Friedel Oscillations Around a Short Range Scatterer: The Case of Graphene

Abstract

We review the basic theory of Friedel oscillations around a well localized impurity using Green's function technique in Born approximation. After a pedagogical presentation of the case of free electrons in various dimensions, we turn our attention to graphene, which is described by a tight-binding scheme. Utilizing the localized nature of the atomic wavefunctions we calculate the change in the electronic density due to the impurity, and identify, in addition to the slowly decaying long wavelength oscillations, a short wavelength pattern as well. The latter, being of opposite sign on the two sublattices of graphene, may cancel the leading inverse square envelope of the long wavelength oscillations, if a probe with resolution worse than a few unit cells is used. We corroborate these findings by exact diagonalization results on a 21 × 21 unit cell graphene sheet.

Friedel oscillations in graphene: Sublattice asymmetry in doping

“…These changes in the electronic scattering manifest as spatial oscillations, called Friedel oscillations (FOs), in quantities like the local density of states (ρ) and the carrier density (n), which radiate away from the location of the symmetry breaking perturbation and decay with the distance from the perturbation, D, with a rate linked directly to the dimensionality of the system and to some extent the resolution of the measurement. Much attention has been focused recently on such symmetry breaking in graphene [2][3][4][5][6][7][8][9] , a hexagonal lattice of sp-2 hybridised carbon with a wide range of unique properties 10 . Fig.…”

“…Previous studies examining the analytical behaviour of ∆n FOs in graphene have generally relied on a linearisation of the electronic bandstructure near the Dirac points and the introduction of a momentum cutoff [2][3][4]8 . In the current work, we present an alternative framework which removes these assumptions and matches numerical results exactly in the long-distance limit and over large energy ranges, paving the way for applications to other electronic quantities and graphene-like materials.…”

Sublattice imbalance of substitutionally doped nitrogen in graphene

“…2). In general terms, LDOS oscillations induce oscillations in other quantities (often referred to as Friedel Oscillations) that depend on the spatial distribution of impurities [24][25][26]. Because the total energy is one such a quantity, C γ 1 ,γ 2 1,2 tends to oscillate between positive and negative values but the commensurability effect masks these oscillations in such a way that impurities on the same (different) sublattices always yield negative (positive) values.…”