Interplay between spin, charge, lattice and orbital degrees of freedom

1. Interplay between spin, charge, lattice and orbital degrees of freedom Lecture notes Les Houches June 2006 George Sawatzky

2. LSDA+U Simplified version : V.I. Anisimov et al., PRB 44, 943 (1991)

3. Czyzk et l PRB 49, 14211(1994) LSDA+U antiferromagnetic S=.8 Bohr magnetons, E gap = 1.65 eV LSDA+U LSDA LSDA+U also has no electron correlation Single Slater det. of Bloch states. No multiplets.

4. Problems with LDA+U for metallic systems Ferromagnetic Note U gap closes with doping No spectral weight transfer

6. Exact diagonalization in 1D • Hubbard 10 sites U=10t • U Gap increases with doping • Spectral weight is transfered from • upper H band to the lower H band Meinders et al, PRB 48, 3916 (1993)

7. N+1 N-1 N-1 N+1 Mott – Hubbard Spectral weight transfer U N N Remove one electron Create two addition states At low energy EF PES PES 2 2 EF EF

8. These particles block 2 or more states Bosons – block 0 states Fermions – block 1 state Integral of the low Energy spectral weight For electron addition if Hole doped (left) and Electron removal for e Doped (right side)

9. Eskes et al PRL 67, (1991) 1035 Meinders et al PRB 48, (1993) 3916

10. Exact diagonalization 1D Hubbard Meinders et al, PRB 48, 3916 (1993)

11. Don’t know of a rigorous Proof of Hubb---t,J (U>>w)

12. Spin charge separation in 1D Antiphase Domain wall Now the charge is free to move

13. Magnons and spinons in 1D Magnon S=1 Two spinons Spinons propagate via J Si+S-1+1

15. Similar is some sense to the 1D case it is proposed that one has 2D rivers of charge separating anti-phase domain walls. Charges can now fluctuate from left to right without costing J Anisimov, Zaanen ,Andersen, Kivelson,Emery-----

16. Closer to real systems

17. Oxides Remember at surfaces U is increased, Madelung is decreased, W is decreased

18. For divalent cations

19. If the charge transfer gap becomes negative we will get a strange metal This seems to happen in CrO2 where the O bands cross the Fermi driving the system to a half metallic ferromagnet Korotin PRL 80 (1998) 4305

20. 3 most frequently used methods • Anderson like impurity in a semiconducting host consisting of full O 2p bands and empty TM 4s bands including all multiplets Developed for oxides in early 1980’s, Zaanen, Sawtzky, Kotani, Gunnarson,----- • Cluster exact diagonalization methods. O cluster of the correct symmetry with TM in the center. Again include all multiplets crystal fields etc Developed for oxides in early 1980’s Fujimori, Sawatzky,Eskes, ------ • Dynamic Mean Field methods, CDMFT, DCA which to date do not include multiplets Developed in the late 1990’s: Kotliar, George, Vollhart---

21. Zaanen et al prl 55 418 (1985) Anderson impurity ansatz Like DMFT but not self consistant But also including all multiplet interactions Kondo resonance

22. To calculate the gap we calculate the ground state of the system with n,n-1, and n+1 d electrons Then the gap isE(Gap)= E(n-1)+E(n+1)-2E(n)

23. Two new complications • d(n) multiplets determined by Slater atomic integrals or Racah parameters A,B,C for d electrons. These determine Hund’s rules and magnetic moments • d-o(2p) hybridization ( d-p hoping int.) and the o(p)-o(p) hoping ( o 2p band width) determine crystal field splitting, superexchange , super transferred hyperfine fields etc.

24. More general multiband model Hamiltonian

25. We usually take U(pp) =0 although it is about 5 eV as • Measured with Auger but the O 2p band is usuallu fiull or • nearly full.

26. Ways to “screen “ or rather reduce Uor F0 U in Cu atom is 18eV in the solid 8eV In a polarizable medium “Solvation” in chemistry

28. Rest comes from bond Polarization involving O 2p and TM 4s states

29. As we remove or add d electrons charge moves from O(2p) to or from TM(4s) reducing the d electron Removal energy as well as the d electron addition energy. Reduces U effectively by about 6-8 eV. Recall though that these effects will yield satellites or incoherent Parts to the spectral function at energies corresponding to the O(2p)-TM(4s) energy splitting.

30. Note that B and C are only slightly reduced in the solid they do not involve changes in the local charge !!!

31. For the N-1 electron states we need d8, d9L, d10L2 where L denotes a hole in O 2p band. The d8 states exhibit multiplets

32. H. Eskes and G.A. Sawatzky PRL 61, 1415 (1988). Anderson Impurity calculation Zhang Rice singlet

33. J. Ghijsen et al Phys. Rev. B. 42, (1990) 2268. Photoemission spectrum of CuO Energy below Ef in eV

35. Example of two cluster calculations to obtain the parameters For a low energy theory ( single band Hubbard or tJ )

36. Eskes etal PRB 44,9656, (1991) 0 to 1 hole spectrum One of the Cu’s is d9 The other d10 in the Final state. Bonding Antibonding splitting Measure d-d hoping 1 hole to 2 holes final State is od9 on both Cu’s Triplet singlet Splitting yields super Exchange J 2 holes to 3 holes final state is d9 for both Cu’s Plus a hole on O forming A singlet (ZR) with one of The Cu’s . Splitting in red Yields the ZR-ZR hoping integral as in tJ

37. Need multiband models to describe TM compounds However numerous studies have shown that this can sometimes be reduced to an effective single band Hubbard model at least for highTc’s BUT ONLY FOR LOW ENERGY EXCITATIONS E<0.5eV Macridin et al Phys. Rev. B 71, 134527 (2005)

39. (Maximize spin)

43. Crystal and ligand field splittings Often about 0.5 eV In Oh symmetry

44. Eg-O2p hoping is 2 times as large as T2g-O-2p hoping Often about 1-2eV In Oxides

46. High Spin – Low Spin transition very common in Co(3+)(d6), as in LaCoO3, not so common in Fe(2+)(d6) Because of the smaller hybridization with O(2p)

47. Mixed valent system could lead to strange effects Such as spin blockade for charge transport and high thermoelectric powers

48. What would happen if 2Jh <10Dq<3Jh If we remove one electron from d6 we would go from S=0 in d6 to S=5/2 in d5. The “hole “ would carry a spin Of 5/2 as it moves in the d6 lattice.