A stoichiometric derivative of graphene with a fluorine atom attached to each carbon is reported. Raman, optical, structural, micromechanical, and transport studies show that the material is qualitatively different from the known graphene-based nonstoichiometric derivatives. Fluorographene is a high-quality insulator (resistivity >10(12) Ω) with an optical gap of 3 eV. It inherits the mechanical strength of graphene, exhibiting a Young's modulus of 100 N m(-1) and sustaining strains of 15%. Fluorographene is inert and stable up to 400 °C even in air, similar to Teflon.

While liquid exfoliation is a powerful technique to produce defect-free nanosheets in large quantities, its usefulness is limited by broad nanosheet thickness distributions and low monolayer contents. Here we demonstrate liquid processing techniques, based on iterative centrifugation cascades, which can be designed to achieve either highly efficient nanosheet size-selection and/or monolayer enrichment. The resultant size-selected dispersions were used to establish quantitative metrics to determine monolayer volume fraction, as well as mean nanosheet size and thickness, from standard spectroscopic measurements. Such metrics allowed us to design and optimize centrifugation cascades to enrich liquid exfoliated WS2 dispersions up to monolayer contents of 75%. Monolayer-rich dispersions show relatively bright photoluminescence with narrow line widths (<35 meV) indicating the high quality of the nanosheets. The enriched dispersions display extinction spectra with distinct features, which also allow the direct estimation of monolayer contents.

We develop a first-principles theory of resonant impurities in graphene and show that a broad range of typical realistic impurities leads to the characteristic sublinear dependence of the conductivity on the carrier concentration. By means of density functional calculations various organic groups as well as adatoms such as H absorbed to graphene are shown to create midgap states within ±0.03  eV around the neutrality point. A low energy tight-binding description is mapped out. Boltzmann transport theory as well as a numerically exact Kubo formula approach yield the conductivity of graphene contaminated with these realistic impurities in accordance with recent experiments.

Liquid phase exfoliation is a powerful and scalable technique to produce defect-free mono- and few-layer graphene. However, samples are typically polydisperse and control over size and thickness is challenging. Notably, high throughput techniques to measure size and thickness are lacking. In this work, we have measured the extinction, absorption, scattering and Raman spectra for liquid phase exfoliated graphene nanosheets of various lateral sizes (90 ≤ 〈L〉 ≤ 810 nm) and thicknesses (2.7 ≤ 〈N〉 ≤ 10.4). We found all spectra to show well-defined dependences on nanosheet dimensions. Measurements of extinction and absorption spectra of nanosheet dispersions showed both peak position and spectral shape to vary with nanosheet thickness in a manner consistent with theoretical calculations. This allows the development of empirical metrics to extract the mean thickness of liquid dispersed nanosheets from an extinction (or absorption) spectrum. While the scattering spectra depended on nanosheet length, poor signal to noise ratios made the resultant length metric unreliable. By analyzing Raman spectra measured on graphene nanosheet networks, we found both the D/G intensity ratio and the width of the G-band to scale with mean nanosheet length allowing us to establish quantitative relationships. In addition, we elucidate the variation of 2D/G band intensities and 2D-band shape with the mean nanosheet thickness, allowing us to establish quantitative metrics for mean nanosheet thickness from Raman spectra.

Artificial crystal lattices can be used to tune repulsive Coulomb interactions between electrons. We trapped electrons, confined as a two-dimensional gas in a gallium arsenide quantum well, in a nanofabricated lattice with honeycomb geometry. We probed the excitation spectrum in a magnetic field, identifying collective modes that emerged from the Coulomb interaction in the artificial lattice, as predicted by the Mott-Hubbard model. These observations allow us to determine the Hubbard gap and suggest the existence of a Coulomb-driven ground state.

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Despite being only one-atom thick, defect-free graphene is considered to be completely impermeable to all gases and liquids [1][2][3][4][5][6][7][8][9][10] . This conclusion is based on theory 3-8 and supported by experiments 1,9,10 that could not detect gas permeation through micrometre-size membranes within a detection limit of 10 5 to 10 6 atoms per second. Here, using small monocrystalline containers tightly sealed with graphene, we show that defect-free graphene is impermeable with an accuracy of eight to nine orders of magnitude higher than in the previous experiments. We could discern permeation of just a few helium atoms per hour, and this detection limit is also valid for all other tested gases (neon, nitrogen, oxygen, argon, krypton and xenon), except for hydrogen. Hydrogen shows noticeable permeation, even though its molecule is larger than helium and should experience a higher energy barrier. The puzzling observation is attributed to a two-stage process that involves dissociation of molecular hydrogen at catalytically active graphene ripples, followed by adsorbed atoms flipping to the other side of the graphene sheet with a relatively low activation energy of about 1.0 electronvolt, a value close to that previously reported for proton transport 11,12 . Our work provides a key reference for the impermeability of two-dimensional materials and is important from a fundamental perspective and for their potential applications.From a theory standpoint, monolayer graphene imposes a very high energy barrier for penetration of atoms and molecules. Density-functional-theory calculations predict that the barrier E is at least several eV 2-6 , which should prohibit any gas permeation under ambient conditions. Indeed, one can estimate that at room temperature T it would take longer than the lifetime of the universe to find an atom energetic enough to pierce a defect-free membrane of any realistic size. These expectations agree with experiments that reported no detectable gas permeation through mechanically-exfoliated graphene. The highest sensitivity was achieved using micrometersize wells etched in oxidized silicon wafers, which were sealed with graphene 1,9,10 . In those measurements, a pressurized gas (e.g., helium) could permeate along the SiO 2 layer and gradually fill the microcontainers making so-called 'nanoballoons'. Their consecutive deflation in air could be monitored using atomic force microscopy (AFM), and it was shown that the leakage occurred only along the SiO 2 surface, within several minutes but independently of the number of graphene layers used for the sealing 1 . These studies allowed a conclusion that graphene membranes were impermeable to all gases, at least with the achieved accuracy of 10 5 -10 6 atoms s -1 . This was further corroborated by creating individual atomic-scale defects in graphene nanoballoons, which resulted in their relatively fast deflation/inflation and confirmed the high sensitivity of the method 9,10 .

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