Transcript Localized surface plasmon resonances

Nanophotonics
Class 2
Metal nanoparticle plasmons
and light scattering
What is a plasmon?
“plasma-oscillation”: density fluctuation of free electrons
p
+
drude

Ne
2
m 0
+ +
Plasmons in the bulk oscillate at p determined by
the free electron density and effective mass
-
-
-
k
+
-
+
Plasmons confined to surfaces that can interact with
light to form propagating “surface plasmon
polaritons (SPP)”
Confinement effects result in resonant SPP modes
in nanoparticles
drude
 particle 
1 Ne
2
3 m 0
Sphere in a uniform static electric field
particle can be considered as a dipole:
in a metal cluster placed in an electric field, the
negative charges are displaced from the positive ones


3   m
p  4 0 R
 m E0
  2 m
electric polarizability of a sphere α
resonant enhancement of p if
 ( )  2  m  minimum
 negative real dielectric constant ε1(ω)
Bohren and Huffman (1983), p.136
  4 0 R
3
  m
  2 m
ε = ε1(ω)+i ε2(ω) =
dielectric constant of
the metal particle
εm = dielectric constant
of the embedding medium
usually real and taken
independent of frequency
Quasi-static approximation

 i k r   t 
V r , t   f r e
,

r  
Einc

 
k  r  2

2
k 



  i t
V r , t   f r e
k
 



i k r   t 
E inc r , t   E 0 e
y
x
Einc
k



 i t
E inc r , t   E 0 e
y
x
Sphere in a uniform static electric field
Equations:
 1  0

2
0
E0
m

r  a 
r  a 
r
q
z
a
Boundary conditions:
1   2
r
 a ,

 1
r
 m
 2
r
r
 a ,
lim  2   E 0 z
r
Jackson (1998), p.157
Bohren and Huffman (1983), p.136
Sphere in a uniform static electric field
Equations:
 1  0

2
0
E0
m
r  a 
r  a 
r

q
z
a
Solution:
   m
 1   E 0 r cos q  
   2
m

3

 3 m
 E 0 r cos q   

   2
m


  m 
E
 2   E 0 r cos q  a 
   2  0
m 

with:


p   0 m E 0
3
  m

  4 a 
   2 m




cos q
r
2

 E 0 r cos q


  E 0 r cos q 
p cos q
4
mr
2
Sphere in electromagnetic field (a << ):
  i t

p   0 m E 0e
Jackson (1998), p.157
Bohren and Huffman (1983), p.136
Measured data and model for Ag:
Drude model:
50
p
2
"
 ' 1
0

-50
-100
Measured data:
'
"
Drude model:
'
"
Modified Drude model:
'
400
600
800
'
1200
Wavelength (nm)
1400
"
,

p
2
 '  
1000
2
3

Modified Drude model:
-150
200

2
p
1600
1800

2
p
2
,
"

3
Contribution of
bound electrons
Ag:    3 . 4
Drude model good, Modified Drude model even better.

Example: silver sphere
10
re a l p a rt  1
im a g in a ry p a rt  2
d ie le c tric c o n s ta n t o f A g
5
d a ta P a lik 's h a n d b o o k
0
-5
-1 0
-1 5
-2 0
200
300
400
500
600
700
800
w a v e le n g th (n m )
 ( )  2  m  minimum
in air
in silica
Other applications of nanoparticles
Old:
New:
(but the same principle)
Different
materials/shapes:
distinct colors
Stained glass
Focusing and
guidance of light
at nanometer
length scales
Nanoparticles in solution
Au colloids in water
(M. Faraday ~1856)
glass containing
Ag clusters
Au colloids in water
Au shell colloids in water
(larger, also scattering)
Metal nanoparticle field enhancement
The resonant excitation of the nanoparticles causes
large local electric fields close to the particle surface,
useful for many applications
n=1.5
20
Au
550nm
Ienh
Coupling in plasmon particle arrays
All particles
are driven by
the external
field and by
each other
René de Waele, Femius Koenderink
Local plasmon array response
• Energy localization
on front or back
side of the array
• Nanoscale
concentration
tunable with
wavelength
• NANOANTENNA
Nano Lett. 7, 2004 (2007)
René de Waele, Femius Koenderink
Interaction between particles
An isolated sphere is
symmetric, so the polarization
direction does not matter.
LONGITUDINAL:
restoring force reduced by coupling to
neighbor
 Resonance shifts to lower frequency
TRANSVERSE:
restoring force increased by coupling to
neighbor
 Resonance shifts to higher frequency
Large local field enhancement between particles
1.65 eV
Light Scattering
•The incident E-field accelerates
charges in the obstacle which radiate
electromagnetic energy in all
directions. This process is called
scattering, thus:
•Scattering = excitation + reradiation
•The excited charges may also
transform their energy into other
forms, like heat. This process is
called absorption. Together the two
processes are called extinction:
•Extinction = scattering + absorption
True for metals
and dielectrics
Rayleigh scattering (d<<λ)
Particles feel homogeneous applied field Ei.
The induced dipole moment in the particle
is:
p  Εi
where α is the polarizability of the
scatterer.
Assume the incident field is time
harmonic:
E i  E 0 cos  0 t
The particle (dipole) will radiate at the
same ω as the applied field with intensity:
E
I s   
p 
2
4
32  c  0
2
3
sin 
2
r
g
2
(see electrodynamics textbook)
p
Electric field lines of oscillating dipole
Blue light is scattered more than red light
Scattering by small gas molecules in the
atmosphere:
d<< λ: Rayleigh scattering
Geometrical scattering (d>>λ)
For particles very large compared
to λ, the incident plane wave can
be subdivided into a large number
of rays which obey Snell’s law and
Fresnel Equations.
For complex , the energy Wabs
absorbed in the sphere depends
on the absorption of the dielectric
and the time the light spends in
the particle (the optical path
length)
Rainbow formation
Metal nanoparticles:
extinction = scattering + absorption
n=1.5
Ienh
20
Au
550nm
At resonance, both scattering and absorption are large
albedo = scattering / extinction = ssca/(sabs+ssca)