# NBTI and Power Reduction Using an Input Vector Control and Supply Voltage Assignment Method
## 2 Preliminaries  
### NBTI-Induced Transistor Aging  
An effective prediction of circuit performance aging depends on an accurate NBTI model; however, NBTI's physical mechanism is still a subject of debate, and different models have been proposed. In general, the Reaction Diffusion (R-D) model and the Trapping Detrapping (T-D) model are two widely accepted physical theories to explain the NBTI effect [20]. The R-D model involves the breaking of Si-H bonds in Si-SiO\({}_{2}\) and the generation of interface traps. The change of threshold voltage (\(\Delta V_{th}\)) follows a power law function of aging time [11]. In comparison, the T-D model involves a charge trapping/detrapping, and \(\Delta V_{th}\) follows a logarithmic function of the stress time [21]. Recent work reveals that there exists a significant amount of permanent degradation from the NBTI effect, and a new kind of Hydrogen Release (H-R) model has been proposed which can explain this phenomenon [22; 23]. In the H-R model, hydrogen is assumed to release from the gate side of the oxide to migrate towards the channel, which in turn increases \(\Delta V_{th}\). The H-R model can explain the degassivation of Si-H bonds and the passivation of channel dopants, as well as the sensitivity of the permanent component to the H concentration introduced during fabrication. The authors also claim that the reaction-limited model is a special case of the H-R model [22]. Different models can explain the NBTI effect and match the experimental data under different measurement conditions. For example, the R-D model has been verified to be effective for a moderate to very long stress time [24]. In contrast, the T-D model is capable of predicting ultrafast measurement data more precisely; moreover, the T-D model can also capture the aging variability of the NBTI effect [21; 25; 26]. Regardless of the argument for NBTI's mechanism, some features about the NBTI effect are widely known and accepted. For example, NBTI increases with an increase in negative stress gate bias; therefore, the NBTI-induced \(\Delta V_{th}\) is dependent on the actual workload of the circuit. In addition, NBTI increases at an elevated temperature and shows Arrhenius \(T\) activation; NBTI recovers quickly after the stress is removed, and measured \(\Delta V_{th}\) and related parameters are sensitive to measurement delay [24]. In this paper, we will not cast our focus on NBTI's physical model but on a method to alleviate NBTI-induced performance degradation while reducing power dissipation to the greatest extent. Since the R-D model can describe the NBTI effect for a long period of stress time, in this paper, we use the R-D-based long-term predictive model for NBTI. In our future work, we will discuss the effectiveness of our proposed method when using the other popular NBTI models. The long-term NBTI model for calculating the threshold voltage shift for the pMOS transistor is expressed as follows [11]:  
\[|\Delta V_{th}|=\left(\sqrt{K_{V}^{2}\times T_{clk}\times\alpha}/(1-\beta_{t}^ {1/2n})\right)^{2n} \tag{1}\]\[\beta_{t}=1-\left(2\xi_{1}\times t_{e}+\sqrt{\xi_{2}\times C\times(1-\alpha)\times T _{clk}}\right)/\left(2t_{ox}+\sqrt{C\times t}\right) \tag{2}\]  
\[K_{V}=\left(q\times t_{ox}/\varepsilon_{ox}\right)^{3}\times K^{2}\times C_{ox }\times\left(V_{gs}-V_{th}\right)\times\sqrt{C}\times e^{2E_{ox}/E_{0}} \tag{3}\]  
where \(T_{clk}\) is the period of a single stress and recovery cycle, \(\xi_{1}\) = 0.9 and \(\xi_{2}\) = 0.5 are two constants, \(t_{ox}\) is the oxide thickness, \(t_{e}\) either equals \(t_{ox}\) or the diffusion distance of hydrogen at the initial stage of recovery, \(n\) is a time exponent and is equal to 1/6 for an H\({}_{2}\) diffusion model, \(q\) is the electron charge, \(K\) = 8 \(\times\) 10\({}^{4}\), \(C_{ox}\) is the oxide capacitance per unit area, \(E_{ox}=V_{gs}/T_{ox}\), \(C=1/T_{0}\) exp\((-E_{a}/kT)\), the temperature \(T\) is set to 300 K, \(E_{a}\) = 0.49 eV, \(T_{0}\) = 10\({}^{-8}\), and \(E_{0}\) = 0.335 V/nm.