Main

We investigated the reflectance R(ω) and transmission T(ω) of graphene samples on a SiO2/Si substrate (inset of Fig. 1a) as a function of gate voltage Vg at 45 K (see the Methods section). We start with data taken at the charge-neutrality point VCN: the gate voltage corresponding to the minimum d.c. conductivity and zero total charge density (inset of Fig. 1c). Figure 1a shows R(ω) of a graphene gated structure (graphene/SiO2/Si) at VCN=3 V normalized by reflectance of the substrate Rsub(ω). Rsub(ω) is dominated by a minimum around 5,500 cm−1 due to interference effects in SiO2. A remarkable observation is that a monolayer of undoped graphene markedly modifies the interference minimum of the substrate leading to a suppression of Rsub(ω) by as much as 15%. This observation is significant because it enables us to evaluate the conductivity of graphene near the interference structure, as will be discussed below.

Figure 1: The reflectance R(ω) and transmission T(ω) of a graphene device under applied gate voltages.
figure 1

a, The reflectance of the graphene device (graphene/SiO2/Si) R(ω) normalized by that of the SiO2/Si substrate Rsub(ω) at the charge-neutrality voltage VCN. A set of R(ω)/Rsub(ω) spectra generated from the multilayer model using a constant σ1(ω,VCN) in the range of (1±0.15)πe2/2h are shown as the shaded area. The upper and lower boundary of the shaded area are defined by σ1(ω,VCN) with values of 0.85*πe2/2h and 1.15*πe2/2h, respectively. Inset: A photograph of a graphene device together with a schematic diagram of the focused synchrotron beam (red dot). b,c, R(V)/R(VCN) (b) and T(V)/T(VCN) (c) spectra of the graphene device at several voltages corresponding to EF on the electron side, where V =Vg−VCN. The inset of c shows the smoothed d.c. conductivity data of the sample as a function of gate voltage Vg.

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Both reflectance and transmission spectra of graphene structures can be modified by a gate voltage. Figure 1b,c shows these modifications at various gate voltages normalized by data at VCN: R(V)/R(VCN) and T(V)/T(VCN), where V =Vg–VCN. These data correspond to the Fermi energy EF on the electron side and similar behaviour was observed with EF on the hole side (not shown). At low voltages (<17 V), we found a dip in R(V)/R(VCN) spectra. With increasing bias, this feature evolves into a peak–dip structure and systematically shifts to higher frequency. The T(V)/T(VCN) spectra reveal a peak at all voltages, which systematically hardens with increasing bias. A voltage-induced increase in transmission (T(V)/T(VCN)>1) signals a decrease of the absorption with bias. Most interestingly, we observed that the frequencies of the main features in R(V)/R(VCN) and T(V)/T(VCN) all evolve approximately as .

To explore the quasiparticle dynamics under applied voltages, it is imperative to first discuss the two-dimensional (2D) optical conductivity of charge-neutral graphene, σ1(ω,VCN)+i σ2(ω,VCN), extracted from a multilayer analysis of the devices (see the Methods section). Theoretical analysis6,7,8 predicts a constant ‘universal’ 2D conductivity σ1(ω,VCN)=πe2/2h for ideal undoped graphene. Our R(ω)/Rsub(ω) data are consistent with this prediction. Figure 1a shows a comparison between the experimental R(ω)/Rsub(ω) spectrum and model spectra generated assuming constant σ1(ω,VCN) values. The constant universal conductivity offers a good agreement (within ±15%) with the experimental spectra in the range 4,000–6,500 cm−1. Outside this spectral region, our infrared measurements do not allow us to unambiguously determine the absolute value of σ1(ω,VCN); therefore, the uncertainty of σ1(ω,VCN) increases as shown by the shaded region weighted around the πe2/2h value in Fig. 2a. However, recent infrared studies of graphene revealed a constant conductivity σ1(ω,VCN)=πe2/2h between 2,400 and 24,000 cm−1 (ref. 9 and Mak, K.F. & Heinz, T. 2008 APS March Meeting, Abstract: L29.00006, unpublished). The universal conductivity is only weakly modified in bulk highly ordered pyrolytic graphite10 and extends down to 800 cm−1. Thus, in the following discussion, we will assume σ1(ω,VCN)=πe2/2h throughout the entire range of our data.

Figure 2: The optical conductivity of graphene at different voltages.
figure 2

a, The real part of the 2D optical conductivity σ1(ω) at VCN and 71 V. The solid red line shows the region where our data support the universal result. The uncertainty of σ1(ω,VCN) is shown by the shaded area with the theoretical σ1(ω)=πe2/2h plotted as the red dashed line. The blue dashed line is σ1(ω) at 71 V evaluated for the theoretical spectra: σ1(ω,VCN)=πe2/2h (red dashed line). The key spectral features of σ1(ω,V) are independent of uncertainties in σ1(ω,VCN) indicated by the shaded area, as discussed in the text. Black square on the left axis: d.c. conductivity at VCN. b,c, σ1(ω) (b) and σ2(ω) (c) of graphene at several voltages with respect to VCN corresponding to EF on the electron side based on σ1(ω,VCN)=πe2/2h. The absolute values of the σ2(ω) spectra in c have uncertainties due to the uncertainties of σ1(ω,VCN) as discussed in the text. The inset of b shows the band structure of graphene near the Dirac point and the interband transition at 2EF.

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Electrostatic doping of graphene introduces two fundamental changes in the optical conductivity σ1(ω,V)+i σ2(ω,V): a strong Drude component formed in the far-infrared with σ1(ω→0)=4–100 πe2/2h accompanied by a shift in the onset of interband transitions at 2EF, as schematically shown in the inset of Fig. 2b. To investigate these effects, we obtained σ1(ω,V)+i σ2(ω,V) (Fig. 2b,c) from voltage-dependent reflectance and transmission spectra (see the Methods section). The key features in the conductivity spectra are independent of uncertainties in σ1(ω,VCN) discussed above. Regardless of the choice of σ1(ω,VCN), under applied biases, we observe a suppression of the conductivity compared with σ1(ω,VCN) and a well-defined threshold structure above which the conductivity recovers the universal value πe2/2h. The energy of the threshold structure systematically increases with voltage, a natural expectation for a transition occurring at 2EF. With a scattering rate 1/τ=30 cm−1 at 71 V independently obtained from transport data, the Drude mode is narrow and confined below the low-ω cutoff of our measurements. We stress that the two voltage-induced transformations of the conductivity, the intraband mode and the onset of interband absorption at 2EF, are interdependent as suggested by our data. Indeed, assuming the intraband component can be described with a simple Drude formula σ1(ω)=σd.c./(1+ω2τ2) using σd.c. and 1/τ obtained from transport measurements, we find that the spectral weight removed from ω<2EF is recovered under the Drude structure, such that the total oscillator strength given by is conserved at any bias with a cutoff frequency Ωc=8,000 cm−1.

Next, we extracted Fermi energy values from the 2EF threshold using two different methods (see the Methods section). We found that the 2EF values (Fig. 3a) are symmetric for biases delivering either holes or electrons to graphene. Moreover, 2EF increases with voltage approximately as (deviations from the square-root law at small biases will be discussed below). Note that EF of Dirac fermions scales with the 2D carrier density N as (refs 2, 3), where vFis the Fermi velocity. In our devices, N=CgV/e, where Cg=115 aF μm−2 is the gate capacitance per unit area. Therefore, the observed dependence of 2EFsubstantiates that graphene samples integrated in gated devices are governed by Dirac quasiparticles.

Figure 3: The Fermi energy EF and the ratio of EF to the Fermi wave vector EF/ℏkF.
figure 3

The magnitude of EF/ℏkF is closely related to the Fermi velocity vFas discussed in the text. a, The magnitude of 2EF plotted as a function of V 1/2 and kF for the electron and hole sides with respect to the charge-neutrality voltage. Red solid symbols: 2EF extracted from the minimum in the σ2(ω,V) spectra. Blue open symbols: 2EF extracted from the centre of the 2EF threshold in σ1(ω,V). The uncertainties of the 2EF values discussed in the Methods section do not exceed the size of the symbols. Solid lines are theoretical 2EFvalues using vF=1.11×106 m S−1. b, Symbols: EF/ℏkF as a function of V 1/2 and kF. The blue line corresponds to a vF value of 1.11×106 m S−1. The error bars reflect uncertainties in the determination of the Fermi energy 2EF from the conductivity data δ(EF) discussed in the Methods section and are calculated as .

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Interestingly, the 2EF threshold in σ1(ω,V) shows a width of about 1,400 cm−1 that is independent of gate voltage and therefore of carrier density N, irrespective of a sevenfold enhancement of N between 10 and 71 V. This effect is much stronger than the theoretical estimate for thermal smearing of the 2EF feature at 45 K (refs 7, 8), which is about 500 cm−1. A recent theoretical study11 showed that disorder effects and electron–phonon coupling are needed to account for the width of the 2EF threshold in our data. Apart from that, a spatial variation of local EF values observed in graphene on SiO2/Si substrates (ref. 12 and Brar, V. et al., 2008 APS March Meeting, Abstract: U29.00003, unpublished) will inevitably lead to a broadening of the absorption onset at 2EF in σ1(ω), because infrared measurements register the absorption averaged over a large area (a few micrometres in our experiments). The origin of the inhomogeneity of EF in graphene is still an open question12, which needs to be explored using spatially resolved probes such as near-field infrared conductivity studies capable of probing the response of a material with nanometre resolution over a large area13.

Our study has uncovered several new properties of graphene that are beyond the ideal Dirac fermion picture14. First, our study revealed unexpected features of σ1(ω,V) below 2EF. The band structure of ideal graphene implies that the interband transition at 2EF is the lowest electronic excitation in the system apart from the Drude response at ω=0. Therefore, we anticipate finding σ1(ω,V)≈0 up to the 2EF threshold, provided the Drude scattering rate is much smaller than 2EF. This latter condition is fulfilled for all data in Fig. 2, and yet we registered significant conductivity below 2EF (see the Supplementary Information). This result has not been anticipated by theories developed for Dirac fermions6,7,8. Both extrinsic and intrinsic effects may give rise to the residual conductivity in Fig. 2. Among the former, charged impurities and unitary scatterers (edge defects, cracks, vacancies and so on) were shown to induce considerable residual conductivity below 2EF (ref. 11). However, the theoretical residual absorption in ref. 11 is systematically suppressed with voltage, whereas this suppression was not observed in our data. In addition, the magnitude of the theoretical residual absorption is smaller compared with experimental values in Fig. 2. Therefore, it is likely that other mechanisms are also responsible for the residual conductivity in our data. One intriguing interpretation of the residual conductivity is in terms of many-body interactions, which are known to produce a strong frequency-dependent quasiparticle scattering rate 1/τ(ω). It is predicted theoretically that 1/τ(ω) in graphene increases with frequency owing to electron–electron15,16 and electron–phonon interactions17. The energy-dependent scattering rate initiates a marked enhancement of the conductivity compared with the lorentzian form prescribed by the Drude model. Such an enhancement in mid-infrared frequencies has been observed in many systems18,19,20.

A closer inspection of the voltage dependence of the 2EF feature uncovers marked departures from the behaviour anticipated within a single particle picture of graphene. To highlight these deviations, we plot 2EF as a function of the Fermi wave vector kF based on , as shown in Fig. 3a. The EF(kF) plot has a clear physical meaning: it is a direct representation of the band dispersion. We then examine the ratio EF/ℏkF, which is directly related to the Fermi velocity vF. The EF/ℏkF plot as a function of V1/2 and kF in Fig. 3b reveals a departure from linear dispersion with a single value of EF/ℏkF expected within a single particle picture. Moreover, EF/ℏkF increases systematically with decreasing kF values compared with that at high kF values. The observed systematic enhancement of EF/ℏkF at low kF is indicative of many-body interactions14,21,22. Signatures of band renormalization were also observed in a previous magneto-optical study of graphene4. Importantly, even the smallest EF/ℏkF values in Fig. 3b are higher than that of the bulk graphite23 (∼0.9×106 m S−1), which also supports the hypothesis of EF/ℏkF renormalization in graphene. Complementary information on the EF/ℏkF renormalization in graphene can be obtained from photoemission, which is another potent probe of many-body effects in solids. Currently available photoemission data were all collected for epitaxial graphene grown on SiC (refs 24, 25). This complicates a direct comparison with infrared results for exfoliated samples on SiO2/Si substrates reported here. We conclude by noting that the strong deviations of the experimental electromagnetic response from a simple single particle picture of graphene reported in our study challenge current theoretical conceptions of fundamental properties of this interesting form of carbon and also have implications for its potential applications in opto-electronics.

Note added in proof. After the submission of our paper, we became aware of another work on gate tunable infrared properties of graphene26.

Methods

Sample fabrication and infrared measurements

In the graphene devices studied here, monolayer graphene mechanically cleaved from Kish graphite was deposited onto an infrared transparent SiO2(300 nm)/Si substrate2,3, which also serves as the gate electrode. Then, standard fabrication procedures were used to define multiple Cr/Au (3/35 nm) contacts to the sample. The devices studied here exhibit mobility as high as 8,700 cm2 V−1 s−1 measured at carrier densities of ∼2×1012 cm−2. The characteristic half-integer quantum Hall effect is observed in these samples2,3, confirming the single layer nature of our specimen. Infrared experiments were carried out using an infrared microscope operating with a synchrotron source at the Advanced Light Source in the frequency range of 700–8,000 cm−1. The synchrotron beam is focused in a diffraction-limited spot, which is smaller than the sample. We measured the reflectance R(ω) and transmission T(ω) of the graphene devices as a function of gate voltage Vg with simultaneous monitoring of the d.c. resistivity.

Temperature of the graphene sample

Data reported here were obtained in a micro-cryostat with the sample mounted on a coldfinger in vacuum. The temperature of our graphene sample is warmer than that of the coldfinger, owing to thermal radiation from room-temperature KBr optical windows and electrical isolation of the devices from the coldfinger that compromises thermal contact. A sensor mounted in the immediate proximity to the Si substrate of the devices read T=45 K at the lowest temperatures attainable at the coldfinger. Because both the temperature sensor and the device are in a nearly identical environment, we assumed this reading to be accurate for graphene as well.

Extracting the optical constants of graphene

The graphene device contains four layers: (1) graphene with 2D optical conductivity σ(ω)=σ1(ω)+i σ2(ω), (2) SiO2 gate insulator, (3) Si accumulation layer that forms at the interface of SiO2/Si under the applied bias and (4) Si substrate. Properties of layers 2 and 4 are independent of the gate voltage, whereas layers 1 and 3 are systematically modified by Vg. In our analysis of these multilayer structures, we followed the protocol detailed in ref. 27. Specifically, we carried out reflection, transmission and ellipsometric measurements on the Si substrates and SiO2/Si wafers used in our devices and thus obtained the optical constants of layers 2 and 4. We then investigated infrared properties of test devices Ti/SiO2/Si as a function of gate voltage and thus extracted the optical constants of the Si accumulation layer in wafers used for graphene devices. We find that the response of the Si accumulation layer is confined to far-infrared frequencies28 and gives negligible contribution to mid-infrared data in Fig. 1. Finally, we used a multi-oscillator fitting procedure27 to account for the contribution of σ(ω) of graphene to the reflectance and transmission spectra shown in Fig. 1 using standard methods for multilayered structures.

Extracting Fermi energy EF from conductivity spectra

Because of the broadening of the 2EF threshold in σ1(ω,V), the EFvalues can be determined most accurately from the imaginary part of the optical conductivity spectra σ2(ω,V) shown in Fig. 2c. Indeed, these spectra reveal a sharp minimum at ω=2EF in agreement with a previous theoretical prediction29. The minimum in the σ2(ω,V) spectrum is found from the frequency where the derivative of σ2(ω,V) with respect to frequency is zero. The uncertainties of 2EF obtained from this method are related to the accuracy in defining the minimum in the σ2(ω,V) spectrum. Alternatively, 2EF values can be extracted from the centre frequency of the 2EF threshold in σ1(ω,V). The second method has larger uncertainties, as shown in Fig. 3, due to the ambiguity of defining the centre of the 2EF threshold in σ1(ω,V).