Anisotropic Short-Wavelength Friedel Oscillation with Adjustable Envelope in Graphene

We investigate theoretically the intervalley charge density oscillation and the screened ionic potential in graphene caused by the intervalley scattering. We demonstrate that the contribution from the intervalley scattering is comparable with that from the intravalley scattering, and oscillation rather than decaying dominates at a large distance away from the external impurity. We show that the intervalley oscillation is strongly anisotropic because of the inequivalency between neighboring valleys. The anisotropic oscillation consists of an anisotropic short-wavelength oscillation with an anisotropic fixed wavelength and an isotropic long-wavelength envelop with an isotropic wavelength modulated by doping, making an adjustable-widthed wave-packet propagation. One weakens the screening and gets anisotropic short-wavelength oscillation by introducing short-range mechanism in graphene and graphene-like materials.