Presentation on theme: "Theory of probing orbitons with RIXS Luuk Ament Lorentz Institute, Leiden, the Netherlands Giniyat Khaliullin Max-Planck-Institute FKF Stuttgart, Germany."— Presentation transcript:
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
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
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
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.
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
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.
–‘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
YTiO 3 - superexchange 3d t 2g Ti Superexchange interaction dependent on bond direction. xz xy yz Ti y-direction
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.
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
RIXS data on YTiO 3 Low energy part for 3 momentum transfers q along -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
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:
RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: Mechanism 2: superexchange bond is modified 2p 3d t 2g 3d e g U
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)
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)
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)
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
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
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!
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).
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
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
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).
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.
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.
2 competing scenario’s Local excitations: No dispersion Superexchange 3d t 2g Jahn-Teller Vs. Collective excitations: Strong dispersion