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.