Method of Determining Statistically-Correct Spatial Interpolation Modeling Parameters from Measurements (02-Feb-2010)

Method of Determining Statistically-Correct Spatial Interpolation Modeling Parameters from Measurements

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This disclosure describes a method of determining Statistically-Corrected Spatial Interpolation (SCSI) modeling parameters based on on-wafer measurement data. This method allows the determination of the distance between equilateral the triangle points used in SCSI methodology. The method allows the hardware-accurate representation of measured spatially-dependent statistical correlated on-chip process variables. An 130nm BiCMOS technology example of this method in practice is presented here. This disclosure relates to filed patent application: BUR920070269US1: "Methods for Distributing a Random Variable Using Statistically-Correct Spatial Interpolation" which describes the general SCSI modeling methodology. A flow chart of the described method of converting on-wafer measurements to SCSI modeling parameters is shown in Figure1.

    Hardware Measurements of identical structures at different locations from many wafers

Figure 1

Figure 2 shows the general statistical correlation behavior of a SCSI modeled, regular array of on-wafer process data points. The SCSI method, in this case, has divided the modeled region into smaller equilateral triangle regions. After SCSI modeling, the regular array of process data

p

 oints exhibits the statistical correlation (ρ) versus on-wafer distance (d) behavior shown in the scatter plot. There is a straight line going from ρ=1 at d=0μm to ρ=0 at d0=100μm. Also shown in the figure is a good-fit mathematical relationship of the statistical-correlation versus distance:

1

Sort measurement data by measurement location

Remove outliers and determine μ and σ

     Determine statistical correlation between locations across many wafer measurements removing any data pairs from two different locations if either location contains an outlier for a given wafer measurement

Determine "best-fit" curve that minimizes the sum of squared error of measured statistical correlation data

   Determine d0 from" best fit" curve using SCSI simulation "best-fit" relationship to the curve known empirically from simulations

ρ(d)=cos^2(πd/2r_max). Also shown in the figure is the computer simulation obtained relationship d0=0.7*r_max. A more accurate relationship between d0 and r_max was actually found to be d0=(2/3)*r_max ord0=0.667*r_max. So, what is needed to properly model on-wafer/on-chip statistical correlation behavior is the value of rmax. Once r_max is known, the distance between the vertices of the SCSI equilateral triangles, d0, can be determined and the required on-wafer/on-chip statistical behavior can be accurately modeled. The method described here allows r_max, and hence, d0=0.667*r_max to...