SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

282 CHAPTER 6. BIPOLAR JUNCTION TRANSISTORS

where

VA=

∫Wbn

0 p(x)dx

p(Wbn)∂W∂VBCbn

(6.6.18)

is defined as the Early voltage. It is clear thatVAis a bias-dependent quantity and hence is best
defined for a particular base-collector voltage, which is chosen to beVBC=0. It turns out that
for heavily doped base regions (used in most high performance devices), the variation ofVAwith
VBCis small.
Let us now study the expression forVA. The quantity
∫Wbn

0

p(x)dx

was defined as the Gummel numberQG(cm−^2 ). If we take the derivative ofQGwith respect to
VBC,weget
∂QG
∂VBC

=

∂

∂VBC

(∫W

bn

0

p(x)dx

]

=p(Wbn)

(

∂Wbn

∂VBC

)

(6.6.19)

SoVAcan be rewritten as

VA=

QG

∂QG/∂VBC

(6.6.20)

The change in base charge with respect toVBCis by definition the base-collector depletion
capacitanceCBC,or

CBC=

e∂QG

∂VBC

(Fcm−^2 ) (6.6.21)

ThusVAcan be written as

VA=

eQG

CBC

(6.6.22)

BothQGandCBCare measured atVBC=0. Variations inQGdue to changes inVBC
are considered negligible, giving a constantVAindependent of bias. In actuality, the output
conductance always increases withVBCbecause the decrease inCBCwith bias tends to be
smaller than the decrease inQG,sinceCBCis determined dominantly by the depletion layer
thickness in the collector.
To minimize the output conductance, or equivalently increase the Early voltage, one must
increase the Gummel numberQGand decreaseCBC. The path with least penalty is to increase
QG, because a decrease inCBCis equivalent to an increase in the collector depletion region
thickness, which in high frequency transistors may result in an unacceptable collector transit
delay.

6.6.4 Drift Effects in the Base: Nonuniform Doping...............

We have assumed so far that the base doping is uniform and consequently there is no built-in
electric field in the base region. In real devices the doping can be quite nonuniform, especially if
the doping is done by ion implantation. The nonuniform doping causes a built-in field that can