Orbital Physics in Transition Metal Oxides

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Orbital Physics in Transition Metal Oxides

• Wei-Cheng Lee

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Outline

• Background knowledge of transition metal oxides
• Modulation doping of Mott-Insulator-Mott-insulator
  heterostructure
• Orbital Physics

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d-Orbtials
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Cubic Perovskite AMO3
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Hubbard Model

• Full electronic Hamiltonian
• Considering only nearest-neighbor hopping and
  onsite Coulomb interaction on a lattice leads to
  Hubbard Model

G,A,C-type AFM, FM, PM
VERY COMPLICATED!!!!!!!
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Interesting Experiment
SrTiO3 ? Band Insulator LaTiO3 ? Mott
Instulator Both are AMO3 perovskite. Lattice
constants are almost the same.
Insulator Insulator Metal !!!!!
A. Ohtomo, et. al., Nature 419, 378 (2002)
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Theoretical Explanation
S. Okamoto and A.J. Millis, Nature (2004), PRB
(2005)
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Dynamical Mean Field Theory (DMFT)
Self-consistent conditions
Successes Mott transition, spectral function,
dynamical properties, etc.
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Mott-Insulator-Mott-Insulator Heterostructure
M
d1 Mott Insulator with smaller U
d1 Mott Insulator with larger U
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Types of Modulation Doping
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Hartree-Fock Results
G-type AFM
FM
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Thomas-Fermi Theory

• Thomas-Fermi relation

G-type AFM
FM
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Paramagnetic State
DMFT
HFT
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Summary

• Modulation doping induced by correlation gap
  occurs in this Mott-insulator-Mott-insulator
  heterostructure.
• Electron density distribution depends on both the
  Coulomb field and the on-site correlation.
• Reference Wei-Cheng Lee and A.H. MacDonald,
  Phys. Rev. B 74, 075106 (2006)

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The idea of modulation doping seems to work. Then
whats next?
Nothing but a lot of complications
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Details Neglected
Cubic Perovskite

• Lattice distortion (GdFeO3-type, JT)

Single Band
?Orbital degeneracy
Hopping between t2g
?Hopping through Oxygen
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Full d-p Model
T. Mizokawa and A. Fujimori, Phys. Rev. B 54,
5368 (1996)
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First Step

• Including orbital degeneracy first.
• First order lifting of the degeneracy caused by
  the crystal field.

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Example CMR Materials

• Pseudospin representation for eg orbitals
• Kugel-Khomskii spin-orbital model (neglecting
  Hunds coupling)

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Order From Disorder Mechanism

• Mean Field Theory
• Classical 3-d Neel order -gt energetically bad for
  orbital degrees of freedom.
• Solution -gt lower effective dimension spin order
  (increase Quantum fluctuations) to result in
  orbital ordering.

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t2g System

• Crystal field is large enough so that we just
  need to consider t2g orbitals
• There is something about the hopping terms

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Inactive Axis
Each orbital has two active axes and one inactive
axis.
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Systems with t2g Orbital Degeneracy

• Pseudospin representation for t2g orbitals

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Spin-Orbital Model for 3d1 t2g Systems

• Neglect the Hunds Coupling and lattice
  distortion
• Strong quantum fluctuations between spin and
  orbitals lead to complicated ground states ?
  Still not completely understood.

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GdFeO3-type Structure
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Details Matters
Distorted
Cubic
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Role of Oxygen Atoms
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LDA Results I Hopping Matrix
Effect of GdFeO3 lattice distortion 1. Activate
the inactive channels 2. Created anisotropic
crystal field (thus orbital order). 3. Reducing
the Rms value of hopping matrix (indicating Mott
transition)
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LDA Results II Low Energy Eigenstates
Close to atomic orbital
CaVO3
LaTiO3
More like orbital liquid
YTiO3
Orbital order with yz and xy
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LDA Results III Spin Exchange
LDADMFT shows that the occupation number of
lowest crystal field orbital 1gt is 0.91 for
LaTiO3 and 0.96 for YTiO3.
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LDA Results IV Effect of JT I
3 JT disappear in the pressure between 9- 14Gpa
YTiO3 with proper structure
JT changes crystal field a little bit but not
crucial. Thus JT is not that important in orbital
order.
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LDA Results IV JT Effect II
JT is of crucial importance for the magnetic
order.
More details about this LDA calculation can be
found at E. Pavarini, A. Yamasaki, J. Nuss, and
O.K. Andersen, New Journal of Physics 7, 188
(2005)
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Mean-Field Study I
Consider spin-Ferromagnetic state.
Ground State energy require Aij is negative along
each bond? Quadrupole moment is introduced.
G. Khaliullin and S. Okamoto, Phys. Rev. B 68,
205109 (2003)
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Mean-Field Study I - Contd
After doing local transformation and rotating the
quantization axis to 111
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Mean-Field Study I - Contd
Putting the above condensate operators into the
Hamiltonian and expanding the remaining operators
up to quadratic terms, we can obtain ground state
energy and orbiton dispersions.
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Mean-Field Study I - Contd
Results

• The small values of order parameters indicate
  strong quantum fluctuation.
• Stabilization of spin ferromagnetic state
  requires something else.
• Hoppings between different t2g bands induced by
  lattice distortion lead to orbital gap which
  stabilizes orbital orders.

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Mean-Field Study II
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General Structure of Spin-Orbital Model

• Superexchange model
• Mean-Field solution decouple spin and orbital
  degree of freedoms. ? Is this always true?

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Violation of Goodenough-Kanamori Rules
3d1
3d2
Exact solution of 4 sites on a chain PRL 96,
147205 (2006)
3d9
Spin-orbital entanglement is important in t2g
systems!!!
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Conclusions

• Delicate balance of energy scales in strongly
  correlation system makes them sensitive to
  details ? Experimentally interesting but
  theoretically annoying.
• A theory accounting for those details is
  necessary. An approach to find entangled
  classically spin-orbital ordered states is highly
  desired (maybe this could never be found).