In this article, a scanning probe method based on nanoscale capacitance measurements was used to investigate the lateral homogeneity of the electron mean free path both in pristine and ion-irradiated graphene. The local variations in the electronic transport properties were explained taking into account the scattering of electrons by charged impurities and point defects (vacancies). Electron mean free path is mainly limited by charged impurities in unirradiated graphene, whereas an important role is played by lattice vacancies after irradiation. The local density of the charged impurities and vacancies were determined for different irradiated ion fluences.
Graphene, a two-dimensional (2D) sheet of carbon atoms in a honeycomb lattice, attracted the interest of the nanoelectronics scientific community for its remarkable carrier transport properties [1, 2]. Ideally, in a free-standing graphene sheet without lattice defects and adsorbed impurities, charge carriers can exhibit a giant intrinsic mobility  and can travel for micrometers without scattering at room temperature. As a matter of fact, very high values of mobility (>2 × 105 cm2 V-1s-1) and electron mean free path have been observed only in vacuum and at low temperature (5 K) in "suspended" graphene sheets obtained by mechanical exfoliation of highly oriented pyrolytic graphite (HOPG) . The mobility values measured at room temperature commonly reported in the literature range from approximately 2 to 2 × 104 cm2 V-1s-1, depending on the graphene synthesis methods [1, 4], on the kind of substrate on which it is deposited , and on the processing conditions used to fabricate the test patterns for electrical characterization. This large variability is a clear indication that the intrinsically outstanding transport properties of graphene are severely limited by extrinsic factors, like the presence of charged impurities, lattice defects and, more generally, by lattice disorder (including local strain). Single layers of graphene (SLG) obtained by mechanical exfoliation of HOPG  typically exhibit a very high crystalline order, whereas a high-defect density is present both in epitaxial graphene growth by thermal decomposition of SiC  and in graphene obtained by chemical reduction of graphene oxide .
Recently, the intentional production of defects in selected areas of a graphene sheet has also been proposed as a method to locally modulate the transport properties. Several methods, like plasma treatments , and electron  or ion irradiation , have been used for this aim. Recently, it has been reported that graphene hydrogenation by exposure to atomic hydrogen resulted in the conversion of graphene, a zero bandgap semiconductor, to graphane, a two-dimensional insulator . Among all these methods, ion irradiation allows a better control through a precise definition on the ion energy and fluence. Spectroscopic characterization methods, like micro Raman spectroscopy (μR), are the commonly used techniques to evaluate the density of defects in a graphene sheet. The characteristic D line at 1360 cm-1 in the Raman spectra is a fingerprint of defects/disorder in the crystalline lattice of graphitic materials. However, the lateral resolution of μR is limited by the laser spot size (typically in the order of 0.5-1 μm). In this article, we present a scanning probe method based on nanoscale capacitance measurements to determine locally (on 10-100 nm scale) the electron mean free path in pristine and in ion-irradiated graphene with different ion fluences. The impurity and vacancy densities on the probed area were extracted by fitting the experimental results with models of electron scattering by Coulomb impurities and lattice defects.
Graphene samples obtained by mechanical exfoliation of HOPG were deposited on a n+-Si substrate covered with 100 nm SiO2. Optical microscopy, tapping mode atomic force microscopy (AFM) and μR spectroscopy were used to identify SLG . Some of the as-deposited (pristine) samples were then irradiated with C+ ions at 500 keV. Irradiations of the samples with C+ ions were carried out under high vacuum conditions (10-6 Torr) to minimize surface contaminations. At 500 keV energy, the projected range of the C+ ions is approximately 1 μm, quite deep into the n+-Si substrate. This minimizes the damage both in the 100 nm SiO2 layer and at the interface between SiO2 and n+ Si. Infact, a quality of SiO2 and SiO2/Si interface comparable to that of non-irradiated samples is crucial for the capacitance measurements discussed later. Different C+ ion fluences, ranging from 1 × 1013 to 1 × 1014 ions/cm2, were used for irradiation .
The lateral homogeneity of the electronic transport properties both in pristine and ion-irradiated graphene was investigated by local capacitance measurements on the graphene/SiO2/n+Si stack, using scanning capacitance spectroscopy (SCS) [12, 15].
Scanning capacitance spectroscopy (SCS) was performed at room temperature using a DI3100 AFM by Veeco equipped with Nanoscope V electronics and with the scanning capacitance microscopy (SCM) head. SCS is an extension of the conventional SCM [16–19]. In SCS, the conductive AFM tip is placed on a discrete array of positions, lifting the tip by 20 nm at every interval. This "step and measure" approach eliminates the lateral (shear) force usually present when tip is scanned on a surface. Moreover, the vertical contact force can be suitably minimized to get a good electrical contact to the graphene layers while avoiding damage at the same time. A modulating bias ΔV = Vg/2(1 + sin(ωt)), with amplitude Vg in the range from -1.2 to 1.2 V and frequency ω = 100 kHz, was applied between the Si n+ backgate and the nanometric contact on graphene represented by a Pt-coated Si tip (see schematic in Figure 1). The ultra-high-sensitive (10-21 F/Hz1/2) capacitance sensor connected to the conductive AFM tip measures, through a lock-in system, the capacitance variation ΔC induced by the modulating bias.
Results and discussion
In Figure 2, capacitance-voltage curves measured on fixed positions on bare SiO2 and on graphene-coated SiO2 are reported for a sample not subjected to ion irradiation. The tip positions are indicated in the AFM image in the inset of Figure 2a. When the tip is in contact on bare SiO2, a typical capacitance-voltage curve for a metal-oxide-semiconductor (MOS) capacitor from accumulation (at negative sample bias) to depletion (at positive sample bias) is measured (see Figure 2a). The area of the MOS capacitor is represented by the tip contact area Atip, as illustrated in the insert of Figure 2c. When tip is in contact on graphene, the measured capacitance is minimum around zero bias and increases both for negative and positive bias (see Figure 2b). At Vg = 0, the Fermi level in graphene is almost coincident with the Dirac point. A positive modulating bias between the substrate and the tip locally induces a shift of the graphene quasi-Fermi energy EF in the conduction band, and, hence, an accumulation of electrons at the nanometric tip/graphene contact. On the contrary, a negative bias induces a shift of EF in the valence band, and, hence, an accumulation of holes at the tip/graphene contact. The carrier density n induced by the gate bias Vg can be expressed as n = Cox'Vg/q, where q is the electron charge, and Cox' is the oxide capacitance per unit area (Cox' = εoxε0/tox, being ε0 the vacuum permittivity, εox = 3.9 and tox are the relative permittivity and the thickness of the SiO2 film, respectively). The value of EF can be related to the applied bias as EF = ħvFkF, being kF = (πn)1/2, ħ the reduced Planck's constant, and vF = 1 × 106 m/s, the electron Fermi velocity in graphene. The induced charge n spreads over an area, Aeff, which can be thought as the tip-graphene-insulator-semiconductor capacitor effective area (as schematically illustrated in the insert of Figure 2c). The effective area Aeff can be evaluated from the ratio of the capacitance measured with the probe on graphene-coated regions (|ΔCgr|) and on bare SiO2 regions (|ΔCox|) , i.e., Aeff = Atip|ΔCgr|/|ΔCox|, where the tip contact area Atip can be independently determined by scanning electron microscopy (Atip = 80 nm2 in the present case). The evaluated Aeff is reported as a function of the gate bias in Figure 2c. Except for Vg = 0, Aeff increases linearly with |Vg| both for negative and positive Vg values.
It has been recently demonstrated that the effective area Aeff obtained by local capacitance measurements is related to the local electron mean free path l in graphene by Aeff = πl2. In Figure 3, l is reported versus the evaluated Fermi energy. It can be noted that l is almost independent of EF close to the Dirac point. The behavior close to the Dirac point is consistent with the common adopted picture of the 2DEG split in a landscape of adjacent "electron-hole puddles" . Close to the Dirac point, the effect of a gate bias is limited to a redistribution of carriers between the electrons and holes puddles without significantly changing the total carrier density. Figure 3 shows also that, for |EF| > 25 meV, l increases linearly with EF both in the hole and electron branches. This linear dependence gives indication on the main scattering mechanisms limiting l in our graphene samples.
Recently, expressions of the energy dependence of l have been determined for the different scattering mechanisms in the framework of a semiclassical model based on the Boltzmann transport theory . The electron mean free path limited by scattering with graphene acoustic phonons (lphon) can be expressed as 
where ρ is the graphene density (ρ = 7.6 × 10-7 kg/m2) , DA is the acoustic deformation potential (DA = 18 eV) , vs is the sound velocity in graphene , kB is the Boltzmann constant, and T is the absolute temperature.
The electron mean free path limited by Coulomb scattering with charged impurities (lci) can be expressed as 
where ε = 2.4 is the average between εox and the vacuum relative dielectric constant, Z is the net charge of the impurity (it will be assumed Z = 1), and Nci is the density of impurities.
Finally, the electron mean free path for scattering by vacancies (lvac) can be expressed as 
where Nvac is the density of vacancies in graphene and R0 is the vacancy radius, that we assumed to be coincident with the C-C distance in the graphene plane (approximately 0.14 nm).
The experimentally determined linear dependence of l on EF, far from the Dirac point, suggests that scattering with charged impurities and/or point defects, e.g., vacancies, can be assumed as the main mechanisms limiting electron mean free path.
In this pristine graphene sample, the density of defects is negligible, as confirmed by the absence of the characteristic D peak in micro-Raman spectra. Hence, charged impurities, either adsorbed on graphene surface, or located at the interface with SiO2 substrate, can be assumed as the main scattering source liming l. The density of charged impurities in the probed position can be estimated by fitting the experimental curves in Figure 3 with Equation 2. The best fit (red line) is obtained with Nci = 49 × 1010 cm-2 both for the holes and the electron branch.
In Figure 4a, l versus EF measured on an array of 5 × 5 tip positions on pristine graphene is reported. By fitting each curve of the array with Equation 2, the local density Nci for each probed position can be extracted. The histogram of the charged impurity density on the analyzed area is reported in Figure 5a. It exhibits a Gaussian distribution peaked at 〈Nci〉 = 50 × 1010 cm-2 and with FWHM of 4 × 1010 cm-2.
In Figure 4b,c, the measured l versus EF is reported for two arrays of tip positions on graphene samples irradiated with two different ion fluences, i.e., Φ = 1 × 1013 cm-2 and Φ = 1 × 1014 cm-2. Comparing the set of curves in Figure 4a, i.e., for pristine sample, with those on Figure 4b,c, it is evident that the lateral inhomogeneity in the l values increases with the irradiated fluence. However, it is worth noting that two groups of l-EF curves can be distinguished for irradiated samples: (i) a first group, with l values comparable to those in the pristine sample, (ii) a second group with reduced mean free path. We assumed that C irradiation causes the formation of point defects (vacancies), whereas the density of charged impurities adsorbed on the graphene surface or at the interface with the substrate remains almost unchanged. Hence, the first group of curves in Figure 4b,c can be associated to the probed positions on the graphene surface without or with a very low density of point defects, whereas the second group associated to the probed positions with point defects. For the first group of curves, l can be fitted using Equation 2. The histograms of the Nci values determined in the probed positions is reported in Figure 4b,c, red bars, for the lowest and highest doses, respectively. It is worth noting that the Nci distributions in irradiated samples are very similar to those of non-irradiated sample. For the second group of curves in Figure 4b,c, l is limited both by charged impurities and vacancies scattering, i.e.,
For simplicity, an average value of the charged impurities density will be assumed in those positions (〈Nci〉 = 50 × 1010 cm-2), and the local vacancy density was determined from Equations 2-4 using Nvac as the fitting parameter. The distributions of the vacancy densities in the probed positions are reported in Figure 5b,c, blue bar, for the two fluences. It is worth noting, that, while in graphene irradiated with the lowest fluence Nvac is higher than 2.5 × 1010 cm-2 (i.e. more than one vacancy on the probed area at Vg = 1V) on only 16% of the probed positions, in graphene irradiated with the highest fluence Nvac > 2.5 × 1010 cm-2 on more than 75% of the probed positions.
For each fluence, the weighted average of the vacancy density on the probed area can be obtained by , being Nvac,ithe values of the vacancy densities in the histograms and fithe associated frequencies. The obtained 〈Nvac〉 exhibits a linear increase as a function of fluence, as reported in Figure 6. This trend can be fitted by the following relation:
where 〈Nvac,0〉 is the extrapolation of the average vacancy density at Φ = 0, σ is the cross section for direct C-C collisions, Ngr is the C density in a graphene sheet (Ngr = 4 × 1015 cm-2), and ν is the vacancy generation efficiency. By linear fitting the data in Figure 6, 〈Nvac,0〉 = (1.59 ± 0.04) × 1010 cm-2 and νσNgr = (8.55 ± 0.06) × 10-4 are obtained. For the calculated values of the C-C scattering cross section σ, ranging from 2 × 10-17 to 7 × 10-17 cm2, a very low vacancy generation efficiency (ranging approximately from 0.3 to 1.1%) is obtained for graphene irradiation with 500 keV C+ ions. It might be associated to a dynamical annealing, e.g. vacancy-interstitial recombination, during irradiation.
In summary, the authors propose an innovative method based on local capacitance measurements to probe the local changes in graphene electron mean free path, due to the presence of charged impurities or point defects, e.g., vacancies. Irradiation with 500 keV C+ ions at fluences ranging from 1 × 1013 to 1 × 1014 cm-2 was used to introduce defects in SLG deposited on a SiO2/n+Si substrate. The local charged impurity and vacancy density distributions were determined for the different irradiation fluences, and a low efficiency of vacancy generation (approximately from 0.3 to 1.1%) was demonstrated.
highly oriented pyrolytic graphite
scanning capacitance microscopy
scanning capacitance spectroscopy
single layers of graphene.
The authors want to acknowledge S. Di Franco and A. Marino from CNR-IMM, Catania, for their expert assistance in sample preparation and ion irradiation experiments. This study has been supported, in part, by the European Science Foundation (ESF) under the EUROCORE program EuroGRAPHENE, within GRAPHIC-RF coordinated project.
CNR-IMM, Strada VIII
Scuola Superiore di Catania
Department of Physics and Astronomy, University of Catania
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