The Local Density of States in Monolayer and Bilayer Graphene in the Presence of an Impurity

We analyze the effect of a single localized impurity on the local density of states in monoand bilayer undoped graphene. We show that for monolayer graphene the Friedel oscillations generated by intranodal scattering of quasiparticles obey an inverse-square law, while those generated by internodal scattering obey an inverse law. Unlike the former, the latter oscillations may break rotational symmetry to a discrete subgroup. For bilayer graphene we find that both oscillations obey an inverse law. We discuss how these signatures can be observed in Fourier transform scanning tunneling spectroscopy (FTSTS) experiments, in particular how one can experimentally distinguish between monolayer and bilayer graphene, as well as between localized and extended impurities. We also show that in general FTSTS spectra contain information not only about the band structure, but also about the underlying Hamiltonian of two-dimensional systems. Graphene has been studied extensively in the recent years, especially after it became possible to fabricate monolayer and few-layer samples [1, 2]. Its most fascinating aspect is the existence of linearly dispersing gapless excitations in the vicinity of the Fermi points. This gives rise to very interesting electronic properties such as Friedel oscillations in the local density of states at low energy which decay as 1/r [3, 4] instead the usual 1/r characteristic to twodimensional systems [5, 6]. It is very important to understand well the physics of impurity scattering in graphene, by studying for example the local density of states (LDOS) in the presence of single impurity scattering [3, 4, 5, 7]. Comparison with experiments can provide information about graphene’s fundamental physics, and about the nature of the impurities. In this Letter we analyze the Fourier transform of the density of states measurable by Fourier transform scanning tunneling spectroscopy (FTSTS). Such measurements have recently been developed for graphene [8], as well as for other two dimensional materials such as ErSi2 [9] and the cuprates [10]. Our first observation is that the FTSTS spectra can be used to distinguish between monolayer and bilayer graphene. In particular, for monolayer graphene with a localized (deltafunction) impurity potential, at low STM bias, the Friedel oscillations generated by intranodal scattering processes decay as 1/r, consistent with previous analysis [3, 4]. In the FTSTS