Lithographic image simulation for the 21st century with 19th-century tools

Abstract

Simulation of lithographic processes in semiconductor manufacturing has gone from a crude learning tool 20 years ago to a critical part of yield enhancement strategy today. Although many disparate models, championed by equally disparate communities, exist to describe various photoresist development phenomena, these communities would all agree that the one piece of the simulation picture that can, and must, be computed accurately is the image intensity in the photoresist.The imaging of a photomask onto a thin-film stack is the only phenomenon in the lithographic process that is described fully by well-known, definitive physical laws. Although many approximations are made in the derivation of the Fourier transform relations between the mask object, the pupil, and the image, these and their impacts are well-understood and need little further investigation.The imaging process in optical lithography is modeled as a partially-coherent, Köhler illumination system. As Hopkins has shown, we can separate the computation into 2 pieces: one that takes information about the illumination source, the projection lens pupil, the resist stack, and the mask size or pitch, and the other that only needs the details of the mask structure. As the latter piece of the calculation can be expressed as a fast Fourier transform, it is the first piece that dominates.This piece involves computation of a potentially large number of complex quantities called Transmission Cross Coefficients (TCCs), which are correlations of the pupil function weighted with the illumination intensity distribution. The advantage of performing the image calculations this way is that the computation of these TCCs represents an up-front cost, not to be repeated if one is only interested in changing the mask features, which is the case in Model-Based Optical Proximity Correction (MBOPC).1 The down side, however, is that the number of these expensive double integrals that must be performed increases as the square of the mask unit cell area; this number can cause even the fastest computers to balk if one needs to study medium-or long-range effects. One can reduce this computational burden by approximating with a smaller area, but accuracy is a concern, especially when building a model that will purportedly represent a manufacturing process. This work will review the current methodologies used to simulate the intensity distribution in air above the resist and address the above problems. More to the point, a methodology has been developed to eliminate the expensive numerical integrations in the TCC calculations, as the resulting integrals in many cases of interest can be either evaluated analytically, or replaced by analytical functions accurate to within machine precision. With the burden of computing these numbers lightened, more accurate representations of the image field can be realized, and better overall models are then possible.

Fast aerial image simulations for partially coherent systems by transmission cross coefficient decomposition with analytical kernels

“…There are many types of radial functions can be used for circle sampling, and here we select a Bessel function based formula as R n;s aq ð Þ ¼ 4z n;s J nÀ1 ðaqÞ À J nþ1 ðaqÞ J n ðaqÞ ðaqÞ 2 À z 2 n;s ; (9) where J n is the nth order Bessel function of the first kind, and z n,s is the sth zero-point of the Bessel function J n . There are many types of radial functions can be used for circle sampling, and here we select a Bessel function based formula as R n;s aq ð Þ ¼ 4z n;s J nÀ1 ðaqÞ À J nþ1 ðaqÞ J n ðaqÞ ðaqÞ 2 À z 2 n;s ; (9) where J n is the nth order Bessel function of the first kind, and z n,s is the sth zero-point of the Bessel function J n .…”

Advanced model and fast algorithm for aerial image computation with well controlled accuracy

“…Thus, intensity is computed as six-fold (3 x 2D) quadrature. However, it is possible to reduce the computational task by taking specifics of the lithography tasks and environment into account [11,12] . Analytical solutions for some of the quadratures from (2) dramatically simplify the computations and/or increase the accuracy of the results.…”