Optical Processes and Excitons:A Brief Introduction Physics Notes | EduRev

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Physics : Optical Processes and Excitons:A Brief Introduction Physics Notes | EduRev

 Page 1


Optical processes and excitons
• Dielectric function and reflectance
• Kramers-Kronig relations
• Excitons
• Frenkel exciton
• Mott-Wannier exciton
• Raman spectroscopy
Page 2


Optical processes and excitons
• Dielectric function and reflectance
• Kramers-Kronig relations
• Excitons
• Frenkel exciton
• Mott-Wannier exciton
• Raman spectroscopy
Dielectric function, reflectivity r, and reflectance R
Response of a crystal to an EM field is characterized by
? (k, ? ), (k ? 0 compared to G/2)
Experimentalists prefer to measure reflectivity r
(normal incidence)
r
E
E
( )
'
?
?
?
?
( ) i kx t
E e
?
?
?
( )
'
i kx t
E e
?
?
? ?
Prob.3
where the refractive index ( ) ( ) n ? ? ? ?
r
n
n
R e
i
( ) ( )
( )
? ?
? ?
?
?
?
?
1
1
It is easier to measure R than to measure ?
?measure R( ? ) for ? ? ? ? ( ? ) (with the help of KK relations)
? n( ? )
? ? ( ? )
Page 3


Optical processes and excitons
• Dielectric function and reflectance
• Kramers-Kronig relations
• Excitons
• Frenkel exciton
• Mott-Wannier exciton
• Raman spectroscopy
Dielectric function, reflectivity r, and reflectance R
Response of a crystal to an EM field is characterized by
? (k, ? ), (k ? 0 compared to G/2)
Experimentalists prefer to measure reflectivity r
(normal incidence)
r
E
E
( )
'
?
?
?
?
( ) i kx t
E e
?
?
?
( )
'
i kx t
E e
?
?
? ?
Prob.3
where the refractive index ( ) ( ) n ? ? ? ?
r
n
n
R e
i
( ) ( )
( )
? ?
? ?
?
?
?
?
1
1
It is easier to measure R than to measure ?
?measure R( ? ) for ? ? ? ? ( ? ) (with the help of KK relations)
? n( ? )
? ? ( ? )
Kramers-Kronig relations (1926)
( )
( )
' ( )
j E
P E
E r E
? ?
? ?
? ?
? ?
? ?
?
?
?
?
Ohm ’ s law
reflective EM wave
polarization
KK relation connects real part of the
response function with the imaginary part
m
d
dt
d
dt
x F e F Z eE
x t x e
j i j j j
i t
j j
j j
i t
2
2
2
? ?
F
H
G
I
K
J
? ? ?
?
?
?
? ?
?
?
? ?
?
?
,
( )
steady state
electric dipole
examples of response
function:
p Z ex E
Z e
m i
j j j j
j
j
j j j
? ? ?
? ?
? ?
? ? ? ?
? ? ?
? ?
? ?
( )
( )
2 2
2 2
d i
Example: Response of charged (independent) oscillators
For the j-th oscillator (atom or molecule with Z bound charges),
• does not depend on any dynamic
detail of the interaction
• the necessary and sufficient
condition for its validity is causality
Page 4


Optical processes and excitons
• Dielectric function and reflectance
• Kramers-Kronig relations
• Excitons
• Frenkel exciton
• Mott-Wannier exciton
• Raman spectroscopy
Dielectric function, reflectivity r, and reflectance R
Response of a crystal to an EM field is characterized by
? (k, ? ), (k ? 0 compared to G/2)
Experimentalists prefer to measure reflectivity r
(normal incidence)
r
E
E
( )
'
?
?
?
?
( ) i kx t
E e
?
?
?
( )
'
i kx t
E e
?
?
? ?
Prob.3
where the refractive index ( ) ( ) n ? ? ? ?
r
n
n
R e
i
( ) ( )
( )
? ?
? ?
?
?
?
?
1
1
It is easier to measure R than to measure ?
?measure R( ? ) for ? ? ? ? ( ? ) (with the help of KK relations)
? n( ? )
? ? ( ? )
Kramers-Kronig relations (1926)
( )
( )
' ( )
j E
P E
E r E
? ?
? ?
? ?
? ?
? ?
?
?
?
?
Ohm ’ s law
reflective EM wave
polarization
KK relation connects real part of the
response function with the imaginary part
m
d
dt
d
dt
x F e F Z eE
x t x e
j i j j j
i t
j j
j j
i t
2
2
2
? ?
F
H
G
I
K
J
? ? ?
?
?
?
? ?
?
?
? ?
?
?
,
( )
steady state
electric dipole
examples of response
function:
p Z ex E
Z e
m i
j j j j
j
j
j j j
? ? ?
? ?
? ?
? ? ? ?
? ? ?
? ?
? ?
( )
( )
2 2
2 2
d i
Example: Response of charged (independent) oscillators
For the j-th oscillator (atom or molecule with Z bound charges),
• does not depend on any dynamic
detail of the interaction
• the necessary and sufficient
condition for its validity is causality
(1) ? ( ? ) has no pole above (including) x-axis.
(2) along (upper) infinite semi-circle
(3) ? ’ ( ? ) is even in ? , ? ’ ’ ( ? ) is odd in ? .
Some properties
of ? ( ? ):
d ?
? ?
?
( )
z
? 0
Collection of
oscillators
P
V
Z ex E
V
Z e
m i
nZ e m
i
j j
j
j
j j j
j
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ?
?
? ?
? ?
1 1
2 2
2 2
2 2
0
2 2
( ) ( )
/
, (1)
d i
for identical
oscillators
 0
?
1 4 ? ? ? ? ?
Page 5


Optical processes and excitons
• Dielectric function and reflectance
• Kramers-Kronig relations
• Excitons
• Frenkel exciton
• Mott-Wannier exciton
• Raman spectroscopy
Dielectric function, reflectivity r, and reflectance R
Response of a crystal to an EM field is characterized by
? (k, ? ), (k ? 0 compared to G/2)
Experimentalists prefer to measure reflectivity r
(normal incidence)
r
E
E
( )
'
?
?
?
?
( ) i kx t
E e
?
?
?
( )
'
i kx t
E e
?
?
? ?
Prob.3
where the refractive index ( ) ( ) n ? ? ? ?
r
n
n
R e
i
( ) ( )
( )
? ?
? ?
?
?
?
?
1
1
It is easier to measure R than to measure ?
?measure R( ? ) for ? ? ? ? ( ? ) (with the help of KK relations)
? n( ? )
? ? ( ? )
Kramers-Kronig relations (1926)
( )
( )
' ( )
j E
P E
E r E
? ?
? ?
? ?
? ?
? ?
?
?
?
?
Ohm ’ s law
reflective EM wave
polarization
KK relation connects real part of the
response function with the imaginary part
m
d
dt
d
dt
x F e F Z eE
x t x e
j i j j j
i t
j j
j j
i t
2
2
2
? ?
F
H
G
I
K
J
? ? ?
?
?
?
? ?
?
?
? ?
?
?
,
( )
steady state
electric dipole
examples of response
function:
p Z ex E
Z e
m i
j j j j
j
j
j j j
? ? ?
? ?
? ?
? ? ? ?
? ? ?
? ?
? ?
( )
( )
2 2
2 2
d i
Example: Response of charged (independent) oscillators
For the j-th oscillator (atom or molecule with Z bound charges),
• does not depend on any dynamic
detail of the interaction
• the necessary and sufficient
condition for its validity is causality
(1) ? ( ? ) has no pole above (including) x-axis.
(2) along (upper) infinite semi-circle
(3) ? ’ ( ? ) is even in ? , ? ’ ’ ( ? ) is odd in ? .
Some properties
of ? ( ? ):
d ?
? ?
?
( )
z
? 0
Collection of
oscillators
P
V
Z ex E
V
Z e
m i
nZ e m
i
j j
j
j
j j j
j
? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ? ?
? ?
?
? ?
? ?
1 1
2 2
2 2
2 2
0
2 2
( ) ( )
/
, (1)
d i
for identical
oscillators
 0
?
1 4 ? ? ? ? ?
(2)
? ?
?
?
?
? ?
?
?
?
?
?
?
?
?
? ?
?
?
?
( )
( )
'( )
''( )
''( ) ''( )
'( )
''( )
?
?
? ?
?
?
?
?
?
F
H
G
I
K
J
?
?
? ?
?
? ?
?
? ?
?
z
z
z z
z
1
1
1
2
0 0
2 2
0
i
P
s
s
ds
P
s
s
ds
i
P
s
s
ds
s
s
ds
s s
s
ds
property (3) used
Also,
From (1), (2), we have
( ?can be ? , or ? , or... )
? ?
?
?
?
?
''( )
'( )
? ?
?
?
z
2
2 2
0
s
s
ds (3)
A few sum rules: 2
0
0
?
?
?
''( )
'( )
s
s
ds
?
z
?
An example of the “ fluctuation-
dissipation ” relation
2
0
?
? ? ? ?
?
'( ) lim ''( ) s ds
?
? ?
z
= , (4)
From (3), ? >>1
(Prob. 4):
From (1) and (2),
? >>1 (Prob. 2):
2 1
0
2 2
?
? s s ds
V
f f
Z e
m
j
j
j
j
''( )
?
z
?
? = ,
Thomas-Reiche-
Kuhn sum rule
and many more …, e.g.,
(4)+(5) (next page):
? ? '( ) s ds
p
?
?
z
1
8
2
0
?
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