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 temperaturetime approach an important similarity becomes apparent: E.g. the NBTI recovery data in Fig. 5.12 analyzed with the temperaturetime 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 lognormally distributed, it follows that recovery has to obey [Gra+11a]
where is the maximum drift after the given stress, are the parameters of the normal distribution of the energy barriers and is the complementary error function [PG13b]. In Fig. 5.31 a comparison of this model with the accelerated recovery data is given.
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 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 lognormal distribution function as
where 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 lognormally distributed time constants.
Fig. 5.32 b) is the semilogarithmic 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 lognormal distribution is evident. Fig. 5.32 c) is the data plotted on loglog 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 longterm behavior of BTI. Bear in mind that the Arrhenius temperature activation of the time constants (5.3) together with the experimentally obtained value of 10^{−9} s allows transforming the axis to an activation energy axis as shown in Fig. 5.32 a). Also, a mean capture time constant value of 10^{12} 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 . Consequently, during the lifetime of the product under use conditions only the tail of the distribution of all defects becomes charged.
To conclude, the temperaturetime 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 lognormally distributed. A possible reason for the lognormal 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 temperaturetime 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; for the temperaturetime approach and for the distributed activation energy approach.
Fig. 5.33 shows a comparison of the activation energy values obtained by the temperaturetime and the distributed activation energy methods over the stress oxide field.
The colored areas around the data points indicate the confidence interval of the fit of the distribution but miss the impact of or 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 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 temperaturetime data lacks the point of inflection in the 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 values.

• Both methods rely on the experimentally determined constants and which are mean values representing distributed variables. The particular choice of or has a certain impact on the extracted value which could affect the conclusions. However, for the distributed approach different values of mainly shift the 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 values do not exactly point to dissociation energies of certain defect precursors in the SiSiO2 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].