Two-Stage Model

8.4 Two-Stage Model

For a microscopic model describing the stress during BTI and the relaxation afterwards the oxygen vacancy and the silicon dangling bond are the most likely candidates. The reason for choosing these two defect configurations is that they have been frequently reported to be involved in reliability issues. According to Lenahan silicon dangling bond defects dominate deep levels in the oxide [139]. The oxygen vacancy $Si–Si$ has been used to explain radiation damage [140, 141] and flicker ( $1∕f$ ) noise [136, 142] so far.

In the models presented in [143, 124, 136] holes can be captured via a thermally activated multi-phonon emission (MPE) process into states deep in energy but close to the interface, named border traps [144, 99]. Since the MPE process, which will be elaborately explained in Appendix D, was originally derived for bulk semiconductors [125], it cannot be directly used.

Therefore, Grasser et al. proposed a two-stage model in [98], which contains an extended MPE process, named multi-phonon field assisted tunneling (MPFAT) process which is similar to the one used in [126, 145]: After [125, 145] the probability of a thermionic transition of a hole over a barrier $ΔEB$ is $exp(− βΔEB )$ . When applying an oxide electric field, the transition probability is further found to be increased by $2 2 exp(Fox∕Fox,ref)$ , with the oxide electric field $Fox$ and a scaling factor $Fox,ref$ [98]. This MPFAT process is schematically depicted in the first stage of Fig. 8.4 (state $1 → 2$ ). From a defect point of view, the initially assumed neutral oxygen vacancy is charged positively via hole capture. The excess energy of the defect system is subsequently released by structural relaxation [98, 146].

Figure 8.4: The two-stage model starts with a neutral precursor in (state 1). Upon hole capture, the $Si–Si$ bond breaks and a positively charged $E ′$ center is created (state 2). Upon hole emission (electron capture) the $E ′$ center is neutralized (state 3). Then being in this state, there are two options. Either hole capture again moves the defect back to (state 2), making it act as switching trap, or the defect structure totally relaxes back to its equilibrium configuration (state 1). The transition between stage-one and stage-two is assumed to be via hydrogen transition between state 2 and 4. The dangling bond of the switching trap in (state 2) can capture a hydrogen atom, leaving back a dangling bond at the interface. This passivation by the hydrogen effectively locks the defect state in the positive charge (state 4).

In this way positive $E ′$ centers (state 2) are created which can now emit a hole and transfer to (state 3). Being at (state 3) the neutralized defect has either the choice to capture a hole and act as a switching trap by hopping between state 2 and 3 [147], or to fully relax back to its initial precursor state again (state $2 → 3 → 1$ ). Each path finally leads to (state 1). Therefore, this stage-one describes the recoverable part of the charge trapping.

When the two-stage model is fitted to experimental data of $SiO2$ , $SiON$ , and high-k devices, the strong (quadratic) voltage and (linear) temperature dependence is predicted correctly supporting the theory of a broad distribution of energetic defects. By involving both oxide charges and interface states contributing to BTI, the model is furthermore able to describe the asymmetric behavior during stress and recovery and the strong bias sensitivity during recovery [98].

The coupling is established via the transition of a hydrogen atom located at the interface between state 2 and 4. This transition is determined by the hole concentration (positive $E′$ centers) and by the number of hydrogen passivated silicon dangling bonds both available at the interface, i.e. the occupancy of (state 2). When the defect is moved from (state $2 → 4$ ), the dangling bond of the oxide defect becomes passivated, leaving back a $P b$ center [139]. Since a $P b$ center is a rather stable configuration compared to the switching trap, i.e. the transition rates (state $2 ↔ 4$ ) are larger than the switching trap rates (state $2 ↔ 3$ ), the positive defect is hence locked. Consequently, an increased number of defects being in (state 2) favours the creation of permanent states. Mathematically the transition between state 2 and 4 is modeled by thermal activation over a field-dependent barrier, which besides the different ground states $E2$ and $E4$ , and its corresponding dissociation barrier $E d$ heavily depends on the applied field $F ox$ , cf. Fig. 8.5 [98].

Figure 8.5: The hydrogen transition between state 2 and 4 is modeled by assuming a field-dependent thermal transition over a barrier $Ed$ [98]. Without applied electric field this results in the solid defect configuration. When applying an electric field $Fox$ , the barriers are altered. Consequently the transition rates are changed by the factor $exp (± βγFox )$ with respect to the zero-field case. Here ‘ $+$ ’ holds for the transition ( $2 → 4$ ) and ‘ $−$ ’ for ( $4 → 2$ ), with $γ$ being the dipole moment [62]. The transition rates are highlighted by differently thick arrows, indicating that the transition ( $2 → 4$ ) is favored, while that of ( $4 → 2$ ) is suppressed by the higher $Fox$ .

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