1. Optical processes and excitons
Dielectric function and reflectance
Kramers-Kronig relations
Excitons
Frenkel exciton
Mott-Wannier exciton
Raman spectroscopy
M.C. Chang
Dept of Phys

2. 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)
Prob.3
It is easier to measure R than to measure 
 measure R() for   () (with the help of KK relations)
 n()
 ()

3. Kramers-Kronig relations (1926)
examples of response function:
KK relation connects real part of the response function with the imaginary part
does not depend on any dynamic detail of the interaction
the necessary and sufficient condition for its validity is causality
Ohm’s law
polarization
reflective EM wave
Example: Response of charged (independent) oscillators
For the j-th oscillator (atom or molecule with Z bound charges),
steady state
electric dipole

4. Collection of oscillators
for identical oscillators
Figs from
(1) () has no pole above (including) x-axis.
(2) along (upper) infinite semi-circle
(3) ’() is even in , ’’() is odd in .
Some properties of ():

5. ω A few sum rules: From (1), (2), we have ( can be , or , or... )
property (3) used
Also,
A few sum rules:
An example of the “fluctuation-dissipation” relation
From (1) and (2), >>1 (Prob. 2):
Thomas-Reiche-Kuhn sum rule
From (3), >>1 (Prob. 4):
and many more…, e.g., (4)+(5) (next page):
By Ferrel and Glover (1958)

6. KK relation for reflection
Why take log? See Yu and Cardona for related discussion
constant R(s) doesn’t contribute
s>>, s<< don’t contribute
ref: Cardona and Yu

7. Conductivity of free electron gas (j=0, with little damping)
(KK relations can be checked easily in this case)
Recall that (chap 10)
Real part of DC conductivity diverges (compare with ne2/m)
Drude weight D
Other sum rules (Mahan, p358):
Related to the energy loss of a passing charged particle

8. Dielectric function and the semiconductor energy gap (prob.5)
band gap absorption (due to nonzero ’’) can be very roughly approximated by
Si Ge GaAs InP GaP ’
Eg (theo)
Ex (exp’t)
(ref: Cardona and Yu)
Figs from Ibach and Luth

9. Interband absorption for GaAs at 21 K
Excitons (e-h pairs) in insulators
Tight bound Frenkel exciton:
(excited state of a single atom)
alkali halide crystals
crystalline inert gases

10. Weakly bound Mott-Wannier excitons
Similar to the impurity states in doped semiconductor:
CuO2

11. Raman effect in crystals
1930