Multi-State Defect Model

9.2 Multi-State Defect Model

Although the non-radiative multi-phonon (NMP) model of Chapter 8.5.3 appears to be the best modeling approach so far, it still suffers from some limitation when used for BTI. First it is not able to fully explain the uncorrelated behavior of the capture and emission times as observed experimentally and second it does not give a stronger than linear field dependence of the capture time constants of the defects though these are observed [169].

When modeling random telegraph noise (RTN) twenty years ago Uren et al. [170] suggested that individual states can exist in more than one charge-equivalent, so-called metastable states. Based on this idea the two possible states of the NMP model are now extended to a four-state defect system [111]. Such a multi-state defect model is depicted in Fig. 9.3 (top) for the oxygen vacancy. It contains two metastable defect states $1′$ and $2′$ which belong to the already used stable states $1$ and $2$ , respectively. As can be seen the transitions between $1$ and $2$ now have to proceed over one of the metastable states. This picture is similar to that already used in the two-stage model, cf. Chapter 8.4, where the oxygen vacancy upon hole capture (state 3) was regarded to be in a kind of metastable state with the choice to either structurally relax or to recapture a hole again. In the multi-state defect model, again the oxygen vacancy is used as switching oxide trap [147, 111].

With the help of a schematic reaction coordinate diagram in Fig. 9.3 (bottom) the transitions of a single defect state are now explained. The neutral defect state $1$ and the charged defect state $2$ are depicted together with their corresponding metastable states $1′$ and $2′$ . Upon the application of a stress bias, the charge transfer reaction from state $1$ to $′ 2$ is favored. This is indicated by the dashed upwards shift of the parabola in Fig. 9.3 (bottom) and leads to a strong electric field dependence of the barrier [111, 157]. As $2′$ is metastable it can relax into its stable form $2$ afterwards.

Figure 9.3: Top: The improved non-radiative multi-phonon (NMP) model includes a metastable defect state for both charge states. The possible transitions exhibit a charge exchange and/or a structural relaxation. Bottom: Schematic reaction coordinate diagram model for a single defect. The different configurational potentials for each defect define the NMP process. The varying bias conditions (stress/relaxation) further change the relative position of the potentials and determine the transition rates from $1$ to $2$ (over the metastable $1 ′$ and $2′$ ) and back.

By performing DFT calculations in crystalline $SiO2$ , Schanovsky et al. concluded that the oxygen vacancy does not fulfill all requirements of the multi-state model. This is because its thermodynamic energy level of around $1eV$ above the $SiO2$ valence band results in a very high barrier for the capture process, which can only be surmounted at very high oxide electric fields. Unfortunately the necessary fields are around $20MV ∕cm$ when assuming the defect to be localized $1nm$ inside the oxide[151]. Another problem of the oxygen vacancy is that its neutralized puckered state $′ 1$ is too unstable to allow a switching trap behavior between state $1′$ and $2$ , as the defect would rather relax back to its initial state $1$ immediately $2$ [138].

For the sake of completeness also the hydrogen bridge is briefly discussed. According to first principle calculations its energy level was determined to lie within the Si bandgap, meaning that the defect configuration is already positively charged prior to stress. Therefore the hydrogen bridge is ruled out as possible defect state when dealing with BTI as well [151].

However, no matter what exact defect configuration is responsible for BTI, the multi-state defect model captures the essence of BTI very accurately and will therefore be used in the following. Its hole capture and emission rates are derived similarly to Chapter 8.1, with the barriers based on the NMP formalism, cf. Appendix D.2

k12′ = σpvth,ppexp−βϵ12′, k2′1 = σpvth,pNv exp−βϵ2′1, (9.1) ′ −βϵ1′2 ′ −βϵ21′ k12 = σpvth,ppexp , k21 = σpvth,pNv exp (9..2)

The indices of the barriers $ϵij$ in the rates $kij$ hold for the corresponding transitions from $i$ to $j$ with the cross section $σp$ and the thermal velocity of holes $vth,p$ . Furthermore, $p$ and $Nv$ denote the hole concentration and the effective valence band weight, respectively. Since the stable and their corresponding metastable states are only separated by a thermal barrier $ϵ ′ ii$ , these barriers are bias independent. Therefore $ϵii′$ is not calculated via the intersection of the parabolas, as it needs to be done for (9.1) and (9.2), but is an explicit parameter together with an attempt frequency $σ0 ≈ 1e13s− 1$ after [111]

−βϵ′ −β(ϵ′ +E ′− E1) k1′1 = σ0 exp 11, k11′ = σ0 exp 11 1 , (9.3) k2′2 = σ0 exp−βϵ2′2, k22′ = σ0 exp−β(ϵ2′2+E2′− E2) (.9.4)

The rates correspond to the barriers and energies in Fig. 9.4. When the effective transition from state $1$ to $2$ over the metastable state $′ 2$ is considered this exhibits a two step process, which is proportional to the product of the first rate $k12′$ times the second rate $k2′2$ which has to be divided by the sum of all rates contributing to state $2′$ , $k12′ + k2′1 + k2′2 + k22′$ [130]. Neglecting the last rate $k22′$ because it will be smaller than $k2′2$ due to the fixed thermal barrier at all times and adding the path over the metastable state $1′$ in the same manner yields

k = ----k12′k2′2-----+ ----k11′k1′2-----. 12,eff k12′ + k2′2 + k2′1 k11′ + k1′2 + k1′1

(9.5)

Analogously the effective transition from $2$ to $1$ over the metastable states can be derived to be

-----k22′k2′1----- ----k21′k1′1------ k21,eff = k22′ + k2′1 + k2′2 + k21′ + k1′1 + k1′2

(9.6)

These reaction rates then define the specific capture and emission time constants $τc = 1∕k12,eff$ and $τe = 1 ∕k21,eff$ , respectively, within which the single defect is charged and discharged on average.

During stress the effective forward rate $k12,eff$ can be approximated to the transition over state $2′$ , as already indicated in Fig. 9.3 (bottom). However, during relaxation transitions over both metastable states contribute, also indicated by arrows. After Grasser et al. [111] switching trap behavior in the multi-state defect model is only observed when both barriers between state $2$ and $1′$ are rather small compared to $ϵ22′$ . Consequently, switching traps favor the backward process over $1′$ . For defects featuring a large barrier $ϵ21′$ on the other hand no such switching trap behavior can be observed, since they practically never reach state $1′$ , they have to recover via $2′$ .

Figure 9.4: The quantities used in the multi-state defect model, taken from [111]. Since the reaction coordinates describing the transition $1 ↔ 2′$ differ from those describing $′ 2 ↔ 1$ , the harmonic potentials describing states $1$ and $′ 1$ (blue) are plotted twice.

9.2.1 Distribution of Defects
9.2.2 Reservoir of Holes - Classical vs. Quantum Mechanical Description

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