The influence of sub-100 nm scattering on high-energy electron beam lithography Page: 5 of 21
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on the plating base wafer extends well into the 100KV backscatter range and the
traditional gaussian functional form is observed when re-plotting this data in the
form of log(dose) versus radius2. Figure 5 shows the 100KV dot data re-plotted to
highlight the gaussian functional form which will be a straight line since:
Log(Dose) - (r/p)2 = C. The slope of the straight line gives the backscatter range
of f = 26pm. The data for 50KV is plotted in figure 6 to highlight the backscatter
range with a derived range, (3 = 8pm.
Fitting the measured data
The dot size versus dose data is well described with a gaussian function for the
long range scattering as seen by the straight line segments of figures 5 and 6. For
analytical convenience, a sum of gaussian terms is a desirable fit of the scattering
data. From the mathematics of the central limit theorem we would expect gaussian
distributions from large numbers of independent scattering events even if each
individual scattering event has a non-gaussian distribution. However, the data is
not well approximated by a sum of gaussian functions. By visual inspection, the
log(dose) is linear in log(r) over a range of values and so an expression of the
form, Log(dose) = ao + allog(r) + a2(log(r))2 + a3(log(r))3 + a4r2 was used.
Note that the last term, a4r2 dominates the expression for large values of r and so
this approximation will yield a gaussian functional form as expected from physical
considerations. Figure 7 shows the data and fitting function for 50KV data while
figure 4 shows the 100KV data. The coefficients are determined using the linear
algebra method of singular value decomposition. The first coefficient ao,
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Anderson, Erik H.; Olynick, Deidre L.; Chao, Weilun; Harteneck, Bruce & Veklerov, Eugene. The influence of sub-100 nm scattering on high-energy electron beam lithography, article, August 1, 2001; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc740262/m1/5/: accessed July 21, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.