The Reaction-Diffusion (RD) model for the Negative Bias Temperature Instability (NBTI) of MOS transistors is still popular amongst reliability engineers for the interpretation of experimental data. This popularity comes mainly from its intuitive concept and the straight-forward prediction of the observed power-law degradation. The model accounts the dominant part of the long-term NBTI induced degradation to silicon dangling bonds at the semiconductor-oxide interface. These dangling bonds are formed during the oxidation and are passivated by hydrogen in the manufacturing process. The RD model assumes that the NBTI stress causes a depassivation of the silicon dangling bonds while concurrently liberating hydrogen atoms at the Si-SiO2 interface. The liberated hydrogen atoms form H2 and diffuse into the oxide. In order to explain the power-law degradation, one has to assume that the diffusive flux of hydrogen molecules away from the interface determines the macroscopically observed degradation.
In its standard formulation, the RD model is described macroscopically, using a reaction rate equation and diffusion equations. We have developed an atomistic version of the RD model based on the chemical master equation of stochastic chemistry, which accounts for the granularity of the diffusing hydrogen atoms and molecules. Initial comparisons of the microscopic and the macroscopic RD model showed large differences between the predicted degradation curves, with the microscopic RD model showing a delay in the onset of the experimentally important power-law regime that is not obtained from the macroscopic model. These differences could be traced back to the inappropriate representation of the bimolecular reactions (repassivation and dimerization) in the macroscopic RD model. As NBTI degradation usually resembles a power-law over the whole measurement window, a delayed onset of this regime is incompatible with the experimental data. The magnitude of the power-law delay strongly depends on the parameters of the model and the microscopic topology of the host material. Our initial calculations were based on a parameter set that was designed to study the basic model behavior and was not related to experimental data. In order to investigate the microscopic model in a more realistic context, we have repeated this comparison based on a published parameter set that has been calibrated to an NBTI degradation measurement. The straight-forward application of this parameter set in our microscopic RD model results in a degradation that strongly deviates from the macroscopic prediction over the whole simulation range (see figure). The power-law regime is shifted far beyond the measurement window and is in stark contrast to the experimental data. Furthermore, it is not possible to explain the experimental data using the microscopic RD model with a physically reasonable parameter set. These findings support our previous conclusions that the RD mechanism in its usually proposed form is physically inconsistent and that the RD model for NBTI is either incomplete or incorrect.