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

Abstract

The massless Dirac spectrum of electrons in single-layer graphene has been thoroughly studied both theoretically and experimentally. Although a subject of considerable theoretical interest, experimental investigations of the richer electronic structure of few-layer graphene (FLG) have been limited. Here we examine FLG graphene crystals with Bernal stacking of layer thicknesses N = 1,2,3,…8 prepared using the mechanical exfoliation technique. For each layer thickness N, infrared conductivity measurements over the spectral range of 0.2–1.0 eV have been performed and reveal a distinctive band structure, with different conductivity peaks present below 0.5 eV and a relatively flat spectrum at higher photon energies. The principal transitions exhibit a systematic energy-scaling behavior with N. These observations are explained within a unified zone-folding scheme that generates the electronic states for all FLG materials from that of the bulk 3D graphite crystal through imposition of appropriate boundary conditions. Using the Kubo formula, we find that the complete infrared conductivity spectra for the different FLG crystals can be reproduced reasonably well within the framework a tight-binding model.

The unique electronic properties of graphene, a single-monolayer of sp²-hybridized carbon, have attracted much attention (1). Graphene’s few-layer counterparts have also recently been the subject of much interest, since this broader class of materials offers the potential for further control of electronic states by interlayer interactions (2–6). Indeed, theoretical investigations have predicted dramatic changes in the electronic properties in few-layer graphene (FLG) compared with single-layer graphene (SLG) (7–17): When two or more layers of graphene are present in ordered FLG, the characteristic linearly dispersing bands of the single layer are either replaced or augmented by pairs of split hyperbolic bands. These new bands correspond to fermions of finite mass, unlike the electrons present in SLG that behave as massless fermions. Further, the characteristics of the electrons in FLG are expected to change sensitively with increasing layer number N, before ultimately approaching the bulk limit of graphite. Despite these fascinating predictions, experimental investigations have been limited to SLG and a few studies of the electronic properties of bilayer graphene (18–24). We examine these predictions experimentally by probing the electronic structure of FLG graphene samples for layer thickness up to N = 8 using infrared conductivity spectroscopy. For each thickness, we find well-defined and distinct peaks arising from the critical points for transitions between valence and conduction bands of the relevant FLG material. The position and shape of these features in the experimental spectra, recorded over a photon-energy range of 0.2–0.9 eV, provide direct information about key features of the band structure of FLG materials.

We are able to compare the positions of these features with the predictions of the zone-folding model of FLG electron structure that is introduced below. From the analysis, one can decompose the electronic structures of FLG into chiral massless and massive components with characteristics (10, 12–14), such as masses and Fermi velocities, that depend critically on layer thickness. In particular, if N is odd, both massless and massive components are present, whereas if N is even, only massive components exist. This behavior is broadly analogous to that known to occur for the model 1D system of single-walled carbon nanotubes. In this case, zone-folding of the 2D graphene band structure implies that certain nanotube physical structures have metallic character with massless electrons, but others with slightly different chiral indices have semiconducting character and only massive electrons (25). Our investigations show that FLG, whereas preserving many of the unusual features of SLG, provides a richness and flexibility of electronic structure that should find many novel applications.

Results

FLG samples of layer thicknesses N = 1,2,3,…8 were prepared using mechanical exfoliation of kish graphite. To facilitate analysis of the optical data, the samples were deposited onto transparent fused quartz substrates. (The procedure for sample preparation and identification of the layer thickness is described in Materials and Methods.) The FLG conductivity spectra were obtained using broadband infrared radiation from the Brookhaven National Synchrotron Light Source (NSLS) synchrotron and a FTIR spectrometer to analyze the spectral content of the radiation. To determine the infrared sheet conductivity σ(ℏω) of the FLG samples as function of the photon energy ℏω, we measured the reflectance spectrum of the bare substrate and that of the substrate covered with a FLG sample. For a layer of material, like these samples, with weak absorption and a thickness much less than a wavelength of light supported by a transparent substrate, the change in reflectance is directly proportional to the infrared conductivity σ(ℏω) (26). As discussed in Materials and Methods, the factor of proportionality is given by a function by the refractive index of the substrate.

The influence of layer thickness on the electronic properties of FLG is already clear in a comparison of the conductivity spectrum for SLG and bilayer graphene (Fig. 1). The conductivity σ(ℏω) for SLG is featureless, as reported previously (26, 27). At photon energies ℏω approaching 1 eV, the sheet conductivity σ(ℏω) is largely independent of energy, with a value of σ ≈ πe²/2h (or, equivalently, an absorbance of A = πα, where α denotes the fine-structure constant). Such universal behavior is intrinsic to the massless fermionic character of the structure for the π electrons near the K-point of the Brillouin zone (Fig. 1, Inset). The numerical value of the conductivity follows from these general properties in 2D (28), once the 4-fold spin and valley degeneracy present in SLG is taken into account. A departure from this universal behavior can be seen for photon energies ℏω < 0.5 eV, an effect attributed to Pauli blocking from unintentional doping and finite temperature effects (26).

Fig. 1.

Measured infrared sheet conductivity spectra σ(ℏω) of single- and bilayer graphene samples. In addition to the calibration of the sheet conductivity in units of πe²/2h, the equivalent absorbance is indicated in units of πα = 2.29%. The optical transitions responsible for the main features observed are shown as inserts.

For bilayer graphene, rather than the featureless spectrum of SLG, a sharp resonance is observed in the infrared conductivity σ(ℏω) at ℏω = 0.37 eV (Fig. 1). This change can be readily understood in terms of the significantly altered low-energy 2D band structure of the bilayer. Instead of the massless fermions found in SLG, electrons in bilayer graphene have finite masses and are described by a pair of hyperbolic bands (7, 8) (Fig. 1, Inset). No band gap is opened, but the pairs of valence and conduction bands are split. Their separation, within the tight-binding (TB) model described below is given by interlayer coupling strength γ₁. These hyperbolic bands lead to a step singularity, characteristic of 2D massive particles, in the joint density of states at the onset of interband transitions. Considering the presence of spontaneous doping of the sample, we should also observe optical transitions from the lower conduction band to the upper conduction band, as well as from the upper valence band to the upper conduction band (Fig. 1, Inset). These transitions give rise to a prominent peak in σ(ℏω) at ℏω = γ₁ (15, 29, 30). For photon energies well above this value (ℏω≫γ₁), the two layers become largely decoupled and infrared conductivity σ(ℏω) approaches twice of the universal value of πe²/2h found in SLG (Fig. 1).

Whereas we can understand the characteristic features of the band structure and infrared conductivity for bilayer graphene rather directly from considering two interacting graphene layers, the behavior is expected to become increasingly complex as the layer number N grows. Indeed, the data for the infrared conductivity σ(ℏω) for FLG up to 8 layers (Fig. 2A) show highly structured and readily distinguishable spectra for each value of N* (31). We note that the absolute strengths exhibited by the various resonances do not decrease significantly with layer number N. However, if we consider infrared conductivity spectra normalized per layer, σ(ℏω)/N, as we discuss below, we see a clear convergence toward the behavior of bulk graphite. In the thicker samples, we observe the emergence of the single broad feature present in graphite at approximately twice the energy of the peak in the bilayer sample.

Fig. 2.

Infrared conductivity spectra σ(ℏω) of FLG samples of layer thickness N = 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, σ(ℏω)/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.

The trends in the data for different layer thicknesses can be identified from a contour plot of the normalized conductivity spectra for all the FLG samples (Fig. 2B). Rather than the seemingly random variation in the energies of the principal optical 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. To carry 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- and bilayer graphene (7, 8). Here we present a generalization that readily generates the band structure of single- and bilayer graphene, but also extends to FLG of arbitrary layer thickness.

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 k_z 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.

We consider the 3D bands of graphite within a model that includes the nearest in-plane coupling coefficient of γ₀ = 3.16 eV and couplings between atoms in adjacent planes characterized by strengths of γ₁ = 370 meV, γ₃ = 315 meV, and γ₄ = 44 meV (10, 13, 32). In this discussion, we ignore the next nearest interlayer coupling of γ₂ = -20 meV (10, 13, 32) and the A/B on-site potential difference (absent in SLG) of Δ = 5 meV (10, 13, 32). Because these latter quantities are significantly smaller than the energy scale and line widths of the optical transitions (Figs. 1 and 2), they are not expected to be important in describing our absorption measurements. Within this description of the graphite band structure, the following are the key features. Along the H–K(z) direction, there are four bands (Fig. 3): two degenerate bands (E₃) without dispersion and two dispersive bands (E₁ and E₂) with energies of ± 2γ₁ cos(k_zc/2). The bands all become degenerate at the H point, where the in-plane dispersion of the (doubly degenerate) bands is linear and isotropic. For all other planes perpendicular to the H–K direction, the in-plane dispersion is described by split pairs of nearly hyperbolic conduction and valence bands (Fig. 3). Because of the trigonal warping associated with the finite value for γ₃, the higher valence and lower conduction bands exhibit slight crossings (over a few meV range of energy). Whereas this effect is critically important for defining the Fermi surface, it does not influence the optical spectra significantly. (For further discussion, please refer to Materials and Methods.)

With only interactions between adjacent planes and neglecting structural relaxation effects, the electronic states of the Hamiltonian for FLG are simply a subset of those obtained for bulk graphite (see Materials and Methods for justification). The additional zone-folding criterion that applies to the N-layer sample is that the wavefunctions must vanish at the position where the graphene planes lying immediately beyond the physical material; i.e., at the positions of the 0th and (N + 1)th layer. Because of the presence of mirror symmetry for odd N and inversion symmetry for even N, this can be accomplished by forming standing waves with momenta perpendicular to the graphene planes quantized as

[1]

Here c/2 is the interlayer separation (0.34 nm); the allowed values for the index are n =  ± 1, ± 2, ± 3,…. ± □(N + 1)/2□, where the symbol □ denotes the integer part of the quantity. Independent (standing-wave) states are generated only for positive values of n for which k_z ≤ π/c, corresponding to the positive half of the graphite Brillouin zone. For N-layer graphene, the number of independent 2D planes in the 3D Brillouin zone (i.e., the number of new sets of 2D bands being created) is thus given by □(N + 1)/2□, as represented by the red planes in Fig. 3.

Following this scheme, we see that the electronic structure of FLG materials is composed of chiral massless and chiral massive components, with a behavior that depends critically on the layer thickness N. For even N, the zone-folding planes never pass through the H point, and we obtain N/2 sets of chiral massive components. Because each of these components corresponds to two bands, we have a total of N conduction (and valence) bands. This behavior is analogous to the generation of semiconducting nanotubes by cutting lines that pass neither through the K nor through the K′ points in the 2D graphene Brillouin zone (Fig. 3). In contrast, for odd values of N, there is always a zone-folding plane that passes through the H point. We therefore generate 1 set of chiral massless component and (N - 1)/2 sets of chiral massive components. Because there is only one distinct band associated with the massless component, we again produce a total of 1 + 2 × (N - 1)/2 = N conduction (and valence) bands. This situation is analogous to generating metallic nanotubes where one of the cutting lines pass through either the K or the K’ points in the 2D graphene Brillouin zone (Fig. 3). We illustrate this decomposition scheme for FLG for values of N = 1,2,…5 (Fig. 4A). This analysis of the electronic structure of FLG was presented earlier by Koshino and Ando in a slightly different formulation (12).

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 = 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 σ(ℏω)/N for layer thickness N = 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 = 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 = 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.

From the dispersion relation for the graphite bands and quantization conditions given above, we find that the energy spacing of the hyperbolic bands in each component of FLG is given (for positive integers n ≤ N/2) by

[2]

The corresponding effective mass for each component is , where v_F is the Fermi velocity for SLG (v_F = √3aγ₀/2ℏ in terms of the interlayer coupling and the lattice constant a). Eq. 2 generates the energy differences between the in-plane bands and, hence, the critical points for transitions in FLG. We expect that these energies will be reflected in the optical conductivity spectra. Indeed, the N-dependent transition energies that we have identified empirically in the infrared conductivity spectra (Fig. 2B) are described by Eq. 2. The three families of transitions correspond to values of n = 1,2,3 in the equation, and the overall energy scale is given by γ₁ = 0.37 eV (32) for the interlayer coupling strength. We note that although this analysis only explicitly involves the interlayer coupling parameter γ₁, the inclusion of parameters γ₃ and γ₄ would not influence the predicted transition energies. The critical points for the optical transitions occur at the K-point in the 2D Brillouin zone, where the energies are not affected by these particular parameters (32, 33).

From the zone-folding construction of the electronic states of few-layer graphene, we can readily calculate the material’s infrared sheet conductivity σ(ℏω). Because only transitions between states with the same value of k_z can contribute to the in-plane response measured here^†, we can simply sum up the contributions of the various massive components and, for FLG with odd N, an additional contribution from the massless component (Fig. 4A). The results of such a calculation using the Kubo formula reproduce the locations of the peaks observed experimentally for samples of layer thicknesses N = 1,2,3,…,8 (Fig. 4B). Some differences are apparent with respect to line shape. At least for the case of bilayer graphene, they are attributed largely to the effect of doping (see SI Text). Other corrections to the theory would include the influence of additional coupling parameters in the tight-binding model, such as out-of-plane couplings γ₃ and γ₄ (23, 33), and the effect of many-body corrections (34). (For further details, please refer to Materials and Methods.)

How do the results for FLG samples converge to the behavior of bulk graphite? As we can see from Fig. 2A, distinctive transitions are still clearly present up to graphene samples of 8-layer thickness. The electronic structure for such a sample consists of four massive components corresponding to different planes perpendicular to the H–K direction. Because the typical broadening of the peaks is about 50 meV, the appearance of distinctive features for 8-layer graphene is to be expected. We anticipate that the individual optical transitions arising from, say, more than twenty subbands (i.e., N > 40 layers) would be blurred together and that response would become indistinguishable from that of graphite. Indeed, we are able to reproduce the experimental optical conductivity for bulk graphite (35) by carrying out the Kubo calculation for few-layer graphene with N = 40 layers (Fig. 4 B and C). In this limit, although the peaks from individual 2D interband transitions wash out, the absorption spectrum retains a broad maximum around 0.75 eV (Fig. 4C). Within our description of FLG, this feature arises from the energy distribution of the different 2D subband transitions in Eq. 2, which shows that there are many transitions near the maximum energy of 2γ₁. This results in a broad peak in the conductivity around that energy. The increased conductivity corresponds, in the language of the bulk response, to the saddle-point singularity along the H–K direction in the 3D graphite band structure.

Conclusions

The present work demonstrates that significant control of the low-energy electronic states of graphene can be achieved by interlayer interactions in few-layer samples. For each layer thickness N, carriers of differing masses are produced, with differing splittings of the conduction and valence bands. For odd values of the layer thickness N, massless carriers are also present, as in single-layer graphene. Understanding of the key features of the 2D band structure in N-layer graphene, and the associated infrared conductivity spectra, can be achieved on the basis of a simple and precisely defined zone folding of the 3D graphite bands. This situation is analogous to the standard description of the 1D bands of single-walled carbon nanotubes in which the electronic structure of the panoply of different nanotubes can be generated by zone folding of the 2D electronic structure of graphene. Just as for carbon nanotubes, the additional control of the electronic properties as a function structure for FLG should extend the range of distinctive physical phenomena and applications of this material system.

Materials and Methods

Sample Preparation.

We deposited FLG samples by mechanical exfoliation of kish graphite (Toshiba) on high-purity SiO₂ substrates (Chemglass, Inc.) that had been carefully cleaning by sonication in methanol. The typical area of the FLG graphene samples was several hundreds to thousands of μm². We established the thickness of each of the deposited graphene samples using absorption spectroscopy in the visible spectral range, where each graphene layer absorbs approximately 2.3% (26, 27, 36) of the light.

Reflectance Measurements.

The infrared conductance measurements were performed using the NSLS at Brookhaven National Laboratory (U2B beamline) as a bright source of broadband infrared radiation. We detected the optical radiation reflected near normal incidence with a FTIR spectrometer equipped with a HgCdTe detector under nitrogen purge. The synchrotron radiation was focused to a spot size below 10 μm with a 32× reflective objective. The reflectance spectra of the graphene samples were obtained by normalizing the sample spectrum by that from the bare substrate, as in an earlier investigation of single-layer graphene (26). In that study, the effects of finite beam divergence were also considered and shown to be insignificant.

Conversion of the Measured Reflectance to the Infrared Conductivity σ(ℏω).

For a sufficiently thin sample on a transparent substrate, its optical sheet conductivity σ(ℏω) is related to the reflectance in a direct manner (26): The fractional change in reflectance associated with the presence of the thin-film sample is proportional to the real part of its optical sheet conductivity σ(ℏω), or equivalently, to its absorbance A = (4π/c)σ(ℏω). We can therefore convert the measured reflectance spectra into σ(ℏω) by multiplication of a suitable numerical factor determined by the (frequency-dependent) refractive index of the substrate (26). With increasing film thickness, propagation effects begin to play a role, and the relation between the conductivity (or dielectric 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, we performed full calculations of the reflectance. Even for the thickest film considered in our measurements, the 8-layer graphene sample, we did not see significant changes in the shape of the inferred σ(ℏω); errors in the inferred magnitude of σ(ℏω) were limited to 15%. In the interest of simplicity, we consequently present all data based on the thin-film analysis.

Theoretical Basis for the Zone-Folding Scheme.

In the text above, we considered a TB Hamiltonian for N-layer graphene with arbitrary couplings between atoms in the same or adjacent layers of graphene. In this model, the Bloch wavefunction is approximated as , where the summation extends over all lattice sites R_i and ϕ(r) represents the carbon p_z orbital. In this treatment, we consider the structural properties of each layer in the sample (including the outer layers) as identical. This assumption is reasonable for graphene, given the saturated character of its in-plane bonds and the weakness of its out-of-plane interactions. Under these assumptions, the eigenstates for N-layer graphene can be taken from those of the bulk graphite by constructing standing wave that satisfies the boundary condition of Eq. 1.

Influence of Other Interlayer Coupling Parameters.

In our calculation of optical conductivities using the Kubo formula, only a single in-plane interaction (γ₀) and a single out-of-plane interaction (γ₁) have been considered. In addition to the γ₁ coupling, the dominant interactions for atoms in adjacent graphene planes are described by the parameters γ₃ and γ₄. Their inclusion has three effects on the band structure of FLG (13, 32): (i) They introduce overlaps (of a few meV) between the lower conduction and upper valence bands; (ii) they produce a weak asymmetry between the electron and hole masses; and (iii) they lead to trigonal warping for the dispersion away from the K-point of the 2D Brillouin zone. Effects (i) and (ii) do not alter the optical conductivities, because the relevant energy scales are too small. For undoped or slightly doped FLG, effect (iii), controlled by coupling γ₃, is also minor. The absorption features are dominated by transitions near the K-point where the joint density of states is high. The inclusion of γ₃ ≠ 0 will slightly modify the 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 scale) electronic bands and the Fermi surfaces of both FLG and bulk graphite, the couplings mentioned above must be considered. In addition, two further parameters are very important: the second-layer coupling strength γ₂ and the A/B sublattice asymmetry parameter Δ (10, 13, 32). The parameter γ₂ introduces extra band overlaps (beyond the effect of γ₃), and Δ ≠ 0 leads to a small band gap for the 2D dispersion at the H point. The zone-folding scheme that we have successfully applied to describe the optical transitions of FLG in terms of the properties of graphite can, however, be justified only when all interactions beyond those of adjacent graphene layers can be neglected. Thus, the construction cannot be rigorously extended to include the case of γ₂ ≠ 0. The zone-folding procedure has, correspondingly, limited validity for predicting the details of the Fermi surfaces in FLG, which will generally depend sensitively on the details of the low-energy band structure of graphite. For further discussion of the nature of the low-energy bands and Fermi surfaces of FLG, we refer the reader to the literature (10, 13).