International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
The Feshbach resonance and nanoscale phase separation in a
polaron liquid near the quantum critical point for a polaron
Wigner crystal
M Fratini, N Poccia and A Bianconi
Department of Physics, University of Rome "La Sapienza", P. le A. Moro 2, 00185
Roma, Italy
E-mail: [email protected]
Abstract. The additional long range order parameter that competes with the high Tc
superconductivity long range order is identified as an electronic crystal of pseudo Jahn-Teller
polarons beyond the critical value of the electron lattice interaction. We show that the region of
quantum critical fluctuations in the two variables phase diagram of cuprates: the doping δ and
the chemical pressure (i.e., the tolerance factor, or the average ionic radius of A-site cations)
can be measured via the microstrain ε of the Cu-O length in the CuO2 lattice. The fluctuating
order in the proximity of the microstrain quantum critical point that competes with the
superconducting long range order is the polaron electronic crystalline phase called a Wigner
polaron crystal and the variation of the spin gap energy as a function of microstrain provides a
strong experimental support for this proposal.
1. Introduction
Quantum phase transitions (QPT) have been identified in different systems going from magnetic
materials, heavy fermions [1, 2] and ultra cold gases in optical lattice [3]. The macroscopic phase
transition in the ground state of a many-body system occurs when the relative strength of two
competing energy terms is varied across a critical value of a coupling term [4]. For a superfluid system
at a temperature of absolute zero, by tuning a generic coupling at a critical value gc, a quantum critical
point (QCP) appears where the superfluid long range order competes with a second different long
range order. At the QCP the thermal fluctuations of standard phase transitions are replaced by
quantum fluctuations driven by the Heisenberg uncertainty principle in the ground state.
The search for the mechanism, that allows a quantum macroscopic condensate to resist to the decoherence effects of high temperature, has been focusing on the identification of the critical point of a
quantum phase transition [5]. The scientific community has been puzzled by the questions: what is the
nature of the long range order that competes with the high Tc superconducting (HTcS) phase? Which
is the coupling g that drives the system to the quantum critical point at gc?
There are two main proposals for high Tc superconductivity: the importance of a strong Jahn-Teller
electron phonon interaction with the formation of anti Jahn-Teller bipolarons was the Müller driving
idea for the discovery of the high Tc superconductivity [6.7], while the importance of electron-electron
interaction (the on site Hubbard repulsion U) driving the electronic system to the border of a Mott
insulator was stressed by Anderson [8].
Following the Anderson proposal, many authors have addressed their interest to the magnetic spin
ordered phases near a Mott insulator and have been looking for the quantum critical point QCP at a
c 2008 IOP Publishing Ltd
1
International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
critical density of doped holes per Cu site (doping) in the phase diagram where the critical temperature
is plotted versus the doping. The superconducting phase has been related with a quantum phase
transition to a Mott antiferromagnetic insulating phase [5], with spin fluctuations investigated mainly
by inelastic neutron scattering [9], the critical behavior of the spin fluctuations has been measured [10,
11] and other authors have associated the pairing mechanism with spin fluctuations [12].
On the other side following the Muller proposal that the electron lattice interaction is the driving
force, the additional long range order (in competition with the HTcS order) has been identified with a
commensurate polaron crystal, CPC, that can be described as a generalized Wigner polaron
commensurate CDW or a Wigner polaron crystal that shows up at a critical electron-lattice interaction
and a critical charge density [13-16].
The classical physics of polarons for the high electron lattice interaction was developed [17-19] for
a single polaron or bipolaron (a small polaron at strong electron lattice interaction or a large polaron at
weak electron lattice interaction). However at high polaron density the physics of a polaron or
bipolaron liquid relevant for HTcS was not clear. Increasing the polaron density several scenarios
could show up such as crystal instability, polaron dissociation, phase separation, polaron strings,
polaron bubbles, polaron pinning at defects, and the polaron electronic crystalline phases like
commensurate or incommensurate CDW and Wigner crystals that could be formed in the presence of
long range Coulomb interaction V [20].
2. Polarons in cuprates
The local lattice fluctuations due to polarons in cuprates have been determined by the fast and local
experimental probes: EXAFS and XANES that probe the instantaneous (with measuring time 10-15 s)
local (in the range of 500 pm) lattice distribution without time or spatial averaging. The solid state
scientific community dealing with homogenous crystalline solids was not so familiar with these novel
methods developed in the eighties using synchrotron radiation to probe complex inhomogeneous
fluctuating systems like glasses, liquids, and biological macromolecules. These methods have been
detected the polarons confirming the Muller predictions on the importance of the Jahn-Teller
interaction, but experimental results have provided several unexpected results and surprises
1) The undoped cuprate perovskites are correlated charge transfer oxides (where the correlation
gap is controlled by the Coulomb repulsion between a hole on copper and another one on the
nearest oxygen Udp ≈ 2 eV) and they are not Mott insulators (where the correlation gap is Udd ≈
6 eV between two holes in the same copper ion) ;
2) The dopant holes go into the O(2p) orbital and not into the Cu(3d) orbital as it was expected.
3) The polarons are pseudo Jahn-Teller (pJT) polarons involving the Q2 Jahn-Teller mode called
also the half-breathing mode of the oxygen motions that are coupled sterically with the Q4/Q5
tilts modes and are not of anti Jahn-Teller type.
4) The polarons are in the intermediate coupling regine where the lattice distortion involves a
domain of about 8 Cu sites while they were expected to be small polarons in the strong
coupling limit localized on a single site.
5) The metallic and superconducting phase shows an unusual nanoscale phase separation and not
an homogeneous phase
6) There is coexistence of 1) pJT intermediate polarons condensed in a one dimensional
incommensurate charge density wave and 2) itinerant large polarons.
7) The nanoscale phase separation gives an heterogeneous nanoscale material with the nanoscale
architecture having a spatial periodicity as large as the superconducting coherence length.
8) A polaron Wigner crystal was observed in a particular perovskite family at hole doping 1/8.
In 1987, XANES experiments have shown that the dopant holes do not form the expected 3d8
itinerant states (giving anti-Jahn-Teller polarons) but the dopant holes go in the oxygen 2p orbitals,
called ligand holes L(k) [21], providing a scenario where the O(2p5) holes move in a 2D network of
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
metallic oxygen diagonal wires intercalated by the antiferromagnetic 2D spin lattice in of the static
holes Cu(3d9) in the Cu sites. In 1988-1990 polarons associated with the 3d 9 L singlets have been
found to be of pseudo Jahn-Teller (pJT) type [22-27] where the hybridization between the d x 2 − y 2 L(b1 )
and dz 2 −r 2 L(a1 ) orbital is associated with the rhombic distortion of the CuO4 square plane. The pseudo
Jahn-Teller polarons in the cuprates are strongly affected by the relevant electronic correlation
energies that determine the charge transfer gap in the parent undoped compounds: the interatomic
Hubbard repulsion Udl between the dopant hole in the O(2p5) orbitals and the hole in the Cu(3d9)
dz 2 −r 2 or dx 2 −y 2 orbitals.
In the years 1990-1993 the focus of the research shifted to probe directly the dynamical fast local
lattice fluctuations between different Cu site conformations associated with pJT polarons in metallic
cuprates focusing on the difference ∆Rapical between the Cu-O(apical) and the Cu-O(planar) bond
length that determines the JT energy splitting ∆JT between dz 2 −r 2 and dx 2 −y 2 orbital. In fact the
electron-lattice interaction of the pseudo JT polaron type is a complex function of several lattice
parameters λ = g ∆R apical f (Q )h(β ) , where Q is the conformational parameter for the tilts of the
(
)
CuO4 square planes, and β is the dimpling angle that measures the displacement of the Cu ion from the
plane of oxygen ions. The conformational parameter showing the formation of pJT polarons is the
splitting ∆Rplanar of the in plane Cu-O(planar) bond lengths. The results of EXAFS experiments have
provided evidence for nanoscale inhomogeneity with the segregation of stripes of localized pseudo
Jahn-Teller polarons and stripes of itinerant carriers [28-32, 13-16]. These unexpected results have
been confirmed by further experimental investigations such as thermopower [33-35], high resolution
EXAFS [36-43], resonant x-ray diffraction [44] the electron paramagnetic resonance (EPR) of Mn2+
doped cuprates [45], copper NQR spectra, demonstrating the existence of a second anomalous copper
site in lanthanum cuprate whose character is independent of the method of doping, and systematic
NMR/NQR experiments [46, 47], susceptibility measurements [48], local structure investigation using
the atomic pair distribution function (PDF) analysis of neutron powder-diffraction [49], isotope effects
on T* [50-52] and an ultrafast real-space probe of atomic displacements (with sub-picometre
resolution), the MeV helium ion channelling, to probe incoherent lattice fluctuations [53], and recently
by the anomalous energy distribution curves E(k) (that strongly deviate from band structure
calculations) in ARPES and the Fermi arcs measured by k scanning ARPES. [53-66]. There is
therefore now a compelling evidence for polarons and nanoscale phase separation in cuprates that
make these materials a realization of a particular pseudo Jahn-Teller polaron fluid showing a
nanoscale phase separation and local lattice fluctuations [67-71] typical of complex systems and
biological systems [72-73].
In 1995 after the compelling evidence for polarons in cuprates was found by fast and local probes
the scientific community started to investigate the physics of fluids made of Jahn-Teller polarons
focusing on doped manganites that show colossal magnetoresistance [74-85]. It was recognized clearly
that these systems show a nanoscale phase separation as in cuprates [35, 86, 87, 88]. The local lattice
fluctuations are similar but larger than in cuprates [81]. The energy dispersion and pseudogap in
ARPES [64] are similar to the experimental features of ARPES of cuprates and finally in particular
materials and at particular dopings x=0.5 electronic crystalline phases called Wigner polaron crystal
[82] or commensurate polaron charge density waves [89] show up as it was shown before in particular
cuprate families [14-16] and the theory of polaron Wigner crystals is now rapidly developing [90-93].
In the superconducting materials the experimental results show a nanoscale phase separation with
the coexistence of incommensurate stripes with fluctuating pseudo Jahn-Teller polarons and
superconductivity that have been recently confirmed [94-96].
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
3. Feshbach resonances
For a metal confined in a single membrane Blatt proposed in 1963 [97, 98] that the critical temperature
is amplified by tuning the chemical potential at an electronic critical point of the electronic structure
(where the Fermi surface of a new subband given by quantum size effects appears or disappears) by a
shape resonance in the interband BCS coupling that is similar to the Feshbach resonances in nuclear
physics, Recently the shape resonance predicted by Blatt has been observed in a thin Pb films [99-101].
The investigation of Bi2212 superconductor has given provided evidence in 1993 that the high Tc
superconductivity occurs in heterostructure at atomic limit where a superconducting material is
intercalated by a different material with different electronic structure forming a superlattice of
quantum stripes. It was reported that in cuprates a shape resonance (called also Feshbach resonance)
where the dimensionality of Fermi surface of one subband changes from 2D to 1D that is a critical
point of the electronic structure, driven by the charge density and the architecture of the superlattice.
At this critical point a large amplification of the critical temperature is realized driven by the role of
the exchange like interband pairing term [102-113]. The Feshbach resonance has been now shown to
be the driving mechanism for high Tc in magnesium diboride [114] that is made of a superlattice of
graphene-like boron monolayers intracalated by magnesium of aluminum.
In 1993 the Feshbach resonance in ultra cold gases was proposed independently by Teisinga et al,
[115] for increasing the bose condensation temperature. Later it was also proposed to realize Feshbach
resonances in optical lattices that has interesting similarities with the Feshbach resonances in metallic
quantum superlattices [116].
The recent theoretical finding that the Feshbach resonance occurs near the quantum critical point in.
atomic Bose gases [117] supports the similarity with the exchange pairing at a shape resonance in
metallic superlattices of quantum wires proposed in cuprates [103-114]. In fact there is an increasing
number of experiments supporting the idea that in the copper oxide layers of cuprates there is a
quantum phase transition from a polaron liquid to an electronic crystalline phase of ordered polarons
that was identified as a polaron Wigner crystal by increasing the electron lattice interaction at a critical
value λ(εc) and at constant charge density δc=1/8. Therefore at constant doping increasing the electron
lattice polaronic interaction we reach a critical point where a quantum phase transition occurs [13] and
the Feshbach resonance occurs in the regime dominated by quantum local lattice fluctuations [118,
119, 120].
4. The chemical pressure in new phase diagram
The cuprate perovskites are heterogeneous materials formed by three different building blocks
[BO](AO) CuO2: the metallic bcc CuO2 layers, the insulating AO layers (rock-salt fcc layers in hole
doped cuprates) (A=Ba, Sr, La, Nd, Ca, Y etc.) and the charge reservoir BO layers. In the perovskite
materials like La2CuO4 (or [LaO]2CuO2) and manganites AMnO3 (or [AO]MnO2) the lattice matching
between the transition metal oxide layer MO (M=Cu or Mn) and that of the rare earth (A=La, Y, Ba,
Sr) oxide (AO) is realized by rotating the crystalline axis by 450 and a good matching occurs if the
ratio between the interatomic distances rA-O and rM-O (the respective bond lengths in homogeneous
isolated parent materials AO and MO2) is
rA −O
rM −O 2
= 1 The chemical pressure (or internal pressure)
due to the interlayer lattice mismatch across the block-layer interface is usually measured by
rM −0 − (< rA > +rO ) 2
.
rM −0
r
Where the Goldschmidt tolerance factor is t = A−O . In doped manganites with a simple
2 rM −O
η = 1− t =
perovskite A1.xA’xMnO3 structure the tolerance factor is usually calculated using the average A cation
size < rA > [76] and keeping fixed the ionic radii of oxygen and copper to get rM-O.
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
Figure 1. The Cu-O micro-strain ε=((197-CuO(pm) ) /197) ) normalized to its
critical value (4%) as a function of the average ion size in the rock-salt layers
that are first neighbours of the CuO2 plane. The static commensurate pJT
polaron crystalline phase in the LTT phase occurs for ε/εc>1.
There is a large experimental agreement that the magnetic and electronic phases of the polaron
fluid in manganites depend not only on the density of the polarons but also on the chemical pressure
[76, 86, 121].
In hole doped cuprate superconductors there is agreement that the stress exerted on the CuO2 plane
by the lattice mismatch induces a compressive CuO2 microstrain of the Cu-O distance and a tensile
microstrain in the rocksalt layer. It is well known that the cuprate perovskite lattice is stable only up to
a critical value of the lattice tolerance factor between the CuO2 layer and the rocksalt layers in the
La1.xAxCuO2 family and when the average ionic radius in the rocksalt layers is smaller than 118 pm the
system go into the T’ phase where the intercalated layers have a fluorite structure [122-127].
The increasing lattice mismatch induces rotation, tilting and dimpling of the CuO4 square planes
i.e., corrugations of the flat CuO2 plane. The ordering of these tilts induced by lattice mismatch gives
the transitions from tetragonal (HTT) to orthorhombic (LTO) and finally to the LTT phase in cuprate
superconductors.
Studies of the effects of the tolerance factor within a single cuprate family have been reported [122127], mainly for the La124 family, but it was not possible to extend the calculation of the tolerance
factor to other complex perovskite families of cuprates (like Y123, Bi2212 Hg1212 and others) where
there are multiple different ions and multiple intercalated different layers.
To overcome these difficulties we have proposed in 1998 at the second stripes conference [128] to
measure the chemical pressure due to the interlayer mismatch by measuring the average copperoxygen bond length < Cu − O > by EXAFS or by refinement of the XRD data avoiding the use of the
average cation size <rA>, taken from the Shannon tables.
η = 1− t ∝ k
rM −O − < Cu − O >
.
rM −O
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
Figure 2. The critical superconducting temperature function of doping δ/ δc and of pressure via
microstrain ε/ εc normalized at the critical doping (δc=1/8) and the critical Cu.O microstrain (εc=4%)
for the formation of the commensurate polaron crystal (CPC). The superconducting long range order
parameter clearly competes with the commensurate pJT polaron crystal formed at (1, 1) that has been
described as a paired Wignet polaron crystal.
where the pressure is measurd via the microstrain in the CuO2 lattice ε = (197 − Cu − O ) 197 and the
<Cu-O> bond length is measured in picometers thatt is plotted in Fig. 1.
The idea is based on the use of the unrelaxed equilibrium value for the Cu-O bond length
rCu −O = 197 pm measured for the planar CuO4 units made by the free Cu2+ ions in water measured by
EXAFS. The constant k is determined by the difference of the elastic constants between the copper
oxide plane and the intercalated planes (it was taken to be about 2 in previous works). There is a
critical value of the microstrain εc = 4 ± 0.3% (i.e., a critical average bond length
Cu − O c = 189 ± 0.5 pm ). For the values of the microstrain larger than the critical microstrain the
crystallographic LTT phase appears and the commensurate static spin modulation appears at doping
δc=1/8 in neutron scattering experiments [9], The critical microstrain is controlled within the range
3.7-4.3%.by the lattice disorder induced by the variance of the ionic radii of intercalated ions [129].
The lattice interlayer mismatch or chemical pressure measured by the Cu-O microstrain is related
with the average ion size in the rocksalt layers in contact with the CuO2 plane. In Fig. 1 the values of
the microstrain normalized to the critical value εc=4% as a function of the average ion size of the
cations in the nearest rockalt intercalated monolayers is plotted.
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
The superconducting critical temperature Tc(δ,ε) of many cuprate perovskites as a function of
normalized doping δ/δ c and the normalized CuO2 microstrain ε/ εc is plotted in Fig. 2 at ambient
pressure. The maximum Tc of 130 K occurs in the mercury cuprate family at doping δ/δc =1.3 (δ=16%)
and microstrain ε/εc =0.55 (i.e. at ε =2.2 % or at the Cu-O bond length of 192.65 pm). The cuprates
La0.6Nd0.4La1-xSrxCuO4 with microstrain ε/ εc >1 are in the region where Wigner polaron crystal
suppresses the superconducting critical temperature around the quantum critical point at (1,1)
Figure 3. Lower panel: The spin-gap energy as a function of normalized
microstrain. In different superconducting cuprates at optimum doping from
ref.[133, 137,138, 139]. Upper panel: the superconducting critical temperature
as a function of the microstrain at constant doping (1/8) and the Wigner
polaron crystal in Nd doped LCO cuprate perovkites from Ref. 13.
as it can be seem clearly in the figure. Therefore the electron-phonon coupling, controlled by the
chemical pressure, is the variable that drives the system to localization and there is a quantum critical
point where an electronic solid with long range order competes with the superconducting order as was
required by several theories [130].
The formation of a polaron Wigner crystal [90-93] was first proposed in 1993 [31, 14-16] for a
critical polaron density δc and a critical electron-phonon interaction gc to explain the drop of Tc in
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International Symposium on Lattice Effects in Cuprate High Temperature Superconductors
IOP Publishing
Journal of Physics: Conference Series 108 (2008) 012036
doi:10.1088/1742-6596/108/1/012036
La0.6Nd0.4La1-xSrxCuO4. and La1.875Ba0.125CuO4. Later it was found that these systems show static spin
stripe order [131, 132, 133] characterized by four magnetic diffraction spots at (π ± δ, π ) and
(π , π ± δ ) where δ is close to 0.125 that have been called the “static stripe phase" that we assign to a
striped polaron Wigner crystal in fact the magnetic order is related with the lattice structure[134-135].
Starting from La1.875Sr0.125CuO4 where we have a polaron liquid and spin fluctuations the Wigner
crystallization can be induced by increasing the microstrain (i.e. decreasing the average ion size) in the
case of La1.875.xNdxSr 0.125CuO4 while in the case of La1.875Ba0.125CuO4 the system close to the critical
point is pushed to localization by decreasing the critical strain value by increasing the disorder via
increasing the ion size variance The polaron Wigner crystals have been observed also in the nichelate
La2NiO4.25 [136] and in manganites LaSr0.5Ca0.5MnO3 [82] at a critical value (εc, δc).
In order to understand the relevance of the quantum critical point for the formation of the polaron
Wigner crystal on the superconducting phase we have plotted in Fig. 3 the energy of the spin gap for
the onset of spin fluctuations in the metallic phase of the polaron fluid for values of the microstrain
smaller than the critical value. The cuprates show an universal distribution of magnetic fluctuations. A
spin gap is observed at low energy in different cuprate families.
Fig. 3 shows the spin-gap energy for several different cuprates near optimal doping. La2-xSrxCuO4
(x=0.16) (LSCO) from reference [137], forYBa2Cu3O6.85 (YBCO) from reference [138];
Bi2Sr2CaCu2O8+y BSCO (estimated by scaling with respect to YBa2Cu3O6+x from reference [139] and
finally the zero gap for La2 -xBax CuO4 (LBCO) from reference [133]. The experimental results show a
correlation between magnetic excitations and microstrain that applies to a variety of cuprates. This
trend makes clear that the magnetic excitations show the typical behavior near a critical point of a
quantum phase transition where the high Tc superconductivity occurs in the region of a quantum
paramagnetism near the onset of quantum fluctuations.
In conclusion we have shown that the recent experimental results support the scenario proposed in
ref. [13] that the high Tc superconductivity occur near a Quantum Phase Transition for a polaron fluid
where the superconducting long range order competes with the long range order of polaron Wigner
crystal.
Acknowledgments
We acknowledge financial support from European STREP project 517039 "Controlling Mesoscopic
Phase Separation" (COMEPHS) (2005).
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