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
>
Ted Yu < tedhyu@wag.caltech.edu>
Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved
1

Remaining Course schedule
Wednesday, Mar. 10 no class (wag at conference in Chicago)
Thursday Mar. 11, 2pm, L27 TODAY (make up for Mar. 10)
Last lecture
Final available Friday March 12
Final due back Friday March 19
Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved
2

Last time
Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved
3

Each N has a doubly occupied sp 2 s orbital pointing at it.
The heme group
The net charge of the
Fe-heme is zero. The VB structure shown is one of several, all of which lead to two neutral N and two negative N.
Thus we consider that the Fe is Fe 2+ with a d 6 configuration
4

x 2 -y 2
Energies of the five Fe 2+ d orbitals z 2 =2z 2 -x 2 -y 2 yz xz xy
5

Exchange stabilizations
6

Ferrous Fe II x 2 -y 2 destabilized by heme N lone pairs z 2 destabilized by
5 th ligand imidazole or 6 th ligand CO y x 7

Out of plane motion of Fe – 4 coordinate
8

Add axial base
N-N Nonbonded interactions push
Fe out of plane is antibonding
9

Free atom to 4 coord to 5 coord
Net effect due to five N ligands is to squish the q, t, and s states by a factor of 3
This makes all three available as possible ground states depending on the 6 th ligand 10

Bonding of O
2 with O to form ozone
O
2 has available a p s orbital for a s bond to a p s orbital of the O atom
And the 3 electron p system for a p bond to a p p orbital of the O atom
11

Bond O
2 to Mb
Simple VB structures  get S=1 or triplet state
In fact MbO
2 singlet
Why?
is
12

change in exchange terms when Bond O
2 to Mb
O
2 p s
O
2 p p
Assume perfect
VB spin pairing
Then get 4 cases
10 K dd
5*4/2 up spin
Thus average K dd is
(10+7+7+6)/4 =7.5
down spin
7 K dd
4*3/2
+
2*1/2
7 K dd
4*3/2
+
2*1/2
6 K dd
3*2/2
+
3*2/2
13

Bonding O
2 to Mb
Exchange loss on bonding O
2
14

Modified exchange energy for q state
But expected t binding to be 2*22 = 44 kcal/mol stronger than q
What happened?
Binding to q would have D H = -33 + 44 = + 11 kcal/mol
Instead the q state retains the high spin pairing so that there is no exchange loss, but now the coupling of Fe to O
2 does not gain the full VB strength, leading to bond of only 8kcal/mol instead of
33
15

Bond CO to Mb
H
2
O and N
2 do not bond strongly enough to promote the Fe to an excited state, thus get S=2
16

compare bonding of CO and O2 to Mb
17

New material
18

Consider Co-heme
Co
Doublet S=1/2
Quartet S=3/2 d q 2
1
-44 kcal/mol
Exchange stabilizations
Up spin down spin total
S=1/2 4*3/2 3*2/2 = 9
S=3/2 5*4/2 2*1/2 = 11
44 kcal/mol
Co stabilization for high spin 44 kcal/mol; Fe 88 kcal/mol
Result: Co-heme is low spin d state
19

Bond O
2 to Co-heme
Co
Co
Before bond
Co
Cannot make, xz doubly occupied
After bond
Co before bond
S=1/2 with net spin on Co
After bond
S=1/2 with net on O
2 p
20

Exchange stabilizations for ferric Fe, Fe III doublet S=1/2 d
1 quartet S=3/2
Sextet S=5/2 q s 1
1
Exchange stabilizations
Up spin down spin total
S=1/2 3*2/2 2*1/2 = 4
S=3/2 4*3/2 0 = 6
S=5/2 5*4/2 0 = 10
2Kdd = 44 kcal/mol
6Kdd = 132 kcal/mol
Fe III stabilization for high spin 132 kcal/mol; Fe II 88 kcal/mol
Result: FeIII cannot bind O
2
, must reduce to Fe II first
21

Hypervalent compounds
It was quite a surprize to most chemists in 1962 when Neil
Bartlett reported the formation of a compound involving Xe-
F bonds.
But this was quickly followed by the synthesis of XeF
4
Xe and F
XeF
6
.
2 at high temperature and XeF
2
(from in 1962 and later
Indeed Pauling had predicted in 1933 that XeF stable, but no one tried to make it.
6 would be
Later compounds such as ClF
3 and ClF
5 were synthesized
These compounds violate simple octet rules and are call hypervalent
22

Noble gas dimers
Recall that there is no chemical bonding in He
2
, Ne
2 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
( s g
) 2 ( s u
) 2
23

Noble gas dimer positive ions
On the other hand the positive ions are strongly bound
This is explained in MO theory as due to one less antibonding electron than bonding, leading to a three electron bond for He
2
+ of
55 kcal/mol, the same strength as the one electron bond of H
2
+
The VB explanation is less straightforward.
( s g
) 2 ( s u
) 1
Using ( s g
) = L+R and ( s u
)=L-R
We consider that there are two equivalent VB structures
Leads to (with negative sign neither of which leads to much bonding, but superimposing them leads to resonance stabilization
-
24

Re-examine the bonding of HeH
Why not describe HeH as ( s g
) 2 ( s u
) 1 where
( s g
) = L+R and ( s u
)=L-R
Would this lead to bonding?
The answer is no, as easily seen with the VB form, where the right structure is 23.6-0.7=23.9 eV above the left.
Thus the energy for the ( s u
) 1 state would be +12.0 – 2.5 =
9.5 eV unbound at R= ∞ g
) 2 ( s
Adding in ionic stabilization lowers the energy by 14.4/2.0 = 7.2 eV (overestimate because of shielding) , still unbound by 2.3 eV
-
He H
He +
IP=+24.6 eV
H-
EA = 0.7 eV 25

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
26

Examine the bonding in XeF
2
The energy to form Xe + F is +1.45 eV
Now consider, the impact of putting a 2 nd on the back side of the Xe +
F
Xe +
Since Xe + has a singly occupied pz orbital pointing directly at this
2 nd 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
27

Stability of XeF
2
Ignoring resonance we predict that XeF
2 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 XeF
2 molecule is stable by 2.7 eV with respect to Xe + F
2
But to assess whether one could make and store XeF
2
, 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 XeF
2 by the chain reaction
XeF
2
+ F  {XeF + F
2
}  Xe + F
2
+ F
Since the bond energy of F
2 is 1.6 eV, this reaction is endothermic by 2.7-1.6 = 1.1 eV, suggesting the XeF
2 relatively stable. is
Indeed it is used with F
2 to synthesize XeF
4 and XeF
6
.
28

XeF
4
Putting 2 additional F to overlap the Xe p y 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 XeF
4
XeF
2 relative to 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.
29

XeF
6
Since XeF
4 still has a pz pair, we can form a third hypervalent bond in this direction to obtain an octahedral XeF
6 molecule.
Indeed XeF
6 is stable with this structure
Here we expect a stability a little less than 8.1 eV.
Pauling in 1933 suggested that XeF
6 years in advance of the experiments. would be stabile, 30
He also suggested that XeF
8 is stable.
However this prediction is wrong
30

Estimated stability of other Nobel gas fluorides (eV)
Using the same method as for
XeF
2
, we can estimate the binding energies for the other
Noble metals.
1.3
-2.9
1.3
-5.3
1.3
-0.1
1.3
1.0
1.3
2.7
KrF
2 is predicted to be stable by
0.7 eV, which makes it susceptible to decomposition by F radicals
1.3
3.9
RnF
2 is quite stable, by 3.6 eV, but I do not know if it has been observed 31

XeCl
2
Since
EA(Cl)=3.615 eV
R(XeCl + )=2.32A
De(XeCl + )=2.15eV,
We estimate that XeCl
2
Xe + Cl
2
. is stable by 1.14 eV with respect to
However since the bond energy of Cl
2 is 2.48 eV, the energy of the chain decomposition process is exothermic by
2.48-1.14=1.34 eV, suggesting at most a small barrier
Thus XeCl
2 would be difficult to observe
32

Halogen Fluorides, ClF n
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 ClF
3 and ClF
5
.
Stability of ClF
3 relative to ClF + 2F
We estimate that ClF
3 by 2.8 eV.
is stable
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
33

Geometry of ClF
3
34

We estimate that
ClHF
2
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 F
2
ClH  ClF + HF is exothermic by 1.4 eV
F
2
ClH has not been observed
35

ClF
5
36

BrF n and IF n
37

SF n
38

SF
6
The VB rationalization for octahedral SF
6 would be to assume that S is promoted from (3s) 2 (3p) 4 to (3s) 0 (3p) 6 which would lead to 3 hypervalent bonds in the x, y, and z directions.
With an “empty” 3s orbital, the EA for SF
6 would be very high
39

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
40

Donor-acceptor bonds to oxygen
41

Ozone, O
3
The simple VB description of ozone is, where the terminal p p 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
42

leading to
Diazomethane
43

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)
Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved
44

Hypervalent iodine assumes many metallic personalities
Oxidations
Hypervalent I alternative
O
Cr O
3
/H
2
SO
4
O
O
I
OH
Radical cyclizations
OAc
I
OAc
Sn Bu
3
Cl
Electrophilic alkene activation
OH
I
OTs
Hg Cl
2
C

C bond formation
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
Ch120a-Goddard-L27 © copyright 2010 William A. Goddard III, all rights reserved
45

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

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
Today
Cuprate
Era
A15 Metal Alloy
Era
Metal
Era
Theoretical Limit
(Anderson)
BCS Theory (1957)
2020
47

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)
1.14 K 1933 Aluminum
Metals Era
16.1 K 1941 NbN (Ascherman, Frederick, Justi, and Kramer, Berlin)
17.1 K 1953 V
3
Si (Hardy and Hulm, U Chicago)
18.1 K 1954 Nb
3
Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs)
A15 Metal
9.8 K 1962 Nb
0.6
Ti
0.4
(First commercial wire, Westinghouse)
Alloy Era
23.2 K 1973 Nb
3
Ge (Gavaler, Janocho, and Jones, Westinghouse)
30 K 1986 (LaBa)
2
CuO
4
(Müller and Bednorz, IBM Rüschlikon, Switzerland)
92 K 1987 YBa
2
Cu
3
O
7
(Wu, Ashburn, and Torng (Alabama), Hor, Meng, Gao, Huang,
Wang, and Chu (Houston)) The Cuprate Era
105 K 1988 Bi
2
Sr
2
CaCu
2
O
8
(Maeda, Tanaka, Fukutomi, Asano, Tsukuba Laboratory)
120 K 1988 Tl
2
Ba
2
Ca
2
Cu
3
O
10
(Hermann and Sheng, U. Arkansas)
133 K 1993 HgBa
2
Ca
2
Cu
3
O
8
(Schilling, Cantoni, Guo, Ott, Zurich, Switzerland)
138 K 1994 (Hg
0.8
Tl
0.2
)Ba
2
Ca
2
Cu
3
O
8.33
(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?
48

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.
49

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
50

(La
0.85
Z
0.15
)
2
CuO
4
: Tc = 38K (Z=Ba), 35K (Z=Sr)
1986 first cuprate superconductor, (LaBa)
2
CuO
4
(Müller and Bednorz) Nobel Prize
Isolated CuO
2 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)
CuO
6 octahedra
51

YBa
2
Cu
3
O
7
– d
Tc=92K ( d
=0.07)
Per formula unit: two CuO
2 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
52

Tc depends strongly on the number of CuO2 layers:
Bi
2
Sr
2
Ca n-1
Cu n
O
4+2n double sheet CuO
2
Tc= 85 K single sheet CuO
2
Tc= 10 K
Triple sheet CuO
2
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 53

Dependence of Tc on layers is not monotonic n= 2 Tc= 103 K
TlBa
2
Ca n-1
Cu n
O
2n+3 n= 3 Tc= 123 K n= 4 Tc= 112 K n= 5 Tc= 107K
CuO
2
CuO
2
CuO
2
CuO
2
Double sheet CuO
2 a = 3.86 Å c = 12.75 Å
CuO
2
Triple sheet CuO
2 a = 3.84 Å c = 15.87 Å
4 sheet CuO
2 a = 3.85 Å c = 19.15 Å
CuO
2
CuO
2
CuO
2
CuO
2
CuO
2
CuO
2
CuO
2
CuO
2
CuO
2
5 sheet CuO
2 a = 3.85 Å

Reining Champion since 1994: Tc=138K
(Hg
0.8
Tl
0.2
)Ba
2
Ca
2
Cu
3
O
8.33
This has the same structure as
TlBa
2
Ca
2
Cu
3
O
9 n= 3 Tc= 123 K CuO
2
1994
Dai, Chakoumakos (ORNL)
Sun, Wong (U Kansas)
Xin, Lu (Midwest
Superconductivity Inc.),
Goldfarb (NIST)
CuO
2
CuO
2 a = 3.84 Å c = 15.87 Å
Triple sheet CuO
2
55

Isolated layer can be great
O
2n+4
CuO
2
CuO
2
CuO
2
Tl
2
Ba
2
Ca n-1
Cu n n= 1 Tc= 95 K
CuO
2 n= 4 Tc= 112 K
CuO
2
CuO
2
CuO
2
CuO
2 single sheet CuO
2
CuO
2 a = 3.86 Å c = 23.14 Å
CuO
2
CuO
2 c = 41.98 Å 4 sheet CuO 56
2

single sheet CuO
2
(Ba,Sr)CuO
2
Tc=90K
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) 57

Some cuprates lead to electron doping not holes
(Nd,Ce)
2
CuO
4
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
CuO
2 single sheet CuO
2
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)
CuO
2
CuO
2
58

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
59

Structural Characteristics of HighTc Superconductors:
Start with Undoped Antiferromagnet
Cu O Cu O Cu O
O O O
Cu O Cu O Cu O
Cu
O
Cu
2D CuO
2 square lattice (xy plane)
Oxidation state of Cu: Cu II or d 9 .
(xy) 2 (xz) 2 (yz) 2 (z 2 ) 2 (x 2 -y 2 ) 1 d 9 hole is 3d (x 2 –y 2 ).
O O O O
Cu O Cu O Cu O Cu
Oxidation state of O: O 2–
(p
σ
) 2 (p p
) 2 (p p z
) 2 .
or p 6 .
O O O
Cu O Cu O Cu O
O
Cu
Cu – O bond = 1.90 – 1.95 Å
Cu can have 5 th or 6 th apical O (

2.
4 Å) to form an octahedron or half-octahedron
Undoped system antiferromagnetic with
T
Neel
= 325 K for La
2
CuO
4
.
Describe states as a Heisenberg AntiFerromagnet with J dd
= 0.13 eV for La
2
CuO
4
:
H dd

J dd

S i

S j
Superexchange
J dd
AF coupling
60

Superexchange coupling of two Cu d 9 sites exactly the same as the hypervalent XeF
2
Two Cu d 9 separated by 4 Å 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 J dd
= 0.13 eV = 1500K for LaCuO
4
)
Explanation: a small amount of charge transfer from O to right Cu
Cu(x 2 -y 2 ) 1 -O(px) 2 Cu(x 2 -y 2 ) 1  Cu(x 2 -y 2 ) 1 -O(px) 1 -Cu(x 2 -y 2 ) 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

Characteristic of High Tc Superconductors: Doping
The undoped La
2
La
2
CuO
4
CuO
4 is an insulator (band gap = 2.0 eV)
(Undoped): La 3+ , Sr 2+ , O 2– , Cu 2+
Thus cation holes = 3*2 + 2 = 8 and anion electrons = 4*2 = 8
Cu 2+  d 9  local spin  antiferromagnetic coupling
To get a metal requires doping to put holes in the valence band
Doping (oxidation) La
2-x
Sr x
CuO
4
:
Assuming 4 O 2requires 8 cation holes.
But La
2-x
Sr x

6-x holes, thus must have x Cu 3+ and 1 – x Cu 2+
Second possibility: assume that excitation from Cu 2+ to Cu 3+ is too high, then must have hole on O 2– leading to O
–
This leads to x O
– and (4 – x) O 2– per formula unit.
YBa
2
Cu
3
O
7
:
Assume that all 7 O are O 2–
Must have 14 cation holes: since Y 3+ +2 Ba 2+ leads to +7, then we must have
1 Cu 3+ and 2 Cu 2+
The second possibility is that all Cu are Cu 2+ requiring that there be 1 O
– and 6 O 2–
62

Essential characteristic of all cuprate superconductors is oxidation
(doping)
Typical phase diagram
La
2-x
Sr x
CuO
4
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 system becomes a normal metal.
63

Summary: Central Characteristics of cuprate superconductors, square CuO
2
CuO
2 plane
Cu O Cu O Cu O Cu lattice, 16% holes
La
2
CuO
4
(Undoped): La 3+ , Sr 2+ , O 2– , Cu 2+ d 9 Cu 2+  spin, with antiferromagnetic coupling
O O O O
Cu O Cu O Cu O pπ
O O O
Cu
O
Doping (oxidation) La
2-x
Sr x
CuO
4
:
Hole  x Cu 3+ and 1 – x Cu 2+ , Or
Hole  x O
– and 4 – x O 2–
Cu O Cu O Cu O pσ
O O O
Cu O Cu O Cu O
Cu
O
Cu
YBa
2
Cu
3
O
7
:
Y 3+ , Ba 2+ , O 2–  1 Cu 3+ and 2 Cu 2+ , Or
Y 3+ , Ba 2+ , Cu 2+  1 O
– and 6 O 2–
Where are the Doped Holes?
Cu III or d 8 : Anderson, Science 235 , 1196 (1987), but Cu II  Cu III 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 (Cu
3
O
8
)
64

Which is right: p s or p p holes ?
Goddard et al. carried out GVB calculations on Cu similar E for p s
)
3
O
10
+ 998 point charges and found p p holes (found
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) undoped doped
65

p p holes
Goddard et al showed if the ground state has p p 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, p p holes does not correct provide an explanation of superconductivity in cuprates.
66

p s holes
Emery and most physicists assumed p s 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 p s provides the basis for a correct explanation of superconductivity.
Goddard believed that if p s holes were correct, then it would lead to strong bonding to the singly occupied d x2-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 67

Current canonical HighTc Hamiltonian
Op s hole  The t –J Model
Undoped
Cu d 9 hole is 3d x 2 –y 2 . O 2p
σ doubly occupied.
Heisenberg AF with J dd
= 0.13 eV for La
2
CuO
4
.
H dd

J dd

S i

S j
Doping creates hole in O p
σ that bonds with x 2 –y to form a bonded singlet (doubly occupied hole).
2
Singlet hole hopping through lattice prefers adjacent sites are same spin, this frustrates the normal AF coupling of d 9 spins.
Doped t
J
J
Because of Coulomb repulsion cannot have doubly
68 occupied holes.

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~T 2 ).
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).
69

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?
70

Basis for all theories of cuprate superconductors
LDA Band calculations of La
LDA and PBE lead to a half filled p s
-d x2-y2 band; predicting that
Un-Occupied
2
CuO
(
0,p)
4
Un-Occupied
( p,p)
La
2
CuO
4 is metallic!
This is Fundamentally Wrong
Experimental Band Gap is 2 eV
Occupied Occupied
G(0,0)
Occupied Occupied
( p,0)
Empty Un-Occupied Un-Occupied
Fermi Energy p s
-d x2-y2 band
LDA: Freeman 1987,
Mattheiss 1987,
Pickett (1989)
Occupied
71

Units: eV
Method
HF
D
H f
G2(MAD)
IP EA PA
H-Ne TM
E tot
D
E xc
6.47 1.036 1.158 0.15 4.49 1.09
G2 or best ab initio 0.07
a
0.053
a
0.057
a
0.05
a
1.59
a
0.19
He
2
Ne
2
(H
2
O)
2
We want to do QM on cuprate
D
E(R e
)
Unbound
D
E(R e
)
Unbound
D e
(R
O..O
) superconductors. Which
0.161(3.048)
DFT functional Is most accurate?
0.0011(2.993
) 0.0043( 3.125
) 0.218(2.912) d
LDA (SVWN)
GGA
BP86
BLYP
BPW91
PW91PW91 mPWPW f
3.94
a
0.88
a
0.31
a
0.665
0.175
0.187
0.749
0.212
0.27
0.05
0.106 0.08
6.67
0.54
e
0.0109(2.377) 0.0231(2.595) 0.391(2.710)
Validation of accuracy of DFT for
148 molecules (the G2 Set) with very
0.19 0.46 Unbound Unbound 0.194(2.889)
0.19 0.37
e accurate experimental data
Unbound Unbound 0.181(2.952)
0.34
a
0.163
0.094
0.05
0.16 0.60
D
H f
0.156(2.946)
0.77 0.164 0.141 0.06 0.35 0.52 IP = Ionization Potential
0.65 0.161 0.122 0.05 0.16 0.38 0.0052(2.823) 0.194( 2.911
)
PBEPBE g
XLYP k
Hybrid Methods l
B3P86
m
0.74
g
0.156
0.101
0.06
1.25 0.34 0.0032(2.752) 0.0048
(3.097) PA = Proton Affinity 0.222(2.899)
0.33 0.186 0.117 0.09 0.95 0.24 E tot
= total atomic energy
0.94 0.207 0.247 0.07 0.08 0.72
0.78
a
0.636
0.593
0.03
2.80 0.34
Data is the mean average
Unbound Unbound 0.214( 2.905
) deviation from experiment
Unbound Unbound 0.206(2.878)
B3LYP n
B3PW91
o
PW1PW p mPW1PW q
PBE1PBE r
X3LYP
s
Exptl.
0.13
a
0.15
a
0.168
0.161
0.103
0.100
0.06
0.03
0.38 0.25
e
0.24 0.38
Unbound
Unbound
0.198(2.926)
0.175(2.923)
0.23 0.160 0.114 0.04
0.30 0.30 0.0066(2.660) 0.0095( 3.003
) 0.227(2.884)
0.17 0.160 0.118 0.04 0.16 0.31 0.0020( 3.052
)
0.21
g
0.162
0.126
0.04
0.0023(3.254) 0.199(2.898)
1.09
B3LYP and X3LYP are most generally
0.12 0.154 0.087 0.07 0.11 0.22 0.0010
accurate DFT methods
0.00 0.000 0.000 0.00 0.00 0.00 0.0010(2.970) h
0.0036(3.091) h
0.236
i
(2.948 j
) 72

U-B3LYP calculations of La
2
CuO
4
U-B3LYP leads to an insulator (2eV band gap) with a doubled unit cell
(one with up-spin Cu and the other down-spin)
Empty k y
Fermi Energy Band gap
Occupied
LDA
PBE
PW91
Hartree-Fock
0.0
0.0
0.0
17.0
Freeman et al. 1987, Mattheiss 1987, Pickett 1989
Tahir-Kheli and Goddard, 2006
Tahir-Kheli and Goddard, 2006
Harrison et al. 1999
B3LYP (unrestricted) 2.0 Perry, Tahir-Kheli, Goddard Phys. Rev. B 63 ,144510(2001)
Experiment 2.0 (Ginder et al. 1988) 73

B3LYP leads to excellent band gaps for La
2
CuO
4 metal!
whereas LDA predicts a
LDA
PBE
PW91
0.0
(Freeman et al. 1987, Mattheiss 1987, Pickett 1989)
0.0
Tahir-Kheli and Goddard, 2006
0.0
Tahir-Kheli and Goddard, 2006
Hartree-Fock 17.0
(Harrison et al. 1999)
B3LYP (unrestricted) 2.0
(Perry, Tahir-Kheli, and Goddard 2001)
Experiment 2.0 (Ginder et al. 1988)
Conclusion #1: To describe the states of the La
2
CuO
4 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).
74

Si
Diamond
GaAs
ZnO
Al
2
O
3
Cr
2
O
3
MgO
MnO
NiO
TiO
2
FeS
2
ZnS
3.4

9.0
3.3
7.8
3.6
3.8
3.0
1.0
3.7
Expt.

3.5
5.5
1.4
B3LYP Works Well for Crystals with Transition Metals Too
B3LYP
8.5
3.4
7.3
3.8
3.9
3.4
2.0
3.5
3.8
5.8
1.5
3.2
Si B3LYP band structure
Points are Expt., GW, and QMC
1.2eV
3.5eV
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).
75

Undoped x=0.0
Band gap 2eV
HOMO LUMO
Band gap x=0.125
Density of States for
Explicitly Doped
La
2 –x
Sr x
CuO
4 using UB3LYP
Use superlattice (2√2 x 2√2 x 1) with 8 primitive cells La
15
SrCu
8
O
32
This allows antiferromagnetic coupling of Cu atoms.
Allows hole to localize (but not forced to)
The down-spin states show a clear localized hole with Opz-Cudz 2 character
Note exactly 0.125 doping leads to ordered supercell with small gap.
Real system  disordered holes + d 9 spins
Becomes conductor for x>0.06
Doped UB3LYP: Perry, Tahir-Kheli, and Goddard,
Phys Rev B 65 , 144501 (2002) 76

Now use B3LYP for
La
2 –x
Sr x
CuO
4
B3LYP leads to a hole along the Sr-
O-Cu Apical axis (z) the apical
Polaron
Use superlattices with 8 primitive cells La
15
SrCu
8
O
32 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.
Cu z 2 /O pz
This state has Cu dz 2 character on the Cu and
Opz character on the bridging O atom.
hole
Because d z2 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 dz 2 and Cu dx 2 -y 2 on the same atom are the same (d 8 high spin)
The singlet state is ~2.0 eV higher.
0.11Å
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)].
77

Nature of the new hole induced by La  Sr: the Apical
Polaron
Mulliken Populations
4 Planar O1 in
CuO
6 near Sr
O2
O1
O2’
2 O1 per hole
Located on apical Cu–O just below the Sr. We call this the Apical Polaron .
78

2 nd La  Sr polaron from Ab-Initio DFT: the Plaquette Polaron
The Plaquette Polaron state is localized on the four-site Cu plaquette above the Sr. It has apical O pz, Cu dz 2 , and planar O pσ
Apical O pz +
Cu z 2 hole de-localized character over the plane of four Cu atoms.
The Plaquette Polaron state is calculated to over plaquette for low doping be 0.053
eV per 8 formula units above the apical polaron state this is
0.007 eV = 0.2 kcal/mol per Cu in the
La
0.875
Sr
0.125
CuO
4 cell.
0.09 Å
0.1 Å
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.
79

The Plaquette Polaron States
Consider 4 Cu-O d z2 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).
80

Real orbital view of Plaquette Polarons get two degerate states
Sr
Main hole character
Cu d z2
O pz
= p s
Cu d z2
O pz
= p s
Sr
81

Coupling of plaquette spin to neighboring d 9 antiferromagnetic lattice (Ising)
The Px plaquette is compatable with the left antiferromagnetic coupling of the d 9 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

S i

S j
82

The Chiral Coupling Term

J
CH
S z 2

( S
1

S
2
)

J
CH
S z 2

( S
2

S
1
)
Chiral coupling twists the spins into a right or left-handed system.
83

An unusual degeneracy: complex linear combinations
(e
+
=P x
+ i P y and e
-
=P x
- i P y
)
It is natural to assume the hole wavefunction is P x or P the two.
y or some real linear combination of
The interaction of the polaron with the background d 9 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
84

We obtain 3 types of Electrons
1. “Undoped” Cu AF d 9 sites
2. 4-site polarons (out of plane)
• Two types of polarons a) Surface polarons
(neighboring) AF d 9 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 d 9 AF
85

Total Spin Hamiltonian
Time-Reversed Chiral Polarons
The total spin Hamiltonian is, 
J
CH
S z 2

( S
1

S
2
)
H tot
= H dd
+ H pd
+ H
CH .
(AF d 9 )–(AF d 9 ) spin coupling
(AF d 9 )–(polaron) spin coupling
Chiral coupling
| P x

 iP y

; s  
| P x

 iP y

;
 s 
J
CH
S z



J
CH

S z
.
,
H
CH
H dd
H pd is invariant under polaron time-reversal.
is polaron time-reversal invariant.
is not invariant. J pd
S z
•S d
 – J pd
S z
•S d
H pd splits the energy between time-reversed chiral polarons. The energy difference is on the order of J pd
~ (8/4)J dd
=2J dd
~ 0.28 eV . The energy difference between polarons with the same spin but different chiralities is on the order of 4J
CH
~ 1.1 eV .
The energy splitting between time-reversed polarons is largest for low doping because there are more d 9 spins to induce the splitting.
86

Isolated Plaquette Polaron in a d 9 sea.
Dopant
The Plaquette is pinned down by the Sr dopant. The
Cu d 9 sea leads to an insulator, just as for undoped
87

As increase doping, weaken AF coupling among Cud 9 states. Above 5% get conductor.
1. “Undoped” Cu AF d 9 sites
2. 4-site polarons (out of plane)
• Two types of polarons a) Surface polarons
(neighboring) AF d 9 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 d 9 AF
88

Superconducting Pairing only on Surface Plaquettes
89

Assume Optimal T c
 Maximum Surface Polarons per Volume experiment
≈ 0.16
1000 x 1000 lattice S p
200 ensembles prediction
Percolation threshold
Becomes metallic
Surface area of pairing leads to correct optimal doping
90

Predict maximum Tc
– 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
91

The Goal
– Presumably, will find the prior argument of Tc peak near 0.16 to be correct with random dopings
– How much higher?
– Increasing surface area by punching holes into metallic swath should increase Tc
92

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),
(r↑, r’↓, P↓)
 (r↓, r’↓, P’↑) 
(r↓, r’↑, P↓)

(+) sign
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 (E ground
–E
I
).
Net (+) coupling
 attractive singlet
(−)
 repulsive for singlet
+
− r↓
−i
1 i
1
P’↑
P
P’
P↓
−1 i
−1
−i r’↑
P, P’ energy splitting ~ J dd
Debye energy in BCS Tc.
maps to
D-Wave
− + r↑
P↓ r’↓
93

Nature of Pairing
−V −V
−V −V
94

Gap Equation in BCS Superconductors k
  k
 pairs with
 k
 k

Scatters into ( k
 
,
 k
 
)
Matrix element V k k

 k

 k
 
D k
D k
 s
 wave D k
 d x2 y2
95

BCS Ground State Wavefunction
H
  k s
(
 k
 
) c
 k s c k s

1
2 k

' k
V k ' k c k

'
 c

 k '
 c
 k
 c k

|

G
 
( u k
 v k c k
 
'
 c
 k '

) | 0
 u k
2  v k
2 
1
D k
  k '
V kk '
 
G
| c
 k
 c k

|

G

H
HF
  k s
(
 k
 
) c
 k s c k s

1
2 k
 k '
(
D k c k

'
 c

 k '

 D  k c
 k
 c k

)
96

Solution (Min F=E-TS)
 k
  k
 
E k
  k
2  D 2 k u k v k

D k
2 E k u k
2 
1
2

 1


E k k

 v k
2 
1
2

 1

 k
E k


D k
  
V kk '
( u k ' v k '
) tanh
 k '
2 kT
D k
  
V kk '
D k '
2 E k ' tanh
 k '
2 kT
97

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
98

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 d 9 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
99

B-dG Equations
H
 
R s

D
RR '

R c

R s c
R s
 
RR ' t
RR ' c

R s c
R ' s
(

V
RR '
)
 c
R
 c
R '


1
2

' RR
V
RR ' c
R

 c

R '
 c
R '
 c
R

 D
R ' R
−V −V
E n u
R n
E n v
R n 

H
0
RR ' u n
R '

H
0
RR ' v
R n
'


 D
' R
RR ' v
R n
'
 D
' R n
RR ' u
R '
D
RR '

(

V
RR '
)
 1
2
 u n
R ' v
R n
−V −V
 u
R n v
R n
'
 tanh


E n
2 kT


100

Computational Approaches
– 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
101

Estimate of Maximum Tc
Chemical Physics Letters 472 (2009) 153 –165
To estimate Tc, use the formula from BCS theory T c ħω
D is Debye energy,
N(0) is the density of states at the Fermi level, and
= 1.13 ħω
D exp(-1/N(0)V)
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 d 9 spins get
~ 2J dd
= 0.26 eV ~ 3000K.
Expect range from J dd
/2 for one-side with d 9 spin neighbors to 3J dd
/2 for case with three-side interfacing d 9 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 0.05J
dd
0.15J
dd or 150K to 450K.
Current maximum of 138K may be 0.05J
dd case.
Expect that Tc of ~ 300K might be attainable..
to
Using 100x100 supercell, self-consistent calculations for 100 random 16% doping cases we adjusted the d 9 -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

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.
1
2
(
P
 x ' iP y '
)
Assumption 2: A band is formed by Cu x 2 –y 2 /O pσ on the polaron sites when the polaron plaquettes percolate through the crystal. Leads to ARPES background.
Assumption 3a: Interaction of the d 9 undoped AF spins with the chiral polarons breaks the energy degeneracy between time-reversed chiral polarons. This leads to the
D-wave superconducting pairing.
E
(
E
(
P x '
P x '
 iP y '
; s
)

 iP y '
;
 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.
103

Cuprate Superconductivity Puzzles
Must all be explained by any correct theory
Exp. Couples to Electron Spin
Exp. Couples to Electron Charge
Neutron spin incommensurability
Linear Resistivity ρ ~ T
Drude scattering 1/τ ~ max(ω,T)
Neutron spin ω/T scaling
(expect ω/J dd or ω/E
F
)
Cu, O different NMR relaxations
Excess Mid-IR absorption
Low temperature resistivity ~ log(T)
Superconductivity
Negative Magnetoresistance low T
“Semi-conducting” c-axis resistivity
Phase transition to superconductivity
Dx 2 –y 2 Gap Symmetry
Hall Effect ~ 1/T (expect ~ constant)
Evolution of Tc with doping
Co-existence of magnetism and
Hall Effect R
H
~ const for field in CuO
2 plane.
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
Chiral plaquette polaron theory of cuprate superconductivity
Tahir-Kheli, Goddard; Phys. Rev. B 76 : 014514 ( 2007 )
Explains each of these phenomena 104

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 CH
4
, benzene, O
2
, 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 + ,
(PH
3
)
2
Pt(CH
3
)
2
, (bpym)Pt(Cl)(CH
3
), 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 and the number of valence electrons.
105

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
106

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 107

Metals vs inulators
108

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)
109

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
110

Body centered cubic (bcc), A2
A2
111

Face-centered cubic (fcc), A1
112

Alternative view of fcc
113

Closest packing layer
114

Stacking of 2 closest packed layers
115

Hexagonal closest packed
(hcp) structure,
A3
116

Cubic closest packing
117

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
118

Structures of elemental metals cc some correlation of structure with number of valence electrons
119

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:
Li
2 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 Li
2
+ , which is 33.7 kcal/mol per valence electron 120

GVB orbitals of ring M
10 molecules
Get 10 valence electrons each localized in a bond midpoint
R=2 a
0 note H
10 is very different, get orbital localized on atom, not bond midpoint
Calculations treated all 11 valence electrons of Cu, Ag, Au using effective core potential.
All electrons for H and Li 121

122

Bonding in alkalis
123

The bonding in column 11
Get trend similar to alkalis
124

Geometries of Li
4 clusters
For H
4
, the electrons are in 1s orbitals centered on each atom
Thus spin pair across sides. Orthogonalization cases distortion to rectangle
For Li
4
, the electrons are in orbitals centered on each bond midpoint
Thus spin pair between bond midpoint. Orthogonalization cases distortion to rhombus
125

Geometries of Li
6 cluster
For H
6
, the electrons are in
1s orbitals centered on each atom
Thus spin pair across sides. Orthogonalization cases distortion to D3h hexagone
For Li
6
, the electrons are in orbitals centered on each bond midpoint
Thus spin pair between bond midpoint.
Orthogonalization cases distortion to triangular structure
126

Geometries of Li
8 cluster
For Li
8
, the electrons are in orbitals centered on each bond midpoint
Thus spin pair between bond midpoint.
Orthogonalization cases distortion to out-of-plane D
2d structure
127

Li
10 get closest packed structure
128

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 overlap = 0.52
129

Crystalline properties of B column
130

Binding of CH3 to Pt clusters
131

Binding of alkyl CH
3-x
Me x to (111) surfaces
Prefers on-top site
\
Decreased binding with increasing x due to steric interactions with other atoms of Pt (111) surface 132

Binding of alkylidene CH
2-x
Me x
Prefers bridge binding site to (111) surfaces
Decreased binding with increasing x due to steric interactions with other atoms of Pt (111) surface
133

Binding of alkylidyne CH
2-x
Me x to (111) surfaces
134

Average bond strength in CHx/M
8 cluster
135

Comparison of bonding energies of CH x and C
2
Hx
136

Energy barriers for CH
4 dehydrogenation on Pt
137

Energy barriers for benzene dehydrogenation on Pt
138

Geometries and Energetics of ethyl binding to M
8
139

Geometries and Energetics of ethylene binding to M
8
140

Geometries and Energetics of vinyl binding to M
8
141

Geometries and Energetics of ethylidene binding to M
8
142

Geometries and Energetics of vinylidene binding to M
8
143

Geometries and Energetics of dicarbond binding to M
8
144

Geometries and Energetics of ethynel binding to M
8
145

Geometries and Energetics of acetylene binding to M
8
Confirmed experimentally by
Wilson Ho
146

Heats of formation for C
2
H x and CH x species for Pt
Most stable is to form CH ad 147

Energetics for C2Hx on Pt
148

149

150

151