Theory of probing orbitons with RIXS Luuk Ament Lorentz Institute, Leiden, the Netherlands Giniyat...
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![Page 1: Theory of probing orbitons with RIXS Luuk Ament Lorentz Institute, Leiden, the Netherlands Giniyat Khaliullin Max-Planck-Institute FKF Stuttgart, Germany.](https://reader035.fdocuments.in/reader035/viewer/2022062511/551a752555034643688b5237/html5/thumbnails/1.jpg)
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
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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
LaMnO3
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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 ~ t2/U
3d e2g
x2-y2
3z2-r2
3d e2g
x2-y2
3z2-r2
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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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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
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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.
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–‘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
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YTiO3 - superexchange
3d t2g
Ti Ti
•Superexchange interaction dependent on bond direction.
xzxyyz
Ti
y-direction
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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.
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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
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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
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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:
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RIXS - mechanismsTwo mechanisms couple RIXS core hole to orbitons:
Mechanism 2: superexchange bond is modified
2p
3d t2g
3d eg
U
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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)
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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)
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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)
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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
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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
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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!
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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).
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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
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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:
€
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
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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).
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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 YTiO3 best explained with orbitons. Lattice distortion scenario doesn’t work.
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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.
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2 competing scenario’s
• Local excitations:No dispersion
Superexchange
3d t2g
Jahn-Teller
Vs.
• Collective excitations:Strong dispersion