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 for low and intermediate Fermi energies. Numerical calculations have been performed within the random phase approximation. It was demonstrated that for intra-valley transitions the contribution of the different Dirac points in the wave-number dependent quantities is determined by the orientation of the wave-number 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 could give the correct physics only at long 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. Increasing the Rashba interaction strength slightly modifies the Friedel oscillations in this family of materials. Anisotropy of the dielectric function in k-space is the clear manifestation of band anisotropy in the graphene-like systems.