optical properties of metallic nanoestructures jorge o. tocho centro de investigaciones Ópticas...

Optical properties of Optical properties of metallic nanoestructuresmetallic nanoestructures

Jorge O. Tocho

Centro de Investigaciones Ópticas (CONICET-CIC) and Departamento de Física, Facultad de Ciencias Exactas, Universidad Nacional de La Plata


JCE Classroom Activity # 62. Color My Nanoworld. W. Adam et al.

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11 Optical properties of materials. Optical properties of materials. Refractive index and dielectric Refractive index and dielectric function. Dielectrics and metals. function. Dielectrics and metals.


22 Size effects in metals. Spheres. Size effects in metals. Spheres. Coated spheres. Bimetallic Coated spheres. Bimetallic particles. particles.



33 NanowiresNanowires




44 Thin filmsThin films

55 SummarySummary









55 SummarySummary

Complex index of refractionComplex index of refraction

The salient optical properties of a material are specified by its The salient optical properties of a material are specified by its complex complex

index of refractionindex of refraction; which differs from the common index of refraction, ; which differs from the common index of refraction,

by incorporating an imaginary part related with the extinction coefficient, by incorporating an imaginary part related with the extinction coefficient,

Dielectric materialsDielectric materials typically have typically have k k equal to zero and equal to zero and nn which varies which varies

little with the wavelength of light. They are often computed using a single little with the wavelength of light. They are often computed using a single

n n value,value,

MetallicMetallic or conductive materialsor conductive materials have non-zero have non-zero kk and and n n which can vary which can vary

wildly with the wavelength, wildly with the wavelength,

Optics is wavelength basedOptics is wavelength based*, so, *, so, nn and and kk will be studied as function of will be studied as function of

wavelength. wavelength.

* Aubrey Jaffer, FreeSnell, Thin-Film Optical * Aubrey Jaffer, FreeSnell, Thin-Film Optical Simulator. Simulator.

N = n + ikN = n + ik

Dielectric functionDielectric functionFor real refractive index, ,

n and k of the complex refractive index are linked to the complex dielectric function,

The two set are related,

For nonmagnetic materials, 0

““Optics is Optics is wavelength based” wavelength based”

Optical spectroscopy is a great invent!

refractive index

dielectric function

Gold

wavelength, nm

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200-200

-150

-100

-50

0

50

Water

wavelength, nm

0 500 1000 1500 2000

0.0

0.5

1.0

1.5

2.0

2.5

3.0

´

´´

´´

´

Gold

wavelength, nm

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

0

2

4

6

8

10

12

14

16

Water

wavelength, nm

0 500 1000 1500 2000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

k

n

n

k

Reflectance

22

222

1

111

kn

knNN

R

Gold

wavelength, nm

0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200

R

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Water

wavelength, nm

0 500 1000 1500 2000

R

0.00

0.02

0.04

0.06

0.08

0.10

in vacuum or air,

K

Dissipative medium, F = -bv

The Lorentz modelThe Lorentz model(damping harmonic oscillator)(damping harmonic oscillator)

EE

m x´´ = e E - K x - b x´m x´´ = e E - K x - b x´

E(t) = E exp (iE(t) = E exp (it), t),

)(/

)( 220

tEi

metx

mbmK

;20

Dipolar moment and Dipolar moment and dielectric functiondielectric function

Dipolar moment for each oscillator is, p = ex.

For N oscillators per unit volume, the dipolar moment per unit volume is, P = Nex.

As P = 0 ( +1)E, results,

= 1 + p2 / (0

2 - 2 - i)

Where Where p = [Ne2 / m 0 ]½ is called “plasma is called “plasma

frecuency”frecuency”

Dielectric functionDielectric functionand refractive indexand refractive index

00

00

Important properties Important properties of dielectric function (1)of dielectric function (1)´ and ´´ satisfy the Kramers-Kronig relations,

No absorption

(´´´´ = 0 for all frequencies)

´́ = 1 for all frequencies

Vacuum or any material

optically equal to vacuum

Important properties Important properties of dielectric function (2)of dielectric function (2)

Dielectric function is additive,

1111

11

22

22

Dielectric function. Dielectric function.

IsolatorIsolator •Ultraviolet: only the lightest sub-component of atoms (i.e., electrons) can respond to “fast” or high frequency radiation. Electron orbital transitions occur in response to ultraviolet radiation.•Infrared: more dense matter (i.e., entire atom) can respond. Oscillations are more sluggish and frequencies associated with atomic vibration are lower.•Microwave: slowest oscillations occur when molecular dipole moments respond and orient themselves parallel to incident E-field (Debye relaxation).

Electrons near Fermi level (free Electrons near Fermi level (free electrons) can be excited with electrons) can be excited with light of low frequencylight of low frequency

Dielectric function. Dielectric function. Metals. Drude modelMetals. Drude model

Dielectric function. Dielectric function. Metals. Drude modelMetals. Drude model

““clipping the springs” clipping the springs” (K = 0) gives (K = 0) gives 0 0 = 0.= 0.

The dielectric function The dielectric function for free electrons is,for free electrons is,

Quantum modelsQuantum models

EE

Gives similar results for Gives similar results for dielectric function if,dielectric function if,

0 0 is not related with any is not related with any

spring, spring, 0 0 = 2= 2 E/hE/h

is related with the is related with the lifetime of the excited lifetime of the excited state,state,

Gold dielectric functionGold dielectric function

(a) real part

Photon energy (eV)

0 1 2 3 4 5 6 7

Die

lect

ric f

unct

ion

-200

-150

-100

-50

0

50

Experimental J&C Experimental T Calculated, Eq. 3

(b) imaginary part

Photon energy (eV)

0 1 2 3 4 5 6 7

Die

lect

ric f

unct

ion

0

5

10

15

20

25

30

Experimental J&C Experimental T Calculated, Eq. 4


It is clear that free electrons do not describe precisely It is clear that free electrons do not describe precisely the optical properties of gold and in fact for any noble the optical properties of gold and in fact for any noble metal where there is a substantial bound electron metal where there is a substantial bound electron component. component. Since the dielectric function is additive, it can be Since the dielectric function is additive, it can be decomposed into two terms, a free-electron term and an decomposed into two terms, a free-electron term and an interband, or bound-electron term [Bohren, 1983]. interband, or bound-electron term [Bohren, 1983].

)()()( freebound

Fermi energyFermi energy

Band

gap

s - p band

d - band

Bound Free electrons electrons

Simplified electrons energy bands of noble metals.


)()()( freebound

Size dependence of refractive index of gold nanoparticles, Lucía B. Scaffardi and Jorge O. Tocho. Nanotechnology, vol.17, no.5, 14 March 2006, pp. 1309-15. Publisher: IOP





55 SummarySummary

Size effects in metals.Size effects in metals.(a) Free electrons (a) Free electrons

For free electrons in bulk, For free electrons in bulk, is limited for is limited for collisions between: electron – electron; collisions between: electron – electron; electron – phonon; electron – defects, etc. electron – phonon; electron – defects, etc.

)()()( freebound

ip

bound

2

2

1)()(

We can take We can take as the as the inverse of the collision inverse of the collision time for conduction electrons. time for conduction electrons.

Size effects in metals Size effects in metals

For For size limited objectssize limited objects, particles, wires or , particles, wires or films, films, damping constant is increased for damping constant is increased for additional collisions with the boundariesadditional collisions with the boundaries (Doyle, Doremus, Kreibig, Granqvist, and (Doyle, Doremus, Kreibig, Granqvist, and others)others)..

Size effects in metals Size effects in metals

GOLDGOLD

bulkbulk = 1.1 x 10 = 1.1 x 101414 Hz Hz

vvFF = 14.1 x 10 = 14.1 x 101414 nm/s nm/s

dd vvFF / d / d

100 nm100 nm 0.14 x 100.14 x 101414

10 nm10 nm 1.41 x 101.41 x 101414

1 nm1 nm 14.1 x 1014.1 x 101414

Only free electrons contribution is corrected, Only free electrons contribution is corrected,

(R)(R) = = bulkbulk + + CC vvF F / R/ R

Dielectric function or refractive index are not Dielectric function or refractive index are not measured for small particles, rather than, measured for small particles, rather than, extinction spectra is quite easily determined.extinction spectra is quite easily determined.

After that, experimental results are fitted with After that, experimental results are fitted with Mie calculations corresponding to different Mie calculations corresponding to different dielectric functions.dielectric functions.

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.6 nm ATS-10

C = 0.8

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.34 nm ATS-6

C = 0.8

(b)

(c)

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

AES-20 R = 0.6 nm R = 0.8 nm

(c)

C = 0.8

C = 0.8

(a)

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.6 nm ATS-10

C = 0.8

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.34 nm ATS-6

C = 0.8

(b)

(c)

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

AES-20 R = 0.6 nm R = 0.8 nm

(c)

C = 0.8

C = 0.8

(a)

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.6 nm ATS-10

C = 0.8

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 1.34 nm ATS-6

C = 0.8

(b)

(c)

Wavelength (nm)

400 500 600 700 800 900

Op

tica

l e

xtin

ctio

n (

a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

AES-20 R = 0.6 nm R = 0.8 nm

(c)

C = 0.8

C = 0.8

(a)

Wavelength (nm)

400 500 600 700 800 900

Opt

ical

ext

inct

ion

(arb

. uni

ts)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

10 nm

Radius (nm)0 1 2 3 4 5

Cou

nts

Scaffardi LB, Pellegri N, de Sanctis O, Tocho JO. Sizing gold Scaffardi LB, Pellegri N, de Sanctis O, Tocho JO. Sizing gold nanoparticles by optical extinction spectroscopy. nanoparticles by optical extinction spectroscopy. Nanotechnology, vol.16, no.1, Jan. 2005, pp. 158-63. Nanotechnology, vol.16, no.1, Jan. 2005, pp. 158-63.

For tiny particles, contribution from bound For tiny particles, contribution from bound electrons must be corrected as well, electrons must be corrected as well,

)()1()( 0/ ReR freeboundRR

GOLD. GOLD. RR00 = 0.35 nm = 0.35 nm Scaffardi et al.,Nanotechnology, 2006 Scaffardi et al.,Nanotechnology, 2006

Size effects in metals.Size effects in metals.(b) Bound electrons (b) Bound electrons

Wavelength (nm)

400 500 600 700 800 900

Opt

ical

ext

inct

ion

(a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 0.4 nmR = 0.6 nmR = 0.8 nmAPS-6

(a) R0 = 0.35 nm

C = 0.8

b = 0.16 eV

EF = 2.5 eV

Eg = 2.1 eV

Wavelength (nm)

400 500 600 700 800 900

Opt

ical

ext

inct

ion

(a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 0.2 nmR = 0.4 nmR = 0.6 nmAPS-20

(b) R0 = 0.35 nm

C = 0.8

b = 0.16 eV

EF = 2.5 eV

Eg = 2.1 eV

Wavelength (nm)

400 500 600 700 800 900

Opt

ical

ext

inct

ion

(a.

u.)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

R = 0.2 nmR = 0.3 nmR = 0.5 nmAPS-10

(c) R0 = 0.35 nm

C = 0.8

b = 0.16 eV

EF = 2.5 eV

Eg = 2.1 eV

Radius (nm)

0.3 0.4 0.6 0.8 1 1.3 1.6 2 2.5

Con

tras

t

0.01

0.03

0.05

0.07

0.1

0.3 ATS-20

ATS-6 ATS-10

AES-20

AES-10

APS-6

APS-20

aluminioaluminioBimetallic particlesBimetallic particles

Core-coat For core-coat particles we use the extinction cross section derived f or acoated ellipsoid in the limit of very small particle (dipolar approximation) [6],

where a is the radius of the core with dielectric f unction 1 ; b is the radius of the particle; 2 is the dielectric function of the coat; m is the dielectric function of the surrounding media and is the wavelength in the extinction spectrum.

,2222

22Im

2)(

212

3

212

221

3

212

mm

mm

ext

babab

Q

**

Absorption spectra of tiny gold and silver objects. Absorption spectra of tiny gold and silver objects. Lucía B. Scaffardi and Jorge O. Tocho, to be Lucía B. Scaffardi and Jorge O. Tocho, to be published J. Luminescence 2007 published J. Luminescence 2007

**

For gold–silver alloys, we considered a mean dielectric function,

where is the fraction of gold. Extinction cross section is,

Alloys

,1 sga

Core-coat Alloy

Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a. u

.)

0.000

0.002

0.004

0.006

0.008

r = 2.5 nm

80 % gold

100 % gold

50 % gold

40 % gold

Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a.

u.)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

a = 0.8 nmb = 1 nm

silver (core), gold (coat)

gold (core), silver (coat)

Core-coat Alloy

Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a. u

.)

0.000

0.002

0.004

0.006

0.008

r = 2.5 nm

80 % gold

100 % gold

50 % gold

40 % gold

Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a. u

.)

0.000

0.002

0.004

0.006

0.008

r = 2.5 nm

80 % gold

100 % gold

50 % gold

40 % gold

Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a.

u.)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

a = 0.8 nmb = 1 nm



Wavelength (nm)300 400 500 600 700 800

Op

tica

l ext

inct

ion

(a.

u.)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

a = 0.8 nmb = 1 nm



J. M

ate

r. C

hem

., 1

2 (

2002)

15

57

JAC

S,

vol.

127

(200

5) 1

019

JAC

S,

vol.

127

(200

5) 1

019





55 SummarySummary

Silver nanowires

L B Scaffardi, M Lester, D Skigin and J O

Tocho. Nanotechnology 18 (2007) 315402

Optical extinction spectroscopy used to characterize metallic nanowires.

P-polarization

S-polarization

Inverse dichroism in metallic nanowires

Single crystal silver nanowires prepared by the metal amplification method. Mladen Barbic, Jack J. Mock, D. R. Smith, and S. Schultz, J. APPL. PHYS. 91 (2002) 9341





55 SummarySummary

Size effects in thin films Size effects in thin films

nn

kk

Size effects in thin films Size effects in thin films

nn

kk

Gold

wavelength, nm

0 500 1000 1500 2000 2500

0

2

4

6

8

10

12

14

16

k

nbulk

size limited

bulk

size limited

Colors of thin films Colors of thin films on glasson glass

Silver Films Gold Films

Light yellow 1 nm

Golden yellow 2.1 nm

Orange yellow 3.2 nm

Ruddy orange 4.3 nm

Crimson 5.2 nm

Indigo 6.0 nm

(colors disappear) 7.0 nm

Ruddy-purple 1.5 nm

Indigo 2.0 nm

Blue 2.7 nm

Green 3.2 nm

Yellow-green 4.0 nm

Golden yellow > 4.0 nm

Heavens, O. S., "Optical Properties of Thin Solid Films", Dover, 1991

Silver Gold

Colors of films FreeSnell computes from 1 nm to 100 nm in thickness with q (volume ratio of metal particles to substrate) from 0.55 to 1.0 under D65 illumination. Color squares with "X" through them are outside of the sRGB gamut.





55 SummarySummary

For tiny objects, contribution from bound For tiny objects, contribution from bound electrons must be corrected as well. electrons must be corrected as well.

When “mean free path” for free electrons is When “mean free path” for free electrons is modified by collisions with boundaries, the modified by collisions with boundaries, the contribution from free electrons to refractive contribution from free electrons to refractive index must be corrected. index must be corrected.

Bulk dielectric function or refractive index can Bulk dielectric function or refractive index can explain many “size effects” in metals. explain many “size effects” in metals.

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optical properties of metallic nanoestructures jorge o. tocho centro de investigaciones Ópticas...

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