Nano-Photonics (2)

1
Nano-Photonics (2)

• W. P. Huang
• Department of Electrical and Computer Engineering
• McMaster University
• Hamilton, Canada

2
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

3
Optical Properties of Metal
4
Basic Relations for Refractive Indices
Complex Refractive Index
Relative Dielectric Constant
Relationship between Dielectric Constant and
Refractive Index
If
If
then
then
5
Microscopic Models
Models Light Matter
Classical Dipole Oscillator Classical EM Wave Classical Atoms
Semi-classical Inter/intra-band transition Classical EM Wave Quantum Atoms
Quantum Photon and Atom Interaction Quantum Photons Quantum Atoms
6
Element Copper Under Different Magnifications
7
The Atomic Structure
100 pm
8
Models For Atoms
9
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
10
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 Fkx
• Natural frequency
• Natural frequencies lie in visible, infrared and
  UV range
• Free Electrons (metals, doped semiconductors)
• k0 so that natural frequency0

11
Atoms and Bounds
One atom, e.g. H.
Two atoms bond formation
Every electron contributes one state
Equilibrium distance d (after reaction)
12
Formation of Energy Bands
1 eV
Pauli principle Only 2 electrons in the same
electronic state (one spin one spin )
13
Energy Band Characteristics
Outermost electrons interact
Form bands
Electrons in inner shells do not interact
Do not form bands
14
Band Diagrams Electron Filling
Energy
Empty band
Partially full band
Metal

• Electrons filled from low to high energies
  till we run out of electrons

15
Color of Metals
Silver
Energy
Empty band
3.1 eV (violet)
gt 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)
16
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 (gt 95)

17
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
18
Low Frequency Response by Drude Model
If ?? ltlt 1
Constant of Frequency, Negligible at low Frequency
Inverse Proportional to ?, Dominant at Low
Frequency
At low frequencies, metals (material with large
concentration of free carriers) is a perfect
reflector
19
High Frequency Response by Drude Model
If ?? gtgt 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
20
Plasma Frequency in Drude Model
For Free Electrons
At the Plasma frequency
The real part of the dielectric function vanishes
At the Plasma frequency
21
Validation of Drude Model
Measured data and model for Ag
Drude model
Modified Drude model
Contribution of bound electrons Ag
22
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
23
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
24
Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp.1099-
1120,1983
25
Improved Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp.4493
-4500,1985
26
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
27
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

28
Refractive Index of Aluminum (Al)
Band-Transition Peak
29
Classical Lorentz Model
Electron Clouds
e-,m
E L
Ion Core
k, ?
x
ro
e
-
p
x
r

Ion Core
Potential Energy
Repulsion Force
Newtons 2nd Law
Damping Force
Electric Force
Repulsion Force
30
Atomic Polarizability by Lorentz Model
Define atomic polarizability
Resonance frequency
Damping term
31
Characteristics of Atomic Polarizability
Response of matter is not instantaneous
?-dependent response
Atomic polarizability
Amplitude
Amplitude
Phase lag of ? with E
180
smaller ?
90
Phase lag
0
32
Correction to Drude Model Due to Band Transition
for Bound Electrons
Brendel-Bormann (BB) Model
Lorentz-Drude (LD) Model
33
Refractive Index of Al from Classical Drude Model
34
Refractive Index of Al from Modified Drude Model
Considering Band-Transition Effects
35
Dielectric Functions for Silver (Ag) By Different
Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271
-5283,1998
36
Dielectric Functions for Gold (Au) By Different
Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271
-5283,1998
37
Dielectric Functions for Copper (Cu) By Different
Models
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271
-5283,1998
38
Dielectric Functions for Aluminum (Al)
A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271
-5283,1998
39
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
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Size Effects of Metal Particles
41
Size Effects of Metal Particles
42
Localized SPP
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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.

44
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.

45
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.

46
Electric Potential by Sub-Wavelength Particle
For the radius of the particle much smaller than
the optical wavelength, i.e., altlt?, the electric
quasi-static approximation is valid.
Governing Equations
a
Eo(t)
?
z
?in???
?out
General Solutions
Boundary Conditions
47
Electric Potential by Sub-Wavelength Particle
a
Eo(t)
?
z
?in???
?out
Induced Dipole
Eo(t)
?
p
z
?out
48
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
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Scattering Field Distribution of Small Metal
Particle
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Electric Field Induced by Sub-Wavelength Particle
n
r
p
Field Inside Weakened for Positive Re(?in) and
Enhanced for Negative Re(?in)
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Field Radiated by Induced Dipole
n
r
p
Near Field (Static Field)
Intermediate Field (Induction Field)
Far Field (Radiation Field)
52
EM Field by An Electric Dipole
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Near Field Approximation Static Field
For krltlt1, the static field dominates
54
Far Field Approximation Radiation Field
For krgtgt1, the radiation field dominates
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Scattering Cross-Section
Time-Average Power Flow Density
Total Radiation Power
Power Flow Density for External Field
Scattering Cross Section
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Absorption Cross-Section
Time-Average Absorbed Power Density
Power Flow Density for External Field
Polarization Vector
Absorption Cross Section
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Extinction Cross-Section
Extinction Cross- Section for a Silver Sphere in
Air (Black) and Silica (Grey), Respectively
58
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
59
Classical Mies 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
60
Classical Mies 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
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Classical Mies 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.
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Classical Mies Theory EM Fields
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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,

64
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,

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Multi-pole Approximation
The Polarizability of a sphere of volume V
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Absorption Spectra of A Nano-Particle of Small
Size
A Single Silver Nano-Particle in Matrix of Index
1
67
Absorption Spectra of Nano-Particle of Large Size
A Single Silver Nano-Particle in Matrix of Index
1
68
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
69
Normalized Scattering Cross-Section for a Gold
Sphere in Air
Note Normalized by the radius6
70
Normalized Absorption Cross-Section for a Gold
Sphere in Air
Note Normalized by the volume V
71
Normalized Scattering and Absorption
Cross-Sections for a Silver Sphere in Air
72
Absorption Efficiency of a 20 nm Gold Sphere for
Different Ambient Refractive Indices
73
Field Patterns for Different Wavelength-Radius
Ratio
altlt?
a2?
agtgt?
a?
74
Effects of Geometrical Shape Ellipsoid
The Polarizabilities along the principal axes
a3
a2
a1
75
Special Cases
Prolate spheroid
Oblate spheroid
Long axes are equal, abgtc disk-shaped
Short axes are equal, agtbc cigar-shaped
76
Shift of Resonance Peaks Due to Geometric Shape

• Normalized absorption cross-section for a gold
  ellipsoid in the air

Oblate
Prolate
77
Shift of Resonance Peaks Due to Geometric Shape

• Normalized absorption cross-section for a
  silver ellipsoid in the air
• Prolate
  Oblate

78
Effects of Geometrical Shape
79
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

80
Split Resonance Frequencies Due to Coupling In
the Nano-Particle Chain
81
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)
82
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.

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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