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Plasmonics: Fundamentals and Applications
SS2012
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Plasmonics:
Fundamentals and
Applications
Lecture notes
by
Dr. E. Margapoti
Summer semester 2012 – TU München
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Plasmonics: Fundamentals and Applications
Introduction
The study of Plasmonics is a brunch of Nanophonics. It studies how the electromagnetic (EM)
field can be confined over a dimension of the order or smaller than the wavelength. When the EM
field interacts with the conductive electrons at the metal interface or in metallic nanostructures an
enhanced optical near field of sub-wavelength is achieved.
In this course the aim is to give a broad overview of the properties of Plasmons going from the
study of volume plasmons, towards the investigation of localized surface plasmons passing
through the understanding of surface plasmon polaritons and its ways of investigation. In order to
achieve this insight we start by introducing the classical Maxwell’s equations, deriving the
dielectric function by using the free electron gas theory.
1 Definition of a Plasmon
Before going in details, we should ask our self what is a Plasmon and how it can be generated.
The most common definition of a Plasmon is
“A plasmon is a quantum of plasma oscillation”
 The quantization of the collective longitudinal excitation of a conductive electron gas in
a metal is known as Plasmon.
The excitation of a Plasmon requires the interaction of an electron passing through a thin metal or
by reflecting an electron or a photon from the metallic film. In Fig.1 is shown a scheme of an
incident electron interacting with a thin metal film, resulting in a scattered electron and one or
two plasmons.
 Electron incident energy: ~ 1-10 KeV
 Plasmon energy: ~ 10 eV
Later we will go more in detail showing how one can detect the surface plasmon polaritons.
Plasmonics: Fundamentals and Applications
SS2012
Fig1. Creation of a Plasmon in a metal film by inelastic
scattering of an electron.
Electromagnetic of a Metals
The interaction of metals with EM fields can be understood using the classical Maxwell’s
equations.
 Even metallic nanostructure with a size on the order of few nanometers can be described
without the need of quantum mechanics. Due to the high density of free carriers, the
energy spacing between the electron energy levels is much smaller than the thermal
excitation kBT.
 Optical properties strong dependent from frequency
Fig.2
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Plasmonics: Fundamentals and Applications
 For frequency up to the visible spectrum, metals are highly reflective and no EM waves
can propagate through them.
 At higher frequencies: in the near-infrared and visible part of the spectrum, field
penetration increases, leading to an increased dissipation.
 In the UV-region, metals acquired a dielectric character, allowing the penetration of the
EM waves. The level of attenuation depends from the electronic band structures.

Alkali metals have almost a free-electron-like behaviour, showing an ultraviolet
transparency.

For noble metals (e.g. Au, Ag), transitions between electronic bands lead to strong
absorption in the UV regime.
The dispersive properties of an EM waves into a metals is described by the complex dielectric
function ɛ(ω).
Before going more in details with the optical properties of metals lets introduce the
Macroscopic Maxwell’s equations:
 ⃗ ⃗
electric charge is the source of the electric field
 ⃗
there are no source of magnetic field
⃗
 ⃗
⃗
 ⃗
⃗⃗
⃗⃗
change of B-field with time leads to a rotating E-field
⃗⃗
electric currents give rise to rotating magnetizing field
In these equation there are four macroscopic fields ⃗ the dielectric displacements, ⃗ the electric
field, ⃗⃗ the magnetic field and ⃗ the magnetic flux density. In addition
density, while
is the external charge
is the external current density. The total charge and current densities are written
as a sum of the internal and external components, that are
and
respectively.
Let’s define now the fields:
 ⃗
⃗
⃗
⃗
( ⃗ is the polarization, the dielectric function )
 ⃗
⃗
⃗⃗ ( ⃗⃗ is the magnetization)
,
Plasmonics: Fundamentals and Applications
SS2012
Since we are dealing only with no magnetic effect, from now we will be concentrate only on the
electric polarization effects.
⃗ describes the sum of the electric dipole moments per unit volume

1-electron in ⃗ -field is displaced by a distance and leads to

n-electrons in a volume V leads to
⃗
Thanking into account last two equations we found the relationship between the polarization ⃗ and
internal charge density:
⃗
Introducing this last equation into the displaced equation above we obtain:
⃗
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Plasmonics: Fundamentals and Applications
2 Optical properties of metals
The optical properties of a metal is described via dielectric function ɛ(ω), as above defined.
 The frequency dependence of ɛ(ω) describe the dispersive properties of metals.
The dielectric function is mainly determined by:
(i)
The freely moving conduction electrons.
(ii)
Interband excitation of bound electrons if Ephot > EgapMetal
Let’s consider a piece of metal:
Fig.3
 In absence of an external field the valence electrons are free to move in the metal, as in
gas of particles (see Drude model)
 If an external field is applied (
≠ 0), the valence electrons will be displaced of an
amount, , leading to a macroscopic polarization ⃗ , ⃗
density (n = N/V).
The dielectric function is connected to the polarization ⃗ by:
⃗
⃗
⃗
⃗
⃗
with
( )
( )
⃗
⃗
( )⃗
, with n the electrons
Plasmonics: Fundamentals and Applications
SS2012
Drude Model
In the earliest 1900, Drude constructed his theory of electrical and thermal conduction by applying the
kinetic theory of gasses to a metal, considered as a gas of electrons. In its simplest form the Kinetic
theory treats the molecules of a gas as identical solid spheres, which move in straight lines until they
collide with one another. Although there is only one type of particle in the simplest gases, in metal there
must be at least two. The electrons must be compensated by positive carriers to leave the metal
electrically neutral. Drude assumed that the compensating positive charge was attached to much heavier
particles, which he considers to be immobile.
At this time was not much knowledge of light and mobile electrons and immobile positive charges.
We should simply assume that when atoms of a metallic elements are brought together to form a metal,
the valence electrons become detached and wander freely through the metal, while the metallic ions
remain intact and play the role of the immobile positive particles in the Drude theory (See Fig.4)
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Fig.4 a) Schematic picture of an isolated atom. b) In a metal the nucleus and ion core retain their configuration in
the free atom, but the valence electrons leave the atom to form the electron gas.
In a single isolated atom we have the following configuration:
 Nucleus of charge eZa, with Za atomic number and e the magnitude of the electronic charge,
Plasmonics: Fundamentals and Applications
 -eZa is the total charge surrounding the nucleus,
 -eZ are the relatively weak bound valence electrons,
 The remaining e( Za –Z) are the tightly bound electrons (known as core electrons).
When these isolated atoms condense to form a metal, the core electrons remain bounded to the nucleus
to form the metallic ions, but the valence electrons are free to move around and away from their parent
atom. They are called conduction electrons.
By calculating the electrons density in a metal one recognizes that the value is much larger than those of
classical gas at normal temperature and pressure. In spite of this and in spite of the strong electronelectron and electron-ion electromagnetic interactions, the Drude model treats the dense metallic
electrons by following the Kinetic theory of a neutral diluted gas. These the basic assumptions:
1. Between collisions the interactions of the electron with the others and with the ions are
neglected. In absence of externally applied fields each electron is moving in a straight line.
Instead in the presence of an applied field the electron will act following the Newton’s law
(
) neglecting the extra fields emerging from other electrons and ions.
2. Collisions in Drude’s model are taken as in Kinetic theory, which is as instantaneous events that
abruptly alter the velocity of an electron. This simplest picture is visualized in Fig.5.
Fig.5 Trajectory of a conduction electron
scattering off the ions according to the naïve
picture of Drude.
3. An electron experiences a collision with a probability per unit time 1/τ. With τ known as the
relaxation time, the mean free time. We can describe the probability of an electron undergoing a
collision in a time interval dt as dt/τ. In the Drude model, τ is taken to be independent of an
electron’s position and velocity.
4. Electrons achieve thermal equilibrium with their surroundings by collisions.
Plasmonics: Fundamentals and Applications
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Definition of the Dielectric Function and Plasma Optics
The equation of motion of a free electron in an electric field is:
⃗
If we assume a harmonic time dependence ⃗ ( )
⃗
of the driving field, the solution of
this equation describing the oscillation of the electron is ( )
. Any phase shift
between the driving field and response is given as:
( )
(
)
⃗ ( ).
The polarization defined as the dipole moment per unit volume is:
⃗
( )
(
)
⃗ ( ).
Let’s now define the dielectric function at the frequency ω as:
( )
( )
( )
( )
( )
Let’s defining the frequency:
,
which is named Plasma frequency. Plasma is a medium with equal concentration of positive and
negative charges, of which at least one is mobile. Therefore the dielectric function of the free
electron gas can be rewritten as:
( )
where γ = 1/τ. We will define later both γ and τ.
,
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Plasmonics: Fundamentals and Applications
For the moment let’s limit our study to the case of no damping. For larger frequencies, close to
the plasma frequency, ωτ
, the damping can be neglected. In this case the dielectric function
can be written again as:
( )
In Fig.6 a scheme of the dielectric function versus the frequency is shown. Electromagnetic
waves propagate without damping when (
when (
) is positive and real, while is totally reflected
) is negative.
Fig.6 Dielectric function ɛ(ω) of a free
electron gas versus frequency in unit of the
plasma frequency.
In general, the dielectric function can be written as: ɛ(ω) = ɛ1(ω) + iɛ2(ω), where ɛ1(ω) is the real
part and ɛ2(ω) the imaginary part. At optical frequencies, ɛ(ω) can be experimentally determined
via reflectivity studies.
We can also define the complex refractive index as ̃( )
̃( )
√ ( )
( )
( ), also defined as
, known as extinction coefficient and determine the optical absorption of
electromagnetic waves propagating trough the medium:
( )
Known as Beer-Lambert law.
( )
,
Plasmonics: Fundamentals and Applications
SS2012
The reflectivity is defined as:
|
̃
̃
(
(
|
)
)
1. ω < ωp : ̃ is imaginary  n = 0, R = 1
2. ω = ωp : ̃
R=1
3. ω > ωp : ̃ is real  ω
(
:
)
̃
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If we limit ourself in the range of low frequency, ωτ
, hence
and
. In this
region metals are mainly absorbing.
The absorption coefficient is
metal as
, where
(
is the skin depth,
)
. For low frequencies the field fall off inside the
.
This description is valid as long as the mean free path of the electrons
the Fermi velocity.
, where
is
Plasmonics: Fundamentals and Applications
Taking into account
| | the Beer law is obtained:
( )
( )
(
)
Beer-Lambert Law
At visible frequencies the free-electron model doesn’t hold anymore due to interband transitions
 ε2 increases
The Drude model describes the optical response of a metal only below the threshold of transitions
between electronic bands. The figure below shows the real and imaginary part
and
of the
dielectric function of a silver and gold [Johnson and Christy, 1972]. The Drude model fit well the
data only at lower frequency.
Plasmonics: Fundamentals and Applications
SS2012
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In the case of Gold, this validity is already broken at the boundary between visible and nearinfrared. The optical properties of Gold and Silver at visible frequencies are described by the
function:
⃗
Interband transitions are described using the classical picture of a bound electron with resonance
.
 Combination of Drude model and Lorentz-oscillator model generates a good agreement
between theory and experiments.