The Maximum Entropy Principle: General Extended Hydrodynamic Approach for Dynamic High-Field Transport in Monolayer Graphene

Abstract

Within the maximum entropy principle, we present a general theory able to describe in a dynamical context the transport properties of hot carriers in monolayer graphene under electric fields of arbitrary strength. Therefore, we obtain a closed extended hyperbolic system of hydrodynamic (HD) equations in which all the unknown constitutive functions are completely determined. In particular, we consider the different scattering mechanisms used in the literature in the kinetic approaches. The closed extended HD system is applied to monolayer graphene at 300 K and is validated by comparing numerical calculations with ensemble Monte Carlo simulations.

Keywords:

• Maximum entropy principle
• extended thermodynamics
• monolayer graphene

Acknowledgments

This research is partially supported by Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale per la Fisica Matematica (GNFM), by national project PRIN of the Italian Ministry for University and Research, grant number 2017YBKNCE, and by grant PIACERI “AsDeA” of the University of Catania.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Notes

^a The kinetic fields $u_{⟨i_1}{u}_{i_2}\cdots u_{i_s⟩}$ can be obtained from kinetic fields used in this paper ${\epsilon }^p{u}_{⟨i_1}{u}_{i_2}\cdots u_{i_s⟩}$ simply by setting $p=0;$ therefore, in this case, we trivially will assume for the corresponding macroscopic moments that $F_{(0)|⟨i_1\cdots i_s⟩}=F_{⟨i_1\cdots i_s⟩}$ and analogously for the other similar tensorial quantities $P_{(0)|⟨i_1\cdots i_s⟩}=P_{⟨i_1\cdots i_s⟩}.$

^b Indeed, in the HD Eq. (23)(23) $\begin{array}{rl}& \frac{\mathrm{\partial }{F}_{\left(p\right)|⟨i_1{i}_2\cdots i_s⟩}}{\mathrm{\partial }t}+\frac{v_F^2}{2}\frac{\mathrm{\partial }{F}_{\left(p\right)|⟨⟨i_1\cdots i_{s-1}⟩}}{\mathrm{\partial }{x}_{i_s⟩}}+\frac{\mathrm{\partial }{F}_{\left(p\right)|⟨i_1\cdots i_{s}k⟩}}{\mathrm{\partial }{x}_k}\\ & =-e\left(p-s\right)F_{\left(p-1\right)|⟨i_1\cdots i_{s}k⟩}{E}_k\\ & -\frac{v_F^2}{2}e\left(p+s\right)F_{\left(p-1\right)|⟨⟨i_1\cdots i_{s-1}⟩}{E}_{i_s⟩}+P_{\left(p\right)|⟨i_1\cdots i_s⟩},\\ & p=0,1,\cdots ,N;s=0,1,\cdots ,M,\end{array}$(23) (for the more general set of moments), the flux of each balance equation is strictly correlated both with the variable of the previous equation and with the variable of the successive equation.

^c For example, for any single simulation under stationary conditions and for about $100$ values of electric fields in the range $[0;80]$ kV/cm, also using the extended HD model with a large number of moments, all the HD results are obtained with computational times of about 2 to 3 min on a standard workstation.