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Hiroshima Nov 2006 Physics of S*L in Magnetic States If spin orientations are fixed due to magnetic ordering, S*L ~ *L acts like Zeeman field and polarize the orbital states. Some orbitals are more occupied than others. The results may be polarization of the electronic wave function.
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Hiroshima Nov
2006
Electric Polarization induced by Magnetic order
Jung Hoon Han Sung Kyun Kwan U. (SKKU)
Korea
Collaboration Chenglong Jia (KIAS)
Shigeki Onoda (U. Tokyo)Naoto Nagaosa (U. Tokyo)
Cond-mat/0608411, PRB
Hiroshima Nov
2006
Spin-orbit coupling is usually a minor, relativistic phenomenon in solids. However, some things can only happen due to SO. Examples are
• Anomalous Hall Effect (->Spintronics)• Spin Hall Effect (->Spintronics)• Etc.
Most recently we might have a new item on the list:
• Magnetism-induced Electric Polarization
S*L in solid state physics
Hiroshima Nov
2006
Physics of S*L in Magnetic States
If spin orientations are fixed due to magnetic ordering,
S*L ~ <S>*L
acts like Zeeman field and polarize the orbital states. Some orbitals are more occupied than others.
The results may be polarization of the electronic wave function.
Hiroshima Nov
2006
Ni3V2O8
Lawes et al PRL 05
TbMnO3
Kenzelman et al PRL 05
Collinear to non-collinear spin transition accompanied by onset of polarization
Hiroshima Nov
2006
CoCr2O4
Tokura group PRL 06
Co spins have ferromagnetic + spiral (conical) componentsEmergence of spiral component accompanied by P
Hiroshima Nov
2006
Phenomenon seems rather general.
Can we understand this on simple theoretical ground?
What is the origin/mechanism of non-collinear spin inducing electric polarization?
What is polarization due to? Ionic shift, or wave function polarization?
Key Issues of Multiferroic Materials
Magnetism-induced Ferroelectricity
Hiroshima Nov
2006
Step 1:
Ginzburg-Landau theory
Hiroshima Nov
2006
Spin <S> and polarization <R> break different symmetries:
<S> breaks time-inversion symmetry <R> breaks space-inversion symmetry
Naively, lowest-order coupling occurs at <S>2 <R>2 .
Lower-order terms involving spatial gradient, <S>2 <grad R> or <S><grad S><R>, are possible.
Spin-polarization Coupling via GL Theory
Hiroshima Nov
2006
Spin-polarization coupling via GL theory
Generally one can write down GL terms like
that result in the induced polarization
Mostovoy PRL 06
in the presence of magnetic ordering
For instance…
Hiroshima Nov
2006
Spin-polarization Coupling via GL Theory
For spiral spins
induced polarization has a uniform component given by
Mostovoy PRL 06
Hiroshima Nov
2006
Uniform induced polarization depends on the product M1 M2
- Collinear (M1M2=0) spin cannot induce polarization - Only non-collinear, spiral spins have a chance
Quite consistent with experimental facts
No microscopic derivation of Mostovoy’s free energy exists yet.
Spin-polarization Coupling via GL Theory
Hiroshima Nov
2006
Step 2:
Microscopic theory &Mechanism
Hiroshima Nov
2006
At the microscopic level, one can write down spin-polarization interaction terms such as
Fully consistent with symmetries
Consistent with GL theory
Pij
Si Sj
Hiroshima Nov
2006
Where does it come from?
Hiroshima Nov
2006
Microscopic Theory of Spin-induced Polarization
A linear chain consisting of alternating M(agnetic) and O(xygen) atoms is a reasonable model for magneto-electric insulators.
M O M O M O M O M O M O M O M
The building block is a single M-O-M cluster.
One tries to solve this model as exactly as possible to see if noncollinear-spin-induced polarization can be understood.
Hiroshima Nov
2006
Microscopic Theory – Further Details
In magnetic atoms, d-orbital electrons are responsible for magnetism.Keep the outermost d-orbitals and truncate out the rest.
Five-fold d-orbitals are further split into 3 t2g and 2 eg orbitals with a large energy gap of a few eV due to crystal field effects. Keep the t2g or eg levels only.
M O M
Crystal Field
d
eg
t2g
Magnetic
Oxygen
Hiroshima Nov
2006
Microscopic Theory – Further Details
Electrons can “hop” between M and O sites as represented by a hopping integral V.
KEY ELEMENT: SPIN-ORBIT INTERACTION
Each magnetic site is subject to spin-orbit interaction. If the spin state is polarized, so is the orbital state.
M O M
Hiroshima Nov
2006
Theory of Katsura, Nagaosa, Balatsky (KNB)
The cluster Hamiltonian
KNB Hamiltonian is solved assuming (spin-orbit) > U (Hund)
,, , ,
, ,
, , , ,
. .
SO M O V
SO
M a a la r l l xy yz zx
O p b bb x y z
V l xy y l zx z r xy y r zx z
H H H H HH S L
H U m S
H E p p
H V d p d p d p d p h c
KNB PRL 05
Hiroshima Nov
2006
Results of KNB
Polarization orthogonal to the spin rotation axis and modulation wave vector develops
Hiroshima Nov
2006
Results of KNB
The results may be generalized to the lattice case; consistent with phenomenological theories
RED: spin orientationBLACK: polarization
Hiroshima Nov
2006
Our Recent Results
We revisited the KNB Hamiltonian in the limit of large Hund coupling U and small spin-orbit interaction , which is presumably more realistic.
A new (longitudinal) component of the polarization is found which was absent in past theories which only predicted transverse polarization.
BEFORE
AFTER
Hiroshima Nov
2006
Our Results
Spontaneous polarization exists ALONG the bond direction - LONGITUDINAL.
Only possible for non-collinear spins
Hiroshima Nov
2006
Numerical Approach
KNB and our results probe different regions of parameter space.
We decided to compute polarization numerically without ANY APPROXIMATION
Exact diagonalization of the cluster Hamiltonian (only 16 dimensional) for arbitrary parameters (/V,U/V)
Both longitudinal and transverse polarizations were found !
Hiroshima Nov
2006
Numerical Results for Polarization
Transverse and longitudinal components exist which we were able to fit using very simple empirical formulas:
KNB
Px (longitudinal) Py (transverse)
JONH
Hiroshima Nov
2006
Uniform vs. non-uniform
KNB term is responsible for macroscopic polarization
JONH term averages out
Locally, JONH >> KNBDetecting such local ordering of polarization will be interesting.
Hiroshima Nov
2006
Consideration of eg orbitals
Real Manganese oxides have inert t2g levels
e.g. TbMnO3 (t32g e1
g)Inert t2g orbitals cannot produce electronic polarization
eg orbital alone is insensitive to spin-orbit coupling due to angular momentum mismatch
A way out:
(1) t2g-eg level mixing due to octahedral tilting (2) Assuming oxygens carry the spin-orbit interaction
(According to some LDA calculations, SO(oxygen) is nontrivial fraction of SO(Mn))
Hiroshima Nov
2006
Model for TbMnO3
Ingredients:
(1)Orbital ordering takes place at high temperature
(2)Only one eg orbital level is assumed, which hybridize with oxygen p orbital (2) Spin-orbit coupling at oxygen site mixes all three p orbitals
Hiroshima Nov
2006
Results for eg orbitals and oxygen SO
Transverse polarization still possible with oxygen SO
We propose this as a possible mechanism of polarization for RMnO3
Jia, Onoda, Han, To be published
Hiroshima Nov
2006
Summary
Microscopic mechanism for magnetism-induced electronic polarization is investigated.
We show that spin-orbit interaction alone gives rise to electronic polarizations of both transverse and longitudinal types
Both t2g and eg orbitals can produce such polarization
Comparison with real multiferroic material is in progress
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