SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

6.5. BJT DESIGN LIMITATIONS: NEED FOR BAND TAILORING 269

6.5.1 The Generalized Moll-Ross Relationship .................

This very important relationship first developed by Moll and Ross, and subsequently gener-
alized by Kroemer, is derived in this section. The Moll-Ross Relationship links the collector
current density to the applied base-emitter voltageVBEand to the Gummel numberQG.Itisa
very powerful relationship, since it shows that the nature of the doping in the base is inconse-
quential as far as the output current is concerned. Rather, the total number of dopants in the base
is the controlling factor. Kroemer’s generalization expands this to heterostructure devices.
Let us assume ann-p-ntransistor with a high current gain such that the hole currentJp≈ 0.
In the base of the transistor, we can write

Jn=μnn

dEFn

dx

(6.5.11)

and

Jp=μpp

dEFp

dx

(6.5.12)

SinceJpis assumed to be approximately zero,dEdxFp≈ 0 , and so equation 6.5.12 can be rewritten
as

Jn=μnn

d

dx

(EFn−EFp) (6.5.13)

Inserting Einstein’s relationship

μn=Dn

e

kBT

(6.5.14)

and using the relations

EFn−Ei=kBTln

(

n

ni

)

(6.5.15)

and

Ei−EFp=kBTln

(

p

ni

)

(6.5.16)

we get
d
dx

[

ln

(

np

n^2 i

)]

=

Jn

eDnn

(6.5.17)

or

d

[

ln

(

np

n^2 i

)]

=

Jn

e

·

p

Dnn^2 i

·dx (6.5.18)

Let us integrate equation 6.5.18 fromx=0to the edge of the neutral basex=Wbn.

np

n^2 i

∣

∣∣

∣W

bn

−

np

n^2 i

∣

∣∣

∣x=0=−

np

n^2 i

∣

∣∣

∣x=0=

Jn

e

∫Wbn

0

p(x)

Dnn^2 i

dx (6.5.19)

Here, because of Shockley boundary conditions, we have assumednp/n^2 i

∣

∣

Wbncan be neglected.
The quantitynp/n^2 i

∣∣

x=0is given by the law of the junction

np

n^2 i

∣∣

∣∣

x=0

=exp

(

eVBE

kBT

)

(6.5.20)