ISE Integrated Systems Engineering Release 9.0. Part 12 MESH

Transcription

1 ISE Integrated Systems Engineering Release 9.0 Part 12 MESH

2 Contents MESH 1 Introduction About this manual Scope of the manual Terms and conventions Getting started About MESH Introductory three-dimensional example Boundary description file Command file Using MESH Running MESH Setting default parameters Setting the maximum size of elements Setting analytical profiles Setting adaptation according to impurities and minimum size of elements More complex doping profiles Statistics for tutorial examples Defining more complicated geometries One-dimensional example: NMOS Two-dimensional example: NMOS Three-dimensional example: MOS-controlled thyristor (MCT) Three-dimensional example: ECL transistor Three-dimensional example: Incorporating external profiles Additional notes for arbitrary 3D geometries Troubleshooting Early termination MESH runs for unexpectedly long time Three-dimensional geometry and topology checker Error messages and invalid geometries MESH options Boundary description Overview Compatibility with other programs Input file in bound format Command file description Overview Description of general function evaluator Controls section Definitions section Defining refinement regions Defining multibox refinement regions Defining submeshes Defining constant profiles Defining analytic profiles iii

3 Contents 12.iv 5.4 Placements section Geometrical elements Placing refinement regions Placing multibox regions Placing profiles Placing submeshes Referencing constant profiles Referencing analytic profiles Examples Using refinement polygons Regionwise and materialwise refinement Using analytic functions for refinement I Using analytic functions for refinement II Using analytic functions for doping specification Reading old command files Instructions not supported Formulas for analytic profiles Overview General concepts Local coordinate systems, valid domains, and reference regions General implantation models Other parameters of interest Available models along the primary direction Gaussian functions Error functions Constant function External 1D profile Lateral or decay functions Lateral Gaussian function Lateral error function No lateral function Technical aspects Valid polygons and polyhedra Numeric considerations Mesh refinement algorithm Constructing the first coarse grid Adaptation according to external data Obtaining a conforming final grid Delaunization module Algorithm Active and total concentrations in MESH Appendix A Examples A.1 Tutorial: Diode example A.1.1 Boundary file: diode.bnd A.1.2 Command file: Abrupt diode A.1.3 Command file: Default parameters A.1.4 Command file: Maximum size of elements A.1.5 Command file: Modifying the n-type region

4 Contents A.1.6 Command file: Doping adaptation A.1.7 Command file: Minimum size of elements A.1.8 Command file: Using values at the junction A.2 Complex examples A.2.1 One-dimensional NMOS A.2.2 Two-dimensional NMOS A.2.3 Three-dimensional: MOS-controlled thyristor A.2.4 Three-dimensional example: ECL transistor A.2.5 Three-dimensional example: Incorporating external profiles A.2.6 Three-dimensional example Bibliography v

5 1 Introduction 1 Introduction 1.1 About this manual This manual describes the ISE TCAD mesh generator MESH. It is a dimension-independent tool, which incorporates several different meshing engines, using different meshing techniques and algorithms. MESH is a command-line tool, where the different meshing options are available by using switches and command-line options. MESH generates meshes that are suitable for semiconductor device simulation. Local mesh refinement is performed by using the doping and refinement information prescribed in the MESH command file (.cmd). 1.2 Scope of the manual This manual is intended for users of the MESH software package. The main chapters are: Chapter 2 describes the MESH application and provides an example of the MESH functionality. Chapter 3 describes how to run MESH and outlines the function of the command file. Chapter 4 describes the bound format of the input file and how MESH is compatible with other programs. Chapter 5 describes the MESH command file. Chapter 6 explains the analytic models implemented by MESH. Chapter 7 describes some technical aspects of the MESH application. 1.3 Terms and conventions Table 12.1 Term Click Double-click Select Standard terms Explanation Using the mouse, point to an item, press and release the left mouse button. Using the mouse, point to an item and in rapid succession, click the left mouse button twice. Using the mouse, point to an icon, a button, or other item and click the left mouse button. 12.1

6 1 Introduction Table 12.2 Convention Blue Bold code Italics NOTE Typographic conventions Definition or type of information Identifies a cross-reference. Identifies a selectable icon, button, menu, or tab, for example, the OK button. It also indicates the name of a field, window, dialog box, or panel. Identifies text that is displayed on the screen, or text that the user must enter. Used to emphasize text or identifies a component of an equation or a formula. Alerts the user to important information. 12.2

7 2 Getting started 2 Getting started 2.1 About MESH The mesh generator MESH is a dimension independent and modular Delaunay grid generator, which is suitable for semiconductor device simulation. MESH generates high quality spatial discretizations for 1D, 2D, and 3D devices using a predefined set of mixed-element types. Taking some of the best meshing algorithms available, and innovative ideas and procedures, MESH is a modular, dimension-independent mesh generation tool kit. A modular organization of the basic steps allows the incorporation of new strategies. In 2D, complex devices that are generated by process simulation can be handled. In 2D and 3D, grids for complex nonplanar devices can be generated. MESH is also the grid generation engine that is used inside MDRAW. The integrated MDRAW mesher is equivalent with the command-line mesh tool, when the -M switch is used (mesh -M). By default, MDRAW generates a mesh without obtuse angles, and MESH produces less restrictive Delaunay meshes (see Chapter 7 on page 12.55). MESH is fully integrated into the ISE environment. The DF ISE boundary representation is used as input for MESH. Impurity concentrations and user-required element sizes can be described using dimension-independent syntax. The same description of analytical profiles can be used in 1D, 2D, and 3D. The grid can be adapted to analytical profiles or profiles generated by DIOS, in 2D, and DIP, in 3D. (All references to concentrations in this document imply active or substitutional concentrations, as DESSIS calculations use them in such a form.) MESH can read old OMEGA boundary files and transform them into the DF ISE boundary syntax, using the -n switch. The user must ensure the conformity of the old boundary files. Old command files can also be used by specifying the -oldcmd file switch, in which case, the old command files are converted to the new syntax. NOTE Rename the newly generated boundary and command files and use them for any future work. The fitting of the geometries is no longer performed by using tensor product grids. A new strategy, which subdivides the domain into macroelements that are subsequently split into elements, allows for the meshing of more general devices with thin non-coplanar layers. The desired point densities, according to the supplied concentration profiles, are obtained by refining the elements in an anisotropic way, generating fine elements for the critical parts of the device and coarse elements in the bulk regions. Unnecessary point propagation due to quadtrees, octrees, or tensor product grid techniques is avoided. Finally, adding a delaunization step after macroelement tessellation allows MESH to obtain high quality conforming Delaunay grids, suitable for control volume discretization methods, used in the device simulation. For more information, refer to the literature [1][2][3][4]. 12.3

8 2 Getting started 2.2 Introductory three-dimensional example The functionality of MESH is demonstrated by using a simple diode example. Figure 12.1 shows three different approximations of a real diode. For the 1D and 2D descriptions, the most important features of the device are represented. The 1D approximation is a line cut along the x-axis, passing through both contacts. The 2D case is a plane cut along the z-axis passing through both contacts. Figure 12.1 Geometry of a simple diode Grid generation in MESH is controlled by two files. One file contains the device geometry description (.bnd) and the other (.cmd) contains information about profiles, for example, doping profiles and grid adaptation criteria. Input files for the entire tutorial are in the Examples Library Boundary description file The first step is creating the boundary description file for the grid generator. MESH identifies the dimensionality from the boundary file. Two boundary input formats are available, but only one is shown in the tutorial. For more information on the available formats, see Chapter 4 on page The device geometry description files have the extension.bnd. For this diode example, the file name is diode.bnd. The geometry of the diode shown in Figure 12.1 is described as follows: For 1D: Silicon "substrate" line [(0) (2)] Contact "anode" point [(0)] Contact "cathode" point [(2)] # silicon region, named substrate # Contact area # Contact area 12.4

9 2 Getting started For 2D: Silicon "substrate" rectangle [(0,0) (2,3)] Contact "anode" line [(0,0) (0,3)] Contact "cathode" line [(2,1) (2,2)] # silicon region, named substrate # Contact area # Contact area For 3D: Silicon "substrate" cuboid [(0,0,0) (2,3,2)] Contact "anode" rectangle [(0,0,0) (0,3,2)] Contact "cathode" rectangle [(2,1,1) (2,2,2)] # silicon region, named substrate # Contact area # Contact area NOTE The boundary file diode.bnd is used as a reference for the remainder of the tutorial Command file The command file controls the grid adaptation by specifying profiles, for example, doping profiles and refinement specifications. The command file must have the extension.cmd. Comment lines start with * or #. According to Figure 12.1 on page 12.4, two types of doping regions are required: a p-type region and an n- type region. The simplest way to define these regions is to use two constant profiles. For the moment, no refinement specification is given and grid generation relies on their default values. For more information about these default values, see Section on page Command file for a simple diode This section describes the command file diode.cmd that is used as an example for the rest of the tutorial. The command file contains two types of information: dimension-independent data and dimension-dependent data. The dimension-independent part of the command file diode.cmd, for our example, is: Title "minimal example: simple diode" Definitions # Profiles Constant "n-type region" Species = "PhosphorusActiveConcentration" Value = 1e+18 Constant "p-type region" Species = "BoronActiveConcentration" Value = 1e+17 The optional keyword Title is used for a short description of the device and mesh. The section Definitions specifies the dimension-independent part of the command file and can be used for all dimensions without modifications. Two constant profiles for doping are described using the keyword Constant followed by the profile name in quotation marks. The keyword Species is used to declare the doping species used in the region. The constant concentration is specified by the number following the keyword Value. The sign is intrinsic to the species. Now, the doping profiles must be placed in the device. The placement of these profiles depends on the device geometry. 12.5

10 2 Getting started Since, we want to fill solid regions with constant doping in this example, the following instructions are added to the command file: Placements # Profiles Constant "n-type region instance" Reference = "n-type region" EvaluateWindow Element = cuboid [(1 0 0), (2 3 2)] # for 3D Constant "p-type region instance" Reference = "p-type region" EvaluateWindow Element = cuboid [(0 0 0), (1 3 2)] # for 3D The keyword Placements starts the dimension-dependent section where the instances of the definitions given in the Definitions section are defined. The keyword Reference specifies a profile defined in the Definitions section. EvaluateWindow defines the valid domain for the profiles. In this example, the valid domains are lines in 1D, rectangles in 2D, and cuboids in 3D. If EvaluateWindow is not defined in the file, the profile is valid in the entire domain of the device. For the 3D case, the valid domain of the p-type region is the lower half of the device given by the cuboid [(0 0 0), (1 2 3)]. In 2D, this domain is given by the rectangle [(0 0), (1 2)] and in 1D, by the line [(0), (1)]. However, the doping profile defined for 3D can be used for the lower dimensions and, for the rest of the tutorial, only the command file for the 3D case will be used. In the example, there is an abrupt decay function between the two constant profiles. The doping associated to points outside the EvaluateWindow is zero. This situation can be modified if the parameter DecayLength is used. By setting the keyword DecayLength in EvaluateWindow, an error function can be used as a decay profile. 12.6

11 . 3 Using MESH 3 Using MESH 3.1 Running MESH After creating the first two basic input files, diode.bnd and diode.cmd, the first mesh can be created using the command: mesh diode MESH automatically adds the extensions.bnd and.cmd to the base name diode. MESH creates the output files diode_msh.grd and diode_msh.dat that contain mesh geometry information and doping information, respectively. Another file, diode_msh.log, is created and used as the log file for the grid generation. The resulting mesh for the simple diode is shown in Figure D Diode 2-D Diode 2-D Diode Net Doping /cm3 +1.0e e e e e+17 Net Doping /cm3 +1.0e e e e e+17 Net Doping /cm3 +1.0e e e e e+17 Figure 12.2 Fitting only the geometry of a diode Table 12.3 on page provides statistics for all meshes shown in the tutorial. See Section A.1.2 on page for the complete diode.cmd used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/abrupt Setting default parameters Figure 12.2 shows the result when only the geometry is fitted and no adaptation parameter is given to MESH. However, it is possible to define a set of default parameters to control the grid generation for the entire device Example To set the default adaptation parameters for the entire device, the following changes are introduced to the previous example: Definitions # Refinement regions Refinement "default region" MaxElementSize = 1 MinElementSize = 0.02 # Profiles same as previous command file 12.7

12 3 Using MESH Placements # Refinement regions Refinement "default region instance" Reference = "default region" # Default region # Profiles same as previous command file These will be the default values taken when an adaptation parameter is not present in the definition of a refinement region. Figure 12.3 shows the meshes created using the above parameter values. 1-D Diode 2-D Diode Net Doping /cm3 +1.0e e e e e+17 Net Doping /cm3 +1.0e e e e e+17 3-D Diode Net Doping /cm3 +1.0e e e e e+17 Figure 12.3 Grids created using some default adaptation parameters See Section A.1.3 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/defaultAdaptation Setting the maximum size of elements Figure 12.2 on page 12.7 and Figure 12.3 show grids too coarse even for an abrupt diode. To change the density and shape of the grid, add new mesh refinement specifications to the Definitions and Placements sections Example To demonstrate MESH flexibility, we will generate meshes with different density in one corner of the device. First, we demonstrate the effect of specifying a smaller maximum element side length MaxElementSize. In the following Definitions section, the information about doping is omitted for brevity. Title "Demonstration of effect of MaxElementSize" Definitions # Refinement regions section Refinement "one corner" MaxElementSize = ( ) Refinement "rest of the device" MaxElementSize = (0.4, 0.8, 0.5) # Profiles same as previous command file In the above example, the upper-left front corner of the device is more refined than the rest of the device. For both refinement regions, MaxElementSize is specified with a vector so that the mesh density varies between the axes. 12.8

13 3 Using MESH The Placements section for this example in 3D is: Placements # Refinement regions section Refinement "one corner instance" Reference = "one corner" RefineWindow = cuboid [( ), ( )] Refinement "rest of the device" Reference = "rest of the device" # Profiles same as previous command file The refinement region instance "rest of the device" does not have the RefineWindow element associated with it. For MESH, this type of refinement region is considered valid for the entire device as the default refinement region. When more than one region instance does not have RefineWindow, the last refinement instance is considered as the default region. RefineWindow can be replaced for the appropriate 2D and 1D geometrical elements (rectangles and lines, respectively). The cuboid definitions can also be kept for the 1D and 2D cases. The meshes generated by MESH for the different number of dimensions are shown in Figure It is observed that the abrupt junctions are better adapted using finer grid elements. 1-D Diode 2-D Diode Net Doping /cm3 +1.0e e e e e+17 Net Doping /cm3 +1.0e e e e e+17 3-D Diode Net Doping /cm3 +1.0e e e e e+17 Figure 12.4 Changing the maximum side of elements See Section A.1.4 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/maximumSize Setting analytical profiles In the previous examples, the doping distribution changes abruptly at the p-n junction in the middle of the device. A more realistic doping profile can be specified using an analytical description in the MESH command file. For this example, the adaptation parameters are the same as those presented in Section on page Command file for a more realistic diode As in the simple diode example, we specify the doping concentration inside cuboids, rectangles, or along lines. Throughout the device, we define a p-type doping of cm 3 using the abrupt profile Constant inside 12.9

14 3 Using MESH a cuboid. A well of n-type doping replaces the abrupt constant profile shown in the previous example. The profile is defined using an error function along the xˆ -axis plus a lateral function. The profile has the maximum at the surface of the top contact. Keeping the same definition for the p-type doping profile, the new profile is described in the Definitions section as: AnalyticalProfile "n-type well" Species = "PhosphorusActiveConcentration" Function = Erf(SymPos = 0.5, PeakVal = 1e+18, Length = 0.1) LateralFunction = Gauss(Factor = 0.8) The keyword AnalyticalProfile starts a description of an analytical profile. The keyword Function represents the function component along the primary direction, and the keyword LateralFunction represents the function component along the lateral direction. Figure on page and Figure on page show the schemes for 2D and 3D profiles, respectively. The syntax to place this analytical profile in the Placements section is: AnalyticalProfile "n-type well instance" Reference = "n-type well" ReferenceElement Element = rectangle [(2, 1, 0) (2, 2, 1)] ReferenceElement defines the origin of the local coordinate system for the primary function. All mesh points are projected to this element and the computed distance is used to evaluate Function. Using the keyword Direction, it is possible to evaluate only along the positive or negative side of the element (see Section on page 12.36). Figure 12.5 shows the resulting doping profiles. Since we have defined a doping of /cm 3 throughout the device and /cm 3 in the well, the resulting doping in the well is /cm 3. The grid is not well adapted to the doping, and the grid adaptation is necessary to resolve the specified doping profile. 1-D Diode 2-D Diode Net Doping /cm3 +9.0e e e e e+17 Net Doping /cm3 +9.0e e e e e+17 3-D Diode Net Doping /cm3 +9.0e e e e e+17 Figure 12.5 Modifying the n-type region See Section A.1.5 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/doping

15 3 Using MESH Setting adaptation according to impurities and minimum size of elements An important parameter in the Definitions section allows for the control of adaptation depending on doping profiles around p-n junctions. It is called RefineFunction. While refinements can occur on many user-required quantities, two different functions can be attached to this keyword. In our example, we use the asinh (keyword MaxTransDiff) of the doping concentration difference (Variable = DopingConcentration) between the mesh points to adapt the grid. In this example, if the difference is greater than 1, the edge of the macroelements used to build the mesh is refined. However, the adaptation needs to be controlled with respect to the smallest element size. Whereas MaxElementSize is the upper bound for the edge length of mesh elements, the lower bound is specified with the keyword MinElementSize. When MinElementSize is not present in the definition of a region, the default value of 0.02 is taken. As before, the influence of these new parameters is shown using two refinement regions. However, to highlight the influence of the new parameters, MaxElementSize is set to 0.5 for both regions. NOTE There are no values associated to RefineFunction when the keyword is not present in the refinement definition statement Example: Command file for doping-dependent adaptation Keeping the same profiles as the previous example, the new statements for the refinement regions are: Definitions # Refinement regions Refinement "one corner" RefineFunction = MaxTransDiff(Variable = "DopingConcentration", Value = 1) Refinement "default region" MaxElementSize = 0.5 # or ( ) # Profiles same as previous command file Placements # Refinement regions Refinement "one corner instance" Reference = "one corner" RefineWindow = cuboid [( ), ( )] Refinement "default region instance" Reference = "default region" # Profiles same as previous command file 12.11

16 3 Using MESH Figure 12.6 shows the grids created with the above command file. 1-D Diode 2-D Diode Net Doping /cm3 +9.0e e e e e+17 Net Doping /cm3 +9.0e e e e e+17 3-D Diode Net Doping /cm3 +9.0e e e e e+17 Figure 12.6 Grid adaptation according to doping The adaptation for the 1D case is fine since the cuboid defining the corner includes the 1D cut, which was used as a boundary description. The 2D and 3D cases show the effect of not having criteria for the doping adaptation. Only the elements in the left corner are refined, which is why the doping profile is resolved better at this corner than in the right corner. See Section A.1.6 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/adaptation Example: Command file with minimum element size The effect of the keyword MinElementSize can be seen in this example. The RefineFunction given in the previous command file is taken for the default region. For the default region, we set MinElementSize to the vector (0.04, 0.1, 0.1), while for the corner, the default value 0.02 is used. Definitions # Refinement regions Refinement "one corner" MinElementSize = 0.02 Refinement "rest of the device" MinElementSize = (0.04, 0.1, 0.1) MaxElementSize = 0.5 RefineFunction = MaxTransDiff(Variable = "DopingConcentration", Value = 1) # Profiles same as previous command file # Placements section same as previous command file Figure 12.7 on page shows the effect of increasing the minimum size of the elements for the default region. The adaptation for the corner is much finer because MinElementSize is smaller than for the rest of the device

17 3 Using MESH 1-D Diode 2-D Diode Net Doping /cm3 +9.0e e e e e+17 Net Doping /cm3 +9.0e e e e e+17 3-D Diode Net Doping /cm3 +9.0e e e e e+17 Figure 12.7 Effect of setting the minimum element size See Section A.1.7 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/minimumSize More complex doping profiles Finally, two more profiles are added under the contact regions. In this case, values at the junction are used to define the profiles along the primary direction instead of standard deviation for Gaussian profiles or diffusion length for error functions. To complete the profiles, the keywords ValueAtDepth and Depth are required. ValueAtDepth is the concentration at the junction and Depth is the distance from the junction with respect to the peak position. NOTE Depth is a local distance from the peak position. One default refinement region is used for the entire device Example: Command file using values at the junction The syntax for the new analytical profiles in the Definitions section is: Definitions # Substrate doping as in the previous example # n-type well doping as in the previous example AnalyticalProfile "high doping for Ohmic contact in n-region" Species = "PhosphorusActiveConcentration" Function = Gauss(PeakPos = 0, PeakVal = 1e+20, ValueAtDepth = 1e+18, Depth = 0.2) LateralFunction = Gauss(Factor = 0.8) AnalyticalProfile "high doping for Ohmic contact in p-region" Species = "BoronActiveConcentration" Function = Gauss(PeakPos = 0, PeakVal = 1e+20, ValueAtDepth = 1e+17, Depth = 0.2) LateralFunction = Gauss(Factor = 0.8) Refinement "entire device" MinElementSize = (0.04, 0.02, 0.02) MaxElementSize = ( ) RefineFunction = MaxTransDiff(Variable = "DopingConcentration", Value = 1) 12.13

18 3 Using MESH and in the Placements section: Placements # Substrate doping as in the previous example # n-type well doping as in the previous example AnalyticalProfile "high doping for Ohmic contact in n-region instance" Reference = "high doping for Ohmic contact in n-region" ReferenceElement Element = rectangle [(2, 1, 0), (2, 2, 1)] AnalyticalProfile "high doping for Ohmic contact in p-region instance" Reference = "high doping for Ohmic contact in p-region" ReferenceElement Element = rectangle [(0 0 0), (0 3 2)] # Refinement regions Refinement "entire device instance" Reference = "entire device" Figure 12.8 shows the resulting meshes. 1-D Diode 2-D Diode Net Doping /cm3 +1.0e e e e e+20 Net Doping /cm3 +1.0e e e e e+20 3-D Diode Net Doping /cm3 +1.0e e e e+16 Figure 12.8 Using values at the junction -1.0e+20 NOTE The curvatures of the doping profiles along the lateral directions are better resolved because of a smaller value for the minimum element size. See Section A.1.8 on page for the complete diode.cmd file used for this example. In the Examples Library, the example is under the project Tool/Mesh/Tutorial/final Statistics for tutorial examples Table 12.3 on page presents statistics for the tutorial examples that correspond to the previous examples in the order they were presented

19 3 Using MESH Table 12.3 Diode Case name Command file (see tutorial directory) Figure Statistics: Range of point count Simple abrupt Default defaultadaptation n-well adaptation Doping doping Max size maximumsize Min size minimumsize Complex final In general, dfisetools -r can be used to generate information on the number of points and elements. NOTE The number of grid elements in Table 12.3 does not include the elements used to define the contact areas only 3D elements are counted for 3D; 2D elements, for 2D; and 1D, elements for 1D. They will vary from version to version as algorithmic changes occur. 3.2 Defining more complicated geometries More complex examples are included in the Examples Library. This section presents a short analysis of the performance of MESH with different examples. A comparison is made between MESH and the rest of the grid generators available in the ISE TCAD environment One-dimensional example: NMOS Figure 12.9 shows the resulting mesh for the example. While the grids produced by MESH and GRID1D are identical, it is interesting to analyze the difference in the command file syntax. In Section A.2.1 on page 12.67, both command files are included for further comparison. N+ NMOS : MESH Net Doping /cm3-6.8e e e e e e+20 P NMOS : MESH (a) (b) Figure D NMOS with MESH: (a) Doping and (b) grid (175 points, 174 1D elements) In the Examples Library, the example is under the project Tool/Mesh/1D/nmos

20 3 Using MESH Two-dimensional example: NMOS This example shows the difference between Delaunay grids and obtuse angle free grids. By default, MESH produces 2D Delaunay grids suitable for device simulation. However, MESH can be forced to produce grids without obtuse angles, that is, the center of the circumscribe circle (Voronoï center) of each grid element lies inside the element, including the element edges. To do this, the option -s must be used. Conversely, by default, MDRAW produces obtuse angle free grids. However, using the option -delaunayall, you can obtain the same type of grid as default MESH grids. For more information, see the MDRAW manual. In Figure 12.10, (a) shows a Delaunay grid for an NMOS example and (b) shows an obtuse angle free grid for the same example. In the first case, the grid is composed of 663 points and 808 2D elements. The second grid is composed of 823 points and 995 2D elements. Although both grids are suitable for device simulation, the first is more desirable due to its smaller size. The size reduction is achieved because the propagation of the green points is stopped earlier in the first case than in the second. NMOS : MESH NMOS : MESH -s -2e+16-2e+15-9e+13 +3e+14 +4e+15 +6e+16 +8e+17 +1e+19 +2e+20 Net Doping /cm3-2e+16-2e+15-9e+13 +3e+14 +4e+15 +6e+16 +8e+17 +1e+19 +2e+20 Net Doping 1/cm**3 Figure (a) (b) Delaunay meshes (a) and obtuse angle free meshes (b) in 2D NOTE The option -s is available only in 2D. To compare MESH and MDRAW command files, refer to the files in Section A.2.2 on page In the Examples Library, the example is under the project Tool/Mesh/2D/nmos Three-dimensional example: MOS-controlled thyristor (MCT) The device in this case is very long (500 µm) and very narrow (10 µm). The most interesting part is located on the top of the device, so the figures show the upper part of the example. MESH and OMEGA both created a valid spatial discretization for this device. Figure on page shows both grids. The number of points has been reduced by over 40% using MESH. Conversely, the number of 3D elements in the MESH grid is greater than in the OMEGA grid. The reason for this increase is the different element composition of the grids

21 3 Using MESH MCT : MESH MCT : OMEGA Net Doping 1/cm**3 +3.8e+19 Net Doping 1/cm**3 +3.8e e e e e e e e e+20 (a) Figure (b) MOS-controlled thyristor created by MESH (a) and OMEGA (b) The command file used to create the MESH grid is included in Section A.2.3 on page In the Examples Library, the example is under the project Tool/Mesh/3D/mct Three-dimensional example: ECL transistor Figure shows both grids for this example. Net Doping 1/cm**3 ECL :: MESH MESH ECL -1.2e e e e e+17 Net Doping 1/cm**3 ECL : OMEGA +1.2e e e e e e e x z Figure y x z y (b) (a) ECL transistor created by MESH (a) and OMEGA (b) In the mesh created by MESH, less point propagation around the insulator layer is observed. An intersectionbased approach can fit the geometry using fewer elements than a typical tensor product grid used in OMEGA. Moreover, OMEGA again produces a grid with green lines, while with the option -w, it fails to create a grid. Therefore, only MESH can produce a grid that is green line free. The command files for MESH are included in Section A.2.4 on page In the Examples Library, the example is under the project Tool/Mesh/3D/ecl

22 3 Using MESH Three-dimensional example: Incorporating external profiles In 3D, the doping profiles can be incorporated using DIP. For both grid generators, the option -D must be used. The script for DIP is read from the file named filename.dip. Figure shows the grids created by MESH and OMEGA. DIP : MESH DIP : OMEGA N /cm3 +4.9e e e e+14 N 1/cm**3 +4.9e e e e e+17 (a) (b) Figure Using DIP as doping profile generator in MESH (a) and OMEGA (b) -4.4e+17 Although the grids have a similar number of points, MESH does not propagate points from the geometry through the entire device as can been seen in the OMEGA grid. The command file for MESH describes the refinement regions for the drain and source regions using the same reference to the Definitions section. This property allows us to re-use refinement regions in the entire device and to ensure similar grid quality for the critical parts of the device without too much effort. The command file is included in Section A.2.5 on page In the Examples Library, the example is under the project Tool/Mesh/3D/dip Additional notes for arbitrary 3D geometries MESH supports arbitrary 3D geometries. The following notes are relevant when creating non-manhattan type geometries: The keyword polyhedron can be used to create closed solids with arbitrary polygons acting as faces. For rectangles that are not parallel to the three coordinate planes, the keyword polygon should be used to describe the four points. When inputting polygons in 3D, it is preferred that polygonal vertices are specified such that the face normal points outside the polyhedra space or material space. Face normal is computed by traversing the polygon nodes; the order of points in the polygon definition should be counterclockwise to achieve the outward-facing normal property

23 3 Using MESH 3.3 Troubleshooting Early termination One common problem that causes MESH to quit before generating the mesh and doping files (_msh.grd and _msh.dat) is an error in the command file (.cmd). A common mistake is not having the correct path for loading submeshes in 2D MESH runs for unexpectedly long time Handling the green points and green lines is the most critical step in MESH. Many loops may be required in order to make this step. There are two possible reasons for excessive computing time: The complexity of the input geometry. In this case, it is recommended that you start with very coarse grids and that you mesh only the geometry using the command-line option -m. The mesh density. Depending on the specification of the mesh density in the command file, a substantial number of mesh points may be inserted during the adaptation. The insertion of the refinement regions in a stepwise or staged manner can help find the critical region that is responsible for the excessive number of points. The option -w is also very helpful in debugging MESH grids. MESH displays the number of elements that will be checked at each step of the program. During the handling of the green points and green lines, the number of elements to check should decrease after each loop Three-dimensional geometry and topology checker Incorrectly constructed input geometry has been a predominant source of 3D MESH failures. Incorrect constructions can take many forms, namely, two regions intruding into each other in 3D, unclosed faces, unclosed polyhedra, incorrect face orientations, and so on. Automatic generation of input files fails to capture some of these errors. To help users, MESH has a topology checker, (-Cin command-line option), which looks at the edges, faces, and polyhedron for several of the above issues. While it provides some clues to correct the structure at the end of its operation, it is currently not very user-friendly Error messages and invalid geometries Chapter 7 on page explains the technical assumptions in creating a valid geometry. Some errors occurring at different stages of the program execution are: If the faces of a polyhedron are not closed, MESH sends the message: Error: invalid combination for closed polygon in LD3DPolygonFace::LD3DPolygonFace If the points in a polygon are not coplanar, the following message is sent to the standard output: Warning: no co-planar points in IISgeoEdgesPolygon::IISgeoEdgesPolygon, dot product = value. Any dot product between the normal vector of the face and two points on the face must be zero, that is, the vectors are perpendicular (value = 0)

24 3 Using MESH If the faces of a polyhedron are open (that is, it is a cube without a lid), MESH gives the following warning and, in some cases, may fail to finish unless the geometry is corrected: Warning : The non-contact region (polyhedron) 'foo' is open and may cause mesh failure. Please check your geometry. 3.4 MESH options The usage is: Mesh [ options ] cmd-filename -v Print version number. -h Print options. -t Generate only simplex elements: triangles in 2D and tetrahedrons in 3D. -s Create a more restrictive Delaunay grid. The maximum angle allowed for triangles is π 2 in 2D. Thus, the Voronoï center of every grid element is completely inside the element. -f Fit MaxElementSize first. MESH creates an initial set of macroelements according to the specifications of the user before fitting the geometry. -m Generate grids using only the device geometry as the refinement criterion and ignore doping specifications. Always using this option is recommended as it helps to debug the geometry for complex devices. -w Skip delaunization. Fit only user requirements and build a mesh, not suitable for device simulation. Valid for 2D and 3D. This option is useful to check if the grid is too coarse or too fine after the adaptation. Moreover, it helps to debug the last step in 3D Delaunay procedures. -Cin -Cout Perform a topology check of the input geometry (3D only). Perform a topology check of the resulting output mesh (3D only). -R Remove unnecessary points, edges, or faces from boundary description. -D Use DIP as doping incorporation functions. The file with the script for DIP must have the extension.dip. Valid only for 3D. -I Do not use profiles; skip all datasets. Neither analytical profiles nor external profiles are taken in account. The file filename_msh.dat is not created. -c <int> Restrict point connectivity for general patterns. -g Build mesh in gas. -G General pattern inserting point inside. -i With analytical implantation (only 3D). -n Translate the input files into the newest syntax. -r Read mesh files specified in command file

25 3 Using MESH -P New (default) delaunization algorithm to finish the mesh (3D only). -p Switch to old delaunization algorithm to finish the mesh (3D only). -noffset -numdecimals <integer> -rounding-off -discontinuousdata -interfaceadapt -presurfacesmooth -compress -binary -voronoioutput -AllActive -AllTotal -DopingAsIs -extract_in -interfaceadapt -iterative -nop -noshrinkdata -presurfacesmooth -shortestedge <value> -rounding-off -oldcmd Generate boundary conformal/orthogonal meshes using the ISE normal offsetting technique (2D only). Round input values from the GUI. Default value is 5. This is a critical parameter and it must be selected depending on the example that the user is working with. If the value is large enough, the input geometry can no longer be valid. For example, the coplanarity of face points can be destroyed in 3D. Disable rounding of input boundary points. Write discontinuous datasets. Enable refinement near material interfaces. Do NOT smooth the boundary conformal surface mesh points before performing delaunization (3D only). Compress output files into.gz format. Write binary output files. Write out Voronoï dual mesh (3D only). Convert total concentrations to active concentrations (default). Convert active concentrations to total concentrations. Use concentrations as in files (for expert user). Extract boundary for regions specified in the control section from input boundary. Enable refinement at interfaces. Use iterative refinement scheme. Do not generate mesh. Read input and write output files (can be used for topology check). Do not check variable validity when continuous data is stored. Do not smooth surface before delaunization (3D only). Size of the shortest allowable edge in the final mesh (2D only). Do not round coordinates from input boundary. Allow the loading of command files that use the old syntax. Use also the -n option to convert the old command file to the new syntax, and use the new command file for any future work

26 4 Boundary description 4 Boundary description 4.1 Overview MESH reads the boundary description from the file filename.bnd. Two formats are supported: bound format, which is a user-friendly geometry description format. DF ISE format, for boundary description (see Utilities, Chapter 2 on page 6.3). 4.2 Compatibility with other programs MESH can read the formats used for GRID1D and MDRAW in 1D and 2D, respectively. For 3D, the formats bound and DF ISE bnd are supported. 4.3 Input file in bound format The most important feature of this format is that it can be easily created and changed manually. In 2D, the bound format can be exported from the standard DF ISE format using MDRAW. A device is described through a list of solid regions and contact areas specified by the user. Solid regions are defined as regions R d, where d is the device dimension and contact areas are defined as regions R d 1. Each solid region is given a material type, region name, and geometrical element defined by the user. General syntax for a solid region is: materialtype "region name" arbitrary list of solid regions Valid materials for the solid regions are listed in the file datexcodes.txt [DATEX], which includes Silicon, Oxide, Copper, and Aluminum. In the MDRAW GUI, valid materials can be found in the Materials menu. Contact areas consist of a list of one or more (d 1)-D surfaces. The keyword for contacts is Contact. General syntax for a contact area is: Contact "contact name" arbitrary list of N-D surfaces NOTE For each Contact, a name must be given using the above syntax. In 1D, the solid regions are lines and the contact areas are points. In 2D, the solid regions are polygons, which can be represented by a polygon or rectangle. A polygon is defined as a set of connected segments or lines. The contact areas are segments or lines. In 3D, the solid regions are polyhedra. A polyhedron can be described as either a cuboid (cuboid) or a general polyhedron (polyhedron), using the old bound format. A polyhedron is a set of connected 2D polygons. In 3D, the contact areas are polygons (see Section on page 12.32)

27 5 Command file description 5 Command file description 5.1 Overview This chapter gives a full description of the MESH command file (.cmd) format and describes the new command file format. The old command file format is still supported for input; however, an additional switch -oldcmd also must be specified, to load a command file that is using the old syntax. In the MESH command file filename.cmd, users can specify different parameters for the grid generation. White spaces separate keywords, strings are presented with quotation marks, and blocks are delimited by opening and closing braces. Comment lines start with * or #. Keywords used in the command file are not case sensitive. Several different types of information can be given in the command file. The user can specify the default meshing engine, refinement information, and doping profile information. In addition, meshing parameters for NOFFSET meshers are stored in the command file. Refinement information is required to control the grid generation procedure according to user requirements (local element size). This information is given using a set of parameters defined in the Definitions section. Profile information is required to define the profiles, for example, doping profiles, which are used in grid adaptation. Doping profiles can be specified with three types of information: External simulation results Constant data Analytic formulas and predefined functions describing a profile The command file is split into several different sections: Controls, Definitions, Placements, and Offsetting. The main blocks and keywords are: Title Controls Definitions Placements Offsetting The command file can start with an optional title statement. The keyword Title followed by a quoted string constitutes the title statement. This keyword starts the control section of the command file. The default meshing engine can be specified in this section. Defines sets of refinement parameters and profile definitions to be used in the Placements section. These sets are referenced using their unique reference names. Defines instances of the definitions given in the Definitions section placed with respect to the current device. Parameters for 2D or 3D NOFFSET meshing engines can be specified in this section. The 2D NOFFSET meshing engine can be accessed using mesh -noffset; while the 3D NOFFSET meshing engine can be accessed as a separate binary, noffset3d. The following syntax is used: Title "example name" Controls meshing engine information Definitions defining information Placements placing information Offsetting offsetting information 12.25

28 5 Command file description 5.2 Description of general function evaluator Apart from the predefined (Gaussian, Erf, ) functions, users can also define general analytic functions that can be used in analytic profiles for defining primary and lateral functions. The syntax is: or: Function = Eval(init = "...", function = "...", value =...) LateralFunction = Eval(init = "...", function = "...") Function = General(init = "...", function = "...", value =...) LateralFunction = General(init = "...", function = "...") Both Eval and General have the same syntax for the argument list. The difference between Eval and General is that General uses spatial coordinates, while Eval uses coordinates that are measured in the primary and lateral directions for profiles. The argument list can be defined in the following way: init function value This is a semicolon-separated list of assignments for variables that are used later, for example, init = "a=2;b=4". This string is evaluated only once. This is an expression that is evaluated for every query. The variable that replaces the primary or lateral distance must be called x, for example, function = "sin(x)", function = "exp(4*x)*sin(x)". In general, 1D, 2D, and 3D analytic functions can specified here, using a valid C syntax. The variables x, y, and z refer to the respective spatial coordinates for General. This is the default return value if the evaluation fails. The default is zero and 1 for LateralFunction. Notes: There is only one name space for all variables, that is, if init="a=1" is set in one instance, it is known for all instances. You can freely mix Eval with Gaussian, Erf, and 1D. The symbols "pi" and "e" can be used in the expressions. The following functions can be used: "sin", "cos", "tan", "asin", "acos", "atan", "sinh", "cosh", "tanh", "exp", "log", "log10", "sqrt", "floor", "ceil", "abs", "hypot", "deg", "rad". Numeric constants must be specified as "2*10^18". As an extension to the Eval function evaluator, the General evaluator assesses device coordinates directly, (x, y) and (x, y, z) and does not use primary and lateral distances. Any lateral functions and reference geometries (in the Placements section) are ignored Controls section The Controls section of the command file records the default meshing engine. The syntax of the Controls section is: Controls meshengine = "meshing engine name" 12.26

29 5 Command file description where the meshing engine name can be specified as mesh or noffset. The meshing engine (mesh or noffset) is chosen with the following priorities: If the command-line switch mesh or noffset is specified in the MESH command line, the respective meshing engine is invoked, even if the command file defines a different meshing in the Controls section. If no meshing engine switch is specified in the MESH command line, but a Controls section is defined in the command file, the specified meshing engine from the command line is invoked. 5.3 Definitions section The Definitions section is composed of sets of refinement and profile blocks. Each block consists of the reference name, an opening brace, the specification of the corresponding parameters, and a closing brace. The order of the definitions inside the block has not effect since these definitions are used as references in the Placements section. The syntax of this section is: Definitions Refinement "reference name" # set of parameters Multibox "reference name" # set of parameters SubMesh "reference name" # set of parameters Constant "reference name" # set of parameters AnalyticalProfile "reference name" # set of parameters Defining refinement regions The syntax for each refinement region is: Refinement "reference name" MaxElementSize = value vector MinElementSize = value vector RefineFunction = MaxTransDifference(parameters) MaxGradient(parameters) where: MaxElementSize Controls the maximum size of grid elements. A real number or a vector x = [ x 1,, x n ] can be specified, where d is the dimension. x d represents the maximum edge lengths along the coordinate axes. A vector can be used to refine nonisotropically. Only values greater than zero are considered. The default for all vector components is

30 5 Command file description MinElementSize A real number or a vector x = [ x 1,, x d ] can be specified. x d represents the minimum edge lengths along the coordinate axes. Grid elements can be refined in one direction if their edge length in this direction is greater than the specified value. Only values greater than zero are considered. The default for all vector components is RefineFunction Two different functions can be used to select grid elements for refinement: MaxTransDifference and MaxGradient. Variable MaxGradient MaxTransDifference Defines the dataset used to adapt the grid. The grid can be adapted according to species or any type of variable defined in the DF ISE dataset files. The values are computed from the analytic formulas, constant data, and external simulation results defined in the command file. Hence, the names for Variable must be taken from the DF ISE dataset files, and these names must be enclosed in quotation marks. The gradient of a profile (keyword variable) in the element is evaluated. If the gradient is greater than value and the edge lengths are large enough, the element is refined. The maximum difference of the transformed values of a profile at the vertices of the element is evaluated. If the difference is greater than value and the edge lengths are large enough, the element is refined. The transformation (linear, logarithmic, arsinh) is defined in the DF ISE dataset files for each variable. The syntax is: RefineFunction = MaxGradient(Variable = "DFISE Dataset Name", Value = value) RefineFunction = MaxTransDifference(Variable = "DFISE Dataset Name", Value = value) RefineFunction can be repeated for different variables in the same Refinement block. If Variable is not defined, DopingConcentration is taken as the default. If Value is not specified, it defaults to 1; however, there is no RefineFunction assigned by default Defining multibox refinement regions The multibox is a special refinement box, which is implemented for 2D only. You can specify the required minimum and maximum element sizes in both directions, and an additional refinement ratio in both directions. The created mesh is graded using the specified ratios (also observing the minimum and maximum element sizes). The syntax to define a multibox refinement region is: Multibox "multibox reference name" MaxElementSize = value vector MinElementSize = value vector Ratio = (ratio_width, ratio_height) where MaxElementSize and MinElementSize are the same as for the refinement region definitions. Ratio controls the grading of the element sizes. ratio_width is the grading factor in the x-direction and ratio_height is the grading factor in the y-direction Defining submeshes External simulation results given on a mesh can be used to define profiles in the device. The external mesh must have the same spatial dimension as the device. The datasets defined on the external mesh are interpolated 12.28

31 5 Command file description to the newly generated mesh. The external profiles are called submeshes. The syntax for submeshes in the Definitions block is: SubMesh "reference name" Geofile = "filename" Datafile = "filename", Mode... Datafile = "filename", Mode where: Geofile Datafile Mode The string indicates the name of a file with an external mesh. This file must be in DF ISE format. The dimension of the external mesh must be the same as the dimension of the device. The string indicates the name of a file with datasets defined on the external mesh. This file must be in DF ISE format. Several Datafiles can be defined for each Geofile. The string indicates the mode in which the data files are handled after generating the mesh. The possible modes are: 'w' or "write" After generating the mesh write a file containing the same datasets as Datafile. By default, the name of the output file is composed of the base name of the command file and the extension of the Datafile. 'w' or "write" = filename This has the same definition as write, except that the output file name can be specified. If filename has no extension, the extension of the Datafile is used. 'r' or "read" Read only mode. The information from the file is used for mesh adaptation but is not saved in the output file. 'i' or "incremental" Incremental mode. If the file name (without the extension) ends with a number, the corresponding output file has the same name with the counter increased by one. If the file name does not end with a number, then zero (0) is appended to the file name. 'o' or "overwrite" Overwrite mode. After mesh generation, the Datafile is overwritten. The default mode is write Defining constant profiles The syntax for constant profiles in the Definitions block is: Constant "reference name" Species = "DFISE Dataset Name" Value = value 12.29

32 5 Command file description where: Species Value Selects the species or variable for the constant profile. The list of available species and variables is given in the DF ISE dataset files. No default value is assumed. Value of the constant profile. No default value is assumed. NOTE Only Species can be used to define a constant profile Defining analytic profiles Profiles can be defined using simple analytic expressions. These expressions are composed of two components. The first component called Function represents the values along a direction defined as the normal direction of the ReferenceElement. These values are smoothed along the direction perpendicular to the normal using the second component LateralFunction. The formulas used for these analytic profiles are described in Chapter 6 on page Apart from the predefined formulas, general analytic functions can also be used. The syntax for analytic profiles in the Definitions section is: AnalyticalProfile "reference name" Function = gauss(parameters) erf(parameters) submesh1d(parameters) General(parameters) Eval(parameters) Species = "DF ISE Dataset Name" LateralFunction = gauss(parameters) erf(parameters) General(parameters) Eval(parameters) where: Species Function Selects the species or variables for the analytic profile. The list of available species and variables is given in the DF ISE dataset files. No default value is assumed. Indicates the type of the component and the parameters used along the direction normal to the ReferenceElement. Gaussian functions, error functions, or external profiles can be used as predefined functions. Apart from the predefined functions (Gaussian, Erf, 1D external profiles), the general function evaluator can also be used to define a general analytic function. A Gaussian profile can be specified as: Function = Gauss(PeakPosition = value, PeakValue = value, StandardDeviation = value) Function = Gauss(PeakPosition = value, Dose = value, StandardDeviation = value) Function = Gauss(PeakPosition = value, PeakValue = value, Length = value) Function = Gauss(PeakPosition = value, Dose = value, Length = value) Function = Gauss(PeakPosition = value, PeakValue = value, ValueAtDepth = value, Depth = value) Function = Gauss(PeakPosition = value, Dose = value, ValueAtDepth = value, Depth = value) By default, PeakPosition is equal to zero and there are no default values for the other parameters. An error function can be defined as: Function = Erf(SymmetryPosition = value, MaxValue = value, Length = value) Function = Erf(SymmetryPosition = value, Dose = value, Length = value) Function = Erf(SymmetryPosition = value, MaxValue = value, ValueAtDepth = value, Depth = value) Function = Erf(SymmetryPosition = value, Dose = value, ValueAtDepth = value, Depth = value) 12.30

33 5 Command file description By default SymmetryPosition is equal to zero. A general analytic function can be defined as: Function = General/Eval(init = "...", function = "...", value = value) For the usage of general functions, see the examples in Section 5.5 on page For the incorporation of 1D external profiles, the syntax is: Function = submesh1d(datafile = "filename", Scale = value, Range = line [(x1), (x2)]) Datafile is a file in the XGRAPH format. More than one profile can be included in Datafile. The keyword Scale allows us to scale the coordinates values from the file. By default, Scale is equal to 1. Range selects a range of values from the file. The keywords x1 and x2 must be given in the file coordinates system. Range is applied to all profiles inside the file. By default, the entire data range is selected. LateralFunction Defines the lateral component of the analytic profile. Either a Gaussian or an error function can be used, as well as a general analytic function can be used, by specifying the keyword General, with a proper definition of the analytic function: LateralFunction = Gauss(Factor = value) LateralFunction = Gauss(StandardDeviation = value) LateralFunction = Gauss(Length = value) LateralFunction = Erf(Factor = value) LateralFunction = Erf(Length = value) LateralFunction = General/Eval(...) By default, an error function is used as LateralFunction. For Function = submesh1d, by default StandardDeviation = 0.8 is chosen, and for Function = Gauss and Function = Erf by default, Factor = Placements section The Placements section is composed of sets of refinement and profile instances. Their positions in the device must be specified and they must reference a definition given in the Definitions section. In other words, each instance or block consists of the instance name, an opening brace, a reference, the specification of the corresponding parameters, and a closing brace. The order of the refinement regions in this section is important. MESH selects the conditions to apply among all refinement regions described in this section. The order of the profile instances in the Placements section is important only when the parameter Replace is present. The syntax for this section is: Placements Refinement "reference name" # set of parameters Multibox "reference name" # set of parameters SubMesh "reference name" # set of parameters Constant "reference name" 12.31

34 5 Command file description # set of parameters AnalyticalProfile "reference name" # set of parameters Geometrical elements The specification of placement blocks requires the use of geometrical elements. These elements are geometrical objects used to select or locate data, and they are not part of the grid elements. The coordinates of these objects are defined relative to the coordinates of the device, except for one section in the SubMesh placement. The allowed geometrical elements and the number of coordinate values that must be specified depend on the dimension n of the device. Let x = [ x 1,, x d ] denote a point. The following geometrical elements are defined: point( x 1 ), line( x 1, x 2 ), rectangle( x 1, x 2 ), polygon( x 1,, x m ) m > 2, cuboid( x 1, x 2 ), polyhedron polygon 1 ( x 1,, x m ),, polygon p ( x 1,, x m ) m > 2 Only simple polygons are allowed. The polygons are closed internally by adding the line segment between x 1 = [ x 1,, x d ] and x m = [ x 1,, x d ]. Only simple closed polyhedra are allowed. All their faces must be described. Table 12.4 on page provides a detailed list of elements and the functions for which these elements can be used. In describing a polyhedron with arbitrary-oriented faces, use polygons instead of rectangles. Apart from the above-defined geometrical elements, the keywords material or region can also be used to perform regionwise or materialwise refinement or both refinements. For material, the argument is a valid DATEX material name in brackets. For region, the argument is a valid (existing) region name in brackets (see Section on page 12.40) Placing refinement regions In the Placements section, a refinement instance is specified by the name, an opening brace, the specification of the reference name, the keyword RefineWindow, and a closing brace. Several refinement instances can refer to the same set of refinement parameters. If no RefineWindow is specified, the refinement instance is used as the default region for the entire device. The syntax for a refinement instance is: Refinement "instance name" Reference = "reference name" RefineWindow = geometrical element material region where: Reference Defines the reference to one of the previously defined refinements

35 5 Command file description RefineWindow Defines the location of the refinement instance in the device. By default, RefineWindow is the bounding box of the device. For the list of elements that can be used as RefineWindow (see Table 12.4 on page 12.36). Two other kinds of refine window have been added to the syntax. These additions allow the user to specify regionwise or materialwise refinement or both refinements (see Section on page 12.40). NOTE Where two or more refinement instances overlap, the one defined first in the command file is taken (see Chapter 7 on page 12.55) Placing multibox regions In the Placements section, a multibox instance is specified by the Multibox keyword, followed by the name of the Multibox window and an opening brace. After the specification of the name of the multibox region (Reference = "") and the size of the Multibox window (Refinewindow=), a closing brace is placed. Several multibox instances can refer to the same set of multibox parameters. If no RefineWindow is specified, the refinement instance is used as the default region for the entire device. The syntax for a refinement instance is: Multibox "instance name" Reference = "multibox name" RefineWindow = geometrical element where: Reference RefineWindow Defines the reference to one of the previously defined multiboxes. Defines the location of the refinement instance in the device. By default, RefineWindow is the bounding box of the device. Since the multibox feature is implemented only in 2D, the geometrical element must be rectangle. NOTE Where two or more multibox instances overlap, the multibox defined first in the command file is taken (see Chapter 7 on page 12.55) Placing profiles A set of common parameters is used for placing submesh (external simulation result), constant, and analytic profiles. These parameters are: EvaluateWindow Element = geometrical element material region DecayLength = value Replace 12.33

36 5 Command file description where: EvaluateWindow Defines the domain where the profile is evaluated and the decay length of an error function is applied in the vicinity of the window boundaries. The domain can be specified by using a geometrical element, as well as by referring to regions or materials. The decay function is used to reduce round-off errors. However, if EvaluateWindow is not defined, the transition between the profiles is abrupt. If DecayLength = 0, no decay function is applied and the transition between EvaluateWindow and its vicinity is abrupt. If DecayLength is negative, the profile is not applied to points on the border of Element. The default DecayLength is equal to 0 for all the profiles. For analytic and constant profiles, the default Element is the bounding box of the device and, for submeshes, the default Element is the bounding box of the submesh. See the equations in Chapter 6 on page for details. Replace In general, the values for each profile at each point of the newly generated mesh are computed as the sum over all profile instances, defined in the Placements section. The instances are inspected in the same order as they are defined in the command file. If the variable Replace is specified for a given instance, all current summed-up values are replaced by the value corresponding to the given profile instance. NOTE Avoid using EvaluateWindow when the profile is valid in the entire device and no decay function is required. The evaluation of a geometrical element is a time-consuming task Placing submeshes The syntax for references to submeshes in the Placements block is: SubMesh Reference = "reference name" SelectWindow Element = element, AttachPoint = x1, ToPoint = x2 Rotation axis = axis, angle = angle Reflection = X Y Z EvaluateWindow Element = geometrical element material region, decaylength = value Replace where: Reference SelectWindow Indicates the reference SubMesh to use. Only references to profiles that are defined as SubMesh are allowed. Can be used to select a part of the external mesh to work with. Element must be specified with respect to the coordinate system used in the Geofile. The selected part of the external mesh can be placed in the device by selecting any point in the submesh (AttachPoint) and defining its position in the device (ToPoint)

37 5 Command file description The syntax is: SelectWindow Element = line [x1, x2] rectangle [x1, x2] cuboid [x1, x2], AttachPoint = x1, ToPoint = x2 The allowed element type depends on the dimension of the device. By default, Element is defined as the bounding box of the submesh. By default, AttachPoint and ToPoint are selected as the corner of the submesh bounding box with minimum coordinates x 1 = x 2 = [ x min 1 xmin n ]. Reflection Rotation Ignoremat Indicates a reflection perpendicular to the specified coordinate axis. The allowed axes depend on the dimension of the device. The reflection point (or line or plane) is placed in the middle of the bounding box of the submesh. Performs a rotation around axis. The rotation is counterclockwise. The center of the rotation is the mid-point of the submesh bounding box. By default, Angle is equal to zero and, for 2D and 3D, Axis is set to Z. In 1D, Rotation is not supported. Ignores material in submeshes. The standard behavior of submesh interpolation is that the interpolated value is only accepted if the point is in a region with the same material. The flag "Ignoremat" allows the code to always accept the interpolation. (By default, if the materials are not matched, the closest region with the correct material is searched.) The default behavior is not checked. Placements SubMesh "NoName_0" Reference = "NoName_0" Ignoremat NOTE SelectWindow, Reflection, and Rotation are performed in the order they appear in the command file. The resultant submesh depends on this order. MESH and DIP support submeshes in 3D Referencing constant profiles The syntax for referencing constant profiles in the Placements block is: Constant Reference = "reference name" EvaluateWindow Element = geometrical element material region DecayLength = value Replace where Reference is the string indicating the reference constant to use. Only references to constant profiles are allowed

38 5 Command file description Referencing analytic profiles The syntax for referencing analytic profiles in the Placements block is: AnalyticalProfile "profile Name" Reference = "reference profile Name" ReferenceElement Element = element, Direction = direction EvaluateWindow Element = geometrical element material region DecayLength = value Replace where: Reference ReferenceElement String indicating the analytic profile to use. Only references to analytic profiles are allowed. The direction of the normal to the ReferenceElement defines the direction of the analytic profile. When evaluating the function values, the mesh points of the newly generated mesh are projected to the element. The distance in the normal direction is used to evaluate the Function. The distance of the projection to the boundary of Element is used to compute the LateralFunction. By default, values are computed on both sides of the element. If Direction is specified, function values are computed only on the positive or negative side of the element. The syntax is: ReferenceElement Element = element, Direction = positive/negative In 1D devices, Element is a point, and the positive and negative directions are given by the coordinate axis. In 2D devices, Element is a line, and the positive direction is taken to the right of the line. In 3D devices, Element can be either a rectangle or polygon. The normal for a rectangle must be one of the coordinate axes. The positive and negative directions are defined from this axis. A (planar) polygon can be arbitrarily oriented in 3D. The direction is defined by the order of the points defining the polygon. A polygon is considered correctly oriented, if the side of the polygon, which is surrounded by the points in positive orientation, defines the positive direction. There is no default value for Element and Direction. Table 12.4 List of elements Function 1D 2D 3D RefineWindow in Placements section for refinements Line Rectangle Cuboid EvaluateWindow in Placements section for profiles Line Rectangle Polygon Cuboid Polyhedron SelectWindow in Placements section for submeshes Line Rectangle Cuboid 12.36

39 5 Command file description Table 12.4 List of elements Function 1D 2D 3D ReferenceElement in Placements for analytic profiles Point Line Rectangle Polygon region for regionwise refinement material for materialwise refinement Region name DATEX material name Table 12.5 Accelerators Keyword Accelerator Default value AnalyticalProfile AnaProf Angle 0 Refinement AttachPoint AttachP Axis Z Constant Datafile DecayLength DecayLen 0 DefineProfiles DefineRefinements Depth Direction Dose Element Erf EvaluateWindow EvalWin Bounding box of the device for constant and analytic profiles. Bounding box of the submesh for submeshes. Factor 0.8 for Gaussian and error functions Function Gauss Geofile LateralFunction LatFunc Error Function Length MaxElementSize MaxElemSize 1 MaxGradient MaxTransDifference MaxGrad MaxTransDiff 12.37

40 5 Command file description Table 12.5 Accelerators Keyword Accelerator Default value MaxValue MaxVal MinElementSize MinElemSize 0.02 Mode read Negative PeakPosition PeakPos 0 PeakValue PeakVal Positive Profile Profiles Range Complete submesh1d ReferenceElement RefElem RefineFunction MaxTransDiff Refinement Refinements RefineWindow Reflection RefineWin Reflect Replace FALSE Rotation Scale 1 SelectWindow SelWin Bounding box of submesh Species StandardDeviation StdDev 0.8 for submesh1d SubMesh SubMesh1D SymmetryPosition SymPos 0 Title "" ToPoint Value 1 ValueAtDepth ValAtDepth Variable "DopingConcentration" 12.38

41 5 Command file description 5.5 Examples The following examples illustrate the use of extensions in the command file syntax Using refinement polygons Figure on page illustrates the use of polygonal domains, both for specifying a polygonal RefineWindow and for using a polygonal domain as an EvaluateWindow. The domain is a simple rectangular boundary and the command file is: Title "Refinement Polygon" Definitions Refinement "global" MaxElementSize = (4, 4) MinElementSize = (.04.04) RefineFunction = MaxTransDiff(Variable="DopingConcentration", Value=0.5) Refinement "refpol" MaxElementSize = ( ) Constant "bor" Species = "BoronConcentration" Value=1e+17 Placements Refinement "global" Reference = "global" RefineWindow = rectangle [( -2-2 ), ( )] Refinement "refpol" Reference = "refpol" RefineWindow = polygon [( 1 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 6 ) ( ) ( ) ( 8 5 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 9 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 5 2 ) ( ) ( ) ( ) ( ) ( 3 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 )] Constant "bor" Reference = "bor" EvaluateWindow Element = polygon [( 1 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 6 ) ( ) ( ) ( 8 5 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 9 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 7 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 5 2 ) ( ) ( ) ( ) ( ) ( 3 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( 1 2 )] 12.39

42 5 Command file description Figure Polygonal refinement Regionwise and materialwise refinement Figure illustrates the effect of using regionwise and materialwise refinement. The following command file segment shows the relevant part of the command file: Placements Refinement "A" Reference = "A" RefineWindow = region ["Ox_Region"] Refinement "B" Reference = "B" RefineWindow = material ["Oxide"] Figure Regionwise and materialwise refinement 12.40

43 5 Command file description Using analytic functions for refinement I Figure illustrates the usage of general analytic functions to specify profiles. The function 0.1sin( x) sin( y) was used as a profile, and linear interpolation ("ElectrostaticPotential") was used to compute the required local element size. The following command file code segment illustrates the syntax: Definitions Refinement "Region_1" MaxElementSize = (1 1) MinElementSize = ( ) RefineFunction = MaxTransDiff(Variable = "ElectrostaticPotential", Value = 0.01) AnalyticalProfile "Profile_1" Species = "ElectrostaticPotential" Function = General(init="a=0.1",function = "a*sin(x)*sin(y)",value = 0) Placements Refinement "Region_1" Reference = "Region_1" AnalyticalProfile "Profile_1" Reference = "Profile_1" EvaluateWindow Element = rectangle [ ( 0 0 ), ( )] Analytic function Generated mesh Figure Usage of analytic refinement functions 12.41

44 5 Command file description Using analytic functions for refinement II This example illustrates the usage of a general analytic function to prescribe 3D refinement, based on a 3D analytic function. The domain is a cube. Figure shows the generated mesh. Definitions Refinement "Region_1" MaxElementSize = (4 4 4) MinElementSize = ( ) RefineFunction = MaxTransDiff(Variable = "ElectrostaticPotential", Value = ) AnalyticalProfile "Profile_1" Species = "ElectrostaticPotential" Function = General(init="a=0.1",function = "a*x*x*y*y*z*z",value = 0) Placements Refinement "Region_1" Reference = "Region_1" AnalyticalProfile "Profile_1" Reference = "Profile_1" EvaluateWindow Element = cuboid [ ( ), ( )] Figure Usage of analytic refinement functions Using analytic functions for doping specification This example illustrates the usage of general analytic functions for defining doping profiles. In order to use the primary and lateral directions as x and y, the keyword Eval must be specified, instead of General, that is, by using global spatial coordinates. Figure on page shows the generated meshes for the example. AnalyticalProfile "NoName_0" Species = "BoronActiveConcentration" Function = Eval(init="a=10",function = "a*sin(x)*cos(y)",value = 0) ReferenceElement Element = line [( 0 0 ), ( 10 10)] # Element = line [( 0 5 ), ( 10 5)] # Element = line [( 5 0 ), ( 5 10)] 12.42

45 5 Command file description Element=line[(0 0),(10 10)] Element=line[(0 5),(10 5)] Element=line[(5 0),(5 10)] Figure Usage of analytic refinement functions for doping 5.6 Reading old command files MESH can transform old syntax when the old commands have a meaning in the MESH multidimensional and flexible syntax. The old command file format is still supported for input; however, an additional switch -oldcmd must be added to load a command file, that is, using the old syntax. It is recommended that the user checks the new file that is generated to verify whether the translation is performed correctly Instructions not supported Although MESH automatically translates old command files, there are some instructions that cannot be transformed into the new syntax: Unsupported commands from GRID1D and MDRAW Unsupported commands from OMEGA Unsupported commands from GRID1D and MDRAW The following commands are not supported in MESH: AlphaGaussFunction Alpha particles profiles. *lifetime The special treatment performed in MDRAW and GRID1D for these species can only be simulated if the concentrations are transformed as follows: 1 C peak = C Life max = C peak Life const = C max Life atdepth = const Life atdepth [Eq. 12.1] *Elifetime or Hlifetime 12.43

46 6 Formulas for analytic profiles 6.1 Overview 6 Formulas for analytic profiles MESH implements a complete set of analytic models to describe a wide range of different situations. The reason for implementing analytical profiles is to have a flexible tool to substitute process simulation results without much effort and within a reasonable time. Due to the wide spectrum of functions to be considered, the organization of this chapter is: Section 6.2 analyzes the general concepts. Section 6.3 on page describes the models along the primary direction. Section 6.4 on page describes the models along the lateral direction. Although the formulas are designed according to the models associated with impurity concentrations, the analytic profiles can be used for any type of variables defined in the DF ISE dataset files. 6.2 General concepts In general, the impurity concentrations can be represented by a set of 1D, 2D, and 3D analytical models. To describe each of the analytical models, we define two main directions: the primary direction that is perpendicular to the reference region, and the lateral direction that is parallel to the reference region. Along each direction, one function is defined, that is, the primary function and lateral function. A proper combination of both functions allows us to have an analytical description of a species concentration Local coordinate systems, valid domains, and reference regions The valid domain for the analytical models depends on the reference region, which is defined using a dimension-dependent geometrical element, and it is placed along the lateral direction. By combining the reference region and primary direction, it is possible to define a local coordinate system for each analytical function One-dimensional profiles One-dimensional profiles require only the definition of the primary function. The primary function is applied along the only available axis, xˆ. The primary direction and valid domain are defined using a vector. The reference region for a profile is defined by using a geometrical element, that is, a point. Figure on page shows the scheme used for the 1D case

47 6 Formulas for analytic profiles Reference Point Valid Domain and Direction Figure Primary direction in 1D Two-dimensional profiles For 2D profiles, the reference region is defined using a baseline. The primary direction is the normal vector to the baseline and the lateral direction is parallel to the baseline. Figure shows a general scheme of the local coordinate system and the valid domain. The valid domain for both the primary and lateral functions is defined by sweeping the primary direction vector along the lateral direction. Reference Line or Baseline Lateral Direction Primary Direction Lateral Direction Primary Domain Lateral Domain Figure Primary and lateral directions in 2D Three-dimensional profiles For 3D profiles, the reference region is defined using a surface. The primary direction is the normal vector to the surface and the lateral direction is the plane perpendicular to the primary direction. Figure on page shows a general scheme of the local coordinate system and the valid domain. The valid domain for both primary and lateral functions is defined by sweeping the primary direction vector along the surface

48 6 Formulas for analytic profiles Reference Surface Lateral Direction Lateral Direction Primary Direction Figure Primary and lateral directions in 3D General implantation models In general, impurity concentrations can be expressed as: doping( x p, x l ) = g x p f x l [Eq. 12.2] where: g x p f x l represents the primary function in the local coordinate system. represents the lateral function in the local coordinate system. The most important functions used as models are Gaussian functions and error functions. For the rest of this chapter, functions along the primary direction are referred to as g y and functions along the lateral direction, as f x. The indices y and x are important to distinguish parameters among the different directions. Each model is defined by the minimum set of parameters. This section presents a basic formulation of each model, by using the minimum set of parameters. The next sections show how to obtain this minimum set from different inputs or initial conditions Gaussian function The minimum set of parameters to define a Gaussian function is: Peak concentration (C peak ) [cm 3 ] Peak position (y peak ) [µm] Length (GLength y ) [µm] or standard deviation (stddev y ) [µm] Using these parameters, the Gaussian is defined by: 1 y y peak gy ( ) C peak exp 2 stddev y y y peak C peak = = exp GLength y [Eq. 12.3] 12.47

49 6 Formulas for analytic profiles Figure shows the model schematically. Cpeak Concentration(y) [cm -3 ] Cpeak * exp(-1) 4 2 difflength ypeak Figure General shape of Gaussian functions y value [um] Error function The minimum set of parameters to define an error function as doping profile is: Maximum concentration (C max ) [cm 3 ] Symmetry position (y sym ) [µm] Length (ELength y ) [µm] gy ( ) = C max 1+ erf 2 y sym y ELength y = C max 1 erf 2 y y sym ELength y [Eq. 12.4] The function is symmetric with respect to the inflection point. Figure shows the feature. Concentration(y) [cm -3 ] Cmax Cmax/ yin Figure General shape for error functions y value [um] Other parameters of interest In order to have flexible models, some special parameters must be considered. These are not included in the standard formulation. However, by applying some definitions, the basic set can be obtained from them

50 6 Formulas for analytic profiles Dose From a process simulation perspective, implantation functions are determined giving the dose concentration of the profiles. The peak concentration value can be obtained from the Dose (see Section 6.3 on page 12.50). The general definition of Dose is: Dose = 0 g( y) dy [Eq. 12.5] For Gaussian functions, the Dose is represented as: 1 y y peak Dose = C peak exp stddev y dy 0 [Eq. 12.6] Dose = C peak π stddev y erf 2 y peak stddev y [Eq. 12.7] For error functions, the Dose is defined as: Dose = 0 C max 1+ erf 2 y sym y dy ELength y [Eq. 12.8] Dose = y sym C max ELength y y sym 1+ erf ELength y ELength y y sym exp π ELength y 2 [Eq. 12.9] NOTE Dose is given in atoms per cm Values at the junction Junction Concentration and Depth are parameters used to define either Gaussian or error functions. A complete description of these parameters and how they can replace the standard deviation in the basic formulation is explained in Section 6.3 on page Length For Gaussian functions, the GLength represents the distance between the peak position and a place where the concentration decays in a factor of exp(-1) (36%) with respect to the peak concentration (see Figure on page 12.48). The relationship between the length and standard deviation for Gaussian functions is: GLength y = 2 stddev y [Eq ] 12.49

51 6 Formulas for analytic profiles 6.3 Available models along the primary direction The following models in MESH are applied along the primary direction: Gaussian function Error function Constant function External 1D profile Gaussian functions The basic set for Gaussian functions is formed by C peak, y peak, and stddev y. According to the input by the user, the basic set of parameters can be specified in six different ways depending on the parameters used to calculate C peak and stddev y : Peak Concentration and Standard Deviation The basic set is complete (see [Eq. 12.3]), and there are no basic parameters to compute. Peak Concentration and Length Standard Deviation is computed from the GLength using: stddev y = GLength y 2 [Eq ] Dose and Standard Deviation Given Dose and Standard Deviation, the Peak Concentration value can be calculated using: Dose factor 2 C peak = y peak π stddev y 1 + erf stddev y [Eq ] where factor = 10 4 because Dose is in cm 2. Dose and Length Given Dose and GLength, the Standard Deviation is computed from [Eq ] and the Peak Concentration from [Eq ]. Peak Concentration and values at the junction Standard Deviation is computed from the values at the junction using: y depth y peak stddev y = ln( C atdepth C peak ) [Eq ] NOTE C peak must be greater than C atdepth

52 6 Formulas for analytic profiles Dose and values at the junction First, Standard Deviation is computed from: y peak C atdepth π stddev y 1+ erf stddev y Dose factor = 1 y -- depth y exp peak 2 stddev y [Eq ] Second, using stddev y, Peak Concentration is computed as in C. NOTE [Eq ] is an implicit equation and Dose is in cm Error functions For error functions, the basic set of parameters includes C max, y sym, and ELength y and can be computed in the following four ways: Maximum Concentration and Length The basic set is complete; there are no parameters to compute (see [Eq. 12.4]). Dose and Length Maximum Concentration is computed from Dose using: 2 Dose = factor ELength y y sym y sym 1 erf y sym ELength y ELength y + exp π ELength y 2 1 C max [Eq ] where factor = 10 4 because Dose is in cm 2. Maximum Concentration and values at the junction ELength can be computed from: erf y sym y depth ELength y = 2 C atdepth 1 C max [Eq ] NOTE [Eq ] is an implicit equation. Dose and values at the junction Maximum Concentration and ELength are computed using the following implicit equations, which follow from [Eq ] and: Dose factor 1 erf y sym y depth = C ELength y atdepth ELength y y sym y sym 1 erf y sym ELength y ELength y + π exp ELength y 2 [Eq ] 12.51

53 6 Formulas for analytic profiles 2 Dose = factor ELength y y sym y sym 1 erf y sym ELength y ELength y + exp π ELength y 2 1 C max [Eq ] Constant function Constant functions are useful to define substrate doping mathematically: gy ( ) = Constant [Eq ] External 1D profile Real 1D process simulation results can be read along the primary direction. To complete the 2D profile and 3D profile, an analytical lateral function is added. The values that do not appear in the file are interpolated using an interpolation function. Every species has a corresponding interpolation function predefined on the datexcodes file (see Chapter 7 on page 12.55). These functions can be linear, arsinh, or logarithmic. If h is an interpolation function, the value at point y is computed from an external 1D profile as follows: gy ( ) data i y = y i = h 1 y y i y y i + 1 hdata ( y i+ 1 y i + 1 ) hdata + ( i y i + 1 y i ) i y < y < y i i + 1 [Eq ] 6.4 Lateral or decay functions The lateral or decay functions are evaluated on the valid lateral domain (see Figure on page and Figure on page 12.47). They are defined as the decay along the lateral direction and depend on the distance from the valid primary domain of the point to evaluate. For 2D, this distance is calculated using the baseline as reference. For 3D, the distance is computed using the surface as reference. There are three available models to apply: Gaussian function Error function No function NOTE Lateral or decay functions are not valid for 1D

54 6 Formulas for analytic profiles Lateral Gaussian function The equation applied is: fx ( ) = x closestp exp 1 x stddev x [Eq ] According to [Eq ], the required value from the user is the Standard Deviation, stddev x, along the lateral direction. There are three ways to define it: Provide the value explicitly. Provide a factor with respect to the standard deviation along the primary direction: stddev x = Factor x stddev y [Eq ] Giving the length of the Gaussian function: GLength x = stddev x 2 [Eq ] By using this function, the decay begins outside the primary domain, that is, the overlap between primary, lateral, and decay domains is zero. Figure shows this effect. baseline (window mask) Figure Using Gaussian function as lateral function in 2D Lateral error function The equation applied is: fx ( ) = erf x closestp x ELength x [Eq ] According to [Eq ], the required value from the user is the length for the error function, Elength x, along the lateral direction. There are two ways to define it: Provide the value explicitly. Provide a factor with respect to the length along the primary direction: ELength x = Factor x ELength y [Eq ] 12.53

55 6 Formulas for analytic profiles For this model, the overlap of primary, lateral, and decay domains is not zero. The lateral decay starts inside the primary domain as shown in Figure baseline (window mask) Figure Using error function as lateral function in 2D No lateral function This property is valid when Factor is equal to zero. For this case, the value of the lateral function is given by the expression: fx ( ) = 1 x PrimaryDomain 0 x PrimaryDomain [Eq ] For this case, the lateral domain is null

56 7 Technical aspects 7.1 Valid polygons and polyhedra 7 Technical aspects There are some constraints for the polygons and polyhedra that are used to define the geometry: Only simple polygons are allowed. A polygon is simple if there is no pair of nonconsecutive edges sharing a point. A single-connected polygon is described using a single closed line, see Figure (a). For MDRAW, it is necessary to describe a hole in a polygon connecting the hole with a double line to the polygon, see Figure (b). Figure (a) Types of polygon: Simple (a) and nonsimple (b) (b) Figure (a) Single-connected polygons (a) and nonsingle-connected polygon (b) No polygon or polyhedron can overlap another one. The detection of overlapping regions, particularly in 3D, is not a simple task and is not performed in MESH. Only the coplanarity of points defining a face is checked. (b) 7.2 Numeric considerations MESH is a Delaunay mesh generator for 1D, 2D, and 3D devices. Grids produced by the program are suitable for the control volume integration method based on the Voronoï diagrams. The Delaunay condition ensures non-overlapping control volumes and the spatial discretization includes the entire device domain. Hence, the effective facets for the Voronoï diagrams are positive. MESH can produce a special type of Delaunay mesh in 2D. This type of grid is more restrictive than the default grid because all the edge-element facet contributions are positive. The condition is reached when the grid elements are free of obtuse angles. The condition guarantees that the Voronoï center of every element lies inside or on the border of the element, and they are called self-contained elements. For rectangular triangles, the Voronoï centers lie on one of the edges. For acute triangles, the Voronoï centers lie inside the elements

57 7 Technical aspects For more information about the control volume method and the assembling of equations, refer to the literature [4]. 7.3 Mesh refinement algorithm MESH is a grid generator based on macroelements, which are refined until the geometry can be well represented or the required density is reached. The refinement point for each macroelement is selected according to: 1. The intersection points between the macroelement and the geometry to fit. 2. Points inserted on the edges and faces of the macroelement after the refinement of the neighborhood. This set is only valid for 2D and 3D. 3. Midpoints of the edges; valid for 2D and 3D. Applying this approach, the macroelements can have more than two neighbors per face or edge, and because of this property, they are called n-connected elements. During grid generation, the following sequence of three steps is performed Constructing the first coarse grid The starting grid is generated by building a set of macroelements according to the geometry description. For complex 2D and 3D devices, the initial grid is a critical step. The refinement of macroelements in this step is performed in a way that the point propagation along the entire device is avoided Adaptation according to external data The adaptation is performed by refining the macroelements along their edges controlling two aspects: 1. The gradient of the profiles. The macroelements are refined according to the values at the points given by either the analytical profiles described in the command files or the external data loaded, using submeshes in 2D, or DIP in 3D. 2. The required minimum and maximum sizes of the edges Obtaining a conforming final grid In this step, the n-connected elements are refined until they can be tessellated into a set of valid known elements: triangles and rectangles in 2D, and tetrahedra, rectangular pyramids, prisms, and bricks in 3D. The Delaunay conditions are fulfilled at this stage of the program. For 2D cases, the condition is achieved element by element and, for 3D cases, the condition is achieved by giving the entire grid to the delaunization module. For more information, refer to the literature [2][3][5]

58 7 Technical aspects 7.4 Delaunization module A delaunization algorithm is available for 3D models in MESH. The module handles green points using an improved algorithm, and it is more successful at meshing nonaxis-aligned structures than the previous delaunization algorithm. The algorithm can be called by using the MESH option -P. Details are in the literature [6], but some technical aspects of the algorithm are discussed here. MESH is a mesh generator based on a modified version of the octree method. The octree method starts from a rectangular box enclosing the device to mesh. This box is successively refined using axis-aligned cuts until the resulting boxes meet size specifications specified by the user. When these size specifications are met, the boxes are triangulated and the triangulation is passed over to the delaunization module. The main difficulty in MESH is precisely the triangulation of the final boxes. After the refinement is performed, some boxes may contain certain fragments of the input boundary of the device. Since the geometry of these boundary fragments can be complicated, they cannot be easily triangulated. The strategy in MESH is to further refine these boxes in the hope that the subboxes contain simpler interface fragments, simple enough that the box can be triangulated using a simple template. Finding a match between the tessellated boxes and the implemented templates is not always possible. Therefore, MESH may perform unnecessary refinement or not stop refining at all (see Figure 12.28). Figure Example of nonstop refinement in MESH; after refinement, the pattern in upper-right corner repeats itself Algorithm The algorithm (called by the option -P from the MESH command line) bypasses the template triangulation inside MESH. Instead of directly triangulating the boxes, they are passed over to a module called the Delaunay refinement module. The idea is to use two independent structures: a set of surface faces (for example, the input boundary plus some isosurfaces or rectangular faces coming from user-defined refinement inside MESH) and a background three-dimensional generic Delaunay triangulation. The generic Delaunay triangulation is used to store the set of points coming from the octree refinement. The algorithm works in the following way: The input surfaces are triangulated, and a 2D surface delaunization algorithm is applied. Each surface face containing a point inside its minimum circumsphere is then refined and the resulting faces in the surface are delaunized. This step is repeated until the minimum circumspheres of all the surface faces are point free (see Figure on page (b)). After the surface faces are refined, a generic Delaunay triangulation is built for the complete set of points (including the bulk points). Since the minimum circumspheres of the surface faces are point free, the surfaces must be completely contained in the generic Delaunay triangulation (see Figure (c)). A material is assigned to each tetrahedron in the final triangulation (see Figure (d))

59 7 Technical aspects Figure Two-dimensional example of Delaunay refinement algorithm In Figure 12.29, the following graphics are displayed: a) Input boundary and some constraints b) Refined boundary and constraints c) Resulting points that are triangulated using a generic Delaunay triangulation algorithm d) Materials that are assigned The advantages are: Only known algorithms are used. The surface refinement algorithm has been successfully used in the former delaunization module. The generic delaunization algorithm is well studied and efficient algorithms exist. The algorithm is suitable for many types of mesh generators, not only octree-based generators. The only input required is a set of surfaces and a point distribution. 7.5 Active and total concentrations in MESH Data handling during meshing has changed. Previous versions automatically changed total concentrations (such as BoronConcentration) to active concentrations (such as BoronActiveConcentration), which were added to the net doping (DopingConcentration). As this behavior is not very transparent, data of concentrations is no longer changed by the meshing tools. Some existing projects employ total concentrations to describe doping. In these cases, that is, when no active concentrations are present, the total concentrations are used to compute net doping. The log files signal this behavior with the comment: Using total concentrations for net doping. For new projects, it is strongly recommended that active concentrations are used from the outset. In addition, ISE provides a script (dopconv.sh) to convert existing projects to the use of active concentrations