Dirac-equation description of the electronic states of graphene with a line defect: Wave-function connection condition

Abstract

The presence of an extended line defect in graphene brings about some interesting electronic properties to such a truly two-dimensional (2D) carbon material, such as the energy-band engineering and valley filtering. By establishing an appropriate connection condition for the spinor wave function across the line defect, we find that the massless Dirac equation is still a valid theoretical model to describe low-energy electronic properties of the line defect embedded graphene structure. To check the validity of the wave-function connection condition, we take two kinds of line defect embedded graphene structures as examples to study the low-energy electronic states by solving the Dirac equation. First, for a line defect embedded zigzag-edged graphene nanoribbon, we obtain analytical results about the subband dispersion and eigenwave function, which coincide well with the numerical results from the tight-binding approach. Then, for a 2D graphene embedded with an extended line defect, we get an exact expression about the valley polarized electronic transmission probability, which demonstrates the simple result estimated previously in the zero-energy limit. More interestingly, our analytical result indicates that in such a 2D graphene structure a quasi-one-dimensional (1D) electronic state occurs along the line defect. And the electronic group velocity in this quasi-1D electronic state can be readily modulated by applying a strain field around the line defect.