Quantum energy-transport and drift-diffusion models for electron transport in graphene: an approach by the wigner function

The present work aims at formulating quantum energy-transport and drift-diffusion equations for charge transport in graphene from a quantum hydrodynamic model proposed in Luca and Romano (Ann. Phys. 406:30–53, 2019), obtained from the Wigner-Boltzmann equation via the moment method. Here, we only sketch the main ideas and show the most relevant formulas. Further developments are under current investigation by the authors. The interested reader can find the omitted details in the references. In analogy with the semiclassical case, we are confident that the energy-transport and drift-diffusion models have mathematical properties which allow an easier numerical treatment.