106
Strain-Engineered MOSFETs
Therefore, the local dilation Δr is given by
( )
. ( )
( )
r
V
V
V
u
u
u
x
u r
i
i
ii
i
i
i
i
∑ ∑ ∑
=
′ −
=
=
=
∂
∂
=
(4.40)
The above-mentioned local dilation relation is equivalent to displacement
of the atoms, and hence equivalent to a local change of lattice parameter.
Therefore, it induces a modification of both bands: (1) conduction band (E C )
and (2) valance band (E V ). The interaction potential H e ph
AC
−
of the acoustic phonon with the lattice depends on the variation of the conduction band and
valance band edges. Since phonons deform the crystal in three dimensions,
for small stress and for an isotropic crystal, interaction potential is given by
. ( )
H
V
V
u r
e ph
AC
= Ξ
δ = Ξ
−
(4.41)
where Ξ is the so-called deformation potential. The lattice displacement u(r)
for long wavelength phonon is given by [34]
( )
2
1/2
.
†
.
u r
w
ae
a e
q
q
q
q
iq r
q
iq r
∑
(
)
=
ρω
+
( )
(
)
−
(4.42)
where ρ is the semiconductor density and w q
is the polarisation vector. For
longitudinal phonons, the polarisation vector is
ˆ
w
q
q
q
q
= = . Thus the acoustic deformation potential is written as
. ( )
( . ) 2
2
1/2
.
†
.
H
V
V
u r i
w q
ae
a e
e ph
AC
q
q
q
q
iq r
q
iq r
∑
(
)
= Ξ
δ = Ξ
=
Ξ
ρω
−
( )
(
)
−
−
(4.43)
4.9.3 Transition Probability for Acoustic Phonon Scattering
The scattering rate of electrons or holes by lattice vibration in the presence of strain may be explained in terms of the corpuscular model with
the aid of the phonon concept. Charge carriers colliding with phonons
exchange energy and quasi-momentum with it. Since the number of phonons depends on temperature, the charge scattering should be temperature dependent. However, in order to calculate the quantum transitions
of electrons and holes from state to state, the perturbation generated by
deformation potential due to strain should be applied. Once we have the
acoustic deformation potential, one may obtain the scattering rate using
Fermi’s golden rule:
2
(
)
2
f H
i
E E
i f
e ph
AC
f
i
Γ
=
π
δ
−
→
−
(4.44)
Strain-Engineered MOSFETs
Therefore, the local dilation Δr is given by
( )
. ( )
( )
r
V
V
V
u
u
u
x
u r
i
i
ii
i
i
i
i
∑ ∑ ∑
=
′ −
=
=
=
∂
∂
=
(4.40)
The above-mentioned local dilation relation is equivalent to displacement
of the atoms, and hence equivalent to a local change of lattice parameter.
Therefore, it induces a modification of both bands: (1) conduction band (E C )
and (2) valance band (E V ). The interaction potential H e ph
AC
−
of the acoustic phonon with the lattice depends on the variation of the conduction band and
valance band edges. Since phonons deform the crystal in three dimensions,
for small stress and for an isotropic crystal, interaction potential is given by
. ( )
H
V
V
u r
e ph
AC
= Ξ
δ = Ξ
−
(4.41)
where Ξ is the so-called deformation potential. The lattice displacement u(r)
for long wavelength phonon is given by [34]
( )
2
1/2
.
†
.
u r
w
ae
a e
q
q
q
q
iq r
q
iq r
∑
(
)
=
ρω
+
( )
(
)
−
(4.42)
where ρ is the semiconductor density and w q
is the polarisation vector. For
longitudinal phonons, the polarisation vector is
ˆ
w
q
q
q
q
= = . Thus the acoustic deformation potential is written as
. ( )
( . ) 2
2
1/2
.
†
.
H
V
V
u r i
w q
ae
a e
e ph
AC
q
q
q
q
iq r
q
iq r
∑
(
)
= Ξ
δ = Ξ
=
Ξ
ρω
−
( )
(
)
−
−
(4.43)
4.9.3 Transition Probability for Acoustic Phonon Scattering
The scattering rate of electrons or holes by lattice vibration in the presence of strain may be explained in terms of the corpuscular model with
the aid of the phonon concept. Charge carriers colliding with phonons
exchange energy and quasi-momentum with it. Since the number of phonons depends on temperature, the charge scattering should be temperature dependent. However, in order to calculate the quantum transitions
of electrons and holes from state to state, the perturbation generated by
deformation potential due to strain should be applied. Once we have the
acoustic deformation potential, one may obtain the scattering rate using
Fermi’s golden rule:
2
(
)
2
f H
i
E E
i f
e ph
AC
f
i
Γ
=
π
δ
−
→
−
(4.44)