Presentation on theme: "Theory of probing orbitons with RIXS"— Presentation transcript:
1 Theory of probing orbitons with RIXS Luuk Ament Lorentz Institute, Leiden, the NetherlandsGiniyat Khaliullin Max-Planck-Institute FKF Stuttgart, GermanyFiona Forte Salerno University Salerno, ItalyJeroen van den Brink Lorentz Institute Leiden, the Netherlands
2 Orbital ordering Why do orbitals order? LaMnO3 Orbital order in plane Goodenough (1963)Orbital order in planeWhy do orbitals order?Lattice distortion (Jahn-Teller)2. Orbital and spin dependent superexchange
3 Kugel-Khomskii modelSuperexchange interaction involving spins and orbitals.Orbitals are degenerate, no coupling to the lattice.Orbitals determine overlap t J ~ t2/Ux2-y23d e2gx2-y23z2-r23d e2g3z2-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 orbitonsSuperexchange: spins and orbitals entangle. Jahn-Teller: spins and orbitals decouple, orbitals frozen out at low T.
5 YTiO3 A 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.C. Ulrich et al., PRL 97, (2006)LA & G. Khaliullin, to be published
6 YTiO3 Ti has 3d t2g1 configuration Ferromagnetic Mott insulator at low temperature: spin and charge degrees of freedom frozen outTiOYTwo 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 with very different orbital excitations.
7 YTiO3 - superexchangeWhat 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.xzTixzTiTiOTiOYOyOyExpand in t/U: Superexchange interaction, dependent on bond direction.
8 YTiO3 - superexchangeSuperexchange interaction dependent on bond direction.y-directionxzxy3d t2gTiyzTi
9 YTiO3 - superexchangeSuperexchange Hamiltonian has an orbitally ordered ground state with 4 sublattices:Condense:In analogy to magnons: collective excitations (orbitons) on top of the ordered ground state.Orbiton gap induced by octahedron tilting: t_2g to e_g hopping -> nonconservation of orbital type.Pictures from E. Saitoh et al., Nature 410, 180 (2001) and Khaliullin et al., Phys. Rev. B68, (2003).
10 Indirect RIXS off YTiO3 YTiO3 wres (~460 eV) Measure energy and momentum transferYTiO3Ti 3d eg levelSay: electron into e_g, not t_2gwres (~460 eV)Ti 2p levelCore hole couples to valence electrons via core hole potential
11 RIXS data on YTiO3Low energy part for 3 momentum transfers q along -direction:C. Ulrich, et al., to be publishedSpectral 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, (2008)
12 RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons. 3d egIf core hole potential is not of A1g symmetry:3d t2g2pCore holeMechanism 1: core hole potential shakes up t2g electronsS. Ishihara et al., PRB 62, 2338 (2000)
13 RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: U 3d t2g3d egUMechanism 2: superexchange bond is modified
14 RIXS - mechanisms Two mechanisms couple RIXS core hole to orbitons: 3d t2g3d egCore hole potential effectively lowers Hubbard U:U-UcCore holeMechanism 2: superexchange bond is modifiedF. Forte et al., PRL 101, (2008) S. Ishihara et al., PRB 62, 2338 (2000)Magnons: J. Hill et al., PRL 100, (2008) J. Van den Brink, EPL 80, (2007)
15 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, (2007)Two RIXS mechanisms:1. Coulomb-induced shakeupPolarizationMultiplet structurefor example if = t2g yz:Transferred momentumH from three-band Hubbard modelcan be obtained by cluster calculation. We take all equal.Mechanism applicable to both J-T and superexchange models.
16 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, (2007)2. Superexchange bond modificationTwo RIXS mechanisms:Hamiltonian, two-orbiton onlyEnhanced fluctuations, create one- and two-orbitonsApplies only to superexchange model of YTiO3.
17 ? Results RIXS Mechanism Superexchange modification Local orbital flip C. Ulrich et al., to be publishedLattice distortions: (local dd-excitations)E. Pavarini et al., New J. Phys. 7, 188 (2005)Orbiton physics:RIXS MechanismPhysics of YTiO3Lattice distortionsSuper- exchangeSuperexchange modificationLocal orbital flip?2-orbiton continuum2-orbiton continuum1-orbiton shoulder
18 RIXS data on YTiO3 Temperature dependence Say: 250 meV peak strongly enhanced at low T - orbital dynamics strongly influenced by magnetic correlations, natural in Kugel-Khomskii model. Local dd-excitations are not expected to show strong T-dependence.Low-energy peak is magnon peak (corresponds to 16 meV magnons)Large increase of spectral weight in low-T ferromagnetic statePeaks sharpen at low temperatureC. Ulrich et al., to be published
19 LaMnO3 Mn 3d4, high-spin configuration: eg t2g 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!
20 LaMnO3 - Superexchange eg orbitals order ‘antiferro-orbitally’: eg t2g 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).
21 LaMnO3 - Single orbitons InitialIntermediateFinalegStress contrast with indirect magnetic RIXS: no change in S^z.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 Hamiltonian for superexchange modification:q=0 gives 0 intensity to first order in UCL, but not to second order.withF. Forte, LA and J. van den Brink, Phys. Rev. Lett. 101, (2008). S. Ishihara and S. Maekawa, PRB 62, 2338 (2000)
23 Results Orbiton RIXS spectrum for LaMnO3 Two-orbiton continuum Experiments: see nothing at low energy (down to ~100 meV).Look at dispersion of dd-excitations?One-orbiton peakF. Forte, L. Ament and J. van den Brink, Phys. Rev. Lett. 101, (2008).
24 ConclusionRIXS 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 LaMnO3 Probably 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 Vs. 2 competing scenario’s Jahn-Teller Superexchange 3d t2gVs.Jahn-TellerSuperexchangeKugel-Khomskii: strong coupling between spins and orbitals, leads to sensitivity of orbital sector to changes in magnetic sector.Local excitations: No dispersionCollective excitations: Strong dispersion