Localized surface plasmon resonance

Localized surface plasmon resonance

1. Localized Surface Plasmon Resonance

R.Gandhimathi

2. Outline

= Light scattering

= Quasi static approximation

= Localized surface plasmon resonance
= Dielectric Optical Response

= Localized Electric field enhancement

= LSPR response of Spheroids

3. Scattering

= Tllumination of metal nanoparticles (MNPs) by an electromagnetic wave, excites electric
charges in the MNPs and the oscillating charge radiates electromagnetic energy in all
directions; this secondary radiation is called as scattering

= Light scattering plays key roles in control, manipulation and use of light in many
technologies

= Light scattering by small particles involves with the electric dipole concept and is
addressed by solving the Laplace equation

= Incident light induces a polarization, which in turn results in light scattering

= Polarization (i.e. the induced dipole) of materials is determined by the dielectric function

4. Optical scattering

OR >>2)

= Particle size
large

= Scattering angle is small =
(30° or less)

NZ7 >
O

is relatively

=>

ANS >

Scattering types

Mie scattering

2 GR-à)

Size of the particles is

greater than the
wavelength of the light
Scattering is much

dependent the size of the

particle DA
LS

Rayleigh scattering

e
QR <<A)

= Particle size is much
smaller than Wavelength

= Scattered radiation
occurs in all directions
uniformly and it is
wavelength dependent

x

5. Qua
Derivation of the optical properties of small MNPs embedded in a dielectric matrix is
performed by QSA
In the Quasistatic regime, the field distribution in the nanoparticle is supposed to be

tatic approximation (QSA)

homogeneous and size of the particle 2R<<A

Also, it is valid for particles that are smaller in size than the skin depth d of metal

In quasi static limit the electric field is represented by a potential E=-V&

The potential has to satisfy the Laplace equation V24-0 and the boundary conditions

Electrostatics Electro dynamics
Laplace equation Helmholtz equation
V?4(r)=0 V24(r)Hk24(r)=0

6. Localized Surface Plasmons

= Non-propagating excitations of electrons of MNPs coupled to the incident
electromagnetic field

= Metal nanoparticles in an incident electromagnetic (EM) field exhibit localized
surface plasmon resonance (LSPR)

Amplitude of the oscillating dipole is obtained from the dependence of its potential (1,0) on
the distance r from the centre of the sphere and on the polar angle 0 with respect to the
larization direction of the incident field

7. Electric potential outside the sphere(E,)
—&, cos(0)

3e, ,
(1,0) =-E, —2—reos(0) — f.(r,0)=E,rcos(8)+ E, REE “05
. gjt2e, Fr

Electric potential inside the sphere(E;)

F1 2
E,=—Vo, E, =-Vb»
3 E,-External field 5
E, =E,—2— EE +43) pr
£, +26, 2 o y?
Quasi static Polarizability Field outside the sphere is the sum of the incident
- field and the field scattered by the particle
3 €,—-&
P =4x1€,€,R" ———E,
é, +28,

Induced dipole moment P=a,,Eo

8. Polarizability of a spherical nanoparticle as well as When polarizability is
its internal and external field show ‘a resonance maximum, resonance will
behavior whenever it satisfies the condition Kerr

Localized surface plasmon

&+28,=0

Electro magnetic wave

Ag, P=00

E, = 28,

Metal Sphere

ee
Electron cloud

3 € >
P = 426@,¢,R° 2
& +28,

If 22 <<a, only the dipole oscillation (I=1) contributes to the extinction cross
section which is a sum of both scattering and absorption

9. Dielectric Optical Response

From Drude’s formula
Resonance frequency of

Dielectric function E
Localized surface plasmons

o?
Po

2 w
E2 + Eo = 7
29" 2

Plasma frequency (@,)

Frequency above which the negative real part of the e, becomes positive and the metal starts
to behave like a dielectric

10. Localized Electric field enhancement

a

enhancement in the vicinity of their surface and strongly depends on the polarization of

(en surface plasmons are confined to a nano structure, there exist

strong field

the excitation laser light

If Polarization direction of the external field is

= Perpendicular to metal surface, the charges oscillate along the polarization direction
of the external driven field

= Parallel to the metal surface, charge oscillation will be along the polarization

direction and no significant field enhancement is achieved (no charge accumulation

L at the poles)
A ci

11. Spatially decaying oscillating in time Hot spot

MEN

Near field Enhancement

A field enhancement
localized in a very small
> spatial region of plasmonic
nanoparticles is called hot
spot
No field enhancement 7

<a>

Enhancement is due to the collective oscillation of conduction band

electrons in the confined region

12. A stronger electric field outside the metal sphere

Field enhancement E 1

factor u E

inc

oc

where E,,, and E;, are the magnitudes of the localized and incident fields

Ein =Eo

3r(pr)- pr?
r

Scattered electric field, FE, = E, = E +k

loc

13. Spheroid
+ Ellipsoids are the most general non-spherical but regular shaped smooth particles

* Spheroids are ellipsoids with two equal semi-major axes (Rx & Ry ) and a semi-minor
axis (Rz) (Rx =Ry)> Rz

+ The equation of a standard axis aligned ellipsoid in an XYZ cartesian coordinate
system is

Long axis

Short axis

14. Types of Spheroid

4 4 4
Oblate spheroid Prolate spheroid R | Axis of 2 R,
when the equal axes when the equal axes are 3 < Symmetry
are larger than the smaller than the third ‘
third one one
” > R,
The aspect ratio of the 2°!

spheroid is defined by =

Ry) Bix .
Oblate (pancake) Prolate (Cigar)
The field inside the spheroid E, = E, — LP (Ry =2,)>2, (Ry =R,)<Rz
o a _ L L;-Geometrical factor that
Depolarization factor La 47% +. describe the influence of the
4

€, —] Eo non-spherical shape

15. Dipole moments induced by the External field

38, +3L, (s, -8£,)=0
Pryz = xy Es

€, =8,(1- E,
Polarizability L,
&-& From Drude Model
AR RR o
3e, +3L,(& —£,) E, =€,- 2
resonance-condition 10)
L=Ly
Resonance condition 0 L )=e a

El E
LSPR frequency of Spheroid OL
36, +3L,(& -&,)=0 we

When 1, = 5-96 +25, =0

€, =2€,

16. Field polarization is parallel to z axis

° 0 Of

(Rx =Ry)<Rz

Extinction Efficiency

The aspect ratio Rz >1
X

Wavelength (nm)

The LSP resonance shifts to the
longer wavelength range with
the increase in aspect ratio

the greatest sensitivity to external dielectric

| Highest aspect ratio nanoparticles demonstrate
environment

17. »

>

u

a

Fe |

9.

10. Comprehensive Nanoscience and Technology, David. L.Andrews,

References

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