1.5 A Modeling Perspective

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

            ΔQ (t)+ΔQ  (t)
ΔVth(t)  = - --it-Cox-ox--                 (1.11)
with Cox  being the areal gate capacitance. The change in interface charges ΔQit(t)  is given by
ΔQit(t)  = q0  ΔDit(Et,t)fit(Ef,Et,t)dEt .          (1.12)
ΔDit (Et,t)  denotes the change in the time-dependent density of interface states at an energy level Et  and fit  corresponds to their occupancy. In equation (1.12) it has been presumed that the temporal charging and discharging of the interface states are negligible since both processes already occur on much shorter timescales than that typically covered by NBTI. The interface traps can hold up to two electrons and exhibit two distinct energy levels, namely one acceptor and one donor level. Accordingly, they are also referred to as amphoteric traps [64965], which behave as either a donor or an acceptor by definition (cf. Fig. 1.6). Starting from midgap, a shift of the Fermi level towards the conduction band charges the interface states negatively, while they become positively charged if the the Fermi level is moved in the opposite direction.


Figure 1.6: A schematic representation of the donor and acceptor distribution within the bandgap of Si  . Energy levels for + ∕0  transitions, termed donor states, are located below midgap while energy levels for 0∕- transitions, acceptor states, are predominantly found above midgap. The spread of the peaks in the trap distribution [1038] has been speculated to originate from the disorder of the atomic structure at the interface. Note that in addition to the interface states density, a band tail states density exponentially decays into the bandgap. They are ascribed to stretched Si  -Si  bonds due to the disorder at the Si∕SiO2  interface[6667].

In addition to interface charges, also oxide-trapped charges [686970] 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 ΔVth  can be evaluated using

            ∫ ∫
ΔQox(t) = q0    Dox(x,Et,t)Δfox(x,Et,t)(1- x∕tox)dxdEt,   (1.13)
where Dox(x,Et,t)  is the density of trap states and Δfox  represents the change in the trap occupancy. It is commonly assumed that oxide traps have larger time constants compared to interface states. For instance, this can be related to the depth of the trap location or the energetical position within the bandgap. Then the structural disorder of a - SiO2  gives rise to a wide spread of trap levels assumed in several NBTI models [24].

1.5.1 Reaction-Diffusion Model

First serious modeling attempts date back to the so-called reaction-diffusion (RD) model [7172], which has been refined successfully in later studies [2568737460]. It relies on an interface reaction involving the interfacial Si  dangling bonds (present in the form of Pb  centers) together with some sort of hydrogen species. Initially, nearly all of the Pb  centers are supposed to be passivated through a hydrogen anneal step. This means that their unsaturated Si  atoms has established a bond to a nearby hydrogen atom H  , thereby shifting the electrically active trap levels out of the substrate bandgap. Upon application of stress, the Si- H  bonds (PbH  ) can break due to the presence of an electric field, thereby activating the forward reaction of

PbH  ⇌  Pb + H .                      (1.14)
With the breakage of the Si- H  bonds, the trap levels associated with the unsaturated Si  atoms are shifted back into the bandgap. Since the released hydrogen atoms can easily rebond to the Pb  centers, the reaction (1.14) is in equilibrium, resulting in a fixed ratio of the concentration of Pb  centers, hydrogen atoms, and Si- H  bonds. The released hydrogen atoms can also diffuse away and are thus not available for the passivation of the interfacial Si  dangling bonds according to reverse reaction of (1.14). This results in a temporally increasing concentration of Pb  centers, measured as a degradation in NBTI. After the removal of stress, the forward reaction of (1.14) is suppressed while the reverse mode dominates the reaction dynamics. It is important to note here that the dynamics in the RD model are eventually governed by the hydrogen diffusion but not by the interface reaction, which has been assumed to be in equilibrium. An alternative reaction involves molecular hydrogen H2   [757677] and presumes that atomic hydrogen dimerizes instantaneously right at the interface according to
 P H    ⇌  P  + H ,                  (1.15)
   b         b
H + H   ⇌  H2 .                      (1.16)
Thus, the generation of interface states is described by the electro-chemical reaction
∂tNit = kf(Nit,0 - Nit) - krNitX1∕a .            (1.17)
Nit  and Nit,0  denote the surface concentration of interface states and its initial concentration, respectively. kf  and kr  stand for the field and temperature dependent forward and the reverse rate of the interface reaction. The kinetic exponent a  determines the migrating hydrogen species X  , that is, 1 for atomic H+ and H0  , and 2 for molecular H2  . Within the RD framework, the interface reaction (1.17) is assumed to be in equilibrium and thus determines the ratio between X  at the interface and Nit  . However, the basis of the RD model is the transport equation (1.18) for the migrating species X  . It is described by the drift-diffusion equation
∂tX  = DX ∂2xX  + ZXμXFox ∂xX  ,                (1.18)
which is coupled to the interface reaction via the boundary condition
a∂tNit = DX ∂xX + ZXμXFoxX .                (1.19)
DX  , μX  , and ZX  are the diffusion coefficient, the mobility, and the charge state of the species X  , respectively. Within the time regime of interest, the dynamics of the interface reaction are governed by the interfacial hydrogen concentration, which in turn is controlled by hydrogen diffusion to and from the interface.

A solution of equation (1.17)-(1.19) can be found as

ΔVth  ≈ t    ,                      (1.20)
where nRD  is referred to as the time exponent. In the case of  0
H  , the RD model yields a time exponent nRD  of 1∕4  , which is only compatible to measurements obtained with a relatively long delay. In more recent studies with a shorter delay, a time exponents close to 1∕6  is obtained, as predicted by the RD model for H2  . However, thorough examinations of the stress/relaxation curves show large discrepancies between the RD theory and the universal relaxation (cf. Fig. 1.7). While the recovery in experiments covers at least 12 decades, the RD model is limited to about 3 decades. This rules out this model as a reasonable explanation for NBTI. Some attempts to remedy this deficiency have been put forward:

Experimentally, the most convincing proof that NBTI is not explained by the RD model comes from the TDDS [5351]. 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 [80818283] 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.  [838182] proposed that the interface reaction can be initiated by protons originating from the substrate.


Figure 1.7: A comparison between simulations, analytical results, and measurement data following universal relaxation [60]. The circles mark measurement data while lines corresponds to simulations. Irrespective of the particular hydrogen species, the recovery according to RD theory extends to only 3 decades which is in strong contrast to experimental findings.

1.5.2 Dispersive Transport

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 [858687] and was combined with the interfacial hydrogen reaction to the reaction dispersive diffusion (RDD) model [6873]. The overall hydrogen H (x,t)  concentration is split into a contribution of free hydrogen Hc (x,t)  in a conduction state and hydrogen ρH(x,Et,t)  residing at traps with an energy level Et  :

H (x,t)  = H (x,t)+ ∫ ρ (x,E ,t)dE  .            (1.21)
           c         H    t     t
The trap dynamics are expressed by balance equations for each trap
                                       (       )
∂tρH(Et)  = Nνc0,H(g(Et)- ρH(Et))Hc - ν0exp - Eck-BETt .    (1.22)
with ν0  being the attempt frequency, Ec  the conduction state for hydrogen, and Nc,H  the effective density of conduction states. g(Et)  stands for an exponential trap distribution. Only the free hydrogen as a migrating species X  is accounted for in the diffusion equation:
∂H    = D ∂2H (x,t)+ Z μ F  ∂ H (x,t)+ ∫ ∂ ρ (E )dE  .   (1.23)
 t c     c x  c       c c ox x c         t H  t   t
Here, the last term reflects the generation of free hydrogen, which escaped from their traps. The particular variants of the RDD model primarily differ in the postulate whether the free hydrogen Hc  in the conduction state [8825] or the total hydrogen concentration H   [8937] can enter the interface reaction (1.19). As pointed out in [31], neither variant of the RDD model can explain the long relaxation tails, irrespective of the assumed hydrogen species (see Fig. 1.8).


Figure 1.8: Simulated recovery curves for the RDD model based on either H  (red) or Hc  (blue) for various dispersion parameters α  . In the case of Hc  the recovery is predicted to end too early while it sets in too late for the Hc  . As a consequence, this model is not capable of reproducing the long lasting relaxation of NBTI.

1.5.3 Reaction-Limited Models

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 SiO2  , the Si- H  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 Eds   [38695890] 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

                   (  Eds)
kf(Eds) = kf,0Nit,0exp - kBT  ,                (1.24)
where k
 f,0  is an attempt frequency. Neglecting the corresponding reverse rate, simple first-order interface kinetics deliver
Nit = Nit,0(1- exp(- kf(Eds)t)) .             (1.25)
With a Fermi-derivative function for the distributions of dissociation barriers
                        ( E-Edsm-)
g(E,E   ,σ ) = -1----exp---σds-------,           (1.26)
      dsm  d    σds(      ( E-Edsm))2
                   1+ exp    σds
the interface state generation follows
ΔNit ≈ Nit,0----t--α-                    (1.27)
           1+ (τ)
τ  = 1∕kf,0 exp(Edsm(Fox)∕kBT) ,           (1.28)
α  = kBT∕σds .                            (1.29)
The field dependence is incorporated in the mean dissociation barrier Edm  , while the temperature activation originates from the spread σds  of the distribution (1.26). Due to a missing reverse rate kr  , this model cannot explain relaxation and must thus be assigned to the permanent component of NBTI. In an improved variant of this model [60], the rate equation (1.25) was extended by a reverse reaction with distributed barriers. Then the universal relaxation behavior can be accurately reproduced but the degradation during the stress phase is drastically underestimated (n ~ 0.03  ). Therefore, this model falls short of capturing both the stress and the relaxation phase at the same time.

1.5.4 Triple-Well Model

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 [919293] motivated the idea of the triple-well model (TWM) [9495] 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.


Figure 1.9: The schematic of the TWM. The double well represents the Born Oppenheimer surface with three stable configuration (states 1  , 3  , and 5  with the energies V1,0  , V3,0  , and V5,0  ) and their separating barriers (V2,0  and V4,0  ). Transitions between two configurations or rather states are indicated by the arrows. Upon application of stress, the energies Vi  are shifted down in energy according to Vi = Vi0 - i× Δ(T)  with Δ(T)  being a parameter and the defects move from state 1  to 5  . When switching back to equilibrium conditions (recovery), the defects return to the state 1  .

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 SiO2  , the first transition mimics the interface reaction involving the hydrogen atom initially bonded to a Pb  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.

1.5.5 Combined Models

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 Si- H  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.


Figure 1.10: Schematic of the coupled double well model (CDW). The left well corresponds to the first step, which is described by a temperature-activated hole trapping process (V1  -V2  -V3  ). The distribution of V3  represents the spread in the hole trap levels, where V2  reflects the required activation energies for this process. The field acceleration of the hole capture process was modeled by the term exp(- Δ1(Fox))  , where the temperature-independent Δ1  referred to as the stress parameter. The second step (V4  -V5  -V6  ) mimics the release of hydrogen analogously to the description of the TWM.