Plasmonics Chapter 1

Plasmonics: Fundamentals and Applications SS2012

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Plasmonics:

Fundamentals and

Applications

Lecture notes

by

Dr. E. Margapoti

Summer semester 2012 TU Mnchen

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 Maxwells 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.

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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 Maxwells

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


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 ().

Before going more in details with the optical properties of metals lets introduce the

Macroscopic Maxwells 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 is the external charge

density, while 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.

Lets define now the fields:

( is the polarization, the dielectric function )

( is the magnetization)


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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:


2 Optical properties of metals

The optical properties of a metal is described via dielectric function (), as above defined.

The frequency dependence of () 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

Lets 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 , , with n the electrons

density (n = N/V).

The dielectric function is connected to the polarization by:

( )

with

( ) ( )


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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)

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,


-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 electron-

electron 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 Newtons law

( ) neglecting the extra fields emerging from other electrons and ions.

2. Collisions in Drudes 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 nave

picture of Drude.

3. An electron experiences a collision with a probability per unit time 1/. With 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/. In the Drude model, is taken to be independent of an

electrons position and velocity.

4. Electrons achieve thermal equilibrium with their surroundings by collisions.


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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:

( )

( ) ( ).

Lets now define the dielectric function at the frequency as:

( ) ( )

( )

( )

( )

Lets 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 = 1/. We will define later both and .


For the moment lets limit our study to the case of no damping. For larger frequencies, close to

the plasma frequency, , 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 ( ) is positive and real, while is totally reflected

when ( ) is negative.

Fig.6 Dielectric function () of a free

electron gas versus frequency in unit of the

plasma frequency.

In general, the dielectric function can be written as: () = 1() + i2(), where 1() is the real

part and 2() the imaginary part. At optical frequencies, () 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.


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The reflectivity is defined as:

|

|

( )

( )

1. < p : is imaginary n = 0, R = 1

2. = p : R = 1

3. > p : is real : ( )

If we limit ourself in the range of low frequency, , hence and . In this

region metals are mainly absorbing.

The absorption coefficient is (

)

. For low frequencies the field fall off inside the

metal as , where is the skin depth,

.

This description is valid as long as the mean free path of the electrons , where is

the Fermi velocity.


Taking into account | | the Beer law is obtained:

( ) (

)

( ) Beer-Lambert Law

At visible frequencies the free-electron model doesnt hold anymore due to interband transitions

2 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.


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In the case of Gold, this validity is already broken at the boundary between visible and near-

infrared. 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.

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