Power-Law Stress Behavior

6.4 Power-Law Stress Behavior

In contrast to Chapter 6.3, where the degradation of the drain current is directly fit by (6.1), the drain current is now first converted to an approximate threshold voltage shift using the simple OTF1 relation derived in Appendix A.1

ID(tstr)−-ID(t0,ref)- ΔVTH (tstr∕t0,ref) ≈ ID(t0,ref) (VG − VTH,0) (6.2) ΔI (t ) = ---D--str-(VG − VTH,0). (6.3) ID(t0,ref)

Note that $ID,0$ is obtained at stress-level with a delay $t0,ref$ and is thus not equal to $ID (0)$ [40], resulting in an offset of the relative degradation. Also, the conversion (6.3) ignores any potential degradation in the mobility and is thus affected by an as-of-yet unknown error [108, 41]. The threshold voltage is extracted at $ID = 70nA ⋅W ∕L$ , which yields $VTH ≈ − 0.3V$ . Then $ΔVTH$ is fit by

ΔVTH (tstr∕t0,ref) ≈ B log 10(tstr∕t0,ref)+ C.

(6.4)

In order to circumvent issues with the logarithmic fit caused by offset data due to the uncertainty in $ID,0$ , the parameter $C$ is included. Besides, it is tried to fit the data to a power-law of the form

ΔVTH (tstr∕t0,ref) ≈ A (tstr∕t0,ref)n + D.

(6.5)

Again, the parameter $D$ is introduced to account for the offset in $I D,0$ .

Interestingly, it turns out that the logarithmic fit (6.4) is always possible, while the power-law fit (6.5) produces reasonable results for high temperatures and high $VG,str$ only. In that high-stress regime, power-law exponents around $0.04$ are obtained. For weaker stresses, the exponent $n$ in (6.5) tends towards zero, which corresponds to a first-order Taylor expansion of (6.5) on a logarithmic scale. As such, in this regime the power-law fit (6.5) becomes equivalent to the logarithmic fit (6.4).

Figure 6.15: At the highest stress condition ( $VG,str = − 2.50V$ , $T = 175∘C$ ) the recorded data slightly deviates from a logarithmic dependence and can be nicely fit using a power-law.

Figure 6.16: By contrast, data recorded using a lower stress condition ( $VG,str = − 2.00V$ , $T = 25 ∘C$ ) nearly perfectly follows a logarithmic behavior and cannot be properly fitted using a power-law.

This behavior is illustrated in Fig. 6.15 and Fig. 6.16. The data obtained from the harshest stress conditions ( $VG,str = 2.50V$ , $T = 175∘C$ , and $tox = 1.8nm$ ) gives a stable fit with $n = 0.041$ . For the other extreme case ( $VG,str = 1.75V$ , $T = 25∘C$ , and $tox = 1.8nm$ ) the fitting algorithm gives an exponent $n$ of practically zero. For the case of the non-converging exponent $n$ the logarithmic and power-law fits coincide.

Consequently, the power-law fit only makes sense for high temperatures and/or high $VG,str$ , as displayed in Fig. 6.17. There, the extracted $n ≈ 0.04$ for short-term stress is roughly one third of the often reported $n ≈ 0.12$ of the long-term behavior.

Figure 6.17: Top: Only data recorded during heavier stress yield a reasonable power-law exponent $n$ . Bottom: Using the (arbitrary) value of $n = 0.01$ as a threshold criterion, a high-stress region, where a deviation from the logarithmic behavior is observed, can be clearly identified.

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