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A parallel multigrid solver applied to the simulation of thermal oxidation and diffusion processes
Computational Materials Science 11 Ž1998. 105–108
A parallel multigrid solver applied to the simulation of thermal oxidation and diffusion processes M.G. Hackenberg a , W. Joppich a
a,)
, T. Sontowski a , S. Mijalkovic´
b
GMD – German National Research Center for Information Technology, Sankt Augustin, Germany b Faculty of Electronic Engineering, UniÕersity of Nis, ˇ Nis, ˇ YugoslaÕia
Abstract The complexity of the future processing models and the need for three-dimensional simulation require both efficient numerical methods and fast computers. In this paper, a general parallel PDE solver is presented, which has been adapted to the solution of so-called coupled problems. The numerical solution strategy exploits the optimal behavior of multigrid methods and the grid partitioning for an easy parallelization. As the typical thermal processes in VLSI process simulation can be considered as a coupled problem, the program has been applied to this important class of applications. The flow of the silicon dioxide is modeled as viscous flow. The impurity diffusion takes oxidation enhancement into account. q 1998 Elsevier Science B.V.
1. Introduction Many technical problems belong to the class of so-called coupled problems. A very interesting example is the movement of cardiac valves and their effect on the flow of the blood, and as a consequence of this flow the effect on the properties of the blood. Of course, there are many other important examples, for instance in the design of modern airplanes or ships. Coupled problems are characterized by several typical features. The single subproblems are often described by completely different models, for instance one part of the problem is described by
)
Corresponding author.
models from computational fluid dynamics, whereas another part of the problem comes from structural mechanics. As a consequence, the solution techniques and the data structures used may differ for each part. This principle is shown in Fig. 1. In many cases the computational sub-domains are time-dependent and there are moving interfaces between the parts of the different subproblems. Existing software usually cannot be used for such problems. Most packages are specialized for a certain class of problems. Therefore, it is important to provide interfaces which allow the coupling of application codes. Several projects worldwide work on such problems. Our project has chosen the thermal oxidation of silicon and the diffusion of dopants in silicon from VLSI process simulation in order to study coupling strategies and especially their numerical behavior. This application is sufficiently com-
0927-0256r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 Ž 9 7 . 0 0 2 0 0 - 0
106
M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
Fig. 1. Possible grid structures for coupled applications and the idea of coupling.
plex, but as explained below, it can be considered as a special case.
2. Oxidation and diffusion models In the oxide region, V O in Fig. 2, the transport of the oxidant is modeled by a stationary diffusion equation. The growing oxide V O , a highly viscous material w1x, is described by the Stokes equations.
The diffusion of dopant material both in V O and V S is modeled with the help of non-linear diffusion equations w3x. If the mask material V M is considered, too, the movement of this material is described by the Stokes equations, again. The geometries are time-dependent. Two additional equations are solved on each sub-domain for the creation of new boundary fitted grids. Therefore, seven partial differential equations have to be solved in the silicon-dioxide, five within the masking area and only three equations have to be considered in the silicon. Obviously, this type of unbalanced numerical work requires special strategies for the parallel solution. The boundary conditions are rather complicated and incorporate normal fluxes, tangential and normal velocities and even the curvature of the free boundary Gf . At the interface G V between silicon and silicon dioxide the concentration of dopants has to satisfy the segregation condition. The implementation of the models has been verified by comparing the computational results to both analytical solutions of model problems and to results obtained by simulations using PROMIS of the Technical University of Vienna w3x. As shown in w2x the agreement is excellent.
3. Modifications of the software for coupled problems
Fig. 2. Geometry of the bird’s beak application.
As each subproblem can be solved on structured grids, it is appropriate to use a general PDE-solver for the solution and to modify it in such a way, that an internal treatment of coupled problems with all the necessary features becomes possible. Parallel processing for future applications is indispensable. Therefore the parallel multigrid environment L i SS w5x developed at the GMD, has been modified appropriately. L i SS solves partial differential equations on general 2D-domains using boundary fitted block structured grids. The discretization on these grids uses finite volume techniques. The core of the solver is a parallel multigrid algorithm. Multigrid methods w4x are asymptotically optimal methods: the numerical work for the solution is linearly proportional to the number of grid points. The parallelization follows the idea of grid partitioning and distributes the
M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
work corresponding to a block among the different parallel processes. The blocks are enlarged by an overlap area which has to be updated regularly within the parallel solution. This communication between the processes is simplified using a library of special communication subroutines for grid oriented problems. The use of structured grids for each single part of the application avoids the technically complicated interpolation at the interface. Due to the oxidation of the silicon, the computational domain is time-dependent and there exists a moving interface between the different materials. At this interface a coupling condition for the doping material, the segregation condition, holds. The multigrid method allows a large number of coupling strategies to be investigated. It is possible to perform the coupling on the finest grid at the end of the time-steps, on the finest grid after each multigrid cycle, or within the multigrid algorithm even on the complete hierarchy of grid-levels. The chosen application requires some features which are also of interest for the general situation. Due to the treatment of time-dependent geometries and free boundaries the possible penetration of boundary segments Žcontact problem. has to be avoided. This requirement especially arises when geometries are used as they are typical for trench oxidation. For the parallel computation the almost equal distribution of numerical work determines the efficiency of the approach. Within our approach the computational domain of each subproblem is subdi-
107
vided into as many blocks as there are parallel processes within the solution. Each process has to work on one part of all subproblems. In order to avoid an external grid generation for each time-step the grid generation has been integrated into the solver. For any complex application the proper choice of the multigrid components is essential. With full weighing of the residuals both for internal grid points and for boundary points combined with alternating line relaxation the convergence rate for the oxidation problem ŽStokes equations. using a cheap VŽ1, 1.cycle is 0.07.
4. Application The described principles have been applied to typical situations in today’s process simulation applications. In w2x the oxidation of a planar silicon domain which is covered by a thin initial layer of oxide has been presented. This application has been extended to the coupled calculation including a nitride mask on top of the oxide. The recent improvement concerns the parallel contact treatment which is of special interest for trench geometries. Especially at the corners of the trench a penetration of the free boundaries is possible. This is shown in the left part of Fig. 3. The parallel contact treatment provides a free surface which is free of collisions. Then a re-meshing is performed using standard grid genera-
Fig. 3. No contact treatment Žleft. and the benefit of contact treatment and re-meshing Žright..
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M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
Fig. 4. Initial boron concentration Žleft. and after 30 min Žright..
tion techniques. The benefit of this approach is obvious, see the right part of Fig. 3. The coupled simulation of oxidation and diffusion starts from a rather realistic trench geometry. The initial profile has been produced using the Monte Carlo implantation module of PROMIS with a tilt angle of 78, an energy of 70 keV and a dose of 10 13 cmy2 . The initial situation and the state after 30 min. are given in Fig. 4, left and right, respectively. The 6 blocks for each sub-domain indicate the use of six parallel processes for this test case. After 30 min the trench oxide has changed its shape considerably. Nevertheless, the changing grid has not influenced the block structure of the complete application. The segregation condition causes a jump of the concentration at the interface.
5. Conclusion It has been shown that a general parallel PDE solver has been modified to solve coupled problems. Although this program will not be a process simulator, coupled thermal processes of VLSI fabrication are solved successfully. For the bird’s beak simulation three different sub-problems Žnitride, oxide and silicon. have been solved in a coupled way. The
trench oxidation has been used to demonstrate the proper treatment of free surfaces.
Acknowledgements This work was supported within the GRISSLi project by the German Federal Ministry for Research and Education under contract no. 01IS512C.
References w1x D. Chin, S.Y. Oh, S.M. Hu, R.W. Dutton, J.L. Moll, Two-dimensional oxidation, IEEE Trans Electron Dev. 30 Ž7. Ž1983. 744–749. w2x M. G. Hackenberg, W. Joppich, S. Mijalkovic, ´ T. Sontowski, Simulation of Thermal Oxidation and Diffusion Processes By Parallel PDE Solver LiSS. Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices – SISPAD’ 96, Also GMD-Arbeitspapier No. 1039. w3x G. Hobler, P. Pichler, K. Wimmer, PROMIS 1.6: User’s guide, Technical report, Technical University Vienna, Vienna, June 1991. w4x W. Joppich, S. Mijalkovic, ´ Multigrid methods for process simulation, Springer-Verlag, Wien, 1993. w5x H. Ritzdorf, A. Schuller, B. Steckel, K. Stuben, Li SS – An ¨ ¨ environment for the parallel multigrid solution of partial differential equations on general 2D domains, Parallel Comput. 20 Ž1994. 1559–1570.
A parallel multigrid solver applied to the simulation of thermal oxidation and diffusion processes M.G. Hackenberg a , W. Joppich a
a,)
, T. Sontowski a , S. Mijalkovic´
b
GMD – German National Research Center for Information Technology, Sankt Augustin, Germany b Faculty of Electronic Engineering, UniÕersity of Nis, ˇ Nis, ˇ YugoslaÕia
Abstract The complexity of the future processing models and the need for three-dimensional simulation require both efficient numerical methods and fast computers. In this paper, a general parallel PDE solver is presented, which has been adapted to the solution of so-called coupled problems. The numerical solution strategy exploits the optimal behavior of multigrid methods and the grid partitioning for an easy parallelization. As the typical thermal processes in VLSI process simulation can be considered as a coupled problem, the program has been applied to this important class of applications. The flow of the silicon dioxide is modeled as viscous flow. The impurity diffusion takes oxidation enhancement into account. q 1998 Elsevier Science B.V.
1. Introduction Many technical problems belong to the class of so-called coupled problems. A very interesting example is the movement of cardiac valves and their effect on the flow of the blood, and as a consequence of this flow the effect on the properties of the blood. Of course, there are many other important examples, for instance in the design of modern airplanes or ships. Coupled problems are characterized by several typical features. The single subproblems are often described by completely different models, for instance one part of the problem is described by
)
Corresponding author.
models from computational fluid dynamics, whereas another part of the problem comes from structural mechanics. As a consequence, the solution techniques and the data structures used may differ for each part. This principle is shown in Fig. 1. In many cases the computational sub-domains are time-dependent and there are moving interfaces between the parts of the different subproblems. Existing software usually cannot be used for such problems. Most packages are specialized for a certain class of problems. Therefore, it is important to provide interfaces which allow the coupling of application codes. Several projects worldwide work on such problems. Our project has chosen the thermal oxidation of silicon and the diffusion of dopants in silicon from VLSI process simulation in order to study coupling strategies and especially their numerical behavior. This application is sufficiently com-
0927-0256r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 9 2 7 - 0 2 5 6 Ž 9 7 . 0 0 2 0 0 - 0
106
M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
Fig. 1. Possible grid structures for coupled applications and the idea of coupling.
plex, but as explained below, it can be considered as a special case.
2. Oxidation and diffusion models In the oxide region, V O in Fig. 2, the transport of the oxidant is modeled by a stationary diffusion equation. The growing oxide V O , a highly viscous material w1x, is described by the Stokes equations.
The diffusion of dopant material both in V O and V S is modeled with the help of non-linear diffusion equations w3x. If the mask material V M is considered, too, the movement of this material is described by the Stokes equations, again. The geometries are time-dependent. Two additional equations are solved on each sub-domain for the creation of new boundary fitted grids. Therefore, seven partial differential equations have to be solved in the silicon-dioxide, five within the masking area and only three equations have to be considered in the silicon. Obviously, this type of unbalanced numerical work requires special strategies for the parallel solution. The boundary conditions are rather complicated and incorporate normal fluxes, tangential and normal velocities and even the curvature of the free boundary Gf . At the interface G V between silicon and silicon dioxide the concentration of dopants has to satisfy the segregation condition. The implementation of the models has been verified by comparing the computational results to both analytical solutions of model problems and to results obtained by simulations using PROMIS of the Technical University of Vienna w3x. As shown in w2x the agreement is excellent.
3. Modifications of the software for coupled problems
Fig. 2. Geometry of the bird’s beak application.
As each subproblem can be solved on structured grids, it is appropriate to use a general PDE-solver for the solution and to modify it in such a way, that an internal treatment of coupled problems with all the necessary features becomes possible. Parallel processing for future applications is indispensable. Therefore the parallel multigrid environment L i SS w5x developed at the GMD, has been modified appropriately. L i SS solves partial differential equations on general 2D-domains using boundary fitted block structured grids. The discretization on these grids uses finite volume techniques. The core of the solver is a parallel multigrid algorithm. Multigrid methods w4x are asymptotically optimal methods: the numerical work for the solution is linearly proportional to the number of grid points. The parallelization follows the idea of grid partitioning and distributes the
M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
work corresponding to a block among the different parallel processes. The blocks are enlarged by an overlap area which has to be updated regularly within the parallel solution. This communication between the processes is simplified using a library of special communication subroutines for grid oriented problems. The use of structured grids for each single part of the application avoids the technically complicated interpolation at the interface. Due to the oxidation of the silicon, the computational domain is time-dependent and there exists a moving interface between the different materials. At this interface a coupling condition for the doping material, the segregation condition, holds. The multigrid method allows a large number of coupling strategies to be investigated. It is possible to perform the coupling on the finest grid at the end of the time-steps, on the finest grid after each multigrid cycle, or within the multigrid algorithm even on the complete hierarchy of grid-levels. The chosen application requires some features which are also of interest for the general situation. Due to the treatment of time-dependent geometries and free boundaries the possible penetration of boundary segments Žcontact problem. has to be avoided. This requirement especially arises when geometries are used as they are typical for trench oxidation. For the parallel computation the almost equal distribution of numerical work determines the efficiency of the approach. Within our approach the computational domain of each subproblem is subdi-
107
vided into as many blocks as there are parallel processes within the solution. Each process has to work on one part of all subproblems. In order to avoid an external grid generation for each time-step the grid generation has been integrated into the solver. For any complex application the proper choice of the multigrid components is essential. With full weighing of the residuals both for internal grid points and for boundary points combined with alternating line relaxation the convergence rate for the oxidation problem ŽStokes equations. using a cheap VŽ1, 1.cycle is 0.07.
4. Application The described principles have been applied to typical situations in today’s process simulation applications. In w2x the oxidation of a planar silicon domain which is covered by a thin initial layer of oxide has been presented. This application has been extended to the coupled calculation including a nitride mask on top of the oxide. The recent improvement concerns the parallel contact treatment which is of special interest for trench geometries. Especially at the corners of the trench a penetration of the free boundaries is possible. This is shown in the left part of Fig. 3. The parallel contact treatment provides a free surface which is free of collisions. Then a re-meshing is performed using standard grid genera-
Fig. 3. No contact treatment Žleft. and the benefit of contact treatment and re-meshing Žright..
108
M.G. Hackenberg et al.r Computational Materials Science 11 (1998) 105–108
Fig. 4. Initial boron concentration Žleft. and after 30 min Žright..
tion techniques. The benefit of this approach is obvious, see the right part of Fig. 3. The coupled simulation of oxidation and diffusion starts from a rather realistic trench geometry. The initial profile has been produced using the Monte Carlo implantation module of PROMIS with a tilt angle of 78, an energy of 70 keV and a dose of 10 13 cmy2 . The initial situation and the state after 30 min. are given in Fig. 4, left and right, respectively. The 6 blocks for each sub-domain indicate the use of six parallel processes for this test case. After 30 min the trench oxide has changed its shape considerably. Nevertheless, the changing grid has not influenced the block structure of the complete application. The segregation condition causes a jump of the concentration at the interface.
5. Conclusion It has been shown that a general parallel PDE solver has been modified to solve coupled problems. Although this program will not be a process simulator, coupled thermal processes of VLSI fabrication are solved successfully. For the bird’s beak simulation three different sub-problems Žnitride, oxide and silicon. have been solved in a coupled way. The
trench oxidation has been used to demonstrate the proper treatment of free surfaces.
Acknowledgements This work was supported within the GRISSLi project by the German Federal Ministry for Research and Education under contract no. 01IS512C.
References w1x D. Chin, S.Y. Oh, S.M. Hu, R.W. Dutton, J.L. Moll, Two-dimensional oxidation, IEEE Trans Electron Dev. 30 Ž7. Ž1983. 744–749. w2x M. G. Hackenberg, W. Joppich, S. Mijalkovic, ´ T. Sontowski, Simulation of Thermal Oxidation and Diffusion Processes By Parallel PDE Solver LiSS. Proceedings of the International Conference on Simulation of Semiconductor Processes and Devices – SISPAD’ 96, Also GMD-Arbeitspapier No. 1039. w3x G. Hobler, P. Pichler, K. Wimmer, PROMIS 1.6: User’s guide, Technical report, Technical University Vienna, Vienna, June 1991. w4x W. Joppich, S. Mijalkovic, ´ Multigrid methods for process simulation, Springer-Verlag, Wien, 1993. w5x H. Ritzdorf, A. Schuller, B. Steckel, K. Stuben, Li SS – An ¨ ¨ environment for the parallel multigrid solution of partial differential equations on general 2D domains, Parallel Comput. 20 Ž1994. 1559–1570.