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

2.2.2.2 Band Alignment in SiGe Heterostructures 
The SiGe quantum well structures developed in this work include thin layers of SiGe 
alloys with different Ge compositions. The large lattice constant mismatch between Si 
and Ge (4.2%) makes the different-composition SiGe layer strained. The strain can 
affect the band structure and the effective mass of electrons and holes and split the 
valence bands and Δ valleys. [53, 62, 64] 
When two semiconductors form a heterojunction, discontinuities can occur in the 
valence bands and in the conduction bands. If one of the semiconductors is coherently 
strained on the other, two additional effects on the band structure have to be 
considered: hydrostatic strain shifts the energy position of bands and uniaxial or 
biaxial strain splits degenerate bands. The total change in the band is expressed as 
a
h
s
E
E
E
E
    
(2.15) 
ΔE
a
stands for the alloy effect for the unstrained material, ΔE
h
represents the shift 
due to hydrostatic strain and ΔE
s
is the possible splitting due to uniaxial strain. The 
hydrostatic tensile (or compressive) strain lowers (or raises) all conduction bands and 
raises (or lowers) all valence bands. On the other hand, the uniaxial stress does not 
affect the average band energies, it only breaks the degeneracy of the valence bands. 
The valence band splittings in heavy hole (HH), light hole (LH), and spin-orbit-split 

---

 
 
 
26 
hole (SO) bands are determined by using the values for the spin–orbit splittings and in 
the strained Si
1−x
Ge
x
layers. The Δ bands split into the four equivalent in-plane valleys 
Δ
4
and the two valleys along the growth direction Δ
2
.
The band gap energy associated with the Δ valley might decrease or increase with 
the biaxial strain due to the complex conduction band structure, but the band gap 
energy associated with the L or Г valley (more relevant to our interest in Ge-rich SiGe 
structures) would increase (or decrease) with the compressive (or tensile) biaxial 
strain. 
(a)
(b)
Figure 2.10: (a) heteroepitaxy of strained Si
1-x
Ge
x
layer on relaxed Si
1-y
Ge
y
buffer. (b) Typical band 
alignment (when x>y). 
In this work, we are focusing on a strained Si
1-x
Ge
x
layer deposited on a relaxed 
Si
1-y
Ge
y
buffer, as shown in Fig 2.10 (a) above. Fig 2.10(b) shows the band alignment. 
Previous work [12] shows that most of the band offset is in the valence band and the 
valence band maximum is always in the SiGe layer with higher Ge concentration. That 
indicates weak electron quantum confinement, which is not favorable for quantum 
well device design. Therefore further band engineering needs to be done.