Chapter 11.
8
“Continuum Theory”
1
“Atomic Structure of Solids”
“Quantum Mechanics”
Summary:
1. In the case of no conductivity:
c
n    r1/2 , n 2   r
v
2. When there is a conductivity: From Maxwell equation of EM wave.

nˆ 2   r 
i  (n 2  k 2 )  2nki  ˆ  1  i 2
2 0
1   r  n 2  k 2

 2  2nk 
2 0
W (penetration depth)=
1


c
2 k

Reflectivity:
R
2
(n  1)
nˆ  1
(n  1)  k


(n  1) 2 nˆ  1
(n  1) 2  k 2
2
2
2
2
c
4 k

nc 0


4 k

Hagen-Rubens equation
R  1 4

 0
0
For dc conductivity
8.1 Survey
Reflectivity of (a) metals and (b) dielectrics
Hagen- Rubens Model
Continuum theory:
limited to frequencies for which the
atomistic structure of solids does not
play a major role
Drude Model
Some electrons in a metal can be considered to
be free and can be accelerated by an external
electric field.
Moving electrons colliding with certain metal
atoms (Friction force):
Can’t explain fluctuation of reflectivity
(absorption band )
Lorentz Model
Electric dipole:
Presented each atom as an electric dipole
Alternating electric field to the atoms cause forced vibrations.
This vibration thought to harmonic oscillator.
An oscillator is known to absorb a maximal amount of energy
when excited near its resonance frequency.
3
8.2 Free Electrons Without Damping
We consider the simplest case at first and assume that the free electrons are excited to
perform forced but undamped vibrations under the influence of an external alternating field.
Momentary value of the field strength of a plane-polarized light wave:
(forced harmonic vibration)
,
(9.14 and 9.15)
( D   r 0E   0E  P)
4
since, nˆ 2  ˆ  1  i 2
(1)
>1
. Then n̂ is imaginary and real part is zero, (for large frequency),
The reflectivity is 100%
(2)
<1
. Then
n̂
is real and imaginary part is zero (for large frequency)
The material is essentially transparent for these
wavelengths
5
1
0
: the condition for a plasma oscillation
Plasma frequency
p  (
6
N f e2
 0m
)1/2
Neff can be obtained by measuring n and k in the red or IR spectrum
and by applying (in a frequency range without absorption bands)
  n 2  k 2 (10.10)
ˆ =1 
7
e2 N f
4  0 m
2
2
(11.6)
8.3 Free Electrons With Damping
(Classical Free Electron Theory of Metals)
We postulate that the velocity is reduced by collisions of the electrons
with atoms of a nonideal lattice.
Equation of motion forced oscillation
With Damping
The damping is depicted to be a friction force which counteracts the electron motion.
At drift velocity , V’=constant (saturation drift velocity)
8
Put this solution to the above equation
m x 
2
9
N f e2
0
(i ) x  eE
x
N f e
0
E
m 2
i
e
=
Damping frequency
nˆ  n  ik
2

nˆ 2  n 2  k 2  (2nk )i  1  2 1
  2i
10
Where
plasma frequency
11
damping frequency
8.4 Special Cases
For the UV, visible, and near IR regions, the frequency varies between 1014
and 1015 s-1, while the average damping frequency, u2, is 5x1012 s-1.
Thus,
, with u~u1
(Table 11-1)
When
 2  2nk 
2 
2 0 12
0

2 0
(10.14)
 2 0 12 0 (11.23)
Thus, in the far IR the a.c. conductivity and the d.c. conductivity may be considered to be
identical.
12
8.5 Reflectivity
The reflectivity of metals is calculated using 10.29 in conjunction with 11.26 and 11.27.
13
8.6 Bound Electrons (Lorentz)
(Classical Electron Theory of Dielectric Materials)
Stationary solution for the above equation
x
14
eE0
m 2 (02   2 )   '2  2
exp[i (t   )]
Where
(Resonance frequency of the oscillator)
(Supplement)
d 2x
dx
m 2   '  kx  eE0 exp(it )
dt
dt
let ,
x  x0 exp(it )
[m(i ) 2   '(i )  k ]x  eE0 exp(it )
eE0 exp(it )
[( m 2  k )  i ( '  )]
x

2
[( m  k )  i ( '  )] [( m 2  k )  i ( '  )]
eE exp(it )
2
 2 2 0 2 2

[(

m

 k )  i ( '  )]
2
2
[m (0   )   '  ]


eE0 exp(it )
2
2
2 2
2
2

[
m
(



)


'

] exp(i )
0
2
2
2 2
2
2
[m (0   )   '  ]
eE0 exp[i (t   )]
[m 2 (02   2 ) 2   '2  2 ]
 '
[m 2 (02   2 ) 2   '2  2 ]
 '
sin 
tan  


m(02   2 ) cos  m(02   2 )
[m 2 (02   2 ) 2   '2  2 ]
(Na is the number of all dipoles)
ˆ  n 2  k 2  2nki
 1
Since,
16
e2 N a
 0 m (   )   ' 
r  1
2
P
0E
2
0
2
2
2
exp(i )
where,
17
Both of these equations reduce
to the Drude equation when 0 is
zero.
18
Resembles the dispersion curve for the
index of refraction.
Depicts the absorption product in the
vicinity of the resonance frequency.
8.7 Discussion of the Lorentz Equations
for Special Cases
8.7.1 High Frequencies
From the figures of 1 and 2, 2 approaches 0 at high frequencies.
And, 1=n2-k2=1. Thus, n assumes a constant value 1.- No refraction
8.7.2 Small Damping:
When radiation-induced energy loss is small,
namely, ’ is small,
2 2
2 2
2
2
From eq’n 11.45
 '   4 m ( 0  ) 2
We observe that for small damping, 1 (and
thus essentially n2=(c/v)2 ) approaches
infinity near the resonance frequency.
19
So what?
8.7.3 Absorption Near
Electrons absorb most energy from light at the
resonance frequency.
which shows that the absorption becomes large for small damping
8.7.4 More Than One Oscillator
Each atom has to be associated with a number of i oscillators, each having an
oscillator strength, resonance frequency damping constant
20
8.8 Contributions of Free Electrons and
Harmonic Oscillators to the Optical Constants
21
free electrons
bound electrons