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