Absence of Jahn−Teller transition in the hexagonal Ba3CuSb2O9 ...
Orbital Dynamics coupled with Jahn-Teller phonons in ...€¦ · Jahn-Teller Effect Adiabatic...
Transcript of Orbital Dynamics coupled with Jahn-Teller phonons in ...€¦ · Jahn-Teller Effect Adiabatic...
TOHOKU UNIVERSITY
Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System
1
Department of Physics, Tohoku University Joji Nasu
In collaboration with Sumio Ishihara
The 5th Scienceweb GCOE International Symposium
TOHOKU UNIVERSITY
Strongly correlated electron system
2
Charge
Spin Orbital
Lattice
Interplay
Internal degrees of freedom of electron
High-Tc superconductivity Colossal magneto-resistance effect Multi-ferroics Quantum spin liquid etc.
Exotic phenomena
TOHOKU UNIVERSITY
Orbital Degree of Freedom
3d
Crystalline field with high symmetry Orbital degeneracy
Orbital degree of freedom
(e.g. Perovskite Manganites)
eg orbitals
t2g orbitals
3
・Exchange interaction of electronic spins ・Anisotropy of electric conductance ・Coupling with lattice distortion
Anisotropy of electronic distribution
TOHOKU UNIVERSITY
Orbital Excitation -Orbiton-
4
Orbital order
Orbital wave (orbiton)
Spin order
Spin wave(magnon) up down
Spin Degree of freedom
up down
eg orbital degree of freedom
Orbital Pseudo spin
TOHOKU UNIVERSITY
Example for materials with orbital degree
5
LaMnO3: End material of colossal magnetoresistance manganites
KCuF3: the material with one-dimensional magnetism
eg
t2g
Mn3+
eg
t2g
Cu2+
hole
staggered orbital order
Superexchange interaction
TOHOKU UNIVERSITY
Observation of “Orbiton”
6
K. Ishii, et.al., PRB83,241101(R) (2011)
In KCuF3, Resonant Inelastic X-ray Scattering In LaMnO3, Raman scattering
E. Saitoh, et. al., Nature 410 180 (2001)
The collective excitation from orbital order
“Orbiton” has been observed.
TOHOKU UNIVERSITY
Difference between “Magnon” and “Orbiton”
7
Anisotropy of electronic distribution
1) Anisotropy of orbital interaction
2) Strong coupling with lattice distortion
“Orbiton” dispersion is anisotropic.
S. Ishihara and S. Maekawa, PRB62, 2338(2000)
In LaMnO3
Jahn-Teller effect
Orbital degree of freedom
In particular, Dynamical aspect of Jahn-Teller effect might affect orbital dynamics.
TOHOKU UNIVERSITY
Jahn-Teller Effect Ligand
Energy loss ∝ Q2 Q
Electron-lattice coupling
Lattice potential
Energy gain∝ Q
Lattice must be distorted!
8
“Jahn-Teller effect” = Coupling of orbital and lattice distortion
Metal ion
Dynamics of lattice Kinetic energy ∝ ∂2/∂Q2
Orbital dynamics
TOHOKU UNIVERSITY
Jahn-Teller Effect
Adiabatic potentials
9
“Any nonlinear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy”
Jahn-Teller Theorem
Orbital essentially couples with lattice
Low-energy dynamics along rotational direction is expected.
Vibronic Hamiltonian
Kinetic energy of lattice Lattice potential Electron-lattice coupling
TOHOKU UNIVERSITY
Weak coupling approach
10
Vibronic Hamiltonian
Neglected Orbiton-phonon hybridization
Repulsion between orbiton and phonon modes
J. van den Brink, PRL87, 19(2001)
Assumed ground state is unstable.
Energy of orbiton-phonon mode deceases.
Hybridization picture is not correct in strong coupling case.
Orbiton
Phonon
TOHOKU UNIVERSITY
Purpose
11
Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System
On-site dynamical Jahn-Teller effect Inter-site superexchange interaction
Unique excitation due to Jahn-Teller effect
originating from Kinetic energy of lattice vibration
TOHOKU UNIVERSITY
Model
12
:Orbital pseudo-spin operator
eg orbital degree of freedom + 2 kinds of phonons
Exchange interaction
Jahn-Teller coupling is local.
TOHOKU UNIVERSITY
Symmetry of Hamiltonian
13
If , the Hamiltonian is invariant under
and
Existence of Goldstone mode
Symmetry for infinitesimal rotation
Exchange interaction
TOHOKU UNIVERSITY
Our approach
14
Hamiltonian
On site part
Exchange part:
:Off-site part :On-site part
On-site Off-site
: Coordination number
Generalized spin wave approximation with mean-field approximation
Mean-field approximation under Ferro orbital order
TOHOKU UNIVERSITY
Solution of on-site Hamiltonian
15
On-site Off-site
Solve self-consistently
Eigenstates in
with
Exact evaluation of Jahn-Teller coupling
Exact diagonalization by using Householder method ~3000 dimensions ( )
TOHOKU UNIVERSITY
Inter-site Hamiltonian
16
:Projection operator
is expanded by using basis of
Eigenstates of
The states involving ground state are taken into account.
Approximation by bosons
Approximation for orbital operator
F. P. Onufrieva, Sov. Phys. JETP 89, 2270 (1985). N. Papanicolaou, Nucl. Phys. B 305, 367 (1988).
Generalization of spin-wave approximation
Introduction of N~3000 kinds of boson
TOHOKU UNIVERSITY
Diagonalization of Hamiltonian
17
Bosonic Hamiltonian
J. Colpa, Physica A93,327 (1978)
Generalized Bogoliubov transformation
:Bogoliubov boson
Diagonalization for system consisting of kinds of bosons
TOHOKU UNIVERSITY
Condition
18
Ferro orbital order
On one dimensional chain
:Bogoliubov vacuum
Dynamical susceptibility
We can easily generalize its dimension and assumed order.
: There should be Goldstone mode
TOHOKU UNIVERSITY
Static properties
19
• By introducing Jahn-Teller coupling, dynamical JT effect reduces orbital moment. (Ham’s reduction effect)
• In strong coupling limit, dynamical JT effect is proportional to 1/g2.
Non-monotonic behavior
• In all parameter region, the ferro-orbital order is stabilized.
Orbital order is the same regardless of JT coupling
TOHOKU UNIVERSITY
Weak coupling case
20
Gapped excitation
Violation of Goldstone theorem
Gapless excitation: consistent with Goldstone theorem
Orbital excitation mode by our method agrees with that by conventional spin-wave approximation
except for gapless feature.
White lines: Conventional spin-wave approximation where orbiton is hybridized with phonon
J. van den Brink, PRL87, 19(2001)
TOHOKU UNIVERSITY
Strong coupling case
21
Adiabatic planes
Frank-C
on
do
n
excitation
s
Gound state
Exci
ted
sta
tes
Local picture
Phonon sidebands appear around JT energy The flat dispersion Local excitation
TOHOKU UNIVERSITY
Dispersion of Phonon sidebands
22
Phonon sidebands
Large J
Dispersive sidebands
Amplitude is determined by exchange interaction J.
Excitations between adiabatic planes
Propagation of local excitations between adiabatic planes
Extent is determined by JT coupling g.
TOHOKU UNIVERSITY
Low-energy excitation
23
Adiabatic plane
Frank-Condon excitation:Orbital excitation with phonon sidebands
Low-energy excitation:Collective excitation involving lattice vibration
Orbital excitation involving lattice vibration within the lowest adiabatic plane
Frank-Condon excitation
TOHOKU UNIVERSITY
Low-energy effective model
24
Effective model on the lowest adiabatic plane in strong coupling limit (Born-Oppenheimer approximation)
Kinetic energy of rotational mode
Lattice dynamics is strongly coupled with orbital degree of freedom
Born-Oppenheimer approx.: Electronic wave-function
Vibration wave-function
TOHOKU UNIVERSITY
Low-energy effective model
25
Original model Effective model
Good agreement
Low-energy mode corresponds to collective mode of orbital-lattice coupled excitation
TOHOKU UNIVERSITY
Origin of low-energy excitation
26
Degeneracy along rotational direction
Adiabatic plane
Frank-Condon excitation
Orbital excitation involving lattice vibration
Degeneracy along rotational direction
Characteristic of Exe Jahn-Teller system with two kinds of phonons
Adiabatic potential
Orbital excitation involving lattice vibration
TOHOKU UNIVERSITY
Degree of freedom of phonon
27
2 kinds of phonons, Exe JT:
Degree of freedom of phonon Low-energy excitation
1 kind of phonon, Exb1 JT:
TOHOKU UNIVERSITY
Conclusion
28
Orbital Dynamics coupled with Jahn-Teller phonons in Strongly Correlated Electron System
Orbital excitation involving lattice vibration
Frank-Condon excitation with phonon sidebands
There are two kinds of excitations
• Dispersive due to superexchange interaction
• Orbital and lattice are strongly coupled.
• This excitation originates from degeneracy of adiabatic plane.