212
Compact Models for Integrated Circuit Design
where:
I dsat is the saturation drain current
If M is the avalanche multiplication factor due to impact ionization, then I sub
can be expressed as
I
M
I
sub
d sat
= (
)
− 1
(5.109)
where M is given by
M
dy
M
M
dy
n
n
=
−
− =
∫
∫
1
1
1
α
α
or
(5.110)
where:
α n is the electron impact ionization coefficient per unit length and is a
strong function of the channel electric field E
In order to derive a generalized expression for I sub , we replace I dsat by I ds . Then
from Equations 5.109 and 5.110, we can show
I
I M
dy
sub
d s
n
=
∫
α
(5.111)
Since I sub resulting from the channel hot electrons impact ionization process is 3–5 orders of magnitude smaller than the drain current I ds , it can be
considered as a low-level avalanche current. For low-level multiplication
M ≈ 1, and therefore, Equation 5.111 becomes
I
I
dy
sub
d s
n
li
=
∫
α
0
(5.112)
where y is the distance along the channel with y = 0 representing the start of
the impact ionization region, and l i is the length of the drain section where
impact ionization takes place as shown in Figure 5.17. Several forms for α n
have been proposed but most commonly used form is
α n
i
i
A
B
E
=
−




 
exp
(5.113)
where:
A i and B i are called the impact ionization coefficients