388 CHAPTER 10 Fundamentals of the Metal–Oxide–Semiconductor Field-Effect Transistor
10.1.6 Threshold Voltage
The threshold voltage is defi ned as the applied gate voltage required to achieve the
threshold inversion point. The threshold inversion point, in turn, is defi ned as the
con dition when the surface potential is !s ! 2!fp for the p-type semiconductor and
!s ! 2!fn for the n-type semiconductor. These conditions are shown in Figures 10.9
and 10.10. The threshold voltage will be derived in terms of the electrical and geo-
metrical properties of the MOS capacitor.
Figure 10.19 shows the charge distribution through the MOS device at the
threshold inversion point for a p-type semiconductor substrate. The space charge
width has reached its maximum value. We will assume that there is an equiva lent
oxide charge Q
ss
"
and the positive charge on the metal gate at threshold is Q
mT
"
. The
prime on the charge terms indicates charge per unit area. Even though we are assum-
ing that the surface has been inverted, we will neglect the inversion layer charge at
this threshold inversion point. From conservation of charge, we can write
Q
mT
"
# Q
ss
"
!
!
Q
SD
"
(max)
!
(10.26)
where
!
Q
SD
"
(max)
!
! eNa xdT (10.27)
Figure 10.19 | Charge distribution in a
MOS capacitor with a p-type substrate at
the threshold inversion point.
Q"
mT
Metal Oxide
p-type
semiconductor
Q"
ss
xdT
$Q"
SD(max)$ ! eNaxd
T
■ Comment
The applied gate voltage required to achieve the fl at-band condition for this p-type substrate
is negative. If the amount of fi xed oxide charge increases, the fl at-band voltage becomes even
more negative.
■ EXERCISE PROBLEM
Ex 10.3 Repeat Example 10.3 for a doping concentration of Na ! 2 % 1015 cm&3, an oxide
thickness of tox ! 4 nm ! 40 Å, and Q
ss
"
! 2 % 1010 electronic charges per cm2.
What is the value of the metal–semiconductor work function difference?
(Ans. !ms " &1.03 V, VFB ! &1.034 V)
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