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Degradation of Electrical Parameters of Power Semiconductor Devices – Process Influences and Modeling

5.4 Other interpretations

In this Section a few other approaches to understand the temperature activation of BTI stress and recovery are given.

5.4.1 Direct interpretation

The common approach is to draw BTI data measured at \( \gls {Trec}=\gls {Tstr} \) in an Arrhenius plot as (math image) over \( 1/(k_\tn {B}\gls {T}) \) and to extract the activation energy by fitting a linear function to the data. This assumes an Arrhenius equation for as

(5.15) \begin{equation} -\gls {dVth} = -\Delta V_{\tn {TH},0} \exp \left ( \frac {E_\tn {A}}{k_\tn {B}T} \right ), \end{equation}

which actually predicts a negative activation energy value for increasing degradation with increasing temperature.

Such an interpretation leads to absolute activation energy values in the range of 0.019 eV to 0.58 eV [JS77; OS95; MKA04; AWL05]. It is remarked that all these values are obtained through measurements where \( \gls {Tstr}=\gls {Trec} \), a limitation which has been already shown in the previous Sections to cause questionable results. However, even for analyses of \( \gls {Tstr}\neq \gls {Trec} \) data as e.g. from Fig. 5.4, (math image) values within a 0.13 eV and 0.14 eV range are obtained as shown in Fig. 5.26.


Fig. 5.26: Standard approach for the determination of the activation energy extracted from Fig. 5.4 at stress times 1 s and 2 × 104 s between 35 °C and 255 °C. (math image) is constant and only (math image) is varied.

The in this way extracted activation energies are rather low and thus in stark contrast to chemical activation energies for the dissociation of atomic bonds in the Si-SiO2 system of about 1.5 eV to 2.91 eV [Ste00].

5.4.2 Ensemble of time constants

Motivated by the temperature activation of individual defects as described in Section 5.2.1, the \( \gls {Tstr}\neq \gls {Trec} \) data could also be interpreted in the following way (cf. also Fig. 5.27):


Fig. 5.27: Threshold voltage drift in an MSM experiment with constant (math image). The three arrows show the decrease of the time constants for certain levels of (math image).

On a micrometer-sized device the individual charging or discharging event of a single defect is not resolvable since the step size is too small and there are simply too many defects to distinguish. Still, a group of defects with short capture time constants will shift the (math image), e.g. up to 16 mV, before a second group of defects with slightly larger capture time constants will shift the (math image) another 10 mV to a total drift of 26 mV. When the experiment is repeated at a higher (math image) the first group of defects will, provided the two groups have a similar activation energy, still be charged before the second group. Also, the (math image) of the first group will not change. Consequently, by searching for the occurrence of \( \gls {dVth}=\SI {16}{\milli \volt } \) in the \( \gls {dVth}(\gls {tstr}) \) data at both stress temperatures it is possible to extract the change of the time values of the group of defects with temperature. Fig. 5.28 shows the dependence of the stress time needed for 16 mV, 26 mV and 40 mV of drift over stress temperature.


Fig. 5.28: Change of the time constants of Fig. 5.27 for three example values of (math image) in an Arrhenius plot. A good correlation to the Arrhenius equation is evident.

The decrease of the time constants for the three groups indeed follows an Arrhenius equation as suggested from individual defects experiments.

This idea can now be applied with a finer grid on the (math image) axis and the activation energy (math image) as well as the minimum time constant (math image) can be extracted for every group. Fig. 5.29 shows the results of such an interpretation for two example experiments for the activation energy (math image) and Fig. 5.30 for the minimum time constant (math image), respectively.


Fig. 5.29: Extracted activation energy (math image) for numerous different (math image) values of the MSM experiments of Fig. 5.27 (blue) and Fig. 5.4 (green). The shaded areas indicate the borders of a 95 % confidence interval of each fit.


Fig. 5.30: Extracted (math image) values for a large number of different (math image) values from the data of Fig. 5.27 (blue) and Fig. 5.4 (green). The thin lines are the borders of 95 % confidence intervals.

The (math image) and the (math image) values are comparable to the values of individual defects as described in Section 5.2.1: (math image) from 0.4 eV to 0.99 eV, (math image) from 10−12 s to 10−7 s. Considering the less intense experiment (blue characteristic in Fig. 5.29 or 5.30) the (math image) and (math image) values are rather well-defined and the error becomes larger only at the border regions of low or high (math image) where only a few data points are available. In contrast, for the higher temperature experiment (green characteristic in Fig. 5.29 or 5.30) the confidence interval of the fit is always rather large. This is because in the Arrhenius plot the wide range data shows a certain degree of curvature which can only be approximated by a linear fit. This is another suggestion that the processes occurring during NBTS do not show a single valued activation energy but rather a distribution of activation energies. If several values of (math image) are evident the in this Subsection presented concept does not apply any more because the occurrence of a group of defects can invisibly exchange with another group, making it impossible to identify an individual group of defects solely by the (math image) value.

However, the approach shows that approximately the same activation energy and minimal time constant values can be obtained on micrometer sized devices and not only on nanometer sized devices [Gra+10a].