9.4. CAPACITANCE-VOLTAGE CHARACTERISTICS OF THE MOS 453
Example 9.5Derive the relation for the semiconductor capacitance per unit area of the
MOS capacitor at the flat band condition. The charge density near the flat band is
δρ(z)=
eδV(z)po
kBT
=
eNaδV(z)
kBT
The Poisson equation then gives us
d^2 δV(z)
dz^2
=−
eNaδV(z)
kBT
The solution is a simple exponentially decaying function:
δV(z)=δVsexp (−bz)
where
b=
√
eNa
kBT
The charge density is now
δρ(z)=
eNa
kBT
The areal charge density is obtained by integrating this from the interface into the bulk,
with the result
|δQs|=
∫∞
o
ρ(z)dz=
eNa
bkBT
δVs
The capacitance is now
Cs=
δQs
δVs
=
eNa
bkBT
which gives the result given in equation 9.4.7 when the value ofbis used.
Example 9.6Consider a MOS capacitor made on ap-type substrate with doping of
1016 cm−^3. The SiO 2 thickness is 500A and the metal gate is made from aluminum. ̊
Calculate the oxide capacitance, the capacitance at the flat band, and the minimum
capacitance at threshold.
The oxide capacitance is simply given by
Cox=
ox
dox
=
3. 9 × 8. 85 × 10 −^14
500 × 10 −^8
=6. 9 × 10 −^8 F/cm^2
To find the minimum capacitance, we need to find the maximum depletion width at the
threshold voltage. The value ofφFis given by
φF =0.026 V ln
(
Na
ni
)
=0.026 ln
(
1016
1. 5 × 1010
)
=0.347 V
