Field distribution in thin superconductors with secondary peak in magnetisation

Physica C 304 Ž1998. 202–212

Field distribution in thin superconductors with secondary peak in magnetisation Mahesh Chandran

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Low Temperature Physics Group, Centre for AdÕanced Technology, P.O. CAT, Indore 452013, India Received 28 May 1998; accepted 19 June 1998

Abstract The high-Tc superconductors show non-monotonic dependence of the critical current density Jc on the magnetic field B which manifests in the magnetisation curves as a peak at intermediate fields Ž‘fish-tail’ effect.. The field distribution in the non-monotonic region is simulated by exploiting the analogy of a two dimensional inductive Josephson junction array to that of a hard type-II superconductor in an applied magnetic field. The simulation is carried out for demagnetisation factor N s 0 and N ) 0.9, corresponding to the case of an infinite slab parallel to and a thin sample transverse to an applied field, respectively. We observe that the increase in Jc Ž B . at an intermediate field is accommodated by a change in the sign of the curvature of the field profile. This gives a point of inflexion where the slope of the screening current also changes sign. For N f 1, the peak-effect region in Jc Ž B . is experienced even for small applied fields due to large field induced at the edge. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Superconductors; Secondary peak; Magnetization

1. Introduction The irreversible magnetisation of a hard type-II superconductor is analyzed within the critical state model ŽCSM.. Over the years, this have been studied theoretically in two extreme limits: for an infinite slab parallel to an applied magnetic field for which N s 0, and for a thin sample in transverse magnetic field for which N f 1. The case of a slab which is simple though unrealistic have served as an useful paradigm for understanding CSM. On the other hand, the case N ™ 1 which is commonly met in experi-

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Fax: q91-731-488300; E-mail: [email protected]

ments and is complicated by non-local relation between screening current J Ž r . and field induction B Ž r . have been studied in detail only in recent years w1x. Analytical solutions were obtained for field independent critical current density Jc for a thin disk and an infinitely long rectangular strip w2–4x. The solutions are obtained in the 2D limit, i.e., drW ™ 0 where d and W are the dimensions parallel and transverse to the applied field. In this limit, only the screening current induced by the bending of the field lines ŽA B= = n, ˆ where B s Bnˆ . is considered whereas that arising due to gradient ŽA =B = nˆ . is neglected. The solution gives the normal component of the field on the surface Bn s 0 in the Meissner region rather than B s 0. The above formalism have

0921-4534r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 3 4 Ž 9 8 . 0 0 3 0 4 - 9

M. Chandranr Physica C 304 (1998) 202–212

been extended for an arbitrary Jc Ž B . and was solved for a monotonically decreasing function w5x for a strip. In another approach, the CSM have been solved for N f 1 by approximating the rectangular crosssections with inscribed ellipse. Exact solutions were obtained for spheroids and elliptical cylinders in transverse magnetic field w6,7x. The solution satisfies B Ž r . s 0 in the Meissner region of the sample. This approach was also extended for arbitrary Jc Ž B . and was solved for a monotonically decreasing function w8x. With such a function, either approach gives M Ž H . which decreases monotonically with increasing applied field above HI f 0 Žthe zero-field peak or central peak.. This is a generic behaviour of all low-Tc superconductors. An intriguing feature of the irreversible magnetisation M Ž H . of high-Tc superconductors is the occurrence of a secondary peak at an applied field Ha between Hc 1 and the irreversibility field Hirr . This ‘peak-effect’ or ‘fish-tail’ magnetisation have been studied over a wide region of the H–T phase diagram for a large class of high-Tc superconductors. Within the CSM, Jc Ž B . A D M Ž H . where D M Ž H . is the width of the envelope hysteresis curve Žthe curve obtained by cycling Ha between ŽyHamax , Hamax . with < Hamax < ) 2 H ), where H ) is the field for full penetration of the sample.. This implies that for samples showing fish-tail magnetisation Jc Ž B . is a non-monotonic function of B which is in sharp contrast with that observed in low-Tc superconductors. The physical origin of increase in Jc Ž B . at intermediate applied fields in high-Tc superconductors is not clear though various mechanism have been envisaged. A crossover due to change in relaxation dynamics w9,10x, oxygen inhomogeneties leading to matching effect w11,12x and crossover from surface barrier to bulk pinning w13,14x have been invoked to interpret experimental observation. More recently, the onset of peak has been attributed to a phase transition in the vortex lattice as shown for YBaCuO w15x, BiSrCaCuO w16x and NdCeCuO w17x. The peak in M Ž H . at intermediate fields can be accounted within CSM by assuming a non-monotonic Jc Ž B .. The motive of solving CSM is not merely to obtain the M Ž H . curves, but to extract the field and current distribution within the sample. Johansen et al. w18x have solved the CSM with a

203

non-monotonic Jc Ž B . for a slab parallel to the magnetic field Ž N s 0.. Since experiments are performed generally on thin samples for which N f 1, it becomes necessary to solve CSM for such a case. In experiments on BiSrCaCuO w16x and NdCeCuO w17x, the local field component Bz Ž x . normal to the surface was measured using micro-Hall probe close to the onset of peak-effect region Žfrom the reported size of the samples, the geometrical N ) 0.9.. Though these experiments are concerned with the change in field distribution close to the transition region, the change in the field-distribution inside the peak effect region has not been addressed yet for thin samples. It is important to note that the peak-effect in bulk critical current density Jc Ž B . implies that the field distribution within the CSM must change from that obtained with monotonic Jc Ž B .. This paper addresses the problem of field distribution in thin superconductors in a transverse magnetic field with a non-monotonic critical current density Jc Ž B .. We present results from direct numerical simulation of a model system which takes the effect of large N into consideration. The only phenomenological input into the simulation is the non-monotonic function dependence of Jc Ž B .. The results are compared with that of a monotonic Jc Ž B .. We also solve analytical expressions for field and current distribution w5x in a thin strip with Jc Ž B . showing peak effect. The results are compared with that obtained from direct numerical simulation. The paper is organized as follows; in Section 2, we solve the critical state model for a two-dimensional strip in transverse magnetic field as given by McDonald and Clem, with a non-monotonic Jc Ž B .. In Section 3, the model for numerical simulation is described, and Section 4 discusses the results.

2. Field and current distribution in a thin strip with a non-monotonic Jc ( B ) The CSM for a strip infinite along y-axis, of thickness dŽ5 z-axis. and transverse width 2W in a magnetic field Ba s Ba eˆ z have been solved by McDonald and Clem w5x for an arbitrary Jc Ž B . following procedure of Ref. w2x. The solutions are exact only in the 2D limit, i.e., Ž d .rŽW . ™ 0. The field

M. Chandranr Physica C 304 (1998) 202–212

204

Bz Ž x . is obtained as Fredholm equation with a nonlinear integral kernel Bz Ž x . s Bf < x <'x 2 y a2 d xX

Jc Ž Bz Ž xX . .

Ž x 2 y xX 2 . (xX 2 y a2

Jc 0

W

=

H

a

a-< x
,

We now discuss the results obtained with B0 s 0.7, B1 s 2.0, B2 s 0.9 and j1 s 0.8. The function Jc Ž B ., which is shown in the inset of Fig. 1Ža., attains a minimum at B f 0.9 and peaks at B f 2.0. Applied fields Ba for which the field distribution is shown in Fig. 1Ža. are marked on the Jc Ž B . curve. Since Bz Ž x . is symmetric around x s 0, only the

Ž 1.

where Bf s Ž m 0 Jc0 d .rŽp ., and the co-ordinate of the flux-front a is related to applied field Ba through the following equation W

Ba s Bf

H ( x a

d xX X2

Jc Ž Bz Ž xX . . Jc 0

y a2

.

Ž 2.

The above two equations must be solved self-consistently to obtain the field distribution Bz Ž x . at an applied field Ba . The current density integrated over the thickness J y Ž x . is given by Jy Ž x .

° ~ ¢

2

y

p

W

'

x a2 y x 2

Ha

s

y

dx

X

'

Ž xX 2 y x 2 . x X 2 y a 2

X

Jc Ž B z Ž x . . ,

x < x<

for < x < - a

for a - < x < - W

Jc Ž B z Ž x . . ,

Ž 3. In Ref. w5x, the above equation was solved for Jc Ž B . s 1rŽ1 q < B
s exp Ž y< B
q j1exp y Ž B y B1 . r Ž 2 B22 . ,

ž

where B0 , B1 , B2 and j1 are constants.

/

Ž 4.

Fig. 1. Ža. The field distribution B z Ž x . in the positive half of a strip for a non-monotonic Jc Ž B . Žinset.. Applied field Ba r Bf increases from 0.25 to 2.25 in steps of 0.25. At large applied fields, the slope of B z Ž x . differs drastically across the dotted vertical line Žsee Section 4.. Žb. The current distribution J y Ž x .r Jc0 Ž Jc0 ' Jc Ž B s 0.. for some of the applied fields Ba r Bf . Inset shows B z Ž x . and J y Ž x . for Ba r Bf s 2.0 on the same coordinate scale.

M. Chandranr Physica C 304 (1998) 202–212

positive half of the strip is shown. A large diverging Bz Ž x . at x s W and diverging slope at the flux-front x s a are characteristics of 2D geometry. As a consequence of this large field at the edge, the sample experiences non-monotonic Jc even for an arbitrarily small applied field. The Bz Ž x . changes curvature at a point of inflexion at which the current distribution J y Ž x . also shows a peak Žinset of Fig. 1Žb... Thus, the non-monotonic Jc Ž B . is reflected in J y Ž x . in the flux carrying region. Note that d J y Ž x .rd x changes sign and is consistent with the change in curvature of Bz Ž x .. The J y Ž x . in flux-free region is similar to that for monotonic Jc Ž B . which terminates at the flux-front with a cusp singularity. Fig. 2 shows the evolution of flux-front coordinate a with increasing BarBf . For comparison, the case of monotonic Jc Ž B . Žonly the exponential term. is also shown. The peak effect delays the flux-entry as expected. It is necessary to point out differences between experimental observation and the solution obtained above. As shown in Ref. w3,4x, for a field independent Jc , the flux-front is given by a s Wr coshŽ BarBf .. Thus, the entire sample goes into the critical state, i.e., a ™ 0, only in the limit Ba ™ ` while experimentally, complete field penetration oc-

Fig. 2. Evolution of the flux-front for monotonic and non-monotonic critical current density Jc Ž B .. The monotonic curve is obtained by retaining only the first term in Eq. Ž3..

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curs at moderate applied field B ) . This artifact persists even for a field dependent Jc Ž B .. The artifact is due to an assumption that the screening current J y Ž x . is calculated as solely due to discontinuity in field B x Ž z . at z s "dr2, and the screening current driven by the gradient in Bz Ž x . is neglected. Thus, the CSM does not predict Bz Ž x . for Ba ) B ) . The problem is severe for non-monotonic Jc Ž B . where the peak in Jc Ž B . occurs above B ) . Any change in Bz Ž x . thus remains elusive analytically in that regime Žthis problem does not arise in thin elliptic strips where B ) is finite w8x. Calculation for this case with non-monotonic Jc Ž B . is currently underway w20x. To address this regime which is observed experimentally, we use direct numerical simulation to obtain Bz Ž x . as discussed below.

3. 2D Josephson junction array In recent years, two dimensional arrays of Josephson junction ŽJJA. have been studied as a model for simulating irreversible magnetisation of a hard type-II superconductor w21–23x. The superconducting islands are assumed to form a regular 2D lattice, thus creating a network of Josephson junctions. The junctions are located at each bond connecting a superconducting island to its four nearest neighbours. The dynamics of the array is governed by the phase difference f between the islands across the junction. The screening current arising due to an applied field is included explicitly in the dynamical equation through the geometrical inductance of the array. As shown in Ref. w21,22x, keeping only the self-inductance of the plaquette Žan unit cell. simulate the N s 0 geometry, and the field and current distribution inside the array is similar to that given by Bean’s critical state model. Including the mutual inductances makes the relation between screening currents and field non-local w24,25x, and in the limit of large size of the array N f 1. The current distribution of a single vortex in the array decays as ; 1rr 2 away from the core similar to Pearl’s solution for a vortex in thin films. The field and current distribution in the array follows critical state model for thin superconductors in transverse magnetic field, as shown Ref. w24–26x in great detail. It is important to

M. Chandranr Physica C 304 (1998) 202–212

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mention here that the irreversibility in magnetisation, field and current distribution in the array arises due to an energy barrier for the motion of vortices w27,28x. This effectively pins the vortices at the plaquette centre and is intrinsic to the dynamics of the array.

3.1. Equation of motion for JJA We consider a 2D JJA lying in the xy plane formed by Nx = Ny plaquettes Žor cells.. The junctions are formed by the superconducting islands located at the corners of each of the plaquette with a lattice constant p. The time evolution of the phase difference across the ith junction, f i is assumed to be governed by the overdamped resistively-shunted junction ŽRSJ. equation w29x

f 0 d fi Ž t . 2p R

dt

q Ic sin f i Ž t . s Iib Ž t . ,

Ž 5.

where f i is assumed to be gauge-invariant Ži.e., f i s f ia y f ib y Ž2p .rŽF 0 .HabA P d l .. In Eq. Ž5., R and Ic are the normal state resistance and critical current of the junction respectively, and F 0 Žs hcr2 e . denotes a quantum of flux. The current flowing through the junction is denoted by Iib. The applied flux f s FextrF 0 is assumed to be along the z-axis and transverse to the array. From the fluxquantisation condition, the magnetic flux through the ith plaquette, F i , is given by the directed sum of f k ’s around the plaquette Žtaken in anti-clockwise sense.

where I jm is the current around the jth plaquette. The junction current Iib is the difference in the plaquette currents I m neighbouring the junction. The L i j are the elements of the inductance matrix of the array and is dependent only on the geometry. Assuming f , F , Ib and Im as representing the column vector of phase differences, fluxes, junction and plaquette currents, respectively, in the array, Eqs. Ž5. – Ž7. can be simplified df dt

s Ib y i c Ž F . sin f ,

F Mf s 2 p n y 2 p

F0

F s Fext q LIm , where t s Ž2p RIc .rŽF 0 . t is the dimensionless time. The field Žflux. dependence of the critical current of the junction i c ŽF . s Ic ŽF .rIc Ž0. is shown explicitly. The matrix M, referred as the loop-sum-operator matrix w28x, carries out the required lattice curl operation for the square array considered here. It is also easy to verify that the transpose of the matrix M relates Im and Ib , i.e., M T Im s Ib . Combining the last of the two equations in Eq. Ž8., and substituting for Ib , the dynamics of current and phase-differences in the array is governed by df dt

s M T IM y sin f ,

Mf s 2 p Ž n y f . y X

fi

Ý f k s 2p n y 2p f k

,

ij

1

l2J

Ž 9.

˜ m, LI

Ž 6.

0

where X denotes summation over f k ’s forming the ith plaquette Žthis is equivalent to lattice curl operation.. On the other hand, the flux F i is also related to the screening currents flowing around the plaquettes, and is given by

f i s fext q Ý L i j Ijm ,

Ž 8.

,

Ž 7.

where l J s ŽF 0r2p L0 Ic .1r2 Ž L0 is the self-inductance of a plaquette., is the dimensionless Josephson penetration depth of the array. The matrix L˜ s LrL0 is the reduced inductance matrix such that L˜ i i s 1 and y1 - L˜ i j - 0. When the full inductance matrix is involved, the effective penetration depth l s l2JrŽ m 0 prL0 . is much larger than l J , similar to an enhanced penetration depth for thin films in transverse magnetic field. The elements of the mutual inductance matrix is dependent on the geometry of

M. Chandranr Physica C 304 (1998) 202–212

the superconducting island and is calculated for ‘q’-shape the details of which is given in Refs. w24,25x. Note that only the functional dependence LŽ r,rX . s LŽ< r y rX <. finally enters the simulation which for the 2D geometry is observed to be of the form ; 1r< r y rX < 3 for distances beyond four or five lattice constants. Eq. Ž5. is solved numerically using FFT-accelerated matrix multiplication w30x. The phase-difference f and current Im is obtained for the stationary state. The magnetisation Žmagnetic moment per unit area. is calculated using the Ampere’s law for magnetic moment 1 Ms

2

Hr = J P d r .

Ž 10 .

The i c ŽF . is chosen to be of the same form as used in Section 2; a monotonically decreasing exponential term, and a Gaussian term representing the peak i c Ž F˜ . s exp Ž y
q i 1exp y F˜ y F˜ 1 r Ž 2F˜ 2 . ,

ž

ž

/

/

Ž 11 .

where F˜ s FrF 0 . The constants i 1 , F˜ 1 and F˜ 2 decides the height, position and width of the peak in i c ŽF˜ ..

4. Results and discussion The simulations were carried out for an array of size 63 = 63 and all the results presented below are for the same. The flux and current distribution in the array is obtained through the central row so as to minimize the influence of corners w24,25x. The value of dimensionless penetration depth l2J Žs 0.05 for L˜ i j s d i j , and 0.01 for full L˜ matrix. is chosen in the strong pinning regime where the critical state behaviour was previously studied w24,25x. We first consider the case L˜ i j s d i j , thus retaining only the self-inductance of the plaquette. The induced flux is locally related to the screening current,

207

and therefore simulates N s 0 behaviour. Fig. 3Ža. shows the M–H curve as f is cycled between 220 and y220. The value of constants in Eq. Ž11. were chosen as F ) s 50, F 1 s 120, F 2 s 500 and i 1 s 0.7. A minimum at f f 80 and a peak at f f 140 ) f ) s 70 Žwhere f ) is the applied flux at which flux-front reaches the central plaquette. in the M–H curve is indicative of non-monotonic critical current Ic . The flux distribution F Ž n.rF 0 as a function of plaquette co-ordinate n is shown in Fig. 3Žb. for f across the peak effect region. The corresponding branch current I Ž n.rIc Ž0. is shown in Fig. 3Žc.. Only the positive half of the distribution is shown, since F˜ Ž n. and I Ž n. are mirror symmetric and inversion symmetric around the central plaquette of the array, respectively. The flux F˜ at which i c shows a minimum and a peak is indicated by dotted lines in Fig. 3Žb.. A point of inflexion is seen in F˜ Ž n. where Žd Ib .rŽd n. changes sign as shown in the inset of Fig. 3Žc.. This point moves towards the centre of the array with increasing f. Above the inflexion point where Ib Ž n. has a negative slope, the flux-distribution is concaÕe, whereas below the inflexion point, Žd Ib Ž n..rd n is negative and F˜ Ž n. is conÕex. It is important to note that the though the overall flux profile is similar to that observed for monotonic i c ŽF˜ ., the difference occurs locally in that region which experience positive slope in i c ŽF˜ . curve. We next consider the case where the full inductance matrix L˜ is used. As shown in Ref. w24,25x, this makes the electromagnetic property of the array non-local and its continuum equivalent is a thin superconducting sample in transverse magnetic field. For array of size Nx s Ny s 63, the slope of the magnetisation curve at f s 0 gives N f 0.96 reflecting 2D behaviour. Fig. 4Ža. shows the magnetisation curve for a non-monotonic i c ŽF˜ . which is shown on the same horizontal axis. The constants in Eq. Ž11. are chosen to be F˜ ) s 40, F˜ 1 s 140, F˜ 2 s 1000 and i 1 s 3.0. The magnetisation curve shows a pronounced peak at the same applied field f at which the critical current shows a peak. The flux distribution F˜ Ž n. for f along the initial descending part of the i c ŽF˜ . curve is shown in Fig. 4Žb.. The flux front reaches the central plaquette for f ) f 20. The flux profile has a large slope close to the edge and the flux front in qualitative agreement with the prediction of the CSM.

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M. Chandranr Physica C 304 (1998) 202–212

Fig. 3. Ža. The M–H curve for a non-monotonic critical current of the junction Ic ŽF .. Only the self-inductance of the plaquette is retained for this calculation. Note that the peak lies above f ) Žapplied flux at which the virgin curve meets the envelope hysteresis curve.. Žb. The flux distribution F Ž n.rF 0 in the positive half of the array for f across the peak-effect region. Žc. The current distribution I Ž n.rIc Ž0. corresponding to Žb.. Note the change in the sign of slope of d I Ž n.rd n. At that point, the flux distribution changes curvature Žpoint of inflexion. which is evident from the inset where the F Ž n. and I Ž n. is plotted on the same co-ordinate axis.

The F˜ Ž n. in the peak effect region is shown in Fig. 4Žc.. The curves are normalized by the applied f and shifted vertically by 0.05 units starting from the lowermost curve. Along the ascending branch of the i c ŽF˜ . curve, the slope close to the edge increases with increasing f which is expected due to large current flowing parallel to the edge. The flux-profile becomes conÕex in response to the peak in the i c as evident from the F˜ Ž n. for f s 100. In a recent experiment, the local gradient in the field was measured directly using micro Hall probes to stress the thermodynamic nature of the phase transition at the

onset of the peak in magnetisation w16x. We have calculated the gradient in flux distribution which is shown in Fig. 4Žd. as a function of applied flux f for three different co-ordinates. The slope at the centre of the array Ž n s 1. is the largest in the peak effect region which drops sharply where the peak effect disappears. The inset shows the same for other coordinates in the peak-effect region. Though the peak in dF˜ rd n occurs at different f for different coordinates, the curves shows an interesting correlation on the ascending part of the i c ŽF˜ . curve. Excluding the dF˜ rd n for the centre and the edge of the array,

M. Chandranr Physica C 304 (1998) 202–212

209

Fig. 4. Ža. The M–H curve showing pronounced peak effect when the full matrix L˜ is used. The i c ŽF˜ . dependence is shown on the same horizontal axis. The constants chosen are F˜ ) s 40, F˜ 1 s 140, F˜ 2 s 1000 and i1 s 3.0. Žb. The flux distribution F˜ Ž n. for f along the initial descending part of the i c ŽF˜ . curve whereas Žc. shows the same in the peak-effect region. All the curves are normalized by f, and in Žc. the curves are displaced vertically by 0.05 units for clarity. Applied flux f is given on the right side. Žd. The gradient of the flux distribution EF˜ Ž n.rE n as a function of f for three co-ordinates. The lines are polynomial fit. The inset shows the same in the peak effect region for n differing by 5 lattice constants from the edge to the centre. The curves fall into two well separated curve as shown encircled. All the curves except for n s 1 Žcentre. is multiplied by a factor 5.

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M. Chandranr Physica C 304 (1998) 202–212

curves for all other co-ordinates fall on two curves which are well separated. This is an interesting result which implies that there are effectively two regions with different gradients at any f in the peak-effect region. Looking back into Fig. 1Ža., one indeed observes two region with differing slopes at large values of BarBf as demarcated by a vertical dotted line. Thus, the CSM can give qualitatively correct picture of the peak effect also at high-fields. Fig. 5 shows the branch current distribution Ib Ž n. in the array. For small f, the cusp singularity is observed ŽFig. 5Ža.. in agreement with the CSM. The current distribution in the flux-penetrated region is given by the i c ŽF˜ Ž n.. and hence reflects the nonmonotonic dependence on F˜ Ž n.. The Ib Ž n. at edge Ži.e., n s 30. increases with increasing f before changing sign of the slope for f close to peak in i c ŽF˜ . curve Žsee Fig. 5Žb. and Žc... Note that the

cusp at the flux front in small f disappears at large f as i c ŽF˜ s 0. - i c ŽF˜peak .. The cusp would be smeared also when F at the centre is close to the F˜peak . Experimentally, some observations indicate w31x that the peak effect could occur at low fields such that the Jc Ž B . initially increases before decreasing monotonically. In Fig. 6Ža. the flux distribution F Ž n. for such a case is shown. The constants in Eq. Ž11. are chosen as F˜ ) s 40, F˜ 1 s 40, F˜ 2 s 1000 and i 1 s 3.0. The M–H curve and the functional dependence of i c ŽF˜ . is shown in Fig. 6Žb. on the same horizontal axis, whereas the current distribution Ib Ž n. is shown in Fig. 6Žc.. The flux penetration is delayed considerably as can be seen by comparing f ) f 70 which is larger than f ) f 20 in Fig. 4Žc.. Comparing the F˜ Ž n. in Fig. 5Ža. with that in Fig. 4Žb., the slope at the flux-front tends to be linear. The Ib Ž n. increases inside the flux carrying region from the

Fig. 5. Ža. The branch current distribution Ib Ž n.rIc0 for f below the peak-effect region Ž Ic 0 ' Ic ŽF˜ s 0., the critical current in zero flux.. Note the cusp which is analytically predicted in the CSM. Žb. and Žc. Same as in Ža. on the positive slope and negative slope of i c ŽF˜ . curve, respectively. The slope of the current distribution in the flux-penetrated changes sign reflecting the i c ŽF˜ ..

M. Chandranr Physica C 304 (1998) 202–212

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Fig. 6. The F˜ Ž n. for i c which increases initially before decreasing monotonically, as shown in Žb.. The M–H curve is shown on the same horizontal scale in Žb.. Žc. The Ib Ž n. for the flux distribution shown in Ža.. The constants chosen are F˜ ) s 40, F˜ 1 s 40, F˜ 2 s 1000 and i1 s 3.0. The inset shows the fit Žshown by lines. to current distribution obtained from simulation Žshown by symbols. in the Meissner region using Eq. Ž3. for f s 10, 20 and 30.

flux-front to the edge. When the applied flux is just greater than the field at which the peak occurs in i c ŽF˜ ., a point of inflexion occurs close to the edge which moves inwards with increasing f. This is shown for f s 60 which coincides with the peak in Ib Ž n. Žsee Fig. 6Žc... For other fields, this point is smeared due to broadness of the peak in i c ŽF˜ .. The

inset of Fig. 6Žc. shows the current distribution obtained from Eq. Ž3. Žshown by lines. in the Meissner region for f s 10, 20 and 30. In the integral kernel, the current distribution in the flux-penetrated region obtained from simulation was used. The curves obtained from simulation is shown by symbols. The results are in good agreement to Eq. Ž3. though

212

M. Chandranr Physica C 304 (1998) 202–212

noting that it is derived under approximations mentioned earlier.

w5x w6x w7x w8x w9x

5. Conclusions w10x

In summary, we have analyzed the irreversible field and current distribution in a thin superconductor showing non-monotonic critical current density Jc Ž B . in a transverse magnetic field. The results are relevant for samples showing ‘fish-tail’ magnetisation and peak effect. The analytical results of Ref. w5x gives qualitatively correct description in the peak effect region. The field distribution shows a point of inflexion at which the corresponding current distribution changes sign of slope. The peak effect in Jc Ž B . thus manifests in the flux penetrated region where the current distribution is non-monotonic. Around the point of inflexion, the flux profile is convex in the region where the J y Ž x . increases towards the edge. Thus, the CSM accomodates the increase in Jc at intermediate applied fields B by locally changing the curvature of the flux profile.

Acknowledgements The author acknowledges useful discussions and suggestions with P. Chaddah during the course of the work. The author also acknowledge Dr. S. Krishnagopal for computational time on workstations. The author is financially supported by CSIR ŽIndia..

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