Electrical Simulation of Liquid Crystals

Transcription

1 Engineered Excellence A Journal for Process and Device Engineers Electrical Simulation of Liquid Crystals Introduction Liquid Crystals (LCs) are state of matter intermediate between that of a crystalline and a liquid. The optical, mechanical, electrical and magnetic properties of LC medium are defined by the orientation order of the constituent anisotropic molecules. Due to the anisotropy of the electrical properties, the orientation of the LC molecules is effectively controlled by electric fields. As a result, LCs exhibit very specific electrooptical phenomena because of their large birefringence. All of these are important to the functional devices based on LCs, for example, flat panel displays that have been commercialized for decades. The constituents of LCs are elongated or rod-like molecules and disk-like molecules. The average direction of the molecular long axes defines the director n, which gives the direction of the preferred orientation of LC molecules. The LC molecules reorient in externally applied electric fields because of their dielectric anisotropy. The electric energy of a LC depends on the orientation of the director in the applied electric field. Under a given electric field, the LC will be in the equilibrium state, where the total free energy is minimized. Theory The field-induced reorientation of the LC is now able to be calculated in Clever. The coupled equations governing the LC physics and the electrostatic potential are solved. The major variables are the voltage and the director. To calculate a director configuration of the LC, it is necessary to express the free energy of the system. Let n(x,y,z) = (n x (x,y,z), n y (x,y,z), n z (x,y,z))be the director, where (x,y,z) is a point in the LC region. The Frank-Oseen free energy density is given as [1]: f(x,y,z) = K 11 ( n) 2 + K 22 (n n) 2 + K 33 (n n) 2 - (K 22 + K 24 ) (n n + n n) q 0 K 22 (n n) where K 11, K 22, and K 33 are the splay, twist, and bend elastic constants of the LC respectively. q 0 is the chirality of the LC. The electric free energy density is given by: g= ε 0 ε[n ( ν)] 2 where ε 0 is the permittivity of vacuum, ε = ε ε is the difference between the dielectric constant parallel and vertical to the director, and ν=ν(x,y,z)is the voltage at point (x,y,z). The Gibbs free energy density is then defined as: F = f g The total free energy is the volume integration of the free energy density within the LC domain. The total free energy is minimized by a method based on a variational approach to the Oseen-Frank free energy formulation considering three elastic constants. The vector representation of the director field is used with the constraint n 2 = n 2 x + n2 y + n2 z = 1. Simulation Examples As an example, a twisted nematic (TN) geometry with the LC layer thickness of 3 µm is simulated. The elastic constants of the LC are K 11 = N, K 22 = N, and K 33 = N. The anisotropic relative permittivity Continued on page 2... INSIDE Performance Improvement by MPI Parallelization in 3D Device Simulation...5 Improvements and Features of the Updated DeckBuild 2 GUI...8 Hints and Tips: How can I do local conformal mesh refinement in Victory Process? Volume 25, Number 3, July, August, September 2015 July, August, September 2015 Page 1 The Simulation Standard

2 Figure 1. The TN-LC cell structure biased at (a) 0V and (b) 4V, respectively. Arrows indicate the orientation of the director. is ε//=10.7 and ε =3.7. LC material parameters can be specified in the SetLC statement in Clever, as follows: setlc epsparadir=10.7 epsvertdir=3.7 splay=14.4e-12 twist=6.9e-12 bend=18.3e-12 The pretilt angles at the top and the bottom boundary of the LC layer are both 1. The initial twist angle is 90. These alignment conditions can be defined in the following commands. LCbndaryT partition(0 10) rubangle(0) tiltangle(1) LCbndaryB partition(0 10) rubangle(90) tiltangle(1) where LCbndaryT and LCbndaryB denote the statement for the top and the bottom boundary, respectively. The rubangle sets the rubbing angle with respect to the x axis and tiltangle specifies the tilt angle measured from the x-y plane. The LC cell structure consisting of a LC layer sandwiched by two electrodes was created in Victory Process and exported with conformal tetrahedral mesh, as shown in Figure 1. The bias voltage on the electrode Pix was increased linearly from 0 to 6V in a 0.25V bias step through the following command in Clever: Interconnect Capacitance \ domainboundarycondition=cyclic \ contact= Pix uservoltage=6 stevolt=0.25 strcture= tn The solution was stored after each voltage step. The structure filename was generated in such a way that the current bias voltage is appended to the root file name tn. The director orientation of the TN-LC structure at 0 and 4V bias voltage is shown in Figure1(a) and 1(b), respectively. The arrows in the LC layer indicate the director orientation. Since Clever uses strong anchoring boundary condition for the LC simulation, the director orientation at the top and bottom boundaries will never change with the bias. Therefore, we can see as the bias is increased from 0 to 4V the director remains the same at the top and bottom boundaries while it becomes highly tilted in the middle of the LC layer. The z component of the director n z was extracted along a line parallel to the z axis across the entire LC layer. The profile of n z at various bias voltages is shown in Figure 2. We can observe quantitatively how the director changes with increased bias. A nominal threshold voltage exists between 1.5V and 2V in this case. In order to identify the threshold voltage more clearly, the nz in the middle x-y plane (in terms of z axis) of the LC layer was extracted. The data was plotted as a function of the bias voltage with different twist angles of the director in Figure 3. We can see the transition region, i.e., the region where the nz goes from nearly 0 to nearly 1 becomes narrower and narrower with increased twist angle. Therefore the Figure 2. The profile of the director component nz along the z direction with different bias voltages. Figure 3. The director component nz in the middle x-y plane of the LC layer as a function of bias voltages with different twist angles. The Simulation Standard Page 2 July, August, September 2015

3 Figure 4. The IPS structure with zigzag electrodes. threshold voltage becomes more distinct. This is a wellknown phenomenon in TN LC. Because of their steep transition, TNs with twist angle larger than 90 (known as super-tns) are used to make multiplexed displays on passive matrices. Figure 5. The n x contour in a x - z plane at (a) 4V and (b) 6V. Figure 7. The nx profile along the z direction with different bias voltages. The electrode plane is on the left side. Another example is an in-plane switching (IPS) cell with two coplanar zigzag electrodes shown in Figure 4. The LC layer of the structure has a lateral size of 10µm 10µm and a thickness of 3m. The whole structure was created in Victory Process with conformal tetrahedral mesh. The LC material parameters are the same as those used in the TN example. The initial rubbing angle of the director is 90 with respect to the x axis. As the applied voltage at the Pix electrode is increased, the director inbetween two electrodes turns to the x direction gradually due to the increased electric field along the x axis. The director above electrodes twists in a much smaller amount because the electric field is almost vertical to the electrode surface. Shown in Figure 5 is the contour plot of the x component of the LC director, nx, in a x-z plane located at y=7.5 m at the (a) 4V and (b) 6V bias voltage. Shown in Figure 6 is the contour plot of the nx in a x-y plane 1m to the electrode plane at the (a) 4V and (b) 6V. We can see clearly from these two figures how the director reorients with bias in different regions of the LC layer. The n x profile was extracted along a line parallel to the z axis across the LC layer at x=4.5 m and y=5 m, the middle point between two zigzag electrodes. The result is shown in Figure 7 where the electrodes are located in the left. The threshold voltage is estimated to be around 3V from the curve shape. Figure 6. The n x contour in a x - y plane at (a) 4V and (b) 6V. Figure 8. The capacitance-voltage curve of the IPS cell. July, August, September 2015 Page 3 The Simulation Standard

4 The capacitance of the LC cell between two electrodes can be obtained during the voltage ramping simulation. The calculation takes into account the anisotropic permittivity of the LC director. To save the capacitance, one needs to add the following command after the Interconnect statement: save spice= ips.net The data in the netlist file can be viewed in Tonyplot. The capacitance-voltage (CV) curve of the IPS cell is plotted in Figure 8. The threshold voltage is a bit smaller than 3V, in consistence with the director result shown in Figure 7. Conclusion In summary, three-dimensional static simulation of the liquid crystal with user-definable material parameters can be carried out in Clever. All vector components of the LC director in the presence of the external electric field (applied voltage) are solved by the finite element method. The director data can be output at any stage of the simulation for visualization or post-simulation processing. The simulation on the TN and IPS structures has been demonstrated. Other LC geometries such as vertical alignment (VA) and fringe field switching (FFS) can be analyzed as well. References [1] H. Mori et al, Multidimensional Director Modeling Using the Q Tensor Representation in a Liquid Crystal Cell and Its Application to the π Cell with Patterned Electrodes, Jpn. J. Appl. Phys. Vol. 38 (1999) pp The Simulation Standard Page 4 July, August, September 2015

5 Performance Improvement by MPI Parallelization in 3D Device Simulation Introduction As the design technology for power devices, such as MOSFET, GTO, and IGBT has matured, the importance of large domain 3D TCAD simulation has increased rapidly. Distributed computing is one of the attractive solutions for such simulations, because the system s performance and capability is not limited by the number of CPUs or the total amount of memory on a specific computer. This advantage of distributed computing is expected to be increasingly advantageous, as the size and mesh point count for these devices becomes ever larger. Silvaco s TCAD applications provide the user with the distributed computing feature which is supported in the solution of linear systems using the PAM solver [1, 2]. The PAM solver is a domain decomposition type solver that runs in parallel using MPI (Message Passing Interface). The user can set up the distributed computing feature with MPI parallelization easily, with the addition of a few simple settings on a Linux operating system [1]. In this article, we demonstrate good performance from the PAM solver with MPI parallelization using Victory Device on a blade server with a total of 120 threads. In addition, we verify the dependence of performance improvement by MPI parallelization, on the device size and number of mesh points. Simulation Conditions Table 1 shows the specification of the server used in this work. It consists of 6 nodes of a Dell PowerEdge C8220/ C8220x blade server, with 20 threads of execution per node and 64 Gbytes of memory available. To achieve optimal performance in MPI parallelization, it is important to use a high speed network interconnect between the cluster nodes. InfiniBand FDR (Fourteen Data Rate, 14Gb/s data rate per lane) was utilized for the interconnection here. On this cluster system, we carried out a DC I c -V c simulation for the multiple cell array of a standard punch-through (PT) type 3D IGBT using the PAM solver in Victory Device. As the physical model in this simulation, we considered the field and concentration dependent mobility, SRH and Auger recombination, together with impact ionization. The PAM solver is a domain decomposition type linear solver specially designed for very large sparse linear systems. Figure 1 shows a schematic diagram of the simulation method using the PAM solver with MPI parallelization. Each MPI process handles the solution of one part of the linear system and the MPI processes are run in parallel on each CPU thread. After each MPI process finishes with it s part of the linear system, the solution is sent back to the main MPI process, and the solution to the global linear system is re-formed and returned [2]. We carried out the same device simulation by using the various numbers of threads, up to 120 on the cluster system shown in Table 1. MPI parallelization was applied to every simulation with 2 threads and more. This means that the MPI processes by default, are spawned across the individual server nodes, even if the number of threads specified in the simulation is less than that on one server node. For example, if the user specifies parallelization with 3 threads, the first process is assigned to node #1, the next to the node #2, and the last to the node #3, respectively. In order to verify the dependence of performance improvement by MPI parallelization on the device size and number of mesh points, we ran device simulations for two cell sizes of IGBT array. One was a 2 2 cell structure resulting in 134K mesh nodes and the other was a 6 5 cell structure with 977K mesh nodes. Table 1. Specification of the server used in this work. July, August, September 2015 Page 5 The Simulation Standard

6 Figure 1. A schematic diagram of the simulation method using the PAM solver with MPI parallelization. Results and Discussions Figure 2 shows the dependence of simulation time on the number of threads parallelized by MPI. For relatively small numbers of threads the simulation time was drastically reduced as the number of threads increased, but the improvement saturated when using a much larger number of threads. The number of threads that gave the fastest simulation time was 32 for the 2 2 cells and 80 for the 6 5 cells, respectively. As expected, the number of parallel threads for the fastest calculation tends to increase as the device size and/or the number of mesh increases. This feature can expand the device size for which the device simulation is completed in a practical time range. It will certainly be useful for the user who wants to simulate a large device especially in power device design. Figure 3 shows the dependence of speed-up rate on the number of threads for the 2 2 cells structure. The speedup rate is defined as the ratio of the single thread simulation time to the parallel thread simulation time. For the number of threads less than the optimal value, it was observed that the speed-up rate exceeded the ideal line on the assumption of the proportion to the number of threads. The speed-up rate reached the peak value of 50x at the optimal 32 threads. The speed-up rate, however, saturated or decreased gradually for the number of threads more than this optimal value. It is not so easy to clarify the reason because this is complicatedly related to not only the software factor like solver performance or mesh quality, but also the hardware factor like CPU capability or network interconnecting speed. More comprehensive design of the cluster system can probably help the user achieve better MPI performance. Figure 2. Dependence of simulation time on the number of threads in parallel. Figure 3. Dependence of speed-up rate on the number of threads in parallel. The Simulation Standard Page 6 July, August, September 2015

7 Acknowledgments We gratefully acknowledge the support of Dell Solution Center Tokyo and HPC Solutions Inc. that provided the server and the equipment used in this work. References [1] Hints, Tips and Solutions, Simulation Standard, Volume 24, Number 1, January, February, March silvaco.com/tech_lib_tcad/simulationstandard/2014/jan_ feb_mar/hints2/hints2.html Figure 4. I c -V c characteristics simulated with the various numbers of threads in parallel. As a result, it can be confirmed from this result that Victory Device has the capability to improve the simulation time by orders of magnitude by using the distributed computing feature. [2] State of the Art 3D SiC Process and Device Simulation, Simulation Standard, Volume 23, Number 1, January, February, March 2013, simulationstandard/2013/jan_feb_mar/a1/state_of_the_ art_3d_sic_process_and_device_simulation_a1.html Figure 4 shows the overlay plot of the Ic-Vc characteristics simulated with the various numbers of threads in parallel. It can be seen from this figure that all the curves are identical with each other and therefore the accuracy of the simulation result is kept independently of the number of parallel threads. MPI parallelization by the PAM solver can provide the user with performance improvement without decreasing the accuracy. Conclusion We have demonstrated a 3D device simulation parallelized by MPI using Victory Device on a blade server with a total of 120 threads. As a result, the simulation time was drastically reduced without decreasing the accuracy as the number of threads increased and the speed-up rate reached the peak value of 50 at the optimal number of threads. Moreover, it has been confirmed that the number of parallel threads for the fastest calculation tends to increase as the device size and/or the number of mesh increases. Silvaco s TCAD having the distributed computing feature can expand the device size for which the device simulation is completed in a practical time range. We believe that it can help the user who wants to do 3D TCAD simulation with a large domain especially in power device design. July, August, September 2015 Page 7 The Simulation Standard

8 Improvements and Features of the Updated DeckBuild 2 GUI Introduction In this article we will emphasize the new features and improvements of the DeckBuild 2 deck editing environment. We will start by illustrating the examples section, followed by the basic execution modes of DeckBuild and a description of how an Athena deck can be automatically converted to be run in Victory Process. The article will also demonstrate how the visualization tools Tony- Plot and TonyPlot3D are integrated and available directly from the various parts of DeckBuild. Deckbuild DeckBuild is an input deck file development environment within which all of Silvaco s TCAD and several other EDA products can run. It offers various modes of how deck can be executed, debugged and how results can be obtained and visualized. DeckBuild contains an extensive library of hundreds of pre-run example decks which cover many technologies and materials. Examples Database DeckBuild is shipped with a comprehensive set of over 500 TCAD examples. Figure 1 displays how the examples dialog can be opened from the File menu of DeckBuild. The initial examples dialog gives a hierarchical view at the examples, similar from what is available in the previous version DeckBuild 1. You are presented with a tree-like view, which allows you to effectively browse through the different sections of the examples database. By clicking on a particular example, the lower part of the examples dialog is updated with the description of the selected example. The example description itself was extended by screen-shots and pictures to better illustrate the textual description of the example. Once an example was selected you get the choice of either loading the deck only or to load the whole example inclusive of all available structure files or screen-shots. The idea is to allow you to quickly browse through a deck without the need to copy large structure files. An interesting extension to the hierarchical view can be found at the very top of the examples dialog, which contains a search field. By entering text in the search field and hitting the enter button (or clicking on search) an index of the examples database is queried. Figure 2 displays the results when for instance - you search for the term stress. The hierarchical view has changed to display a list of results. In this example a total of 27 hits is shown as is indicated in the top right of the examples dialog. Search strings are not limited to simple words but can be complex boolean expressions as illustrated in Figure 3. If we extend the search term from stress to stress and title:substrate then only results which additionally have the word substrate in the title are shown. The hits are now reduced to a total of five. This example also illustrates how a search string can be limited to be effective in a particular section only. Here the word substrate is only matched against the title section of an example, whereas the word stress is matched against any part of an example. The home button at the very left of the search field brings back the hierarchical view should it be desired. Figure 1. Examples menu and popup. Figure 2. Search results for search string stress. The Simulation Standard Page 8 July, August, September 2015

9 Apart from executing deck by using Stop Points, you also have the choice of running a single command at a time only. This so-called single-stepping feature is available via the Run menu as well as via a button on the toolbar and via a keyboard shortcut. Finally, if you want to run through the whole deck without ever stopping, you can simply use the Run button. Execution will then Stop after the last line has been executed (provided that no Stop points have been defined in the deck). Tracking Variables and Results Figure 3. Boolean search phrase. Execution Modes One fundamental concept of DeckBuild is to allow the user to execute a simulation deck in various ways. The following basic options are offered: DeckBuild allows you to keep track of all variables and files, which are created during a simulation run. Two separate panes can be opened, one to view all variables and extracts and a second one to view generated files. Run the whole deck without stopping Run line-by-line, halting execution at every line Run to a pre-defined Stop point and halt execution at the stop point. The execution modes are made available via the Run menu as well as via the toolbar shown in Figure 4 and via configurable keyboard shortcuts. Figure 4. Deck execution buttons on toolbar. DeckBuild 2 allows you to define an arbitrary number of stop points in the deck. Figure 5 displays a portion of the deck which has three Stop Points defined. If you execute this deck by using the Run->Run/Continue menu entry or the corresponding button on the tool bar, execution will stop right at the first Stop Point. After that, when invoking Run/Continue again execution will continue and stop at the 2nd Stop Point and so forth. Figure 6. Toolbar buttons for variable and output tracking. Figure 6 displays the buttons on the toolbar, which are used to open or close respectively, the variables and outputs tracking windows of DeckBuild. Figure 7 displays a screen-shot of the two tracking windows. At the top right the pane titled Variables history shows a view at all extracted values. The line number in the deck where the corresponding extract (or set) statement appeared is given in parentheses. By clicking on the value of a variable, you can change its value. This can be useful if you are debugging a deck and want to temporarily assign a different value to a variable. The change is effective immediately and will effect any statement, which takes as input the variable value and is executed after the variable was changed. Figure 5. Deck with Stop Points. Figure 7. Variables and Outputs tracking. July, August, September 2015 Page 9 The Simulation Standard

10 Figure 8. Plotting structure and log files. At the bottom right the pane called Outputs shows all files that have been created in due course of the simulation. To limit the number or kind of displayed files, you can enter a filter string. Only files matching the filter are then displayed. Files shown in the Outputs pane are visualized in the exact same way as files that are encountered in the deck or runtime output by simply right-clicking on them. Figure 8 displays the context menu that opens upon right-clicking. The first option Plot append allows you to load a file into a previously started TonyPlot window. It is only displayed if you have started TonyPlot before. The 2nd option Plot, will load the file into a new TonyPlot window. Appending is helpful for instance if you want to create an overlay plot of several curves in TonyPlot. Figure 10. Converted deck with Athena commands commented out. Conclusion In this article we gave an overview of the new features of the DeckBuild software. We presented the new look and functionality of the examples section and the various modes of execution. The new variables and outputs tracking facility was illustrated and the Athena to Victory Process converter was demonstrated. Future work on DeckBuild will include features to navigate between the various components of DeckBuild (runtime output, deck, variables, outputs) at a simple mouse click, will offer improved editor capabilities, and will also allow to run DOE and optimization experiments known from VWF. Victory Process Deck Conversion As of version 4.2.0, DeckBuild comes with a built-in Athena to Victory Process converter. A loaded Athena deck can thereby be converted to execute in the Victory Process simulator. Figure 9. Toolbar buttons for the Syntax converter. Figure 9 displays the two toolbar buttons to use the Athena to Victory Process syntax converter. The converter is used by first loading an Athena deck and then hitting the left of the two buttons. This will then initiate the conversion and open the converted deck in a new window. The original Athena deck commands will still be visible in the new deck for your reference, but have been commented out as shown in Figure 10. The right button on the toolbar shown in Figure 9 can be used to strip the commented Athena commands and only keep the converted Victory Process commands. The Simulation Standard Page 10 July, August, September 2015

11 Hints, Tips and Solutions How can I do local conformal mesh refinement in Victory Process? Introduction The Victory Process conformal export is generated from the volume planes specified in a user deck. The accuracy and resolution of the export is currently controlled by these planes. Local refinement gives a further level of user control that allows the mesh density to be increased near regions of interest. We now support: interface, junction, box and global refinement schemes. These refinement schemes behave similarly to the Delaunay export, with the exception that the distance parameter is calculated automatically. This is necessary since the conformal nature of the mesh would be lost if the wrong distance is used. Global The uniform refinement syntax is: export victory(conformal) \ structure= global.str \ max.size=0.25 This will refine until all elements within the mesh have a maximum feature size of 0.25 microns. The feature size is defined to be the radius of the circumsphere of the elements (tetrahedra). An example is shown in Figure 2. Conformal refinement is achieved by subdividing an element into eight. The element shape is maintained by the subdivision. In the case of a path simplex within the conformal mesh (six tetrahedra forming a cube), a single level of refinement will create 48 new elements. These new elements will form eight smaller path simplices, where each cube is now 1/8 of the original cubes volume. An example of our input structure is shown in Figure 1. This was generated using the victory(conformal) process mode export. Figure 2. Victory (conformal) export with max.size refinement. Shape The shape refinement can be used to specify a 3D cuboid within which the mesh will be refined. The syntax is: export victory(conformal) \ structure= box.str \ box.min= 5, 5, 3 box.max= 7, 7, 2 \ max.box.size = Figure 1. Victory(conformal) process mode example export. July, August, September 2015 Page 11 The Simulation Standard

12 export victory(conformal) \ structure= sphere.str \ sphere.center= 6.5, 6.5, 1.0 \ sphere.radius=1.0 max.sphere.size=0.25 Interface The interface refinement syntax is: export victory(conformal) \ structure= interface.str \ distance.interface \ material= SiliconDioxide \ max.interface.size=0.1 Figure 3. Victory(conformal) box refinement example. In this case we have a box from (5, 5, -3) to (7, 7, -2). Unlike the Delaunay refinement, it is necessary to create grading elements outside of the refined section. The grading elements are calculated automatically, and will result in a small level of refinement outside of the given box. This is to maintain the mesh quality between refined/nonrefined mesh regions. In this case the elements at the interface of Silicon Dioxide will be refined until they have a maximum feature size of 0.1 microns. An example of the interface refinement is given in Figure 4. It should also be noted that grading elements can only occur in perfect path simplex cubes (six tetrahedra forming the cube). If the box refinement is close to an interface, the algorithm may need to refine the interface in order to ensure smooth grading levels. An example of the box refinement is shown in Figure 3. Cone, cylinder and sphere refinement are also supported. The respective export commands are: export victory(conformal) \ structure= cone.str \ cone.start= 6.5, 6.5, 1.0 \ cone.end= 6.5, 6.5, 4 \ start.cone.radius=1.0 \ end.cone.radius=0.05 \ max.cone.size=0.25 export victory(conformal) \ structure= cylinder.str \ cylinder.start= 6.5, 6.5, 1.0 \ cylinder.end= 6.5, 6.5, 4 \ cylinder.radius=1.0 max.cylinder.size=0.25 Figure 4. Victory(conformal) interface refinement example. Junction The junction refinement specifies the maximum feature size for elements that have a minimum containment center with zero distance from the junction. An example of junction refinement is shown in Figure 5. The Simulation Standard Page 12 July, August, September 2015

13 In Figure 6 and 7, an example of the unstructured Delaunay junction refinement, in comparison to the structured conformal junction refinement on the same device, is shown. The export statements used were: export victory(conformal) \ structure= conformal.str \ max.junction.size=0.4 export victory(delaunay) \ structure= delaunay.str \ max.junction.size=0.4 \ max.junction.distance=9.5 \ Figure 5. Victory(conformal) junction refinement example. max.size=2 Delaunay and Conformal Comparison The conformal export already provides a means to refine regions of the mesh through the placement of the volume planes. However, in many cases regular refinement is insufficient. The refinement schemes demonstrated in this document allow refinement of localized regions, without the requirement to refine along an entire axis plane. However, the placement of the volume planes is paramount to achieve a quality refined mesh. In this section we will demonstrate how the volume planes should be placed to achieve refinement in regions of interest. The following conditions must be taken into account when using the structured conformal refinement: 1. In a conformal mesh, assuming no interfaces, a single block (path simplex), is comprised of 6 tetrahedra. A single refinement level subdivides each edge once. The 6 tetrahedra become 48. Figure 6. Structured conformal junction refinement. 2. We only place vertices on edges (i.e. subdivide). In comparison, the unstructured Delaunay refinement will place vertices inside tetrahedra. 3. The user must be careful where the initial volume planes are placed in order to ensure sufficient volume to grade within (further details on this point are given below). 4. If we wish for refinement along an entire axis plane, the best option is to use a volume plane. If we wish for refinement along a junction or interface, the volume planes must be placed to ensure the refinement can grade correctly. Figure 7. Unstructured Delaunay junction refinement. July, August, September 2015 Page 13 The Simulation Standard

14 Further details on grading: 1. The Delaunay refinement requires a user specified junction distance within which the refinement is graded (9.5 microns in this example). 2. The conformal refinement determines this distance based on the existing volume planes, and therefore a distance parameter is redundant. 3. Each level of structured conformal refinement will introduce another level of grading. 4. The conformal refinement junction distance is effectively the number of refinement bisection levels multiplied by the volume plane spacing at that region of the mesh. 5. In the example, we must bisect a maximum of two levels to meet the 0.4 microns size requirement, so we have two levels of grading, i.e. the distance is two times the volume mesh planes spacing (approximately 9.5 microns at the center of the structure). Further, it should also be noted that the unstructured Delaunay refinement will only refine the junction elements that require it. In the structured conformal refinement, we may refine additional junction elements that already meet the 0.4 size requirement. This can be seen at the top of the structure. This behavior is necessary to maintain the conformal constraints of the export. Call for Questions If you have hints, tips, solutions or questions to contribute, please contact our Applications and Support Department Phone: +1 (408) Fax: +1 (408) support@silvaco.com Hints, Tips and Solutions Archive Check out our Web Page to see more details of this example plus an archive of previous Hints, Tips, and Solutions The Simulation Standard Page 14 July, August, September 2015

15 USA Headquarters: Silvaco, Inc Patrick Henry Drive, Bldg. 2 Santa Clara, CA USA Phone: Fax: sales@silvaco.com Worldwide Offices: Silvaco Japan jpsales@silvaco.com Silvaco Korea krsales@silvaco.com Silvaco Taiwan twsales@silvaco.com Silvaco Singapore sgsales@silvaco.com Silvaco Europe eusales@silvaco.com July, August, September 2015 Page 15 The Simulation Standard