Patent Publication Number: US-8539390-B2

Title: Determining manufacturability of lithographic mask based on manufacturing shape penalty of aspect ratio of edge that takes into account pair of connected edges of the edge

Description:
RELATED PATENT APPLICATIONS 
     The present patent application is related to the following patent applications, which are hereby incorporated by reference: 
     (1) The patent application entitled “determining manufacturability of lithographic mask by reducing target edge pairs used in determining a manufacturing penalty of the lithographic mask,” filed on Dec. 14, 2008, and assigned Ser. No. 12/334,482, now U.S. Pat. No. 8,056,023; 
     (2) The patent application entitled “determining manufacturability of lithographic mask by selecting target edge pairs used in determining a manufacturing penalty of the lithographic mask,” filed on Dec. 14, 2008, and assigned Ser. No. 12/334,485, now U.S. Pat. No. 8,056,026; 
     (3) The patent application entitled “determining manufacturability of lithographic mask using continuous derivatives characterizing the manufacturability on a continuous scale,” filed on Dec. 14, 2008, and assigned Ser. No. 12/334,488, now U.S. Pat. No. 8,028,254. 
     FIELD OF THE INVENTION 
     The present invention relates generally to determining the manufacturability of a lithographic mask employed in fabricating instances of a semiconductor device, and more particularly to determining such manufacturability based on a manufacturing shape penalty of an aspect ratio of an edge. The manufacturing shape penalty takes into account a pair of connected edges of this edge. 
     BACKGROUND OF THE INVENTION 
     Semiconductor devices include semiconductor processors, semiconductor memories, such as static random-access memories (SRAM&#39;s), and other types of semiconductor devices. A common semiconductor device fabrication process is photolithography. In photolithography, a semiconductor surface is selectively exposed to light through a lithographic mask. The semiconductor surface is developed, and the areas that were exposed to light (or the areas that were not exposed to light) are removed. 
     Therefore, to employ photolithography in fabricating instances of a given semiconductor device, a lithographic mask first has to be manufactured. However, depending on various aspects of the semiconductor device, such as its complexity, the lithographic mask can be relatively difficult (if not impossible) to manufacture, or relatively easy to manufacture. As such, it can be important to assess the manufacturability of a lithographic mask before the mask is actually made. 
     SUMMARY OF THE INVENTION 
     A method of an embodiment of the invention is for determining manufacturability of a lithographic mask employed in fabricating instances of a semiconductor device. The method can be implemented as one or more computer programs stored on a computer-readable medium, such as a non-transitory computer-readable data storage medium. The computer programs are thus executed by the processor to perform the method. 
     The method selects target edge pairs from mask layout data of the lithographic mask for determining a manufacturing penalty in making the lithographic mask. The mask layout data includes polygons that each have edges. Each target edge pair is defined by two of the edges of one or more of the polygons. The method determines the manufacturability of the lithographic mask, including determining the manufacturing penalty in making the lithographic mask, where determining the manufacturing penalty is based on the target edge pairs as selected. The method outputs the manufacturability of the lithographic mask, where the manufacturability of the lithographic mask is dependent on the manufacturing penalty in making the lithographic mask. 
     Determining the manufacturability of the lithographic mask, including determining the manufacturing penalty in making the lithographic mask, includes the following. For a selected edge pair having a first edge and a second edge that are at least substantially parallel to one another, a manufacturing shape penalty owing to an aspect ratio of the first edge relative to a size of a gap between the first edge and the second edge is determined. The manufacturing shape penalty takes into account a pair of connected edges of the first edge that are at least substantially parallel to the first edge. 
     An optimization process is performed in relation to the lithographic mask. The optimization process includes performing one or more iterations of the following. One or more selected edges of the polygons are shrunk to a smaller size. Thereafter, the polygons are cleaned up to reduce a number of the selected edges of the polygons, Still other aspects and embodiments of the invention will become apparent by reading the detailed description that follows, and by referring to the following drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The drawings referenced herein form a part of the specification. Features shown in the drawing are meant as illustrative of only some embodiments of the invention, and not of all embodiments of the invention, unless otherwise explicitly indicated, and implications to the contrary are otherwise not to be made. 
         FIG. 1A  is a diagram of a representative lithographic mask layout, according to an embodiment of the invention. 
         FIG. 1B  is a diagram depicting the symmetry of the layout of  FIG. 1A , according to an embodiment of the invention. 
         FIG. 1C  is a diagram depicting the replication of a cell of the layout of  FIG. 1A , according to an embodiment of the invention. 
         FIG. 2  is a diagram depicting representative mask layout data for a portion of a lithographic mask layout, according to an embodiment of the invention. 
         FIGS. 3A ,  3 B, and  3 C are diagrams representing three different types of manufacturing penalties that can occur when determining the manufacturability of a lithographic mask, according to an embodiment of the invention. 
         FIG. 4  is a flowchart of a method, according to an embodiment of the invention. 
         FIG. 5  is a diagram illustratively depicting an approach to selecting target edges and target edge pairs, according to an embodiment of the invention. 
         FIG. 6  is a diagram of a mask layout data polygon having edges that represent a jog, according to an embodiment of the invention. 
         FIGS. 7A ,  7 B, and  7 C are diagrams of mask layout data polygons having edges that represent a jog, a tab-like jog, and a stair, respectively, according to an embodiment of the invention. 
         FIGS. 8A ,  8 B,  8 C, and  8 D are diagrams of smooth, continuous, and differentiable functions that can be used to determine manufacturing penalties such as a shape penalty and/or a gap penalty, according to varying embodiments of the invention. 
         FIG. 8E  is a diagram of a scale assessment factor function that is used to turn off manufacturability (i.e., manufacturing) penalties during optimization, when a jog becomes small in all directions, according to an embodiment of the invention. 
         FIG. 9  is a diagram of a mask layout data polygon, according to an embodiment of the invention. 
         FIG. 10  is a diagram of a function that can be used to determine an aspect ratio shape penalty, according to an embodiment of the invention. 
         FIG. 11  is a diagram of mask layout data polygons having edges in relation to which a manufacturing gap penalty can be determined, according to an embodiment of the invention. 
         FIGS. 12A ,  12 B,  12 C, and  12 D are diagrams of mask layout data polygons having edges in relation to which a manufacturing crossing penalty can be determined, according to an embodiment of the invention. 
         FIGS. 13A and 13B  are diagrams of smooth, continuous, and differentiable functions that can be used to determine manufacturing penalties such as a crossing penalty, according to varying embodiments of the invention. 
         FIG. 14  is a diagram of a representative mask layout data cell in relation to which lithographic mask optimization can be performed, according to an embodiment of the present invention. 
         FIG. 15  is a diagram depicting an exemplary spatial domain diffraction orders error function, according to an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION OF THE DRAWINGS 
     In the following detailed description of exemplary embodiments of the invention, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration specific exemplary embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention. Other embodiments may be utilized, and logical, mechanical, and other changes may be made without departing from the spirit or scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined only by the appended claims. 
     Problem Background and Solution Overview 
     There is increasing interest in lithographic optimization methods that attempt to optimally manage the fundamental resolution constraints imposed by state-of-the-art exposure tools. One such approach is known as source mask optimization (SMO). In SMO, the set of diffraction orders obtained from a mask, which is referred to as a wavefront, is directly optimized, permitting this approach to efficiently gauge the true degrees of freedom of the process, and thus to determine an optimal set of image-forming waves that can be propagated through a finite optical system to print a desired pattern. 
     To achieve this, SMO typically defers consideration of purely indirect constraints that are imposed by mask-shape topology and that are not fundamental to the imaging process. More specifically, SMO addresses mask shape constraints typically during a concluding wavefront engineering step that is carried out after the optimum wavefront has been determined. This divide-and-conquer approach prevents the unnecessary penalization of imaging performance that can result from ad-hoc mixing of manufacturability and process window terms in a single relatively cumbersome objective. 
     The wavefront engineering process of SMO has two stages. The first stage generates an initial approximation to a manufacturable mask by synthesizing a bitmap mask. In the second stage, the bitmap is converted to polygons, and the corner positions of these corner positions are optimized. One requirement during the second stage optimization is that the mask polygons be made manufacturable. While manufacturability is also important to the first stage—which is referred to as gridline optimization—existing technologies for assessing mask manufacturability are ill-suited to the solution of mask optimization problems. 
     It is noted that polygon corner optimization can be performed on mask polygons that are obtained by approaches other than first stage wavefront engineering. Such approaches include model-based optical proximity correction (MBOPC), for instance. As a matter of nomenclature used herein, polygon corner optimization does not have to be referred to as a stage of wavefront engineering, particularly if the SMO process does not employ a first stage of wavefront engineering. 
     Until recently, lithographic practice retained a partial correspondence between mask shapes and target designs. For instance, the known MBOPC approach exploits this correspondence to correct printed edge positioning by using separate sequential applications of a derivative-free shooting or feedback method, in which each printed edge is treated as being coupled to a corresponding and controlling mask edge. This model-driven approach itself has replaced earlier rule-based OPC methods for determining edge correction. 
     However, in state-of-the-art applications, the corrector framework of MBOPC can become problematic for a number of reasons. The underlying cause of these reasons is the necessity of pushing very aggressively against the resolution limits of exposure tools where performance has reached technologic limits, by such techniques as superimposing multiple exposures of masks that have pattern densities very near the resolution limit. Such aggressive measures pose certain difficulties. For example, increasing edge density by summing across multiple exposures can make it less meaningful to posit specific connections between individual edges in the masks and individual edges in the target pattern. Such identification can even become more undesirable, since improved performance can be obtained using mask patterns that have greater complexity than the original design. 
     In addition, the complex optimization approaches that are being increasingly resorted to in the absence of direct resolution improvements can give rise to topologically complex mask shapes having edge fragments that often have very little geometrical correlation with individual edge segments of features of the original design. This is a consequence of the success with which advanced mask design approaches make possible the printing of image features that are considerably smaller than the classical resolution limit of the exposure tool. When such methods are pursued aggressively, it usually becomes desirable to fit a relatively large number of mask feature elements within each resolution patch of the lens, and these features are sometimes phase-shifted. As a result, there may no longer be a simple one-to-one correspondence between the mask features and the original design features. 
     Conversely, the mask manufacturability rules that have conventionally been used have their origin in rules for laying out circuit designs. As such, these rules are relatively simple, involving features and separations of main features, or involve decorations to these features, such as serifs or hammerheads. This simple division also allows existing mask manufacturability rules to distinguish requirements for rendering features that predominantly correspond to primary circuit features from those which take the form of unresolved OPC decorations on the primary features. Such OPC decorations, which may also be serifs or hammerheads, are referred to as assist features, and rules relating to them may be considered a third category of rules. Such assist features are unresolved lines which are separate from the primary features, but are not typically adjusted during conventional OPC techniques. 
     Unfortunately, advanced lithographic optimization approaches can give rise to masks having features that follow no such simple divisions, and in fact the features need not fall into any clear-cut discrete categories. Elements that resemble decorations can themselves have decorations, and in general the features in these newer types of heavily optimized advanced masks exhibit a more uniform continuity in their structures across different scales that has been traditionally seen. 
     Moreover, conventional mask manufacturability rules only serve to classify a set of patterns as manufacturable or non-manufacturable, and do so based on specific topological categories into which the pattern features fall. However, small motions can sometimes change the character of a feature, and as a result difficulties arise in advanced applications when one attempts to employ existing rule-based techniques that apply classification rules to ascertain manufacturability. In particular, it can be difficult to apply conventional rule-based techniques to the advanced task of obtaining a quantitative metric as to the severity of non-manufacturability in patterns that conventional approaches simply classify as being non-manufacturable. Such a metric is nevertheless desirable for optimization, since powerful techniques are available to improve features of interest that can be expressed as smooth continuous functions, including the case where the improvements must respect constraint requirements that can likewise be expressed in terms of smooth and continuous functions. 
     In addition, conventional classification of patterns as manufacturable (or not) is ultimately an oversimplification. Typically what is meant is that when working with patterns that are derived using standard OPC techniques, the mask maker has found that if the patterns obey a set of empirically deduced manufacturability rules, the patterns can very likely be fabricated within a quoted set of tolerances. Although useful as a practical matter, such a formulation ignores the possibility that the body of experience that is used to determine the set points for the manufacturability rules will likely fail to provide proper guidance when the set points are applied to patterns that are atypical in some way. 
     Furthermore, even if correct, conventional prior art mask manufacturability rules may not provide the user with an adequate picture of the tradeoffs involved in fabricating a set of patterns. A simple certification that the feature dimensions will allow specified tolerances to be met is limited in this respect. In some cases, there can be potential benefits to the user from achieving significantly tightened tolerances by using features that do not push close to the manufacturability limits, assuming that such features can still provide adequate performance within the lithographic image that produce. The conventional rules thus provide no quantitative guidance as to whether a contemplated change in a manufacturable shape improves manufacturability or degrades it. 
     Another consideration in advanced mask designs is that feature decorations and even some features themselves, such as assist features that are desirably subject to optimization along with other features, are not resolvable by the projection lens, but are included on the mask only for their corrective effect on the imaging behavior of neighboring features. It is noted that the patterns in lithographic masks are commonly polygonal, and often can be assumed to consist of only horizontal and vertical edges, which is referred to as being “Manhattan.” 
     However, with state-of-the-art features that are heavily directed for OPC, it can be impossible and unnecessary to faithfully reproduce shallow fragments and sharp corners of such polygons as literal shapes. Nonetheless, such shapes must usually be successfully registered into the mask film with high yield as rounded apertures. Although fragmented edge decorations, which can also be referred to as “jogs,” in such features cannot be reproduced with high fidelity, they should still register as appropriate deviations within the mask aperture&#39;s contour at the positions of the jogs. 
     The patent application entitled “determining manufacturability of lithographic mask using continuous derivatives characterizing the manufacturability on a continuous scale,” filed on Dec. 14, 2008, and assigned Ser. No. 12/334,488, solves these foregoing issues and problems. In particular, this patent application provides a smoothly differentiable metric that expresses the manufacturability of mask features in terms of the above-identified requirements for lithographic masks. Such a metric can be used in any mask optimization or refinement procedure in which manufacturability tradeoffs are usefully assessed using a quantitative metric. An example of such a procedure is SMO, as has been described above. 
     In general, the patent application having the Ser. No. 12/334,448 generates a manufacturability metric for a set of Manhattan patterns by applying smooth assessment functions to the pattern coordinates, and specifically to the coordinates of pairs of edges in the patterns. If the polygon edges are traversed in a specific handedness, such as the well-known right-handed rule, each edge has a start coordinate and an end coordinate, allowing the edges to be treated as signed vector quantities. This in turn permits the distinction between close edges that form what is referred to herein as a “stair,” as is also described in detail later in the detailed description, and close edges that form a narrow gap, where the former are easier to manufacture than the latter. 
     Furthermore, the difficulty in manufacturing a narrow gap can be eased if the polygon portion in question is actually elongated in such a way as to be narrower in the perpendicular direction to form what is referred to herein as a “tab-like jog,” which is also described in detail later in the detailed description. As noted above, the mask writer need not reproduced such a jog with high fidelity, but nevertheless has to register an appropriate deviation in the feature contour at the jog&#39;s position. The manufacturability metric provided encapsulates these considerations using smoothly varying assessment functions, as is also described in detail later in the detailed description. 
     The patent application having the Ser. No. 12/334,448 provides for different assessment functions for different types of manufacturability constraints, such as shape, gap, and crossing constraints, and that permit smooth convergence. A particular structure of these functions, such as piece-wise quadratic, can be selected in a way that renders the functions not only continuous, but also differentiable. This is advantageous, because optimization algorithms can be arbitrarily sensitive to slope discontinuities. Jog and stair assessment functions can be combined in a uniform manner, which provides for a smooth transition between these two otherwise topologically distinct configurations. 
     However, the patent application having the Ser. No. 12/334,448, which has issued a U.S. Pat. No. 8,028,254, has certain shortcomings. While the approach disclosed in this patent application works well with a non-linear optimizer, it does not easily accommodate various sizes of shapes and optical responses having the same order. Specifically, the approach has difficulties with smaller than a specified scale of smoothness. While an aspect ratio tolerance can be relaxed to overcome these difficulties, this results in unacceptable shapes. 
     Embodiments of the present invention thus build on the above-identified patent application that has since issued as a patent as noted above to overcome these shortcomings. In particular, determining the manufacturability of a lithographic mask, including determining the manufacturing penalty in making the mask, includes the following for a selected edge pair having first and second edges that are at least substantially parallel to one another. Specifically, a manufacturing shape penalty owing to an aspect ratio of the first edge relative to a size of a gap between the first and second edges is determined such that this penalty takes into account a pair of connected edges of the first edge that are at least substantially parallel to the first edge. A manufacturing shape penalty owing to an aspect ratio of the second edge relative to a size of the gap can also be determined that takes into account a pair of connected edges of the second edge that are at least substantially parallel to the second edge. 
     In this way, embodiments of the present invention employ a manufacturability metric that is based on edge information that is contextual. In addition to target edge pairs, data regarding edges connected to the edges of the edge pairs is also employed. As such, a more precise expression of manufacturability requirements is achieved, as is better performance with respect to convergence on optical results. 
     Furthermore, another difficulty is with jogs that are smaller than a specified scale of smoothness in the manufacturability metrics. To alleviate this problem, embodiments of the present invention add scale assessment factor functions, and clean up shorter edges during optimization. In this way, embodiments of the invention can smoothly switch off manufacturability penalties during optimization when the jogs become very small in all dimensions, so that the optimization process converges quickly. The subsequent clean-up routine is then run to remove these small (shorter) edges. 
     Technical Background and Overview 
       FIG. 1A  shows a representative lithographic mask layout  100 , according to an embodiment of the invention. The lithographic mask layout  100  includes a cell  102  that is symmetrically duplicated throughout the layout  100 . In particular,  FIG. 1B  shows the symmetry of the cell  102  within the lithographic mask layout  100 , according to an embodiment of the invention. The letter F is used in  FIG. 1B  to denote the cell  102  in the layout  100  of  FIG. 1A , and the various iterations of the letter F in  FIG. 1B  show how the cell  102  is symmetrically duplicated throughout the layout  100  in  FIG. 1A . The symmetry in  FIG. 1B  can be referred to as a staggered symmetry. To assess the manufacturability of the lithographic mask layout  100 , particularly at the boundaries of the cell  102 , the mask layout data underlying the layout  100  is replicated over n-by-n units, where n may be equal to three. For instance,  FIG. 1C  shows such a replication  104  based on the cell  102 , according to an embodiment of the invention. 
       FIG. 2  shows representative mask layout data for a portion  202  of the lithographic mask layout  100 , according to an embodiment of the invention. It is noted that the lithographic mask layout  100  includes a number of polygons. Each polygon has a number of edges. An edge pair is defined as a pair of different edges, of the same or different polygons. For example, an edge pair may include two different edges of the same polygon, or an edge pair may include two edges of different polygons. 
     Furthermore, each edge of each polygon can have a number of attributes: polynum, edgenum, dir, pos, start, end, last_edgenum, and next_edgenum. Polynum is the identification for of the polygon to which the edge in question belongs. Edgenum is the identification of the edge in question within this polygon. Dir is the edge direction, where a first value, like one, may specify the horizontal direction, and a second value, like two, may specify the vertical direction. Pos is the edge position, including the x-coordinate for a vertical edge and a y-coordinate for a horizontal hedge. Start is the smaller coordinate of the end of the edge, which is the lower coordinate for a vertical edge, and the left coordinate for a horizontal edge. End is the larger coordinate of the end of the edge, which is the upper coordinate for a vertical edge, and the right coordinate for a horizontal edge. Last_edgenum is the identification number of the previously connected edge within the same polygon, in a prespecified direction (i.e., clockwise or counter-clockwise). Similarly, next_edgenum is the identification number of the next connected edge within the same polygon, in the prespecified direction. 
       FIGS. 3A ,  3 B, and  3 C represent three different types of manufacturing penalties that can occur when determining the manufacturability of a lithographic mask, according to varying embodiments of the invention.  FIG. 3A  particularly shows a penalty for the shape of a single polygon, which is referred to as a shape penalty.  FIG. 3B  particularly shows a penalty for the gap between two different polygons, which is referred to as a gap penalty.  FIG. 3C  particularly shows a penalty for crossing edges between two different polygons, which is referred to as a crossing penalty. 
     In  FIG. 3A , the polygon  302  has six edges. Each different pair of parallel edges within this polygon  302  makes up a target edge pair that represents a corresponding shape penalty for the polygon  302 , as depicted on the right-hand side of  FIG. 3A . In  FIG. 3B , the polygons  304  and  306  each has four edges. Each different pair of parallel edges of the polygons  304  and  306  (i.e., with one edge from the polygon  304  and one edge from the polygon  306 ) makes up a target edge pair that represents a corresponding gap penalty for the two polygons  304  and  306 , as depicted on the right-hand side of  FIG. 3B . In  FIG. 3C , the polygons  308  and  310  each has four edges. Each different pair of non-parallel, or crossing, edges of the polygons  308  and  310  (i.e., with one edge from the polygon  308  and one edge from the polygon  310 ) makes up a target edge pair that represents a corresponding crossing penalty for the two polygons  308  and  310 . 
     A shape penalty is a manufacturing penalty incurred in manufacturing the lithographic mask due to the shape of a single polygon, owing to the difficulty in making the shape. Thus, a target edge pair can define the two edges of a given polygon that represent a shape penalty. A gap penalty is a manufacturing penalty incurred in manufacturing the lithographic mask due to the gap between the two edges of two different polygons, owing to the difficulty in maintaining this gap. Thus, a target edge pair can define the two edges of two different polygons that represent a gap penalty. A crossing penalty is a manufacturing penalty incurred in manufacturing the lithographic mask due to the potential for overlap between the two edges of two different polygons and a bow tie shape by one polygon, owning to the difficulty in insuring that such overlap does not occur. Thus, a target edge pair can define the two edges of two different polygons that represent a crossing penalty. 
       FIG. 4  shows a method  400  for determining the manufacturability of a lithographic mask used to fabricate instances of a semiconductor device, according to an embodiment of the invention. Target edges are selected from mask layout data representing a lithographic mask ( 402 ). Thereafter, target edge pairs are selected ( 404 ). The target edge pairs are selected in one embodiment to represent each potential instance of a manufacturing penalty, such as a shape penalty, a gap penalty, or a crossing penalty. That is, in at least some embodiments, target edge pairs are identified in part  402  that represent all potential instances of manufacturing penalties like shape, gap, and crossing penalties. 
     Furthermore, in one embodiment, target edges are selected from mask layout data in part  402 , and target edge pairs are selected in part  404  as described in a subsequent section of the detailed description. However, various other approaches to how to select target edges and target edge pairs can be employed as well. For instance, a greedy approach may be followed in which all target edges of all polygons are selected, and all unique target edge pairs of these target edges are selected, although this approach may be computationally intractable. Furthermore, the target edges and/or the target edge pairs may be selected and/or reduced in number as described in the patent applications referenced above and that have been incorporated by reference. 
     Thereafter, the manufacturability of the lithographic mask is determined ( 406 ), which includes determining the manufacturing penalty in making the lithographic mask, based on the target edge pairs. The manufacturing penalty in making the lithographic mask can include the shape, gap, and crossing penalties that have been described. In one embodiment of the invention, the manufacturability of the lithographic mask and the manufacturing penalty in making the lithographic mask are determined as representatively described in detail in subsequent sections of the detailed description in relation to particular types of manufacturing penalties. 
     In particular, the shape, gap, and crossing penalties for all target edge pairs may each be constrained to lie below a specified threshold of manufacturability, which for the shape and gap penalties might correspond to the penalty level reached by edges which are separated by the minimum resolution of the mask making process, while for the crossing penalty the constraint threshold may be set at the level reached by edges which just touch, i.e. just begin to cross. Alternatively, in relation to the manufacturing penalties that are determined for specified mask layout data, an overall manufacturability may be determined by summing these penalties and/or by averaging them, in various embodiments of the invention. 
     Once the manufacturability of the lithographic mask has been determined, it is output ( 408 ). For instance, the manufacturability may be displayed on a display device of a computer for viewing by a user. The method  400  of  FIG. 4 , including at least parts  402 ,  404 ,  406 , and  408 , may be performed by one or more computer programs, which may be executed by one or more processors of one or more computer devices, as can be appreciated by those of ordinary skill within the art. 
     Ultimately, the lithographic mask may have its design optimized, based on the manufacturability determine, so that it is in fact easier to manufacture ( 410 ). In this respect, parts  402 ,  404 ,  406 ,  408 , and  410  of the method  400  may be iteratively performed until a lithographic mask having a desired manufacturability difficulty has been achieved. In one approach, variables and constraints for the optimization problem are first set up, and then a nonlinear optimization problem. More information regarding such an optimization problem solution is described later in the detailed description, along with an example of one particular type of optimization that can be implemented. 
     It is noted that the optimization process of part  410  may be performed as has been described above. That is, one or more selected edges of the polygons may be shrunk to a smaller sized, and then the polygons may be cleaned up to reduce the number of these edges. This shrinking and reduction process may be iteratively performed a number of times as well, as has been noted above. Finally, once a design for the lithographic mask has been approved, the lithographic mask may be made ( 412 ), and instances of a semiconductor device fabricated using the lithographic mask ( 414 ). 
     Selecting Target Edges and Target Edge Pairs 
       FIG. 5  illustratively depicts an approach to selecting target edges and target edge pairs, according to an embodiment of the invention. Mask layout data is divided into a number of cells. For a given cell being analyzed, such as the cell  502 , the surrounding eight cells  504  are considered in addition to the cell  502 . In particular, a predetermined boundary threshold around the cell  502  is considered, as indicated by the dotted line box  506  in  FIG. 5 . 
     Thus, in one embodiment, the target edges that are selected in part  402  of the method  400  of  FIG. 4  for the cell  502  include all the edges that lie within the dotted line box  506 . Thereafter, the target edge pairs that are selected in part  404  of the method  400  for the cell  502  include each unique pair of the target edges in one embodiment of the invention. In other embodiments, the target edge pairs may be selected in different ways, such as, for example, where at least one of the target edges within each target edge pair is within the cell  502  (such that the other edge of each target edge pair may be within the cell  502  or outside the cell  502  but within the dotted line box  506 ). As such, edges outside the dotted line box  506 , such as the edges  508 A and  508 B, are not selected and are not part of any target edge pairs. 
     Determining Manufacturing Shape Penalty to Determine Manufacturability, Part I 
       FIGS. 6 ,  7 A,  7 B, and  7 C show representative polygons  600 ,  700 ,  750 , and  770 , respectively, in relation to which how a manufacturing shape penalty to determine manufacturability is determined, according to varying embodiments of the invention. In  FIG. 6 , the polygon  600  has its edges ordered starting from edge i in a counter-clockwise manner around the polygon  600 . Thus, the edge i points in the opposite direction as compared to the edge j. 
     In  FIG. 7A , the polygon  700  also has its edges ordered starting from edge i in a counter-clockwise manner around the polygon  700 . The edge i points in the opposite direction as compared to the edge j. Furthermore, it is noted that the polygon  700  has a jog shape. A jog shape can be described in relation to two parallel edges having opposite directions and separated by perpendicular edges. 
     In  FIG. 7B , the polygon  750  has its edges ordered in a counter-clockwise manner around the polygon  750 . The edge i points in the opposite direction as compared to the edge j. Furthermore, it is noted that the polygon  750  has a tab-like jog shape. A tab-like jog shape can be described in relation to two parallel edges having opposite directions and separated by perpendicular edges. In addition, these two parallel edges are short enough that they do not need to be considered as a penalized shape. 
     In  FIG. 7C , the polygon  770  has its edges ordered in a counter-clockwise manner around the polygon  770 . The edge i points in the same direction as compared to the edge j. Furthermore, it is noted that the polygon  750  has a stair shape. A stair shape can be described in relation to two parallel edges having the same direction and separated by a perpendicular edge. It is noted, therefore, that one difference of the edges i and j in  FIG. 7C  as compared to in  FIGS. 7A and 7B  is that these edges have the same direction in  FIG. 7C , while they have opposite directions in  FIGS. 7A and 7B . 
     To determine the manufacturing shape penalty in part  406  of the method  400  of  FIG. 4  for a selected target edge pair having a first edge and a second edge that belong to the same polygon and that are at least substantially parallel to one another, the following expression is evaluated:
 
 P   shape   =P   gap     —     size   ×P   jog     —     stair   ×P   aspect     —     ratio     —     i     —     a   ×P   aspect     —     ratio     —     j     —     a   ×P   overlap  
 
The shape penalty P shape  is defined as a product of four assessment functions: a gap size factor function P gap     —     size ; a jog stair detection function P jog     —     stair ; two aspect ratio factor functions P aspect     —     ratio     —     i     —     a  and P aspect     —     ratio     —     j     —     a ; and an overlap detection factor function P overlap . P shape , P gap     —     size , P jog     —     stair , P aspect     —     ratio     —     i     —     a , P aspect     —     ratio     —     j     —     a , and P overlap  are smoothly varying continuous functions, as opposed to non-smoothly varying discontinuous functions. These functions are desirably differentiable, and may be defined as piece-wise quadratic functions or sigmoid functions in varying embodiments of the invention.
 
       FIG. 8A  illustratively depicts the gap size factor function P gap     —     size , according to an embodiment of the invention. P gap     —     size  denotes a manufacturing penalty owing to a size of a gap between the first and the second edges in question. In  FIG. 8A , w gap  denotes a weight coefficient for this assessment factor, and a ij  denotes the gap between the edges. Thus, rather being a non-smoothly varying discontinuous function where P gap     —     size  forces a yes or no answer depending on whether there is indeed a gap between the two parallel edges in question, P gap     —     size  in  FIG. 8A  is a smoothly varying continuous functions where it is able to take on a number of different values as a function of the gap between the parallel edges. The function P gap     —     size  thus varies depending on the gap a ij  between the edges. 
       FIG. 8B  illustratively depicts the jog stair detection function P jog     —     stair , according to an embodiment of the invention. P jog     —     stair  denotes a manufacturing penalty owing to whether the first and the second edges define a jog shape or a stair shape. In  FIG. 8B , A denotes a threshold for judging a jog shape, and i·j denotes the dot product of the two edges (where these edges are vectors having both directions and magnitudes). The usual shape classification of whether two edges are forming a jog shape or a stair shape forces a discrete binary decision, yes or no. To avoid such a discontinuous outcome, the function of  FIG. 8B  provides a detection zone  804 B of a tab-like jog shape between the jog shape zone  802 B the and stair shapes zone  806 B. An intermediate assessment value between 0 and 1 is provided to such a tab-like jog shape. The function P jog     —     stair  thus varies between 0 and 1 based on the value of the dot product of the two edges. 
     It is noted that while  FIG. 7A  shows a jog shape and  FIG. 7C  shows a stair shape,  FIG. 7B  shows a tab-like jog shape. It is further noted that as to the edges i and j in  FIG. 7A , the inner product of the edge vectors i and j is negative. As a result, a P jog     —     stair  value is one. By comparison, as to the edges i and j in  FIG. 7B , the inner product of the edge vectors i and j is small negative value, such that a P jog     —     stair  value is an intermediate value. By comparison, as to the edges i and j in  FIG. 7C , the inner product of the edge vectors i and j is positive, such that a zero P jog     —     stair  value results. 
     An advantage of the approach used in embodiments of the invention is that it robustly encapsulates the manufacturability of complex pattern topologies by a regular and repetitive pair-wise assessment of edges. For example, two edges can be assessed as stair-like, even though they contain intermediate edges connecting them. As such, the possibility of narrow jogs in the pattern as a whole is taken care of when the intermediate edge pairs themselves are assessed. 
       FIGS. 8C and 8D  illustratively depict functions ƒ minus (u) and ƒ plus (u) that are used in evaluating the overlap detection factor function P overlap , according to an embodiment of the invention. P overlap  denotes a manufacturing penalty owing to a degree of overlap between the first and the second edges. P overlap  is determined in one embodiment as:
 
 P   overlap =(1 −v   1 )·(1 −v   2 ),
 
where:
 
 v   1 =ƒ minus ( x   1   −x   3 )·ƒ minus ( x   1   −x   4 )·ƒ minus ( x   2   −x   3 )·ƒ minus ( x   2   −x   4 ) and
 
 v   2   =f   plus ( x   1   −x   3 )·ƒ plus ( x   1   −x   4 )·ƒ plus ( x   2   −x   3 )·ƒ plus ( x   2   −x   4 ).
 
As denoted in  FIG. 6 , x 1  is the starting x coordinate of edge i, x 2  is the ending x coordinate of edge i, x 3  is the starting x coordinate of edge j, and x 4  is the ending x coordinate of edge j. Thus, smooth functions are again used to assess whether two edges overlap or not. A buffer distance, d buf , is provided to handle intermediate conditions of overlap.
 
       FIG. 8E  illustratively depicts a scale assessment factor function P scale  that is used to turn off manufacturability penalties during optimization, when a jog becomes small in all directions, according to an embodiment of the invention. When a jog has a size i between t and −t, P scale  is thus zero. When a jog has a size i greater than u or less than −u, P scale  is equal to one. Finally, when a jog has a size i between t and u or between −t and −u, then P scale  is between zero and one. As noted above, turning off manufacturability penalties permits the optimization process to converge quickly. A clean-up routine can thereafter be run to remove these small (shorter) jogs. 
     It is often found that the optimization process shrinks unimportant edges to a very small size. As such, as part of the optimization process, a clean-up may be performed in which the number of edges in the polygons are benignly reduced. This reduction in turn typically permits the mask to be fabricated at lower cost. To enhance the degree of edge simplification, the optimization process may be reperformed after clean-up has been accomplished, to potentially shrinking additional edges to a size that is trimmable during a subsequent clean-up. This cycle can be repeated for additional iterations, where the clean-up can include performing a complementary operation in which edges that are longer than some threshold (such as larger than about four times the mask resolution limit) are divided into two or more shorter edges. It is noted that this iterative process of performing clean-up followed by again performing optimization may be carried out without implementing the scale assessment factor P scale . Instead, each polygon optimization can terminate after a fixed number of adjustment steps, or after a solution has been found to contain new short edges during several recent adjustment steps. 
     Determining Manufacturing Shape Penalty to Determine Manufacturability, Part II 
     Determining the manufacturability of the lithographic mask, including determining the manufacturing penalty in making the lithographic mask, in part  406  of the method  400  can include the following. For a selected edge pair having a first edge and a second edge that are at least substantially parallel to one another, a manufacturing shape penalty owing to an aspect ratio of the first edge relative to a size of a gap between the first and second edges is determined. Specifically, this manufacturing shape penalty takes into account a pair of connected edges of the first edge that are at least substantially parallel to the first edge. Likewise, a manufacturing shape penalty owing to an aspect ratio of the second edge relative to a size of the gap between the first and second edges is determined. Specifically, this manufacturing shape penalty takes into account a pair of connected edges of the second edge that are at least substantially parallel to the second edge. 
       FIG. 9  shows a representative polygon  900 , according to an embodiment of the invention. The polygon  900  includes two selected edges i and j that are at least substantially parallel to one another, and in relation to which a manufacturing shape penalty owing to an aspect ratio of each selected edge relative to a size of the gap a between the two edges is determined. The edge i has a pair of connected edges that includes a previous edge i prev  and a next edge i next . Likewise, the edge j has a pair of connected edges that includes a previous edge j prev  and a next edge j next . It is noted that the edge i next  is the same edge as the edge j prev  in the example of  FIG. 9 . 
     The manufacturing shape penalty owing to the aspect ratio of the edge i is determined to take into account its next and previous edges, via: 
               P     aspect   ⁢           ⁢   _   ⁢           ⁢   ratio   ⁢           ⁢   _   ⁢           ⁢   i   ⁢           ⁢   _   ⁢           ⁢   a       =     f   (           i   →     2     +       L   excursion     ⁡     (       i   →     ,       i   →     prev       )       +       L   excursion     ⁡     (       i   →     ,       i   →     next       )               a   →     2     +   δ       )           
In this equation, {right arrow over (i)} is a vector representing the edge i, {right arrow over (i)} prev  is a vector representing a previous connected edge of the pair of connected edges of the edge i, and {right arrow over (i)} next  is a vector representing a subsequent connected edge of the pair of connected edges of the edge i. Furthermore, L excursion  ({right arrow over (x)}, {right arrow over (j)}) is a smoothly varying continuous function providing a value between zero and |{right arrow over (y)}| 2  as a function of gap between {right arrow over (x)} and {right arrow over (y)}, and can also be considered as, for example, a piece-wise quadratic of the gap between {right arrow over (x)} and {right arrow over (y)}. The vector {right arrow over (a)} is a vector between the edges i and j that represents the perpendicular distance between these two edges, and δ is a constant.
 
       FIG. 10  shows smoothly varying continuous function L excursion  ({right arrow over (x)}, {right arrow over (y)}) according to an embodiment of the invention. To the left and the right of the origin, the function in the example of  FIG. 10  is symmetrical. The x-axis denotes the gap between {right arrow over (x)} and {right arrow over (y)}. Where the gap is between 0 and a threshold t, or between 0 and a threshold −t, then the value of the function is |{right arrow over (y)}| 2 . Where the gap is greater than a threshold u or is less than a threshold −u, then the value of the function is zero. Where the gap is between the thresholds t and u or is between the thresholds −t and −u, then the value of the function substantially linearly is between zero and |{right arrow over (y)}| 2 . 
     The manufacturing shape penalty owing to the aspect ratio of the edge j is similarly determined to take into account its next and previous edges, via: 
               P     aspect   ⁢           ⁢   _   ⁢           ⁢   ratio   ⁢           ⁢   _   ⁢           ⁢   j   ⁢           ⁢   _   ⁢           ⁢   a       =     f   (           j   →     2     +       L   excursion     ⁡     (       j   →     ,       j   →     prev       )       +       L   excursion     ⁡     (       j   →     ,       j   →     next       )               a   →     2     +   δ       )           
In this equation, {right arrow over (j)} is a vector representing the edge j, {right arrow over (j)} prev  is a vector representing a previous connected edge of the pair of connected edges of the edge j, and {right arrow over (j)} next  is a vector representing a subsequent connected edge of the pair of connected edges of the edge j. Furthermore, L excursion  ({right arrow over (x)}, {right arrow over (y)}) is as has been described.
 
Determining Manufacturing Gap Penalty to Determine Manufacturability
 
       FIG. 11  shows representative polygons  902  and  904  in relation to which how a manufacturing gap penalty to determine manufacturability is determined, according to an embodiment of the invention. The edge i is located within the polygon  902 , whereas the edge j is located within the polygon  904 . Thus, the edges i and j are in different polygons, and are at least substantially parallel to one another. The manufacturing gap penalty relates to the penalty incurred in manufacturing the lithographic mask due to the gap between the polygons  902  and  904 , and specifically due to the gap between these edges. 
     To determine the manufacturing gap penalty in part  406  of the method  400  of  FIG. 4  for a selected target edge pair having a first edge and a second edge that are at least substantially parallel to one another and are located in different polygons, the following expression is evaluated:
 
 P   gap   =P   gap     —     size   ×P   overlap  
 
The gap penalty P gap  is defined as a product of two assessment functions: a gap size factor function P gap     —     size  and an overlap detection factor function P overlap .
 
     The gap size factor function P gap     —     size  can be defined as has been described in the preceding section of the detailed description in relation to  FIG. 8A . The overlap detection factor function P overlap  can be defined as has been described in the preceding section of the detailed description in relation to  FIGS. 8C and 8D . Thus, both of these functions, as well as the function P gap , are smoothly varying continuous functions, as opposed to non-smoothly varying discontinuous functions. The functions are desirably differentiable, and may be defined as piece-wise quadratic functions or sigmoid functions in varying embodiments of the invention. 
     Determining Manufacturing Crossing Penalty to Determine Manufacturability 
       FIGS. 12A ,  12 B,  12 C, and  12 D show representative polygons in relation to which how a manufacturing crossing penalty to determine manufacturability is determined, according to an embodiment of the invention.  FIGS. 12A and 12B  show different polygons case. In  FIGS. 12A and 12B , the edge i are located within the polygon  1002 A and  1002 B, whereas the edge j are located within the polygon  1004 A and  1004 B.  FIGS. 12C and 12D  show one polygon case. Thus, the edges i and j are at least substantially perpendicular to one another. The manufacturing crossing penalty relates to the penalty incurred in manufacturing the lithographic mask due to the potential overlap between different polygons and a bow tie shape of one polygon. Thus,  FIGS. 12A and 12C  are not penalized and  FIGS. 12B and 12D  are heavily penalized. 
     To determine the manufacturing crossing penalty in part  406  of the method  400  of  FIG. 4  for a selected target edge pair having a first edge and a second edge that are at least substantially perpendicular to one another, the following expression is evaluated:
 
 P   crossing   =w   crossing   ×P   crossing     —     i     —     j   ×P   crossing     —     j     —     i  
 
The crossing penalty P crossing  is defined as a product of a constant w crossing  and two assessment functions P crossing     —     i     —     j  and P crossing     —     j     —     i . The constant w crossing  refers to a weight coefficient. The functions P crossing , P crossing     —     i     —     j , and P crossing     —     j     —     i  are smoothly varying continuous functions, as opposed to non-smoothly varying discontinuous functions. The functions are desirably differentiable, and may be defined as piece-wise quadratic functions or sigmoid functions in varying embodiments of the invention.
 
       FIGS. 13A and 13B  illustratively depict the assessment functions P crossing     —     i     —     j  and P crossing     —     j     —     i , according to an embodiment of the invention. P crossing     —     i     —     j  denotes a manufacturing penalty owing to the edge i crossing the second edge j, whereas P crossing     —     j     —     i  denotes a manufacturing penalty owing to the edge j crossing the edge i. Thus, P crossing     —     i     —     j  focuses on the edge i in relation to the edge j, whereas P crossing     —     j     —     i  focuses on the edge j in relation to the edge i. 
     It is noted that for the polygons  1002 A and  1004 A of  FIG. 12A , the edge i is a horizontal edge having starting coordinates (x 1 ,y i ) and ending coordinates (x 2 ,y i ), and the edge j is a vertical edge having starting coordinates (x 3 ,y 2 ) and ending coordinates (x 3 ,y 3 ). As such, the value u used to determine P crossing     —     i     —     j    FIG. 13A  is u=(x 1 −x 3 )·(x 2 −x 3 ). By comparison, the value v used to determined P crossing     —     j     —     i  in  FIG. 13B  is v=(y 1 −y 2 )·(y 1 −y 3 ). 
     Optimizing Lithographic Mask 
     Once the manufacturability of the lithographic mask has been determined in part  408  of the method  400  of  FIG. 4 , it can be optimized in part  410  as has been summarized above. An example of a relatively straightforward optimization is to modify a polygon that has two edges defining a jog shape to instead define a tab-like job shape. Tab-like jog shapes are easier to manufacture than jog shapes, and as such this optimization reduces the manufacturing penalty in manufacturing the lithographic mask in question. 
     For example, a polygon may have a first edge and a second edge. Initially, where these edges define a jog shape, the first and the second edges are relatively long, where the edges have opposite directions. After modification, the edges define a tab-like jog shape, in which the two edges are relatively short. 
     More generally, as has been described above, optimization can include setting up variables and constraints for an optimization problem, and then solving the resulting nonlinear optimization problem. In particular, independent variables can be extracted with consideration of the Manhattan requirement as well as of different symmetries along cell boundaries. For instance,  FIG. 14  shows a representative cell  1200 , according to an embodiment of the invention. The cell  1200  includes polygons  1202  and  1204 . 
     In  FIG. 14 , X i  are independent variables used in optimization, and (x i ,y i ) are the coordinates of each corner of the polygons  1202  and  1204 . For each corner, at most just one of the x or y coordinates is assigned as an independent variable, due to the Manhattan requirement. When an edge is on the boundary of the cell  1200 , one coordinate can be skipped because this edge cannot move away from the boundary due to symmetry considerations. For example, the point  1206  can be skipped if even minor symmetry is assumed along this boundary of the cell  1200 . Such extraction of independent variables reduces the total computation and improves the convergence of manufacturability optimization. 
     Other lithographic metrics can be considered during optimization using the manufacturability determined in part  408  of the method  400 . For example, different types of diffraction orders error functions can be implemented, such as a sum of the square of diffraction orders error function, a maximum diffraction orders error function, and a spatial domain diffraction orders error function.  FIG. 15  particularly shows an example of a spatial domain diffraction orders error function, according to an embodiment of the invention. In particular,  FIG. 15  shows an inverse Fourier transform of a diffraction orders error function. Furthermore, weighted diffraction order error functions can be implemented to provide different tolerances to different diffraction orders, in accordance with their effects on the final image of the photolithographic mask in question. Spatial domain diffraction order errors and worst manufacturable region detection may also be employed to identify problematic areas within the mask layout data that may deserve special attention. 
     Conclusion 
     It is noted that, although specific embodiments have been illustrated and described herein, it will be appreciated by those of ordinary skill in the art that any arrangement calculated to achieve the same purpose may be substituted for the specific embodiments shown. This application is intended to cover any adaptations or variations of embodiments of the present invention. For example, a write-back cache may or may not be employed. Therefore, it is manifestly intended that this invention be limited only by the claims and equivalents thereof.