6.6. SECONDARY EFFECTS IN REAL DEVICES 275
as np,C→ 0
ve→∞wherenp,Cis the electron concentration at the base-collector junction. Since in reality the bulk
velocity in the base will saturate at some valuevs,np,Ccannot drop to zero, but instead drops to
avalue
np,C=JC
evs(6.6.6)
Because electrons in the depleted collector all travel at the saturated velocityvs, the injected
carrier concentration in the collector will everywhere be equal tonp,C, and the net charge density
in the collector
NC,net=Nd,C−np,C (6.6.7)
The resulting charge profile, as well as electric field profile in the collector region, for a device
under dc bias with collector dopingNd,Cis shown in figure 6.17. Here we have assumed that
the device is biased such that then−collector region is fully depleted (this is typically the case
for bipolars in modern circuits). As indicated in figure 6.17c, the slope of the electric field in the
collector region is given by
dE
dx
=
eNC,net
(6.6.8)
We will now consider the effect that increasing the collector current densityJChas on the
charge distribution and electric field in the structure. We assume that the voltage across the base-
collector junction maintains a constant valueVCB, implying that the total potential drop across
the junction isVCB+Vbi. Under this assumption, the area underneath the electric field curve in
figure 6.17c always maintains a constant valueVCB+Vbi. The voltage drops in thep+andn+
regions at the edge of the base-collector depletion region are very small relative to the voltage in
then−layer, and hence the area of the shaded region in figure 6.17c is assumed to beVCB+Vbi.
Equivalently, the total base-collector depletion depthwd,BCcan be assumed to be approximately
equal to the collector widthwC.
AsJCincreases, the injected charge density in the collectornp,Cmust increase to maintain
current continuity, as indicated by equation 6.6.6. This causes the net charge in the collector
NC,netto decrease (equation 6.6.7). Hence the slope of the electric field profile in the collector,
which is proportional toNC,net, decreases. At the critical current density
Jcrit=eNd,Cve (6.6.9)the injected mobile chargenp,Cexactly balances the ionized donor chargeNd,C, resulting in
zero net charge as well as a constant electric field in the collector (see figure 6.18). Concur-
rently, the depletion region depth at the base edgexpCdecreases, since the electric field at the
base edge of the collector must decrease in order for the shaded area in figure 6.17c to remain
constant. Similarly, the depletion region in then+subcollectorxnCincreases, as the region has
to terminate a higher electric field.
AsJC>Jcrit, the slope of the electric field reverses sign, asNC,net=Nd,C−np,Cis now
negative. This process continues until another critical current threshold is reached, when the