Maharaj Mukherjee

The problem of optimal placement of sub-resolution assist features (SRAF)

In this paper, we present a formulation of the Sub-Resolution Assist Feature (SRAF) placement problem as a geometric optimization problem. We present three independent geometric methodologies that use the above formulation to optimize SRAF placements under mask and lithographic process constraints. Traditional rules-based methodology, are mainly one dimensional in nature. These methods, though apparently very simple, has proven to be inadequate for complex two-dimensional layouts. The methodologies presented in this paper, on the other hand, are inherently two-dimensional and attempt to maximize SRAF coverage on real and complex designs, and minimizes mask rule and lithographic violations.

A polynomial-time optimization algorithm for a rectilinear partitioning problem with applications in VLSI design automation

* Corresponding author. E-mail addresses: mmukherj@us.ibm.com (M. Mukherjee), kanadc@agere.com (K. Chakraborty). 1 The work presented in this paper was done when this author was at the EDA Laboratory, IBM Microelectronics. 2 In this paper we use the terms rectilinear and isothetic interchangeably. (i) localizing pins of high-fanout nets for buffer insertion; (ii) congestion analysis; (iii) insertion of fill geometries during post-processing of VLSI layouts; (iv) localizing terminals of multiple supply voltages to minimize interference and crosstalk; and (v) localization of defect areas in mask inspection. The above problem belongs to the general class of rectilinearp-center andp-piercing problems. The Rectilinear p-Center Problem is defined as follows:

Optical rule checking for proximity-corrected mask shapes

Optical Rule Checking (ORC) is an important vehicle to predict the failure of wafer shapes due to the process proximity effects. Optical Proximity Correction (OPC) if not aided by ORC may cause severe failures affecting the yield in manufacturing. However, it is fairly complicated to do ORC on mask shapes that are pre-corrected either by rules-based or by model-based OPC. ORC is also a good tool to capture the problems that may occur at multi-layer interactions. We present a methodology to use both geometric directives and limited optical simulation to detect potential failures using ORC. We extend our methodology to multi-layer interactions. In case of multi-layer ORC, we present several approaches that deal with how to judiciously mix the geometric directives and the optical simulations for different layers. We show the ORC can help us design better rules for OPC.

A Randomized Greedy Method for Rectangular-Pattern Fill Problems

In this paper, a class of problems of physical-design optimization with rectangular shapes is presented. Efficient solutions for this class of optimization problems are needed for rectangular metal fill/negative-fill insertion during postrouting optimization. These problems are also shown to be closely related to the problem of floorplanning with rectangular macrocells. The solution must satisfy certain constraints for minimum and maximum pattern densities within a moving rectangular window while optimizing a given objective function such as the area or the aspect ratio of the resultant bounding box. It is shown in this paper that the general class of such problems defined with a gridless formulation is at least NP-hard. This proof, with minor modifications, also extends to the formulation in the gridded domain. A greedy randomized algorithm for one of our problems in the gridded domain is proposed, along with a proof that this algorithm can achieve the given objective with a very high probability while satisfying the constraints. Experimental results based on an implementation of our algorithm are provided and are shown to agree with our theoretical proof.

Novel algorithms for placement of rectangular covers for mask inspection in advanced lithography and other VLSI design applications

The continuous drive of very large scale integrated (VLSI) chip manufacturers to meet Moores law has spurred the development of novel resolution enhancement techniques (RETs) and optical proximity correction (OPC) methodologies in optical microlithography. These RET and OPC methods have increased the complexity of mask-manufacturing manifold and have, at the same time, put added emphasis on the mask inspection procedure. A technique to simplify mask inspection is to identify rectangular regions on the mask that do not require inspection. Such a region is referred to as a do not inspect region (DNIR). A novel and practical algorithm to place DNIR rectangles on the mask is presented. It is shown that the most general DNIR placement problem is at least NP-Hard (Garey and Johnson, 1979). However, under certain relaxed criteria, there exists a polynomial-time algorithm for DNIR placement using dynamic programming. However, the optimal algorithm has very-high-degree polynomial bounds on its runtime and space complexities. On the other hand, a very simple greedy algorithm extended by lookahead and randomization, or by simulated annealing, can greatly improve the performance of the DNIR placement and produce near-optimal results. Although the algorithm developed in this work is targeted primarily toward DNIR placement, it has many other VLSI design applications.

Improved line-end foreshortening and corner-rounding control in optical proximity correction using radius of curvature method

We describe how to generate better Optical Proximity Corrections (OPC) for line-ends and corners by using rounded anchors and serifs. These rounded serifs and anchors can be made smaller in size and shape than the traditional rectilinear anchors and serifs. The smaller size of the serifs tend to have less problems in satisfying mask-rule constraints. They also have less adverse effects on the printability of neighboring shapes. We refer to these rounded anchors and serifs as Mouse-Ears. The rounding is done by circles which are regular octagons with Ortho-45 straight lines. The main idea of this paper stems from the physical description of the lithographic process, which can be conceptualized as a low-pass filter. The low-pass filter eliminates the sharp corners of the feature which are made of high spatial-frequency components and retains the low spatial-frequency components. Since the rounded anchors and serifs have fewer high-frequency components than their rectilinear counterparts they get less deformed in the lithographic process.

Integer Approximation to the Intersection of Three Planes with Planar Constraints

The intersection point of three planes specified with rational coordinates is approximated by a point with integer-valued coordinates so that the integer point is constrained within the pyramid specified by three half-spaces of the planes. An O(N log N) algorithm guarantees that the obtained integer point is closest to the apex of the pyramid. An O(log N) algorithm, not yet proven to guarantee the closest point, yields promising results.

A Randomized Greedy Algorithm for the Pattern Fill Problem for DFM Applications

This work deals with the dummy fill and negative fill insertion problem with constraints on the minimum and maximum pattern density within a moving rectangular window. It is shown that the general class of such problems is at least NP-hard. A greedy randomized algorithm for this problem is proposed, and its proof of convergence and some experimental results are presented. More detailed account of our implementation and results are available in [7].

Satisfying coplanarity constraints on points sets in three dimensions with finite precision arithmetic

Geometric operations that form the very heart of many commercial solid modelers are susceptible to failures without any prior warning due to round-off errors in their computations. The consequences of these unwarranted failures can sometimes be really severe. Collinearity and coplanarity of point sets are important properties that need to be preserved in many applications. We present algorithms that preserve the collinearity and coplanarity relationships among point sets in three dimensions. We further extend these algorithms for approximating three-dimensional cellular objects with planar surfaces while maintaining the planarity of component faces. Partial expansions of continued fraction expansions of rational numbers underlies the basic strategy of our approximation paradigm. The solutions are guaranteed to be correct and they work for a large class of engineering objects.