The last years have witnessed a great interest for 2D-materials due to their promising applications. One of the most investigated is graphene which is considered as a potential new material to be exploited in nano-electronic and optoelectronic devices.
Charge transport in graphene can be described with several degrees of physical complexity . At quantum level an accurate model is represented by the Wigner equation but in several cases its semiclassical limit, the Boltzmann equation, constitutes a fully acceptable model. However, the numerical difficulties encountered in the direct solution of both the Wigner and the semiclassical Boltzmann equation has prompted the development of hydrodynamical, energy transport and drift diffusion models, in view of the design of a future generation of electron devices where graphene replaces standard semiconductors like silicon and gallium arsenide. Moreover, thermal effects in low dimensional structures play a relevant role and, therefore, phonon transport must be also included.
Interesting new mathematical issues related to the peculiar features of graphene arise. The main aspects will be discussed and recent results [2-13] illustrated in the perspective of future developments, in particular the design and optimization of graphene field effect transistors.
 Camiola V. D, Mascali G., Romano V.: Charge transport in low dimensional semiconductor structures, Springer, Switzerland (2020)
 Nastasi G., Romano V.: Mathematical aspects and simulation of electron–electron scattering in graphene, Z. Angew. Math. Phys. 74:28 (2023)
 Nastasi G., Camiola V. D., Romano V.: Direct Simulation of Charge Transport in Graphene Nanoribbons. Communications in Computational Physics 31 n. 2, 449--494 (2022)
 Nastasi G., Romano V.: An efficient GFET structure. IEEE Transaction on Electron Devices 68 n. 9, 4729 (2021)
 Barletti L., Nastasi G., Negulescu C., RomanoV.: Mathematical modelling of charge transport in graphene heterojunction. Kinetic and Related Models 14, 407--427 (2021)
 Nastasi G., Romano V.: A full coupled drift-diffusion-Poisson simulation of a GFET. Commun. Nonlinear Sci. Numer. Simulat. 87, 105300 (2020)
 Mascali G., Romano V.: A hierarchy of macroscopic models for phonon transport in graphene. Physica A 548, 124489 (2020)
 Luca L., Romano V.: Quantum corrected hydrodynamic models for charge transport in graphene. Ann. of Phys. 406, 50--33 (2019)
 Majorana A., Nastasi G., Romano V.: Simulation of bipolar charge transport in graphene by using a discontinuous Galerkin method. Comm in Comp. Phys. 26, 114--134 (2019)
 Mascali G., Romano V.: Charge transport in graphene including thermal effects. SIAM J. Applied Math. 77 n. 2, 593--613 (2017)
 Romano V, Majorana A., Coco M.: DSMC method consistent with the Pauli exclusion principle and comparison with deterministic solutions for charge transport in graphene. J. Comput. Phys. 302, 267--284 (2015)
 Camiola V. D., Romano V.: Hydrodynamical Model for Charge Transport in Graphene. J. Stat. Phys. 157, 1114--1137 (2014)