High-field mobility in graphene on substrate with a proper inclusion of the Pauli exclusion principle

The aim of this work is to simulate charge transport in a monolayer graphene on different substrates. This requires the inclusion of the scatterings of charge carriers with impurities and phonons of the substrate, besides the interaction mechanisms already present in the graphene layer. As mathematical model, the semiclassical Boltzmann equation is assumed and the results are based on Direct Simulation Monte Carlo (DSMC) method. A crucial point is the correct inclusion of the Pauli Exclusion Principle (PEP). Most simulations use the approach proposed by Jacoboni e Lugli which, however, allows an occupation number greater than one with an evident violation of PEP. Here the Monte Carlo scheme devised by Romano et al. (J. Comput. Phys. 302, 267--284, 2015) is employed. It predicts occupation numbers consistent with PEP and therefore is physically more accurate. Two different substrates are investigated: SiO_2 and hexagonal boron nitride (h-BN). We adopt the model for charge-impurities scattering described by E. H. Hwang and S. Das Sarma (Phys. Rev. B 75, 205418, 2007). In such a model a crucial parameter is the distance d between the graphene layer and the impurities of the substrate. Usually d is considered constant. Here we assume that $d$ is a random variable in order to take into account the roughness of the substrate and the randomness of the location of the impurities. Our results confirm that h-BN is one of the most promising substrate also for the high-field mobility on account of the reduced degradation of the velocity due to the remote impurities. This is in agreement with results shown by Hirai et al., (J. Appl. Phys. 116, 083703, 2014) where only the low-field mobility has been investigated.