A discontinuous Galerkin approach for simulating graphene-based electron devices via the Boltzmann transport equation

TL;DR

This work develops a discontinuous Galerkin framework for the bipolar semiclassical Boltzmann transport equation in graphene, coupled to Poisson, using linear in-space elements and a polar momentum-space grid. The method employs a maximum-principle limiter and UNO-based fluxes to ensure physical distribution values while maintaining stability and accuracy. Through two benchmarks—suspended graphene and GFET—it demonstrates robustness, accurate capture of charge transport in graphene, and provides high-quality benchmarks for evaluating macroscopic models like drift-diffusion and hydrodynamics. The approach offers a computationally efficient pathway for device-scale graphene simulations and rigorous analysis of transport phenomena in 2D materials.

Abstract

Electron devices based on graphene have lately received a considerable interest; in fact, they could represent the ultimate miniaturization, since the active area is only one atom tick. However, the gapless dispersion relation of graphene at the Dirac points limits the possibility of using pristine graphene instead of traditional semiconductors in Field Effect Transistors (FET). For such a reason very accurate simulations are needed. In Nastasi & Romano, IEEE TED (2021) a graphene field effect transistor (GFET) has been proposed and simulated adopting a drift-diffusion model. Here, electron devices whose active area is made of monolayer graphene are simulated adopting as mathematical model the semiclassical Boltzmann transport equations (BTEs) in the bipolar case, coupled with the Poisson equation for the electric field. The system is solved by means of a discontinuous Galerkin (DG) approach (see Cockburn & Shu, J. Comp. Phys. (1998); Hesthaven & Warburton, 2008) with linear elements in the spatial coordinate and constant approximation for the wave-vector space, discretized with a polar mesh. The correct physical range for the distribution function is preserved with the maximum-principle-satisfying scheme introduced in Zhang & Shu, J. Comp. Phys. (2010). The adopted method reveals very robust and possesses a good degree of accuracy, making it particularly well suited for capturing the complex charge transport dynamics inherent to graphene-based devices. The results for suspended monolayer graphene and GFET constitute benchmark solutions for a rigorous assessment of the validity of macroscopic models, such as drift-diffusion and hydrodynamic ones.

Figures (11)

• Figure 1: Contacted monolayer graphene.
• Figure 2: GFET.
• Figure 3: Polar grid in the space of wave-vectors.
• Figure 4: Discretization with respect to $x$ where $\varepsilon$ and $\theta$ are fixed.
• Figure 5: Representation of a generic cell for the computation of the numerical fluxes.