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Theory of probing orbitons with RIXS

Theory of probing orbitons with RIXS. Luuk Ament Lorentz Institute, Leiden, the Netherlands. Fiona Forte Salerno University Salerno, Italy. Jeroen van den Brink Lorentz Institute Leiden, the Netherlands. Giniyat Khaliullin Max-Planck-Institute FKF Stuttgart, Germany. Orbital ordering.

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Theory of probing orbitons with RIXS

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  1. Theory of probing orbitonswith RIXS Luuk AmentLorentz Institute, Leiden, the Netherlands Fiona ForteSalerno UniversitySalerno, Italy Jeroen van den BrinkLorentz InstituteLeiden, the Netherlands Giniyat KhaliullinMax-Planck-Institute FKFStuttgart, Germany

  2. Orbital ordering LaMnO3 Orbital order in plane Goodenough (1963) Why do orbitals order? • Lattice distortion (Jahn-Teller) • 2. Orbital and spin dependent superexchange

  3. x2-y2 3d e2g 3z2-r2 Kugel-Khomskii model • Superexchange interaction involving spins and orbitals. • Orbitals are degenerate, no coupling to the lattice. • Orbitals determine overlap t  J ~ t2/U x2-y2 3d e2g 3z2-r2

  4. Jahn-Teller Vs. Superexchange • Both lead to orbital order, so why is it interesting? • Excitations are very different!Local crystal field excitations Vs. dispersing orbitons • Superexchange: spins and orbitals entangle.Jahn-Teller: spins and orbitals decouple, orbitals frozen out at low T.

  5. C. Ulrich et al., PRL 97, 157401 (2006) LA & G. Khaliullin, to be published YTiO3A good candidate for orbitons. Why? • t2g orbitals: directed away from oxygen ions. • No cooperative JT phase transition seen. • TiO6 octahedra are tilted, but only slightly deformed. • Spin wave spectrum is isotropic. • Raman data: temperature dependence.

  6. Ti O Y YTiO3 • Ti has 3d t2g1 configuration • Ferromagnetic Mott insulator atlow temperature: spin and chargedegrees of freedom frozen out • Two scenario’s: • Lattice distortions split t2g orbitals. • Orbital fluctuations dominate over Jahn-Teller distortions.Degenerate t2g orbitals with superexchange interactions. • Both models lead to orbital order, but withvery different orbital excitations.

  7. z z y y O Ti Ti x Ti O Y Ti O O x Ti YTiO3 - superexchange • What are the possible hopping processes via oxygen? • ‘Out-of-plane’ hopping is symmetry forbidden. • ‘In-plane’ hopping: only via one of the two 2p’s allowed. • Result: t2g orbitals are conserved and confined to their plane. • Expand in t/U: Superexchange interaction, dependent on bond direction.

  8. 3d t2g Ti Ti YTiO3 - superexchange • Superexchange interaction dependent on bond direction. y-direction xz xy yz Ti

  9. In analogy to magnons: collective excitations (orbitons) on top of the ordered ground state. YTiO3 - superexchange • Superexchange Hamiltonian has an orbitally ordered ground state with 4 sublattices: Condense: Pictures from E. Saitoh et al., Nature 410, 180 (2001)and Khaliullin et al., Phys. Rev. B68, 205109 (2003).

  10. Indirect RIXS off YTiO3 Measure energy and momentum transfer YTiO3 Ti 3d eg level wres (~460 eV) Ti 2p level Core hole couples to valence electrons via core hole potential

  11. RIXS data on YTiO3 Low energy part for 3 momentum transfers q along [001]-direction: C. Ulrich, et al., to be published • Spectral weight increases with larger q. • Maximum of 250 meV peak shows little dispersion. • Multi-phonons? Multi-magnons? Orbital excitations? C. Ulrich, G. Ghiringhelli, L. Braicovich et al., PRB 77, 113102 (2008)

  12. RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons. 3d eg If core hole potential is not of A1g symmetry: 3d t2g 2p Core hole Mechanism 1: core hole potential shakes up t2g electrons S. Ishihara et al., PRB 62, 2338 (2000)

  13. 3d eg 3d t2g 2p RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: U Mechanism 2: superexchange bond is modified

  14. 3d eg Core hole potential effectively lowers Hubbard U: 3d t2g 2p RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: U-Uc Core hole Mechanism 2: superexchange bond is modified F. Forte et al., PRL101,106406 (2008) S. Ishihara et al., PRB 62, 2338 (2000) Magnons: J. Hill et al., PRL 100, 097001 (2008) J. Van den Brink, EPL 80, 47003 (2007)

  15. Two RIXS mechanisms: 1. Coulomb-induced shakeup for example if  = t2g yz: Transferred momentum Polarization Multiplet structure • can be obtained by cluster calculation. We take all equal. Results • Calculate effective scattering operator (UCL): J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007) • Mechanism applicable to both J-T and superexchange models.

  16. Two RIXS mechanisms: 2. Superexchange bond modification Hamiltonian,two-orbiton only Enhanced fluctuations,create one- and two-orbitons Results • Calculate effective scattering operator (UCL): J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007) • Applies only to superexchange model of YTiO3.

  17. 2-orbitoncontinuum 1-orbitonshoulder RIXS Mechanism Superexchange modification Local orbital flip ? Lattice distortions Orbiton physics: Physics of YTiO3 2-orbitoncontinuum ? ? Super-exchange Lattice distortions:(local dd-excitations) E. Pavarini et al., New J. Phys. 7, 188 (2005) Results C. Ulrich et al., to be published

  18. RIXS data on YTiO3 Temperature dependence • Low-energy peak is magnon peak (corresponds to 16 meV magnons) • Large increase of spectral weight in low-T ferromagnetic state • Peaks sharpen at low temperature C. Ulrich et al., to be published

  19. Mn O La • Kugel-Khomskii model without Hund’s rule coupling: • To first order,orbitals of different layers decouple! LaMnO3 • Mn 3d4, high-spin configuration: eg t2g • Mott insulator, A-type AFM at low temperature (FM layers).

  20. Excitations: eg orbital waves (orbitons) E. Saitoh et al., Nature 410, 180 (2001) J. van den Brink, F. Mack, P. Horsch and A. Oles, Phys. Rev. B. 59, 6795 (1999). LaMnO3 - Superexchange • eg orbitals order ‘antiferro-orbitally’: eg t2g

  21. LaMnO3 - Single orbitons Final Initial Intermediate eg Looks like Heisenberg, but no conservation of Tz. This leads to single orbiton excitations. J. van den Brink, P. Horsch, F. Mack & A. M. Oles, PRB 59, 6795 (1999)

  22. Orbitons in indirect RIXS Orbital Hamiltonian: J. van den Brink, F. Mack, P. Horsch and A. Oles, Phys. Rev. B. 59, 6795 (1999). Intermediate state Hamiltonianfor superexchange modification: with F. Forte, LA and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008).S. Ishihara and S. Maekawa, PRB 62, 2338 (2000)

  23. Results Orbiton RIXS spectrum for LaMnO3 Two-orbiton continuum One-orbiton peak F. Forte, L. Ament and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008).

  24. Conclusion • RIXS is an excellent probe of orbital excitations, discrimination between Jahn-Teller and superexchange driven order is possible. • RIXS data for YTiO3 best explained with orbitons. Lattice distortion scenario doesn’t work.

  25. LaMnO3Probably Jahn-Teller dominated • eg orbitals: directed towards oxygen ions leads to higher Jahn-Teller coupling than t2g orbitals. • Cooperative JT phase transition around T = 800 K.2-sublattice orbital order below 800 K.Magnetic order sets in only below TN = 140 K. • JT splitting EJT = 0.7 eV.Classical orbitals describe experimental data well.

  26. 3d t2g 2 competing scenario’s • Local excitations:No dispersion Vs. Jahn-Teller Superexchange • Collective excitations:Strong dispersion

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