Since both the measurement equipment and the pulse generator are operated at their limits, a few points have to be carefully considered during the final data extraction.
-0.082
Figure 6.3: Left: Different amplification factors in the DSO settings are responsible for the vertical offset (top).
This has to be corrected to make the stress sequences coincide. Merged stress sample (bottom) using a log-fit and shifted to the reference time t0,ref = 2µs. Right: Different reference times t0,ref result in different degradation. It can be seen that fort0,ref = 50µs about 25 % of the ∆ID/ID,0 are missed.
On the other hand, too shortt0,refare not reasonable and result in a spurious shift by a not-yet steady measurement signal (t0,ref= 0.2µs,1µs). Compare with Fig. 6.4.
6.2.1 Offset
Acquisition of 25 kSamples yields 3 to 4 usable decades in time for each sequence. The combined sequences result in 5 to 6 decades in time, with a possibly too large deviation of VG,str from the reference VG,strref set at the DSO during the first decade1. In the remaining decades the data can be either fit by a logarithmic time-dependence
∆ID(tstr)
ID,0 = ID(tstr)−ID,0
ID,0 =−Blog10(tstr/t0,ref) (6.1) withID,0=ID(t0,ref), or a power-law −A(tstr/t0,ref)n with a very small exponentn≈0.04. ID,0 is obtained at stress-level with a delayt0,ref and thus not equal to ID(0) [40] and results in an offset of the relative degradation, see Fig. 6.3 (right).
6.2.2 Initial Measurement as Reference
Unfortunately, the transition from the end of stress to the following recovery is always accompanied by some delay and finite transition times. Effects faster than 1µs are not visible in the experiments.
The delay of the first measurement point serving as an initial reference is often discussed in literature [15, 30, 40, 109]. Some authors [42, 102, 104] argue 1µs to be sufficiently short. However, while a reference time t0,ref taken at 0 s would be the ideal case, the real t0,ref >0 s always depends on the used equipment. Furthermore, different t0,ref strongly influence the following stress behavior, cf.
Fig. 6.3 (right).
1The reason why this happens and its consequences will be explained in more detail in Chapter 6.2.3.
Chapter 6. Short-Term NBTI 55
Figure 6.4: Left: The main graph is enlarged to make the transient and the overshoot of different stress pulses visible which are shown in the inset. This is due to the limited switching speed of the pulse generator when moving fromVG,reltoVG,strand back. The employed error criterion|(VG,str−VG,strref )/VG,strref | ≤ǫis displayed forǫ= 0.3 %. The first (last) proper values of the pulse for each sequence are marked by circles (squares). The noise is apparent in all three sequences and limitsǫto extremely small values. Right:
Logarithmically weighting in time and skipping the first data points corresponding to a parametertskip
does affect the shifting stability but only slightly changes the shift ∆ID/ID,0 (1 %).
6.2.3 Gate Voltage Criteria
In this section it is demonstrated that, in general, fast NBTI measurements have to be taken with a grain of salt. This is largely due to difficulties with synchronization between the stimulus and the actual measurement. So even when the experiment is free of systematic synchronization errors, i.e.
switching of the gate voltage and recording ofID start at the same time, the finite settling time of real signals makes ex-post time zero adjustments necessary. Hence, the time evolution of the actual waveform has to be checked carefully [18]. It turned out that the pulse length is around 0.3 % longer than originally set by the pulse generator. This factor has to be accounted for and the real stress times tstr of the sequences need to be extracted using the applied gate pulse. As shown in Fig. 6.4 the pulse is affected by the transient behavior and a possible overshoot due to the non-instantaneous switching between VG,rel, which is applied in-between the pulses, and VG,str. Therefore, after the transition regime, a steady state value of VG,str is determined and set as VG,strref (usually taken at tstr/2). Then an error criterion, i.e. |(VG,str−VG,strref )/VG,strref | ≤ǫis employed. Since noise is apparent in all three sequences, ǫhas to be chosen large enough to not disrupt the pulse, usually in the range of ǫ ≈ 0.3 %. Starting at tstr/2 and moving as well to lower (to the beginning of the pulse) and higher (to the end of the pulse) times sets new borders of our accepted stress time tstr.
The treatment of the relaxation phase is more complex. It is argued that the noise level is the same during stress and relaxation (the DSO continuously records, using the same settings), and the settling time of the pulse generator in theory is equal regardless if switching from VG,rel to VG,str or vice versa occurred. The criterion for the relaxation phase could then be established as ‘all points extending to both sides of t= 2tstr that fulfill |(VG−VG,relref )/VG,strref | ≤ǫ’. This effectively uses the
10-6 10-5 10-4 10-3
Figure 6.5: Left: The extracted change inIDfor different values ofǫof the relative and absolute truncation criterion is depicted. The logarithmic dependence for longer relaxation times is indicated, too. Right: The main graph is enlarged to make the transient and the overshoot visible. The bounds due to both ‘ǫabs-criterion’
and the ‘ǫrel-criterion’ are displayed for ǫ= 0.3 %, with the first points of the relaxation pulse (after tstr= 1 ms) marked by circles.
sameabsolute allowed deviation from VG,relref as was used during determination of the stress phase, hence this method will be referred to as the ‘ǫabs-criterion’. On the other hand, the relative error inID (and hence in ∆VTH) that would erroneously be attributed to NBTI is given by the relative deviation of VG, asking for a criterion |(VG−VG,relref )/VG,relref | ≤ǫ. This method, which is tighter by a factor of |VG,str/VG,rel| ≈7, is referred to as the ‘ǫrel-criterion’. Both methods were investigated thoroughly, and the relative method was chosen.
6.2.4 Brute-Force Truncation of the Transient
A second possibility to determine tstr is to skip the first data points during the transient until a specific time tskip. This method, displayed in Fig. 6.4 (right), is far easier to implement and gives stable results for various values of tskip. Unfortunately, tskip has to be adjusted manually for every measurement. Hence, the first method is chosen.
6.2.5 Final Setting of Parameters
The finally extracted data is more or less sensitive to the values of the parameters t0,ref and ǫ. For ǫ a value of 0.3 % is used for the stress case, while for relaxation larger values have to be chosen to account for the instantly beginning relaxation. This issue will be dealt with in Chapter 6.5. As can be seen in Fig. 6.4 a t0,ref slightly after the first value should be selected to both eliminate the influence of the first noisy points and delay time. Hence, t0,ref = 2µs appears a reasonable compromise.
Chapter 6. Short-Term NBTI 57
−2.25 V, and −2.50 V) of ∆ID/ID,0 degradation. Scaling to the dotted lines works perfectly for var-ious stress voltages at equal temperatures, while different temperatures lead to a small deviation for tstr>10 ms. The scaling factors are also given. Right: ∆ID/ID,0for different oxide thicknesses (1.8 nm, 2.2 nm, and 5.0 nm) can be scaled as well. Only the thick device is affected by noise due to the low degradation. The graph at the very bottom combines the three dependencies.