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Nanophotonics Class 2 Metal nanoparticle plasmons and light scattering What is a plasmon? “plasma-oscillation”: density fluctuation of free electrons p + drude Ne2 m 0 + + Plasmons in the bulk oscillate at p determined by the free electron density and effective mass - - - k + - + Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)” Confinement effects result in resonant SPP modes in nanoparticles 1 Ne2 particle 3 m 0 drude Sphere in a uniform static electric field particle can be considered as a dipole: in a metal cluster placed in an electric field, the negative charges are displaced from the positive ones 3 m p 40 R m E0 2 m electric polarizability of a sphere α resonant enhancement of p if ( ) 2 m minimum negative real dielectric constant ε1(ω) Bohren and Huffman (1983), p.136 4 0 R 3 m 2 m ε = ε1(ω)+i ε2(ω) = dielectric constant of the metal particle εm = dielectric constant of the embedding medium usually real and taken independent of frequency Quasi-static approximation i kr t 2 V r ,t f r e , k r k r 2 V r , t f r e it Einc k i kr t E inc r ,t E 0 e y x Einc k it Einc r ,t E 0 e y x Sphere in a uniform static electric field Equations: 1 0 2 0 E0 r a r a m r q z a Boundary conditions: 1 2 r a , 1 m 2 r r r a , lim 2 E 0 z r Jackson (1998), p.157 Bohren and Huffman (1983), p.136 Sphere in a uniform static electric field Equations: 1 0 2 0 E0 r a r a m r q z a Solution: m 3 m E 0r cos q E 0r cos q 1 E 0r cos q 2 m 2 m cos q p cos q 3 m E 0 2 E 0r cos q 2 E 0r cos q a 2 2 r 4 r m m with: p 0 mE 0 m 2 m 4a 3 Sphere in electromagnetic field (a << ): it p 0 mE0e Jackson (1998), p.157 Bohren and Huffman (1983), p.136 Measured data and model for Ag: Drude model: 50 p2 p2 ' 1 2 , " 3 " 0 -50 -100 Measured data: ' " Drude model: ' " Modified Drude model: ' Modified Drude model: ' p2 p2 ' 2 , " 3 -150 200 400 600 800 1000 1200 Wavelength (nm) 1400 1600 1800 Contribution of bound electrons Ag: 3.4 Drude model good, Modified Drude model even better. Example: silver sphere 10 real part 1 imaginary part 2 data Palik's handbook dielectric constant of Ag 5 0 -5 -10 -15 -20 200 300 400 500 600 700 800 wavelength (nm) ( ) 2 m minimum in air in silica Other applications of nanoparticles Old: New: (but the same principle) Different materials/shapes: distinct colors Stained glass Focusing and guidance of light at nanometer length scales Nanoparticles in solution Au colloids in water (M. Faraday ~1856) glass containing Ag clusters Au colloids in water Au shell colloids in water (larger, also scattering) Metal nanoparticle field enhancement The resonant excitation of the nanoparticles causes large local electric fields close to the particle surface, useful for many applications n=1.5 20 Au 550nm Ienh Coupling in plasmon particle arrays All particles are driven by the external field and by each other René de Waele, Femius Koenderink Local plasmon array response • Energy localization on front or back side of the array • Nanoscale concentration tunable with wavelength • NANOANTENNA Nano Lett. 7, 2004 (2007) René de Waele, Femius Koenderink Interaction between particles An isolated sphere is symmetric, so the polarization direction does not matter. LONGITUDINAL: restoring force reduced by coupling to neighbor Resonance shifts to lower frequency TRANSVERSE: restoring force increased by coupling to neighbor Resonance shifts to higher frequency Large local field enhancement between particles 1.65 eV Light Scattering •The incident E-field accelerates charges in the obstacle which radiate electromagnetic energy in all directions. This process is called scattering, thus: •Scattering = excitation + reradiation •The excited charges may also transform their energy into other forms, like heat. This process is called absorption. Together the two processes are called extinction: •Extinction = scattering + absorption True for metals and dielectrics Rayleigh scattering (d<<λ) Particles feel homogeneous applied field Ei. The induced dipole moment in the particle is: p Εi where α is the polarizability of the scatterer. Assume the incident field is time harmonic: Ei E0 cos 0t The particle (dipole) will radiate at the same ω as the applied field with intensity: E p sin I s 32 2c 3 0 r 2 2 4 2 (see electrodynamics textbook) g p Electric field lines of oscillating dipole Blue light is scattered more than red light Scattering by small gas molecules in the atmosphere: d<< λ: Rayleigh scattering Geometrical scattering (d>>λ) For particles very large compared to λ, the incident plane wave can be subdivided into a large number of rays which obey Snell’s law and Fresnel Equations. For complex , the energy Wabs absorbed in the sphere depends on the absorption of the dielectric and the time the light spends in the particle (the optical path length) Rainbow formation Metal nanoparticles: extinction = scattering + absorption n=1.5 Ienh 20 Au 550nm At resonance, both scattering and absorption are large albedo = scattering / extinction = ssca/(sabs+ssca)