Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ
The thin twins theory consistently applied to the dislocation description of twins in the high-temperature superconductor YBa₂Cu₃O₇₋δ(YBCO). Quantitative analysis of the twin shape shows an agreement with theoretical consideration and gives new method to evaluate the twin boundary energy γtw. Fric...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1173332017-05-23T03:03:07Z Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ Boyko, V.S. Свеpхпpоводимость, в том числе высокотемпеpатуpная The thin twins theory consistently applied to the dislocation description of twins in the high-temperature superconductor YBa₂Cu₃O₇₋δ(YBCO). Quantitative analysis of the twin shape shows an agreement with theoretical consideration and gives new method to evaluate the twin boundary energy γtw. Friction stress Sfr acting on a twinning dislocation is also determined. The formation of a twin and twin microstructures are analyzed. The principles of microstructure design by twinning in YBCO crystals for enhanced Jc at high magnetic fields are discussed. 2008 Article Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ / V.S. Boyko // Физика низких температур. — 2008. — Т. 34, № 7. — С. 639–644. — Бібліогр.: 49 назв. — англ. 0132-6414 PACS: 74.72.Bk;74.25.Qt;61.72.Lk;61.72.Mm http://dspace.nbuv.gov.ua/handle/123456789/117333 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная |
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Свеpхпpоводимость, в том числе высокотемпеpатуpная Свеpхпpоводимость, в том числе высокотемпеpатуpная Boyko, V.S. Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ Физика низких температур |
| description |
The thin twins theory consistently applied to the dislocation description of twins in the high-temperature
superconductor YBa₂Cu₃O₇₋δ(YBCO). Quantitative analysis of the twin shape shows an agreement with
theoretical consideration and gives new method to evaluate the twin boundary energy γtw. Friction stress Sfr
acting on a twinning dislocation is also determined. The formation of a twin and twin microstructures are analyzed.
The principles of microstructure design by twinning in YBCO crystals for enhanced Jc at high magnetic
fields are discussed. |
| format |
Article |
| author |
Boyko, V.S. |
| author_facet |
Boyko, V.S. |
| author_sort |
Boyko, V.S. |
| title |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ |
| title_short |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ |
| title_full |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ |
| title_fullStr |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ |
| title_full_unstemmed |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ |
| title_sort |
dislocation description of twins in high-temperature superconductor yba₂cu₃o₇₋δ |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Свеpхпpоводимость, в том числе высокотемпеpатуpная |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117333 |
| citation_txt |
Dislocation description of twins in high-temperature superconductor YBa₂Cu₃O₇₋δ / V.S. Boyko // Физика низких температур. — 2008. — Т. 34, № 7. — С. 639–644. — Бібліогр.: 49 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT boykovs dislocationdescriptionoftwinsinhightemperaturesuperconductoryba2cu3o7d |
| first_indexed |
2025-07-08T12:03:03Z |
| last_indexed |
2025-07-08T12:03:03Z |
| _version_ |
1837080180139491328 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7, p. 639–644
Dislocation description of twins in high-temperature
superconductor YBa Cu O
2 3 7��
V.S. Boyko
Physics Department, New York City College of Technology of the City University of New York, Brooklyn, NY, 11201, USA
E-mail: vboyko@CityTech.Cuny.Edu
Received January 11, 2008
The thin twins theory consistently applied to the dislocation description of twins in the high-temperature
superconductor YBa Cu O2 3 7–� (YBCO). Quantitative analysis of the twin shape shows an agreement with
theoretical consideration and gives new method to evaluate the twin boundary energy �tw. Friction stress S fr
acting on a twinning dislocation is also determined. The formation of a twin and twin microstructures are an-
alyzed. The principles of microstructure design by twinning in YBCO crystals for enhanced Jc at high mag-
netic fields are discussed.
PACS: 74.72.Bk Y-based cuprates;
74.25.Qt Vortex lattices, flux pinning, flux creep;
61.72.Lk Linear defects: dislocations, disclinations;
61.72.Mm Grain and twin boundaries.
Keywords: twin boundaries, dislocations, flux pinning.
Introduction
Mechanical twinning is one of the main forms of plas-
tic deformation of crystals and, because of this, mechani-
cal properties related to twinning have been important
materials issues [1,2]. Dislocation description of twining
turned out to be very productive. It could be attributed, in
particular, to dislocation thin twins theory developed by
A.M. Kosevich and his disciples (for details see review
[3] and books [4,5]). The development of this theory was
initiated by an article of A.M. Kosevich and L.A. Pastur
[6] published in 1961. It was a long-term interest in
Kharkov scientific community to the investigation of
twinning. Exactly 70 years ago, in 1938, R.I. Garber dis-
covered elastic twinning in calcite. Twins in calcite at-
tracted attention of prominent physicists including
Huygens, Brewster, Kelvin, and others. In 1890, Voigt
found that the calculated strength of calcite with respect
to twinning 1000 times greater than real strength. There
was no any explanation to this dramatic discrepancy be-
tween theory and experiment for abut 40 years. This prob-
lem attracted an attention of I.V. Obreimov and he sug-
gested R.I. Garber to work in direction of Voigt problem
solution. R.I. Garber developed new technique of
twinning and in fine experiments discovered a new phe-
nomenon — elastic twinning. This discovery explained
Voigt paradox and stimulated formation of a twinning dis-
location concept. I.M. Lifshitz and I.V. Obreimov as well
as T.A. Kontorova and Ya.M. Frenkel analyzed the
twinning process at the atomic level. K.V. Vladimirskii
used a macroscopic approach. He considered elastic twin
as a surface at which there is rapture of stresses and esti-
mated the ratio of twin thickness to twin length. I.M.
Lifshitz developed a general macroscopic theory of twins
using a nonlinear theory of elasticity [7]. The most impor-
tant result of K.V. Vladimirskii work was the introduction
of the concept of twinning dislocation in 1947 [8]. This
concept was independently introduced by Frank and van
der Merwe in 1949 [9]. Small value of the ratio of the twin
thickness h to its length L allowed to consider all disloca-
tions of an elastic twin as distributed in a single plane (the
twinning plane) [10]. In an article of A.M. Kosevich and
L.A. Pastur [6], that initiates the thin twins theory, the dis-
tribution of twinning dislocations was described by den-
sity of dislocations �( )x that is the continuous function of
coordinate x. The condition of equilibrium of a single
twinning dislocation in this plane pile-up of twinning dis-
locations under the action of elastic and inelastic forces
used as integral equation for determination of �( )x and, as
a result, to the description of a twin shape in a medium. In
the frame of this approach, A.M. Kosevich and L.A.
Pastur considered twins in an unbounded isotropic and
© V.S. Boyko, 2008
anisotropic solids, twins perpendicular to the plane sur-
face of isotropic and anisotropic solids and so on
[6,11–13]. The development of a twin in the isotropic
plane-parallel plate and near the boundary between two
anisotropic media was considered by L.A. Pastur, V.S.
Boyko, and E.P. Feldman in [14,15]. A.M. Kosevich and
L.A. Pastur analyzed in [11] relationship between the thin
twins theory and I.M. Lifshitz general macroscopic the-
ory of twins. A.M. Kosevich made two important steps
that made possible the direct comparison of the thin twin
theory with experimental observations. He considered
hysteresis under elastic twinning conditions [16] and in-
troduced phenomenological parameters of the theory and
developed approach of determination of the twin length in
terms of these parameters [17]. Dynamics of a twin in the
frame of the thin twins theory was considered by A.M.
Kosevich, V.S. Boyko, and A.A. Slutskin [18–20]. The
description of the dynamic motion of the boundary of a
residual twin was considered by V.S. Boyko [21]. Thus,
all basic stages of the twins development in crystals found
adequate theoretical description in the frames of the thin
twins theory. Theoretical results of the thin twins theory
were in qualitative agreement with experimental
observations in calcite made by R.I. Garber, I.V.
Obreimov, V.I. Startsev, V.Z. Bengus, E.I. Stepina, and
some observations of twinning in metals and alloys made
by V.I. Startsev, V.M. Kosevich, V.P. Soldatov, S.V.
Lubenets, L.S. Fomenko, I.A. Gindin, V.M. Finkel, V.I.
Bashmakov, and others.
V.S. Boyko, R.I. Garber, and L.F. Krivenko [22,23] de-
veloped the new experimental methods of formation of
thin twins as macroscopic pile-ups of rectilinear twinning
dislocations under the influence of distributed load ap-
plied accordingly to theoretically calculated external
elastic field. In the cycle of quantitative investigations
performed by V.S. Boyko, R.I. Garber, L.F. Krivenko, and
V.F. Kivshik (for details see [4,5]), the basic results of
thin twins theory related to the statics and dynamics of
elastic twinning of calcite were experimentally con-
firmed. As a result, the complete quantitative description
of elastic twinning in calcite on the dislocation level was
achieved. Generalization of the approach of thin twins
theory to the case of pile-ups of transformation disloca-
tions in the external elastic, thermal, electric, and mag-
netic fields allowed A.M. Kosevich and V.S. Boyko to de-
velop a new unified approach for description of the basic
manifestations of reversible plasticity of crystals: elastic
twinning, thermoelastic martensitic transformation,
superelasticity, shape memory effect, reversible behavior
of ferroelastic domains (for details see [4,5,24–26]).
In the twentieth century, most of twin description in-
vestigations was dedicated to the plasticity of structural
materials by twinning. The thin twins theory turned out to
be very productive in this activity. Now, due to
microdesign and nanotechnology, the new materials are
exploited more and more not as a construction materials
but as an essential part of devices. The new materials sat-
isfying the requirements of new technology very often
have mechanical properties different from those of tradi-
tional materials. As a rule, they have low-symmetry crys-
tal lattices. These materials frequently do not exhibit slip,
in contrast to the traditional materials, but do exhibit
twinning [27]. Good examples are high-Tc superconduc-
tors. The detailed analysis of mechanical properties of
high-Tc superconductors were done in the review [28].
The major problem for industrial applications of high
temperature superconductors is their low critical current
density J c caused by poor current transmission across
high angle grain boundaries (see for details [29]). Con-
trary to these boundaries, twin boundaries in YBCO im-
prove superconductive properties (see, for example,
[30]). Therefore, the description of twin microstructure
formation in YBCO has risen as an important problem.
Some aspects of applicability of the thin twins theory to
the description of twins in YBCO were successfully dem-
onstrated in our joint investigations with group of Profes-
sor Chan from Columbia University [31–37]. In this arti-
cle, we describe the application of the thin twins theory
developed mainly for description of mechanical
twinning, to the description of the twin structures in the
YBa Cu O2 3 7–�. The influence of these structures on
physical properties of YBCO will be discussed as well.
1. Shape of a twin
There is no appreciable mechanical twinning in the
YBa Cu O2 3 7–� at conditions of loading material by exter-
nal elastic field. Therefore the verification of thin twins
theory could be done by analysis of the shape of the twins.
Consider a lenticular twin column located along the Y
axes with the two wedges pointing along � X directions
for each twin. The d is the distance between the twins
along the Y axes. In each individual twin, the negative
edge twinning dislocations are located in the region
� � �L x 0, while positive edge twinning dislocations are
located in the region 0 � �x L. The twin is infinitely long
along the Z axis, while the thickness of the twin lies along
the Y axis. The thickness of the twin h x( ) is supposed to
be small relatively to the length i.e. h /L( )0 1�� . In such a
case, the twin can be modeled as a pile-up of dislocations
in a plane. It is assumed that the distribution of disloca-
tions can be described by a continuous dislocation density
function �( )x which defines the profile of the twin. In the
case of large frictional force and a comparatively long
twin, the dislocation density function �( )x does not
change after twin formation. If the density of dislocations
is known, we can determine the shape of the twin by the
thickness function:
640 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7
V.S. Boyko
h x a x dx
x
L
( ) ( ) ,�
� (1)
where a is the interplanar distance of glide planes of
twinning dislocations.
There are three regions of a twin along X -axis for
which meaningful expressions for comparison can be de-
rived [34]. They are the twin tip, the root of twin and the
intermediate region. We will discuss each region as
follows.
At the twin tip*
h x
a M
b L
L x( )
( )
�
�
�
4 1 2 2�
�
, (2)
where b is the Burgers vector, � is the shear modulus, � is
the Poisson ratio, and S xst ( ) is the surface tension stress.
This stress always tends to shrink the twin and deter-
mines M
M
S x dx
L x
L
�
�
2
0
st ( )
.
M is a phenomenological parameter of the thin twin theory.
Twin shape in the intermediate region away from the
tip is determined by the external stress distribution of
�ext ( )x just before unloading. Since this distribution is
unknown, we can only qualitatively describe the shape of
the twin at the region adjacent to the tip as
h x
x
L
x
L
L L x
x
( ) – ln� �
�
�
�
�
� �
� ��
�
�
�
�
�
�
�
2
2 2
arcsin . (3)
Considering the shape of the twin at the root (h x h( ) ( ))� 0
we should take into account that twins are in lamellae col-
onies. Central region of the twin could be closer to the
neighboring twins than the twin tips. We use a semi-quan-
titative result [24] to give an approximation and write for
the function h x( ) at the root of the twin
h x
a L x
bd
( )
( ) ( ) ( )
�
� �
� �
�
4 2 21
3
0ext . (4)
In [34], V.S. Boyko, S.W. Chan, and M. Chopra investi-
gated with electron microscopy the twin shape in
YBa Cu O2 3 7–� matrix of melt-textured pellets which con-
tain dispersed particles of Y BaCuO2 5. We compared the
recorded images with the shape predicted from the thin
twins theory for three regions (twin tip, root and interme-
diate region) and found good agreement between the the-
ory and experiment. Thus, the verification of this theory
for twin description in YBa Cu O2 3 7�� was done.
2. Determination of the phenomenological parame-
ters of the thin twins theory
The thin twins theory considers both elastic and in-
elastic forces acting on twinning dislocations. These
forces are incorporated in the thin twin theory as pheno-
menological parameters M and S fr — forces of surface
tension (i.e. twin boundary tension) and lattice friction
(i.e. Peierls force) correspondingly experienced by a
single twinning dislocation. If these phenomenological
parameters are known, this theory allows to give com-
plete description of mechanical twin behavior including
the equilibrium length of a twin and its static shape. Me-
thods for determining these forces were developed in
[22,23]. They used the data of experimental measure-
ments of the dependences of the elastic twin length on the
external applied stress, of a hysteresis of length, of the
critical length in bounded crystals. In YBCO crystals,
however, all these methods could not be used because the
twin length does not change in these crystals after the
elastic stresses has been relaxed and as samples were
slowly cooled to room temperature. Therefore, we need
to develop other approach for determination of pheno-
menological parameters of the thin twins theory for the
new materials in which mechanical twinning does not oc-
cur and twin structure is formed during phase transforma-
tion. In this case, we can consider and estimate tension
force acting on a single twinning dislocation (pheno-
menological parameter M of the theory) from the shape of
a twin. Comparison of the precise twin tip measurement
data obtained by electron microscopy measurement and
the dislocation thin twins theory predictions (see Sec. 1)
opens possibility of the phenomenological parameter M
and twin boundary energy � tw estimate [34]. These twin
shape investigations yielded twin boundary energy
� tw
2mJ / m� �( . . )60 0 21 0 . Addition of Pt proves to de-
crease twin boundary energy to ( . . )26 8 9 5� mJ / m 2 as
evaluated by the twin tip shape method. Thus, in [34],
V.S. Boyko, S.W. Chan, and M. Chopra suggested a new
method (twin tip shape method) of determination of twin
boundary energy using electron microscopy measure-
ments of twin tip shape and relating them to the prediction
from the dislocation theory of thin twins. This method can
be applied to the general case of twin structure when twin
tips are observable in addition to the traditional twin
Dislocation description of twins in high-temperature superconductor YBa2Cu3O7–�
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 641
* The thin twins theory predicts a sharp «beak» at the very end of the twin. The size of this «beak» is determined by the action of
molecular forces. Therefore this «beak» is very small, and will not be discussed here (possibly, this region could be traced in
the observations made by V.S. Boyko, R.I. Garber, A.I. Spol'nik, and L.I. Fedorova in calcite (for details see [5])).
spacing method (see, for example, [37]). It was also used
for investigation of the temperature dependence of M and
twin boundary energy in YBCO [39].
In [34], we proposed also method of determination of
the second phenomenological parameter of the thin twins
theory — frictional force S fr , acting on a single twinning
dislocation in YBCO. This quantity can be determined by
experimental measurement of a critical plane-parallel
twin thickness hc — the smallest thickness of plane paral-
lel twin lamella that can exist in the crystal. The hc is de-
fined [5] as the plane-parallel twin lamella thickness at
which the work done against the lattice frictional forces
on the twin dislocations during detwinning equals the sur-
face energy of the two twin boundaries. Hence with h hc�
twin lamella would disappear while with h hc� twins
would be stable. The magnitude of S fr can be estimated
from the relationship:
S
a
bhc
fr
tw�
2 �
. (5)
In experiments [34], we did not observe plane-parallel
twins thinner than 80 nm in the crystals without Pt and
plane-parallel twins thinner than 30 nm in ones with Pt.
Substitution value of � tw in equation (5) yielded
S fr MPa� 40 in crystals without Pt and S fr MPa� 42 in
ones with Pt. S fr determines the stability of twin
microstructure and gives possibility to estimate natural
limit of its reduction (see for details [34]). The theoretical
consideration and experimental investigations show va-
lidity of proposed methods of determination of the twin
boundary energy and S fr [34,35,39,40]. After determina-
tion of numerical values of phenomenological parameters
in YBCO, one can use the dislocation thin twin theory to
the full extent. By means of this theory, we will analyze
the formation of an individual twin and twin microstruc-
tures. The influence of these microstructutes on the
superconductive properties of YBCO, will be analyzed as
well.
3. Formation of a twin in the presence
of an embedded particle
Hoping to improve superconductive properties of
polycrystal YBCO we can make twin microstructure finer
by decrease grain size, but this increases total area of
grain boundaries and drops J c . The possibility was found
to govern twin microstructure large grain YBCO without
changing the grain size by incorporating in YBCO matrix
the large volume fractions of dispersed addition of (211)
in the form of small particles [30,31], dopants [34], and
additives such as yttrium nanoparticles [40]. In those
cases, the samples were heavily twinned and demonstrate
better superconductive properties.
Presence of all these particles embedded in the matrix
gives highly nonuniform stress distribution creating
twins. The stresses� 0 corresponding to the appearance of
a wedged twin can be estimated (see, for example, [5]) as
follows: � �0 � tw /b. The wedged twin appears in the
point of a crystal where this condition is fulfilled. The
length of a twin is determined according to [5] by equa-
tion
F L
M
L
S( ) � � fr , (6)
where
F L
x dx
L x
e
L
( )
( )
�
�
2
2 2
0
�
, (7)
� e x( ) is the stress in the matrix near the immediate vicin-
ity of the embedded particle; L is the length of the twin.
Integration is hold along the twin. Twin is originated at
the origin of coordinates and is situated along X -axis. The
real situation is very complicated but qualitative descrip-
tion can be obtained by considering the stress distribution
around stiff inclusion. We will use the consideration of
elastic equilibrium of a plate with a circular hole in which
a circular disk is inserted [41]. When the materials of disk
and plate have essentially different elastic constants (it is
actually so in our case), we can treat disk to be absolutely
rigid. Then, the spatial distribution of the shear stress
causing twin appearance can be estimated as
� �e
a
r
� 0
2
2
, (8)
where a is the radius of the inclusion, r is the distance
from the center of inclusion. In the condition of twin for-
mation (comparatively high temperatures and short
twins) we can neglect S fr in (6) that yields F L M/ L( ) � .
Using (6), (7), and (8), one can get expression for length
of the twin appeared in the vicinity of the inclusion
L
a
Mb
�
�
�
�
�
�
�
2
2�
tw . (9)
If we take values of data from [34], then for the particle
with radius a � �10 6 m we will get L � �10 4 m, that corre-
lates by the order of magnitude with the experimental ob-
servation made in [32].
4. Principles of microstructure design by twinning for
enhanced critical currents at high magnetic fields
The influence of twin structures of traditional and
high-Tc superconductors on their superconductive pa-
rameters were discussed in the frames of thin twins theory
[4,5,42,43] but with respect to reversible plasticity of su-
642 Fizika Nizkikh Temperatur, 2008, v. 34, No. 7
V.S. Boyko
perconductors that may change the ratio of supercon-
ductive and nonsuperconductive phases. There is addi-
tional possibility of twin microstructure influence on the
superconductive properties — the pinning vortices by
elements of the twin microstructure. The resources of
twin microstructure design in YBCO by twining were
analyzed by V.S. Boyko and S.W. Chan in [37] by com-
paring pinning capability of different microstructures:
system of plane-parallel twin lamellas; system of wedge-
shaped twins; microstructure composed of twin intersec-
tions. The twin intersections are very strong pinning cen-
ters [36,47]. They tend to pin a fluxoid simultaneously in
both directions with the twin boundaries behaving as
high-energy barriers, which prevent vortex motion. Such
defects possess a very intense elastic field and would tend
to suppress the order parameter locally hence improving
flux pinning.
Twins in YBCO stem from phase transformation from
non-superconducting tetragonal phase to the supercon-
ducting orthorhombic phase. In this case, a crystal is pen-
etrated by parallel twin lamellas. Situation was consid-
ered us ing energy approach (see , for example ,
[38,44–46]).The distance between lamellas can be esti-
mated from this approach from a relationship:
T
N R
w
t
�
�
�
tw col
2
, (10)
where Tw is the distance between centers of lamellas, Rcol
is the size of the twin colony, t is the twin strain, � is the
elastic modulus, N is a numerical factor.
The scenario of formation of this type of microstruc-
ture could be as follows. The appearance of first twins ini-
tiates the appearance of twin colonies around it. The
lengths of twins in each colony will increase until they
come into a collision with twins of the neighbor colony.
Inside each of colony the intertwin distance can be esti-
mated from relationship (10). If each twin intersection
can be considered as pinning one vortex, we can estimate
trapped magnetic field H tr by analogy with [48] as fol-
lows:
H tr int� !0� , (11)
where!0 is elementary magnetic flux (! �0 2 07 10 15. " � Wb),
� int — density of twin intersections per unit area. If we
assume that � int � �( )Tw
2 1, then
H T
N R
w
t
tr
tw col
� ! � !0
2
0
2
/
�
�
. (12)
One can expect that there are two different regimes of for-
mation of intersected areas. In the first regime, concentra-
tion of embedded particles is high, and the distance be-
tween particles determines the size of a colony. Second
regime will be occurred when the concentration of parti-
cles is small and the size of colony is smaller than the dis-
tance between particles. As a result, R Lcol � and we will get
H
b
a
t
tr
tw
� !0
2 2 2
2 2
�
�
(13)
I f we take t
2 310� � , � tw
2J / m� �10 2 , b � �10 11 m,
� �1011 Pa then for the particle with radius a � �10 6 m we
will get H tr mT Gs� �10 100 , that correlates by the order
of magnitude with the experimental data obtained in [47]
for the samples with the small concentration of particles.
From (13) one can deduce that H tr tw� 1 2/ � . Thus we can
substantially increase trapped magnetic field H tr by de-
creasing twin boundary energy � tw . It is shown in [34]
that addition of PtO2 to YBCO decreases � tw by two
times. Therefore we should expect that samples with PtO2
addition should increase trapped magnetic field by four
times in comparison with samples with CeO2 (the ad-
dition of CeO2 does not substantially affect the � tw ).
Exactly this enhancement of H tr was experimentally ob-
served by [49].
Now we will consider regime for which concentration
of embedded particles is high, and the size of colony Rcol
is determined by the distance between particles R p :
R R pcol � . In this case
H
R
t
p
tr
tw
� !0
2�
�
. (14)
If we take � �1011 Pa, t
2 310� � , � tw
2J / m� �10 2 , R p �
� �10 5 m, !0 = 2 07 10 15. " � Wb we will get H tr T�1 . It is
agreed with the estimate of the maximum trapped mag-
netic field in YBCO containing a completely intersected
microstructure at 77 K (4.8 T) [47].
Analyzing relationship (14) we can formulate the key
ideas of microstructure design for enhanced J c at high
magnetic fields. The twin boundary energy and twin co-
lony size are important factors for engineering twin mor-
phology for the strong flux pinning and high J c . To im-
prove magnet ic proper t ies of YBCO we should
investigate possibilities to decrease � tw , Rcol . A pinning
capability of corresponding basic units of the twin
microstructure also should be taken into account (for
details see [37]).
Conclusion
The thin twins theory consistently applied to the dislo-
cation description of twins in the high-temperature super-
conductor YBa Cu O2 3 7–�. The shape of a twin in un-
loaded medium is determined by the elastic interaction
between twinning dislocations and by the inelastic forces
(surface tension and friction forces) acting on them.
These forces are evaluated by using experimental data of
twin shape and relationships of the theory. Quantitative
analysis of the twin shape shows an agreement with theo-
Dislocation description of twins in high-temperature superconductor YBa2Cu3O7–�
Fizika Nizkikh Temperatur, 2008, v. 34, No. 7 643
retical consideration and gives new method to evaluate
the twin boundary energy � tw . Friction stress S fr acting
on twinning dislocation is also determined. The formation
of a twin and twin microstructures are analyzed. The prin-
ciples of microstructure design by twinning in YBCO
crystals for enhanced J c at high magnetic fields are
developed.
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