Parinov I.A. Microstructure and Properties of High-Temperature Superconductors

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10.3 Vortex Structures and Current Lines in HTSC with Defects 443

10.3 Vortex Structures and Current Lines

in HTSC with Defects

The observed inﬂuences of defects on ﬂux pattern lead to a classiﬁcation of

the defects into two groups [557]:

(1) The “extended” defects, formed by a weakly superconducting material

(typical size between some micrometers up to millimeters), which force

the currents to ﬂow along them, thus leading to a facilitation of ﬂux

penetration along the defects (these defects form channels for easy ﬂux

penetration into the sample).

(2) The small obstacles of non-superconducting material (typical size of

micrometers or even less), where the currents can ﬂow around them. There

is no enhancement of the ﬂux penetration, but a parabolic discontinuity

line of the current is formed.

In Figs. 10.3 and 10.4, schematic current patterns (the critical currents are

shown by the arrows) for both defect classes are presented. The characteristic

feature of the ﬁrst type (see Fig. 10.3) is an extended defect line, where ﬂux

penetrates the sample preferentially. This leads to the formation of a ﬁnger-

like ﬂux pattern. Flux density gradients always are pointed away from the

defect line and at the end of the ﬂux front the currents are forced to a U-turn.

At the beginning of the defect line, the currents form a broad turn towards

the defect line as the currents always ﬂow along the sample edges [949] and

also along the defect lines.

Figure 10.4 shows the current ﬂow around a small obstacle of the second

type, here symbolized as a round object. When the currents come close to

this defect, they are forced to a slight turn and now follow the circumference

of the obstacle. All the points, where the current lines turn around, lie on

Pore

Fig. 10.3. A defect of linear type, symbolized by the dashed curve (current ﬂow

around cylindrical pore), leads to increasing ﬂux penetration along it [120]. There-

fore, the current ﬂow is found by the ﬂux density gradients. In contrast to the defect,

shown in Fig. 10.4, there is no bottleneck between the normal ﬂux front and the ﬂux

penetration along the defect line

444 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

(a) (b)

(c)

Fig. 10.4. A schematic drawing of the current distribution in the vicinity of a

cylindrical cavity near the superconductor edge, located at the bottom of the ﬁgure.

The current density spatially constant is shown by equidistant current lines (dashed

curves). When the magnetic ﬂux, penetrating from the sample edge, reaches the

cavity, the current lines form sharp bends in order to ﬂow around the hole on arcs

of concentric circles. These bending points lie on a parabola, but the penetrating

ﬂux to the inside of the parabola always passes the defect, forming a “bottleneck”

for the vortices [557]

a parabola, which can be seen in the ﬂux distributions. The defect forms a

“bottleneck” for the ﬂux penetration that may trigger ﬂux jumps [949]. If the

small obstacle has a rectangular shape, the beginning of the parabola will be

linear (forming an angle of 45

◦

[949]), which in some distance from the defect

again becomes as in Fig. 10.4.

The suﬃciently large density of the defects localizes an area of interaction

of two adjacent defects (see Fig. 10.5). At small ﬁelds, the current ﬂow is sim-

ilar to Fig. 10.3. During ﬂux penetration in an increasing external magnetic

ﬁeld the ﬂux fronts meet each other in some moment. This leads to complete

(a) (b)

(c)

Fig. 10.5. At high defect density, two adjacent defects can be considered in sample

(a). The ﬂux penetration and current ﬂow are identical to those of Fig. 10.3 as long

as the external magnetic ﬁeld is small (b). With rising ﬁeld, the vortices penetrating

the sample from both sides may merge together (c). This causes the current ﬂow

along both defect lines, leading to formation of a current loop [557]

10.3 Vortex Structures and Current Lines in HTSC with Defects 445

change of the current pattern, because the currents can now ﬂow along both

defect lines and form new smaller current loops. In this case, the length scale

of the currents changes and the sample is divided magnetically into smaller

pieces. If the external magnetic ﬁeld is raised further, the parts of the sample

will behave completely independent from each other. This leads to magneti-

cally induced granularity of the sample, with the defect lines acting as weak

links at higher magnetic ﬁelds.

Figure 10.6 presents schematically a crystal with many defects. At mod-

erate ﬁelds, where Meissner’s phase still resides in the sample, the current

ﬂow is the same as that shown in Fig. 10.3, and many “ﬂux ﬁngers” are

formed along the defect lines. With suﬃciently large external magnetic ﬁeld,

many current sub-loops are formed. For this sample, the critical state model,

assuming a homogeneous ﬂux distribution is not used. One will severely un-

derestimate the critical current density of such sample, if the critical current is

(a)

(b)

(c)

Fig. 10.6. A schematic representation of many defect lines in a rectangular sample

(a). In moderate external ﬁelds (b), the ﬂux penetrates easily along the defect lines

and only a small ﬂux penetration along the sample edges occurred. The critical cur-

rents (shown by dashed curves) ﬂow around the defects. If the external magnetic ﬁeld

is suﬃciently large (c), the ﬂux penetration along the defect lines merges together

and current loops are formed, both for the critical currents and for the eventually

remaining shielding currents [557]

446 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

determined from its magnetic moment (with the tacit assumption of a single

current loop, shielding the whole sample). It should be noted that, once a

ﬁeld or current distribution is disturbed, the pattern stays like this even up

to high external magnetic ﬁelds.

The observed various ﬂux distributions permit to classify the defects in

HTSC through defect inﬂuence on the ﬂux patterns [557]. In all cases, in-

trinsic (crystallographic) defects should be distinguished from extrinsic ones,

which are caused by external inﬂuences (i.e., by cracks, scratches, cutting and

etching).

(1) To the extended defects of the ﬁrst type belong the intrinsic defects, such

as twin boundaries [210, 211, 934, 1087, 1088, 1120] and the tilt bound-

aries found in Bi-2212 single crystals, caused by intergrowths of Bi-2223

phase [560]. Moreover, extrinsic defects, such as cracks in the sample sur-

face, which are typical defects for most Bi-2212 single crystals and of

melt-processed YBCO sample [403, 743], and also the defects caused by

inhomogeneous ﬁlm growth on substrates, containing scratches, belong to

this type. However, the growth steps, observed in many single crystals,

do not inﬂuence the current patterns [560]. Grain boundaries also belong

to the extended defects, but they act as channels for easy ﬂux penetra-

tion even at very small external magnetic ﬁelds. This separates a sample

magnetically into smaller pieces.

(2) Above-described small obstacles of the second type in HTSC are caused by

small ﬂux droplets, observed in thin ﬁlms and single crystals, by islands of

the a-axis-oriented growth in the c-axis-oriented ﬁlms, by small particles

of foreign phases and by extrinsic defects, created owing to irregular cut

or etching of the sample.

The presented classiﬁcation characterizes all defects found in HTSCs.

However, this approach is only valid for nearly homogeneous, monocrystalline

samples, where a homogeneous ﬂux front will appear in the ideal case. If an in-

ternal granularity is presented in the sample, the ﬂux distribution is disturbed

by the immediate ﬂux penetration along the grain boundaries. Therefore, the

analysis can only be fulﬁlled for individual grains.

Then, the ﬂux pinning in superconductors with high critical current den-

sity is also found by a dense network of planar crystalline defects. In this

case, there are two essential features of HTSCs, namely (i) the most eﬀective

pinning can be caused by the dense network of planar defects parallel to the

ﬂux lines, however (ii) unlike the case of arbitrary distributed point pins, the

network of planar defects can block or divert the macroscopic current ﬂow,

if the tunneling superconducting current density (j

) through these defects is

smaller than J

, determined by ﬂux pinning. For J

, the pinning struc-

ture can lead to the magnetic granularity, which manifests itself in a drop of

the transport J

due to the appearance of closed current loops within macro-

scopic crystalline grains, where densities of circulating magnetization currents

become larger than J

[171, 557].

10.3 Vortex Structures and Current Lines in HTSC with Defects 447

Note also the importance of geometry of the planar defect network. For

example, in twin domains with various orientations of the crystallographic

axes, the planar defects can divert the local current ﬂow, limiting the macro-

scopic value of J

owing to reduction of the current-carrying cross-section area

[288, 593, 622]. On the other hand, in a random network of planar defects with

the density of the pinning centers, exceeding the percolation threshold, the

supercurrent must cross several defects, which would increase J

owing to

ﬂux pinning. This arbitrary defect network can have a complicated topologi-

cal structure, in which only a small fraction of the defects belongs to the path

of percolation [1013]. Generally, both regimes, in which the planar defects act

as pinning centers or block (or divert) the current ﬂow co-exist. Their relative

contributions depend on both j

and the geometry of the pinning network.

The theoretical study of ﬂux pinning due to a network of planar high-j

crys-

talline defects, which are modeled by the corresponding Josephson contacts

is carried out in [366]. A solution of equations of the non-local Josephson

electrodynamics [362, 371] for a vortex parallel to the planar defect leads to

the calculation of the magnetic ﬁeld distribution and the transversal pinning

force between the vortex and defect. The longitudinal pinning force of vortices

along the defect is determined by both their magnetic interaction with pinned

transgranular ﬂuxons and local inhomogeneities of the defect. It is shown that

the longitudinal component is much smaller than the transversal one, leading

to the preferential ﬂux motion along the paths of percolation formed by planar

defects.

In highly anisotropic-layered superconductors or artiﬁcial superlattices,

the interlayer superconducting coupling can be so weak that the coherence

length in the direction perpendicular to the layers (ξ

) becomes smaller than

the interlayer spacing (s). In this case, the discreteness of the superconduc-

tor at the atomic level directly deﬁnes the structure of the vortex core [103,

1121]. There are several models, taking into account the layered structure of

superconductor, for example Lawrence–Doniach’s model depicting a stack of

thin superconducting layers connected by Josephson interaction [607], or the

models of S-N-S superlattices consisting of alternating superconducting (S)

and normal (N) layers [159, 576, 1121]. An important feature of these models

is that, due to weak Josephson interlayer coupling, the maximum supercurrent

density (j

) between the layers is much smaller than the interlayer de-pairing

current density (j

). As a result, the maximum current density that can be

locally generated by a vortex parallel to the layers is limited by j

.Dueto

the short coherence length (ξ

<s) in layered HTSCs planar defects paral-

lel to the ab-planes can signiﬁcantly reduce interlayer coupling, limiting the

current ﬂow along the c-axis and rising the magnetic ﬁeld penetration along

the defects. In this case, the methods of non-local Josephson electrodynamics

can be used to study both static and moving vortex (including non-linear area

of Josephson core) at planar defect in layered HTSC and for the description

of their structure [365].

448 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

The eﬀect of macroscopic defects on localized magnetic ﬂux (ﬂuxons) and

critical force of de-pinning is found by the defect size and type, and also by

its disposition at grain boundary (or at Josephson junction). The case of the

defect width of order of the Josephson penetration depth is considered in

[1014]. Using the sine-Gordon model for defective Josephson junction, critical

current as a function of magnetic ﬁeld in the case of asymmetrically disposed

defect is calculated. Moreover, the investigation of the interaction between

ﬂuxons and defects deﬁnes coercive forces of pinning or de-pinning a ﬂuxon

from a defect. Consideration, using an analysis of the diﬀerent mode stability

in dependence on the defect disposition at zero magnetic ﬁeld, ﬁnds the bounds

of these modes due to two factors, namely (i) the instability at the junction

boundaries away from the defect and (ii) the instability due to the ﬂuxon

trapping or de-trapping by the defect. When the defect localizes near one of

the junction edges, both criteria contribute (in deﬁnition) to the instability.

In general case, there is coupling between defects and junction edges (surface

defects), especially in the case of the numerous defects.

10.4 Non-Linear Current in Superconductors

with Obstacles

Following [367], consider features of non-linear current in HTSC systems with

existence of obstacles. Macroscopic electrodynamics of type II superconduc-

tors in the mixed state is found by the pinning and thermally activated creep of

vortex structures, increasing a weakly dissipative critical current, irreversible

magnetization and slow current relaxation (ﬂux creep). These phenomena

manifest themselves on length scales much longer than the Larkin’s pinning

correlation length (L

), on which the critical state is formed [71]. On this

macroscopic scale (L

), the material property, deﬁning the behavior of

superconductor in electromagnetic ﬁelds, is the next non-linear local relation

between the electric ﬁeld, E, and current density, J:

J =(E/E)J(E) , (10.2)

here E(r,t)andJ(r,t) correspond to the macroscopic electric ﬁeld and current

density, averaged over all relevant intrinsic scales of pinned vortex structure.

Consider an isotropic E–J dependence stated by (10.2), that, in particular,

model the nearly two-dimensional current ﬂow in the ab-plane of layered

HTSCs. Equation (10.2) combined with the Maxwell equations

∂

B = −∇ × E ; ∇×H = J(E) , (10.3)

permit to calculate the evolution of heterogeneous distributions E(r,t)and

B(r,t) and, thus, to describe macroscopic magnetic, transport and relaxation

phenomena in superconductors.

10.4 Non-Linear Current in Superconductors with Obstacles 449

Fig. 10.7. E(J) characteristic of a type II superconductor

A type II superconductor in the mixed state demonstrates a highly non-

linear E(J) dependence below a critical current density of J

, which sepa-

rates regimes of magnetic ﬂux ﬂow (at J>J

) and its creep (at J<J

)

(see Fig. 10.7). The critical behavior of E(J)atJ<J

is determined by

thermally activated vortex creep as

E(J)=E

exp



−

U(J, T, B)



, (10.4)

where U (J, T, B) is the activation barrier, depending on the current density,

J, temperature, T , and magnetic induction, B. Here, E

is the conditional

criterion for electric ﬁeld, which deﬁnes corresponding value of J

through

the relation, U (J

,T,B)=0.Forexample,thevortex glass/collective creep

models [71] determines a dependence, U = U

[(J

/J)

− 1], for small values

of J<<J

, which is observed in transport and magnetization measurements

in HTSCs [1173]. Similar, though less singular logarithmic dependence U =

ln (J

/J) corresponds to power-law E–J dependencies, E = E

(J/J

)

with n(T,B)=U

/T ∼ 3 − 30, to being a well approximation, especially for

layered BSCCO [767, 1103, 1204].

When the distribution J(r) changes on spatial scales greater than L

superconductor can be considered as a highly non-linear, heterogeneous con-

ductor with a local characteristic of E(J, r). For example, (10.3) describes

non-linear transport current in superconductors with macroscopic obstacles

or current percolation in HTSC polycrystals with grain size >> L

(see

Fig. 10.8). This case is important to understand the current-limiting mecha-

nisms of HTSCs, which in addition to grain boundaries, often contain other

macrodefects, for example, secondary phases, microcracks and areas of local

non-stoichiometry on scales of order 10–1000 μm [597, 598]. These obstacles,

blocking current ﬂow, cause macroscopically heterogeneous distributions of

transport and magnetization currents [115, 832, 854, 861, 950, 951, 952].

In turn, even relatively weak heterogeneity of local current density, J(r),

can lead to exponentially strong variation of the electric ﬁeld, E(r) ∝ J

exp[−U(J)/T ], radically changing global properties of HTSC,

J(E,B,T), ob-

served in experiments. Thus, the behavior of

J(E,B,T) in superconductor can

diﬀer from local characteristics, described by (10.2) and deﬁned by thermally

450 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

Fig. 10.8. Percolation of current ﬂow in HTSC polycrystals. High-angle grain

boundaries, blocking the current ﬂow, are shown by solid lines, but low-angle grain

boundaries, “transparent” for current ﬂow, are denoted by dotted lines

activated vortex dynamics and pinning on mesoscopic scales (at L<L

This diﬀerence increases due to the non-linearity of E(J), makes the eﬀective

current-carrying cross-section dependent on T , B and E [339, 360, 361, 369,

374, 398, 424, 611, 1006, 1018].

As examples of two-dimensional current ﬂows for various cases, which

could be considered as elements of a more general percolation network, pre-

sented in Fig. 10.8, we note proper cases, shown in Fig. 10.9 [367]. These

geometries are standard in experimental studies of resistive states in supercon-

ductors, thin-ﬁlm superconducting electronic circuits and HTSC conductors

for power applications. In all cases presented in Figs. 10.8 and 10.9, the electric

ﬁeld and current density are heterogeneous on macroscales and demonstrate

singularities near the sharp edges and corners. Calculation of the global E–J

characteristic demands the solution of highly non-linear (10.2) and (10.3).

For large values of n, even weak spatial changes of J(r) around planar defects

in Fig. 10.8 lead to power variation of electric ﬁeld, E(r)=E

[J(r)/J

]

(a) (b)

(d)(c)

Fig. 10.9. Characteristic cases of two-dimensional non-linear current ﬂow:

(a) superconducting ﬁlm with strong current-limiting planar defect (microcrack at

high-angle grain boundary), (b) current ﬂow through faceted grain boundary with

alternating segments of various J

values, (c) current injector (or magnetic ﬂux

transformer) and (d) microbridge

10.4 Non-Linear Current in Superconductors with Obstacles 451

Together with singularities of J(r)andE(r) at the edges, this signiﬁcantly

complicate numerical analysis of (10.2) and (10.3).

An approximate method for the deﬁnition of current distribution in super-

conductor is given by Bean’s critical state model [56, 120], which replaces the

real curve E(J) in (10.2) by a step-wise dependence J = EJ

/E for E>0

and E =0forJ<J

(see Fig. 10.7). This model can be considered as a limit

of the power-law dependence, E = E

(J/J

)

for n →∞[85-90]. It enables

analytical solutions for some simple cases, presented in Fig. 10.10. These so-

lutions demonstrate characteristic “discontinuity” lines (d-lines), along which

the current abruptly alters direction. Another feature of this model is that the

defects with ﬁnite size can disturb current in inﬁnite range, as it take places

for current ﬂow around a cylindrical void, in which parabolic d-lines extend to

inﬁnity [120, 953] (see Fig. 10.10b). Both the zero thickness of the d-lines and

the inﬁnite spatial scale of current perturbations are followed from the zero

resistivity for the idealized E–J characteristic in the whole range, 0 <J≤ J

In this case (10.3), describing steady state, become an ill-deﬁned mathemat-

ical problem, which reduces to the only condition of current continuity, div

J =0,whenJ ≤ J

. These conditions can be satisﬁed by many various cur-

rent distributions for a given sample geometry, because the selection of Bean’s

solutions for J(r) is found by corresponding initial conditions.

The inherent hysteresis of the critical state makes the Bean model

inappropriate for calculations of steady-state transport current ﬂow around

planar obstacles, presented in Figs. 10.8 and 10.9, for which the current-

carrying cross-section varies along a superconductor. In this case, the cur-

rent path inevitably breaks into the regions of critical state, J = J

and in

Pore

(a)

(b)

Fig. 10.10. Magnetization current ﬂow, predicted by Bean model: (a)longslabin

parallel magnetic ﬁeld and (b) current ﬂow around cylindrical void [120]. Dashed

lines show parabolic d-lines, on which J sharply changes direction. Current stream-

lines between the d-lines are circular and centered in the void

452 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

sub-critical areas of 0 <J<J

. At the same time, the speciﬁc distribution

of these regions depends on initial conditions, and thus cannot be calculated

by solving the steady-state equations, div J =0,andJ ≤ J

. This unphysi-

cal situation is caused became Bean’s model ignores the necessary condition,

∇ × E = 0, which can only be satisﬁed taking into account the highly non-

linear E–J characteristic stated by (10.2). The non-zero electric ﬁeld, although

being small due to the power-law dependence at J<J

, plays an important

role, because now the solution of (10.2) can be calculated into framework of a

well-deﬁned mathematical problem, for which each non-zero value of J corre-

sponds to a certain value of E. The account of E eliminates the inﬁnite extent

of current perturbations around local inhomogeneities (see Fig. 10.10b), the

zero thickness of the d-lines and multiple solutions of the Bean model, ﬁxing

a unique steady-state transport current distribution for given boundary con-

ditions. However, the account of the non-linear dependence, E(J), depends

on the solution of the next non-linear equation for scalar potential, ϕ [367]:

div



∇ϕ

|∇ϕ|

J (|∇ϕ|)



=0, (10.5)

which is obtained, substituting E = −∇ϕ into (10.2) and then into equation,

div J =0.

An analytical distribution of two-dimensional steady-state transport cur-

rent ﬂow in superconductors, taking into account their nonlinear E(J)char-

acteristics, can be found using a hodograph transformation [363, 367, 370],

initially used to describe compressible gas ﬂow [138, 596, 703, 955]. In [363],

the hodograph method was applied to describe current ﬂow in anisotropic su-

perconductors. Using this method, analytical steady-state solutions of (10.2)

and (10.3) for current ﬂow past a planar defect were obtained for the power-

law E–J characteristics [370]. The essence of this method is the following

[367]. Instead of considering highly non-linear equation for ϕ(r)inthecoor-

dinate space, the variables are altered and ϕ(r)isexpressedasafunctionof

the electric ﬁeld, E:

ϕ(r) → ϕ(E) . (10.6)

The hodograph transformation (10.6) reduces the non-linear (9.5) for ϕ(r)

to a linear one for ϕ(E). In turn, the equation for ϕ(E) can be further reduced

to other well-stated linear partial diﬀerential equations, such as Thomas–

Fermi or London equations, whose known solutions can be used to obtain exact

solutions for two-dimensional current ﬂows in superconductors with defects

[367, 370].

The hodograph method enables to resolve the ambiguities of the Bean’s

model and to study analytically the features of two-dimensional non-linear

transport current ﬂow. In particular, it gives an unique steady-state distribu-

tion, J(r) for a given sample geometry and shows that current ﬂow breaks into

domains with diﬀerent orientations of J, separated by current domain walls

reminiscent of the d-lines of the Bean’s model. However, there are important