Numerical quasi-conformal transformations for electron dynamics on strained graphene surfaces

TL;DR

This paper develops numerical methods using quasi-conformal transformations to analyze electron dynamics on strained graphene surfaces, enabling better understanding of electron scattering and focusing effects due to surface deformations.

Contribution

It introduces two strategies, including a quasi-conformal transformation approach with a finite-element scheme, to simplify and solve the Dirac equation on curved graphene surfaces.

Findings

Electron wave packets can be focused by local strained regions.

The quasi-conformal transformation approach effectively simplifies the Dirac equation.

Numerical schemes accurately simulate electron scattering on deformed graphene.

Abstract

The dynamics of low energy electrons in general static strained graphene surface is modelled mathematically by the Dirac equation in curved space-time. In Cartesian coordinates, a parametrization of the surface can be straightforwardly obtained, but the resulting Dirac equation is intricate for general surface deformations. Two different strategies are introduced to simplify this problem: the diagonal metric approximation and the change of variables to isothermal coordinates. These coordinates are obtained from quasi-conformal transformations characterized by the Beltrami equation, whose solution gives the mapping between both coordinate systems. To implement this second strategy, a least square finite-element numerical scheme is introduced to solve the Beltrami equation. The Dirac equation is then solved via an accurate pseudo-spectral numerical method in the pseudo-Hermitian representation that is endowed with explicit unitary evolution and conservation of the norm. The two approaches are compared and applied to the scattering of electrons on Gaussian shaped graphene surface deformations. It is demonstrated that electron wave packets can be focused by these local strained regions.