OPTICAL PROPERTIES OF OPTICAL PROPERTIES OF METALLIC NANOPARTICLES, METALLIC NANOPARTICLES,

MOLECULES AND POLYMERSMOLECULES AND POLYMERS

Dr. Mica GrujicicDr. Mica Grujicic

April, 2004April, 2004Department of Mechanical EngineeringDepartment of Mechanical Engineering

Mie Theory Mie Theory --Dilute Colloidal Dilute Colloidal Solution LimitSolution Limit

Spherical ParticlesSpherical Particles

Ref: Ref: C. F.C. F.Bohren Bohren and D. R. Huffman, Absorption and D. R. Huffman, Absorption and Scattering of Light by Small Particles, and Scattering of Light by Small Particles,

Wiley: New York, 1983Wiley: New York, 1983..

Extinction CrossExtinction Cross--section of section of Spherical ParticlesSpherical Particles

( ) 222/332

224

pmp

pmpext

RC

++

=

mm Dielectric Function of the MediumDielectric Function of the Medium

RRpp -- Particle Radius Particle Radius -- Incident WavelengthIncident Wavelength

pp Real Part of Dielectric Function of ParticlesReal Part of Dielectric Function of Particles

pp Imaginary Part of Dielectric Function of ParticlesImaginary Part of Dielectric Function of Particles

ppp i +=

Dielectric Function of the NanoparticlesDielectric Function of the Nanoparticles

Complex Dielectric Function Complex Dielectric Function For Bulk MaterialFor Bulk Material

( ) ( ) ( ) ( )

bulkbulke

Pbulk iim

+=+

= 22

1

Excitation Angular FrequencyExcitation Angular Frequency

mmee Mass of ElectronMass of Electron

1

==e

F

lv

vvFF FermiFermi VelocityVelocity

02 eP mne=

nn Density of Free ElectronsDensity of Free Electrons

ee Electron ChargeElectron Charge

00 Permittivity of Free SpacePermittivity of Free Space

llee Mean Free PathMean Free Path

Relaxation TimeRelaxation Time

Damping Damping FrequencyFrequency

Bulk Bulk PlasmonPlasmon FrequencyFrequency

( ) 222

1+

=

Pbulk

Real Part of Dielectric Function of Bulk MaterialReal Part of Dielectric Function of Bulk Material

( ) ( )222

+

=

Pbulk

Imaginary Part of Dielectric Function of Bulk MaterialImaginary Part of Dielectric Function of Bulk Material

( ) 222

+=

Pbulk

Corrected Real Part of Dielectric Function of Bulk MaterialCorrected Real Part of Dielectric Function of Bulk Material

High Frequency Dielectric ConstantHigh Frequency Dielectric Constant

peeff Rll111

+=

Particle Size Effective MeanParticle Size Effective Mean--Free PathFree Path

eff

Feff l

v=

Effective Damping FrequencyEffective Damping Frequency

Effect of the Small Particle SizeEffect of the Small Particle Size

( ) 222

eff

Pp +

=

( ) ( )222

eff

effPp +

=

FreeFree--electron Real Part of the electron Real Part of the Dielectric Function of Spherical ParticlesDielectric Function of Spherical Particles

FreeFree--electron Imaginary Part of the electron Imaginary Part of the Dielectric Function of Spherical ParticlesDielectric Function of Spherical Particles

Effect of Effect of Intrabound Intrabound TransitionsTransitions

( ) ( ) ( ) ( )pbulkierbandifreeii ,int =+= Total Complex Dielectric FunctionTotal Complex Dielectric Function

( ) ( ) ( ) ( ) freebulkbulkfreepp +=Dielectric Constant in Metallic NanoparticlesDielectric Constant in Metallic Nanoparticles

Nextddl C =

DiluteDilute--dispersion Limit dispersion Limit Adsorption CoefficientAdsorption Coefficient

NN Number Density of ParticlesNumber Density of Particles

NonNon--spherical Particlesspherical Particles

Ref: R. Ref: R. GansGans, Ann. Phys., 47 (1915) 270, Ann. Phys., 47 (1915) 270

Extinction Cross Section of Extinction Cross Section of NonNon--Spherical ParticlesSpherical Particles

( )cbaj

PP

PRCj

pmj

jp

pjmext ,,

1

13

8

2

2

22332

=

+

+

=

21

;111ln

211

2

2a

cbaP

PPrr

rrrP

==

+

=

( )21 abr =

Depolarization Vector for Nanorod (a>b=c)Depolarization Vector for Nanorod (a>b=c)

wherewhere

Gold

SilverGold

Silver

Input: Real and Imaginary Parts of the Dielectric Constants For Input: Real and Imaginary Parts of the Dielectric Constants For Gold and Silver as a Function of the Photon WavelengthGold and Silver as a Function of the Photon Wavelength 43704370

Ref: P. B. Johnson and R. W. Christy, Phys. Rev. B, 6 (1972) 4370

Wavelength, nm

Extin

ctio

nC

oeffi

cien

t,M

-1cm

-1

300 400 500 600 7000

1000

2000

3000

4000

5000

Calculated Absorption Spectra of Au Particles in Water

n = 1.334

ResultsResults

Dielectric Constant of

Water

Spherical Spherical ParticlesParticles

Wavelength, nm

Nor

mal

ized

Abso

rban

ce

500 525 550 575 6000

0.5

1

1.5

2

Calculated Absorption Spectra of Au Particles Media with DifferentDielectric Constant

n=1.602

1.3341.376

1.421

1.471

H2O (n=1.334)Cyclohexane (n=1.376)

Dodecane (n=1.421)Decalin (n=1.471)

CS2 (n=1.602)

Medium Dielectric Constant

Spherical Spherical ParticlesParticles

n=1.334n=1.334 n=1.407n=1.407 n=1.481n=1.481 n=1.525n=1.525 n=1.583n=1.583

Ref: S. Underwood and P. Mulvaney,

Langmuir, 10 (1994) 3427-3430

15 nm Au Spherical Particles in Water and in

Mixtures of Butyl Acetate and

Carbon Disulfide

Spherical Gold ParticlesEffect of Dielectric Constant of the Medium

Mie Mie Theory Theory Transmission Transmission

ColorsColors

Spherical Spherical ParticlesParticles

ExperimentExperiment

TheoryTheory

Elongated Ellipsoidal ParticlesElongated Ellipsoidal Particles

Wavelength, nm

Abso

rban

ce,a

.u

400 500 600 700 800 9000

50

100

150

200

250

300

350

400

450

Particle Aspect Ratio

3.63.33.1

2.9

2.6Longitudinal Longitudinal

PlasmonsPlasmons, Red Shift, Red ShiftTransverse Transverse PlasmonsPlasmons, Blue Shift, Blue Shift

Medium Dielectric Constant = 4.0

Wavelength, nm

Abso

rban

ce,a

.u

400 500 600 700 800 9000

50

100

150

200

250

300

350

400

450

Elongated Ellipsoidal ParticlesElongated Ellipsoidal Particles

Medium Dielectric Constant

3.0

2.5 Longitudinal Longitudinal PlasmonsPlasmons, Red Shift, Red ShiftTransverse Transverse

PlasmonsPlasmons, Red Shift, Red Shift

3.54.0 4.5

Aspect Ratio = 3.3

Regression Analysis of the Regression Analysis of the Wavelength at the Wavelength at the

LongitudinalLongitudinal PlasmonPlasmon PeakPeak

( ) 31.47231.4634.33max += mR

Maxwell Garnett TheoryMaxwell Garnett TheoryNonNon--Dilute Colloidal SolutionsDilute Colloidal Solutions

Ref: J. C. Maxwell Garnett, Ref: J. C. Maxwell Garnett, PhilosPhilos. Trans. R. Soc. . Trans. R. Soc. London,203 (1904) 385. London,203 (1904) 385.

Au Core

SiO2 Shell

(a)(a) (b)(b)

(a) Silica Coated Gold Particle; (b) Ideal Packing of Silica Coated Gold Particles in the Film to Form FCC Lattice with Volume Fraction 0.74.

Average Electric Field in Composite MaterialAverage Electric Field in Composite Material

( ) pmav EEE += 1

Particle Volume FractionParticle Volume Fraction

RRAuAu Radius of the Gold CoreRadius of the Gold Core

( )33

2

74.0

SiOAu

Au

RRR

+=

RRSiO2 SiO2 Thickness of SiOThickness of SiO22 ShellShell

EEmm Electrical Field in the Matrix MaterialElectrical Field in the Matrix Material

EEpp Electrical Field in the ParticleElectrical Field in the Particle

( )( ) ( ) ( ) avavppmmav EEEP 000 1111 =+=Average Polarization in Composite MaterialAverage Polarization in Composite Material

mmp

mp EE

2

3+

=

Electric Field Inside the ParticlesElectric Field Inside the Particles(Lorentz Cavity Field)(Lorentz Cavity Field)

mm Dielectric Function of the Matrix MaterialDielectric Function of the Matrix Material

Final Form of the EquationsFinal Form of the Equations

( ) mmp

mmav EEE

23

1+

+=

Average Electric Field in Composite MaterialAverage Electric Field in Composite Material

Average Dielectric Function in Composite MaterialAverage Dielectric Function in Composite Material

( ) ( )( ) ( )

++

++=

211221

mp

mpmav

( ) av

av

avav

knc

4Im==

Average Absorption Coefficient in Composite MaterialAverage Absorption Coefficient in Composite Material

Complex Dielectric FunctionComplex Dielectric Function

( ) ( )avddliikni iiiii ,2 =+=+=

( )avddlin iiii ,2

21

22

=

++=

( )avddlik iiii .2

21

22

=

+=

Complex Refractory IndexComplex Refractory Index

Optical ReflectanceOptical Reflectance

( )( )

( )avddliknkn

Rii

ii ,11

22

22

=++

+=

( )( ) ( ) ( )

( )avddliRhhR

RRTii

,2cos2expexp