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

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A.12 Carbon Superconductors 503

In 2000–2001, J. H. Sch¨on et al. announced that, using an intercalation

with CHCl

and CHBr

of C

single crystals, directed to expansion of the

lattice and formation of high densities of electrons and holes, they reached ﬁrst

the critical temperature, T

= 52 K [943], and then T

= 117 K in hole-doped

/CHBr

[944]. Figure A.8 presents the variation in critical temperature as

a function of charge carrier density for electron- and hole-doped C

crystals

in diﬀerent types of intercalation. Obviously, these results rendered doubtful.

120

100

Electron-Doping

Hole-Doping

/CHCl

/CHBr

Holes/Electrons per C

Molecule

Transition Temperature, T

(K)

0.5

4.03.53.02.52.01.51.0

Fig. A.8. Variation in T

as a function of charge carrier density for electron- and

hole-doped C

crystals in incorporation of CHCl

and CHBr

[944]

The critical papers [96,964,965] about J. H. Sch¨on et al. results [940–942,

945–947] have been published in 2002 (see also http://www.lucent.com/

news

events/researchreview/). The papers [940–942, 945–947] have been devoted

to the use of organic molecules, segregated on thin ﬁlm, as “molecular switches”,

and also to the use of ﬁeld transistors to change the charge carrier concentration in

diﬀerent organic matters, providing a regulation to their electric properties (from

dielectric to semiconductor, from metal to superconductor and so on). Suppos-

edly, this rendered to induce high-T

superconductivity of the fullerenes C

electric ﬁeld. However, ﬁrst, total identity of the ﬁgures (right up to accidental

noise inevitably accompanying any experiment) in [947] and [946] has been found.

The same ﬁgure has been also presented in [940], devoted to another type of mi-

504 A Classiﬁcation of Superconductors

Superconductivity with T

= 7 K has been found in C

monocrystal with

size of about 1 mm [319]. Obviously, carbon nanotubes (1D molecular con-

ductors) are ideal candidates for research of 1D superconductivity. Today,

maximum critical temperature, T

= 15 K, for one-wall nanotubes [1047], and

c, on

∼ 11.5 K (beginning of resistive transition), T

(R =0)=7.8K, for

multiwall nanotubes [1040] have been reached.

The attempts to switch graphite into superconducting state, using its dop-

ing by diﬀerent chemical elements, led to superconductivity with T

< 1Kfor

cases of K and Na [36, 388]. Introduction of ytterbium and calcium atoms

between graphite layers led to the fabrication of superconducting compounds

with T

=6.5 and 11.5 K, respectively [1139]. In this case, the role of Yb and

Ca atoms is that they supply free charge carriers in the graphite layers.

Then, the oriented (111) boron-doped diamond thin ﬁlms (with boron con-

centration 0.53%) have been grown at (001) silicon substrates, using MPCVD

technique (modiﬁcation method of the chemical vapor deposition) [1038].

These ﬁlms demonstrated T

c, on

=7.4K and T

(R =0)=4.2K at H =0.

The linear extrapolation gives H

(0) = 10.4T and H

irr

(0) = 5.12 T, the

critical current density J

= 200 A/cm

at H = 0. First signs of super-

conductivity arose at boron concentration of 0.18%. Finally, the materials,

combining superconductivity, superhardness and high strength (these ma-

terials could be used for research of electric and superconducting proper-

ties under pressure) have been synthesized at high static pressures (up to

7.7 GPa) and temperatures (up to 2173 K) in the following systems [226]: (i)

diamond/Nb, T

=12.6K(ΔT =1.5K), H

(4.2K) = 1.25 T (diamond ma-

trix and superconducting channels from niobium carbide); (ii) diamond/Mo,

=9.3K, ΔT = 5 K (diamond matrix and superconducting channels from

molybdenum carbide); (iii) composites with matrices from superhard mate-

rials (80 wt%) and MgB

channels, namely diamond/MgB

=37K)and

cubic boron nitride/MgB

=36.1 K), frame of these composites has mi-

crohardness in the range of 57–95 GPa.

A.13 MgB

and Related Superconductors

The superconducting system MgB

discovered in 2001 demonstrates highest

volume superconductivity (critical temperature, T

= 39 K) among non-

copper oxide conductors [756].

Crystal structure of MgB

,havingspatial

croelectronic devices. In this case, the coincidence has not been in the whole,

but separate parts of the plots coincided totally with one another. Then, eight

cases of the ﬁgure coincidence (total or fragmentary) have been found in six pa-

pers of J. H. Sch¨on, devoted to diﬀerent types of devices for various materials

and temperatures. After that, 100 scientiﬁc groups in the world have attempted

unsuccessfully to repeat the results of J. H. Sch¨on et al.

monocrystals, doped by Na, demonstrate high-temperature superconduc-

tivity with T

=91Kinthesurface layer with content of Na

0.05

[417].

A.13 MgB

and Related Superconductors 505

Fig. A.9. Crystal structure of MgB

[756]

group of symmetry P 6/mmm, is shown in Fig. A.9 (where a =3.086

A, and

c =3.524

A are the parameters of hexagonal unit cell). It is most surprising

that simplicity and accessibility of this material, known from the beginning of

1950 [496], combines with complexity of the superconductivity phenomenon.

Even in non-textured multiphase MgB

samples, superconducting currents

ﬂow in all volume of the sample and are not sensitive to weak magnetic

ﬁeld (which should suppress J

at the existence of Josephson links between

grains). The temperature scaling of the pinning strength practically in all su-

perconducting H–T plane supports the concept that critical current density

is found by the magnetic ﬂux pinning [602]. It has been shown in experiments

that both the transport (i.e., intergranular) j

and its dependence on H and

T , totally coincide with those for inductive (i.e., intragranular) j

.Thetest

results prove that grain boundaries in MgB

are transparent for supercon-

ducting current [194]. It is observed small, but evident anisotropy of critical

ﬁeld, H

=1.1 in the bulks [387]. At the same time, it is found that

=1.8–2.0inMgB

ﬁlms with c-axis, oriented perpendicular to the

The nature of the observed superconductivity shows that in this case, supercur-

rents ﬂow on the crystal surface but not into its volume.

506 A Classiﬁcation of Superconductors

ﬁlm surface [837]. The high-qualitative-oriented MgB

ﬁlms has critical cur-

rent density, j

=1.6 × 10

A/cm

at T =15KandH = 0 [543]. It is stated

that T

and residual electric resistivity, ρ

, depend only on thickness, d,of

the epitaxial MgB

ﬁlms, but are independent of the ﬁlm growth rate. An

increase in d leads to increase of T

and decreasing of ρ

.Now,thevaluesof

=41.8K and ρ

=0.28μΩ · cm at d>300 nm are maximal and minimal

magnitudes, respectively [853]. The record value of H

=52TatT =4.2K

has been reached in one sample of thin MgB

ﬁlms in the ﬁeld parallel to the

ﬁlm surface [274].

The irreversibility ﬁeld, H

irr

,inNb

Sn, Nb-Ti and MgB is equal to 20,

10 and 14 T at T =4.2 K, respectively [254]. Generally, the characteristics

of the superconductors MgB

,BKBOandA − 15 (see Table A.8) [737] show

similar values. Therefore, this group of superconductors can be considered

as intermediate between the conventional superconductors, subjected to BCS

theory, and the superconductors with high critical temperature.

The attained values of critical current density for MgB

-coated conduc-

tors in Fe-sheath [490]: J

> 85 kA/cm

(T =4.2K and H =0)and

=23kA/cm

(T =20KandH = 0) are nearly suﬃcient for current-

carrying cables.

The required components Mg, B and Fe are lightly acces-

sible chemical elements and method of “oxide powder in tube” used in the

conductor fabrication be very well tested at preparation both low- and high-

temperature large-scale superconductors. Now, this technique can supply high

temperatures of industrial processing. All these factors open a tempting per-

spective of fastest application of MgB

systems and MgB

-based supercon-

ducting products.

Several compounds, related to MgB

, reach the following critical tem-

peratures: BeB

=0.79 K [1182]); Re

B(T

=4.7 K [1020]); ZrB

=5.5 K [309]); ReB

=6.7 K [1020]); TaB

=9.5 K [506]);

0.76

=9.2K atP = 5 GPa [1162]); CaAlSi (T

=7.8 K [645]); SrAlSi

=5.1 K [645]); MgCNi

= 8 K [401]); MgAlB

= 12 K [623]). The

calculations of electronic structure and constant of electron–phonon interac-

tion predict that AgB

and AuB

should be high-temperature superconduc-

tors with T

= 59 and 72K, respectively [590].

Table A.8. Characteristics of Nb

Ge (A-15 compound), BKBO (x ≈ 0.4) and

MgB

No. Compound T

(K) v

(10

cm/s) ξ(

A) 2Δ/k

(T)

1Nb

Ge 23 2.2 35–50 4.2 38

2 BKBO 31 3.0 35–50 4.5 32

3MgB

39 4.8 35–50 4.5 39

is the critical temperature; v

is Fermi rate; ξ is the coherence length; Δ is the

energy gap; k

is Boltzmann constant; and B

is the upper critical magnetic ﬁeld.

MgB

-coated conductor (or tape), fabricated by laser evaporation on ﬂexible

metallic tape has J

=1.1 × 10

A/cm

(T =4.2K and H = 10 T) [563].

A.14 Room-Temperature Superconductivity 507

A.14 Room-Temperature Superconductivity

There are at least two papers reporting superconductivity above room temper-

ature. In the ﬁrst paper, superconductivity is observed in a thin surface layer

of the complex compound Ag

(0.7 <β<1) at 240–340 K [207]. The

second paper claims that superconductivity exists in carbon-based multiwall

nanotubes at T>400 K [1197].

Finite Element Implementation

of Carbon-Induced

Embrittlement Model

Consider the ﬁnite element implementation of the governing equations (4.14)

and (4.27) for carbon diﬀusion and non-mechanical energy ﬂow, respectively

[810]. The next initial and boundary conditions supplement the governing

equations:

= C

,T= T

, at t =0; (B.1)

= C

, on S

; J

= ϕ

, on S

;(B.2)

T = T

, on S

; −k

∂T

∂x

= ϕ

, on S

, (B.3)

where C

and T

are the initial carbon concentration and temperature,

which may vary within material volume V .IfC

is larger than carbon

terminal solid solubility, the initial carbon concentration in carbonate equals

the terminal solubility of carbon in carbonate, and the initial carbon volume

fraction is calculated according to (4.15). A similar comment is valid for C

which is prescribed carbon concentration on S

, a part of the bounding surface

S. For the calculation of C

, stress and temperature distributions are taken

into account. ϕ

is the prescribed carbon ﬂux on S

; ϕ

is the prescribed heat

ﬂux on S

and T

is the prescribed temperature on S

.NotethatS

∪ S

∪S

= S. The quantities C

, ϕ

, T and ϕ

may vary with time.

The ﬁnite element equations are derived from variational descriptions of

diﬀusion and energy ﬂow. For this purpose variations of carbon concentra-

tion, δC

, and temperature, δT, are considered, which satisfy the boundary

conditions. Therefore,

δC

=0 onS

(B.4)

δT =0 onS

(B.5)

Relation (4.14) is multiplied by a carbon concentration variation satisfying

(B.4). Subsequently, it is integrated over the volume V . Then, taking into

510 B Finite Element Implementation of Carbon-Induced Embrittlement Model

account (4.11), (4.12) and (4.31), the following expression is derived, which is

valid at any time t:



δC

dV = −



δC

dS −



∂C

∂x

∂(δC

)

∂x

dV −

−





−

3RT

∂σ

∂x

∂T

∂x



∂(δC

)

∂x

dV.

(B.6)

The above relation is simpliﬁed if, within a time increment, Δt, the change

of carbonate volume fraction is included in C

. Value of Δt is order of a

characteristic time, introduced by diﬀusion and carbonate size, or smaller.

Then

= f

. (B.7)

Note that C

is no longer the carbon concentration in carbonate, but the

carbon concentration in a part of the material, which at time t is in the form

of carbonate and has volume f

V .

Spatial discretization is obtained by introducing the usual ﬁnite element

interpolation for carbon concentration, carbonate volume fraction, tempera-

ture and stress trace. For example, carbon concentration is calculated from

nodal values as

= a

, (B.8)

where a

and C

are the interpolation function and the nodal carbon con-

centration value for q-node, respectively. Substitution of (B.7) into (B.6) and

use of spatial discretization lead to the ﬁnite element equations for carbon

diﬀusion:



+ D



= F

, (B.9)

where



dV, D



∂a

∂x

∂a

∂x

dV ; (B.10)





−

3RT

∂a

∂x

∂a

∂x



∂a

∂x

dV ;

= −



dS. (B.11)

The time derivative of carbon concentration is approximated by

C,t+Δt

Δt



C,t+Δt

− C

C,t



. (B.12)

B Finite Element Implementation of Carbon-Induced Embrittlement Model 511

Relation (B.9) is taken at time t +Δt. Equation(B.12) is substituted into

(B.9) and the D

-term is transferred to the right hand, leading to the next

relation (similar to the one developed in [1003] for hydrogen diﬀusion due to

concentration and stress gradient):



Δt

+ D



C,t+Δt

= F

− D

C,t

Δt

C,t

. (B.13)

The matrices C

, D

and D

are calculated by using the known nodal

values of carbonate volume fraction, temperature and stress from the previous

calculation step. The value of carbonate volume fraction corresponds to time

t. However, the values of temperature and stress correspond to time t +Δt,

according to the discussion at the end of Appendix B. Vector F

is calculated

from the boundary conditions at time t +Δt. Note that the solution of (B.13)

provides a preliminary carbon concentration value, C

C,pr

t+Δt

, which according to

(B.7) may include carbon in carbonate. This preliminary value can be used

for the determination of the total carbon concentration

t+Δt

= f

C,pr

t+Δt

, (B.14)

as well as the new carbonate volume fraction, f

t+Δt

− C

t+Δt

− C

t+Δt

; f

t+Δt



0,f

< 0;

, 0 ≤ f

≤ 1 ,

(B.15)

where C

t+Δt

is calculated based on the results for temperature and stress

from the previous calculation step.

The new carbon concentration in the carbonate is derived, based on (B.15),

t+Δt



C,pr

t+Δt

< 0,

t+Δt

, 0 ≤ f

≤ 1 .

(B.16)

Then, the ﬁnite element implementation of the governing equations for

non-mechanical energy ﬂow will be obtained. Relation (4.27) is multiplied by

a temperature variation, satisfying (B.5). Subsequently, it is integrated over

volume V and the following expression is derived, which is valid at any time t:



δTρc

dV +



δT

car

= −



δTϕ

dS −



∂T

∂x

∂(δT)

∂x

dV −



δTJ

∂μ

∂x

dV. (B.17)

As in the case of the carbon diﬀusion, spatial discretization is introduced

and the following ﬁnite element equations are derived:

+ K

=Φ

+Φ

− L

, (B.18)

512 B Finite Element Implementation of Carbon-Induced Embrittlement Model

where



ρc

dV ; K



∂a

∂x

∂a

∂x

dV ; (B.19)

= −



dS;Φ

= −



∂μ

∂x

dV ; L



ΔH

car

dV.

(B.20)

Assuming that temperature time derivative is given by

t+Δt

Δt



t+Δt

− T



, (B.21)

and following an approach similar to that for carbon diﬀusion, one may derive

the next algebraic system from (B.18)



Δt

+ K



t+Δt

=Φ

+Φ

− L

Δt

, (B.22)

where Φ

is calculated from the boundary conditions at time t+Δt.Inorderto

estimate Φ

, the nodal values of temperature, carbon concentration, carbonate

volume fraction and stress from the previous calculation step are used.

A complete calculation cycle is as follows. At time t, all ﬁeld quantities are

known: u

,ε

,σ

, dC

/dt, df

/dt, dT

/dt,whereu

are the compo-

nents of the displacement vector of a material particle. A time increment Δt

is considered and the following calculation steps are performed:

(1) Material deformation problem is solved ﬁrst. The boundary conditions of

the applied traction and/or displacements are deﬁned at time t+Δt.The

isotropic expansion strain rate due to carbon dissolution, carbonate for-

mation and thermal expansion is calculated by using the values and time

rates of carbon concentration, carbonate volume fraction and tempera-

ture at time t. The parameters of the de-cohesion model are also derived

from carbonate volume fraction and temperature distribution at time t.

By performing calculation step (1), u

t+Δt

,ε

t+Δt

and σ

t+Δt

are calculated.

(2) The energy ﬂow problem is solved next. The boundary conditions of the

applied surface temperature and/or heat ﬂux are deﬁned at time t +Δt.

-term is calculated, based on the distributions of temperature, carbon

concentration and carbonate volume fraction, at time t,aswellasonthe

distribution of stress, σ

t+Δt

, calculated within step (1). The carbonate

volume fraction rate, at time t, is used for the calculation of L

-term. By

performing calculation step (2), T

t+Δt

is calculated.

(3) The carbon diﬀusion problem is solved at the end. The boundary con-

ditions of the applied surface carbon concentration and/or carbon ﬂux

are deﬁned at time t +Δt.Inallterms,f

is used. D

-term is calcu-

lated, based on the values σ

t+Δt

and T

t+Δt

, derived using steps (1) and

B Finite Element Implementation of Carbon-Induced Embrittlement Model 513

(2), respectively. By performing calculation step (3), C

t+Δt

and f

t+Δt

are

calculated.

The material deformation problem is solved, assuming a constant value

of Young’s modulus. The error in the calculations is further minimized, con-

sidering the value of E for the temperature in the crack-tip region, where

embrittlement and fracture processes operate. When the variation of temper-

ature, either in space or in time, is signiﬁcant, its eﬀect on elastic modules

should be taken into account (see, e.g., [865]).

Adequate numerical results by using the ﬁnite element implementation

can be obtained after carrying out preliminary tests, estimating properties of

carbon, cuprate, carbonate and YBCO superconductor, which are necessary

for calculation.