Pinning via intrinsic weak links: A possible scenario and applications

Pinning via intrinsic weak links: A possible scenario and applications

I. INTRODUCTION

Sergei A. SergeenkoP) Institute of Physics, University of Liege, B-4000, Liege, Belgium

(Received 24 January 1995; accepted for publication 15 March 1995)

A possible scenario for critical current enhancement via intrinsic weak-link-assisted pinning in isotropic high-T, superconductors is proposed. By considering a subtle balance between different forces acting upon extended (dislocations) and point (oxygen vacancies) defects to stabilize their equilibrium position inside a crystal, the mode1 suggests a qualitative description of some anomalous magnetic properties observed in dislocated, deoxygenated, and irradiated YBazCu307-8 superconductors. 0 1995 American Institute of Physics.

Due to the smallness of the coherence length, practically any defects (imperfections) may contribute to both the weak- link properties and the flux pinning in high-T, superconductors (HTS). Such an inevitable dualism brings about a lot of interesting peculiarities and anomalies in these materials.‘-26 According to Deutscher and Mullet-’ there are serious argu- ments to consider the twinning boundary (TB) in HTS as insulating regions of the Josephson SIS-type structure. Addi- tional evidence in favor of this conclusion is due to observed deviations in stoichiometry at the TB. As was shown by Zhu d aZ.” for YBa2(Cu1 -XMX)307-s samples, the TB thickness is changed from 7 A (for M=Ni, x=0.02, and 6=0) to 26 A (for M=Al, x=0.02, and S=O), while for pure YBa2Cu307 (YBCO) it gives ==lO A. If after Zhu and co-workers’ we assume that the physical thickness of the TB is of the order of an interplane distance (atomic scale), then such a boundary should rather rapidly move via the movement of the twinning dislocations. And this type of motion indeed has been observed by the electron-microscopic images in HTS.6-s As a result of the structural phase transition (if spe- cial precautions have not been done) a HTS sample is di- vided into a large number of twinning blocks,’ i.e., the sample is crossed by a large number of insulating layers (twin boundaries). An average distance between boundaries is essentially less than the grain size. This net of layers induces the net of Josephson junctions inside a single grain. In addition, the well-known processes of the oxygen ordering in HTS leads to the continuous change of the lattice period along TB with the change of the oxygen content.tO”’ Agassi and Kasowski’* considered the ideal TB atomic configura- tion and the oxygen-disordered TB model. The latter model with oxygen dislocations implies percolative conduction in the c-axis direction parallel to the TB planes. Babcock and Larbalestiert3 observed the regular networks of localized grain-boundary dislocations (GBDs) with the spacing rahged from 10 to 100 nm. They concluded that closely spaced GBDs may produce effectively continuous normal or insulating barriers at the boundary which then behave like a planar SNS or SIS junction. Shang et a1.i4 observed dislocations inside the grains as well as coherent intragranular boundaries

“Permanent address: Frank Laboratory of Neutron Physics, Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia, CIS; Elec- tronic mails [email protected]

acting as weak links. By imaging epitaxial YBa2Cu307+ films with scanning tunneling microscopy, Gerber et aZ.15 have directly observed screw dislocations with densities of alO9 cme2 sufficient to account for a major part of the pinning forces and for weak links contribution in their films. Chen and co-worker@ found the strong correlation between high density of the TBs and critical-current-density enhancement in YBazCu30,-B films. A more direct evidence as for influence of dislocations on the pinning ability of HTS has been obtained by Mannhart et al. l7 in screw-dislocated HTS single crystals. They detected a nearly linear increase of the critical current density with the number of dislocations as well as a substantial improvement of j,(N) in applied magnetic field (at least up to 0.05 T). The use of the same screw dislocated thin films in the so-called “superconducting field- effect transistor” (SuFET) devices allowed Mannhart et al. I8 to detect the electric-field-induced modulation of critical current density in their films. Depending on the gate polarity, a quite tangible change (increase or decrease) of j,(E) has been observed. Yang et al. l9 found out that dislocation networks provide effective pinning centers when the characteristic length scale of network matches that of the vortex spacing. They stressed that in HTS, oxygen vacancies are effective flux-pinning centers at low temperatures while the high-temperature spatial variation of the oxygen ordering may well account for the so-called anomalous “fishtail” magnetization feature in YBa2Cu307-s crystals.20-24 Daeum- ling and co-workers” discussed a rather intriguing correlation between defect pinning, intragrain weak links, and oxygen deficiency in YBa2Cu30,-, single crystals. In particular, they have found that “as the nominal oxygen deficiency S decreases toward zero, the flux pinning declines and the crystals lose their explicitly granular signature.” The above anomalous magnetic-field behavior has been attributed to the “field-induced intragrain granularity” in oxygen-depleted materiaIs.“0-24 A “phase diagram” H,( 8, T) that demarcates the multigrain onset as a function of temperature and oxygen deficiency reconstructed by Osofsky et aL2* allowed them to confirm that their single crystals exhibited behavior characteristic of homogeneous superconductors for H < H, and inhomogeneous superconductors for H> H, . The granular behavior for H> H,,, has been related to the clusters of oxygen defects (within the CuO plane) that restrict supercurrent flow and allow excess flux to enter the crystal. The field at which the fishtail anomaly occurs was found19*22*23 to decrease both

1114 J. Appl. Phys. 78 (2), 15 July 1995 0021-8979/95/78(2)/l 114/9/$6.00 Q 1995 American institute of Physics

with increasing the temperature and oxygen deficiency. This in turn suggests that the characteristic scale of the defect network structure is strongly dependent on oxygen stoichiometry. On the other hand, Ullrich et ~1.~~~~ have observed a rather substantial critical current enhancement in stoichiometric melt+textured crystallites with the dislocation density of =2X10” cm-‘. They argue that the strained region sur- rounding a dislocation can cause the fishtail anomaly as well. Indeed, since the strained regions near the dislocation core as well as oxygen-deficient regions are both result in a lower B,, (the upper critical field) compared to the YBa2Cus07-6 matrix, they become (normal conducting) pinning centers if the external magnetic field exceeds their upper critical field.

Another important way to improve pinning ability of HTS single crystals is to use irradiation to incorporate (in a controllable way) different types of defects into these materials.“-37 Civale et aZ.27*28 achieved the pinning enhancement at high fields and high temperature for vortex confinement by columnar defects in YBa2Cus07-.-6 crystals. Aligned columns of damaged material 50 A in diameter and 15 pm long were produced by 580 Mev Sn ion irradiation. The pinning obtained was much greater than that produced by random point defects, and caused a considerable enlarge- ment of the irreversibility region in the H-T plane. Hardy et al.2g>30 have measured magnetic properties of the Pb ion irradiated YBa,Cu,O,-, crystals and found a rather strong fluence dependence of irradiation-induced critical current enhancement j,(@,H). The maximum of j,(@,H) was found to shift to higher fluences with increasing the applied field and to higher fields with increasing the irradiation fuence. An important correlation between the extent of radiation damage and the oxygen stoichiometry of the sample has been observed by Zhu et aL31 They found that a decrease of S significantly reduces the radiation damage.

To take advantage of the above-mentioned dualism of (extended) defects (as pinning centers and weak links) in HTS, in the present article the concept of vortex pinning via defect-induced intragrain weak links is proposed. By considering a balance between different forces acting upon extended (dislocations) and point (oxygen vacancies) defects to stabilize their equilibrium position inside a crystal, the model allows us to introduce external fields which control the defect behavior of the superconducting sample. The model of intrinsic weak links is used then to discuss qualitatively some general features of the above-mentioned anomalous properties of HTS crystals. It should be stressed that in the present article we restrict ourselves to the description of a more simple and rather isotropic YBCO-like superconductors only since in highly anisotropic Bi- and Tl-based layered materials the two-dimensional structure of vortices as well as the Josephson coupling between CuO planes become significant and thus should be incorporated into any realistic scenario from the very beginning. It is also worthwhile to mention that a similar concept of weak-link-induced vortex pinning has been proposed recently by Lee, Stroud, and Girvin.38 Using a Monte Carlo technique within a 3D model of Jo- sephson junction arrays, they found a substantial increase of critical current densities due to the flux pinning by columnar defects.

P(X) t

1: I ’

L 4

FIG. 1. Sketch (out of scaIej ?f the dislocation-induced Josephson contact geometry. The magnetic field is applied normally to the contact area Ld (L is the length of the contact, and d=X,, +h,,+r with t being an insulator thickness). The dislocation density profile near the twin boundary tip of length L and thickness t is hatched. The dislocations are lying parallel to the applied magnetic field and distributed along the x axis with density p(x) [see Es. WI.

II. DISLOCATION-INDUCED JOSEPHSON EFFECT

A. Single Josephson junction

To realize a scenario for the dislocation-induced atomic scale (intragrain) Josephson effect in YBCO-like HTS single crystals, let us consider the model of small Josephson con- tacts with length L<X, (hJ= dm is the Josephson penetration depth) in a strong enough magnetic field (which is applied normally to the contact area EC) such that H>4,,/2rrXJl, where l=hL,+X,2+t [X,,(z) is the London penetration depth of the first (second) junction, and t is an insulator thickness]. Assuming that twinning dislocations, lying along the z axis (which coincides with the direction of the applied magnetic field H) and distributed along the x axis with dimensionless density p(x) (see Fig. l), give rise to an (inhomogeneous) Josephson supercurrent density j,[ p(x) 1, the critical current density through such an inhomogeneous single Josephson contact reads3974o

jf(p,Hj=k Ldx j,[p(x)]sin(kx+c). I 0

Here k~2s-lHlq&, c is the zero-field phase difference between two superconductors forming a SE-type Josephson contact, and we have assumed that the length of the contact L is the TB length and the (local) insulator thickness t(x) corresponds to the (local) TB thickness (see Fig. 1). Notice that the local insulator thickness of the dislocation-induced Josephson junction t(x) is related to p(x) as follows:40-42

t(x) = dx’ p(x’).

To proceed further, we postulate, for simplicity, a &functional form of dislocation-induced critical current den: sity assuming that j,[p(x)] is centered around an average (dimensionless) dislocation density p,

~cMx)l=~coe -"""*qp(xj-p]. (3)

J. Appl. Phys., Vol. 78, No. 2, 15 July 1995 Sergei A. Sergeenkov 1115

Here jCO obeys the usual Ambegoakar-Baratoff expression for Josephson critical current,39 the exponential dependence of j&(x)] accounts for the probability of Josephson tunneling on the local TB of thickness t(x), and 5, = h u F/ rrkBTc is the decay length of the Josephson, contact. Taking into account the known property of the Dirac delta function

where p’(x)=dp/dx and x0(p) is the solution of the equation p(xo> =p, Eqs. (l)-(3) g ive for the dislocation-induced single-junction critical current density

j$p,H) =j,WJbin

where

~,(P~o)=j,o ew[-G#&l

LlP’CXo>l

and

f+ ” 27hdP>.

We neglected a possible (via t[x,(p)]) p dependence of 1 in Bq. (6), since as usual it is assumed that XL4t(p).

6. Josephson-junction arrays

To model a real situation of an inhomogeneous (due to the intragrain granularity) yet isotropic single crystal, let us consider a random network of dislocation-induced Josephson junctions (JJ). Assuming, for simplicity, that JJ are randomly distributed according to the exponential law P(Z) = ( l/Zo)exp( - III,), with Ia being an average “sand- wich” thickness, Eq. (4) results in the maximum (with c = T/ 2) critical current density for a “granular” superconductor,43

~,(P,H) = I

mdZ P(Z)j;(p,H)= l”$;;;;2, (7) 0 P

where HP = qbo/2 n-Eoxo( p). As is seen, Eq. (7). at least formally, describes the ordi-

nary decrease of the critical current density with H; but ac- tually this is not the case in view of the implicit dependence of a characteristic Josephson field HP on the number of dislocations and on external sources (wo, see below) affecting the distribution of defects inside a crystal.

111. APPLICATIONS

A. Screw dislocated thin films

Supposing, for simplicity, the TB as the only weak-link sources, we are’able to use for their treatment the dislocation model of elastic twinning.g~41 According to this model, the distribution of dislocations with density p(x) along the elastic thin TB is defined by the equilibrium condition under the influence of the external forces o(x),~‘~’

- p(x’>= w(x),

where

(8)

Here, L is the twinning length, p is the shear modulus of a dislocation, b is the magnitude of the total Burgers vector, and f(x) is a linear force affecting the distribution of dislocations within the TB.

Besides, p(x) is supposed to obey the normalization condition,44

J 1,

dx p(x)=b. 0

(10)

To make our consideration more definitive, let us discuss a concrete profile of the TB, i.e., solve the integral IQ. (8). If the dislocations are constrained only by homogeneous external forces [with o(x) 3 a,,], the dislocation density distribution inside the TB can be approximated by the expression (an example of inhomogeneous external forces, resulting in a “self-tuned” Josephson effect, is considered in Ref. 40)

( i l/2

P(X)==Po I- ; ) pp:.

The above expression is valid for a,<x<l, where a0 is of the order of interatomic spacing. To describe correctly the vicinity of the TB origin (x<ao) a more accurate treatment of Eq. (8) is needkd.g In view of Eqs. (5) and (lo), distribution (11) results in the following explicit p dependencies of the zero-field critical current density,

&L~Pm=2j,o(,~)~np[ -( ;)( g3]?

and characteristic Josephson field,

where Ho = cpo/3 rrb I,. In view of Eq. (12), j,(p,O) increases with p for p<pm,

has a maximum at p= pm~po~n/3b, and then expo- nentially decreases for p>pm. Using HTS parameters, uF=2X105 m/s, T,s90 K, and bzl nm, we get pm= 1.2~~. On the other hand, according to the validity of distribution (11 j and taking into account that usually9 ao/L<10-2, we tind that Eqs. (12) and (13) are valid for OGp/po<l only. It means that up to the highest admissible dislocation density (i.e., up to p/po=l) we are always in the linear region (with p < p,). Hence, in what follows the insig- nificant (in this region) exponential dependence of j,(p,O) will be neglected. Figure 2 shows the behavior of the dislocation-induced critical current density [calculated according to Eqs. (7), (12), and (13)] j,(p,H)/j,(p,O) versus applied magnetic field H/Ho for various number of dislocations p/pa. At least in a qualitative agreement with the experimental observations in screw-dislocated single crystals,17 Eqs. (7), (12), and (13) suggest a linear (for small p) increase of the critical current density as well as a rather strong weak- ening of its field dependence (via the increase of the characteristic Josephson field HP) with the number of defects. It is worthwhile to mention that in view of Eqs. (10) and (11) the

1116 J. Appl. Phys., Vol. 78, No. 2, 15 July 1995 Sergei A. Sergeenkov

5 0.8 .

CL ~~ 0.6 b ; 0.4

l

,” 2 0.2

1 0 I

0 2 4 6 8 10

H/H,

B. Fishtail anomaly 1. Stoichiometric melt-textured crystals

FIG. 2. Field dependence of the normalized dislocation-induced critical current density, calculated according to Eqs. (7), (12), and (13), for various values of the number of dislocations: (1) p=O.lpo; (2) p=0.5p0; and (3) p=o.sp, .

Let us apply our scenario to discuss a possible origin of the so-called fishtail anomaly of the critical current density versus applied magnetic field in stoichiometric melt-textured crystals possessing a large number of dislocations.25’26

length of the JJ contact L will be altered by external forces ,~a, namely: L= 3b/2po. In turn, due to the equilibrium equation (9), it means that within our scenario the number of dislocations p does not depend on external forces p. and is considered as a constant parameter. To estimate pa, let us consider the pinning mechanism due to the screw dislocations. According to McElfresh et a1.,“5 the strain tield of a dislocation can provide a mechanism by which the superconducting order parameter can be reduced, making it a possible site for core pinning. The elastic pinning mechanism could also be responsible for the large pinning forces associated with dislocation defects. There are two types of elastic pinning mechanisms possible, a first-order (parelastic) interaction and a second-order (dielastic) interaction. In the case of a screw plane defect, the parelastic pinning can be shown to be negIigible;s5 however, the dielastic interaction comes about because the self-energy of a defect depends on the elastic constants of the material in which it forms. Since the crystalline material in a vortex core is stiffer, there is a higher energy bound up in the defect. For a tetragonal system with a screw dislocation, the energy density due to the defect strain is ii),&&, where ,u is the shear modulus and Ed is the shear strain. For a screw plane, q4=blr, where r is the distance from the center of the defect and b is the Burgers vector of the defect. The interaction energy density between the vortex and the screw plane is just ($)A,&&, where A,u is the difference in shear modulus between superconducting and normal regions. For a vortex a distance r away from the defect, the interaction energy (per unit length of dislocation) re?ds’5 ‘

&(r)=~A&&r~;). (14)

Setting r=&, we get finally for the elastic force affecting the distribution of dislocations inside a crystal,

As is knownF41 any external force leading to the redistribution of dislocations in the crystal changes the volume content of the martensite (superconducting) phase in HTS sample. Under an influence of external magnetic field, for instance, the dislocation will be affected by additional stress fields (T,, due to the differences between the magnetic inductions of two phases. When one of these phases falls into the superconducting state, its magnetic properties will change drastically in the external magnetic field. If B, and B, are the magnetic inductions of the defect (normal) and superconducting regions, respectively, then such a dislocation will be influenced, in nonzero magnetic field H, by a force9,40,4’,47-49

fm=bcm =$ ABH.

Here dB=B,-B,$, B,=,u”H, B,=,u,,H+4~M(H), and h=4rrX10B7 N/A’. To choose an appropriate form of the vortex lattice magnetization M(H) to describe the magnetic anomalies at the =l T region, we note that according to our scenario (see also Ref. 25), the strained regions near dislocation cores result in a lower Hz2(T) as compared to the YBCO matrix Hc2( T). Indeed,* the order parameter near the TB Ars is supposed to be reduced as compared to its bulk value As, as follows: ATB=AO(bl&), where &, is the decay length of the TB-induced Josephson contact. In turn, this leads to a similar depression of the superconducting coherence length .&==&,(&Jb) and the upper critical field

fel=aP( &) 3 (15)

which in view of Eq. (11) results in p. = 2f,,l,ub = (bl&,) (A,uu/p). Taking the recently reported45 values for YBCO of p=lOt” N/m’, Ap=10W5p, and b= &-,= 1 nm, we find that f,tz5X 1 OF5 N/m and pas 1 OW5.

where H,,(T>=Hc2(0)( 1 -T’/Tz), H,,(O)~~,/~~, and &=fiu F/rAO. Since for YBCO crystals,= &sl nm, ~0H,2(0)=140 T, and T,=90 K, we get for T=70 K that Hz2(T = 70 K)z2 T. Thus, with a good accuracy we can use the high-field limit for vortex lattice magnetization (valid for H =G H,*): 47&?(H) = - yo(H,*2 - H)/~K”~,

where K* = Ko(b/&,)2 is the Ginzburg-Landau parameter near the TB [for YBCO ~~~90 (Ref. 23)]. According to Eqs. (7), (12), and (13), dislocation-induced critical current density j,(p,H), driven by the magnetic-type external force (16), will exhibit a maximum at some field H*(p) which is defined as the solution of the balance equation p = po(H*), where

J. Appt. Phys., Vol. 78, No. 2, 15 July 1995 Sergei A. Sergeenkov

It is interesting to mention that a pinning force per unit length fp =2 X 10 -’ N/m (which is of the same order of magnitude as the above-considered elastic force f,t acting upon dislocations), calculated in the single-vortex limit, .fp=jcOfjl corresponds to a critical current density of j, = 10 A/m’, in reasonable agreement with the jcO found in dislocated YBCO thin film~.‘~,~~*~~

(16)

1117

In view of the above-given explicit form of p”(H), the balance equation produces the following [dislocation- dependent) threshold field:

where Hd = dGP-%-&. The existence of distinct magnetic properties along both

sides of the TB leads, in nonzero magnetic field, to the grow- ing of this twin through the crystal. This process can be stopped by the friction of dislocation via the crystalline lattice or by generation of an excess nonequilibrium concentration of (oxygen) vacancies (see the following section). When the applied magnetic fieId reaches the threshold field H*(p), the moving dislocation is bIocked, thus leading to the adjust- ment of the defect structure to the penetrating vortex lines and, as a result, to the enhancement of the pinning forces and the recovery of the critical current density. When magnetic field overcomes this barrier, the pinning becomes less effective, and a rather smooth falloff of the critical current is restored. Using B * = /.L~H” s 1 T for the typical field region where anomalous fishtail behavior of critical currents has been observed,z5V26 we can estimate a minimal number of dislocations needed to produce a peak effect in stoichiometric melt-textured samples. Namely, taking bs 1 nm, @lO” N/m’, ~oH~2(0)=5 T, and K*=& we get ,uoHd= 1 T and

n,,==4ptilmb2~2X 10” cmM2

which is only 1% of the maximum number of dislocations found in the melt-textured materials.25’26 Finally, using the above parameters, we can estimate the magnitude of the magnetic force f, . The result for Bz 1 T is f,=~2 X 10m5 N/m which is quite comparable with the elementary pinning force density fP and elastic force f,r .

2. Deoxygenated single crystals

To apply the above-considered scenario for the description of the anomalous magnetic- behavior observed in the oxygen-deficient single crystals, we should take into account the existence of a rather important force of inelastic origin (so-called osmotic force f,) which, together with the above- mentioned magnetic force f, , will define the ultimate defect structure of these crystals. The origin of this force in deoxygenated crystals is due to an excess nonequilibrium concentration of oxygen vacancies CB co. It is important to point out that unlike the ordinary oxygen diffusion which is knownso to be extremely slow in YBCO crystals, the osmotic (pumping) mechanism can play a crucial role in our scenario. The osmotic force in (oxygen-deficient) dislocated crystals is strictly related to the chemical potential of (oxygen) vacancies. Indeed, when in such a crystal there exists a nonequilibrium concentration of vacancies (c # co), dislocation is moved for atomic distance (w b) by adding excess vacancies to the extraplane edge and as a result the force f, produces the work bf D . This work is equal to the chemical potential of (#l/b) added vacancies, that is bf,=&b, where ,z, = - k,T ln( c/c,) is the chemical potential of a single va-

cancy. Therefore, in the oxygen-deficient case the dislocation will be affected by the osmotic force,47-4gV51,52

~BT kBT j-,=-,-z ln(l-8)--p- 3.

Here we have introduced the oxygen deficiency parameter S which is related to the oxygen concentration c as follows: c/c, = 1 - 8, and assumed that & 1.

Let us consider influence of two competitive forces, f, ad j-0, on the magnetic-field critical current behavior. In accordance with Eqs. (8), (ll), and (16), the external-force- dependent part of the dislocation density po, reads (for &l)

where

HB3 ( 4k;;;;2) 1’2a (3,)

Thus, the nonzero oxygen deficiency (which is responsible for the sample nonstoichiometry) results in an intrinsic characteristic field Hs, which reflects a subtle baIance between extended (dislocations) and point (oxygen vacancies) defects in a given sample. In view of Eqs. (7), (12), and (13), the above po( 8,H) brings about the following explicit dependence of the dislocation-induced critical current density:

+(3’[4&-i’1’)‘. (20

Here we have taken into account that according to the observationsZ3 for small 8, j,( s>=j,,(O)( 1 - S,. Figure 3 illustrates the evolution of j,(H)lj,(O) versus applied magnetic field HIH, with increasing the oxygen deficiency 6 [with p=po(S,O)]. As is seen, below some threshold concentration of oxygen defects S, , the critical current density ex- hibits a peaklike behavior. The peak is shifted to lower fields with increasing the oxygen deficiency and disappears at 8= a,, in agreement with the observations. Due to Eqs. (19)-(21), j,(H) passes through the maximum above some threshold field H”(S,T), which is defined via the balance equation p=po( &H*). Taking into account the explicit form of po(S,H) [see Eqs. (19) and (20)], the threshold field reads

or alternativeIy

H*j&T)=(;)H:,j I--[ +j2( 1-$)]“2], (3)

where

,ub’p 8,s - t.cb3p 2kBT, To=---- 2kgS’

and Hd is defined in Eq. (17).

CW


0.8

0 2 4 6 8 10 0 2 4 6 8 10

H/Ho H/Ho

0.6

0.4

0.2

n

-0 2 4 6 8 10 0 2 4 6 8 10

H/H, H/H,

0 2 4 6 8 10 H/Ho

“0 2 4 6 8 10 H/Ho

0.8

0.6

0.4

0.2

0

FIG. 3. The behavior of the normalized critical current density vs reduced magnetic field, calculated according to Eq. (21), for various values of the oxygen deficiency parameter: (a) S=O.OlS,; (b) S=O.lS,; (c) ~V0.36,; (d) 6=0.56,; (e) 6=0.75,; (f) S=S,.

Thus, in addition to the strong 6 dependence [see Eq. rather strong correlation between dislocations and oxygen (22)], Eq. (23) predicts a decrease of the peak-effect thresh- defects is seen through the p dependence of the critical oxy- old field H*(S,Z’) with increasing the temperature as gen deficiency parameter SC [see Eq. (24)]. The latter param- well.‘g-u Taking b= 1 nm, c’?,-O.23,” poHd=l T, and eter has a simple physical meaning. Namely,‘“‘*’ for S>S, ,uoHzz (T=70 K)=2 T, for S=O.O5 and T=70 K we obtain oxygen-rich superconducting grains are separated by the estimate of the threshold field B” = ,uoH* s 1 T, which is oxygen-poor insulating boundaries so that there is no super- very close to the fields where the anomalous phenomena in conducting path through the sample. Notice also [by compar- YBa&u307-d single crystals have been observed.“’ Accord- ing the low-field regions in Figs. 3(a) and 3(f)] that at a given ing to Eq. (18), the above estimates predict f,-5 X 10m5 number of dislocations, increase of oxygen defects leads to a N/m for the osmotic force magnitude. Notice that in contrast more flat behavior of the critical current at small fields indi- to the melt-textured case, in oxygen-depleted single-crystal cating that point defects are the major pinning centers in this dislocations play an auxiliary (as compared to the point de- region.lg In a similar way, To plays a role of the phase- fects and oxygen vacancies) yet important role, providing a locking temperature above which the superconductivity be- background for oxygen defect redistribution inside a crystal tween oxygen-rich grains is suppressed and the fishtail phe- and thus adjusting the pinning ability of the material. A nomenon disappears.


C. Irradiated single crystals

There are two possibilities to adapt our scenario for a qualitative description of the anomalous magnetic phenomena observed in irradiated HTS single crystals:“-37 We can either consider the well-aligned tracks of damaged material, produced by particle irradiation, as potential sources of intragrain weak links (provided that the track’s diameter is of the order of the coherence length”) or, alternatively, exploit the well-known53P54 fact that irradiation produces a large density of extended defects (in particular dislocations) in these materials. Recent experiments have revealed that there is some threshold magnetic field H*(q), which is almost linearly increases with the fluence Q and above which a substantial increase of the critical current density j,(@,H) has been observed.“9730 Besides, it was found that the magnitude of fluence Q*(H), which optimizes j,(@,H), increases almost linearly with the applied field. As it was observed,31 the strain and structural disorder of the amorphous region propa- gates into the crystal Iattice in a direction normal to the ion track. This means that, for an isolated amorphous track, a radially symmetric displacement field ~(r-)=E,ff(R~Ir-) (where R is the radius of the amorphous track and, for YBCO, I&.+0.02) occurs in the matrix around the ion track (cf. the appearance of a displacement field around a screw dislocation). According to Weaver et al.,54 irradiation of fluence cf, induces the network of dislocations of density

n(d?)=n(O)-tn,[l-exp(-a@)],

where n(O) is the dislocation density of a virgin (preirradiated) sample, n, the maximum density after irradiation, and a is the so-called displacement cross section which is related to the number of displaced atoms per particle per unit length. Assuming that &Ml (which is usually the case for real experiments), in the dimensionless units the above equation reads p(@)~p(O)+&D, where p(@)=n(@)ln, and p(O)-n(O)ln,. Let us consider the magnetic-field behavior of the critical current density, j,( @,H), upon irradiation. Ac- cording to our scenario for weak-link-mediated critical current enhancement in defected samples, the irradiation- induced behavior of j,(@,H) will show the same peculiarities we have discussed earlier considering the tish- tail anomaly in melt-textured and deoxygenated crystals (see Sec. III B). Indeed, in view of the fluence dependence of the characteristic field H, [through p(Q),), see Bq. (13)], it is easy to verify that j,(@, H) will have a maximum at some threshold field, H*( a,@), which is defined [by analogy with the threshold field for oxygen-deficient sample, see Eqs. (22) and (23)] as the solution of the bafance equation p(Q) =po( S,H*). Taking into account the definitions of po(G,H), given by Eqs. (19) and (20), and p(a), the above balance equation results in the following threshold field:

H*(S,O)=(;)H:;{ I-[ l-(sj2

x( l+ @*;,o))]1’2}.

where

1120 J. Appl. Phys., Vol. 78, No. 2, 15 July 1995

(25)

3.0

z 2.5 0' -w 2.0 .-

. 1.5 2 s‘ 1.0 -ln -- 0.5

0.0

17.5

;; 15.0

0' 12.5 -w 7 10.0

I 7.5 e' zw 5-O

2.5

0.0

80

0 0' 60 -VI .-

L 40 2 e* ZJI 20

0

,i 0 2 4 6 8 10 12 14

a/o,

(b)

1 ta --.-L-J-d 0 2 4 6 8 10 12 14

a/o0

O 2 4 6 8 10 12 14 m&

ETG. 4. The behavior of the normalized critical current density vs reduced irradiation fluence, calculated according to Eq. (28), for various values of the reduced magnetic field: (a) H/H,= 1; (b) HIH,-,=2; (c) H/H,=3.

~*(6,0)=(6,-s)qf, a$=%.

Here Hd( s) = Hddm with Hd and 8, governed by Eqs. (21) and (24) with p=;p(O).

Alternatively, we can introduce an optimum value of the fluence, cP*( 8, H), as the solution of the equation ~(@*)=poW,s>,

@*(S,H)=Q*(S,O) (27)

Turning to the discussion of the above-obtained model predictions, we notice that they cover practically all the pe-

Sergei A. Sergeenkov

culiarities observed in irradiated single crystals. Indeed, in view of Eqs. (7), (12). and (13), the irradiation-induced critical current density reads

x[ +$%)2rl-*- (28)

Here we have taken into account the possibility of deteriora- tion of the preirradiated weak links upon irradiation. Within the same accuracy (@Ml), experiments5’ revealed that jco(S) = j,*(O)(l - @). Let us consider first the case of nearly fully oxygenated preirradiated samples (with 6~0). Figure 4 shows the behavior of j,(@,H)/j,(O,O) versus irradiation fluence @‘/a0 [with @o=p(0)la;l for different values of the applied magnetic tield H/H,, calculated according to Eqs. (28) and (19). Notice a tremendous increase of the critical current density with fluence (especially at high magnetic fields where the irradiation-induced defect structure is supposed to match optimally the vortex lattice structure). The magnitude of the peak a*(H) is seen to shift to higher fluences with increasing the field, in agreement with observations.29*30 In turn, Fig. 5 depicts the calculated dependence of the critical current density j,(@,H)/j,(O,H) upon applied magnetic field H/Ho for various values of the irradiation fluence @/a,. As is seen, a pronounced peak appears which is shifted to higher magnetic fields with increasing the irradiation fuence. And this model prediction is also in agreement with the observations. Let us briefly consider the case when the preirradiated sample is in a highly oxygen- deficient state (with S- 8,). Since @*(S,O) = (3, - sp;, it means that a lesser value of the fluence is re- quired to be applied to the deoxygenated sample to get the same (as for fully oxygenated sample) optimum properties of the critical current density in irradiated sample. This conclusion is in agreement with some recent experimental observations.31*37 Namely, in oxygenated samples the irradiation-induced damage was found3’ to be much less than that in oxygen-deficient samples.

In conclusion, a possible scenario for a qualitative description of the critical-current-density peculiarities in dislocated, deoxygenated, and irradiated isotropic YBCO-like superconductors was proposed based on the intrinsic Josephson effect which is supposed to be active in these materials. The model is based on the existence of various competitive forces affecting a rather delicate balance between extended (dislocations) and point (oxygen vacancies) defects inside a crystal. The scenario incorporates the idea of a quite strong correlation between the intragrain weak links and vortex pinning, and implies that practically any treatment of the superconducting sample (such as heat treatment,

’ oxygenation/deoxygenation process, particle irradiation, ap- plication of high magnetic fields, etc.) will inevitably result in the rearrangement of the pretreated defect strncture of the material to optimize its critical parameters. Regarding the applicability of the scenario proposed to the description of some peculiarities observed in highly anisotropic Bi- and Tl-based materials (see, e.g., Ref. 56), it should be empha-

1.8

1.6

0 2 4 6 8 10 H/H,

25 I 0‘ 20

3 . 15

z 2 10

-u, *- 5

0 0 2 4 6 8 10

H/H,

0 2 4 6 a 10 H/H,

F’IG. 5. The behavior of the normalized critical current density vs reduced magnetic field, calculated according to Eq. i28), for various values of the reduced irradiation fluence: (aj Q/Q,= 1; (b) @X+,=5; (c) @/@,= 10.

sized that the so-called Lawrence-Doniach57 or interlayer tunneling58 models [rather than an isotropic single-junction Josephson model (1) used in the present paper to discuss YBCO-like materials] are more appropriate for these layered systems, which should be described as a set of discrete superconducting layers whose order parameters are coupled by Josephson interaction. Besides, to account for a strong angu- lar dependence of pinning force density and the two- dimensional (pancakelike) vortex structure in these anisotropic materials, the TB misorientation angle dependence of the dislocation density distribution along the c axis has to be taken into account.


ACKNOWLEDGMENTS

Useful discussions with Marcel Ausloos, John Clem, Guy Deutscher, Ted Geballe, Ken Gray, Jorge Jose, David Larbalestier, Pranc Nabarro, Terry Orlando, James Thomp- son, and Michael Tinkham are highly appreciated. Part of this work has been financially supported through the Impulse Program on High Temperature Superconductors of Belgium Federal Services for Scientific, Technological and Cultural (SSTC) Affairs under Contract No. SUIOUO13.

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Pinning via intrinsic weak links: A possible scenario and applications

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