INVERSE DESIGN FOR NANO-PHOTONICS Yablonovitch Group, UC Berkeley June 4 th 2013.
Nano-Photonics (2)
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1
Department of Electrical and Computer Engineeringhotonicsresearchaboratory
Nano-Photonics (2)Nano-Photonics (2)W. P. Huang
Department of Electrical and Computer Engineering
McMaster University
Hamilton, Canada
2
AgendaAgenda
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 MetalOptical Properties of Metal
Basic Relations for Refractive Indices
R In n jn
Complex Refractive Index
12 2 2 1
2[ ]
2( )R I R
Rn
R Ij
Relative Dielectric Constant
12 2 2 1
2[ ]
2( )R I R
In
Relationship between Dielectric Constant and Refractive Index
2n n
2 2R R In n
2I R In n
R In n
0R
If
then 12( )R Rn 1
22( )I
I
R
n
I R If
then
5
Microscopic ModelsMicroscopic 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 MagnificationsElement Copper Under Different Magnifications
7
The Atomic StructureThe Atomic Structure
~100 pm
8
Models For AtomsModels For Atoms
9
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”
Electron
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
2nd shell: 8 electrons
3rd shell: 18 electrons
3s
3p
3dSub-shells
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: 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
mkωo
11
• One atom, e.g. H.
• Two atoms: bond formation
E
H+
H+ H+?
• Equilibrium distance d (after reaction)
Every electron contributes one state
Atoms and Bounds
12
~ 1 eV
• Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin )
Formation of Energy Bands
13
Empty
outer orbitals
Partly filled
valence orbitals
FilledInnershells
Distance between atoms
Ene
rgy
Outermost electrons interact
Electrons in inner shells do not interact
Form bands
Do not form bands
Energy Band Characteristics
14
Band Diagrams & Electron Filling
Electrons filled from low to high energies till we run out of electrons
Empty band
Energy
Partially full band
Metal
Empty band
Full band
Gap ( ~ 1 eV)
Semiconductor
Empty band
Full band
Gap ( > 5 eV)
Insulator
15
Color of Metals
Empty band
Energy
Partially full band
> 3.1 eV3.1 eV (violet)
1.7 eV (red)2.4 eV (yellow)
Silver
All colors absorbed and immediately re-emitted; this is why silver is white (or silvery)
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”
16
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%)
Optical Processes in Metals
17
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.
Drude Model: Drude Model: Free Carrier Contributions to Optical PropertiesFree Carrier Contributions to Optical Properties
Bound electrons Conduction electrons
EEH ωεjωωσωjω EFFoBo εεε
ωτj1
σωσ o
m
τneσ
2
o
ωε
ωσjωεωε
oBEFF
teEtvτ
mtv
dt
dm
m
ωeEωv
τ
1jω
ωEωσωnevωJ
2
o
o2
o
oBEFF
1
1
ωε
σj
1
1
ε
σεωε
ωτωτ
τ
ωτj1
1
ωε
σjεωε
o
oBEFF
18
Low Frequency Response by Drude Model
If << 1:
Constant of Frequency, Negligible at low Frequency
Inverse Proportional to ω, Dominant at Low Frequency
ωτωτ
ωτ
ωτj1σ
1
j1σ
j1
σωσ o2
oo
ωε
σj
ε
τσεjω1
ωε
σjεωε
o
o
o
oB
o
oBEFF
τ
o
oEFF ε
σ
ω
1jωε
ωε
σj
ε
τσεωε
o
o
o
oBEFF
IREFF jnnεωn
R
2I
2R
2I
2R
n
21
n1n
n1nωR
ωε
σnn
o
oIR
At low frequencies, metals (material with large concentration of free carriers) is a perfect reflector
m
τneσ
2
o
19
If >> 1:
Plasma Frequency:
ωτωτωτ
ωτ
ωτo
2o
2oo σ
jσ
1
j1σ
j1
σωσ
ωε
ωσjεωε
oBEFF
τω
ωj
ω
ωεωε
3
2P
2
2P
BEFF mε
ne
τε
σω
o
2
o
o2P
(about 10eV for metals)
High Frequency Response by Drude Model
2
2P
BEFF ω
ωεωε
2
2P
BREFF ω
ωεnωεn
2R
2R
1n
1nωR
At high frequencies, the contribution of free carriers is negligible and metals behaves like an insulator
32
o
o2
o
oBEFF ωε
σj
ωε
σεωε
ττ
As the frequency is very high
20
2
o
2
2o
22
IREFF1
1
ωmε
τnej
1
1
mε
τne1ωjεωεωε
ωτωτ
Plasma Frequency in Drude Model
0τω1
1
mε
τne10ωε
2po
22
pR
1mε
τne
τ
1ω
o
22
p o
2
p mε
neω
pωω At the Plasma frequency
The real part of the dielectric function vanishes
p
pIpEFF τω
1jωjεωε
ωε R
pωω At the Plasma frequency
2
2p
2
2p
EFF ω
ω
ωτ
1j
ω
ω1ωε
For Free Electrons
21
Measured data and model for Ag:
τ
1
ω
ωε,
ω
ω1ε
3
2p
I2
2p
R
τω
ωε
ω
ωεε
3
2p
I
2
2p
BR
Drude model:
Modified Drude model:
Contribution of bound electrons
Ag: 3.4ε B
200 400 600 800 1000 1200 1400 1600 1800-150
-100
-50
0
50
Measured data: ' "
Drude model: ' "
Modified Drude model: '
"
Wavelength (nm)
'Rε
Iε
Iε
Iε
Rε
Rε
Rε
Validation of Drude Model
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
24M.A.Ordal,et.al., Appl. Opt., vol.22,no.7,pp.1099-1120,1983
Model Parameters
25
Improved Model Parameters
M.A.Ordal,et.al., Appl. Opt., vol.24,no.24,pp.4493-4500,1985
26M.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
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
Ion Core
e-,m
+
k, xIon Core ro
r
E L
Electron Clouds
ep x
Classical Lorentz Model
Potential Energy
kxrrkUF 0
Repulsion Force
tEdt
dxmγkx
dt
xdm L2
2
e
Newton’s 2nd Law Damping Force
Electric Force
20rrk2
1U
Repulsion Force
30
2 2
1L
o
eX E
m j
2
2 L
d x dxm kx m eE t
dt dt
j tx t X e
j tL LE t E e
2
2 2
1e L
o
ep q X E
m j
Atomic Polarizability by Lorentz ModelAtomic Polarizability by Lorentz Model
Resonance frequency
Define atomic polarizability:
Damping term
2
2 2
1
o L o o
p e
E m j
31
Characteristics of Atomic PolarizabilityCharacteristics of Atomic Polarizability
• Atomic polarizability:
2
1/ 222 2 2 200
1eA
m
Response of matter is not instantaneous
• Amplitude
• Phase lag of with E:
12 20
tan
Am
plitu
deP
hase
lag
00
180
90
smaller
smaller
-dependent response
2
2 2
1exp
o o
eA j
m j
32
Correction to Drude Model Due to Band Transition for Bound Electrons
f bEFF
1
Mb
kk
1
11
pf jj
2
2 2
i pk
k k
f
j
Lorentz-Drude (LD) Model Brendel-Bormann (BB) Model
2 2
2 2 2
1exp
22i pk
kkk k
fd
j
2expw z z erfc jz
2
2 2 2k p k k k k
k
k k k k
f a aj w w
a
21exp
zerfc z t dt
33
Refractive Index of Al from Classical Drude ModelRefractive 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
36A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998
Dielectric Functions for Gold (Au) By Different Models
37A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998
Dielectric Functions for Copper (Cu) By Different Models
38A. D. Rakic,et.al.,Appl.Opt., vol.37,no.22,pp.5271-5283,1998
Dielectric Functions for Aluminum (Al)
39
Correction to Drude Model Due to Size Effect
f bEFF 1
11
pf jj
bulk FAv a
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 A constant whose value depends on the shape of the particle and close to unity
:Fv The Fermi velocity of electrons
:a The radius of the metal particle
40
Size Effects of Metal ParticlesSize Effects of Metal Particles
41
Size Effects of Metal ParticlesSize Effects of Metal Particles
42
Localized SPPLocalized SPP
43
Ideal metal particle under static electric fieldIdeal 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 fieldIdeal 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 particleNano-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 ParticleElectric Potential by Sub-Wavelength Particle
Boundary Conditions
z
inout
Eo(t)
0
1
0
, cos
, cos
lin l l
l
llout l l l
l
r A r P
r B r C r P
, ,
, ,
in out
in in out out
a a
a ar r
General Solutions
Governing Equations
2
22
2 2 2 2 2
, 0
1 1 1sin 0
sin sin
r
rr r r r r
For the radius of the particle much smaller than the optical wavelength, i.e., a<<, the electric quasi-static approximation is valid.
a
47
Electric Potential by Sub-Wavelength ParticleElectric Potential by Sub-Wavelength Particle
z
inout
Eo(t)
3
3
3, cos
2
, cos 12
outin o
out in
out inout o
out in
r E r
ar E r
r
3, cos
4out oin out
r E rr
p r
in out op E34
2out in
out in
a
Induced Dipole
z
pout
Eo(t)
a
48
Polarizability of Sub-Wavelength ParticlePolarizability of Sub-Wavelength Particle
342
out in
out in
a
max
2 0out in
If the following condition is satisfied,
then we have localized SPP resonance
0, 0out in
For meal particles in dielectric materials
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
49
Scattering Field Distribution of Small Metal ParticleScattering Field Distribution of Small Metal Particle
50
Electric Field Induced by Sub-Wavelength ParticleElectric Field Induced by Sub-Wavelength Particle
3
3 1
4 in out r
out o
n p n - pE E
3
2out
in out
in oE E
p
n
r
1.0, 10.0out in
Field Inside Weakened for Positive Re(in)
and Enhanced for Negative Re(in)
51
Field Radiated by Induced DipoleField Radiated by Induced Dipole
p
n
r expin outt j t op E
22
1 1
4
jkr
in out
jk ek
r r r
E n p n 3n n p p
2 1
14
jkrck e
jkr r
H n p
Far Field
(Radiation Field)
Near Field
(Static Field)
Intermediate Field
(Induction Field)
342
out in
out in
a
52
EM Field by An Electric DipoleEM Field by An Electric Dipole
53
Near Field Approximation: Static FieldNear Field Approximation: Static FieldFor kr<<1, the static field dominates
in out op E
3
1
4
jkr
in out
e
r
E 3n n p p
0H
3a
r
E
342
out in
out in
a
54
Far Field Approximation: Radiation FieldFar Field Approximation: Radiation Field
For kr>>1, the radiation field dominates
2
4
jkrck e
r
H n p jkr
o
in out
e
r
E H n
out
o
H E
2 2 sin2
jkrout in
out in
eE k a
r
in out op E34
2out in
out in
a
55
Scattering Cross-SectionScattering Cross-Section
Total Radiation Power
2 2 sin2
jkrout in
out in
eE k a
r
22
0 0
2
2 3
1sin
2
4
3 2
r
out out ino
o out in
P r d d E H
k a E
242 4 68
6 3 2out inr
scao out in
P kC k a
S
Scattering Cross Section
Time-Average Power Flow Density
1 1 ˆˆ2 2 rE H E H
S E H r θ out
o
H E
21
2out
o oo
S E
Power Flow Density for External Field
56
Absorption Cross-SectionAbsorption Cross-Section
342
a out inabs
o out in
PC k a k
S
Polarization Vector
1
2a oP P E
Absorption Cross Section
Time-Average Absorbed Power Density
21
2out
o oo
S E
Power Flow Density for External Field
in out oP E34
2out in
out in
a
57
Extinction Cross-SectionExtinction Cross-Section
24
2 4 6 384
6 3 2 2out in out in
extout in out in
kC k k a a k
ext sca absC C C
3 2
2 292
iin
ext outr iin out in
C Vc
Extinction Cross- Section for a Silver Sphere in Air (Black) and Silica (Grey), Respectively
58
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
Beyond Quasi-Static Limit: Multipole EffectsBeyond Quasi-Static Limit: Multipole Effects
59
Classical Mie’s Theory: General FormulationsClassical 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
mk
M r
N M
2 2 2 0k m
v u
u v
j
m j
E M N
H M N
The EM fields can be expressed in terms of the scalar potential function
60
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
1
1
1
1
2 1exp( )cos ( ) (cos ) ( )
( 1)
2 1exp( )sin ( ) (cos ) ( )
( 1)
nn n
n
nn n
n
nu j t j P j kmr
n n
nv j t j P j kmr
n n
1 (2)
1
1 (2)
1
2 1exp( )cos ( ) (cos ) ( )
( 1)
2 1exp( )sin ( ) (cos ) ( )
( 1)
nn n n
n
nn n n
n
nu j t a j P h kmr
n n
nv j t b j P h kmr
n n
11
1
11
1
2 1exp( )cos ( ) (cos ) ( )
( 1)
2 1exp( )sin ( ) (cos ) ( )
( 1)
nn n n
n
nn n n
n
nu j t m c j P j kmr
n n
nv j t m d j P j kmr
n n
Even solution:
Odd solution:
61
Classical Mie’s Theory: Expansion Coefficients
Match the boundary conditions on the interface to determine the coefficients,
'( ) ( ) ( ) '( )( , )
'( ) ( ) ( ) '( )
'( ) ( ) ( ) '( )( , )
'( ) ( ) ( ) '( )
n n n nn
n n n n
n n n nn
n n n n
y x m y xa x y
y x m y x
m y x y xb x y
m y x y x
2 in
in
out
ax n
ny x
n
1/ 2
(2)1/ 2
( ) ( )2
( ) ( )2
n n
n n
J
H
: the radius of metal particle;
: the wavelength in vacuum;
: the refractive index of the metal particle;
: the refractive index of the matrix.
a
in inn
out outn
62
2
1
exp( )cos ( )
exp( )sin ( )
jE H jkmz j t S
krj
E H jkmz j t Skr
11
21
2 1( ) (cos ) (cos )
( 1)
2 1( ) (cos ) (cos )
( 1)
n n n nn
n n n nn
nS a b
n n
nS b a
n n
1
1
1(cos ) (cos )
sin
(cos ) (cos )
n n
n n
P
dP
d
Classical Mie’s Theory: EM Fields
63
Efficiency Factors and Cross-SectionsEfficiency 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,
/
/
/
ext ext
sca sca
abs abs
Q C G
Q C G
Q C G
2G a
abs ext scaQ Q Q
2
4Re (0)extC S
k
1 21
1(0) (0) (0) (2 1)
2 n nn
S S S n a b
21
2(2 1) Reext n n
n
Q n a bx
2 2
21
2(2 1)sca n n
n
Q n a bx
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,
1outn x
3
1
2
3
jxa P
2in out
in out
P
1 1
2 1
3
23
cos2
S a
S a
428
4 Im 4 Im9abs ext sca
xQ Q Q x P P x P
2 1/ 2 34Im
2in out
abs abs outin out
C a Q ac
65
Multi-pole Approximation
2
22
3
10
3 2413 30 3
1
10out
in out
zout in
Vzout in out
Vj
az
The Polarizability of a sphere of volume V
66
Absorption Spectra of A Nano-Particle of Small SizeAbsorption 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 SizeAbsorption Spectra of Nano-Particle of Large Size
A Single Silver Nano-Particle in Matrix of Index 1
68
Broadening of Absorption Spectrum Broadening of Absorption Spectrum Due to Quantum Size EffectDue to Quantum Size Effect
2
( ) 1( ( ))
ps
i R
( ) FAR
R
Increase of Damping Leads to Broadening as Size of the Particle Decreases
Modified Drude Model
Additional Damping Due to Size Effect
69
Normalized Scattering Cross-Section Normalized Scattering Cross-Section for a Gold Sphere in Airfor a Gold Sphere in Air
Note: Normalized by the radius^6
70
Normalized Absorption Cross-Section Normalized Absorption Cross-Section for a Gold Sphere in Airfor a Gold Sphere in Air
Note: Normalized by the volume V
71
Normalized Scattering and Absorption Cross-Sections Normalized Scattering and Absorption Cross-Sections for a Silver Sphere in Airfor 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 RatioField Patterns for Different Wavelength-Radius Ratio
a<<
a≈
a≈2
a>>
74
Effects of Geometrical Shape: Ellipsoid
22 2
1 2 3
1x y z
a a a
1 2 343 3
in outi
out i in out
a a aL
a1
a2a3
1 2 3
2 2 2 20 1 2 3
2i
i
a a a dqL
a q a q a q a q
3
1
1ii
L
The Polarizabilities along the principal axes:
75
Special CasesProlate spheroid Oblate spheroid
Short axes are equal, a>b=c; cigar-shaped Long axes are equal, a=b>c; disk-shaped
76
Shift of Resonance Peaks Due to Geometric ShapeShift of Resonance Peaks Due to Geometric Shape
Normalized absorption cross-section for a gold ellipsoid in the air
Prolate Oblate
77
Normalized absorption cross-section for a silver ellipsoid in the air
Prolate Oblate
Shift of Resonance Peaks Due to Geometric ShapeShift of Resonance Peaks Due to Geometric Shape
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 Split Resonance Frequencies Due to Coupling In the Nano-Particle ChainIn the Nano-Particle Chain
81
Propagation Modes along SPP ChainPropagation Modes along SPP Chain
Calculated dispersions relations for gold nanoparticle chain, including only dipole-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)]dipole coupling in quasistatic approximation [S. A. Maier et al, Adv. Mat. 13, 1501 (2001)]
(L and T denote longitudinal and transverse modes)
82
SummarySummary 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.
83
Project Topics: Choose 1 of 2Project Topics: Choose 1 of 2
Topic A: SPP Waveguides and ApplicationsTopic 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