Pauli principle and the Monte Carlo method for charge transport in graphene

An attempt to include the Pauli principle in the Monte Carlo method by also acting on the free-flight step and not only at the end of each collision is investigated. The charge transport in suspended monolayer graphene is considered as a test case. The results are compared with those obtained with the standard ensemble Monte Carlo technique and with the updated direct simulation Monte Carlo algorithm which is able to correctly handle with Pauli's principle. The physical aspects of the investigated approach are analyzed as well.

Figure 1

View of the particle distribution along the $x$ axis for the different values of the particle number $n_P$.

Figure 2

(a) Mean energy and (b) velocity for the SEMC, NEMC, and FFMC procedures, with ${\varepsilon }_F=0.6$ eV and $E=20$ kV/cm.

Figure 3

(a) Mean energy and (b) mean velocity in the FFMC case when ${\varepsilon }_F=0.6$ eV and $E=0$ kV/cm.

Figure 4

(a) Mean energy and (b) mean velocity in the NEMC case when ${\varepsilon }_F=0.6$ and $E=0$ kV/cm.

Figure 5

(a) Mean energy and (b) mean velocity in the EMC case when ${\varepsilon }_F=0.6$ eV and $E=0$ kV/cm

Figure 6

Mean energy in the NEMC, SEMC, FFMC cases, with FD and MB initial distributions, when (a) ${\varepsilon }_F=0.6$ eV and $E=10$ kV/cm and (b) ${\varepsilon }_F=0.6$ eV and $E=20$ kV/cm.

Figure 7

Mean velocity in the NEMC, SEMC, FFMC cases, with FD and MB initial distributions, when (a) ${\varepsilon }_F=0.6$ eV and $E=10$ kV/cm and (b) ${\varepsilon }_F=0.6$ eV and $E=20$ kV/cm.

Figure 8

Mean energy and velocity in the (a) and (b) NEMC and (c) and (d) SEMC cases when the FD or the MB initial distribution is used; ${\varepsilon }_F=0.6$ eV, and $E=20$ kV/cm.

Figure 9

FFMC scheme. (a) Initial Fermi-Dirac and (b) charge distributions at 5 ps. (c) Initial Maxwell-Boltzmann and (d) charge distributions at 5 ps. ${\varepsilon }_F=0.6$ eV and $E=20$ kV/cm.

Figure 10

Number of accepted free flights (a) for SEMC, with the initial FD, and (b) for FFMC when the FD and the MB are used.

Figure 11

Ratio of the number of accepted free flights with respect to their total number for the FFMC, with the FD (blue dots) and the MB (red dots) initial conditions.

Figure 12

Ratio of the number of accepted scatterings with respect to their total number for the FFMC scheme, with the FD and MB initial conditions, and the SEMC and NEMC schemes.

Figure 13

Ratio of the number of emitted (a) longitudinal-optical and (b) $K$ scatterings, whose final velocity is negative, with respect to their total number for the NEMC approach.

Figure 14

Mean velocity when (a) only the LO scatterings and (b) only they $K$ scatterings are included for the FFMC, with the FD and the MB initial conditions.

Figure 15

Total mean velocity and velocity due to the free flight when all the scatterings are neglected for the FFMC with the (a) FD and (b) MB initial conditions.

Figure 16

Percentage of particles whose velocity does not vary at each time step for the FFMC, with the FD and the MB initial conditions.

Figure 17

Contribution of each scattering type to the mean velocity for the FFMC with the (a) FD and (b) MB initial conditions.

Figure 18

Scattering rates evaluated with the Bose-Einstein equilibrium distributions for phonons at a temperature of 300 K.