A Design-of-Experiments Approach
size between 1 and 4 nm. In each case the total line dose was held fixed (at 4.8 nC/cm) by adjusting the number of writing repeats of the pattern. Center points are included in the design to allow modeling of non-linear behavior. Te step size center point was set to 2 nm (rather than 2.5 nm) to avoid aliasing artifacts. Tis DOE generated the set of 10 experiments listed in Table 1. A Si wafer was taken as substrate to perform the experiments.
Results Table 1 shows the 10 parameter combinations that were
executed in the experiment. Because we looked at 7 different pitches (defined as the programmed spacing between the line pairs) in our line pairs, a total of 70 experiments were carried out. Figure 1 shows as an example result from experiment 2, for 6 of the 7 line pairs. Not shown is the smallest (8 nm) pitch, where the two lines were obviously melded into one; these data were not fully processed. Te selected range of the pitch evidently covers the transition from isolated to merged lines.
Te line width can now be expressed as a function of the parameters:
W(nm) = C + A⋅(step size) + B⋅(dwell time) + AB⋅(step size)⋅(dwell time) + AA⋅(step size)2 + BB⋅(dwell time)2
Figure (1)
where W is line width (nm), C is a constant, A is a prefactor for step size, B is a prefactor for dwell time, AB is a prefactor for step size times dwell time, AA is a prefactor for step size squared, and BB is a prefactor for dwell time squared. Line-width measurements were made from HIM images.
Line scans were taken in four places for each line pair. Each profile scan is actually an average of 17 adjacent rows of pixels, corresponding to a 24.9-nm distance along the lines. Te edge position was determined manually as the position at which the gray value was halfway between the peak and the baseline. See Figure 2 for an example. It was found in the measurements that the image gray level did not return to the base line for the 12-nm pitch lines, from which we conclude that these lines were not fully separated when written. Te analysis was
Table 1: The 10 experimental runs carried out for each of 7 pairs of lines of different spacing in order to determine the influencers on line width.
Factor Row # 1 2 3 4 5 6 7 8 9
10 24 A
Step Size 1 1 4 4 2 2 1 4 2 2
B
Dwell Time 0.2
19.8 0.2
19.8 10 10 10 10
0.2 19.8
Figure 2: Example line scan (experiment 2, 20-nm pitch) used for measurement of line and gap width. Line width is measured as the full width at half maximum signal. In each of 4 positions along the lines, a scan like this was generated by averaging together 17 adjacent rows. The line and gap widths were determined using the coordinates noted where the vertical reference marks are here. The asymmetric appearance of the left line in particular is attributed to the off-axis signal collection.
www.microscopy-today.com • 2011 May 1: Pairs of Pt lines from
experiment 2. Writing was carried out in each case in a 10-µm field of view, with a raster perpendicular to the line direction. The programmed pitch between the lines pairs, as laid out from left to right in this montage, are 48 nm, 28 nm, 24 nm, 20 nm, 16 nm, and 12 nm. The lines at 8-nm pitch overlapped and therefore are not shown.
thus ended at the 16-nm pitch. Figure 3 shows a survey of all the results with line width plotted as a function of pitch for all ten parameter sets. Te reader can promptly observe that experiment 2 consistently provided the narrowest lines. Although the factor analysis has been carried out for all the experiments, we will show just some illustrative examples here. We take as an
example the 48-nm pitch line pair experiments. Te values for the coefficients in Equation (1) were derived from the factor analysis, and a confidence level in their significance was calculated. A numer- ical figure of merit, P, should generally be below 0.05. Te reader is referred to Reference [6] for a definition of this metric. Te positive value of A obtained (0.70) indicates that increasing step size
increases line width, whereas the negative value of B (–1.58) shows that a larger dwell time is favorable for decreasing line width. Tis is also observed graphically by the Marginal Means plots in Figure 4. We note in this experiment the value for AB (0.66) has a marginal P value of 0.02, leading to an initial conclusion that there is some level of interaction between step size and dwell time. Te coefficient of determination (R2 = 0.5907) gives an indication that the fit is somewhat low in this case; a value of 0.9 signifies high
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