A Three-Dimensional Physical Model for $V_{\rm th}$ Variations Considering the Combined Effect of NBTI and RDF

Aggressive transistor scaling affects the threshold voltage V_th in two ways: space and time. Transistors on the same die can have different V_th due to random dopant fluctuations (RDFs). On the other hand, V_th of a specific transistor can randomly shift in time due to random position of dangling bonds and random motion of atomic H or H₂ molecules in the oxide [negative bias temperature instability (NBTI)]. These two random effects are not totally independent because the dissociation rate of Si-H bonds at the silicon-oxide interface depends on the nonuniform electric field. In this paper, we describe the combined effect of RDF and NBTI on V_th using stochastic differential equation. In this paper, we are able to examine the effect of RDF on critical NBTI parameters such as k_f, k_r, and temperature and the correlation among these parameters. The efficacy of the proposed model is evaluated by performing Monte Carlo simulations on various transistors under different direct-current stress and obtained distributions for N_IT and V_th.