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FT-2: Magneto-optics and Magneto-plasmonics Part 1 P. Vavassori -Ikerbasque, Basque Fundation for Science and CIC nanoGUNE Consolider, San Sebastian, Spain. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September

  1. FT-2: Magneto-optics and Magneto-plasmonics Part 1 P. Vavassori -Ikerbasque, Basque Fundation for Science and CIC nanoGUNE Consolider, San Sebastian, Spain. P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  2. Outline Magneto-optics Brief overview of the Magneto-optical Kerr effects (MOKE) Advanced MOKE: vector magnetometry Magnetic nanostructures micro-MOKE Approach 1: focused beam Approach 2: microscopy MOKE from diffracted beams: simple theory of diffracted MOKE in conjunction with micromagnetics and MFM from magnetometry to magnetic imaging P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  3. The MagnetoOptical Effect z z s s r r   θ θ θ θ p p s s p s → → R =   s s p p   r r   s p p p → → x x y y But, what happens if we applied a magnetic field?? p Reflected Light p z s s θ Polarization Plane x p Sample John Kerr Michael Faraday 1824 - 1907 1791 - 1867 s Transmitted Light P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  4. The Magneto-Optical Effect Polar and Longitudinal Configuration Transverse Configuration z z s s θ θ θ θ p p s s p p H z H x x x H y y y Dielectric Tensor Reflectivity matrix      r r    xx xy xz  s s p s → → R =          r r yx yy yz   s p p p → →        zx zy zz P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  5. MOKE The magneto-optic Kerr effect (MOKE) is widely used in studying technologically relevant magnetic materials. It relies on small, magnetization induced changes in the optical properties which modify the polarization or the intensity of the reflected light.     −  i i    0 0 0 z y   Macroscopically, magneto-optic effects arise from 0    = −    ˆ  =  ˆ i i   the antisymmetric, off-diagonal elements in the 0 0   z 0 x 0 dielectric tensor.       −     0 0 i i   0 y x 0  x =  0 Q m x ;  y =  0 Q m y ;  z =  0 Q m z ; • Non-destructive; Fresnell reflection coefficients • High sensitivity;   r r M  m y r pp = r 0 pp + r pp   • Finite penetration depth (~ 10 nm); pp ps Sample   r ps  - m x - m z • Fast (time resolved measurements); r r   r sp  m x -m z sp ss • Laterally resolved (microscopy); • Can be easily used in vacuum and E E E E r = r = r = r = rTM rTM rTE rTE cryogenic systems; pp ps sp ss E E E E iTM iTE iTM iTE J. Kerr, Philosophical Magazine 3 321 (1877) Z. Q. Qui and S. D. Bader, Rev. Sci. Instrum. 71, 1243 (2000) P. Vavassori, APL 77 1605 (2000) P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  6. MOKE origin: classical picture Lorentz force P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  7. Electron theory of Magneto-Optics Microscopically, the coupling between the electric field of the propagating light and the electron spin in a magnetic medium occurs through the spin-orbit interaction splitting of optical absorption lines (Zeeman effect). Here, x ( r ) is the spin-orbit parameter or coupling constant, which depends on the gradient of the electrostatic potential of the nuclear charges. Its values are of the order of 10-100meV and, thus, the spin-orbit interaction is much weaker than the exchange interaction ( ≈ 1eV). P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  8. Electron theory of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  9. Microscopic origin of Magneto-Optics Optical transitions between the d-orbitals and the p-orbitals s + and s - Selection rules Only transition from m =  1 to m = 0 are considered for simplicity P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  10. Microscopic origin of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  11. Microscopic origin of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  12. Microscopic origin of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  13. Microscopic origin of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  14. Microscopic origin of Magneto-Optics P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  15. Electron theory of Magneto-Optics MOKE results from a lifting of orbital degeneracy due to spin-orbit interaction (SOI) in the presence of spontaneous spin polarization. • Magnetization →Splitting of spin -states (Exchange) – 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 P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  16. A simple case: M || z     0 xx xy    xy = i  0 Q m z ; ~  = −     0 xy xx      0 0 xx ( ) ~    2 ( ) ( ) 0  +  = rot rot E E Maxwell Equation  2 2 c t ( ) ( ) −  −  = ˆ  = ( ) −  n E n n E E 0 2 E E e i nk z t 0 0 −  −      n 0 E 2     xx xy x  −  = Eigenequation     n 0 E 0 2 xy xx y     −      0 0 E Therefore, incident zz z light becomes =    elliptically polarized n 2 i Eigenvalue Eigenmodes ： LCP and RCP  xx xy after propagation in a MO active Different modes :different speed and attenuation material Without off-diagonal terms ： No difference between LCP & RCP P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  17. Phenomenology of MO effect Linearly polarized light can be decomposed to LCP and RCP Difference in phase causes rotation of Different speed the direction of Linear polarization (phase lag) Difference in amplitudes makes Different attenuation Elliptically polarized light In general, elliptically polarized light With the principal axis rotated P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  18. General case: Oblique incidence and arbitrary direction of M   r r   pp ps   r r   sp ss −  n 2 n 2 n 2 sin 2  = −  = −  = 0 1 0 cos 1 sin 1 sin 2 2 1 1 n n 2 1 1 M  m y r pp = r 0 pp + r pp  xy = i  1 Q m z ;  xz = -i  1 Q m y ;  yz = i  1 Q m x ;  - m x - m z r ps  xy = -  yx ;  zx = -  xz ;  zy = -  yz ;  m x -m z r sp P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  19. Summary of phenomenology E i E r E r E i Polarization conversion  M = 0 yx = p y p p x y ( )  −  p x p y p x xx 0 M P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  20. Longitudinal and polar Kerr effect y E oy E (z,t) Normalized representation b     1 1   r     ~  K pp r r = = = a Elliptically polarized light   ( ) E sp sp  +       i r e cos i sin  r  x sp      r   r  E ox  K pp pp    2 r r cos 2 b cos 2 cos r  = pp sp = tan 2 sp     tan 2 2 K − 2 2 − 2 1 b K K r r r pp sp r sp << r pp pp    2 r r sin 2 r sin 2 b sin  = pp sp =     sp sin 2 sin 2 2 K K K + 2 2 + 2 1 b r r r pp pp sp     r r r r sp = sp  sp sp   =    Im sin Re cos      r  r  r  r pp pp pp pp         r r r r  K  K ps ps  sp   sp      Re Re Im Im          r   r  r r pp pp ss ss P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

  21. Measurement of ellipticity y x’ y’ E ’ E 0 sin h y h E 0 E r E 0 E x h x Optic E’ r M axis     − i = h + h E 0 cos h 2 E ' E (cos i ie sin j ) 0   l /4plate ( ) = h + h E cos i sin j 0     = E i ' = h + h E E 0 (cos i i sin j ) 0 P. VAVASSORI European School on Magnetism (ESM-2018), Krakow 17-28 September 2018

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