# Unified Victory Conformal Export for 2D Process Mode

**Introduction**

Victory Process can operate in either 1, 2 or 3-dimensions and can produce one of two different geometric representations: cell mode and process mode. Cell mode structures are generally composed of large, flat geometric parts but process mode structures may be smoothly varying and only very locally flat. Device meshes of cell mode structures resolve the input shape precisely but this is undesirable for the case of process mode structures as it would result in very finely sampled meshes which would be unsuitable for device simulation.

These differences present a challenge for the development of device meshing algorithms as a naive strategy may produce a separate algorithm for each combination of dimension and geometric representation. Alternatively, a more effective approach would have the goal of producing a unified device meshing algorithm which could be applied to all of the various necessary types of input. This allows development time to be minimised while maintenance and improvement effort is focused in one area which simultaneously benefits all contexts in which the unified algorithm is used. The difficulty here is that it may be more technically-challenging to write abstract dimension-independent meshing algorithms.

This document describes some recent work on the unified conformal algorithm, as demonstrated for the 2-dimensional process mode case.

**Removal of Fan Triangulations in Device Mesh**

Traditional Delaunay triangulation is based on an isotropic or Euclidean measurement of distance. This treats all directions as equal and is ideal for generating meshes from vertices which are sampled with a locally-uniform isotropic density. The Cartesian volume grids used as the basis for the conformal export may be strongly anisotropic which means isotropic Delaunay triangulation can produce undesirable results. These can include fan-like triangulations which negatively affect the appearance of the mesh as well as the subsequent device simulation.

The solution is to use an anisotropic Delaunay triangulation where the local measurement of distance is inherited from shape of the volume grid elements in that region. This helps to ensure that the structure of the underlying Cartesian mesh is reproduced in the device mesh. Figure 1 shows the difference in results between isotropic Delaunay meshing and the anisotropic approach used by default in the unified conformal algorithms. The difference is even more pronounced compared to the existing export 2D algorithm which attempts to capture the geometric detail of the input to a higher fidelity by including more vertices on the interfaces between pairs of materials.

Figure 1. Difference between isotropic and anisotropic Delaunay meshing. |

**Feature Detection in Input Geometry**

Reasonable approaches for device meshing involve the adequate reproduction of geometric features of various intrinsic dimensions; in 2D, there are 0-dimensional feature vertices and 1-dimensional feature curves which occur at the interfaces of pairs of layers. The analogue of these feature curves in 3D are the 2-dimensional interface surfaces. An effective method for device meshing needs to firstly identify these features in the input and then subsequently ensure that these are reasonably resolved in the output.

The problem of geometric feature detection is straightforward for cell mode structures which are mainly composed of flat constituents, such as polygons and line segments. The situation is less clear with process mode structures as these may be smoothly varying and lean-cut features may not be present due to the approximate nature of its geometric representation. This presents a problem for meshing algorithms even if they have the ability to resolve multidimensional features; these features must be uncovered in the input if they are to be included properly within the resulting device mesh.

In 2-dimensions, the unified conformal meshing algorithms operate in terms of rectangles defined by the volume grid. If an interface curve between a pair of layers passes through a rectangle, then that curve is locally approximated by a line-segment with vertices on the boundary of the rectangle. This means that if this curve has a corner within the rectangle then this feature is lost in the device mesh. On-the-other-hand, if the corner is identified as a 0-dimensional feature then it will be adequately resolved as the point on the curve is inherited in the remesh.

The unified conformal meshing algorithm now has the ability to identify features in the input geometry to ensure that the distance of approximation of the device mesh is within some tolerance of the input. This is an adaptive procedure which identifies sections of the input geometry which, when added to the device mesh, result in the maximum reduction of this approximation. This is performed iteratively until the user-specified maximum approximation distance is reached. Figure 2 shows the difference between the standard conformal 2D meshes and those produced with a tolerance of 0.01.

Figure 2. Difference between no feature detection and feature detection. |

The export statement used for the structure on the right of Figure 2 was:

export victory(conformal) name=”test4_conformal2d_md_001” max.distance=0.01

**Smoothing of Input Geometry**

The implicit geometric representation used in process mode may unequivocally locate the interface between a pair of layers but it only roughly approximates the 0-dimensional features which occur when these interfaces meet. The feature vertices are subject to a form of quantisation according to the local resolution of the grid used by Victory Process to represent the geometry. This can lead to “beak”-like artifacts which affect the appearance of the mesh and also reduce the quality of the device mesh if these are faithfully reproduced.

The 2D version of the unified conformal algorithm can precisely identify which vertices of the input geometry are suspicious and take steps to smooth the associated artifacts in the output mesh. This smoothing occurs for vertices which lie in three of the internal “layers” that Victory Process uses and takes two forms depending on whether the vertex lies in two or three materials.

**Two Material Case**

The smoothing which occurs for triple-layer vertices shared between two materials is related to traditional “Laplacian” mesh-smoothing where the vertex is moved to the average of its neighbours in the input. The result is similar to the existing smoothing that is produced by Victory Process as a result of the uncertainty in its geometric representation. Figure 3 shows the difference between the conformal device meshes without and with the smoothing algorithm activated for the case of triple-layers vertices shared between two materials.

Figure 3. Difference between no smoothing and double material smoothing. |

**Three Material Case**

Triple-layer vertices shared between three materials are dealt with by intersecting a ray between neighboring non-suspicious vertices onto the line defined other reasonable neighboring vertices. Figure 4 shows the difference between the conformal device meshes without and with the smoothing algorithm activated for the case of triple-layers vertices shared between three materials.

The export statement used for the structure on the right of Figures 4 and 5 was:

export victory(conformal) \ name=”test4_conformal2d_md_001_s” \ max.distance=0.01 smooth

Figure 4. Difference between no smoothing and triple material smoothing. |

Figure 5. Difference between default mesh and one with an obtuseness bounded by one. |

**Reduction of Level of Obtuseness of Device Mesh**

The ultimate aim of a device meshing algorithm is to allow the accurate and robust simulation of the associated characteristics of the device. The shape of the elements of the mesh may be crucial for stability of the device simulation and it may not be sufficient to generate a mesh which appears to be reasonable.

The anisotropic Delaunay meshing algorithm used for the 2D conformal algorithm has the by-product of introducing a number of more obtuse triangles than would occur with isotropic Delaunay meshing. It is commonly known that such triangles can lead to convergence problems when running a device simulation and so the conformal 2D algorithm includes an optional post-processing step to reduce the level of obtuseness of the final mesh.

The post-processing mesh quality improvement step operates by replacing an obtuse triangle with a cascade of right-angled triangles which terminates with a less obtuse triangle than that which triggered it. This produces a mesh whose elements all have a level of obtuseness less than the user-specified value. Figure 5 shows the differences between the default results and those with an obtuseness bounded to one.

The export statement used to generate the mesh on the right of Figure 5 was:

export victory(conformal) \ name=”test4_conformal2d_md_001_s_mo_1” \ max.distance=0.01 smooth max.obtuseness=1

The improvement in convergence or accuracy of the device simulation as a result of bounding the obtuseness of the mesh may be significant.

Figure 6 shows an example of the final results of the full conformal 2D device meshing algorithm on a CMOS structure with trenches.

Figure 6. Example of the final results of the full conformal 2D device meshing algorithm on a CMOS structure with trenches. |

**Conclusions**

This document describes four enhancements to the unified conformal device meshing algorithm as applied to the 2-dimensional process mode case:

- Removal of fan triangulations
- Adaptive feature detection in input
- Double and triple material smoothing
- Reduction of obtuseness of device mesh

These extensions have been demonstrated to improve the overall stability and quality of the mesh generated and as a consequence allows good convergence and accuracy of subsequent device simulation.

These additional features will also be available for the 3D and cell mode instantiations of the unified Victory Conformal device meshing algorithm in the future.