Boundary conditions

  The simulation domain is a finite region determined by boundaries at the periphery of the simulation domain. For a full description of differential equations the boundary conditions at this external boundaries are essential. They affecting the solution variables inside the simulation domain. Figure 4.1-9 shows a discretization box around the boundary grid point i. The spatial discretization has already accounted for the box volume during discretization of the divergence operator. The boundary conditions can now be specified via a flux across the interface area in the direction of or by simply specifying the value of at the interface grid point. A general formulation of a boundary condition is given in (4.1-20).

 

  
Figure 4.1-9: Boundary discretization box for the box integration method. The boundary condition rely upon the flux across the interface area , the boundary concentration or the next inner concentration .

It depends on the coefficients and , which kind of boundary condition is used. The following boundary conditions can be deduced from (4.1-20):

NEUMANN:

; specifies the flux across the interface area. If , we have homogeneous Neumann boundary condition, which leads to dopant conservation within the simulation domain.

DIRICHLET:

; specifies a specific value to be fixed for the solution on the boundary point. These Dirichlet points represent the exact solutions at these boundary points. Hence, they are eliminated from the system matrix prior to solving the linear system.

CAUCHY:

; mixed boundary condition comprising particle flux and concentration value at the boundary. This is often used to specify particle transport over the boundary, i.e. interface kinetic, oxidizing surface conditions.

SECOND ORDER:

; boundary points as well as inner points are used to specify the dopant flux across the interface. Generally, homogeneous Neumann boundary conditions are assumed for the artificial boundaries of the simulation domain to ensure dopant conservation. The simulation domain must be selected large enough that this Neumann boundary conditions have no impact on the solution. If point defects are involved in the diffusion process the simulation domain has to be extended to several hundred microns due to the high diffusivity of the point defects. To avoid these large simulation domains a second order boundary condition can be used to approximate the dopant concentrations at the interface by expression (4.1-21), where k denotes a prefactor (mostly depending on the diffusivity D), the equilibrium dopant concentration and h the perpendicular distance to the interface.