Ronald L. Gordon

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

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 one of the only phenomena 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, Kohler 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 numbers 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). 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 usually 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.

Exact computation of scalar, 2D aerial imagery

An exact formulation of the problem of imaging a 2D object through a Koehler illumination system is presented; the accurate simulation of a real layout is then not time- limited but memory-limited. The main idea behind the algorithm is that the boundary of the region that comprise a typical TCC Is made up of circular arcs, and therefore the area - which determines the value of the TCC - should be exactly computable in terms of elementary analytical functions. A change to integration around the boundary leads to an expression with minimal dependence on expensive functions such as arctangents and square roots. Numerical comparisons are made for a simple 2D structure.

Taming pattern and focus variation in VLSI design

Tight ACLV control has become increasingly diffcult due to the diminishing process constant, K1. Focus variation and pitch variation are two major systematic components of ACLV. In this paper, we demonstrate these systematic effects and propose a design flow which exploits the systematic effect. We demonstrate the systematic ACLV by showing a Bossung plot for a nominal 90nm technology node. The plot is generated by simulation with lithographic parameters closely resembling a production technology node. Traditionally, tight CD control is achieved by sophisticated RET such as OPC, SRAF, AltPSM and more recently the Dense Template Design. The CD variation is specified in the design manual and the circuit designs will ensure functionality by building in enough margin to account for the variability. Even though, the systematic components of CD variation are understood, they have always been considered together with other random components as being random. This approach has left design performance on the table. We propose a holistic design flow by integrating the technology development process, design process and the manufacturing process. This holistic approach is aiming to tame the systematic through-pitch and through-focus CD variation. We quantify the design timing benefit using this approach by circuit design experiments. Results of our experiments show that timing uncertainty can be reduced by up to 30%. We also discuss other possibilities which are infeasible to carry out in traditional approach with silos of technology development, design and manufacturing.

Through-pitch correction of scattering effects in 193-nm alternating phase-shift masks

A methodology to study the bias and phase correction of strong phase-shifting masks is introduced. Isolated apertures are simulated to investigate the influence of aperture size, undercut etch, and quartz sidewall angle on aperture transmission. The simulations match well with experimental results that are measured with an ArF microlithography simulation microscope. For alternating apertures, electromagnetic calculations are done to solve for the diffracted fields. An analytical method is derived to deduce aperture bias and phase error from the diffracted orders. This method can be used as an easy way to optimize the cross section of the phase shifting mask. The method is demonstrated for the example of a single trench alternating phase shifting mask. A constant bias that minimizes the asymmetry and phase error without the need for an undercut etch is found. Such bias works for both the case of equal lines and spaces through pitch and constant linewidth through pitch. Because this bias is easy to design into a mask, the design and manufacturing of alternating phase shifting can be simplified.