Multiscale models and hybrid numerical methods for semiconductors – MoHyCon

Regarding the Asymptotic Preserving approach, our aim is to construct schemes for the linear Boltzmann equation for semiconductors, asymptotic preserving at the limit given by the drift-diffusion model. We will start with a very simplified model, with a linearized BGK collision operator. The constructed scheme will tend to an implicit discretization of Scharfetter-Gummel type for the drift-diffusion equation. The main objective will then be to lead a complete and rigorous study of the AP property by adapting to the discrete framework some continuous techniques: establish a discrete dissipation property yielding uniform estimates on the approximate solution which will allow to pass to the diffusion limit in the scheme.
Concerning the second approach (domain decomposition), the aim is to build a hybrid model coupling the kinetic equation on weakly collisional regions with macroscopic models on the remaining domain. We
will first study specifically discretizations for energy-transport model, since we are then going to couple them with kinetic and other macroscopic schemes in our hybrid method. A particular attention will be paid to the implementation in order to obtain a robust and efficient code.

Concerning the analysis of numerical schemes for energy transport model, M. Bessemoulin-Chatard, C. Chainais-Hillairet and H. Mathis proposed a finite volume scheme compatible with the entropic variables. This change of variables allows in particular to obtain a discrete entropy-dissipation estimate from which we can deduce a priori estimates useful to study the scheme. A 1D code corresponding to this scheme is being written.

Concerning the design and analysis of AP schemes for simplified models, M. Bessemoulin-Chatard, M. Herda (INRIA Lille) and T. Rey constructed and studied a 1D finite volume scheme for linear kinetic equations (of Fokker-Planck and linear BGK types). They established rigorously the asymptotic preserving property of the scheme at the diffusion limit towards the heat equation, as well as the hypocoercivity property (exponential convergence to the equilibrium as time tends to infinity). An article about these results has been submitted in December 2018, and a Jupyter Notebook including the developed code is available online.

Concerning the numerical schemes for the energy transport model, implementation of a 2D code on unstructured admissible mesh is planned in the coming weeks. The writing of an article describing construction and analysis of this scheme will be done before the end of 2019. With the arrival of Giulia Lissoni as a postdoctoral fellow in October 2019, we plan to extend this scheme to the DDFV framework, in order to apply it on more general meshes. We expect to submit a proceedings on the subject for the FVCA9 conference (Bergen, Norway, June 2020).

Concerning the construction and analysis of AP schemes, the natural perspective is to extend our first results to linear kinetic equations coupled with a Poisson equation describing the electric field.

Finally, concerning the full multiscale model, a working group on hybrid methods and domain decomposition, including Mr. Bessemoulin-Chatard, A. Crestetto and H. Mathis, is planned from September 2019 to make a precise current state of the art.

Details are available on the project webpage: 6 articles published or to be published in peer-reviewed international journals, 2 papers submitted, 2 conference proceedings, 6 communications without proceedings in international conferences. A Jupyter Notebook page with the developed AP scheme and corresponding test cases is available online.

The MoHyCon project is related to the analysis and simulation of numerical methods for multiscale models of semiconductors. As almost all current electronic technology involves the use of semiconductors, there is a strong interest for modeling and simulating the behavior of such devices, which was recently reinforced by the development of organic semiconductors used for example in solar panels or in mobile phones and television screens (among others).

There exists a hierarchy of semiconductors models, including mainly three classes, which correspond to different scales of observation: microscopic, mesoscopic and macroscopic. At the microscopic scale, the particles are described one by one, leading to a huge system almost impossible to study, both theoretically and numerically. Within MoHyCon, we are then rather interested in the two other scales. The considered models at the mesoscopic scale are kinetic, of Boltzmann type, describing a distribution of particles submitted to an electric field. These models describe accurately the behavior of the semiconductor, but can be intricate and highly time and resource consuming to solve numerically. Thus, when the mean free path becomes small, it is preferable to consider fluid models, describing macroscopic quantities. Depending on the considered number of moments, various models can be obtained. The more common ones are the energy transport model, describing the densities of electron and energy, and the more simple drift-diffusion model, where the temperature is assumed to be a given function of the electron density.

In this project and provided this context, our aim is to construct and study rigorously numerical methods for these multiscale models. To this end, we will consider two distinct approaches: ``Asymptotic Preserving'' (AP) methods and coupling methods. The idea of AP methods is to design only one scheme which will be able to treat accurately every scale, without imposing restrictive stability conditions on the discretization parameters. Regarding the coupling methods, they consist in decomposing the domain into different regions on which the more relevant model (kinetic or macroscopic) will be considered. After locating the kinetic and fluid domains, the main difficulty is to obtain correct coupling conditions at each interface between two regions.

Considering the AP approach, our aim is to construct schemes for the linear Boltzmann equation for semiconductors, asymptotic preserving at the limit given by the drift-diffusion model. We will start with a very simplified model, with a linearized BGK collision operator. The constructed scheme will tend to an implicit discretization of Scharfetter-Gummel type for the drift-diffusion equation. The main objective will be then to lead a complete and rigorous study of the AP property by adapting to the discrete framework some continuous techniques: establish a discrete dissipation property yielding uniform estimates on the approximate solution which will allow to pass to the diffusion limit in the scheme.

As regards the second approach, the aim is to build a hybrid model coupling the kinetic equation on weakly collisional regions with macroscopic models on the remaining domain. We will first study specifically discretizations for energy-transport model, since we are then going to couple them with kinetic and other macroscopic schemes in our hybrid method. A particular attention will be paid to the implementation in order to obtain a robust and efficient code.