Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation
Title Development of efficient parallel methods for the simulation and optimization of CVD processes
Participiants Thomas Schiekofer Michael Griebel
Keywords CVD, Finite Differences, Parallelisation, Sparse Grids, SMP
Description This BMBF-Project is a collaboration of the University of Erlangen-Nürnberg (Prof. Dr. Dr. h.c. F. Durst, Institute of Fluid Mechanics, Technical Faculty), the University of Bonn (Prof. Dr. Michael Griebel , Department of Applied Mathematics, Division for Scientific Computing and Numerical Simulation), the Technical University of Munich (Prof. Dr. A. Bode, LRR, Department of Informatics) and AIXTRON AG in Aachen.

The aim of this project is the develoment of mathematical models, efficient numerical techniques and modern software tools for the study and optimization of the growth of layers of compound semiconductor of AlxGa1-xAs, GaxIn1-xP and InxGa1-xN obtained by MOVPE (Metalorganic Vapor Phase Epitaxy). MOVPE is a special kind of CVD where single crystal thin films of compound semiconductor layers with metal organic compounds as basic ingrediences are produced. Fields of applications of this technique are, e.g., the production of microelectronic and opto-electronic components which are based on III-V-semiconductor layers such as GaAs or InP (Recently, the development of blue light emitting LED's and blue lasers using MOVPE based on (Al,Ga,In)N rose interest amongst experts on this field). For a more detailed introduction in this topic compare [3], [4] [9] and [10].

There exists some different types of CVD reactor which are different in their behaviour concerning fluid flow and chemical reactions. Widespread in the application of MOVPE and also the subject of this BMBF-project are linear horizontal reactors and radial symmetric, horizontal multiwaver reactors such as the AIX 2400/2600G3. A rough sketch of these types of reactors can be found in Figure 1.

Figure 1: types of reactors for MOVPE: (a) linear horizontal reactor, (b) radial symmetric, horizontal multiwaver reactor (compare also [4])

From a mathematical point of view we have to deal with laminar fluid flow of a chemically reacting mixture of gas with strongly coupled natural and enforced convection. Here, we face low Reynold numbers (about 1-100), low Mach numbers (about 10,-2), average to high Rayleigh numbers (about 102 to 105) and high temperature gradients. The equations of interest are the unsteady Navier Stokes equations together with equations for the chemical compounds (compare [1], [4] and [9]):

Standard discretizations of the above equations normally use full grids where the total number of grid points is of the order O(h3). Exploiting the benefits of sparse grids leads to grids that possess O(h (log h-1)2) grid points and hence have substantially less grid points compared to full grids. Nevertheless, the accuracies on sparse and full grids measured in different norms are compareable (for a more detailed desription, compare those Related Projects that deal with Finite Differences on Sparse Grids at the bottom of this page). To study the impact of the rotation of the susceptor within numerical simulations we focused the following problem: fluid was flowing through a linear horizontal CVD reactor and was heated from below. Due to the rotation of the susceptor there are jumps in the boundary condition for the velocities. Due to the rotation a non-symmetric temperature distribution as well as non-symmetric velocities are obtained. Within the numerical simulation the jumps of the velocities have to be resolved which can be done either by a kind of deformed full grid or adaptivity. In our example, we use adaptively refined sparse grids. A rough sketch of the CVD reactor with the domain of interest as well as the used adaptive sparse grid can be seen in Figures 2 and 3.

Figure 2: sketch of the CVD reactor

Figure 3: adaptive sparse grid of the purple marked region of the CVD reactor

In Figure 4 the temperature distribution 20% above the susceptor can be seen. Moreover, Figure 5 shows the impact of the rotation of the susceptor to the u-velocity which is the velocity in the main direction of the fluid flow.

Figure 4: temperature distribution 20% above the susceptor

Figure 5: impact of the rotation of the susceptor to the u-velocity

  • [1] R. B. Bird, W. E. Steward and E. N. Lightfood, Transport phenomena, John Wiley & Sons, New York, 1960
  • [2] A. J. Chorin , Numerical Solution of the Navier Stokes Equations, J. Math. Comput., Vol. 22, pp 745-762, 1968
  • [3] M. Dauelsberg , Entwicklung mathematischer Modelle und numerische Simulation der MOCVD von Verbindungshalbleitern, Diplomarbeit, Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, 1995
  • [4] M. Dauelsberg, F. Durst, L. Kadinski, Y. Makarov, G. Strauch, H. Jürgensen, T. Schiekofer, M. Griebel, A. Bode, P. Luksch and M. May, PAR-CVD: Entwicklung leistungsfähiger paralleler Berechnungsverfahren zur Untersuchung und Optimierung von CVD-Prozessen, in Proceedings der BMWF-Tagung HPSC97, München, 1997
  • [5] J. H. Ferzinger and M. Peric, Computational Methods for Fluid Dynamics, Springer, Berlin/Heidelberg 1996
  • [6] P. M. Gresho and R. L. Sani, On Pressure Boundary Conditions for the Incompressible Navier-Stokes-Equations, International Journal for Numerical Methods in Fluids, also: Lawrence Livermore National Laberatory, Preprint UCRL-96471
  • [7] M.Griebel, , Adaptive sparse grid multilevel methods for elliptic PDEs based on finite differences, Computing, 1998
  • [8] M.Griebel, T. Schiekofer An adaptive sparse grid Navier Stokes solver in 3D based on the finite difference approach, Proceedings of ENUMATH 97, Wiley, 1998 (to appear)
  • [9] L. Kadinski, Mathematische Modellierung und numerische Simulation von CVD-Prozessen in der Halbleitertechnik, Dissertation, Universität Erlangen-Nürnberg, 1996
  • [10] C. R. Kleijn, Chemical Vapor Deposition Processes, in M. Meyyappen, Computational modelling in semiconductor processing, Artech House, 1995, chapter 4, pp. 97
  • [11] T. Schiekofer, Die Methode der Finiten Differenzen auf Dünnen Gittern zur adaptiven Lösung partieller Differentialgleichungen, Dissertation Universität Bonn, Institut für Angewandte Mathematik, to appear 1998
  • [12] T. Schiekofer and M. May, An Abstract Data Type for Parallel Simulations based on Sparse Grids, in Proceedings of the Third European PVM Conference, Munich, Germany, October 7-9th 1996, Springer Verlag, Lecture Notes in Computer Science, Vol. 1156, 1997
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