Effect of a single impurity on the local density of states in monolayer and bilayer graphene

We use the T-matrix approximation to analyze the effect of a localized impurity on the local density of states in monoand bilayer graphene. For monolayer graphene the Friedel oscillations generated by intranodal scattering obey an inverse-square law, while the internodal ones obey an inverse law. In the Fourier transform this translates into a filled circle of high intensity in the center of the Brillouin zone, and empty circular contours around its corners. For bilayer graphene both types of oscillations obey an inverse law. If the impurity is extended, the internodal scattering is greatly reduced. These features give information not only on the band structure but also on the Hamiltonian, and should also be investigated for other 2D systems such as the cuprates. 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 Dirac 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, the Friedel oscillations generated by intranodal scattering decay as 1/r at low STM bias, consistent with previous analysis [3, 4]. In the FTSTS spectra this is manifested by a filled circle of high intensity in the center of the Brillouin zone (BZ), with a radius proportional to the STM bias. On the other hand, the Friedel oscillations generated by the scattering of quasiparticles between different Dirac points decay as 1/r. In the FTSTS spectra these oscillations are translated into circular contours of high intensity centered