The models proposed for the high-field mobility are based on the mobility expression of the form [352]:
For the DD high-field mobility two different models are available. The first is the convenient model used for Silicon, referred as Model A. The second one is a modified model which can account for negative differential mobility (NDM), referred as Model B. The latter is especially tailored to describe the transport properties of electrons in Nitrides. Further, based on Model B a corresponding HD model can be synthesized.
The basic high-field mobility model has the form:
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The model is derived from (4.66), with =1/2 and
using the driving force
instead of the electric field
. It offers
excellent convergence behavior and a straight-forward calibration
method. However, it cannot account for the velocity decrease at higher
electric fields. Yet it offers a reasonable agreement with the
experimental data and MC simulation results for electric fields below
the maximum velocity for all three materials (Fig. 4.16,
Fig. 4.17, and Fig. 4.18) using the parameter
setup provided in Table 4.8.
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The model is based on the same expression for the
mobility (4.66), with =1 and
=1/2. In
order to approximate the mobility decay due to intervalley transfer at
high fields, two sets of
are used. One describes the mobility
in the lower valley and the other one in the higher
(while
and U are the lowest two valleys only
in GaN, this notation is retained for the other materials for
simplicity):
This model allows to set a lower electron mobility and velocity in the higher conduction band. Thus, the characteristic electron velocity decrease of most Nitrides can be described well.
Since all MC simulations and experiments, on which we rely to calibrate
the low-field mobility in GaN, were performed at low electric fields, we
set
=
as calculated by the low-field mobility
model. Using a down-scaled mobility (
supported by MC data) and velocity in the higher band results in a
decrease of the electron velocity at higher fields. With the parameters
listed in Table 4.9, the model delivers an acceptable
approximation in comparison to MC simulations, accounting for as much as
six bands [355] (Fig. 4.16). Results from
different groups vary widely (e.g. peak velocity from
cm/s to
cm/s), therefore our goal is not really a
perfect agreement with this particular MC simulation. The model (and the
corresponding HD model) is a carefully chosen trade-off. On the one hand
they provide a velocity-field characteristics close to the one obtained
by MC simulation, while on the other hand they maintain low calculation
complexity and a good convergence behavior. An extension accounting for
three valleys is possible, however, it was ruled out due to the
downgraded convergence of the solution process.
For AlN, the model cannot be applied due to the very slow increase of
the velocity versus electric field. In order to model the latter
properly =0.45 is needed, however, while being straightforward
for the DD model, this will increase considerably the complexity of the
corresponding HD model. Therefore, no parameter setup is given here
for AlN.
Based on the recent MC simulation studies for InN (accounting for the
re-evaluated band gap), a parameter setup is extracted
(Table 4.9). Due to the value of close to 1, a
good agreement can be achieved (Fig. 4.18) for velocity
characteristics below the maximum, while a low value of
accounts
for the intervalley transfer at fields starting at 50 kV/cm.