OrbialOrbial Anisotropy in Anisotropy in MultiferroicMultiferroic Systems: Systems: XX--ray Absorption & Magnetic ray Absorption & Magnetic CicularCicular DichroismDichroism

JaeJae--Hoon ParkHoon Park(POSTECH/PAL)(POSTECH/PAL)

Contributors

B.-G. Park, K.-J. Noh, T.-Y. Khim, H. Jang, K.-T. Ko, J.-S. Lee (POSTECH, Physics Department)

J.-Y. Kim, T. Y. KooPohang Acceleration Laboratory (PAL)

H.-J. Noh, Y. K. Bang (Chonnam National University)

D.-Y. Cho, S.-J. Oh J. H. Lee, P. Murugavel, T. W. Noh(Seoul National University)

S.-B. Kim, J. S. Ahn, H. Ishibashi, S-W. Cheong(l_PEM & Rutgers University)

A. Tanaka, T. Jo (Hiroshima University)

D. H. Kim (Oak Ridge National Laboratory)

Multiferroic• Coexistance of Magnetism and Ferroelectricity• Magnetoelectric Effects

“Multiferroics might hold the future for the ultimate memory device.” J.F. Scott, Nature Materials, 2007

M

H

MultiFerroic

− + − + − +− + − + − +P

E

H

M

M

H

Ferromagnetic

Magnetic Storage Devices

H-field switching M

E

PFerroelectric

− + − + − +− + − + − +P

E

SensorsFeRAM

E-field switching P

H,E-field switching M, P Multiple-state Memories

- Electrical Polarization Reversal by Magnetic field

TbMn2O5-Nature (2004) Cheong’s group CoCr2O4

-PRL (2006) Tokura’s group

Magnetoelectric effects in Multiferroic Materials

Eerenstein et al, Nature 442, 759 (2006)

NiO, CoO, MnO

LaMnO3

LaFeO3

CoCr2O4

LuFe2O4

(La,Bi)MnO3

NiFe2O4

CoFe2O4

EuOTbFeO3

BaTiO3

PbTiO3

PZT

TbMnO3

TbMn2O5

YbMnO3

BiFeO3

Cr2O3

BiCrO3

PbZrO3

Multiferroic and Magnetoelectric Materials

Orbital Anisotropy vs. Multiferroic

Magnetism (Spin)Internal Coordinates

Electricity (Polarization)Real Coordinates

Orbital Angular Momentum

L·S coupling

• Dzyloshinskii-Moriya (DM) Interaction - Spin-Orbit coupling (derived from Relativistic Dirac equation)

SLslrH ii

iiSO

rrrrr⋅=⋅= ∑ λζ )(

- Effective local spin Hamiltonian

∑∑

∑∑

Λ−=

−Γ⋅′Γ′

−=

Γ−

+Γ=

′Γ′ Γ′Γ′

μννμμν

μμμ

γ γγμμμ

λλ

γλγλ

γγ

SSSL

EESL

SL

HHE

HHH SOSO

SOSOeff

2

2||

1

rr

Magnetic AnisotropyOrient local spin axis

DM InteractionAnisotropic Superexchange

jji

iDM ssDH rrr×⋅= ∑

,)(λ

Orbital Anisotropy vs. Multiferroic

• Ferroelectricity (Polarization)

Break of Central-Symmetry

L ≠ 0 Orbital Anisotropy

• Multiferroic (Spin & Polarization)

L ≠ 0 Orbital Anisotropy

L·S coupling Magnetic Anisotropy

0

0

=

=

L

Pr

r

Central SymmetryCentral Symmetry

00ˆ

, ≠=

=

yx

z

LL

zPPr

Ferroelectric

S

Orbital Anisotropy is an essential factor in Multiferroic!!

In normal cases MPrr

⊥

-a Powerful tool determining the Ground State Symmetry

hν

EF

Core level

Dipole Selection rule3d Transition Metal L2,3-edge: 2p → 3d

4f Rare Earth M4,5-edge: 3d → 3fΟ K-edge : 1s → 2p

Etc.

Two Particle Excitation Spectral Function(Quasi-one Particle Excitation Spectrum)

ρ∝|<ΨfN* |ψv

†ψc| ΨGN>|2δ(hν − (Εf − ΕG))

Importance of Atomic MultipletsFinger Print of Ground State Symmetry

O K-edge XAS : Show Conduction Bands

Soft X-ray Absorption Spectroscopy (XAS)

Multiplet Calculations of Mn L2,3-edge XAS for Different Valence States

F.M.F. de Groot et al, PRB (1990)

Different multiplet structure for different valence states

Finger Print of the Ground StatesGround State Valence

Ground State Symmetry (Spin State)

V 2p63d2 →2p53d3 Mn 2p63d5 →2p53d6 Co 2p63d7 →2p53d8

Multiplet structure: determined by 2p53dn+1 configurationdepends on the valence, crystal field, spin state

Multiplet structure vs ionic multiplet with crystal field

eg

t2g

Cu2+(d9) Ni2+(d8)

Co2+(d7) Fe2+(d6) Mn2+(d5)

Cr3+(d3) V3+(d2) Ti3+(d1)

eg

t2g

eg

t2g

O 1s XAS in TMOs (Conduction band) Due to TM 3d – O 2p Hybridization

O 1s XAS

Photon Energy (eV)

525 530 535 540 545

Inte

nsity

(Arb

. Uni

t)

CuO

NiO

CoO

FeO

MnO

Cr2O3

V2O3

Ti2O3 Ti 4s/4pTi 3d

V 4s/4p

Cr 4s/4p

Mn 4s/4p

Fe 4s/4p

Ni 4s/4p

Co 4s/4p

Cu 4s/4p

Cr 3d

Mn 3d

Fe 3d

Co 3d

Ni 3d

Cu 3d

V 3d

3d region

Photon Energy (eV)

530 535

10Dq

J

Transmission yield

• direct measurement• sample thickness limit (≤1000Å): due to large absorption cross sectionin soft x-ray region

Total electron yield

• indirect measurement• rather surface sensitivity: small probing depth (≈50Å)

Fluorescence yield

• indirect measurement• self-absorption problem: difficult at 3d transition metal L-edge, 4f rare-earh M-edge

t

I0 It = I0e-μt

Ae-

I0

Ie = I0μLe

If = I0μLf

I0

XAS Measurement Modes

Direct Measurements on Orbital Anisotropy & Momentum

-XAS using linearly polarized light on a system with geometrical AnisotropyOrbital Anisotropy/Directional Lattice distortion

• Selective Dipole selection rule depending on the Orbital Symmetryi.e. Polarization vector E // c vs E ⊥ c

⇒ Information on the Orbital states and anisotropy

yz, zxxy

3z2-r2

x2-y2

Cu2+ Cu3+

Chen et al. PRL (1992)

d-orbitals under Oh crystal field None-zero transition matrix elements in TM L-edge ; 2p → 3d

2p: px, py, pz

3d: xy, yz, zx, x2-y2, 2z2-x2-y2 (3z2-r2)

E // x 1 1 1/2 1/6

1 1 1/6E // y 1/2

E // z 1 1 2/3

O 2p – TM 3d orbital bondings

zx(yz) : pz, px (py)xy : px, py

3z2-r2 : pz, (weak px, py)x2-y2 : px, py

O K-edge ; 1s → 2p (px, py, pz)

Orbital Selection in Polarization dependent XAS

Sum rule:

( ) ( )α+×σΔ−σΔ∝σΔ≡σ−σ∝ −+

14223 LLS

L

MM

qp

p

L

L

−=σΔ

=σΔ

μ≡σ ∫ ±±

2

3

Chen et al. PRL (1995)

Element specificOrbital and Spin Moments

Determination of Orbital & Spin Moment in XMCD

qpq

mm

s

o

692−

=

Multiferroic Systems under Research

• GaFeO3 (Polar Ferrimagnet – Piezo-electric)Fe3+ (d5) ions : half-full, L = 0 (?)

• LuFe2O4 (Ferrimagnet– Ferroelectric)Fe2+ (d6) – Fe3+ (d5) Mixed valence in hexagonal, charge-orderingHuge Coercivity (HC > 10 T)

• BiFeO3 (Antiferromagnet-Huge Ferroelectricity)Fe3+ (d5) half-full : Orbital Anisotropy and Magnetization/ Stain effect in films

• BiFeO3 (Antiferromagnet-Huge Ferroelectricity)Fe3+ (d5) half-full : Orbital Anisotropy and Magnetization/ Stain effect in films

• YMnO3 (Antiferromagnet - Ferroelectricity) Mn3+ (d4) in hexagonal : Orbital Anisotropy and origin of ferroelectricity

I. Polar Ferrimagnet GaFeO3

Kim et al. PRL (2006)

Kubota et al., PRL (2004)

X-ray nonreciplocal dichroism (XNDD) Optical Magnetoelectric EffectJung et al., PRL (2004)

Ferrimagnetic + Piezoelectric

What is the mechanism?

Need microscopic information.Mostly Phenomenological Studies

-Magnetoelectric effect-Switching Magnetization

affects the electric polarization.

Magnetoelectric phenomena in GaFeO3

GaFeO3 Fe3+ (d5) half-full

a

c

Ga2Fe2

Ga2Fe2

Ga1Fe1

a

b

c

O

• Polar-Ferrimagnet (Ferrimagnet + Piezoelectric)TC = 200K ~ 300K (depends on Growth)M = 0.6 ~ 0.9 μB/Fe (due to the site disorder)4 different Cation sites (Ga1, Fe1, Ga2, Fe2)

Ga1: Tetrahedral sites Ga2, Fe1, Fe2: Octahedral Sites

Double HCP Closed-Pack(Orthorhombic)

B

A

A

Cc : Magnetic easy axis (chain direction)a : Interchain directionb : piezo direction

A

Fe fraction on Cation site

fGa1 = 0 (Ga3+ only)

fFe1 = 0.825 (Down-spin)

fGa2 = 0.35 (Up-spin)

fFe2 = 0.825 (Up-spin)

=> M = 0.86 μB/Fe

• Expected Fe3+ fractional distributions on 4 different cation sitesGilleo Classic Model for Ga2-xFexO3i) fGa1 = 0ii) fFe1 = fFe2 = 3x/7 + 0.8 ( 1 for x = 1.4)iii) fGa2 = 8x/7 – 0.8 ( 0 for x = 0.7)

• GaFeO3 (x =1)Ga1 (T3d) : fGa1 = 0 (Ga3+ only)Fe1 (Oh) : fFe1 = 0.825 (Down-spin)Ga2 (Oh) : fGa2 = 0.35 (Up-spin)Fe2 (Oh) : fFe2 = 0.825 (Up-spin)

It is important to check whether fGa1 ≈ 0

μ = 0.86 μB/ Fe

Gilleo’s Classic Model for Ga2-xFexO3

Very slim hysteresiscurve along c-axis

a

b

c

Grown by Flux methodM(5K) = 0.87μB/Feagree to the expected Cation site Fe fractionTC ~ 250Kc : easy axis (Strong Magnetic Anisotropy)b : hardest axis

GaFeO3 crystal in this study

Fe L2,3-edge XAS and XMCD

-XAS line shape is nearly identical to that of α-Fe2O3 (Oh Fe3+ only)-MCD line shape indicates almost no Fe3+(Td)(γ-Fe2O3: FeT

3/4FeO5/4O3)

Fe3+ fraction at Ga1(Td) site is negligible

Fe L2,3-edge XMCD

Very large Titling of latticeAnisotropic Bonding with O 2p states

MO/MS ~ 0.046MO ~ 0.23 μB for S = 5/2 (5 μB)Large Orbital Magnetic Moment

(d5 at Oh/T3d : MO ≈ 0)

~ MO

MO ≈ 0

-Movements in ab-direction and largest b-directionFe1: mainly b-direction L along a and cFe2: along a & b-direction L along c

Origin of Large Orbital Moments and Magnetic Anisotropy

Crystal Off-Centered Movements

easy axis : chardest axis : bLargest L along c

Off-centering Movements Parity Symm. Breaking Non-vanishing L normal to the Movement

EMC ~ − ζΔL•S

x

y

z

[111]

yx

zX

YZ

eg

t2g

a1g : Along Z-direction (crystal b-direction)eg

π : In XY-Plane (crystal ca-plane)

egσ : Toward Ligands

10Dq

Oh D3d

3K

3K : + or − depends on the trigonal tilting

Crystal Field under Trigonal Symmetry (D3d)

O K-edge XAS (Fe 3d state conduction band)

eg

t2geg

π

a1g

egσ

~0.4eV

~0.04eV

~ 1.4eV

Anisotropic Bonding with O 2p the Orbital Momentum

& Magnetoelectric Effect

egπ

a1g

Largest Hybrid. along b-axis

AdditionalAtomic movements

- Fe3+ fraction at the tetrahedral Ga1 site is negligible, consistent with Gilleo’s classic model.

- A large orbital moment (MO/MS ~ 0.05) is obtained from XMCD.

- The large orbital moment results from off-centering atomic movements , which explains the magnetic anisotropy.

- O K-edge XAS shows a strong polarization dependence of Fe 3d states; Orbital Anisotrpy Orbital Moment

- Magnetoelectric effects is probably due to the anisotropic bonding and the orbital anisotropy.

Conclusions & Summary

II. Hexagonal Manganite YMnO3

Cho et al. PRL (2007)

Crystal structure of Manganites ReMnO3

Large ionic radius Orthorhombic Re = La, Ce, Pr, Nd, Eu, Gd, Tb, Dy, Bi

P63cm Pbnm

Small ionic radius HexagonalRe = Y, Ho, Er, Tm, Yb, Lu

Hexagonal Orthorhombic

MnO5

Bi-pyramid

MnO6

Hexagonal Manganites ReMnO3

Acta Cryst 16, 957(1963)

Ferroelectric TC

~ 1000K~ 1000K~ 900K~ 600K

~ 1100K~ 600K

Spont. Pol (P//c)

5.5 μC/cm2

5.6 μC/cm2

5.6 μC/cm2

5.6 μC/cm2

Magnetic order: Antiferromagnetic (A-type) (TN ~ 80K)Geometrically spin frustration

Mn3+

► Coexistence of Ferroelectricity & Anti-ferromagnetism

= “Multiferroicity” (Structure driven Multiferroic)

With High ferroelectric TC vs Low Magnetic TN

Mechanisms of Ferroelectric Distortions

In general, short-range inter-atomic repulsion and lattice energy obstructs a ferroelectric distortion.

Need a stabilizing mechanism with soft phonon modes

• Prototype ferroelectricity : the so-called “d0-ness” (e.g. BaTiO3)Off-center movement of Ti in TiO6 lowers the energy throughthe Ti 3d-O 2p enhanced hybridization (rehybridization).

– Cohen, Nature 358 136(1992)

• PbTiO3 or BiMnO3 : Lone pair of Pb2+ or Bi3+ (6s2) => Ferroelectric Distortion

• TbMnO3 : Spin frustration of sinusoidal AFM ordering – Kimura et al., Nature 426 55 (2003)

Ferroelectricity of YMnO3 ?

• Mn3+ (3d4) ; In-Planar Spins (AFM) Not d0-ness

• Unlike Bi3+ in BiMnO3, Y3+ ion; no lone pair of electrons.

• Hexagonal Unit Cell -- dZ20-ness due to the peculiar Crystal field (D3h)

Directional Mn d0– ness - Filippetti et al. PRB 65, 195120 (2002)

MnO5 Bi-pyramid

NO t2g-eg thing!!

NOT a Jahn-Teller but an extreme limit of JT.

Cubic Hexagonal

eg

t2g

z2

x2-y2/xy

yz/zx

x2-y2

z2

yz/zxxy

Jahn-Teller

Ferroelectric distortions in YMnO3

B.B. van Aken et al. Nature Mat. 3 164 (2004)

• Actual ferroelectric displacements at rather Y sites, not Mn sites.

• Y considered to be almost purely ionic (no re-hybridization).

Suggest electro-static dipole-dipole interaction (no anomalous dynamic charge)

(B.B. van Aken et al. Nature Mat. 3 164 (2004))

• Y sites truly ionic?

• What is the origin of the ferroelectric distortion in YMnO3?

Mn3+ ion itself responsible for Magnetism & Ferroeletricity?

O K-edge Polarized XAS

-Strong Hybridization for both Mn 3d and Y 4d

-Strong Anisotropy for both Mn 3d and Y 4d

Strong covalency(strong hybridization)Large transferred O 2p holes in Y 4d

Against ionic Y

Is it Is it ““Y dY d00--nessness””??

Born Effective Charge Z* (Dynamical Charge)(Born and Goppert-Mayer 1933): responsible for the electric polarization P

can be several times larger than the static charge e.g. BaTiO3 : Z*Ti≈ +7.5 vs ZTi≈+2.5

Z*O||≈ −6 vs ZO≈−1.5

Dynamical charge = Static charge + Dynamical contributionCf) Anomalous contribution : Z*(u)− <Z(u)>

uuZuuZ

uupuZ

∂∂

+=∂

∂=

)()()()(*

• For interatomic displacement u : dipole moment p(u) = uZ(u) Change of polarization: displacement current

• Harrison’s bond orbital model (simple tight binding estimation) in 1st order

p- d hybridization strength V ~ d-7/2, Z(u) = Zionic − ΔZ (~ Veff2/(εd−εp) ~ d-7)

)2()(7)()(×Δ×−=

Δ= Z

uZu

uuZu

δδ

δδ

Dynamic contribution propor. to Transferred Charge (ΔZ)!!

Very large dynamic contribution(anomalous)

Local Structure of MnO5 and YO8

• Average hybridization change

Y1-O : 8% , Y2-O : 4%

(BaTiO3 : Ti-O : 5%)

•Y d0-ness lower the energy through the rehybridizationdue to the ferroelectric distortion as in BaTiO3 (Ti d0-ness)!!

• Hybridization change along c

YMnO3 : Op-Y1-Op : 70% , Op-Y2-OP : 35%

Off-centering Movements of Y-ions

Polarization Dependent Mn L-edge and O K-edge XASHexagonal vs. Orthorhombic Manganites

Summary & Conclusions

• Orbital and Bonding Anisotropy not only in Mn 3d but also in Y 4d

• Strong Y 4d-O 2p HybridizationTransferred charge to Y 4d is even larger than that to Mn 3d.

• Large hybridization enhancement (Rehybridization) of Y 4d-O 2p comparable to Ti 3d-O 2p in BaTiO3

large anomalous contribution on Born effective charges of Y and Op

• Large ferroelectric off-center displacements in Y-O ions in polar. direction even larger than those in BaTiO3.

• Y d0-ness with rehybridization is responsible for the ferroelectric instability.

• Re d0-ness is extended to the ferroelectricity of hexagonal ReMnO3

III. Giant Coercive Ferrimagnet LuFe2O4

LuFe2O4 – Ferrimagnet + Ferroelectric, Huge coercivity (HC > 10 T)

• RFe2O4 family Rhombohedral structure (R-3m)*

Fe2+(d6) and Fe3+(d5) mixed valence

• Bipyramidal local symmetry (D3h)

• Hexagonal bi-layered structureTriangular Network Frustration (charge & spin)Polar charge ordering Ferroelectricity† c = 25.25 Å

a = 3.44 Å

W-layer

Fe

O

* M. Isobe et al., Acta Cryst. C46, 1917 (1990)† N. Ikeda et al., Nature 436 1136 (2005)

1Fe2+ : 2Fe3+

2Fe2+ : 1Fe3+

Fe2+

Fe3+P

Physical Phenomenon of LuFe2O4

330 K260 K

2D – CDW3D – CDWFerrimagnetic Ordering

Fe2+

Fe3+

Charge and Spin Ordering

M. A. Subramanian et al., Adv. Mat. 18 1737 N. Ikeda et al., Nature 436 1136

M : ~ 2.9 μB HC > 10T @ 4.2K

J. Iida et al., JPSJ 62 1723 Y. Yamada et al., JPSJ 66 3733

P = 28 μC/cm2

The charge ordering are strongly correlated with magnetic ordering.

Polar charge order in the W-layer

Large coercivity & strange saturate magnetic moment

Giant magneto-dielectric effect : field induced charge freezing

ferrimagneticMagnetic order enhances ferroelectricity

Charge and Spin Ordering of LuFe2O4

• (⅓, ⅓, L+0.5) charge ordering• slightly incommensurate• (0.030, 0.030, 1.5) : 3D CDW

• Y. Yamada et al., JPSJ 66 3733 (1997)• N. Ikeda et al., Nature 436 1136 (2005)

• (⅓, ⅓, 0) magnetic spin ordering• slightly incommensurate (δ, δ, 0) • δ = 0.027• basically (⅓, ⅓, l) rod

• J. Iida et al., JPSJ 62 1723 (1993)• A. D. Christianson et al., PRL 100 107601(2008)

Frustration of Charge and Spin orderings in LuFe2O4

• The triangular lattice causes the charge frustration.

1Fe2+ : 2Fe3+ (Fe3+ rich)

2Fe2+ : 1Fe3+ (Fe2+ rich)

Polar Charge Ordering(1/3, 1/3, l) CO

P

• The triangular lattice causes spin frustration.Fe2+

Fe3+

undetermined spin

almostdegenerateda

(1/3, 1/3) spin ordering

• (1/3 1/3) spin ordering cannot be determined uniquely.

M. Naka et al., Phys. Rev. B 77 224441 (2008)

Issues related to the Orbital Anisotropy in LuFe2O4

• Orbital Occupation and Energy levels under the D3h symmetry

a1g (3z2-r2)e'' (x2-y2/xy)

e' (yz/zx)

Fe2+ (d6)

or

a1g (3z2-r2)

e'' (x2-y2/xy)

e' (yz/zx)

-GGA + U Band Calculation (Xiang et al., PRL. 98 246403 (2007))without Charge Ordering : e'' (x2-y2/xy) is the lowest.

-Under the assumption of the lowest e'' (x2-y2/xy). (Nagano et al., PRL. 99 217202 (2007))(1/3,1/3, l) CO produced in Monte Carlo Simulation But Spin order inconsistent with the neutron results.

Charge order Spin order

• Mössbauer suggests three different Fe3+ sites.

three Fe3+ ionsdifferent magnetic responsea

a. M. Tanaka et al., J. Phys. Soc. Jpn. 58 1433 (1989) ,b. S. Nakamura et al., J. Alloys Compd. 275 574 (1998) c. B. K. Bang et al., J. Appl. Phys. 103 07E307 (2008)

Fe2+ Fe3+

P C M

↑ ↑ ↑ ↓ ↓ ↑

Suggested Magnetic Configurations

Magnetic Moment ( 4μB × 3 + 5μB ) / 3 = 2.33 μB << Mobs ~ 2.9 μB

• Others- What is the origin of the large magnetic anisotropy

and coersivity?- Orbital anisotropy?

2.5

2.0

1.5

1.0

0.5

0.0

-0.5

Inte

nsi

ty (A

rb. U

nit)

750740730720710700Photon Energy (eV)

MCD (x5, R-L) Integrated

R L

• Ferrimagnetic spin ordering• Large unquenched orbital moment ~ 0.8 μB/f.u.

XMCD measurement

05.035.069

2±=

−=

qpq

mm

s

o

MFe2+ AF MFe3+

Mtotal // MFe2+

MFe2+ : ↑↑↑

MFe3+ : ↑↓↓

../3.23/)543(

ufm

B

BBs

μμμ

=−×=

../8.035.0 ufmm Bso μ≈×=

../1.3 ufm Btot μ≈

LuFe2O4 XMCD at 220K

Assume Ionic Spins

Fe2+

Fe3+

p q

( M ~ 2.9μB )

Linear Dichroism

• Linear dichroism shows that the x2-y2 and xy orbitals are unoccupied.1.4

1.2

1.0

0.8

XA

S In

ten

sity

(A

rb.

Uni

t)

750740730720710700Photon Energy (eV)

E//a E//c LD

in-plane

out-of-plane

a1g (3z2-r2)e'' (x2-y2/xy)

e' (yz/zx)

Against the GGA + U band calculation results.Need the band calculation with CO.

Not

a1g (3z2-r2)

e'' (x2-y2/xy)

e' (yz/zx)

0.30

0.25

0.20

0.15

0.10

0.05

0.00

XA

S I

nte

nsi

ty (

Arb

. U

nit)

550545540535530Photon Energy (eV)

Fe3+

Fe2+

E//a E//c

eg'eg'' a1g eg' eg'' a1g a1g ( 3z2-r2 )

eg'' ( xy , x2-y2 )eg' ( zx , yz )

0.3 eV

1.5 eV

Fe2+

• zx, yz is the lowest state again in Fe sites• degenerated zx, yz levels origin of MO

O K-edge Spectrum

Hybridization and dissolution of Orbital mixing

3d

Eg’ {a(yz)+b(xy) , -a(zx)+b(x2-y2)}0.087 eV

0.341 eV

Eg’’ {-a(yz)+b(xy) , a(zx)+b(x2-y2)}

A1g {3z2-r2}

( b=0.783 and a=0.622 )

C3v crystal field

Eg u (3z2-r2) 0.994 0.104 A1g

Eg v (x2-y2) 0.994 0.107 Eg’

T2g a (yz) 0.994 0.501 Eg’’

T2g b (zx) 0.994 0.501 Eg’’

T2g c (xy) 0.994 0.107 Eg’

full multiplet CI calculation (Fe2+)

( mS = 1.83 mO = 0.81 )

0.3 eV

1.5 eV

0.26

0.24

0.22

0.20

0.18

0.16

XA

S I

nte

nsity

(A

rb. U

nit)

537536535534533Photon Energy (eV)

E//a E//c

O K-edge Spectrum (Fe2+ region)

xy, x2-y2exp.

cal.yz, zx

hybridization

( the level structure is similar to that of Nagano et al. and Naka et al. )

2.20 Å

1.96 Å

2.00 Å

3z2-r2

Fe2+ (d6)

SOC d1 level

d-1 level

yz, zx Large orbital moment

But …

• The calculation reproduce the overall XMCD spectral line.

• MCD line shape shows an unexpected low energy side-peak

• There seem to be distinguishable Fe2+ and Fe3+ sites.

30

20

10

0

-10

Inte

nsity

(A

rb. U

nit)

725720715710705700

Photon Energy (eV)

Calculation

Experiment

R L MCD (x5 , R-L) MCD (calculated)

2:1 Fe3+ assumption

Polar Charge Ordering in LuFe2O4

• Two Different sites for both Fe2+ and Fe3+

1Fe2+ : 2Fe3+ (Fe3+ rich)

2Fe2+ : 1Fe3+ (Fe2+ rich)

- Different Madelung potentials - Different Charge environments

• Ferroelectric or Antiferroelectric

<13.4meV/f.u.a

c

2/3c = 3/2c*

a. M. Angst et al., arXiv:0807.3527v2 (2008)

(1/3 1/3 3/2) charge ordering is consistent with the anti-ferroelectric ordering.a

Madelung Potential Calculation

Antiferroelectic FerroelecticeÅ–1

Fe2+ Fe3+ Lu3+ Fe2+ Fe3+ Lu3+

Fe2+ rich -1.9426 -2.3908 -2.4110 -1.8701 -2.3184

Fe3+ rich -1.6995 -2.1476 -1.9124 -1.7718 -2.2201-2.1617

Least Charge Fluctuation Gap 2.951 eV / 4 = 0.7378 eV 5.0378 eV / 4 = 1.259 eV

( unit : eA-1 ; 14.39 eV )Antiferroelectric charge order is more appropriate to explain the size of the 0.7eV insulating gap of LuFe2O4.

0.7 eV< optical measurementa >

a. X. S. Xu et al., arXiv:0809.4483v1 (2008)

Fe2+

Fe3+

0.7eV

Two type of Fe3+ spin ordering

• Two possible ferrimagnetic orderings.

a b

• Take into account for different Fe3+ & Fe2+ sites

2Fe2+↑: 1Fe3+↓

1Fe2+↑: 1Fe3+↓: 1Fe3+↑a.

2Fe2+↑: 1Fe3+ ↑

1Fe2+↑: 2Fe3+↓b.

MFe2+ : ↑↑↑

MFe3+ : ↑↓↓

Three different Fe3+ ions as observed in Mössbauer

10

5

0

-5

Inte

nsity

(A

rb.

Un

it)

725720715710705700Photon Energy (eV)

MCD (x5 , R-L) MCD (calculated)

10

5

0

-5

Inte

nsity

(A

rb.

Un

it)

725720715710705700Photon Energy (eV)

MCD (x5 , R-L) MCD (calculated)

XMCD and Calculation

• CI calculation successfully reproduce XMCD

• All of the Fe2+ spins are parallel, two of the Fe3+ spins are anti-parallel and the other one is opposite direction.

• (1/3 1/3 0) magnetic ordering • Take into account for different Fe2+ and

Fe3+ sites.• Determine the spin configuration.

a. b.

30

20

10

0In

tens

ity (

Arb

. Uni

t)

725720715710705700

Photon Energy (eV)

Calculation

Experiment

R L MCD (x5 , R-L) MCD (calculated)

Summary

• Fe2+ possesses the large orbital moment.– Large crystalline anisotropy increases the coercivity with the

collective freezing of magnetic domains a

– Origin of the giant magneto-electric effect– Explain the saturated magnetic momenta,b M ~ 2.9μB

< magnetization curvea >

a. W. Wu et al., Phys. Rev. Lett. 101 137203 (2008)b. J. Iida et al., J. Phys. Soc. Jpn. 62 1723 (1993)

mspin of Fe2+ morbit of Fe2+ mspin of Fe3+

3.67 μB 0.81 μB 4.50 μB

mo/ms ≈ 0.35 (measured ratio)

< calculated spin and orbital magnetic moment >

Mtotal = (3×(3.67+0.81)-4.50)/3 = 2.98 μB

Mspin = (3×3.67-4.50)/3 = 2.17 μB

Morbit / Mspin = 0.81/2.17 = 0.37

• Orbital occupation and level splittings a1g (3z2-r2)

e'' (x2-y2/xy)e' (yz/zx)

e' (yz/zx) lowest energy state orbital=> contribute the large orbital moment of Fe2+

Fe3+ Fe2+