High speed, low driving voltage vertical cavity germanium-silicon modulators for optical

2.1.4 Franz-Keldysh Effect 
The change in absorption in the semiconductor with a strong electric field applied is 
called the Franz-Keldysh effect [30-31]. Under applied bias, the wavefunctions of 
electrons and holes turn from oscillatory to decaying behavior, which is declined by 
Airy function. In the absence of a photon, the valence electron has to tunnel through a 
triangular barrier of height E
g
and thickness d, given by d= E
g
/qE. With the assistance 
of an absorbed photon with energy less than the band-gap energy, the tunneling barrier 
thickness is reduced to d*= (E
g
-hω)/qE, the overlap of the wavefunctions increases 
further and the valence electron can tunnel more easily to the conduction band. The 
whole process absorbs a photon with energy less than the band gap while providing 
conservation of momentum. The relationship of the absorption coefficient and the 
electric-field can be written as follows: [50] 

---

 
 
 
17 
)
3
4
e x p (
8
1
)
'
(
2
3
2
1





E
K
(2.8) 
Where 
'
)
2
(
'
3
1
*
2
2
2
E
E
m
E
q
E
g
r







(2.9) 
And K is a material-dependent parameter that has the value of 5×10
4
cm
-1
(eV)
-1/2
in 
GaAs. From a quantitative examination of the various terms in equation 2.8, the 
exponential term is the transmission coefficient of an electron through a triangular 
barrier and can be obtained from the Wentzel-Kramers-Brillouin (WKB) 
approximation. The other factors are related to the upward transition of an electron 
due to photon absorption. Based on that, in GaAs or SiGe materials, the magnitude of 
absorption is relatively small unless the electric field is higher than 10
5
V/cm [12]; 
thus it is not an efficient modulation mechanism.