166
Competing interactions in unconventional superconductors
does not have such a topological constraint. On the experimental side, there are
measurements of out-of-plane resistivity ρ
c
[152, 153] which show the metallic-
like temperature dependence of ρ
c
and the mean-free path in the c-direction
comparable with the interplane spacing in a number of high-T
c
cuprates. Well-
developed interlayer tunnelling invalidates these and any other low-dimensional
concepts (see figure 5.2), which definitely fail to account for dρ
c
/dT > 0and
high-T
c
of highly oxygenated cuprates, doped fullerenes and MgB
2
.
Another problem of the microscopic Hamiltonians (equations (5.1) and
(5.2)), which are behind the RVB concept, is that they neglect the long-
range Coulomb and electon-phonon interactions, which are essential in novel
superconductors (section 5.2). As a result, some exact solutions of various limits
and numerical studies on t–J and Hubbard models led to the conclusion that
the electronic structure in cuprates is much more prone to inhomogeneity and
intermediate-scale structures such as stripes of hole-rich domains separated by
insulating antiferromagnetic domains. Moreover, it has been proposed that the
stripes are essential to a high-T
c
mechanism [141], especially in the underdoped
regime. In this scenario, stripe formation permits hole delocalization in one
direction, but hole motion transverse to the stripe is still restricted. It is thus
favourable for the holes to pair so that the pairs can spread out somewhat into
the antiferromagnetic neighbourhood of the stripe, where their interaction is
allegedly attractive due to spin fluctuations. The proximity effect in conventional
superconductor–normal metal structures is a prototypical example of such a
mechanism of pairing: when the BCS superconductor and a normal metal are
placed in contact with each other, electrons in the metal pair even if the interaction
between them is repulsive. As we discuss in section 8.6, this ‘stripe’ scenario is
incompatible with Coulomb’s law: there are no stripes if the long-range Coulomb
repulsion is properly taken into account. On the experimental side, recent neutron
[154] and X-ray [155] spectroscopic studies did not find any bulk charge/spin
segregation in the normal state of a few cuprates suggesting an absence of in-
plane carrier density modulations in these materials above T
c
.
Strong repulsive correlations between holes could provide another novel
mechanism for Cooper pairing, i.e. the kinetic-energy-driven mechanism of
superconductivity proposed by Hirsch [156] (see figure 5.2). Such a mechanism
does not require any dynamic attraction between holes. The qualitative
explanation is as follows: electrons in metals are ‘dressed’ by a cloud of other
electrons with which they interact and form the so-called electronic polarons.
The dressing causes an increase in the electron’s effective mass and when the
dressing is large, the metal is a poor conductor. If, however, the electrons
manage to ‘undress’ when the temperature is lowered, their effective mass will be
reduced and electricity will flow easily. A model Hamiltonian, which describes
the ‘undressing’, is that of small polarons with a nonlinear interaction with a
background bosonic degree of freedom which gives rise to an effective mass
enhancement that depends on the local charge occupation. The undressing process
can only occur if the carriers are ‘holes’ rather than electrons and when two hole