Localised Surface Plasmon
 We have seen how the interaction of EM waves with electrons at an extended metal
surface (smooth via prism or periodically nanostructured) leads to the excitation of
SPP modes
 We now need to look at the interaction of EM waves with metal nanoparticles and
see how that differs from the extended film case.
 We have seen that SPPs are propagating, dispersive electromagnetic waves coupled
to the electron plasma of a conductor at a dielectric interface. Localized surface
plasmons on the other hand are non-propagating excitations of the conduction
electrons of metallic nanostructures coupled to the electromagnetic field.
Localised Surface Plasmon
 We will start by considering spherical particles of dimensions much
smaller than the wavelength:
𝑑≪𝜆
 In this situation the E-field can be
considered to be the same across the whole particle
Localised Surface Plasmon
 In this case, the electromagnetic wave is practically constant over
the particle volume, so we can treat it as a simplified problem: a
particle in an electrostatic field.
 The well-known harmonic time dependence can then be added to the
solution once the field distributions are known.
 This is therefore called the
quasi-static approximation
Localised Surface Plasmon
 Let us consider the following geometry:
 a spherical metal particle, with dielectric constant 𝜀 𝜔
 in a dielectric medium with dielectric constant 𝜀𝑑
ො
 with a static E-field in the z direction: 𝑬 = 𝐸0 𝒛
S. A. Maier. Plasmonics: Fundamentals and Applications
Localised Surface Plasmon
 In the electrostatic approach, we need to find a solution of the
Laplace equation for the potential, 𝜙 :
 𝛻2𝜙 = 0
 From which we will can calculate the electric field:
 𝑬 = −𝛻𝜙
 You do not need to know how to do this, that’s for Advanced
Photonics perhaps next year.
 But we will look at the important results.
Localised Surface Plasmon
 We find that the polarisation (dipole moment) can be written as:
𝜀(𝜔)−𝜀𝑑
3
4𝜋𝜀0 𝜀𝑚 𝑎
𝑬
𝜀(𝜔)+2𝜀𝑑 𝟎

𝒑=

This can be written in the form 𝒑 = 𝜀0 𝜀𝑑 𝛼𝑬𝟎
 Alpha is the the polarisability:
 𝛼 = 4𝜋𝑎 3
𝜀(𝜔)−𝜀𝑑
𝜀(𝜔)+2𝜀𝑑
 Since 𝜀 is a complex
function of the frequency,
so is the polarizability 𝛼
 On the right is an example
for a silver nanoparticle
Localised Surface Plasmon
 The polarizability is a measure of how strongly the charges are
separated in the nanoparticle
𝛼=
4𝜋𝑎3
𝜀(𝜔) − 𝜀𝑑
𝜀(𝜔) + 2𝜀𝑑
 Examining this equation, we can see that it will be a maximum if the
denominator is zero (resonance).
 The resonance is therefore:
𝜀(𝜔) + 2𝜀𝑑 = 0
with 𝜀(𝜔) = 𝜀 ′ (𝜔) + 𝑖𝜀 ′′ (𝜔)
So 𝜀 ′ (𝜔) + 𝜀𝑑 2 + 𝜀 ′′ (𝜔) 2 = 0
 For 𝜀 ′′ ≪ 1, this condition is met for 𝜀 ′ (𝜔) = −2𝜀𝑑
 This is known as the Fröhlich condition
Localised Surface Plasmon
 We can also recover the electric field due to the particle.
 Using 𝑬 = −𝛻𝜙, it can be shown that the field is:
3𝜀𝑑
 𝑬𝒊𝒏 =
𝑬
𝜀+2𝜀𝑑 𝟎
 𝑬𝒐𝒖𝒕 =
 Where
3𝒏 𝒏∙𝒑 −𝒑 1
𝑬𝟎 +
4𝜋𝜀0 𝜀𝑑 𝑟 3
𝒓
𝒏 = 𝑟 is the unit vector
Additional field
enhancement
in direction of the point of interest
 This additional electromagnetic field can
make the total field outside the particle many
times stronger than the incident field.
Localised Surface Plasmon
 The corresponding localized oscillating electromagnetic mode, i.e. in
an oscillating E-field) is referred to as the dipole surface plasmon.
E0
Localised Surface Plasmon
 The field-enhancement at the plasmon resonance is the basis of many of
the prominent applications of metal nanoparticles in optical devices and
sensors
 Also note the effect of the dielectric
environment on the polarizability
and therefore the plasmonic
resonance
 𝛼 = 4𝜋𝑎 3
𝜀−𝜀𝑑
𝜀+2𝜀𝑑
The dielectric environment
affects resonance
Localised Surface Plasmon
 What makes these particles so useful as sensors?
We can see that the penetration depth of the field
into the dielectric medium is many times smaller
than that for an SPP.
E0
This means that it have the potential to be more
sensitive to smaller objects than SPPs!
~ 20 nm
Localised Surface Plasmon
 The colour of metal nanoparticles solutions depends on the spectral
position of the plasmonic resonances
30 nm radius, silver nanoparticle
30 nm radius, gold nanoparticle
Localised Surface Plasmon
 The range of colours available using plasmonic nanoparticles was
used in ancient art
 Lycurgus cup (4th century BC)
 Stained glass
Localised Surface Plasmon
 The shape of the nanoparticle can strongly affect the polarizability
of the particle.
 If we consider the electrons
oscillations, they are clearly
subject to a coulomb restoring
force when oscillating with a
metallic particle.
 Change in both the size, and the
shape can alter this restoring
force, and therefore the
resonance. e.g. picture the
electrons squeezed into the
point of the blue star!
Localised Surface Plasmon
 Of course, the electrons are also moved in the direction of the
incident electric field.
 So particles (e.g. triangle again!)
can have multiple plasmonic
resonances, sometimes
separated by 100’s of
nanometres in wavelength!
Localised Surface Plasmon
 Going beyond the quasi-static approximation
 The quasi-static approximation breaks down when the particle size is
large enough that the incident field is not homogeneous across the
whole particle
 In this case, we talk about “retardation effects”
Localised Surface Plasmon
 The retardation effects induce a red-shift and a broadening of the
localised surface plasmon resonance
 There exists extensions of models to account for them
Summary
 Plasmons exist in nanoparticle due to the resonant oscillation of the
free charges in the nanoparticle.
 When this resonant condition is satisfied by the light wavelength, a
strong and confined electromagnetic dipole is created.
 In the quasistatic approximation, the resonance condition is known
as the Fröhlich, depends only on the permittivity of the metal and
the surrounding dielectric.
Summary
 Plasmonic nanoparticles are now being used in many aspects of
research and technology. They allow to the diffraction limit of light
to be smashed, and enable light manipulation at the nano-scale.
 They have geometrically adjustable optical properties with
sensitivity to:
 Polarisation
 Wavelength
 Refractive index
• They also have an enhanced electromagnetic
field close to the particle surface, which is
highly localised in a nanometric volume