between simulation resuits and tape samples is explained by the fact that the present model represents a grain layer in a c-axis oriented samples, i.e. the model has an equivalent ordered structure as the tape.

4.3. Magnetic relaxation Finally we study the influence of magnetic relaxation on the flux distribution. This case comprises /3ij(0) = 10 arrays, where I j ( B ) is constant and where Ij is B dependent. The natural time unit of the model is L / R , equal to 1 0 - 1 ° - 1 0 -12 s, depending on parameter values. Due to the short unit time experimental relaxation times cannot be achieved as they would require enormous CPU times. Therefore we concentrate on the qualitative time dependence and present the results in L / R units. Relaxing flux-density profiles of the B independent [3ij-- 10 system are shown in Fig. 8(a) corresponding to the cases of partial (B e = 0.25 mT) and full (B e = 0.50 mT) flux penetration. The maximum Josephson current, I j - - 0.3 mA, results in an energy ratio of E j / k B T = 90. However, then the barrier preventing thermally activated flux creep is so high that only small changes could be observed in the array magnetization during realistic computation

flij(O)

1.5

I

(a)

21

1.5

1.0

(c)

I

I

I

L0

'~ 0.5

0.0 1.5

0.0

I

i

I

i

I

1.5

(b)

1.0

~,4 o~ 0.0

~

1.0

~

0.5

(a)

I

~l

I

i

I

0.0 I 0

i

I 25

junctions

i 50

0

25

5o

junctions

Fig. 7. Penetration of a magnetic field into two Ij(B) dependent 50 × 50 arrays consisting of 5 × 5 grain blocks. Zero-field intra-/inter-block parameters of the arrays are: (a, b) flij(O)= 40/10, and (c, d) flij(O)= 100/10. The profiles in (a) and (c) are taken at an interblock array (i, j) = (25, j) where f125.j(0)= 10. The profiles in (b) and (d) are taken at the next nearest array (i, j) = (23, j).

J. Paasi et a l . / Physica C 259 (1996) 10-26

22 .

vt

(a)

I

I

!

25

50

0.25

0.0

0.5

(bl

~o.z5 0.0 0

junctions Fig. 8. Relaxing fiux-density profiles of arrays at times t = O L / R , IO00L/R, and 1 0 0 0 0 L / R after establishing the constant Be: (a) profiles of a B independent flij = 10 array at B e = 0.25 raT and 0.50 mT; (b) profiles of a B dependent f l i j ( 0 ) = 10 array at B~ = 0.25 raT, 0.33 m T and 0.50 mT.

times. Therefore we reduced the ratio to Ej/kBT = 9. The relaxation at constant B e was preceded by ZFC and a flux change of sweep rate d q S J d t = O.01qbo/(L/R). The thermal oscillation option was also active during the sweep. From the profiles at B e = 0.25 m T we can see a clear difference between the t=O and the t= IO000L/R cases. For full penetration, at B e = 0.50 mT, the difference between the profiles is even stronger. The M(t) curves reveal that the decay is actually stronger at B e - - 0 . 5 0 mT than at 0.25 mT, see Fig. 9(a). This is explained by the simple fact that the number of JJ loops under relaxation is higher in the full than in the partial penetration case. The abrupt change of dq~Jdt at t = 0 causes a diffusion redistribution of the magnetic flux and a transient change before the steadystate flux creep. This explains the initial curvature of M(t) on a logarithmic time scale. An analogous situation has been described in detail in Ref. [32] for type-II superconductors. Relaxing profiles of the field dependent flij(0) = I0 array are presented in Fig. 8(b) in the cases of B e = 0.25 mT (partial penetration) and B e = 0.33 mT and 0.50 mT (full penetration). Corresponding time decays of M are shown in Fig. 9(b). The field dependence of 1j is reflected in Ej/kBT= Ij(B)qbo/(21rkBT), for which we used an initial

value Ej(O)/kBT--- 90 based on Ij0 = 0.3 mA. From the simulation results one can see that the relaxation is, roughly speaking, as strong as in the I j ( B ) = constant array, despite an essentially different Ej(O)/kBT. Furthermore, the difference in the relaxation rates between the B e = 0.25 m T and 0.50 mT cases is remarkable. The difference is essentially larger than in the/3;j = 10 system, hence it cannot be explained solely by the different amount of JJ loops under relaxation. The main reason for the behavior is found from the influence of Ijij(B) on relaxation dynamics, i.e. the barrier AUij, which prevents thermally activated flux creeping, strongly decreases as B increases (see Appendix A). In the case of partial penetration there exist junctions near the boundary of the penetrated flux where ]Jij is high due to a low local flux density. This leads to a high AUij and good resistance against flux creep. When B >> 0, the height of AU,.j has decreased and the relaxation is enhanced. The fact, that the relaxation rate in the case of partial penetration (B e = 0.25 mT) is only about half of that for full penetration (B e = 0.50 mT), is in good accordance with the experimental results of intergranular screening current relaxation

[6]. The B dependent flij(O)= 10 array has still a couple of interesting features. The M(t) characteris.......

i

........

I

.......

'1

.......

't

~ ~

.......

(a)

1.0

0.8 .o.. 0.6 0.4

0.2 0.0 1.0 0.8

|,,~,,[I

I

~ ....... .......

I I

~ ,,,,,,,J ..... '1

~

~ ~,~,,~,

(b)

......~.-..-.

0.6

~

~, ....... I ' ' "''l

0.4

0.2 1).0 1C

i Is ,--!

l0~

........

I

Io2

. iil..J

.......

103

J

10"

. ,..

i t5

t (I_JR) Fig. 9. Time decay of average magnetization: (a) the B independent ~ij = 10 array at B e = 0.25 m T ( ) and B e = 0.50 m T ( . . . . . . ); (b) the B dependent /3ij(0) = 10 army at B e = 0 . 2 5 naT ( ), B e = 0 . 3 3 m T (- • -), and B e = 0 . 5 0 m T ( . . . . . . ). M(t) is normalized by the value of the m o m e n t when a constant B e was established.

J. Paasi et al./Physica C 259 (1996) 10-26

tics are curved also during the steady-state flux creep (t > 200 L/R), see Fig. 9(b). The curvature is especially pronounced at Be = 0.33 mT and 0.50 mT. We believe that this behavior is a direct consequence of the nonlinear relation between AUu and L Furthermore, from Fig. 8(b) one can see that the full penetration B e value is much lower than for the non-relaxing case, Fig. 5(a). The difference is explained by the continuous relaxation of screening currents during the increase of B e, because thermal oscillations were also active. This allows the additional flux penetration in comparison to the non-relaxing case. We also did simulations with arrays where /3u(0) >> 10. In these cases the relaxation rate was strongly reduced with respect to the /3u(0)= 10 array. We believe that the reduction was caused by enhanced flux pinning due to higher Iji](O).

5. Concluding r e m a r k s The intergranular current system of a real BSCCO-2223 conductor is very complex because of the large spatial variations in almost all the important parameters. Parameter variations occur in grain orientation, grain connectivity, phase purity, intergranular defect sizes, grain sizes, intergranular junction widths, etc. In the simulations we had to restrict ourselves to just a few cases. The most important restriction done was the assumption that all junctions in the array are narrow. In real samples there are both narrow and wide junctions and the influence of the wide grain boundary junctions on the overall electrodynamie performance of the sample is not negligible. However, the proportion of narrow intergranular junctions is often large enough to validate the narrow-junction assumption. The simulations showed the importance of intergranular defects on the intergranular magnetization and flux pinning. A defect means here a non-superconducting region consisting of a secondary-phase particle or a void in the middle of an array mesh. Since there were no wide junctions in the array providing pinning for intergranular flux, the pinning was solely due to the defects. The flux-pinning capacity of a defect region is related to the product /3= 27rLIj/qb o, where L is the inductance of the

23

current loop around the defect and Ij is the maximum Josephson current of the weakest junction in the loop. The pinning results in magnetic flux density gradients, which are coupled with the macroscopic intergranular screening current density according to Ampere's law, V X B = / x 0 J . The simulations showed that, when/3 >> 1, the array magnetization was strong and in many ways equivalent to the magnetization of hard type-II superconductors. An analogy has also been observed in the experiments of intergranular flux dynamics of BSCCO-2223 samples [6]. It is therefore appropriate to propose that, like in the critical-state model of type-II superconductors [28], a current density according to Ampere's law equals the critical current density, Jc, of the array. The results suggest that, in order to increase the intergranular flux pinning required for high overall Jc, there should be non-superconducting regions with diameters of the p,m order, homogeneously distributed among the superconducting grains. We did not, however, optimize the exact grain and defect dimensions in order to obtain the maximum pinning and Jc. It is evident that the most suitable defect shape is not square since it does not optimize the relation /3/a of a mesh (a is here the mesh dimension perpendicular to the current). Much better results could be obtained by long and narrow defects with the long side parallel to the direction of transport current. Such defects are typically present in BSCCO-2223/Ag tapes among the "railwayswitch" network of grains [33,34]. Even though the overall array Jc was not optimized, it became evident that the J~ of a narrow junction array will be essentially lower than the Jj of individual junctions. This is equivalent to type-II superconductors, where Jc can be orders of magnitude lower than the depairing current density of the material. The result proposes that the maximum Josephson current density Jj in the best parts of high-quality B S C C O / A g tapes could be as high as 106 A//cm 2 at 77 K and self-field. In such parts grain boundaries are not directly limiting the transport Jc but the Jc is controlled by flux pinning in the grains as well as in the intergranular system. This is in agreement with recent experiments of the critical current and magnetization of BSCCO-2223/Ag tapes [30], where it was found that the overall level

24

J. Paasi et a l . / Physica C 259 (1996) 10-26

of J¢ was determined by intergranular flux pinning but the decrease of the J¢ in increasing magnetic fields was strongly influenced by flux pinning in superconducting grains. In summary, we have studied the intergranular magnetization of BSCCO-2223 superconductors by means of numerical simulation. As a model for the intergranular current system we used Josephson junction arrays, where the junctions consisted of resistively shunted, narrow junctions. The study focused on the following subjects: the influence of the array disorder and defects on the intergranular magnetization and flux pinning, the influence of the magneticfield dependence of Josephson currents on the magnetic-flux distribution and hysteresis of the arrays, and the effects of thermal fluctuations on the array magnetization (magnetic relaxation). The best correspondence between the simulation results and the experiments of the intergranular magnetization of BSCCO-2223 [6] was obtained when the simulation model included large non-superconducting regions among the superconducting grains. Such regions could act effectively as pinning centers for the intergranular flux enhancing the overall critical current density, allowing strong magnetization, and decreasing the relaxation rate.

Acknowledgement This research was supported by Imatran Voima Foundation and Technology Development Centre of Finland (NEMO 2-program).

Appendix A The usual method to include thermal fluctuations would be to add a noise term to the mesh current equations of the JJ array, Eqs. (10a) and (10b), and then to do the time integration followed by a thermal average [8,22,35]. In the present work the favor is for a term which could be directly applied in the flux equations, Eqs. (13a) and (13b), and which includes parameters characterizing the properties of a JJ loop. Therefore we developed an alternative procedure based on the current relaxation equation of a single JJ loop, Eq. (8). By a proper choice of the level and

statistical features of the noise current term, both approaches, however, would give the same results. If we multiply Eq. (8) by 2wL/qb o and write it for a loop junction, we obtain d(-

rid

sin (Oi,j)) = -2"rrt°0 e-av/kBr" dt

(A.1)

An analogical equation holds for horizontal junctions. In Eq. (A.1) we have approximated the junction current by accounting for only the Josephson tunneling current. This can be done because in overdamped dynamics the duration of resistive currents due to a 2Tr phase-slip process is short compared to too. In an array the mesh currents are influenced by changes in neighboring loops. While in a single loop the sign of dI/dt remains the same during a relaxation process in constant q6e, that is not valid for an array mesh. In an individual loop the flux either increases or decreases depending on the value of the ~be. In the case of an array mesh this is not so straightforward because the mesh flux is affected by current relaxation in neighboring loops. For instance, a thermally activated movement of a flux quantum from a left-hand loop into the right-hand loop can increase the current in the left loop but decrease the right-hand loop current. Beside this, the flux movement has a long-range influence on the phase differences in neighboring junctions. All these must be accounted for the equations of the relaxation dynamics of an JJ array. Therefore, the addition of Eq. (A.1) to the array model, Eqs. (13-15), is inadequate in describing the dynamics of the system. One solution to the problem is to use a coefficient, Kij(t), which describes the ratio: (tunneling current at t = t~)/(tunneling current at t = 0). For a vertical junction Kij(t) can be determined as

Koi,j (tl) =1+

1

[ t l d( 13 sin(Oi,j))

} ~i,j(~)) sin(Oi,j)t=o I Jo

dt

1 I /3i,j(th) sin(~i,j)t=0 ]

>(fot127rtooe-Av/kBr dt.

(A.2)

J. Paasi et al./ Physica C 259 (1996) 10-26 Table 1 Additional computation parameters for Figs. 4 - 7 Fig.

/3(0)

4(a), (b)

Be.roBx (roT)

0.1 ± 20%

(c), (d) (e), (f) (g), (h) (j) 5(a) (b) (c) 6

1 ± 20% 10 ± 20% 100 ± 20% 40/10 10 20 100 10 40/10 100/10 40/10 40/10 100/10 100/10

7(a) (b) (c) (d)

When the /3 of an array mesh is much higher than one, a flux quantum is well localized in a single mesh and the barrier AU in Eqs. (20) and (A.2) is given by Eq, (6). However, when /3 < 1, a flux quantum is spread over several loops and the longrange interaction decreases the effective barrier AU [14]. In general, the potential barrier preventing the transitions in a two-dimensional array is given by [36,37]

Cycle time (X 103 L / R )

0.53

4O

1.8 3.6 26 3.6 1.5 1.5 2.0 1.5 1.5 1.5 1.5 1.5 1.5 1.5

25

40 40 40 80 120 80 80 120 240 240 240 240 240 240

AU,.j = F ( f l ) E j ( B ) G ( T ) ,

(A.4)

where F ( / 3 ) is an adimensional weighting function, Ej(B) is the Josephson coupling energy, and G(T) is a function giving the reduction of the energy barrier due to a renormalized bias current, y. G(T) is determined by G ( T ) = ~/1 -- y z - 3, c o s - ' T,

The result of the equation is then used in the system Eq. (13a) to replace the tunneling current term by Koi,j(tl)/Bi.j(qb)sin Oi,j. To minimize the numerical error the integration time of the above integral should be short. It is convenient to set it equal to the integration step of the solving method used, tl = A t. This leads to Eq. (20). The current coefficient after the previous iteration round, K o i , j ( t - - A t ) , is then given by

Koi,j ( t - A t) R =

with 2( li,j -- l i - 1 , j )

27rdPi-l,j

2"rrqbi,j

dt qbo qbo /3i,j(qb ) s i n ( O i , j ) , = , _ a ,

The weighting function can be approximated by [37] F(/3) = 1 +

(/32_/3o)(xA/3o +x./32)

lim F ( / 3 ) = 2.0,

fl--,~

Fig. 8(a)

1 2 3 4 5 6

AB e (mT) At ( × 103 L/R) Relaxation (>( 10 3 L/R) AB e (mT) At (X 103 L/R) Relaxation (X 103 L/R) A Be (mT) At (X 103 L/R) Relaxation (X 10 3 L/R)

0.25

5 10 0.25

8(b)

9(a)

9(b)

0.25 5 10 0.08

0.25 5 10 0.25

0.25 5 10 0.08

5

1.7

5

1.7

10 -

10 0.17 5

10 -

10 0.17 5 10

-

(A.8)

which leads to parameter values: x A = 0.8, x B = 1.0, and /30 = 5.4. In the upper limit (/3 >> 1) Eq. (A.4) then reduces to Eq. (6).

Table 2 Additional computation parameters for Figs. 8 and 9 Step

(A.7)

For a square array the limiting conditions are [16] /3-,0

(A.3)

(A.6)

F(/3)ljij(a ) "

7=

lim F ( / 3 ) = 0.2,

L dOid --+

(A.5)

10

-

26

J. Paasi et al./Physica C 259 (1996) 10-26

Appendix B In this Appendix we give additional computation parameters, not mentioned in the text, belonging to the simulations of Figs. 4-9. External magnetic-field amplitudes and simulation times (durations of a cycle) used in Figs. 4 - 7 are given in Table 1. The flux distribution in Fig. 4(i) was computed with a simulation time of 10 O00L/R after applying a stepwise external field. Computation parameters for Figs. 8 and 9 are given in Table 2 describing the procedure of the simulations. The external magnetic field was first swept from zero to B e = 0.25 mT within the time of 5000L/R (step 1). Then B e was kept constant until the desired relaxation time, IO000L/R, was passed (step 2). After that B e was increased further according to the given values (step 3), and new relaxation cases were computed (step 4). For the (b) figures the procedure still continued through steps 5 and 6. The integration step was constant throughout the simulations, A t -----0.01 L / R .

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