Download OPTICAL PROPERTIES OF METALS

OPTICAL PROPERTIES OF MATERIALS
WHY STUDY THE OPTICAL PROPERTIES OF MATERIALS?
•
•
When materials are exposed to electromagnetic radiation, it is sometimes important to be able
to predict and alter their responses. This is possible when we are familiar with their optical
properties and understand the mechanisms responsible for their optical behaviours.
By “optical property” is meant a material’s response to exposure to electromagnetic radiation
and, in particular, to visible light.
LIGHT INTERACTIONS WITH SOLIDS
•
•
•
When light proceeds from one medium into another (e.g., from air into a solid substance),
several things happen. Some of the light radiation may be transmitted through the medium,
some will be absorbed, and some will be reflected at the interface between the two media.
The intensity Io of the beam incident to the surface of the solid medium must equal to the sum
of the intensities of the transmitted, absorbed, and reflected beams,
denoted as;
Io = IR + IT +IA
An alternate form of Equation; T+A+R = 1
where T, A, and R represent, respectively, the
IT
IR
I
A
transmissivity I ,absorptivity and reflectivity
or
I
O
O
IO
the fractions of incident light.
LIGHT INTERACTIONS WITH SOLIDS
•
•
If the material is not perfectly transparent, then light intensity decreases exponentially with distance.
Let us consider a material of small thickness, δx. If the light intensity drop in δx is δI, then:
dI
 I
dx
-1
δI = -α δx I, where α is an absorption coefficient (m ). In the case of δx → 0, we get:
Integrating both sides, we have: I = I0 eαx
Or, in logarithmical form: αx = - ln (I/I0 ), which is known as Lambert’s Law, or
Beer’s Law, or Bouger’s Law, or Beer-Lambert’s Law, or Bouger-LambertBeer’s Law (too many names for just one law, so do not get misguided by
different names combinations in the literature - they all refer to the same law!)
• Thus, if we plot -ln(I/I0) vs. x, we can find α from the angle. Depending on the
material and the wavelength, light can be absorbed by the nuclei (all materials)
or by the electrons (metals and narrow Eg semiconductors).
OPTICAL PROPERTIES OF METALS
Metals are opaque because the incident radiation having frequencies within the visible
range excites electrons into unoccupied energy states above the Fermi energy.
Total absorption is within a very thin outer layer, usually less than 0.1 μm thus only
metallic films thinner than 0.1 μm are capable of transmitting visible light. All
frequencies of visible light are absorbed by metals because of the continuously available
empty electron states, which permit electron transitions.
(a) Schematic representation of the mechanism of photon absorption for metallic materials in which an electron is
excited into a higher-energy unoccupied state. The change in energy of the electron is equal to the energy of the
photon. (b) Reemission of a photon of light by the direct transition of an electron from a high to a low energy state.
OPTICAL PROPERTIES OF METALS
•
Penetration depths (I/I0 = 1/e) for some materials: H20: 32
cm; glass: 29 cm; graphite: 0.6 μm; gold: 0.15 μm.
What happens to the excited electrons in the surface layers
of metal?
They recombine again, emitting the light.
The energy lost by the electron is exactly the same as that
was spent to excite it. That is why metals reflect the light
very well (about 95%). Thus, metals are both opaque and
reflective (the remaining energy is usually lost as heat).
DEFINITION OF INDEX OF REFRACTION
In uniform isotropic linear media, the wave equation is:
2


E

E
0
2

t
2
2


H

H

0

t2
2
They are satisfied by plane wave
y=A e i(k r- wt)
kkω
y can be any Cartesian components of E and H
 1
The phase velocity of plane wave travels in the direction of k is vω
k

OPTICAL PROPERTIES OF NON-METALS
•
By virtue of their electron energy band structures, nonmetallic materials may be transparent to visible light.
Therefore, in addition to reflection and absorption, refraction and transmission phenomena also need to be
considered. The index of refraction n of a material is defined as the ratio of the velocity in a vacuum c to
c
the velocity in the medium v;
n
v
and
;
c 
v
1

n


r
r
v 

o
o
Where εr and μr are the dielectric constant and relative magnetic permeability, since, most of
the substances are slightly magnetic μr  1 i.e.
where symbols have their usual meaning.
n  r
Thus, for transparent materials, there is a relation between the index of refraction and the dielectric
constant. As already mentioned, the phenomenon of refraction is related to electronic polarization at the
relatively high frequencies for visible light; thus, the electronic component of the dielectric constant may be
determined from index of refraction measurements
OPTICAL PROPERTIES OF NON-METALS
•
•
Dielectrics and semiconductors behave essentially the same way, and as you know, the only
difference is the bandgap width.
In direct bandgap semiconductors, electron simply drops from the bottom of CB to the top of VB,
and excess energy is emitted as a photon (GaAs is an example). Photons with the energy
exceeding Eg are absorbed by giving their energy to electron-hole pairs. They may or may not reemit the light during the recombination depending on whether the gap is direct or indirect.
In indirect bandgap semiconductors, direct electron-hole
recombination does not take place. Recombination process
occurs in two stages via recombination centers (usually
defects) in the bandgap: electron falls from the bottom of
CB to the defect level, and then further down to the top
of VB. Electron energy is therefore lost in two portions
by the emission of phonons (lattice vibrations). This process is also called non-radiative
recombination. (Si and Ge are the examples of indirect bandgap semiconductors).
REFLECTION AND TRANSMISSION IN NON-METALS
Reflection occurs at the interface between two materials and is therefore related to
index of refraction
Reflectivity, R = IR/I0, where the I’s are intensities
Assuming the light is normally incident to the interface:
§ where n1 and n2 are the indices for the two materials
Optical lenses are frequently coated with antireflection layers such as MgF2
which work by reducing the overall reflectivity. some lenses have multiple
coatings for different wavelengths
For an incident beam of intensity Io on the
surface of a non-metal, the transmittivity
will be given by;
Io
n1
n2, α, x
2

x
IT
Io(
1

R
)e
IT
IR
Where R is the reflectivity, α is the absorption coefficient and x is the thickness of
the non-metal .
2
n2n1
R
n n 

 2 1
Quest: Determine the expression between IT and I0.
Ans:
Io
R1

α1,x1,n1

x

x
2
2
1
1
2
2
I

I
(
1

R
)
(
1

R
)
e
T
o
1
2
IT
R2
α2,x2,n2
INTERPRETATION OF OPTICAL PROPERTIES IN TERMS OF
SIMPLIFIED ELECTRON BAND STRUCTURE
OPTICAL ABSORPTION PROCESSES
(a)Interband transitions
(i) The highest energy transitions are those from the bottom of the
valance band to the top of the conduction band.
→absorption of light at different energies
→ a frequency dependence of the attenuation coefficient.
(ii) Other lower energy interband transitions from the top of the valence
band to the bottom of the conduction band also occur.
The ‘absorption edge’ occurs at   Eg , the gap energy.
The attenuation coefficient  in semiconductors is usually in the
range 1017 – 108 m-1 for energies above the band gap energy Eg.
However decreases by several orders of magnitude once the energy drops below the
band gap energy Eg because there are no longer energy states for the excited
electrons to occupy so they cannot absorb the energy of the incident photons.
(iii) Another electronic process is known as exciton generation.
Excitation produces a bound electron-hole pair. The electron is trapped in a
localized energy level in the band gap while the hole remains mobile in the valence
band. The exciton can dissociate into independent free carriers or can recombine
with the emission of a photon of phonon.
(iv) Excitation of electrons from localized trap sites in the band gap into the conduction
band can occur at energies lower than Eg. This usually occurs from optical
absorption, although it can also arise from thermal excitation.
The optical absorption arising from this process is much lower than for other interband transitions
because there are relatively few trapped electrons compared with electrons in the valence band.
(b) Intraband transitions
In metals the absorption of photons by electrons occurs over a continuous wide range of
energies beginning effectively from zero energy. This usually involves the absorption or
emission of phonons to conserve momentum. In this process the electrons move between
energy states in the same band. The intraband transitions can only occur in metals, and they are
responsible for the high reflectivity of metals at low energies.
PHOTOLUMINESCENCE AND ELECTROLUMINESCENCE
Photoluminescence : phosphorescence and fluorescence
Whenever an electron is excited into a higher energy state it must eventually revert to a
lower unoccupied state, and this occurs with the emission of a photon. If the initial
excitation is by incident light → photoluminescence.
The lifetime of the electrons in the higher energy states determines the duration of the
emission process. If the lifetime is short then the emission of photons occurs almost
immediately and the luminescence stops when the light source is switched off. This
process is called fluorescence. If the lifetime extends over a period of several seconds, a
few minutes or even hours then the luminescence continues even after the light source is
removed. → phosphorescence.
Electroluminescence
Consider injection electroluminescence in a single p-n junction of a
semiconductor. When the n-type and p-type materials are placed in contact,
electrons flow into the p-type material leaving it with a negative charge and
the n-type with a positive charge.
If the p-type side of such a junction is then connected to the positive
terminal of a voltage supply, current is carried by the flow of electrons into the
p-type material where there are already free holes in equilibrium.
Recombination of electrons and holes can take place and this results in
emission of photons.
The Classical Drude Model
Classical Electron Model ( Lorentz Model)
The simplest possible theory of the response of the solid
to an oscillating external EM field was provided by
Drude.
Let the electric field of optical wave in an atom be
E=E0e-iwt as also x (i.e. the time dependence form of exp
(-iωt) ) the electron obeys the following equation of
motion
ma

E

E

E
(
harm
)
el
local
damping
spring
2
d
X m
m2 dx

Cx


e
E1.1
dtdt
If C=0, then it is known as the
DRUDE model
x is the position of the electron relative to the atom, m is the mass of the
electron, τ is the relaxation time (= i.e. the damping coefficient)
1

Thus, equation 1.1 becomes
i

xm

m

x



eE
2

eE
x
m
2 i
(m


)

To obtain the optical conductivity we may assume that x  v ; where
v is the mean velocity and τ is the relaxation time
The current density is: j=nev=nex/τ
2
ne eE
j
ne
 
(
)
j (
) 
i
i
E m
 m

(

)

(
)


In the limits   1 , i.e. when the collision frequency 1/τ is small as compared with the
frequency ω of the light wave the equation reduces to;

2
2
ne
eE
j
ne
(
x
 2)
 
(
) 2 with
i
m
m
E m

(

)




This is known as the DRUDE formula for optical conductivity; The real part of the dielectric
constant may be calculated as;
The electric polarization Por
2
ne
E

E
o asP
P


nex

 2 
oE
m

For ε = ε1 + iε2 (includes the complex and real part), the real part
2
ne
p2


1



[
1
 2 ]Or 1 [1 2 ]
1
m



o
2 1
ne
]2 is the electron plasma frequency
Where, p [
mo
Here 2 
2 

i.e.  2o
o
1
 o
ne 2
ne 2
2
[
] with  p 
2
m 
m 0
 p2  o
 p2
2 
[
] 3
2
 o  
 
1
 p2
 p2
   1  i 2  1  2  i 3

 
 p2
i
  1  2 [1 
]


 p2
 p2
i2
1
[
1

]
[
1

]
2
2
2
2

( )

( )
 1 
1 
[1 
i

]
[1 
i

]
When ωτ>>1 i.e. when the collision frequency 1/τ is small as compared
with the frequency of light wave, 1/(ωτ)2 will be very small as compared to 1.
Thus,
  1
 p2
2
[1 
i

]



i
.
e
.


i

1


1

i
i
[
1
 ]

[

]


2
p
2
1
2
The real part of the dielectric constant,
the optical conductivity and imaginary
part (negative) are shown:
2
2

p 
p



i


1


i
1 2
2
3



2
p
The Physical significance
2
2

p 
p



i


1


i
1 2
2
3



EM wave incident on a medium can be propagated through
the medium only if the real part of the refractive index is +ve,
but they are totally reflected from the medium if the real part
of the medium is –ve.
Alkali metals (free electron gas, 1028 electrons/m3 , ωp = 1016
rad/sec {calculate λp }) are transparent to UV light but reflects
visible light.
For quantitative observation, we need to form the dispersion
relation using Maxwell wave equations:
2

E

E
2

E


 2


o
O

t

t
The last term of the equation describes the absorption of EM radiation is often termed
as dissipation term. Having a solution of the form:

E

E
exp(
i
(
k
.
r
)
t
)
O
We find dispersion relation as:
2
2

k





i


o
o
Letting the value of C2 =1/μoεo, we get
2
εω
iωω
ω ε
iσ 
2
k = 2 + 2 ork =  +

εo c εo c
c  εo εo ω 
The standard relation k=ωc= 2πν/c= 2π/λ, is thus modified by complex index of refraction,
1
2

i 

nc nni







o
 o

i 
2
2
2  


(nni ) n 2nni ni 


 o o
equating
realandimaginary
parts
;
kc
n2 ni2 


and2nni 
o
o
k is a complex propagation vector, In free space, k=ωc, as the ordinary light; but in a
dispersive medium there is dispersion. In such cases the dielectric constant is given by;
 i


 




o 
o
With the assumption;
(kc
/
)2
Thus the dispersion takes the form;






kc
2
2
restrictin
g
our
attention
to
real
part
;



c
k
2
2
p
22
The polarization defined as the dipole moment per unit volume is
2
ne
P


nex


E
2
m

The dielectric function at frequency ω is
D
(

)
P
(

)

(

)


1


E
(

) 
E
(

)
o
o
Comparing with;
2
ne
P


nex

2
E
m
2
we
get
(
)
1


ne
om

2
And
the
plasma
frequency

given
by
;
pis
2
ne

p
om

eE
as
(
x

)
2
m
Photonic Band Gap
Crystals
What are Photonic Crystals / Photonic Band Gap
Materials?
A photonic crystal is a periodic arrangement of a dielectric material that exhibits
strong interaction with light.
Engineered material in which the refractive index (RI) is periodically
modulated on a length scale comparable to the desired operational
wavelength.
(a)
(b)
(c)
Schematic diagram of photonic crystal. (a) 1-D, (b) 2-D and (c) 3-D.
The periodic arrangement of refractive indices in a photonic crystal can be in onedimension (1D), two-dimensions (2D) or three-dimensions (3D). Fig. above shows
examples of 1D, 2D and 3D photonic crystals. Here two materials with different
refractive indices, represented by red and yellow colors are taken, and arranged
alternately in one, two or three dimensions to form a stack. The lattice period (i.e.
the size of the basic unit cell) must be comparable to the wavelength of the light
incident on the structure.
Yablonovitch and John (PRL, 1987) used a combination of electromagnetism
and solid state physics to describe photonic band gaps. These two works are
considered as the origin of the research field known as “photonic crystals or
photonic band gap materials”.
 Photonic
Since photonic crystals are designed to affect the
propagation properties of photons.
 Crystal
Because it is formed by a periodic arrangement of basic
building blocks.
Examples:
1D: Bragg Reflector
2D: Si pillar crystal
3D: colloidal crystal
Photonic Crystals- Semiconductors of Light
Semiconductors
Photonic Crystals
Periodic array of atoms
Periodic variation of dielectric
constant
Atomic length scales
Length scale ~ 
Natural structures
Artificial structures
Control electron flow
Control e.m. wave
propagation
1950’s electronic revolution
New frontier in modern optics
Natural Photonic Crystals:
Structural Colours through Photonic Crystals
Natural opals
Periodic structure  striking colour effect even in the absence of pigments
Important crystal features and parameters.
In order to design a photonic crystal with a complete PBG or a specific band
behaviour there are some crystal features and parameters that must be
engineered.
Dimensionality: The periodicity of the refractive index will determine the dimensioality
of the photonic crystal. We may have one (1D), two (2D) or three (3D) dimensional
lattices. Stacks of planes, sets of columns and any of the Bravais lattices are examples of
each of those structures (see Fig. 1.1 a-d)
Symmetry: The position of the building blocks of our photonic crystal will set the
symmetry of our lattice. Examples of several three-dimensional symmetries can be
found in Bravais lattices:
Topology: A lattice with a given symmetry may present variations in its topology that
will importantly affect the photonic band structure. The topology can be varied by
interpenetrating the building blocks (network topology) or isolating them (Cermet
topology).
Filling fraction and effective refractive index: The relative amount of material
composing the scattering building block is called filling fraction (ff). The effective
refractive index (neff) is usually calculated as the square root of the average dielectric
constant (eav). As happens with the lattice parameter, the range of wavelengths of the
optical spectrum where our photonic crystal works will also depend on neff.
PHOTONIC BAND GAP CRYSTAL STRUCTURE
A 2D structure made of cylinders and forming a square lattice is shown in Fig. 1.3. The distance between cylinders is the lattice parameter a. Vectors  andin this figure point in
directions which present a different RI modulation. Therefore, the dispersion relation for each impinging angle should differ. Fortunately thanks to the symmetry of the lattice there are
directions which are equivalent, as shown in Fig. 1.3 b for  and ’. When two directions are equivalent, the periodicity of the RI is the same and the dispersion relation is the same
for both.
 Photonic Crystals
 complex dielectric environment that controls the flow of radiation
 designer vacuum for the emission and absorption of radiation
Photonic crystal waveguides
Why Employ Photonics?
•
Signals travel at the speed of light
•
Photons, unlike electrons have no weight and create no resistance.
•
Focused light generated by lasers constitutes the highest concentration of
energy known on earth
•
A pulse of photons can be as short as one-millionth of a billionth of a
second, the dimension of time in which molecular and atomic reactions
take place
•
Light beams are well-suited not only to help us see, but also to hold and
manipulate atoms
•
As light acts virtually contact-free; it can be used as a tool even under
extreme conditions.
Photonic Crystals Based Light Bulbs
C. Cornelius, J. Dowling, PRA 59, 4736 (1999)
“Modification of Planck blackbody radiation by photonic band-gap structures”
3D Complete Photonic Band Gap
Suppress blackbody radiation in the infrared and redirect and enhance thermal energy into visible
Solid Tungsten Filament
3D Tungsten
Photonic Crystal
Filament
S. Y. Lin et al., Appl. Phys. Lett. (2003)
 Light bulb efficiency may raise from 5 percent to 60 percent
Solar Cell Applications
– Funneling of thermal radiation of larger wavelength (orange area) to thermal radiation
of shorter wavelength (grey area).
– Spectral and angular control over the thermal radiation.
Other Applications of PBG
Device
Description
Status
Optical Fibers
2-D band gap material stretched along the Early version
third dimension.
commercialized
Nanoscale
Lasers
World’s tiniest optical cavities and tiniest Demonstrated in lab
lasers: formed in a thin film 2-D band gap
material.
already
Radio-frequency Uses inductors and capacitors in place of Demonstrated
for
antennas,
ordinary dielectric materials.
magnetic
resonance
reflectors
imaging and antennas
Ultra-white
pigment
Incomplete 3-D band gap material , usually Demonstrated, low cost
patterned a opal structure.
manufacturing methods
under development.
LEDs
PBG structure can extract light very efficiently
( better than 50%)
Photonic
Integrated
circuits
2-D thin films can be patterned like Under development
conventional integrated circuits to make
channel filters, modulators, couplers etc
Demonstrated, but must
complete with other
methods of achieving the
same goal.
Naturally occurring photonic crystals
Photonic crystal structures are found in
Beetles
Butterflies
Peacock feathers
Opals
etc.
Cleopatra Beetle
The microstructures are responsible for
the iridescent colors in all of the naturally
occurring photonic crystals.
Feather of a peacock
Brown butterfly
ENERGY DISPERSION RELATIONS (ELECTRON)
Electronic Band Gap (EBG) can be categorized as the band gaps
associated with the Electromagnetic Matter Particle (ELECTRON)
Crystal
Periodicity
Free electrons
ENERGY DISPERSION RELATIONS (PHOTON)
Photonic Band Gap (PBG) can be categorized as the band gaps
associated with the Electromagnetic Field Particle (PHOTON)
Photon in
vacuum
Photon in
periodic
dielectrics
Negative refractive-index materials are of great interest for a variety of potential applications. Because natural negative refractive-index
materials do not exist, artificial structures have been proposed and fabricated that exhibit an effective negative index over limited frequency
ranges.
The two principal approaches to the realization of negative refraction are metamaterials and photonic crystals. Metamaterials typically use
metallic structures to provide a negative permittivity and use resonant structures (inductor-capacitor tank circuits) with a scale much
smaller than the wavelength to provide a negative permeability leading to negative refraction while photonic crystals exhibit negative
refraction as a consequence of band-folding effects.