Lecture 27 March 9, 2011
Cuprates, metals
Nature of the Chemical Bond
with applications to catalysis, materials
science, nanotechnology, surface science,
bioinorganic chemistry, and energy
William A. Goddard, III, wag@wag.caltech.edu
316 Beckman Institute, x3093
Charles and Mary Ferkel Professor of Chemistry,
Materials Science, and Applied Physics,
California Institute of Technology
Teaching Assistants: Wei-Guang Liu <wgliu@wag.caltech.edu>
Caitlin Scott <cescott@caltech.edu>
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© copyright 2011 William A. Goddard III, all rights reserved
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Last time
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Examine the bonding of XeF
Consider the energy to form the charge transfer complex
Xe
Xe+
The energy to form Xe+ F- can be estimated from
Using IP(Xe)=12.13eV, EA(F)=3.40eV, and R(IF)=1.98 A,
we get E(Xe+ F-) = 1.45eV (unbound)
Thus there is no covalent bond for XeF, which has a weak
bond of ~ 0.1 eV and a long bond
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Examine the bonding in XeF2
The energy to form Xe+F- is +1.45 eV
Now consider, the impact of putting a 2nd F
on the back side of the Xe+
Xe+
Since Xe+ has a singly occupied pz orbital pointing directly at this
2nd F, we can now form a covalent bond to it
How strong would the bond be?
Probably the same as for IF, which is 2.88 eV.
Thus we expect F--Xe+F- to have a bond strength of ~2.88 – 1.45
= 1.43 eV!
Of course for FXeF we can also form an equivalent bond for
F-Xe+--F. Thus we get a resonance, which we estimate below
We will denote this 3 center – 4 electron charge transfer bond as
FXeF
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Estimate stability of XeF2 (eV)
Energy form F Xe+ F- at R=∞
F-Xe+ covalent bond length (from IF)
Energy form F Xe+ F- at R=Re
F-Xe+ covalent bond energy (from IF)
Net bond strength of F--Xe+ F-
1.3
Resonance due to F- Xe+--F
Net bond strength of XeF2
2.7
XeF2 is stable with respect to the free atoms by 2.7 eV
Bond energy F2 is 1.6 eV.
Thus stability of XeF2 with respect to Xe + F2 is 1.1 eV
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Stability of gas of XeF2
The XeF2 molecule is stable by 1.1 eV with respect to Xe + F2
But to assess whether one could make and store XeF2, say in a
bottle, we have to consider other modes of decomposition.
The most likely might be that light or surfaces might generate F
atoms, which could then decompose XeF2 by the chain reaction
XeF2 + F  {XeF + F2}  Xe + F2 + F
Since the bond energy of F2 is 1.6 eV, this reaction is
endothermic by 2.7-1.6 = 1.1 eV, suggesting the XeF2 is
relatively stable.
Indeed XeF2 is used with F2 to synthesize XeF4 and XeF6.
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XeF4
Putting 2 additional F to overlap
the Xe py pair leads to the
square planar structure, which
allows 3 center – 4 electron
charge transfer bonds in both
the x and y directions.
The VB analysis indicates that the stability for XeF4 relative to
XeF2 should be ~ 2.7 eV, but maybe a bit weaker due to the
increased IP of the Xe due to the first hypervalent bond and
because of some possible F---F steric interactions.
There is a report that the bond energy is 6 eV, which seems
too high, compared to our estimate of 5.4 eV.
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XeF6
Since XeF4 still has a pz pair, we can form
a third hypervalent bond in this direction to
obtain an octahedral XeF6 molecule.
Indeed XeF6 is stable with this structure
Here we expect a stability a little less than 8.1 eV.
Pauling in 1933 suggested that XeF6 would be stabile, 30
years in advance of the experiments.
He also suggested that XeF8 is stable.
However this prediction is wrong
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Estimated stability of other Nobel gas fluorides (eV)
Using the same
method as for
XeF2, we can
estimate the
binding energies
for the other
Noble metals.
KrF2 is predicted
to be stable by
0.7 eV, which
makes it
susceptible to
decomposition
by F radicals
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1.3
1.3
1.3
1.3
1.3
1.3
-2.9
-5.3
-0.1
1.0
2.7
3.9
RnF2 is quite stable, by 3.6
eV, but I do not know if it
9
has been
observed
© copyright 2011 William A. Goddard
III, all rights
reserved
Halogen Fluorides, ClFn
The IP of ClF is 12.66 eV
comparable to the IP of 12.13
for Xe.
This suggests that the px and
py pairs of Cl could be used to
form hypervalent bonds leading
to ClF3 and ClF5.
Stability of ClF3 relative to ClF + 2F
We estimate that ClF3 is stable
by 2.8 eV.
Indeed the experiment energy
for ClF3  ClF + 2F is 2.6 eV,
quite similar to XeF2.
Thus ClF3  is endothermic by
2.6 -1.6 = 1.0 ©eV
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Geometry of ClF3
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ClHF2
We estimate that
Is stable to ClH + 2F by 2.7 eV
This is stable with respect to ClH + F2 by 1.1 ev
But D(HF) = 5.87 eV, D(HCl)=4.43 eV, D(ClF) = 2.62 eV
Thus F2ClH  ClF + HF is exothermic by 1.4 eV
F2ClH has not been observed
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ClF5
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SFn
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PFn
The VB view is that the PF3 was distorted into a planar
geometry, leading the 3s lone pair to become a 3pz pair,
which can then form a hypervalent bond to two additional F
atoms to form PF5
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Donor-acceptor bonds to oxygen
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Ozone, O3
The simple VB description of ozone is,
where the terminal pp electrons are not
doing much
This is analogous to the s system in the
covalent description of XeF2.
Thus we can look at the p system of
ozone as hypervalent, leading to
charge transfer to form
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Diazomethane
leading to
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Application of hypervalent concepts
Origin of reactivity in the hypervalent
reagent o-iodoxybenzoic acid (IBX)
Hypervalent O-I-O
linear bond
Enhancing 2-iodoxybenzoic acid reactivity by exploiting a hypervalent twist
Su JT, Goddard WA; J. Am. Chem. Soc., 127 (41): 14146-14147 (2005)
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Hypervalent iodine assumes many metallic personalities
Hypervalent I
alternative
O
Oxidations
O
O
OH
I
CrO3/H2SO4
OAc
Radical
cyclizations
SnBu3Cl
I
OAc
OH
Electrophilic
alkene activation
CC bond
formation
I
HgCl2
OTs
O
I
Pd(OAc)2
this remarkable chemistry of iodine can be
understood in terms of hypervalent concepts
Martin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514
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New material
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Bonding in metallic solids
Most of the systems discussed so far in this course have been
covalent, with the number of bonds to an atom related to the
number of valence electrons.
Thus we have discussed the bonding of molecules such as CH4,
benzene, O2, and Ozone. The solids with covalent bonding, such
as diamond, silicon, GaAs, are generally insulators or
semiconductors
We also considered covalent bonds to metals such as FeH+,
(PH3)2Pt(CH3)2, (bpym)Pt(Cl)(CH3), The Grubbs Ru catalysts
We have also discussed the bonding in ionic materials such as
(NaCl)n, NaCl crystal, and BaTiO3, where the atoms are best
modeled as ions with the bonding dominated by electrostatics
Next we consider the bonding in bulk metals, such as iron, Pt, Li,
etc. where there is little connection between the number of bonds
22
and the number of valence electrons.
Elementary ideas about metals and insulators
The first attempts to develop
quantum theory started with the
Bohr model H atom with
electrons in orbits around
the nucleus.
With Schrodinger QM came the
idea that the electrons were in
distinct orbitals (s, p, d..), leading
to a universal Aufbau diagram
which is filled with 2 electrons in
each of the lowest orbitals
For example:
O (1s)2(2s)2(2p)4
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Bringing atoms together to form the solid
As we bring atoms together to
form the solid, the levels broaden
into energy bands, which may
overlap . Thus for Cu we obtain
Energy
Fermi energy
(HOMO and
LUMO
Thus Cu does
not have a
band gap at
ordinary
distances
Density states
24
Metals vs inulators
25
conductivity
For systems with a band
gap, there is no current
until excite an electron
from the occupied valence
band to the empty
conduction band
The population of electrons
in the conduction band
and holes in the valence
bond is proportional to
exp(-Egap/RT).
Thus conductivity
incresses with T (resistivity
decreases)
26
The elements leading to metallic binding
There is not yet a conceptual description for metals of a
quality comparable to that for non-metals. However
there are some trends, as will be described
27
Body centered cubic (bcc), A2
A2
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Face-centered cubic (fcc), A1
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Alternative view of fcc
30
Closest packing layer
31
Stacking of 2 closest packed layers
32
Hexagonal
closest packed
(hcp) structure,
A3
33
Cubic closest packing
34
Double hcp
The hexagonal lanthanides mostly exhibit a
packing of closest packed layers in the
sequence
ABAC ABAC ABAC
This is called the double hcp structure
35
mis
fcc
hcp
b cc
Structures of elemental metals
some correlation of
structure with number of
valence electrons
36
Binding in metals
Li has the bcc structure with 8 nearest neighbor atoms, but there
is only one valence electron per atom.
Similarly fcc and hcp have 12 nearest neighbor atoms, but Al with
fcc has only three valence electrons per atom while Mg with hcp
has only 2.
Clearly the bonding is very different than covalent
One model (Pauling) resonating valence bonds
One problem is energetics:
Li2 bond energy = 24 kcal/mol  12 kcal/mol per valence electron
Cohesive energy of Li (energy to atomize the crystal is 37.7
kcal/mol per valence electron. Too much to explain with resonance
New paradigm: Interstitial Electron Model (IEM). Each valence
electron localizes in a tetrahedron between four Li nuclei.
Bonding like in Li2+, which is 33.7 kcal/mol per valence electron
37
GVB orbitals of ring M10 molecules
Get 10 valence electrons each
localized in a bond midpoint
note H10 is very different, get
orbital localized on atom, not
bond midpoint
R=2 a0
Calculations treated all 11
valence electrons of Cu, Ag, Au
using effective core potential.
All electrons for H and Li
38
39
Bonding in alkalis
40
The bonding in column 11
Get trend similar to alkalis
41
Geometries of Li4 clusters
For H4, the electrons are in 1s orbitals centered on each atom
Thus spin pair across sides. Orthogonalization cases distortion to
rectangle
For Li4, the electrons are in orbitals centered on each bond
midpoint
Thus spin pair between bond midpoint. Orthogonalization cases
distortion to rhombus
42
Geometries of Li6 cluster
For H6, the electrons are in
1s orbitals centered on
each atom
Thus spin pair across
sides. Orthogonalization
cases distortion to D3h
hexagone
For Li6, the electrons are in
orbitals centered on each
bond midpoint
Thus spin pair between
bond midpoint.
Orthogonalization cases
distortion to triangular
structure
43
Geometries of Li8 cluster
For Li8, the electrons are in orbitals centered on each bond
midpoint
Thus spin pair between bond midpoint.
Orthogonalization cases distortion to out-of-plane D2d structure
44
Li10 get closest packed structure
45
Li two dimensional
Electrons localize into triangular
interstitial regions
Closest packed structure has 2 triangles per electron
One occupied and one empty
Spin pair adjacent triangles but leave others empty to avoid Pauli
Repulsion
Calculation periodic cell with 8 electrons or 4 GVB pairs with
46
overlap = 0.52
Crystalline properties of B column
47
Binding of CH3 to Pt clusters
48
Binding of alkyl CH3-xMex to (111) surfaces
Prefers on-top site
Decreased binding with increasing x due to steric interactions with
49
other atoms of Pt (111) surface
Binding of alkylidene CH2-xMex to (111) surfaces
Prefers bridge binding site
Decreased binding with increasing x due to steric interactions with
50
other atoms of Pt (111) surface
Binding of alkylidyne CH2-xMex to (111) surfaces
51
Average bond strength in CHx/M8 cluster
52
Comparison of bonding energies of CHx and C2Hx
53
Energy barriers for CH4 dehydrogenation on Pt
54
Energy barriers for benzene dehydrogenation on Pt
55
Geometries and Energetics of ethyl binding to M8
56
Geometries and Energetics of ethylene binding to M8
57
Geometries and Energetics of vinyl binding to M8
58
Geometries and Energetics of ethylidene binding to M8
59
Geometries and Energetics of vinylidene binding to M8
60
Geometries and Energetics of dicarbond binding to M8
61
Geometries and Energetics of ethynel binding to M8
62
Geometries and Energetics of acetylene binding to M8
Confirmed experimentally by
Wilson Ho
63
Heats of formation for C2Hx and CHx species for Pt
Most stable is to
form CHad
64
Energetics for C2Hx on Pt
65
66
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68
High Temperature Superconductors Cuprates
Chiral plaquette polaron theory of cuprate superconductivity;
Tahir-Kheli J, Goddard WA
Phys. Rev. B 76 (1): Art. No. 014514 (2007)
The Chiral Plaquette Polaron Paradigm (CPPP) for high
temperature cuprate superconductors; Tahir-Kheli J, Goddard
WA; Chem. Phys. Lett. (4-6) 153-165 (2009)
Plaquette model of the phase diagram, thermopower, and
neutron resonance peak of cuprate superconductors; Jamil
Tahir-Kheli and William A. Goddard III, J. Phys. Chem. Lett,
submitted
Jamil
69
Tahir-Kheli
Superconducting Tc; A Story of Punctuated Evolution
All Serendipity
Discovery is made. Then all combinations
tried. Then stagnation until next discovery.
Theory has never successfully predicted a
new higher temperature material.
Embarrassing state for Theorists.
To ensure progress we need to learn the
fundamental mechanism in terms of the
atomistic interactions
Metal
Era
A15 Metal Alloy
Era
Today
Cuprate
Era
Theoretical Limit
(Anderson)
2020
BCS Theory (1957)
70
Short history of superconductivity
4.15 K 1911 Hg (Heike Kamerlingh Onnes, Leiden U, Netherlands)
3.69 K 1913 Tin (Onnes)
7.26 K 1913 Lead (Onnes)
9.2 K 1930 Niobium (Meissner, Berlin)
Metals Era
1.14 K 1933 Aluminum
16.1 K 1941 NbN (Ascherman, Frederick, Justi, and Kramer, Berlin)
17.1 K 1953 V3Si (Hardy and Hulm, U Chicago)
A15 Metal
18.1 K 1954 Nb3Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs)
Alloy Era
9.8 K 1962 Nb0.6Ti0.4 (First commercial wire, Westinghouse)
23.2 K 1973 Nb3Ge (Gavaler, Janocho, and Jones, Westinghouse)
30 K 1986 (LaBa)2CuO4 (Müller and Bednorz, IBM Rüschlikon, Switzerland)
92 K 1987 YBa2Cu3O7 (Wu, Ashburn, and Torng (Alabama), Hor, Meng, Gao, Huang,
Wang, and Chu (Houston))
The Cuprate Era
105 K 1988 Bi2Sr2CaCu2O8 (Maeda, Tanaka, Fukutomi, Asano, Tsukuba Laboratory)
120 K 1988 Tl2Ba2Ca2Cu3O10 (Hermann and Sheng, U. Arkansas)
133 K 1993 HgBa2Ca2Cu3O8 (Schilling, Cantoni, Guo, Ott, Zurich, Switzerland)
138 K 1994 (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 (Dai, Chakoumakos, (ORNL) Sun, Wong,
(U Kansas) Xin, Lu (Midwest Superconductivity Inc.), Goldfarb, NIST)
after a 15 year drought, the next generation is due soon, what will it be?
71
Fundamental Goals in Our Research on Cuprate
Superconductivity
Determine the fundamental mechanism in order to
have a sound basis for designing improved systems.
Criterion for any proposed mechanism of
superconductivity:
Does it explain the unusual properties of the normal
and superconducting state for cuprates?
There is no precedent for a theory of superconductivity
that actually predicts new materials.
Indeed we know of no case of a theorist successfully
predicting a new improved superconducting material!
Bernt Matthias always claimed that before trying new
compositions for superconductors he would ask his Bell
Labs theorists what to try and then he would always do
just the opposite.
72
Perovskites
Perovskite (CaTiO3) first described in the 1830s
by the geologist Gustav Rose, who named it
after the famous Russian mineralogist Count Lev
Aleksevich von Perovski
crystal lattice appears cubic, but it is actually
orthorhombic in symmetry due to a slight
distortion of the structure.
Characteristic chemical formula of a perovskite
ceramic: ABO3,
A atom +2 charge. 12 coordinate at the corners
of a cube.
B atom +4 charge.
Octahedron of O ions on the faces of that cube
centered on a B ions at the center of the cube.
Together A and B form an FCC structure
73
(La0.85Z0.15)2CuO4: Tc = 38K (Z=Ba), 35K (Z=Sr)
1986 first cuprate superconductor, (LaBa)2CuO4 (Müller and Bednorz) Nobel Prize
Isolated CuO2 sheets
with apical O on both
sides of Cu to form an
elongated octahedron
Structure type: 0201
Crystal system: Tetragonal
Lattice constants:
a = 3.7873 Å
c = 13.2883 Å
Space group: I4/mmm
Atomic positions:
La,Ba at (0, 0, 0.3606)
Cu at (0, 0, 0)
O1 at (0, 1/2, 0)
O2 at (0, 0, 0.1828)
CuO6 octahedra
74
YBa2Cu3O7–d Tc=92K (d=0.07)
Per formula unit:
two CuO2 sheets (five coordinate  pyramid)
one CuO chain (four coordinate  square)
Structure type: 1212C
Crystal system: Orthorhombic
Lattice constants:
a = 3.8227 Å
b = 3.8872 Å
c = 11.6802 Å
Space group: Pmmm
Atomic positions:
Y at (1/2,1/2,1/2)
Ba at (1/2,1/2,0.1843)
Cu1 at (0,0,0)
Cu2 at (0, 0, 0.3556)
O1 at (0, 1/2, 0)
O2 at (1/2,0,0.3779)
O3 at (0,1/2,0.379)
O4 at (0, 0,0.159)
1987: Alabama: Wu, Ashburn, and Torng
Houston: Hor, Meng, Gao, Huang, Wang, Chu
75
Tc depends strongly on the number of CuO2 layers:
Bi2Sr2Can-1CunO4+2n
single sheet CuO2
Tc= 10 K
double sheet CuO2 Tc= 85 K
Triple sheet CuO2
Tc= 110 K
a = 3.85 Å
c = 26.8 Å
a = 3.85 Å
c = 30.9 Å
a = 3.85 Å
c = 36.5 Å
76
Dependence of Tc on layers is not monotonic
TlBa2Can-1CunO2n+3
n= 2 Tc= 103 K
n= 3 Tc= 123 K
n= 4 Tc= 112 K
n= 5 Tc= 107K
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
Double sheet CuO2
a = 3.86 Å
c = 12.75 Å
Triple sheet CuO2
a = 3.84 Å
c = 15.87 Å
4 sheet CuO2
5 sheet CuO2
a = 3.85 Å
c = 19.15 Å
a = 3.85 Å
77
c = 22.25 Å
Reining Champion since 1994: Tc=138K
(Hg0.8Tl0.2)Ba2Ca2Cu3O8.33
This has the
same structure
as
TlBa2Ca2Cu3O9
n= 3 Tc= 123 K
1994
Dai, Chakoumakos (ORNL)
Sun, Wong (U Kansas)
Xin, Lu (Midwest
Superconductivity Inc.),
Goldfarb (NIST)
CuO2
CuO2
CuO2
a = 3.84 Å
c = 15.87 Å
Triple sheet CuO2
78
Isolated layer can be great
CuO2
Tl2Ba2Can-1CunO2n+4
CuO2
n= 1 Tc= 95 K
CuO2
n= 4 Tc= 112 K
CuO2
CuO2
CuO2
CuO2
CuO2
CuO2
single sheet CuO2
CuO2 a = 3.86 Å
c = 23.14 Å
a = 3.85CuO
Å 2
c = 41.98 Å
4 sheet CuO79
2
(Ba,Sr)CuO2 Tc=90K
single sheet CuO2
Structure type: 02"∞ -1"∞
Crystal system: Tetragonal
Lattice constants:
a = 3.93 Å
c = 3.47 Å
Space group: P4/mmm
Atomic positions:
Cu at (0,0,0)
O at (0,1/2,0)
Ba,Sr at (1/2,1/2,1/2)
80
Some cuprates lead to electron doping not holes
(Nd,Ce)2CuO4d Tc =24K
For d=0
2 Nd (+3) and 1 Cu (+2) lead to 8 holes
4 O (2) lead to 8 electrons, get insulator
Dope with Ce (+4) leading to an extra electron
CuO2
single sheet CuO2
Structure type: 0201T '
Crystal system: Tetragonal
Lattice constants:
a = 3.95 Å
c = 12.07 Å
Space group: I4/mmm
Atomic positions:
Nd,Ce at (0,0,0.3513)
Cu at(0,0,0)
O1 at (0, 1/2, 0)
O2 at (0, 1/2,1/4)
CuO2
CuO2
81
Our Goal
Explain which systems lead to high Tc and
which do not
Explain how the number of layers and the
location of holes and electrons affects the Tc
Use this information to design new structures
with higher Tc
82
Structural Characteristics of HighTc Superconductors:
Start with Undoped Antiferromagnet
Cu
O
O
Cu
O
Cu
O
Cu
O
Cu
O
Cu
O
Cu
O
Cu
Cu
O
O
O
O
Cu
O
O
O
O
Cu
O
O
O
Cu
O
O
O
Cu
Cu
Cu
O
O
Cu
2D CuO2 square lattice (xy plane)
Oxidation state of Cu: CuII or d9.
(xy)2(xz)2(yz)2(z2)2(x2-y2)1
d9 hole is 3d (x2–y2).
Oxidation state of O: O2– or p6.
(pσ)2 (pp)2 (ppz)2.
Cu – O bond = 1.90 – 1.95 Å
Cu can have 5th or 6th apical O (2.4 Å)
to form an octahedron or half-octahedron
Undoped system antiferromagnetic with
TNeel = 325 K for La2CuO4.
Describe states as a Heisenberg AntiFerromagnet
with Jdd = 0.13 eV for La2CuO4:
H dd  J dd  Si  S j
Superexchange
Jdd AF coupling
83
Superexchange coupling of two Cu d9 sites
exactly the same as the hypervalent XeF2
Two Cu d9 separated by 4Å leads to no bonding
(ground state singlet and excited triplet separated by 0.0001 eV)
With O in-between get strong bonding
(the singlet is stabilized by Jdd = 0.13 eV = 1500K for LaCuO4)
Explanation: a small amount of charge transfer from O to right Cu
Cu(x2-y2)1-O(px)2-Cu(x2-y2)1  Cu(x2-y2)1-O(px)1-Cu(x2-y2)2
allows bonding of the O to the left Cu, but only for the singlet state
The explanation is referred to as superexchange.
bond
bond
No direct bonding
84
267-14
Characteristic of High Tc Superconductors: Doping
The undoped La2CuO4 is an insulator (band gap = 2.0 eV)
La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+
Thus cation holes = 3*2 + 2 = 8 and anion electrons = 4*2 = 8
Cu2+  d9  local spin  antiferromagnetic coupling
To get a metal requires doping to put holes in the valence band
Doping (oxidation) La2-xSrxCuO4:
Assuming 4 O2- requires 8 cation holes.
But La2-xSrx  6-x holes, thus must have x Cu3+ and 1 – x Cu2+
Second possibility: assume that excitation from Cu2+ to Cu3+ is too high,
then must have hole on O2– leading to O–
This leads to x O– and (4 – x) O2– per formula unit.
YBa2Cu3O7:
Assume that all 7 O are O2–
Must have 14 cation holes: since Y3+ +2 Ba2+ leads to +7, then we must have
1 Cu3+ and 2 Cu2+
The second possibility is that all Cu are Cu2+ requiring that there be 1 O–
and 6 O2–
85
Essential characteristic of all cuprate superconductors is oxidation
(doping)
Typical phase diagram
La2-xSrxCuO4
Superconductor: 0.05 < x < 0.32
Spin Glass: 0.02 < x < 0.05
Antiferromagnetic: 0 < x < 0.02
Minimum doping to obtain
superconductivity, x > 0.05.
Optimum doping for highest Tc=35K at
x ~ 0.15.
Maximum doping above which the
superconductivity disappears and the
86
system becomes a normal metal.
Summary: Central Characteristics of cuprate
superconductors, square CuO2 lattice, 16% holes
CuO2 plane
Cu
O
O
Cu
O
Cu
O
Cu
O
O
pσ
O
O
Cu
Cu
O
O
O
O
Cu
O
O
O
Cu
Cu
O
Cu
O
Cu
O
pπ
O
Cu
O
Cu
O
O
La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+
Cu d9 Cu2+  spin, with antiferromagnetic coupling
Cu
O
O
Cu
Doping (oxidation) La2-xSrxCuO4:
Hole  x Cu3+ and 1 – x Cu2+, Or
Hole  x O– and 4 – x O2–
YBa2Cu3O7:
Y3+, Ba2+, O2–  1 Cu3+ and 2 Cu2+, Or
Y3+, Ba2+, Cu2+  1 O– and 6 O2–
Where are the Doped Holes?
CuIII or d8: Anderson, Science 235, 1196 (1987), but CuII CuIII IP = 36.83 eV
O pσ: Emery, Phys. Rev. Lett. 58, 2794 (1987).
O pπ: Goddard et al., Science 239, 896, 899 (1988).
O pσ: Freeman et al. (1987), Mattheiss (1987), Pickett (1989).
All wrong: based on simple QM (LDA) or clusters (Cu3O8)
87
Which is right: ps or pp holes?
Goddard et al. carried out GVB
calculations on Cu3O10 + 998 point
charges and found pp holes (found
similar E for ps)
undoped
Electronic Structure and Valence Bond Band
Structure of Cuprate Superconducting Materials;
Y. Guo, J-M. Langlois, and W. A. Goddard III
Science 239, 896 (1988)
The Magnon Pairing Mechanism of
Superconductivity in Cuprate Ceramics
G. Chen and W. A. Goddard III; Science 239, 899
(1988)
doped
88
pp holes
Goddard et al showed if the ground state has pp holes there is an
attractive pair that leads to triplet P-wave Cooper pairs, and hence
superconductivity.
The Superconducting Properties of Copper Oxide High Temperature Superconductors;
G. Chen, J-M. Langlois, Y. Guo, and W. A. Goddard III; Proc. Nat. Acad. Sci. USA 86,
3447 (1989)
The Quantum Chemistry View of High Temperature Superconductors; W. A. Goddard III,
Y. Guo, G. Chen, H. Ding, J-M. Langlois, and G. Lang; In High Temperature
Superconductivity Proc. 39th Scottish Universities Summer School in Physics, St.
Andrews, Scotland, D.P. Tunstall, W. Barford, and P. Osborne Editors, 1991
However experiment shows that the systems are singlet D-wave.
Thus, pp holes does not correct provide an explanation of
superconductivity in cuprates.
89
ps holes
Emery and most physicists assumed ps holes on the oxygen.
Simplifying to t–J model, calculations with on-site Coulomb
repulsion suggest that if the system leads to a
superconductor it should be singlet D-wave.
Thus most physicists believe that ps provides the basis for a
correct explanation of superconductivity.
Goddard believed that if ps holes
were correct, then it would lead to
strong bonding to the singly occupied
dx2-y2 orbitals on the adjacent Cu
atoms, leading to a distortions that
localize the state.
This would cause a barrier to
hopping to adjacent sites and hence
would not be superconducting
90
Current canonical HighTc Hamiltonian
Ops holeThe t–J Model
Undoped
Cu d9 hole is 3d x2–y2. O 2pσ doubly occupied.
Heisenberg AF with Jdd = 0.13 eV for La2CuO4.
H dd  J dd  Si  S j
Doped
Doping creates hole in O pσ that bonds with x2–y2
to form a bonded singlet (doubly occupied hole).
Singlet hole hopping through lattice prefers
adjacent sites are same spin, this frustrates the
normal AF coupling of d9 spins.
J
t
J
Because of Coulomb repulsion cannot have doubly
occupied holes.
91
Summary of the t–J model
Coulomb repulsion of singlet holes leads to singlet d-wave Cooper
pairing.
d-wave is observed in phase sensitive Josephson tunneling, in
NMR spin relaxation (no Hebel-Slichter coherence peak), and in
the temperature dependence of the penetration depth (λ~T2).
t-J predicts an ARPES (angle-resolved photoemission) pseudogap
which may have the right qualitative dependence.
The t–J model has difficulty explaining most of the normal state
properties (linear T resistivity, non-standard Drude relaxation,
temperature dependent Hall effect, mid-IR optical absorption,
and neutron ω/T scaling).
92
Universal Superconducting Tc Curve
(What we must explain to have a credible theory)
≈ 0.16
Superconducting
Phase
≈ 0.05
≈ 0.27
Where do these three special doping values come from?
93
Basis for all theories of cuprate superconductors
LDA Band calculations of La2CuO4
LDA and PBE lead to a half filled
ps-dx2-y2 band; predicting that
La2CuO4 is metallic!
This is Fundamentally Wrong
Experimental Band Gap is 2 eV
Empty
Fermi Energy
(0,p)
Un-Occupied
Un-Occupied
Occupied Occupied
G(0,0)
(p,0)
Occupied Occupied
Un-Occupied
ps-dx2-y2 band
(p,p)
Un-Occupied
LDA: Freeman 1987,
Mattheiss 1987,
Pickett (1989)
Occupied
94
G2(MAD)
H-Ne
TM
He2
Ne2
(H2O)2
0.0109(2.377)
0.0231(2.595)
0.391(2.710)
Units:
eV
Method
Hf
IP
EA
PA
Etot
HF
6.47
1.036
1.158
0.15
4.49
G2 or best ab initio 0.07a
0.053a
0.057a
0.05a
1.59a
WeE(R
want
to do
QM onDcuprate
E(Re)
e(RO..O)
e)
superconductors.
Which
1.09
Unbound
Unbound
0.161(3.048)
DFT functional Is most accurate?
0.19b 0.0011(2.993)c 0.0043(3.125)c 0.218(2.912)d
3.94a
0.665
0.749
0.27
6.67
0.54e
BP86
0.88a
0.175
0.212
0.05
0.19
BLYP
0.31a
0.187
0.106
0.08
0.19
Validation of accuracy of DFT for
148 molecules
(the G2 0.194(2.889)
Set) with very
0.46
Unbound
Unbound
accurate experimental
data
0.37e
Unbound
Unbound
0.181(2.952)
BPW91
0.34a
0.163
0.094
0.05
0.16
0.60
PW91PW91
0.77
0.164
0.141
0.06
0.35
0.52
OLYP
mPWPWf
0.20
0.65
0.185
0.161
0.133
0.122
1.38
0.05
0.16
0.38
PBEPBEg
0.74g
0.156
0.101
0.06
1.25
0.34
XLYPk
0.33
0.186
0.117
0.09
0.95
0.24
0.18
0.94
0.139
0.207
0.107
0.247
1.13
0.07
0.08
0.72
Unbound
B3P86 m
0.78a
0.636
0.593
0.03
2.80
0.34
Unbound
B3LYP n
0.13a
0.168
0.103
0.06
0.38
e
0.25LDA
Unbound
(3.9 eV
B3PW91 o
0.15a
0.161
0.100
0.03
0.24
0.38
PW1PW p
0.23
0.160
0.114
0.04
0.30
0.30
0.0066(2.660)
0.0095(3.003)
0.227(2.884)
mPW1PW q
0.17
0.160
0.118
0.04
0.16
0.31
0.0020(3.052)
0.0023(3.254)
0.199(2.898)
PBE1PBE r
0.21g
0.162
0.126
0.04
1.09
B3LYP
and X3LYP
are most
generally
0.30
0.0018(2.818)
0.0026(3.118)
0.216(2.896)
X3LYP s
0.12
0.154
0.087
0.07
0.11
0.22
accurate
DFT methods
0.0010(2.726)
0.0028(2.904)
0.216(2.908)
Exptl.
0.00
0.000
0.000
0.00
0.00
0.00
0.0010(2.970)h 0.0036(3.091)h 0.236i(2.948 j)
LDA (SVWN)
GGA
Exc
Unbound
Unbound
0.156(2.946)
H
f = Heat of Formation (298K)
0.0100(2.645)
0.0137(3.016)
0.235(2.886)
IP = Ionization
Potential
0.0052(2.823)
0.0076(3.178)
0.194(2.911)
EA = Electron
affinity
0.0032(2.752)
0.0048(3.097)
PA = Proton
Affinity 0.222(2.899)
0.0010(2.805)
0.192(2.953)
Etot = total0.0030(3.126)
atomic energy
Hybrid Methods
O3LYP
BH&HLYPl
Data isUnbound
the mean 0.214(2.905)
average
deviation from experiment
Unbound
0.206(2.878)
Unbound
0.198(2.926)
error)
and HF
(6.5 eV error)
Unbound
Unbound
0.175(2.923)
Useless for
thermochemistry
95
U-B3LYP calculations of La2CuO4
U-B3LYP leads to an insulator (2eV
band gap) with a doubled unit cell
(one with up-spin Cu and the other
down-spin)
ky
Empty
Fermi Energy
Band gap
Occupied
LDA
0.0
PBE
0.0
PW91
0.0
Hartree-Fock
17.0
B3LYP (unrestricted) 2.0
Experiment
2.0
Freeman et al. 1987, Mattheiss 1987, Pickett 1989
Tahir-Kheli and Goddard, 2006
Tahir-Kheli and Goddard, 2006
Harrison et al. 1999
Perry, Tahir-Kheli, Goddard Phys. Rev. B 63,144510(2001)
96
(Ginder et al. 1988)
B3LYP leads to excellent band gaps for La2CuO4 whereas LDA predicts a
metal!
LDA
0.0
PBE
0.0
PW91
0.0
Hartree-Fock
17.0
B3LYP (unrestricted) 2.0
Experiment
2.0
(Freeman et al. 1987, Mattheiss 1987, Pickett 1989)
Tahir-Kheli and Goddard, 2006
Tahir-Kheli and Goddard, 2006
(Harrison et al. 1999)
(Perry, Tahir-Kheli, and Goddard 2001)
(Ginder et al. 1988)
Conclusion #1: To describe the states of the La2CuO4 antiferromagnet we
must include some amount of true (Hartree-Fock) exchange.
Plane wave based periodic codes do not allow this (Castep, CPMD, VASP,
Vienna, Siesta).
LDA or GGA is not sufficient (e.g. PBE, PW91, BLYP).
Crystal does allow B3LYP and X3LYP.
Conclusion #2: Unrestricted B3LYP (U-B3LYP) DFT gives an excellent
description of the band gap.
Hence, B3LYP should be useful for describing doped cuprates (both hole
type and electron type).
97
B3LYP Works Well for Crystals with Transition Metals Too
Expt.
B3LYP
Si
3.5
3.8
Diamond
5.5
5.8
GaAs
1.4
1.5
ZnO
3.4
3.2
Al2O3
9.0
8.5
Cr2O3
3.3
3.4
MgO
7.8
7.3
MnO
3.6
3.8
NiO
3.8
3.9
TiO2
3.0
3.4
FeS2
1.0
2.0
ZnS
3.7
3.5
Si B3LYP band structure
1.2eV
3.5eV
Points are Expt., GW, and QMC
Direct (Vertical) Bandgaps in eV
B3LYP accurate for transition metals
and band structures
J. Muscat, A. Wander, and N. M. Harrison,
Chem. Phys. Lett. 342, 397 (2001). 98
Undoped
x=0.0
Band gap 2eV
Density of States for
Explicitly Doped
La2–xSrxCuO4 using UB3LYP
Use superlattice (2√2 x 2√2 x 1) with 8
primitive cells La15SrCu8O32 This allows
antiferromagnetic coupling of Cu atoms.
Allows hole to localize (but not forced to)
HOMO
LUMO
The down-spin states show a clear
localized hole with Opz-Cudz2
character
Note exactly 0.125 doping leads to ordered
supercell with small gap.
Band
gap
Real system  disordered holes + d9 spins
x=0.125
Becomes conductor for x>0.06
Doped UB3LYP: Perry, Tahir-Kheli, and Goddard,
Phys Rev B 65, 144501 (2002)
99
Now use B3LYP for
La2–xSrxCuO4
B3LYP leads to a hole along the SrO-Cu Apical axis (z) the apical
Polaron
Use superlattices with 8 primitive cells La15SrCu8O32
allowing antiferromagnetic coupling of Cu atoms.
Allows hole to localize (but not forced to)
0.26 Å
Find extra hole localized on apical Cu
and O atoms below the Sr site.
2/O pz
Cu
z
This state has Cu
character on the Cu and
hole
Opz character on the bridging O atom.
dz2
Because dz2 not doubly occupied, the top O –
Cu bond goes from 2.40 to 2.14 Å and O – Cu
bond below Cu decreases 0.11 Å to 2.29 Å.
The spin of the Cu dz2 and Cu dx2-y2 on the
same atom are the same (d8 high spin)
The singlet state is ~2.0 eV higher.
Bottom line: the hole is NOT in the Cu-O
plane (as assumed in ALL previous
attempts to explain superconductivity of
cuprates)
Sr
Perry, Tahir-Kheli, and Goddard; Phys. Rev. B 65, 144501 (2002)].
0.11Å
100
Nature of the new hole induced by LaSr: the Apical
Polaron
Two holes
Mulliken Populations
4 Planar O1 in
CuO6 near Sr
O2
O1
One hole
O2’
2 O1 per
hole
Located on apical Cu–O just below the Sr. We call this the Apical Polaron.
101
2nd LaSr polaron from Ab-Initio DFT:
the Plaquette Polaron
The Plaquette Polaron state is localized on Apical O pz +
the four-site Cu plaquette above the Sr. It has Cu z2 hole
de-localized
apical O pz, Cu dz2, and planar O pσ
character over the plane of four Cu atoms. over plaquette
The Plaquette Polaron state is calculated to for low doping
be 0.053 eV per 8 formula units above the
0.09 Å
apical polaron state this is
0.1 Å
0.007 eV = 0.2 kcal/mol per Cu in the
La0.875Sr0.125CuO4 cell.
The apical O below the Sr shifts up 0.1 Å to
a Cu – O bond distance of 2.50 Å (seen in Sr
XAFS) leading to a plaquette state.
The apical O below the plaquette Cu
distance optimizes to a Cu – O bond distance
of 2.29 Å.
Sr
Assumption: LaSr Doping leads to Plaquette Polarons.
102
The Plaquette Polaron States
Consider 4 Cu-O dz2 orbitals
and the 4 Ops orbitals. For the
undoped system, there are 16
electrons in these 8 orbitals
In the Plaquette Polaron, one
electron is removed. This leads
to a hole in either the Px or Py
orbital (degenerate).
103
Real orbital view of Plaquette Polarons
get two degerate states
Main hole character
Cu dz2
O pz = ps
Sr
Cu dz2
O pz = ps
Sr
104
Coupling of plaquette spin to neighboring d9
antiferromagnetic lattice (Ising)
The Px plaquette is compatable with the left
antiferromagnetic coupling of the d9 regions
while the Py plaquette is compatable with the
right antiferromagnetic coupling of the d9
regions.
This Ising-like description is over-simplified.
Must find ground state of Heisenberg system,
including the Plaquette spin
H dd  J dd  Si  S j
105
The Chiral Coupling Term
 J CH S z 2  ( S1  S2 )
 J CH S z 2  ( S2  S1 )
Chiral coupling twists the spins into a right or left-handed system.
106
We obtain 3 types of Electrons
1. “Undoped” Cu AF d9 sites
2. 4-site polarons (out of plane)
• Two types of polarons
a) Surface polarons
(neighboring) AF d9
sites
b) Interior polarons
(surrounded by
other polarons)
3. x2-y2/pσ band electrons
inside the percolating
polaron swath
(the “Doped” Cu sites)
x2-y2/pσ
band
Surface
Polaron
Interior
Polaron
d9 AF
107
Total Spin Hamiltonian
Time-Reversed Chiral Polarons
 J CH S z 2  ( S1  S2 )
The total spin Hamiltonian is,
Htot = Hdd + Hpd + HCH.
d9)–(AF
d9)
(AF
spin coupling
(AF d9)–(polaron) spin coupling
Chiral coupling
| Px  iPy ; s   | Px  iPy ;  s 
J CH   J CH ,
Sz  Sz .
HCH is invariant under polaron time-reversal.
Hdd is polaron time-reversal invariant.
Hpd is not invariant. JpdSz•Sd – JpdSz•Sd
Hpd splits the energy between time-reversed chiral polarons. The energy difference
is on the order of Jpd ~ (8/4)Jdd=2Jdd ~ 0.28 eV. The energy difference between
polarons with the same spin but different chiralities is on the order of 4JCH ~ 1.1 eV.
The energy splitting between time-reversed polarons is largest for low doping
because there are more d9 spins to induce the splitting.
108
Isolated Plaquette Polaron in a d9 sea.
Dopant
The Plaquette is pinned down by the Sr dopant. The
Cu d9 sea leads to an insulator, just as for undoped
109
As increase doping, weaken AF coupling among Cud9
states. Above 5% get conductor.
1. “Undoped” Cu AF d9 sites
2. 4-site polarons (out of plane)
• Two types of polarons
a) Surface polarons
(neighboring) AF d9
sites
b) Interior polarons
(surrounded by
other polarons)
3. x2-y2/pσ band electrons
inside the percolating
polaron swath
(the “Doped” Cu sites)
x2-y2/pσ
band
Surface
Polaron
Interior
Polaron
d9 AF
110
Superconducting Pairing only on Surface Plaquettes
111
Assume Optimal Tc  Maximum Surface Polarons
per Volume
experiment
1000 x 1000 lattice
200 ensembles
≈ 0.16
Sp
prediction
Percolation threshold
Becomes metallic
Surface area of
pairing leads to
correct optimal
doping
112
Predict maximum Tc
• Given our Plaquette Theory of Cuprate
Superconductors
– Find a way to calculate Tc for different
plaquette arrangements as a function of
doping
• The hope is to predict dopant configurations that
could lead to increased Tc
– Here, we show some math for the gap
equations in the hope that there will be
interest in the group to attack this large
computational problem
113
The Goal
• Compute Tc for different arrangements of
dopings
– Presumably, will find the prior argument of Tc
peak near 0.16 to be correct with random
dopings
• The question is, is there a non-random
doping distribution that can lead to a
higher Tc prediction?
– How much higher?
– Increasing surface area by punching holes
into metallic swath should increase Tc
114
D-Wave Pairing
Local singlet Cooper pairing within a plaquette.
The intermediate state of the plaquette is the time-reversed partner
(P↓  P’↑). Only coupling that leads to pairing is spin-exchange
coupling with x2y2/pσ electron.
Sign of the wavefunction (from Pauli principle part),
−i
−1
P
1
i
(r↑, r’↓, P↓)  (r↓, r’↓, P’↑)  (r↓, r’↑, P↓)  (+) sign i
−1
Spin-exchange matrix element part,
If r and r’ on same diagonal, then (+) sign.
If r and r’ along Cu-O bond, then (−) sign
because P and P’ are time-reversed!
One more (−) sign due to denominator in
second-order perturbation (Eground–EI).
Net (+) coupling  attractive singlet +
P’
−i
1
P↓ r’↑
r↓
−
P’↑
(−)  repulsive for singlet
P, P’ energy splitting ~ Jdd maps to
Debye energy in BCS Tc.
D-Wave
−
+ r↑
P↓
r’↓
115
Nature of Pairing
−V
−V
−V
−V
−V
−V
116
Gap Equation in BCS Superconductors
k 
k  pairs with  k 
k
Scatters into (k ,k )
Matrix element V
kk 
k 
k  

k
 swave
k
 d
k x2-y2
117
BCS Ground State Wavefunction
1
H   ( k   )c c  Vk 'k ck'ck ' ck  ck 
2 k 'k
ks

ks ks
| G  (uk  v c c


k k ' k '
)|0
u  v 1
2
k
2
k
 k  Vkk '  G | ck ck  | G 
k'
H HF
(
1
  ( k   )c c    k ck' ck '  k ck  ck 
2 k 'k
ks

ks ks
)
118
Solution (Min F=E-TS)
k   k  
k
uk vk 
2 Ek
Ek   k2  2k
1  k 

u  1 
2  Ek 
2
k
1  k
v  1 
2  Ek
2
k



 k ' 
 k  Vkk ' (uk 'vk ' ) tanh

 2kT 
k '
 k ' 
 k  Vkk '
tanh

2Ek '
 2kT 
119
Going beyond the BCS Gap equation: the
Bogolubov-De Gennes Equations
In essence, rewrite the BCS pairing
equations in Real-Space.
This was originally developed to address
the questions of non-uniform magnetic
fields in type-II superconductors and also
impurities (magnetic and non-magnetic).
Leads to Ginzburg-Landau
phenomenological theory of
superconductivity in complex magnetic
fields and disorder.
−V
−V
−V
−V
−V
−V
120
The Bogolubov-De Gennes Equations
Allows us to have a pairing in real-space,
V(r’,r) that leads to a gap that is a function
of position Δ(r’,r).
We can incorporate a spatially varying
pairing at the interface between the d9
spins and the metallic swath and determine
Tc as a function of doping by solving the
real-space gap equation.
If this leads to correct Tc(x) curve, it
shows that the strong coupling
Eliashberg formulation is unnecessary.
-V neighboring
pair attraction
121
B-dG Equations
H    c cRs   t c cR 's
Rs

R Rs

RR ' Rs
RR '
1
 VRR'cR cR ' cR ' cR
2 RR'
 RR'  (VRR' )  cRcR'   R'R
−V
−V
−V
E u  H u   v
n
n R
0
n
RR' R '
n
RR' R '
R'
−V
−V
−V
E v  H v    u
n
n R
0
n
RR' R '
n
RR' R '
R'
 RR'


1 n n
 En 
n n
 (VRR' ) uR 'vR  uR vR ' tanh

2
 2kT 
122
Computational Approaches
• 2D need 1000 x 1000 lattice
– Could start from 0.25 doping of plaquettes
where there is no surface (Δ=0) and lower
doping (Periodic Boundary Conditions)
• Only need to get down to ~ 0.16 doping
• If need 3D lattice, then could do 100 x 100
x 100, but this may be too discrete for
band structure
• Is there a Greens function approach?
123
Estimate of Maximum Tc
Chemical Physics Letters 472 (2009) 153–165
To estimate Tc, use the formula from BCS theory Tc = 1.13 ħωD exp(-1/N(0)V)
ħωD is Debye energy,
N(0) is the density of states at the Fermi level, and
V is the strength of the attractive coupling.
In CPPP, the Debye energy is replaced by the scale of the energy splitting
between opposite chirality plaquettes.
For a plaquette surrounded on all four sides by d9 spins get
~ 2Jdd = 0.26 eV ~ 3000K.
Expect range from Jdd/2 for one-side with d9 spin neighbors
to 3Jdd/2 for case with three-side interfacing d9 spin neighbors
Assume exponential term is ~ 1/10 as for A15 superconductors (Tc ~ 23K)
Expect that Maximum Tc for a cuprate superconductor is in range of 0.05Jdd to
0.15Jdd or 150K to 450K.
Current maximum of 138K may be 0.05Jdd case.
Expect that Tc of ~ 300K might be attainable..
Using 100x100 supercell, self-consistent calculations for 100 random 16% doping cases
we adjusted the d9-plaquette coupling to give gap  Tc ~138K, then we chose specific
doping patterns and calculate Tc. We have found cases with Tc > 200K. We expect to
124
predict optimum doping structure to have Tc > 200K. May be a challenge to synthesize.
The Three Assumptions of the Chiral Plaquette Polaron Model
Assumption 1: A polaron hole due to doping will be in a
chiral combination of Px’ and Py’. Each chiral polaron
has an orbital symmetry and a spin. Leads to neutron
incommensurability and Hall effect.
Assumption 2: A band is formed by Cu x2–y2/O pσ on
the polaron sites when the polaron plaquettes percolate
through the crystal. Leads to ARPES background.
d9
Assumption 3a: Interaction of the undoped AF spins
with the chiral polarons breaks the energy degeneracy
between time-reversed chiral polarons. This leads to the
D-wave superconducting pairing.
1
(Px '  iPy ' )
2
E (Px '  iPy ' ;s )

E (Px '  iPy ' ;  s ).
Assumption 3b: Since the environment of each polaron
is different, the distribution of energy splittings between
the polaron states is uniform. Yields neutron scaling
and linear resistivity.
125
Cuprate Superconductivity Puzzles
Must all be explained by any correct theory
Exp. Couples to Electron Charge
Linear Resistivity ρ ~ T
Drude scattering 1/τ ~ max(ω,T)
Excess Mid-IR absorption
Low temperature resistivity ~ log(T)
Negative Magnetoresistance low T
Superconductivity
“Semi-conducting” c-axis resistivity
Phase transition to superconductivity
Hall Effect ~ 1/T (expect ~ constant)
2
2
Dx –y Gap Symmetry
Hall Effect RH ~ const for field
Evolution of Tc with doping
in CuO2 plane.
Co-existence of magnetism and
Photoemission Pseudogap
superconductivity
Photoemission Background Large
A successful theory must explain experiments from each category.
Previous theories leave many of the very puzzling properties unexplained. The
chiral plaquette paradigm based on out-of-plane holes explains all of these
Exp. Couples to Electron Spin
Neutron spin incommensurability
Neutron spin ω/T scaling
(expect ω/Jdd or ω/EF)
Cu, O different NMR relaxations
Chiral plaquette polaron theory of cuprate superconductivity
Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007)
Explains each of these phenomena
126
Summary Ch120a
Fundamental concept of QM KE = (Ћ2/2me)<(Φ Φ>
Keeps atom from collapsing (best PE = e2/Rave
Leads to bond formation: decrease gradient for sum of two orbitals
H atom states 1s, 2s, 2p, 3s, 3p, 3d corrected for shielding leads to
Aufbau principle atoms 1s<<2s<2p<<3s<3p<3d ~ 4s<4p etc
Get covalent bond when have unpaired spins on bond atoms
Start with ground config of atoms and find maximum spin pairs
Two bonds to p orbitals  90 degree bond angle (but HH
repulsion  H2O is 104.5°
Bonds to Be,B,C columns: 2s pair  2s ±2p hybrids
Transition metals start with Ground config of atoms Pt (6s)(5d)9
Explain structures, reactions, properties organometallics, solids,
127
etc
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