SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

7.5. HIGH-FREQUENCY BEHAVIOR OF A BJT 337

Following the analysis in section 7.5 (equation 7.5.16 through equation 7.5.26) and using

ΔJc

v(x)

=eΔn(x)·Δx (7.5.51)

we see that the charge induced on the base is

ΔQc=(1−

x

Wc

)·eΔn(x)·Δx (7.5.52)

and the image on the collector is by charge neutrality

ΔQ

′

c=Δn(x)Δx−ΔQc=eΔn(x)Δx·

x

Wc

(7.5.53)

The displacement current flowing in the external circuit,ΔJcis given by

ΔJc=

d

dt

ΔQ

′

c=eΔn(x)Δx·

dx

dt

·

1

Wc

(7.5.54)

Usingdxdt=vswe arrive at an important relationship also known as the Ramo-Shockley theorem

ΔJc=

eΔn(x)Δx

τ

(7.5.55)

The current carrying electrons in the collector can be assumed to comprise of several sheet
charges of magnitudeΔn(x)Δx. Hence the net induced current due to the electrons will be a
sum (integral) of all the induced currents. The total current per unit area is therefore obtained by
integration over all sheets:

J=−e·(v/w)·

∫

n(X)·dX (7.5.56)

We apply (equation 7.5.56) to a traveling electron wave of the form

n(x, t)=n 0 exp [jω(t−x/v)] (7.5.57)

It clearly corresponds to a wave of (angular) frequencyωtraveling with a uniform speedv.
Theconvectioncurrent in the planex=0is evidently

iω(wB)=−en 0 v·exp (jωt) (7.5.58)

this is the current density that would be flowing if the capacitor were infinitesimally thin and the
transit time of the electrons through the capacitor were zero. With the help of equation 7.5.58
we may write equation 7.5.57 as

n(x, t)=−

(

iω(wB)

ev

)

exp (−jωx/v) (7.5.59)