268
Compact Models for Integrated Circuit Design
where
N
v
q
C
C
C
kT
ox
dep
i t
* =
+
+
(
)
(7.18)
In order to evaluate δ µ δ
∆
∆
eff
t
N , Matthiessen’s rule is used so that
1
1
1
1
µ
µ
µ
µ
α
eff
n
ox
n
t
N
=
+
=
+
(7.19)
where:
μ ox is the mobility limited by oxide charge scattering
α is the scattering coefficient and is a function of the local carrier density
due to channel charge screening effect [31]
It can be shown from Equation 7.19 that
δµ
δ
αµ
eff
t
eff
eff
N
W y
∆
∆
= −
2
(7.20)
Substituting Equations 7.15 and 7.20 in Equation 7.14 yields
δ
αµ
δ
I
I
R
N
N
W y
ds
ds
n
eff
t
eff
=
±

 

  .
∆
∆
(7.21)
Thus, PSD of the local current fluctuations is
S y f
I
W y
R
N
S y f
Id
ds
eff
n
eff
N t
∆
∆
∆
( , )
( , )
=


 


 
±






2
2
αµ
(7.22)
where:
S ΔNt (y, f ) is the PSD of the fluctuations in the number of the occupied traps
over the area W eff Δy
and is given by
S y f
N E x y z yf
f
E x y z
Nt
T
W
t
t
E
E
t
ox
eff
v
c
∆
∆
( , )
(, , , )
( , , ,
=
−
( )⋅
∫
∫
∫
4
1
0
0
τ
) )
( , , , )
1
2
+
⋅ ⋅
ω τ E x y z
dx dz dE (7.23)
where:
N t (E,x,y,z) is the distribution of the traps in the oxide and over the energy
band
τ(E,x,y,z) is the trapping time constant