Introduction to Magneto-Optics

1. Introduction toMagneto-Optics Katsuaki Sato Department of Applied Physics Tokyo University of Agriculture & Technology ISOM2000 Tutorial

2. 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

3. 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

4. 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)

5. 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

6. 3.1 Faraday & Voigt Configurations • (a) Faraday Configuration: • Magnetization // Light Vector • (b)Voigt Configuration: • Magnetization  Light Vector

7. 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）

8. 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

9. Materials rotation (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 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) 3.3 Faraday rotation of magnetic materials

10. 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）

11. aterials rotation Photon energy temperature field 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) 3.5 MO Kerr rotation of magnetic materials * "a-" means "amorphous".

12. 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

13. 4.1 Dielectric tensor Isotromic media；M//z Invariant C4 for 90°rotation around z-axis

14. 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

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

16. 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

17. 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

18. 5.1 Microscopic concepts of electronic polarization E - + + Wavefunction perturbed by electric field Unperturbed wavefunction + - +　・・ = + + S-like P-like Expansion by unperturbed orbitals

19. 5.2 Orbital angular momentum-selection rules and circular dichroism py-orbital px-orbital p+=px+ipy Lz=+1 Lz=-1 p-=px-ipy s-like Lz=0

20. 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

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

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

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

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

25. P +F B ID S D F A P 6.2 Vibrating polarizer technique

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

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

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

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

30. 6.7 Measurement of ellipticity y x’ y’ E’ E0sinh y h E0 E x h x Optic axis E0cosh l/4plate

31. 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

32. Theory and experiment of MO spectra in Fe Katayama theory

33. (c) (b) (a) MO spectra of PtMnSb 誘電率対角成分 誘電率非対角成分 カー回転と楕円率

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

35. Wavelength (nm) 300 400 500 600 700 0 -0.2 Polar Kerr rotation (deg) -0.4 -0.6 5 4 3 2 Photon Energy (eV) MO spectra in RE-TM(2)

36. 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

37. 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.