We investigate the nonlinear optical properties of graphene flakes using four-wave mixing. The corresponding third-order optical susceptibility is found to be remarkably large and only weakly dependent on the wavelength in the near-infrared frequency range. The magnitude of the response is in good agreement with our calculations based on the nonlinear quantum response theory. DOI: 10.1103/PhysRevLett.105.097401 PACS numbers: 78.67.Wj, 42.65.Ky, 78.47.nj Graphene, a single sheet of carbon atoms in a hexagonal lattice, is the basic building block for all graphitic materials. Although it has been known as a theoretical concept for some time [1], a layer of graphene has only recently been isolated from bulk graphite and deposited on a dielectric substrate [2]. The great interest in studying graphene is driven by its linear, massless band structureand many unusual electrical, thermal, mechanical, and optical properties [3,4] [here the upper (lower) sign corresponds to the electron (hole) band, p is the quasimomentum, and V % 10 6 m=s is the Fermi velocity]. For example, the optical absorption of graphene has been shown to be wavelength independent ('2:3% per layer) in a broad range of optical frequencies [5][6][7]. Recently, it has been predicted that the linear dispersion described by Eq. (1) should lead to strongly nonlinear optical behavior at microwave and terahertz frequencies [8]. At higher, optical frequencies one can also expect an enhanced optical nonlinearity as, due to graphene's band structure, interband optical transitions occur at all photon energies. Here we report on the first observation of the coherent nonlinear optical response of graphene at visible and nearinfrared frequencies. We show that graphene has an exceptionally high nonlinear response, described by the effective nonlinear susceptibility j ð3Þ j $ 10 À7 esu (electrostatic units). This nonlinearity is shown to be essentially dispersionless over the wavelength range in our experiments (emission at e ' 760-840 nm). These results are in good agreement with predictions derived from nonlinear quantum response theory. The large optical nonlinearity of graphene can be used for exceptionally high-contrast imaging of single and multilayered graphene flakes.Single-and few-layer graphene samples are fabricated using the method of mechanical exfoliation [2] and deposited onto a 100 m thick glass cover slip. Prior to investigation in the nonlinear microscope, the layer thickness is estimated via contrast measurements under an optical microscope, using a method similar to Ref. [9]. To investigate the nonlinear response of graphene flakes, we employ the four-wave mixing technique [10]. This involves the generation of mixed optical frequency harmonics 2! 1 À ! 2 under irradiation by two monochromatic waves with the frequencies ! 1 and ! 2 .Figure 1(a) illustrates the principle of the method: two incident pump laser beams with wavelengths 1 (tunable from 670 nm to 980 nm) and 2 (1130 nm to 1450 nm) are focused collinearly onto a sample and mix together...

We show that the manifestation of quantum interference in graphene is very different from that in conventional two-dimensional systems. Due to the chiral nature of charge carriers, it is sensitive not only to inelastic, phase-breaking scattering, but also to a number of elastic scattering processes. We study weak localization in different samples and at different carrier densities, including the Dirac region, and find the characteristic rates that determine it. We show how the shape and quality of graphene flakes affect the values of the elastic and inelastic rates and discuss their physical origin.

We have fabricated transistor structures using fluorinated single-layer graphene flakes and studied their electronic properties at different temperatures. Compared with pristine graphene, fluorinated graphene has very large and strongly temperature dependent resistance in the electro-neutrality region. We show that fluorination creates a mobility gap in graphene's spectrum where electron transport takes place via localised electron states

We report the observation of a metal-insulator transition at B 0 in a high mobility two dimensional hole gas in a GaAs-AlGaAs heterostructure. A clear critical point separates the insulating phase from the metallic phase, demonstrating the existence of a well defined minimum metallic conductivity s min 2e 2 ͞h. The s͑T͒ data either side of the transition can be "scaled" onto one curve with a single parameter T 0 . The application of a parallel magnetic field increases s min and broadens the transition. We argue that strong electron-electron interactions (r s Ӎ 10) suppress quantum interference corrections to the conductivity. [S0031-9007(98)05325-3] PACS numbers: 73.20.Dx, 71.30. + h, 73.20.Fz In the mid-1970s experiments on silicon inversion layers produced considerable evidence for the existence of a metal-insulator transition in 2D and a minimum metallic conductance, s min [1-3]. The decay constants of localized state wave functions were investigated, and it was shown that when the number of localized electrons exceeded 2 3 10 11 cm 22 the location of the mobility edge was determined by electron-electron interactions and increased with increasing carrier concentration. Subsequent theoretical work in 1979 suggested that all states in 2D were localized [4] and that phase incoherent scattering imposed a cutoff to a localized wave function giving a logarithmic correction to metallic conduction (weak localization) which was widely observed and used to obtain very detailed information on the various types of electronelectron scattering in all three dimensions [5,6]. However, in order to investigate the logarithmic correction at low, but accessible, temperatures it was necessary to use samples with low mobility so that the elastic scattering length l was small [7]. In view of the success of the theory it was then assumed that the earlier high mobility samples did not show a logarithmic correction because the phase coherence length l f was not greater than the elastic scattering length, but that if experiments could be performed at much lower temperatures (beyond the capability of cryogenics) then the logarithmic correction would be found.Recent experimental results have raised this issue again and indicate that states in 2D are not always localized with strong evidence for a metal-insulator transition in high mobility Si metal-oxide-semiconductor field-effect transistors (MOSFETs) [8]. It was found that the resistivity on both the metallic and insulating sides of the transition varied exponentially with decreasing temperature, and that a single scaling parameter could be used to collapse the data on both sides of the transition onto a single curve. While the exact nature of the transition is presently not understood, there have been several reports of similar scaling and duality between the resistivity (and conductivity) on opposite sides of the transition, both for electrons in Si MOSFETs [9,10] and for holes in SiGe quantum wells [11]. In all of these reports electron-electron interactions are known to be import...

We have fabricated graphene devices with a top gate separated from the graphene layer by an air gap-a design which does not decrease the mobility of charge carriers under the gate. This gate is used to realise p-n-p structures where the conducting properties of chiral carriers are studied. The band profile of the structures is calculated taking into account the specifics of the graphene density of states and is used to find the resistance of the p-n junctions expected for chiral carriers. We show that ballistic p-n junctions have larger resistance than diffusive ones. This is caused by suppressed transmission of chiral carriers at angles away from the normal to the junction.