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

Indirect Conduction Bands: The indirect band gap alignment is not critical to the 
optical absorption in this work. The band alignment has been shown in Fig 2.14. The 
indirect conduction band contour is shown in Fig 2.12 for SiGe strained layers grown 
on SiGe relaxed buffer layers with different Ge composition in each layers. 

---

 
 
 
33 
Effective Masses: The effective masses of Si
1-x
Ge
x
are linearly interpolated between 
the values for Si and Ge. Their values along the growth direction at the Г point are 
0.041m
o
+0.115(1-x)m
o 
[50]
, 
0.28m
o
+0.21(1-x)m
o
[58], and 0.044+0.116(1-x)m
o
[58] 
for the electron, heavy hole, and light hole respectively and m
o
is the electron rest 
mass. It should be noted that there is an uncertainty in the electron effective mass at 
the zone center where fewer experimental studies have been done for silicon.
It can be seen that all the calculations are based on linear interpolation, excluding he 
strain effect for the conduction band. However, previous simulation [12] has shown 
that the uncertainties from strain have only negligible effect on the quantum well 
energy shift, because of the large conduction band offset in the direct bandgap. 
2.4.3 Previous Simulation 
Theoretical calculation and simulation of band alignment were previously done in 
order to understand how the design parameters in the quantum well structure impact 
the performance of the material [12]. 
The simulation procedure includes three steps [66]: (i) first design the potential 
line-up of the quantum well and divide it into small slices along the growth direction, 
(ii) build up carrier transfer matrices for each slice and heterojunction based on the 
electric field, well/barrier thicknesses, and band alignment as well as carrier effective 
masses, (iii) multiply the transfer matrixes and then extract the tunneling resonance 
energy under different electric fields.
Simulation shows the following trend in quantum well design: (i) the quantum well 
energies of the electron and heavy hole are lowered by increasing electric field, (ii) 
quantum well thickness affects the transition energy and stark shift significantly, (iii) 
barrier and buffer composition have weak impact on QCSE (iv) conduction band 
offset does not affect the quantum well energy very much. 

---

 
 
 
34 
With the conclusions above, we finalize our design to be: Si
0.1
Ge
0.9 
buffer layer and 
Si
0.15
Ge
0.85
barrier with pure Ge quantum wells. The quantum well thickness is 10 nm, 
and the barrier thickness is 16.5nm.