The Evolution of Electronic Structure in Few-Layer Graphene Revealed by Optical Spectroscopy

The evolution of electronic structure in few-layer graphene revealed by optical spectroscopy

Kin Fai Maka, Matthew Y. Sfeirb, James A. Misewichb, and Tony F. Heinza,1 aDepartments of Physics and Electrical Engineering, Columbia University, New York, NY 10027; and bBrookhaven National Laboratory, Upton, NY 11973

Edited by Mildred Dresselhaus, Massachusetts Institute of Technology, Cambridge, MA, and approved July 8, 2010 (received for review April 5, 2010)

The massless Dirac spectrum of electrons in single-layer graphene are present, whereas if N is even, only massive components exist. has been thoroughly studied both theoretically and experimen- This behavior is broadly analogous to that known to occur for the tally. Although a subject of considerable theoretical interest, model 1D system of single-walled carbon nanotubes. In this case, experimental investigations of the richer electronic structure of zone-folding of the 2D graphene band structure implies that cer- few-layer graphene (FLG) have been limited. Here we examine tain nanotube physical structures have metallic character with FLG graphene crystals with Bernal stacking of layer thicknesses massless electrons, but others with slightly different chiral indices N ¼ 1,2,3,…8 prepared using the mechanical exfoliation technique. have semiconducting character and only massive electrons (25). For each layer thickness N, infrared conductivity measurements Our investigations show that FLG, whereas preserving many of over the spectral range of 0.2–1.0 eV have been performed and the unusual features of SLG, provides a richness and flexibility reveal a distinctive band structure, with different conductivity of electronic structure that should find many novel applications. peaks present below 0.5 eV and a relatively flat spectrum at higher photon energies. The principal transitions exhibit a systematic Results energy-scaling behavior with N. These observations are explained FLG samples of layer thicknesses N ¼ 1;2;3;…8 were prepared within a unified zone-folding scheme that generates the electronic using mechanical exfoliation of kish graphite. To facilitate states for all FLG materials from that of the bulk 3D graphite crystal analysis of the optical data, the samples were deposited onto through imposition of appropriate boundary conditions. Using the transparent fused quartz substrates. (The procedure for sample PHYSICS Kubo formula, we find that the complete infrared conductivity preparation and identification of the layer thickness is described spectra for the different FLG crystals can be reproduced reasonably in Materials and Methods.) The FLG conductivity spectra were well within the framework a tight-binding model. obtained using broadband infrared radiation from the Brookha- ven National Synchrotron Light Source (NSLS) synchrotron electronic structures ∣ infrared spectroscopy ∣ zone-folding method and a FTIR spectrometer to analyze the spectral content of the radiation. To determine the infrared sheet conductivity σðℏωÞ he unique electronic properties of graphene, a single-mono- of the FLG samples as function of the photon energy ℏω, we mea- 2 Tlayer of sp -hybridized carbon, have attracted much attention sured the reflectance spectrum of the bare substrate and that of (1). Graphene’s few-layer counterparts have also recently been the substrate covered with a FLG sample. For a layer of material, the subject of much interest, since this broader class of materials like these samples, with weak absorption and a thickness much less offers the potential for further control of electronic states by in- than a wavelength of light supported by a transparent substrate, terlayer interactions (2–6). Indeed, theoretical investigations the change in reflectance is directly proportional to the infrared have predicted dramatic changes in the electronic properties conductivity σðℏωÞ (26). As discussed in Materials and Methods, in few-layer graphene (FLG) compared with single-layer gra- the factor of proportionality is given by a function by the refractive – phene (SLG) (7 17): When two or more layers of graphene are index of the substrate. present in ordered FLG, the characteristic linearly dispersing The influence of layer thickness on the electronic properties of bands of the single layer are either replaced or augmented by FLG is already clear in a comparison of the conductivity spectrum pairs of split hyperbolic bands. These new bands correspond for SLG and bilayer graphene (Fig. 1). The conductivity σðℏωÞ to fermions of finite mass, unlike the electrons present in SLG for SLG is featureless, as reported previously (26, 27). At photon that behave as massless fermions. Further, the characteristics of energies ℏω approaching 1 eV, the sheet conductivity σðℏωÞ is the electrons in FLG are expected to change sensitively with in- largely independent of energy, with a value of σ ≈ πe2∕2h (or, creasing layer number N, before ultimately approaching the bulk equivalently, an absorbance of A ¼ πα, where α denotes the limit of graphite. Despite these fascinating predictions, experi- fine-structure constant). Such universal behavior is intrinsic to mental investigations have been limited to SLG and a few studies the massless fermionic character of the structure for the π elec- of the electronic properties of bilayer graphene (18–24). We ex- trons near the K-point of the Brillouin zone (Fig. 1, Inset). The amine these predictions experimentally by probing the electronic structure of FLG graphene samples for layer thickness up to N ¼ numerical value of the conductivity follows from these general 8 using infrared conductivity spectroscopy. For each thickness, we properties in 2D (28), once the 4-fold spin and valley degeneracy find well-defined and distinct peaks arising from the critical present in SLG is taken into account. A departure from this uni- ℏω 0 5 points for transitions between valence and conduction bands versal behavior can be seen for photon energies < . eV, an of the relevant FLG material. The position and shape of these effect attributed to Pauli blocking from unintentional doping and features in the experimental spectra, recorded over a photon- finite temperature effects (26). energy range of 0.2–0.9 eV, provide direct information about key features of the band structure of FLG materials. Author contributions: K.F.M., M.Y.S., J.A.M., and T.F.H. designed research; K.F.M. and M.Y.S. We are able to compare the positions of these features with the performed research; K.F.M. and T.F.H. analyzed data; and K.F.M., M.Y.S., J.A.M., and T.F.H. predictions of the zone-folding model of FLG electron structure wrote the paper. that is introduced below. From the analysis, one can decompose The authors declare no conflict of interest. the electronic structures of FLG into chiral massless and massive This article is a PNAS Direct Submission. components with characteristics (10, 12–14), such as masses 1To whom correspondence should be addressed. E-mail: [email protected]. and Fermi velocities, that depend critically on layer thickness. This article contains supporting information online at www.pnas.org/lookup/suppl/ In particular, if N is odd, both massless and massive components doi:10.1073/pnas.1004595107/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1004595107 PNAS ∣ August 24, 2010 ∣ vol. 107 ∣ no. 34 ∣ 14999–15004 transitions, we observe a systematic evolution in the energies of the principal transitions as a function of the layer thickness N. Three families of transitions, each exhibiting a smooth energy- scaling behavior with N, can be identified (solid curves in Fig. 2B). Discussion To understand these observations, we introduce a description of FLG in which the electronic states are obtained by zone folding of the 3D band structure of the parent graphite crystal from which any FLG sample can be thought of as having been formed (Fig. 3). This approach is similar to the construction of the electronic states of different 1D carbon nanotubes from zone folding of the 2D bands of graphene. Tocarry out the scheme, we work within a TB description of the electronic structure of bulk graphite. This type of model was introduced many years ago to describe its valence and conduction bands (32). This treatment has been the basis of widely adopted descriptions of the bands of both single- Fig. 1. Measured infrared sheet conductivity spectra σðℏωÞ of single- and and bilayer graphene (7, 8). Here we present a generalization bilayer graphene samples. In addition to the calibration of the sheet conduc- that readily generates the band structure of single- and bilayer tivity in units of πe2∕2h, the equivalent absorbance is indicated in units πα ¼ 2 29% graphene, but also extends to FLG of arbitrary layer thickness. of . . The optical transitions responsible for the main features We consider the 3D bands of graphite within a model that in- observed are shown as inserts. cludes the nearest in-plane coupling coefficient of γ0 ¼ 3.16 eV and couplings between atoms in adjacent planes characterized by For bilayer graphene, rather than the featureless spectrum of strengths of γ1 ¼ 370 meV, γ3 ¼ 315 meV, and γ4 ¼ 44 meV (10, SLG, a sharp resonance is observed in the infrared conductivity 13, 32). In this discussion, we ignore the next nearest interlayer σðℏωÞ at ℏω ¼ 0.37 eV (Fig. 1). This change can be readily un- coupling of γ2 ¼ −20 meV (10, 13, 32) and the A∕B on-site derstood in terms of the significantly altered low-energy 2D band potential difference (absent in SLG) of Δ ¼ 5 meV (10, 13, 32). structure of the bilayer. Instead of the massless fermions found in Because these latter quantities are significantly smaller than the SLG, electrons in bilayer graphene have finite masses and are Inset energy scale and line widths of the optical transitions (Figs. 1 described by a pair of hyperbolic bands (7, 8) (Fig. 1, ). and 2), they are not expected to be important in describing No band gap is opened, but the pairs of valence and conduction our absorption measurements. Within this description of the gra- bands are split. Their separation, within the tight-binding (TB) phite band structure, the following are the key features. Along the model described below is given by interlayer coupling strength H–KðzÞ direction, there are four bands (Fig. 3): two degenerate γ1. These hyperbolic bands lead to a step singularity, character- bands (E3) without dispersion and two dispersive bands (E1 and istic of 2D massive particles, in the joint density of states at the E2) with energies of 2γ1 cosðkzc∕2Þ. The bands all become onset of interband transitions. Considering the presence of spon- degenerate at the H point, where the in-plane dispersion of taneous doping of the sample, we should also observe optical the (doubly degenerate) bands is linear and isotropic. For all transitions from the lower conduction band to the upper conduc- other planes perpendicular to the H–K direction, the in-plane tion band, as well as from the upper valence band to the upper dispersion is described by split pairs of nearly hyperbolic conduc- conduction band (Fig. 1, Inset). These transitions give rise to a tion and valence bands (Fig. 3). Because of the trigonal warping prominent peak in σðℏωÞ at ℏω ¼ γ1 (15, 29, 30). For photon associated with the finite value for γ3, the higher valence and energies well above this value (ℏω ≫ γ1), the two layers become lower conduction bands exhibit slight crossings (over a few meV largely decoupled and infrared conductivity σðℏωÞ approaches 2 range of energy). Whereas this effect is critically important for twice of the universal value of πe ∕2h found in SLG (Fig. 1). defining the Fermi surface, it does not influence the optical spec- Whereas we can understand the characteristic features of the tra significantly. (For further discussion, please refer to Materials band structure and infrared conductivity for bilayer graphene and Methods.) rather directly from considering two interacting graphene layers, With only interactions between adjacent planes and neglecting the behavior is expected to become increasingly complex as the structural relaxation effects, the electronic states of the Hamilto- layer number N grows. Indeed, the data for the infrared conduc- nian for FLG are simply a subset of those obtained for bulk tivity σðℏωÞ for FLG up to 8 layers (Fig. 2A) show highly graphite (see Materials and Methods for justification). The addi- structured and readily distinguishable spectra for each value of tional zone-folding criterion that applies to the N-layer sample is N* (31). We note that the absolute strengths exhibited by the that the wavefunctions must vanish at the position where the various resonances do not decrease significantly with layer num- graphene planes lying immediately beyond the physical material; ber N. However, if we consider infrared conductivity spectra i.e., at the positions of the 0th and ðN þ 1Þth layer. Because of the normalized per layer, σðℏωÞ∕N, as we discuss below, we see a presence of mirror symmetry for odd N and inversion symmetry clear convergence toward the behavior of bulk graphite. In the N thicker samples, we observe the emergence of the single broad for even , this can be accomplished by forming standing waves feature present in graphite at approximately twice the energy with momenta perpendicular to the graphene planes quantized as of the peak in the bilayer sample. 2πn The trends in the data for different layer thicknesses can be kz ¼ : [1] identified from a contour plot of the normalized conductivity ðN þ 1Þc spectra for all the FLG samples (Fig. 2B). Rather than the see- mingly random variation in the energies of the principal optical Here c∕2 is the interlayer separation (0.34 nm); the allowed values for the index are n ¼1; 2; 3;…: □ðN þ 1Þ∕2□, where the symbol □ denotes the integer part of the quantity. *Significantly different infrared conductivity spectra for samples with N>2 were Independent (standing-wave) states are generated only for posi- observed occasionally. These spectra arose from FLG samples that did not have Bernal n k ≤ π∕c (AB) crystallographic stacking. The properties of these samples will be presented tive values of for which z , corresponding to the positive elsewhere. half of the graphite Brillouin zone. For N-layer graphene, the

15000 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1004595107 Mak et al. Fig. 2. Infrared conductivity spectra σðℏωÞ of FLG samples of layer thickness N ¼ 1;2;3;…8.(A) The experimental data for the eight different layer thickness, presented as in Fig. 1. The data for different samples have not been offset. (B) A contour plot of the infrared conductivity per layer, σðℏωÞ∕N, as a function of photon energy and N. The dots indentify the positions of the peaks in experimental infrared conductivity. These transition energies are seen to follow three well-defined energy-scaling relations. The solid curves are the theoretical predictions based on the zone-folding model, as described in the text. number of independent 2D planes in the 3D Brillouin zone generate 1 set of chiral massless component and ðN − 1Þ∕2 sets of (i.e., the number of new sets of 2D bands being created) is thus chiral massive components. Because there is only one distinct given by □ðN þ 1Þ∕2□, as represented by the red planes band associated with the massless component, we again produce in Fig. 3. a total of 1 þ 2 × ðN − 1Þ∕2 ¼ N conduction (and valence) bands. Following this scheme, we see that the electronic structure of This situation is analogous to generating metallic nanotubes FLG materials is composed of chiral massless and chiral massive where one of the cutting lines pass through either the K or components, with a behavior that depends critically on the layer the K’ points in the 2D graphene Brillouin zone (Fig. 3). We N N thickness . For even , the zone-folding planes never pass illustrate this decomposition scheme for FLG for values of through the H point, and we obtain N∕2 sets of chiral massive N ¼ 1;2;…5 (Fig. 4A). This analysis of the electronic structure

components. Because each of these components corresponds PHYSICS of FLG was presented earlier by Koshino and Ando in a slightly to two bands, we have a total of N conduction (and valence) different formulation (12). bands. This behavior is analogous to the generation of semiconducting nanotubes by cutting lines that pass neither through the K From the dispersion relation for the graphite bands and quan- nor through the K′ points in the 2D graphene Brillouin zone tization conditions given above, we find that the energy spacing (Fig. 3). In contrast, for odd values of N, there is always a of the hyperbolic bands in each component of FLG is given (for n ≤ N∕2 zone-folding plane that passes through the H point. We therefore positive integers )by

Fig. 3. Application of zone folding to obtain the electronic states of FLG from graphite compared with the generation of the electronic states of carbon nanotubes by zone folding of graphene. The left column shows the generation of 2D chiral massless and massive fermions in FLG from zone folding the 3D Brillouin zone of bulk graphite. The upper panel displays the band structure of bulk graphite in two spatial dimensions and the zone-folding scheme that generates planes cutting at specific values of kz satisfying Eq. 1. The lower panel presents the resulting fundamental building blocks of the electronic structure of FLG: the massless and the massive components. For comparison, the right column displays the standard procedure for generating 1D metallic and semiconducting carbon nanotubes from zone folding of the 2D Brillouin zone of graphene. The upper panel is a schematic representation of the 2D electronic structure of SLG and the corresponding zone-folding scheme that generates states satisfying the periodic boundary conditions for nanotubes. The lower panel presents the resulting states: metallic and semiconducting nanotubes.

Mak et al. PNAS ∣ August 24, 2010 ∣ vol. 107 ∣ no. 34 ∣ 15001 Fig. 4. Schematic band structure of the components of FLG and a comparison between experiment and theory for the normalized conductivity spectra. (A) Representation of the low-energy band structure of FLG for N ¼ 1;2;…5 layers in terms of the massless and massive components generated by the zone-folding construction described in the text. (B) Experimental data for infrared conductivity per layer σðℏωÞ∕N for layer thickness N ¼ 1;2;…8. The normalized data are offset for clarity by half a unit of πe2∕2h per layer. The spectrum for bulk graphite (per layer, offset from the N ¼ 8 spectrum) is taken from ref. 35. In this presentation, we see the convergence toward the bulk graphite response with increasing thickness, although distinctive peaks are still observed up to N ¼ 8.(C) The calculated conductivity spectra corresponding to the data in (B) obtained from the zone-folding construction of the FLG electronic states and the Kubo formula, as described in the text.

nπ response measured here†, we can simply sum up the contributions γ ¼ 2γ : [2] N;n 1 cos ðN þ 1Þ of the various massive components and, for FLG with odd N,an additional contribution from the massless component (Fig. 4A). The corresponding effective mass for each component is The results of such a calculation using the Kubo formula repro- 2 mN;n ¼ γN;n∕2vF , where vF is the Fermi velocity for SLG duce the locations of the peaks observed experimentally for sam- p N ¼ 1;2;3;…;8 B (vF ¼ 3aγ0∕2ℏ in terms of the interlayer coupling and the ples of layer thicknesses (Fig. 4 ). Some lattice constant a). Eq. 2 generates the energy differences between differences are apparent with respect to line shape. At least the in-plane bands and, hence, the critical points for transitions in for the case of bilayer graphene, they are attributed largely to FLG. We expect that these energies will be reflected in the optical the effect of doping (see SI Text). Other corrections to the theory conductivity spectra. Indeed, the N-dependent transition energies would include the influence of additional coupling parameters in that we have identified empirically in the infrared conductivity the tight-binding model, such as out-of-plane couplings γ3 and γ4 spectra (Fig. 2B) are described by Eq. 2. The three families of (23, 33), and the effect of many-body corrections (34). (For transitions correspond to values of n ¼ 1;2;3 in the equation, further details, please refer to Materials and Methods.) and the overall energy scale is given by γ1 ¼ 0.37 eV (32) for How do the results for FLG samples converge to the behavior the interlayer coupling strength. We note that although this ana- of bulk graphite? As we can see from Fig. 2A, distinctive transi- lysis only explicitly involves the interlayer coupling parameter γ1, tions are still clearly present up to graphene samples of 8-layer the inclusion of parameters γ3 and γ4 would not influence the pre- thickness. The electronic structure for such a sample consists dicted transition energies. The critical points for the optical tran- of four massive components corresponding to different planes sitions occur at the K-point in the 2D Brillouin zone, where the perpendicular to the H–K direction. Because the typical broad- energies are not affected by these particular parameters (32, 33). ening of the peaks is about 50 meV, the appearance of distinctive From the zone-folding construction of the electronic states of ’ few-layer graphene, we can readily calculate the material s infra- † σðℏωÞ For light polarization perpendicular to the plane, transitions between components red sheet conductivity . Because only transitions between with kz differing by 2π∕ðN þ 1Þc are dipole allowed and the analysis in terms of a simple states with the same value of kz can contribute to the in-plane superposition of the constituents must be modified.

15002 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1004595107 Mak et al. features for 8-layer graphene is to be expected. We anticipate that the measured reflectance spectra into σðℏωÞ by multiplication of a suitable the individual optical transitions arising from, say, more than numerical factor determined by the (frequency-dependent) refractive index twenty subbands (i.e., N>40 layers) would be blurred together of the substrate (26). With increasing film thickness, propagation effects and that response would become indistinguishable from that begin to play a role, and the relation between the conductivity (or dielectric of graphite. Indeed, we are able to reproduce the experimental function) of the film and the measured properties becomes more complex. To determine the validity of the thin-film approximation in analysis of our data, optical conductivity for bulk graphite (35) by carrying out the N ¼ 40 we performed full calculations of the reflectance. Even for the thickest film Kubo calculation for few-layer graphene with layers considered in our measurements, the 8-layer graphene sample, we did not B C (Fig. 4 and ). In this limit, although the peaks from individual see significant changes in the shape of the inferred σðℏωÞ; errors in the 2D interband transitions wash out, the absorption spectrum inferred magnitude of σðℏωÞ were limited to 15%. In the interest of simplicity, retains a broad maximum around 0.75 eV (Fig. 4C). Within we consequently present all data based on the thin-film analysis. our description of FLG, this feature arises from the energy dis- tribution of the different 2D subband transitions in Eq. 2, which Theoretical Basis for the Zone-Folding Scheme. In the text above, we consid- shows that there are many transitions near the maximum energy ered a TB Hamiltonian for N-layer graphene with arbitrary couplings of 2γ1. This results in a broad peak in the conductivity around that between atoms in the same or adjacent layers of graphene. In this model, ψ ðrÞ¼∑ eikRϕðr − R Þ energy. The increased conductivity corresponds, in the language the Bloch wavefunction is approximated as k i i , where R ϕðrÞ of the bulk response, to the saddle-point singularity along the the summation extends over all lattice sites i and represents the carbon pz orbital. In this treatment, we consider the structural properties of each H–K direction in the 3D graphite band structure. layer in the sample (including the outer layers) as identical. This assumption Conclusions is reasonable for graphene, given the saturated character of its in-plane bonds and the weakness of its out-of-plane interactions. Under these The present work demonstrates that significant control of the assumptions, the eigenstates for N-layer graphene can be taken from low-energy electronic states of graphene can be achieved by those of the bulk graphite by constructing standing wave that satisfies the interlayer interactions in few-layer samples. For each layer thick- boundary condition of Eq. 1. ness N, carriers of differing masses are produced, with differing splittings of the conduction and valence bands. For odd values of Influence of Other Interlayer Coupling Parameters. In our calculation of optical the layer thickness N, massless carriers are also present, as in conductivities using the Kubo formula, only a single in-plane interaction (γ0) single-layer graphene. Understanding of the key features of and a single out-of-plane interaction (γ1) have been considered. In addition the 2D band structure in N-layer graphene, and the associated to the γ1 coupling, the dominant interactions for atoms in adjacent graphene

γ γ PHYSICS infrared conductivity spectra, can be achieved on the basis of a planes are described by the parameters 3 and 4. Their inclusion has three i simple and precisely defined zone folding of the 3D graphite effects on the band structure of FLG (13, 32): ( ) They introduce overlaps bands. This situation is analogous to the standard description (of a few meV) between the lower conduction and upper valence bands; (ii) they produce a weak asymmetry between the electron and hole masses; of the 1D bands of single-walled carbon nanotubes in which and (iii) they lead to trigonal warping for the dispersion away from the the electronic structure of the panoply of different nanotubes K-point of the 2D Brillouin zone. Effects (i)and(ii) do not alter the optical can be generated by zone folding of the 2D electronic structure conductivities, because the relevant energy scales are too small. For undoped of graphene. Just as for carbon nanotubes, the additional control or slightly doped FLG, effect (iii), controlled by coupling γ3, is also minor. The of the electronic properties as a function structure for FLG absorption features are dominated by transitions near the K-point where the should extend the range of distinctive physical phenomena and joint density of states is high. The inclusion of γ3 ≠ 0 will slightly modify the applications of this material system. line shape of the optical transitions away from the critical points (23, 33). We note that for an accurate description of the low-energy (∼10 meV Materials and Methods scale) electronic bands and the Fermi surfaces of both FLG and bulk graphite, Sample Preparation. We deposited FLG samples by mechanical exfoliation of the couplings mentioned above must be considered. In addition, two further γ kish graphite (Toshiba) on high-purity SiO2 substrates (Chemglass, Inc.) that parameters are very important: the second-layer coupling strength 2 and the had been carefully cleaning by sonication in methanol. The typical area of A∕B sublattice asymmetry parameter Δ (10, 13, 32). The parameter γ2 intro- the FLG graphene samples was several hundreds to thousands of μm2.We duces extra band overlaps (beyond the effect of γ3), and Δ ≠ 0 leads to a small established the thickness of each of the deposited graphene samples using band gap for the 2D dispersion at the H point. The zone-folding scheme that absorption spectroscopy in the visible spectral range, where each graphene we have successfully applied to describe the optical transitions of FLG in layer absorbs approximately 2.3% (26, 27, 36) of the light. terms of the properties of graphite can, however, be justified only when all interactions beyond those of adjacent graphene layers can be neglected. Reflectance Measurements. The infrared conductance measurements were Thus, the construction cannot be rigorously extended to include the case of performed using the NSLS at Brookhaven National Laboratory (U2B beam- γ2 ≠ 0. The zone-folding procedure has, correspondingly, limited validity line) as a bright source of broadband infrared radiation. We detected the for predicting the details of the Fermi surfaces in FLG, which will generally optical radiation reflected near normal incidence with a FTIR spectrometer depend sensitively on the details of the low-energy band structure of equipped with a HgCdTe detector under nitrogen purge. The synchrotron graphite. For further discussion of the nature of the low-energy bands radiation was focused to a spot size below 10 μm with a 32× reflective and Fermi surfaces of FLG, we refer the reader to the literature (10, 13). objective. The reflectance spectra of the graphene samples were obtained by normalizing the sample spectrum by that from the bare substrate, as ACKNOWLEDGMENTS. We thank Drs. Mikito Koshino, Mark S. Hybertsen, and in an earlier investigation of single-layer graphene (26). In that study, the Sami Rosenblatt for valuable discussions. The authors at Columbia University effects of finite beam divergence were also considered and shown to be acknowledge support from the Nanoscale Science and Engineering Initiative insignificant. of the National Science Foundation under Grant CHE-06-41523, from the New York State Office of Science, Technology, and Academic Research (NYSTAR), and from the Office of Naval Research under the Multidisciplinary Conversion of the Measured Reflectance to the Infrared Conductivity σðℏωÞ. For University Research Initiative (MURI) program; the authors at Brookhaven a sufficiently thin sample on a transparent substrate, its optical sheet conduc- were supported under contract DE-AC02-98CH10886 with the US Depart- σðℏωÞ tivity is related to the reflectance in a direct manner (26): The fractional ment of Energy. The synchrotron studies were supported by the NSLS at change in reflectance associated with the presence of the thin-film sample Brookhaven and the Center for Synchrotron Biosciences, Case Western is proportional to the real part of its optical sheet conductivity σðℏωÞ,or Reserve University, under P41-EB-01979 with the National Institute for equivalently, to its absorbance A ¼ð4π∕cÞσðℏωÞ. We can therefore convert Biomedical Imaging and Bioengineering.

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15004 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1004595107 Mak et al.