Institute for Mathematics and its Applications

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1 Institute for Mathematics and its Applications University of Minnesota 400 Lind Hall, 207 Church St. SE, Minneapolis, MN Numerical Simulation and Control of Sublimation Growth of SiC Bulk Single Crystals: Modeling, Finite Volume Method, Analysis and Results Peter Philip Show & Tell at IMA Minneapolis, September 15, 2004

2 Joint work with: Jürgen Geiser, Olaf Klein, Jürgen Sprekels, Krzysztof Wilmański (Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin) (modeling, finite volume method) Christian Meyer, Fredi Tröltzsch (TU Berlin, Department of Mathematics) (optimal control) Cooperation with: Klaus Böttcher, Detev Schulz, Dietmar Siche (Institute of Crystal Growth (IKZ), Berlin) (growth experiments) Supported by: Research Center Matheon: Mathematics for Key Technologies of the German Science Foundation (DFG) ( , extension until 2010 pending) German Ministry for Education and Research (BMBF) ( )

3 SiC growth by physical vapor transport (PVT) SiC seed crystal K insulated graphite crucible SiC source powder coil for induction heating polycrystalline SiC powder sublimates inside induction-heated graphite crucible at K and 20 hpa a gas mixture consisting of Ar (inert gas), Si, SiC 2, Si 2 C,... is created an SiC single crystal grows on a cooled seed

4 Goal: Stationary and transient optimal control of process, using mathematical modeling, numerical simulation. Heat Transport Model Nonlinear heat conduction in material j: ε j t + div q j = f j, q j = κ j T, ε j : internal energy, T : absolute temperature, q j : heat flux, κ j : thermal conductivity, f j : power density of heat sources (induction heating). Interface Conditions Continuity of the heat flux: Between solids: q j1 n j1 = q j2 n j1 on γ j1,j 2. Between gas and solid j: q gas n gas R+J = q j n gas on γ j,gas, n j, n gas : outer unit normal, R: radiosity, J: irradiation. Continuity of temperature throughout apparatus.

5 Outer Boundary Conditions Emission according to Stefan-Boltzmann law: (κ j T ) n j = σɛ j (T ) ( T 4 Troom) 4, ɛ j : emissivity, T room = 293 K. On surfaces of open cavities: q j n j R + J = 0.

6 Finite Volume Scheme General theory for finite volume methods for systems of nonlinear evolution equations in space-time domains Ω [0, t f ], Ω = N j=1 Ω j, with disjoint polytopes (bounded polyhedral sets) Ω j. Type and/or form of the PDEs may vary from subdomain to subdomain. Typical form: t b j (u j, x, t) + v j (u j, x, t) (k j (u j, x, t) u j ) = f j (u j, x, t). The unknown functions u j on Ω j are connected by interface conditions between adjacent subdomains. Typical example: u j = T j = temperature on the subdomain Ω j (gas, different solid components of the growth apparatus). Possible interface conditions between Ω j1, Ω j2 : u j1 = u j2 (continuity) k j1 (u j1, x, t) u j1 n pj1 = ξ {j1,j 2 } (u j1 u j2 ), ξ {j1,j 2 } > 0 (jump condition) k j2 (u j2, x, t) u j2 n pj2 k j1 (u j1, x, t) u j1 n pj1 = A γ (u 1,..., u N )(x) (nonlocal operator; typically: radiation)

7 Outer boundary conditions (on Ω): Dirichlet, Neumann, Robin, emission, nonlocal radiation. Discrete Existence Result: Assume that: b j 0, b j (, x, t), b j (0, x, ) ; L > 0 : b j (u, x, t) b j (ũ, x, t) L u ũ j. f j (, x, t) locally Lipschitz; f j (0, x, t) 0. k j (, x, t) locally Lipschitz; k j 0. Functions in interface and boundary conditions are locally Lipschitz and have the right monotonicity properties (valid for heat conduction). v(u, x, t) = v 1 (u, x, t) v 2 (x, t), where v 1 (0, x, t) = 0, v 1 (, x, t), v 1 is locally Lipschitz and bounded from below. The discretization of nonlocal operators satisfies a technical condition (satisfied for suitable discretization of radiation operators). Then the finite volume discretization has a unique solution in [0, M] n, provided that the time step is sufficiently small (n: number of discrete unknowns, M: independent of time discretization).

8 Stationary optimal control problem for the temperature field Γ SiC-C Ω gas Ω SiC-S Known fact: Crystal surface forms along isotherms. Goal: Radially constant isotherms during growth. R ş T ť 2 Control: w(z) r (r, z) d(r, z) min. Ω gas PDEs (v gas = 0, f(x, T, P ) = f(x, P )): div ą κ (Ar) (T ) T ć = 0 in Ω gas, div ą κ(x, T ) T ć = f(x, P ) in Ω \ Ω gas. Constraints: T room T T max in Ω, T min,sic-c T T max,sic-c on Γ SiC-C (need right polytype), T ΩSiC-S T ΓSiC-C +δ, δ > 0 (source temp. seed temp. +δ), 0 P P max (bounds for heating power P (control parameter)).

9 Numerical results: Optimization of temperature field (a): T (P = 10.0 kw, z rim = 24.0 cm, f = 10.0 khz) SiC crystal 3002 K 3007 K 3012 K 3022 K 3042 K SiC powder (b): T (P = 7.98 kw, z rim = 22.7 cm, f = 165 khz) Nelder-Mead res. for F r,2 (T ) SiC crystal 2304 K SiC powder 2314 K 2334 K (c): T (P = 10.3 kw, z rim = 12.9 cm, f = 84.9 khz), Nelder-Mead res. for F r,2(t ) F z,2 (T ) 2 SiC powder SiC crystal 2299 K 2304 K 2314 K 2324 K 2364 K

10 Selected Publications O. KLEIN, P. PHILIP: Transient conductive-radiative heat transfer: Discrete existence and uniqueness for a finite volume scheme, accepted for publication in Mathematical Models and Methods in Applied Sciences. O. KLEIN, P. PHILIP, J. SPREKELS: Modeling and simulation of sublimation growth of SiC bulk single crystals, Interfaces and Free Boundaries 6 (2004), P. PHILIP: Transient Numerical Simulation of Sublimation Growth of SiC Bulk Single Crystals. Modeling, Finite Volume Method, Results. Humboldt University of Berlin, Report No. 22, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin. More Publications / Information: philip/sic/#publications philip/sic/ Extended 1-hour talk tomorrow, Applied Mathematics and Numerical Analysis Seminar, School of Mathematics Thu, Sep 16, 11:15 a.m., Vincent Hall 570.