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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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
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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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The neutron as a probe of condensed matter
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
The neutron as a probe of condensed matter
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
The neutron as a probe of condensed matter
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
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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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Scattering processes
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Fermi’s Golden rule
Energy conservation
Interaction potential
Energy conservation
Sum of nuclear and magnetic scattering
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Scattering processes
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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
Scattering processes
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
with the scattering amplitude with the scattering amplitude
magnetic form factor of the free ion
Scattering processes
p= 0.2696x10-12 cm
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
electron i
neutron
with the scattering amplitude with the scattering amplitude
Scattering processes
= 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∗
j (0)Aj(t)e−iQRj (0)ei
QRj(t)e−iωtdt
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Separation elastic/inelastic: Keeps only the time-independent (static) terms in the cross-section and integrate over energy elastic scattering
ESMF2010, L’Aquila
Scattering processes
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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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Outline of the lecture
Scattering processes Diffraction by a crystal
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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)
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Nuclear diffraction
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Diffraction from a crystal
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
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Magnetic diffraction
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Diffraction from a crystal
magnetic ordering may not have same periodicity as nuclear one propagation vector periodicity and propagation direction propagation vector
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Magnetic diffraction
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Diffraction from a crystal
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 diffraction
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Diffraction from a crystal
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
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Magnetic diffraction: classification of magnetic structures
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Diffraction from a crystal
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
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Non Bravais lattice =0 ⇒ ferromagnetic structure or not Arrangement of moments in cell intensities of magnetic peaks
Magnetic diffraction: classification of magnetic structures
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Diffraction from a crystal
If =0, magnetic/nuclear structures same periodicity Bragg peaks at reciprocal lattice nodes
If =0, magnetic/nuclear structures same periodicity
=0
Direct space Reciprocal space
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Magnetic diffraction: classification of magnetic structures
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Diffraction from a crystal
Ex. , antiferromagnetic structure
If ≠0, magnetic satellites at If
Ex. , antiferromagnetic structure
Irrational , incommensurate magnetic structure , incommensurate magnetic structure
Direct space Reciprocal space
Rational , commensurate magnetic structure
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Diffraction from a crystal
Magnetic diffraction: classification of magnetic structures
and
Sine wave modulated and spiral structures
Direct space Reciprocal space Direct space Reciprocal space
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Diffraction from a crystal
Magnetic diffraction: classification of magnetic structures
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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Diffraction from a crystal
Magnetic diffraction: classification of magnetic structures
Single- and multi- magnetic structures :
ex canted arrangement
Single- and multi- magnetic structures : Single- and multi- magnetic structures :
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Diffraction from a crystal
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
Diffraction from a crystal
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
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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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Diffraction from a crystal
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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Diffraction from a crystal
Example: YMnO3 Mn-O distance variation Coupling spin lattice at TN
Lee et al., PRB 2005
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Diffraction from a crystal
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
Diffraction from a crystal
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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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Outline of the lecture
Inelastic scattering
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Inelastic scattering from a crystal
Reminder on phonons
ESMF2010, L’Aquila
Displacement from equilibrium harmonic forces
Normal modes (stationary waves) Characterized by wave vector Frequency Polarization
longitudinal transverse
Frequency related to by dispersion relation
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Inelastic scattering from a crystal
Reminder on phonons
ESMF2010, L’Aquila
diatomic chain
Acoustic modes
Optic modes Optic modes
Displacement from equilibrium harmonic forces
Normal modes (stationary waves) Characterized by wave vector Frequency Polarization
longitudinal transverse
Frequency related to by dispersion relation
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Inelastic scattering from a crystal
Reminder on phonons
ESMF2010, L’Aquila
Displacement from equilibrium harmonic forces
3p-3 optic branches
3 acoustic branches
ω-scans at constant at constant
-scans at constant ω -scans
p atoms/unit cell
Normal modes (stationary waves) Characterized by wave vector Frequency Polarization
longitudinal transverse
Frequency related to by dispersion relation
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Inelastic scattering from a crystal
Reminder on phonons
ESMF2010, L’Aquila
Quantum description: modes=quasi-particles called phonon
Creation/annihilation processes in cross-section
dynamical nuclear structure factor
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Inelastic scattering from a crystal
ESMF2010, L’Aquila
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
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Inelastic scattering from a crystal
Crystal with p atoms/unit cell: p branches
ESMF2010, L’Aquila
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
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ESMF2010, L’Aquila
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
Inelastic scattering from a crystal
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Inelastic scattering from a crystal
Instruments: triple-axis
Measure scattered neutrons as a function of and as a function of
Detector Analyser
Monochromator
≠
ESMF2010, L’Aquila
Sample
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Inelastic scattering from a crystal
ESMF2010, L’Aquila
Instruments: time-of-flight
Neutron pulses chopped from a monochromatic beam Arrival time and position on multi detector give final energy and Q
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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
Inelastic scattering from a crystal
ESMF2010, L’Aquila
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
ESMF2010, L’Aquila
Inelastic scattering from a crystal
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Inelastic scattering from a crystal
ESMF2010, L’Aquila
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
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Inelastic scattering from a crystal
ESMF2010, L’Aquila
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)
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Inelastic scattering from a crystal
ESMF2010, L’Aquila
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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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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ESMF2010, L’Aquila
Outline of the lecture
Examples Polarized neutrons
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ESMF2010, L’Aquila
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
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ESMF2010, L’Aquila
Use of polarized neutrons
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
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ESMF2010, L’Aquila
Use of polarized 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
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ESMF2010, L’Aquila ESMF2010, L’Aquila
Use of polarized neutrons
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
ESMF2010, L’Aquila
Use of polarized neutrons
Polarization matrix P final polarization
incident polarization
Spherical polarization analysis
CRYOPAD Zero magnetic field chamber
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ESMF2010, L’Aquila
Use of polarized neutrons
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
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ESMF2010, L’Aquila
Chirality and ferroelectricity
Use of polarized neutrons
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)
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ESMF2010, L’Aquila
Example: Ba3NbFe2Si2O14
Static Chirality
Chirality and ferroelectricity
Use of polarized neutrons
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)
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ESMF2010, L’Aquila
Calculated vs measured final polarization
Spherical polarization analysis with CRYOPAD
A single possibility is selected (+1,-1)
Chirality and ferroelectricity
Use of polarized neutrons
Link with electrical polarization?
single chirality
Marty et al. PRL (2008)
Example: Ba3NbFe2Si2O14
Static Chirality
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ESMF2010, L’Aquila
Totally chiral spin wave branch measured by longitudinal polarization analysis
Chirality and ferroelectricity
Use of polarized neutrons
measured calculated Chiral scattering
Example: Ba3NbFe2Si2O14
Dynamical Chirality
Loire et al. submitted
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Chirality and ferroelectricity
Use of polarized neutrons
Triangular magnetic structure ⇒ 2 domains of triangular chiralities ∃ PE at TN ⇒ proportional to the chirality domain unbalance
Kenzelmann et al. PRL (2007)
ESMF2010, L’Aquila
RbFe(MoO4)2
TbMnO3: electric control of spin helicity in a magnetic ferroelectric
Yamasaki et al. PRL (2007) (polarized neutrons)
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ESMF2010, L’Aquila
Domains in multiferroics/magnetoelectrics
Use of polarized neutrons
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
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ESMF2010, L’Aquila
Use of polarized neutrons
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
Domains in multiferroics/magnetoelectrics
TN=78 K
allows linear ME effect
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ESMF2010, L’Aquila
Use of polarized neutrons
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
Domains in multiferroics/magnetoelectrics
Example: MnPS3
Spherical polarization analysis
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ESMF2010, L’Aquila
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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ESMF2010, L’Aquila
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)