Virtual probe: A statistically optimal framework for minimum-cost silicon characterization of nanoscale integrated circuits

In this paper, we propose a new technique, referred to as virtual probe (VP), to efficiently measure, characterize and monitor both inter-die and spatially-correlated intra-die variations in nanoscale manufacturing process. VP exploits recent breakthroughs in compressed sensing to accurately predict spatial variations from an exceptionally small set of measurement data, thereby reducing the cost of silicon characterization. By exploring the underlying sparse structure in (spatial) frequency domain, VP achieves substantially lower sampling frequency than the well-known (spatial) Nyquist rate. In addition, VP is formulated as a linear programming problem and, therefore, can be solved both robustly and efficiently. Our industrial measurement data demonstrate that by testing the delay of just 50 chips on a wafer, VP accurately predicts the delay of the other 219 chips on the same wafer. In this example, VP reduces the estimation error by up to 10× compared to other traditional methods.