Assumption of a Permanent Component

4.2 Assumption of a Permanent Component

Grasser et al. refined the approach of [6, 88] to obtain universality irrespective of the amount of $P(t ) str$ . In [61] they presented a correction scheme for (4.3), which is based on the determination of the accumulated degradation due to stress after the delay time $tM$ as

S (t ,t ) = R (t ,t )+ P (t ). M str M M str M str

(4.4)

Here, $SM$ splits up into $RM$ , the recoverable amount of degradation monitored at $tM$ , and a permanent components $P$ which is regarded as independent of $tM$ . In [29], $P(tstr)$ was supposed to follow a power-law of the form $Atnstr$ . In order to characterize the temperature and bias dependence of the components of (4.4), plasma-nitrided-oxide (PNO) devices with an effective oxide thickness (EOT) of $1.4nm$ and $2.2nm$ were characterized. Therefore the OTF-method [17] and the fast- $V TH$ -method developed by [11] were used. The latter method is embedded into an eMSM-sequence which is carried out with $N$ stress/relaxation-subsequences, already described in Chapter 2.1. A typical eMSM-measurement is shown in Fig. 4.3.

Figure 4.3: Top: The measurement data (symbols) of an MSM-sequence with very short delay of $1μs$ is matched on the relaxation model (4.5) with $N + 2$ parameters. It yields perfect universality over more than $10$ decades in time (lines). Bottom Left: After subtracting the single $Pi$ of each relaxation sequence (marked above), all data can be fit to a single line. The universality of the relaxing component $R$ is clearly visible. Bottom Right: Without considering $Pi$ , data at large stress times does not conform to the universality.

For the extraction of $R$ and $P$ , the yet unknown permanent contributions of the single relaxation phases $Pi$ have to be determined simultaneously. The remaining non-permanent parts of the relaxation sequences are then fit to the universal relaxation law (4.2). Altogether this yields a relaxation model with $N + 2$ parameters

SM (tstr,i,trel) = R (tstr,i,tM)r(trel∕tstr,i)+ Pi r(tM ∕tstr,i) 1+ B (tM ∕tstr,i)β = R (tstr,i,tM)---------------β + Pi. (4.5) 1 + B (trel∕tstr,i)

Therein, $B$ and $β$ are fit parameters for the universal recoverable component $R$ , and the $Pi$ with $i = 1 ...N$ denote the $N$ relaxation sequences which have to be optimized. The results of the optimization loop are then illustrated in Fig. 4.4, clearly showing the existence of a permanent (or slowly relaxing) component [30], when the recovery levels off. In contrast to $R$ , which can be fitted by a power-law or $nlog(1+ Atstr)$ , $P$ behaves like a power-law for shorter stress times only. It clearly shows signs of saturation at longer stress times, which is fundamental for lifetime extrapolation. Without considering such a permanent component, universality is not given, like shown in Fig. 4.3 (bottom left).

Figure 4.4: The original measured data $SM (tstr,tM ≈ 1μs)$ and the extracted recoverable and permanent components $R$ and $P$ . By back-extrapolating $SM(tstr,trel = tM )$ the “real” total degradation $S (tstr,trel = 0)$ is obtained, consisting of $R + P$ (compare to the true degradation depicted left). The relaxation data in-between the stress sequences is indicated by dash-dotted lines on a relative time scale $tstr,i + trel$ . The recoverable component $R$ can be well fit by either a power-law or $nlog(1+ Atstr)$ (used in the following) while $P$ closely follows $Pmax ∕(1 + (tstr∕τ)−α)$ after [6].

Moreover, it is of utmost importance to study wide relaxation periods, as the data gained that way yields a much more reliable basis for modeling, compared to other measurements done on commercial equipment: While Alam et al. covered about $3$ decades in time [85, 49], the widest recovery behavior observed with commercial equipment so far accounted for $5$ decades time [6, 66, 40]. Using their dedicated equipment Reisinger et al. [11] were able to measure BTI relaxation periods of $10$ to $12$ decades in time with the shortest available delay time of $1μs$ , cf. Fig. 4.3.

Applying the universality on various pMOS/nMOS-NBTI/PBTI-combinations yields different quantitative, but all in all consistent results. Surprisingly, this also applies to the negative shift of the threshold voltage they all have in common, for details refer to Fig. 4.5 and Fig. 4.6.

Figure 4.5: The universal relaxation during PBTI/NBTI stress depicted for pMOS transistors (symbols: data, lines: model). The samples have a much smaller EOT of $1.4nm$ compared to the devices of Fig. 4.3 and Fig. 4.4 with $2.2nm$ , a considerably larger $R$ but a comparable $P$ component. The recoverable component during PBTI stress is qualitatively the same but shifted by a factor of $3$ to smaller values, while the permanent component is quite similar compared to NBTI.

Figure 4.6: The universal relaxation during PBTI/NBTI stress depicted for pMOS and nMOS transistors (symbols: data, lines: model). The samples have a much smaller EOT of $1.4nm$ compared to the devices of Fig. 4.3 and Fig. 4.4 with $2.2nm$ , a considerably larger $R$ but a comparable $P$ component. Comparison of pMOS/NBTI and nMOS/PBTI stress. The recoverable component of the nMOS is very small ( $6mV$ ). Hence, the degradation is dominated by $P$ from an early stage when comparing it to pMOS.

4.2.1 Temperature and Voltage Dependence of Universal Law
4.2.2 Measurement Delay

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