Nonlinear Optics on Ferroics & Multiferroics

Nonlinear Optics on Ferroics & Multiferroics Introduction Part I – Symmetry Part II – Nonlinear Optics Part III – Experimental Techniques Part IV – Multiferroic RMnO3 Thomas Lottermoser

Principle of Nonlinear Optical Structure Analysis Material structure

point group

Symmetry

selection rules

Symmetry

Nonlinear optical signal

polarization

Symmetry and Multiferroics Time

Space

invariant

invariant

change

ferroelastic

ferroelectric

ferromagnetic ferrotoroidic changeFerroic properties Time

determine Space & time invariant change space symmetry +- ++- +-

invariant change

N S

Symmetry and Nonlinear Optics Case 1:

IR light

• Crystal in a paraelectric phase • Inversion symmetry not broken

+++++

• Crystal in a ferroelectric phase • Inversion symmetry broken

-----

Case 2:

IR light IR light + visible light

Literature • General introductions to nonlinear Optics: Y.R. Shen, A. Yariv, or R.W. Boyd • K.H. Bennemann: Nonlinear Optics in Metals • R.R. Birss: Symmetry and Magnetism

Part I - Symmetry • • • •

Symmetry Operations Space & Time Inversion Tensors Point Groups

What’s Symmetry? Nature:

Art:

Architecture:

I. Symmetry is a property related to harmony and aesthetics

II. Symmetry describes the behaviour Geometry: Astronomy: Crystallography: under certain transformations

Pictures: en.wikipedia.org & commons.wikimedia.org

What’s Symmetry? Symmetry: Conservation of shape under applying a symmetry operation Asymmetry: Change of shape under applying a symmetry operation

e.g. mirror symmetry

Symmetry Operations in Space Inversion: I

Rotation:

Mirroring: (= I • R)

Mathematical Matrix Representation Inversion: I

Rotation:

− 1 0 0  I =  0 − 1 0  ≡ 1  0 0 − 1

[]

 cos(ϕ y ) 0 sin(ϕ y )    R(ϕ y ) =  0 1 0  − sin(ϕ y ) 0 cos(ϕ y )   − 1 0 0  ⇒ R (180°) =  0 1 0  ≡ 2 y  0 0 − 1

[ ]

Mirroring: (= I • R)

1 0 0 M y = 1 ⋅ 2 y = 0 − 1 0 ≡ 2 y 0 0 1

[ ][ ]

[ ]

Point Symmetries in Crystals Only discrete symmetries in crystals: • Identity operation: [1] • Inversion:

[1 ]

• Rotations:

[1x ], [2 x ], [3x ], [4 x ], [6 x ], etc .

• Mirror planes:

[2 ], [2 ], [2 ], etc . x

y

z

⇒ System of 32 point groups with different subsets of symmetry operations

Symmetry and Physics P. Curie, 1894: "C'est la dissymmétrie qui crée le phénomène" e.g.: Ferroelectricity: I. Paraelectric phase:

II. Ferroelectric phase:

Cubic crystal with inversion symmetry

+++++

transition

-----

Phase

Polar tetragonal crystal without inversion symmetry

Curie’s Principle The symmetry of a crystal exhibiting a certain effect is the intersection of the symmetries of the bare crystal and the effect itself

Crystal symmetry = Gc GC ∩ GE ≡ GCE Effect symmetry = GE

Symmetry of crystal + effect

Neumann’s Principle The physical properties of a crystal have at least t hh ee s sy m y m emt rey t royf toh fe ct rhyes t acl r iyt s et laf !l Example: Ferroelectricity ---

P ---

Inversion

-P +++

+++

Electric polarization breaks inversion symmetry ⇒ No ferroelectricity in crystals possessing inversion symmetry!

Property & Field Tensors Physical effects in crystals can be described by the equation: E(ffect) = P(roperty) · C(ause) E, C: Field tensors P: Property tensor

Anisotropic quantities Symmetry!

Examples: • Dielectric displacement • Magnetic induction

D∝ε·E B∝µ·H

Neumann’s Principle & Property Tensors Property tensors must be invariant under all symmetry operations of the crystal Example: Second harmonic generation (SHG) P(2ω) ∝ χ(2) E(ω) E(ω) Inversion I ⇒ -P(2ω) ∝ χ’(2) (-E(ω))(- E(ω)) ⇒

χ’(2) = -χ(2)

if inversion is symmetry operation

⇒ No SHG in crystals possessing inversion symmetry!

Classification of Tensors: Parity Operations

Symmetry operations with eigenvalues ±1 Spatial inversion Ι Time inversion Τ

Parity Operations: Spatial Inversion

⇒ Polar tensors of odd rank are equal to zero in centrosymmetric crystals!

+++

Ι

P

-P +++

    IP( r , t ) = −P( − r , t)

---

Polar tensors:

---

Axial tensors:

    IM( r , t ) = M( − r , t) ⇒ Axial tensors of even rank are equal to zero in centrosymmetric crystals!

M

Ι j

M j

Parity Operations: Time Inversion i-tensors:

    TP( r , t ) = P( r ,− t)

+++

+++

Τ

P ---

P ---

c-tensors:

    TM( r , t ) = −M( r ,− t) ⇒ c-tensors in non-magnetic crystals are equal to zero!

M

Τ j

-M -j

Time Inversion = Going Back in Time? Answer: NO! Here: Only static effects ↔ no increase of entropy! e.g. electric current: ji = σij Ej ji: c-tensor Ei: i-tensor

⇒

Neumann’s principle: σij must also be a c-tensor!

Only current flow in magnetic crystals!

In this context: Time reversal equivalent to spin reversal!

General Classification of Symmetry Operations 1-, 2-, 3-, 4-, and 6-fold rotations ⇒ 11 Laue-groups + Spatial inversion I

⇒ 32 crystallographic point groups

+ Time inversion T

⇒ 122 magnetic point groups

+ Translation

⇒ 230 crystallographic space groups & 1651 magnetic space groups

Optical regime: λ >> a

Nomenclature of Point Groups System after Hermann-Mauguin: Directly derived from the allowed symmetry operations, but only ‘significant’ subset is used. I. Crystallographic point groups e.g. 3m 3-fold rotation

II. Magnetic point groups e.g. 3m 3-fold rotation Time inversion

Inversion Mirror plane

3m Inversion Mirror plane

3m

‘Forbidden’ Effects What symmetry can not do: • No definite prediction of a certain effect • No microscopic/quantitative description of physical effects What symmetry can do: • A prediction which effects are possible • A prediction which effects are definitely forbidden e.g. no ferroelectricity in centrosymmetric crystals

The Magnetoelectric (ME) Effect 1 Pi = ε 0 χ E j + α ijH j c e ij

1 Mi = χ H j + α ijE j µ0c m ij

Pi, Ej = first rank polar i-tensors, Mi, Hj = first rank axial c-tensors ⇒ αij = second rank axial c-tensor Only allowed in non-centrosymmetric crystals ME forbidden in at least 53 of the 122 magnetic point groups!

Only allowed in magnetic crystals

How to Derive Tensor Components? 1. Be smart and solve for each allowed symmetry operation 3^n equations of the type:

dijk ...n = σ ipσ jqσ kr ...σ nu dpqr ...u 2. Be even smarter and look them up in the book of Birss:

Magnetoelectric Effect in Cr2O3 1. Symmetry and symmetry operations for Cr2O3

The magnetoelectric tensor in Cr2O3: 2. The magnetoelectric tensor α

0   α xx 0   α =  0 α xx 0   0  0 α zz  

3. The components of α

Part II – Nonlinear Optics • • • •

Introduction & overview Second harmonic generation (SHG) Determination of tensor components SHG & (multi-)ferroic order

Nonlinear Optics Ei(ω)

Ea (ω)

χˆ L

Linear optics:

Ei(ω) Nonlinear optics:

Ea (2ω)

χˆ NL

Nonlinear Optics ( 3) Pi (ω ) ∝ χ ij(1) E j (ω1 ) + χ ijk( 2 ) E j (ω1 ) Ek (ω2 ) + χ ijkl E j (ω1 ) Ek (ω2 ) El (ω3 ) + ...

PNL

PL Quadratic effects:

Pockels effect Cubic effects:

ω = 2ω1

Frequency doubling

Pi (ω ) ∝ χ ijk( 2 )E j (ω )Ek (0)

Sum frequency generation ω = ω1 + ω2 + ω3 Difference frequency

ω = 2ω1 − ω2

But: χ(1) ≈ 1, χ(2) ≈ 10-10 cm / V, χ(3) ≈ 10-17 cm2 / V2 Conventional light sources E ≈ 1 V/cm ⇒ Laser

First Observation of Optical Harmonic Generation P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich Generation of Optical Harmonics Phys. Rev. Lett. 7, 118 (1961)

Optical Second Harmonic Generation (SHG) Excited state

SH-source term: SNL(2ω) ∝ PNL(2ω) ∝ χ(2ω)E(ω)E(ω)

E (ω) SNL (2ω) SH intensity: Ground state

ISH ∝ |SNL|2 ∝ |χ EE|2 = |χ|2 I02

PNL(2ω), E(ω): polar tensors of first rank ⇒ χ(2ω) polar tensor of third rank No SHG in centrosymmetric crystals!

The Nonlinear Susceptibility χ

|i〉

For SHG follows from perturbation theory: χˆ (2ω ) ∝ ∑ i, f

|g〉

g | Hˆ (2ω ) | f

(E

f

f | Hˆ (ω ) | i i | Hˆ (ω ) | g

− E g − 2ω )(Ei − E g − ω )

States 〈 i | are real states that are excited with large energy mismatch: ∆E·∆t ~ 

SHG is a coherent, resonant process! e.g. SHG spectra of antiferromagnetic Cr2O3

Phys. Rev. Lett. 73, 2127 (1994)

|f〉

Nonlinear Multipole Contributions Light-matter interaction Hamiltonian:

  ˆ ∝ p⋅A H

with

  ik r e . . A=∑ A c c + k  k

p: Electron impulse operator, A: Light field vector potential

 −1 In crystals usually λ ∝ k >> a (= lattice constant)   ⇒ exp(ik r ) ≅ 1 + ik r + ...

ˆ =H ˆ +H ˆ +H ˆ ⇒ H ED MD EQ Zero order

First order

Nonlinear Multipole Contributions Three nonlinear contributions: Electric dipole (ED): Magnetic dipole (MD):

 NL P ( 2ω ) ∝ χ ED ( 2ω )E(ω )E(ω )  NL M ( 2ω ) ∝ χ MD ( 2ω )E(ω )E(ω )

ˆ NL ( 2ω ) ∝ χ EQ ( 2ω )E(ω )E(ω ) Electric quadrupole (EQ): Q

 ⇒ Multipole expansion of source term S for SHG:  NL  NL ˆ NL     ∂ 2Q   ∂M  ∂P   − µ0  ∇ + µ 0  ∇ × S = µ0 2 2   ∂t ∂t  ∂ t    2

Leading contribution of the order α larger, but maybe symmetry forbidden!

SHG in (Multi-)ferroics Sa et al. Eur. Phys. J. B 14 (2000): χ SHG ( T < TO ) = χ ( T > TO )O

Order parameter Susceptibility in the para phase Susceptibility in the ferroic phase • χSHG is a function of the order parameter O • Non-zero components of χSHG obtained from symmetry of the order parameter and the crystal in the para phase → Curie’s principle

Order Parameter O Properties: • Zero for T>TO, non-zero for T TN )Lz

i-axial, third rank: χyyy= -χyxx = -χxyx = -χxxy c-axial, third rank: χyyy= -χyxx = -χxyx = -χxxy

Antiferromagnetic Cr2O3 Tensor components: χyyy= -χyxx = -χxyx = -χxxy

 2 χ yyy (2ω ) E x (ω ) E y (ω )  2 2 ⇒ P (2ω ) ∝  χ yyy (2ω ) E x (ω ) − E y (ω )  0 

(

    

)

k-direction & polarization selection rules: 1. k||x: Only signal for E||y & P||y 2. k||y: No signal 3. k||z: All components allowed

Set of yes or no type rules to determine symmetry and structure

Generalized Description Higher order contributions of O: χ ( T < TO ) = χ 0 ( T > TO ) + χ1 ( T > TO )O + χ 2 ( T > TO )OO + ...

Multiple order parameters O1, O2,… (TO1 TO1 )O1 + ... = χ 00 ( T > TO 2 ) + χ 01 ( T > TO 2 )O2 + ... + χ10 ( T > TO 2 )O1 + χ11 ( T > TO 2 )O2O1 + ...

... Analogue contributions for ED, MD and EQ: Up to 12 χ-tensors for two order parameter compounds!

SHG in a Multiferroic Compound ED contribution for magnetic ferroelectrics:

P (2ω ) = ε 0 [χˆ (0) + χˆ (℘) + χˆ () + χˆ (℘) + ...] E (ω ) E (ω ) NL

χ(0): χ(P ): χ( ): χ(P  ):

Paraelectric paramagnetic contribution (Anti)ferroelectric contribution (Anti)ferromagnetic contribution Magnetoelectric contribution

always allowed allowed below the respective ordering temperature

• SHG allows simultaneous investigation of magnetic and electric structures • Selective access to electric and magnetic sublattices • Ferroelectromagnetic contribution reveals the magneto-electric interaction between the sublattices

Part III - Experimental Techniques • • • • • • • • •

Spectral sensitivity Freedom of k-direction and light polarizations Temperature variation External magnetic and electric fields Optical phase sensitivity Spatial resolution Transmission & Reflection measurements Surface & interface sensitivity Time resolution

Basic Experimental Setup ns-System:

Tunable light source

Nd:Yag-Laser BBO-OPO

Pumplaser

fs-System: Ti:Saphire-Laser BBO-OPA

Lens Cryostat Sample

Glan Prism Polarizer Filter

Filter

Lens

Analyzer

Spectroscopy & Imaging Setup Magnet cryostat

BBO-OPO

CCD camera

Nd:YAG laser

Optical Parametric Oscillator Passive tuneable laser light source in the range 400nm - 3000nm ωp

ωs

Parametric oscillation of transparent nonlinear crystal with high χ(2)coefficients (here: betabarium-borate β-BaB2O4)

ωi

Conservation of energy: Conservation of momentum: n – refractive index

→

ωp = ωs + ωi → ωp = ωs + ωi kp = ks + ki → npωp = nsωs + niωi

frequency tuning by rotation of crystal

Part III - Experimental Techniques

Nonlinear optical phase measurements

Domain Imaging Example: Ferroelectric 180° domains: P = +P E (ω)

+++++

E (2ω, +P)

E (ω)

−−−−−

E (2ω, -P)

P = -P E(2ω, +P) ∝ χ(+P) E(ω)E (ω) = +χ(|P|)E(ω)E(ω) E(2ω, -P) ∝ χ(-P) E(ω)E (ω) = -χ(|P|) E(ω)E(ω) Domains distinguishable by the phase of the nonlinear signal.

180° Phase difference!

Phase Sensitive Measurements Excited state

SH-source term: SNL(2ω) ∝ χ(2ω)E(ω)E(ω)

E (ω) SNL (2ω) SH intensity: Ground state

ISH ∝ |SNL|2 ∝ |χ EE|2 = |χ|2 I02

Problem: Only direct measurement of intensity ⇒ No direct access to the phase! Solution: Interference measurements

Phase Sensitive Measurements Excited state

Sample source term: SS(2ω) ∝ χS(2ω) E(ω)E(ω)

E (ω) SS (2ω) SR (2ω) Reference source term: SR(2ω) ∝ χR(2ω) E(ω)E(ω)

Ground state

Total intensity: ISH ∝ |SS + SR|2 ∝ |χS + Aeiψ χR|2 I02 = (|χS|2 + |AχR|2 + 2χS χR cos ψ) I02(ω) always > 0

interference term

Experimental access to amplitude A and phase ψ!

Experimental Realisation Sample E(ω) ES(2ω)

Reference

Filter

PSU Ψrel

Detector ER(2ω)

PSU = Phase Shifting Unit: Induces phase shift Ψrel between E(ω) and ES(2ω) and therefore between ES(2ω) and ER(2ω) Experimental realisation: • Gas pressure cell • Rotated glass plates or shifted glass wedges • Distance variation

Experimental Realisation Measuring SH-intensity I as function of Ψrel: I(Ψrel) ∝ |ES+ER|2 = |ES|2 + |ER|2 + 2 |ES||ER| cos (Ψ + Ψrel) with Ψ = ΨS + ΨR + Ψ0

For Ψ0 → 0 and if ΨR known:

(Ψ0 by PSU and distance sample ↔ reference)

Absolute measurement of ΨS Spatially resolved:

Interference SH signal

I(ψrel) ~ |ES|2 + |ER|2 + 2|ESHG||ERef|cos(ψ+ψrel)

e.g. AFM domain in YMnO3 ψ = -25° 0

360

720

PSU-controled phase shift ψrel (°)

1080

Disadvantages of the Standard Methods • Only measurements with external reference ( multiferroics) • Distance sample ↔ reference reduces image quality • Loss of coherence due to large sample ↔ reference distance ⇒ week interference signal • Mechanical/optical instabilities due to moving parts

Phase Resolved SH Imaging ψ

y x z

ER

⊥

∆ψ

y ES

ES

ϑ

Fundamental

Sample

Analyzer

ES ES’ ER’

Reference

Filter

Soleil-Babinet: Quartz assembly with tunable birefringence

ER

x

Soleil-Babinet compensator as PSU behind sample & reference: ⇒ Sample ↔ reference distance can be reduced to zero ⇒ Measurements with external or internal reference ES and ER are projected on common direction via an analyzer: ⇒ Optimization of signal contrast

Soleil-Babinet Compensator Quartz assembly made of two wedged crystals (2a, 2b) + a compensation crystal

2b 1

Phase shift Ψ:

2π Ψ = (d2 − d1 )∆n λ d1 : Thickness compensation crystal d2 : Total thickness of the wedges λ : Wavelength ∆n = ne - no : Refrective index missmatch

Incoming beam

A

A

A 2a

A = optical axis

shift

Signal Optimization R R  E = ER0 e iΨ e x S S  E = ES0 e iΨ e y

E (ϑ ) = E e R'

⇒

R 0

y iΨ R

Analyzer

sin ϑ

ES

S

ES' (ϑ ) = ES0 e iΨ cos ϑ

ϑ

Interference:

ES’ ER’

2

I(ϑ ) = ER ' (ϑ ) + ES' (ϑ ) = 2

2

(

= ER0 ' sin ϑ + ES0 ' cos ϑ + 2 ER0 ' ES0 ' sin ϑ cos ϑ cos Ψ R − Ψ S

Imax = 1...∞ Contrast: C = Imin

Visibility: V = Imax − Imin = 0...1 Imax + Imin

)

Maximum for ER’(ϑ0)= ES’(ϑ0)

E

R

x

Phase Resolved SH Imaging (Setup) Sample in GP Pol LPF cryostat

Spherical mirror f = 300 mm Ref SPF

L

SB

Ana

• Measurements with external or internal reference • Reference outside cryostat ⇒ high degree of experimental freedom • Achromatic beam imaging for improved image quality and compensation of loss of (spatial) coherence

Coherence Effects

High γ

Low γ

Interference including the effect of coherence:

For |γ| > 99% possible!

−

Interference signal

Visibility:

1 mm

Coherence: |γ|

0

+ HoMnO3

≈ 180°

− 0

180

+ 360

External phase shift (°)

540

Phase-Resolved SH Imaging (Results) 1.00

Visibility

0,98

Direct

2hν = 2.42 eV ∆ν = 46 µeV

0,96

V=

0,94 0,92 0,90

d

I max − I min I max + I min

Beam imaging

Sample Reference

Sample

∝ degree of coherence -10

-5

0

5

Distance d, d' (mm)

10

d’

Visibility almost 100%

Loss of (spatial) coherence fully compensated!

Reference

Phase-Resolved SH Imaging (Results) Sample + Reference

Sample (a)

(c)

(b)

(d)

0.5mm

Reference

imaging

(e)

direct

Phase-Resolved SH Imaging (Summary) • Large working distances (~1 m) • More experimental freedom • High signal contrast • Improved image quality • Allows use of broadband laser sources with poor beam quality


Nonlinear imaging & the problem of optical resolution

Limit of Optical Resolution Optical resolution is limited by diffraction:

A

with numerical aperture A = n sin ϕ

Typical values of A: • Standard lens (f=200mm, ∅=50mm): A ≈ 0.2 • Photo lens: A ≈ 0.7 • Microscope objective up to A = 1.4

Resolution limit dmin (µm)

dmin = 0.61 λ

3.0 2.5

A = 0.2 A = 0.7 A = 1.4

2.0 1.5 1.0 0.5

0.0 200 400 600 800 1000 1200 1400 1600

Wavelength λ (nm)

⇒ Resolution limit down to some hundred nm

SHG Microscopy

S. Kurimura & Y. Uesu J. Appl. Phys. 81, 369 (1997)

SHG Microscopy Ferroelectric stripe domains in a LiTaO3 QPM device

S. Kurimura & Y. Uesu J. Appl. Phys. 81, 369 (1997)

Going Beyond the Optical Resolution Limit Tip- enhanced near-field microscope

SEM micrographs of Au tips: R ~ 10 nm [Neacsu, Reider, and Raschke, Phys. Rev. B 71, 201402 (2005)]

Imaging of FEL Domains in YMnO3 Tip-sample distance dependence

 E (ω) = ( E x (ω), E y (ω),0)

Pz( 2 ) (2ω ) ≈ χ (zxx2 ) [ Ex + E y ]2

Far-field Near-field self-homodyne enhanced reference contribution

[Neacsu, Reider, and Raschke, Phys. Rev. B 71, 201402 (2005)]

Imaging of FEL Domains in YMnO3 Domain dimensions: > 100 nm wide (y) ~ 1μm long (z)

Ferroelectric domains Extended along the hexagonal axis z

Parallel with pz

[Neacsu et.al., Nature Materials, submitted]


Surfaces & interfaces

First Magnetic SHG Experiment Observation of a second harmonic contribution which depends on the magnetization of a Fe(001) surface

+M

−M

Small signal, but with high contrast → typical for SHG!

Linear Magneto-Optical Effects Dielectric function :

in: ω,k out: ω,k

• Rotation of plane of polarization upon transmission/reflection on magnetized medium • Described by non-diagonal elements of 3×3 matrix • Q