Mask roughness induced LER: a rule of thumb -- paper Page: 3 of 9
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change in line-edge position will be small in comparison to a lower ILS value. Therefore, a higher ILS is desirable, from
the perspective of minimizing sensitivity to mask roughness.
Noting that movements in line-edge, dx, are directly related to the 3-sigma definition of line-edge roughness, we can
formalize the relation between speckle statistics and LER as
LER = 3 x dx = 3 x S x . (2)
Intensity x ILS
3. MASK ROUGHNESS INDUCED LER - AN ANALYTIC SOLUTION
Current methods to predict mask roughness induced LER involve conducting full 2D aerial image simulations in
commercially available software (such as PROLITH  or Panoramic ), followed by extracting the LER from the
computed aerial image through offline analysis using a software package such as SuMMIT . As stated earlier, this
method is time consuming and cumbersome.
Our goal instead, is to simplify the LER modeling process by using simplified form of Eq. 2. The power in this simple
equation is not in any new formulation, but in the method of implementing it. By looking at Eq. 2, we can break-up the
problem into smaller parts. Consider an imaging system of given objective NA. Instead of doing 2D aerial image
simulations for each process parameter individually, we can minimize this time-consuming process by only conducting a
one-time 2D aerial image simulation through focus on a clear-field rough mask to get the illumination- and mask
roughness-specific clear-field speckle. We can then couple this speckle to LER by using fast 1D aerial image
simulations (e.g., using PROLITH) to get the feature- and illumination-specific ILS as it varies through focus. From
here, the full parameter space can be reached by analytic extension. A comparison schematic of the two methods is
shown in Fig. 2.
There are two distinct advantages to such a formulation. Firstly, it provides a straightforward, tractable simplified
solution to computing mask-roughness-induced LER that is fast. Secondly, there is no need for an exhaustive list of 2D
aerial image simulations for every mask pattern to be considered.
4. MODELING APPROACH
To verify the validity of our approach, we need to compute the LER from both methods illustrated in Fig. 2 and compare
them. Both used scalar aerial image modeling software based on the equations of partially coherent image formation
. Commercial software with similar capabilities include PROLITH  and Panoramic . We modeled an
aberration-free optical system with NA = 0.32. Again, the low NA allows for scalar and thin mask modeling. Following
a similar numerical analysis approach used elsewhere , we constructed a statistical representation of a clear rough
mask as a random phase object, whose pure phase distribution is determined from the geometric path length differences
imparted by the rough surface of the mask. We started with a randomly generated mask object that was 1024 x 1024
pixels at 1 nm / pixel with 1:1 imaging to the wafer plane. The standard deviation of the original height map of the mask
object was calculated to verify the RSR value. By taking the FWHM of the autocorrelation of the mask, the correlation
length was found. Using this process we generated an entire set of masks all with an RSR of 50 pm, and with a variety
of correlation lengths (5, 13, 20, 26, 32, 42, 47, 61, 68, 83, 96, 106, 127 nm). In order to build up a significant statistical
ensemble, we created 10 random realizations of the mask for each RSR / correlation length pair. Assuming a
wavelength of 13.5 nm, we converted each topographic surface to a phase perturbation. This set of masks was the
starting point for the two approaches we wish to compare: the traditional method and the proposed simplified one.
To reproduce the traditional method which calls for calculating the LER directly from aerial-image simulations, we
overlaid (multiplied) each clear-field rough mask realization with an ideal binary amplitude 50 nm line-space pattern.
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McClinton, Brittany & Naulleau, Patrick. Mask roughness induced LER: a rule of thumb -- paper, article, March 12, 2010; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc1013103/m1/3/: accessed December 9, 2018), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.