Statistical Approach for Yield Optimization for Minimum Energy Operation in Subthreshold Circuits Considering Variability Issues

The supply voltage (
V

_dd
) and threshold voltage (
V

_th
) are two significant design variables that directly impact the performance and power consumption of circuits. The scaling of these voltages has become a popular option to satisfy performance and low power requirements. Subthreshold operation is a compelling approach for energy-constrained applications where processor speed is less important. However, subthreshold designs show dramatically increased sensitivity to process variations due to the exponential relationship of subthreshold drive current with
V

_th
variation and drastically growing leakage power. If there is uncertainty in the value of the threshold or supply voltage, the power advantages of this very low-voltage operation diminishes. This paper presents a statistical methodology for choosing the optimum
V

_dd
and
V

_th
under manufacturing uncertainties and different operating conditions to minimize energy for a given frequency in subthreshold operation while ensuring yield maximality. Unlike the traditional energy optimization, to find the optimal values for the voltages, we have considered the following factors to make the optimization technique more acceptable: the application-dependent design constraints, variations in the design variables due to manufacturing uncertainty, device sizing, activity factor of the circuit, and power reduction techniques. To maximize the yield, a two-level optimization is employed. First, the design metric is carefully chosen and deterministically optimized to the optimum point in the feasible region. At the second level, a tolerance box is moved over the design space to find the best location in order to maximize the yield. The feasible region, which is application dependent, is constrained by the minimum performance and the maximum ratio of leakage to total power in the
V

_dd
-
V

_th
plane. The center of the tolerance box provides the nominal design values for
V

_dd
and
V

_th
such that the design has a maximum immunity to the variations and maximizes the yield. The yield is estimated directly using the joint cumulative distribution function over the tolerance box requiring no numerical integration and saving considerable computational complexity for multidimensional problems. The optimal designs, verified by Monte Carlo and SPECTRE simulations, demonstrate significant increase in yield. By using this methodology, yield is found to be strongly dependent on the design metrics, circuit switching activity, transistor sizing, and the given constraints.