6.2 The NBTI Time Exponent

The degradation of transistors due to NBTI is often found to follow a power law in time over a wide range of decades as

$$\ensuremath{\Delta V_\textrm{th}}= A \exp\left(\frac{E}{E_\... ...{a}}}{\ensuremath{\textrm{k$_\textrm{B}$}} T}\right) t^n ,$$ (6.1)

where the threshold voltage shift depends exponentially on the electric field stress $E$ and on the time $t$ with an exponent $n$. The symbol $A$ is a pre-factor and $E_\mathrm{ref}$ the reference electric field. The temperature dependence is here modeled to follow Arrhenius' law, $\exp(-\ensuremath {E_\textrm{a}}/(\ensuremath{\textrm{k$_\textrm{B}$}}T))$, with the activation energy .

The exact physical mechanism for the threshold voltage degradation is still not clear. Especially the introduction of the temperature dependence as Arrhenius' law is debated [88].

The most important value researchers try to gain from analytic formalisms as (6.1) is the capability to extrapolate the degradation for longer times and/or lower stress fields. Here, the evaluation of the parameters has to be as precise as possible. A small error in the time exponent $n$, for example, can lead to under- or overestimation of the product lifetime by several years.