A kinetic model for the transport of electrons in a graphene layer

TL;DR

This paper introduces a new kinetic model for electron transport in graphene, derived from quantum principles, and implements a particle-based numerical method to simulate non-adiabatic transitions with energy conservation.

Contribution

The paper develops a coupled Boltzmann equation model with a Landau-Zener collision term for graphene electron transport, including an energy-conserving jump operator.

Findings

The kinetic model accurately captures non-adiabatic transitions.

The particle method effectively simulates the coupled equations.

Numerical experiments validate the model's accuracy and energy conservation.

Abstract

In this article, we propose a new numerical model for computation of the transport of electrons in a graphene device. The underlying quantum model for graphene is a massless Dirac equation, whose eigenvalues display a conical singularity responsible for non adiabatic transitions between the two modes. We first derive a kinetic model which takes the form of two Boltzmann equations coupled by a collision operator modeling the non-adiabatic transitions. This collision term includes a Landau-Zener transfer term and a jump operator whose presence is essential in order to ensure a good energy conservation during the transitions. We propose an algorithmic realization of the semi-group solving the kinetic model, by a particle method. We give analytic justification of the model and propose a series of numerical experiments studying the influences of the various sources of errors between the quantum and the kinetic models.