3.5.1 One-Dimensional Example

Considering the one-dimensional discretization of the continuity equation at the interface (see Fig. 3.8) and inserting (3.11) and (3.12) into (3.8) and (3.9) and assuming the electrostatic potential and dielectric flux as constant at the heterojunction gives

$${\frac{n_{2} - n_{1}}{d}} \alpha \left(\vphantom{v_{3}{\cdot}n_{3} - \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot n_{2}\cdot\exp(-e_{\mathrm{b}})}\right. {\frac{m_{3}}{m_{2}}} \left.\vphantom{v_{3}{\cdot}n_{3} - \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot n_{2}\cdot\exp(-e_{\mathrm{b}})}\right)$$ (3.17)

$${\frac{n_{4} - n_{3}}{d}} \alpha \left(\vphantom{v_{3}{\cdot}n_{3} - \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot n_{2}\cdot\exp(-e_{\mathrm{b}})}\right. {\frac{m_{3}}{m_{2}}} \left.\vphantom{v_{3}{\cdot}n_{3} - \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot n_{2}\cdot\exp(-e_{\mathrm{b}})}\right)$$ (3.18)

This equation system for n₂ and n₃ can be rewritten as

$$\left(\vphantom{\!\!\!\begin{array}{cc} -\alpha\cdot\displaystyle\frac{m_{3}}{... ...m{b}}) &\! -\alpha\cdot v_{3}-\displaystyle\frac{D}{d}\end{array}\!\!\!}\right. \begin{array}{cc} -\alpha\cdot\displaystyle\frac{m_{3}}{m_{2}}\cdot v_{2}\cdot... ...exp(-e_{\mathrm{b}}) &\! -\alpha\cdot v_{3}-\displaystyle\frac{D}{d}\end{array} \left.\vphantom{\!\!\!\begin{array}{cc} -\alpha\cdot\displaystyle\frac{m_{3}}{... ...m{b}}) &\! -\alpha\cdot v_{3}-\displaystyle\frac{D}{d}\end{array}\!\!\!}\right) \left(\vphantom{\!\begin{array}{c}n_{2} \\ n_{3}\end{array}\!}\right. \begin{array}{c}n_{2} \\ n_{3}\end{array} \left.\vphantom{\!\begin{array}{c}n_{2} \\ n_{3}\end{array}\!}\right) \left(\vphantom{\begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}}\right. \begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array} \left.\vphantom{\begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}}\right)$$ (3.19)

Figure 3.8: One-dimensional discretization of the continuity equation across an interface. The boundary values n₁ and n₄ are fixed (Dirichlet boundary condition). For the current across the interface the thermionic field-emission model (3.12) is used.

n₁ and n₄ are the fixed values of the carrier concentration on the left and the right boundary (Dirichlet boundary condition).

The characteristic polynomial of the system matrix is

$$\lambda^{2}_{} \left(\vphantom{2\cdot\frac{D}{d} + \frac{\alpha\cdot m_{3}\cdot v_{2}}{m_{2}\cdot\exp(e_{\mathrm{b}})} + \alpha\cdot v_{3} }\right. {\frac{D}{d}} {\frac{\alpha\cdot m_{3}\cdot v_{2}}{m_{2}\cdot\exp(e_{\mathrm{b}})}} \alpha \left.\vphantom{2\cdot\frac{D}{d} + \frac{\alpha\cdot m_{3}\cdot v_{2}}{m_{2}\cdot\exp(e_{\mathrm{b}})} + \alpha\cdot v_{3} }\right) \lambda {\frac{D^{2}}{d^{2}}} {\frac{\alpha\cdot D\cdot m_{3}\cdot v_{2}}{d\cdot m_{2}\cdot\exp(e_{\mathrm{b}})}} {\frac{\alpha\cdot D\cdot v_{3}}{d}}$$ (3.20)

The solutions of (3.20) are the eigenvalues of (3.19).

$$\lambda_{1}^{} {\frac{D}{d}} \alpha \left(\vphantom{v_{3} + \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot\exp(-e_{\mathrm{b}})}\right. {\frac{m_{3}}{m_{2}}} \left.\vphantom{v_{3} + \frac{m_{3}}{m_{2}}\cdot v_{2}\cdot\exp(-e_{\mathrm{b}})}\right)$$ (3.21)

$$\lambda_{2}^{} {\frac{D}{d}}$$ (3.22)

According to (3.7) the spectral condition number of (3.19) is

$$\kappa_{\mathrm{s}}^{} \alpha {\frac{d}{D}} \left(\vphantom{v_{3}+\frac{m_{3}}{m_{2}}{\cdot}v_{2}\cdot\exp(-e_{\mathrm{b}})}\right. {\frac{m_{3}}{m_{2}}} \left.\vphantom{v_{3}+\frac{m_{3}}{m_{2}}{\cdot}v_{2}\cdot\exp(-e_{\mathrm{b}})}\right)$$ (3.23)

and, therefore, for large $\alpha$ the spectral condition of the system matrix will be poor. Thus, if the internal state of a device results in a large value of $\alpha$ the solver cannot compute the solution of the linear system with sufficient accuracy. The result will be an increase of iteration steps for the Newton scheme, if convergence can be achieved at all.

This problem can be alleviated by transforming the linear system (3.19) as follows. Adding the second equation to the first one and scaling the second equation with ${\frac{1}{\alpha}}$

$$\left(\vphantom{\begin{array}{cc} -\alpha\cdot\displaystyle\frac{... ...mathrm{b}}) &\! -\alpha\cdot v_{3} -\displaystyle\frac{D}{d}\end{array}}\right. \begin{array}{cc} -\alpha\cdot\displaystyle\frac{m_{3}}{m_{2}}\cd... ...xp(-e_{\mathrm{b}}) &\! -\alpha\cdot v_{3} -\displaystyle\frac{D}{d}\end{array} \left.\vphantom{\begin{array}{cc} -\alpha\cdot\displaystyle\frac{... ...mathrm{b}}) &\! -\alpha\cdot v_{3} -\displaystyle\frac{D}{d}\end{array}}\right) \left(\vphantom{\!\begin{array}{c}n_{2} \\ n_{3}\end{array}}\right. \begin{array}{c}n_{2} \\ n_{3}\end{array} \left.\vphantom{\!\begin{array}{c}n_{2} \\ n_{3}\end{array}}\right)$$

$$\left.\vphantom{\left. \left(\begin{array}{c}\displaystyle -\frac... ...playstyle -\frac{D}{d}\cdot n_{4}\end{array}\right)\;\;\right\vert+\;\;}\right. \left.\vphantom{ \left(\begin{array}{c}\displaystyle -\frac{D}{d}... ...{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}\right)\;\;}\right. \left(\vphantom{\begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}}\right. \begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array} \left.\vphantom{\begin{array}{c}\displaystyle -\frac{D}{d}\cdot n_{1}\\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}}\right) \left.\vphantom{ \left(\begin{array}{c}\displaystyle -\frac{D}{d}... ...\ [2mm]\displaystyle -\frac{D}{d}\cdot n_{4}\end{array}\right)\;\;}\right\vert \left.\vphantom{\left. \left(\begin{array}{c}\displaystyle -\frac... ...style -\frac{D}{d}\cdot n_{4}\end{array}\right)\;\;\right\vert+\;\;}\right\vert \begin{array}{c} \\ [4mm]\cdot\frac{1}{\alpha}\end{array}$$ (3.24)

results in the new system

$$\left(\vphantom{\begin{array}{cc} -\displaystyle\frac{D}{d} &-\displaystyle\fr... ...m{b}}) &-v_{3}-\displaystyle\frac{1}{\alpha}\cdot\frac{D}{d}\end{array}}\right. \begin{array}{cc} -\displaystyle\frac{D}{d} &-\displaystyle\frac{D}{d} \\ [2m... ..._{\mathrm{b}}) &-v_{3}-\displaystyle\frac{1}{\alpha}\cdot\frac{D}{d}\end{array} \left.\vphantom{\begin{array}{cc} -\displaystyle\frac{D}{d} &-\displaystyle\fr... ...m{b}}) &-v_{3}-\displaystyle\frac{1}{\alpha}\cdot\frac{D}{d}\end{array}}\right) \left(\vphantom{\begin{array}{c}n_{2} \\ n_{3}\end{array}}\right. \begin{array}{c}n_{2} \\ n_{3}\end{array} \left.\vphantom{\begin{array}{c}n_{2} \\ n_{3}\end{array}}\right) \left(\vphantom{\begin{array}{c}-\displaystyle\frac{D}{d}\cdot\left(n_{1} + n_... ...tyle\frac{1}{\alpha}\cdot\displaystyle\frac{D}{d}\cdot n_{4}\end{array}}\right. \begin{array}{c}-\displaystyle\frac{D}{d}\cdot\left(n_{1} + n_{4}\right)\\ [2... ...displaystyle\frac{1}{\alpha}\cdot\displaystyle\frac{D}{d}\cdot n_{4}\end{array} \left.\vphantom{\begin{array}{c}-\displaystyle\frac{D}{d}\cdot\left(n_{1} + n_... ...tyle\frac{1}{\alpha}\cdot\displaystyle\frac{D}{d}\cdot n_{4}\end{array}}\right)$$ (3.25)

The new system matrix has the eigenvalues

hence, for large values of $\alpha$ the spectral condition number $\kappa_{\mathrm{s}}^{}$ is nearly independent of $\alpha$ because in the limit for $\alpha$ $\rightarrow$ $\infty$ the factors ${\frac{\alpha + 1}{\alpha}}$ and ${\frac{\alpha - 1}{\alpha}}$ are 1.

$$\lim_{\alpha\rightarrow\infty}^{} \lambda_{1,2}^{} {\frac{1}{2}} \left(\vphantom{v_{3} +\frac{D}{d}}\right. {\frac{D}{d}} \left.\vphantom{v_{3} +\frac{D}{d}}\right) \pm \sqrt{\frac{1}{4}\cdot\left(v_{3} - \frac{D}{d}\right)^{2}- \frac{D}{d}\cdot\frac{m_{3}}{m_{2}}\cdot v_{2}\cdot\exp(-e_{\mathrm{b}})}$$ (3.27)

The strong influence of the internal state of the device on the spectral condition of the equation system has been eliminated.