Transcript Probing orbitons in YTiO3 with Resonant Inelastic X

Theory of probing orbitons
with RIXS
Luuk Ament
Lorentz Institute, Leiden, the Netherlands
Giniyat Khaliullin
Fiona Forte
Max-Planck-Institute FKF
Stuttgart, Germany
Salerno University
Salerno, Italy
Jeroen van den Brink
Lorentz Institute
Leiden, the Netherlands
Orbital ordering
LaMnO3
Goodenough (1963)
Orbital order in plane
Why do orbitals order?
1. Lattice distortion (Jahn-Teller)
2. Orbital and spin dependent superexchange
Kugel-Khomskii model
• Superexchange interaction involving spins and
orbitals.
– Orbitals are degenerate, no coupling to the lattice.
– Orbitals determine overlap t  J ~ t2/U
x2-y2
x2-y2
3d e2g
3z2-r2
3d e2g
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 at
low temperature: spin and charge
degrees of freedom frozen out
•
Ti
O
Y
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 with
very different orbital excitations.
YTiO3 - superexchange
• What 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.
z
Ti
O
Y
Ti
•
O y
Ti
x
Expand in t/U: Superexchange interaction,
dependent on bond direction.
O
Ti
YTiO3 - superexchange
•
Superexchange interaction dependent on bond direction.
y-direction
xz
xy
yz
3d t2g
Ti
Ti
Ti
YTiO3 - superexchange
• Superexchange 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.
Pictures from E. Saitoh et al., Nature 410, 180 (2001)
and Khaliullin et al., Phys. Rev. B•68, 205109 (2003).
Indirect RIXS off YTiO3
Measure energy and
momentum transfer
YTiO3
Ti 3d eg level
wres (~460 eV)
Ti 2p level
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:
C. Ulrich, et al., to be published
• 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)
RIXS - mechanisms
Two mechanisms couple RIXS core hole to orbitons.
3d eg
3d t2g
If core hole potential is
not of A1g symmetry:
2p
Core hole
Mechanism 1: core hole potential shakes up t2g electrons
S. Ishihara et al., PRB 62, 2338 (2000)
RIXS - mechanisms
Two mechanisms couple RIXS core hole to orbitons:
3d eg
U
3d t2g
2p
Mechanism 2: superexchange bond is modified
RIXS - mechanisms
Two mechanisms couple RIXS core hole to orbitons:
3d eg
3d t2g
U-Uc
Core hole potential effectively
lowers Hubbard U:
2p
Core hole
Mechanism 2: superexchange bond is modified
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
• Two RIXS mechanisms:
1. Coulomb-induced shakeup
Polarization
•
•
Multiplet
structure
(UCL):
J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006)
L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)
for example if  = t2g yz:
Transferred
momentum
can be obtained by cluster calculation. We take all
equal.
Mechanism applicable to both J-T and superexchange models.
Results
• Calculate effective scattering operator
• Two RIXS mechanisms:
(UCL):
J. van den Brink & M. van Veenendaal, EPL 73, 121 (2006)
L. Ament, F. Forte & J. van den Brink, PRB 75, 115118 (2007)
2. Superexchange bond modification
Enhanced fluctuations,
create one- and two-orbitons
•
Hamiltonian,
two-orbiton only
Applies only to superexchange model of YTiO3.
Results
C. Ulrich et al., to be published
Lattice distortions:
(local dd-excitations)
E. Pavarini et al., New J. Phys. 7, 188 (2005)
Orbiton physics:
RIXS Mechanism
Physics of YTiO3
Local
Lattice
distortions
Superexchange
2-orbiton Superexchange
orbital
flip
continuum
modification
?
?
2-orbiton
continuum
1-orbiton
shoulder
?
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 temperature
C. Ulrich et al., to be published
LaMnO3
• Mn 3d4, high-spin configuration:
Mn
eg
O
La
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!
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).
LaMnO3 - Single orbitons
Initial
Intermediate
Final
eg
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)

Orbitons in indirect RIXS
Orbital Hamiltonian:
Hij0  3TizTjz  TixTjx  3TizTjx  TixTjz 
J. van den Brink, F. Mack, P. Horsch and A. Oles, Phys. Rev. B. 59, 6795 (1999).
H int  H 0  J  Hijcore si si
Intermediate state Hamiltonian
for superexchange modification:
ij
with

H ijcore  1H ij0  2 T jz  Tix 


3 T jz  Tiz 
F. Forte, LA and J. van den Brink, Phys. Rev. Lett. 101, 106406 (2008).
S. Ishihara and S. Maekawa, PRB 62, 2338 (2000)
Results
Orbiton RIXS spectrum for LaMnO3
Two-orbiton continuum
One-orbiton peak
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
3d t2g
Jahn-Teller
• Local excitations:
No dispersion
Vs.
Superexchange
• Collective excitations:
Strong dispersion