1.4 Phenomenological Findings
Before embarking on detailed physical models, the focus is now put on a
phenomenological understanding of NBTI. In this respect, special attention is put on
the functional form of the time evolution of
depending on gate bias and
temperature. Long term extrapolation does not only allow lifetime projection relevant
for industrial purposes but also should be viewed as a touchstone for subsequent
modeling attempts.
Grasser et al. [24] recognized that the recorded threshold voltage curves follow the
same pattern at different stress temperatures and voltages. Therefore, these curves
can be scaled so that they overlap for the stress and the relaxation phase.
Mathematically, this can be described by
The scaling factor
represents the temperature (
) and oxide field
(
) dependence, which is best approximated by
mimics a universally valid shape of the threshold voltage curves. A
series of stress sequences for different temperatures are depicted in Fig. 1.3 in order
to illustrate the common pattern of the threshold voltage data. As a proof for
scalability, all curves line up to one curve by dividing them by their corresponding
. As demonstrated in Fig. 1.3, the initial part of the stress phase shows a
logarithmic behavior up to a stress time
while the subsequent part follows
a power-law with an exponent of
afterwards. The stress phase [24] can be
expressed as where
denotes the first stress measurement point. However, Huard [57] reported
that, for some devices, the scalability property is violated for the long term
degradation with stress times larger
. Depending on the details of the
fabrication process, the long term part has a scaling factor, which is estimated as
with
being an activation energy of
.
In the context of the universal behavior, special attention has been paid to the
recovery phase. This was motivated by the notion that NBTI underlies reversible
reaction kinetics. The degradation during the stress phase was assumed to be caused
by the combination of a forward and a reverse rate. Therefore, the kinetics during
stress were supposed to cover the whole physics, however, the individual
contributions of the forward and the reverse rate are obscured. By contrast, only the
reverse rate is activated during the relaxation phase, which is thus more suited for
analyzing NBTI. In the following, some observations on the relaxation phase are
summarized:
- The recovery already sets in before the shortest measurable relaxation
time of about
.
- The recovery slows down before reaching the pre-stress level.
- The recovery data follow a universal curve when plotted as a function of
.
These observations suggest the following procedure to process the recorded NBTI
recovery data: First, the relaxation curves must be normalized to their last respective
stress points, which are generally unknown but can be obtained using the
back-extrapolation method proposed in [31]. Second, the relaxation times have to
be scaled to their last accumulated stress time. The resulting curve can
be best analytically described by the empirical relation [31] (cf. Fig. 1.4)
with
,
, and
. In a further step, the universal
relaxation is extended by the so-called permanent component [58, 28, 26] attributed
to a mechanism with another physical origin. This permanent component, however, is
ascertained to be a slowly recovering component rather than a constant contribution
and is best represented by a power-law [59]. Separating the permanent from the
recoverable component
, the degradation during stress can be written as
with
being the first stress measurement point. Alternatively, it can also be
formulated as
is the prefactor of a power law with exponent
, where the subscripts
and
refer to the recoverable or the permanent component, respectively. Note that
is larger than
in equation (1.9) indicating that the permanent component
becomes dominant at large stress times [59]. The impact of the permanent
component is demonstrated in Fig. 1.5, where the long term recovery tails deviate
from the universal curve. However, universality is regained by accounting
for the permanent component. For times longer than
, the relaxation
data can be well approximated by a logarithmic behavior [60, 61] using the
expression
with
being a parameter. The first term at the right hand side of equation (1.10)
represents the recovering component, which shows a nearly logarithmic behavior, and
compares well with the short term part of expression (1.7) [60, 31]. In [61] the
prefactors
and
have been extracted from the eMSM data in
Fig. 1.3 for the time range
during stress and
during
relaxation. A comparison of the prefactors has revealed a certain ratio
,
meaning that the stress and relaxation curves have different slopes. This
asymmetry has long remained unrecognized but rules out several proposed NBTI
models.
The logarithmic and power-law-like part of the stress curve along with the two
components during recovery raise the question whether NBTI is governed by two
mechanisms with one of them dominating in each of the time regimes. If
this is the case, each of these mechanisms is highly likely to be subject to
different field and temperature acceleration. Then the transition between
these regimes should be controllable by varying the temperature and the
electric field. However, no such transition has been observed so far. In [63], it
has been argued that NBTI must be caused by either a single process or
two tightly coupled ones. In the latter case, the interplay of both process
enforces a single field dependence and temperature dependence without any
transitions.