Theory of probing orbitons in YTiO

3

with Resonant Inelastic X-ray Scattering

L.J.P. Ament

1

, J. van den Brink

1

, G. Khaliullin

2

1 Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands 2 Max-Planck-Institut f ür Festkörperforschung, Heisenbergstr. 1, D-70569 Stuttgart, Germany

1. What is RIXS?

4. Cross Section: UCL expansion

RIXS = Resonant Inelastic X-ray Scattering Synchrotron X-rays The sum over intermediate states n is hard to compute.

For



> E

n

we can trade the sum over intermediate states n for a perturbation expansion in 1/



: the Ultrashort Core hole Lifetime (UCL) expansion: Valence level Momentum transfer q & Energy loss



Interesting sample

w

res (1-10 keV) Core level Core-hole couples to valence electrons via core-hole potential

2. What are orbitons and YTiO

3

?

Oxygen ions have filled 2p-shell.

Ti Ti ions have 1 electron in 3d t 2g We assume t 2g levels.

levels are degenerate: O Y

5. RIXS coupling mechanisms

How does the core hole affect orbitons? Two mechanisms: 1. Superexchange constant modified at core hole atom 3d t 2g 3d t 2g Core hole potential effectively lowers U: 2p Core hole 2. Core hole potential directly induces orbital flip 2p Core hole Both mechanisms change H. The latter also works for local model.

5. Comparison to experiment

Recently, Ulrich et al.

1

measured RIXS spectra of YTiO 3 at the SLS: YTiO 3 is a ferromagnetic Mott insulator at low T. Only orbital degree of freedom with superexchange interaction: Ti O Ti O Ti Many hopping terms are zero by symmetry of overlap integrals.

Note: direction-dependent superexchange interaction!

For instance, sites i, j are neighbors in x-direction: 2-orbiton continuum 1-orbiton shoulder Superexchange Hamiltonian leads to orbital ordered ground state with collective excitations: orbitons!

Big debate about existence of orbitons.

Opposite picture: local model of YTiO 3 where t 2g orbitals are split by coupling to the lattice, and orbital excitations are local.

3

3. RIXS Cross Section

Kramers-Heisenberg relation: Intermediate states Dipole operator with Core-hole exists a very short time:



≈ 0.5-3 eV Mechanism 2 gives very good agreement, only 1 orbiton ‘peak’ is too strong. For mechanism 1, we neglected polarization dependence.

Agreement of local model with data is very much worse (2-peak structure, temperature dependence, dispersion, intensity trend):

7. Conclusions

Comparing the two extreme models for YTiO 3 (local Vs.

collective excitations), the experimental RIXS spectra clearly favor the collec tive model over the local one. Raman scattering experiments reached exactly the same conclusion.

New RIXS measurements with momentum transfer along the [110]-direction could verify the existence of the 1-orbiton peak.

1 C. Ulrich, G. Ghiringhelli, M. Moretti, N. Pezzotte, T. Schmitt, L. J.P. Ament, G. Khaliullin, J. van den Brink, H. Roth, T. Lorenz and B. Keimer, to be published 2 G. Khaliullin and S. Okamoto, Phys. Rev. B 68 , 205109 (2003) 3 E. Pavarini et al., New J. Phys. 7 , 188 (2005)