Neutron scattering a probe for multiferroics and...

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Neutron scattering a probe for multiferroics and magnetoelectrics

V. Simonet Institut Néel, CNRS/UJF, Grenoble, France

ESMF2010, L’Aquila

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Outline of the lecture

The neutron as a probe of condensed matter Properties Sources Environments Scattering processes

Diffraction by a crystal Nuclear, magnetic structures

Instruments type Examples

Inelastic scattering Phonons, spin waves, localized excitations Instruments type Examples

Polarized neutrons Longitudinal, spherical polarization analysis Examples

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The neutron as a probe of condensed matter

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Properties:

Neutron: particle/plane wave with and

Wavelength of the order of few Å ≈ interatomic distances  diffraction condition

Energies of thermal neutrons ≈ kBT ≈ 25 meV of the order of the excitations in condensed matter

  Neutral: probe volume, nuclear interaction with nuclei

Subatomic particle discovered in 1932 by Chadwik First neutron scattering experiment in 1946 by Shull

Neutron: particle/plane wave with and

Properties:

  carries spin ½: sensitive to spins of unpaired electrons  probe magnetic structure and dynamics

possibility to polarize neutron beam

Better than XR for Light or neighbor elements Neutron needs big samples

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neutrons Scattering length

Z Z

X rays

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Two types of neutron sources for research:

neutron reactor (ex. ILL Grenoble 57 MW), spallation sources (ex. ISIS UK 1.5 MW)

235U


FISSION SPALLATION

U, W, Hg

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Various environments :

Temperature: 15 mK-700 K

High magnetic fields up to 15 T (D23, ILL) Pulsed fields up to 30 T (IN22, ILL)

Pressure (gas, Paris-Edinburgh, clamp cells) up to 30 kbar

Electric field

CRYOPAD zero field chamber for polarization analysis

Pressure (gas, Paris-Edinburgh, clamp cells) up to 30 kbar


D23@ILL 15 T magnet

CRYOPAD

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The neutron as a probe of multiferroics and magnetoelectrics

  Complex magnetic structures spirals, sine waves modulated, incommensurate   Link with atomic positions   Complex phase diagrams (T, P, H, E)   Hybrid excitations: electromagnons   Domains manipulation under E/H fields Chirality determination and manipulation

Microscopic probe of atomic positions magnetic arrangements related excitations Example:

orthorhombic RMnO3

Goto et al. PRL (2005)

ferroelectric

Scattering processes

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Counts neutrons scattered by sample in solid angle dΩ for energy transfer

Cross section in barns (10-24 cm2) after integration on energy and solid angle

Counts neutrons scattered by sample in solid angle dCounts neutrons scattered by sample in solid angle d

Scattering vector

sample

detector

Inelastic scattering Ef2>Ei |kf2|>|ki|

Incident neutrons E, ki

Inelastic scattering Ef1<Ei |kf1|<|ki|

Elastic scattering Ef=Ei |kf|=|ki| Q=2sinθ/λ

2θ

dΩ


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Fermi’s Golden rule

Energy conservation

Interaction potential

Energy conservation

Sum of nuclear and magnetic scattering


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Sum of nuclear and magnetic scattering

Interaction potential very short range isotropic

Interaction potential  Longer range (e- cloud)  Anisotropic

b Scattering length depends on isotope and nuclear spin

dipolar interaction: neutron magnetic moments with magnetic field from unpaired e-

Orbital contribution Spin contribution

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nucleus j

neutron

electron i

neutron

with the scattering amplitude with the scattering amplitude

Projection of the Magnetic moment _ to

maximum intensity for for isotropic


Projection of the Magnetic moment _ to Magnetic moment _ to

d2σ

dΩdE=

kfki

(1

2π )

j,j

∞

−∞A∗

j (0)Aj(t)e−iQRj (0)ei

QRj(t)e−iωtdt

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nucleus j

neutron

Scattering experiment FT of interaction potential

electron i

neutron


magnetic form factor of the free ion


p= 0.2696x10-12 cm

d2σ

dΩdE=

kfki

(1

2π )

j,j

∞

−∞A∗


QRj(t)e−iωtdt

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nucleus j

neutron

electron i

neutron



= Double FT in space and time of the correlation function of the nuclear magnetic |

density in and magnetic |

density in and magnetic |

density in and

d2σ

dΩdE=

kfki

(1

2π )

j,j

∞

−∞A∗


QRj(t)e−iωtdt

Separation elastic/inelastic: Keeps only the time-independent (static) terms in the cross-section and integrate over energy elastic scattering








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Scattering processes Diffraction by a crystal

reciprocal lattice

Nuclear diffraction

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Diffraction from a crystal

Coherent elastic scattering from crystal Bragg peaks at nodes of reciprocal lattice

crystal lattice

diffraction condition (lattice)

Nuclear diffraction

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with

Nuclear structure factor Information on atomic arrangement inside unit cell

for a Bravais lattice: 1 atom/unit cell

Non Bravais lattice: ν atoms/unit cell

FN ( Q) = b FN ( Q) =

ν

bνeiQ.rν

Magnetic diffraction

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magnetic ordering may not have same periodicity as nuclear one propagation vector periodicity and propagation direction propagation vector


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magnetic ordering may not have same periodicity as nuclear one propagation vector periodicity and propagation direction

for a non-Bravais lattice

propagation vector

diffraction condition

Bragg peaks at satellites positions


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magnetic ordering may not have same periodicity as nuclear one propagation vector periodicity and propagation direction

for a non-Bravais lattice

propagation vector

magnetic structure factor: info. on magnetic arrangement in unit cell

Fourier component associated to Magnetic moment of

atom ν in nth unit cell

p= 0.2696x10-12 cm FM ( Q = H + τ) = p

fν( Q)mν,τe

iQ.rν

Magnetic diffraction: classification of magnetic structures

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If =0, magnetic/nuclear structures same periodicity Bragg peaks at reciprocal lattice nodes

If =0, magnetic/nuclear structures same periodicity

Bravais lattice =0 ⇒ ferromagnetic structure

Direct space Reciprocal space

Non Bravais lattice =0 ⇒ ferromagnetic structure or not Arrangement of moments in cell intensities of magnetic peaks


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If =0, magnetic/nuclear structures same periodicity Bragg peaks at reciprocal lattice nodes

If =0, magnetic/nuclear structures same periodicity

=0



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Ex. , antiferromagnetic structure

If ≠0, magnetic satellites at If

Ex. , antiferromagnetic structure

Irrational , incommensurate magnetic structure , incommensurate magnetic structure


Rational , commensurate magnetic structure

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and

Sine wave modulated and spiral structures

Direct space Reciprocal space Direct space Reciprocal space

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and

Kenzelmann et al., PRL 2007 Example TbMnO3

28 K<T<41 K : incommensurate sine wave modulated paraelectric

T<28 K : commensurate spiral (cycloid) ferroelectric

Sine wave modulated and spiral structures

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Single- and multi- magnetic structures :

ex canted arrangement

Single- and multi- magnetic structures : Single- and multi- magnetic structures :

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Solving a magnetic structure

Help with group theory and representation analysis Use of rotation/inversion symmetries to infer possible magnetic

arrangements compatible with the symmetry group that leaves invariant (pgm like basireps…) (cf. L. Chapon’s lecture)

Find the propagation vector (periodicity of magnetic structure): powder diffraction difference between measurements < and > Tc. Indexing magnetic Bragg reflections with

Refining Bragg peaks intensities moment values and Magnetic arrangement of atoms in the cell (using scaling factor from nuclear structure refinement) with programs like Fullprof, CCSL, MXD…

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Techniques: powder diffractometers

D2B@ILL


Fixed θ, varying λ: time-of-flight diffractometer (spallation source)

Fixed λ, varying θ (or multidetector): 2-axes diffractometer (neutron reactor)

Bragg’s law

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Techniques: single-crystal diffractometers Diffraction from a crystal

Bring a reciprocal lattice node in coincidence with then measure its integrated intensity (rocking curve)

4-circles mode lifting arm (bulky environments)

detector

Each reciprocal space point accessible by set up positioning

Usually better signal/noise

More information to solve complex magnetic structures

Sometimes problem with magnetic domains

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Techniques: difference powder/single-crystal diffraction Diffraction from a crystal

All reciprocal space accessible in same acquisition

Best to find propagation vector

Sometimes not enough information to solve structure (Several Bragg peaks at at same positions)

accessible by set up positioning

(Several Bragg peaks at at same positions)

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Purpose : solve nuclear and magnetic structures, distortion under external parameters, variation of unit cell…

Example: YMnO3 Influence of AF transition on on ferroelectric order Influence of AF transition on

Powder diffraction : Nuclear & magnetic structures Mn-O distance variation

Lee et al., PRB 2005

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Example: YMnO3 Mn-O distance variation Coupling spin lattice at TN

Lee et al., PRB 2005

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b

a

Example: Ba3NbFe2Si2O14 powder and single-crystal diffraction Complex magnetic structure

Powder diffraction Propagation vector =(0, 0, ≈1/7) Propagation vector =(0, 0,

triangular lattice of Fe3+ triangles, S=5/2

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Example: Ba3NbFe2Si2O14

120° moments on triangles in (a, b) plane

Helices propagating along c

Single-crystal diffraction

Marty et al., PRL 2008







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Inelastic scattering

Inelastic scattering from a crystal

Reminder on phonons


Displacement from equilibrium harmonic forces

Normal modes (stationary waves) Characterized by wave vector Frequency Polarization

longitudinal transverse

Frequency related to by dispersion relation


Reminder on phonons


diatomic chain

Acoustic modes

Optic modes Optic modes






Reminder on phonons



3p-3 optic branches

3 acoustic branches

ω-scans at constant at constant

-scans at constant ω -scans

p atoms/unit cell





Reminder on phonons


Quantum description: modes=quasi-particles called phonon

Creation/annihilation processes in cross-section

dynamical nuclear structure factor



Reminder on spin waves

Spin waves as elementary excitations of magnetic compounds

Spin waves in ferromagnet

Characterized by wave vector Frequency Certain spin components involved Frequency related to by dispersion relation

= transverse oscillations in relative orientation of the spins

1 atom/unit cell


Crystal with p atoms/unit cell: p branches


Quantum description: spin wave mode=quasi-particle called magnon Creation/annihilation processes in cross-section

1 atom/unit cell

dynamical magnetic structure factor

(0 1 l)

Reminder on spin waves

Spin waves in antiferromagnet


Intensity max for and zero for

Form factor Form factor with

Intensity maximum for Intensity maximum for

Phonons and magnons

Intensity max for

Transition between energy levels : non dispersive signal Example crystal field excitations in rare-earth ions

Localized excitations

Nuclear excitations Magnetic excitations



Instruments: triple-axis

Measure scattered neutrons as a function of and as a function of

Detector Analyser

Monochromator

≠


Sample



Instruments: time-of-flight

Neutron pulses chopped from a monochromatic beam Arrival time and position on multi detector give final energy and Q

Powder (sometimes single-crystal)

t-scans at fixed 2θ

Multidetector (all reciprocal space accessible in one acquisition)

Resolution 1%

Used for localized excitations Often not enough to analyze collective excitations



Instruments: Time-of-Flight versus Triple-Axis

Single-crystal (sometimes powder)

Positioning at each points single-detector

Scattering plane accessible only

Resolution 5%

Use to study collective excitations

Resolution 1% Resolution 5%

Positioning at each points single-detector

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Purposes of inelastic scattering experiments:

Nuclear: Information on elastic constants, sound velocity, Structural instabilities…

Magnetic: Information on magnetic interactions and microscopic mechanisms yielding the magnetic order…

In multiferroics: Spin-lattice coupling, hybrid modes ex. electromagnons





Fabrèges et al., PRL 2009 Example: hexagonal RMnO3 When spin waves helps solving magnetic structures When spin waves helps solving magnetic structuresDifferent possible Mn 120° magnetic orders: impossible to distinguish with unpolarized neutrons diffraction

depends on the two Jz exchange interactions: Given by spin waves

R crystal field levels transitions



Petit et al., PRL 2008

Oxygens

Mn

Spin wave dispersion

Phonon mode polarized along the ferroelectric c axis

Example: YMnO3 Spin and lattice excitations

(q 0 12)

(q 0 6)



Example: YMnO3 Petit et al., PRL 2008

Opening of a gap in phonon mode below TN locked on the spin wave mode

Resonant interaction between both modes = Strong spin-lattice coupling






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Examples Polarized neutrons


Use of polarized neutrons

Cross section depends on the spin state of the neutron. Polarized neutron experiment uses this spin state and its change upon scattering process to obtain additional information

Polarization P = two spin state populations’ difference



Blume-Maleyev equations

Information about correlations between nuclear-magnetic terms, between different spin components: ex. chiral term

P0 initial polarization Pf final polarization

Unpolarized neutrons



Select spin state and direction of incident neutron

Principle of an experiment

Analyze polarization of scattered beam in same direction: longitudinal polarization analysis

Analyze polarization of scattered beam in any direction: spherical polarization analysis

Z

Y X

ESMF2010, L’Aquila ESMF2010, L’Aquila


Longitudinal polarization analysis

Separation magnetic/nuclear access to spin components Mx, My, Mz Separation coherent/incoherent Access to chirality and nuclear/magnetic interference terms

Helmotz coils

N N N

C

Incident beam

sample Scattered beam

Detector

Heussler

Heussler monochromator

Analyzed beam

NSF

SF

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→ refine complex magnetic structure → Access to non-diagonal terms of the polarization matrix → probe domains



Polarization matrix P final polarization

incident polarization

Spherical polarization analysis

CRYOPAD Zero magnetic field chamber



Example: YMnO3 Spin flip: magnetism

Non Spin flip: nuclear

Hybrid Goldstone mode of multiferroic phase “spin waves dresses with atomic fluctuations”

T<TN

Pailhès et al., PRB 2009

Longitudinal polarization analysis T>TN


Chirality and ferroelectricity


Chirality: sense of spin precession in spiral or triangular arrangements

Magnetic spiral order ⇒ ferroelectricity with Katsura et al. PRL (2005)

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



Static Chirality



2 chiralities associated to helices and triangular arrangements 4 possible chiralities (+1,+1), (-1,-1), (-1,+1), (+1,-1)

Unpolarized neutron single-crystal diffraction 2 possible magnetic structures (+1,-1) and (-1,+1)


Calculated vs measured final polarization

Spherical polarization analysis with CRYOPAD

A single possibility is selected (+1,-1)



Link with electrical polarization?

single chirality

Marty et al. PRL (2008)


Static Chirality


Totally chiral spin wave branch measured by longitudinal polarization analysis



measured calculated Chiral scattering


Dynamical Chirality

Loire et al. submitted

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Triangular magnetic structure ⇒ 2 domains of triangular chiralities ∃  PE at TN ⇒ proportional to the chirality domain unbalance

Kenzelmann et al. PRL (2007)


RbFe(MoO4)2

TbMnO3: electric control of spin helicity in a magnetic ferroelectric

Yamasaki et al. PRL (2007) (polarized neutrons)


Domains in multiferroics/magnetoelectrics


Cabrera, PRL 2009

Ni3V2O8: coupled magnetic and ferroelectric domains (longitudinal polarization analysis)

YMn2O5: Electric field switching of antiferromagnetic domains (spherical polarization analysis)

Radaelli, PRL 2008

PYX



Example: MnPS3

with non diagonal tensor

in the (a, b, c*) frame

Magnetic group 2’/m Collinear antiferromagnet with =0 allows ferrotoroidicity Collinear antiferromagnet with =0

Brown JPCM 1998 cf. antiferromagnetic domains handling in linear magnetoelectric Cr2O3


TN=78 K

allows linear ME effect



sample

electrodes

Antiferromagnetic/ferrotoroidic domains manipulated in crossed E and H fields when cooled through TN

Ressouche et al. PRB (2010)

Rela

tive

dom

ain

popu

lati

on


Example: MnPS3

Spherical polarization analysis

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What I did not speak about:

Debye-Waller factor omitted in the scattering expressions Incoherent versus coherent Diffuse scattering (short-range order) reflectometry and small angle scattering: heterostructures flipping ratio technique: magnetization density map …

Thanks to B. Grenier and E. Ressouche (CEA-Grenoble, France) Institut Néel coll. M. Loire (PhD), R. Ballou, S. de Brion, E. Lhotel

Also possible to study films Neutron diffraction study of hexagonal manganite YMnO3, HoMnO3, and ErMnO3 epitaxial films Gélard et al APL (2008)

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Bibliography

“thermal neutrons scattering”, G. L. Squires, Cambridge University Press (1978).

“Neutrons Polarisés”, Es. Kernavanois, Ressouche, Schober, Soubeyroux, EDP Sciences, (in english).

“Neutron Physics” J. Rossat-Mignod, Eds Skold and Price, Academic Press (1987)

“Theory of Neutron scattering from Condensed Matter”, S. W. Lovesey, Oxford, Clarendon Press (1984)

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