108
Strain-Engineered MOSFETs
4.9.5 Relaxation Time for Acoustic Phonon Scattering
The well-known relation to describe relaxation time is given by [35]
1
2
1
cos
cos
3
V
k
k
dk
k k
∫
( )
τ
= π
−
′
′
θ
θ

 

  Γ
′
→ ′
(4.52)
Replacing and combining Equations (4.46), (4.48), and (4.52), we get the
following:
1
1
(2 )
(
1)
1
cos
cos
3
2
2
2
n
q E
E
n q E
E
k
k
dk
q
q
k
k
q
q
q
k
k
q
∫
(
)
(
)
τ
= π
π Ξ
ρ

 

 
+
ω
δ
− + ω + ω
δ
− − ω






× −
′
′
θ
θ

 

  ′
′
′
(4.53)
where n q represents the occupation number of acoustic phonons with wave
vector q
. Since an electron or hole may change its state from k by emission or
absorption of an acoustic phonon of energy
q
ω , there are two terms involved
in k k
Γ → ′ corresponding to these two types of transitions. Since k k q
′ = + , the
integral may also be carried out in q space. τ may therefore be written as
1
1
(2 )
(
1) (
)
(
)
1
cos
cos
3
2
2
n
E
E
n E
E
q
k
k
dq
q
k
k
q
q
k
k
q
q
∫
τ
= π
π Ξ
ρ

 

 
+ δ
− + ω + δ
− − ω
 
 
× ω
−
′
′
θ
θ

 

 
′
′
(4.54)
The element of volume dq
in q space may be expressed in the spherical
coordinate system as shown in Figure 4.8.
sin
2
dq q
d d dq
=
β β φ
(4.55)
p
k
k´
θ´
θ
q
p
k
k´
θ´
θ
q
β
FIGURE 4.8
Reference coordinate system.