234
Strain-Engineered MOSFETs
where is Planck’s constant. E k and E k
/ denote the initial and final energies of the mobile charge being scattered. Scattering of inversion layer mobile
charges takes place due to Coulombic interactions with occupied traps at the
interface and also with fixed charges distributed in the oxide. We define the
2D charge density N 2D δ(z i ) at depth z i inside the oxide as the combination of
the fixed charge N f and trapped charge N it as
( )= ( )
( )
<
+
=
N z
D
i
N z
z
N N
z
f i
i
it
f
i
{
2
,
0
0 ,
0
(8.6)
Using the above approximation, one obtains the total transition rate. Since
Coulombic scattering is an elastic scattering mechanism, the scattering rate
or, equivalently, the inverse of the momentum relaxation time is then calculated as
N z
e
k
q q
z
z e
dz
E E
k
m
D i
s
i
j
q z z
k
k
i
1
( )
(2 )
.
2
2
1
(
)
( ) ( )
(
)1 cos
2
2
2
2
0
2
.
2
/
∫∫
∫
(
)
τ
=
π
π
ε

 

 
+
ξ ξ

 

 
δ
−
−
θ δ
−
−
(8.7)
Using the above relaxation time, one obtains the mobility of the ith subband as
∑
∫
∫
µ =
τ ε
∂ ε
∂ε
ε
ε
∂ ε
∂ε
ε
e
m
f
d
f
d
i
m
i
( )
( )
*
0
0
(8.8)
The average mobility, ,
µ is then given by [11]
∑
∑
µ =
µ
µ
p
p
i i
i
i i
i
2
(8.9)
where p i is the hole concentration in the ith subband. Taking into account the
different scattering mechanism and using Matthiessen’s rule, one obtains
the total mobility µ. In the presence of the NBTI effect N 2D (z i ) changes to
ΔN 2D (z i , t). This NBTI-induced change of interface traps degrades the mobility of the carriers in the channel of the MOS device and leads to a reduction in channel conductance and transconductance. The mobility model
described above has been implemented in the SDevice simulator. To activate