3. Discovery of Surface Plasmon
• Anomalous spectrum by Wood (1902)
– Could not be explained by old diffraction
theory
• Partial explanation by Rayleigh (1907)
• Detailed explanation by Fano (1940)
• Ritchie predicted plasmons: collective
oscillation of electron (1950)
4. Localized surface plasmon
resonance (LSPR)
Localized surface plasmon: Surface plasmons excited
in metallic nanoparticles
Observations at resonance
Strong absorption and scattering
Strong enhancement near the particles (“Hot spots”)
5. Conditions of LSPR
Resonance condition
If we apply a field E0 to a sphere with radius , the internal field
is
If is real, when
Polarization:
gets very large strong external field just outside the surface
We measure the resonance by
6. Materials for LSPR
Why metal?
Negative real part of the dielectric function
Usually Gold (Au) and Silver (Ag)
Why?
Resonance is in visible range.
9. Properties of LSPR
Size dependence
Spheroid, aspect ratio = 2
Size parameter L: radius of equal-volume sphere
10. Applications of LSPR
Strong absorption and enhancement of field
leads to some applications, e.g.,
1. Light trapping on solar cells
2. Photothermal therapy
11. Plasmonic soler cell
Light trapped within the waveguide by
nanoparticle
[1] H. A. Atwater and A. Polman,
"Plasmonics For Improved
Photovoltaic Devices," Nature Mater.
9, 865 (2010).
[2] M. A. Green and S. Pillai,
"Harnessing Plasmonics For Solar
Cells," Nature Photon. 6, 130 (2012).
14. Theoretical Methods for LSPR
Differential Equations
Frequency domain
Time domain (FDTD)
Integral Equations
Exact methods
T-Matrix
Approximation
Analytic Approximation (AA) (assume constant internal
field)
15. Differential Equation vs Integral
Equation
Problems of differential equation
Scalar wave equation:
Boundary condition is needed for
Particle surface
At infinity (Difficult!)
Discretized equation: connects one points with the neighboring
points
Internal field External near field External far
field
Complicated! Simple
16. Why integral equation?
Why integral equation?
No Infinity boundary condition to be imposed
numerically
Bypassing the external near field
Implicit outgoing
boundary
condition
17. Goals
1. Introduce integral equation
2. Evaluate the features of scattering by
nanoparticles numerically by T-Matrix
3. Indicate the problems of FDTD
4. Develop AA
1. Check the validity
2. Investigate the shape dependence
19. Integral Equation for Scalar
Field
Scalar wave equation:
The integral equation solution for homogeneous particle:
where
(Reduces to solution to Laplace Equation when k 0)
24. T-Matrix
Make use of the integral equation.
Goal: obtain the internal field as an expansion
of spherical Bessel functions and
spherical harmonics , where
26. T-Matrix for vector field
Formalism (Vector)
For the integral equation solution:
We obtain
27. Results
Apply to different particles
• measure and the internal
field
• Size of particle L: the radius of equal-
volume sphere
28. Results
Size dependence (aspect ratio=2)
The size dependence is weak (L<1 nm does not shift)
consider small particle case only for a qualitative
understanding of LSPR
30. Comparison
E0
k E0
k
when aspect ratio
increases:
• Marked red-shift in
resonance peak
position
• Drastic increases of
the peak value.
when aspect ratio
increases:
• No shift in resonance
peak position
• Slight decreases of
the peak value.
vs
Higher sensitivity vs. aspect ratio!
31. Results
Internal field (2 nm : 1 nm particles)
E0
k
E0
k (AA not
accurate)
Constant
Rapidly
varying
32. Problems of T-Matrix method
Complicated computation
Coupled shape and frequency dependence
Separate by AA!
34. FDTD
Advantages:
Analytically the same for all geometry
Does not involve matrix inversion, which is
troublesome for large system
Find the optical response of a range of
frequencies
Commercial package available
35. FDTD
Principle
Maxwell equations
In the time domain,
FDTD solves D(t) and H(t) for an impulse
Fourier transform frequency domain
Need for all
frequencies fit
the data with
analytic model
where
36. FDTD
Dielectric function
<10% difference in
general
No difference at a
given wavelength
Broadband fitting (For finding
spectrum)
Fit a single point (For finding the
internal field more accurately.)
39. Comparing FDTD and T-Matrix
• Computational effort
Two cases:
Small
spheroi
d
Small
cylinder
40. Comparing FDTD and T-Matrix
Convergence
FDTD
1. d=0.1 nm
2. d=0.05 nm
3. d=0.025 nm
T-Matrix
4. 1 x 1
5. 3 x 3
6. 5 x 5
41. FDTD vs. T-Matrix
T-Matrix converges more quickly
FDTD can cause significant error near the
surface of particle.
T-Matrix results are used as the exact
numerical results.
But, Both are complicated!
43. Analytic Approximation
Assumption:
Small particle, quasi-static case (kL 0)
constant internal field and incident field
Consequence: simple formula for internal fied:
dependence on frequency and shape are
separated.
Shape factor
Dielectric constant: implicit
frequency dependence
44. Shape factor
For rotationally symmetric particle,
Where
1
2
3
Follows from
46. Validity
Spheroid of aspect ratio 2
• Exactly agrees with T-
Matrix
Cylinder of aspect ratio 2
• AA (1) does not agree
with T-Matrix (3)
• Modified AA (2) better
agrees with T-Matrix (3)
modified
51. Summary
T-Matrix
Size dependence of LSPR: weak
Aspect ratio dependence of LSPR position: large shift
for field along the long axis
FDTD
significant error near the particle surface
Poor converge around the surface
Analytic Approximation
exact for small spheroid
not good for small cylinder, unless with modified
shape factor
Explain the aspect ratio dependence
53. External Field
From the integral equation:
Methods
a) Direct substitution
b) Multipole expansion
Known
54. Result (2 nm: 1 nm gold spheroid at resonance)
- 2
0
2
xênm
- 2
0
2
zênm
5
10
14-fold enhancement
E on x-z plane
External Field
55. Enhancement around the tips:
• The typical scale of the enhancement regions is small (~ 0.5 nm),
compared with
– = 551 nm
– particle size (~ 1 to 2 nm).
• ``Hot spot”: Important for applications
56. Multipole expansion of the external
field
Obtain an expansion for the scattered field as:
Advantages:
Avoid the singular point at |x| = |y|.
Consistent with the formulation of the T-Matrix method.
Problem
Poor convergence for the near field
57. Issues
• Poor convergence rate for external near field
compared with:
Internal Field External far field
2 nm
1 nm
0.5 nm
Field at this
point
58. Summary of external field
Strong enhancement field is reproduced using
integral equation method.
T-Matrix method:
useful for internal field
Not good for external near field.
59. Further Investigation
Extend analytic approximation to and Check
its validity for
Particles of other shapes
Many particles system
The solution to those problems may smoothen
the process of development of applications
