Electron-longitudinal acoustic phonon interaction is an equation concerning atoms.
The equation of motions of the atoms of mass M which locates in the periodic lattice is
M
d
2
d
t
2
u
n
=
−
k
0
(
u
n
−
1
+
u
n
+
1
−
2
u
n
)
,
where
u
n
is the displacement of the nth atom from their equilibrium positions.
If we define the displacement
u
l
of the nth atom by
u
l
=
x
l
−
l
a
, where
x
l
is the coordinates of the lth atom and a is the lattice size,
the displacement is given by
u
n
=
A
e
i
q
l
a
−
ω
t
Using Fourier transform, we can define
Q
q
=
1
N
∑
l
u
l
e
−
i
q
a
l
and
u
l
=
1
N
∑
q
Q
q
e
i
q
a
l
.
Since
u
l
is a Hermite operator,
u
l
=
1
2
N
∑
q
(
Q
q
e
i
q
a
l
+
Q
q
†
e
−
i
q
a
l
)
From the definition of the creation and annihilation operator
a
q
†
=
q
2
M
ℏ
ω
q
(
M
ω
q
Q
−
q
−
i
P
q
)
,
a
q
=
q
2
M
ℏ
ω
q
(
M
ω
q
Q
−
q
+
i
P
q
)
Q
q
is written as
Q
q
=
ℏ
2
M
ω
q
(
a
−
q
†
+
a
q
)
Then
u
l
expressed as
u
l
=
∑
q
ℏ
2
M
N
ω
q
(
a
q
e
i
q
a
l
+
a
q
†
e
−
i
q
a
l
)
Hence, when we use continuum model, the displacement for the 3-dimensional case is
u
(
r
)
=
∑
q
ℏ
2
M
N
ω
q
e
q
[
a
q
e
i
q
⋅
r
+
a
q
†
e
−
i
q
⋅
r
]
,
where
e
q
is the unit vector along the displacement direction.
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
H
e
l
H
e
l
=
D
a
c
δ
V
V
=
D
a
c
d
i
v
u
(
r
)
,
where
D
a
c
is the deformation potential for electron scattering by acoustic phonons.
Inserting the displacement vector to the Hamiltonian results to
H
e
l
=
D
a
c
∑
q
ℏ
2
M
N
ω
q
(
i
e
q
⋅
q
)
[
a
q
e
i
q
⋅
r
−
a
q
†
e
−
i
q
⋅
r
]
The scattering probability for electrons from
|
k
⟩
to
|
k
′
⟩
states is
P
(
k
,
k
′
)
=
2
π
ℏ
∣
⟨
k
′
,
q
′
|
H
e
l
|
k
,
q
⟩
∣
2
δ
[
ε
(
k
′
)
−
ε
(
k
)
∓
ℏ
ω
q
]
=
2
π
ℏ
|
D
a
c
∑
q
ℏ
2
M
N
ω
q
(
i
e
q
⋅
q
)
n
q
+
1
2
∓
1
2
1
L
3
∫
d
3
r
u
k
′
∗
(
r
)
u
k
(
r
)
e
i
(
k
−
k
′
±
q
)
⋅
r
|
2
δ
[
ε
(
k
′
)
−
ε
(
k
)
∓
ℏ
ω
q
]
Replace the integral over the whole space with a summation of unit cell integrations
P
(
k
,
k
′
)
=
2
π
ℏ
(
D
a
c
∑
q
ℏ
2
M
N
ω
q
|
q
|
n
q
+
1
2
∓
1
2
I
(
k
,
k
′
)
δ
k
′
,
k
±
q
)
2
δ
[
ε
(
k
′
)
−
ε
(
k
)
∓
ℏ
ω
q
]
,
where
I
(
k
,
k
′
)
=
Ω
∫
Ω
d
3
r
u
k
′
∗
(
r
)
u
k
(
r
)
,
Ω
is the volume of a unit cell.
P
(
k
,
k
′
)
=
{
2
π
ℏ
D
a
c
2
ℏ
2
M
N
ω
q
|
q
|
2
n
q
(
k
′
=
k
+
q
;
absorption
)
,
2
π
ℏ
D
a
c
2
ℏ
2
M
N
ω
q
|
q
|
2
(
n
q
+
1
)
(
k
′
=
k
−
q
;
emission
)
.
