A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: Performance and comparisons with Monte Carlo methods. In this paper we develop a deterministic high order accurate finite-difference WENO solver to the solution of the 1-D Boltzmann-Poisson system for semiconductor devices. We follow the work in E. Fatemi and F. Odeh [J. Comput. Phys. 108, 209–217 (1993; Zbl 0792.65110)] and in A. Majorana and R. Pidatella [J. Comput. Phys. 174, 649–668 (2001; Zbl 0992.82047)] to formulate the Boltzmann-Poisson system in a spherical coordinate system using the energy as one of the coordinate variables, thus reducing the computational complexity to two dimensions in phase space and dramatically simplifying the evaluations of the collision terms. The solver is accurate in time hence potentially useful for time-dependent simulations, although in this paper we only test it for steady-state devices. The high order accuracy and nonoscillatory properties of the solver allow us to use very coarse meshes to get a satisfactory resolution, thus making it feasible to develop a 2-D solver (which will be five dimensional plus time when the phase space is discretized) on today’s computers. The computational results have been compared with those by a Monte Carlo simulation and excellent agreements have been found. The advantage of the current solver over a Monte Carlo solver includes its faster speed, noise-free resolution, and easiness for arbitrary moment evaluations. This solver is thus a useful benchmark to check on the physical validity of various hydrodynamic and energy transport models. Some comparisons have been included in this paper.

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  1. Bresten, Christopher; Gottlieb, Sigal; Grant, Zachary; Higgs, Daniel; Ketcheson, David I.; Németh, Adrian: Explicit strong stability preserving multistep Runge-Kutta methods (2017)
  2. He, Yuan; Gamba, Irene M.; Lee, Heung-Chan; Ren, Kui: On the modeling and simulation of reaction-transfer dynamics in semiconductor-electrolyte solar cells (2015)
  3. Li, Ruo; Lu, Tiao; Yao, Wenqi: Discrete kernel preserving model for 1D electron-optical phonon scattering (2015)
  4. Vecil, Francesco; Mantas, José M.; Cáceres, María J.; Sampedro, Carlos; Godoy, Andrés; Gámiz, Francisco: A parallel deterministic solver for the Schrödinger-Poisson-Boltzmann system in ultra-short DG-MOSFETs: Comparison with Monte-Carlo (2014)
  5. Carrillo, José A.; Yan, Bokai: An asymptotic preserving scheme for the diffusive limit of kinetic systems for chemotaxis (2013)
  6. Jin, Shi; Wang, Li: Asymptotic-preserving numerical schemes for the semiconductor Boltzmann equation efficient in the high field regime (2013)
  7. Cheng, Yingda; Gamba, Irene M.; Proft, Jennifer: Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations (2012)
  8. Alekseenko, A.M.: Numerical properties of high order discrete velocity solutions to the BGK kinetic equation (2011)
  9. Cheng, Yingda; Gamba, Irene M.; Majorana, Armando; Shu, Chi-Wang: A brief survey of the discontinuous Galerkin method for the Boltzmann-Poisson equations (2011)
  10. Cheng, Yingda; Gamba, Irene M.; Ren, Kui: Recovering doping profiles in semiconductor devices with the Boltzmann-Poisson model (2011)
  11. Ketcheson, David I.; Gottlieb, Sigal; Macdonald, Colin B.: Strong stability preserving two-step Runge-Kutta methods (2011)
  12. Lu, Tiao; Du, Gang; Liu, Xiaoyan; Zhang, Pingwen: A finite volume method for the multi subband Boltzmann equation with realistic 2D scattering in double gate MOSFETs (2011)
  13. Yang, Jaw-Yen; Muljadi, Bagus Putra: Simulation of shock wave diffraction over $90^\circ $ sharp corner in gases of arbitrary statistics (2011)
  14. Besse, C.; Goudon, T.: Derivation of a non-local model for diffusion asymptotics -- application to radiative transfer problems (2010)
  15. Ben Abdallah, N.; Cáceres, M.J.; Carrillo, J.A.; Vecil, F.: A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs (2009)
  16. Gottlieb, Sigal; Ketcheson, David I.; Shu, Chi-Wang: High order strong stability preserving time discretizations (2009)
  17. La Rosa, Salvatore; Mascali, Giovanni; Romano, Vittorio: Exact maximum entropy closure of the hydrodynamical model for si semiconductors: the 8-moment case (2009)
  18. Mantas, José M.; Cáceres, María J.: Efficient deterministic parallel simulation of 2D semiconductor devices based on WENO-Boltzmann schemes (2009)
  19. Shu, Chi-Wang: High order weighted essentially nonoscillatory schemes for convection dominated problems (2009)
  20. Baker, Lowell L.; Hadjiconstantinou, Nicolas G.: Variance-reduced Monte Carlo solutions of the Boltzmann equation for low-speed gas flows: a discontinuous Galerkin formulation (2008)

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