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

2.4.2 Band Parameters 
Through previous band structure calculations [62], we calculate the band offset by 
using the following equations. 
Valence Bands: The valence band offsets for a strained Si
1-x
Ge
x
layer on a relaxed 
Si
1-y
Ge
y
layer can be expressed as [62-64]: 
 



,
0.74 0.07
hh
E
x y
y x
y




(2.16) 
 








2
3
2
3
2
3
2
3
2
3
2
,
0.3
0.289
0.142
0.683 2.58
3.21
1.24
0.354 3.77
8.79
2.46
0.435 0.704
2.439
1.295
1 2.7
28.1
lh
E
x y
y
y
y
y
y
y
x
y
y
y
y
y
y
x
x
y
y

 

















(2.17) 
if z > 0.5 and 
0.5
x z
 
, which covers the SiGe composition range of our interest.
The valence band energy positions of the strained Ge well and SiGe barrier 
relative to the relaxed Si
1-z
Ge
z
buffer can be calculated using Eq. (2.16) for the heavy 
hole and Eq. (2.17) for the light hole;, thus the offsets of the heavy hole and light hole 
valence bands, ΔE
v,hh
and ΔE
v,lh
, between the well and barrier can be extracted. 
Direct Conduction Bands: The direct band gap energies (with the relevant conduction 
minimum at the Г’
2 
point) of bulk Ge and Si are 0.8 eV and 4.175 eV respectively at 
room temperature [65]. The direct band gap energy of SiGe is linearly interpolated 
between the value of bulk Ge and Si here, thus the band offset between the Ge well 
and Si
1-x
Ge
x
barrier can be expressed as
h h
v
c
E
x
E
,
,
)
8
.
0
175
.
4
(






. 
(2.10) 
This interpolation does not consider the strain effect on the conduction band.