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10 Advanced Step Detection Algorithm

In the previous sections the TDDS and the measurement setup based on the TMI has been discussed in detail. Furthermore, it has been mentioned that in nanoscale transistors the recovery after the device has been subjected to NBTI or PBTI stress typically proceeds in discrete steps. To further analyze the individual constituents contributing to the recovery behavior, an elaborate step detection algorithm is required. For that the advanced bootstrapping and cumulative sum (BCSUM) algorithm is presented in this section. After the requirements are briefly introduced, the cumulative sum charts [167, 168] and the bootstrapping mechanism which are the key components of the algorithm are discussed. Finally, the functional interplay is presented.

10.1 Requirements

To study the trapping kinetics of single traps, the recovery traces have to be analyzed for discrete steps. Thus the algorithm has to be able to detect abrupt changes in various signals which

  • 1. are uniformly or non-uniformly sampled, and thus

  • 2. need not to exhibit a constant variance over time, i.e. a constant signal noise level, and which

  • 3. contain positive and negative discrete steps.

Furthermore, the algorithm should be configurable with just a few parameters and has to be executable with reasonable computational effort.

In general, the measurement window of the TDDS can be very large. The emission times of the single traps contributing to BTI have widely distributed time constants, from microseconds over days up to weeks or even years. A measurement window spreading over several decades in time goes hand in hand with the necessity of finding a trade off between the sample intervals and the amount of measurement data recorded. An adjustment of the sampling time intervals to higher values for larger recovery times is therefore necessary. Thus, for recording the recovery traces a non-uniform sampling scheme with usually 200 datapoints per decade is preferred to uniformly sampled data. The non-uniform sampling provides a high sampling rate immediately after switching from stress to recovery bias conditions to avoid the loss of any important information. For very long recovery traces a feasible number of data samples is achieved. In order to link the sampling intervals with the signal bandwidth the scaling property of the Fourier transform [169]

(10.1) \begin{equation} r(at) \xLeftrightarrow [\mbox {}]{\quad \mathcal {F}\quad } \frac {1}{a} R\left (\frac {\omega }{a}\right ) \end{equation}

has to be considered. Using this relation a compression in the time is transferred domain into a dilatation in the frequency domain and vice versa, \( \mathcal {F} \) denotes the Fourier transform operator and \( R(\omega ) = \mathcal {F}\left \{r(t)\right \} \) is the Fourier transform of the time signal \( r(t) \). An decade-wise adjustment of the sampling rate directly decreases or increases the noise power of the measurement signal when the sampling rate is decreased or increased, respectively. It is therefore necessary that the step detection of the advanced algorithm can operate on non-uniformly sampled measurement data as well.

The experimental investigation over the last years performed on different technologies showed that a large number of single traps tend to produce fast RTN signals with \( \tauc \approx \taue     \). Such RTN signals are typically observed over many decades showing the same capture and emission time. By using a non-uniform sampling scheme, the sampling time intervals get increased and as a consequence the RTN signal is visible as very short pulses in the traces at high recovery times. Furthermore, in most cases peaks in the measurement data caused due to RTN consists of just a few datapoints, which makes the detection of RTN very difficult. Nonetheless, the introduced BCSUM algorithm is able to detect such short pulses present in our measurement data.

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