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

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10.5 Current Percolation and Pinning of Magnetic Flux in HTSC 483

0.00

0.81

(0.71)

0.15

(a) (b)

(c)

0.59

(0.23)

0.40

0.65

(0.19)

0.38

(0.28)

0.54

(0.44)

0.52

0.38

(0.09)

0.60

(0.50)

0.85

(0.75)

Fig. 10.32. An example of percolating cluster growing in BSCCO bulk. Numbers de-

note probabilities of cell occupation (in brackets there are considered probabilities).

A cross corresponds to the cluster cell at previous step. Microcracks are presented

at GBs

grains. The modiﬁcation of the ant into maze algorithm for this example is

the following. When the considered cell is separated from the main one by an

intercrystalline boundary and the grains have the same orientation, then its

probable number is decreased by 0.1 (for designation of the predominant clus-

ter growth within the grain). If these grains have diﬀerent orientations, then

an additional generated probable number is multiplied by the corresponding

value of the considered cell, so taking into account the grain connectivity. An

example of the cluster growth is shown in Fig. 10.32 and the obtained results

are presented in Table 10.4 [577].

The percolation properties have been calculated for t = 100. The com-

puted relative increasing of the mean quadratic displacement, R/R

,andthe

joining cluster area, S/S

(where R

and S

are corresponding values for the

case D/δ =2.40), together with a decreasing of total GBs length (l

)inthe

case of the larger grain growth coincide with test data (e.g., for large-grain

Table 10.4. Numerical results

D/δ l

R/R

S/S

2.40 1.00 1.00 1.00

2.84 0.86 1.85 1.52

484 10 Modeling of Electromagnetic and Superconducting Properties of HTSC

melt-processed HTSCs, considered in detail in Sect. 3.4). A comparison of

the obtained results with estimations of the strength properties, presented

in Sect. 8.6, shows that the denser lattice of the intergranular boundaries

and, especially, the misorientation of adjacent grains aﬀect the degradation of

transport properties in a greater degree in the case of ﬁne-grain ceramic com-

pared to some increased microcracking of larger-grain structures. Thus, the

greater toughening of the Bi-2223 bulks by ductile Ag particles (a ﬁne-grain

case) and the increased conductivity (a large-grain case) demand optimization

of both contributions in the estimation of the HTSC properties.

Classiﬁcation of Superconductors

Some main properties (including superconducting ones) of diﬀerent chemical

elements are shown in color inset. Here, a classiﬁcation of known supercon-

ducting compounds is carried out. One includes 13 types of compounds,

namely organic superconductors, A-15 compounds, magnetic superconduc-

tors, heavy fermions, oxides without copper, pyrochlore oxides, rutheno-

cuprates, high-temperature superconductors, rare-earth borocarbides, silicon

superconductors, chalcogens, carbon superconductors, MgB

and related com-

pounds. The last section is devoted to room-temperature superconductivity.

A.1 Organic Superconductors

The organo-metallic compounds, discovered in 1979, have double bonds and

can provide electrons to form Cooper pairs. There are vibration modes so

that a phonon-induced mechanism is feasible. Moreover, the electrons may

become attractive due to strong correlations, in which a phonon mech-

anism is not required. Majority of organic superconductors demonstrate

co-existence of superconductivity and antiferromagnetism [300]. The crit-

ical temperatures, T

, for BT=BEDT-TTF (BEDT = bis(ethylenedithio),

TTF = tetrathiafulvalene) compounds [993] are given in Table A.1. The prop-

erties of these superconductors depend on pressure and on annealing proce-

dure. The critical temperatures of most of these compounds are < 10 K.

The mechanical loading leads to change of Fermi surface. The stress par-

allel to conducting planes of quasi-two-dimensional organic superconductor

k-(BEDT-TTF)

Cu(SCN)

, causes an increase of T

and H

, but the perpen-

dicular stress leads to their decrease. As it is known, superconductivity can

be destroyed by a suﬃciently high magnetic ﬁeld. However, this is not always

true. The organic superconductor k-(BEDT-TTF)

Cu(SCN)

demonstrates

Fulde-Ferrell-Larkin–Ovchinnikov (FFLO) eﬀect, namely superconductivity

could be observed only in strong magnetic ﬁeld. FFLO superconductivity ini-

tiates when an electron with spin “up” interacts with a hole with spin “down.”

486 A Classiﬁcation of Superconductors

Table A.1. The properties of several organo-metallic compounds with

BT = BEDT − TTF [BEDT = bis(ethylenedithio), TTF = tetrathiafulvalene]

No. Formula Phase Pressure (kbar) T

(K)Remarks

1(BT)

ReO

–4 2.0

2(BT)

α 03.3I

doped

3(BT)

01.5

4(BT)

0 2.0 Annealed

5(BT)

0.5 8.1

6(BT)

08.1

7(BT)

IBr

β 0 2.2–3.0

8(BT)

AuI

β 0 3.4–5.0

9(BT)

3−2.5

γ 02.5

10 (BT)

θ 03.6

11 (BT)

k 03.6

12 (BT)

2.89

–0 4.0

13 (BT)

2.89

–12 1.8

14 (BT)

–16 2.0

15 (BT)

Cu(NCS)

k 0 10.4

16 (BT)

Cu[N(CN)

]Br k 0 11.5

In this case, “Cooper pair” of a new type forms, the necessary condition being

an external magnetic ﬁeld directed strictly along BEDT-TTF layers [994].

In other quasi-two-dimensional organic conductor λ-(BETS)

FeCl

,where

BETS = bis(ethylenedithio)tetraselenafulvalene, the superconducting phase

is induced by a magnetic ﬁeld exceeding 18 T [1093]. Crystalline λ-(BETS)

FeCl

consists of layers of highly conducting BETS sandwiched between in-

sulating layers of iron chloride. The ﬁeld is applied parallel to the conducting

layers. At the same time this compound, at zero ﬁeld, is an antiferromagnetic

insulator below 8.5 K. By increasing the applied magnetic ﬁeld value above

41 T, this compound becomes metallic at 0.8 K. The dependence T

(B)hasa

bell-like shape with a maximum T

≈ 4.2 K near 33 T. It is interesting to note

that a magnetic ﬁeld of only 0.1 T applied perpendicular to conducting BETS

planes destroys the superconducting state. Thus, in this organic conductor,

there is a strong superconductivity along the c-axis and weak in the ab-plane

(in cuprates, it is the other way round).

A.2 A-15 Compounds

Chemical compounds of binary composition A

3−x

1+x

crystallize into many

diﬀerent structures, depending on the value of x, temperature and pressure.

One of the structures existing near x =0,A

B (where A =Nb,V,Ta,Zr

and B = Sn, Ge, Al, Ga, Si) has the structure of beta-tungsten, designated in

crystallography by the symbol A-15, and is superconducting. Hardy and Hulm

A.2 A-15 Compounds 487

ﬁrst discovered A-15 superconductor (V

Si) in 1954. Intermetallic compound

Ge has a critical temperature T

=23.2 K, while Nb

Ga shows T

20.7K, Nb

Al,T

=19.1K and Nb

Sn,T

=18.3 K. Figure A.1 shows the

structure of the binary A

B compound. The transition temperatures of several

A − 15 structures are given in Table A.2.

The critical temperature and second critical ﬁeld increase in Nb

Al com-

pound, adding Ge and Cu in Nb/Al at the initial stage of the rapid heat-

ing, quenching and transforming (RQHT) process [463], and they attain for

Al-Ge(T

=19.4K,H

=39.5T) and for Nb

Al-Cu(T

=18.2K,H

28.7 T) [462]. Moreover, the superconductors Nb

Al-Ge, Cu have highest crit-

ical current densities among all metallic multiﬁlamentary superconductors at

H>20 T and T =4.2 K. Only signiﬁcantly more expensive HTSC, based on

Bi-2223 and Bi-2212, have the near values of supercurrent at T<20 K.

Fig. A.1. Crystalline structure of A

B compound (A − 15 superconductors). The

atoms A form one-dimensional chains on each face of the cube. Chains on the

opposite faces are parallel, while on the neighboring faces they are orthogonal to

each other

488 A Classiﬁcation of Superconductors

Table A.2. The critical temperatures, T

(K), of A-15 compounds

Sb 6.5 Ti

Ir 4.2

0.92 Ti

Pt 0.5

Zr-Pb 0.76 V

8.4

∼3

3.4 V

5.7

V-Al

11.8 V

≈ 1

Ga 15.9 V

1.7

Si 17.0 V

∼3

Pd 0.08

∼3

Ge 6.0–7.5 V

Pt 3.7

∼3

6.0–11.0 Nb

1.0

∼79

∼21

4.3 Nb

2.6

V-Sn

7.0–17.0 Nb

3.2

0.2 Nb

Pt 11.0

0.8 Ta

0.4

Al 19.1 Cr

3.4

Be 10.0 Cr

4.7

Ga 20.7 Cr

0.07

Pb 5.6 Cr

0.75

∼3

8.0–9.2 Mo

13.4

4.4 Mo

∼65

∼35

≈ 15 (A-15)

Nb-Si

9.3 Mo

12.7

Nb-Si

4.0–8.0 Mo

8.5

Nb-Si

11.0–17.0 Mo

4.6

Nb-Ge

6.0–17.0 W

∼60

∼40

11.0

Nb-Ge

23.2 Ta

∼80

0.55

Sn 18.3 Zr

Au 0.9

Nb-Sb 2.0 V

3.0

∼3

3.0 Nb

∼3

Au 11.5

∼3

8.0

∼3

Sn 8.3

∼3

Sb 0.7

Al 0.58

Ga 0.76

1.7

1.8

The left column has no transition metals; the right column has transition metals;

three alloys of gold are also included. Note: q = quenching, p = pressure, f=ﬁlm

A.3 Magnetic Superconductors (Chevrel Phases)

In 1971, Chevrel and co-workers discovered a new class of ternary molybdenum

sulﬁdes having the general chemical formula M

,whereM stands for

a large number of metals and rare earths. These superconductors have un-

usually high values of the upper critical ﬁeld, B

, given in Table A.3 [737].

The superconducting compounds REMo

(where RE =Gd,Tb,Dy,Er

and X =S,Se,Te),andRERh

(where RE = Nd, Sm, Tm) are usually

A.3 Magnetic Superconductors (Chevrel Phases) 489

Table A.3. The critical temperature and the upper critical magnetic ﬁeld of Chevrel

phases

No. Compound T

(K) B

(T)

1PbMo

15 60

2LaMo

7 44.5

3 SnMo

12 36

related to the Chevrel phases [1028]. Magnetic superconductors demonstrate

some novel features not found in conventional type-I superconductors. Upon

cooling from the superconducting phase (T<T

), the material becomes

normal again at a low temperature and the superconductivity is destroyed.

Very often this normal phase is magnetically ordered. Upon cooling from the

normal state the system becomes superconducting below T

, and upon further

cooling it becomes magnetically ordered below Neel temperature, T

(where

). Thus, the superconducting phase occurs only in a limited range

of temperatures, T

<T <T

. The interaction of conduction electrons with

magnetic atoms leads to the formation of a bound state of the electron with the

magnetic atom below a certain characteristic temperature called the Kondo

temperature. It was found that the resistivity of a magnetic alloy, such as Fe

impurities in Cu shows a minimum in the resistivity as a function of tempera-

ture due to the interaction of conduction electrons of the metal atom with the

magnetic moment of the magnetic atom. Such a Kondo eﬀect exists in super-

conductors containing magnetic impurity atoms at low temperatures, because

the superconductivity is destroyed. In the case of magnetic superconductors,

there is a sub-lattice of magnetic atoms in addition to the lattice of metallic

atoms, so that magnetically ordered phase exists at low temperatures below

the superconducting phase, T

While ferromagnetic superconductors (e.g., ErRh

and HoMo

)

demonstrate the above behavior, the antiferromagnetic superconductors (e.g.,

REMo

,whereRE = Gd, Tb, Dy and Er and also RERh

,where

RE = Nd, Sm and Tm) present examples of the co-existence of the two phases

together with anomalous behavior of the second critical ﬁeld. In ErRh

,pos-

sessing tetragonal structure with a =5.299

Aandc =7.588

A, a plot of the

ac-magnetic susceptibility (χ

) and electric resistancedepictsanormalto

superconducting transition at T

=8.7 K, followed by a loss of susceptibility

at T

=0.9 K together with the appearance of the ferromagnetic long-range

order. This result takes place at zero magnetic ﬁeld. At non-zero ﬁnite ﬁeld

also, the resistance disappears at some temperature interval within the above

domain. The compound HoMo

becomes superconducting at T

=1.3K

in zero ﬁeld. On further cooling, it becomes normal at T

=0.6 K together

with the appearance of ferromagnetic long-range order.

The antiferromagnetic superconductors provide the most striking case of

the co-existence of the two kinds of order. These systems (e.g., REMo

490 A Classiﬁcation of Superconductors

where RE =Gd,Tb,Dy,ErandalsoRERh

,whereRE = Nd, Sm, Tm)

demonstrate near T

the antiferromagnetic alignment of rare-earth magnetic

moments in the superconducting state of the system. The most important

result is the anomalous behavior of the upper critical ﬁeld as a function of

temperature near T

.Inparticular,(RE = Gd, Tb and Dy) demonstrate

anomalous decreasing of H

near, but below T

. The crystal structure of

REMo

is presented in Fig. A.2.

An attempt of T

increasing in REMo

owing to change of RE ion

radius leads to the structure transition, as a result of which the dielectric gap

opens at Fermi level and the superconducting properties disappear together

with metallic ones. Based on BCS model, this is explained that the growth of

is caused by the approaching Fermi level to the peak of the electron state

density, N(E). However, it is disadvantageous energetically for the structure,

therefore it suﬀers phase transition, which suppresses superconductivity.

The superconducting carbosulﬁde Nb

[912] with layered crystalline

lattice has critical temperature T

=5Katx =0.8−1. The superconductivity

demonstrates volume character, and Nb atom octahedron is the key structure

element, in the center of which C atom locates.

S Mo

Fig. A.2. Structure of REMo

, which have Mo octahedron inside sulphur cube,

which are inside a rare-earth cube [986]

A.4 Heavy Fermion Superconductors 491

A.4 Heavy Fermion Superconductors

Co-existence of superconductivity and antiferromagnetic order is found in

U-based heavy fermions (UPt

, U(Pt

1−x

)

, U

1−x

, URu

, UNi

, UPd

, U

Co and U

Fe) [456, 497, 526, 927] and in the heavy fermions:

RERh

(where RE =LaandY),Cr

1−x

, CeRu

, CeIn

, CePd

CeNi

, CeCoIn

and Ce

3n+2m

(where T =RhorIr,n = 1 or 2;

m = 1) [674, 842, 874, 1054]. In CeCoIn

critical temperature T

=2.3 K [874].

At the same time, the co-existence of superconductivity and ferromag-

netism is discovered in the compounds UGe

[930], URhGe [25] and ZrZn

[843]. In ZrZn

, the maximum critical temperature is slightly less than 3 K at

ambient pressure and decreases with increasing pressure. The superconduc-

tor YNi

1.9

1.2

demonstrates one of the maximum T

≈ 14 K among known

triple compounds with ferromagnetic component [870]. Another supercon-

ducting phase YNiB

has T

∼ 4.5 K (tetragonal lattice with parameters of

a =0.3782 nm,c=1.1347 nm, spatial group of symmetry P 4/mmm) [871].

As it is known, the electronic heat capacity of a solid varies linearly with

temperature at low temperatures, c = γT, where the constant of proportional-

ity, γ, determines the mass of the electron in the conduction band. It is found

that in many compounds of Ce or U, this band mass is very large. Not only the

eﬀective mass but also Fermi rate diﬀers by two orders of magnitude from the

free electron values found in ordinary metals. For example in CeCu

,the

Fermi rate, v

=8.7 ×10

m/s and the band mass, m

= 220m

,wherem

the free electron mass. The critical temperature is T

≈ 0.5 K and the London

penetration depth approaches a high value of λ

=0.2 μm at zero temper-

ature. The large speciﬁc heat is caused by the strong correlations between

electrons. In UBe

, a sharp superconducting transition occurs at 0.97 K. The

variation in T

from sample to sample is in the range of 0.88–0.97K. When Np

is added to UBe

,theT

reduces. For Np

0.68

0.32

, the transition temper-

ature is about 0.8 K. The compound UPt

has a superconducting transition

temperature of 0.54 K. The extreme low temperature indicates that there are

strong magnetic correlations [986].

The superconductor PuCoGa

with T

∼ 18 K [922] occupies intermediate

place between heavy fermion superconductors with T

∼ 1K and HTSC with

∼ 100 K. Besides “exotic” superconducting state, the common property for

all these materials is the very strong Coulomb correlations between electrons

of atomic f-shells (in CeCoIn

and PuCoGa

)ord-shells (in HTSC), which

lead to sharp increase of the eﬀective electron mass or to transition in the

state of Mott’s dielectric, respectively. Co-existence of the localized and de-

localized states, which are close in energy, favors the magnetic mechanism of

the electron pairing [169].

Iron is the ferromagnetic, which demonstrates superconductivity at very

high pressure. The superconducting transition is registered at 15 GPa <P <

30GPa by both resistive and magnetic (Meissner’s eﬀect) methods [983].

492 A Classiﬁcation of Superconductors

The dependence T

(P ) is “bell-like” (similar to the dependence of T

on hole

concentration in HTSC) with maximum value of T

=2KatP =21GPa.

Lithium is the metal at normal pressure. Under pressure of P ≈ 30 GPa,

it becomes a superconductor and T

attains 20 K at P = 48 GPa. This is the

highest T

for “simple” one-element superconductors [982].

A.5 Oxide Superconductors without Copper

The superconductors of A

type have hexagonal structure of the tung-

sten bronze, where as A, the large alkaline ions of K, Rb and Cs are used

more often (see Fig. A.3a). Many superconducting materials from this fam-

ily demonstrate T

≈ 2−7 K. The monocrystals of perovskite dielectric WO

doped by Na, demonstrate in surface layer of Na

0.05

high-temperature

superconductivity with T

= 91 K [417]. In this case, Na segregates at the

grain surface and takes the structure of the WO

surface layer.

The superconductor without copper, Ba

1−x

BiO

(BKBO) demonstrates

maximum critical temperature (T

= 31 K) among “old” oxide supercon-

ductors [737]. The oxide superconductor, LiTi

, without Bi and Cu, also

possessing high transition temperature, has the spinel crystalline structure.

Two other oxide superconductors: NbO and TiO, showing T

≤ 1K, demon-

strate clearly the expanded links of the “metal-metal” type as the previous

oxide. The oxide superconductor BaPb

0.75

0.25

with the critical tempera-

ture T

= 12 K has simple perovskite structure, depicted in Fig. A.3b.

Recently, the layered superconductor Na

CoO

·1.3H

OwithT

=4Khas

been fabricated using the chemical oxidation technique [1035]. The conductive

layers (CoO

) in this superconductor alternate with the dielectric buﬀer layers

(Na

· 1.3H

O), which carry out a function of reservoirs for electric charge.

SimilartoHTSC,thevalueofT

is highest at the optimum level of doping

and decreases in overdoped and underdoped samples [931].

(a) (b)

Fig. A.3. (a) Hexagonal structure of the tungsten bronze. One elementary layer

presented, which is formed by the octahedrons MO

with large ions in cavities.

(b) Perovskite structure [130]