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A robust hybridizable discontinuous Galerkin scheme with harmonic averaging technique for steady state of real-world semiconductor devices

Qingyuan Shi, Yongyong Cai, Chijie Zhuang, Bo Lin, Dan Wu, Rong Zeng, Weizhu Bao

Abstract

Solving real-world nonlinear semiconductor device problems modeled by the drift-diffusion equations coupled with the Poisson equation (also known as the Poisson-Nernst-Planck equations) necessitates an accurate and efficient numerical scheme which can avoid non-physical oscillations even for problems with extremely sharp doping profiles. In this paper, we propose a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging (HA) technique to tackle these challenges. The proposed HDG-HA scheme combines the robustness of finite volume Scharfetter-Gummel (FVSG) method with the high-order accuracy and $hp$-flexibility offered by the locally conservative HDG scheme. The coupled Poisson equation and two drift-diffusion equations are simultaneously solved by the Newton method. Indicators based on the gradient of net doping $N$ and solution variables are proposed to switch between cells with HA technique and high-order conventional HDG cells, utilizing the flexibility of HDG scheme. Numerical results suggest that the proposed scheme does not exhibit oscillations or convergence issues, even when applied to heavily doped and sharp PN-junctions. Devices with circular junctions and realistic doping profiles are simulated in two dimensions, qualifying this scheme for practical simulation of real-world semiconductor devices.

A robust hybridizable discontinuous Galerkin scheme with harmonic averaging technique for steady state of real-world semiconductor devices

Abstract

Solving real-world nonlinear semiconductor device problems modeled by the drift-diffusion equations coupled with the Poisson equation (also known as the Poisson-Nernst-Planck equations) necessitates an accurate and efficient numerical scheme which can avoid non-physical oscillations even for problems with extremely sharp doping profiles. In this paper, we propose a flexible and high-order hybridizable discontinuous Galerkin (HDG) scheme with harmonic averaging (HA) technique to tackle these challenges. The proposed HDG-HA scheme combines the robustness of finite volume Scharfetter-Gummel (FVSG) method with the high-order accuracy and -flexibility offered by the locally conservative HDG scheme. The coupled Poisson equation and two drift-diffusion equations are simultaneously solved by the Newton method. Indicators based on the gradient of net doping and solution variables are proposed to switch between cells with HA technique and high-order conventional HDG cells, utilizing the flexibility of HDG scheme. Numerical results suggest that the proposed scheme does not exhibit oscillations or convergence issues, even when applied to heavily doped and sharp PN-junctions. Devices with circular junctions and realistic doping profiles are simulated in two dimensions, qualifying this scheme for practical simulation of real-world semiconductor devices.
Paper Structure (22 sections, 39 equations, 17 figures, 3 tables)

This paper contains 22 sections, 39 equations, 17 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The model equations and their nondimensionalization
  3. The governing equations of semiconductor devices in two dimensions (2D)
  4. Nondimensionless form
  5. The hybridizable discontinuous Galerkin scheme
  6. Discretization of the hybridizable discontinuous Galerkin scheme
  7. The $hp$-adaptivity of the HDG scheme
  8. The Newton method and static condensation of HDG scheme
  9. Local post-processing of the HDG scheme
  10. HDG scheme with harmonic averaging technique
  11. Harmonic averaging technique and the HDG discretization
  12. HA-adaptivity and its indicator
  13. Numerical experiments
  14. Stability of HDG schemes on abrupt semiconductor junctions
  15. Numerical accuracy of various HDG schemes
  16. ...and 7 more sections

Figures (17)

  • Figure 1: Structure of a power BJT: SEM cross-section of multiple cells (left) and schematic representation of a single cell (right).
  • Figure 2: Illustration of $hp$-adaptivity on a quadrilateral mesh.
  • Figure 3: The oscillating numerical solution of $\psi$ using the conventional HDG scheme with 100 cells. The settings are taken from test case 3 in subsection \ref{['sec:convergeHA']}. Reference solution is obtained with FVSG scheme using 10000 cells.
  • Figure 4: Comparisons of different schemes: solutions of $\psi$ (left column) and solutions of $n$ (right column). Top row: case 1 using $100$ cells. Middle row: case 3 using $100$ cells. Bottom row: case 5 where both HDG-HA and FVSG schemes use $100$ cells, and HDG-P3 scheme used $10000$ cells.
  • Figure 5: The doping and solution profiles of the smooth PN-junction test case, where solutions are computed using the FVSG scheme with $10^6$ cells.
  • ...and 12 more figures