3/1/2017
NANOPHOTONICS 1
1 Z Adelpour
TEXT AND REFERENCES
1
3/1/2017
INTRODUCTION
ELECTROMAGNETICS OF METALS
Maxwell’s Equations and Electromagnetic Wave
Propagation
The Dielectric Function of the Free Electron Gas
The Dispersion of the Free Electron Gas and Volume
Plasmons
Real Metals and Interband Transitions
WAVELENGTH REGIONS
2
3/1/2017
GENERAL VIEW
THE INTERACTION OF METALS AND EM FIELDS
Maxwell’s equations
Quantum mechanics
3
3/1/2017
DISPERSIVE CHARACTERISTICS OF METALS
frequencies up to the visible part of the spectrum:
•highly reflective
•do not allow electromagnetic waves to propagate through them
•cladding layers for the construction of waveguides and resonators for
electromagnetic radiation at microwave and far-infrared frequencies
•perfect or good conductor approximation
frequencies towards the near-infrared and visible part of the spectrum
•field penetration increases significantly
at ultraviolet frequencies
•dielectric character and allow the propagation of electromagnetic
waves 7
MAXWELL’S EQUATIONS
D : dielectric displacement
E : electric field
H : magnetic field
B : magnetic induction or magnetic flux density
ρext : external charge densities
Jext : external current densities 8
4
3/1/2017
INTERNAL & EXTERNAL SETS
External :
Internal :
o The external set drives the system, while the internal set responds to
the external effects.
ELECTRIC & MAGNETIC POLARIZATION
P : electric polarization
M : magnetic polarization
ε0 : the electric permittivity of vacuum
μ0 : the magnetic permeability of vacuum
10
5
3/1/2017
ELECTRIC POLARIZATION
P describes the electric dipole moment per unit
volume inside the material, caused by the
alignment of microscopic dipoles with the electric
field.
internal charge density
current densities
11
The great advantage of this approach is that the macroscopic
electric field includes all polarization effects: In other words, both
the external and the induced fields are absorbed into it.
12
6
3/1/2017
dielectric susceptibility
13
RELATIONSHIP BETWEEN AND
at low frequencies preference is given to the conductivity,
while
Experimentalists usually express observations at optical frequencies in
terms of the dielectric constant.
14
7
3/1/2017
RELATIONSHIP BETWEEN AND
At low frequencies,
ε is usually used for the description of the response of bound charges to a
driving field, leading to an electric polarization,
while σ describes the contribution of free charges to the current flow.
At optical frequencies however,
the distinction between bound and free charges is blurred.
15
COMPLEX VALUES
dielectric constant
conductivity
refractive index
κ is called the extinction coefficient and determines the optical
absorption of electromagnetic waves propagating through the medium.
16
8
3/1/2017
EXTINCTION COEFFICIENT
κ is called the extinction coefficient and determines the optical
absorption of electromagnetic waves propagating through the medium.
the imaginary part ε2 of the dielectric function determines
the amount of absorption inside the medium.
17
THE DIELECTRIC FUNCTION OF THE
FREE ELECTRON GAS
Over a wide frequency range
the optical properties of metals can be explained by a plasma model
where
a gas of free electrons of number density n moves against a fixed
background of positive ion cores.
For alkali metals,
this range extends up to the ultraviolet,
For noble metals
interband transitions occur at visible frequencies, limiting the
validity of this approach.
18
9
3/1/2017
PLASMA MODEL
The electrons oscillate in response to the applied electromagnetic
field, and their motion is damped via collisions occurring with a
characteristic collision frequency γ = 1/τ.
τ is known as the relaxation time of the free electron gas, which is
typically on the order of 10−14 s at room temperature, corresponding
to γ = 100 THz.
19
PLASMA MODEL
equation of motion for an electron of the plasma sea
subjected to an external electric field E:
Then the
If
solution of the
equation is
20
10
3/1/2017
PLASMA MODEL
21
plasma frequency of the free electron gas
PLASMA MODEL
ε(ω) = ε1(ω) + iε2(ω)
22
τ is the relaxation time of the free electron gas
ωp is the plasma frequency of the free electron gas.
11
3/1/2017
PLASMA MODEL FOR
In this region metals retain their metallic character.
For large frequencies close to ωp, the product ωτ >> 1, leading to
negligible damping. ε(ω) is predominantly real.
ω → ωp : interband transitions!!!
23
PLASMA MODEL FOR VERY LOW FREQUENCIES
In this region, metals are mainly absorbing.
real and the imaginary part of the complex refractive index are of
comparable magnitude
absorption coefficient :
24
12
3/1/2017
EXTENSION TO PLASMA MODEL
ideal free-electron metal
25
Drude model
26
13
3/1/2017
COMPARISON (GOLD)
free electron gas (solid line)
dielectric data for gold [Johnson and Christy, 1972] (dots).
27
Interband transitions limit the validity of this model
at visible and higher frequencies.
COMPARISON (SILVER)
free electron gas (solid line)
dielectric data for gold [Johnson and Christy, 1972] (dots).
28
Interband transitions limit the validity of this model
at visible and higher frequencies.
14
3/1/2017
REAL METALS AND INTERBAND TRANSITIONS
Interband transitions are described using the classical
picture of a bound electron with resonance frequency ω0
Lorentz-oscillator term
29
Plasma Extension Drude-
to plasma Lorentz
model model model
30
15
