INTERACTIONS IN MATTER Alexander A. Iskandar Physics of Magnetism - PDF document

08/01/2018 FI 3221 ELECTROMAGNETIC INTERACTIONS IN MATTER Alexander A. Iskandar Physics of Magnetism and Photonics • Surface Plasmon • Propagation of Surface SURFACE PLASMON Plasmon • Localized Plasmon • SPR Spectroscopy Alexander A. Iskandar Electromagnetic Interactions in Matter 2 1

08/01/2018 REFERENCES  Main ▪ S.A. Maier : Section 2.1 – 2.3, 5.1  Supplementary ▪ J. A. Dionne et.al., Phys. Rev. B72, 075405 (2005) Alexander A. Iskandar Electromagnetic Interactions in Matter 3 WAVE EQUATION AT METAL/INSULATOR INTERFACES  Consider the x z y direction as the dielectric effective propagation direction and by x metal symmetry, there is no y -dependence.  E    ( ) i x t ( , , , ) ( ) E x y z t z e  Substituting to the Maxwell’s eq.s, we obtain    2 ( z ) E      2 2 0 k E  0 2 z And similar equation for the H -field. Alexander A. Iskandar Electromagnetic Interactions in Matter 4 2

08/01/2018 WAVE EQUATION AT METAL/INSULATOR INTERFACES  Consider the two special polarization ▪ TE : with only H x , H z and E y are nonzero ▪ TM : with only E x , E z and H y are nonzero  For the TE mode, we need to solve only for the E y field component from the wave equation,  2   E      y 2 2 0 k E  0 y 2 z  And, the H x , H z field components can be obtained from the Maxwell’s eq.   E 1   y H i , H E    x z y z 0 0 Alexander A. Iskandar Electromagnetic Interactions in Matter 5 WAVE EQUATION AT METAL/INSULATOR INTERFACES  Similarly, for the TM mode, we need to solve only for the H y field component from the wave equation,  2   H      y 2 2 0 k H  0 y 2 z  And, the E x , E z field components can be obtained from the Maxwell’s eq.   1 H     y , E i E H    x z y z 0 0 Alexander A. Iskandar Electromagnetic Interactions in Matter 6 3

08/01/2018 TE SURFACE-WAVE SOLUTION  Consider the TE mode, solution for region of z > 0 is     i ( x k z ) E ( z ) A e , Re[ k ] 0 d 2 y d  k        ( ) ( ) i x k z i x i x k z ( ) d , ( ) H z A e H z A e e d d   x 2 z 2 0 0  And for region z < 0 , we obtain     ( ) i x k z ( ) , Re[ ] 0 E z A e k m y 1 m  k        i ( x k z ) i ( x k z ) m H ( z ) A e , H ( z ) A e m m 1  1  x z 0 0 Alexander A. Iskandar Electromagnetic Interactions in Matter 7 TE SURFACE-WAVE SOLUTION  Applying continuity condition to the E y and H x field components, we arrive at the condition      0 A A and A k k 1 2 1 m d  Since confinement condition requires that Re[ k m ] > 0 and Re[ k d ] > 0 , the above condition can only be satisfied with A 1 = 0 , which yield a trivial solution. Alexander A. Iskandar Electromagnetic Interactions in Matter 8 4

08/01/2018 TM SURFACE-WAVE SOLUTION  On the other hand, for the TM mode, solution for region of z > 0 is     ( ) i x k z ( ) , Re[ ] 0 H z A e k d y 2 d  k         i ( x k z ) i ( x k z ) ( ) d , ( ) E z A e E z A e d d 2   2   x z d 0 d 0  And for region z < 0 , we obtain     ( ) i x k z ( ) , Re[ ] 0 H z A e k m y 1 m  k        ( ) ( ) i x k z i x k z ( ) m , ( ) E z A e E z A e m m     x 1 z 1 0 0 m m Alexander A. Iskandar Electromagnetic Interactions in Matter 9 TM SURFACE-WAVE SOLUTION  Applying continuity condition to the H y and E x field components, we arrive at the conditions  k    d d A A and  1 2 k m m  Hence confinement condition requires that Re[ k m ] > 0 and Re[ k d ] > 0 , and this can be fulfilled by       Re 0 0 if m d  Further, the wave number in each region is given by         2 2 2 2 2 2 k k and k k 0 0 m m d d Alexander A. Iskandar Electromagnetic Interactions in Matter 10 5

08/01/2018 SURFACE PLASMON WAVE  Combining with the previous condition we have the following dispersion relation for the propagation wavenumber     m d k 0    m d  This surface wave is called Surface Plasmon Wave, that exist only for TM polarization. Alexander A. Iskandar Electromagnetic Interactions in Matter 11 SURFACE PLASMON WAVE  Plasmons: ▪ collective oscillations of the “free electron gas” density, often at optical frequencies.  Surface Plasmons: ▪ plasmons confined to surface (interface) and interact with light resulting in polaritons. ▪ propagating electron density waves occurring at the interface between metal and dielectric.  Surface Plasmon Resonance: ▪ light (  ) in resonance with surface plasmon oscillation Alexander A. Iskandar Electromagnetic Interactions in Matter 13 6

08/01/2018 SURFACE PLASMON DISPERSION RELATION  ck x Radiative modes real    2  ′ m > 0)    real k z ( k m ) d p 1 m  2  p 1 / 2 Quasi-bound modes      imaginary      m d    d <  ′ m < 0)    real k z   c  m d   p sp   1 d     2 2 2 k k 0 m m Dielectric:  d real  z Bound modes imaginary k z x (  ′ m <  d ) Metal:  m =  ′ m + i  ″ m  ″ m << 1 Re  (or k x ) Alexander A. Iskandar Electromagnetic Interactions in Matter 14 SURFACE PLASMON DISPERSION RELATION     2 2 2 k k m 0 m  Radiative solution ( k x and k m real) lies the left of the light-line. This happens for  >  p .  Bound solution ( k x real and k m imaginary) lies to the right of the light-line.  Between the regime of the bound and radiative modes, a frequency gap region with purely imaginary  prohibiting propagation exists. Alexander A. Iskandar Electromagnetic Interactions in Matter 15 7

08/01/2018 EXAMPLE OF Ag-SiO 2 INTERFACE Kurva Dispersi - Model Gas Elektron Bebas Kurva Dispersi - Komponen Re[kx] dengan Data Empiris 6 6 5.82 eV kx' / 213 nm     8 . 85 10 15 1 light line 5.5 s 5.5 P 5 5 4.5 4.5 Modus Radiatif 3.73 eV / (RPP) 332 nm 4 4 Energi (eV) Energi (eV)    1   Modus Quasi-bound (QB) 3.5 SP P 2 3.5 3.28 eV / 378 nm 3 3 Modus Terikat 3.44 eV / (SPP) 360 nm 2.5 2.5 2 2 Re[kx] 1.5 1.5 Im[kx] 1 1 light line 0 0.05 0.1 0.15 0.2 0.25 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 kx (1/nm) kx' (1/nm) Alexander A. Iskandar Electromagnetic Interactions in Matter 16 PROPAGATION LENGTH AND SKIN DEPTH  For real metals,  ″ m is not negligible, hence  (= k x ) Kurva Propagasi untuk Tiga Konstanta Dielektrik yang Berbeda will be a complex quantity. -2 10 -3  The propagation of surface 10 plasmon wave will be -4 10 Panjang Propagasi (m) attenuated. Define -5 10 propagation length L as -6 RPP 10 the distance when the -7 10 intensity has become 1/ e -8 SiO2 10 of the initial intensity. Udara SPP Si -9 10 1 200 400 600 800 1000 1200 1400 1600 1800 2000  Lambda Vakum (nm) L  2 Im[ ] Alexander A. Iskandar Electromagnetic Interactions in Matter 17 8

08/01/2018 PROPAGATION LENGTH AND SKIN DEPTH Skin Depth SiO2 Vs. Lambda  Confinement of the 2500 SiO 2 surface plasmon 2000 wave near the Skin Depth 1500 interface can be (nm) 1000 360 nm (3.44 eV) characterize by its 500 skin depth, defined 0 200 400 600 800 1000 1200 1400 1600 1800 2000 as the length d Panjang Gelombang Vakum (nm) Skin Depth Ag when the intensity 0 has become 1/ e of -50 the initial intensity. 360 nm (3.44 eV) -100 Skin Depth 1 (nm) d  -150   Ag Im k -200 i Alexander A. Iskandar Electromagnetic Interactions in Matter 18 -250 200 400 600 800 1000 1200 1400 1600 1800 2000 Panjang Gelombang Vakum (nm) E x DISTRIBUTION AND ENERGY DENSITY Distribusi Medan Ex, untuk Lambda = 476 nm 1 0.5 Illumination with  = 476 nm (2.61 eV) Ex(N/C) 0 – SPP mode -0.5 Ag -1 SiO2 200 100 200 150 0 100 -100 50 -200 0 z(nm) x(nm) Distribusi Medan Ex, untuk Lambda = 476 nm Distribusi Rapat Energi pada x = 0 200 5 SiO2 Ag Ag 4.5 150 4 100 3.5 50 3 z(nm) 0 U 2.5 -50 2 1.5 -100 1 -150 0.5 SiO2 -200 0 0 20 40 60 80 100 120 140 160 180 200 -200 -150 -100 -50 0 50 100 150 200 x(nm) z(nm) Alexander A. Iskandar Electromagnetic Interactions in Matter 19 9