Mourachkine A. Room-Temperature Superconductivity

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124 ROOM-TEMPERATURE SUPERCONDUCTIVITY

the magnetic order are found to be homogeneous. Chu and his collaborates

suggested that a bulk Meissner effect does not exist in the ruthenocuprates.

Indeed, the value of the penetration depth is very large, λ ∼ 30–50 µm. In

principle, this can happen if metallic RuO

layers remain normal below T

The situation with the ruthenocuprates is even worse than that with Sr

RuO

If in the Sr ruthenate, there is disagreement only on the type of spin ﬂuctua-

tions that mediate superconductivity, in the ruthenocuprates there are two ma-

jor problems. First, the type of ordering at T

still remains controversial. Ear-

lier experimental studies suggested a homogeneous ferromagnetic ordering of

the Ru moments, while the latest ones report that the magnetic order of the Ru

spins is predominantly antiferromagnetic. Second, from the beginning it was

assumed that superconductivity occurs exclusively in the CuO

planes. How-

ever, recent NMR studies reveal that the superconducting gap develops also at

the magnetically ordered RuO

planes with a ferromagnetic component.

There is a consensus that in the ruthenocuprate, there is a small ferromag-

netic component; however, there is no agreement on its origin. It may origi-

nate not only from the Ru moments but also, for example, from the Gd spins.

There are many reports on this issue, which often contradict one another. In an

attempt to reconcile these discrepancies, it was suggested that the RuO

lay-

ers ordered ferromagnetically couple antiferromagnetically. In RuSr

RCu

with R = Gd and Y, neutron scattering studies found that these two compounds

have an antiferromagnetic ground state with a very small canting ferromag-

netic component, and that an external magnetic ﬁeld can tune the ﬁeld-induced

ferromagnetic component that coexists with superconductivity in a high ﬁeld.

3.12 MgCNi

MgCNi

is the second most recent superconductor described in this chapter,

after Cd

(see the following subsection). Superconductivity in MgCNi

was discovered in 2001 by Cava and co-workers, a few months later than that

in MgB

. The crystal structure of MgCNi

is cubic-perovskite, and similar to

that of BKBO (see Fig. 3.2). The perovskite MgCNi

is special in that it is

neither an oxide nor does it contain any copper. Since Ni is ferromagnetic, the

discovery of superconductivity in MgCNi

was surprising. The critical tem-

perature is near 8 K. MgCNi

is metallic, and the charge carriers are electrons

which are derived predominantly from Ni.

The estimated values of the coherence length and upper critical ﬁeld in

MgCNi

are ξ ≈ 46 A

◦

and H

 15 T, respectively. Penetration-depth mea-

surements at microwave frequencies show unambiguously that superconduc-

tivity in MgCNi

is not of the BCS type, and λ(0) = 2480 A

◦

. In Andreev-

reﬂection measurements performed on polycrystalline samples (single crystals

of MgCNi

are not yet available), a zero-bias conductance peak was observed.

In analogy with the cuprates, the presence of this peak in a conductance indi-

Superconducting materials 125

cates a d-wave symmetry of the energy gap. The gap ratio obtained in these

Andreev-reﬂection measurements, 2∆/k

∼ 10, is very large. However,

this value is only an estimate because it is obtained in polycrystalline samples.

The Debye temperature obtained from speciﬁc-heat measurements is Θ 

284 K, and the value of speciﬁc-heat jump at T

is β  2.1. At low tem-

perature, the Ginzburg-Landau parameter is k = 54.

Structural studies of MgCNi

reveal structural inhomogeneity. Apparently,

the perovskite cubic structure of MgCNi

is modulated locally by the variable

stoichiometry on the C sites.

3.13 Cd

is the most recent superconductor described in this chapter. Al-

though Cd

was synthesized in 1965, its physical properties remained

almost unstudied. Unexpectedly, superconductivity in Cd

was discov-

ered in the second half of 2001 by Sakai and co-workers. The critical tem-

perature of Cd

is low, T

=1–1.5 K. This compound is the ﬁrst super-

conductor found among the large family of pyrochlore oxides with the formula

, where A is either a rare earth or a late transition metal, and B is a

transition metal. In this structure, the A and B cations are 4- and 6-coordinated

by oxygen anions. The A-O

tetrahedra are connected as a pyrochlore lattice

with straight A-O-A bonds, while B-O

octahedra form a pyrochlore lattice

with the bent B-O-B bonds with an angle of 110–140

◦

. Assuming that elec-

tronic structure in Cd

as formally Cd

and Re

, the

electronic and magnetic properties are primarily dominated by the Re 5d elec-

trons. Cd

shows an anomaly at 200 K in electrical resistivity, magnetic

susceptibility, speciﬁc heat and Hall coefﬁcient: there is a structural phase tran-

sition near 200 K. Another structural phase transition occurs around 1.5 K, just

above the superconducting transition.

Oxide superconductors with non-perovskite structure are rare. Previous

studies indicate that the pyrochlores, like the spinels, are geometrically frus-

trated. The effect of geometric frustration on the physical properties of spinel

materials is drastic, resulting in, for example, heavy-fermion behavior in

LiV

. Another spinel compound LiTi

is a superconductor below T

13.7 K. Indeed, x-ray diffractionstudies performed underhigh pressure showed

that superconductivity in Cd

is detected only for the phases with a

structural distortion. It was suggested that the charge ﬂuctuations of Re ions

play a crucial role in determining the electronic properties of Cd

Between 2 and 60 K, the resistivity in Cd

exhibits a T

dependence

indicative of the Fermi-liquid behavior. The value of the speciﬁc-heat jump at

, 1.29, is close to the weak coupling BCS value. In comparison with other

pyrochlores, the value of the Sommerfeld constant γ for Cd

is large,

suggesting that the electrons in Cd

are strongly correlated with the en-

126 ROOM-TEMPERATURE SUPERCONDUCTIVITY

hanced effective mass, resulting possibly from geometric frustration. The Re

nuclear quadrupole resonance (NQR) measurements performed in zero mag-

netic ﬁeld below 100 K rule out any magnetic or charge order. Speciﬁcheat

and Re NQR measurements suggest that the superconducting gap in Cd

is almost isotropic.

The value of the coherence length in Cd

is ξ

∼ 260 A

◦

. The value

of the penetration depth is very large, λ(0) ∼ 7500 A

◦

. The lower and upper

magnetic ﬁelds are H

≤ 0.002 T and H

≈ 0.85 T, respectively.

3.14 Hydrides and deuterides

In addition to the nitrides and carbides from the second group of supercon-

ductors, another class of superconducting compounds that also has the NaCl

structure are hydrides and deuterides (i.e. compounds containing hydrogen or

deuterium). However, in contrast to the nitrides and carbides, superconduct-

ing hydrides and deuterides are magnetic. In the seventies it was discovered

that some metals and alloys, not being superconducting in pure form, become

relatively good superconductors when they form alloys or compounds with hy-

drogen or deuterium. These metals include the transition elements palladium

(Pd) and thorium (Th) that have unoccupied 4d- and 5f-electron shells, respec-

tively.

In 1972, Skoskewitz discovered that the transition element Pd which has

a small magnetic moment normally preventing the pairing of electrons, joins

hydrogen and forms the PdH compound that superconducts at T

= 9 K. This

compounds has the NaCl cubic structure, thus it is a B1 compound. Later on, it

was found that by doping such a system with noble metals the critical temper-

ature increases up to 17 K. Interestingly, the palladium-deuterium compound

also superconducts, and its critical temperature equal to 11 K is higher than

that of PdH. So the hydrogen isotope effect in PdH is reverse (negative), which

is similar to that observed in some organic superconductors. In contrast, the

critical temperatures of the ThH and ThD compounds do not differ drastically

from each other like those of PdH and PdD. Probably, the higher atomic mass

of thorium is the cause of this discrepancy.

The experimental studies of the Pd

1−x

hydrides, where M = Al, Pb, In

and Cu, showed that, as one increases the M concentration from zero, their crit-

ical temperatures as a function of x ﬁrst increase and then drop quite sharply.

This study was performed under the most favorable hydrogen concentrations

that correspond to the maximum value of T

with ﬁxed x. The critical temper-

ature of the Pd

1−x

deuteride as a function of x also exhibits the same

tendency as those of the doped palladium hydrides. However, in the doped hy-

drides and deuterides, their critical temperatures reach maximums at different

dopant concentrations. This is due to the fact that different metals have differ-

ent abilities to donate electrons. However, such a bell-like shape of the T

(p)

Superconducting materials 127

dependence, where p is the doping level, is typical for all the compounds of

the third group of superconductors.

The hydrides and deuterides have two conduction bands as the heavy fermi-

ons do: the wide valence band of s- and p-electrons and the very narrow band

generated by electrons in the inner 4d- and 5f -subshells incompletely occupied

by electrons. The existence of two conduction bands is also typical for the

second-group superconductors.

3.15 Oxides

Superconducting oxides are a special family of superconductors. The class

of superconducting compounds containing the element oxygen O is probably

the largest family among all superconducting materials. Furthermore, they ex-

hibit the highest critical temperatures (cuprates). Representatives of this family

of superconductors are members of either the second (BKBO, SrTiO

) or third

group of superconductors (cuprates, ruthenates, etc.). NbO was the ﬁrst su-

perconducting oxide discovered in 1965 by Miller and his collaborators. The

oxide series, beginning from NbO, is shown in Fig. 1.2.

Usually, oxides are associated with insulators, while superconductors with

the best conductors such as Cu, Ag and Au. However, the opposite is true.

In materials with a weak phonon-electron interaction, like Cu, Ag and Au,

superconductivity is absent, while materials with a moderately strong phonon-

electron interaction, like oxides, exhibit sometimes superconductivity. This

fact shows the importance of the electron-lattice interaction for superconduc-

tivity.

Chapter 4

PRINCIPLES OF SUPERCONDUCTIVITY

The issue of room-temperature superconductivity is the main topic of this

book. Even if this subject was raised for the ﬁrst time before the development

of the BCS theory and later by Little in 1964 [2], from the standpoint of prac-

tical realization, this issue is still a new, “untouched territory.” To go there, we

need to know Nature’s basic rules for arrangement of matter over there. Other-

wise, this journey will face a ﬁasco. To have the microscopic BCS theory in a

bag is very useful, but not enough. It is clear to everyone by now that a room-

temperature superconductor can not be of the BCS type. Therefore, we need to

know more general rules, principles of superconductivity that incorporate also

the BCS-type superconductivity as a particular case.

The purpose of this chapter is to discuss the main principles of supercon-

ductivity as a phenomenon, valid for every superconductor independently of

its characteristic properties and material. The underlying mechanisms of su-

perconductivity can be different for various materials, but certain principles

must be satisﬁed. One should however realize that the principles of supercon-

ductivity are not limited to those discussed in this chapter: it is possible that

there are others which we do not know yet about.

The ﬁrst three principles of superconductivity were introduced in [19].

1. First principle of superconductivity

The microscopic theory of superconductivity for conventional superconduc-

tors, the BCS theory, is based on Leon Cooper’s work published in 1956. This

paper was the ﬁrst major breakthrough for understanding the phenomenon of

superconductivity on a microscopic scale. Cooper showed that electrons in a

solid would always form pairs if an attractive potential was present. It did not

matter if this potential was very weak. It is interesting that, during his cal-

129

130 ROOM-TEMPERATURE SUPERCONDUCTIVITY

culations, Cooper was not looking for pairs—they just “dropped out” of the

mathematics. Later it became clear that the interaction of electrons with the

lattice allowed them to attract each other despite their mutual Coulomb repul-

sion. These electron pairs are now known as Cooper pairs.

An important note

: in this chapter and further throughout the book, we

shall use the term “a Cooper pair” more generally than its initial meaning. In

the framework of the BCS theory, the Cooper pairs are formed in momentum

space, not in real space. Further, we shall consider the case of electron pairing

in real space. For simplicity, we shall sometimes call electron pairs formed in

real space also as Cooper pairs.

In solids, superconductivity as a quantum state cannot occur without the

presence of bosons. Fermions are not suitable for forming a quantum state

since they have spin and, therefore, they obey the Pauli exclusion principle

according to which two identical fermions cannot occupy the same quantum

state. Electrons are fermions with a spin of 1/2, while Cooper pairs are already

composite bosons since the value of their total spin is either 0 or 1. Therefore,

the electron pairing is an inseparable part of the phenomenon of superconduc-

tivity and, in any material, superconductivity cannot occur without electron

pairing.

In some unconventional superconductors, the charge carriers are not elec-

trons but holes with a charge of +|e| and spin of 1/2. The reasoning used

above for electrons is valid for holes as well. Thus, in the general case, it

is better to use the term “quasiparticles” which also reﬂects the fact that the

electrons and holes are in a medium.

The ﬁrst principle of superconductivity:

Principle 1:

Superconductivity requires quasiparticle pairing

In paying tribute to Cooper, the ﬁrst principle of superconductivity can be

called the Cooper principle.

In the framework of the BCS theory, the quasiparticle (electron) pairing oc-

curs in momentum space, not in real space. Indeed in the next section, we shall

see that the electron pairing in conventional superconductors cannot occur in

real space because the onset of long-range phase coherence in classical super-

conductors occurs due to the overlap of Cooper-pair wavefunctions, as shown

in Fig. 4.1. As a consequence, the order parameter and the Cooper-pair wave-

functions in conventional superconductors are the same: the order parameter

is a “magniﬁed” version of the Cooper-pair wavefunctions. However, in un-

conventional superconductors, the electron pairing is not restricted by the mo-

mentum space because the order parameter in unconventional superconductors

has nothing to do with the Cooper-pair wavefunctions. Generally speaking, the

electron pairing in unconventional superconductors may take place not only in

Principles of superconductivity 131

Pair

phase

(

)

Cooper-pair

wavefunction

(

)

Figure 4.1. In conventional superconductors, the superconducting ground state is composed

by a very large number of overlapping Cooper-pair wavefunctions, ψ(r). To avoid confusion,

only three Cooper-pair wavefunctions are shown in the sketch; the other are depicted by open

circles. The phases of the wavefunctions are locked together since this minimizes the free

energy. The Cooper-pair phase Θ(r), illustrated in the sketch, is also the phase of the order

parameter Ψ(r).

momentum space but also in real space. We shall discuss such a possibility in

the following section.

The electron pairing in momentum space can be considered as a collective

phenomenon, while that in real space as individual. We already know that

the density of free (conduction) electrons in conventional superconductors is

relatively high (∼ 5 × 10

−3

); however, only a small fraction of them

participate in electron pairing (∼ 0.01%). In unconventional superconductors

it is just the other way round: the electron density is low (∼ 5 × 10

−3

)

but a relatively large part of them participate in the electron pairing (∼ 10%).

Independently of the space where they are paired—momentum or real—two

electrons can form a bound state only if the net force acting between them is

attractive.

2. Second principle of superconductivity

After the development of the BCS theory in 1957, the issue of long-range

phase coherence in superconductors was not discussed widely in the literature

because, in conventional superconductors, the pairing and the onset of phase

coherence take place simultaneously at T

. The onset of phase coherence in

conventional superconductors occurs due to the overlap of Cooper-pair wave-

132 ROOM-TEMPERATURE SUPERCONDUCTIVITY

functions, as shown in Fig. 4.1. Only after 1986 when high-T

superconduc-

tors were discovered, the question of electron pairing above T

appeared. So, it

was then realized that it is necessary to consider the two processes—the elec-

tron pairing and the onset of phase coherence—separately and independently

of one another [12].

In many unconventional superconductors, quasiparticles become paired

above T

and start forming the superconducting condensate only at T

. Su-

perconductivity requires both the electron pairing and the Cooper-pair conden-

sation. Thus, the second principle of superconductivity deals with the Cooper-

pair condensation taking place at T

. This process is also known as the onset

of long-range phase coherence.

Principle 2:

The transition into the superconducting state is

the Bose-Einstein-like condensation and occurs

in momentum space

Let us ﬁrst start with one main difference between fermions and bosons.

Figure 4.2 schematically shows an ensemble of fermions and an ensemble of

bosons at T  0andT = 0. In Fig. 4.2 one can see that, at high temperatures,

both types of particles behave in a similar manner by distributing themselves in

their energy levels somewhat haphazardly but with more of them toward lower

energies. At absolute zero, the two types of particles rearrange themselves in

their lowest energy conﬁguration. Fermions obey the Pauli exclusion principle.

Therefore, at absolute zero, each level from the bottom up to the Fermi energy

is occupied by two electrons, one with spin up and the other with spin

down, as shown in Fig. 4.2. At absolute zero, all energy levels above the Fermi

level are empty. In contrast to this, bosons do not conform to the exclusion

principle, therefore, at absolute zero, they all consolidate in their lowest energy

state, as shown in Fig. 4.2. Since all the bosons are in the same quantum

state, they form a quantum condensate (which is similar to a superconducting

condensate). In practice, however, absolute zero is not accessible.

We are now ready to discuss the so-called Bose-Einstein condensation.In

the 1920s, Einstein predicted that if an ideal gas of identical atoms, i.e. bosons,

at thermal equilibrium is trapped in a box, at sufﬁciently low temperatures

the particles can in principle accumulate in the lowest energy level (see Fig.

4.2). This may take place only if the quantum wave packets of the particles

overlap. In other words, the wavelengths of the matter waves associated with

the particles—the Broglie waves—become similar in size to the mean particle

distances in the box. If this happens, the particles condense, almost motionless,

Principles of superconductivity 133

Fermions

>> 0

= 0

Bosons

>> 0

= 0

Figure 4.2. Sketch of the occupation of energy levels for fermions and bosons at high temper-

atures and absolute zero. Arrows indicate the spin direction of the fermions. For simplicity, the

spin of the bosons is chosen to be zero. E

is the Fermi level for the fermions.

into the lowest quantum state, forming a Bose-Einstein condensate. So, the

Bose-Einstein condensation is a macroscopic quantum phenomenon and, thus,

similar to the superconducting condensation.

For many decades physicists dreamt of cooling a sufﬁciently large number

of ordinary atoms to low enough temperatures to undergo the Bose-Einstein

condensation spontaneously. During 1995 this was accomplished by three

groups acting independently. The ﬁrst Bose-Einstein condensate was formed

by using rubidium atoms cooled to 2 × 10

−9

The superconducting and Bose-Einstein condensates have much in common

but also a number of differences. Let us start with their similarities. Firstly, the

superconducting and Bose-Einstein condensations are both quantum phenom-

ena occurring on a macroscopic scale. Thus, every Bose-Einstein condensate

exhibits most of the superconducting-state properties described in Chapter 2.

Secondly, the superconducting and Bose-Einstein condensations both occur in

momentum space, not in real space. What is the difference between a con-

densation in momentum space and one in real space? For example, the vapor-

liquid transition is a condensation in ordinary space. After the transition, the

average distance between particles (atoms or molecules) is changed—becomes

smaller when the vapor condenses and larger when the liquid evaporates. So, if

a condensation takes place in real space, there may be some noticeable changes