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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
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Optical properties of Optical properties of metallic nanoestructuresmetallic nanoestructures
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Optical properties of Optical properties of metallic nanoestructuresmetallic nanoestructures
JCE Classroom Activity # 62. Color My Nanoworld. W. Adam et al.
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outlookoutlook
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.
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.
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
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
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
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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
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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
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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
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““Optics is Optics is wavelength based” wavelength based”
Optical spectroscopy is a great invent!
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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
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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,
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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
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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”
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Dielectric functionDielectric functionand refractive indexand refractive index
00
00
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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
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Important properties Important properties of dielectric function (2)of dielectric function (2)
Dielectric function is additive,
1111
11
22
22
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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).
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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
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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,
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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,
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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
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Gold dielectric functionGold dielectric function
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.
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Gold dielectric functionGold dielectric function
)()()( 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
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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
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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.
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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)..
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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
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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.
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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.
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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
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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
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aluminioaluminioBimetallic particlesBimetallic particles
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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
**
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For gold–silver alloys, we considered a mean dielectric function,
where is the fraction of gold. Extinction cross section is,
Alloys
,1 sga
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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
silver (core), gold (coat)
gold (core), silver (coat)
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)
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
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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
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Silver nanowires
L B Scaffardi, M Lester, D Skigin and J O
Tocho. Nanotechnology 18 (2007) 315402
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Optical extinction spectroscopy used to characterize metallic nanowires.
![Page 38: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/38.jpg)
P-polarization
![Page 39: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/39.jpg)
S-polarization
![Page 40: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/40.jpg)
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
![Page 41: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/41.jpg)
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
![Page 42: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/42.jpg)
Size effects in thin films Size effects in thin films
nn
kk
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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
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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.
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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
![Page 46: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/46.jpg)
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.
![Page 47: Optical properties of metallic nanoestructures Jorge O. Tocho Centro de Investigaciones Ópticas (CONICET- CIC) and Departamento de Física, Facultad de](https://reader035.vdocument.in/reader035/viewer/2022062320/56649d0c5503460f949dff75/html5/thumbnails/47.jpg)
www.ciop.unlp.edu.arwww.ciop.unlp.edu.ar
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www.ciop.unlp.edu.arwww.ciop.unlp.edu.ar
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