We have measured a strictly linear π plasmon dispersion along the axis of individualized single wall carbon nanotubes, which is completely different from plasmon dispersions of graphite or bundled single wall carbon nanotubes. Comparative ab initio studies on graphene based systems allow us to reproduce the different dispersions. This suggests that individualized nanotubes provide viable experimental access to collective electronic excitations of graphene, and it validates the use of graphene to understand electronic excitations of carbon nanotubes. In particular, the calculations reveal that local field effects (LFE) cause a mixing of electronic transitions, including the 'Dirac cone', resulting in the observed linear dispersion. PACS numbers: 73.20.Mf,78.20.Bh Single-wall carbon nanotubes (SWNT) and its parent compound graphene are archetypes of low dimensional systems with strongly anisotropic and unique electronic properties which make them interesting for both fundamental research and as building blocks in nanoelectronic applications [1]. Their electronic bandstructure is frequently studied. In graphene, the linear band dispersion at the Fermi level, the 'Dirac cone', leads to unique characteristics in nanoelectronic devices [2]. One can expect a strong analogy between graphene and isolated SWNT for excitations along the sheet and along the tube axis, respectively. Within the zone-folding model, i.e. neglecting curvature effects, the graphene bandstructure is sliced along parallel lines when the sheet is rolled up into a cylinder. The result are characteristic van Hove singularities (VHS) in the density of states (DOS) [3]. Bulk (i.e. bundled) SWNT show an optical absorption peak at ∼ 4.5 eV due to transitions of the π electrons [4]. In vertically aligned SWNT (VA-SWNT) one finds the same peak position for onaxis polarization and an additional peak for perpendicular polarization at ∼ 5.2 eV [5]. Further information can be obtained from collective electronic excitations (plasmons) beyond the optical limit [6] (i.e. momentum transfer q > 0). Angle resolved electron energy loss spectroscopy (EELS) assesses the detailed plasmon dispersion [7,8], but it is so far missing for freestanding isolated sp 2 carbon systems. Models based on the homogeneous electron gas [9], or the tight-binding scheme [10,11] have been used to describe these excitations. The former are however bound to metallic systems. The latter have provided valuable insight and predictions for the properties of isolated sheets, tubes, and assemblies of these objects; in particular, they have predicted an almost linear plasmon dispersion for isolated systems. However, the tight binding results neglect screening beyond the π bands, and they depend on parameters that hide the underlying complexity. No realistic parameter-free calculations have been performed to predict the plasmon dispersion in these systems, nor has its origin been analyzed. Instead, ab initio spectroscopy calculations have dealt with absorption spectra (q → 0) for SWNT [12,13,14],...

International audienceIn this joint experimental and theoretical work, we investigate collective electronic excitations (plasmons) in free-standing, single-layer graphene. The energy- and momentum-dependent electron energy-loss function was measured up to 50eV along two independent in-plane symmetry directions (ΓM and ΓK) over the first Brillouin zone by momentum-resolved electron energy-loss spectroscopy in a transmission electron microscope. We compare our experimental results with corresponding time-dependent density-functional theory calculations. For finite momentum transfers, good agreement with experiments is found if crystal local-field effects are taken into account. In the limit of small and vanishing momentum transfers, we discuss differences between calculations and the experimentally obtained electron energy-loss functions of graphene due to a finite momentum resolution and out-of-plane excitations

High-energy collective electronic excitations (plasmons) in freestanding multilayer graphene are studied by momentum-resolved electron energy-loss spectroscopy (EELS). For normal incidence, only the high-energy plasmon band is excited and we measure a blueshift of the π-plasmon dispersion with increasing thickness. The observed transition between two-dimensional and three-dimensional behavior is explained using a layeredelectron-gas (LEG) model. We propose a method to measure all individual plasmon bands by tilting the sample with respect to the electron beam. As a proof of concept, EELS experiments for three-layer graphene are compared with predictions from the LEG model.

The electron energy-loss function of graphite is studied for momentum transfers q beyond the first Brillouin zone. We find that near Bragg reflections the spectra can change drastically for very small variations in q. The effect is investigated by means of first principle calculations in the random phase approximation and confirmed by inelastic x-ray scattering measurements of the dynamic structure factor S(q, omega). We demonstrate that this effect is governed by crystal local field effects and the stacking of graphite. It is traced back to a strong coupling between excitations at small and large momentum transfers.

The optical design and analysis of modern micro-optical elements with high index contrasts and large numerical apertures is still challenging, as fast and accurate wave-optical simulations beyond the thin-element-approximation are required. We introduce a modified formulation of the wave-propagation-method and assess its performance in comparison to different beam-propagation-methods with respect to accuracy, required sampling densities, and computational performance. For typical micro-optical components, the wave-propagation-method is found to be considerably faster and more accurate at even lower sampling densities compared to the different beam-propagation-methods. This enables realistic wave-optical simulations beyond the thin-element-approximation for micro-optical components. As an example, the modified wave-propagation-method is applied for in-line holographic measurements of strongly diffracting objects. From a direct comparison of experimental results and corresponding simulations, the geometric parameters of a test object could be retrieved with high accuracy.

High-energy electronic excitations of graphene and MoS 2 heterostructures are investigated by momentumresolved electron energy-loss spectroscopy in the range of 1 to 35 eV. The interplay of excitations on different sheets is understood in terms of long-range Coulomb interactions and is simulated using a combination of ab initio and dielectric model calculations. In particular, the layered electron-gas model is extended to thick layers by including the spatial dependence of the dielectric response in the direction perpendicular to the sheets. We apply this model to the case of graphene/MoS 2 /graphene heterostructures and discuss the possibility of extracting the dielectric properties of an encapsulated monolayer from measurements of the entire stack.

Transmission electron microscopy has been a promising candidate for mapping atomic orbitals for a long time. Here, we explore its capabilities by a first-principles approach. For the example of defected graphene, exhibiting either an isolated vacancy or a substitutional nitrogen atom, we show that three different kinds of images are to be expected, depending on the orbital character. To judge the feasibility of visualizing orbitals in a real microscope, the effect of the optics' aberrations is simulated. We demonstrate that, by making use of energy filtering, it should indeed be possible to map atomic orbitals in a state-of-the-art transmission electron microscope.

Decomposition of a general arbitrary field into a set of Gaussian beams has been one of the challenges in the Gaussian beam decomposition method for field propagation through optical systems. The most commonly used method in this regard is the Gabor expansion, which decomposes initial fields into shifted and rotated Gaussian beams in a plane. Since the Gaussian beams used have zero initial curvatures, the Gabor expansion method does not utilize the ability of the Gaussian beams to represent the quadratic behavior of the local wavefront. In this paper, we describe an alternative method of decomposing an arbitrary field with smooth wavefront into a set of Gaussian beams with non-zero initial curvatures. The individual Gaussian beams are used to represent up to the quadratic term in the Taylor expansion of the local wavefront. This significantly reduces the number of Gaussian beams required for the decomposition of the field with smooth wavefront and gives more accurate decomposition results. The proposed method directly gives the five ray sets representing the parabasal Gaussian beams, which can then be directly used for propagation of the Gaussian beams through optical systems. To demonstrate the application of the method, we have presented results for the decomposition of fields with strongly curved spherical wavefronts, a cone shaped wavefront, and a wavefront with large spherical aberration. The numerical comparison of the input field with the field reconstructed after the decomposition shows very good agreement in both amplitude and phase profiles. We also show results for the far field intensity distributions of the decomposed wavefronts by propagating in free space using the Gaussian beam propagation method.