Multiscale Modeling of Low Pressure Chemical Vapor Deposition K. Kumar, I.S. Pop 07-April-2010 1/30
Motivation 2/30
Silicon Wafer 3/30
Trenches in wafer 4/30
Motivation JJ J N I II /centre for analysis, scientific computing and applications 5/30
Trench Geometry and Deposition JJ J N I II /centre for analysis, scientific computing and applications 6/30
Trench Geometry:Modeling t u + (qu D u) = 0 in Ω ν D u = Ku Γ at Γ Γ Interface Ω t u T + (qu T D T u T ) = 0 in Ω T ν D T u T = Ku T at Γ T Ω T Γ T 7/30
Equations: Trench Geometry t u + (qu D u) = 0 t u T + (qu T D T u T ) = 0 ν D u = r(u, w) ν D T u T = r(u T, w) t w = r(u, w, u T ) at Ɣ in at Ɣ T at Ɣ Ɣ T in T ν (qu D u) = ν (qu T D T u T ); and u = u T at Ɣ inter f ace 8/30
Multiscale Computation Schematics for the multiscale computation 9/30
Macroscale Geometry 10/30
Macroscale Geometry 11/30
Intermediate Scale: Mesoscale 12/30
Intermediate Scale: Mesoscale 13/30
Microscale: trench Geometry 14/30
Experimental versus simulation results 15/30
Experimental versus simulation results 16/30
Further Approach 17/30
Free Boundary Formulation u ε ε t = (D u ε q ε u ε ), (C-D eqn) (t)(1) (ρq) = 0, in ε (t) (Cont. eqn) ρ t q + ρq q = µ q P. (Momentum eqn) For the evolution of the free boundary St ε + V S = 0, in g(w, V, H) = 0, in Ɣ ε (t), (Surface evolution, curvature and normal velocity) Here, Ɣ := {x : S(x) = 0} characterizes the free boundary. 18/30
Free Boundary Evolution for 30 µm trench 19/30
Free Boundary Evolution for 30 µm trench 20/30
Free Boundary Evolution for 10 µm trench 21/30
Free Boundary Evolution for 10 µm trench 22/30
Upscaling of the Boundary condition ν D u = ku ν D u = ku k = k 1 0 f (x, ξ)dξ; where f (x, ξ) describes the periodically oscillating boundary curve. 23/30
Upscaling of the Boundary condition Γ t Γ w Ω ε y x Γ a 24/30
Asymptotic Analysis of CVD model on a microstructured surface Gobbert and Ringhofer [98], T L van Noorden [09]. Boundary conditions: t u = z F + R(u, z, t), in ε, F = D z u in ε. Ɣ a = {(x, y) : y = h ε (x)}. y = h ε (x) = εh(x, x ) = εh(x, ξ). ε h(x, ξ + e 1 ) = h(x, ξ), h is periodic ν F = S(u, z, t), on z Ɣ a, u(z, t) = u D (z, t), on z Ɣ t, ν F(z, t) = 0 on z Ɣ w. 25/30
Asymptotic Expansion 0 := {(x, y), 0 < y < 1, 0 < x < 1} u(z, t) = ũ ε (z, t) + ū ε (x, x ε, y ε, t), = ũ ε (z, t) }{{} Outer Solution F(z, t) = F ε (z, t) }{{} Outer Solution + } ū ε (x, {{ ξ, η, t) }. Inner Solution + F ε (x, ξ, η, t), }{{} Inner Solution ũ ε (z, t) = ũ 0 (z, t) + εũ 1 (z, t) + O(ε 2 ), ū ε (x, ξ, η, t) = ū 0 (x, ξ, η, t) + εū 1 (x, ξ, η, t) + O(ε 2 ), F ε (z, t) = F 0 (z, t) + ε F 1 (z, t) + O(ε 2 ), F ε (x, ξ, η, t) = 1 ε F 1 (x, ξ, η, t) + F 0 (x, ξ, η, t) + O(ε 2 ). 26/30
Outer solution t ũ 0 = z F 0 + R(ũ 0, z, t), t ũ 1 = z F 1 + R(ũ 1, z, t). Inner Solution ξ,η F 1 = 0, ξ,η F 0 + x F 1 = 0. Normal y = εh(x, ξ, t), ( ) ( ) ν = ε x h ξ h + 0 1 }{{}}{{} 1 1 + (ε x h) 2 + ( ξ h) 2 εν 1 ν 0 27/30
Boundary condition (ν 0 + εν 1 ) ( 1 ε F 1 + F 0 + F 0 ) = S, which provides Movement of free boundary ν 0 F 1 = 0, ν 0 F 0 + ν 1 F 1 + ν 0 F 0 = S. t h = S 1 + ( ξ h) 2. 28/30
Upscaled equations t ũ 0 = F 0 + R in 0, F 0 = D z ũ 0, in 0 ũ 0 = u D, on Ɣ t ν F 0 = 0, on Ɣ w e 2 F 0 = σ }{{} Correction factor 1 σ (x) = σ 0 (x, ξ)dξ, 0 where σ 0 = 1 + ( ξ h) 2, t h = S 1 + ( ξ h) 2. S(ũ 0, y = 0, t), z on Ɣ a, 29/30
Acknowledgement STW for funding the project. Tycho van Noorden. Jos Oudenhoven. Mark van Helvoort. Mark Peletier 30/30