In order to obtain a consistent hydrodynamic mobility expression, the local energy balance equation:
4.4.3 High-Field Mobility for HD Equations
is solved for
, which is then inserted into
(4.66). This is performed again with =1/2 for
both models, and with =2 for the first model and =1 for
the second model, respectively.
is the energy relaxation time.
The expression obtained with the chosen values for and is
identical with the one proposed by Hänsch et al.
. In order to account for NDM effects it is modified by
introducing two parameters ( and )
In the standard Hänsch model
corresponds to the
in (4.66)). However, due to the powered temperature
in the numerator the velocity is steadily
decreasing at high-fields. Hence,
does not describe the
saturation velocity as a physical quantity, although it does affect
the high-field transport characteristics. The parameter has
a more pronounced effect at low fields, while influences
primarily the high-field mobility, though their impact cannot be
isolated to a specific field region. The conventional Hänsch model
corresponds to the parameter set
in order to approximate the simulation and experimental data, a set
is chosen for GaN
(Table 4.10). It delivers good agreement with the
velocity-field characteristics obtained using the DD Model B
(Fig. 4.16). A similar good match with DD Model A can be
achieved for AlN, too (Fig. 4.17). Only for InN it is not
possible to model the very strong NDM effect.
Parameters for HD high-field electron mobility Model A.
Inserting (4.72) into (4.66) with
=1/2 and =2 gives the following expression for the
In order to approximate the intervalley transfer at high fields, two
are used as in DD Model B. In this model too,
does not denote the saturation velocity. The weighted mean
The expression for
is analogous to that for
Fig. 4.19 compares the valley occupancy as a function of
the electric field as calculated in this model and MC simulation in
GaN. The used parameter setup is the same as for DD Model B
(Table 4.9). The only additional values needed, are the
scaled energy relaxation times listed in Table 4.11 (the
base relaxation times
discussed in Section 4.4.4. An excellent agreement between
all models is achieved for GaN (Fig. 4.16). For InN the
model can describe the abrupt decay in velocity very well
(Fig. 4.18), while the maximum velocity value is slightly
higher than the one achieved by the DD model (which is possibly due to
the high energy relaxation times used).
GaN valley occupancy as a function of the electric field.
While the models deliver consistent results, the two approaches expose
some differences. HD Model A is close to already established models
and offers a straightforward calibration with only two auxiliary
parameters (within a narrow value range). HD Model B is more complex,
however, it allows for a more flexible calibration. Its parameters are
derived from physical quantities.
Parameters for the energy relaxation times for HD high-field
electron mobility Model B [ps].
S. Vitanov: Simulation of High Electron Mobility Transistors