N. Lu, N. Zamdmer, R. Wachnik
Keywords: statistical variability, mismatch, modeling of across-chip variations, modeling of distance-dependent spatial correlation, compact model of distance-dependent mismatch
Summary:We study distance-dependent mismatch and spatial correlation problems both experimentally and theoretically. The two problems are intimately connected. We present our hardware results and, using the relationship between mismatch and correlation, show that the relationship between mismatch and distance must deviate from the well-known Pelgrom quadratic relationship when the distance is large. For the first time, we present a compact and complete solution of modeling both distance-dependent mismatch and spatial correlation problems. We show how to model an arbitrarily given shape of mismatch or correlation vs. distance relation. Explicit analytic solution for modeling each of Gaussian, exponential, Lorentz, and linear decreasing types of spatial correlation is given. There are no grid points, no brackets, and no matrix solution in our method. Both model-generated spatial correlation and mismatch are translational invariant and continuous. The correlation range of spatial correlation can be much smaller than chip size, be about the chip size, or be much larger than chip size. Our method produces a very compact Monte Carlo model for SPICE or TCAD simulation, and it enables a simultaneous modeling of N(N-1)/2 pairs of mismatch relations among N devices/circuits, regardless of the physical locations of the N devices/circuits. Our method is very compact, since we do not use a matrix or a set of eigen solutions to represent correlations among a group of devices. Characterization of Across-Chip Variations. To measure and characterize across-chip variations (ACV), dozens ring oscillators or FETs were placed on SOI technology chips. The delay of ROs at various chip locations was measured on many wafers. The delay mismatch between each pair of ROs was calculated, and then was plotted vs. their separation. We show how a correlation coefficient vs. distance relation was obtained. We also show scatter plots of mismatch and correlation coefficient of FinFET’s linear threshold voltage vs. distance for 14nm SOI FinFET technology, along with characterized distance-dependent relations. Modeling of Across-Chip Variations. For a semiconductor device/circuit parameter P, we want to model the spatial correlations among P(x_i), i = 1, 2, 3, …, where x_i is the location of the ith device/circuit. For an arbitrarily given spatial correlation for the parameter P, our ACV model is very compact, and it uses two random variables Gx and Gy inside sine and cosine functions. When netlisting the ith device, none of other device’s coordinates are needed. This is the compactness of our models for being able to model distance-dependent mismatch even without knowing other device’s locations when the ith device is specified in a SPICE netlist or in a TCAD deck. The joint probability distribution for both random variables Gx and Gy is a 2D integration over the product of the given spatial correlation and two cosine functions, which is symmetric in both Gx and Gy. We provide analytic solutions of several often discussed spatial correlation functions. For example, for the linear decreasing type, the distribution of random variable Gx is proportional to the square of a sinc function.