7.6 Emission Time Constants

The degraded VTH   in small-area transistors with only a few defects relaxes in discrete steps. Each step reveals a hole emission event at the emission time τe,i = τ0exp(EA,i∕kBT )  of a particular defect [115111]. Larger devices contain a larger number of defects, which in combination with a nearly uniform distribution of the activation energies EA,i  yields a log-like recovery behavior as displayed in Fig. 7.12. As there are many different pairs of τc,i  and τe,i  within the device, their extraction from the experimental data is discussed first.


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Figure 7.12: If there are few defects with emission times τi  like in small-area transistors [115111], the relaxation after BTI exhibits discrete jumps. Enlarging the area (more defects) and assuming a uniform distribution of them adds up to a log (t)  behavior, instead.


By subtracting two recovery traces after stress times ts,i  and ts,i+1   , the fraction of defects with capture time constants with ts,i < τc < ts,i+1   is determined first [116], which is shown in Fig. 7.13. By dividing the difference trace into intervals [tr,i,tr,i+1 ]  , the fraction of defects having ts,i < τc < ts,i+1   and t  <  τ < t
 r,i   e    r,i+1   is obtained.


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Figure 7.13: Evaluation of the shift between two relaxation curves after stress times ts,i  and ts,i+1   yields the fraction of defects with capture time constants with ts,i < τc < ts,i+1   These ranges of capture time constants of certain defects are depicted as function of trel   . The contour lines below the three graphs emphasize the amount of defects contributing to ΔVTH   . For NBTI with an E
 ox   of − 6MV  ∕cm  , the characteristics of t
 rel   are not changed with increasing τc   , despite some shift along the positive ΔVTH   -axis. The maximum ΔVTH   values for all τc   -ranges are obtained for small values of trel   . This implies fast relaxation. On the contrary, PBTI (6MV  ∕cm  ) yields a larger degradation and additionally moves the characteristics of trel   towards increasing τc   . For the largest available τc   , which covers time constants between 103s  and 104s  , the maximum of ΔV
   TH   is moved away from the minimum t
rel   . This maximum marks the beginning of the change of emission time constants τe   depicted in Fig. 7.15 and is even more pronounced for 8MV ∕cm  .


To be able to describe the frequency of occurrence of capture time constants τc   and emission time constants τe   properly, a large set of long recovery traces with varying tstr   is needed. The experiments performed cover τc   from 10− 6s  up to 104s  and τe   intervals between 10 −6s  and 103s  . This allows for an extraction of the time constants as exemplarily depicted in Fig. 7.14.


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Figure 7.14: Map of time constants of capture and emission split into decades of time.


It is now possible to explain the above mentioned effect with the varying oxide electric field on the basis of Fig. 7.15, where the fraction of ΔVTH   due to defects with τc   and τe   is plotted as smoothed surface over τc   and τe   .

For NBTI with an E
 ox   of − 6MV  ∕cm  the surface shows two peaks. One peak covers τc   and τe   smaller than 1μs  , while the other more pronounced one clearly illustrates that the largest part of the degradation was due to defects with τc   larger than 1s  , which is highlighted by the contour lines below the graph. When comparing the different Eox   for PBTI for τc   covering time constants between 102s  and 103s  , the peak of 6MV ∕cm  mainly consists of τe > 10s  , while it is widened for 8MV  ∕cm  towards smaller τe   . This supports the hypothesis of decreased τe   for higher E
 ox   after PBTI stress, which appears as faster long-term recovery.


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Figure 7.15: The fraction of defects having ts,i < τc < ts,i+1   and tr,i < τe < tr,i+1   is depicted for three different oxide electric fields. The contour lines below the graphs highlight the biggest changes of ΔVTH   . Both surface and contour lines are smoothed for a better visualization. It is shown that the oxide electric field is related to the magnitude of τe   . Increasing E
  ox   yields a shift of the peak towards smaller τ
 e   , which corresponds to our monitored increased recovery at larger trel   . Note that only for 6MV ∕cm  a full set of τc   and τe   is available and therefore the map has to be truncated in order to be comparable with the case 8MV  ∕cm  .