Parinov I.A. Microstructure and Properties of High-Temperature Superconductors
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Contents xiii
6 Modeling of BSCCO Systems and Composites .............269
6.1 Transformation of Bi-2212 to Bi-2223 Phase . . . . . . . . . . . . . . . . 269
6.1.1 Edge Dislocations as Channels for Fast Ion Diffusion . . . 270
6.1.2 TheLayer-RigidityModel ..........................272
6.1.3 Dynamics of Bi-2223 Phase Growth . . . . . . . . . . . . . . . . . . 276
6.1.4 Formation Energy of Bi-2223 Phase . . . . . . . . . . . . . . . . . . 278
6.1.5 Effect of Deformation on Bi-2212/Bi-2223
Transformation ...................................280
6.2 Modeling of Preparation Processes for BSCCO/Ag Tapes . . . . . 283
6.2.1 Sample Texturing by External Magnetic Field . . . . . . . . . 283
6.2.2 DeformationatTapeCooling .......................286
6.2.3 EffectsofMechanicalLoading.......................287
6.2.4 Finite-Element Modeling of Deformation Processes . . . . 295
7 Modeling of YBCO Oxide Superconductors ................301
7.1 Modeling of 123 Phase Solidification from Liquid . . . . . . . . . . . . 301
7.1.1 HeterogeneousMechanism..........................301
7.1.2 Models Based on Yttrium Diffusion in Liquid . . . . . . . . . 301
7.1.3 ModelsBased on InterfacePhenomena...............309
7.1.4 Models of Platelets-Like Growth of 123 Phase . . . . . . . . . 315
7.1.5 Modeling ofSolidificationKinetics...................323
7.1.6 Multi-phaseField Method ..........................326
7.2 Stress–Strain State of HTSC in Applied Magnetic Fields . . . . . 331
8 Computer Simulation of HTSC Microstructure
and Toughening Mechanisms ..............................337
8.1 YBCOCeramicSintering andFracture.....................337
8.1.1 Sintering Model of Superconducting Ceramic . . . . . . . . . 337
8.1.2 CeramicCrackingDuringCooling ...................341
8.1.3 Formation of Microcracks Around 211 Particles
in 123Matrix.....................................343
8.1.4 FractureFeaturesatExternalLoading ...............347
8.1.5 Microcracking Process Zone near Macrocrack . . . . . . . . . 349
8.1.6 CrackBranching ..................................353
8.1.7 CrackBridging....................................356
8.1.8 SomeNumerical Results............................363
8.2 Twinning Processes in Ferroelastics and Ferroelectrics . . . . . . . . 370
8.2.1 Domain Structure and Fracture
ofFerroelectricCeramic............................371
8.2.2 Fracture Features in Domain Structure
ofFerroelectric....................................374
8.2.3 Thermodynamics of Martensitic Transformation
inHTSC .........................................378
8.2.4 About Toughening of Superconducting Ceramics . . . . . . 381
8.3 Toughening Mechanisms for Large-Grain YBCO . . . . . . . . . . . . . 385
xiv Contents
8.3.1 ModelRepresentations.............................385
8.3.2 Effect of 211 Particles on YBCO Fracture . . . . . . . . . . . . 387
8.3.3 SomeNumerical Results............................391
8.4 SmallCyclic FatigueofYBCO Ceramics ...................392
8.4.1 ModelRepresentations.............................392
8.4.2 Microstructure Dissimilitude Effect . . . . . . . . . . . . . . . . . . 393
8.4.3 Fracture Energy and Microstructure Features . . . . . . . . . 395
8.4.4 SomeNumerical Results............................398
8.5 ResidualThermalStressesin YBCO/AgComposite..........399
8.6 Toughening of Bi-2223 Bulk, Fabricated by Hot-Pressing
Method ................................................401
8.6.1 MicrostructureFormationbyProcessing..............401
8.6.2 Bi-2223 Toughening by Silver Dispersion . . . . . . . . . . . . . 404
9 Mechanical Destructions of HTSC Josephson Junctions
and Composites ...........................................407
9.1 InterfaceFracture .......................................408
9.2 Thin Films on Substrates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413
9.3 Step-EdgeJunctions .....................................417
9.4 TransversalFracture .....................................420
9.5 HTSCSystemsofS-N-SType.............................424
9.6 TougheningMechanisms..................................426
9.7 ChartsofMaterialPropertiesandFracture .................428
9.8 ConcludingRemarks.....................................432
10 Modeling of Electromagnetic and Superconducting
Properties of HTSC .......................................435
10.1 Modeling of Intercrystalline Dislocations . . . . . . . . . . . . . . . . . . . 435
10.2 Current-Limiting Mechanisms and Grain Boundary Pinning . . 440
10.3 Vortex Structures and Current Lines in HTSC with Defects . . . 443
10.4 Non-Linear Current in Superconductors with Obstacles . . . . . . . 448
10.5 Current Percolation and Pinning of Magnetic Flux in HTSC . . 457
10.5.1 ModelofNon-linearResistorNetwork................458
10.5.2 Simulation of Current Percolation and Magnetic Flux
in YBCO Coated Conductors . . . . . . . . . . . . . . . . . . . . . . . 466
10.5.3 Modeling of Electromagnetic Properties
ofBSCCO/AgTapes ..............................469
10.5.4 AgingatMechanicalLoading .......................474
10.5.5 Effective Electrical Conductivity
of Superconducting Oxide Systems . . . . . . . . . . . . . . . . . . 477
A Classification of Superconductors ..........................485
A.1 Organic Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
A.2 A-15 Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486
A.3 Magnetic Superconductors (Chevrel Phases) . . . . . . . . . . . . . . . . 488
Contents xv
A.4 Heavy Fermion Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
A.5 Oxide Superconductors without Copper . . . . . . . . . . . . . . . . . . . . 492
A.6 PyrochloreOxides.......................................493
A.7 Rutheno-Cuprates.......................................493
A.8 High-Temperature Superconductors . . . . . . . . . . . . . . . . . . . . . . . . 494
A.9 Rare-EarthBorocarbides .................................500
A.10 Silicon-Based Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
A.11Chalcogens .............................................501
A.12 Carbon Superconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
A.13 MgB
2
and Related Superconductors . . . . . . . . . . . . . . . . . . . . . . . 504
A.14 Room-Temperature Superconductivity . . . . . . . . . . . . . . . . . . . . . 507
B Finite Element Implementation of Carbon-Induced
Embrittlement Model ......................................509
C Macrostructure Modeling of Heat Conduction .............515
C.1 Method of Summary Approximation
for Quasi-Linear Equation of Heat Conduction . . . . . . . . . . . . . . 515
C.2 Heat Conduction of Heterogeneous Systems . . . . . . . . . . . . . . . . . 518
C.2.1 Effective Heat Conduction of Mixes and Composites . . . 518
C.2.2 Polystructural Model of Granular Material . . . . . . . . . . . 520
C.2.3 Model of Granular System with Chaotic Structure . . . . . 523
C.2.4 HeatFluxThrough AveragedElement ...............526
DEdenModel...............................................531
References .....................................................535
Index ..........................................................571
1
Superconductors and Superconductivity:
General Issues
1.1 Superconductivity Discovery
The discovery of superconductivity, that is, the phenomenon by which current
flow in material occurs without noticeable energy dissipation, has been recog-
nized as one of the greatest scientific achievements of twentieth century. This
phenomenon is accompanied by a sudden drop of the electrical resistance to
zero
1
by cooling below the critical temperature (T
c
), which is the temperature
of superconducting transition that is defined for every specific material.
In 1911, Heike Kamerlingh-Onnes when researching the properties of some
metals in the vicinity of liquid helium (4.2 K) had found that mercury cooled to
∼ 4.25 K losing its electrical resistance, that is, transformed into superconduc-
tor [782]. In the next years, the superconductivity of some other metals, sev-
eral alloys and intermetallic compounds had also been discovered (Fig. 1.1)
2
;
however, they demonstrated very low critical temperatures (maximal value
T
c
=23.2K for Nb
3
Ge), some increasing the liquid helium temperature.
This circumstance impeded the practical applications of superconductors in
tremendous degree due to the high cost of liquid helium (∼ $25 per liter) and
difficulties in its preparation. The long absence of noticeable successes in the
increasing of critical temperature (last record was achieved in 1973 for com-
pound Nb
3
Ge) raised the highly restrained moods of scientists, who worked
in this field in the middle of 1980. So, experimentalists discussed the issue of
the perspectives of the T
c
increasing for Nb
3
Sn upto a value of 30 K with the
application of very exotic techniques [193], and theorists predicted a 40 K T
c
ceiling [416].
The situation changed dramatically in 1986 with the discovery of the
so-called high-temperature superconductivity (HTSC) in non-traditional
1
On the basis of the sensitivity of modern equipment, it may be argued that the
resistivity of superconductors is no more 8 ×10
−25
Ω cm. For comparison, we note
that the resistivity of high-purity copper is of the order 10
−9
Ωcm at 4.2K.
2
The total classification of known superconductors is presented in Appendix A.
2 1 Superconductors and Superconductivity
1900 1960 1980 2000 203019401920
0
20
40
60
80
100
120
140
160
Year
Liquid CF
4
••••
Liquid H
2
••••
Liquid He
••••
••••
Liquid N
2
Hg
Pb
Nb
NbC
NbN
V
3
Si
Nb
3
Sn
Nb
3
Ge
Nb-Al-Ge
(LaBa) -214
Y-123
Bi-2223
Tl-2223
Hg-1223
April, 1986
December, 1986 (under pressure)
December, 1986 (under pressure)
January, 1987
January, 1988
September, 1992 (under pressure)
February, 1988
April, 1993
September1993
(under pressure)
August, 1993 (under pressure)
•
•
•
•
•
•
•
•
•
•
•
??
?
•
293
Room
Temperature
T
c
(K)
•
•
•
•
•
•
•
•
BaKBiO
3
MgB
2
2001
Na
0.05
WO
3
•
Hg-1223F
2004 (under pressure)
Room
Temperature
1999
Fig. 1.1. History of superconductors’ discovery and of T
c
-increasing
compounds, namely cuprates. In spite of the long time and tremendous
forces of world scientific society, the mechanism of superconductivity for
these materials is not yet stated. The distinctive feature of HTSC is that
all these compounds have atomic CuO
2
plane, playing key role as the ori-
gin of superconductivity. The first from HTSC was La
2−x
Ba
x
CuO
4
with
T
c
= 30 K (George Bednorz and Alex M¨uller, April 1986) [58]. The main
peak of discoveries took place between December 1986 and March 1987, when
the existence of the critical temperature T
c
> 30 K was confirmed by Chu’s
group and Kitazawa’s group, independently [152], and then the supercon-
ductivity in YBa
2
Cu
3
O
7
at 93 K, attained jointly by Chu’s group and Wu’s
group [1150], was published. These achievements opened a new epoch in the
1.1 Superconductivity Discovery 3
superconductivity investigations as they brought down the liquid nitrogen
temperature barrier of 77 K. Low cost (∼ $0.5 per liter), simple conditions for
its preparation and utilization led to a considerable progress, in the next years,
in the development, manufacture and initial application of high-temperature
superconductors. The discovery of HTSC initiated real ecstasy in scientific
society (the number of researchers in this field had increased more than order
in 1987), and also the tremendous interest of press and public. In one year,
the critical temperature increased on 70 K, whereas, for the previous 75 years
of superconductivity researches the growth of T
c
was 20 K only! Scientists, as
simple people, called 1987 “the year of progress in physics.” Public interest
caused a sharp increase in financing of HTSC studies in the world. In the fol-
lowing years, other compounds were discovered (see Fig. 1.1 and Appendix A)
with critical temperature also above the liquid nitrogen temperature, namely
Bi
2
Sr
2
Ca
2
Cu
3
O
10+x
(T
c
= 110 K) [653], Tl
2
Ba
2
Ca
2
Cu
3
O
10
(T
c
= 125 K)
[972] and HgBa
2
Ca
2
Cu
3
O
8
(T
c
= 134 K) [980]. The record critical temper-
ature, 164 K, was achieved at high pressure (30 GPa) in HgBa
2
Ca
2
Cu
3
O
8+x
family in September 1993 [153].
In 1988, Sheng et al. presented results of measurements of the electri-
cal resistance in Tl
2
Ba
2
Ca
x−1
Cu
x
O
2x+4
samples [971, 972]. The results and
data obtained by Hazen et al. [400] demonstrated that at increasing num-
ber x the critical temperature T
c
(x)growsbythenextway:T
c
(1) = 90 K,
T
c
(2) = 110 K, T
c
(3) = 125 K. By the linear dependence on x,thetempera-
ture T
c
= 300 K is attained at x = 10. However, the values of T
c
> 125 K for
thallium oxides were not defined. Analogous predictions for the behavior of
T
c
on increasing x had been fulfilled for Bi
2
Sr
2
Ca
x−1
Cu
x
O
2x+4
family [438].
Regretfully, the dependence also destroys at x>3.
Thus, numerous studies of HTSC have stated that maximal critical tem-
perature, T
c
, is reached in compounds with three CuO
2
layers per elementary
cell. Moreover, it is necessary to carry out two conditions: (i) to reach the
small distance, d
Cu–O
= a/2, between atoms of copper and oxygen (where a
is the period of two-dimensional lattice in the CuO
2
plane) and (ii) the hole
concentration in CuO
2
layers should be near the optimal value of p =0.16
(in account per copper atom). All these conditions have been realized in mer-
cury HTSC Hg-1223 with fluorine additives (Hg-1223F) in 2004. The maximal
T
c
= 138 K (at P = 0) has been attained in samples with a =0.38496 nm,
and a record T
c
= (166 ± 1.5) K at P = 23 GPa [721]. In mercurial HTSC,
a linear dependence of T
c
on the lattice constant, a, is observed, namely: T
c
increases with decreasing of a. The critical temperature T
c
∼ 100 K in Hg-
1201 at a =0.388 nm, and T
c
= 138 K in Hg-1223F at a =0.38496 nm. Then,
in order to reach T
c
= T
room
= 293 K, it is necessary that a ≈ 0.374. However,
a decreasing of T
c
is observed at increasing number of the CuO
2
layers, x>3,
that obviously occurs due to a buckling of the CuO
2
layers at diminishing
a. All the above-mentioned cuprates are hole-doped. In 1989, single cuprate
family was discovered to be electron-doped: (Nd,Pr,Sm)CeCuO with critical
temperature T
c
=24K.
4 1 Superconductors and Superconductivity
Periodically appeared information on the sightings of superconductivity
at temperatures even above 300 K today are doubtful because they do not
satisfy the four criteria to determine the superconductivity existence, stated
in 1987 by the researchers who discovered the HTSC [152]: (i) zero resistiv-
ity, (ii) Meissner’s effect marked (when on the decrease of temperature and
magnetic field below the critical values, the total displacement of magnetic
flux from conductor, transforming into superconductor, is observed), (iii) high
reproducibility of results and (iv) high stability of effect. Today, the mistakes
of the superconductivity effect statement at temperatures above 130 K are
linked usually with two general problems, namely:
(1) with experimental mistakes of non-practiced experimentalists having
insufficient knowledge of the modern measuring methods or with what
should be demonstrated to prove the superconductivity of material (the
last cause is met very seldom by development of this field);
(2) with discovery of “new” superconductors really being known supercon-
ducting chemical compounds involving some additional components.
Note that the last problem is very difficult even for experienced researchers.
The main cause that decreased the interest of scientific society consid-
erably to HTSC in the middle of 1990 and decreased considerably financial
supporting (in particular, in USA), directed to these aims, concluded that
the high-temperature superconductors could not replace low-temperature
superconductors and attain sufficiently wide applications. The main obsta-
cle is the material brittleness, which is intrinsic for HTSC compositions. It
forms microstructure defects (voids, microcracks, damage, etc.) during sample
preparation and loading that can be rapidly transformed to macrodefects and
degrade the superconductivity properties. Moreover, the critical current den-
sity, J
c
, was found to be the key parameter for engineering application than the
value of T
c
. Its value is the limit magnitude of direct undamped electric current
in superconducting sample, above which the sample transforms to normal (i.e.,
non-superconducting) state. In particular, the (Bi, Pb)
2
Sr
2
Ca
2
Cu
3
O
10+x
/Ag
tapes possess critical current density up to 80 kA/cm
2
(at 77 K and 0 T) that
changes in dependence on the conductor length and shape [287]. At a temper-
ature of 4 K and in the absence of the magnetic field, their value of J
c
increases
three times. The volume critical current density, j
c
=(2− 5) × 10
6
A/cm
2
in
Bi
2
Sr
2
CaCu
2
O
8+x
monocrystal, found on the basis of the measurement of the
critical current in Cu
2
O
4
surface layer (i.e., pair of parallel CuO
2
layers) at
T =4.5 K [1052], evidently, is also the limit for Bi-2212 thin films and tapes.
Melt-processed Y(RE)BCO bulks (where RE is the rare-earth elements),
the most prospective for applications, demonstrate values of J
c
> 100 kA/cm
2
(at 77 K and 0 T) that decrease rapidly with increase of temperature and
magnetic field [908]. For comparison, the low-temperature superconductors of
Nb–Ti family possess J
c
= 300 kA/cm
2
(at 4 K and 5 T), but Nb
3
Sn samples
demonstrate J
c
= 100–200 kA/cm
2
(at 4 K and 10 T) [598].
1.2 Progress and Prognosis of Superconductivity Applications 5
Remarkable discoveries were also made in another (no cuprate) super-
conducting systems. In particular, in 2001 in broadly accessible and very
cheap MgB
2
system, T
c
= 39 K [756] was obtained. These achievements again
increased sharply an interest to superconductivity in the world.
1.2 Progress and Prognosis of Superconductivity
Applications
At the beginning of 1990, the epoch of engineering applications of HTSC
(Fig. 1.2) was started marked by considerable technical achievements that
stated the use of HTSC in specific products and devices. Growth of HTSC
First HTSC motor
BSCCO/Ag wire (10
m)
First HTSC induction coil
HTSC current lead (2
kA)
BSCCO/Ag wire (100 m)
Motor (0.03 h.f.)
HTSC cable(1
m)
Induction coil (2.5
T)
Synchronous motor with
HTSC windings (5 h.f., 77
K)
Cable (2.3 kA, 1 m)
TBCCO wire (>300 kA/cm
2
)
Induction coil (3 T)
YBCO flexible wire (10
6
A/cm
2
)
Motor (200 h.f.)
Current lead (13
kA)
First thin solenoid (2
T)
Fault current limiter (2.4
kVA)
Transformer (800
kVA, 66 K)
Underground HTSC
electrical power cable (50
m)
1990
1991
1992
1993
1994
1995
1996
1997
1998
Underground HTSC electrical
power cable (120 m)
Hysteresis motor (40 kW)
1999
Superconducting
bearings (10
kWh)
2000
Motor (1000 h.f.)
Transformer (1
MVA)
Motor (5000 h.f.)
2001
Fault current limiter (15
kVA)
EBIS (77
K)
2002
Underground HTSC
electrical power cable (200 m)
Transformer (30 MVA)
BSCCO/Ag wire
(>20
kA/cm
2
, 1 km)
First magnet for field
generation (7 T)
Gilbert-spectrometer
(60 GHz−2.5 THz)
BSCCO/Ag wire (1000
m)
Hysteresis motor (>10
kW)
Leads-in wire (20 kA)
Information systems (RSFQ-scheme)
Transformer (0.5
MW)
Foam and cloth from YBCO ceramic
Hysteresis motor (100 kW)
NDC of defects in
integral schemes
2003
2004
Prototypes all electric technical devices
2005
51-channel SQUID system for
magneto-cardiography
First tomograph
Fig. 1.2. Progress attained in the development and manufacture of HTSC products
6 1 Superconductors and Superconductivity
applications attained and predicted in perspective to comparison with low-
temperature superconductivity (LTSC) is present in Fig. 1.3. The areas of
existing and potential applications of superconductivity (both HTSC and
LTSC) could be depicted in the form of symbolic “tree” (Fig. 1.4):
I. Technical directions. These are present at the “roots” of the tree. How-
ever, the set of the disciplines pointed does not cover all possible applications.
The experience shows that practically every scientific and technical direction –
and also most social ones – can impact upon and be impacted positively by
the future of superconductivity. Then we proceed clockwise, considering the
branches of the tree.
II. Electronics. In this area, thin-film and SQUID-based systems,
3
as well
as cellular and satellite communicating gear, are the most close to achieving
commercial success. The next development of superconductivity applications
in the area of electronic and information systems promises very fruitful compe-
tition between semiconductor and superconductor technologies, a competition
that involves the state of development of cost-effective refrigeration and pack-
aging of circuits and devices. It may be assumed that existing advantages of
Relative Practical Application
0
2000 2005 2010 2020 2030 2040
1
2
3
4
5
LTSC
HTSC
Magnetic Shield
Flow Control
Micro SMES
SQUIDs
Magnet for
Accelerator
MRI
Microwave Devices
Wires
Magnetic Bearings
Thin Films
Analog Signal
Digital Signal Processing
Fault Current Limiters
Micro SMES
Digital and Analog
Signals Processing
Cable (2 GVA)
Fault Current
Limiters
Flow
Controllers
Fly-wheels
Motors
Power Cables
Wires
MagLev Train
Electric Ship
SMES - Large
Generators (MVA)
Electric Motors (MW)
Transformers (1 GVA)
Electromagnetic Ships
Fusion Power
MagLev Train
(long distance)
Motors (MW)
Generators (MV)
Electromagnetic
Ship Propulsion
Fusion Power
SMES (100 kWh)
Transformers (GVA)
Cable (5 GVA)
Years
Fig. 1.3. HTSC and LTSC applications reached and predicted [898]
3
SQUID is an acronym of superconducting quantum interference device.
1.2 Progress and Prognosis of Superconductivity Applications 7
Advanced
Welfare Society
Magnetic
Resonance
Imaging
Magnetic
Shields
Magneto-
cardiogram
Magneto-
encephalogram
Motors
Constructions
& Transportation
Machines
Magnetic
Bearings
Synchrotrons
Accelerators
Big Science
Technologies
Electrical
Power
Cables
Electrical
Power
Reservoirs
Electrical
Power
Generators
Saving Energy & New
Energy Resources
Electrical
Ships
Magnetic
Levitation
Trains
Space
Station
Elevators
Cellular
& Satellite
Communi-
cations
Super-
conductivity
Computing
Super-
conductivity
Electronic
Devices
SQUIDs
Current
Leads
New Traffic
Systems matched with
Environments
Infrastructure for
Information Systems
Applied
Physics
Electrical
Engineering
Material
Science
Mechanical
Engineering
Electronic
Engineering
ELECTRONICS
TRANSPORTATION
MECHANICS
HIGH ENERGY PHYSICS
ELECTRICAL POWER & ENERGY
Superconductivity
Technologies
Fig. 1.4. Symbolic “tree” representing wide perspectives of already existing and
potential applications of superconductivity [1045]
semiconductors will decrease more and more with solution of problems intrin-
sic for HTSC and taking into account unique properties of superconducting
devices, namely, high rate of switching, small consumed power, weakened noise
and high sensitivity to external electromagnetic radiation.
Today, the application of superconducting electronics, in particular,
includes passive super-high-frequency (SHF) filters for system of communi-
cations, Josephson standards of voltage, hot electron bolometers (HEB). One
of the main goals is R&D of superconducting computer. Superconducting
Josephson qubits are two-level quantum systems, in which electric charge
or magnetic flux is the degree of freedom. Josephson contacts, that is, two
superconducting layers with dielectric buffer layer, present their basis. In