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The electron-LA phonon interaction is an interaction that can take place between an electron and a longitudinal acoustic (LA) phonon in a material such as a semiconductor .
Displacement operator of the LA phonon [ ]
The equations of motion 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
)
{\displaystyle M{\frac {d^{2}}{dt^{2}}}u_{n}=-k_{0}(u_{n-1}+u_{n+1}-2u_{n})}
,
where
u
n
{\displaystyle u_{n}}
is the displacement of the n th atom from their equilibrium positions.
Defining the displacement
u
ℓ
{\displaystyle u_{\ell }}
of the
ℓ
{\displaystyle \ell }
th atom by
u
ℓ
=
x
ℓ
−
ℓ
a
{\displaystyle u_{\ell }=x_{\ell }-\ell a}
, where
x
ℓ
{\displaystyle x_{\ell }}
is the coordinates of the
ℓ
{\displaystyle \ell }
th atom and
a
{\displaystyle a}
is the lattice constant ,
the displacement is given by
u
l
=
A
e
i
(
q
ℓ
a
−
ω
t
)
{\displaystyle u_{l}=Ae^{i(q\ell a-\omega t)}}
Then using Fourier transform :
Q
q
=
1
N
∑
ℓ
u
ℓ
e
−
i
q
a
ℓ
{\displaystyle Q_{q}={\frac {1}{\sqrt {N}}}\sum _{\ell }u_{\ell }e^{-iqa\ell }}
and
u
ℓ
=
1
N
∑
q
Q
q
e
i
q
a
ℓ
{\displaystyle u_{\ell }={\frac {1}{\sqrt {N}}}\sum _{q}Q_{q}e^{iqa\ell }}
.
Since
u
ℓ
{\displaystyle u_{\ell }}
is a Hermite operator,
u
ℓ
=
1
2
N
∑
q
(
Q
q
e
i
q
a
ℓ
+
Q
q
†
e
−
i
q
a
ℓ
)
{\displaystyle u_{\ell }={\frac {1}{2{\sqrt {N}}}}\sum _{q}(Q_{q}e^{iqa\ell }+Q_{q}^{\dagger }e^{-iqa\ell })}
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
)
{\displaystyle a_{q}^{\dagger }={\frac {q}{\sqrt {2M\hbar \omega _{q}}}}(M\omega _{q}Q_{-q}-iP_{q}),\;a_{q}={\frac {q}{\sqrt {2M\hbar \omega _{q}}}}(M\omega _{q}Q_{-q}+iP_{q})}
Q
q
{\displaystyle Q_{q}}
is written as
Q
q
=
ℏ
2
M
ω
q
(
a
−
q
†
+
a
q
)
{\displaystyle Q_{q}={\sqrt {\frac {\hbar }{2M\omega _{q}}}}(a_{-q}^{\dagger }+a_{q})}
Then
u
ℓ
{\displaystyle u_{\ell }}
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
)
{\displaystyle u_{\ell }=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(a_{q}e^{iqa\ell }+a_{q}^{\dagger }e^{-iqa\ell })}
Hence, using the continuum model, the displacement operator 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
]
{\displaystyle u(r)=\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}e_{q}[a_{q}e^{iq\cdot r}+a_{q}^{\dagger }e^{-iq\cdot r}]}
,
where
e
q
{\displaystyle e_{q}}
is the unit vector along the displacement direction.
Interaction Hamiltonian [ ]
The electron-longitudinal acoustic phonon interaction Hamiltonian is defined as
H
el
{\displaystyle H_{\text{el}}}
H
el
=
D
ac
δ
V
V
=
D
ac
÷
u
(
r
)
{\displaystyle H_{\text{el}}=D_{\text{ac}}{\frac {\delta V}{V}}=D_{\text{ac}}\,\div \,u(r)}
,
where
D
ac
{\displaystyle D_{\text{ac}}}
is the deformation potential for electron scattering by acoustic phonons .[1]
Inserting the displacement vector to the Hamiltonian results to
H
el
=
D
ac
∑
q
ℏ
2
M
N
ω
q
(
i
e
q
⋅
q
)
[
a
q
e
i
q
⋅
r
−
a
q
†
e
−
i
q
⋅
r
]
{\displaystyle H_{\text{el}}=D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q)[a_{q}e^{iq\cdot r}-a_{q}^{\dagger }e^{-iq\cdot r}]}
Scattering probability [ ]
The scattering probability for electrons from
|
k
⟩
{\displaystyle |k\rangle }
to
|
k
′
⟩
{\displaystyle |k'\rangle }
states is
P
(
k
,
k
′
)
=
2
π
ℏ
∣
⟨
k
′
,
q
′
|
H
el
|
k
,
q
⟩
∣
2
δ
[
ε
(
k
′
)
−
ε
(
k
)
∓
ℏ
ω
q
]
{\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\mid \langle k',q'|H_{\text{el}}|\ k,q\rangle \mid ^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}
=
2
π
ℏ
|
D
ac
∑
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
]
{\displaystyle ={\frac {2\pi }{\hbar }}\left|D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}(ie_{q}\cdot q){\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,{\frac {1}{L^{3}}}\int d^{3}r\,u_{k'}^{\ast }(r)u_{k}(r)e^{i(k-k'\pm q)\cdot r}\right|^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}]}
Replace the integral over the whole space with a summation of unit cell integrations
P
(
k
,
k
′
)
=
2
π
ℏ
(
D
ac
∑
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
]
,
{\displaystyle P(k,k')={\frac {2\pi }{\hbar }}\left(D_{\text{ac}}\sum _{q}{\sqrt {\frac {\hbar }{2MN\omega _{q}}}}|q|{\sqrt {n_{q}+{\frac {1}{2}}\mp {\frac {1}{2}}}}\,I(k,k')\delta _{k',k\pm q}\right)^{2}\delta [\varepsilon (k')-\varepsilon (k)\mp \hbar \omega _{q}],}
where
I
(
k
,
k
′
)
=
Ω
∫
Ω
d
3
r
u
k
′
∗
(
r
)
u
k
(
r
)
{\displaystyle I(k,k')=\Omega \int _{\Omega }d^{3}r\,u_{k'}^{\ast }(r)u_{k}(r)}
,
Ω
{\displaystyle \Omega }
is the volume of a unit cell .
P
(
k
,
k
′
)
=
{
2
π
ℏ
D
ac
2
ℏ
2
M
N
ω
q
|
q
|
2
n
q
(
k
′
=
k
+
q
;
absorption
)
,
2
π
ℏ
D
ac
2
ℏ
2
M
N
ω
q
|
q
|
2
(
n
q
+
1
)
(
k
′
=
k
−
q
;
emission
)
.
{\displaystyle P(k,k')={\begin{cases}{\frac {2\pi }{\hbar }}D_{\text{ac}}^{2}{\frac {\hbar }{2MN\omega _{q}}}|q|^{2}n_{q}&(k'=k+q;{\text{absorption}}),\\{\frac {2\pi }{\hbar }}D_{\text{ac}}^{2}{\frac {\hbar }{2MN\omega _{q}}}|q|^{2}(n_{q}+1)&(k'=k-q;{\text{emission}}).\end{cases}}}
See also [ ]
Notes [ ]
References [ ]