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In this section a model is proposed which was implemented into the numerical
device simulator Minimos-NT. Figure 6.16 gives an
overview of the involved processes. The model is able to achieve excellent
agreement with measurement data on thick, pure
dielectrics of
high-voltage devices from our industry partner (Section 7.1) for both,
the stress and the relaxation phase of NBTI and for different temperatures.
As the model comprises of many models and physical assumptions described in different parts of this work, I will collect all important equations in this section for the sake of clarity.
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(6.31) |
Assuming electrically neutral hydrogen,
, or hydrogen protons,
,
the kinetics exponent
equals
. In this case, for every generated
a mobile hydrogen,
, is released. To assume molecular hydrogen,
,
as the mobile species the kinetics exponent equals
.
The amount of interface states
is strongly dependent on the wafer
orientation and the process technology. It can greatly differ for different
interfaces. Reasonable values are often in the range of
-
cm
as described in Section 3.1.
For reliable device operation it is, of course, of highest interest to keep
this number as low as possible.
The forward reaction rate is dependent on the local electric field,
, and
the local hole concentration
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(6.32) |
The reverse reaction rate is also introduced as Arrhenius activated with the
activation energy
and the prefactor
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(6.33) |
As described in Section 3.1, the energy distribution of
electrically active interface traps
can be very important. When the
Fermi-level is not very close to one of the band edges, the charge state of
these traps differs depending on their density-of-states (DOS). The density of
is given as the integral of
across the whole band-gap
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(6.34) |
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(6.35) |
Several models for different DOS are available in Minimos-NT, where the most plausible according to literature are two peaks of Gaussian form as found in Section 3.1.2.
The convection of the electrically neutral species
and
is
a purely diffusive process due to the concentration gradient
For the case of proton transport,
, an additional drift term has to be
added to (6.36). In the typical NBT stress condition
where a negative voltage is applied to the gate contact, this leads to quick
removal of the positively charged protons from the interface. As a consequence
more interface traps can break and the degradation is increased.
At each trap center in the dielectric a balance equation is solved to account
for the trapping and de-trapping effects. The trap occupancy
is
considered at each trap energy level
as
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(6.37) |
The trap energy
is zero at the mobility edge and below zero at the trap
levels with more negative values for deeper traps. Therefore, with decreasing
temperature the trap release rate decreases (
) and de-trapping
becomes more unlikely. The trapping probability does not change in this model.
Within the model the charge state of trapped hydrogen can be chosen to model the formation of positively charged trapped protons, as described in [107].
The DOS of the trap levels can be selected from the model. Two plausible densities are schematically illustrated on the right hand side of Figure 6.16. An exponential distribution with the highest DOS at the band edge [88] or with an additional Gaussian peak in deeper states forming deep and therefore ``slow'' traps [106,107,108].