Band Gaps and Wavefunctions of Electrons Coupled to Pseudo Electromagnetic Waves in Rippled Graphene

TL;DR

This paper investigates how sinusoidal out-of-plane ripples in graphene influence electron wavefunctions and energy band gaps, revealing resonance conditions and diffraction effects through a non-perturbative Dirac equation approach.

Contribution

It introduces a non-perturbative method to analyze electron-ripple coupling in graphene, deriving band structures and wavefunctions using Mathieu equations and resonance conditions.

Findings

Band gaps occur at energies approximately equal to n times the Fermi velocity times Planck's constant times the ripple wave-vector.

Electron diffraction in phase with the ripple causes the formation of energy gaps.

Wavefunctions are expressed in Mathieu cosine/sine functions or Bessel functions depending on the potential type.

Abstract

The effects of a propagating sinusoidal out-of-plane flexural deformation in the electronic properties of a tense membrane of graphene are considered within a non-perturbative approach, leading to an electron-ripple coupling. The deformation is taken into account by introducing its corresponding pseudo-vector and pseudo-scalar potentials in the Dirac equation. By using a transformation to the time-cone of the strain wave, the Dirac equation is reduced to an ordinary second-order differential Matthieu equation, i.e., to a parametric pendulum, giving a spectrum of bands and gaps determined by resonance conditions between the electron and ripple wave-vector (G), and their incidence angles. The location of the nth gap is thus determined by $E\approx n v_{F}\hbar G$, where $v_F$ is the Fermi velocity. Physically, gaps are produced by diffraction of electrons in phase with the wave. The propagation is mainly in the direction of the ripple. In the case of a pure pseudoelectric field and for energies lower than a certain threshold, we found a different kind of equation. Its analytical solutions are in excellent agreement with the numerical solutions. The wavefunctions can be expressed in terms of the Matthieu cosine and sine functions, and for the case of a pure pseudo-electric potential, as a combination of Bessel functions.