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

5.5 Comparison of the interpretation methods

The above described interpretation approaches for the temperature activation of NBTI stress and recovery appear to be comparable. Through a more close analysis of the result of the temperature-time approach an important similarity becomes apparent: E.g. the NBTI recovery data in Fig. 5.12 analyzed with the temperature-time of Section 5.2 shows one single recovery trace which seems to follow a cumulative normal distribution function. Indeed, assuming that the emission time constants of the defects activated during NBTS are log-normally distributed, it follows that recovery has to obey [Gra+11a]

(5.16) \begin{equation} \gls {dVth}^\tn {rec} = \frac {\gls {dVthmax}}{2} \operatorname {erfc}\left ( \frac { k_\tn {B}T \ln \left (\frac {\vartheta _\tn {rec}}{\gls {tau0}}\right )-\mu }{\sqrt {2}\sigma }
\right ), \label {eq:dVthtrec} \end{equation}

where (math image) is the maximum drift after the given stress, \( \mu , \sigma \) are the parameters of the normal distribution of the energy barriers and \( \operatorname {erfc} \) is the complementary error function [PG13b]. In Fig. 5.31 a comparison of this model with the (math image) accelerated recovery data is given.


Fig. 5.31: Cumulative normal distribution function (5.16) fitted to the (math image) accelerated NBTI recovery data of Fig. 5.12 [PG13b].

The mean and the variance of the distribution are given in eV. The Arrhenius equation for the time constants (5.3) allows transforming the mean of 0.71 eV at 100 °C with the experimentally determined (math image) value of 10−15 s to a mean emission time constant value of 10−5 s. This means a stress with 10 s duration charges defects with a mean emission time constant of 10−5 s and most experiments with the first measurement point around 10 ms after termination of stress do not capture the mean but only the long time constant part of the distribution. This is due to the large variance of the emission time constant distribution. The approach assumes that the distribution of the emission time constants is due to a distribution of emission activation energies, an assumption which cannot be unequivocally verified at the time.

Similarly, also for the stress time dependence of NBTI a cumulative log-normal distribution function as

(5.17) \begin{equation} \gls {dVth}^\tn {str} = \frac {\gls {dVthmax}}{2} \operatorname {erfc}\left ( -\frac {k_\tn {B}T \ln \left (\frac {\vartheta _\tn {str}}{\gls {tau0}}\right ) - \mu }{\sqrt {2}\sigma
} \right ), \label {eq:dVthtstr} \end{equation}

where (math image) is the maximum drift capability of the device for a given stress oxide field, is evident. Fig. 5.32 compares a large data set with a wide range of time and temperature values with the prediction of the reaction–diffusion (RD) theory [AM05; Mah+11] and the assumption of log-normally distributed time constants.


Fig. 5.32: Comparison of (math image) accelerated NBTS data with the assumption of log-normally distributed capture time constants and the RD model [PG13b]. From top to bottom are plotted: a) The derivative of \( \gls {dVth}(\lg \gls {tstr}) \) giving the probability density function, b) the semi-logarithmic \( \gls {dVth}(\lg \gls {tstr}) \) showing the cumulative distribution function (5.17), c) the log-log diagram common for NBTI research and d) its derivative giving the change of the power law coefficient with stress duration.

Fig. 5.32 b) is the semi-logarithmic plot of the degradation versus the stress time. The derivative of this function gives the probability density function of the distribution in Fig. 5.32 a). A good match between the data and the log-normal distribution is evident. Fig. 5.32 c) is the data plotted on log-log scale which shows a power law for short stress times. For larger stress times the data clearly deviates from this power law. A slow but continuous decrease of the power law coefficient is observed as shown in Fig. 5.32 d). It is mandatory for the degradation to saturate from a microscopic perspective, since eventually all possible defect sites have become charged at a certain time. But the details on how the degradation saturates is a very interesting observation which has been mentioned only rarely [HDP06; RMY03]. This is primarily due to the logarithmic nature of NBTI degradation and the consequential extreme need of measurement time. Consequently, the approach enables a convenient study of the long-term behavior of BTI. Bear in mind that the Arrhenius temperature activation of the time constants (5.3) together with the experimentally obtained (math image) value of 10−9 s allows transforming the (math image) axis to an activation energy axis as shown in Fig. 5.32 a). Also, a mean capture time constant value of 1012 s at 30 °C can be calculated from the mean value of the distribution. This large value indicates that most of the defects of the MOSFET, which can be activated for a given stress gate voltage, have a capture time constant which is much larger than the intended lifetime of the device of ten years or \( \approx \SI {3e8}{\second } \). Consequently, during the lifetime of the product under use conditions only the tail of the distribution of all defects becomes charged.

To conclude, the temperature-time approach described in Section 5.2 is able to handle arbitrary distributions of time constants. By applying this approach to NBTS data measured at high temperatures and long time ranges it is observed that the emission and capture time constants for NBTI defects are log-normally distributed. A possible reason for the log-normal distribution of the time constants could be a normal distribution of activation energies due to the Arrhenius temperature activation of the defect time constants (5.3). In accordance, the temperature-time approach and the distributed activation energies approach give equivalent results. This is not surprising since both methods rely on the similar, though important, assumption of constant scaling factors for all temperatures; (math image) for the temperature-time approach and (math image) for the distributed activation energy approach.

Fig. 5.33 shows a comparison of the activation energy values obtained by the temperature-time and the distributed activation energy methods over the stress oxide field.


Fig. 5.33: Stress electric oxide field dependence of the activation energy for MOSFETs with 30 nm SiO2 or 2.2 nm SiON gate dielectrics. The extraction using the temperature time approach is compared with measuring the (math image) distribution directly.

The colored areas around the data points indicate the confidence interval of the fit of the distribution but miss the impact of (math image) or (math image) on the mean and variance values. All technologies available in this thesis are considered, including Si based MOSFETs with either 30 nm SiO2 or 2.2 nm SiON gate dielectrics. Also a comparison to previously published data [HDP06] is included. Both methods and all technologies give comparable mean activation energy values. However, there are still differences between the methods. Especially for the SiO2 data no difference is expected because the measurements are performed on the same technology. Naturally, a better agreement between the two methods can be achieved when the same data sets are used. However, the in Fig. 5.33 observed deviations may arise from the following origins:

  • • The two MSM experiments were recorded at different chuck temperatures. The distributed (math image) measurement was recorded at 30 °C while the temperature time measurement was recorded at 180 °C. Both measurements had the same time delay of 10 ms and are differently affected by recovery. Provided the emission and capture activation energies are correlated [Gra+11a], the particular amount of recovery could affect the extracted mean capture activation energies.

  • • The two data sets are measured on two different devices, which always includes small device variations, as well as on two slightly differently processed wafers. The impact of processing variants on the mean activation energy could unfortunately not be analyzed in this thesis.

  • • The temperature-time data lacks the point of inflection in the \( \gls {dVth}(\lg \gls {tstr}) \) plot which indicates the mean of the time constants distribution. Consequently, the mean value of the distribution is not fully fixed and also largely dependent on measurement errors of the last (math image) values.

  • • Both methods rely on the experimentally determined constants (math image) and (math image) which are mean values representing distributed variables. The particular choice of (math image) or (math image) has a certain impact on the extracted (math image) value which could affect the conclusions. However, for the distributed (math image) approach different values of (math image) mainly shift the \( \gls {Ea}(\gls {Eox}) \) dependence upwards or downwards, respectively. Considering Fig. 5.33, this would not allow to fully match the two methods.

Despite the uncertainty regarding the exact activation energy values, the approaches presented in this Section show that NBTI is not an intrinsic low energy effect as occasionally stated [MKA04; AM05; AWL05]. On the contrary, NBTS only activates the low energy fraction of a wide distribution of activation energies [Tya+09]. The obtained (math image) values do not exactly point to dissociation energies of certain defect precursors in the Si-SiO2 system which would identify the defect precursors responsible for NBTI. But the values are all within the expected range of 1.5 eV to 2.8 eV [Bro90; Sta95a; Ste00], which is consistent with other recent results [Yon13].