Simulation of Bipolar Charge Transport in Graphene by Using a Discontinuous Galerkin Method

Abstract

Charge transport in suspended monolayer graphene is simulated by a numerical deterministic approach, based on a discontinuous Galerkin (DG) method, for solving the semiclassical Boltzmann equation for electrons. Both the conduction and valence bands are included and the interband scatterings are taken into account. The use of a Direct Simulation Monte Carlo (DSMC) approach, which properly describes the interband scatterings, is computationally very expensive because the valence band is very populated and a huge number of particles is needed. Also the choice of simulating holes instead of electrons does not overcome the problem because there is a certain degree of ambiguity in the generation and recombination terms of electronhole pairs. Often, direct solutions of the Boltzmann equations with a DSMC neglect the interband scatterings on the basis of physical arguments. The DG approach does not suffer from the previous drawbacks and requires a reasonable computing effort. In the present paper the importance of the interband scatterings is accurately evaluated for several values of the Fermi energy, addressing the issue related to the validity of neglecting the generation-recombination terms. It is found out that the inclusion of the interband scatterings produces huge variations in the average values, as the current, with zero Fermi energy while, as expected, the effect of the interband scattering becomes negligible by increasing the absolute value of the Fermi energy.

Improved mobility models for charge transport in graphene

“…The integral involving the collisional operator can be solved analytically after performing the discretization. For more details about the numerical method one can refer to [1,3] for the unipolar case and to [5] for the treatment of the coupled collisional terms in the bipolar case while all the details about the treatment of the drift term can be found in [3]. The spatial gradient ∇ x f s (t, x, k) can be discretized in the same way as the drift one.…”

“…The integrals involving the electric field are derived by means of the same technique (see [3]). More details about the collisional term of interband transtions can be found in [5].…”

Mathematical modelling of charge transport in graphene heterojunctions

“…hold for all couples (s, s ) satisfying the conservation of energy (2) for some p and p in a nonzero measure set, where n i,∞ s are the asymptotic densities of the solution to the Milne problem (21)-(23) (see Theorem 2.1). Moreover, condition (25) is not affected by the particular choice of the solution to (21)- (23).…”

“…Evaluation of the asymptotic densities. Solving the Milne problem (21)- (23), which is needed in order to obtain the asymptotic densities n i,∞ s , implies that a "kinetic stage" is still present in our diffusive model. This is not very appealing, when looking for a simple and numerically treatable model.…”

“…However, as remarked above, since the aim of this paper is mainly to illustrate the mathematical method of quantum transmission conditions, rather than to simulate a particular device, we preferred to use a simple model for the bulk transport and to focus on the treatment of the quantum interface. Of course, a more detailed description of the electron scattering [11,20,23], more refined expressions for the mobility [24,26], a self-consistent potential model [25,26] or even quantum drift-diffusion equations [21,29,32], could be used to improve the model (see also Ref. [7] for a general reference).…”