3.5.2 Transformation for Two-Dimensional Discretization

Applying the transformation described in the previous section to the two-dimensional situation, the equations (3.8) and (3.9) for box i₁ and i₂ can be rewritten such as

$$\sum_{j_{2}}^{} \sum_{j_{1}}^{}$$ box i₂ :          (3.28)

$$\sum_{j_{1}}^{} \perp$$ box i₁ :          (3.29)

Equation (3.28) is Kirchhoff's law for the compound of box i₁ and i₂ (i.e., the sum of (3.8) and (3.9)) and determines the electron concentration n₂, (3.29) does the same for box i₁ and determines n₁. The problem of $\alpha$ becoming large can be solved by scaling (3.29) with ${\frac{1}{\alpha}}$ as proposed for the one-dimensional case:

$$\tilde{\alpha} \sum_{j_{1}}^{} \tilde{\alpha} {\frac{1}{\alpha}}$$ (3.30)

For $\lim_{e_{\mathrm{b}}\rightarrow0}^{}$$\tilde{\alpha}$(e_b) = 0 follows f (n₁, n₂) = 0 which is equivalent to the Dirichlet boundary condition

$$\left(\vphantom{\frac{m_{1}}{m_{2}}}\right. {\frac{m_{1}}{m_{2}}} \left.\vphantom{\frac{m_{1}}{m_{2}}}\right)^{\frac{3}{2}}_{} \left(\vphantom{\frac{T_{2}}{T_{1}}}\right. {\frac{T_{2}}{T_{1}}} \left.\vphantom{\frac{T_{2}}{T_{1}}}\right)^{\frac{1}{2}}_{}$$ (3.31)

Instead of large values of $\alpha$ the simulator has now to cope with small values of $\tilde{\alpha}$. Furthermore, the spectral condition of the system matrix is not deteriorated by large values of $\alpha$.