30
Compact Models for Integrated Circuit Design
Similarly, for a p-type silicon with a shallow acceptor concentration N a , the
Fermi level is given by
E E
kT
N
N
f
v
v
a
− =






ln
(2.22)
In this case, the hole density is p N
N
a
a
=
=
−
, and the electron density is
n
n
N
i
a
=
2
(2.23)
Instead of using Equations 2.20 and 2.22, we can express these in terms of E f
and E i using Equations 2.9 and 2.10. From Equation 2.9, the intrinsic carrier
concentration can be shown as
n N
E E
kT
i
c
c
i
≅
−
−






exp
(2.24)
Or,
E
E kT
N
n
c
i
c
i
= +






ln
(2.25)
Then substituting for E c from Equation 2.25 into Equation 2.20, we get for an
n-type silicon,
E E kT
N
n
f
i
d
i
− =






ln
(2.26)
Similarly, using Equation 2.10, we can express Equation 2.22 for a p-type
silicon by
E E
kT
N
n
i
f
a
i
− =






ln
(2.27)
Equations 2.26 and 2.27 are the measure of the Fermi level with reference
to the mid-gap energy level for the n-type and p-type semiconductors,
respectively.
2.2.4.2 Fermi Level in Degenerately Doped Semiconductor
For heavily doped silicon, the impurity concentration N d or N a can exceed
the effective density of states N c or N v , so that E f   ≥  E c and E f   ≤  E v according
to Equations 2.20 and 2.22. In other words, the Fermi level moves into the CB
for n+ silicon, and into VB for the p+ silicon. In addition, when the impurity
concentration is higher than 10
18
 cm –3 , the donor (or acceptor) levels broaden