william a. goddard, iii, [email protected]

EEWS-90.502-Goddard-L15 1© copyright 2009 William A. Goddard III, all rights reserved

Nature of the Chemical Bond with applications to catalysis, materials

science, nanotechnology, surface science, bioinorganic chemistry, and energy

William A. Goddard, III, [email protected] Professor at EEWS-KAIST and

Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics,

California Institute of Technology

Course number: KAIST EEWS 80.502 Room E11-101Hours: 0900-1030 Tuesday and Thursday

Senior Assistant: Dr. Hyungjun Kim: [email protected] of Center for Materials Simulation and Design (CMSD)

Teaching Assistant: Ms. Ga In Lee: [email protected] assistant: Tod Pascal:[email protected]

Lecture 27, December 14, 2009


Schedule changesDec. 14, Monday, 2pm, L27, additional lecture, room 101Dec. 15, Final exam 9am-noon, room 101


Last time


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 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


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

(g)2(u)2

It is explained in MO theory as due to filled bonding and antibonding orbitals


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

+ (g)2(u)1

-

The VB explanation is a little less straightforward. Here we consider that there are two equivalent VB structures neither of which leads to much bonding, but superimposing them leads to resonance stabilization

Using (g) = L+R and (u)=L-R

Leads to (with negative sign


Examine the bonding of XeF

Xe Xe+

The energy to form Xe+ F- can be estimated from

Consider the energy to form the charge transfer complex

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


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

FXeF


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.


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.

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


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


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

1.3 1.3 1.3 1.3 1.3 1.3

2.71.0 3.9-5.3-2.9 -0.1

RnF2 is quite stable, by 3.6 eV, but I do not know if it has been observed


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.


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+


Donor-acceptor bonds to O atom


Ozone

We saw earlier that bonding O to O2 removes most of the resonane of the O2 ,leading to the VB configuration


Diazomethane

Leading to


Julius Su and William A. Goddard IIIMaterials and Process Simulation Center,

California Institute of Technology

Origin of reactivity in the hypervalentreagent o-iodoxybenzoic acid (IBX)

Hypervalent O-I-O linear bond


Hypervalent iodine assumes many metallic personalities

IOH

O O

O

I

OAc

OAc

I

OH

OTs

I

O

Oxidations

Radicalcyclizations

CC bondformation

Electrophilicalkene activation

CrO3/H2SO4

Pd(OAc)2

SnBu3Cl

HgCl2

Can we understand this remarkable chemistry of iodineMartin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514

Hypervalent I alternative


New

25

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

High Temperature Superconductors Cuprates

Jamil Tahir-Kheli

26

Superconducting Tc; A Story of Punctuated Evolution All Serendipity

Today

MetalEra

A15 Metal AlloyEra

CuprateEra

2020

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

BCS Theory (1957)

Theoretical Limit(Anderson)

27

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 16.1 K 1941 NbN (Ascherman, Frederick, Justi, and Kramer, Berlin) 17.1 K 1953 V3Si (Hardy and Hulm, U Chicago) 18.1 K 1954 Nb3Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs) 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)) 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)

Metals Era

A15 Metal Alloy Era

The Cuprate Era

Short history of superconductivity

after a 15 year drought, the next generation is due soon, what will it be?

28

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?

Fundamental Goals in Our Research on Cuprate Superconductivity

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.

29

PerovskitesPerovskite (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

30

(La0.85Z0.15)2CuO4: Tc = 38K (Z=Ba), 35K (Z=Sr)

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

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

31

YBa2Cu3O7– Tc=92K (=0.07)

Structure type: 1212CCrystal system: Orthorhombic

Lattice constants: a = 3.8227 Åb = 3.8872 Åc = 11.6802 Å

Space group: PmmmAtomic 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

Per formula unit: two CuO2 sheets (five coordinate pyramid)one CuO chain (four coordinate square)

32

Tc depends strongly on the number of CuO2 layers: Bi2Sr2Can-1CunO4+2n

a = 3.85 Åc = 26.8 Å

a = 3.85 Åc = 30.9 Å

a = 3.85 Åc = 36.5 Å

Triple sheet CuO2

Tc= 110 K

single sheet CuO2

Tc= 10 K

double sheet CuO2 Tc= 85 K

33

Dependence of Tc on layers is not monotonic TlBa2Can-1CunO2n+3

CuO2

CuO2

n= 2 Tc= 103 K n= 3 Tc= 123 K n= 4 Tc= 112 K n= 5 Tc= 107K

Double sheet CuO2Triple sheet CuO2 4 sheet CuO2 5 sheet CuO2

a = 3.86 Åc = 12.75 Å

a = 3.84 Åc = 15.87 Å

a = 3.85 Åc = 19.15 Å

a = 3.85 Åc = 22.25 Å

CuO2

CuO2CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

34

Reining Champion since 1994: Tc=138K (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33

This has the same structure

asTlBa2Ca2Cu3O9

n= 3 Tc= 123 K

CuO2

a = 3.84 Åc = 15.87 Å

Triple sheet CuO2

CuO2

CuO2

1994 Dai, Chakoumakos (ORNL)

Sun, Wong (U Kansas) Xin, Lu (Midwest

Superconductivity Inc.), Goldfarb (NIST)

35

Isolated layer can be greatTl2Ba2Can-1CunO2n+4

n= 1 Tc= 95 K n= 4 Tc= 112 K

CuO2

CuO2

single sheet CuO2 4 sheet CuO2

a = 3.86 Åc = 23.14 Å

a = 3.85 Åc = 41.98 Å

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

CuO2

36

(Ba,Sr)CuO2 Tc=90K

Structure type: 02"∞ -1"∞Crystal system: Tetragonal

Lattice constants: a = 3.93 Åc = 3.47 Å

Space group: P4/mmmAtomic positions:

Cu at (0,0,0)O at (0,1/2,0)

Ba,Sr at (1/2,1/2,1/2)

single sheet CuO2

37

Some cuprates lead to electron doping not holes (Nd,Ce)2CuO4 Tc =24K

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)

For =0 2 Nd (+3) and 1 Cu (+2) lead to 8 holes 4 O (2) lead to 8 electrons, get insulatorDope with Ce (+4) leading to an extra electron

CuO2

single sheet CuO2

CuO2

CuO2

38

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

39

Structural Characteristics of HighTc Superconductors: Start with Undoped Antiferromagnet

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

Undoped system antiferromagnetic withTNeel = 325 K for La2CuO4.Describe states as a Heisenberg AntiFerromagnetwith Jdd = 0.13 eV for La2CuO4:

jidddd SSJH

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 (p)2 (pz)2.

Cu – O bond = 1.90 – 1.95 ÅCu can have 5th or 6th apical O (2.Å) to form an octahedron or half-octahedron

Superexchange Jdd AF coupling

40

Superexchange coupling of two Cu d9 sitesexactly the same as the hypervalent XeF2

267-14

Two Cu d9 separated by 4Å leads to no bonding (ground state singlet and excited triplet separated by 0.0001 eV)

With O in-between get strong bonding(the singlet is stabilized by Jdd = 0.13 eV = 1500K for LaCuO4)

Explanation: a small amount of charge transfer from O to right CuCu(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 stateThe explanation is referred to as superexchange.

No direct bonding

bond

bond

41

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 = 8Cu2+ d9 local spin antiferromagnetic couplingTo get a metal requires doping to put holes in the valence band

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–

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.

42

Essential characteristic of all cuprate superconductors is oxidation (doping)

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.Antiferromagnetic: 0 < x < 0.02

Spin Glass: 0.02 < x < 0.05

Superconductor: 0.05 < x < 0.32

Typical phase diagram

La2-xSrxCuO4

43

Summary: Central Characteristics of cuprate superconductors, square CuO2 lattice, 16% holes

CuO2 plane La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+

d9 Cu2+ spin, with antiferromagnetic coupling

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 eVO 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).

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

O O OO

CuCu O Cu O Cu O

Doping (oxidation) La2-xSrxCuO4:Hole x Cu3+ and 1 – x Cu2+, OrHole x O– and 4 – x O2–

pσ

pπ

All wrong: based on simple QM (LDA) or clusters (Cu3O8)

44

Goddard et al. carried out GVB calculations on Cu3O10 + 998 point charges and found p holes (found similar E for p)

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)

Which is right: p or pholes?

undoped

doped

45

pholes

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

46

p holes

Emery and most physicists assumed p 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 provides the basis for a correct explanation of superconductivity.

Goddard believed that if p 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

47

Current canonical HighTc Hamiltonian Op holeThe t–J Model

Cu d9 hole is 3d x2–y2. O 2pσ doubly occupied.

Heisenberg AF with Jdd = 0.13 eV for La2CuO4.

Undoped

jidddd SSJH

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.Doped

J

tJ

Because of Coulomb repulsion cannot have doubly occupied holes.

48

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).

49

Universal Superconducting Tc Curve(What we must explain to have a credible theory)

Where do these three special doping values come from?

≈ 0.05

≈ 0.16

≈ 0.27

SuperconductingPhase

50

Basis for all theories of cuprate superconductors LDA Band calculations of La2CuO4

LDA and PBE lead to a half filled p-dx2-y2 band; predicting that La2CuO4 is metallic!

This is Fundamentally Wrong

Experimental Band Gap is 2 eV

LDA: Freeman 1987, Mattheiss 1987, Pickett (1989)

(

Occupied

Un-Occupied

Occupied

OccupiedOccupied

Un-Occupied

Un-Occupied

Un-Occupied

(

(

Fermi Energy

Occupied

Empty

p-dx2-y2 band

51

Validation of accuracy of DFT for 148 molecules (the G2 Set) with very

accurate experimental data

Hf = Heat of Formation (298K)IP = Ionization PotentialEA = Electron affinityPA = Proton AffinityEtot = total atomic energy

Units: eV

Data is the mean average deviation from experiment

O3LYP 0.18 0.139 0.107 1.13

OLYP 0.20 0.185 0.133 1.38

LDA (3.9 eV error) and HF (6.5 eV error) Useless for thermochemistry

B3LYP and X3LYP are most generally accurate DFT methods

We want to do QM on cuprate superconductors. Which

DFT functional Is most accurate?

52

Occupied

U-B3LYP calculations of La2CuO4

Fermi Energy

EmptyU-B3LYP leads to an insulator (2eV band gap) with a doubled unit cell (one with up-spin Cu and the other down-spin)

Band gapky

LDA 0.0 Freeman et al. 1987, Mattheiss 1987, Pickett 1989PBE 0.0 Tahir-Kheli and Goddard, 2006PW91 0.0 Tahir-Kheli and Goddard, 2006Hartree-Fock 17.0 Harrison et al. 1999B3LYP (unrestricted) 2.0 Perry, Tahir-Kheli, Goddard Phys. Rev. B 63,144510(2001)Experiment 2.0 (Ginder et al. 1988)

53

B3LYP leads to excellent band gaps for La2CuO4 whereas LDA predicts a metal!

LDA 0.0 (Freeman et al. 1987, Mattheiss 1987, Pickett 1989)PBE 0.0 Tahir-Kheli and Goddard, 2006PW91 0.0 Tahir-Kheli and Goddard, 2006Hartree-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 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).

54

B3LYP Works Well for Crystals with Transition Metals Too

J. Muscat, A. Wander, and N. M. Harrison, Chem. Phys. Lett. 342, 397 (2001).

3.7

1.0

3.0

3.8

3.6

7.8

3.3

9.0

3.4

1.4

5.5

3.5

Expt.

3.5ZnS

2.0FeS2

3.4TiO2

3.9NiO

3.8MnO

7.3MgO

3.4Cr2O3

8.5Al2O3

3.2ZnO

1.5GaAs

5.8Diamond

3.8Si

B3LYP

Direct (Vertical) Bandgaps in eV

Si B3LYP band structure

Points are Expt., GW, and QMC

B3LYP accurate for transition metals and band structures

1.2eV

3.5eV

55

Density of States for Explicitly Doped

La2–xSrxCuO4 using UB3LYP

Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002)

The down-spin states show a clear localized hole with Opz-Cudz2 character

Undopedx=0.0

x=0.125

HOMO LUMO

Band gap

Note exactly 0.125 doping leads to ordered supercell with small gap.

Real system disordered holes + d9 spins

Becomes conductor for x>0.06

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)

Band gap 2eV

56

Find extra hole localized on apical Cu and O atoms below the Sr site.

Sr

0.11Å

0.26 Å

Cu z2/O pz hole

B3LYP leads to a hole along the Sr-O-Cu Apical axis (z) the

apical Polaron

Bottom line: the hole is NOT in the Cu-O plane (as assumed in ALL previous attempts to explain superconductivity of cuprates)

This state has Cu dz2 character on the Cu and Opz character on the bridging O atom.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.

Perry, Tahir-Kheli, and Goddard; Phys. Rev. B 65, 144501 (2002)].

Now use B3LYP for La2–xSrxCuO4

Use superlattices with 8 primitive cells La15SrCu8O32

allowing antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to)

57

Nature of the new hole induced by LaSr: the Apical Polaron

Mulliken Populations4 Planar O1 in CuO6 near Sr

On

e h

ole

Tw

o h

oles

2 O1 per hole

Located on apical Cu–O just below the Sr. We call this the Apical Polaron.

O1

O2

O2’

58

The Plaquette Polaron state is localized on the four-site Cu plaquette above the Sr. It has apical O pz, Cu dz2, and planar O pσ character over the plane of four Cu atoms.The Plaquette Polaron state is calculated to 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 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.

2nd LaSr polaron from Ab-Initio DFT: the Plaquette Polaron

Sr

Apical O pz + Cu z2 hole

de-localizedover plaquettefor low doping

0.09 Å0.1 Å

Assumption: LaSr Doping leads to Plaquette Polarons.

59

The Plaquette Polaron StatesConsider 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).

60

Real orbital view of Plaquette Polaronsget two degerate states

Sr Sr

Cu dz2

O pz = p

Main hole character

Cu dz2

O pz = p

61

Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice (Ising)

jidddd SSJH

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

62

The Chiral Coupling Term

)( 212 SSSJ zCH )( 122 SSSJ zCH

Chiral coupling twists the spins into a right or left-handed system.

63

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.

Px-iPy

Px+iPy

-i

-i

i

i

1

-1

-1

1

3D Quantum character of spins leads to Chiral Polarons.

64

We obtain 3 types of Electrons

1. “Undoped” Cu AF d9 sites2. 4-site polarons (out of plane)

• Two types of polaronsa) 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)

SurfacePolaron

InteriorPolaron d9 AFx2-y2/pσ

band

65

Total Spin HamiltonianTime-Reversed Chiral Polarons

)( 212 SSSJ zCH

.

,

;| ;|

zz

CHCH

yxyx

SS

JJ

iPPiPP

The total spin Hamiltonian is,

Htot = Hdd + Hpd + HCH.

(AF d9)–(AF d9) spin coupling

(AF d9)–(polaron) spin coupling

Chiral coupling

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.

66

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

67

As increase doping, weaken AF coupling among Cud9 states. Above 5% get conductor.

1. “Undoped” Cu AF d9 sites2. 4-site polarons (out of plane)

• Two types of polaronsa) 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)

SurfacePolaron

InteriorPolaron d9 AFx2-y2/pσ

band

68

Superconducting Pairing only on Surface Plaquettes

69

Assume Optimal Tc Maximum Surface Polarons per Volume

1000 x 1000 lattice200 ensembles

≈ 0.16

Surface area of pairing leads to correct optimal doping

Sp

Percolation threshold

Becomes metallic

prediction

experiment

70

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

71

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

72

D-Wave PairingLocal 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.

1 i

−1

−1

1 −i

−i

i

P

P’

P↓

P’↑

P↓

r↑ r’↓

r↓ r’↑

Sign of the wavefunction (from Pauli principle part),

(r↑, r’↓, P↓) (r↓, r’↓, P’↑) (r↓, r’↑, P↓) (+) signSpin-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(−) repulsive for singlet

−

−

+

+

D-Wave

P, P’ energy splitting ~ Jdd maps to Debye energy in BCS Tc.

73

Nature of Pairing

−V−V −V

−V−V −V

−V−V −V

−V−V −V

74

Gap Equation in BCS Superconductors

kk withpairs

k

k

k

k

kkV

kk

element Matrix

),( into Scatters

k

wavesk y2-x2

d k

75

BCS Ground State Wavefunction

kk

kkkkkkk

kkk ccccVccH'

'''2

1)(

0|)(|

'' kkkkG ccvu

GkkGk

kkk ccV ||'

'

kk

kkkkkkk

kkkHF ccccccH'

''2

1)(

122 kk vu

76

Solution (Min F=E-TS)

kk22kkkE

k

kkk E

vu2

k

kk E

u

12

12

k

kk E

v

12

12

kTvuV k

kkkkk 2tanh)( '

'''

kTE

V k

k

kkkk 2

tanh2

'

'

''

77

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

−V−V −V

−V−V −V

78

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

79

B-dG Equations

'

''''

'' 2

1

RRRRRRRR

RRRRRR

RRRR ccccVcctccH

−V−V −V

−V−V −V

−V−V −V

−V−V −V

RRRRRRRR ccV '''' )(

'

'''0

'R

nRRR

nRRR

nRn vuHuE

kT

EvuvuV nn

RnR

nR

nRRRRR 2

tanh2

1)( ''''

'

'''0

'R

nRRR

nRRR

nRn uvHvE

80

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?

81

Estimate of Maximum TcChemical 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 predict optimum doping structure to have Tc > 200K. May be a challenge to synthesize.

82

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.

Assumption 3a: Interaction of the d9 undoped AF spins with the chiral polarons breaks the energy degeneracy between time-reversed chiral polarons. This leads to the D-wave superconducting pairing.

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.

''21

yx iPP

. ;

;

''

''

yx

yx

iPPE

iPPE

83

Cuprate Superconductivity Puzzles Must all be explained by any correct theory

Exp. Couples to Electron SpinNeutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF)Cu, O different NMR relaxations

Exp. Couples to Electron ChargeLinear Resistivity ρ ~ TDrude scattering 1/τ ~ max(ω,T)Excess Mid-IR absorptionLow temperature resistivity ~ log(T)Negative Magnetoresistance low T“Semi-conducting” c-axis resistivityHall Effect ~ 1/T (expect ~ constant)Hall Effect RH ~ const for field in CuO2 plane.Photoemission PseudogapPhotoemission Background Large

SuperconductivityPhase transition to superconductivityDx2–y2 Gap SymmetryEvolution of Tc with dopingCo-existence of magnetism and superconductivity

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

84

Extra slides

85

What Characterizes a Superconductor? (The GAP)

kk withpairs

k

k

k

k

kkV

kk

element Matrix

),( into Scatters

Cooper PairingSuperconducting Gap, Δk

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.

Bardeen, Cooper, Schrieffer (BCS) showed in 1957 that coupling through phonons can lead to a net attraction Electron distorts nuclei and moves away opposite momentum electron is attracted to distortion.

wavesk

86

The Superconducting Gap Symmetry

kk withpairs

k

k

k

k

kkV

kk

element Matrix

),( into Scatters

Cooper PairingSuperconducting Gap, Δk

If Vk’k = –V < 0 (attraction), then can make “bonding” linear combinations of Cooper pairs in the ground state, Ψ ~ (k, –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 symmetry.

wavesk y2-x2

d k

(

(

k

k’+V

Ψ ~ (k, –k) – (k’, –k’) D-Wave “anti-bonding”

87

Matrix Element for On-Site Coulomb Scattering|(π,0), (Px - iPy)> |(0,π), (Px + iPy)>

1

+i -1

-i

Px + iPy

1

+i -1

-i

Px - iPy

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

-11

-11

(π,0)

[(x+iy) (x-iy) [(x-iy) , (x+iy) ]

1

-1 -1

1

(0,π)

88

Cuprate Superconductivity PuzzlesExp. Couples to Electron SpinNeutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF)Cu, O different NMR relaxations

Exp. Couples to Electron ChargeLinear Resistivity ρ ~ TDrude scattering 1/τ ~ max(ω,T)Excess Mid-IR absorptionLow temperature resistivity ~ log(T)Negative Magnetoresistance low T“Semi-conducting” c-axis resistivityHall Effect ~ 1/T (expect ~ constant)Hall Effect RH ~ const for field in CuO2 plane.Photoemission PseudogapPhotoemission Background Large

SuperconductivityPhase transition to superconductivityDx2–y2 Gap SymmetryEvolution of Tc with dopingCo-existence of magnetism and superconductivity

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

Chiral plaquette polaron theory of cuprate superconductivity Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007) Explains each of these phenomena

89

Experimental Neutron Data

Doping

shift δ from ()

Raw Datax = 0.14

Corrected Data

For antiferromagnet expect peak in neutron scattering at ()

((

(

Instead the peak is shifted by ±(2/a)and increases with doping x

90

Calculated spins (x = 0.10, JCH = 3Jdd)

Red AF d9 regions, Green plaquette polarons

91

Computed Neutron Structure Factor

x = 0.10 256 256 lattice. 5,000 ensembles.

.1)(1

ion,Normalizat k

kSN

Pure AF would be peaked at (π,π)

92

Computed Neutron Structure Factor

256 256 lattice. 5,000 ensembles.

x = 0.075 x = 0.10

x = 0.125

Incommensurability ring around (π, π) of

magnitude

xa

2

.x

a

Correlation length approximately

((

(

93

The Hall Effect in Metals and Semi-conductors

Lorentz Force .1

Bv

cEe

dtdp

.11

, ,

, ,

, E,

,

22

2

necBJE

R

BEmce

Em

eEv

EBmc

evB

cv

emv

mne

nevJ

eEmv

H

Drude picture

.1

Bv

cEe

dtdk

Band picture

v

vT

Magnetic field deflects charge

In metals, n = constant RH = constant.In semi-conductors, n ~ T3/2 e–Δ/T.

FermiSurface

94

Temperature Dependent Hall Effect in Cuprates

n=0.31 (per cell)

n=0.16 (per cell)

n=0.63 (per cell)

La2–xSrxCuO4

x>0.34 get RH = constant.

x=0.15 get strong T dependence to

RH.

95

Skew Scattering from Chiral Plaquette Polarons

x2-y2 band scattering from a chiral polaron leads to a left-right scattering asymmetry that is an additive contribution to the Hall effect.

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.

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

Px + i Py

Current flow

96

Calculating the Skew Scattering

Number of polarons Difference between ± polarons

The chiral polaron difference is proportional to the magnetic field, B.

Δmax is the maximum time-reversed polaron energy separation and Δmin is the minimum separation.

Δmin and Δmax are larger for lower dopings where there are more undoped d9 spins to cause splitting.

11

)(

e

f

Δ Δ+2gpμBB

97

Comparison of Theory to Experiment

Answer #1: Skew scattering from chiral polarons leads to the observed temperature dependence.

La2–xSrxCuO4

Fitting experimental data to the form,

11

)(

e

f

1.05106.3K11.6K0.25

3.81378.9K0.0K0.15

8.82454.1K16.7K0.10

15.4791.5K12.6K0.05

R0ΔmaxΔminx

RH = R0[f(Δmin) – f(Δmax)]

98

The Magnitude of Skew Scattering

./cm 104 Experiment

,/cm 1039.5

,4 1 ,1

eV, 5.1 eV, 0.1

,eV 5.2)eV 1(4x1

U,10.0

33

33

maxmax

C

CcR

xn,mg

x

H

p

DS

Answer #2: The magnitude is right too.

Δmin = 16.7K and Δmax = 454.1K from experimental fit.

Computed value

U is enhanced because fewer electrons to screen the repulsion.

99

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 “anti-clockwise” plaquette energies.

Leads to no temperature dependence of Hall effect for in-plane magnetic fields.

Experimentally observed. Px – i Py

Px + i Py

Current flow

100

Conclusions

Experiments

Exp. Couples to Electron Spin

Neutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF)Cu, O different 0NMR relaxations

Superconductivity

Phase transition to superconductivityDx2–y2 Gap SymmetryEvolution of Tc with dopingCo-existence of magnetism and superconductivity

A successful theory must explain experiments from each category. Current theories do not, leaving many of the very puzzling properties unexplained.

Exp. Couples to Electron Charge

Linear Resistivity ρ ~ TDrude scattering 1/τ ~ max(ω,T)Excess Mid-IR absorptionLow temperature resistivity ~ log(T)Negative Magnetoresistance low T“Semi-conducting” c-axis resistivityHall Effect ~ 1/T (expect ~ constant)Hall Effect RH ~ const for field in CuO2 plane.Photoemission PseudogapPhotoemission Background Large

PRB vol 76, 014514 (2007)

101

The Chiral Polaron Magnetic Susceptibility

occupation

occupation

<P(ω)> = )( )()( Wff.

11

)(

e

f

By Fermi’s Golden Rule, the absorption rate is,

The power absorbed is,

Probability distributionof polaron energy splittings.

Bottom Line: Uniform ρ = Scaling

102

.x

a

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,

103

The Total Magnetic Susceptibility in the Random Phase Approximation (RPA)

Summing the diagrams,

and solving,

Scaling term

104

The Total Magnetic Susceptibility in the Random Phase Approximation (RPA)

Using the integrals,

Satisfies ω/T scaling and has the correct functional form.

a1 = 0.43 a3 = 10.5

Experiment LSCO x = 0.04

105

Solving the Heisenberg Spin Hamiltonian in terms of two sets of spins A and B (non-linear sigma model)

ij

Bj

Ai

ij

Bj

Ai SSJSSJH )(

A B A BB A B AA B A BB A B A

Staggered magnetization (r)

Effective Continuum Hamiltonian

. ),1,0,0()(

,1

,

222

2222

rr

rdH

zyx

zyxeff

MinimizeConstraint

Boundary Condition

Map AF system onto Ferromagnetic system

This converts the problem of solving for the ground state into a problem of minimizing a continuous field with constraints

106

Topological Invariants

. ),1,0,0()(

,1

,

222

2222

rr

rdH

zyx

zyxeffMinimize

Boundary Condition

Topological Invariant, ), ( 41 2 yxrdQ

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

n = 0n = 1

.)(2

1

az

dzi

n

a

.||8 2222 QrdH zyxeff

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.

Constraint

107

Heuristic Derivation of Incommensurability Magnitude Q=(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 x)= x.

Net rotation per spin is Np . If area chosen such that net rotation per spin is 2π, then A=(a/x)2.

A translation of L with L2=A returns to identical patch. L = a/x and Q=(a)x.

Chiral coupling dominant

108

Coulomb Scattering of x2–y2 Band Electronswith Polarons

Absorb pσ orbitals into “effective” A1 orbitals on Cu sites for simplicity.

(U > K > 0)

x2y2 band k state spin σ

Polaron state spin s

On-site with no spin exchange

On-site with spin exchange

Coefficients αR determine polaron state

Fundamental Reason for Superconductivity is thatPolaron is spread out over more than one site.

109

Matrix Element for Coulomb Scattering|k, S> |k’, Px + iPy>

e i(kx+ky)a/2

e i(kx-ky)a/2e -i(kx+ky)a/2

e -i(kx-ky)a/2

+1

+1 +1

+1

S

e i(kx’+ky’)a/2

e i(kx’-ky’)a/2e -i(kx’+ky’)a/2

e -i(kx’-ky’)a/2

-1

+i +1

-i

Px+iPy

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 ]

S or D

q = k’ - k

Max for q = (π, 0), (0, π)

110

Px – i Py Px + i Py

Matrix Element for Coulomb Scattering|k, Px - iPy> |k’, Px + iPy>

e i(kx+ky)a/2


e -i(kx-ky)a/2

-1

-i +1

+i

Px - iPy

e i(kx’+ky’)a/2



-1

+i +1

-i

Px+iPy

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

Max for q = (π, π)

111

Coulomb Cooper Pairing Diagrams

k –k

k’ –k’

Direct Coulomb band repulsion

Intermediate Polaron change with no spin exchange

Intermediate Polaron change with spin exchange

Matrix element +V Repulsion of pairs

112

Matrix Elements for no spin exchange: Attraction

q = k’ – kM(q) depends on polaron states I and I’

and is the key to D-wave pairing.

Attraction for states near Fermi level

113

Matrix Elements for spin exchange: Repulsion

q = k’ + k

M(q) depends on polaron states I and I’

Minus sign due to interchange of (k’, –k’)

in final state

Δ = EI’ – EI

Repulsion for states near Fermi level

114

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 = I’ = Px’ ± i Py’

Only for spin exchange

I = Px’ i Py’

Time-Reversed intermediate polaron

115

Superconducting Pairing Possibilities

(

Γ

Spin exch only

k

k’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 (±π, ± π)

116

Debye Energy in BCS the Maximum Energy Splitting of Time-Reversed Polarons

Δ = EI’ – EI

The pairing occurs for electrons within the Maximum Energy Splitting betweenTime-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.

117

Evolution of Tc with Doping

Typical phase diagram

La2-xSrxCuO4

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.

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.

118

Resistivity Temperature Dependence

Polaron energy splitting Δmax

If ρ(Δ) ~Δμ, then resistivity ~Tμ+1

119

Optical Absorption

Scattering rate

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.

22

2

/1/1

4)(Re

p

Simple Drude

Added oscillators

one-component fit

)(/1

22220

2

2

/

/

120

A Simple Model for Skew Scattering

Consider x2y2 band scattering with a complex matrix element Vk’k.

Inversion and Time-Reversal symmetry guarantees no skew scattering.w(kk’) = w(–k–k’) from inversion and w(–k–k’) = w(k’k) from time-reversal. An applied magnetic field breaks time-reversal.

Scattering Hamiltonian

Scattering T-Matrix

Fermi’s Golden Rule

Third order term changes sign whenk and k’ are interchanged.

121

The Magnitude of Skew Scattering

./cm 104 Experiment

,/cm 1039.5

,4 1 ,1

eV, 5.1 eV, 0.1

,eV 5.2)eV 1(4x1

U,10.0

33

33

maxmax

C

CcR

xn,mg

x

H

p

DS

Question #2: Does it have the right magnitude? YES

Δmin = 16.7K and Δmax = 454.1K from experimental fit.

Computed value

U is enhanced because fewer electrons to screen the repulsion.

122


Scattering with Chiral Polaron Hole in P±

Expanding to second order

First and second order terms

Mod-squared for scattering rate

123


Invariant under interchangek’ k. Does not contribute to skew scattering term.

For hole-doped cuprates, the polarons are holes and U < 0.

For electron-doped cuprates, the polarons are electrons and U > 0.

q = k’ – k

124


Extracting the skew scattering contributions,

Ak,k’ = –Ak’,k

q = k’ – 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.

125


11

Mean skew scattering

ρS = 1/ΔSmax = constant.

ρD = 1/ΔDmax = constant.

λ = 0.19 for LSCO at x = 0.10

Must take mean over chiral polaron

energy splittings

126

Temperature Dependent Hall Effect

Band Structureshave hole-like Fermi surfacesleading to RH > 0

(

Occupied

Un-Occupied

Fermi Surface

127

Log(T) Low-Temperature Resistivity

Negative Magnetoresistance

Tesla

60 T magnetic field to suppress superconductivity

log(T)

No log(T)

La2–xSrxCuO4 x = 0.08 and 0.13

“semi-conducting” ρc

128

Kondo Effect

S+ S–S+S–

kp

kp

q q

log(T) resistivityNegative magnetoresistance

Suppressed for T < 80K in 60T field

Chirality + spin = effective “spin” impurity. If splitting in 60T small (small g-factor), then resistivity obtained

B = 0.671 Kelvin/Tesla

80 K 60 T

Spin impurity in metal

g = 2

No Kondo effect for “ferromagnetic” Coulomb coupling with polarons.

129

Kondo spin-flip from x2y2 band hopping on and off polaron

A1 orbitals

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 k-vectors 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-of-plane than in-plane leading to the “semi-conducting” ρc.

No Kondo effect for “ferromagnetic” Coulomb coupling with polarons.

130

2D Plaquette Percolation and Polaron Islands

Loss of Kondo effect (Insulator to Metal transition)

131

Angle-Resolved Photoemission (ARPES)

(k||, kz)Surface

h (k||, kz’)

Parallel MomentumConserved

Spectral Functionsfor k passing through

Fermi level

Occupied

Empty

Fermi Level

ARPES cuts off spectral function with Fermi factor

ARPES on cuprates finds a Fermi surface with a broad background even for k states far above the Fermi energy.

132

ARPES Pseudogap Underdoped Cuprates

83K underdoped Bi2Sr2CaCu2O8+δ

ARPES taken at this k-vector

Pt reference spectra

Pseudogap forunderdoped

Gap

Gap closed

Pseudogap exists above Tc and closes as thetemperature increases. Pseudogap is D-wave.

133

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).

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.

(

(

k

k+q

q (π,π)

V MatrixElement

134

Polarization Matrix Elements (Electric Dipole)

y

x

kx = ky

E

Allowed

kx = ky

E

Not Allowed

kx = ky

E

Not Allowed

kx = ky

E

Allowed

z2

z2

E

E

kx = 0

kx = 0

Allowed

Not Allowed

135

Photoemission Process (Energy)

136

Computed Percolation Values

Experiment: LSCO = 0.05 YBCO = 0.35

137

(LaSr)CuO4 Doping Constraints

No Sr above and below plaquette

No neighboring Sr in plane

No Sr separted by two lattice spacings

No Sr separted by two lattice spacings on opposite sides of

plane

No out-of-plane Sr neighbors

138

YBa2Cu3O7– Doping Constraints

Chain O

always dopes same Cu

triple

Chain O dopes Cu

triple randomly, but not

twice

Adjacent chain O doping.

Adjacent Cu triples

not allowed

Experiment: YBCO at x=0.35Tc drops from ~92 to ~65K

139

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.

140

The Chiral Polaron Magnetic Susceptibility

The low-energy excitation of a chiral plaquettepolaron 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).

).,(

1;|)()(|;

q

iPPqSqSiPP yxzz

yx

occupation

occupation

.1

1)(

ef

141

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.

142

Thermopower

143

Undopedx=0.0

x=0.125

x=0.25

x=0.5

Density of States for Explicitly Doped La2–xSrxCuO4 using UB3LYP

Highest hole states Lowest electron states

Band gap

Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002)

The down-spin states show a clear localized hole with Opz-Cudz2 character

Band gap

144

2D Plaquette Percolation and Polaron Islands

Loss of Kondo effect (Insulator to Metal transition)

145

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 out-of-plane polarons (two-components), 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 two-component theories? NO.

146


Nandor et al PRB 60, 6907 (1999)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 one-component models, requiring some antiferromagnetic correlations larger than the Cu-Cu distance.

Pennington et al PRB 63,054513 (2001)

147


Nandor et al PRB 60, 6907 (1999)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.

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

148

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 out-of-plane polarons (two-components), the data appear to contradict our assumptions.

Currently, no theory explains this temperature dependence.

Takigawa et al, PRB 43, 247 (1991)



149


Nandor et al PRB 60, 6907 (1999)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 one-component models requiring some antiferromagnetic correlations larger than the Cu-Cu distance.

Pennington et al PRB 63,054513 (2001)

150


Nandor et al PRB 60, 6907 (1999)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.

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

151

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.

152

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?

153

Spin-1/2 states along different axes

Θ

φ

|2

sin|2

cos),(| ie

|2

cos|2

sin),(| ie

x

y

z

α-spin along x-axis = 2

1)0,

2(|

α-spin along y-axis = i2

1)

2,

2(|

This extra factor of i is due to quantum nature of spin and is reason for chirality.

154

Simplifying Plaquette Polaron orbitals

t’

t

155

Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice

++1

++1

+1+

+1+

+ pointing out-of-pagez-axis

pointing in xy-planeof page

+1 z2/x2y2 triplet

++1

++1

++1

+1+

H

H

156

Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice

++1

++1

++1

+1+

H

H

++1

++1

++1 +1 +

= t

= t

|2

sin|2

cos),(| ie

α-spin along x-axis = 2

1)0,

2(|

α-spin along y-axis = i2

1)

2,

2(|

~(1+i )t

~(1-i )t

Px + i Py will lowerenergy.

Jahn-Teller Distortion!

157

Net Effect of Chiral Polarons

++1

++1

+1+

+1+

+ pointing out-of-pagez-axis

pointing in xy-planeof page

+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.

158

Chiral Polarons are distributed randomly throughout the crystal

.1

1)(

ef

Probability distributionof polaron energy splittings.

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

occupationΔ

Neutron inelastically scatters losing energy ħω

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!

159

Neutron Spin ω/T Scaling

a1 = 0.43 a3 = 10.5

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.

160

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

161

Coulomb Cooper Pairing Diagrams

Scattering through time-reversed polaron intermediate state

162

Px – i Py Px + i Py

Matrix Element for Coulomb Scattering|k, Px - iPy> |k’, Px + iPy>

e i(kx+ky)a/2


e -i(kx-ky)a/2

-1

-i +1

+i

Px - iPy

e i(kx’+ky’)a/2



-1

+i +1

-i

Px+iPy

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

Max for q = (π, π) D-Wave Superconductor!

163

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.

164

Resistivity Temperature Dependence from Coulomb Scattering with Polarons

fi EE 21

x2-y2 electronenergies Polaron

energiesif EE

1

1)(

),( of Occupation

e

f

fEi

)0(for 1

)0(||2

,)()](1)[()()0(||2

)(1

11

02

1222

1

1

Te

NM

ffdNM

There are T polaronsavailable for scattering.

Band electron lifetime

Polaron energy splitting density

Constant density

If ρ(Δ) ~Δμ, then resistivity ~Tμ+1. ρ(Δ) is doping dependent.

ρ(Δ) = # polarons with energy splitting Δ. Constant Linear resistivity.

william a. goddard, iii, [email protected]

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