luigi paolasini paolasini@esrf · x-ray scattering (3-30 kev) - surface sensitivity (high absorp.,...

Luigi Paolasini [email protected]

L. Paolasini - LECTURES ON MAGNETISM- LECT.9

LECTURE 9: “X-ray magnetic scattering”

-  Elements of scattering theory -  X-rays-photon interactions. - Non-resonant magnetic x-ray scattering. - Resonant magnetic scattering.

References: - J. Als-Nielsen and D. McMorrow, “Elements of Modern X-ray Physics” (Wiley, New York, 2001). - M. Blume,”Resonant anomalous x-ray scattering”, ed. G. Materlik, (1994), p. 494-512, Elsevier Science B.V., Amsterdam.


- Determination of temporal evolution of the physical state of a large number of particle at the thermal equilibrium. -  Measurement of the probe dynamical variables Ni and Nf after the interaction with the sample

Transition probability per unit time of from an initial ⎟Si Ni> and a final state ⎟Sf Nf> of the sample+probe system

Fermi's Golden rule

Magnetic probe Energy ΔE= ENi-ENf = ħω Momentum Q=k-k’ Spin σ => σ’

Sample Transition |Si> => |Sf>


In scattering experiments we are interested in the changes of the probe states Ni and Nf:

The Fermi’s Golden Rule can be expressed in term of the spectral density G(ω) associated to the autocorrelation function of the matrix elements of the interaction potential V.

Matrix elements of interaction potential V

This expression relates the change of the state of the probe with the time-evolution of the sample state

Spectral density G(ω)


Measurable quantity in the scattering experiments The energy of the probe is comparable with the sample eigenstates

Ratio between the number of scattered particles with energies Ef in the solid angle ΔΩ(θ, φ) and the incident flux density Ω0 per unit of solid angle dΩ and energy dE

Total scattering cross-section

Differential scattering cross-section

Partial differential scattering cross-section


Neutron scattering (Cold-Thermal) -  Bulk sensitivity (low absorp., ~10 cm) -  Amplitudes: Nuclear/Magnetic ~ 1 -  High E-resolution -  Unpolarized source -  Soft interaction neutron-sample -  Well established sample environment

X-ray scattering (3-30 keV) -  Surface sensitivity (high absorp., ~10 µm) -  Amplitudes: Charge/Magnetic ~ 105 -  High Q-resolution -  Polarized source -  Easy Focusing -  Hard probe (T-heating, sample damage)…

Neutron and x-ray magnetic scattering are a powerfull probe to study the magnetic interactions at atomic scale in bulk and nanostructured materials


H

E

k

E(r,t) = ε E0 e i(k . r - ωt) ε

Transverse EM waves

k=2π/λ wavenumber

ε polarization

ν=ω/2π frequency

Energy = ħω = 12.398 [keV] / λ [Å]

Intensity I0(ω) = <E0

2> =N(ω) ħω

ex. 1 Å = 12.398 keV

λ wavelength


Photo-electric absorption

Photon absorbed and electron emitted in the continuum

Fluorescent emission

An electron from the outer shell fill the hole and emit a photon

Auger electron emission

The atom relax into the ground state by emitting an electron

Absorption and emission processes doesn’t conserve the number of photons


XANES (X-ray Absorption Near-Edge Structure) -  Oxidation state -  Link with photoemission -  Polarization and angular dependece

EXAFS (Extended X-ray Absorption Fine Structure) - Sensitivity to the local structure -  The most intense features are due to electric-dipole allowed transitions (i.e. Δℓ = ± 1) to unfilled orbitals.

Amorphous solids and liquid systems Solid solutions Local distortions of crystal lattices Catalysis Vibrational dynamics


The electric field Ein of the incident electromagnetic wave forces the motion of the electron which radiates a spherical wave Erad.

Radiated field Erad : -  anti-phase with respect Ein - - proportional to the electron acceleration -  decreases with cos(ψ)

Thomson scattering length:

Thomson scattering cross section


The differential cross section for the Thomson scattering depends from the incident and scattered photon polarizations


... a classical interpretation from de Bergevin and Brunel (Acta Cryst, 1981)

Thomson scattering

Orbital dependent

Spin dependent

Spin dependent

Spin dependent (Larmor precession)


one atom

Scattering processes conserve the number of photons If the photon energy is conserved, the scattering is elastic

a molecule

a crystal

Structure factors


-  The Laue’s condition the scattering intensities are non-vanishing when the scattering vector Q=k-k' coincides with a reciprocal lattice vector Ghkl:

- The Bragg’s law interference condition between an incident and a reflected waves by a periodic arrangement of atoms.

k’

kGhkl =Q=(113)

2θ

Ghkl =Q=(440)


-  The Structure factor F(hkl) describes the interference between the resultatant waves diffused from each atom in the unit cell for any given reciprocal lattice vector Q=Ghkl i.e.:

N° unit cell unit cell volume Laue’s condition

fs = atomic scattering amplitude for the atoms s related to the Fourier transform of the atomic electron density

Structure factor: information about the atom distribution inside the unit cell

L. Paolasini - LECTURES ON MAGNETISM- LECT.9 5/10/2009 Advances in crystallographic methods for the study of magnetism 17

Theory photon-electron interaction -  Klein & Nishina: Compton scattering (1929) -  Bohr (1932) -  Gell-Mann & Goldberg (1954) -  Lowe (1955)

Pre-synchrotron works -  Platzman & Tzoar: Theory (1972) -  de Bergevin & Brunel: Theory and Exp.NiO, Fe2O3, MnO (1972,1981,1984)

2nd generation synchrotron works Resonant Exchange scattering -  Blume (1985) -  Gibbs, Bohr, Moncton: Exp. Ho (1985,1986) -  Namikawa: Exp. Ni (1985) -  Hannon, Trammel, Blume, Gibbs: theory (1988) -  Vettier, Isaac, McWhan: Exp. UAs (1989) Magnetic dichroism -  Shutz: Exp. Fe (1988) -  Carra & Altarelli: Theory (1989)

…Many experiments done at 2nd generation synchrotron sources

Vettier,Isaac,McWhan: UAs (1989) (1989)

Gibbs, Bohr, Moncton: Exp. Ho (1985) Gibbs, Bohr, Moncton: Exp. Ho (1985) Gibbs, Bohr, Moncton: Exp. Ho (1985)

Bergevin & Brunel: Exp.NiO (1972)

M. Blume,”Resonant anomalous x-ray scattering”, ed. G. Materlik, (1994), p. 494-512, Elsevier Science B.V., Amsterdam. M. Blume and D. Gibbs, Phys. Rev. B 37, 1779 (1988)


1)  Write Hamiltonian to describe charges and photons 2)  Separate the electric field generated by the charges from that

generated by the radiation field 3)  Expansion of spin-orbit term by A(r) (transv.plane waves) 4)  Exploit the Fermi’s Golden rule (2nd order) 5)  Distinguish non-resonant and resonant x-ray scattering regimes

Interaction (weak) Non-interacting electrons Non interacting photons


•  Non-interacting electrons Hel

•  Non-interacting photons Hph

•  Interaction terms H’

1st order

2st order

H’1

H’4

H’2

H’3


•  Elastic scattering processes conserves the number of photons •  => A(r) is linear in c+(k,ε) and c-(k,ε) •  For the 1st order perturbation only the terms QUADRATIC in A(r) are

retained, i.e. H’1 and H’4

•  The linear terms in A(r) are contribute to the second order perturbation

1st order

2st order

Notice that the in general both non-resonant and resonant terms must be considered, and the resonant terms could be neglected only when ħω>>|En-Ea|


Thomson Charge density

Non-resonant magnetic Orbital and spin magnetization densities

L(Q) = orbital form factor S(Q) = spin form factor

Resonant terms Energy dependent terms

Core-hole virtual transition Anomalous tensor susceptibility

High order multipole terms

Γc = core-hole lifetime ħωk= incident photon energy

Current operators J(k)

Polarization dependent terms

Total x-rays scattering amplitude


Coherent elastic scattering cross-section for periodic crystals

n= unit cell atomic site

Site selectivity -  Space group symmetries -  Extinction rules

Atomic scattering amplitudes -  Atomic properties (photon-electron interactions) -  Electronic order parameters

Atomic scattering amplitudes

MAGNETIC E=3.0-10 keV

!ω/mc2 Zmagn. r0 ~ 0.001-0.03 r0

Magnetic structure determination

MAGNETIC RESONANT E=3-20 keV

Zres. r0 ~ 0.01-100. r0

Valence electronic anisotropies

CHARGE (Thomson) E=10-50 keV Z r0 ~ 1-95 r0

CHARGE (Thomson) CHARGE (Thomson) CHARGE (Thomson)

Structural characterization


High-Q quality samples are required to detect the weak magnetic reflections

~ 10-6 @ 9 keV

CHARGE (Thomson)

Ich.~105-109 cts/s

MAGNETIC

Imagn.~100-102 cts/s

RESONANT Ires.~103-105 cts/s

10 cm

100 m

10 km


Azimuthal diffractometer Normal beam diffractometer

k

k’

k k’

Polarization analyzer

Low temperature displex

Polarization analyzer

Be windows In-vacuum

flight path

Be dome

10T vertical split pair magnet


X-rays Polarization analyser -  Thomson selection rules (ε.ε’) -  Bragg diffraction by a crystal analyzer 2θp~90° -  η = rotation about scattered wavevector k’

2θp~90°

π'

Detector

σ'

σ'

η

k’ η

Half wave plate mode (Δα=π) - Rotation of 90° of liner polarization (when χ=45°)

Quarter wave plate mode (Δα=π/2) - Circular left/right polarizations (~98-99%)

Linear Polarization Scan (Δα=π) - Continous rotation of χ

Phase plate retarder Phase shift Δα between the transmitted and incident beam in the dynamical diffraction limit


Fourier transforms of spin and orbital magnetization densities

σ-σ’ π-σ’

σ-π’ π-π’

•  Jones’s matrices for NRMS:

S3

S2,L2 ,L2 S1,L1

Μ


0.0

1.0 103

2.0 103

3.0 103

4.0 103

5.0 103

6.0 103

0 60 120 180 240 300 360

Phot

ons/m

on (a

rb.u

nits

)ψ (deg.)

Q=(2,2,2)Azimuthal scans

0.0

1.0 103

2.0 103

3.0 103

4.0 103

5.0 103

6.0 103

0 60 120 180 240 300 360

Phot

ons/m

on (a

rb.u

nits

)

η (deg.)

Q=(2,2,2)σ' (η= 0º)

π' (η= 90º)

Polarization analysis

Charge scattering

0.0

1.0 10-4

2.0 10-4

3.0 10-4

4.0 10-4

5.0 10-4

0 60 120 180 240 300 360

Phot

ons/m

on (a

rb.u

nits

)

ψ (deg.)

Q=(2.5,2.5,2.5)

σ-π'

σ-σ'

Azimuthal scans

0.0

1.0 10-4

2.0 10-4

3.0 10-4

4.0 10-4

5.0 10-4

0 60 120 180 240 300 360

Phot

ons/m

on (a

rb.u

nits

)η (deg.)

Q=(2.5,2.5,2.5) (at ψ=90º)

π' (η= 90º)

σ' (η= 0º)

Polarization analysis ψ=90°

Mσσ ∼ S2 sin2θ, Mσπ ∼ 2 sin2θ cosθ (L1+S1) Magnetic scattering

S1,L1=0 S2=m S3=0 @ψ=0


Azimuthal scan : • Single magnetic domain probed •  Moment lies along the directions <110>

Polarization analysis: • Determination of L/S ratio

µL/S=0.29(5)

From neutrons: µtot=µL+µS =0.49 µB/Cu2+

From NRXS: µL/µS ~ L/2S ~ 0.14

µS = 0.43 mB/ Cu2+ µL = 0.06 mB/ Cu2+

R. Caciuffo, L. Paolasini, A. Sollier, P. Ghigna, E. Pavarini, J. van der Brink and M. Altarelli. Phys. Rev. B 63 (2002) 174425.


“Cycloidal helicity” vs Ferro-electric domains Antisymmetric Dzyaloshinskii-Moriya exchange interaction

+

Electric polarisation P :

P ∝ ∑i γ ei,i+1 x (Si x Si+1 ) γ = constant (superexchange and spin-orbit coupling)

Magnetic “spin current” js :

js ∝ ∑i Si x Si+1

Katsura et al. PRL (2005) Mostovoy PRL (2006) Sergienko et al. PRB (2006)

Anti-Clockwise cycloid

js= . P = =

Clockwise cycloid

js= X P = =

- Incommensurate magnetic structures - Non-collinear magnetic order

Helical non-coll.

Sinusoidal collinear

Cycloidal non-coll.

js=0 jsjsj =0

js=

js= =


Magnetic scattering

0

1

2

3

4

5RCPLCP

0 30 60 90 120 150 180

Phot

on in

tens

ity I(η

) (a

rb. u

nits

)

P1 = 0P2 = ± sin2θ

η (deg)

Interference Ch.-Mag.

0

1

2

3

4

0 30 60 90 120 150 180

RCPLCP

η (deg)

Thom./Magn.=0.2

Phot

on in

tens

ity I(η)

(a

rb. u

nits

)

I(η)=I0/2 (1+P1sin(2η)+P2cos(2η)) Stokes-Poincaré parameters: P1= linear polarization P2= oblique linear polarization P3=1-√(P1

2+P22) = circular polarization

P2~1 P3~1

Charge scattering

0

1

2

3

4

5RCPLCP

0 30 60 90 120 150 180

P1=(1-cos22θ)/(1+cos22θ) P2 = 0

Phot

on in

tens

ity I(η

) (a

rb. u

nits

)

η (deg)

RCP= Right Circular Polarization LCP= Left Circular Polarization


F. Fabrizi, H.C. Walker, L. Paolasini, F, de Bergevin, A.T. Boothroyd, D. Prabhakaran and D. McMorrow, Phys. Rev. Letters 102 (2009) 237205

Scientific background -  Model system for typoe-II multiferroics -  Magnetic moments lie in {b,c} plane, P//c -  Incommensurate and non collinear cycloidal order -  Flop of magnetic cycloidal plane at high H (P//c -> P//a) Experimental results -  Switching of magnetic domains by E-field -  Precise sublattice magnetization and phase relationship between Mn and Tb moments


H. Walker et al., SCIENCE 333, 1273 (2011)

Magnetic field induced femtoscale displacements in in TbMnO3 revealed the strong coupling between magnetism and ferroelectricity

H=0 H= Circularly polarized x-ray magnetic diffraction under applied magnetic field

Magnetic field induces a charge displacement wave with the same periodicity of magnetic cycloidal structure (magneto-elastic coupling)

Exploiting the interference interference between charge and magnetic x-ray scattering, the oriented atomic displacements could be measured with an unprecedented accuracy.

A breakthrough in understanding materials for next-generation electronic devices.


E1 = Electric dipole transitions (L=1) E2 = Electric quadrupole transitions (L=2) k’, ε’ k, ε

!ω !ω

L3 edge

E1: 2p3/2 -> 5d5/2 E2: 2p3/2 -> 4f7/2

- Enhancement of scattering amplitude near absorption edge - Excitation of a inner-shell electron into an empty valence state - Sensitivity to the local degeneration of valence-electron states

•  Local symmetries of bound electrons •  Tensorial structure factor •  Forbidden lattice reflections •  Polarization effects (links with magneto-optics) •  Mixing diffraction and atomic spectroscopy


J.P. Hannon, G.T. Trammel, M. Blume, and D. Gibbs, Phys. Rev. Lett. 61 (1988) 1245;62 (1989) 263 (E).

- Expansion of spatial part of vector potential in spherical harmonics YLM - Spherical symmetry SU(2) broken by an axial vector - Cubic and centro-symmetric local symmetries

Resonant strength Geometrical and polarization dependence

L=1 => Electric dipole E1 L=2 => Electric quadrupole E2

Matrix elements


Dominant terms in RXS amplitudes

Charge scattering

Magnetic dipole

Electric quadrupole

- Polarization dependence for magnetic dipole: •  Vertical scattering geometry •  σ-incident polarization

z1 (z3) magnetic dipole component along u1 (u3)

M z3

z1 zz


NpRu2Si2

U0.5Np0.5Ru2Si2

AF(0,0,4.808) T=1.55K

Actinide Sample: mounted on 2x2 mm2 Ge(111) wafer volume 0.1 mm3, 30 µg Np Element selectivity and sublattice magnetization Np and U at M4,5 edges

Branching ratios between M4-M5 edges Electronic ground state Exchange and spin-orbit coupling

Ge single crystal wafer fixed onto the Cu stopper

Resonant magnetic scattering in (U0.5Np0.5)Ru2Si2 solid solution E. Lidstrom et al., Phys. Rev. B 61, 1375 (2000)


E=7.109keVσ−π, MgO(222)

FeK-edge

-150 -100 -50 0 50 100 150Azimuthal angle ψ (deg)

0.00

0.01

0.02

0.03

7.100 7.105 7.110 7.115 7.120Energy (keV)

Phot

/mon

*2*

µ(E)

Inte

grat

ed in

tens

ities

(arb

. uni

ts)

0

2

4

6

8

0

5

10

15

20

E=5.719 keVσ−π, LiF(220)

CeL3-edge

0

1

2

5.70 5.71 5.72 5.73 5.74 5.75

Phot

/mon

*2*

µ(E)

Energy (keV)

( )52

52

52

( )52

52

52

( )52

52

32•  Laves Phase structure (Fd-4m)

•  In pure CeFe2 AF short range fluctuations cohexist with a nominal F state

•  Co doping stabilize AF ground state Experimental results •  Azimuthal dependence at Ce L3-edge and

at Fe K-edge •  Individual sublattice magnetization and

non collinear magnetic structure of Fe •  Geometrical frustration of Fe sublattice

(pyroclore sublattice)

On the magnetic ground state of Ce(CoxFe1-x)2

L. Paolasini, et al., Phys. Rev. Lett. 90 (2003) 57201; ibid: Phys. Rev. B 77 (2008) 094433


E2 = 1s -> 3d E1 = 1s -> 4p

Interplay between orbital and magnetic ordering in KCuF3 R. Caciuffo, et al., Phys. Rev. B 63 (2002) 174425; ibid. L. Paolasini, Phys. Rev. Letters 88 (2002) 106403.

Cu K-edge

Scientific background - Mott-Hubbard insulator - Model system for orbital ordering

Experimental results - Orbital and AF order strictly related. - OO of dy

2-z

2 - dx2-z

2 type with qOO=<111>. - ATS due to the difference in the 2px(y) DOS (Jahn-Teller distortion)

Violation of the extinction rules

f(N1)+f(N2)exp[iπh]

N2

N1


Direct observation of charge order in epitaxial NdNiO3 films Staub U. et al., Phys. Rev. Letters 88 (2002) 126402.

- Prototype of bandwidth-controlled metal-insulator -  Metal/insulator transition TMI=150-170K

Experimental results - Strong enhancement of RXS at Ni K-edge on the

forbidden charge reflection (105) - ATS due to a charge disproportionation at Ni3+ site

Nd3±δ where δ~0.45±0.04

Violation of the extinction rules

f(N1)≠f(N2)

N2

N1


RMXS Gd L2

Pol. Neutrons

E. Kravtsov, et al., Phys. Rev B 79 (2009) 134438. Complementary polarized neutron and resonant x-ray magnetic reflectometry

RMX reflectivity (circular polarization) Polarized neutron reflectivity

Element-specific magnetization profile in the multilayer (Fe35 Å /Gd50 Å)5

Asymmetric termination Fe-top, Gd-bottom lead to unique low-temperature magnetic phases.

Significant twisting of Fe and Gd magnetic moments

Nonuniform distribution of vectorial magnetization within Gd layers. A

Dipole-Dipole (E1-E1)

Dipole-Quadrupole (E1-E2)

Quadrupole-Quadrupole (E2-E2)


P. Carra and B. T. Thole, Rev. Mod. Phys. 66, 1509 (1994) Extension of RXS to all the local symmetries E2-E2 Interpretation of forbidden reflection of Fe2O3 in term of charge multipoles

I. Marri and P. Carra, Phys. Rev. B 69, 113101 (2004) E1-E2 events for non centrosymmetric systems Interpretation of dichroic signals (parity breaking symmetries)

Symmetry transformation of coordinate system Space inversion (parity): •  Tensors D and Q are even and I odd => Dipole-Quadrupole transitions (E1-E2) allowed for “resonant” atom breaking the site inversion symmetry Rotation: •  Decomposition of tensor elements (Dαβ, Iαβψ or Qαβψδ) in its irreducible components T(j) (with dimension 2j+1)

Ex: For Dipole Dipole (E1-E1)

Time reversal: (exchange k and k’ wavevector directions)

⇒ Exchange of αβγδ indexes • j=1,3 pure magnetic terms: antisymmetric with respect to the time-reversal symmetry


Irrotational (rotor free) Solenoidal (divergenceless)


Permanent currents j(x): Magnetic energy

Charge multipoles

(

Stationary states Stationary states Stationary states

Toroidal moment M Poloidal moment T

ψ(x) scalar χ(x) pseudo-scalar

Dubovik, V.M. & Tugushev, V.V., Physics Reports 187, 145-202 (1990) Di Matteo, J. Phys. D: Appl. Phys. 45 163001 (2012).

Electrostatic energy Charge distribution ρ(x):


S. Di Matteo, Y. Joly, C.R. Natoli, Phys. Rev. B 72, 144406 (2005)

Product of irreducible spherical tensors Xq and Fq. The rank q depends on the order of multipole in the EM field expansion:

P-T- Magneto-electric P-T- Magneto-electric Magneto-electric

P+T- Magnetic P+T- Magnetic Magnetic

P+/-T+ Electric/charge P+/-T+ Electric/charge Electric/charge

Experiments: L. Paolasini, S. Di Matteo, C. Vettier, F. de Bergevin, A. Sollier, W. Neubeck, F. Yakhou, P.A. Metcalf and J. Honig, J. Electron Spectrosc. Relat. Phenom. 120, 1 (2001) Theory: S. Di Matteo, Y. Joly, A. Bombardi, L. Paolasini, F.de Bergevin, and C.R. Natoli, Phys. Rev. Lett. 91, 257402 (2003) Y. Joly, S. Di Matteo and C.R. Natoli, Phys. Rev. B 69, 224401 (2004)

AF reflections km+lm=even and hm=even

Special reflections km+lm=even and hm=odd

ATS (h0.-h)m hm=2n+1

Electric quadrupole F2(E1-E1) (dominant contribution)

Magnetic dipole and magnetic quadrupole F1(E1-E1)+F2-(E2-E2)

Polar toroidals and magnetic quadrupole F(1,3)-(E1-E2) and F2(E2-E2)

ATS (00.l)H lH=3(2n+1)

Electric octupole and hexadecapole F3(E1-E2)+F4(E2-E2)


, Phys. Rev. B 69, 224401 (2004)

magnetic quadrupole


X-ray diffraction: - Disentangle the individual site contributions - High Q-resolution to resolve small splitting of magnetic Bragg peaks - Small scattering volumes (small samples, thin films) - Versatility of scattering geometry (structure factor)

Non-resonant magnetic x-ray scattering: - Determination of L/S ratio of ordered magnetic phases - Single magnetic domain imaging

Resonant X-ray scattering: - Chemical and shell selectivity - RXS: unique information about electronic and magnetic order parameters - Orbital selectivity and tensorial structure factor - High order multipoles, orbital ordering, local site anisotropies

X-ray polarization analysis and control: - Single out the individual scattering amplitude contributions (f0, fmagn, fRXS) - Circular polarimetry: - Handedness of allows the determination of helicity, absolute phase - Linear polarimetry: - Disentangle the magnetic and RXS contributions

luigi paolasini paolasini@esrf · x-ray scattering (3-30 kev) - surface sensitivity (high absorp.,...

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