Optical processes and excitons
•Dielectric function and reflectance
•Kramers-Kronig relations
•Excitons
•Frenkel exciton
•Mott-Wannier exciton
•Raman spectroscopy
Dept of Phys
M.C. Chang
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)
E '
r () 
E
Prob.3
n 1
( )
r () 
 R () ei
n 1
Eei ( kx t )
E ' ei ( kx t )
where the refractive index n()  
( )
It is easier to measure R than to measure 
 measure R() for  () (with the help of KK relations)
 n()

()
Kramers-Kronig relations (1926)
KK relation connects real part of the
response function with the imaginary part
examples of response
function:
j 
( ) E
Ohm’
s law
P 
( ) E
polarization
•does not depend on any dynamic
detail of the interaction
E ' r () E reflective EM wave
•the necessary and sufficient
condition for its validity is causality
Example: Response of charged (independent) oscillators
For the j-th oscillator (atom or molecule with Z bound charges),
Fd
mG
H dt
2
j
2
i
IJ
K
d
2j x j Fje it ,
dt
x j (t ) x je it
Fj Z j eE
steady state
p j Z j ex j j () E
electric dipole
Z 2j e 2
 j () 
m j 2j 2 i
j
d
i
Figs from http://www.physics.uc.edu/~sitko/LightColor/10-Optics1/optics1.htm
Collection of
oscillators
Z 2j e 2
1
1
P  Z j ex j 
( ) E  
( ) 
, (1)
V j
V j m j 2j 2 i
j
d
2 2
nZ e / m
for identical
 2
oscillators
0 2 i
1 4
Some properties
of ():
0
(1) () has no pole above (including) x-axis.
( )
(2) d 
0 along (upper) infinite semi-circle

(3) ’
() is even in , ’
’
() is odd in .
z
i
1
( ) P
From (1), (2), we have 
i
(can be , or , or... )
z
z
( s)
ds
 s 


 ' ' ( s)
1
 '()  P
ds
  s 
' '( s)
' ' ( s)
1
 P
ds 
ds
0 s
0 s
i


2  s' ' ( s)
' () 
ds (2)
 0 s2 2
z
Also,
FG z
H
2
' ' () 

z
z

0
IJ
K
ω
property (3) used
'( s)
ds (3)
s2 2
z
z
z
A few sum rules:
2 ' ' ( s)
ds ' (0)
0

s
An example of the “
fluctuationdissipation”relation
From (1) and (2),
>>1 (Prob. 2):
2 
1
s' ' ( s)ds = f j ,
0
V j
Z 2j e 2 Thomas-Reichefj 
m Kuhn sum rule
From (3), >>1
(Prob. 4):
and many more…, e.g.,
(4)+(5) (next page):
2 
' ( s) ds = lim ' ' (), (4)
0


z

0
1 2
' ( s)ds  p
8
By Ferrel and Glover (1958)
KK relation for reflection E ' r () E
ln r () ln R1/ 2 () i
( )
Why take log? See Yu and
Cardona for related discussion
  ln R( s)
then 
( )   2
ds
 0 s 2
1  s  d ln R
 ln
ds
2 0
s  ds
•constant R(s) doesn’
t contribute
•s>>, s<<don’
t contribute
ref: Cardona and Yu
Conductivity of free electron gas (j=0, with little damping)
1
e2

( ) 
V j m 2j 2 i
j
d
i
LM
N
ne 2
1
ne 2 1
 
( ) =  lim

i
( )
m0 i
m 
OP
Q
(KK relations can be
checked easily in this case)
2p

( ) 1 4
( ) 1  2 1 i
( )

Recall that
(chap 10)
=1+
4
i

 ne 2
 
1 
4i
m
Real part of DC
conductivity diverges
(compare with ne2
/m)
i



(
)

, (5)




Drude weight D
Other sum rules (Mahan, p358):
For a clean conductor at T =0
(or superconductor),
1 
 2
s
Im
ds


p



0
( s) 
2


( ) D
( ) reg (),
1 1 

Im
ds




0 s
( s ) 
2


then D lim ''()



Related to the energy loss of
a passing charged particle
Dielectric function and the semiconductor energy gap (prob.5)
band gap absorption (due to
nonzero 
’
’
) can be very
roughly approximated by

''()  
( )

p2

(g )
2
2
p
p2
 '()=1  2
 g2
p2
 '(0)=1  2
g
Si

’
Ge
GaAs InP GaP
12.0 16.0 10.9
Eg (theo) 4.8
4.3
5.2
Ex (exp’
t) 4.44 4.49 5.11
(ref: Cardona and Yu)
Figs from Ibach and Luth
9.6
9.1
5.2
5.75
5.05 5.21
Interband absorption
for GaAs at 21 K
Tight bound Frenkel exciton:
(excited state of a single atom)
•alkali halide crystals
•crystalline inert gases
Excitons (e-h pairs)
in insulators
Weakly bound Mott-Wannier excitons
Similar to the impurity states in
doped semiconductor:
e 4
En E g  2 2 2 ,
2n
1
1
1
 
 me mh
CuO2
Raman effect in crystals
I ()  nk 1 u nk
2
nk 1
I ()  nk 1 u nk
2
nk
I ()
n
 k e / k B T
I () nk 1
1930