So far, only empirical relations, emerging from an experimental perspective but lacking any profound physical justification, have been presented. Now the focus is put on an in-depth microscopic understanding of the NBTI phenomenon. Hence, a series of modeling approaches are discussed in the following, where each of them is traced back to the creation of either interface charges and/or oxide charges.
The relation between the threshold voltage shift and the created charges can be expressed as
In addition to interface charges, also oxide-trapped charges [68, 69, 70] impact the
threshold voltage shift. According to the current understanding of oxide-traps,
charges are stored in preexisting defects whose occupation is governed by the
quantum mechanical trapping dynamics. The contribution of the oxide traps to
can be evaluated using
First serious modeling attempts date back to the so-called reaction-diffusion (RD)
model [71, 72], which has been refined successfully in later studies [25, 68, 73, 74, 60].
It relies on an interface reaction involving the interfacial dangling bonds (present
in the form of
centers) together with some sort of hydrogen species. Initially,
nearly all of the
centers are supposed to be passivated through a hydrogen
anneal step. This means that their unsaturated
atoms has established a bond to
a nearby hydrogen atom
, thereby shifting the electrically active trap levels out of
the substrate bandgap. Upon application of stress, the
bonds (
) can
break due to the presence of an electric field, thereby activating the forward reaction
of
A solution of equation (1.17)-(1.19) can be found as
Experimentally, the most convincing proof that NBTI is not explained by the RD model comes from the TDDS [53, 51]. The spectral maps show clusters which are fixed on the emission time axis for a certain temperature and evolve with increasing stress time. By contrast, the RD model predicts clusters that extend towards larger emission times for rising stress times. Theoretical first-principles calculations of the Pantelides group [80, 81, 82, 83] predict too high dissociation barrier for the interface reactions (1.14) and (1.15-1.16). In contradiction to the assumption of the RD model, Tsetseris et al. [83, 81, 82] proposed that the interface reaction can be initiated by protons originating from the substrate.
In order to explain the long recovery tails seen in experiments, a refinement of the
hydrogen transport in the RD model has been proposed. Due to the exposure to a
hydrogen ambient during device fabrication, a large background concentration of
hydrogen has to be expected. However, this background concentration would strongly
enhance the reverse mode of the interface reaction so that no device degradation
could occur. According to dispersive transport, a large fraction of the hydrogen
particles is bonded to traps and thus cannot participate in the interface reaction.
The retarded release of the strongly bonded particles during recovery [84]
should bring the required long recovery tails. The hydrogen transport has
been modeled to proceed over single trap levels, in which the particles dwell
most of their time. Diffusion only takes place when the hydrogen atoms are
released from their traps. This kind of transport is referred to as dispersive
transport. Its formulation relies on multiple trapping theory [85, 86, 87]
and was combined with the interfacial hydrogen reaction to the reaction
dispersive diffusion (RDD) model [68, 73]. The overall hydrogen
concentration is split into a contribution of free hydrogen
in a conduction
state and hydrogen
residing at traps with an energy level
:
The previous models rest on the assumption that hydrogen diffusion ultimately
governs the generation of interface states. Another modeling approach assumes the
interface reaction as the rate-limiting step. Due to the amorphous structure of ,
the
bonds at the interface are subjected to a wide spread of bond lengths
and angles, which are both related to large variations of bond strengths. In order to
account for this fact, the associated dissociation barriers
[38, 69, 58, 90] are
taken to be distributed rather than single-valued. According to transition state
theory, the bond breakage rates follow an Arrhenius law and can be expressed as
So far, the prolonged degradation and recovery are ascribed to the dispersive nature of either the interface reaction or the hydrogen transport in the oxide. Since both modeling attempts remained fruitless, a new model has been developed, which combines the dispersive interface reaction with a diffusion-like mechanism. The concept of the Born Oppenheimer energy surface [91, 92, 93] motivated the idea of the triple-well model (TWM) [94, 95] where the stable sites of hydrogen along with their separating barriers are represented in one common energy diagram (see Fig. 1.9). The dynamics of this system are expressed by coupled rate equations with Arrhenius-like expressions for transition rates following transition state theory. In a simplified mathematical model, there exist three states corresponding to an equilibrium, an intermediate and a lock-in configuration, which are connected in series. While the temperature activation is already incorporated in Arrhenius-type transition rates, the field acceleration is assumed to be due to an energetical downward shift of the intermediate and the lock-in states along with their connecting barriers (see Fig. 1.9). For instance, this shift can be related to breaking bonds with a dipole moment whose energy contribution depends on the oxide field.
During stress the hydrogen particles travel from the equilibrium towards the lock-in
configuration, where a considerable fraction remains in the intermediate state. After
the stress is removed, particles from the intermediate state first return to the
equilibrium configuration. This fraction of particles correspond to the recoverable
component of NBTI. The return of the other particles from the lock-in configuration
occur at longer timescales and corresponds to the permanent or rather the slowly
recoverable component of NBTI. In , the first transition mimics the
interface reaction involving the hydrogen atom initially bonded to a
center,
while the lock-in reflects the hydrogen diffusion away from the interface. In
contrast to previous models, the triple-well model cannot only reproduce the
complicated stress/relaxation pattern but also exhibits the correct temperature
activation.
Investigations of the universal recovery (see Section 1.4) have revealed that there
exists a permanent in addition to a recoverable component, where each of them are
caused by their own physical mechanism. As a result, some sort of hole trapping into
defects was suggested as the recoverable component and assumed to be due to elastic
tunneling of holes into preexisting traps [58]. By contrast, a hydrogen reaction like
the bond breakage in the RD model was ascribed to the permanent
component. However, both mechanisms were assumed to be tightly coupled and
therefore do not take place independently according the argumentation in
Section 1.4.
A more promising approach assumes hole trapping “triggering” a hydrogen reaction as illustrated in Fig. 1.10. This model [24] relies on thermally activated tunneling into precursor defects. The captured charge weakens the hydrogen bond to the defect and thus causes the release of hydrogen. The last step corresponds to the permanent component of NBTI since the reaction of the defect with hydrogen requires considerably larger times compared to the hole trapping or detrapping process. Even though unprecedented accuracy is achieved for the stress and the relaxation phase at different temperatures and gate voltages, the field dependence of the stress parameter was phenomenologically introduced but has not been justified so far.