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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.

The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at $$v=0$$ , we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at $$v=0$$ , and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful.

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.

By the moment closure of the Boltzmann transport equation, the extended hydrodynamic models for electron transport have been derived in Cai et al. (J Math Phys 53:103503, 2012). With the numerical scheme developed in Li et al. (2012) recently, it has been demonstrated that the derived extended hydrodynamic models can capture the major features of the solution of kinetic equations. As the application of the models and the numerical scheme proposed therein, we in this paper carry out the numerical simulation to investigate the carrier transport in $$n^{+}$$ n + - $$n$$ n - $$n^{+}$$ n + structures by solving the moment system derived from the Boltzmann---Poisson equations. Without any additional empirical parameters than that used in directly solving the kinetic equations, we obtain numerical results by the extended hydrodynamic models with very satisfied agreement with the solution of the kinetic equations, even in case that the length of the channel is as short as 50 nm.