You don’t need 1nm contours for curvilinear shapes: pixel-based computing is the answer

Abstract

Enabled by multi-beam mask writing [1], curvilinear free-form ILT [2], and GPU acceleration [3], curvilinear masks are quickly becoming the norm in leading edge masks, whether for 193i or for EUV, particularly for contact and via layers. An industry standard for compactly representing curvilinear shapes is being developed for SEMI through an industry working group. In it, Bezier, and B-spline "Multigon" formats are proposed to augment the piecewise linear polygons that are supported today [4]. Whether these infinite-resolution curvilinear formats are used or piecewise linear polygons are used, there is a question of what constitutes a high enough vertex density to be of some pre-defined accuracy requirement. With these infinite-resolution curvilinear formats, the vertex density would be lower than with piecewise linear polygons for a particular accuracy requirement. But it is still useful to know what density is theoretically sufficient. This paper explores the concept of rasterization and the mathematical dual between contours and pixel dose arrays given a particular known resolution limit. The paper further argues that curvilinear ILT, practically speaking, is all computed in the pixel domain. And all curvilinear masks, with the notable exception of MWCO masks for 193i [5], are written with multi-beam machines using pixel dose arrays. The paper further argues that all images taken of the resulting masks, whether for inspection, disposition, or for metrology are pictures taken as pixel dose arrays of some resolution with some image processing afterwards. Information theory is a branch of computer science that, among other things, gives insight on how much data is sufficient to represent any particular information content [6,7,8]. More generally, the field covers the idea of digitizing the analog world to some known limit of resolution. Rasterization is digitalization of images that converts from contours, be it piecewise linear polygons, or some infinite resolution curves, to pixel doses of some pixel size and dose range. Contouring is the converse, going from pixel doses to geometric space. By understanding information theory, how curvilinear mask shapes are computed, and how curvilinear mask shapes are generated on

Curvilinear mask process correction embedded on multi-beam mask writer

“…• It is certain that MPC in geometry domain has infinite resolution. But the pixel sizes and the bits of dose levels of pixels of a multi-beam mask writer are clearly the most resolution that is meaningfully transferrable to the resists on the mask 5 . The precision in the pixel dose manipulation is the best and the most adequate data representation possible without any loss or addition of information.…”

“…The other solution is to apply MPC to the bitmap in the pixel domain inside multi-beam mask writer. In this paper, we call MPC in the pixel domain as pixel-level dose correction 4 , 5 (PLDC). In Sec.…”

Leaping into the curvy world with GPU accelerated O(p) computing

“…Representing curvy shapes is challenging if we continue using traditional piecewise linear formats. The paper [7] talks about how using a curve format provide us similar information content with less data if we us a pixel-based computing aware piecewise curvy format. As shown in Figure 4, the red piecewise linear representation requires a lot of vertices to represent the curve.…”

“…The paper [7] talks about how the intuition behind Nyquist criteria in pixel domain can be seen from grid alignment. In fact, if the input mask shape is sampled into pixels with sufficient resolution, the resulting shapes on masks are always preserved.…”

“…The mask industry is tackling this issue using a Bezier and B-spline based "Multigon" format [6] to replace the traditional piecewise linear polygon-based formats. Pixel-based computing and consequently a representation of curves that is aware of the mask writer pixel size [7] can further assuage the problems of data path and computational overhead in using curvilinear photomasks. This paper demonstrates the inherent advantages of pixel-based computing for curvilinear photomasks, when using a GPUbased platform, through comprehensive analysis and empirical evidence.…”