## Discontinuous Galerkin methods for Boltzmann - Poisson models of electron transport in semiconductors

##### Date

2016-08##### Metadata

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The work presented in this dissertation is related to several lines of research in the area of Discontinuous Galerkin (DG) Methods for computational electronic transport in semiconductor devices using Boltzmann - Poisson (BP) models. The first line of research is the use of EPM related energy bands in a DG solver for BP where we consider a n⁺ -- n -- n⁺ diode problem, in order to increase the accuracy of the physical modeling of the energy band structure and its derivatives, via a spherical average of the EPM band structure and the spline interpolation of its derivatives, as these functions are involved in the collision mechanism, such as electron - phonon scattering in silicon, and transport via the electron group velocity. The balance of these two mechanisms is the core of the modeling of electron transport in semiconductors by means of Boltzmann - Poisson. The more physically accurate values of the spherical average EPM energy band and its derivatives interpolated by splines give a quantitative difference in kinetic moments related to the energy band model, such as average velocity, energy, and particularly the current given by our numerical solver. This highlights the importance of band models and features such as anisotropy and derivative interpolation in the BP numerical modeling of electron transport via DG schemes. The second line of research is related to the mathematical and numerical modeling of Reflective Boundary Conditions (BC) in 2D devices and their implementation in DG-BP schemes. We have studied the specular, diffusive and mixed reflection BC on the boundaries of the position domain of the device. We developed a numerical equivalent of the pointwise zero flux condition at the position domain insulating boundaries for the case of a more general mixed reflection with a momentum dependant specularity parameter p([k with left to right arrow above it]). We obtain this numerical zero flux condition by formulating the general mixed reflection BC as the solution of the problem of finding a function and parameter that balance the incident and reflected microscopic probability ow at each point of the insulating boundary. We compared the influence of the different reflection BC cases in the computational prediction of moments after implementing numerical BC equivalent to the respective reflective BC. There are expected effects due to the inclusion of diffusive reflection boundary conditions over the moments of the probability density function and over the electric field and potential, whose influence is not only restricted to the boundaries but actually to the whole domain. We observe in our simulations effects in kinetic moments of the inclusion of diffusion in the BC, such as the increase of the density close to the reflecting boundary, the decrease of the mean energy over the domain and the increase of the momentum x-component over the domain. The third line of research is related to the development of positivity preserving DG schemes for BP semiconductor models. We pose the Boltzmann Equation for electron transport in curvilinear coordinates for the momentum. We consider the 1D diode problem with azimuthal symmetry, which is a 3D plus time problem. We choose for this problem the spherical coordinate system [mathematical symbols], slightly different to the choice in previous DG solvers for BP, because its DG formulation gives simpler integrals involving just piecewise polynomial functions for both transport and collision terms. Applying the strategy of Zhang & Shu, [63], [64], Cheng, Gamba, Proft, [36], and Endeve et al. [65], we treat the collision operator as a source term, and find convex combinations of the transport and collision terms which guarantee the positivity of the cell average of our numerical probability density function at the next time step. The positivity of the numerical solution to the pdf in the whole domain is guaranteed by applying the limiters in [63], [64] that preserve the cell average but modify the slope of the piecewise linear solutions in order to make the function non - negative. In addition of the proofs of positivity preservation in the DG scheme, we prove the stability of the semi-discrete DG scheme under an entropy norm, using the dissipative properties of our collisional operator given by its entropy inequalities. The entropy inequality we use depends on an exponential of the Hamiltonian rather than the Maxwellian associated just to the kinetic energy.