Gaussian Quadrature and Lattice Discretization of the Fermi-Dirac Distribution for Graphene

TL;DR

This paper develops a lattice kinetic scheme for simulating electronic flow in graphene using Gaussian quadrature and lattice discretization of the Fermi-Dirac distribution, validated through Riemann problem solutions.

Contribution

It introduces a novel orthogonal polynomial basis and a discretization method that reduces momentum space points to 18, enabling efficient numerical modeling of graphene's electronic flow.

Findings

Accurate simulation of electronic flow in graphene using the proposed lattice scheme.

Validation of the model through Riemann problem solutions showing excellent agreement.

Demonstration that increased chemical potential effectively raises the electronic fluid's viscosity.

Abstract

We construct a lattice kinetic scheme to study electronic flow in graphene. For this purpose, we first derive a basis of orthogonal polynomials, using as weight function the ultrarelativistic Fermi-Dirac distribution at rest. Later, we use these polynomials to expand the respective distribution in a moving frame, for both cases, undoped and doped graphene. In order to discretize the Boltzmann equation and make feasible the numerical implementation, we reduce the number of discrete points in momentum space to 18 by applying a Gaussian quadrature, finding that the family of representative wave (2+1)-vectors, that satisfies the quadrature, reconstructs a honeycomb lattice. The procedure and discrete model are validated by solving the Riemann problem, finding excellent agreement with other numerical models. In addition, we have extended the Riemann problem to the case of different dopings, finding that by increasing the chemical potential, the electronic fluid behaves as if it increases its effective viscosity.