Hydrodynamic modeling of electron transport in graphene

Semi-classical hydrodynamic models for charge transport in graphene have been presented. They are deduced as moment equations of the semiclassical Boltzmann equation with the needed closure relations obtained by resorting to the Maximum Entropy Principle. The models differ in the choice of the moments to assume as basic field variables. Both linear and nonlinear closure relations are analyzed. The validity of all the semi-classical models presented is assessed by comparing the mean values of energy and velocity with those obtained from the direct solutions of the Boltzmann equation in the simple case of suspended monolayer graphene. It has been found that it is crucial to include- among the field variables- the deviatoric part of the stress tensor to maintain a good accuracy in a wider range of applied electric fields. Moreover apparently the results confirm that the nonlinearity is not critical for accuracy. Then, to take into account quantum phenomena, in the last part of this work a quantum hydrodynamic model for charge transport in graphene is derived from a moment expansion of the Wigner-Boltzmann equation. The needed closure relations are obtained by adding to the semiclassical ones quantum corrections based on the equilibrium Wigner function. The latter is obtained from the Bloch equation by taking into account the appropriate energy band of graphene. Furthermore, quantum energy-transport and drift-diffusion models have been formally derived from the quantum hydrodynamic equations in the long time asymptotic limit. 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.