UNIT 3_Metallic Nanoparticle property.ppt

useful for postgraduates

1. EM Absorption in
Metallic Nano-particles

2. Consider a time dependent electric field E(t) acting on a metal.
Take the case when the wavelength of the field is large compared to
the electron mean free path between collisions:
 >> 
In this limit, the conduction electrons will “see” a  homogeneous
field when moving between collisions.
E(t) = E(ω)e
-iωt

That is, assume a harmonic dependence on frequency.
AC Electrical Conductivity in Metals
(considering free & independent electron approximation)
Application to the Propagation of Electromagnetic Radiation in a Metal
E is time dependent

3. electric field leads to
electric current jconductivity j/E metals
polarization P polarizability P/
0Edielectrics
dielectric function 
r
Response to the electric fieldmainly historically
both in metals and dielectrics described by used
mainly for
   
E
E
EED
















im
ne
im
ne
r 2
0
2
02
2
00 1
~
eExmxmxm 
2
0
 Lorentz oscillator model for bound electron
eExmxm  for free electron in a viscous medium
ti
eeExmxm



 )( for free electron in presence of AC field
PEED 
00

r
 

im
ne
r


22
0
2
00
~ E
EED

4. Free-electrons Dielectric function for metal
 
1)(
~
2
0
2


im
ne
r


  )1(
)(
~
and
1
1)(
~
where1)(
~
)(
~
)(
~
22
2
22
22
2
2















P
r
P
r
P
rrr
i
i
m
ne
pr
P
r
0
2
2
2
where0)(
~
and 1)(
~




 
At high frequencies or for very large  i.e. for free electrons, 
>>1
Plasma Frequency
A plasma is a medium with positive & negative charges & at least one charge
type is mobile.

5. /
p
cK/
p
forbidden
frequency
gap
Longitudinal Plasma
OscillationsA charge density oscillation, or a longitudinal plasma oscillation, or a plasmon
m
ne
p
0
2
2


Oscillations at the
Plasma Frequency
 01
2
2

p
L
L



Longitudinal Plasma Oscillations

L
= 
p

Transverse Electromagnetic Waves
The Nature of Plasma Oscillations: Correspond to a
displacement of the entire electron gas a distance d with
respect to the positive ion background. This creates surface
charges  = nde & thus an electric field E = nde/
0
.
Longitudinal Plasma Oscillations
0
2
2
2
0
2
2














d
t
d
nde
NeNeE
t
d
Nm
p

Equation of
Motion

6. Transverse optical modes in a plasma
222
),( kcK
Dispersion relation for
electromagnetic waves
2
2
0
2
2
1)(





p
p
m
ne


2222
kc
p

(1) For > 
p
→ k
2
> 0, k is real, waves
with > 
p propagate in the media with
the dispersion relation:
The electron gas is transparent.
(2) For < 
p → k
2
< 0, k is imaginary,
waves with < 
p
incident on the medium
do not propagate, but are totally reflected
rK
eE


E&M waves are totally
reflected from the medium
when 
E&M waves propagate
with no damping when
 > 0 and real
cK/
p
2222
kc
p

/
p

7. Plasma Frequency
Ionosphere
Semiconductors
Metals
Ultraviolet Transparency of Metals
2
2
0
2
2
1)(





p
p
m
ne

Plasma Frequency 
p
& Free Space Wavelength 
p
= 2c/
p
Range Metals SemiconductorsIonosphere
n, cm
-3
10
22
10
18
10
10

p, Hz 5.7×10
15
5.7×10
13
5.7×10
9

p
, cm 3.3×10
-5
3.3×10
-3
33
spectral rangeUV IF radio
The Electron Gas is Transparent when > 
p i.e.
 < 
p
The reflection of
light from a metal
is similar to the
reflection of radio
waves from the
Ionosphere!
reflectstransparent for
metal visibleUV
ionosphere radiovisible

8. Dispersion Relation for EM Waves in Electron Gas (bulk metal): Bulk Plasmon
222
p
2
kcωω 
Reflectance
Permittivity
1
< λ
Dispersion Relation for EM Waves at dielectric-metal interface: Surface Plasmon

21
dm
dm
x
εε
εε
c
ω
ωk













21
dm
2
d md m
z
εε
ε
c
ω
ωk










Propagating along the interface: real k
x,
Exponentially decaying away from it:
imaginary k
z

9. 
21
dm
dm
x
εε
εε
c
ω
ωk










ω
ω
1ε
2
2
p
m
 
 

21
2
p
2
d
d
2
p
2
x
ω -ω ε1
ε ω-ω
c
ω
ωk










•At low   , for to be real  
21
dx
ε
c
ω
k ωk
x
0ε
d

•for to be real  ωk
x pdmdm
ωω 0εε 0εε  and 1. Case
p
dmm
ddm
dm
ωω
εε 0ε
0ε , 0εε
0εε




for i.e.
andso
but
and

2. Case

xk



dx
εckω
sp
ω
xckω
pω
real k
x
real k
z
real k
x
Imag k
z
imag k
x
real k
z










d
p
spx
ε1
ω
ωω k when
0εε0εωωω
mmmpsp
 and ,



21
dm
2
d md m
z
εε
ε
c
ω
ωk











10. Summary

11. •An approximate surface area to volume ratio of > 10
7
:1 which
is significantly larger than a macro sized particle.
•As the surface area to volume ratio increases  the
percentage of atoms at the surface increases and surface
forces become more dominant.
•In contrast to Surface Plasmon (SP), the surface fields in the
metallic nanoparticles do not propagate and are confined to the
nanoparticles  Localized Surface Plasmons (LSP)
•These have a dramatic effect on the optical properties at the
nanoscale.
Surface Area/ Volume ratio

12. Resonance in metal NP:
Localized Surface Plasmons
(LSP)

19. Size dependence
•As the percentage of atoms at the surface increases, the optical
and electronic properties change.
–For example optical properties (color) of gold and silver change, when
the spatial dimensions are reduced and the concentration is changed.

20. Martin, Olivier J.F. "Plasmons". Plasmons. 22 Mar. 2006. Ecole Polytechnique Fédérale de Lausanne. 26 Jan. 2003.
Surface Plasmons: Shape dependence of absorption spectra
•The amount of light that is scattered into the far field is described by the
scattering cross section (SCS).
•Triangular shaped nanoparticles produce plasmons with altered frequency
and magnitude.
Absorption spectra: shape dependence

21. Martin, Olivier J.F. “Spectral response of plasmon resonant nanoparticles with non-regular shape”. Optics Express Col. 6. No. 11 May
2000
Surface Plasmons: angle dependence of absorption spectra
•The arrows indicate the illumination angle, and their colors correspond
to the different plot lines.
Absorption spectra: angle dependence

22. •The reduction of materials' dimension has pronounced effects
on the optical properties.
•The size dependence can be generally classified into two
groups.
–One is due to the increased energy level spacing as the system
becomes more confined. (in semiconductor nanoparticles)
–Other is related to Surface Plasmon Resonance: SPR (in metallic
nanoparticles).
In a nutshell

23. END