Universality of BTI recovery

4.1 Universality of BTI recovery

It has been shown that the reaction-diffusion theory is not able to predict the different observed recovery characteristics following from different stress conditions. Therefore scaling and normalization routines are empirically used to characterize the recovery traces [14, 33, 82, 61].

In Fig. 4.1 the normalization routine is depicted step by step. First, the relative recovery² after different $tstr,i$ is normalized to the last corresponding stress value. Furthermore, all relaxation times are divided by the last stress times, yielding $ξ = t ∕t rel str$ . The normalized data now lie on top of each other and feature the same “universal” behavior. Based on [61], this universal behavior can be described by the universal relaxation function $r(ξ)$ which reads

r(ξ) =---R(tstr,trel)-- = R-(tstr,trel). S (tstr) − P(tstr) R(tstr,0)

(4.1)

Here $R (tstr,trel)$ denotes the relaxation depending on the total stress time $tstr$ and the relaxation time $trel$ , $S(tstr)$ the total degradation observed right at $tstr$ , $P(tstr)$ the permanent component, which only depends on the total stress time, and $R(tstr,0)$ stands for the total amount the device is going to recover from $trel = 0$ . As such the term universality in NBTI was not a new finding in 2007, but was already reported by Denais et al., who claimed that the relative recovery follows the same pattern when plotted over the stress time $ξ = t ∕t rel str$ [82, 61].

Figure 4.1: Demonstration of how universal relaxation works for OTF data of Denais et al.[28]. Starting with the data normalized to the first relaxation value, the second step is to refer the relaxation curves to the last stress values. By then dividing their relaxation time by their corresponding stress time yields perfect universality for all three stress times.

Unfortunately yet no generally accepted and valid theory exists for BTI, which means that the exact form of $r(ξ)$ remains speculative. Empirically derived expressions have therefore been presented so far: Kakalios et al. for example used a stretched-exponential of the form $r(ξ) = exp(− Bξβ )$ to describe “relaxation in disordered systems” [83]. Other empirical “log-like” expressions are $r(ξ) = 1− β log (1 + B ξ)$ [82] or $r(ξ) = β log(1+ B ∕ξ)$ [84]. Also a generalized power-law-like expression after [61]

r(ξ) = 1 ∕(1 + B ξβ),

(4.2)

amongst many others [85, 17, 86, 87], can be used. Whereas equation (4.2) can be used to fit the OTF-data from [82], the first recovery point $R (trel = 0)$ of the MSM-data, depicted in Fig. 4.2, is unknown due to the instant recovery. The first MSM point is obtained with a delay time $tM$ and according to [61, 11] even a $tM = 1μs$ is still too slow. Thus back-extrapolation to reconstruct the “true” initial degradation has been suggested. With $ξM = tM∕tstr$ we get

r(ξ) 1 + BξβM r(ξ-)-= 1-+-B-ξβ . M

(4.3)

Provided that there is a set of relaxation data with different stress times, $B$ and $β$ can be determined, as shown in Fig. 4.2 taken from [11, 61]³ . The slight deviation for $ξ > 102$ is due to the existence of a non-negligible permanent component $P (tstr)$ , which is additionally present here and consequently has to be considered too.

Figure 4.2: When fitting the MSM data of [11] by (4.3), many relaxation traces are required to solve for $B$ and $β$ since the first recovery point $R (trel = 0)$ is unknown due to the measurement delay. The linear behavior is depicted by $1∕r(ξ)− 1$ on a log/log plot (Top). Note the slight deviation for larger $ξ$ , which is due to an existing permanent component and in fact makes the correction of the universal curve (4.2) unavoidable.

[next] [front] [up]