SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.4. TRANSPORT UNDER AN ELECTRIC FIELD 113

This gives a transit time of 0.123 ps.

The transit time, if the saturation velocity (which is the correct velocity value) is used, is

τtr=

L

v

=

1 × 10 −^5

107

=1ps

This example shows that in short channel devices, ballistic effects can be very strong.

Very high field transport: breakdown phenomena

When the electric field becomes extremely high (∼ 100 kV cm−^1 ), the semiconductor suffers
a “breakdown” in which the current has a “runaway” behavior. The breakdown occurs due to car-
rier multiplication, which arises from the two sources discussed below. By carrier multiplication
we mean that the number of electrons and holes that can participate in current flow increases. Of
course, the total number of electrons is always conserved.

Avalanche breakdown

In the transport considered in the previous subsections, the electron (hole) remains in the
same band during the transport. At very high electric fields, this does not hold true. In the
impact ionization process shown schematically in figure 3.11, an electron, which is “very hot”
(i.e., has a very high energy due to the applied field) scatters with an electron in the valence
band via Coulombic interaction, and knocks it into the conduction band. The initial electron
must provide enough energy to bring the valence-band electron up into the conduction band.
Thus theinitialelectronshouldhaveenergyslightlylargerthanthebandgap (measured from the
conduction-band minimum). In the final state we now have two electrons in the conduction band
and one hole in the valence band. Thus the number of current carrying charges have multiplied,
and the process is often calledavalanching. Note that the same could happen to “hot holes” and
thus could then trigger the avalanche.
Once avalanching starts, the carrier density in a device changes as

dn(z)

dz

=αimpn (3.4.12)

wherenis the carrier density andαimprepresents the average rate of ionization per unit distance.
The coefficientsαimpfor electrons andβimpfor holes depend upon the bandgap of the material
in a very strong manner. This is because, as discussed above, the process can start only if the
initial electron has a kinetic energy equal to a certain threshold (roughly equal to the bandgap).
This is achieved for lower electric fields in narrow gap materials.
If the electric field is constant so thatαimpis constant, the number of times an initial electron
will suffer impact ionization after traveling a distancexis

n(x)=exp (αimpz) (3.4.13)

A critical breakdown fieldEcritis defined whereαimporβimpapproaches 10^4 cm−^1 .When
αimp(βimp) approaches 10^4 cm−^1 , there is about one impact ionization when a carrier travels