Modeling of stretched-exponential and stretched-hyperbola time dependence of threshold voltage shift in thin-film transistors

To gain insight into the underlying properties of the well-known stretched-exponential and stretched-hyperbola time dependence models, I propose a numerical method to study threshold voltage (V_t) shift caused by both defect creation and charge trapping using a generic kinetic equation. It is shown that during the early phase of V_t shift, the time evolution of the shift is determined by the density of barrier states or trap sites which exponentially increases with barrier energy or trap location from a channel and is characterized by the dispersion parameter β in the models. The later phase is effectively determined by reaction rates in the kinetic equation in addition to β. In the case of the stretched-hyperbola model, the later phase is distinguished by the backward reaction and characterized by the fitting parameter α in the model. It is shown that V_t shifts in which backward reactions dominate during the later phase are represented by the stretched-exponential model and the rest is represented by the stretched-hyperbola model. The proposed method is also used to analyze the logarithmic time dependence model and cases when two instances of the models coexist in order to show that it is useful to study V_t shift of arbitrary shape. It is concluded that the shape of V_t shift is determined by the reaction rates and the density of barrier states or trap sites.