Positivity preserving discretization of time dependent semiconductor drift-diffusion equations

Positivity preserving discretization of the semiconductor drift-diffusion equations is considered. The equations are spatially discretized by mixed hybrid finite elements leading to a positive ODE or DAE system with index of at most one. For time discretization a second-order splitting technique based on a combination of explicit exponential integration and implicit one-step methods is proposed. This allows for positivity preservation with larger time steps than the corresponding one-step methods. An algorithm is presented coupling the splitting technique with the Gummel iteration scheme allowing for efficient positivity preserving device simulation. Numerical results for a one-dimensional pn-diode are given, showing that the proposed scheme allows for runtime acceleration.

[1]  Pierre Degond,et al.  On a hierarchy of macroscopic models for semiconductors , 1996 .

[2]  Cleve B. Moler,et al.  Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later , 1978, SIAM Rev..

[3]  S. Selberherr.  Analysis and simulation of semiconductor devices , 1984 .

[4]  P. Raviart,et al.  A mixed finite element method for 2-nd order elliptic problems , 1977 .

[5]  J. Verwer,et al.  Numerical solution of time-dependent advection-diffusion-reaction equations , 2003 .

[6]  C. Schmeiser,et al.  Semiconductor equations , 1990 .

[7]  T. Fujimoto,et al.  TWO CHARACTERIZATIONS OF INVERSE-POSITIVE MATRICES: THE HAWKINS-SIMON CONDITION AND THE LE CHATELIER-BRAUN PRINCIPLE ∗ , 2004 .

[8]  Ansgar Jüngel,et al.  Numerical Discretization of Energy-Transport Models for Semiconductors with Nonparabolic Band Structure , 2000, SIAM J. Sci. Comput..

[9]  C. Bolley,et al.  Conservation de la positivité lors de la discrétisation des problèmes d'évolution paraboliques , 1978 .

[10]  Martin Arnold.  Stability of numerical methods for differential-algebraic equations of higher index , 1993 .

[11]  Elena Virnik,et al.  Analysis of positive descriptor systems , 2008 .

[12]  Paola Pietra,et al.  New mixed finite element schemes for current continuity equations , 1990 .

[13]  A. Jüngel.  Transport Equations for Semiconductors , 2009 .

[14]  Franco Brezzi,et al.  Numerical simulation of semiconductor devices , 1989 .

[15]  Lawrence F. Shampine,et al.  The MATLAB ODE Suite , 1997, SIAM J. Sci. Comput..

[16]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[17]  Ansgar Jüngel,et al.  Numerical Coupling of Electric Circuit Equations and Energy-Transport Models for Semiconductors , 2008, SIAM J. Sci. Comput..

[18]  Paola Pietra,et al.  Two-dimensional exponential fitting and applications to drift-diffusion models , 1989 .