Quantum MEP Hydrodynamical Model for Charge Transport

A well known procedure to get quantum hydrodynamical models for charge transport is to resort to the Wigner equations and deduce the hierarchy of the moment equations as in the semiclassical approach. If one truncates the moment hierarchy to a finite order, the resulting set of balance equations requires some closure assumption because the number of unknowns exceed the number of equations. In the classical and semiclassical kinetic theory a sound approach to get the desired closure relations is that based on the Maximum Entropy Principle (MEP) (Jaynes in Phys Rev 106:620–630, 1957) [see Camiola et al. (Charge transport in low dimensional semiconductor structures, the maximum entropy approach. Springer, Cham, 2020) for charge transport in semiconductors]. In Romano (J Math Phys 48:123504, 2007) a quantum MEP hydrodynamical model has been devised for charge transport in the parabolic band approximation by introducing quantum correction based on the equilibrium Wigner function (Wigner in Phys Rev 40:749–749, 1932). An extension to electron moving in pristine graphene has been obtained in Luca and Romano (in: Atti della Accademia Peloritana dei Pericolanti—Classe di Scienze Fisiche, Matematiche e Naturali, [S.l.], p. A5, 2018, https://doi.org/10.1478/AAPP.96S1A5). Here we present a quantum hydrodynamical model which is valid for a general energy band considering a closure of the moment system deduced by the Wigner equation resorting to a quantum version of MEP. Explicit formulas for quantum correction at order \(\hbar ^2\) are obtained with the aid of the Moyal calculus for silicon and graphene removing the limitation that the quantum corrections are based on the equilibrium Wigner function as in Romano (J Math Phys 48:123504, 2007), Luca and Romano (in: Atti della Accademia Peloritana dei Pericolanti—Classe di Scienze Fisiche, Matematiche e Naturali, [S.l.], p. A5, 2018, https://doi.org/10.1478/AAPP.96S1A5). As an application, quantum correction to the mobilities are deduced.