Proje t C9Numeri al simulation and ontrol ofsublimation growth of semi ondu tor bulksingle rystalsO. Klein, P. Philip, J. Sprekels, F. Tröltzs h, I. YouseptDFG Resear h Center MATHEONMathemati s for key te hnologiesC-Day, Berlin, May 21, 2007

Appli ations of Semi ondu tor CrystalsLight-emitting diodes:Lifetime: ≈ 10 yearsLight extra tion e ien y > 32 %(light bulb: ≈ 10 %) Blue laser:Its use in DVD players admits up to10-fold apa ity of dis SiC-based ele troni s still works at600 C; SiC sensors pla ed lose to ar engines save resour es and osts 2 / 11

Physi al Vapor Transport MethodSiCseed rystalHHHHHj20003000K HHHHHj

SiCsour epowder

insu-latedgraphite ru ible

oil forindu tionheatingAAAK

⊲ poly rystalline SiC powdersublimates insideindu tion-heated graphite ru ible at 2000 3000 Kand ≈ 20 hPa⊲ a gas mixture onsisting ofAr (inert gas), Si, SiC2,Si2C, . . . is reated⊲ an SiC single rystal growson a ooled seed 3 / 11

Software Produ t WIAS-HiTNIHSComputes and optimizes temperature and magneti elds inaxisymmetri apparatus.⊲ Computation of temp T a ounts foranisotropi ondu tion [Geiser, Klein,Philip, 2006/7, radiation, andele tromagneti heating.⊲ Numeri al optimization of T eld ingrowth apparatus:

Small radial T gradient on rystalsurfa e avoids defe ts. Large verti al T gradient betweensour e and rystal in reases growthrate. State onstraints: Need pres ribed Trange on seed, sour e, and apparatus.

Numeri al Results:Optimization of Temperature Field(a): Generic (Unoptimized) Temperature Field

SiC crystal

SiC source powder

3002 K

3007 K

3012 K

3022 K

3042 K

(b): Minimized Radial Gradient on Crystal Surface

SiC crystal

SiC source powder

2304 K

2314 K

2334 K

(c): Minimized Radial Gradient on Crystal Surface

& Maximized Vertical Gradient Between Source and Seed

SiC crystal

SiC source powder

2299 K

2304 K

2314 K

2324 K

2364 K4 / 11

Model analysis⊲ A fairly simplied model for the seeded sublimation growth geometry:

Γr Γ0Ωg

Ωs

⊲ Optimization of the gradient temperature ∇y in the gas phase Ωg by ontrolling the heatsour e u in the solid phase Ωs :(P) minimize J(u, y) :=

12Z

Ωg |∇y − yd |2 dx +β2 Z

Ωs u2 dx .

⊲ The temperature distribution y is given by the solution of the stationary heat equation:(SL)

8

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:

−div(κs ∇y) = u in Ωs−div(κg ∇y) = 0 in Ωg

κg (∂y∂nr )g − κs( ∂y

∂nr )s = Gσ |y |3y on Γrκs ∂y

∂n0 + εσ |y |3y = εσ y40 on Γ0. 5 / 11

⊲ We impose inequality state onstraints to avoid melting in Ωs and to ensure sublimation inΩg : y(x) ≤ ym(x) a.e. in Ωs ,ya(x) ≤ y(x) ≤ yb(x) a.e. in Ωg .

⊲ Additionally, we onsider the following ontrol- onstraints:ua(x) ≤ u(x) ≤ ub(x) a.e. in Ωswhere ua and ub ree t the minimum and maximum heating power.Theorem (C. Meyer, J. Rehberg and I. Yousept, 2007)For every u ∈ L2(Ωs ), the state equation (SL) admits a unique solutiony = y(u) ∈ H1(Ω) ∩ C(Ω) and there exists a onstant > 0 independent of u su h that‖y‖H1(Ω) + ‖y‖C(Ω) ≤ `1 + ‖u‖L2(Ωs ) + ‖u‖4L2(Ωs )). 6 / 11

⊲ Based on the ontinuity of y , we established rst-order ne essary and se ond-order su ient onditions for (P).⊲ Lagrange multipliers asso iated to the pointwise state onstraints of (P) are in general Borelmeasures ⇒ Regularization is ne essary.⊲ Utilizing a "Moreau-Yosida" type regularization to the optimal ontrol problem (P):

(Pγ)

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:

minu∈L2(Ωs ) f (u) := J(u, y(u)) + γ2 (‖max(0, y(u) − yb)‖2L2(Ωg )

+‖max(0, ya − y(u))‖2L2(Ωg )+ ‖max(0, y(u) − ym)‖2L2(Ωs )

),subje t to ua(x) ≤ u(x) ≤ ub(x) a.e. in Ωs .Theorem (C. Meyer and I. Yousept, 2007)Let u be a lo al solution of (P) satisfying the se ond-order optimality onditions. Then, there exists a sequen e (uγ)γ>0 of lo al solutions to (Pγ) onverging strongly in L2(Ωs)to u as γ → ∞. 7 / 11

Numeri al resultFigure: Control uh Figure: State yh

Figure: Lagrange multiplier µah Figure: Lagrange multiplier µ

bh 8 / 11

Further resear hFurther resear h⊲ In luding Maxwell's equations in the model analysis.⊲ Study of optimal ontrol of indu tion heating based on theMaxwell's equations: First- and se ond-order optimality onditions,numeri al analysis and numeri al simulation.

9 / 11

Sele ted Publi ations⊲ M. Hintermüller, F. Tröltzs h, I. Yousept: Mesh-independen e of semismoothNewton methods for Lavrentiev-regularized state onstrained nonlinear optimal ontrol problems, submitted.⊲ F. Tröltzs h, I. Yousept: Lavrentiev type regularization for optimal boundary ontrol problems with pointwise state onstraints, submitted.⊲ J. Geiser, O. Klein, P. Philip: Numeri al simulation of heat transfer inmaterials with anisotropi thermal ondu tivity: A nite volume s heme tohandle omplex geometries, submitted.⊲ J. Rehberg, C. Meyer, I. Yousept: State- onstrained optimal ontrol ofsemilinear ellipti equations with nonlo al radiation interfa e onditions,submitted.⊲ J. Geiser, O. Klein, P. Philip: Transient numeri al study of temperaturegradients during sublimation growth of SiC: Dependen e on apparatus design,J. Crystal Growth 297 (2006), 20 32. 10 / 11

Sele ted Publi ations⊲ J. Geiser, O. Klein, P. Philip: Numeri al simulation of temperature eldsduring the sublimation growth of SiC single rystals, using WIAS-HiTNIHS,J. Crystal Growth 303 (2007), 352 356.⊲ C. Meyer, I. Yousept: Optimal ontrol of temperature distribution in seededsublimation growth pro esses of semi ondu tor single rystal, submitted.⊲ F. Tröltzs h, I. Yousept: A regularization method for the numeri al solution ofellipti boundary ontrol problems with pointwise state onstraints, Comp.Opt. Appl. 2007 (in press).⊲ J. Geiser, O. Klein, P. Philip: Inuen e of anisotropi thermal ondu tivity inthe apparatus insulation for sublimation growth of SiC: Numeri al investigationof heat transfer, Crystal Growth & Design 6 (2006), 2021 2028.⊲ C. Meyer, P. Philip, F. Tröltzs h: Optimal Control of a Semilinear PDE withNonlo al Radiation Interfa e Conditions, SICON 45 (2006), 699 721.⊲ C. Meyer, P. Philip: Optimizing the temperature prole during sublimationgrowth of SiC single rystals: Control of heating power, frequen y, and oilposition, Crystal Growth & Design 5 (2005), 1145 1156. 11 / 11