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Next: 3.6 Application to a Up: 3.5 The New Method Previous: 3.5.1 One-Dimensional Example


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

box i2 :          $\displaystyle \sum_{j_{2}}^{}$Jj2 + $\displaystyle \sum_{j_{1}}^{}$Jj1 = 0, (3.28)
box i1 :          $\displaystyle \sum_{j_{1}}^{}$Jj1 = Ji $\scriptstyle \perp$. (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}}^{}$Jj1 = f (n1, n2, eb),    $ \tilde{\alpha}$ = $ {\frac{1}{\alpha}}$. (3.30)

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

n1 = n2 . $ \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$.

next up previous
Next: 3.6 Application to a Up: 3.5 The New Method Previous: 3.5.1 One-Dimensional Example
Martin Rottinger
1999-05-31