Understanding DC Behavior of Subthreshold CMOS Logic through Closed-Form Analysis

In this paper, the DC behavior of subthreshold CMOS logic is analyzed in a closed form for the first time in the literature. To this aim, simplified large-signal and small-signal models of MOS transistors in subthreshold region are first developed. After replacing transistors with these equivalent models, analysis of the main DC parameters of CMOS logic gates is performed. In particular, the change in the DC characteristics shape due to operation at ultra-low voltages is analyzed in detail, evaluating analytically the degradation in the logic swing, the symmetry and the steepness of the transition region, as well as the change in the unity-gain points position. The resulting expressions permit to gain an insight into the basic dependencies of DC behavior on design and device parameters. The noise margin is explicitly evaluated and modeled with a very simple expression. Interestingly, analysis shows that the noise margin deviates from the ideal half-swing value by an amount that linearly depends on the logarithm of the pn-ratio. Analysis permits to evaluate the minimum supply voltage that ensures correct operation of CMOS logic (i.e., positive noise margin). Previously proposed rule of thumbs to evaluate minimum voltage are also theoretically justified. Moreover, the impact of PMOS/NMOS unbalancing on DC characteristics is analyzed from a design perspective. Considerations on the impact of process/voltage/temperature variations are also introduced. Results are validated through extensive simulations in a 65-nm CMOS technology.