General Information about stripe phases in Super Conductors is below
Refer to picture in
http://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=33690&rendertype=figure&id=F1
The work that Patrick Lee and Xiao-Gang did is detailed in
web.mit.edu/cmse/www/Lee99.pdf
Abstract
Stripe phases are predicted and observed to occur in a class of
strongly correlated materials describable as doped antiferromagnets,
of which the copper-oxide superconductors are the most prominent
representatives. The existence of stripe correlations necessitates the
development of new principles for describing charge transport and
especially superconductivity in these materials.
Thirteen years ago, the discovery (1) of superconductivity in layered
copper?oxide compounds came as a great surprise, not only because of
the record-high transition temperatures, but also because these
materials are relatively poor conductors in the ?normal? (that is,
nonsuperconducting) state. Indeed, these superconductors are obtained
by electronically doping ?parent? compounds that are antiferromagnetic
Mott insulators?materials in which both the antiferromagnetism and the
insulating behavior are the result of strong electron?electron
interactions. Because local magnetic correlations survive in the
metallic compounds, it is necessary to view these materials as doped
antiferromagnets. A number of other related materials, such as the
layered nickelates (which remain insulating when doped) and manganites
(the ?colossal? magnetoresistance materials), are also doped
antiferromagnets in this sense.
The conventional quantum theory of the electronic structure of solids
(2), which has been outstandingly successful at describing the
properties of good electrical conductors (metals such as Cu and Al)
and semiconductors (such as Si and Ge), treats the electronic
excitations as a weakly interacting gas. This approach, known as the
?Fermi liquid theory,? breaks down when applied to doped
antiferromagnets. New principles must be developed to deal with these
problems, which are at the core of the study of ?strongly correlated
electronic systems,? one of the central and most intellectually rich
branches of contemporary physics. One idea that has evolved over the
last decade, and which offers a framework for interpreting a broad
range of experimental results on copper-oxide superconductors and
related systems, is the concept of a stripe phase. A stripe phase is
one in which the doped charges are concentrated along spontaneously
generated domain walls between antiferromagnetic insulating regions.
Stripe phases occur as a compromise between the antiferromagnetic
interactions among magnetic ions and the Coulomb interactions between
charges (both of which favor localized electrons) and the zero-point
kinetic energy of the doped holes (which tends to delocalize charge).
Experimentally, stripe phases are most clearly detected in insulating
materials (where the stripe order is relatively static), but there is
increasingly strong evidence of fluctuating stripe correlations in
metallic and superconducting compounds. The existence of dynamic
stripes, in turn, forces one to consider new mechanisms for charge
transport and for superconductivity. More generally, we will show that
the concept of electronic stripe phases developed for transition-metal
oxides is applicable to a broad range of materials.
Theoretical Background. Doped antiferromagnets are a particularly
important and well studied class of strongly correlated electronic
materials. Here, the parent compound is insulating, even at elevated
temperatures, because of the strong short-range repulsion between
electrons. At sufficiently low temperatures, antiferromagnetic order
develops in which there is a nonzero average magnetic moment on each
site pointing in a direction that alternates from site to site (see
Fig. 1). Frequently the doping process, ?hole doping,? involves
chemically modifying the material so that a small fraction of
electrons is removed from the insulating antiferromagnet. Whereas the
charge distribution in a doped semiconductor is homogeneous, in a
doped antiferromagnet the added charge forms clumps?solitons in one
dimension, linear ?rivers of charge? in two dimensions, and planes of
charge in three dimensions, as exemplified by organic conductors,
cuprates or nickelates, and manganites, respectively. Typically, these
clumps form what are known as ?topological defects,? across which
there is a change in the phase of the background spins or orbital
degrees of freedom. In d dimensions, the defects are (d-1)?dimensional
extended objects (3). Stripes in a two-dimensional system are
illustrated schematically in Fig. 1.
Self-organized local inhomogeneities were predicted theoretically
(4?7). These inhomogeneities arise because the electrons tend to
cluster in regions of suppressed antiferromagnetism (8), which
produces a strong short-range tendency to phase separation (9?11) that
is frustrated by the long-range Coulomb interaction. The best
compromise (7, 12) between these competing imperatives is achieved by
allowing the doped holes to be delocalized along linear stripes, while
the intervening regions remain more or less in the undoped correlated
insulating state.
Experimental Evidence for Stripes. The most direct evidence for stripe
phases in doped antiferromagnets has come from neutron scattering
studies. Diffraction of a neutron beam by long-period spin and charge
density modulations, extending over a few unit cells as indicated in
Fig. 1, yields extra Bragg peaks. The position of such a
superstructure peak measures the spatial period and orientation of the
corresponding density modulation, whereas the intensity provides a
measure of the modulation amplitude. Because neutrons have no charge,
they do not scatter directly from the modulated electron density, but
instead are scattered by the ionic displacements induced by the charge
modulation. The lattice modulation is also measurable with electron
and x-ray diffraction.
The antiferromagnetic order found in the parent compounds of the
cuprate superconductors is destroyed rapidly as holes are introduced
by doping. The first indications of long-period (?incommensurate?)
spin-density modulations were provided by inelastic neutron scattering
(13) of superconducting La2?xSrxCuO4 and by related measurements on
the insulating nickelate analog (14). After the discovery of
?incommensurate? charge ordering in the latter system by electron
diffraction (15), the proper connection between the magnetic and
charge-order peaks was determined in a neutron diffraction study (16)
of La2NiO4.125. The positions of the observed peaks indicate that the
charge stripes run diagonally through the NiO2 layers (as opposed to
the vertical stripes shown in Fig. 1). More recent experiments (17,
18) on La2?xSrxNiO4 have shown that the diagonal stripe ordering
occurs for doping levels up to x ? 1/2 (corresponding to a hole
density of 1 for every two Ni sites), with the maximum ordering
temperatures occurring at x = 1/3.
It is significant that the charge ordering is always observed at a
higher temperature than the magnetic ordering, which is characteristic
(19) of a transition that is driven by the charge. It is also
important to note that the period of the charge order is generally
temperature dependent, which means that the hole concentration along
each stripe also varies with temperature; this is characteristic (20)
of structures that arise from competing interactions. These
observations are consistent with the idea that the stripes are
generated by the competition between the clustering tendency of the
holes and the long-range Coulomb interactions. [Weak density?wave
order can occur in conventional solids under special conditions
(?nested Fermi surfaces?), but the transitions tend to be ?spin
driven? and occur at a fixed ?nesting? wave vector (5)].
Charge order is most easily detected when stripes are static, but
perfect static charge order can be shown (21) to be incompatible with
the metallic behavior of the cuprates. Nevertheless, to get a better
experimental handle on the charge order, one might hope to pin down
fluctuating stripes with a suitably anisotropic distortion of the
crystal structure. Just such a distortion of the La2?xSrxCuO4
structure is obtained by partial substitution of Nd for La. Neutron
diffraction measurements (22) x ?1/8 a Nd-doped crystal with the
special Sr concentration of x ?1/8 revealed charge and spin order
consistent with the vertical stripes of Fig. 1. (An anomalous
suppression of superconductivity, associated with the lattice
distortion, is maximum for x ?1/8.) The charge order has since been
confirmed by high-energy x-ray diffraction (23). As in the nickelates,
the spin ordering occurs at lower temperatures than the charge order,
and the hole concentration on a stripe varies as a function of the Sr
concentration, x.
Although it has been difficult to observe a direct signature of charge
stripes in other cuprate families, the existing neutron scattering
studies of magnetic correlations are certainly most easily understood
in terms of the stripe-phase concept. The doping dependence of dynamic
magnetic correlations (24) in Nd-free La2-xSrxCuO4 is found to be
essentially the same as the static correlations in Nd-doped samples
(22), and a comprehensive study (25) of a Nd-free sample near
?optimum? doping (that is, maximum superconducting transition
temperature) indicates that ordering may be prevented by quantum
fluctuations. To keep things interesting, static magnetic order has
been observed (26) to set in near the superconducting transition
temperature in La2CuO4+?. Finally, a beautiful experiment (27) on
superconducting YBa2Cu3O6+x has shown that the low-energy magnetic
correlations in that system have strong similarities to those in
La2?xSrxCuO4.
An example of planar domain walls in a three-dimensional system occurs
in nearly cubic La1?xCaxMnO3 with x = 0.5. Charge order has been
imaged by transmission electron microscopy (28). The ordering
phenomena are somewhat more complex in this case because the occupied
Mn 3d orbitals are degenerate. As a consequence, charge, spin, and
orbital ordering are all involved, although, again, charge order sets
in at a higher temperature than magnetic order.
Electronic Liquid Crystals. Once the idea of stripe phases of a
two-dimensional doped insulator has been established, a major question
arises: How can a stripe phase become a high-temperature
superconductor, as in the cuprates, rather than an insulator, as in
the nickelates? Typically, interactions drive quasi one-dimensional
metals to an insulating ordered charge density wave (CDW) state at low
temperatures (29) (and quenched disorder only enhances the insulating
tendency). However, we have shown (21) that the CDW instability is
eliminated and superconductivity is enhanced if the transverse stripe
fluctuations have a large enough amplitude. To satisfy this condition,
the stripes could oscillate in time or be static and meandering. They
are then electronic (and quantum-mechanical) analogues of classical
liquid crystals and, as such, they constitute new states of matter,
which can be either high-temperature superconductors or
two-dimensional anisotropic unconventional metals.
Classical liquid crystals are phases that are intermediate between a
liquid and a solid and spontaneously break the symmetries of free
space. Electronic liquid crystals are quantum analogues of these
phases in which the ground state is intermediate between a liquid,
where quantum fluctuations are large, and a crystal, where they are
small. Because the electrons exist in a solid, it is the symmetry of
the host crystal that is spontaneously broken, rather than the
symmetry of free space. An electronic liquid crystal has the following
phases: (i) a liquid, which breaks no spatial symmetries and, in the
absence of disorder, is a conductor or a superconductor; (ii) a
nematic, or anisotropic liquid, which breaks the rotation symmetry of
the lattice and has an axis of orientation; (iii) a smectic, which
breaks translational symmetry in one direction and otherwise is an
electron liquid; (iv) an insulator with the character of an electronic
solid or glass. These classifications applied to stripe phases make
the stripe notion, which is based on local electronic correlations,
macroscopically precise. Neutron and x-ray scattering experiments give
direct evidence of electronic liquid crystal phases (conducting stripe
ordered phases) in the cuprate superconductors.
Charge Transport. In the standard theory of solids, the electron's
kinetic energy is treated as the largest energy in the problem, and
the effects of electron?electron interactions are introduced as an
afterthought. As a consequence, the electronic states in normal solids
are highly structured in momentum space (k-space), and therefore,
according to the uncertainty principle, they are highly homogeneous in
real space. Moreover, as the ?normal? (metallic) state is continuously
connected to the ground state of the kinetic energy, any phase
transition to a low-temperature ordered phase is necessarily (30)
driven by the potential energy, inasmuch as it involves a gain in the
interaction energy between electrons at a smaller cost of kinetic
energy. For transport properties, the central concept of a mean free
path l, that is, the distance an electron travels between collisions,
is well defined so long as l is much larger than the electron's de
Broglie wavelength, ?F, at the Fermi energy.
A number of interesting synthetic metals, discovered in the past few
decades, seem to violate the conventional theory. They are ?bad
metals? (31, 32), in the sense that their resistivities, ?(T), have a
metallic temperature dependence [?(T) increases with the temperature
T] but the mean free path, inferred from the data by a conventional
analysis, is shorter than ?F, so the concept of a state in momentum
space would be ill defined. Among the materials in question are the
cuprate high-temperature superconductors; other oxides, including the
ruthenates, the nickelates, and the ?colossal magnetoresistance
materials? (manganites), organic conductors, and alkali-doped C60.
Most of these materials are doped correlated insulators, in which the
short-range repulsive interaction between electrons is the largest
energy in the system. However, the ground state of this part of the
Hamiltonian is not unique, so the kinetic energy cannot simply be
treated as a perturbation; such materials display substantial
structure in both real space and momentum space. As a consequence, the
conventional theory (2) must be abandoned. Neither the kinetic energy
nor the potential energy is totally dominant, and they must be treated
on an equal footing.
Superconductivity. The highly successful theory of superconductivity
(33) developed by Bardeen, Cooper, and Schrieffer in the 1950s was
designed for good metals, not for doped insulators. A key issue,
therefore, is the relation of stripes to the mechanism of
high-temperature superconductivity. In fact, there is a strong
empirical case for an intimate relation between these phenomena: (i)
strongly condensed stripe order can suppress superconductivity (as it
does in La1.6?yNdySrxCuO4); (ii) weak stripe ordering can, at times,
appear at the superconducting transition temperature Tc (as it does in
La2CuO4+?; (iii) there is a simple linear relation between the inverse
stripe spacing and the superconducting Tc observed in several
materials (24, 34) (including La2?xSrxCuO4 and YBa2Cu3O6+x); and (iv)
stripe structure and other features of the doped insulator, together
with high-temperature superconductivity, disappear as the materials
emerge from the doped-insulator regime (?overdoping?). Moreover, there
is a clear indication that the optimal situation for high-temperature
superconductivity is stripe correlations that are not too static or
strongly condensed, but also are not too ethereal or wildly
fluctuating. We have argued (21, 35, 36) that the driving force for
the physics of the doped insulator is the reduction of the zero-point
kinetic energy. This proceeds in three steps: (i) the development of
an array of metallic stripes lowers the kinetic energy along a stripe;
(ii) hopping of pairs of electrons perpendicular to a stripe in the
CuO2 planes creates spin pairs on and in the immediate neighborhood of
a stripe; and (iii) at a lower temperature, pair hopping between
stripes creates the phase coherence that is essential for
superconductivity. Steps ii and iii lower the kinetic energy of motion
perpendicular to a stripe.
Generality of the Stripe Concept. The physics of charge clustering in
doped correlated insulators is general and robust, so one might expect
that local stripe structures would appear in other related systems.
Indeed, topological doping has long been documented in the case of
quasi one-dimensional charge-density-wave systems, such as
polyacetylene (29, 37); it is an interesting open question whether it
occurs in other higher dimensional systems. One recent fascinating
discovery is the observation (38, 39) that, under appropriate
circumstances, quantum Hall systems (that is, an ultra-clean
two-dimensional electron gas in a high magnetic field) spontaneously
develop a large transport anisotropy on cooling below 150 mK. It is
likely that this anisotropy is related to stripe formation on
short-length scales (40, 41), and it apparently reflects the existence
of an electronic nematic phase in this system (42).
Stripe-like structures have also been observed (ref. 43 and references
therein) in many other systems with competing interactions, on widely
differing length scales. Beyond this generality, the existence of
spontaneously generated local structures is clearly important for
understanding all of the electronic properties of synthetic metals,
including the anomalous charge transport and the mechanism of
high-temperature superconductivity. Many of these implications have
already been explored in considerable detail, but many remain to be
discovered. Here we content ourselves with a few general observations.
The phenomena described above represent a form of ?dynamical dimension
reduction? whereby, over a substantial range of temperatures and
energies, a synthetic metal will behave, electronically, as if it were
of lower dimensionality. This observation has profound implications
because conventional charge transport occurs in a high-dimensional
state, and fluctuation effects are systematically more important in
lower dimensions. In particular, in the quasi two-dimensional
high-temperature superconductors, stripes provide a mechanism for the
appearance of quasi one-dimensional electronic physics, where
conventional transport theory fails, and is replaced by such key
notions as separation of charge and spin and solitonic quasi-particles
(29). At the highest temperatures (up to 1,000 K), in what is often
called the ?normal state? of the high-temperature superconductors,
where coherent stripe-like structures are unlikely to occur, it is
still probable that local charge inhomogeneities occur because of the
strong tendency of holes in an antiferromagnet to phase separate (44).
This behavior can lead to quasi zero-dimensional physics (quantum
impurity model physics), which also produces a host of interesting and
well documented quantum critical phenomena and may be at the heart of
much of the anomalous normal-state behavior of these systems.
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