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# Discrete Kernel Preserving Model for 1D Electron---Optical Phonon Scattering

J. Sci. Comput., no. 2 (2015): 317-335

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Abstract

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 sca...More

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Introduction

- As the development of the miniature technology, low-dimensional semiconductor devices such as nanowires and nanotubes have drowned interest of more and more researchers [9].
- It has been pointed out in [3,5] that as the channel length is as short as 50nm, the relaxation time approximation of electron scattering fails.

Highlights

- As the development of the miniature technology, low-dimensional semiconductor devices such as nanowires and nanotubes have drowned interest of more and more researchers [9]
- We propose a finite volume method to discretize an electron– optical phonon scattering integral, where the grid is symmetric in velocity purposely
- We prove that the algebraic multiplicity of eigenvalue λ = 0 of M is equal to the number of the strongly connected components of a directed graph, which depends only on the energy grids used in the finite volume discretization
- We proposed a discrete scattering model for a 1D electron–optical phonon scattering operator
- Criteria have been given to judge whether a discrete scattering matrix is irreducible from the grid partition

Results

- As electrons are scattered from one state to another by absorbing or emitting an optical phonon with energy p, uniform energy grids with E = p/n, n ∈ N + is often considered.
- When n > 1, n ∈ N, the scattering matrix M derived from a uniform energy grid is reducible because there exist Ei = (i −1) E, i = 1, .

Conclusion

- The authors proposed a discrete scattering model for a 1D electron–optical phonon scattering operator.
- For an irreducible scattering matrix, the eigenvalue λ = 0 is proved to be semisimple and the real part of all the other eigenvalues are negative.
- Criteria have been given to judge whether a discrete scattering matrix is irreducible from the grid partition.
- This provided further understanding to the discretization of the scattering operator

Funding

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

Reference

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- 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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