soft x-ray absorption and resonant...

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 2008, SPring8, JapanOct. 3

[email protected]

1


1. Soft X-ray AbsorptionBasicExperimental SetupApplications- Chemical analysis- Orbital polarization- Magnetic Circular Dichroism

2. Resonant Soft X-ray Scattering IntroductionBasic of resonant X-ray scatteringExamples:Magnetic Transition of a Quantum Multiferroic LiCu2O2

2

Soft x-ray: 250 eV ~ a few keV3

Interaction of photons with matter:

• lattice structure: arrangement of atoms • electronic states• magnetic order• excitations (electronic states or phonons)

Photoelectric effectSynchrotronRadiation

Transmission(absorption)

Inelastic Scattering

photoemissionReflected Photons

Fluorescence

Crystal

Bragg Diffraction

Scattering/diffraction

Photoabsorption

4

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

5

( )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=

( )2f i

d f i E E hd

σ δ ν∝ ⋅ ⋅ − −Ω ∑ Pε

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 rQh h

ˆ[ , ] , if [ ( ), ] 0H V ri m

= =Pr rh

If ⋅k r < < 1 ,

1s

2s

3s

K

L1

M2,3

L2

L32p1/2

2p3/2

3p1/2, 3/2M1

( )2f if i E E hδ ν∝ ⋅ − −⋅∑ ε r

6

ˆijM f i∝ ⋅ε r)(2 2

ifij EEMW +−= ωδπh

h

2p

3d

Dipole transition

1Δ ≡ − = ±f il l lΔ ≡ −( )l f im m m

( )θφθφθ cos,sinsin,cossinr̂ =

),(cos 0,134 φθθ π Y= ),(sin 1,13

8 φθθ πφ±

± =⋅ Ye i m

41,1 1, 1 1,2 03 2

r̂ ( )xx y yiz

i Y Y Yεε επ ε ε− +−

+⋅ = + +ε

sin cos sin sin cr̂ osx 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:

7

•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. 8

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

9

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

10




11

Incident light

transmitted light

Measurement of Soft X-ray absorption

electrometer

mesh

detector

I0

12

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

13

Auger electronshν

M1

L2

L3

Photoabsorption

2p

d

3s

Auger electrons

secondary electrons

Kinetic energy

Inte

nsity

(log

)

secondary electrons

PhotoelectronsAuger electrons

14

Auger electrons(20 Å)

Fluorescence(1000 Å)

Incident light

transmitted light

electrometer

total electrons(40 ~ 50 Å)

Measurement of Soft X-ray absorption

15




16

L-edge XAS provides information on the chemical state, orbital symmetry, and spin state of materials.

17

Polarization of Synchrotron Radiation

Linearly polarized Left-handed circularly polarized

Right-handed circularly polarized

18

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

19

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.1)

2) Fixing the helicity of light, measure XAS with two opposite directions of M.

20

Chen et al., PRB 48, 642 (1993)

21

Nature 416, 301 (2001)

22

23




24

Spin, Charge, and 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

modulation vector k = 2π/λ

25

Elastic x-ray scattering

2

qfdd

∝Ωσ

scattering form factor

kv

'kv

q k' k= −v vv

h h h

momentum transferqv

2 sin4 sin

q k θπ θλ

=

=

θ2

A volume element at will contribute an amount to the scattering field with a phase factor .

r3vd rvre iq ⋅v v jr

j(r )eiqq

jf ρ ⋅≡ ∑

v vv

Fourier transform of charge distribution.

rv

rk vv⋅ r'k vv ⋅

' kkvv

=k

q 'k

θ2θ

26

0( , ) ( ) '( ) ''( )q qω ω ω= + +h h hsf f f i f

Resonant scattering

As the photon energy approaches the binding energy of one of the core-level electrons,

ωh

dispersion corrections

2222

222

)()()('Γ+−

−=

ωωωωωω

s

ssf

2222

2

)()(''

Γ+−Γ

−=ωωω

ωω

s

sf

Absorptionωh

'f

''f

' ''f f i f≡ +

4 Im( )fkπσ = −

27

• With ω =E3d − E2p, fres enhanced dramatically and ordering of Ψ3d focused.

0, k εv v

, 'k εv v

2 pΨ

3dΨ

'q k k≡ −v vv

momentum transfer

modulation vector

h

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∑v vv vv v v v

h

Advantage of Resonant Soft X-ray Scattering

TM

28

1.5 GeVNSRRC, Taiwan

UHV two-circlediffractometer

EPU

Setup of Soft X-ray Scattering

29

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+ ⋅ − ⋅∇ ×=∂

− ⋅ ×∂∑ ∑∑ ∑ hh

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

⎛ ⎞= ⎜ ⎟

⎝ ⎠− ⋅ ×⋅∑ ∑ h

h

h

h

kinetic energy m.B SO

Non-resonant

mag 32

charge

~ ~ 10f

f mcω −h 6

e

mag 10~ −

σσ

for ω ~ 600 eV

X-ray magnetic scattering

Resonant

int i2

nt2 ( )n i n

i f

f H n n HE

iE

E Eπσ δ

ω −−

+∑h

h

Blume, J. Appl. Phys. (1985)

30

h

Resonant X-ray magnetic scattering

1±=Δ lm

electric dipole transitions

1=Δ lm

0ε ε

F1,1 F1,-1

'kv

kvq 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

31

Resonant soft x-ray magnetic scattering:

•Cross section comparable to that of neutron scattering.

•Good k resolution (Δk < 0.0005 Å-1)

•Spectroscopic information.

•Limited to a small k space, less bulk sensitive.

32




33

Magnetism: ordering of spins

Ferroelectricity: polar arrangement of chargesMagnetization can be induced by H field

Electric polarization can be induced by E field

34

Induction of magnetization by an electric field; induction of polarization by a magnetic field.

- first presumed to exist by Pierre Curie in 1894 on the basis of symmetry considerations

However, the effects are typically too small to be useful in applications!

Magnetoelectric effect

Multiferroics: materials exhibiting ME couplingCr2O3BiMnO3BiFeO3…..

35

Two contrasting order parameters

:

⇒ − ⇒ −

⇒

v

v

v

vt t M M

P P

Magnetization: time-reversal symmetry broken

Polarization: inversion symmetry broken

: , ⇒ − ⇒ − ⇒v vv v v v

r P MP Mr P qr=v r

36

2 0

2 5

3 0

3 5

0 1 0 2 0 3 0 4 0

-4 0

-2 0

0

2 0

4 0

0

( b )

( a )

0 T 1 T 3 T 5 T 7 T 9 T

E //b , H / /a ( F C )1 k H z

w a r m in g

ε

P1

P2T

P

0

40

-80

40

P1

P2T

P

0

40

-80

40

E p o le / /+ bH //a ( Z F C )

P (n

C/cm

2 )

T ( K )

• 3 transitions on cooling. • Magnetic field induces a sign reversal

of the electric polarization.

TbMn2O5 Nature, 429, 392 (2004)

37

Recently discovery in the coexistence and gigantic coupling of antiferromagnetism and ferroelectricity in frustrated spin systems such RMnO3 and RMn2O5(R=Tb, Ho , …)

revived interest in “multiferroicity”

•TC < TN• Frustrated magnetic systems.

TbMnO3: Kimura et al., Nature 426, 55, (2003)TbMn2O5: Hur et al., Nature 429, 392 (2004)

Magnetism and ferroelectricity coexist in materials called “multiferroics.”

38

LiCu2O2O2-

Cu2+

Cu1+

Li1+

orthorhombic, Pnmaa=5.734 Å, b=2.856 Å, c=12.415 Å

Qausi-1D spin ½ chain: •Characterization of the ground state?•Multiferroicity

(Park et al. PRL98, 057601; Seki et al., cond-mat 0810.2533)

39

Seki et al., PRL (2008)

Two transitions ?

C (J

/mol

K)

Masuda et al., PRL (2004)

40


(1/ 2, ,0)q ζ=r

a

b

J1

J2

94°

1( )n ne S SP +∝ × ×rrr r

along the a axis(1/ 2, ,0)q ζ=

r

in units of 2 2 2( , , )a b cπ π π

41

2p3/2

930 eV

3d

Cu2+

LiCu2O2 (100) single crystal

Resonant soft x-ray magnetic scattering

5 mm

Correlation length ξ = 1/HWHM

T = 10K: ξa = 690 Å ξb = 2100 Å42


(1/ 2, ,0)q ζ=r

qa (2p/a)

qb (2p/b)

non-zero scatteringintensity above TN

43

cleaved LiCu2O2(001) 5 mm

23.5 K transition temperature of the precursor phase

onset of the long-rangespin ordering and induced P

21.5 K

ξab: in-planecorrelation length

intensity( ,1)abq q=

r

ξc: correlation length along the c-axis

SW Huang et al., PRL (2008)44

45

,( ,1)a bq q=rq1

q2

•There is an in-plane short-range AFM ordering above TN

•extension of the zero-temperature AFM ordering

• spin coupling along the c axis is essential for inducing P.

46

2 /( ) e πρξ ∝ S Bk TT

•The ground state of LiCu2O2 exhibits a long range AFM ordering.

Above TN, in-plane correlation

averageJ ~ 4.25 meV

• The spin coupling along the c axis is essential for inducing P.

renormalized classical behavior

47

END

48

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 -3/2L3

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

49

↑= 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 ± ±∝ ⋅

m

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 m

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 jQ

∑+

−=

=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 jQ 2,21,11,1 YYY = 50

↑= 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 ± ±∝ ⋅

m

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, −= = = =Q jY Y j m Y Y Y

∑+

−=

=21

21

2211)|( 2211

ll

llllmmlml YlmmlmlYY

2211

21

),|( 2211 mlmlmm

lm YYmlmlmlY ∑=


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, −

− − − −= = = =Q jY Y j m Y Y Y

.....1,221

0,11,1 += −− YYY51

↑= 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= = = =Q

∑+

−=

=21

21

2211)|( 2211

ll

llllmmlml YlmmlmlYY

2211

21

),|( 2211 mlmlmm

lm YYmlmlmlY ∑=


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 −= = = =Q

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= = = =Q

52

soft x-ray absorption and resonant...

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