107
Electronic Properties of Strain-Engineered Semiconductors
where f and i refer to final and initial states. The energy of lattice vibrations
under electrons or holes phonon interaction may change by the creation or
annihilation of a phonon. Hence, in collisions the initial and final energies of
the electron/hole-phonon system are
1
E E n
E E
n
n n
i
k
q
q
f
k
q
q
q
q
= +
ω
=
+ ′ ω
′ = ±
′
(4.45)
Hence, transition probability for the state E k to E k
/ in an electron/hole-phonon
collision involving a phonon in the wave vector q is
2
(
)
2
k H
k
E E
k k
e ph
AC
k
k
q
Γ
=
π ′
δ
−
± ω
→ ′
−
′
(4.46)
4.9.4 Strain-Induced Scattering Matrix
If
( )
r
k
ψ
is the eigenfunction for the unstrained condition and
( , )
r
k
ψ
ε is the
eigenfunction for the strained condition, then
( , )
( , )
( )
( )
.
r
u r e
r
r
k
ik r
ψ ε =
ε
= ψ + δψ
(4.47)
Now the scattering matrix elements for long-wavelength acoustic phonon
scattering in isotropic material are given as [28]
( , )
( . ) 2
2
1/2
* *
.
†
.
k k
k H
k
i
w q
a e
a e
d V
e ph
AC
q
q
q
k n
q
iq r
q
iq r
k n
q
q
∑
∫
(
)
ℑ ′ = ′
=
Ξ
ρω
ψ φ
−
ψφ
( )
(
)
−
′
′
−
(4.48)
Now, we consider the following:
( )
,
()
.
*
.
u r e
u r e
k
k
ik r
k
k
ik r
ψ =
ψ = ′
− ′
(4.49)
,
(
1)
†
1
1
a
n
a
n
q n
q n
q n
q
n
q
q
q
q
φ =
φ
φ =
+ φ
−
+
(4.50)
where
†
a q and a q are the creation and annihilation operators, and k
ψ and q
φ
are the wave functions of the electron and lattice vibration mode of vector q
.
Now substituting Equations (4.49) and (4.50) in Equation (4.48), one obtains
( , )
( . ) 2
( )
( )
1
2
1/2
* *
1
(
) .
* *
1
(
) .
k k i
w q
u
u r
n e
u
u r
n
e
dV
q
q
q
k n k
n
q
i k q k r
k n k
n
q
i k q k r
q
q
q
q
∑
∫
ℑ ′ =
Ξ
ρω
φ
φ
− φ
φ
+
{
}
{
}
′
′
−
+ − ′
′
′
+
− − ′
(4.51)
Electronic Properties of Strain-Engineered Semiconductors
where f and i refer to final and initial states. The energy of lattice vibrations
under electrons or holes phonon interaction may change by the creation or
annihilation of a phonon. Hence, in collisions the initial and final energies of
the electron/hole-phonon system are
1
E E n
E E
n
n n
i
k
q
q
f
k
q
q
q
q
= +
ω
=
+ ′ ω
′ = ±
′
(4.45)
Hence, transition probability for the state E k to E k
/ in an electron/hole-phonon
collision involving a phonon in the wave vector q is
2
(
)
2
k H
k
E E
k k
e ph
AC
k
k
q
Γ
=
π ′
δ
−
± ω
→ ′
−
′
(4.46)
4.9.4 Strain-Induced Scattering Matrix
If
( )
r
k
ψ
is the eigenfunction for the unstrained condition and
( , )
r
k
ψ
ε is the
eigenfunction for the strained condition, then
( , )
( , )
( )
( )
.
r
u r e
r
r
k
ik r
ψ ε =
ε
= ψ + δψ
(4.47)
Now the scattering matrix elements for long-wavelength acoustic phonon
scattering in isotropic material are given as [28]
( , )
( . ) 2
2
1/2
* *
.
†
.
k k
k H
k
i
w q
a e
a e
d V
e ph
AC
q
q
q
k n
q
iq r
q
iq r
k n
q
q
∑
∫
(
)
ℑ ′ = ′
=
Ξ
ρω
ψ φ
−
ψφ
( )
(
)
−
′
′
−
(4.48)
Now, we consider the following:
( )
,
()
.
*
.
u r e
u r e
k
k
ik r
k
k
ik r
ψ =
ψ = ′
− ′
(4.49)
,
(
1)
†
1
1
a
n
a
n
q n
q n
q n
q
n
q
q
q
q
φ =
φ
φ =
+ φ
−
+
(4.50)
where
†
a q and a q are the creation and annihilation operators, and k
ψ and q
φ
are the wave functions of the electron and lattice vibration mode of vector q
.
Now substituting Equations (4.49) and (4.50) in Equation (4.48), one obtains
( , )
( . ) 2
( )
( )
1
2
1/2
* *
1
(
) .
* *
1
(
) .
k k i
w q
u
u r
n e
u
u r
n
e
dV
q
q
q
k n k
n
q
i k q k r
k n k
n
q
i k q k r
q
q
q
q
∑
∫
ℑ ′ =
Ξ
ρω
φ
φ
− φ
φ
+
{
}
{
}
′
′
−
+ − ′
′
′
+
− − ′
(4.51)