Having discussed some principal properties of bulk material
within the DD approach impact ionization modeling of submicron FETs is addressed in this section.
The model by Quade et al. [225] used here is
based on the assumption of a heated Maxwellian energetic
distribution of the carriers. As impact ionization is a
generically a hydrodynamic effect based on the carrier
temperature the generation rates for direct semiconductors
written as:
exp erfc
(3.73)
and
exp erfc
(3.74)
For indirect semiconductors, however, the formula reads:
erfc
(3.75)
and:
erfc
(3.76)
using:
(3.77)
The two sets of equations (3.76)(3.77)
and (3.78)(3.79) are derived as two limiting cases
from the same basic model as explained in [225]. The
difference arises from the allowed energies after the impact
ionization process in the case of a direct and indirect
semiconductor. erfc() is the complementary error function 1-erf().
Special care has to be taken for the implementation of the direct
model to avoid numerical uncertainties, that may arise due to
often encountered poor implementations of the complementary error
function [223].
The lattice temperature dependence is modeled in the
prefactor [147] using four parameters. This allows to
include the two different temperature represented by GaAs and
InGaAs.
(3.78)
is the threshold energy at
= 300 K
modeled as given above. For indirect semiconductors, the energy
is the approximately the band gap, while for direct
semiconductors, there is the difference, as shown in
Table 3.31.
Aspects of the parameter extraction are described in
Section 3.6. Basically, the impact ionization generation
rates are found by one-dimensional MC simulation as a function of
the corresponding carrier temperatures using the potential
profile obtained from a HEMT channel at high bias. The model
parameters in (3.81) are then fitted to the generation
rates obtained by the MC model as a function of carrier
temperature. The model parameters thus contain all the
assumptions made by the HD impact ionization model, which
suggests
to be a single valued function of the average
carrier temperature
. The values found useful can be found
in Table 3.31.
Table 3.31:
Modeled HD impact ionization rates for electrons and
holes in III-V semiconductors.
The data for the lattice-temperature dependence are
obtained by comparison with the approach in the previous
section, which suggests to model the threshold energy. For
InGaAs the lattice temperature dependence is
changed by the sign of the parameter , which leads to the
increase of the impact ionization with lattice temperature.
Figure 3.20:
Modeling of the impact
ionization rate vs. carrier temperature for GaAs with the lattice temperature as a
parameter.
Figure 3.21:
Modeling of the impact
ionization rate vs. carrier temperature for InGaAs with the lattice temperature
as a parameter.
Fig. 3.20 and Fig. 3.21 show the
principal difference of impact ionization for GaAs representative
for AlGaAs and InAlAs for all material compositions, and for
InGaAs, while InGaAs shows a
different behaviour (Fig. 3.21). It can be seen that for GaAs
with increasing lattice temperatures
the slope of
drops. This leads to a crossing of the curves for carrier
temperatures about
= 20000 K, i.e., for average carrier
energy of 2 eV. As will be shown in Fig. 3.30, some
carriers reach energies 1 eV already for relatively low
bias of
= 2 V. For InGaAs, an opposite
behavior is found: The generation increases as a function of
, which causes hot electron reliability problems. If
material compositions other than given in Table 3.31 are
needed they can be interpolated between the materials.