Electric Polarization induced by Magnetic order

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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
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SL in solid state physics

• 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 (-gtSpintronics)
• Spin Hall Effect (-gtSpintronics)
• Etc.
• Most recently we might have a new item on the
  list
• Magnetism-induced Electric Polarization

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Physics of SL in Magnetic States
If spin orientations are fixed due to magnetic
ordering, SL
ltSgtL 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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Ni3V2O8
TbMnO3

Collinear to non-collinear spin transition
accompanied by onset of polarization
Lawes et al PRL 05
Kenzelman et al PRL 05
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CoCr2O4
Co spins have ferromagnetic spiral (conical)
components Emergence of spiral component
accompanied by P
Tokura group PRL 06
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Magnetism-induced Ferroelectricity
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
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Step 1 Ginzburg-Landau theory
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Spin-polarization Coupling via GL Theory
Spin ltSgt and polarization ltRgt break different
symmetries ltSgt breaks time-inversion
symmetry ltRgt breaks space-inversion
symmetry Naively, lowest-order coupling occurs
at ltSgt2 ltRgt2 . Lower-order terms involving
spatial gradient, ltSgt2 ltgrad Rgt or ltSgtltgrad
SgtltRgt, are possible.
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Spin-polarization coupling via GL theory
Generally one can write down GL terms like
that result in the induced polarization
in the presence of magnetic ordering For
instance
Mostovoy PRL 06
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Spin-polarization Coupling via GL Theory
For spiral spins
induced polarization has a uniform component
given by
Mostovoy PRL 06
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Spin-polarization Coupling via GL Theory

• Uniform induced polarization depends on the
  product M1 M2
• - Collinear (M1M20) spin cannot induce
  polarization
• - Only non-collinear, spiral spins have a chance
• Quite consistent with experimental facts
• No microscopic derivation of Mostovoys free
  energy exists yet.

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Step 2 Microscopic theory Mechanism
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At the microscopic level, one can write down
spin-polarization interaction terms such
as Fully consistent with symmetries Consiste
nt with GL theory
Pij
Si
Sj
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Where does it come from?
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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.
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.
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Microscopic Theory Further Details
Magnetic
Oxygen
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.
eg
d
Crystal Field
t2g
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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.
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Theory of Katsura, Nagaosa, Balatsky (KNB)
The cluster Hamiltonian
KNB PRL 05
KNB Hamiltonian is solved assuming ?
(spin-orbit) gt U (Hund)
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Results of KNB
Polarization orthogonal to the spin rotation axis
and modulation wave vector develops
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Results of KNB
The results may be generalized to the lattice
case consistent with phenomenological theories
RED spin orientation BLACK polarization
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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
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Our Results
Spontaneous polarization exists ALONG the bond
direction - LONGITUDINAL. Only possible for
non-collinear spins
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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 !
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Numerical Results for Polarization
Transverse and longitudinal components exist
which we were able to fit using very simple
empirical formulas
Px (longitudinal)
Py (transverse)
JONH
KNB
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Uniform vs. non-uniform
KNB term is responsible for macroscopic
polarization JONH term averages out Locally,
JONH gtgt KNB Detecting such local ordering of
polarization will be interesting.
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Consideration of eg orbitals
Real Manganese oxides have inert t2g levels e.g.
TbMnO3 (t32g e1g) 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))
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Model for TbMnO3

• Ingredients
• Orbital ordering takes place at high temperature
• 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

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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
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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