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Lecture 21
Optical properties
Incoming light
Absorbed light
Reflected light
Heat
Transmitted light
Light impinging onto an object (material) can be absorbed, reflected, or transmitted.
Reflectivity (R) =
Reflected fraction
Incident fraction
If the medium is transparent (no absorption):
Transmitivity (T) =
T  R 1
Transmitted fraction
Incident fraction
In more detail:
n1
n2
Reflection: the incident and exit angle
with respect to the normal to the surface
are identical.
Transmission: the refractive index change
from one material to the next determines
the change in direction from outside to
inside the medium.
Scattering: on a rough surface, locally the
surface normal varies, resulting in a broad
macroscopic distribution of “reflected”
light called scattering.
Absorption: the incoming light partially
penetrates the material, transfers energy
to electron and/or lattice excitations.
These in turn may relax back to the
ground state by emitting light and or
phonons.
Transmission
n
c
v
Refractive index n is the ratio of the vacuum speed of light c to the speed of light in the medium v. Frequency is the constant.
n1 sin 1  n2 sin 2
Reflection
n
c
v
The amplitude of the reflected light depends on the polarization of the light and the dielectric properties of the material.
The components of E parallel and perpendicular to the plane of incidence.
1   2
r 
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos  2
r|| 
n2 cos 1  n1 cos  2
n2 cos 1  n1 cos  2
Absorption
A finite fraction of the light intensity dI is absorbed over a small distance dx into the material. The absorption coefficient 
is a material property.
I    x

I ( x )  I o e  ( x  x o )
I o  I ( xo )
In a semiconductor  is proportional to the frequency of visible light and a material property  .
  0 means that there is no absorption.
  4
f

c
Windows
Mirrors
X-rays for detectors
The parameters T, R, and A (for absorption) along with n and  are optical material constants.
T  R 1
n
c
v
1   2
When there is no absorption A=0 is zero.
n1 sin 1  n2 sin 2
r 
I ( x )  I o e  ( x  x o )
Transmission
n1 cos 1  n2 cos  2
n1 cos 1  n2 cos  2
A(x)  e x
r|| 
n2 cos 1  n1 cos  2
n2 cos 1  n1 cos  2
Absorption
Depth profiling of shallow water
Reflection
Reflection data for Al, Ag, Au, and Cu
Band diagram for Al
Energy
Silicon
Aluminum
Silver
Copper
Fermi level
2 eV
4 eV
Energy from 3d levels
Al
Cu
Au
Ag
2.0 eV
2.3 eV
4.0 eV
Density of states per energy
more
absorption
Background: Light moving in z-direction with electrical polarization in x-direction
2
2
 
c
E


E

Ex
x
x
z 2
t 2
 o t
2
Ex  Eo e
nˆ 2    i
From Maxwell’s equations
z
i ( t  n )
c
“trial” solution (note: complex)


  i
 o
2o
nˆ  n  i
ˆ   1  i 2
Complex index of refraction
nˆ 2  n 2   2  i 2n    i

1  n 2   2

1  n 2   2
Ex  Eo e
z
i ( t  nˆ )
c
 Eo  e

 2

c


 o
  4n o

c
z
damping
From I  Ê 2
2 

 o

nc o
e
z
i ( t  n )
c
plane wave
 2  2n
Reflectivity
I
nˆ  1
(n  1) 2   2
R r 

I o nˆ  1
(n  1) 2   2
2
In a metal with low frequencies and dielectric values less than 10:
nˆ 2    i


  i
 o
2o
  n2   2

1017
 13  
2   o 10
n2 

2
2o
R  1 4

o

  4n o
Hagan-Rubens relation
I r nˆ  1
n 2  2n  1   2 2n 2  2n  1  4n
n

 2

 1 2 2
2
2
I o nˆ  1
n  2n  1  
2n  2n  1
n  n 1/ 2
2
R
Theoretical
Classical Drude model Electrons are free within a band and can be accelerated by external fields and can
loose energy by scattering. They transition between energy states within a band. Typically phonons are
involved. Transitions are caused by phonon-electron scattering or by impurity-electron scattering (in
insulators).
The phenomenological model by Drude includes these contributions

m
E  Re( Eˆ  eit )
vˆ  vˆ  qEˆ
t

m
1

 q
results in imvˆ  vˆ   i    mvˆ  qEˆ
vˆ  



m
m
And from there to the complex current density
v  Re( vˆ  eit )
1

 Eˆ

 1  i
2
ˆj  nqvˆ   nq    1  Eˆ
 m  1  i


o
 ( ) 
1  i
The real and imaginary parts are
Re  ( )  
o
1   2 2
Im ( )  
 o
1   2 2
complex
Estimate of scale

At  =1THz=1012/s   0.024  1
2
ˆj  nqvˆ   nq 
 m

Re  ( )  





2
 2.4 1014 s
and

 nq 2  ˆ
1
ˆ
 
  E   o  Eˆ
 E  
 1  i
 m 
o
1   2 2
Conductivity

m o
9.11031 kg  5.8 107 S

nq 2
8.5 1028 m3  1.6 1019 C
and for the non-complex versions:
Im ( )  
 o
1   2 2
Damping
Al
p 
Au
e2 N p
 o mo
Ag
5um
2um
1um
500nm
200nm
60
150
300
600
1500
THz
Theoretical background
Lorentz theory of bound electrons
electrons are bound to nuclei by “springs”, which determine the natural frequency.
Recall the harmonic oscillator
F  kx
o 
k
m
An external electric field displaces charges and creates a dipole. This is assumed to be
the oscillator. Vibrations are forced by an external AC field.
At the resonance frequency the maximum amount of energy is absorbed.
The combination explains free electrons with high absorption (R near zero) for low frequencies ( o  0 )
for the IR region of the light spectrum. The bound electrons, oscillator explain the absorption bands.
Insulators and semiconductors are explained by the harmonic oscillator of bound electrons.
Failures
Why should electrons be free at low frequencies and bound at higher frequencies?
Quantum Mechanics
Solves the dilemma and explains the absorption (or not) of light with intra-band and inter-band
transitions as well as direct and indirect energy gaps.
This is mostly observed in metals, where small energy transitions are possible due to partially filled
bands. In semiconductors and insulators this does not occur. Exceptions may be highly doped
semiconductors where the Fermi Energy is right at the conduction (or valence) band.
At low frequencies this effect dominates (not quantized).
Window coatings with materials such as ITO are used to transmit visible light but reflect IR light. Loss of
heat is minimized in the winter or the room temperature remains cooler in the summer.
Conclusions
Inter-band transitions correspond to the bound-state version of the Lorentz model, while
Intra-band transitions correspond to free electron effects of the Drude model.
The sharp absorption lines from atoms (i.e. single resonance frequencies) give way to broad bands and
hence absorption bands and not absorption frequencies. The plasma frequency corresponds to the
edge where the reflectance of a metal turns up.
Platinum
12
Silver
Tungsten
Copper
14
Threshold frequency (10 Hz)
14
10
Zinc
8
6
Sodium
Lithium
Potassium
Rubidium
Caesium
4
2
0
0
1
2
3
4
5
Work function (eV)
6
7
Materials
Work function
(eV)
threshold
frequency (1014/s
Ceasium
1.91
4.62
Rubidium
2.17
5.25
Potassium
2.24
5.42
Lithium
2.28
5.51
Sodium
2.46
5.95
Zinc
3.57
8.63
Copper
4.16
10.06
Tungsten
4.54
10.98
Silver
4.74
11.46
Platinum
6.30
15.23
Inter-band transitions
Electrons transition from one band to another (usually from the valence to the conduction band)
Direct gap GaN
vs
Indirect gap Si
Direct: Electrons transition “vertically” without the “assistance” of phonons. The momentum vector k
remains constant (the momentum of the photon is very much smaller and insignificant here). There is
a vast number of near continuous transitions possible. The band gap merely represents the lower
minimum.
Indirect: Phonons are created or absorbed to accommodate the required change on momentum
vector k. In metals these transitions play a miniscule role (100 to 1000 times smaller in intensity).
However, in semiconductors they play a big role. Keep in mind that jumps to higher bands (not shown)
are also possible.