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

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.