Lecture 27, December 14, 2009
Nature of the Chemical Bond
with applications to catalysis, materials
science, nanotechnology, surface science,
bioinorganic chemistry, and energy
Course number: KAIST EEWS 80.502 Room E11-101
Hours: 0900-1030 Tuesday and Thursday
William A. Goddard, III, [email protected]
WCU Professor at EEWS-KAIST and
Charles and Mary Ferkel Professor of Chemistry,
Materials Science, and Applied Physics,
California Institute of Technology
Senior Assistant: Dr. Hyungjun Kim: [email protected]
Manager of Center for Materials Simulation and Design (CMSD)
Teaching Assistant: Ms. Ga In Lee: [email protected]
Special assistant:
Tod Pascal:[email protected]
EEWS-90.502-Goddard-L15
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2009 William A. Goddard III, all rights reserved
1
Schedule changes
Dec. 14, Monday, 2pm, L27, additional lecture, room 101
Dec. 15, Final exam 9am-noon, room 101
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Last time
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Hypervalent compounds
It was quite a surprize to most chemists in 1962 when Neil
bartlett reported the formation of a compound involving XeF bonds. But this was quickly folllowed by the synthesis of
XeF4 (from Xe and F2 at high temperature and XeF2 in
1962 and later XeF6.
Indeed Pauling had predicted in 1933 that XeF6 would be
stable, but noone tried to make it.
Later compounds such as ClF3 and ClF5 were synthesized
These compounds violate simple octet rules and are call
hypervalent
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Noble gas dimers
Recall from L17 that there is no
chemical bonding in He2, Ne2 etc
This is explained in VB theory as due
to repulsive Pauli repulsion from the
overlap of doubly occupied orbitals
It is explained in MO theory as due
to filled bonding and antibonding
orbitals
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(sg)2(su)2
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Noble gas dimer positive ions
On the other hand the positive ions
are strongly bound (L17)
This is explained in MO theory
as due to one less antibonding
electron than bonding, leading to
a three electron bond for He2+ of
2.5 eV, the same strength as the
one electron bond of H2+
(sg)2(su)1
The VB explanation is a little
Using (sg) = L+R and (su)=L-R
less straightforward. Here
we consider that there are
Leads to (with negative sign
two equivalent VB
structures neither of which
leads to much bonding, but
superimposing them leads
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to
resonance stabilization
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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
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
We saw that the energy to form Xe+F-,
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 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
We will denote this 3 center – 4 electron charge transfer bond as
FXeF
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Stability of XeF2
Ignoring resonance we predict that XeF2 is stable by 1.43 eV. In
fact the experimental bond energy is 2.69 eV suggesting that the
resonance energy is ~ 1.3 eV.
The XeF2 molecule is stable by 2.7 eV with respect to Xe + F2
But to assess where someone 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 it 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, leading to
a square planar structure
The VB analysis would indicate 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.
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.
Here we see that
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
12
has been
observed
© copyright 2009 William A. Goddard
III, all rights
reserved
Halogen Fluorides, ClFn
The IP of ClF is 12.66 eV which
compares well 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.
Indeed these estimates suggest
that ClF3 and ClF5 are stable.
Indeed the experiment energy
for ClF3  ClF +F2 is 2.6 eV,
quite similar to XeF2.
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PF5
Think of this as planar PF3+, which
has a pz pair making a
hypervalent bond to one F (+ z
direction) with F- in the –z
direction + the resonance with the
other state
Adding F-, leafs to PF6-, analogous to SF6,
which is stabilize with appropriate cation,
e.g. Li+
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Donor-acceptor bonds to O atom
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Ozone
We saw earlier that bonding O to O2
removes most of the resonane of the
O2 ,leading to the VB configuration
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Diazomethane
Leading to
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Origin of reactivity in the hypervalent
reagent o-iodoxybenzoic acid (IBX)
Hypervalent O-I-O
linear bond
Julius Su and William A. Goddard III
Materials and Process Simulation Center,
California Institute of Technology
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22
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
HgCl2
I
OTs
O
I
Pd(OAc)2
Can we understand this remarkable chemistry of iodine
Martin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514
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New
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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, Phys Rev Lett,
submitted
Jamil
25
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)
26
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?
27
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.
28
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
29
(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
30
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
31
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
32
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
33
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
34
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 CuO35
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)
36
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
37
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
38
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
39
Superexchange coupling of two Cu d9 sites
exactly the same
as the hypervalent XeF2
9
Two Cu d 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
40
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–
41
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
42
system becomes a normal metal.
Summary: Central Characteristics of cuprate
superconductors, square CuO2 lattice, 16%
holes
CuO2 plane
La CuO (Undoped): La3+, Sr2+, O2–, Cu2+
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
Cu
O
Cu
O
pπ
O
Cu
O
Cu
Cu
4
2+
Cu 
spin, with antiferromagnetic coupling
O
O
O
Cu
2
d9
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)
43
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
44
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.
45
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
46
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.
47
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).
48
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?
49
Basis for all theories of cuprate
superconductors
LDA Band calculations of La
2CuO4
(0,p)
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
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
50
G2(MAD)
H-Ne
TM
He2
(H O)
We want to doNe QM on
cuprate
D (R )
E
E(R )
E(R )
superconductors.
Which
1.09
Unbound
Unbound
0.161(3.048)
DFT
Is most
accurate?
0.19 functional
0.0011(2.993) 0.0043(3.125)
0.218(2.912)
2
2
2
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
3.94a
0.665
0.749
0.27
6.67
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
xc
e
e
e
O..O
b
c
c
d
0.54e
0.0109(2.377)
0.0231(2.595)
0.391(2.710)
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
51
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)
52
(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).
53
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). 54
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)
55
Now use B3LYP for
La2–xSrxCuO4
B3LYP leads to a hole along
the Sr-O-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
56
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.
57
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.
58
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).
59
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
60
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
61
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.
62
An unusual degeneracy: complex linear combinations
(e+=Px + i Py and e-=Px - i Py)
It is natural to assume the hole wavefunction
is Px or Py or some real linear combination of
the two.
The interaction of the polaron with the
background d9 spins chooses the complex
combinations.
This implies each polaron has a clockwise or
counter-clockwire current.
This result is surprising because we usually
think of spins as Ising spins rather as 3D
quantum spins.
-i
-1
Px+iPy
1
i
i
-1
3D Quantum character of spins leads
to Chiral Polarons.
Px-iPy
1
-i
63
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
64
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.
65
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
66
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
67
Superconducting Pairing only on Surface
Plaquettes
68
Assume Optimal Tc  Maximum Surface
Polarons per Volume experiment
≈ 0.16
1000 x 1000 lattice
Sp
200 ensembles
prediction
Percolation threshold
Becomes metallic
Surface area of
pairing leads to
correct optimal
doping
69
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
70
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
71
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’↓
72
Nature of Pairing
−V
−V
−V
−V
−V
−V
73
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
74
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
uk2  vk2  1
 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
)
75
Solution (Min F=E-TS)
k   k  
k
u k vk 
2 Ek
1  k
u  1 
2  Ek
2
k
Ek   k2  2k



1  k
v  1 
2  Ek
2
k



 k ' 
 k  Vkk ' (uk 'vk ' ) tanh 

 2kT 
k '
 k ' 
 k  Vkk '
tanh 

2 Ek '
 2kT 
76
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
77
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
78
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' ) u R 'vR  u R vR ' tanh 

2
 2kT 
79
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?
80
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
81
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.
82
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
83
Extra slides
84
What Characterizes a Superconductor? (The GAP)
Metals are characterized by a Fermi surface and the
existence of excited states of infinitesimal energy
(LUMO-HOMO energy separation is zero).
If Vk’k = –V < 0 (attraction), then can make
“bonding” linear combinations of Cooper pairs in
the ground state, Ψ ~ (k, –k) + (k’, –k’) + …
that lowers the energy and forms a ground state
that separates from the excited states.
Leads to a gap Δ in the excitation spectrum.
Coulomb repulsion is isotropic with VC > 0. Lower
energy state cannot be formed.
k  pairs with  k 
k 
k
k 
k 
Scatters into (k  ,k  )
Matrix element V
Bardeen, Cooper, Schrieffer (BCS) showed in 1957
kk 
that coupling through phonons can lead to a net
attraction  Electron distorts nuclei and moves
away  opposite momentum electron is
attracted to distortion.
Cooper Pairing
Superconducting Gap, Δk
 s wave
k
85
The Superconducting Gap Symmetry
If Vk’k = –V < 0 (attraction), then can make
“bonding” linear combinations of Cooper pairs in
the ground state, Ψ ~ (k, –k) + (k’, –k’) + …
k  pairs with  k 
k 
k
The symmetry of the gap, Δk, is determined by the
phase of the coefficients in Ψ. Δk = const =S-wave.
A strong anisotropic repulsion for momentum
change (π, π) can lead to a D-wave gap of x2y2
(p,p)
symmetry.
k’
+V
k 
k 
Scatters into (k  ,k  )
Matrix element V
kk 
Cooper Pairing
Superconducting Gap, Δk
k
(0,0)
 s wave
k
 d
k x2- y2
Ψ ~ (k, –k) – (k’, –k’) D-Wave “anti-bonding” 86
Matrix Element for On-Site Coulomb Scattering
|(π,0), (Px - iPy)>  |(0,π), (Px + iPy)>
Px + iPy
Px - iPy
-1
+i
-1
-1
+i
1
-1
(π,0)
(0,π)
1
1
-1
1
1
-i
1
-1
-i
M ~ { (+1)(+1)* (+1) (+1) + (+1)(-i)* (-1) (-i) + (-1)(-1)* (-1) (-1) +
(-1)(+i)* (+1) (+i) } = 4 (Max Value)
Maximum repulsion occurs for (π,0)(0,π) for spin-exchange
pairing through time-reversed polaron. D-WAVE
[(x+iy) a, (x-iy) b  [(x-iy) b, (x+iy) a]
87
Cuprate Superconductivity Puzzles
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.
Current theories leave many of the very puzzling properties unexplained.
The chiral plaquette theory 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
88
Experimental Neutron
Data
x = 0.14
Raw Data
Corrected Data
For antiferromagnet expect peak in
neutron scattering at (,p)
Instead the peak is
shifted by ±(2p/a)d
and d increases with
doping x
(0,p)
(p,p)
shift δ from (,p)
Doping
G(0,0)
(p,0)
89
Calculated spins (x = 0.10, JCH = 3Jdd)
Red  AF d9 regions, Green  plaquette polarons
90
Computed Neutron Structure Factor
x = 0.10
256  256 lattice. 5,000 ensembles.
Pure AF would be
peaked at (π,π)
1
Normalizat ion,  S ( k )  1.
N k
91
Computed Neutron Structure Factor
256  256 lattice. 5,000 ensembles.
Incommensurability
ring around (π, π) of
magnitude
2p 

  x
 a 
(0,p)
G(0,0)
x = 0.075
(p,p)
(p,0)
x = 0.10
Correlation length
approximately
a

.
x
x = 0.125
92
The Hall Effect in Metals and Semi-conductors
Lorentz Force
dp
1

 e E  v  B .
dt
c


Magnetic field
deflects charge
vT
v
Drude picture
Band picture
mv  eE ,
J  nev  sE,
s
ne 
,
m
2
v
e 2 2
mv   e B, v  
EB,
c
mc
eE
e
v     , E     BE ,
m
 mc 
E
1
1
RH      
.
 J  B  nec
dk
1

  e E  v  B .
dt
c


Fermi
Surface
In metals, n = constant  RH = constant.
In semi-conductors, n ~ T3/2 e–Δ/T.
93
Temperature Dependent Hall Effect in Cuprates
La2–xSrxCuO4
x=0.15 get strong
T dependence to
RH.
n=0.63 (per cell)
n=0.16 (per cell)
n=0.31 (per cell)
x>0.34 get RH = constant.
94
Skew Scattering from Chiral Plaquette Polarons
x2-y2 band scattering from a chiral polaron leads to a leftright scattering asymmetry that is an additive contribution
to the Hall effect.
Current flow
When there is no magnetic field, B = 0, the number of
“plus” and “minus” chiral polarons is equal and there is no
net skew scattering.
For non-zero B, the contribution to the Hall effect is
proportional to the difference between the number of
“plus” and “minus” chiral polarons and this is ~ 1/T.
Px – i Py
With chiral plaquette polarons, skew scattering definitely
exists.
Question #1: What is the temperature dependence of
the skew scattering from chiral polarons?
Question #2: What is its magnitude?
Px + i Py
95
Calculating the Skew Scattering
The chiral polaron difference is
proportional to the magnetic
field, B.
Δ
Δmax is the maximum timereversed polaron energy
separation and Δmin is the
minimum separation.
Number of polarons
Δmin and Δmax are larger for
lower dopings where there are
more undoped d9 spins to cause
splitting.
Δ+2gpμBB
Difference between ± polarons
1
f ( )  b
e 1
96
Comparison of Theory to
Experiment
Fitting experimental data to the form,
RH = R0[f(Δmin) – f(Δmax)]
1
f ( )  b
e 1
x
Δmin
Δmax
R0
0.05
12.6K 791.5K
15.4
0.10
16.7K 454.1K
8.82
0.15
0.0K
378.9K
3.81
0.25
11.6K 106.3K
1.05
Answer #1: Skew scattering from
chiral polarons leads to the observed
temperature dependence.
La2–xSrxCuO4
97
The Magnitude of Skew Scattering
Δmin = 16.7K and
Δmax = 454.1K from
experimental fit.
1

x  0.10, U   ( 1 eV )  2.5 eV ,
 4x 
Smax  1.0 eV, Dmax  1.5 eV,
U is enhanced because fewer
electrons to screen the repulsion.
Computed value
g p  1, m  1, n  4 x,
RH c  5.39  103 cm 3 / C ,
Experiment  4  103 cm 3 / C.
Answer #2: The magnitude is right too.
98
Hall Effect for Magnetic Field in the CuO2 Plane
Since the chiral current is in the plane, a magnetic field in
the plane will not split the “clockwise” and “anticlockwise” plaquette energies.
Current flow
Leads to no temperature dependence of Hall effect for
in-plane magnetic fields.
Experimentally observed.
Px – i Py
Px + i Py
99
Conclusions
PRB vol 76, 014514 (2007)
Experiments
Exp. Couples to Electron Spin
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
Hall Effect ~ 1/T (expect ~ constant)
Phase transition to superconductivity Hall Effect RH ~ const for field
Dx2–y2 Gap Symmetry
in CuO2 plane.
Evolution of Tc with doping
Photoemission Pseudogap
Co-existence of magnetism and
Photoemission Background Large
superconductivity
Neutron spin incommensurability
Neutron spin ω/T scaling
(expect ω/Jdd or ω/EF)
Cu, O different 0NMR relaxations
A successful theory must explain experiments from each category. Current
theories do not, leaving many of the very puzzling properties unexplained.
100
The Chiral Polaron Magnetic Susceptibility
By Fermi’s Golden Rule, the absorption rate is,
occupation
The power absorbed is,
<P(ω)> =   f ( )  f ( )W ( )
Bottom Line: Uniform
ρ = Scaling
occupation
1
f ( )  b
.
e 1
Probability distribution
of polaron energy splittings.
101
Adding in the d9 and x2–y2 terms to obtain the total Magnetic
Susceptibility
χ(q, ω) ~ ω/EF for a metal and can be neglected.
χ"(q, ω) ~ ω/Jdd for AF spins. Thus, the imaginary part of the d9 susceptibility
can be neglected. The real part is constant for ω less than Jdd and may be
taken to be constant for neutron scattering.
All of the ω dependence of the total susceptibility comes from the polaron
term, χp"(ω).
The q dependence of the susceptibility is incommensurate from our
calculations and of the form,
a

.
x
102
The Total Magnetic Susceptibility in the Random Phase
Approximation (RPA)
Summing the diagrams,
and solving,
Scaling term
103
The Total Magnetic Susceptibility in the Random Phase
Approximation (RPA)
Using the integrals,
Satisfies ω/T scaling and has the
correct functional form.
Experiment LSCO x = 0.04
a1 = 0.43
a3 = 10.5
104
Solving the Heisenberg Spin Hamiltonian in terms of two sets
of spins A and B (non-linear sigma model)
Map AF system onto Ferromagnetic system
A
B
A
B
B
A
B
A
A
B
A
B
B
A
B
A
H  J  SiA  S Bj   J  SiA  (  S Bj )
ij 
ij 
Staggered magnetization W(r)
Effective Continuum Hamiltonian
This converts the problem of solving for the ground state into a
problem of minimizing a continuous field with constraints
H eff   d 2 r (W x )  (W y )  (W z ) ,
2
Minimize
Constraint
2
2
W 2x  W 2y  W 2z  1,
W( r )  (0,0,1), r  .
105
Boundary Condition
Topological Invariants
H eff   d 2 r (W x )  (W y )  (W z ) ,
2
Minimize
2
2
W 2x  W 2y  W 2z  1,
Constraint
W( r )  (0,0,1), r  .
Boundary Condition
Q
Topological Invariant,
1
2
d
r W  ( x W   y W),

4p
States partition into discrete sets each characterized by an Integer Q described as
the number of times the spin field covers sphere. This is a Winding number
Fields with different Q cannot be continuously deformed into one another
1
dz
n
.

2pi ( z  a )
2
2
2
Px+iPy has Q=+1 and cannot be deformed to
another Q. Px is a mixture of Q’s and can be
deformed to Q=0.
a
n=0
H eff   d 2 r (W x )  (W y )  (W z )  8p | Q | .
n=1
106
Heuristic Derivation of Incommensurability
Magnitude dQ=(2p/a)x
Our calculations find that the minimum spin
configuration consists of undoped patches of d9
spins aligned antiferromagnetically, with the
polarons acting to rotate the direction of the AF
alignment of adjacent patches.
Consider area A. The number of polarons in this
area is Np=Ax/a2. The polaron rotates each
adjacent spin on opposite sides of polaron by π
or 2π total. If this is rigidly transmitted to patch,
then rotation per spin is dq  2p/(1/x)= 2px.
Net rotation per spin is Np dq. If area chosen
such that net rotation per spin is 2π, then
A=(a/x)2.
Chiral
coupling
dominant
A translation of L with L2=A returns to identical
patch. L = a/x and dQ=(2p/a)x.
107
Coulomb Scattering of x2–y2 Band Electrons
with Polarons
Absorb pσ orbitals into “effective” A1 orbitals on Cu sites for simplicity.
x2y2 band k state spin σ
Polaron state spin s
On-site with no spin exchange
On-site with spin exchange
(U > K > 0)
Coefficients αR determine polaron state
Fundamental Reason for Superconductivity is that
Polaron is spread out over more than one site.
108
Matrix Element for Coulomb Scattering
|k, S>  |k’, Px + iPy>
e i(kx+ky)a/2
+1
e -i(kx-ky)a/2
+1
e -i(kx’-ky’)a/2
+i
S
e i(kx’+ky’)a/2
+1
Px+iPy
+1
+1
e -i(kx+ky)a/2
e i(kx-ky)a/2
-1
e -i(kx’+ky’)a/2
-i
e i(kx’-ky’)a/2
M(q) ~ [e i(qx+qy)a/2 - e -i(qx+qy)a/2] - i [e i(qx-qy)a/2 - e -i(qx-qy)a/2]
~ 2i [ sin(qx+qy)a/2 – i sin(qx-qy)a/2 ]
q = k’ - k
S or D 
Max for q = (π, 0), (0, π)
109
Matrix Element for Coulomb Scattering
|k, Px - iPy>  |k’, Px + iPy>
e i(kx+ky)a/2
+1
e -i(kx-ky)a/2
-i
e -i(kx’-ky’)a/2
+i
Px - iPy
e i(kx’+ky’)a/2
+1
Px+iPy
-1
+i
e -i(kx+ky)a/2
e i(kx-ky)a/2
-1
e -i(kx’+ky’)a/2
-i
e i(kx’-ky’)a/2
M(q) ~ [e i(qx+qy)a/2 + e -i(qx+qy)a/2] - [e i(qx-qy)a/2 + e -i(qx-qy)a/2]
~ 2[ cos(qx+qy)a/2 – cos(qx-qy)a/2 ]
q = k’ - k
Px – i Py 
Px + i Py
Max for q = (π, π)
110
Coulomb Cooper Pairing Diagrams
k’
k
–k’
–k
Intermediate Polaron change with no spin exchange
Direct Coulomb
band repulsion
Matrix element +V
Repulsion of pairs
111
Intermediate Polaron change with spin exchange
Matrix Elements for no spin exchange:
Attraction
Attraction for states
near Fermi level
q = k’ – k
M(q) depends on polaron states I and I’
and is the key to D-wave pairing.
112
Matrix Elements for spin exchange: Repulsion
Minus sign due to
interchange of (k’, –k’)
in final state
q = k’ + k
Repulsion for states
near Fermi level
Δ = EI’ – EI
M(q) depends on
polaron states I and I’
113
Anisotropic Pairing term |M(q)|2
Initial state, I = S, D, or Px’ ± i Py’ (initially occupied)
Intermediate state, I’ = Px’ ± i Py’ a chiral polaron (unoccupied)
I = S or D
I = Px’  i Py’
Time-Reversed intermediate polaron
I = I’ = Px’ ± i Py’
Only for spin exchange
114
Superconducting Pairing Possibilities
Spin exch
only
(p,p)
No Spin Exchange: q = k’ – k, Attraction plus isotropic
Repulsion +V. For S, D, and ± combinations, leads to
net repulsion for q  0. No superconductivity.
Spin Exchange: q = k’ + k, Repulsion plus isotropic
Repulsion +V. For S, D net repulsion for q  (π,0). No
good Fermi surface nesting. For ++, repulsion for q  0
or k’  –k. P-wave triplet pairing not possible with one
continuous Fermi surface.
Time-Reversed intermediate polaron, ±, extra repulsion
for k’+k  (±π, ± π) is compatible with D-wave.
k’
k
Γ
k’+k  (±π, ± π)
115
Debye Energy in BCS  the Maximum Energy Splitting of
Time-Reversed Polarons
The pairing occurs for electrons within the Maximum Energy Splitting between
Time-Reversed Polarons.
This splitting arises due to coupling with the undoped d9 spin background and is
on the order of Jdd. The time-reversed energy splitting is equivalent to the Debye
energy in BCS superconductors.
Δ = EI’ – EI
116
Evolution of Tc with Doping
With Increasing Doping:
The number of chiral polarons increases 
increase of Tc.
The number of undoped d9 spins decreases 
decrease of the time-reversed polaron splitting 
decrease of Tc.
The number of x2y2 band electrons near the
Fermi level (Density of States) that can pair
increases  increase of Tc.
Typical phase
diagram
La2-xSrxCuO4
Increase x2y2 band electrons increases the
screening of the Coulomb repulsion  decrease
of Tc.
Chiral Plaquette Polarons crowd together
forming 1-Cu “Apical Polarons” with no
superconducting pairing  decrease of Tc.
117
Resistivity Temperature Dependence
Polaron energy
splitting  Δmax
If ρ(Δ) ~Δμ, then resistivity ~Tμ+1
118
Optical Absorption
2

 p  1/
Res ( )    2
2
4
p


1
/

 
 2 /
 2
2 2
(  0 )   2 /  2
Scattering rate
Simple Drude
Added oscillators
1/τ(ω) ~ T, for ω << T,
1/τ(ω) ~ ω, for ω >> T.
In addition, there should be absorption
due to polarons absorbing the photon
and filling the unoccupied chiral state.
Mid-IR.
one-component fit
1 /  ( )
119
A Simple Model for Skew Scattering
Inversion and Time-Reversal symmetry
guarantees no skew scattering.
w(kk’) = w(–k–k’) from inversion and
w(–k–k’) = w(k’k) from time-reversal.
An applied magnetic field breaks time-reversal.
Consider x2y2 band scattering with a
complex matrix element Vk’k.
Scattering Hamiltonian
Scattering T-Matrix
Fermi’s Golden Rule
Third order term changes sign when
k and k’ are interchanged.
120
The Magnitude of Skew Scattering
Δmin = 16.7K and
Δmax = 454.1K from
experimental fit.
1

x  0.10, U   ( 1 eV )  2.5 eV ,
 4x 
Smax  1.0 eV, Dmax  1.5 eV,
U is enhanced because fewer
electrons to screen the repulsion.
Computed value
g p  1, m  1, n  4 x,
RH c  5.39  103 cm 3 / C ,
Experiment  4  103 cm 3 / C.
Question #2: Does it have the right magnitude? YES
121
Calculating the Skew Scattering
Scattering with Chiral Polaron Hole in P±
Expanding to second order
First and second order terms
Mod-squared for scattering rate
122
Calculating the Skew Scattering
For hole-doped cuprates, the
polarons are holes and
U < 0.
For electron-doped cuprates,
the polarons are electrons
and U > 0.
Invariant under interchange
k’ k. Does not
contribute to skew scattering
term.
q = k’ – k
123
Calculating the Skew Scattering
Extracting the skew
scattering contributions,
Ak,k’ = –Ak’,k
E± << ES, ED because E± ~ Jdd
(time-reverse polaron energy
difference) and ES, ED ~
hopping energies ~ 0.5-1 eV.
Allows simplification of
mean skew scattering.
q = k’ – k
124
Calculating the Skew Scattering
Mean skew scattering
ρS = 1/ΔSmax = constant.
ρD = 1/ΔDmax = constant.
Must take mean
over chiral polaron
energy splittings
λ = 0.19 for LSCO at x = 0.10
1    1
125
Temperature Dependent Hall Effect
(p,p)
Un-Occupied
Occupied
G
Band Structures
have hole-like
Fermi surfaces
leading to RH > 0
Fermi Surface
126
Log(T) Low-Temperature Resistivity
La2–xSrxCuO4 x = 0.08 and 0.13
Tesla
Negative Magnetoresistance
60 T magnetic field to
suppress superconductivity
“semi-conducting” ρc
No log(T)
log(T)
127
Kondo Effect
Spin impurity in metal
k
p
k
p
q
q
S+
S–
S–
S+
B = 0.671 Kelvin/Tesla
log(T) resistivity
Negative magnetoresistance
Suppressed for T < 80K in 60T field
Chirality + spin = effective “spin” impurity. If splitting
in 60T small (small g-factor), then resistivity obtained
60 T
80 K
g=2
No Kondo effect for
“ferromagnetic” Coulomb
coupling with polarons.
128
Kondo spin-flip from x2y2 band hopping on
and off polaron
Matrix element for hopping of k-state
x2y2 onto A1 ~ cos(kx) – cos(ky).
Second order band coupling to polaron
spin is antiferromagnetic with maximum
coupling for k  (π, 0) and minimum for k
along the diagonals.
Resistivity out-of-plane is dominated by kvectors near (π, 0) and planar resistivity is
dominated by k-vectors along the
diagonal.
Therefore, the Kondo resistivity will
appear at a higher temperature for out-ofplane than in-plane leading to the “semiconducting” ρc.
A1 orbitals
No Kondo effect for
“ferromagnetic” Coulomb
coupling with polarons.
129
2D Plaquette Percolation and Polaron Islands
Loss of Kondo effect (Insulator to Metal transition)
130
Angle-Resolved Photoemission (ARPES)
Spectral Functions
for k passing through
Fermi level
(k||, kz’)
hn
Occupied
Surface
Empty
ARPES cuts off
spectral function
with Fermi factor
(k||, kz)
Parallel Momentum
Conserved
Fermi Level
ARPES on cuprates finds a Fermi surface
with a broad background even for k states
far above the Fermi energy.
131
ARPES Pseudogap Underdoped Cuprates
Pt reference spectra
Gap
ARPES taken
at this k-vector
Pseudogap for
underdoped
Gap closed
83K underdoped Bi2Sr2CaCu2O8+δ
Pseudogap exists above Tc and closes as the
temperature increases. Pseudogap is D-wave.
132
Pseudogap from undoped d9 AF coupling
Antiferromagnetic fluctuations with momentum q
of the undoped d9 spins induce mixing between k
and k ± q.
This leads to a decrease in the density of states for
k-vectors at the Fermi level with either k+q or k–q
near the Fermi level (nesting). This occurs for k
near (π,0).
q  (π,π)
(p,p)
k+q
Matrix
Element
V
k
(0,0)
As doping increases, the number of d9 spins
decreases, thereby decreasing the strength of the
mixing V. This leads to a pseudogap that can only
exist for underdoped cuprates.
At fixed doping, AF fluctuations of the d9 spins
decrease with increasing temperature, or V
decreases to 0. The pseudogap closes.
With percolating metallic paths, k
is not a good quantum number
leading to a large background. 133
Polarization Matrix Elements (Electric Dipole)
kx = ky
y
kx = ky
x
kx = 0
Not Allowed
Not Allowed
E
E
E
Allowed
z2
kx = ky
Not Allowed
kx = ky
Allowed
E
E
Allowed
kx = 0
E
z2
134
Photoemission Process (Energy)
135
Computed Percolation Values
Experiment: LSCO = 0.05 YBCO = 0.35
136
(LaSr)CuO4 Doping Constraints
No Sr above and
below plaquette
No Sr separted by two lattice
spacings on opposite sides of
plane
No neighboring
Sr in plane
No out-of-plane
Sr neighbors
No Sr separted by
two lattice spacings
137
YBa2Cu3O7–d Doping Constraints
Chain
O
always
dopes
same
Cu
triple
Experiment:
YBCO at x=0.35
Tc drops from ~92 to ~65K
Chain O
dopes Cu
triple
randomly
, but not
twice
Adjacent
chain O
doping.
Adjacent
Cu triples
not
allowed
138
x2–y2/pσ Band Energy due to
Incommensurability
d9 spin ordering of momentum q hybridizes
band electrons of momentum k and k+q with
coupling energy V on the order of Jdd~0.1eV.
This mixing perturbs the band energies and
the total ground state energy. We calculate
the energy change for the ring of
incommensurate q vectors to find minimum.
x = 0.10, ΓS = ΓD = 0.01 eV.
eV
q vector for Cu – O bond directions on BZ edge. Additional
Umklapp scattering makes this direction favorable.
139
The Chiral Polaron Magnetic Susceptibility
The low-energy excitation of a chiral plaquette
polaron is the time-reversed polaron. For the
q dependence of χp"(q, ω), the matrix element
between the initial and final polaron state of
the q momentum spin operator, Sz(–q)Sz(q),
must be evaluated.
This is largest for q = (π,π).
Since the polaron is localized over a four-site
plaquette, the q = (π,π) peak is substantially
broadened. In the vicinity of (π,π), where the
neutron spin scattering is largest, we may
assume the polaron susceptibility, χp"(q, ω), is
momentum independent, χp"(q, ω)  χp"(ω).
occupation
occupation
1
f ( )  b
.
e 1
 Px  iPy ;s | S (  q) S ( q) | Px  iPy ;s   1
z
z
q  (p , p ).
140
Pseudogap from undoped d9 AF coupling
Antiferromagnetic fluctuations with momentum q
of the undoped d9 spins induce mixing between k
and k ± q. The incommensurate q along the Cu–
O bond direction has the lowest band energy and
explained the observed neutron spin peaks.
The same mixing leads to a decrease in the
density of states for k-vectors at the Fermi level
with either k+q or k–q near the Fermi level
(nesting). This occurs for k near (π,0).
As doping increases, the number of d9 spins
decreases, thereby decreasing the strength of the
mixing V. This leads to a pseudogap that can only
exist for underdoped cuprates.
At fixed doping, AF fluctuations of the d9 spins
decrease with increasing temperature, or V
decreases to 0. The pseudogap closes.
With percolating metallic paths, k
is not a good quantum number
leading to a large background. 141
Thermopower
142
Density of States for Explicitly Doped
La2–xSrxCuO4 using UB3LYP
Undoped
x=0.0
x=0.25
Band gap
The down-spin states show a clear
localized hole with Opz-Cudz2 character
Highest hole states
Lowest electron states
x=0.5
Band
gap
x=0.125
Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002)
143
2D Plaquette Percolation and Polaron Islands
Loss of Kondo effect (Insulator to Metal transition)
144
NMR Knight Shift Temperature Dependence: One-Component
vs. Two-Component Theories
For underdoped YBa2Cu3O6.63, the
planar Cu and O Knight shifts have
the same temperature dependence.
Optimally doped YBa2Cu3O7 has
constant Knight shifts.
This leads to the conclusion that
there is only one electronic
component that hyperfine couples
to the planar Cu and O atoms
leading to the same temperature
dependence. Since our model has
x2y2 band electrons and chiral outof-plane polarons (twocomponents), the data appear to
contradict our assumptions.
Currently, no theory explains this
temperature dependence.
Takigawa et al, PRB 43, 247 (1991)
Question 1: Does the data unequivocally
imply one-component? NO.
Question 2: Does the NMR kill twocomponent theories? NO.
145
Question 1: Does the data unequivocally
imply one-component? NO.
More recent data on optimally doped
YBa2Cu3O7 finds the Y spin relaxation
rate and Knight shifts have different
temperature dependencies contradicting
one-component models.
Also, the ratio of the Cu-O to Cu-Cu
nuclear coupling is too large to fit onecomponent models, requiring some
antiferromagnetic correlations larger than
the Cu-Cu distance.
Nandor et al PRB 60, 6907 (1999)
Pennington et al PRB 63,054513 (2001)
146
Question 2: Does the NMR kill two-component
theories? NO.
Chiral plaquette polarons lead to a
susceptibility as seen in the Hall
effect. This is compatible with high
temperature Y shift. The x2y2 band
electrons have a temperature
independent Knight shift. Since the
polarons are out-of-plane, their
hyperfine coupling to the planar Cu
and O are smaller than the x2y2 band.
Nandor et al PRB 60, 6907 (1999)
We expect a temperature dependence
of the Knight shift in underdoped
YBa2Cu3O6.63 due to the pseudogap in
x2y2 band from coupling to our
undoped d9 spins.
Our NMR explanation is still
incomplete at present.
Computed Hall Effect
147
NMR Knight Shift Temperature Dependence: One-Component
vs. Two-Component Theories
For underdoped YBa2Cu3O6.63, the
planar Cu and O Knight shifts have
the same temperature dependence.
Optimally doped YBa2Cu3O7 has
constant Knight shifts.
This leads to the conclusion that
there is only one electronic
component that hyperfine couples
to the planar Cu and O atoms
leading to the same temperature
dependence. Since our model has
x2y2 band electrons and chiral outof-plane polarons (twocomponents), the data appear to
contradict our assumptions.
Currently, no theory explains this
temperature dependence.
Takigawa et al, PRB 43, 247 (1991)
Question 1: Does the data unequivocally
imply one-component? NO.
Question 2: Does the NMR kill twocomponent theories? NO.
148
Question 1: Does the data unequivocally
imply one-component? NO.
More recent data on optimally doped
YBa2Cu3O7 finds the Y spin relaxation
rate and Knight shifts have different
temperature dependencies contradicting
one-component models.
Also, the ratio of the Cu-O to Cu-Cu
nuclear coupling is too large to fit onecomponent models requiring some
antiferromagnetic correlations larger than
the Cu-Cu distance.
Nandor et al PRB 60, 6907 (1999)
Pennington et al PRB 63,054513 (2001)
149
Question 2: Does the NMR kill two-component
theories? NO.
Chiral plaquette polarons lead to a
susceptibility as seen in the Hall
effect. This is compatible with high
temperature Y shift. The x2y2 band
electrons have a temperature
independent Knight shift. Since the
polarons are out-of-plane, their
hyperfine coupling to the planar Cu
and O are smaller than the x2y2 band.
Nandor et al PRB 60, 6907 (1999)
The temperature dependence of the
Knight shift in underdoped
YBa2Cu3O6.63 is due to pseudogap in
x2y2 band from coupling to our
undoped d9 spins.
Our NMR explanation is still
incomplete at present.
Computed Hall Effect
150
More Physical Percolation near 0.25
If we assume plaquettes are laid down
uniformly, then leads to 0.25.
If we assume more realistic scenario that
plaquette are not as uniform, then will
exhaust constraints before 0.25.
Exhausted at 0.187.
Then fill 3-site sites to 0.226.
Then fill 2-site sites to 0.271!
This is where any AF coupling disappears.
1-site exhausted at 0.316.
151
What is “Bold” about this Talk? And why is it a “Minefield”
BOLD:
We propose that out-of-plane (CuO2) hole orbitals are formed in the
superconducting cuprates.
We demonstrate this fact using ab-initio DFT calculations using B3LYP.
We show that out-of-planes orbitals play a major role in the
superconducting mechanism of these materials and can explain a large
number of the “anomalous” properties.
MINEFIELD:
The field has assumed for the past 20 years that only in-plane orbitals are
relevant. All theories are built on this assumption.
Are chemist functionals are superior to physicist functionals for
characterizing the electronic structure of these complex materials?
152
Spin-1/2 states along different axes
z
| a (q ,  )  cos
q
2
i
| a  e sin
q
|b 
Θ
2
q
q
i
| b (q ,  )  e sin | a   cos | b 
2
2
p
φ
y
x
1
(a  b )
α-spin along x-axis = | a (q  ,   0) 
2
2
p
p
1
(a  ib )
α-spin along y-axis =| a (q  ,   ) 
2
2
2
This extra factor of i is due to quantum nature of
spin and is reason for chirality.
153
Simplifying Plaquette Polaron orbitals
t’
t
154
Coupling of plaquette spin to neighboring d9
antiferromagnetic lattice
+
+1
pointing out-of-page
z-axis
pointing in xy-plane
of page
+
+1
+1
z2/x2y2 triplet
+
+1
H
+
+1
+1
+
+
+1
+
H
+1
+
+1
+
155
Coupling of plaquette spin to neighboring d9
antiferromagnetic lattice
+
+1
+
+1
H
H
+
=t
+1
+1
=t
+
| a (q ,  )  cos
q
2
+
+1
+
+1 +
+1
i
| a  e sin
p
+
+1
q
2
~(1+i )t
~(1-i )t
|b 
1
(a  b )
α-spin along x-axis =| a (q  ,   0) 
2
2
p
p
1
(a  ib )
α-spin along y-axis =| a (q  ,   ) 
2
2
2
Px + i Py will lower
energy.
Jahn-Teller
Distortion!
156
Net Effect of Chiral Polarons
+
+1
pointing out-of-page
z-axis
pointing in xy-plane
of page
+
+
+1
+1
z2/x2y2 triplet
Px + i Py plaquette polaron
moves d9 spins off z-axis
and into xy-plane. Creates
a right-handed coordinate
system.
+1
+
+1
+
157
Chiral Polarons are distributed randomly throughout the
crystal
Δ = Energy difference between a polaron and
its time-reversed partner.
The random environment of the polarons
leads to random energy   the probability
distribution of the energy splitting is
uniform.
Leads to an observable effect with neutron
scattering.
occupation
Δ
occupation
1
f ( )  b
.
e 1
Neutron inelastically scatters losing energy ħω
Probability distribution
of polaron energy splittings.
If ħω <kT, then no absorption of energy since
“up” and “down” polarons have equal
occupations. Absorption when ħω>kT.
Neutron energy absorption
should scale as ω/T!
158
Neutron Spin ω/T Scaling
Neutron spin scattering with
momentum and energy
change q and ω measures the
imaginary part of the
magnetic susceptibility,
χ"(q, ω) (Absorption).
χ"(q, ω) ~ ω/Jdd for
antiferromagnetic spins.
χ"(q, ω) ~ ω/EF for a metal.
∫d2q χ"(q, ω) ~ ω/T is
observed.
This is the local
susceptibility.
a1 = 0.43
a3 = 10.5
159
Chiral Polarons take AF order 180o out-of-phase
A large JCH determines the local spin
structure around the polaron.
Further changes in the parameters
does not affect the spin structure.
Should lead to an observable effect in
neutron spin scattering structure factor
160
Coulomb Cooper Pairing Diagrams
Scattering through time-reversed
polaron intermediate state
161
Matrix Element for Coulomb Scattering
|k, Px - iPy>  |k’, Px + iPy>
e i(kx+ky)a/2
+1
e -i(kx-ky)a/2
-i
e -i(kx’-ky’)a/2
+i
Px - iPy
e i(kx’+ky’)a/2
+1
Px+iPy
-1
+i
e -i(kx+ky)a/2
e i(kx-ky)a/2
-1
-i
e -i(kx’+ky’)a/2
e i(kx’-ky’)a/2
M(q) ~ [e i(qx+qy)a/2 + e -i(qx+qy)a/2] - [e i(qx-qy)a/2 + e -i(qx-qy)a/2]
~ 2[ cos(qx+qy)a/2 – cos(qx-qy)a/2 ]
q = k’ - k
Px – i Py 
Px + i Py
Max for q = (π, π)
D-Wave Superconductor!
162
Resistivity Temperature and
Doping Dependence
Normal metal resistivity is usually dominated by
phonons and is ~ T for high temperatures and
~ T5 for low temperatures (Bloch-Gruneisen
form). There is an additional constant term due
to scattering from impurities.
Underdoped cuprates are non-linear for low T
becoming linear at higher T. Near optimal
doping, the resistivity is ~ T and approximately
extrapolates to the origin (small constant
impurity term ~ 10 μΩ-cm ).
Overdoped cuprates become non-linear in T and
acquire a constant impurity term.
The magnitude of the resistivity is large,
~ 200 μΩ-cm for x = 0.15 at 300K. A typical
metal is ~ 1 μΩ-cm at 300K.
163
Resistivity Temperature Dependence from Coulomb
Scattering with Polarons
1  Ei   2  E f
x2-y2 electron
energies
  E f  Ei
ρ(Δ) = # polarons with
energy splitting Δ. Constant
 Linear resistivity.
Polaron
energies
Occupation of Ei  f (  ),
Band electron lifetime
Polaron energy
splitting density
f ( ) 
1
e b  1
There are 0T polarons
available for scattering.
1
2p

  | M |2 N (0)  d () f ()[1  f ( 2 )]d ( 2  1  ),
 (1 )  

Constant density
2
2p
1 


| M | N ( 0)  0 
 T (for 1  0)
 b1 

1  e

If ρ(Δ) ~Δμ, then resistivity ~Tμ+1. ρ(Δ) is doping dependent.
164

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