98
Strain-Engineered MOSFETs
and LH (J = 3/2, M J = ±1/2) bands at the Γ point. The degeneracy between
HH and LH bands is lifted due to application of strain on silicon crystal
[27]. The Γ point at k = 0 has the full crystal symmetry. Uniaxial stress
breaks the crystal symmetry by shortening (under compressive stress) and
elongating (under tensile stress) and thus HH and LH bands are lifted.
The HH and LH bands have negligible warping because they do not mix
since the rotation symmetry is unchanged. Longitudinal uniaxial stress
along <110> destroys the crystal symmetry more because the <100> axis
has higher symmetry than the <110> axis. The lifting of energy bands in
<100> and <110> directions is shown in Figure 4.7(a) and (b), respectively.
VB-lifting: Due to the introduction of strain, the valence band warping for
unstrained Si, Si under uniaxial compression is shown in Figure 4.7. The
Si conduction band edges are located along the Δ valley and are sixfold
degenerate.
Due to the strain induced by the uniaxial stress, the sixfold degenerate Δ
valley splits into two valleys, i.e., Δ 2 and Δ 4 valleys, in either case. The splitting of energy and the ordering of the Δ 2 and Δ 4 valleys depend on the type
and magnitude of the stress. Bir and Pikus [21] proposed another model for
the strain-induced change in the energy of carrier subvalleys in silicon where
they ignore the shear strain for electrons and suggest nonlinear dependence
for holes:
,
1 1
2 2
3 3
C i
d
U ii
(
)
Ε = Ξ ′
ε + ′
ε + ′
ε +Ξ ′
ε
(4.16)
,
1 1
2 2
3 3
a
V i
(
)
Ε = − ′
ε + ′
ε + ′
ε ±δΕ
(4.17)
where
((
) (
) (
) )
(
)
2
11
22
2
22
33
2
11
33
2
2
2
11
2
22
2
33
2
d
b
δΕ =
′
ε − ′
ε
+ ′
ε − ′
ε
+ ′
ε − ′
ε
+
′
ε + ′
ε + ′
ε , and
Ξ d , Ξ u , a, b, and d are other deformation potentials that correspond to the Bir
and Pikus model. The sign ± separates heavy-hole and light-hole subvalleys
of silicon in Equation (4.17). The above deformation potentials were used in
FIGURE 4.7
Simplified hole valance band structure for uniaxial strain in (a) <100> and (b) <110> directions.
Strain-Engineered MOSFETs
and LH (J = 3/2, M J = ±1/2) bands at the Γ point. The degeneracy between
HH and LH bands is lifted due to application of strain on silicon crystal
[27]. The Γ point at k = 0 has the full crystal symmetry. Uniaxial stress
breaks the crystal symmetry by shortening (under compressive stress) and
elongating (under tensile stress) and thus HH and LH bands are lifted.
The HH and LH bands have negligible warping because they do not mix
since the rotation symmetry is unchanged. Longitudinal uniaxial stress
along <110> destroys the crystal symmetry more because the <100> axis
has higher symmetry than the <110> axis. The lifting of energy bands in
<100> and <110> directions is shown in Figure 4.7(a) and (b), respectively.
VB-lifting: Due to the introduction of strain, the valence band warping for
unstrained Si, Si under uniaxial compression is shown in Figure 4.7. The
Si conduction band edges are located along the Δ valley and are sixfold
degenerate.
Due to the strain induced by the uniaxial stress, the sixfold degenerate Δ
valley splits into two valleys, i.e., Δ 2 and Δ 4 valleys, in either case. The splitting of energy and the ordering of the Δ 2 and Δ 4 valleys depend on the type
and magnitude of the stress. Bir and Pikus [21] proposed another model for
the strain-induced change in the energy of carrier subvalleys in silicon where
they ignore the shear strain for electrons and suggest nonlinear dependence
for holes:
,
1 1
2 2
3 3
C i
d
U ii
(
)
Ε = Ξ ′
ε + ′
ε + ′
ε +Ξ ′
ε
(4.16)
,
1 1
2 2
3 3
a
V i
(
)
Ε = − ′
ε + ′
ε + ′
ε ±δΕ
(4.17)
where
((
) (
) (
) )
(
)
2
11
22
2
22
33
2
11
33
2
2
2
11
2
22
2
33
2
d
b
δΕ =
′
ε − ′
ε
+ ′
ε − ′
ε
+ ′
ε − ′
ε
+
′
ε + ′
ε + ′
ε , and
Ξ d , Ξ u , a, b, and d are other deformation potentials that correspond to the Bir
and Pikus model. The sign ± separates heavy-hole and light-hole subvalleys
of silicon in Equation (4.17). The above deformation potentials were used in
FIGURE 4.7
Simplified hole valance band structure for uniaxial strain in (a) <100> and (b) <110> directions.