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## Nano-Photonics (2)

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**Nano-Photonics (2)**W. P. Huang Department of Electrical and Computer Engineering McMaster University Hamilton, Canada**Agenda**• Optical Properties of Metals • Classical Drude model for free electrons • Modifications due to band-tranistions for bound electrons • Modifications due to quantum size effects • Confinement and resonance of light at nano-scale • Scattering of Light by Metal Particles • Surface plasma polariton resonators • Light-matter interaction in nano-crystals • Optical properties of nano-crystals**Basic Relations for Refractive Indices**Complex Refractive Index Relative Dielectric Constant Relationship between Dielectric Constant and Refractive Index If If then then**The Atomic Structure**~100 pm**Nature of Electrons in Atoms**Electron • Electron energy levels are quantized • Energy for transition can be thermal or light (electromagnetic), both of which are quantized resulting in “quantum leap” • Electrons arranged in shells around the nucleus • Each shell can contain 2n2 electrons, where n is the number of the shell • Within each shell there are sub-shells 3rd shell: 18 electrons 3d Sub-shells 2nd shell: 8 electrons 3p 3s**Classical Model of Atoms**Classical Model: Electrons are bound to the nucleus by springs which determine the natural frequencies Bound Electrons (insulators, intrinsic semiconductors) • Restoring force for small displacements: F=–kx • Natural frequency • Natural frequencies lie in visible, infrared and UV range Free Electrons (metals, doped semiconductors) • k=0 so that natural frequency=0**E**+ H ? + + H H Atoms and Bounds • One atom, e.g. H. • Two atoms: bond formation Every electron contributes one state • Equilibrium distance d (after reaction)**Formation of Energy Bands**~ 1 eV • Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin )**Empty**outer orbitals Partly filled valence orbitals Energy Filled Inner shells Distance between atoms Energy Band Characteristics Outermost electrons interact Form bands Electrons in inner shells do not interact Do not form bands**Empty band**Empty band Gap ( ~ 1 eV) Gap ( > 5 eV) Full band Full band Semiconductor Insulator Band Diagrams & Electron Filling Electrons filled from low to high energies till we run out of electrons Energy Empty band Partially full band Metal**Gold**Empty band 3.1 eV (violet) 2.4 eV (yellow) 1.7 eV (red) Partially full band Only colors up to yellow absorbed and immediately re-emitted; blue end of spectrum goes through, and gets “lost” Color of Metals Silver Energy Empty band 3.1 eV (violet) > 3.1 eV 2.4 eV (yellow) 1.7 eV (red) Partially full band All colors absorbed and immediately re-emitted; this is why silver is white (or silvery)**Optical Processes in Metals**• Macroscopic Views: • The field of the radiation causes the free electrons in metal to move and a moving charge emits electromagnetic radiation • Microscopic Views: • Large density of empty, closely spaced electron energy states above the Fermi level lead to wide range of wavelength readily absorbed by conduction band electron • Excited electrons within the thin layer close to the surface of the metal move to higher energy levels, relax and emit photons (light) • Some excited electrons collide with lattice ions and dissipate energy in form of phonons (heat) • Metal reflects the light very well (> 95%)**Drude Model: Free Carrier Contributions to Optical**Properties Paul Drude (1863-1906) A highly respected physicist, who performed pioneering work on the optics of absorbing media and connected the optical with the electrical and thermal properties of solids. Bound electrons Conduction electrons**Low Frequency Response by Drude Model**If << 1: Constant of Frequency, Negligible at low Frequency Inverse Proportional to ω, Dominant at Low Frequency At low frequencies, metals (material with large concentration of free carriers) is a perfect reflector**High Frequency Response by Drude Model**If >> 1: Plasma Frequency: (about 10eV for metals) As the frequency is very high At high frequencies, the contribution of free carriers is negligible and metals behaves like an insulator**Plasma Frequency in Drude Model**For Free Electrons At the Plasma frequency The real part of the dielectric function vanishes At the Plasma frequency**Validation of Drude Model**Measured data and model for Ag: Drude model: Modified Drude model: Contribution of bound electrons Ag:**Dielectric Functions of Aluminum (Al) and Copper (Cu)Drude**Model M. A. Ordal, et.al., Appl. Opt., vol.22, no.7, pp.1099-1120, 1983**Dielectric Functions of Gold (Au) and Silver (Ag) Drude**Model M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp.1099-1120,1983**Model Parameters**M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp.1099-1120,1983**Improved Model Parameters**M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp.4493-4500,1985**Dielectric Functions of Copper (Cu)Drude Model with Improved**Model Parameters M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp.4493-4500,1985**Limitation of Drude Model**• Drude model considers only free electron contributions to the optical properties • The band structures of the solids are not considered • Inter-band transitions, which are important at higher frequencies, are not accounted for • When the dimension of the metal decreases such that the size of the metal particle becomes smaller than the mean free path of the free electrons, the electrons collide with the boundary of the particle, which leads to quantum-size effects**Refractive Index of Aluminum (Al)**Band-Transition Peak**Classical Lorentz Model**Electron Clouds e-,m E L Ion Core k, ro x p =- e x r + Ion Core Potential Energy Repulsion Force Newton’s 2nd Law Damping Force Electric Force Repulsion Force**Atomic Polarizability by Lorentz Model**Define atomic polarizability: Resonance frequency Damping term**smaller **Characteristics of Atomic Polarizability -dependent response Response of matter is not instantaneous •Atomic polarizability: •Amplitude Amplitude •Phase lag of with E: 180 smaller 90 Phase lag 0**Correction to Drude Model Due to Band Transition for Bound**Electrons Brendel-Bormann (BB) Model Lorentz-Drude (LD) Model**Refractive Index of Al from Modified Drude Model Considering**Band-Transition Effects**Dielectric Functions for Silver (Ag) By Different Models**A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998**Dielectric Functions for Gold (Au) By Different Models**A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998**Dielectric Functions for Copper (Cu) By Different Models**A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998**Dielectric Functions for Aluminum (Al)**A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998**Correction to Drude Model Due to Size Effect**For nano-particles with dimensions comparable to free electron mean-free-path (i.e., 10nm), the particle surface puts restriction to the movement of the free electrons, leading to Surface Damping Effect . A constant whose value depends on the shape of the particle and close to unity The Fermi velocity of electrons The radius of the metal particle**Ideal metal particle under static electric field**• The original electric field induces surface charge on the metal particle, of which the induced scattering electric field cancels the original field inside the metallic particle and enhances the outer field.**Ideal metal particle in quasi-static field**• In case of quasi-static field, which means the incident field is slow-varying, the scattering field of induced charge and its movement inside the ball follows the incident field.**If the dimension of the particle is much smaller than the**incident wavelength, it can be considered as quasi-static case. Comparing to the light wavelength in scale of μm, we choose nm scale for the radius of the metallic particle, to obey the quasi-static condition. In this case, the metallic ball can be seen equivalent to an oscillating electric dipole. Nano-metallic particle**Electric Potential by Sub-Wavelength Particle**For the radius of the particle much smaller than the optical wavelength, i.e., a<<, the electric quasi-static approximation is valid. Governing Equations a Eo(t) z in out General Solutions Boundary Conditions**Electric Potential by Sub-Wavelength Particle**a Eo(t) z in out Induced Dipole Eo(t) p z out**Polarizability of Sub-Wavelength Particle**For meal particles in dielectric materials If the following condition is satisfied, The SPP resonance is due to the interaction between EM field and localized plasma and determined by the geometric and material properties of the sub-wavelength particle, independent of its size then we have localized SPP resonance**Electric Field Induced by Sub-Wavelength Particle**n r p Field Inside Weakened for Positive Re(in) and Enhanced for Negative Re(in)