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Discrete Kernel Preserving Model for 1D Electron–Optical Phonon Scattering


We investigate the discretization of of an electron–optical phonon scattering using a finite volume method. The discretization is conservative in mass and is essentially based on an energy point of view. This results in a discrete scattering system with elegant mathematical features, which are fully clarified. Precisely the discrete scattering matrix is thoroughly studied, including its sparsity pattern and its symmetries, the structure of its eigenvalues and eigenvectors. It makes us reveal the strategy to setup grid points, so that the proper scattering matrix can be obtained to preserve the unique discrete scattering kernel. Numerical results are presented to validate these theoretical findings.

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  1. Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system in one dimensional space. Comm. Math Sci. 11(2), 547–571 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cai, Z., Fan, Y., Li, R.: Globally hyperbolic regularization of Grad’s moment system. Comm. Pure Appl. Math. 67(3), 1747–1771 (2014)

    Article  MathSciNet  Google Scholar 

  3. Carrillo, J.A., Gamba, I.M., Majorana, A., Shu, C.-W.: A WENO-solver for the transient of devices: perfomance and comparisons with Monte Carlo methods. J. Comput. Phys. 184, 498–525 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. De Martino, A., Egger, R.: Acoustic phonon exchange, attractive interactions, and the Wentzel–Bardeen singularity in single-wall nanotubes. Phys. Rev. B 67, 235418 (Jun 2003)

    Google Scholar 

  5. Hu, Z., Li, R., Lu, T., Wang, Y., Yao, W.: Simulation of an \(n^{+}\)-\(n\)-\(n^{+}\) diode by using globally-hyperbolically-closed high-order moment models. J. Sci. Comput. (accepted for publication) (2013)

  6. Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, New York (1985)

    MATH  Google Scholar 

  7. Lenzi, M., Palestri, P., Gnani, E., Reggiani, S., Gnudi, A., Esseni, D., Selmi, L., Baccarani, G.: Investigation of the transport properties of silicon nanowires using deterministic and Monte Carlo approaches to the solution of the Boltzmann transport equation. IEEE Trans. Electron Devices 55(8), 2086–2096 (2008)

    Article  Google Scholar 

  8. Lu, T., Du, G., Liu, X., Zhang, P.: A finite volume method for the multi subband Boltzmann equation with realistic 2D scattering in DG MOSFETs. Commun. Comput. Phys. 10, 305–338 (2011)

    Google Scholar 

  9. Wu, Y., Xiang, J., Yang, C., Lu, W., Lieber, C.M.: Single-crystal metallic nanowires and metal/semiconducotor nanowire heterostructures. Nature 430, 61–65 (2004)

    Article  Google Scholar 

  10. Yao, W., Li, R., Lu, T., Liu, X., Du, G., Zhao, K.: Globally hyperbolic moment method for BTE including phonon scattering. In: 2013 International Conference on Simulation of Semiconductor Processes and Devices (SISPAD), pp. 300–303, Glasgow, Scotland, UK (Sept. 3–5, 2013)

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This research was supported in part by the National Basic Research Program of China (2011CB309704) and NSFC (91230107, 11325102).

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Correspondence to Tiao Lu.

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Li, R., Lu, T. & Yao, W. Discrete Kernel Preserving Model for 1D Electron–Optical Phonon Scattering. J Sci Comput 62, 317–335 (2015).

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  • Eigenvalues
  • Eigenspaces
  • Electron–phonon scattering integral operator
  • Semiconductor device simulation
  • Finite volume method