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Soft X-ray Absorption and Resonant Scattering
Di-Jing HuangNat’l Synchrotron Radiation Research Center, TaiwanDept. of Physics, Nat’l Tsing Hua University, Taiwan
Chairon 2007, SPring8, JapanSept. 16
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Soft x-ray: 250 eV ~ a few keV
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Interaction of photons with matter:
• lattice structure: arrangement of atoms • electronic states• magnetic order• excitations (electronic states or phonons)
Photoelectric effectPhotoabsorptionScattering/diffraction
SynchrotronRadiation
Transmission(absorption)
Inelastic Scattering
photoemissionReflected Photons
Fluorescence
Crystal
Bragg Diffraction
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
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X-ray absorption
( ) ( ) μ− =dI z I z dz
( ) ( )- dI z I zdz
μ= zIzI 0e)( μ−=
absorption coefficient μ
dz
0I )(zIΔΩΦ
=Ω
ΔΩ
0
Wddσ
00 A
I=Φ
( ) ( )aW I z dz I z dzρ σ μ= =# of absorption events
# of atoms/area aμ ρσ=
1 δ β≡ − +n i (α and β are real numbers)
kzkzinkzi βδ −−= eeEeE )1(0
)(0
The wave propagating in the medium is
2μ β=
absorption cross section
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EM waves scattering by an electron
0 0e , ˆE E E //i tin xω−=
r'z
x
y
rR
X
ψ
( + )a E v B= − ×in inm e
The dipole approximation 'r r− ≈ r 20
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(r, ) (r ', / ) r 'A Jπε
≈ −∫c r Vt t r c d
AE ∂= − Φ −
∂t∇
Far away from the electron:ctt /'rr' −−=2
0
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(r ', ')(r, ) r 'r r '
JAπε
=−∫c V
tt dThe retarded vector potential
0(t) e cos(t)
i
in
rr
ψ⋅
= −k rE
E
Thomson scattering length(the classical electron radius)
The scattering involves a phase shift of π
2
0 20
5
4
2.82 10 Å
ermcπε
−
=
= ×2 2
0 cosd rd
σ ψ=Ω
2 2 20 0
8cos3T r d rπσ ψ= Ω =∫
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X-rays scattered by an atom
'q k k≡ −k 'k
r-k r⋅ 'k r⋅
'k k= = k
kq 'k
θ2θ
2 sinq k θ=
elastic scattering
A volume element at will contribute an amount to the
scattered field with a phase factor .
3rd re q r⋅i
30 ( ) rrρ−r d
Momentum transfer (in units of )
2 2
2 2 2
2
2 2
'
2 'cos 22 (1 cos 2 )4 sin
q k k
θ
θ
θ
= −
= − +
= −
=
q k kk kkk
e q r⋅i
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X-rays scattering
∫ ⋅≡ r)er()q( rq0 df iρ
Scattering amplitude: )q(re)r( 00
rq0 frd-r i −=∫ ⋅ρ
Atomic scattering factor
Z)q( ,0 0 =→ fq (the number of electrons in the atom)
All of the different volume elements scatter in phase; each electron contributes -r0 to the scattered field.
f 0 is independent on photon energy ; it is the scattering amplitude in units of –r0.
f 0 is the Fourier transform of the charge distribution.
ω
04 4'' Im( )r f fk kπ πσ = − = −
Optical theorem
absorption cross-section
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The forced charge oscillator: a classical model of an electron bound in an atom.
'em κ= − −− Γx x xEdamping force
02 E esi tx x x e
mωω −−+ Γ + = mms /' ,/ Γ≡Γ≡ κω
)(1 ,e 22
000 Γ−−
−== −
ωωωω
imeExxx
s
ti
How does the scattering amplitude f change if the photon energy approaches an absorption edge?
2
0 2 2
e( , )( )
ikr
ss
ca inrrr t
iω
ω ω ω−
− + Γ=E E
Resonant scattering amplitude in units of -r0:
2
2 2( )ss
fi
ωω ω ω
=− + Γ
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0( , ) ( ) '( ) ''( )q qω ω ω= + +sf f f i f
Resonant scattering
As the photon energy approaches the binding energy of one of the core-level electrons,
ω
dispersion corrections
2222
222
)()()('Γ+−
−=
ωωωωωω
s
ssf
2222
2
)()(''
Γ+−Γ
−=ωωω
ωω
s
sf
Absorptionω
'f
''f
' ''f f i f= +
4 Im( )fkπσ = −
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
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core level
Continuum
Photon energy
Abs
orpt
ion
Atom
core level
Continuum
Solid
EF
EV
Photon energy
Abs
orpt
ion
Band resonance + atomic multiplet
atomicbackground
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( )2
f if i
d E E hd
σ δ ν∝ ⋅ ⋅ − −Ω ∑ Ψ ΨPAPhoto-absorption
2 3p d→
1s np→ K-edge XAS can be accurately described with single-particle methods.
L-edge XAS: the single-particle approximation breaks down and the pre-edge structure is affected by the core hole wave function. The multiplet effect exists.
Dipole approximation: 1 1ie i⋅ ≈ + ⋅ ≈k r k r( )i t i te eω ω⋅ − −≈k rA ε ε=
( ) ( )2 2f i f i
d f i E E h f i E E hd
σ δ ν δ ν∝ ⋅ ⋅ − − ∝ ⋅ − −Ω
⋅∑ ∑ ε rε P
polarization of x-ray
ε ε εˆ[ , ] ( )f iim imf i f H i E E f i im f iω⋅ ≈ ⋅ = − ⋅ = ⋅ε P r r r∵
ˆ[ , ] , if [ ( ), ] 0H V ri m
= =Pr r
If ⋅k r 1,
1s
2s
3s
K
L1
M2,3
L2
L32p1/2
2p3/2
3p1/2, 3/2M1
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ˆijM f i∝ ⋅ε r)(2 2
ifij EEMW +−= ωδπ
2p
3d
Dipole transition
1f il l lΔ ≡ − = ±
Δ ≡ −( )l f im m m
( )θφθφθ cos,sinsin,cossinr̂ =
),(cos 0,134 φθθ π Y= ),(sin 1,13
8 φθθ πφ±
± =⋅ Ye i ∓
41,1 1, 1 1,2 03 2
r̂ ( )xx y yiz
i Y Y Yεε επ ε ε− +−
+⋅ = + +ε
r̂ sin cos sin sin cosx y ze e e⋅ = + +ε θ φ θ φ θ
Absorption probability:
L. circularly polarized
Y1,1: Δ ml = +1R. circularly polarized
Y1,-1: Δ ml = - 1
Δ ml = 0
5 1 62 3 2 3n nijM p d p d+∝ ⋅ε rFor 2p 3d:
Selection rule:
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•Two absorption “peaks” from 2p 3d:
3 22 /p
L3
d
1 22 /p
L2
•Absorption cross section is proportional to numbers of d holes and density of states in the core level.
L3
L2
Fe
L-edge XAS provides information on the chemical state, orbital symmetry, and spin state of materials.
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2 64 3s d
2 74 3s d2 84 3s d 1 104 3s d
Each element has specific absorption energies. “finger print” element specific spectroscopy
Fe
CoNi
Cu
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental Setup: how to measure XAS spectra?Applications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
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Incident light
transmitted light
Measurement of Soft X-ray absorption
electrometer
mesh
detector
I0
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Photoelectric absorption
2p1/2
2p3/2
3p1/2
3p3/2
3d
1s
2s
3s
K
L1
M
L2
L3
3d
K
L1
M
L2
L3
Fluorescent x-ray emission
3d
K
L1
M
L2
L3
Auger electron emission
Photoexcitation and Relaxation
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Photoelectrons or Auger electrons
hν
M1
L2
L3
Photoabsorption
2p
d
3s
Auger electrons
secondary electrons
Kinetic energy
Inte
nsity
(log
)
secondary electrons
PhotoelectronsAuger electrons
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Auger electrons(20 Å)
Fluorescence(1000 Å)
Incident light
transmitted light
electrometer
total electrons(40 ~ 50 Å)
Measurement of Soft X-ray absorption
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
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soft x-ray absorption & scattering
TM: 2p 3dO: 1s 2p
direct, element-specific probing of electronic structure of TMO
TM 2p
In transition-metal oxides
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L-edge XAS provides information on the chemical state, orbital symmetry, and spin state of materials.
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•Coulomb interactions of 3d•Interaction between core hole and valence states
53d
5 62 3p d
cdU
43d
5 52 3p d'cdU
+2Mn3Mn +
2 54 3Mns d
hole doping:LaMnO3 La1-xSrxMnO3
La0.7Ce0.3MnO3 electron doping?
3 4Mn Mn+ +→
+2Mn
+2Mn 3Mn +
MnO2
MnO
4Mn +
LaMnO33Mn +
Mitra et al., PRB 67, 92404 (2003)
3 2Mn Mn+ +→
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L-edge XAS provides information on the chemical state, orbital symmetry, and spin state of materials.
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CuO
La/Sr
High Temperature Superconducting Cu oxides
Sr doping: La2CuO4 La2-xSrxCuO4
x2-y2
3z2-r2
eg
t2g
3d crystal field
dxy
dyz, dzx
hole doping: 3d9 3d9 + O 2p hole
La2CuO4(Cu2+ : 3d9)
28Chen et al. PRL 68, 2543 (1992)
Orbital Characters of Doped Holes in La2-xSrxCuO4
3z2-r2
eg
t2g
3d
dxy
dyz, dzx
L2
L3
ˆ ˆσ σ⊥c E cE
x2-y2
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O 1s X-Ray Absorption on TMO
O 1s
O 2p
⋅⋅⋅++=Φ + Ldd nnG
1βαIn a cluster approach
O 1s XAS
1+→ nn dd
⋅⋅⋅+=Φ +1' ndα
(neglecting 1s core hole)
A: d 9L d 9
spectral weight arising from doped holes ( α 2 << β 2)
B: d 9 d 10
upper Hubbard band
EF
Chen et al.,PRL (1991)
Photon Energy (eV)
A B
La2-xSrxCuO4
d 9 d 10L
d 10
Spectral weight transfer:a fingerprint of strong electron correlations
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
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ferromagnetism (FM) ferrimagnetismanti-ferromagnetism (AFM)
Magnetic materials:In some solids, individual ions have non vanishing average vector moments below a critical temperature. Such solids are called magnetically ordered.
If a solid exhibits a spontaneous magnetization, its ordered state is described as ferromagnetic or ferrimagnetic. A solid with magnetic ordering has no spontaneous called antiferromagnetic.
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Fe Cu
Density of states Density of states
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Polarization of Synchrotron Radiation
Linearly polarized Left-handed circularly polarized
Right-handed circularly polarized
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Soft X-Ray Magnetic Circular Dichroism in Absorption
MCD is defined as the difference in absorption intensity of magnetic systems excited by left and right circularly polarized light.
−+ −≡ σσMCD
Right-handedcircularly polarized light preferentially excites spin-downelectrons.
Left-handed circularly polarized light preferentially excites spin-upelectrons.
For 2p3/2 3d
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MCD sum rules
Experimental confirmation:Chen et al., PRL 75, 152 (1995)
L3
L23
( )L
A μ μ+ −≡ −∫
2
( )L
B μ μ+ −≡ −∫
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Soft X-Ray Magnetic Circular Dichroism in Absorption
Soft X-ray MCD in absorption providesa unique means to probe:
element-specific magnetic hysteresisorbital and spin momentsmagnetic coupling.
There are two ways to obtain a MCD spectrum:1) Fixing M, measure XAS with left and right
circular lights.2) Fixing the helicity of light, measure XAS with
two opposite directions of M.
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ps →MCD in K-edge or L1-edge XAS is not sensitive to spin magnetic moments. Instead, the MCD measures orbital moments.(H. W.)
2sL1
2p
L1
L2
L3
lm10
-1
1=Δ lm
0=−σ
0≠+σ
d
2s
dp →
MCD in L-edge XAS measures both spin and orbital magnetic moments because of the spin-orbit interaction in the core levels.
p
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Chen et al., PRB 48, 642 (1993)
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Nature 416, 301 (2001)
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
42
T
R B = 0
B = 15 T
T
R
• Variety of fascinating macroscopic phenomena• Tunable electronic and magnetic properties
T
R
MITHTSC CMR
Correlated-Electron Materials
High Temperature SuperConductivity
Metal-to-Insulator Transition
Colossal Magneto-Resistance
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spin
charge
orbital
lattice
charge ordering: spatial localization of the charge carriers on certain sites
2+
3+
spin ordering: long range ordering of local magnetic moments
orbital ordering:periodic arrangement of specific electron orbitals
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Charge-orbital ordering in correlated electron systems
Sternlieb et al. (1996)
La0.5Sr1.5MnO4
La1.6-xNd0.4SrxCuO4
J. Tranquda et al. (1995)
Moritomo et al. (1995)
Murakami et al.(1998)
C. H. Chen et al. (1998)
La1/3Ca2/3MnO3
Small valencedisproportionation !
Δq/qtotal << 1
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• With ω =E3d − E2p, fres enhanced dramatically and ordering of Ψ3d focused.
0, k ε , 'k ε2 pΨ
3dΨ
'q k k≡ −momentum transfer
modulation vector
0
3 3 2
32 23'
~( )
dik r ik r
resd d
p pd
p
re ref
E E iε ε
ω
⋅ ⋅⋅ ⋅ΨΨ Ψ
−
Ψ
− − Γ∑
Advantage of Resonant Soft X-ray Scattering
TM
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Elastic x-ray scattering
2
qfdd
∝Ωσ
scattering form factor
k 'k
q k' k= −momentum transfer
q
θλπθ sin4sin2 == kq
θ2
A volume element at will contribute an amount to the scattering field with a phase factor .
r3d rreiq⋅
jrj(r )eiq
qj
f ρ ⋅≡ ∑ Fourier transform of charge distribution.
Bragg condition:q = modulation vector of
charge, spin , or orbital order
rrk ⋅ r'k ⋅
' kk =elastic scattering
Fourier transform of spin distribution.
jrj(r )eiq
qj
S S ⋅≡ ∑
2
qSdd
∝Ωσ
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2 2
2 2 22
2 22 )int (( ) ( )e e
j jmc me e
j jmc mcj
cj j j
AA r A P AH At
s s+ ⋅ − ⋅∇×=∂
− ⋅ ×∂∑ ∑∑ ∑
2
222
int
2
2 2
2
2 2 ' 'j jiq
j
rq r
j
ijf ec e
V mi f
f H i
mcce s ii ε ε
πσ
π π ε εωω
⎛ ⎞= ⎜ ⎟
⎝ ⎠− ⋅ ×⋅∑ ∑ ii
kinetic energy m.B SO
Non-resonant
mag 32
charge
~ ~ 10f
f mcω − 6
e
mag 10~ −
σσ
X-ray magnetic scattering
Resonant
int i2
nt2 ( )n i n
i f
f H n n HE
iE
E Eπσ δ
ω −−
+∑
Blume, J. Appl. Phys. (1985)
for ~ 600 eVω
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Resonant X-ray magnetic scattering
1±=Δ lm
electric dipole transitions
1=Δ lm
0ε ε
F1,1 F1,-1
'k
kq σ=0ε
π=ε
[ ]001
[ ]100
[ ]010
σπ ×
scattering amplitudes )(ˆ)εε(83
1,11,10*
−−⋅×−= FFzif resmag π
λHannon et al., PRL(1988)
As a result of spin-orbit and exchange interactions,magnetic ordering manifests itself in resonant scattering.
1−=Δ lm
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Resonant Soft X-ray Scattering
Charge-orbital ordering in manganite and magnetite• Wilkins et al., PRL (2003)• Thomas et al., PRL (2004)• Dhesi et al., PRL (2004)• Huang et al., PRL (2006)
…….
CDW or charge stripes in cuprate and nickelate• Abbamonte et al., Nature (2004)
• Abbamonte et al., Nature Physics (2005)• Schüßler-Langeheine et al., PRL (2005)
……
Antiferromagnetic ordering of manganite• Wilkins et al., PRL (2003)
• Okamoto et al., PRL(2007)
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Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering AFM of La0.5Sr1.5MnO4
26
51
Inverted spinel structure (cubic)1/3: tetrahedral (A-site) Fe3+
2/3: octahedral (B-site) Fe3+, Fe2+
The Verwey transition of magnetite (Fe3O4)
oxygen
Fe3O4 is believed to be a classic example of charge ordering.
1st order
A-siteFe
B-siteFe
Verwey model:
charge order-disorder transition of B-site Fe3+ & Fe2+ with TV ~ 120 K
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• Charge ordering deduced from the Fe-O distance.
• Valence of B-site Fe: 2.4 & 2.6 (rather than 2 & 3)
• (0 0 1)c and (0 0 1/2)ccharge modulation along the c-axis
Refinement of x-ray and neutron diffraction
Wright, Attfield, and Radaelli, PRL (2001), PRB (2002)
2a
Fe2+ Fe3+B-site
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LDA+U calculations: charge-orbital orderingJeng, Guo, and Huang, PRL (2004)
z = 0
a2
2/a
2/a
OFe3+A Fe3+B Fe3+
B Fe2+ O
electron-rich Fe4O4 cube(3 B-Fe2+ and 1 B-Fe3+)
electron-poor Fe4O4 cube(3 B-Fe3+ and 1 B-Fe2+)
Fe2+
Fe3+ Fe3+
Fe3+
Fe3+
Breakdown of Anderson’s criterion can be explained by the existence of orbital ordering.
Leonov et al., PRL (2004)
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Fe 3d O 2p
O 1s
≈ 1
R is the factor accounting for the enhancement of the form factor due to the O K-edge resonance.
The change in the structure factor with q=(0 0 ½)c around O K-edgeis negligible:
(0 0 L)
Γ
c
Correlation length ξ≡1/Γ = 900 Å
(0 0 ½)c resonant scattering at O K-edge of Fe3O4
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(0 0 ½)c resonant scattering at O K-edge of Fe3O4
ξ ~ 900 Å
ρ
(0 0 ½)
Huang et al., PRL (2004)Evidence for the ordering of electronic states associated with the Verwey transition.
56
Soft X-ray Absorption and Resonant Scattering
1. Basics of x-ray scattering and absorption2. Soft X-ray Absorption
Experimental SetupApplications
- Chemical analysis- Orbital polarization- Magnetic Circular Dichroisn
3. Resonant Soft X-ray Scattering BasicsExamples
- Verwey transition of Fe3O4- Charge-Orbital ordering and Quasi-2D
magnetic ordering of La0.5Sr1.5MnO4
29
57
thermallydisordered(renormalized classical)
quantum disordered
quantum critical
Low-dimensional quantum magnetism
QCP
non-thermal parameter g(P, B, doping, …)
T
ordered at T=0
Chakravarty et al., PRL 60, 1057 (1988)
2D quantum Heisenbergantiferromagnet on a square lattice with
2 /( ) e S Bk TT πρξ ∝spin correlation
12s =
Mermin-Wagner (1966):Pure 2D magnetic order only exists at T=0. renormalized
classical
ordered
classical critical,
1JJ⊥
1C
TT
νξ
−−∼
quasi 2D
58Greven et al., PRL(1994)
Cu2+, S=1/2, TN=256 KSr2CuO2Cl2(1/2, 1/2, L/2)
TN
410JJ
−⊥ ∼
2 /( ) e S Bk TT πρξ ∝
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MnO
La/Sr
LaSrMnO4 La0.5Sr1.5MnO4
Half-doped single-layered manganitedoping Sr
Mn3+ (d4)
AFM(1/4, 1/4, 1/2)
TN ~ 110 K
C.O.(1/4, 1/4, 0)TCO ~ 220 K
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2cosI φ∝
(¼, ¼, 0)orbital ordering
0φ =
Mn L-edge (2p 3d)
(¼, ¼, ½)AFM ordering
q 001
(r. l
. u.)
q110 (r. l. u.)
Resonant soft x-ray scattering on La0.5Sr1.5MnO4
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61
La0.5Sr1.5MnO4(1/4, 1/4, 1/2 )
116 K
113 K
( 1) 0.75 0.04νξ ν−∝ − = ±, N
TT
T=123 KT=114 K
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La0.5Sr1.5MnO4
116 K
2 /( ) e S Bk TT πρξ ∝semi- classical
(1/4, 1/4, 1/2 )113 K
Extension of zero-temperature AFM to T > TN
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Appendix: Basic of Magnetic Circular Dichroism in X-ray Absorption
Considering L-edge 2p3/2 3d absorption, and ignoring the spin-orbit interaction in the 3d bands,
2p3/2
3d
1/3 1/3 1/6
m j = 3/2 1/2 -1/2 -3/2L3
m l = 2 1 0 -1 -2
σ+=5/6
22
1,1( )2
,lm jx yi
R r Y Y j mε ε
σ +
−∝ ⋅
2ˆ, r ,l jl m j mσ ∝ ⋅ε
1 1/3 1/18
m j = 3/2 1/2 -1/2 -3/2
L3
m l = 2 1 0 -1 -2
Left-handed circularly polarized light preferentially excites spin-up electrons.
For left-handed circular light
radial part1/6 1/3 1/3
1/18 1/3 1
σ−=25/18
22
1, 1( )2
,x ylm jR r Y Y j m
iε εσ −−
+∝ ⋅
For light-handed circular light
Right-handed circularly polarized light preferentially excites spin-down electrons.
σ+=25/18σ−=5/6
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↑= 1,123
23 , Y
↓+↑= 1,131
0,132
21
23 , YY
↓+↑= −−
0,132
1,131
21
23 , YY
↓= −−
1,123
23 , Y
22
1, 1( ) ,2
x ylm j
iR r Y Y j m
ε εσ ± ±∝ ⋅
∓
2
3 1 12,2 1,1 2 2 3,
2x y
j
iY Y j m
ε εσ +
−∝ = = =
2ˆ, r ,l jl m j m∝ ⋅εσ
( )θφθφθ cos,sinsin,cossinr̂ =
),(cos 0,134 φθθ π Y= ),(sin 1,13
8 φθθ πφ±
± =⋅ Ye i ∓
41,1 1, 1 1,03 2 2
r̂ ( )x y x yi izY Y Yε ε ε επ ε− + +
−⋅ = + +ε
r̂ sin cos sin sin cosx y ze e e⋅ = + +ε θ φ θ φ θ
31
1,11,12,231
21
23
1,12,2 , ==== YYYmjYY j∵
∑+
−=
=21
21
2211)|( 2211
ll
llllmmlml YlmmlmlYY
2211
21
),|( 2211 mlmlmm
lm YYmlmlmlY ∑=
Clesbsch-Gordan coefficients
1/3 1/3 1/6
1/18 1/3 1
m j = 3/2 1/2 -1/2 -3/2
j=3/2
L3
m l = 2 1 0 -1 -2
2
3 1 12,0 1, 1 2 2 18,
2x y
j
iY Y j m
ε εσ − −
+∝ = = =
.....0,261
1,11,1 +=− YYY181
1,11,10,231
21
23
1,10,2 , ==== −− YYYmjYY j∵ 2,21,11,1 YYY =
33
65
↑= 1,123
23 , Y
↓+↑= 1,131
0,132
21
23 , YY
↓+↑= −−
0,132
1,131
21
23 , YY
↓= −−
1,123
23 , Y
22
1, 1( ) ,2
x ylm j
iR r Y Y j m
ε εσ ± ±∝ ⋅
∓
2
3 1 12,1 1,1 2 2 3,
2x y
j
iY Y j m −
+
−∝ = = =
ε εσ
2ˆ, r ,l jl m j mσ ∝ ⋅ε
41,1 1, 1 1,03 2 2
r̂ ( )x y x yi izY Y Yε ε ε επ ε− + +
−⋅ = + +ε
3 1 2 12,1 1,1 2,1 1,1 1,02 2 3 3, −= = = =∵ jY Y j m Y Y Y
∑+
−=
=21
21
2211)|( 2211
ll
llllmmlml YlmmlmlYY
2211
21
),|( 2211 mlmlmm
lm YYmlmlmlY ∑=
Clesbsch-Gordan coefficients
1/3 1/3 1/6
1/18 1/3 1
m j = 3/2 1/2 -1/2 -3/2
j=3/2
L3
m l = 2 1 0 -1 -2
2
3 1 12, 1 1, 1 2 2 3,
2x y
j
iY Y j m
ε εσ −
− − −
+∝ = = =
.....1,221
0,11,1 += YYY
3 1 2 12, 1 1, 1 2, 1 1, 1 1,02 2 3 3, −
− − − −= = = =∵ jY Y j m Y Y Y
.....1,221
0,11,1 += −− YYY
66
↑= 1,123
23 , Y
↓+↑= 1,131
0,132
21
23 , YY
↓+↑= −−
0,132
1,131
21
23 , YY
↓= −−
1,123
23 , Y
22
1,1( ) ,2
x ylm j
iR r Y Y j m
ε εσ +
−∝ ⋅
2
3 1 12,1 1,1 2 2 3,
2x y
j
iY Y j m
ε εσ +
−∝ = = =
3 1 2 12,1 1,1 2,1 1,1 1,02 2 3 3, jY Y j m Y Y Y= = = =∵
∑+
−=
=21
21
2211)|( 2211
ll
llllmmlml YlmmlmlYY
2211
21
),|( 2211 mlmlmm
lm YYmlmlmlY ∑=
Clesbsch-Gordan coefficients
1 1/3 1/18
m j = 3/2 1/2 -1/2 -3/2
j=3/2
L3
m l = 2 1 0 -1 -2
.....0,261
1,11,1 +=− YYY3 1 1 1
2,0 1,1 2,0 1,1 1, 12 2 3 18, jY Y j m Y Y Y −= = = =∵2,21,11,1 YYY =
1/3 1/3 1/6
m j = 3/2 1/2 -1/2 -3/2
j=3/2
L3
m l = 2 1 0 -1 -2
.....1,221
0,11,1 += YYY2
3 1 12,0 1,1 2 2 18,
2x y
j
iY Y j m
ε εσ −
+
−∝ = = =
11,1 1, 1 2,06 .....Y Y Y− = +
2
3 32,2 1,1 2 2, 1
2x y
j
iY Y j m
ε εσ +
−∝ = = =
3 32,2 1,1 2,1 1,1 1,12 2, 1jY Y j m Y Y Y= = = =∵