Theory of probing orbitonswith RIXS

Luuk AmentLorentz Institute, Leiden, the Netherlands

Giniyat KhaliullinMax-Planck-Institute FKF

Stuttgart, Germany

Jeroen van den BrinkLorentz InstituteLeiden, the Netherlands

Fiona ForteSalerno 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

LaMnO3

Kugel-Khomskii model

• Superexchange interaction involving spins and orbitals.– Orbitals are degenerate, no coupling to the lattice.– Orbitals determine overlap t J ~ t2/U

3d e2g

x2-y2

3z2-r2

3d e2g

x2-y2

3z2-r2

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.

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, 157401 (2006) LA & G. Khaliullin, to be published

YTiO3

• Ti has 3d t2g1 configuration

• Ferromagnetic Mott insulator atlow temperature: spin and chargedegrees of freedom frozen out

Ti

OY

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.

–‘In-plane’ hopping: only via one of the two 2p’s allowed.

x

z

Ti

YTiO3 - superexchange

• What are the possible hopping processes via oxygen?– ‘Out-of-plane’ hopping is symmetry forbidden.

Ti TiO

•Expand in t/U: Superexchange interaction, dependent on bond direction. O

y

x

z

Ti O

y

–Result: t2g orbitals are conserved and confined to their plane.

Ti

OY

YTiO3 - superexchange

3d t2g

Ti Ti

•Superexchange interaction dependent on bond direction.

xzxyyz

Ti

y-direction

YTiO3 - 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. B �68, 205109 (2003).

Condense:

• In analogy to magnons: collective excitations (orbitons) on top of the ordered ground state.

Indirect RIXS off YTiO3

Ti 2p level

Ti 3d eg level

res (~460 eV)

YTiO3

Measure energy and momentum transfer

Core hole couples to valence electrons via core hole potential

RIXS data on YTiO3

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

RIXS - mechanismsTwo mechanisms couple RIXS core hole to orbitons.

2p

3d t2g

Core hole

Mechanism 1: core hole potential shakes up t2g electrons

3d eg

S. Ishihara et al., PRB 62, 2338 (2000)

If core hole potential is not of A1g symmetry:

RIXS - mechanismsTwo mechanisms couple RIXS core hole to orbitons:

Mechanism 2: superexchange bond is modified

2p

3d t2g

3d eg

U

RIXS - mechanismsTwo mechanisms couple RIXS core hole to orbitons:

Core hole2p

3d t2g

3d eg

Core hole potential effectively lowers Hubbard U:

Mechanism 2: superexchange bond is modified

U-Uc

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

Polarization Multiplet structure

for example if = t2g 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 onlyEnhanced fluctuations,

create one- and two-orbitons

•Applies only to superexchange model of YTiO3.

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

Ph

ysic

s o

f YT

iO3

Lattice distortions

Super-exchange

Superexchange modification

Local orbital flip

?

??

Results

2-orbitoncontinuum

1-orbitonshoulder

Lattice distortions:(local dd-excitations)

E. Pavarini et al., New J. Phys. 7, 188 (2005)

Orbiton physics:

2-orbitoncontinuum

C. Ulrich et al., to be published

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 temperatureC. Ulrich et al., to be published

LaMnO3

eg

t2g

Mn

OLa

• Mn 3d4, 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!

LaMnO3 - Superexchange

eg

t2g

• eg orbitals order ‘antiferro-orbitally’:

• 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 - Single orbitons

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)

eg

Initial FinalIntermediate

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:

€

H ij0 = 3Ti

zTjz + Ti

xTjx ± 3 Ti

zTjx + Ti

xTjz

( )

Intermediate state Hamiltonianfor superexchange modification:

€

H int = H 0 + J H ijcore

ij

∑ si si+

with

€

H ijcore =η1H ij

0 +η 2 Tjz −Ti

x( ) m 3 Tj

z −Tiz

( )[ ]

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 LaMnO3

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 YTiO3 best explained with orbitons. Lattice distortion scenario doesn’t work.

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.

2 competing scenario’s

• Local excitations:No dispersion

Superexchange

3d t2g

Jahn-Teller

Vs.

• Collective excitations:Strong dispersion