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