Lecture 27, December 14, 2009 Nature of the Chemical Bond with applications to catalysis, materials science, nanotechnology, surface science, bioinorganic chemistry, and energy Course number: KAIST EEWS 80.502 Room E11-101 Hours: 0900-1030 Tuesday and Thursday William A. Goddard, III, [email protected] WCU Professor at EEWS-KAIST and Charles and Mary Ferkel Professor of Chemistry, Materials Science, and Applied Physics, California Institute of Technology Senior Assistant: Dr. Hyungjun Kim: [email protected] Manager of Center for Materials Simulation and Design (CMSD) Teaching Assistant: Ms. Ga In Lee: [email protected] Special assistant: Tod Pascal:[email protected] EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 1 Schedule changes Dec. 14, Monday, 2pm, L27, additional lecture, room 101 Dec. 15, Final exam 9am-noon, room 101 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 2 Last time EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 3 Hypervalent compounds It was quite a surprize to most chemists in 1962 when Neil bartlett reported the formation of a compound involving XeF bonds. But this was quickly folllowed by the synthesis of XeF4 (from Xe and F2 at high temperature and XeF2 in 1962 and later XeF6. Indeed Pauling had predicted in 1933 that XeF6 would be stable, but noone tried to make it. Later compounds such as ClF3 and ClF5 were synthesized These compounds violate simple octet rules and are call hypervalent EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 4 Noble gas dimers Recall from L17 that there is no chemical bonding in He2, Ne2 etc This is explained in VB theory as due to repulsive Pauli repulsion from the overlap of doubly occupied orbitals It is explained in MO theory as due to filled bonding and antibonding orbitals EEWS-90.502-Goddard-L15 (sg)2(su)2 © copyright 2009 William A. Goddard III, all rights reserved 5 Noble gas dimer positive ions On the other hand the positive ions are strongly bound (L17) This is explained in MO theory as due to one less antibonding electron than bonding, leading to a three electron bond for He2+ of 2.5 eV, the same strength as the one electron bond of H2+ (sg)2(su)1 The VB explanation is a little Using (sg) = L+R and (su)=L-R less straightforward. Here we consider that there are Leads to (with negative sign two equivalent VB structures neither of which leads to much bonding, but superimposing them leads EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved to resonance stabilization 6 Examine the bonding of XeF Consider the energy to form the charge transfer complex Xe Xe+ The energy to form Xe+ F- can be estimated from Using IP(Xe)=12.13eV, EA(F)=3.40eV, and R(IF)=1.98 A, we get E(Xe+ F-)=1.45eV Thus there is no covalent bond for XeF, which has a weak bond of ~ 0.1 eV and a long bond EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 7 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 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 8 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. EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 9 XeF4 Putting 2 additional F to overlap the Xe py pair leads to the square planar structure, which allows 3 center – 4 electron charge transfer bonds in both the x and y directions, leading to a square planar structure The VB analysis would indicate that the stability for XeF4 relative to XeF2 should be ~ 2.7 eV, but maybe a bit weaker due to the increased IP of the Xe due to the first hypervalent bond and because of some possible F---F steric interactions. There is a report that the bond energy is 6 eV, which seems too high, compared to our estimate of 5.4 eV. EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 10 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 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 11 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 EEWS-90.502-Goddard-L15 1.3 1.3 1.3 1.3 1.3 1.3 -2.9 -5.3 -0.1 1.0 2.7 3.9 RnF2 is quite stable, by 3.6 eV, but I do not know if it 12 has been observed © copyright 2009 William A. Goddard III, all rights reserved Halogen Fluorides, ClFn The IP of ClF is 12.66 eV which compares well to the IP of 12.13 for Xe. This suggests that the px and py pairs of Cl could be used to form hypervalent bonds leading to ClF3 and ClF5. Indeed these estimates suggest that ClF3 and ClF5 are stable. Indeed the experiment energy for ClF3 ClF +F2 is 2.6 eV, quite similar to XeF2. EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 13 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 14 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 15 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 16 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 17 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+ EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 18 Donor-acceptor bonds to O atom EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 19 Ozone We saw earlier that bonding O to O2 removes most of the resonane of the O2 ,leading to the VB configuration EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 20 Diazomethane Leading to EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 21 Origin of reactivity in the hypervalent reagent o-iodoxybenzoic acid (IBX) Hypervalent O-I-O linear bond Julius Su and William A. Goddard III Materials and Process Simulation Center, California Institute of Technology EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 22 Hypervalent iodine assumes many metallic personalities Hypervalent I alternative O Oxidations O O OH I CrO3/H2SO4 OAc Radical cyclizations SnBu3Cl I OAc OH Electrophilic alkene activation CC bond formation HgCl2 I OTs O I Pd(OAc)2 Can we understand this remarkable chemistry of iodine Martin, J. C. organo-nonmetallic chemistry – Science 1983 221(4610):509-514 EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 23 New EEWS-90.502-Goddard-L15 © copyright 2009 William A. Goddard III, all rights reserved 24 High Temperature Superconductors Cuprates Chiral plaquette polaron theory of cuprate superconductivity; Tahir-Kheli J, Goddard WA Phys. Rev. B 76 (1): Art. No. 014514 (2007) The Chiral Plaquette Polaron Paradigm (CPPP) for high temperature cuprate superconductors; Tahir-Kheli J, Goddard WA; Chem. Phys. Lett. (4-6) 153-165 (2009) Plaquette model of the phase diagram, thermopower, and neutron resonance peak of cuprate superconductors; Jamil Tahir-Kheli and William A. Goddard III, Phys Rev Lett, submitted Jamil 25 Tahir-Kheli Superconducting Tc; A Story of Punctuated Evolution All Serendipity Discovery is made. Then all combinations tried. Then stagnation until next discovery. Theory has never successfully predicted a new higher temperature material. Embarrassing state for Theorists. To ensure progress we need to learn the fundamental mechanism in terms of the atomistic interactions Metal Era A15 Metal Alloy Era Today Cuprate Era Theoretical Limit (Anderson) 2020 BCS Theory (1957) 26 Short history of superconductivity 4.15 K 1911 Hg (Heike Kamerlingh Onnes, Leiden U, Netherlands) 3.69 K 1913 Tin (Onnes) 7.26 K 1913 Lead (Onnes) 9.2 K 1930 Niobium (Meissner, Berlin) Metals Era 1.14 K 1933 Aluminum 16.1 K 1941 NbN (Ascherman, Frederick, Justi, and Kramer, Berlin) 17.1 K 1953 V3Si (Hardy and Hulm, U Chicago) A15 Metal 18.1 K 1954 Nb3Sn (Matthias, Gebelle, Geller, and Corenzwit, Bell Labs) Alloy Era 9.8 K 1962 Nb0.6Ti0.4 (First commercial wire, Westinghouse) 23.2 K 1973 Nb3Ge (Gavaler, Janocho, and Jones, Westinghouse) 30 K 1986 (LaBa)2CuO4 (Müller and Bednorz, IBM Rüschlikon, Switzerland) 92 K 1987 YBa2Cu3O7 (Wu, Ashburn, and Torng (Alabama), Hor, Meng, Gao, Huang, Wang, and Chu (Houston)) The Cuprate Era 105 K 1988 Bi2Sr2CaCu2O8 (Maeda, Tanaka, Fukutomi, Asano, Tsukuba Laboratory) 120 K 1988 Tl2Ba2Ca2Cu3O10 (Hermann and Sheng, U. Arkansas) 133 K 1993 HgBa2Ca2Cu3O8 (Schilling, Cantoni, Guo, Ott, Zurich, Switzerland) 138 K 1994 (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 (Dai, Chakoumakos, (ORNL) Sun, Wong, (U Kansas) Xin, Lu (Midwest Superconductivity Inc.), Goldfarb, NIST) after a 15 year drought, the next generation is due soon, what will it be? 27 Fundamental Goals in Our Research on Cuprate Superconductivity Determine the fundamental mechanism in order to have a sound basis for designing improved systems. Criterion for any proposed mechanism of superconductivity: Does it explain the unusual properties of the normal and superconducting state for cuprates? There is no precedent for a theory of superconductivity that actually predicts new materials. Indeed we know of no case of a theorist successfully predicting a new improved superconducting material! Bernt Matthias always claimed that before trying new compositions for superconductors he would ask his Bell Labs theorists what to try and then he would always do just the opposite. 28 Perovskites Perovskite (CaTiO3) first described in the 1830s by the geologist Gustav Rose, who named it after the famous Russian mineralogist Count Lev Aleksevich von Perovski crystal lattice appears cubic, but it is actually orthorhombic in symmetry due to a slight distortion of the structure. Characteristic chemical formula of a perovskite ceramic: ABO3, A atom +2 charge. 12 coordinate at the corners of a cube. B atom +4 charge. Octahedron of O ions on the faces of that cube centered on a B ions at the center of the cube. Together A and B form an FCC structure 29 (La0.85Z0.15)2CuO4: Tc = 38K (Z=Ba), 35K (Z=Sr) 1986 first cuprate superconductor, (LaBa)2CuO4 (Müller and Bednorz) Nobel Prize Isolated CuO2 sheets with apical O on both sides of Cu to form an elongated octahedron Structure type: 0201 Crystal system: Tetragonal Lattice constants: a = 3.7873 Å c = 13.2883 Å Space group: I4/mmm Atomic positions: La,Ba at (0, 0, 0.3606) Cu at (0, 0, 0) O1 at (0, 1/2, 0) O2 at (0, 0, 0.1828) CuO6 octahedra 30 YBa2Cu3O7–d Tc=92K (d=0.07) Per formula unit: two CuO2 sheets (five coordinate pyramid) one CuO chain (four coordinate square) Structure type: 1212C Crystal system: Orthorhombic Lattice constants: a = 3.8227 Å b = 3.8872 Å c = 11.6802 Å Space group: Pmmm Atomic positions: Y at (1/2,1/2,1/2) Ba at (1/2,1/2,0.1843) Cu1 at (0,0,0) Cu2 at (0, 0, 0.3556) O1 at (0, 1/2, 0) O2 at (1/2,0,0.3779) O3 at (0,1/2,0.379) O4 at (0, 0,0.159) 1987: Alabama: Wu, Ashburn, and Torng Houston: Hor, Meng, Gao, Huang, Wang, Chu 31 Tc depends strongly on the number of CuO2 layers: Bi2Sr2Can-1CunO4+2n single sheet CuO2 Tc= 10 K double sheet CuO2 Tc= 85 K Triple sheet CuO2 Tc= 110 K a = 3.85 Å c = 26.8 Å a = 3.85 Å c = 30.9 Å a = 3.85 Å c = 36.5 Å 32 Dependence of Tc on layers is not monotonic TlBa2Can-1CunO2n+3 n= 2 Tc= 103 K n= 3 Tc= 123 K n= 4 Tc= 112 K n= 5 Tc= 107K CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 Double sheet CuO2 a = 3.86 Å c = 12.75 Å Triple sheet CuO2 a = 3.84 Å c = 15.87 Å 4 sheet CuO2 5 sheet CuO2 a = 3.85 Å c = 19.15 Å a = 3.85 Å 33 c = 22.25 Å Reining Champion since 1994: Tc=138K (Hg0.8Tl0.2)Ba2Ca2Cu3O8.33 This has the same structure as TlBa2Ca2Cu3O9 n= 3 Tc= 123 K 1994 Dai, Chakoumakos (ORNL) Sun, Wong (U Kansas) Xin, Lu (Midwest Superconductivity Inc.), Goldfarb (NIST) CuO2 CuO2 CuO2 a = 3.84 Å c = 15.87 Å Triple sheet CuO2 34 Isolated layer can be great CuO2 Tl2Ba2Can-1CunO2n+4 CuO2 n= 1 Tc= 95 K CuO2 n= 4 Tc= 112 K CuO2 CuO2 CuO2 CuO2 CuO2 CuO2 single sheet CuO2 CuO2 a = 3.86 Å c = 23.14 Å a = 3.85CuO Å 2 c = 41.98 Å 4 sheet CuO35 2 (Ba,Sr)CuO2 Tc=90K single sheet CuO2 Structure type: 02"∞ -1"∞ Crystal system: Tetragonal Lattice constants: a = 3.93 Å c = 3.47 Å Space group: P4/mmm Atomic positions: Cu at (0,0,0) O at (0,1/2,0) Ba,Sr at (1/2,1/2,1/2) 36 Some cuprates lead to electron doping not holes (Nd,Ce)2CuO4d Tc =24K For d=0 2 Nd (+3) and 1 Cu (+2) lead to 8 holes 4 O (2) lead to 8 electrons, get insulator Dope with Ce (+4) leading to an extra electron CuO2 single sheet CuO2 Structure type: 0201T ' Crystal system: Tetragonal Lattice constants: a = 3.95 Å c = 12.07 Å Space group: I4/mmm Atomic positions: Nd,Ce at (0,0,0.3513) Cu at(0,0,0) O1 at (0, 1/2, 0) O2 at (0, 1/2,1/4) CuO2 CuO2 37 Our Goal Explain which systems lead to high Tc and which do not Explain how the number of layers and the location of holes and electrons affects the Tc Use this information to design new structures with higher Tc 38 Structural Characteristics of HighTc Superconductors: Start with Undoped Antiferromagnet Cu O O Cu O Cu O Cu O Cu O Cu O Cu O Cu Cu O O O O Cu O O O O Cu O O O Cu O O O Cu Cu Cu O O Cu 2D CuO2 square lattice (xy plane) Oxidation state of Cu: CuII or d9. (xy)2(xz)2(yz)2(z2)2(x2-y2)1 d9 hole is 3d (x2–y2). Oxidation state of O: O2– or p6. (pσ)2 (pp)2 (ppz)2. Cu – O bond = 1.90 – 1.95 Å Cu can have 5th or 6th apical O (2.4 Å) to form an octahedron or half-octahedron Undoped system antiferromagnetic with TNeel = 325 K for La2CuO4. Describe states as a Heisenberg AntiFerromagnet with Jdd = 0.13 eV for La2CuO4: H dd J dd Si S j Superexchange Jdd AF coupling 39 Superexchange coupling of two Cu d9 sites exactly the same as the hypervalent XeF2 9 Two Cu d separated by 4Å leads to no bonding (ground state singlet and excited triplet separated by 0.0001 eV) With O in-between get strong bonding (the singlet is stabilized by Jdd = 0.13 eV = 1500K for LaCuO4) Explanation: a small amount of charge transfer from O to right Cu Cu(x2-y2)1-O(px)2-Cu(x2-y2)1 Cu(x2-y2)1-O(px)1-Cu(x2-y2)2 allows bonding of the O to the left Cu, but only for the singlet state The explanation is referred to as superexchange. bond bond No direct bonding 40 267-14 Characteristic of High Tc Superconductors: Doping The undoped La2CuO4 is an insulator (band gap = 2.0 eV) La2CuO4 (Undoped): La3+, Sr2+, O2–, Cu2+ Thus cation holes = 3*2 + 2 = 8 and anion electrons = 4*2 = 8 Cu2+ d9 local spin antiferromagnetic coupling To get a metal requires doping to put holes in the valence band Doping (oxidation) La2-xSrxCuO4: Assuming 4 O2- requires 8 cation holes. But La2-xSrx 6-x holes, thus must have x Cu3+ and 1 – x Cu2+ Second possibility: assume that excitation from Cu2+ to Cu3+ is too high, then must have hole on O2– leading to O– This leads to x O– and (4 – x) O2– per formula unit. YBa2Cu3O7: Assume that all 7 O are O2– Must have 14 cation holes: since Y3+ +2 Ba2+ leads to +7, then we must have 1 Cu3+ and 2 Cu2+ The second possibility is that all Cu are Cu2+ requiring that there be 1 O– and 6 O2– 41 Essential characteristic of all cuprate superconductors is oxidation (doping) Typical phase diagram La2-xSrxCuO4 Superconductor: 0.05 < x < 0.32 Spin Glass: 0.02 < x < 0.05 Antiferromagnetic: 0 < x < 0.02 Minimum doping to obtain superconductivity, x > 0.05. Optimum doping for highest Tc=35K at x ~ 0.15. Maximum doping above which the superconductivity disappears and the 42 system becomes a normal metal. Summary: Central Characteristics of cuprate superconductors, square CuO2 lattice, 16% holes CuO2 plane La CuO (Undoped): La3+, Sr2+, O2–, Cu2+ Cu O O Cu O Cu O Cu O O pσ O O Cu Cu O O O O Cu O O O Cu Cu Cu O Cu O pπ O Cu O Cu Cu 4 2+ Cu spin, with antiferromagnetic coupling O O O Cu 2 d9 O O Cu Doping (oxidation) La2-xSrxCuO4: Hole x Cu3+ and 1 – x Cu2+, Or Hole x O– and 4 – x O2– YBa2Cu3O7: Y3+, Ba2+, O2– 1 Cu3+ and 2 Cu2+, Or Y3+, Ba2+, Cu2+ 1 O– and 6 O2– Where are the Doped Holes? CuIII or d8: Anderson, Science 235, 1196 (1987), but CuII CuIII IP = 36.83 eV O pσ: Emery, Phys. Rev. Lett. 58, 2794 (1987). O pπ: Goddard et al., Science 239, 896, 899 (1988). O pσ: Freeman et al. (1987), Mattheiss (1987), Pickett (1989). All wrong: based on simple QM (LDA) or clusters (Cu3O8) 43 Which is right: ps or pp holes? Goddard et al. carried out GVB calculations on Cu3O10 + 998 point charges and found pp holes (found similar E for ps) undoped Electronic Structure and Valence Bond Band Structure of Cuprate Superconducting Materials; Y. Guo, J-M. Langlois, and W. A. Goddard III Science 239, 896 (1988) The Magnon Pairing Mechanism of Superconductivity in Cuprate Ceramics G. Chen and W. A. Goddard III; Science 239, 899 (1988) doped 44 pp holes Goddard et al showed if the ground state has pp holes there is an attractive pair that leads to triplet P-wave Cooper pairs, and hence superconductivity. The Superconducting Properties of Copper Oxide High Temperature Superconductors; G. Chen, J-M. Langlois, Y. Guo, and W. A. Goddard III; Proc. Nat. Acad. Sci. USA 86, 3447 (1989) The Quantum Chemistry View of High Temperature Superconductors; W. A. Goddard III, Y. Guo, G. Chen, H. Ding, J-M. Langlois, and G. Lang; In High Temperature Superconductivity Proc. 39th Scottish Universities Summer School in Physics, St. Andrews, Scotland, D.P. Tunstall, W. Barford, and P. Osborne Editors, 1991 However experiment shows that the systems are singlet D-wave. Thus, pp holes does not correct provide an explanation of superconductivity in cuprates. 45 ps holes Emery and most physicists assumed ps holes on the oxygen. Simplifying to t–J model, calculations with on-site Coulomb repulsion suggest that if the system leads to a superconductor it should be singlet D-wave. Thus most physicists believe that ps provides the basis for a correct explanation of superconductivity. Goddard believed that if ps holes were correct, then it would lead to strong bonding to the singly occupied dx2-y2 orbitals on the adjacent Cu atoms, leading to a distortions that localize the state. This would cause a barrier to hopping to adjacent sites and hence would not be superconducting 46 Current canonical HighTc Hamiltonian Ops holeThe t–J Model Undoped Cu d9 hole is 3d x2–y2. O 2pσ doubly occupied. Heisenberg AF with Jdd = 0.13 eV for La2CuO4. H dd J dd Si S j Doped Doping creates hole in O pσ that bonds with x2–y2 to form a bonded singlet (doubly occupied hole). Singlet hole hopping through lattice prefers adjacent sites are same spin, this frustrates the normal AF coupling of d9 spins. J t J Because of Coulomb repulsion cannot have doubly occupied holes. 47 Summary of the t–J model Coulomb repulsion of singlet holes leads to singlet d-wave Cooper pairing. d-wave is observed in phase sensitive Josephson tunneling, in NMR spin relaxation (no Hebel-Slichter coherence peak), and in the temperature dependence of the penetration depth (λ~T2). t-J predicts an ARPES (angle-resolved photoemission) pseudogap which may have the right qualitative dependence. The t–J model has difficulty explaining most of the normal state properties (linear T resistivity, non-standard Drude relaxation, temperature dependent Hall effect, mid-IR optical absorption, and neutron ω/T scaling). 48 Universal Superconducting Tc Curve (What we must explain to have a credible theory) ≈ 0.16 Superconducting Phase ≈ 0.05 ≈ 0.27 Where do these three special doping values come from? 49 Basis for all theories of cuprate superconductors LDA Band calculations of La 2CuO4 (0,p) LDA and PBE lead to a half filled ps-dx2-y2 band; predicting that La2CuO4 is metallic! This is Fundamentally Wrong Experimental Band Gap is 2 eV Empty Fermi Energy Un-Occupied Un-Occupied Occupied Occupied G(0,0) (p,0) Occupied Occupied Un-Occupied ps-dx2-y2 band (p,p) Un-Occupied LDA: Freeman 1987, Mattheiss 1987, Pickett (1989) Occupied 50 G2(MAD) H-Ne TM He2 (H O) We want to doNe QM on cuprate D (R ) E E(R ) E(R ) superconductors. Which 1.09 Unbound Unbound 0.161(3.048) DFT Is most accurate? 0.19 functional 0.0011(2.993) 0.0043(3.125) 0.218(2.912) 2 2 2 Units: eV Method Hf IP EA PA Etot HF 6.47 1.036 1.158 0.15 4.49 G2 or best ab initio 0.07a 0.053a 0.057a 0.05a 1.59a 3.94a 0.665 0.749 0.27 6.67 BP86 0.88a 0.175 0.212 0.05 0.19 BLYP 0.31a 0.187 0.106 0.08 0.19 Validation of accuracy of DFT for 148 molecules (the G2 0.194(2.889) Set) with very 0.46 Unbound Unbound accurate experimental data 0.37e Unbound Unbound 0.181(2.952) BPW91 0.34a 0.163 0.094 0.05 0.16 0.60 PW91PW91 0.77 0.164 0.141 0.06 0.35 0.52 OLYP mPWPWf 0.20 0.65 0.185 0.161 0.133 0.122 1.38 0.05 0.16 0.38 PBEPBEg 0.74g 0.156 0.101 0.06 1.25 0.34 XLYPk 0.33 0.186 0.117 0.09 0.95 0.24 0.18 0.94 0.139 0.207 0.107 0.247 1.13 0.07 0.08 0.72 Unbound B3P86 m 0.78a 0.636 0.593 0.03 2.80 0.34 Unbound B3LYP n 0.13a 0.168 0.103 0.06 0.38 e 0.25LDA Unbound (3.9 eV B3PW91 o 0.15a 0.161 0.100 0.03 0.24 0.38 PW1PW p 0.23 0.160 0.114 0.04 0.30 0.30 0.0066(2.660) 0.0095(3.003) 0.227(2.884) mPW1PW q 0.17 0.160 0.118 0.04 0.16 0.31 0.0020(3.052) 0.0023(3.254) 0.199(2.898) PBE1PBE r 0.21g 0.162 0.126 0.04 1.09 B3LYP and X3LYP are most generally 0.30 0.0018(2.818) 0.0026(3.118) 0.216(2.896) X3LYP s 0.12 0.154 0.087 0.07 0.11 0.22 accurate DFT methods 0.0010(2.726) 0.0028(2.904) 0.216(2.908) Exptl. 0.00 0.000 0.000 0.00 0.00 0.00 0.0010(2.970)h 0.0036(3.091)h 0.236i(2.948 j) LDA (SVWN) GGA xc e e e O..O b c c d 0.54e 0.0109(2.377) 0.0231(2.595) 0.391(2.710) Unbound Unbound 0.156(2.946) H f = Heat of Formation (298K) 0.0100(2.645) 0.0137(3.016) 0.235(2.886) IP = Ionization Potential 0.0052(2.823) 0.0076(3.178) 0.194(2.911) EA = Electron affinity 0.0032(2.752) 0.0048(3.097) PA = Proton Affinity 0.222(2.899) 0.0010(2.805) 0.192(2.953) Etot = total0.0030(3.126) atomic energy Hybrid Methods O3LYP BH&HLYPl Data isUnbound the mean 0.214(2.905) average deviation from experiment Unbound 0.206(2.878) Unbound 0.198(2.926) error) and HF (6.5 eV error) Unbound Unbound 0.175(2.923) Useless for thermochemistry 51 U-B3LYP calculations of La2CuO4 U-B3LYP leads to an insulator (2eV band gap) with a doubled unit cell (one with up-spin Cu and the other down-spin) ky Empty Fermi Energy Band gap Occupied LDA 0.0 PBE 0.0 PW91 0.0 Hartree-Fock 17.0 B3LYP (unrestricted) 2.0 Experiment 2.0 Freeman et al. 1987, Mattheiss 1987, Pickett 1989 Tahir-Kheli and Goddard, 2006 Tahir-Kheli and Goddard, 2006 Harrison et al. 1999 Perry, Tahir-Kheli, Goddard Phys. Rev. B 63,144510(2001) 52 (Ginder et al. 1988) B3LYP leads to excellent band gaps for La2CuO4 whereas LDA predicts a metal! LDA 0.0 PBE 0.0 PW91 0.0 Hartree-Fock 17.0 B3LYP (unrestricted) 2.0 Experiment 2.0 (Freeman et al. 1987, Mattheiss 1987, Pickett 1989) Tahir-Kheli and Goddard, 2006 Tahir-Kheli and Goddard, 2006 (Harrison et al. 1999) (Perry, Tahir-Kheli, and Goddard 2001) (Ginder et al. 1988) Conclusion #1: To describe the states of the La2CuO4 antiferromagnet we must include some amount of true (Hartree-Fock) exchange. Plane wave based periodic codes do not allow this (Castep, CPMD, VASP, Vienna, Siesta). LDA or GGA is not sufficient (e.g. PBE, PW91, BLYP). Crystal does allow B3LYP and X3LYP. Conclusion #2: Unrestricted B3LYP (U-B3LYP) DFT gives an excellent description of the band gap. Hence, B3LYP should be useful for describing doped cuprates (both hole type and electron type). 53 B3LYP Works Well for Crystals with Transition Metals Too Expt. B3LYP Si 3.5 3.8 Diamond 5.5 5.8 GaAs 1.4 1.5 ZnO 3.4 3.2 Al2O3 9.0 8.5 Cr2O3 3.3 3.4 MgO 7.8 7.3 MnO 3.6 3.8 NiO 3.8 3.9 TiO2 3.0 3.4 FeS2 1.0 2.0 ZnS 3.7 3.5 Si B3LYP band structure 1.2eV 3.5eV Points are Expt., GW, and QMC Direct (Vertical) Bandgaps in eV B3LYP accurate for transition metals and band structures J. Muscat, A. Wander, and N. M. Harrison, Chem. Phys. Lett. 342, 397 (2001). 54 Undoped x=0.0 Band gap 2eV Density of States for Explicitly Doped La2–xSrxCuO4 using UB3LYP Use superlattice (2√2 x 2√2 x 1) with 8 primitive cells La15SrCu8O32 This allows antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to) HOMO LUMO The down-spin states show a clear localized hole with Opz-Cudz2 character Note exactly 0.125 doping leads to ordered supercell with small gap. Band gap Real system disordered holes + d9 spins x=0.125 Becomes conductor for x>0.06 Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002) 55 Now use B3LYP for La2–xSrxCuO4 B3LYP leads to a hole along the Sr-O-Cu Apical axis (z) the apical Polaron Use superlattices with 8 primitive cells La15SrCu8O32 allowing antiferromagnetic coupling of Cu atoms. Allows hole to localize (but not forced to) 0.26 Å Find extra hole localized on apical Cu and O atoms below the Sr site. 2/O pz Cu z This state has Cu character on the Cu and hole Opz character on the bridging O atom. dz2 Because dz2 not doubly occupied, the top O – Cu bond goes from 2.40 to 2.14 Å and O – Cu bond below Cu decreases 0.11 Å to 2.29 Å. The spin of the Cu dz2 and Cu dx2-y2 on the same atom are the same (d8 high spin) The singlet state is ~2.0 eV higher. Bottom line: the hole is NOT in the Cu-O plane (as assumed in ALL previous attempts to explain superconductivity of cuprates) Sr Perry, Tahir-Kheli, and Goddard; Phys. Rev. B 65, 144501 (2002)]. 0.11Å 56 Nature of the new hole induced by LaSr: the Apical Polaron Two holes Mulliken Populations 4 Planar O1 in CuO6 near Sr O2 O1 One hole O2’ 2 O1 per hole Located on apical Cu–O just below the Sr. We call this the Apical Polaron. 57 2nd LaSr polaron from Ab-Initio DFT: the Plaquette Polaron The Plaquette Polaron state is localized on Apical O pz + the four-site Cu plaquette above the Sr. It has Cu z2 hole de-localized apical O pz, Cu dz2, and planar O pσ character over the plane of four Cu atoms. over plaquette The Plaquette Polaron state is calculated to for low doping be 0.053 eV per 8 formula units above the 0.09 Å apical polaron state this is 0.1 Å 0.007 eV = 0.2 kcal/mol per Cu in the La0.875Sr0.125CuO4 cell. The apical O below the Sr shifts up 0.1 Å to a Cu – O bond distance of 2.50 Å (seen in Sr XAFS) leading to a plaquette state. The apical O below the plaquette Cu distance optimizes to a Cu – O bond distance of 2.29 Å. Sr Assumption: LaSr Doping leads to Plaquette Polarons. 58 The Plaquette Polaron States Consider 4 Cu-O dz2 orbitals and the 4 Ops orbitals. For the undoped system, there are 16 electrons in these 8 orbitals In the Plaquette Polaron, one electron is removed. This leads to a hole in either the Px or Py orbital (degenerate). 59 Real orbital view of Plaquette Polarons get two degerate states Main hole character Cu dz2 O pz = ps Sr Cu dz2 O pz = ps Sr 60 Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice (Ising) The Px plaquette is compatable with the left antiferromagnetic coupling of the d9 regions while the Py plaquette is compatable with the right antiferromagnetic coupling of the d9 regions. This Ising-like description is over-simplified. Must find ground state of Heisenberg system, including the Plaquette spin H dd J dd Si S j 61 The Chiral Coupling Term J CH S z 2 ( S1 S2 ) J CH S z 2 ( S2 S1 ) Chiral coupling twists the spins into a right or left-handed system. 62 An unusual degeneracy: complex linear combinations (e+=Px + i Py and e-=Px - i Py) It is natural to assume the hole wavefunction is Px or Py or some real linear combination of the two. The interaction of the polaron with the background d9 spins chooses the complex combinations. This implies each polaron has a clockwise or counter-clockwire current. This result is surprising because we usually think of spins as Ising spins rather as 3D quantum spins. -i -1 Px+iPy 1 i i -1 3D Quantum character of spins leads to Chiral Polarons. Px-iPy 1 -i 63 We obtain 3 types of Electrons 1. “Undoped” Cu AF d9 sites 2. 4-site polarons (out of plane) • Two types of polarons a) Surface polarons (neighboring) AF d9 sites b) Interior polarons (surrounded by other polarons) 3. x2-y2/pσ band electrons inside the percolating polaron swath (the “Doped” Cu sites) x2-y2/pσ band Surface Polaron Interior Polaron d9 AF 64 Total Spin Hamiltonian Time-Reversed Chiral Polarons J CH S z 2 ( S1 S2 ) The total spin Hamiltonian is, Htot = Hdd + Hpd + HCH. d9)–(AF d9) (AF spin coupling (AF d9)–(polaron) spin coupling Chiral coupling | Px iPy ; s | Px iPy ; s J CH J CH , Sz Sz . HCH is invariant under polaron time-reversal. Hdd is polaron time-reversal invariant. Hpd is not invariant. JpdSz•Sd – JpdSz•Sd Hpd splits the energy between time-reversed chiral polarons. The energy difference is on the order of Jpd ~ (8/4)Jdd=2Jdd ~ 0.28 eV. The energy difference between polarons with the same spin but different chiralities is on the order of 4JCH ~ 1.1 eV. The energy splitting between time-reversed polarons is largest for low doping because there are more d9 spins to induce the splitting. 65 Isolated Plaquette Polaron in a d9 sea. Dopant The Plaquette is pinned down by the Sr dopant. The Cu d9 sea leads to an insulator, just as for undoped 66 As increase doping, weaken AF coupling among Cud9 states. Above 5% get conductor. 1. “Undoped” Cu AF d9 sites 2. 4-site polarons (out of plane) • Two types of polarons a) Surface polarons (neighboring) AF d9 sites b) Interior polarons (surrounded by other polarons) 3. x2-y2/pσ band electrons inside the percolating polaron swath (the “Doped” Cu sites) x2-y2/pσ band Surface Polaron Interior Polaron d9 AF 67 Superconducting Pairing only on Surface Plaquettes 68 Assume Optimal Tc Maximum Surface Polarons per Volume experiment ≈ 0.16 1000 x 1000 lattice Sp 200 ensembles prediction Percolation threshold Becomes metallic Surface area of pairing leads to correct optimal doping 69 Predict maximum Tc • Given our Plaquette Theory of Cuprate Superconductors – Find a way to calculate Tc for different plaquette arrangements as a function of doping • The hope is to predict dopant configurations that could lead to increased Tc – Here, we show some math for the gap equations in the hope that there will be interest in the group to attack this large computational problem 70 The Goal • Compute Tc for different arrangements of dopings – Presumably, will find the prior argument of Tc peak near 0.16 to be correct with random dopings • The question is, is there a non-random doping distribution that can lead to a higher Tc prediction? – How much higher? – Increasing surface area by punching holes into metallic swath should increase Tc 71 D-Wave Pairing Local singlet Cooper pairing within a plaquette. The intermediate state of the plaquette is the time-reversed partner (P↓ P’↑). Only coupling that leads to pairing is spin-exchange coupling with x2y2/pσ electron. Sign of the wavefunction (from Pauli principle part), −i −1 P 1 i (r↑, r’↓, P↓) (r↓, r’↓, P’↑) (r↓, r’↑, P↓) (+) sign i −1 Spin-exchange matrix element part, If r and r’ on same diagonal, then (+) sign. If r and r’ along Cu-O bond, then (−) sign because P and P’ are time-reversed! One more (−) sign due to denominator in second-order perturbation (Eground–EI). Net (+) coupling attractive singlet + P’ −i 1 P↓ r’↑ r↓ − P’↑ (−) repulsive for singlet P, P’ energy splitting ~ Jdd maps to Debye energy in BCS Tc. D-Wave − + r↑ P↓ r’↓ 72 Nature of Pairing −V −V −V −V −V −V 73 Gap Equation in BCS Superconductors k k pairs with k k Scatters into (k ,k ) Matrix element V kk k k k s wave k d k x2-y2 74 BCS Ground State Wavefunction 1 H ( k )c c Vk 'k ck' ck ' ck ck 2 k 'k ks ks ks | G (uk v c c k k ' k ' )|0 uk2 vk2 1 k Vkk ' G | ck ck | G k' H HF ( 1 ( k )c c k ck' ck ' k ck ck 2 k 'k ks ks ks ) 75 Solution (Min F=E-TS) k k k u k vk 2 Ek 1 k u 1 2 Ek 2 k Ek k2 2k 1 k v 1 2 Ek 2 k k ' k Vkk ' (uk 'vk ' ) tanh 2kT k ' k ' k Vkk ' tanh 2 Ek ' 2kT 76 Going beyond the BCS Gap equation: the Bogolubov-De Gennes Equations In essence, rewrite the BCS pairing equations in Real-Space. This was originally developed to address the questions of non-uniform magnetic fields in type-II superconductors and also impurities (magnetic and non-magnetic). Leads to Ginzburg-Landau phenomenological theory of superconductivity in complex magnetic fields and disorder. −V −V −V −V −V −V 77 The Bogolubov-De Gennes Equations Allows us to have a pairing in real-space, V(r’,r) that leads to a gap that is a function of position Δ(r’,r). We can incorporate a spatially varying pairing at the interface between the d9 spins and the metallic swath and determine Tc as a function of doping by solving the real-space gap equation. If this leads to correct Tc(x) curve, it shows that the strong coupling Eliashberg formulation is unnecessary. -V neighboring pair attraction 78 B-dG Equations H c cRs t c cR 's Rs R Rs RR ' Rs RR ' 1 VRR'cR cR ' cR ' cR 2 RR' RR' (VRR' ) cRcR ' R 'R −V −V −V E u H u v n n R 0 n RR ' R ' n RR ' R ' R' −V −V −V E v H v u n n R 0 n RR ' R ' n RR ' R ' R' RR' 1 n n En n n (VRR' ) u R 'vR u R vR ' tanh 2 2kT 79 Computational Approaches • 2D need 1000 x 1000 lattice – Could start from 0.25 doping of plaquettes where there is no surface (Δ=0) and lower doping (Periodic Boundary Conditions) • Only need to get down to ~ 0.16 doping • If need 3D lattice, then could do 100 x 100 x 100, but this may be too discrete for band structure • Is there a Greens function approach? 80 Estimate of Maximum Tc Chemical Physics Letters 472 (2009) 153–165 To estimate Tc, use the formula from BCS theory Tc = 1.13 ħωD exp(-1/N(0)V) ħωD is Debye energy, N(0) is the density of states at the Fermi level, and V is the strength of the attractive coupling. In CPPP, the Debye energy is replaced by the scale of the energy splitting between opposite chirality plaquettes. For a plaquette surrounded on all four sides by d9 spins get ~ 2Jdd = 0.26 eV ~ 3000K. Expect range from Jdd/2 for one-side with d9 spin neighbors to 3Jdd/2 for case with three-side interfacing d9 spin neighbors Assume exponential term is ~ 1/10 as for A15 superconductors (Tc ~ 23K) Expect that Maximum Tc for a cuprate superconductor is in range of 0.05Jdd to 0.15Jdd or 150K to 450K. Current maximum of 138K may be 0.05Jdd case. Expect that Tc of ~ 300K might be attainable.. Using 100x100 supercell, self-consistent calculations for 100 random 16% doping cases we adjusted the d9-plaquette coupling to give gap Tc ~138K, then we chose specific doping patterns and calculate Tc. We have found cases with Tc > 200K. We expect to 81 predict optimum doping structure to have Tc > 200K. May be a challenge to synthesize. The Three Assumptions of the Chiral Plaquette Polaron Model Assumption 1: A polaron hole due to doping will be in a chiral combination of Px’ and Py’. Each chiral polaron has an orbital symmetry and a spin. Leads to neutron incommensurability and Hall effect. Assumption 2: A band is formed by Cu x2–y2/O pσ on the polaron sites when the polaron plaquettes percolate through the crystal. Leads to ARPES background. d9 Assumption 3a: Interaction of the undoped AF spins with the chiral polarons breaks the energy degeneracy between time-reversed chiral polarons. This leads to the D-wave superconducting pairing. 1 (Px ' iPy ' ) 2 E (Px ' iPy ' ;s ) E (Px ' iPy ' ; s ). Assumption 3b: Since the environment of each polaron is different, the distribution of energy splittings between the polaron states is uniform. Yields neutron scaling and linear resistivity. 82 Cuprate Superconductivity Puzzles Must all be explained by any correct theory Exp. Couples to Electron Charge Linear Resistivity ρ ~ T Drude scattering 1/τ ~ max(ω,T) Excess Mid-IR absorption Low temperature resistivity ~ log(T) Negative Magnetoresistance low T Superconductivity “Semi-conducting” c-axis resistivity Phase transition to superconductivity Hall Effect ~ 1/T (expect ~ constant) 2 2 Dx –y Gap Symmetry Hall Effect RH ~ const for field Evolution of Tc with doping in CuO2 plane. Co-existence of magnetism and Photoemission Pseudogap superconductivity Photoemission Background Large A successful theory must explain experiments from each category. Previous theories leave many of the very puzzling properties unexplained. The chiral plaquette paradigm based on out-of-plane holes explains all of these Exp. Couples to Electron Spin Neutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF) Cu, O different NMR relaxations Chiral plaquette polaron theory of cuprate superconductivity Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007) Explains each of these phenomena 83 Extra slides 84 What Characterizes a Superconductor? (The GAP) Metals are characterized by a Fermi surface and the existence of excited states of infinitesimal energy (LUMO-HOMO energy separation is zero). If Vk’k = –V < 0 (attraction), then can make “bonding” linear combinations of Cooper pairs in the ground state, Ψ ~ (k, –k) + (k’, –k’) + … that lowers the energy and forms a ground state that separates from the excited states. Leads to a gap Δ in the excitation spectrum. Coulomb repulsion is isotropic with VC > 0. Lower energy state cannot be formed. k pairs with k k k k k Scatters into (k ,k ) Matrix element V Bardeen, Cooper, Schrieffer (BCS) showed in 1957 kk that coupling through phonons can lead to a net attraction Electron distorts nuclei and moves away opposite momentum electron is attracted to distortion. Cooper Pairing Superconducting Gap, Δk s wave k 85 The Superconducting Gap Symmetry If Vk’k = –V < 0 (attraction), then can make “bonding” linear combinations of Cooper pairs in the ground state, Ψ ~ (k, –k) + (k’, –k’) + … k pairs with k k k The symmetry of the gap, Δk, is determined by the phase of the coefficients in Ψ. Δk = const =S-wave. A strong anisotropic repulsion for momentum change (π, π) can lead to a D-wave gap of x2y2 (p,p) symmetry. k’ +V k k Scatters into (k ,k ) Matrix element V kk Cooper Pairing Superconducting Gap, Δk k (0,0) s wave k d k x2- y2 Ψ ~ (k, –k) – (k’, –k’) D-Wave “anti-bonding” 86 Matrix Element for On-Site Coulomb Scattering |(π,0), (Px - iPy)> |(0,π), (Px + iPy)> Px + iPy Px - iPy -1 +i -1 -1 +i 1 -1 (π,0) (0,π) 1 1 -1 1 1 -i 1 -1 -i M ~ { (+1)(+1)* (+1) (+1) + (+1)(-i)* (-1) (-i) + (-1)(-1)* (-1) (-1) + (-1)(+i)* (+1) (+i) } = 4 (Max Value) Maximum repulsion occurs for (π,0)(0,π) for spin-exchange pairing through time-reversed polaron. D-WAVE [(x+iy) a, (x-iy) b [(x-iy) b, (x+iy) a] 87 Cuprate Superconductivity Puzzles Exp. Couples to Electron Charge Linear Resistivity ρ ~ T Drude scattering 1/τ ~ max(ω,T) Excess Mid-IR absorption Low temperature resistivity ~ log(T) Negative Magnetoresistance low T Superconductivity “Semi-conducting” c-axis resistivity Phase transition to superconductivity Hall Effect ~ 1/T (expect ~ constant) 2 2 Dx –y Gap Symmetry Hall Effect RH ~ const for field Evolution of Tc with doping in CuO2 plane. Co-existence of magnetism and Photoemission Pseudogap superconductivity Photoemission Background Large A successful theory must explain experiments from each category. Current theories leave many of the very puzzling properties unexplained. The chiral plaquette theory based on out-of-plane holes explains all of these Exp. Couples to Electron Spin Neutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF) Cu, O different NMR relaxations Chiral plaquette polaron theory of cuprate superconductivity Tahir-Kheli, Goddard; Phys. Rev. B 76: 014514 (2007) Explains each of these phenomena 88 Experimental Neutron Data x = 0.14 Raw Data Corrected Data For antiferromagnet expect peak in neutron scattering at (,p) Instead the peak is shifted by ±(2p/a)d and d increases with doping x (0,p) (p,p) shift δ from (,p) Doping G(0,0) (p,0) 89 Calculated spins (x = 0.10, JCH = 3Jdd) Red AF d9 regions, Green plaquette polarons 90 Computed Neutron Structure Factor x = 0.10 256 256 lattice. 5,000 ensembles. Pure AF would be peaked at (π,π) 1 Normalizat ion, S ( k ) 1. N k 91 Computed Neutron Structure Factor 256 256 lattice. 5,000 ensembles. Incommensurability ring around (π, π) of magnitude 2p x a (0,p) G(0,0) x = 0.075 (p,p) (p,0) x = 0.10 Correlation length approximately a . x x = 0.125 92 The Hall Effect in Metals and Semi-conductors Lorentz Force dp 1 e E v B . dt c Magnetic field deflects charge vT v Drude picture Band picture mv eE , J nev sE, s ne , m 2 v e 2 2 mv e B, v EB, c mc eE e v , E BE , m mc E 1 1 RH . J B nec dk 1 e E v B . dt c Fermi Surface In metals, n = constant RH = constant. In semi-conductors, n ~ T3/2 e–Δ/T. 93 Temperature Dependent Hall Effect in Cuprates La2–xSrxCuO4 x=0.15 get strong T dependence to RH. n=0.63 (per cell) n=0.16 (per cell) n=0.31 (per cell) x>0.34 get RH = constant. 94 Skew Scattering from Chiral Plaquette Polarons x2-y2 band scattering from a chiral polaron leads to a leftright scattering asymmetry that is an additive contribution to the Hall effect. Current flow When there is no magnetic field, B = 0, the number of “plus” and “minus” chiral polarons is equal and there is no net skew scattering. For non-zero B, the contribution to the Hall effect is proportional to the difference between the number of “plus” and “minus” chiral polarons and this is ~ 1/T. Px – i Py With chiral plaquette polarons, skew scattering definitely exists. Question #1: What is the temperature dependence of the skew scattering from chiral polarons? Question #2: What is its magnitude? Px + i Py 95 Calculating the Skew Scattering The chiral polaron difference is proportional to the magnetic field, B. Δ Δmax is the maximum timereversed polaron energy separation and Δmin is the minimum separation. Number of polarons Δmin and Δmax are larger for lower dopings where there are more undoped d9 spins to cause splitting. Δ+2gpμBB Difference between ± polarons 1 f ( ) b e 1 96 Comparison of Theory to Experiment Fitting experimental data to the form, RH = R0[f(Δmin) – f(Δmax)] 1 f ( ) b e 1 x Δmin Δmax R0 0.05 12.6K 791.5K 15.4 0.10 16.7K 454.1K 8.82 0.15 0.0K 378.9K 3.81 0.25 11.6K 106.3K 1.05 Answer #1: Skew scattering from chiral polarons leads to the observed temperature dependence. La2–xSrxCuO4 97 The Magnitude of Skew Scattering Δmin = 16.7K and Δmax = 454.1K from experimental fit. 1 x 0.10, U ( 1 eV ) 2.5 eV , 4x Smax 1.0 eV, Dmax 1.5 eV, U is enhanced because fewer electrons to screen the repulsion. Computed value g p 1, m 1, n 4 x, RH c 5.39 103 cm 3 / C , Experiment 4 103 cm 3 / C. Answer #2: The magnitude is right too. 98 Hall Effect for Magnetic Field in the CuO2 Plane Since the chiral current is in the plane, a magnetic field in the plane will not split the “clockwise” and “anticlockwise” plaquette energies. Current flow Leads to no temperature dependence of Hall effect for in-plane magnetic fields. Experimentally observed. Px – i Py Px + i Py 99 Conclusions PRB vol 76, 014514 (2007) Experiments Exp. Couples to Electron Spin Exp. Couples to Electron Charge Linear Resistivity ρ ~ T Drude scattering 1/τ ~ max(ω,T) Excess Mid-IR absorption Low temperature resistivity ~ log(T) Negative Magnetoresistance low T Superconductivity “Semi-conducting” c-axis resistivity Hall Effect ~ 1/T (expect ~ constant) Phase transition to superconductivity Hall Effect RH ~ const for field Dx2–y2 Gap Symmetry in CuO2 plane. Evolution of Tc with doping Photoemission Pseudogap Co-existence of magnetism and Photoemission Background Large superconductivity Neutron spin incommensurability Neutron spin ω/T scaling (expect ω/Jdd or ω/EF) Cu, O different 0NMR relaxations A successful theory must explain experiments from each category. Current theories do not, leaving many of the very puzzling properties unexplained. 100 The Chiral Polaron Magnetic Susceptibility By Fermi’s Golden Rule, the absorption rate is, occupation The power absorbed is, <P(ω)> = f ( ) f ( )W ( ) Bottom Line: Uniform ρ = Scaling occupation 1 f ( ) b . e 1 Probability distribution of polaron energy splittings. 101 Adding in the d9 and x2–y2 terms to obtain the total Magnetic Susceptibility χ(q, ω) ~ ω/EF for a metal and can be neglected. χ"(q, ω) ~ ω/Jdd for AF spins. Thus, the imaginary part of the d9 susceptibility can be neglected. The real part is constant for ω less than Jdd and may be taken to be constant for neutron scattering. All of the ω dependence of the total susceptibility comes from the polaron term, χp"(ω). The q dependence of the susceptibility is incommensurate from our calculations and of the form, a . x 102 The Total Magnetic Susceptibility in the Random Phase Approximation (RPA) Summing the diagrams, and solving, Scaling term 103 The Total Magnetic Susceptibility in the Random Phase Approximation (RPA) Using the integrals, Satisfies ω/T scaling and has the correct functional form. Experiment LSCO x = 0.04 a1 = 0.43 a3 = 10.5 104 Solving the Heisenberg Spin Hamiltonian in terms of two sets of spins A and B (non-linear sigma model) Map AF system onto Ferromagnetic system A B A B B A B A A B A B B A B A H J SiA S Bj J SiA ( S Bj ) ij ij Staggered magnetization W(r) Effective Continuum Hamiltonian This converts the problem of solving for the ground state into a problem of minimizing a continuous field with constraints H eff d 2 r (W x ) (W y ) (W z ) , 2 Minimize Constraint 2 2 W 2x W 2y W 2z 1, W( r ) (0,0,1), r . 105 Boundary Condition Topological Invariants H eff d 2 r (W x ) (W y ) (W z ) , 2 Minimize 2 2 W 2x W 2y W 2z 1, Constraint W( r ) (0,0,1), r . Boundary Condition Q Topological Invariant, 1 2 d r W ( x W y W), 4p States partition into discrete sets each characterized by an Integer Q described as the number of times the spin field covers sphere. This is a Winding number Fields with different Q cannot be continuously deformed into one another 1 dz n . 2pi ( z a ) 2 2 2 Px+iPy has Q=+1 and cannot be deformed to another Q. Px is a mixture of Q’s and can be deformed to Q=0. a n=0 H eff d 2 r (W x ) (W y ) (W z ) 8p | Q | . n=1 106 Heuristic Derivation of Incommensurability Magnitude dQ=(2p/a)x Our calculations find that the minimum spin configuration consists of undoped patches of d9 spins aligned antiferromagnetically, with the polarons acting to rotate the direction of the AF alignment of adjacent patches. Consider area A. The number of polarons in this area is Np=Ax/a2. The polaron rotates each adjacent spin on opposite sides of polaron by π or 2π total. If this is rigidly transmitted to patch, then rotation per spin is dq 2p/(1/x)= 2px. Net rotation per spin is Np dq. If area chosen such that net rotation per spin is 2π, then A=(a/x)2. Chiral coupling dominant A translation of L with L2=A returns to identical patch. L = a/x and dQ=(2p/a)x. 107 Coulomb Scattering of x2–y2 Band Electrons with Polarons Absorb pσ orbitals into “effective” A1 orbitals on Cu sites for simplicity. x2y2 band k state spin σ Polaron state spin s On-site with no spin exchange On-site with spin exchange (U > K > 0) Coefficients αR determine polaron state Fundamental Reason for Superconductivity is that Polaron is spread out over more than one site. 108 Matrix Element for Coulomb Scattering |k, S> |k’, Px + iPy> e i(kx+ky)a/2 +1 e -i(kx-ky)a/2 +1 e -i(kx’-ky’)a/2 +i S e i(kx’+ky’)a/2 +1 Px+iPy +1 +1 e -i(kx+ky)a/2 e i(kx-ky)a/2 -1 e -i(kx’+ky’)a/2 -i e i(kx’-ky’)a/2 M(q) ~ [e i(qx+qy)a/2 - e -i(qx+qy)a/2] - i [e i(qx-qy)a/2 - e -i(qx-qy)a/2] ~ 2i [ sin(qx+qy)a/2 – i sin(qx-qy)a/2 ] q = k’ - k S or D Max for q = (π, 0), (0, π) 109 Matrix Element for Coulomb Scattering |k, Px - iPy> |k’, Px + iPy> e i(kx+ky)a/2 +1 e -i(kx-ky)a/2 -i e -i(kx’-ky’)a/2 +i Px - iPy e i(kx’+ky’)a/2 +1 Px+iPy -1 +i e -i(kx+ky)a/2 e i(kx-ky)a/2 -1 e -i(kx’+ky’)a/2 -i e i(kx’-ky’)a/2 M(q) ~ [e i(qx+qy)a/2 + e -i(qx+qy)a/2] - [e i(qx-qy)a/2 + e -i(qx-qy)a/2] ~ 2[ cos(qx+qy)a/2 – cos(qx-qy)a/2 ] q = k’ - k Px – i Py Px + i Py Max for q = (π, π) 110 Coulomb Cooper Pairing Diagrams k’ k –k’ –k Intermediate Polaron change with no spin exchange Direct Coulomb band repulsion Matrix element +V Repulsion of pairs 111 Intermediate Polaron change with spin exchange Matrix Elements for no spin exchange: Attraction Attraction for states near Fermi level q = k’ – k M(q) depends on polaron states I and I’ and is the key to D-wave pairing. 112 Matrix Elements for spin exchange: Repulsion Minus sign due to interchange of (k’, –k’) in final state q = k’ + k Repulsion for states near Fermi level Δ = EI’ – EI M(q) depends on polaron states I and I’ 113 Anisotropic Pairing term |M(q)|2 Initial state, I = S, D, or Px’ ± i Py’ (initially occupied) Intermediate state, I’ = Px’ ± i Py’ a chiral polaron (unoccupied) I = S or D I = Px’ i Py’ Time-Reversed intermediate polaron I = I’ = Px’ ± i Py’ Only for spin exchange 114 Superconducting Pairing Possibilities Spin exch only (p,p) No Spin Exchange: q = k’ – k, Attraction plus isotropic Repulsion +V. For S, D, and ± combinations, leads to net repulsion for q 0. No superconductivity. Spin Exchange: q = k’ + k, Repulsion plus isotropic Repulsion +V. For S, D net repulsion for q (π,0). No good Fermi surface nesting. For ++, repulsion for q 0 or k’ –k. P-wave triplet pairing not possible with one continuous Fermi surface. Time-Reversed intermediate polaron, ±, extra repulsion for k’+k (±π, ± π) is compatible with D-wave. k’ k Γ k’+k (±π, ± π) 115 Debye Energy in BCS the Maximum Energy Splitting of Time-Reversed Polarons The pairing occurs for electrons within the Maximum Energy Splitting between Time-Reversed Polarons. This splitting arises due to coupling with the undoped d9 spin background and is on the order of Jdd. The time-reversed energy splitting is equivalent to the Debye energy in BCS superconductors. Δ = EI’ – EI 116 Evolution of Tc with Doping With Increasing Doping: The number of chiral polarons increases increase of Tc. The number of undoped d9 spins decreases decrease of the time-reversed polaron splitting decrease of Tc. The number of x2y2 band electrons near the Fermi level (Density of States) that can pair increases increase of Tc. Typical phase diagram La2-xSrxCuO4 Increase x2y2 band electrons increases the screening of the Coulomb repulsion decrease of Tc. Chiral Plaquette Polarons crowd together forming 1-Cu “Apical Polarons” with no superconducting pairing decrease of Tc. 117 Resistivity Temperature Dependence Polaron energy splitting Δmax If ρ(Δ) ~Δμ, then resistivity ~Tμ+1 118 Optical Absorption 2 p 1/ Res ( ) 2 2 4 p 1 / 2 / 2 2 2 ( 0 ) 2 / 2 Scattering rate Simple Drude Added oscillators 1/τ(ω) ~ T, for ω << T, 1/τ(ω) ~ ω, for ω >> T. In addition, there should be absorption due to polarons absorbing the photon and filling the unoccupied chiral state. Mid-IR. one-component fit 1 / ( ) 119 A Simple Model for Skew Scattering Inversion and Time-Reversal symmetry guarantees no skew scattering. w(kk’) = w(–k–k’) from inversion and w(–k–k’) = w(k’k) from time-reversal. An applied magnetic field breaks time-reversal. Consider x2y2 band scattering with a complex matrix element Vk’k. Scattering Hamiltonian Scattering T-Matrix Fermi’s Golden Rule Third order term changes sign when k and k’ are interchanged. 120 The Magnitude of Skew Scattering Δmin = 16.7K and Δmax = 454.1K from experimental fit. 1 x 0.10, U ( 1 eV ) 2.5 eV , 4x Smax 1.0 eV, Dmax 1.5 eV, U is enhanced because fewer electrons to screen the repulsion. Computed value g p 1, m 1, n 4 x, RH c 5.39 103 cm 3 / C , Experiment 4 103 cm 3 / C. Question #2: Does it have the right magnitude? YES 121 Calculating the Skew Scattering Scattering with Chiral Polaron Hole in P± Expanding to second order First and second order terms Mod-squared for scattering rate 122 Calculating the Skew Scattering For hole-doped cuprates, the polarons are holes and U < 0. For electron-doped cuprates, the polarons are electrons and U > 0. Invariant under interchange k’ k. Does not contribute to skew scattering term. q = k’ – k 123 Calculating the Skew Scattering Extracting the skew scattering contributions, Ak,k’ = –Ak’,k E± << ES, ED because E± ~ Jdd (time-reverse polaron energy difference) and ES, ED ~ hopping energies ~ 0.5-1 eV. Allows simplification of mean skew scattering. q = k’ – k 124 Calculating the Skew Scattering Mean skew scattering ρS = 1/ΔSmax = constant. ρD = 1/ΔDmax = constant. Must take mean over chiral polaron energy splittings λ = 0.19 for LSCO at x = 0.10 1 1 125 Temperature Dependent Hall Effect (p,p) Un-Occupied Occupied G Band Structures have hole-like Fermi surfaces leading to RH > 0 Fermi Surface 126 Log(T) Low-Temperature Resistivity La2–xSrxCuO4 x = 0.08 and 0.13 Tesla Negative Magnetoresistance 60 T magnetic field to suppress superconductivity “semi-conducting” ρc No log(T) log(T) 127 Kondo Effect Spin impurity in metal k p k p q q S+ S– S– S+ B = 0.671 Kelvin/Tesla log(T) resistivity Negative magnetoresistance Suppressed for T < 80K in 60T field Chirality + spin = effective “spin” impurity. If splitting in 60T small (small g-factor), then resistivity obtained 60 T 80 K g=2 No Kondo effect for “ferromagnetic” Coulomb coupling with polarons. 128 Kondo spin-flip from x2y2 band hopping on and off polaron Matrix element for hopping of k-state x2y2 onto A1 ~ cos(kx) – cos(ky). Second order band coupling to polaron spin is antiferromagnetic with maximum coupling for k (π, 0) and minimum for k along the diagonals. Resistivity out-of-plane is dominated by kvectors near (π, 0) and planar resistivity is dominated by k-vectors along the diagonal. Therefore, the Kondo resistivity will appear at a higher temperature for out-ofplane than in-plane leading to the “semiconducting” ρc. A1 orbitals No Kondo effect for “ferromagnetic” Coulomb coupling with polarons. 129 2D Plaquette Percolation and Polaron Islands Loss of Kondo effect (Insulator to Metal transition) 130 Angle-Resolved Photoemission (ARPES) Spectral Functions for k passing through Fermi level (k||, kz’) hn Occupied Surface Empty ARPES cuts off spectral function with Fermi factor (k||, kz) Parallel Momentum Conserved Fermi Level ARPES on cuprates finds a Fermi surface with a broad background even for k states far above the Fermi energy. 131 ARPES Pseudogap Underdoped Cuprates Pt reference spectra Gap ARPES taken at this k-vector Pseudogap for underdoped Gap closed 83K underdoped Bi2Sr2CaCu2O8+δ Pseudogap exists above Tc and closes as the temperature increases. Pseudogap is D-wave. 132 Pseudogap from undoped d9 AF coupling Antiferromagnetic fluctuations with momentum q of the undoped d9 spins induce mixing between k and k ± q. This leads to a decrease in the density of states for k-vectors at the Fermi level with either k+q or k–q near the Fermi level (nesting). This occurs for k near (π,0). q (π,π) (p,p) k+q Matrix Element V k (0,0) As doping increases, the number of d9 spins decreases, thereby decreasing the strength of the mixing V. This leads to a pseudogap that can only exist for underdoped cuprates. At fixed doping, AF fluctuations of the d9 spins decrease with increasing temperature, or V decreases to 0. The pseudogap closes. With percolating metallic paths, k is not a good quantum number leading to a large background. 133 Polarization Matrix Elements (Electric Dipole) kx = ky y kx = ky x kx = 0 Not Allowed Not Allowed E E E Allowed z2 kx = ky Not Allowed kx = ky Allowed E E Allowed kx = 0 E z2 134 Photoemission Process (Energy) 135 Computed Percolation Values Experiment: LSCO = 0.05 YBCO = 0.35 136 (LaSr)CuO4 Doping Constraints No Sr above and below plaquette No Sr separted by two lattice spacings on opposite sides of plane No neighboring Sr in plane No out-of-plane Sr neighbors No Sr separted by two lattice spacings 137 YBa2Cu3O7–d Doping Constraints Chain O always dopes same Cu triple Experiment: YBCO at x=0.35 Tc drops from ~92 to ~65K Chain O dopes Cu triple randomly , but not twice Adjacent chain O doping. Adjacent Cu triples not allowed 138 x2–y2/pσ Band Energy due to Incommensurability d9 spin ordering of momentum q hybridizes band electrons of momentum k and k+q with coupling energy V on the order of Jdd~0.1eV. This mixing perturbs the band energies and the total ground state energy. We calculate the energy change for the ring of incommensurate q vectors to find minimum. x = 0.10, ΓS = ΓD = 0.01 eV. eV q vector for Cu – O bond directions on BZ edge. Additional Umklapp scattering makes this direction favorable. 139 The Chiral Polaron Magnetic Susceptibility The low-energy excitation of a chiral plaquette polaron is the time-reversed polaron. For the q dependence of χp"(q, ω), the matrix element between the initial and final polaron state of the q momentum spin operator, Sz(–q)Sz(q), must be evaluated. This is largest for q = (π,π). Since the polaron is localized over a four-site plaquette, the q = (π,π) peak is substantially broadened. In the vicinity of (π,π), where the neutron spin scattering is largest, we may assume the polaron susceptibility, χp"(q, ω), is momentum independent, χp"(q, ω) χp"(ω). occupation occupation 1 f ( ) b . e 1 Px iPy ;s | S ( q) S ( q) | Px iPy ;s 1 z z q (p , p ). 140 Pseudogap from undoped d9 AF coupling Antiferromagnetic fluctuations with momentum q of the undoped d9 spins induce mixing between k and k ± q. The incommensurate q along the Cu– O bond direction has the lowest band energy and explained the observed neutron spin peaks. The same mixing leads to a decrease in the density of states for k-vectors at the Fermi level with either k+q or k–q near the Fermi level (nesting). This occurs for k near (π,0). As doping increases, the number of d9 spins decreases, thereby decreasing the strength of the mixing V. This leads to a pseudogap that can only exist for underdoped cuprates. At fixed doping, AF fluctuations of the d9 spins decrease with increasing temperature, or V decreases to 0. The pseudogap closes. With percolating metallic paths, k is not a good quantum number leading to a large background. 141 Thermopower 142 Density of States for Explicitly Doped La2–xSrxCuO4 using UB3LYP Undoped x=0.0 x=0.25 Band gap The down-spin states show a clear localized hole with Opz-Cudz2 character Highest hole states Lowest electron states x=0.5 Band gap x=0.125 Doped UB3LYP: Perry, Tahir-Kheli, and Goddard, Phys Rev B 65, 144501 (2002) 143 2D Plaquette Percolation and Polaron Islands Loss of Kondo effect (Insulator to Metal transition) 144 NMR Knight Shift Temperature Dependence: One-Component vs. Two-Component Theories For underdoped YBa2Cu3O6.63, the planar Cu and O Knight shifts have the same temperature dependence. Optimally doped YBa2Cu3O7 has constant Knight shifts. This leads to the conclusion that there is only one electronic component that hyperfine couples to the planar Cu and O atoms leading to the same temperature dependence. Since our model has x2y2 band electrons and chiral outof-plane polarons (twocomponents), the data appear to contradict our assumptions. Currently, no theory explains this temperature dependence. Takigawa et al, PRB 43, 247 (1991) Question 1: Does the data unequivocally imply one-component? NO. Question 2: Does the NMR kill twocomponent theories? NO. 145 Question 1: Does the data unequivocally imply one-component? NO. More recent data on optimally doped YBa2Cu3O7 finds the Y spin relaxation rate and Knight shifts have different temperature dependencies contradicting one-component models. Also, the ratio of the Cu-O to Cu-Cu nuclear coupling is too large to fit onecomponent models, requiring some antiferromagnetic correlations larger than the Cu-Cu distance. Nandor et al PRB 60, 6907 (1999) Pennington et al PRB 63,054513 (2001) 146 Question 2: Does the NMR kill two-component theories? NO. Chiral plaquette polarons lead to a susceptibility as seen in the Hall effect. This is compatible with high temperature Y shift. The x2y2 band electrons have a temperature independent Knight shift. Since the polarons are out-of-plane, their hyperfine coupling to the planar Cu and O are smaller than the x2y2 band. Nandor et al PRB 60, 6907 (1999) We expect a temperature dependence of the Knight shift in underdoped YBa2Cu3O6.63 due to the pseudogap in x2y2 band from coupling to our undoped d9 spins. Our NMR explanation is still incomplete at present. Computed Hall Effect 147 NMR Knight Shift Temperature Dependence: One-Component vs. Two-Component Theories For underdoped YBa2Cu3O6.63, the planar Cu and O Knight shifts have the same temperature dependence. Optimally doped YBa2Cu3O7 has constant Knight shifts. This leads to the conclusion that there is only one electronic component that hyperfine couples to the planar Cu and O atoms leading to the same temperature dependence. Since our model has x2y2 band electrons and chiral outof-plane polarons (twocomponents), the data appear to contradict our assumptions. Currently, no theory explains this temperature dependence. Takigawa et al, PRB 43, 247 (1991) Question 1: Does the data unequivocally imply one-component? NO. Question 2: Does the NMR kill twocomponent theories? NO. 148 Question 1: Does the data unequivocally imply one-component? NO. More recent data on optimally doped YBa2Cu3O7 finds the Y spin relaxation rate and Knight shifts have different temperature dependencies contradicting one-component models. Also, the ratio of the Cu-O to Cu-Cu nuclear coupling is too large to fit onecomponent models requiring some antiferromagnetic correlations larger than the Cu-Cu distance. Nandor et al PRB 60, 6907 (1999) Pennington et al PRB 63,054513 (2001) 149 Question 2: Does the NMR kill two-component theories? NO. Chiral plaquette polarons lead to a susceptibility as seen in the Hall effect. This is compatible with high temperature Y shift. The x2y2 band electrons have a temperature independent Knight shift. Since the polarons are out-of-plane, their hyperfine coupling to the planar Cu and O are smaller than the x2y2 band. Nandor et al PRB 60, 6907 (1999) The temperature dependence of the Knight shift in underdoped YBa2Cu3O6.63 is due to pseudogap in x2y2 band from coupling to our undoped d9 spins. Our NMR explanation is still incomplete at present. Computed Hall Effect 150 More Physical Percolation near 0.25 If we assume plaquettes are laid down uniformly, then leads to 0.25. If we assume more realistic scenario that plaquette are not as uniform, then will exhaust constraints before 0.25. Exhausted at 0.187. Then fill 3-site sites to 0.226. Then fill 2-site sites to 0.271! This is where any AF coupling disappears. 1-site exhausted at 0.316. 151 What is “Bold” about this Talk? And why is it a “Minefield” BOLD: We propose that out-of-plane (CuO2) hole orbitals are formed in the superconducting cuprates. We demonstrate this fact using ab-initio DFT calculations using B3LYP. We show that out-of-planes orbitals play a major role in the superconducting mechanism of these materials and can explain a large number of the “anomalous” properties. MINEFIELD: The field has assumed for the past 20 years that only in-plane orbitals are relevant. All theories are built on this assumption. Are chemist functionals are superior to physicist functionals for characterizing the electronic structure of these complex materials? 152 Spin-1/2 states along different axes z | a (q , ) cos q 2 i | a e sin q |b Θ 2 q q i | b (q , ) e sin | a cos | b 2 2 p φ y x 1 (a b ) α-spin along x-axis = | a (q , 0) 2 2 p p 1 (a ib ) α-spin along y-axis =| a (q , ) 2 2 2 This extra factor of i is due to quantum nature of spin and is reason for chirality. 153 Simplifying Plaquette Polaron orbitals t’ t 154 Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice + +1 pointing out-of-page z-axis pointing in xy-plane of page + +1 +1 z2/x2y2 triplet + +1 H + +1 +1 + + +1 + H +1 + +1 + 155 Coupling of plaquette spin to neighboring d9 antiferromagnetic lattice + +1 + +1 H H + =t +1 +1 =t + | a (q , ) cos q 2 + +1 + +1 + +1 i | a e sin p + +1 q 2 ~(1+i )t ~(1-i )t |b 1 (a b ) α-spin along x-axis =| a (q , 0) 2 2 p p 1 (a ib ) α-spin along y-axis =| a (q , ) 2 2 2 Px + i Py will lower energy. Jahn-Teller Distortion! 156 Net Effect of Chiral Polarons + +1 pointing out-of-page z-axis pointing in xy-plane of page + + +1 +1 z2/x2y2 triplet Px + i Py plaquette polaron moves d9 spins off z-axis and into xy-plane. Creates a right-handed coordinate system. +1 + +1 + 157 Chiral Polarons are distributed randomly throughout the crystal Δ = Energy difference between a polaron and its time-reversed partner. The random environment of the polarons leads to random energy the probability distribution of the energy splitting is uniform. Leads to an observable effect with neutron scattering. occupation Δ occupation 1 f ( ) b . e 1 Neutron inelastically scatters losing energy ħω Probability distribution of polaron energy splittings. If ħω <kT, then no absorption of energy since “up” and “down” polarons have equal occupations. Absorption when ħω>kT. Neutron energy absorption should scale as ω/T! 158 Neutron Spin ω/T Scaling Neutron spin scattering with momentum and energy change q and ω measures the imaginary part of the magnetic susceptibility, χ"(q, ω) (Absorption). χ"(q, ω) ~ ω/Jdd for antiferromagnetic spins. χ"(q, ω) ~ ω/EF for a metal. ∫d2q χ"(q, ω) ~ ω/T is observed. This is the local susceptibility. a1 = 0.43 a3 = 10.5 159 Chiral Polarons take AF order 180o out-of-phase A large JCH determines the local spin structure around the polaron. Further changes in the parameters does not affect the spin structure. Should lead to an observable effect in neutron spin scattering structure factor 160 Coulomb Cooper Pairing Diagrams Scattering through time-reversed polaron intermediate state 161 Matrix Element for Coulomb Scattering |k, Px - iPy> |k’, Px + iPy> e i(kx+ky)a/2 +1 e -i(kx-ky)a/2 -i e -i(kx’-ky’)a/2 +i Px - iPy e i(kx’+ky’)a/2 +1 Px+iPy -1 +i e -i(kx+ky)a/2 e i(kx-ky)a/2 -1 -i e -i(kx’+ky’)a/2 e i(kx’-ky’)a/2 M(q) ~ [e i(qx+qy)a/2 + e -i(qx+qy)a/2] - [e i(qx-qy)a/2 + e -i(qx-qy)a/2] ~ 2[ cos(qx+qy)a/2 – cos(qx-qy)a/2 ] q = k’ - k Px – i Py Px + i Py Max for q = (π, π) D-Wave Superconductor! 162 Resistivity Temperature and Doping Dependence Normal metal resistivity is usually dominated by phonons and is ~ T for high temperatures and ~ T5 for low temperatures (Bloch-Gruneisen form). There is an additional constant term due to scattering from impurities. Underdoped cuprates are non-linear for low T becoming linear at higher T. Near optimal doping, the resistivity is ~ T and approximately extrapolates to the origin (small constant impurity term ~ 10 μΩ-cm ). Overdoped cuprates become non-linear in T and acquire a constant impurity term. The magnitude of the resistivity is large, ~ 200 μΩ-cm for x = 0.15 at 300K. A typical metal is ~ 1 μΩ-cm at 300K. 163 Resistivity Temperature Dependence from Coulomb Scattering with Polarons 1 Ei 2 E f x2-y2 electron energies E f Ei ρ(Δ) = # polarons with energy splitting Δ. Constant Linear resistivity. Polaron energies Occupation of Ei f ( ), Band electron lifetime Polaron energy splitting density f ( ) 1 e b 1 There are 0T polarons available for scattering. 1 2p | M |2 N (0) d () f ()[1 f ( 2 )]d ( 2 1 ), (1 ) Constant density 2 2p 1 | M | N ( 0) 0 T (for 1 0) b1 1 e If ρ(Δ) ~Δμ, then resistivity ~Tμ+1. ρ(Δ) is doping dependent. 164

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