Mourachkine A. Room-Temperature Superconductivity

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84 ROOM-TEMPERATURE SUPERCONDUCTIVITY

Table 3.1. Characteristics of A-15 superconductors: the critical temperature T

; the upper

critical magnetic ﬁeld H

; and the gap ratio

2∆

inferred from infrared measurements.

Compound T

(K) H

(T)

2∆

Ge 23.2 38 4.2

Ga 20.3 34 -

Al 18.9 33 4.4

Sn 18.3 24 4.2-4.4

Si 17.1 23 3.8

Ga 15.4 23 -

work of the BCS theory. The coherence length of the A-15 superconductors is

very short, ξ

 35–200 A

◦

. They have extraordinary soft acoustic and optical

phonon modes. A lattice instability preceding a structural phase transition and

then followed by superconductivity is typical for the A-15 compounds. This

structural phase transition is called martensite. The presence of densely packed

one-dimensional chains is believed to be responsible for this crystalline insta-

bility. The phase-transition temperature T

for V

Si and Nb

Sn is 20.5 K and

43 K, respectively. In V

Ga and Nb

Al, this transition occurs at about 50 K

and 80 K, respectively. The symmetry transition, from cubic to tetragonal, is

B atom

A atom

Figure 3.1. Crystal structure of A

B compounds (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 orthogonal to each other.

Superconducting materials 85

accompanied only by a rearrangement of the crystal lattice, with the volume

of the crystal remaining unchanged. Immediately after the transition, there is a

softening of the elastic coefﬁcients of the lattice, as observed in acoustic mea-

surements. The Debye temperature of the A-15 superconductors is moderate,

300–500 K. It has been established that the more metastable lattice exhibits the

higher T

value. One of the main factors in increasing the T

value is the soft-

ening of the phonon spectrum, i.e. the phonon spectrum shifts to lower phonon

frequencies.

It is worth noting that the search of high-temperature superconductors in

materials exhibiting structural instabilities was already proposed in 1971 [28].

Even at that time, it was already known that metals with strong electron-lattice

coupling tend towards structural instabilities.

In spite of the fact that the A-15 compounds exhibit high critical temper-

atures and upper magnetic ﬁelds (see Table 3.1), they are not widely used in

applications because they are too brittle and therefore not ﬂexible enough to

be drawn into wires. In contrast, the alloy niobium-titanium (NbTi) with lower

= 9.5 K is easily drawn into wires; hence it is much more useful for appli-

cations. This problem was however solved for Nb

Sn and V

Ga by the use

of a technique called the bronze process. Magnets made from these wires can

produce magnetic ﬁelds of about 20 T at 4.2 K.

2.2 Metal oxide Ba

−x

BiO

In 1975, Sleight and co-workers discovered superconductivity in the metal

oxide BaPb

1−x

with a maximum T

 13.7 K at x = 0.25. Other

members of this family, BaPb

0.75

0.25

= 0.3 K) and Ba

1−x

BiO

(BKBO), were discovered in 1988. The metal oxide BKBO is an exceptionally

interesting material and the ﬁrst oxide superconductor without copper with a

critical temperature above that of all the A-15 compounds. Its critical temper-

ature is T

 32Katx = 0.4. At the moment of writing, BKBO still exhibits

the highest T

known for an oxide other than the cuprates. For the potassium

concentration x ≥ 0.35, this compound has the regular cubic perovskite struc-

ture sketched in Fig. 3.2. However, in one recent study the crystal structure of

BKBO has been found to be non-cubic and of the layered nature, having the

lattice parameters a ≈ a

and c ≈ 2a

, where a

is a simple cubic perovskite

cell parameter shown in Fig. 3.2.

Materials called perovskites are minerals (hard ceramics) whose chemical

formula is ABX

or AB

. Thus, perovskites contain three elements A, B, X

in the proportion 1:1:3 or 1:2:3. The atoms A are metal cations, and the atoms

B and X are nonmetal anions. The element X is often represented by oxygen.

The compound BKBO is a perovskite of type 1:1:3 with a part of the barium

atoms replaced by potassium atoms.

86 ROOM-TEMPERATURE SUPERCONDUCTIVITY

BiO

Figure 3.2. Cubic perovskite unit cell of Ba

1−x

BiO

Superconductivity in BKBO is observed near a metal-insulator transition.

The undoped parent compound for BKBO is BaBiO

containing an insulating

charge-density-wavephase formed of an ordered arrangement of non-equivalent

bismuth ions referred to as Bi

and Bi

. Superconducting BKBO with low

potassium content also exhibits a charge-density-wave ordering. The density of

charge carriers in BKBO is very low (see Fig. 3.6). Depending on x, the effec-

tive mass of charge carriers in BKBO is small, and can be even smaller than the

electron mass, m

∗

. Various evidence suggests that the electron-phonon

coupling is responsible for superconductivity in BKBO. For example, the iso-

tope effect in BKBO is sufﬁciently large: upon partial replacement of

O with

O in BKBO, the critical temperature is shifted down, and the isotope expo-

nent is about α ≈ 0.4. Above and below T

, BKBO exhibits a normal-state

pseudogap caused most likely by the electron-phonon coupling. A two-band

model applied to BKBO accounts very well for all the available data on BKBO.

Acoustic measurements performed in BKBO show that many physical prop-

erties of BKBO are quite similar to those of the A-15 superconductors. Exam-

ples are some structural instabilities just above the superconducting state, soft

phonon modes (acoustic in A-15 compounds and optical in BKBO), an an-

harmonicity of some phonons and large softening of the elastic constants. In

the A-15 compounds and BKBO, there is a structural phase transition slightly

above T

(martensite transition). For comparison, Table 3.2 lists some charac-

teristics of Nb

Ge (A-15 compound) and BKBO. In Table 3.2, one can see that

the characteristics of these superconductors have similar values.

2.3 Magnesium diboride MgB

In January 2001, magnesium diboride MgB

was found to superconduct at

= 39 K. The discovery was made by the group of Akimitsu in Tokyo. At

the moment of writing, the intermetallic MgB

has the highest critical temper-

Superconducting materials 87

Table 3.2. Characteristics of Nb

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

: the

critical temperature T

; the energy of the highest phonon peak ω

in the Eliashberg function

F (ω); the Fermi velocity v

; the coherence length ξ

; the gap ratio

2∆

and the upper

critical magnetic ﬁeld H

Compound T

(K) ω

(meV) v

(10

cm/s) ξ

◦

)

2∆

(T)

Ge 23 25 2.2 35–50 4.2 38

BKBO 32 70 3 35–50 4.5 32

MgB

39 90 4.8 35–50 4.5 39

ature at ambient pressure among all superconductors with the exception of the

cuprates. As shown in Fig. 3.3, the crystal structure of MgB

is very sim-

ple: it is composed of layers of boron and magnesium, alternating along the c

axis. Each boron layer has a hexagonal lattice similar to that of graphite. The

magnesium atoms are arranged between the boron layers in the centers of the

hexagons.

The physical properties of MgB

are also quite unique. The density of states

in MgB

is small. MgB

has a very low normal-state resistance: at 42 K the

resistivity of MgB

is more than 20 times smaller than that of Nb

Ge (A-15

compound) in its normal state. In the superconducting state, MgB

has a highly

anisotropic critical magnetic ﬁeld (∼ 7 times), and exhibits two energy gaps.

The gap ratio 2∆/(k

) for the larger gap ∆

is given in Table 3.2. For the

smaller gap ∆

, this ratio is around 1.7, so that ∆

/∆

2.7. Seemingly, both

the energy gaps have s-wave symmetries: the larger gap is highly anisotropic,

while the smaller one is either isotropic or slightly anisotropic. Band-structure

calculations of MgB

show that there are at least two types of bands at the

Fermi surface. The ﬁrst one is a heavy hole band, built up of boron σ orbitals.

MgB

View from the top

View from one side

Figure 3.3. Crystal structure of MgB

. Boron atoms form honeycomb planes, and magnesium

atoms occupy the centers of the hexagons in between boron planes.

88 ROOM-TEMPERATURE SUPERCONDUCTIVITY

The second one is a broader band with a smaller effective mass, built up mainly

of π boron orbitals. The larger energy gap ∆

occurs in the σ-orbital band,

while ∆

in the π-orbital band.

In MgB

, superconductivityoccurs in the boron layers. The electron-phonon

interaction seems to be responsible for the occurrence of superconductivity in

MgB

. For example, the boron isotope effect is sufﬁciently large, α  0.3 (the

Mg isotope effect is very small). The muon relaxation rate in MgB

is about

8–10 µs

−1

. So, in the Uemura plot (see Fig. 3.6), MgB

is literally situated be-

tween the large group of unconventional superconductors and the conventional

superconductor Nb.

In Table 3.2 which is presented for comparison of some characteristics of

the A-15 compounds, BKBO and MgB

, one can see that the characteristics of

these superconductors have similar values.

From the standpoint of practical application, magnesium diboride is very

attractive because MgB

has a very high critical temperature, and it is inex-

pensive to produce in large quantities since it is made from elements that are

abundant in nature. Because magnesium and diboride atoms are light, MgB

is light-weight.

2.4 Binary compounds

There are a large number of binary superconductors. Non-magnetic binary

compounds exhibiting high values of T

and H

most likely belong to the

second group of superconductors.

2.4.1 Nitrides and carbides

There is a number of superconducting binary compounds AB with the sodium

chloride structure shown in Fig. 3.4. The NaCl structure is a cubic face-

centered structure with alternating A and B elements in all directions. In crys-

tallography, such a structure is denoted as B1. In AB superconductors with

the NaCl structure, the A atom is one of the transition elements of the III, IV,

V and VI subgroups of the periodic table, and the B atom is a nontransitional

element. The highest critical temperature is observed in the binary compounds

with transition metals of the IV, V and VI subgroups: Zr, Nb, Mo, Ta and W,

which have incomplete 4d- and 5d-shells when they join nitrogen (nitrides) or

carbon (carbides). Like the A-15 compounds, these nitrides and carbides have

extraordinary properties in the normal and superconducting states. Table 3.3

gives the values of T

for some nitrides and carbides.

In fact, some nitrides and carbides do not have precisely the 1:1 stoichiom-

etry. For example, the NbN nitride listed in Table 3.3 cannot be prepared with

1:1 stoichiometry. Its exact formula is NbN

0.92

, so the structure has many

vacancies. Another example from Table 3.3 is vanadium nitride which is in

reality VN

0.75

. Vanadium carbide also has the non-exact 1:1 stoichiometry,

Superconducting materials 89

Table 3.3. Critical temperature T

for some nitrides and carbides

Nitride T

(K) Carbide T

(K)

NbN 17.3 MoC 14.3

ZrN 10.7

NbC 12.0

HfN 8.8

TaC 10.4

VN 8.5

WC 10.0

TaN 6.5

TlC 3.4

0.84

. Interestingly, if the vacancies in NbN

0.92

are ﬁlled by carbon to form

NbC

0.1

0.9

, the critical temperature increases to 17.8 K. The latter B1 com-

pound was synthesized by Matthias in 1953. During the fourteen years that

followed, it was one of the available superconductors with the highest T

.Itis

worth to mention that the ﬁrst nitride discovered to superconduct was NbN, in

1941.

As all compounds of the second group of superconductors, the B1 com-

pounds have two conduction bands formed by the d-electrons and sp-electrons.

The d-electron band is very narrow, while the sp-electron band is sufﬁciently

wide. Tunneling measurements carried out in some B1 compounds showed

that the electron-phonon interaction is mainly responsible for the occurrence

of superconductivity in these materials, and the dominant contribution to the

electron-phonon interaction parameter comes from acoustic phonons. The neu-

tron scattering studies of the phonon spectrum performed in the stoichiometric

carbides HfC (T

= 0.25 K) and TaC (T

= 10.3 K) conﬁrmed the dominant

role of the electron-phonon interaction for the occurrence of superconductivity

in these materials.

As opposed to the A-15 superconductors, the binary compounds with the

NaCl lattice are very stable and less sensitive to mechanical defects: the B1

B atom

A atom

Figure 3.4. Crystal structure of AB compounds (B1 superconductors). NaCl has the same

crystal structure.

90 ROOM-TEMPERATURE SUPERCONDUCTIVITY

materials are far more resistant to radiation and disorder than the A-15 com-

pounds. Nonetheless, the nitrides and carbides are also quite brittle and difﬁ-

cult to fabricate into wires, but they have been successfully exploited as thin

ﬁlms for superconducting electronics.

2.4.2 Laves phases

There are several dozen metallic AB

compounds called the laves phases

that superconduct. Some of them have a critical temperature above 10 K and

a high critical ﬁeld H

. For example, Zr

0.5

has T

= 10.1 K and

=24T, and the same compound with a different Zr:Hf ratio has sim-

ilar T

and H

values and the critical current density J

≈ 4 × 10

A/cm

The critical temperatures of the laves phases CaIr

and ZrV

are 6.2 K and

9.6 K, respectively. The AB

materials also have the advantage of not being so

hard and brittle as some other compounds and alloys with comparable critical

temperatures.

2.5 Semiconductors

There are a few semiconductors that become superconducting at very low

temperatures. The carrier concentration in semiconductors is much lower than

that in metals. Since semiconductors are not magnetic, they most likely belong

to the second group of superconductors. GeTe was the ﬁrst superconduct-

ing semiconductor discovered in 1964 , having a very low critical temperature

 0.1 K. This was followed by the discovery of superconductivity in the

perovskite SrTiO

with T

 0.3 K. The crystal structure of SrTiO

is cubic-

perovskite and similar to that of BKBO (see Fig. 3.2). However, by lowering

the temperature, it undergoes some phase transitions which destroy its cubic

symmetry. Doping SrTiO

by carriers, one can tune its critical temperature,

and the dependence T

(p) has a shape similar to that of the hyperbolic sine,

sinhp, with a sharp maximum of 0.3 K located at p ≈ 10

−3

. The criti-

cal temperature of some superconducting semiconductors can dramatically be

increased by applying an external pressure. In SrTiO

, the Ginzburg-Landau

parameter k is of the order of k ∼ 10, thus, SrTiO

is a type-II superconductor

and has H

(0) ∼ 400 Oe.

The tunneling studies carried out in Nb-doped SrTiO

with different Nb

concentrations clearly showed the presence of two-band superconductivity in

this compound.

The semiconductor SnTe also superconducts at low temperatures.

3. Third group of superconducting materials

The third group of superconductors is the largest and incorporates so-called

unconventional superconductors. A distinctive characteristic of unconventional

Superconducting materials 91

superconductors is that they are low-dimensional and magnetic, or at least,

these compounds have strong magnetic correlations. Furthermore, the den-

sity of charge carriers in these superconductors is very low. In unconventional

superconductors, spin ﬂuctuations mediate the onset of long-range phase co-

herence. In the majority of unconventional superconductors, the magnetic cor-

relations favor an antiferromagnetic ordering. In contrast to antiferromagnetic

superconductors, ferromagnetic ones usually have a low critical temperature.

Independently of the type of magnetic ordering, the pairing mechanism in un-

conventional superconductors is due to the electron-phonon interaction which

is moderately strong and non-linear. We shall discuss the mechanism of un-

conventional superconductivity in detail in Chapter 6. In all superconductors

belonging to this group, the coherence length is very short, while the pen-

etration depth is very large, so that all unconventional superconductors are of

type-II. They havea very largeupper critical magnetic ﬁeld. As a consequence,

many superconductors from this group are used for practical applications. We

start with the so-called Chevrel phases.

3.1 Chevrel phases

In 1971, Chevrel and co-workersdiscovereda new class of ternary molybde-

num sulﬁdes, having the general chemical formula M

, where M stands

for a large number of metals and rare earths (nearly 40), and x = 1 or 2. They

were called the Chevrel phases. The Chevrel phases with S substituted by Se or

Te also display superconductivity. Before the discovery of high-T

supercon-

ductivity in cuprates in 1986, the A-15 superconductors had the highest values

of T

, but the Chevrel phases were the record holders in exhibiting the highest

values of upper critical magnetic ﬁeld H

, listed in Table 3.4. The Chevrel

phases are of great interest, largely because of their striking superconducting

properties.

Table 3.4. Critical temperature and the upper critical magnetic ﬁeld of Chevrel phases

Compound T

(K) H

(T)

PbMo

15 60

LaMo

7 44.5

SnMo

12 36

LaMo

11 5

PbMo

3.6 3.8

The crystal structure of Chevrel phases, shown in Fig. 3.5, is quite interest-

ing. These compounds crystallize in a hexagonal-rhombohedral structure. The

building blocks of the Chevrel-phase crystal structure are the M elements and

molecular clusters. Each Mo

is a slightly deformed cube with X

92 ROOM-TEMPERATURE SUPERCONDUCTIVITY

PbMo

Figure 3.5. Crystal structure of the Chevrel phase PbMo

. Each lead atom is surrounded by

eight Mo

units (only four Mo

units are shown).

atoms at the corners, and Mo atoms at the face centers (these distortions of the

cubes are not shown in Fig. 3.5). Such a crystal structure leads to materials

particularly brittle, which give problems in the fabrication of wires. The elec-

tronic and superconducting properties of these compounds depend mainly on

the Mo

group, with the M ion having very little effect.

Superconductivity in the Chevrel phases coexists with antiferromagnetism

of the rare earth elements (in fact, superconductivity in the Chevrel phases is

mediated by magnetic ﬂuctuations). For example, a long-range antiferromag-

netic order of the rare earth elements RE = Gd, Tb, Dy and Er in (RE)Mo

setting in respectively at T

= 0.84, 0.9, 0.4 and 0.15 K, coexists with super-

conductivity occurring at T

= 1.4, 1.65, 2.1 and 1.85 K, respectively. T

the Ne



el temperature of an antiferromagnetic ordering. In HoMo

, for ex-

ample, the magnetic correlations of rare earth elements result in a long-range

ferromagnetic ordering. First, a non-uniform ferromagnetic phase appears in

the superconducting state of HoMo

. Then, on further cooling, a long-range

ferromagnetic order develops, destroying superconductivity. HoMo

is su-

perconducting only between two critical temperatures2Kand0.65 K. This

is called reentrant superconductivity. Below 0.65 K, the material is ferromag-

netic.

The superconductivity in the Chevrel phases is primarily associated with the

mobile 4d-shell electrons of Mo, while the magnetic order involves the local-

ized 4f-shell electrons of the rare earth atoms which occupy regular positions

throughout the lattice. In the normal state, the Chevrel phases exhibit a strong

Superconducting materials 93

softening in the elastic constants as a function of temperature for the longitu-

dinal and transverse modes.

The Chevrel phases and all superconductors of this group have one distinc-

tive characteristic: they all have a very low superﬂuid density. Furthermore,

the critical temperature of unconventional superconductors depends linearly

on superﬂuid density, and this dependence is universal for all superconductors

of this group, including the Chevrel phases. Let us consider this dependence.

From Eq. (2.9), the superﬂuid density n

directly relates to the penetration

depth, n

∝ 1/λ

. So, penetration-depth measurements are able to provide the

value of superﬂuid density. Carrying out muon-Spin-Relaxation (µSR) mea-

surements, Uemura and co-workers showed that the critical temperature in un-

conventional superconductors ﬁrst depends linearly on the superﬂuid density,

as shown in Fig. 3.6. However, on further increasing the superﬂuid density,

follows a “boomerang” path shown in the lower inset of Fig. 3.6. The

Uemura plot shows also that the concentration of charge carriers in unconven-

tional superconductors is more than one order of magnitude lower than that in

the metallic Nb superconductor. We shall continue to discuss the Uemura plot

to the end of this section.

HTS

0.05

0.1

0.5

1.5

UPt

2223

123

2212

214

BKBO

Chevrel

BEDT

150

100

(K)

Relaxation rate,

→

0) (

-1

)

UBe

Opt

Figure 3.6. Critical temperature versus muon-spin-relaxation rate σ(T → 0) for various su-

perconductors (σ ∝ 1/λ

∝ n

∗

). The cuprates are marked by 214, 123, 2212 and 2223

(see Table 3.5). BEDT is a layered organic superconductor. Heavy fermions are shown in

the upper inset (HTS = high-T

superconductors). For the cuprates, the lower inset shows the

“boomerang” path with increasing doping: the underdoped (UD), optimally doped (Opt) and

overdoped (OD) regions [29].