4.6. HIGH-VOLTAGE EFFECTS IN DIODES 177
given bynnoexp (−e(Vbi−Vr)/kBT)ornp 0 exp (eVr/kBT), but sinceVris negative this is
very small. The linear decrease in the value of EFnfrom thep−bulk to the edge of the junc-
tion reflects the exponential decrease in the carrier concentration because of diffusion toward the
junction as shown in equation 4.5.7. The reader should be aware that this picture reflects the case
of no velocity saturation of either electrons or holes in the materials. If that happens, the electron
concentration at the edge of the junction cannot decrease arbitrarily and the problem has to be
solved such that the carriers allow current to be continuous through the structure and is left to
the reader as an exercise.
4.6 HIGH-VOLTAGE EFFECTS IN DIODES.....................
In deriving the current-voltage relation we have made two important assumptions: i) the excess
carrier density injected across the depletion region is small compared to the majority charge
density; ii) the reverse current saturates since it is due to the carriers drifting across the depletion
region and is limited by the diffusive flux of minority carriers to the junction.
4.6.1 ForwardBias:HighInjectionRegion ...................
We have so far assumed that the injection density of minority carriers was low so that the
voltage all dropped across the depletion region. However, as the forward bias is increased, the
injection level increases and eventually the injected minority carrier density becomes comparable
to the majority carrier density. When this happens, an increasingly larger fraction of the external
bias drops across the undepleted region. The diode current will then saturate, as shown as Region
3 in figure 4.14. The minority carriers transport is not only due to diffusion, but also due to the
electric field that is now present in the undepleted region. As the forward bias increases, the
devices start to behave like a resistor, where the current-voltage relation is given by a simple
linear expression. The current is now controlled by the resistance of then-andp-type regions as
well as the contact resistance.
4.6.2 Reverse Bias: Impact Ionization ......................
We have noticed that under reverse bias conditions the electric field across the depletion region
increases. As a result electrons and holes forming the reverse current can acquire very high
energies. Once this excess energy reaches the value of the bandgap we can have impact ionization
as discussed in chapter 3. The final result is that one initial electron can create two electrons in
the conduction band and one hole in the valence band. This results in current multiplication and
the initial current reverse biasIobecomes
I′
o=M(V)Io (4.6.1)hereMis a factor that depends upon the impact ionization rate which we now derive.