A coarse-graining approach to rate equations of the composite AC-NBTI model

For ac negative bias temperature instability (NBTI), the composite model that combines the reaction-diffusion (R-D) model for permanent component and the trap-detrap (T-D) model for recoverable component has almost been accepted. However, simple analytical formulas, which are useful for product qualification, have not yet been established. In this paper, we present a coarse-graining approach to analyze the rate equations in the T-D and R-D schemes. Here, the time-averaged rate equations are derived from the equations for stress and recovery phases. The analytical solutions obtained by solving this set of equations can explain the essential features of ac-NBTI, such as the duty-cycle dependence and the memory effect of hole-trapping determined by the standby condition. For the T-D scheme, the exact analytical solutions of the original equations verify this approach, whereas for the R-D scheme, the double-interface R-D model rather than the conventional R-D model captures the duty-cycle dependence observed in the experimental universal curve. Our experimental data on the ac-NBTI are also consistent with the developed analytical model providing a general validity of the approach presented here.