A singular perturbation analysis of the fundamental semiconductor device equations.

*(English)*Zbl 0568.35007The author presents a singular perturbation analysis of the fundamental semiconductor device equations reformulated as an elliptic nonlinear system, subject to mixed Neumann-Dirichlet boundary conditions. The minimal normed Debye-length \(\lambda\) of the device appears as a multiplier of the Laplace operator in Poisson’s equation and plays the role of the perturbation parameter. Under the assumption of zero- generation-recombination, a priori estimates independent of \(\lambda\) are derived and lead to the existence of weak solutions.

To investigate the asymptotic behaviour of the solutions as \(\lambda\) \(\to 0\), the author employs the method of matched asymptotic expansions. It turns out that the regular expansion (in power series of \(\lambda)\) has to be supplemented by interior layer terms locally about semiconductor-oxide interfaces and functions. No boundary layers occur at ohmic contacts and insulating elements.

The author gives existence results for the reduced problems, for the outer solution as well as for the layer terms. It is proved that the sum of the inner and outer solutions approximates the full solution uniformly for small \(\lambda\) if the device is in thermal equilibrium.

The singular perturbation analysis provides a tool for the construction of meshes for the numerical solution. A numerical simulation for a 2- dimensional diode is presented with several interesting figures.

These results appear to extend previous one-dimensional results by the same author with others [see P. A. Markowich and C. Ringhofer, ibid. 44, 231-256 (1984; Zbl 0559.34058)].

To investigate the asymptotic behaviour of the solutions as \(\lambda\) \(\to 0\), the author employs the method of matched asymptotic expansions. It turns out that the regular expansion (in power series of \(\lambda)\) has to be supplemented by interior layer terms locally about semiconductor-oxide interfaces and functions. No boundary layers occur at ohmic contacts and insulating elements.

The author gives existence results for the reduced problems, for the outer solution as well as for the layer terms. It is proved that the sum of the inner and outer solutions approximates the full solution uniformly for small \(\lambda\) if the device is in thermal equilibrium.

The singular perturbation analysis provides a tool for the construction of meshes for the numerical solution. A numerical simulation for a 2- dimensional diode is presented with several interesting figures.

These results appear to extend previous one-dimensional results by the same author with others [see P. A. Markowich and C. Ringhofer, ibid. 44, 231-256 (1984; Zbl 0559.34058)].

Reviewer: C.M.Brauner

##### MSC:

35B25 | Singular perturbations in context of PDEs |

35J65 | Nonlinear boundary value problems for linear elliptic equations |

78A30 | Electro- and magnetostatics |

35C20 | Asymptotic expansions of solutions to PDEs |

35D05 | Existence of generalized solutions of PDE (MSC2000) |