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7.1.3 Diffusion and Reaction Model

The chemistry model is set up with AMIGOS [56], an Analytical Model Interface & General Object-Oriented Solver. The Analytical Model Interface is an interpreter for translating mathematical expressions into a one-, two-, or three-dimensional discretized numerical representation. It provides an analytic input language for the formulation of the dicretization scheme (finite elements, finite boxes) and for the considered partial differential equations. Since the differential equations are set up in an object-oriented formulation separately from the discretization scheme, the dimension of the model can be changed easily. Multiple volume and boundary models can be specified and assigned separately to different grids or boundaries. Afterwards the global equation system is assembled according to the dicretization scheme and the global system matrix is passed on to the General Object-Oriented Solver, a nonlinear numerical solver. The Analytical Model Interface allows a great flexibility in choosing the involved gas species and chemical reactions.

For the presented formulation of the HPCVD model which is derived from [35] a finite element discretization scheme was used. The governing principle for continuum transport determined chemical vapor deposition is species balance, i.e., time dependent diffusion of gas species in the gas phase including homogeneous (volume) reactions

$$\frac{\partial c_i}{\partial t} = \nabla (D_i \nabla c_i) + R_j^{\mathrm{hom}}$$ (7.1)

and heterogeneous (surface) reactions

$$-D_i \frac{\partial c_i}{\partial n} = R_k^{\mathrm{het}}$$ (7.2)

with multiple species $i$, their concentrations $c_i$ and their effective diffusivities $D_i$ in the mixture. $R_j$ and $R_k$ represent the reaction rates of different chemical reactions $j$ and $k$ which may occur either in the gas phase ($\mathrm{hom}$) or at the wafer surface ($\mathrm{het}$).

Moreover we assume a constant concentration of the $i$th species at the top of the simulation domain

$$c_i = c_i^0$$ (7.3)

and no flux across the side walls of the domain

$$\frac{\partial c_i}{\partial n} = n \nabla c_i = 0.$$ (7.4)

(7.1) is formulated in an AMIGOS volume model assigned to the complete simulation domain. (7.2) is set up in a boundary model description applied only to the wafer surface. (7.3) and (7.4) are handled as Dirichlet boundary condition at the top of the simulation domain and as Neumann condition at the domain sidewalls, respectively. The diffusion and reaction equations contain the concentrations of all involved gas species which are coupled by the stoichiometry of the chemical reactions. Consequently, transport of gas molecules from the plasma above the wafer into the feature competes with surface reactions which transform the reactants to a solid material forming the deposited layer. This competition leads to a steady state equilibrium and a geometry specific species concentration distribution depending on the ratio between gas diffusivities and surface reaction rates. For the steady state it is sufficient to solve the time independent formulation

$$\nabla (D_i \nabla c_i) + R_j^{\mathrm{hom}} = 0.$$ (7.5)

Putting the resulting steady state concentrations at the boundary into the reaction rate equations leads to specific local deposition velocities which are passed on to the topography module.

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