SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

270 CHAPTER 6. BIPOLAR JUNCTION TRANSISTORS

This leads to the generalized Moll-Ross relation

Jn=−

e·exp

(

eVBE

kBT

)

∫Wbn

0

(

p(x)

Dnn^2 i

)

dx

(6.5.21)

where the integration is over the extent of the neutral base. In the case whereniandDnare
constant throughout the base, or equivalently the material is homogeneous,

Jn=−

eDnn^2 i

∫Wbn

0 p(x)dx

exp

(

eVBE

kBT

)

=−

eDnn^2 i

QG

exp

(

eVBE

kBT

)

(6.5.22)

where

QG=

∫Wbn

0

p(x)dx

∫Wbn

0

Nadx (6.5.23)

is the Gummel number, defined as the total number of acceptor atoms in the neutral base.

6.5.2 How muchβdo we need? .........................

This question is very important, but it really has no universal answer. Different applications
have different minimum tolerances forβ. This will be illustrated in the four examples shown
below. Because an understanding of these applications requires some knowledge of bipolar
frequency response, it is recommended that the reader examine chapter 7 before reading this
section. We thank Prof. Mark Rodwell for discussions on this topic.

Microwave power amplifiers

In figure 6.13, we show a basic BJT small-signal model. As derived in chapter 7,

Cin=Cπ+CBE=

τB+τC

re

+CBE (6.5.24)

We will assume thatCπ>> CBE,sothatCincan be written as

Cin≈

τB+τC

re

=(τB+τC)gm (6.5.25)

At a small signal frequencyω, the input currentIinis given by

Iin=

{

jω[(τB+τC)gm]+

gm

β

}

Vin (6.5.26)

For an efficient transistor, one wants the first term in this expression to dominate, or

ω(τB+τC)>

1

β

(6.5.27)