Discrete Kernel Preserving Model for 3D Electron-Optical Phonon Scattering Under Arbitrary Band Structures
Discrete Kernel Preserving Model for 3D Electron-Optical Phonon Scattering Under Arbitrary Band Structures
复制标题
任意能带结构下3D电子光声子散射的离散核保持模型
DOI：
10.1007/s10915-019-01082-2
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
Lu Tiao
中科院分区：
文献类型：
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作者：
Yao Wenqi;Lu Tiao
在Li et al. （J Sci computer 62:317-335, 2015）中，我们深入研究了离散一维（1D）非极性光学声子(NPOP)-电子散射矩阵的核空间结构，并提出了一种设置网格点的策略，以保持离散散射核的唯一性。在本文中，我们将上述工作扩展到三维（3D）情况，并研究了极性光学声子(POP)-电子的情况。在数值离散化中，得到一个尽可能多地保持连续散射算子性质的离散化散射矩阵是非常重要的。我们证明了对于三维npop -电子散射，(i)离散散射矩阵的核空间维数为1，（ii）只要能量区间上的网格符合Li et al.（2015）提出的规则，平衡分布相对于角坐标是恒定的。对于pop -电子散射，(i)也保持在相同的条件下，但要保持（ii）是一项具有挑战性的任务，因为通常在方位角和极角坐标上简单的均匀网格不能保持（ii）。基于柏拉图立体和正则金字塔的高度对称性，我们提出了两个条件，并证明了它们足以保证（ii）。数值实验有力地支持了我们的理论发现。
In Li et al. (J Sci Comput 62:317–335, 2015), we thoroughly investigated the structure of the kernel space of the discrete one-dimensional (1D) non-polar optical phonon (NPOP)-electron scattering matrix, and proposed a strategy to setup grid points so that the uniqueness of the discrete scattering kernel is preserved. In this paper, we extend the above work to the three dimensional (3D) case, and also investigate the polar optical phonon (POP)-electron case. In numerical discretization, it is important to get a discretization scattering matrix that keeps as many properties of the continuous scattering operator as possible. We prove that for the 3D NPOP-electron scattering, (i) the dimension of the kernel space of the discrete scattering matrix is one and (ii) the equilibrium distribution is constant with respect to the angular coordinates as long as the mesh over the energy interval obeys the rule proposed in Li et al. (2015). For the POP-electron scattering, (i) is also kept under the same condition, but to keep (ii) becomes a challenging task, since generally a simple uniform mesh in the azimuth and polar angular coordinates will not preserve (ii). Based on high degree of symmetry of the Platonic solids and regular pyramids, we propose two conditions and prove they are sufficient to guarantee (ii). Numerical experiments strongly support our theoretical findings.