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Hydrodynamical Model for Charge Transport in Graphene

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Abstract

A hydrodynamical model for simulating charge transport in graphene is formulated by using of the maximum entropy principle. Both electrons in the conduction band and holes in the valence band are considered and it is assumed a linear dispersion relation for the energy bands around the equivalent Dirac points. The closure relations do not contain any fitting parameters except the ones already present in the kinetic description.

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Notes

  1. For a couple of functions \(f(x)\) and \(h(x)\), \( f(x) \sim h(x)\) means \(f(x)/h(x) \rightarrow 1\) as \( x \rightarrow + \infty \).

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Acknowledgments

The authors acknowledge the financial support by P.R.A. University of Catania, the P.R.I.N. project 2010 “Kinetic and macroscopic models for particle transport in gases and semiconductors: analytical and computational aspects”, by the project MIUR PON “AMBITION POWER”.

Author information

Authors and Affiliations

  1. NEST, Istituto Nanoscienze-CNR, Scuola Normale Superiore, Piazza San Silvestro 12, 56127, Pisa, PI, Italy

    V. D. Camiola

  2. Dipartimento di Matematica e Informatica, Universitá degli Studi di Catania, Viale A. Doria 6, 95125, Catania, CT, Italy

    V. Romano

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Correspondence to V. Romano.

Appendix

Appendix

Here we get the closure relation for the fluxes (39–40). For the sake of simplifying the notation the index indicating electron and holes is omitted because no difference arises. Let \(\mathbf{n}\) be the unit vector of \(\mathbf{k}\). One has \(\mathbf{v} = v_F \mathbf{n}\). The relations

$$\begin{aligned} \int _{S_1} n_i n_j d \, \varphi = \pi \delta _{ij} \quad \text{ and } \quad \int _{S_1} n_{i_1} n_{i_2} \cdots n_{i_r} \, d\, \varphi = 0 \quad \text{ if } \,\, r \, \text{ odd } \end{aligned}$$

hold, \(n_i\)’s being the components of \(\mathbf{n}\) in a base and \(d \varphi \) the elementary angle of the unit circle \(S_1\) of \(\mathbb {R}^2\). From the kinetic definitions (20) by approximating the distribution function with the linearized MEP one (25) we have

$$\begin{aligned}&\rho _A \, \left( \begin{array}{c} F^{(0)}_{ij} \\ F^{(1)}_{ij} \end{array} \right) \approx \frac{2}{(2 \, \pi )^2} \int _{\mathbb {R}^2} \left( \begin{array}{c} 1\\ \varepsilon \end{array} \right) v_i \, v_j f_{MEP} (\mathbf{r},\mathbf{k},t) \,d^2 \mathbf{k}\\&\quad =\frac{2 v_F^2}{(2 \, \pi \, \hbar v_F)^2} \int _{S_1} n_i n_j \, d\, \varphi \int _0^{+\infty } \left( \begin{array}{c} 1\\ \varepsilon \end{array} \right) \frac{\varepsilon }{1+ e^{\lambda _A+\lambda _{W_A} \varepsilon }} \, d\, \varepsilon = \frac{\delta _{ij}}{2 \, \pi \, \hbar ^2} \left( \begin{array}{c} I_1\\ I_2 \end{array} \right) . \end{aligned}$$

Note that the integral of the anisotropic part of \(f_{MEP}\) vanishes because the integrand contains only odd powers in the components of the velocity. The calculations to get (21) are similar.

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Camiola, V.D., Romano, V. Hydrodynamical Model for Charge Transport in Graphene. J Stat Phys 157, 1114–1137 (2014). https://doi.org/10.1007/s10955-014-1102-z

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