Distribution of Defects

9.2.1 Distribution of Defects

Compared to the very small devices described in [116, 111], where a discussion of a single defect makes sense and one is indeed able to resolve the state in which the defect currently is, this is not the case for the large area devices considered in this chapter, where a large number of differently behaving defects is located in the oxide. Therefore, a distribution of defects regarding their characteristics has to be assumed, with all the defects featuring different energies $E1$ , $E1 ′$ and $ϵ1′1$ . Furthermore, different positions $xT$ inside the oxide and relaxation energies $S ℏω1$ and $Sℏ ω2$ are assumed.

As a reasonable discretization of the defect characteristics is not feasible, the defect properties are considered using a statistical approach, which is shown in Fig. 9.5. Whereas $E1$ , $E1 ′$ , and $ϵ1′1$ are distributed gaussian, $xT$ , $S ℏω1$ , and $Sℏω2$ are distributed uniformly within the depicted ranges. All other parameters of the model in (9.1) to (9.4) are taken to be constant.

By chosing 1000 different defects a gaussian profile can already be well approximated, cf. Fig. 9.5 (top left). Further increasing the number of defects (top right) to 100000 yields a nearly perfect gaussian profile. The same numerical improvement is observed for the relaxation energies in Fig. 9.5 (bottom right). However, as will be shown later, a number of 1000 representative defects in a defect band, like depicted in Fig. 9.5 (bottom left), is sufficient to describe the measurement results properly.

Figure 9.5: Distributed quantities of the defects. Top Left: The determining energies and barriers of state $1$ are assumed to be gaussian distributed for 1000 defects. Note that the gaussian is intentionally cut off above $6eV$ , since barriers above can be not surmounted within the monitored timescale. Top Right: When the number of defects is increased by two decades a nearly perfect gaussian distribution is obtained. Bottom Left: The defects are uniformly distributed within around $0.8nm$ from the interface. Note the already charged defect above the Fermi level for the depicted equilibrium condition. Bottom Right: The relaxation energies for both stable defect states $Sℏω1$ and $S ℏω2$ are uniformly distributed within $0eV$ and $1.1eV$ or $1.5eV$ , respectively.

When applying NBTI stress, the defect band is shifted with respect to the valence band. Hence only a part of all present defects is able to contribute to the degradation depending on the applied bias conditions. As an example of which fraction of the defects can become charged, the occupied defects are marked as filled within the defect band in Fig. 9.6 (top) after $2000s$ of stress. When switching back to relaxation the defect band is lowered again and the defects can become discharged.

However, the defect occupancy is not only determined by $E1$ and $xT$ anymore as it is the case after the SRH-like approach of Kirton et al. [124], where all defects up to a certain distance $xT$ become charged as a direct consequence of the tunneling front due to the WKB factor. The addition of a barrier $ΔEB$ for all defects only changes the level above and below which the defects can become charged and discharged, respectively. In contrast to this there are also empty defects inbetween the filled ones in terms of both $E1$ and $xT$ after the multi-state defect model, cf. Fig. 9.6 (top). Unfortunately this means that it is no longer possible to estimate the time dependence of the total degradation as an analytic expression, as it was possible for the SRH approach on basis of $xT$ in first order. Due to the superposition of many defects, which are further distributed in different quantities, no kind of degradation estimation does make sense for the large devices investigated in this chapter.

Before further discussing the repeateded charging and discharging of the switching traps and their time dependence within the performed stress/relaxation cycles, the reservoir conditions of the substrate providing the holes have to be set.

Figure 9.6: The oxide defects are distributed over energy and depth and are marked filled when occupied. When switching from relaxation (Center and Bottom) to stress (Top) the defect band is shifted upwards. The two graphs describe the last relaxation sequence for $∘ 125 C$ and $Vstr = 1.83V$ . Top: At the end of stress more than $9%$ of the possible defects are filled. Center: After only $0.1s$ of relaxation around $3%$ of the defect are still filled. Bottom: After about $400s$ relaxation only $1%$ are left. This corresponds to nearly complete relaxation of the oxide traps.

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