Modeling within-field gate length spatial variation for process-design co-optimization

Pelgrom's model [1] suggests that a spatial correlation structure is inherent in the physical properties of semiconductor devices; specifically, devices situated closely together will be subject to a higher degree of correlation than devices separated by larger distances. Since correlation of device gate length values caused by systematic variations in microlithographic processing is known to carry a significant impact on the variability of circuit performance, we attempt to extract and understand the nature of spatial correlation across an entire die. Based on exhaustive, full-wafer critical dimension measurements collected using electrical linewidth metrology for wafers processed in a standard 130nm lithography cell [8], we calculate a spatial correlation metric of gate length over a full-field range in both horizontal and vertical orientations. Using a rudimentary model fit to these results, we investigate the impact of correlation in the spatial distribution on the variability of circuit performance using a series of Monte Carlo analyses in HSPICE; it is confirmed that this correlation does indeed present a significant influence on performance variability. From the same dataset, we also extract both the across-wafer (AW) and within-field (WIF) contributions to systematic variation. We find that the spatial correlation model's shape is strongly related to these two components of variation (both in magnitude as well as by spatial fingerprint). By artificially reducing each of these components of systematic variation - thereby simulating the effects of active, across-field process compensation - we show that spatial correlation is significantly reduced, nearly to zero. This implies that Pelgrom's model may not apply to die-scale separation distances, and that a more accurate correlation theory would combine Pelgrom's model over very short separation distances with a systematic variation model that captures variability over longer distances by means of non-stationary distributions.