Anisotropic Friedel oscillations in graphene-like materials: The Dirac point approximation in wave-number dependent quantities revisited

Friedel oscillations of the graphene-like materials are investigated theoretically beyond the Dirac point-approximation. Numerical calculations have been performed within the random phase approximation (RPA). For intra-valley transitions it was demonstrated that the contribution of the different Dirac points in the wave-number dependent quantities, such as dielectric function $\epsilon(q)$, has been determined by the orientation of the wave-number with respect to the Dirac point position vector in $k$-space. Therefore identical contribution of the different Dirac points is not automatically guaranteed by the degeneracy of the Hamiltonian at these points. Meanwhile it was shown that the contribution of the inter-valley transitions is always anisotropic even when the Dirac points coincide with the Fermi level ($E_F=0$). This means that the Dirac point approximation based studies give the correct physics only at high wave length limit. The anisotropy of the static dielectric function reveals different contribution of the each Dirac point. Additionally, the anisotropic $k$-space dielectric function results in anisotropic Friedel oscillations in graphene-like materials. Calculations have also been performed in the presence of the Rashba interaction. It was shown that increasing the Rashba interaction strength slightly modifies the Friedel oscillations in graphene-like materials. Therefore the anisotropic dielectric function in $k$-space is the clear manifestation of band anisotropy in the graphene-like systems.