1. Models in CM physics uses and misuses George Sawatzky ubc

2. Although we know the exact theory i.e. all interactions and elementary particles of importance in CM physics the many body nature of the problem makes a solution impossible and we resort to models to try to get an understanding of the diversity of physical properties. This is not only because of curiosity but also because we would like to optimize properties

3. Solids exhibit a Wide diversity of properties Metals: CrO2, Fe3O4 T>120K Insulators: Cr2O3, SrTiO3,CoO Semiconductors: Cu2O Semiconductor –metal: VO2,V2O3, Ti4O7 Superconductors: La(Sr)2CuO4, LiTiO4,NaxCoO2 Piezo and Ferroelectric: BaTiO3 Catalysts: Fe,Co,Ni Oxides Ferro and Ferri magnets: CrO2, gammaFe2O3 Antiferromagnets: alfa Fe2O3, MnO,NiO --- Properties depend in detail on composition and structure Take for example only the transition metal oxides

4. Atoms in a periodic array in solids We are interested in the potential Produced by the nuclei and the inner electrons on the outermost “Valence” electrons K2π/wave length Ef is the Fermi level up to which Each k state is filled with 2 electrons ONLY METALS !! Bloch Wilson 1937

5. More atomic like states for atoms in solids with large inter-atomic spacing compared to orbital radius Electrons can quantum mechanically Tunnel from atom to atom forming again Waves and bands of states but now the Bands are finite in width. If such a band is full ( 2 electrons per atom for S orbitals the material will be an insulator Because of a forbidden gap to the next band of states INSULATOR OR SEMICONDUCTOR Still rather boring since we have no magnetism and systems With an odd number of electrons per atom would all be metallic

6. One electron band theory Electrons are in delocalized states labeled by a wave vector k forming bands There are two electrons per k state ( spin up and down) (non magnetic) An even number of electrons per unit cell could yield either an insulating or metallic state but an odd number would always yield a metal Bloch Wilson theory of 1937 already falsified in 1937 Verwey and de Boer ( CoO is an insulator) and explained by Peierls ( stay at home principle for the d electrons coined by Herring )

7. Surely a lattice of H atoms separated by say 1 cm would not behave like a metal What have we forgotten ? The electron electron repulsive interaction

8. The hole can freely Propagate leading to A width The electron can freely Propagate leading to a width Largest coulomb Interaction is on site U Simplest model single band Hubbard Row of H atoms 1s orbitals only E gap = 12.9eV-W The actual motion of the Particles will turn out to be more complicated

9. For large U>>W One electron per site ----Insulator Low energy scale physics contains no charge fluctuations Spin fluctuations determine the low energy scale properties Can we project out the high energy scale? Heisenberg Spin Hamiltonian

10. NN EFEF PES U EFEF N-1 2 EFEF N+1 N-1 2 Doping a Mott – Hubbard system (1-x)/2x

11. x=0.0 x=0.1 x=0.2 x=0.3 x=0.4 x=0.5 x=0.6 x=0.7 x=0.8 x=0.9 Meinders et al, PRB 48, 3916 (1993) These states would be visible in a two particle addition spectral function

12. These particles block 2 or more states Bosons – block 0 states Fermions – block 1 state

13. Elfimov unpublished What would a mean field theory give you?

14. Sometimes we get so involved in the beauty and complexity of the model that we forget what the validating conditions were and use them outside of the range of validity

15. Remember that Transition metal compounds Consist of atoms on a lattice not a jelium The charge carriers and spins live on atoms The atoms or ions can be strongly polarizable Polarizability is very non uniform i.e. O2- is highly polarizable Cu2+ is not We cannot use conventional screening models to screen short range interactions

16. Hossain et al., Nature Physics 4, 527 (2008)

17. Correlated Electrons in a Solid J.Hubbard, Proc. Roy. Soc. London A 276, 238 (1963) ZSA, PRL 55, 418 (1985) If Δ < (W+w)/2  Self doped metal d n d n  d n-1 d n+1 U : p 6 d n  p 5 d n+1 Δ : U = E I TM – E A TM - Epol Δ = E I O – E A TM - Epol + δE M E I ionization energy E A electron affinity energy E M Madelung energy Cu (d 9 ) O (p 6 ) Epol depends on surroundings!!!

18. Cu2+ (d9) Impurity in CuO2 lattice Eskes PRL 61,1475 (1988) Zhang rice singlet Forms the lowest energy band for a lattice of “impurities” Other symmetry States at about 0.4 eV Below ZR

21. Is single band Hubbard justified for Cuprates? Zhang Rice PRB 1988 37,3759

22. Problem with ZR singlets The combination of O 2p states is not compatible with a band structure state The wave functions are non orthogonal From ZR PRL 37,3759 Note it goes to infinity at k=0, should we see it at Gamma in ARPES? Luckly it goes to 1 for K= Pi/2,Pi/2 and along the antiferromagnetic zone boundary where the doped holes go at low doping

23. Problems with ZR singlets As we dope the system the integrety of the ZR states disappears As we dope the system the ZR states strongly overlap forbidden by Pauli so they must change.

24. Effective Hamiltonians can be misleading Hubbard like models are based on the assumption that longer range coulomb interactions are screened and the short range on site interactions remain However U for the atom is about 20 eV but U as measured in the solid is only of order 5 eV HOW IS THIS POSSIBLE?

25. Coulomb interactions in solids How large is U ? How are short range interactions screened in solids?

26. I will show that The polarizability of anions results in a strong reduction of the Hubbard on site U The charged carriers living on transition metal ions are dressed by virtual electron hole excitations on the anions resulting in electronic polarons The nearest neighbor coulomb interactions can be either screened or antiscreened depending on the details of the structure

27. polarizability in TM compounds is very non uniform The dielectric constant is a function of r,r’,w and not only r-r’,w and so Is a function of q,q’,w Strong local field corrections for short range interactions Meinders et al PRB 52, 2484 (1995) Van den Brink et al PRL 75, 4658 (1995)

28. Reduction of onsite interactions and changing the nearest neighbor interactions with polarizable ions in a lattice We assume that the hole and electron move slowly compared to the response time of the polarizability of the atoms. Note the oppositely polarized atoms next to the hole and extra electron

29. So the reduction of the Hubbard U in a polarizable medium like this introduces a strong nextnn repulsive interaction. This changes our model!!

31. Note short range interactions are reduced “screened” and intermediate range interactions are enhanced or antiscreened-quite opposite to conventional wisdom in solid state physics Jeroen van den Brink Thesis U of Groningen 1997

32. Homogeneous Maxwell Equations  (r,r ’ ) —>  (r – r ’ ) —>  (q) Ok if polarizability is uniform In most correlated electron systems and molecular solids the polarizability is actually Very NONUNIFORM

33. In many solids the plarizability is very non uniform Short range interactions cannot be described in terms of Є(r-r’) but rather of Є(r,r’) and so we cannot use Є(q) to screen Rather than working with Є go back work in real space with polarizability Atomic plarizabilities are high frequency i.e. of order 5 or more eV. Most correlated systems involve narrow bands i.e. less than 2 eV and so the response of atomic polarizability to the motion of a charge in a narrow band is instantaneous. i.e Electrons are dressed by the polarizable medium and move like heavier polarons

34. + e ћ ћ  ћ ћ  e — PES (E I ) IPES (E A ) Full polarization can develop provided that Dynamic Response Time of the polarizable medium is faster than hopping time of the charge  E (polarizability) > W ;  E  MO energy splitting in molecules, plasma frequency in metals----- A Picture of Solvation of ions in a polarizable medium

35. Reduction of U due to polarizability of O2- (SOLVATION) U = E I TM – E A TM -2Epol E I ionization energy E A electron affinity energy Epol = 2 For 6 nn of O2- ~ 13eV For 4 nn As3- ~17eV ELECTONIC POLARON

36. What about intersite interaction V? For pnictides the Fe-As-Fe nn bond angle is ~70 degrees Therefore the contribution to V is attractive ~4 eV Fro the cuprates the Cu-O-Cu bond angle is 180 degrees therefor the repulsive interaction is enhanced.

37. Polarization cloud For Two charges on Neighboring Fe “ELECTRONIC BIPOLARON

38. Rough estimate Atomic or ionic polarizability ~volume Consider atom = nucleus at the center of a uniformly charge sphere of electrons In a field E a dipole moment is induced P=αE For Z=1 and 1 electron restoring force =

39. Concluding remarks Models are great but on applying them to real systems one should be aware of the approximations made to get to them In testing models one has to remain within the energy range excluding contributions from other states not included. Non uniform polarizabilities can introduce surprises with regard to short range coulomb interactions We would all be dead if it was not for solvation and so would weakly correlated electron systems

40. Single band model is only valid at low energy scales i.e. less than.5 eV!!! In doped systems

41. polarizability in TM compounds is very inhomogeneous The dielectric constant is a function of r,r’,w and not only r-r’,w and so Is a function of q,q’,w Strong local field corrections for short range interactions Meinders et al PRB 52, 2484 (1995) Van den Brink et al PRL 75, 4658 (1995) J. van den Brink and G.A. Sawatzky Non conventional screening of the Coulomb interaction In low dimensional and finite size systems. Europhysics Letters 50, 447 (2000) arXiv:0808.1390arXiv:0808.1390 Heavy anion solvation of polarity fluctuations G.A. Sawatzky, I.S. Elfimov, J. van den Brink, J. ZaanenG.A. SawatzkyI.S. ElfimovJ. van den BrinkJ. Zaanen arXiv: 0811.0214v1 Electronic polarons and bipolarons Mona Berciu, Ilya Elfimov and George A sawatzky

42. U for C60 Gas phase : Smalley I = 7.6 eV A = 2.65 eV E = 1.6 eV T 1u -H u U = I – A – E = 3.4 eV U [‘atomic’] = 3.4 eV Solid  Screening ---Solvation Z=12 [FCC] but smaller at surface E I = E I 0 – E p E A = E A 0 +E p effect: reduction I increase A U [‘solid’] = 1.6 eV Now: Compares well with our experiments !

43. polarizability in TM compounds is very non uniform The dielectric constant is a function of r,r’,w and not only r-r’,w and so Is a function of q,q’,w Strong local field corrections for short range interactions Meinders et al PRB 52, 2484 (1995) Van den Brink et al PRL 75, 4658 (1995) arXiv:0808.1390arXiv:0808.1390 Heavy anion solvation of polarity fluctuations in Pnictides G.A. Sawatzky, I.S. Elfimov, J. van den Brink, J. ZaanenG.A. SawatzkyI.S. ElfimovJ. van den BrinkJ. Zaanen arXiv:08110214v Electronic polarons and bipolarons in Fe-based superconductors Mona Berciu, Ilya Elfimov and George A. Sawatzky