ISOM2000 Tutorial

Introduction to Magneto-Optics

Katsuaki SatoDepartment of Applied Physics

Tokyo University of Agriculture & Technology

CONTENTS

1. Introduction2. Light and Magnetism3. What is the Magneto-Optical Effect?4. Electromagnetism and Magneto-Optics5. Electronic Theory6. Measurement of Magneto-Optical Effect7. Magneto-Optical Spectra 8. Recent Advances in Magneto-Optics9. Summary

1. Introduction

• Magneto-Optical Effect： Discovered by Faraday on 1845

• Phenomenon： Change of Linear Polarization to Elliptically Polarized Light Accompanied by Rotation of Principal Axis

• Cause： Difference of Optical Response between LCP and RCP

• Application：– Magneto-Optical Disk

– Optical Isolator

– Current Sensors

– Observation Technique

2. Light and Magnetism

• Light→Magnetism ： Photomagentic Effect– Thermomagnetic Effect ： Curie pt. recording→MO disk– Light-induced Magnetization ： ruby, DMS– Light-induced spin reorientation→Optical motor

• Magnetism→Light ： Magneto-Optical Effect– Shift or splitting of optical absorption line(Zeeman eff.)– Magnetic resonance ： ESR, magneto-plasma effect– Magneto-optical effect(Faraday, Kerr, Cotton Mouton)

3.What is the Magneto-Optical Effect?

• MO Effect in Wide MeaningAny change of optical response induced by magnetizatio

n

• MO Effect in Narrow MeaningChange of intensity or polarization induced by magentizat

ion – Faraday effect– MOKE(Magneto-optical Kerr effect)– Cotton-Mouton effect

3.1 Faraday & Voigt Configurations

• (a) Faraday Configuration: – Magnetization // Light Vector

• (b)Voigt Configuration:– Magnetization Light Vector

3.2 Faraday Effect• MO effect for optical transmission

– Magnetic rotation （ Faraday rotation ） F

– Magnetic Circular Dichroism （ Faraday Ellipticity ） F

• Comparison to Natural Optical Rotation– Faraday Effect is Nonreciprocal (Double rotation for round tr

ip)

– Natural rotation is Reciprocal (Zero for round trip)

• Verdet Constant F=VlH (For paramagnetic and diamagnetic materials ）

Illustration of Faraday Effect

For linearly polarized light incidence,

•　 Elliptically polarized light goes out (MCD)

• With the principal axis rotated (Magnetic rotation)

Linearly polarized light

EllipticallyPolarized light

Rotation of Principal axis

3.3 Faraday rotation of magnetic materialsMaterials rotation

(deg)　 figure of

merit(deg/dB)wavelength

(nm)temperat

ure(K)

Mag. field(T)

literature

Fe 3.825 ･ 105   578 RT 2.4 1.11)

Co 1.88 ･ 105   546 〃 2 1.11)

Ni 1.3 ･ 105   826 120 K 0.27 1.11)

Y3Fe5O12 250   1150 100 K   1.12)

Gd2BiFe5O12 1.01 ･ 104 44 800 RT   1.13)

MnSb 2.8 ･ 105   500 〃   1.14)

MnBi 5.0 ･ 105 1.43 633 〃   1.15)

YFeO3 4.9 ･ 103   633 〃   1.16)

NdFeO3 4.72 ･ 104   633 〃   1.17)

CrBr3 1.3 ･ 105   500 1.5K   1.18)

EuO 5 ･ 105 104 660 4.2 K 2.08 1.19)

CdCr2S4 3.8 ･ 103 35(80K) 1000 4K 0.6 1.20)

3.4 Magneto-Optical Kerr Effect

• Three kinds of MO Kerr effects– Polar Kerr （ Magnetization is oriented perpen

dicular to the suraface ）– Longitudinal Kerr （ Magnetization is in plane

and is parallel to the plane of incidence ）– Transverse Kerr （ Magnetization is in plane

and is perpendicular to the plane of incidence ）

3.5 MO Kerr rotation of magnetic materialsaterials rotation Photon

energytemperat

urefield literature

  (deg) (eV) (K) (T)  

Fe 0.87 0.75 RT   1.21)

Co 0.85 0.62 〃   1.21)

Ni 0.19 3.1 〃   1.21)

Gd 0.16 4.3 〃   1.22)

Fe3O4 0.32 1 〃   1.23)

MnBi 0.7 1.9 〃   1.24)

PtMnSb 2.0 1.75 〃 1.7 1.8)

CoS2 1.1 0.8 4.2 0.4 1.25)

CrBr3 3.5 2.9 4.2   1.26)

EuO 6 2.1 12   1.27)

USb0.8Te0

.2

9.0 0.8 10 4.0 1.28)

CoCr2S4 4.5 0.7 80   1.29)

a-GdCo *

0.3 1.9 RT   1.30)

CeSb 90   2   1.31)

4. Electromagnetism and Magnetooptics

• Light is the electromagnetic wave.• Transmission of EM wave ： Maxwell equation• Medium is regareded as continuum→dielectric permeabi

lity tensor– Effect of Magnetic field→mainly to off-diagonal element

• Eigenequation• →Complex refractive index ： two eigenvalues

eigenfunctions ： right and left circularpolarization– Phase difference between RCP and LCP→rotation– Amplitude difference →circular dichroism

4.1 Dielectric tensor

ED 0~ ε

zzzyzx

yzyyyx

xzxyxx~

ijijij

Isotromic media ； M//zInvariant C4 for 90°rotation around z-axis

zzzxzy

xzxxxy

yzyxyy

CC 41

4~~

0

zyzxyzxz

xyyx

yyxx

zz

xxxy

xyxx

00

0

0~

4.2 MO Equations (1)

0~

2

2

2

Etc

Erotrot

0

00

0ˆ0ˆ

2

2

z

y

x

zz

xxxy

xyxx

E

E

E

N

N

xyxx iN 2ˆEigenvalue

Eigenfunction ： LCP and RCP

Without off-diagonal terms： No difference between LCP & RCP

No magnetooptical effect

Maxwell Equation

Eigenequation

MO Equations (2)

xx

yxyxxxyxxx iiiNNN

ˆˆˆ

2)2(21)0(

)1(

ˆ

M

Mi

iN

xxxx

xy

xx

yxF

Both diagonal and off-diagonal terms contribute toMagneto-optical effect

4.3 Phenomenology of MO effectLinearly polarized light can be decomposed to LCP and RCP

Difference in phase causes rotation ofthe direction of Linear polarization

Difference in amplitudes makes Elliptically polarized light

In general, elliptically polarized lightWith the principal axis rotated

5. Electron theory of Magneto-Optics

• Magnetization→Splitting of spin-states– No direct cause of difference of optical response

between LCP and RCP

• Spin-orbit interaction→Splitting of orbital states– Absorption of circular polarization→Induction of circular

motion of electrons

• Condition for large magneto-optical response– Presence of strong (allowed) transitions– Involving elements with large spin-orbit interaction– Not directly related with Magnetization

5.1 Microscopic concepts of electronic polarization

= +++ + 　・・

+ + -

-

Unperturbed wavefunction

Wavefunction perturbed by electric field

E

S-like P-like

Expansion by unperturbed orbitals

5.2 Orbital angular momentum-selection rules and circular dichroism

Lz=0

Lz=+1

Lz=-1

s-like

p-=px-ipy

p+=px+ipy

px-orbitalpy-orbital

5.3 Role of Spin-Orbit Interaction

L=1

L=0

LZ=+1,0,-1

LZ=0

Jz=-3/2Jz=-1/2

Jz=+1/2Jz=+3/2

Jz=-1/2

Jz=+1/2

Exchange splitting

Exchange

+spin-orbit

Without magnetization

5.4 MO lineshapes (1)

Excited state

Ground state

0 1 2

Without magnetization

With magnetization

Lz=0

Lz=+1

Lz=-1

1+2

Photon energy Photon energy

’xy ”xy

1.Diamagnetic lineshape

5.4 MO lineshapes (2)

excited state

ground state

f+ f-

f=f+ - f-

0

without magneticfield

with magneticfield

’xy

”xy

photon energy

(a) (b)d

iele

ctri

c co

nst

ant

6. Measurement of MO effect

1. Cross-polarizer technique

2. Vibrating polarizer technique

3. Rotating analyzer technique

4. Faraday modulation technique

5. Optical retardation modulation

6. Measuring system for MO spectrum

7. Measurement of elleipticity

L

P B A

D

PF A I

P=A+/2

/4 rotation

/2 rotation

rotation

B

(a)

(b)

S

6.1 Cross-Nicol technique

P

B

P

F

+F

AD

ID

S

6.2 Vibrating polarizer technique

PA

DS

BEF

A=pt

ID

6.3 Rotating analyzer technique

Faraday modulator

P

=0+sin pt

B

S

A

DI=I0+ I sin pt

F ID

6.4 Faraday modulation technique

Zero method

i

j

/4

P

PEM A

D

quartz Isotropicmedium

B

fused silica CaF2

Ge etc.

Piezoelectriccrystal

amplitude

position

l

Retardation=(2/)nl sin pt =0sin pt

6.5 Retardation modulation technique

L MC

P

AC (f Hz)

M1

M2

PEM(p Hz) S

Electromagnet

D

Preamplifier

LA1 (f Hz)

LA2 (p Hz)

LA3 (2p Hz)

6.6 Spectral measurement

x

y

x’y’

/4plate

E0

E0sin

E0cos

E E i i j 0 (cos sin )

Opticaxis

E E i i e j

E i j

E i

i' (cos sin )

cos sin

'

02

0

0

x

y

E’

E

6.7 Measurement of ellipticity

7. MO spectra of materials

• Magnetic garnets• Metallic ferromagnet ： Fe, Co, Ni• Intermetallic compounds and alloys ： PtMnSb et

c.• Magnetic semiconductor ： CdMnTe etc.• Superlattices ： Pt/Co, Fe/Au etc.• Amorphous ： TbFeCo, GdFeCo etc.• Granular ： Al2O3:Co など

Theory and experiment of MO spectra in Fe

Katayama

theory

(a) (b) (c)

MO spectra of PtMnSb

カー回転と楕円率 誘電率対角成分 誘電率非対角成分

xxxx

xyK

1

Wavelength (nm)P

ola

r K

err

ro

tatio

n (

min

)

MO spectra in RE-TM (1)

5 4 3 2

Photon Energy (eV)

0

-0.2

-0.4

-0.6

Pol

ar

Ker

r ro

tatio

n (d

eg)

Wavelength (nm)

300 400 500 600 700

MO spectra in RE-TM(2)

Recent Advances in Magneto-Optics

• Scanning Near Field Magneto-Optical Microscope (MO-SNOM)

• Nonlinear Magneto-Optics

• Sagnac Magneto-Optical Microscope

• X-ray Magneto-Optical Imaging

SUMMARY

• Basic concept of magneto-optics is described.

• Macroscopic and microscopic origins of magneto-optics are described.

• Some of the recent development of magneto-optics is also given.