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

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Orbital ordering Goodenough (1963) Why do orbitals order? 1.Lattice distortion (Jahn-Teller) 2. Orbital and spin dependent superexchange Orbital order in plane LaMnO 3

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Kugel-Khomskii model Superexchange interaction involving spins and orbitals. –Orbitals are degenerate, no coupling to the lattice. –Orbitals determine overlap t J ~ t 2 /U 3d e 2g x 2 -y 2 3z 2 -r 2 3d e 2g x 2 -y 2 3z 2 -r 2

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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.

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YTiO 3 A good candidate for orbitons. Why? t 2g orbitals: directed away from oxygen ions. No cooperative JT phase transition seen. TiO 6 octahedra are tilted, but only slightly deformed. Spin wave spectrum is isotropic. Raman data: temperature dependence. C. Ulrich et al., PRL 97, 157401 (2006)LA & G. Khaliullin, to be published

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YTiO 3 Ti has 3d t 2g 1 configuration Ferromagnetic Mott insulator at low temperature: spin and charge degrees of freedom frozen out Ti O Y Two scenario’s: Lattice distortions split t 2g orbitals. Orbital fluctuations dominate over Jahn-Teller distortions. Degenerate t 2g orbitals with superexchange interactions. Both models lead to orbital order, but with very different orbital excitations.

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–‘In-plane’ hopping: only via one of the two 2p ’s allowed. x z Ti YTiO 3 - superexchange What are the possible hopping processes via oxygen? –‘Out-of-plane’ hopping is symmetry forbidden. Ti O Expand in t/U: Superexchange interaction, dependent on bond direction. O y x z Ti O y –Result: t 2g orbitals are conserved and confined to their plane. Ti O Y

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YTiO 3 - superexchange 3d t 2g Ti Superexchange interaction dependent on bond direction. xz xy yz Ti y-direction

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YTiO 3 - superexchange Superexchange Hamiltonian has an orbitally ordered ground state with 4 sublattices: Pictures from E. Saitoh et al., Nature 410, 180 (2001) and Khaliullin et al., Phys. Rev. B68, 205109 (2003). Condense: In analogy to magnons: collective excitations (orbitons) on top of the ordered ground state.

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Indirect RIXS off YTiO 3 Ti 2p level Ti 3d e g level w res (~460 eV) YTiO 3 Measure energy and momentum transfer Core hole couples to valence electrons via core hole potential

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RIXS data on YTiO 3 Low energy part for 3 momentum transfers q along [001]-direction: 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) C. Ulrich, et al., to be published

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

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RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: Mechanism 2: superexchange bond is modified 2p 3d t 2g 3d e g U

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RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: Core hole 2p 3d t 2g 3d e g Core hole potential effectively lowers Hubbard U: Mechanism 2: superexchange bond is modified U-U c F. Forte et al., PRL 101, 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)

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Results Calculate effective scattering operator (UCL): Two RIXS mechanisms: 1. Coulomb-induced shakeup PolarizationMultiplet structure for example if = t 2g yz: Transferred momentum Mechanism applicable to both J-T and superexchange models. can be obtained by cluster calculation. We take all equal. J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)

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Results Calculate effective scattering operator (UCL): 2. Superexchange bond modification Two RIXS mechanisms: Hamiltonian, two-orbiton only Enhanced fluctuations, create one- and two-orbitons Applies only to superexchange model of YTiO 3. J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006) L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)

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RIXS Mechanism Physics of YTiO 3 Lattice distortions Super- exchange Superexchange modification Local orbital flip ? ?? Results 2-orbiton continuum 1-orbiton shoulder Lattice distortions: (local dd-excitations) E. Pavarini et al., New J. Phys. 7, 188 (2005) Orbiton physics: 2-orbiton continuum C. Ulrich et al., to be published

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RIXS data on YTiO 3 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

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LaMnO 3 egeg t 2g Mn O La Mn 3d 4, high-spin configuration: Mott insulator, A-type AFM at low temperature (FM layers). Kugel-Khomskii model without Hund’s rule coupling: To first order, orbitals of different layers decouple!

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LaMnO 3 - Superexchange egeg t 2g e g orbitals order ‘antiferro-orbitally’: Excitations: e g 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).

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LaMnO 3 - Single orbitons Looks like Heisenberg, but no conservation of T z. This leads to single orbiton excitations. J. van den Brink, P. Horsch, F. Mack & A. M. Oles, PRB 59, 6795 (1999) egeg Initial Final Intermediate

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

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One-orbiton peak Two-orbiton continuum Orbiton RIXS spectrum for LaMnO 3 Results F. Forte, L. Ament and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008).

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Conclusion RIXS is an excellent probe of orbital excitations, discrimination between Jahn-Teller and superexchange driven order is possible. RIXS data for YTiO 3 best explained with orbitons. Lattice distortion scenario doesn’t work.

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LaMnO 3 Probably Jahn-Teller dominated e g orbitals: directed towards oxygen ions leads to higher Jahn-Teller coupling than t 2g orbitals. Cooperative JT phase transition around T = 800 K. 2-sublattice orbital order below 800 K. Magnetic order sets in only below T N = 140 K. JT splitting E JT = 0.7 eV. Classical orbitals describe experimental data well.

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2 competing scenario’s Local excitations: No dispersion Superexchange 3d t 2g Jahn-Teller Vs. Collective excitations: Strong dispersion

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