Introduction to Magneto-Optics Katsuaki Sato Department of Applied Physics Tokyo University of Agriculture & Technology ISOM2000 Tutorial

CONTENTS Introduction Light and Magnetism What is the Magneto-Optical Effect? Electromagnetism and Magneto-Optics Electronic Theory Measurement of Magneto-Optical Effect Magneto-Optical Spectra Recent Advances in Magneto-Optics 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 Meaning Any change of optical response induced by magnetization MO Effect in Narrow Meaning Change of intensity or polarization induced by magentization 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 trip) 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) Rotation of Principal axis Elliptically Polarized light Linearly polarized light

3.3 Faraday rotation of magnetic materials (deg) 　figure of merit(deg/dB) wavelength (nm) temperature (K) Mag. field (T) literature Fe 3.825･105   578 RT 2.4 1.11) Co 1.88･105 546 〃 2 Ni 1.3･105 826 120 K 0.27 Y3Fe5O12 250 1150 100 K 1.12) Gd2BiFe5O12 1.01･104 44 800 1.13) MnSb 2.8･105 500 1.14) MnBi 5.0･105 1.43 633 1.15) YFeO3 4.9･103 1.16) NdFeO3 4.72･104 1.17) CrBr3 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 perpendicular 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 materials Photon energy temperature field literature   (deg) (eV) (K) (T) Fe 0.87 0.75 RT 1.21) Co 0.85 0.62 〃 Ni 0.19 3.1 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 1.26) EuO 6 2.1 12 1.27) USb0.8Te0.2 9.0 10 4.0 1.28) CoCr2S4 4.5 80 1.29) a-GdCo * 0.3 1.30) CeSb 90 2 1.31) * "a-" means "amorphous".

4. Electromagnetism and Magnetooptics Light is the electromagnetic wave. Transmission of EM wave：Maxwell equation Medium is regareded as continuum→dielectric permeability 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 Isotromic media；M//z Invariant C4 for 90°rotation around z-axis

4.2 MO Equations (1) Maxwell Equation Eigenequation Eigenvalue Eigenfunction：LCP and RCP Without off-diagonal terms：No difference between LCP & RCP No magnetooptical effect

MO Equations (2) Both diagonal and off-diagonal terms contribute to Magneto-optical effect

4.3 Phenomenology of MO effect Linearly polarized light can be decomposed to LCP and RCP Difference in phase causes rotation of the direction of Linear polarization Difference in amplitudes makes Elliptically polarized light In general, elliptically polarized light With 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 + + - Wavefunction perturbed by electric field Unperturbed wavefunction + - = + + +　・・ S-like P-like Expansion by unperturbed orbitals

5.2 Orbital angular momentum-selection rules and circular dichroism py-orbital px-orbital p+=px+ipy Lz=+1 Lz=-1 p-=px-ipy Lz=0 s-like

5.3 Role of Spin-Orbit Interaction Jz=-3/2 Jz=-1/2 L=1 Jz=+1/2 LZ=+1,0,-1 Jz=+3/2 Jz=-1/2 L=0 Jz=+1/2 LZ=0 Exchange +spin-orbit Without magnetization Exchange splitting

5.4 MO lineshapes (1) 1.Diamagnetic lineshape Excited state Ground state 0 1 2  Without magnetization With Lz=0 Lz=+1 Lz=-1 1+2 Photon energy ’xy ”xy 1.Diamagnetic lineshape

5.4 MO lineshapes (2) excited state ground state f+ f-  f=f+ - f- 0 without magnetic field with magnetic ’xy ”xy photon energy (a) (b) dielectric constant

6. Measurement of MO effect Cross-polarizer technique Vibrating polarizer technique Rotating analyzer technique Faraday modulation technique Optical retardation modulation Measuring system for MO spectrum Measurement of elleipticity

6.1 Cross-Nicol technique P B A D P F A I P=A+/2 /4 rotation /2 rotation  rotation (a) (b) S

6.2 Vibrating polarizer technique F +F A D ID S

6.3 Rotating analyzer technique P A D S B E F A=pt ID

6.4 Faraday modulation technique Faraday modulator P =0+sin pt B S A D I=I0+ I sin pt F ID Zero method

6.5 Retardation modulation technique j /4 P PEM A D quartz Isotropic medium B fused silica CaF2 Ge etc. Piezoelectric crystal amplitude position l Retardation =(2/)nl sin pt =0sin pt

6.6 Spectral measurement M1 L MC PEM S P C (f Hz) M2 A D LA1 (f Hz) (p Hz) S Electromagnet D Preamplifier LA1 (f Hz) LA2 (p Hz) LA3 (2p Hz)

6.7 Measurement of ellipticity x’ y’ E’ E0sinh y E0 h E x h x Optic axis E0cosh l/4plate

7. MO spectra of materials Magnetic garnets Metallic ferromagnet：Fe, Co, Ni Intermetallic compounds and alloys：PtMnSb etc. 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

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 (a) (b) (c)

MO spectra in RE-TM (1) Wavelength (nm) Polar Kerr rotation (min)

MO spectra in RE-TM(2) Photon Energy (eV) -0.2 -0.4 -0.6 5 4 3 2 Photon Energy (eV) -0.2 -0.4 -0.6 Polar Kerr rotation (deg) Wavelength (nm) 300 400 500 600 700

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.