Department of Electrical and Computer Engineering 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

Optical Properties of Metal

Basic Relations for Refractive Indices Complex Refractive Index Relative Dielectric Constant Relationship between Dielectric Constant and Refractive Index If If then then

Microscopic Models Models Light Matter Classical: Dipole Oscillator Classical EM Wave Classical Atoms Semi-classical: Inter/intra-band transition Quantum Atoms Quantum: Photon and Atom Interaction Quantum Photons

Element Copper Under Different Magnifications

The Atomic Structure ~100 pm

Models For Atoms Bohr Model: The energy of the electron is proportional to its distance from the nucleus. It takes energy to move an electron from a region close to the nucleus to one that is further away. Only a limited number of regions with certain energies are allowed. As a result, the energy of the electron in a hydrogen atom is quantized. When light is absorbed an electron jumps from a lower energy state to a higher energy state. The energy of this radiation is equal to the difference between the energies of the two states. Light is emitted when an electron falls from a higher energy state into a lower energy state, the energy of the radiation is equal to the difference.

Nature of Electrons in Atoms 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

Atoms and Bounds • One atom, e.g. H. E + • 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 )

Energy Band Characteristics Empty outer orbitals Partly filled valence orbitals Filled Inner shells Distance between atoms Energy Outermost electrons interact Form bands Distance between atoms determines whether we have an insulator, a metal or a semiconductor Electrons in inner shells do not interact Do not form bands

Band Diagrams & Electron Filling Energy Empty band Empty band Full band Gap ( ~ 1 eV) Semiconductor Empty band Full band Gap ( > 5 eV) Insulator Partially full band Metal Electrons filled from low to high energies till we run out of electrons

Color of Metals Silver Gold Empty band Partially full band 3.1 eV (violet) 1.7 eV (red) 2.4 eV (yellow) Gold Only colors up to yellow absorbed and immediately re-emitted; blue end of spectrum goes through, and gets “lost” Energy Empty band 3.1 eV (violet) 2.4 eV (yellow) > 3.1 eV 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 ro k,  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

Characteristics of Atomic Polarizability Response of matter is not instantaneous -dependent response • Atomic polarizability: • Amplitude smaller  Amplitude • Phase lag of  with E: 180 At low frequencies charges can follow. At high frequencies they cannot Any complex number you can always write as an amplitude times an exponent with a phase factor smaller  Phase lag 90

Correction to Drude Model Due to Band Transition for Bound Electrons Brendel-Bormann (BB) Model Lorentz-Drude (LD) Model

Refractive Index of Al from Classical Drude 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

Size Effects of Metal Particles

Size Effects of Metal Particles

Localized SPP

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.

Nano-metallic particle 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.

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)  in z out General Solutions Boundary Conditions

Electric Potential by Sub-Wavelength Particle Eo(t) a  in z 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

Scattering Field Distribution of Small Metal Particle

Electric Field Induced by Sub-Wavelength Particle Field Inside Weakened for Positive Re(in) and Enhanced for Negative Re(in)

Field Radiated by Induced Dipole Near Field (Static Field) Far Field (Radiation Field) Intermediate Field (Induction Field)

EM Field by An Electric Dipole

Near Field Approximation: Static Field For kr<<1, the static field dominates

Far Field Approximation: Radiation Field For kr>>1, the radiation field dominates

Scattering Cross-Section Time-Average Power Flow Density Total Radiation Power Power Flow Density for External Field Scattering Cross Section

Absorption Cross-Section Time-Average Absorbed Power Density Power Flow Density for External Field Polarization Vector Absorption Cross Section

Extinction Cross-Section Extinction Cross- Section for a Silver Sphere in Air (Black) and Silica (Grey), Respectively

Beyond Quasi-Static Limit: Multipole Effects When radius of the particle increases and becomes large compared with the optical wavelength, the distribution of the induced charge and current as well as the phase change or retardation effect of the field need to be considered. The distribution of charge and current can be decomposed to two sequences of electric and magnetic multi-poles. First four are showed as below, As the radius increases, higher order multi-poles occurs in sequence. Electric and magnetic dipoles and quadrupoles

Classical Mie’s Theory: General Formulations Exact solutions to Maxwell equations in terms of vector spherical harmonics. The wave equations for the scalar potential The EM fields are expressed by The EM fields can be expressed in terms of the scalar potential function

Classical Mie’s Theory: Scalar Potential In spherical coordinates, the incident wave can expand as the series of Legendre polynomials and spherical Bessel functions The scattered wave and the wave inside the sphere are given by Even solution: Odd solution:

Classical Mie’s Theory: Expansion Coefficients Match the boundary conditions on the interface to determine the coefficients, : the radius of metal particle; : the wavelength in vacuum; : the refractive index of the metal particle; : the refractive index of the matrix.

Classical Mie’s Theory: EM Fields

Efficiency Factors and Cross-Sections The efficiency factors and cross sections for extinction, scattering and absorption are related as below, where G is the geometrical cross section of the particle, for instance, for a sphere of radius a, Due to the fundamental extinction formula,

Small-Particle Limit: Static Approximation The small-particle limit is indicated as, Under this limit, only one of the Mie coefficients remains non-zero value,

Multi-pole Approximation The Polarizability of a sphere of volume V

Absorption Spectra of A Nano-Particle of Small Size A Single Silver Nano-Particle in Matrix of Index 1

Absorption Spectra of Nano-Particle of Large Size A Single Silver Nano-Particle in Matrix of Index 1

Broadening of Absorption Spectrum Due to Quantum Size Effect Modified Drude Model Additional Damping Due to Size Effect Increase of Damping Leads to Broadening as Size of the Particle Decreases

Normalized Scattering Cross-Section for a Gold Sphere in Air Note: Normalized by the radius^6

Normalized Absorption Cross-Section for a Gold Sphere in Air Note: Normalized by the volume V

Normalized Scattering and Absorption Cross-Sections for a Silver Sphere in Air

Absorption Efficiency of a 20 nm Gold Sphere for Different Ambient Refractive Indices

Field Patterns for Different Wavelength-Radius Ratio

Effects of Geometrical Shape: Ellipsoid The Polarizabilities along the principal axes: a3 a2 a1

Long axes are equal, a=b>c; disk-shaped Special Cases Prolate spheroid Oblate spheroid Short axes are equal, a>b=c; cigar-shaped Long axes are equal, a=b>c; disk-shaped

Shift of Resonance Peaks Due to Geometric Shape Normalized absorption cross-section for a gold ellipsoid in the air Prolate Oblate

Shift of Resonance Peaks Due to Geometric Shape Normalized absorption cross-section for a silver ellipsoid in the air Prolate Oblate

Effects of Geometrical Shape

Coupling between Spheres in a Particle Chain In the dipole approximation, there are three SP modes on each sphere, two polarized perpendicular to chain, and one polarized parallel. The propagating waves are linear combinations of these modes on different spheres

Split Resonance Frequencies Due to Coupling In the Nano-Particle Chain

Propagation Modes along SPP Chain Calculated dispersions relations for gold nanoparticle chain, including only dipole-dipole coupling in quasistatic approximation [S. A. Maier et al, Adv. Mat. 13, 1501 (2001)] (L and T denote longitudinal and transverse modes)

Summary Localized SPP Resonance occurs at the frequency in which the negative real part of dielectric constant of the metal is equal to positive real part of dielectric constant for the surround materials For small particles, the SPP resonance frequency is dependent on the geometrical shape of the particle as well as the material properties of the metal and surrounding material, but independent of the size As the dimension of the particle increases, the multi-pole effects become important, whereas for ultra-small dimension the surface damping effect is more pronounced Near-field dipole-dipole coupling is important as an efficient energy transfer mechanism for nano-photonic materials and devices.

Project Topics: Choose 1 of 2 Topic A: SPP Waveguides and Applications Topic B: SPP Resonators and Applications Requirements: Write a general review for the working principles and potential applications of SPP waveguides or resonators Submit your project report in MS word format to the instructor