Full text of DOI:10.1007/PL00011068

Eur. Phys. J. B 17, 85–94 (2000)
THE EUROPEAN
PHYSICAL JOURNAL B
EDP Sciences
Societa` Italiana di Fisica
Springer-Verlag 2000
c
ꢀ
On electron pairing in unconventional superconductors
P. Brovettoa, V. Maxia, and M. Salis
Istituto Nazionale di Fisica della Materia, 09124 Cagliari, Italy
and
Istituto di Fisica Superiore dell’Universit`a di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy
Received 3 November 1999 and Received in final form 18 May 2000
Abstract. This paper presents a study whose aim is to decide whether the basic mechanism of electron
pairing is the same for all unconventional superconductors such as perovskites, cuprates, organic com-
pounds and alkaly-stuffed fullerene. By analyzing the features of these dissimilar materials, arguments
are singled out showing that superconduction is originated by electrons combined in weakly bound pairs
running in regions bordering on certain lattice discontinuities which are common to the structure of the
superconductors dealt with. Special attention is devoted to the properties of the YBCO cuprate.
PACS. 74.20.-z Theories and models of superconducting state
1 Introduction
quite low temperatures, that is 0.25 - 0.28 K [5]. In 1975,
the mixed oxide BaPb0.7Bi0.3O3 showing 13 K super-
conduction was singled out [6]. After the discovery of
cuprates in 1986 [7], superconduction was detected at
30 K in the Ba0.6K0.4BiO3 compound [8]. Recently, an
analogous result was reached with Sr1−xKxBiO3 (x =
0.45 - 0.6) and Sr0.5Rb0.5BiO3 compounds, which su-
perconduct at about 12 K and 13 K, respectively [9].
Organic superconductors are quite different materials.
Let us mention, in this connection, the Bechgaard salt,
that is, tetramethyl-tetraselena-fulvalene hexafluorophos-
phate (TMTSF)2PF6, which superconducts at about
1 K [10]. But the most striking case is that of alkali-stuffed
fullerene. With KxC60 and RbxC60 (x ≤ 3), superconduc-
tion was observed at 18 K and 28 K, respectively [11]. Of
course, the previous list of superconductors is completed
with cuprates. On the whole, the superconductors men-
tioned in Table 1 can be divided into four groups, that is,
1 - 5 perovskites, 6 organic compounds, 7 - 8 fullerenes,
9 - 10 cuprates. Therefore, four kinds of superconduction
mechanisms should be considered in principle. It is obvi-
ous, however, that a unitary explanation is preferable. It
is to be searched for in something that is common to all
materials notwithstanding their very different natures. For
this reason, some special features of the superconductors
dealt with are now examined.
Till now, no definitive explanation has been found for high
Tc superconduction in cuprates [1]. The most accredited
opinion is that superconduction in these materials arises
from singlet electron pairs as in the BCS theory, but with
a pairing mechanism different and more effective than
electron-phonon coupling. Such a mechanism should con-
vert the electron-lattice interaction into a binding force
that stabilizes the pairs which are thus allowed to move
freely through the lattice as do Cooper’s pairs. According
to BCS theory, the superconducting ground state is sep-
arated from the lowest excited state by an energy gap of
amplitude depending on binding force strength [2]. This
result follows directly from the Bogolyubov-Valatin (BV)
transformation, which shows that an interacting electron
system is equivalent to a set of non-interacting quasi-
particles [3,4]. The BV transformation is quite general and
can be applied to different fermion systems independently
of the actual pairing mechanism. As a consequence, if a
suitable pairing mechanism is singled out, the outcome of
BCS theory, that is, the existence of a gap, the critical
temperature-gap amplitude relationship, the intrinsic co-
herence length and so on, remain valid for cuprates as well.
Cuprates surely represent the most interesting super-
conductors as they show the highest Tc so far recorded.
But various unconventional superconductors other than
cuprates are known. Although these materials show low
critical temperatures, they are worthy of consideration be-
cause their peculiarities may supply data useful in finding
a general explanation of the electron pairing mechanism
suitable for all unconventional superconductors. Table 1
below gives a short list of the superconductors dealt with.
Actually, superconduction has been observed since
1964 in reduced strontium titanate, SrTiO3−δ, but at
2 Features common to unconventional
superconductors
2.1 Electronic configuration
Let us first consider the electronic configurations of atoms
and ions sharing in the material compositions. The re-
duced strontium titanate cell consists of a perovskitic cube
a
e-mail: brovetto@vaxcal.unica.it

86
The European Physical Journal B
Table 1. Some unconventional superconductors.
Superconductor Tc (K) Electronic configuration
Structure
1
2
SrTiO3−δ
BaPb0.7Bi0.3O3
Ba0.6K0.4BiO3
Sr0.5K0.5BiO3
Sr0.5Rb0.5BiO3
(TMTSF)2PF6
KxC60
0.28
13
Ti+3 → [Ar] 3d1
Dishomogeneous perovskite
”
Bi+4 → [Xe] 4f145d106s1
3
30
12
”
”
4
”
”
”
5
13
”
(TMTSF)+1 free radical
K → [Ar] 4s1
Rb → [Kr] 5s1
Cu+2 → [Ar] 3d9
”
6
∼ 1
18
Molecular stackings
Stacking of C60 spheres
”
7
8
RbxC60
28
9
10
La2CuO4
YBa2Cu3O7
35
92
Layered (K2NiF4)
Layered perovskite
with a strontium ion at the centre. The Ti+4 ions at the ichiometries or with complex anisotropic cells. Actually,
cube vertices show the [Ar] configuration, but, owing to a fractional cell stoichiometry is common to all the per-
the partial lack of oxygen, a number of Ti+3 ions with ovskitic oxides mentioned previously, that is, SrTiO3−δ
,
the [Ar] 3d configuration, showing an excess 3d-electron, BaPb0.7Bi0.3O3 and so on. In Bechgaard salt we find
is present as well. This means that unpaired electrons stackings of strongly bound molecules with a much weaker
are found in some cells. In the mixed perovskitic oxide intermolecular bonding in directions transverse to the
BaPb0.7Bi0.3O3, bismuth is forced to assume valence four stackings. Alkaly stuffed fullerene, in turn, shows a stack
like lead. But, while Pb+4 ions show the [Xe] 4f145d10 of C60 balls in which links between carbons in contigu-
configuration, Bi+4 ions show the [Xe] 4f145d106s con- ous balls are weaker than links between carbons in the
figuration, in which an unpaired 6s-electron is present. same ball. As for cuprates, the La2CuO4 compound shows
Analogous arguments apply to Ba0.6K0.4BiO3 oxide in a layered structure like that of K2NiF4, while YBCO is
which 60% of bismuth ions are tetravalent. The situation characterized by a layered structure showing couples of
is quite similar for compounds with the barium-strontium contiguous CuO2 planes intercalated with barium-centred
and potassium-rubidium substitutions, which again hold perovskitic cubes. It follows that the presence of faults
tetravalent bismuth. Unpaired electrons are found in all in structure continuity is indeed a feature common to all
these perovskitic oxides. In the Bechgaard salt one elec- the superconductors considered above. In reality, leaving
tron is transferred from one TMTSF molecule to one fluo- metallic solids aside, no superconductor showing a simple
rine atom so that (TMTSF)+1 cations and (PF6)−1 anions regular lattice is known. Let us mention, in this connec-
appear. Since in neutral TMTSF molecules all electrons tion, the divalent copper oxide which is characterized by a
are coupled in σ− or π−bonds, one unpaired electron is monoclinic lattice (S.G. C12/c1, monoclinic no. 15). This
present in the (TMTSF)+1 cation, which consequently be- material, which can be regarded as the simplest cuprate,
haves as a free radical. In the alkaly stuffed fullerene, un- is not a superconductor. This state of affairs leads us to
paired electrons are inserted in the material by the al- conclude that faults in structure continuity are essential
kaly atoms themselves. Potassium and rubidium are in for the appearance of superconduction in the materials
fact characterized by the [Ar] 4s and [Kr] 5s configura- dealt with.
tions showing unpaired 4s- and 5s-electrons, respectively.
Cuprates consist of a wide variety of metal oxides in which
copper is kept in perovskitic structures. For brevity’s
3 Superconducting electron pairs
sake, we will consider only the first compound discovered,
La2CuO4 [7] and the YBCO cuprate, YBa2Cu3O7, which
is perhaps the most representative material. Their stoi-
chiometry includes divalent copper showing the [Ar] 3d9
configuration in which an unpaired electron is found in a
half-filled 3d-state. The conclusion follows that the pres-
ence of unpaired electrons is indeed a feature common to
all materials considered up to now. What is more, inert
compounds, such as BaPbO3 and C60, become supercon-
ductors when treated in such a way that unpaired elec-
trons appear in them.
A further point which is to be considered, in order to sin-
gle out the right electron pairing mechanism, is the strong
effect on critical temperature of some atomic substitutions
shown by the superconducting compounds in Table 1. In-
deed, partial substitution of Ba with K and of Pb with Bi
between superconductors 2 and 3 originates an increase of
Tc from 13 K to 30 K. Likewise, substitution of Ba with Sr
between superconductors 3 and 4 decreases Tc from 30 K
to 12 K. A similar effect is originated by the K to Rb sub-
stitution in the alkaly-stuffed fullerene, which increases Tc
from 18 K to 28 K. This means that in these supercon-
ductors electrons are tightly coupled to the atoms present
in the lattice structure. These electrons, therefore, can
be conveniently represented by tight-binding (TB) wave
2.2 Structure
But another notable feature equalises these materials. All functions, that is, linear combinations of atomic orbitals
show uneven lattices with a blend of cells of different sto- with appropriate wavy phase factors. Keeping in mind

P. Brovetto et al.: On electron pairing in unconventional superconductors
87
that atomic orbitals with unpaired electrons tend to form where plus or minus signs and S (1, 2) or T(1, 2) spin fac-
spin-singlet bonds, the previous considerations induce us tors mean spin singlet or triplet states, respectively. The
to argue that at low temperature electrons running in a re- pair energy is given by the expectation value of the two-
gion bordering on a lattice discontinuity originate bound electron Hamiltonian
pairs with electrons running in the region bordering on
N
N
Zb e2
Za e2
e2
X
X
the opposite side of the lattice discontinuity. Bond stabil-
ity depends on exchange integrals which, in turn, depend
on Coulomb interactions of electrons with the lattice. Con-
sequently, the electron-lattice interaction is shut up within
the pairs, which are thus allowed to move freely through
the lattice. Below are presented some simple equations
dealing with this pairing mechanism.
b
Ha, b(r1, r2) = −
−
+
·
| r1 − vq |
| r2 − up | r1, 2
q=1
p=1
(5)
Therefore, by taking into account only the most significant
exchange integrals, that is,
Let us consider, within a lattice composed of ions of
both signs, two regions a and b separated by a lattice
discontinuity. By assuming that electrons moving in these
regions interact mainly with ions of lattice vectors up and
vq, respectively, their TB wave functions, φa and φb, can
be written as
N
Za e2
X
Ka = − hφb(kb, r2) |
Kb = − hφa(ka, r1) |
| φa(ka, r2)i,
| φb(kb, r1)i
| r2 − up |
p=1
N
Zb e2
X
(6)
| r1 − vq |
q=1
N
X
1
and
a·up
√
φa(ka, r1) =
φb(kb, r2) =
eik
a(r1 − up),
N
N
e2
p=1
Ka0 , b = hφa(ka, r1)φb(kb, r2) |
| φa(ka, r2)φb(kb, r1)i,
N
X
r1, 2
1
eik ·v b(r2 − vq),
(1)
b
q
√
(7)
q=1
we have
a(r1 − up) and b(r2 − vq) standing for the atomic orbitals
and ka and kb for the electron wave vectors. The directions
of ka and kb are obviously related to the geometry of a and
b regions. With a layered geometry, ka and kb lie parallel
to the layers. Normalization factors are determined by as-
suming that there is not much overlap between orbitals in
neighbouring lattice positions. Energies of TB functions
are given by the expectation values of Hamiltonians
b
Wpair = hφa(ka, r1) φb(kb, r2) | Ha, b(1, 2)
× | φa(ka, r2)φb(kb, r1)i + h.c.
ꢂ
ꢃ
= ꢀ R e hφa | φbiKa + hφb | φaiKb + Ka0 , b . (8)
Obviously, electron pairing occurs if Wpair is negative.
Pairing energy is essentially an exchange or resonance en-
ergy. Its actual value depends on the distance between
a and b regions. But also the electron momenta play a
role. We point out, in this connection, that TB functions,
neglecting a slight modulation like that of Bloch func-
tions, can be approximated by plane waves. In fact, by
taking into account that orbitals a(r1 − up) are closely lo-
calized at the lattice positions up, only those orbitals for
which up ' r1 give a significant contribution in the sum
of equation (1). Therefore, by letting such orbitals lie in
the interval p0 ≤ p ≤ p00, by putting r1 − up = ꢀp and by
considering that | ꢀp |ꢁ| r1 |, we get
N
2
X
pb
Za e2
1
b
Ha =
−
−
+ Vlat(r1),
+ Vlat(r2),
2m
| r1 − up |
p=1
N
2
pb
Zb e2
X
2
b
Hb =
(2)
2m
| r2 − vq |
q=1
that is,
ꢀ
ꢀ
ꢀ
ꢀ
b
W (k ) = hφ (k , r ) H φ (k , r )i,
ꢀ
ꢀ
a
a
a
a
1
a
a
a
1


p=p00
ꢀ
ꢀ
X
ꢀ
ꢀ
1
b
ika·r1
W (k ) = hφ (k , r ) H φ (k , r )i,
(3)


ꢀ
ꢀ
b
b
b
b
2
b
b
b
2
√
φa(ka, r1) =
a(ꢀp)
e
,
(9)
N
p=p0
Vlat(r) standing for the lattice potential due to ions in
positions other than up and vq. On this ground, in anal-
ogy with the Heitler-London treatment of the hydrogen
molecule [12] and disregarding the squared overlap inte-
grals hφa | φbi of TB functions against unity, the pair wave
function can be written as
where the factor in brackets is nearly constant.
Consequently
hφa | φbi ∝ δk
,
(10)
a,kb
which, owing to equation (8), means that only electrons
with alike k, that is, alike momenta, allow for significant
pairing energies. In the following, for simplicity, we restrict
ourselves to this case.
It is clear that the pairs considered here have nothing
to do with the Cooper’s pairs which concern electrons with
1
2
√
Ψ(r1, r2) =
[φa(ka, r1)φb(kb, r2)
ꢁ
S(1, 2)
T(1, 2)
ꢀφa(ka, r2)φb(kb, r1)]
(4)

88
The European Physical Journal B
∼
opposite momenta. On the contrary, some likeness exists
with the nucleon-nucleon pairs in nuclear matter. Actu-
ally, the nuclear forces, after separation of a self-consistent
part, leave a residual interaction between nucleons which
is rather weak, but which plays an important role in
various nuclear properties. This residual interaction is es-
sentially an exchange interaction owing to the peculiar
nature of nuclear forces (Majorana forces). It couples nu-
cleons with like momenta but with opposite projections
of angular momenta along the quantization axis. There-
fore, by dropping the boldface types, the coupled nuclear
states can be marked with the indexes: k1+, k2−, instead
of −k, +k as the Cooper’s pairs. An exhaustive treatment
of pairing correlations in nuclear matter was performed by
Belyaev [13] on the basis of the BV transformation1. This
treatment can be directly applied to the electron pairs
dealt with provided that components σ = + and σ = −
of angular momentum are reinterpreted as the labels a
and b of the neighbouring lattice regions2. This trick is al-
lowed by the merely formal character of the BV transfor-
mation which leaves the physical nature of the pairs out of
consideration. Therefore, by performing the substitution
k a, k b ⇒ k σ and by taking into account equation (8), we
get, in line with reference [13]
εk
and
is the electron energy modified by the self-
consistent field due to electron pairing interactions. In
equation (15), matrix elements have the form
hk k | G | k1k1i = hka kb | G | k1b k1ai
− hka kb | G | k1a k1bi,
(16)
in which dependence on conjugated states a and b is shown
explicitly. In this way, by taking into account diagonal
terms α+k αk and βk+βk in the transformed Hamiltonian,
the energy of quasi-particles is found to be
q
∼
ε
Ek =
(
−µ)2 + ∆20 ,
(17)
k
as in BCS theory. Likewise, by applying Fermi’s statistic,
the critical temperature law is obtained in the form
kTc = 1.14 δe exp(−1/NF V0 )
(18)
δe standing for the half-amplitude of the energy region
around the Fermi level in which the matrix element (16)
is different from zero, NF for the density of states in
this region and V0 for the average value of element (16).
With Cooper pairs, δe is related to Debye energy by
δe = ~ ωD [16]. This result is a consequence of the fact
that phonon coupling is active only for energies less than
Debye energy. In general, for a system of weakly interact-
ing fermions with attraction between particles other than
Wpair ⇒ hk1 σ1 k2 σ2 | G | k20 σ20 k10 σ10 i ·
(11)
In this way, with the substitutions Wa(ka), Wb(kb) ⇒ εk, phonon coupling, we have 2 δe = µ [17]. These results au-
the Hamiltonian of the interacting electron system can be
written as
thorize the conclusion that the pairing mechanism dealt
with leads to the appearance of an energy gap ∆0 in the
quasi-particles levels, as occurs with Cooper’s pairs.
The different kinds of fermion pairs mentioned above
are represented in Figure 1. Besides Cooper’s pairs,
Majorana’s pairs are also shown, that is, nucleons coupled
with identical linear momenta, but with opposite angu-
lar momenta. Electrons coupled in unconventional super-
conductors are also characterized by like linear momenta.
Since these electrons lie in neighbouring lattice layers, the
pairs are similar to the pairs of electrons lying in neigh-
bouring atoms that were accounted for by Lewis in 1916
to explain molecular bonds. These pairs, therefore, can be
referred to as Lewis’s pairs [18].
X
H =
(εk − µ) (c+k+ck+ + ck+−ck−
)
k
X
1
−
hk1σ1k2σ2 | G | k20 σ20 k10 σ10 i c+ ck+ σ ck0 σ0
c
2
,
k10 σ10
k1σ1
2
2
2
2
kσ
(12)
µ standing for the chemical potential and c+k σ, ck σ for the
fermion creation and destruction operators, respectively.
Assuming Uk2 + Vk2 = 1 and inserting the transformation
ck+ = Ukαk + Vkβk+,
ck− = Ukβk − Vkα+k (13)
in equation (12), coefficient Uk and Vk can be determined
by the diagonalizing condition
h
i
X
∼
4 Discussion
ε
(
−µ) 2UkVk − ∆k(Uk2 − Vk2) (αk+βk+ + βkαk) = 0,
k
k
Arguments advanced thus far show how superconducting
electron pairs may be originated in the materials dealt
with. There is a wide variety of lattice structures, so that
a special investigation is required for each material. We
first consider YBCO cuprate which represents perhaps the
most interesting case, owing to the great amount of per-
tinent experimental data now available.
(14)
where
X
∆k =
hk k | G | k1k1iUk Vk
(15)
1
1
k1
1
An equivalent treatment of pairing correlations in nuclear
matter was performed by Gor’kov [14] and Alekseev [15] uti-
lizing the Green function technique.
2
The sole difference between nuclear and superconductor
4.1 The YBCO cuprate
treatments lies in the fact that for superconductors the prelim-
inary separation of the self-consistent field due to the average
particle interaction is unnecessary.
The YBCO orthorhombic cell consists of three super-
imposed perovskitic cubes, the middle one containing

P. Brovetto et al.: On electron pairing in unconventional superconductors
89
Fig. 1. Different kinds of pairs in fermion systems. A) in metallic solids electrons e1 and e2 with opposite linear momenta are
coupled as Cooper’s pairs. B) in nuclear matter nucleons n1 and n2 with like linear momenta but opposite angular momenta are
coupled as Majorana’s pairs. C) in unconventional superconductors electrons e1 and e2 running in neighbouring lattice layers
are coupled with like linear momenta as Lewis’s pairs.
yttrium at its center and the others barium. Divalent cop- fact
per ions at the yttrium cube vertices are kept in lacunar
Z Z
N
N
X X
oxygen octhaedra and lie on CuO2 planes orthogonal to
the c-axis. The oxygen lacunae are placed between diva-
lent copper ions on edges of the yttrium cube parallel to
the c-axis. Trivalent copper ions on the cell basal planes
are separated alternately by oxygens and oxygen lacunae.
With this symmetric geometry, regions a and b are iden-
tified with the CuO2 planes facing each other at opposite
sides of the yttrium cubes. Each copper ion in region b is
separated from a corresponding ion in region a by length
λ of the yttrium cube edges parallel to the c-axis, that is,
vq −up = λ. Consequently, considering TB wave functions
normalized as in equations (1) and taking into account
that only nearest orbitals allow for significant overlap in-
tegrals, we obtain
1
Ka0 , b
=
a(r1 − up)b(r1 − up − λ)
N2
p=1
h=1
ꢄ
ꢅ
e2
1
N
(23)
×
a(r2 − uh)b(r2 − uh − λ)d3r1d3r2 = O
,
r1, 2
since significant contributions arise only when r1 ' r2 and
this in turn entails up ' uh so that the sums on p and h
are coupled. Eventually, taking into account that Za = Zb,
it follows from equation (8)
Wpair = 2 Sa, b Eb, a
(24)
which, owing to the negative value of exchange integral
Eb, a, corresponds to a singlet state. In this way, the Wpair
found is independent of k so it remains constant for the
whole electron distribution. It can therefore be directly
identified with V0 in equation (18).
Orbitals involved in previous equations are 3d orbitals
characterized by prominent lobes arranged symmetrically
around the ion centers. It is to be considered, however,
that the electron distribution of copper ions is warped
by the electric field due to neighbouring oxygen ions. In
equations (2), potential Vlat has been introduced to ac-
count for this field. Indeed, the lacunar oxygen octhaedra,
that is, the pyramids CuO5 of five oxygens closest to di-
valent copper ions, originate the electric field
Z
N
X
1
N
hφa | φbi =
eik·λ
a(r1 − up) b(r1 − up − λ)d3r1
p=1
ik·λ
= e
S
,
a,b
(19)
where
Z
Sa,b
=
a(ρ) b(ρ − λ) d3ρ.
(20)
As for exchange integrals, we have analogously
1
Ka = − e− ik·λ
N
2e
Flat
=
τ,
(25)
(L / 2)2
Z
N
Za e2
X
×
a(r2 − ul)
b(r2 − ul − λ)d3r2
τ standing for a unit vector parallel to the c-axis and di-
rected from copper to the pyramid apex and L = 3.9 A for
˚
| r2 − ul |
l=1
the length of the barium cube edge, that is, twice the dis-
tance between copper and the pyramid apex. This field
polarizes the electron distributions of a and b copper ions
which are stretched toward each other, thus greatly en-
hancing overlap of a and b orbitals and, as a consequence,
the values of Sa,b and Eb,a integrals. Nevertheless, owing
= e− i k·λEb,a
(21)
where
Z
Za e2
Eb, a = − a(ρ)
b(ρ − λ)d3ρ ·
(22)
| ρ |
˚
to the large separation | λ |= 3.2 A of copper ions, pair-
It is easy to see that, with the same approximation, the ing energy surely remains small with respect to current
exchange integral Ka0 , b is vanishingly small. We have in covalent bond energies.

90
The European Physical Journal B
Table 2. Binding energies of oxygen ions in the different sites of the YBCO cell originated by their Coulomb interactions with
the neighbouring ions. Items “Pyramid apex” (1) and (2) mark oxygen sites with neighbourhoods of Cu ions of different charges.
√
Distance of oxygen from neighbouring Cu ions is L/2, from O−2, Ba+2 and Y+3 ions is L/ 2.
Oxygen ion site
Cell base
Pyramid apex (1) Pyramid apex (2)
2 Cu+2
Pyramid base
2 Cu+2
Neighbouring ions
Cu+2 ; Cu+3
4 O−2; 4 Ba+2
−20e2/L
Cu+2; Cu+3
6 O−2; 4 Ba+2
6 O−2; 4 Ba+2
6 O−2; 2 Ba+2; 2 Y+3
√
√
√
Energy
eV
−(16 − 8 2)e2/L −(20 − 8 2)e2/L
−(16 − 4 2)e2/L
−38.2
−73.8
−17.8
−32.1
Concerning the comparison of these results with ex- the presence of oxygen lacunae on the basal planes, which
periments, the most significant observations are perhaps reduces Coulomb repulsion between oxygens, and on the
the large effects on critical temperature of pressure and binding energy of trivalent yttrium, which exceeds that
oxygen content. Actually, various data on dTc/dP have of divalent barium. It follows that oxygen removal mostly
been reported. In particular, with fully-oxidized samples takes place at the pyramid apex, thus leaving copper in a
and with various substitutions on barium and yttrium, a neighbourhood of four oxygens arranged in a square con-
value dTc/dP = 0.96 K/GPa was found for pressures up figuration3. With this configuration, the polarizing field
to 0.6 GPa [19,20]. As for the oxygen content, compounds Flat drops to zero, thus greatly reducing overlap of cop-
III
of stoichiometry YBa2CuI2I+2xCu
O7−x with x < 0.2 per orbitals. Since two oxygens, out of the seven present in
1−2x
show Tc = 92 K, a figure that decreases to 60 K for the fully oxidized cell, lie at the pyramid apexes, in com-
x ' 0.4 [21]. To analyze these effects, it is useful to write pounds with x ' 0.4 about 20% of a or b copper ions are
equation (18) in differential form. Taking into account that unpolarized so that the corresponding Sa,b and Eb,a inte-
2 δe = µ [17], we have
grals are substantially smaller. Neglecting these integrals
allows us to replace equations (19, 21) with
ꢄ
ꢅ
δ Tc
0.57 µ δV0
= log
,
(26)
ik·λ
hφa | φbi ' 0.8 e
S
a,b
,
Ka ' 0.8 e− ik·λEb,a
,
Tc
kTc
V0
(27)
in which the logarithmic factor is positive since, as follows
from equation (18), kTc is exponentially small compared respectively. In this way, even if the logarithmic factor in
with µ, On the basis of this equation, the effect of pres- equation (26) is not large with respect to unity, the ob-
sure is readily understood. In fact, owing to the presence served decrease of Tc can be justified. We get, in fact,
of oxygen lacunae, pressure easily reduces distance λ be- δTc/Tc ' δV0/V0 ≡ δWpair/Wpair ' 0.82 − 1 = − 0.36,
tween the CuO2 planes thus enhancing overlap and ex- to be compared with the experimental result δTc/Tc
=
change integrals Sa,b and Eb,a. This in turn allows for a (60 − 92)/ 92 = − 0.35. In spite of its merely qualita-
larger Wpair, that is, V0 and, according to equation (26), tive nature, the previous discussion can be regarded as
for an increased Tc.
a reasonable explanation of the effect of oxygen removal.
The effect of the oxygen content requires more at- The conclusion follows that in YBCO cuprate the oxygen
tentive consideration. When an oxygen is removed, two doping increases the average overlap of copper orbitals
trivalent copper ions are reduced to divalent and a further thus increasing Tc. To complete this analysis, we point
oxygen lacuna is introduced into the cell. To determine the out that also the orthorhombic-tetragonal transition for
position of this lacuna, binding energies of oxygen ions in x ≥ 0.5 can be ascribed to the removal of apex oxygens.
the different sites of YBCO cell are to be compared. Three In fact, apex oxygen lacunae allow for higher binding en-
kinds of sites have to be considered, that is, sites on the ergies of oxygens placed in the four sites of the cell basal
cell bases and sites on the CuO5 pyramid apices and bases, planes surrounding the copper ions close to these lacunae.
respectively. Owing to the essentially ionic character of the Consequently, oxygens are gathered around these coppers,
YBCO lattice, these energies are originated by Coulomb thus breaking the laying out of oxygens and lacunae in
interactions of oxygen with neighbouring ions. In Table 2, basal planes, which is responsible for the orthorhombic
oxygen binding energies are reported as evaluated by con- symmetry.
sidering the nearest neighbouring ions and by assuming
3
Mossbauer spectroscopy of 57Fe-doped YBCO has shown
that on the cell bases divalent copper alternates with triva-
lent copper. Moreover, for simplicity’s sake, all edges of
barium and yttrium perovskitic cubes are assumed to have
that in the orthorhombic fully-oxidized phase Fe+4 ions
substitute Cu ions both on the cell basal planes (sites Cu1)
and on the yttrium cube vertices (sites Cu2). When oxygen is
removed, Fe+3 ions are found both in Cu1 and Cu2 sites [22].
That is: oxygen removal affects iron independently of its posi-
tion. This means that removal takes place at the pyramid apex
which lies just half-way between the copper sites. Removal from
cell base or from pyramid base would have different effects on
˚
the same length L ' 3.9 A. It appears that oxygens at
the cell bases are the most tightly bound, while oxygens
at the pyramid apices show the least binding energy. This
result remains unchanged if ions placed at increasing dis-
tances and different arrangements of Cu+2 and Cu+3 ions
on the cell bases are considered. It merely depends on iron in Cu1 and Cu2 sites.

P. Brovetto et al.: On electron pairing in unconventional superconductors
91
where
Opq = ha(r − up) | a(r − uq)i
(30)
are the overlap integrals of Cu ion orbitals. The sum for
ꢆ ꢇ
N
2
q > p in equation (29) includes
terms. However, only
the 2N terms with the minimum uq −up separation along
the square diagonals give significant contributions. The
terms along the square sides make no contribution owing
to the presence of O−2 ions. By dropping labels p, q and
putting
Fig. 2. Polar plots, for φ = π and different values of overlap
integral O, of the superconducting energy gap as a function of
angle θ between the electron wave vector k and the cell a-axis.
The four maxima of the energy gap are lined up with a- and
b-axes.
uq − up = δ,
(31)
we have

−1

X
K2 = N−1 1 + O
cos(k · δ)
,
(32)
There is another question which deserves attention
and which has long been debated in the last few years,
that is, the question of the “s/d” symmetry of the or-
der parameter. Different types of experiments have given
conflicting data, so that a final conclusion has not been
reached [23,24]. It is to be pointed out, however, that
recent measurements by scanning tunnel microscopy on
Bi2Sr2CaCu2O8+δ crystals containing zinc atom impu-
rities have revealed the presence of four-fold symmetric
quasi-particle clouds aligned with the Cu–O bonds on the
CuO2 planes [25,26]. This means that the superconduct-
ing energy gap is largest along the cell a- and b-axes. Such
a result supports the d-symmetry option in cuprates. In
the following, a simple argument is presented dealing with
the symmetry of the energy gap in YBCO.
Owing to the orthorhombic but almost tetragonal sym-
metry of the YBCO cell, copper ions on the CuO2 planes
form a square grid with Cu+2 ions at the square ver-
tices and O−2 ions at the middle of the square sides. The
Coulomb field of O−2 ions cuts down the electron charge
density of Cu+2 ions along the square sides, thus increas-
ing density along the square diagonals. Consequently, the
Cu+2 ion charge distribution becomes like that of 3dxy
orbitals4. By letting up and uq be the lattice vectors and
K a constant factor, normalization of TB wave functions
yields

→
δ
√
˚
where | δ |= 2L = 5.5 A. By assuming I parallel to the
cell a-axis, taking into account that
√
δ = (δ/ 2)(ꢀI ꢀ J),
(33)
(34)
k = k cos θ I + k sin θ J
and putting K2N = Φ(θ), we obtain
Φ(θ) = {1 + 2O cos [φ (cos θ + sin θ)]
+2O cos [φ (cos θ − sin θ)]}−1
(35)
√
where φ = kδ/ 2 is a phase angle. In this equation, over-
lap integral O accounts for the electron charge distribution
on the CuO2 planes, while angle φ accounts for electron
momenta. It follows that, when applying equations (8, 24),
a further normalization factor Φ2(θ) is to be inserted in
the expression of Wpair. Consequently, by letting the phase
√
angle assume the value φ = kFδ/ 2 at the Fermi level, the
angular dependence of the superconducting energy gap is
found to be
∆(θ) ∝ Φ2(θ) ∆0.
(36)
hφ∗(k, r) | φ(k, ri =
*
+
N
N
This angular dependence corresponds to a four-fold sym-
metry. In fact, it can be seen from equation (35) that,
while for φ = 0, factor Φ2(θ) is independent of θ, for
X
X
K2
e− ik·u a(r − up) |
eik·u a(r − uq) = 1
p
q
p=1
q=1
(28) φ = π the ratios Φ2(nπ/4)/Φ2(0) (n = 1, 2, 3, 4) are less
than unity and attain a common minimum value. For in-
termediate values of φ, four-fold symmetry is still present,
although less evident. It is interesting to note that for
φ = π, taking into account that kF = 2π/λF, we have
which leads to
(
)
−1
N
X
K2 = N + 2
cos [k · (uq − up)] Opq
,
(29)
λF/2 = L. Owing to the 3dxy character of Cu ion orbitals,
the maximun value of overlap integral is expected to be
1/4. This occurs when one lobe out of four of each or-
bital is wholly overlapped with that of the other. When
q> p=1
4
Owing to the electric field Flat, these orbitals are warped
towards the YBCO cell centre, so that the electron charge dis-
tribution is not symmetrical with respect to the sides of the
CuO2 plane.
O tends to 1/4, Φ2(θ) tends to infinity. Therefore, only
values of O less than 1/4 are significant. In Figure 2,
Φ2(θ) is plotted for some values of O by assuming φ = π.

92
The European Physical Journal B
The four-fold symmetry of the energy gap is evident. As for equation (24), pairing energy is negative, but its
These results, although only of a qualitative nature, yield actual value, which depends in this case on k, is expected
evidence in favour of d-symmetry.
to be smaller than that for the YBCO cuprate owing to the
Let us quote finally the following important experi- larger Cu ion separations. This entails a smaller V0 and,
mental result which clearly substantiates the hypothesis according to equation (26), a smaller critical temperature.
that superconduction depends on interaction between the
CuO2 planes. Measurements of the Hall effect on YBCO
single crystals have shown that the zero-temperature
Ginzburg-Landau coherence length along the c-axis is
4.3 Perovskites
˚
about 1.5 A. Since the spacing of superconducting lay-
˚
ers is near the c-axis lattice parameter 11.68 A this indi-
Perovskites with fractional stoichiometries are character-
ized by disordered lattices. Therefore, to identify the
conjugated a and b regions, special hypotheses have to
be introduced. Actually, all superconductors reported in
cates that “the two copper oxide planes, which are spaced
˚
3.2 A, are tightly coupled and act as a single superconduct-
ing layer” [27].
Table 1, items 2 to 5, are constituted of MeIIBiO3 (MeII
=
Ba, Sr) cells, each holding an unpaired electron, embed-
ded in BaPbO3 or MeIBiO3 (MeI = K, Rb) perovskitic
lattices. We assume that the MeIIBiO3 cells are joined
in form of chains or layers. Two contiguous sets of joined
cells are recognized as the conjugated regions. We assume,
moreover, that the unpaired electrons are placed on the
MeII ions, so that Ba+1 or Sr+1 ions are found in the cells
and the atomic orbitals appearing in the TB functions of
equations (1) have to be identified with the s-orbitals of Ba
or Sr atoms. As a consequence, separation λ of a and b or-
bitals equals the edge of the MeIIBiO3 perovskitic cube. In
this way, the notable variation of Tc originated by the sub-
stitution of barium with strontium in the Ba0.6K0.4BiO3
compound (see Tab. 1) is at once understood. Indeed, the
radius of the Sr+1 ion is smaller than that of the Ba+1
ion. Owing to the almost equal sizes of the Ba and Sr
perovskitic cubes, the s-electron separation is larger with
Sr+1 ions, thus reducing pairing energy and critical tem-
perature from 30 K to 12-13 K. Similar considerations can
be applied to the reduced strontium titanate SrTiO3−δ as
well (see Tab. 1 item 1).
4.2 The La2CuO4 cuprate
This compound consists of a stacking of LaCuO3 per-
ovskitic layers intercalated with LaO layers of the NaCl
structure. The subsequent LaCuO3 layers are displaced
sidewise with respect to the stacking by half a perovskitic
cube edge. With this geometry, regions a and b are iden-
tified with the copper ion planes lying at the sides of the
LaO layers. Each copper ion in region a shows four nearest
ions in region b with separations vp+n − up = λ1+n(n =
0, 1, 2, 3). Only these ions allow for significant contribu-
tions in overlap and exchange integrals. By proceeding as
for equation (19), we thus have
4
X
eik·λ
S
a, b
,
(37)
1+n
hφa | φbi =
n=0
where
Z
Sa, b
=
a(ρ) b(ρ − λ1)d3ρ,
(38)
since overlap integrals are independent of n owing to the
square symmetry of separations λ1+n. Likewise, we have
4.4 Bechgaard salt
4
X
eik·λ
E
a, b
,
(39)
(40)
1+n
Ka =
In Bechgaard salt, TMTSF molecules and (TMTSF)+1
cations, together with (PF6)−1 anions, form stacks show-
ing high metal-like conductivity parallel to the a-axis [10].
In directions transverse to the stackings, intermolecular
coupling and conductivity are much smaller. Therefore,
the conjugated a and b regions can be identified with
contiguous stackings of these molecules. The geometry is,
in some respect, similar to that of YBCO, but in equa-
tions (1) atomic orbitals should be substituted by the
molecular orbitals of (TMTSF)+1 cations. The unpaired
electrons present in these cations originate intermolecular
bonds between the conjugated regions. Pairing energy is
expected to be rather weak, owing to the separation of the
stackings. It is interesting to note that some anomalies ob-
served with this material in connection with its magnetic
behavior suggest the possibility of triplet pairing, instead
of the usual singlet pairing [28].
n=0
where
Z
Za e2
Ea, b = − a(ρ)
b(ρ − λ1)d3ρ,
| ρ |
while, as in equation (23), integral Ka0 , b is negligible. In
this way, by putting
4
6
X
X
1+n−λ1+m
)
Θ(k) =
eik·(λ
= 4 + 2
cos(k · σj)
n,m=0
j=1
(41)
where σj = λ1+n − λ1+m (n = m) are the sides and the
diagonals of the copper ion squares found at the bases of
the LaCuO3 cubes, we obtain
Wpair = 2 Θ(k) Sa, b Eb, a
.
(42)

P. Brovetto et al.: On electron pairing in unconventional superconductors
93
4.5 Fullerene
also considering compounds with different transition met-
als5, are probably still more advantageous.
As for the alkaly-stuffed fullerene, the interpretation is as
follows. Sixty π-electrons are coupled in the C60 balls, thus
originating thirty π − π bonds, as occurs in the benzene
molecule in which six π-electrons originate three π − π
bonds. In presence of the unpaired 3s- or 4s-electron of
potassium or rubidium atom, one π − π bond is bro-
ken, thus giving rise to a 3s-π or 4s-π bond between the
C60 ball and the alkaly atoms and leaving an unpaired
π-electron on the ball. The MeIC60 radicals thus origi-
nated can be assembled to form the conjugated regions.
Lacking reliable data on the actual stacking of the C60
balls and the placing of the alkaly atoms, it seems risky
to advance a more definite model of superconduction in
this material. Despite this, fullerene is interesting since it
provides clear evidence on the role of unpaired electrons
in originating superconductivity.
It is to be pointed out that the superconduction mech-
anism we propose is quite different from that advanced by
Anderson for YBCO cuprate [29]. In fact, the former as-
cribes superconduction to exchange interactions between
electrons running in contiguous CuO2 planes, while the
latter appeals to superexchange interactions between cop-
per ions mediated by oxygens in a network of –Cu–O–Cu–
chains. Accordingly, each single CuO2 plane shows super-
conduction. We note that, if this mechanism is taken into
account, it becomes difficult to understand why mono-
clinic CuO, in which –Cu–O–Cu– chains are present as
well, is not a superconductor.
In reality, the energy involved in the superconduct-
ing pairs is very weak. It is likely that, even in materials
with the highest Tc, it attains at the most some tens of
meV. Consequently, different mechanisms can be devised
in principle, each sufficient to justify the pairing energy
required. This is perhaps the main difficulty of the super-
conduction problem and explains why a large number of
superconduction models have been proposed so far.
5 Final remarks
In our opinion, the arguments advanced here yield
considerable clues that the basic mechanism of electron
pairing is essentially the same for all the unconventional
superconductors considered above. The mechanism sin-
gled out relies on an electron exchange interaction show-
ing a clear likeness with the ordinary exchange interaction
that accounts for covalent bonds. In practice, the mech-
anism proposed can be regarded as a molecular mecha-
nism in which the main feature is the coupling between
electrons and atoms present in the lattice structure. It fol-
lows that the peculiarities of the Fermi surface are of little
moment as regards the pairing mechanism. On the other
hand, it should be taken into account that the supercon-
ductors dealt with range from perovskites to cuprates to
organic compounds to fullerenes, which are quite dissim-
ilar materials certainly characterized by different Fermi
surfaces. Consequently, if the Fermi surface plays a signif-
icant role its variability would rule out the possibility of a
pairing mechanism common to all superconductors.
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