1. Physics of correlated electron materials: Experiments with photoelectron spectroscopy Ralph Claessen U Würzburg, Germany e-e- h Summer School on Ab-initio Many-Body Theory, San Sebastian, 25-07-2007

2. Outline: Photoemission of interacting electron systems Mott-Hubbard physics in transition metal oxides Correlation effects in 1D TiOCl: Challenges for ab initio many-body theory e-e- h

3. Angle-resolved photoelectron spectroscopy non-interacting electrons ARPES  band structure E(k) interacting electrons ARPES  spectral function

4. Photoemission: many-body effects E kin h electron-electron interaction photoelectron: "loss" of kinetic energy due to excitation energy stored in the remaining interacting system !

5. Many-body theory of photoemission Fermi´s Golden Rule for N -particle states: with N -electron ground state of energy E N, 0 N -electron excited state of energy E N, s, consisting of N-1 electrons in the solid and one free photoelectron of momentum and energy  in second quantization one-particle matrix element

6. Many-body theory of photoemission Fermi´s Golden Rule for N -particle states: SUDDEN APPROXIMATION: Factorization ! photoelectron s th eigenstate of remaining N-1 electron system Physical meaning: photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in

7. Many-body theory of photoemission Fermi´s Golden Rule for N -particle states: SUDDEN APPROXIMATION: Factorization ! photoelectron s th eigenstate of remaining N-1 electron system Physical meaning: photoelectron decouples from remaining system immediately after photoexcitation, before relaxation sets in

8. Many-body theory of photoemission If additionally M if ~ const in energy and k-range of interest: The ARPES signal is directly proportional to the single-particle spectral function single-particle Green´s function other electrons phonons spin excitations ?

9. L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970) Many-body effects in photoemission Example: Photoemission from the H 2 molecule E kin H2H2 E gg u*u*

10. L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970) Many-body effects in photoemission Example: Photoemission from the H 2 molecule E kin H2H2 E electrons couple to proton dynamics ! photoemission intensity: electronic-vibrational eigenstates of H 2 + : gg u*u*

11. L. Åsbrink, Chem. Phys. Lett. 7, 549 (1970) Many-body effects in photoemission Example: Photoemission from the H 2 molecule E kin H2H2 E Franck-Condon principle proton distance eneregy v' = 0 v = 0 v = 1 v = 2 ħ0ħ0 gg u*u*

12. Caveat: Effect of photoelectron lifetime ARPES intensity actually convolution of photohole and photoelectron spectral function h hh ee energy kk slope  tot assuming Lorentzian lineshapes the total width is given by ~ meV ~ eV spectrum dominated by photo- electron linewidth unless  low-dim systems !

13. Outline: Photoemission of interacting electron systems Mott-Hubbard physics in transition metal oxides Correlation effects in 1D TiOCl: Challenges for ab initio many-body theory e-e- h

14. Transition metal oxides oxides of the 3d transition metals: M = Ti, V, …,Ni, Cu basic building blocks: MO 6 octahedra electronic configuration: O 2s 2 p 6 = [Ne] M 3d n cubic perovskites perovskite-like anatas rutile spinel O 2- quasi-atomic, strongly localized

15. Hubbard model t U kinetic energy, itinerancy local Coulomb energy, localization

16. k-integrated spectral function for limiting cases (non-interacting bandwidth W  t ): U/W << 1 U/W >> 1  Hubbard model with half-filled band (n=1) d 1 configuration (Ti 3+, V 4+ ) A()A()   one-electron conduction band: metal  U atomic limit: Mott insulator W

17. Photoemission of a Mott insulator TiOCl O 2p / Cl 3p Ti 3d 1  U  d 1  d 0 LHB d 1  d 2 UHB

18. Bandwidth-controlled Mott transition dynamical mean-field theory band metal insulator evolution of quasiparticle peak for local self-energy  (  ) correlated metal dynamical mean-field theory of the Hubbard model

19. Photoemission of a correlated d 1 metal A. Fujimori et al., PRL 1992 LHB QP O 2p V 3d 1 incoherent weight coherent excitations LHB QP

20. Spectral evolution through the Mott transition A. Fujimori et al., PRL 1992 DMFT photoemission QP LHB QP LHB UHB

21. Surface effects in photoemission photoelectron mean free path ( E kin ) E kin ~ h A. Sekiyama et al., PRL 2004 CaVO 3 40 eV 275 eV 900 eV LHB QP surface bulk h (E kin )

22. Surface effects in photoemission A. Sekiyama et al., PRL 2004 CaVO 3 LHB QP  at surface reduced atomic coordination  effective bandwidth smaller: W surf < W bulk  surface stronger correlated: U / W surf >U / W bulk

23. Surface versus bulk: V 2 O 3 S.K. Mo et al., PRL 90, 186403 (2003) unit cell d ~ 8 Å (40 eV) ~ 5 Å surface (800 eV) ~ 15 Å (6 keV) ~ 50 Å G. Panaccione et al., PRL 97, 116401 (2006) soft x-ray PES (h ~ several 100 eV) hard x-ray PES (h ~ several keV)

24. Outline: Photoemission of interacting electron systems Mott-Hubbard physics in transition metal oxides Correlation effects in 1D TiOCl: Challenges for ab initio many-body theory e-e- h

25. Spectral function of a Fermi liquid Fermi liquid dressed quasiparticles non-interacting electrons bare particles EF=0EF=0 k0k0 kk energy k kFkF A(k,)A(k,)

26. E 0(k)0(k) k EFEF kFkF charge spin Electron-electron interaction in 1D metals  E F density of states 0.125 2 1.5 1 0.5  =  ~   charge spin Voit (1995) Schönhammer and Meden (1995) Tomonaga-Luttinger model:

27. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy tt U -J

28. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy strong coupling U >> t

29. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

30. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

31. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

32. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

33. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

34. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

35. Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy t J strong coupling U >> t

36. spinonholon Strongly coupled electrons: 1D Hubbard model t – hopping integral 1D atomic (or molecular) chain U – local Coulomb energy J  t 2 /U - magnetic exchange energy J strong coupling U >> t

37. Strongly coupled electrons: 1D Hubbard model t – hopping integral U – local Coulomb energy J  t 2 /U - magnetic exchange energy J J J strong coupling U >> t

38. Strongly coupled electrons: 1D Hubbard model t – hopping integral U – local Coulomb energy J  t 2 /U - magnetic exchange energy in D>1: heavy hole (quasiparticle) strong coupling U >> t QP

39. Strongly coupled electrons: 1D Hubbard model t – hopping integral U – local Coulomb energy J  t 2 /U - magnetic exchange energy spinonholon in 1D: spin-charge separation strong coupling U >> t

40. 1D Hubbard-Model: spectral function A(k,  ) energy relative to E F spinon holon charge ~ O (t) ~ O (J) spin momentum -  /2 -k F kFkF 3k F   /2 0 0 K. Penc et al. (1996): tJ-model J.M.P. Carmelo et al. (2002 / 2003): Bethe ansatz E. Jeckelmann et al. (2003): dynamical DMRG

41. TTF-TCNQ: An organic 1D metal strongly anisotropic conductivity  b /  a   b /  c ~1000

42. a d b c TCNQ-band: ARPES versus 1D Hubbard model band theory photoemissionmodel dynamical DMRG E. Jeckelmann et al., PRL 92, 256401 (2004) model parameters for TCNQ band: n = 0.59 (<1) U/t = 4.9 t  2t LDA (?)

43. TTF-TCNQ: low energy behavior ? ARPES @ k F Binding energy (eV) ~E 1/8 Tomonaga-Luttinger model: power law exponent for 1D Hubbard model: α  1/8 (~0.04) experiment: α ~ 1  electron-phonon interaction ?  long-range Coulomb interaction ?

44. TCNQ-band: non-local interaction L. Cano-Cortés et al., Eur. Phys. J. B 56, 173 (2007) on-site Coulomb energy U (screened):1.7 eV Hubbard model fit of PES data:1.9 eV BUT: nearest neighbor interaction V :0.9 eV  extended Hubbard model: V induces larger "band width", i.e. mimicks larger t ! also: Maekawa et al, PRB (2006) local spectral function:

45. Spin-charge separation in 1D Mott insulators B.J. Kim et al., Nature Physics 2, 397 (2006) ARPES on SrCuO 2 1D Hubbard model (n=1) H. Benthien and E. Jeckelmann, in Phys. Rev. B 72, 125127 (2005)

46. Outline: Photoemission of interacting electron systems Mott-Hubbard physics in transition metal oxides Correlation effects in 1D TiOCl: Challenges for ab initio many-body theory e-e- h

47. TiOCl: A low-dimensional Mott insulator configuration: Ti 3d 1  1e - /atom: Mott insulator  local spin s=1/2 Ti O Cl a b c b a (a) (b) t t´

48. TiOCl: A low-dimensional Mott insulator ? configuration: Ti 3d 1  1e - /atom: Mott insulator  local spin s=1/2  frustrated magnetism, resonating valence bond (RVB) physics ? Ti O Cl a b c b a (a) (b) t t´

49. Magnetic susceptibility: 1D physics High TBonner-Fisher behavior characteristic for 1D AF spin ½ chains Low Tspin gap formation of spin singlets due to a spin-Peierls transition ?

50. TiOCl: Electronic origin of 1D physics Seidel et al. (2003) Valenti et al. (2004) band theory (LDA+U):

51. Valence band: Photoemission vs. theory PRB 72, 125127 (2005) with T. Saha-Dasgupta, R. Valenti et al. O 2p / Cl 3p Ti 3d T = 370 K

52. Ti 3d PDOS: photoemission vs. theory PRB 72, 125127 (2005) cluster = Ti dimer T. Saha-Dasgupta, R. Valenti, A. Lichtenstein et al., submitted T = 370 K (QMC, T=1400K)

53. ARPES on Ti 3d band PRB 72, 125127 (2005) k T = 370 K

54. ARPES on Ti 3d band PRB 72, 125127 (2005) 1D Hubbard model DDMRG H. Benthien, E. Jeckelmann

55. TiOCl vs. TiOBr: effective dimensionality? TiOCl: W b ~ 4 x W a WaWa WbWb TiOBr: W b ~ W a

56. Doping a Mott insulator Oxide-based electronics 2DEG SrTiO 3 LaTiO 3 High-T c superconductors  field effect transistor (FET) doping x temperature metal insulator e.g., La 2-x Sr x CuO 4

57. Doping a Mott insulator: TiOCl Doping by intercalation van der Waals-gap Na, K

58. doped Hubbard model In situ doping of TiOCl with Na new states in the Mott gap Na exposure [min]

59. In situ doping of TiOCl with Na new states in the Mott gap but not metallic (?)  XXXX k || energy relative to chem. potential (eV) pristine TiOClNa-doped ARPES multiorbital and/or lattice (polaronic) effects ? t 2g U  cf U +  cf - J H

60. Summary Photoemission of interacting electron systems - (AR)PES probes single-particle excitation spectrum - Im G(k,  ) (generalized Franck-Condon effect) - required: Sudden Approximation, low dimensionality, constant matrix elements - pitfalls: surface effects, charging Transition metal oxides: - Hubbard model good starting point Correlation effects in 1D: - spin-charge separation on high energy scale Additional challenges for real materials: - orbital degrees of freedom - electron/spin-lattice coupling - magnetic frustration - doping of Mott insulators (  oxide-based electronics, FET,…) other electrons phonons spin excitations ?

61. Reading Photoemission of interacting electron systems: Theory L. Hedin and S. Lundqvist Effects of electron-electron and electron-phonon interactions on the one-elecron states of solids Vol. 23 of Solid State Physics Academic Press (1970) C.-O. Almbladh and L. Hedin Beyond the one-electron model / Many-body effects in atoms, molecules and solids in Vol. 1 of Handbook on Synchrotron Radiation North-Holland (1983) Photoemission of interacting electrons systens: Examples S. Hüfner (ed.) Very High Resolution Photoelectron Spectroscopy Springer (2007)